LMFDB - Embedded newform 6300.2.dd.b.1349.7

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			1349.7
Character	$\chi$	$=$	6300.1349
Dual form			6300.2.dd.b.4049.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0290059 + 2.64559i) q^{7} +O(q^{10})$	$q+(0.0290059 + 2.64559i) q^{7} +(-1.55024 + 0.895031i) q^{11} -2.19816 q^{13} +(-0.810754 + 0.468089i) q^{17} +(-0.366307 - 0.211487i) q^{19} +(-2.04359 + 3.53961i) q^{23} -8.43097i q^{29} +(-9.20510 + 5.31456i) q^{31} +(7.25340 + 4.18775i) q^{37} +6.29118 q^{41} -9.56621i q^{43} +(-2.63936 - 1.52383i) q^{47} +(-6.99832 + 0.153476i) q^{49} +(3.32045 + 5.75120i) q^{53} +(-6.36526 - 11.0249i) q^{59} +(1.68487 + 0.972758i) q^{61} +(-6.20381 + 3.58177i) q^{67} +4.91826i q^{71} +(-2.30565 - 3.99351i) q^{73} +(-2.41285 - 4.07534i) q^{77} +(2.78022 - 4.81548i) q^{79} -15.1448i q^{83} +(2.15103 - 3.72570i) q^{89} +(-0.0637596 - 5.81543i) q^{91} +4.55329 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * 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

Display 100 coefficients 10 coefficients

Character values

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

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{5}{6}\right)$

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
$7$	0.0290059	+	2.64559i	0.0109632	+	0.999940i
$7$	0.0290059	+	2.64559i	0.0109632	+	0.999940i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.55024	+	0.895031i	−0.467415	+	0.269862i	−0.715157	−	0.698964i	$-0.753645\pi$
$11$	−1.55024	+	0.895031i	−0.467415	+	0.269862i	0.247742	+	0.968826i	$0.420311\pi$
$12$		0			0
$13$	−2.19816			−0.609659			−0.304830	−	0.952407i	$-0.598600\pi$
$13$	−2.19816			−0.609659			−0.304830	+	0.952407i	$0.598600\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.810754	+	0.468089i	−0.196637	+	0.113528i	−0.595086	−	0.803662i	$-0.702882\pi$
$17$	−0.810754	+	0.468089i	−0.196637	+	0.113528i	0.398449	+	0.917190i	$0.369549\pi$
$18$		0			0
$19$	−0.366307	−	0.211487i	−0.0840366	−	0.0485186i	0.457393	−	0.889265i	$-0.348783\pi$
$19$	−0.366307	−	0.211487i	−0.0840366	−	0.0485186i	−0.541429	+	0.840746i	$0.682117\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.04359	+	3.53961i	−0.426119	+	0.738059i	−0.996524	−	0.0833039i	$-0.973453\pi$
$23$	−2.04359	+	3.53961i	−0.426119	+	0.738059i	0.570405	+	0.821363i	$0.306786\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	8.43097i		−	1.56559i	−0.622278	−	0.782796i	$-0.713793\pi$
$29$		−	8.43097i		−	1.56559i	0.622278	−	0.782796i	$-0.286207\pi$
$30$		0			0
$31$	−9.20510	+	5.31456i	−1.65328	+	0.954524i	−0.677575	+	0.735454i	$0.736969\pi$
$31$	−9.20510	+	5.31456i	−1.65328	+	0.954524i	−0.975709	+	0.219070i	$0.929698\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.25340	+	4.18775i	1.19245	+	0.688463i	0.958862	−	0.283873i	$-0.0916193\pi$
$37$	7.25340	+	4.18775i	1.19245	+	0.688463i	0.233590	+	0.972335i	$0.424953\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.29118			0.982518			0.491259	−	0.871014i	$-0.336537\pi$
$41$	6.29118			0.982518			0.491259	+	0.871014i	$0.336537\pi$
$42$		0			0
$43$		−	9.56621i		−	1.45883i	−0.684069	−	0.729417i	$-0.739791\pi$
$43$		−	9.56621i		−	1.45883i	0.684069	−	0.729417i	$-0.260209\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.63936	−	1.52383i	−0.384990	−	0.222274i	0.294997	−	0.955498i	$-0.404681\pi$
$47$	−2.63936	−	1.52383i	−0.384990	−	0.222274i	−0.679987	+	0.733224i	$0.738015\pi$
$48$		0			0
$49$	−6.99832	+	0.153476i	−0.999760	+	0.0219251i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.32045	+	5.75120i	0.456100	+	0.789988i	0.998751	−	0.0499706i	$-0.0159128\pi$
$53$	3.32045	+	5.75120i	0.456100	+	0.789988i	−0.542651	+	0.839958i	$0.682579\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.36526	−	11.0249i	−0.828686	−	1.43533i	−0.899069	−	0.437806i	$-0.855756\pi$
$59$	−6.36526	−	11.0249i	−0.828686	−	1.43533i	0.0703834	−	0.997520i	$-0.477578\pi$
$60$		0			0
$61$	1.68487	+	0.972758i	0.215725	+	0.124549i	0.603969	−	0.797008i	$-0.293585\pi$
$61$	1.68487	+	0.972758i	0.215725	+	0.124549i	−0.388244	+	0.921557i	$0.626918\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.20381	+	3.58177i	−0.757916	+	0.437583i	−0.828547	−	0.559920i	$-0.810832\pi$
$67$	−6.20381	+	3.58177i	−0.757916	+	0.437583i	0.0706311	+	0.997503i	$0.477499\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.91826i			0.583690i	0.956466	+	0.291845i	$0.0942691\pi$
$71$			4.91826i			0.583690i	−0.956466	+	0.291845i	$0.905731\pi$
$72$		0			0
$73$	−2.30565	−	3.99351i	−0.269856	−	0.467405i	0.698968	−	0.715153i	$-0.253643\pi$
$73$	−2.30565	−	3.99351i	−0.269856	−	0.467405i	−0.968825	+	0.247748i	$0.920310\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.41285	−	4.07534i	−0.274970	−	0.464428i
$78$		0			0
$79$	2.78022	−	4.81548i	0.312799	−	0.541784i	−0.666168	−	0.745802i	$-0.732067\pi$
$79$	2.78022	−	4.81548i	0.312799	−	0.541784i	0.978967	+	0.204018i	$0.0654000\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	15.1448i		−	1.66236i	−0.556007	−	0.831178i	$-0.687667\pi$
$83$		−	15.1448i		−	1.66236i	0.556007	−	0.831178i	$-0.312333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.15103	−	3.72570i	0.228009	−	0.394924i	−0.729209	−	0.684291i	$-0.760112\pi$
$89$	2.15103	−	3.72570i	0.228009	−	0.394924i	0.957218	+	0.289368i	$0.0934450\pi$
$90$		0			0
$91$	−0.0637596	−	5.81543i	−0.00668382	−	0.609623i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.55329			0.462317			0.231158	−	0.972916i	$-0.425748\pi$
$97$	4.55329			0.462317			0.231158	+	0.972916i	$0.425748\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.40748	−	7.63398i	−0.438561	−	0.759610i	0.559018	−	0.829156i	$-0.311178\pi$
$101$	−4.40748	−	7.63398i	−0.438561	−	0.759610i	−0.997579	+	0.0695459i	$0.977845\pi$
$102$		0			0
$103$	−0.808903	+	1.40106i	−0.0797036	+	0.138051i	−0.903122	−	0.429384i	$-0.858731\pi$
$103$	−0.808903	+	1.40106i	−0.0797036	+	0.138051i	0.823418	+	0.567435i	$0.192064\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.22361	−	5.58345i	0.311638	−	0.539773i	−0.667079	−	0.744987i	$-0.732456\pi$
$107$	3.22361	−	5.58345i	0.311638	−	0.539773i	0.978717	+	0.205214i	$0.0657891\pi$
$108$		0			0
$109$	−1.39335	−	2.41335i	−0.133458	−	0.231157i	0.791549	−	0.611106i	$-0.209275\pi$
$109$	−1.39335	−	2.41335i	−0.133458	−	0.231157i	−0.925008	+	0.379949i	$0.875942\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.71522			0.725787			0.362894	−	0.931831i	$-0.381789\pi$
$113$	7.71522			0.725787			0.362894	+	0.931831i	$0.381789\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.26189	−	2.13135i	−0.115677	−	0.195380i
$120$		0			0
$121$	−3.89784	+	6.75125i	−0.354349	+	0.613750i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$			1.80732i			0.160374i	0.996780	+	0.0801869i	$0.0255517\pi$
$127$			1.80732i			0.160374i	−0.996780	+	0.0801869i	$0.974448\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.53153	−	9.58089i	0.483292	−	0.837086i	−0.516524	−	0.856273i	$-0.672774\pi$
$131$	5.53153	−	9.58089i	0.483292	−	0.837086i	0.999816	+	0.0191864i	$0.00610758\pi$
$132$		0			0
$133$	0.548885	−	0.975233i	0.0475943	−	0.0845635i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.16520	+	15.8746i	0.783035	+	1.35626i	0.930166	+	0.367140i	$0.119663\pi$
$137$	9.16520	+	15.8746i	0.783035	+	1.35626i	−0.147130	+	0.989117i	$0.547004\pi$
$138$		0			0
$139$			5.62390i			0.477013i	0.971141	+	0.238507i	$0.0766579\pi$
$139$			5.62390i			0.477013i	−0.971141	+	0.238507i	$0.923342\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.40767	−	1.96742i	0.284964	−	0.164524i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.8179	−	7.40039i	−1.05008	−	0.606264i	−0.127408	−	0.991850i	$-0.540666\pi$
$149$	−12.8179	−	7.40039i	−1.05008	−	0.606264i	−0.922672	+	0.385587i	$0.873999\pi$
$150$		0			0
$151$	−3.87557	−	6.71269i	−0.315390	−	0.546271i	0.664131	−	0.747616i	$-0.268802\pi$
$151$	−3.87557	−	6.71269i	−0.315390	−	0.546271i	−0.979520	+	0.201346i	$0.935468\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2.25617	−	3.90780i	−0.180062	−	0.311876i	0.761840	−	0.647766i	$-0.224296\pi$
$157$	−2.25617	−	3.90780i	−0.180062	−	0.311876i	−0.941901	+	0.335889i	$0.890963\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−9.42364	−	5.30385i	−0.742687	−	0.418002i
$162$		0			0
$163$	−17.5876	−	10.1542i	−1.37757	−	0.795340i	−0.385703	−	0.922623i	$-0.626041\pi$
$163$	−17.5876	−	10.1542i	−1.37757	−	0.795340i	−0.991866	+	0.127283i	$0.959374\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.32784i		−	0.334898i	−0.985881	−	0.167449i	$-0.946447\pi$
$167$		−	4.32784i		−	0.334898i	0.985881	−	0.167449i	$-0.0535530\pi$
$168$		0			0
$169$	−8.16810			−0.628316
$170$		0			0
$171$		0			0
$172$		0			0
$173$	21.9327	+	12.6629i	1.66752	+	0.962740i	0.968971	+	0.247173i	$0.0795016\pi$
$173$	21.9327	+	12.6629i	1.66752	+	0.962740i	0.698544	+	0.715567i	$0.253832\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	14.9074	−	8.60678i	1.11423	−	0.643301i	0.174308	−	0.984691i	$-0.444231\pi$
$179$	14.9074	−	8.60678i	1.11423	−	0.643301i	0.939922	+	0.341390i	$0.110898\pi$
$180$		0			0
$181$		−	3.08246i		−	0.229118i	−0.993416	−	0.114559i	$-0.963455\pi$
$181$		−	3.08246i		−	0.229118i	0.993416	−	0.114559i	$-0.0365454\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.837909	−	1.45130i	0.0612740	−	0.106130i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.82256	+	2.20696i	0.276591	+	0.159690i	0.631879	−	0.775067i	$-0.282284\pi$
$191$	3.82256	+	2.20696i	0.276591	+	0.159690i	−0.355288	+	0.934757i	$0.615617\pi$
$192$		0			0
$193$	18.7242	−	10.8104i	1.34780	−	0.778153i	0.359863	−	0.933005i	$-0.382823\pi$
$193$	18.7242	−	10.8104i	1.34780	−	0.778153i	0.987938	+	0.154852i	$0.0494901\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.65937			−0.189472			−0.0947362	−	0.995502i	$-0.530201\pi$
$197$	−2.65937			−0.189472			−0.0947362	+	0.995502i	$0.530201\pi$
$198$		0			0
$199$	−10.1095	+	5.83671i	−0.716641	+	0.413753i	−0.813515	−	0.581544i	$-0.802449\pi$
$199$	−10.1095	+	5.83671i	−0.716641	+	0.413753i	0.0968739	+	0.995297i	$0.469116\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	22.3049	−	0.244548i	1.56550	−	0.0171639i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.757152			0.0523733
$210$		0			0
$211$	−5.93357			−0.408484			−0.204242	−	0.978920i	$-0.565473\pi$
$211$	−5.93357			−0.408484			−0.204242	+	0.978920i	$0.565473\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−14.3272	−	24.1988i	−0.972592	−	1.64272i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.78217	−	1.02893i	0.119881	−	0.0692136i
$222$		0			0
$223$	16.2789			1.09011			0.545057	−	0.838399i	$-0.316508\pi$
$223$	16.2789			1.09011			0.545057	+	0.838399i	$0.316508\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.636719	−	0.367610i	0.0422605	−	0.0243991i	−0.478721	−	0.877967i	$-0.658899\pi$
$227$	0.636719	−	0.367610i	0.0422605	−	0.0243991i	0.520981	+	0.853568i	$0.325566\pi$
$228$		0			0
$229$	−25.4384	−	14.6868i	−1.68101	−	0.970534i	−0.960989	−	0.276585i	$-0.910797\pi$
$229$	−25.4384	−	14.6868i	−1.68101	−	0.970534i	−0.720025	−	0.693948i	$-0.755870\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.1993	−	21.1298i	0.799203	−	1.38426i	−0.120932	−	0.992661i	$-0.538588\pi$
$233$	12.1993	−	21.1298i	0.799203	−	1.38426i	0.920136	−	0.391600i	$-0.128078\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	6.84320i		−	0.442650i	−0.975200	−	0.221325i	$-0.928962\pi$
$239$		−	6.84320i		−	0.442650i	0.975200	−	0.221325i	$-0.0710381\pi$
$240$		0			0
$241$	−0.0607315	+	0.0350634i	−0.00391206	+	0.00225863i	−0.501955	−	0.864894i	$-0.667386\pi$
$241$	−0.0607315	+	0.0350634i	−0.00391206	+	0.00225863i	0.498043	+	0.867153i	$0.334052\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.805201	+	0.464883i	0.0512337	+	0.0295798i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−14.5728			−0.919824			−0.459912	−	0.887965i	$-0.652119\pi$
$251$	−14.5728			−0.919824			−0.459912	+	0.887965i	$0.652119\pi$
$252$		0			0
$253$		−	7.31632i		−	0.459973i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.70543	−	5.60343i	−0.605408	−	0.349532i	0.165758	−	0.986166i	$-0.446993\pi$
$257$	−9.70543	−	5.60343i	−0.605408	−	0.349532i	−0.771166	+	0.636634i	$0.780326\pi$
$258$		0			0
$259$	−10.8687	+	19.3110i	−0.675348	+	1.19993i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.325239	−	0.563331i	−0.0200551	−	0.0347365i	0.855824	−	0.517268i	$-0.173051\pi$
$263$	−0.325239	−	0.563331i	−0.0200551	−	0.0347365i	−0.875879	+	0.482531i	$0.839717\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.26420	+	3.92171i	0.138051	+	0.239111i	0.926759	−	0.375657i	$-0.122583\pi$
$269$	2.26420	+	3.92171i	0.138051	+	0.239111i	−0.788708	+	0.614768i	$0.789250\pi$
$270$		0			0
$271$	6.11430	+	3.53009i	0.371417	+	0.214438i	0.674077	−	0.738661i	$-0.264541\pi$
$271$	6.11430	+	3.53009i	0.371417	+	0.214438i	−0.302660	+	0.953098i	$0.597875\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	15.0030	−	8.66200i	0.901444	−	0.520449i	0.0237759	−	0.999717i	$-0.492431\pi$
$277$	15.0030	−	8.66200i	0.901444	−	0.520449i	0.877669	+	0.479268i	$0.159098\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	26.9791i		−	1.60944i	−0.593655	−	0.804720i	$-0.702316\pi$
$281$		−	26.9791i		−	1.60944i	0.593655	−	0.804720i	$-0.297684\pi$
$282$		0			0
$283$	2.52554	+	4.37436i	0.150128	+	0.260029i	0.931274	−	0.364319i	$-0.118698\pi$
$283$	2.52554	+	4.37436i	0.150128	+	0.260029i	−0.781147	+	0.624348i	$0.785365\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.182482	+	16.6439i	0.0107715	+	0.982459i
$288$		0			0
$289$	−8.06179	+	13.9634i	−0.474223	+	0.821378i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			9.78083i			0.571402i	0.958319	+	0.285701i	$0.0922264\pi$
$293$			9.78083i			0.571402i	−0.958319	+	0.285701i	$0.907774\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.49214	−	7.78062i	0.259787	−	0.449965i
$300$		0			0
$301$	25.3083	−	0.277477i	1.45875	−	0.0159935i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−26.9503			−1.53813			−0.769067	−	0.639168i	$-0.779279\pi$
$307$	−26.9503			−1.53813			−0.769067	+	0.639168i	$0.779279\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6.35136	−	11.0009i	−0.360152	−	0.623802i	0.627833	−	0.778348i	$-0.283942\pi$
$311$	−6.35136	−	11.0009i	−0.360152	−	0.623802i	−0.987986	+	0.154546i	$0.950609\pi$
$312$		0			0
$313$	−11.9840	+	20.7569i	−0.677375	+	1.17325i	0.298394	+	0.954443i	$0.403549\pi$
$313$	−11.9840	+	20.7569i	−0.677375	+	1.17325i	−0.975769	+	0.218805i	$0.929784\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.202548	−	0.350824i	0.0113762	−	0.0197042i	−0.860281	−	0.509820i	$-0.829712\pi$
$317$	0.202548	−	0.350824i	0.0113762	−	0.0197042i	0.871657	+	0.490116i	$0.163045\pi$
$318$		0			0
$319$	7.54598	+	13.0700i	0.422494	+	0.731781i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.395980			0.0220329
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.95488	−	7.02686i	0.218040	−	0.387403i
$330$		0			0
$331$	−4.18929	+	7.25607i	−0.230264	+	0.398830i	−0.957886	−	0.287149i	$-0.907292\pi$
$331$	−4.18929	+	7.25607i	−0.230264	+	0.398830i	0.727621	+	0.685979i	$0.240626\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$			13.8266i			0.753181i	0.926380	+	0.376590i	$0.122904\pi$
$337$			13.8266i			0.753181i	−0.926380	+	0.376590i	$0.877096\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	9.51340	−	16.4777i	0.515180	−	0.892317i
$342$		0			0
$343$	−0.609027	−	18.5102i	−0.0328843	−	0.999459i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.18483	−	10.7124i	−0.332019	−	0.575074i	0.650889	−	0.759173i	$-0.274397\pi$
$347$	−6.18483	−	10.7124i	−0.332019	−	0.575074i	−0.982908	+	0.184099i	$0.941063\pi$
$348$		0			0
$349$		−	6.67915i		−	0.357527i	−0.983892	−	0.178763i	$-0.942790\pi$
$349$		−	6.67915i		−	0.357527i	0.983892	−	0.178763i	$-0.0572096\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.93658	−	2.85014i	0.262748	−	0.151697i	−0.362840	−	0.931852i	$-0.618193\pi$
$353$	4.93658	−	2.85014i	0.262748	−	0.151697i	0.625587	+	0.780154i	$0.284859\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−21.8381	−	12.6082i	−1.15257	−	0.665436i	−0.203057	−	0.979167i	$-0.565088\pi$
$359$	−21.8381	−	12.6082i	−1.15257	−	0.665436i	−0.949512	+	0.313731i	$0.898421\pi$
$360$		0			0
$361$	−9.41055	−	16.2995i	−0.495292	−	0.857871i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	14.1371	+	24.4862i	0.737952	+	1.27817i	0.953416	+	0.301658i	$0.0975400\pi$
$367$	14.1371	+	24.4862i	0.737952	+	1.27817i	−0.215465	+	0.976512i	$0.569127\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−15.1190	+	8.95139i	−0.784940	+	0.464733i
$372$		0			0
$373$	−14.2458	−	8.22479i	−0.737617	−	0.425863i	0.0835851	−	0.996501i	$-0.473363\pi$
$373$	−14.2458	−	8.22479i	−0.737617	−	0.425863i	−0.821202	+	0.570637i	$0.806696\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			18.5326i			0.954478i
$378$		0			0
$379$	−6.16869			−0.316864			−0.158432	−	0.987370i	$-0.550644\pi$
$379$	−6.16869			−0.316864			−0.158432	+	0.987370i	$0.550644\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.76747	+	5.06190i	0.447997	+	0.258651i	0.706984	−	0.707230i	$-0.250055\pi$
$383$	8.76747	+	5.06190i	0.447997	+	0.258651i	−0.258987	+	0.965881i	$0.583389\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	23.7572	−	13.7162i	1.20454	−	0.695441i	0.242978	−	0.970032i	$-0.421876\pi$
$389$	23.7572	−	13.7162i	1.20454	−	0.695441i	0.961561	+	0.274591i	$0.0885425\pi$
$390$		0			0
$391$		−	3.82634i		−	0.193506i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	12.3851	−	21.4517i	0.621592	−	1.07663i	−0.367598	−	0.929985i	$-0.619820\pi$
$397$	12.3851	−	21.4517i	0.621592	−	1.07663i	0.989189	−	0.146643i	$-0.0468470\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	20.8748	+	12.0521i	1.04244	+	0.601851i	0.920522	−	0.390690i	$-0.127764\pi$
$401$	20.8748	+	12.0521i	1.04244	+	0.601851i	0.121914	+	0.992541i	$0.461097\pi$
$402$		0			0
$403$	20.2343	−	11.6822i	1.00794	−	0.581934i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−14.9927			−0.743160
$408$		0			0
$409$	−21.1709	+	12.2230i	−1.04683	+	0.604388i	−0.921761	−	0.387760i	$-0.873249\pi$
$409$	−21.1709	+	12.2230i	−1.04683	+	0.604388i	−0.125071	+	0.992148i	$0.539916\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	28.9829	−	17.1597i	1.42615	−	0.844372i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.05848			−0.295976			−0.147988	−	0.988989i	$-0.547280\pi$
$419$	−6.05848			−0.295976			−0.147988	+	0.988989i	$0.547280\pi$
$420$		0			0
$421$	−19.0817			−0.929986			−0.464993	−	0.885314i	$-0.653943\pi$
$421$	−19.0817			−0.929986			−0.464993	+	0.885314i	$0.653943\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.52465	+	4.48568i	−0.122176	+	0.217077i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−29.4339	+	16.9937i	−1.41778	+	0.818557i	−0.996104	−	0.0881879i	$-0.971892\pi$
$431$	−29.4339	+	16.9937i	−1.41778	+	0.818557i	−0.421679	+	0.906745i	$0.638559\pi$
$432$		0			0
$433$	−19.2496			−0.925075			−0.462537	−	0.886600i	$-0.653061\pi$
$433$	−19.2496			−0.925075			−0.462537	+	0.886600i	$0.653061\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.49717	−	0.864389i	0.0716191	−	0.0413493i
$438$		0			0
$439$	−29.0204	−	16.7549i	−1.38507	−	0.799668i	−0.392312	−	0.919832i	$-0.628325\pi$
$439$	−29.0204	−	16.7549i	−1.38507	−	0.799668i	−0.992754	+	0.120164i	$0.961658\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.38952	+	2.40673i	−0.0660183	+	0.114347i	−0.897145	−	0.441736i	$-0.854363\pi$
$443$	−1.38952	+	2.40673i	−0.0660183	+	0.114347i	0.831127	+	0.556083i	$0.187696\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	40.7948i		−	1.92523i	−0.270881	−	0.962613i	$-0.587315\pi$
$449$		−	40.7948i		−	1.92523i	0.270881	−	0.962613i	$-0.412685\pi$
$450$		0			0
$451$	−9.75284	+	5.63081i	−0.459243	+	0.265144i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.94014	+	1.12014i	0.0907558	+	0.0523979i	0.544691	−	0.838637i	$-0.316647\pi$
$457$	1.94014	+	1.12014i	0.0907558	+	0.0523979i	−0.453935	+	0.891035i	$0.649980\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	36.5014			1.70004			0.850020	−	0.526751i	$-0.176590\pi$
$461$	36.5014			1.70004			0.850020	+	0.526751i	$0.176590\pi$
$462$		0			0
$463$		−	6.02692i		−	0.280095i	−0.990145	−	0.140047i	$-0.955275\pi$
$463$		−	6.02692i		−	0.280095i	0.990145	−	0.140047i	$-0.0447254\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−23.6814	−	13.6725i	−1.09585	−	0.632687i	−0.160720	−	0.987000i	$-0.551382\pi$
$467$	−23.6814	−	13.6725i	−1.09585	−	0.632687i	−0.935127	+	0.354313i	$0.884715\pi$
$468$		0			0
$469$	−9.65585	−	16.3089i	−0.445866	−	0.753073i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8.56206	+	14.8299i	0.393684	+	0.681881i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−18.4048	−	31.8781i	−0.840937	−	1.45655i	−0.889103	−	0.457706i	$-0.848671\pi$
$479$	−18.4048	−	31.8781i	−0.840937	−	1.45655i	0.0481665	−	0.998839i	$-0.484662\pi$
$480$		0			0
$481$	−15.9441	−	9.20534i	−0.726989	−	0.419728i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−27.5060	+	15.8806i	−1.24642	+	0.719619i	−0.970393	−	0.241532i	$-0.922350\pi$
$487$	−27.5060	+	15.8806i	−1.24642	+	0.719619i	−0.276023	+	0.961151i	$0.589017\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	0.651553i		−	0.0294042i	−0.999892	−	0.0147021i	$-0.995320\pi$
$491$		−	0.651553i		−	0.0294042i	0.999892	−	0.0147021i	$-0.00467999\pi$
$492$		0			0
$493$	3.94645	+	6.83545i	0.177739	+	0.307853i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−13.0117	+	0.142659i	−0.583655	+	0.00639911i
$498$		0			0
$499$	10.9966	−	19.0467i	0.492276	−	0.852647i	−0.507684	−	0.861543i	$-0.669498\pi$
$499$	10.9966	−	19.0467i	0.492276	−	0.852647i	0.999960	+	0.00889594i	$0.00283170\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	26.7369i		−	1.19214i	−0.802933	−	0.596070i	$-0.796728\pi$
$503$		−	26.7369i		−	1.19214i	0.802933	−	0.596070i	$-0.203272\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.01326	+	12.1473i	−0.310857	+	0.538421i	−0.978548	−	0.206018i	$-0.933950\pi$
$509$	−7.01326	+	12.1473i	−0.310857	+	0.538421i	0.667691	+	0.744439i	$0.267283\pi$
$510$		0			0
$511$	10.4983	−	6.21565i	0.464418	−	0.274964i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	5.45551			0.239933
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.0733	−	26.1077i	−0.660372	−	1.14380i	−0.980518	−	0.196429i	$-0.937065\pi$
$521$	−15.0733	−	26.1077i	−0.660372	−	1.14380i	0.320146	−	0.947368i	$-0.396268\pi$
$522$		0			0
$523$	−0.361210	+	0.625635i	−0.0157946	+	0.0273571i	−0.873815	−	0.486259i	$-0.838361\pi$
$523$	−0.361210	+	0.625635i	−0.0157946	+	0.0273571i	0.858020	+	0.513616i	$0.171694\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.97538	−	8.61761i	0.216731	−	0.375389i
$528$		0			0
$529$	3.14745	+	5.45154i	0.136846	+	0.237023i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−13.8290			−0.599001
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	10.7117	−	6.50164i	0.461386	−	0.280045i
$540$		0			0
$541$	−6.31658	+	10.9406i	−0.271571	+	0.470375i	−0.969264	−	0.246022i	$-0.920877\pi$
$541$	−6.31658	+	10.9406i	−0.271571	+	0.470375i	0.697693	+	0.716397i	$0.254210\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			10.0662i			0.430401i	0.976570	+	0.215201i	$0.0690405\pi$
$547$			10.0662i			0.430401i	−0.976570	+	0.215201i	$0.930959\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.78304	+	3.08832i	−0.0759603	+	0.131567i
$552$		0			0
$553$	12.8204	+	7.21565i	0.545181	+	0.306841i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7.01135	+	12.1440i	0.297081	+	0.514559i	0.975467	−	0.220147i	$-0.0706537\pi$
$557$	7.01135	+	12.1440i	0.297081	+	0.514559i	−0.678386	+	0.734706i	$0.737320\pi$
$558$		0			0
$559$			21.0280i			0.889392i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−25.0329	+	14.4528i	−1.05501	+	0.609112i	−0.924048	−	0.382275i	$-0.875141\pi$
$563$	−25.0329	+	14.4528i	−1.05501	+	0.609112i	−0.130964	+	0.991387i	$0.541807\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.728101	−	0.420369i	−0.0305236	−	0.0176228i	0.484661	−	0.874702i	$-0.338943\pi$
$569$	−0.728101	−	0.420369i	−0.0305236	−	0.0176228i	−0.515184	+	0.857080i	$0.672276\pi$
$570$		0			0
$571$	22.6922	+	39.3040i	0.949638	+	1.64482i	0.746187	+	0.665736i	$0.231882\pi$
$571$	22.6922	+	39.3040i	0.949638	+	1.64482i	0.203451	+	0.979085i	$0.434784\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	9.97704	+	17.2807i	0.415350	+	0.719406i	0.995465	−	0.0951277i	$-0.0303260\pi$
$577$	9.97704	+	17.2807i	0.415350	+	0.719406i	−0.580116	+	0.814534i	$0.696993\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	40.0669	−	0.439288i	1.66226	−	0.0182247i
$582$		0			0
$583$	−10.2950	−	5.94382i	−0.426375	−	0.246168i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	24.0763i		−	0.993733i	−0.867827	−	0.496867i	$-0.834484\pi$
$587$		−	24.0763i		−	0.993733i	0.867827	−	0.496867i	$-0.165516\pi$
$588$		0			0
$589$	4.49586			0.185248
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−31.0711	−	17.9389i	−1.27594	−	0.736663i	−0.299839	−	0.953990i	$-0.596933\pi$
$593$	−31.0711	−	17.9389i	−1.27594	−	0.736663i	−0.976099	+	0.217327i	$0.930266\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−21.4343	+	12.3751i	−0.875783	+	0.505633i	−0.869266	−	0.494345i	$-0.835408\pi$
$599$	−21.4343	+	12.3751i	−0.875783	+	0.505633i	−0.00651720	+	0.999979i	$0.502075\pi$
$600$		0			0
$601$		−	35.0712i		−	1.43058i	−0.698826	−	0.715291i	$-0.746294\pi$
$601$		−	35.0712i		−	1.43058i	0.698826	−	0.715291i	$-0.253706\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1.15887	−	2.00722i	0.0470371	−	0.0814706i	−0.841548	−	0.540182i	$-0.818355\pi$
$607$	1.15887	−	2.00722i	0.0470371	−	0.0814706i	0.888585	+	0.458711i	$0.151689\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.80172	+	3.34963i	0.234713	+	0.135511i
$612$		0			0
$613$	11.1266	−	6.42396i	0.449400	−	0.259461i	−0.258177	−	0.966098i	$-0.583122\pi$
$613$	11.1266	−	6.42396i	0.449400	−	0.259461i	0.707577	+	0.706636i	$0.249788\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	29.7437			1.19743			0.598717	−	0.800960i	$-0.295677\pi$
$617$	29.7437			1.19743			0.598717	+	0.800960i	$0.295677\pi$
$618$		0			0
$619$	37.6945	−	21.7629i	1.51507	−	0.874726i	0.515226	−	0.857054i	$-0.327708\pi$
$619$	37.6945	−	21.7629i	1.51507	−	0.874726i	0.999844	−	0.0176719i	$-0.00562543\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9.91908	+	5.58269i	0.397400	+	0.223666i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7.84097			−0.312640
$630$		0			0
$631$	4.21974			0.167985			0.0839925	−	0.996466i	$-0.473233\pi$
$631$	4.21974			0.167985			0.0839925	+	0.996466i	$0.473233\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	15.3834	−	0.337364i	0.609513	−	0.0133668i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−38.7681	+	22.3828i	−1.53125	+	0.884066i	−0.531943	+	0.846780i	$0.678538\pi$
$641$	−38.7681	+	22.3828i	−1.53125	+	0.884066i	−0.999305	+	0.0372866i	$0.988129\pi$
$642$		0			0
$643$	−27.0494			−1.06672			−0.533362	−	0.845887i	$-0.679072\pi$
$643$	−27.0494			−1.06672			−0.533362	+	0.845887i	$0.679072\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−16.6837	+	9.63234i	−0.655904	+	0.378686i	−0.790714	−	0.612185i	$-0.790291\pi$
$647$	−16.6837	+	9.63234i	−0.655904	+	0.378686i	0.134811	+	0.990871i	$0.456957\pi$
$648$		0			0
$649$	19.7354	+	11.3942i	0.774680	+	0.447262i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.0711	+	26.1038i	−0.589776	+	1.02152i	0.404485	+	0.914544i	$0.367451\pi$
$653$	−15.0711	+	26.1038i	−0.589776	+	1.02152i	−0.994261	+	0.106978i	$0.965883\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			21.2084i			0.826164i	0.910694	+	0.413082i	$0.135548\pi$
$659$			21.2084i			0.826164i	−0.910694	+	0.413082i	$0.864452\pi$
$660$		0			0
$661$	−37.3887	+	21.5864i	−1.45425	+	0.839614i	−0.998719	−	0.0506043i	$-0.983885\pi$
$661$	−37.3887	+	21.5864i	−1.45425	+	0.839614i	−0.455535	+	0.890218i	$0.650552\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	29.8423	+	17.2295i	1.15550	+	0.667128i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.48259			−0.134444
$672$		0			0
$673$		−	31.7115i		−	1.22239i	−0.791480	−	0.611195i	$-0.790689\pi$
$673$		−	31.7115i		−	1.22239i	0.791480	−	0.611195i	$-0.209311\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	29.6292	+	17.1064i	1.13874	+	0.657453i	0.946119	−	0.323820i	$-0.104967\pi$
$677$	29.6292	+	17.1064i	1.13874	+	0.657453i	0.192623	+	0.981273i	$0.438301\pi$
$678$		0			0
$679$	0.132072	+	12.0462i	0.00506848	+	0.462289i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	23.9838	+	41.5412i	0.917715	+	1.58953i	0.802877	+	0.596144i	$0.203301\pi$
$683$	23.9838	+	41.5412i	0.917715	+	1.58953i	0.114838	+	0.993384i	$0.463365\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.29888	−	12.6420i	−0.278065	−	0.481623i
$690$		0			0
$691$	24.7315	+	14.2787i	0.940831	+	0.543189i	0.890221	−	0.455529i	$-0.150550\pi$
$691$	24.7315	+	14.2787i	0.940831	+	0.543189i	0.0506104	+	0.998718i	$0.483883\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−5.10060	+	2.94484i	−0.193199	+	0.111544i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			25.7765i			0.973564i	0.873523	+	0.486782i	$0.161829\pi$
$701$			25.7765i			0.973564i	−0.873523	+	0.486782i	$0.838171\pi$
$702$		0			0
$703$	−1.77132	−	3.06801i	−0.0668064	−	0.115712i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	20.0686	−	11.8818i	0.754756	−	0.446862i
$708$		0			0
$709$	21.7874	−	37.7369i	0.818243	−	1.41724i	−0.0887325	−	0.996055i	$-0.528282\pi$
$709$	21.7874	−	37.7369i	0.818243	−	1.41724i	0.906976	−	0.421183i	$-0.138385\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	43.4432i		−	1.62696i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−0.203564	+	0.352584i	−0.00759166	+	0.0131491i	−0.869796	−	0.493411i	$-0.835750\pi$
$719$	−0.203564	+	0.352584i	−0.00759166	+	0.0131491i	0.862205	+	0.506560i	$0.169083\pi$
$720$		0			0
$721$	−3.73010	−	2.09939i	−0.138916	−	0.0781853i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	17.8033			0.660288			0.330144	−	0.943931i	$-0.392903\pi$
$727$	17.8033			0.660288			0.330144	+	0.943931i	$0.392903\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.47784	+	7.75585i	0.165619	+	0.286860i
$732$		0			0
$733$	1.73460	−	3.00442i	0.0640690	−	0.110971i	−0.832212	−	0.554458i	$-0.812926\pi$
$733$	1.73460	−	3.00442i	0.0640690	−	0.110971i	0.896281	+	0.443487i	$0.146259\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.41159	−	11.1052i	0.236174	−	0.409066i
$738$		0			0
$739$	−12.4022	−	21.4812i	−0.456222	−	0.790201i	0.542535	−	0.840033i	$-0.317465\pi$
$739$	−12.4022	−	21.4812i	−0.456222	−	0.790201i	−0.998758	+	0.0498326i	$0.984131\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	29.8141			1.09377			0.546887	−	0.837206i	$-0.315813\pi$
$743$	29.8141			1.09377			0.546887	+	0.837206i	$0.315813\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	14.8650	+	8.36640i	0.543157	+	0.305702i
$750$		0			0
$751$	−16.9247	+	29.3145i	−0.617592	+	1.06970i	0.372332	+	0.928100i	$0.378558\pi$
$751$	−16.9247	+	29.3145i	−0.617592	+	1.06970i	−0.989924	+	0.141601i	$0.954775\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			1.19627i			0.0434791i	0.999764	+	0.0217395i	$0.00692045\pi$
$757$			1.19627i			0.0434791i	−0.999764	+	0.0217395i	$0.993080\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	24.3373	−	42.1534i	0.882226	−	1.52806i	0.0333649	−	0.999443i	$-0.489378\pi$
$761$	24.3373	−	42.1534i	0.882226	−	1.52806i	0.848861	−	0.528616i	$-0.177289\pi$
$762$		0			0
$763$	6.34432	−	3.75623i	0.229680	−	0.135985i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	13.9918	+	24.2346i	0.505216	+	0.875060i
$768$		0			0
$769$			9.21143i			0.332173i	0.986111	+	0.166086i	$0.0531131\pi$
$769$			9.21143i			0.332173i	−0.986111	+	0.166086i	$0.946887\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−38.1373	+	22.0186i	−1.37170	+	0.791954i	−0.991143	−	0.132801i	$-0.957603\pi$
$773$	−38.1373	+	22.0186i	−1.37170	+	0.791954i	−0.380562	+	0.924755i	$0.624269\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.30451	−	1.33051i	−0.0825675	−	0.0476703i
$780$		0			0
$781$	−4.40199	−	7.62448i	−0.157516	−	0.272825i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−21.0220	−	36.4112i	−0.749354	−	1.29792i	−0.948133	−	0.317875i	$-0.897031\pi$
$787$	−21.0220	−	36.4112i	−0.749354	−	1.29792i	0.198778	−	0.980044i	$-0.436303\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.223787	+	20.4113i	0.00795695	+	0.725743i
$792$		0			0
$793$	−3.70360	−	2.13827i	−0.131519	−	0.0759324i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	33.6692i		−	1.19263i	−0.802752	−	0.596313i	$-0.796632\pi$
$797$		−	33.6692i		−	1.19263i	0.802752	−	0.596313i	$-0.203368\pi$
$798$		0			0
$799$	2.85316			0.100938
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.14863	+	4.12726i	0.252270	+	0.145648i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	8.22735	−	4.75006i	0.289258	−	0.167003i	−0.348349	−	0.937365i	$-0.613258\pi$
$809$	8.22735	−	4.75006i	0.289258	−	0.167003i	0.637607	+	0.770362i	$0.279924\pi$
$810$		0			0
$811$		−	14.8210i		−	0.520435i	−0.965550	−	0.260217i	$-0.916206\pi$
$811$		−	14.8210i		−	0.520435i	0.965550	−	0.260217i	$-0.0837942\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.02313	+	3.50417i	−0.0707805	+	0.122595i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.6109	−	9.01294i	−0.544823	−	0.314554i	0.202208	−	0.979343i	$-0.435188\pi$
$821$	−15.6109	−	9.01294i	−0.544823	−	0.314554i	−0.747031	+	0.664789i	$0.768522\pi$
$822$		0			0
$823$	22.4821	−	12.9801i	0.783677	−	0.452456i	−0.0540548	−	0.998538i	$-0.517215\pi$
$823$	22.4821	−	12.9801i	0.783677	−	0.452456i	0.837732	+	0.546082i	$0.183881\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	24.4040			0.848612			0.424306	−	0.905519i	$-0.360518\pi$
$827$	24.4040			0.848612			0.424306	+	0.905519i	$0.360518\pi$
$828$		0			0
$829$	8.99187	−	5.19146i	0.312301	−	0.180307i	−0.335655	−	0.941985i	$-0.608958\pi$
$829$	8.99187	−	5.19146i	0.312301	−	0.180307i	0.647956	+	0.761678i	$0.275624\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	5.60207	−	3.40027i	0.194100	−	0.117812i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	29.7021			1.02543			0.512715	−	0.858559i	$-0.328640\pi$
$839$	29.7021			1.02543			0.512715	+	0.858559i	$0.328640\pi$
$840$		0			0
$841$	−42.0813			−1.45108
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−17.9741	−	10.1163i	−0.617598	−	0.347599i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−29.6460	+	17.1161i	−1.01625	+	0.586734i
$852$		0			0
$853$	−47.4739			−1.62548			−0.812738	−	0.582629i	$-0.802024\pi$
$853$	−47.4739			−1.62548			−0.812738	+	0.582629i	$0.802024\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−29.0287	+	16.7597i	−0.991602	+	0.572501i	−0.905753	−	0.423807i	$-0.860694\pi$
$857$	−29.0287	+	16.7597i	−0.991602	+	0.572501i	−0.0858490	+	0.996308i	$0.527360\pi$
$858$		0			0
$859$	29.1235	+	16.8144i	0.993680	+	0.573702i	0.906372	−	0.422480i	$-0.138840\pi$
$859$	29.1235	+	16.8144i	0.993680	+	0.573702i	0.0873078	+	0.996181i	$0.472174\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10.0735	+	17.4477i	−0.342904	+	0.593928i	−0.984971	−	0.172721i	$-0.944744\pi$
$863$	−10.0735	+	17.4477i	−0.342904	+	0.593928i	0.642066	+	0.766649i	$0.278077\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$			9.95353i			0.337650i
$870$		0			0
$871$	13.6369	−	7.87330i	0.462070	−	0.266776i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−24.6721	−	14.2444i	−0.833118	−	0.481001i	0.0218011	−	0.999762i	$-0.493060\pi$
$877$	−24.6721	−	14.2444i	−0.833118	−	0.481001i	−0.854919	+	0.518761i	$0.826393\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	27.5545			0.928336			0.464168	−	0.885747i	$-0.346353\pi$
$881$	27.5545			0.928336			0.464168	+	0.885747i	$0.346353\pi$
$882$		0			0
$883$			48.1856i			1.62158i	0.585340	+	0.810788i	$0.300961\pi$
$883$			48.1856i			1.62158i	−0.585340	+	0.810788i	$0.699039\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−41.5470	−	23.9872i	−1.39501	−	0.805410i	−0.401147	−	0.916014i	$-0.631388\pi$
$887$	−41.5470	−	23.9872i	−1.39501	−	0.805410i	−0.993865	+	0.110604i	$0.964722\pi$
$888$		0			0
$889$	−4.78143	+	0.0524230i	−0.160364	+	0.00175821i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.644543	+	1.11638i	0.0215688	+	0.0373583i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	44.8069	+	77.6079i	1.49440	+	2.58837i
$900$		0			0
$901$	−5.38414	−	3.10854i	−0.179372	−	0.103560i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	19.6639	−	11.3530i	0.652931	−	0.376970i	−0.136647	−	0.990620i	$-0.543633\pi$
$907$	19.6639	−	11.3530i	0.652931	−	0.376970i	0.789578	+	0.613650i	$0.210299\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			6.95948i			0.230578i	0.993332	+	0.115289i	$0.0367794\pi$
$911$			6.95948i			0.230578i	−0.993332	+	0.115289i	$0.963221\pi$
$912$		0			0
$913$	13.5551	+	23.4780i	0.448607	+	0.777010i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	25.5076	+	14.3563i	0.842335	+	0.474086i
$918$		0			0
$919$	−18.7082	+	32.4036i	−0.617128	+	1.06890i	0.372879	+	0.927880i	$0.378371\pi$
$919$	−18.7082	+	32.4036i	−0.617128	+	1.06890i	−0.990007	+	0.141017i	$0.954963\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	10.8111i		−	0.355852i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17.1344	+	29.6776i	−0.562160	+	0.973689i	0.435148	+	0.900359i	$0.356696\pi$
$929$	−17.1344	+	29.6776i	−0.562160	+	0.973689i	−0.997308	+	0.0733304i	$0.976637\pi$
$930$		0			0
$931$	2.59599	+	1.42384i	0.0850802	+	0.0466644i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−10.0892			−0.329600			−0.164800	−	0.986327i	$-0.552698\pi$
$937$	−10.0892			−0.329600			−0.164800	+	0.986327i	$0.552698\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.02257	+	5.23524i	0.0985328	+	0.170664i	0.911078	−	0.412235i	$-0.135252\pi$
$941$	3.02257	+	5.23524i	0.0985328	+	0.170664i	−0.812545	+	0.582899i	$0.801918\pi$
$942$		0			0
$943$	−12.8566	+	22.2683i	−0.418669	+	0.725157i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	10.6982	−	18.5297i	0.347643	−	0.602136i	−0.638187	−	0.769881i	$-0.720315\pi$
$947$	10.6982	−	18.5297i	0.347643	−	0.602136i	0.985830	+	0.167746i	$0.0536487\pi$
$948$		0			0
$949$	5.06819	+	8.77836i	0.164520	+	0.284958i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−29.6828			−0.961519			−0.480760	−	0.876852i	$-0.659639\pi$
$953$	−29.6828			−0.961519			−0.480760	+	0.876852i	$0.659639\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−41.7318	+	24.7078i	−1.34759	+	0.797857i
$960$		0			0
$961$	40.9892	−	70.9954i	1.32223	−	2.29017i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	41.7092i		−	1.34128i	−0.741783	−	0.670639i	$-0.766020\pi$
$967$		−	41.7092i		−	1.34128i	0.741783	−	0.670639i	$-0.233980\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−10.7837	+	18.6779i	−0.346065	+	0.599402i	−0.985547	−	0.169405i	$-0.945816\pi$
$971$	−10.7837	+	18.6779i	−0.346065	+	0.599402i	0.639482	+	0.768806i	$0.279149\pi$
$972$		0			0
$973$	−14.8786	+	0.163126i	−0.476985	+	0.00522960i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	15.6923	+	27.1798i	0.502040	+	0.869558i	0.999997	+	0.00235693i	$0.000750236\pi$
$977$	15.6923	+	27.1798i	0.502040	+	0.869558i	−0.497957	+	0.867201i	$0.665916\pi$
$978$		0			0
$979$			7.70097i			0.246124i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−48.4572	+	27.9768i	−1.54555	+	0.892322i	−0.547073	+	0.837085i	$0.684258\pi$
$983$	−48.4572	+	27.9768i	−1.54555	+	0.892322i	−0.998473	+	0.0552365i	$0.982409\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	33.8607	+	19.5495i	1.07671	+	0.621637i
$990$		0			0
$991$	31.1869	+	54.0173i	0.990685	+	1.71592i	0.613271	+	0.789873i	$0.289854\pi$
$991$	31.1869	+	54.0173i	0.990685	+	1.71592i	0.377415	+	0.926044i	$0.376813\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	14.5208	+	25.1507i	0.459878	+	0.796532i	0.998954	−	0.0457254i	$-0.0145599\pi$
$997$	14.5208	+	25.1507i	0.459878	+	0.796532i	−0.539076	+	0.842257i	$0.681227\pi$
$998$		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.1349.7		24
3.2	odd	2		6300.2.dd.c.1349.7		24
5.2	odd	4		1260.2.cg.b.341.1	yes	12
5.3	odd	4		6300.2.ch.b.1601.6		12
5.4	even	2	inner	6300.2.dd.b.1349.6		24
7.3	odd	6		6300.2.dd.c.4049.6		24
15.2	even	4		1260.2.cg.a.341.1	✓	12
15.8	even	4		6300.2.ch.c.1601.6		12
15.14	odd	2		6300.2.dd.c.1349.6		24
21.17	even	6	inner	6300.2.dd.b.4049.6		24
35.2	odd	12		8820.2.d.a.881.6		12
35.3	even	12		6300.2.ch.c.4301.6		12
35.12	even	12		8820.2.d.b.881.6		12
35.17	even	12		1260.2.cg.a.521.1	yes	12
35.24	odd	6		6300.2.dd.c.4049.7		24
105.2	even	12		8820.2.d.b.881.7		12
105.17	odd	12		1260.2.cg.b.521.1	yes	12
105.38	odd	12		6300.2.ch.b.4301.6		12
105.47	odd	12		8820.2.d.a.881.7		12
105.59	even	6	inner	6300.2.dd.b.4049.7		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.cg.a.341.1	✓	12	15.2	even	4
1260.2.cg.a.521.1	yes	12	35.17	even	12
1260.2.cg.b.341.1	yes	12	5.2	odd	4
1260.2.cg.b.521.1	yes	12	105.17	odd	12
6300.2.ch.b.1601.6		12	5.3	odd	4
6300.2.ch.b.4301.6		12	105.38	odd	12
6300.2.ch.c.1601.6		12	15.8	even	4
6300.2.ch.c.4301.6		12	35.3	even	12
6300.2.dd.b.1349.6		24	5.4	even	2	inner
6300.2.dd.b.1349.7		24	1.1	even	1	trivial
6300.2.dd.b.4049.6		24	21.17	even	6	inner
6300.2.dd.b.4049.7		24	105.59	even	6	inner
6300.2.dd.c.1349.6		24	15.14	odd	2
6300.2.dd.c.1349.7		24	3.2	odd	2
6300.2.dd.c.4049.6		24	7.3	odd	6
6300.2.dd.c.4049.7		24	35.24	odd	6
8820.2.d.a.881.6		12	35.2	odd	12
8820.2.d.a.881.7		12	105.47	odd	12
8820.2.d.b.881.6		12	35.12	even	12
8820.2.d.b.881.7		12	105.2	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)

\(f(q)\)	\(=\)	\(q+(0.0290059 + 2.64559i) q^{7} +O(q^{10})\)	\(q+(0.0290059 + 2.64559i) q^{7} +(-1.55024 + 0.895031i) q^{11} -2.19816 q^{13} +(-0.810754 + 0.468089i) q^{17} +(-0.366307 - 0.211487i) q^{19} +(-2.04359 + 3.53961i) q^{23} -8.43097i q^{29} +(-9.20510 + 5.31456i) q^{31} +(7.25340 + 4.18775i) q^{37} +6.29118 q^{41} -9.56621i q^{43} +(-2.63936 - 1.52383i) q^{47} +(-6.99832 + 0.153476i) q^{49} +(3.32045 + 5.75120i) q^{53} +(-6.36526 - 11.0249i) q^{59} +(1.68487 + 0.972758i) q^{61} +(-6.20381 + 3.58177i) q^{67} +4.91826i q^{71} +(-2.30565 - 3.99351i) q^{73} +(-2.41285 - 4.07534i) q^{77} +(2.78022 - 4.81548i) q^{79} -15.1448i q^{83} +(2.15103 - 3.72570i) q^{89} +(-0.0637596 - 5.81543i) q^{91} +4.55329 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * 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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