LMFDB - Embedded newform 2268.2.t.c.1781.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(1781,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.1781"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.t (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,-8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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			1781.4
Character	$\chi$	$=$	2268.1781
Dual form			2268.2.t.c.2105.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00175 - 1.73509i) q^{5} +(-2.64494 + 0.0655283i) q^{7} +(4.15726 + 2.40020i) q^{11} +0.998235i q^{13} +(0.0445736 - 0.0772037i) q^{17} +(3.68849 - 2.12955i) q^{19} +(-0.839792 + 0.484854i) q^{23} +(0.492982 - 0.853871i) q^{25} +3.38338i q^{29} +(-4.35640 - 2.51517i) q^{31} +(2.76327 + 4.52356i) q^{35} +(-0.0675641 - 0.117024i) q^{37} -11.2438 q^{41} +7.33558 q^{43} +(1.76803 + 3.06231i) q^{47} +(6.99141 - 0.346637i) q^{49} +(5.31849 + 3.07063i) q^{53} -9.61761i q^{55} +(4.88252 - 8.45678i) q^{59} +(11.2876 - 6.51692i) q^{61} +(1.73203 - 0.999985i) q^{65} +(7.57741 - 13.1245i) q^{67} +4.56985i q^{71} +(3.73647 + 2.15725i) q^{73} +(-11.1530 - 6.07596i) q^{77} +(-5.94084 - 10.2898i) q^{79} -3.24989 q^{83} -0.178607 q^{85} +(-1.06316 - 1.84146i) q^{89} +(-0.0654127 - 2.64027i) q^{91} +(-7.38992 - 4.26657i) q^{95} -1.19543i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7} - 16 q^{25} + 24 q^{31} - 4 q^{37} + 8 q^{43} - 4 q^{49} + 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} - 24 q^{85} - 36 q^{91}+O(q^{100}) $ 32 * q - 8 * q^7 - 16 * q^25 + 24 * q^31 - 4 * q^37 + 8 * q^43 - 4 * q^49 + 12 * q^61 + 4 * q^67 - 36 * q^73 + 28 * q^79 - 24 * q^85 - 36 * q^91

Character values

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

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

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	0.550258	−	0.834995i	$-0.314529\pi$
$5$	−1.00175	−	1.73509i	−0.447998	−	0.775954i	−0.998256	+	0.0590402i	$0.981196\pi$
$6$		0			0
$7$	−2.64494	+	0.0655283i	−0.999693	+	0.0247674i
$7$	−2.64494	+	0.0655283i	−0.999693	+	0.0247674i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.15726	+	2.40020i	1.25346	+	0.723686i	0.971795	−	0.235826i	$-0.0757795\pi$
$11$	4.15726	+	2.40020i	1.25346	+	0.723686i	0.281666	+	0.959512i	$0.409113\pi$
$12$		0			0
$13$			0.998235i			0.276861i	0.990372	+	0.138430i	$0.0442057\pi$
$13$			0.998235i			0.276861i	−0.990372	+	0.138430i	$0.955794\pi$
$14$		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	$-0.698903\pi$
$23$	−0.839792	+	0.484854i	−0.175109	+	0.101099i	0.409884	+	0.912138i	$0.365569\pi$
$24$		0			0
$25$	0.492982	−	0.853871i	0.0985965	−	0.170774i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			3.38338i			0.628278i	0.949377	+	0.314139i	$0.101716\pi$
$29$			3.38338i			0.628278i	−0.949377	+	0.314139i	$0.898284\pi$
$30$		0			0
$31$	−4.35640	−	2.51517i	−0.782433	−	0.451738i	0.0548586	−	0.998494i	$-0.482529\pi$
$31$	−4.35640	−	2.51517i	−0.782433	−	0.451738i	−0.837292	+	0.546756i	$0.815863\pi$
$32$		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
$41$	−11.2438			−1.75598			−0.877992	−	0.478676i	$-0.841117\pi$
$41$	−11.2438			−1.75598			−0.877992	+	0.478676i	$0.841117\pi$
$42$		0			0
$43$	7.33558			1.11867			0.559333	−	0.828943i	$-0.311057\pi$
$43$	7.33558			1.11867			0.559333	+	0.828943i	$0.311057\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.76803	+	3.06231i	0.257893	+	0.446684i	0.965677	−	0.259745i	$-0.0836385\pi$
$47$	1.76803	+	3.06231i	0.257893	+	0.446684i	−0.707784	+	0.706429i	$0.750305\pi$
$48$		0			0
$49$	6.99141	−	0.346637i	0.998773	−	0.0495195i
$50$		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
$59$	4.88252	−	8.45678i	0.635650	−	1.10098i	−0.350727	−	0.936478i	$-0.614065\pi$
$59$	4.88252	−	8.45678i	0.635650	−	1.10098i	0.986377	−	0.164501i	$-0.0526013\pi$
$60$		0			0
$61$	11.2876	−	6.51692i	1.44523	−	0.834407i	0.447043	−	0.894513i	$-0.352477\pi$
$61$	11.2876	−	6.51692i	1.44523	−	0.834407i	0.998192	+	0.0601058i	$0.0191438\pi$
$62$		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	0.135343	−	0.990799i	$-0.456786\pi$
$67$	7.57741	−	13.1245i	0.925728	−	1.60341i	0.790386	−	0.612610i	$-0.209880\pi$
$68$		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
$77$	−11.1530	−	6.07596i	−1.27100	−	0.692420i
$78$		0			0
$79$	−5.94084	−	10.2898i	−0.668397	−	1.15770i	−0.978352	−	0.206946i	$-0.933647\pi$
$79$	−5.94084	−	10.2898i	−0.668397	−	1.15770i	0.309955	−	0.950751i	$-0.399686\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.24989			−0.356722			−0.178361	−	0.983965i	$-0.557079\pi$
$83$	−3.24989			−0.356722			−0.178361	+	0.983965i	$0.557079\pi$
$84$		0			0
$85$	−0.178607			−0.0193726
$86$		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
$95$	−7.38992	−	4.26657i	−0.758189	−	0.437741i
$96$		0			0
$97$		−	1.19543i		−	0.121377i	−0.998157	−	0.0606885i	$-0.980670\pi$
$97$		−	1.19543i		−	0.121377i	0.998157	−	0.0606885i	$-0.0193296\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.99395	−	10.3818i	0.596420	−	1.03303i	−0.396925	−	0.917851i	$-0.629923\pi$
$101$	5.99395	−	10.3818i	0.596420	−	1.03303i	0.993345	−	0.115179i	$-0.0367440\pi$
$102$		0			0
$103$	8.25554	−	4.76634i	0.813442	−	0.469641i	−0.0347074	−	0.999398i	$-0.511050\pi$
$103$	8.25554	−	4.76634i	0.813442	−	0.469641i	0.848150	+	0.529756i	$0.177717\pi$
$104$		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
$113$		−	12.2761i		−	1.15484i	−0.816448	−	0.577419i	$-0.804060\pi$
$113$		−	12.2761i		−	1.15484i	0.816448	−	0.577419i	$-0.195940\pi$
$114$		0			0
$115$	1.68253	+	0.971408i	0.156896	+	0.0905842i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.112835	+	0.207120i	−0.0103436	+	0.0189866i
$120$		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	0.540214	−	0.841528i	$-0.318343\pi$
$131$	−5.24981	−	9.09293i	−0.458678	−	0.794453i	−0.998891	+	0.0470748i	$0.985010\pi$
$132$		0			0
$133$	−9.61629	+	5.87424i	−0.833838	+	0.509361i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.3253	+	7.69335i	1.13845	+	0.657287i	0.946047	−	0.324028i	$-0.105037\pi$
$137$	13.3253	+	7.69335i	1.13845	+	0.657287i	0.192407	+	0.981315i	$0.438371\pi$
$138$		0			0
$139$		−	14.5920i		−	1.23768i	−0.785518	−	0.618838i	$-0.787604\pi$
$139$		−	14.5920i		−	1.23768i	0.785518	−	0.618838i	$-0.212396\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.39596	+	4.14993i	−0.200360	+	0.347034i
$144$		0			0
$145$	5.87046	−	3.38931i	0.487515	−	0.281467i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.4588	+	7.19307i	−1.02066	+	0.589279i	−0.914295	−	0.405048i	$-0.867255\pi$
$149$	−12.4588	+	7.19307i	−1.02066	+	0.589279i	−0.106366	+	0.994327i	$0.533921\pi$
$150$		0			0
$151$	7.75751	−	13.4364i	0.631297	−	1.09344i	−0.355990	−	0.934490i	$-0.615856\pi$
$151$	7.75751	−	13.4364i	0.631297	−	1.09344i	0.987287	−	0.158949i	$-0.0508104\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			10.0783i			0.809510i
$156$		0			0
$157$	−3.02449	−	1.74619i	−0.241381	−	0.139361i	0.374431	−	0.927255i	$-0.377838\pi$
$157$	−3.02449	−	1.74619i	−0.241381	−	0.139361i	−0.615811	+	0.787894i	$0.711172\pi$
$158$		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
$167$	−3.90503			−0.302180			−0.151090	−	0.988520i	$-0.548278\pi$
$167$	−3.90503			−0.302180			−0.151090	+	0.988520i	$0.548278\pi$
$168$		0			0
$169$	12.0035			0.923348
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.96624	+	15.5300i	0.681690	+	1.18072i	0.974465	+	0.224541i	$0.0720882\pi$
$173$	8.96624	+	15.5300i	0.681690	+	1.18072i	−0.292774	+	0.956182i	$0.594578\pi$
$174$		0			0
$175$	−1.24796	+	2.29074i	−0.0943366	+	0.173164i
$176$		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
$185$	−0.135365	+	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	−0.742012	−	0.670387i	$-0.766128\pi$
$191$	−7.35854	+	4.24846i	−0.532446	+	0.307408i	0.209566	+	0.977794i	$0.432795\pi$
$192$		0			0
$193$	−6.79801	+	11.7745i	−0.489331	+	0.847547i	−0.999925	−	0.0122755i	$-0.996092\pi$
$193$	−6.79801	+	11.7745i	−0.489331	+	0.847547i	0.510593	+	0.859822i	$0.329426\pi$
$194$		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
$203$	−0.221707	−	8.94884i	−0.0155608	−	0.628086i
$204$		0			0
$205$	11.2635	+	19.5089i	0.786676	+	1.36256i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	20.4454			1.41424
$210$		0			0
$211$	−16.8830			−1.16227			−0.581136	−	0.813807i	$-0.697391\pi$
$211$	−16.8830			−1.16227			−0.581136	+	0.813807i	$0.697391\pi$
$212$		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
$223$			4.42723i			0.296469i	0.988952	+	0.148235i	$0.0473591\pi$
$223$			4.42723i			0.296469i	−0.988952	+	0.148235i	$0.952641\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.69552	+	11.5970i	−0.444397	+	0.769718i	−0.998010	−	0.0630557i	$-0.979915\pi$
$227$	−6.69552	+	11.5970i	−0.444397	+	0.769718i	0.553613	+	0.832774i	$0.313249\pi$
$228$		0			0
$229$	6.62507	−	3.82498i	0.437797	−	0.252762i	−0.264866	−	0.964285i	$-0.585328\pi$
$229$	6.62507	−	3.82498i	0.437797	−	0.252762i	0.702663	+	0.711523i	$0.251994\pi$
$230$		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
$239$			16.7472i			1.08329i	0.840608	+	0.541644i	$0.182198\pi$
$239$			16.7472i			1.08329i	−0.840608	+	0.541644i	$0.817802\pi$
$240$		0			0
$241$	−26.5411	−	15.3235i	−1.70967	−	0.987076i	−0.934960	−	0.354753i	$-0.884565\pi$
$241$	−26.5411	−	15.3235i	−1.70967	−	0.987076i	−0.774705	−	0.632323i	$-0.782102\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−7.60511	−	11.7835i	−0.485873	−	0.752818i
$246$		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	0.893736	+	0.448592i	$0.148074\pi$
$257$	13.3918	+	23.1954i	0.835361	+	1.44689i	−0.0583759	+	0.998295i	$0.518592\pi$
$258$		0			0
$259$	0.186371	+	0.305095i	0.0115805	+	0.0189577i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−18.1138	−	10.4580i	−1.11695	−	0.644869i	−0.176326	−	0.984332i	$-0.556421\pi$
$263$	−18.1138	−	10.4580i	−1.11695	−	0.644869i	−0.940619	+	0.339463i	$0.889755\pi$
$264$		0			0
$265$		−	12.3041i		−	0.755833i
$266$		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	−0.895070	−	0.445925i	$-0.852875\pi$
$277$	−1.02112	+	1.76863i	−0.0613531	+	0.106267i	0.833717	+	0.552191i	$0.186208\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	16.9288i		−	1.00989i	−0.863152	−	0.504945i	$-0.831513\pi$
$281$		−	16.9288i		−	1.00989i	0.863152	−	0.504945i	$-0.168487\pi$
$282$		0			0
$283$	9.97089	+	5.75670i	0.592708	+	0.342200i	0.766168	−	0.642641i	$-0.222161\pi$
$283$	9.97089	+	5.75670i	0.592708	+	0.342200i	−0.173460	+	0.984841i	$0.555495\pi$
$284$		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
$293$	20.8111			1.21580			0.607898	−	0.794015i	$-0.292013\pi$
$293$	20.8111			1.21580			0.607898	+	0.794015i	$0.292013\pi$
$294$		0			0
$295$	−19.5643			−1.13908
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.483998	−	0.838310i	−0.0279903	−	0.0484807i
$300$		0			0
$301$	−19.4022	+	0.480688i	−1.11832	+	0.0277064i
$302$		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	$-0.753929\pi$
$311$	4.35374	−	7.54089i	0.246878	−	0.427605i	0.962658	+	0.270721i	$0.0872620\pi$
$312$		0			0
$313$	9.54198	−	5.50906i	0.539345	−	0.311391i	−0.205469	−	0.978664i	$-0.565872\pi$
$313$	9.54198	−	5.50906i	0.539345	−	0.311391i	0.744813	+	0.667273i	$0.232539\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.92196	−	5.72844i	0.557273	−	0.321742i	−0.194777	−	0.980847i	$-0.562398\pi$
$317$	9.92196	−	5.72844i	0.557273	−	0.321742i	0.752050	+	0.659106i	$0.229065\pi$
$318$		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
$329$	−4.87699	−	7.98377i	−0.268877	−	0.440159i
$330$		0			0
$331$	−2.70596	−	4.68685i	−0.148733	−	0.257613i	0.782027	−	0.623245i	$-0.214186\pi$
$331$	−2.70596	−	4.68685i	−0.148733	−	0.257613i	−0.930759	+	0.365632i	$0.880853\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−30.3628			−1.65890
$336$		0			0
$337$	24.3709			1.32757			0.663783	−	0.747926i	$-0.268950\pi$
$337$	24.3709			1.32757			0.663783	+	0.747926i	$0.268950\pi$
$338$		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	0.693247	−	0.720700i	$-0.256180\pi$
$347$	7.74410	+	4.47106i	0.415725	+	0.240019i	−0.277522	+	0.960719i	$0.589513\pi$
$348$		0			0
$349$			11.1848i			0.598706i	0.954142	+	0.299353i	$0.0967709\pi$
$349$			11.1848i			0.598706i	−0.954142	+	0.299353i	$0.903229\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.75651	−	11.7026i	0.359613	−	0.622867i	−0.628284	−	0.777984i	$-0.716242\pi$
$353$	6.75651	−	11.7026i	0.359613	−	0.622867i	0.987896	+	0.155117i	$0.0495756\pi$
$354$		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
$365$		−	8.64413i		−	0.452454i
$366$		0			0
$367$	−0.922490	−	0.532600i	−0.0481536	−	0.0278015i	0.475730	−	0.879591i	$-0.342184\pi$
$367$	−0.922490	−	0.532600i	−0.0481536	−	0.0278015i	−0.523884	+	0.851790i	$0.675517\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−14.2683	−	7.77313i	−0.740774	−	0.403561i
$372$		0			0
$373$	2.16655	+	3.75257i	0.112180	+	0.194301i	0.916649	−	0.399694i	$-0.130883\pi$
$373$	2.16655	+	3.75257i	0.112180	+	0.194301i	−0.804469	+	0.593994i	$0.797550\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.37741			−0.173946
$378$		0			0
$379$	2.43862			0.125263			0.0626317	−	0.998037i	$-0.480051\pi$
$379$	2.43862			0.125263			0.0626317	+	0.998037i	$0.480051\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	18.7552	+	32.4849i	0.958344	+	1.65990i	0.726524	+	0.687141i	$0.241135\pi$
$383$	18.7552	+	32.4849i	0.958344	+	1.65990i	0.231819	+	0.972759i	$0.425532\pi$
$384$		0			0
$385$	0.630226	+	25.4380i	0.0321193	+	1.29644i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5.97057	−	3.44711i	−0.302720	−	0.174775i	0.340944	−	0.940083i	$-0.389253\pi$
$389$	−5.97057	−	3.44711i	−0.302720	−	0.174775i	−0.643664	+	0.765308i	$0.722587\pi$
$390$		0			0
$391$			0.0864467i			0.00437180i
$392$		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	$-0.820424\pi$
$401$	−16.1101	+	9.30115i	−0.804498	+	0.464477i	0.0405433	+	0.999178i	$0.487091\pi$
$402$		0			0
$403$	2.51073	−	4.34872i	0.125069	−	0.216625i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	0.648668i		−	0.0321533i
$408$		0			0
$409$	−1.89129	−	1.09193i	−0.0935181	−	0.0539927i	0.452512	−	0.891759i	$-0.350528\pi$
$409$	−1.89129	−	1.09193i	−0.0935181	−	0.0539927i	−0.546030	+	0.837766i	$0.683861\pi$
$410$		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
$419$	29.8666			1.45908			0.729540	−	0.683938i	$-0.239734\pi$
$419$	29.8666			1.45908			0.729540	+	0.683938i	$0.239734\pi$
$420$		0			0
$421$	−1.34630			−0.0656147			−0.0328073	−	0.999462i	$-0.510445\pi$
$421$	−1.34630			−0.0656147			−0.0328073	+	0.999462i	$0.510445\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.0439480	−	0.0761201i	−0.00213179	−	0.00369237i
$426$		0			0
$427$	−29.4281	+	17.9765i	−1.42413	+	0.869945i
$428$		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
$437$	−2.06504	+	3.57676i	−0.0987844	+	0.171100i
$438$		0			0
$439$	3.49248	−	2.01638i	0.166687	−	0.0962367i	−0.414336	−	0.910124i	$-0.635986\pi$
$439$	3.49248	−	2.01638i	0.166687	−	0.0962367i	0.581023	+	0.813887i	$0.302653\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−29.3085	+	16.9212i	−1.39249	+	0.803953i	−0.993590	−	0.113043i	$-0.963940\pi$
$443$	−29.3085	+	16.9212i	−1.39249	+	0.803953i	−0.398897	+	0.916996i	$0.630607\pi$
$444$		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	$-0.730563\pi$
$449$		−	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
$455$	−4.51558	+	2.75840i	−0.211693	+	0.129316i
$456$		0			0
$457$	17.7802	+	30.7963i	0.831724	+	1.44059i	0.896670	+	0.442700i	$0.145979\pi$
$457$	17.7802	+	30.7963i	0.831724	+	1.44059i	−0.0649459	+	0.997889i	$0.520687\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	32.5894			1.51784			0.758919	−	0.651185i	$-0.225728\pi$
$461$	32.5894			1.51784			0.758919	+	0.651185i	$0.225728\pi$
$462$		0			0
$463$	−21.2795			−0.988941			−0.494471	−	0.869194i	$-0.664638\pi$
$463$	−21.2795			−0.988941			−0.494471	+	0.869194i	$0.664638\pi$
$464$		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
$473$	30.4959	+	17.6068i	1.40221	+	0.809563i
$474$		0			0
$475$		−	4.19933i		−	0.192678i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.26814	−	9.12468i	0.240707	−	0.416917i	−0.720209	−	0.693758i	$-0.755954\pi$
$479$	5.26814	−	9.12468i	0.240707	−	0.416917i	0.960916	+	0.276840i	$0.0892873\pi$
$480$		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
$491$			7.79146i			0.351624i	0.984424	+	0.175812i	$0.0562550\pi$
$491$			7.79146i			0.351624i	−0.984424	+	0.175812i	$0.943745\pi$
$492$		0			0
$493$	0.261210	+	0.150809i	0.0117643	+	0.00679211i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.299454	−	12.0870i	−0.0134324	−	0.542175i
$498$		0			0
$499$	11.8950	+	20.6027i	0.532493	+	0.922305i	0.999280	+	0.0379356i	$0.0120782\pi$
$499$	11.8950	+	20.6027i	0.532493	+	0.922305i	−0.466787	+	0.884370i	$0.654589\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−15.1692			−0.676363			−0.338182	−	0.941081i	$-0.609812\pi$
$503$	−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	0.991179	−	0.132533i	$-0.0423112\pi$
$509$	8.59151	+	14.8809i	0.380812	+	0.659586i	−0.610367	+	0.792119i	$0.708978\pi$
$510$		0			0
$511$	−10.0241	−	5.46095i	−0.443440	−	0.241578i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−16.5400	−	9.54939i	−0.728840	−	0.420796i
$516$		0			0
$517$			16.9744i			0.746535i
$518$		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
$533$		−	11.2239i		−	0.486163i
$534$		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
$545$	−3.61121			−0.154687
$546$		0			0
$547$	−31.9694			−1.36691			−0.683456	−	0.729992i	$-0.739524\pi$
$547$	−31.9694			−1.36691			−0.683456	+	0.729992i	$0.739524\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7.20509	+	12.4796i	0.306947	+	0.531648i
$552$		0			0
$553$	16.3874	+	26.8267i	0.696865	+	1.14079i
$554$		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
$563$	14.0447	−	24.3261i	0.591913	−	1.02522i	−0.402062	−	0.915613i	$-0.631706\pi$
$563$	14.0447	−	24.3261i	0.591913	−	1.02522i	0.993975	−	0.109611i	$-0.0349605\pi$
$564$		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	−0.957501	−	0.288430i	$-0.906867\pi$
$569$	−28.3016	+	16.3399i	−1.18646	+	0.685005i	−0.228963	+	0.973435i	$0.573533\pi$
$570$		0			0
$571$	8.71183	−	15.0893i	0.364579	−	0.631469i	−0.624130	−	0.781321i	$-0.714546\pi$
$571$	8.71183	−	15.0893i	0.364579	−	0.631469i	0.988708	+	0.149852i	$0.0478797\pi$
$572$		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
$581$	8.59576	−	0.212960i	0.356612	−	0.00883506i
$582$		0			0
$583$	14.7402	+	25.5309i	0.610479	+	1.05738i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.0388			−0.579445			−0.289722	−	0.957111i	$-0.593563\pi$
$587$	−14.0388			−0.579445			−0.289722	+	0.957111i	$0.593563\pi$
$588$		0			0
$589$	−21.4248			−0.882792
$590$		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
$599$	9.51980	+	5.49626i	0.388969	+	0.224571i	0.681713	−	0.731620i	$-0.261235\pi$
$599$	9.51980	+	5.49626i	0.388969	+	0.224571i	−0.292745	+	0.956191i	$0.594569\pi$
$600$		0			0
$601$		−	35.9629i		−	1.46696i	−0.679712	−	0.733479i	$-0.737895\pi$
$601$		−	35.9629i		−	1.46696i	0.679712	−	0.733479i	$-0.262105\pi$
$602$		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	$-0.495181\pi$
$607$	21.8936	−	12.6403i	0.888634	−	0.513053i	0.873495	+	0.486832i	$0.161848\pi$
$608$		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
$617$			9.14453i			0.368145i	0.982913	+	0.184072i	$0.0589281\pi$
$617$			9.14453i			0.368145i	−0.982913	+	0.184072i	$0.941072\pi$
$618$		0			0
$619$	−21.4791	−	12.4010i	−0.863317	−	0.498437i	0.00180436	−	0.999998i	$-0.499426\pi$
$619$	−21.4791	−	12.4010i	−0.863317	−	0.498437i	−0.865122	+	0.501562i	$0.832759\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.93267	+	4.80087i	0.117495	+	0.192343i
$624$		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
$637$	0.346025	+	6.97908i	0.0137100	+	0.276521i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19.9823	+	11.5368i	0.789253	+	0.455676i	0.839700	−	0.543051i	$-0.182731\pi$
$641$	19.9823	+	11.5368i	0.789253	+	0.455676i	−0.0504463	+	0.998727i	$0.516064\pi$
$642$		0			0
$643$			18.2567i			0.719975i	0.932957	+	0.359988i	$0.117219\pi$
$643$			18.2567i			0.719975i	−0.932957	+	0.359988i	$0.882781\pi$
$644$		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	$-0.794930\pi$
$653$	−17.3561	+	10.0206i	−0.679197	+	0.392134i	0.120356	+	0.992731i	$0.461597\pi$
$654$		0			0
$655$	−10.5180	+	18.2177i	−0.410973	+	0.711826i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	34.3056i		−	1.33636i	−0.744002	−	0.668178i	$-0.767075\pi$
$659$		−	34.3056i		−	1.33636i	0.744002	−	0.668178i	$-0.232925\pi$
$660$		0			0
$661$	−24.6383	−	14.2249i	−0.958320	−	0.553286i	−0.0626645	−	0.998035i	$-0.519960\pi$
$661$	−24.6383	−	14.2249i	−0.958320	−	0.553286i	−0.895656	+	0.444748i	$0.853293\pi$
$662$		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
$671$	62.5676			2.41540
$672$		0			0
$673$	−25.9411			−0.999954			−0.499977	−	0.866039i	$-0.666658\pi$
$673$	−25.9411			−0.999954			−0.499977	+	0.866039i	$0.666658\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.1894	−	31.5050i	−0.699075	−	1.21083i	−0.968787	−	0.247893i	$-0.920262\pi$
$677$	−18.1894	−	31.5050i	−0.699075	−	1.21083i	0.269712	−	0.962941i	$-0.413071\pi$
$678$		0			0
$679$	0.0783342	+	3.16183i	0.00300619	+	0.121340i
$680$		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
$689$	−3.06522	+	5.30911i	−0.116775	+	0.202261i
$690$		0			0
$691$	31.1043	−	17.9581i	1.18326	−	0.683157i	0.226495	−	0.974012i	$-0.427273\pi$
$691$	31.1043	−	17.9581i	1.18326	−	0.683157i	0.956767	+	0.290856i	$0.0939400\pi$
$692$		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	$0.285569\pi$
$701$			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
$707$	−15.1733	+	27.8521i	−0.570652	+	1.04748i
$708$		0			0
$709$	9.22204	+	15.9730i	0.346341	+	0.599880i	0.985596	−	0.169114i	$-0.0540907\pi$
$709$	9.22204	+	15.9730i	0.346341	+	0.599880i	−0.639256	+	0.768994i	$0.720757\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.87796			0.182681
$714$		0			0
$715$	9.60064			0.359044
$716$		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
$725$	2.88897	+	1.66795i	0.107294	+	0.0619460i
$726$		0			0
$727$		−	32.8216i		−	1.21728i	−0.793445	−	0.608642i	$-0.791714\pi$
$727$		−	32.8216i		−	1.21728i	0.793445	−	0.608642i	$-0.208286\pi$
$728$		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	$-0.758616\pi$
$733$	−13.3583	+	7.71244i	−0.493402	+	0.284866i	0.232583	+	0.972577i	$0.425282\pi$
$734$		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
$743$		−	27.2228i		−	0.998709i	−0.866398	−	0.499355i	$-0.833570\pi$
$743$		−	27.2228i		−	0.998709i	0.866398	−	0.499355i	$-0.166430\pi$
$744$		0			0
$745$	24.9612	+	14.4113i	0.914507	+	0.527991i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−42.2794	+	25.8269i	−1.54485	+	0.943694i
$750$		0			0
$751$	−10.6549	−	18.4549i	−0.388804	−	0.673428i	0.603485	−	0.797374i	$-0.293778\pi$
$751$	−10.6549	−	18.4549i	−0.388804	−	0.673428i	−0.992289	+	0.123946i	$0.960445\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−31.0844			−1.13128
$756$		0			0
$757$	43.2079			1.57042			0.785209	−	0.619231i	$-0.212556\pi$
$757$	43.2079			1.57042			0.785209	+	0.619231i	$0.212556\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	24.8852	+	43.1024i	0.902087	+	1.56246i	0.824780	+	0.565453i	$0.191299\pi$
$761$	24.8852	+	43.1024i	0.902087	+	1.56246i	0.0773067	+	0.997007i	$0.475368\pi$
$762$		0			0
$763$	−2.28139	+	4.18771i	−0.0825920	+	0.151605i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.44185	+	4.87391i	0.304818	+	0.175987i
$768$		0			0
$769$			17.4769i			0.630232i	0.949053	+	0.315116i	$0.102043\pi$
$769$			17.4769i			0.630232i	−0.949053	+	0.315116i	$0.897957\pi$
$770$		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
$785$			6.99700i			0.249734i
$786$		0			0
$787$	−21.8422	−	12.6106i	−0.778591	−	0.449519i	0.0573400	−	0.998355i	$-0.481738\pi$
$787$	−21.8422	−	12.6106i	−0.778591	−	0.449519i	−0.835931	+	0.548835i	$0.815071\pi$
$788$		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
$797$	−29.6641			−1.05075			−0.525377	−	0.850869i	$-0.676076\pi$
$797$	−29.6641			−1.05075			−0.525377	+	0.850869i	$0.676076\pi$
$798$		0			0
$799$	0.315229			0.0111520
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10.3557	+	17.9365i	0.365443	+	0.632966i
$804$		0			0
$805$	−4.51384	−	2.45906i	−0.159092	−	0.0866705i
$806$		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
$815$	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	0.00162745	−	0.999999i	$-0.499482\pi$
$821$	24.8842	−	14.3669i	0.868465	−	0.501409i	0.866838	+	0.498590i	$0.166149\pi$
$822$		0			0
$823$	25.3472	−	43.9026i	0.883547	−	1.53035i	0.0361761	−	0.999345i	$-0.488482\pi$
$823$	25.3472	−	43.9026i	0.883547	−	1.53035i	0.847371	−	0.531002i	$-0.178184\pi$
$824$		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
$833$	0.284871	−	0.555214i	0.00987018	−	0.0192370i
$834$		0			0
$835$	3.91188	+	6.77557i	0.135376	+	0.234478i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12.3657			0.426913			0.213456	−	0.976953i	$-0.431528\pi$
$839$	12.3657			0.426913			0.213456	+	0.976953i	$0.431528\pi$
$840$		0			0
$841$	17.5527			0.605266
$842$		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
$853$		−	23.4951i		−	0.804457i	−0.915539	−	0.402229i	$-0.868236\pi$
$853$		−	23.4951i		−	0.804457i	0.915539	−	0.402229i	$-0.131764\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.75960	+	15.1721i	−0.299222	+	0.518268i	−0.975958	−	0.217958i	$-0.930060\pi$
$857$	−8.75960	+	15.1721i	−0.299222	+	0.518268i	0.676736	+	0.736226i	$0.263394\pi$
$858$		0			0
$859$	−11.9224	+	6.88338i	−0.406786	+	0.234858i	−0.689408	−	0.724373i	$-0.742129\pi$
$859$	−11.9224	+	6.88338i	−0.406786	+	0.234858i	0.282622	+	0.959231i	$0.408796\pi$
$860$		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
$869$		−	57.0368i		−	1.93484i
$870$		0			0
$871$	13.1013	+	7.56404i	0.443921	+	0.256298i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	31.7205	−	0.785875i	1.07235	−	0.0265674i
$876$		0			0
$877$	25.9895	+	45.0152i	0.877604	+	1.52005i	0.853963	+	0.520334i	$0.174193\pi$
$877$	25.9895	+	45.0152i	0.877604	+	1.52005i	0.0236415	+	0.999721i	$0.492474\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−37.6723			−1.26921			−0.634607	−	0.772835i	$-0.718838\pi$
$881$	−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	−0.796602	−	0.604504i	$-0.793371\pi$
$887$	−27.4541	−	47.5518i	−0.921817	−	1.59663i	−0.125215	−	0.992130i	$-0.539962\pi$
$888$		0			0
$889$	19.4547	−	0.481990i	0.652490	−	0.0161654i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	13.0427	+	7.53020i	0.436457	+	0.251989i
$894$		0			0
$895$		−	13.7311i		−	0.458981i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.50979	−	14.7394i	0.283817	−	0.491586i
$900$		0			0
$901$	0.474128	−	0.273738i	0.0157955	−	0.00911954i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−20.9965	+	12.1223i	−0.697946	+	0.402959i
$906$		0			0
$907$	4.32671	−	7.49408i	0.143666	−	0.248837i	−0.785208	−	0.619232i	$-0.787444\pi$
$907$	4.32671	−	7.49408i	0.143666	−	0.248837i	0.928874	+	0.370395i	$0.120778\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	11.1238i		−	0.368547i	−0.982875	−	0.184273i	$-0.941007\pi$
$911$		−	11.1238i		−	0.368547i	0.982875	−	0.184273i	$-0.0589932\pi$
$912$		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
$923$	−4.56178			−0.150153
$924$		0			0
$925$	−0.133232			−0.00438063
$926$		0			0
$927$		0			0
$928$		0			0
$929$	17.9035	+	31.0097i	0.587394	+	1.01740i	0.994572	+	0.104048i	$0.0331795\pi$
$929$	17.9035	+	31.0097i	0.587394	+	1.01740i	−0.407178	+	0.913349i	$0.633487\pi$
$930$		0			0
$931$	25.0496	−	16.1671i	0.820967	−	0.529857i
$932$		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	−0.415357	−	0.909658i	$-0.636343\pi$
$941$	17.7953	−	30.8223i	0.580109	−	1.00478i	0.995466	−	0.0951193i	$-0.0303233\pi$
$942$		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	−0.876708	−	0.481022i	$-0.840266\pi$
$947$	−27.6494	+	15.9634i	−0.898485	+	0.518741i	−0.0217769	+	0.999763i	$0.506932\pi$
$948$		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	$-0.886135\pi$
$953$		−	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
$959$	−35.7487	−	19.4753i	−1.15438	−	0.628889i
$960$		0			0
$961$	−2.84783	−	4.93258i	−0.0918653	−	0.159115i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	27.2397			0.876877
$966$		0			0
$967$	−15.5887			−0.501298			−0.250649	−	0.968078i	$-0.580644\pi$
$967$	−15.5887			−0.501298			−0.250649	+	0.968078i	$0.580644\pi$
$968$		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
$977$	−37.6105	−	21.7144i	−1.20327	−	0.694706i	−0.241986	−	0.970280i	$-0.577799\pi$
$977$	−37.6105	−	21.7144i	−1.20327	−	0.694706i	−0.961280	+	0.275573i	$0.911132\pi$
$978$		0			0
$979$		−	10.2072i		−	0.326224i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	17.4704	−	30.2596i	0.557218	−	0.965130i	−0.440509	−	0.897748i	$-0.645202\pi$
$983$	17.4704	−	30.2596i	0.557218	−	0.965130i	0.997727	−	0.0673821i	$-0.0214646\pi$
$984$		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
$995$			27.2514i			0.863927i
$996$		0			0
$997$	−43.7264	−	25.2454i	−1.38483	−	0.799531i	−0.392102	−	0.919922i	$-0.628252\pi$
$997$	−43.7264	−	25.2454i	−1.38483	−	0.799531i	−0.992727	+	0.120391i	$0.961585\pi$
$998$		0			0
$999$		0			0

Twists

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

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

\(f(q)\)	\(=\)	\(q+(-1.00175 - 1.73509i) q^{5} +(-2.64494 + 0.0655283i) q^{7} +(4.15726 + 2.40020i) q^{11} +0.998235i q^{13} +(0.0445736 - 0.0772037i) q^{17} +(3.68849 - 2.12955i) q^{19} +(-0.839792 + 0.484854i) q^{23} +(0.492982 - 0.853871i) q^{25} +3.38338i q^{29} +(-4.35640 - 2.51517i) q^{31} +(2.76327 + 4.52356i) q^{35} +(-0.0675641 - 0.117024i) q^{37} -11.2438 q^{41} +7.33558 q^{43} +(1.76803 + 3.06231i) q^{47} +(6.99141 - 0.346637i) q^{49} +(5.31849 + 3.07063i) q^{53} -9.61761i q^{55} +(4.88252 - 8.45678i) q^{59} +(11.2876 - 6.51692i) q^{61} +(1.73203 - 0.999985i) q^{65} +(7.57741 - 13.1245i) q^{67} +4.56985i q^{71} +(3.73647 + 2.15725i) q^{73} +(-11.1530 - 6.07596i) q^{77} +(-5.94084 - 10.2898i) q^{79} -3.24989 q^{83} -0.178607 q^{85} +(-1.06316 - 1.84146i) q^{89} +(-0.0654127 - 2.64027i) q^{91} +(-7.38992 - 4.26657i) q^{95} -1.19543i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} - 16 q^{25} + 24 q^{31} - 4 q^{37} + 8 q^{43} - 4 q^{49} + 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} - 24 q^{85} - 36 q^{91}+O(q^{100}) \) 32 * q - 8 * q^7 - 16 * q^25 + 24 * q^31 - 4 * q^37 + 8 * q^43 - 4 * q^49 + 12 * q^61 + 4 * q^67 - 36 * q^73 + 28 * q^79 - 24 * q^85 - 36 * q^91

Introduction

L-functions

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