LMFDB - Embedded newform 6300.2.v.e.5993.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6300,2,Mod(1457,6300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6300, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6300.1457"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.v (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,-8,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 6x^{10} - 24x^{9} + 18x^{8} + 40x^{7} - 82x^{6} + 12x^{5} + 228x^{4} - 284x^{3} + 124x^{2} - 16x + 2 $ x^12 + 6x^10 - 24x^9 + 18x^8 + 40x^7 - 82x^6 + 12x^5 + 228x^4 - 284x^3 + 124x^2 - 16x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5993.5
Root			$0.0572576 - 0.138232i$ of defining polynomial
Character	$\chi$	$=$	6300.5993
Dual form			6300.2.v.e.1457.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{7} -2.16195i q^{11} +(0.457310 + 0.457310i) q^{13} +(0.704639 + 0.704639i) q^{17} +2.08189i q^{19} +(-0.819154 + 0.819154i) q^{23} -5.75906 q^{29} -4.16978 q^{31} +(7.23073 - 7.23073i) q^{37} -4.82963i q^{41} +(3.98689 + 3.98689i) q^{43} +(3.24035 + 3.24035i) q^{47} -1.00000i q^{49} +(7.01566 - 7.01566i) q^{53} +4.69577 q^{59} +0.627320 q^{61} +(8.71058 - 8.71058i) q^{67} -10.3048i q^{71} +(0.228279 + 0.228279i) q^{73} +(-1.52873 - 1.52873i) q^{77} +2.49466i q^{79} +(2.75267 - 2.75267i) q^{83} -15.2684 q^{89} +0.646733 q^{91} +(-5.18100 + 5.18100i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{13} - 8 q^{17} + 8 q^{23} - 32 q^{29} + 16 q^{31} - 4 q^{37} - 8 q^{43} - 8 q^{47} + 8 q^{53} - 16 q^{59} + 12 q^{73} + 56 q^{83} - 72 q^{89} - 20 q^{97}+O(q^{100}) $ 12 * q + 12 * q^13 - 8 * q^17 + 8 * q^23 - 32 * q^29 + 16 * q^31 - 4 * q^37 - 8 * q^43 - 8 * q^47 + 8 * q^53 - 16 * q^59 + 12 * q^73 + 56 * q^83 - 72 * q^89 - 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times$.

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$-1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.707107	−	0.707107i	0.267261	−	0.267261i
$7$	0.707107	−	0.707107i	0.267261	−	0.267261i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.16195i		−	0.651852i	−0.945395	−	0.325926i	$-0.894324\pi$
$11$		−	2.16195i		−	0.651852i	0.945395	−	0.325926i	$-0.105676\pi$
$12$		0			0
$13$	0.457310	+	0.457310i	0.126835	+	0.126835i	0.767675	−	0.640840i	$-0.221414\pi$
$13$	0.457310	+	0.457310i	0.126835	+	0.126835i	−0.640840	+	0.767675i	$0.721414\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.704639	+	0.704639i	0.170900	+	0.170900i	0.787375	−	0.616475i	$-0.211440\pi$
$17$	0.704639	+	0.704639i	0.170900	+	0.170900i	−0.616475	+	0.787375i	$0.711440\pi$
$18$		0			0
$19$			2.08189i			0.477619i	0.971066	+	0.238809i	$0.0767571\pi$
$19$			2.08189i			0.477619i	−0.971066	+	0.238809i	$0.923243\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.819154	+	0.819154i	−0.170805	+	0.170805i	−0.787333	−	0.616528i	$-0.788539\pi$
$23$	−0.819154	+	0.819154i	−0.170805	+	0.170805i	0.616528	+	0.787333i	$0.288539\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.75906			−1.06943			−0.534715	−	0.845033i	$-0.679581\pi$
$29$	−5.75906			−1.06943			−0.534715	+	0.845033i	$0.679581\pi$
$30$		0			0
$31$	−4.16978			−0.748914			−0.374457	−	0.927244i	$-0.622171\pi$
$31$	−4.16978			−0.748914			−0.374457	+	0.927244i	$0.622171\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.23073	−	7.23073i	1.18872	−	1.18872i	0.211304	−	0.977420i	$-0.432229\pi$
$37$	7.23073	−	7.23073i	1.18872	−	1.18872i	0.977420	−	0.211304i	$-0.0677709\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	4.82963i		−	0.754261i	−0.926160	−	0.377130i	$-0.876911\pi$
$41$		−	4.82963i		−	0.754261i	0.926160	−	0.377130i	$-0.123089\pi$
$42$		0			0
$43$	3.98689	+	3.98689i	0.607994	+	0.607994i	0.942422	−	0.334427i	$-0.108543\pi$
$43$	3.98689	+	3.98689i	0.607994	+	0.607994i	−0.334427	+	0.942422i	$0.608543\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.24035	+	3.24035i	0.472654	+	0.472654i	0.902772	−	0.430119i	$-0.141528\pi$
$47$	3.24035	+	3.24035i	0.472654	+	0.472654i	−0.430119	+	0.902772i	$0.641528\pi$
$48$		0			0
$49$		−	1.00000i		−	0.142857i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.01566	−	7.01566i	0.963675	−	0.963675i	−0.0356883	−	0.999363i	$-0.511362\pi$
$53$	7.01566	−	7.01566i	0.963675	−	0.963675i	0.999363	+	0.0356883i	$0.0113623\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.69577			0.611337			0.305668	−	0.952138i	$-0.401120\pi$
$59$	4.69577			0.611337			0.305668	+	0.952138i	$0.401120\pi$
$60$		0			0
$61$	0.627320			0.0803201			0.0401601	−	0.999193i	$-0.487213\pi$
$61$	0.627320			0.0803201			0.0401601	+	0.999193i	$0.487213\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	8.71058	−	8.71058i	1.06417	−	1.06417i	0.0663712	−	0.997795i	$-0.478858\pi$
$67$	8.71058	−	8.71058i	1.06417	−	1.06417i	0.997795	−	0.0663712i	$-0.0211421\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.3048i		−	1.22295i	−0.791263	−	0.611477i	$-0.790576\pi$
$71$		−	10.3048i		−	1.22295i	0.791263	−	0.611477i	$-0.209424\pi$
$72$		0			0
$73$	0.228279	+	0.228279i	0.0267181	+	0.0267181i	0.720340	−	0.693622i	$-0.243986\pi$
$73$	0.228279	+	0.228279i	0.0267181	+	0.0267181i	−0.693622	+	0.720340i	$0.743986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.52873	−	1.52873i	−0.174215	−	0.174215i
$78$		0			0
$79$			2.49466i			0.280671i	0.990104	+	0.140336i	$0.0448181\pi$
$79$			2.49466i			0.280671i	−0.990104	+	0.140336i	$0.955182\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.75267	−	2.75267i	0.302145	−	0.302145i	−0.539708	−	0.841853i	$-0.681465\pi$
$83$	2.75267	−	2.75267i	0.302145	−	0.302145i	0.841853	+	0.539708i	$0.181465\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−15.2684			−1.61845			−0.809226	−	0.587498i	$-0.800113\pi$
$89$	−15.2684			−1.61845			−0.809226	+	0.587498i	$0.800113\pi$
$90$		0			0
$91$	0.646733			0.0677961
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−5.18100	+	5.18100i	−0.526051	+	0.526051i	−0.919392	−	0.393342i	$-0.871319\pi$
$97$	−5.18100	+	5.18100i	−0.526051	+	0.526051i	0.393342	+	0.919392i	$0.371319\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			0.107633i			0.0107099i	0.999986	+	0.00535494i	$0.00170454\pi$
$101$			0.107633i			0.0107099i	−0.999986	+	0.00535494i	$0.998295\pi$
$102$		0			0
$103$	3.11840	+	3.11840i	0.307266	+	0.307266i	0.843848	−	0.536582i	$-0.180285\pi$
$103$	3.11840	+	3.11840i	0.307266	+	0.307266i	−0.536582	+	0.843848i	$0.680285\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.32368	−	3.32368i	−0.321313	−	0.321313i	0.527958	−	0.849271i	$-0.322958\pi$
$107$	−3.32368	−	3.32368i	−0.321313	−	0.321313i	−0.849271	+	0.527958i	$0.822958\pi$
$108$		0			0
$109$			16.5587i			1.58604i	0.609197	+	0.793019i	$0.291492\pi$
$109$			16.5587i			1.58604i	−0.609197	+	0.793019i	$0.708508\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.60144	+	1.60144i	−0.150651	+	0.150651i	−0.778409	−	0.627758i	$-0.783973\pi$
$113$	−1.60144	+	1.60144i	−0.150651	+	0.150651i	0.627758	+	0.778409i	$0.283973\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.996510			0.0913500
$120$		0			0
$121$	6.32598			0.575089
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−2.55293	+	2.55293i	−0.226536	+	0.226536i	−0.811244	−	0.584708i	$-0.801209\pi$
$127$	−2.55293	+	2.55293i	−0.226536	+	0.226536i	0.584708	+	0.811244i	$0.301209\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	20.0896i		−	1.75524i	−0.479361	−	0.877618i	$-0.659132\pi$
$131$		−	20.0896i		−	1.75524i	0.479361	−	0.877618i	$-0.340868\pi$
$132$		0			0
$133$	1.47212	+	1.47212i	0.127649	+	0.127649i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.78161	+	3.78161i	0.323084	+	0.323084i	0.849949	−	0.526865i	$-0.176633\pi$
$137$	3.78161	+	3.78161i	0.323084	+	0.323084i	−0.526865	+	0.849949i	$0.676633\pi$
$138$		0			0
$139$		−	18.6382i		−	1.58087i	−0.612544	−	0.790436i	$-0.709854\pi$
$139$		−	18.6382i		−	1.58087i	0.612544	−	0.790436i	$-0.290146\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.988680	−	0.988680i	0.0826776	−	0.0826776i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.91874			−0.648729			−0.324364	−	0.945932i	$-0.605150\pi$
$149$	−7.91874			−0.648729			−0.324364	+	0.945932i	$0.605150\pi$
$150$		0			0
$151$	2.97265			0.241911			0.120956	−	0.992658i	$-0.461404\pi$
$151$	2.97265			0.241911			0.120956	+	0.992658i	$0.461404\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2.29162	+	2.29162i	−0.182892	+	0.182892i	−0.792615	−	0.609723i	$-0.791281\pi$
$157$	−2.29162	+	2.29162i	−0.182892	+	0.182892i	0.609723	+	0.792615i	$0.291281\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			1.15846i			0.0912994i
$162$		0			0
$163$	−1.49547	−	1.49547i	−0.117134	−	0.117134i	0.646110	−	0.763244i	$-0.276395\pi$
$163$	−1.49547	−	1.49547i	−0.117134	−	0.117134i	−0.763244	+	0.646110i	$0.776395\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.95353	−	1.95353i	−0.151169	−	0.151169i	0.627471	−	0.778640i	$-0.284090\pi$
$167$	−1.95353	−	1.95353i	−0.151169	−	0.151169i	−0.778640	+	0.627471i	$0.784090\pi$
$168$		0			0
$169$		−	12.5817i		−	0.967826i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.54559	−	8.54559i	0.649709	−	0.649709i	−0.303214	−	0.952923i	$-0.598060\pi$
$173$	8.54559	−	8.54559i	0.649709	−	0.649709i	0.952923	+	0.303214i	$0.0980596\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.6139			−1.16704			−0.583519	−	0.812099i	$-0.698325\pi$
$179$	−15.6139			−1.16704			−0.583519	+	0.812099i	$0.698325\pi$
$180$		0			0
$181$	1.76020			0.130835			0.0654174	−	0.997858i	$-0.479162\pi$
$181$	1.76020			0.130835			0.0654174	+	0.997858i	$0.479162\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1.52339	−	1.52339i	0.111402	−	0.111402i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	10.1718i		−	0.736007i	−0.929824	−	0.368004i	$-0.880041\pi$
$191$		−	10.1718i		−	0.736007i	0.929824	−	0.368004i	$-0.119959\pi$
$192$		0			0
$193$	−11.4344	−	11.4344i	−0.823068	−	0.823068i	0.163479	−	0.986547i	$-0.447728\pi$
$193$	−11.4344	−	11.4344i	−0.823068	−	0.823068i	−0.986547	+	0.163479i	$0.947728\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.2591	−	10.2591i	−0.730931	−	0.730931i	0.239873	−	0.970804i	$-0.422894\pi$
$197$	−10.2591	−	10.2591i	−0.730931	−	0.730931i	−0.970804	+	0.239873i	$0.922894\pi$
$198$		0			0
$199$		−	21.3619i		−	1.51430i	−0.653238	−	0.757152i	$-0.726590\pi$
$199$		−	21.3619i		−	1.51430i	0.653238	−	0.757152i	$-0.273410\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.07227	+	4.07227i	−0.285817	+	0.285817i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.50094			0.311337
$210$		0			0
$211$	13.3621			0.919885			0.459943	−	0.887949i	$-0.347870\pi$
$211$	13.3621			0.919885			0.459943	+	0.887949i	$0.347870\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.94848	+	2.94848i	−0.200156	+	0.200156i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			0.644477i			0.0433522i
$222$		0			0
$223$	17.8563	+	17.8563i	1.19574	+	1.19574i	0.975429	+	0.220314i	$0.0707083\pi$
$223$	17.8563	+	17.8563i	1.19574	+	1.19574i	0.220314	+	0.975429i	$0.429292\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.7375	−	10.7375i	−0.712674	−	0.712674i	0.254420	−	0.967094i	$-0.418115\pi$
$227$	−10.7375	−	10.7375i	−0.712674	−	0.712674i	−0.967094	+	0.254420i	$0.918115\pi$
$228$		0			0
$229$			1.70765i			0.112845i	0.998407	+	0.0564224i	$0.0179693\pi$
$229$			1.70765i			0.112845i	−0.998407	+	0.0564224i	$0.982031\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.2261	−	11.2261i	0.735449	−	0.735449i	−0.236245	−	0.971694i	$-0.575917\pi$
$233$	11.2261	−	11.2261i	0.735449	−	0.735449i	0.971694	+	0.236245i	$0.0759167\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	17.2424			1.11532			0.557658	−	0.830071i	$-0.311700\pi$
$239$	17.2424			1.11532			0.557658	+	0.830071i	$0.311700\pi$
$240$		0			0
$241$	29.6047			1.90701			0.953503	−	0.301382i	$-0.0974480\pi$
$241$	29.6047			1.90701			0.953503	+	0.301382i	$0.0974480\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−0.952069	+	0.952069i	−0.0605787	+	0.0605787i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			15.1380i			0.955500i	0.878496	+	0.477750i	$0.158548\pi$
$251$			15.1380i			0.955500i	−0.878496	+	0.477750i	$0.841452\pi$
$252$		0			0
$253$	1.77097	+	1.77097i	0.111340	+	0.111340i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−18.0010	−	18.0010i	−1.12287	−	1.12287i	−0.991308	−	0.131564i	$-0.958000\pi$
$257$	−18.0010	−	18.0010i	−1.12287	−	1.12287i	−0.131564	−	0.991308i	$-0.542000\pi$
$258$		0			0
$259$		−	10.2258i		−	0.635400i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	15.2668	−	15.2668i	0.941390	−	0.941390i	−0.0569851	−	0.998375i	$-0.518149\pi$
$263$	15.2668	−	15.2668i	0.941390	−	0.941390i	0.998375	+	0.0569851i	$0.0181487\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.38784			−0.328502			−0.164251	−	0.986419i	$-0.552521\pi$
$269$	−5.38784			−0.328502			−0.164251	+	0.986419i	$0.552521\pi$
$270$		0			0
$271$	4.37925			0.266021			0.133010	−	0.991115i	$-0.457536\pi$
$271$	4.37925			0.266021			0.133010	+	0.991115i	$0.457536\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	8.72854	−	8.72854i	0.524447	−	0.524447i	−0.394464	−	0.918911i	$-0.629070\pi$
$277$	8.72854	−	8.72854i	0.524447	−	0.524447i	0.918911	+	0.394464i	$0.129070\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			1.73523i			0.103515i	0.998660	+	0.0517575i	$0.0164823\pi$
$281$			1.73523i			0.103515i	−0.998660	+	0.0517575i	$0.983518\pi$
$282$		0			0
$283$	21.4946	+	21.4946i	1.27772	+	1.27772i	0.941941	+	0.335778i	$0.108999\pi$
$283$	21.4946	+	21.4946i	1.27772	+	1.27772i	0.335778	+	0.941941i	$0.391001\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.41506	−	3.41506i	−0.201585	−	0.201585i
$288$		0			0
$289$		−	16.0070i		−	0.941586i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7.87213	+	7.87213i	−0.459895	+	0.459895i	−0.898621	−	0.438726i	$-0.855430\pi$
$293$	−7.87213	+	7.87213i	−0.459895	+	0.459895i	0.438726	+	0.898621i	$0.355430\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.749214			−0.0433282
$300$		0			0
$301$	5.63831			0.324987
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−22.4429	+	22.4429i	−1.28088	+	1.28088i	−0.340719	+	0.940165i	$0.610671\pi$
$307$	−22.4429	+	22.4429i	−1.28088	+	1.28088i	−0.940165	+	0.340719i	$0.889329\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	23.1376i		−	1.31202i	−0.754754	−	0.656008i	$-0.772244\pi$
$311$		−	23.1376i		−	1.31202i	0.754754	−	0.656008i	$-0.227756\pi$
$312$		0			0
$313$	−12.3533	−	12.3533i	−0.698249	−	0.698249i	0.265784	−	0.964033i	$-0.414369\pi$
$313$	−12.3533	−	12.3533i	−0.698249	−	0.698249i	−0.964033	+	0.265784i	$0.914369\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.04030	−	2.04030i	−0.114595	−	0.114595i	0.647484	−	0.762079i	$-0.275821\pi$
$317$	−2.04030	−	2.04030i	−0.114595	−	0.114595i	−0.762079	+	0.647484i	$0.775821\pi$
$318$		0			0
$319$			12.4508i			0.697110i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.46698	+	1.46698i	−0.0816251	+	0.0816251i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.58255			0.252644
$330$		0			0
$331$	28.8778			1.58727			0.793634	−	0.608395i	$-0.208186\pi$
$331$	28.8778			1.58727			0.793634	+	0.608395i	$0.208186\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−4.18958	+	4.18958i	−0.228221	+	0.228221i	−0.811949	−	0.583728i	$-0.801593\pi$
$337$	−4.18958	+	4.18958i	−0.228221	+	0.228221i	0.583728	+	0.811949i	$0.301593\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			9.01485i			0.488181i
$342$		0			0
$343$	−0.707107	−	0.707107i	−0.0381802	−	0.0381802i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	22.9891	+	22.9891i	1.23412	+	1.23412i	0.962368	+	0.271751i	$0.0876027\pi$
$347$	22.9891	+	22.9891i	1.23412	+	1.23412i	0.271751	+	0.962368i	$0.412397\pi$
$348$		0			0
$349$		−	0.0763218i		−	0.00408541i	−0.999998	−	0.00204271i	$-0.999350\pi$
$349$		−	0.0763218i		−	0.00408541i	0.999998	−	0.00204271i	$-0.000650214\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.69583	−	6.69583i	0.356383	−	0.356383i	−0.506095	−	0.862478i	$-0.668911\pi$
$353$	6.69583	−	6.69583i	0.356383	−	0.356383i	0.862478	+	0.506095i	$0.168911\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.2111			−0.644480			−0.322240	−	0.946658i	$-0.604436\pi$
$359$	−12.2111			−0.644480			−0.322240	+	0.946658i	$0.604436\pi$
$360$		0			0
$361$	14.6657			0.771881
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−11.4045	+	11.4045i	−0.595310	+	0.595310i	−0.939061	−	0.343751i	$-0.888303\pi$
$367$	−11.4045	+	11.4045i	−0.595310	+	0.595310i	0.343751	+	0.939061i	$0.388303\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	9.92164i		−	0.515106i
$372$		0			0
$373$	−2.94462	−	2.94462i	−0.152467	−	0.152467i	0.626752	−	0.779219i	$-0.284384\pi$
$373$	−2.94462	−	2.94462i	−0.152467	−	0.152467i	−0.779219	+	0.626752i	$0.784384\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.63367	−	2.63367i	−0.135641	−	0.135641i
$378$		0			0
$379$		−	15.3970i		−	0.790892i	−0.918489	−	0.395446i	$-0.870590\pi$
$379$		−	15.3970i		−	0.790892i	0.918489	−	0.395446i	$-0.129410\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.22336	+	7.22336i	−0.369097	+	0.369097i	−0.867148	−	0.498051i	$-0.834049\pi$
$383$	−7.22336	+	7.22336i	−0.369097	+	0.369097i	0.498051	+	0.867148i	$0.334049\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−32.8823			−1.66720			−0.833599	−	0.552371i	$-0.813723\pi$
$389$	−32.8823			−1.66720			−0.833599	+	0.552371i	$0.813723\pi$
$390$		0			0
$391$	−1.15442			−0.0583814
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	16.4822	−	16.4822i	0.827216	−	0.827216i	−0.159915	−	0.987131i	$-0.551122\pi$
$397$	16.4822	−	16.4822i	0.827216	−	0.827216i	0.987131	+	0.159915i	$0.0511220\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	24.6094i		−	1.22894i	−0.788942	−	0.614468i	$-0.789371\pi$
$401$		−	24.6094i		−	1.22894i	0.788942	−	0.614468i	$-0.210629\pi$
$402$		0			0
$403$	−1.90688	−	1.90688i	−0.0949885	−	0.0949885i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−15.6325	−	15.6325i	−0.774872	−	0.774872i
$408$		0			0
$409$		−	26.8405i		−	1.32718i	−0.748098	−	0.663589i	$-0.769033\pi$
$409$		−	26.8405i		−	1.32718i	0.748098	−	0.663589i	$-0.230967\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.32041	−	3.32041i	0.163387	−	0.163387i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	30.5542			1.49267			0.746336	−	0.665570i	$-0.231811\pi$
$419$	30.5542			1.49267			0.746336	+	0.665570i	$0.231811\pi$
$420$		0			0
$421$	−5.58675			−0.272282			−0.136141	−	0.990690i	$-0.543470\pi$
$421$	−5.58675			−0.272282			−0.136141	+	0.990690i	$0.543470\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.443582	−	0.443582i	0.0214665	−	0.0214665i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	12.9160i		−	0.622141i	−0.950387	−	0.311071i	$-0.899312\pi$
$431$		−	12.9160i		−	0.622141i	0.950387	−	0.311071i	$-0.100688\pi$
$432$		0			0
$433$	22.6837	+	22.6837i	1.09011	+	1.09011i	0.995516	+	0.0945949i	$0.0301556\pi$
$433$	22.6837	+	22.6837i	1.09011	+	1.09011i	0.0945949	+	0.995516i	$0.469844\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.70539	−	1.70539i	−0.0815799	−	0.0815799i
$438$		0			0
$439$			28.3899i			1.35498i	0.735533	+	0.677489i	$0.236932\pi$
$439$			28.3899i			1.35498i	−0.735533	+	0.677489i	$0.763068\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.98110	−	2.98110i	0.141636	−	0.141636i	−0.632733	−	0.774370i	$-0.718067\pi$
$443$	2.98110	−	2.98110i	0.141636	−	0.141636i	0.774370	+	0.632733i	$0.218067\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−14.5517			−0.686735			−0.343367	−	0.939201i	$-0.611568\pi$
$449$	−14.5517			−0.686735			−0.343367	+	0.939201i	$0.611568\pi$
$450$		0			0
$451$	−10.4414			−0.491667
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	14.1881	−	14.1881i	0.663690	−	0.663690i	−0.292558	−	0.956248i	$-0.594506\pi$
$457$	14.1881	−	14.1881i	0.663690	−	0.663690i	0.956248	+	0.292558i	$0.0945063\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	2.48749i		−	0.115854i	−0.998321	−	0.0579270i	$-0.981551\pi$
$461$		−	2.48749i		−	0.115854i	0.998321	−	0.0579270i	$-0.0184491\pi$
$462$		0			0
$463$	2.52717	+	2.52717i	0.117448	+	0.117448i	0.763388	−	0.645940i	$-0.223535\pi$
$463$	2.52717	+	2.52717i	0.117448	+	0.117448i	−0.645940	+	0.763388i	$0.723535\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.34428	−	8.34428i	−0.386127	−	0.386127i	0.487176	−	0.873304i	$-0.338027\pi$
$467$	−8.34428	−	8.34428i	−0.386127	−	0.386127i	−0.873304	+	0.487176i	$0.838027\pi$
$468$		0			0
$469$		−	12.3186i		−	0.568821i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8.61944	−	8.61944i	0.396322	−	0.396322i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	18.5818			0.849026			0.424513	−	0.905422i	$-0.360445\pi$
$479$	18.5818			0.849026			0.424513	+	0.905422i	$0.360445\pi$
$480$		0			0
$481$	6.61336			0.301543
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	9.95319	−	9.95319i	0.451022	−	0.451022i	−0.444672	−	0.895694i	$-0.646680\pi$
$487$	9.95319	−	9.95319i	0.451022	−	0.451022i	0.895694	+	0.444672i	$0.146680\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	21.3648i		−	0.964182i	−0.876121	−	0.482091i	$-0.839878\pi$
$491$		−	21.3648i		−	0.964182i	0.876121	−	0.482091i	$-0.160122\pi$
$492$		0			0
$493$	−4.05806	−	4.05806i	−0.182766	−	0.182766i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7.28658	−	7.28658i	−0.326848	−	0.326848i
$498$		0			0
$499$			12.7852i			0.572342i	0.958179	+	0.286171i	$0.0923826\pi$
$499$			12.7852i			0.572342i	−0.958179	+	0.286171i	$0.907617\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	30.6628	−	30.6628i	1.36718	−	1.36718i	0.502757	−	0.864428i	$-0.332319\pi$
$503$	30.6628	−	30.6628i	1.36718	−	1.36718i	0.864428	−	0.502757i	$-0.167681\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−15.0914			−0.668913			−0.334456	−	0.942411i	$-0.608553\pi$
$509$	−15.0914			−0.668913			−0.334456	+	0.942411i	$0.608553\pi$
$510$		0			0
$511$	0.322836			0.0142814
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	7.00547	−	7.00547i	0.308100	−	0.308100i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			21.7185i			0.951506i	0.879579	+	0.475753i	$0.157824\pi$
$521$			21.7185i			0.951506i	−0.879579	+	0.475753i	$0.842176\pi$
$522$		0			0
$523$	4.46406	+	4.46406i	0.195200	+	0.195200i	0.797938	−	0.602739i	$-0.205924\pi$
$523$	4.46406	+	4.46406i	0.195200	+	0.195200i	−0.602739	+	0.797938i	$0.705924\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.93819	−	2.93819i	−0.127990	−	0.127990i
$528$		0			0
$529$			21.6580i			0.941651i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.20863	−	2.20863i	0.0956666	−	0.0956666i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.16195			−0.0931217
$540$		0			0
$541$	−34.5916			−1.48721			−0.743604	−	0.668621i	$-0.766885\pi$
$541$	−34.5916			−1.48721			−0.743604	+	0.668621i	$0.766885\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	9.60954	−	9.60954i	0.410874	−	0.410874i	−0.471169	−	0.882043i	$-0.656168\pi$
$547$	9.60954	−	9.60954i	0.410874	−	0.410874i	0.882043	+	0.471169i	$0.156168\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	11.9897i		−	0.510779i
$552$		0			0
$553$	1.76399	+	1.76399i	0.0750125	+	0.0750125i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	20.2677	+	20.2677i	0.858772	+	0.858772i	0.991193	−	0.132422i	$-0.0422753\pi$
$557$	20.2677	+	20.2677i	0.858772	+	0.858772i	−0.132422	+	0.991193i	$0.542275\pi$
$558$		0			0
$559$			3.64648i			0.154230i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.00112	+	5.00112i	−0.210772	+	0.210772i	−0.804595	−	0.593823i	$-0.797618\pi$
$563$	−5.00112	+	5.00112i	−0.210772	+	0.210772i	0.593823	+	0.804595i	$0.297618\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−28.4657			−1.19334			−0.596671	−	0.802486i	$-0.703510\pi$
$569$	−28.4657			−1.19334			−0.596671	+	0.802486i	$0.703510\pi$
$570$		0			0
$571$	−22.3373			−0.934786			−0.467393	−	0.884050i	$-0.654807\pi$
$571$	−22.3373			−0.934786			−0.467393	+	0.884050i	$0.654807\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−12.6382	+	12.6382i	−0.526134	+	0.526134i	−0.919417	−	0.393284i	$-0.871339\pi$
$577$	−12.6382	+	12.6382i	−0.526134	+	0.526134i	0.393284	+	0.919417i	$0.371339\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	3.89286i		−	0.161503i
$582$		0			0
$583$	−15.1675	−	15.1675i	−0.628173	−	0.628173i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	22.8827	+	22.8827i	0.944469	+	0.944469i	0.998537	−	0.0540682i	$-0.0172188\pi$
$587$	22.8827	+	22.8827i	0.944469	+	0.944469i	−0.0540682	+	0.998537i	$0.517219\pi$
$588$		0			0
$589$		−	8.68103i		−	0.357695i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	10.2439	−	10.2439i	0.420665	−	0.420665i	−0.464768	−	0.885433i	$-0.653862\pi$
$593$	10.2439	−	10.2439i	0.420665	−	0.420665i	0.885433	+	0.464768i	$0.153862\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.16453			−0.129299			−0.0646496	−	0.997908i	$-0.520593\pi$
$599$	−3.16453			−0.129299			−0.0646496	+	0.997908i	$0.520593\pi$
$600$		0			0
$601$	−11.3100			−0.461346			−0.230673	−	0.973031i	$-0.574093\pi$
$601$	−11.3100			−0.461346			−0.230673	+	0.973031i	$0.574093\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	27.1529	−	27.1529i	1.10210	−	1.10210i	0.107946	−	0.994157i	$-0.465573\pi$
$607$	27.1529	−	27.1529i	1.10210	−	1.10210i	0.994157	−	0.107946i	$-0.0344274\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			2.96369i			0.119898i
$612$		0			0
$613$	−4.12198	−	4.12198i	−0.166485	−	0.166485i	0.618947	−	0.785433i	$-0.287559\pi$
$613$	−4.12198	−	4.12198i	−0.166485	−	0.166485i	−0.785433	+	0.618947i	$0.787559\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−26.6408	−	26.6408i	−1.07252	−	1.07252i	−0.997156	−	0.0753620i	$-0.975989\pi$
$617$	−26.6408	−	26.6408i	−1.07252	−	1.07252i	−0.0753620	−	0.997156i	$-0.524011\pi$
$618$		0			0
$619$			38.8564i			1.56177i	0.624674	+	0.780886i	$0.285232\pi$
$619$			38.8564i			1.56177i	−0.624674	+	0.780886i	$0.714768\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.7964	+	10.7964i	−0.432549	+	0.432549i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	10.1901			0.406306
$630$		0			0
$631$	15.7400			0.626601			0.313300	−	0.949654i	$-0.398565\pi$
$631$	15.7400			0.626601			0.313300	+	0.949654i	$0.398565\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.457310	−	0.457310i	0.0181193	−	0.0181193i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			10.5538i			0.416852i	0.978038	+	0.208426i	$0.0668340\pi$
$641$			10.5538i			0.416852i	−0.978038	+	0.208426i	$0.933166\pi$
$642$		0			0
$643$	−9.80987	−	9.80987i	−0.386863	−	0.386863i	0.486704	−	0.873567i	$-0.338199\pi$
$643$	−9.80987	−	9.80987i	−0.386863	−	0.386863i	−0.873567	+	0.486704i	$0.838199\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	19.3542	+	19.3542i	0.760892	+	0.760892i	0.976484	−	0.215591i	$-0.0691679\pi$
$647$	19.3542	+	19.3542i	0.760892	+	0.760892i	−0.215591	+	0.976484i	$0.569168\pi$
$648$		0			0
$649$		−	10.1520i		−	0.398501i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−34.0108	+	34.0108i	−1.33095	+	1.33095i	−0.426420	+	0.904525i	$0.640225\pi$
$653$	−34.0108	+	34.0108i	−1.33095	+	1.33095i	−0.904525	+	0.426420i	$0.859775\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15.8401			−0.617042			−0.308521	−	0.951218i	$-0.599834\pi$
$659$	−15.8401			−0.617042			−0.308521	+	0.951218i	$0.599834\pi$
$660$		0			0
$661$	−34.0035			−1.32258			−0.661291	−	0.750129i	$-0.729991\pi$
$661$	−34.0035			−1.32258			−0.661291	+	0.750129i	$0.729991\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	4.71756	−	4.71756i	0.182664	−	0.182664i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1.35623i		−	0.0523568i
$672$		0			0
$673$	11.9081	+	11.9081i	0.459025	+	0.459025i	0.898335	−	0.439310i	$-0.144777\pi$
$673$	11.9081	+	11.9081i	0.459025	+	0.459025i	−0.439310	+	0.898335i	$0.644777\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	27.4842	+	27.4842i	1.05630	+	1.05630i	0.998317	+	0.0579868i	$0.0184681\pi$
$677$	27.4842	+	27.4842i	1.05630	+	1.05630i	0.0579868	+	0.998317i	$0.481532\pi$
$678$		0			0
$679$			7.32704i			0.281186i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.9803	−	11.9803i	0.458413	−	0.458413i	−0.439721	−	0.898134i	$-0.644923\pi$
$683$	11.9803	−	11.9803i	0.458413	−	0.458413i	0.898134	+	0.439721i	$0.144923\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.41666			0.244455
$690$		0			0
$691$	6.79041			0.258319			0.129160	−	0.991624i	$-0.458772\pi$
$691$	6.79041			0.258319			0.129160	+	0.991624i	$0.458772\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	3.40314	−	3.40314i	0.128903	−	0.128903i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	41.4657i		−	1.56614i	−0.621935	−	0.783069i	$-0.713653\pi$
$701$		−	41.4657i		−	1.56614i	0.621935	−	0.783069i	$-0.286347\pi$
$702$		0			0
$703$	15.0536	+	15.0536i	0.567757	+	0.567757i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.0761079	+	0.0761079i	0.00286233	+	0.00286233i
$708$		0			0
$709$			10.5840i			0.397491i	0.980051	+	0.198745i	$0.0636867\pi$
$709$			10.5840i			0.397491i	−0.980051	+	0.198745i	$0.936313\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.41569	−	3.41569i	0.127919	−	0.127919i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−14.0588			−0.524304			−0.262152	−	0.965027i	$-0.584432\pi$
$719$	−14.0588			−0.524304			−0.262152	+	0.965027i	$0.584432\pi$
$720$		0			0
$721$	4.41009			0.164240
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−32.1631	+	32.1631i	−1.19286	+	1.19286i	−0.216603	+	0.976260i	$0.569498\pi$
$727$	−32.1631	+	32.1631i	−1.19286	+	1.19286i	−0.976260	+	0.216603i	$0.930502\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			5.61863i			0.207813i
$732$		0			0
$733$	−23.8284	−	23.8284i	−0.880121	−	0.880121i	0.113426	−	0.993546i	$-0.463818\pi$
$733$	−23.8284	−	23.8284i	−0.880121	−	0.880121i	−0.993546	+	0.113426i	$0.963818\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−18.8318	−	18.8318i	−0.693679	−	0.693679i
$738$		0			0
$739$			32.9821i			1.21327i	0.794981	+	0.606634i	$0.207481\pi$
$739$			32.9821i			1.21327i	−0.794981	+	0.606634i	$0.792519\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−18.9516	+	18.9516i	−0.695265	+	0.695265i	−0.963385	−	0.268120i	$-0.913598\pi$
$743$	−18.9516	+	18.9516i	−0.695265	+	0.695265i	0.268120	+	0.963385i	$0.413598\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.70040			−0.171749
$750$		0			0
$751$	26.1199			0.953129			0.476564	−	0.879140i	$-0.341882\pi$
$751$	26.1199			0.953129			0.476564	+	0.879140i	$0.341882\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	30.7144	−	30.7144i	1.11633	−	1.11633i	0.124057	−	0.992275i	$-0.460409\pi$
$757$	30.7144	−	30.7144i	1.11633	−	1.11633i	0.992275	−	0.124057i	$-0.0395906\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	7.13933i		−	0.258801i	−0.991592	−	0.129400i	$-0.958695\pi$
$761$		−	7.13933i		−	0.258801i	0.991592	−	0.129400i	$-0.0413052\pi$
$762$		0			0
$763$	11.7088	+	11.7088i	0.423886	+	0.423886i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.14742	+	2.14742i	0.0775388	+	0.0775388i
$768$		0			0
$769$		−	27.7870i		−	1.00203i	−0.865440	−	0.501013i	$-0.832961\pi$
$769$		−	27.7870i		−	1.00203i	0.865440	−	0.501013i	$-0.167039\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−21.5414	+	21.5414i	−0.774791	+	0.774791i	−0.978940	−	0.204149i	$-0.934557\pi$
$773$	−21.5414	+	21.5414i	−0.774791	+	0.774791i	0.204149	+	0.978940i	$0.434557\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	10.0548			0.360249
$780$		0			0
$781$	−22.2784			−0.797185
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	10.9055	−	10.9055i	0.388738	−	0.388738i	−0.485499	−	0.874237i	$-0.661362\pi$
$787$	10.9055	−	10.9055i	0.388738	−	0.388738i	0.874237	+	0.485499i	$0.161362\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			2.26478i			0.0805264i
$792$		0			0
$793$	0.286880	+	0.286880i	0.0101874	+	0.0101874i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	25.5137	+	25.5137i	0.903743	+	0.903743i	0.995758	−	0.0920143i	$-0.0293305\pi$
$797$	25.5137	+	25.5137i	0.903743	+	0.903743i	−0.0920143	+	0.995758i	$0.529331\pi$
$798$		0			0
$799$			4.56656i			0.161553i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.493528	−	0.493528i	0.0174162	−	0.0174162i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.53780			0.124383			0.0621913	−	0.998064i	$-0.480191\pi$
$809$	3.53780			0.124383			0.0621913	+	0.998064i	$0.480191\pi$
$810$		0			0
$811$	−18.3184			−0.643246			−0.321623	−	0.946868i	$-0.604228\pi$
$811$	−18.3184			−0.643246			−0.321623	+	0.946868i	$0.604228\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−8.30026	+	8.30026i	−0.290389	+	0.290389i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	24.3167i		−	0.848658i	−0.905508	−	0.424329i	$-0.860510\pi$
$821$		−	24.3167i		−	0.848658i	0.905508	−	0.424329i	$-0.139490\pi$
$822$		0			0
$823$	−5.89626	−	5.89626i	−0.205531	−	0.205531i	0.596834	−	0.802365i	$-0.296425\pi$
$823$	−5.89626	−	5.89626i	−0.205531	−	0.205531i	−0.802365	+	0.596834i	$0.796425\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	28.1324	+	28.1324i	0.978259	+	0.978259i	0.999769	−	0.0215101i	$-0.00684739\pi$
$827$	28.1324	+	28.1324i	0.978259	+	0.978259i	−0.0215101	+	0.999769i	$0.506847\pi$
$828$		0			0
$829$		−	9.44361i		−	0.327990i	−0.986461	−	0.163995i	$-0.947562\pi$
$829$		−	9.44361i		−	0.327990i	0.986461	−	0.163995i	$-0.0524381\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.704639	−	0.704639i	0.0244143	−	0.0244143i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	11.1959			0.386525			0.193263	−	0.981147i	$-0.438093\pi$
$839$	11.1959			0.386525			0.193263	+	0.981147i	$0.438093\pi$
$840$		0			0
$841$	4.16672			0.143680
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	4.47314	−	4.47314i	0.153699	−	0.153699i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			11.8462i			0.406081i
$852$		0			0
$853$	20.8836	+	20.8836i	0.715040	+	0.715040i	0.967585	−	0.252545i	$-0.0812676\pi$
$853$	20.8836	+	20.8836i	0.715040	+	0.715040i	−0.252545	+	0.967585i	$0.581268\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−15.9264	−	15.9264i	−0.544037	−	0.544037i	0.380673	−	0.924710i	$-0.375692\pi$
$857$	−15.9264	−	15.9264i	−0.544037	−	0.544037i	−0.924710	+	0.380673i	$0.875692\pi$
$858$		0			0
$859$		−	30.2920i		−	1.03355i	−0.856122	−	0.516775i	$-0.827133\pi$
$859$		−	30.2920i		−	1.03355i	0.856122	−	0.516775i	$-0.172867\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	11.6185	−	11.6185i	0.395497	−	0.395497i	−0.481144	−	0.876641i	$-0.659779\pi$
$863$	11.6185	−	11.6185i	0.395497	−	0.395497i	0.876641	+	0.481144i	$0.159779\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	5.39333			0.182956
$870$		0			0
$871$	7.96686			0.269947
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	30.3134	−	30.3134i	1.02361	−	1.02361i	0.0238969	−	0.999714i	$-0.492393\pi$
$877$	30.3134	−	30.3134i	1.02361	−	1.02361i	0.999714	−	0.0238969i	$-0.00760734\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			46.5757i			1.56917i	0.620019	+	0.784587i	$0.287125\pi$
$881$			46.5757i			1.56917i	−0.620019	+	0.784587i	$0.712875\pi$
$882$		0			0
$883$	−29.4266	−	29.4266i	−0.990285	−	0.990285i	0.00966815	−	0.999953i	$-0.496922\pi$
$883$	−29.4266	−	29.4266i	−0.990285	−	0.990285i	−0.999953	+	0.00966815i	$0.996922\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	31.3384	+	31.3384i	1.05224	+	1.05224i	0.998558	+	0.0536830i	$0.0170960\pi$
$887$	31.3384	+	31.3384i	1.05224	+	1.05224i	0.0536830	+	0.998558i	$0.482904\pi$
$888$		0			0
$889$			3.61039i			0.121088i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.74606	+	6.74606i	−0.225748	+	0.225748i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	24.0140			0.800911
$900$		0			0
$901$	9.88701			0.329384
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−31.9163	+	31.9163i	−1.05976	+	1.05976i	−0.0616666	+	0.998097i	$0.519642\pi$
$907$	−31.9163	+	31.9163i	−1.05976	+	1.05976i	−0.998097	+	0.0616666i	$0.980358\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			57.4801i			1.90440i	0.305474	+	0.952200i	$0.401185\pi$
$911$			57.4801i			1.90440i	−0.305474	+	0.952200i	$0.598815\pi$
$912$		0			0
$913$	−5.95113	−	5.95113i	−0.196954	−	0.196954i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−14.2055	−	14.2055i	−0.469106	−	0.469106i
$918$		0			0
$919$			18.5220i			0.610985i	0.952194	+	0.305492i	$0.0988211\pi$
$919$			18.5220i			0.610985i	−0.952194	+	0.305492i	$0.901179\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4.71248	−	4.71248i	0.155113	−	0.155113i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.95373			0.129718			0.0648588	−	0.997894i	$-0.479340\pi$
$929$	3.95373			0.129718			0.0648588	+	0.997894i	$0.479340\pi$
$930$		0			0
$931$	2.08189			0.0682312
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−15.7125	+	15.7125i	−0.513304	+	0.513304i	−0.915537	−	0.402233i	$-0.868234\pi$
$937$	−15.7125	+	15.7125i	−0.513304	+	0.513304i	0.402233	+	0.915537i	$0.368234\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			42.0600i			1.37112i	0.728017	+	0.685559i	$0.240442\pi$
$941$			42.0600i			1.37112i	−0.728017	+	0.685559i	$0.759558\pi$
$942$		0			0
$943$	3.95621	+	3.95621i	0.128832	+	0.128832i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−27.1406	−	27.1406i	−0.881951	−	0.881951i	0.111782	−	0.993733i	$-0.464344\pi$
$947$	−27.1406	−	27.1406i	−0.881951	−	0.881951i	−0.993733	+	0.111782i	$0.964344\pi$
$948$		0			0
$949$			0.208789i			0.00677757i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−22.2671	+	22.2671i	−0.721302	+	0.721302i	−0.968870	−	0.247569i	$-0.920368\pi$
$953$	−22.2671	+	22.2671i	−0.721302	+	0.721302i	0.247569	+	0.968870i	$0.420368\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.34800			0.172696
$960$		0			0
$961$	−13.6129			−0.439127
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−11.2401	+	11.2401i	−0.361459	+	0.361459i	−0.864350	−	0.502891i	$-0.832270\pi$
$967$	−11.2401	+	11.2401i	−0.361459	+	0.361459i	0.502891	+	0.864350i	$0.332270\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1.03205i			0.0331202i	0.999863	+	0.0165601i	$0.00527148\pi$
$971$			1.03205i			0.0331202i	−0.999863	+	0.0165601i	$0.994729\pi$
$972$		0			0
$973$	−13.1792	−	13.1792i	−0.422506	−	0.422506i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−30.1067	−	30.1067i	−0.963198	−	0.963198i	0.0361488	−	0.999346i	$-0.488491\pi$
$977$	−30.1067	−	30.1067i	−0.963198	−	0.963198i	−0.999346	+	0.0361488i	$0.988491\pi$
$978$		0			0
$979$			33.0096i			1.05499i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	33.0384	−	33.0384i	1.05376	−	1.05376i	0.0552903	−	0.998470i	$-0.482392\pi$
$983$	33.0384	−	33.0384i	1.05376	−	1.05376i	0.998470	−	0.0552903i	$-0.0176084\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6.53175			−0.207698
$990$		0			0
$991$	−27.6554			−0.878504			−0.439252	−	0.898364i	$-0.644756\pi$
$991$	−27.6554			−0.878504			−0.439252	+	0.898364i	$0.644756\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−20.3948	+	20.3948i	−0.645909	+	0.645909i	−0.952002	−	0.306093i	$-0.900978\pi$
$997$	−20.3948	+	20.3948i	−0.645909	+	0.645909i	0.306093	+	0.952002i	$0.400978\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.v.e.5993.5		12
3.2	odd	2		6300.2.v.f.5993.5		12
5.2	odd	4		6300.2.v.f.1457.5		12
5.3	odd	4		1260.2.v.a.197.5	✓	12
5.4	even	2		1260.2.v.b.953.2	yes	12
15.2	even	4	inner	6300.2.v.e.1457.5		12
15.8	even	4		1260.2.v.b.197.2	yes	12
15.14	odd	2		1260.2.v.a.953.5	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.v.a.197.5	✓	12	5.3	odd	4
1260.2.v.a.953.5	yes	12	15.14	odd	2
1260.2.v.b.197.2	yes	12	15.8	even	4
1260.2.v.b.953.2	yes	12	5.4	even	2
6300.2.v.e.1457.5		12	15.2	even	4	inner
6300.2.v.e.5993.5		12	1.1	even	1	trivial
6300.2.v.f.1457.5		12	5.2	odd	4
6300.2.v.f.5993.5		12	3.2	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.v (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{7} -2.16195i q^{11} +(0.457310 + 0.457310i) q^{13} +(0.704639 + 0.704639i) q^{17} +2.08189i q^{19} +(-0.819154 + 0.819154i) q^{23} -5.75906 q^{29} -4.16978 q^{31} +(7.23073 - 7.23073i) q^{37} -4.82963i q^{41} +(3.98689 + 3.98689i) q^{43} +(3.24035 + 3.24035i) q^{47} -1.00000i q^{49} +(7.01566 - 7.01566i) q^{53} +4.69577 q^{59} +0.627320 q^{61} +(8.71058 - 8.71058i) q^{67} -10.3048i q^{71} +(0.228279 + 0.228279i) q^{73} +(-1.52873 - 1.52873i) q^{77} +2.49466i q^{79} +(2.75267 - 2.75267i) q^{83} -15.2684 q^{89} +0.646733 q^{91} +(-5.18100 + 5.18100i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{13} - 8 q^{17} + 8 q^{23} - 32 q^{29} + 16 q^{31} - 4 q^{37} - 8 q^{43} - 8 q^{47} + 8 q^{53} - 16 q^{59} + 12 q^{73} + 56 q^{83} - 72 q^{89} - 20 q^{97}+O(q^{100}) \) 12 * q + 12 * q^13 - 8 * q^17 + 8 * q^23 - 32 * q^29 + 16 * q^31 - 4 * q^37 - 8 * q^43 - 8 * q^47 + 8 * q^53 - 16 * q^59 + 12 * q^73 + 56 * q^83 - 72 * q^89 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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