LMFDB - Embedded newform 2736.2.s.z.1873.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(577,2736)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2736.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.s (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.954288.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 $ x^6 - x^5 - 2x^4 + 3x^3 - 6x^2 - 9x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 57)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1873.2
Root			$1.71903 + 0.211943i$ of defining polynomial
Character	$\chi$	$=$	2736.1873
Dual form			2736.2.s.z.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.675970 - 1.17081i) q^{5} -0.351939 q^{7} +O(q^{10})$	$q+(0.675970 - 1.17081i) q^{5} -0.351939 q^{7} +5.52420 q^{11} +(2.58613 + 4.47931i) q^{13} +(2.43807 + 3.61328i) q^{19} +(-4.41016 - 7.63862i) q^{23} +(1.58613 + 2.74726i) q^{25} +(1.35194 + 2.34163i) q^{29} +0.524200 q^{31} +(-0.237900 + 0.412055i) q^{35} -1.00000 q^{37} +(-1.35194 + 2.34163i) q^{41} +(-3.26210 + 5.65012i) q^{43} +(3.00000 + 5.19615i) q^{47} -6.87614 q^{49} +(-2.02791 - 3.51244i) q^{53} +(3.73419 - 6.46781i) q^{55} +(2.76210 - 4.78410i) q^{59} +(0.938069 + 1.62478i) q^{61} +6.99258 q^{65} +(5.99629 + 10.3859i) q^{67} +(-2.52420 + 4.37204i) q^{71} +(-3.85194 + 6.67175i) q^{73} -1.94418 q^{77} +(3.91016 - 6.77260i) q^{79} +8.34452 q^{83} +(-2.32403 - 4.02534i) q^{89} +(-0.910161 - 1.57644i) q^{91} +(5.87854 - 0.412055i) q^{95} +(6.90645 - 11.9623i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 6 * q + 2 * q^5 + 2 * q^7	$ 6 q + 2 q^{5} + 2 q^{7} + q^{13} - 4 q^{19} - 14 q^{23} - 5 q^{25} + 4 q^{29} - 30 q^{31} - 18 q^{35} - 6 q^{37} - 4 q^{41} - 3 q^{43} + 18 q^{47} - 4 q^{49} - 6 q^{53} + 12 q^{55} - 13 q^{61} - 12 q^{65} + 9 q^{67} + 18 q^{71} - 19 q^{73} - 24 q^{77} + 11 q^{79} - 8 q^{83} - 16 q^{89} + 7 q^{91} + 2 q^{95} + 2 q^{97}+O(q^{100}) $ 6 * q + 2 * q^5 + 2 * q^7 + q^13 - 4 * q^19 - 14 * q^23 - 5 * q^25 + 4 * q^29 - 30 * q^31 - 18 * q^35 - 6 * q^37 - 4 * q^41 - 3 * q^43 + 18 * q^47 - 4 * q^49 - 6 * q^53 + 12 * q^55 - 13 * q^61 - 12 * q^65 + 9 * q^67 + 18 * q^71 - 19 * q^73 - 24 * q^77 + 11 * q^79 - 8 * q^83 - 16 * q^89 + 7 * q^91 + 2 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

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
$5$	0.675970	−	1.17081i	0.302303	−	0.523604i	−0.674354	−	0.738408i	$-0.735578\pi$
$5$	0.675970	−	1.17081i	0.302303	−	0.523604i	0.976657	+	0.214804i	$0.0689113\pi$
$6$		0			0
$7$	−0.351939			−0.133021			−0.0665103	−	0.997786i	$-0.521187\pi$
$7$	−0.351939			−0.133021			−0.0665103	+	0.997786i	$0.521187\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.52420			1.66561			0.832804	−	0.553567i	$-0.186734\pi$
$11$	5.52420			1.66561			0.832804	+	0.553567i	$0.186734\pi$
$12$		0			0
$13$	2.58613	+	4.47931i	0.717263	+	1.24234i	0.962080	+	0.272767i	$0.0879389\pi$
$13$	2.58613	+	4.47931i	0.717263	+	1.24234i	−0.244817	+	0.969569i	$0.578728\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	2.43807	+	3.61328i	0.559331	+	0.828944i
$19$	2.43807	+	3.61328i	0.559331	+	0.828944i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.41016	−	7.63862i	−0.919582	−	1.59276i	−0.800051	−	0.599932i	$-0.795194\pi$
$23$	−4.41016	−	7.63862i	−0.919582	−	1.59276i	−0.119531	−	0.992830i	$-0.538139\pi$
$24$		0			0
$25$	1.58613	+	2.74726i	0.317226	+	0.549452i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.35194	+	2.34163i	0.251049	+	0.434829i	0.963815	−	0.266573i	$-0.0858912\pi$
$29$	1.35194	+	2.34163i	0.251049	+	0.434829i	−0.712766	+	0.701402i	$0.752558\pi$
$30$		0			0
$31$	0.524200			0.0941490			0.0470745	−	0.998891i	$-0.485010\pi$
$31$	0.524200			0.0941490			0.0470745	+	0.998891i	$0.485010\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.237900	+	0.412055i	−0.0402125	+	0.0696500i
$36$		0			0
$37$	−1.00000			−0.164399			−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000			−0.164399			−0.0821995	+	0.996616i	$0.526194\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.35194	+	2.34163i	−0.211137	+	0.365701i	−0.952071	−	0.305878i	$-0.901050\pi$
$41$	−1.35194	+	2.34163i	−0.211137	+	0.365701i	0.740933	+	0.671579i	$0.234383\pi$
$42$		0			0
$43$	−3.26210	+	5.65012i	−0.497466	+	0.861636i	−0.999996	−	0.00292406i	$-0.999069\pi$
$43$	−3.26210	+	5.65012i	−0.497466	+	0.861636i	0.502530	+	0.864560i	$0.332403\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	0.997503	−	0.0706177i	$-0.0224970\pi$
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	−0.559908	+	0.828554i	$0.689164\pi$
$48$		0			0
$49$	−6.87614			−0.982306
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.02791	−	3.51244i	−0.278555	−	0.482471i	0.692471	−	0.721446i	$-0.256522\pi$
$53$	−2.02791	−	3.51244i	−0.278555	−	0.482471i	−0.971026	+	0.238975i	$0.923189\pi$
$54$		0			0
$55$	3.73419	−	6.46781i	0.503518	−	0.872119i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.76210	−	4.78410i	0.359595	−	0.622836i	−0.628298	−	0.777972i	$-0.716248\pi$
$59$	2.76210	−	4.78410i	0.359595	−	0.622836i	0.987893	+	0.155136i	$0.0495816\pi$
$60$		0			0
$61$	0.938069	+	1.62478i	0.120107	+	0.208032i	0.919810	−	0.392364i	$-0.128343\pi$
$61$	0.938069	+	1.62478i	0.120107	+	0.208032i	−0.799702	+	0.600397i	$0.795009\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.99258			0.867323
$66$		0			0
$67$	5.99629	+	10.3859i	0.732564	+	1.26884i	0.955784	+	0.294069i	$0.0950096\pi$
$67$	5.99629	+	10.3859i	0.732564	+	1.26884i	−0.223221	+	0.974768i	$0.571657\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.52420	+	4.37204i	−0.299567	+	0.518866i	−0.976037	−	0.217605i	$-0.930176\pi$
$71$	−2.52420	+	4.37204i	−0.299567	+	0.518866i	0.676470	+	0.736471i	$0.263509\pi$
$72$		0			0
$73$	−3.85194	+	6.67175i	−0.450835	+	0.780870i	−0.998438	−	0.0558687i	$-0.982207\pi$
$73$	−3.85194	+	6.67175i	−0.450835	+	0.780870i	0.547603	+	0.836738i	$0.315541\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.94418			−0.221560
$78$		0			0
$79$	3.91016	−	6.77260i	0.439927	−	0.761977i	−0.557756	−	0.830005i	$-0.688337\pi$
$79$	3.91016	−	6.77260i	0.439927	−	0.761977i	0.997683	+	0.0680283i	$0.0216708\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.34452			0.915930			0.457965	−	0.888970i	$-0.348578\pi$
$83$	8.34452			0.915930			0.457965	+	0.888970i	$0.348578\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.32403	−	4.02534i	−0.246347	−	0.426685i	0.716163	−	0.697933i	$-0.245897\pi$
$89$	−2.32403	−	4.02534i	−0.246347	−	0.426685i	−0.962509	+	0.271248i	$0.912564\pi$
$90$		0			0
$91$	−0.910161	−	1.57644i	−0.0954108	−	0.165256i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.87854	−	0.412055i	0.603126	−	0.0422760i
$96$		0			0
$97$	6.90645	−	11.9623i	0.701244	−	1.21459i	−0.266786	−	0.963756i	$-0.585962\pi$
$97$	6.90645	−	11.9623i	0.701244	−	1.21459i	0.968030	−	0.250834i	$-0.0807049\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.17226	−	2.03041i	−0.116644	−	0.202034i	0.801792	−	0.597604i	$-0.203880\pi$
$101$	−1.17226	−	2.03041i	−0.116644	−	0.202034i	−0.918436	+	0.395570i	$0.870547\pi$
$102$		0			0
$103$	16.1042			1.58680			0.793398	−	0.608703i	$-0.208310\pi$
$103$	16.1042			1.58680			0.793398	+	0.608703i	$0.208310\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.592243			−0.0572543			−0.0286272	−	0.999590i	$-0.509114\pi$
$107$	−0.592243			−0.0572543			−0.0286272	+	0.999590i	$0.509114\pi$
$108$		0			0
$109$	6.79001	−	11.7606i	0.650365	−	1.12647i	−0.332669	−	0.943043i	$-0.607949\pi$
$109$	6.79001	−	11.7606i	0.650365	−	1.12647i	0.983034	−	0.183422i	$-0.0587173\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.9926			1.22224			0.611120	−	0.791538i	$-0.290719\pi$
$113$	12.9926			1.22224			0.611120	+	0.791538i	$0.290719\pi$
$114$		0			0
$115$	−11.9245			−1.11197
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	19.5168			1.77425
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0484			0.988199
$126$		0			0
$127$	5.35194	+	9.26983i	0.474908	+	0.822564i	0.999587	−	0.0287355i	$-0.00914807\pi$
$127$	5.35194	+	9.26983i	0.474908	+	0.822564i	−0.524679	+	0.851300i	$0.675815\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.64806	−	2.85453i	0.143992	−	0.249401i	−0.785005	−	0.619490i	$-0.787339\pi$
$131$	1.64806	−	2.85453i	0.143992	−	0.249401i	0.928996	+	0.370089i	$0.120673\pi$
$132$		0			0
$133$	−0.858052	−	1.27166i	−0.0744026	−	0.110267i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.52420	+	14.7643i	0.728272	+	1.26140i	0.957613	+	0.288057i	$0.0930094\pi$
$137$	8.52420	+	14.7643i	0.728272	+	1.26140i	−0.229342	+	0.973346i	$0.573657\pi$
$138$		0			0
$139$	−8.96598	−	15.5295i	−0.760484	−	1.31720i	−0.942601	−	0.333921i	$-0.891628\pi$
$139$	−8.96598	−	15.5295i	−0.760484	−	1.31720i	0.182117	−	0.983277i	$-0.441705\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	14.2863	+	24.7446i	1.19468	+	2.06925i
$144$		0			0
$145$	3.65548			0.303571
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.972091	−	1.68371i	0.0796368	−	0.137935i	−0.823456	−	0.567379i	$-0.807957\pi$
$149$	0.972091	−	1.68371i	0.0796368	−	0.137935i	0.903093	+	0.429444i	$0.141291\pi$
$150$		0			0
$151$	−13.6406			−1.11006			−0.555030	−	0.831830i	$-0.687293\pi$
$151$	−13.6406			−1.11006			−0.555030	+	0.831830i	$0.687293\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.354343	−	0.613740i	0.0284615	−	0.0492968i
$156$		0			0
$157$	−8.32032	+	14.4112i	−0.664034	+	1.15014i	0.315512	+	0.948921i	$0.397824\pi$
$157$	−8.32032	+	14.4112i	−0.664034	+	1.15014i	−0.979546	+	0.201219i	$0.935510\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.55211	+	2.68833i	0.122323	+	0.211870i
$162$		0			0
$163$	1.99258			0.156071			0.0780355	−	0.996951i	$-0.475135\pi$
$163$	1.99258			0.156071			0.0780355	+	0.996951i	$0.475135\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.29372	−	2.24078i	−0.100111	−	0.173397i	0.811619	−	0.584187i	$-0.198586\pi$
$167$	−1.29372	−	2.24078i	−0.100111	−	0.173397i	−0.911730	+	0.410790i	$0.865253\pi$
$168$		0			0
$169$	−6.87614	+	11.9098i	−0.528934	+	0.916140i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.82032	−	15.2772i	0.670597	−	1.16151i	−0.307139	−	0.951665i	$-0.599372\pi$
$173$	8.82032	−	15.2772i	0.670597	−	1.16151i	0.977735	−	0.209843i	$-0.0672952\pi$
$174$		0			0
$175$	−0.558221	−	0.966868i	−0.0421976	−	0.0730883i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.22808			0.166534			0.0832672	−	0.996527i	$-0.473465\pi$
$179$	2.22808			0.166534			0.0832672	+	0.996527i	$0.473465\pi$
$180$		0			0
$181$	−4.73419	−	8.19986i	−0.351890	−	0.609491i	0.634691	−	0.772766i	$-0.281127\pi$
$181$	−4.73419	−	8.19986i	−0.351890	−	0.609491i	−0.986581	+	0.163275i	$0.947794\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.675970	+	1.17081i	−0.0496983	+	0.0860799i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−22.5726			−1.63330			−0.816648	−	0.577136i	$-0.804170\pi$
$191$	−22.5726			−1.63330			−0.816648	+	0.577136i	$0.804170\pi$
$192$		0			0
$193$	6.93807	−	12.0171i	0.499413	−	0.865009i	−0.500587	−	0.865686i	$-0.666882\pi$
$193$	6.93807	−	12.0171i	0.499413	−	0.865009i	1.00000		0.000677488i	$0.000215651\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.30354			−0.591603			−0.295801	−	0.955249i	$-0.595587\pi$
$197$	−8.30354			−0.591603			−0.295801	+	0.955249i	$0.595587\pi$
$198$		0			0
$199$	−3.44178	−	5.96134i	−0.243981	−	0.422588i	0.717863	−	0.696184i	$-0.245120\pi$
$199$	−3.44178	−	5.96134i	−0.243981	−	0.422588i	−0.961845	+	0.273596i	$0.911787\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.475800	−	0.824110i	−0.0333946	−	0.0578412i
$204$		0			0
$205$	1.82774	+	3.16574i	0.127655	+	0.221105i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	13.4684	+	19.9605i	0.931627	+	1.38070i
$210$		0			0
$211$	−4.31792	+	7.47885i	−0.297258	+	0.514865i	−0.975508	−	0.219966i	$-0.929405\pi$
$211$	−4.31792	+	7.47885i	−0.297258	+	0.514865i	0.678250	+	0.734831i	$0.262739\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.41016	+	7.63862i	0.300770	+	0.520950i
$216$		0			0
$217$	−0.184486			−0.0125238
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	6.13824	−	10.6317i	0.411047	−	0.711954i	−0.583958	−	0.811784i	$-0.698497\pi$
$223$	6.13824	−	10.6317i	0.411047	−	0.711954i	0.995004	+	0.0998301i	$0.0318299\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.475800			0.0315800			0.0157900	−	0.999875i	$-0.494974\pi$
$227$	0.475800			0.0315800			0.0157900	+	0.999875i	$0.494974\pi$
$228$		0			0
$229$	−5.17226			−0.341793			−0.170896	−	0.985289i	$-0.554666\pi$
$229$	−5.17226			−0.341793			−0.170896	+	0.985289i	$0.554666\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.34452	+	9.25698i	−0.350131	+	0.606445i	−0.986272	−	0.165128i	$-0.947196\pi$
$233$	−5.34452	+	9.25698i	−0.350131	+	0.606445i	0.636141	+	0.771573i	$0.280530\pi$
$234$		0			0
$235$	8.11164			0.529145
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.475800			−0.0307770			−0.0153885	−	0.999882i	$-0.504899\pi$
$239$	−0.475800			−0.0307770			−0.0153885	+	0.999882i	$0.504899\pi$
$240$		0			0
$241$	0.320321	+	0.554813i	0.0206337	+	0.0357386i	0.876158	−	0.482024i	$-0.160098\pi$
$241$	0.320321	+	0.554813i	0.0206337	+	0.0357386i	−0.855524	+	0.517763i	$0.826765\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.64806	+	8.05068i	−0.296954	+	0.514339i
$246$		0			0
$247$	−9.87985	+	20.2653i	−0.628640	+	1.28945i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.52420	+	4.37204i	0.159326	+	0.275961i	0.934626	−	0.355633i	$-0.115735\pi$
$251$	2.52420	+	4.37204i	0.159326	+	0.275961i	−0.775300	+	0.631593i	$0.782401\pi$
$252$		0			0
$253$	−24.3626	−	42.1973i	−1.53166	−	2.65292i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.3724	+	17.9656i	0.647014	+	1.12066i	0.983832	+	0.179092i	$0.0573160\pi$
$257$	10.3724	+	17.9656i	0.647014	+	1.12066i	−0.336818	+	0.941570i	$0.609351\pi$
$258$		0			0
$259$	0.351939			0.0218684
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−8.52420	+	14.7643i	−0.525625	+	0.910409i	0.473930	+	0.880563i	$0.342835\pi$
$263$	−8.52420	+	14.7643i	−0.525625	+	0.910409i	−0.999555	+	0.0298460i	$0.990498\pi$
$264$		0			0
$265$	−5.48322			−0.336831
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.55211	−	7.88448i	0.277547	−	0.480725i	−0.693228	−	0.720719i	$-0.743812\pi$
$269$	4.55211	−	7.88448i	0.277547	−	0.480725i	0.970775	+	0.239993i	$0.0771453\pi$
$270$		0			0
$271$	13.2207	−	22.8989i	0.803098	−	1.39101i	−0.114470	−	0.993427i	$-0.536517\pi$
$271$	13.2207	−	22.8989i	0.803098	−	1.39101i	0.917568	−	0.397580i	$-0.130150\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.76210	+	15.1764i	0.528374	+	0.915171i
$276$		0			0
$277$	26.1574			1.57165			0.785824	−	0.618451i	$-0.212239\pi$
$277$	26.1574			1.57165			0.785824	+	0.618451i	$0.212239\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.2002	−	21.1313i	−0.727801	−	1.26059i	−0.957811	−	0.287400i	$-0.907209\pi$
$281$	−12.2002	−	21.1313i	−0.727801	−	1.26059i	0.230010	−	0.973188i	$-0.426124\pi$
$282$		0			0
$283$	−0.172260	+	0.298364i	−0.0102398	+	0.0177359i	−0.871100	−	0.491106i	$-0.836593\pi$
$283$	−0.172260	+	0.298364i	−0.0102398	+	0.0177359i	0.860860	+	0.508842i	$0.169926\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.475800	−	0.824110i	0.0280856	−	0.0486457i
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−17.2813			−1.00958			−0.504792	−	0.863241i	$-0.668431\pi$
$293$	−17.2813			−1.00958			−0.504792	+	0.863241i	$0.668431\pi$
$294$		0			0
$295$	−3.73419	−	6.46781i	−0.217413	−	0.376570i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	22.8105	−	39.5089i	1.31917	−	2.28486i
$300$		0			0
$301$	1.14806	−	1.98850i	0.0661731	−	0.114615i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.53643			0.145235
$306$		0			0
$307$	4.43807	−	7.68696i	0.253294	−	0.438718i	−0.711137	−	0.703054i	$-0.751819\pi$
$307$	4.43807	−	7.68696i	0.253294	−	0.438718i	0.964431	+	0.264336i	$0.0851527\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−11.1648			−0.633100			−0.316550	−	0.948576i	$-0.602525\pi$
$311$	−11.1648			−0.633100			−0.316550	+	0.948576i	$0.602525\pi$
$312$		0			0
$313$	2.25839	+	3.91165i	0.127652	+	0.221099i	0.922766	−	0.385360i	$-0.125923\pi$
$313$	2.25839	+	3.91165i	0.127652	+	0.221099i	−0.795115	+	0.606459i	$0.792589\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.50371	−	4.33655i	−0.140622	−	0.243565i	0.787109	−	0.616814i	$-0.211577\pi$
$317$	−2.50371	−	4.33655i	−0.140622	−	0.243565i	−0.927731	+	0.373249i	$0.878244\pi$
$318$		0			0
$319$	7.46838	+	12.9356i	0.418149	+	0.724256i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−8.20388	+	14.2095i	−0.455069	+	0.788203i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.05582	−	1.82873i	−0.0582091	−	0.100821i
$330$		0			0
$331$	10.1797			0.559526			0.279763	−	0.960069i	$-0.409744\pi$
$331$	10.1797			0.559526			0.279763	+	0.960069i	$0.409744\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	16.2132			0.885824
$336$		0			0
$337$	−11.7584	+	20.3661i	−0.640520	+	1.10941i	0.344796	+	0.938677i	$0.387948\pi$
$337$	−11.7584	+	20.3661i	−0.640520	+	1.10941i	−0.985317	+	0.170736i	$0.945385\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.89578			0.156815
$342$		0			0
$343$	4.88356			0.263687
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.23790	+	10.8044i	−0.334868	+	0.580008i	−0.983459	−	0.181128i	$-0.942025\pi$
$347$	−6.23790	+	10.8044i	−0.334868	+	0.580008i	0.648591	+	0.761137i	$0.275358\pi$
$348$		0			0
$349$	6.23550			0.333778			0.166889	−	0.985976i	$-0.446628\pi$
$349$	6.23550			0.333778			0.166889	+	0.985976i	$0.446628\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−29.8081			−1.58652			−0.793262	−	0.608880i	$-0.791619\pi$
$353$	−29.8081			−1.58652			−0.793262	+	0.608880i	$0.791619\pi$
$354$		0			0
$355$	3.41256	+	5.91073i	0.181120	+	0.313709i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.87614	−	11.9098i	0.362909	−	0.628576i	−0.625529	−	0.780201i	$-0.715117\pi$
$359$	6.87614	−	11.9098i	0.362909	−	0.628576i	0.988438	+	0.151624i	$0.0484503\pi$
$360$		0			0
$361$	−7.11164	+	17.6189i	−0.374297	+	0.927309i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5.20759	+	9.01981i	0.272578	+	0.472118i
$366$		0			0
$367$	9.25468	+	16.0296i	0.483090	+	0.836737i	0.999811	−	0.0194166i	$-0.00618090\pi$
$367$	9.25468	+	16.0296i	0.483090	+	0.836737i	−0.516721	+	0.856154i	$0.672848\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.713701	+	1.23617i	0.0370535	+	0.0641785i
$372$		0			0
$373$	6.53162			0.338194			0.169097	−	0.985599i	$-0.445915\pi$
$373$	6.53162			0.338194			0.169097	+	0.985599i	$0.445915\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.99258	+	12.1115i	−0.360136	+	0.623774i
$378$		0			0
$379$	−24.8687			−1.27742			−0.638710	−	0.769447i	$-0.720532\pi$
$379$	−24.8687			−1.27742			−0.638710	+	0.769447i	$0.720532\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.99018	−	8.64324i	0.254986	−	0.441649i	−0.709906	−	0.704297i	$-0.751262\pi$
$383$	4.99018	−	8.64324i	0.254986	−	0.441649i	0.964892	+	0.262648i	$0.0845957\pi$
$384$		0			0
$385$	−1.31421	+	2.27628i	−0.0669783	+	0.116010i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.54469	−	14.7998i	−0.433233	−	0.750382i	0.563917	−	0.825832i	$-0.309294\pi$
$389$	−8.54469	−	14.7998i	−0.433233	−	0.750382i	−0.997150	+	0.0754502i	$0.975961\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5.28630	−	9.15614i	−0.265983	−	0.460695i
$396$		0			0
$397$	0.821627	−	1.42310i	0.0412363	−	0.0714233i	−0.844671	−	0.535286i	$-0.820204\pi$
$397$	0.821627	−	1.42310i	0.0412363	−	0.0714233i	0.885907	+	0.463863i	$0.153537\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.67597	+	6.36697i	−0.183569	+	0.317951i	−0.943093	−	0.332528i	$-0.892098\pi$
$401$	−3.67597	+	6.36697i	−0.183569	+	0.317951i	0.759524	+	0.650479i	$0.225432\pi$
$402$		0			0
$403$	1.35565	+	2.34805i	0.0675297	+	0.116965i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.52420			−0.273824
$408$		0			0
$409$	−1.03773	−	1.79740i	−0.0513125	−	0.0888758i	0.839228	−	0.543779i	$-0.183007\pi$
$409$	−1.03773	−	1.79740i	−0.0513125	−	0.0888758i	−0.890541	+	0.454903i	$0.849674\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.972091	+	1.68371i	−0.0478335	+	0.0828500i
$414$		0			0
$415$	5.64064	−	9.76988i	0.276888	−	0.479585i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8.22808			0.401968			0.200984	−	0.979595i	$-0.435586\pi$
$419$	8.22808			0.401968			0.200984	+	0.979595i	$0.435586\pi$
$420$		0			0
$421$	−2.79001	+	4.83244i	−0.135977	+	0.235519i	−0.925970	−	0.377597i	$-0.876751\pi$
$421$	−2.79001	+	4.83244i	−0.135977	+	0.235519i	0.789993	+	0.613115i	$0.210084\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.330143	−	0.571825i	−0.0159768	−	0.0276726i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−14.0484	−	24.3325i	−0.676688	−	1.17206i	−0.975973	−	0.217893i	$-0.930081\pi$
$431$	−14.0484	−	24.3325i	−0.676688	−	1.17206i	0.299285	−	0.954164i	$-0.403252\pi$
$432$		0			0
$433$	−14.1661	−	24.5365i	−0.680782	−	1.17915i	−0.974743	−	0.223331i	$-0.928307\pi$
$433$	−14.1661	−	24.5365i	−0.680782	−	1.17915i	0.293961	−	0.955817i	$-0.405026\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	16.8482	−	34.5587i	0.805960	−	1.65316i
$438$		0			0
$439$	−5.16855	+	8.95219i	−0.246681	+	0.427265i	−0.962603	−	0.270916i	$-0.912673\pi$
$439$	−5.16855	+	8.95219i	−0.246681	+	0.427265i	0.715922	+	0.698181i	$0.246007\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.475800	+	0.824110i	0.0226060	+	0.0391547i	0.877107	−	0.480295i	$-0.159470\pi$
$443$	0.475800	+	0.824110i	0.0226060	+	0.0391547i	−0.854501	+	0.519450i	$0.826137\pi$
$444$		0			0
$445$	−6.28390			−0.297885
$446$		0			0
$447$		0			0
$448$		0			0
$449$	21.8639			1.03182			0.515911	−	0.856642i	$-0.327454\pi$
$449$	21.8639			1.03182			0.515911	+	0.856642i	$0.327454\pi$
$450$		0			0
$451$	−7.46838	+	12.9356i	−0.351672	+	0.609114i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.46096			−0.115372
$456$		0			0
$457$	21.7645			1.01810			0.509050	−	0.860737i	$-0.329997\pi$
$457$	21.7645			1.01810			0.509050	+	0.860737i	$0.329997\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.5521	−	18.2768i	0.491461	−	0.851235i	−0.508491	−	0.861067i	$-0.669796\pi$
$461$	10.5521	−	18.2768i	0.491461	−	0.851235i	0.999952	+	0.00983244i	$0.00312981\pi$
$462$		0			0
$463$	−5.16745			−0.240152			−0.120076	−	0.992765i	$-0.538314\pi$
$463$	−5.16745			−0.240152			−0.120076	+	0.992765i	$0.538314\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−36.8007			−1.70293			−0.851466	−	0.524410i	$-0.824286\pi$
$467$	−36.8007			−1.70293			−0.851466	+	0.524410i	$0.824286\pi$
$468$		0			0
$469$	−2.11033	−	3.65520i	−0.0974460	−	0.168781i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−18.0205	+	31.2124i	−0.828583	+	1.43515i
$474$		0			0
$475$	−6.05953	+	12.4291i	−0.278030	+	0.570288i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7.28870	−	12.6244i	−0.333029	−	0.576824i	0.650075	−	0.759870i	$-0.274737\pi$
$479$	−7.28870	−	12.6244i	−0.333029	−	0.576824i	−0.983104	+	0.183046i	$0.941404\pi$
$480$		0			0
$481$	−2.58613	−	4.47931i	−0.117917	−	0.204239i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.33710	−	16.1723i	−0.423976	−	0.734348i
$486$		0			0
$487$	−7.04840			−0.319393			−0.159697	−	0.987166i	$-0.551052\pi$
$487$	−7.04840			−0.319393			−0.159697	+	0.987166i	$0.551052\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.22808	+	9.05530i	−0.235940	+	0.408660i	−0.959545	−	0.281554i	$-0.909150\pi$
$491$	−5.22808	+	9.05530i	−0.235940	+	0.408660i	0.723606	+	0.690214i	$0.242484\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.888365	−	1.53869i	0.0398486	−	0.0690198i
$498$		0			0
$499$	2.23659	−	3.87390i	0.100124	−	0.173419i	−0.811612	−	0.584197i	$-0.801409\pi$
$499$	2.23659	−	3.87390i	0.100124	−	0.173419i	0.911735	+	0.410778i	$0.134743\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−11.5242	−	19.9605i	−0.513839	−	0.889995i	−0.999871	−	0.0160539i	$-0.994890\pi$
$503$	−11.5242	−	19.9605i	−0.513839	−	0.889995i	0.486032	−	0.873941i	$-0.338444\pi$
$504$		0			0
$505$	−3.16965			−0.141048
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−13.1723	−	22.8150i	−0.583850	−	1.01126i	−0.995018	−	0.0996984i	$-0.968212\pi$
$509$	−13.1723	−	22.8150i	−0.583850	−	1.01126i	0.411168	−	0.911560i	$-0.365121\pi$
$510$		0			0
$511$	1.35565	−	2.34805i	0.0599704	−	0.103872i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.8860	−	18.8550i	0.479693	−	0.830852i
$516$		0			0
$517$	16.5726	+	28.7046i	0.728862	+	1.26243i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−42.6332			−1.86780			−0.933898	−	0.357540i	$-0.883615\pi$
$521$	−42.6332			−1.86780			−0.933898	+	0.357540i	$0.883615\pi$
$522$		0			0
$523$	−5.94047	−	10.2892i	−0.259759	−	0.449915i	0.706418	−	0.707794i	$-0.250310\pi$
$523$	−5.94047	−	10.2892i	−0.259759	−	0.449915i	−0.966177	+	0.257879i	$0.916976\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−27.3990	+	47.4565i	−1.19126	+	2.06333i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−13.9852			−0.605765
$534$		0			0
$535$	−0.400338	+	0.693406i	−0.0173081	+	0.0299786i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−37.9852			−1.63614
$540$		0			0
$541$	−16.9610	−	29.3773i	−0.729209	−	1.26303i	−0.957218	−	0.289368i	$-0.906555\pi$
$541$	−16.9610	−	29.3773i	−0.729209	−	1.26303i	0.228009	−	0.973659i	$-0.426778\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.17968	−	15.8997i	−0.393214	−	0.681067i
$546$		0			0
$547$	−4.49760	−	7.79007i	−0.192303	−	0.333079i	0.753710	−	0.657207i	$-0.228262\pi$
$547$	−4.49760	−	7.79007i	−0.192303	−	0.333079i	−0.946013	+	0.324128i	$0.894929\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−5.16484	+	10.5940i	−0.220030	+	0.451319i
$552$		0			0
$553$	−1.37614	+	2.38354i	−0.0585194	+	0.101359i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.94418	+	8.56358i	0.209492	+	0.362850i	0.951555	−	0.307480i	$-0.0994857\pi$
$557$	4.94418	+	8.56358i	0.209492	+	0.362850i	−0.742063	+	0.670330i	$0.766152\pi$
$558$		0			0
$559$	−33.7449			−1.42726
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−36.4413			−1.53582			−0.767909	−	0.640559i	$-0.778703\pi$
$563$	−36.4413			−1.53582			−0.767909	+	0.640559i	$0.778703\pi$
$564$		0			0
$565$	8.78259	−	15.2119i	0.369486	−	0.639969i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	27.2717			1.14329			0.571644	−	0.820502i	$-0.306306\pi$
$569$	27.2717			1.14329			0.571644	+	0.820502i	$0.306306\pi$
$570$		0			0
$571$	16.1042			0.673940			0.336970	−	0.941515i	$-0.390598\pi$
$571$	16.1042			0.673940			0.336970	+	0.941515i	$0.390598\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	13.9902	−	24.2317i	0.583431	−	1.01053i
$576$		0			0
$577$	−27.3323			−1.13786			−0.568929	−	0.822387i	$-0.692642\pi$
$577$	−27.3323			−1.13786			−0.568929	+	0.822387i	$0.692642\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.93676			−0.121838
$582$		0			0
$583$	−11.2026	−	19.4034i	−0.463963	−	0.803608i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.0508	+	22.6047i	−0.538664	+	0.932994i	0.460312	+	0.887757i	$0.347738\pi$
$587$	−13.0508	+	22.6047i	−0.538664	+	0.932994i	−0.998976	+	0.0452367i	$0.985596\pi$
$588$		0			0
$589$	1.27803	+	1.89408i	0.0526605	+	0.0780443i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.55211	−	7.88448i	−0.186933	−	0.323777i	0.757293	−	0.653075i	$-0.226521\pi$
$593$	−4.55211	−	7.88448i	−0.186933	−	0.323777i	−0.944226	+	0.329298i	$0.893188\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	13.2937	+	23.0254i	0.543167	+	0.940792i	0.998720	+	0.0505836i	$0.0161081\pi$
$599$	13.2937	+	23.0254i	0.543167	+	0.940792i	−0.455553	+	0.890209i	$0.650559\pi$
$600$		0			0
$601$	31.2691			1.27549			0.637746	−	0.770247i	$-0.279867\pi$
$601$	31.2691			1.27549			0.637746	+	0.770247i	$0.279867\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.1928	−	22.8505i	0.536362	−	0.929006i
$606$		0			0
$607$	−18.8687			−0.765858			−0.382929	−	0.923778i	$-0.625085\pi$
$607$	−18.8687			−0.765858			−0.382929	+	0.923778i	$0.625085\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−15.5168	+	26.8759i	−0.627742	+	1.08728i
$612$		0			0
$613$	9.61033	−	16.6456i	0.388158	−	0.672309i	−0.604044	−	0.796951i	$-0.706445\pi$
$613$	9.61033	−	16.6456i	0.388158	−	0.672309i	0.992202	+	0.124642i	$0.0397783\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	11.2608	+	19.5043i	0.453343	+	0.785212i	0.998591	−	0.0530621i	$-0.0168981\pi$
$617$	11.2608	+	19.5043i	0.453343	+	0.785212i	−0.545249	+	0.838274i	$0.683565\pi$
$618$		0			0
$619$	−5.99258			−0.240862			−0.120431	−	0.992722i	$-0.538428\pi$
$619$	−5.99258			−0.240862			−0.120431	+	0.992722i	$0.538428\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.817917	+	1.41667i	0.0327692	+	0.0567579i
$624$		0			0
$625$	−0.462269	+	0.800673i	−0.0184908	+	0.0320269i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.700169	−	1.21273i	−0.0278733	−	0.0482780i	0.851752	−	0.523945i	$-0.175540\pi$
$631$	−0.700169	−	1.21273i	−0.0278733	−	0.0482780i	−0.879626	+	0.475667i	$0.842207\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.4710			0.574264
$636$		0			0
$637$	−17.7826	−	30.8003i	−0.704572	−	1.22035i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5.30354	−	9.18600i	0.209477	−	0.362825i	−0.742073	−	0.670319i	$-0.766157\pi$
$641$	5.30354	−	9.18600i	0.209477	−	0.362825i	0.951550	+	0.307494i	$0.0994905\pi$
$642$		0			0
$643$	−21.9963	+	38.0987i	−0.867449	+	1.50247i	−0.00285431	+	0.999996i	$0.500909\pi$
$643$	−21.9963	+	38.0987i	−0.867449	+	1.50247i	−0.864595	+	0.502470i	$0.832425\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.11164			−0.318901			−0.159451	−	0.987206i	$-0.550972\pi$
$647$	−8.11164			−0.318901			−0.159451	+	0.987206i	$0.550972\pi$
$648$		0			0
$649$	15.2584	−	26.4283i	0.598944	−	1.03740i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−34.6890			−1.35749			−0.678744	−	0.734375i	$-0.737475\pi$
$653$	−34.6890			−1.35749			−0.678744	+	0.734375i	$0.737475\pi$
$654$		0			0
$655$	−2.22808	−	3.85914i	−0.0870582	−	0.150789i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15.5218	−	26.8845i	−0.604643	−	1.04727i	−0.992108	−	0.125388i	$-0.959982\pi$
$659$	−15.5218	−	26.8845i	−0.604643	−	1.04727i	0.387464	−	0.921885i	$-0.373351\pi$
$660$		0			0
$661$	7.34452	+	12.7211i	0.285669	+	0.494793i	0.972771	−	0.231768i	$-0.0744510\pi$
$661$	7.34452	+	12.7211i	0.285669	+	0.494793i	−0.687102	+	0.726561i	$0.741118\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.06889	+	0.145018i	−0.0802281	+	0.00562357i
$666$		0			0
$667$	11.9245	−	20.6539i	0.461720	−	0.799722i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.18208	+	8.97563i	0.200052	+	0.346500i
$672$		0			0
$673$	−12.3567			−0.476318			−0.238159	−	0.971226i	$-0.576544\pi$
$673$	−12.3567			−0.476318			−0.238159	+	0.971226i	$0.576544\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−32.7858			−1.26006			−0.630031	−	0.776570i	$-0.716958\pi$
$677$	−32.7858			−1.26006			−0.630031	+	0.776570i	$0.716958\pi$
$678$		0			0
$679$	−2.43065	+	4.21001i	−0.0932798	+	0.161565i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	41.3977			1.58404			0.792020	−	0.610495i	$-0.209030\pi$
$683$	41.3977			1.58404			0.792020	+	0.610495i	$0.209030\pi$
$684$		0			0
$685$	23.0484			0.880634
$686$		0			0
$687$		0			0
$688$		0			0
$689$	10.4889	−	18.1673i	0.399594	−	0.692117i
$690$		0			0
$691$	42.9368			1.63339			0.816696	−	0.577069i	$-0.195803\pi$
$691$	42.9368			1.63339			0.816696	+	0.577069i	$0.195803\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−24.2429			−0.919586
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.03533	+	1.79324i	−0.0391038	+	0.0677297i	−0.884915	−	0.465753i	$-0.845784\pi$
$701$	−1.03533	+	1.79324i	−0.0391038	+	0.0677297i	0.845811	+	0.533482i	$0.179117\pi$
$702$		0			0
$703$	−2.43807	−	3.61328i	−0.0919535	−	0.136278i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.412564	+	0.714582i	0.0155161	+	0.0268746i
$708$		0			0
$709$	6.28259	+	10.8818i	0.235948	+	0.408673i	0.959548	−	0.281546i	$-0.0908473\pi$
$709$	6.28259	+	10.8818i	0.235948	+	0.408673i	−0.723600	+	0.690220i	$0.757514\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.31180	−	4.00416i	−0.0865778	−	0.149957i
$714$		0			0
$715$	38.6284			1.44462
$716$		0			0
$717$		0			0
$718$		0			0
$719$	25.1550	−	43.5698i	0.938124	−	1.62488i	0.169158	−	0.985589i	$-0.445895\pi$
$719$	25.1550	−	43.5698i	0.938124	−	1.62488i	0.768966	−	0.639289i	$-0.220771\pi$
$720$		0			0
$721$	−5.66771			−0.211076
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.28870	+	7.42825i	−0.159278	+	0.275878i
$726$		0			0
$727$	−16.6672	+	28.8685i	−0.618154	+	1.07067i	0.371668	+	0.928366i	$0.378786\pi$
$727$	−16.6672	+	28.8685i	−0.618154	+	1.07067i	−0.989822	+	0.142308i	$0.954548\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−25.8129			−0.953421			−0.476711	−	0.879060i	$-0.658171\pi$
$733$	−25.8129			−0.953421			−0.476711	+	0.879060i	$0.658171\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	33.1247	+	57.3737i	1.22016	+	2.11339i
$738$		0			0
$739$	24.2802	−	42.0545i	0.893161	−	1.54700i	0.0570970	−	0.998369i	$-0.481816\pi$
$739$	24.2802	−	42.0545i	0.893161	−	1.54700i	0.836064	−	0.548632i	$-0.184851\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.33470	+	12.7041i	−0.269084	+	0.466067i	−0.968626	−	0.248525i	$-0.920054\pi$
$743$	−7.33470	+	12.7041i	−0.269084	+	0.466067i	0.699542	+	0.714592i	$0.253388\pi$
$744$		0			0
$745$	−1.31421	−	2.27628i	−0.0481489	−	0.0833963i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.208434			0.00761600
$750$		0			0
$751$	3.87243	+	6.70724i	0.141307	+	0.244751i	0.927989	−	0.372607i	$-0.121536\pi$
$751$	3.87243	+	6.70724i	0.141307	+	0.244751i	−0.786682	+	0.617358i	$0.788203\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−9.22066	+	15.9707i	−0.335574	+	0.581231i
$756$		0			0
$757$	15.8626	−	27.4748i	0.576536	−	0.998590i	−0.419337	−	0.907831i	$-0.637737\pi$
$757$	15.8626	−	27.4748i	0.576536	−	0.998590i	0.995873	−	0.0907593i	$-0.0289294\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	37.2255			1.34942			0.674711	−	0.738082i	$-0.264268\pi$
$761$	37.2255			1.34942			0.674711	+	0.738082i	$0.264268\pi$
$762$		0			0
$763$	−2.38967	+	4.13903i	−0.0865119	+	0.149843i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	28.5726			1.03170
$768$		0			0
$769$	11.2645	+	19.5107i	0.406208	+	0.703574i	0.994461	−	0.105103i	$-0.0335174\pi$
$769$	11.2645	+	19.5107i	0.406208	+	0.703574i	−0.588253	+	0.808677i	$0.700184\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−25.5726	−	44.2930i	−0.919782	−	1.59311i	−0.799744	−	0.600341i	$-0.795032\pi$
$773$	−25.5726	−	44.2930i	−0.919782	−	1.59311i	−0.120038	−	0.992769i	$-0.538302\pi$
$774$		0			0
$775$	0.831449	+	1.44011i	0.0298665	+	0.0517303i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−11.7571	+	0.824110i	−0.421241	+	0.0295268i
$780$		0			0
$781$	−13.9442	+	24.1520i	−0.498962	+	0.864228i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11.2486	+	19.4831i	0.401479	+	0.695381i
$786$		0			0
$787$	1.16745			0.0416152			0.0208076	−	0.999783i	$-0.493376\pi$
$787$	1.16745			0.0416152			0.0208076	+	0.999783i	$0.493376\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4.57260			−0.162583
$792$		0			0
$793$	−4.85194	+	8.40381i	−0.172297	+	0.298428i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−37.1600			−1.31628			−0.658138	−	0.752897i	$-0.728656\pi$
$797$	−37.1600			−1.31628			−0.658138	+	0.752897i	$0.728656\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−21.2789	+	36.8561i	−0.750915	+	1.30062i
$804$		0			0
$805$	4.19671			0.147915
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−30.7597			−1.08145			−0.540727	−	0.841198i	$-0.681851\pi$
$809$	−30.7597			−1.08145			−0.540727	+	0.841198i	$0.681851\pi$
$810$		0			0
$811$	−17.1042	−	29.6254i	−0.600610	−	1.04029i	−0.992729	−	0.120373i	$-0.961591\pi$
$811$	−17.1042	−	29.6254i	−0.600610	−	1.04029i	0.392118	−	0.919915i	$-0.371742\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.34692	−	2.33294i	0.0471807	−	0.0817194i
$816$		0			0
$817$	−28.3687	+	1.98850i	−0.992496	+	0.0695688i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	14.4610	+	25.0471i	0.504691	+	0.874151i	0.999985	+	0.00542532i	$0.00172694\pi$
$821$	14.4610	+	25.0471i	0.504691	+	0.874151i	−0.495294	+	0.868725i	$0.664940\pi$
$822$		0			0
$823$	3.24030	+	5.61237i	0.112950	+	0.195635i	0.916958	−	0.398983i	$-0.130637\pi$
$823$	3.24030	+	5.61237i	0.112950	+	0.195635i	−0.804008	+	0.594618i	$0.797303\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.2281	+	35.0361i	0.703399	+	1.21832i	0.967266	+	0.253764i	$0.0816687\pi$
$827$	20.2281	+	35.0361i	0.703399	+	1.21832i	−0.263867	+	0.964559i	$0.584998\pi$
$828$		0			0
$829$	−51.5019			−1.78874			−0.894368	−	0.447331i	$-0.852374\pi$
$829$	−51.5019			−1.78874			−0.894368	+	0.447331i	$0.852374\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−3.49806			−0.121055
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−16.8105	+	29.1166i	−0.580363	+	1.00522i	0.415073	+	0.909788i	$0.363756\pi$
$839$	−16.8105	+	29.1166i	−0.580363	+	1.00522i	−0.995436	+	0.0954300i	$0.969577\pi$
$840$		0			0
$841$	10.8445	−	18.7833i	0.373949	−	0.647699i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.29612	+	16.1014i	0.319796	+	0.553903i
$846$		0			0
$847$	−6.86872			−0.236012
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.41016	+	7.63862i	0.151178	+	0.261849i
$852$		0			0
$853$	25.3007	−	43.8221i	0.866279	−	1.50044i	0.000506763	−	1.00000i	$-0.499839\pi$
$853$	25.3007	−	43.8221i	0.866279	−	1.50044i	0.865772	−	0.500439i	$-0.166828\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.00000	+	5.19615i	−0.102478	+	0.177497i	−0.912705	−	0.408619i	$-0.866010\pi$
$857$	−3.00000	+	5.19615i	−0.102478	+	0.177497i	0.810227	+	0.586116i	$0.199344\pi$
$858$		0			0
$859$	11.0447	+	19.1300i	0.376840	+	0.652706i	0.990601	−	0.136786i	$-0.0436773\pi$
$859$	11.0447	+	19.1300i	0.376840	+	0.652706i	−0.613761	+	0.789492i	$0.710344\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	10.6890			0.363859			0.181930	−	0.983312i	$-0.441766\pi$
$863$	10.6890			0.363859			0.181930	+	0.983312i	$0.441766\pi$
$864$		0			0
$865$	−11.9245	−	20.6539i	−0.405446	−	0.702254i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	21.6005	−	37.4132i	0.732747	−	1.26916i
$870$		0			0
$871$	−31.0144	+	53.7185i	−1.05088	+	1.82018i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.88836			−0.131451
$876$		0			0
$877$	−15.5968	+	27.0144i	−0.526666	+	0.912213i	0.472851	+	0.881143i	$0.343225\pi$
$877$	−15.5968	+	27.0144i	−0.526666	+	0.912213i	−0.999517	+	0.0310705i	$0.990108\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	9.69646			0.326682			0.163341	−	0.986570i	$-0.447773\pi$
$881$	9.69646			0.326682			0.163341	+	0.986570i	$0.447773\pi$
$882$		0			0
$883$	−19.0750	−	33.0389i	−0.641925	−	1.11185i	−0.985003	−	0.172540i	$-0.944803\pi$
$883$	−19.0750	−	33.0389i	−0.641925	−	1.11185i	0.343078	−	0.939307i	$-0.388531\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4.99018	+	8.64324i	0.167554	+	0.290212i	0.937559	−	0.347826i	$-0.113080\pi$
$887$	4.99018	+	8.64324i	0.167554	+	0.290212i	−0.770005	+	0.638037i	$0.779747\pi$
$888$		0			0
$889$	−1.88356	−	3.26242i	−0.0631725	−	0.109418i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−11.4610	+	23.5084i	−0.383527	+	0.786680i
$894$		0			0
$895$	1.50611	−	2.60866i	0.0503438	−	0.0871980i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	0.708686	+	1.22748i	0.0236360	+	0.0409388i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12.8007			−0.425509
$906$		0			0
$907$	10.2329	−	17.7239i	0.339777	−	0.588512i	−0.644613	−	0.764509i	$-0.722982\pi$
$907$	10.2329	−	17.7239i	0.339777	−	0.588512i	0.984391	+	0.175997i	$0.0563149\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−27.1696			−0.900171			−0.450085	−	0.892986i	$-0.648606\pi$
$911$	−27.1696			−0.900171			−0.450085	+	0.892986i	$0.648606\pi$
$912$		0			0
$913$	46.0968			1.52558
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.580017	+	1.00462i	−0.0191539	+	0.0331755i
$918$		0			0
$919$	−33.1378			−1.09311			−0.546557	−	0.837422i	$-0.684062\pi$
$919$	−33.1378			−1.09311			−0.546557	+	0.837422i	$0.684062\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−26.1116			−0.859475
$924$		0			0
$925$	−1.58613	−	2.74726i	−0.0521516	−	0.0903293i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−24.3166	+	42.1176i	−0.797802	+	1.38183i	0.123242	+	0.992377i	$0.460671\pi$
$929$	−24.3166	+	42.1176i	−0.797802	+	1.38183i	−0.921045	+	0.389457i	$0.872663\pi$
$930$		0			0
$931$	−16.7645	−	24.8454i	−0.549434	−	0.814276i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	24.8626	+	43.0633i	0.812226	+	1.40682i	0.911303	+	0.411737i	$0.135078\pi$
$937$	24.8626	+	43.0633i	0.812226	+	1.40682i	−0.0990768	+	0.995080i	$0.531589\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23.9245	+	41.4385i	0.779918	+	1.35086i	0.931989	+	0.362487i	$0.118072\pi$
$941$	23.9245	+	41.4385i	0.779918	+	1.35086i	−0.152071	+	0.988370i	$0.548594\pi$
$942$		0			0
$943$	23.8491			0.776633
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6.69646	+	11.5986i	−0.217606	+	0.376904i	−0.954075	−	0.299566i	$-0.903158\pi$
$947$	−6.69646	+	11.5986i	−0.217606	+	0.376904i	0.736470	+	0.676470i	$0.236491\pi$
$948$		0			0
$949$	−39.8465			−1.29347
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.72437	+	4.71875i	−0.0882510	+	0.152855i	−0.906772	−	0.421622i	$-0.861461\pi$
$953$	−2.72437	+	4.71875i	−0.0882510	+	0.152855i	0.818521	+	0.574477i	$0.194794\pi$
$954$		0			0
$955$	−15.2584	+	26.4283i	−0.493750	+	0.855200i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.00000	−	5.19615i	−0.0968751	−	0.167793i
$960$		0			0
$961$	−30.7252			−0.991136
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.37985	−	16.2464i	−0.301948	−	0.522989i
$966$		0			0
$967$	16.6066	−	28.7635i	0.534033	−	0.924972i	−0.465177	−	0.885218i	$-0.654009\pi$
$967$	16.6066	−	28.7635i	0.534033	−	0.924972i	0.999209	−	0.0397542i	$-0.0126575\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	26.5652	−	46.0122i	0.852517	−	1.47660i	−0.0264121	−	0.999651i	$-0.508408\pi$
$971$	26.5652	−	46.0122i	0.852517	−	1.47660i	0.878929	−	0.476952i	$-0.158258\pi$
$972$		0			0
$973$	3.15548	+	5.46545i	0.101160	+	0.175214i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	10.9219			0.349423			0.174712	−	0.984620i	$-0.444101\pi$
$977$	10.9219			0.349423			0.174712	+	0.984620i	$0.444101\pi$
$978$		0			0
$979$	−12.8384	−	22.2368i	−0.410317	−	0.710690i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	19.6941	−	34.1111i	0.628143	−	1.08798i	−0.359781	−	0.933037i	$-0.617149\pi$
$983$	19.6941	−	34.1111i	0.628143	−	1.08798i	0.987924	−	0.154939i	$-0.0495180\pi$
$984$		0			0
$985$	−5.61294	+	9.72190i	−0.178843	+	0.309765i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	57.5455			1.82984
$990$		0			0
$991$	7.19406	−	12.4605i	0.228527	−	0.395820i	−0.728845	−	0.684679i	$-0.759943\pi$
$991$	7.19406	−	12.4605i	0.228527	−	0.395820i	0.957372	+	0.288859i	$0.0932758\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−9.30615			−0.295025
$996$		0			0
$997$	−17.9758	−	31.1350i	−0.569299	−	0.986055i	−0.996635	−	0.0819625i	$-0.973881\pi$
$997$	−17.9758	−	31.1350i	−0.569299	−	0.986055i	0.427336	−	0.904093i	$-0.359452\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.s.z.1873.2		6
3.2	odd	2		912.2.q.l.49.2		6
4.3	odd	2		171.2.f.b.163.3		6
12.11	even	2		57.2.e.b.49.1	yes	6
19.7	even	3	inner	2736.2.s.z.577.2		6
57.26	odd	6		912.2.q.l.577.2		6
76.7	odd	6		171.2.f.b.64.3		6
76.11	odd	6		3249.2.a.y.1.1		3
76.27	even	6		3249.2.a.t.1.3		3
228.11	even	6		1083.2.a.l.1.3		3
228.83	even	6		57.2.e.b.7.1	✓	6
228.179	odd	6		1083.2.a.o.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.e.b.7.1	✓	6	228.83	even	6
57.2.e.b.49.1	yes	6	12.11	even	2
171.2.f.b.64.3		6	76.7	odd	6
171.2.f.b.163.3		6	4.3	odd	2
912.2.q.l.49.2		6	3.2	odd	2
912.2.q.l.577.2		6	57.26	odd	6
1083.2.a.l.1.3		3	228.11	even	6
1083.2.a.o.1.1		3	228.179	odd	6
2736.2.s.z.577.2		6	19.7	even	3	inner
2736.2.s.z.1873.2		6	1.1	even	1	trivial
3249.2.a.t.1.3		3	76.27	even	6
3249.2.a.y.1.1		3	76.11	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.675970 - 1.17081i) q^{5} -0.351939 q^{7} +O(q^{10})\)	\(q+(0.675970 - 1.17081i) q^{5} -0.351939 q^{7} +5.52420 q^{11} +(2.58613 + 4.47931i) q^{13} +(2.43807 + 3.61328i) q^{19} +(-4.41016 - 7.63862i) q^{23} +(1.58613 + 2.74726i) q^{25} +(1.35194 + 2.34163i) q^{29} +0.524200 q^{31} +(-0.237900 + 0.412055i) q^{35} -1.00000 q^{37} +(-1.35194 + 2.34163i) q^{41} +(-3.26210 + 5.65012i) q^{43} +(3.00000 + 5.19615i) q^{47} -6.87614 q^{49} +(-2.02791 - 3.51244i) q^{53} +(3.73419 - 6.46781i) q^{55} +(2.76210 - 4.78410i) q^{59} +(0.938069 + 1.62478i) q^{61} +6.99258 q^{65} +(5.99629 + 10.3859i) q^{67} +(-2.52420 + 4.37204i) q^{71} +(-3.85194 + 6.67175i) q^{73} -1.94418 q^{77} +(3.91016 - 6.77260i) q^{79} +8.34452 q^{83} +(-2.32403 - 4.02534i) q^{89} +(-0.910161 - 1.57644i) q^{91} +(5.87854 - 0.412055i) q^{95} +(6.90645 - 11.9623i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 6 * q + 2 * q^5 + 2 * q^7	\( 6 q + 2 q^{5} + 2 q^{7} + q^{13} - 4 q^{19} - 14 q^{23} - 5 q^{25} + 4 q^{29} - 30 q^{31} - 18 q^{35} - 6 q^{37} - 4 q^{41} - 3 q^{43} + 18 q^{47} - 4 q^{49} - 6 q^{53} + 12 q^{55} - 13 q^{61} - 12 q^{65} + 9 q^{67} + 18 q^{71} - 19 q^{73} - 24 q^{77} + 11 q^{79} - 8 q^{83} - 16 q^{89} + 7 q^{91} + 2 q^{95} + 2 q^{97}+O(q^{100}) \) 6 * q + 2 * q^5 + 2 * q^7 + q^13 - 4 * q^19 - 14 * q^23 - 5 * q^25 + 4 * q^29 - 30 * q^31 - 18 * q^35 - 6 * q^37 - 4 * q^41 - 3 * q^43 + 18 * q^47 - 4 * q^49 - 6 * q^53 + 12 * q^55 - 13 * q^61 - 12 * q^65 + 9 * q^67 + 18 * q^71 - 19 * q^73 - 24 * q^77 + 11 * q^79 - 8 * q^83 - 16 * q^89 + 7 * q^91 + 2 * q^95 + 2 * q^97

Introduction

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