LMFDB - Embedded newform 3276.2.bi.c.2449.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3276,2,Mod(1945,3276)] 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("3276.1945"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3276, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 2, 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.bi (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,-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:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 36x^{14} + 472x^{12} + 2912x^{10} + 8914x^{8} + 13164x^{6} + 8828x^{4} + 2648x^{2} + 289 $ x^16 + 36x^14 + 472x^12 + 2912x^10 + 8914x^8 + 13164x^6 + 8828x^4 + 2648*x^2 + 289
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 364)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2449.4
Root			$0.536261i$ of defining polynomial
Character	$\chi$	$=$	3276.2449
Dual form			3276.2.bi.c.1945.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0659529 - 0.0659529i) q^{5} +(2.36200 + 1.19204i) q^{7} +(0.538581 + 0.538581i) q^{11} +(2.42961 - 2.66401i) q^{13} -1.53626 q^{17} +(-3.03032 - 3.03032i) q^{19} -3.99130i q^{23} -4.99130i q^{25} +2.55404 q^{29} +(-1.47031 - 1.47031i) q^{31} +(-0.0771620 - 0.234399i) q^{35} +(3.65536 + 3.65536i) q^{37} +(-5.69433 - 5.69433i) q^{41} -2.75668i q^{43} +(7.88955 - 7.88955i) q^{47} +(4.15807 + 5.63120i) q^{49} -4.71150 q^{53} -0.0710419i q^{55} +(7.33875 - 7.33875i) q^{59} +4.99114i q^{61} +(-0.335939 + 0.0154593i) q^{65} +(-2.07716 + 2.07716i) q^{67} +(-7.80412 + 7.80412i) q^{71} +(-3.96588 + 3.96588i) q^{73} +(0.630116 + 1.91414i) q^{77} +4.35140 q^{79} +(1.13342 + 1.13342i) q^{83} +(0.101321 + 0.101321i) q^{85} +(11.7657 - 11.7657i) q^{89} +(8.91436 - 3.39619i) q^{91} +0.399716i q^{95} +(5.23833 + 5.23833i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{7} - 12 q^{11} - 20 q^{29} + 40 q^{35} - 28 q^{53} + 20 q^{65} + 8 q^{67} + 4 q^{71} + 4 q^{79} + 4 q^{85} + 10 q^{91}+O(q^{100}) $ 16 * q - 2 * q^7 - 12 * q^11 - 20 * q^29 + 40 * q^35 - 28 * q^53 + 20 * q^65 + 8 * q^67 + 4 * q^71 + 4 * q^79 + 4 * q^85 + 10 * q^91

Character values

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

$n$	$1639$	$2017$	$2341$	$2549$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$	$1$

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
$5$	−0.0659529	−	0.0659529i	−0.0294950	−	0.0294950i	0.692205	−	0.721700i	$-0.256639\pi$
$5$	−0.0659529	−	0.0659529i	−0.0294950	−	0.0294950i	−0.721700	+	0.692205i	$0.756639\pi$
$6$		0			0
$7$	2.36200	+	1.19204i	0.892751	+	0.450550i
$7$	2.36200	+	1.19204i	0.892751	+	0.450550i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.538581	+	0.538581i	0.162388	+	0.162388i	0.783624	−	0.621236i	$-0.213369\pi$
$11$	0.538581	+	0.538581i	0.162388	+	0.162388i	−0.621236	+	0.783624i	$0.713369\pi$
$12$		0			0
$13$	2.42961	−	2.66401i	0.673854	−	0.738865i
$13$	2.42961	−	2.66401i	0.673854	−	0.738865i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.53626			−0.372598			−0.186299	−	0.982493i	$-0.559649\pi$
$17$	−1.53626			−0.372598			−0.186299	+	0.982493i	$0.559649\pi$
$18$		0			0
$19$	−3.03032	−	3.03032i	−0.695203	−	0.695203i	0.268169	−	0.963372i	$-0.413581\pi$
$19$	−3.03032	−	3.03032i	−0.695203	−	0.695203i	−0.963372	+	0.268169i	$0.913581\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.99130i		−	0.832244i	−0.909309	−	0.416122i	$-0.863389\pi$
$23$		−	3.99130i		−	0.832244i	0.909309	−	0.416122i	$-0.136611\pi$
$24$		0			0
$25$		−	4.99130i		−	0.998260i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.55404			0.474273			0.237137	−	0.971476i	$-0.423791\pi$
$29$	2.55404			0.474273			0.237137	+	0.971476i	$0.423791\pi$
$30$		0			0
$31$	−1.47031	−	1.47031i	−0.264075	−	0.264075i	0.562632	−	0.826707i	$-0.309789\pi$
$31$	−1.47031	−	1.47031i	−0.264075	−	0.264075i	−0.826707	+	0.562632i	$0.809789\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.0771620	−	0.234399i	−0.0130428	−	0.0396207i
$36$		0			0
$37$	3.65536	+	3.65536i	0.600938	+	0.600938i	0.940561	−	0.339624i	$-0.110300\pi$
$37$	3.65536	+	3.65536i	0.600938	+	0.600938i	−0.339624	+	0.940561i	$0.610300\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.69433	−	5.69433i	−0.889305	−	0.889305i	0.105151	−	0.994456i	$-0.466467\pi$
$41$	−5.69433	−	5.69433i	−0.889305	−	0.889305i	−0.994456	+	0.105151i	$0.966467\pi$
$42$		0			0
$43$		−	2.75668i		−	0.420390i	−0.977659	−	0.210195i	$-0.932590\pi$
$43$		−	2.75668i		−	0.420390i	0.977659	−	0.210195i	$-0.0674099\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.88955	−	7.88955i	1.15081	−	1.15081i	0.164418	−	0.986391i	$-0.447425\pi$
$47$	7.88955	−	7.88955i	1.15081	−	1.15081i	0.986391	−	0.164418i	$-0.0525747\pi$
$48$		0			0
$49$	4.15807	+	5.63120i	0.594010	+	0.804457i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.71150			−0.647174			−0.323587	−	0.946198i	$-0.604889\pi$
$53$	−4.71150			−0.647174			−0.323587	+	0.946198i	$0.604889\pi$
$54$		0			0
$55$		−	0.0710419i		−	0.00957929i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.33875	−	7.33875i	0.955424	−	0.955424i	−0.0436241	−	0.999048i	$-0.513890\pi$
$59$	7.33875	−	7.33875i	0.955424	−	0.955424i	0.999048	+	0.0436241i	$0.0138904\pi$
$60$		0			0
$61$			4.99114i			0.639049i	0.947578	+	0.319525i	$0.103523\pi$
$61$			4.99114i			0.639049i	−0.947578	+	0.319525i	$0.896477\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.335939	+	0.0154593i	−0.0416682	+	0.00191749i
$66$		0			0
$67$	−2.07716	+	2.07716i	−0.253766	+	0.253766i	−0.822513	−	0.568747i	$-0.807428\pi$
$67$	−2.07716	+	2.07716i	−0.253766	+	0.253766i	0.568747	+	0.822513i	$0.307428\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.80412	+	7.80412i	−0.926178	+	0.926178i	−0.997456	−	0.0712781i	$-0.977292\pi$
$71$	−7.80412	+	7.80412i	−0.926178	+	0.926178i	0.0712781	+	0.997456i	$0.477292\pi$
$72$		0			0
$73$	−3.96588	+	3.96588i	−0.464171	+	0.464171i	−0.900020	−	0.435849i	$-0.856448\pi$
$73$	−3.96588	+	3.96588i	−0.464171	+	0.464171i	0.435849	+	0.900020i	$0.356448\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.630116	+	1.91414i	0.0718084	+	0.218136i
$78$		0			0
$79$	4.35140			0.489571			0.244785	−	0.969577i	$-0.421283\pi$
$79$	4.35140			0.489571			0.244785	+	0.969577i	$0.421283\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.13342	+	1.13342i	0.124409	+	0.124409i	0.766570	−	0.642161i	$-0.221962\pi$
$83$	1.13342	+	1.13342i	0.124409	+	0.124409i	−0.642161	+	0.766570i	$0.721962\pi$
$84$		0			0
$85$	0.101321	+	0.101321i	0.0109898	+	0.0109898i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	11.7657	−	11.7657i	1.24716	−	1.24716i	0.290197	−	0.956967i	$-0.406279\pi$
$89$	11.7657	−	11.7657i	1.24716	−	1.24716i	0.956967	−	0.290197i	$-0.0937209\pi$
$90$		0			0
$91$	8.91436	−	3.39619i	0.934479	−	0.356018i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			0.399716i			0.0410100i
$96$		0			0
$97$	5.23833	+	5.23833i	0.531871	+	0.531871i	0.921129	−	0.389258i	$-0.127268\pi$
$97$	5.23833	+	5.23833i	0.531871	+	0.531871i	−0.389258	+	0.921129i	$0.627268\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.25699			−0.523090			−0.261545	−	0.965191i	$-0.584232\pi$
$101$	−5.25699			−0.523090			−0.261545	+	0.965191i	$0.584232\pi$
$102$		0			0
$103$	−10.6561			−1.04997			−0.524986	−	0.851111i	$-0.675930\pi$
$103$	−10.6561			−1.04997			−0.524986	+	0.851111i	$0.675930\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	19.8561			1.91956			0.959779	−	0.280757i	$-0.0905853\pi$
$107$	19.8561			1.91956			0.959779	+	0.280757i	$0.0905853\pi$
$108$		0			0
$109$	−5.28970	+	5.28970i	−0.506661	+	0.506661i	−0.913500	−	0.406839i	$-0.866631\pi$
$109$	−5.28970	+	5.28970i	−0.506661	+	0.506661i	0.406839	+	0.913500i	$0.366631\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.5509			0.992546			0.496273	−	0.868167i	$-0.334702\pi$
$113$	10.5509			0.992546			0.496273	+	0.868167i	$0.334702\pi$
$114$		0			0
$115$	−0.263238	+	0.263238i	−0.0245470	+	0.0245470i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.62865	−	1.83129i	−0.332637	−	0.167874i
$120$		0			0
$121$		−	10.4199i		−	0.947260i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.658955	+	0.658955i	−0.0589387	+	0.0589387i
$126$		0			0
$127$			7.11121i			0.631018i	0.948923	+	0.315509i	$0.102175\pi$
$127$			7.11121i			0.631018i	−0.948923	+	0.315509i	$0.897825\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	7.28355i		−	0.636367i	−0.948029	−	0.318183i	$-0.896927\pi$
$131$		−	7.28355i		−	0.636367i	0.948029	−	0.318183i	$-0.103073\pi$
$132$		0			0
$133$	−3.54534	−	10.7699i	−0.307420	−	0.933867i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.16978	−	4.16978i	−0.356249	−	0.356249i	0.506180	−	0.862428i	$-0.331057\pi$
$137$	−4.16978	−	4.16978i	−0.356249	−	0.356249i	−0.862428	+	0.506180i	$0.831057\pi$
$138$		0			0
$139$		−	11.5916i		−	0.983188i	−0.870825	−	0.491594i	$-0.836414\pi$
$139$		−	11.5916i		−	0.983188i	0.870825	−	0.491594i	$-0.163586\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.74333	−	0.126243i	0.229409	−	0.0105570i
$144$		0			0
$145$	−0.168446	−	0.168446i	−0.0139887	−	0.0139887i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.76344	+	8.76344i	−0.717929	+	0.717929i	−0.968181	−	0.250252i	$-0.919487\pi$
$149$	−8.76344	+	8.76344i	−0.717929	+	0.717929i	0.250252	+	0.968181i	$0.419487\pi$
$150$		0			0
$151$	6.37242	+	6.37242i	0.518581	+	0.518581i	0.917142	−	0.398561i	$-0.130490\pi$
$151$	6.37242	+	6.37242i	0.518581	+	0.518581i	−0.398561	+	0.917142i	$0.630490\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			0.193942i			0.0155778i
$156$		0			0
$157$		−	16.5507i		−	1.32089i	−0.750876	−	0.660443i	$-0.770369\pi$
$157$		−	16.5507i		−	1.32089i	0.750876	−	0.660443i	$-0.229631\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.75780	−	9.42744i	0.374967	−	0.742987i
$162$		0			0
$163$	−0.960382	−	0.960382i	−0.0752229	−	0.0752229i	0.668494	−	0.743717i	$-0.266939\pi$
$163$	−0.960382	−	0.960382i	−0.0752229	−	0.0752229i	−0.743717	+	0.668494i	$0.766939\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.02201	+	5.02201i	−0.388615	+	0.388615i	−0.874193	−	0.485578i	$-0.838609\pi$
$167$	−5.02201	+	5.02201i	−0.388615	+	0.388615i	0.485578	+	0.874193i	$0.338609\pi$
$168$		0			0
$169$	−1.19394	−	12.9451i	−0.0918417	−	0.995774i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	18.3849			1.39778			0.698888	−	0.715232i	$-0.253679\pi$
$173$	18.3849			1.39778			0.698888	+	0.715232i	$0.253679\pi$
$174$		0			0
$175$	5.94984	−	11.7894i	0.449766	−	0.891198i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	16.9728i		−	1.26861i	−0.773083	−	0.634305i	$-0.781286\pi$
$179$		−	16.9728i		−	1.26861i	0.773083	−	0.634305i	$-0.218714\pi$
$180$		0			0
$181$	17.9274			1.33253			0.666266	−	0.745714i	$-0.267891\pi$
$181$	17.9274			1.33253			0.666266	+	0.745714i	$0.267891\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	0.482163i		−	0.0354493i
$186$		0			0
$187$	−0.827401	−	0.827401i	−0.0605056	−	0.0605056i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.25202			−0.0905928			−0.0452964	−	0.998974i	$-0.514423\pi$
$191$	−1.25202			−0.0905928			−0.0452964	+	0.998974i	$0.514423\pi$
$192$		0			0
$193$	14.0907	+	14.0907i	1.01427	+	1.01427i	0.999897	+	0.0143727i	$0.00457514\pi$
$193$	14.0907	+	14.0907i	1.01427	+	1.01427i	0.0143727	+	0.999897i	$0.495425\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.16302	+	1.16302i	−0.0828620	+	0.0828620i	−0.747323	−	0.664461i	$-0.768661\pi$
$197$	−1.16302	+	1.16302i	−0.0828620	+	0.0828620i	0.664461	+	0.747323i	$0.268661\pi$
$198$		0			0
$199$	12.7568			0.904309			0.452154	−	0.891940i	$-0.350656\pi$
$199$	12.7568			0.904309			0.452154	+	0.891940i	$0.350656\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.03264	+	3.04452i	0.423408	+	0.213684i
$204$		0			0
$205$			0.751115i			0.0524601i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	3.26414i		−	0.225786i
$210$		0			0
$211$	−8.92766			−0.614605			−0.307303	−	0.951612i	$-0.599426\pi$
$211$	−8.92766			−0.614605			−0.307303	+	0.951612i	$0.599426\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.181811	+	0.181811i	−0.0123994	+	0.0123994i
$216$		0			0
$217$	−1.72020	−	5.22553i	−0.116775	−	0.354732i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.73252	+	4.09262i	−0.251077	+	0.275299i
$222$		0			0
$223$	−12.6540	−	12.6540i	−0.847374	−	0.847374i	0.142431	−	0.989805i	$-0.454508\pi$
$223$	−12.6540	−	12.6540i	−0.847374	−	0.847374i	−0.989805	+	0.142431i	$0.954508\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−16.6511	−	16.6511i	−1.10517	−	1.10517i	−0.993776	−	0.111397i	$-0.964468\pi$
$227$	−16.6511	−	16.6511i	−1.10517	−	1.10517i	−0.111397	−	0.993776i	$-0.535532\pi$
$228$		0			0
$229$	10.5907	−	10.5907i	0.699852	−	0.699852i	−0.264527	−	0.964378i	$-0.585216\pi$
$229$	10.5907	−	10.5907i	0.699852	−	0.699852i	0.964378	+	0.264527i	$0.0852157\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			26.9728i			1.76705i	0.468384	+	0.883525i	$0.344836\pi$
$233$			26.9728i			1.76705i	−0.468384	+	0.883525i	$0.655164\pi$
$234$		0			0
$235$	−1.04068			−0.0678863
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.07716	−	6.07716i	0.393099	−	0.393099i	−0.482692	−	0.875790i	$-0.660341\pi$
$239$	6.07716	−	6.07716i	0.393099	−	0.393099i	0.875790	+	0.482692i	$0.160341\pi$
$240$		0			0
$241$	−10.4829	+	10.4829i	−0.675262	+	0.675262i	−0.958924	−	0.283662i	$-0.908451\pi$
$241$	−10.4829	+	10.4829i	−0.675262	+	0.675262i	0.283662	+	0.958924i	$0.408451\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.0971572	−	0.645631i	0.00620714	−	0.0412478i
$246$		0			0
$247$	−15.4353	+	0.710304i	−0.982126	+	0.0451956i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10.3581			−0.653798			−0.326899	−	0.945059i	$-0.606004\pi$
$251$	−10.3581			−0.653798			−0.326899	+	0.945059i	$0.606004\pi$
$252$		0			0
$253$	2.14964	−	2.14964i	0.135147	−	0.135147i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	12.4086			0.774029			0.387015	−	0.922074i	$-0.373506\pi$
$257$	12.4086			0.774029			0.387015	+	0.922074i	$0.373506\pi$
$258$		0			0
$259$	4.27661	+	12.9913i	0.265736	+	0.807240i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	23.4904			1.44848			0.724240	−	0.689548i	$-0.242191\pi$
$263$	23.4904			1.44848			0.724240	+	0.689548i	$0.242191\pi$
$264$		0			0
$265$	0.310737	+	0.310737i	0.0190884	+	0.0190884i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			7.77426i			0.474005i	0.971509	+	0.237003i	$0.0761650\pi$
$269$			7.77426i			0.474005i	−0.971509	+	0.237003i	$0.923835\pi$
$270$		0			0
$271$	4.02617	−	4.02617i	0.244572	−	0.244572i	−0.574166	−	0.818739i	$-0.694674\pi$
$271$	4.02617	−	4.02617i	0.244572	−	0.244572i	0.818739	+	0.574166i	$0.194674\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.68822	−	2.68822i	0.162106	−	0.162106i
$276$		0			0
$277$		−	16.3340i		−	0.981415i	−0.871324	−	0.490707i	$-0.836738\pi$
$277$		−	16.3340i		−	0.981415i	0.871324	−	0.490707i	$-0.163262\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8.93516	+	8.93516i	0.533027	+	0.533027i	0.921472	−	0.388445i	$-0.126988\pi$
$281$	8.93516	+	8.93516i	0.533027	+	0.533027i	−0.388445	+	0.921472i	$0.626988\pi$
$282$		0			0
$283$	−16.8486			−1.00155			−0.500773	−	0.865579i	$-0.666951\pi$
$283$	−16.8486			−1.00155			−0.500773	+	0.865579i	$0.666951\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.66212	−	20.2379i	−0.393253	−	1.19460i
$288$		0			0
$289$	−14.6399			−0.861171
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11.8997	−	11.8997i	0.695186	−	0.695186i	−0.268182	−	0.963368i	$-0.586423\pi$
$293$	11.8997	−	11.8997i	0.695186	−	0.695186i	0.963368	+	0.268182i	$0.0864229\pi$
$294$		0			0
$295$	−0.968023			−0.0563605
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.6329	−	9.69732i	−0.614915	−	0.560811i
$300$		0			0
$301$	3.28608	−	6.51128i	0.189407	−	0.375304i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.329180	−	0.329180i	0.0188488	−	0.0188488i
$306$		0			0
$307$	1.00355	−	1.00355i	0.0572756	−	0.0572756i	−0.677889	−	0.735164i	$-0.737105\pi$
$307$	1.00355	−	1.00355i	0.0572756	−	0.0572756i	0.735164	+	0.677889i	$0.237105\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−31.1170			−1.76448			−0.882242	−	0.470796i	$-0.843967\pi$
$311$	−31.1170			−1.76448			−0.882242	+	0.470796i	$0.843967\pi$
$312$		0			0
$313$			23.3439i			1.31948i	0.751496	+	0.659738i	$0.229333\pi$
$313$			23.3439i			1.31948i	−0.751496	+	0.659738i	$0.770667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.29332	−	8.29332i	0.465799	−	0.465799i	−0.434751	−	0.900551i	$-0.643164\pi$
$317$	8.29332	−	8.29332i	0.465799	−	0.465799i	0.900551	+	0.434751i	$0.143164\pi$
$318$		0			0
$319$	1.37556	+	1.37556i	0.0770164	+	0.0770164i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.65536	+	4.65536i	0.259031	+	0.259031i
$324$		0			0
$325$	−13.2969	−	12.1269i	−0.737579	−	0.672682i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	28.0398	−	9.23043i	1.54588	−	0.508890i
$330$		0			0
$331$	−5.72576	+	5.72576i	−0.314716	+	0.314716i	−0.846734	−	0.532017i	$-0.821434\pi$
$331$	−5.72576	+	5.72576i	−0.314716	+	0.314716i	0.532017	+	0.846734i	$0.321434\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.273990			0.0149696
$336$		0			0
$337$		−	3.48727i		−	0.189963i	−0.995479	−	0.0949817i	$-0.969721\pi$
$337$		−	3.48727i		−	0.189963i	0.995479	−	0.0949817i	$-0.0302793\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	1.58376i		−	0.0857654i
$342$		0			0
$343$	3.10873	+	18.2575i	0.167856	+	0.985812i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.43620			0.130782			0.0653911	−	0.997860i	$-0.479171\pi$
$347$	2.43620			0.130782			0.0653911	+	0.997860i	$0.479171\pi$
$348$		0			0
$349$	−10.3614	+	10.3614i	−0.554631	+	0.554631i	−0.927774	−	0.373143i	$-0.878280\pi$
$349$	−10.3614	+	10.3614i	−0.554631	+	0.554631i	0.373143	+	0.927774i	$0.378280\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	14.7542	+	14.7542i	0.785287	+	0.785287i	0.980718	−	0.195430i	$-0.0626104\pi$
$353$	14.7542	+	14.7542i	0.785287	+	0.785287i	−0.195430	+	0.980718i	$0.562610\pi$
$354$		0			0
$355$	1.02941			0.0546353
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−16.1785	−	16.1785i	−0.853868	−	0.853868i	0.136739	−	0.990607i	$-0.456338\pi$
$359$	−16.1785	−	16.1785i	−0.853868	−	0.853868i	−0.990607	+	0.136739i	$0.956338\pi$
$360$		0			0
$361$		−	0.634335i		−	0.0333860i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.523122			0.0273814
$366$		0			0
$367$			27.9868i			1.46090i	0.682966	+	0.730450i	$0.260690\pi$
$367$			27.9868i			1.46090i	−0.682966	+	0.730450i	$0.739310\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−11.1285	−	5.61630i	−0.577765	−	0.291584i
$372$		0			0
$373$	27.6051			1.42934			0.714669	−	0.699463i	$-0.246577\pi$
$373$	27.6051			1.42934			0.714669	+	0.699463i	$0.246577\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.20533	−	6.80400i	0.319591	−	0.350424i
$378$		0			0
$379$	18.9209	−	18.9209i	0.971901	−	0.971901i	−0.0277146	−	0.999616i	$-0.508823\pi$
$379$	18.9209	−	18.9209i	0.971901	−	0.971901i	0.999616	+	0.0277146i	$0.00882297\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.63137	+	9.63137i	0.492140	+	0.492140i	0.908980	−	0.416840i	$-0.136862\pi$
$383$	9.63137	+	9.63137i	0.492140	+	0.492140i	−0.416840	+	0.908980i	$0.636862\pi$
$384$		0			0
$385$	0.0846849	−	0.167801i	0.00431594	−	0.00855192i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			26.0896i			1.32280i	0.750035	+	0.661398i	$0.230037\pi$
$389$			26.0896i			1.32280i	−0.750035	+	0.661398i	$0.769963\pi$
$390$		0			0
$391$			6.13168i			0.310092i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.286987	−	0.286987i	−0.0144399	−	0.0144399i
$396$		0			0
$397$	19.1101	−	19.1101i	0.959110	−	0.959110i	−0.0400860	−	0.999196i	$-0.512763\pi$
$397$	19.1101	−	19.1101i	0.959110	−	0.959110i	0.999196	+	0.0400860i	$0.0127632\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.03092	−	7.03092i	−0.351107	−	0.351107i	0.509414	−	0.860521i	$-0.329862\pi$
$401$	−7.03092	−	7.03092i	−0.351107	−	0.351107i	−0.860521	+	0.509414i	$0.829862\pi$
$402$		0			0
$403$	−7.48920	+	0.344639i	−0.373064	+	0.0171677i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.93742i			0.195170i
$408$		0			0
$409$	12.5117	+	12.5117i	0.618664	+	0.618664i	0.945189	−	0.326525i	$-0.105878\pi$
$409$	12.5117	+	12.5117i	0.618664	+	0.618664i	−0.326525	+	0.945189i	$0.605878\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	26.0822	−	8.58602i	1.28342	−	0.422490i
$414$		0			0
$415$		−	0.149504i		−	0.00733886i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			8.63967i			0.422076i	0.977478	+	0.211038i	$0.0676843\pi$
$419$			8.63967i			0.422076i	−0.977478	+	0.211038i	$0.932316\pi$
$420$		0			0
$421$	−18.0464	+	18.0464i	−0.879527	+	0.879527i	−0.993485	−	0.113959i	$-0.963647\pi$
$421$	−18.0464	+	18.0464i	−0.879527	+	0.879527i	0.113959	+	0.993485i	$0.463647\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			7.66794i			0.371950i
$426$		0			0
$427$	−5.94964	+	11.7891i	−0.287923	+	0.570512i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.679520	−	0.679520i	0.0327313	−	0.0327313i	−0.690552	−	0.723283i	$-0.742632\pi$
$431$	0.679520	−	0.679520i	0.0327313	−	0.0327313i	0.723283	+	0.690552i	$0.242632\pi$
$432$		0			0
$433$	−19.7892			−0.951009			−0.475505	−	0.879713i	$-0.657734\pi$
$433$	−19.7892			−0.951009			−0.475505	+	0.879713i	$0.657734\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.0949	+	12.0949i	−0.578578	+	0.578578i
$438$		0			0
$439$	13.7761			0.657496			0.328748	−	0.944418i	$-0.393373\pi$
$439$	13.7761			0.657496			0.328748	+	0.944418i	$0.393373\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.7691			−0.939260			−0.469630	−	0.882863i	$-0.655613\pi$
$443$	−19.7691			−0.939260			−0.469630	+	0.882863i	$0.655613\pi$
$444$		0			0
$445$	−1.55197			−0.0735702
$446$		0			0
$447$		0			0
$448$		0			0
$449$	13.1765	+	13.1765i	0.621840	+	0.621840i	0.946002	−	0.324162i	$-0.105082\pi$
$449$	13.1765	+	13.1765i	0.621840	+	0.621840i	−0.324162	+	0.946002i	$0.605082\pi$
$450$		0			0
$451$		−	6.13372i		−	0.288826i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.811916	−	0.363939i	−0.0380632	−	0.0170617i
$456$		0			0
$457$	−18.7207	+	18.7207i	−0.875717	+	0.875717i	−0.993088	−	0.117371i	$-0.962553\pi$
$457$	−18.7207	+	18.7207i	−0.875717	+	0.875717i	0.117371	+	0.993088i	$0.462553\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.81589	−	7.81589i	−0.364022	−	0.364022i	0.501269	−	0.865291i	$-0.332867\pi$
$461$	−7.81589	−	7.81589i	−0.364022	−	0.364022i	−0.865291	+	0.501269i	$0.832867\pi$
$462$		0			0
$463$	−16.7412	−	16.7412i	−0.778031	−	0.778031i	0.201465	−	0.979496i	$-0.435430\pi$
$463$	−16.7412	−	16.7412i	−0.778031	−	0.778031i	−0.979496	+	0.201465i	$0.935430\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−27.7818			−1.28559			−0.642795	−	0.766038i	$-0.722225\pi$
$467$	−27.7818			−1.28559			−0.642795	+	0.766038i	$0.722225\pi$
$468$		0			0
$469$	−7.38232	+	2.43019i	−0.340884	+	0.112216i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.48470	−	1.48470i	0.0682664	−	0.0682664i
$474$		0			0
$475$	−15.1252	+	15.1252i	−0.693993	+	0.693993i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−24.7968	+	24.7968i	−1.13299	+	1.13299i	−0.143316	+	0.989677i	$0.545777\pi$
$479$	−24.7968	+	24.7968i	−1.13299	+	1.13299i	−0.989677	+	0.143316i	$0.954223\pi$
$480$		0			0
$481$	18.6191	−	0.856813i	0.848956	−	0.0390673i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	0.690965i		−	0.0313751i
$486$		0			0
$487$	5.51442	−	5.51442i	0.249882	−	0.249882i	−0.571040	−	0.820922i	$-0.693460\pi$
$487$	5.51442	−	5.51442i	0.249882	−	0.249882i	0.820922	+	0.571040i	$0.193460\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	7.63990i		−	0.344784i	−0.985028	−	0.172392i	$-0.944850\pi$
$491$		−	7.63990i		−	0.344784i	0.985028	−	0.172392i	$-0.0551496\pi$
$492$		0			0
$493$	−3.92367			−0.176713
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−27.7362	+	9.13048i	−1.24414	+	0.409558i
$498$		0			0
$499$	31.2660	−	31.2660i	1.39966	−	1.39966i	0.598646	−	0.801013i	$-0.295706\pi$
$499$	31.2660	−	31.2660i	1.39966	−	1.39966i	0.801013	−	0.598646i	$-0.204294\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			21.7334i			0.969045i	0.874779	+	0.484522i	$0.161007\pi$
$503$			21.7334i			0.969045i	−0.874779	+	0.484522i	$0.838993\pi$
$504$		0			0
$505$	0.346713	+	0.346713i	0.0154285	+	0.0154285i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.03840	+	7.03840i	0.311972	+	0.311972i	0.845673	−	0.533701i	$-0.179199\pi$
$509$	7.03840	+	7.03840i	0.311972	+	0.311972i	−0.533701	+	0.845673i	$0.679199\pi$
$510$		0			0
$511$	−14.0949	+	4.63990i	−0.623521	+	0.205257i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.702797	+	0.702797i	0.0309690	+	0.0309690i
$516$		0			0
$517$	8.49832			0.373756
$518$		0			0
$519$		0			0
$520$		0			0
$521$			43.8483i			1.92103i	0.278231	+	0.960514i	$0.410252\pi$
$521$			43.8483i			1.92103i	−0.278231	+	0.960514i	$0.589748\pi$
$522$		0			0
$523$		−	43.5958i		−	1.90631i	−0.302480	−	0.953156i	$-0.597815\pi$
$523$		−	43.5958i		−	1.90631i	0.302480	−	0.953156i	$-0.402185\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.25878	+	2.25878i	0.0983939	+	0.0983939i
$528$		0			0
$529$	7.06952			0.307370
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−29.0048	+	1.33475i	−1.25634	+	0.0578143i
$534$		0			0
$535$	−1.30956	−	1.30956i	−0.0566174	−	0.0566174i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.793400	+	5.27232i	−0.0341742	+	0.227095i
$540$		0			0
$541$	21.6609	+	21.6609i	0.931276	+	0.931276i	0.997786	−	0.0665097i	$-0.0211863\pi$
$541$	21.6609	+	21.6609i	0.931276	+	0.931276i	−0.0665097	+	0.997786i	$0.521186\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.697741			0.0298879
$546$		0			0
$547$	−4.75355			−0.203247			−0.101624	−	0.994823i	$-0.532404\pi$
$547$	−4.75355			−0.203247			−0.101624	+	0.994823i	$0.532404\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.73956	−	7.73956i	−0.329716	−	0.329716i
$552$		0			0
$553$	10.2780	+	5.18705i	0.437065	+	0.220576i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.58602	+	4.58602i	0.194316	+	0.194316i	0.797558	−	0.603242i	$-0.206125\pi$
$557$	4.58602	+	4.58602i	0.194316	+	0.194316i	−0.603242	+	0.797558i	$0.706125\pi$
$558$		0			0
$559$	−7.34384	−	6.69768i	−0.310611	−	0.283281i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−30.9842			−1.30583			−0.652915	−	0.757432i	$-0.726454\pi$
$563$	−30.9842			−1.30583			−0.652915	+	0.757432i	$0.726454\pi$
$564$		0			0
$565$	−0.695862	−	0.695862i	−0.0292752	−	0.0292752i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			9.08273i			0.380768i	0.981710	+	0.190384i	$0.0609733\pi$
$569$			9.08273i			0.380768i	−0.981710	+	0.190384i	$0.939027\pi$
$570$		0			0
$571$			26.9103i			1.12616i	0.826403	+	0.563080i	$0.190384\pi$
$571$			26.9103i			1.12616i	−0.826403	+	0.563080i	$0.809616\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−19.9218			−0.830796
$576$		0			0
$577$	−0.774239	−	0.774239i	−0.0322320	−	0.0322320i	0.690807	−	0.723039i	$-0.257255\pi$
$577$	−0.774239	−	0.774239i	−0.0322320	−	0.0322320i	−0.723039	+	0.690807i	$0.757255\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.32605	+	4.02821i	0.0550137	+	0.167118i
$582$		0			0
$583$	−2.53752	−	2.53752i	−0.105093	−	0.105093i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15.4018	+	15.4018i	0.635702	+	0.635702i	0.949492	−	0.313791i	$-0.101599\pi$
$587$	15.4018	+	15.4018i	0.635702	+	0.635702i	−0.313791	+	0.949492i	$0.601599\pi$
$588$		0			0
$589$			8.91101i			0.367172i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5.29898	+	5.29898i	−0.217603	+	0.217603i	−0.807488	−	0.589884i	$-0.799173\pi$
$593$	−5.29898	+	5.29898i	−0.217603	+	0.217603i	0.589884	+	0.807488i	$0.299173\pi$
$594$		0			0
$595$	0.118541	+	0.360098i	0.00485970	+	0.0147626i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−37.3124			−1.52454			−0.762272	−	0.647257i	$-0.775916\pi$
$599$	−37.3124			−1.52454			−0.762272	+	0.647257i	$0.775916\pi$
$600$		0			0
$601$			33.9825i			1.38618i	0.720853	+	0.693088i	$0.243750\pi$
$601$			33.9825i			1.38618i	−0.720853	+	0.693088i	$0.756250\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.687220	+	0.687220i	−0.0279394	+	0.0279394i
$606$		0			0
$607$		−	11.2547i		−	0.456815i	−0.973566	−	0.228408i	$-0.926648\pi$
$607$		−	11.2547i		−	0.456815i	0.973566	−	0.228408i	$-0.0733519\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.84930	−	40.1864i	−0.0748148	−	1.62577i
$612$		0			0
$613$	1.03649	−	1.03649i	0.0418633	−	0.0418633i	−0.685865	−	0.727729i	$-0.740576\pi$
$613$	1.03649	−	1.03649i	0.0418633	−	0.0418633i	0.727729	+	0.685865i	$0.240576\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−25.6965	+	25.6965i	−1.03450	+	1.03450i	−0.0351203	+	0.999383i	$0.511181\pi$
$617$	−25.6965	+	25.6965i	−1.03450	+	1.03450i	−0.999383	+	0.0351203i	$0.988819\pi$
$618$		0			0
$619$	−19.7578	+	19.7578i	−0.794132	+	0.794132i	−0.982163	−	0.188031i	$-0.939789\pi$
$619$	−19.7578	+	19.7578i	−0.794132	+	0.794132i	0.188031	+	0.982163i	$0.439789\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	41.8159	−	13.7654i	1.67532	−	0.551498i
$624$		0			0
$625$	−24.8696			−0.994783
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.61559	−	5.61559i	−0.223908	−	0.223908i
$630$		0			0
$631$	10.8370	+	10.8370i	0.431413	+	0.431413i	0.889109	−	0.457696i	$-0.151325\pi$
$631$	10.8370	+	10.8370i	0.431413	+	0.431413i	−0.457696	+	0.889109i	$0.651325\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.469005	−	0.469005i	0.0186119	−	0.0186119i
$636$		0			0
$637$	25.1041	+	2.60449i	0.994661	+	0.103194i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	31.3496i		−	1.23823i	−0.785299	−	0.619117i	$-0.787491\pi$
$641$		−	31.3496i		−	1.23823i	0.785299	−	0.619117i	$-0.212509\pi$
$642$		0			0
$643$	−8.96980	−	8.96980i	−0.353734	−	0.353734i	0.507763	−	0.861497i	$-0.330473\pi$
$643$	−8.96980	−	8.96980i	−0.353734	−	0.353734i	−0.861497	+	0.507763i	$0.830473\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−46.2110			−1.81674			−0.908371	−	0.418165i	$-0.862673\pi$
$647$	−46.2110			−1.81674			−0.908371	+	0.418165i	$0.862673\pi$
$648$		0			0
$649$	7.90502			0.310299
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.5287			0.725084			0.362542	−	0.931968i	$-0.381909\pi$
$653$	18.5287			0.725084			0.362542	+	0.931968i	$0.381909\pi$
$654$		0			0
$655$	−0.480371	+	0.480371i	−0.0187696	+	0.0187696i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	26.5533			1.03437			0.517185	−	0.855874i	$-0.326980\pi$
$659$	26.5533			1.03437			0.517185	+	0.855874i	$0.326980\pi$
$660$		0			0
$661$	−6.04014	+	6.04014i	−0.234934	+	0.234934i	−0.814749	−	0.579814i	$-0.803125\pi$
$661$	−6.04014	+	6.04014i	−0.234934	+	0.234934i	0.579814	+	0.814749i	$0.303125\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.476479	+	0.944129i	−0.0184771	+	0.0366118i
$666$		0			0
$667$		−	10.1939i		−	0.394711i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.68813	+	2.68813i	−0.103774	+	0.103774i
$672$		0			0
$673$		−	23.6614i		−	0.912079i	−0.889960	−	0.456039i	$-0.849268\pi$
$673$		−	23.6614i		−	0.912079i	0.889960	−	0.456039i	$-0.150732\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			8.81908i			0.338945i	0.985535	+	0.169472i	$0.0542063\pi$
$677$			8.81908i			0.338945i	−0.985535	+	0.169472i	$0.945794\pi$
$678$		0			0
$679$	6.12861	+	18.6172i	0.235195	+	0.714463i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.0462	+	10.0462i	0.384409	+	0.384409i	0.872688	−	0.488279i	$-0.162375\pi$
$683$	10.0462	+	10.0462i	0.384409	+	0.384409i	−0.488279	+	0.872688i	$0.662375\pi$
$684$		0			0
$685$			0.550018i			0.0210151i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−11.4471	+	12.5515i	−0.436101	+	0.478174i
$690$		0			0
$691$	21.6571	+	21.6571i	0.823874	+	0.823874i	0.986661	−	0.162787i	$-0.0520484\pi$
$691$	21.6571	+	21.6571i	0.823874	+	0.823874i	−0.162787	+	0.986661i	$0.552048\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.764500	+	0.764500i	−0.0289991	+	0.0289991i
$696$		0			0
$697$	8.74798	+	8.74798i	0.331353	+	0.331353i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	27.5520i		−	1.04062i	−0.853976	−	0.520312i	$-0.825816\pi$
$701$		−	27.5520i		−	1.04062i	0.853976	−	0.520312i	$-0.174184\pi$
$702$		0			0
$703$		−	22.1538i		−	0.835547i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12.4170	−	6.26655i	−0.466989	−	0.235678i
$708$		0			0
$709$	−32.0203	−	32.0203i	−1.20255	−	1.20255i	−0.973389	−	0.229157i	$-0.926403\pi$
$709$	−32.0203	−	32.0203i	−1.20255	−	1.20255i	−0.229157	−	0.973389i	$-0.573597\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.86844	+	5.86844i	−0.219775	+	0.219775i
$714$		0			0
$715$	−0.189257	−	0.172604i	−0.00707780	−	0.00645504i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	40.8147			1.52213			0.761066	−	0.648675i	$-0.224676\pi$
$719$	40.8147			1.52213			0.761066	+	0.648675i	$0.224676\pi$
$720$		0			0
$721$	−25.1696	−	12.7025i	−0.937364	−	0.473065i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	12.7480i		−	0.473448i
$726$		0			0
$727$	−24.8863			−0.922980			−0.461490	−	0.887145i	$-0.652685\pi$
$727$	−24.8863			−0.922980			−0.461490	+	0.887145i	$0.652685\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			4.23498i			0.156637i
$732$		0			0
$733$	24.5115	+	24.5115i	0.905351	+	0.905351i	0.995893	−	0.0905416i	$-0.0288598\pi$
$733$	24.5115	+	24.5115i	0.905351	+	0.905351i	−0.0905416	+	0.995893i	$0.528860\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.23744			−0.0824172
$738$		0			0
$739$	−13.8382	−	13.8382i	−0.509045	−	0.509045i	0.405188	−	0.914233i	$-0.367206\pi$
$739$	−13.8382	−	13.8382i	−0.509045	−	0.509045i	−0.914233	+	0.405188i	$0.867206\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	13.9158	−	13.9158i	0.510522	−	0.510522i	−0.404164	−	0.914686i	$-0.632438\pi$
$743$	13.9158	−	13.9158i	0.510522	−	0.510522i	0.914686	+	0.404164i	$0.132438\pi$
$744$		0			0
$745$	1.15595			0.0423507
$746$		0			0
$747$		0			0
$748$		0			0
$749$	46.9000	+	23.6693i	1.71369	+	0.864856i
$750$		0			0
$751$		−	18.3010i		−	0.667812i	−0.942606	−	0.333906i	$-0.891633\pi$
$751$		−	18.3010i		−	0.667812i	0.942606	−	0.333906i	$-0.108367\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	0.840559i		−	0.0305911i
$756$		0			0
$757$	46.3896			1.68606			0.843029	−	0.537868i	$-0.180770\pi$
$757$	46.3896			1.68606			0.843029	+	0.537868i	$0.180770\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−17.6974	+	17.6974i	−0.641532	+	0.641532i	−0.950932	−	0.309400i	$-0.899872\pi$
$761$	−17.6974	+	17.6974i	−0.641532	+	0.641532i	0.309400	+	0.950932i	$0.399872\pi$
$762$		0			0
$763$	−18.7998	+	6.18871i	−0.680598	+	0.224046i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.72020	−	37.3809i	−0.0621127	−	1.34975i
$768$		0			0
$769$	24.6803	+	24.6803i	0.889993	+	0.889993i	0.994522	−	0.104529i	$-0.0333334\pi$
$769$	24.6803	+	24.6803i	0.889993	+	0.889993i	−0.104529	+	0.994522i	$0.533333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.85711	−	1.85711i	−0.0667955	−	0.0667955i	0.672920	−	0.739715i	$-0.265040\pi$
$773$	−1.85711	−	1.85711i	−0.0667955	−	0.0667955i	−0.739715	+	0.672920i	$0.765040\pi$
$774$		0			0
$775$	−7.33875	+	7.33875i	−0.263616	+	0.263616i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			34.5113i			1.23650i
$780$		0			0
$781$	−8.40630			−0.300801
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.09156	+	1.09156i	−0.0389595	+	0.0389595i
$786$		0			0
$787$	−35.1911	+	35.1911i	−1.25443	+	1.25443i	−0.300710	+	0.953716i	$0.597224\pi$
$787$	−35.1911	+	35.1911i	−1.25443	+	1.25443i	−0.953716	+	0.300710i	$0.902776\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	24.9212	+	12.5771i	0.886097	+	0.447191i
$792$		0			0
$793$	13.2965	+	12.1265i	0.472171	+	0.430626i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.23702			−0.150083			−0.0750415	−	0.997180i	$-0.523909\pi$
$797$	−4.23702			−0.150083			−0.0750415	+	0.997180i	$0.523909\pi$
$798$		0			0
$799$	−12.1204	+	12.1204i	−0.428789	+	0.428789i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−4.27189			−0.150752
$804$		0			0
$805$	−0.935557	+	0.307977i	−0.0329741	+	0.0108547i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−35.5595			−1.25021			−0.625103	−	0.780543i	$-0.714943\pi$
$809$	−35.5595			−1.25021			−0.625103	+	0.780543i	$0.714943\pi$
$810$		0			0
$811$	−7.76026	−	7.76026i	−0.272499	−	0.272499i	0.557606	−	0.830106i	$-0.311720\pi$
$811$	−7.76026	−	7.76026i	−0.272499	−	0.272499i	−0.830106	+	0.557606i	$0.811720\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			0.126680i			0.00443740i
$816$		0			0
$817$	−8.35363	+	8.35363i	−0.292256	+	0.292256i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	31.0191	−	31.0191i	1.08257	−	1.08257i	0.0863048	−	0.996269i	$-0.472494\pi$
$821$	31.0191	−	31.0191i	1.08257	−	1.08257i	0.996269	−	0.0863048i	$-0.0275059\pi$
$822$		0			0
$823$		−	0.483504i		−	0.0168539i	−0.999964	−	0.00842694i	$-0.997318\pi$
$823$		−	0.483504i		−	0.0168539i	0.999964	−	0.00842694i	$-0.00268241\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	31.8967	+	31.8967i	1.10916	+	1.10916i	0.993261	+	0.115897i	$0.0369743\pi$
$827$	31.8967	+	31.8967i	1.10916	+	1.10916i	0.115897	+	0.993261i	$0.463026\pi$
$828$		0			0
$829$	3.98250			0.138318			0.0691589	−	0.997606i	$-0.477968\pi$
$829$	3.98250			0.138318			0.0691589	+	0.997606i	$0.477968\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−6.38788	−	8.65100i	−0.221327	−	0.299739i
$834$		0			0
$835$	0.662432			0.0229244
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−9.04288	+	9.04288i	−0.312195	+	0.312195i	−0.845759	−	0.533564i	$-0.820852\pi$
$839$	−9.04288	+	9.04288i	−0.312195	+	0.312195i	0.533564	+	0.845759i	$0.320852\pi$
$840$		0			0
$841$	−22.4769			−0.775065
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.775020	+	0.932507i	−0.0266615	+	0.0320792i
$846$		0			0
$847$	12.4209	−	24.6117i	0.426788	−	0.845668i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	14.5896	−	14.5896i	0.500127	−	0.500127i
$852$		0			0
$853$	−32.7204	+	32.7204i	−1.12032	+	1.12032i	−0.128632	+	0.991692i	$0.541059\pi$
$853$	−32.7204	+	32.7204i	−1.12032	+	1.12032i	−0.991692	+	0.128632i	$0.958941\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−34.0669			−1.16370			−0.581852	−	0.813295i	$-0.697672\pi$
$857$	−34.0669			−1.16370			−0.581852	+	0.813295i	$0.697672\pi$
$858$		0			0
$859$		−	19.3050i		−	0.658679i	−0.944212	−	0.329339i	$-0.893174\pi$
$859$		−	19.3050i		−	0.658679i	0.944212	−	0.329339i	$-0.106826\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−33.6096	+	33.6096i	−1.14408	+	1.14408i	−0.156389	+	0.987696i	$0.549985\pi$
$863$	−33.6096	+	33.6096i	−1.14408	+	1.14408i	−0.987696	+	0.156389i	$0.950015\pi$
$864$		0			0
$865$	−1.21253	−	1.21253i	−0.0412274	−	0.0412274i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.34358	+	2.34358i	0.0795005	+	0.0795005i
$870$		0			0
$871$	0.486885	+	10.5803i	0.0164975	+	0.358500i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.34195	+	0.770948i	−0.0791724	+	0.0260628i
$876$		0			0
$877$	−35.6483	+	35.6483i	−1.20376	+	1.20376i	−0.230745	+	0.973014i	$0.574116\pi$
$877$	−35.6483	+	35.6483i	−1.20376	+	1.20376i	−0.973014	+	0.230745i	$0.925884\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.6467			−0.493460			−0.246730	−	0.969084i	$-0.579356\pi$
$881$	−14.6467			−0.493460			−0.246730	+	0.969084i	$0.579356\pi$
$882$		0			0
$883$			39.9426i			1.34417i	0.740472	+	0.672087i	$0.234602\pi$
$883$			39.9426i			1.34417i	−0.740472	+	0.672087i	$0.765398\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	17.8142i		−	0.598143i	−0.954231	−	0.299071i	$-0.903323\pi$
$887$		−	17.8142i		−	0.598143i	0.954231	−	0.299071i	$-0.0966769\pi$
$888$		0			0
$889$	−8.47686	+	16.7967i	−0.284305	+	0.563342i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−47.8157			−1.60009
$894$		0			0
$895$	−1.11941	+	1.11941i	−0.0374177	+	0.0374177i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.75523	−	3.75523i	−0.125244	−	0.125244i
$900$		0			0
$901$	7.23809			0.241136
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.18236	−	1.18236i	−0.0393031	−	0.0393031i
$906$		0			0
$907$		−	27.9235i		−	0.927183i	−0.886049	−	0.463592i	$-0.846560\pi$
$907$		−	27.9235i		−	0.927183i	0.886049	−	0.463592i	$-0.153440\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−47.4556			−1.57227			−0.786137	−	0.618052i	$-0.787922\pi$
$911$	−47.4556			−1.57227			−0.786137	+	0.618052i	$0.787922\pi$
$912$		0			0
$913$			1.22087i			0.0404050i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.68230	−	17.2037i	0.286715	−	0.568117i
$918$		0			0
$919$	33.7160			1.11219			0.556094	−	0.831119i	$-0.312299\pi$
$919$	33.7160			1.11219			0.556094	+	0.831119i	$0.312299\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.82928	+	39.7513i	0.0602114	+	1.30843i
$924$		0			0
$925$	18.2450	−	18.2450i	0.599892	−	0.599892i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	27.4521	+	27.4521i	0.900673	+	0.900673i	0.995494	−	0.0948212i	$-0.0302279\pi$
$929$	27.4521	+	27.4521i	0.900673	+	0.900673i	−0.0948212	+	0.995494i	$0.530228\pi$
$930$		0			0
$931$	4.46405	−	29.6646i	0.146303	−	0.972219i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			0.109139i			0.00356922i
$936$		0			0
$937$		−	40.2604i		−	1.31525i	−0.753346	−	0.657624i	$-0.771561\pi$
$937$		−	40.2604i		−	1.31525i	0.753346	−	0.657624i	$-0.228439\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	15.1968	+	15.1968i	0.495403	+	0.495403i	0.910004	−	0.414600i	$-0.136079\pi$
$941$	15.1968	+	15.1968i	0.495403	+	0.495403i	−0.414600	+	0.910004i	$0.636079\pi$
$942$		0			0
$943$	−22.7278	+	22.7278i	−0.740119	+	0.740119i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	32.4587	+	32.4587i	1.05477	+	1.05477i	0.998411	+	0.0563564i	$0.0179483\pi$
$947$	32.4587	+	32.4587i	1.05477	+	1.05477i	0.0563564	+	0.998411i	$0.482052\pi$
$948$		0			0
$949$	0.929598	+	20.2007i	0.0301760	+	0.655743i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	43.5348i		−	1.41023i	−0.709092	−	0.705116i	$-0.750895\pi$
$953$		−	43.5348i		−	1.41023i	0.709092	−	0.705116i	$-0.249105\pi$
$954$		0			0
$955$	0.0825741	+	0.0825741i	0.00267204	+	0.00267204i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.87846	−	14.8196i	−0.157534	−	0.478549i
$960$		0			0
$961$		−	26.6764i		−	0.860529i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1.85864i		−	0.0598318i
$966$		0			0
$967$	4.57989	−	4.57989i	0.147279	−	0.147279i	−0.629622	−	0.776901i	$-0.716790\pi$
$967$	4.57989	−	4.57989i	0.147279	−	0.147279i	0.776901	+	0.629622i	$0.216790\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			0.928153i			0.0297858i	0.999889	+	0.0148929i	$0.00474074\pi$
$971$			0.928153i			0.0297858i	−0.999889	+	0.0148929i	$0.995259\pi$
$972$		0			0
$973$	13.8177	−	27.3794i	0.442975	−	0.877742i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	18.0147	−	18.0147i	0.576342	−	0.576342i	−0.357552	−	0.933893i	$-0.616388\pi$
$977$	18.0147	−	18.0147i	0.576342	−	0.576342i	0.933893	+	0.357552i	$0.116388\pi$
$978$		0			0
$979$	12.6736			0.405050
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−25.4789	+	25.4789i	−0.812651	+	0.812651i	−0.985031	−	0.172380i	$-0.944854\pi$
$983$	−25.4789	+	25.4789i	−0.812651	+	0.812651i	0.172380	+	0.985031i	$0.444854\pi$
$984$		0			0
$985$	0.153409			0.00488803
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−11.0027			−0.349867
$990$		0			0
$991$	−10.9837			−0.348907			−0.174454	−	0.984665i	$-0.555816\pi$
$991$	−10.9837			−0.348907			−0.174454	+	0.984665i	$0.555816\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.841351	−	0.841351i	−0.0266726	−	0.0266726i
$996$		0			0
$997$			13.8232i			0.437784i	0.975749	+	0.218892i	$0.0702442\pi$
$997$			13.8232i			0.437784i	−0.975749	+	0.218892i	$0.929756\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3276.2.bi.c.2449.4		16
3.2	odd	2		364.2.n.a.265.3	yes	16
7.6	odd	2	inner	3276.2.bi.c.2449.5		16
13.8	odd	4	inner	3276.2.bi.c.1945.5		16
21.20	even	2		364.2.n.a.265.6	yes	16
39.8	even	4		364.2.n.a.125.3	✓	16
91.34	even	4	inner	3276.2.bi.c.1945.4		16
273.125	odd	4		364.2.n.a.125.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.n.a.125.3	✓	16	39.8	even	4
364.2.n.a.125.6	yes	16	273.125	odd	4
364.2.n.a.265.3	yes	16	3.2	odd	2
364.2.n.a.265.6	yes	16	21.20	even	2
3276.2.bi.c.1945.4		16	91.34	even	4	inner
3276.2.bi.c.1945.5		16	13.8	odd	4	inner
3276.2.bi.c.2449.4		16	1.1	even	1	trivial
3276.2.bi.c.2449.5		16	7.6	odd	2	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:	\( N \)	\(=\)	\( 3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3276.bi (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.0659529 - 0.0659529i) q^{5} +(2.36200 + 1.19204i) q^{7} +(0.538581 + 0.538581i) q^{11} +(2.42961 - 2.66401i) q^{13} -1.53626 q^{17} +(-3.03032 - 3.03032i) q^{19} -3.99130i q^{23} -4.99130i q^{25} +2.55404 q^{29} +(-1.47031 - 1.47031i) q^{31} +(-0.0771620 - 0.234399i) q^{35} +(3.65536 + 3.65536i) q^{37} +(-5.69433 - 5.69433i) q^{41} -2.75668i q^{43} +(7.88955 - 7.88955i) q^{47} +(4.15807 + 5.63120i) q^{49} -4.71150 q^{53} -0.0710419i q^{55} +(7.33875 - 7.33875i) q^{59} +4.99114i q^{61} +(-0.335939 + 0.0154593i) q^{65} +(-2.07716 + 2.07716i) q^{67} +(-7.80412 + 7.80412i) q^{71} +(-3.96588 + 3.96588i) q^{73} +(0.630116 + 1.91414i) q^{77} +4.35140 q^{79} +(1.13342 + 1.13342i) q^{83} +(0.101321 + 0.101321i) q^{85} +(11.7657 - 11.7657i) q^{89} +(8.91436 - 3.39619i) q^{91} +0.399716i q^{95} +(5.23833 + 5.23833i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{7} - 12 q^{11} - 20 q^{29} + 40 q^{35} - 28 q^{53} + 20 q^{65} + 8 q^{67} + 4 q^{71} + 4 q^{79} + 4 q^{85} + 10 q^{91}+O(q^{100}) \) 16 * q - 2 * q^7 - 12 * q^11 - 20 * q^29 + 40 * q^35 - 28 * q^53 + 20 * q^65 + 8 * q^67 + 4 * q^71 + 4 * q^79 + 4 * q^85 + 10 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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