LMFDB - Embedded newform 378.4.k.d.269.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(215,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

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

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

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.k (of order $6$, degree $2$, 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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} + 2 x^{18} - 32 x^{17} - 669 x^{16} + 1752 x^{15} - 1654 x^{14} + 13878 x^{13} + \cdots + 2458624 $ x^20 - 2x^19 + 2x^18 - 32x^17 - 669x^16 + 1752x^15 - 1654x^14 + 13878x^13 + 494085x^12 - 780096x^11 + 556152x^10 - 2898760x^9 - 1242580x^8 + 9104368x^7 - 4109728x^6 + 12152000x^5 + 28375312x^4 - 23774016x^3 + 7683200x^2 - 6146560*x + 2458624
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{18}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.9
Root			$1.29489 + 4.83258i$ of defining polynomial
Character	$\chi$	$=$	378.269
Dual form			378.4.k.d.215.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(3.43124 + 5.94308i) q^{5} +(-18.5152 - 0.434813i) q^{7} -8.00000i q^{8} +O(q^{10})$	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(3.43124 + 5.94308i) q^{5} +(-18.5152 - 0.434813i) q^{7} -8.00000i q^{8} +(11.8862 + 6.86248i) q^{10} +(-28.6032 - 16.5141i) q^{11} +65.9110i q^{13} +(-32.5040 + 17.7620i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(7.20650 - 12.4820i) q^{17} +(-44.9254 + 25.9377i) q^{19} +27.4499 q^{20} -66.0562 q^{22} +(-174.131 + 100.535i) q^{23} +(38.9532 - 67.4689i) q^{25} +(65.9110 + 114.161i) q^{26} +(-38.5365 + 63.2688i) q^{28} +227.304i q^{29} +(-24.5744 - 14.1881i) q^{31} +(-27.7128 - 16.0000i) q^{32} -28.8260i q^{34} +(-60.9458 - 111.529i) q^{35} +(-101.081 - 175.078i) q^{37} +(-51.8754 + 89.8508i) q^{38} +(47.5446 - 27.4499i) q^{40} -34.6120 q^{41} -483.052 q^{43} +(-114.413 + 66.0562i) q^{44} +(-201.070 + 348.263i) q^{46} +(95.7687 + 165.876i) q^{47} +(342.622 + 16.1012i) q^{49} -155.813i q^{50} +(228.322 + 131.822i) q^{52} +(75.4532 + 43.5629i) q^{53} -226.655i q^{55} +(-3.47850 + 148.121i) q^{56} +(227.304 + 393.702i) q^{58} +(-292.449 + 506.536i) q^{59} +(303.713 - 175.349i) q^{61} -56.7523 q^{62} -64.0000 q^{64} +(-391.714 + 226.156i) q^{65} +(315.762 - 546.916i) q^{67} +(-28.8260 - 49.9281i) q^{68} +(-217.090 - 132.228i) q^{70} +932.935i q^{71} +(191.352 + 110.477i) q^{73} +(-350.156 - 202.163i) q^{74} +207.501i q^{76} +(522.412 + 318.197i) q^{77} +(228.057 + 395.007i) q^{79} +(54.8998 - 95.0893i) q^{80} +(-59.9498 + 34.6120i) q^{82} -1363.85 q^{83} +98.9088 q^{85} +(-836.670 + 483.052i) q^{86} +(-132.112 + 228.825i) q^{88} +(-358.658 - 621.215i) q^{89} +(28.6589 - 1220.35i) q^{91} +804.279i q^{92} +(331.752 + 191.537i) q^{94} +(-308.299 - 177.997i) q^{95} -9.90916i q^{97} +(609.540 - 314.734i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 40 q^{4} + 4 q^{7}+O(q^{10}) $ 20 * q + 40 * q^4 + 4 * q^7	$ 20 q + 40 q^{4} + 4 q^{7} - 24 q^{10} - 160 q^{16} + 408 q^{19} + 240 q^{22} - 646 q^{25} + 56 q^{28} - 102 q^{31} + 194 q^{37} - 96 q^{40} - 2332 q^{43} - 624 q^{46} + 2840 q^{49} - 648 q^{52} + 96 q^{58} + 1878 q^{61} - 1280 q^{64} - 386 q^{67} + 3672 q^{70} + 1788 q^{73} + 814 q^{79} - 672 q^{82} - 4560 q^{85} + 480 q^{88} + 2724 q^{91} - 1536 q^{94}+O(q^{100}) $ 20 * q + 40 * q^4 + 4 * q^7 - 24 * q^10 - 160 * q^16 + 408 * q^19 + 240 * q^22 - 646 * q^25 + 56 * q^28 - 102 * q^31 + 194 * q^37 - 96 * q^40 - 2332 * q^43 - 624 * q^46 + 2840 * q^49 - 648 * q^52 + 96 * q^58 + 1878 * q^61 - 1280 * q^64 - 386 * q^67 + 3672 * q^70 + 1788 * q^73 + 814 * q^79 - 672 * q^82 - 4560 * q^85 + 480 * q^88 + 2724 * q^91 - 1536 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	1.73205	−	1.00000i	0.612372	−	0.353553i
$2$	1.73205	−	1.00000i	0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	2.00000	−	3.46410i	0.250000	−	0.433013i
$5$	3.43124	+	5.94308i	0.306899	+	0.531565i	0.977682	−	0.210088i	$-0.0673752\pi$
$5$	3.43124	+	5.94308i	0.306899	+	0.531565i	−0.670783	+	0.741654i	$0.734042\pi$
$6$		0			0
$7$	−18.5152	−	0.434813i	−0.999724	−	0.0234777i
$7$	−18.5152	−	0.434813i	−0.999724	−	0.0234777i
$8$		−	8.00000i		−	0.353553i
$9$		0			0
$10$	11.8862	+	6.86248i	0.375873	+	0.217011i
$11$	−28.6032	−	16.5141i	−0.784017	−	0.452652i	0.0538351	−	0.998550i	$-0.482855\pi$
$11$	−28.6032	−	16.5141i	−0.784017	−	0.452652i	−0.837852	+	0.545898i	$0.816189\pi$
$12$		0			0
$13$			65.9110i			1.40619i	0.711097	+	0.703093i	$0.248198\pi$
$13$			65.9110i			1.40619i	−0.711097	+	0.703093i	$0.751802\pi$
$14$	−32.5040	+	17.7620i	−0.620504	+	0.339079i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	7.20650	−	12.4820i	0.102814	−	0.178078i	−0.810029	−	0.586390i	$-0.800549\pi$
$17$	7.20650	−	12.4820i	0.102814	−	0.178078i	0.912843	+	0.408311i	$0.133882\pi$
$18$		0			0
$19$	−44.9254	+	25.9377i	−0.542452	+	0.313185i	−0.746072	−	0.665865i	$-0.768063\pi$
$19$	−44.9254	+	25.9377i	−0.542452	+	0.313185i	0.203620	+	0.979050i	$0.434729\pi$
$20$	27.4499			0.306899
$21$		0			0
$22$	−66.0562			−0.640147
$23$	−174.131	+	100.535i	−1.57865	+	0.911433i	−0.583600	+	0.812041i	$0.698356\pi$
$23$	−174.131	+	100.535i	−1.57865	+	0.911433i	−0.995048	+	0.0993915i	$0.968310\pi$
$24$		0			0
$25$	38.9532	−	67.4689i	0.311626	−	0.539752i
$26$	65.9110	+	114.161i	0.497162	+	0.861110i
$27$		0			0
$28$	−38.5365	+	63.2688i	−0.260097	+	0.427024i
$29$			227.304i			1.45549i	0.685847	+	0.727746i	$0.259432\pi$
$29$			227.304i			1.45549i	−0.685847	+	0.727746i	$0.740568\pi$
$30$		0			0
$31$	−24.5744	−	14.1881i	−0.142377	−	0.0822017i	0.427119	−	0.904195i	$-0.359528\pi$
$31$	−24.5744	−	14.1881i	−0.142377	−	0.0822017i	−0.569497	+	0.821994i	$0.692862\pi$
$32$	−27.7128	−	16.0000i	−0.153093	−	0.0883883i
$33$		0			0
$34$		−	28.8260i		−	0.145400i
$35$	−60.9458	−	111.529i	−0.294335	−	0.538624i
$36$		0			0
$37$	−101.081	−	175.078i	−0.449126	−	0.777910i	0.549203	−	0.835689i	$-0.314931\pi$
$37$	−101.081	−	175.078i	−0.449126	−	0.777910i	−0.998329	+	0.0577792i	$0.981598\pi$
$38$	−51.8754	+	89.8508i	−0.221455	+	0.383572i
$39$		0			0
$40$	47.5446	−	27.4499i	0.187937	−	0.108505i
$41$	−34.6120			−0.131841			−0.0659206	−	0.997825i	$-0.520998\pi$
$41$	−34.6120			−0.131841			−0.0659206	+	0.997825i	$0.520998\pi$
$42$		0			0
$43$	−483.052			−1.71313			−0.856566	−	0.516038i	$-0.827406\pi$
$43$	−483.052			−1.71313			−0.856566	+	0.516038i	$0.827406\pi$
$44$	−114.413	+	66.0562i	−0.392008	+	0.226326i
$45$		0			0
$46$	−201.070	+	348.263i	−0.644480	+	1.11627i
$47$	95.7687	+	165.876i	0.297219	+	0.514799i	0.975499	−	0.220005i	$-0.0706075\pi$
$47$	95.7687	+	165.876i	0.297219	+	0.514799i	−0.678280	+	0.734804i	$0.737274\pi$
$48$		0			0
$49$	342.622	+	16.1012i	0.998898	+	0.0469424i
$50$		−	155.813i		−	0.440705i
$51$		0			0
$52$	228.322	+	131.822i	0.608897	+	0.351547i
$53$	75.4532	+	43.5629i	0.195553	+	0.112902i	0.594579	−	0.804037i	$-0.297319\pi$
$53$	75.4532	+	43.5629i	0.195553	+	0.112902i	−0.399027	+	0.916939i	$0.630652\pi$
$54$		0			0
$55$		−	226.655i		−	0.555675i
$56$	−3.47850	+	148.121i	−0.00830061	+	0.353456i
$57$		0			0
$58$	227.304	+	393.702i	0.514594	+	0.891303i
$59$	−292.449	+	506.536i	−0.645315	+	1.11772i	0.338913	+	0.940818i	$0.389941\pi$
$59$	−292.449	+	506.536i	−0.645315	+	1.11772i	−0.984229	+	0.176901i	$0.943393\pi$
$60$		0			0
$61$	303.713	−	175.349i	0.637483	−	0.368051i	−0.146162	−	0.989261i	$-0.546692\pi$
$61$	303.713	−	175.349i	0.637483	−	0.368051i	0.783644	+	0.621210i	$0.213359\pi$
$62$	−56.7523			−0.116251
$63$		0			0
$64$	−64.0000			−0.125000
$65$	−391.714	+	226.156i	−0.747480	+	0.431558i
$66$		0			0
$67$	315.762	−	546.916i	0.575768	−	0.997260i	−0.420190	−	0.907436i	$-0.638036\pi$
$67$	315.762	−	546.916i	0.575768	−	0.997260i	0.995958	−	0.0898233i	$-0.0286302\pi$
$68$	−28.8260	−	49.9281i	−0.0514068	−	0.0890392i
$69$		0			0
$70$	−217.090	−	132.228i	−0.370675	−	0.225775i
$71$			932.935i			1.55942i	0.626139	+	0.779712i	$0.284634\pi$
$71$			932.935i			1.55942i	−0.626139	+	0.779712i	$0.715366\pi$
$72$		0			0
$73$	191.352	+	110.477i	0.306795	+	0.177128i	0.645491	−	0.763768i	$-0.276653\pi$
$73$	191.352	+	110.477i	0.306795	+	0.177128i	−0.338697	+	0.940896i	$0.609986\pi$
$74$	−350.156	−	202.163i	−0.550065	−	0.317580i
$75$		0			0
$76$			207.501i			0.313185i
$77$	522.412	+	318.197i	0.773174	+	0.470934i
$78$		0			0
$79$	228.057	+	395.007i	0.324790	+	0.562553i	0.981470	−	0.191616i	$-0.0613729\pi$
$79$	228.057	+	395.007i	0.324790	+	0.562553i	−0.656680	+	0.754170i	$0.728040\pi$
$80$	54.8998	−	95.0893i	0.0767248	−	0.132891i
$81$		0			0
$82$	−59.9498	+	34.6120i	−0.0807360	+	0.0466129i
$83$	−1363.85			−1.80364			−0.901822	−	0.432107i	$-0.857770\pi$
$83$	−1363.85			−1.80364			−0.901822	+	0.432107i	$0.857770\pi$
$84$		0			0
$85$	98.9088			0.126214
$86$	−836.670	+	483.052i	−1.04907	+	0.605684i
$87$		0			0
$88$	−132.112	+	228.825i	−0.160037	+	0.277192i
$89$	−358.658	−	621.215i	−0.427165	−	0.739872i	0.569455	−	0.822023i	$-0.307154\pi$
$89$	−358.658	−	621.215i	−0.427165	−	0.739872i	−0.996620	+	0.0821507i	$0.973821\pi$
$90$		0			0
$91$	28.6589	−	1220.35i	0.0330140	−	1.40580i
$92$			804.279i			0.911433i
$93$		0			0
$94$	331.752	+	191.537i	0.364018	+	0.210166i
$95$	−308.299	−	177.997i	−0.332956	−	0.192232i
$96$		0			0
$97$		−	9.90916i		−	0.0103724i	−0.999987	−	0.00518620i	$-0.998349\pi$
$97$		−	9.90916i		−	0.0103724i	0.999987	−	0.00518620i	$-0.00165083\pi$
$98$	609.540	−	314.734i	0.628294	−	0.324417i
$99$		0			0
$100$	−155.813	−	269.876i	−0.155813	−	0.269876i
$101$	47.7022	−	82.6226i	0.0469955	−	0.0813985i	−0.841571	−	0.540147i	$-0.818369\pi$
$101$	47.7022	−	82.6226i	0.0469955	−	0.0813985i	0.888566	+	0.458748i	$0.151702\pi$
$102$		0			0
$103$	983.832	−	568.016i	0.941163	−	0.543381i	0.0508385	−	0.998707i	$-0.483811\pi$
$103$	983.832	−	568.016i	0.941163	−	0.543381i	0.890325	+	0.455326i	$0.150477\pi$
$104$	527.288			0.497162
$105$		0			0
$106$	174.252			0.159668
$107$	1799.33	−	1038.84i	1.62568	−	0.938584i	0.640313	−	0.768114i	$-0.278805\pi$
$107$	1799.33	−	1038.84i	1.62568	−	0.938584i	0.985363	−	0.170470i	$-0.0545287\pi$
$108$		0			0
$109$	59.1994	−	102.536i	0.0520208	−	0.0901027i	−0.838842	−	0.544374i	$-0.816767\pi$
$109$	59.1994	−	102.536i	0.0520208	−	0.0901027i	0.890863	+	0.454272i	$0.150100\pi$
$110$	−226.655	−	392.577i	−0.196461	−	0.340280i
$111$		0			0
$112$	142.096	+	260.032i	0.119882	+	0.219381i
$113$		−	112.853i		−	0.0939496i	−0.998896	−	0.0469748i	$-0.985042\pi$
$113$		−	112.853i		−	0.0939496i	0.998896	−	0.0469748i	$-0.0149580\pi$
$114$		0			0
$115$	−1194.97	−	689.918i	−0.968972	−	0.559436i
$116$	787.403	+	454.608i	0.630246	+	0.363873i
$117$		0			0
$118$			1169.80i			0.912614i
$119$	−138.857	+	227.973i	−0.106966	+	0.175616i
$120$		0			0
$121$	−120.072	−	207.970i	−0.0902117	−	0.156251i
$122$	350.697	−	607.426i	0.260251	−	0.450768i
$123$		0			0
$124$	−98.2978	+	56.7523i	−0.0711887	+	0.0411008i
$125$	1392.44			0.996349
$126$		0			0
$127$	−857.420			−0.599084			−0.299542	−	0.954083i	$-0.596834\pi$
$127$	−857.420			−0.599084			−0.299542	+	0.954083i	$0.596834\pi$
$128$	−110.851	+	64.0000i	−0.0765466	+	0.0441942i
$129$		0			0
$130$	−452.313	+	783.429i	−0.305157	+	0.528548i
$131$	−357.900	−	619.902i	−0.238702	−	0.413443i	0.721640	−	0.692268i	$-0.243388\pi$
$131$	−357.900	−	619.902i	−0.238702	−	0.413443i	−0.960342	+	0.278825i	$0.910055\pi$
$132$		0			0
$133$	843.078	−	460.706i	0.549655	−	0.300363i
$134$		−	1263.05i		−	0.814259i
$135$		0			0
$136$	−99.8562	−	57.6520i	−0.0629603	−	0.0363501i
$137$	−114.971	−	66.3786i	−0.0716982	−	0.0413950i	0.463722	−	0.885981i	$-0.346514\pi$
$137$	−114.971	−	66.3786i	−0.0716982	−	0.0413950i	−0.535421	+	0.844586i	$0.679847\pi$
$138$		0			0
$139$		−	2907.60i		−	1.77424i	−0.461540	−	0.887119i	$-0.652703\pi$
$139$		−	2907.60i		−	1.77424i	0.461540	−	0.887119i	$-0.347297\pi$
$140$	−508.239	−	11.9356i	−0.306815	−	0.00720528i
$141$		0			0
$142$	932.935	+	1615.89i	0.551339	+	0.954948i
$143$	1088.46	−	1885.27i	0.636514	−	1.10247i
$144$		0			0
$145$	−1350.88	+	779.933i	−0.773688	+	0.446689i
$146$	441.908			0.250497
$147$		0			0
$148$	−808.651			−0.449126
$149$	−2517.63	+	1453.55i	−1.38424	+	0.799192i	−0.992659	−	0.120951i	$-0.961406\pi$
$149$	−2517.63	+	1453.55i	−1.38424	+	0.799192i	−0.391583	+	0.920143i	$0.628072\pi$
$150$		0			0
$151$	−451.859	+	782.642i	−0.243521	+	0.421791i	−0.961715	−	0.274052i	$-0.911636\pi$
$151$	−451.859	+	782.642i	−0.243521	+	0.421791i	0.718193	+	0.695843i	$0.244969\pi$
$152$	207.501	+	359.403i	0.110728	+	0.191786i
$153$		0			0
$154$	1223.04	+	28.7221i	0.639971	+	0.0150292i
$155$		−	194.730i		−	0.100911i
$156$		0			0
$157$	2157.41	+	1245.58i	1.09669	+	0.633174i	0.935350	−	0.353725i	$-0.115085\pi$
$157$	2157.41	+	1245.58i	1.09669	+	0.633174i	0.161340	+	0.986899i	$0.448418\pi$
$158$	790.013	+	456.114i	0.397785	+	0.229661i
$159$		0			0
$160$		−	219.599i		−	0.108505i
$161$	3267.78	−	1785.70i	1.59961	−	0.874119i
$162$		0			0
$163$	727.153	+	1259.47i	0.349417	+	0.605208i	0.986146	−	0.165879i	$-0.0530463\pi$
$163$	727.153	+	1259.47i	0.349417	+	0.605208i	−0.636729	+	0.771088i	$0.719713\pi$
$164$	−69.2241	+	119.900i	−0.0329603	+	0.0570890i
$165$		0			0
$166$	−2362.27	+	1363.85i	−1.10450	+	0.637685i
$167$	1402.20			0.649733			0.324866	−	0.945760i	$-0.394681\pi$
$167$	1402.20			0.649733			0.324866	+	0.945760i	$0.394681\pi$
$168$		0			0
$169$	−2147.26			−0.977362
$170$	171.315	−	98.9088i	0.0772898	−	0.0446233i
$171$		0			0
$172$	−966.103	+	1673.34i	−0.428283	+	0.741808i
$173$	−1458.60	−	2526.36i	−0.641012	−	1.11027i	−0.985207	−	0.171367i	$-0.945182\pi$
$173$	−1458.60	−	2526.36i	−0.641012	−	1.11027i	0.344195	−	0.938898i	$-0.388152\pi$
$174$		0			0
$175$	−750.561	+	1232.26i	−0.324212	+	0.532287i
$176$			528.450i			0.226326i
$177$		0			0
$178$	−1242.43	−	717.317i	−0.523169	−	0.302052i
$179$	2921.84	+	1686.93i	1.22005	+	0.704396i	0.964928	−	0.262513i	$-0.0845512\pi$
$179$	2921.84	+	1686.93i	1.22005	+	0.704396i	0.255121	+	0.966909i	$0.417885\pi$
$180$		0			0
$181$		−	3857.00i		−	1.58392i	−0.610576	−	0.791958i	$-0.709062\pi$
$181$		−	3857.00i		−	1.58392i	0.610576	−	0.791958i	$-0.290938\pi$
$182$	−1170.71	−	2142.37i	−0.476808	−	0.872545i
$183$		0			0
$184$	804.279	+	1393.05i	0.322240	+	0.558136i
$185$	693.669	−	1201.47i	0.275673	−	0.477480i
$186$		0			0
$187$	−412.258	+	238.017i	−0.161215	+	0.0930777i
$188$	766.149			0.297219
$189$		0			0
$190$	−711.987			−0.271858
$191$	−1849.52	+	1067.82i	−0.700662	+	0.404527i	−0.807594	−	0.589739i	$-0.799231\pi$
$191$	−1849.52	+	1067.82i	−0.700662	+	0.404527i	0.106932	+	0.994266i	$0.465897\pi$
$192$		0			0
$193$	566.229	−	980.737i	0.211182	−	0.365777i	−0.740903	−	0.671612i	$-0.765602\pi$
$193$	566.229	−	980.737i	0.211182	−	0.365777i	0.952085	+	0.305835i	$0.0989355\pi$
$194$	−9.90916	−	17.1632i	−0.00366720	−	0.00635177i
$195$		0			0
$196$	741.020	−	1154.67i	0.270051	−	0.420800i
$197$		−	1972.82i		−	0.713490i	−0.934202	−	0.356745i	$-0.883886\pi$
$197$		−	1972.82i		−	0.713490i	0.934202	−	0.356745i	$-0.116114\pi$
$198$		0			0
$199$	1006.36	+	581.025i	0.358489	+	0.206974i	0.668418	−	0.743786i	$-0.266972\pi$
$199$	1006.36	+	581.025i	0.358489	+	0.206974i	−0.309929	+	0.950760i	$0.600305\pi$
$200$	−539.752	−	311.626i	−0.190831	−	0.110176i
$201$		0			0
$202$		−	190.809i		−	0.0664616i
$203$	98.8346	−	4208.56i	0.0341716	−	1.45509i
$204$		0			0
$205$	−118.762	−	205.702i	−0.0404620	−	0.0700822i
$206$	1136.03	−	1967.66i	0.384228	−	0.665503i
$207$		0			0
$208$	913.290	−	527.288i	0.304448	−	0.175773i
$209$	1713.35			0.567055
$210$		0			0
$211$	−3561.70			−1.16207			−0.581037	−	0.813877i	$-0.697353\pi$
$211$	−3561.70			−1.16207			−0.581037	+	0.813877i	$0.697353\pi$
$212$	301.813	−	174.252i	0.0977764	−	0.0564512i
$213$		0			0
$214$	2077.68	−	3598.65i	0.663679	−	1.14953i
$215$	−1657.46	−	2870.81i	−0.525759	−	0.910641i
$216$		0			0
$217$	448.831	+	273.379i	0.140408	+	0.0855217i
$218$		−	236.797i		−	0.0735686i
$219$		0			0
$220$	−785.155	−	453.309i	−0.240614	−	0.138919i
$221$	822.703	+	474.988i	0.250412	+	0.144575i
$222$		0			0
$223$			479.700i			0.144050i	0.997403	+	0.0720249i	$0.0229461\pi$
$223$			479.700i			0.144050i	−0.997403	+	0.0720249i	$0.977054\pi$
$224$	506.150	+	308.292i	0.150976	+	0.0919583i
$225$		0			0
$226$	−112.853	−	195.467i	−0.0332162	−	0.0575321i
$227$	−2054.69	+	3558.83i	−0.600769	+	1.04056i	0.391935	+	0.919993i	$0.371806\pi$
$227$	−2054.69	+	3558.83i	−0.600769	+	1.04056i	−0.992705	+	0.120570i	$0.961528\pi$
$228$		0			0
$229$	−1965.47	+	1134.77i	−0.567170	+	0.327456i	−0.756018	−	0.654550i	$-0.772858\pi$
$229$	−1965.47	+	1134.77i	−0.567170	+	0.327456i	0.188848	+	0.982006i	$0.439525\pi$
$230$	−2759.67			−0.791162
$231$		0			0
$232$	1818.43			0.514594
$233$	292.043	−	168.611i	0.0821133	−	0.0474081i	−0.458381	−	0.888756i	$-0.651571\pi$
$233$	292.043	−	168.611i	0.0821133	−	0.0474081i	0.540494	+	0.841348i	$0.318237\pi$
$234$		0			0
$235$	−657.210	+	1138.32i	−0.182433	+	0.315983i
$236$	1169.80	+	2026.15i	0.322658	+	0.558859i
$237$		0			0
$238$	−12.5339	+	533.718i	−0.00341367	+	0.145360i
$239$			1119.50i			0.302990i	0.988458	+	0.151495i	$0.0484088\pi$
$239$			1119.50i			0.302990i	−0.988458	+	0.151495i	$0.951591\pi$
$240$		0			0
$241$	3778.92	+	2181.76i	1.01005	+	0.583152i	0.911206	−	0.411951i	$-0.135153\pi$
$241$	3778.92	+	2181.76i	1.01005	+	0.583152i	0.0988426	+	0.995103i	$0.468486\pi$
$242$	−415.941	−	240.144i	−0.110486	−	0.0637893i
$243$		0			0
$244$		−	1402.79i		−	0.368051i
$245$	1079.93	+	2091.48i	0.281608	+	0.545386i
$246$		0			0
$247$	−1709.58	−	2961.08i	−0.440396	−	0.762789i
$248$	−113.505	+	196.596i	−0.0290627	+	0.0503380i
$249$		0			0
$250$	2411.78	−	1392.44i	0.610137	−	0.352263i
$251$	3878.06			0.975222			0.487611	−	0.873061i	$-0.337868\pi$
$251$	3878.06			0.975222			0.487611	+	0.873061i	$0.337868\pi$
$252$		0			0
$253$	6640.95			1.65025
$254$	−1485.09	+	857.420i	−0.366863	+	0.211808i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	−300.225	−	520.005i	−0.0728698	−	0.126214i	0.827288	−	0.561778i	$-0.189882\pi$
$257$	−300.225	−	520.005i	−0.0728698	−	0.126214i	−0.900158	+	0.435564i	$0.856549\pi$
$258$		0			0
$259$	1795.41	+	3285.55i	0.430739	+	0.788240i
$260$			1809.25i			0.431558i
$261$		0			0
$262$	−1239.80	−	715.801i	−0.292349	−	0.168788i
$263$	3050.60	+	1761.27i	0.715240	+	0.412944i	0.812998	−	0.582266i	$-0.197834\pi$
$263$	3050.60	+	1761.27i	0.715240	+	0.412944i	−0.0977581	+	0.995210i	$0.531167\pi$
$264$		0			0
$265$			597.899i			0.138599i
$266$	999.549	−	1641.04i	0.230399	−	0.378267i
$267$		0			0
$268$	−1263.05	−	2187.66i	−0.287884	−	0.498630i
$269$	−2938.40	+	5089.46i	−0.666012	+	1.15357i	0.312998	+	0.949754i	$0.398667\pi$
$269$	−2938.40	+	5089.46i	−0.666012	+	1.15357i	−0.979010	+	0.203813i	$0.934666\pi$
$270$		0			0
$271$	2902.80	−	1675.93i	0.650673	−	0.375666i	−0.138041	−	0.990427i	$-0.544080\pi$
$271$	2902.80	−	1675.93i	0.650673	−	0.375666i	0.788714	+	0.614760i	$0.210747\pi$
$272$	−230.608			−0.0514068
$273$		0			0
$274$	−265.515			−0.0585413
$275$	−2228.37	+	1286.55i	−0.488640	+	0.282116i
$276$		0			0
$277$	−2396.31	+	4150.53i	−0.519785	+	0.900293i	0.479951	+	0.877295i	$0.340654\pi$
$277$	−2396.31	+	4150.53i	−0.519785	+	0.900293i	−0.999736	+	0.0229980i	$0.992679\pi$
$278$	−2907.60	−	5036.11i	−0.627288	−	1.08649i
$279$		0			0
$280$	−892.232	+	487.566i	−0.190432	+	0.104063i
$281$			6132.70i			1.30194i	0.759102	+	0.650971i	$0.225638\pi$
$281$			6132.70i			1.30194i	−0.759102	+	0.650971i	$0.774362\pi$
$282$		0			0
$283$	−2487.31	−	1436.05i	−0.522458	−	0.301641i	0.215482	−	0.976508i	$-0.430868\pi$
$283$	−2487.31	−	1436.05i	−0.522458	−	0.301641i	−0.737940	+	0.674867i	$0.764201\pi$
$284$	3231.78	+	1865.87i	0.675250	+	0.389856i
$285$		0			0
$286$		−	4353.83i		−	0.900166i
$287$	640.847	+	15.0498i	0.131805	+	0.00309533i
$288$		0			0
$289$	2352.63	+	4074.88i	0.478859	+	0.829408i
$290$	−1559.87	+	2701.77i	−0.315857	+	0.547080i
$291$		0			0
$292$	765.407	−	441.908i	0.153397	−	0.0885640i
$293$	−4741.31			−0.945360			−0.472680	−	0.881234i	$-0.656713\pi$
$293$	−4741.31			−0.945360			−0.472680	+	0.881234i	$0.656713\pi$
$294$		0			0
$295$	−4013.85			−0.792187
$296$	−1400.62	+	808.651i	−0.275033	+	0.158790i
$297$		0			0
$298$	−2907.10	+	5035.25i	−0.565114	+	0.978806i
$299$	−6626.35	−	11477.2i	−1.28165	−	2.21987i
$300$		0			0
$301$	8943.77	+	210.037i	1.71266	+	0.0402204i
$302$			1807.43i			0.344391i
$303$		0			0
$304$	718.806	+	415.003i	0.135613	+	0.0782962i
$305$	2084.22	+	1203.33i	0.391286	+	0.225909i
$306$		0			0
$307$		−	6632.39i		−	1.23300i	−0.787355	−	0.616499i	$-0.788550\pi$
$307$		−	6632.39i		−	1.23300i	0.787355	−	0.616499i	$-0.211450\pi$
$308$	2147.09	−	1173.29i	0.397214	−	0.217060i
$309$		0			0
$310$	−194.730	−	337.283i	−0.0356773	−	0.0617948i
$311$	−3785.11	+	6556.00i	−0.690140	+	1.19536i	0.281651	+	0.959517i	$0.409118\pi$
$311$	−3785.11	+	6556.00i	−0.690140	+	1.19536i	−0.971792	+	0.235841i	$0.924215\pi$
$312$		0			0
$313$	−5933.44	+	3425.67i	−1.07150	+	0.618628i	−0.928590	−	0.371108i	$-0.878978\pi$
$313$	−5933.44	+	3425.67i	−1.07150	+	0.618628i	−0.142906	+	0.989736i	$0.545645\pi$
$314$	4982.33			0.895443
$315$		0			0
$316$	1824.46			0.324790
$317$	−4568.52	+	2637.63i	−0.809443	+	0.467332i	−0.846762	−	0.531971i	$-0.821451\pi$
$317$	−4568.52	+	2637.63i	−0.809443	+	0.467332i	0.0373193	+	0.999303i	$0.488118\pi$
$318$		0			0
$319$	3753.71	−	6501.61i	0.658832	−	1.14113i
$320$	−219.599	−	380.357i	−0.0383624	−	0.0664456i
$321$		0			0
$322$	3874.26	−	6360.71i	0.670510	−	1.10083i
$323$			747.679i			0.128799i
$324$		0			0
$325$	4446.95	+	2567.45i	0.758992	+	0.438204i
$326$	2518.93	+	1454.31i	0.427947	+	0.247075i
$327$		0			0
$328$			276.896i			0.0466129i
$329$	−1701.05	−	3112.87i	−0.285051	−	0.521635i
$330$		0			0
$331$	3089.55	+	5351.25i	0.513042	+	0.888614i	0.999886	+	0.0151252i	$0.00481468\pi$
$331$	3089.55	+	5351.25i	0.513042	+	0.888614i	−0.486844	+	0.873489i	$0.661852\pi$
$332$	−2727.71	+	4724.53i	−0.450911	+	0.781001i
$333$		0			0
$334$	2428.68	−	1402.20i	0.397878	−	0.229715i
$335$	4333.82			0.706811
$336$		0			0
$337$	−3792.02			−0.612951			−0.306475	−	0.951879i	$-0.599150\pi$
$337$	−3792.02			−0.612951			−0.306475	+	0.951879i	$0.599150\pi$
$338$	−3719.17	+	2147.26i	−0.598509	+	0.345550i
$339$		0			0
$340$	197.818	−	342.630i	0.0315534	−	0.0546522i
$341$	468.605	+	811.648i	0.0744175	+	0.128895i
$342$		0			0
$343$	−6336.70	−	447.093i	−0.997520	−	0.0703813i
$344$			3864.41i			0.605684i
$345$		0			0
$346$	−5052.73	−	2917.19i	−0.785076	−	0.453264i
$347$	6380.27	+	3683.65i	0.987062	+	0.569881i	0.904395	−	0.426696i	$-0.140323\pi$
$347$	6380.27	+	3683.65i	0.987062	+	0.569881i	0.0826675	+	0.996577i	$0.473656\pi$
$348$		0			0
$349$			9408.31i			1.44302i	0.692402	+	0.721512i	$0.256553\pi$
$349$			9408.31i			1.44302i	−0.692402	+	0.721512i	$0.743447\pi$
$350$	−67.7494	+	2884.90i	−0.0103467	+	0.440584i
$351$		0			0
$352$	528.450	+	915.302i	0.0800184	+	0.138596i
$353$	1411.89	−	2445.47i	0.212883	−	0.368723i	−0.739733	−	0.672901i	$-0.765048\pi$
$353$	1411.89	−	2445.47i	0.212883	−	0.368723i	0.952615	+	0.304177i	$0.0983815\pi$
$354$		0			0
$355$	−5544.51	+	3201.12i	−0.828935	+	0.478586i
$356$	−2869.27			−0.427165
$357$		0			0
$358$	6747.71			0.996167
$359$	−9971.93	+	5757.30i	−1.46601	+	0.846402i	−0.999278	−	0.0379953i	$-0.987903\pi$
$359$	−9971.93	+	5757.30i	−1.46601	+	0.846402i	−0.466734	+	0.884398i	$0.654569\pi$
$360$		0			0
$361$	−2083.97	+	3609.55i	−0.303830	+	0.526250i
$362$	−3857.00	−	6680.52i	−0.559999	−	0.969946i
$363$		0			0
$364$	−4170.11	−	2539.98i	−0.600475	−	0.365745i
$365$			1516.29i			0.217442i
$366$		0			0
$367$	−7817.42	−	4513.39i	−1.11190	−	0.641954i	−0.172577	−	0.984996i	$-0.555209\pi$
$367$	−7817.42	−	4513.39i	−1.11190	−	0.641954i	−0.939320	+	0.343042i	$0.888543\pi$
$368$	2786.10	+	1608.56i	0.394662	+	0.227858i
$369$		0			0
$370$		−	2774.67i		−	0.389861i
$371$	−1378.09	−	839.382i	−0.192848	−	0.117462i
$372$		0			0
$373$	1982.76	+	3434.23i	0.275236	+	0.476723i	0.970195	−	0.242327i	$-0.0779105\pi$
$373$	1982.76	+	3434.23i	0.275236	+	0.476723i	−0.694958	+	0.719050i	$0.744577\pi$
$374$	−476.034	+	824.515i	−0.0658159	+	0.113996i
$375$		0			0
$376$	1327.01	−	766.149i	0.182009	−	0.105083i
$377$	−14981.8			−2.04669
$378$		0			0
$379$	−12103.7			−1.64044			−0.820221	−	0.572047i	$-0.806149\pi$
$379$	−12103.7			−1.64044			−0.820221	+	0.572047i	$0.806149\pi$
$380$	−1233.20	+	711.987i	−0.166478	+	0.0961162i
$381$		0			0
$382$	−2135.64	+	3699.03i	−0.286044	+	0.495443i
$383$	2263.10	+	3919.81i	0.301930	+	0.522958i	0.976573	−	0.215186i	$-0.0690358\pi$
$383$	2263.10	+	3919.81i	0.301930	+	0.522958i	−0.674643	+	0.738144i	$0.735703\pi$
$384$		0			0
$385$	−98.5523	+	4196.55i	−0.0130459	+	0.555521i
$386$		−	2264.92i		−	0.298656i
$387$		0			0
$388$	−34.3263	−	19.8183i	−0.00449138	−	0.00259310i
$389$	2027.54	+	1170.60i	0.264268	+	0.152575i	0.626280	−	0.779598i	$-0.284577\pi$
$389$	2027.54	+	1170.60i	0.264268	+	0.152575i	−0.362012	+	0.932173i	$0.617910\pi$
$390$		0			0
$391$			2898.02i			0.374831i
$392$	128.810	−	2740.98i	0.0165966	−	0.353164i
$393$		0			0
$394$	−1972.82	−	3417.02i	−0.252257	−	0.436922i
$395$	−1565.04	+	2710.72i	−0.199356	+	0.345294i
$396$		0			0
$397$	9683.92	−	5591.01i	1.22424	−	0.706813i	0.258418	−	0.966033i	$-0.416799\pi$
$397$	9683.92	−	5591.01i	1.22424	−	0.706813i	0.965818	+	0.259220i	$0.0834655\pi$
$398$	2324.10			0.292705
$399$		0			0
$400$	−1246.50			−0.155813
$401$	−9808.28	+	5662.81i	−1.22145	+	0.705205i	−0.965227	−	0.261412i	$-0.915812\pi$
$401$	−9808.28	+	5662.81i	−1.22145	+	0.705205i	−0.256224	+	0.966617i	$0.582479\pi$
$402$		0			0
$403$	935.150	−	1619.73i	0.115591	−	0.200209i
$404$	−190.809	−	330.490i	−0.0234977	−	0.0406993i
$405$		0			0
$406$	−4037.38	−	7388.28i	−0.493526	−	0.903139i
$407$			6677.06i			0.813193i
$408$		0			0
$409$	9068.41	+	5235.65i	1.09634	+	0.632974i	0.935258	−	0.353967i	$-0.115168\pi$
$409$	9068.41	+	5235.65i	1.09634	+	0.632974i	0.161084	+	0.986941i	$0.448501\pi$
$410$	−411.404	−	237.524i	−0.0495556	−	0.0286109i
$411$		0			0
$412$		−	4544.12i		−	0.543381i
$413$	5634.98	−	9251.44i	0.671379	−	1.10226i
$414$		0			0
$415$	−4679.71	−	8105.50i	−0.553537	−	0.958755i
$416$	1054.58	−	1826.58i	0.124291	−	0.215278i
$417$		0			0
$418$	2967.60	−	1713.35i	0.347249	−	0.200484i
$419$	10355.8			1.20743			0.603717	−	0.797199i	$-0.293686\pi$
$419$	10355.8			1.20743			0.603717	+	0.797199i	$0.293686\pi$
$420$		0			0
$421$	2613.96			0.302604			0.151302	−	0.988488i	$-0.451653\pi$
$421$	2613.96			0.302604			0.151302	+	0.988488i	$0.451653\pi$
$422$	−6169.05	+	3561.70i	−0.711623	+	0.410855i
$423$		0			0
$424$	348.503	−	603.626i	0.0399170	−	0.0691383i
$425$	−561.432	−	972.430i	−0.0640788	−	0.110988i
$426$		0			0
$427$	−5699.53	+	3114.55i	−0.645948	+	0.352983i
$428$		−	8310.73i		−	0.938584i
$429$		0			0
$430$	−5741.63	−	3314.93i	−0.643920	−	0.371768i
$431$	3344.94	+	1931.20i	0.373829	+	0.215830i	0.675130	−	0.737699i	$-0.264088\pi$
$431$	3344.94	+	1931.20i	0.373829	+	0.215830i	−0.301301	+	0.953529i	$0.597421\pi$
$432$		0			0
$433$			17569.5i			1.94997i	0.222280	+	0.974983i	$0.428650\pi$
$433$			17569.5i			1.94997i	−0.222280	+	0.974983i	$0.571350\pi$
$434$	1050.78	+	24.6766i	0.116219	+	0.00272930i
$435$		0			0
$436$	−236.797	−	410.145i	−0.0260104	−	0.0450514i
$437$	5215.28	−	9033.13i	0.570894	−	0.988817i
$438$		0			0
$439$	−9969.18	+	5755.71i	−1.08383	+	0.625752i	−0.931928	−	0.362643i	$-0.881874\pi$
$439$	−9969.18	+	5755.71i	−1.08383	+	0.625752i	−0.151906	+	0.988395i	$0.548541\pi$
$440$	−1813.24			−0.196461
$441$		0			0
$442$	1899.95			0.204460
$443$	−6386.93	+	3687.49i	−0.684993	+	0.395481i	−0.801734	−	0.597681i	$-0.796089\pi$
$443$	−6386.93	+	3687.49i	−0.684993	+	0.395481i	0.116740	+	0.993162i	$0.462755\pi$
$444$		0			0
$445$	2461.28	−	4263.07i	0.262193	−	0.454132i
$446$	479.700	+	830.865i	0.0509293	+	0.0882121i
$447$		0			0
$448$	1184.97	+	27.8280i	0.124966	+	0.00293471i
$449$			10388.6i			1.09192i	0.837813	+	0.545958i	$0.183834\pi$
$449$			10388.6i			1.09192i	−0.837813	+	0.545958i	$0.816166\pi$
$450$		0			0
$451$	990.015	+	571.585i	0.103366	+	0.0596783i
$452$	−390.934	−	225.706i	−0.0406814	−	0.0234874i
$453$		0			0
$454$			8218.77i			0.849616i
$455$	7350.99	−	4017.00i	0.757406	−	0.413890i
$456$		0			0
$457$	−1264.70	−	2190.53i	−0.129454	−	0.224220i	0.794011	−	0.607903i	$-0.207989\pi$
$457$	−1264.70	−	2190.53i	−0.129454	−	0.224220i	−0.923465	+	0.383683i	$0.874656\pi$
$458$	−2269.53	+	3930.94i	−0.231546	+	0.401050i
$459$		0			0
$460$	−4779.89	+	2759.67i	−0.484486	+	0.279718i
$461$	19385.0			1.95845			0.979227	−	0.202767i	$-0.0649934\pi$
$461$	19385.0			1.95845			0.979227	+	0.202767i	$0.0649934\pi$
$462$		0			0
$463$	7386.83			0.741458			0.370729	−	0.928741i	$-0.379108\pi$
$463$	7386.83			0.741458			0.370729	+	0.928741i	$0.379108\pi$
$464$	3149.61	−	1818.43i	0.315123	−	0.181936i
$465$		0			0
$466$	337.223	−	584.087i	0.0335226	−	0.0580629i
$467$	−6019.62	−	10426.3i	−0.596477	−	1.03313i	−0.993337	−	0.115249i	$-0.963233\pi$
$467$	−6019.62	−	10426.3i	−0.596477	−	1.03313i	0.396860	−	0.917879i	$-0.370100\pi$
$468$		0			0
$469$	−6084.19	+	9988.93i	−0.599023	+	0.983467i
$470$			2628.84i			0.257999i
$471$		0			0
$472$	4052.29	+	2339.59i	0.395173	+	0.228153i
$473$	13816.8	+	7977.14i	1.34312	+	0.775453i
$474$		0			0
$475$			4041.42i			0.390386i
$476$	512.008	+	936.960i	0.0493022	+	0.0902216i
$477$		0			0
$478$	1119.50	+	1939.04i	0.107123	+	0.185543i
$479$	7469.93	−	12938.3i	0.712546	−	1.23417i	−0.251352	−	0.967896i	$-0.580875\pi$
$479$	7469.93	−	12938.3i	0.712546	−	1.23417i	0.963898	−	0.266271i	$-0.0857915\pi$
$480$		0			0
$481$	11539.6	−	6662.38i	1.09389	−	0.631556i
$482$	8727.04			0.824701
$483$		0			0
$484$	−960.574			−0.0902117
$485$	58.8909	−	34.0007i	0.00551360	−	0.00318328i
$486$		0			0
$487$	2645.81	−	4582.68i	0.246187	−	0.426409i	−0.716278	−	0.697815i	$-0.754156\pi$
$487$	2645.81	−	4582.68i	0.246187	−	0.426409i	0.962465	+	0.271407i	$0.0874889\pi$
$488$	−1402.79	−	2429.70i	−0.130126	−	0.225384i
$489$		0			0
$490$	3961.96	+	2542.62i	0.365272	+	0.234416i
$491$		−	14095.0i		−	1.29552i	−0.761845	−	0.647759i	$-0.775706\pi$
$491$		−	14095.0i		−	1.29552i	0.761845	−	0.647759i	$-0.224294\pi$
$492$		0			0
$493$	2837.21	+	1638.06i	0.259192	+	0.149644i
$494$	−5922.16	−	3419.16i	−0.539373	−	0.311407i
$495$		0			0
$496$			454.018i			0.0411008i
$497$	405.652	−	17273.4i	0.0366116	−	1.55899i
$498$		0			0
$499$	−4303.22	−	7453.40i	−0.386050	−	0.668658i	0.605865	−	0.795568i	$-0.292827\pi$
$499$	−4303.22	−	7453.40i	−0.386050	−	0.668658i	−0.991914	+	0.126910i	$0.959494\pi$
$500$	2784.88	−	4823.56i	0.249087	−	0.431432i
$501$		0			0
$502$	6716.99	−	3878.06i	0.597199	−	0.344793i
$503$	−9541.90			−0.845830			−0.422915	−	0.906169i	$-0.638993\pi$
$503$	−9541.90			−0.845830			−0.422915	+	0.906169i	$0.638993\pi$
$504$		0			0
$505$	654.710			0.0576915
$506$	11502.5	−	6640.95i	1.01057	−	0.583451i
$507$		0			0
$508$	−1714.84	+	2970.19i	−0.149771	+	0.259411i
$509$	−4949.39	−	8572.60i	−0.430998	−	0.746510i	0.565962	−	0.824432i	$-0.308505\pi$
$509$	−4949.39	−	8572.60i	−0.430998	−	0.746510i	−0.996960	+	0.0779213i	$0.975172\pi$
$510$		0			0
$511$	−3494.87	−	2128.70i	−0.302552	−	0.184282i
$512$			512.000i			0.0441942i
$513$		0			0
$514$	−1040.01	−	600.450i	−0.0892469	−	0.0515267i
$515$	6751.52	+	3897.99i	0.577685	+	0.333526i
$516$		0			0
$517$		−	6326.12i		−	0.538148i
$518$	6395.29	+	3895.33i	0.542458	+	0.330407i
$519$		0			0
$520$	1809.25	+	3133.72i	0.152579	+	0.264274i
$521$	−4465.58	+	7734.61i	−0.375510	+	0.650402i	−0.990403	−	0.138208i	$-0.955866\pi$
$521$	−4465.58	+	7734.61i	−0.375510	+	0.650402i	0.614893	+	0.788610i	$0.289199\pi$
$522$		0			0
$523$	−7532.64	+	4348.97i	−0.629789	+	0.363609i	−0.780670	−	0.624943i	$-0.785122\pi$
$523$	−7532.64	+	4348.97i	−0.629789	+	0.363609i	0.150882	+	0.988552i	$0.451789\pi$
$524$	−2863.20			−0.238702
$525$		0			0
$526$	7045.06			0.583991
$527$	−354.191	+	204.492i	−0.0292767	+	0.0169029i
$528$		0			0
$529$	14131.0	−	24475.6i	1.16142	−	2.01164i
$530$	597.899	+	1035.59i	0.0490020	+	0.0848740i
$531$		0			0
$532$	90.2243	−	3841.92i	0.00735285	−	0.313099i
$533$		−	2281.32i		−	0.185393i
$534$		0			0
$535$	12347.8	+	7129.02i	0.997837	+	0.576102i
$536$	−4375.33	−	2526.10i	−0.352585	−	0.203565i
$537$		0			0
$538$			11753.6i			0.941884i
$539$	−9534.18	−	6118.62i	−0.761904	−	0.488957i
$540$		0			0
$541$	−3981.40	−	6895.99i	−0.316402	−	0.548025i	0.663332	−	0.748325i	$-0.269142\pi$
$541$	−3981.40	−	6895.99i	−0.316402	−	0.548025i	−0.979735	+	0.200300i	$0.935808\pi$
$542$	3351.86	−	5805.60i	0.265636	−	0.460096i
$543$		0			0
$544$	−399.425	+	230.608i	−0.0314801	+	0.0181751i
$545$	812.508			0.0638606
$546$		0			0
$547$	−10238.1			−0.800276			−0.400138	−	0.916455i	$-0.631038\pi$
$547$	−10238.1			−0.800276			−0.400138	+	0.916455i	$0.631038\pi$
$548$	−459.885	+	265.515i	−0.0358491	+	0.0206975i
$549$		0			0
$550$	−2573.10	+	4456.74i	−0.199486	+	0.345520i
$551$	−5895.73	−	10211.7i	−0.455838	−	0.789534i
$552$		0			0
$553$	−4050.76	−	7412.77i	−0.311493	−	0.570024i
$554$			9585.24i			0.735086i
$555$		0			0
$556$	−10072.2	−	5815.19i	−0.768268	−	0.443560i
$557$	−2902.01	−	1675.47i	−0.220758	−	0.127454i	0.385543	−	0.922690i	$-0.374014\pi$
$557$	−2902.01	−	1675.47i	−0.220758	−	0.127454i	−0.606301	+	0.795235i	$0.707347\pi$
$558$		0			0
$559$		−	31838.4i		−	2.40898i
$560$	−1057.82	+	1736.72i	−0.0798236	+	0.131053i
$561$		0			0
$562$	6132.70	+	10622.1i	0.460306	+	0.797274i
$563$	−1477.19	+	2558.58i	−0.110580	+	0.191529i	−0.916004	−	0.401169i	$-0.868604\pi$
$563$	−1477.19	+	2558.58i	−0.110580	+	0.191529i	0.805425	+	0.592698i	$0.201937\pi$
$564$		0			0
$565$	670.693	−	387.225i	0.0499403	−	0.0288331i
$566$	−5744.21			−0.426585
$567$		0			0
$568$	7463.48			0.551339
$569$	10620.5	−	6131.74i	0.782485	−	0.451768i	−0.0548252	−	0.998496i	$-0.517460\pi$
$569$	10620.5	−	6131.74i	0.782485	−	0.451768i	0.837310	+	0.546728i	$0.184127\pi$
$570$		0			0
$571$	6836.24	−	11840.7i	0.501029	−	0.867808i	−0.498970	−	0.866619i	$-0.666288\pi$
$571$	6836.24	−	11840.7i	0.501029	−	0.867808i	0.999999	−	0.00118894i	$-0.000378452\pi$
$572$	−4353.83	−	7541.06i	−0.318257	−	0.551237i
$573$		0			0
$574$	1125.03	−	614.780i	0.0818081	−	0.0447046i
$575$			15664.6i			1.13610i
$576$		0			0
$577$	21055.8	+	12156.6i	1.51918	+	0.877096i	0.999745	+	0.0225837i	$0.00718923\pi$
$577$	21055.8	+	12156.6i	1.51918	+	0.877096i	0.519431	+	0.854513i	$0.326144\pi$
$578$	8149.76	+	4705.27i	0.586480	+	0.338604i
$579$		0			0
$580$			6239.47i			0.446689i
$581$	25252.0	+	593.021i	1.80315	+	0.0423454i
$582$		0			0
$583$	−1438.80	−	2492.08i	−0.102211	−	0.177035i
$584$	883.815	−	1530.81i	0.0626242	−	0.108468i
$585$		0			0
$586$	−8212.19	+	4741.31i	−0.578912	+	0.334235i
$587$	20911.9			1.47040			0.735200	−	0.677851i	$-0.237088\pi$
$587$	20911.9			1.47040			0.735200	+	0.677851i	$0.237088\pi$
$588$		0			0
$589$	1472.02			0.102977
$590$	−6952.19	+	4013.85i	−0.485114	+	0.280080i
$591$		0			0
$592$	−1617.30	+	2801.25i	−0.112282	+	0.194477i
$593$	−2269.12	−	3930.24i	−0.157136	−	0.272168i	0.776699	−	0.629872i	$-0.216893\pi$
$593$	−2269.12	−	3930.24i	−0.157136	−	0.272168i	−0.933835	+	0.357705i	$0.883559\pi$
$594$		0			0
$595$	−1831.31	−	43.0068i	−0.126179	−	0.00296321i
$596$			11628.4i			0.799192i
$597$		0			0
$598$	−22954.4	−	13252.7i	−1.56969	−	0.906260i
$599$	−19553.9	−	11289.4i	−1.33380	−	0.770072i	−0.347924	−	0.937523i	$-0.613113\pi$
$599$	−19553.9	−	11289.4i	−1.33380	−	0.770072i	−0.985880	+	0.167451i	$0.946447\pi$
$600$		0			0
$601$			3449.47i			0.234121i	0.993125	+	0.117061i	$0.0373471\pi$
$601$			3449.47i			0.234121i	−0.993125	+	0.117061i	$0.962653\pi$
$602$	15701.1	−	8579.98i	1.06301	−	0.580887i
$603$		0			0
$604$	1807.43	+	3130.57i	0.121761	+	0.210896i
$605$	823.990	−	1427.19i	0.0553718	−	0.0959068i
$606$		0			0
$607$	4570.23	−	2638.63i	0.305601	−	0.176439i	−0.339355	−	0.940658i	$-0.610209\pi$
$607$	4570.23	−	2638.63i	0.305601	−	0.176439i	0.644956	+	0.764219i	$0.276875\pi$
$608$	1660.01			0.110728
$609$		0			0
$610$	4813.30			0.319484
$611$	−10933.1	+	6312.21i	−0.723903	+	0.417946i
$612$		0			0
$613$	−5405.00	+	9361.73i	−0.356127	+	0.616830i	−0.987310	−	0.158804i	$-0.949236\pi$
$613$	−5405.00	+	9361.73i	−0.356127	+	0.616830i	0.631183	+	0.775634i	$0.282570\pi$
$614$	−6632.39	−	11487.6i	−0.435931	−	0.755054i
$615$		0			0
$616$	2545.58	−	4179.30i	0.166500	−	0.273358i
$617$		−	19744.4i		−	1.28830i	−0.764900	−	0.644149i	$-0.777212\pi$
$617$		−	19744.4i		−	1.28830i	0.764900	−	0.644149i	$-0.222788\pi$
$618$		0			0
$619$	948.715	+	547.741i	0.0616027	+	0.0355664i	0.530485	−	0.847694i	$-0.322010\pi$
$619$	948.715	+	547.741i	0.0616027	+	0.0355664i	−0.468882	+	0.883261i	$0.655343\pi$
$620$	−674.566	−	389.461i	−0.0436955	−	0.0252276i
$621$		0			0
$622$			15140.4i			0.976006i
$623$	6370.50	+	11657.8i	0.409677	+	0.749697i
$624$		0			0
$625$	−91.3572	−	158.235i	−0.00584686	−	0.0101271i
$626$	−6851.35	+	11866.9i	−0.437436	+	0.757662i
$627$		0			0
$628$	8629.65	−	4982.33i	0.548345	−	0.316587i
$629$	−2913.77			−0.184705
$630$		0			0
$631$	4564.95			0.287999			0.144000	−	0.989578i	$-0.454004\pi$
$631$	4564.95			0.287999			0.144000	+	0.989578i	$0.454004\pi$
$632$	3160.05	−	1824.46i	0.198893	−	0.114831i
$633$		0			0
$634$	−5275.27	+	9137.03i	−0.330454	+	0.572363i
$635$	−2942.01	−	5095.71i	−0.183859	−	0.318452i
$636$		0			0
$637$	−1061.25	+	22582.6i	−0.0660098	+	1.40464i
$638$		−	15014.8i		−	0.931729i
$639$		0			0
$640$	−760.714	−	439.198i	−0.0469842	−	0.0271263i
$641$	−4810.24	−	2777.20i	−0.296401	−	0.171127i	0.344424	−	0.938814i	$-0.388074\pi$
$641$	−4810.24	−	2777.20i	−0.296401	−	0.171127i	−0.640825	+	0.767687i	$0.721408\pi$
$642$		0			0
$643$			27961.1i			1.71490i	0.514570	+	0.857449i	$0.327952\pi$
$643$			27961.1i			1.71490i	−0.514570	+	0.857449i	$0.672048\pi$
$644$	349.710	−	14891.3i	0.0213983	−	0.911182i
$645$		0			0
$646$	747.679	+	1295.02i	0.0455372	+	0.0788728i
$647$	10177.2	−	17627.5i	0.618406	−	1.07111i	−0.371371	−	0.928485i	$-0.621112\pi$
$647$	10177.2	−	17627.5i	0.618406	−	1.07111i	0.989777	−	0.142626i	$-0.0455546\pi$
$648$		0			0
$649$	16729.9	−	9659.03i	1.01188	−	0.584207i
$650$	10269.8			0.619714
$651$		0			0
$652$	5817.22			0.349417
$653$	−10991.1	+	6345.71i	−0.658674	+	0.380286i	−0.791772	−	0.610817i	$-0.790841\pi$
$653$	−10991.1	+	6345.71i	−0.658674	+	0.380286i	0.133097	+	0.991103i	$0.457508\pi$
$654$		0			0
$655$	2456.08	−	4254.06i	0.146515	−	0.253771i
$656$	276.896	+	479.599i	0.0164802	+	0.0285445i
$657$		0			0
$658$	−6059.17	−	3690.59i	−0.358983	−	0.218654i
$659$			19546.6i			1.15543i	0.816239	+	0.577714i	$0.196055\pi$
$659$			19546.6i			1.15543i	−0.816239	+	0.577714i	$0.803945\pi$
$660$		0			0
$661$	−22849.3	−	13192.0i	−1.34453	−	0.776264i	−0.357060	−	0.934081i	$-0.616221\pi$
$661$	−22849.3	−	13192.0i	−1.34453	−	0.776264i	−0.987468	+	0.157818i	$0.949554\pi$
$662$	10702.5	+	6179.09i	0.628345	+	0.362775i
$663$		0			0
$664$			10910.8i			0.637685i
$665$	5630.82	+	3429.69i	0.328351	+	0.199996i
$666$		0			0
$667$	−22851.9	−	39580.7i	−1.32658	−	2.29771i
$668$	2804.40	−	4857.36i	0.162433	−	0.281343i
$669$		0			0
$670$	7506.39	−	4333.82i	0.432832	−	0.249896i
$671$	−11582.9			−0.666396
$672$		0			0
$673$	829.227			0.0474953			0.0237477	−	0.999718i	$-0.492440\pi$
$673$	829.227			0.0474953			0.0237477	+	0.999718i	$0.492440\pi$
$674$	−6567.97	+	3792.02i	−0.375354	+	0.216711i
$675$		0			0
$676$	−4294.53	+	7438.34i	−0.244340	+	0.423210i
$677$	−745.967	−	1292.05i	−0.0423484	−	0.0733495i	0.844074	−	0.536226i	$-0.180151\pi$
$677$	−745.967	−	1292.05i	−0.0423484	−	0.0733495i	−0.886423	+	0.462877i	$0.846817\pi$
$678$		0			0
$679$	−4.30863	+	183.470i	−0.000243520	+	0.0103695i
$680$		−	791.271i		−	0.0446233i
$681$		0			0
$682$	1623.30	+	937.210i	0.0911425	+	0.0526212i
$683$	6008.93	+	3469.26i	0.336640	+	0.194359i	0.658785	−	0.752331i	$-0.271071\pi$
$683$	6008.93	+	3469.26i	0.336640	+	0.194359i	−0.322145	+	0.946690i	$0.604404\pi$
$684$		0			0
$685$		−	911.044i		−	0.0508163i
$686$	−11422.6	+	5562.31i	−0.635737	+	0.309577i
$687$		0			0
$688$	3864.41	+	6693.36i	0.214141	+	0.370904i
$689$	−2871.28	+	4973.20i	−0.158762	+	0.274984i
$690$		0			0
$691$	−150.567	+	86.9299i	−0.00828921	+	0.00478578i	−0.504139	−	0.863623i	$-0.668190\pi$
$691$	−150.567	+	86.9299i	−0.00828921	+	0.00478578i	0.495850	+	0.868408i	$0.334857\pi$
$692$	−11668.8			−0.641012
$693$		0			0
$694$	14734.6			0.805933
$695$	17280.1	−	9976.66i	0.943123	−	0.544513i
$696$		0			0
$697$	−249.432	+	432.028i	−0.0135551	+	0.0234781i
$698$	9408.31	+	16295.7i	0.510186	+	0.883668i
$699$		0			0
$700$	2767.55	+	5064.54i	0.149434	+	0.273460i
$701$		−	13902.8i		−	0.749076i	−0.927211	−	0.374538i	$-0.877801\pi$
$701$		−	13902.8i		−	0.749076i	0.927211	−	0.374538i	$-0.122199\pi$
$702$		0			0
$703$	9082.24	+	5243.63i	0.487259	+	0.281319i
$704$	1830.60	+	1056.90i	0.0980021	+	0.0565815i
$705$		0			0
$706$		−	5647.58i		−	0.301061i
$707$	−919.138	+	1509.03i	−0.0488936	+	0.0802728i
$708$		0			0
$709$	−1902.55	−	3295.32i	−0.100778	−	0.174553i	0.811227	−	0.584731i	$-0.198800\pi$
$709$	−1902.55	−	3295.32i	−0.100778	−	0.174553i	−0.912006	+	0.410178i	$0.865467\pi$
$710$	−6402.25	+	11089.0i	−0.338411	+	0.586146i
$711$		0			0
$712$	−4969.72	+	2869.27i	−0.261584	+	0.151026i
$713$	5705.58			0.299685
$714$		0			0
$715$	14939.0			0.781382
$716$	11687.4	−	6747.71i	0.610025	−	0.352198i
$717$		0			0
$718$	−11514.6	+	19943.9i	−0.598497	+	1.03663i
$719$	15869.0	+	27485.9i	0.823107	+	1.42566i	0.903358	+	0.428888i	$0.141094\pi$
$719$	15869.0	+	27485.9i	0.823107	+	1.42566i	−0.0802513	+	0.996775i	$0.525572\pi$
$720$		0			0
$721$	−18462.8	+	10089.1i	−0.953661	+	0.521135i
$722$			8335.89i			0.429681i
$723$		0			0
$724$	−13361.0	−	7714.01i	−0.685856	−	0.395979i
$725$	15335.9	+	8854.21i	0.785604	+	0.453569i
$726$		0			0
$727$		−	3362.88i		−	0.171558i	−0.996314	−	0.0857789i	$-0.972662\pi$
$727$		−	3362.88i		−	0.171558i	0.996314	−	0.0857789i	$-0.0273379\pi$
$728$	−9762.82	−	229.272i	−0.497025	−	0.0116722i
$729$		0			0
$730$	1516.29	+	2626.29i	0.0768773	+	0.133155i
$731$	−3481.11	+	6029.46i	−0.176133	+	0.305072i
$732$		0			0
$733$	−26902.9	+	15532.4i	−1.35564	+	0.782676i	−0.989032	−	0.147701i	$-0.952813\pi$
$733$	−26902.9	+	15532.4i	−1.35564	+	0.782676i	−0.366603	+	0.930377i	$0.619479\pi$
$734$	−18053.6			−0.907860
$735$		0			0
$736$	6434.23			0.322240
$737$	−18063.6	+	10429.0i	−0.902824	+	0.521246i
$738$		0			0
$739$	−222.377	+	385.168i	−0.0110694	+	0.0191727i	−0.871507	−	0.490383i	$-0.836857\pi$
$739$	−222.377	+	385.168i	−0.0110694	+	0.0191727i	0.860438	+	0.509556i	$0.170190\pi$
$740$	−2774.67	−	4805.88i	−0.137837	−	0.238740i
$741$		0			0
$742$	−3226.30	−	75.7669i	−0.159624	−	0.00374864i
$743$			9748.08i			0.481322i	0.970609	+	0.240661i	$0.0773643\pi$
$743$			9748.08i			0.481322i	−0.970609	+	0.240661i	$0.922636\pi$
$744$		0			0
$745$	−17277.1	−	9974.97i	−0.849645	−	0.490543i
$746$	6868.47	+	3965.51i	0.337094	+	0.194622i
$747$		0			0
$748$			1904.14i			0.0930777i
$749$	−33766.5	+	18451.9i	−1.64726	+	0.900159i
$750$		0			0
$751$	−8765.46	−	15182.2i	−0.425907	−	0.737693i	0.570598	−	0.821230i	$-0.306712\pi$
$751$	−8765.46	−	15182.2i	−0.425907	−	0.737693i	−0.996505	+	0.0835371i	$0.973378\pi$
$752$	1532.30	−	2654.02i	0.0743048	−	0.128700i
$753$		0			0
$754$	−25949.3	+	14981.8i	−1.25334	+	0.723615i
$755$	−6201.74			−0.298946
$756$		0			0
$757$	4717.30			0.226490			0.113245	−	0.993567i	$-0.463875\pi$
$757$	4717.30			0.226490			0.113245	+	0.993567i	$0.463875\pi$
$758$	−20964.3	+	12103.7i	−1.00456	+	0.579984i
$759$		0			0
$760$	−1423.97	+	2466.39i	−0.0679644	+	0.117718i
$761$	16185.3	+	28033.8i	0.770983	+	1.33538i	0.937025	+	0.349262i	$0.113568\pi$
$761$	16185.3	+	28033.8i	0.770983	+	1.33538i	−0.166042	+	0.986119i	$0.553099\pi$
$762$		0			0
$763$	−1140.67	+	1872.73i	−0.0541219	+	0.0888565i
$764$			8542.55i			0.404527i
$765$		0			0
$766$	7839.62	+	4526.20i	0.369787	+	0.213497i
$767$	−33386.3	−	19275.6i	−1.57172	−	0.907434i
$768$		0			0
$769$		−	15910.1i		−	0.746076i	−0.927816	−	0.373038i	$-0.878316\pi$
$769$		−	15910.1i		−	0.746076i	0.927816	−	0.373038i	$-0.121684\pi$
$770$	4025.85	+	7367.18i	0.188418	+	0.344798i
$771$		0			0
$772$	−2264.92	−	3922.95i	−0.105591	−	0.182889i
$773$	−7751.27	+	13425.6i	−0.360665	+	0.624690i	−0.988070	−	0.154003i	$-0.950784\pi$
$773$	−7751.27	+	13425.6i	−0.360665	+	0.624690i	0.627406	+	0.778693i	$0.284117\pi$
$774$		0			0
$775$	−1914.51	+	1105.34i	−0.0887369	+	0.0512323i
$776$	−79.2732			−0.00366720
$777$		0			0
$778$	4682.40			0.215774
$779$	1554.96	−	897.756i	0.0715176	−	0.0412907i
$780$		0			0
$781$	15406.5	−	26684.9i	0.705877	−	1.22261i
$782$	2898.02	+	5019.51i	0.132523	+	0.229536i
$783$		0			0
$784$	−2517.87	−	4876.32i	−0.114699	−	0.222135i
$785$			17095.6i			0.777283i
$786$		0			0
$787$	−25807.5	−	14899.9i	−1.16892	−	0.674874i	−0.215491	−	0.976506i	$-0.569135\pi$
$787$	−25807.5	−	14899.9i	−1.16892	−	0.674874i	−0.953424	+	0.301632i	$0.902469\pi$
$788$	−6834.05	−	3945.64i	−0.308950	−	0.178373i
$789$		0			0
$790$			6260.15i			0.281932i
$791$	−49.0699	+	2089.49i	−0.00220572	+	0.0939237i
$792$		0			0
$793$	11557.4	+	20018.0i	0.517548	+	0.896420i
$794$	11182.0	−	19367.8i	0.499792	−	0.865666i
$795$		0			0
$796$	4025.46	−	2324.10i	0.179244	−	0.103487i
$797$	−34495.2			−1.53310			−0.766550	−	0.642184i	$-0.778028\pi$
$797$	−34495.2			−1.53310			−0.766550	+	0.642184i	$0.778028\pi$
$798$		0			0
$799$	2760.63			0.122233
$800$	−2159.01	+	1246.50i	−0.0954155	+	0.0550882i
$801$		0			0
$802$	−11325.6	+	19616.6i	−0.498655	+	0.863697i
$803$	−3648.84	−	6319.98i	−0.160355	−	0.277743i
$804$		0			0
$805$	21825.1	+	13293.5i	0.955571	+	0.582031i
$806$		−	3740.60i		−	0.163470i
$807$		0			0
$808$	−660.980	−	381.617i	−0.0287787	−	0.0166154i
$809$	8073.09	+	4661.00i	0.350846	+	0.202561i	0.665058	−	0.746792i	$-0.268407\pi$
$809$	8073.09	+	4661.00i	0.350846	+	0.202561i	−0.314212	+	0.949353i	$0.601740\pi$
$810$		0			0
$811$		−	13684.8i		−	0.592524i	−0.955107	−	0.296262i	$-0.904260\pi$
$811$		−	13684.8i		−	0.592524i	0.955107	−	0.296262i	$-0.0957401\pi$
$812$	−14381.2	−	8759.50i	−0.621530	−	0.378569i
$813$		0			0
$814$	6677.06	+	11565.0i	0.287507	+	0.497977i
$815$	−4990.07	+	8643.05i	−0.214472	+	0.371476i
$816$		0			0
$817$	21701.3	−	12529.2i	0.929292	−	0.536527i
$818$	20942.6			0.895160
$819$		0			0
$820$	−950.097			−0.0404620
$821$	−34689.6	+	20028.1i	−1.47464	+	0.851381i	−0.999591	−	0.0285809i	$-0.990901\pi$
$821$	−34689.6	+	20028.1i	−1.47464	+	0.851381i	−0.475044	+	0.879962i	$0.657568\pi$
$822$		0			0
$823$	3961.13	−	6860.88i	0.167772	−	0.290589i	−0.769864	−	0.638208i	$-0.779676\pi$
$823$	3961.13	−	6860.88i	0.167772	−	0.290589i	0.937636	+	0.347618i	$0.113009\pi$
$824$	−4544.12	−	7870.65i	−0.192114	−	0.332751i
$825$		0			0
$826$	508.642	−	21658.9i	0.0214260	−	0.912362i
$827$		−	15771.5i		−	0.663155i	−0.943428	−	0.331577i	$-0.892419\pi$
$827$		−	15771.5i		−	0.663155i	0.943428	−	0.331577i	$-0.107581\pi$
$828$		0			0
$829$	30793.3	+	17778.5i	1.29010	+	0.744840i	0.978672	−	0.205429i	$-0.0658589\pi$
$829$	30793.3	+	17778.5i	1.29010	+	0.744840i	0.311429	+	0.950269i	$0.399192\pi$
$830$	−16211.0	−	9359.42i	−0.677942	−	0.391410i
$831$		0			0
$832$		−	4218.31i		−	0.175773i
$833$	2670.08	−	4160.58i	0.111060	−	0.173056i
$834$		0			0
$835$	4811.28	+	8333.38i	0.199403	+	0.345375i
$836$	3426.69	−	5935.20i	0.141764	−	0.245542i
$837$		0			0
$838$	17936.8	−	10355.8i	0.739399	−	0.426892i
$839$	24413.9			1.00460			0.502301	−	0.864693i	$-0.332487\pi$
$839$	24413.9			1.00460			0.502301	+	0.864693i	$0.332487\pi$
$840$		0			0
$841$	−27278.0			−1.11846
$842$	4527.50	−	2613.96i	0.185307	−	0.106987i
$843$		0			0
$844$	−7123.41	+	12338.1i	−0.290519	+	0.503193i
$845$	−7367.77	−	12761.4i	−0.299952	−	0.519531i
$846$		0			0
$847$	2132.72	+	3902.81i	0.0865184	+	0.158326i
$848$		−	1394.01i		−	0.0564512i
$849$		0			0
$850$	−1944.86	−	1122.86i	−0.0784801	−	0.0453105i
$851$	35202.9	+	20324.4i	1.41803	+	0.818697i
$852$		0			0
$853$		−	32086.8i		−	1.28796i	−0.765041	−	0.643981i	$-0.777281\pi$
$853$		−	32086.8i		−	1.28796i	0.765041	−	0.643981i	$-0.222719\pi$
$854$	−6757.33	+	11094.1i	−0.270762	+	0.444534i
$855$		0			0
$856$	−8310.73	−	14394.6i	−0.331840	−	0.574763i
$857$	−6399.64	+	11084.5i	−0.255085	+	0.441820i	−0.964919	−	0.262549i	$-0.915437\pi$
$857$	−6399.64	+	11084.5i	−0.255085	+	0.441820i	0.709834	+	0.704369i	$0.248770\pi$
$858$		0			0
$859$	−42677.2	+	24639.7i	−1.69514	+	0.978690i	−0.744899	+	0.667177i	$0.767502\pi$
$859$	−42677.2	+	24639.7i	−1.69514	+	0.978690i	−0.950242	+	0.311513i	$0.899164\pi$
$860$	−13259.7			−0.525759
$861$		0			0
$862$	7724.82			0.305230
$863$	10317.0	−	5956.55i	0.406948	−	0.234952i	−0.282529	−	0.959259i	$-0.591174\pi$
$863$	10317.0	−	5956.55i	0.406948	−	0.234952i	0.689478	+	0.724307i	$0.257840\pi$
$864$		0			0
$865$	10009.6	−	17337.1i	0.393452	−	0.681479i
$866$	17569.5	+	30431.2i	0.689417	+	1.19411i
$867$		0			0
$868$	1844.68	−	1008.04i	0.0721341	−	0.0394182i
$869$		−	15064.6i		−	0.588069i
$870$		0			0
$871$	36047.8	+	20812.2i	1.40233	+	0.809638i
$872$	−820.290	−	473.595i	−0.0318561	−	0.0183921i
$873$		0			0
$874$		−	20861.1i		−	0.807366i
$875$	−25781.2	−	605.451i	−0.996075	−	0.0233920i
$876$		0			0
$877$	−8249.14	−	14287.9i	−0.317621	−	0.550136i	0.662370	−	0.749177i	$-0.269551\pi$
$877$	−8249.14	−	14287.9i	−0.317621	−	0.550136i	−0.979991	+	0.199041i	$0.936217\pi$
$878$	−11511.4	+	19938.4i	−0.442473	+	0.766386i
$879$		0			0
$880$	−3140.62	+	1813.24i	−0.120307	+	0.0694593i
$881$	21721.2			0.830655			0.415327	−	0.909672i	$-0.363667\pi$
$881$	21721.2			0.830655			0.415327	+	0.909672i	$0.363667\pi$
$882$		0			0
$883$	−30605.7			−1.16644			−0.583219	−	0.812315i	$-0.698207\pi$
$883$	−30605.7			−1.16644			−0.583219	+	0.812315i	$0.698207\pi$
$884$	3290.81	−	1899.95i	0.125206	−	0.0722876i
$885$		0			0
$886$	−7374.99	+	12773.9i	−0.279647	+	0.484363i
$887$	−2826.08	−	4894.91i	−0.106979	−	0.185293i	0.807566	−	0.589777i	$-0.200784\pi$
$887$	−2826.08	−	4894.91i	−0.106979	−	0.185293i	−0.914545	+	0.404484i	$0.867451\pi$
$888$		0			0
$889$	15875.3	+	372.817i	0.598919	+	0.0140651i
$890$		−	9845.14i		−	0.370798i
$891$		0			0
$892$	1661.73	+	959.400i	0.0623754	+	0.0360124i
$893$	−8604.89	−	4968.04i	−0.322454	−	0.186169i
$894$		0			0
$895$			23153.0i			0.864715i
$896$	2080.26	−	1136.77i	0.0775630	−	0.0423849i
$897$		0			0
$898$	10388.6	+	17993.7i	0.386051	+	0.668659i
$899$	3225.00	−	5585.86i	0.119644	−	0.207229i
$900$		0			0
$901$	1087.51	−	627.872i	0.0402110	−	0.0232158i
$902$	2286.34			0.0843978
$903$		0			0
$904$	−902.823			−0.0332162
$905$	22922.5	−	13234.3i	0.841954	−	0.486103i
$906$		0			0
$907$	17938.9	−	31071.2i	0.656728	−	1.13749i	−0.324729	−	0.945807i	$-0.605273\pi$
$907$	17938.9	−	31071.2i	0.656728	−	1.13749i	0.981458	−	0.191680i	$-0.0613935\pi$
$908$	8218.77	+	14235.3i	0.300385	+	0.520282i
$909$		0			0
$910$	8715.29	−	14308.6i	0.317482	−	0.521238i
$911$			45908.2i			1.66960i	0.550554	+	0.834800i	$0.314417\pi$
$911$			45908.2i			1.66960i	−0.550554	+	0.834800i	$0.685583\pi$
$912$		0			0
$913$	39010.6	+	22522.8i	1.41409	+	0.816424i
$914$	−4381.06	−	2529.40i	−0.158548	−	0.0915375i
$915$		0			0
$916$			9078.12i			0.327456i
$917$	6357.04	+	11633.2i	0.228929	+	0.418933i
$918$		0			0
$919$	21716.4	+	37613.9i	0.779498	+	1.35013i	0.932231	+	0.361862i	$0.117859\pi$
$919$	21716.4	+	37613.9i	0.779498	+	1.35013i	−0.152734	+	0.988267i	$0.548808\pi$
$920$	−5519.34	+	9559.78i	−0.197791	+	0.342583i
$921$		0			0
$922$	33575.7	−	19385.0i	1.19930	−	0.692418i
$923$	−61490.7			−2.19284
$924$		0			0
$925$	−15749.8			−0.559837
$926$	12794.4	−	7386.83i	0.454048	−	0.262145i
$927$		0			0
$928$	3636.86	−	6299.23i	0.128648	−	0.222826i
$929$	−19732.4	−	34177.5i	−0.696877	−	1.20703i	−0.969544	−	0.244917i	$-0.921239\pi$
$929$	−19732.4	−	34177.5i	−0.696877	−	1.20703i	0.272667	−	0.962108i	$-0.412094\pi$
$930$		0			0
$931$	−15810.0	+	8163.46i	−0.556556	+	0.287376i
$932$		−	1348.89i		−	0.0474081i
$933$		0			0
$934$	−20852.6	−	12039.2i	−0.730532	−	0.421773i
$935$	−2829.11	−	1633.39i	−0.0989537	−	0.0571309i
$936$		0			0
$937$			28000.6i			0.976242i	0.872776	+	0.488121i	$0.162318\pi$
$937$			28000.6i			0.976242i	−0.872776	+	0.488121i	$0.837682\pi$
$938$	−549.189	+	23385.5i	−0.0191169	+	0.814035i
$939$		0			0
$940$	2628.84	+	4553.29i	0.0912163	+	0.157991i
$941$	−22909.8	+	39680.9i	−0.793664	+	1.37467i	0.130019	+	0.991511i	$0.458496\pi$
$941$	−22909.8	+	39680.9i	−0.793664	+	1.37467i	−0.923684	+	0.383156i	$0.874837\pi$
$942$		0			0
$943$	6027.04	−	3479.72i	0.208131	−	0.120164i
$944$	9358.36			0.322658
$945$		0			0
$946$	31908.6			1.09666
$947$	21397.3	−	12353.7i	0.734234	−	0.423910i	−0.0857353	−	0.996318i	$-0.527324\pi$
$947$	21397.3	−	12353.7i	0.734234	−	0.423910i	0.819969	+	0.572408i	$0.193991\pi$
$948$		0			0
$949$	−7281.65	+	12612.2i	−0.249075	+	0.431411i
$950$	4041.42	+	6999.95i	0.138022	+	0.239062i
$951$		0			0
$952$	1823.78	+	1110.85i	0.0620895	+	0.0378183i
$953$		−	48776.9i		−	1.65796i	−0.559275	−	0.828982i	$-0.688920\pi$
$953$		−	48776.9i		−	1.65796i	0.559275	−	0.828982i	$-0.311080\pi$
$954$		0			0
$955$	−12692.3	−	7327.88i	−0.430065	−	0.248298i
$956$	3878.08	+	2239.01i	0.131199	+	0.0757476i
$957$		0			0
$958$		−	29879.7i		−	1.00769i
$959$	2099.85	+	1279.00i	0.0707066	+	0.0430669i
$960$		0			0
$961$	−14492.9	−	25102.4i	−0.486486	−	0.842618i
$962$	13324.8	−	23079.2i	0.446577	−	0.773495i
$963$		0			0
$964$	15115.7	−	8727.04i	0.505024	−	0.291576i
$965$	7771.47			0.259246
$966$		0			0
$967$	1312.29			0.0436406			0.0218203	−	0.999762i	$-0.493054\pi$
$967$	1312.29			0.0436406			0.0218203	+	0.999762i	$0.493054\pi$
$968$	−1663.76	+	960.574i	−0.0552432	+	0.0318947i
$969$		0			0
$970$	68.0013	−	117.782i	0.00225092	−	0.00389871i
$971$	20988.7	+	36353.5i	0.693676	+	1.20148i	0.970625	+	0.240597i	$0.0773433\pi$
$971$	20988.7	+	36353.5i	0.693676	+	1.20148i	−0.276949	+	0.960885i	$0.589323\pi$
$972$		0			0
$973$	−1264.26	+	53834.6i	−0.0416550	+	1.77375i
$974$		−	10583.2i		−	0.348161i
$975$		0			0
$976$	−4859.40	−	2805.58i	−0.159371	−	0.0920127i
$977$	−48936.5	−	28253.5i	−1.60247	−	0.925188i	−0.990991	−	0.133931i	$-0.957240\pi$
$977$	−48936.5	−	28253.5i	−1.60247	−	0.925188i	−0.611483	−	0.791258i	$-0.709427\pi$
$978$		0			0
$979$			23691.6i			0.773430i
$980$	9404.94	+	441.978i	0.306561	+	0.0144066i
$981$		0			0
$982$	−14095.0	−	24413.3i	−0.458035	−	0.793340i
$983$	11084.9	−	19199.5i	0.359667	−	0.622961i	−0.628238	−	0.778021i	$-0.716224\pi$
$983$	11084.9	−	19199.5i	0.359667	−	0.622961i	0.987905	+	0.155060i	$0.0495571\pi$
$984$		0			0
$985$	11724.6	−	6769.21i	0.379267	−	0.218970i
$986$	6552.26			0.211629
$987$		0			0
$988$	−13676.6			−0.440396
$989$	84114.4	−	48563.5i	2.70443	−	1.56140i
$990$		0			0
$991$	17799.4	−	30829.4i	0.570550	−	0.988222i	−0.425959	−	0.904742i	$-0.640063\pi$
$991$	17799.4	−	30829.4i	0.570550	−	0.988222i	0.996509	−	0.0834796i	$-0.0266033\pi$
$992$	454.018	+	786.382i	0.0145313	+	0.0251690i
$993$		0			0
$994$	−16570.8	−	30324.1i	−0.528767	−	0.967629i
$995$			7974.54i			0.254080i
$996$		0			0
$997$	−16868.4	−	9738.99i	−0.535836	−	0.309365i	0.207554	−	0.978224i	$-0.433450\pi$
$997$	−16868.4	−	9738.99i	−0.535836	−	0.309365i	−0.743390	+	0.668859i	$0.766783\pi$
$998$	−14906.8	−	8606.45i	−0.472812	−	0.272978i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.k.d.269.9	yes	20
3.2	odd	2	inner	378.4.k.d.269.2	yes	20
7.5	odd	6	inner	378.4.k.d.215.2	✓	20
21.5	even	6	inner	378.4.k.d.215.9	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.k.d.215.2	✓	20	7.5	odd	6	inner
378.4.k.d.215.9	yes	20	21.5	even	6	inner
378.4.k.d.269.2	yes	20	3.2	odd	2	inner
378.4.k.d.269.9	yes	20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(3.43124 + 5.94308i) q^{5} +(-18.5152 - 0.434813i) q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(3.43124 + 5.94308i) q^{5} +(-18.5152 - 0.434813i) q^{7} -8.00000i q^{8} +(11.8862 + 6.86248i) q^{10} +(-28.6032 - 16.5141i) q^{11} +65.9110i q^{13} +(-32.5040 + 17.7620i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(7.20650 - 12.4820i) q^{17} +(-44.9254 + 25.9377i) q^{19} +27.4499 q^{20} -66.0562 q^{22} +(-174.131 + 100.535i) q^{23} +(38.9532 - 67.4689i) q^{25} +(65.9110 + 114.161i) q^{26} +(-38.5365 + 63.2688i) q^{28} +227.304i q^{29} +(-24.5744 - 14.1881i) q^{31} +(-27.7128 - 16.0000i) q^{32} -28.8260i q^{34} +(-60.9458 - 111.529i) q^{35} +(-101.081 - 175.078i) q^{37} +(-51.8754 + 89.8508i) q^{38} +(47.5446 - 27.4499i) q^{40} -34.6120 q^{41} -483.052 q^{43} +(-114.413 + 66.0562i) q^{44} +(-201.070 + 348.263i) q^{46} +(95.7687 + 165.876i) q^{47} +(342.622 + 16.1012i) q^{49} -155.813i q^{50} +(228.322 + 131.822i) q^{52} +(75.4532 + 43.5629i) q^{53} -226.655i q^{55} +(-3.47850 + 148.121i) q^{56} +(227.304 + 393.702i) q^{58} +(-292.449 + 506.536i) q^{59} +(303.713 - 175.349i) q^{61} -56.7523 q^{62} -64.0000 q^{64} +(-391.714 + 226.156i) q^{65} +(315.762 - 546.916i) q^{67} +(-28.8260 - 49.9281i) q^{68} +(-217.090 - 132.228i) q^{70} +932.935i q^{71} +(191.352 + 110.477i) q^{73} +(-350.156 - 202.163i) q^{74} +207.501i q^{76} +(522.412 + 318.197i) q^{77} +(228.057 + 395.007i) q^{79} +(54.8998 - 95.0893i) q^{80} +(-59.9498 + 34.6120i) q^{82} -1363.85 q^{83} +98.9088 q^{85} +(-836.670 + 483.052i) q^{86} +(-132.112 + 228.825i) q^{88} +(-358.658 - 621.215i) q^{89} +(28.6589 - 1220.35i) q^{91} +804.279i q^{92} +(331.752 + 191.537i) q^{94} +(-308.299 - 177.997i) q^{95} -9.90916i q^{97} +(609.540 - 314.734i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 40 q^{4} + 4 q^{7}+O(q^{10}) \) 20 * q + 40 * q^4 + 4 * q^7	\( 20 q + 40 q^{4} + 4 q^{7} - 24 q^{10} - 160 q^{16} + 408 q^{19} + 240 q^{22} - 646 q^{25} + 56 q^{28} - 102 q^{31} + 194 q^{37} - 96 q^{40} - 2332 q^{43} - 624 q^{46} + 2840 q^{49} - 648 q^{52} + 96 q^{58} + 1878 q^{61} - 1280 q^{64} - 386 q^{67} + 3672 q^{70} + 1788 q^{73} + 814 q^{79} - 672 q^{82} - 4560 q^{85} + 480 q^{88} + 2724 q^{91} - 1536 q^{94}+O(q^{100}) \) 20 * q + 40 * q^4 + 4 * q^7 - 24 * q^10 - 160 * q^16 + 408 * q^19 + 240 * q^22 - 646 * q^25 + 56 * q^28 - 102 * q^31 + 194 * q^37 - 96 * q^40 - 2332 * q^43 - 624 * q^46 + 2840 * q^49 - 648 * q^52 + 96 * q^58 + 1878 * q^61 - 1280 * q^64 - 386 * q^67 + 3672 * q^70 + 1788 * q^73 + 814 * q^79 - 672 * q^82 - 4560 * q^85 + 480 * q^88 + 2724 * q^91 - 1536 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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