LMFDB - Embedded newform 3276.2.gj.b.17.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3276,2,Mod(17,3276)] 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("3276.17"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3276, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3276.gj (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,2] 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:	$26.1589917022$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.1
Root			$1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3276.17
Dual form			3276.2.gj.b.1349.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 2.59808i) q^{7} +(2.50000 + 2.59808i) q^{13} -7.34847 q^{17} +(0.500000 - 0.866025i) q^{19} +8.48528i q^{23} +(-2.50000 + 4.33013i) q^{25} +(3.67423 + 2.12132i) q^{29} +(1.00000 - 1.73205i) q^{31} +8.66025i q^{37} +(-7.34847 - 4.24264i) q^{41} +(-0.500000 - 0.866025i) q^{43} +(3.67423 - 2.12132i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(7.34847 + 4.24264i) q^{53} +12.7279i q^{59} +(10.5000 + 6.06218i) q^{61} +(6.00000 - 3.46410i) q^{67} +(-3.67423 - 6.36396i) q^{71} +(5.50000 - 9.52628i) q^{73} +(7.00000 + 12.1244i) q^{79} +(8.00000 - 5.19615i) q^{91} +(-2.50000 - 4.33013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{7} + 10 q^{13} + 2 q^{19} - 10 q^{25} + 4 q^{31} - 2 q^{43} - 26 q^{49} + 42 q^{61} + 24 q^{67} + 22 q^{73} + 28 q^{79} + 32 q^{91} - 10 q^{97}+O(q^{100}) $ 4 * q + 2 * q^7 + 10 * q^13 + 2 * q^19 - 10 * q^25 + 4 * q^31 - 2 * q^43 - 26 * q^49 + 42 * q^61 + 24 * q^67 + 22 * q^73 + 28 * q^79 + 32 * q^91 - 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3276\mathbb{Z}\right)^\times$.

$n$	$1639$	$2017$	$2341$	$2549$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$-1$

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		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$	0.500000	−	2.59808i	0.188982	−	0.981981i
$7$	0.500000	−	2.59808i	0.188982	−	0.981981i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	2.50000	+	2.59808i	0.693375	+	0.720577i
$13$	2.50000	+	2.59808i	0.693375	+	0.720577i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−7.34847			−1.78227			−0.891133	−	0.453743i	$-0.850089\pi$
$17$	−7.34847			−1.78227			−0.891133	+	0.453743i	$0.850089\pi$
$18$		0			0
$19$	0.500000	−	0.866025i	0.114708	−	0.198680i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	0.500000	−	0.866025i	0.114708	−	0.198680i	0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			8.48528i			1.76930i	0.466252	+	0.884652i	$0.345604\pi$
$23$			8.48528i			1.76930i	−0.466252	+	0.884652i	$0.654396\pi$
$24$		0			0
$25$	−2.50000	+	4.33013i	−0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.67423	+	2.12132i	0.682288	+	0.393919i	0.800717	−	0.599043i	$-0.204452\pi$
$29$	3.67423	+	2.12132i	0.682288	+	0.393919i	−0.118428	+	0.992963i	$0.537786\pi$
$30$		0			0
$31$	1.00000	−	1.73205i	0.179605	−	0.311086i	−0.762140	−	0.647412i	$-0.775851\pi$
$31$	1.00000	−	1.73205i	0.179605	−	0.311086i	0.941745	+	0.336327i	$0.109185\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			8.66025i			1.42374i	0.702313	+	0.711868i	$0.252151\pi$
$37$			8.66025i			1.42374i	−0.702313	+	0.711868i	$0.747849\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.34847	−	4.24264i	−1.14764	−	0.662589i	−0.199328	−	0.979933i	$-0.563876\pi$
$41$	−7.34847	−	4.24264i	−1.14764	−	0.662589i	−0.948311	+	0.317344i	$0.897209\pi$
$42$		0			0
$43$	−0.500000	−	0.866025i	−0.0762493	−	0.132068i	0.825380	−	0.564578i	$-0.190961\pi$
$43$	−0.500000	−	0.866025i	−0.0762493	−	0.132068i	−0.901629	+	0.432511i	$0.857628\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.67423	−	2.12132i	0.535942	−	0.309426i	−0.207491	−	0.978237i	$-0.566530\pi$
$47$	3.67423	−	2.12132i	0.535942	−	0.309426i	0.743433	+	0.668811i	$0.233196\pi$
$48$		0			0
$49$	−6.50000	−	2.59808i	−0.928571	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.34847	+	4.24264i	1.00939	+	0.582772i	0.911013	−	0.412378i	$-0.135302\pi$
$53$	7.34847	+	4.24264i	1.00939	+	0.582772i	0.0983769	+	0.995149i	$0.468635\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$			12.7279i			1.65703i	0.559964	+	0.828517i	$0.310815\pi$
$59$			12.7279i			1.65703i	−0.559964	+	0.828517i	$0.689185\pi$
$60$		0			0
$61$	10.5000	+	6.06218i	1.34439	+	0.776182i	0.987448	−	0.157945i	$-0.0504869\pi$
$61$	10.5000	+	6.06218i	1.34439	+	0.776182i	0.356939	+	0.934128i	$0.383820\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.00000	−	3.46410i	0.733017	−	0.423207i	−0.0865081	−	0.996251i	$-0.527571\pi$
$67$	6.00000	−	3.46410i	0.733017	−	0.423207i	0.819525	+	0.573044i	$0.194238\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.67423	−	6.36396i	−0.436051	−	0.755263i	0.561329	−	0.827592i	$-0.310290\pi$
$71$	−3.67423	−	6.36396i	−0.436051	−	0.755263i	−0.997381	+	0.0723293i	$0.976957\pi$
$72$		0			0
$73$	5.50000	−	9.52628i	0.643726	−	1.11497i	−0.340868	−	0.940111i	$-0.610721\pi$
$73$	5.50000	−	9.52628i	0.643726	−	1.11497i	0.984594	−	0.174855i	$-0.0559458\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	7.00000	+	12.1244i	0.787562	+	1.36410i	0.927457	+	0.373930i	$0.121990\pi$
$79$	7.00000	+	12.1244i	0.787562	+	1.36410i	−0.139895	+	0.990166i	$0.544677\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	8.00000	−	5.19615i	0.838628	−	0.544705i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−2.50000	−	4.33013i	−0.253837	−	0.439658i	0.710742	−	0.703452i	$-0.248359\pi$
$97$	−2.50000	−	4.33013i	−0.253837	−	0.439658i	−0.964579	+	0.263795i	$0.915026\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.67423	+	6.36396i	0.365600	+	0.633238i	0.988872	−	0.148767i	$-0.0475305\pi$
$101$	3.67423	+	6.36396i	0.365600	+	0.633238i	−0.623272	+	0.782005i	$0.714197\pi$
$102$		0			0
$103$	4.50000	−	2.59808i	0.443398	−	0.255996i	−0.261640	−	0.965166i	$-0.584263\pi$
$103$	4.50000	−	2.59808i	0.443398	−	0.255996i	0.705038	+	0.709170i	$0.250930\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			4.24264i			0.410152i	0.978746	+	0.205076i	$0.0657441\pi$
$107$			4.24264i			0.410152i	−0.978746	+	0.205076i	$0.934256\pi$
$108$		0			0
$109$	−13.5000	−	7.79423i	−1.29307	−	0.746552i	−0.313869	−	0.949466i	$-0.601625\pi$
$109$	−13.5000	−	7.79423i	−1.29307	−	0.746552i	−0.979196	+	0.202915i	$0.934959\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.34847	+	4.24264i	−0.691286	+	0.399114i	−0.804094	−	0.594503i	$-0.797349\pi$
$113$	−7.34847	+	4.24264i	−0.691286	+	0.399114i	0.112808	+	0.993617i	$0.464016\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.67423	+	19.0919i	−0.336817	+	1.75015i
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	5.50000	−	9.52628i	0.488046	−	0.845321i	−0.511859	−	0.859069i	$-0.671043\pi$
$127$	5.50000	−	9.52628i	0.488046	−	0.845321i	0.999905	+	0.0137486i	$0.00437646\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	11.0227	+	19.0919i	0.963058	+	1.66807i	0.714744	+	0.699386i	$0.246543\pi$
$131$	11.0227	+	19.0919i	0.963058	+	1.66807i	0.248314	+	0.968680i	$0.420124\pi$
$132$		0			0
$133$	−2.00000	−	1.73205i	−0.173422	−	0.150188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.34847			−0.627822			−0.313911	−	0.949452i	$-0.601639\pi$
$137$	−7.34847			−0.627822			−0.313911	+	0.949452i	$0.601639\pi$
$138$		0			0
$139$	−6.00000	+	3.46410i	−0.508913	+	0.293821i	−0.732387	−	0.680889i	$-0.761594\pi$
$139$	−6.00000	+	3.46410i	−0.508913	+	0.293821i	0.223474	+	0.974710i	$0.428260\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.34847	+	12.7279i	−0.602010	+	1.04271i	0.390506	+	0.920600i	$0.372300\pi$
$149$	−7.34847	+	12.7279i	−0.602010	+	1.04271i	−0.992516	+	0.122112i	$0.961033\pi$
$150$		0			0
$151$	18.0000	+	10.3923i	1.46482	+	0.845714i	0.999228	−	0.0392861i	$-0.0125084\pi$
$151$	18.0000	+	10.3923i	1.46482	+	0.845714i	0.465591	+	0.885000i	$0.345842\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	16.5000	+	9.52628i	1.31684	+	0.760280i	0.983220	−	0.182426i	$-0.0583951\pi$
$157$	16.5000	+	9.52628i	1.31684	+	0.760280i	0.333624	+	0.942706i	$0.391728\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	22.0454	+	4.24264i	1.73742	+	0.334367i
$162$		0			0
$163$	−10.5000	−	6.06218i	−0.822423	−	0.474826i	0.0288280	−	0.999584i	$-0.490822\pi$
$163$	−10.5000	−	6.06218i	−0.822423	−	0.474826i	−0.851251	+	0.524758i	$0.824156\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.67423	+	2.12132i	0.284321	+	0.164153i	0.635378	−	0.772201i	$-0.280844\pi$
$167$	3.67423	+	2.12132i	0.284321	+	0.164153i	−0.351057	+	0.936354i	$0.614178\pi$
$168$		0			0
$169$	−0.500000	+	12.9904i	−0.0384615	+	0.999260i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.67423	−	6.36396i	0.279347	−	0.483843i	−0.691876	−	0.722017i	$-0.743215\pi$
$173$	3.67423	−	6.36396i	0.279347	−	0.483843i	0.971223	+	0.238174i	$0.0765487\pi$
$174$		0			0
$175$	10.0000	+	8.66025i	0.755929	+	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.67423	−	2.12132i	0.274625	−	0.158555i	−0.356362	−	0.934348i	$-0.615983\pi$
$179$	3.67423	−	2.12132i	0.274625	−	0.158555i	0.630988	+	0.775793i	$0.282650\pi$
$180$		0			0
$181$			12.1244i			0.901196i	0.892727	+	0.450598i	$0.148789\pi$
$181$			12.1244i			0.901196i	−0.892727	+	0.450598i	$0.851211\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.0227	−	6.36396i	−0.797575	−	0.460480i	0.0450476	−	0.998985i	$-0.485656\pi$
$191$	−11.0227	−	6.36396i	−0.797575	−	0.460480i	−0.842622	+	0.538505i	$0.818989\pi$
$192$		0			0
$193$	4.50000	−	2.59808i	0.323917	−	0.187014i	−0.329220	−	0.944253i	$-0.606786\pi$
$193$	4.50000	−	2.59808i	0.323917	−	0.187014i	0.653137	+	0.757240i	$0.273452\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.67423	+	6.36396i	−0.261778	+	0.453413i	−0.966715	−	0.255857i	$-0.917642\pi$
$197$	−3.67423	+	6.36396i	−0.261778	+	0.453413i	0.704936	+	0.709271i	$0.250976\pi$
$198$		0			0
$199$			22.5167i			1.59616i	0.602549	+	0.798082i	$0.294152\pi$
$199$			22.5167i			1.59616i	−0.602549	+	0.798082i	$0.705848\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.34847	−	8.48528i	0.515761	−	0.595550i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−5.50000	+	9.52628i	−0.378636	+	0.655816i	−0.990864	−	0.134865i	$-0.956940\pi$
$211$	−5.50000	+	9.52628i	−0.378636	+	0.655816i	0.612228	+	0.790681i	$0.290273\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−4.00000	−	3.46410i	−0.271538	−	0.235159i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−18.3712	−	19.0919i	−1.23578	−	1.28426i
$222$		0			0
$223$	5.00000	−	8.66025i	0.334825	−	0.579934i	−0.648626	−	0.761107i	$-0.724656\pi$
$223$	5.00000	−	8.66025i	0.334825	−	0.579934i	0.983451	+	0.181173i	$0.0579895\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			21.2132i			1.40797i	0.710215	+	0.703985i	$0.248598\pi$
$227$			21.2132i			1.40797i	−0.710215	+	0.703985i	$0.751402\pi$
$228$		0			0
$229$	−12.5000	−	21.6506i	−0.826023	−	1.43071i	−0.901135	−	0.433539i	$-0.857265\pi$
$229$	−12.5000	−	21.6506i	−0.826023	−	1.43071i	0.0751115	−	0.997175i	$-0.476069\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−22.0454	+	12.7279i	−1.44424	+	0.833834i	−0.998129	−	0.0611424i	$-0.980526\pi$
$233$	−22.0454	+	12.7279i	−1.44424	+	0.833834i	−0.446114	+	0.894976i	$0.647192\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−7.34847			−0.475333			−0.237666	−	0.971347i	$-0.576383\pi$
$239$	−7.34847			−0.475333			−0.237666	+	0.971347i	$0.576383\pi$
$240$		0			0
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3.50000	−	0.866025i	0.222700	−	0.0551039i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.67423	−	6.36396i	−0.231916	−	0.401690i	0.726456	−	0.687213i	$-0.241166\pi$
$251$	−3.67423	−	6.36396i	−0.231916	−	0.401690i	−0.958372	+	0.285523i	$0.907833\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.34847			−0.458385			−0.229192	−	0.973381i	$-0.573609\pi$
$257$	−7.34847			−0.458385			−0.229192	+	0.973381i	$0.573609\pi$
$258$		0			0
$259$	22.5000	+	4.33013i	1.39808	+	0.269061i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−14.6969	+	8.48528i	−0.906252	+	0.523225i	−0.879223	−	0.476410i	$-0.841938\pi$
$263$	−14.6969	+	8.48528i	−0.906252	+	0.523225i	−0.0270287	+	0.999635i	$0.508605\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−19.0000			−1.15417			−0.577084	−	0.816685i	$-0.695809\pi$
$271$	−19.0000			−1.15417			−0.577084	+	0.816685i	$0.695809\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−13.0000			−0.781094			−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000			−0.781094			−0.390547	+	0.920583i	$0.627714\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.6969			0.876746			0.438373	−	0.898793i	$-0.355555\pi$
$281$	14.6969			0.876746			0.438373	+	0.898793i	$0.355555\pi$
$282$		0			0
$283$	13.5000	−	7.79423i	0.802492	−	0.463319i	−0.0418500	−	0.999124i	$-0.513325\pi$
$283$	13.5000	−	7.79423i	0.802492	−	0.463319i	0.844342	+	0.535805i	$0.179992\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−14.6969	+	16.9706i	−0.867533	+	1.00174i
$288$		0			0
$289$	37.0000			2.17647
$290$		0			0
$291$		0			0
$292$		0			0
$293$	25.7196	−	14.8492i	1.50256	−	0.867502i	0.502562	−	0.864541i	$-0.332391\pi$
$293$	25.7196	−	14.8492i	1.50256	−	0.867502i	0.999996	−	0.00296079i	$-0.000942451\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−22.0454	+	21.2132i	−1.27492	+	1.22679i
$300$		0			0
$301$	−2.50000	+	0.866025i	−0.144098	+	0.0499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−26.0000			−1.48390			−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000			−1.48390			−0.741949	+	0.670456i	$0.766098\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	7.34847	−	12.7279i	0.416693	−	0.721734i	−0.578911	−	0.815391i	$-0.696522\pi$
$311$	7.34847	−	12.7279i	0.416693	−	0.721734i	0.995605	+	0.0936564i	$0.0298555\pi$
$312$		0			0
$313$	4.50000	−	2.59808i	0.254355	−	0.146852i	−0.367402	−	0.930062i	$-0.619753\pi$
$313$	4.50000	−	2.59808i	0.254355	−	0.146852i	0.621757	+	0.783210i	$0.286419\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.34847	−	12.7279i	−0.412731	−	0.714871i	0.582456	−	0.812862i	$-0.302092\pi$
$317$	−7.34847	−	12.7279i	−0.412731	−	0.714871i	−0.995187	+	0.0979908i	$0.968758\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.67423	+	6.36396i	−0.204440	+	0.354100i
$324$		0			0
$325$	−17.5000	+	4.33013i	−0.970725	+	0.240192i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.67423	−	10.6066i	−0.202567	−	0.584761i
$330$		0			0
$331$	4.50000	+	2.59808i	0.247342	+	0.142803i	0.618547	−	0.785748i	$-0.287722\pi$
$331$	4.50000	+	2.59808i	0.247342	+	0.142803i	−0.371204	+	0.928551i	$0.621055\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−31.0000			−1.68868			−0.844339	−	0.535810i	$-0.820006\pi$
$337$	−31.0000			−1.68868			−0.844339	+	0.535810i	$0.820006\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			21.2132i			1.13878i	0.822066	+	0.569392i	$0.192821\pi$
$347$			21.2132i			1.13878i	−0.822066	+	0.569392i	$0.807179\pi$
$348$		0			0
$349$	2.50000	−	4.33013i	0.133822	−	0.231786i	−0.791325	−	0.611396i	$-0.790608\pi$
$349$	2.50000	−	4.33013i	0.133822	−	0.231786i	0.925147	+	0.379610i	$0.123942\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	11.0227	−	6.36396i	0.586679	−	0.338719i	−0.177104	−	0.984192i	$-0.556673\pi$
$353$	11.0227	−	6.36396i	0.586679	−	0.338719i	0.763783	+	0.645473i	$0.223340\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	11.0227	+	19.0919i	0.581756	+	1.00763i	0.995271	+	0.0971339i	$0.0309675\pi$
$359$	11.0227	+	19.0919i	0.581756	+	1.00763i	−0.413515	+	0.910497i	$0.635699\pi$
$360$		0			0
$361$	9.00000	+	15.5885i	0.473684	+	0.820445i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−4.50000	+	2.59808i	−0.234898	+	0.135618i	−0.612830	−	0.790215i	$-0.709969\pi$
$367$	−4.50000	+	2.59808i	−0.234898	+	0.135618i	0.377932	+	0.925834i	$0.376635\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	14.6969	−	16.9706i	0.763027	−	0.881068i
$372$		0			0
$373$	10.0000	−	17.3205i	0.517780	−	0.896822i	−0.482006	−	0.876168i	$-0.660092\pi$
$373$	10.0000	−	17.3205i	0.517780	−	0.896822i	0.999787	−	0.0206542i	$-0.00657489\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.67423	+	14.8492i	0.189233	+	0.764775i
$378$		0			0
$379$	30.0000	+	17.3205i	1.54100	+	0.889695i	0.998776	+	0.0494584i	$0.0157495\pi$
$379$	30.0000	+	17.3205i	1.54100	+	0.889695i	0.542220	+	0.840236i	$0.317584\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−14.6969	−	8.48528i	−0.750978	−	0.433578i	0.0750689	−	0.997178i	$-0.476082\pi$
$383$	−14.6969	−	8.48528i	−0.750978	−	0.433578i	−0.826047	+	0.563601i	$0.809416\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.34847	−	4.24264i	−0.372582	−	0.215110i	0.302004	−	0.953307i	$-0.402344\pi$
$389$	−7.34847	−	4.24264i	−0.372582	−	0.215110i	−0.674586	+	0.738196i	$0.735678\pi$
$390$		0			0
$391$		−	62.3538i		−	3.15337i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	3.50000	−	6.06218i	0.175660	−	0.304252i	−0.764730	−	0.644351i	$-0.777127\pi$
$397$	3.50000	−	6.06218i	0.175660	−	0.304252i	0.940389	+	0.340099i	$0.110461\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	36.7423			1.83483			0.917413	−	0.397937i	$-0.130274\pi$
$401$	36.7423			1.83483			0.917413	+	0.397937i	$0.130274\pi$
$402$		0			0
$403$	7.00000	−	1.73205i	0.348695	−	0.0862796i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−13.0000			−0.642809			−0.321404	−	0.946942i	$-0.604155\pi$
$409$	−13.0000			−0.642809			−0.321404	+	0.946942i	$0.604155\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	33.0681	+	6.36396i	1.62718	+	0.313150i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	18.3712	−	31.8198i	0.897491	−	1.55450i	0.0667989	−	0.997766i	$-0.478721\pi$
$419$	18.3712	−	31.8198i	0.897491	−	1.55450i	0.830692	−	0.556733i	$-0.187945\pi$
$420$		0			0
$421$		−	3.46410i		−	0.168830i	−0.996431	−	0.0844150i	$-0.973098\pi$
$421$		−	3.46410i		−	0.168830i	0.996431	−	0.0844150i	$-0.0269021\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	18.3712	−	31.8198i	0.891133	−	1.54349i
$426$		0			0
$427$	21.0000	−	24.2487i	1.01626	−	1.17348i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	14.6969	+	25.4558i	0.707927	+	1.22616i	0.965625	+	0.259939i	$0.0837026\pi$
$431$	14.6969	+	25.4558i	0.707927	+	1.22616i	−0.257698	+	0.966225i	$0.582964\pi$
$432$		0			0
$433$	−15.0000	+	8.66025i	−0.720854	+	0.416185i	−0.815067	−	0.579367i	$-0.803300\pi$
$433$	−15.0000	+	8.66025i	−0.720854	+	0.416185i	0.0942129	+	0.995552i	$0.469967\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.34847	+	4.24264i	0.351525	+	0.202953i
$438$		0			0
$439$			22.5167i			1.07466i	0.843372	+	0.537331i	$0.180567\pi$
$439$			22.5167i			1.07466i	−0.843372	+	0.537331i	$0.819433\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.67423	+	2.12132i	−0.174568	+	0.100787i	−0.584738	−	0.811222i	$-0.698803\pi$
$443$	−3.67423	+	2.12132i	−0.174568	+	0.100787i	0.410170	+	0.912009i	$0.365469\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−7.34847	−	12.7279i	−0.346796	−	0.600668i	0.638883	−	0.769304i	$-0.279397\pi$
$449$	−7.34847	−	12.7279i	−0.346796	−	0.600668i	−0.985678	+	0.168636i	$0.946064\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			24.2487i			1.13431i	0.823612	+	0.567153i	$0.191955\pi$
$457$			24.2487i			1.13431i	−0.823612	+	0.567153i	$0.808045\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	25.7196	−	14.8492i	1.19788	−	0.691598i	0.237800	−	0.971314i	$-0.423574\pi$
$461$	25.7196	−	14.8492i	1.19788	−	0.691598i	0.960083	+	0.279716i	$0.0902403\pi$
$462$		0			0
$463$			5.19615i			0.241486i	0.992684	+	0.120743i	$0.0385276\pi$
$463$			5.19615i			0.241486i	−0.992684	+	0.120743i	$0.961472\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−11.0227	−	19.0919i	−0.510070	−	0.883467i	−0.999932	−	0.0116670i	$-0.996286\pi$
$467$	−11.0227	−	19.0919i	−0.510070	−	0.883467i	0.489862	−	0.871800i	$-0.337047\pi$
$468$		0			0
$469$	−6.00000	−	17.3205i	−0.277054	−	0.799787i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	2.50000	+	4.33013i	0.114708	+	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−25.7196	+	14.8492i	−1.17516	+	0.678479i	−0.954890	−	0.296960i	$-0.904027\pi$
$479$	−25.7196	+	14.8492i	−1.17516	+	0.678479i	−0.220270	+	0.975439i	$0.570694\pi$
$480$		0			0
$481$	−22.5000	+	21.6506i	−1.02591	+	0.987184i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$			19.0526i			0.863354i	0.902028	+	0.431677i	$0.142078\pi$
$487$			19.0526i			0.863354i	−0.902028	+	0.431677i	$0.857922\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−29.3939	−	16.9706i	−1.32653	−	0.765871i	−0.341766	−	0.939785i	$-0.611025\pi$
$491$	−29.3939	−	16.9706i	−1.32653	−	0.765871i	−0.984761	+	0.173914i	$0.944358\pi$
$492$		0			0
$493$	−27.0000	−	15.5885i	−1.21602	−	0.702069i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−18.3712	+	6.36396i	−0.824060	+	0.285463i
$498$		0			0
$499$	16.5000	−	9.52628i	0.738641	−	0.426455i	−0.0829337	−	0.996555i	$-0.526429\pi$
$499$	16.5000	−	9.52628i	0.738641	−	0.426455i	0.821575	+	0.570100i	$0.193096\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.67423	+	6.36396i	0.163826	+	0.283755i	0.936238	−	0.351367i	$-0.114283\pi$
$503$	3.67423	+	6.36396i	0.163826	+	0.283755i	−0.772412	+	0.635122i	$0.780950\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	4.24264i		−	0.188052i	−0.995570	−	0.0940259i	$-0.970026\pi$
$509$		−	4.24264i		−	0.188052i	0.995570	−	0.0940259i	$-0.0299736\pi$
$510$		0			0
$511$	−22.0000	−	19.0526i	−0.973223	−	0.842836i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	11.0227	−	19.0919i	0.482913	−	0.836431i	−0.516894	−	0.856049i	$-0.672912\pi$
$521$	11.0227	−	19.0919i	0.482913	−	0.836431i	0.999808	+	0.0196188i	$0.00624525\pi$
$522$		0			0
$523$			8.66025i			0.378686i	0.981911	+	0.189343i	$0.0606359\pi$
$523$			8.66025i			0.378686i	−0.981911	+	0.189343i	$0.939364\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.34847	+	12.7279i	−0.320104	+	0.554437i
$528$		0			0
$529$	−49.0000			−2.13043
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.34847	−	29.6985i	−0.318298	−	1.28638i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−22.5000	+	12.9904i	−0.967351	+	0.558500i	−0.898428	−	0.439122i	$-0.855290\pi$
$541$	−22.5000	+	12.9904i	−0.967351	+	0.558500i	−0.0689231	+	0.997622i	$0.521956\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−13.0000			−0.555840			−0.277920	−	0.960604i	$-0.589645\pi$
$547$	−13.0000			−0.555840			−0.277920	+	0.960604i	$0.589645\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3.67423	−	2.12132i	0.156528	−	0.0903713i
$552$		0			0
$553$	35.0000	−	12.1244i	1.48835	−	0.515580i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−22.0454	−	38.1838i	−0.934094	−	1.61790i	−0.776242	−	0.630435i	$-0.782876\pi$
$557$	−22.0454	−	38.1838i	−0.934094	−	1.61790i	−0.157852	−	0.987463i	$-0.550457\pi$
$558$		0			0
$559$	1.00000	−	3.46410i	0.0422955	−	0.146516i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.34847			0.309701			0.154851	−	0.987938i	$-0.450510\pi$
$563$	7.34847			0.309701			0.154851	+	0.987938i	$0.450510\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			12.7279i			0.533582i	0.963754	+	0.266791i	$0.0859634\pi$
$569$			12.7279i			0.533582i	−0.963754	+	0.266791i	$0.914037\pi$
$570$		0			0
$571$	5.50000	−	9.52628i	0.230168	−	0.398662i	−0.727690	−	0.685907i	$-0.759406\pi$
$571$	5.50000	−	9.52628i	0.230168	−	0.398662i	0.957857	+	0.287244i	$0.0927391\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−36.7423	−	21.2132i	−1.53226	−	0.884652i
$576$		0			0
$577$	9.50000	−	16.4545i	0.395490	−	0.685009i	−0.597673	−	0.801740i	$-0.703908\pi$
$577$	9.50000	−	16.4545i	0.395490	−	0.685009i	0.993164	+	0.116731i	$0.0372414\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	29.3939	+	16.9706i	1.21322	+	0.700450i	0.963458	−	0.267858i	$-0.0863160\pi$
$587$	29.3939	+	16.9706i	1.21322	+	0.700450i	0.249757	+	0.968309i	$0.419649\pi$
$588$		0			0
$589$	−1.00000	−	1.73205i	−0.0412043	−	0.0713679i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−36.7423	+	21.2132i	−1.50883	+	0.871122i	−0.508880	+	0.860837i	$0.669940\pi$
$593$	−36.7423	+	21.2132i	−1.50883	+	0.871122i	−0.999947	+	0.0102845i	$0.996726\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	25.7196	+	14.8492i	1.05088	+	0.606724i	0.922894	−	0.385053i	$-0.125817\pi$
$599$	25.7196	+	14.8492i	1.05088	+	0.606724i	0.127982	+	0.991777i	$0.459150\pi$
$600$		0			0
$601$	−34.5000	−	19.9186i	−1.40728	−	0.812496i	−0.412159	−	0.911112i	$-0.635225\pi$
$601$	−34.5000	−	19.9186i	−1.40728	−	0.812496i	−0.995126	+	0.0986160i	$0.968558\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	34.5000	+	19.9186i	1.40031	+	0.808470i	0.994424	−	0.105453i	$-0.0336291\pi$
$607$	34.5000	+	19.9186i	1.40031	+	0.808470i	0.405887	+	0.913923i	$0.366962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	14.6969	+	4.24264i	0.594574	+	0.171639i
$612$		0			0
$613$	−19.5000	+	11.2583i	−0.787598	+	0.454720i	−0.839116	−	0.543952i	$-0.816927\pi$
$613$	−19.5000	+	11.2583i	−0.787598	+	0.454720i	0.0515185	+	0.998672i	$0.483594\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14.6969	−	25.4558i	−0.591676	−	1.02481i	−0.994007	−	0.109319i	$-0.965133\pi$
$617$	−14.6969	−	25.4558i	−0.591676	−	1.02481i	0.402330	−	0.915495i	$-0.368200\pi$
$618$		0			0
$619$	18.5000	−	32.0429i	0.743578	−	1.28791i	−0.207279	−	0.978282i	$-0.566461\pi$
$619$	18.5000	−	32.0429i	0.743578	−	1.28791i	0.950856	−	0.309633i	$-0.100206\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	63.6396i		−	2.53748i
$630$		0			0
$631$	−34.5000	+	19.9186i	−1.37342	+	0.792946i	−0.991357	−	0.131189i	$-0.958121\pi$
$631$	−34.5000	+	19.9186i	−1.37342	+	0.792946i	−0.382066	+	0.924135i	$0.624787\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−9.50000	−	23.3827i	−0.376404	−	0.926456i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	33.9411i		−	1.34059i	−0.742093	−	0.670297i	$-0.766167\pi$
$641$		−	33.9411i		−	1.34059i	0.742093	−	0.670297i	$-0.233833\pi$
$642$		0			0
$643$	2.50000	+	4.33013i	0.0985904	+	0.170764i	0.911101	−	0.412182i	$-0.135233\pi$
$643$	2.50000	+	4.33013i	0.0985904	+	0.170764i	−0.812511	+	0.582946i	$0.801900\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−14.6969	−	25.4558i	−0.577796	−	1.00077i	−0.995732	−	0.0922957i	$-0.970579\pi$
$647$	−14.6969	−	25.4558i	−0.577796	−	1.00077i	0.417935	−	0.908477i	$-0.362754\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			16.9706i			0.664109i	0.943260	+	0.332055i	$0.107742\pi$
$653$			16.9706i			0.664109i	−0.943260	+	0.332055i	$0.892258\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.67423	−	2.12132i	0.143128	−	0.0826349i	−0.426726	−	0.904381i	$-0.640333\pi$
$659$	3.67423	−	2.12132i	0.143128	−	0.0826349i	0.569854	+	0.821746i	$0.307000\pi$
$660$		0			0
$661$	−8.00000	−	13.8564i	−0.311164	−	0.538952i	0.667451	−	0.744654i	$-0.267385\pi$
$661$	−8.00000	−	13.8564i	−0.311164	−	0.538952i	−0.978615	+	0.205702i	$0.934052\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−18.0000	+	31.1769i	−0.696963	+	1.20717i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−9.50000	+	16.4545i	−0.366198	+	0.634274i	−0.988968	−	0.148132i	$-0.952674\pi$
$673$	−9.50000	+	16.4545i	−0.366198	+	0.634274i	0.622770	+	0.782405i	$0.286007\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.67423	−	6.36396i	−0.141212	−	0.244587i	0.786741	−	0.617283i	$-0.211767\pi$
$677$	−3.67423	−	6.36396i	−0.141212	−	0.244587i	−0.927953	+	0.372696i	$0.878433\pi$
$678$		0			0
$679$	−12.5000	+	4.33013i	−0.479706	+	0.166175i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	44.0908			1.68709			0.843544	−	0.537060i	$-0.180465\pi$
$683$	44.0908			1.68709			0.843544	+	0.537060i	$0.180465\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7.34847	+	29.6985i	0.279954	+	1.13142i
$690$		0			0
$691$	−1.00000			−0.0380418			−0.0190209	−	0.999819i	$-0.506055\pi$
$691$	−1.00000			−0.0380418			−0.0190209	+	0.999819i	$0.506055\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	54.0000	+	31.1769i	2.04540	+	1.18091i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	33.9411i		−	1.28194i	−0.767567	−	0.640969i	$-0.778533\pi$
$701$		−	33.9411i		−	1.28194i	0.767567	−	0.640969i	$-0.221467\pi$
$702$		0			0
$703$	7.50000	+	4.33013i	0.282868	+	0.163314i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	18.3712	−	6.36396i	0.690919	−	0.239341i
$708$		0			0
$709$	−28.5000	−	16.4545i	−1.07034	−	0.617961i	−0.142066	−	0.989857i	$-0.545374\pi$
$709$	−28.5000	−	16.4545i	−1.07034	−	0.617961i	−0.928274	+	0.371896i	$0.878708\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	14.6969	+	8.48528i	0.550405	+	0.317776i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	14.6969	−	25.4558i	0.548103	−	0.949343i	−0.450301	−	0.892877i	$-0.648683\pi$
$719$	14.6969	−	25.4558i	0.548103	−	0.949343i	0.998405	−	0.0564661i	$-0.0179833\pi$
$720$		0			0
$721$	−4.50000	−	12.9904i	−0.167589	−	0.483787i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−18.3712	+	10.6066i	−0.682288	+	0.393919i
$726$		0			0
$727$			24.2487i			0.899335i	0.893196	+	0.449667i	$0.148458\pi$
$727$			24.2487i			0.899335i	−0.893196	+	0.449667i	$0.851542\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3.67423	+	6.36396i	0.135896	+	0.235380i
$732$		0			0
$733$	10.0000	+	17.3205i	0.369358	+	0.639748i	0.989465	−	0.144770i	$-0.0462441\pi$
$733$	10.0000	+	17.3205i	0.369358	+	0.639748i	−0.620107	+	0.784517i	$0.712911\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	28.5000	−	16.4545i	1.04839	−	0.605288i	0.126191	−	0.992006i	$-0.459725\pi$
$739$	28.5000	−	16.4545i	1.04839	−	0.605288i	0.922198	+	0.386718i	$0.126391\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	7.34847	−	12.7279i	0.269589	−	0.466942i	−0.699167	−	0.714959i	$-0.746445\pi$
$743$	7.34847	−	12.7279i	0.269589	−	0.466942i	0.968756	+	0.248017i	$0.0797788\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	11.0227	+	2.12132i	0.402761	+	0.0775114i
$750$		0			0
$751$	25.0000			0.912263			0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000			0.912263			0.456131	+	0.889912i	$0.349235\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−10.0000	+	17.3205i	−0.363456	+	0.629525i	−0.988527	−	0.151043i	$-0.951737\pi$
$757$	−10.0000	+	17.3205i	−0.363456	+	0.629525i	0.625071	+	0.780568i	$0.285070\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−7.34847	−	4.24264i	−0.266382	−	0.153796i	0.360860	−	0.932620i	$-0.382483\pi$
$761$	−7.34847	−	4.24264i	−0.266382	−	0.153796i	−0.627242	+	0.778824i	$0.715816\pi$
$762$		0			0
$763$	−27.0000	+	31.1769i	−0.977466	+	1.12868i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−33.0681	+	31.8198i	−1.19402	+	1.14895i
$768$		0			0
$769$	23.5000	−	40.7032i	0.847432	−	1.46779i	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	23.5000	−	40.7032i	0.847432	−	1.46779i	0.883493	−	0.468445i	$-0.155186\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	42.4264i		−	1.52597i	−0.646415	−	0.762986i	$-0.723733\pi$
$773$		−	42.4264i		−	1.52597i	0.646415	−	0.762986i	$-0.276267\pi$
$774$		0			0
$775$	5.00000	+	8.66025i	0.179605	+	0.311086i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.34847	+	4.24264i	−0.263286	+	0.152008i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−43.0000			−1.53278			−0.766392	−	0.642373i	$-0.777950\pi$
$787$	−43.0000			−1.53278			−0.766392	+	0.642373i	$0.777950\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	7.34847	+	21.2132i	0.261281	+	0.754255i
$792$		0			0
$793$	10.5000	+	42.4352i	0.372866	+	1.50692i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−22.0454	−	38.1838i	−0.780888	−	1.35254i	−0.931425	−	0.363934i	$-0.881433\pi$
$797$	−22.0454	−	38.1838i	−0.780888	−	1.35254i	0.150536	−	0.988604i	$-0.451900\pi$
$798$		0			0
$799$	−27.0000	+	15.5885i	−0.955191	+	0.551480i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−25.7196	+	14.8492i	−0.904254	+	0.522072i	−0.878578	−	0.477599i	$-0.841507\pi$
$809$	−25.7196	+	14.8492i	−0.904254	+	0.522072i	−0.0256764	+	0.999670i	$0.508174\pi$
$810$		0			0
$811$	−17.0000			−0.596951			−0.298475	−	0.954417i	$-0.596478\pi$
$811$	−17.0000			−0.596951			−0.298475	+	0.954417i	$0.596478\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.00000			−0.0349856
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$	−4.00000			−0.139431			−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000			−0.139431			−0.0697156	+	0.997567i	$0.522209\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−36.7423			−1.27766			−0.638828	−	0.769349i	$-0.720581\pi$
$827$	−36.7423			−1.27766			−0.638828	+	0.769349i	$0.720581\pi$
$828$		0			0
$829$	4.50000	−	2.59808i	0.156291	−	0.0902349i	−0.419815	−	0.907610i	$-0.637905\pi$
$829$	4.50000	−	2.59808i	0.156291	−	0.0902349i	0.576106	+	0.817375i	$0.304572\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	47.7650	+	19.0919i	1.65496	+	0.661495i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	47.7650	−	27.5772i	1.64903	−	0.952069i	0.671575	−	0.740936i	$-0.265618\pi$
$839$	47.7650	−	27.5772i	1.64903	−	0.952069i	0.977457	−	0.211133i	$-0.0677154\pi$
$840$		0			0
$841$	−5.50000	−	9.52628i	−0.189655	−	0.328492i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−22.0000	−	19.0526i	−0.755929	−	0.654654i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−73.4847			−2.51902
$852$		0			0
$853$	−4.00000			−0.136957			−0.0684787	−	0.997653i	$-0.521815\pi$
$853$	−4.00000			−0.136957			−0.0684787	+	0.997653i	$0.521815\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	7.50000	−	4.33013i	0.255897	−	0.147742i	−0.366565	−	0.930393i	$-0.619466\pi$
$859$	7.50000	−	4.33013i	0.255897	−	0.147742i	0.622461	+	0.782651i	$0.286133\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	3.67423	+	6.36396i	0.125072	+	0.216632i	0.921761	−	0.387758i	$-0.126750\pi$
$863$	3.67423	+	6.36396i	0.125072	+	0.216632i	−0.796689	+	0.604390i	$0.793417\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	24.0000	+	6.92820i	0.813209	+	0.234753i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	9.00000	+	5.19615i	0.303908	+	0.175462i	0.644197	−	0.764859i	$-0.277192\pi$
$877$	9.00000	+	5.19615i	0.303908	+	0.175462i	−0.340289	+	0.940321i	$0.610525\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.6969	+	25.4558i	−0.495152	+	0.857629i	−0.999984	−	0.00558847i	$-0.998221\pi$
$881$	−14.6969	+	25.4558i	−0.495152	+	0.857629i	0.504832	+	0.863218i	$0.331554\pi$
$882$		0			0
$883$	−43.0000			−1.44707			−0.723533	−	0.690290i	$-0.757483\pi$
$883$	−43.0000			−1.44707			−0.723533	+	0.690290i	$0.757483\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	29.3939			0.986950			0.493475	−	0.869760i	$-0.335726\pi$
$887$	29.3939			0.986950			0.493475	+	0.869760i	$0.335726\pi$
$888$		0			0
$889$	−22.0000	−	19.0526i	−0.737856	−	0.639002i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	4.24264i		−	0.141975i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	7.34847	−	4.24264i	0.245085	−	0.141500i
$900$		0			0
$901$	−54.0000	−	31.1769i	−1.79900	−	1.03865i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−2.50000	−	4.33013i	−0.0830111	−	0.143780i	0.821531	−	0.570164i	$-0.193120\pi$
$907$	−2.50000	−	4.33013i	−0.0830111	−	0.143780i	−0.904542	+	0.426385i	$0.859787\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	50.9117i		−	1.68678i	−0.537302	−	0.843390i	$-0.680557\pi$
$911$		−	50.9117i		−	1.68678i	0.537302	−	0.843390i	$-0.319443\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	55.1135	−	19.0919i	1.82001	−	0.630470i
$918$		0			0
$919$	18.5000	−	32.0429i	0.610259	−	1.05700i	−0.380938	−	0.924601i	$-0.624399\pi$
$919$	18.5000	−	32.0429i	0.610259	−	1.05700i	0.991197	−	0.132398i	$-0.0422678\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	7.34847	−	25.4558i	0.241878	−	0.837889i
$924$		0			0
$925$	−37.5000	−	21.6506i	−1.23299	−	0.711868i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	25.7196	+	14.8492i	0.843834	+	0.487188i	0.858566	−	0.512704i	$-0.171356\pi$
$929$	25.7196	+	14.8492i	0.843834	+	0.487188i	−0.0147316	+	0.999891i	$0.504689\pi$
$930$		0			0
$931$	−5.50000	+	4.33013i	−0.180255	+	0.141914i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			8.66025i			0.282918i	0.989944	+	0.141459i	$0.0451794\pi$
$937$			8.66025i			0.282918i	−0.989944	+	0.141459i	$0.954821\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−18.3712	−	10.6066i	−0.598883	−	0.345765i	0.169719	−	0.985493i	$-0.445714\pi$
$941$	−18.3712	−	10.6066i	−0.598883	−	0.345765i	−0.768602	+	0.639727i	$0.779047\pi$
$942$		0			0
$943$	36.0000	−	62.3538i	1.17232	−	2.03052i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	29.3939			0.955173			0.477586	−	0.878585i	$-0.341512\pi$
$947$	29.3939			0.955173			0.477586	+	0.878585i	$0.341512\pi$
$948$		0			0
$949$	38.5000	−	9.52628i	1.24976	−	0.309236i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	14.6969	−	8.48528i	0.476081	−	0.274865i	−0.242701	−	0.970101i	$-0.578033\pi$
$953$	14.6969	−	8.48528i	0.476081	−	0.274865i	0.718782	+	0.695236i	$0.244700\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.67423	+	19.0919i	−0.118647	+	0.616509i
$960$		0			0
$961$	13.5000	+	23.3827i	0.435484	+	0.754280i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1.73205i			0.0556990i	0.999612	+	0.0278495i	$0.00886592\pi$
$967$			1.73205i			0.0556990i	−0.999612	+	0.0278495i	$0.991134\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18.3712	+	31.8198i	−0.589559	+	1.02115i	0.404731	+	0.914436i	$0.367365\pi$
$971$	−18.3712	+	31.8198i	−0.589559	+	1.02115i	−0.994290	+	0.106710i	$0.965968\pi$
$972$		0			0
$973$	6.00000	+	17.3205i	0.192351	+	0.555270i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−11.0227	−	19.0919i	−0.352648	−	0.610803i	0.634065	−	0.773280i	$-0.281385\pi$
$977$	−11.0227	−	19.0919i	−0.352648	−	0.610803i	−0.986712	+	0.162476i	$0.948052\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18.3712	−	10.6066i	−0.585949	−	0.338298i	0.177545	−	0.984113i	$-0.443185\pi$
$983$	−18.3712	−	10.6066i	−0.585949	−	0.338298i	−0.763494	+	0.645815i	$0.776518\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.34847	−	4.24264i	0.233668	−	0.134908i
$990$		0			0
$991$	−18.5000	−	32.0429i	−0.587672	−	1.01788i	−0.994537	−	0.104389i	$-0.966711\pi$
$991$	−18.5000	−	32.0429i	−0.587672	−	1.01788i	0.406865	−	0.913488i	$-0.366622\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$			15.5885i			0.493691i	0.969055	+	0.246846i	$0.0793941\pi$
$997$			15.5885i			0.493691i	−0.969055	+	0.246846i	$0.920606\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3276.2.gj.b.17.1	✓	4
3.2	odd	2	inner	3276.2.gj.b.17.2	yes	4
7.5	odd	6		3276.2.hp.b.2357.1	yes	4
13.10	even	6		3276.2.hp.b.2285.2	yes	4
21.5	even	6		3276.2.hp.b.2357.2	yes	4
39.23	odd	6		3276.2.hp.b.2285.1	yes	4
91.75	odd	6	inner	3276.2.gj.b.1349.2	yes	4
273.257	even	6	inner	3276.2.gj.b.1349.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3276.2.gj.b.17.1	✓	4	1.1	even	1	trivial
3276.2.gj.b.17.2	yes	4	3.2	odd	2	inner
3276.2.gj.b.1349.1	yes	4	273.257	even	6	inner
3276.2.gj.b.1349.2	yes	4	91.75	odd	6	inner
3276.2.hp.b.2285.1	yes	4	39.23	odd	6
3276.2.hp.b.2285.2	yes	4	13.10	even	6
3276.2.hp.b.2357.1	yes	4	7.5	odd	6
3276.2.hp.b.2357.2	yes	4	21.5	even	6

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:	\( N \)	\(=\)	\( 3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3276.gj (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 2.59808i) q^{7} +(2.50000 + 2.59808i) q^{13} -7.34847 q^{17} +(0.500000 - 0.866025i) q^{19} +8.48528i q^{23} +(-2.50000 + 4.33013i) q^{25} +(3.67423 + 2.12132i) q^{29} +(1.00000 - 1.73205i) q^{31} +8.66025i q^{37} +(-7.34847 - 4.24264i) q^{41} +(-0.500000 - 0.866025i) q^{43} +(3.67423 - 2.12132i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(7.34847 + 4.24264i) q^{53} +12.7279i q^{59} +(10.5000 + 6.06218i) q^{61} +(6.00000 - 3.46410i) q^{67} +(-3.67423 - 6.36396i) q^{71} +(5.50000 - 9.52628i) q^{73} +(7.00000 + 12.1244i) q^{79} +(8.00000 - 5.19615i) q^{91} +(-2.50000 - 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{7} + 10 q^{13} + 2 q^{19} - 10 q^{25} + 4 q^{31} - 2 q^{43} - 26 q^{49} + 42 q^{61} + 24 q^{67} + 22 q^{73} + 28 q^{79} + 32 q^{91} - 10 q^{97}+O(q^{100}) \) 4 * q + 2 * q^7 + 10 * q^13 + 2 * q^19 - 10 * q^25 + 4 * q^31 - 2 * q^43 - 26 * q^49 + 42 * q^61 + 24 * q^67 + 22 * q^73 + 28 * q^79 + 32 * q^91 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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