LMFDB - Embedded newform 2268.2.bm.j.1025.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(593,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1, 5])) N = Newforms(chi, 2, names="a")

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

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

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,4,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1025.4
Character	$\chi$	$=$	2268.1025
Dual form			2268.2.bm.j.593.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00351 q^{5} +(1.37922 + 2.25782i) q^{7} +4.80039i q^{11} +(-0.864497 - 0.499118i) q^{13} +(-0.0445736 + 0.0772037i) q^{17} +(3.68849 - 2.12955i) q^{19} +0.969708i q^{23} -0.985965 q^{25} +(-2.93010 + 1.69169i) q^{29} +(4.35640 - 2.51517i) q^{31} +(-2.76327 - 4.52356i) q^{35} +(-0.0675641 - 0.117024i) q^{37} +(-5.62189 + 9.73740i) q^{41} +(-3.66779 - 6.35280i) q^{43} +(-1.76803 + 3.06231i) q^{47} +(-3.19551 + 6.22806i) q^{49} +(-5.31849 - 3.07063i) q^{53} -9.61761i q^{55} +(-4.88252 - 8.45678i) q^{59} +(-11.2876 - 6.51692i) q^{61} +(1.73203 + 0.999985i) q^{65} +(7.57741 + 13.1245i) q^{67} -4.56985i q^{71} +(3.73647 + 2.15725i) q^{73} +(-10.8384 + 6.62079i) q^{77} +(-5.94084 + 10.2898i) q^{79} +(-1.62494 - 2.81449i) q^{83} +(0.0893034 - 0.154678i) q^{85} +(1.06316 + 1.84146i) q^{89} +(-0.0654127 - 2.64027i) q^{91} +(-7.38992 + 4.26657i) q^{95} +(-1.03527 + 0.597713i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 4 q^{7} - 12 q^{13} + 32 q^{25} - 24 q^{31} - 4 q^{37} - 4 q^{43} - 16 q^{49} - 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} + 12 q^{85} - 36 q^{91} - 12 q^{97}+O(q^{100}) $ 32 * q + 4 * q^7 - 12 * q^13 + 32 * q^25 - 24 * q^31 - 4 * q^37 - 4 * q^43 - 16 * q^49 - 12 * q^61 + 4 * q^67 - 36 * q^73 + 28 * q^79 + 12 * q^85 - 36 * q^91 - 12 * q^97

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.00351			−0.895995			−0.447998	−	0.894035i	$-0.647863\pi$
$5$	−2.00351			−0.895995			−0.447998	+	0.894035i	$0.647863\pi$
$6$		0			0
$7$	1.37922	+	2.25782i	0.521296	+	0.853376i
$7$	1.37922	+	2.25782i	0.521296	+	0.853376i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.80039i			1.44737i	0.690129	+	0.723686i	$0.257554\pi$
$11$			4.80039i			1.44737i	−0.690129	+	0.723686i	$0.742446\pi$
$12$		0			0
$13$	−0.864497	−	0.499118i	−0.239768	−	0.138430i	0.375302	−	0.926903i	$-0.377539\pi$
$13$	−0.864497	−	0.499118i	−0.239768	−	0.138430i	−0.615070	+	0.788472i	$0.710872\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.0445736	+	0.0772037i	−0.0108107	+	0.0187246i	−0.871380	−	0.490608i	$-0.836775\pi$
$17$	−0.0445736	+	0.0772037i	−0.0108107	+	0.0187246i	0.860569	+	0.509333i	$0.170108\pi$
$18$		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
$23$			0.969708i			0.202198i	0.994876	+	0.101099i	$0.0322359\pi$
$23$			0.969708i			0.202198i	−0.994876	+	0.101099i	$0.967764\pi$
$24$		0			0
$25$	−0.985965			−0.197193
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.93010	+	1.69169i	−0.544105	+	0.314139i	−0.746741	−	0.665115i	$-0.768383\pi$
$29$	−2.93010	+	1.69169i	−0.544105	+	0.314139i	0.202636	+	0.979254i	$0.435049\pi$
$30$		0			0
$31$	4.35640	−	2.51517i	0.782433	−	0.451738i	−0.0548586	−	0.998494i	$-0.517471\pi$
$31$	4.35640	−	2.51517i	0.782433	−	0.451738i	0.837292	+	0.546756i	$0.184137\pi$
$32$		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	−0.0244505	+	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
$47$	−1.76803	+	3.06231i	−0.257893	+	0.446684i	−0.965677	−	0.259745i	$-0.916362\pi$
$47$	−1.76803	+	3.06231i	−0.257893	+	0.446684i	0.707784	+	0.706429i	$0.249695\pi$
$48$		0			0
$49$	−3.19551	+	6.22806i	−0.456501	+	0.889723i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.31849	−	3.07063i	−0.730551	−	0.421784i	0.0880726	−	0.996114i	$-0.471929\pi$
$53$	−5.31849	−	3.07063i	−0.730551	−	0.421784i	−0.818624	+	0.574330i	$0.805263\pi$
$54$		0			0
$55$		−	9.61761i		−	1.29684i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.88252	−	8.45678i	−0.635650	−	1.10098i	−0.986377	−	0.164501i	$-0.947399\pi$
$59$	−4.88252	−	8.45678i	−0.635650	−	1.10098i	0.350727	−	0.936478i	$-0.385935\pi$
$60$		0			0
$61$	−11.2876	−	6.51692i	−1.44523	−	0.834407i	−0.447043	−	0.894513i	$-0.647523\pi$
$61$	−11.2876	−	6.51692i	−1.44523	−	0.834407i	−0.998192	+	0.0601058i	$0.980856\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.73203	+	0.999985i	0.214831	+	0.124033i
$66$		0			0
$67$	7.57741	+	13.1245i	0.925728	+	1.60341i	0.790386	+	0.612610i	$0.209880\pi$
$67$	7.57741	+	13.1245i	0.925728	+	1.60341i	0.135343	+	0.990799i	$0.456786\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.56985i		−	0.542341i	−0.962531	−	0.271171i	$-0.912589\pi$
$71$		−	4.56985i		−	0.542341i	0.962531	−	0.271171i	$-0.0874108\pi$
$72$		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
$77$	−10.8384	+	6.62079i	−1.23515	+	0.754509i
$78$		0			0
$79$	−5.94084	+	10.2898i	−0.668397	+	1.15770i	0.309955	+	0.950751i	$0.399686\pi$
$79$	−5.94084	+	10.2898i	−0.668397	+	1.15770i	−0.978352	+	0.206946i	$0.933647\pi$
$80$		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	0.916856	−	0.399218i	$-0.130718\pi$
$89$	1.06316	+	1.84146i	0.112695	+	0.195194i	−0.804161	+	0.594412i	$0.797385\pi$
$90$		0			0
$91$	−0.0654127	−	2.64027i	−0.00685711	−	0.276776i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.38992	+	4.26657i	−0.758189	+	0.437741i
$96$		0			0
$97$	−1.03527	+	0.597713i	−0.105116	+	0.0606885i	−0.551636	−	0.834085i	$-0.685996\pi$
$97$	−1.03527	+	0.597713i	−0.105116	+	0.0606885i	0.446521	+	0.894773i	$0.352663\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.9879			1.19284			0.596420	−	0.802673i	$-0.296589\pi$
$101$	11.9879			1.19284			0.596420	+	0.802673i	$0.296589\pi$
$102$		0			0
$103$			9.53268i			0.939282i	0.882857	+	0.469641i	$0.155617\pi$
$103$			9.53268i			0.939282i	−0.882857	+	0.469641i	$0.844383\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.2170	+	9.36287i	−1.56775	+	0.905143i	−0.571323	+	0.820725i	$0.693570\pi$
$107$	−16.2170	+	9.36287i	−1.56775	+	0.905143i	−0.996430	+	0.0844177i	$0.973097\pi$
$108$		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	−0.0918351	−	0.995774i	$-0.529273\pi$
$113$	−10.6314	−	6.13804i	−1.00012	−	0.577419i	−0.908283	+	0.418356i	$0.862607\pi$
$114$		0			0
$115$		−	1.94282i		−	0.181168i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.235789	+	0.00584166i	−0.0216147	+	0.000535504i
$120$		0			0
$121$	−12.0438			−1.09489
$122$		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
$131$	−10.4996			−0.917355			−0.458678	−	0.888603i	$-0.651677\pi$
$131$	−10.4996			−0.917355			−0.458678	+	0.888603i	$0.651677\pi$
$132$		0			0
$133$	9.89539	+	5.39084i	0.858039	+	0.467445i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			15.3867i			1.31457i	0.753640	+	0.657287i	$0.228296\pi$
$137$			15.3867i			1.31457i	−0.753640	+	0.657287i	$0.771704\pi$
$138$		0			0
$139$	12.6370	+	7.29600i	1.07186	+	0.618838i	0.928689	−	0.370860i	$-0.120937\pi$
$139$	12.6370	+	7.29600i	1.07186	+	0.618838i	0.143171	+	0.989698i	$0.454270\pi$
$140$		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
$149$			14.3861i			1.17856i	0.807930	+	0.589279i	$0.200588\pi$
$149$			14.3861i			1.17856i	−0.807930	+	0.589279i	$0.799412\pi$
$150$		0			0
$151$	−15.5150			−1.26259			−0.631297	−	0.775541i	$-0.717477\pi$
$151$	−15.5150			−1.26259			−0.631297	+	0.775541i	$0.717477\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.72808	+	5.03916i	−0.701056	+	0.404755i
$156$		0			0
$157$	3.02449	−	1.74619i	0.241381	−	0.139361i	−0.374431	−	0.927255i	$-0.622162\pi$
$157$	3.02449	−	1.74619i	0.241381	−	0.139361i	0.615811	+	0.787894i	$0.288828\pi$
$158$		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	$-0.881612\pi$
$167$	−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
$173$	−8.96624	+	15.5300i	−0.681690	+	1.18072i	0.292774	+	0.956182i	$0.405422\pi$
$173$	−8.96624	+	15.5300i	−0.681690	+	1.18072i	−0.974465	+	0.224541i	$0.927912\pi$
$174$		0			0
$175$	−1.35986	−	2.22613i	−0.102796	−	0.168280i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.93535	−	3.42678i	−0.443629	−	0.256129i	0.261507	−	0.965202i	$-0.415781\pi$
$179$	−5.93535	−	3.42678i	−0.443629	−	0.256129i	−0.705136	+	0.709072i	$0.749114\pi$
$180$		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
$185$	0.135365	+	0.234459i	0.00995223	+	0.0172378i
$186$		0			0
$187$	−0.370608	−	0.213971i	−0.0271015	−	0.0156471i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.35854	−	4.24846i	−0.532446	−	0.307408i	0.209566	−	0.977794i	$-0.432795\pi$
$191$	−7.35854	−	4.24846i	−0.532446	−	0.307408i	−0.742012	+	0.670387i	$0.766128\pi$
$192$		0			0
$193$	−6.79801	−	11.7745i	−0.489331	−	0.847547i	0.510593	−	0.859822i	$-0.329426\pi$
$193$	−6.79801	−	11.7745i	−0.489331	−	0.847547i	−0.999925	+	0.0122755i	$0.996092\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	23.5565i		−	1.67833i	−0.543878	−	0.839164i	$-0.683045\pi$
$197$		−	23.5565i		−	1.67833i	0.543878	−	0.839164i	$-0.316955\pi$
$198$		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
$203$	−7.86078	−	4.28242i	−0.551718	−	0.300567i
$204$		0			0
$205$	11.2635	−	19.5089i	0.786676	−	1.36256i
$206$		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
$221$	0.0770674	−	0.0444949i	0.00518412	−	0.00299305i
$222$		0			0
$223$	3.83410	−	2.21362i	0.256750	−	0.148235i	−0.366101	−	0.930575i	$-0.619308\pi$
$223$	3.83410	−	2.21362i	0.256750	−	0.148235i	0.622851	+	0.782340i	$0.285974\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.3910			−0.888794			−0.444397	−	0.895830i	$-0.646582\pi$
$227$	−13.3910			−0.888794			−0.444397	+	0.895830i	$0.646582\pi$
$228$		0			0
$229$			7.64997i			0.505524i	0.967528	+	0.252762i	$0.0813390\pi$
$229$			7.64997i			0.505524i	−0.967528	+	0.252762i	$0.918661\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−12.1646	+	7.02323i	−0.796929	+	0.460107i	−0.842396	−	0.538859i	$-0.818856\pi$
$233$	−12.1646	+	7.02323i	−0.796929	+	0.460107i	0.0454674	+	0.998966i	$0.485522\pi$
$234$		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	0.889382	−	0.457165i	$-0.151135\pi$
$239$	14.5035	+	8.37362i	0.938156	+	0.541644i	0.0487739	+	0.998810i	$0.484469\pi$
$240$		0			0
$241$			30.6471i			1.97415i	0.160255	+	0.987076i	$0.448768\pi$
$241$			30.6471i			1.97415i	−0.160255	+	0.987076i	$0.551232\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.40222	−	12.4780i	0.409023	−	0.797187i
$246$		0			0
$247$	−4.25159			−0.270522
$248$		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
$257$	26.7837			1.67072			0.835361	−	0.549702i	$-0.185259\pi$
$257$	26.7837			1.67072			0.835361	+	0.549702i	$0.185259\pi$
$258$		0			0
$259$	0.171034	−	0.313950i	0.0106276	−	0.0195079i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	20.9160i		−	1.28974i	−0.764293	−	0.644869i	$-0.776912\pi$
$263$		−	20.9160i		−	1.28974i	0.764293	−	0.644869i	$-0.223088\pi$
$264$		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	$-0.531640\pi$
$269$	13.3200	−	23.0709i	0.812133	−	1.40666i	0.911368	−	0.411592i	$-0.135027\pi$
$270$		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
$275$		−	4.73302i		−	0.285412i
$276$		0			0
$277$	2.04224			0.122706			0.0613531	−	0.998116i	$-0.480458\pi$
$277$	2.04224			0.122706			0.0613531	+	0.998116i	$0.480458\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.6608	−	8.46441i	0.874590	−	0.504945i	0.00571889	−	0.999984i	$-0.498180\pi$
$281$	14.6608	−	8.46441i	0.874590	−	0.504945i	0.868871	+	0.495039i	$0.164846\pi$
$282$		0			0
$283$	−9.97089	+	5.75670i	−0.592708	+	0.342200i	−0.766168	−	0.642641i	$-0.777839\pi$
$283$	−9.97089	+	5.75670i	−0.592708	+	0.342200i	0.173460	+	0.984841i	$0.444505\pi$
$284$		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
$299$	0.483998	−	0.838310i	0.0279903	−	0.0484807i
$300$		0			0
$301$	9.28480	−	17.0431i	0.535167	−	0.982349i
$302$		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
$311$	−4.35374	−	7.54089i	−0.246878	−	0.427605i	0.715780	−	0.698326i	$-0.246071\pi$
$311$	−4.35374	−	7.54089i	−0.246878	−	0.427605i	−0.962658	+	0.270721i	$0.912738\pi$
$312$		0			0
$313$	−9.54198	−	5.50906i	−0.539345	−	0.311391i	0.205469	−	0.978664i	$-0.434128\pi$
$313$	−9.54198	−	5.50906i	−0.539345	−	0.311391i	−0.744813	+	0.667273i	$0.767461\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.92196	+	5.72844i	0.557273	+	0.321742i	0.752050	−	0.659106i	$-0.229065\pi$
$317$	9.92196	+	5.72844i	0.557273	+	0.321742i	−0.194777	+	0.980847i	$0.562398\pi$
$318$		0			0
$319$	−8.12078	−	14.0656i	−0.454677	−	0.787523i
$320$		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
$329$	−9.35264	+	0.231711i	−0.515628	+	0.0127747i
$330$		0			0
$331$	−2.70596	+	4.68685i	−0.148733	+	0.257613i	−0.930759	−	0.365632i	$-0.880853\pi$
$331$	−2.70596	+	4.68685i	−0.148733	+	0.257613i	0.782027	+	0.623245i	$0.214186\pi$
$332$		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
$347$	7.74410	−	4.47106i	0.415725	−	0.240019i	−0.277522	−	0.960719i	$-0.589513\pi$
$347$	7.74410	−	4.47106i	0.415725	−	0.240019i	0.693247	+	0.720700i	$0.256180\pi$
$348$		0			0
$349$	9.68628	−	5.59238i	0.518495	−	0.299353i	−0.217824	−	0.975988i	$-0.569896\pi$
$349$	9.68628	−	5.59238i	0.518495	−	0.299353i	0.736319	+	0.676635i	$0.236562\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	13.5130			0.719225			0.359613	−	0.933102i	$-0.382909\pi$
$353$	13.5130			0.719225			0.359613	+	0.933102i	$0.382909\pi$
$354$		0			0
$355$			9.15572i			0.485935i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−29.5673	+	17.0707i	−1.56050	+	0.900957i	−0.563297	+	0.826254i	$0.690467\pi$
$359$	−29.5673	+	17.0707i	−1.56050	+	0.900957i	−0.997206	+	0.0747023i	$0.976199\pi$
$360$		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
$367$			1.06520i			0.0556029i	0.999613	+	0.0278015i	$0.00885062\pi$
$367$			1.06520i			0.0556029i	−0.999613	+	0.0278015i	$0.991149\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.402427	−	16.2433i	−0.0208929	−	0.843309i
$372$		0			0
$373$	−4.33310			−0.224359			−0.112180	−	0.993688i	$-0.535783\pi$
$373$	−4.33310			−0.224359			−0.112180	+	0.993688i	$0.535783\pi$
$374$		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
$383$	37.5103			1.91669			0.958344	−	0.285618i	$-0.0921988\pi$
$383$	37.5103			1.91669			0.958344	+	0.285618i	$0.0921988\pi$
$384$		0			0
$385$	21.7149	−	13.2648i	1.10669	−	0.676037i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	6.89422i		−	0.349551i	−0.984608	−	0.174775i	$-0.944080\pi$
$389$		−	6.89422i		−	0.349551i	0.984608	−	0.174775i	$-0.0559199\pi$
$390$		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
$401$			18.6023i			0.928955i	0.885585	+	0.464477i	$0.153758\pi$
$401$			18.6023i			0.928955i	−0.885585	+	0.464477i	$0.846242\pi$
$402$		0			0
$403$	−5.02147			−0.250137
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.561763	−	0.324334i	0.0278456	−	0.0160766i
$408$		0			0
$409$	1.89129	−	1.09193i	0.0935181	−	0.0539927i	−0.452512	−	0.891759i	$-0.649472\pi$
$409$	1.89129	−	1.09193i	0.0935181	−	0.0539927i	0.546030	+	0.837766i	$0.316139\pi$
$410$		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
$425$	0.0439480	−	0.0761201i	0.00213179	−	0.00369237i
$426$		0			0
$427$	−0.854086	−	34.4737i	−0.0413321	−	1.66830i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	22.9854	+	13.2706i	1.10717	+	0.639223i	0.938094	−	0.346381i	$-0.112590\pi$
$431$	22.9854	+	13.2706i	1.10717	+	0.639223i	0.169072	+	0.985604i	$0.445923\pi$
$432$		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
$437$	2.06504	+	3.57676i	0.0987844	+	0.171100i
$438$		0			0
$439$	−3.49248	−	2.01638i	−0.166687	−	0.0962367i	0.414336	−	0.910124i	$-0.364014\pi$
$439$	−3.49248	−	2.01638i	−0.166687	−	0.0962367i	−0.581023	+	0.813887i	$0.697347\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−29.3085	−	16.9212i	−1.39249	−	0.803953i	−0.398897	−	0.916996i	$-0.630607\pi$
$443$	−29.3085	−	16.9212i	−1.39249	−	0.803953i	−0.993590	+	0.113043i	$0.963940\pi$
$444$		0			0
$445$	−2.13006	−	3.68937i	−0.100974	−	0.174893i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			31.7395i			1.49788i	0.662638	+	0.748940i	$0.269437\pi$
$449$			31.7395i			1.49788i	−0.662638	+	0.748940i	$0.730563\pi$
$450$		0			0
$451$	−46.7433	−	26.9873i	−2.20106	−	1.27078i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.131055	+	5.28980i	0.00614394	+	0.247990i
$456$		0			0
$457$	17.7802	−	30.7963i	0.831724	−	1.44059i	−0.0649459	−	0.997889i	$-0.520687\pi$
$457$	17.7802	−	30.7963i	0.831724	−	1.44059i	0.896670	−	0.442700i	$-0.145979\pi$
$458$		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	0.851240	+	0.524777i	$0.175851\pi$
$467$	19.0189	+	32.9417i	0.880090	+	1.52436i	0.0288500	+	0.999584i	$0.490815\pi$
$468$		0			0
$469$	−19.1818	+	35.2100i	−0.885732	+	1.62584i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	30.4959	−	17.6068i	1.40221	−	0.809563i
$474$		0			0
$475$	−3.63672	+	2.09966i	−0.166864	+	0.0963392i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10.5363			0.481415			0.240707	−	0.970598i	$-0.422621\pi$
$479$	10.5363			0.481415			0.240707	+	0.970598i	$0.422621\pi$
$480$		0			0
$481$			0.134890i			0.00615044i
$482$		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	0.644469	−	0.764630i	$-0.277078\pi$
$491$	6.74760	+	3.89573i	0.304515	+	0.175812i	−0.339954	+	0.940442i	$0.610412\pi$
$492$		0			0
$493$		−	0.301619i		−	0.0135842i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.3179	−	6.30282i	0.462821	−	0.282720i
$498$		0			0
$499$	−23.7900			−1.06499			−0.532493	−	0.846434i	$-0.678745\pi$
$499$	−23.7900			−1.06499			−0.532493	+	0.846434i	$0.678745\pi$
$500$		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
$509$	17.1830			0.761624			0.380812	−	0.924652i	$-0.375644\pi$
$509$	17.1830			0.761624			0.380812	+	0.924652i	$0.375644\pi$
$510$		0			0
$511$	0.282722	+	11.4116i	0.0125069	+	0.504819i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	19.0988i		−	0.841592i
$516$		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	−0.609782	−	0.792570i	$-0.708743\pi$
$521$	8.70778	−	15.0823i	0.381495	−	0.660768i	0.991276	+	0.131802i	$0.0420762\pi$
$522$		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
$527$			0.448441i			0.0195344i
$528$		0			0
$529$	22.0597			0.959116
$530$		0			0
$531$		0			0
$532$		0			0
$533$	9.72022	−	5.61197i	0.421029	−	0.243081i
$534$		0			0
$535$	32.4908	−	18.7586i	1.40470	−	0.811004i
$536$		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	0.973919	+	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
$551$	−7.20509	+	12.4796i	−0.306947	+	0.531648i
$552$		0			0
$553$	−31.4263	+	0.778586i	−1.33638	+	0.0331089i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.37676	−	3.10427i	−0.227821	−	0.131532i	0.381746	−	0.924267i	$-0.375323\pi$
$557$	−5.37676	−	3.10427i	−0.227821	−	0.131532i	−0.609566	+	0.792735i	$0.708656\pi$
$558$		0			0
$559$			7.32264i			0.309715i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−14.0447	−	24.3261i	−0.591913	−	1.02522i	−0.993975	−	0.109611i	$-0.965040\pi$
$563$	−14.0447	−	24.3261i	−0.591913	−	1.02522i	0.402062	−	0.915613i	$-0.368294\pi$
$564$		0			0
$565$	21.3001	+	12.2976i	0.896101	+	0.517364i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−28.3016	−	16.3399i	−1.18646	−	0.685005i	−0.228963	−	0.973435i	$-0.573533\pi$
$569$	−28.3016	−	16.3399i	−1.18646	−	0.685005i	−0.957501	+	0.288430i	$0.906867\pi$
$570$		0			0
$571$	8.71183	+	15.0893i	0.364579	+	0.631469i	0.988708	−	0.149852i	$-0.0478797\pi$
$571$	8.71183	+	15.0893i	0.364579	+	0.631469i	−0.624130	+	0.781321i	$0.714546\pi$
$572$		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
$581$	4.11345	−	7.55063i	0.170655	−	0.313253i
$582$		0			0
$583$	14.7402	−	25.5309i	0.610479	−	1.05738i
$584$		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	0.788670	+	0.614817i	$0.210770\pi$
$593$	22.5686	+	39.0900i	0.926782	+	1.60523i	0.138112	+	0.990417i	$0.455897\pi$
$594$		0			0
$595$	0.472404	−	0.0117038i	0.0193667	−	0.000479809i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9.51980	−	5.49626i	0.388969	−	0.224571i	−0.292745	−	0.956191i	$-0.594569\pi$
$599$	9.51980	−	5.49626i	0.388969	−	0.224571i	0.681713	+	0.731620i	$0.261235\pi$
$600$		0			0
$601$	−31.1448	+	17.9815i	−1.27042	+	0.733479i	−0.975068	−	0.221908i	$-0.928772\pi$
$601$	−31.1448	+	17.9815i	−1.27042	+	0.733479i	−0.295356	+	0.955387i	$0.595438\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	24.1298			0.981015
$606$		0			0
$607$			25.2806i			1.02611i	0.858357	+	0.513053i	$0.171486\pi$
$607$			25.2806i			1.02611i	−0.858357	+	0.513053i	$0.828514\pi$
$608$		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	0.650868	−	0.759191i	$-0.274405\pi$
$617$	7.91940	+	4.57227i	0.318823	+	0.184072i	−0.332045	+	0.943264i	$0.607739\pi$
$618$		0			0
$619$			24.8019i			0.996873i	0.866926	+	0.498437i	$0.166092\pi$
$619$			24.8019i			0.996873i	−0.866926	+	0.498437i	$0.833908\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.69134	+	4.94021i	−0.107826	+	0.197925i
$624$		0			0
$625$	−19.0980			−0.763922
$626$		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
$635$	14.7367			0.584807
$636$		0			0
$637$	5.87104	−	3.78920i	0.232619	−	0.150134i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			23.0736i			0.911351i	0.890146	+	0.455676i	$0.150602\pi$
$641$			23.0736i			0.911351i	−0.890146	+	0.455676i	$0.849398\pi$
$642$		0			0
$643$	−15.8108	−	9.12837i	−0.623517	−	0.359988i	0.154720	−	0.987958i	$-0.450552\pi$
$643$	−15.8108	−	9.12837i	−0.623517	−	0.359988i	−0.778237	+	0.627971i	$0.783886\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.59782	+	2.76751i	−0.0628168	+	0.108802i	−0.895723	−	0.444612i	$-0.853342\pi$
$647$	−1.59782	+	2.76751i	−0.0628168	+	0.108802i	0.832907	+	0.553413i	$0.186675\pi$
$648$		0			0
$649$	40.5959	−	23.4380i	1.59353	−	0.920023i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			20.0411i			0.784269i	0.919908	+	0.392134i	$0.128263\pi$
$653$			20.0411i			0.784269i	−0.919908	+	0.392134i	$0.871737\pi$
$654$		0			0
$655$	21.0360			0.821946
$656$		0			0
$657$		0			0
$658$		0			0
$659$	29.7095	−	17.1528i	1.15732	−	0.668178i	0.206658	−	0.978413i	$-0.433741\pi$
$659$	29.7095	−	17.1528i	1.15732	−	0.668178i	0.950660	+	0.310235i	$0.100408\pi$
$660$		0			0
$661$	24.6383	−	14.2249i	0.958320	−	0.553286i	0.0626645	−	0.998035i	$-0.480040\pi$
$661$	24.6383	−	14.2249i	0.958320	−	0.553286i	0.895656	+	0.444748i	$0.146707\pi$
$662$		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
$677$	18.1894	−	31.5050i	0.699075	−	1.21083i	−0.269712	−	0.962941i	$-0.586929\pi$
$677$	18.1894	−	31.5050i	0.699075	−	1.21083i	0.968787	−	0.247893i	$-0.0797381\pi$
$678$		0			0
$679$	−2.77739	−	1.51307i	−0.106586	−	0.0580665i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.25354	+	0.723729i	0.0479652	+	0.0276927i	0.523791	−	0.851847i	$-0.324517\pi$
$683$	1.25354	+	0.723729i	0.0479652	+	0.0276927i	−0.475826	+	0.879540i	$0.657851\pi$
$684$		0			0
$685$		−	30.8273i		−	1.17785i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.06522	+	5.30911i	0.116775	+	0.202261i
$690$		0			0
$691$	−31.1043	−	17.9581i	−1.18326	−	0.683157i	−0.226495	−	0.974012i	$-0.572727\pi$
$691$	−31.1043	−	17.9581i	−1.18326	−	0.683157i	−0.956767	+	0.290856i	$0.906060\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−25.3184	−	14.6176i	−0.960381	−	0.554476i
$696$		0			0
$697$	−0.501175	−	0.868061i	−0.0189834	−	0.0328802i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	41.3851i		−	1.56309i	−0.623847	−	0.781547i	$-0.714431\pi$
$701$		−	41.3851i		−	1.56309i	0.623847	−	0.781547i	$-0.285569\pi$
$702$		0			0
$703$	−0.498419	−	0.287762i	−0.0187982	−	0.0108532i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	16.5339	+	27.0665i	0.621822	+	1.01794i
$708$		0			0
$709$	9.22204	−	15.9730i	0.346341	−	0.599880i	−0.639256	−	0.768994i	$-0.720757\pi$
$709$	9.22204	−	15.9730i	0.346341	−	0.599880i	0.985596	+	0.169114i	$0.0540907\pi$
$710$		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	0.882729	−	0.469882i	$-0.155703\pi$
$719$	0.923351	+	1.59929i	0.0344352	+	0.0596435i	−0.848294	+	0.529525i	$0.822370\pi$
$720$		0			0
$721$	−21.5231	+	13.1476i	−0.801561	+	0.489644i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.88897	−	1.66795i	0.107294	−	0.0619460i
$726$		0			0
$727$	−28.4243	+	16.4108i	−1.05420	+	0.608642i	−0.923822	−	0.382822i	$-0.874952\pi$
$727$	−28.4243	+	16.4108i	−1.05420	+	0.608642i	−0.130378	+	0.991464i	$0.541619\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.653946			0.0241871
$732$		0			0
$733$		−	15.4249i		−	0.569731i	−0.958567	−	0.284866i	$-0.908051\pi$
$733$		−	15.4249i		−	0.569731i	0.958567	−	0.284866i	$-0.0919490\pi$
$734$		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	0.000744979	−	1.00000i	$-0.499763\pi$
$743$	−23.5757	−	13.6114i	−0.864908	−	0.499355i	−0.865653	+	0.500645i	$0.833096\pi$
$744$		0			0
$745$		−	28.8227i		−	1.05598i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−43.5064	−	23.7016i	−1.58969	−	0.866036i
$750$		0			0
$751$	21.3098			0.777607			0.388804	−	0.921321i	$-0.372888\pi$
$751$	21.3098			0.777607			0.388804	+	0.921321i	$0.372888\pi$
$752$		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
$761$	49.7704			1.80417			0.902087	−	0.431554i	$-0.142035\pi$
$761$	49.7704			1.80417			0.902087	+	0.431554i	$0.142035\pi$
$762$		0			0
$763$	4.76736	−	0.118111i	0.172590	−	0.00427591i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			9.74781i			0.351973i
$768$		0			0
$769$	−15.1354	−	8.73843i	−0.545797	−	0.315116i	0.201628	−	0.979462i	$-0.435377\pi$
$769$	−15.1354	−	8.73843i	−0.545797	−	0.315116i	−0.747425	+	0.664346i	$0.768710\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	20.7020	−	35.8569i	0.744598	−	1.28968i	−0.205784	−	0.978597i	$-0.565974\pi$
$773$	20.7020	−	35.8569i	0.744598	−	1.28968i	0.950382	−	0.311085i	$-0.100692\pi$
$774$		0			0
$775$	−4.29526	+	2.47987i	−0.154290	+	0.0890796i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			47.8884i			1.71578i
$780$		0			0
$781$	21.9371			0.784970
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.05958	+	3.49850i	−0.216276	+	0.124867i
$786$		0			0
$787$	21.8422	−	12.6106i	0.778591	−	0.449519i	−0.0573400	−	0.998355i	$-0.518262\pi$
$787$	21.8422	−	12.6106i	0.778591	−	0.449519i	0.835931	+	0.548835i	$0.184929\pi$
$788$		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
$803$	−10.3557	+	17.9365i	−0.365443	+	0.632966i
$804$		0			0
$805$	4.38653	−	2.67957i	0.154605	−	0.0944423i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−5.88592	−	3.39824i	−0.206938	−	0.119476i	0.392950	−	0.919560i	$-0.371455\pi$
$809$	−5.88592	−	3.39824i	−0.206938	−	0.119476i	−0.599888	+	0.800084i	$0.704788\pi$
$810$		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
$815$	−13.8626	−	24.0107i	−0.485585	−	0.841059i
$816$		0			0
$817$	−27.0572	−	15.6215i	−0.946613	−	0.546527i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	24.8842	+	14.3669i	0.868465	+	0.501409i	0.866838	−	0.498590i	$-0.166149\pi$
$821$	24.8842	+	14.3669i	0.868465	+	0.501409i	0.00162745	+	0.999999i	$0.499482\pi$
$822$		0			0
$823$	25.3472	+	43.9026i	0.883547	+	1.53035i	0.847371	+	0.531002i	$0.178184\pi$
$823$	25.3472	+	43.9026i	0.883547	+	1.53035i	0.0361761	+	0.999345i	$0.488482\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	3.19233i		−	0.111008i	−0.998458	−	0.0555041i	$-0.982323\pi$
$827$		−	3.19233i		−	0.111008i	0.998458	−	0.0555041i	$-0.0176766\pi$
$828$		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
$833$	−0.338394	−	0.524312i	−0.0117246	−	0.0181663i
$834$		0			0
$835$	3.91188	−	6.77557i	0.135376	−	0.234478i
$836$		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
$851$	0.113479	−	0.0655174i	0.00389003	−	0.00224591i
$852$		0			0
$853$	−20.3474	+	11.7476i	−0.696680	+	0.402229i	−0.806110	−	0.591766i	$-0.798431\pi$
$853$	−20.3474	+	11.7476i	−0.696680	+	0.402229i	0.109429	+	0.993995i	$0.465098\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−17.5192			−0.598444			−0.299222	−	0.954183i	$-0.596727\pi$
$857$	−17.5192			−0.598444			−0.299222	+	0.954183i	$0.596727\pi$
$858$		0			0
$859$		−	13.7668i		−	0.469716i	−0.972030	−	0.234858i	$-0.924537\pi$
$859$		−	13.7668i		−	0.469716i	0.972030	−	0.234858i	$-0.0754625\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−22.0835	+	12.7499i	−0.751732	+	0.434013i	−0.826319	−	0.563202i	$-0.809569\pi$
$863$	−22.0835	+	12.7499i	−0.751732	+	0.434013i	0.0745872	+	0.997214i	$0.476236\pi$
$864$		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
$871$		−	15.1281i		−	0.512596i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	16.5409	+	27.0779i	0.559183	+	0.915399i
$876$		0			0
$877$	−51.9791			−1.75521			−0.877604	−	0.479386i	$-0.840859\pi$
$877$	−51.9791			−1.75521			−0.877604	+	0.479386i	$0.840859\pi$
$878$		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
$887$	−54.9081			−1.84363			−0.921817	−	0.387625i	$-0.873295\pi$
$887$	−54.9081			−1.84363			−0.921817	+	0.387625i	$0.873295\pi$
$888$		0			0
$889$	−10.1448	−	16.6073i	−0.340245	−	0.556990i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			15.0604i			0.503977i
$894$		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
$905$			24.2446i			0.805919i
$906$		0			0
$907$	−8.65342			−0.287332			−0.143666	−	0.989626i	$-0.545889\pi$
$907$	−8.65342			−0.287332			−0.143666	+	0.989626i	$0.545889\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.63346	−	5.56188i	0.319171	−	0.184273i	−0.331852	−	0.943331i	$-0.607673\pi$
$911$	9.63346	−	5.56188i	0.319171	−	0.184273i	0.651023	+	0.759058i	$0.274340\pi$
$912$		0			0
$913$	13.5106	−	7.80037i	0.447137	−	0.258155i
$914$		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
$929$	−17.9035	+	31.0097i	−0.587394	+	1.01740i	0.407178	+	0.913349i	$0.366513\pi$
$929$	−17.9035	+	31.0097i	−0.587394	+	1.01740i	−0.994572	+	0.104048i	$0.966820\pi$
$930$		0			0
$931$	1.47636	+	29.7772i	0.0483858	+	0.975907i
$932$		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
$941$	−17.7953	−	30.8223i	−0.580109	−	1.00478i	−0.995466	−	0.0951193i	$-0.969677\pi$
$941$	−17.7953	−	30.8223i	−0.580109	−	1.00478i	0.415357	−	0.909658i	$-0.363657\pi$
$942$		0			0
$943$	−9.44243	−	5.45159i	−0.307488	−	0.177528i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−27.6494	−	15.9634i	−0.898485	−	0.518741i	−0.0217769	−	0.999763i	$-0.506932\pi$
$947$	−27.6494	−	15.9634i	−0.898485	−	0.518741i	−0.876708	+	0.481022i	$0.840266\pi$
$948$		0			0
$949$	−2.15344	−	3.72988i	−0.0699038	−	0.121077i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			21.6179i			0.700273i	0.936699	+	0.350136i	$0.113865\pi$
$953$			21.6179i			0.700273i	−0.936699	+	0.350136i	$0.886135\pi$
$954$		0			0
$955$	14.7429	+	8.51181i	0.477069	+	0.275436i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−34.7404	+	21.2216i	−1.12183	+	0.685282i
$960$		0			0
$961$	−2.84783	+	4.93258i	−0.0918653	+	0.159115i
$962$		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	0.945488	−	0.325656i	$-0.105585\pi$
$971$	5.94294	+	10.2935i	0.190718	+	0.330334i	−0.754770	+	0.655989i	$0.772252\pi$
$972$		0			0
$973$	0.956188	+	38.5949i	0.0306540	+	1.23730i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−37.6105	+	21.7144i	−1.20327	+	0.694706i	−0.961280	−	0.275573i	$-0.911132\pi$
$977$	−37.6105	+	21.7144i	−1.20327	+	0.694706i	−0.241986	+	0.970280i	$0.577799\pi$
$978$		0			0
$979$	−8.83971	+	5.10361i	−0.282518	+	0.163112i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	34.9407			1.11444			0.557218	−	0.830366i	$-0.311869\pi$
$983$	34.9407			1.11444			0.557218	+	0.830366i	$0.311869\pi$
$984$		0			0
$985$			47.1955i			1.50377i
$986$		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
$997$			50.4909i			1.59906i	0.600625	+	0.799531i	$0.294918\pi$
$997$			50.4909i			1.59906i	−0.600625	+	0.799531i	$0.705082\pi$
$998$		0			0
$999$		0			0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.00351 q^{5} +(1.37922 + 2.25782i) q^{7} +4.80039i q^{11} +(-0.864497 - 0.499118i) q^{13} +(-0.0445736 + 0.0772037i) q^{17} +(3.68849 - 2.12955i) q^{19} +0.969708i q^{23} -0.985965 q^{25} +(-2.93010 + 1.69169i) q^{29} +(4.35640 - 2.51517i) q^{31} +(-2.76327 - 4.52356i) q^{35} +(-0.0675641 - 0.117024i) q^{37} +(-5.62189 + 9.73740i) q^{41} +(-3.66779 - 6.35280i) q^{43} +(-1.76803 + 3.06231i) q^{47} +(-3.19551 + 6.22806i) q^{49} +(-5.31849 - 3.07063i) q^{53} -9.61761i q^{55} +(-4.88252 - 8.45678i) q^{59} +(-11.2876 - 6.51692i) q^{61} +(1.73203 + 0.999985i) q^{65} +(7.57741 + 13.1245i) q^{67} -4.56985i q^{71} +(3.73647 + 2.15725i) q^{73} +(-10.8384 + 6.62079i) q^{77} +(-5.94084 + 10.2898i) q^{79} +(-1.62494 - 2.81449i) q^{83} +(0.0893034 - 0.154678i) q^{85} +(1.06316 + 1.84146i) q^{89} +(-0.0654127 - 2.64027i) q^{91} +(-7.38992 + 4.26657i) q^{95} +(-1.03527 + 0.597713i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 4 q^{7} - 12 q^{13} + 32 q^{25} - 24 q^{31} - 4 q^{37} - 4 q^{43} - 16 q^{49} - 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} + 12 q^{85} - 36 q^{91} - 12 q^{97}+O(q^{100}) \) 32 * q + 4 * q^7 - 12 * q^13 + 32 * q^25 - 24 * q^31 - 4 * q^37 - 4 * q^43 - 16 * q^49 - 12 * q^61 + 4 * q^67 - 36 * q^73 + 28 * q^79 + 12 * q^85 - 36 * q^91 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more