LMFDB - Embedded newform 6300.2.dd.c.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.c.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	−0.0382018	−	0.999270i	$-0.512163\pi$
$11$	−3.06024	−	1.76683i	−0.922696	−	0.532719i	−0.884494	+	0.466551i	$0.845496\pi$
$12$		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	0.310955	−	0.950424i	$-0.399351\pi$
$17$	−1.47054	−	0.849017i	−0.356658	−	0.205917i	−0.667614	+	0.744508i	$0.732684\pi$
$18$		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	0.640007	−	0.768369i	$-0.278932\pi$
$23$	−1.65660	−	2.86931i	−0.345424	−	0.598292i	−0.985431	+	0.170077i	$0.945598\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	0.456069i		−	0.0846898i	−0.999103	−	0.0423449i	$-0.986517\pi$
$29$		−	0.456069i		−	0.0846898i	0.999103	−	0.0423449i	$-0.0134828\pi$
$30$		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	$-0.511888\pi$
$41$	−0.478149			−0.0746743			−0.0373372	+	0.999303i	$0.511888\pi$
$42$		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	−0.931186	−	0.364544i	$-0.881225\pi$
$47$	−7.41148	+	4.27902i	−1.08108	+	0.624159i	−0.149889	+	0.988703i	$0.547892\pi$
$48$		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	−0.695871	−	0.718167i	$-0.744981\pi$
$53$	1.99486	−	3.45520i	0.274015	−	0.474609i	0.969886	+	0.243558i	$0.0783147\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.30307	+	2.25698i	−0.169645	+	0.293834i	−0.938295	−	0.345836i	$-0.887595\pi$
$59$	−1.30307	+	2.25698i	−0.169645	+	0.293834i	0.768650	+	0.639669i	$0.220929\pi$
$60$		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	$-0.993637\pi$
$71$		−	0.336875i		−	0.0399797i	0.999800	−	0.0199898i	$-0.00636339\pi$
$72$		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	$-0.647862\pi$
$83$		−	16.2901i		−	1.78807i	0.447996	−	0.894036i	$-0.352138\pi$
$84$		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	$-0.974805\pi$
$89$	−5.34822	−	9.26339i	−0.566910	−	0.981918i	0.429959	−	0.902849i	$-0.358528\pi$
$90$		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	−0.991561	−	0.129645i	$-0.958616\pi$
$101$	−3.85417	+	6.67563i	−0.383505	+	0.664250i	0.608056	+	0.793894i	$0.291950\pi$
$102$		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	$0.249655\pi$
$107$	9.98870	+	17.3009i	0.965645	+	1.67255i	0.257771	+	0.966206i	$0.417012\pi$
$108$		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	$-0.258563\pi$
$113$	14.6235			1.37566			0.687830	+	0.725871i	$0.258563\pi$
$114$		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	$-0.920415\pi$
$131$	−7.99734	−	13.8518i	−0.698731	−	1.21024i	0.270175	−	0.962811i	$-0.412918\pi$
$132$		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	−0.584819	−	0.811163i	$-0.698835\pi$
$137$	4.79985	−	8.31358i	0.410078	−	0.710277i	0.994898	+	0.100887i	$0.0321680\pi$
$138$		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	−0.997717	−	0.0675373i	$-0.978486\pi$
$149$	−17.5541	+	10.1349i	−1.43809	+	0.830279i	−0.440369	+	0.897817i	$0.645153\pi$
$150$		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	$-0.874286\pi$
$167$		−	9.94429i		−	0.769512i	0.923018	−	0.384756i	$-0.125714\pi$
$168$		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	−0.136059	−	0.990701i	$-0.543444\pi$
$173$	8.60049	−	4.96550i	0.653883	−	0.377520i	0.789942	+	0.613181i	$0.210110\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−16.3233	−	9.42424i	−1.22006	−	0.704400i	−0.255127	−	0.966907i	$-0.582117\pi$
$179$	−16.3233	−	9.42424i	−1.22006	−	0.704400i	−0.964930	+	0.262507i	$0.915451\pi$
$180$		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	−0.760422	−	0.649429i	$-0.775008\pi$
$191$	−7.99103	+	4.61363i	−0.578211	+	0.333830i	0.182211	+	0.983259i	$0.441675\pi$
$192$		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	$-0.312228\pi$
$197$	15.6156			1.11256			0.556282	+	0.830994i	$0.312228\pi$
$198$		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	0.947933	−	0.318471i	$-0.103169\pi$
$227$	17.2677	+	9.96949i	1.14609	+	0.661698i	0.198162	+	0.980169i	$0.436503\pi$
$228$		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	$-0.830930\pi$
$233$	−13.2765	−	22.9956i	−0.869775	−	1.50650i	−0.00754927	−	0.999972i	$-0.502403\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	6.76076i		−	0.437317i	−0.975801	−	0.218659i	$-0.929832\pi$
$239$		−	6.76076i		−	0.437317i	0.975801	−	0.218659i	$-0.0701681\pi$
$240$		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	$-0.315665\pi$
$251$	17.3410			1.09455			0.547276	+	0.836952i	$0.315665\pi$
$252$		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	−0.708115	−	0.706097i	$-0.750454\pi$
$257$	−7.22486	+	4.17127i	−0.450674	+	0.260197i	0.257441	+	0.966294i	$0.417121\pi$
$258$		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	−0.995293	−	0.0969137i	$-0.969103\pi$
$263$	−6.70936	+	11.6209i	−0.413717	+	0.716578i	0.581576	+	0.813492i	$0.302436\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.98145	+	6.89607i	−0.242753	+	0.420461i	−0.961497	−	0.274814i	$-0.911384\pi$
$269$	−3.98145	+	6.89607i	−0.242753	+	0.420461i	0.718744	+	0.695274i	$0.244717\pi$
$270$		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	$0.247180\pi$
$281$			23.4956i			1.40163i	−0.713343	+	0.700815i	$0.752820\pi$
$282$		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	$-0.858329\pi$
$293$		−	14.7388i		−	0.861048i	0.902579	−	0.430524i	$-0.141671\pi$
$294$		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	−0.903908	−	0.427727i	$-0.859314\pi$
$311$	−1.43782	+	2.49037i	−0.0815310	+	0.141216i	0.822377	+	0.568943i	$0.192648\pi$
$312$		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	$-0.970402\pi$
$317$	−10.2955	−	17.8323i	−0.578252	−	1.00156i	0.417428	−	0.908710i	$-0.362932\pi$
$318$		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	−0.999909	−	0.0135233i	$-0.995695\pi$
$347$	−9.09495	+	15.7529i	−0.488243	+	0.845661i	0.511666	+	0.859185i	$0.329029\pi$
$348$		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	0.614667	−	0.788787i	$-0.289290\pi$
$353$	4.48835	+	2.59135i	0.238891	+	0.137924i	−0.375776	+	0.926711i	$0.622624\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	15.7893	−	9.11596i	0.833327	−	0.481122i	−0.0216632	−	0.999765i	$-0.506896\pi$
$359$	15.7893	−	9.11596i	0.833327	−	0.481122i	0.854991	+	0.518644i	$0.173563\pi$
$360$		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	$-0.277072\pi$
$383$	31.8783	−	18.4049i	1.62891	−	0.940449i	0.984420	−	0.175832i	$-0.0562616\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−27.4003	−	15.8196i	−1.38925	−	0.802084i	−0.396020	−	0.918242i	$-0.629609\pi$
$389$	−27.4003	−	15.8196i	−1.38925	−	0.802084i	−0.993231	+	0.116157i	$0.962942\pi$
$390$		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	−0.696768	−	0.717296i	$-0.745379\pi$
$401$	−8.48970	+	4.90153i	−0.423956	+	0.244771i	0.272813	+	0.962067i	$0.412046\pi$
$402$		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	$-0.678639\pi$
$419$	−21.7882			−1.06443			−0.532213	+	0.846611i	$0.678639\pi$
$420$		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	−0.0876688	−	0.996150i	$-0.527942\pi$
$431$	−20.6400	−	11.9165i	−0.994194	−	0.573998i	−0.906525	+	0.422151i	$0.861275\pi$
$432$		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	$-0.947570\pi$
$443$	−13.3701	−	23.1578i	−0.635235	−	1.10026i	0.351230	−	0.936289i	$-0.385763\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	32.8394i		−	1.54979i	−0.632091	−	0.774895i	$-0.717803\pi$
$449$		−	32.8394i		−	1.54979i	0.632091	−	0.774895i	$-0.282197\pi$
$450$		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	$-0.566529\pi$
$461$	−8.90991			−0.414976			−0.207488	+	0.978238i	$0.566529\pi$
$462$		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	−0.123097	−	0.992395i	$-0.539283\pi$
$467$	14.5824	−	8.41916i	0.674794	−	0.389592i	0.797891	+	0.602802i	$0.205949\pi$
$468$		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	−0.576227	−	0.817290i	$-0.695476\pi$
$479$	9.18515	−	15.9091i	0.419680	−	0.726907i	0.995907	+	0.0903824i	$0.0288089\pi$
$480$		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	$-0.839019\pi$
$491$		−	21.4694i		−	0.968902i	0.874818	−	0.484451i	$-0.160981\pi$
$492$		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	$-0.776629\pi$
$503$		−	28.9563i		−	1.29110i	0.763719	−	0.645548i	$-0.223371\pi$
$504$		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	$0.0571744\pi$
$509$	14.5897	+	25.2700i	0.646675	+	1.12007i	−0.337237	+	0.941420i	$0.609492\pi$
$510$		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	$0.400699\pi$
$521$	−15.3104	+	26.5184i	−0.670761	+	1.16179i	−0.977689	+	0.210059i	$0.932635\pi$
$522$		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	−0.983028	−	0.183458i	$-0.941271\pi$
$557$	−7.85045	+	13.5974i	−0.332634	+	0.576139i	0.650393	+	0.759598i	$0.274604\pi$
$558$		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	0.958977	−	0.283485i	$-0.0914905\pi$
$563$	28.3061	+	16.3425i	1.19296	+	0.688756i	0.233983	+	0.972241i	$0.424824\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	20.0566	−	11.5797i	0.840817	−	0.485446i	−0.0167248	−	0.999860i	$-0.505324\pi$
$569$	20.0566	−	11.5797i	0.840817	−	0.485446i	0.857542	+	0.514414i	$0.171991\pi$
$570$		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	$0.00246726\pi$
$587$			0.375586i			0.0155021i	−0.999970	+	0.00775105i	$0.997533\pi$
$588$		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	−0.0308124	−	0.999525i	$-0.509809\pi$
$593$	19.9536	−	11.5202i	0.819396	−	0.473078i	0.850208	+	0.526447i	$0.176476\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−7.46233	−	4.30838i	−0.304902	−	0.176036i	0.339741	−	0.940519i	$-0.389661\pi$
$599$	−7.46233	−	4.30838i	−0.304902	−	0.176036i	−0.644643	+	0.764484i	$0.722994\pi$
$600$		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	$-0.687816\pi$
$617$	−27.6411			−1.11279			−0.556395	+	0.830918i	$0.687816\pi$
$618$		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	0.272863	−	0.962053i	$-0.412029\pi$
$641$	−10.7315	−	6.19582i	−0.423868	−	0.244720i	−0.696731	+	0.717333i	$0.745363\pi$
$642$		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	0.881986	−	0.471275i	$-0.156206\pi$
$647$	23.2701	+	13.4350i	0.914843	+	0.528185i	0.0328565	+	0.999460i	$0.489540\pi$
$648$		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	0.859875	−	0.510504i	$-0.170541\pi$
$653$	−0.311046	−	0.538748i	−0.0121722	−	0.0210828i	−0.872047	+	0.489422i	$0.837208\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			17.9996i			0.701167i	0.936532	+	0.350583i	$0.114017\pi$
$659$			17.9996i			0.701167i	−0.936532	+	0.350583i	$0.885983\pi$
$660$		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	−0.790349	−	0.612657i	$-0.790101\pi$
$677$	−17.0412	+	9.83874i	−0.654947	+	0.378134i	0.135402	+	0.990791i	$0.456767\pi$
$678$		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	−0.985318	−	0.170731i	$-0.945387\pi$
$683$	−9.01113	+	15.6077i	−0.344801	+	0.597213i	0.640516	+	0.767945i	$0.278720\pi$
$684$		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	$0.215364\pi$
$701$			33.1556i			1.25227i	−0.779715	+	0.626134i	$0.784636\pi$
$702$		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	$0.0248810\pi$
$719$	15.1795	+	26.2916i	0.566098	+	0.980510i	−0.430849	+	0.902424i	$0.641786\pi$
$720$		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	$-0.463737\pi$
$743$	6.19725			0.227355			0.113678	+	0.993518i	$0.463737\pi$
$744$		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	0.843461	−	0.537191i	$-0.180514\pi$
$761$	−1.19974	−	2.07800i	−0.0434904	−	0.0753275i	−0.886951	+	0.461863i	$0.847181\pi$
$762$		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	0.808446	−	0.588570i	$-0.200309\pi$
$773$	19.5441	+	11.2838i	0.702953	+	0.405850i	−0.105493	+	0.994420i	$0.533642\pi$
$774$		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	$0.0392677\pi$
$797$			6.94772i			0.246101i	−0.992400	+	0.123050i	$0.960732\pi$
$798$		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	0.913197	−	0.407518i	$-0.133606\pi$
$809$	28.9229	+	16.6986i	1.01687	+	0.587093i	0.103677	+	0.994611i	$0.466939\pi$
$810$		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	−0.538889	−	0.842377i	$-0.681156\pi$
$821$	−2.25824	+	1.30380i	−0.0788132	+	0.0455028i	0.460076	+	0.887880i	$0.347822\pi$
$822$		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	$-0.425515\pi$
$827$	13.3362			0.463744			0.231872	+	0.972746i	$0.425515\pi$
$828$		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	$-0.388187\pi$
$839$	19.9335			0.688181			0.344090	+	0.938937i	$0.388187\pi$
$840$		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	0.991801	−	0.127793i	$-0.0407894\pi$
$857$	40.3119	+	23.2741i	1.37703	+	0.795028i	0.385228	+	0.922821i	$0.374123\pi$
$858$		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	0.956765	−	0.290863i	$-0.0939423\pi$
$863$	6.65350	+	11.5242i	0.226488	+	0.392288i	−0.730277	+	0.683151i	$0.760609\pi$
$864$		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	$-0.696222\pi$
$881$	−34.3203			−1.15628			−0.578141	+	0.815937i	$0.696222\pi$
$882$		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	0.250732	−	0.968056i	$-0.419329\pi$
$887$	36.1697	−	20.8826i	1.21446	−	0.701169i	0.963728	+	0.266888i	$0.0859953\pi$
$888$		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	$-0.689973\pi$
$911$		−	49.9301i		−	1.65426i	0.562012	−	0.827129i	$-0.310027\pi$
$912$		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	$-0.783928\pi$
$929$	−28.4346	−	49.2502i	−0.932910	−	1.61585i	−0.154590	−	0.987979i	$-0.549406\pi$
$930$		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	$0.450449\pi$
$941$	−23.8667	+	41.3384i	−0.778033	+	1.34759i	−0.933074	+	0.359684i	$0.882885\pi$
$942$		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	$0.0145335\pi$
$947$	16.5870	+	28.7296i	0.539006	+	0.933587i	−0.459951	+	0.887944i	$0.652133\pi$
$948$		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	$-0.650748\pi$
$953$	−28.1591			−0.912164			−0.456082	+	0.889938i	$0.650748\pi$
$954$		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	$0.0622390\pi$
$971$	20.5266	+	35.5531i	0.658729	+	1.14095i	−0.322216	+	0.946666i	$0.604428\pi$
$972$		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	−0.536491	−	0.843906i	$-0.680251\pi$
$977$	14.4594	−	25.0445i	0.462598	−	0.801244i	0.999090	+	0.0426622i	$0.0135839\pi$
$978$		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	0.908836	−	0.417153i	$-0.136972\pi$
$983$	31.4152	+	18.1376i	1.00199	+	0.578499i	0.0931532	+	0.995652i	$0.470305\pi$
$984$		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.c.4049.12		24
3.2	odd	2		6300.2.dd.b.4049.12		24
5.2	odd	4		1260.2.cg.a.521.3	yes	12
5.3	odd	4		6300.2.ch.c.4301.4		12
5.4	even	2	inner	6300.2.dd.c.4049.1		24
7.5	odd	6		6300.2.dd.b.1349.1		24
15.2	even	4		1260.2.cg.b.521.3	yes	12
15.8	even	4		6300.2.ch.b.4301.4		12
15.14	odd	2		6300.2.dd.b.4049.1		24
21.5	even	6	inner	6300.2.dd.c.1349.1		24
35.12	even	12		1260.2.cg.b.341.3	yes	12
35.17	even	12		8820.2.d.a.881.11		12
35.19	odd	6		6300.2.dd.b.1349.12		24
35.32	odd	12		8820.2.d.b.881.11		12
35.33	even	12		6300.2.ch.b.1601.4		12
105.17	odd	12		8820.2.d.b.881.2		12
105.32	even	12		8820.2.d.a.881.2		12
105.47	odd	12		1260.2.cg.a.341.3	✓	12
105.68	odd	12		6300.2.ch.c.1601.4		12
105.89	even	6	inner	6300.2.dd.c.1349.12		24

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