LMFDB - Embedded newform 2268.2.bm.j.593.3

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)

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);

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")

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			593.3
Character	$\chi$	$=$	2268.593
Dual form			2268.2.bm.j.1025.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.86458 q^{5} +(-2.00236 - 1.72932i) q^{7} -1.25482i q^{11} +(-4.54703 + 2.62523i) q^{13} +(3.07356 + 5.32357i) q^{17} +(-2.62182 - 1.51371i) q^{19} -5.71702i q^{23} +3.20580 q^{25} +(3.23174 + 1.86584i) q^{29} +(-5.87105 - 3.38965i) q^{31} +(5.73592 + 4.95377i) q^{35} +(3.85796 - 6.68219i) q^{37} +(4.53443 + 7.85386i) q^{41} +(-4.33067 + 7.50094i) q^{43} +(2.97597 + 5.15454i) q^{47} +(1.01892 + 6.92545i) q^{49} +(7.17959 - 4.14514i) q^{53} +3.59453i q^{55} +(1.48584 - 2.57354i) q^{59} +(2.24837 - 1.29810i) q^{61} +(13.0253 - 7.52018i) q^{65} +(5.31994 - 9.21441i) q^{67} -2.62965i q^{71} +(8.25342 - 4.76512i) q^{73} +(-2.16998 + 2.51261i) q^{77} +(5.49170 + 9.51191i) q^{79} +(7.30919 - 12.6599i) q^{83} +(-8.80446 - 15.2498i) q^{85} +(-5.95534 + 10.3150i) q^{89} +(13.6447 + 2.60660i) q^{91} +(7.51040 + 4.33613i) q^{95} +(6.19217 + 3.57505i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

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

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	$-0.721289\pi$
$5$	−2.86458			−1.28108			−0.640539	+	0.767926i	$0.721289\pi$
$6$		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	−0.973291	−	0.229574i	$-0.926267\pi$
$13$	−4.54703	+	2.62523i	−1.26112	+	0.728108i	−0.287829	+	0.957682i	$0.592933\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.07356	+	5.32357i	0.745448	+	1.29115i	0.949985	+	0.312295i	$0.101098\pi$
$17$	3.07356	+	5.32357i	0.745448	+	1.29115i	−0.204537	+	0.978859i	$0.565569\pi$
$18$		0			0
$19$	−2.62182	−	1.51371i	−0.601486	−	0.347268i	0.168140	−	0.985763i	$-0.446224\pi$
$19$	−2.62182	−	1.51371i	−0.601486	−	0.347268i	−0.769626	+	0.638495i	$0.779557\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	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	$-0.220711\pi$
$29$	3.23174	+	1.86584i	0.600118	+	0.346478i	−0.168970	+	0.985621i	$0.554044\pi$
$30$		0			0
$31$	−5.87105	−	3.38965i	−1.05447	−	0.608800i	−0.130574	−	0.991439i	$-0.541682\pi$
$31$	−5.87105	−	3.38965i	−1.05447	−	0.608800i	−0.923898	+	0.382639i	$0.875015\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.73592	+	4.95377i	0.969548	+	0.837339i
$36$		0			0
$37$	3.85796	−	6.68219i	0.634245	−	1.09854i	−0.352429	−	0.935838i	$-0.614644\pi$
$37$	3.85796	−	6.68219i	0.634245	−	1.09854i	0.986675	−	0.162706i	$-0.0520223\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.53443	+	7.85386i	0.708159	+	1.22657i	0.965539	+	0.260257i	$0.0838073\pi$
$41$	4.53443	+	7.85386i	0.708159	+	1.22657i	−0.257380	+	0.966310i	$0.582859\pi$
$42$		0			0
$43$	−4.33067	+	7.50094i	−0.660421	+	1.14388i	0.320084	+	0.947389i	$0.396289\pi$
$43$	−4.33067	+	7.50094i	−0.660421	+	1.14388i	−0.980505	+	0.196494i	$0.937044\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.97597	+	5.15454i	0.434090	+	0.751866i	0.997221	−	0.0745015i	$-0.0237366\pi$
$47$	2.97597	+	5.15454i	0.434090	+	0.751866i	−0.563131	+	0.826368i	$0.690403\pi$
$48$		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	$-0.473851\pi$
$53$	7.17959	−	4.14514i	0.986193	−	0.569379i	0.904134	+	0.427249i	$0.140517\pi$
$54$		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	$-0.771369\pi$
$59$	1.48584	−	2.57354i	0.193439	−	0.335047i	0.946388	+	0.323033i	$0.104702\pi$
$60$		0			0
$61$	2.24837	−	1.29810i	0.287874	−	0.166204i	−0.349109	−	0.937082i	$-0.613516\pi$
$61$	2.24837	−	1.29810i	0.287874	−	0.166204i	0.636983	+	0.770878i	$0.280182\pi$
$62$		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	0.0679788	−	0.997687i	$-0.478345\pi$
$73$	8.25342	−	4.76512i	0.965990	−	0.557715i	0.898011	+	0.439972i	$0.145012\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.16998	+	2.51261i	−0.247293	+	0.286338i
$78$		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
$83$	7.30919	−	12.6599i	0.802287	−	1.38960i	−0.115820	−	0.993270i	$-0.536949\pi$
$83$	7.30919	−	12.6599i	0.802287	−	1.38960i	0.918107	−	0.396332i	$-0.129717\pi$
$84$		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	0.356029	+	0.934475i	$0.384131\pi$
$89$	−5.95534	+	10.3150i	−0.631265	+	1.09338i	−0.987293	+	0.158908i	$0.949203\pi$
$90$		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	0.780256	−	0.625460i	$-0.215089\pi$
$97$	6.19217	+	3.57505i	0.628720	+	0.362992i	−0.151536	+	0.988452i	$0.548422\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.89419			0.287983			0.143991	−	0.989579i	$-0.454006\pi$
$101$	2.89419			0.287983			0.143991	+	0.989579i	$0.454006\pi$
$102$		0			0
$103$			6.44681i			0.635224i	0.948221	+	0.317612i	$0.102881\pi$
$103$			6.44681i			0.635224i	−0.948221	+	0.317612i	$0.897119\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	14.4017	+	8.31483i	1.39227	+	0.803825i	0.993566	−	0.113256i	$-0.0361279\pi$
$107$	14.4017	+	8.31483i	1.39227	+	0.803825i	0.398701	+	0.917081i	$0.369461\pi$
$108$		0			0
$109$	−1.54822	−	2.68159i	−0.148292	−	0.256850i	0.782304	−	0.622897i	$-0.214044\pi$
$109$	−1.54822	−	2.68159i	−0.148292	−	0.256850i	−0.930596	+	0.366047i	$0.880711\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.5691	+	8.41149i	−1.37055	+	0.791286i	−0.990997	−	0.133884i	$-0.957255\pi$
$113$	−14.5691	+	8.41149i	−1.37055	+	0.791286i	−0.379551	+	0.925171i	$0.623922\pi$
$114$		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	$-0.341744\pi$
$131$	10.9178			0.953888			0.476944	+	0.878934i	$0.341744\pi$
$132$		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	−0.962762	−	0.270352i	$-0.912860\pi$
$139$	−14.2658	+	8.23638i	−1.21001	+	0.698600i	−0.247249	+	0.968952i	$0.579527\pi$
$140$		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	0.744262	−	0.667887i	$-0.232801\pi$
$157$	6.74094	+	3.89189i	0.537986	+	0.310606i	−0.206276	+	0.978494i	$0.566135\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−9.88655	+	11.4476i	−0.779169	+	0.902194i
$162$		0			0
$163$	3.80101	−	6.58354i	0.297718	−	0.515663i	−0.677896	−	0.735158i	$-0.737108\pi$
$163$	3.80101	−	6.58354i	0.297718	−	0.515663i	0.975613	+	0.219496i	$0.0704412\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.18568	+	3.78570i	0.169133	+	0.292947i	0.938115	−	0.346323i	$-0.112570\pi$
$167$	2.18568	+	3.78570i	0.169133	+	0.292947i	−0.768982	+	0.639270i	$0.779237\pi$
$168$		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	$-0.0503160\pi$
$173$	4.70140	+	8.14307i	0.357441	+	0.619106i	−0.630092	+	0.776521i	$0.716983\pi$
$174$		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	$-0.681303\pi$
$179$	−1.06513	+	0.614954i	−0.0796116	+	0.0459638i	0.459666	+	0.888092i	$0.347969\pi$
$180$		0			0
$181$		−	5.11079i		−	0.379882i	−0.981795	−	0.189941i	$-0.939170\pi$
$181$		−	5.11079i		−	0.379882i	0.981795	−	0.189941i	$-0.0608297\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−11.0514	+	19.1416i	−0.812517	+	1.40732i
$186$		0			0
$187$	6.68012	−	3.85677i	0.488498	−	0.282035i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.04399	+	2.33480i	−0.292613	+	0.168940i	−0.639120	−	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	0.171590	−	0.985168i	$-0.445110\pi$
$199$	15.6665	−	9.04503i	1.11057	−	0.641185i	0.938976	+	0.343983i	$0.111776\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−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	−0.987303	−	0.158845i	$-0.949223\pi$
$211$	−9.16894	−	15.8811i	−0.631216	−	1.09330i	0.356088	−	0.934453i	$-0.384110\pi$
$212$		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	0.255185	−	0.966892i	$-0.417864\pi$
$223$	−6.78826	−	3.91920i	−0.454576	−	0.262449i	−0.709761	+	0.704443i	$0.751197\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−16.1759			−1.07363			−0.536816	−	0.843699i	$-0.680373\pi$
$227$	−16.1759			−1.07363			−0.536816	+	0.843699i	$0.680373\pi$
$228$		0			0
$229$		−	7.43757i		−	0.491488i	−0.969335	−	0.245744i	$-0.920968\pi$
$229$		−	7.43757i		−	0.491488i	0.969335	−	0.245744i	$-0.0790323\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	19.8568	+	11.4643i	1.30086	+	0.751051i	0.980551	−	0.196262i	$-0.0628803\pi$
$233$	19.8568	+	11.4643i	1.30086	+	0.751051i	0.320308	+	0.947314i	$0.396214\pi$
$234$		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	$-0.521052\pi$
$239$	11.8266	−	6.82808i	0.764998	−	0.441672i	0.831087	+	0.556142i	$0.187719\pi$
$240$		0			0
$241$			8.13750i			0.524182i	0.965043	+	0.262091i	$0.0844121\pi$
$241$			8.13750i			0.524182i	−0.965043	+	0.262091i	$0.915588\pi$
$242$		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	$-0.601763\pi$
$251$	−9.95828			−0.628561			−0.314281	+	0.949330i	$0.601763\pi$
$252$		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	$-0.238292\pi$
$257$	23.4900			1.46526			0.732632	+	0.680625i	$0.238292\pi$
$258$		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	$-0.155895\pi$
$269$	0.554894	+	0.961104i	0.0338325	+	0.0585995i	−0.848613	+	0.529014i	$0.822562\pi$
$270$		0			0
$271$	11.9571	+	6.90342i	0.726341	+	0.419353i	0.817082	−	0.576521i	$-0.195590\pi$
$271$	11.9571	+	6.90342i	0.726341	+	0.419353i	−0.0907411	+	0.995875i	$0.528924\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	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	−0.995063	−	0.0992419i	$-0.968358\pi$
$281$	−26.4612	−	15.2774i	−1.57854	−	0.911371i	−0.583478	−	0.812129i	$-0.698308\pi$
$282$		0			0
$283$	−12.7910	−	7.38491i	−0.760349	−	0.438988i	0.0690722	−	0.997612i	$-0.477996\pi$
$283$	−12.7910	−	7.38491i	−0.760349	−	0.438988i	−0.829421	+	0.558624i	$0.811329\pi$
$284$		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	$-0.153244\pi$
$293$	0.721612	+	1.24987i	0.0421570	+	0.0730181i	−0.844177	+	0.536065i	$0.819910\pi$
$294$		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	$0.214765\pi$
$307$			21.8901i			1.24933i	−0.780891	+	0.624667i	$0.785235\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.25630	+	9.10417i	−0.298057	+	0.516250i	−0.975691	−	0.219149i	$-0.929672\pi$
$311$	−5.25630	+	9.10417i	−0.298057	+	0.516250i	0.677634	+	0.735399i	$0.263005\pi$
$312$		0			0
$313$	2.85633	−	1.64910i	0.161449	−	0.0932127i	−0.417099	−	0.908861i	$-0.636953\pi$
$313$	2.85633	−	1.64910i	0.161449	−	0.0932127i	0.578548	+	0.815649i	$0.303620\pi$
$314$		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
$329$	2.95486	−	15.4677i	0.162906	−	0.852760i
$330$		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
$335$	−15.2394	+	26.3954i	−0.832617	+	1.44213i
$336$		0			0
$337$	9.03282	+	15.6453i	0.492049	+	0.852254i	0.999958	−	0.00915682i	$-0.00291475\pi$
$337$	9.03282	+	15.6453i	0.492049	+	0.852254i	−0.507909	+	0.861411i	$0.669581\pi$
$338$		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
$349$	19.9926	+	11.5427i	1.07018	+	0.617869i	0.928231	−	0.372005i	$-0.121329\pi$
$349$	19.9926	+	11.5427i	1.07018	+	0.617869i	0.141950	+	0.989874i	$0.454663\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−19.7318			−1.05022			−0.525110	−	0.851034i	$-0.675976\pi$
$353$	−19.7318			−1.05022			−0.525110	+	0.851034i	$0.675976\pi$
$354$		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	−0.986022	−	0.166617i	$-0.946716\pi$
$359$	−30.7577	−	17.7579i	−1.62333	−	0.937228i	−0.637305	−	0.770611i	$-0.719951\pi$
$360$		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	$-0.798186\pi$
$367$		−	22.6970i		−	1.18477i	0.805654	−	0.592386i	$-0.201814\pi$
$368$		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	$-0.150837\pi$
$383$	34.8278			1.77962			0.889810	+	0.456331i	$0.150837\pi$
$384$		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	−0.364742	−	0.931108i	$-0.618843\pi$
$397$	−26.9678	−	15.5699i	−1.35348	−	0.781430i	−0.988735	+	0.149678i	$0.952176\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	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	0.744769	−	0.667322i	$-0.232560\pi$
$409$	10.9054	+	6.29621i	0.539236	+	0.311328i	−0.205534	+	0.978650i	$0.565893\pi$
$410$		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	$-0.0594581\pi$
$419$	6.76463	+	11.7167i	0.330474	+	0.572398i	−0.652131	+	0.758106i	$0.726125\pi$
$420$		0			0
$421$	−1.48749	+	2.57641i	−0.0724959	+	0.125567i	−0.899995	−	0.435901i	$-0.856430\pi$
$421$	−1.48749	+	2.57641i	−0.0724959	+	0.125567i	0.827499	+	0.561468i	$0.189763\pi$
$422$		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	$-0.746660\pi$
$431$	−8.94179	+	5.16254i	−0.430711	+	0.248671i	0.268939	+	0.963157i	$0.413327\pi$
$432$		0			0
$433$		−	10.9117i		−	0.524384i	−0.965016	−	0.262192i	$-0.915555\pi$
$433$		−	10.9117i		−	0.524384i	0.965016	−	0.262192i	$-0.0844453\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.65390	+	14.9890i	−0.413972	+	0.717021i
$438$		0			0
$439$	10.1562	−	5.86369i	0.484729	−	0.279859i	−0.237656	−	0.971349i	$-0.576379\pi$
$439$	10.1562	−	5.86369i	0.484729	−	0.279859i	0.722385	+	0.691491i	$0.243046\pi$
$440$		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
$455$	−39.0862	−	7.46682i	−1.83239	−	0.350050i
$456$		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
$461$	−5.94968	+	10.3051i	−0.277104	+	0.479958i	−0.970664	−	0.240441i	$-0.922708\pi$
$461$	−5.94968	+	10.3051i	−0.277104	+	0.479958i	0.693560	+	0.720399i	$0.256041\pi$
$462$		0			0
$463$	13.7464	+	23.8094i	0.638847	+	1.10652i	0.985686	+	0.168591i	$0.0539217\pi$
$463$	13.7464	+	23.8094i	0.638847	+	1.10652i	−0.346839	+	0.937925i	$0.612745\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.57357	−	2.72550i	0.0728161	−	0.126121i	−0.827318	−	0.561733i	$-0.810135\pi$
$467$	1.57357	−	2.72550i	0.0728161	−	0.126121i	0.900134	+	0.435612i	$0.143468\pi$
$468$		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	$-0.858071\pi$
$479$	−39.4926			−1.80446			−0.902231	+	0.431253i	$0.858071\pi$
$480$		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	0.955816	+	0.293966i	$0.0949754\pi$
$487$	16.1647	+	27.9980i	0.732490	+	1.26871i	−0.223326	+	0.974744i	$0.571691\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.4585	+	7.19290i	−0.562242	+	0.324611i	−0.754045	−	0.656823i	$-0.771900\pi$
$491$	−12.4585	+	7.19290i	−0.562242	+	0.324611i	0.191803	+	0.981434i	$0.438567\pi$
$492$		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	$-0.589714\pi$
$503$	−12.4755			−0.556256			−0.278128	+	0.960544i	$0.589714\pi$
$504$		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	$-0.710528\pi$
$509$	−27.7147			−1.22843			−0.614217	+	0.789137i	$0.710528\pi$
$510$		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	−0.999390	−	0.0349216i	$-0.988882\pi$
$521$	−12.0961	−	20.9510i	−0.529938	−	0.917880i	0.469452	−	0.882958i	$-0.344451\pi$
$522$		0			0
$523$	−20.8779	−	12.0539i	−0.912927	−	0.527078i	−0.0315550	−	0.999502i	$-0.510046\pi$
$523$	−20.8779	−	12.0539i	−0.912927	−	0.527078i	−0.881372	+	0.472424i	$0.843379\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	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	−0.00552468	+	0.999985i	$0.501759\pi$
$541$	−20.2072	+	34.9999i	−0.868775	+	1.50476i	−0.863250	+	0.504777i	$0.831575\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.43499	+	7.68163i	0.189974	+	0.329045i
$546$		0			0
$547$	−11.1424	+	19.2992i	−0.476415	+	0.825175i	−0.999635	−	0.0270226i	$-0.991397\pi$
$547$	−11.1424	+	19.2992i	−0.476415	+	0.825175i	0.523220	+	0.852198i	$0.324731\pi$
$548$		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	$-0.858553\pi$
$557$	−23.1768	+	13.3811i	−0.982031	+	0.566976i	−0.0791483	+	0.996863i	$0.525220\pi$
$558$		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	$-0.708808\pi$
$563$	9.04743	−	15.6706i	0.381304	−	0.660438i	0.991249	+	0.132006i	$0.0421418\pi$
$564$		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	−0.286323	−	0.958133i	$-0.592433\pi$
$577$	9.61514	−	5.55131i	0.400284	−	0.231104i	0.686606	+	0.727029i	$0.259100\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−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	0.0887810	+	0.996051i	$0.471703\pi$
$587$	−19.8238	+	34.3358i	−0.818215	+	1.41719i	−0.906996	+	0.421139i	$0.861630\pi$
$588$		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	0.422151	+	0.906525i	$0.361275\pi$
$593$	−13.9778	+	24.2102i	−0.573998	+	0.994194i	−0.996150	+	0.0876691i	$0.972058\pi$
$594$		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
$601$	−12.9722	−	7.48950i	−0.529147	−	0.305503i	0.211522	−	0.977373i	$-0.432158\pi$
$601$	−12.9722	−	7.48950i	−0.529147	−	0.305503i	−0.740669	+	0.671870i	$0.765491\pi$
$602$		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	$0.164645\pi$
$607$			24.3659i			0.988982i	−0.869183	+	0.494491i	$0.835355\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−27.0637	−	15.6252i	−1.09488	−	0.632129i
$612$		0			0
$613$	10.9343	+	18.9387i	0.441631	+	0.764927i	0.997811	−	0.0661347i	$-0.0210667\pi$
$613$	10.9343	+	18.9387i	0.441631	+	0.764927i	−0.556180	+	0.831062i	$0.687733\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	35.7489	−	20.6396i	1.43920	−	0.830921i	0.441403	−	0.897309i	$-0.354481\pi$
$617$	35.7489	−	20.6396i	1.43920	−	0.830921i	0.997794	+	0.0663885i	$0.0211477\pi$
$618$		0			0
$619$		−	40.8583i		−	1.64223i	−0.570760	−	0.821117i	$-0.693351\pi$
$619$		−	40.8583i		−	1.64223i	0.570760	−	0.821117i	$-0.306649\pi$
$620$		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	0.0346728	−	0.999399i	$-0.488961\pi$
$643$	23.2658	−	13.4325i	0.917514	−	0.529727i	0.882841	+	0.469672i	$0.155628\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.988791	+	1.71264i	0.0388734	+	0.0673306i	0.884807	−	0.465957i	$-0.154290\pi$
$647$	0.988791	+	1.71264i	0.0388734	+	0.0673306i	−0.845934	+	0.533287i	$0.820956\pi$
$648$		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	$-0.184905\pi$
$659$	19.9901	+	11.5413i	0.778706	+	0.449586i	−0.0572657	+	0.998359i	$0.518238\pi$
$660$		0			0
$661$	2.42515	+	1.40016i	0.0943273	+	0.0544599i	0.546422	−	0.837510i	$-0.315990\pi$
$661$	2.42515	+	1.40016i	0.0943273	+	0.0544599i	−0.452094	+	0.891970i	$0.649323\pi$
$662$		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	−0.357343	−	0.933973i	$-0.616317\pi$
$673$	16.3481	−	28.3158i	0.630173	−	1.09149i	0.987516	−	0.157519i	$-0.0503496\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	22.7293	+	39.3684i	0.873559	+	1.51305i	0.858290	+	0.513166i	$0.171527\pi$
$677$	22.7293	+	39.3684i	0.873559	+	1.51305i	0.0152697	+	0.999883i	$0.495139\pi$
$678$		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	0.677757	−	0.735286i	$-0.262952\pi$
$683$	43.2107	−	24.9477i	1.65341	−	0.954598i	0.975655	−	0.219312i	$-0.0703811\pi$
$684$		0			0
$685$		−	21.3192i		−	0.814565i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−21.7639	+	37.6962i	−0.829138	+	1.43611i
$690$		0			0
$691$	−27.1274	+	15.6620i	−1.03198	+	0.595812i	−0.917550	−	0.397620i	$-0.869836\pi$
$691$	−27.1274	+	15.6620i	−1.03198	+	0.595812i	−0.114426	+	0.993432i	$0.536503\pi$
$692$		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	−0.992960	−	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
$713$	−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	$-0.775762\pi$
$719$	4.82338	−	8.35434i	0.179882	−	0.311564i	0.941840	+	0.336062i	$0.109095\pi$
$720$		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	0.968618	−	0.248553i	$-0.0799550\pi$
$727$	33.3713	+	19.2670i	1.23767	+	0.714572i	0.269056	+	0.963125i	$0.413288\pi$
$728$		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	$0.106781\pi$
$733$			17.8259i			0.658414i	−0.944258	+	0.329207i	$0.893219\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.5624	−	6.67557i	−0.425907	−	0.245898i
$738$		0			0
$739$	4.27651	+	7.40713i	0.157314	+	0.272476i	0.933899	−	0.357536i	$-0.116383\pi$
$739$	4.27651	+	7.40713i	0.157314	+	0.272476i	−0.776585	+	0.630012i	$0.783050\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	14.5617	−	8.40718i	0.534216	−	0.308430i	−0.208516	−	0.978019i	$-0.566863\pi$
$743$	14.5617	−	8.40718i	0.534216	−	0.308430i	0.742731	+	0.669589i	$0.233530\pi$
$744$		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	$-0.242660\pi$
$761$	39.9019			1.44644			0.723222	+	0.690616i	$0.242660\pi$
$762$		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	−0.344549	−	0.938768i	$-0.611968\pi$
$769$	8.21315	−	4.74186i	0.296174	−	0.170996i	0.640723	+	0.767772i	$0.278635\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.60509	+	11.4403i	0.237568	+	0.411481i	0.960016	−	0.279945i	$-0.0903163\pi$
$773$	6.60509	+	11.4403i	0.237568	+	0.411481i	−0.722448	+	0.691426i	$0.756983\pi$
$774$		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	0.618079	−	0.786116i	$-0.287911\pi$
$787$	6.91018	+	3.98960i	0.246321	+	0.142214i	−0.371757	+	0.928330i	$0.621245\pi$
$788$		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	−0.979794	−	0.200009i	$-0.935903\pi$
$797$	−18.7204	−	32.4247i	−0.663110	−	1.14854i	0.316684	−	0.948531i	$-0.397431\pi$
$798$		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	$-0.744990\pi$
$809$	−12.0001	+	6.92824i	−0.421900	+	0.243584i	0.273990	+	0.961733i	$0.411656\pi$
$810$		0			0
$811$			17.6343i			0.619223i	0.950863	+	0.309611i	$0.100199\pi$
$811$			17.6343i			0.619223i	−0.950863	+	0.309611i	$0.899801\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10.8883	+	18.8591i	−0.381400	+	0.660604i
$816$		0			0
$817$	22.7085	−	13.1107i	0.794469	−	0.458687i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	42.6604	−	24.6300i	1.48886	−	0.859593i	0.488940	−	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	−0.351305	−	0.936261i	$-0.614262\pi$
$829$	8.17327	−	4.71884i	0.283869	−	0.163892i	0.635174	+	0.772369i	$0.280928\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−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	$-0.858700\pi$
$839$	−2.30587	+	3.99389i	−0.0796075	+	0.137884i	0.823473	+	0.567355i	$0.192033\pi$
$840$		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	−0.376623	−	0.926367i	$-0.622915\pi$
$853$	−39.9304	−	23.0538i	−1.36719	−	0.789348i	−0.990568	+	0.137019i	$0.956248\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	43.5523			1.48772			0.743859	−	0.668336i	$-0.232993\pi$
$857$	43.5523			1.48772			0.743859	+	0.668336i	$0.232993\pi$
$858$		0			0
$859$			15.5165i			0.529418i	0.964328	+	0.264709i	$0.0852758\pi$
$859$			15.5165i			0.529418i	−0.964328	+	0.264709i	$0.914724\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	15.3098	+	8.83910i	0.521151	+	0.300887i	0.737405	−	0.675450i	$-0.236051\pi$
$863$	15.3098	+	8.83910i	0.521151	+	0.300887i	−0.216255	+	0.976337i	$0.569384\pi$
$864$		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	$-0.419104\pi$
$881$	14.9248			0.502830			0.251415	+	0.967879i	$0.419104\pi$
$882$		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	$-0.448382\pi$
$887$	9.61689			0.322903			0.161452	+	0.986881i	$0.448382\pi$
$888$		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	$-0.508427\pi$
$911$	−27.3284	−	15.7781i	−0.905431	−	0.522751i	−0.878958	+	0.476899i	$0.841761\pi$
$912$		0			0
$913$	−15.8859	−	9.17171i	−0.525746	−	0.303539i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−21.8613	−	18.8803i	−0.721924	−	0.623481i
$918$		0			0
$919$	−23.5531	+	40.7951i	−0.776944	+	1.34571i	0.156751	+	0.987638i	$0.449898\pi$
$919$	−23.5531	+	40.7951i	−0.776944	+	1.34571i	−0.933695	+	0.358069i	$0.883435\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	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	0.842965	+	0.537968i	$0.180808\pi$
$929$	27.0468	+	46.8464i	0.887377	+	1.53698i	0.0444117	+	0.999013i	$0.485859\pi$
$930$		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	$-0.674245\pi$
$937$		−	52.2751i		−	1.70775i	0.520476	−	0.853876i	$-0.325755\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−22.1565	+	38.3761i	−0.722280	+	1.25103i	0.237804	+	0.971313i	$0.423572\pi$
$941$	−22.1565	+	38.3761i	−0.722280	+	1.25103i	−0.960084	+	0.279712i	$0.909761\pi$
$942$		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
$959$	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	0.991996	−	0.126267i	$-0.0402995\pi$
$967$	12.0234	+	20.8252i	0.386648	+	0.669694i	−0.605348	+	0.795961i	$0.706966\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	3.25010	−	5.62934i	0.104301	−	0.180654i	−0.809152	−	0.587600i	$-0.800073\pi$
$971$	3.25010	−	5.62934i	0.104301	−	0.180654i	0.913452	+	0.406946i	$0.133406\pi$
$972$		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
$979$	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	$-0.324118\pi$
$983$	32.9116			1.04972			0.524859	+	0.851189i	$0.324118\pi$
$984$		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	0.690756	−	0.723088i	$-0.257278\pi$
$991$	−8.84072	−	15.3126i	−0.280835	−	0.486420i	−0.971591	+	0.236668i	$0.923945\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−44.8778	+	25.9102i	−1.42272	+	0.821409i
$996$		0			0
$997$			38.7450i			1.22706i	0.789669	+	0.613532i	$0.210252\pi$
$997$			38.7450i			1.22706i	−0.789669	+	0.613532i	$0.789748\pi$
$998$		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.593.3		32
3.2	odd	2	inner	2268.2.bm.j.593.14		32
7.3	odd	6		2268.2.w.j.269.3		32
9.2	odd	6		2268.2.t.c.2105.3	yes	32
9.4	even	3		2268.2.w.j.1349.14		32
9.5	odd	6		2268.2.w.j.1349.3		32
9.7	even	3		2268.2.t.c.2105.14	yes	32
21.17	even	6		2268.2.w.j.269.14		32
63.31	odd	6	inner	2268.2.bm.j.1025.14		32
63.38	even	6		2268.2.t.c.1781.14	yes	32
63.52	odd	6		2268.2.t.c.1781.3	✓	32
63.59	even	6	inner	2268.2.bm.j.1025.3		32

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
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

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