LMFDB - Embedded newform 4140.2.s.a.737.19

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			737.19
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.a.2393.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.19878 + 0.406643i) q^{5} +(1.26371 + 1.26371i) q^{7} +O(q^{10})$	$q+(2.19878 + 0.406643i) q^{5} +(1.26371 + 1.26371i) q^{7} -3.93389i q^{11} +(1.46857 - 1.46857i) q^{13} +(-0.467340 + 0.467340i) q^{17} +4.88799i q^{19} +(0.707107 + 0.707107i) q^{23} +(4.66928 + 1.78824i) q^{25} -0.536884 q^{29} +3.91417 q^{31} +(2.26474 + 3.29250i) q^{35} +(-0.291992 - 0.291992i) q^{37} +11.7420i q^{41} +(-2.67235 + 2.67235i) q^{43} +(7.55616 - 7.55616i) q^{47} -3.80607i q^{49} +(-0.0492456 - 0.0492456i) q^{53} +(1.59969 - 8.64977i) q^{55} +9.00685 q^{59} +3.93000 q^{61} +(3.82624 - 2.63188i) q^{65} +(0.268143 + 0.268143i) q^{67} -15.6969i q^{71} +(-4.49050 + 4.49050i) q^{73} +(4.97130 - 4.97130i) q^{77} -1.65653i q^{79} +(5.31845 + 5.31845i) q^{83} +(-1.21762 + 0.837537i) q^{85} -6.48418 q^{89} +3.71169 q^{91} +(-1.98767 + 10.7476i) q^{95} +(11.1170 + 11.1170i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 12 q^{7}+O(q^{10}) $ 44 * q - 12 * q^7	$ 44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100}) $ 44 * q - 12 * q^7 - 4 * q^13 + 24 * q^25 - 48 * q^37 + 8 * q^43 + 40 * q^55 - 96 * q^61 - 44 * q^67 + 76 * q^73 + 72 * q^85 - 48 * q^91 - 20 * 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
$5$	2.19878	+	0.406643i	0.983325	+	0.181856i
$5$	2.19878	+	0.406643i	0.983325	+	0.181856i
$6$		0			0
$7$	1.26371	+	1.26371i	0.477637	+	0.477637i	0.904375	−	0.426738i	$-0.140337\pi$
$7$	1.26371	+	1.26371i	0.477637	+	0.477637i	−0.426738	+	0.904375i	$0.640337\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.93389i		−	1.18611i	−0.805161	−	0.593056i	$-0.797921\pi$
$11$		−	3.93389i		−	1.18611i	0.805161	−	0.593056i	$-0.202079\pi$
$12$		0			0
$13$	1.46857	−	1.46857i	0.407308	−	0.407308i	−0.473491	−	0.880799i	$-0.657006\pi$
$13$	1.46857	−	1.46857i	0.407308	−	0.407308i	0.880799	+	0.473491i	$0.157006\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.467340	+	0.467340i	−0.113346	+	0.113346i	−0.761505	−	0.648159i	$-0.775539\pi$
$17$	−0.467340	+	0.467340i	−0.113346	+	0.113346i	0.648159	+	0.761505i	$0.275539\pi$
$18$		0			0
$19$			4.88799i			1.12138i	0.828025	+	0.560691i	$0.189464\pi$
$19$			4.88799i			1.12138i	−0.828025	+	0.560691i	$0.810536\pi$
$20$		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
$25$	4.66928	+	1.78824i	0.933857	+	0.357648i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.536884			−0.0996968			−0.0498484	−	0.998757i	$-0.515874\pi$
$29$	−0.536884			−0.0996968			−0.0498484	+	0.998757i	$0.515874\pi$
$30$		0			0
$31$	3.91417			0.703005			0.351503	−	0.936187i	$-0.385671\pi$
$31$	3.91417			0.703005			0.351503	+	0.936187i	$0.385671\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.26474	+	3.29250i	0.382812	+	0.556534i
$36$		0			0
$37$	−0.291992	−	0.291992i	−0.0480033	−	0.0480033i	0.682698	−	0.730701i	$-0.260807\pi$
$37$	−0.291992	−	0.291992i	−0.0480033	−	0.0480033i	−0.730701	+	0.682698i	$0.760807\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			11.7420i			1.83380i	0.399118	+	0.916899i	$0.369316\pi$
$41$			11.7420i			1.83380i	−0.399118	+	0.916899i	$0.630684\pi$
$42$		0			0
$43$	−2.67235	+	2.67235i	−0.407529	+	0.407529i	−0.880876	−	0.473347i	$-0.843046\pi$
$43$	−2.67235	+	2.67235i	−0.407529	+	0.407529i	0.473347	+	0.880876i	$0.343046\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.55616	−	7.55616i	1.10218	−	1.10218i	0.108032	−	0.994147i	$-0.465545\pi$
$47$	7.55616	−	7.55616i	1.10218	−	1.10218i	0.994147	−	0.108032i	$-0.0344550\pi$
$48$		0			0
$49$		−	3.80607i		−	0.543725i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.0492456	−	0.0492456i	−0.00676441	−	0.00676441i	0.703716	−	0.710481i	$-0.251523\pi$
$53$	−0.0492456	−	0.0492456i	−0.00676441	−	0.00676441i	−0.710481	+	0.703716i	$0.751523\pi$
$54$		0			0
$55$	1.59969	−	8.64977i	0.215702	−	1.16633i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.00685			1.17259			0.586296	−	0.810097i	$-0.300586\pi$
$59$	9.00685			1.17259			0.586296	+	0.810097i	$0.300586\pi$
$60$		0			0
$61$	3.93000			0.503185			0.251592	−	0.967833i	$-0.419046\pi$
$61$	3.93000			0.503185			0.251592	+	0.967833i	$0.419046\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.82624	−	2.63188i	0.474587	−	0.326444i
$66$		0			0
$67$	0.268143	+	0.268143i	0.0327589	+	0.0327589i	0.723297	−	0.690538i	$-0.242626\pi$
$67$	0.268143	+	0.268143i	0.0327589	+	0.0327589i	−0.690538	+	0.723297i	$0.742626\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	15.6969i		−	1.86288i	−0.363890	−	0.931442i	$-0.618552\pi$
$71$		−	15.6969i		−	1.86288i	0.363890	−	0.931442i	$-0.381448\pi$
$72$		0			0
$73$	−4.49050	+	4.49050i	−0.525573	+	0.525573i	−0.919249	−	0.393676i	$-0.871203\pi$
$73$	−4.49050	+	4.49050i	−0.525573	+	0.525573i	0.393676	+	0.919249i	$0.371203\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.97130	−	4.97130i	0.566532	−	0.566532i
$78$		0			0
$79$		−	1.65653i		−	0.186374i	−0.995649	−	0.0931871i	$-0.970295\pi$
$79$		−	1.65653i		−	0.186374i	0.995649	−	0.0931871i	$-0.0297055\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.31845	+	5.31845i	0.583775	+	0.583775i	0.935939	−	0.352163i	$-0.114554\pi$
$83$	5.31845	+	5.31845i	0.583775	+	0.583775i	−0.352163	+	0.935939i	$0.614554\pi$
$84$		0			0
$85$	−1.21762	+	0.837537i	−0.132069	+	0.0908437i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.48418			−0.687321			−0.343661	−	0.939094i	$-0.611667\pi$
$89$	−6.48418			−0.687321			−0.343661	+	0.939094i	$0.611667\pi$
$90$		0			0
$91$	3.71169			0.389091
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.98767	+	10.7476i	−0.203930	+	1.10268i
$96$		0			0
$97$	11.1170	+	11.1170i	1.12876	+	1.12876i	0.990379	+	0.138381i	$0.0441897\pi$
$97$	11.1170	+	11.1170i	1.12876	+	1.12876i	0.138381	+	0.990379i	$0.455810\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	5.43796i		−	0.541097i	−0.962706	−	0.270548i	$-0.912795\pi$
$101$		−	5.43796i		−	0.541097i	0.962706	−	0.270548i	$-0.0872050\pi$
$102$		0			0
$103$	−4.98246	+	4.98246i	−0.490937	+	0.490937i	−0.908601	−	0.417665i	$-0.862849\pi$
$103$	−4.98246	+	4.98246i	−0.490937	+	0.490937i	0.417665	+	0.908601i	$0.362849\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.0289	−	10.0289i	0.969529	−	0.969529i	−0.0300201	−	0.999549i	$-0.509557\pi$
$107$	10.0289	−	10.0289i	0.969529	−	0.969529i	0.999549	+	0.0300201i	$0.00955714\pi$
$108$		0			0
$109$		−	0.864343i		−	0.0827890i	−0.999143	−	0.0413945i	$-0.986820\pi$
$109$		−	0.864343i		−	0.0827890i	0.999143	−	0.0413945i	$-0.0131800\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.87009	+	2.87009i	0.269995	+	0.269995i	0.829098	−	0.559103i	$-0.188854\pi$
$113$	2.87009	+	2.87009i	0.269995	+	0.269995i	−0.559103	+	0.829098i	$0.688854\pi$
$114$		0			0
$115$	1.26723	+	1.84231i	0.118170	+	0.171797i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.18116			−0.108277
$120$		0			0
$121$	−4.47550			−0.406864
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.53956	+	5.83068i	0.853244	+	0.521512i
$126$		0			0
$127$	−2.90838	−	2.90838i	−0.258077	−	0.258077i	0.566195	−	0.824272i	$-0.308415\pi$
$127$	−2.90838	−	2.90838i	−0.258077	−	0.258077i	−0.824272	+	0.566195i	$0.808415\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	10.9357i		−	0.955460i	−0.878507	−	0.477730i	$-0.841460\pi$
$131$		−	10.9357i		−	0.955460i	0.878507	−	0.477730i	$-0.158540\pi$
$132$		0			0
$133$	−6.17700	+	6.17700i	−0.535614	+	0.535614i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.796106	−	0.796106i	0.0680159	−	0.0680159i	−0.672281	−	0.740296i	$-0.734685\pi$
$137$	0.796106	−	0.796106i	0.0680159	−	0.0680159i	0.740296	+	0.672281i	$0.234685\pi$
$138$		0			0
$139$		−	3.22146i		−	0.273240i	−0.990624	−	0.136620i	$-0.956376\pi$
$139$		−	3.22146i		−	0.273240i	0.990624	−	0.136620i	$-0.0436240\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.77719	−	5.77719i	−0.483113	−	0.483113i
$144$		0			0
$145$	−1.18049	−	0.218320i	−0.0980343	−	0.0181305i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.43637			−0.691134			−0.345567	−	0.938394i	$-0.612313\pi$
$149$	−8.43637			−0.691134			−0.345567	+	0.938394i	$0.612313\pi$
$150$		0			0
$151$	−11.7645			−0.957380			−0.478690	−	0.877984i	$-0.658888\pi$
$151$	−11.7645			−0.957380			−0.478690	+	0.877984i	$0.658888\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.60640	+	1.59167i	0.691282	+	0.127846i
$156$		0			0
$157$	6.90704	+	6.90704i	0.551242	+	0.551242i	0.926799	−	0.375557i	$-0.122549\pi$
$157$	6.90704	+	6.90704i	0.551242	+	0.551242i	−0.375557	+	0.926799i	$0.622549\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			1.78716i			0.140848i
$162$		0			0
$163$	−7.97138	+	7.97138i	−0.624367	+	0.624367i	−0.946645	−	0.322278i	$-0.895551\pi$
$163$	−7.97138	+	7.97138i	−0.624367	+	0.624367i	0.322278	+	0.946645i	$0.395551\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.76681	+	2.76681i	−0.214102	+	0.214102i	−0.806008	−	0.591905i	$-0.798376\pi$
$167$	−2.76681	+	2.76681i	−0.214102	+	0.214102i	0.591905	+	0.806008i	$0.298376\pi$
$168$		0			0
$169$			8.68661i			0.668201i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.48348	+	6.48348i	0.492930	+	0.492930i	0.909228	−	0.416298i	$-0.136673\pi$
$173$	6.48348	+	6.48348i	0.492930	+	0.492930i	−0.416298	+	0.909228i	$0.636673\pi$
$174$		0			0
$175$	3.64080	+	8.16043i	0.275219	+	0.616871i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.02924			0.151673			0.0758363	−	0.997120i	$-0.475837\pi$
$179$	2.02924			0.151673			0.0758363	+	0.997120i	$0.475837\pi$
$180$		0			0
$181$	−0.306297			−0.0227669			−0.0113834	−	0.999935i	$-0.503624\pi$
$181$	−0.306297			−0.0227669			−0.0113834	+	0.999935i	$0.503624\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.523291	−	0.760764i	−0.0384731	−	0.0559325i
$186$		0			0
$187$	1.83846	+	1.83846i	0.134442	+	0.134442i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	9.81264i		−	0.710018i	−0.934863	−	0.355009i	$-0.884478\pi$
$191$		−	9.81264i		−	0.710018i	0.934863	−	0.355009i	$-0.115522\pi$
$192$		0			0
$193$	11.6079	−	11.6079i	0.835553	−	0.835553i	−0.152717	−	0.988270i	$-0.548802\pi$
$193$	11.6079	−	11.6079i	0.835553	−	0.835553i	0.988270	+	0.152717i	$0.0488022\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	7.99076	−	7.99076i	0.569318	−	0.569318i	−0.362619	−	0.931937i	$-0.618117\pi$
$197$	7.99076	−	7.99076i	0.569318	−	0.569318i	0.931937	+	0.362619i	$0.118117\pi$
$198$		0			0
$199$			26.9413i			1.90982i	0.296900	+	0.954909i	$0.404047\pi$
$199$			26.9413i			1.90982i	−0.296900	+	0.954909i	$0.595953\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.678465	−	0.678465i	−0.0476189	−	0.0476189i
$204$		0			0
$205$	−4.77482	+	25.8182i	−0.333488	+	1.80322i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	19.2288			1.33009
$210$		0			0
$211$	−17.7365			−1.22103			−0.610515	−	0.792004i	$-0.709038\pi$
$211$	−17.7365			−1.22103			−0.610515	+	0.792004i	$0.709038\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.96260	+	4.78922i	−0.474846	+	0.326622i
$216$		0			0
$217$	4.94637	+	4.94637i	0.335782	+	0.335782i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			1.37264i			0.0923338i
$222$		0			0
$223$	0.393884	−	0.393884i	0.0263764	−	0.0263764i	−0.693796	−	0.720172i	$-0.744063\pi$
$223$	0.393884	−	0.393884i	0.0263764	−	0.0263764i	0.720172	+	0.693796i	$0.244063\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.46778	−	9.46778i	0.628398	−	0.628398i	−0.319267	−	0.947665i	$-0.603437\pi$
$227$	9.46778	−	9.46778i	0.628398	−	0.628398i	0.947665	+	0.319267i	$0.103437\pi$
$228$		0			0
$229$			26.3328i			1.74012i	0.492943	+	0.870061i	$0.335921\pi$
$229$			26.3328i			1.74012i	−0.492943	+	0.870061i	$0.664079\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.36032	+	3.36032i	0.220142	+	0.220142i	0.808558	−	0.588416i	$-0.200248\pi$
$233$	3.36032	+	3.36032i	0.220142	+	0.220142i	−0.588416	+	0.808558i	$0.700248\pi$
$234$		0			0
$235$	19.6870	−	13.5417i	1.28424	−	0.883363i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	17.5102			1.13264			0.566320	−	0.824186i	$-0.308367\pi$
$239$	17.5102			1.13264			0.566320	+	0.824186i	$0.308367\pi$
$240$		0			0
$241$	−19.4650			−1.25385			−0.626924	−	0.779080i	$-0.715686\pi$
$241$	−19.4650			−1.25385			−0.626924	+	0.779080i	$0.715686\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.54771	−	8.36873i	0.0988798	−	0.534658i
$246$		0			0
$247$	7.17835	+	7.17835i	0.456747	+	0.456747i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	1.27863i		−	0.0807063i	−0.999185	−	0.0403531i	$-0.987152\pi$
$251$		−	1.27863i		−	0.0807063i	0.999185	−	0.0403531i	$-0.0128483\pi$
$252$		0			0
$253$	2.78168	−	2.78168i	0.174883	−	0.174883i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−15.8009	+	15.8009i	−0.985630	+	0.985630i	−0.999898	−	0.0142682i	$-0.995458\pi$
$257$	−15.8009	+	15.8009i	−0.985630	+	0.985630i	0.0142682	+	0.999898i	$0.495458\pi$
$258$		0			0
$259$		−	0.737987i		−	0.0458563i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.53542	−	3.53542i	−0.218003	−	0.218003i	0.589653	−	0.807657i	$-0.299264\pi$
$263$	−3.53542	−	3.53542i	−0.218003	−	0.218003i	−0.807657	+	0.589653i	$0.799264\pi$
$264$		0			0
$265$	−0.0882550	−	0.128306i	−0.00542146	−	0.00788176i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.02647			−0.0625852			−0.0312926	−	0.999510i	$-0.509962\pi$
$269$	−1.02647			−0.0625852			−0.0312926	+	0.999510i	$0.509962\pi$
$270$		0			0
$271$	18.2657			1.10956			0.554780	−	0.831997i	$-0.312802\pi$
$271$	18.2657			1.10956			0.554780	+	0.831997i	$0.312802\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.03474	−	18.3685i	0.424211	−	1.10766i
$276$		0			0
$277$	−6.20977	−	6.20977i	−0.373109	−	0.373109i	0.495499	−	0.868608i	$-0.334985\pi$
$277$	−6.20977	−	6.20977i	−0.373109	−	0.373109i	−0.868608	+	0.495499i	$0.834985\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	20.3062i		−	1.21137i	−0.795706	−	0.605683i	$-0.792900\pi$
$281$		−	20.3062i		−	1.21137i	0.795706	−	0.605683i	$-0.207100\pi$
$282$		0			0
$283$	19.5140	−	19.5140i	1.15999	−	1.15999i	0.175511	−	0.984478i	$-0.443842\pi$
$283$	19.5140	−	19.5140i	1.15999	−	1.15999i	0.984478	−	0.175511i	$-0.0561577\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−14.8385	+	14.8385i	−0.875891	+	0.875891i
$288$		0			0
$289$			16.5632i			0.974305i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−11.7998	−	11.7998i	−0.689351	−	0.689351i	0.272738	−	0.962088i	$-0.412071\pi$
$293$	−11.7998	−	11.7998i	−0.689351	−	0.689351i	−0.962088	+	0.272738i	$0.912071\pi$
$294$		0			0
$295$	19.8041	+	3.66257i	1.15304	+	0.213243i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.07687			0.120108
$300$		0			0
$301$	−6.75415			−0.389303
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8.64121	+	1.59811i	0.494794	+	0.0915073i
$306$		0			0
$307$	−17.1493	−	17.1493i	−0.978761	−	0.978761i	0.0210185	−	0.999779i	$-0.493309\pi$
$307$	−17.1493	−	17.1493i	−0.978761	−	0.978761i	−0.999779	+	0.0210185i	$0.993309\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	8.69692i		−	0.493157i	−0.969123	−	0.246578i	$-0.920694\pi$
$311$		−	8.69692i		−	0.493157i	0.969123	−	0.246578i	$-0.0793063\pi$
$312$		0			0
$313$	6.48845	−	6.48845i	0.366749	−	0.366749i	−0.499541	−	0.866290i	$-0.666498\pi$
$313$	6.48845	−	6.48845i	0.366749	−	0.366749i	0.866290	+	0.499541i	$0.166498\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	20.0512	−	20.0512i	1.12619	−	1.12619i	0.135395	−	0.990792i	$-0.456770\pi$
$317$	20.0512	−	20.0512i	1.12619	−	1.12619i	0.990792	−	0.135395i	$-0.0432302\pi$
$318$		0			0
$319$			2.11204i			0.118252i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.28435	−	2.28435i	−0.127105	−	0.127105i
$324$		0			0
$325$	9.48331	−	4.23101i	0.526040	−	0.234694i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	19.0976			1.05288
$330$		0			0
$331$	−1.50008			−0.0824521			−0.0412260	−	0.999150i	$-0.513126\pi$
$331$	−1.50008			−0.0824521			−0.0412260	+	0.999150i	$0.513126\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.480550	+	0.698628i	0.0262553	+	0.0381701i
$336$		0			0
$337$	−15.6110	−	15.6110i	−0.850384	−	0.850384i	0.139797	−	0.990180i	$-0.455355\pi$
$337$	−15.6110	−	15.6110i	−0.850384	−	0.850384i	−0.990180	+	0.139797i	$0.955355\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	15.3979i		−	0.833843i
$342$		0			0
$343$	13.6557	−	13.6557i	0.737341	−	0.737341i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.0235	−	15.0235i	0.806505	−	0.806505i	−0.177598	−	0.984103i	$-0.556833\pi$
$347$	15.0235	−	15.0235i	0.806505	−	0.806505i	0.984103	+	0.177598i	$0.0568328\pi$
$348$		0			0
$349$		−	4.40148i		−	0.235606i	−0.993037	−	0.117803i	$-0.962415\pi$
$349$		−	4.40148i		−	0.235606i	0.993037	−	0.117803i	$-0.0375851\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	22.6196	+	22.6196i	1.20392	+	1.20392i	0.972964	+	0.230956i	$0.0741852\pi$
$353$	22.6196	+	22.6196i	1.20392	+	1.20392i	0.230956	+	0.972964i	$0.425815\pi$
$354$		0			0
$355$	6.38305	−	34.5141i	0.338777	−	1.83182i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.29969			−0.226929			−0.113465	−	0.993542i	$-0.536195\pi$
$359$	−4.29969			−0.226929			−0.113465	+	0.993542i	$0.536195\pi$
$360$		0			0
$361$	−4.89244			−0.257497
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−11.6997	+	8.04760i	−0.612388	+	0.421231i
$366$		0			0
$367$	−4.83823	−	4.83823i	−0.252553	−	0.252553i	0.569463	−	0.822017i	$-0.307151\pi$
$367$	−4.83823	−	4.83823i	−0.252553	−	0.252553i	−0.822017	+	0.569463i	$0.807151\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	0.124464i		−	0.00646187i
$372$		0			0
$373$	−19.7501	+	19.7501i	−1.02262	+	1.02262i	−0.0228809	+	0.999738i	$0.507284\pi$
$373$	−19.7501	+	19.7501i	−1.02262	+	1.02262i	−0.999738	+	0.0228809i	$0.992716\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.788450	+	0.788450i	−0.0406073	+	0.0406073i
$378$		0			0
$379$			10.8544i			0.557551i	0.960356	+	0.278776i	$0.0899286\pi$
$379$			10.8544i			0.557551i	−0.960356	+	0.278776i	$0.910071\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	19.6060	+	19.6060i	1.00182	+	1.00182i	0.999998	+	0.00182390i	$0.000580565\pi$
$383$	19.6060	+	19.6060i	1.00182	+	1.00182i	0.00182390	+	0.999998i	$0.499419\pi$
$384$		0			0
$385$	12.9523	−	8.90925i	0.660112	−	0.454058i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5.01117			−0.254076			−0.127038	−	0.991898i	$-0.540547\pi$
$389$	−5.01117			−0.254076			−0.127038	+	0.991898i	$0.540547\pi$
$390$		0			0
$391$	−0.660918			−0.0334241
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.673617	−	3.64235i	0.0338933	−	0.183266i
$396$		0			0
$397$	21.0975	+	21.0975i	1.05885	+	1.05885i	0.998156	+	0.0606979i	$0.0193326\pi$
$397$	21.0975	+	21.0975i	1.05885	+	1.05885i	0.0606979	+	0.998156i	$0.480667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			29.6297i			1.47964i	0.672807	+	0.739818i	$0.265088\pi$
$401$			29.6297i			1.47964i	−0.672807	+	0.739818i	$0.734912\pi$
$402$		0			0
$403$	5.74822	−	5.74822i	0.286339	−	0.286339i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.14867	+	1.14867i	−0.0569373	+	0.0569373i
$408$		0			0
$409$			24.7731i			1.22495i	0.790490	+	0.612474i	$0.209826\pi$
$409$			24.7731i			1.22495i	−0.790490	+	0.612474i	$0.790174\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	11.3820	+	11.3820i	0.560074	+	0.560074i
$414$		0			0
$415$	9.53140	+	13.8568i	0.467878	+	0.680204i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−28.8500			−1.40941			−0.704707	−	0.709498i	$-0.748922\pi$
$419$	−28.8500			−1.40941			−0.704707	+	0.709498i	$0.748922\pi$
$420$		0			0
$421$	−24.9134			−1.21420			−0.607102	−	0.794624i	$-0.707668\pi$
$421$	−24.9134			−1.21420			−0.607102	+	0.794624i	$0.707668\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.01786	+	1.34643i	−0.146387	+	0.0653113i
$426$		0			0
$427$	4.96638	+	4.96638i	0.240340	+	0.240340i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	8.14115i		−	0.392145i	−0.980589	−	0.196073i	$-0.937181\pi$
$431$		−	8.14115i		−	0.392145i	0.980589	−	0.196073i	$-0.0628188\pi$
$432$		0			0
$433$	16.7018	−	16.7018i	0.802637	−	0.802637i	−0.180870	−	0.983507i	$-0.557891\pi$
$433$	16.7018	−	16.7018i	0.802637	−	0.802637i	0.983507	+	0.180870i	$0.0578913\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.45633	+	3.45633i	−0.165339	+	0.165339i
$438$		0			0
$439$			0.830843i			0.0396540i	0.999803	+	0.0198270i	$0.00631154\pi$
$439$			0.830843i			0.0396540i	−0.999803	+	0.0198270i	$0.993688\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.41440	−	9.41440i	−0.447291	−	0.447291i	0.447162	−	0.894453i	$-0.352435\pi$
$443$	−9.41440	−	9.41440i	−0.447291	−	0.447291i	−0.894453	+	0.447162i	$0.852435\pi$
$444$		0			0
$445$	−14.2573	−	2.63674i	−0.675860	−	0.124994i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9.90480			−0.467436			−0.233718	−	0.972304i	$-0.575089\pi$
$449$	−9.90480			−0.467436			−0.233718	+	0.972304i	$0.575089\pi$
$450$		0			0
$451$	46.1919			2.17509
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.16119	+	1.50933i	0.382603	+	0.0707586i
$456$		0			0
$457$	−24.7528	−	24.7528i	−1.15789	−	1.15789i	−0.984929	−	0.172957i	$-0.944668\pi$
$457$	−24.7528	−	24.7528i	−1.15789	−	1.15789i	−0.172957	−	0.984929i	$-0.555332\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			6.52853i			0.304064i	0.988376	+	0.152032i	$0.0485817\pi$
$461$			6.52853i			0.304064i	−0.988376	+	0.152032i	$0.951418\pi$
$462$		0			0
$463$	2.02407	−	2.02407i	0.0940666	−	0.0940666i	−0.658508	−	0.752574i	$-0.728812\pi$
$463$	2.02407	−	2.02407i	0.0940666	−	0.0940666i	0.752574	+	0.658508i	$0.228812\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.0755	−	20.0755i	0.928982	−	0.928982i	−0.0686583	−	0.997640i	$-0.521872\pi$
$467$	20.0755	−	20.0755i	0.928982	−	0.928982i	0.997640	+	0.0686583i	$0.0218718\pi$
$468$		0			0
$469$			0.677711i			0.0312938i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.5127	+	10.5127i	0.483376	+	0.483376i
$474$		0			0
$475$	−8.74089	+	22.8234i	−0.401060	+	1.04721i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.2808			−0.561123			−0.280561	−	0.959836i	$-0.590521\pi$
$479$	−12.2808			−0.561123			−0.280561	+	0.959836i	$0.590521\pi$
$480$		0			0
$481$	−0.857622			−0.0391042
$482$		0			0
$483$		0			0
$484$		0			0
$485$	19.9232	+	28.9645i	0.904666	+	1.31521i
$486$		0			0
$487$	−0.770287	−	0.770287i	−0.0349050	−	0.0349050i	0.689439	−	0.724344i	$-0.257857\pi$
$487$	−0.770287	−	0.770287i	−0.0349050	−	0.0349050i	−0.724344	+	0.689439i	$0.757857\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			26.0756i			1.17677i	0.808579	+	0.588387i	$0.200237\pi$
$491$			26.0756i			1.17677i	−0.808579	+	0.588387i	$0.799763\pi$
$492$		0			0
$493$	0.250907	−	0.250907i	0.0113003	−	0.0113003i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	19.8364	−	19.8364i	0.889783	−	0.889783i
$498$		0			0
$499$		−	0.0541912i		−	0.00242593i	−0.999999	−	0.00121297i	$-0.999614\pi$
$499$		−	0.0541912i		−	0.00242593i	0.999999	−	0.00121297i	$-0.000386099\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12.1445	−	12.1445i	−0.541495	−	0.541495i	0.382472	−	0.923967i	$-0.375073\pi$
$503$	−12.1445	−	12.1445i	−0.541495	−	0.541495i	−0.923967	+	0.382472i	$0.875073\pi$
$504$		0			0
$505$	2.21131	−	11.9569i	0.0984019	−	0.532074i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.19346			−0.0528994			−0.0264497	−	0.999650i	$-0.508420\pi$
$509$	−1.19346			−0.0528994			−0.0264497	+	0.999650i	$0.508420\pi$
$510$		0			0
$511$	−11.3494			−0.502067
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12.9814	+	8.92926i	−0.572030	+	0.393470i
$516$		0			0
$517$	−29.7251	−	29.7251i	−1.30731	−	1.30731i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			27.3453i			1.19802i	0.800742	+	0.599010i	$0.204439\pi$
$521$			27.3453i			1.19802i	−0.800742	+	0.599010i	$0.795561\pi$
$522$		0			0
$523$	−2.62272	+	2.62272i	−0.114684	+	0.114684i	−0.762120	−	0.647436i	$-0.775841\pi$
$523$	−2.62272	+	2.62272i	−0.114684	+	0.114684i	0.647436	+	0.762120i	$0.275841\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.82924	+	1.82924i	−0.0796832	+	0.0796832i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	17.2440	+	17.2440i	0.746920	+	0.746920i
$534$		0			0
$535$	26.1295	−	17.9732i	1.12968	−	0.777047i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−14.9727			−0.644919
$540$		0			0
$541$	−15.2452			−0.655442			−0.327721	−	0.944775i	$-0.606281\pi$
$541$	−15.2452			−0.655442			−0.327721	+	0.944775i	$0.606281\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.351479	−	1.90050i	0.0150557	−	0.0814085i
$546$		0			0
$547$	−1.10303	−	1.10303i	−0.0471620	−	0.0471620i	0.683132	−	0.730294i	$-0.260617\pi$
$547$	−1.10303	−	1.10303i	−0.0471620	−	0.0471620i	−0.730294	+	0.683132i	$0.760617\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	2.62428i		−	0.111798i
$552$		0			0
$553$	2.09337	−	2.09337i	0.0890193	−	0.0890193i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−16.3180	+	16.3180i	−0.691417	+	0.691417i	−0.962544	−	0.271126i	$-0.912604\pi$
$557$	−16.3180	+	16.3180i	−0.691417	+	0.691417i	0.271126	+	0.962544i	$0.412604\pi$
$558$		0			0
$559$			7.84905i			0.331980i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.2319	−	19.2319i	−0.810530	−	0.810530i	0.174183	−	0.984713i	$-0.444271\pi$
$563$	−19.2319	−	19.2319i	−0.810530	−	0.810530i	−0.984713	+	0.174183i	$0.944271\pi$
$564$		0			0
$565$	5.14359	+	7.47780i	0.216393	+	0.314593i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−27.4879			−1.15235			−0.576175	−	0.817326i	$-0.695455\pi$
$569$	−27.4879			−1.15235			−0.576175	+	0.817326i	$0.695455\pi$
$570$		0			0
$571$	29.9272			1.25242			0.626208	−	0.779656i	$-0.284606\pi$
$571$	29.9272			1.25242			0.626208	+	0.779656i	$0.284606\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.03721	+	4.56616i	0.0849574	+	0.190422i
$576$		0			0
$577$	11.9441	+	11.9441i	0.497239	+	0.497239i	0.910577	−	0.413339i	$-0.135637\pi$
$577$	11.9441	+	11.9441i	0.497239	+	0.497239i	−0.413339	+	0.910577i	$0.635637\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			13.4419i			0.557666i
$582$		0			0
$583$	−0.193727	+	0.193727i	−0.00802335	+	0.00802335i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.3438	+	14.3438i	−0.592033	+	0.592033i	−0.938180	−	0.346147i	$-0.887490\pi$
$587$	−14.3438	+	14.3438i	−0.592033	+	0.592033i	0.346147	+	0.938180i	$0.387490\pi$
$588$		0			0
$589$			19.1324i			0.788337i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.53829	+	9.53829i	0.391691	+	0.391691i	0.875290	−	0.483599i	$-0.160671\pi$
$593$	9.53829	+	9.53829i	0.391691	+	0.391691i	−0.483599	+	0.875290i	$0.660671\pi$
$594$		0			0
$595$	−2.59712	−	0.480312i	−0.106472	−	0.0196909i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−22.6483			−0.925385			−0.462692	−	0.886519i	$-0.653117\pi$
$599$	−22.6483			−0.925385			−0.462692	+	0.886519i	$0.653117\pi$
$600$		0			0
$601$	−38.2388			−1.55979			−0.779897	−	0.625908i	$-0.784729\pi$
$601$	−38.2388			−1.55979			−0.779897	+	0.625908i	$0.784729\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9.84065	−	1.81993i	−0.400079	−	0.0739907i
$606$		0			0
$607$	−30.3726	−	30.3726i	−1.23279	−	1.23279i	−0.962889	−	0.269896i	$-0.913011\pi$
$607$	−30.3726	−	30.3726i	−1.23279	−	1.23279i	−0.269896	−	0.962889i	$-0.586989\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	22.1935i		−	0.897852i
$612$		0			0
$613$	15.0192	−	15.0192i	0.606619	−	0.606619i	−0.335442	−	0.942061i	$-0.608886\pi$
$613$	15.0192	−	15.0192i	0.606619	−	0.606619i	0.942061	+	0.335442i	$0.108886\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−13.0154	+	13.0154i	−0.523979	+	0.523979i	−0.918771	−	0.394792i	$-0.870817\pi$
$617$	−13.0154	+	13.0154i	−0.523979	+	0.523979i	0.394792	+	0.918771i	$0.370817\pi$
$618$		0			0
$619$			42.1327i			1.69346i	0.532026	+	0.846728i	$0.321431\pi$
$619$			42.1327i			1.69346i	−0.532026	+	0.846728i	$0.678569\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.19412	−	8.19412i	−0.328290	−	0.328290i
$624$		0			0
$625$	18.6044	+	16.6996i	0.744176	+	0.667983i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.272919			0.0108820
$630$		0			0
$631$	−16.3587			−0.651229			−0.325614	−	0.945503i	$-0.605571\pi$
$631$	−16.3587			−0.651229			−0.325614	+	0.945503i	$0.605571\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.21222	−	7.57756i	−0.206841	−	0.300706i
$636$		0			0
$637$	−5.58948	−	5.58948i	−0.221463	−	0.221463i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			5.37462i			0.212285i	0.994351	+	0.106142i	$0.0338499\pi$
$641$			5.37462i			0.212285i	−0.994351	+	0.106142i	$0.966150\pi$
$642$		0			0
$643$	−27.1071	+	27.1071i	−1.06900	+	1.06900i	−0.0715655	+	0.997436i	$0.522799\pi$
$643$	−27.1071	+	27.1071i	−1.06900	+	1.06900i	−0.997436	+	0.0715655i	$0.977201\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−14.4388	+	14.4388i	−0.567646	+	0.567646i	−0.931468	−	0.363822i	$-0.881472\pi$
$647$	−14.4388	+	14.4388i	−0.567646	+	0.567646i	0.363822	+	0.931468i	$0.381472\pi$
$648$		0			0
$649$		−	35.4320i		−	1.39083i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−32.7417	−	32.7417i	−1.28128	−	1.28128i	−0.939940	−	0.341340i	$-0.889119\pi$
$653$	−32.7417	−	32.7417i	−1.28128	−	1.28128i	−0.341340	−	0.939940i	$-0.610881\pi$
$654$		0			0
$655$	4.44694	−	24.0453i	0.173756	−	0.939528i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.31312			−0.323833			−0.161917	−	0.986804i	$-0.551768\pi$
$659$	−8.31312			−0.323833			−0.161917	+	0.986804i	$0.551768\pi$
$660$		0			0
$661$	8.50698			0.330883			0.165442	−	0.986220i	$-0.447095\pi$
$661$	8.50698			0.330883			0.165442	+	0.986220i	$0.447095\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−16.0937	+	11.0700i	−0.624087	+	0.429278i
$666$		0			0
$667$	−0.379634	−	0.379634i	−0.0146995	−	0.0146995i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	15.4602i		−	0.596834i
$672$		0			0
$673$	31.2238	−	31.2238i	1.20359	−	1.20359i	0.230520	−	0.973067i	$-0.425957\pi$
$673$	31.2238	−	31.2238i	1.20359	−	1.20359i	0.973067	−	0.230520i	$-0.0740429\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.9835	+	18.9835i	−0.729596	+	0.729596i	−0.970539	−	0.240943i	$-0.922543\pi$
$677$	−18.9835	+	18.9835i	−0.729596	+	0.729596i	0.240943	+	0.970539i	$0.422543\pi$
$678$		0			0
$679$			28.0973i			1.07828i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.92562	−	2.92562i	−0.111946	−	0.111946i	0.648915	−	0.760861i	$-0.275223\pi$
$683$	−2.92562	−	2.92562i	−0.111946	−	0.111946i	−0.760861	+	0.648915i	$0.775223\pi$
$684$		0			0
$685$	2.07419	−	1.42673i	0.0792508	−	0.0545126i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.144641			−0.00551039
$690$		0			0
$691$	13.8247			0.525917			0.262958	−	0.964807i	$-0.415302\pi$
$691$	13.8247			0.525917			0.262958	+	0.964807i	$0.415302\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.30998	−	7.08328i	0.0496905	−	0.268684i
$696$		0			0
$697$	−5.48752	−	5.48752i	−0.207855	−	0.207855i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			12.4348i			0.469656i	0.972037	+	0.234828i	$0.0754527\pi$
$701$			12.4348i			0.469656i	−0.972037	+	0.234828i	$0.924547\pi$
$702$		0			0
$703$	1.42726	−	1.42726i	0.0538300	−	0.0538300i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.87200	−	6.87200i	0.258448	−	0.258448i
$708$		0			0
$709$			10.8759i			0.408452i	0.978924	+	0.204226i	$0.0654677\pi$
$709$			10.8759i			0.408452i	−0.978924	+	0.204226i	$0.934532\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.76773	+	2.76773i	0.103652	+	0.103652i
$714$		0			0
$715$	−10.3535	−	15.0520i	−0.387200	−	0.562914i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3.47194			−0.129482			−0.0647408	−	0.997902i	$-0.520622\pi$
$719$	−3.47194			−0.129482			−0.0647408	+	0.997902i	$0.520622\pi$
$720$		0			0
$721$	−12.5928			−0.468979
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.50686	−	0.960076i	−0.0931025	−	0.0356563i
$726$		0			0
$727$	−29.8844	−	29.8844i	−1.10835	−	1.10835i	−0.993368	−	0.114983i	$-0.963319\pi$
$727$	−29.8844	−	29.8844i	−1.10835	−	1.10835i	−0.114983	−	0.993368i	$-0.536681\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	2.49779i		−	0.0923840i
$732$		0			0
$733$	−2.60946	+	2.60946i	−0.0963825	+	0.0963825i	−0.753654	−	0.657271i	$-0.771711\pi$
$733$	−2.60946	+	2.60946i	−0.0963825	+	0.0963825i	0.657271	+	0.753654i	$0.271711\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.05485	−	1.05485i	0.0388558	−	0.0388558i
$738$		0			0
$739$		−	42.7511i		−	1.57262i	−0.617830	−	0.786312i	$-0.711988\pi$
$739$		−	42.7511i		−	1.57262i	0.617830	−	0.786312i	$-0.288012\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−26.9393	−	26.9393i	−0.988309	−	0.988309i	0.0116238	−	0.999932i	$-0.496300\pi$
$743$	−26.9393	−	26.9393i	−0.988309	−	0.988309i	−0.999932	+	0.0116238i	$0.996300\pi$
$744$		0			0
$745$	−18.5497	−	3.43059i	−0.679610	−	0.125687i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	25.3472			0.926167
$750$		0			0
$751$	8.64546			0.315477			0.157739	−	0.987481i	$-0.449580\pi$
$751$	8.64546			0.315477			0.157739	+	0.987481i	$0.449580\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−25.8675	−	4.78394i	−0.941416	−	0.174106i
$756$		0			0
$757$	−28.7226	−	28.7226i	−1.04394	−	1.04394i	−0.998989	−	0.0449510i	$-0.985687\pi$
$757$	−28.7226	−	28.7226i	−1.04394	−	1.04394i	−0.0449510	−	0.998989i	$-0.514313\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	40.2064i		−	1.45748i	−0.684789	−	0.728741i	$-0.740106\pi$
$761$		−	40.2064i		−	1.45748i	0.684789	−	0.728741i	$-0.259894\pi$
$762$		0			0
$763$	1.09228	−	1.09228i	0.0395431	−	0.0395431i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	13.2272	−	13.2272i	0.477605	−	0.477605i
$768$		0			0
$769$			20.1636i			0.727117i	0.931571	+	0.363559i	$0.118438\pi$
$769$			20.1636i			0.727117i	−0.931571	+	0.363559i	$0.881562\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	25.2663	+	25.2663i	0.908765	+	0.908765i	0.996173	−	0.0874080i	$-0.0278584\pi$
$773$	25.2663	+	25.2663i	0.908765	+	0.908765i	−0.0874080	+	0.996173i	$0.527858\pi$
$774$		0			0
$775$	18.2763	+	6.99946i	0.656506	+	0.251428i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−57.3950			−2.05639
$780$		0			0
$781$	−61.7501			−2.20959
$782$		0			0
$783$		0			0
$784$		0			0
$785$	12.3784	+	17.9958i	0.441803	+	0.642297i
$786$		0			0
$787$	−19.0070	−	19.0070i	−0.677527	−	0.677527i	0.281913	−	0.959440i	$-0.409031\pi$
$787$	−19.0070	−	19.0070i	−0.677527	−	0.677527i	−0.959440	+	0.281913i	$0.909031\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			7.25391i			0.257920i
$792$		0			0
$793$	5.77147	−	5.77147i	0.204951	−	0.204951i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	37.2392	−	37.2392i	1.31908	−	1.31908i	0.404575	−	0.914505i	$-0.367420\pi$
$797$	37.2392	−	37.2392i	1.31908	−	1.31908i	0.914505	−	0.404575i	$-0.132580\pi$
$798$		0			0
$799$			7.06259i			0.249856i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	17.6651	+	17.6651i	0.623389	+	0.623389i
$804$		0			0
$805$	−0.726734	+	3.92957i	−0.0256140	+	0.138499i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−27.4373			−0.964643			−0.482322	−	0.875994i	$-0.660206\pi$
$809$	−27.4373			−0.964643			−0.482322	+	0.875994i	$0.660206\pi$
$810$		0			0
$811$	21.1952			0.744263			0.372132	−	0.928180i	$-0.378627\pi$
$811$	21.1952			0.744263			0.372132	+	0.928180i	$0.378627\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−20.7688	+	14.2858i	−0.727501	+	0.500411i
$816$		0			0
$817$	−13.0624	−	13.0624i	−0.456996	−	0.456996i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	31.5332i		−	1.10052i	−0.834994	−	0.550258i	$-0.814529\pi$
$821$		−	31.5332i		−	1.10052i	0.834994	−	0.550258i	$-0.185471\pi$
$822$		0			0
$823$	−14.1329	+	14.1329i	−0.492643	+	0.492643i	−0.909138	−	0.416495i	$-0.863258\pi$
$823$	−14.1329	+	14.1329i	−0.492643	+	0.492643i	0.416495	+	0.909138i	$0.363258\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.8020	−	20.8020i	0.723357	−	0.723357i	−0.245930	−	0.969288i	$-0.579093\pi$
$827$	20.8020	−	20.8020i	0.723357	−	0.723357i	0.969288	+	0.245930i	$0.0790934\pi$
$828$		0			0
$829$			8.15733i			0.283316i	0.989916	+	0.141658i	$0.0452433\pi$
$829$			8.15733i			0.283316i	−0.989916	+	0.141658i	$0.954757\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.77873	+	1.77873i	0.0616293	+	0.0616293i
$834$		0			0
$835$	−7.20872	+	4.95851i	−0.249468	+	0.171596i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−25.0363			−0.864348			−0.432174	−	0.901790i	$-0.642253\pi$
$839$	−25.0363			−0.864348			−0.432174	+	0.901790i	$0.642253\pi$
$840$		0			0
$841$	−28.7118			−0.990061
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.53235	+	19.1000i	−0.121517	+	0.657059i
$846$		0			0
$847$	−5.65573	−	5.65573i	−0.194333	−	0.194333i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	0.412940i		−	0.0141554i
$852$		0			0
$853$	−16.5195	+	16.5195i	−0.565616	+	0.565616i	−0.930897	−	0.365281i	$-0.880973\pi$
$853$	−16.5195	+	16.5195i	−0.565616	+	0.565616i	0.365281	+	0.930897i	$0.380973\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.0281	+	20.0281i	−0.684145	+	0.684145i	−0.960932	−	0.276786i	$-0.910731\pi$
$857$	−20.0281	+	20.0281i	−0.684145	+	0.684145i	0.276786	+	0.960932i	$0.410731\pi$
$858$		0			0
$859$		−	6.96022i		−	0.237480i	−0.992925	−	0.118740i	$-0.962115\pi$
$859$		−	6.96022i		−	0.237480i	0.992925	−	0.118740i	$-0.0378855\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.93309	−	3.93309i	−0.133884	−	0.133884i	0.636989	−	0.770873i	$-0.280180\pi$
$863$	−3.93309	−	3.93309i	−0.133884	−	0.133884i	−0.770873	+	0.636989i	$0.780180\pi$
$864$		0			0
$865$	11.6193	+	16.8922i	0.395068	+	0.574352i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6.51661			−0.221061
$870$		0			0
$871$	0.787574			0.0266859
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.68695	+	19.4235i	0.158448	+	0.656635i
$876$		0			0
$877$	−1.99539	−	1.99539i	−0.0673797	−	0.0673797i	0.672614	−	0.739994i	$-0.265171\pi$
$877$	−1.99539	−	1.99539i	−0.0673797	−	0.0673797i	−0.739994	+	0.672614i	$0.765171\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			20.3258i			0.684794i	0.939555	+	0.342397i	$0.111239\pi$
$881$			20.3258i			0.684794i	−0.939555	+	0.342397i	$0.888761\pi$
$882$		0			0
$883$	28.0650	−	28.0650i	0.944461	−	0.944461i	−0.0540759	−	0.998537i	$-0.517221\pi$
$883$	28.0650	−	28.0650i	0.944461	−	0.944461i	0.998537	+	0.0540759i	$0.0172213\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−9.67659	+	9.67659i	−0.324908	+	0.324908i	−0.850646	−	0.525738i	$-0.823789\pi$
$887$	−9.67659	+	9.67659i	−0.324908	+	0.324908i	0.525738	+	0.850646i	$0.323789\pi$
$888$		0			0
$889$		−	7.35069i		−	0.246534i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	36.9344	+	36.9344i	1.23596	+	1.23596i
$894$		0			0
$895$	4.46186	+	0.825176i	0.149143	+	0.0275826i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.10145			−0.0700873
$900$		0			0
$901$	0.0460288			0.00153344
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.673480	−	0.124553i	−0.0223872	−	0.00414030i
$906$		0			0
$907$	25.5870	+	25.5870i	0.849601	+	0.849601i	0.990083	−	0.140482i	$-0.0448653\pi$
$907$	25.5870	+	25.5870i	0.849601	+	0.849601i	−0.140482	+	0.990083i	$0.544865\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			22.9378i			0.759963i	0.924994	+	0.379982i	$0.124070\pi$
$911$			22.9378i			0.759963i	−0.924994	+	0.379982i	$0.875930\pi$
$912$		0			0
$913$	20.9222	−	20.9222i	0.692424	−	0.692424i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	13.8196	−	13.8196i	0.456363	−	0.456363i
$918$		0			0
$919$		−	25.2018i		−	0.831330i	−0.909518	−	0.415665i	$-0.863549\pi$
$919$		−	25.2018i		−	0.831330i	0.909518	−	0.415665i	$-0.136451\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−23.0520	−	23.0520i	−0.758767	−	0.758767i
$924$		0			0
$925$	−0.841243	−	1.88555i	−0.0276599	−	0.0619964i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−47.7088			−1.56528			−0.782638	−	0.622477i	$-0.786126\pi$
$929$	−47.7088			−1.56528			−0.782638	+	0.622477i	$0.786126\pi$
$930$		0			0
$931$	18.6041			0.609723
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3.29478	+	4.78998i	0.107751	+	0.156649i
$936$		0			0
$937$	−14.8444	−	14.8444i	−0.484946	−	0.484946i	0.421761	−	0.906707i	$-0.361412\pi$
$937$	−14.8444	−	14.8444i	−0.484946	−	0.484946i	−0.906707	+	0.421761i	$0.861412\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	5.91556i		−	0.192842i	−0.995341	−	0.0964208i	$-0.969261\pi$
$941$		−	5.91556i		−	0.192842i	0.995341	−	0.0964208i	$-0.0307395\pi$
$942$		0			0
$943$	−8.30288	+	8.30288i	−0.270379	+	0.270379i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.92080	−	2.92080i	0.0949131	−	0.0949131i	−0.658056	−	0.752969i	$-0.728621\pi$
$947$	2.92080	−	2.92080i	0.0949131	−	0.0949131i	0.752969	+	0.658056i	$0.228621\pi$
$948$		0			0
$949$			13.1892i			0.428140i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−8.49975	−	8.49975i	−0.275334	−	0.275334i	0.555909	−	0.831243i	$-0.312370\pi$
$953$	−8.49975	−	8.49975i	−0.275334	−	0.275334i	−0.831243	+	0.555909i	$0.812370\pi$
$954$		0			0
$955$	3.99024	−	21.5759i	0.129121	−	0.698178i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.01209			0.0649739
$960$		0			0
$961$	−15.6793			−0.505784
$962$		0			0
$963$		0			0
$964$		0			0
$965$	30.2435	−	20.8029i	0.973571	−	0.669670i
$966$		0			0
$967$	7.77441	+	7.77441i	0.250008	+	0.250008i	0.820974	−	0.570966i	$-0.193431\pi$
$967$	7.77441	+	7.77441i	0.250008	+	0.250008i	−0.570966	+	0.820974i	$0.693431\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	42.2633i		−	1.35629i	−0.734926	−	0.678147i	$-0.762783\pi$
$971$		−	42.2633i		−	1.35629i	0.734926	−	0.678147i	$-0.237217\pi$
$972$		0			0
$973$	4.07099	−	4.07099i	0.130510	−	0.130510i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	8.77663	−	8.77663i	0.280789	−	0.280789i	−0.552634	−	0.833424i	$-0.686377\pi$
$977$	8.77663	−	8.77663i	0.280789	−	0.280789i	0.833424	+	0.552634i	$0.186377\pi$
$978$		0			0
$979$			25.5080i			0.815241i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	5.23102	+	5.23102i	0.166844	+	0.166844i	0.785590	−	0.618747i	$-0.212359\pi$
$983$	5.23102	+	5.23102i	0.166844	+	0.166844i	−0.618747	+	0.785590i	$0.712359\pi$
$984$		0			0
$985$	20.8193	−	14.3205i	0.663359	−	0.456291i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.77927			−0.120174
$990$		0			0
$991$	−39.7583			−1.26296			−0.631482	−	0.775391i	$-0.717553\pi$
$991$	−39.7583			−1.26296			−0.631482	+	0.775391i	$0.717553\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−10.9555	+	59.2380i	−0.347312	+	1.87797i
$996$		0			0
$997$	9.63062	+	9.63062i	0.305005	+	0.305005i	0.842968	−	0.537963i	$-0.180806\pi$
$997$	9.63062	+	9.63062i	0.305005	+	0.305005i	−0.537963	+	0.842968i	$0.680806\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.s.a.737.19	yes	44
3.2	odd	2	inner	4140.2.s.a.737.4	✓	44
5.3	odd	4	inner	4140.2.s.a.2393.4	yes	44
15.8	even	4	inner	4140.2.s.a.2393.19	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.a.737.4	✓	44	3.2	odd	2	inner
4140.2.s.a.737.19	yes	44	1.1	even	1	trivial
4140.2.s.a.2393.4	yes	44	5.3	odd	4	inner
4140.2.s.a.2393.19	yes	44	15.8	even	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)

\(f(q)\)	\(=\)	\(q+(2.19878 + 0.406643i) q^{5} +(1.26371 + 1.26371i) q^{7} +O(q^{10})\)	\(q+(2.19878 + 0.406643i) q^{5} +(1.26371 + 1.26371i) q^{7} -3.93389i q^{11} +(1.46857 - 1.46857i) q^{13} +(-0.467340 + 0.467340i) q^{17} +4.88799i q^{19} +(0.707107 + 0.707107i) q^{23} +(4.66928 + 1.78824i) q^{25} -0.536884 q^{29} +3.91417 q^{31} +(2.26474 + 3.29250i) q^{35} +(-0.291992 - 0.291992i) q^{37} +11.7420i q^{41} +(-2.67235 + 2.67235i) q^{43} +(7.55616 - 7.55616i) q^{47} -3.80607i q^{49} +(-0.0492456 - 0.0492456i) q^{53} +(1.59969 - 8.64977i) q^{55} +9.00685 q^{59} +3.93000 q^{61} +(3.82624 - 2.63188i) q^{65} +(0.268143 + 0.268143i) q^{67} -15.6969i q^{71} +(-4.49050 + 4.49050i) q^{73} +(4.97130 - 4.97130i) q^{77} -1.65653i q^{79} +(5.31845 + 5.31845i) q^{83} +(-1.21762 + 0.837537i) q^{85} -6.48418 q^{89} +3.71169 q^{91} +(-1.98767 + 10.7476i) q^{95} +(11.1170 + 11.1170i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 12 q^{7}+O(q^{10}) \) 44 * q - 12 * q^7	\( 44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100}) \) 44 * q - 12 * q^7 - 4 * q^13 + 24 * q^25 - 48 * q^37 + 8 * q^43 + 40 * q^55 - 96 * q^61 - 44 * q^67 + 76 * q^73 + 72 * q^85 - 48 * q^91 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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