LMFDB - Embedded newform 6300.2.dd.d.1349.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6300,2,Mod(1349,6300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [40,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1349.3
Character	$\chi$	$=$	6300.1349
Dual form			6300.2.dd.d.4049.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.14507 - 1.54876i) q^{7} +(-1.40119 + 0.808980i) q^{11} +6.11640 q^{13} +(-4.66363 + 2.69255i) q^{17} +(0.667037 + 0.385114i) q^{19} +(1.86812 - 3.23568i) q^{23} -5.62291i q^{29} +(0.201585 - 0.116385i) q^{31} +(-0.411626 - 0.237653i) q^{37} -0.682087 q^{41} +3.64192i q^{43} +(-3.37027 - 1.94583i) q^{47} +(2.20268 + 6.64441i) q^{49} +(-6.43189 - 11.1404i) q^{53} +(5.41271 + 9.37509i) q^{59} +(-3.26629 - 1.88579i) q^{61} +(2.72158 - 1.57131i) q^{67} +13.9116i q^{71} +(-3.71204 - 6.42944i) q^{73} +(4.25858 + 0.434792i) q^{77} +(-0.416272 + 0.721004i) q^{79} -0.682087i q^{83} +(5.18151 - 8.97464i) q^{89} +(-13.1201 - 9.47283i) q^{91} +16.8308 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 12 q^{31} - 8 q^{49} + 12 q^{61} - 4 q^{79} - 80 q^{91}+O(q^{100}) $ 40 * q + 12 * q^31 - 8 * q^49 + 12 * q^61 - 4 * q^79 - 80 * q^91

Character values

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

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

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
$7$	−2.14507	−	1.54876i	−0.810762	−	0.585376i
$7$	−2.14507	−	1.54876i	−0.810762	−	0.585376i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.40119	+	0.808980i	−0.422476	+	0.243917i	−0.696136	−	0.717910i	$-0.745099\pi$
$11$	−1.40119	+	0.808980i	−0.422476	+	0.243917i	0.273660	+	0.961826i	$0.411766\pi$
$12$		0			0
$13$	6.11640			1.69638			0.848192	−	0.529689i	$-0.177691\pi$
$13$	6.11640			1.69638			0.848192	+	0.529689i	$0.177691\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.66363	+	2.69255i	−1.13110	+	0.653039i	−0.944211	−	0.329342i	$-0.893173\pi$
$17$	−4.66363	+	2.69255i	−1.13110	+	0.653039i	−0.186886	+	0.982382i	$0.559840\pi$
$18$		0			0
$19$	0.667037	+	0.385114i	0.153029	+	0.0883512i	0.574559	−	0.818463i	$-0.305174\pi$
$19$	0.667037	+	0.385114i	0.153029	+	0.0883512i	−0.421530	+	0.906814i	$0.638507\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.86812	−	3.23568i	0.389530	−	0.674686i	−0.602856	−	0.797850i	$-0.705971\pi$
$23$	1.86812	−	3.23568i	0.389530	−	0.674686i	0.992386	+	0.123164i	$0.0393041\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	5.62291i		−	1.04415i	−0.852900	−	0.522074i	$-0.825159\pi$
$29$		−	5.62291i		−	1.04415i	0.852900	−	0.522074i	$-0.174841\pi$
$30$		0			0
$31$	0.201585	−	0.116385i	0.0362057	−	0.0209034i	−0.481788	−	0.876288i	$-0.660012\pi$
$31$	0.201585	−	0.116385i	0.0362057	−	0.0209034i	0.517994	+	0.855384i	$0.326679\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.411626	−	0.237653i	−0.0676709	−	0.0390698i	0.465783	−	0.884899i	$-0.345773\pi$
$37$	−0.411626	−	0.237653i	−0.0676709	−	0.0390698i	−0.533454	+	0.845829i	$0.679106\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.682087			−0.106524			−0.0532621	−	0.998581i	$-0.516962\pi$
$41$	−0.682087			−0.106524			−0.0532621	+	0.998581i	$0.516962\pi$
$42$		0			0
$43$			3.64192i			0.555388i	0.960670	+	0.277694i	$0.0895701\pi$
$43$			3.64192i			0.555388i	−0.960670	+	0.277694i	$0.910430\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.37027	−	1.94583i	−0.491605	−	0.283828i	0.233635	−	0.972324i	$-0.424938\pi$
$47$	−3.37027	−	1.94583i	−0.491605	−	0.283828i	−0.725240	+	0.688496i	$0.758271\pi$
$48$		0			0
$49$	2.20268	+	6.64441i	0.314669	+	0.949201i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.43189	−	11.1404i	−0.883488	−	1.53025i	−0.847437	−	0.530895i	$-0.821856\pi$
$53$	−6.43189	−	11.1404i	−0.883488	−	1.53025i	−0.0360503	−	0.999350i	$-0.511478\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.41271	+	9.37509i	0.704675	+	1.22053i	0.966809	+	0.255501i	$0.0822405\pi$
$59$	5.41271	+	9.37509i	0.704675	+	1.22053i	−0.262134	+	0.965032i	$0.584426\pi$
$60$		0			0
$61$	−3.26629	−	1.88579i	−0.418205	−	0.241451i	0.276104	−	0.961128i	$-0.410957\pi$
$61$	−3.26629	−	1.88579i	−0.418205	−	0.241451i	−0.694309	+	0.719677i	$0.744290\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.72158	−	1.57131i	0.332494	−	0.191966i	−0.324454	−	0.945902i	$-0.605180\pi$
$67$	2.72158	−	1.57131i	0.332494	−	0.191966i	0.656948	+	0.753936i	$0.271847\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			13.9116i			1.65100i	0.564403	+	0.825500i	$0.309106\pi$
$71$			13.9116i			1.65100i	−0.564403	+	0.825500i	$0.690894\pi$
$72$		0			0
$73$	−3.71204	−	6.42944i	−0.434462	−	0.752510i	0.562790	−	0.826600i	$-0.309728\pi$
$73$	−3.71204	−	6.42944i	−0.434462	−	0.752510i	−0.997252	+	0.0740904i	$0.976395\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.25858	+	0.434792i	0.485311	+	0.0495492i
$78$		0			0
$79$	−0.416272	+	0.721004i	−0.0468342	+	0.0811193i	−0.888492	−	0.458892i	$-0.848247\pi$
$79$	−0.416272	+	0.721004i	−0.0468342	+	0.0811193i	0.841658	+	0.540011i	$0.181580\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	0.682087i		−	0.0748688i	−0.999299	−	0.0374344i	$-0.988081\pi$
$83$		−	0.682087i		−	0.0748688i	0.999299	−	0.0374344i	$-0.0119185\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.18151	−	8.97464i	0.549239	−	0.951310i	−0.449088	−	0.893487i	$-0.648251\pi$
$89$	5.18151	−	8.97464i	0.549239	−	0.951310i	0.998327	−	0.0578220i	$-0.0184156\pi$
$90$		0			0
$91$	−13.1201	−	9.47283i	−1.37536	−	0.993023i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	16.8308			1.70891			0.854456	−	0.519524i	$-0.173891\pi$
$97$	16.8308			1.70891			0.854456	+	0.519524i	$0.173891\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.15721	−	5.46846i	−0.314155	−	0.544132i	0.665103	−	0.746752i	$-0.268388\pi$
$101$	−3.15721	−	5.46846i	−0.314155	−	0.544132i	−0.979257	+	0.202620i	$0.935054\pi$
$102$		0			0
$103$	3.78865	−	6.56213i	0.373307	−	0.646586i	−0.616765	−	0.787147i	$-0.711557\pi$
$103$	3.78865	−	6.56213i	0.373307	−	0.646586i	0.990072	+	0.140561i	$0.0448906\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.81684	−	11.8071i	0.659009	−	1.14144i	−0.321864	−	0.946786i	$-0.604309\pi$
$107$	6.81684	−	11.8071i	0.659009	−	1.14144i	0.980873	−	0.194650i	$-0.0623572\pi$
$108$		0			0
$109$	−7.03461	−	12.1843i	−0.673793	−	1.16704i	−0.976820	−	0.214062i	$-0.931330\pi$
$109$	−7.03461	−	12.1843i	−0.673793	−	1.16704i	0.303027	−	0.952982i	$-0.402003\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.55242			−0.428255			−0.214128	−	0.976806i	$-0.568691\pi$
$113$	−4.55242			−0.428255			−0.214128	+	0.976806i	$0.568691\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	14.1739	+	1.44713i	1.29932	+	0.132658i
$120$		0			0
$121$	−4.19110	+	7.25920i	−0.381009	+	0.659927i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$			3.39541i			0.301294i	0.988588	+	0.150647i	$0.0481357\pi$
$127$			3.39541i			0.301294i	−0.988588	+	0.150647i	$0.951864\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.4721	+	18.1382i	−0.914952	+	1.58474i	−0.107979	+	0.994153i	$0.534438\pi$
$131$	−10.4721	+	18.1382i	−0.914952	+	1.58474i	−0.806972	+	0.590590i	$0.798895\pi$
$132$		0			0
$133$	−0.834394	−	1.85918i	−0.0723511	−	0.161211i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.29576	−	16.1007i	−0.794190	−	1.37558i	−0.923352	−	0.383955i	$-0.874562\pi$
$137$	−9.29576	−	16.1007i	−0.794190	−	1.37558i	0.129161	−	0.991624i	$-0.458771\pi$
$138$		0			0
$139$			13.0251i			1.10478i	0.833587	+	0.552388i	$0.186283\pi$
$139$			13.0251i			1.10478i	−0.833587	+	0.552388i	$0.813717\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.57027	+	4.94805i	−0.716682	+	0.413776i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.27299	−	5.35376i	−0.759673	−	0.438597i	0.0695052	−	0.997582i	$-0.477858\pi$
$149$	−9.27299	−	5.35376i	−0.759673	−	0.438597i	−0.829178	+	0.558984i	$0.811191\pi$
$150$		0			0
$151$	−5.18256	−	8.97645i	−0.421751	−	0.730493i	0.574360	−	0.818603i	$-0.305251\pi$
$151$	−5.18256	−	8.97645i	−0.421751	−	0.730493i	−0.996111	+	0.0881092i	$0.971918\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−7.49197	−	12.9765i	−0.597924	−	1.03564i	−0.993127	−	0.117042i	$-0.962659\pi$
$157$	−7.49197	−	12.9765i	−0.597924	−	1.03564i	0.395203	−	0.918594i	$-0.370674\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−9.01855	+	4.04750i	−0.710761	+	0.318988i
$162$		0			0
$163$	−3.81589	−	2.20310i	−0.298883	−	0.172560i	0.343058	−	0.939314i	$-0.388537\pi$
$163$	−3.81589	−	2.20310i	−0.298883	−	0.172560i	−0.641941	+	0.766754i	$0.721871\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	8.49296i		−	0.657205i	−0.944468	−	0.328603i	$-0.893422\pi$
$167$		−	8.49296i		−	0.657205i	0.944468	−	0.328603i	$-0.106578\pi$
$168$		0			0
$169$	24.4103			1.87772
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.26949	−	0.732940i	−0.0965175	−	0.0557244i	0.450964	−	0.892542i	$-0.351080\pi$
$173$	−1.26949	−	0.732940i	−0.0965175	−	0.0557244i	−0.547482	+	0.836818i	$0.684413\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.0846	+	7.55441i	−0.977991	+	0.564643i	−0.901663	−	0.432439i	$-0.857653\pi$
$179$	−13.0846	+	7.55441i	−0.977991	+	0.564643i	−0.0763281	+	0.997083i	$0.524320\pi$
$180$		0			0
$181$		−	20.1371i		−	1.49678i	−0.663261	−	0.748388i	$-0.730828\pi$
$181$		−	20.1371i		−	1.49678i	0.663261	−	0.748388i	$-0.269172\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	4.35644	−	7.54557i	0.318574	−	0.551787i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−14.2314	−	8.21652i	−1.02975	−	0.594527i	−0.112837	−	0.993613i	$-0.535994\pi$
$191$	−14.2314	−	8.21652i	−1.02975	−	0.594527i	−0.916913	+	0.399087i	$0.869327\pi$
$192$		0			0
$193$	21.5944	−	12.4675i	1.55440	−	0.897431i	0.556621	−	0.830767i	$-0.312098\pi$
$193$	21.5944	−	12.4675i	1.55440	−	0.897431i	0.997775	−	0.0666645i	$-0.0212357\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.28956			−0.376865			−0.188433	−	0.982086i	$-0.560341\pi$
$197$	−5.28956			−0.376865			−0.188433	+	0.982086i	$0.560341\pi$
$198$		0			0
$199$	−4.91550	+	2.83797i	−0.348451	+	0.201178i	−0.664003	−	0.747730i	$-0.731144\pi$
$199$	−4.91550	+	2.83797i	−0.348451	+	0.201178i	0.315552	+	0.948908i	$0.397810\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−8.70853	+	12.0616i	−0.611219	+	0.846555i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.24620			−0.0862013
$210$		0			0
$211$	0.400950			0.0276025			0.0138013	−	0.999905i	$-0.495607\pi$
$211$	0.400950			0.0276025			0.0138013	+	0.999905i	$0.495607\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.612668	−	0.0625521i	−0.0415906	−	0.00424631i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−28.5246	+	16.4687i	−1.91877	+	1.10780i
$222$		0			0
$223$	−3.55830			−0.238281			−0.119141	−	0.992877i	$-0.538014\pi$
$223$	−3.55830			−0.238281			−0.119141	+	0.992877i	$0.538014\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−23.3751	+	13.4956i	−1.55146	+	0.895736i	−0.553437	+	0.832891i	$0.686684\pi$
$227$	−23.3751	+	13.4956i	−1.55146	+	0.895736i	−0.998023	+	0.0628447i	$0.979983\pi$
$228$		0			0
$229$	0.914543	+	0.528011i	0.0604347	+	0.0348920i	0.529913	−	0.848052i	$-0.322225\pi$
$229$	0.914543	+	0.528011i	0.0604347	+	0.0348920i	−0.469478	+	0.882944i	$0.655558\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.53695	−	6.12617i	0.231713	−	0.401339i	−0.726599	−	0.687062i	$-0.758900\pi$
$233$	3.53695	−	6.12617i	0.231713	−	0.401339i	0.958312	+	0.285723i	$0.0922336\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	18.3849i		−	1.18922i	−0.804013	−	0.594612i	$-0.797306\pi$
$239$		−	18.3849i		−	1.18922i	0.804013	−	0.594612i	$-0.202694\pi$
$240$		0			0
$241$	1.41894	−	0.819225i	0.0914019	−	0.0527709i	−0.453602	−	0.891204i	$-0.649861\pi$
$241$	1.41894	−	0.819225i	0.0914019	−	0.0527709i	0.545004	+	0.838433i	$0.316528\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	4.07986	+	2.35551i	0.259595	+	0.149877i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−11.4769			−0.724418			−0.362209	−	0.932097i	$-0.617977\pi$
$251$	−11.4769			−0.724418			−0.362209	+	0.932097i	$0.617977\pi$
$252$		0			0
$253$			6.04509i			0.380052i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−20.3310	−	11.7381i	−1.26821	−	0.732203i	−0.293563	−	0.955940i	$-0.594841\pi$
$257$	−20.3310	−	11.7381i	−1.26821	−	0.732203i	−0.974649	+	0.223737i	$0.928174\pi$
$258$		0			0
$259$	0.514902	+	1.14729i	0.0319945	+	0.0712893i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−10.0368	−	17.3843i	−0.618897	−	1.07196i	−0.989687	−	0.143244i	$-0.954247\pi$
$263$	−10.0368	−	17.3843i	−0.618897	−	1.07196i	0.370791	−	0.928716i	$-0.379087\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.16917	−	15.8815i	−0.559054	−	0.968311i	−0.997576	−	0.0695898i	$-0.977831\pi$
$269$	−9.16917	−	15.8815i	−0.559054	−	0.968311i	0.438521	−	0.898721i	$-0.355502\pi$
$270$		0			0
$271$	−18.6007	−	10.7391i	−1.12991	−	0.652356i	−0.186001	−	0.982550i	$-0.559553\pi$
$271$	−18.6007	−	10.7391i	−1.12991	−	0.652356i	−0.943913	+	0.330193i	$0.892886\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	7.15706	−	4.13213i	0.430026	−	0.248276i	−0.269332	−	0.963047i	$-0.586803\pi$
$277$	7.15706	−	4.13213i	0.430026	−	0.248276i	0.699358	+	0.714772i	$0.253469\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			19.9460i			1.18988i	0.803771	+	0.594939i	$0.202824\pi$
$281$			19.9460i			1.18988i	−0.803771	+	0.594939i	$0.797176\pi$
$282$		0			0
$283$	−0.429791	−	0.744420i	−0.0255484	−	0.0442512i	0.852969	−	0.521962i	$-0.174800\pi$
$283$	−0.429791	−	0.744420i	−0.0255484	−	0.0442512i	−0.878517	+	0.477711i	$0.841467\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.46313	+	1.05639i	0.0863657	+	0.0623567i
$288$		0			0
$289$	5.99964	−	10.3917i	0.352920	−	0.611275i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	14.0627i		−	0.821549i	−0.911737	−	0.410775i	$-0.865258\pi$
$293$		−	14.0627i		−	0.821549i	0.911737	−	0.410775i	$-0.134742\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	11.4262	−	19.7907i	0.660793	−	1.14453i
$300$		0			0
$301$	5.64046	−	7.81219i	0.325111	−	0.450287i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	13.1301			0.749377			0.374689	−	0.927151i	$-0.377750\pi$
$307$	13.1301			0.749377			0.374689	+	0.927151i	$0.377750\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6.85906	−	11.8802i	−0.388942	−	0.673667i	0.603366	−	0.797465i	$-0.293826\pi$
$311$	−6.85906	−	11.8802i	−0.388942	−	0.673667i	−0.992307	+	0.123798i	$0.960493\pi$
$312$		0			0
$313$	−4.12696	+	7.14811i	−0.233270	+	0.404035i	−0.958768	−	0.284189i	$-0.908276\pi$
$313$	−4.12696	+	7.14811i	−0.233270	+	0.404035i	0.725499	+	0.688224i	$0.241609\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.15857	+	2.00670i	−0.0650717	+	0.112707i	−0.896726	−	0.442587i	$-0.854061\pi$
$317$	−1.15857	+	2.00670i	−0.0650717	+	0.112707i	0.831654	+	0.555294i	$0.187394\pi$
$318$		0			0
$319$	4.54882	+	7.87879i	0.254685	+	0.441127i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.14775			−0.230787
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.21586	+	9.39369i	0.232428	+	0.517891i
$330$		0			0
$331$	9.65316	−	16.7198i	0.530585	−	0.919001i	−0.468778	−	0.883316i	$-0.655305\pi$
$331$	9.65316	−	16.7198i	0.530585	−	0.919001i	0.999363	−	0.0356847i	$-0.0113612\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	21.8405i		−	1.18973i	−0.803826	−	0.594865i	$-0.797205\pi$
$337$		−	21.8405i		−	1.18973i	0.803826	−	0.594865i	$-0.202795\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.188307	+	0.326157i	−0.0101974	+	0.0176624i
$342$		0			0
$343$	5.56568	−	17.6642i	0.300518	−	0.953776i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.85138	+	11.8669i	0.367801	+	0.637050i	0.989221	−	0.146427i	$-0.0467773\pi$
$347$	6.85138	+	11.8669i	0.367801	+	0.637050i	−0.621420	+	0.783477i	$0.713444\pi$
$348$		0			0
$349$		−	5.28656i		−	0.282983i	−0.989939	−	0.141491i	$-0.954810\pi$
$349$		−	5.28656i		−	0.282983i	0.989939	−	0.141491i	$-0.0451898\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.58179	−	2.06795i	0.190640	−	0.110066i	−0.401642	−	0.915797i	$-0.631561\pi$
$353$	3.58179	−	2.06795i	0.190640	−	0.110066i	0.592282	+	0.805731i	$0.298227\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	26.7125	+	15.4225i	1.40983	+	0.813967i	0.995372	−	0.0961010i	$-0.0306372\pi$
$359$	26.7125	+	15.4225i	1.40983	+	0.813967i	0.414460	+	0.910068i	$0.363971\pi$
$360$		0			0
$361$	−9.20337	−	15.9407i	−0.484388	−	0.838985i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−2.78235	−	4.81917i	−0.145237	−	0.251559i	0.784224	−	0.620478i	$-0.213061\pi$
$367$	−2.78235	−	4.81917i	−0.145237	−	0.251559i	−0.929462	+	0.368919i	$0.879728\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.45686	+	33.8583i	−0.179471	+	1.75784i
$372$		0			0
$373$	9.05036	+	5.22523i	0.468610	+	0.270552i	0.715658	−	0.698451i	$-0.246127\pi$
$373$	9.05036	+	5.22523i	0.468610	+	0.270552i	−0.247048	+	0.969003i	$0.579460\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	34.3919i		−	1.77127i
$378$		0			0
$379$	−22.4235			−1.15182			−0.575910	−	0.817513i	$-0.695352\pi$
$379$	−22.4235			−1.15182			−0.575910	+	0.817513i	$0.695352\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−24.3597	−	14.0641i	−1.24472	−	0.718642i	−0.274672	−	0.961538i	$-0.588569\pi$
$383$	−24.3597	−	14.0641i	−1.24472	−	0.718642i	−0.970052	+	0.242896i	$0.921903\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.65189	+	2.10842i	−0.185158	+	0.106901i	−0.589714	−	0.807612i	$-0.700759\pi$
$389$	−3.65189	+	2.10842i	−0.185158	+	0.106901i	0.404556	+	0.914513i	$0.367426\pi$
$390$		0			0
$391$			20.1200i			1.01751i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.757060	+	1.31127i	−0.0379957	+	0.0658106i	−0.884398	−	0.466734i	$-0.845431\pi$
$397$	−0.757060	+	1.31127i	−0.0379957	+	0.0658106i	0.846402	+	0.532544i	$0.178764\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	28.8743	+	16.6706i	1.44191	+	0.832490i	0.997978	−	0.0635679i	$-0.0202479\pi$
$401$	28.8743	+	16.6706i	1.44191	+	0.832490i	0.443937	+	0.896058i	$0.353581\pi$
$402$		0			0
$403$	1.23297	−	0.711858i	0.0614188	−	0.0354602i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.769025			0.0381191
$408$		0			0
$409$	−2.82101	+	1.62871i	−0.139490	+	0.0805346i	−0.568121	−	0.822945i	$-0.692329\pi$
$409$	−2.82101	+	1.62871i	−0.139490	+	0.0805346i	0.428631	+	0.903480i	$0.358996\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.90910	−	28.4933i	0.143147	−	1.40206i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	38.9059			1.90068			0.950340	−	0.311215i	$-0.100736\pi$
$419$	38.9059			1.90068			0.950340	+	0.311215i	$0.100736\pi$
$420$		0			0
$421$	−37.7183			−1.83828			−0.919139	−	0.393934i	$-0.871114\pi$
$421$	−37.7183			−1.83828			−0.919139	+	0.393934i	$0.871114\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4.08579	+	9.10385i	0.197725	+	0.440566i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11.1785	+	6.45391i	−0.538449	+	0.310874i	−0.744450	−	0.667678i	$-0.767288\pi$
$431$	−11.1785	+	6.45391i	−0.538449	+	0.310874i	0.206001	+	0.978552i	$0.433955\pi$
$432$		0			0
$433$	5.59318			0.268791			0.134396	−	0.990928i	$-0.457091\pi$
$433$	5.59318			0.268791			0.134396	+	0.990928i	$0.457091\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.49221	−	1.43888i	0.119219	−	0.0688309i
$438$		0			0
$439$	31.5177	+	18.1968i	1.50426	+	0.868484i	0.999988	+	0.00493767i	$0.00157172\pi$
$439$	31.5177	+	18.1968i	1.50426	+	0.868484i	0.504270	+	0.863546i	$0.331762\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.20695	+	9.01870i	−0.247390	+	0.428491i	−0.962801	−	0.270212i	$-0.912906\pi$
$443$	−5.20695	+	9.01870i	−0.247390	+	0.428491i	0.715411	+	0.698704i	$0.246239\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			2.28525i			0.107848i	0.998545	+	0.0539239i	$0.0171728\pi$
$449$			2.28525i			0.107848i	−0.998545	+	0.0539239i	$0.982827\pi$
$450$		0			0
$451$	0.955737	−	0.551795i	0.0450039	−	0.0259830i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	28.4192	+	16.4078i	1.32940	+	0.767527i	0.985206	−	0.171373i	$-0.0548205\pi$
$457$	28.4192	+	16.4078i	1.32940	+	0.767527i	0.344189	+	0.938900i	$0.388154\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.56922			−0.212810			−0.106405	−	0.994323i	$-0.533934\pi$
$461$	−4.56922			−0.212810			−0.106405	+	0.994323i	$0.533934\pi$
$462$		0			0
$463$		−	29.2197i		−	1.35795i	−0.734159	−	0.678977i	$-0.762423\pi$
$463$		−	29.2197i		−	1.35795i	0.734159	−	0.678977i	$-0.237577\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−14.4431	−	8.33872i	−0.668347	−	0.385870i	0.127103	−	0.991889i	$-0.459432\pi$
$467$	−14.4431	−	8.33872i	−0.668347	−	0.385870i	−0.795450	+	0.606019i	$0.792765\pi$
$468$		0			0
$469$	−8.27157	−	0.844510i	−0.381946	−	0.0389958i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.94624	−	5.10304i	−0.135468	−	0.234638i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.36679	−	4.09940i	−0.108141	−	0.187306i	0.806876	−	0.590721i	$-0.201157\pi$
$479$	−2.36679	−	4.09940i	−0.108141	−	0.187306i	−0.915017	+	0.403415i	$0.867823\pi$
$480$		0			0
$481$	−2.51767	−	1.45358i	−0.114796	−	0.0662774i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−34.7408	+	20.0576i	−1.57426	+	0.908898i	−0.578620	+	0.815597i	$0.696409\pi$
$487$	−34.7408	+	20.0576i	−1.57426	+	0.908898i	−0.995638	+	0.0933009i	$0.970258\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	17.9082i		−	0.808186i	−0.914718	−	0.404093i	$-0.867587\pi$
$491$		−	17.9082i		−	0.808186i	0.914718	−	0.404093i	$-0.132413\pi$
$492$		0			0
$493$	15.1400	+	26.2232i	0.681869	+	1.18103i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	21.5457	−	29.8413i	0.966456	−	1.33857i
$498$		0			0
$499$	−15.1488	+	26.2384i	−0.678152	+	1.17459i	0.297385	+	0.954758i	$0.403885\pi$
$499$	−15.1488	+	26.2384i	−0.678152	+	1.17459i	−0.975537	+	0.219836i	$0.929448\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	0.181490i		−	0.00809225i	−0.999992	−	0.00404613i	$-0.998712\pi$
$503$		−	0.181490i		−	0.00809225i	0.999992	−	0.00404613i	$-0.00128793\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−16.0302	+	27.7651i	−0.710525	+	1.23067i	0.254136	+	0.967169i	$0.418209\pi$
$509$	−16.0302	+	27.7651i	−0.710525	+	1.23067i	−0.964660	+	0.263496i	$0.915124\pi$
$510$		0			0
$511$	−1.99506	+	19.5407i	−0.0882564	+	0.864429i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	6.29655			0.276922
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10.8863	+	18.8557i	0.476939	+	0.826083i	0.999651	−	0.0264264i	$-0.00841276\pi$
$521$	10.8863	+	18.8557i	0.476939	+	0.826083i	−0.522711	+	0.852510i	$0.675079\pi$
$522$		0			0
$523$	10.9494	−	18.9650i	0.478785	−	0.829280i	−0.520919	−	0.853606i	$-0.674411\pi$
$523$	10.9494	−	18.9650i	0.478785	−	0.829280i	0.999704	+	0.0243264i	$0.00774409\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.626746	+	1.08556i	−0.0273015	+	0.0472875i
$528$		0			0
$529$	4.52025	+	7.82930i	0.196533	+	0.340404i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.17192			−0.180706
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.46159	−	7.52819i	−0.364466	−	0.324262i
$540$		0			0
$541$	−9.53205	+	16.5100i	−0.409815	+	0.709820i	−0.994869	−	0.101174i	$-0.967740\pi$
$541$	−9.53205	+	16.5100i	−0.409815	+	0.709820i	0.585054	+	0.810994i	$0.301073\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	26.7493i		−	1.14372i	−0.820352	−	0.571859i	$-0.806222\pi$
$547$		−	26.7493i		−	1.14372i	0.820352	−	0.571859i	$-0.193778\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.16546	−	3.75068i	0.0922517	−	0.159785i
$552$		0			0
$553$	2.00960	−	0.901902i	0.0854567	−	0.0383528i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.0561	+	27.8099i	0.680318	+	1.17834i	0.974884	+	0.222714i	$0.0714916\pi$
$557$	16.0561	+	27.8099i	0.680318	+	1.17834i	−0.294566	+	0.955631i	$0.595175\pi$
$558$		0			0
$559$			22.2754i			0.942151i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	25.2830	−	14.5972i	1.06555	−	0.615197i	0.138590	−	0.990350i	$-0.455743\pi$
$563$	25.2830	−	14.5972i	1.06555	−	0.615197i	0.926963	+	0.375153i	$0.122410\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.84837	−	1.06716i	−0.0774876	−	0.0447375i	0.460756	−	0.887527i	$-0.347578\pi$
$569$	−1.84837	−	1.06716i	−0.0774876	−	0.0447375i	−0.538243	+	0.842790i	$0.680912\pi$
$570$		0			0
$571$	−15.7635	−	27.3032i	−0.659682	−	1.14260i	−0.980698	−	0.195529i	$-0.937358\pi$
$571$	−15.7635	−	27.3032i	−0.659682	−	1.14260i	0.321016	−	0.947074i	$-0.395976\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−10.0048	−	17.3288i	−0.416504	−	0.721406i	0.579081	−	0.815270i	$-0.303411\pi$
$577$	−10.0048	−	17.3288i	−0.416504	−	0.721406i	−0.995585	+	0.0938639i	$0.970078\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.05639	+	1.46313i	−0.0438264	+	0.0607008i
$582$		0			0
$583$	18.0247	+	10.4065i	0.746505	+	0.430995i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	5.77666i		−	0.238428i	−0.992869	−	0.119214i	$-0.961963\pi$
$587$		−	5.77666i		−	0.238428i	0.992869	−	0.119214i	$-0.0380375\pi$
$588$		0			0
$589$	0.179286			0.00738736
$590$		0			0
$591$		0			0
$592$		0			0
$593$	11.2417	+	6.49041i	0.461642	+	0.266529i	0.712734	−	0.701434i	$-0.247457\pi$
$593$	11.2417	+	6.49041i	0.461642	+	0.266529i	−0.251093	+	0.967963i	$0.580790\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	26.0753	−	15.0546i	1.06541	−	0.615113i	0.138484	−	0.990365i	$-0.455777\pi$
$599$	26.0753	−	15.0546i	1.06541	−	0.615113i	0.926923	+	0.375251i	$0.122444\pi$
$600$		0			0
$601$			31.0797i			1.26777i	0.773428	+	0.633884i	$0.218540\pi$
$601$			31.0797i			1.26777i	−0.773428	+	0.633884i	$0.781460\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	20.1315	−	34.8689i	0.817114	−	1.41528i	−0.0906858	−	0.995880i	$-0.528906\pi$
$607$	20.1315	−	34.8689i	0.817114	−	1.41528i	0.907800	−	0.419404i	$-0.137761\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−20.6139	−	11.9015i	−0.833950	−	0.481482i
$612$		0			0
$613$	13.4523	−	7.76670i	0.543334	−	0.313694i	−0.203095	−	0.979159i	$-0.565100\pi$
$613$	13.4523	−	7.76670i	0.543334	−	0.313694i	0.746429	+	0.665465i	$0.231767\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−24.1475			−0.972144			−0.486072	−	0.873919i	$-0.661571\pi$
$617$	−24.1475			−0.972144			−0.486072	+	0.873919i	$0.661571\pi$
$618$		0			0
$619$	42.7110	−	24.6592i	1.71670	−	0.991138i	0.791930	−	0.610611i	$-0.209076\pi$
$619$	42.7110	−	24.6592i	1.71670	−	0.991138i	0.924770	−	0.380526i	$-0.124257\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−25.0143	+	11.2263i	−1.00218	+	0.449774i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.55956			0.102057
$630$		0			0
$631$	25.7293			1.02427			0.512134	−	0.858906i	$-0.328855\pi$
$631$	25.7293			1.02427			0.512134	+	0.858906i	$0.328855\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	13.4725	+	40.6399i	0.533800	+	1.61021i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−28.1334	+	16.2428i	−1.11120	+	0.641554i	−0.939141	−	0.343532i	$-0.888377\pi$
$641$	−28.1334	+	16.2428i	−1.11120	+	0.641554i	−0.172063	+	0.985086i	$0.555043\pi$
$642$		0			0
$643$	−29.7863			−1.17466			−0.587328	−	0.809349i	$-0.699820\pi$
$643$	−29.7863			−1.17466			−0.587328	+	0.809349i	$0.699820\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−28.2328	+	16.3002i	−1.10995	+	0.640828i	−0.938815	−	0.344420i	$-0.888075\pi$
$647$	−28.2328	+	16.3002i	−1.10995	+	0.640828i	−0.171131	+	0.985248i	$0.554742\pi$
$648$		0			0
$649$	−15.1685	−	8.75755i	−0.595417	−	0.343764i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.666558	+	1.15451i	−0.0260844	+	0.0451795i	−0.878773	−	0.477240i	$-0.841637\pi$
$653$	−0.666558	+	1.15451i	−0.0260844	+	0.0451795i	0.852689	+	0.522420i	$0.174971\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			42.8917i			1.67082i	0.549626	+	0.835411i	$0.314770\pi$
$659$			42.8917i			1.67082i	−0.549626	+	0.835411i	$0.685230\pi$
$660$		0			0
$661$	23.7428	−	13.7079i	0.923486	−	0.533175i	0.0387407	−	0.999249i	$-0.487665\pi$
$661$	23.7428	−	13.7079i	0.923486	−	0.533175i	0.884746	+	0.466074i	$0.154332\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−18.1939	−	10.5043i	−0.704472	−	0.406727i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6.10227			0.235576
$672$		0			0
$673$		−	18.1821i		−	0.700870i	−0.936587	−	0.350435i	$-0.886034\pi$
$673$		−	18.1821i		−	0.700870i	0.936587	−	0.350435i	$-0.113966\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.58420	+	1.49199i	0.0993190	+	0.0573419i	0.548837	−	0.835930i	$-0.315071\pi$
$677$	2.58420	+	1.49199i	0.0993190	+	0.0573419i	−0.449518	+	0.893271i	$0.648404\pi$
$678$		0			0
$679$	−36.1034	−	26.0669i	−1.38552	−	1.00036i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.1604	+	19.3304i	0.427041	+	0.739657i	0.996609	−	0.0822877i	$-0.0262226\pi$
$683$	11.1604	+	19.3304i	0.427041	+	0.739657i	−0.569568	+	0.821945i	$0.692889\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−39.3400	−	68.1388i	−1.49873	−	2.59588i
$690$		0			0
$691$	8.89912	+	5.13791i	0.338539	+	0.195455i	0.659626	−	0.751594i	$-0.270715\pi$
$691$	8.89912	+	5.13791i	0.338539	+	0.195455i	−0.321087	+	0.947050i	$0.604048\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	3.18100	−	1.83655i	0.120489	−	0.0695644i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	33.7536i		−	1.27486i	−0.770510	−	0.637428i	$-0.779999\pi$
$701$		−	33.7536i		−	1.27486i	0.770510	−	0.637428i	$-0.220001\pi$
$702$		0			0
$703$	−0.183047	−	0.317046i	−0.00690373	−	0.0119576i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.69687	+	16.6200i	−0.0638173	+	0.625060i
$708$		0			0
$709$	4.03582	−	6.99025i	0.151568	−	0.262524i	−0.780236	−	0.625485i	$-0.784901\pi$
$709$	4.03582	−	6.99025i	0.151568	−	0.262524i	0.931804	+	0.362961i	$0.118234\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	0.869686i		−	0.0325700i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	11.6144	−	20.1168i	0.433145	−	0.750229i	−0.563997	−	0.825777i	$-0.690737\pi$
$719$	11.6144	−	20.1168i	0.433145	−	0.750229i	0.997142	+	0.0755475i	$0.0240704\pi$
$720$		0			0
$721$	−18.2901	+	8.20855i	−0.681159	+	0.305702i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−30.7743			−1.14136			−0.570678	−	0.821174i	$-0.693320\pi$
$727$	−30.7743			−1.14136			−0.570678	+	0.821174i	$0.693320\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9.80605	−	16.9846i	−0.362690	−	0.628197i
$732$		0			0
$733$	−6.92980	+	12.0028i	−0.255958	+	0.443332i	−0.965155	−	0.261678i	$-0.915724\pi$
$733$	−6.92980	+	12.0028i	−0.255958	+	0.443332i	0.709197	+	0.705010i	$0.249058\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.54231	+	4.40341i	−0.0936472	+	0.162202i
$738$		0			0
$739$	0.182348	+	0.315835i	0.00670777	+	0.0116182i	0.869360	−	0.494180i	$-0.164532\pi$
$739$	0.182348	+	0.315835i	0.00670777	+	0.0116182i	−0.862652	+	0.505798i	$0.831198\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.3910			0.674701			0.337351	−	0.941379i	$-0.390469\pi$
$743$	18.3910			0.674701			0.337351	+	0.941379i	$0.390469\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−32.9090	+	14.7695i	−1.20247	+	0.539665i
$750$		0			0
$751$	22.2501	−	38.5383i	0.811918	−	1.40628i	−0.0996021	−	0.995027i	$-0.531757\pi$
$751$	22.2501	−	38.5383i	0.811918	−	1.40628i	0.911520	−	0.411256i	$-0.134910\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			24.4357i			0.888130i	0.895994	+	0.444065i	$0.146464\pi$
$757$			24.4357i			0.888130i	−0.895994	+	0.444065i	$0.853536\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	13.2293	−	22.9138i	0.479561	−	0.830624i	−0.520164	−	0.854066i	$-0.674129\pi$
$761$	13.2293	−	22.9138i	0.479561	−	0.830624i	0.999725	+	0.0234421i	$0.00746255\pi$
$762$		0			0
$763$	−3.78080	+	37.0311i	−0.136874	+	1.34062i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	33.1063	+	57.3418i	1.19540	+	2.07049i
$768$		0			0
$769$		−	10.9377i		−	0.394424i	−0.980361	−	0.197212i	$-0.936811\pi$
$769$		−	10.9377i		−	0.394424i	0.980361	−	0.197212i	$-0.0631887\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	14.0944	−	8.13738i	0.506939	−	0.292681i	−0.224636	−	0.974443i	$-0.572119\pi$
$773$	14.0944	−	8.13738i	0.506939	−	0.292681i	0.731574	+	0.681762i	$0.238786\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.454977	−	0.262681i	−0.0163012	−	0.00941153i
$780$		0			0
$781$	−11.2542	−	19.4928i	−0.402706	−	0.697508i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−6.47320	−	11.2119i	−0.230745	−	0.399661i	0.727283	−	0.686338i	$-0.240783\pi$
$787$	−6.47320	−	11.2119i	−0.230745	−	0.399661i	−0.958027	+	0.286676i	$0.907450\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9.76527	+	7.05060i	0.347213	+	0.250691i
$792$		0			0
$793$	−19.9779	−	11.5342i	−0.709436	−	0.409593i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	48.9463i		−	1.73377i	−0.498510	−	0.866884i	$-0.666119\pi$
$797$		−	48.9463i		−	1.73377i	0.498510	−	0.866884i	$-0.333881\pi$
$798$		0			0
$799$	20.9570			0.741404
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10.4026	+	6.00594i	0.367099	+	0.211945i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−23.4901	+	13.5620i	−0.825869	+	0.476816i	−0.852436	−	0.522831i	$-0.824876\pi$
$809$	−23.4901	+	13.5620i	−0.825869	+	0.476816i	0.0265672	+	0.999647i	$0.491542\pi$
$810$		0			0
$811$		−	23.7669i		−	0.834568i	−0.908776	−	0.417284i	$-0.862982\pi$
$811$		−	23.7669i		−	0.834568i	0.908776	−	0.417284i	$-0.137018\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.40255	+	2.42930i	−0.0490692	+	0.0849903i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	21.7449	+	12.5544i	0.758903	+	0.438153i	0.828902	−	0.559394i	$-0.188966\pi$
$821$	21.7449	+	12.5544i	0.758903	+	0.438153i	−0.0699988	+	0.997547i	$0.522300\pi$
$822$		0			0
$823$	−46.9627	+	27.1139i	−1.63702	+	0.945132i	−0.655163	+	0.755488i	$0.727400\pi$
$823$	−46.9627	+	27.1139i	−1.63702	+	0.945132i	−0.981853	+	0.189644i	$0.939267\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−33.6990			−1.17183			−0.585914	−	0.810373i	$-0.699265\pi$
$827$	−33.6990			−1.17183			−0.585914	+	0.810373i	$0.699265\pi$
$828$		0			0
$829$	35.0353	−	20.2276i	1.21683	−	0.702535i	0.252588	−	0.967574i	$-0.418718\pi$
$829$	35.0353	−	20.2276i	1.21683	−	0.702535i	0.964238	+	0.265039i	$0.0853848\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−28.1629	−	25.0562i	−0.975787	−	0.868147i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12.6553			0.436909			0.218455	−	0.975847i	$-0.429898\pi$
$839$	12.6553			0.436909			0.218455	+	0.975847i	$0.429898\pi$
$840$		0			0
$841$	−2.61708			−0.0902441
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	20.2330	−	9.08051i	0.695214	−	0.312010i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.53794	+	0.887927i	−0.0527197	+	0.0304378i
$852$		0			0
$853$	−21.0912			−0.722150			−0.361075	−	0.932537i	$-0.617590\pi$
$853$	−21.0912			−0.722150			−0.361075	+	0.932537i	$0.617590\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.76402	−	1.59581i	0.0944173	−	0.0545118i	−0.452048	−	0.891994i	$-0.649306\pi$
$857$	2.76402	−	1.59581i	0.0944173	−	0.0545118i	0.546465	+	0.837482i	$0.315973\pi$
$858$		0			0
$859$	−20.7568	−	11.9839i	−0.708212	−	0.408886i	0.102187	−	0.994765i	$-0.467416\pi$
$859$	−20.7568	−	11.9839i	−0.708212	−	0.408886i	−0.810399	+	0.585879i	$0.800749\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−22.4082	+	38.8121i	−0.762784	+	1.32118i	0.178626	+	0.983917i	$0.442835\pi$
$863$	−22.4082	+	38.8121i	−0.762784	+	1.32118i	−0.941410	+	0.337263i	$0.890499\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1.34702i		−	0.0456946i
$870$		0			0
$871$	16.6463	−	9.61073i	0.564038	−	0.325647i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−10.7665	−	6.21606i	−0.363560	−	0.209901i	0.307081	−	0.951683i	$-0.400648\pi$
$877$	−10.7665	−	6.21606i	−0.363560	−	0.209901i	−0.670641	+	0.741782i	$0.733981\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.2360			1.01868			0.509338	−	0.860566i	$-0.329890\pi$
$881$	30.2360			1.01868			0.509338	+	0.860566i	$0.329890\pi$
$882$		0			0
$883$			25.6124i			0.861925i	0.902370	+	0.430963i	$0.141826\pi$
$883$			25.6124i			0.861925i	−0.902370	+	0.430963i	$0.858174\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.9156	−	6.30210i	−0.366508	−	0.211604i	0.305424	−	0.952217i	$-0.401202\pi$
$887$	−10.9156	−	6.30210i	−0.366508	−	0.211604i	−0.671932	+	0.740613i	$0.734535\pi$
$888$		0			0
$889$	5.25868	−	7.28341i	0.176371	−	0.244278i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.49873	−	2.59588i	−0.0501531	−	0.0868677i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.654423	−	1.13349i	−0.0218262	−	0.0378041i
$900$		0			0
$901$	59.9919	+	34.6363i	1.99862	+	1.15390i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−37.7588	+	21.8001i	−1.25376	+	0.723859i	−0.971854	−	0.235582i	$-0.924300\pi$
$907$	−37.7588	+	21.8001i	−1.25376	+	0.723859i	−0.281907	+	0.959442i	$0.590967\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	23.8501i		−	0.790188i	−0.918641	−	0.395094i	$-0.870712\pi$
$911$		−	23.8501i		−	0.790188i	0.918641	−	0.395094i	$-0.129288\pi$
$912$		0			0
$913$	0.551795	+	0.955737i	0.0182618	+	0.0316303i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	50.5552	−	22.6890i	1.66948	−	0.749258i
$918$		0			0
$919$	−19.5865	+	33.9249i	−0.646100	+	1.11908i	0.337946	+	0.941165i	$0.390268\pi$
$919$	−19.5865	+	33.9249i	−0.646100	+	1.11908i	−0.984046	+	0.177913i	$0.943066\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			85.0887i			2.80073i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.21214	−	2.09950i	0.0397692	−	0.0688822i	−0.845456	−	0.534046i	$-0.820671\pi$
$929$	1.21214	−	2.09950i	0.0397692	−	0.0688822i	0.885225	+	0.465163i	$0.154004\pi$
$930$		0			0
$931$	−1.08958	+	5.28035i	−0.0357096	+	0.173056i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	38.5715			1.26007			0.630037	−	0.776565i	$-0.283040\pi$
$937$	38.5715			1.26007			0.630037	+	0.776565i	$0.283040\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12.6354	−	21.8852i	−0.411904	−	0.713438i	0.583194	−	0.812333i	$-0.301803\pi$
$941$	−12.6354	−	21.8852i	−0.411904	−	0.713438i	−0.995098	+	0.0988946i	$0.968469\pi$
$942$		0			0
$943$	−1.27422	+	2.20702i	−0.0414944	+	0.0718703i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	7.95004	−	13.7699i	0.258341	−	0.447461i	−0.707456	−	0.706757i	$-0.750157\pi$
$947$	7.95004	−	13.7699i	0.258341	−	0.447461i	0.965798	+	0.259297i	$0.0834907\pi$
$948$		0			0
$949$	−22.7043	−	39.3250i	−0.737014	−	1.27654i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−21.1062			−0.683696			−0.341848	−	0.939755i	$-0.611053\pi$
$953$	−21.1062			−0.683696			−0.341848	+	0.939755i	$0.611053\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.99607	+	48.9342i	−0.161332	+	1.58017i
$960$		0			0
$961$	−15.4729	+	26.7999i	−0.499126	+	0.864512i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			5.22722i			0.168096i	0.996462	+	0.0840480i	$0.0267849\pi$
$967$			5.22722i			0.168096i	−0.996462	+	0.0840480i	$0.973215\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−5.03879	+	8.72744i	−0.161702	+	0.280077i	−0.935479	−	0.353381i	$-0.885032\pi$
$971$	−5.03879	+	8.72744i	−0.161702	+	0.280077i	0.773777	+	0.633458i	$0.218365\pi$
$972$		0			0
$973$	20.1728	−	27.9398i	0.646709	−	0.895710i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	28.4901	+	49.3463i	0.911479	+	1.57873i	0.811976	+	0.583691i	$0.198392\pi$
$977$	28.4901	+	49.3463i	0.911479	+	1.57873i	0.0995039	+	0.995037i	$0.468274\pi$
$978$		0			0
$979$			16.7670i			0.535874i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	36.7314	−	21.2069i	1.17155	−	0.676394i	0.217505	−	0.976059i	$-0.430208\pi$
$983$	36.7314	−	21.2069i	1.17155	−	0.676394i	0.954044	+	0.299665i	$0.0968749\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	11.7841	+	6.80355i	0.374712	+	0.216340i
$990$		0			0
$991$	15.0586	+	26.0822i	0.478352	+	0.828529i	0.999692	−	0.0248194i	$-0.00790108\pi$
$991$	15.0586	+	26.0822i	0.478352	+	0.828529i	−0.521340	+	0.853349i	$0.674568\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	21.5074	+	37.2519i	0.681146	+	1.17978i	0.974631	+	0.223816i	$0.0718514\pi$
$997$	21.5074	+	37.2519i	0.681146	+	1.17978i	−0.293486	+	0.955963i	$0.594815\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.dd.d.1349.3		40
3.2	odd	2	inner	6300.2.dd.d.1349.4		40
5.2	odd	4		6300.2.ch.d.1601.7	✓	20
5.3	odd	4		6300.2.ch.e.1601.3	yes	20
5.4	even	2	inner	6300.2.dd.d.1349.17		40
7.3	odd	6	inner	6300.2.dd.d.4049.18		40
15.2	even	4		6300.2.ch.d.1601.8	yes	20
15.8	even	4		6300.2.ch.e.1601.4	yes	20
15.14	odd	2	inner	6300.2.dd.d.1349.18		40
21.17	even	6	inner	6300.2.dd.d.4049.17		40
35.3	even	12		6300.2.ch.e.4301.4	yes	20
35.17	even	12		6300.2.ch.d.4301.8	yes	20
35.24	odd	6	inner	6300.2.dd.d.4049.4		40
105.17	odd	12		6300.2.ch.d.4301.7	yes	20
105.38	odd	12		6300.2.ch.e.4301.3	yes	20
105.59	even	6	inner	6300.2.dd.d.4049.3		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6300.2.ch.d.1601.7	✓	20	5.2	odd	4
6300.2.ch.d.1601.8	yes	20	15.2	even	4
6300.2.ch.d.4301.7	yes	20	105.17	odd	12
6300.2.ch.d.4301.8	yes	20	35.17	even	12
6300.2.ch.e.1601.3	yes	20	5.3	odd	4
6300.2.ch.e.1601.4	yes	20	15.8	even	4
6300.2.ch.e.4301.3	yes	20	105.38	odd	12
6300.2.ch.e.4301.4	yes	20	35.3	even	12
6300.2.dd.d.1349.3		40	1.1	even	1	trivial
6300.2.dd.d.1349.4		40	3.2	odd	2	inner
6300.2.dd.d.1349.17		40	5.4	even	2	inner
6300.2.dd.d.1349.18		40	15.14	odd	2	inner
6300.2.dd.d.4049.3		40	105.59	even	6	inner
6300.2.dd.d.4049.4		40	35.24	odd	6	inner
6300.2.dd.d.4049.17		40	21.17	even	6	inner
6300.2.dd.d.4049.18		40	7.3	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-2.14507 - 1.54876i) q^{7} +(-1.40119 + 0.808980i) q^{11} +6.11640 q^{13} +(-4.66363 + 2.69255i) q^{17} +(0.667037 + 0.385114i) q^{19} +(1.86812 - 3.23568i) q^{23} -5.62291i q^{29} +(0.201585 - 0.116385i) q^{31} +(-0.411626 - 0.237653i) q^{37} -0.682087 q^{41} +3.64192i q^{43} +(-3.37027 - 1.94583i) q^{47} +(2.20268 + 6.64441i) q^{49} +(-6.43189 - 11.1404i) q^{53} +(5.41271 + 9.37509i) q^{59} +(-3.26629 - 1.88579i) q^{61} +(2.72158 - 1.57131i) q^{67} +13.9116i q^{71} +(-3.71204 - 6.42944i) q^{73} +(4.25858 + 0.434792i) q^{77} +(-0.416272 + 0.721004i) q^{79} -0.682087i q^{83} +(5.18151 - 8.97464i) q^{89} +(-13.1201 - 9.47283i) q^{91} +16.8308 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 12 q^{31} - 8 q^{49} + 12 q^{61} - 4 q^{79} - 80 q^{91}+O(q^{100}) \) 40 * q + 12 * q^31 - 8 * q^49 + 12 * q^61 - 4 * q^79 - 80 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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