LMFDB - Embedded newform 3060.2.w.f.2177.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,2,Mod(953,3060)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3060.953"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3, 0])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [28,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$24.4342230185$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2177.12
Character	$\chi$	$=$	3060.2177
Dual form			3060.2.w.f.953.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.05353 + 0.884870i) q^{5} +(0.130751 + 0.130751i) q^{7} +1.92102i q^{11} +(4.58124 - 4.58124i) q^{13} +(0.707107 - 0.707107i) q^{17} +4.85426i q^{19} +(-1.87618 - 1.87618i) q^{23} +(3.43401 + 3.63422i) q^{25} +2.18605 q^{29} +5.07036 q^{31} +(0.152805 + 0.384200i) q^{35} +(-2.23510 - 2.23510i) q^{37} +0.0637839i q^{41} +(-0.0618727 + 0.0618727i) q^{43} +(-1.55630 + 1.55630i) q^{47} -6.96581i q^{49} +(3.22520 + 3.22520i) q^{53} +(-1.69986 + 3.94489i) q^{55} +8.32179 q^{59} +1.32701 q^{61} +(13.4615 - 5.35393i) q^{65} +(-6.19130 - 6.19130i) q^{67} +9.55399i q^{71} +(4.57376 - 4.57376i) q^{73} +(-0.251177 + 0.251177i) q^{77} +7.02471i q^{79} +(0.282320 + 0.282320i) q^{83} +(2.07777 - 0.826371i) q^{85} +1.50076 q^{89} +1.19801 q^{91} +(-4.29539 + 9.96839i) q^{95} +(-6.73088 - 6.73088i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 8 q^{13} + 16 q^{25} - 24 q^{31} - 4 q^{37} + 36 q^{43} + 40 q^{55} - 40 q^{61} + 64 q^{67} + 28 q^{73} - 4 q^{85} - 120 q^{91} + 44 q^{97}+O(q^{100}) $ 28 * q + 8 * q^13 + 16 * q^25 - 24 * q^31 - 4 * q^37 + 36 * q^43 + 40 * q^55 - 40 * q^61 + 64 * q^67 + 28 * q^73 - 4 * q^85 - 120 * q^91 + 44 * q^97

Character values

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

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

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
$5$	2.05353	+	0.884870i	0.918369	+	0.395726i
$5$	2.05353	+	0.884870i	0.918369	+	0.395726i
$6$		0			0
$7$	0.130751	+	0.130751i	0.0494194	+	0.0494194i	0.731385	−	0.681965i	$-0.238874\pi$
$7$	0.130751	+	0.130751i	0.0494194	+	0.0494194i	−0.681965	+	0.731385i	$0.738874\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.92102i			0.579211i	0.957146	+	0.289605i	$0.0935240\pi$
$11$			1.92102i			0.579211i	−0.957146	+	0.289605i	$0.906476\pi$
$12$		0			0
$13$	4.58124	−	4.58124i	1.27061	−	1.27061i	0.324836	−	0.945770i	$-0.394691\pi$
$13$	4.58124	−	4.58124i	1.27061	−	1.27061i	0.945770	−	0.324836i	$-0.105309\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.707107	−	0.707107i	0.171499	−	0.171499i
$17$	0.707107	−	0.707107i	0.171499	−	0.171499i
$18$		0			0
$19$			4.85426i			1.11364i	0.830632	+	0.556822i	$0.187979\pi$
$19$			4.85426i			1.11364i	−0.830632	+	0.556822i	$0.812021\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.87618	−	1.87618i	−0.391211	−	0.391211i	0.483908	−	0.875119i	$-0.339217\pi$
$23$	−1.87618	−	1.87618i	−0.391211	−	0.391211i	−0.875119	+	0.483908i	$0.839217\pi$
$24$		0			0
$25$	3.43401	+	3.63422i	0.686802	+	0.726844i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.18605			0.405938			0.202969	−	0.979185i	$-0.434941\pi$
$29$	2.18605			0.405938			0.202969	+	0.979185i	$0.434941\pi$
$30$		0			0
$31$	5.07036			0.910663			0.455331	−	0.890322i	$-0.349521\pi$
$31$	5.07036			0.910663			0.455331	+	0.890322i	$0.349521\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.152805	+	0.384200i	0.0258287	+	0.0649417i
$36$		0			0
$37$	−2.23510	−	2.23510i	−0.367448	−	0.367448i	0.499098	−	0.866546i	$-0.333665\pi$
$37$	−2.23510	−	2.23510i	−0.367448	−	0.367448i	−0.866546	+	0.499098i	$0.833665\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			0.0637839i			0.00996137i	0.999988	+	0.00498069i	$0.00158541\pi$
$41$			0.0637839i			0.00996137i	−0.999988	+	0.00498069i	$0.998415\pi$
$42$		0			0
$43$	−0.0618727	+	0.0618727i	−0.00943549	+	0.00943549i	−0.711809	−	0.702373i	$-0.752124\pi$
$43$	−0.0618727	+	0.0618727i	−0.00943549	+	0.00943549i	0.702373	+	0.711809i	$0.252124\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.55630	+	1.55630i	−0.227009	+	0.227009i	−0.811442	−	0.584433i	$-0.801317\pi$
$47$	−1.55630	+	1.55630i	−0.227009	+	0.227009i	0.584433	+	0.811442i	$0.301317\pi$
$48$		0			0
$49$		−	6.96581i		−	0.995115i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.22520	+	3.22520i	0.443015	+	0.443015i	0.893024	−	0.450009i	$-0.148579\pi$
$53$	3.22520	+	3.22520i	0.443015	+	0.443015i	−0.450009	+	0.893024i	$0.648579\pi$
$54$		0			0
$55$	−1.69986	+	3.94489i	−0.229209	+	0.531929i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.32179			1.08340			0.541702	−	0.840570i	$-0.317780\pi$
$59$	8.32179			1.08340			0.541702	+	0.840570i	$0.317780\pi$
$60$		0			0
$61$	1.32701			0.169906			0.0849529	−	0.996385i	$-0.472926\pi$
$61$	1.32701			0.169906			0.0849529	+	0.996385i	$0.472926\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	13.4615	−	5.35393i	1.66970	−	0.664073i
$66$		0			0
$67$	−6.19130	−	6.19130i	−0.756388	−	0.756388i	0.219275	−	0.975663i	$-0.429631\pi$
$67$	−6.19130	−	6.19130i	−0.756388	−	0.756388i	−0.975663	+	0.219275i	$0.929631\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			9.55399i			1.13385i	0.823769	+	0.566925i	$0.191867\pi$
$71$			9.55399i			1.13385i	−0.823769	+	0.566925i	$0.808133\pi$
$72$		0			0
$73$	4.57376	−	4.57376i	0.535318	−	0.535318i	−0.386832	−	0.922150i	$-0.626431\pi$
$73$	4.57376	−	4.57376i	0.535318	−	0.535318i	0.922150	+	0.386832i	$0.126431\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.251177	+	0.251177i	−0.0286242	+	0.0286242i
$78$		0			0
$79$			7.02471i			0.790342i	0.918608	+	0.395171i	$0.129315\pi$
$79$			7.02471i			0.790342i	−0.918608	+	0.395171i	$0.870685\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.282320	+	0.282320i	0.0309886	+	0.0309886i	0.722431	−	0.691443i	$-0.243025\pi$
$83$	0.282320	+	0.282320i	0.0309886	+	0.0309886i	−0.691443	+	0.722431i	$0.743025\pi$
$84$		0			0
$85$	2.07777	−	0.826371i	0.225365	−	0.0896325i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.50076			0.159080			0.0795401	−	0.996832i	$-0.474655\pi$
$89$	1.50076			0.159080			0.0795401	+	0.996832i	$0.474655\pi$
$90$		0			0
$91$	1.19801			0.125585
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.29539	+	9.96839i	−0.440697	+	1.02274i
$96$		0			0
$97$	−6.73088	−	6.73088i	−0.683417	−	0.683417i	0.277351	−	0.960769i	$-0.410543\pi$
$97$	−6.73088	−	6.73088i	−0.683417	−	0.683417i	−0.960769	+	0.277351i	$0.910543\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			9.69256i			0.964446i	0.876048	+	0.482223i	$0.160171\pi$
$101$			9.69256i			0.964446i	−0.876048	+	0.482223i	$0.839829\pi$
$102$		0			0
$103$	6.58985	−	6.58985i	0.649317	−	0.649317i	−0.303511	−	0.952828i	$-0.598159\pi$
$103$	6.58985	−	6.58985i	0.649317	−	0.649317i	0.952828	+	0.303511i	$0.0981588\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.00885	+	7.00885i	−0.677571	+	0.677571i	−0.959450	−	0.281879i	$-0.909042\pi$
$107$	−7.00885	+	7.00885i	−0.677571	+	0.677571i	0.281879	+	0.959450i	$0.409042\pi$
$108$		0			0
$109$		−	5.35304i		−	0.512728i	−0.966580	−	0.256364i	$-0.917475\pi$
$109$		−	5.35304i		−	0.512728i	0.966580	−	0.256364i	$-0.0825246\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.16703	+	1.16703i	0.109785	+	0.109785i	0.759865	−	0.650080i	$-0.225265\pi$
$113$	1.16703	+	1.16703i	0.109785	+	0.109785i	−0.650080	+	0.759865i	$0.725265\pi$
$114$		0			0
$115$	−2.19263	−	5.51298i	−0.204464	−	0.514088i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.184910			0.0169507
$120$		0			0
$121$	7.30967			0.664515
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.83605	+	10.5017i	0.343107	+	0.939296i
$126$		0			0
$127$	14.9837	+	14.9837i	1.32959	+	1.32959i	0.905728	+	0.423859i	$0.139325\pi$
$127$	14.9837	+	14.9837i	1.32959	+	1.32959i	0.423859	+	0.905728i	$0.360675\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			10.2813i			0.898283i	0.893461	+	0.449141i	$0.148270\pi$
$131$			10.2813i			0.898283i	−0.893461	+	0.449141i	$0.851730\pi$
$132$		0			0
$133$	−0.634701	+	0.634701i	−0.0550355	+	0.0550355i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.19705	+	4.19705i	−0.358578	+	0.358578i	−0.863289	−	0.504710i	$-0.831599\pi$
$137$	−4.19705	+	4.19705i	−0.358578	+	0.358578i	0.504710	+	0.863289i	$0.331599\pi$
$138$		0			0
$139$		−	12.2971i		−	1.04302i	−0.853244	−	0.521511i	$-0.825368\pi$
$139$		−	12.2971i		−	1.04302i	0.853244	−	0.521511i	$-0.174632\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.80067	+	8.80067i	0.735949	+	0.735949i
$144$		0			0
$145$	4.48912	+	1.93437i	0.372801	+	0.160640i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−18.0446			−1.47827			−0.739135	−	0.673557i	$-0.764765\pi$
$149$	−18.0446			−1.47827			−0.739135	+	0.673557i	$0.764765\pi$
$150$		0			0
$151$	1.29172			0.105119			0.0525593	−	0.998618i	$-0.483262\pi$
$151$	1.29172			0.105119			0.0525593	+	0.998618i	$0.483262\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	10.4122	+	4.48660i	0.836324	+	0.360373i
$156$		0			0
$157$	8.52405	+	8.52405i	0.680293	+	0.680293i	0.960066	−	0.279773i	$-0.0902592\pi$
$157$	8.52405	+	8.52405i	0.680293	+	0.680293i	−0.279773	+	0.960066i	$0.590259\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	0.490627i		−	0.0386668i
$162$		0			0
$163$	3.13148	−	3.13148i	0.245276	−	0.245276i	−0.573752	−	0.819029i	$-0.694513\pi$
$163$	3.13148	−	3.13148i	0.245276	−	0.245276i	0.819029	+	0.573752i	$0.194513\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.81766	−	4.81766i	0.372802	−	0.372802i	−0.495695	−	0.868497i	$-0.665087\pi$
$167$	4.81766	−	4.81766i	0.372802	−	0.372802i	0.868497	+	0.495695i	$0.165087\pi$
$168$		0			0
$169$		−	28.9754i		−	2.22888i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.14933	+	9.14933i	0.695611	+	0.695611i	0.963461	−	0.267850i	$-0.0863132\pi$
$173$	9.14933	+	9.14933i	0.695611	+	0.695611i	−0.267850	+	0.963461i	$0.586313\pi$
$174$		0			0
$175$	−0.0261777	+	0.924181i	−0.00197885	+	0.0698615i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.7852			0.806126			0.403063	−	0.915172i	$-0.367946\pi$
$179$	10.7852			0.806126			0.403063	+	0.915172i	$0.367946\pi$
$180$		0			0
$181$	−17.2396			−1.28141			−0.640706	−	0.767787i	$-0.721358\pi$
$181$	−17.2396			−1.28141			−0.640706	+	0.767787i	$0.721358\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.61208	−	6.56762i	−0.192044	−	0.482861i
$186$		0			0
$187$	1.35837	+	1.35837i	0.0993338	+	0.0993338i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			11.1687i			0.808137i	0.914729	+	0.404068i	$0.132404\pi$
$191$			11.1687i			0.808137i	−0.914729	+	0.404068i	$0.867596\pi$
$192$		0			0
$193$	0.594617	−	0.594617i	0.0428015	−	0.0428015i	−0.685382	−	0.728184i	$-0.740365\pi$
$193$	0.594617	−	0.594617i	0.0428015	−	0.0428015i	0.728184	+	0.685382i	$0.240365\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.34066	−	3.34066i	0.238012	−	0.238012i	−0.578014	−	0.816027i	$-0.696172\pi$
$197$	3.34066	−	3.34066i	0.238012	−	0.238012i	0.816027	+	0.578014i	$0.196172\pi$
$198$		0			0
$199$		−	12.6261i		−	0.895043i	−0.894273	−	0.447521i	$-0.852307\pi$
$199$		−	12.6261i		−	0.895043i	0.894273	−	0.447521i	$-0.147693\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.285828	+	0.285828i	0.0200612	+	0.0200612i
$204$		0			0
$205$	−0.0564404	+	0.130982i	−0.00394197	+	0.00914821i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−9.32515			−0.645034
$210$		0			0
$211$	−14.7883			−1.01807			−0.509034	−	0.860747i	$-0.669997\pi$
$211$	−14.7883			−1.01807			−0.509034	+	0.860747i	$0.669997\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.181807	+	0.0723084i	−0.0123991	+	0.00493139i
$216$		0			0
$217$	0.662956	+	0.662956i	0.0450044	+	0.0450044i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	6.47885i		−	0.435814i
$222$		0			0
$223$	3.10164	−	3.10164i	0.207701	−	0.207701i	−0.595588	−	0.803290i	$-0.703081\pi$
$223$	3.10164	−	3.10164i	0.207701	−	0.207701i	0.803290	+	0.595588i	$0.203081\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.2619	−	12.2619i	0.813851	−	0.813851i	−0.171358	−	0.985209i	$-0.554815\pi$
$227$	12.2619	−	12.2619i	0.813851	−	0.813851i	0.985209	+	0.171358i	$0.0548154\pi$
$228$		0			0
$229$			6.41680i			0.424034i	0.977266	+	0.212017i	$0.0680032\pi$
$229$			6.41680i			0.424034i	−0.977266	+	0.212017i	$0.931997\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.94134	−	2.94134i	−0.192694	−	0.192694i	0.604165	−	0.796859i	$-0.293507\pi$
$233$	−2.94134	−	2.94134i	−0.192694	−	0.192694i	−0.796859	+	0.604165i	$0.793507\pi$
$234$		0			0
$235$	−4.57303	+	1.81879i	−0.298312	+	0.118645i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.77199			−0.373359			−0.186679	−	0.982421i	$-0.559773\pi$
$239$	−5.77199			−0.373359			−0.186679	+	0.982421i	$0.559773\pi$
$240$		0			0
$241$	−19.1919			−1.23626			−0.618131	−	0.786075i	$-0.712110\pi$
$241$	−19.1919			−1.23626			−0.618131	+	0.786075i	$0.712110\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.16383	−	14.3045i	0.393793	−	0.913883i
$246$		0			0
$247$	22.2385	+	22.2385i	1.41500	+	1.41500i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			4.98760i			0.314815i	0.987534	+	0.157407i	$0.0503136\pi$
$251$			4.98760i			0.314815i	−0.987534	+	0.157407i	$0.949686\pi$
$252$		0			0
$253$	3.60419	−	3.60419i	0.226594	−	0.226594i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	17.4010	−	17.4010i	1.08544	−	1.08544i	0.0894506	−	0.995991i	$-0.471489\pi$
$257$	17.4010	−	17.4010i	1.08544	−	1.08544i	0.995991	−	0.0894506i	$-0.0285111\pi$
$258$		0			0
$259$		−	0.584484i		−	0.0363181i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.59811	+	7.59811i	0.468519	+	0.468519i	0.901435	−	0.432915i	$-0.142515\pi$
$263$	7.59811	+	7.59811i	0.468519	+	0.468519i	−0.432915	+	0.901435i	$0.642515\pi$
$264$		0			0
$265$	3.76917	+	9.47693i	0.231539	+	0.582163i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	28.1423			1.71587			0.857934	−	0.513759i	$-0.171748\pi$
$269$	28.1423			1.71587			0.857934	+	0.513759i	$0.171748\pi$
$270$		0			0
$271$	−4.07728			−0.247677			−0.123839	−	0.992302i	$-0.539521\pi$
$271$	−4.07728			−0.247677			−0.123839	+	0.992302i	$0.539521\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.98143	+	6.59682i	−0.420996	+	0.397803i
$276$		0			0
$277$	−10.6683	−	10.6683i	−0.640997	−	0.640997i	0.309804	−	0.950801i	$-0.399737\pi$
$277$	−10.6683	−	10.6683i	−0.640997	−	0.640997i	−0.950801	+	0.309804i	$0.899737\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			12.7632i			0.761389i	0.924701	+	0.380695i	$0.124315\pi$
$281$			12.7632i			0.761389i	−0.924701	+	0.380695i	$0.875685\pi$
$282$		0			0
$283$	6.99277	−	6.99277i	0.415677	−	0.415677i	−0.468034	−	0.883711i	$-0.655037\pi$
$283$	6.99277	−	6.99277i	0.415677	−	0.415677i	0.883711	+	0.468034i	$0.155037\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.00833983	+	0.00833983i	−0.000492285	+	0.000492285i
$288$		0			0
$289$		−	1.00000i		−	0.0588235i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.2879	−	16.2879i	−0.951548	−	0.951548i	0.0473316	−	0.998879i	$-0.484928\pi$
$293$	−16.2879	−	16.2879i	−0.951548	−	0.951548i	−0.998879	+	0.0473316i	$0.984928\pi$
$294$		0			0
$295$	17.0891	+	7.36370i	0.994965	+	0.428731i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−17.1905			−0.994150
$300$		0			0
$301$	−0.0161799			−0.000932592
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.72506	+	1.17423i	0.156036	+	0.0672361i
$306$		0			0
$307$	18.8850	+	18.8850i	1.07782	+	1.07782i	0.996704	+	0.0811189i	$0.0258494\pi$
$307$	18.8850	+	18.8850i	1.07782	+	1.07782i	0.0811189	+	0.996704i	$0.474151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			20.3205i			1.15227i	0.817355	+	0.576135i	$0.195440\pi$
$311$			20.3205i			1.15227i	−0.817355	+	0.576135i	$0.804560\pi$
$312$		0			0
$313$	8.60033	−	8.60033i	0.486119	−	0.486119i	−0.420960	−	0.907079i	$-0.638307\pi$
$313$	8.60033	−	8.60033i	0.486119	−	0.486119i	0.907079	+	0.420960i	$0.138307\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.57134	+	5.57134i	−0.312918	+	0.312918i	−0.846039	−	0.533121i	$-0.821019\pi$
$317$	−5.57134	+	5.57134i	−0.312918	+	0.312918i	0.533121	+	0.846039i	$0.321019\pi$
$318$		0			0
$319$			4.19945i			0.235124i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.43248	+	3.43248i	0.190988	+	0.190988i
$324$		0			0
$325$	32.3812	+	0.917210i	1.79619	+	0.0508776i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.406976			−0.0224373
$330$		0			0
$331$	−5.16271			−0.283768			−0.141884	−	0.989883i	$-0.545316\pi$
$331$	−5.16271			−0.283768			−0.141884	+	0.989883i	$0.545316\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.23556	−	18.1926i	−0.395321	−	0.993965i
$336$		0			0
$337$	5.63119	+	5.63119i	0.306750	+	0.306750i	0.843648	−	0.536897i	$-0.180404\pi$
$337$	5.63119	+	5.63119i	0.306750	+	0.306750i	−0.536897	+	0.843648i	$0.680404\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			9.74028i			0.527466i
$342$		0			0
$343$	1.82605	−	1.82605i	0.0985973	−	0.0985973i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.86564	+	6.86564i	−0.368567	+	0.368567i	−0.866954	−	0.498388i	$-0.833926\pi$
$347$	−6.86564	+	6.86564i	−0.368567	+	0.368567i	0.498388	+	0.866954i	$0.333926\pi$
$348$		0			0
$349$		−	11.3458i		−	0.607326i	−0.952780	−	0.303663i	$-0.901790\pi$
$349$		−	11.3458i		−	0.607326i	0.952780	−	0.303663i	$-0.0982096\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	17.6986	+	17.6986i	0.942000	+	0.942000i	0.998408	−	0.0564076i	$-0.0179646\pi$
$353$	17.6986	+	17.6986i	0.942000	+	0.942000i	−0.0564076	+	0.998408i	$0.517965\pi$
$354$		0			0
$355$	−8.45404	+	19.6195i	−0.448694	+	1.04129i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.97599			−0.368178			−0.184089	−	0.982910i	$-0.558934\pi$
$359$	−6.97599			−0.368178			−0.184089	+	0.982910i	$0.558934\pi$
$360$		0			0
$361$	−4.56382			−0.240201
$362$		0			0
$363$		0			0
$364$		0			0
$365$	13.4395	−	5.34519i	0.703458	−	0.279780i
$366$		0			0
$367$	6.51370	+	6.51370i	0.340013	+	0.340013i	0.856372	−	0.516359i	$-0.172713\pi$
$367$	6.51370	+	6.51370i	0.340013	+	0.340013i	−0.516359	+	0.856372i	$0.672713\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			0.843397i			0.0437870i
$372$		0			0
$373$	−6.49881	+	6.49881i	−0.336496	+	0.336496i	−0.855047	−	0.518551i	$-0.826472\pi$
$373$	−6.49881	+	6.49881i	−0.336496	+	0.336496i	0.518551	+	0.855047i	$0.326472\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	10.0148	−	10.0148i	0.515788	−	0.515788i
$378$		0			0
$379$			15.2605i			0.783878i	0.919991	+	0.391939i	$0.128196\pi$
$379$			15.2605i			0.783878i	−0.919991	+	0.391939i	$0.871804\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−26.5979	−	26.5979i	−1.35909	−	1.35909i	−0.875039	−	0.484053i	$-0.839164\pi$
$383$	−26.5979	−	26.5979i	−1.35909	−	1.35909i	−0.484053	−	0.875039i	$-0.660836\pi$
$384$		0			0
$385$	−0.738058	+	0.293541i	−0.0376149	+	0.0149602i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.385928			0.0195673			0.00978365	−	0.999952i	$-0.496886\pi$
$389$	0.385928			0.0195673			0.00978365	+	0.999952i	$0.496886\pi$
$390$		0			0
$391$	−2.65332			−0.134184
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.21596	+	14.4255i	−0.312759	+	0.725826i
$396$		0			0
$397$	−17.4628	−	17.4628i	−0.876435	−	0.876435i	0.116729	−	0.993164i	$-0.462759\pi$
$397$	−17.4628	−	17.4628i	−0.876435	−	0.876435i	−0.993164	+	0.116729i	$0.962759\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			4.38328i			0.218891i	0.993993	+	0.109445i	$0.0349075\pi$
$401$			4.38328i			0.218891i	−0.993993	+	0.109445i	$0.965093\pi$
$402$		0			0
$403$	23.2285	−	23.2285i	1.15709	−	1.15709i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.29368	−	4.29368i	0.212830	−	0.212830i
$408$		0			0
$409$			14.4117i			0.712615i	0.934369	+	0.356307i	$0.115964\pi$
$409$			14.4117i			0.712615i	−0.934369	+	0.356307i	$0.884036\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.08809	+	1.08809i	0.0535412	+	0.0535412i
$414$		0			0
$415$	0.329937	+	0.829569i	0.0161960	+	0.0407219i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−37.9153			−1.85228			−0.926141	−	0.377178i	$-0.876894\pi$
$419$	−37.9153			−1.85228			−0.926141	+	0.377178i	$0.876894\pi$
$420$		0			0
$421$	−24.5715			−1.19754			−0.598771	−	0.800920i	$-0.704344\pi$
$421$	−24.5715			−1.19754			−0.598771	+	0.800920i	$0.704344\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.99800	+	0.141570i	0.242438	+	0.00686715i
$426$		0			0
$427$	0.173508	+	0.173508i	0.00839664	+	0.00839664i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	28.1976i		−	1.35823i	−0.734032	−	0.679115i	$-0.762364\pi$
$431$		−	28.1976i		−	1.35823i	0.734032	−	0.679115i	$-0.237636\pi$
$432$		0			0
$433$	6.75767	−	6.75767i	0.324753	−	0.324753i	−0.525834	−	0.850587i	$-0.676247\pi$
$433$	6.75767	−	6.75767i	0.324753	−	0.324753i	0.850587	+	0.525834i	$0.176247\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9.10747	−	9.10747i	0.435670	−	0.435670i
$438$		0			0
$439$		−	34.9645i		−	1.66876i	−0.551188	−	0.834381i	$-0.685825\pi$
$439$		−	34.9645i		−	1.66876i	0.551188	−	0.834381i	$-0.314175\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.1657	−	17.1657i	−0.815569	−	0.815569i	0.169894	−	0.985462i	$-0.445658\pi$
$443$	−17.1657	−	17.1657i	−0.815569	−	0.815569i	−0.985462	+	0.169894i	$0.945658\pi$
$444$		0			0
$445$	3.08186	+	1.32798i	0.146094	+	0.0629521i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10.5014			0.495591			0.247795	−	0.968812i	$-0.420294\pi$
$449$	10.5014			0.495591			0.247795	+	0.968812i	$0.420294\pi$
$450$		0			0
$451$	−0.122530			−0.00576973
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.46015	+	1.06008i	0.115333	+	0.0496973i
$456$		0			0
$457$	5.51700	+	5.51700i	0.258074	+	0.258074i	0.824271	−	0.566196i	$-0.191586\pi$
$457$	5.51700	+	5.51700i	0.258074	+	0.258074i	−0.566196	+	0.824271i	$0.691586\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	26.9321i		−	1.25435i	−0.778877	−	0.627177i	$-0.784210\pi$
$461$		−	26.9321i		−	1.25435i	0.778877	−	0.627177i	$-0.215790\pi$
$462$		0			0
$463$	−29.4341	+	29.4341i	−1.36792	+	1.36792i	−0.504521	+	0.863399i	$0.668331\pi$
$463$	−29.4341	+	29.4341i	−1.36792	+	1.36792i	−0.863399	+	0.504521i	$0.831669\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−23.2994	+	23.2994i	−1.07817	+	1.07817i	−0.0814935	+	0.996674i	$0.525969\pi$
$467$	−23.2994	+	23.2994i	−1.07817	+	1.07817i	−0.996674	+	0.0814935i	$0.974031\pi$
$468$		0			0
$469$		−	1.61904i		−	0.0747604i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.118859	−	0.118859i	−0.00546514	−	0.00546514i
$474$		0			0
$475$	−17.6415	+	16.6696i	−0.809445	+	0.764853i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.3713			−0.473877			−0.236938	−	0.971525i	$-0.576144\pi$
$479$	−10.3713			−0.473877			−0.236938	+	0.971525i	$0.576144\pi$
$480$		0			0
$481$	−20.4790			−0.933763
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.86615	−	19.7781i	−0.357183	−	0.898075i
$486$		0			0
$487$	−7.06812	−	7.06812i	−0.320287	−	0.320287i	0.528590	−	0.848877i	$-0.322721\pi$
$487$	−7.06812	−	7.06812i	−0.320287	−	0.320287i	−0.848877	+	0.528590i	$0.822721\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			10.8915i			0.491527i	0.969330	+	0.245763i	$0.0790386\pi$
$491$			10.8915i			0.491527i	−0.969330	+	0.245763i	$0.920961\pi$
$492$		0			0
$493$	1.54577	−	1.54577i	0.0696179	−	0.0696179i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.24920	+	1.24920i	−0.0560342	+	0.0560342i
$498$		0			0
$499$			14.1598i			0.633880i	0.948446	+	0.316940i	$0.102655\pi$
$499$			14.1598i			0.633880i	−0.948446	+	0.316940i	$0.897345\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−31.1703	−	31.1703i	−1.38982	−	1.38982i	−0.825685	−	0.564131i	$-0.809211\pi$
$503$	−31.1703	−	31.1703i	−1.38982	−	1.38982i	−0.564131	−	0.825685i	$-0.690789\pi$
$504$		0			0
$505$	−8.57666	+	19.9040i	−0.381656	+	0.885717i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	39.5322			1.75223			0.876116	−	0.482100i	$-0.160126\pi$
$509$	39.5322			1.75223			0.876116	+	0.482100i	$0.160126\pi$
$510$		0			0
$511$	1.19605			0.0529101
$512$		0			0
$513$		0			0
$514$		0			0
$515$	19.3636	−	7.70133i	0.853264	−	0.339361i
$516$		0			0
$517$	−2.98968	−	2.98968i	−0.131486	−	0.131486i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	6.57493i		−	0.288053i	−0.989574	−	0.144027i	$-0.953995\pi$
$521$		−	6.57493i		−	0.288053i	0.989574	−	0.144027i	$-0.0460051\pi$
$522$		0			0
$523$	10.7967	−	10.7967i	0.472106	−	0.472106i	−0.430490	−	0.902596i	$-0.641659\pi$
$523$	10.7967	−	10.7967i	0.472106	−	0.472106i	0.902596	+	0.430490i	$0.141659\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.58528	−	3.58528i	0.156177	−	0.156177i
$528$		0			0
$529$		−	15.9599i		−	0.693908i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	0.292209	+	0.292209i	0.0126570	+	0.0126570i
$534$		0			0
$535$	−20.5948	+	8.19099i	−0.890392	+	0.354128i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	13.3815			0.576381
$540$		0			0
$541$	−40.9112			−1.75891			−0.879454	−	0.475984i	$-0.842092\pi$
$541$	−40.9112			−1.75891			−0.879454	+	0.475984i	$0.842092\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.73674	−	10.9927i	0.202900	−	0.470873i
$546$		0			0
$547$	−30.9996	−	30.9996i	−1.32545	−	1.32545i	−0.909293	−	0.416156i	$-0.863377\pi$
$547$	−30.9996	−	30.9996i	−1.32545	−	1.32545i	−0.416156	−	0.909293i	$-0.636623\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			10.6116i			0.452071i
$552$		0			0
$553$	−0.918491	+	0.918491i	−0.0390582	+	0.0390582i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	24.1937	−	24.1937i	1.02512	−	1.02512i	0.0254453	−	0.999676i	$-0.491900\pi$
$557$	24.1937	−	24.1937i	1.02512	−	1.02512i	0.999676	−	0.0254453i	$-0.00810035\pi$
$558$		0			0
$559$			0.566907i			0.0239776i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−16.2297	−	16.2297i	−0.683999	−	0.683999i	0.276900	−	0.960899i	$-0.410693\pi$
$563$	−16.2297	−	16.2297i	−0.683999	−	0.683999i	−0.960899	+	0.276900i	$0.910693\pi$
$564$		0			0
$565$	1.36387	+	3.42920i	0.0573783	+	0.144268i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−32.9088			−1.37961			−0.689804	−	0.723996i	$-0.742303\pi$
$569$	−32.9088			−1.37961			−0.689804	+	0.723996i	$0.742303\pi$
$570$		0			0
$571$	15.1322			0.633262			0.316631	−	0.948549i	$-0.397448\pi$
$571$	15.1322			0.633262			0.316631	+	0.948549i	$0.397448\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.375631	−	13.2613i	0.0156649	−	0.553034i
$576$		0			0
$577$	−4.28605	−	4.28605i	−0.178431	−	0.178431i	0.612241	−	0.790671i	$-0.290268\pi$
$577$	−4.28605	−	4.28605i	−0.178431	−	0.178431i	−0.790671	+	0.612241i	$0.790268\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			0.0738273i			0.00306287i
$582$		0			0
$583$	−6.19568	+	6.19568i	−0.256599	+	0.256599i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−33.5989	+	33.5989i	−1.38677	+	1.38677i	−0.554769	+	0.832005i	$0.687193\pi$
$587$	−33.5989	+	33.5989i	−1.38677	+	1.38677i	−0.832005	+	0.554769i	$0.812807\pi$
$588$		0			0
$589$			24.6128i			1.01415i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−21.1407	−	21.1407i	−0.868146	−	0.868146i	0.124121	−	0.992267i	$-0.460389\pi$
$593$	−21.1407	−	21.1407i	−0.868146	−	0.868146i	−0.992267	+	0.124121i	$0.960389\pi$
$594$		0			0
$595$	0.379720	+	0.163622i	0.0155670	+	0.00670783i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−37.1224			−1.51678			−0.758391	−	0.651800i	$-0.774014\pi$
$599$	−37.1224			−1.51678			−0.758391	+	0.651800i	$0.774014\pi$
$600$		0			0
$601$	−40.9887			−1.67196			−0.835981	−	0.548758i	$-0.815101\pi$
$601$	−40.9887			−1.67196			−0.835981	+	0.548758i	$0.815101\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	15.0107	+	6.46810i	0.610270	+	0.262966i
$606$		0			0
$607$	27.8738	+	27.8738i	1.13136	+	1.13136i	0.989951	+	0.141412i	$0.0451643\pi$
$607$	27.8738	+	27.8738i	1.13136	+	1.13136i	0.141412	+	0.989951i	$0.454836\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			14.2595i			0.576879i
$612$		0			0
$613$	−11.5237	+	11.5237i	−0.465439	+	0.465439i	−0.900433	−	0.434994i	$-0.856750\pi$
$613$	−11.5237	+	11.5237i	−0.465439	+	0.465439i	0.434994	+	0.900433i	$0.356750\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.78036	+	6.78036i	−0.272967	+	0.272967i	−0.830293	−	0.557327i	$-0.811827\pi$
$617$	−6.78036	+	6.78036i	−0.272967	+	0.272967i	0.557327	+	0.830293i	$0.311827\pi$
$618$		0			0
$619$		−	20.3290i		−	0.817093i	−0.912737	−	0.408547i	$-0.866036\pi$
$619$		−	20.3290i		−	0.817093i	0.912737	−	0.408547i	$-0.133964\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.196226	+	0.196226i	0.00786164	+	0.00786164i
$624$		0			0
$625$	−1.41513	+	24.9599i	−0.0566053	+	0.998397i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.16091			−0.126034
$630$		0			0
$631$	−11.2748			−0.448844			−0.224422	−	0.974492i	$-0.572049\pi$
$631$	−11.2748			−0.448844			−0.224422	+	0.974492i	$0.572049\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	17.5109	+	44.0281i	0.694899	+	1.74720i
$636$		0			0
$637$	−31.9120	−	31.9120i	−1.26440	−	1.26440i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	2.17279i		−	0.0858202i	−0.999079	−	0.0429101i	$-0.986337\pi$
$641$		−	2.17279i		−	0.0858202i	0.999079	−	0.0429101i	$-0.0136629\pi$
$642$		0			0
$643$	11.9797	−	11.9797i	0.472435	−	0.472435i	−0.430267	−	0.902702i	$-0.641581\pi$
$643$	11.9797	−	11.9797i	0.472435	−	0.472435i	0.902702	+	0.430267i	$0.141581\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.97383	−	9.97383i	0.392112	−	0.392112i	−0.483328	−	0.875439i	$-0.660572\pi$
$647$	9.97383	−	9.97383i	0.392112	−	0.392112i	0.875439	+	0.483328i	$0.160572\pi$
$648$		0			0
$649$			15.9864i			0.627520i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−27.0978	−	27.0978i	−1.06042	−	1.06042i	−0.998053	−	0.0623662i	$-0.980135\pi$
$653$	−27.0978	−	27.0978i	−1.06042	−	1.06042i	−0.0623662	−	0.998053i	$-0.519865\pi$
$654$		0			0
$655$	−9.09762	+	21.1130i	−0.355474	+	0.824955i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−33.5888			−1.30843			−0.654217	−	0.756307i	$-0.727002\pi$
$659$	−33.5888			−1.30843			−0.654217	+	0.756307i	$0.727002\pi$
$660$		0			0
$661$	35.4249			1.37787			0.688934	−	0.724824i	$-0.258079\pi$
$661$	35.4249			1.37787			0.688934	+	0.724824i	$0.258079\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.86501	+	0.741753i	−0.0723219	+	0.0287639i
$666$		0			0
$667$	−4.10142	−	4.10142i	−0.158808	−	0.158808i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			2.54921i			0.0984113i
$672$		0			0
$673$	0.143629	−	0.143629i	0.00553649	−	0.00553649i	−0.704333	−	0.709870i	$-0.748754\pi$
$673$	0.143629	−	0.143629i	0.00553649	−	0.00553649i	0.709870	+	0.704333i	$0.248754\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−20.5972	+	20.5972i	−0.791614	+	0.791614i	−0.981757	−	0.190142i	$-0.939105\pi$
$677$	−20.5972	+	20.5972i	−0.791614	+	0.791614i	0.190142	+	0.981757i	$0.439105\pi$
$678$		0			0
$679$		−	1.76014i		−	0.0675481i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	7.59093	+	7.59093i	0.290459	+	0.290459i	0.837261	−	0.546803i	$-0.184155\pi$
$683$	7.59093	+	7.59093i	0.290459	+	0.290459i	−0.546803	+	0.837261i	$0.684155\pi$
$684$		0			0
$685$	−12.3326	+	4.90495i	−0.471206	+	0.187409i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	29.5508			1.12579
$690$		0			0
$691$	−20.8881			−0.794620			−0.397310	−	0.917684i	$-0.630056\pi$
$691$	−20.8881			−0.794620			−0.397310	+	0.917684i	$0.630056\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	10.8813	−	25.2524i	0.412751	−	0.957879i
$696$		0			0
$697$	0.0451020	+	0.0451020i	0.00170836	+	0.00170836i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			51.4969i			1.94501i	0.232882	+	0.972505i	$0.425184\pi$
$701$			51.4969i			1.94501i	−0.232882	+	0.972505i	$0.574816\pi$
$702$		0			0
$703$	10.8497	−	10.8497i	0.409206	−	0.409206i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.26732	+	1.26732i	−0.0476623	+	0.0476623i
$708$		0			0
$709$			22.4477i			0.843040i	0.906819	+	0.421520i	$0.138503\pi$
$709$			22.4477i			0.843040i	−0.906819	+	0.421520i	$0.861497\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9.51291	−	9.51291i	−0.356261	−	0.356261i
$714$		0			0
$715$	10.2850	+	25.8599i	0.384638	+	0.967106i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−46.4554			−1.73250			−0.866248	−	0.499615i	$-0.833475\pi$
$719$	−46.4554			−1.73250			−0.866248	+	0.499615i	$0.833475\pi$
$720$		0			0
$721$	1.72326			0.0641777
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.50691	+	7.94457i	0.278799	+	0.295054i
$726$		0			0
$727$	27.4296	+	27.4296i	1.01731	+	1.01731i	0.999848	+	0.0174613i	$0.00555838\pi$
$727$	27.4296	+	27.4296i	1.01731	+	1.01731i	0.0174613	+	0.999848i	$0.494442\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			0.0875012i			0.00323635i
$732$		0			0
$733$	34.1523	−	34.1523i	1.26145	−	1.26145i	0.311053	−	0.950393i	$-0.399318\pi$
$733$	34.1523	−	34.1523i	1.26145	−	1.26145i	0.950393	−	0.311053i	$-0.100682\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	11.8936	−	11.8936i	0.438108	−	0.438108i
$738$		0			0
$739$		−	25.0012i		−	0.919684i	−0.888001	−	0.459842i	$-0.847906\pi$
$739$		−	25.0012i		−	0.919684i	0.888001	−	0.459842i	$-0.152094\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−9.44863	−	9.44863i	−0.346637	−	0.346637i	0.512219	−	0.858855i	$-0.328824\pi$
$743$	−9.44863	−	9.44863i	−0.346637	−	0.346637i	−0.858855	+	0.512219i	$0.828824\pi$
$744$		0			0
$745$	−37.0552	−	15.9671i	−1.35760	−	0.584990i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.83283			−0.0669702
$750$		0			0
$751$	−35.6958			−1.30256			−0.651280	−	0.758838i	$-0.725768\pi$
$751$	−35.6958			−1.30256			−0.651280	+	0.758838i	$0.725768\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.65259	+	1.14300i	0.0965377	+	0.0415982i
$756$		0			0
$757$	6.96239	+	6.96239i	0.253052	+	0.253052i	0.822221	−	0.569169i	$-0.192735\pi$
$757$	6.96239	+	6.96239i	0.253052	+	0.253052i	−0.569169	+	0.822221i	$0.692735\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	41.7707i		−	1.51419i	−0.653306	−	0.757094i	$-0.726618\pi$
$761$		−	41.7707i		−	1.51419i	0.653306	−	0.757094i	$-0.273382\pi$
$762$		0			0
$763$	0.699917	−	0.699917i	0.0253387	−	0.0253387i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	38.1241	−	38.1241i	1.37658	−	1.37658i
$768$		0			0
$769$		−	30.6035i		−	1.10359i	−0.833980	−	0.551795i	$-0.813943\pi$
$769$		−	30.6035i		−	1.10359i	0.833980	−	0.551795i	$-0.186057\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−31.8815	−	31.8815i	−1.14670	−	1.14670i	−0.987198	−	0.159500i	$-0.949012\pi$
$773$	−31.8815	−	31.8815i	−1.14670	−	1.14670i	−0.159500	−	0.987198i	$-0.550988\pi$
$774$		0			0
$775$	17.4117	+	18.4268i	0.625445	+	0.661910i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.309624			−0.0110934
$780$		0			0
$781$	−18.3534			−0.656738
$782$		0			0
$783$		0			0
$784$		0			0
$785$	9.96176	+	25.0471i	0.355550	+	0.893969i
$786$		0			0
$787$	21.4363	+	21.4363i	0.764122	+	0.764122i	0.977065	−	0.212943i	$-0.0683048\pi$
$787$	21.4363	+	21.4363i	0.764122	+	0.764122i	−0.212943	+	0.977065i	$0.568305\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			0.305181i			0.0108510i
$792$		0			0
$793$	6.07933	−	6.07933i	0.215883	−	0.215883i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4.33182	−	4.33182i	0.153441	−	0.153441i	−0.626212	−	0.779653i	$-0.715396\pi$
$797$	4.33182	−	4.33182i	0.153441	−	0.153441i	0.779653	+	0.626212i	$0.215396\pi$
$798$		0			0
$799$			2.20094i			0.0778635i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8.78630	+	8.78630i	0.310062	+	0.310062i
$804$		0			0
$805$	0.434141	−	1.00752i	0.0153014	−	0.0355104i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.02188			−0.0359272			−0.0179636	−	0.999839i	$-0.505718\pi$
$809$	−1.02188			−0.0359272			−0.0179636	+	0.999839i	$0.505718\pi$
$810$		0			0
$811$	54.2466			1.90486			0.952428	−	0.304762i	$-0.0985771\pi$
$811$	54.2466			1.90486			0.952428	+	0.304762i	$0.0985771\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	9.20155	−	3.65965i	0.322316	−	0.128192i
$816$		0			0
$817$	−0.300346	−	0.300346i	−0.0105078	−	0.0105078i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	51.2683i		−	1.78928i	−0.446792	−	0.894638i	$-0.647434\pi$
$821$		−	51.2683i		−	1.78928i	0.446792	−	0.894638i	$-0.352566\pi$
$822$		0			0
$823$	−26.5591	+	26.5591i	−0.925792	+	0.925792i	−0.997431	−	0.0716383i	$-0.977177\pi$
$823$	−26.5591	+	26.5591i	−0.925792	+	0.925792i	0.0716383	+	0.997431i	$0.477177\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−13.8434	+	13.8434i	−0.481381	+	0.481381i	−0.905572	−	0.424192i	$-0.860558\pi$
$827$	−13.8434	+	13.8434i	−0.481381	+	0.481381i	0.424192	+	0.905572i	$0.360558\pi$
$828$		0			0
$829$			34.5011i			1.19827i	0.800647	+	0.599136i	$0.204489\pi$
$829$			34.5011i			1.19827i	−0.800647	+	0.599136i	$0.795511\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.92557	−	4.92557i	−0.170661	−	0.170661i
$834$		0			0
$835$	14.1562	−	5.63024i	0.489897	−	0.194842i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−52.9597			−1.82837			−0.914185	−	0.405297i	$-0.867168\pi$
$839$	−52.9597			−1.82837			−0.914185	+	0.405297i	$0.867168\pi$
$840$		0			0
$841$	−24.2212			−0.835214
$842$		0			0
$843$		0			0
$844$		0			0
$845$	25.6395	−	59.5021i	0.882025	−	2.04693i
$846$		0			0
$847$	0.955749	+	0.955749i	0.0328399	+	0.0328399i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			8.38690i			0.287499i
$852$		0			0
$853$	−7.66228	+	7.66228i	−0.262351	+	0.262351i	−0.826009	−	0.563657i	$-0.809394\pi$
$853$	−7.66228	+	7.66228i	−0.262351	+	0.262351i	0.563657	+	0.826009i	$0.309394\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.3854	−	25.3854i	0.867147	−	0.867147i	−0.125008	−	0.992156i	$-0.539896\pi$
$857$	25.3854	−	25.3854i	0.867147	−	0.867147i	0.992156	+	0.125008i	$0.0398957\pi$
$858$		0			0
$859$		−	31.3361i		−	1.06917i	−0.845114	−	0.534586i	$-0.820467\pi$
$859$		−	31.3361i		−	1.06917i	0.845114	−	0.534586i	$-0.179533\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−8.15463	−	8.15463i	−0.277587	−	0.277587i	0.554558	−	0.832145i	$-0.312887\pi$
$863$	−8.15463	−	8.15463i	−0.277587	−	0.277587i	−0.832145	+	0.554558i	$0.812887\pi$
$864$		0			0
$865$	10.6925	+	26.8844i	0.363556	+	0.914098i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−13.4946			−0.457775
$870$		0			0
$871$	−56.7276			−1.92214
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.871537	+	1.87467i	−0.0294633	+	0.0633756i
$876$		0			0
$877$	17.0696	+	17.0696i	0.576399	+	0.576399i	0.933909	−	0.357510i	$-0.116374\pi$
$877$	17.0696	+	17.0696i	0.576399	+	0.576399i	−0.357510	+	0.933909i	$0.616374\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			14.0462i			0.473230i	0.971604	+	0.236615i	$0.0760380\pi$
$881$			14.0462i			0.473230i	−0.971604	+	0.236615i	$0.923962\pi$
$882$		0			0
$883$	−18.9044	+	18.9044i	−0.636185	+	0.636185i	−0.949612	−	0.313427i	$-0.898523\pi$
$883$	−18.9044	+	18.9044i	−0.636185	+	0.636185i	0.313427	+	0.949612i	$0.398523\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	30.1353	−	30.1353i	1.01185	−	1.01185i	0.0119168	−	0.999929i	$-0.496207\pi$
$887$	30.1353	−	30.1353i	1.01185	−	1.01185i	0.999929	−	0.0119168i	$-0.00379331\pi$
$888$		0			0
$889$			3.91828i			0.131415i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.55467	−	7.55467i	−0.252807	−	0.252807i
$894$		0			0
$895$	22.1478	+	9.54352i	0.740321	+	0.319005i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	11.0840			0.369673
$900$		0			0
$901$	4.56112			0.151953
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−35.4022	−	15.2548i	−1.17681	−	0.507087i
$906$		0			0
$907$	−25.8749	−	25.8749i	−0.859162	−	0.859162i	0.132077	−	0.991239i	$-0.457835\pi$
$907$	−25.8749	−	25.8749i	−0.859162	−	0.859162i	−0.991239	+	0.132077i	$0.957835\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	56.0681i		−	1.85762i	−0.370556	−	0.928810i	$-0.620833\pi$
$911$		−	56.0681i		−	1.85762i	0.370556	−	0.928810i	$-0.379167\pi$
$912$		0			0
$913$	−0.542343	+	0.542343i	−0.0179489	+	0.0179489i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.34430	+	1.34430i	−0.0443926	+	0.0443926i
$918$		0			0
$919$		−	35.5420i		−	1.17242i	−0.810158	−	0.586212i	$-0.800619\pi$
$919$		−	35.5420i		−	1.17242i	0.810158	−	0.586212i	$-0.199381\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	43.7691	+	43.7691i	1.44068	+	1.44068i
$924$		0			0
$925$	0.447489	−	15.7982i	0.0147134	−	0.519441i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−8.26157			−0.271053			−0.135527	−	0.990774i	$-0.543273\pi$
$929$	−8.26157			−0.271053			−0.135527	+	0.990774i	$0.543273\pi$
$930$		0			0
$931$	33.8138			1.10820
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.58748	+	3.99144i	0.0519161	+	0.130534i
$936$		0			0
$937$	15.5291	+	15.5291i	0.507315	+	0.507315i	0.913701	−	0.406386i	$-0.133211\pi$
$937$	15.5291	+	15.5291i	0.507315	+	0.507315i	−0.406386	+	0.913701i	$0.633211\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			40.9925i			1.33632i	0.744019	+	0.668158i	$0.232917\pi$
$941$			40.9925i			1.33632i	−0.744019	+	0.668158i	$0.767083\pi$
$942$		0			0
$943$	0.119670	−	0.119670i	0.00389700	−	0.00389700i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.3892	−	12.3892i	0.402594	−	0.402594i	−0.476552	−	0.879146i	$-0.658114\pi$
$947$	12.3892	−	12.3892i	0.402594	−	0.402594i	0.879146	+	0.476552i	$0.158114\pi$
$948$		0			0
$949$		−	41.9069i		−	1.36036i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−8.40834	−	8.40834i	−0.272373	−	0.272373i	0.557682	−	0.830055i	$-0.311691\pi$
$953$	−8.40834	−	8.40834i	−0.272373	−	0.272373i	−0.830055	+	0.557682i	$0.811691\pi$
$954$		0			0
$955$	−9.88282	+	22.9353i	−0.319801	+	0.742168i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.09754			−0.0354414
$960$		0			0
$961$	−5.29148			−0.170693
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.74723	−	0.694909i	0.0562452	−	0.0223699i
$966$		0			0
$967$	19.7383	+	19.7383i	0.634741	+	0.634741i	0.949253	−	0.314512i	$-0.101841\pi$
$967$	19.7383	+	19.7383i	0.634741	+	0.634741i	−0.314512	+	0.949253i	$0.601841\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	8.03812i		−	0.257955i	−0.991647	−	0.128978i	$-0.958830\pi$
$971$		−	8.03812i		−	0.257955i	0.991647	−	0.128978i	$-0.0411696\pi$
$972$		0			0
$973$	1.60786	−	1.60786i	0.0515455	−	0.0515455i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.41768	+	5.41768i	−0.173327	+	0.173327i	−0.788439	−	0.615113i	$-0.789111\pi$
$977$	−5.41768	+	5.41768i	−0.173327	+	0.173327i	0.615113	+	0.788439i	$0.289111\pi$
$978$		0			0
$979$			2.88299i			0.0921409i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.52079	−	6.52079i	−0.207981	−	0.207981i	0.595428	−	0.803409i	$-0.296983\pi$
$983$	−6.52079	−	6.52079i	−0.207981	−	0.207981i	−0.803409	+	0.595428i	$0.796983\pi$
$984$		0			0
$985$	9.81621	−	3.90411i	0.312770	−	0.124395i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.232169			0.00738254
$990$		0			0
$991$	−53.7026			−1.70592			−0.852959	−	0.521978i	$-0.825194\pi$
$991$	−53.7026			−1.70592			−0.852959	+	0.521978i	$0.825194\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	11.1725	−	25.9282i	0.354191	−	0.821979i
$996$		0			0
$997$	−9.73139	−	9.73139i	−0.308196	−	0.308196i	0.536013	−	0.844209i	$-0.319930\pi$
$997$	−9.73139	−	9.73139i	−0.308196	−	0.308196i	−0.844209	+	0.536013i	$0.819930\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.w.f.2177.12	yes	28
3.2	odd	2	inner	3060.2.w.f.2177.3	yes	28
5.3	odd	4	inner	3060.2.w.f.953.3	✓	28
15.8	even	4	inner	3060.2.w.f.953.12	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3060.2.w.f.953.3	✓	28	5.3	odd	4	inner
3060.2.w.f.953.12	yes	28	15.8	even	4	inner
3060.2.w.f.2177.3	yes	28	3.2	odd	2	inner
3060.2.w.f.2177.12	yes	28	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(2.05353 + 0.884870i) q^{5} +(0.130751 + 0.130751i) q^{7} +1.92102i q^{11} +(4.58124 - 4.58124i) q^{13} +(0.707107 - 0.707107i) q^{17} +4.85426i q^{19} +(-1.87618 - 1.87618i) q^{23} +(3.43401 + 3.63422i) q^{25} +2.18605 q^{29} +5.07036 q^{31} +(0.152805 + 0.384200i) q^{35} +(-2.23510 - 2.23510i) q^{37} +0.0637839i q^{41} +(-0.0618727 + 0.0618727i) q^{43} +(-1.55630 + 1.55630i) q^{47} -6.96581i q^{49} +(3.22520 + 3.22520i) q^{53} +(-1.69986 + 3.94489i) q^{55} +8.32179 q^{59} +1.32701 q^{61} +(13.4615 - 5.35393i) q^{65} +(-6.19130 - 6.19130i) q^{67} +9.55399i q^{71} +(4.57376 - 4.57376i) q^{73} +(-0.251177 + 0.251177i) q^{77} +7.02471i q^{79} +(0.282320 + 0.282320i) q^{83} +(2.07777 - 0.826371i) q^{85} +1.50076 q^{89} +1.19801 q^{91} +(-4.29539 + 9.96839i) q^{95} +(-6.73088 - 6.73088i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 8 q^{13} + 16 q^{25} - 24 q^{31} - 4 q^{37} + 36 q^{43} + 40 q^{55} - 40 q^{61} + 64 q^{67} + 28 q^{73} - 4 q^{85} - 120 q^{91} + 44 q^{97}+O(q^{100}) \) 28 * q + 8 * q^13 + 16 * q^25 - 24 * q^31 - 4 * q^37 + 36 * q^43 + 40 * q^55 - 40 * q^61 + 64 * q^67 + 28 * q^73 - 4 * q^85 - 120 * q^91 + 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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