LMFDB - Embedded newform 3060.2.z.g.829.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,2,Mod(829,3060)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 2, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3060.829"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3060.z (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [40,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$24.4342230185$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 1020)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			829.8
Character	$\chi$	$=$	3060.829
Dual form			3060.2.z.g.2809.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.725140 - 2.11522i) q^{5} +(0.920356 + 0.920356i) q^{7} +(2.91237 - 2.91237i) q^{11} +1.15938i q^{13} +(-3.13193 + 2.68160i) q^{17} +6.96915i q^{19} +(0.453675 + 0.453675i) q^{23} +(-3.94834 + 3.06767i) q^{25} +(1.79476 + 1.79476i) q^{29} +(4.62604 + 4.62604i) q^{31} +(1.27937 - 2.61415i) q^{35} +(-4.42242 + 4.42242i) q^{37} +(3.18157 - 3.18157i) q^{41} +12.6350 q^{43} +9.92920i q^{47} -5.30589i q^{49} +6.07145 q^{53} +(-8.27219 - 4.04844i) q^{55} -5.65133i q^{59} +(-7.40360 + 7.40360i) q^{61} +(2.45234 - 0.840709i) q^{65} +1.99215i q^{67} +(-8.29652 - 8.29652i) q^{71} +(7.41073 - 7.41073i) q^{73} +5.36084 q^{77} +(10.1183 - 10.1183i) q^{79} -11.7211 q^{83} +(7.94328 + 4.68020i) q^{85} -9.32011 q^{89} +(-1.06704 + 1.06704i) q^{91} +(14.7413 - 5.05361i) q^{95} +(-1.46372 + 1.46372i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 4 q^{5} + 8 q^{11} - 24 q^{29} - 16 q^{31} - 8 q^{35} + 8 q^{41} + 28 q^{55} + 16 q^{61} - 56 q^{71} + 16 q^{79} - 40 q^{85} - 32 q^{89} + 64 q^{91} - 64 q^{95}+O(q^{100}) $ 40 * q - 4 * q^5 + 8 * q^11 - 24 * q^29 - 16 * q^31 - 8 * q^35 + 8 * q^41 + 28 * q^55 + 16 * q^61 - 56 * q^71 + 16 * q^79 - 40 * q^85 - 32 * q^89 + 64 * q^91 - 64 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times$.

$n$	$1261$	$1361$	$1531$	$1837$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$-1$

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
$5$	−0.725140	−	2.11522i	−0.324292	−	0.945957i
$5$	−0.725140	−	2.11522i	−0.324292	−	0.945957i
$6$		0			0
$7$	0.920356	+	0.920356i	0.347862	+	0.347862i	0.859313	−	0.511451i	$-0.170892\pi$
$7$	0.920356	+	0.920356i	0.347862	+	0.347862i	−0.511451	+	0.859313i	$0.670892\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.91237	−	2.91237i	0.878113	−	0.878113i	−0.115226	−	0.993339i	$-0.536759\pi$
$11$	2.91237	−	2.91237i	0.878113	−	0.878113i	0.993339	+	0.115226i	$0.0367594\pi$
$12$		0			0
$13$			1.15938i			0.321553i	0.986991	+	0.160776i	$0.0513998\pi$
$13$			1.15938i			0.321553i	−0.986991	+	0.160776i	$0.948600\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.13193	+	2.68160i	−0.759605	+	0.650384i
$17$	−3.13193	+	2.68160i	−0.759605	+	0.650384i
$18$		0			0
$19$			6.96915i			1.59883i	0.600778	+	0.799416i	$0.294858\pi$
$19$			6.96915i			1.59883i	−0.600778	+	0.799416i	$0.705142\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.453675	+	0.453675i	0.0945977	+	0.0945977i	0.752822	−	0.658224i	$-0.228692\pi$
$23$	0.453675	+	0.453675i	0.0945977	+	0.0945977i	−0.658224	+	0.752822i	$0.728692\pi$
$24$		0			0
$25$	−3.94834	+	3.06767i	−0.789669	+	0.613533i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.79476	+	1.79476i	0.333279	+	0.333279i	0.853831	−	0.520551i	$-0.174273\pi$
$29$	1.79476	+	1.79476i	0.333279	+	0.333279i	−0.520551	+	0.853831i	$0.674273\pi$
$30$		0			0
$31$	4.62604	+	4.62604i	0.830862	+	0.830862i	0.987635	−	0.156773i	$-0.0501091\pi$
$31$	4.62604	+	4.62604i	0.830862	+	0.830862i	−0.156773	+	0.987635i	$0.550109\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.27937	−	2.61415i	0.216253	−	0.441871i
$36$		0			0
$37$	−4.42242	+	4.42242i	−0.727041	+	0.727041i	−0.970029	−	0.242988i	$-0.921872\pi$
$37$	−4.42242	+	4.42242i	−0.727041	+	0.727041i	0.242988	+	0.970029i	$0.421872\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.18157	−	3.18157i	0.496878	−	0.496878i	−0.413587	−	0.910465i	$-0.635724\pi$
$41$	3.18157	−	3.18157i	0.496878	−	0.496878i	0.910465	+	0.413587i	$0.135724\pi$
$42$		0			0
$43$	12.6350			1.92682			0.963411	−	0.268029i	$-0.0863724\pi$
$43$	12.6350			1.92682			0.963411	+	0.268029i	$0.0863724\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			9.92920i			1.44832i	0.689630	+	0.724161i	$0.257773\pi$
$47$			9.92920i			1.44832i	−0.689630	+	0.724161i	$0.742227\pi$
$48$		0			0
$49$		−	5.30589i		−	0.757984i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.07145			0.833978			0.416989	−	0.908912i	$-0.363085\pi$
$53$	6.07145			0.833978			0.416989	+	0.908912i	$0.363085\pi$
$54$		0			0
$55$	−8.27219	−	4.04844i	−1.11542	−	0.545892i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	5.65133i		−	0.735741i	−0.929877	−	0.367870i	$-0.880087\pi$
$59$		−	5.65133i		−	0.735741i	0.929877	−	0.367870i	$-0.119913\pi$
$60$		0			0
$61$	−7.40360	+	7.40360i	−0.947934	+	0.947934i	−0.998710	−	0.0507757i	$-0.983831\pi$
$61$	−7.40360	+	7.40360i	−0.947934	+	0.947934i	0.0507757	+	0.998710i	$0.483831\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.45234	−	0.840709i	0.304175	−	0.104277i
$66$		0			0
$67$			1.99215i			0.243379i	0.992568	+	0.121690i	$0.0388313\pi$
$67$			1.99215i			0.243379i	−0.992568	+	0.121690i	$0.961169\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.29652	−	8.29652i	−0.984616	−	0.984616i	0.0152672	−	0.999883i	$-0.495140\pi$
$71$	−8.29652	−	8.29652i	−0.984616	−	0.984616i	−0.999883	+	0.0152672i	$0.995140\pi$
$72$		0			0
$73$	7.41073	−	7.41073i	0.867360	−	0.867360i	−0.124819	−	0.992180i	$-0.539835\pi$
$73$	7.41073	−	7.41073i	0.867360	−	0.867360i	0.992180	+	0.124819i	$0.0398350\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.36084			0.610924
$78$		0			0
$79$	10.1183	−	10.1183i	1.13840	−	1.13840i	0.149657	−	0.988738i	$-0.452183\pi$
$79$	10.1183	−	10.1183i	1.13840	−	1.13840i	0.988738	−	0.149657i	$-0.0478170\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−11.7211			−1.28656			−0.643280	−	0.765631i	$-0.722427\pi$
$83$	−11.7211			−1.28656			−0.643280	+	0.765631i	$0.722427\pi$
$84$		0			0
$85$	7.94328	+	4.68020i	0.861570	+	0.507639i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.32011			−0.987930			−0.493965	−	0.869482i	$-0.664453\pi$
$89$	−9.32011			−0.987930			−0.493965	+	0.869482i	$0.664453\pi$
$90$		0			0
$91$	−1.06704	+	1.06704i	−0.111856	+	0.111856i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	14.7413	−	5.05361i	1.51243	−	0.518489i
$96$		0			0
$97$	−1.46372	+	1.46372i	−0.148618	+	0.148618i	−0.777501	−	0.628882i	$-0.783513\pi$
$97$	−1.46372	+	1.46372i	−0.148618	+	0.148618i	0.628882	+	0.777501i	$0.283513\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	12.2223			1.21616			0.608081	−	0.793875i	$-0.291939\pi$
$101$	12.2223			1.21616			0.608081	+	0.793875i	$0.291939\pi$
$102$		0			0
$103$			9.75626i			0.961313i	0.876909	+	0.480657i	$0.159602\pi$
$103$			9.75626i			0.961313i	−0.876909	+	0.480657i	$0.840398\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.82873	+	2.82873i	−0.273463	+	0.273463i	−0.830493	−	0.557029i	$-0.811941\pi$
$107$	−2.82873	+	2.82873i	−0.273463	+	0.273463i	0.557029	+	0.830493i	$0.311941\pi$
$108$		0			0
$109$	9.18746	−	9.18746i	0.879999	−	0.879999i	−0.113535	−	0.993534i	$-0.536217\pi$
$109$	9.18746	−	9.18746i	0.879999	−	0.879999i	0.993534	+	0.113535i	$0.0362175\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.03145	+	9.03145i	0.849607	+	0.849607i	0.990084	−	0.140477i	$-0.0448636\pi$
$113$	9.03145	+	9.03145i	0.849607	+	0.849607i	−0.140477	+	0.990084i	$0.544864\pi$
$114$		0			0
$115$	0.630646	−	1.28860i	0.0588080	−	0.120163i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.35052	−	0.414464i	−0.490482	−	0.0379938i
$120$		0			0
$121$		−	5.96381i		−	0.542164i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.35190	+	6.12715i	0.836460	+	0.548029i
$126$		0			0
$127$	13.9130			1.23458			0.617290	−	0.786736i	$-0.288231\pi$
$127$	13.9130			1.23458			0.617290	+	0.786736i	$0.288231\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.27286	+	9.27286i	0.810173	+	0.810173i	0.984660	−	0.174486i	$-0.0558265\pi$
$131$	9.27286	+	9.27286i	0.810173	+	0.810173i	−0.174486	+	0.984660i	$0.555826\pi$
$132$		0			0
$133$	−6.41410	+	6.41410i	−0.556173	+	0.556173i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			18.4292i			1.57452i	0.616624	+	0.787258i	$0.288500\pi$
$137$			18.4292i			1.57452i	−0.616624	+	0.787258i	$0.711500\pi$
$138$		0			0
$139$	−0.0744117	−	0.0744117i	−0.00631152	−	0.00631152i	0.703944	−	0.710255i	$-0.251421\pi$
$139$	−0.0744117	−	0.0744117i	−0.00631152	−	0.00631152i	−0.710255	+	0.703944i	$0.751421\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.37653	+	3.37653i	0.282360	+	0.282360i
$144$		0			0
$145$	2.49487	−	5.09778i	0.207188	−	0.423348i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.11888			−0.665125			−0.332562	−	0.943081i	$-0.607913\pi$
$149$	−8.11888			−0.665125			−0.332562	+	0.943081i	$0.607913\pi$
$150$		0			0
$151$		−	7.22211i		−	0.587727i	−0.955847	−	0.293864i	$-0.905059\pi$
$151$		−	7.22211i		−	0.587727i	0.955847	−	0.293864i	$-0.0949411\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.43059	−	13.1396i	0.516517	−	1.05540i
$156$		0			0
$157$			20.0539i			1.60048i	0.599681	+	0.800239i	$0.295294\pi$
$157$			20.0539i			1.60048i	−0.599681	+	0.800239i	$0.704706\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			0.835084i			0.0658139i
$162$		0			0
$163$	−6.35679	−	6.35679i	−0.497902	−	0.497902i	0.412882	−	0.910784i	$-0.364522\pi$
$163$	−6.35679	−	6.35679i	−0.497902	−	0.497902i	−0.910784	+	0.412882i	$0.864522\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	16.9499	−	16.9499i	1.31162	−	1.31162i	0.391399	−	0.920221i	$-0.371991\pi$
$167$	16.9499	−	16.9499i	1.31162	−	1.31162i	0.920221	−	0.391399i	$-0.128009\pi$
$168$		0			0
$169$	11.6558			0.896604
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.145437	+	0.145437i	−0.0110573	+	0.0110573i	−0.712614	−	0.701556i	$-0.752489\pi$
$173$	−0.145437	+	0.145437i	−0.0110573	+	0.0110573i	0.701556	+	0.712614i	$0.252489\pi$
$174$		0			0
$175$	−6.45723	−	0.810537i	−0.488121	−	0.0612709i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			17.0893i			1.27732i	0.769490	+	0.638659i	$0.220510\pi$
$179$			17.0893i			1.27732i	−0.769490	+	0.638659i	$0.779490\pi$
$180$		0			0
$181$	4.77844	−	4.77844i	0.355179	−	0.355179i	−0.506853	−	0.862032i	$-0.669191\pi$
$181$	4.77844	−	4.77844i	0.355179	−	0.355179i	0.862032	+	0.506853i	$0.169191\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	12.5613	+	6.14754i	0.923524	+	0.451976i
$186$		0			0
$187$	−1.31153	+	16.9312i	−0.0959084	+	1.23813i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	11.6178			0.840632			0.420316	−	0.907378i	$-0.361919\pi$
$191$	11.6178			0.840632			0.420316	+	0.907378i	$0.361919\pi$
$192$		0			0
$193$	−7.26385	−	7.26385i	−0.522864	−	0.522864i	0.395572	−	0.918435i	$-0.370547\pi$
$193$	−7.26385	−	7.26385i	−0.522864	−	0.522864i	−0.918435	+	0.395572i	$0.870547\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.3378	+	10.3378i	0.736539	+	0.736539i	0.971907	−	0.235367i	$-0.0756293\pi$
$197$	10.3378	+	10.3378i	0.736539	+	0.736539i	−0.235367	+	0.971907i	$0.575629\pi$
$198$		0			0
$199$	−3.63355	−	3.63355i	−0.257575	−	0.257575i	0.566492	−	0.824067i	$-0.308300\pi$
$199$	−3.63355	−	3.63355i	−0.257575	−	0.257575i	−0.824067	+	0.566492i	$0.808300\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			3.30365i			0.231870i
$204$		0			0
$205$	−9.03681	−	4.42265i	−0.631158	−	0.308891i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	20.2967	+	20.2967i	1.40395	+	1.40395i
$210$		0			0
$211$	0.480180	−	0.480180i	0.0330570	−	0.0330570i	−0.690385	−	0.723442i	$-0.742559\pi$
$211$	0.480180	−	0.480180i	0.0330570	−	0.0330570i	0.723442	+	0.690385i	$0.242559\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.16215	−	26.7259i	−0.624854	−	1.82269i
$216$		0			0
$217$			8.51521i			0.578050i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.10899	−	3.63109i	−0.209133	−	0.244253i
$222$		0			0
$223$	−5.43069			−0.363666			−0.181833	−	0.983329i	$-0.558203\pi$
$223$	−5.43069			−0.363666			−0.181833	+	0.983329i	$0.558203\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.35688	+	7.35688i	0.488293	+	0.488293i	0.907767	−	0.419474i	$-0.137785\pi$
$227$	7.35688	+	7.35688i	0.488293	+	0.488293i	−0.419474	+	0.907767i	$0.637785\pi$
$228$		0			0
$229$		−	25.9282i		−	1.71339i	−0.515826	−	0.856693i	$-0.672515\pi$
$229$		−	25.9282i		−	1.71339i	0.515826	−	0.856693i	$-0.327485\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	14.9626	−	14.9626i	0.980231	−	0.980231i	−0.0195771	−	0.999808i	$-0.506232\pi$
$233$	14.9626	−	14.9626i	0.980231	−	0.980231i	0.999808	+	0.0195771i	$0.00623198\pi$
$234$		0			0
$235$	21.0025	−	7.20006i	1.37005	−	0.469680i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.8457			−0.701550			−0.350775	−	0.936460i	$-0.614082\pi$
$239$	−10.8457			−0.701550			−0.350775	+	0.936460i	$0.614082\pi$
$240$		0			0
$241$	2.68742	+	2.68742i	0.173112	+	0.173112i	0.788345	−	0.615233i	$-0.210938\pi$
$241$	2.68742	+	2.68742i	0.173112	+	0.173112i	−0.615233	+	0.788345i	$0.710938\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−11.2231	+	3.84751i	−0.717020	+	0.245809i
$246$		0			0
$247$	−8.07986			−0.514109
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−22.0499			−1.39178			−0.695889	−	0.718149i	$-0.744990\pi$
$251$	−22.0499			−1.39178			−0.695889	+	0.718149i	$0.744990\pi$
$252$		0			0
$253$	2.64254			0.166135
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.19266			−0.323909			−0.161955	−	0.986798i	$-0.551780\pi$
$257$	−5.19266			−0.323909			−0.161955	+	0.986798i	$0.551780\pi$
$258$		0			0
$259$	−8.14040			−0.505820
$260$		0			0
$261$		0			0
$262$		0			0
$263$	10.1590			0.626428			0.313214	−	0.949683i	$-0.398594\pi$
$263$	10.1590			0.626428			0.313214	+	0.949683i	$0.398594\pi$
$264$		0			0
$265$	−4.40265	−	12.8425i	−0.270453	−	0.788907i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.44432	−	5.44432i	−0.331946	−	0.331946i	0.521379	−	0.853325i	$-0.325418\pi$
$269$	−5.44432	−	5.44432i	−0.331946	−	0.331946i	−0.853325	+	0.521379i	$0.825418\pi$
$270$		0			0
$271$	−8.33133			−0.506092			−0.253046	−	0.967454i	$-0.581432\pi$
$271$	−8.33133			−0.506092			−0.253046	+	0.967454i	$0.581432\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.56486	+	20.4332i	−0.154667	+	1.23217i
$276$		0			0
$277$	11.6068	−	11.6068i	0.697385	−	0.697385i	−0.266461	−	0.963846i	$-0.585854\pi$
$277$	11.6068	−	11.6068i	0.697385	−	0.697385i	0.963846	+	0.266461i	$0.0858543\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			12.2079i			0.728263i	0.931348	+	0.364131i	$0.118634\pi$
$281$			12.2079i			0.728263i	−0.931348	+	0.364131i	$0.881366\pi$
$282$		0			0
$283$	−22.4914	−	22.4914i	−1.33698	−	1.33698i	−0.898975	−	0.438001i	$-0.855687\pi$
$283$	−22.4914	−	22.4914i	−1.33698	−	1.33698i	−0.438001	−	0.898975i	$-0.644313\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.85635			0.345690
$288$		0			0
$289$	2.61801	−	16.7972i	0.154000	−	0.988071i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			26.1428i			1.52728i	0.645642	+	0.763640i	$0.276590\pi$
$293$			26.1428i			1.52728i	−0.645642	+	0.763640i	$0.723410\pi$
$294$		0			0
$295$	−11.9538	+	4.09801i	−0.695979	+	0.238595i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.525979	+	0.525979i	−0.0304182	+	0.0304182i
$300$		0			0
$301$	11.6287	+	11.6287i	0.670268	+	0.670268i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	21.0289	+	10.2916i	1.20411	+	0.589297i
$306$		0			0
$307$			3.71962i			0.212290i	0.994351	+	0.106145i	$0.0338508\pi$
$307$			3.71962i			0.212290i	−0.994351	+	0.106145i	$0.966149\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−11.6637	−	11.6637i	−0.661388	−	0.661388i	0.294319	−	0.955707i	$-0.404907\pi$
$311$	−11.6637	−	11.6637i	−0.661388	−	0.661388i	−0.955707	+	0.294319i	$0.904907\pi$
$312$		0			0
$313$	15.7319	+	15.7319i	0.889219	+	0.889219i	0.994448	−	0.105229i	$-0.0335576\pi$
$313$	15.7319	+	15.7319i	0.889219	+	0.889219i	−0.105229	+	0.994448i	$0.533558\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.89666	+	2.89666i	0.162692	+	0.162692i	0.783758	−	0.621066i	$-0.213300\pi$
$317$	2.89666	+	2.89666i	0.162692	+	0.162692i	−0.621066	+	0.783758i	$0.713300\pi$
$318$		0			0
$319$	10.4540			0.585314
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−18.6885	−	21.8269i	−1.03986	−	1.21448i
$324$		0			0
$325$	−3.55658	−	4.57761i	−0.197283	−	0.253920i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−9.13840	+	9.13840i	−0.503816	+	0.503816i
$330$		0			0
$331$			10.4885i			0.576502i	0.957555	+	0.288251i	$0.0930737\pi$
$331$			10.4885i			0.576502i	−0.957555	+	0.288251i	$0.906926\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.21384	−	1.44459i	0.230227	−	0.0789261i
$336$		0			0
$337$	−8.12749	+	8.12749i	−0.442733	+	0.442733i	−0.892929	−	0.450197i	$-0.851354\pi$
$337$	−8.12749	+	8.12749i	−0.442733	+	0.442733i	0.450197	+	0.892929i	$0.351354\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	26.9455			1.45918
$342$		0			0
$343$	11.3258	−	11.3258i	0.611536	−	0.611536i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−14.7752	−	14.7752i	−0.793175	−	0.793175i	0.188834	−	0.982009i	$-0.439529\pi$
$347$	−14.7752	−	14.7752i	−0.793175	−	0.793175i	−0.982009	+	0.188834i	$0.939529\pi$
$348$		0			0
$349$			11.9614i			0.640282i	0.947370	+	0.320141i	$0.103730\pi$
$349$			11.9614i			0.640282i	−0.947370	+	0.320141i	$0.896270\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			18.9492i			1.00856i	0.863539	+	0.504282i	$0.168243\pi$
$353$			18.9492i			1.00856i	−0.863539	+	0.504282i	$0.831757\pi$
$354$		0			0
$355$	−11.5329	+	23.5651i	−0.612101	+	1.25071i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			9.24214i			0.487781i	0.969803	+	0.243891i	$0.0784238\pi$
$359$			9.24214i			0.487781i	−0.969803	+	0.243891i	$0.921576\pi$
$360$		0			0
$361$	−29.5690			−1.55626
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−21.0492	−	10.3015i	−1.10176	−	0.539207i
$366$		0			0
$367$	−0.751175	−	0.751175i	−0.0392110	−	0.0392110i	0.687229	−	0.726440i	$-0.258827\pi$
$367$	−0.751175	−	0.751175i	−0.0392110	−	0.0392110i	−0.726440	+	0.687229i	$0.758827\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.58790	+	5.58790i	0.290109	+	0.290109i
$372$		0			0
$373$			7.67593i			0.397445i	0.980056	+	0.198722i	$0.0636792\pi$
$373$			7.67593i			0.397445i	−0.980056	+	0.198722i	$0.936321\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.08081	+	2.08081i	−0.107167	+	0.107167i
$378$		0			0
$379$	3.46061	+	3.46061i	0.177760	+	0.177760i	0.790378	−	0.612619i	$-0.209884\pi$
$379$	3.46061	+	3.46061i	0.177760	+	0.177760i	−0.612619	+	0.790378i	$0.709884\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.0930			−0.617926			−0.308963	−	0.951074i	$-0.599982\pi$
$383$	−12.0930			−0.617926			−0.308963	+	0.951074i	$0.599982\pi$
$384$		0			0
$385$	−3.88736	−	11.3394i	−0.198118	−	0.577908i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	15.2339i		−	0.772390i	−0.922417	−	0.386195i	$-0.873789\pi$
$389$		−	15.2339i		−	0.772390i	0.922417	−	0.386195i	$-0.126211\pi$
$390$		0			0
$391$	−2.63745	−	0.204303i	−0.133382	−	0.0103321i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−28.7396	−	14.0653i	−1.44605	−	0.707700i
$396$		0			0
$397$	14.7337	+	14.7337i	0.739464	+	0.739464i	0.972474	−	0.233010i	$-0.0748575\pi$
$397$	14.7337	+	14.7337i	0.739464	+	0.739464i	−0.233010	+	0.972474i	$0.574857\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.04136	+	5.04136i	−0.251753	+	0.251753i	−0.821689	−	0.569936i	$-0.806968\pi$
$401$	−5.04136	+	5.04136i	−0.251753	+	0.251753i	0.569936	+	0.821689i	$0.306968\pi$
$402$		0			0
$403$	−5.36332	+	5.36332i	−0.267166	+	0.267166i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			25.7595i			1.27685i
$408$		0			0
$409$	12.5044			0.618302			0.309151	−	0.951013i	$-0.399955\pi$
$409$	12.5044			0.618302			0.309151	+	0.951013i	$0.399955\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.20124	−	5.20124i	0.255936	−	0.255936i
$414$		0			0
$415$	8.49946	+	24.7928i	0.417222	+	1.21703i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.8752	+	12.8752i	−0.628995	+	0.628995i	−0.947815	−	0.318820i	$-0.896713\pi$
$419$	−12.8752	+	12.8752i	−0.628995	+	0.628995i	0.318820	+	0.947815i	$0.396713\pi$
$420$		0			0
$421$	−11.2740			−0.549459			−0.274729	−	0.961522i	$-0.588588\pi$
$421$	−11.2740			−0.549459			−0.274729	+	0.961522i	$0.588588\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.13969	−	20.1956i	0.200804	−	0.979631i
$426$		0			0
$427$	−13.6279			−0.659501
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.551302	+	0.551302i	−0.0265553	+	0.0265553i	−0.720260	−	0.693704i	$-0.755977\pi$
$431$	−0.551302	+	0.551302i	−0.0265553	+	0.0265553i	0.693704	+	0.720260i	$0.255977\pi$
$432$		0			0
$433$	1.85198			0.0890004			0.0445002	−	0.999009i	$-0.485830\pi$
$433$	1.85198			0.0890004			0.0445002	+	0.999009i	$0.485830\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.16173	+	3.16173i	−0.151246	+	0.151246i
$438$		0			0
$439$	24.0840	+	24.0840i	1.14947	+	1.14947i	0.986657	+	0.162811i	$0.0520561\pi$
$439$	24.0840	+	24.0840i	1.14947	+	1.14947i	0.162811	+	0.986657i	$0.447944\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	17.4319i		−	0.828214i	−0.910228	−	0.414107i	$-0.864094\pi$
$443$		−	17.4319i		−	0.828214i	0.910228	−	0.414107i	$-0.135906\pi$
$444$		0			0
$445$	6.75839	+	19.7141i	0.320378	+	0.934539i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18.6026	+	18.6026i	−0.877913	+	0.877913i	−0.993318	−	0.115406i	$-0.963183\pi$
$449$	−18.6026	+	18.6026i	−0.877913	+	0.877913i	0.115406	+	0.993318i	$0.463183\pi$
$450$		0			0
$451$		−	18.5318i		−	0.872629i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.03078	+	1.48327i	0.142085	+	0.0695369i
$456$		0			0
$457$	−17.4794			−0.817653			−0.408826	−	0.912612i	$-0.634062\pi$
$457$	−17.4794			−0.817653			−0.408826	+	0.912612i	$0.634062\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	32.6942i		−	1.52272i	−0.648330	−	0.761360i	$-0.724532\pi$
$461$		−	32.6942i		−	1.52272i	0.648330	−	0.761360i	$-0.275468\pi$
$462$		0			0
$463$			1.96866i			0.0914913i	0.998953	+	0.0457457i	$0.0145664\pi$
$463$			1.96866i			0.0914913i	−0.998953	+	0.0457457i	$0.985434\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.96756			−0.461244			−0.230622	−	0.973043i	$-0.574076\pi$
$467$	−9.96756			−0.461244			−0.230622	+	0.973043i	$0.574076\pi$
$468$		0			0
$469$	−1.83348	+	1.83348i	−0.0846625	+	0.0846625i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	36.7978	−	36.7978i	1.69197	−	1.69197i
$474$		0			0
$475$	−21.3790	−	27.5166i	−0.980937	−	1.26255i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−13.9579	−	13.9579i	−0.637753	−	0.637753i	0.312248	−	0.950001i	$-0.398918\pi$
$479$	−13.9579	−	13.9579i	−0.637753	−	0.637753i	−0.950001	+	0.312248i	$0.898918\pi$
$480$		0			0
$481$	−5.12725	−	5.12725i	−0.233782	−	0.233782i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.15750	+	2.03470i	0.188782	+	0.0923908i
$486$		0			0
$487$	15.0332	+	15.0332i	0.681218	+	0.681218i	0.960275	−	0.279057i	$-0.0900218\pi$
$487$	15.0332	+	15.0332i	0.681218	+	0.681218i	−0.279057	+	0.960275i	$0.590022\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	3.51034i		−	0.158419i	−0.996858	−	0.0792097i	$-0.974760\pi$
$491$		−	3.51034i		−	0.158419i	0.996858	−	0.0792097i	$-0.0252397\pi$
$492$		0			0
$493$	−10.4339	−	0.808236i	−0.469920	−	0.0364011i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	15.2715i		−	0.685021i
$498$		0			0
$499$	−6.45699	+	6.45699i	−0.289055	+	0.289055i	−0.836706	−	0.547652i	$-0.815522\pi$
$499$	−6.45699	+	6.45699i	−0.289055	+	0.289055i	0.547652	+	0.836706i	$0.315522\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.81392	−	3.81392i	−0.170054	−	0.170054i	0.616949	−	0.787003i	$-0.288368\pi$
$503$	−3.81392	−	3.81392i	−0.170054	−	0.170054i	−0.787003	+	0.616949i	$0.788368\pi$
$504$		0			0
$505$	−8.86287	−	25.8529i	−0.394392	−	1.15044i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	11.9248			0.528555			0.264278	−	0.964447i	$-0.414866\pi$
$509$	11.9248			0.528555			0.264278	+	0.964447i	$0.414866\pi$
$510$		0			0
$511$	13.6410			0.603443
$512$		0			0
$513$		0			0
$514$		0			0
$515$	20.6367	−	7.07466i	0.909361	−	0.311747i
$516$		0			0
$517$	28.9175	+	28.9175i	1.27179	+	1.27179i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	29.1089	−	29.1089i	1.27528	−	1.27528i	0.332008	−	0.943277i	$-0.392274\pi$
$521$	29.1089	−	29.1089i	1.27528	−	1.27528i	0.943277	−	0.332008i	$-0.107726\pi$
$522$		0			0
$523$		−	37.8707i		−	1.65597i	−0.560749	−	0.827986i	$-0.689487\pi$
$523$		−	37.8707i		−	1.65597i	0.560749	−	0.827986i	$-0.310513\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−26.8937	−	2.08324i	−1.17151	−	0.0907475i
$528$		0			0
$529$		−	22.5884i		−	0.982103i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.68863	+	3.68863i	0.159772	+	0.159772i
$534$		0			0
$535$	8.03461	+	3.93217i	0.347366	+	0.170002i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−15.4527	−	15.4527i	−0.665596	−	0.665596i
$540$		0			0
$541$	18.5764	+	18.5764i	0.798663	+	0.798663i	0.982885	−	0.184221i	$-0.0589763\pi$
$541$	18.5764	+	18.5764i	0.798663	+	0.798663i	−0.184221	+	0.982885i	$0.558976\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−26.0957	−	12.7713i	−1.11782	−	0.547064i
$546$		0			0
$547$	−22.7641	+	22.7641i	−0.973321	+	0.973321i	−0.999653	−	0.0263321i	$-0.991617\pi$
$547$	−22.7641	+	22.7641i	−0.973321	+	0.973321i	0.0263321	+	0.999653i	$0.491617\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12.5080	+	12.5080i	−0.532858	+	0.532858i
$552$		0			0
$553$	18.6248			0.792009
$554$		0			0
$555$		0			0
$556$		0			0
$557$			7.68311i			0.325544i	0.986664	+	0.162772i	$0.0520435\pi$
$557$			7.68311i			0.325544i	−0.986664	+	0.162772i	$0.947957\pi$
$558$		0			0
$559$			14.6487i			0.619575i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−38.6723			−1.62984			−0.814922	−	0.579571i	$-0.803220\pi$
$563$	−38.6723			−1.62984			−0.814922	+	0.579571i	$0.803220\pi$
$564$		0			0
$565$	12.5545	−	25.6526i	0.528171	−	1.07921i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	13.0075i		−	0.545304i	−0.962113	−	0.272652i	$-0.912099\pi$
$569$		−	13.0075i		−	0.545304i	0.962113	−	0.272652i	$-0.0879008\pi$
$570$		0			0
$571$	−26.4593	+	26.4593i	−1.10729	+	1.10729i	−0.113781	+	0.993506i	$0.536296\pi$
$571$	−26.4593	+	26.4593i	−1.10729	+	1.10729i	−0.993506	+	0.113781i	$0.963704\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.18299	−	0.399541i	−0.132740	−	0.0166620i
$576$		0			0
$577$		−	36.9260i		−	1.53725i	−0.639699	−	0.768626i	$-0.720941\pi$
$577$		−	36.9260i		−	1.53725i	0.639699	−	0.768626i	$-0.279059\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10.7876	−	10.7876i	−0.447546	−	0.447546i
$582$		0			0
$583$	17.6823	−	17.6823i	0.732327	−	0.732327i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−26.1089			−1.07763			−0.538814	−	0.842425i	$-0.681127\pi$
$587$	−26.1089			−1.07763			−0.538814	+	0.842425i	$0.681127\pi$
$588$		0			0
$589$	−32.2396	+	32.2396i	−1.32841	+	1.32841i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−7.06479			−0.290116			−0.145058	−	0.989423i	$-0.546337\pi$
$593$	−7.06479			−0.290116			−0.145058	+	0.989423i	$0.546337\pi$
$594$		0			0
$595$	3.00319	+	11.6181i	0.123119	+	0.476296i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	25.4369			1.03932			0.519662	−	0.854372i	$-0.326058\pi$
$599$	25.4369			1.03932			0.519662	+	0.854372i	$0.326058\pi$
$600$		0			0
$601$	13.0754	−	13.0754i	0.533355	−	0.533355i	−0.388214	−	0.921569i	$-0.626908\pi$
$601$	13.0754	−	13.0754i	0.533355	−	0.533355i	0.921569	+	0.388214i	$0.126908\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.6148	+	4.32459i	−0.512864	+	0.175820i
$606$		0			0
$607$	−33.2572	+	33.2572i	−1.34987	+	1.34987i	−0.464067	+	0.885800i	$0.653610\pi$
$607$	−33.2572	+	33.2572i	−1.34987	+	1.34987i	−0.885800	+	0.464067i	$0.846390\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−11.5117			−0.465713
$612$		0			0
$613$		−	30.0418i		−	1.21338i	−0.794940	−	0.606688i	$-0.792498\pi$
$613$		−	30.0418i		−	1.21338i	0.794940	−	0.606688i	$-0.207502\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−13.1004	+	13.1004i	−0.527402	+	0.527402i	−0.919797	−	0.392395i	$-0.871647\pi$
$617$	−13.1004	+	13.1004i	−0.527402	+	0.527402i	0.392395	+	0.919797i	$0.371647\pi$
$618$		0			0
$619$	−17.8124	+	17.8124i	−0.715940	+	0.715940i	−0.967771	−	0.251832i	$-0.918967\pi$
$619$	−17.8124	+	17.8124i	−0.715940	+	0.715940i	0.251832	+	0.967771i	$0.418967\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.57782	−	8.57782i	−0.343663	−	0.343663i
$624$		0			0
$625$	6.17885	−	24.2244i	0.247154	−	0.968976i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.99155	−	25.7099i	0.0794082	−	1.02512i
$630$		0			0
$631$		−	37.9978i		−	1.51267i	−0.654186	−	0.756333i	$-0.726989\pi$
$631$		−	37.9978i		−	1.51267i	0.654186	−	0.756333i	$-0.273011\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−10.0889	−	29.4291i	−0.400365	−	1.16786i
$636$		0			0
$637$	6.15152			0.243732
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−23.2894	−	23.2894i	−0.919874	−	0.919874i	0.0771457	−	0.997020i	$-0.475419\pi$
$641$	−23.2894	−	23.2894i	−0.919874	−	0.919874i	−0.997020	+	0.0771457i	$0.975419\pi$
$642$		0			0
$643$	−12.3729	+	12.3729i	−0.487939	+	0.487939i	−0.907655	−	0.419716i	$-0.862130\pi$
$643$	−12.3729	+	12.3729i	−0.487939	+	0.487939i	0.419716	+	0.907655i	$0.362130\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	17.8600i		−	0.702151i	−0.936347	−	0.351075i	$-0.885816\pi$
$647$		−	17.8600i		−	0.702151i	0.936347	−	0.351075i	$-0.114184\pi$
$648$		0			0
$649$	−16.4588	−	16.4588i	−0.646064	−	0.646064i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−19.9032	−	19.9032i	−0.778873	−	0.778873i	0.200766	−	0.979639i	$-0.435657\pi$
$653$	−19.9032	−	19.9032i	−0.778873	−	0.778873i	−0.979639	+	0.200766i	$0.935657\pi$
$654$		0			0
$655$	12.8901	−	26.3383i	0.503656	−	1.02912i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	24.7532			0.964250			0.482125	−	0.876103i	$-0.339865\pi$
$659$	24.7532			0.964250			0.482125	+	0.876103i	$0.339865\pi$
$660$		0			0
$661$			25.2845i			0.983452i	0.870750	+	0.491726i	$0.163634\pi$
$661$			25.2845i			0.983452i	−0.870750	+	0.491726i	$0.836366\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	18.2184	+	8.91614i	0.706478	+	0.345753i
$666$		0			0
$667$			1.62848i			0.0630549i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			43.1241i			1.66479i
$672$		0			0
$673$	17.2029	+	17.2029i	0.663122	+	0.663122i	0.956115	−	0.292993i	$-0.0946511\pi$
$673$	17.2029	+	17.2029i	0.663122	+	0.663122i	−0.292993	+	0.956115i	$0.594651\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6.81527	+	6.81527i	−0.261932	+	0.261932i	−0.825839	−	0.563907i	$-0.809298\pi$
$677$	−6.81527	+	6.81527i	−0.261932	+	0.261932i	0.563907	+	0.825839i	$0.309298\pi$
$678$		0			0
$679$	−2.69429			−0.103397
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9.96792	+	9.96792i	−0.381412	+	0.381412i	−0.871611	−	0.490199i	$-0.836924\pi$
$683$	−9.96792	+	9.96792i	−0.381412	+	0.381412i	0.490199	+	0.871611i	$0.336924\pi$
$684$		0			0
$685$	38.9820	−	13.3638i	1.48942	−	0.510604i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			7.03909i			0.268168i
$690$		0			0
$691$	23.0027	−	23.0027i	0.875062	−	0.875062i	−0.117956	−	0.993019i	$-0.537634\pi$
$691$	23.0027	−	23.0027i	0.875062	−	0.875062i	0.993019	+	0.117956i	$0.0376343\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.103439	+	0.211356i	−0.00392365	+	0.00801720i
$696$		0			0
$697$	−1.43276	+	18.4962i	−0.0542695	+	0.700592i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−5.92797			−0.223896			−0.111948	−	0.993714i	$-0.535709\pi$
$701$	−5.92797			−0.223896			−0.111948	+	0.993714i	$0.535709\pi$
$702$		0			0
$703$	−30.8205	−	30.8205i	−1.16242	−	1.16242i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	11.2489	+	11.2489i	0.423057	+	0.423057i
$708$		0			0
$709$	29.5924	+	29.5924i	1.11137	+	1.11137i	0.992966	+	0.118400i	$0.0377767\pi$
$709$	29.5924	+	29.5924i	1.11137	+	1.11137i	0.118400	+	0.992966i	$0.462223\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			4.19744i			0.157195i
$714$		0			0
$715$	4.69366	−	9.59058i	0.175533	−	0.358667i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	14.8911	+	14.8911i	0.555344	+	0.555344i	0.927978	−	0.372634i	$-0.121545\pi$
$719$	14.8911	+	14.8911i	0.555344	+	0.555344i	−0.372634	+	0.927978i	$0.621545\pi$
$720$		0			0
$721$	−8.97924	+	8.97924i	−0.334404	+	0.334404i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−12.5921	−	1.58061i	−0.467658	−	0.0587024i
$726$		0			0
$727$		−	10.2119i		−	0.378738i	−0.981906	−	0.189369i	$-0.939356\pi$
$727$		−	10.2119i		−	0.378738i	0.981906	−	0.189369i	$-0.0606442\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−39.5720	+	33.8821i	−1.46362	+	1.25317i
$732$		0			0
$733$	−13.2721			−0.490215			−0.245107	−	0.969496i	$-0.578823\pi$
$733$	−13.2721			−0.490215			−0.245107	+	0.969496i	$0.578823\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.80187	+	5.80187i	0.213715	+	0.213715i
$738$		0			0
$739$		−	27.0478i		−	0.994968i	−0.867473	−	0.497484i	$-0.834257\pi$
$739$		−	27.0478i		−	0.994968i	0.867473	−	0.497484i	$-0.165743\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	27.8098	−	27.8098i	1.02024	−	1.02024i	0.0204515	−	0.999791i	$-0.493490\pi$
$743$	27.8098	−	27.8098i	1.02024	−	1.02024i	0.999791	−	0.0204515i	$-0.00651036\pi$
$744$		0			0
$745$	5.88732	+	17.1732i	0.215695	+	0.629179i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.20687			−0.190255
$750$		0			0
$751$	5.97276	+	5.97276i	0.217949	+	0.217949i	0.807634	−	0.589684i	$-0.200748\pi$
$751$	5.97276	+	5.97276i	0.217949	+	0.217949i	−0.589684	+	0.807634i	$0.700748\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−15.2764	+	5.23704i	−0.555964	+	0.190595i
$756$		0			0
$757$	40.6471			1.47734			0.738672	−	0.674065i	$-0.235453\pi$
$757$	40.6471			1.47734			0.738672	+	0.674065i	$0.235453\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	32.7541			1.18733			0.593667	−	0.804711i	$-0.297680\pi$
$761$	32.7541			1.18733			0.593667	+	0.804711i	$0.297680\pi$
$762$		0			0
$763$	16.9115			0.612236
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6.55202			0.236580
$768$		0			0
$769$	19.5789			0.706032			0.353016	−	0.935617i	$-0.385156\pi$
$769$	19.5789			0.706032			0.353016	+	0.935617i	$0.385156\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−37.6385			−1.35376			−0.676881	−	0.736092i	$-0.736669\pi$
$773$	−37.6385			−1.35376			−0.676881	+	0.736092i	$0.736669\pi$
$774$		0			0
$775$	−32.4564	−	4.07405i	−1.16587	−	0.146344i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	22.1728	+	22.1728i	0.794424	+	0.794424i
$780$		0			0
$781$	−48.3251			−1.72921
$782$		0			0
$783$		0			0
$784$		0			0
$785$	42.4186	−	14.5419i	1.51398	−	0.519023i
$786$		0			0
$787$	−24.7987	+	24.7987i	−0.883978	+	0.883978i	−0.993936	−	0.109959i	$-0.964928\pi$
$787$	−24.7987	+	24.7987i	−0.883978	+	0.883978i	0.109959	+	0.993936i	$0.464928\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			16.6243i			0.591092i
$792$		0			0
$793$	−8.58356	−	8.58356i	−0.304811	−	0.304811i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.98758			−0.105826			−0.0529128	−	0.998599i	$-0.516851\pi$
$797$	−2.98758			−0.105826			−0.0529128	+	0.998599i	$0.516851\pi$
$798$		0			0
$799$	−26.6262	−	31.0976i	−0.941967	−	1.10015i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	43.1656i		−	1.52328i
$804$		0			0
$805$	1.76639	−	0.605553i	0.0622571	−	0.0213429i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−16.6706	+	16.6706i	−0.586108	+	0.586108i	−0.936575	−	0.350467i	$-0.886023\pi$
$809$	−16.6706	+	16.6706i	−0.586108	+	0.586108i	0.350467	+	0.936575i	$0.386023\pi$
$810$		0			0
$811$	14.3540	+	14.3540i	0.504039	+	0.504039i	0.912690	−	0.408652i	$-0.134001\pi$
$811$	14.3540	+	14.3540i	0.504039	+	0.504039i	−0.408652	+	0.912690i	$0.634001\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8.83647	+	18.0556i	−0.309528	+	0.632460i
$816$		0			0
$817$			88.0553i			3.08066i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.57512	+	6.57512i	0.229473	+	0.229473i	0.812473	−	0.582999i	$-0.198121\pi$
$821$	6.57512	+	6.57512i	0.229473	+	0.229473i	−0.582999	+	0.812473i	$0.698121\pi$
$822$		0			0
$823$	−11.4869	−	11.4869i	−0.400408	−	0.400408i	0.477969	−	0.878377i	$-0.341373\pi$
$823$	−11.4869	−	11.4869i	−0.400408	−	0.400408i	−0.878377	+	0.477969i	$0.841373\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−20.7914	−	20.7914i	−0.722988	−	0.722988i	0.246225	−	0.969213i	$-0.420810\pi$
$827$	−20.7914	−	20.7914i	−0.722988	−	0.722988i	−0.969213	+	0.246225i	$0.920810\pi$
$828$		0			0
$829$	−48.9429			−1.69986			−0.849929	−	0.526898i	$-0.823355\pi$
$829$	−48.9429			−1.69986			−0.849929	+	0.526898i	$0.823355\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	14.2283	+	16.6177i	0.492981	+	0.575769i
$834$		0			0
$835$	−48.1438	−	23.5617i	−1.66608	−	0.815388i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	29.7911	−	29.7911i	1.02850	−	1.02850i	0.0289208	−	0.999582i	$-0.490793\pi$
$839$	29.7911	−	29.7911i	1.02850	−	1.02850i	0.999582	−	0.0289208i	$-0.00920706\pi$
$840$		0			0
$841$		−	22.5576i		−	0.777850i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.45212	−	24.6547i	−0.290762	−	0.848148i
$846$		0			0
$847$	5.48883	−	5.48883i	0.188598	−	0.188598i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.01268			−0.137553
$852$		0			0
$853$	−14.4227	+	14.4227i	−0.493823	+	0.493823i	−0.909508	−	0.415685i	$-0.863542\pi$
$853$	−14.4227	+	14.4227i	−0.493823	+	0.493823i	0.415685	+	0.909508i	$0.363542\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	23.5643	+	23.5643i	0.804940	+	0.804940i	0.983863	−	0.178923i	$-0.0572613\pi$
$857$	23.5643	+	23.5643i	0.804940	+	0.804940i	−0.178923	+	0.983863i	$0.557261\pi$
$858$		0			0
$859$		−	7.83039i		−	0.267170i	−0.991037	−	0.133585i	$-0.957351\pi$
$859$		−	7.83039i		−	0.267170i	0.991037	−	0.133585i	$-0.0426488\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			11.8712i			0.404099i	0.979375	+	0.202050i	$0.0647603\pi$
$863$			11.8712i			0.404099i	−0.979375	+	0.202050i	$0.935240\pi$
$864$		0			0
$865$	0.413093	+	0.202169i	0.0140456	+	0.00687395i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	58.9363i		−	1.99928i
$870$		0			0
$871$	−2.30965			−0.0782594
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.96793	+	14.2462i	0.100334	+	0.481611i
$876$		0			0
$877$	1.45075	+	1.45075i	0.0489882	+	0.0489882i	0.731177	−	0.682188i	$-0.238972\pi$
$877$	1.45075	+	1.45075i	0.0489882	+	0.0489882i	−0.682188	+	0.731177i	$0.738972\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	5.33607	+	5.33607i	0.179777	+	0.179777i	0.791259	−	0.611482i	$-0.209426\pi$
$881$	5.33607	+	5.33607i	0.179777	+	0.179777i	−0.611482	+	0.791259i	$0.709426\pi$
$882$		0			0
$883$		−	48.5959i		−	1.63538i	−0.575657	−	0.817691i	$-0.695254\pi$
$883$		−	48.5959i		−	1.63538i	0.575657	−	0.817691i	$-0.304746\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−25.1417	+	25.1417i	−0.844177	+	0.844177i	−0.989399	−	0.145222i	$-0.953610\pi$
$887$	−25.1417	+	25.1417i	−0.844177	+	0.844177i	0.145222	+	0.989399i	$0.453610\pi$
$888$		0			0
$889$	12.8049	+	12.8049i	0.429463	+	0.429463i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−69.1981			−2.31563
$894$		0			0
$895$	36.1478	−	12.3922i	1.20829	−	0.414224i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			16.6053i			0.553818i
$900$		0			0
$901$	−19.0154	+	16.2812i	−0.633494	+	0.542406i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−13.5725	−	6.64244i	−0.451166	−	0.220802i
$906$		0			0
$907$	−23.3183	−	23.3183i	−0.774272	−	0.774272i	0.204578	−	0.978850i	$-0.434418\pi$
$907$	−23.3183	−	23.3183i	−0.774272	−	0.774272i	−0.978850	+	0.204578i	$0.934418\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−40.0778	+	40.0778i	−1.32784	+	1.32784i	−0.420580	+	0.907256i	$0.638173\pi$
$911$	−40.0778	+	40.0778i	−1.32784	+	1.32784i	−0.907256	+	0.420580i	$0.861827\pi$
$912$		0			0
$913$	−34.1363	+	34.1363i	−1.12975	+	1.12975i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			17.0687i			0.563657i
$918$		0			0
$919$	−5.02824			−0.165866			−0.0829332	−	0.996555i	$-0.526429\pi$
$919$	−5.02824			−0.165866			−0.0829332	+	0.996555i	$0.526429\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	9.61879	−	9.61879i	0.316606	−	0.316606i
$924$		0			0
$925$	3.89473	−	31.0277i	0.128058	−	1.02019i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17.4499	+	17.4499i	−0.572514	+	0.572514i	−0.932830	−	0.360316i	$-0.882669\pi$
$929$	−17.4499	+	17.4499i	−0.572514	+	0.572514i	0.360316	+	0.932830i	$0.382669\pi$
$930$		0			0
$931$	36.9775			1.21189
$932$		0			0
$933$		0			0
$934$		0			0
$935$	36.7643	−	9.50329i	1.20232	−	0.310791i
$936$		0			0
$937$	−42.1294			−1.37631			−0.688154	−	0.725565i	$-0.741579\pi$
$937$	−42.1294			−1.37631			−0.688154	+	0.725565i	$0.741579\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−27.5722	+	27.5722i	−0.898828	+	0.898828i	−0.995333	−	0.0965045i	$-0.969234\pi$
$941$	−27.5722	+	27.5722i	−0.898828	+	0.898828i	0.0965045	+	0.995333i	$0.469234\pi$
$942$		0			0
$943$	2.88679			0.0940070
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15.1175	−	15.1175i	0.491253	−	0.491253i	−0.417448	−	0.908701i	$-0.637075\pi$
$947$	15.1175	−	15.1175i	0.491253	−	0.491253i	0.908701	+	0.417448i	$0.137075\pi$
$948$		0			0
$949$	8.59182	+	8.59182i	0.278902	+	0.278902i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			28.4337i			0.921058i	0.887645	+	0.460529i	$0.152340\pi$
$953$			28.4337i			0.921058i	−0.887645	+	0.460529i	$0.847660\pi$
$954$		0			0
$955$	−8.42450	−	24.5742i	−0.272611	−	0.795201i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−16.9615	+	16.9615i	−0.547714	+	0.547714i
$960$		0			0
$961$			11.8005i			0.380662i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−10.0974	+	20.6320i	−0.325046	+	0.664167i
$966$		0			0
$967$	−10.9648			−0.352604			−0.176302	−	0.984336i	$-0.556414\pi$
$967$	−10.9648			−0.352604			−0.176302	+	0.984336i	$0.556414\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			23.7882i			0.763400i	0.924286	+	0.381700i	$0.124661\pi$
$971$			23.7882i			0.763400i	−0.924286	+	0.381700i	$0.875339\pi$
$972$		0			0
$973$		−	0.136971i		−	0.00439107i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.33721			−0.0747738			−0.0373869	−	0.999301i	$-0.511903\pi$
$977$	−2.33721			−0.0747738			−0.0373869	+	0.999301i	$0.511903\pi$
$978$		0			0
$979$	−27.1436	+	27.1436i	−0.867514	+	0.867514i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	30.1083	−	30.1083i	0.960306	−	0.960306i	−0.0389361	−	0.999242i	$-0.512397\pi$
$983$	30.1083	−	30.1083i	0.960306	−	0.960306i	0.999242	+	0.0389361i	$0.0123969\pi$
$984$		0			0
$985$	14.3704	−	29.3632i	0.457880	−	0.935588i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.73218	+	5.73218i	0.182273	+	0.182273i
$990$		0			0
$991$	−11.1150	−	11.1150i	−0.353080	−	0.353080i	0.508174	−	0.861254i	$-0.330321\pi$
$991$	−11.1150	−	11.1150i	−0.353080	−	0.353080i	−0.861254	+	0.508174i	$0.830321\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5.05094	+	10.3206i	−0.160126	+	0.327185i
$996$		0			0
$997$	−30.1544	−	30.1544i	−0.954998	−	0.954998i	0.0440317	−	0.999030i	$-0.485980\pi$
$997$	−30.1544	−	30.1544i	−0.954998	−	0.954998i	−0.999030	+	0.0440317i	$0.985980\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.z.g.829.8		40
3.2	odd	2		1020.2.y.a.829.15	yes	40
5.4	even	2	inner	3060.2.z.g.829.3		40
15.14	odd	2		1020.2.y.a.829.6	yes	40
17.4	even	4	inner	3060.2.z.g.2809.3		40
51.38	odd	4		1020.2.y.a.769.6	✓	40
85.4	even	4	inner	3060.2.z.g.2809.8		40
255.89	odd	4		1020.2.y.a.769.15	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1020.2.y.a.769.6	✓	40	51.38	odd	4
1020.2.y.a.769.15	yes	40	255.89	odd	4
1020.2.y.a.829.6	yes	40	15.14	odd	2
1020.2.y.a.829.15	yes	40	3.2	odd	2
3060.2.z.g.829.3		40	5.4	even	2	inner
3060.2.z.g.829.8		40	1.1	even	1	trivial
3060.2.z.g.2809.3		40	17.4	even	4	inner
3060.2.z.g.2809.8		40	85.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3060.z (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.725140 - 2.11522i) q^{5} +(0.920356 + 0.920356i) q^{7} +(2.91237 - 2.91237i) q^{11} +1.15938i q^{13} +(-3.13193 + 2.68160i) q^{17} +6.96915i q^{19} +(0.453675 + 0.453675i) q^{23} +(-3.94834 + 3.06767i) q^{25} +(1.79476 + 1.79476i) q^{29} +(4.62604 + 4.62604i) q^{31} +(1.27937 - 2.61415i) q^{35} +(-4.42242 + 4.42242i) q^{37} +(3.18157 - 3.18157i) q^{41} +12.6350 q^{43} +9.92920i q^{47} -5.30589i q^{49} +6.07145 q^{53} +(-8.27219 - 4.04844i) q^{55} -5.65133i q^{59} +(-7.40360 + 7.40360i) q^{61} +(2.45234 - 0.840709i) q^{65} +1.99215i q^{67} +(-8.29652 - 8.29652i) q^{71} +(7.41073 - 7.41073i) q^{73} +5.36084 q^{77} +(10.1183 - 10.1183i) q^{79} -11.7211 q^{83} +(7.94328 + 4.68020i) q^{85} -9.32011 q^{89} +(-1.06704 + 1.06704i) q^{91} +(14.7413 - 5.05361i) q^{95} +(-1.46372 + 1.46372i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 4 q^{5} + 8 q^{11} - 24 q^{29} - 16 q^{31} - 8 q^{35} + 8 q^{41} + 28 q^{55} + 16 q^{61} - 56 q^{71} + 16 q^{79} - 40 q^{85} - 32 q^{89} + 64 q^{91} - 64 q^{95}+O(q^{100}) \) 40 * q - 4 * q^5 + 8 * q^11 - 24 * q^29 - 16 * q^31 - 8 * q^35 + 8 * q^41 + 28 * q^55 + 16 * q^61 - 56 * q^71 + 16 * q^79 - 40 * q^85 - 32 * q^89 + 64 * q^91 - 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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