LMFDB - Embedded newform 2268.2.w.j.1349.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(269,2268)] 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, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.269"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.w (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			1349.13
Character	$\chi$	$=$	2268.1349
Dual form			2268.2.w.j.269.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00175 + 1.73509i) q^{5} +(1.26572 + 2.32335i) q^{7} +(4.15726 + 2.40020i) q^{11} +(0.864497 + 0.499118i) q^{13} +(-0.0445736 - 0.0772037i) q^{17} +(3.68849 + 2.12955i) q^{19} +(-0.839792 + 0.484854i) q^{23} +(0.492982 - 0.853871i) q^{25} +(2.93010 - 1.69169i) q^{29} -5.03034i q^{31} +(-2.76327 + 4.52356i) q^{35} +(-0.0675641 + 0.117024i) q^{37} +(-5.62189 + 9.73740i) q^{41} +(-3.66779 - 6.35280i) q^{43} +3.53605 q^{47} +(-3.79590 + 5.88142i) q^{49} +(-5.31849 + 3.07063i) q^{53} +9.61761i q^{55} +9.76504 q^{59} -13.0338i q^{61} +1.99997i q^{65} -15.1548 q^{67} +4.56985i q^{71} +(3.73647 - 2.15725i) q^{73} +(-0.314561 + 12.6968i) q^{77} +11.8817 q^{79} +(-1.62494 - 2.81449i) q^{83} +(0.0893034 - 0.154678i) q^{85} +(1.06316 - 1.84146i) q^{89} +(-0.0654127 + 2.64027i) q^{91} +8.53314i q^{95} +(1.03527 - 0.597713i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 4 q^{7} + 12 q^{13} - 16 q^{25} - 4 q^{37} - 4 q^{43} + 20 q^{49} - 8 q^{67} - 36 q^{73} - 56 q^{79} + 12 q^{85} - 36 q^{91} + 12 q^{97}+O(q^{100}) $ 32 * q + 4 * q^7 + 12 * q^13 - 16 * q^25 - 4 * q^37 - 4 * q^43 + 20 * q^49 - 8 * q^67 - 36 * q^73 - 56 * q^79 + 12 * q^85 - 36 * q^91 + 12 * q^97

Character values

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

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

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
$5$	1.00175	+	1.73509i	0.447998	+	0.775954i	0.998256	−	0.0590402i	$-0.0188040\pi$
$5$	1.00175	+	1.73509i	0.447998	+	0.775954i	−0.550258	+	0.834995i	$0.685471\pi$
$6$		0			0
$7$	1.26572	+	2.32335i	0.478397	+	0.878143i
$7$	1.26572	+	2.32335i	0.478397	+	0.878143i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.15726	+	2.40020i	1.25346	+	0.723686i	0.971795	−	0.235826i	$-0.0757795\pi$
$11$	4.15726	+	2.40020i	1.25346	+	0.723686i	0.281666	+	0.959512i	$0.409113\pi$
$12$		0			0
$13$	0.864497	+	0.499118i	0.239768	+	0.138430i	0.615070	−	0.788472i	$-0.289128\pi$
$13$	0.864497	+	0.499118i	0.239768	+	0.138430i	−0.375302	+	0.926903i	$0.622461\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.0445736	−	0.0772037i	−0.0108107	−	0.0187246i	0.860569	−	0.509333i	$-0.170108\pi$
$17$	−0.0445736	−	0.0772037i	−0.0108107	−	0.0187246i	−0.871380	+	0.490608i	$0.836775\pi$
$18$		0			0
$19$	3.68849	+	2.12955i	0.846198	+	0.488553i	0.859366	−	0.511361i	$-0.170858\pi$
$19$	3.68849	+	2.12955i	0.846198	+	0.488553i	−0.0131681	+	0.999913i	$0.504192\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.839792	+	0.484854i	−0.175109	+	0.101099i	−0.584993	−	0.811039i	$-0.698903\pi$
$23$	−0.839792	+	0.484854i	−0.175109	+	0.101099i	0.409884	+	0.912138i	$0.365569\pi$
$24$		0			0
$25$	0.492982	−	0.853871i	0.0985965	−	0.170774i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.93010	−	1.69169i	0.544105	−	0.314139i	−0.202636	−	0.979254i	$-0.564951\pi$
$29$	2.93010	−	1.69169i	0.544105	−	0.314139i	0.746741	+	0.665115i	$0.231617\pi$
$30$		0			0
$31$		−	5.03034i		−	0.903476i	−0.892151	−	0.451738i	$-0.850804\pi$
$31$		−	5.03034i		−	0.903476i	0.892151	−	0.451738i	$-0.149196\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.76327	+	4.52356i	−0.467078	+	0.764621i
$36$		0			0
$37$	−0.0675641	+	0.117024i	−0.0111075	+	0.0192387i	−0.871526	−	0.490350i	$-0.836869\pi$
$37$	−0.0675641	+	0.117024i	−0.0111075	+	0.0192387i	0.860418	+	0.509589i	$0.170202\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.62189	+	9.73740i	−0.877992	+	1.52073i	−0.0244505	+	0.999701i	$0.507784\pi$
$41$	−5.62189	+	9.73740i	−0.877992	+	1.52073i	−0.853541	+	0.521025i	$0.825550\pi$
$42$		0			0
$43$	−3.66779	−	6.35280i	−0.559333	−	0.968793i	−0.997552	−	0.0699250i	$-0.977724\pi$
$43$	−3.66779	−	6.35280i	−0.559333	−	0.968793i	0.438219	−	0.898868i	$-0.355609\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.53605			0.515786			0.257893	−	0.966173i	$-0.416972\pi$
$47$	3.53605			0.515786			0.257893	+	0.966173i	$0.416972\pi$
$48$		0			0
$49$	−3.79590	+	5.88142i	−0.542272	+	0.840203i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.31849	+	3.07063i	−0.730551	+	0.421784i	−0.818624	−	0.574330i	$-0.805263\pi$
$53$	−5.31849	+	3.07063i	−0.730551	+	0.421784i	0.0880726	+	0.996114i	$0.471929\pi$
$54$		0			0
$55$			9.61761i			1.29684i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.76504			1.27130			0.635650	−	0.771977i	$-0.280732\pi$
$59$	9.76504			1.27130			0.635650	+	0.771977i	$0.280732\pi$
$60$		0			0
$61$		−	13.0338i		−	1.66881i	−0.551149	−	0.834407i	$-0.685810\pi$
$61$		−	13.0338i		−	1.66881i	0.551149	−	0.834407i	$-0.314190\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			1.99997i			0.248066i
$66$		0			0
$67$	−15.1548			−1.85146			−0.925728	−	0.378189i	$-0.876547\pi$
$67$	−15.1548			−1.85146			−0.925728	+	0.378189i	$0.876547\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.56985i			0.542341i	0.962531	+	0.271171i	$0.0874108\pi$
$71$			4.56985i			0.542341i	−0.962531	+	0.271171i	$0.912589\pi$
$72$		0			0
$73$	3.73647	−	2.15725i	0.437321	−	0.252487i	−0.265140	−	0.964210i	$-0.585418\pi$
$73$	3.73647	−	2.15725i	0.437321	−	0.252487i	0.702460	+	0.711723i	$0.252085\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.314561	+	12.6968i	−0.0358476	+	1.44693i
$78$		0			0
$79$	11.8817			1.33679			0.668397	−	0.743805i	$-0.266981\pi$
$79$	11.8817			1.33679			0.668397	+	0.743805i	$0.266981\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.62494	−	2.81449i	−0.178361	−	0.308930i	0.762958	−	0.646448i	$-0.223746\pi$
$83$	−1.62494	−	2.81449i	−0.178361	−	0.308930i	−0.941319	+	0.337518i	$0.890413\pi$
$84$		0			0
$85$	0.0893034	−	0.154678i	0.00968631	−	0.0167772i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.06316	−	1.84146i	0.112695	−	0.195194i	−0.804161	−	0.594412i	$-0.797385\pi$
$89$	1.06316	−	1.84146i	0.112695	−	0.195194i	0.916856	+	0.399218i	$0.130718\pi$
$90$		0			0
$91$	−0.0654127	+	2.64027i	−0.00685711	+	0.276776i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			8.53314i			0.875482i
$96$		0			0
$97$	1.03527	−	0.597713i	0.105116	−	0.0606885i	−0.446521	−	0.894773i	$-0.647337\pi$
$97$	1.03527	−	0.597713i	0.105116	−	0.0606885i	0.551636	+	0.834085i	$0.314004\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.99395	+	10.3818i	−0.596420	+	1.03303i	0.396925	+	0.917851i	$0.370077\pi$
$101$	−5.99395	+	10.3818i	−0.596420	+	1.03303i	−0.993345	+	0.115179i	$0.963256\pi$
$102$		0			0
$103$	−8.25554	+	4.76634i	−0.813442	+	0.469641i	−0.848150	−	0.529756i	$-0.822283\pi$
$103$	−8.25554	+	4.76634i	−0.813442	+	0.469641i	0.0347074	+	0.999398i	$0.488950\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.2170	−	9.36287i	−1.56775	−	0.905143i	−0.996430	−	0.0844177i	$-0.973097\pi$
$107$	−16.2170	−	9.36287i	−1.56775	−	0.905143i	−0.571323	−	0.820725i	$-0.693570\pi$
$108$		0			0
$109$	0.901223	+	1.56096i	0.0863215	+	0.149513i	0.905954	−	0.423377i	$-0.139155\pi$
$109$	0.901223	+	1.56096i	0.0863215	+	0.149513i	−0.819632	+	0.572890i	$0.805822\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.6314	+	6.13804i	1.00012	+	0.577419i	0.908283	−	0.418356i	$-0.137393\pi$
$113$	10.6314	+	6.13804i	1.00012	+	0.577419i	0.0918351	+	0.995774i	$0.470727\pi$
$114$		0			0
$115$	−1.68253	−	0.971408i	−0.156896	−	0.0905842i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.122953	−	0.201278i	0.0112711	−	0.0184511i
$120$		0			0
$121$	6.02189	+	10.4302i	0.547444	+	0.948201i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.9929			1.07268
$126$		0			0
$127$	−7.35545			−0.652690			−0.326345	−	0.945251i	$-0.605817\pi$
$127$	−7.35545			−0.652690			−0.326345	+	0.945251i	$0.605817\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.24981	+	9.09293i	0.458678	+	0.794453i	0.998891	−	0.0470748i	$-0.0149899\pi$
$131$	5.24981	+	9.09293i	0.458678	+	0.794453i	−0.540214	+	0.841528i	$0.681657\pi$
$132$		0			0
$133$	−0.279092	+	11.2651i	−0.0242003	+	0.976806i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.3253	+	7.69335i	1.13845	+	0.657287i	0.946047	−	0.324028i	$-0.105037\pi$
$137$	13.3253	+	7.69335i	1.13845	+	0.657287i	0.192407	+	0.981315i	$0.438371\pi$
$138$		0			0
$139$	−12.6370	−	7.29600i	−1.07186	−	0.618838i	−0.143171	−	0.989698i	$-0.545730\pi$
$139$	−12.6370	−	7.29600i	−1.07186	−	0.618838i	−0.928689	+	0.370860i	$0.879063\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.39596	+	4.14993i	0.200360	+	0.347034i
$144$		0			0
$145$	5.87046	+	3.38931i	0.487515	+	0.281467i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.4588	+	7.19307i	−1.02066	+	0.589279i	−0.914295	−	0.405048i	$-0.867255\pi$
$149$	−12.4588	+	7.19307i	−1.02066	+	0.589279i	−0.106366	+	0.994327i	$0.533921\pi$
$150$		0			0
$151$	7.75751	−	13.4364i	0.631297	−	1.09344i	−0.355990	−	0.934490i	$-0.615856\pi$
$151$	7.75751	−	13.4364i	0.631297	−	1.09344i	0.987287	−	0.158949i	$-0.0508104\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.72808	−	5.03916i	0.701056	−	0.404755i
$156$		0			0
$157$		−	3.49238i		−	0.278722i	−0.990242	−	0.139361i	$-0.955495\pi$
$157$		−	3.49238i		−	0.278722i	0.990242	−	0.139361i	$-0.0445049\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.18943	−	1.33744i	−0.172551	−	0.105405i
$162$		0			0
$163$	6.91917	−	11.9844i	0.541951	−	0.938687i	−0.456841	−	0.889548i	$-0.651019\pi$
$163$	6.91917	−	11.9844i	0.541951	−	0.938687i	0.998792	−	0.0491384i	$-0.0156475\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.95252	+	3.38186i	−0.151090	+	0.261696i	−0.931629	−	0.363412i	$-0.881612\pi$
$167$	−1.95252	+	3.38186i	−0.151090	+	0.261696i	0.780538	+	0.625108i	$0.214945\pi$
$168$		0			0
$169$	−6.00176	−	10.3954i	−0.461674	−	0.799643i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	17.9325			1.36338			0.681690	−	0.731641i	$-0.261245\pi$
$173$	17.9325			1.36338			0.681690	+	0.731641i	$0.261245\pi$
$174$		0			0
$175$	2.60782	+	0.0646086i	0.197132	+	0.00488395i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.93535	+	3.42678i	−0.443629	+	0.256129i	−0.705136	−	0.709072i	$-0.749114\pi$
$179$	−5.93535	+	3.42678i	−0.443629	+	0.256129i	0.261507	+	0.965202i	$0.415781\pi$
$180$		0			0
$181$			12.1011i			0.899468i	0.893163	+	0.449734i	$0.148481\pi$
$181$			12.1011i			0.899468i	−0.893163	+	0.449734i	$0.851519\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.270730			−0.0199045
$186$		0			0
$187$		−	0.427941i		−	0.0312942i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	8.49691i		−	0.614815i	−0.951578	−	0.307408i	$-0.900539\pi$
$191$		−	8.49691i		−	0.614815i	0.951578	−	0.307408i	$-0.0994615\pi$
$192$		0			0
$193$	13.5960			0.978663			0.489331	−	0.872098i	$-0.337241\pi$
$193$	13.5960			0.978663			0.489331	+	0.872098i	$0.337241\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			23.5565i			1.67833i	0.543878	+	0.839164i	$0.316955\pi$
$197$			23.5565i			1.67833i	−0.543878	+	0.839164i	$0.683045\pi$
$198$		0			0
$199$	−11.7796	+	6.80093i	−0.835030	+	0.482105i	−0.855572	−	0.517684i	$-0.826794\pi$
$199$	−11.7796	+	6.80093i	−0.835030	+	0.482105i	0.0205416	+	0.999789i	$0.493461\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.63907	+	4.66643i	0.536158	+	0.327519i
$204$		0			0
$205$	−22.5270			−1.57335
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.2227	+	17.7062i	0.707118	+	1.22476i
$210$		0			0
$211$	8.44148	−	14.6211i	0.581136	−	1.00656i	−0.414209	−	0.910182i	$-0.635942\pi$
$211$	8.44148	−	14.6211i	0.581136	−	1.00656i	0.995345	−	0.0963750i	$-0.0307248\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.34844	−	12.7279i	0.501160	−	0.868034i
$216$		0			0
$217$	11.6872	−	6.36701i	0.793382	−	0.432221i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	0.0889898i		−	0.00598610i
$222$		0			0
$223$	−3.83410	+	2.21362i	−0.256750	+	0.148235i	−0.622851	−	0.782340i	$-0.714026\pi$
$223$	−3.83410	+	2.21362i	−0.256750	+	0.148235i	0.366101	+	0.930575i	$0.380692\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.69552	−	11.5970i	0.444397	−	0.769718i	−0.553613	−	0.832774i	$-0.686751\pi$
$227$	6.69552	−	11.5970i	0.444397	−	0.769718i	0.998010	+	0.0630557i	$0.0200846\pi$
$228$		0			0
$229$	−6.62507	+	3.82498i	−0.437797	+	0.252762i	−0.702663	−	0.711523i	$-0.748006\pi$
$229$	−6.62507	+	3.82498i	−0.437797	+	0.252762i	0.264866	+	0.964285i	$0.414672\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−12.1646	−	7.02323i	−0.796929	−	0.460107i	0.0454674	−	0.998966i	$-0.485522\pi$
$233$	−12.1646	−	7.02323i	−0.796929	−	0.460107i	−0.842396	+	0.538859i	$0.818856\pi$
$234$		0			0
$235$	3.54225	+	6.13535i	0.231071	+	0.400226i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−14.5035	−	8.37362i	−0.938156	−	0.541644i	−0.0487739	−	0.998810i	$-0.515531\pi$
$239$	−14.5035	−	8.37362i	−0.938156	−	0.541644i	−0.889382	+	0.457165i	$0.848865\pi$
$240$		0			0
$241$	26.5411	+	15.3235i	1.70967	+	0.987076i	0.934960	+	0.354753i	$0.115435\pi$
$241$	26.5411	+	15.3235i	1.70967	+	0.987076i	0.774705	+	0.632323i	$0.217898\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−14.0073	−	0.694489i	−0.894896	−	0.0443692i
$246$		0			0
$247$	2.12579	+	3.68198i	0.135261	+	0.234279i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.56378			0.288063			0.144032	−	0.989573i	$-0.453993\pi$
$251$	4.56378			0.288063			0.144032	+	0.989573i	$0.453993\pi$
$252$		0			0
$253$	−4.65498			−0.292656
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.3918	−	23.1954i	−0.835361	−	1.44689i	−0.893736	−	0.448592i	$-0.851926\pi$
$257$	−13.3918	−	23.1954i	−0.835361	−	1.44689i	0.0583759	−	0.998295i	$-0.481408\pi$
$258$		0			0
$259$	−0.357406	−	0.00885471i	−0.0222081	−	0.000550205i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−18.1138	−	10.4580i	−1.11695	−	0.644869i	−0.176326	−	0.984332i	$-0.556421\pi$
$263$	−18.1138	−	10.4580i	−1.11695	−	0.644869i	−0.940619	+	0.339463i	$0.889755\pi$
$264$		0			0
$265$	−10.6556	−	6.15203i	−0.654570	−	0.377916i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	13.3200	+	23.0709i	0.812133	+	1.40666i	0.911368	+	0.411592i	$0.135027\pi$
$269$	13.3200	+	23.0709i	0.812133	+	1.40666i	−0.0992351	+	0.995064i	$0.531640\pi$
$270$		0			0
$271$	17.6224	+	10.1743i	1.07048	+	0.618045i	0.928314	−	0.371798i	$-0.121258\pi$
$271$	17.6224	+	10.1743i	1.07048	+	0.618045i	0.142171	+	0.989842i	$0.454592\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.09891	−	2.36651i	0.247174	−	0.142706i
$276$		0			0
$277$	−1.02112	+	1.76863i	−0.0613531	+	0.106267i	−0.895070	−	0.445925i	$-0.852875\pi$
$277$	−1.02112	+	1.76863i	−0.0613531	+	0.106267i	0.833717	+	0.552191i	$0.186208\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−14.6608	+	8.46441i	−0.874590	+	0.504945i	−0.868871	−	0.495039i	$-0.835154\pi$
$281$	−14.6608	+	8.46441i	−0.874590	+	0.504945i	−0.00571889	+	0.999984i	$0.501820\pi$
$282$		0			0
$283$			11.5134i			0.684400i	0.939627	+	0.342200i	$0.111172\pi$
$283$			11.5134i			0.684400i	−0.939627	+	0.342200i	$0.888828\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−29.7391	−	0.736786i	−1.75544	−	0.0434911i
$288$		0			0
$289$	8.49603	−	14.7155i	0.499766	−	0.865621i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.4055	−	18.0229i	0.607898	−	1.05291i	−0.383689	−	0.923463i	$-0.625346\pi$
$293$	10.4055	−	18.0229i	0.607898	−	1.05291i	0.991586	−	0.129447i	$-0.0413203\pi$
$294$		0			0
$295$	9.78216	+	16.9432i	0.569539	+	0.986471i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.967997			−0.0559807
$300$		0			0
$301$	10.1174	−	16.5624i	0.583156	−	0.954643i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	22.6149	−	13.0567i	1.29492	−	0.747624i
$306$		0			0
$307$		−	2.00298i		−	0.114316i	−0.998365	−	0.0571580i	$-0.981796\pi$
$307$		−	2.00298i		−	0.114316i	0.998365	−	0.0571580i	$-0.0182039\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.70747			0.493756			0.246878	−	0.969047i	$-0.420595\pi$
$311$	8.70747			0.493756			0.246878	+	0.969047i	$0.420595\pi$
$312$		0			0
$313$		−	11.0181i		−	0.622782i	−0.950282	−	0.311391i	$-0.899205\pi$
$313$		−	11.0181i		−	0.622782i	0.950282	−	0.311391i	$-0.100795\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			11.4569i			0.643483i	0.946828	+	0.321742i	$0.104268\pi$
$317$			11.4569i			0.643483i	−0.946828	+	0.321742i	$0.895732\pi$
$318$		0			0
$319$	16.2416			0.909353
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	0.379687i		−	0.0211263i
$324$		0			0
$325$	0.852364	−	0.492113i	0.0472806	−	0.0272975i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.47565	+	8.21548i	0.246751	+	0.452934i
$330$		0			0
$331$	5.41191			0.297466			0.148733	−	0.988877i	$-0.452481\pi$
$331$	5.41191			0.297466			0.148733	+	0.988877i	$0.452481\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−15.1814	−	26.2949i	−0.829448	−	1.43665i
$336$		0			0
$337$	−12.1854	+	21.1058i	−0.663783	+	1.14971i	0.315831	+	0.948815i	$0.397717\pi$
$337$	−12.1854	+	21.1058i	−0.663783	+	1.14971i	−0.979614	+	0.200890i	$0.935617\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	12.0738	−	20.9125i	0.653834	−	1.13247i
$342$		0			0
$343$	−18.4691	−	1.37497i	−0.997240	−	0.0742413i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	8.94211i		−	0.480038i	−0.970768	−	0.240019i	$-0.922846\pi$
$347$		−	8.94211i		−	0.480038i	0.970768	−	0.240019i	$-0.0771536\pi$
$348$		0			0
$349$	−9.68628	+	5.59238i	−0.518495	+	0.299353i	−0.736319	−	0.676635i	$-0.763438\pi$
$349$	−9.68628	+	5.59238i	−0.518495	+	0.299353i	0.217824	+	0.975988i	$0.430104\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.75651	+	11.7026i	−0.359613	+	0.622867i	−0.987896	−	0.155117i	$-0.950424\pi$
$353$	−6.75651	+	11.7026i	−0.359613	+	0.622867i	0.628284	+	0.777984i	$0.283758\pi$
$354$		0			0
$355$	−7.92908	+	4.57786i	−0.420832	+	0.242968i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−29.5673	−	17.0707i	−1.56050	−	0.900957i	−0.997206	−	0.0747023i	$-0.976199\pi$
$359$	−29.5673	−	17.0707i	−1.56050	−	0.900957i	−0.563297	−	0.826254i	$-0.690467\pi$
$360$		0			0
$361$	−0.430017	−	0.744810i	−0.0226324	−	0.0392006i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.48604	+	4.32206i	0.391837	+	0.226227i
$366$		0			0
$367$	0.922490	+	0.532600i	0.0481536	+	0.0278015i	0.523884	−	0.851790i	$-0.324483\pi$
$367$	0.922490	+	0.532600i	0.0481536	+	0.0278015i	−0.475730	+	0.879591i	$0.657816\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−13.8659	−	8.47015i	−0.719881	−	0.439748i
$372$		0			0
$373$	2.16655	+	3.75257i	0.112180	+	0.194301i	0.916649	−	0.399694i	$-0.130883\pi$
$373$	2.16655	+	3.75257i	0.112180	+	0.194301i	−0.804469	+	0.593994i	$0.797550\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.37741			0.173946
$378$		0			0
$379$	2.43862			0.125263			0.0626317	−	0.998037i	$-0.480051\pi$
$379$	2.43862			0.125263			0.0626317	+	0.998037i	$0.480051\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−18.7552	−	32.4849i	−0.958344	−	1.65990i	−0.726524	−	0.687141i	$-0.758865\pi$
$383$	−18.7552	−	32.4849i	−0.958344	−	1.65990i	−0.231819	−	0.972759i	$-0.574468\pi$
$384$		0			0
$385$	−22.3451	+	12.1732i	−1.13881	+	0.620404i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5.97057	−	3.44711i	−0.302720	−	0.174775i	0.340944	−	0.940083i	$-0.389253\pi$
$389$	−5.97057	−	3.44711i	−0.302720	−	0.174775i	−0.643664	+	0.765308i	$0.722587\pi$
$390$		0			0
$391$	0.0748650	+	0.0432233i	0.00378609	+	0.00218590i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	11.9025	+	20.6158i	0.598880	+	1.03729i
$396$		0			0
$397$	−33.7783	−	19.5019i	−1.69529	−	0.978774i	−0.950114	−	0.311902i	$-0.899034\pi$
$397$	−33.7783	−	19.5019i	−1.69529	−	0.978774i	−0.745172	−	0.666872i	$-0.767633\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−16.1101	+	9.30115i	−0.804498	+	0.464477i	−0.845042	−	0.534700i	$-0.820424\pi$
$401$	−16.1101	+	9.30115i	−0.804498	+	0.464477i	0.0405433	+	0.999178i	$0.487091\pi$
$402$		0			0
$403$	2.51073	−	4.34872i	0.125069	−	0.216625i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.561763	+	0.324334i	−0.0278456	+	0.0160766i
$408$		0			0
$409$		−	2.18387i		−	0.107985i	−0.998541	−	0.0539927i	$-0.982805\pi$
$409$		−	2.18387i		−	0.107985i	0.998541	−	0.0539927i	$-0.0171948\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	12.3598	+	22.6876i	0.608187	+	1.11638i
$414$		0			0
$415$	3.25559	−	5.63884i	0.159810	−	0.276800i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.9333	−	25.8653i	0.729540	−	1.26360i	−0.227538	−	0.973769i	$-0.573067\pi$
$419$	14.9333	−	25.8653i	0.729540	−	1.26360i	0.957078	−	0.289831i	$-0.0935992\pi$
$420$		0			0
$421$	0.673150	+	1.16593i	0.0328073	+	0.0568240i	0.881963	−	0.471319i	$-0.156222\pi$
$421$	0.673150	+	1.16593i	0.0328073	+	0.0568240i	−0.849156	+	0.528143i	$0.822889\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.0878959			−0.00426358
$426$		0			0
$427$	30.2822	−	16.4972i	1.46546	−	0.798356i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	22.9854	−	13.2706i	1.10717	−	0.639223i	0.169072	−	0.985604i	$-0.445923\pi$
$431$	22.9854	−	13.2706i	1.10717	−	0.639223i	0.938094	+	0.346381i	$0.112590\pi$
$432$		0			0
$433$		−	27.4682i		−	1.32004i	−0.751250	−	0.660018i	$-0.770549\pi$
$433$		−	27.4682i		−	1.32004i	0.751250	−	0.660018i	$-0.229451\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.13009			−0.197569
$438$		0			0
$439$		−	4.03277i		−	0.192473i	−0.995358	−	0.0962367i	$-0.969319\pi$
$439$		−	4.03277i		−	0.192473i	0.995358	−	0.0962367i	$-0.0306806\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	33.8425i		−	1.60791i	−0.594693	−	0.803953i	$-0.702726\pi$
$443$		−	33.8425i		−	1.60791i	0.594693	−	0.803953i	$-0.297274\pi$
$444$		0			0
$445$	4.26011			0.201949
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	31.7395i		−	1.49788i	−0.662638	−	0.748940i	$-0.730563\pi$
$449$		−	31.7395i		−	1.49788i	0.662638	−	0.748940i	$-0.269437\pi$
$450$		0			0
$451$	−46.7433	+	26.9873i	−2.20106	+	1.27078i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.64663	+	2.53140i	−0.217837	+	0.118674i
$456$		0			0
$457$	−35.5605			−1.66345			−0.831724	−	0.555189i	$-0.812646\pi$
$457$	−35.5605			−1.66345			−0.831724	+	0.555189i	$0.812646\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.2947	+	28.2232i	0.758919	+	1.31449i	0.943402	+	0.331650i	$0.107605\pi$
$461$	16.2947	+	28.2232i	0.758919	+	1.31449i	−0.184483	+	0.982836i	$0.559061\pi$
$462$		0			0
$463$	10.6397	−	18.4286i	0.494471	−	0.856448i	−0.505509	−	0.862821i	$-0.668695\pi$
$463$	10.6397	−	18.4286i	0.494471	−	0.856448i	0.999980	+	0.00637294i	$0.00202858\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	19.0189	−	32.9417i	0.880090	−	1.52436i	0.0288500	−	0.999584i	$-0.490815\pi$
$467$	19.0189	−	32.9417i	0.880090	−	1.52436i	0.851240	−	0.524777i	$-0.175851\pi$
$468$		0			0
$469$	−19.1818	−	35.2100i	−0.885732	−	1.62584i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	35.2137i		−	1.61913i
$474$		0			0
$475$	3.63672	−	2.09966i	0.166864	−	0.0963392i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.26814	+	9.12468i	−0.240707	+	0.416917i	−0.960916	−	0.276840i	$-0.910713\pi$
$479$	−5.26814	+	9.12468i	−0.240707	+	0.416917i	0.720209	+	0.693758i	$0.244046\pi$
$480$		0			0
$481$	−0.116818	+	0.0674448i	−0.00532644	+	0.00307522i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.07417	+	1.19752i	0.0941831	+	0.0543766i
$486$		0			0
$487$	−4.92393	−	8.52850i	−0.223125	−	0.386463i	0.732631	−	0.680627i	$-0.238292\pi$
$487$	−4.92393	−	8.52850i	−0.223125	−	0.386463i	−0.955755	+	0.294163i	$0.904959\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.74760	−	3.89573i	−0.304515	−	0.175812i	0.339954	−	0.940442i	$-0.389588\pi$
$491$	−6.74760	−	3.89573i	−0.304515	−	0.175812i	−0.644469	+	0.764630i	$0.722922\pi$
$492$		0			0
$493$	−0.261210	−	0.150809i	−0.0117643	−	0.00679211i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−10.6174	+	5.78415i	−0.476253	+	0.259455i
$498$		0			0
$499$	11.8950	+	20.6027i	0.532493	+	0.922305i	0.999280	+	0.0379356i	$0.0120782\pi$
$499$	11.8950	+	20.6027i	0.532493	+	0.922305i	−0.466787	+	0.884370i	$0.654589\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	15.1692			0.676363			0.338182	−	0.941081i	$-0.390188\pi$
$503$	15.1692			0.676363			0.338182	+	0.941081i	$0.390188\pi$
$504$		0			0
$505$	−24.0178			−1.06878
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8.59151	−	14.8809i	−0.380812	−	0.659586i	0.610367	−	0.792119i	$-0.291022\pi$
$509$	−8.59151	−	14.8809i	−0.380812	−	0.659586i	−0.991179	+	0.132533i	$0.957689\pi$
$510$		0			0
$511$	9.74137	+	5.95064i	0.430933	+	0.263241i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−16.5400	−	9.54939i	−0.728840	−	0.420796i
$516$		0			0
$517$	14.7003	+	8.48722i	0.646518	+	0.373267i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8.70778	+	15.0823i	0.381495	+	0.660768i	0.991276	−	0.131802i	$-0.0420762\pi$
$521$	8.70778	+	15.0823i	0.381495	+	0.660768i	−0.609782	+	0.792570i	$0.708743\pi$
$522$		0			0
$523$	25.5361	+	14.7432i	1.11661	+	0.644677i	0.940534	−	0.339699i	$-0.110325\pi$
$523$	25.5361	+	14.7432i	1.11661	+	0.644677i	0.176079	+	0.984376i	$0.443658\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.388361	+	0.224220i	−0.0169173	+	0.00976719i
$528$		0			0
$529$	−11.0298	+	19.1042i	−0.479558	+	0.830619i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−9.72022	+	5.61197i	−0.421029	+	0.243081i
$534$		0			0
$535$		−	37.5171i		−	1.62201i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−29.8971	+	15.3397i	−1.28776	+	0.660728i
$540$		0			0
$541$	3.51522	−	6.08854i	0.151131	−	0.261767i	−0.780512	−	0.625140i	$-0.785042\pi$
$541$	3.51522	−	6.08854i	0.151131	−	0.261767i	0.931644	+	0.363373i	$0.118375\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.80561	+	3.12740i	−0.0773437	+	0.133963i
$546$		0			0
$547$	15.9847	+	27.6863i	0.683456	+	1.18378i	0.973919	+	0.226894i	$0.0728572\pi$
$547$	15.9847	+	27.6863i	0.683456	+	1.18378i	−0.290463	+	0.956886i	$0.593809\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14.4102			0.613894
$552$		0			0
$553$	15.0389	+	27.6053i	0.639519	+	1.17390i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.37676	+	3.10427i	−0.227821	+	0.131532i	−0.609566	−	0.792735i	$-0.708656\pi$
$557$	−5.37676	+	3.10427i	−0.227821	+	0.131532i	0.381746	+	0.924267i	$0.375323\pi$
$558$		0			0
$559$		−	7.32264i		−	0.309715i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	28.0894			1.18383			0.591913	−	0.806002i	$-0.298373\pi$
$563$	28.0894			1.18383			0.591913	+	0.806002i	$0.298373\pi$
$564$		0			0
$565$			24.5952i			1.03473i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	32.6798i		−	1.37001i	−0.728538	−	0.685005i	$-0.759800\pi$
$569$		−	32.6798i		−	1.37001i	0.728538	−	0.685005i	$-0.240200\pi$
$570$		0			0
$571$	−17.4237			−0.729157			−0.364579	−	0.931173i	$-0.618787\pi$
$571$	−17.4237			−0.729157			−0.364579	+	0.931173i	$0.618787\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			0.956098i			0.0398720i
$576$		0			0
$577$	29.9406	−	17.2862i	1.24645	−	0.719635i	0.276047	−	0.961144i	$-0.410976\pi$
$577$	29.9406	−	17.2862i	1.24645	−	0.719635i	0.970399	+	0.241509i	$0.0776423\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.48231	−	7.33767i	0.185958	−	0.304418i
$582$		0			0
$583$	−29.4805			−1.22096
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.01942	−	12.1580i	−0.289722	−	0.501814i	0.684021	−	0.729462i	$-0.260230\pi$
$587$	−7.01942	−	12.1580i	−0.289722	−	0.501814i	−0.973743	+	0.227648i	$0.926896\pi$
$588$		0			0
$589$	10.7124	−	18.5544i	0.441396	−	0.764520i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	22.5686	−	39.0900i	0.926782	−	1.60523i	0.138112	−	0.990417i	$-0.455897\pi$
$593$	22.5686	−	39.0900i	0.926782	−	1.60523i	0.788670	−	0.614817i	$-0.210770\pi$
$594$		0			0
$595$	0.472404	+	0.0117038i	0.0193667	+	0.000479809i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	10.9925i		−	0.449142i	−0.974458	−	0.224571i	$-0.927902\pi$
$599$		−	10.9925i		−	0.449142i	0.974458	−	0.224571i	$-0.0720981\pi$
$600$		0			0
$601$	31.1448	−	17.9815i	1.27042	−	0.733479i	0.295356	−	0.955387i	$-0.404562\pi$
$601$	31.1448	−	17.9815i	1.27042	−	0.733479i	0.975068	+	0.221908i	$0.0712285\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.0649	+	20.8970i	−0.490507	+	0.849584i
$606$		0			0
$607$	−21.8936	+	12.6403i	−0.888634	+	0.513053i	−0.873495	−	0.486832i	$-0.838152\pi$
$607$	−21.8936	+	12.6403i	−0.888634	+	0.513053i	−0.0151385	+	0.999885i	$0.504819\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.05691	+	1.76491i	0.123669	+	0.0714004i
$612$		0			0
$613$	1.24691	+	2.15971i	0.0503621	+	0.0872298i	0.890108	−	0.455751i	$-0.150629\pi$
$613$	1.24691	+	2.15971i	0.0503621	+	0.0872298i	−0.839745	+	0.542980i	$0.817296\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.91940	−	4.57227i	−0.318823	−	0.184072i	0.332045	−	0.943264i	$-0.392261\pi$
$617$	−7.91940	−	4.57227i	−0.318823	−	0.184072i	−0.650868	+	0.759191i	$0.725595\pi$
$618$		0			0
$619$	21.4791	+	12.4010i	0.863317	+	0.498437i	0.865122	−	0.501562i	$-0.167241\pi$
$619$	21.4791	+	12.4010i	0.863317	+	0.498437i	−0.00180436	+	0.999998i	$0.500574\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.62401	+	0.139335i	0.225321	+	0.00558233i
$624$		0			0
$625$	9.54902	+	16.5394i	0.381961	+	0.661576i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.0120463			0.000480317
$630$		0			0
$631$	8.66061			0.344773			0.172387	−	0.985029i	$-0.444852\pi$
$631$	8.66061			0.344773			0.172387	+	0.985029i	$0.444852\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.36834	−	12.7623i	−0.292404	−	0.506458i
$636$		0			0
$637$	−6.21707	+	3.18987i	−0.246329	+	0.126387i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19.9823	+	11.5368i	0.789253	+	0.455676i	0.839700	−	0.543051i	$-0.182731\pi$
$641$	19.9823	+	11.5368i	0.789253	+	0.455676i	−0.0504463	+	0.998727i	$0.516064\pi$
$642$		0			0
$643$	15.8108	+	9.12837i	0.623517	+	0.359988i	0.778237	−	0.627971i	$-0.216114\pi$
$643$	15.8108	+	9.12837i	0.623517	+	0.359988i	−0.154720	+	0.987958i	$0.549448\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.59782	−	2.76751i	−0.0628168	−	0.108802i	0.832907	−	0.553413i	$-0.186675\pi$
$647$	−1.59782	−	2.76751i	−0.0628168	−	0.108802i	−0.895723	+	0.444612i	$0.853342\pi$
$648$		0			0
$649$	40.5959	+	23.4380i	1.59353	+	0.920023i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−17.3561	+	10.0206i	−0.679197	+	0.392134i	−0.799552	−	0.600596i	$-0.794930\pi$
$653$	−17.3561	+	10.0206i	−0.679197	+	0.392134i	0.120356	+	0.992731i	$0.461597\pi$
$654$		0			0
$655$	−10.5180	+	18.2177i	−0.410973	+	0.711826i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−29.7095	+	17.1528i	−1.15732	+	0.668178i	−0.950660	−	0.310235i	$-0.899592\pi$
$659$	−29.7095	+	17.1528i	−1.15732	+	0.668178i	−0.206658	+	0.978413i	$0.566259\pi$
$660$		0			0
$661$		−	28.4499i		−	1.10657i	−0.832991	−	0.553286i	$-0.813374\pi$
$661$		−	28.4499i		−	1.10657i	0.832991	−	0.553286i	$-0.186626\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−19.8255	+	10.8006i	−0.768798	+	0.418828i
$666$		0			0
$667$	−1.64045	+	2.84134i	−0.0635183	+	0.110017i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	31.2838	−	54.1851i	1.20770	−	2.09179i
$672$		0			0
$673$	12.9705	+	22.4656i	0.499977	+	0.865986i	1.00000		2.64053e-5i	$-8.40506e-6\pi$
$673$	12.9705	+	22.4656i	0.499977	+	0.865986i	−0.500023	+	0.866012i	$0.666675\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−36.3788			−1.39815			−0.699075	−	0.715048i	$-0.746405\pi$
$677$	−36.3788			−1.39815			−0.699075	+	0.715048i	$0.746405\pi$
$678$		0			0
$679$	2.69906	+	1.64875i	0.103580	+	0.0632734i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.25354	−	0.723729i	0.0479652	−	0.0276927i	−0.475826	−	0.879540i	$-0.657851\pi$
$683$	1.25354	−	0.723729i	0.0479652	−	0.0276927i	0.523791	+	0.851847i	$0.324517\pi$
$684$		0			0
$685$			30.8273i			1.17785i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.13043			−0.233551
$690$		0			0
$691$		−	35.9161i		−	1.36631i	−0.730272	−	0.683157i	$-0.760607\pi$
$691$		−	35.9161i		−	1.36631i	0.730272	−	0.683157i	$-0.239393\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	29.2351i		−	1.10895i
$696$		0			0
$697$	1.00235			0.0379667
$698$		0			0
$699$		0			0
$700$		0			0
$701$			41.3851i			1.56309i	0.623847	+	0.781547i	$0.285569\pi$
$701$			41.3851i			1.56309i	−0.623847	+	0.781547i	$0.714431\pi$
$702$		0			0
$703$	−0.498419	+	0.287762i	−0.0187982	+	0.0108532i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−31.7073	−	0.785546i	−1.19247	−	0.0295435i
$708$		0			0
$709$	−18.4441			−0.692682			−0.346341	−	0.938109i	$-0.612576\pi$
$709$	−18.4441			−0.692682			−0.346341	+	0.938109i	$0.612576\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.43898	+	4.22444i	0.0913406	+	0.158207i
$714$		0			0
$715$	−4.80032	+	8.31440i	−0.179522	+	0.310941i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	0.923351	−	1.59929i	0.0344352	−	0.0596435i	−0.848294	−	0.529525i	$-0.822370\pi$
$719$	0.923351	−	1.59929i	0.0344352	−	0.0596435i	0.882729	+	0.469882i	$0.155703\pi$
$720$		0			0
$721$	−21.5231	−	13.1476i	−0.801561	−	0.489644i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	3.33590i		−	0.123892i
$726$		0			0
$727$	28.4243	−	16.4108i	1.05420	−	0.608642i	0.130378	−	0.991464i	$-0.458381\pi$
$727$	28.4243	−	16.4108i	1.05420	−	0.608642i	0.923822	+	0.382822i	$0.125048\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.326973	+	0.566334i	−0.0120935	+	0.0209466i
$732$		0			0
$733$	13.3583	−	7.71244i	0.493402	−	0.284866i	−0.232583	−	0.972577i	$-0.574718\pi$
$733$	13.3583	−	7.71244i	0.493402	−	0.284866i	0.725985	+	0.687711i	$0.241384\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−63.0026	−	36.3746i	−2.32073	−	1.33987i
$738$		0			0
$739$	−11.6142	−	20.1165i	−0.427237	−	0.739996i	0.569390	−	0.822068i	$-0.307180\pi$
$739$	−11.6142	−	20.1165i	−0.427237	−	0.739996i	−0.996626	+	0.0820719i	$0.973846\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	23.5757	+	13.6114i	0.864908	+	0.499355i	0.865653	−	0.500645i	$-0.166904\pi$
$743$	23.5757	+	13.6114i	0.864908	+	0.499355i	−0.000744979		1.00000i	$0.500237\pi$
$744$		0			0
$745$	−24.9612	−	14.4113i	−0.914507	−	0.527991i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.22707	−	49.5285i	0.0448360	−	1.80973i
$750$		0			0
$751$	−10.6549	−	18.4549i	−0.388804	−	0.673428i	0.603485	−	0.797374i	$-0.293778\pi$
$751$	−10.6549	−	18.4549i	−0.388804	−	0.673428i	−0.992289	+	0.123946i	$0.960445\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	31.0844			1.13128
$756$		0			0
$757$	43.2079			1.57042			0.785209	−	0.619231i	$-0.212556\pi$
$757$	43.2079			1.57042			0.785209	+	0.619231i	$0.212556\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−24.8852	−	43.1024i	−0.902087	−	1.56246i	−0.824780	−	0.565453i	$-0.808701\pi$
$761$	−24.8852	−	43.1024i	−0.902087	−	1.56246i	−0.0773067	−	0.997007i	$-0.524632\pi$
$762$		0			0
$763$	−2.48597	+	4.06960i	−0.0899981	+	0.147329i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.44185	+	4.87391i	0.304818	+	0.175987i
$768$		0			0
$769$	15.1354	+	8.73843i	0.545797	+	0.315116i	0.747425	−	0.664346i	$-0.231290\pi$
$769$	15.1354	+	8.73843i	0.545797	+	0.315116i	−0.201628	+	0.979462i	$0.564623\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	20.7020	+	35.8569i	0.744598	+	1.28968i	0.950382	+	0.311085i	$0.100692\pi$
$773$	20.7020	+	35.8569i	0.744598	+	1.28968i	−0.205784	+	0.978597i	$0.565974\pi$
$774$		0			0
$775$	−4.29526	−	2.47987i	−0.154290	−	0.0890796i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−41.4726	+	23.9442i	−1.48591	+	0.857891i
$780$		0			0
$781$	−10.9685	+	18.9981i	−0.392485	+	0.679804i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.05958	−	3.49850i	0.216276	−	0.124867i
$786$		0			0
$787$		−	25.2212i		−	0.899039i	−0.893271	−	0.449519i	$-0.851595\pi$
$787$		−	25.2212i		−	0.899039i	0.893271	−	0.449519i	$-0.148405\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−0.804431	+	32.4695i	−0.0286023	+	1.15448i
$792$		0			0
$793$	6.50543	−	11.2677i	0.231014	−	0.400129i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−14.8320	+	25.6898i	−0.525377	+	0.909980i	0.474186	+	0.880425i	$0.342743\pi$
$797$	−14.8320	+	25.6898i	−0.525377	+	0.909980i	−0.999563	+	0.0295555i	$0.990591\pi$
$798$		0			0
$799$	−0.157614	−	0.272996i	−0.00557600	−	0.00965791i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	20.7113			0.730886
$804$		0			0
$805$	0.127309	−	5.13863i	0.00448706	−	0.181113i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−5.88592	+	3.39824i	−0.206938	+	0.119476i	−0.599888	−	0.800084i	$-0.704788\pi$
$809$	−5.88592	+	3.39824i	−0.206938	+	0.119476i	0.392950	+	0.919560i	$0.371455\pi$
$810$		0			0
$811$			0.00414872i			0.000145681i	1.00000		7.28407e-5i	$2.31859e-5\pi$
$811$			0.00414872i			0.000145681i	−1.00000		7.28407e-5i	$0.999977\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	27.7252			0.971171
$816$		0			0
$817$		−	31.2430i		−	1.09305i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			28.7338i			1.00282i	0.865211	+	0.501409i	$0.167185\pi$
$821$			28.7338i			1.00282i	−0.865211	+	0.501409i	$0.832815\pi$
$822$		0			0
$823$	−50.6943			−1.76709			−0.883547	−	0.468343i	$-0.844851\pi$
$823$	−50.6943			−1.76709			−0.883547	+	0.468343i	$0.844851\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			3.19233i			0.111008i	0.998458	+	0.0555041i	$0.0176766\pi$
$827$			3.19233i			0.111008i	−0.998458	+	0.0555041i	$0.982323\pi$
$828$		0			0
$829$	−41.4645	+	23.9395i	−1.44012	+	0.831454i	−0.997857	−	0.0654311i	$-0.979158\pi$
$829$	−41.4645	+	23.9395i	−1.44012	+	0.831454i	−0.442264	+	0.896885i	$0.645824\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.623264	+	0.0309017i	0.0215948	+	0.00107068i
$834$		0			0
$835$	−7.82375			−0.270752
$836$		0			0
$837$		0			0
$838$		0			0
$839$	6.18287	+	10.7091i	0.213456	+	0.369717i	0.952794	−	0.303618i	$-0.0981945\pi$
$839$	6.18287	+	10.7091i	0.213456	+	0.369717i	−0.739337	+	0.673335i	$0.764861\pi$
$840$		0			0
$841$	−8.77636	+	15.2011i	−0.302633	+	0.524176i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.0246	−	20.8272i	0.413658	−	0.716476i
$846$		0			0
$847$	−16.6110	+	27.1927i	−0.570761	+	0.934352i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	0.131035i		−	0.00449182i
$852$		0			0
$853$	20.3474	−	11.7476i	0.696680	−	0.402229i	−0.109429	−	0.993995i	$-0.534902\pi$
$853$	20.3474	−	11.7476i	0.696680	−	0.402229i	0.806110	+	0.591766i	$0.201569\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	8.75960	−	15.1721i	0.299222	−	0.518268i	−0.676736	−	0.736226i	$-0.736606\pi$
$857$	8.75960	−	15.1721i	0.299222	−	0.518268i	0.975958	+	0.217958i	$0.0699395\pi$
$858$		0			0
$859$	11.9224	−	6.88338i	0.406786	−	0.234858i	−0.282622	−	0.959231i	$-0.591204\pi$
$859$	11.9224	−	6.88338i	0.406786	−	0.234858i	0.689408	+	0.724373i	$0.257871\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−22.0835	−	12.7499i	−0.751732	−	0.434013i	0.0745872	−	0.997214i	$-0.476236\pi$
$863$	−22.0835	−	12.7499i	−0.751732	−	0.434013i	−0.826319	+	0.563202i	$0.809569\pi$
$864$		0			0
$865$	17.9639	+	31.1144i	0.610791	+	1.05792i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	49.3953	+	28.5184i	1.67562	+	0.967420i
$870$		0			0
$871$	−13.1013	−	7.56404i	−0.443921	−	0.256298i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	15.1797	+	27.8637i	0.513167	+	0.941966i
$876$		0			0
$877$	25.9895	+	45.0152i	0.877604	+	1.52005i	0.853963	+	0.520334i	$0.174193\pi$
$877$	25.9895	+	45.0152i	0.877604	+	1.52005i	0.0236415	+	0.999721i	$0.492474\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	37.6723			1.26921			0.634607	−	0.772835i	$-0.281162\pi$
$881$	37.6723			1.26921			0.634607	+	0.772835i	$0.281162\pi$
$882$		0			0
$883$	−37.4259			−1.25948			−0.629741	−	0.776805i	$-0.716839\pi$
$883$	−37.4259			−1.25948			−0.629741	+	0.776805i	$0.716839\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	27.4541	+	47.5518i	0.921817	+	1.59663i	0.796602	+	0.604504i	$0.206629\pi$
$887$	27.4541	+	47.5518i	0.921817	+	1.59663i	0.125215	+	0.992130i	$0.460038\pi$
$888$		0			0
$889$	−9.30994	−	17.0893i	−0.312245	−	0.573156i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	13.0427	+	7.53020i	0.436457	+	0.251989i
$894$		0			0
$895$	−11.8915	−	6.86557i	−0.397490	−	0.229491i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−8.50979	−	14.7394i	−0.283817	−	0.491586i
$900$		0			0
$901$	0.474128	+	0.273738i	0.0157955	+	0.00911954i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−20.9965	+	12.1223i	−0.697946	+	0.402959i
$906$		0			0
$907$	4.32671	−	7.49408i	0.143666	−	0.248837i	−0.785208	−	0.619232i	$-0.787444\pi$
$907$	4.32671	−	7.49408i	0.143666	−	0.248837i	0.928874	+	0.370395i	$0.120778\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−9.63346	+	5.56188i	−0.319171	+	0.184273i	−0.651023	−	0.759058i	$-0.725660\pi$
$911$	−9.63346	+	5.56188i	−0.319171	+	0.184273i	0.331852	+	0.943331i	$0.392327\pi$
$912$		0			0
$913$		−	15.6007i		−	0.516309i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−14.4813	+	23.7062i	−0.478214	+	0.782849i
$918$		0			0
$919$	22.9971	−	39.8322i	0.758605	−	1.31394i	−0.184957	−	0.982747i	$-0.559215\pi$
$919$	22.9971	−	39.8322i	0.758605	−	1.31394i	0.943562	−	0.331196i	$-0.107452\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2.28089	+	3.95062i	−0.0750765	+	0.130036i
$924$		0			0
$925$	0.0666158	+	0.115382i	0.00219031	+	0.00379373i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	35.8070			1.17479			0.587394	−	0.809301i	$-0.300154\pi$
$929$	35.8070			1.17479			0.587394	+	0.809301i	$0.300154\pi$
$930$		0			0
$931$	−26.5260	+	13.6100i	−0.869353	+	0.446050i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.742515	−	0.428691i	0.0242828	−	0.0140197i
$936$		0			0
$937$		−	1.01234i		−	0.0330718i	−0.999863	−	0.0165359i	$-0.994736\pi$
$937$		−	1.01234i		−	0.0330718i	0.999863	−	0.0165359i	$-0.00526377\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	35.5905			1.16022			0.580109	−	0.814539i	$-0.303010\pi$
$941$	35.5905			1.16022			0.580109	+	0.814539i	$0.303010\pi$
$942$		0			0
$943$		−	10.9032i		−	0.355056i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	31.9268i		−	1.03748i	−0.854932	−	0.518741i	$-0.826401\pi$
$947$		−	31.9268i		−	1.03748i	0.854932	−	0.518741i	$-0.173599\pi$
$948$		0			0
$949$	4.30689			0.139808
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	21.6179i		−	0.700273i	−0.936699	−	0.350136i	$-0.886135\pi$
$953$		−	21.6179i		−	0.700273i	0.936699	−	0.350136i	$-0.113865\pi$
$954$		0			0
$955$	14.7429	−	8.51181i	0.477069	−	0.275436i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.00826	+	40.6969i	−0.0325585	+	1.31417i
$960$		0			0
$961$	5.69565			0.183731
$962$		0			0
$963$		0			0
$964$		0			0
$965$	13.6199	+	23.5903i	0.438439	+	0.759398i
$966$		0			0
$967$	7.79433	−	13.5002i	0.250649	−	0.434137i	−0.713056	−	0.701107i	$-0.752689\pi$
$967$	7.79433	−	13.5002i	0.250649	−	0.434137i	0.963705	+	0.266971i	$0.0860227\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5.94294	−	10.2935i	0.190718	−	0.330334i	−0.754770	−	0.655989i	$-0.772252\pi$
$971$	5.94294	−	10.2935i	0.190718	−	0.330334i	0.945488	+	0.325656i	$0.105585\pi$
$972$		0			0
$973$	0.956188	−	38.5949i	0.0306540	−	1.23730i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			43.4289i			1.38941i	0.719294	+	0.694706i	$0.244466\pi$
$977$			43.4289i			1.38941i	−0.719294	+	0.694706i	$0.755534\pi$
$978$		0			0
$979$	8.83971	−	5.10361i	0.282518	−	0.163112i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17.4704	+	30.2596i	−0.557218	+	0.965130i	0.440509	+	0.897748i	$0.354798\pi$
$983$	−17.4704	+	30.2596i	−0.557218	+	0.965130i	−0.997727	+	0.0673821i	$0.978535\pi$
$984$		0			0
$985$	−40.8725	+	23.5977i	−1.30231	+	0.751887i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.16036	+	3.55669i	0.195888	+	0.113096i
$990$		0			0
$991$	2.17783	+	3.77211i	0.0691811	+	0.119825i	0.898541	−	0.438889i	$-0.144628\pi$
$991$	2.17783	+	3.77211i	0.0691811	+	0.119825i	−0.829360	+	0.558715i	$0.811295\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−23.6004	−	13.6257i	−0.748183	−	0.431964i
$996$		0			0
$997$	43.7264	+	25.2454i	1.38483	+	0.799531i	0.992727	−	0.120391i	$-0.0384149\pi$
$997$	43.7264	+	25.2454i	1.38483	+	0.799531i	0.392102	+	0.919922i	$0.371748\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.w.j.1349.13		32
3.2	odd	2	inner	2268.2.w.j.1349.4		32
7.3	odd	6		2268.2.bm.j.1025.13		32
9.2	odd	6		2268.2.bm.j.593.13		32
9.4	even	3		2268.2.t.c.2105.13	yes	32
9.5	odd	6		2268.2.t.c.2105.4	yes	32
9.7	even	3		2268.2.bm.j.593.4		32
21.17	even	6		2268.2.bm.j.1025.4		32
63.31	odd	6		2268.2.t.c.1781.4	✓	32
63.38	even	6	inner	2268.2.w.j.269.13		32
63.52	odd	6	inner	2268.2.w.j.269.4		32
63.59	even	6		2268.2.t.c.1781.13	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.t.c.1781.4	✓	32	63.31	odd	6
2268.2.t.c.1781.13	yes	32	63.59	even	6
2268.2.t.c.2105.4	yes	32	9.5	odd	6
2268.2.t.c.2105.13	yes	32	9.4	even	3
2268.2.w.j.269.4		32	63.52	odd	6	inner
2268.2.w.j.269.13		32	63.38	even	6	inner
2268.2.w.j.1349.4		32	3.2	odd	2	inner
2268.2.w.j.1349.13		32	1.1	even	1	trivial
2268.2.bm.j.593.4		32	9.7	even	3
2268.2.bm.j.593.13		32	9.2	odd	6
2268.2.bm.j.1025.4		32	21.17	even	6
2268.2.bm.j.1025.13		32	7.3	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.00175 + 1.73509i) q^{5} +(1.26572 + 2.32335i) q^{7} +(4.15726 + 2.40020i) q^{11} +(0.864497 + 0.499118i) q^{13} +(-0.0445736 - 0.0772037i) q^{17} +(3.68849 + 2.12955i) q^{19} +(-0.839792 + 0.484854i) q^{23} +(0.492982 - 0.853871i) q^{25} +(2.93010 - 1.69169i) q^{29} -5.03034i q^{31} +(-2.76327 + 4.52356i) q^{35} +(-0.0675641 + 0.117024i) q^{37} +(-5.62189 + 9.73740i) q^{41} +(-3.66779 - 6.35280i) q^{43} +3.53605 q^{47} +(-3.79590 + 5.88142i) q^{49} +(-5.31849 + 3.07063i) q^{53} +9.61761i q^{55} +9.76504 q^{59} -13.0338i q^{61} +1.99997i q^{65} -15.1548 q^{67} +4.56985i q^{71} +(3.73647 - 2.15725i) q^{73} +(-0.314561 + 12.6968i) q^{77} +11.8817 q^{79} +(-1.62494 - 2.81449i) q^{83} +(0.0893034 - 0.154678i) q^{85} +(1.06316 - 1.84146i) q^{89} +(-0.0654127 + 2.64027i) q^{91} +8.53314i q^{95} +(1.03527 - 0.597713i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 4 q^{7} + 12 q^{13} - 16 q^{25} - 4 q^{37} - 4 q^{43} + 20 q^{49} - 8 q^{67} - 36 q^{73} - 56 q^{79} + 12 q^{85} - 36 q^{91} + 12 q^{97}+O(q^{100}) \) 32 * q + 4 * q^7 + 12 * q^13 - 16 * q^25 - 4 * q^37 - 4 * q^43 + 20 * q^49 - 8 * q^67 - 36 * q^73 - 56 * q^79 + 12 * q^85 - 36 * q^91 + 12 * q^97

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