LMFDB - Embedded newform 54.4.c.b.19.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,4,Mod(19,54)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("54.19");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	54.c (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:	$3.18610314031$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-35})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} - 9x + 81 $ x^4 - x^3 - 8x^2 - 9x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			19.2
Root			$-2.31174 - 1.91203i$ of defining polynomial
Character	$\chi$	$=$	54.19
Dual form			54.4.c.b.37.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.43521 + 9.41407i) q^{5} +(-12.4352 + 21.5384i) q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.43521 + 9.41407i) q^{5} +(-12.4352 + 21.5384i) q^{7} +8.00000 q^{8} -21.7409 q^{10} +(-21.3704 + 37.0147i) q^{11} +(-7.56479 - 13.1026i) q^{13} +(-24.8704 - 43.0768i) q^{14} +(-8.00000 + 13.8564i) q^{16} +13.8704 q^{17} +143.352 q^{19} +(21.7409 - 37.6563i) q^{20} +(-42.7409 - 74.0293i) q^{22} +(9.56479 + 16.5667i) q^{23} +(3.41692 - 5.91828i) q^{25} +30.2591 q^{26} +99.4817 q^{28} +(113.046 - 195.802i) q^{29} +(-29.6944 - 51.4321i) q^{31} +(-16.0000 - 27.7128i) q^{32} +(-13.8704 + 24.0243i) q^{34} -270.352 q^{35} -84.1860 q^{37} +(-143.352 + 248.293i) q^{38} +(43.4817 + 75.3125i) q^{40} +(101.630 + 176.028i) q^{41} +(-162.945 + 282.229i) q^{43} +170.963 q^{44} -38.2591 q^{46} +(5.47180 - 9.47744i) q^{47} +(-137.769 - 238.623i) q^{49} +(6.83384 + 11.8366i) q^{50} +(-30.2591 + 52.4104i) q^{52} +140.186 q^{53} -464.611 q^{55} +(-99.4817 + 172.307i) q^{56} +(226.093 + 391.605i) q^{58} +(57.3704 + 99.3685i) q^{59} +(377.528 - 653.898i) q^{61} +118.777 q^{62} +64.0000 q^{64} +(82.2325 - 142.431i) q^{65} +(383.723 + 664.627i) q^{67} +(-27.7409 - 48.0486i) q^{68} +(270.352 - 468.264i) q^{70} -335.854 q^{71} +167.279 q^{73} +(84.1860 - 145.814i) q^{74} +(-286.704 - 496.586i) q^{76} +(-531.492 - 920.570i) q^{77} +(12.6578 - 21.9239i) q^{79} -173.927 q^{80} -406.518 q^{82} +(143.861 - 249.174i) q^{83} +(75.3887 + 130.577i) q^{85} +(-325.890 - 564.458i) q^{86} +(-170.963 + 296.117i) q^{88} +860.817 q^{89} +376.279 q^{91} +(38.2591 - 66.2668i) q^{92} +(10.9436 + 18.9549i) q^{94} +(779.149 + 1349.53i) q^{95} +(201.075 - 348.272i) q^{97} +551.076 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 8 q^{4} - 9 q^{5} - 19 q^{7} + 32 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 - 8 * q^4 - 9 * q^5 - 19 * q^7 + 32 * q^8	$ 4 q - 4 q^{2} - 8 q^{4} - 9 q^{5} - 19 q^{7} + 32 q^{8} + 36 q^{10} - 24 q^{11} - 61 q^{13} - 38 q^{14} - 32 q^{16} - 6 q^{17} + 266 q^{19} - 36 q^{20} - 48 q^{22} + 69 q^{23} - 263 q^{25} + 244 q^{26} + 152 q^{28} + 237 q^{29} - 211 q^{31} - 64 q^{32} + 6 q^{34} - 774 q^{35} + 524 q^{37} - 266 q^{38} - 72 q^{40} + 468 q^{41} + 86 q^{43} + 192 q^{44} - 276 q^{46} + 483 q^{47} + 33 q^{49} - 526 q^{50} - 244 q^{52} - 300 q^{53} - 1674 q^{55} - 152 q^{56} + 474 q^{58} + 168 q^{59} + 1049 q^{61} + 844 q^{62} + 256 q^{64} - 747 q^{65} + 1166 q^{67} + 12 q^{68} + 774 q^{70} + 624 q^{71} - 622 q^{73} - 524 q^{74} - 532 q^{76} - 1173 q^{77} - 349 q^{79} + 288 q^{80} - 1872 q^{82} + 1221 q^{83} + 486 q^{85} + 172 q^{86} - 192 q^{88} + 984 q^{89} + 214 q^{91} + 276 q^{92} + 966 q^{94} + 1764 q^{95} + 128 q^{97} - 132 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 - 8 * q^4 - 9 * q^5 - 19 * q^7 + 32 * q^8 + 36 * q^10 - 24 * q^11 - 61 * q^13 - 38 * q^14 - 32 * q^16 - 6 * q^17 + 266 * q^19 - 36 * q^20 - 48 * q^22 + 69 * q^23 - 263 * q^25 + 244 * q^26 + 152 * q^28 + 237 * q^29 - 211 * q^31 - 64 * q^32 + 6 * q^34 - 774 * q^35 + 524 * q^37 - 266 * q^38 - 72 * q^40 + 468 * q^41 + 86 * q^43 + 192 * q^44 - 276 * q^46 + 483 * q^47 + 33 * q^49 - 526 * q^50 - 244 * q^52 - 300 * q^53 - 1674 * q^55 - 152 * q^56 + 474 * q^58 + 168 * q^59 + 1049 * q^61 + 844 * q^62 + 256 * q^64 - 747 * q^65 + 1166 * q^67 + 12 * q^68 + 774 * q^70 + 624 * q^71 - 622 * q^73 - 524 * q^74 - 532 * q^76 - 1173 * q^77 - 349 * q^79 + 288 * q^80 - 1872 * q^82 + 1221 * q^83 + 486 * q^85 + 172 * q^86 - 192 * q^88 + 984 * q^89 + 214 * q^91 + 276 * q^92 + 966 * q^94 + 1764 * q^95 + 128 * q^97 - 132 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/54\mathbb{Z}\right)^\times$.

$n$	$29$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−1.00000	+	1.73205i	−0.353553	+	0.612372i
$2$	−1.00000	+	1.73205i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.250000	−	0.433013i
$5$	5.43521	+	9.41407i	0.486140	+	0.842020i	0.999873	−	0.0159306i	$-0.00507109\pi$
$5$	5.43521	+	9.41407i	0.486140	+	0.842020i	−0.513733	+	0.857950i	$0.671738\pi$
$6$		0			0
$7$	−12.4352	+	21.5384i	−0.671438	+	1.16297i	0.306058	+	0.952013i	$0.400990\pi$
$7$	−12.4352	+	21.5384i	−0.671438	+	1.16297i	−0.977496	+	0.210953i	$0.932343\pi$
$8$	8.00000			0.353553
$9$		0			0
$10$	−21.7409			−0.687506
$11$	−21.3704	+	37.0147i	−0.585766	+	1.01458i	0.409013	+	0.912528i	$0.365873\pi$
$11$	−21.3704	+	37.0147i	−0.585766	+	1.01458i	−0.994779	+	0.102048i	$0.967460\pi$
$12$		0			0
$13$	−7.56479	−	13.1026i	−0.161392	−	0.279539i	0.773976	−	0.633215i	$-0.218265\pi$
$13$	−7.56479	−	13.1026i	−0.161392	−	0.279539i	−0.935368	+	0.353676i	$0.884932\pi$
$14$	−24.8704	−	43.0768i	−0.474779	−	0.822341i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	13.8704			0.197887			0.0989433	−	0.995093i	$-0.468454\pi$
$17$	13.8704			0.197887			0.0989433	+	0.995093i	$0.468454\pi$
$18$		0			0
$19$	143.352			1.73091			0.865454	−	0.500989i	$-0.167030\pi$
$19$	143.352			1.73091			0.865454	+	0.500989i	$0.167030\pi$
$20$	21.7409	−	37.6563i	0.243070	−	0.421010i
$21$		0			0
$22$	−42.7409	−	74.0293i	−0.414199	−	0.717414i
$23$	9.56479	+	16.5667i	0.0867129	+	0.150191i	0.906120	−	0.423021i	$-0.139030\pi$
$23$	9.56479	+	16.5667i	0.0867129	+	0.150191i	−0.819407	+	0.573212i	$0.805697\pi$
$24$		0			0
$25$	3.41692	−	5.91828i	0.0273353	−	0.0473462i
$26$	30.2591			0.228243
$27$		0			0
$28$	99.4817			0.671438
$29$	113.046	−	195.802i	0.723869	−	1.25378i	−0.235569	−	0.971858i	$-0.575695\pi$
$29$	113.046	−	195.802i	0.723869	−	1.25378i	0.959438	−	0.281920i	$-0.0909714\pi$
$30$		0			0
$31$	−29.6944	−	51.4321i	−0.172041	−	0.297983i	0.767092	−	0.641537i	$-0.221703\pi$
$31$	−29.6944	−	51.4321i	−0.172041	−	0.297983i	−0.939133	+	0.343553i	$0.888369\pi$
$32$	−16.0000	−	27.7128i	−0.0883883	−	0.153093i
$33$		0			0
$34$	−13.8704	+	24.0243i	−0.0699635	+	0.121180i
$35$	−270.352			−1.30565
$36$		0			0
$37$	−84.1860			−0.374056			−0.187028	−	0.982355i	$-0.559886\pi$
$37$	−84.1860			−0.374056			−0.187028	+	0.982355i	$0.559886\pi$
$38$	−143.352	+	248.293i	−0.611968	+	1.05996i
$39$		0			0
$40$	43.4817	+	75.3125i	0.171877	+	0.297699i
$41$	101.630	+	176.028i	0.387119	+	0.670510i	0.992061	−	0.125760i	$-0.0401370\pi$
$41$	101.630	+	176.028i	0.387119	+	0.670510i	−0.604942	+	0.796270i	$0.706804\pi$
$42$		0			0
$43$	−162.945	+	282.229i	−0.577881	+	1.00092i	0.417841	+	0.908520i	$0.362787\pi$
$43$	−162.945	+	282.229i	−0.577881	+	1.00092i	−0.995722	+	0.0923995i	$0.970546\pi$
$44$	170.963			0.585766
$45$		0			0
$46$	−38.2591			−0.122631
$47$	5.47180	−	9.47744i	0.0169818	−	0.0294133i	−0.857410	−	0.514635i	$-0.827928\pi$
$47$	5.47180	−	9.47744i	0.0169818	−	0.0294133i	0.874391	+	0.485221i	$0.161261\pi$
$48$		0			0
$49$	−137.769	−	238.623i	−0.401659	−	0.695694i
$50$	6.83384	+	11.8366i	0.0193290	+	0.0334788i
$51$		0			0
$52$	−30.2591	+	52.4104i	−0.0806959	+	0.139769i
$53$	140.186			0.363321			0.181661	−	0.983361i	$-0.441853\pi$
$53$	140.186			0.363321			0.181661	+	0.983361i	$0.441853\pi$
$54$		0			0
$55$	−464.611			−1.13906
$56$	−99.4817	+	172.307i	−0.237389	+	0.411170i
$57$		0			0
$58$	226.093	+	391.605i	0.511853	+	0.886555i
$59$	57.3704	+	99.3685i	0.126593	+	0.219266i	0.922355	−	0.386345i	$-0.126262\pi$
$59$	57.3704	+	99.3685i	0.126593	+	0.219266i	−0.795761	+	0.605610i	$0.792929\pi$
$60$		0			0
$61$	377.528	−	653.898i	0.792419	−	1.37251i	−0.132047	−	0.991243i	$-0.542155\pi$
$61$	377.528	−	653.898i	0.792419	−	1.37251i	0.924465	−	0.381266i	$-0.124512\pi$
$62$	118.777			0.243302
$63$		0			0
$64$	64.0000			0.125000
$65$	82.2325	−	142.431i	0.156918	−	0.271790i
$66$		0			0
$67$	383.723	+	664.627i	0.699689	+	1.21190i	0.968574	+	0.248725i	$0.0800115\pi$
$67$	383.723	+	664.627i	0.699689	+	1.21190i	−0.268885	+	0.963172i	$0.586655\pi$
$68$	−27.7409	−	48.0486i	−0.0494717	−	0.0856874i
$69$		0			0
$70$	270.352	−	468.264i	0.461618	−	0.799546i
$71$	−335.854			−0.561387			−0.280694	−	0.959797i	$-0.590564\pi$
$71$	−335.854			−0.561387			−0.280694	+	0.959797i	$0.590564\pi$
$72$		0			0
$73$	167.279			0.268199			0.134099	−	0.990968i	$-0.457186\pi$
$73$	167.279			0.268199			0.134099	+	0.990968i	$0.457186\pi$
$74$	84.1860	−	145.814i	0.132249	−	0.229062i
$75$		0			0
$76$	−286.704	−	496.586i	−0.432727	−	0.749505i
$77$	−531.492	−	920.570i	−0.786612	−	1.36245i
$78$		0			0
$79$	12.6578	−	21.9239i	0.0180267	−	0.0312232i	−0.856871	−	0.515530i	$-0.827595\pi$
$79$	12.6578	−	21.9239i	0.0180267	−	0.0312232i	0.874898	+	0.484307i	$0.160928\pi$
$80$	−173.927			−0.243070
$81$		0			0
$82$	−406.518			−0.547469
$83$	143.861	−	249.174i	0.190250	−	0.329523i	−0.755083	−	0.655629i	$-0.772403\pi$
$83$	143.861	−	249.174i	0.190250	−	0.329523i	0.945333	+	0.326107i	$0.105737\pi$
$84$		0			0
$85$	75.3887	+	130.577i	0.0962006	+	0.166624i
$86$	−325.890	−	564.458i	−0.408624	−	0.707757i
$87$		0			0
$88$	−170.963	+	296.117i	−0.207100	+	0.358707i
$89$	860.817			1.02524			0.512620	−	0.858615i	$-0.328675\pi$
$89$	860.817			1.02524			0.512620	+	0.858615i	$0.328675\pi$
$90$		0			0
$91$	376.279			0.433459
$92$	38.2591	−	66.2668i	0.0433564	−	0.0750955i
$93$		0			0
$94$	10.9436	+	18.9549i	0.0120079	+	0.0207984i
$95$	779.149	+	1349.53i	0.841464	+	1.45746i
$96$		0			0
$97$	201.075	−	348.272i	0.210475	−	0.364553i	−0.741389	−	0.671076i	$-0.765832\pi$
$97$	201.075	−	348.272i	0.210475	−	0.364553i	0.951863	+	0.306523i	$0.0991657\pi$
$98$	551.076			0.568032
$99$		0			0
$100$	−27.3353			−0.0273353
$101$	−665.714	+	1153.05i	−0.655852	+	1.13597i	0.325828	+	0.945429i	$0.394357\pi$
$101$	−665.714	+	1153.05i	−0.655852	+	1.13597i	−0.981680	+	0.190540i	$0.938976\pi$
$102$		0			0
$103$	259.252	+	449.038i	0.248009	+	0.429563i	0.962973	−	0.269597i	$-0.0868905\pi$
$103$	259.252	+	449.038i	0.248009	+	0.429563i	−0.714965	+	0.699161i	$0.753557\pi$
$104$	−60.5183	−	104.821i	−0.0570606	−	0.0988319i
$105$		0			0
$106$	−140.186	+	242.809i	−0.128453	+	0.222488i
$107$	−1471.87			−1.32982			−0.664912	−	0.746922i	$-0.731531\pi$
$107$	−1471.87			−1.32982			−0.664912	+	0.746922i	$0.731531\pi$
$108$		0			0
$109$	−643.668			−0.565616			−0.282808	−	0.959176i	$-0.591266\pi$
$109$	−643.668			−0.565616			−0.282808	+	0.959176i	$0.591266\pi$
$110$	464.611	−	804.730i	0.402718	−	0.697528i
$111$		0			0
$112$	−198.963	−	344.615i	−0.167860	−	0.290741i
$113$	511.864	+	886.574i	0.426125	+	0.738069i	0.996525	−	0.0832976i	$-0.0265452\pi$
$113$	511.864	+	886.574i	0.426125	+	0.738069i	−0.570400	+	0.821367i	$0.693212\pi$
$114$		0			0
$115$	−103.973	+	180.087i	−0.0843092	+	0.146028i
$116$	−904.372			−0.723869
$117$		0			0
$118$	−229.482			−0.179030
$119$	−172.482	+	298.747i	−0.132869	+	0.230135i
$120$		0			0
$121$	−247.890	−	429.358i	−0.186244	−	0.322583i
$122$	755.056	+	1307.80i	0.560325	+	0.970511i
$123$		0			0
$124$	−118.777	+	205.729i	−0.0860204	+	0.148992i
$125$	1433.09			1.02544
$126$		0			0
$127$	−31.4481			−0.0219730			−0.0109865	−	0.999940i	$-0.503497\pi$
$127$	−31.4481			−0.0219730			−0.0109865	+	0.999940i	$0.503497\pi$
$128$	−64.0000	+	110.851i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	164.465	+	284.862i	0.110958	+	0.192185i
$131$	−968.678	−	1677.80i	−0.646059	−	1.11901i	−0.984056	−	0.177860i	$-0.943083\pi$
$131$	−968.678	−	1677.80i	−0.646059	−	1.11901i	0.337997	−	0.941147i	$-0.390251\pi$
$132$		0			0
$133$	−1782.61	+	3087.58i	−1.16220	+	2.01299i
$134$	−1534.89			−0.989510
$135$		0			0
$136$	110.963			0.0699635
$137$	579.297	−	1003.37i	0.361261	−	0.625722i	−0.626908	−	0.779093i	$-0.715680\pi$
$137$	579.297	−	1003.37i	0.361261	−	0.625722i	0.988169	+	0.153372i	$0.0490131\pi$
$138$		0			0
$139$	−1155.80	−	2001.90i	−0.705277	−	1.22158i	−0.966591	−	0.256323i	$-0.917489\pi$
$139$	−1155.80	−	2001.90i	−0.705277	−	1.22158i	0.261314	−	0.965254i	$-0.415844\pi$
$140$	540.704	+	936.527i	0.326413	+	0.565364i
$141$		0			0
$142$	335.854	−	581.716i	0.198480	−	0.343778i
$143$	646.651			0.378151
$144$		0			0
$145$	2457.73			1.40761
$146$	−167.279	+	289.736i	−0.0948226	+	0.164238i
$147$		0			0
$148$	168.372	+	291.629i	0.0935141	+	0.161971i
$149$	−1475.11	−	2554.96i	−0.811043	−	1.40477i	−0.912135	−	0.409890i	$-0.865567\pi$
$149$	−1475.11	−	2554.96i	−0.811043	−	1.40477i	0.101092	−	0.994877i	$-0.467766\pi$
$150$		0			0
$151$	863.199	−	1495.10i	0.465206	−	0.805761i	−0.534005	−	0.845482i	$-0.679314\pi$
$151$	863.199	−	1495.10i	0.465206	−	0.805761i	0.999211	+	0.0397208i	$0.0126469\pi$
$152$	1146.82			0.611968
$153$		0			0
$154$	2125.97			1.11244
$155$	322.790	−	559.089i	0.167272	−	0.289723i
$156$		0			0
$157$	−641.694	−	1111.45i	−0.326196	−	0.564988i	0.655558	−	0.755145i	$-0.272434\pi$
$157$	−641.694	−	1111.45i	−0.326196	−	0.564988i	−0.981754	+	0.190157i	$0.939100\pi$
$158$	25.3155	+	43.8478i	0.0127468	+	0.0220781i
$159$		0			0
$160$	173.927	−	301.250i	0.0859383	−	0.148849i
$161$	−475.761			−0.232889
$162$		0			0
$163$	1033.93			0.496831			0.248415	−	0.968654i	$-0.420090\pi$
$163$	1033.93			0.496831			0.248415	+	0.968654i	$0.420090\pi$
$164$	406.518	−	704.110i	0.193559	−	0.335255i
$165$		0			0
$166$	287.721	+	498.347i	0.134527	+	0.233008i
$167$	−141.309	−	244.754i	−0.0654778	−	0.113411i	0.831428	−	0.555632i	$-0.187524\pi$
$167$	−141.309	−	244.754i	−0.0654778	−	0.113411i	−0.896906	+	0.442222i	$0.854190\pi$
$168$		0			0
$169$	984.048	−	1704.42i	0.447905	−	0.775795i
$170$	−301.555			−0.136048
$171$		0			0
$172$	1303.56			0.577881
$173$	−1766.36	+	3059.43i	−0.776266	+	1.34453i	0.157814	+	0.987469i	$0.449555\pi$
$173$	−1766.36	+	3059.43i	−0.776266	+	1.34453i	−0.934080	+	0.357063i	$0.883778\pi$
$174$		0			0
$175$	84.9802	+	147.190i	0.0367080	+	0.0635801i
$176$	−341.927	−	592.235i	−0.146441	−	0.253644i
$177$		0			0
$178$	−860.817	+	1490.98i	−0.362477	+	0.627829i
$179$	4052.74			1.69227			0.846135	−	0.532969i	$-0.178924\pi$
$179$	4052.74			1.69227			0.846135	+	0.532969i	$0.178924\pi$
$180$		0			0
$181$	−2830.97			−1.16257			−0.581283	−	0.813702i	$-0.697449\pi$
$181$	−2830.97			−1.16257			−0.581283	+	0.813702i	$0.697449\pi$
$182$	−376.279	+	651.734i	−0.153251	+	0.265438i
$183$		0			0
$184$	76.5183	+	132.534i	0.0306576	+	0.0531006i
$185$	−457.569	−	792.532i	−0.181844	−	0.314963i
$186$		0			0
$187$	−296.417	+	513.409i	−0.115915	+	0.200771i
$188$	−43.7744			−0.0169818
$189$		0			0
$190$	−3116.60			−1.19001
$191$	2254.62	−	3905.12i	0.854129	−	1.47940i	−0.0233215	−	0.999728i	$-0.507424\pi$
$191$	2254.62	−	3905.12i	0.854129	−	1.47940i	0.877451	−	0.479667i	$-0.159243\pi$
$192$		0			0
$193$	1610.56	+	2789.58i	0.600678	+	1.04040i	0.992719	+	0.120457i	$0.0384359\pi$
$193$	1610.56	+	2789.58i	0.600678	+	1.04040i	−0.392041	+	0.919948i	$0.628231\pi$
$194$	402.149	+	696.543i	0.148828	+	0.257778i
$195$		0			0
$196$	−551.076	+	954.492i	−0.200830	+	0.347847i
$197$	−3784.20			−1.36859			−0.684297	−	0.729204i	$-0.739891\pi$
$197$	−3784.20			−1.36859			−0.684297	+	0.729204i	$0.739891\pi$
$198$		0			0
$199$	−2926.27			−1.04240			−0.521200	−	0.853435i	$-0.674515\pi$
$199$	−2926.27			−1.04240			−0.521200	+	0.853435i	$0.674515\pi$
$200$	27.3353	−	47.3462i	0.00966450	−	0.0167394i
$201$		0			0
$202$	−1331.43	−	2306.10i	−0.463757	−	0.803251i
$203$	2811.51	+	4869.69i	0.972067	+	1.68367i
$204$		0			0
$205$	−1104.76	+	1913.49i	−0.376388	+	0.651923i
$206$	−1037.01			−0.350737
$207$		0			0
$208$	242.073			0.0806959
$209$	−3063.50	+	5306.13i	−1.01391	+	1.75614i
$210$		0			0
$211$	−156.737	−	271.476i	−0.0511385	−	0.0885744i	0.839323	−	0.543633i	$-0.182952\pi$
$211$	−156.737	−	271.476i	−0.0511385	−	0.0885744i	−0.890462	+	0.455059i	$0.849618\pi$
$212$	−280.372	−	485.618i	−0.0908303	−	0.157323i
$213$		0			0
$214$	1471.87	−	2549.35i	0.470164	−	0.814347i
$215$	−3542.57			−1.12373
$216$		0			0
$217$	1477.02			0.462059
$218$	643.668	−	1114.87i	0.199976	−	0.346368i
$219$		0			0
$220$	929.223	+	1609.46i	0.284764	+	0.493226i
$221$	−104.927	−	181.739i	−0.0319373	−	0.0553170i
$222$		0			0
$223$	355.193	−	615.212i	0.106661	−	0.184743i	−0.807754	−	0.589519i	$-0.799317\pi$
$223$	355.193	−	615.212i	0.106661	−	0.184743i	0.914416	+	0.404776i	$0.132651\pi$
$224$	795.854			0.237389
$225$		0			0
$226$	−2047.45			−0.602631
$227$	16.3308	−	28.2858i	0.00477496	−	0.00827046i	−0.863628	−	0.504130i	$-0.831813\pi$
$227$	16.3308	−	28.2858i	0.00477496	−	0.00827046i	0.868403	+	0.495859i	$0.165147\pi$
$228$		0			0
$229$	2751.92	+	4766.47i	0.794114	+	1.37545i	0.923400	+	0.383839i	$0.125398\pi$
$229$	2751.92	+	4766.47i	0.794114	+	1.37545i	−0.129286	+	0.991607i	$0.541268\pi$
$230$	−207.947	−	360.174i	−0.0596156	−	0.103257i
$231$		0			0
$232$	904.372	−	1566.42i	0.255926	−	0.443278i
$233$	6788.81			1.90880			0.954399	−	0.298534i	$-0.0964977\pi$
$233$	6788.81			1.90880			0.954399	+	0.298534i	$0.0964977\pi$
$234$		0			0
$235$	118.962			0.0330221
$236$	229.482	−	397.474i	0.0632966	−	0.109633i
$237$		0			0
$238$	−344.963	−	597.494i	−0.0939523	−	0.162730i
$239$	−214.694	−	371.862i	−0.0581064	−	0.100643i	0.835509	−	0.549477i	$-0.185173\pi$
$239$	−214.694	−	371.862i	−0.0581064	−	0.100643i	−0.893615	+	0.448834i	$0.851840\pi$
$240$		0			0
$241$	2421.82	−	4194.72i	0.647317	−	1.12119i	−0.336444	−	0.941703i	$-0.609224\pi$
$241$	2421.82	−	4194.72i	0.647317	−	1.12119i	0.983761	−	0.179483i	$-0.0574424\pi$
$242$	991.561			0.263388
$243$		0			0
$244$	−3020.23			−0.792419
$245$	1497.61	−	2593.93i	0.390525	−	0.676410i
$246$		0			0
$247$	−1084.43	−	1878.28i	−0.279354	−	0.483856i
$248$	−237.555	−	411.457i	−0.0608256	−	0.105353i
$249$		0			0
$250$	−1433.09	+	2482.18i	−0.362546	+	0.627949i
$251$	−2400.87			−0.603752			−0.301876	−	0.953347i	$-0.597613\pi$
$251$	−2400.87			−0.603752			−0.301876	+	0.953347i	$0.597613\pi$
$252$		0			0
$253$	−817.614			−0.203174
$254$	31.4481	−	54.4698i	0.00776863	−	0.0134557i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	1490.50	+	2581.62i	0.361769	+	0.626602i	0.988252	−	0.152833i	$-0.0488396\pi$
$257$	1490.50	+	2581.62i	0.361769	+	0.626602i	−0.626483	+	0.779435i	$0.715506\pi$
$258$		0			0
$259$	1046.87	−	1813.23i	0.251156	−	0.435015i
$260$	−657.860			−0.156918
$261$		0			0
$262$	3874.71			0.913666
$263$	2780.41	−	4815.81i	0.651891	−	1.12911i	−0.330773	−	0.943710i	$-0.607309\pi$
$263$	2780.41	−	4815.81i	0.651891	−	1.12911i	0.982664	−	0.185398i	$-0.0593573\pi$
$264$		0			0
$265$	761.941	+	1319.72i	0.176625	+	0.305924i
$266$	−3565.23	−	6175.16i	−0.821798	−	1.42340i
$267$		0			0
$268$	1534.89	−	2658.51i	0.349845	−	0.605949i
$269$	−6288.50			−1.42534			−0.712671	−	0.701499i	$-0.752515\pi$
$269$	−6288.50			−1.42534			−0.712671	+	0.701499i	$0.752515\pi$
$270$		0			0
$271$	6854.90			1.53655			0.768275	−	0.640119i	$-0.221115\pi$
$271$	6854.90			1.53655			0.768275	+	0.640119i	$0.221115\pi$
$272$	−110.963	+	192.194i	−0.0247358	+	0.0428437i
$273$		0			0
$274$	1158.59	+	2006.74i	0.255450	+	0.442452i
$275$	146.042	+	252.952i	0.0320242	+	0.0554676i
$276$		0			0
$277$	−449.086	+	777.840i	−0.0974114	+	0.168722i	−0.910612	−	0.413261i	$-0.864390\pi$
$277$	−449.086	+	777.840i	−0.0974114	+	0.168722i	0.813201	+	0.581983i	$0.197723\pi$
$278$	4623.20			0.997413
$279$		0			0
$280$	−2162.82			−0.461618
$281$	119.385	−	206.781i	0.0253448	−	0.0438986i	−0.853075	−	0.521789i	$-0.825265\pi$
$281$	119.385	−	206.781i	0.0253448	−	0.0438986i	0.878420	+	0.477890i	$0.158598\pi$
$282$		0			0
$283$	−1035.58	−	1793.68i	−0.217523	−	0.376760i	0.736527	−	0.676408i	$-0.236464\pi$
$283$	−1035.58	−	1793.68i	−0.217523	−	0.376760i	−0.954050	+	0.299647i	$0.903131\pi$
$284$	671.707	+	1163.43i	0.140347	+	0.243088i
$285$		0			0
$286$	−646.651	+	1120.03i	−0.133697	+	0.231570i
$287$	−5055.14			−1.03971
$288$		0			0
$289$	−4720.61			−0.960841
$290$	−2457.73	+	4256.91i	−0.497664	+	0.861980i
$291$		0			0
$292$	−334.558	−	579.471i	−0.0670497	−	0.116134i
$293$	−3288.88	−	5696.51i	−0.655763	−	1.13581i	−0.981702	−	0.190423i	$-0.939014\pi$
$293$	−3288.88	−	5696.51i	−0.655763	−	1.13581i	0.325939	−	0.945391i	$-0.394319\pi$
$294$		0			0
$295$	−623.641	+	1080.18i	−0.123084	+	0.213188i
$296$	−673.488			−0.132249
$297$		0			0
$298$	5900.42			1.14699
$299$	144.711	−	250.647i	0.0279895	−	0.0484792i
$300$		0			0
$301$	−4052.51	−	7019.16i	−0.776023	−	1.34411i
$302$	1726.40	+	2990.21i	0.328950	+	0.569759i
$303$		0			0
$304$	−1146.82	+	1986.35i	−0.216363	+	0.374752i
$305$	8207.78			1.54091
$306$		0			0
$307$	−5237.30			−0.973644			−0.486822	−	0.873501i	$-0.661844\pi$
$307$	−5237.30			−0.973644			−0.486822	+	0.873501i	$0.661844\pi$
$308$	−2125.97	+	3682.28i	−0.393306	+	0.681226i
$309$		0			0
$310$	645.581	+	1118.18i	0.118279	+	0.204865i
$311$	2852.49	+	4940.67i	0.520097	+	0.900834i	0.999727	+	0.0233635i	$0.00743750\pi$
$311$	2852.49	+	4940.67i	0.520097	+	0.900834i	−0.479630	+	0.877471i	$0.659229\pi$
$312$		0			0
$313$	−2538.74	+	4397.23i	−0.458460	+	0.794076i	−0.998880	−	0.0473193i	$-0.984932\pi$
$313$	−2538.74	+	4397.23i	−0.458460	+	0.794076i	0.540420	+	0.841396i	$0.318265\pi$
$314$	2566.78			0.461311
$315$		0			0
$316$	−101.262			−0.0180267
$317$	1434.46	−	2484.55i	0.254155	−	0.440209i	−0.710511	−	0.703686i	$-0.751536\pi$
$317$	1434.46	−	2484.55i	0.254155	−	0.440209i	0.964666	+	0.263477i	$0.0848694\pi$
$318$		0			0
$319$	4831.70	+	8368.76i	0.848036	+	1.46884i
$320$	347.854	+	602.500i	0.0607675	+	0.105252i
$321$		0			0
$322$	475.761	−	824.042i	0.0823388	−	0.142615i
$323$	1988.36			0.342523
$324$		0			0
$325$	−103.393			−0.0176468
$326$	−1033.93	+	1790.81i	−0.175656	+	0.304245i
$327$		0			0
$328$	813.037	+	1408.22i	0.136867	+	0.237061i
$329$	136.086	+	235.708i	0.0228045	+	0.0394985i
$330$		0			0
$331$	−1015.67	+	1759.20i	−0.168660	+	0.292128i	−0.937949	−	0.346773i	$-0.887277\pi$
$331$	−1015.67	+	1759.20i	−0.168660	+	0.292128i	0.769289	+	0.638901i	$0.220611\pi$
$332$	−1150.88			−0.190250
$333$		0			0
$334$	565.235			0.0925996
$335$	−4171.23	+	7224.78i	−0.680294	+	1.17830i
$336$		0			0
$337$	−4899.14	−	8485.56i	−0.791909	−	1.37163i	−0.924784	−	0.380493i	$-0.875754\pi$
$337$	−4899.14	−	8485.56i	−0.791909	−	1.37163i	0.132875	−	0.991133i	$-0.457579\pi$
$338$	1968.10	+	3408.84i	0.316717	+	0.548570i
$339$		0			0
$340$	301.555	−	522.308i	0.0481003	−	0.0833122i
$341$	2538.32			0.403103
$342$		0			0
$343$	−1677.81			−0.264120
$344$	−1303.56	+	2257.83i	−0.204312	+	0.353879i
$345$		0			0
$346$	−3532.72	−	6118.86i	−0.548903	−	0.950728i
$347$	−2278.28	−	3946.10i	−0.352463	−	0.610483i	0.634218	−	0.773154i	$-0.281322\pi$
$347$	−2278.28	−	3946.10i	−0.352463	−	0.610483i	−0.986680	+	0.162671i	$0.947989\pi$
$348$		0			0
$349$	−1674.44	+	2900.22i	−0.256822	+	0.444829i	−0.965389	−	0.260815i	$-0.916009\pi$
$349$	−1674.44	+	2900.22i	−0.256822	+	0.444829i	0.708567	+	0.705644i	$0.249342\pi$
$350$	−339.921			−0.0519129
$351$		0			0
$352$	1367.71			0.207100
$353$	931.134	−	1612.77i	0.140395	−	0.243170i	−0.787251	−	0.616633i	$-0.788496\pi$
$353$	931.134	−	1612.77i	0.140395	−	0.243170i	0.927645	+	0.373463i	$0.121830\pi$
$354$		0			0
$355$	−1825.44	−	3161.75i	−0.272913	−	0.472699i
$356$	−1721.63	−	2981.96i	−0.256310	−	0.443942i
$357$		0			0
$358$	−4052.74	+	7019.56i	−0.598308	+	1.03630i
$359$	−6179.46			−0.908466			−0.454233	−	0.890883i	$-0.650087\pi$
$359$	−6179.46			−0.908466			−0.454233	+	0.890883i	$0.650087\pi$
$360$		0			0
$361$	13690.8			1.99604
$362$	2830.97	−	4903.38i	0.411029	−	0.711923i
$363$		0			0
$364$	−752.558	−	1303.47i	−0.108365	−	0.187693i
$365$	909.197	+	1574.77i	0.130382	+	0.225829i
$366$		0			0
$367$	−3436.98	+	5953.02i	−0.488852	+	0.846717i	−0.999918	−	0.0128251i	$-0.995918\pi$
$367$	−3436.98	+	5953.02i	−0.488852	+	0.846717i	0.511066	+	0.859542i	$0.329251\pi$
$368$	−306.073			−0.0433564
$369$		0			0
$370$	1830.27			0.257166
$371$	−1743.24	+	3019.38i	−0.243948	+	0.422530i
$372$		0			0
$373$	−635.013	−	1099.87i	−0.0881494	−	0.152679i	0.818580	−	0.574393i	$-0.194762\pi$
$373$	−635.013	−	1099.87i	−0.0881494	−	0.152679i	−0.906729	+	0.421714i	$0.861429\pi$
$374$	−592.834	−	1026.82i	−0.0819645	−	0.141967i
$375$		0			0
$376$	43.7744	−	75.8195i	0.00600397	−	0.0103992i
$377$	−3420.69			−0.467306
$378$		0			0
$379$	2490.54			0.337548			0.168774	−	0.985655i	$-0.446019\pi$
$379$	2490.54			0.337548			0.168774	+	0.985655i	$0.446019\pi$
$380$	3116.60	−	5398.11i	0.420732	−	0.728729i
$381$		0			0
$382$	4509.24	+	7810.24i	0.603960	+	1.04609i
$383$	156.213	+	270.568i	0.0208410	+	0.0360976i	0.876258	−	0.481843i	$-0.160032\pi$
$383$	156.213	+	270.568i	0.0208410	+	0.0360976i	−0.855417	+	0.517940i	$0.826699\pi$
$384$		0			0
$385$	5777.54	−	10007.0i	0.764807	−	1.32468i
$386$	−6442.25			−0.849487
$387$		0			0
$388$	−1608.60			−0.210475
$389$	4821.58	−	8351.23i	0.628442	−	1.08849i	−0.359422	−	0.933175i	$-0.617026\pi$
$389$	4821.58	−	8351.23i	0.628442	−	1.08849i	0.987864	−	0.155319i	$-0.0496404\pi$
$390$		0			0
$391$	132.668	+	229.787i	0.0171593	+	0.0297208i
$392$	−1102.15	−	1908.98i	−0.142008	−	0.245965i
$393$		0			0
$394$	3784.20	−	6554.42i	0.483871	−	0.838089i
$395$	275.191			0.0350540
$396$		0			0
$397$	−2260.32			−0.285749			−0.142874	−	0.989741i	$-0.545634\pi$
$397$	−2260.32			−0.285749			−0.142874	+	0.989741i	$0.545634\pi$
$398$	2926.27	−	5068.44i	0.368544	−	0.638337i
$399$		0			0
$400$	54.6707	+	94.6924i	0.00683384	+	0.0118366i
$401$	5084.64	+	8806.85i	0.633204	+	1.09674i	0.986893	+	0.161378i	$0.0515938\pi$
$401$	5084.64	+	8806.85i	0.633204	+	1.09674i	−0.353689	+	0.935363i	$0.615073\pi$
$402$		0			0
$403$	−449.263	+	778.146i	−0.0555320	+	0.0961842i
$404$	5325.71			0.655852
$405$		0			0
$406$	−11246.1			−1.37471
$407$	1799.09	−	3116.12i	0.219110	−	0.379509i
$408$		0			0
$409$	474.916	+	822.579i	0.0574159	+	0.0994472i	0.893305	−	0.449452i	$-0.148381\pi$
$409$	474.916	+	822.579i	0.0574159	+	0.0994472i	−0.835889	+	0.548899i	$0.815047\pi$
$410$	−2209.51	−	3826.99i	−0.266147	−	0.460979i
$411$		0			0
$412$	1037.01	−	1796.15i	0.124004	−	0.214782i
$413$	−2853.65			−0.339998
$414$		0			0
$415$	3127.65			0.369953
$416$	−242.073	+	419.283i	−0.0285303	+	0.0494160i
$417$		0			0
$418$	−6126.99	−	10612.3i	−0.716940	−	1.24178i
$419$	−5899.63	−	10218.5i	−0.687866	−	1.19142i	−0.972527	−	0.232791i	$-0.925214\pi$
$419$	−5899.63	−	10218.5i	−0.687866	−	1.19142i	0.284660	−	0.958628i	$-0.408119\pi$
$420$		0			0
$421$	3206.46	−	5553.76i	0.371196	−	0.642930i	−0.618554	−	0.785742i	$-0.712281\pi$
$421$	3206.46	−	5553.76i	0.371196	−	0.642930i	0.989750	+	0.142812i	$0.0456145\pi$
$422$	626.948			0.0723207
$423$		0			0
$424$	1121.49			0.128453
$425$	47.3941	−	82.0890i	0.00540930	−	0.00936918i
$426$		0			0
$427$	9389.29	+	16262.7i	1.06412	+	1.84311i
$428$	2943.74	+	5098.71i	0.332456	+	0.575830i
$429$		0			0
$430$	3542.57	−	6135.90i	0.397297	−	0.688138i
$431$	12042.7			1.34589			0.672945	−	0.739693i	$-0.265029\pi$
$431$	12042.7			1.34589			0.672945	+	0.739693i	$0.265029\pi$
$432$		0			0
$433$	7279.83			0.807959			0.403980	−	0.914768i	$-0.367627\pi$
$433$	7279.83			0.807959			0.403980	+	0.914768i	$0.367627\pi$
$434$	−1477.02	+	2558.28i	−0.163363	+	0.282952i
$435$		0			0
$436$	1287.34	+	2229.73i	0.141404	+	0.244919i
$437$	1371.13	+	2374.87i	0.150092	+	0.259967i
$438$		0			0
$439$	1799.35	−	3116.57i	0.195623	−	0.338828i	−0.751482	−	0.659754i	$-0.770661\pi$
$439$	1799.35	−	3116.57i	0.195623	−	0.338828i	0.947104	+	0.320926i	$0.103994\pi$
$440$	−3716.89			−0.402718
$441$		0			0
$442$	419.707			0.0451662
$443$	−7393.18	+	12805.4i	−0.792913	+	1.37337i	0.131243	+	0.991350i	$0.458103\pi$
$443$	−7393.18	+	12805.4i	−0.792913	+	1.37337i	−0.924156	+	0.382016i	$0.875230\pi$
$444$		0			0
$445$	4678.72	+	8103.79i	0.498411	+	0.863273i
$446$	710.386	+	1230.42i	0.0754209	+	0.130633i
$447$		0			0
$448$	−795.854	+	1378.46i	−0.0839298	+	0.145371i
$449$	−114.489			−0.0120336			−0.00601681	−	0.999982i	$-0.501915\pi$
$449$	−114.489			−0.0120336			−0.00601681	+	0.999982i	$0.501915\pi$
$450$		0			0
$451$	−8687.47			−0.907044
$452$	2047.45	−	3546.29i	0.213062	−	0.369035i
$453$		0			0
$454$	32.6616	+	56.5716i	0.00337640	+	0.00584810i
$455$	2045.16	+	3542.31i	0.210722	+	0.364981i
$456$		0			0
$457$	−3155.57	+	5465.60i	−0.323001	+	0.559453i	−0.981106	−	0.193473i	$-0.938025\pi$
$457$	−3155.57	+	5465.60i	−0.323001	+	0.559453i	0.658105	+	0.752926i	$0.271358\pi$
$458$	−11007.7			−1.12305
$459$		0			0
$460$	831.786			0.0843092
$461$	−6872.41	+	11903.4i	−0.694317	+	1.20259i	0.276094	+	0.961131i	$0.410960\pi$
$461$	−6872.41	+	11903.4i	−0.694317	+	1.20259i	−0.970411	+	0.241461i	$0.922373\pi$
$462$		0			0
$463$	−7824.30	−	13552.1i	−0.785369	−	1.36030i	−0.928778	−	0.370636i	$-0.879140\pi$
$463$	−7824.30	−	13552.1i	−0.785369	−	1.36030i	0.143409	−	0.989664i	$-0.454194\pi$
$464$	1808.74	+	3132.84i	0.180967	+	0.313445i
$465$		0			0
$466$	−6788.81	+	11758.6i	−0.674862	+	1.16890i
$467$	7395.79			0.732840			0.366420	−	0.930450i	$-0.380583\pi$
$467$	7395.79			0.732840			0.366420	+	0.930450i	$0.380583\pi$
$468$		0			0
$469$	−19086.7			−1.87919
$470$	−118.962	+	206.048i	−0.0116751	+	0.0202219i
$471$		0			0
$472$	458.963	+	794.948i	0.0447574	+	0.0775221i
$473$	−6964.41	−	12062.7i	−0.677006	−	1.17261i
$474$		0			0
$475$	489.822	−	848.397i	0.0473149	−	0.0819519i
$476$	1379.85			0.132869
$477$		0			0
$478$	858.777			0.0821748
$479$	−3130.53	+	5422.24i	−0.298617	+	0.517221i	−0.975820	−	0.218576i	$-0.929859\pi$
$479$	−3130.53	+	5422.24i	−0.298617	+	0.517221i	0.677202	+	0.735797i	$0.263192\pi$
$480$		0			0
$481$	636.849	+	1103.05i	0.0603697	+	0.104563i
$482$	4843.65	+	8389.45i	0.457722	+	0.792798i
$483$		0			0
$484$	−991.561	+	1717.43i	−0.0931218	+	0.161292i
$485$	4371.54			0.409281
$486$		0			0
$487$	−10314.7			−0.959763			−0.479881	−	0.877333i	$-0.659320\pi$
$487$	−10314.7			−0.959763			−0.479881	+	0.877333i	$0.659320\pi$
$488$	3020.23	−	5231.18i	0.280162	−	0.485255i
$489$		0			0
$490$	2995.22	+	5187.87i	0.276143	+	0.478294i
$491$	−1380.45	−	2391.01i	−0.126881	−	0.219765i	0.795585	−	0.605841i	$-0.207163\pi$
$491$	−1380.45	−	2391.01i	−0.126881	−	0.219765i	−0.922467	+	0.386076i	$0.873830\pi$
$492$		0			0
$493$	1568.00	−	2715.86i	0.143244	−	0.248106i
$494$	4337.71			0.395067
$495$		0			0
$496$	950.220			0.0860204
$497$	4176.41	−	7233.76i	0.376937	−	0.652874i
$498$		0			0
$499$	4793.00	+	8301.71i	0.429988	+	0.744761i	0.996872	−	0.0790369i	$-0.0251845\pi$
$499$	4793.00	+	8301.71i	0.429988	+	0.744761i	−0.566884	+	0.823798i	$0.691851\pi$
$500$	−2866.18	−	4964.37i	−0.256359	−	0.444027i
$501$		0			0
$502$	2400.87	−	4158.43i	0.213459	−	0.369721i
$503$	−8829.60			−0.782689			−0.391344	−	0.920244i	$-0.627990\pi$
$503$	−8829.60			−0.782689			−0.391344	+	0.920244i	$0.627990\pi$
$504$		0			0
$505$	−14473.2			−1.27534
$506$	817.614	−	1416.15i	0.0718328	−	0.124418i
$507$		0			0
$508$	62.8963	+	108.940i	0.00549325	+	0.00951458i
$509$	2370.87	+	4106.47i	0.206458	+	0.357595i	0.950596	−	0.310430i	$-0.100473\pi$
$509$	2370.87	+	4106.47i	0.206458	+	0.357595i	−0.744138	+	0.668025i	$0.767140\pi$
$510$		0			0
$511$	−2080.15	+	3602.92i	−0.180079	+	0.311906i
$512$	512.000			0.0441942
$513$		0			0
$514$	−5961.99			−0.511619
$515$	−2818.18	+	4881.24i	−0.241134	+	0.417656i
$516$		0			0
$517$	233.870	+	405.074i	0.0198947	+	0.0344587i
$518$	2093.74	+	3626.47i	0.177594	+	0.307602i
$519$		0			0
$520$	657.860	−	1139.45i	0.0554790	−	0.0960924i
$521$	2753.22			0.231518			0.115759	−	0.993277i	$-0.463070\pi$
$521$	2753.22			0.231518			0.115759	+	0.993277i	$0.463070\pi$
$522$		0			0
$523$	17115.3			1.43098			0.715489	−	0.698624i	$-0.246204\pi$
$523$	17115.3			1.43098			0.715489	+	0.698624i	$0.246204\pi$
$524$	−3874.71	+	6711.20i	−0.323030	+	0.559504i
$525$		0			0
$526$	5560.82	+	9631.61i	0.460956	+	0.798400i
$527$	−411.873	−	713.386i	−0.0340446	−	0.0589669i
$528$		0			0
$529$	5900.53	−	10220.0i	0.484962	−	0.839978i
$530$	−3047.76			−0.249786
$531$		0			0
$532$	14260.9			1.16220
$533$	1537.61	−	2663.22i	0.124956	−	0.216430i
$534$		0			0
$535$	−7999.93	−	13856.3i	−0.646481	−	1.11974i
$536$	3069.78	+	5317.02i	0.247377	+	0.428470i
$537$		0			0
$538$	6288.50	−	10892.0i	0.503934	−	0.872840i
$539$	11776.7			0.941113
$540$		0			0
$541$	17880.1			1.42093			0.710467	−	0.703731i	$-0.248484\pi$
$541$	17880.1			1.42093			0.710467	+	0.703731i	$0.248484\pi$
$542$	−6854.90	+	11873.0i	−0.543253	+	0.940941i
$543$		0			0
$544$	−221.927	−	384.389i	−0.0174909	−	0.0302951i
$545$	−3498.47	−	6059.53i	−0.274969	−	0.476260i
$546$		0			0
$547$	6534.73	−	11318.5i	0.510795	−	0.884723i	−0.489127	−	0.872213i	$-0.662684\pi$
$547$	6534.73	−	11318.5i	0.510795	−	0.884723i	0.999922	−	0.0125101i	$-0.00398218\pi$
$548$	−4634.38			−0.361261
$549$		0			0
$550$	−584.168			−0.0452891
$551$	16205.5	−	28068.7i	1.25295	−	2.17017i
$552$		0			0
$553$	314.804	+	545.257i	0.0242077	+	0.0419289i
$554$	−898.172	−	1555.68i	−0.0688803	−	0.119304i
$555$		0			0
$556$	−4623.20	+	8007.61i	−0.352639	+	0.610788i
$557$	−9507.62			−0.723251			−0.361626	−	0.932323i	$-0.617778\pi$
$557$	−9507.62			−0.723251			−0.361626	+	0.932323i	$0.617778\pi$
$558$		0			0
$559$	4930.58			0.373061
$560$	2162.82	−	3746.11i	0.163207	−	0.282682i
$561$		0			0
$562$	238.770	+	413.561i	0.0179215	+	0.0310410i
$563$	−10222.3	−	17705.6i	−0.765221	−	1.32540i	−0.940130	−	0.340817i	$-0.889296\pi$
$563$	−10222.3	−	17705.6i	−0.765221	−	1.32540i	0.174909	−	0.984585i	$-0.444037\pi$
$564$		0			0
$565$	−5564.17	+	9637.43i	−0.414313	+	0.717610i
$566$	4142.33			0.307624
$567$		0			0
$568$	−2686.83			−0.198480
$569$	1323.03	−	2291.56i	0.0974770	−	0.168835i	−0.813163	−	0.582036i	$-0.802256\pi$
$569$	1323.03	−	2291.56i	0.0974770	−	0.168835i	0.910640	+	0.413201i	$0.135589\pi$
$570$		0			0
$571$	−878.514	−	1521.63i	−0.0643864	−	0.111521i	0.832035	−	0.554723i	$-0.187176\pi$
$571$	−878.514	−	1521.63i	−0.0643864	−	0.111521i	−0.896422	+	0.443202i	$0.853842\pi$
$572$	−1293.30	−	2240.06i	−0.0945379	−	0.163744i
$573$		0			0
$574$	5055.14	−	8755.76i	0.367592	−	0.636687i
$575$	130.728			0.00948130
$576$		0			0
$577$	−7515.43			−0.542238			−0.271119	−	0.962546i	$-0.587394\pi$
$577$	−7515.43			−0.542238			−0.271119	+	0.962546i	$0.587394\pi$
$578$	4720.61	−	8176.34i	0.339709	−	0.588392i
$579$		0			0
$580$	−4915.45	−	8513.82i	−0.351902	−	0.609512i
$581$	3577.87	+	6197.06i	0.255482	+	0.442508i
$582$		0			0
$583$	−2995.83	+	5188.94i	−0.212821	+	0.368617i
$584$	1338.23			0.0948226
$585$		0			0
$586$	13155.5			0.927388
$587$	−2476.24	+	4288.98i	−0.174115	+	0.301576i	−0.939855	−	0.341575i	$-0.889040\pi$
$587$	−2476.24	+	4288.98i	−0.174115	+	0.301576i	0.765740	+	0.643151i	$0.222373\pi$
$588$		0			0
$589$	−4256.75	−	7372.91i	−0.297787	−	0.515782i
$590$	−1247.28	−	2160.36i	−0.0870335	−	0.150747i
$591$		0			0
$592$	673.488	−	1166.51i	0.0467571	−	0.0809856i
$593$	−17115.3			−1.18523			−0.592613	−	0.805487i	$-0.701904\pi$
$593$	−17115.3			−1.18523			−0.592613	+	0.805487i	$0.701904\pi$
$594$		0			0
$595$	−3749.90			−0.258371
$596$	−5900.42	+	10219.8i	−0.405521	+	0.702384i
$597$		0			0
$598$	289.422	+	501.294i	0.0197916	+	0.0342800i
$599$	4207.06	+	7286.85i	0.286972	+	0.497049i	0.973085	−	0.230445i	$-0.0740182\pi$
$599$	4207.06	+	7286.85i	0.286972	+	0.497049i	−0.686114	+	0.727494i	$0.740685\pi$
$600$		0			0
$601$	−14047.1	+	24330.3i	−0.953399	+	1.65134i	−0.215409	+	0.976524i	$0.569108\pi$
$601$	−14047.1	+	24330.3i	−0.953399	+	1.65134i	−0.737990	+	0.674812i	$0.764225\pi$
$602$	16210.1			1.09746
$603$		0			0
$604$	−6905.59			−0.465206
$605$	2694.67	−	4667.31i	0.181081	−	0.313642i
$606$		0			0
$607$	−715.423	−	1239.15i	−0.0478388	−	0.0828592i	0.841114	−	0.540857i	$-0.181900\pi$
$607$	−715.423	−	1239.15i	−0.0478388	−	0.0828592i	−0.888953	+	0.457998i	$0.848567\pi$
$608$	−2293.63	−	3972.69i	−0.152992	−	0.264990i
$609$		0			0
$610$	−8207.78	+	14216.3i	−0.544793	+	0.943608i
$611$	−165.572			−0.0109629
$612$		0			0
$613$	14438.1			0.951306			0.475653	−	0.879633i	$-0.342212\pi$
$613$	14438.1			0.951306			0.475653	+	0.879633i	$0.342212\pi$
$614$	5237.30	−	9071.27i	0.344235	−	0.596233i
$615$		0			0
$616$	−4251.93	−	7364.56i	−0.278109	−	0.481699i
$617$	12722.5	+	22036.0i	0.830125	+	1.43782i	0.897938	+	0.440121i	$0.145065\pi$
$617$	12722.5	+	22036.0i	0.830125	+	1.43782i	−0.0678130	+	0.997698i	$0.521602\pi$
$618$		0			0
$619$	1739.73	−	3013.30i	0.112966	−	0.195662i	−0.803999	−	0.594631i	$-0.797298\pi$
$619$	1739.73	−	3013.30i	0.112966	−	0.195662i	0.916965	+	0.398968i	$0.130632\pi$
$620$	−2582.32			−0.167272
$621$		0			0
$622$	−11410.0			−0.735528
$623$	−10704.4	+	18540.6i	−0.688386	+	1.19232i
$624$		0			0
$625$	7362.03	+	12751.4i	0.471170	+	0.816091i
$626$	−5077.48	−	8794.45i	−0.324180	−	0.561497i
$627$		0			0
$628$	−2566.78	+	4445.79i	−0.163098	+	0.282494i
$629$	−1167.70			−0.0740208
$630$		0			0
$631$	11151.7			0.703552			0.351776	−	0.936084i	$-0.385578\pi$
$631$	11151.7			0.703552			0.351776	+	0.936084i	$0.385578\pi$
$632$	101.262	−	175.391i	0.00637341	−	0.0110391i
$633$		0			0
$634$	2868.91	+	4969.10i	0.179714	+	0.311275i
$635$	−170.927	−	296.055i	−0.0106820	−	0.0185017i
$636$		0			0
$637$	−2084.39	+	3610.26i	−0.129649	+	0.224559i
$638$	−19326.8			−1.19930
$639$		0			0
$640$	−1391.41			−0.0859383
$641$	−546.388	+	946.373i	−0.0336678	+	0.0583143i	−0.882368	−	0.470559i	$-0.844052\pi$
$641$	−546.388	+	946.373i	−0.0336678	+	0.0583143i	0.848701	+	0.528874i	$0.177385\pi$
$642$		0			0
$643$	−15847.0	−	27447.8i	−0.971922	−	1.68342i	−0.689741	−	0.724056i	$-0.742276\pi$
$643$	−15847.0	−	27447.8i	−0.971922	−	1.68342i	−0.282181	−	0.959361i	$-0.591058\pi$
$644$	951.521	+	1648.08i	0.0582223	+	0.100844i
$645$		0			0
$646$	−1988.36	+	3443.93i	−0.121100	+	0.209752i
$647$	13719.9			0.833672			0.416836	−	0.908982i	$-0.363139\pi$
$647$	13719.9			0.833672			0.416836	+	0.908982i	$0.363139\pi$
$648$		0			0
$649$	−4904.12			−0.296616
$650$	103.393	−	179.082i	0.00623909	−	0.0108064i
$651$		0			0
$652$	−2067.85	−	3581.63i	−0.124208	−	0.215134i
$653$	6642.59	+	11505.3i	0.398078	+	0.689491i	0.993489	−	0.113930i	$-0.0363441\pi$
$653$	6642.59	+	11505.3i	0.398078	+	0.689491i	−0.595411	+	0.803421i	$0.703011\pi$
$654$		0			0
$655$	10529.9	−	18238.4i	0.628151	−	1.08799i
$656$	−3252.15			−0.193559
$657$		0			0
$658$	−544.344			−0.0322504
$659$	−1296.99	+	2246.45i	−0.0766670	+	0.132791i	−0.901810	−	0.432133i	$-0.857761\pi$
$659$	−1296.99	+	2246.45i	−0.0766670	+	0.132791i	0.825143	+	0.564924i	$0.191095\pi$
$660$		0			0
$661$	7937.67	+	13748.4i	0.467079	+	0.809005i	0.999293	−	0.0376052i	$-0.0119729\pi$
$661$	7937.67	+	13748.4i	0.467079	+	0.809005i	−0.532213	+	0.846610i	$0.678640\pi$
$662$	−2031.35	−	3518.40i	−0.119261	−	0.206566i
$663$		0			0
$664$	1150.88	−	1993.39i	0.0672635	−	0.116504i
$665$	−38755.6			−2.25996
$666$		0			0
$667$	4325.06			0.251075
$668$	−565.235	+	979.015i	−0.0327389	+	0.0567054i
$669$		0			0
$670$	−8342.46	−	14449.6i	−0.481041	−	0.833187i
$671$	16135.9	+	27948.2i	0.928344	+	1.60794i
$672$		0			0
$673$	−10717.3	+	18563.0i	−0.613853	+	1.06323i	0.376731	+	0.926323i	$0.377048\pi$
$673$	−10717.3	+	18563.0i	−0.613853	+	1.06323i	−0.990585	+	0.136903i	$0.956285\pi$
$674$	19596.6			1.11993
$675$		0			0
$676$	−7872.38			−0.447905
$677$	−10166.5	+	17608.9i	−0.577150	+	0.999653i	0.418655	+	0.908146i	$0.362502\pi$
$677$	−10166.5	+	17608.9i	−0.577150	+	0.999653i	−0.995804	+	0.0915074i	$0.970832\pi$
$678$		0			0
$679$	5000.81	+	8661.66i	0.282642	+	0.489549i
$680$	603.110	+	1044.62i	0.0340121	+	0.0589106i
$681$		0			0
$682$	−2538.32	+	4396.51i	−0.142518	+	0.246849i
$683$	27149.0			1.52097			0.760487	−	0.649353i	$-0.224960\pi$
$683$	27149.0			1.52097			0.760487	+	0.649353i	$0.224960\pi$
$684$		0			0
$685$	12594.4			0.702493
$686$	1677.81	−	2906.05i	0.0933804	−	0.161740i
$687$		0			0
$688$	−2607.12	−	4515.67i	−0.144470	−	0.250230i
$689$	−1060.48	−	1836.80i	−0.0586371	−	0.101562i
$690$		0			0
$691$	−11355.1	+	19667.6i	−0.625134	+	1.08276i	0.363381	+	0.931641i	$0.381622\pi$
$691$	−11355.1	+	19667.6i	−0.625134	+	1.08276i	−0.988515	+	0.151123i	$0.951711\pi$
$692$	14130.9			0.776266
$693$		0			0
$694$	9113.12			0.498457
$695$	12564.0	−	21761.5i	0.685728	−	1.18771i
$696$		0			0
$697$	1409.65	+	2441.58i	0.0766056	+	0.132685i
$698$	−3348.89	−	5800.45i	−0.181601	−	0.314542i
$699$		0			0
$700$	339.921	−	588.760i	0.0183540	−	0.0317901i
$701$	20079.5			1.08187			0.540936	−	0.841064i	$-0.318070\pi$
$701$	20079.5			1.08187			0.540936	+	0.841064i	$0.318070\pi$
$702$		0			0
$703$	−12068.2			−0.647457
$704$	−1367.71	+	2368.94i	−0.0732207	+	0.126822i
$705$		0			0
$706$	1862.27	+	3225.54i	0.0992739	+	0.171947i
$707$	−16556.6	−	28676.9i	−0.880728	−	1.52547i
$708$		0			0
$709$	14491.8	−	25100.6i	0.767634	−	1.32958i	−0.171209	−	0.985235i	$-0.554767\pi$
$709$	14491.8	−	25100.6i	0.767634	−	1.32958i	0.938843	−	0.344346i	$-0.111899\pi$
$710$	7301.74			0.385957
$711$		0			0
$712$	6886.54			0.362477
$713$	568.040	−	983.875i	0.0298363	−	0.0516780i
$714$		0			0
$715$	3514.69	+	6087.61i	0.183835	+	0.318411i
$716$	−8105.49	−	14039.1i	−0.423067	−	0.732774i
$717$		0			0
$718$	6179.46	−	10703.1i	0.321191	−	0.556320i
$719$	−14496.2			−0.751901			−0.375951	−	0.926640i	$-0.622684\pi$
$719$	−14496.2			−0.751901			−0.375951	+	0.926640i	$0.622684\pi$
$720$		0			0
$721$	−12895.4			−0.666090
$722$	−13690.8	+	23713.2i	−0.705706	+	1.22232i
$723$		0			0
$724$	5661.94	+	9806.77i	0.290641	+	0.503405i
$725$	−772.541	−	1338.08i	−0.0395744	−	0.0685449i
$726$		0			0
$727$	−2500.25	+	4330.56i	−0.127550	+	0.220924i	−0.922727	−	0.385454i	$-0.874045\pi$
$727$	−2500.25	+	4330.56i	−0.127550	+	0.220924i	0.795177	+	0.606378i	$0.207378\pi$
$728$	3010.23			0.153251
$729$		0			0
$730$	−3636.79			−0.184388
$731$	−2260.12	+	3914.64i	−0.114355	+	0.198069i
$732$		0			0
$733$	−8757.79	−	15168.9i	−0.441305	−	0.764362i	0.556482	−	0.830860i	$-0.312151\pi$
$733$	−8757.79	−	15168.9i	−0.441305	−	0.764362i	−0.997787	+	0.0664977i	$0.978817\pi$
$734$	−6873.95	−	11906.0i	−0.345671	−	0.598719i
$735$		0			0
$736$	306.073	−	530.134i	0.0153288	−	0.0265503i
$737$	−32801.3			−1.63942
$738$		0			0
$739$	−20169.2			−1.00397			−0.501985	−	0.864876i	$-0.667397\pi$
$739$	−20169.2			−1.00397			−0.501985	+	0.864876i	$0.667397\pi$
$740$	−1830.27	+	3170.13i	−0.0909219	+	0.157481i
$741$		0			0
$742$	−3486.48	−	6038.77i	−0.172497	−	0.298774i
$743$	−18351.3	−	31785.4i	−0.906116	−	1.56944i	−0.819412	−	0.573205i	$-0.805700\pi$
$743$	−18351.3	−	31785.4i	−0.906116	−	1.56944i	−0.0867044	−	0.996234i	$-0.527634\pi$
$744$		0			0
$745$	16035.0	−	27773.5i	0.788561	−	1.36583i
$746$	2540.05			0.124662
$747$		0			0
$748$	2371.34			0.115915
$749$	18303.0	−	31701.8i	0.892894	−	1.54654i
$750$		0			0
$751$	16660.0	+	28856.0i	0.809499	+	1.40209i	0.913212	+	0.407485i	$0.133594\pi$
$751$	16660.0	+	28856.0i	0.809499	+	1.40209i	−0.103713	+	0.994607i	$0.533072\pi$
$752$	87.5489	+	151.639i	0.00424545	+	0.00735334i
$753$		0			0
$754$	3420.69	−	5924.81i	0.165218	−	0.286166i
$755$	18766.7			0.904622
$756$		0			0
$757$	26515.6			1.27309			0.636543	−	0.771241i	$-0.280364\pi$
$757$	26515.6			1.27309			0.636543	+	0.771241i	$0.280364\pi$
$758$	−2490.54	+	4313.74i	−0.119341	+	0.206705i
$759$		0			0
$760$	6233.20	+	10796.2i	0.297502	+	0.515289i
$761$	−2842.17	−	4922.79i	−0.135386	−	0.234495i	0.790359	−	0.612644i	$-0.209894\pi$
$761$	−2842.17	−	4922.79i	−0.135386	−	0.234495i	−0.925745	+	0.378149i	$0.876561\pi$
$762$		0			0
$763$	8004.14	−	13863.6i	0.379777	−	0.657792i
$764$	−18037.0			−0.854129
$765$		0			0
$766$	−624.851			−0.0294736
$767$	867.990	−	1503.40i	0.0408622	−	0.0707754i
$768$		0			0
$769$	199.189	+	345.005i	0.00934060	+	0.0161784i	0.870658	−	0.491889i	$-0.163693\pi$
$769$	199.189	+	345.005i	0.00934060	+	0.0161784i	−0.861317	+	0.508067i	$0.830360\pi$
$770$	11555.1	+	20014.0i	0.540800	+	0.936694i
$771$		0			0
$772$	6442.25	−	11158.3i	0.300339	−	0.520202i
$773$	8437.17			0.392579			0.196290	−	0.980546i	$-0.437111\pi$
$773$	8437.17			0.392579			0.196290	+	0.980546i	$0.437111\pi$
$774$		0			0
$775$	−405.853			−0.0188112
$776$	1608.60	−	2786.17i	0.0744140	−	0.128889i
$777$		0			0
$778$	9643.17	+	16702.5i	0.444376	+	0.769681i
$779$	14568.8	+	25233.9i	0.670067	+	1.16059i
$780$		0			0
$781$	7177.34	−	12431.5i	0.328842	−	0.569570i
$782$	−530.671			−0.0242669
$783$		0			0
$784$	4408.61			0.200830
$785$	6975.49	−	12081.9i	0.317154	−	0.549327i
$786$		0			0
$787$	−8138.94	−	14097.1i	−0.368643	−	0.638508i	0.620711	−	0.784040i	$-0.286844\pi$
$787$	−8138.94	−	14097.1i	−0.368643	−	0.638508i	−0.989354	+	0.145531i	$0.953511\pi$
$788$	7568.40	+	13108.8i	0.342148	+	0.592618i
$789$		0			0
$790$	−275.191	+	476.644i	−0.0123935	+	0.0214661i
$791$	−25460.5			−1.14447
$792$		0			0
$793$	−11423.7			−0.511560
$794$	2260.32	−	3914.99i	0.101027	−	0.174985i
$795$		0			0
$796$	5852.53	+	10136.9i	0.260600	+	0.451372i
$797$	2556.06	+	4427.22i	0.113601	+	0.196763i	0.917220	−	0.398382i	$-0.130428\pi$
$797$	2556.06	+	4427.22i	0.113601	+	0.196763i	−0.803619	+	0.595145i	$0.797095\pi$
$798$		0			0
$799$	75.8963	−	131.456i	0.00336047	−	0.00582051i
$800$	−218.683			−0.00966450
$801$		0			0
$802$	−20338.6			−0.895485
$803$	−3574.82	+	6191.77i	−0.157102	+	0.272108i
$804$		0			0
$805$	−2585.86	−	4478.84i	−0.113217	−	0.196097i
$806$	−898.526	−	1556.29i	−0.0392670	−	0.0680125i
$807$		0			0
$808$	−5325.71	+	9224.41i	−0.231879	+	0.401626i
$809$	13141.2			0.571100			0.285550	−	0.958364i	$-0.407824\pi$
$809$	13141.2			0.571100			0.285550	+	0.958364i	$0.407824\pi$
$810$		0			0
$811$	−18614.2			−0.805957			−0.402979	−	0.915209i	$-0.632025\pi$
$811$	−18614.2			−0.805957			−0.402979	+	0.915209i	$0.632025\pi$
$812$	11246.1	−	19478.7i	0.486034	−	0.841835i
$813$		0			0
$814$	3598.18	+	6232.23i	0.154934	+	0.268353i
$815$	5619.61	+	9733.45i	0.241529	+	0.418341i
$816$		0			0
$817$	−23358.5	+	40458.2i	−1.00026	+	1.73250i
$818$	−1899.67			−0.0811983
$819$		0			0
$820$	8838.05			0.376388
$821$	−5660.01	+	9803.43i	−0.240604	+	0.416738i	−0.960886	−	0.276943i	$-0.910679\pi$
$821$	−5660.01	+	9803.43i	−0.240604	+	0.416738i	0.720283	+	0.693681i	$0.244012\pi$
$822$		0			0
$823$	−5433.29	−	9410.73i	−0.230125	−	0.398587i	0.727720	−	0.685874i	$-0.240580\pi$
$823$	−5433.29	−	9410.73i	−0.230125	−	0.398587i	−0.957845	+	0.287287i	$0.907247\pi$
$824$	2074.02	+	3592.30i	0.0876843	+	0.151874i
$825$		0			0
$826$	2853.65	−	4942.67i	0.120207	−	0.208205i
$827$	13059.3			0.549114			0.274557	−	0.961571i	$-0.411469\pi$
$827$	13059.3			0.549114			0.274557	+	0.961571i	$0.411469\pi$
$828$		0			0
$829$	−21203.7			−0.888341			−0.444171	−	0.895942i	$-0.646502\pi$
$829$	−21203.7			−0.888341			−0.444171	+	0.895942i	$0.646502\pi$
$830$	−3127.65	+	5417.25i	−0.130798	+	0.226549i
$831$		0			0
$832$	−484.146	−	838.566i	−0.0201740	−	0.0349424i
$833$	−1910.92	−	3309.80i	−0.0794829	−	0.137669i
$834$		0			0
$835$	1536.09	−	2660.58i	0.0636628	−	0.110267i
$836$	24508.0			1.01391
$837$		0			0
$838$	23598.5			0.972790
$839$	−7240.28	+	12540.5i	−0.297929	+	0.516028i	−0.975662	−	0.219280i	$-0.929629\pi$
$839$	−7240.28	+	12540.5i	−0.297929	+	0.516028i	0.677733	+	0.735308i	$0.262962\pi$
$840$		0			0
$841$	−13364.5	−	23148.0i	−0.547973	−	0.949117i
$842$	6412.93	+	11107.5i	0.262475	+	0.454620i
$843$		0			0
$844$	−626.948	+	1085.91i	−0.0255692	+	0.0442872i
$845$	21394.0			0.870979
$846$		0			0
$847$	12330.3			0.500204
$848$	−1121.49	+	1942.47i	−0.0454151	+	0.0786613i
$849$		0			0
$850$	94.7882	+	164.178i	0.00382495	+	0.00662501i
$851$	−805.221	−	1394.68i	−0.0324355	−	0.0561799i
$852$		0			0
$853$	3233.61	−	5600.78i	0.129797	−	0.224815i	−0.793801	−	0.608178i	$-0.791901\pi$
$853$	3233.61	−	5600.78i	0.129797	−	0.224815i	0.923598	+	0.383363i	$0.125234\pi$
$854$	−37557.1			−1.50489
$855$		0			0
$856$	−11775.0			−0.470164
$857$	1947.32	−	3372.85i	0.0776186	−	0.134439i	−0.824603	−	0.565711i	$-0.808602\pi$
$857$	1947.32	−	3372.85i	0.0776186	−	0.134439i	0.902222	+	0.431272i	$0.141935\pi$
$858$		0			0
$859$	18826.9	+	32609.1i	0.747805	+	1.29524i	0.948873	+	0.315659i	$0.102226\pi$
$859$	18826.9	+	32609.1i	0.747805	+	1.29524i	−0.201067	+	0.979577i	$0.564441\pi$
$860$	7085.13	+	12271.8i	0.280931	+	0.486587i
$861$		0			0
$862$	−12042.7	+	20858.6i	−0.475844	+	0.824186i
$863$	−47067.5			−1.85654			−0.928271	−	0.371905i	$-0.878705\pi$
$863$	−47067.5			−1.85654			−0.928271	+	0.371905i	$0.878705\pi$
$864$		0			0
$865$	−38402.2			−1.50950
$866$	−7279.83	+	12609.0i	−0.285657	+	0.494772i
$867$		0			0
$868$	−2954.05	−	5116.56i	−0.115515	−	0.200077i
$869$	541.004	+	937.046i	0.0211189	+	0.0365790i
$870$		0			0
$871$	5805.56	−	10055.5i	0.225848	−	0.391181i
$872$	−5149.34			−0.199976
$873$		0			0
$874$	−5484.53			−0.212262
$875$	−17820.8	+	30866.5i	−0.688517	+	1.19255i
$876$		0			0
$877$	3721.77	+	6446.29i	0.143301	+	0.248205i	0.928738	−	0.370737i	$-0.120895\pi$
$877$	3721.77	+	6446.29i	0.143301	+	0.248205i	−0.785437	+	0.618942i	$0.787562\pi$
$878$	3598.70	+	6233.13i	0.138326	+	0.239588i
$879$		0			0
$880$	3716.89	−	6437.84i	0.142382	−	0.246613i
$881$	13781.9			0.527040			0.263520	−	0.964654i	$-0.415116\pi$
$881$	13781.9			0.527040			0.263520	+	0.964654i	$0.415116\pi$
$882$		0			0
$883$	12230.3			0.466119			0.233060	−	0.972462i	$-0.425126\pi$
$883$	12230.3			0.466119			0.233060	+	0.972462i	$0.425126\pi$
$884$	−419.707	+	726.954i	−0.0159686	+	0.0276585i
$885$		0			0
$886$	−14786.4	−	25610.7i	−0.560674	−	0.971116i
$887$	−837.437	−	1450.48i	−0.0317005	−	0.0549069i	0.849740	−	0.527202i	$-0.176759\pi$
$887$	−837.437	−	1450.48i	−0.0317005	−	0.0549069i	−0.881440	+	0.472295i	$0.843426\pi$
$888$		0			0
$889$	391.064	−	677.343i	0.0147535	−	0.0255538i
$890$	−18714.9			−0.704859
$891$		0			0
$892$	−2841.54			−0.106661
$893$	784.395	−	1358.61i	0.0293939	−	0.0509118i
$894$		0			0
$895$	22027.5	+	38152.8i	0.822680	+	1.42492i
$896$	−1591.71	−	2756.92i	−0.0593473	−	0.102793i
$897$		0			0
$898$	114.489	−	198.302i	0.00425452	−	0.00736905i
$899$	−13427.4			−0.498140
$900$		0			0
$901$	1944.44			0.0718964
$902$	8687.47	−	15047.1i	0.320689	−	0.555449i
$903$		0			0
$904$	4094.91	+	7092.59i	0.150658	+	0.260947i
$905$	−15386.9	−	26650.9i	−0.565170	−	0.978903i
$906$		0			0
$907$	−8544.04	+	14798.7i	−0.312790	+	0.541768i	−0.978965	−	0.204027i	$-0.934597\pi$
$907$	−8544.04	+	14798.7i	−0.312790	+	0.541768i	0.666175	+	0.745795i	$0.267930\pi$
$908$	−130.647			−0.00477496
$909$		0			0
$910$	−8180.63			−0.298006
$911$	10644.2	−	18436.3i	0.387111	−	0.670496i	−0.604949	−	0.796264i	$-0.706806\pi$
$911$	10644.2	−	18436.3i	0.387111	−	0.670496i	0.992060	+	0.125769i	$0.0401397\pi$
$912$		0			0
$913$	6148.72	+	10649.9i	0.222884	+	0.386046i
$914$	−6311.14	−	10931.2i	−0.228396	−	0.395593i
$915$		0			0
$916$	11007.7	−	19065.9i	0.397057	−	0.687723i
$917$	48182.8			1.73516
$918$		0			0
$919$	−10413.4			−0.373784			−0.186892	−	0.982380i	$-0.559841\pi$
$919$	−10413.4			−0.373784			−0.186892	+	0.982380i	$0.559841\pi$
$920$	−831.786	+	1440.70i	−0.0298078	+	0.0516286i
$921$		0			0
$922$	−13744.8	−	23806.7i	−0.490956	−	0.850361i
$923$	2540.66	+	4400.55i	0.0906033	+	0.156930i
$924$		0			0
$925$	−287.657	+	498.236i	−0.0102250	+	0.0177102i
$926$	31297.2			1.11068
$927$		0			0
$928$	−7234.98			−0.255926
$929$	−3411.72	+	5909.26i	−0.120490	+	0.208694i	−0.919961	−	0.392010i	$-0.871780\pi$
$929$	−3411.72	+	5909.26i	−0.120490	+	0.208694i	0.799471	+	0.600704i	$0.205113\pi$
$930$		0			0
$931$	−19749.5	−	34207.1i	−0.695234	−	1.20418i
$932$	−13577.6	−	23517.1i	−0.477199	−	0.826534i
$933$		0			0
$934$	−7395.79	+	12809.9i	−0.259098	+	0.448771i
$935$	−6444.36			−0.225404
$936$		0			0
$937$	41049.8			1.43120			0.715602	−	0.698508i	$-0.246152\pi$
$937$	41049.8			1.43120			0.715602	+	0.698508i	$0.246152\pi$
$938$	19086.7	−	33059.1i	0.664395	−	1.15077i
$939$		0			0
$940$	−237.923	−	412.095i	−0.00825554	−	0.0142990i
$941$	1208.06	+	2092.42i	0.0418508	+	0.0724878i	0.886192	−	0.463318i	$-0.153341\pi$
$941$	1208.06	+	2092.42i	0.0418508	+	0.0724878i	−0.844341	+	0.535806i	$0.820008\pi$
$942$		0			0
$943$	−1944.13	+	3367.33i	−0.0671364	+	0.116284i
$944$	−1835.85			−0.0632966
$945$		0			0
$946$	27857.7			0.957432
$947$	−1197.04	+	2073.33i	−0.0410755	+	0.0711448i	−0.885832	−	0.464006i	$-0.846412\pi$
$947$	−1197.04	+	2073.33i	−0.0410755	+	0.0711448i	0.844757	+	0.535150i	$0.179745\pi$
$948$		0			0
$949$	−1265.43	−	2191.79i	−0.0432851	−	0.0749720i
$950$	979.645	+	1696.79i	0.0334567	+	0.0579487i
$951$		0			0
$952$	−1379.85	+	2389.98i	−0.0469762	+	0.0813651i
$953$	50651.3			1.72168			0.860838	−	0.508879i	$-0.169940\pi$
$953$	50651.3			1.72168			0.860838	+	0.508879i	$0.169940\pi$
$954$		0			0
$955$	49017.4			1.66091
$956$	−858.777	+	1487.45i	−0.0290532	+	0.0503216i
$957$		0			0
$958$	−6261.07	−	10844.5i	−0.211154	−	0.365730i
$959$	14407.4	+	24954.3i	0.485128	+	0.840267i
$960$		0			0
$961$	13132.0	−	22745.3i	0.440804	−	0.763495i
$962$	−2547.40			−0.0853756
$963$		0			0
$964$	−19374.6			−0.647317
$965$	−17507.5	+	30323.9i	−0.584027	+	1.01156i
$966$		0			0
$967$	−12517.5	−	21680.9i	−0.416271	−	0.721003i	0.579290	−	0.815122i	$-0.303330\pi$
$967$	−12517.5	−	21680.9i	−0.416271	−	0.721003i	−0.995561	+	0.0941189i	$0.969997\pi$
$968$	−1983.12	−	3434.87i	−0.0658471	−	0.114050i
$969$		0			0
$970$	−4371.54	+	7571.72i	−0.144703	+	0.250632i
$971$	−49553.7			−1.63775			−0.818875	−	0.573972i	$-0.805402\pi$
$971$	−49553.7			−1.63775			−0.818875	+	0.573972i	$0.805402\pi$
$972$		0			0
$973$	57490.4			1.89420
$974$	10314.7	−	17865.6i	0.339327	−	0.587732i
$975$		0			0
$976$	6040.45	+	10462.4i	0.198105	+	0.343127i
$977$	−17588.1	−	30463.5i	−0.575939	−	0.997556i	−0.995939	−	0.0900320i	$-0.971303\pi$
$977$	−17588.1	−	30463.5i	−0.575939	−	0.997556i	0.419999	−	0.907524i	$-0.362030\pi$
$978$		0			0
$979$	−18396.0	+	31862.9i	−0.600551	+	1.04019i
$980$	−11980.9			−0.390525
$981$		0			0
$982$	5521.79			0.179437
$983$	16674.6	−	28881.3i	0.541036	−	0.937102i	−0.457809	−	0.889051i	$-0.651366\pi$
$983$	16674.6	−	28881.3i	0.541036	−	0.937102i	0.998845	−	0.0480511i	$-0.0153010\pi$
$984$		0			0
$985$	−20567.9	−	35624.7i	−0.665328	−	1.15238i
$986$	3136.01	+	5431.72i	0.101289	+	0.175437i
$987$		0			0
$988$	−4337.71	+	7513.14i	−0.139677	+	0.241928i
$989$	−6234.14			−0.200439
$990$		0			0
$991$	23066.3			0.739378			0.369689	−	0.929156i	$-0.379464\pi$
$991$	23066.3			0.739378			0.369689	+	0.929156i	$0.379464\pi$
$992$	−950.220	+	1645.83i	−0.0304128	+	0.0526765i
$993$		0			0
$994$	8352.82	+	14467.5i	0.266535	+	0.461652i
$995$	−15904.9	−	27548.1i	−0.506752	−	0.877721i
$996$		0			0
$997$	27884.9	−	48298.1i	0.885782	−	1.53422i	0.0409671	−	0.999160i	$-0.486956\pi$
$997$	27884.9	−	48298.1i	0.885782	−	1.53422i	0.844815	−	0.535059i	$-0.179711\pi$
$998$	−19172.0			−0.608095
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.4.c.b.19.2		4
3.2	odd	2		18.4.c.b.7.2	✓	4
4.3	odd	2		432.4.i.b.289.2		4
9.2	odd	6		162.4.a.f.1.2		2
9.4	even	3	inner	54.4.c.b.37.2		4
9.5	odd	6		18.4.c.b.13.2	yes	4
9.7	even	3		162.4.a.g.1.1		2
12.11	even	2		144.4.i.b.97.1		4
36.7	odd	6		1296.4.a.r.1.1		2
36.11	even	6		1296.4.a.l.1.2		2
36.23	even	6		144.4.i.b.49.1		4
36.31	odd	6		432.4.i.b.145.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.4.c.b.7.2	✓	4	3.2	odd	2
18.4.c.b.13.2	yes	4	9.5	odd	6
54.4.c.b.19.2		4	1.1	even	1	trivial
54.4.c.b.37.2		4	9.4	even	3	inner
144.4.i.b.49.1		4	36.23	even	6
144.4.i.b.97.1		4	12.11	even	2
162.4.a.f.1.2		2	9.2	odd	6
162.4.a.g.1.1		2	9.7	even	3
432.4.i.b.145.2		4	36.31	odd	6
432.4.i.b.289.2		4	4.3	odd	2
1296.4.a.l.1.2		2	36.11	even	6
1296.4.a.r.1.1		2	36.7	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:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	54.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.43521 + 9.41407i) q^{5} +(-12.4352 + 21.5384i) q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.43521 + 9.41407i) q^{5} +(-12.4352 + 21.5384i) q^{7} +8.00000 q^{8} -21.7409 q^{10} +(-21.3704 + 37.0147i) q^{11} +(-7.56479 - 13.1026i) q^{13} +(-24.8704 - 43.0768i) q^{14} +(-8.00000 + 13.8564i) q^{16} +13.8704 q^{17} +143.352 q^{19} +(21.7409 - 37.6563i) q^{20} +(-42.7409 - 74.0293i) q^{22} +(9.56479 + 16.5667i) q^{23} +(3.41692 - 5.91828i) q^{25} +30.2591 q^{26} +99.4817 q^{28} +(113.046 - 195.802i) q^{29} +(-29.6944 - 51.4321i) q^{31} +(-16.0000 - 27.7128i) q^{32} +(-13.8704 + 24.0243i) q^{34} -270.352 q^{35} -84.1860 q^{37} +(-143.352 + 248.293i) q^{38} +(43.4817 + 75.3125i) q^{40} +(101.630 + 176.028i) q^{41} +(-162.945 + 282.229i) q^{43} +170.963 q^{44} -38.2591 q^{46} +(5.47180 - 9.47744i) q^{47} +(-137.769 - 238.623i) q^{49} +(6.83384 + 11.8366i) q^{50} +(-30.2591 + 52.4104i) q^{52} +140.186 q^{53} -464.611 q^{55} +(-99.4817 + 172.307i) q^{56} +(226.093 + 391.605i) q^{58} +(57.3704 + 99.3685i) q^{59} +(377.528 - 653.898i) q^{61} +118.777 q^{62} +64.0000 q^{64} +(82.2325 - 142.431i) q^{65} +(383.723 + 664.627i) q^{67} +(-27.7409 - 48.0486i) q^{68} +(270.352 - 468.264i) q^{70} -335.854 q^{71} +167.279 q^{73} +(84.1860 - 145.814i) q^{74} +(-286.704 - 496.586i) q^{76} +(-531.492 - 920.570i) q^{77} +(12.6578 - 21.9239i) q^{79} -173.927 q^{80} -406.518 q^{82} +(143.861 - 249.174i) q^{83} +(75.3887 + 130.577i) q^{85} +(-325.890 - 564.458i) q^{86} +(-170.963 + 296.117i) q^{88} +860.817 q^{89} +376.279 q^{91} +(38.2591 - 66.2668i) q^{92} +(10.9436 + 18.9549i) q^{94} +(779.149 + 1349.53i) q^{95} +(201.075 - 348.272i) q^{97} +551.076 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 8 q^{4} - 9 q^{5} - 19 q^{7} + 32 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 - 8 * q^4 - 9 * q^5 - 19 * q^7 + 32 * q^8	\( 4 q - 4 q^{2} - 8 q^{4} - 9 q^{5} - 19 q^{7} + 32 q^{8} + 36 q^{10} - 24 q^{11} - 61 q^{13} - 38 q^{14} - 32 q^{16} - 6 q^{17} + 266 q^{19} - 36 q^{20} - 48 q^{22} + 69 q^{23} - 263 q^{25} + 244 q^{26} + 152 q^{28} + 237 q^{29} - 211 q^{31} - 64 q^{32} + 6 q^{34} - 774 q^{35} + 524 q^{37} - 266 q^{38} - 72 q^{40} + 468 q^{41} + 86 q^{43} + 192 q^{44} - 276 q^{46} + 483 q^{47} + 33 q^{49} - 526 q^{50} - 244 q^{52} - 300 q^{53} - 1674 q^{55} - 152 q^{56} + 474 q^{58} + 168 q^{59} + 1049 q^{61} + 844 q^{62} + 256 q^{64} - 747 q^{65} + 1166 q^{67} + 12 q^{68} + 774 q^{70} + 624 q^{71} - 622 q^{73} - 524 q^{74} - 532 q^{76} - 1173 q^{77} - 349 q^{79} + 288 q^{80} - 1872 q^{82} + 1221 q^{83} + 486 q^{85} + 172 q^{86} - 192 q^{88} + 984 q^{89} + 214 q^{91} + 276 q^{92} + 966 q^{94} + 1764 q^{95} + 128 q^{97} - 132 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 - 8 * q^4 - 9 * q^5 - 19 * q^7 + 32 * q^8 + 36 * q^10 - 24 * q^11 - 61 * q^13 - 38 * q^14 - 32 * q^16 - 6 * q^17 + 266 * q^19 - 36 * q^20 - 48 * q^22 + 69 * q^23 - 263 * q^25 + 244 * q^26 + 152 * q^28 + 237 * q^29 - 211 * q^31 - 64 * q^32 + 6 * q^34 - 774 * q^35 + 524 * q^37 - 266 * q^38 - 72 * q^40 + 468 * q^41 + 86 * q^43 + 192 * q^44 - 276 * q^46 + 483 * q^47 + 33 * q^49 - 526 * q^50 - 244 * q^52 - 300 * q^53 - 1674 * q^55 - 152 * q^56 + 474 * q^58 + 168 * q^59 + 1049 * q^61 + 844 * q^62 + 256 * q^64 - 747 * q^65 + 1166 * q^67 + 12 * q^68 + 774 * q^70 + 624 * q^71 - 622 * q^73 - 524 * q^74 - 532 * q^76 - 1173 * q^77 - 349 * q^79 + 288 * q^80 - 1872 * q^82 + 1221 * q^83 + 486 * q^85 + 172 * q^86 - 192 * q^88 + 984 * q^89 + 214 * q^91 + 276 * q^92 + 966 * q^94 + 1764 * q^95 + 128 * q^97 - 132 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more