LMFDB - Embedded newform 2268.2.bm.j.1025.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(593,2268)] 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, 1, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.593"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.bm (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,4,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == 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			1025.14
Character	$\chi$	$=$	2268.1025
Dual form			2268.2.bm.j.593.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.86458 q^{5} +(-2.00236 + 1.72932i) q^{7} -1.25482i q^{11} +(-4.54703 - 2.62523i) q^{13} +(-3.07356 + 5.32357i) q^{17} +(-2.62182 + 1.51371i) q^{19} -5.71702i q^{23} +3.20580 q^{25} +(-3.23174 + 1.86584i) q^{29} +(-5.87105 + 3.38965i) q^{31} +(-5.73592 + 4.95377i) q^{35} +(3.85796 + 6.68219i) q^{37} +(-4.53443 + 7.85386i) q^{41} +(-4.33067 - 7.50094i) q^{43} +(-2.97597 + 5.15454i) q^{47} +(1.01892 - 6.92545i) q^{49} +(-7.17959 - 4.14514i) q^{53} -3.59453i q^{55} +(-1.48584 - 2.57354i) q^{59} +(2.24837 + 1.29810i) q^{61} +(-13.0253 - 7.52018i) q^{65} +(5.31994 + 9.21441i) q^{67} -2.62965i q^{71} +(8.25342 + 4.76512i) q^{73} +(2.16998 + 2.51261i) q^{77} +(5.49170 - 9.51191i) q^{79} +(-7.30919 - 12.6599i) q^{83} +(-8.80446 + 15.2498i) q^{85} +(5.95534 + 10.3150i) q^{89} +(13.6447 - 2.60660i) q^{91} +(-7.51040 + 4.33613i) q^{95} +(6.19217 - 3.57505i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 4 q^{7} - 12 q^{13} + 32 q^{25} - 24 q^{31} - 4 q^{37} - 4 q^{43} - 16 q^{49} - 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} + 12 q^{85} - 36 q^{91} - 12 q^{97}+O(q^{100}) $ 32 * q + 4 * q^7 - 12 * q^13 + 32 * q^25 - 24 * q^31 - 4 * q^37 - 4 * q^43 - 16 * q^49 - 12 * q^61 + 4 * q^67 - 36 * q^73 + 28 * q^79 + 12 * q^85 - 36 * q^91 - 12 * q^97

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.86458			1.28108			0.640539	−	0.767926i	$-0.278711\pi$
$5$	2.86458			1.28108			0.640539	+	0.767926i	$0.278711\pi$
$6$		0			0
$7$	−2.00236	+	1.72932i	−0.756822	+	0.653621i
$7$	−2.00236	+	1.72932i	−0.756822	+	0.653621i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.25482i		−	0.378342i	−0.981944	−	0.189171i	$-0.939420\pi$
$11$		−	1.25482i		−	0.378342i	0.981944	−	0.189171i	$-0.0605801\pi$
$12$		0			0
$13$	−4.54703	−	2.62523i	−1.26112	−	0.728108i	−0.287829	−	0.957682i	$-0.592933\pi$
$13$	−4.54703	−	2.62523i	−1.26112	−	0.728108i	−0.973291	+	0.229574i	$0.926267\pi$
$14$		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
$23$		−	5.71702i		−	1.19208i	−0.802954	−	0.596041i	$-0.796740\pi$
$23$		−	5.71702i		−	1.19208i	0.802954	−	0.596041i	$-0.203260\pi$
$24$		0			0
$25$	3.20580			0.641161
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.23174	+	1.86584i	−0.600118	+	0.346478i	−0.769088	−	0.639143i	$-0.779289\pi$
$29$	−3.23174	+	1.86584i	−0.600118	+	0.346478i	0.168970	+	0.985621i	$0.445956\pi$
$30$		0			0
$31$	−5.87105	+	3.38965i	−1.05447	+	0.608800i	−0.923898	−	0.382639i	$-0.875015\pi$
$31$	−5.87105	+	3.38965i	−1.05447	+	0.608800i	−0.130574	+	0.991439i	$0.541682\pi$
$32$		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
$41$	−4.53443	+	7.85386i	−0.708159	+	1.22657i	0.257380	+	0.966310i	$0.417141\pi$
$41$	−4.53443	+	7.85386i	−0.708159	+	1.22657i	−0.965539	+	0.260257i	$0.916193\pi$
$42$		0			0
$43$	−4.33067	−	7.50094i	−0.660421	−	1.14388i	−0.980505	−	0.196494i	$-0.937044\pi$
$43$	−4.33067	−	7.50094i	−0.660421	−	1.14388i	0.320084	−	0.947389i	$-0.396289\pi$
$44$		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
$49$	1.01892	−	6.92545i	0.145559	−	0.989350i
$50$		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	0.752948	−	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	0.636983	−	0.770878i	$-0.280182\pi$
$61$	2.24837	+	1.29810i	0.287874	+	0.166204i	−0.349109	+	0.937082i	$0.613516\pi$
$62$		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	$-0.950127\pi$
$71$		−	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
$77$	2.16998	+	2.51261i	0.247293	+	0.286338i
$78$		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
$83$	−7.30919	−	12.6599i	−0.802287	−	1.38960i	−0.918107	−	0.396332i	$-0.870283\pi$
$83$	−7.30919	−	12.6599i	−0.802287	−	1.38960i	0.115820	−	0.993270i	$-0.463051\pi$
$84$		0			0
$85$	−8.80446	+	15.2498i	−0.954977	+	1.65407i
$86$		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
$97$	6.19217	−	3.57505i	0.628720	−	0.362992i	−0.151536	−	0.988452i	$-0.548422\pi$
$97$	6.19217	−	3.57505i	0.628720	−	0.362992i	0.780256	+	0.625460i	$0.215089\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.89419			−0.287983			−0.143991	−	0.989579i	$-0.545994\pi$
$101$	−2.89419			−0.287983			−0.143991	+	0.989579i	$0.545994\pi$
$102$		0			0
$103$		−	6.44681i		−	0.635224i	−0.948221	−	0.317612i	$-0.897119\pi$
$103$		−	6.44681i		−	0.635224i	0.948221	−	0.317612i	$-0.102881\pi$
$104$		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
$113$	14.5691	+	8.41149i	1.37055	+	0.791286i	0.990997	−	0.133884i	$-0.0427451\pi$
$113$	14.5691	+	8.41149i	1.37055	+	0.791286i	0.379551	+	0.925171i	$0.376078\pi$
$114$		0			0
$115$		−	16.3769i		−	1.52715i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.05175	−	15.9749i	−0.279754	−	1.46441i
$120$		0			0
$121$	9.42543			0.856857
$122$		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
$131$	−10.9178			−0.953888			−0.476944	−	0.878934i	$-0.658256\pi$
$131$	−10.9178			−0.953888			−0.476944	+	0.878934i	$0.658256\pi$
$132$		0			0
$133$	2.63215	−	7.56495i	0.228236	−	0.655964i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			7.44236i			0.635844i	0.948117	+	0.317922i	$0.102985\pi$
$137$			7.44236i			0.635844i	−0.948117	+	0.317922i	$0.897015\pi$
$138$		0			0
$139$	−14.2658	−	8.23638i	−1.21001	−	0.698600i	−0.247249	−	0.968952i	$-0.579527\pi$
$139$	−14.2658	−	8.23638i	−1.21001	−	0.698600i	−0.962762	+	0.270352i	$0.912860\pi$
$140$		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
$149$		−	15.7606i		−	1.29116i	−0.763694	−	0.645578i	$-0.776616\pi$
$149$		−	15.7606i		−	1.29116i	0.763694	−	0.645578i	$-0.223384\pi$
$150$		0			0
$151$	0.678729			0.0552342			0.0276171	−	0.999619i	$-0.491208\pi$
$151$	0.678729			0.0552342			0.0276171	+	0.999619i	$0.491208\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−16.8181	+	9.70993i	−1.35086	+	0.779920i
$156$		0			0
$157$	6.74094	−	3.89189i	0.537986	−	0.310606i	−0.206276	−	0.978494i	$-0.566135\pi$
$157$	6.74094	−	3.89189i	0.537986	−	0.310606i	0.744262	+	0.667887i	$0.232801\pi$
$158$		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
$167$	−2.18568	+	3.78570i	−0.169133	+	0.292947i	−0.938115	−	0.346323i	$-0.887430\pi$
$167$	−2.18568	+	3.78570i	−0.169133	+	0.292947i	0.768982	+	0.639270i	$0.220763\pi$
$168$		0			0
$169$	7.28368	+	12.6157i	0.560283	+	0.970438i
$170$		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
$175$	−6.41919	+	5.54386i	−0.485245	+	0.419076i
$176$		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	$-0.279299\pi$
$191$	4.04399	+	2.33480i	0.292613	+	0.168940i	−0.346507	+	0.938047i	$0.612632\pi$
$192$		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	$0.109056\pi$
$197$			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
$203$	3.24447	−	9.32479i	0.227717	−	0.654472i
$204$		0			0
$205$	−12.9892	+	22.4980i	−0.907207	+	1.57133i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.89943	+	3.28991i	0.131386	+	0.227568i
$210$		0			0
$211$	−9.16894	+	15.8811i	−0.631216	+	1.09330i	0.356088	+	0.934453i	$0.384110\pi$
$211$	−9.16894	+	15.8811i	−0.631216	+	1.09330i	−0.987303	+	0.158845i	$0.949223\pi$
$212$		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
$223$	−6.78826	+	3.91920i	−0.454576	+	0.262449i	−0.709761	−	0.704443i	$-0.751197\pi$
$223$	−6.78826	+	3.91920i	−0.454576	+	0.262449i	0.255185	+	0.966892i	$0.417864\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.1759			1.07363			0.536816	−	0.843699i	$-0.319627\pi$
$227$	16.1759			1.07363			0.536816	+	0.843699i	$0.319627\pi$
$228$		0			0
$229$			7.43757i			0.491488i	0.969335	+	0.245744i	$0.0790323\pi$
$229$			7.43757i			0.491488i	−0.969335	+	0.245744i	$0.920968\pi$
$230$		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
$239$	−11.8266	−	6.82808i	−0.764998	−	0.441672i	0.0660893	−	0.997814i	$-0.478948\pi$
$239$	−11.8266	−	6.82808i	−0.764998	−	0.441672i	−0.831087	+	0.556142i	$0.812281\pi$
$240$		0			0
$241$		−	8.13750i		−	0.524182i	−0.965043	−	0.262091i	$-0.915588\pi$
$241$		−	8.13750i		−	0.524182i	0.965043	−	0.262091i	$-0.0844121\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.91876	−	19.8385i	0.186473	−	1.26743i
$246$		0			0
$247$	15.8953			1.01140
$248$		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
$257$	−23.4900			−1.46526			−0.732632	−	0.680625i	$-0.761708\pi$
$257$	−23.4900			−1.46526			−0.732632	+	0.680625i	$0.761708\pi$
$258$		0			0
$259$	−19.2807	−	6.70852i	−1.19804	−	0.416847i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			5.35144i			0.329984i	0.986295	+	0.164992i	$0.0527598\pi$
$263$			5.35144i			0.329984i	−0.986295	+	0.164992i	$0.947240\pi$
$264$		0			0
$265$	−20.5665	−	11.8741i	−1.26339	−	0.729418i
$266$		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
$275$		−	4.02271i		−	0.242578i
$276$		0			0
$277$	−13.7231			−0.824543			−0.412271	−	0.911061i	$-0.635264\pi$
$277$	−13.7231			−0.824543			−0.412271	+	0.911061i	$0.635264\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	26.4612	−	15.2774i	1.57854	−	0.911371i	0.583478	−	0.812129i	$-0.301692\pi$
$281$	26.4612	−	15.2774i	1.57854	−	0.911371i	0.995063	−	0.0992419i	$-0.0316418\pi$
$282$		0			0
$283$	−12.7910	+	7.38491i	−0.760349	+	0.438988i	−0.829421	−	0.558624i	$-0.811329\pi$
$283$	−12.7910	+	7.38491i	−0.760349	+	0.438988i	0.0690722	+	0.997612i	$0.477996\pi$
$284$		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
$293$	−0.721612	+	1.24987i	−0.0421570	+	0.0730181i	−0.886334	−	0.463046i	$-0.846756\pi$
$293$	−0.721612	+	1.24987i	−0.0421570	+	0.0730181i	0.844177	+	0.536065i	$0.180090\pi$
$294$		0			0
$295$	−4.25629	−	7.37212i	−0.247811	−	0.429221i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−15.0085	+	25.9955i	−0.867964	+	1.50336i
$300$		0			0
$301$	21.6431	+	7.53050i	1.24749	+	0.434051i
$302$		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	0.975691	−	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	0.578548	−	0.815649i	$-0.303620\pi$
$313$	2.85633	+	1.64910i	0.161449	+	0.0932127i	−0.417099	+	0.908861i	$0.636953\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−18.6051	−	10.7416i	−1.04496	−	0.603311i	−0.123729	−	0.992316i	$-0.539485\pi$
$317$	−18.6051	−	10.7416i	−1.04496	−	0.603311i	−0.921236	+	0.389005i	$0.872819\pi$
$318$		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
$329$	−2.95486	−	15.4677i	−0.162906	−	0.852760i
$330$		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
$335$	15.2394	+	26.3954i	0.832617	+	1.44213i
$336$		0			0
$337$	9.03282	−	15.6453i	0.492049	−	0.852254i	−0.507909	−	0.861411i	$-0.669581\pi$
$337$	9.03282	−	15.6453i	0.492049	−	0.852254i	0.999958	+	0.00915682i	$0.00291475\pi$
$338$		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	$-0.929564\pi$
$347$	−23.7199	+	13.6947i	−1.27335	+	0.735169i	−0.297734	+	0.954649i	$0.596231\pi$
$348$		0			0
$349$	19.9926	−	11.5427i	1.07018	−	0.617869i	0.141950	−	0.989874i	$-0.454663\pi$
$349$	19.9926	−	11.5427i	1.07018	−	0.617869i	0.928231	+	0.372005i	$0.121329\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	19.7318			1.05022			0.525110	−	0.851034i	$-0.324024\pi$
$353$	19.7318			1.05022			0.525110	+	0.851034i	$0.324024\pi$
$354$		0			0
$355$		−	7.53285i		−	0.399802i
$356$		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
$365$	23.6426	+	13.6500i	1.23751	+	0.714476i
$366$		0			0
$367$			22.6970i			1.18477i	0.805654	+	0.592386i	$0.201814\pi$
$367$			22.6970i			1.18477i	−0.805654	+	0.592386i	$0.798186\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	21.5444	−	4.11573i	1.11853	−	0.213678i
$372$		0			0
$373$	14.4653			0.748983			0.374492	−	0.927230i	$-0.377817\pi$
$373$	14.4653			0.748983			0.374492	+	0.927230i	$0.377817\pi$
$374$		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
$383$	−34.8278			−1.77962			−0.889810	−	0.456331i	$-0.849163\pi$
$383$	−34.8278			−1.77962			−0.889810	+	0.456331i	$0.849163\pi$
$384$		0			0
$385$	6.21608	+	7.19755i	0.316801	+	0.366821i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			2.04173i			0.103520i	0.998660	+	0.0517600i	$0.0164831\pi$
$389$			2.04173i			0.103520i	−0.998660	+	0.0517600i	$0.983517\pi$
$390$		0			0
$391$	30.4349	+	17.5716i	1.53916	+	0.888635i
$392$		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
$401$		−	13.0195i		−	0.650164i	−0.945686	−	0.325082i	$-0.894608\pi$
$401$		−	13.0195i		−	0.650164i	0.945686	−	0.325082i	$-0.105392\pi$
$402$		0			0
$403$	35.5945			1.77309
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.38494	−	4.84105i	0.415626	−	0.239962i
$408$		0			0
$409$	10.9054	−	6.29621i	0.539236	−	0.311328i	−0.205534	−	0.978650i	$-0.565893\pi$
$409$	10.9054	−	6.29621i	0.539236	−	0.311328i	0.744769	+	0.667322i	$0.232560\pi$
$410$		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
$419$	−6.76463	+	11.7167i	−0.330474	+	0.572398i	−0.982605	−	0.185709i	$-0.940542\pi$
$419$	−6.76463	+	11.7167i	−0.330474	+	0.572398i	0.652131	+	0.758106i	$0.273875\pi$
$420$		0			0
$421$	−1.48749	−	2.57641i	−0.0724959	−	0.125567i	0.827499	−	0.561468i	$-0.189763\pi$
$421$	−1.48749	−	2.57641i	−0.0724959	−	0.125567i	−0.899995	+	0.435901i	$0.856430\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−9.85324	+	17.0663i	−0.477952	+	0.827838i
$426$		0			0
$427$	−6.74687	+	1.28889i	−0.326504	+	0.0623735i
$428$		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	$-0.243046\pi$
$439$	10.1562	+	5.86369i	0.484729	+	0.279859i	−0.237656	+	0.971349i	$0.576379\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−30.1605	−	17.4131i	−1.43297	−	0.827324i	−0.435621	−	0.900130i	$-0.643471\pi$
$443$	−30.1605	−	17.4131i	−1.43297	−	0.827324i	−0.997346	+	0.0728067i	$0.976804\pi$
$444$		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
$455$	39.0862	−	7.46682i	1.83239	−	0.350050i
$456$		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
$461$	5.94968	+	10.3051i	0.277104	+	0.479958i	0.970664	−	0.240441i	$-0.0772920\pi$
$461$	5.94968	+	10.3051i	0.277104	+	0.479958i	−0.693560	+	0.720399i	$0.743959\pi$
$462$		0			0
$463$	13.7464	−	23.8094i	0.638847	−	1.10652i	−0.346839	−	0.937925i	$-0.612745\pi$
$463$	13.7464	−	23.8094i	0.638847	−	1.10652i	0.985686	−	0.168591i	$-0.0539217\pi$
$464$		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
$475$	−8.40504	+	4.85265i	−0.385650	+	0.222655i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	39.4926			1.80446			0.902231	−	0.431253i	$-0.141929\pi$
$479$	39.4926			1.80446			0.902231	+	0.431253i	$0.141929\pi$
$480$		0			0
$481$		−	40.5122i		−	1.84720i
$482$		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
$491$	12.4585	+	7.19290i	0.562242	+	0.324611i	0.754045	−	0.656823i	$-0.228100\pi$
$491$	12.4585	+	7.19290i	0.562242	+	0.324611i	−0.191803	+	0.981434i	$0.561433\pi$
$492$		0			0
$493$		−	22.9391i		−	1.03313i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.54751	+	5.26552i	0.203984	+	0.236191i
$498$		0			0
$499$	41.6827			1.86597			0.932987	−	0.359911i	$-0.117193\pi$
$499$	41.6827			1.86597			0.932987	+	0.359911i	$0.117193\pi$
$500$		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
$509$	27.7147			1.22843			0.614217	−	0.789137i	$-0.289472\pi$
$509$	27.7147			1.22843			0.614217	+	0.789137i	$0.289472\pi$
$510$		0			0
$511$	−24.7668	+	4.73130i	−1.09562	+	0.209301i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	18.4674i		−	0.813771i
$516$		0			0
$517$	6.46801	+	3.73431i	0.284463	+	0.164235i
$518$		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
$527$		−	41.6733i		−	1.81532i
$528$		0			0
$529$	−9.68434			−0.421058
$530$		0			0
$531$		0			0
$532$		0			0
$533$	41.2364	−	23.8078i	1.78615	−	1.03123i
$534$		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
$545$	−4.43499	+	7.68163i	−0.189974	+	0.329045i
$546$		0			0
$547$	−11.1424	−	19.2992i	−0.476415	−	0.825175i	0.523220	−	0.852198i	$-0.324731\pi$
$547$	−11.1424	−	19.2992i	−0.476415	−	0.825175i	−0.999635	+	0.0270226i	$0.991397\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.64868	−	9.78380i	0.240642	−	0.416804i
$552$		0			0
$553$	5.45273	+	28.5432i	0.231874	+	1.21378i
$554$		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	0.609945	−	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	0.862196	−	0.506575i	$-0.169089\pi$
$569$	20.3851	+	11.7693i	0.854588	+	0.493396i	−0.00760850	+	0.999971i	$0.502422\pi$
$570$		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	0.686606	−	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
$581$	36.5286	+	12.7098i	1.51546	+	0.527290i
$582$		0			0
$583$	−5.20140	+	9.00909i	−0.215420	+	0.373119i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	19.8238	+	34.3358i	0.818215	+	1.41719i	0.906996	+	0.421139i	$0.138370\pi$
$587$	19.8238	+	34.3358i	0.818215	+	1.41719i	−0.0887810	+	0.996051i	$0.528297\pi$
$588$		0			0
$589$	10.2619	−	17.7741i	0.422834	−	0.732370i
$590$		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	$-0.870114\pi$
$599$	−25.2868	+	14.5993i	−1.03319	+	0.596513i	−0.115294	+	0.993331i	$0.536781\pi$
$600$		0			0
$601$	−12.9722	+	7.48950i	−0.529147	+	0.305503i	−0.740669	−	0.671870i	$-0.765491\pi$
$601$	−12.9722	+	7.48950i	−0.529147	+	0.305503i	0.211522	+	0.977373i	$0.432158\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	26.9999			1.09770
$606$		0			0
$607$		−	24.3659i		−	0.988982i	−0.869183	−	0.494491i	$-0.835355\pi$
$607$		−	24.3659i		−	0.988982i	0.869183	−	0.494491i	$-0.164645\pi$
$608$		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
$617$	−35.7489	−	20.6396i	−1.43920	−	0.830921i	−0.441403	−	0.897309i	$-0.645519\pi$
$617$	−35.7489	−	20.6396i	−1.43920	−	0.830921i	−0.997794	+	0.0663885i	$0.978852\pi$
$618$		0			0
$619$			40.8583i			1.64223i	0.570760	+	0.821117i	$0.306649\pi$
$619$			40.8583i			1.64223i	−0.570760	+	0.821117i	$0.693351\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−29.7626	−	10.3556i	−1.19241	−	0.414888i
$624$		0			0
$625$	−30.7518			−1.23007
$626$		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
$635$	−31.2992			−1.24207
$636$		0			0
$637$	−22.8139	+	28.8153i	−0.903921	+	1.14171i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			34.8689i			1.37724i	0.725124	+	0.688619i	$0.241783\pi$
$641$			34.8689i			1.37724i	−0.725124	+	0.688619i	$0.758217\pi$
$642$		0			0
$643$	23.2658	+	13.4325i	0.917514	+	0.529727i	0.882841	−	0.469672i	$-0.155628\pi$
$643$	23.2658	+	13.4325i	0.917514	+	0.529727i	0.0346728	+	0.999399i	$0.488961\pi$
$644$		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
$653$		−	20.1457i		−	0.788363i	−0.919033	−	0.394182i	$-0.871028\pi$
$653$		−	20.1457i		−	0.788363i	0.919033	−	0.394182i	$-0.128972\pi$
$654$		0			0
$655$	−31.2747			−1.22201
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−19.9901	+	11.5413i	−0.778706	+	0.449586i	−0.835971	−	0.548773i	$-0.815095\pi$
$659$	−19.9901	+	11.5413i	−0.778706	+	0.449586i	0.0572657	+	0.998359i	$0.481762\pi$
$660$		0			0
$661$	2.42515	−	1.40016i	0.0943273	−	0.0544599i	−0.452094	−	0.891970i	$-0.649323\pi$
$661$	2.42515	−	1.40016i	0.0943273	−	0.0544599i	0.546422	+	0.837510i	$0.315990\pi$
$662$		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
$671$	1.62888	−	2.82130i	0.0628821	−	0.108915i
$672$		0			0
$673$	16.3481	+	28.3158i	0.630173	+	1.09149i	0.987516	+	0.157519i	$0.0503496\pi$
$673$	16.3481	+	28.3158i	0.630173	+	1.09149i	−0.357343	+	0.933973i	$0.616317\pi$
$674$		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
$679$	−6.21658	+	17.8668i	−0.238570	+	0.685665i
$680$		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	$-0.536503\pi$
$691$	−27.1274	−	15.6620i	−1.03198	−	0.595812i	−0.917550	+	0.397620i	$0.869836\pi$
$692$		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
$707$	5.79522	−	5.00497i	0.217952	−	0.188231i
$708$		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
$713$	19.3787	+	33.5649i	0.725739	+	1.25702i
$714$		0			0
$715$	−9.43647	+	16.3444i	−0.352904	+	0.611248i
$716$		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
$727$	33.3713	−	19.2670i	1.23767	−	0.714572i	0.269056	−	0.963125i	$-0.413288\pi$
$727$	33.3713	−	19.2670i	1.23767	−	0.714572i	0.968618	+	0.248553i	$0.0799550\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	53.2424			1.96924
$732$		0			0
$733$		−	17.8259i		−	0.658414i	−0.944258	−	0.329207i	$-0.893219\pi$
$733$		−	17.8259i		−	0.658414i	0.944258	−	0.329207i	$-0.106781\pi$
$734$		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
$743$	−14.5617	−	8.40718i	−0.534216	−	0.308430i	0.208516	−	0.978019i	$-0.433137\pi$
$743$	−14.5617	−	8.40718i	−0.534216	−	0.308430i	−0.742731	+	0.669589i	$0.766470\pi$
$744$		0			0
$745$		−	45.1474i		−	1.65407i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	14.4585	−	41.5545i	0.528301	−	1.51837i
$750$		0			0
$751$	−42.6126			−1.55496			−0.777478	−	0.628910i	$-0.783501\pi$
$751$	−42.6126			−1.55496			−0.777478	+	0.628910i	$0.783501\pi$
$752$		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
$761$	−39.9019			−1.44644			−0.723222	−	0.690616i	$-0.757340\pi$
$761$	−39.9019			−1.44644			−0.723222	+	0.690616i	$0.757340\pi$
$762$		0			0
$763$	−1.53723	−	8.04689i	−0.0556516	−	0.291317i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			15.6027i			0.563379i
$768$		0			0
$769$	8.21315	+	4.74186i	0.296174	+	0.170996i	0.640723	−	0.767772i	$-0.278635\pi$
$769$	8.21315	+	4.74186i	0.296174	+	0.170996i	−0.344549	+	0.938768i	$0.611968\pi$
$770$		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
$779$		−	27.4552i		−	0.983685i
$780$		0			0
$781$	−3.29974			−0.118074
$782$		0			0
$783$		0			0
$784$		0			0
$785$	19.3100	−	11.1486i	0.689202	−	0.397911i
$786$		0			0
$787$	6.91018	−	3.98960i	0.246321	−	0.142214i	−0.371757	−	0.928330i	$-0.621245\pi$
$787$	6.91018	−	3.98960i	0.246321	−	0.142214i	0.618079	+	0.786116i	$0.287911\pi$
$788$		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
$797$	18.7204	−	32.4247i	0.663110	−	1.14854i	−0.316684	−	0.948531i	$-0.602569\pi$
$797$	18.7204	−	32.4247i	0.663110	−	1.14854i	0.979794	−	0.200009i	$-0.0640972\pi$
$798$		0			0
$799$	−18.2937	−	31.6856i	−0.647184	−	1.12095i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5.97936	−	10.3566i	0.211007	−	0.365475i
$804$		0			0
$805$	28.3208	+	32.7924i	0.998177	+	1.15578i
$806$		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	$-0.662616\pi$
$821$	−42.6604	−	24.6300i	−1.48886	−	0.859593i	−0.999919	+	0.0127244i	$0.995950\pi$
$822$		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	$0.288308\pi$
$827$			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
$833$	33.7364	+	26.7101i	1.16890	+	0.925449i
$834$		0			0
$835$	−6.26104	+	10.8444i	−0.216672	+	0.375287i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.30587	+	3.99389i	0.0796075	+	0.137884i	0.903081	−	0.429471i	$-0.141300\pi$
$839$	2.30587	+	3.99389i	0.0796075	+	0.137884i	−0.823473	+	0.567355i	$0.807967\pi$
$840$		0			0
$841$	−7.53726	+	13.0549i	−0.259905	+	0.450169i
$842$		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
$853$	−39.9304	+	23.0538i	−1.36719	+	0.789348i	−0.990568	−	0.137019i	$-0.956248\pi$
$853$	−39.9304	+	23.0538i	−1.36719	+	0.789348i	−0.376623	+	0.926367i	$0.622915\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−43.5523			−1.48772			−0.743859	−	0.668336i	$-0.767007\pi$
$857$	−43.5523			−1.48772			−0.743859	+	0.668336i	$0.767007\pi$
$858$		0			0
$859$		−	15.5165i		−	0.529418i	−0.964328	−	0.264709i	$-0.914724\pi$
$859$		−	15.5165i		−	0.529418i	0.964328	−	0.264709i	$-0.0852758\pi$
$860$		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
$869$	−11.9357	−	6.89110i	−0.404892	−	0.233764i
$870$		0			0
$871$		−	55.8643i		−	1.89289i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	10.2914	−	8.88802i	0.347912	−	0.300470i
$876$		0			0
$877$	15.0758			0.509074			0.254537	−	0.967063i	$-0.418077\pi$
$877$	15.0758			0.509074			0.254537	+	0.967063i	$0.418077\pi$
$878$		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
$887$	−9.61689			−0.322903			−0.161452	−	0.986881i	$-0.551618\pi$
$887$	−9.61689			−0.322903			−0.161452	+	0.986881i	$0.551618\pi$
$888$		0			0
$889$	21.8784	−	18.8950i	0.733777	−	0.633719i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	18.0190i		−	0.602983i
$894$		0			0
$895$	3.05115	+	1.76158i	0.101989	+	0.0588832i
$896$		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
$905$			14.6403i			0.486659i
$906$		0			0
$907$	41.4725			1.37707			0.688536	−	0.725202i	$-0.258254\pi$
$907$	41.4725			1.37707			0.688536	+	0.725202i	$0.258254\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	27.3284	−	15.7781i	0.905431	−	0.522751i	0.0264725	−	0.999650i	$-0.491573\pi$
$911$	27.3284	−	15.7781i	0.905431	−	0.522751i	0.878958	+	0.476899i	$0.158239\pi$
$912$		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
$923$	−6.90345	+	11.9571i	−0.227230	+	0.393574i
$924$		0			0
$925$	12.3679	+	21.4218i	0.406653	+	0.704344i
$926$		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
$931$	7.81169	+	19.6996i	0.256018	+	0.645628i
$932$		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	0.960084	+	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	−0.0384035	−	0.999262i	$-0.512227\pi$
$947$	−28.4036	−	16.3988i	−0.922992	−	0.532890i	−0.884588	+	0.466373i	$0.845561\pi$
$948$		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
$959$	−12.8702	−	14.9023i	−0.415601	−	0.481220i
$960$		0			0
$961$	7.47952	−	12.9549i	0.241275	−	0.417900i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−35.4535	−	61.4073i	−1.14129	−	1.97677i
$966$		0			0
$967$	12.0234	−	20.8252i	0.386648	−	0.669694i	−0.605348	−	0.795961i	$-0.706966\pi$
$967$	12.0234	−	20.8252i	0.386648	−	0.669694i	0.991996	+	0.126267i	$0.0402995\pi$
$968$		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	$-0.711693\pi$
$977$	−7.63263	+	4.40670i	−0.244190	+	0.140983i	0.372911	+	0.927867i	$0.378360\pi$
$978$		0			0
$979$	12.9434	−	7.47288i	0.413673	−	0.238834i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−32.9116			−1.04972			−0.524859	−	0.851189i	$-0.675882\pi$
$983$	−32.9116			−1.04972			−0.524859	+	0.851189i	$0.675882\pi$
$984$		0			0
$985$			27.0142i			0.860745i
$986$		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
$995$	44.8778	+	25.9102i	1.42272	+	0.821409i
$996$		0			0
$997$		−	38.7450i		−	1.22706i	−0.789669	−	0.613532i	$-0.789748\pi$
$997$		−	38.7450i		−	1.22706i	0.789669	−	0.613532i	$-0.210252\pi$
$998$		0			0
$999$		0			0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.bm (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+2.86458 q^{5} +(-2.00236 + 1.72932i) q^{7} -1.25482i q^{11} +(-4.54703 - 2.62523i) q^{13} +(-3.07356 + 5.32357i) q^{17} +(-2.62182 + 1.51371i) q^{19} -5.71702i q^{23} +3.20580 q^{25} +(-3.23174 + 1.86584i) q^{29} +(-5.87105 + 3.38965i) q^{31} +(-5.73592 + 4.95377i) q^{35} +(3.85796 + 6.68219i) q^{37} +(-4.53443 + 7.85386i) q^{41} +(-4.33067 - 7.50094i) q^{43} +(-2.97597 + 5.15454i) q^{47} +(1.01892 - 6.92545i) q^{49} +(-7.17959 - 4.14514i) q^{53} -3.59453i q^{55} +(-1.48584 - 2.57354i) q^{59} +(2.24837 + 1.29810i) q^{61} +(-13.0253 - 7.52018i) q^{65} +(5.31994 + 9.21441i) q^{67} -2.62965i q^{71} +(8.25342 + 4.76512i) q^{73} +(2.16998 + 2.51261i) q^{77} +(5.49170 - 9.51191i) q^{79} +(-7.30919 - 12.6599i) q^{83} +(-8.80446 + 15.2498i) q^{85} +(5.95534 + 10.3150i) q^{89} +(13.6447 - 2.60660i) q^{91} +(-7.51040 + 4.33613i) q^{95} +(6.19217 - 3.57505i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 4 q^{7} - 12 q^{13} + 32 q^{25} - 24 q^{31} - 4 q^{37} - 4 q^{43} - 16 q^{49} - 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} + 12 q^{85} - 36 q^{91} - 12 q^{97}+O(q^{100}) \) 32 * q + 4 * q^7 - 12 * q^13 + 32 * q^25 - 24 * q^31 - 4 * q^37 - 4 * q^43 - 16 * q^49 - 12 * q^61 + 4 * q^67 - 36 * q^73 + 28 * q^79 + 12 * q^85 - 36 * q^91 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more