LMFDB - Embedded newform 4140.2.s.b.737.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(737,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4140.737");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.s (of order $4$, 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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$44$
Relative dimension:	$22$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			737.6
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.b.2393.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.52812 - 1.63243i) q^{5} +(-2.30479 - 2.30479i) q^{7} +O(q^{10})$	$q+(-1.52812 - 1.63243i) q^{5} +(-2.30479 - 2.30479i) q^{7} +3.32736i q^{11} +(-3.34641 + 3.34641i) q^{13} +(-0.668244 + 0.668244i) q^{17} -1.73320i q^{19} +(-0.707107 - 0.707107i) q^{23} +(-0.329672 + 4.98912i) q^{25} +4.39260 q^{29} +2.04377 q^{31} +(-0.240409 + 7.28443i) q^{35} +(4.47227 + 4.47227i) q^{37} -7.40057i q^{41} +(-5.11206 + 5.11206i) q^{43} +(-1.48003 + 1.48003i) q^{47} +3.62415i q^{49} +(-3.98341 - 3.98341i) q^{53} +(5.43168 - 5.08461i) q^{55} -1.36490 q^{59} +3.91418 q^{61} +(10.5765 + 0.349058i) q^{65} +(10.6961 + 10.6961i) q^{67} -10.3416i q^{71} +(-2.13556 + 2.13556i) q^{73} +(7.66887 - 7.66887i) q^{77} -8.87969i q^{79} +(2.21101 + 2.21101i) q^{83} +(2.11202 + 0.0697032i) q^{85} +2.17165 q^{89} +15.4256 q^{91} +(-2.82934 + 2.64855i) q^{95} +(9.81888 + 9.81888i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{7}+O(q^{10}) $ 44 * q - 4 * q^7	$ 44 q - 4 q^{7} - 4 q^{13} - 24 q^{25} + 32 q^{31} + 40 q^{37} - 8 q^{43} - 24 q^{55} + 64 q^{61} + 12 q^{67} - 84 q^{73} - 104 q^{85} - 48 q^{91} + 44 q^{97}+O(q^{100}) $ 44 * q - 4 * q^7 - 4 * q^13 - 24 * q^25 + 32 * q^31 + 40 * q^37 - 8 * q^43 - 24 * q^55 + 64 * q^61 + 12 * q^67 - 84 * q^73 - 104 * q^85 - 48 * q^91 + 44 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$5$	−1.52812	−	1.63243i	−0.683398	−	0.730046i
$5$	−1.52812	−	1.63243i	−0.683398	−	0.730046i
$6$		0			0
$7$	−2.30479	−	2.30479i	−0.871130	−	0.871130i	0.121465	−	0.992596i	$-0.461241\pi$
$7$	−2.30479	−	2.30479i	−0.871130	−	0.871130i	−0.992596	+	0.121465i	$0.961241\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.32736i			1.00324i	0.865089	+	0.501618i	$0.167262\pi$
$11$			3.32736i			1.00324i	−0.865089	+	0.501618i	$0.832738\pi$
$12$		0			0
$13$	−3.34641	+	3.34641i	−0.928128	+	0.928128i	−0.997585	−	0.0694574i	$-0.977873\pi$
$13$	−3.34641	+	3.34641i	−0.928128	+	0.928128i	0.0694574	+	0.997585i	$0.477873\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.668244	+	0.668244i	−0.162073	+	0.162073i	−0.783484	−	0.621411i	$-0.786560\pi$
$17$	−0.668244	+	0.668244i	−0.162073	+	0.162073i	0.621411	+	0.783484i	$0.286560\pi$
$18$		0			0
$19$		−	1.73320i		−	0.397624i	−0.980038	−	0.198812i	$-0.936292\pi$
$19$		−	1.73320i		−	0.397624i	0.980038	−	0.198812i	$-0.0637083\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.707107	−	0.707107i	−0.147442	−	0.147442i
$23$	−0.707107	−	0.707107i	−0.147442	−	0.147442i
$24$		0			0
$25$	−0.329672	+	4.98912i	−0.0659343	+	0.997824i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.39260			0.815686			0.407843	−	0.913052i	$-0.366281\pi$
$29$	4.39260			0.815686			0.407843	+	0.913052i	$0.366281\pi$
$30$		0			0
$31$	2.04377			0.367072			0.183536	−	0.983013i	$-0.441246\pi$
$31$	2.04377			0.367072			0.183536	+	0.983013i	$0.441246\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.240409	+	7.28443i	−0.0406365	+	1.23129i
$36$		0			0
$37$	4.47227	+	4.47227i	0.735237	+	0.735237i	0.971652	−	0.236415i	$-0.0759725\pi$
$37$	4.47227	+	4.47227i	0.735237	+	0.735237i	−0.236415	+	0.971652i	$0.575972\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	7.40057i		−	1.15578i	−0.816116	−	0.577888i	$-0.803877\pi$
$41$		−	7.40057i		−	1.15578i	0.816116	−	0.577888i	$-0.196123\pi$
$42$		0			0
$43$	−5.11206	+	5.11206i	−0.779581	+	0.779581i	−0.979760	−	0.200178i	$-0.935848\pi$
$43$	−5.11206	+	5.11206i	−0.779581	+	0.779581i	0.200178	+	0.979760i	$0.435848\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.48003	+	1.48003i	−0.215884	+	0.215884i	−0.806761	−	0.590877i	$-0.798782\pi$
$47$	−1.48003	+	1.48003i	−0.215884	+	0.215884i	0.590877	+	0.806761i	$0.298782\pi$
$48$		0			0
$49$			3.62415i			0.517736i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.98341	−	3.98341i	−0.547163	−	0.547163i	0.378456	−	0.925619i	$-0.376455\pi$
$53$	−3.98341	−	3.98341i	−0.547163	−	0.547163i	−0.925619	+	0.378456i	$0.876455\pi$
$54$		0			0
$55$	5.43168	−	5.08461i	0.732408	−	0.685609i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.36490			−0.177695			−0.0888473	−	0.996045i	$-0.528318\pi$
$59$	−1.36490			−0.177695			−0.0888473	+	0.996045i	$0.528318\pi$
$60$		0			0
$61$	3.91418			0.501159			0.250580	−	0.968096i	$-0.419379\pi$
$61$	3.91418			0.501159			0.250580	+	0.968096i	$0.419379\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.5765	+	0.349058i	1.31186	+	0.0432953i
$66$		0			0
$67$	10.6961	+	10.6961i	1.30673	+	1.30673i	0.923760	+	0.382972i	$0.125099\pi$
$67$	10.6961	+	10.6961i	1.30673	+	1.30673i	0.382972	+	0.923760i	$0.374901\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.3416i		−	1.22732i	−0.789571	−	0.613660i	$-0.789697\pi$
$71$		−	10.3416i		−	1.22732i	0.789571	−	0.613660i	$-0.210303\pi$
$72$		0			0
$73$	−2.13556	+	2.13556i	−0.249948	+	0.249948i	−0.820949	−	0.571001i	$-0.806555\pi$
$73$	−2.13556	+	2.13556i	−0.249948	+	0.249948i	0.571001	+	0.820949i	$0.306555\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.66887	−	7.66887i	0.873949	−	0.873949i
$78$		0			0
$79$		−	8.87969i		−	0.999044i	−0.866301	−	0.499522i	$-0.833509\pi$
$79$		−	8.87969i		−	0.999044i	0.866301	−	0.499522i	$-0.166491\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.21101	+	2.21101i	0.242690	+	0.242690i	0.817962	−	0.575272i	$-0.195104\pi$
$83$	2.21101	+	2.21101i	0.242690	+	0.242690i	−0.575272	+	0.817962i	$0.695104\pi$
$84$		0			0
$85$	2.11202	+	0.0697032i	0.229081	+	0.00756038i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.17165			0.230194			0.115097	−	0.993354i	$-0.463282\pi$
$89$	2.17165			0.230194			0.115097	+	0.993354i	$0.463282\pi$
$90$		0			0
$91$	15.4256			1.61704
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.82934	+	2.64855i	−0.290284	+	0.271736i
$96$		0			0
$97$	9.81888	+	9.81888i	0.996956	+	0.996956i	0.999995	−	0.00303933i	$-0.000967451\pi$
$97$	9.81888	+	9.81888i	0.996956	+	0.996956i	−0.00303933	+	0.999995i	$0.500967\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	4.82234i		−	0.479841i	−0.970793	−	0.239920i	$-0.922879\pi$
$101$		−	4.82234i		−	0.479841i	0.970793	−	0.239920i	$-0.0771213\pi$
$102$		0			0
$103$	−3.43010	+	3.43010i	−0.337978	+	0.337978i	−0.855606	−	0.517628i	$-0.826815\pi$
$103$	−3.43010	+	3.43010i	−0.337978	+	0.337978i	0.517628	+	0.855606i	$0.326815\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.14994	−	5.14994i	0.497864	−	0.497864i	−0.412909	−	0.910772i	$-0.635487\pi$
$107$	5.14994	−	5.14994i	0.497864	−	0.497864i	0.910772	+	0.412909i	$0.135487\pi$
$108$		0			0
$109$		−	4.69664i		−	0.449856i	−0.974375	−	0.224928i	$-0.927785\pi$
$109$		−	4.69664i		−	0.449856i	0.974375	−	0.224928i	$-0.0722147\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.4656	−	10.4656i	−0.984525	−	0.984525i	0.0153569	−	0.999882i	$-0.495112\pi$
$113$	−10.4656	−	10.4656i	−0.984525	−	0.984525i	−0.999882	+	0.0153569i	$0.995112\pi$
$114$		0			0
$115$	−0.0737570	+	2.23485i	−0.00687787	+	0.208401i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.08033			0.282373
$120$		0			0
$121$	−0.0712980			−0.00648164
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.64818	−	7.08583i	0.773517	−	0.633776i
$126$		0			0
$127$	4.50076	+	4.50076i	0.399378	+	0.399378i	0.878014	−	0.478636i	$-0.158869\pi$
$127$	4.50076	+	4.50076i	0.399378	+	0.399378i	−0.478636	+	0.878014i	$0.658869\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	7.33197i		−	0.640597i	−0.947317	−	0.320299i	$-0.896217\pi$
$131$		−	7.33197i		−	0.640597i	0.947317	−	0.320299i	$-0.103783\pi$
$132$		0			0
$133$	−3.99468	+	3.99468i	−0.346382	+	0.346382i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.943223	−	0.943223i	0.0805850	−	0.0805850i	−0.665665	−	0.746250i	$-0.731852\pi$
$137$	0.943223	−	0.943223i	0.0805850	−	0.0805850i	0.746250	+	0.665665i	$0.231852\pi$
$138$		0			0
$139$		−	10.8696i		−	0.921948i	−0.887414	−	0.460974i	$-0.847500\pi$
$139$		−	10.8696i		−	0.921948i	0.887414	−	0.460974i	$-0.152500\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.1347	−	11.1347i	−0.931131	−	0.931131i
$144$		0			0
$145$	−6.71245	−	7.17063i	−0.557438	−	0.595488i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.38387			0.522987			0.261494	−	0.965205i	$-0.415785\pi$
$149$	6.38387			0.522987			0.261494	+	0.965205i	$0.415785\pi$
$150$		0			0
$151$	−4.15277			−0.337948			−0.168974	−	0.985621i	$-0.554045\pi$
$151$	−4.15277			−0.337948			−0.168974	+	0.985621i	$0.554045\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.12314	−	3.33632i	−0.250857	−	0.267980i
$156$		0			0
$157$	6.94797	+	6.94797i	0.554509	+	0.554509i	0.927739	−	0.373230i	$-0.121750\pi$
$157$	6.94797	+	6.94797i	0.554509	+	0.554509i	−0.373230	+	0.927739i	$0.621750\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			3.25947i			0.256882i
$162$		0			0
$163$	13.0078	−	13.0078i	1.01885	−	1.01885i	0.0190341	−	0.999819i	$-0.493941\pi$
$163$	13.0078	−	13.0078i	1.01885	−	1.01885i	0.999819	−	0.0190341i	$-0.00605912\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.40719	−	5.40719i	0.418421	−	0.418421i	−0.466238	−	0.884659i	$-0.654391\pi$
$167$	5.40719	−	5.40719i	0.418421	−	0.418421i	0.884659	+	0.466238i	$0.154391\pi$
$168$		0			0
$169$		−	9.39694i		−	0.722841i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.48767	+	7.48767i	0.569277	+	0.569277i	0.931926	−	0.362649i	$-0.118128\pi$
$173$	7.48767	+	7.48767i	0.569277	+	0.569277i	−0.362649	+	0.931926i	$0.618128\pi$
$174$		0			0
$175$	12.2587	−	10.7391i	0.926672	−	0.811797i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.99794			0.523051			0.261525	−	0.965197i	$-0.415774\pi$
$179$	6.99794			0.523051			0.261525	+	0.965197i	$0.415774\pi$
$180$		0			0
$181$	−12.8405			−0.954426			−0.477213	−	0.878788i	$-0.658353\pi$
$181$	−12.8405			−0.954426			−0.477213	+	0.878788i	$0.658353\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.466494	−	14.1349i	0.0342973	−	1.03922i
$186$		0			0
$187$	−2.22349	−	2.22349i	−0.162597	−	0.162597i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	21.6720i		−	1.56813i	−0.620678	−	0.784066i	$-0.713143\pi$
$191$		−	21.6720i		−	1.56813i	0.620678	−	0.784066i	$-0.286857\pi$
$192$		0			0
$193$	1.21213	−	1.21213i	0.0872509	−	0.0872509i	−0.662134	−	0.749385i	$-0.730349\pi$
$193$	1.21213	−	1.21213i	0.0872509	−	0.0872509i	0.749385	+	0.662134i	$0.230349\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	15.8261	−	15.8261i	1.12756	−	1.12756i	0.136990	−	0.990572i	$-0.456257\pi$
$197$	15.8261	−	15.8261i	1.12756	−	1.12756i	0.990572	−	0.136990i	$-0.0437427\pi$
$198$		0			0
$199$		−	17.0772i		−	1.21057i	−0.796010	−	0.605284i	$-0.793060\pi$
$199$		−	17.0772i		−	1.21057i	0.796010	−	0.605284i	$-0.206940\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−10.1240	−	10.1240i	−0.710569	−	0.710569i
$204$		0			0
$205$	−12.0809	+	11.3090i	−0.843769	+	0.789854i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.76698			0.398911
$210$		0			0
$211$	19.4031			1.33576			0.667881	−	0.744268i	$-0.267201\pi$
$211$	19.4031			1.33576			0.667881	+	0.744268i	$0.267201\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	16.1569	+	0.533229i	1.10189	+	0.0363659i
$216$		0			0
$217$	−4.71048	−	4.71048i	−0.319768	−	0.319768i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	4.47244i		−	0.300849i
$222$		0			0
$223$	−0.624745	+	0.624745i	−0.0418360	+	0.0418360i	−0.727715	−	0.685879i	$-0.759418\pi$
$223$	−0.624745	+	0.624745i	−0.0418360	+	0.0418360i	0.685879	+	0.727715i	$0.259418\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.9001	+	12.9001i	−0.856209	+	0.856209i	−0.990889	−	0.134680i	$-0.956999\pi$
$227$	−12.9001	+	12.9001i	−0.856209	+	0.856209i	0.134680	+	0.990889i	$0.456999\pi$
$228$		0			0
$229$			16.6374i			1.09943i	0.835353	+	0.549714i	$0.185263\pi$
$229$			16.6374i			1.09943i	−0.835353	+	0.549714i	$0.814737\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.3269	+	10.3269i	0.676536	+	0.676536i	0.959215	−	0.282679i	$-0.0912231\pi$
$233$	10.3269	+	10.3269i	0.676536	+	0.676536i	−0.282679	+	0.959215i	$0.591223\pi$
$234$		0			0
$235$	4.67771	+	0.154379i	0.305140	+	0.0100706i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	17.3787			1.12414			0.562068	−	0.827091i	$-0.310006\pi$
$239$	17.3787			1.12414			0.562068	+	0.827091i	$0.310006\pi$
$240$		0			0
$241$	−21.6714			−1.39598			−0.697990	−	0.716108i	$-0.745922\pi$
$241$	−21.6714			−1.39598			−0.697990	+	0.716108i	$0.745922\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.91618	−	5.53815i	0.377971	−	0.353820i
$246$		0			0
$247$	5.80001	+	5.80001i	0.369046	+	0.369046i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	11.4963i		−	0.725642i	−0.931859	−	0.362821i	$-0.881814\pi$
$251$		−	11.4963i		−	0.725642i	0.931859	−	0.362821i	$-0.118186\pi$
$252$		0			0
$253$	2.35280	−	2.35280i	0.147919	−	0.147919i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.2208	+	10.2208i	−0.637557	+	0.637557i	−0.949952	−	0.312395i	$-0.898869\pi$
$257$	−10.2208	+	10.2208i	−0.637557	+	0.637557i	0.312395	+	0.949952i	$0.398869\pi$
$258$		0			0
$259$		−	20.6153i		−	1.28097i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.174750	−	0.174750i	−0.0107756	−	0.0107756i	0.701698	−	0.712474i	$-0.252425\pi$
$263$	−0.174750	−	0.174750i	−0.0107756	−	0.0107756i	−0.712474	+	0.701698i	$0.752425\pi$
$264$		0			0
$265$	−0.415501	+	12.5898i	−0.0255241	+	0.773384i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	14.7405			0.898742			0.449371	−	0.893345i	$-0.351648\pi$
$269$	14.7405			0.898742			0.449371	+	0.893345i	$0.351648\pi$
$270$		0			0
$271$	10.2858			0.624821			0.312410	−	0.949947i	$-0.398864\pi$
$271$	10.2858			0.624821			0.312410	+	0.949947i	$0.398864\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−16.6006	−	1.09694i	−1.00105	−	0.0661477i
$276$		0			0
$277$	10.7038	+	10.7038i	0.643128	+	0.643128i	0.951323	−	0.308195i	$-0.0997249\pi$
$277$	10.7038	+	10.7038i	0.643128	+	0.643128i	−0.308195	+	0.951323i	$0.599725\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			33.1367i			1.97677i	0.151971	+	0.988385i	$0.451438\pi$
$281$			33.1367i			1.97677i	−0.151971	+	0.988385i	$0.548562\pi$
$282$		0			0
$283$	−16.3141	+	16.3141i	−0.969771	+	0.969771i	−0.999556	−	0.0297850i	$-0.990518\pi$
$283$	−16.3141	+	16.3141i	−0.969771	+	0.969771i	0.0297850	+	0.999556i	$0.490518\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−17.0568	+	17.0568i	−1.00683	+	1.00683i
$288$		0			0
$289$			16.1069i			0.947465i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.68206	+	2.68206i	0.156688	+	0.156688i	0.781097	−	0.624410i	$-0.214660\pi$
$293$	2.68206	+	2.68206i	0.156688	+	0.156688i	−0.624410	+	0.781097i	$0.714660\pi$
$294$		0			0
$295$	2.08573	+	2.22810i	0.121436	+	0.129725i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.73254			0.273690
$300$		0			0
$301$	23.5645			1.35823
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.98135	−	6.38963i	−0.342491	−	0.365869i
$306$		0			0
$307$	5.34372	+	5.34372i	0.304982	+	0.304982i	0.842959	−	0.537977i	$-0.180811\pi$
$307$	5.34372	+	5.34372i	0.304982	+	0.304982i	−0.537977	+	0.842959i	$0.680811\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	5.73509i		−	0.325207i	−0.986691	−	0.162604i	$-0.948011\pi$
$311$		−	5.73509i		−	0.325207i	0.986691	−	0.162604i	$-0.0519892\pi$
$312$		0			0
$313$	13.3170	−	13.3170i	0.752721	−	0.752721i	−0.222265	−	0.974986i	$-0.571345\pi$
$313$	13.3170	−	13.3170i	0.752721	−	0.752721i	0.974986	+	0.222265i	$0.0713450\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	14.1119	−	14.1119i	0.792602	−	0.792602i	−0.189314	−	0.981917i	$-0.560627\pi$
$317$	14.1119	−	14.1119i	0.792602	−	0.792602i	0.981917	+	0.189314i	$0.0606265\pi$
$318$		0			0
$319$			14.6158i			0.818325i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.15820	+	1.15820i	0.0644441	+	0.0644441i
$324$		0			0
$325$	−15.5924	−	17.7989i	−0.864912	−	0.987303i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.82231			0.376126
$330$		0			0
$331$	−21.6832			−1.19182			−0.595909	−	0.803052i	$-0.703208\pi$
$331$	−21.6832			−1.19182			−0.595909	+	0.803052i	$0.703208\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.11569	−	33.8055i	0.0609564	−	1.84699i
$336$		0			0
$337$	16.6800	+	16.6800i	0.908619	+	0.908619i	0.996161	−	0.0875421i	$-0.0279012\pi$
$337$	16.6800	+	16.6800i	0.908619	+	0.908619i	−0.0875421	+	0.996161i	$0.527901\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			6.80036i			0.368260i
$342$		0			0
$343$	−7.78064	+	7.78064i	−0.420115	+	0.420115i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.3065	−	11.3065i	0.606967	−	0.606967i	−0.335185	−	0.942152i	$-0.608799\pi$
$347$	11.3065	−	11.3065i	0.606967	−	0.606967i	0.942152	+	0.335185i	$0.108799\pi$
$348$		0			0
$349$			11.3389i			0.606958i	0.952838	+	0.303479i	$0.0981482\pi$
$349$			11.3389i			0.606958i	−0.952838	+	0.303479i	$0.901852\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	13.7921	+	13.7921i	0.734082	+	0.734082i	0.971426	−	0.237344i	$-0.0762768\pi$
$353$	13.7921	+	13.7921i	0.734082	+	0.734082i	−0.237344	+	0.971426i	$0.576277\pi$
$354$		0			0
$355$	−16.8819	+	15.8032i	−0.896000	+	0.838748i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	28.8705			1.52373			0.761863	−	0.647738i	$-0.224285\pi$
$359$	28.8705			1.52373			0.761863	+	0.647738i	$0.224285\pi$
$360$		0			0
$361$	15.9960			0.841895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.74955	+	0.222756i	0.353288	+	0.0116596i
$366$		0			0
$367$	22.8404	+	22.8404i	1.19226	+	1.19226i	0.976432	+	0.215825i	$0.0692442\pi$
$367$	22.8404	+	22.8404i	1.19226	+	1.19226i	0.215825	+	0.976432i	$0.430756\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			18.3619i			0.953300i
$372$		0			0
$373$	−2.25000	+	2.25000i	−0.116501	+	0.116501i	−0.762954	−	0.646453i	$-0.776252\pi$
$373$	−2.25000	+	2.25000i	−0.116501	+	0.116501i	0.646453	+	0.762954i	$0.276252\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−14.6995	+	14.6995i	−0.757061	+	0.757061i
$378$		0			0
$379$			6.64580i			0.341372i	0.985325	+	0.170686i	$0.0545984\pi$
$379$			6.64580i			0.341372i	−0.985325	+	0.170686i	$0.945402\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.1236	−	12.1236i	−0.619486	−	0.619486i	0.325914	−	0.945400i	$-0.394328\pi$
$383$	−12.1236	−	12.1236i	−0.619486	−	0.619486i	−0.945400	+	0.325914i	$0.894328\pi$
$384$		0			0
$385$	−24.2379	−	0.799925i	−1.23528	−	0.0407680i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.47535			−0.0748033			−0.0374017	−	0.999300i	$-0.511908\pi$
$389$	−1.47535			−0.0748033			−0.0374017	+	0.999300i	$0.511908\pi$
$390$		0			0
$391$	0.945040			0.0477927
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−14.4955	+	13.5693i	−0.729348	+	0.682744i
$396$		0			0
$397$	−18.4685	−	18.4685i	−0.926910	−	0.926910i	0.0705954	−	0.997505i	$-0.477510\pi$
$397$	−18.4685	−	18.4685i	−0.926910	−	0.926910i	−0.997505	+	0.0705954i	$0.977510\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			25.3860i			1.26772i	0.773448	+	0.633859i	$0.218530\pi$
$401$			25.3860i			1.26772i	−0.773448	+	0.633859i	$0.781470\pi$
$402$		0			0
$403$	−6.83931	+	6.83931i	−0.340690	+	0.340690i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−14.8808	+	14.8808i	−0.737616	+	0.737616i
$408$		0			0
$409$			25.4786i			1.25983i	0.776662	+	0.629917i	$0.216911\pi$
$409$			25.4786i			1.25983i	−0.776662	+	0.629917i	$0.783089\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.14581	+	3.14581i	0.154795	+	0.154795i
$414$		0			0
$415$	0.230626	−	6.98802i	0.0113210	−	0.343028i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−16.6109			−0.811497			−0.405748	−	0.913985i	$-0.632989\pi$
$419$	−16.6109			−0.811497			−0.405748	+	0.913985i	$0.632989\pi$
$420$		0			0
$421$	29.0022			1.41348			0.706742	−	0.707472i	$-0.250164\pi$
$421$	29.0022			1.41348			0.706742	+	0.707472i	$0.250164\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.11365	−	3.55425i	−0.151034	−	0.172406i
$426$		0			0
$427$	−9.02138	−	9.02138i	−0.436575	−	0.436575i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	27.1768i		−	1.30906i	−0.756036	−	0.654530i	$-0.772867\pi$
$431$		−	27.1768i		−	1.30906i	0.756036	−	0.654530i	$-0.227133\pi$
$432$		0			0
$433$	21.9196	−	21.9196i	1.05339	−	1.05339i	0.0548950	−	0.998492i	$-0.482518\pi$
$433$	21.9196	−	21.9196i	1.05339	−	1.05339i	0.998492	−	0.0548950i	$-0.0174824\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.22556	+	1.22556i	−0.0586265	+	0.0586265i
$438$		0			0
$439$			3.35097i			0.159933i	0.996798	+	0.0799665i	$0.0254813\pi$
$439$			3.35097i			0.159933i	−0.996798	+	0.0799665i	$0.974519\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.0480	+	10.0480i	0.477397	+	0.477397i	0.904298	−	0.426901i	$-0.140395\pi$
$443$	10.0480	+	10.0480i	0.477397	+	0.477397i	−0.426901	+	0.904298i	$0.640395\pi$
$444$		0			0
$445$	−3.31855	−	3.54507i	−0.157314	−	0.168052i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−24.2261			−1.14330			−0.571651	−	0.820497i	$-0.693697\pi$
$449$	−24.2261			−1.14330			−0.571651	+	0.820497i	$0.693697\pi$
$450$		0			0
$451$	24.6243			1.15951
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−23.5722	−	25.1812i	−1.10508	−	1.18051i
$456$		0			0
$457$	8.91796	+	8.91796i	0.417164	+	0.417164i	0.884225	−	0.467061i	$-0.154687\pi$
$457$	8.91796	+	8.91796i	0.417164	+	0.417164i	−0.467061	+	0.884225i	$0.654687\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			4.48346i			0.208816i	0.994535	+	0.104408i	$0.0332947\pi$
$461$			4.48346i			0.208816i	−0.994535	+	0.104408i	$0.966705\pi$
$462$		0			0
$463$	14.9675	−	14.9675i	0.695598	−	0.695598i	−0.267860	−	0.963458i	$-0.586316\pi$
$463$	14.9675	−	14.9675i	0.695598	−	0.695598i	0.963458	+	0.267860i	$0.0863163\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.50750	+	6.50750i	−0.301131	+	0.301131i	−0.841456	−	0.540325i	$-0.818301\pi$
$467$	−6.50750	+	6.50750i	−0.301131	+	0.301131i	0.540325	+	0.841456i	$0.318301\pi$
$468$		0			0
$469$		−	49.3044i		−	2.27667i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−17.0096	−	17.0096i	−0.782104	−	0.782104i
$474$		0			0
$475$	8.64716	+	0.571388i	0.396759	+	0.0262171i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−27.6866			−1.26503			−0.632516	−	0.774548i	$-0.717978\pi$
$479$	−27.6866			−1.26503			−0.632516	+	0.774548i	$0.717978\pi$
$480$		0			0
$481$	−29.9321			−1.36479
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.02419	−	31.0331i	0.0465060	−	1.40914i
$486$		0			0
$487$	6.45385	+	6.45385i	0.292452	+	0.292452i	0.838048	−	0.545596i	$-0.183697\pi$
$487$	6.45385	+	6.45385i	0.292452	+	0.292452i	−0.545596	+	0.838048i	$0.683697\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			40.6025i			1.83237i	0.400759	+	0.916183i	$0.368746\pi$
$491$			40.6025i			1.83237i	−0.400759	+	0.916183i	$0.631254\pi$
$492$		0			0
$493$	−2.93533	+	2.93533i	−0.132201	+	0.132201i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−23.8352	+	23.8352i	−1.06916	+	1.06916i
$498$		0			0
$499$		−	3.65115i		−	0.163448i	−0.996655	−	0.0817239i	$-0.973957\pi$
$499$		−	3.65115i		−	0.163448i	0.996655	−	0.0817239i	$-0.0260426\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−28.5058	−	28.5058i	−1.27101	−	1.27101i	−0.945559	−	0.325450i	$-0.894484\pi$
$503$	−28.5058	−	28.5058i	−1.27101	−	1.27101i	−0.325450	−	0.945559i	$-0.605516\pi$
$504$		0			0
$505$	−7.87214	+	7.36913i	−0.350306	+	0.327922i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	32.9364			1.45988			0.729940	−	0.683511i	$-0.239548\pi$
$509$	32.9364			1.45988			0.729940	+	0.683511i	$0.239548\pi$
$510$		0			0
$511$	9.84404			0.435475
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.8410	+	0.357787i	0.477713	+	0.0157660i
$516$		0			0
$517$	−4.92458	−	4.92458i	−0.216583	−	0.216583i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	15.2827i		−	0.669549i	−0.942298	−	0.334775i	$-0.891340\pi$
$521$		−	15.2827i		−	0.669549i	0.942298	−	0.334775i	$-0.108660\pi$
$522$		0			0
$523$	−13.2400	+	13.2400i	−0.578946	+	0.578946i	−0.934613	−	0.355667i	$-0.884254\pi$
$523$	−13.2400	+	13.2400i	−0.578946	+	0.578946i	0.355667	+	0.934613i	$0.384254\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.36574	+	1.36574i	−0.0594925	+	0.0594925i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	24.7654	+	24.7654i	1.07271	+	1.07271i
$534$		0			0
$535$	−16.2767	−	0.537181i	−0.703703	−	0.0232244i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.0588			−0.519411
$540$		0			0
$541$	18.7418			0.805772			0.402886	−	0.915250i	$-0.368007\pi$
$541$	18.7418			0.805772			0.402886	+	0.915250i	$0.368007\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7.66694	+	7.17705i	−0.328416	+	0.307431i
$546$		0			0
$547$	−6.35624	−	6.35624i	−0.271773	−	0.271773i	0.558040	−	0.829814i	$-0.311553\pi$
$547$	−6.35624	−	6.35624i	−0.271773	−	0.271773i	−0.829814	+	0.558040i	$0.811553\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	7.61328i		−	0.324336i
$552$		0			0
$553$	−20.4659	+	20.4659i	−0.870297	+	0.870297i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.88154	+	1.88154i	−0.0797233	+	0.0797233i	−0.745844	−	0.666121i	$-0.767954\pi$
$557$	−1.88154	+	1.88154i	−0.0797233	+	0.0797233i	0.666121	+	0.745844i	$0.267954\pi$
$558$		0			0
$559$		−	34.2141i		−	1.44710i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.64046	−	3.64046i	−0.153427	−	0.153427i	0.626220	−	0.779647i	$-0.284601\pi$
$563$	−3.64046	−	3.64046i	−0.153427	−	0.153427i	−0.779647	+	0.626220i	$0.784601\pi$
$564$		0			0
$565$	−1.09165	+	33.0773i	−0.0459261	+	1.39157i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−18.6487			−0.781793			−0.390897	−	0.920435i	$-0.627835\pi$
$569$	−18.6487			−0.781793			−0.390897	+	0.920435i	$0.627835\pi$
$570$		0			0
$571$	−8.07667			−0.337998			−0.168999	−	0.985616i	$-0.554053\pi$
$571$	−8.07667			−0.337998			−0.168999	+	0.985616i	$0.554053\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.76095	−	3.29473i	0.156843	−	0.137400i
$576$		0			0
$577$	−8.11139	−	8.11139i	−0.337682	−	0.337682i	0.517813	−	0.855494i	$-0.326746\pi$
$577$	−8.11139	−	8.11139i	−0.337682	−	0.337682i	−0.855494	+	0.517813i	$0.826746\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	10.1918i		−	0.422829i
$582$		0			0
$583$	13.2542	−	13.2542i	0.548933	−	0.548933i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15.1304	−	15.1304i	0.624499	−	0.624499i	−0.322180	−	0.946679i	$-0.604415\pi$
$587$	15.1304	−	15.1304i	0.624499	−	0.624499i	0.946679	+	0.322180i	$0.104415\pi$
$588$		0			0
$589$		−	3.54227i		−	0.145957i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−13.6097	−	13.6097i	−0.558883	−	0.558883i	0.370106	−	0.928989i	$-0.379321\pi$
$593$	−13.6097	−	13.6097i	−0.558883	−	0.558883i	−0.928989	+	0.370106i	$0.879321\pi$
$594$		0			0
$595$	−4.70713	−	5.02843i	−0.192973	−	0.206145i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	21.2874			0.869780			0.434890	−	0.900484i	$-0.356787\pi$
$599$	21.2874			0.869780			0.434890	+	0.900484i	$0.356787\pi$
$600$		0			0
$601$	31.3467			1.27866			0.639329	−	0.768933i	$-0.279212\pi$
$601$	31.3467			1.27866			0.639329	+	0.768933i	$0.279212\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.108952	+	0.116389i	0.00442954	+	0.00473190i
$606$		0			0
$607$	−6.66528	−	6.66528i	−0.270535	−	0.270535i	0.558780	−	0.829316i	$-0.311269\pi$
$607$	−6.66528	−	6.66528i	−0.270535	−	0.270535i	−0.829316	+	0.558780i	$0.811269\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	9.90556i		−	0.400736i
$612$		0			0
$613$	−0.392499	+	0.392499i	−0.0158529	+	0.0158529i	−0.714989	−	0.699136i	$-0.753568\pi$
$613$	−0.392499	+	0.392499i	−0.0158529	+	0.0158529i	0.699136	+	0.714989i	$0.253568\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.2561	−	18.2561i	0.734964	−	0.734964i	−0.236635	−	0.971599i	$-0.576044\pi$
$617$	18.2561	−	18.2561i	0.734964	−	0.734964i	0.971599	+	0.236635i	$0.0760444\pi$
$618$		0			0
$619$			14.8938i			0.598631i	0.954154	+	0.299316i	$0.0967584\pi$
$619$			14.8938i			0.598631i	−0.954154	+	0.299316i	$0.903242\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.00520	−	5.00520i	−0.200529	−	0.200529i
$624$		0			0
$625$	−24.7826	−	3.28954i	−0.991305	−	0.131582i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.97714			−0.238324
$630$		0			0
$631$	−28.5944			−1.13833			−0.569163	−	0.822225i	$-0.692733\pi$
$631$	−28.5944			−1.13833			−0.569163	+	0.822225i	$0.692733\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.469466	−	14.2249i	0.0186302	−	0.564498i
$636$		0			0
$637$	−12.1279	−	12.1279i	−0.480525	−	0.480525i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			24.3383i			0.961304i	0.876911	+	0.480652i	$0.159600\pi$
$641$			24.3383i			0.961304i	−0.876911	+	0.480652i	$0.840400\pi$
$642$		0			0
$643$	13.5022	−	13.5022i	0.532474	−	0.532474i	−0.388834	−	0.921308i	$-0.627122\pi$
$643$	13.5022	−	13.5022i	0.532474	−	0.532474i	0.921308	+	0.388834i	$0.127122\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−10.7471	+	10.7471i	−0.422511	+	0.422511i	−0.886067	−	0.463557i	$-0.846573\pi$
$647$	−10.7471	+	10.7471i	−0.422511	+	0.422511i	0.463557	+	0.886067i	$0.346573\pi$
$648$		0			0
$649$		−	4.54150i		−	0.178269i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14.0250	+	14.0250i	0.548840	+	0.548840i	0.926105	−	0.377265i	$-0.123136\pi$
$653$	14.0250	+	14.0250i	0.548840	+	0.548840i	−0.377265	+	0.926105i	$0.623136\pi$
$654$		0			0
$655$	−11.9689	+	11.2042i	−0.467665	+	0.437783i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	42.4088			1.65201			0.826006	−	0.563661i	$-0.190607\pi$
$659$	42.4088			1.65201			0.826006	+	0.563661i	$0.190607\pi$
$660$		0			0
$661$	−0.734309			−0.0285613			−0.0142806	−	0.999898i	$-0.504546\pi$
$661$	−0.734309			−0.0285613			−0.0142806	+	0.999898i	$0.504546\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.6254	+	0.416677i	0.489592	+	0.0161580i
$666$		0			0
$667$	−3.10604	−	3.10604i	−0.120266	−	0.120266i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			13.0239i			0.502781i
$672$		0			0
$673$	−16.1053	+	16.1053i	−0.620816	+	0.620816i	−0.945740	−	0.324924i	$-0.894661\pi$
$673$	−16.1053	+	16.1053i	−0.620816	+	0.620816i	0.324924	+	0.945740i	$0.394661\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.09950	−	9.09950i	0.349722	−	0.349722i	−0.510284	−	0.860006i	$-0.670460\pi$
$677$	9.09950	−	9.09950i	0.349722	−	0.349722i	0.860006	+	0.510284i	$0.170460\pi$
$678$		0			0
$679$		−	45.2610i		−	1.73696i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.71416	−	3.71416i	−0.142118	−	0.142118i	0.632468	−	0.774586i	$-0.282042\pi$
$683$	−3.71416	−	3.71416i	−0.142118	−	0.142118i	−0.774586	+	0.632468i	$0.782042\pi$
$684$		0			0
$685$	−2.98111	−	0.0983858i	−0.113902	−	0.00375913i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	26.6602			1.01567
$690$		0			0
$691$	−4.44413			−0.169063			−0.0845313	−	0.996421i	$-0.526939\pi$
$691$	−4.44413			−0.169063			−0.0845313	+	0.996421i	$0.526939\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−17.7439	+	16.6101i	−0.673065	+	0.630058i
$696$		0			0
$697$	4.94539	+	4.94539i	0.187320	+	0.187320i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			50.9289i			1.92356i	0.273824	+	0.961780i	$0.411711\pi$
$701$			50.9289i			1.92356i	−0.273824	+	0.961780i	$0.588289\pi$
$702$		0			0
$703$	7.75136	−	7.75136i	0.292348	−	0.292348i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−11.1145	+	11.1145i	−0.418004	+	0.418004i
$708$		0			0
$709$		−	39.9766i		−	1.50135i	−0.660670	−	0.750677i	$-0.729728\pi$
$709$		−	39.9766i		−	1.50135i	0.660670	−	0.750677i	$-0.270272\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.44517	−	1.44517i	−0.0541219	−	0.0541219i
$714$		0			0
$715$	−1.16144	+	35.1919i	−0.0434354	+	1.31610i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	31.9771			1.19255			0.596273	−	0.802782i	$-0.296648\pi$
$719$	31.9771			1.19255			0.596273	+	0.802782i	$0.296648\pi$
$720$		0			0
$721$	15.8114			0.588846
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.44812	+	21.9152i	−0.0537817	+	0.813911i
$726$		0			0
$727$	7.30489	+	7.30489i	0.270923	+	0.270923i	0.829472	−	0.558549i	$-0.188642\pi$
$727$	7.30489	+	7.30489i	0.270923	+	0.270923i	−0.558549	+	0.829472i	$0.688642\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	6.83220i		−	0.252698i
$732$		0			0
$733$	−31.4176	+	31.4176i	−1.16044	+	1.16044i	−0.176056	+	0.984380i	$0.556334\pi$
$733$	−31.4176	+	31.4176i	−1.16044	+	1.16044i	−0.984380	+	0.176056i	$0.943666\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−35.5896	+	35.5896i	−1.31096	+	1.31096i
$738$		0			0
$739$		−	39.9278i		−	1.46877i	−0.678734	−	0.734384i	$-0.737471\pi$
$739$		−	39.9278i		−	1.46877i	0.678734	−	0.734384i	$-0.262529\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.2758	+	12.2758i	0.450354	+	0.450354i	0.895472	−	0.445118i	$-0.146838\pi$
$743$	12.2758	+	12.2758i	0.450354	+	0.450354i	−0.445118	+	0.895472i	$0.646838\pi$
$744$		0			0
$745$	−9.75535	−	10.4212i	−0.357408	−	0.381805i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−23.7391			−0.867409
$750$		0			0
$751$	27.4987			1.00344			0.501721	−	0.865029i	$-0.332700\pi$
$751$	27.4987			1.00344			0.501721	+	0.865029i	$0.332700\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.34595	+	6.77912i	0.230953	+	0.246717i
$756$		0			0
$757$	−17.3031	−	17.3031i	−0.628890	−	0.628890i	0.318899	−	0.947789i	$-0.396687\pi$
$757$	−17.3031	−	17.3031i	−0.628890	−	0.628890i	−0.947789	+	0.318899i	$0.896687\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			39.6747i			1.43821i	0.694902	+	0.719104i	$0.255448\pi$
$761$			39.6747i			1.43821i	−0.694902	+	0.719104i	$0.744552\pi$
$762$		0			0
$763$	−10.8248	+	10.8248i	−0.391883	+	0.391883i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.56751	−	4.56751i	0.164923	−	0.164923i
$768$		0			0
$769$			5.77095i			0.208106i	0.994572	+	0.104053i	$0.0331811\pi$
$769$			5.77095i			0.208106i	−0.994572	+	0.104053i	$0.966819\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9.24325	−	9.24325i	−0.332457	−	0.332457i	0.521062	−	0.853519i	$-0.325536\pi$
$773$	−9.24325	−	9.24325i	−0.332457	−	0.332457i	−0.853519	+	0.521062i	$0.825536\pi$
$774$		0			0
$775$	−0.673774	+	10.1966i	−0.0242027	+	0.366274i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.8267			−0.459564
$780$		0			0
$781$	34.4101			1.23129
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.724730	−	21.9595i	0.0258667	−	0.783767i
$786$		0			0
$787$	−35.1273	−	35.1273i	−1.25215	−	1.25215i	−0.954752	−	0.297402i	$-0.903880\pi$
$787$	−35.1273	−	35.1273i	−1.25215	−	1.25215i	−0.297402	−	0.954752i	$-0.596120\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			48.2423i			1.71530i
$792$		0			0
$793$	−13.0985	+	13.0985i	−0.465140	+	0.465140i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.0455	−	17.0455i	0.603784	−	0.603784i	−0.337531	−	0.941314i	$-0.609592\pi$
$797$	17.0455	−	17.0455i	0.603784	−	0.603784i	0.941314	+	0.337531i	$0.109592\pi$
$798$		0			0
$799$		−	1.97804i		−	0.0699780i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.10576	−	7.10576i	−0.250757	−	0.250757i
$804$		0			0
$805$	5.32087	−	4.98088i	0.187536	−	0.175553i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	11.1490			0.391977			0.195988	−	0.980606i	$-0.437208\pi$
$809$	11.1490			0.391977			0.195988	+	0.980606i	$0.437208\pi$
$810$		0			0
$811$	15.0660			0.529039			0.264519	−	0.964380i	$-0.414787\pi$
$811$	15.0660			0.529039			0.264519	+	0.964380i	$0.414787\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−41.1120	−	1.35682i	−1.44009	−	0.0475275i
$816$		0			0
$817$	8.86023	+	8.86023i	0.309980	+	0.309980i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			43.9646i			1.53437i	0.641423	+	0.767187i	$0.278344\pi$
$821$			43.9646i			1.53437i	−0.641423	+	0.767187i	$0.721656\pi$
$822$		0			0
$823$	−20.8602	+	20.8602i	−0.727139	+	0.727139i	−0.970049	−	0.242910i	$-0.921898\pi$
$823$	−20.8602	+	20.8602i	−0.727139	+	0.727139i	0.242910	+	0.970049i	$0.421898\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9.61299	−	9.61299i	0.334276	−	0.334276i	−0.519932	−	0.854208i	$-0.674043\pi$
$827$	9.61299	−	9.61299i	0.334276	−	0.334276i	0.854208	+	0.519932i	$0.174043\pi$
$828$		0			0
$829$			53.7708i			1.86754i	0.357879	+	0.933768i	$0.383500\pi$
$829$			53.7708i			1.86754i	−0.357879	+	0.933768i	$0.616500\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.42182	−	2.42182i	−0.0839110	−	0.0839110i
$834$		0			0
$835$	−17.0897	−	0.564013i	−0.591414	−	0.0195185i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−26.5784			−0.917590			−0.458795	−	0.888542i	$-0.651719\pi$
$839$	−26.5784			−0.917590			−0.458795	+	0.888542i	$0.651719\pi$
$840$		0			0
$841$	−9.70503			−0.334656
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−15.3399	+	14.3597i	−0.527707	+	0.493988i
$846$		0			0
$847$	0.164327	+	0.164327i	0.00564635	+	0.00564635i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	6.32475i		−	0.216810i
$852$		0			0
$853$	38.2333	−	38.2333i	1.30908	−	1.30908i	0.387004	−	0.922078i	$-0.373510\pi$
$853$	38.2333	−	38.2333i	1.30908	−	1.30908i	0.922078	−	0.387004i	$-0.126490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.94602	−	1.94602i	0.0664749	−	0.0664749i	−0.673088	−	0.739563i	$-0.735032\pi$
$857$	1.94602	−	1.94602i	0.0664749	−	0.0664749i	0.739563	+	0.673088i	$0.235032\pi$
$858$		0			0
$859$		−	50.5755i		−	1.72561i	−0.505533	−	0.862807i	$-0.668704\pi$
$859$		−	50.5755i		−	1.72561i	0.505533	−	0.862807i	$-0.331296\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−22.6077	−	22.6077i	−0.769577	−	0.769577i	0.208455	−	0.978032i	$-0.433156\pi$
$863$	−22.6077	−	22.6077i	−0.769577	−	0.769577i	−0.978032	+	0.208455i	$0.933156\pi$
$864$		0			0
$865$	0.781024	−	23.6652i	0.0265556	−	0.804641i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	29.5459			1.00228
$870$		0			0
$871$	−71.5868			−2.42563
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−36.2637	−	3.60090i	−1.22594	−	0.121733i
$876$		0			0
$877$	−26.5411	−	26.5411i	−0.896228	−	0.896228i	0.0988722	−	0.995100i	$-0.468476\pi$
$877$	−26.5411	−	26.5411i	−0.896228	−	0.896228i	−0.995100	+	0.0988722i	$0.968476\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			7.54565i			0.254219i	0.991889	+	0.127110i	$0.0405700\pi$
$881$			7.54565i			0.254219i	−0.991889	+	0.127110i	$0.959430\pi$
$882$		0			0
$883$	13.6779	−	13.6779i	0.460297	−	0.460297i	−0.438456	−	0.898753i	$-0.644474\pi$
$883$	13.6779	−	13.6779i	0.460297	−	0.460297i	0.898753	+	0.438456i	$0.144474\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−17.9261	+	17.9261i	−0.601899	+	0.601899i	−0.940816	−	0.338917i	$-0.889939\pi$
$887$	−17.9261	+	17.9261i	−0.601899	+	0.601899i	0.338917	+	0.940816i	$0.389939\pi$
$888$		0			0
$889$		−	20.7466i		−	0.695820i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.56519	+	2.56519i	0.0858407	+	0.0858407i
$894$		0			0
$895$	−10.6937	−	11.4237i	−0.357452	−	0.381851i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.97749			0.299416
$900$		0			0
$901$	5.32377			0.177361
$902$		0			0
$903$		0			0
$904$		0			0
$905$	19.6219	+	20.9612i	0.652253	+	0.696775i
$906$		0			0
$907$	−26.3530	−	26.3530i	−0.875038	−	0.875038i	0.117978	−	0.993016i	$-0.462359\pi$
$907$	−26.3530	−	26.3530i	−0.875038	−	0.875038i	−0.993016	+	0.117978i	$0.962359\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			33.9264i			1.12403i	0.827126	+	0.562016i	$0.189974\pi$
$911$			33.9264i			1.12403i	−0.827126	+	0.562016i	$0.810026\pi$
$912$		0			0
$913$	−7.35681	+	7.35681i	−0.243475	+	0.243475i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−16.8987	+	16.8987i	−0.558044	+	0.558044i
$918$		0			0
$919$		−	34.6588i		−	1.14329i	−0.820501	−	0.571645i	$-0.806306\pi$
$919$		−	34.6588i		−	1.14329i	0.820501	−	0.571645i	$-0.193694\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	34.6072	+	34.6072i	1.13911	+	1.13911i
$924$		0			0
$925$	−23.7871	+	20.8383i	−0.782115	+	0.685160i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−46.0724			−1.51159			−0.755794	−	0.654810i	$-0.772749\pi$
$929$	−46.0724			−1.51159			−0.755794	+	0.654810i	$0.772749\pi$
$930$		0			0
$931$	6.28139			0.205864
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.231928	+	7.02745i	−0.00758484	+	0.229822i
$936$		0			0
$937$	12.4255	+	12.4255i	0.405923	+	0.405923i	0.880314	−	0.474391i	$-0.157332\pi$
$937$	12.4255	+	12.4255i	0.405923	+	0.405923i	−0.474391	+	0.880314i	$0.657332\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	52.6602i		−	1.71667i	−0.513088	−	0.858336i	$-0.671498\pi$
$941$		−	52.6602i		−	1.71667i	0.513088	−	0.858336i	$-0.328502\pi$
$942$		0			0
$943$	−5.23299	+	5.23299i	−0.170410	+	0.170410i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.94840	+	2.94840i	−0.0958103	+	0.0958103i	−0.753387	−	0.657577i	$-0.771581\pi$
$947$	−2.94840	+	2.94840i	−0.0958103	+	0.0958103i	0.657577	+	0.753387i	$0.271581\pi$
$948$		0			0
$949$		−	14.2929i		−	0.463967i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	28.6925	+	28.6925i	0.929442	+	0.929442i	0.997670	−	0.0682275i	$-0.0217344\pi$
$953$	28.6925	+	28.6925i	0.929442	+	0.929442i	−0.0682275	+	0.997670i	$0.521734\pi$
$954$		0			0
$955$	−35.3781	+	33.1175i	−1.14481	+	1.07166i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.34787			−0.140400
$960$		0			0
$961$	−26.8230			−0.865258
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.83100	−	0.126435i	−0.123324	−	0.00407008i
$966$		0			0
$967$	−20.0078	−	20.0078i	−0.643408	−	0.643408i	0.307984	−	0.951392i	$-0.400346\pi$
$967$	−20.0078	−	20.0078i	−0.643408	−	0.643408i	−0.951392	+	0.307984i	$0.900346\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			24.5560i			0.788040i	0.919102	+	0.394020i	$0.128916\pi$
$971$			24.5560i			0.788040i	−0.919102	+	0.394020i	$0.871084\pi$
$972$		0			0
$973$	−25.0522	+	25.0522i	−0.803137	+	0.803137i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−21.2875	+	21.2875i	−0.681046	+	0.681046i	−0.960236	−	0.279190i	$-0.909934\pi$
$977$	−21.2875	+	21.2875i	−0.681046	+	0.681046i	0.279190	+	0.960236i	$0.409934\pi$
$978$		0			0
$979$			7.22585i			0.230939i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.2091	+	16.2091i	0.516990	+	0.516990i	0.916659	−	0.399669i	$-0.130875\pi$
$983$	16.2091	+	16.2091i	0.516990	+	0.516990i	−0.399669	+	0.916659i	$0.630875\pi$
$984$		0			0
$985$	−50.0192	−	1.65079i	−1.59375	−	0.0525985i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.22954			0.229886
$990$		0			0
$991$	45.0424			1.43082			0.715410	−	0.698705i	$-0.246240\pi$
$991$	45.0424			1.43082			0.715410	+	0.698705i	$0.246240\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−27.8773	+	26.0960i	−0.883770	+	0.827299i
$996$		0			0
$997$	31.6571	+	31.6571i	1.00259	+	1.00259i	0.999997	+	0.00259275i	$0.000825300\pi$
$997$	31.6571	+	31.6571i	1.00259	+	1.00259i	0.00259275	+	0.999997i	$0.499175\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.s.b.737.6	✓	44
3.2	odd	2	inner	4140.2.s.b.737.17	yes	44
5.3	odd	4	inner	4140.2.s.b.2393.17	yes	44
15.8	even	4	inner	4140.2.s.b.2393.6	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.b.737.6	✓	44	1.1	even	1	trivial
4140.2.s.b.737.17	yes	44	3.2	odd	2	inner
4140.2.s.b.2393.6	yes	44	15.8	even	4	inner
4140.2.s.b.2393.17	yes	44	5.3	odd	4	inner

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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.52812 - 1.63243i) q^{5} +(-2.30479 - 2.30479i) q^{7} +O(q^{10})\)	\(q+(-1.52812 - 1.63243i) q^{5} +(-2.30479 - 2.30479i) q^{7} +3.32736i q^{11} +(-3.34641 + 3.34641i) q^{13} +(-0.668244 + 0.668244i) q^{17} -1.73320i q^{19} +(-0.707107 - 0.707107i) q^{23} +(-0.329672 + 4.98912i) q^{25} +4.39260 q^{29} +2.04377 q^{31} +(-0.240409 + 7.28443i) q^{35} +(4.47227 + 4.47227i) q^{37} -7.40057i q^{41} +(-5.11206 + 5.11206i) q^{43} +(-1.48003 + 1.48003i) q^{47} +3.62415i q^{49} +(-3.98341 - 3.98341i) q^{53} +(5.43168 - 5.08461i) q^{55} -1.36490 q^{59} +3.91418 q^{61} +(10.5765 + 0.349058i) q^{65} +(10.6961 + 10.6961i) q^{67} -10.3416i q^{71} +(-2.13556 + 2.13556i) q^{73} +(7.66887 - 7.66887i) q^{77} -8.87969i q^{79} +(2.21101 + 2.21101i) q^{83} +(2.11202 + 0.0697032i) q^{85} +2.17165 q^{89} +15.4256 q^{91} +(-2.82934 + 2.64855i) q^{95} +(9.81888 + 9.81888i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{7}+O(q^{10}) \) 44 * q - 4 * q^7	\( 44 q - 4 q^{7} - 4 q^{13} - 24 q^{25} + 32 q^{31} + 40 q^{37} - 8 q^{43} - 24 q^{55} + 64 q^{61} + 12 q^{67} - 84 q^{73} - 104 q^{85} - 48 q^{91} + 44 q^{97}+O(q^{100}) \) 44 * q - 4 * q^7 - 4 * q^13 - 24 * q^25 + 32 * q^31 + 40 * q^37 - 8 * q^43 - 24 * q^55 + 64 * q^61 + 12 * q^67 - 84 * q^73 - 104 * q^85 - 48 * q^91 + 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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