LMFDB - Embedded newform 4140.2.s.a.737.1

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			737.1
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.a.2393.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.23603 + 0.0138422i) q^{5} +(2.36435 + 2.36435i) q^{7} +O(q^{10})$	$q+(-2.23603 + 0.0138422i) q^{5} +(2.36435 + 2.36435i) q^{7} +6.45508i q^{11} +(-1.76494 + 1.76494i) q^{13} +(-4.25553 + 4.25553i) q^{17} -2.13613i q^{19} +(0.707107 + 0.707107i) q^{23} +(4.99962 - 0.0619028i) q^{25} -4.27645 q^{29} -7.22430 q^{31} +(-5.31947 - 5.25401i) q^{35} +(-2.32074 - 2.32074i) q^{37} +4.23009i q^{41} +(3.77722 - 3.77722i) q^{43} +(-3.19024 + 3.19024i) q^{47} +4.18027i q^{49} +(5.04237 + 5.04237i) q^{53} +(-0.0893522 - 14.4337i) q^{55} +7.56824 q^{59} -1.07814 q^{61} +(3.92202 - 3.97088i) q^{65} +(-5.84385 - 5.84385i) q^{67} -6.64319i q^{71} +(7.79721 - 7.79721i) q^{73} +(-15.2620 + 15.2620i) q^{77} -10.0446i q^{79} +(-10.5880 - 10.5880i) q^{83} +(9.45656 - 9.57437i) q^{85} +11.4239 q^{89} -8.34586 q^{91} +(0.0295686 + 4.77643i) q^{95} +(8.04515 + 8.04515i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.23603	+	0.0138422i	−0.999981	+	0.00619040i
$5$	−2.23603	+	0.0138422i	−0.999981	+	0.00619040i
$6$		0			0
$7$	2.36435	+	2.36435i	0.893639	+	0.893639i	0.994864	−	0.101225i	$-0.0322760\pi$
$7$	2.36435	+	2.36435i	0.893639	+	0.893639i	−0.101225	+	0.994864i	$0.532276\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			6.45508i			1.94628i	0.230215	+	0.973140i	$0.426057\pi$
$11$			6.45508i			1.94628i	−0.230215	+	0.973140i	$0.573943\pi$
$12$		0			0
$13$	−1.76494	+	1.76494i	−0.489506	+	0.489506i	−0.908150	−	0.418644i	$-0.862505\pi$
$13$	−1.76494	+	1.76494i	−0.489506	+	0.489506i	0.418644	+	0.908150i	$0.362505\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.25553	+	4.25553i	−1.03212	+	1.03212i	−0.0326502	+	0.999467i	$0.510395\pi$
$17$	−4.25553	+	4.25553i	−1.03212	+	1.03212i	−0.999467	+	0.0326502i	$0.989605\pi$
$18$		0			0
$19$		−	2.13613i		−	0.490061i	−0.969515	−	0.245030i	$-0.921202\pi$
$19$		−	2.13613i		−	0.490061i	0.969515	−	0.245030i	$-0.0787980\pi$
$20$		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
$25$	4.99962	−	0.0619028i	0.999923	−	0.0123806i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.27645			−0.794116			−0.397058	−	0.917793i	$-0.629969\pi$
$29$	−4.27645			−0.794116			−0.397058	+	0.917793i	$0.629969\pi$
$30$		0			0
$31$	−7.22430			−1.29752			−0.648762	−	0.760992i	$-0.724713\pi$
$31$	−7.22430			−1.29752			−0.648762	+	0.760992i	$0.724713\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.31947	−	5.25401i	−0.899154	−	0.888090i
$36$		0			0
$37$	−2.32074	−	2.32074i	−0.381527	−	0.381527i	0.490125	−	0.871652i	$-0.336951\pi$
$37$	−2.32074	−	2.32074i	−0.381527	−	0.381527i	−0.871652	+	0.490125i	$0.836951\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			4.23009i			0.660628i	0.943871	+	0.330314i	$0.107155\pi$
$41$			4.23009i			0.660628i	−0.943871	+	0.330314i	$0.892845\pi$
$42$		0			0
$43$	3.77722	−	3.77722i	0.576020	−	0.576020i	−0.357784	−	0.933804i	$-0.616468\pi$
$43$	3.77722	−	3.77722i	0.576020	−	0.576020i	0.933804	+	0.357784i	$0.116468\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.19024	+	3.19024i	−0.465344	+	0.465344i	−0.900402	−	0.435058i	$-0.856728\pi$
$47$	−3.19024	+	3.19024i	−0.465344	+	0.465344i	0.435058	+	0.900402i	$0.356728\pi$
$48$		0			0
$49$			4.18027i			0.597182i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.04237	+	5.04237i	0.692622	+	0.692622i	0.962808	−	0.270186i	$-0.0870852\pi$
$53$	5.04237	+	5.04237i	0.692622	+	0.692622i	−0.270186	+	0.962808i	$0.587085\pi$
$54$		0			0
$55$	−0.0893522	−	14.4337i	−0.0120482	−	1.94624i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.56824			0.985302			0.492651	−	0.870227i	$-0.336028\pi$
$59$	7.56824			0.985302			0.492651	+	0.870227i	$0.336028\pi$
$60$		0			0
$61$	−1.07814			−0.138042			−0.0690208	−	0.997615i	$-0.521987\pi$
$61$	−1.07814			−0.138042			−0.0690208	+	0.997615i	$0.521987\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.92202	−	3.97088i	0.486467	−	0.492527i
$66$		0			0
$67$	−5.84385	−	5.84385i	−0.713940	−	0.713940i	0.253417	−	0.967357i	$-0.418446\pi$
$67$	−5.84385	−	5.84385i	−0.713940	−	0.713940i	−0.967357	+	0.253417i	$0.918446\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	6.64319i		−	0.788402i	−0.919024	−	0.394201i	$-0.871021\pi$
$71$		−	6.64319i		−	0.788402i	0.919024	−	0.394201i	$-0.128979\pi$
$72$		0			0
$73$	7.79721	−	7.79721i	0.912594	−	0.912594i	−0.0838815	−	0.996476i	$-0.526732\pi$
$73$	7.79721	−	7.79721i	0.912594	−	0.912594i	0.996476	+	0.0838815i	$0.0267317\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−15.2620	+	15.2620i	−1.73927	+	1.73927i
$78$		0			0
$79$		−	10.0446i		−	1.13011i	−0.825054	−	0.565054i	$-0.808855\pi$
$79$		−	10.0446i		−	1.13011i	0.825054	−	0.565054i	$-0.191145\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.5880	−	10.5880i	−1.16218	−	1.16218i	−0.983997	−	0.178187i	$-0.942977\pi$
$83$	−10.5880	−	10.5880i	−1.16218	−	1.16218i	−0.178187	−	0.983997i	$-0.557023\pi$
$84$		0			0
$85$	9.45656	−	9.57437i	1.02571	−	1.03849i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	11.4239			1.21093			0.605464	−	0.795873i	$-0.292988\pi$
$89$	11.4239			1.21093			0.605464	+	0.795873i	$0.292988\pi$
$90$		0			0
$91$	−8.34586			−0.874884
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.0295686	+	4.77643i	0.00303367	+	0.490052i
$96$		0			0
$97$	8.04515	+	8.04515i	0.816862	+	0.816862i	0.985652	−	0.168790i	$-0.0539861\pi$
$97$	8.04515	+	8.04515i	0.816862	+	0.816862i	−0.168790	+	0.985652i	$0.553986\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	15.9966i		−	1.59172i	−0.605480	−	0.795861i	$-0.707019\pi$
$101$		−	15.9966i		−	1.59172i	0.605480	−	0.795861i	$-0.292981\pi$
$102$		0			0
$103$	−0.336452	+	0.336452i	−0.0331516	+	0.0331516i	−0.723488	−	0.690337i	$-0.757462\pi$
$103$	−0.336452	+	0.336452i	−0.0331516	+	0.0331516i	0.690337	+	0.723488i	$0.257462\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.25435	−	7.25435i	0.701304	−	0.701304i	−0.263386	−	0.964691i	$-0.584839\pi$
$107$	7.25435	−	7.25435i	0.701304	−	0.701304i	0.964691	+	0.263386i	$0.0848393\pi$
$108$		0			0
$109$			4.32126i			0.413901i	0.978351	+	0.206951i	$0.0663540\pi$
$109$			4.32126i			0.413901i	−0.978351	+	0.206951i	$0.933646\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.44766	−	3.44766i	−0.324328	−	0.324328i	0.526097	−	0.850425i	$-0.323655\pi$
$113$	−3.44766	−	3.44766i	−0.324328	−	0.324328i	−0.850425	+	0.526097i	$0.823655\pi$
$114$		0			0
$115$	−1.59090	−	1.57132i	−0.148352	−	0.146526i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−20.1231			−1.84468
$120$		0			0
$121$	−30.6680			−2.78800
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.1784	+	0.207622i	−0.999828	+	0.0185702i
$126$		0			0
$127$	1.49268	+	1.49268i	0.132454	+	0.132454i	0.770225	−	0.637772i	$-0.220144\pi$
$127$	1.49268	+	1.49268i	0.132454	+	0.132454i	−0.637772	+	0.770225i	$0.720144\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			17.3426i			1.51523i	0.652700	+	0.757616i	$0.273636\pi$
$131$			17.3426i			1.51523i	−0.652700	+	0.757616i	$0.726364\pi$
$132$		0			0
$133$	5.05054	−	5.05054i	0.437938	−	0.437938i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.61637	−	1.61637i	0.138096	−	0.138096i	−0.634680	−	0.772775i	$-0.718868\pi$
$137$	1.61637	−	1.61637i	0.138096	−	0.138096i	0.772775	+	0.634680i	$0.218868\pi$
$138$		0			0
$139$			3.17935i			0.269669i	0.990868	+	0.134835i	$0.0430503\pi$
$139$			3.17935i			0.269669i	−0.990868	+	0.134835i	$0.956950\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.3928	−	11.3928i	−0.952716	−	0.952716i
$144$		0			0
$145$	9.56224	−	0.0591952i	0.794101	−	0.00491590i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−21.4193			−1.75474			−0.877369	−	0.479816i	$-0.840703\pi$
$149$	−21.4193			−1.75474			−0.877369	+	0.479816i	$0.840703\pi$
$150$		0			0
$151$	16.3100			1.32729			0.663643	−	0.748050i	$-0.269009\pi$
$151$	16.3100			1.32729			0.663643	+	0.748050i	$0.269009\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.1537	−	0.0999999i	1.29750	−	0.00803219i
$156$		0			0
$157$	11.1767	+	11.1767i	0.892002	+	0.892002i	0.994711	−	0.102710i	$-0.0327513\pi$
$157$	11.1767	+	11.1767i	0.892002	+	0.892002i	−0.102710	+	0.994711i	$0.532751\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			3.34369i			0.263520i
$162$		0			0
$163$	−2.84703	+	2.84703i	−0.222996	+	0.222996i	−0.809759	−	0.586763i	$-0.800402\pi$
$163$	−2.84703	+	2.84703i	−0.222996	+	0.222996i	0.586763	+	0.809759i	$0.300402\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.63733	−	5.63733i	0.436230	−	0.436230i	−0.454511	−	0.890741i	$-0.650186\pi$
$167$	5.63733	−	5.63733i	0.436230	−	0.436230i	0.890741	+	0.454511i	$0.150186\pi$
$168$		0			0
$169$			6.76997i			0.520767i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.91489	+	9.91489i	0.753815	+	0.753815i	0.975189	−	0.221374i	$-0.0710541\pi$
$173$	9.91489	+	9.91489i	0.753815	+	0.753815i	−0.221374	+	0.975189i	$0.571054\pi$
$174$		0			0
$175$	11.9672	+	11.6745i	0.904634	+	0.882507i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.18233			−0.611576			−0.305788	−	0.952100i	$-0.598920\pi$
$179$	−8.18233			−0.611576			−0.305788	+	0.952100i	$0.598920\pi$
$180$		0			0
$181$	−16.1648			−1.20152			−0.600760	−	0.799430i	$-0.705135\pi$
$181$	−16.1648			−1.20152			−0.600760	+	0.799430i	$0.705135\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.22135	+	5.15710i	0.383881	+	0.379158i
$186$		0			0
$187$	−27.4698	−	27.4698i	−2.00879	−	2.00879i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	15.3880i		−	1.11344i	−0.830701	−	0.556719i	$-0.812060\pi$
$191$		−	15.3880i		−	1.11344i	0.830701	−	0.556719i	$-0.187940\pi$
$192$		0			0
$193$	−2.89628	+	2.89628i	−0.208479	+	0.208479i	−0.803621	−	0.595142i	$-0.797096\pi$
$193$	−2.89628	+	2.89628i	−0.208479	+	0.208479i	0.595142	+	0.803621i	$0.297096\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.81659	+	6.81659i	−0.485662	+	0.485662i	−0.906934	−	0.421272i	$-0.861584\pi$
$197$	−6.81659	+	6.81659i	−0.485662	+	0.485662i	0.421272	+	0.906934i	$0.361584\pi$
$198$		0			0
$199$			12.5120i			0.886952i	0.896286	+	0.443476i	$0.146255\pi$
$199$			12.5120i			0.886952i	−0.896286	+	0.443476i	$0.853745\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−10.1110	−	10.1110i	−0.709653	−	0.709653i
$204$		0			0
$205$	−0.0585535	−	9.45858i	−0.00408955	−	0.660616i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	13.7889			0.953796
$210$		0			0
$211$	−17.5872			−1.21075			−0.605377	−	0.795939i	$-0.706978\pi$
$211$	−17.5872			−1.21075			−0.605377	+	0.795939i	$0.706978\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.39366	+	8.49823i	−0.572443	+	0.579575i
$216$		0			0
$217$	−17.0808	−	17.0808i	−1.15952	−	1.15952i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	15.0215i		−	1.01046i
$222$		0			0
$223$	−14.4047	+	14.4047i	−0.964613	+	0.964613i	−0.999395	−	0.0347822i	$-0.988926\pi$
$223$	−14.4047	+	14.4047i	−0.964613	+	0.964613i	0.0347822	+	0.999395i	$0.488926\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−20.6214	+	20.6214i	−1.36869	+	1.36869i	−0.506381	+	0.862310i	$0.669017\pi$
$227$	−20.6214	+	20.6214i	−1.36869	+	1.36869i	−0.862310	+	0.506381i	$0.830983\pi$
$228$		0			0
$229$			9.43043i			0.623180i	0.950217	+	0.311590i	$0.100862\pi$
$229$			9.43043i			0.623180i	−0.950217	+	0.311590i	$0.899138\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−15.7587	−	15.7587i	−1.03239	−	1.03239i	−0.999458	−	0.0329296i	$-0.989516\pi$
$233$	−15.7587	−	15.7587i	−1.03239	−	1.03239i	−0.0329296	−	0.999458i	$-0.510484\pi$
$234$		0			0
$235$	7.08929	−	7.17761i	0.462454	−	0.468216i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−16.9285			−1.09501			−0.547507	−	0.836801i	$-0.684423\pi$
$239$	−16.9285			−1.09501			−0.547507	+	0.836801i	$0.684423\pi$
$240$		0			0
$241$	23.4186			1.50852			0.754261	−	0.656575i	$-0.227995\pi$
$241$	23.4186			1.50852			0.754261	+	0.656575i	$0.227995\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.0578640	−	9.34719i	−0.00369679	−	0.597170i
$246$		0			0
$247$	3.77013	+	3.77013i	0.239888	+	0.239888i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			20.5804i			1.29902i	0.760352	+	0.649511i	$0.225026\pi$
$251$			20.5804i			1.29902i	−0.760352	+	0.649511i	$0.774974\pi$
$252$		0			0
$253$	−4.56443	+	4.56443i	−0.286963	+	0.286963i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−12.3156	+	12.3156i	−0.768224	+	0.768224i	−0.977794	−	0.209570i	$-0.932794\pi$
$257$	−12.3156	+	12.3156i	−0.768224	+	0.768224i	0.209570	+	0.977794i	$0.432794\pi$
$258$		0			0
$259$		−	10.9741i		−	0.681894i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.31501	+	2.31501i	0.142750	+	0.142750i	0.774870	−	0.632120i	$-0.217815\pi$
$263$	2.31501	+	2.31501i	0.142750	+	0.142750i	−0.632120	+	0.774870i	$0.717815\pi$
$264$		0			0
$265$	−11.3447	−	11.2051i	−0.696897	−	0.688321i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−23.3035			−1.42084			−0.710421	−	0.703777i	$-0.751495\pi$
$269$	−23.3035			−1.42084			−0.710421	+	0.703777i	$0.751495\pi$
$270$		0			0
$271$	−6.43619			−0.390971			−0.195485	−	0.980707i	$-0.562628\pi$
$271$	−6.43619			−0.390971			−0.195485	+	0.980707i	$0.562628\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.399587	+	32.2729i	0.0240960	+	1.94613i
$276$		0			0
$277$	−9.87513	−	9.87513i	−0.593339	−	0.593339i	0.345193	−	0.938532i	$-0.387813\pi$
$277$	−9.87513	−	9.87513i	−0.593339	−	0.593339i	−0.938532	+	0.345193i	$0.887813\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	28.0175i		−	1.67138i	−0.549200	−	0.835691i	$-0.685067\pi$
$281$		−	28.0175i		−	1.67138i	0.549200	−	0.835691i	$-0.314933\pi$
$282$		0			0
$283$	20.2382	−	20.2382i	1.20304	−	1.20304i	0.229797	−	0.973239i	$-0.426194\pi$
$283$	20.2382	−	20.2382i	1.20304	−	1.20304i	0.973239	−	0.229797i	$-0.0738063\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.0014	+	10.0014i	−0.590363	+	0.590363i
$288$		0			0
$289$		−	19.2190i		−	1.13053i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−14.7998	−	14.7998i	−0.864613	−	0.864613i	0.127257	−	0.991870i	$-0.459383\pi$
$293$	−14.7998	−	14.7998i	−0.864613	−	0.864613i	−0.991870	+	0.127257i	$0.959383\pi$
$294$		0			0
$295$	−16.9228	+	0.104761i	−0.985283	+	0.00609941i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.49600			−0.144348
$300$		0			0
$301$	17.8613			1.02951
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.41075	−	0.0149238i	0.138039	−	0.000854532i
$306$		0			0
$307$	−22.1204	−	22.1204i	−1.26248	−	1.26248i	−0.949888	−	0.312592i	$-0.898803\pi$
$307$	−22.1204	−	22.1204i	−1.26248	−	1.26248i	−0.312592	−	0.949888i	$-0.601197\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			30.3176i			1.71915i	0.511006	+	0.859577i	$0.329273\pi$
$311$			30.3176i			1.71915i	−0.511006	+	0.859577i	$0.670727\pi$
$312$		0			0
$313$	−6.59875	+	6.59875i	−0.372984	+	0.372984i	−0.868563	−	0.495579i	$-0.834956\pi$
$313$	−6.59875	+	6.59875i	−0.372984	+	0.372984i	0.495579	+	0.868563i	$0.334956\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.5488	−	12.5488i	0.704811	−	0.704811i	−0.260628	−	0.965439i	$-0.583930\pi$
$317$	12.5488	−	12.5488i	0.704811	−	0.704811i	0.965439	+	0.260628i	$0.0839298\pi$
$318$		0			0
$319$		−	27.6048i		−	1.54557i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	9.09034	+	9.09034i	0.505800	+	0.505800i
$324$		0			0
$325$	−8.71477	+	8.93328i	−0.483408	+	0.495529i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−15.0857			−0.831699
$330$		0			0
$331$	−0.585481			−0.0321810			−0.0160905	−	0.999871i	$-0.505122\pi$
$331$	−0.585481			−0.0321810			−0.0160905	+	0.999871i	$0.505122\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	13.1479	+	12.9861i	0.718346	+	0.709507i
$336$		0			0
$337$	11.0877	+	11.0877i	0.603983	+	0.603983i	0.941367	−	0.337384i	$-0.109542\pi$
$337$	11.0877	+	11.0877i	0.603983	+	0.603983i	−0.337384	+	0.941367i	$0.609542\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	46.6335i		−	2.52534i
$342$		0			0
$343$	6.66682	−	6.66682i	0.359974	−	0.359974i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.7305	−	14.7305i	0.790772	−	0.790772i	−0.190847	−	0.981620i	$-0.561124\pi$
$347$	14.7305	−	14.7305i	0.790772	−	0.790772i	0.981620	+	0.190847i	$0.0611236\pi$
$348$		0			0
$349$			6.82798i			0.365493i	0.983160	+	0.182747i	$0.0584988\pi$
$349$			6.82798i			0.365493i	−0.983160	+	0.182747i	$0.941501\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	18.3635	+	18.3635i	0.977389	+	0.977389i	0.999750	−	0.0223611i	$-0.00711836\pi$
$353$	18.3635	+	18.3635i	0.977389	+	0.977389i	−0.0223611	+	0.999750i	$0.507118\pi$
$354$		0			0
$355$	0.0919561	+	14.8543i	0.00488052	+	0.788387i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.0714			0.531546			0.265773	−	0.964036i	$-0.414373\pi$
$359$	10.0714			0.531546			0.265773	+	0.964036i	$0.414373\pi$
$360$		0			0
$361$	14.4370			0.759840
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−17.3268	+	17.5427i	−0.906927	+	0.918226i
$366$		0			0
$367$	11.6856	+	11.6856i	0.609982	+	0.609982i	0.942941	−	0.332959i	$-0.108047\pi$
$367$	11.6856	+	11.6856i	0.609982	+	0.609982i	−0.332959	+	0.942941i	$0.608047\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			23.8438i			1.23791i
$372$		0			0
$373$	15.9254	−	15.9254i	0.824586	−	0.824586i	−0.162176	−	0.986762i	$-0.551851\pi$
$373$	15.9254	−	15.9254i	0.824586	−	0.824586i	0.986762	+	0.162176i	$0.0518511\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.54767	−	7.54767i	0.388725	−	0.388725i
$378$		0			0
$379$		−	17.9744i		−	0.923281i	−0.887067	−	0.461641i	$-0.847261\pi$
$379$		−	17.9744i		−	0.923281i	0.887067	−	0.461641i	$-0.152739\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.88129	+	5.88129i	0.300520	+	0.300520i	0.841217	−	0.540697i	$-0.181840\pi$
$383$	5.88129	+	5.88129i	0.300520	+	0.300520i	−0.540697	+	0.841217i	$0.681840\pi$
$384$		0			0
$385$	33.9151	−	34.3376i	1.72847	−	1.75001i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.48646			−0.176771			−0.0883854	−	0.996086i	$-0.528171\pi$
$389$	−3.48646			−0.176771			−0.0883854	+	0.996086i	$0.528171\pi$
$390$		0			0
$391$	−6.01822			−0.304355
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.139039	+	22.4600i	0.00699582	+	1.13009i
$396$		0			0
$397$	−9.09927	−	9.09927i	−0.456679	−	0.456679i	0.440885	−	0.897564i	$-0.354665\pi$
$397$	−9.09927	−	9.09927i	−0.456679	−	0.456679i	−0.897564	+	0.440885i	$0.854665\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	12.9697i		−	0.647675i	−0.946113	−	0.323837i	$-0.895027\pi$
$401$		−	12.9697i		−	0.647675i	0.946113	−	0.323837i	$-0.104973\pi$
$402$		0			0
$403$	12.7505	−	12.7505i	0.635146	−	0.635146i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	14.9805	−	14.9805i	0.742558	−	0.742558i
$408$		0			0
$409$			15.2185i			0.752506i	0.926517	+	0.376253i	$0.122788\pi$
$409$			15.2185i			0.752506i	−0.926517	+	0.376253i	$0.877212\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	17.8940	+	17.8940i	0.880504	+	0.880504i
$414$		0			0
$415$	23.8216	+	23.5285i	1.16936	+	1.15497i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8.13686			0.397511			0.198756	−	0.980049i	$-0.436310\pi$
$419$	8.13686			0.397511			0.198756	+	0.980049i	$0.436310\pi$
$420$		0			0
$421$	7.86105			0.383124			0.191562	−	0.981481i	$-0.438645\pi$
$421$	7.86105			0.383124			0.191562	+	0.981481i	$0.438645\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−21.0126	+	21.5394i	−1.01926	+	1.04482i
$426$		0			0
$427$	−2.54909	−	2.54909i	−0.123359	−	0.123359i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			13.4477i			0.647754i	0.946099	+	0.323877i	$0.104986\pi$
$431$			13.4477i			0.647754i	−0.946099	+	0.323877i	$0.895014\pi$
$432$		0			0
$433$	−12.3089	+	12.3089i	−0.591529	+	0.591529i	−0.938044	−	0.346515i	$-0.887365\pi$
$433$	−12.3089	+	12.3089i	−0.591529	+	0.591529i	0.346515	+	0.938044i	$0.387365\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.51047	−	1.51047i	0.0722555	−	0.0722555i
$438$		0			0
$439$			33.0262i			1.57625i	0.615513	+	0.788126i	$0.288949\pi$
$439$			33.0262i			1.57625i	−0.615513	+	0.788126i	$0.711051\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.94979	+	1.94979i	0.0926372	+	0.0926372i	0.751907	−	0.659270i	$-0.229134\pi$
$443$	1.94979	+	1.94979i	0.0926372	+	0.0926372i	−0.659270	+	0.751907i	$0.729134\pi$
$444$		0			0
$445$	−25.5440	+	0.158131i	−1.21090	+	0.00749612i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−35.9186			−1.69510			−0.847551	−	0.530713i	$-0.821924\pi$
$449$	−35.9186			−1.69510			−0.847551	+	0.530713i	$0.821924\pi$
$450$		0			0
$451$	−27.3055			−1.28577
$452$		0			0
$453$		0			0
$454$		0			0
$455$	18.6616	−	0.115525i	0.874867	−	0.00541588i
$456$		0			0
$457$	8.89878	+	8.89878i	0.416267	+	0.416267i	0.883915	−	0.467648i	$-0.154898\pi$
$457$	8.89878	+	8.89878i	0.416267	+	0.416267i	−0.467648	+	0.883915i	$0.654898\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	2.72607i		−	0.126966i	−0.997983	−	0.0634830i	$-0.979779\pi$
$461$		−	2.72607i		−	0.126966i	0.997983	−	0.0634830i	$-0.0202209\pi$
$462$		0			0
$463$	−4.21245	+	4.21245i	−0.195769	+	0.195769i	−0.798184	−	0.602414i	$-0.794206\pi$
$463$	−4.21245	+	4.21245i	−0.195769	+	0.195769i	0.602414	+	0.798184i	$0.294206\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9.72252	−	9.72252i	0.449904	−	0.449904i	−0.445418	−	0.895323i	$-0.646945\pi$
$467$	9.72252	−	9.72252i	0.449904	−	0.449904i	0.895323	+	0.445418i	$0.146945\pi$
$468$		0			0
$469$		−	27.6338i		−	1.27601i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	24.3822	+	24.3822i	1.12110	+	1.12110i
$474$		0			0
$475$	−0.132232	−	10.6798i	−0.00606723	−	0.490023i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.7824			0.584042			0.292021	−	0.956412i	$-0.405672\pi$
$479$	12.7824			0.584042			0.292021	+	0.956412i	$0.405672\pi$
$480$		0			0
$481$	8.19192			0.373520
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−18.1005	−	17.8778i	−0.821903	−	0.811789i
$486$		0			0
$487$	6.70542	+	6.70542i	0.303852	+	0.303852i	0.842519	−	0.538667i	$-0.181072\pi$
$487$	6.70542	+	6.70542i	0.303852	+	0.303852i	−0.538667	+	0.842519i	$0.681072\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			2.94934i			0.133102i	0.997783	+	0.0665508i	$0.0211995\pi$
$491$			2.94934i			0.133102i	−0.997783	+	0.0665508i	$0.978801\pi$
$492$		0			0
$493$	18.1985	−	18.1985i	0.819621	−	0.819621i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15.7068	−	15.7068i	0.704547	−	0.704547i
$498$		0			0
$499$			36.1855i			1.61989i	0.586509	+	0.809943i	$0.300502\pi$
$499$			36.1855i			1.61989i	−0.586509	+	0.809943i	$0.699498\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−0.220002	−	0.220002i	−0.00980940	−	0.00980940i	0.702185	−	0.711994i	$-0.252208\pi$
$503$	−0.220002	−	0.220002i	−0.00980940	−	0.00980940i	−0.711994	+	0.702185i	$0.752208\pi$
$504$		0			0
$505$	0.221427	+	35.7688i	0.00985339	+	1.59169i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−28.7002			−1.27211			−0.636057	−	0.771642i	$-0.719436\pi$
$509$	−28.7002			−1.27211			−0.636057	+	0.771642i	$0.719436\pi$
$510$		0			0
$511$	36.8706			1.63106
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.747659	−	0.756973i	0.0329458	−	0.0333562i
$516$		0			0
$517$	−20.5932	−	20.5932i	−0.905690	−	0.905690i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	23.2026i		−	1.01652i	−0.861203	−	0.508262i	$-0.830288\pi$
$521$		−	23.2026i		−	1.01652i	0.861203	−	0.508262i	$-0.169712\pi$
$522$		0			0
$523$	−18.2434	+	18.2434i	−0.797730	+	0.797730i	−0.982737	−	0.185007i	$-0.940769\pi$
$523$	−18.2434	+	18.2434i	−0.797730	+	0.797730i	0.185007	+	0.982737i	$0.440769\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	30.7432	−	30.7432i	1.33920	−	1.33920i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.46585	−	7.46585i	−0.323382	−	0.323382i
$534$		0			0
$535$	−16.1205	+	16.3213i	−0.696950	+	0.705632i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−26.9840			−1.16228
$540$		0			0
$541$	8.26015			0.355132			0.177566	−	0.984109i	$-0.443178\pi$
$541$	8.26015			0.355132			0.177566	+	0.984109i	$0.443178\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.0598155	−	9.66244i	−0.00256221	−	0.413893i
$546$		0			0
$547$	−21.1503	−	21.1503i	−0.904320	−	0.904320i	0.0914865	−	0.995806i	$-0.470838\pi$
$547$	−21.1503	−	21.1503i	−0.904320	−	0.904320i	−0.995806	+	0.0914865i	$0.970838\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			9.13503i			0.389165i
$552$		0			0
$553$	23.7490	−	23.7490i	1.00991	−	1.00991i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.73097	+	1.73097i	−0.0733436	+	0.0733436i	−0.742827	−	0.669483i	$-0.766516\pi$
$557$	−1.73097	+	1.73097i	−0.0733436	+	0.0733436i	0.669483	+	0.742827i	$0.266516\pi$
$558$		0			0
$559$			13.3331i			0.563931i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.39228	+	1.39228i	0.0586777	+	0.0586777i	0.735837	−	0.677159i	$-0.236789\pi$
$563$	1.39228	+	1.39228i	0.0586777	+	0.0586777i	−0.677159	+	0.735837i	$0.736789\pi$
$564$		0			0
$565$	7.75677	+	7.66132i	0.326330	+	0.322314i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15.5125			−0.650317			−0.325158	−	0.945660i	$-0.605418\pi$
$569$	−15.5125			−0.650317			−0.325158	+	0.945660i	$0.605418\pi$
$570$		0			0
$571$	15.9699			0.668318			0.334159	−	0.942517i	$-0.391548\pi$
$571$	15.9699			0.668318			0.334159	+	0.942517i	$0.391548\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.57903	+	3.49149i	0.149256	+	0.145605i
$576$		0			0
$577$	15.0282	+	15.0282i	0.625630	+	0.625630i	0.946966	−	0.321335i	$-0.104132\pi$
$577$	15.0282	+	15.0282i	0.625630	+	0.625630i	−0.321335	+	0.946966i	$0.604132\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	50.0674i		−	2.07714i
$582$		0			0
$583$	−32.5489	+	32.5489i	−1.34804	+	1.34804i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9.69513	+	9.69513i	−0.400161	+	0.400161i	−0.878290	−	0.478129i	$-0.841315\pi$
$587$	−9.69513	+	9.69513i	−0.400161	+	0.400161i	0.478129	+	0.878290i	$0.341315\pi$
$588$		0			0
$589$			15.4320i			0.635865i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	21.3785	+	21.3785i	0.877912	+	0.877912i	0.993318	−	0.115407i	$-0.0368171\pi$
$593$	21.3785	+	21.3785i	0.877912	+	0.877912i	−0.115407	+	0.993318i	$0.536817\pi$
$594$		0			0
$595$	44.9957	−	0.278547i	1.84464	−	0.0114193i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−20.5733			−0.840602			−0.420301	−	0.907385i	$-0.638075\pi$
$599$	−20.5733			−0.840602			−0.420301	+	0.907385i	$0.638075\pi$
$600$		0			0
$601$	−20.1920			−0.823649			−0.411825	−	0.911263i	$-0.635108\pi$
$601$	−20.1920			−0.823649			−0.411825	+	0.911263i	$0.635108\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	68.5745	−	0.424512i	2.78795	−	0.0172589i
$606$		0			0
$607$	33.2178	+	33.2178i	1.34827	+	1.34827i	0.887543	+	0.460725i	$0.152410\pi$
$607$	33.2178	+	33.2178i	1.34827	+	1.34827i	0.460725	+	0.887543i	$0.347590\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	11.2612i		−	0.455578i
$612$		0			0
$613$	−9.63906	+	9.63906i	−0.389318	+	0.389318i	−0.874444	−	0.485126i	$-0.838774\pi$
$613$	−9.63906	+	9.63906i	−0.389318	+	0.389318i	0.485126	+	0.874444i	$0.338774\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	31.1128	−	31.1128i	1.25255	−	1.25255i	0.297981	−	0.954572i	$-0.403687\pi$
$617$	31.1128	−	31.1128i	1.25255	−	1.25255i	0.954572	−	0.297981i	$-0.0963133\pi$
$618$		0			0
$619$			32.9375i			1.32387i	0.749561	+	0.661935i	$0.230265\pi$
$619$			32.9375i			1.32387i	−0.749561	+	0.661935i	$0.769735\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	27.0100	+	27.0100i	1.08213	+	1.08213i
$624$		0			0
$625$	24.9923	−	0.618981i	0.999693	−	0.0247592i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	19.7519			0.787561
$630$		0			0
$631$	−30.2569			−1.20451			−0.602255	−	0.798304i	$-0.705731\pi$
$631$	−30.2569			−1.20451			−0.602255	+	0.798304i	$0.705731\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.35832	−	3.31700i	−0.133271	−	0.131631i
$636$		0			0
$637$	−7.37793	−	7.37793i	−0.292324	−	0.292324i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	0.282963i		−	0.0111764i	−0.999984	−	0.00558819i	$-0.998221\pi$
$641$		−	0.282963i		−	0.0111764i	0.999984	−	0.00558819i	$-0.00177878\pi$
$642$		0			0
$643$	27.1006	−	27.1006i	1.06874	−	1.06874i	0.0712880	−	0.997456i	$-0.477289\pi$
$643$	27.1006	−	27.1006i	1.06874	−	1.06874i	0.997456	−	0.0712880i	$-0.0227110\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.23685	−	8.23685i	0.323824	−	0.323824i	−0.526408	−	0.850232i	$-0.676462\pi$
$647$	8.23685	−	8.23685i	0.323824	−	0.323824i	0.850232	+	0.526408i	$0.176462\pi$
$648$		0			0
$649$			48.8536i			1.91767i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.2648	−	15.2648i	−0.597358	−	0.597358i	0.342251	−	0.939609i	$-0.388811\pi$
$653$	−15.2648	−	15.2648i	−0.597358	−	0.597358i	−0.939609	+	0.342251i	$0.888811\pi$
$654$		0			0
$655$	−0.240059	−	38.7785i	−0.00937989	−	1.51520i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	5.47477			0.213267			0.106633	−	0.994298i	$-0.465993\pi$
$659$	5.47477			0.213267			0.106633	+	0.994298i	$0.465993\pi$
$660$		0			0
$661$	−36.2911			−1.41156			−0.705780	−	0.708431i	$-0.749403\pi$
$661$	−36.2911			−1.41156			−0.705780	+	0.708431i	$0.749403\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−11.2232	+	11.3631i	−0.435218	+	0.440640i
$666$		0			0
$667$	−3.02391	−	3.02391i	−0.117086	−	0.117086i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	6.95947i		−	0.268667i
$672$		0			0
$673$	19.3171	−	19.3171i	0.744618	−	0.744618i	−0.228845	−	0.973463i	$-0.573495\pi$
$673$	19.3171	−	19.3171i	0.744618	−	0.744618i	0.973463	+	0.228845i	$0.0734949\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8.02258	−	8.02258i	0.308333	−	0.308333i	−0.535930	−	0.844263i	$-0.680039\pi$
$677$	8.02258	−	8.02258i	0.308333	−	0.308333i	0.844263	+	0.535930i	$0.180039\pi$
$678$		0			0
$679$			38.0431i			1.45996i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	14.1348	+	14.1348i	0.540851	+	0.540851i	0.923778	−	0.382927i	$-0.125084\pi$
$683$	14.1348	+	14.1348i	0.540851	+	0.540851i	−0.382927	+	0.923778i	$0.625084\pi$
$684$		0			0
$685$	−3.59187	+	3.63662i	−0.137238	+	0.138948i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−17.7989			−0.678086
$690$		0			0
$691$	−14.3671			−0.546550			−0.273275	−	0.961936i	$-0.588107\pi$
$691$	−14.3671			−0.546550			−0.273275	+	0.961936i	$0.588107\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.0440091	−	7.10911i	−0.00166936	−	0.269664i
$696$		0			0
$697$	−18.0012	−	18.0012i	−0.681846	−	0.681846i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			29.2024i			1.10296i	0.834188	+	0.551480i	$0.185937\pi$
$701$			29.2024i			1.10296i	−0.834188	+	0.551480i	$0.814063\pi$
$702$		0			0
$703$	−4.95739	+	4.95739i	−0.186971	+	0.186971i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	37.8215	−	37.8215i	1.42242	−	1.42242i
$708$		0			0
$709$			16.8536i			0.632952i	0.948601	+	0.316476i	$0.102500\pi$
$709$			16.8536i			0.632952i	−0.948601	+	0.316476i	$0.897500\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.10835	−	5.10835i	−0.191309	−	0.191309i
$714$		0			0
$715$	25.6324	+	25.3170i	0.958596	+	0.946800i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−11.2396			−0.419165			−0.209582	−	0.977791i	$-0.567210\pi$
$719$	−11.2396			−0.419165			−0.209582	+	0.977791i	$0.567210\pi$
$720$		0			0
$721$	−1.59098			−0.0592512
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−21.3806	+	0.264724i	−0.794056	+	0.00983161i
$726$		0			0
$727$	7.66090	+	7.66090i	0.284127	+	0.284127i	0.834753	−	0.550625i	$-0.185611\pi$
$727$	7.66090	+	7.66090i	0.284127	+	0.284127i	−0.550625	+	0.834753i	$0.685611\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			32.1481i			1.18904i
$732$		0			0
$733$	36.1737	−	36.1737i	1.33611	−	1.33611i	0.436308	−	0.899798i	$-0.356286\pi$
$733$	36.1737	−	36.1737i	1.33611	−	1.33611i	0.899798	−	0.436308i	$-0.143714\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	37.7225	−	37.7225i	1.38953	−	1.38953i
$738$		0			0
$739$			13.9937i			0.514768i	0.966309	+	0.257384i	$0.0828605\pi$
$739$			13.9937i			0.514768i	−0.966309	+	0.257384i	$0.917140\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−30.0988	−	30.0988i	−1.10422	−	1.10422i	−0.993896	−	0.110322i	$-0.964812\pi$
$743$	−30.0988	−	30.0988i	−1.10422	−	1.10422i	−0.110322	−	0.993896i	$-0.535188\pi$
$744$		0			0
$745$	47.8941	−	0.296489i	1.75470	−	0.0108625i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	34.3036			1.25343
$750$		0			0
$751$	−48.2997			−1.76248			−0.881241	−	0.472667i	$-0.843291\pi$
$751$	−48.2997			−1.76248			−0.881241	+	0.472667i	$0.843291\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−36.4695	+	0.225765i	−1.32726	+	0.00821643i
$756$		0			0
$757$	1.92520	+	1.92520i	0.0699725	+	0.0699725i	0.741227	−	0.671254i	$-0.234244\pi$
$757$	1.92520	+	1.92520i	0.0699725	+	0.0699725i	−0.671254	+	0.741227i	$0.734244\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			47.5992i			1.72547i	0.505656	+	0.862735i	$0.331250\pi$
$761$			47.5992i			1.72547i	−0.505656	+	0.862735i	$0.668750\pi$
$762$		0			0
$763$	−10.2169	+	10.2169i	−0.369878	+	0.369878i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−13.3575	+	13.3575i	−0.482311	+	0.482311i
$768$		0			0
$769$			0.0991955i			0.00357708i	0.999998	+	0.00178854i	$0.000569310\pi$
$769$			0.0991955i			0.00357708i	−0.999998	+	0.00178854i	$0.999431\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9.06777	−	9.06777i	−0.326145	−	0.326145i	0.524973	−	0.851119i	$-0.324075\pi$
$773$	−9.06777	−	9.06777i	−0.326145	−	0.326145i	−0.851119	+	0.524973i	$0.824075\pi$
$774$		0			0
$775$	−36.1188	+	0.447205i	−1.29742	+	0.0160641i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9.03600			0.323748
$780$		0			0
$781$	42.8823			1.53445
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−25.1462	−	24.8368i	−0.897506	−	0.886463i
$786$		0			0
$787$	−17.6316	−	17.6316i	−0.628499	−	0.628499i	0.319191	−	0.947690i	$-0.396589\pi$
$787$	−17.6316	−	17.6316i	−0.628499	−	0.628499i	−0.947690	+	0.319191i	$0.896589\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	16.3029i		−	0.579665i
$792$		0			0
$793$	1.90285	−	1.90285i	0.0675722	−	0.0675722i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−11.4214	+	11.4214i	−0.404567	+	0.404567i	−0.879839	−	0.475272i	$-0.842350\pi$
$797$	−11.4214	+	11.4214i	−0.404567	+	0.404567i	0.475272	+	0.879839i	$0.342350\pi$
$798$		0			0
$799$		−	27.1523i		−	0.960579i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	50.3316	+	50.3316i	1.77616	+	1.77616i
$804$		0			0
$805$	−0.0462839	−	7.47658i	−0.00163129	−	0.263515i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	39.8205			1.40001			0.700007	−	0.714136i	$-0.253180\pi$
$809$	39.8205			1.40001			0.700007	+	0.714136i	$0.253180\pi$
$810$		0			0
$811$	−44.8733			−1.57572			−0.787858	−	0.615857i	$-0.788810\pi$
$811$	−44.8733			−1.57572			−0.787858	+	0.615857i	$0.788810\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.32661	−	6.40543i	0.221612	−	0.224372i
$816$		0			0
$817$	−8.06861	−	8.06861i	−0.282285	−	0.282285i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	18.4377i		−	0.643480i	−0.946828	−	0.321740i	$-0.895732\pi$
$821$		−	18.4377i		−	0.643480i	0.946828	−	0.321740i	$-0.104268\pi$
$822$		0			0
$823$	−21.4046	+	21.4046i	−0.746117	+	0.746117i	−0.973748	−	0.227631i	$-0.926902\pi$
$823$	−21.4046	+	21.4046i	−0.746117	+	0.746117i	0.227631	+	0.973748i	$0.426902\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.9378	−	20.9378i	0.728080	−	0.728080i	−0.242157	−	0.970237i	$-0.577855\pi$
$827$	20.9378	−	20.9378i	0.728080	−	0.728080i	0.970237	+	0.242157i	$0.0778548\pi$
$828$		0			0
$829$			7.77000i			0.269863i	0.990855	+	0.134932i	$0.0430815\pi$
$829$			7.77000i			0.269863i	−0.990855	+	0.134932i	$0.956919\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−17.7893	−	17.7893i	−0.616361	−	0.616361i
$834$		0			0
$835$	−12.5272	+	12.6832i	−0.433521	+	0.438922i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−22.8978			−0.790520			−0.395260	−	0.918569i	$-0.629345\pi$
$839$	−22.8978			−0.790520			−0.395260	+	0.918569i	$0.629345\pi$
$840$		0			0
$841$	−10.7120			−0.369379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.0937110	−	15.1378i	−0.00322376	−	0.520757i
$846$		0			0
$847$	−72.5099	−	72.5099i	−2.49147	−	2.49147i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	3.28202i		−	0.112506i
$852$		0			0
$853$	11.8352	−	11.8352i	0.405229	−	0.405229i	−0.474842	−	0.880071i	$-0.657495\pi$
$853$	11.8352	−	11.8352i	0.405229	−	0.405229i	0.880071	+	0.474842i	$0.157495\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	13.0543	−	13.0543i	0.445926	−	0.445926i	−0.448072	−	0.893998i	$-0.647889\pi$
$857$	13.0543	−	13.0543i	0.445926	−	0.445926i	0.893998	+	0.448072i	$0.147889\pi$
$858$		0			0
$859$			38.2891i			1.30641i	0.757183	+	0.653203i	$0.226575\pi$
$859$			38.2891i			1.30641i	−0.757183	+	0.653203i	$0.773425\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	30.2440	+	30.2440i	1.02952	+	1.02952i	0.999551	+	0.0299681i	$0.00954056\pi$
$863$	30.2440	+	30.2440i	1.02952	+	1.02952i	0.0299681	+	0.999551i	$0.490459\pi$
$864$		0			0
$865$	−22.3072	−	22.0327i	−0.758467	−	0.749134i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	64.8388			2.19951
$870$		0			0
$871$	20.6281			0.698956
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−26.9205	−	25.9388i	−0.910080	−	0.876890i
$876$		0			0
$877$	−10.2972	−	10.2972i	−0.347713	−	0.347713i	0.511544	−	0.859257i	$-0.329074\pi$
$877$	−10.2972	−	10.2972i	−0.347713	−	0.347713i	−0.859257	+	0.511544i	$0.829074\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			24.6652i			0.830992i	0.909595	+	0.415496i	$0.136392\pi$
$881$			24.6652i			0.830992i	−0.909595	+	0.415496i	$0.863608\pi$
$882$		0			0
$883$	−38.4286	+	38.4286i	−1.29323	+	1.29323i	−0.360447	+	0.932780i	$0.617376\pi$
$883$	−38.4286	+	38.4286i	−1.29323	+	1.29323i	−0.932780	+	0.360447i	$0.882624\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2.64962	−	2.64962i	0.0889656	−	0.0889656i	−0.661223	−	0.750189i	$-0.729962\pi$
$887$	2.64962	−	2.64962i	0.0889656	−	0.0889656i	0.750189	+	0.661223i	$0.229962\pi$
$888$		0			0
$889$			7.05841i			0.236731i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6.81475	+	6.81475i	0.228047	+	0.228047i
$894$		0			0
$895$	18.2959	−	0.113261i	0.611564	−	0.00378590i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	30.8944			1.03038
$900$		0			0
$901$	−42.9159			−1.42973
$902$		0			0
$903$		0			0
$904$		0			0
$905$	36.1449	−	0.223755i	1.20150	−	0.00743788i
$906$		0			0
$907$	−23.8352	−	23.8352i	−0.791434	−	0.791434i	0.190293	−	0.981727i	$-0.439056\pi$
$907$	−23.8352	−	23.8352i	−0.791434	−	0.791434i	−0.981727	+	0.190293i	$0.939056\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	13.8278i		−	0.458137i	−0.973410	−	0.229069i	$-0.926432\pi$
$911$		−	13.8278i		−	0.458137i	0.973410	−	0.229069i	$-0.0735680\pi$
$912$		0			0
$913$	68.3463	−	68.3463i	2.26193	−	2.26193i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−41.0040	+	41.0040i	−1.35407	+	1.35407i
$918$		0			0
$919$			19.6742i			0.648994i	0.945887	+	0.324497i	$0.105195\pi$
$919$			19.6742i			0.648994i	−0.945887	+	0.324497i	$0.894805\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	11.7248	+	11.7248i	0.385928	+	0.385928i
$924$		0			0
$925$	−11.7465	−	11.4591i	−0.386221	−	0.376774i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.24811			0.106567			0.0532836	−	0.998579i	$-0.483031\pi$
$929$	3.24811			0.106567			0.0532836	+	0.998579i	$0.483031\pi$
$930$		0			0
$931$	8.92959			0.292655
$932$		0			0
$933$		0			0
$934$		0			0
$935$	61.8033	+	61.0429i	2.02119	+	1.99631i
$936$		0			0
$937$	11.6686	+	11.6686i	0.381196	+	0.381196i	0.871533	−	0.490337i	$-0.163126\pi$
$937$	11.6686	+	11.6686i	0.381196	+	0.381196i	−0.490337	+	0.871533i	$0.663126\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	14.0889i		−	0.459285i	−0.973275	−	0.229643i	$-0.926244\pi$
$941$		−	14.0889i		−	0.459285i	0.973275	−	0.229643i	$-0.0737557\pi$
$942$		0			0
$943$	−2.99112	+	2.99112i	−0.0974043	+	0.0974043i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−34.7060	+	34.7060i	−1.12779	+	1.12779i	−0.137256	+	0.990536i	$0.543828\pi$
$947$	−34.7060	+	34.7060i	−1.12779	+	1.12779i	−0.990536	+	0.137256i	$0.956172\pi$
$948$		0			0
$949$			27.5232i			0.893441i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	36.8300	+	36.8300i	1.19304	+	1.19304i	0.976209	+	0.216832i	$0.0695723\pi$
$953$	36.8300	+	36.8300i	1.19304	+	1.19304i	0.216832	+	0.976209i	$0.430428\pi$
$954$		0			0
$955$	0.213003	+	34.4080i	0.00689262	+	1.11342i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.64332			0.246816
$960$		0			0
$961$	21.1906			0.683567
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6.43607	−	6.51625i	0.207184	−	0.209766i
$966$		0			0
$967$	−10.4405	−	10.4405i	−0.335745	−	0.335745i	0.519018	−	0.854763i	$-0.326298\pi$
$967$	−10.4405	−	10.4405i	−0.335745	−	0.335745i	−0.854763	+	0.519018i	$0.826298\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			8.61548i			0.276484i	0.990398	+	0.138242i	$0.0441452\pi$
$971$			8.61548i			0.276484i	−0.990398	+	0.138242i	$0.955855\pi$
$972$		0			0
$973$	−7.51709	+	7.51709i	−0.240987	+	0.240987i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	39.4908	−	39.4908i	1.26342	−	1.26342i	0.313999	−	0.949423i	$-0.398331\pi$
$977$	39.4908	−	39.4908i	1.26342	−	1.26342i	0.949423	−	0.313999i	$-0.101669\pi$
$978$		0			0
$979$			73.7420i			2.35680i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16.5149	−	16.5149i	−0.526744	−	0.526744i	0.392856	−	0.919600i	$-0.371487\pi$
$983$	−16.5149	−	16.5149i	−0.526744	−	0.526744i	−0.919600	+	0.392856i	$0.871487\pi$
$984$		0			0
$985$	15.1477	−	15.3364i	0.482646	−	0.488659i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.34179			0.169859
$990$		0			0
$991$	−13.0249			−0.413750			−0.206875	−	0.978367i	$-0.566329\pi$
$991$	−13.0249			−0.413750			−0.206875	+	0.978367i	$0.566329\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.173193	−	27.9771i	−0.00549059	−	0.886935i
$996$		0			0
$997$	16.8653	+	16.8653i	0.534129	+	0.534129i	0.921798	−	0.387670i	$-0.126720\pi$
$997$	16.8653	+	16.8653i	0.534129	+	0.534129i	−0.387670	+	0.921798i	$0.626720\pi$
$998$		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.737.1	✓	44
3.2	odd	2	inner	4140.2.s.a.737.22	yes	44
5.3	odd	4	inner	4140.2.s.a.2393.22	yes	44
15.8	even	4	inner	4140.2.s.a.2393.1	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.a.737.1	✓	44	1.1	even	1	trivial
4140.2.s.a.737.22	yes	44	3.2	odd	2	inner
4140.2.s.a.2393.1	yes	44	15.8	even	4	inner
4140.2.s.a.2393.22	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)

\(f(q)\)	\(=\)	\(q+(-2.23603 + 0.0138422i) q^{5} +(2.36435 + 2.36435i) q^{7} +O(q^{10})\)	\(q+(-2.23603 + 0.0138422i) q^{5} +(2.36435 + 2.36435i) q^{7} +6.45508i q^{11} +(-1.76494 + 1.76494i) q^{13} +(-4.25553 + 4.25553i) q^{17} -2.13613i q^{19} +(0.707107 + 0.707107i) q^{23} +(4.99962 - 0.0619028i) q^{25} -4.27645 q^{29} -7.22430 q^{31} +(-5.31947 - 5.25401i) q^{35} +(-2.32074 - 2.32074i) q^{37} +4.23009i q^{41} +(3.77722 - 3.77722i) q^{43} +(-3.19024 + 3.19024i) q^{47} +4.18027i q^{49} +(5.04237 + 5.04237i) q^{53} +(-0.0893522 - 14.4337i) q^{55} +7.56824 q^{59} -1.07814 q^{61} +(3.92202 - 3.97088i) q^{65} +(-5.84385 - 5.84385i) q^{67} -6.64319i q^{71} +(7.79721 - 7.79721i) q^{73} +(-15.2620 + 15.2620i) q^{77} -10.0446i q^{79} +(-10.5880 - 10.5880i) q^{83} +(9.45656 - 9.57437i) q^{85} +11.4239 q^{89} -8.34586 q^{91} +(0.0295686 + 4.77643i) q^{95} +(8.04515 + 8.04515i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 12 q^{7}+O(q^{10}) \) 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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