LMFDB - Embedded newform 2268.2.t.c.2105.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(1781,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.1781"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.t (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,-8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2105.3
Character	$\chi$	$=$	2268.2105
Dual form			2268.2.t.c.1781.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.43229 + 2.48080i) q^{5} +(-0.496452 + 2.59876i) q^{7} +(1.08671 - 0.627410i) q^{11} -5.25046i q^{13} +(-3.07356 - 5.32357i) q^{17} +(-2.62182 - 1.51371i) q^{19} +(-4.95109 - 2.85851i) q^{23} +(-1.60290 - 2.77631i) q^{25} +3.73169i q^{29} +(5.87105 - 3.38965i) q^{31} +(-5.73592 - 4.95377i) q^{35} +(3.85796 - 6.68219i) q^{37} +9.06886 q^{41} +8.66134 q^{43} +(-2.97597 + 5.15454i) q^{47} +(-6.50707 - 2.58032i) q^{49} +(-7.17959 + 4.14514i) q^{53} +3.59453i q^{55} +(-1.48584 - 2.57354i) q^{59} +(-2.24837 - 1.29810i) q^{61} +(13.0253 + 7.52018i) q^{65} +(5.31994 + 9.21441i) q^{67} +2.62965i q^{71} +(8.25342 - 4.76512i) q^{73} +(1.09099 + 3.13556i) q^{77} +(5.49170 - 9.51191i) q^{79} +14.6184 q^{83} +17.6089 q^{85} +(5.95534 - 10.3150i) q^{89} +(13.6447 + 2.60660i) q^{91} +(7.51040 - 4.33613i) q^{95} -7.15011i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7} - 16 q^{25} + 24 q^{31} - 4 q^{37} + 8 q^{43} - 4 q^{49} + 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} - 24 q^{85} - 36 q^{91}+O(q^{100}) $ 32 * q - 8 * q^7 - 16 * q^25 + 24 * q^31 - 4 * q^37 + 8 * q^43 - 4 * q^49 + 12 * q^61 + 4 * q^67 - 36 * q^73 + 28 * q^79 - 24 * q^85 - 36 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$-1$

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
$5$	−1.43229	+	2.48080i	−0.640539	+	1.10945i	0.344774	+	0.938686i	$0.387956\pi$
$5$	−1.43229	+	2.48080i	−0.640539	+	1.10945i	−0.985313	+	0.170760i	$0.945378\pi$
$6$		0			0
$7$	−0.496452	+	2.59876i	−0.187641	+	0.982238i
$7$	−0.496452	+	2.59876i	−0.187641	+	0.982238i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.08671	−	0.627410i	0.327654	−	0.189171i	−0.327145	−	0.944974i	$-0.606087\pi$
$11$	1.08671	−	0.627410i	0.327654	−	0.189171i	0.654799	+	0.755803i	$0.272753\pi$
$12$		0			0
$13$		−	5.25046i		−	1.45622i	−0.685462	−	0.728108i	$-0.740400\pi$
$13$		−	5.25046i		−	1.45622i	0.685462	−	0.728108i	$-0.259600\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.07356	−	5.32357i	−0.745448	−	1.29115i	−0.949985	−	0.312295i	$-0.898902\pi$
$17$	−3.07356	−	5.32357i	−0.745448	−	1.29115i	0.204537	−	0.978859i	$-0.434431\pi$
$18$		0			0
$19$	−2.62182	−	1.51371i	−0.601486	−	0.347268i	0.168140	−	0.985763i	$-0.446224\pi$
$19$	−2.62182	−	1.51371i	−0.601486	−	0.347268i	−0.769626	+	0.638495i	$0.779557\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.95109	−	2.85851i	−1.03237	−	0.596041i	−0.114709	−	0.993399i	$-0.536594\pi$
$23$	−4.95109	−	2.85851i	−1.03237	−	0.596041i	−0.917664	+	0.397358i	$0.869927\pi$
$24$		0			0
$25$	−1.60290	−	2.77631i	−0.320580	−	0.555262i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			3.73169i			0.692957i	0.938058	+	0.346478i	$0.112623\pi$
$29$			3.73169i			0.692957i	−0.938058	+	0.346478i	$0.887377\pi$
$30$		0			0
$31$	5.87105	−	3.38965i	1.05447	−	0.608800i	0.130574	−	0.991439i	$-0.458318\pi$
$31$	5.87105	−	3.38965i	1.05447	−	0.608800i	0.923898	+	0.382639i	$0.124985\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.73592	−	4.95377i	−0.969548	−	0.837339i
$36$		0			0
$37$	3.85796	−	6.68219i	0.634245	−	1.09854i	−0.352429	−	0.935838i	$-0.614644\pi$
$37$	3.85796	−	6.68219i	0.634245	−	1.09854i	0.986675	−	0.162706i	$-0.0520223\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	9.06886			1.41632			0.708159	−	0.706053i	$-0.249526\pi$
$41$	9.06886			1.41632			0.708159	+	0.706053i	$0.249526\pi$
$42$		0			0
$43$	8.66134			1.32084			0.660421	−	0.750895i	$-0.270378\pi$
$43$	8.66134			1.32084			0.660421	+	0.750895i	$0.270378\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.97597	+	5.15454i	−0.434090	+	0.751866i	−0.997221	−	0.0745015i	$-0.976263\pi$
$47$	−2.97597	+	5.15454i	−0.434090	+	0.751866i	0.563131	+	0.826368i	$0.309597\pi$
$48$		0			0
$49$	−6.50707	−	2.58032i	−0.929582	−	0.368617i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.17959	+	4.14514i	−0.986193	+	0.569379i	−0.904134	−	0.427249i	$-0.859483\pi$
$53$	−7.17959	+	4.14514i	−0.986193	+	0.569379i	−0.0820587	+	0.996628i	$0.526149\pi$
$54$		0			0
$55$			3.59453i			0.484686i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.48584	−	2.57354i	−0.193439	−	0.335047i	0.752948	−	0.658080i	$-0.228631\pi$
$59$	−1.48584	−	2.57354i	−0.193439	−	0.335047i	−0.946388	+	0.323033i	$0.895298\pi$
$60$		0			0
$61$	−2.24837	−	1.29810i	−0.287874	−	0.166204i	0.349109	−	0.937082i	$-0.386484\pi$
$61$	−2.24837	−	1.29810i	−0.287874	−	0.166204i	−0.636983	+	0.770878i	$0.719818\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	13.0253	+	7.52018i	1.61559	+	0.932763i
$66$		0			0
$67$	5.31994	+	9.21441i	0.649934	+	1.12572i	0.983138	+	0.182865i	$0.0585370\pi$
$67$	5.31994	+	9.21441i	0.649934	+	1.12572i	−0.333204	+	0.942855i	$0.608130\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.62965i			0.312083i	0.987751	+	0.156041i	$0.0498733\pi$
$71$			2.62965i			0.312083i	−0.987751	+	0.156041i	$0.950127\pi$
$72$		0			0
$73$	8.25342	−	4.76512i	0.965990	−	0.557715i	0.0679788	−	0.997687i	$-0.478345\pi$
$73$	8.25342	−	4.76512i	0.965990	−	0.557715i	0.898011	+	0.439972i	$0.145012\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.09099	+	3.13556i	0.124330	+	0.357331i
$78$		0			0
$79$	5.49170	−	9.51191i	0.617865	−	1.07017i	−0.372010	−	0.928229i	$-0.621331\pi$
$79$	5.49170	−	9.51191i	0.617865	−	1.07017i	0.989875	−	0.141944i	$-0.0453354\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	14.6184			1.60457			0.802287	−	0.596938i	$-0.203616\pi$
$83$	14.6184			1.60457			0.802287	+	0.596938i	$0.203616\pi$
$84$		0			0
$85$	17.6089			1.90995
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.95534	−	10.3150i	0.631265	−	1.09338i	−0.356029	−	0.934475i	$-0.615869\pi$
$89$	5.95534	−	10.3150i	0.631265	−	1.09338i	0.987293	−	0.158908i	$-0.0507972\pi$
$90$		0			0
$91$	13.6447	+	2.60660i	1.43035	+	0.273246i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.51040	−	4.33613i	0.770551	−	0.444878i
$96$		0			0
$97$		−	7.15011i		−	0.725983i	−0.931792	−	0.362992i	$-0.881755\pi$
$97$		−	7.15011i		−	0.725983i	0.931792	−	0.362992i	$-0.118245\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.44709	+	2.50644i	0.143991	+	0.249400i	0.928996	−	0.370089i	$-0.120673\pi$
$101$	1.44709	+	2.50644i	0.143991	+	0.249400i	−0.785005	+	0.619490i	$0.787340\pi$
$102$		0			0
$103$	−5.58311	−	3.22341i	−0.550120	−	0.317612i	0.199051	−	0.979989i	$-0.436214\pi$
$103$	−5.58311	−	3.22341i	−0.550120	−	0.317612i	−0.749170	+	0.662377i	$0.769548\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.4017	−	8.31483i	−1.39227	−	0.803825i	−0.398701	−	0.917081i	$-0.630539\pi$
$107$	−14.4017	−	8.31483i	−1.39227	−	0.803825i	−0.993566	+	0.113256i	$0.963872\pi$
$108$		0			0
$109$	−1.54822	−	2.68159i	−0.148292	−	0.256850i	0.782304	−	0.622897i	$-0.214044\pi$
$109$	−1.54822	−	2.68159i	−0.148292	−	0.256850i	−0.930596	+	0.366047i	$0.880711\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			16.8230i			1.58257i	0.611446	+	0.791286i	$0.290588\pi$
$113$			16.8230i			1.58257i	−0.611446	+	0.791286i	$0.709412\pi$
$114$		0			0
$115$	14.1828	−	8.18843i	1.32255	−	0.763575i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	15.3605	−	5.34454i	1.40810	−	0.489934i
$120$		0			0
$121$	−4.71271	+	8.16266i	−0.428429	+	0.742060i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.13961			−0.459701
$126$		0			0
$127$	−10.9263			−0.969551			−0.484775	−	0.874639i	$-0.661099\pi$
$127$	−10.9263			−0.969551			−0.484775	+	0.874639i	$0.661099\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.45888	−	9.45505i	0.476944	−	0.826092i	−0.522707	−	0.852513i	$-0.675078\pi$
$131$	5.45888	−	9.45505i	0.476944	−	0.826092i	0.999651	+	0.0264210i	$0.00841105\pi$
$132$		0			0
$133$	5.23536	−	6.06198i	0.453964	−	0.525641i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.44527	+	3.72118i	−0.550657	+	0.317922i	−0.749387	−	0.662132i	$-0.769652\pi$
$137$	−6.44527	+	3.72118i	−0.550657	+	0.317922i	0.198730	+	0.980054i	$0.436318\pi$
$138$		0			0
$139$		−	16.4728i		−	1.39720i	−0.715512	−	0.698600i	$-0.753807\pi$
$139$		−	16.4728i		−	1.39720i	0.715512	−	0.698600i	$-0.246193\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.29419	−	5.70571i	−0.275474	−	0.477135i
$144$		0			0
$145$	−9.25756	−	5.34485i	−0.768798	−	0.443866i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−13.6491	−	7.88029i	−1.11817	−	0.645578i	−0.177240	−	0.984168i	$-0.556717\pi$
$149$	−13.6491	−	7.88029i	−1.11817	−	0.645578i	−0.940934	+	0.338589i	$0.890050\pi$
$150$		0			0
$151$	−0.339365	−	0.587797i	−0.0276171	−	0.0478342i	0.851887	−	0.523726i	$-0.175459\pi$
$151$	−0.339365	−	0.587797i	−0.0276171	−	0.0478342i	−0.879504	+	0.475892i	$0.842125\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			19.4199i			1.55984i
$156$		0			0
$157$	−6.74094	+	3.89189i	−0.537986	+	0.310606i	−0.744262	−	0.667887i	$-0.767199\pi$
$157$	−6.74094	+	3.89189i	−0.537986	+	0.310606i	0.206276	+	0.978494i	$0.433865\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	9.88655	−	11.4476i	0.779169	−	0.902194i
$162$		0			0
$163$	3.80101	−	6.58354i	0.297718	−	0.515663i	−0.677896	−	0.735158i	$-0.737108\pi$
$163$	3.80101	−	6.58354i	0.297718	−	0.515663i	0.975613	+	0.219496i	$0.0704412\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.37135			0.338266			0.169133	−	0.985593i	$-0.445903\pi$
$167$	4.37135			0.338266			0.169133	+	0.985593i	$0.445903\pi$
$168$		0			0
$169$	−14.5674			−1.12057
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.70140	+	8.14307i	−0.357441	+	0.619106i	−0.987533	−	0.157415i	$-0.949684\pi$
$173$	−4.70140	+	8.14307i	−0.357441	+	0.619106i	0.630092	+	0.776521i	$0.283017\pi$
$174$		0			0
$175$	8.01071	−	2.78725i	0.605553	−	0.210696i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.06513	−	0.614954i	0.0796116	−	0.0459638i	−0.459666	−	0.888092i	$-0.652031\pi$
$179$	1.06513	−	0.614954i	0.0796116	−	0.0459638i	0.539277	+	0.842128i	$0.318697\pi$
$180$		0			0
$181$		−	5.11079i		−	0.379882i	−0.981795	−	0.189941i	$-0.939170\pi$
$181$		−	5.11079i		−	0.379882i	0.981795	−	0.189941i	$-0.0608297\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	11.0514	+	19.1416i	0.812517	+	1.40732i
$186$		0			0
$187$	−6.68012	−	3.85677i	−0.488498	−	0.282035i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.04399	−	2.33480i	−0.292613	−	0.168940i	0.346507	−	0.938047i	$-0.387368\pi$
$191$	−4.04399	−	2.33480i	−0.292613	−	0.168940i	−0.639120	+	0.769107i	$0.720701\pi$
$192$		0			0
$193$	−12.3765	−	21.4368i	−0.890881	−	1.54305i	−0.838821	−	0.544408i	$-0.816754\pi$
$193$	−12.3765	−	21.4368i	−0.890881	−	1.54305i	−0.0520610	−	0.998644i	$-0.516579\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	9.43044i		−	0.671891i	−0.941881	−	0.335946i	$-0.890944\pi$
$197$		−	9.43044i		−	0.671891i	0.941881	−	0.335946i	$-0.109056\pi$
$198$		0			0
$199$	15.6665	−	9.04503i	1.11057	−	0.641185i	0.171590	−	0.985168i	$-0.445110\pi$
$199$	15.6665	−	9.04503i	1.11057	−	0.641185i	0.938976	+	0.343983i	$0.111776\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.69774	−	1.85260i	−0.680648	−	0.130027i
$204$		0			0
$205$	−12.9892	+	22.4980i	−0.907207	+	1.57133i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.79886			−0.262773
$210$		0			0
$211$	18.3379			1.26243			0.631216	−	0.775607i	$-0.282556\pi$
$211$	18.3379			1.26243			0.631216	+	0.775607i	$0.282556\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−12.4055	+	21.4870i	−0.846051	+	1.46540i
$216$		0			0
$217$	5.89419	+	16.9402i	0.400124	+	1.14998i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−27.9512	+	16.1376i	−1.88020	+	1.08553i
$222$		0			0
$223$			7.83841i			0.524899i	0.964946	+	0.262449i	$0.0845303\pi$
$223$			7.83841i			0.524899i	−0.964946	+	0.262449i	$0.915470\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.08795	−	14.0087i	−0.536816	−	0.929793i	−0.999073	−	0.0430471i	$-0.986293\pi$
$227$	−8.08795	−	14.0087i	−0.536816	−	0.929793i	0.462257	−	0.886746i	$-0.347040\pi$
$228$		0			0
$229$	6.44112	+	3.71878i	0.425641	+	0.245744i	0.697488	−	0.716596i	$-0.254301\pi$
$229$	6.44112	+	3.71878i	0.425641	+	0.245744i	−0.271847	+	0.962341i	$0.587634\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.8568	−	11.4643i	−1.30086	−	0.751051i	−0.320308	−	0.947314i	$-0.603786\pi$
$233$	−19.8568	−	11.4643i	−1.30086	−	0.751051i	−0.980551	+	0.196262i	$0.937120\pi$
$234$		0			0
$235$	−8.52491	−	14.7656i	−0.556104	−	0.963200i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	13.6562i		−	0.883344i	−0.897177	−	0.441672i	$-0.854386\pi$
$239$		−	13.6562i		−	0.883344i	0.897177	−	0.441672i	$-0.145614\pi$
$240$		0			0
$241$	7.04728	−	4.06875i	0.453955	−	0.262091i	−0.255544	−	0.966797i	$-0.582255\pi$
$241$	7.04728	−	4.06875i	0.453955	−	0.262091i	0.709499	+	0.704706i	$0.248921\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.7212	−	12.4470i	1.00439	−	0.795207i
$246$		0			0
$247$	−7.94766	+	13.7658i	−0.505698	+	0.875894i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.95828			0.628561			0.314281	−	0.949330i	$-0.398237\pi$
$251$	9.95828			0.628561			0.314281	+	0.949330i	$0.398237\pi$
$252$		0			0
$253$	−7.17383			−0.451015
$254$		0			0
$255$		0			0
$256$		0			0
$257$	11.7450	−	20.3429i	0.732632	−	1.26896i	−0.223122	−	0.974790i	$-0.571625\pi$
$257$	11.7450	−	20.3429i	0.732632	−	1.26896i	0.955754	−	0.294166i	$-0.0950418\pi$
$258$		0			0
$259$	15.4501	+	13.3433i	0.960021	+	0.829112i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.63448	+	2.67572i	−0.285775	+	0.164992i	−0.636035	−	0.771660i	$-0.719427\pi$
$263$	−4.63448	+	2.67572i	−0.285775	+	0.164992i	0.350260	+	0.936652i	$0.386093\pi$
$264$		0			0
$265$		−	23.7481i		−	1.45884i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.554894	−	0.961104i	−0.0338325	−	0.0585995i	0.848613	−	0.529014i	$-0.177438\pi$
$269$	−0.554894	−	0.961104i	−0.0338325	−	0.0585995i	−0.882446	+	0.470414i	$0.844105\pi$
$270$		0			0
$271$	11.9571	+	6.90342i	0.726341	+	0.419353i	0.817082	−	0.576521i	$-0.195590\pi$
$271$	11.9571	+	6.90342i	0.726341	+	0.419353i	−0.0907411	+	0.995875i	$0.528924\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.48377	−	2.01135i	−0.210079	−	0.121289i
$276$		0			0
$277$	6.86156	+	11.8846i	0.412271	+	0.714075i	0.995138	−	0.0984929i	$-0.0314022\pi$
$277$	6.86156	+	11.8846i	0.412271	+	0.714075i	−0.582866	+	0.812568i	$0.698069\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	30.5547i		−	1.82274i	−0.411586	−	0.911371i	$-0.635025\pi$
$281$		−	30.5547i		−	1.82274i	0.411586	−	0.911371i	$-0.364975\pi$
$282$		0			0
$283$	12.7910	−	7.38491i	0.760349	−	0.438988i	−0.0690722	−	0.997612i	$-0.522004\pi$
$283$	12.7910	−	7.38491i	0.760349	−	0.438988i	0.829421	+	0.558624i	$0.188671\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.50225	+	23.5678i	−0.265760	+	1.39116i
$288$		0			0
$289$	−10.3936	+	18.0022i	−0.611386	+	1.05895i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.44322			0.0843141			0.0421570	−	0.999111i	$-0.486577\pi$
$293$	1.44322			0.0843141			0.0421570	+	0.999111i	$0.486577\pi$
$294$		0			0
$295$	8.51259			0.495622
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−15.0085	+	25.9955i	−0.867964	+	1.50336i
$300$		0			0
$301$	−4.29994	+	22.5087i	−0.247845	+	1.29738i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.44063	−	3.71850i	0.368789	−	0.212921i
$306$		0			0
$307$			21.8901i			1.24933i	0.780891	+	0.624667i	$0.214765\pi$
$307$			21.8901i			1.24933i	−0.780891	+	0.624667i	$0.785235\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.25630	+	9.10417i	0.298057	+	0.516250i	0.975691	−	0.219149i	$-0.0703281\pi$
$311$	5.25630	+	9.10417i	0.298057	+	0.516250i	−0.677634	+	0.735399i	$0.736995\pi$
$312$		0			0
$313$	−2.85633	−	1.64910i	−0.161449	−	0.0932127i	0.417099	−	0.908861i	$-0.363047\pi$
$313$	−2.85633	−	1.64910i	−0.161449	−	0.0932127i	−0.578548	+	0.815649i	$0.696380\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.6051	+	10.7416i	1.04496	+	0.603311i	0.921236	−	0.389005i	$-0.127181\pi$
$317$	18.6051	+	10.7416i	1.04496	+	0.603311i	0.123729	+	0.992316i	$0.460515\pi$
$318$		0			0
$319$	2.34130	+	4.05525i	0.131087	+	0.227050i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			18.6099i			1.03548i
$324$		0			0
$325$	−14.5769	+	8.41598i	−0.808581	+	0.466835i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−11.9180	−	10.2928i	−0.657058	−	0.567461i
$330$		0			0
$331$	−8.49354	+	14.7112i	−0.466847	+	0.808603i	−0.999283	−	0.0378676i	$-0.987943\pi$
$331$	−8.49354	+	14.7112i	−0.466847	+	0.808603i	0.532436	+	0.846470i	$0.321277\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−30.4788			−1.66523
$336$		0			0
$337$	−18.0656			−0.984098			−0.492049	−	0.870568i	$-0.663752\pi$
$337$	−18.0656			−0.984098			−0.492049	+	0.870568i	$0.663752\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.25341	−	7.36712i	0.230335	−	0.398952i
$342$		0			0
$343$	9.93606	−	15.6293i	0.536497	−	0.843902i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	23.7199	−	13.6947i	1.27335	−	0.735169i	0.297734	−	0.954649i	$-0.403769\pi$
$347$	23.7199	−	13.6947i	1.27335	−	0.735169i	0.975617	+	0.219480i	$0.0704360\pi$
$348$		0			0
$349$		−	23.0855i		−	1.23574i	−0.786281	−	0.617869i	$-0.787996\pi$
$349$		−	23.0855i		−	1.23574i	0.786281	−	0.617869i	$-0.212004\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.86592	−	17.0883i	−0.525110	−	0.909517i	−0.999572	−	0.0292415i	$-0.990691\pi$
$353$	−9.86592	−	17.0883i	−0.525110	−	0.909517i	0.474462	−	0.880276i	$-0.342643\pi$
$354$		0			0
$355$	−6.52364	−	3.76642i	−0.346239	−	0.199901i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	30.7577	+	17.7579i	1.62333	+	0.937228i	0.986022	+	0.166617i	$0.0532843\pi$
$359$	30.7577	+	17.7579i	1.62333	+	0.937228i	0.637305	+	0.770611i	$0.280049\pi$
$360$		0			0
$361$	−4.91738	−	8.51715i	−0.258809	−	0.448271i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			27.3001i			1.42895i
$366$		0			0
$367$	−19.6561	+	11.3485i	−1.02604	+	0.592386i	−0.915848	−	0.401524i	$-0.868481\pi$
$367$	−19.6561	+	11.3485i	−1.02604	+	0.592386i	−0.110194	+	0.993910i	$0.535147\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.20788	−	20.7159i	−0.374215	−	1.07551i
$372$		0			0
$373$	−7.23264	+	12.5273i	−0.374492	+	0.648639i	−0.990251	−	0.139296i	$-0.955516\pi$
$373$	−7.23264	+	12.5273i	−0.374492	+	0.648639i	0.615759	+	0.787934i	$0.288849\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	19.5931			1.00909
$378$		0			0
$379$	27.2141			1.39790			0.698948	−	0.715173i	$-0.253652\pi$
$379$	27.2141			1.39790			0.698948	+	0.715173i	$0.253652\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	17.4139	−	30.1618i	0.889810	−	1.54120i	0.0497104	−	0.998764i	$-0.484170\pi$
$383$	17.4139	−	30.1618i	0.889810	−	1.54120i	0.840099	−	0.542432i	$-0.182497\pi$
$384$		0			0
$385$	−9.34131	−	1.78451i	−0.476077	−	0.0909471i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.76819	+	1.02087i	−0.0896510	+	0.0517600i	−0.544155	−	0.838985i	$-0.683150\pi$
$389$	−1.76819	+	1.02087i	−0.0896510	+	0.0517600i	0.454504	+	0.890745i	$0.349816\pi$
$390$		0			0
$391$			35.1432i			1.77727i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	15.7314	+	27.2476i	0.791533	+	1.37098i
$396$		0			0
$397$	−26.9678	−	15.5699i	−1.35348	−	0.781430i	−0.364742	−	0.931108i	$-0.618843\pi$
$397$	−26.9678	−	15.5699i	−1.35348	−	0.781430i	−0.988735	+	0.149678i	$0.952176\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.2752	−	6.50976i	−0.563058	−	0.325082i	0.191314	−	0.981529i	$-0.438725\pi$
$401$	−11.2752	−	6.50976i	−0.563058	−	0.325082i	−0.754372	+	0.656447i	$0.772059\pi$
$402$		0			0
$403$	−17.7973	−	30.8257i	−0.886544	−	1.53554i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	9.68210i		−	0.479924i
$408$		0			0
$409$	−10.9054	+	6.29621i	−0.539236	+	0.311328i	−0.744769	−	0.667322i	$-0.767440\pi$
$409$	−10.9054	+	6.29621i	−0.539236	+	0.311328i	0.205534	+	0.978650i	$0.434107\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.42566	−	2.58369i	0.365393	−	0.127135i
$414$		0			0
$415$	−20.9377	+	36.2652i	−1.02779	+	1.78019i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13.5293			0.660948			0.330474	−	0.943815i	$-0.392791\pi$
$419$	13.5293			0.660948			0.330474	+	0.943815i	$0.392791\pi$
$420$		0			0
$421$	2.97498			0.144992			0.0724959	−	0.997369i	$-0.476904\pi$
$421$	2.97498			0.144992			0.0724959	+	0.997369i	$0.476904\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−9.85324	+	17.0663i	−0.477952	+	0.827838i
$426$		0			0
$427$	4.48965	−	5.19852i	0.217269	−	0.251574i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.94179	−	5.16254i	0.430711	−	0.248671i	−0.268939	−	0.963157i	$-0.586673\pi$
$431$	8.94179	−	5.16254i	0.430711	−	0.248671i	0.699649	+	0.714486i	$0.253340\pi$
$432$		0			0
$433$		−	10.9117i		−	0.524384i	−0.965016	−	0.262192i	$-0.915555\pi$
$433$		−	10.9117i		−	0.524384i	0.965016	−	0.262192i	$-0.0844453\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.65390	+	14.9890i	0.413972	+	0.717021i
$438$		0			0
$439$	−10.1562	−	5.86369i	−0.484729	−	0.279859i	0.237656	−	0.971349i	$-0.423621\pi$
$439$	−10.1562	−	5.86369i	−0.484729	−	0.279859i	−0.722385	+	0.691491i	$0.756954\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	30.1605	+	17.4131i	1.43297	+	0.827324i	0.997346	−	0.0728067i	$-0.0231956\pi$
$443$	30.1605	+	17.4131i	1.43297	+	0.827324i	0.435621	+	0.900130i	$0.356529\pi$
$444$		0			0
$445$	17.0595	+	29.5480i	0.808699	+	1.40071i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	4.37875i		−	0.206646i	−0.994648	−	0.103323i	$-0.967052\pi$
$449$		−	4.37875i		−	0.206646i	0.994648	−	0.103323i	$-0.0329475\pi$
$450$		0			0
$451$	9.85518	−	5.68989i	0.464062	−	0.267927i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−26.0096	+	30.1163i	−1.21935	+	1.41187i
$456$		0			0
$457$	−3.63140	+	6.28977i	−0.169870	+	0.294223i	−0.938374	−	0.345622i	$-0.887668\pi$
$457$	−3.63140	+	6.28977i	−0.169870	+	0.294223i	0.768504	+	0.639845i	$0.221001\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11.8994			−0.554208			−0.277104	−	0.960840i	$-0.589375\pi$
$461$	−11.8994			−0.554208			−0.277104	+	0.960840i	$0.589375\pi$
$462$		0			0
$463$	−27.4927			−1.27769			−0.638847	−	0.769334i	$-0.720588\pi$
$463$	−27.4927			−1.27769			−0.638847	+	0.769334i	$0.720588\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.57357	+	2.72550i	−0.0728161	+	0.126121i	−0.900134	−	0.435612i	$-0.856532\pi$
$467$	−1.57357	+	2.72550i	−0.0728161	+	0.126121i	0.827318	+	0.561733i	$0.189865\pi$
$468$		0			0
$469$	−26.5871	+	9.25072i	−1.22768	+	0.427159i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.41233	−	5.43421i	0.432780	−	0.249865i
$474$		0			0
$475$			9.70530i			0.445310i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.7463	−	34.2016i	−0.902231	−	1.56271i	−0.824592	−	0.565728i	$-0.808595\pi$
$479$	−19.7463	−	34.2016i	−0.902231	−	1.56271i	−0.0776390	−	0.996982i	$-0.524738\pi$
$480$		0			0
$481$	−35.0846	−	20.2561i	−1.59972	−	0.923598i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	17.7380	+	10.2410i	0.805439	+	0.465021i
$486$		0			0
$487$	16.1647	+	27.9980i	0.732490	+	1.26871i	0.955816	+	0.293966i	$0.0949754\pi$
$487$	16.1647	+	27.9980i	0.732490	+	1.26871i	−0.223326	+	0.974744i	$0.571691\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			14.3858i			0.649222i	0.945848	+	0.324611i	$0.105233\pi$
$491$			14.3858i			0.649222i	−0.945848	+	0.324611i	$0.894767\pi$
$492$		0			0
$493$	19.8659	−	11.4696i	0.894714	−	0.516563i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.83383	−	1.30550i	−0.306539	−	0.0585596i
$498$		0			0
$499$	−20.8413	+	36.0983i	−0.932987	+	1.61598i	−0.154801	+	0.987946i	$0.549474\pi$
$499$	−20.8413	+	36.0983i	−0.932987	+	1.61598i	−0.778185	+	0.628035i	$0.783860\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.4755			0.556256			0.278128	−	0.960544i	$-0.410286\pi$
$503$	12.4755			0.556256			0.278128	+	0.960544i	$0.410286\pi$
$504$		0			0
$505$	−8.29063			−0.368928
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−13.8574	+	24.0017i	−0.614217	+	1.06385i	0.376304	+	0.926496i	$0.377195\pi$
$509$	−13.8574	+	24.0017i	−0.614217	+	1.06385i	−0.990521	+	0.137359i	$0.956139\pi$
$510$		0			0
$511$	8.28595	+	23.8143i	0.366549	+	1.05348i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	15.9932	−	9.23370i	0.704746	−	0.406885i
$516$		0			0
$517$			7.46862i			0.328470i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.0961	+	20.9510i	0.529938	+	0.917880i	0.999390	+	0.0349216i	$0.0111181\pi$
$521$	12.0961	+	20.9510i	0.529938	+	0.917880i	−0.469452	+	0.882958i	$0.655549\pi$
$522$		0			0
$523$	−20.8779	−	12.0539i	−0.912927	−	0.527078i	−0.0315550	−	0.999502i	$-0.510046\pi$
$523$	−20.8779	−	12.0539i	−0.912927	−	0.527078i	−0.881372	+	0.472424i	$0.843379\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−36.0901	−	20.8366i	−1.57211	−	0.907658i
$528$		0			0
$529$	4.84217	+	8.38689i	0.210529	+	0.364647i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	47.6157i		−	2.06246i
$534$		0			0
$535$	41.2548	−	23.8185i	1.78360	−	1.02976i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.69019	+	1.27856i	−0.374313	+	0.0550713i
$540$		0			0
$541$	−20.2072	+	34.9999i	−0.868775	+	1.50476i	−0.00552468	+	0.999985i	$0.501759\pi$
$541$	−20.2072	+	34.9999i	−0.868775	+	1.50476i	−0.863250	+	0.504777i	$0.831575\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.86998			0.379948
$546$		0			0
$547$	22.2848			0.952830			0.476415	−	0.879220i	$-0.341936\pi$
$547$	22.2848			0.952830			0.476415	+	0.879220i	$0.341936\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.64868	−	9.78380i	0.240642	−	0.416804i
$552$		0			0
$553$	21.9928	+	18.9938i	0.935228	+	0.807699i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	23.1768	−	13.3811i	0.982031	−	0.566976i	0.0791483	−	0.996863i	$-0.474780\pi$
$557$	23.1768	−	13.3811i	0.982031	−	0.566976i	0.902883	+	0.429887i	$0.141447\pi$
$558$		0			0
$559$		−	45.4761i		−	1.92343i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9.04743	−	15.6706i	−0.381304	−	0.660438i	0.609945	−	0.792444i	$-0.291192\pi$
$563$	−9.04743	−	15.6706i	−0.381304	−	0.660438i	−0.991249	+	0.132006i	$0.957858\pi$
$564$		0			0
$565$	−41.7344	−	24.0954i	−1.75578	−	1.01370i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−20.3851	−	11.7693i	−0.854588	−	0.493396i	0.00760850	−	0.999971i	$-0.497578\pi$
$569$	−20.3851	−	11.7693i	−0.854588	−	0.493396i	−0.862196	+	0.506575i	$0.830911\pi$
$570$		0			0
$571$	−6.19412	−	10.7285i	−0.259216	−	0.448975i	0.706816	−	0.707397i	$-0.250131\pi$
$571$	−6.19412	−	10.7285i	−0.259216	−	0.448975i	−0.966032	+	0.258422i	$0.916797\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			18.3277i			0.764316i
$576$		0			0
$577$	9.61514	−	5.55131i	0.400284	−	0.231104i	−0.286323	−	0.958133i	$-0.592433\pi$
$577$	9.61514	−	5.55131i	0.400284	−	0.231104i	0.686606	+	0.727029i	$0.259100\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.25732	+	37.9896i	−0.301084	+	1.57607i
$582$		0			0
$583$	−5.20140	+	9.00909i	−0.215420	+	0.373119i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−39.6476			−1.63643			−0.818215	−	0.574912i	$-0.805036\pi$
$587$	−39.6476			−1.63643			−0.818215	+	0.574912i	$0.805036\pi$
$588$		0			0
$589$	−20.5238			−0.845668
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13.9778	−	24.2102i	0.573998	−	0.994194i	−0.422151	−	0.906525i	$-0.638725\pi$
$593$	13.9778	−	24.2102i	0.573998	−	0.994194i	0.996150	−	0.0876691i	$-0.0279418\pi$
$594$		0			0
$595$	−8.74198	+	45.7613i	−0.358386	+	1.87603i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	25.2868	−	14.5993i	1.03319	−	0.596513i	0.115294	−	0.993331i	$-0.463219\pi$
$599$	25.2868	−	14.5993i	1.03319	−	0.596513i	0.917897	+	0.396818i	$0.129886\pi$
$600$		0			0
$601$			14.9790i			0.611006i	0.952191	+	0.305503i	$0.0988247\pi$
$601$			14.9790i			0.611006i	−0.952191	+	0.305503i	$0.901175\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−13.4999	−	23.3826i	−0.548850	−	0.950637i
$606$		0			0
$607$	−21.1015	−	12.1830i	−0.856483	−	0.494491i	0.00634989	−	0.999980i	$-0.497979\pi$
$607$	−21.1015	−	12.1830i	−0.856483	−	0.494491i	−0.862833	+	0.505489i	$0.831312\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	27.0637	+	15.6252i	1.09488	+	0.632129i
$612$		0			0
$613$	10.9343	+	18.9387i	0.441631	+	0.764927i	0.997811	−	0.0661347i	$-0.0210667\pi$
$613$	10.9343	+	18.9387i	0.441631	+	0.764927i	−0.556180	+	0.831062i	$0.687733\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	41.2793i		−	1.66184i	−0.556391	−	0.830921i	$-0.687814\pi$
$617$		−	41.2793i		−	1.66184i	0.556391	−	0.830921i	$-0.312186\pi$
$618$		0			0
$619$	−35.3843	+	20.4292i	−1.42222	+	0.821117i	−0.996488	−	0.0837339i	$-0.973315\pi$
$619$	−35.3843	+	20.4292i	−1.42222	+	0.821117i	−0.425728	+	0.904851i	$0.639982\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	23.8495	+	20.5974i	0.955510	+	0.825216i
$624$		0			0
$625$	15.3759	−	26.6319i	0.615037	−	1.06527i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−47.4307			−1.89119
$630$		0			0
$631$	26.4215			1.05182			0.525911	−	0.850540i	$-0.323725\pi$
$631$	26.4215			1.05182			0.525911	+	0.850540i	$0.323725\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	15.6496	−	27.1059i	0.621035	−	1.07566i
$636$		0			0
$637$	−13.5479	+	34.1651i	−0.536785	+	1.35367i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−30.1973	+	17.4344i	−1.19272	+	0.688619i	−0.958923	−	0.283666i	$-0.908449\pi$
$641$	−30.1973	+	17.4344i	−1.19272	+	0.688619i	−0.233799	+	0.972285i	$0.575116\pi$
$642$		0			0
$643$			26.8650i			1.05945i	0.848168	+	0.529727i	$0.177706\pi$
$643$			26.8650i			1.05945i	−0.848168	+	0.529727i	$0.822294\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.988791	−	1.71264i	−0.0388734	−	0.0673306i	0.845934	−	0.533287i	$-0.179044\pi$
$647$	−0.988791	−	1.71264i	−0.0388734	−	0.0673306i	−0.884807	+	0.465957i	$0.845710\pi$
$648$		0			0
$649$	−3.22933	−	1.86446i	−0.126762	−	0.0731863i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−17.4467	−	10.0729i	−0.682742	−	0.394182i	0.118145	−	0.992996i	$-0.462305\pi$
$653$	−17.4467	−	10.0729i	−0.682742	−	0.394182i	−0.800888	+	0.598815i	$0.795639\pi$
$654$		0			0
$655$	15.6374	+	27.0847i	0.611003	+	1.05829i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			23.0826i			0.899172i	0.893237	+	0.449586i	$0.148428\pi$
$659$			23.0826i			0.899172i	−0.893237	+	0.449586i	$0.851572\pi$
$660$		0			0
$661$	−2.42515	+	1.40016i	−0.0943273	+	0.0544599i	−0.546422	−	0.837510i	$-0.684010\pi$
$661$	−2.42515	+	1.40016i	−0.0943273	+	0.0544599i	0.452094	+	0.891970i	$0.350677\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7.54000	+	21.6704i	0.292389	+	0.840342i
$666$		0			0
$667$	10.6671	−	18.4759i	0.413030	−	0.715390i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.25776			−0.125764
$672$		0			0
$673$	−32.6962			−1.26035			−0.630173	−	0.776454i	$-0.717016\pi$
$673$	−32.6962			−1.26035			−0.630173	+	0.776454i	$0.717016\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.7293	+	39.3684i	−0.873559	+	1.51305i	−0.0152697	+	0.999883i	$0.504861\pi$
$677$	−22.7293	+	39.3684i	−0.873559	+	1.51305i	−0.858290	+	0.513166i	$0.828473\pi$
$678$		0			0
$679$	18.5814	+	3.54969i	0.713088	+	0.136224i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−43.2107	+	24.9477i	−1.65341	+	0.954598i	−0.677757	+	0.735286i	$0.737048\pi$
$683$	−43.2107	+	24.9477i	−1.65341	+	0.954598i	−0.975655	+	0.219312i	$0.929619\pi$
$684$		0			0
$685$		−	21.3192i		−	0.814565i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	21.7639	+	37.6962i	0.829138	+	1.43611i
$690$		0			0
$691$	27.1274	+	15.6620i	1.03198	+	0.595812i	0.917550	−	0.397620i	$-0.130164\pi$
$691$	27.1274	+	15.6620i	1.03198	+	0.595812i	0.114426	+	0.993432i	$0.463497\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	40.8655	+	23.5937i	1.55012	+	0.894961i
$696$		0			0
$697$	−27.8737	−	48.2787i	−1.05579	−	1.82868i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			13.3593i			0.504576i	0.967652	+	0.252288i	$0.0811829\pi$
$701$			13.3593i			0.504576i	−0.967652	+	0.252288i	$0.918817\pi$
$702$		0			0
$703$	−20.2298	+	11.6797i	−0.762980	+	0.440506i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.23204	+	2.51632i	−0.271989	+	0.0946359i
$708$		0			0
$709$	−15.9513	+	27.6284i	−0.599062	+	1.03761i	0.393897	+	0.919154i	$0.371127\pi$
$709$	−15.9513	+	27.6284i	−0.599062	+	1.03761i	−0.992960	+	0.118452i	$0.962207\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−38.7575			−1.45148
$714$		0			0
$715$	18.8729			0.705808
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.82338	+	8.35434i	−0.179882	+	0.311564i	−0.941840	−	0.336062i	$-0.890905\pi$
$719$	−4.82338	+	8.35434i	−0.179882	+	0.311564i	0.761958	+	0.647626i	$0.224238\pi$
$720$		0			0
$721$	11.1486	−	12.9089i	0.415195	−	0.480751i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	10.3603	−	5.98153i	0.384772	−	0.222148i
$726$		0			0
$727$		−	38.5339i		−	1.42914i	−0.699562	−	0.714572i	$-0.746622\pi$
$727$		−	38.5339i		−	1.42914i	0.699562	−	0.714572i	$-0.253378\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−26.6212	−	46.1092i	−0.984620	−	1.70541i
$732$		0			0
$733$	−15.4377	−	8.91294i	−0.570203	−	0.329207i	0.187027	−	0.982355i	$-0.440115\pi$
$733$	−15.4377	−	8.91294i	−0.570203	−	0.329207i	−0.757230	+	0.653148i	$0.773448\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	11.5624	+	6.67557i	0.425907	+	0.245898i
$738$		0			0
$739$	4.27651	+	7.40713i	0.157314	+	0.272476i	0.933899	−	0.357536i	$-0.116383\pi$
$739$	4.27651	+	7.40713i	0.157314	+	0.272476i	−0.776585	+	0.630012i	$0.783050\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	16.8144i		−	0.616859i	−0.951247	−	0.308430i	$-0.900197\pi$
$743$		−	16.8144i		−	0.616859i	0.951247	−	0.308430i	$-0.0998034\pi$
$744$		0			0
$745$	39.0988	−	22.5737i	1.43247	−	0.827036i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	28.7580	−	33.2986i	1.05079	−	1.21671i
$750$		0			0
$751$	21.3063	−	36.9036i	0.777478	−	1.34663i	−0.155913	−	0.987771i	$-0.549832\pi$
$751$	21.3063	−	36.9036i	0.777478	−	1.34663i	0.933391	−	0.358861i	$-0.116835\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.94427			0.0707593
$756$		0			0
$757$	8.97784			0.326305			0.163153	−	0.986601i	$-0.447834\pi$
$757$	8.97784			0.326305			0.163153	+	0.986601i	$0.447834\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	19.9510	−	34.5561i	0.723222	−	1.25266i	−0.236480	−	0.971636i	$-0.575994\pi$
$761$	19.9510	−	34.5561i	0.723222	−	1.25266i	0.959702	−	0.281021i	$-0.0906730\pi$
$762$		0			0
$763$	7.73742	−	2.69216i	0.280114	−	0.0974627i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−13.5123	+	7.80133i	−0.487901	+	0.281690i
$768$		0			0
$769$			9.48373i			0.341992i	0.985272	+	0.170996i	$0.0546985\pi$
$769$			9.48373i			0.341992i	−0.985272	+	0.170996i	$0.945301\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.60509	−	11.4403i	−0.237568	−	0.411481i	0.722448	−	0.691426i	$-0.243017\pi$
$773$	−6.60509	−	11.4403i	−0.237568	−	0.411481i	−0.960016	+	0.279945i	$0.909684\pi$
$774$		0			0
$775$	−18.8215	−	10.8666i	−0.676087	−	0.390339i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−23.7769	−	13.7276i	−0.851896	−	0.491842i
$780$		0			0
$781$	1.64987	+	2.85766i	0.0590370	+	0.102255i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	22.2972i		−	0.795822i
$786$		0			0
$787$	−6.91018	+	3.98960i	−0.246321	+	0.142214i	−0.618079	−	0.786116i	$-0.712089\pi$
$787$	−6.91018	+	3.98960i	−0.246321	+	0.142214i	0.371757	+	0.928330i	$0.378755\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−43.7188	−	8.35180i	−1.55446	−	0.296956i
$792$		0			0
$793$	−6.81561	+	11.8050i	−0.242029	+	0.419207i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−37.4408			−1.32622			−0.663110	−	0.748522i	$-0.730764\pi$
$797$	−37.4408			−1.32622			−0.663110	+	0.748522i	$0.730764\pi$
$798$		0			0
$799$	36.5873			1.29437
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5.97936	−	10.3566i	0.211007	−	0.365475i
$804$		0			0
$805$	14.2387	+	40.9227i	0.501847	+	1.44234i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	12.0001	−	6.92824i	0.421900	−	0.243584i	−0.273990	−	0.961733i	$-0.588344\pi$
$809$	12.0001	−	6.92824i	0.421900	−	0.243584i	0.695890	+	0.718149i	$0.255010\pi$
$810$		0			0
$811$			17.6343i			0.619223i	0.950863	+	0.309611i	$0.100199\pi$
$811$			17.6343i			0.619223i	−0.950863	+	0.309611i	$0.899801\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.8883	+	18.8591i	0.381400	+	0.660604i
$816$		0			0
$817$	−22.7085	−	13.1107i	−0.794469	−	0.458687i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	42.6604	+	24.6300i	1.48886	+	0.859593i	0.999919	−	0.0127244i	$-0.00405042\pi$
$821$	42.6604	+	24.6300i	1.48886	+	0.859593i	0.488940	+	0.872318i	$0.337384\pi$
$822$		0			0
$823$	17.2854	+	29.9392i	0.602530	+	1.04361i	0.992437	+	0.122759i	$0.0391741\pi$
$823$	17.2854	+	29.9392i	0.602530	+	1.04361i	−0.389906	+	0.920855i	$0.627493\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	45.2579i		−	1.57377i	−0.617099	−	0.786886i	$-0.711692\pi$
$827$		−	45.2579i		−	1.57377i	0.617099	−	0.786886i	$-0.288308\pi$
$828$		0			0
$829$	8.17327	−	4.71884i	0.283869	−	0.163892i	−0.351305	−	0.936261i	$-0.614262\pi$
$829$	8.17327	−	4.71884i	0.283869	−	0.163892i	0.635174	+	0.772369i	$0.280928\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.26340	+	42.5716i	0.217014	+	1.47502i
$834$		0			0
$835$	−6.26104	+	10.8444i	−0.216672	+	0.375287i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−4.61174			−0.159215			−0.0796075	−	0.996826i	$-0.525367\pi$
$839$	−4.61174			−0.159215			−0.0796075	+	0.996826i	$0.525367\pi$
$840$		0			0
$841$	15.0745			0.519811
$842$		0			0
$843$		0			0
$844$		0			0
$845$	20.8647	−	36.1386i	0.717766	−	1.24321i
$846$		0			0
$847$	−18.8731	−	16.2996i	−0.648488	−	0.560060i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−38.2022	+	22.0561i	−1.30955	+	0.756072i
$852$		0			0
$853$			46.1077i			1.57870i	0.613946	+	0.789348i	$0.289581\pi$
$853$			46.1077i			1.57870i	−0.613946	+	0.789348i	$0.710419\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	21.7762	+	37.7174i	0.743859	+	1.28840i	0.950726	+	0.310033i	$0.100340\pi$
$857$	21.7762	+	37.7174i	0.743859	+	1.28840i	−0.206867	+	0.978369i	$0.566327\pi$
$858$		0			0
$859$	−13.4377	−	7.75827i	−0.458489	−	0.264709i	0.252920	−	0.967487i	$-0.418609\pi$
$859$	−13.4377	−	7.75827i	−0.458489	−	0.264709i	−0.711409	+	0.702779i	$0.751942\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−15.3098	−	8.83910i	−0.521151	−	0.300887i	0.216255	−	0.976337i	$-0.430616\pi$
$863$	−15.3098	−	8.83910i	−0.521151	−	0.300887i	−0.737405	+	0.675450i	$0.763949\pi$
$864$		0			0
$865$	−13.4675	−	23.3264i	−0.457910	−	0.793123i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	13.7822i		−	0.467529i
$870$		0			0
$871$	48.3799	−	27.9322i	1.63929	−	0.946445i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.55157	−	13.3566i	0.0862588	−	0.451535i
$876$		0			0
$877$	−7.53791	+	13.0560i	−0.254537	+	0.440871i	−0.964770	−	0.263096i	$-0.915256\pi$
$877$	−7.53791	+	13.0560i	−0.254537	+	0.440871i	0.710233	+	0.703967i	$0.248590\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.9248			−0.502830			−0.251415	−	0.967879i	$-0.580896\pi$
$881$	−14.9248			−0.502830			−0.251415	+	0.967879i	$0.580896\pi$
$882$		0			0
$883$	19.8737			0.668802			0.334401	−	0.942431i	$-0.391466\pi$
$883$	19.8737			0.668802			0.334401	+	0.942431i	$0.391466\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4.80844	−	8.32847i	0.161452	−	0.279643i	−0.773938	−	0.633262i	$-0.781716\pi$
$887$	4.80844	−	8.32847i	0.161452	−	0.279643i	0.935390	+	0.353619i	$0.115049\pi$
$888$		0			0
$889$	5.42438	−	28.3947i	0.181928	−	0.952329i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	15.6049	−	9.00951i	0.522199	−	0.301492i
$894$		0			0
$895$			3.52317i			0.117766i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	12.6491	+	21.9089i	0.421872	+	0.730704i
$900$		0			0
$901$	44.1338	+	25.4807i	1.47031	+	0.848885i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.6788	+	7.32013i	0.421459	+	0.243329i
$906$		0			0
$907$	−20.7363	−	35.9163i	−0.688536	−	1.19258i	−0.972311	−	0.233689i	$-0.924920\pi$
$907$	−20.7363	−	35.9163i	−0.688536	−	1.19258i	0.283775	−	0.958891i	$-0.408413\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	31.5561i		−	1.04550i	−0.852486	−	0.522751i	$-0.824906\pi$
$911$		−	31.5561i		−	1.04550i	0.852486	−	0.522751i	$-0.175094\pi$
$912$		0			0
$913$	15.8859	−	9.17171i	0.525746	−	0.303539i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	21.8613	+	18.8803i	0.721924	+	0.623481i
$918$		0			0
$919$	−23.5531	+	40.7951i	−0.776944	+	1.34571i	0.156751	+	0.987638i	$0.449898\pi$
$919$	−23.5531	+	40.7951i	−0.776944	+	1.34571i	−0.933695	+	0.358069i	$0.883435\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	13.8069			0.454460
$924$		0			0
$925$	−24.7357			−0.813306
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−27.0468	+	46.8464i	−0.887377	+	1.53698i	−0.0444117	+	0.999013i	$0.514141\pi$
$929$	−27.0468	+	46.8464i	−0.887377	+	1.53698i	−0.842965	+	0.537968i	$0.819192\pi$
$930$		0			0
$931$	13.1545	+	16.6149i	0.431122	+	0.544532i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	19.1357	−	11.0480i	0.625805	−	0.361308i
$936$		0			0
$937$		−	52.2751i		−	1.70775i	−0.520476	−	0.853876i	$-0.674245\pi$
$937$		−	52.2751i		−	1.70775i	0.520476	−	0.853876i	$-0.325755\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	22.1565	+	38.3761i	0.722280	+	1.25103i	0.960084	+	0.279712i	$0.0902390\pi$
$941$	22.1565	+	38.3761i	0.722280	+	1.25103i	−0.237804	+	0.971313i	$0.576428\pi$
$942$		0			0
$943$	−44.9007	−	25.9234i	−1.46217	−	0.844183i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	28.4036	+	16.3988i	0.922992	+	0.532890i	0.884588	−	0.466373i	$-0.154439\pi$
$947$	28.4036	+	16.3988i	0.922992	+	0.532890i	0.0384035	+	0.999262i	$0.487773\pi$
$948$		0			0
$949$	−25.0191	−	43.3343i	−0.812153	−	1.40669i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	8.96162i		−	0.290295i	−0.989410	−	0.145148i	$-0.953634\pi$
$953$		−	8.96162i		−	0.290295i	0.989410	−	0.145148i	$-0.0463657\pi$
$954$		0			0
$955$	11.5843	−	6.68821i	0.374860	−	0.216425i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.47067	−	18.5971i	−0.208949	−	0.600531i
$960$		0			0
$961$	7.47952	−	12.9549i	0.241275	−	0.417900i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	70.9070			2.28258
$966$		0			0
$967$	−24.0469			−0.773296			−0.386648	−	0.922227i	$-0.626367\pi$
$967$	−24.0469			−0.773296			−0.386648	+	0.922227i	$0.626367\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.25010	+	5.62934i	−0.104301	+	0.180654i	−0.913452	−	0.406946i	$-0.866594\pi$
$971$	−3.25010	+	5.62934i	−0.104301	+	0.180654i	0.809152	+	0.587600i	$0.199927\pi$
$972$		0			0
$973$	42.8087	+	8.17793i	1.37238	+	0.262172i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	7.63263	−	4.40670i	0.244190	−	0.140983i	−0.372911	−	0.927867i	$-0.621640\pi$
$977$	7.63263	−	4.40670i	0.244190	−	0.140983i	0.617101	+	0.786884i	$0.288307\pi$
$978$		0			0
$979$		−	14.9458i		−	0.477668i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.4558	+	28.5023i	0.524859	+	0.909083i	0.999581	+	0.0289467i	$0.00921530\pi$
$983$	16.4558	+	28.5023i	0.524859	+	0.909083i	−0.474722	+	0.880136i	$0.657451\pi$
$984$		0			0
$985$	23.3950	+	13.5071i	0.745427	+	0.430373i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−42.8831	−	24.7585i	−1.36360	−	0.787276i
$990$		0			0
$991$	−8.84072	−	15.3126i	−0.280835	−	0.486420i	0.690756	−	0.723088i	$-0.257278\pi$
$991$	−8.84072	−	15.3126i	−0.280835	−	0.486420i	−0.971591	+	0.236668i	$0.923945\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			51.8204i			1.64282i
$996$		0			0
$997$	33.5541	−	19.3725i	1.06267	−	0.613532i	0.136500	−	0.990640i	$-0.456415\pi$
$997$	33.5541	−	19.3725i	1.06267	−	0.613532i	0.926169	+	0.377108i	$0.123081\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.t.c.2105.3	yes	32
3.2	odd	2	inner	2268.2.t.c.2105.14	yes	32
7.3	odd	6	inner	2268.2.t.c.1781.14	yes	32
9.2	odd	6		2268.2.w.j.1349.14		32
9.4	even	3		2268.2.bm.j.593.14		32
9.5	odd	6		2268.2.bm.j.593.3		32
9.7	even	3		2268.2.w.j.1349.3		32
21.17	even	6	inner	2268.2.t.c.1781.3	✓	32
63.31	odd	6		2268.2.w.j.269.14		32
63.38	even	6		2268.2.bm.j.1025.14		32
63.52	odd	6		2268.2.bm.j.1025.3		32
63.59	even	6		2268.2.w.j.269.3		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.t.c.1781.3	✓	32	21.17	even	6	inner
2268.2.t.c.1781.14	yes	32	7.3	odd	6	inner
2268.2.t.c.2105.3	yes	32	1.1	even	1	trivial
2268.2.t.c.2105.14	yes	32	3.2	odd	2	inner
2268.2.w.j.269.3		32	63.59	even	6
2268.2.w.j.269.14		32	63.31	odd	6
2268.2.w.j.1349.3		32	9.7	even	3
2268.2.w.j.1349.14		32	9.2	odd	6
2268.2.bm.j.593.3		32	9.5	odd	6
2268.2.bm.j.593.14		32	9.4	even	3
2268.2.bm.j.1025.3		32	63.52	odd	6
2268.2.bm.j.1025.14		32	63.38	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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.43229 + 2.48080i) q^{5} +(-0.496452 + 2.59876i) q^{7} +(1.08671 - 0.627410i) q^{11} -5.25046i q^{13} +(-3.07356 - 5.32357i) q^{17} +(-2.62182 - 1.51371i) q^{19} +(-4.95109 - 2.85851i) q^{23} +(-1.60290 - 2.77631i) q^{25} +3.73169i q^{29} +(5.87105 - 3.38965i) q^{31} +(-5.73592 - 4.95377i) q^{35} +(3.85796 - 6.68219i) q^{37} +9.06886 q^{41} +8.66134 q^{43} +(-2.97597 + 5.15454i) q^{47} +(-6.50707 - 2.58032i) q^{49} +(-7.17959 + 4.14514i) q^{53} +3.59453i q^{55} +(-1.48584 - 2.57354i) q^{59} +(-2.24837 - 1.29810i) q^{61} +(13.0253 + 7.52018i) q^{65} +(5.31994 + 9.21441i) q^{67} +2.62965i q^{71} +(8.25342 - 4.76512i) q^{73} +(1.09099 + 3.13556i) q^{77} +(5.49170 - 9.51191i) q^{79} +14.6184 q^{83} +17.6089 q^{85} +(5.95534 - 10.3150i) q^{89} +(13.6447 + 2.60660i) q^{91} +(7.51040 - 4.33613i) q^{95} -7.15011i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} - 16 q^{25} + 24 q^{31} - 4 q^{37} + 8 q^{43} - 4 q^{49} + 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} - 24 q^{85} - 36 q^{91}+O(q^{100}) \) 32 * q - 8 * q^7 - 16 * q^25 + 24 * q^31 - 4 * q^37 + 8 * q^43 - 4 * q^49 + 12 * q^61 + 4 * q^67 - 36 * q^73 + 28 * q^79 - 24 * q^85 - 36 * q^91

Introduction

L-functions

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