LMFDB - Embedded newform 2268.2.w.j.269.4

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			269.4
Character	$\chi$	$=$	2268.269
Dual form			2268.2.w.j.1349.4

$q$-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 $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

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

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
$5$	−1.00175	+	1.73509i	−0.447998	+	0.775954i	−0.998256	−	0.0590402i	$-0.981196\pi$
$5$	−1.00175	+	1.73509i	−0.447998	+	0.775954i	0.550258	+	0.834995i	$0.314529\pi$
$6$		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	$-0.924220\pi$
$11$	−4.15726	+	2.40020i	−1.25346	+	0.723686i	−0.281666	+	0.959512i	$0.590887\pi$
$12$		0			0
$13$	0.864497	−	0.499118i	0.239768	−	0.138430i	−0.375302	−	0.926903i	$-0.622461\pi$
$13$	0.864497	−	0.499118i	0.239768	−	0.138430i	0.615070	+	0.788472i	$0.289128\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.0445736	−	0.0772037i	0.0108107	−	0.0187246i	−0.860569	−	0.509333i	$-0.829892\pi$
$17$	0.0445736	−	0.0772037i	0.0108107	−	0.0187246i	0.871380	+	0.490608i	$0.163225\pi$
$18$		0			0
$19$	3.68849	−	2.12955i	0.846198	−	0.488553i	−0.0131681	−	0.999913i	$-0.504192\pi$
$19$	3.68849	−	2.12955i	0.846198	−	0.488553i	0.859366	+	0.511361i	$0.170858\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.839792	+	0.484854i	0.175109	+	0.101099i	0.584993	−	0.811039i	$-0.301097\pi$
$23$	0.839792	+	0.484854i	0.175109	+	0.101099i	−0.409884	+	0.912138i	$0.634431\pi$
$24$		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	$-0.435049\pi$
$29$	−2.93010	−	1.69169i	−0.544105	−	0.314139i	−0.746741	+	0.665115i	$0.768383\pi$
$30$		0			0
$31$			5.03034i			0.903476i	0.892151	+	0.451738i	$0.149196\pi$
$31$			5.03034i			0.903476i	−0.892151	+	0.451738i	$0.850804\pi$
$32$		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	0.860418	−	0.509589i	$-0.170202\pi$
$37$	−0.0675641	−	0.117024i	−0.0111075	−	0.0192387i	−0.871526	+	0.490350i	$0.836869\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.62189	+	9.73740i	0.877992	+	1.52073i	0.853541	+	0.521025i	$0.174450\pi$
$41$	5.62189	+	9.73740i	0.877992	+	1.52073i	0.0244505	+	0.999701i	$0.492216\pi$
$42$		0			0
$43$	−3.66779	+	6.35280i	−0.559333	+	0.968793i	0.438219	+	0.898868i	$0.355609\pi$
$43$	−3.66779	+	6.35280i	−0.559333	+	0.968793i	−0.997552	+	0.0699250i	$0.977724\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.53605			−0.515786			−0.257893	−	0.966173i	$-0.583028\pi$
$47$	−3.53605			−0.515786			−0.257893	+	0.966173i	$0.583028\pi$
$48$		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	$-0.194737\pi$
$53$	5.31849	+	3.07063i	0.730551	+	0.421784i	−0.0880726	+	0.996114i	$0.528071\pi$
$54$		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	$-0.719268\pi$
$59$	−9.76504			−1.27130			−0.635650	+	0.771977i	$0.719268\pi$
$60$		0			0
$61$			13.0338i			1.66881i	0.551149	+	0.834407i	$0.314190\pi$
$61$			13.0338i			1.66881i	−0.551149	+	0.834407i	$0.685810\pi$
$62$		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	0.702460	−	0.711723i	$-0.252085\pi$
$73$	3.73647	+	2.15725i	0.437321	+	0.252487i	−0.265140	+	0.964210i	$0.585418\pi$
$74$		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	$-0.776254\pi$
$83$	1.62494	−	2.81449i	0.178361	−	0.308930i	0.941319	+	0.337518i	$0.109587\pi$
$84$		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	$-0.202615\pi$
$89$	−1.06316	−	1.84146i	−0.112695	−	0.195194i	−0.916856	+	0.399218i	$0.869282\pi$
$90$		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	0.551636	−	0.834085i	$-0.314004\pi$
$97$	1.03527	+	0.597713i	0.105116	+	0.0606885i	−0.446521	+	0.894773i	$0.647337\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.99395	+	10.3818i	0.596420	+	1.03303i	0.993345	+	0.115179i	$0.0367440\pi$
$101$	5.99395	+	10.3818i	0.596420	+	1.03303i	−0.396925	+	0.917851i	$0.629923\pi$
$102$		0			0
$103$	−8.25554	−	4.76634i	−0.813442	−	0.469641i	0.0347074	−	0.999398i	$-0.488950\pi$
$103$	−8.25554	−	4.76634i	−0.813442	−	0.469641i	−0.848150	+	0.529756i	$0.822283\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	16.2170	−	9.36287i	1.56775	−	0.905143i	0.571323	−	0.820725i	$-0.306430\pi$
$107$	16.2170	−	9.36287i	1.56775	−	0.905143i	0.996430	−	0.0844177i	$-0.0269030\pi$
$108$		0			0
$109$	0.901223	−	1.56096i	0.0863215	−	0.149513i	−0.819632	−	0.572890i	$-0.805822\pi$
$109$	0.901223	−	1.56096i	0.0863215	−	0.149513i	0.905954	+	0.423377i	$0.139155\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.6314	+	6.13804i	−1.00012	+	0.577419i	−0.908283	−	0.418356i	$-0.862607\pi$
$113$	−10.6314	+	6.13804i	−1.00012	+	0.577419i	−0.0918351	+	0.995774i	$0.529273\pi$
$114$		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	$-0.985010\pi$
$131$	−5.24981	+	9.09293i	−0.458678	+	0.794453i	0.540214	+	0.841528i	$0.318343\pi$
$132$		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	$-0.894963\pi$
$137$	−13.3253	+	7.69335i	−1.13845	+	0.657287i	−0.192407	+	0.981315i	$0.561629\pi$
$138$		0			0
$139$	−12.6370	+	7.29600i	−1.07186	+	0.618838i	−0.928689	−	0.370860i	$-0.879063\pi$
$139$	−12.6370	+	7.29600i	−1.07186	+	0.618838i	−0.143171	+	0.989698i	$0.545730\pi$
$140$		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	$-0.132745\pi$
$149$	12.4588	+	7.19307i	1.02066	+	0.589279i	0.106366	+	0.994327i	$0.466079\pi$
$150$		0			0
$151$	7.75751	+	13.4364i	0.631297	+	1.09344i	0.987287	+	0.158949i	$0.0508104\pi$
$151$	7.75751	+	13.4364i	0.631297	+	1.09344i	−0.355990	+	0.934490i	$0.615856\pi$
$152$		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	$0.0445049\pi$
$157$			3.49238i			0.278722i	−0.990242	+	0.139361i	$0.955495\pi$
$158$		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	0.998792	+	0.0491384i	$0.0156475\pi$
$163$	6.91917	+	11.9844i	0.541951	+	0.938687i	−0.456841	+	0.889548i	$0.651019\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.95252	+	3.38186i	0.151090	+	0.261696i	0.931629	−	0.363412i	$-0.118388\pi$
$167$	1.95252	+	3.38186i	0.151090	+	0.261696i	−0.780538	+	0.625108i	$0.785055\pi$
$168$		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	$-0.738755\pi$
$173$	−17.9325			−1.36338			−0.681690	+	0.731641i	$0.738755\pi$
$174$		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	$-0.250886\pi$
$179$	5.93535	+	3.42678i	0.443629	+	0.256129i	−0.261507	+	0.965202i	$0.584219\pi$
$180$		0			0
$181$		−	12.1011i		−	0.899468i	−0.893163	−	0.449734i	$-0.851519\pi$
$181$		−	12.1011i		−	0.899468i	0.893163	−	0.449734i	$-0.148481\pi$
$182$		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	0.0205416	−	0.999789i	$-0.493461\pi$
$199$	−11.7796	−	6.80093i	−0.835030	−	0.482105i	−0.855572	+	0.517684i	$0.826794\pi$
$200$		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	0.995345	+	0.0963750i	$0.0307248\pi$
$211$	8.44148	+	14.6211i	0.581136	+	1.00656i	−0.414209	+	0.910182i	$0.635942\pi$
$212$		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	0.366101	−	0.930575i	$-0.380692\pi$
$223$	−3.83410	−	2.21362i	−0.256750	−	0.148235i	−0.622851	+	0.782340i	$0.714026\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.69552	−	11.5970i	−0.444397	−	0.769718i	0.553613	−	0.832774i	$-0.313249\pi$
$227$	−6.69552	−	11.5970i	−0.444397	−	0.769718i	−0.998010	+	0.0630557i	$0.979915\pi$
$228$		0			0
$229$	−6.62507	−	3.82498i	−0.437797	−	0.252762i	0.264866	−	0.964285i	$-0.414672\pi$
$229$	−6.62507	−	3.82498i	−0.437797	−	0.252762i	−0.702663	+	0.711523i	$0.748006\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.1646	−	7.02323i	0.796929	−	0.460107i	−0.0454674	−	0.998966i	$-0.514478\pi$
$233$	12.1646	−	7.02323i	0.796929	−	0.460107i	0.842396	+	0.538859i	$0.181144\pi$
$234$		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	$-0.484469\pi$
$239$	14.5035	−	8.37362i	0.938156	−	0.541644i	0.889382	+	0.457165i	$0.151135\pi$
$240$		0			0
$241$	26.5411	−	15.3235i	1.70967	−	0.987076i	0.774705	−	0.632323i	$-0.217898\pi$
$241$	26.5411	−	15.3235i	1.70967	−	0.987076i	0.934960	−	0.354753i	$-0.115435\pi$
$242$		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	$-0.546007\pi$
$251$	−4.56378			−0.288063			−0.144032	+	0.989573i	$0.546007\pi$
$252$		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	−0.0583759	−	0.998295i	$-0.518592\pi$
$257$	13.3918	−	23.1954i	0.835361	−	1.44689i	0.893736	−	0.448592i	$-0.148074\pi$
$258$		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	$-0.443579\pi$
$263$	18.1138	−	10.4580i	1.11695	−	0.644869i	0.940619	+	0.339463i	$0.110245\pi$
$264$		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	0.0992351	+	0.995064i	$0.468360\pi$
$269$	−13.3200	+	23.0709i	−0.812133	+	1.40666i	−0.911368	+	0.411592i	$0.864973\pi$
$270$		0			0
$271$	17.6224	−	10.1743i	1.07048	−	0.618045i	0.142171	−	0.989842i	$-0.454592\pi$
$271$	17.6224	−	10.1743i	1.07048	−	0.618045i	0.928314	+	0.371798i	$0.121258\pi$
$272$		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	0.833717	−	0.552191i	$-0.186208\pi$
$277$	−1.02112	−	1.76863i	−0.0613531	−	0.106267i	−0.895070	+	0.445925i	$0.852875\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.6608	+	8.46441i	0.874590	+	0.504945i	0.868871	−	0.495039i	$-0.164846\pi$
$281$	14.6608	+	8.46441i	0.874590	+	0.504945i	0.00571889	+	0.999984i	$0.498180\pi$
$282$		0			0
$283$		−	11.5134i		−	0.684400i	−0.939627	−	0.342200i	$-0.888828\pi$
$283$		−	11.5134i		−	0.684400i	0.939627	−	0.342200i	$-0.111172\pi$
$284$		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	−0.991586	−	0.129447i	$-0.958680\pi$
$293$	−10.4055	−	18.0229i	−0.607898	−	1.05291i	0.383689	−	0.923463i	$-0.374654\pi$
$294$		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	$0.0182039\pi$
$307$			2.00298i			0.114316i	−0.998365	+	0.0571580i	$0.981796\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.70747			−0.493756			−0.246878	−	0.969047i	$-0.579405\pi$
$311$	−8.70747			−0.493756			−0.246878	+	0.969047i	$0.579405\pi$
$312$		0			0
$313$			11.0181i			0.622782i	0.950282	+	0.311391i	$0.100795\pi$
$313$			11.0181i			0.622782i	−0.950282	+	0.311391i	$0.899205\pi$
$314$		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	−0.979614	−	0.200890i	$-0.935617\pi$
$337$	−12.1854	−	21.1058i	−0.663783	−	1.14971i	0.315831	−	0.948815i	$-0.397717\pi$
$338$		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	0.217824	−	0.975988i	$-0.430104\pi$
$349$	−9.68628	−	5.59238i	−0.518495	−	0.299353i	−0.736319	+	0.676635i	$0.763438\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.75651	+	11.7026i	0.359613	+	0.622867i	0.987896	−	0.155117i	$-0.0495756\pi$
$353$	6.75651	+	11.7026i	0.359613	+	0.622867i	−0.628284	+	0.777984i	$0.716242\pi$
$354$		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	0.563297	−	0.826254i	$-0.309533\pi$
$359$	29.5673	−	17.0707i	1.56050	−	0.900957i	0.997206	−	0.0747023i	$-0.0238007\pi$
$360$		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	−0.475730	−	0.879591i	$-0.657816\pi$
$367$	0.922490	−	0.532600i	0.0481536	−	0.0278015i	0.523884	+	0.851790i	$0.324483\pi$
$368$		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	−0.804469	−	0.593994i	$-0.797550\pi$
$373$	2.16655	−	3.75257i	0.112180	−	0.194301i	0.916649	+	0.399694i	$0.130883\pi$
$374$		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	0.231819	−	0.972759i	$-0.425532\pi$
$383$	18.7552	−	32.4849i	0.958344	−	1.65990i	0.726524	−	0.687141i	$-0.241135\pi$
$384$		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	$-0.610747\pi$
$389$	5.97057	−	3.44711i	0.302720	−	0.174775i	0.643664	+	0.765308i	$0.277413\pi$
$390$		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	−0.745172	+	0.666872i	$0.767633\pi$
$397$	−33.7783	+	19.5019i	−1.69529	+	0.978774i	−0.950114	+	0.311902i	$0.899034\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	16.1101	+	9.30115i	0.804498	+	0.464477i	0.845042	−	0.534700i	$-0.179576\pi$
$401$	16.1101	+	9.30115i	0.804498	+	0.464477i	−0.0405433	+	0.999178i	$0.512909\pi$
$402$		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	$0.0171948\pi$
$409$			2.18387i			0.107985i	−0.998541	+	0.0539927i	$0.982805\pi$
$410$		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	−0.957078	−	0.289831i	$-0.906401\pi$
$419$	−14.9333	−	25.8653i	−0.729540	−	1.26360i	0.227538	−	0.973769i	$-0.426933\pi$
$420$		0			0
$421$	0.673150	−	1.16593i	0.0328073	−	0.0568240i	−0.849156	−	0.528143i	$-0.822889\pi$
$421$	0.673150	−	1.16593i	0.0328073	−	0.0568240i	0.881963	+	0.471319i	$0.156222\pi$
$422$		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	$-0.554077\pi$
$431$	−22.9854	−	13.2706i	−1.10717	−	0.639223i	−0.938094	+	0.346381i	$0.887410\pi$
$432$		0			0
$433$			27.4682i			1.32004i	0.751250	+	0.660018i	$0.229451\pi$
$433$			27.4682i			1.32004i	−0.751250	+	0.660018i	$0.770549\pi$
$434$		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	$0.0306806\pi$
$439$			4.03277i			0.192473i	−0.995358	+	0.0962367i	$0.969319\pi$
$440$		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	0.184483	+	0.982836i	$0.440939\pi$
$461$	−16.2947	+	28.2232i	−0.758919	+	1.31449i	−0.943402	+	0.331650i	$0.892395\pi$
$462$		0			0
$463$	10.6397	+	18.4286i	0.494471	+	0.856448i	0.999980	−	0.00637294i	$-0.00202858\pi$
$463$	10.6397	+	18.4286i	0.494471	+	0.856448i	−0.505509	+	0.862821i	$0.668695\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−19.0189	−	32.9417i	−0.880090	−	1.52436i	−0.851240	−	0.524777i	$-0.824149\pi$
$467$	−19.0189	−	32.9417i	−0.880090	−	1.52436i	−0.0288500	−	0.999584i	$-0.509185\pi$
$468$		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	$-0.0892873\pi$
$479$	5.26814	+	9.12468i	0.240707	+	0.416917i	−0.720209	+	0.693758i	$0.755954\pi$
$480$		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	−0.955755	−	0.294163i	$-0.904959\pi$
$487$	−4.92393	+	8.52850i	−0.223125	+	0.386463i	0.732631	+	0.680627i	$0.238292\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.74760	−	3.89573i	0.304515	−	0.175812i	−0.339954	−	0.940442i	$-0.610412\pi$
$491$	6.74760	−	3.89573i	0.304515	−	0.175812i	0.644469	+	0.764630i	$0.277078\pi$
$492$		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	−0.466787	−	0.884370i	$-0.654589\pi$
$499$	11.8950	−	20.6027i	0.532493	−	0.922305i	0.999280	−	0.0379356i	$-0.0120782\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−15.1692			−0.676363			−0.338182	−	0.941081i	$-0.609812\pi$
$503$	−15.1692			−0.676363			−0.338182	+	0.941081i	$0.609812\pi$
$504$		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	$-0.708978\pi$
$509$	8.59151	−	14.8809i	0.380812	−	0.659586i	0.991179	+	0.132533i	$0.0423112\pi$
$510$		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	$-0.957924\pi$
$521$	−8.70778	+	15.0823i	−0.381495	+	0.660768i	0.609782	+	0.792570i	$0.291257\pi$
$522$		0			0
$523$	25.5361	−	14.7432i	1.11661	−	0.644677i	0.176079	−	0.984376i	$-0.443658\pi$
$523$	25.5361	−	14.7432i	1.11661	−	0.644677i	0.940534	+	0.339699i	$0.110325\pi$
$524$		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	0.931644	−	0.363373i	$-0.118375\pi$
$541$	3.51522	+	6.08854i	0.151131	+	0.261767i	−0.780512	+	0.625140i	$0.785042\pi$
$542$		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	−0.290463	−	0.956886i	$-0.593809\pi$
$547$	15.9847	−	27.6863i	0.683456	−	1.18378i	0.973919	−	0.226894i	$-0.0728572\pi$
$548$		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	$-0.291344\pi$
$557$	5.37676	+	3.10427i	0.227821	+	0.131532i	−0.381746	+	0.924267i	$0.624677\pi$
$558$		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	$-0.701627\pi$
$563$	−28.0894			−1.18383			−0.591913	+	0.806002i	$0.701627\pi$
$564$		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	0.970399	−	0.241509i	$-0.0776423\pi$
$577$	29.9406	+	17.2862i	1.24645	+	0.719635i	0.276047	+	0.961144i	$0.410976\pi$
$578$		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	$-0.739770\pi$
$587$	7.01942	−	12.1580i	0.289722	−	0.501814i	0.973743	+	0.227648i	$0.0731037\pi$
$588$		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	−0.788670	−	0.614817i	$-0.789230\pi$
$593$	−22.5686	−	39.0900i	−0.926782	−	1.60523i	−0.138112	−	0.990417i	$-0.544103\pi$
$594$		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	0.975068	−	0.221908i	$-0.0712285\pi$
$601$	31.1448	+	17.9815i	1.27042	+	0.733479i	0.295356	+	0.955387i	$0.404562\pi$
$602$		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	−0.0151385	−	0.999885i	$-0.504819\pi$
$607$	−21.8936	−	12.6403i	−0.888634	−	0.513053i	−0.873495	+	0.486832i	$0.838152\pi$
$608$		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	−0.839745	−	0.542980i	$-0.817296\pi$
$613$	1.24691	−	2.15971i	0.0503621	−	0.0872298i	0.890108	+	0.455751i	$0.150629\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.91940	−	4.57227i	0.318823	−	0.184072i	−0.332045	−	0.943264i	$-0.607739\pi$
$617$	7.91940	−	4.57227i	0.318823	−	0.184072i	0.650868	+	0.759191i	$0.274405\pi$
$618$		0			0
$619$	21.4791	−	12.4010i	0.863317	−	0.498437i	−0.00180436	−	0.999998i	$-0.500574\pi$
$619$	21.4791	−	12.4010i	0.863317	−	0.498437i	0.865122	+	0.501562i	$0.167241\pi$
$620$		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	$-0.817269\pi$
$641$	−19.9823	+	11.5368i	−0.789253	+	0.455676i	0.0504463	+	0.998727i	$0.483936\pi$
$642$		0			0
$643$	15.8108	−	9.12837i	0.623517	−	0.359988i	−0.154720	−	0.987958i	$-0.549448\pi$
$643$	15.8108	−	9.12837i	0.623517	−	0.359988i	0.778237	+	0.627971i	$0.216114\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.59782	−	2.76751i	0.0628168	−	0.108802i	−0.832907	−	0.553413i	$-0.813325\pi$
$647$	1.59782	−	2.76751i	0.0628168	−	0.108802i	0.895723	+	0.444612i	$0.146658\pi$
$648$		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	$-0.205070\pi$
$653$	17.3561	+	10.0206i	0.679197	+	0.392134i	−0.120356	+	0.992731i	$0.538403\pi$
$654$		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	$-0.100408\pi$
$659$	29.7095	+	17.1528i	1.15732	+	0.668178i	0.206658	+	0.978413i	$0.433741\pi$
$660$		0			0
$661$			28.4499i			1.10657i	0.832991	+	0.553286i	$0.186626\pi$
$661$			28.4499i			1.10657i	−0.832991	+	0.553286i	$0.813374\pi$
$662$		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	−0.500023	−	0.866012i	$-0.666675\pi$
$673$	12.9705	−	22.4656i	0.499977	−	0.865986i	1.00000		2.64053e-5i	$8.40506e-6\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	36.3788			1.39815			0.699075	−	0.715048i	$-0.253595\pi$
$677$	36.3788			1.39815			0.699075	+	0.715048i	$0.253595\pi$
$678$		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	$-0.342149\pi$
$683$	−1.25354	−	0.723729i	−0.0479652	−	0.0276927i	−0.523791	+	0.851847i	$0.675483\pi$
$684$		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	$0.239393\pi$
$691$			35.9161i			1.36631i	−0.730272	+	0.683157i	$0.760607\pi$
$692$		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	$-0.177630\pi$
$719$	−0.923351	−	1.59929i	−0.0344352	−	0.0596435i	−0.882729	+	0.469882i	$0.844297\pi$
$720$		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	0.923822	−	0.382822i	$-0.125048\pi$
$727$	28.4243	+	16.4108i	1.05420	+	0.608642i	0.130378	+	0.991464i	$0.458381\pi$
$728$		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	0.725985	−	0.687711i	$-0.241384\pi$
$733$	13.3583	+	7.71244i	0.493402	+	0.284866i	−0.232583	+	0.972577i	$0.574718\pi$
$734$		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	−0.996626	−	0.0820719i	$-0.973846\pi$
$739$	−11.6142	+	20.1165i	−0.427237	+	0.739996i	0.569390	+	0.822068i	$0.307180\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−23.5757	+	13.6114i	−0.864908	+	0.499355i	−0.865653	−	0.500645i	$-0.833096\pi$
$743$	−23.5757	+	13.6114i	−0.864908	+	0.499355i	0.000744979		1.00000i	$0.499763\pi$
$744$		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	−0.992289	−	0.123946i	$-0.960445\pi$
$751$	−10.6549	+	18.4549i	−0.388804	+	0.673428i	0.603485	+	0.797374i	$0.293778\pi$
$752$		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	0.0773067	−	0.997007i	$-0.475368\pi$
$761$	24.8852	−	43.1024i	0.902087	−	1.56246i	0.824780	−	0.565453i	$-0.191299\pi$
$762$		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	−0.201628	−	0.979462i	$-0.564623\pi$
$769$	15.1354	−	8.73843i	0.545797	−	0.315116i	0.747425	+	0.664346i	$0.231290\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−20.7020	+	35.8569i	−0.744598	+	1.28968i	0.205784	+	0.978597i	$0.434026\pi$
$773$	−20.7020	+	35.8569i	−0.744598	+	1.28968i	−0.950382	+	0.311085i	$0.899308\pi$
$774$		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	$0.148405\pi$
$787$			25.2212i			0.899039i	−0.893271	+	0.449519i	$0.851595\pi$
$788$		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	0.999563	+	0.0295555i	$0.00940919\pi$
$797$	14.8320	+	25.6898i	0.525377	+	0.909980i	−0.474186	+	0.880425i	$0.657257\pi$
$798$		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	$-0.295212\pi$
$809$	5.88592	+	3.39824i	0.206938	+	0.119476i	−0.392950	+	0.919560i	$0.628545\pi$
$810$		0			0
$811$		−	0.00414872i		−	0.000145681i	−1.00000		7.28407e-5i	$-0.999977\pi$
$811$		−	0.00414872i		−	0.000145681i	1.00000		7.28407e-5i	$-2.31859e-5\pi$
$812$		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	−0.442264	−	0.896885i	$-0.645824\pi$
$829$	−41.4645	−	23.9395i	−1.44012	−	0.831454i	−0.997857	+	0.0654311i	$0.979158\pi$
$830$		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	$-0.901805\pi$
$839$	−6.18287	+	10.7091i	−0.213456	+	0.369717i	0.739337	+	0.673335i	$0.235139\pi$
$840$		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	0.806110	−	0.591766i	$-0.201569\pi$
$853$	20.3474	+	11.7476i	0.696680	+	0.402229i	−0.109429	+	0.993995i	$0.534902\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.75960	−	15.1721i	−0.299222	−	0.518268i	0.676736	−	0.736226i	$-0.263394\pi$
$857$	−8.75960	−	15.1721i	−0.299222	−	0.518268i	−0.975958	+	0.217958i	$0.930060\pi$
$858$		0			0
$859$	11.9224	+	6.88338i	0.406786	+	0.234858i	0.689408	−	0.724373i	$-0.257871\pi$
$859$	11.9224	+	6.88338i	0.406786	+	0.234858i	−0.282622	+	0.959231i	$0.591204\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	22.0835	−	12.7499i	0.751732	−	0.434013i	−0.0745872	−	0.997214i	$-0.523764\pi$
$863$	22.0835	−	12.7499i	0.751732	−	0.434013i	0.826319	+	0.563202i	$0.190431\pi$
$864$		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	0.0236415	−	0.999721i	$-0.492474\pi$
$877$	25.9895	−	45.0152i	0.877604	−	1.52005i	0.853963	−	0.520334i	$-0.174193\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−37.6723			−1.26921			−0.634607	−	0.772835i	$-0.718838\pi$
$881$	−37.6723			−1.26921			−0.634607	+	0.772835i	$0.718838\pi$
$882$		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	−0.125215	+	0.992130i	$0.539962\pi$
$887$	−27.4541	+	47.5518i	−0.921817	+	1.59663i	−0.796602	+	0.604504i	$0.793371\pi$
$888$		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	0.928874	−	0.370395i	$-0.120778\pi$
$907$	4.32671	+	7.49408i	0.143666	+	0.248837i	−0.785208	+	0.619232i	$0.787444\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.63346	+	5.56188i	0.319171	+	0.184273i	0.651023	−	0.759058i	$-0.274340\pi$
$911$	9.63346	+	5.56188i	0.319171	+	0.184273i	−0.331852	+	0.943331i	$0.607673\pi$
$912$		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	0.943562	+	0.331196i	$0.107452\pi$
$919$	22.9971	+	39.8322i	0.758605	+	1.31394i	−0.184957	+	0.982747i	$0.559215\pi$
$920$		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	$-0.699846\pi$
$929$	−35.8070			−1.17479			−0.587394	+	0.809301i	$0.699846\pi$
$930$		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	$0.00526377\pi$
$937$			1.01234i			0.0330718i	−0.999863	+	0.0165359i	$0.994736\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−35.5905			−1.16022			−0.580109	−	0.814539i	$-0.696990\pi$
$941$	−35.5905			−1.16022			−0.580109	+	0.814539i	$0.696990\pi$
$942$		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	0.963705	−	0.266971i	$-0.0860227\pi$
$967$	7.79433	+	13.5002i	0.250649	+	0.434137i	−0.713056	+	0.701107i	$0.752689\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−5.94294	−	10.2935i	−0.190718	−	0.330334i	0.754770	−	0.655989i	$-0.227748\pi$
$971$	−5.94294	−	10.2935i	−0.190718	−	0.330334i	−0.945488	+	0.325656i	$0.894415\pi$
$972$		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	0.997727	+	0.0673821i	$0.0214646\pi$
$983$	17.4704	+	30.2596i	0.557218	+	0.965130i	−0.440509	+	0.897748i	$0.645202\pi$
$984$		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	−0.829360	−	0.558715i	$-0.811295\pi$
$991$	2.17783	−	3.77211i	0.0691811	−	0.119825i	0.898541	+	0.438889i	$0.144628\pi$
$992$		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	0.392102	−	0.919922i	$-0.371748\pi$
$997$	43.7264	−	25.2454i	1.38483	−	0.799531i	0.992727	+	0.120391i	$0.0384149\pi$
$998$		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.269.4		32
3.2	odd	2	inner	2268.2.w.j.269.13		32
7.5	odd	6		2268.2.bm.j.593.4		32
9.2	odd	6		2268.2.t.c.1781.13	yes	32
9.4	even	3		2268.2.bm.j.1025.13		32
9.5	odd	6		2268.2.bm.j.1025.4		32
9.7	even	3		2268.2.t.c.1781.4	✓	32
21.5	even	6		2268.2.bm.j.593.13		32
63.5	even	6	inner	2268.2.w.j.1349.4		32
63.40	odd	6	inner	2268.2.w.j.1349.13		32
63.47	even	6		2268.2.t.c.2105.4	yes	32
63.61	odd	6		2268.2.t.c.2105.13	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.t.c.1781.4	✓	32	9.7	even	3
2268.2.t.c.1781.13	yes	32	9.2	odd	6
2268.2.t.c.2105.4	yes	32	63.47	even	6
2268.2.t.c.2105.13	yes	32	63.61	odd	6
2268.2.w.j.269.4		32	1.1	even	1	trivial
2268.2.w.j.269.13		32	3.2	odd	2	inner
2268.2.w.j.1349.4		32	63.5	even	6	inner
2268.2.w.j.1349.13		32	63.40	odd	6	inner
2268.2.bm.j.593.4		32	7.5	odd	6
2268.2.bm.j.593.13		32	21.5	even	6
2268.2.bm.j.1025.4		32	9.5	odd	6
2268.2.bm.j.1025.13		32	9.4	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

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