LMFDB - Embedded newform 135.4.e.c.91.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,4,Mod(46,135)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("135.46");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	135.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.96525785077$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} + 48 x^{12} - 60 x^{11} + 1605 x^{10} - 1800 x^{9} + 23232 x^{8} - 2346 x^{7} + \cdots + 82944 $ x^14 - 2x^13 + 48x^12 - 60x^11 + 1605x^10 - 1800x^9 + 23232x^8 - 2346x^7 + 209529x^6 - 55412x^5 + 765088x^4 + 276096x^3 + 1572480x^2 - 345600*x + 82944
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{13} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			91.6
Root			$-1.52087 + 2.63422i$ of defining polynomial
Character	$\chi$	$=$	135.91
Dual form			135.4.e.c.46.6

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.52087 + 2.63422i) q^{2} +(-0.626094 + 1.08443i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(6.85611 + 11.8751i) q^{7} +20.5251 q^{8} +O(q^{10})$	\(q+(1.52087 + 2.63422i) q^{2} +(-0.626094 + 1.08443i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(6.85611 + 11.8751i) q^{7} +20.5251 q^{8} -15.2087 q^{10} +(15.9034 + 27.5455i) q^{11} +(-29.1216 + 50.4401i) q^{13} +(-20.8545 + 36.1211i) q^{14} +(36.2248 + 62.7431i) q^{16} -109.055 q^{17} +129.695 q^{19} +(-3.13047 - 5.42213i) q^{20} +(-48.3740 + 83.7863i) q^{22} +(39.8342 - 68.9949i) q^{23} +(-12.5000 - 21.6506i) q^{25} -177.161 q^{26} -17.1703 q^{28} +(4.51769 + 7.82486i) q^{29} +(16.6904 - 28.9087i) q^{31} +(-28.0860 + 48.6463i) q^{32} +(-165.858 - 287.275i) q^{34} -68.5611 q^{35} -22.1645 q^{37} +(197.250 + 341.647i) q^{38} +(-51.3127 + 88.8763i) q^{40} +(60.8698 - 105.430i) q^{41} +(-5.07086 - 8.78298i) q^{43} -39.8281 q^{44} +242.331 q^{46} +(-220.746 - 382.343i) q^{47} +(77.4875 - 134.212i) q^{49} +(38.0218 - 65.8556i) q^{50} +(-36.4657 - 63.1604i) q^{52} +593.610 q^{53} -159.034 q^{55} +(140.722 + 243.738i) q^{56} +(-13.7416 + 23.8012i) q^{58} +(221.230 - 383.182i) q^{59} +(-72.2881 - 125.207i) q^{61} +101.536 q^{62} +408.736 q^{64} +(-145.608 - 252.200i) q^{65} +(-431.360 + 747.138i) q^{67} +(68.2786 - 118.262i) q^{68} +(-104.273 - 180.605i) q^{70} +818.541 q^{71} +495.052 q^{73} +(-33.7093 - 58.3863i) q^{74} +(-81.2014 + 140.645i) q^{76} +(-218.071 + 377.710i) q^{77} +(-585.263 - 1013.71i) q^{79} -362.248 q^{80} +370.300 q^{82} +(-212.022 - 367.232i) q^{83} +(272.637 - 472.221i) q^{85} +(15.4242 - 26.7156i) q^{86} +(326.419 + 565.374i) q^{88} -1031.37 q^{89} -798.643 q^{91} +(49.8799 + 86.3945i) q^{92} +(671.452 - 1162.99i) q^{94} +(-324.238 + 561.597i) q^{95} +(-799.356 - 1384.53i) q^{97} +471.394 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8}+O(q^{10}) $ 14 * q - 2 * q^2 - 36 * q^4 - 35 * q^5 - 22 * q^7 + 36 * q^8	$ 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8} + 20 q^{10} - 23 q^{11} - 96 q^{13} + 21 q^{14} - 324 q^{16} + 322 q^{17} + 558 q^{19} - 180 q^{20} - 311 q^{22} - 96 q^{23} - 175 q^{25} - 716 q^{26} + 674 q^{28} + 296 q^{29} - 244 q^{31} + 314 q^{32} - 125 q^{34} + 220 q^{35} + 808 q^{37} - 305 q^{38} - 90 q^{40} + 47 q^{41} - 525 q^{43} + 110 q^{44} + 1434 q^{46} - 164 q^{47} - 1225 q^{49} - 50 q^{50} - 1682 q^{52} + 1012 q^{53} + 230 q^{55} + 981 q^{56} - 1183 q^{58} + 85 q^{59} - 828 q^{61} - 1572 q^{62} + 4472 q^{64} - 480 q^{65} - 1093 q^{67} - 2473 q^{68} + 105 q^{70} + 656 q^{71} + 4170 q^{73} + 1316 q^{74} - 2789 q^{76} - 24 q^{77} - 2110 q^{79} + 3240 q^{80} - 124 q^{82} - 1290 q^{83} - 805 q^{85} + 2569 q^{86} - 2271 q^{88} - 6096 q^{89} + 6676 q^{91} - 2763 q^{92} + 517 q^{94} - 1395 q^{95} - 1787 q^{97} + 2558 q^{98}+O(q^{100}) $ 14 * q - 2 * q^2 - 36 * q^4 - 35 * q^5 - 22 * q^7 + 36 * q^8 + 20 * q^10 - 23 * q^11 - 96 * q^13 + 21 * q^14 - 324 * q^16 + 322 * q^17 + 558 * q^19 - 180 * q^20 - 311 * q^22 - 96 * q^23 - 175 * q^25 - 716 * q^26 + 674 * q^28 + 296 * q^29 - 244 * q^31 + 314 * q^32 - 125 * q^34 + 220 * q^35 + 808 * q^37 - 305 * q^38 - 90 * q^40 + 47 * q^41 - 525 * q^43 + 110 * q^44 + 1434 * q^46 - 164 * q^47 - 1225 * q^49 - 50 * q^50 - 1682 * q^52 + 1012 * q^53 + 230 * q^55 + 981 * q^56 - 1183 * q^58 + 85 * q^59 - 828 * q^61 - 1572 * q^62 + 4472 * q^64 - 480 * q^65 - 1093 * q^67 - 2473 * q^68 + 105 * q^70 + 656 * q^71 + 4170 * q^73 + 1316 * q^74 - 2789 * q^76 - 24 * q^77 - 2110 * q^79 + 3240 * q^80 - 124 * q^82 - 1290 * q^83 - 805 * q^85 + 2569 * q^86 - 2271 * q^88 - 6096 * q^89 + 6676 * q^91 - 2763 * q^92 + 517 * q^94 - 1395 * q^95 - 1787 * q^97 + 2558 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.52087	+	2.63422i	0.537709	+	0.931339i	0.999027	+	0.0441043i	$0.0140434\pi$
$2$	1.52087	+	2.63422i	0.537709	+	0.931339i	−0.461318	+	0.887235i	$0.652623\pi$
$3$		0			0
$3$		0			0
$4$	−0.626094	+	1.08443i	−0.0782617	+	0.135553i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$6$		0			0
$7$	6.85611	+	11.8751i	0.370195	+	0.641197i	0.989595	−	0.143879i	$-0.0459575\pi$
$7$	6.85611	+	11.8751i	0.370195	+	0.641197i	−0.619400	+	0.785075i	$0.712624\pi$
$8$	20.5251			0.907090
$9$		0			0
$10$	−15.2087			−0.480941
$11$	15.9034	+	27.5455i	0.435914	+	0.755026i	0.997370	−	0.0724812i	$-0.0230917\pi$
$11$	15.9034	+	27.5455i	0.435914	+	0.755026i	−0.561455	+	0.827507i	$0.689758\pi$
$12$		0			0
$13$	−29.1216	+	50.4401i	−0.621298	+	1.07612i	0.367946	+	0.929847i	$0.380061\pi$
$13$	−29.1216	+	50.4401i	−0.621298	+	1.07612i	−0.989244	+	0.146272i	$0.953272\pi$
$14$	−20.8545	+	36.1211i	−0.398115	+	0.689555i
$15$		0			0
$16$	36.2248	+	62.7431i	0.566012	+	0.980361i
$17$	−109.055			−1.55586			−0.777932	−	0.628348i	$-0.783731\pi$
$17$	−109.055			−1.55586			−0.777932	+	0.628348i	$0.783731\pi$
$18$		0			0
$19$	129.695			1.56601			0.783004	−	0.622017i	$-0.213687\pi$
$19$	129.695			1.56601			0.783004	+	0.622017i	$0.213687\pi$
$20$	−3.13047	−	5.42213i	−0.0349997	−	0.0606213i
$21$		0			0
$22$	−48.3740	+	83.7863i	−0.468790	+	0.811968i
$23$	39.8342	−	68.9949i	0.361131	−	0.625497i	−0.627017	−	0.779006i	$-0.715724\pi$
$23$	39.8342	−	68.9949i	0.361131	−	0.625497i	0.988147	+	0.153509i	$0.0490575\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$	−177.161			−1.33631
$27$		0			0
$28$	−17.1703			−0.115888
$29$	4.51769	+	7.82486i	0.0289280	+	0.0501048i	0.880127	−	0.474738i	$-0.157457\pi$
$29$	4.51769	+	7.82486i	0.0289280	+	0.0501048i	−0.851199	+	0.524843i	$0.824124\pi$
$30$		0			0
$31$	16.6904	−	28.9087i	0.0966998	−	0.167489i	−0.813617	−	0.581401i	$-0.802505\pi$
$31$	16.6904	−	28.9087i	0.0966998	−	0.167489i	0.910317	+	0.413912i	$0.135838\pi$
$32$	−28.0860	+	48.6463i	−0.155154	+	0.268735i
$33$		0			0
$34$	−165.858	−	287.275i	−0.836602	−	1.44904i
$35$	−68.5611			−0.331113
$36$		0			0
$37$	−22.1645			−0.0984817			−0.0492408	−	0.998787i	$-0.515680\pi$
$37$	−22.1645			−0.0984817			−0.0492408	+	0.998787i	$0.515680\pi$
$38$	197.250	+	341.647i	0.842056	+	1.45848i
$39$		0			0
$40$	−51.3127	+	88.8763i	−0.202831	+	0.351314i
$41$	60.8698	−	105.430i	0.231860	−	0.401593i	−0.726495	−	0.687171i	$-0.758852\pi$
$41$	60.8698	−	105.430i	0.231860	−	0.401593i	0.958356	+	0.285578i	$0.0921855\pi$
$42$		0			0
$43$	−5.07086	−	8.78298i	−0.0179837	−	0.0311487i	0.856894	−	0.515493i	$-0.172391\pi$
$43$	−5.07086	−	8.78298i	−0.0179837	−	0.0311487i	−0.874877	+	0.484345i	$0.839058\pi$
$44$	−39.8281			−0.136462
$45$		0			0
$46$	242.331			0.776733
$47$	−220.746	−	382.343i	−0.685088	−	1.18661i	−0.973409	−	0.229073i	$-0.926430\pi$
$47$	−220.746	−	382.343i	−0.685088	−	1.18661i	0.288321	−	0.957534i	$-0.406903\pi$
$48$		0			0
$49$	77.4875	−	134.212i	0.225911	−	0.391289i
$50$	38.0218	−	65.8556i	0.107542	−	0.186268i
$51$		0			0
$52$	−36.4657	−	63.1604i	−0.0972477	−	0.168438i
$53$	593.610			1.53846			0.769232	−	0.638969i	$-0.220639\pi$
$53$	593.610			1.53846			0.769232	+	0.638969i	$0.220639\pi$
$54$		0			0
$55$	−159.034			−0.389894
$56$	140.722	+	243.738i	0.335800	+	0.581623i
$57$		0			0
$58$	−13.7416	+	23.8012i	−0.0311097	+	0.0538836i
$59$	221.230	−	383.182i	0.488164	−	0.845525i	−0.511743	−	0.859139i	$-0.671000\pi$
$59$	221.230	−	383.182i	0.488164	−	0.845525i	0.999907	+	0.0136133i	$0.00433337\pi$
$60$		0			0
$61$	−72.2881	−	125.207i	−0.151730	−	0.262804i	0.780133	−	0.625613i	$-0.215151\pi$
$61$	−72.2881	−	125.207i	−0.151730	−	0.262804i	−0.931864	+	0.362809i	$0.881818\pi$
$62$	101.536			0.207985
$63$		0			0
$64$	408.736			0.798312
$65$	−145.608	−	252.200i	−0.277853	−	0.481255i
$66$		0			0
$67$	−431.360	+	747.138i	−0.786553	+	1.36235i	0.141514	+	0.989936i	$0.454803\pi$
$67$	−431.360	+	747.138i	−0.786553	+	1.36235i	−0.928067	+	0.372414i	$0.878530\pi$
$68$	68.2786	−	118.262i	0.121765	−	0.210902i
$69$		0			0
$70$	−104.273	−	180.605i	−0.178042	−	0.308378i
$71$	818.541			1.36821			0.684105	−	0.729384i	$-0.260193\pi$
$71$	818.541			1.36821			0.684105	+	0.729384i	$0.260193\pi$
$72$		0			0
$73$	495.052			0.793719			0.396859	−	0.917879i	$-0.370100\pi$
$73$	495.052			0.793719			0.396859	+	0.917879i	$0.370100\pi$
$74$	−33.7093	−	58.3863i	−0.0529545	−	0.0917198i
$75$		0			0
$76$	−81.2014	+	140.645i	−0.122558	+	0.212277i
$77$	−218.071	+	377.710i	−0.322747	+	0.559014i
$78$		0			0
$79$	−585.263	−	1013.71i	−0.833510	−	1.44368i	−0.895238	−	0.445588i	$-0.852995\pi$
$79$	−585.263	−	1013.71i	−0.833510	−	1.44368i	0.0617284	−	0.998093i	$-0.480339\pi$
$80$	−362.248			−0.506256
$81$		0			0
$82$	370.300			0.498693
$83$	−212.022	−	367.232i	−0.280390	−	0.485651i	0.691090	−	0.722768i	$-0.257131\pi$
$83$	−212.022	−	367.232i	−0.280390	−	0.485651i	−0.971481	+	0.237118i	$0.923797\pi$
$84$		0			0
$85$	272.637	−	472.221i	0.347902	−	0.602584i
$86$	15.4242	−	26.7156i	0.0193400	−	0.0334978i
$87$		0			0
$88$	326.419	+	565.374i	0.395413	+	0.684876i
$89$	−1031.37			−1.22837			−0.614183	−	0.789163i	$-0.710514\pi$
$89$	−1031.37			−1.22837			−0.614183	+	0.789163i	$0.710514\pi$
$90$		0			0
$91$	−798.643			−0.920006
$92$	49.8799	+	86.3945i	0.0565254	+	0.0979049i
$93$		0			0
$94$	671.452	−	1162.99i	0.736756	−	1.27610i
$95$	−324.238	+	561.597i	−0.350170	+	0.606512i
$96$		0			0
$97$	−799.356	−	1384.53i	−0.836725	−	1.44925i	−0.892618	−	0.450814i	$-0.851134\pi$
$97$	−799.356	−	1384.53i	−0.836725	−	1.44925i	0.0558930	−	0.998437i	$-0.482199\pi$
$98$	471.394			0.485897
$99$		0			0
$100$	31.3047			0.0313047
$101$	−107.000	−	185.330i	−0.105415	−	0.182584i	0.808493	−	0.588506i	$-0.200284\pi$
$101$	−107.000	−	185.330i	−0.105415	−	0.182584i	−0.913908	+	0.405922i	$0.866950\pi$
$102$		0			0
$103$	−836.160	+	1448.27i	−0.799897	+	1.38546i	0.119786	+	0.992800i	$0.461779\pi$
$103$	−836.160	+	1448.27i	−0.799897	+	1.38546i	−0.919683	+	0.392662i	$0.871554\pi$
$104$	−597.723	+	1035.29i	−0.563573	+	0.976137i
$105$		0			0
$106$	902.804	+	1563.70i	0.827246	+	1.43283i
$107$	600.699			0.542727			0.271363	−	0.962477i	$-0.412525\pi$
$107$	600.699			0.542727			0.271363	+	0.962477i	$0.412525\pi$
$108$		0			0
$109$	−771.570			−0.678009			−0.339005	−	0.940785i	$-0.610090\pi$
$109$	−771.570			−0.678009			−0.339005	+	0.940785i	$0.610090\pi$
$110$	−241.870	−	418.931i	−0.209649	−	0.363123i
$111$		0			0
$112$	−496.722	+	860.348i	−0.419070	+	0.725850i
$113$	−583.338	+	1010.37i	−0.485627	+	0.841131i	−0.999864	−	0.0165177i	$-0.994742\pi$
$113$	−583.338	+	1010.37i	−0.485627	+	0.841131i	0.514237	+	0.857648i	$0.328075\pi$
$114$		0			0
$115$	199.171	+	344.974i	0.161503	+	0.279731i
$116$	−11.3140			−0.00905583
$117$		0			0
$118$	1345.85			1.04996
$119$	−747.692	−	1295.04i	−0.575973	−	0.997615i
$120$		0			0
$121$	159.663	−	276.545i	0.119957	−	0.207772i
$122$	219.882	−	380.846i	0.163173	−	0.282624i
$123$		0			0
$124$	20.8996	+	36.1991i	0.0151358	+	0.0262159i
$125$	125.000			0.0894427
$126$		0			0
$127$	1630.10			1.13896			0.569479	−	0.822006i	$-0.307145\pi$
$127$	1630.10			1.13896			0.569479	+	0.822006i	$0.307145\pi$
$128$	846.322	+	1465.87i	0.584414	+	1.01223i
$129$		0			0
$130$	442.901	−	767.128i	0.298808	−	0.517550i
$131$	129.412	−	224.149i	0.0863115	−	0.149496i	−0.819638	−	0.572882i	$-0.805825\pi$
$131$	129.412	−	224.149i	0.0863115	−	0.149496i	0.905949	+	0.423386i	$0.139159\pi$
$132$		0			0
$133$	889.206	+	1540.15i	0.579729	+	1.00412i
$134$	−2624.17			−1.69175
$135$		0			0
$136$	−2238.36			−1.41131
$137$	526.385	+	911.726i	0.328264	+	0.568569i	0.982167	−	0.188009i	$-0.0602033\pi$
$137$	526.385	+	911.726i	0.328264	+	0.568569i	−0.653904	+	0.756578i	$0.726870\pi$
$138$		0			0
$139$	−1192.25	+	2065.03i	−0.727519	+	1.26010i	0.230410	+	0.973094i	$0.425993\pi$
$139$	−1192.25	+	2065.03i	−0.727519	+	1.26010i	−0.957929	+	0.287006i	$0.907340\pi$
$140$	42.9257	−	74.3495i	0.0259135	−	0.0448834i
$141$		0			0
$142$	1244.89	+	2156.22i	0.735699	+	1.27427i
$143$	−1852.53			−1.08333
$144$		0			0
$145$	−45.1769			−0.0258740
$146$	752.910	+	1304.08i	0.426790	+	0.739221i
$147$		0			0
$148$	13.8771	−	24.0358i	0.00770735	−	0.0133495i
$149$	573.144	−	992.715i	0.315126	−	0.545815i	−0.664338	−	0.747432i	$-0.731286\pi$
$149$	573.144	−	992.715i	0.315126	−	0.545815i	0.979464	+	0.201617i	$0.0646198\pi$
$150$		0			0
$151$	583.058	+	1009.89i	0.314229	+	0.544261i	0.979273	−	0.202543i	$-0.0649207\pi$
$151$	583.058	+	1009.89i	0.314229	+	0.544261i	−0.665044	+	0.746804i	$0.731587\pi$
$152$	2662.01			1.42051
$153$		0			0
$154$	−1326.63			−0.694175
$155$	83.4522	+	144.544i	0.0432454	+	0.0749033i
$156$		0			0
$157$	1737.81	−	3009.97i	0.883390	−	1.53008i	0.0358423	−	0.999357i	$-0.488589\pi$
$157$	1737.81	−	3009.97i	0.883390	−	1.53008i	0.847548	−	0.530719i	$-0.178078\pi$
$158$	1780.22	−	3083.43i	0.896371	−	1.55256i
$159$		0			0
$160$	−140.430	−	243.232i	−0.0693872	−	0.120182i
$161$	1092.43			0.534755
$162$		0			0
$163$	−2223.32			−1.06837			−0.534183	−	0.845369i	$-0.679381\pi$
$163$	−2223.32			−1.06837			−0.534183	+	0.845369i	$0.679381\pi$
$164$	76.2204	+	132.018i	0.0362915	+	0.0628588i
$165$		0			0
$166$	644.915	−	1117.03i	0.301537	−	0.522277i
$167$	−793.991	+	1375.23i	−0.367909	+	0.637238i	−0.989239	−	0.146312i	$-0.953260\pi$
$167$	−793.991	+	1375.23i	−0.367909	+	0.637238i	0.621329	+	0.783550i	$0.286593\pi$
$168$		0			0
$169$	−597.632	−	1035.13i	−0.272022	−	0.471156i
$170$	1658.58			0.748280
$171$		0			0
$172$	12.6993			0.00562974
$173$	753.898	+	1305.79i	0.331317	+	0.573857i	0.982770	−	0.184831i	$-0.0591738\pi$
$173$	753.898	+	1305.79i	0.331317	+	0.573857i	−0.651454	+	0.758689i	$0.725840\pi$
$174$		0			0
$175$	171.403	−	296.878i	0.0740390	−	0.128239i
$176$	−1152.19	+	1995.66i	−0.493465	+	0.854707i
$177$		0			0
$178$	−1568.58	−	2716.85i	−0.660504	−	1.14403i
$179$	−1572.66			−0.656683			−0.328341	−	0.944559i	$-0.606490\pi$
$179$	−1572.66			−0.656683			−0.328341	+	0.944559i	$0.606490\pi$
$180$		0			0
$181$	1984.41			0.814918			0.407459	−	0.913223i	$-0.366415\pi$
$181$	1984.41			0.814918			0.407459	+	0.913223i	$0.366415\pi$
$182$	−1214.63	−	2103.81i	−0.494695	−	0.856838i
$183$		0			0
$184$	817.601	−	1416.13i	0.327578	−	0.567382i
$185$	55.4113	−	95.9751i	0.0220212	−	0.0381418i
$186$		0			0
$187$	−1734.34	−	3003.97i	−0.678223	−	1.17472i
$188$	552.831			0.214465
$189$		0			0
$190$	−1972.50			−0.753158
$191$	−1174.18	−	2033.73i	−0.444819	−	0.770450i	0.553220	−	0.833035i	$-0.313399\pi$
$191$	−1174.18	−	2033.73i	−0.444819	−	0.770450i	−0.998040	+	0.0625853i	$0.980065\pi$
$192$		0			0
$193$	117.085	−	202.797i	0.0436681	−	0.0756354i	−0.843365	−	0.537341i	$-0.819429\pi$
$193$	117.085	−	202.797i	0.0436681	−	0.0756354i	0.887033	+	0.461705i	$0.152762\pi$
$194$	2431.43	−	4211.37i	0.899829	−	1.55855i
$195$		0			0
$196$	97.0288	+	168.059i	0.0353604	+	0.0612460i
$197$	717.846			0.259616			0.129808	−	0.991539i	$-0.458564\pi$
$197$	717.846			0.259616			0.129808	+	0.991539i	$0.458564\pi$
$198$		0			0
$199$	1701.45			0.606094			0.303047	−	0.952976i	$-0.401996\pi$
$199$	1701.45			0.606094			0.303047	+	0.952976i	$0.401996\pi$
$200$	−256.564	−	444.381i	−0.0907090	−	0.157113i
$201$		0			0
$202$	325.467	−	563.725i	0.113365	−	0.196354i
$203$	−61.9475	+	107.296i	−0.0214180	+	0.0370971i
$204$		0			0
$205$	304.349	+	527.148i	0.103691	+	0.179598i
$206$	−5086.77			−1.72045
$207$		0			0
$208$	−4219.69			−1.40665
$209$	2062.60	+	3572.52i	0.682645	+	1.18238i
$210$		0			0
$211$	2469.33	−	4277.01i	0.805667	−	1.39546i	−0.110172	−	0.993912i	$-0.535140\pi$
$211$	2469.33	−	4277.01i	0.805667	−	1.39546i	0.915840	−	0.401544i	$-0.131526\pi$
$212$	−371.656	+	643.727i	−0.120403	+	0.208544i
$213$		0			0
$214$	913.585	+	1582.38i	0.291829	+	0.505463i
$215$	50.7086			0.0160851
$216$		0			0
$217$	457.726			0.143191
$218$	−1173.46	−	2032.49i	−0.364572	−	0.631456i
$219$		0			0
$220$	99.5703	−	172.461i	0.0305137	−	0.0528514i
$221$	3175.85	−	5500.73i	0.966655	−	1.67430i
$222$		0			0
$223$	1526.57	+	2644.10i	0.458416	+	0.794001i	0.998877	−	0.0473684i	$-0.0150835\pi$
$223$	1526.57	+	2644.10i	0.458416	+	0.794001i	−0.540461	+	0.841369i	$0.681750\pi$
$224$	−770.242			−0.229750
$225$		0			0
$226$	−3548.73			−1.04450
$227$	−419.976	−	727.420i	−0.122797	−	0.212690i	0.798073	−	0.602561i	$-0.205853\pi$
$227$	−419.976	−	727.420i	−0.122797	−	0.212690i	−0.920869	+	0.389871i	$0.872520\pi$
$228$		0			0
$229$	−700.753	+	1213.74i	−0.202214	+	0.350245i	−0.949242	−	0.314548i	$-0.898147\pi$
$229$	−700.753	+	1213.74i	−0.202214	+	0.350245i	0.747027	+	0.664793i	$0.231480\pi$
$230$	−605.827	+	1049.32i	−0.173683	+	0.300827i
$231$		0			0
$232$	92.7259	+	160.606i	0.0262403	+	0.0454496i
$233$	2856.99			0.803295			0.401647	−	0.915794i	$-0.368438\pi$
$233$	2856.99			0.803295			0.401647	+	0.915794i	$0.368438\pi$
$234$		0			0
$235$	2207.46			0.612761
$236$	277.021	+	479.815i	0.0764092	+	0.132345i
$237$		0			0
$238$	2274.29	−	3939.18i	0.619412	−	1.07285i
$239$	−854.917	+	1480.76i	−0.231381	+	0.400763i	−0.958215	−	0.286050i	$-0.907658\pi$
$239$	−854.917	+	1480.76i	−0.231381	+	0.400763i	0.726834	+	0.686813i	$0.240991\pi$
$240$		0			0
$241$	2166.23	+	3752.02i	0.579000	+	1.00286i	0.995594	+	0.0937648i	$0.0298902\pi$
$241$	2166.23	+	3752.02i	0.579000	+	1.00286i	−0.416594	+	0.909092i	$0.636776\pi$
$242$	971.309			0.258009
$243$		0			0
$244$	181.036			0.0474987
$245$	387.437	+	671.061i	0.101030	+	0.174990i
$246$		0			0
$247$	−3776.93	+	6541.84i	−0.972957	+	1.68521i
$248$	342.573	−	593.354i	0.0877154	−	0.151927i
$249$		0			0
$250$	190.109	+	329.278i	0.0480941	+	0.0833015i
$251$	1724.73			0.433722			0.216861	−	0.976202i	$-0.430418\pi$
$251$	1724.73			0.433722			0.216861	+	0.976202i	$0.430418\pi$
$252$		0			0
$253$	2534.00			0.629688
$254$	2479.16	+	4294.04i	0.612428	+	1.06076i
$255$		0			0
$256$	−939.349	+	1627.00i	−0.229333	+	0.397217i
$257$	−2926.63	+	5069.07i	−0.710342	+	1.23035i	0.254386	+	0.967103i	$0.418127\pi$
$257$	−2926.63	+	5069.07i	−0.710342	+	1.23035i	−0.964729	+	0.263246i	$0.915207\pi$
$258$		0			0
$259$	−151.962	−	263.206i	−0.0364574	−	0.0631462i
$260$	364.657			0.0869810
$261$		0			0
$262$	787.277			0.185642
$263$	196.180	+	339.794i	0.0459962	+	0.0796677i	0.888107	−	0.459637i	$-0.152020\pi$
$263$	196.180	+	339.794i	0.0459962	+	0.0796677i	−0.842111	+	0.539305i	$0.818687\pi$
$264$		0			0
$265$	−1484.03	+	2570.41i	−0.344011	+	0.595845i
$266$	−2704.73	+	4684.73i	−0.623450	+	1.07985i
$267$		0			0
$268$	−540.144	−	935.557i	−0.123114	−	0.213240i
$269$	4610.33			1.04497			0.522485	−	0.852648i	$-0.325005\pi$
$269$	4610.33			1.04497			0.522485	+	0.852648i	$0.325005\pi$
$270$		0			0
$271$	−1155.72			−0.259058			−0.129529	−	0.991576i	$-0.541347\pi$
$271$	−1155.72			−0.259058			−0.129529	+	0.991576i	$0.541347\pi$
$272$	−3950.49	−	6842.44i	−0.880638	−	1.52531i
$273$		0			0
$274$	−1601.13	+	2773.23i	−0.353020	+	0.611449i
$275$	397.585	−	688.638i	0.0871829	−	0.151005i
$276$		0			0
$277$	304.205	+	526.898i	0.0659852	+	0.114290i	0.897131	−	0.441765i	$-0.145648\pi$
$277$	304.205	+	526.898i	0.0659852	+	0.114290i	−0.831145	+	0.556055i	$0.812314\pi$
$278$	−7253.02			−1.56477
$279$		0			0
$280$	−1407.22			−0.300349
$281$	1374.62	+	2380.91i	0.291825	+	0.505456i	0.974241	−	0.225508i	$-0.0724042\pi$
$281$	1374.62	+	2380.91i	0.291825	+	0.505456i	−0.682416	+	0.730964i	$0.739071\pi$
$282$		0			0
$283$	−510.780	+	884.697i	−0.107289	+	0.185830i	−0.914671	−	0.404199i	$-0.867550\pi$
$283$	−510.780	+	884.697i	−0.107289	+	0.185830i	0.807382	+	0.590029i	$0.200884\pi$
$284$	−512.483	+	887.647i	−0.107078	+	0.185465i
$285$		0			0
$286$	−2817.46	−	4879.98i	−0.582516	−	1.00895i
$287$	1669.32			0.343334
$288$		0			0
$289$	6979.96			1.42071
$290$	−68.7082	−	119.006i	−0.0139127	−	0.0240975i
$291$		0			0
$292$	−309.949	+	536.848i	−0.0621178	+	0.107591i
$293$	−2616.25	+	4531.47i	−0.521648	+	0.903521i	0.478035	+	0.878341i	$0.341349\pi$
$293$	−2616.25	+	4531.47i	−0.521648	+	0.903521i	−0.999683	+	0.0251800i	$0.991984\pi$
$294$		0			0
$295$	1106.15	+	1915.91i	0.218314	+	0.378130i
$296$	−454.929			−0.0893317
$297$		0			0
$298$	3486.71			0.677785
$299$	2320.07	+	4018.48i	0.448739	+	0.777240i
$300$		0			0
$301$	69.5327	−	120.434i	0.0133149	−	0.0230622i
$302$	−1773.51	+	3071.81i	−0.337927	+	0.585307i
$303$		0			0
$304$	4698.18	+	8137.49i	0.886379	+	1.53525i
$305$	722.881			0.135712
$306$		0			0
$307$	−4912.68			−0.913296			−0.456648	−	0.889648i	$-0.650950\pi$
$307$	−4912.68			−0.913296			−0.456648	+	0.889648i	$0.650950\pi$
$308$	−273.066	−	472.964i	−0.0505174	−	0.0874988i
$309$		0			0
$310$	−253.840	+	439.664i	−0.0465069	+	0.0805524i
$311$	−414.430	+	717.813i	−0.0755632	+	0.130879i	−0.901331	−	0.433131i	$-0.857409\pi$
$311$	−414.430	+	717.813i	−0.0755632	+	0.130879i	0.825768	+	0.564010i	$0.190742\pi$
$312$		0			0
$313$	−4618.60	−	7999.66i	−0.834054	−	1.44462i	−0.894798	−	0.446471i	$-0.852681\pi$
$313$	−4618.60	−	7999.66i	−0.834054	−	1.44462i	0.0607442	−	0.998153i	$-0.480653\pi$
$314$	10571.9			1.90003
$315$		0			0
$316$	1465.72			0.260928
$317$	−4549.20	−	7879.45i	−0.806021	−	1.39607i	−0.915599	−	0.402091i	$-0.868283\pi$
$317$	−4549.20	−	7879.45i	−0.806021	−	1.39607i	0.109578	−	0.993978i	$-0.465050\pi$
$318$		0			0
$319$	−143.693	+	248.884i	−0.0252203	+	0.0436828i
$320$	−1021.84	+	1769.88i	−0.178508	+	0.309185i
$321$		0			0
$322$	1661.45	+	2877.71i	0.287543	+	0.498039i
$323$	−14143.9			−2.43649
$324$		0			0
$325$	1456.08			0.248519
$326$	−3381.38	−	5856.72i	−0.574470	−	0.995011i
$327$		0			0
$328$	1249.36	−	2163.95i	0.210318	−	0.364281i
$329$	3026.92	−	5242.78i	0.507233	−	0.878553i
$330$		0			0
$331$	−2981.73	−	5164.50i	−0.495137	−	0.857603i	0.504847	−	0.863209i	$-0.331549\pi$
$331$	−2981.73	−	5164.50i	−0.495137	−	0.857603i	−0.999984	+	0.00560570i	$0.998216\pi$
$332$	530.982			0.0877754
$333$		0			0
$334$	−4830.23			−0.791313
$335$	−2156.80	−	3735.69i	−0.351757	−	0.609261i
$336$		0			0
$337$	−298.427	+	516.890i	−0.0482384	+	0.0835514i	−0.889136	−	0.457642i	$-0.848694\pi$
$337$	−298.427	+	516.890i	−0.0482384	+	0.0835514i	0.840898	+	0.541194i	$0.182027\pi$
$338$	1817.84	−	3148.60i	0.292537	−	0.506690i
$339$		0			0
$340$	341.393	+	591.310i	0.0544548	+	0.0943185i
$341$	1061.74			0.168611
$342$		0			0
$343$	6828.34			1.07492
$344$	−104.080	−	180.272i	−0.0163128	−	0.0282546i
$345$		0			0
$346$	−2293.16	+	3971.87i	−0.356304	+	0.617137i
$347$	−2281.05	+	3950.90i	−0.352891	+	0.611226i	−0.986755	−	0.162219i	$-0.948135\pi$
$347$	−2281.05	+	3950.90i	−0.352891	+	0.611226i	0.633863	+	0.773445i	$0.281468\pi$
$348$		0			0
$349$	−337.633	−	584.798i	−0.0517854	−	0.0896949i	0.838971	−	0.544177i	$-0.183158\pi$
$349$	−337.633	−	584.798i	−0.0517854	−	0.0896949i	−0.890756	+	0.454482i	$0.849825\pi$
$350$	1042.73			0.159246
$351$		0			0
$352$	−1786.65			−0.270536
$353$	−1680.91	−	2911.43i	−0.253445	−	0.438979i	0.711027	−	0.703165i	$-0.248230\pi$
$353$	−1680.91	−	2911.43i	−0.253445	−	0.438979i	−0.964472	+	0.264185i	$0.914897\pi$
$354$		0			0
$355$	−2046.35	+	3544.38i	−0.305941	+	0.529905i
$356$	645.732	−	1118.44i	0.0961341	−	0.166509i
$357$		0			0
$358$	−2391.81	−	4142.74i	−0.353104	−	0.611594i
$359$	−7701.61			−1.13224			−0.566122	−	0.824322i	$-0.691557\pi$
$359$	−7701.61			−1.13224			−0.566122	+	0.824322i	$0.691557\pi$
$360$		0			0
$361$	9961.87			1.45238
$362$	3018.03	+	5227.39i	0.438189	+	0.758965i
$363$		0			0
$364$	500.026	−	866.070i	0.0720013	−	0.124710i
$365$	−1237.63	+	2143.64i	−0.177481	+	0.307406i
$366$		0			0
$367$	−2954.75	−	5117.77i	−0.420263	−	0.727917i	0.575702	−	0.817660i	$-0.304729\pi$
$367$	−2954.75	−	5117.77i	−0.420263	−	0.727917i	−0.995965	+	0.0897425i	$0.971396\pi$
$368$	5771.94			0.817617
$369$		0			0
$370$	337.093			0.0473639
$371$	4069.86	+	7049.20i	0.569532	+	0.986459i
$372$		0			0
$373$	4212.19	−	7295.73i	0.584716	−	1.01276i	−0.410195	−	0.911998i	$-0.634539\pi$
$373$	4212.19	−	7295.73i	0.584716	−	1.01276i	0.994911	−	0.100760i	$-0.0321273\pi$
$374$	5275.42	−	9137.30i	0.729374	−	1.26331i
$375$		0			0
$376$	−4530.84	−	7847.64i	−0.621436	−	1.07636i
$377$	−526.249			−0.0718917
$378$		0			0
$379$	−3510.24			−0.475749			−0.237875	−	0.971296i	$-0.576451\pi$
$379$	−3510.24			−0.475749			−0.237875	+	0.971296i	$0.576451\pi$
$380$	−406.007	−	703.225i	−0.0548098	−	0.0949334i
$381$		0			0
$382$	3571.54	−	6186.09i	0.478367	−	0.828555i
$383$	2866.01	−	4964.08i	0.382367	−	0.662279i	−0.609033	−	0.793145i	$-0.708442\pi$
$383$	2866.01	−	4964.08i	0.382367	−	0.662279i	0.991400	+	0.130866i	$0.0417758\pi$
$384$		0			0
$385$	−1090.36	−	1888.55i	−0.144337	−	0.249999i
$386$	712.283			0.0939229
$387$		0			0
$388$	2001.89			0.261934
$389$	7183.60	+	12442.4i	0.936306	+	1.62173i	0.772288	+	0.635273i	$0.219112\pi$
$389$	7183.60	+	12442.4i	0.936306	+	1.62173i	0.164019	+	0.986457i	$0.447554\pi$
$390$		0			0
$391$	−4344.11	+	7524.23i	−0.561870	+	0.973188i
$392$	1590.44	−	2754.72i	0.204922	−	0.354934i
$393$		0			0
$394$	1091.75	+	1890.97i	0.139598	+	0.241791i
$395$	5852.63			0.745514
$396$		0			0
$397$	−9014.42			−1.13960			−0.569800	−	0.821784i	$-0.692979\pi$
$397$	−9014.42			−1.13960			−0.569800	+	0.821784i	$0.692979\pi$
$398$	2587.69	+	4482.01i	0.325902	+	0.564479i
$399$		0			0
$400$	905.619	−	1568.58i	0.113202	−	0.196072i
$401$	5255.55	−	9102.87i	0.654487	−	1.13361i	−0.327535	−	0.944839i	$-0.606218\pi$
$401$	5255.55	−	9102.87i	0.654487	−	1.13361i	0.982022	−	0.188766i	$-0.0604488\pi$
$402$		0			0
$403$	972.104	+	1683.73i	0.120159	+	0.208121i
$404$	267.969			0.0329998
$405$		0			0
$406$	−376.857			−0.0460667
$407$	−352.491	−	610.533i	−0.0429296	−	0.0743562i
$408$		0			0
$409$	−1659.10	+	2873.65i	−0.200580	+	0.347415i	−0.948715	−	0.316131i	$-0.897616\pi$
$409$	−1659.10	+	2873.65i	−0.200580	+	0.347415i	0.748135	+	0.663546i	$0.230949\pi$
$410$	−925.751	+	1603.45i	−0.111511	+	0.193143i
$411$		0			0
$412$	−1047.03	−	1813.51i	−0.125203	−	0.216857i
$413$	6067.11			0.722864
$414$		0			0
$415$	2120.22			0.250789
$416$	−1635.81	−	2833.31i	−0.192794	−	0.333929i
$417$		0			0
$418$	−6273.89	+	10866.7i	−0.734129	+	1.27155i
$419$	−243.069	+	421.008i	−0.0283406	+	0.0490873i	−0.879848	−	0.475255i	$-0.842356\pi$
$419$	−243.069	+	421.008i	−0.0283406	+	0.0490873i	0.851507	+	0.524343i	$0.175689\pi$
$420$		0			0
$421$	−2525.96	−	4375.10i	−0.292418	−	0.506482i	0.681963	−	0.731387i	$-0.261126\pi$
$421$	−2525.96	−	4375.10i	−0.292418	−	0.506482i	−0.974381	+	0.224904i	$0.927793\pi$
$422$	15022.1			1.73286
$423$		0			0
$424$	12183.9			1.39553
$425$	1363.19	+	2361.11i	0.155586	+	0.269484i
$426$		0			0
$427$	991.230	−	1716.86i	0.112340	−	0.194578i
$428$	−376.094	+	651.414i	−0.0424747	+	0.0735684i
$429$		0			0
$430$	77.1212	+	133.578i	0.00864910	+	0.0149807i
$431$	−6944.24			−0.776084			−0.388042	−	0.921642i	$-0.626848\pi$
$431$	−6944.24			−0.776084			−0.388042	+	0.921642i	$0.626848\pi$
$432$		0			0
$433$	−13738.3			−1.52476			−0.762378	−	0.647131i	$-0.775968\pi$
$433$	−13738.3			−1.52476			−0.762378	+	0.647131i	$0.775968\pi$
$434$	696.142	+	1205.75i	0.0769952	+	0.133360i
$435$		0			0
$436$	483.075	−	836.711i	0.0530622	−	0.0919064i
$437$	5166.31	−	8948.31i	0.565533	−	0.979533i
$438$		0			0
$439$	−4240.92	−	7345.48i	−0.461066	−	0.798589i	0.537949	−	0.842978i	$-0.319199\pi$
$439$	−4240.92	−	7345.48i	−0.461066	−	0.798589i	−0.999014	+	0.0443883i	$0.985866\pi$
$440$	−3264.19			−0.353668
$441$		0			0
$442$	19320.2			2.07912
$443$	−279.098	−	483.411i	−0.0299330	−	0.0518455i	0.850671	−	0.525699i	$-0.176196\pi$
$443$	−279.098	−	483.411i	−0.0299330	−	0.0518455i	−0.880604	+	0.473853i	$0.842863\pi$
$444$		0			0
$445$	2578.42	−	4465.95i	0.274671	−	0.475744i
$446$	−4643.44	+	8042.67i	−0.492989	+	0.853882i
$447$		0			0
$448$	2802.34	+	4853.79i	0.295531	+	0.511875i
$449$	−14775.0			−1.55295			−0.776476	−	0.630147i	$-0.782995\pi$
$449$	−14775.0			−1.55295			−0.776476	+	0.630147i	$0.782995\pi$
$450$		0			0
$451$	3872.15			0.404285
$452$	−730.449	−	1265.17i	−0.0760120	−	0.131657i
$453$		0			0
$454$	1277.46	−	2212.62i	0.132058	−	0.228730i
$455$	1996.61	−	3458.23i	0.205720	−	0.356317i
$456$		0			0
$457$	7103.97	+	12304.4i	0.727154	+	1.25947i	0.958081	+	0.286497i	$0.0924909\pi$
$457$	7103.97	+	12304.4i	0.727154	+	1.25947i	−0.230927	+	0.972971i	$0.574176\pi$
$458$	−4263.02			−0.434930
$459$		0			0
$460$	−498.799			−0.0505579
$461$	−3851.26	−	6670.57i	−0.389091	−	0.673925i	0.603237	−	0.797562i	$-0.293877\pi$
$461$	−3851.26	−	6670.57i	−0.389091	−	0.673925i	−0.992328	+	0.123637i	$0.960544\pi$
$462$		0			0
$463$	−3954.53	+	6849.45i	−0.396939	+	0.687518i	−0.993347	−	0.115163i	$-0.963261\pi$
$463$	−3954.53	+	6849.45i	−0.396939	+	0.687518i	0.596408	+	0.802682i	$0.296594\pi$
$464$	−327.304	+	566.908i	−0.0327472	+	0.0567199i
$465$		0			0
$466$	4345.11	+	7525.96i	0.431939	+	0.748140i
$467$	−11639.8			−1.15337			−0.576685	−	0.816967i	$-0.695654\pi$
$467$	−11639.8			−1.15337			−0.576685	+	0.816967i	$0.695654\pi$
$468$		0			0
$469$	−11829.8			−1.16471
$470$	3357.26	+	5814.95i	0.329487	+	0.570688i
$471$		0			0
$472$	4540.77	−	7864.84i	0.442809	−	0.766967i
$473$	161.288	−	279.359i	0.0156787	−	0.0271563i
$474$		0			0
$475$	−1621.19	−	2807.99i	−0.156601	−	0.271240i
$476$	1872.50			0.180307
$477$		0			0
$478$	−5200.87			−0.497662
$479$	−9347.93	−	16191.1i	−0.891687	−	1.54445i	−0.837853	−	0.545897i	$-0.816189\pi$
$479$	−9347.93	−	16191.1i	−0.891687	−	1.54445i	−0.0538341	−	0.998550i	$-0.517144\pi$
$480$		0			0
$481$	645.465	−	1117.98i	0.0611865	−	0.105978i
$482$	−6589.10	+	11412.7i	−0.622667	+	1.07849i
$483$		0			0
$484$	199.928	+	346.286i	0.0187761	+	0.0325212i
$485$	7993.56			0.748390
$486$		0			0
$487$	−2249.54			−0.209315			−0.104658	−	0.994508i	$-0.533375\pi$
$487$	−2249.54			−0.209315			−0.104658	+	0.994508i	$0.533375\pi$
$488$	−1483.72	−	2569.88i	−0.137633	−	0.238387i
$489$		0			0
$490$	−1178.48	+	2041.19i	−0.108650	+	0.188187i
$491$	994.175	−	1721.96i	0.0913778	−	0.158271i	−0.816713	−	0.577044i	$-0.804206\pi$
$491$	994.175	−	1721.96i	0.0913778	−	0.158271i	0.908091	+	0.418773i	$0.137540\pi$
$492$		0			0
$493$	−492.676	−	853.339i	−0.0450081	−	0.0779563i
$494$	−22976.9			−2.09267
$495$		0			0
$496$	2418.43			0.218933
$497$	5612.01	+	9720.28i	0.506505	+	0.877292i
$498$		0			0
$499$	−1206.95	+	2090.50i	−0.108278	+	0.187543i	−0.915073	−	0.403289i	$-0.867867\pi$
$499$	−1206.95	+	2090.50i	−0.108278	+	0.187543i	0.806795	+	0.590832i	$0.201200\pi$
$500$	−78.2617	+	135.553i	−0.00699994	+	0.0121243i
$501$		0			0
$502$	2623.10	+	4543.34i	0.233216	+	0.403942i
$503$	8758.56			0.776391			0.388196	−	0.921577i	$-0.373098\pi$
$503$	8758.56			0.776391			0.388196	+	0.921577i	$0.373098\pi$
$504$		0			0
$505$	1070.00			0.0942860
$506$	3853.88	+	6675.12i	0.338589	+	0.586453i
$507$		0			0
$508$	−1020.59	+	1767.72i	−0.0891368	+	0.154389i
$509$	−5924.01	+	10260.7i	−0.515869	+	0.893511i	0.483961	+	0.875089i	$0.339198\pi$
$509$	−5924.01	+	10260.7i	−0.515869	+	0.893511i	−0.999830	+	0.0184220i	$0.994136\pi$
$510$		0			0
$511$	3394.13	+	5878.81i	0.293831	+	0.508930i
$512$	7826.64			0.675570
$513$		0			0
$514$	−17804.1			−1.52783
$515$	−4180.80	−	7241.36i	−0.357725	−	0.619597i
$516$		0			0
$517$	7021.23	−	12161.1i	0.597279	−	1.03452i
$518$	462.230	−	800.606i	0.0392070	−	0.0679085i
$519$		0			0
$520$	−2988.62	−	5176.43i	−0.252037	−	0.436542i
$521$	−3816.55			−0.320933			−0.160466	−	0.987041i	$-0.551300\pi$
$521$	−3816.55			−0.320933			−0.160466	+	0.987041i	$0.551300\pi$
$522$		0			0
$523$	−12158.9			−1.01658			−0.508288	−	0.861187i	$-0.669722\pi$
$523$	−12158.9			−1.01658			−0.508288	+	0.861187i	$0.669722\pi$
$524$	162.049	+	280.676i	0.0135098	+	0.0233996i
$525$		0			0
$526$	−596.729	+	1033.57i	−0.0494651	+	0.0856761i
$527$	−1820.17	+	3152.63i	−0.150452	+	0.260590i
$528$		0			0
$529$	2909.97	+	5040.22i	0.239169	+	0.414253i
$530$	−9028.04			−0.739911
$531$		0			0
$532$	−2226.90			−0.181482
$533$	3545.25	+	6140.55i	0.288108	+	0.499018i
$534$		0			0
$535$	−1501.75	+	2601.10i	−0.121357	+	0.210197i
$536$	−8853.71	+	15335.1i	−0.713474	+	1.23577i
$537$		0			0
$538$	7011.72	+	12144.7i	0.561890	+	0.973222i
$539$	4929.26			0.393911
$540$		0			0
$541$	14919.8			1.18568			0.592840	−	0.805321i	$-0.298007\pi$
$541$	14919.8			1.18568			0.592840	+	0.805321i	$0.298007\pi$
$542$	−1757.69	−	3044.42i	−0.139298	−	0.241271i
$543$		0			0
$544$	3062.91	−	5305.12i	0.241399	−	0.418116i
$545$	1928.92	−	3341.00i	0.151607	−	0.262592i
$546$		0			0
$547$	513.578	+	889.543i	0.0401444	+	0.0695322i	0.885400	−	0.464831i	$-0.153885\pi$
$547$	513.578	+	889.543i	0.0401444	+	0.0695322i	−0.845255	+	0.534363i	$0.820552\pi$
$548$	−1318.27			−0.102762
$549$		0			0
$550$	2418.70			0.187516
$551$	585.923	+	1014.85i	0.0453015	+	0.0784646i
$552$		0			0
$553$	8025.26	−	13900.2i	0.617123	−	1.06889i
$554$	−925.313	+	1602.69i	−0.0709617	+	0.122909i
$555$		0			0
$556$	−1492.92	−	2585.81i	−0.113874	−	0.197235i
$557$	10590.8			0.805648			0.402824	−	0.915277i	$-0.368029\pi$
$557$	10590.8			0.805648			0.402824	+	0.915277i	$0.368029\pi$
$558$		0			0
$559$	590.685			0.0446929
$560$	−2483.61	−	4301.74i	−0.187414	−	0.324610i
$561$		0			0
$562$	−4181.23	+	7242.11i	−0.313834	+	0.543576i
$563$	−1096.03	+	1898.38i	−0.0820467	+	0.142109i	−0.904129	−	0.427260i	$-0.859479\pi$
$563$	−1096.03	+	1898.38i	−0.0820467	+	0.142109i	0.822082	+	0.569369i	$0.192812\pi$
$564$		0			0
$565$	−2916.69	−	5051.86i	−0.217179	−	0.376165i
$566$	−3107.32			−0.230761
$567$		0			0
$568$	16800.6			1.24109
$569$	−8284.89	−	14349.8i	−0.610405	−	1.05725i	−0.991172	−	0.132582i	$-0.957673\pi$
$569$	−8284.89	−	14349.8i	−0.610405	−	1.05725i	0.380767	−	0.924671i	$-0.375660\pi$
$570$		0			0
$571$	−3639.66	+	6304.08i	−0.266752	+	0.462027i	−0.968021	−	0.250869i	$-0.919284\pi$
$571$	−3639.66	+	6304.08i	−0.266752	+	0.462027i	0.701269	+	0.712896i	$0.252617\pi$
$572$	1159.86	−	2008.93i	0.0847833	−	0.146849i
$573$		0			0
$574$	2538.82	+	4397.36i	0.184614	+	0.319760i
$575$	−1991.71			−0.144452
$576$		0			0
$577$	−17938.0			−1.29422			−0.647112	−	0.762395i	$-0.724024\pi$
$577$	−17938.0			−1.29422			−0.647112	+	0.762395i	$0.724024\pi$
$578$	10615.6	+	18386.8i	0.763930	+	1.32317i
$579$		0			0
$580$	28.2850	−	48.9910i	0.00202495	−	0.00350731i
$581$	2907.29	−	5035.57i	0.207598	−	0.359571i
$582$		0			0
$583$	9440.42	+	16351.3i	0.670639	+	1.16158i
$584$	10161.0			0.719974
$585$		0			0
$586$	−15915.9			−1.12198
$587$	5754.83	+	9967.67i	0.404646	+	0.700868i	0.994280	−	0.106803i	$-0.0340613\pi$
$587$	5754.83	+	9967.67i	0.404646	+	0.700868i	−0.589634	+	0.807671i	$0.700728\pi$
$588$		0			0
$589$	2164.67	−	3749.32i	0.151433	−	0.262289i
$590$	−3364.62	+	5827.70i	−0.234778	+	0.406648i
$591$		0			0
$592$	−802.904	−	1390.67i	−0.0557418	−	0.0965476i
$593$	9612.80			0.665684			0.332842	−	0.942983i	$-0.391992\pi$
$593$	9612.80			0.665684			0.332842	+	0.942983i	$0.391992\pi$
$594$		0			0
$595$	7476.92			0.515166
$596$	717.684	+	1243.07i	0.0493247	+	0.0854328i
$597$		0			0
$598$	−7057.05	+	12223.2i	−0.482582	+	0.835857i
$599$	3686.00	−	6384.33i	0.251429	−	0.435487i	−0.712491	−	0.701681i	$-0.752433\pi$
$599$	3686.00	−	6384.33i	0.251429	−	0.435487i	0.963919	+	0.266194i	$0.0857663\pi$
$600$		0			0
$601$	12095.3	+	20949.7i	0.820927	+	1.42189i	0.904993	+	0.425427i	$0.139876\pi$
$601$	12095.3	+	20949.7i	0.820927	+	1.42189i	−0.0840654	+	0.996460i	$0.526790\pi$
$602$	423.001			0.0286383
$603$		0			0
$604$	−1460.20			−0.0983684
$605$	798.317	+	1382.72i	0.0536466	+	0.0929186i
$606$		0			0
$607$	658.866	−	1141.19i	0.0440569	−	0.0763088i	−0.843156	−	0.537669i	$-0.819305\pi$
$607$	658.866	−	1141.19i	0.0440569	−	0.0763088i	0.887213	+	0.461360i	$0.152638\pi$
$608$	−3642.62	+	6309.20i	−0.242973	+	0.420842i
$609$		0			0
$610$	1099.41	+	1904.23i	0.0729733	+	0.126393i
$611$	25713.9			1.70257
$612$		0			0
$613$	−4137.52			−0.272615			−0.136307	−	0.990667i	$-0.543523\pi$
$613$	−4137.52			−0.272615			−0.136307	+	0.990667i	$0.543523\pi$
$614$	−7471.56	−	12941.1i	−0.491087	−	0.850588i
$615$		0			0
$616$	−4475.93	+	7752.54i	−0.292760	+	0.507076i
$617$	−3895.81	+	6747.74i	−0.254197	+	0.440281i	−0.964677	−	0.263436i	$-0.915144\pi$
$617$	−3895.81	+	6747.74i	−0.254197	+	0.440281i	0.710480	+	0.703717i	$0.248478\pi$
$618$		0			0
$619$	−10978.2	−	19014.8i	−0.712844	−	1.23468i	−0.963785	−	0.266680i	$-0.914073\pi$
$619$	−10978.2	−	19014.8i	−0.712844	−	1.23468i	0.250941	−	0.968002i	$-0.419260\pi$
$620$	−208.996			−0.0135379
$621$		0			0
$622$	−2521.18			−0.162524
$623$	−7071.17	−	12247.6i	−0.454736	−	0.787625i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$	14048.6	−	24332.9i	0.896957	−	1.55357i
$627$		0			0
$628$	2176.06	+	3769.05i	0.138271	+	0.239493i
$629$	2417.15			0.153224
$630$		0			0
$631$	−11152.7			−0.703618			−0.351809	−	0.936072i	$-0.614433\pi$
$631$	−11152.7			−0.703618			−0.351809	+	0.936072i	$0.614433\pi$
$632$	−12012.6	−	20806.4i	−0.756068	−	1.30955i
$633$		0			0
$634$	13837.5	−	23967.2i	0.866810	−	1.50136i
$635$	−4075.24	+	7058.52i	−0.254679	+	0.441116i
$636$		0			0
$637$	4513.11	+	7816.94i	0.280716	+	0.486214i
$638$	−874.155			−0.0542447
$639$		0			0
$640$	−8463.22			−0.522716
$641$	2404.72	+	4165.10i	0.148176	+	0.256648i	0.930553	−	0.366156i	$-0.119326\pi$
$641$	2404.72	+	4165.10i	0.148176	+	0.256648i	−0.782377	+	0.622805i	$0.785993\pi$
$642$		0			0
$643$	1276.51	−	2210.98i	0.0782904	−	0.135603i	−0.824222	−	0.566267i	$-0.808387\pi$
$643$	1276.51	−	2210.98i	0.0782904	−	0.135603i	0.902512	+	0.430664i	$0.141721\pi$
$644$	−683.964	+	1184.66i	−0.0418509	+	0.0724879i
$645$		0			0
$646$	−21511.0	−	37258.2i	−1.31013	−	2.26920i
$647$	−8446.00			−0.513210			−0.256605	−	0.966516i	$-0.582604\pi$
$647$	−8446.00			−0.513210			−0.256605	+	0.966516i	$0.582604\pi$
$648$		0			0
$649$	14073.2			0.851191
$650$	2214.51	+	3835.64i	0.133631	+	0.231456i
$651$		0			0
$652$	1392.01	−	2411.02i	0.0836122	−	0.144821i
$653$	4047.44	−	7010.37i	0.242555	−	0.420118i	−0.718886	−	0.695128i	$-0.755348\pi$
$653$	4047.44	−	7010.37i	0.242555	−	0.420118i	0.961441	+	0.275010i	$0.0886812\pi$
$654$		0			0
$655$	647.062	+	1120.74i	0.0385997	+	0.0668566i
$656$	8819.97			0.524942
$657$		0			0
$658$	18414.2			1.09097
$659$	1087.67	+	1883.89i	0.0642936	+	0.111360i	0.896380	−	0.443286i	$-0.146187\pi$
$659$	1087.67	+	1883.89i	0.0642936	+	0.111360i	−0.832087	+	0.554645i	$0.812854\pi$
$660$		0			0
$661$	4438.47	−	7687.66i	0.261175	−	0.452368i	−0.705379	−	0.708830i	$-0.749223\pi$
$661$	4438.47	−	7687.66i	0.261175	−	0.452368i	0.966554	+	0.256462i	$0.0825567\pi$
$662$	9069.64	−	15709.1i	0.532480	−	0.922282i
$663$		0			0
$664$	−4351.77	−	7537.48i	−0.254339	−	0.440529i
$665$	−8892.06			−0.518525
$666$		0			0
$667$	719.834			0.0417872
$668$	−994.226	−	1722.05i	−0.0575865	−	0.0997427i
$669$		0			0
$670$	6560.43	−	11363.0i	0.378286	−	0.655211i
$671$	2299.25	−	3982.42i	0.132283	−	0.229120i
$672$		0			0
$673$	4583.96	+	7939.65i	0.262554	+	0.454757i	0.966920	−	0.255080i	$-0.0821019\pi$
$673$	4583.96	+	7939.65i	0.262554	+	0.454757i	−0.704366	+	0.709837i	$0.748769\pi$
$674$	−1815.47			−0.103753
$675$		0			0
$676$	1496.70			0.0851557
$677$	−9438.93	−	16348.7i	−0.535846	−	0.928112i	−0.999122	−	0.0418983i	$-0.986659\pi$
$677$	−9438.93	−	16348.7i	−0.535846	−	0.928112i	0.463276	−	0.886214i	$-0.346674\pi$
$678$		0			0
$679$	10961.0	−	18984.9i	0.619503	−	1.07301i
$680$	5595.90	−	9692.39i	0.315578	−	0.546597i
$681$		0			0
$682$	1614.77	+	2796.86i	0.0906638	+	0.157034i
$683$	27207.5			1.52426			0.762128	−	0.647426i	$-0.224155\pi$
$683$	27207.5			1.52426			0.762128	+	0.647426i	$0.224155\pi$
$684$		0			0
$685$	−5263.85			−0.293608
$686$	10385.0	+	17987.4i	0.577991	+	1.00111i
$687$		0			0
$688$	367.381	−	636.323i	0.0203580	−	0.0352610i
$689$	−17286.9	+	29941.7i	−0.955845	+	1.65557i
$690$		0			0
$691$	8703.62	+	15075.1i	0.479162	+	0.829934i	0.999714	−	0.0238963i	$-0.00760715\pi$
$691$	8703.62	+	15075.1i	0.479162	+	0.829934i	−0.520552	+	0.853830i	$0.674274\pi$
$692$	−1888.04			−0.103718
$693$		0			0
$694$	−13876.7			−0.759011
$695$	−5961.24	−	10325.2i	−0.325356	−	0.563534i
$696$		0			0
$697$	−6638.15	+	11497.6i	−0.360743	+	0.624825i
$698$	1026.99	−	1778.80i	0.0556909	−	0.0964595i
$699$		0			0
$700$	214.628	+	371.747i	0.0115888	+	0.0200725i
$701$	−18543.0			−0.999086			−0.499543	−	0.866289i	$-0.666499\pi$
$701$	−18543.0			−0.999086			−0.499543	+	0.866289i	$0.666499\pi$
$702$		0			0
$703$	−2874.63			−0.154223
$704$	6500.29	+	11258.8i	0.347996	+	0.602746i
$705$		0			0
$706$	5112.91	−	8855.81i	0.272559	−	0.472086i
$707$	1467.21	−	2541.28i	0.0780482	−	0.135184i
$708$		0			0
$709$	−13116.9	−	22719.1i	−0.694803	−	1.20343i	−0.970247	−	0.242117i	$-0.922158\pi$
$709$	−13116.9	−	22719.1i	−0.694803	−	1.20343i	0.275444	−	0.961317i	$-0.411175\pi$
$710$	−12448.9			−0.658029
$711$		0			0
$712$	−21168.9			−1.11424
$713$	−1329.70	−	2303.11i	−0.0698425	−	0.120971i
$714$		0			0
$715$	4631.32	−	8021.69i	0.242240	−	0.419572i
$716$	984.633	−	1705.43i	0.0513931	−	0.0890155i
$717$		0			0
$718$	−11713.2	−	20287.8i	−0.608817	−	1.05450i
$719$	3043.06			0.157840			0.0789199	−	0.996881i	$-0.474853\pi$
$719$	3043.06			0.157840			0.0789199	+	0.996881i	$0.474853\pi$
$720$		0			0
$721$	−22931.2			−1.18447
$722$	15150.7	+	26241.8i	0.780958	+	1.35266i
$723$		0			0
$724$	−1242.43	+	2151.95i	−0.0637769	+	0.110465i
$725$	112.942	−	195.622i	0.00578561	−	0.0100210i
$726$		0			0
$727$	−10971.2	−	19002.7i	−0.559698	−	0.969426i	−0.997521	−	0.0703647i	$-0.977584\pi$
$727$	−10971.2	−	19002.7i	−0.559698	−	0.969426i	0.437823	−	0.899061i	$-0.355750\pi$
$728$	−16392.2			−0.834528
$729$		0			0
$730$	−7529.10			−0.381732
$731$	553.002	+	957.827i	0.0279802	+	0.0484631i
$732$		0			0
$733$	13288.4	−	23016.2i	0.669604	−	1.15979i	−0.308411	−	0.951253i	$-0.599797\pi$
$733$	13288.4	−	23016.2i	0.669604	−	1.15979i	0.978015	−	0.208535i	$-0.0668694\pi$
$734$	8987.58	−	15566.9i	0.451959	−	0.782815i
$735$		0			0
$736$	2237.56	+	3875.57i	0.112062	+	0.194097i
$737$	−27440.4			−1.37148
$738$		0			0
$739$	28787.7			1.43298			0.716489	−	0.697598i	$-0.245748\pi$
$739$	28787.7			1.43298			0.716489	+	0.697598i	$0.245748\pi$
$740$	69.3853	+	120.179i	0.00344683	+	0.00597008i
$741$		0			0
$742$	−12379.5	+	21441.8i	−0.612485	+	1.06086i
$743$	−2251.51	+	3899.73i	−0.111171	+	0.192553i	−0.916243	−	0.400624i	$-0.868793\pi$
$743$	−2251.51	+	3899.73i	−0.111171	+	0.192553i	0.805072	+	0.593177i	$0.202127\pi$
$744$		0			0
$745$	2865.72	+	4963.58i	0.140929	+	0.244096i
$746$	25624.8			1.25763
$747$		0			0
$748$	4343.45			0.212316
$749$	4118.46	+	7133.38i	0.200915	+	0.347995i
$750$		0			0
$751$	−2852.83	+	4941.24i	−0.138617	+	0.240091i	−0.926973	−	0.375128i	$-0.877599\pi$
$751$	−2852.83	+	4941.24i	−0.138617	+	0.240091i	0.788357	+	0.615219i	$0.210932\pi$
$752$	15993.0	−	27700.6i	0.775536	−	1.34327i
$753$		0			0
$754$	−800.356	−	1386.26i	−0.0386568	−	0.0669556i
$755$	−5830.58			−0.281055
$756$		0			0
$757$	17397.6			0.835305			0.417652	−	0.908607i	$-0.362853\pi$
$757$	17397.6			0.835305			0.417652	+	0.908607i	$0.362853\pi$
$758$	−5338.62	−	9246.77i	−0.255815	−	0.443084i
$759$		0			0
$760$	−6655.02	+	11526.8i	−0.317636	+	0.550161i
$761$	−17757.2	+	30756.3i	−0.845857	+	1.46507i	0.0390179	+	0.999239i	$0.487577\pi$
$761$	−17757.2	+	30756.3i	−0.845857	+	1.46507i	−0.884875	+	0.465829i	$0.845756\pi$
$762$		0			0
$763$	−5289.97	−	9162.50i	−0.250996	−	0.434737i
$764$	2940.58			0.139249
$765$		0			0
$766$	17435.3			0.822408
$767$	12885.1	+	22317.7i	0.606591	+	1.05065i
$768$		0			0
$769$	−15631.8	+	27075.1i	−0.733027	+	1.26964i	0.222556	+	0.974920i	$0.428560\pi$
$769$	−15631.8	+	27075.1i	−0.733027	+	1.26964i	−0.955583	+	0.294721i	$0.904773\pi$
$770$	3316.58	−	5744.48i	0.155222	−	0.268853i
$771$		0			0
$772$	146.612	+	253.940i	0.00683508	+	0.0118387i
$773$	6676.47			0.310654			0.155327	−	0.987863i	$-0.450357\pi$
$773$	6676.47			0.310654			0.155327	+	0.987863i	$0.450357\pi$
$774$		0			0
$775$	−834.522			−0.0386799
$776$	−16406.9	−	28417.5i	−0.758985	−	1.31460i
$777$		0			0
$778$	−21850.7	+	37846.5i	−1.00692	+	1.74404i
$779$	7894.53	−	13673.7i	0.363095	−	0.628898i
$780$		0			0
$781$	13017.6	+	22547.1i	0.596422	+	1.03303i
$782$	−26427.3			−1.20849
$783$		0			0
$784$	11227.9			0.511473
$785$	8689.04	+	15049.9i	0.395064	+	0.684271i
$786$		0			0
$787$	−4716.91	+	8169.93i	−0.213646	+	0.370047i	−0.952853	−	0.303432i	$-0.901867\pi$
$787$	−4716.91	+	8169.93i	−0.213646	+	0.370047i	0.739207	+	0.673479i	$0.235201\pi$
$788$	−449.439	+	778.451i	−0.0203180	+	0.0351919i
$789$		0			0
$790$	8901.10	+	15417.2i	0.400869	+	0.694326i
$791$	−15997.7			−0.719107
$792$		0			0
$793$	8420.57			0.377078
$794$	−13709.8	−	23746.0i	−0.612773	−	1.06135i
$795$		0			0
$796$	−1065.27	+	1845.10i	−0.0474340	+	0.0821580i
$797$	−11738.1	+	20331.1i	−0.521689	+	0.903592i	0.477992	+	0.878364i	$0.341365\pi$
$797$	−11738.1	+	20331.1i	−0.521689	+	0.903592i	−0.999682	+	0.0252283i	$0.991969\pi$
$798$		0			0
$799$	24073.4	+	41696.4i	1.06590	+	1.84620i
$800$	1404.30			0.0620618
$801$		0			0
$802$	31972.0			1.40769
$803$	7873.02	+	13636.5i	0.345993	+	0.599278i
$804$		0			0
$805$	−2731.08	+	4730.37i	−0.119575	+	0.207110i
$806$	−2956.89	+	5121.48i	−0.129221	+	0.223817i
$807$		0			0
$808$	−2196.19	−	3803.91i	−0.0956208	−	0.165620i
$809$	33269.8			1.44586			0.722932	−	0.690919i	$-0.242794\pi$
$809$	33269.8			1.44586			0.722932	+	0.690919i	$0.242794\pi$
$810$		0			0
$811$	27892.8			1.20771			0.603853	−	0.797096i	$-0.293631\pi$
$811$	27892.8			1.20771			0.603853	+	0.797096i	$0.293631\pi$
$812$	−77.5699	−	134.355i	−0.00335243	−	0.00580657i
$813$		0			0
$814$	1072.19	−	1857.08i	0.0461672	−	0.0799640i
$815$	5558.30	−	9627.25i	0.238894	−	0.413777i
$816$		0			0
$817$	−657.666	−	1139.11i	−0.0281626	−	0.0487790i
$818$	−10093.1			−0.431415
$819$		0			0
$820$	−762.204			−0.0324601
$821$	−1848.66	−	3201.98i	−0.0785856	−	0.136114i	0.824054	−	0.566511i	$-0.191707\pi$
$821$	−1848.66	−	3201.98i	−0.0785856	−	0.136114i	−0.902640	+	0.430397i	$0.858374\pi$
$822$		0			0
$823$	−3572.25	+	6187.32i	−0.151301	+	0.262061i	−0.931706	−	0.363213i	$-0.881680\pi$
$823$	−3572.25	+	6187.32i	−0.151301	+	0.262061i	0.780405	+	0.625275i	$0.215013\pi$
$824$	−17162.3	+	29725.9i	−0.725578	+	1.25674i
$825$		0			0
$826$	9227.29	+	15982.1i	0.388691	+	0.673232i
$827$	−19866.9			−0.835357			−0.417679	−	0.908595i	$-0.637156\pi$
$827$	−19866.9			−0.835357			−0.417679	+	0.908595i	$0.637156\pi$
$828$		0			0
$829$	−36736.5			−1.53910			−0.769548	−	0.638589i	$-0.779518\pi$
$829$	−36736.5			−1.53910			−0.769548	+	0.638589i	$0.779518\pi$
$830$	3224.58	+	5585.13i	0.134851	+	0.233569i
$831$		0			0
$832$	−11903.0	+	20616.7i	−0.495990	+	0.859079i
$833$	−8450.38	+	14636.5i	−0.351487	+	0.608793i
$834$		0			0
$835$	−3969.96	−	6876.16i	−0.164534	−	0.284981i
$836$	−5165.52			−0.213700
$837$		0			0
$838$	−1478.71			−0.0609560
$839$	15006.1	+	25991.4i	0.617485	+	1.06951i	0.989943	+	0.141466i	$0.0451816\pi$
$839$	15006.1	+	25991.4i	0.617485	+	1.06951i	−0.372458	+	0.928049i	$0.621485\pi$
$840$		0			0
$841$	12153.7	−	21050.8i	0.498326	−	0.863127i
$842$	7683.32	−	13307.9i	0.314471	−	0.544680i
$843$		0			0
$844$	3092.07	+	5355.62i	0.126106	+	0.218422i
$845$	5976.32			0.243304
$846$		0			0
$847$	4378.68			0.177631
$848$	21503.4	+	37245.0i	0.870789	+	1.50825i
$849$		0			0
$850$	−4146.46	+	7181.88i	−0.167320	+	0.289807i
$851$	−882.905	+	1529.24i	−0.0355648	+	0.0616000i
$852$		0			0
$853$	2690.51	+	4660.10i	0.107997	+	0.187056i	0.914959	−	0.403548i	$-0.132223\pi$
$853$	2690.51	+	4660.10i	0.107997	+	0.187056i	−0.806962	+	0.590604i	$0.798890\pi$
$854$	6030.13			0.241624
$855$		0			0
$856$	12329.4			0.492302
$857$	7319.88	+	12678.4i	0.291765	+	0.505352i	0.974227	−	0.225569i	$-0.0724242\pi$
$857$	7319.88	+	12678.4i	0.291765	+	0.505352i	−0.682462	+	0.730921i	$0.739091\pi$
$858$		0			0
$859$	−14770.0	+	25582.4i	−0.586667	+	1.01614i	0.407999	+	0.912983i	$0.366227\pi$
$859$	−14770.0	+	25582.4i	−0.586667	+	1.01614i	−0.994665	+	0.103154i	$0.967107\pi$
$860$	−31.7483	+	54.9897i	−0.00125885	+	0.00218039i
$861$		0			0
$862$	−10561.3	−	18292.7i	−0.417307	−	0.722797i
$863$	−3387.63			−0.133622			−0.0668112	−	0.997766i	$-0.521283\pi$
$863$	−3387.63			−0.133622			−0.0668112	+	0.997766i	$0.521283\pi$
$864$		0			0
$865$	−7538.98			−0.296339
$866$	−20894.1	−	36189.7i	−0.819875	−	1.42007i
$867$		0			0
$868$	−286.580	+	496.370i	−0.0112064	+	0.0194100i
$869$	18615.4	−	32242.8i	0.726678	−	1.25864i
$870$		0			0
$871$	−25123.8	−	43515.7i	−0.977368	−	1.69285i
$872$	−15836.5			−0.615015
$873$		0			0
$874$	31429.2			1.21637
$875$	857.014	+	1484.39i	0.0331113	+	0.0573504i
$876$		0			0
$877$	122.052	−	211.400i	0.00469944	−	0.00813967i	−0.863666	−	0.504064i	$-0.831837\pi$
$877$	122.052	−	211.400i	0.00469944	−	0.00813967i	0.868366	+	0.495925i	$0.165171\pi$
$878$	12899.8	−	22343.1i	0.495838	−	0.858817i
$879$		0			0
$880$	−5760.97	−	9978.30i	−0.220684	−	0.382237i
$881$	−13910.2			−0.531948			−0.265974	−	0.963980i	$-0.585693\pi$
$881$	−13910.2			−0.531948			−0.265974	+	0.963980i	$0.585693\pi$
$882$		0			0
$883$	−8805.87			−0.335607			−0.167803	−	0.985820i	$-0.553667\pi$
$883$	−8805.87			−0.335607			−0.167803	+	0.985820i	$0.553667\pi$
$884$	3976.76	+	6887.95i	0.151304	+	0.262067i
$885$		0			0
$886$	848.943	−	1470.41i	0.0321905	−	0.0557556i
$887$	6640.41	−	11501.5i	0.251368	−	0.435381i	−0.712535	−	0.701637i	$-0.752453\pi$
$887$	6640.41	−	11501.5i	0.251368	−	0.435381i	0.963903	+	0.266255i	$0.0857864\pi$
$888$		0			0
$889$	11176.1	+	19357.6i	0.421637	+	0.730296i
$890$	15685.8			0.590773
$891$		0			0
$892$	−3823.11			−0.143506
$893$	−28629.7	−	49588.2i	−1.07285	−	1.85824i
$894$		0			0
$895$	3931.65	−	6809.82i	0.146839	−	0.254332i
$896$	−11605.0	+	20100.4i	−0.432695	+	0.749449i
$897$		0			0
$898$	−22470.9	−	38920.7i	−0.835036	−	1.44633i
$899$	301.609			0.0111893
$900$		0			0
$901$	−64736.1			−2.39364
$902$	5889.04	+	10200.1i	0.217387	+	0.376526i
$903$		0			0
$904$	−11973.1	+	20738.0i	−0.440507	+	0.762981i
$905$	−4961.03	+	8592.76i	−0.182221	+	0.315616i
$906$		0			0
$907$	542.122	+	938.982i	0.0198466	+	0.0343753i	0.875778	−	0.482714i	$-0.160349\pi$
$907$	542.122	+	938.982i	0.0198466	+	0.0343753i	−0.855932	+	0.517089i	$0.827016\pi$
$908$	1051.78			0.0384411
$909$		0			0
$910$	12146.3			0.442469
$911$	25094.4	+	43464.7i	0.912638	+	1.58073i	0.810324	+	0.585983i	$0.199291\pi$
$911$	25094.4	+	43464.7i	0.912638	+	1.58073i	0.102314	+	0.994752i	$0.467375\pi$
$912$		0			0
$913$	6743.74	−	11680.5i	0.244452	−	0.423404i
$914$	−21608.4	+	37426.9i	−0.781995	+	1.35445i
$915$		0			0
$916$	−877.475	−	1519.83i	−0.0316513	−	0.0548216i
$917$	3549.06			0.127808
$918$		0			0
$919$	30376.1			1.09033			0.545166	−	0.838328i	$-0.316467\pi$
$919$	30376.1			1.09033			0.545166	+	0.838328i	$0.316467\pi$
$920$	4088.00	+	7080.63i	0.146497	+	0.253741i
$921$		0			0
$922$	11714.5	−	20290.2i	0.418435	−	0.724751i
$923$	−23837.2	+	41287.2i	−0.850066	+	1.47236i
$924$		0			0
$925$	277.056	+	479.876i	0.00984817	+	0.0170575i
$926$	−24057.3			−0.853750
$927$		0			0
$928$	−507.534			−0.0179533
$929$	−23098.8	−	40008.3i	−0.815766	−	1.41295i	−0.908776	−	0.417283i	$-0.862982\pi$
$929$	−23098.8	−	40008.3i	−0.815766	−	1.41295i	0.0930102	−	0.995665i	$-0.470351\pi$
$930$		0			0
$931$	10049.8	−	17406.7i	0.353778	−	0.612762i
$932$	−1788.74	+	3098.20i	−0.0628672	+	0.108889i
$933$		0			0
$934$	−17702.6	−	30661.7i	−0.620177	−	1.07418i
$935$	17343.4			0.606621
$936$		0			0
$937$	37004.7			1.29017			0.645085	−	0.764111i	$-0.276822\pi$
$937$	37004.7			1.29017			0.645085	+	0.764111i	$0.276822\pi$
$938$	−17991.6	−	31162.4i	−0.626277	−	1.08474i
$939$		0			0
$940$	−1382.08	+	2393.83i	−0.0479558	+	0.0830618i
$941$	21907.2	−	37944.4i	0.758932	−	1.31451i	−0.184464	−	0.982839i	$-0.559055\pi$
$941$	21907.2	−	37944.4i	0.758932	−	1.31451i	0.943396	−	0.331669i	$-0.107612\pi$
$942$		0			0
$943$	−4849.40	−	8399.41i	−0.167464	−	0.290055i
$944$	32056.0			1.10523
$945$		0			0
$946$	981.191			0.0337223
$947$	2304.86	+	3992.14i	0.0790898	+	0.136988i	0.902857	−	0.429940i	$-0.141465\pi$
$947$	2304.86	+	3992.14i	0.0790898	+	0.136988i	−0.823768	+	0.566928i	$0.808132\pi$
$948$		0			0
$949$	−14416.7	+	24970.5i	−0.493136	+	0.854136i
$950$	4931.24	−	8541.17i	0.168411	−	0.291697i
$951$		0			0
$952$	−15346.5	−	26580.8i	−0.522460	−	0.904927i
$953$	−4281.80			−0.145542			−0.0727708	−	0.997349i	$-0.523184\pi$
$953$	−4281.80			−0.145542			−0.0727708	+	0.997349i	$0.523184\pi$
$954$		0			0
$955$	11741.8			0.397859
$956$	−1070.52	−	1854.19i	−0.0362165	−	0.0627288i
$957$		0			0
$958$	28434.0	−	49249.1i	0.958936	−	1.66093i
$959$	−7217.91	+	12501.8i	−0.243043	+	0.420963i
$960$		0			0
$961$	14338.4	+	24834.8i	0.481298	+	0.833633i
$962$	3926.68			0.131602
$963$		0			0
$964$	−5425.05			−0.181254
$965$	585.424	+	1013.98i	0.0195290	+	0.0338252i
$966$		0			0
$967$	18114.9	−	31375.9i	0.602416	−	1.04341i	−0.390038	−	0.920799i	$-0.627538\pi$
$967$	18114.9	−	31375.9i	0.602416	−	1.04341i	0.992454	−	0.122616i	$-0.0391284\pi$
$968$	3277.10	−	5676.11i	0.108812	−	0.188468i
$969$		0			0
$970$	12157.2	+	21056.8i	0.402416	+	0.697005i
$971$	−844.928			−0.0279249			−0.0139624	−	0.999903i	$-0.504445\pi$
$971$	−844.928			−0.0279249			−0.0139624	+	0.999903i	$0.504445\pi$
$972$		0			0
$973$	−32696.7			−1.07730
$974$	−3421.26	−	5925.80i	−0.112551	−	0.194943i
$975$		0			0
$976$	5237.24	−	9071.16i	0.171762	−	0.297501i
$977$	22922.5	−	39702.9i	0.750619	−	1.30011i	−0.196904	−	0.980423i	$-0.563089\pi$
$977$	22922.5	−	39702.9i	0.750619	−	1.30011i	0.947523	−	0.319687i	$-0.103578\pi$
$978$		0			0
$979$	−16402.2	−	28409.5i	−0.535463	−	0.927449i
$980$	−970.288			−0.0316273
$981$		0			0
$982$	6048.05			0.196539
$983$	−20101.0	−	34815.9i	−0.652210	−	1.12966i	−0.982586	−	0.185811i	$-0.940509\pi$
$983$	−20101.0	−	34815.9i	−0.652210	−	1.12966i	0.330376	−	0.943849i	$-0.392824\pi$
$984$		0			0
$985$	−1794.62	+	3108.37i	−0.0580520	+	0.100549i
$986$	1498.59	−	2595.64i	0.0484025	−	0.0838356i
$987$		0			0
$988$	−4729.43	−	8191.61i	−0.152291	−	0.263775i
$989$	−807.974			−0.0259778
$990$		0			0
$991$	1797.91			0.0576313			0.0288157	−	0.999585i	$-0.490826\pi$
$991$	1797.91			0.0576313			0.0288157	+	0.999585i	$0.490826\pi$
$992$	937.534	+	1623.86i	0.0300068	+	0.0519733i
$993$		0			0
$994$	−17070.3	+	29566.6i	−0.544704	+	0.943455i
$995$	−4253.63	+	7367.50i	−0.135527	+	0.234739i
$996$		0			0
$997$	−4566.04	−	7908.61i	−0.145043	−	0.251222i	0.784346	−	0.620324i	$-0.212999\pi$
$997$	−4566.04	−	7908.61i	−0.145043	−	0.251222i	−0.929389	+	0.369102i	$0.879665\pi$
$998$	−7342.47			−0.232888
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.e.c.91.6		14
3.2	odd	2		45.4.e.c.31.2	yes	14
9.2	odd	6		45.4.e.c.16.2	✓	14
9.4	even	3		405.4.a.n.1.2		7
9.5	odd	6		405.4.a.m.1.6		7
9.7	even	3	inner	135.4.e.c.46.6		14
15.2	even	4		225.4.k.d.49.11		28
15.8	even	4		225.4.k.d.49.4		28
15.14	odd	2		225.4.e.d.76.6		14
45.2	even	12		225.4.k.d.124.4		28
45.4	even	6		2025.4.a.ba.1.6		7
45.14	odd	6		2025.4.a.bb.1.2		7
45.29	odd	6		225.4.e.d.151.6		14
45.38	even	12		225.4.k.d.124.11		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.2	✓	14	9.2	odd	6
45.4.e.c.31.2	yes	14	3.2	odd	2
135.4.e.c.46.6		14	9.7	even	3	inner
135.4.e.c.91.6		14	1.1	even	1	trivial
225.4.e.d.76.6		14	15.14	odd	2
225.4.e.d.151.6		14	45.29	odd	6
225.4.k.d.49.4		28	15.8	even	4
225.4.k.d.49.11		28	15.2	even	4
225.4.k.d.124.4		28	45.2	even	12
225.4.k.d.124.11		28	45.38	even	12
405.4.a.m.1.6		7	9.5	odd	6
405.4.a.n.1.2		7	9.4	even	3
2025.4.a.ba.1.6		7	45.4	even	6
2025.4.a.bb.1.2		7	45.14	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.52087 + 2.63422i) q^{2} +(-0.626094 + 1.08443i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(6.85611 + 11.8751i) q^{7} +20.5251 q^{8} +O(q^{10})\)	\(q+(1.52087 + 2.63422i) q^{2} +(-0.626094 + 1.08443i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(6.85611 + 11.8751i) q^{7} +20.5251 q^{8} -15.2087 q^{10} +(15.9034 + 27.5455i) q^{11} +(-29.1216 + 50.4401i) q^{13} +(-20.8545 + 36.1211i) q^{14} +(36.2248 + 62.7431i) q^{16} -109.055 q^{17} +129.695 q^{19} +(-3.13047 - 5.42213i) q^{20} +(-48.3740 + 83.7863i) q^{22} +(39.8342 - 68.9949i) q^{23} +(-12.5000 - 21.6506i) q^{25} -177.161 q^{26} -17.1703 q^{28} +(4.51769 + 7.82486i) q^{29} +(16.6904 - 28.9087i) q^{31} +(-28.0860 + 48.6463i) q^{32} +(-165.858 - 287.275i) q^{34} -68.5611 q^{35} -22.1645 q^{37} +(197.250 + 341.647i) q^{38} +(-51.3127 + 88.8763i) q^{40} +(60.8698 - 105.430i) q^{41} +(-5.07086 - 8.78298i) q^{43} -39.8281 q^{44} +242.331 q^{46} +(-220.746 - 382.343i) q^{47} +(77.4875 - 134.212i) q^{49} +(38.0218 - 65.8556i) q^{50} +(-36.4657 - 63.1604i) q^{52} +593.610 q^{53} -159.034 q^{55} +(140.722 + 243.738i) q^{56} +(-13.7416 + 23.8012i) q^{58} +(221.230 - 383.182i) q^{59} +(-72.2881 - 125.207i) q^{61} +101.536 q^{62} +408.736 q^{64} +(-145.608 - 252.200i) q^{65} +(-431.360 + 747.138i) q^{67} +(68.2786 - 118.262i) q^{68} +(-104.273 - 180.605i) q^{70} +818.541 q^{71} +495.052 q^{73} +(-33.7093 - 58.3863i) q^{74} +(-81.2014 + 140.645i) q^{76} +(-218.071 + 377.710i) q^{77} +(-585.263 - 1013.71i) q^{79} -362.248 q^{80} +370.300 q^{82} +(-212.022 - 367.232i) q^{83} +(272.637 - 472.221i) q^{85} +(15.4242 - 26.7156i) q^{86} +(326.419 + 565.374i) q^{88} -1031.37 q^{89} -798.643 q^{91} +(49.8799 + 86.3945i) q^{92} +(671.452 - 1162.99i) q^{94} +(-324.238 + 561.597i) q^{95} +(-799.356 - 1384.53i) q^{97} +471.394 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8}+O(q^{10}) \) 14 * q - 2 * q^2 - 36 * q^4 - 35 * q^5 - 22 * q^7 + 36 * q^8	\( 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8} + 20 q^{10} - 23 q^{11} - 96 q^{13} + 21 q^{14} - 324 q^{16} + 322 q^{17} + 558 q^{19} - 180 q^{20} - 311 q^{22} - 96 q^{23} - 175 q^{25} - 716 q^{26} + 674 q^{28} + 296 q^{29} - 244 q^{31} + 314 q^{32} - 125 q^{34} + 220 q^{35} + 808 q^{37} - 305 q^{38} - 90 q^{40} + 47 q^{41} - 525 q^{43} + 110 q^{44} + 1434 q^{46} - 164 q^{47} - 1225 q^{49} - 50 q^{50} - 1682 q^{52} + 1012 q^{53} + 230 q^{55} + 981 q^{56} - 1183 q^{58} + 85 q^{59} - 828 q^{61} - 1572 q^{62} + 4472 q^{64} - 480 q^{65} - 1093 q^{67} - 2473 q^{68} + 105 q^{70} + 656 q^{71} + 4170 q^{73} + 1316 q^{74} - 2789 q^{76} - 24 q^{77} - 2110 q^{79} + 3240 q^{80} - 124 q^{82} - 1290 q^{83} - 805 q^{85} + 2569 q^{86} - 2271 q^{88} - 6096 q^{89} + 6676 q^{91} - 2763 q^{92} + 517 q^{94} - 1395 q^{95} - 1787 q^{97} + 2558 q^{98}+O(q^{100}) \) 14 * q - 2 * q^2 - 36 * q^4 - 35 * q^5 - 22 * q^7 + 36 * q^8 + 20 * q^10 - 23 * q^11 - 96 * q^13 + 21 * q^14 - 324 * q^16 + 322 * q^17 + 558 * q^19 - 180 * q^20 - 311 * q^22 - 96 * q^23 - 175 * q^25 - 716 * q^26 + 674 * q^28 + 296 * q^29 - 244 * q^31 + 314 * q^32 - 125 * q^34 + 220 * q^35 + 808 * q^37 - 305 * q^38 - 90 * q^40 + 47 * q^41 - 525 * q^43 + 110 * q^44 + 1434 * q^46 - 164 * q^47 - 1225 * q^49 - 50 * q^50 - 1682 * q^52 + 1012 * q^53 + 230 * q^55 + 981 * q^56 - 1183 * q^58 + 85 * q^59 - 828 * q^61 - 1572 * q^62 + 4472 * q^64 - 480 * q^65 - 1093 * q^67 - 2473 * q^68 + 105 * q^70 + 656 * q^71 + 4170 * q^73 + 1316 * q^74 - 2789 * q^76 - 24 * q^77 - 2110 * q^79 + 3240 * q^80 - 124 * q^82 - 1290 * q^83 - 805 * q^85 + 2569 * q^86 - 2271 * q^88 - 6096 * q^89 + 6676 * q^91 - 2763 * q^92 + 517 * q^94 - 1395 * q^95 - 1787 * q^97 + 2558 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more