LMFDB - Embedded newform 2268.2.t.c.1781.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(1781,2268)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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			1781.3
Character	$\chi$	$=$	2268.1781
Dual form			2268.2.t.c.2105.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.43229 - 2.48080i) q^{5} +(-0.496452 - 2.59876i) q^{7} +O(q^{10})$	$q+(-1.43229 - 2.48080i) q^{5} +(-0.496452 - 2.59876i) q^{7} +(1.08671 + 0.627410i) q^{11} +5.25046i q^{13} +(-3.07356 + 5.32357i) q^{17} +(-2.62182 + 1.51371i) q^{19} +(-4.95109 + 2.85851i) q^{23} +(-1.60290 + 2.77631i) q^{25} -3.73169i q^{29} +(5.87105 + 3.38965i) q^{31} +(-5.73592 + 4.95377i) q^{35} +(3.85796 + 6.68219i) q^{37} +9.06886 q^{41} +8.66134 q^{43} +(-2.97597 - 5.15454i) q^{47} +(-6.50707 + 2.58032i) q^{49} +(-7.17959 - 4.14514i) q^{53} -3.59453i q^{55} +(-1.48584 + 2.57354i) q^{59} +(-2.24837 + 1.29810i) q^{61} +(13.0253 - 7.52018i) q^{65} +(5.31994 - 9.21441i) q^{67} -2.62965i q^{71} +(8.25342 + 4.76512i) q^{73} +(1.09099 - 3.13556i) q^{77} +(5.49170 + 9.51191i) q^{79} +14.6184 q^{83} +17.6089 q^{85} +(5.95534 + 10.3150i) q^{89} +(13.6447 - 2.60660i) q^{91} +(7.51040 + 4.33613i) q^{95} +7.15011i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7}+O(q^{10}) $ 32 * q - 8 * q^7	$ 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

Display 100 coefficients 10 coefficients

Character values

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

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

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	−0.985313	−	0.170760i	$-0.945378\pi$
$5$	−1.43229	−	2.48080i	−0.640539	−	1.10945i	0.344774	−	0.938686i	$-0.387956\pi$
$6$		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	0.654799	−	0.755803i	$-0.272753\pi$
$11$	1.08671	+	0.627410i	0.327654	+	0.189171i	−0.327145	+	0.944974i	$0.606087\pi$
$12$		0			0
$13$			5.25046i			1.45622i	0.685462	+	0.728108i	$0.259600\pi$
$13$			5.25046i			1.45622i	−0.685462	+	0.728108i	$0.740400\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.07356	+	5.32357i	−0.745448	+	1.29115i	0.204537	+	0.978859i	$0.434431\pi$
$17$	−3.07356	+	5.32357i	−0.745448	+	1.29115i	−0.949985	+	0.312295i	$0.898902\pi$
$18$		0			0
$19$	−2.62182	+	1.51371i	−0.601486	+	0.347268i	−0.769626	−	0.638495i	$-0.779557\pi$
$19$	−2.62182	+	1.51371i	−0.601486	+	0.347268i	0.168140	+	0.985763i	$0.446224\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.95109	+	2.85851i	−1.03237	+	0.596041i	−0.917664	−	0.397358i	$-0.869927\pi$
$23$	−4.95109	+	2.85851i	−1.03237	+	0.596041i	−0.114709	+	0.993399i	$0.536594\pi$
$24$		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	$-0.887377\pi$
$29$		−	3.73169i		−	0.692957i	0.938058	−	0.346478i	$-0.112623\pi$
$30$		0			0
$31$	5.87105	+	3.38965i	1.05447	+	0.608800i	0.923898	−	0.382639i	$-0.124985\pi$
$31$	5.87105	+	3.38965i	1.05447	+	0.608800i	0.130574	+	0.991439i	$0.458318\pi$
$32$		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	0.986675	+	0.162706i	$0.0520223\pi$
$37$	3.85796	+	6.68219i	0.634245	+	1.09854i	−0.352429	+	0.935838i	$0.614644\pi$
$38$		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	0.563131	−	0.826368i	$-0.309597\pi$
$47$	−2.97597	−	5.15454i	−0.434090	−	0.751866i	−0.997221	+	0.0745015i	$0.976263\pi$
$48$		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	−0.0820587	−	0.996628i	$-0.526149\pi$
$53$	−7.17959	−	4.14514i	−0.986193	−	0.569379i	−0.904134	+	0.427249i	$0.859483\pi$
$54$		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	−0.946388	−	0.323033i	$-0.895298\pi$
$59$	−1.48584	+	2.57354i	−0.193439	+	0.335047i	0.752948	+	0.658080i	$0.228631\pi$
$60$		0			0
$61$	−2.24837	+	1.29810i	−0.287874	+	0.166204i	−0.636983	−	0.770878i	$-0.719818\pi$
$61$	−2.24837	+	1.29810i	−0.287874	+	0.166204i	0.349109	+	0.937082i	$0.386484\pi$
$62$		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	−0.333204	−	0.942855i	$-0.608130\pi$
$67$	5.31994	−	9.21441i	0.649934	−	1.12572i	0.983138	−	0.182865i	$-0.0585370\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.62965i		−	0.312083i	−0.987751	−	0.156041i	$-0.950127\pi$
$71$		−	2.62965i		−	0.312083i	0.987751	−	0.156041i	$-0.0498733\pi$
$72$		0			0
$73$	8.25342	+	4.76512i	0.965990	+	0.557715i	0.898011	−	0.439972i	$-0.145012\pi$
$73$	8.25342	+	4.76512i	0.965990	+	0.557715i	0.0679788	+	0.997687i	$0.478345\pi$
$74$		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	0.989875	+	0.141944i	$0.0453354\pi$
$79$	5.49170	+	9.51191i	0.617865	+	1.07017i	−0.372010	+	0.928229i	$0.621331\pi$
$80$		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	0.987293	+	0.158908i	$0.0507972\pi$
$89$	5.95534	+	10.3150i	0.631265	+	1.09338i	−0.356029	+	0.934475i	$0.615869\pi$
$90$		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	$0.118245\pi$
$97$			7.15011i			0.725983i	−0.931792	+	0.362992i	$0.881755\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.44709	−	2.50644i	0.143991	−	0.249400i	−0.785005	−	0.619490i	$-0.787340\pi$
$101$	1.44709	−	2.50644i	0.143991	−	0.249400i	0.928996	+	0.370089i	$0.120673\pi$
$102$		0			0
$103$	−5.58311	+	3.22341i	−0.550120	+	0.317612i	−0.749170	−	0.662377i	$-0.769548\pi$
$103$	−5.58311	+	3.22341i	−0.550120	+	0.317612i	0.199051	+	0.979989i	$0.436214\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.4017	+	8.31483i	−1.39227	+	0.803825i	−0.993566	−	0.113256i	$-0.963872\pi$
$107$	−14.4017	+	8.31483i	−1.39227	+	0.803825i	−0.398701	+	0.917081i	$0.630539\pi$
$108$		0			0
$109$	−1.54822	+	2.68159i	−0.148292	+	0.256850i	−0.930596	−	0.366047i	$-0.880711\pi$
$109$	−1.54822	+	2.68159i	−0.148292	+	0.256850i	0.782304	+	0.622897i	$0.214044\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	16.8230i		−	1.58257i	−0.611446	−	0.791286i	$-0.709412\pi$
$113$		−	16.8230i		−	1.58257i	0.611446	−	0.791286i	$-0.290588\pi$
$114$		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	0.999651	−	0.0264210i	$-0.00841105\pi$
$131$	5.45888	+	9.45505i	0.476944	+	0.826092i	−0.522707	+	0.852513i	$0.675078\pi$
$132$		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	0.198730	−	0.980054i	$-0.436318\pi$
$137$	−6.44527	−	3.72118i	−0.550657	−	0.317922i	−0.749387	+	0.662132i	$0.769652\pi$
$138$		0			0
$139$			16.4728i			1.39720i	0.715512	+	0.698600i	$0.246193\pi$
$139$			16.4728i			1.39720i	−0.715512	+	0.698600i	$0.753807\pi$
$140$		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	−0.940934	−	0.338589i	$-0.890050\pi$
$149$	−13.6491	+	7.88029i	−1.11817	+	0.645578i	−0.177240	+	0.984168i	$0.556717\pi$
$150$		0			0
$151$	−0.339365	+	0.587797i	−0.0276171	+	0.0478342i	−0.879504	−	0.475892i	$-0.842125\pi$
$151$	−0.339365	+	0.587797i	−0.0276171	+	0.0478342i	0.851887	+	0.523726i	$0.175459\pi$
$152$		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	0.206276	−	0.978494i	$-0.433865\pi$
$157$	−6.74094	−	3.89189i	−0.537986	−	0.310606i	−0.744262	+	0.667887i	$0.767199\pi$
$158$		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	0.975613	−	0.219496i	$-0.0704412\pi$
$163$	3.80101	+	6.58354i	0.297718	+	0.515663i	−0.677896	+	0.735158i	$0.737108\pi$
$164$		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	0.630092	−	0.776521i	$-0.283017\pi$
$173$	−4.70140	−	8.14307i	−0.357441	−	0.619106i	−0.987533	+	0.157415i	$0.949684\pi$
$174$		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	0.539277	−	0.842128i	$-0.318697\pi$
$179$	1.06513	+	0.614954i	0.0796116	+	0.0459638i	−0.459666	+	0.888092i	$0.652031\pi$
$180$		0			0
$181$			5.11079i			0.379882i	0.981795	+	0.189941i	$0.0608297\pi$
$181$			5.11079i			0.379882i	−0.981795	+	0.189941i	$0.939170\pi$
$182$		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	−0.639120	−	0.769107i	$-0.720701\pi$
$191$	−4.04399	+	2.33480i	−0.292613	+	0.168940i	0.346507	+	0.938047i	$0.387368\pi$
$192$		0			0
$193$	−12.3765	+	21.4368i	−0.890881	+	1.54305i	−0.0520610	+	0.998644i	$0.516579\pi$
$193$	−12.3765	+	21.4368i	−0.890881	+	1.54305i	−0.838821	+	0.544408i	$0.816754\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			9.43044i			0.671891i	0.941881	+	0.335946i	$0.109056\pi$
$197$			9.43044i			0.671891i	−0.941881	+	0.335946i	$0.890944\pi$
$198$		0			0
$199$	15.6665	+	9.04503i	1.11057	+	0.641185i	0.938976	−	0.343983i	$-0.111776\pi$
$199$	15.6665	+	9.04503i	1.11057	+	0.641185i	0.171590	+	0.985168i	$0.445110\pi$
$200$		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	$-0.915470\pi$
$223$		−	7.83841i		−	0.524899i	0.964946	−	0.262449i	$-0.0845303\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.08795	+	14.0087i	−0.536816	+	0.929793i	0.462257	+	0.886746i	$0.347040\pi$
$227$	−8.08795	+	14.0087i	−0.536816	+	0.929793i	−0.999073	+	0.0430471i	$0.986293\pi$
$228$		0			0
$229$	6.44112	−	3.71878i	0.425641	−	0.245744i	−0.271847	−	0.962341i	$-0.587634\pi$
$229$	6.44112	−	3.71878i	0.425641	−	0.245744i	0.697488	+	0.716596i	$0.254301\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.8568	+	11.4643i	−1.30086	+	0.751051i	−0.980551	−	0.196262i	$-0.937120\pi$
$233$	−19.8568	+	11.4643i	−1.30086	+	0.751051i	−0.320308	+	0.947314i	$0.603786\pi$
$234$		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	$0.145614\pi$
$239$			13.6562i			0.883344i	−0.897177	+	0.441672i	$0.854386\pi$
$240$		0			0
$241$	7.04728	+	4.06875i	0.453955	+	0.262091i	0.709499	−	0.704706i	$-0.248921\pi$
$241$	7.04728	+	4.06875i	0.453955	+	0.262091i	−0.255544	+	0.966797i	$0.582255\pi$
$242$		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	0.955754	+	0.294166i	$0.0950418\pi$
$257$	11.7450	+	20.3429i	0.732632	+	1.26896i	−0.223122	+	0.974790i	$0.571625\pi$
$258$		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	0.350260	−	0.936652i	$-0.386093\pi$
$263$	−4.63448	−	2.67572i	−0.285775	−	0.164992i	−0.636035	+	0.771660i	$0.719427\pi$
$264$		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	−0.882446	−	0.470414i	$-0.844105\pi$
$269$	−0.554894	+	0.961104i	−0.0338325	+	0.0585995i	0.848613	+	0.529014i	$0.177438\pi$
$270$		0			0
$271$	11.9571	−	6.90342i	0.726341	−	0.419353i	−0.0907411	−	0.995875i	$-0.528924\pi$
$271$	11.9571	−	6.90342i	0.726341	−	0.419353i	0.817082	+	0.576521i	$0.195590\pi$
$272$		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	−0.582866	−	0.812568i	$-0.698069\pi$
$277$	6.86156	−	11.8846i	0.412271	−	0.714075i	0.995138	+	0.0984929i	$0.0314022\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			30.5547i			1.82274i	0.411586	+	0.911371i	$0.364975\pi$
$281$			30.5547i			1.82274i	−0.411586	+	0.911371i	$0.635025\pi$
$282$		0			0
$283$	12.7910	+	7.38491i	0.760349	+	0.438988i	0.829421	−	0.558624i	$-0.188671\pi$
$283$	12.7910	+	7.38491i	0.760349	+	0.438988i	−0.0690722	+	0.997612i	$0.522004\pi$
$284$		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	$-0.785235\pi$
$307$		−	21.8901i		−	1.24933i	0.780891	−	0.624667i	$-0.214765\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.25630	−	9.10417i	0.298057	−	0.516250i	−0.677634	−	0.735399i	$-0.736995\pi$
$311$	5.25630	−	9.10417i	0.298057	−	0.516250i	0.975691	+	0.219149i	$0.0703281\pi$
$312$		0			0
$313$	−2.85633	+	1.64910i	−0.161449	+	0.0932127i	−0.578548	−	0.815649i	$-0.696380\pi$
$313$	−2.85633	+	1.64910i	−0.161449	+	0.0932127i	0.417099	+	0.908861i	$0.363047\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.6051	−	10.7416i	1.04496	−	0.603311i	0.123729	−	0.992316i	$-0.460515\pi$
$317$	18.6051	−	10.7416i	1.04496	−	0.603311i	0.921236	+	0.389005i	$0.127181\pi$
$318$		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	0.532436	−	0.846470i	$-0.321277\pi$
$331$	−8.49354	−	14.7112i	−0.466847	−	0.808603i	−0.999283	+	0.0378676i	$0.987943\pi$
$332$		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	0.975617	−	0.219480i	$-0.0704360\pi$
$347$	23.7199	+	13.6947i	1.27335	+	0.735169i	0.297734	+	0.954649i	$0.403769\pi$
$348$		0			0
$349$			23.0855i			1.23574i	0.786281	+	0.617869i	$0.212004\pi$
$349$			23.0855i			1.23574i	−0.786281	+	0.617869i	$0.787996\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.86592	+	17.0883i	−0.525110	+	0.909517i	0.474462	+	0.880276i	$0.342643\pi$
$353$	−9.86592	+	17.0883i	−0.525110	+	0.909517i	−0.999572	+	0.0292415i	$0.990691\pi$
$354$		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	0.637305	−	0.770611i	$-0.280049\pi$
$359$	30.7577	−	17.7579i	1.62333	−	0.937228i	0.986022	−	0.166617i	$-0.0532843\pi$
$360$		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	−0.110194	−	0.993910i	$-0.535147\pi$
$367$	−19.6561	−	11.3485i	−1.02604	−	0.592386i	−0.915848	+	0.401524i	$0.868481\pi$
$368$		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	0.615759	−	0.787934i	$-0.288849\pi$
$373$	−7.23264	−	12.5273i	−0.374492	−	0.648639i	−0.990251	+	0.139296i	$0.955516\pi$
$374$		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	0.840099	+	0.542432i	$0.182497\pi$
$383$	17.4139	+	30.1618i	0.889810	+	1.54120i	0.0497104	+	0.998764i	$0.484170\pi$
$384$		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	0.454504	−	0.890745i	$-0.349816\pi$
$389$	−1.76819	−	1.02087i	−0.0896510	−	0.0517600i	−0.544155	+	0.838985i	$0.683150\pi$
$390$		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	−0.988735	−	0.149678i	$-0.952176\pi$
$397$	−26.9678	+	15.5699i	−1.35348	+	0.781430i	−0.364742	+	0.931108i	$0.618843\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.2752	+	6.50976i	−0.563058	+	0.325082i	−0.754372	−	0.656447i	$-0.772059\pi$
$401$	−11.2752	+	6.50976i	−0.563058	+	0.325082i	0.191314	+	0.981529i	$0.438725\pi$
$402$		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	0.205534	−	0.978650i	$-0.434107\pi$
$409$	−10.9054	−	6.29621i	−0.539236	−	0.311328i	−0.744769	+	0.667322i	$0.767440\pi$
$410$		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	0.699649	−	0.714486i	$-0.253340\pi$
$431$	8.94179	+	5.16254i	0.430711	+	0.248671i	−0.268939	+	0.963157i	$0.586673\pi$
$432$		0			0
$433$			10.9117i			0.524384i	0.965016	+	0.262192i	$0.0844453\pi$
$433$			10.9117i			0.524384i	−0.965016	+	0.262192i	$0.915555\pi$
$434$		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	−0.722385	−	0.691491i	$-0.756954\pi$
$439$	−10.1562	+	5.86369i	−0.484729	+	0.279859i	0.237656	+	0.971349i	$0.423621\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	30.1605	−	17.4131i	1.43297	−	0.827324i	0.435621	−	0.900130i	$-0.356529\pi$
$443$	30.1605	−	17.4131i	1.43297	−	0.827324i	0.997346	+	0.0728067i	$0.0231956\pi$
$444$		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	$0.0329475\pi$
$449$			4.37875i			0.206646i	−0.994648	+	0.103323i	$0.967052\pi$
$450$		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	0.768504	−	0.639845i	$-0.221001\pi$
$457$	−3.63140	−	6.28977i	−0.169870	−	0.294223i	−0.938374	+	0.345622i	$0.887668\pi$
$458$		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	0.827318	−	0.561733i	$-0.189865\pi$
$467$	−1.57357	−	2.72550i	−0.0728161	−	0.126121i	−0.900134	+	0.435612i	$0.856532\pi$
$468$		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	−0.0776390	+	0.996982i	$0.524738\pi$
$479$	−19.7463	+	34.2016i	−0.902231	+	1.56271i	−0.824592	+	0.565728i	$0.808595\pi$
$480$		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	−0.223326	−	0.974744i	$-0.571691\pi$
$487$	16.1647	−	27.9980i	0.732490	−	1.26871i	0.955816	−	0.293966i	$-0.0949754\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	14.3858i		−	0.649222i	−0.945848	−	0.324611i	$-0.894767\pi$
$491$		−	14.3858i		−	0.649222i	0.945848	−	0.324611i	$-0.105233\pi$
$492$		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	−0.778185	−	0.628035i	$-0.783860\pi$
$499$	−20.8413	−	36.0983i	−0.932987	−	1.61598i	−0.154801	−	0.987946i	$-0.549474\pi$
$500$		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	−0.990521	−	0.137359i	$-0.956139\pi$
$509$	−13.8574	−	24.0017i	−0.614217	−	1.06385i	0.376304	−	0.926496i	$-0.377195\pi$
$510$		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	−0.469452	−	0.882958i	$-0.655549\pi$
$521$	12.0961	−	20.9510i	0.529938	−	0.917880i	0.999390	−	0.0349216i	$-0.0111181\pi$
$522$		0			0
$523$	−20.8779	+	12.0539i	−0.912927	+	0.527078i	−0.881372	−	0.472424i	$-0.843379\pi$
$523$	−20.8779	+	12.0539i	−0.912927	+	0.527078i	−0.0315550	+	0.999502i	$0.510046\pi$
$524$		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	−0.863250	−	0.504777i	$-0.831575\pi$
$541$	−20.2072	−	34.9999i	−0.868775	−	1.50476i	−0.00552468	−	0.999985i	$-0.501759\pi$
$542$		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	0.902883	−	0.429887i	$-0.141447\pi$
$557$	23.1768	+	13.3811i	0.982031	+	0.566976i	0.0791483	+	0.996863i	$0.474780\pi$
$558$		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	−0.991249	−	0.132006i	$-0.957858\pi$
$563$	−9.04743	+	15.6706i	−0.381304	+	0.660438i	0.609945	+	0.792444i	$0.291192\pi$
$564$		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	−0.862196	−	0.506575i	$-0.830911\pi$
$569$	−20.3851	+	11.7693i	−0.854588	+	0.493396i	0.00760850	+	0.999971i	$0.497578\pi$
$570$		0			0
$571$	−6.19412	+	10.7285i	−0.259216	+	0.448975i	−0.966032	−	0.258422i	$-0.916797\pi$
$571$	−6.19412	+	10.7285i	−0.259216	+	0.448975i	0.706816	+	0.707397i	$0.250131\pi$
$572$		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	0.686606	−	0.727029i	$-0.259100\pi$
$577$	9.61514	+	5.55131i	0.400284	+	0.231104i	−0.286323	+	0.958133i	$0.592433\pi$
$578$		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	0.996150	+	0.0876691i	$0.0279418\pi$
$593$	13.9778	+	24.2102i	0.573998	+	0.994194i	−0.422151	+	0.906525i	$0.638725\pi$
$594$		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	0.917897	−	0.396818i	$-0.129886\pi$
$599$	25.2868	+	14.5993i	1.03319	+	0.596513i	0.115294	+	0.993331i	$0.463219\pi$
$600$		0			0
$601$		−	14.9790i		−	0.611006i	−0.952191	−	0.305503i	$-0.901175\pi$
$601$		−	14.9790i		−	0.611006i	0.952191	−	0.305503i	$-0.0988247\pi$
$602$		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	−0.862833	−	0.505489i	$-0.831312\pi$
$607$	−21.1015	+	12.1830i	−0.856483	+	0.494491i	0.00634989	+	0.999980i	$0.497979\pi$
$608$		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	−0.556180	−	0.831062i	$-0.687733\pi$
$613$	10.9343	−	18.9387i	0.441631	−	0.764927i	0.997811	+	0.0661347i	$0.0210667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			41.2793i			1.66184i	0.556391	+	0.830921i	$0.312186\pi$
$617$			41.2793i			1.66184i	−0.556391	+	0.830921i	$0.687814\pi$
$618$		0			0
$619$	−35.3843	−	20.4292i	−1.42222	−	0.821117i	−0.425728	−	0.904851i	$-0.639982\pi$
$619$	−35.3843	−	20.4292i	−1.42222	−	0.821117i	−0.996488	+	0.0837339i	$0.973315\pi$
$620$		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	−0.233799	−	0.972285i	$-0.575116\pi$
$641$	−30.1973	−	17.4344i	−1.19272	−	0.688619i	−0.958923	+	0.283666i	$0.908449\pi$
$642$		0			0
$643$		−	26.8650i		−	1.05945i	−0.848168	−	0.529727i	$-0.822294\pi$
$643$		−	26.8650i		−	1.05945i	0.848168	−	0.529727i	$-0.177706\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.988791	+	1.71264i	−0.0388734	+	0.0673306i	−0.884807	−	0.465957i	$-0.845710\pi$
$647$	−0.988791	+	1.71264i	−0.0388734	+	0.0673306i	0.845934	+	0.533287i	$0.179044\pi$
$648$		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	−0.800888	−	0.598815i	$-0.795639\pi$
$653$	−17.4467	+	10.0729i	−0.682742	+	0.394182i	0.118145	+	0.992996i	$0.462305\pi$
$654$		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	$-0.851572\pi$
$659$		−	23.0826i		−	0.899172i	0.893237	−	0.449586i	$-0.148428\pi$
$660$		0			0
$661$	−2.42515	−	1.40016i	−0.0943273	−	0.0544599i	0.452094	−	0.891970i	$-0.350677\pi$
$661$	−2.42515	−	1.40016i	−0.0943273	−	0.0544599i	−0.546422	+	0.837510i	$0.684010\pi$
$662$		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	−0.858290	−	0.513166i	$-0.828473\pi$
$677$	−22.7293	−	39.3684i	−0.873559	−	1.51305i	−0.0152697	−	0.999883i	$-0.504861\pi$
$678$		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	−0.975655	−	0.219312i	$-0.929619\pi$
$683$	−43.2107	−	24.9477i	−1.65341	−	0.954598i	−0.677757	−	0.735286i	$-0.737048\pi$
$684$		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	0.114426	−	0.993432i	$-0.463497\pi$
$691$	27.1274	−	15.6620i	1.03198	−	0.595812i	0.917550	+	0.397620i	$0.130164\pi$
$692$		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	$-0.918817\pi$
$701$		−	13.3593i		−	0.504576i	0.967652	−	0.252288i	$-0.0811829\pi$
$702$		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	−0.992960	−	0.118452i	$-0.962207\pi$
$709$	−15.9513	−	27.6284i	−0.599062	−	1.03761i	0.393897	−	0.919154i	$-0.371127\pi$
$710$		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	0.761958	−	0.647626i	$-0.224238\pi$
$719$	−4.82338	−	8.35434i	−0.179882	−	0.311564i	−0.941840	+	0.336062i	$0.890905\pi$
$720$		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	$0.253378\pi$
$727$			38.5339i			1.42914i	−0.699562	+	0.714572i	$0.746622\pi$
$728$		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	−0.757230	−	0.653148i	$-0.773448\pi$
$733$	−15.4377	+	8.91294i	−0.570203	+	0.329207i	0.187027	+	0.982355i	$0.440115\pi$
$734$		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	−0.776585	−	0.630012i	$-0.783050\pi$
$739$	4.27651	−	7.40713i	0.157314	−	0.272476i	0.933899	+	0.357536i	$0.116383\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			16.8144i			0.616859i	0.951247	+	0.308430i	$0.0998034\pi$
$743$			16.8144i			0.616859i	−0.951247	+	0.308430i	$0.900197\pi$
$744$		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	0.933391	+	0.358861i	$0.116835\pi$
$751$	21.3063	+	36.9036i	0.777478	+	1.34663i	−0.155913	+	0.987771i	$0.549832\pi$
$752$		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	0.959702	+	0.281021i	$0.0906730\pi$
$761$	19.9510	+	34.5561i	0.723222	+	1.25266i	−0.236480	+	0.971636i	$0.575994\pi$
$762$		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	$-0.945301\pi$
$769$		−	9.48373i		−	0.341992i	0.985272	−	0.170996i	$-0.0546985\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.60509	+	11.4403i	−0.237568	+	0.411481i	−0.960016	−	0.279945i	$-0.909684\pi$
$773$	−6.60509	+	11.4403i	−0.237568	+	0.411481i	0.722448	+	0.691426i	$0.243017\pi$
$774$		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	0.371757	−	0.928330i	$-0.378755\pi$
$787$	−6.91018	−	3.98960i	−0.246321	−	0.142214i	−0.618079	+	0.786116i	$0.712089\pi$
$788$		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	0.695890	−	0.718149i	$-0.255010\pi$
$809$	12.0001	+	6.92824i	0.421900	+	0.243584i	−0.273990	+	0.961733i	$0.588344\pi$
$810$		0			0
$811$		−	17.6343i		−	0.619223i	−0.950863	−	0.309611i	$-0.899801\pi$
$811$		−	17.6343i		−	0.619223i	0.950863	−	0.309611i	$-0.100199\pi$
$812$		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	0.488940	−	0.872318i	$-0.337384\pi$
$821$	42.6604	−	24.6300i	1.48886	−	0.859593i	0.999919	+	0.0127244i	$0.00405042\pi$
$822$		0			0
$823$	17.2854	−	29.9392i	0.602530	−	1.04361i	−0.389906	−	0.920855i	$-0.627493\pi$
$823$	17.2854	−	29.9392i	0.602530	−	1.04361i	0.992437	−	0.122759i	$-0.0391741\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			45.2579i			1.57377i	0.617099	+	0.786886i	$0.288308\pi$
$827$			45.2579i			1.57377i	−0.617099	+	0.786886i	$0.711692\pi$
$828$		0			0
$829$	8.17327	+	4.71884i	0.283869	+	0.163892i	0.635174	−	0.772369i	$-0.280928\pi$
$829$	8.17327	+	4.71884i	0.283869	+	0.163892i	−0.351305	+	0.936261i	$0.614262\pi$
$830$		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	$-0.710419\pi$
$853$		−	46.1077i		−	1.57870i	0.613946	−	0.789348i	$-0.289581\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	21.7762	−	37.7174i	0.743859	−	1.28840i	−0.206867	−	0.978369i	$-0.566327\pi$
$857$	21.7762	−	37.7174i	0.743859	−	1.28840i	0.950726	−	0.310033i	$-0.100340\pi$
$858$		0			0
$859$	−13.4377	+	7.75827i	−0.458489	+	0.264709i	−0.711409	−	0.702779i	$-0.751942\pi$
$859$	−13.4377	+	7.75827i	−0.458489	+	0.264709i	0.252920	+	0.967487i	$0.418609\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−15.3098	+	8.83910i	−0.521151	+	0.300887i	−0.737405	−	0.675450i	$-0.763949\pi$
$863$	−15.3098	+	8.83910i	−0.521151	+	0.300887i	0.216255	+	0.976337i	$0.430616\pi$
$864$		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	0.710233	−	0.703967i	$-0.248590\pi$
$877$	−7.53791	−	13.0560i	−0.254537	−	0.440871i	−0.964770	+	0.263096i	$0.915256\pi$
$878$		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	0.935390	−	0.353619i	$-0.115049\pi$
$887$	4.80844	+	8.32847i	0.161452	+	0.279643i	−0.773938	+	0.633262i	$0.781716\pi$
$888$		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	0.283775	+	0.958891i	$0.408413\pi$
$907$	−20.7363	+	35.9163i	−0.688536	+	1.19258i	−0.972311	+	0.233689i	$0.924920\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			31.5561i			1.04550i	0.852486	+	0.522751i	$0.175094\pi$
$911$			31.5561i			1.04550i	−0.852486	+	0.522751i	$0.824906\pi$
$912$		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	−0.933695	−	0.358069i	$-0.883435\pi$
$919$	−23.5531	−	40.7951i	−0.776944	−	1.34571i	0.156751	−	0.987638i	$-0.449898\pi$
$920$		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	−0.842965	−	0.537968i	$-0.819192\pi$
$929$	−27.0468	−	46.8464i	−0.887377	−	1.53698i	−0.0444117	−	0.999013i	$-0.514141\pi$
$930$		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	$0.325755\pi$
$937$			52.2751i			1.70775i	−0.520476	+	0.853876i	$0.674245\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	22.1565	−	38.3761i	0.722280	−	1.25103i	−0.237804	−	0.971313i	$-0.576428\pi$
$941$	22.1565	−	38.3761i	0.722280	−	1.25103i	0.960084	−	0.279712i	$-0.0902390\pi$
$942$		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	0.0384035	−	0.999262i	$-0.487773\pi$
$947$	28.4036	−	16.3988i	0.922992	−	0.532890i	0.884588	+	0.466373i	$0.154439\pi$
$948$		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	$0.0463657\pi$
$953$			8.96162i			0.290295i	−0.989410	+	0.145148i	$0.953634\pi$
$954$		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	0.809152	−	0.587600i	$-0.199927\pi$
$971$	−3.25010	−	5.62934i	−0.104301	−	0.180654i	−0.913452	+	0.406946i	$0.866594\pi$
$972$		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	0.617101	−	0.786884i	$-0.288307\pi$
$977$	7.63263	+	4.40670i	0.244190	+	0.140983i	−0.372911	+	0.927867i	$0.621640\pi$
$978$		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	−0.474722	−	0.880136i	$-0.657451\pi$
$983$	16.4558	−	28.5023i	0.524859	−	0.909083i	0.999581	−	0.0289467i	$-0.00921530\pi$
$984$		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	−0.971591	−	0.236668i	$-0.923945\pi$
$991$	−8.84072	+	15.3126i	−0.280835	+	0.486420i	0.690756	+	0.723088i	$0.257278\pi$
$992$		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	0.926169	−	0.377108i	$-0.123081\pi$
$997$	33.5541	+	19.3725i	1.06267	+	0.613532i	0.136500	+	0.990640i	$0.456415\pi$
$998$		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.1781.3	✓	32
3.2	odd	2	inner	2268.2.t.c.1781.14	yes	32
7.5	odd	6	inner	2268.2.t.c.2105.14	yes	32
9.2	odd	6		2268.2.bm.j.1025.3		32
9.4	even	3		2268.2.w.j.269.3		32
9.5	odd	6		2268.2.w.j.269.14		32
9.7	even	3		2268.2.bm.j.1025.14		32
21.5	even	6	inner	2268.2.t.c.2105.3	yes	32
63.5	even	6		2268.2.bm.j.593.14		32
63.40	odd	6		2268.2.bm.j.593.3		32
63.47	even	6		2268.2.w.j.1349.3		32
63.61	odd	6		2268.2.w.j.1349.14		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.t.c.1781.3	✓	32	1.1	even	1	trivial
2268.2.t.c.1781.14	yes	32	3.2	odd	2	inner
2268.2.t.c.2105.3	yes	32	21.5	even	6	inner
2268.2.t.c.2105.14	yes	32	7.5	odd	6	inner
2268.2.w.j.269.3		32	9.4	even	3
2268.2.w.j.269.14		32	9.5	odd	6
2268.2.w.j.1349.3		32	63.47	even	6
2268.2.w.j.1349.14		32	63.61	odd	6
2268.2.bm.j.593.3		32	63.40	odd	6
2268.2.bm.j.593.14		32	63.5	even	6
2268.2.bm.j.1025.3		32	9.2	odd	6
2268.2.bm.j.1025.14		32	9.7	even	3

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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-1.43229 - 2.48080i) q^{5} +(-0.496452 - 2.59876i) q^{7} +O(q^{10})\)	\(q+(-1.43229 - 2.48080i) q^{5} +(-0.496452 - 2.59876i) q^{7} +(1.08671 + 0.627410i) q^{11} +5.25046i q^{13} +(-3.07356 + 5.32357i) q^{17} +(-2.62182 + 1.51371i) q^{19} +(-4.95109 + 2.85851i) q^{23} +(-1.60290 + 2.77631i) q^{25} -3.73169i q^{29} +(5.87105 + 3.38965i) q^{31} +(-5.73592 + 4.95377i) q^{35} +(3.85796 + 6.68219i) q^{37} +9.06886 q^{41} +8.66134 q^{43} +(-2.97597 - 5.15454i) q^{47} +(-6.50707 + 2.58032i) q^{49} +(-7.17959 - 4.14514i) q^{53} -3.59453i q^{55} +(-1.48584 + 2.57354i) q^{59} +(-2.24837 + 1.29810i) q^{61} +(13.0253 - 7.52018i) q^{65} +(5.31994 - 9.21441i) q^{67} -2.62965i q^{71} +(8.25342 + 4.76512i) q^{73} +(1.09099 - 3.13556i) q^{77} +(5.49170 + 9.51191i) q^{79} +14.6184 q^{83} +17.6089 q^{85} +(5.95534 + 10.3150i) q^{89} +(13.6447 - 2.60660i) q^{91} +(7.51040 + 4.33613i) q^{95} +7.15011i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7}+O(q^{10}) \) 32 * q - 8 * q^7	\( 32 q - 8 q^{7} - 16 q^{25} + 24 q^{31} - 4 q^{37} + 8 q^{43} - 4 q^{49} + 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} - 24 q^{85} - 36 q^{91}+O(q^{100}) \) 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

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