LMFDB - Embedded newform 6300.2.dd.b.4049.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6300,2,Mod(1349,6300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6300, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))

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

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

chi := DirichletCharacter("6300.1349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.dd (of order $6$, degree $2$, not 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:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4049.12
Character	$\chi$	$=$	6300.4049
Dual form			6300.2.dd.b.1349.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.63285 - 0.260926i) q^{7} +(3.06024 + 1.76683i) q^{11} +0.599777 q^{13} +(1.47054 + 0.849017i) q^{17} +(6.74071 - 3.89175i) q^{19} +(1.65660 + 2.86931i) q^{23} +0.456069i q^{29} +(-0.914390 - 0.527923i) q^{31} +(0.334501 - 0.193124i) q^{37} +0.478149 q^{41} -4.21237i q^{43} +(7.41148 - 4.27902i) q^{47} +(6.86384 - 1.37396i) q^{49} +(-1.99486 + 3.45520i) q^{53} +(1.30307 - 2.25698i) q^{59} +(-9.60378 + 5.54474i) q^{61} +(2.48915 + 1.43711i) q^{67} +0.336875i q^{71} +(-1.54239 + 2.67151i) q^{73} +(8.51816 + 3.85331i) q^{77} +(-6.80447 - 11.7857i) q^{79} +16.2901i q^{83} +(5.34822 + 9.26339i) q^{89} +(1.57913 - 0.156497i) q^{91} -2.18092 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{11} + 12 q^{19} - 12 q^{31} + 16 q^{41} - 44 q^{49} - 28 q^{79} - 40 q^{89} + 20 q^{91}+O(q^{100}) $ 24 * q - 24 * q^11 + 12 * q^19 - 12 * q^31 + 16 * q^41 - 44 * q^49 - 28 * q^79 - 40 * q^89 + 20 * q^91

Character values

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

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.63285	−	0.260926i	0.995125	−	0.0986206i
$7$	2.63285	−	0.260926i	0.995125	−	0.0986206i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.06024	+	1.76683i	0.922696	+	0.532719i	0.884494	−	0.466551i	$-0.154504\pi$
$11$	3.06024	+	1.76683i	0.922696	+	0.532719i	0.0382018	+	0.999270i	$0.487837\pi$
$12$		0			0
$13$	0.599777			0.166348			0.0831742	−	0.996535i	$-0.473494\pi$
$13$	0.599777			0.166348			0.0831742	+	0.996535i	$0.473494\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.47054	+	0.849017i	0.356658	+	0.205917i	0.667614	−	0.744508i	$-0.267316\pi$
$17$	1.47054	+	0.849017i	0.356658	+	0.205917i	−0.310955	+	0.950424i	$0.600649\pi$
$18$		0			0
$19$	6.74071	−	3.89175i	1.54642	−	0.892829i	0.548014	−	0.836469i	$-0.315384\pi$
$19$	6.74071	−	3.89175i	1.54642	−	0.892829i	0.998411	−	0.0563594i	$-0.0179493\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.65660	+	2.86931i	0.345424	+	0.598292i	0.985431	−	0.170077i	$-0.0544018\pi$
$23$	1.65660	+	2.86931i	0.345424	+	0.598292i	−0.640007	+	0.768369i	$0.721068\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			0.456069i			0.0846898i	0.999103	+	0.0423449i	$0.0134828\pi$
$29$			0.456069i			0.0846898i	−0.999103	+	0.0423449i	$0.986517\pi$
$30$		0			0
$31$	−0.914390	−	0.527923i	−0.164229	−	0.0948178i	0.415633	−	0.909533i	$-0.363560\pi$
$31$	−0.914390	−	0.527923i	−0.164229	−	0.0948178i	−0.579862	+	0.814715i	$0.696894\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.334501	−	0.193124i	0.0549917	−	0.0317494i	−0.472252	−	0.881464i	$-0.656559\pi$
$37$	0.334501	−	0.193124i	0.0549917	−	0.0317494i	0.527244	+	0.849714i	$0.323225\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.478149			0.0746743			0.0373372	−	0.999303i	$-0.488112\pi$
$41$	0.478149			0.0746743			0.0373372	+	0.999303i	$0.488112\pi$
$42$		0			0
$43$		−	4.21237i		−	0.642381i	−0.947015	−	0.321190i	$-0.895917\pi$
$43$		−	4.21237i		−	0.642381i	0.947015	−	0.321190i	$-0.104083\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.41148	−	4.27902i	1.08108	−	0.624159i	0.149889	−	0.988703i	$-0.452108\pi$
$47$	7.41148	−	4.27902i	1.08108	−	0.624159i	0.931186	+	0.364544i	$0.118775\pi$
$48$		0			0
$49$	6.86384	−	1.37396i	0.980548	−	0.196280i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.99486	+	3.45520i	−0.274015	+	0.474609i	−0.969886	−	0.243558i	$-0.921685\pi$
$53$	−1.99486	+	3.45520i	−0.274015	+	0.474609i	0.695871	+	0.718167i	$0.255019\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.30307	−	2.25698i	0.169645	−	0.293834i	−0.768650	−	0.639669i	$-0.779071\pi$
$59$	1.30307	−	2.25698i	0.169645	−	0.293834i	0.938295	+	0.345836i	$0.112405\pi$
$60$		0			0
$61$	−9.60378	+	5.54474i	−1.22964	+	0.709932i	−0.966954	−	0.254950i	$-0.917941\pi$
$61$	−9.60378	+	5.54474i	−1.22964	+	0.709932i	−0.262684	+	0.964882i	$0.584608\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.48915	+	1.43711i	0.304097	+	0.175571i	0.644282	−	0.764788i	$-0.277156\pi$
$67$	2.48915	+	1.43711i	0.304097	+	0.175571i	−0.340185	+	0.940359i	$0.610490\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			0.336875i			0.0399797i	0.999800	+	0.0199898i	$0.00636339\pi$
$71$			0.336875i			0.0399797i	−0.999800	+	0.0199898i	$0.993637\pi$
$72$		0			0
$73$	−1.54239	+	2.67151i	−0.180524	+	0.312676i	−0.942059	−	0.335447i	$-0.891113\pi$
$73$	−1.54239	+	2.67151i	−0.180524	+	0.312676i	0.761535	+	0.648123i	$0.224446\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.51816	+	3.85331i	0.970735	+	0.439125i
$78$		0			0
$79$	−6.80447	−	11.7857i	−0.765562	−	1.32599i	−0.939949	−	0.341316i	$-0.889127\pi$
$79$	−6.80447	−	11.7857i	−0.765562	−	1.32599i	0.174386	−	0.984677i	$-0.444206\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			16.2901i			1.78807i	0.447996	+	0.894036i	$0.352138\pi$
$83$			16.2901i			1.78807i	−0.447996	+	0.894036i	$0.647862\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.34822	+	9.26339i	0.566910	+	0.981918i	0.996869	+	0.0790691i	$0.0251948\pi$
$89$	5.34822	+	9.26339i	0.566910	+	0.981918i	−0.429959	+	0.902849i	$0.641472\pi$
$90$		0			0
$91$	1.57913	−	0.156497i	0.165537	−	0.0164054i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−2.18092			−0.221439			−0.110719	−	0.993852i	$-0.535315\pi$
$97$	−2.18092			−0.221439			−0.110719	+	0.993852i	$0.535315\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.85417	−	6.67563i	0.383505	−	0.664250i	−0.608056	−	0.793894i	$-0.708050\pi$
$101$	3.85417	−	6.67563i	0.383505	−	0.664250i	0.991561	+	0.129645i	$0.0413837\pi$
$102$		0			0
$103$	−0.367281	−	0.636149i	−0.0361893	−	0.0626816i	0.847363	−	0.531013i	$-0.178189\pi$
$103$	−0.367281	−	0.636149i	−0.0361893	−	0.0626816i	−0.883553	+	0.468332i	$0.844855\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.98870	−	17.3009i	−0.965645	−	1.67255i	−0.707873	−	0.706339i	$-0.750345\pi$
$107$	−9.98870	−	17.3009i	−0.965645	−	1.67255i	−0.257771	−	0.966206i	$-0.582988\pi$
$108$		0			0
$109$	−0.221512	+	0.383671i	−0.0212170	+	0.0367490i	−0.876439	−	0.481513i	$-0.840087\pi$
$109$	−0.221512	+	0.383671i	−0.0212170	+	0.0367490i	0.855222	+	0.518262i	$0.173421\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.6235			−1.37566			−0.687830	−	0.725871i	$-0.741437\pi$
$113$	−14.6235			−1.37566			−0.687830	+	0.725871i	$0.741437\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.09325	+	1.85164i	0.375227	+	0.169739i
$120$		0			0
$121$	0.743363	+	1.28754i	0.0675785	+	0.117049i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	0.961155i		−	0.0852887i	−0.999090	−	0.0426444i	$-0.986422\pi$
$127$		−	0.961155i		−	0.0852887i	0.999090	−	0.0426444i	$-0.0135782\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.99734	+	13.8518i	0.698731	+	1.21024i	0.968907	+	0.247427i	$0.0795850\pi$
$131$	7.99734	+	13.8518i	0.698731	+	1.21024i	−0.270175	+	0.962811i	$0.587082\pi$
$132$		0			0
$133$	16.7318	−	12.0052i	1.45083	−	1.04099i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.79985	+	8.31358i	−0.410078	+	0.710277i	−0.994898	−	0.100887i	$-0.967832\pi$
$137$	−4.79985	+	8.31358i	−0.410078	+	0.710277i	0.584819	+	0.811163i	$0.301165\pi$
$138$		0			0
$139$		−	9.13902i		−	0.775162i	−0.921836	−	0.387581i	$-0.873311\pi$
$139$		−	9.13902i		−	0.775162i	0.921836	−	0.387581i	$-0.126689\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.83546	+	1.05970i	0.153489	+	0.0886169i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	17.5541	−	10.1349i	1.43809	−	0.830279i	0.440369	−	0.897817i	$-0.354847\pi$
$149$	17.5541	−	10.1349i	1.43809	−	0.830279i	0.997717	+	0.0675373i	$0.0215142\pi$
$150$		0			0
$151$	4.00516	−	6.93714i	0.325935	−	0.564536i	−0.655766	−	0.754964i	$-0.727654\pi$
$151$	4.00516	−	6.93714i	0.325935	−	0.564536i	0.981701	+	0.190428i	$0.0609876\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−4.66593	+	8.08163i	−0.372382	+	0.644984i	−0.989931	−	0.141548i	$-0.954792\pi$
$157$	−4.66593	+	8.08163i	−0.372382	+	0.644984i	0.617550	+	0.786532i	$0.288125\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.11025	+	7.12222i	0.402744	+	0.561309i
$162$		0			0
$163$	−11.8245	+	6.82685i	−0.926163	+	0.534720i	−0.885596	−	0.464457i	$-0.846250\pi$
$163$	−11.8245	+	6.82685i	−0.926163	+	0.534720i	−0.0405668	+	0.999177i	$0.512916\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			9.94429i			0.769512i	0.923018	+	0.384756i	$0.125714\pi$
$167$			9.94429i			0.769512i	−0.923018	+	0.384756i	$0.874286\pi$
$168$		0			0
$169$	−12.6403			−0.972328
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.60049	+	4.96550i	−0.653883	+	0.377520i	−0.789942	−	0.613181i	$-0.789890\pi$
$173$	−8.60049	+	4.96550i	−0.653883	+	0.377520i	0.136059	+	0.990701i	$0.456556\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	16.3233	+	9.42424i	1.22006	+	0.704400i	0.964930	−	0.262507i	$-0.0845493\pi$
$179$	16.3233	+	9.42424i	1.22006	+	0.704400i	0.255127	+	0.966907i	$0.417883\pi$
$180$		0			0
$181$		−	20.0424i		−	1.48974i	−0.667211	−	0.744869i	$-0.732512\pi$
$181$		−	20.0424i		−	1.48974i	0.667211	−	0.744869i	$-0.267488\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	3.00013	+	5.19639i	0.219392	+	0.379997i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.99103	−	4.61363i	0.578211	−	0.333830i	−0.182211	−	0.983259i	$-0.558325\pi$
$191$	7.99103	−	4.61363i	0.578211	−	0.333830i	0.760422	+	0.649429i	$0.224992\pi$
$192$		0			0
$193$	22.5467	+	13.0173i	1.62294	+	0.937007i	0.986127	+	0.165992i	$0.0530827\pi$
$193$	22.5467	+	13.0173i	1.62294	+	0.937007i	0.636817	+	0.771015i	$0.280251\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.6156			−1.11256			−0.556282	−	0.830994i	$-0.687772\pi$
$197$	−15.6156			−1.11256			−0.556282	+	0.830994i	$0.687772\pi$
$198$		0			0
$199$	−5.29961	−	3.05973i	−0.375679	−	0.216899i	0.300257	−	0.953858i	$-0.402927\pi$
$199$	−5.29961	−	3.05973i	−0.375679	−	0.216899i	−0.675937	+	0.736960i	$0.736261\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.119000	+	1.20076i	0.00835216	+	0.0842770i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	27.5042			1.90251
$210$		0			0
$211$	26.7224			1.83964			0.919822	−	0.392337i	$-0.128333\pi$
$211$	26.7224			1.83964			0.919822	+	0.392337i	$0.128333\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.54520	−	1.15136i	−0.172780	−	0.0781592i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.881997	+	0.509221i	0.0593295	+	0.0342539i
$222$		0			0
$223$	20.3668			1.36386			0.681932	−	0.731416i	$-0.261140\pi$
$223$	20.3668			1.36386			0.681932	+	0.731416i	$0.261140\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−17.2677	−	9.96949i	−1.14609	−	0.661698i	−0.198162	−	0.980169i	$-0.563497\pi$
$227$	−17.2677	−	9.96949i	−1.14609	−	0.661698i	−0.947933	+	0.318471i	$0.896831\pi$
$228$		0			0
$229$	16.4015	−	9.46944i	1.08384	−	0.625758i	0.151914	−	0.988394i	$-0.451456\pi$
$229$	16.4015	−	9.46944i	1.08384	−	0.625758i	0.931931	+	0.362636i	$0.118123\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.2765	+	22.9956i	0.869775	+	1.50650i	0.862226	+	0.506524i	$0.169070\pi$
$233$	13.2765	+	22.9956i	0.869775	+	1.50650i	0.00754927	+	0.999972i	$0.497597\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			6.76076i			0.437317i	0.975801	+	0.218659i	$0.0701681\pi$
$239$			6.76076i			0.437317i	−0.975801	+	0.218659i	$0.929832\pi$
$240$		0			0
$241$	−10.4654	−	6.04219i	−0.674134	−	0.389211i	0.123507	−	0.992344i	$-0.460586\pi$
$241$	−10.4654	−	6.04219i	−0.674134	−	0.389211i	−0.797641	+	0.603132i	$0.793919\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	4.04292	−	2.33418i	0.257245	−	0.148521i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−17.3410			−1.09455			−0.547276	−	0.836952i	$-0.684335\pi$
$251$	−17.3410			−1.09455			−0.547276	+	0.836952i	$0.684335\pi$
$252$		0			0
$253$			11.7077i			0.736055i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.22486	−	4.17127i	0.450674	−	0.260197i	−0.257441	−	0.966294i	$-0.582879\pi$
$257$	7.22486	−	4.17127i	0.450674	−	0.260197i	0.708115	+	0.706097i	$0.249546\pi$
$258$		0			0
$259$	0.830302	−	0.595748i	0.0515924	−	0.0370180i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.70936	−	11.6209i	0.413717	−	0.716578i	−0.581576	−	0.813492i	$-0.697564\pi$
$263$	6.70936	−	11.6209i	0.413717	−	0.716578i	0.995293	+	0.0969137i	$0.0308971\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.98145	−	6.89607i	0.242753	−	0.420461i	−0.718744	−	0.695274i	$-0.755283\pi$
$269$	3.98145	−	6.89607i	0.242753	−	0.420461i	0.961497	+	0.274814i	$0.0886162\pi$
$270$		0			0
$271$	−6.46871	+	3.73471i	−0.392946	+	0.226867i	−0.683436	−	0.730011i	$-0.739515\pi$
$271$	−6.46871	+	3.73471i	−0.392946	+	0.226867i	0.290490	+	0.956878i	$0.406182\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−27.0734	−	15.6308i	−1.62668	−	0.939166i	−0.985073	−	0.172135i	$-0.944933\pi$
$277$	−27.0734	−	15.6308i	−1.62668	−	0.939166i	−0.641610	−	0.767031i	$-0.721733\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	23.4956i		−	1.40163i	−0.713343	−	0.700815i	$-0.752820\pi$
$281$		−	23.4956i		−	1.40163i	0.713343	−	0.700815i	$-0.247180\pi$
$282$		0			0
$283$	−9.25634	+	16.0325i	−0.550232	+	0.953030i	0.448025	+	0.894021i	$0.352127\pi$
$283$	−9.25634	+	16.0325i	−0.550232	+	0.953030i	−0.998257	+	0.0590093i	$0.981206\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.25890	−	0.124761i	0.0743103	−	0.00736442i
$288$		0			0
$289$	−7.05834	−	12.2254i	−0.415196	−	0.719141i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			14.7388i			0.861048i	0.902579	+	0.430524i	$0.141671\pi$
$293$			14.7388i			0.861048i	−0.902579	+	0.430524i	$0.858329\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.993588	+	1.72095i	0.0574607	+	0.0995248i
$300$		0			0
$301$	−1.09912	−	11.0906i	−0.0633520	−	0.639249i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−17.6583			−1.00781			−0.503906	−	0.863758i	$-0.668104\pi$
$307$	−17.6583			−1.00781			−0.503906	+	0.863758i	$0.668104\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.43782	−	2.49037i	0.0815310	−	0.141216i	−0.822377	−	0.568943i	$-0.807352\pi$
$311$	1.43782	−	2.49037i	0.0815310	−	0.141216i	0.903908	+	0.427727i	$0.140686\pi$
$312$		0			0
$313$	10.0193	+	17.3539i	0.566324	+	0.980901i	0.996925	+	0.0783592i	$0.0249681\pi$
$313$	10.0193	+	17.3539i	0.566324	+	0.980901i	−0.430602	+	0.902542i	$0.641699\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.2955	+	17.8323i	0.578252	+	1.00156i	0.995680	+	0.0928516i	$0.0295982\pi$
$317$	10.2955	+	17.8323i	0.578252	+	1.00156i	−0.417428	+	0.908710i	$0.637068\pi$
$318$		0			0
$319$	−0.805795	+	1.39568i	−0.0451159	+	0.0781430i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	13.2166			0.735394
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	18.3968	−	13.1999i	1.01425	−	0.727733i
$330$		0			0
$331$	11.4497	+	19.8315i	0.629335	+	1.09004i	0.987685	+	0.156453i	$0.0500060\pi$
$331$	11.4497	+	19.8315i	0.629335	+	1.09004i	−0.358350	+	0.933587i	$0.616661\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	17.7305i		−	0.965840i	−0.875664	−	0.482920i	$-0.839576\pi$
$337$		−	17.7305i		−	0.965840i	0.875664	−	0.482920i	$-0.160424\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.86550	−	3.23114i	−0.101022	−	0.174976i
$342$		0			0
$343$	17.7130	−	5.40838i	0.956411	−	0.292025i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9.09495	−	15.7529i	0.488243	−	0.845661i	−0.511666	−	0.859185i	$-0.670971\pi$
$347$	9.09495	−	15.7529i	0.488243	−	0.845661i	0.999909	+	0.0135233i	$0.00430473\pi$
$348$		0			0
$349$		−	4.44213i		−	0.237782i	−0.992907	−	0.118891i	$-0.962066\pi$
$349$		−	4.44213i		−	0.237782i	0.992907	−	0.118891i	$-0.0379339\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.48835	−	2.59135i	−0.238891	−	0.137924i	0.375776	−	0.926711i	$-0.377376\pi$
$353$	−4.48835	−	2.59135i	−0.238891	−	0.137924i	−0.614667	+	0.788787i	$0.710710\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.7893	+	9.11596i	−0.833327	+	0.481122i	−0.854991	−	0.518644i	$-0.826437\pi$
$359$	−15.7893	+	9.11596i	−0.833327	+	0.481122i	0.0216632	+	0.999765i	$0.493104\pi$
$360$		0			0
$361$	20.7914	−	36.0118i	1.09429	−	1.89536i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	10.0298	−	17.3722i	0.523553	−	0.906821i	−0.476071	−	0.879407i	$-0.657939\pi$
$367$	10.0298	−	17.3722i	0.523553	−	0.906821i	0.999624	−	0.0274140i	$-0.00872725\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.35063	+	9.61755i	−0.225873	+	0.499319i
$372$		0			0
$373$	23.0267	−	13.2945i	1.19228	−	0.688361i	0.233454	−	0.972368i	$-0.424997\pi$
$373$	23.0267	−	13.2945i	1.19228	−	0.688361i	0.958822	+	0.284007i	$0.0916638\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			0.273540i			0.0140880i
$378$		0			0
$379$	18.3623			0.943210			0.471605	−	0.881810i	$-0.343675\pi$
$379$	18.3623			0.943210			0.471605	+	0.881810i	$0.343675\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−31.8783	+	18.4049i	−1.62891	+	0.940449i	−0.644485	+	0.764617i	$0.722928\pi$
$383$	−31.8783	+	18.4049i	−1.62891	+	0.940449i	−0.984420	+	0.175832i	$0.943738\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	27.4003	+	15.8196i	1.38925	+	0.802084i	0.993231	−	0.116157i	$-0.0370577\pi$
$389$	27.4003	+	15.8196i	1.38925	+	0.802084i	0.396020	+	0.918242i	$0.370391\pi$
$390$		0			0
$391$			5.62591i			0.284515i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	13.4929	+	23.3704i	0.677190	+	1.17293i	0.975824	+	0.218560i	$0.0701358\pi$
$397$	13.4929	+	23.3704i	0.677190	+	1.17293i	−0.298634	+	0.954368i	$0.596531\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	8.48970	−	4.90153i	0.423956	−	0.244771i	−0.272813	−	0.962067i	$-0.587954\pi$
$401$	8.48970	−	4.90153i	0.423956	−	0.244771i	0.696768	+	0.717296i	$0.254621\pi$
$402$		0			0
$403$	−0.548430	−	0.316636i	−0.0273193	−	0.0157728i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.36487			0.0676541
$408$		0			0
$409$	−7.43046	−	4.28998i	−0.367413	−	0.212126i	0.304915	−	0.952380i	$-0.401372\pi$
$409$	−7.43046	−	4.28998i	−0.367413	−	0.212126i	−0.672328	+	0.740254i	$0.734705\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.84188	−	6.28230i	0.139840	−	0.309132i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	21.7882			1.06443			0.532213	−	0.846611i	$-0.321361\pi$
$419$	21.7882			1.06443			0.532213	+	0.846611i	$0.321361\pi$
$420$		0			0
$421$	−24.2280			−1.18080			−0.590400	−	0.807111i	$-0.701030\pi$
$421$	−24.2280			−1.18080			−0.590400	+	0.807111i	$0.701030\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−23.8386	+	17.1044i	−1.15363	+	0.827738i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	20.6400	+	11.9165i	0.994194	+	0.573998i	0.906525	−	0.422151i	$-0.138725\pi$
$431$	20.6400	+	11.9165i	0.994194	+	0.573998i	0.0876688	+	0.996150i	$0.472058\pi$
$432$		0			0
$433$	−1.39596			−0.0670855			−0.0335427	−	0.999437i	$-0.510679\pi$
$433$	−1.39596			−0.0670855			−0.0335427	+	0.999437i	$0.510679\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	22.3333	+	12.8941i	1.06834	+	0.616809i
$438$		0			0
$439$	10.2953	−	5.94397i	0.491366	−	0.283690i	−0.233775	−	0.972291i	$-0.575108\pi$
$439$	10.2953	−	5.94397i	0.491366	−	0.283690i	0.725141	+	0.688600i	$0.241775\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.3701	+	23.1578i	0.635235	+	1.10026i	0.986465	+	0.163970i	$0.0524301\pi$
$443$	13.3701	+	23.1578i	0.635235	+	1.10026i	−0.351230	+	0.936289i	$0.614237\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			32.8394i			1.54979i	0.632091	+	0.774895i	$0.282197\pi$
$449$			32.8394i			1.54979i	−0.632091	+	0.774895i	$0.717803\pi$
$450$		0			0
$451$	1.46325	+	0.844807i	0.0689017	+	0.0397804i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	30.3502	−	17.5227i	1.41972	−	0.819677i	0.423448	−	0.905920i	$-0.360820\pi$
$457$	30.3502	−	17.5227i	1.41972	−	0.819677i	0.996274	+	0.0862436i	$0.0274863\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8.90991			0.414976			0.207488	−	0.978238i	$-0.433471\pi$
$461$	8.90991			0.414976			0.207488	+	0.978238i	$0.433471\pi$
$462$		0			0
$463$		−	10.2000i		−	0.474036i	−0.971505	−	0.237018i	$-0.923830\pi$
$463$		−	10.2000i		−	0.474036i	0.971505	−	0.237018i	$-0.0761700\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−14.5824	+	8.41916i	−0.674794	+	0.389592i	−0.797891	−	0.602802i	$-0.794051\pi$
$467$	−14.5824	+	8.41916i	−0.674794	+	0.389592i	0.123097	+	0.992395i	$0.460717\pi$
$468$		0			0
$469$	6.92853	+	3.13421i	0.319930	+	0.144725i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.44254	−	12.8909i	0.342208	−	0.592722i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.18515	+	15.9091i	−0.419680	+	0.726907i	−0.995907	−	0.0903824i	$-0.971191\pi$
$479$	−9.18515	+	15.9091i	−0.419680	+	0.726907i	0.576227	+	0.817290i	$0.304524\pi$
$480$		0			0
$481$	0.200626	−	0.115832i	0.00914777	−	0.00528147i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−10.1012	−	5.83194i	−0.457729	−	0.264270i	0.253360	−	0.967372i	$-0.418464\pi$
$487$	−10.1012	−	5.83194i	−0.457729	−	0.264270i	−0.711089	+	0.703102i	$0.751798\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			21.4694i			0.968902i	0.874818	+	0.484451i	$0.160981\pi$
$491$			21.4694i			0.968902i	−0.874818	+	0.484451i	$0.839019\pi$
$492$		0			0
$493$	−0.387210	+	0.670668i	−0.0174391	+	0.0302054i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.0878993	+	0.886942i	0.00394282	+	0.0397848i
$498$		0			0
$499$	−7.41090	−	12.8361i	−0.331757	−	0.574621i	0.651099	−	0.758993i	$-0.274308\pi$
$499$	−7.41090	−	12.8361i	−0.331757	−	0.574621i	−0.982857	+	0.184372i	$0.940975\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			28.9563i			1.29110i	0.763719	+	0.645548i	$0.223371\pi$
$503$			28.9563i			1.29110i	−0.763719	+	0.645548i	$0.776629\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.5897	−	25.2700i	−0.646675	−	1.12007i	−0.983912	−	0.178654i	$-0.942826\pi$
$509$	−14.5897	−	25.2700i	−0.646675	−	1.12007i	0.337237	−	0.941420i	$-0.390508\pi$
$510$		0			0
$511$	−3.36384	+	7.43614i	−0.148807	+	0.328955i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	30.2412			1.33001
$518$		0			0
$519$		0			0
$520$		0			0
$521$	15.3104	−	26.5184i	0.670761	−	1.16179i	−0.306928	−	0.951733i	$-0.599301\pi$
$521$	15.3104	−	26.5184i	0.670761	−	1.16179i	0.977689	−	0.210059i	$-0.0673655\pi$
$522$		0			0
$523$	−6.04118	−	10.4636i	−0.264162	−	0.457542i	0.703182	−	0.711010i	$-0.251762\pi$
$523$	−6.04118	−	10.4636i	−0.264162	−	0.457542i	−0.967344	+	0.253468i	$0.918429\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.896431	−	1.55266i	−0.0390492	−	0.0676351i
$528$		0			0
$529$	6.01138	−	10.4120i	0.261365	−	0.452697i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	0.286783			0.0124219
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	23.4325	+	7.92258i	1.00931	+	0.341250i
$540$		0			0
$541$	−2.77440	−	4.80539i	−0.119281	−	0.206600i	0.800202	−	0.599730i	$-0.204725\pi$
$541$	−2.77440	−	4.80539i	−0.119281	−	0.206600i	−0.919483	+	0.393130i	$0.871392\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	4.60440i		−	0.196870i	−0.995143	−	0.0984349i	$-0.968616\pi$
$547$		−	4.60440i		−	0.196870i	0.995143	−	0.0984349i	$-0.0313836\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.77491	+	3.07423i	0.0756135	+	0.130966i
$552$		0			0
$553$	−20.9904	−	29.2545i	−0.892601	−	1.24403i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7.85045	−	13.5974i	0.332634	−	0.576139i	−0.650393	−	0.759598i	$-0.725396\pi$
$557$	7.85045	−	13.5974i	0.332634	−	0.576139i	0.983028	+	0.183458i	$0.0587292\pi$
$558$		0			0
$559$		−	2.52649i		−	0.106859i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−28.3061	−	16.3425i	−1.19296	−	0.688756i	−0.233983	−	0.972241i	$-0.575176\pi$
$563$	−28.3061	−	16.3425i	−1.19296	−	0.688756i	−0.958977	+	0.283485i	$0.908509\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−20.0566	+	11.5797i	−0.840817	+	0.485446i	−0.857542	−	0.514414i	$-0.828009\pi$
$569$	−20.0566	+	11.5797i	−0.840817	+	0.485446i	0.0167248	+	0.999860i	$0.494676\pi$
$570$		0			0
$571$	1.95247	−	3.38177i	0.0817083	−	0.141523i	−0.822276	−	0.569090i	$-0.807296\pi$
$571$	1.95247	−	3.38177i	0.0817083	−	0.141523i	0.903984	+	0.427567i	$0.140629\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−3.83961	+	6.65040i	−0.159845	+	0.276860i	−0.934813	−	0.355141i	$-0.884433\pi$
$577$	−3.83961	+	6.65040i	−0.159845	+	0.276860i	0.774968	+	0.632001i	$0.217766\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.25050	+	42.8895i	0.176341	+	1.77935i
$582$		0			0
$583$	−12.2095	+	7.04916i	−0.505666	+	0.291946i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	0.375586i		−	0.0155021i	−0.999970	−	0.00775105i	$-0.997533\pi$
$587$		−	0.375586i		−	0.0155021i	0.999970	−	0.00775105i	$-0.00246726\pi$
$588$		0			0
$589$	−8.21818			−0.338624
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−19.9536	+	11.5202i	−0.819396	+	0.473078i	−0.850208	−	0.526447i	$-0.823524\pi$
$593$	−19.9536	+	11.5202i	−0.819396	+	0.473078i	0.0308124	+	0.999525i	$0.490191\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.46233	+	4.30838i	0.304902	+	0.176036i	0.644643	−	0.764484i	$-0.277006\pi$
$599$	7.46233	+	4.30838i	0.304902	+	0.176036i	−0.339741	+	0.940519i	$0.610339\pi$
$600$		0			0
$601$			27.4954i			1.12156i	0.827965	+	0.560780i	$0.189499\pi$
$601$			27.4954i			1.12156i	−0.827965	+	0.560780i	$0.810501\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	17.9638	+	31.1143i	0.729130	+	1.26289i	0.957251	+	0.289257i	$0.0934084\pi$
$607$	17.9638	+	31.1143i	0.729130	+	1.26289i	−0.228121	+	0.973633i	$0.573258\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.44524	−	2.56646i	0.179835	−	0.103828i
$612$		0			0
$613$	−36.2468	−	20.9271i	−1.46400	−	0.845238i	−0.464803	−	0.885414i	$-0.653875\pi$
$613$	−36.2468	−	20.9271i	−1.46400	−	0.845238i	−0.999193	+	0.0401755i	$0.987208\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	27.6411			1.11279			0.556395	−	0.830918i	$-0.312184\pi$
$617$	27.6411			1.11279			0.556395	+	0.830918i	$0.312184\pi$
$618$		0			0
$619$	11.4818	+	6.62901i	0.461492	+	0.266442i	0.712671	−	0.701498i	$-0.247485\pi$
$619$	11.4818	+	6.62901i	0.461492	+	0.266442i	−0.251179	+	0.967941i	$0.580818\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	16.4981	+	22.9937i	0.660984	+	0.921222i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.655863			0.0261510
$630$		0			0
$631$	−12.2455			−0.487485			−0.243742	−	0.969840i	$-0.578375\pi$
$631$	−12.2455			−0.487485			−0.243742	+	0.969840i	$0.578375\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	4.11677	−	0.824069i	0.163112	−	0.0326508i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.7315	+	6.19582i	0.423868	+	0.244720i	0.696731	−	0.717333i	$-0.254637\pi$
$641$	10.7315	+	6.19582i	0.423868	+	0.244720i	−0.272863	+	0.962053i	$0.587971\pi$
$642$		0			0
$643$	42.8847			1.69121			0.845604	−	0.533811i	$-0.179241\pi$
$643$	42.8847			1.69121			0.845604	+	0.533811i	$0.179241\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−23.2701	−	13.4350i	−0.914843	−	0.528185i	−0.0328565	−	0.999460i	$-0.510460\pi$
$647$	−23.2701	−	13.4350i	−0.914843	−	0.528185i	−0.881986	+	0.471275i	$0.843794\pi$
$648$		0			0
$649$	7.97539	−	4.60459i	0.313061	−	0.180746i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.311046	+	0.538748i	0.0121722	+	0.0210828i	0.872047	−	0.489422i	$-0.162792\pi$
$653$	0.311046	+	0.538748i	0.0121722	+	0.0210828i	−0.859875	+	0.510504i	$0.829459\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	17.9996i		−	0.701167i	−0.936532	−	0.350583i	$-0.885983\pi$
$659$		−	17.9996i		−	0.701167i	0.936532	−	0.350583i	$-0.114017\pi$
$660$		0			0
$661$	−26.0446	−	15.0369i	−1.01302	−	0.584866i	−0.100944	−	0.994892i	$-0.532186\pi$
$661$	−26.0446	−	15.0369i	−1.01302	−	0.584866i	−0.912074	+	0.410026i	$0.865519\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.30860	+	0.755521i	−0.0506693	+	0.0292539i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−39.1864			−1.51278
$672$		0			0
$673$			39.4637i			1.52121i	0.649213	+	0.760606i	$0.275098\pi$
$673$			39.4637i			1.52121i	−0.649213	+	0.760606i	$0.724902\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17.0412	−	9.83874i	0.654947	−	0.378134i	−0.135402	−	0.990791i	$-0.543233\pi$
$677$	17.0412	−	9.83874i	0.654947	−	0.378134i	0.790349	+	0.612657i	$0.209899\pi$
$678$		0			0
$679$	−5.74204	+	0.569057i	−0.220359	+	0.0218384i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.01113	−	15.6077i	0.344801	−	0.597213i	−0.640516	−	0.767945i	$-0.721280\pi$
$683$	9.01113	−	15.6077i	0.344801	−	0.597213i	0.985318	+	0.170731i	$0.0546129\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.19647	+	2.07235i	−0.0455820	+	0.0789503i
$690$		0			0
$691$	−23.5609	+	13.6029i	−0.896298	+	0.517478i	−0.875997	−	0.482316i	$-0.839796\pi$
$691$	−23.5609	+	13.6029i	−0.896298	+	0.517478i	−0.0203004	+	0.999794i	$0.506462\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.703137	+	0.405957i	0.0266332	+	0.0153767i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	33.1556i		−	1.25227i	−0.779715	−	0.626134i	$-0.784636\pi$
$701$		−	33.1556i		−	1.25227i	0.779715	−	0.626134i	$-0.215364\pi$
$702$		0			0
$703$	1.50318	−	2.60359i	0.0566936	−	0.0981962i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8.40563	−	18.5816i	0.316126	−	0.698833i
$708$		0			0
$709$	0.989596	+	1.71403i	0.0371650	+	0.0643717i	0.884010	−	0.467469i	$-0.154834\pi$
$709$	0.989596	+	1.71403i	0.0371650	+	0.0643717i	−0.846845	+	0.531840i	$0.821501\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	3.49822i		−	0.131009i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15.1795	−	26.2916i	−0.566098	−	0.980510i	−0.996947	−	0.0780862i	$-0.975119\pi$
$719$	−15.1795	−	26.2916i	−0.566098	−	0.980510i	0.430849	−	0.902424i	$-0.358214\pi$
$720$		0			0
$721$	−1.13298	−	1.57905i	−0.0421945	−	0.0588071i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−17.5883			−0.652313			−0.326156	−	0.945316i	$-0.605754\pi$
$727$	−17.5883			−0.652313			−0.326156	+	0.945316i	$0.605754\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3.57638	−	6.19446i	0.132277	−	0.229111i
$732$		0			0
$733$	−18.4722	−	31.9948i	−0.682287	−	1.18176i	−0.974281	−	0.225336i	$-0.927652\pi$
$733$	−18.4722	−	31.9948i	−0.682287	−	1.18176i	0.291994	−	0.956420i	$-0.405681\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.07825	+	8.79578i	0.187060	+	0.323997i
$738$		0			0
$739$	−2.52958	+	4.38136i	−0.0930520	+	0.161171i	−0.908794	−	0.417245i	$-0.862996\pi$
$739$	−2.52958	+	4.38136i	−0.0930520	+	0.161171i	0.815742	+	0.578416i	$0.196329\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6.19725			−0.227355			−0.113678	−	0.993518i	$-0.536263\pi$
$743$	−6.19725			−0.227355			−0.113678	+	0.993518i	$0.536263\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−30.8131	−	42.9445i	−1.12588	−	1.56916i
$750$		0			0
$751$	18.9100	+	32.7531i	0.690037	+	1.19518i	0.971825	+	0.235703i	$0.0757392\pi$
$751$	18.9100	+	32.7531i	0.690037	+	1.19518i	−0.281788	+	0.959477i	$0.590928\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			13.7436i			0.499519i	0.968308	+	0.249760i	$0.0803516\pi$
$757$			13.7436i			0.499519i	−0.968308	+	0.249760i	$0.919648\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.19974	+	2.07800i	0.0434904	+	0.0753275i	0.886951	−	0.461863i	$-0.152819\pi$
$761$	1.19974	+	2.07800i	0.0434904	+	0.0753275i	−0.843461	+	0.537191i	$0.819486\pi$
$762$		0			0
$763$	−0.483100	+	1.06795i	−0.0174894	+	0.0386623i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.781550	−	1.35368i	0.0282201	−	0.0488787i
$768$		0			0
$769$		−	9.42300i		−	0.339802i	−0.985461	−	0.169901i	$-0.945655\pi$
$769$		−	9.42300i		−	0.339802i	0.985461	−	0.169901i	$-0.0543448\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.5441	−	11.2838i	−0.702953	−	0.405850i	0.105493	−	0.994420i	$-0.466358\pi$
$773$	−19.5441	−	11.2838i	−0.702953	−	0.405850i	−0.808446	+	0.588570i	$0.799691\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.22306	−	1.86084i	0.115478	−	0.0666714i
$780$		0			0
$781$	−0.595200	+	1.03092i	−0.0212979	+	0.0368891i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−12.9573	+	22.4427i	−0.461878	+	0.799997i	−0.999055	−	0.0434733i	$-0.986158\pi$
$787$	−12.9573	+	22.4427i	−0.461878	+	0.799997i	0.537176	+	0.843470i	$0.319491\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−38.5015	+	3.81564i	−1.36895	+	0.135668i
$792$		0			0
$793$	−5.76013	+	3.32561i	−0.204548	+	0.118096i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	6.94772i		−	0.246101i	−0.992400	−	0.123050i	$-0.960732\pi$
$797$		−	6.94772i		−	0.246101i	0.992400	−	0.123050i	$-0.0392677\pi$
$798$		0			0
$799$	14.5318			0.514100
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9.44019	+	5.45029i	−0.333137	+	0.192337i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−28.9229	−	16.6986i	−1.01687	−	0.587093i	−0.103677	−	0.994611i	$-0.533061\pi$
$809$	−28.9229	−	16.6986i	−1.01687	−	0.587093i	−0.913197	+	0.407518i	$0.866394\pi$
$810$		0			0
$811$			28.9701i			1.01728i	0.860979	+	0.508640i	$0.169851\pi$
$811$			28.9701i			1.01728i	−0.860979	+	0.508640i	$0.830149\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−16.3935	−	28.3944i	−0.573536	−	0.993393i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.25824	−	1.30380i	0.0788132	−	0.0455028i	−0.460076	−	0.887880i	$-0.652178\pi$
$821$	2.25824	−	1.30380i	0.0788132	−	0.0455028i	0.538889	+	0.842377i	$0.318844\pi$
$822$		0			0
$823$	−7.32393	−	4.22848i	−0.255296	−	0.147395i	0.366891	−	0.930264i	$-0.380422\pi$
$823$	−7.32393	−	4.22848i	−0.255296	−	0.147395i	−0.622187	+	0.782869i	$0.713756\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−13.3362			−0.463744			−0.231872	−	0.972746i	$-0.574485\pi$
$827$	−13.3362			−0.463744			−0.231872	+	0.972746i	$0.574485\pi$
$828$		0			0
$829$	−10.0783	−	5.81869i	−0.350033	−	0.202091i	0.314667	−	0.949202i	$-0.398107\pi$
$829$	−10.0783	−	5.81869i	−0.350033	−	0.202091i	−0.664700	+	0.747111i	$0.731440\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	11.2601	+	3.80705i	0.390138	+	0.131907i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−19.9335			−0.688181			−0.344090	−	0.938937i	$-0.611813\pi$
$839$	−19.9335			−0.688181			−0.344090	+	0.938937i	$0.611813\pi$
$840$		0			0
$841$	28.7920			0.992828
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	2.29312	+	3.19595i	0.0787925	+	0.109814i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.10827	+	0.639858i	0.0379909	+	0.0219340i
$852$		0			0
$853$	2.08233			0.0712975			0.0356487	−	0.999364i	$-0.488650\pi$
$853$	2.08233			0.0712975			0.0356487	+	0.999364i	$0.488650\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−40.3119	−	23.2741i	−1.37703	−	0.795028i	−0.385228	−	0.922821i	$-0.625877\pi$
$857$	−40.3119	−	23.2741i	−1.37703	−	0.795028i	−0.991801	+	0.127793i	$0.959211\pi$
$858$		0			0
$859$	−4.62888	+	2.67248i	−0.157935	+	0.0911840i	−0.576885	−	0.816826i	$-0.695732\pi$
$859$	−4.62888	+	2.67248i	−0.157935	+	0.0911840i	0.418949	+	0.908010i	$0.362398\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−6.65350	−	11.5242i	−0.226488	−	0.392288i	0.730277	−	0.683151i	$-0.239391\pi$
$863$	−6.65350	−	11.5242i	−0.226488	−	0.392288i	−0.956765	+	0.290863i	$0.906058\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	48.0893i		−	1.63132i
$870$		0			0
$871$	1.49293	+	0.861945i	0.0505861	+	0.0292059i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	17.1339	−	9.89224i	0.578570	−	0.334037i	−0.181995	−	0.983299i	$-0.558256\pi$
$877$	17.1339	−	9.89224i	0.578570	−	0.334037i	0.760565	+	0.649262i	$0.224922\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	34.3203			1.15628			0.578141	−	0.815937i	$-0.303778\pi$
$881$	34.3203			1.15628			0.578141	+	0.815937i	$0.303778\pi$
$882$		0			0
$883$			28.1109i			0.946006i	0.881061	+	0.473003i	$0.156830\pi$
$883$			28.1109i			0.946006i	−0.881061	+	0.473003i	$0.843170\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−36.1697	+	20.8826i	−1.21446	+	0.701169i	−0.963728	−	0.266888i	$-0.914005\pi$
$887$	−36.1697	+	20.8826i	−1.21446	+	0.701169i	−0.250732	+	0.968056i	$0.580671\pi$
$888$		0			0
$889$	−0.250790	−	2.53058i	−0.00841122	−	0.0848730i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	33.3058	−	57.6873i	1.11453	−	1.93043i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	0.240769	−	0.417025i	0.00803010	−	0.0139085i
$900$		0			0
$901$	−5.86705	+	3.38734i	−0.195460	+	0.112849i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−6.85979	−	3.96050i	−0.227776	−	0.131506i	0.381770	−	0.924257i	$-0.375315\pi$
$907$	−6.85979	−	3.96050i	−0.227776	−	0.131506i	−0.609546	+	0.792751i	$0.708648\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			49.9301i			1.65426i	0.562012	+	0.827129i	$0.310027\pi$
$911$			49.9301i			1.65426i	−0.562012	+	0.827129i	$0.689973\pi$
$912$		0			0
$913$	−28.7818	+	49.8516i	−0.952539	+	1.64985i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	24.6701	+	34.3831i	0.814679	+	1.13543i
$918$		0			0
$919$	−0.132073	−	0.228757i	−0.00435668	−	0.00754598i	0.863839	−	0.503768i	$-0.168053\pi$
$919$	−0.132073	−	0.228757i	−0.00435668	−	0.00754598i	−0.868196	+	0.496222i	$0.834720\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			0.202050i			0.00665055i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	28.4346	+	49.2502i	0.932910	+	1.61585i	0.778320	+	0.627868i	$0.216072\pi$
$929$	28.4346	+	49.2502i	0.932910	+	1.61585i	0.154590	+	0.987979i	$0.450594\pi$
$930$		0			0
$931$	40.9200	−	35.9738i	1.34110	−	1.17899i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	37.3673			1.22074			0.610369	−	0.792118i	$-0.291021\pi$
$937$	37.3673			1.22074			0.610369	+	0.792118i	$0.291021\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23.8667	−	41.3384i	0.778033	−	1.34759i	−0.155042	−	0.987908i	$-0.549551\pi$
$941$	23.8667	−	41.3384i	0.778033	−	1.34759i	0.933074	−	0.359684i	$-0.117115\pi$
$942$		0			0
$943$	0.792099	+	1.37196i	0.0257943	+	0.0446770i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−16.5870	−	28.7296i	−0.539006	−	0.933587i	−0.998958	−	0.0456424i	$-0.985467\pi$
$947$	−16.5870	−	28.7296i	−0.539006	−	0.933587i	0.459951	−	0.887944i	$-0.347867\pi$
$948$		0			0
$949$	−0.925094	+	1.60231i	−0.0300298	+	0.0520131i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	28.1591			0.912164			0.456082	−	0.889938i	$-0.349252\pi$
$953$	28.1591			0.912164			0.456082	+	0.889938i	$0.349252\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.4681	+	23.1408i	−0.338031	+	0.747256i
$960$		0			0
$961$	−14.9426	−	25.8813i	−0.482019	−	0.834882i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			53.8983i			1.73325i	0.498959	+	0.866626i	$0.333716\pi$
$967$			53.8983i			1.73325i	−0.498959	+	0.866626i	$0.666284\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−20.5266	−	35.5531i	−0.658729	−	1.14095i	−0.980945	−	0.194286i	$-0.937761\pi$
$971$	−20.5266	−	35.5531i	−0.658729	−	1.14095i	0.322216	−	0.946666i	$-0.395572\pi$
$972$		0			0
$973$	−2.38460	−	24.0617i	−0.0764469	−	0.771383i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−14.4594	+	25.0445i	−0.462598	+	0.801244i	−0.999090	−	0.0426622i	$-0.986416\pi$
$977$	−14.4594	+	25.0445i	−0.462598	+	0.801244i	0.536491	+	0.843906i	$0.319749\pi$
$978$		0			0
$979$			37.7976i			1.20802i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−31.4152	−	18.1376i	−1.00199	−	0.578499i	−0.0931532	−	0.995652i	$-0.529695\pi$
$983$	−31.4152	−	18.1376i	−1.00199	−	0.578499i	−0.908836	+	0.417153i	$0.863028\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.0866	−	6.97820i	0.384331	−	0.221894i
$990$		0			0
$991$	19.6101	−	33.9656i	0.622934	−	1.07895i	−0.366002	−	0.930614i	$-0.619274\pi$
$991$	19.6101	−	33.9656i	0.622934	−	1.07895i	0.988936	−	0.148340i	$-0.0473930\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−6.20678	+	10.7505i	−0.196571	+	0.340471i	−0.947414	−	0.320009i	$-0.896314\pi$
$997$	−6.20678	+	10.7505i	−0.196571	+	0.340471i	0.750844	+	0.660480i	$0.229647\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.dd.b.4049.12		24
3.2	odd	2		6300.2.dd.c.4049.12		24
5.2	odd	4		1260.2.cg.b.521.3	yes	12
5.3	odd	4		6300.2.ch.b.4301.4		12
5.4	even	2	inner	6300.2.dd.b.4049.1		24
7.5	odd	6		6300.2.dd.c.1349.1		24
15.2	even	4		1260.2.cg.a.521.3	yes	12
15.8	even	4		6300.2.ch.c.4301.4		12
15.14	odd	2		6300.2.dd.c.4049.1		24
21.5	even	6	inner	6300.2.dd.b.1349.1		24
35.12	even	12		1260.2.cg.a.341.3	✓	12
35.17	even	12		8820.2.d.b.881.2		12
35.19	odd	6		6300.2.dd.c.1349.12		24
35.32	odd	12		8820.2.d.a.881.2		12
35.33	even	12		6300.2.ch.c.1601.4		12
105.17	odd	12		8820.2.d.a.881.11		12
105.32	even	12		8820.2.d.b.881.11		12
105.47	odd	12		1260.2.cg.b.341.3	yes	12
105.68	odd	12		6300.2.ch.b.1601.4		12
105.89	even	6	inner	6300.2.dd.b.1349.12		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.cg.a.341.3	✓	12	35.12	even	12
1260.2.cg.a.521.3	yes	12	15.2	even	4
1260.2.cg.b.341.3	yes	12	105.47	odd	12
1260.2.cg.b.521.3	yes	12	5.2	odd	4
6300.2.ch.b.1601.4		12	105.68	odd	12
6300.2.ch.b.4301.4		12	5.3	odd	4
6300.2.ch.c.1601.4		12	35.33	even	12
6300.2.ch.c.4301.4		12	15.8	even	4
6300.2.dd.b.1349.1		24	21.5	even	6	inner
6300.2.dd.b.1349.12		24	105.89	even	6	inner
6300.2.dd.b.4049.1		24	5.4	even	2	inner
6300.2.dd.b.4049.12		24	1.1	even	1	trivial
6300.2.dd.c.1349.1		24	7.5	odd	6
6300.2.dd.c.1349.12		24	35.19	odd	6
6300.2.dd.c.4049.1		24	15.14	odd	2
6300.2.dd.c.4049.12		24	3.2	odd	2
8820.2.d.a.881.2		12	35.32	odd	12
8820.2.d.a.881.11		12	105.17	odd	12
8820.2.d.b.881.2		12	35.17	even	12
8820.2.d.b.881.11		12	105.32	even	12

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:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.dd (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.63285 - 0.260926i) q^{7} +(3.06024 + 1.76683i) q^{11} +0.599777 q^{13} +(1.47054 + 0.849017i) q^{17} +(6.74071 - 3.89175i) q^{19} +(1.65660 + 2.86931i) q^{23} +0.456069i q^{29} +(-0.914390 - 0.527923i) q^{31} +(0.334501 - 0.193124i) q^{37} +0.478149 q^{41} -4.21237i q^{43} +(7.41148 - 4.27902i) q^{47} +(6.86384 - 1.37396i) q^{49} +(-1.99486 + 3.45520i) q^{53} +(1.30307 - 2.25698i) q^{59} +(-9.60378 + 5.54474i) q^{61} +(2.48915 + 1.43711i) q^{67} +0.336875i q^{71} +(-1.54239 + 2.67151i) q^{73} +(8.51816 + 3.85331i) q^{77} +(-6.80447 - 11.7857i) q^{79} +16.2901i q^{83} +(5.34822 + 9.26339i) q^{89} +(1.57913 - 0.156497i) q^{91} -2.18092 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{11} + 12 q^{19} - 12 q^{31} + 16 q^{41} - 44 q^{49} - 28 q^{79} - 40 q^{89} + 20 q^{91}+O(q^{100}) \) 24 * q - 24 * q^11 + 12 * q^19 - 12 * q^31 + 16 * q^41 - 44 * q^49 - 28 * q^79 - 40 * q^89 + 20 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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