LMFDB - Embedded newform 1512.2.bs.a.1097.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1512,2,Mod(521,1512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1512.521"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.bs (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1097.1
Character	$\chi$	$=$	1512.1097
Dual form			1512.2.bs.a.521.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.05742 - 3.56356i) q^{5} +(1.89056 - 1.85089i) q^{7} +(5.04147 + 2.91069i) q^{11} +(2.52259 + 1.45642i) q^{13} +(1.58162 + 2.73945i) q^{17} +(0.722488 + 0.417129i) q^{19} +(6.14668 - 3.54879i) q^{23} +(-5.96595 + 10.3333i) q^{25} +(1.91234 - 1.10409i) q^{29} -4.23424i q^{31} +(-10.4854 - 2.92907i) q^{35} +(1.82507 - 3.16112i) q^{37} +(2.04811 - 3.54743i) q^{41} +(0.155460 + 0.269265i) q^{43} -1.00467 q^{47} +(0.148439 - 6.99843i) q^{49} +(-1.94801 + 1.12469i) q^{53} -23.9541i q^{55} -5.03496 q^{59} -4.60348i q^{61} -11.9859i q^{65} -9.99453 q^{67} +11.4186i q^{71} +(-3.04990 + 1.76086i) q^{73} +(14.9186 - 3.82834i) q^{77} -1.15870 q^{79} +(-7.57669 - 13.1232i) q^{83} +(6.50813 - 11.2724i) q^{85} +(4.82266 - 8.35309i) q^{89} +(7.46478 - 1.91558i) q^{91} -3.43284i q^{95} +(-5.06969 + 2.92699i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 12 q^{23} - 24 q^{25} - 18 q^{29} - 6 q^{41} - 6 q^{43} - 36 q^{47} + 6 q^{49} - 12 q^{53} + 36 q^{77} - 12 q^{79} - 18 q^{89} + 6 q^{91}+O(q^{100}) $ 48 * q + 12 * q^23 - 24 * q^25 - 18 * q^29 - 6 * q^41 - 6 * q^43 - 36 * q^47 + 6 * q^49 - 12 * q^53 + 36 * q^77 - 12 * q^79 - 18 * q^89 + 6 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$1$

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
$5$	−2.05742	−	3.56356i	−0.920106	−	1.59367i	−0.799248	−	0.601001i	$-0.794769\pi$
$5$	−2.05742	−	3.56356i	−0.920106	−	1.59367i	−0.120858	−	0.992670i	$-0.538564\pi$
$6$		0			0
$7$	1.89056	−	1.85089i	0.714565	−	0.699569i
$7$	1.89056	−	1.85089i	0.714565	−	0.699569i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.04147	+	2.91069i	1.52006	+	0.877607i	0.999720	+	0.0236471i	$0.00752781\pi$
$11$	5.04147	+	2.91069i	1.52006	+	0.877607i	0.520339	+	0.853960i	$0.325806\pi$
$12$		0			0
$13$	2.52259	+	1.45642i	0.699641	+	0.403938i	0.807214	−	0.590259i	$-0.200975\pi$
$13$	2.52259	+	1.45642i	0.699641	+	0.403938i	−0.107573	+	0.994197i	$0.534308\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.58162	+	2.73945i	0.383600	+	0.664415i	0.991574	−	0.129542i	$-0.0413508\pi$
$17$	1.58162	+	2.73945i	0.383600	+	0.664415i	−0.607974	+	0.793957i	$0.708017\pi$
$18$		0			0
$19$	0.722488	+	0.417129i	0.165750	+	0.0956959i	0.580580	−	0.814203i	$-0.302826\pi$
$19$	0.722488	+	0.417129i	0.165750	+	0.0956959i	−0.414830	+	0.909899i	$0.636159\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.14668	−	3.54879i	1.28167	−	0.739973i	0.304518	−	0.952507i	$-0.401505\pi$
$23$	6.14668	−	3.54879i	1.28167	−	0.739973i	0.977154	+	0.212533i	$0.0681714\pi$
$24$		0			0
$25$	−5.96595	+	10.3333i	−1.19319	+	2.06667i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.91234	−	1.10409i	0.355113	−	0.205025i	−0.311822	−	0.950141i	$-0.600939\pi$
$29$	1.91234	−	1.10409i	0.355113	−	0.205025i	0.666935	+	0.745116i	$0.267606\pi$
$30$		0			0
$31$		−	4.23424i		−	0.760493i	−0.924885	−	0.380246i	$-0.875839\pi$
$31$		−	4.23424i		−	0.760493i	0.924885	−	0.380246i	$-0.124161\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−10.4854	−	2.92907i	−1.77236	−	0.495103i
$36$		0			0
$37$	1.82507	−	3.16112i	0.300040	−	0.519684i	−0.676105	−	0.736806i	$-0.736333\pi$
$37$	1.82507	−	3.16112i	0.300040	−	0.519684i	0.976145	+	0.217121i	$0.0696667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.04811	−	3.54743i	0.319861	−	0.554016i	−0.660598	−	0.750740i	$-0.729697\pi$
$41$	2.04811	−	3.54743i	0.319861	−	0.554016i	0.980459	+	0.196724i	$0.0630303\pi$
$42$		0			0
$43$	0.155460	+	0.269265i	0.0237074	+	0.0410625i	0.877636	−	0.479328i	$-0.159120\pi$
$43$	0.155460	+	0.269265i	0.0237074	+	0.0410625i	−0.853928	+	0.520391i	$0.825786\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.00467			−0.146546			−0.0732731	−	0.997312i	$-0.523344\pi$
$47$	−1.00467			−0.146546			−0.0732731	+	0.997312i	$0.523344\pi$
$48$		0			0
$49$	0.148439	−	6.99843i	0.0212055	−	0.999775i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.94801	+	1.12469i	−0.267580	+	0.154487i	−0.627787	−	0.778385i	$-0.716039\pi$
$53$	−1.94801	+	1.12469i	−0.267580	+	0.154487i	0.360207	+	0.932872i	$0.382706\pi$
$54$		0			0
$55$		−	23.9541i		−	3.22997i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.03496			−0.655496			−0.327748	−	0.944765i	$-0.606290\pi$
$59$	−5.03496			−0.655496			−0.327748	+	0.944765i	$0.606290\pi$
$60$		0			0
$61$		−	4.60348i		−	0.589416i	−0.955587	−	0.294708i	$-0.904778\pi$
$61$		−	4.60348i		−	0.589416i	0.955587	−	0.294708i	$-0.0952224\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	11.9859i		−	1.48666i
$66$		0			0
$67$	−9.99453			−1.22103			−0.610513	−	0.792006i	$-0.709037\pi$
$67$	−9.99453			−1.22103			−0.610513	+	0.792006i	$0.709037\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			11.4186i			1.35514i	0.735458	+	0.677571i	$0.236967\pi$
$71$			11.4186i			1.35514i	−0.735458	+	0.677571i	$0.763033\pi$
$72$		0			0
$73$	−3.04990	+	1.76086i	−0.356964	+	0.206093i	−0.667748	−	0.744387i	$-0.732742\pi$
$73$	−3.04990	+	1.76086i	−0.356964	+	0.206093i	0.310784	+	0.950480i	$0.399408\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	14.9186	−	3.82834i	1.70013	−	0.436280i
$78$		0			0
$79$	−1.15870			−0.130364			−0.0651820	−	0.997873i	$-0.520763\pi$
$79$	−1.15870			−0.130364			−0.0651820	+	0.997873i	$0.520763\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.57669	−	13.1232i	−0.831650	−	1.44046i	−0.896729	−	0.442579i	$-0.854063\pi$
$83$	−7.57669	−	13.1232i	−0.831650	−	1.44046i	0.0650797	−	0.997880i	$-0.479270\pi$
$84$		0			0
$85$	6.50813	−	11.2724i	0.705905	−	1.22266i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.82266	−	8.35309i	0.511201	−	0.885426i	−0.488715	−	0.872444i	$-0.662534\pi$
$89$	4.82266	−	8.35309i	0.511201	−	0.885426i	0.999916	−	0.0129824i	$-0.00413254\pi$
$90$		0			0
$91$	7.46478	−	1.91558i	0.782521	−	0.200808i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	3.43284i		−	0.352201i
$96$		0			0
$97$	−5.06969	+	2.92699i	−0.514749	+	0.297190i	−0.734784	−	0.678302i	$-0.762716\pi$
$97$	−5.06969	+	2.92699i	−0.514749	+	0.297190i	0.220035	+	0.975492i	$0.429383\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.92632	+	5.06853i	−0.291180	+	0.504338i	−0.974089	−	0.226165i	$-0.927381\pi$
$101$	−2.92632	+	5.06853i	−0.291180	+	0.504338i	0.682909	+	0.730503i	$0.260714\pi$
$102$		0			0
$103$	−0.675732	+	0.390134i	−0.0665818	+	0.0384410i	−0.532921	−	0.846165i	$-0.678906\pi$
$103$	−0.675732	+	0.390134i	−0.0665818	+	0.0384410i	0.466340	+	0.884606i	$0.345573\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.15538	−	4.13116i	−0.691737	−	0.399374i	0.112526	−	0.993649i	$-0.464106\pi$
$107$	−7.15538	−	4.13116i	−0.691737	−	0.399374i	−0.804262	+	0.594274i	$0.797439\pi$
$108$		0			0
$109$	3.63584	+	6.29745i	0.348250	+	0.603187i	0.985939	−	0.167107i	$-0.0534427\pi$
$109$	3.63584	+	6.29745i	0.348250	+	0.603187i	−0.637689	+	0.770294i	$0.720109\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	16.0816	+	9.28474i	1.51283	+	0.873435i	0.999887	+	0.0150145i	$0.00477945\pi$
$113$	16.0816	+	9.28474i	1.51283	+	0.873435i	0.512947	+	0.858421i	$0.328554\pi$
$114$		0			0
$115$	−25.2926	−	14.6027i	−2.35855	−	1.36171i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8.06057	+	2.25169i	0.738911	+	0.206412i
$120$		0			0
$121$	11.4443	+	19.8220i	1.04039	+	1.80200i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	28.5237			2.55124
$126$		0			0
$127$	4.38363			0.388984			0.194492	−	0.980904i	$-0.437694\pi$
$127$	4.38363			0.388984			0.194492	+	0.980904i	$0.437694\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.85419	+	13.6039i	0.686224	+	1.18857i	0.973051	+	0.230592i	$0.0740662\pi$
$131$	7.85419	+	13.6039i	0.686224	+	1.18857i	−0.286827	+	0.957982i	$0.592600\pi$
$132$		0			0
$133$	2.13797	−	0.548636i	0.185385	−	0.0475728i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	15.9901	+	9.23190i	1.36613	+	0.788734i	0.990431	−	0.138008i	$-0.0440701\pi$
$137$	15.9901	+	9.23190i	1.36613	+	0.788734i	0.375697	+	0.926743i	$0.377403\pi$
$138$		0			0
$139$	−3.80522	−	2.19695i	−0.322755	−	0.186343i	0.329865	−	0.944028i	$-0.392997\pi$
$139$	−3.80522	−	2.19695i	−0.322755	−	0.186343i	−0.652620	+	0.757685i	$0.726330\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.47837	+	14.6850i	0.708997	+	1.22802i
$144$		0			0
$145$	−7.86898	−	4.54316i	−0.653483	−	0.377289i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−17.7580	+	10.2526i	−1.45479	+	0.839922i	−0.998747	−	0.0500363i	$-0.984066\pi$
$149$	−17.7580	+	10.2526i	−1.45479	+	0.839922i	−0.456041	+	0.889959i	$0.650733\pi$
$150$		0			0
$151$	−2.70776	+	4.68998i	−0.220354	+	0.381665i	−0.954916	−	0.296877i	$-0.904055\pi$
$151$	−2.70776	+	4.68998i	−0.220354	+	0.381665i	0.734561	+	0.678542i	$0.237388\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−15.0890	+	8.71162i	−1.21197	+	0.699734i
$156$		0			0
$157$		−	11.2537i		−	0.898140i	−0.893497	−	0.449070i	$-0.851755\pi$
$157$		−	11.2537i		−	0.898140i	0.893497	−	0.449070i	$-0.148245\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.05227	−	18.0860i	0.398175	−	1.42538i
$162$		0			0
$163$	−1.25819	+	2.17924i	−0.0985487	+	0.170691i	−0.911084	−	0.412220i	$-0.864753\pi$
$163$	−1.25819	+	2.17924i	−0.0985487	+	0.170691i	0.812535	+	0.582912i	$0.198087\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.38259	−	16.2511i	0.726047	−	1.25755i	−0.232495	−	0.972598i	$-0.574689\pi$
$167$	9.38259	−	16.2511i	0.726047	−	1.25755i	0.958542	−	0.284952i	$-0.0919777\pi$
$168$		0			0
$169$	−2.25769	−	3.91043i	−0.173668	−	0.300803i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.71715			−0.738781			−0.369391	−	0.929274i	$-0.620434\pi$
$173$	−9.71715			−0.738781			−0.369391	+	0.929274i	$0.620434\pi$
$174$		0			0
$175$	7.84683	+	30.5781i	0.593165	+	2.31149i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.24569	+	5.33800i	−0.691055	+	0.398981i	−0.804007	−	0.594619i	$-0.797303\pi$
$179$	−9.24569	+	5.33800i	−0.691055	+	0.398981i	0.112952	+	0.993600i	$0.463969\pi$
$180$		0			0
$181$		−	4.34901i		−	0.323259i	−0.986851	−	0.161630i	$-0.948325\pi$
$181$		−	4.34901i		−	0.323259i	0.986851	−	0.161630i	$-0.0516750\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−15.0198			−1.10427
$186$		0			0
$187$			18.4145i			1.34660i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	24.7918i		−	1.79387i	−0.442159	−	0.896937i	$-0.645787\pi$
$191$		−	24.7918i		−	1.79387i	0.442159	−	0.896937i	$-0.354213\pi$
$192$		0			0
$193$	26.6512			1.91840			0.959198	−	0.282736i	$-0.0912419\pi$
$193$	26.6512			1.91840			0.959198	+	0.282736i	$0.0912419\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			6.11068i			0.435368i	0.976019	+	0.217684i	$0.0698502\pi$
$197$			6.11068i			0.435368i	−0.976019	+	0.217684i	$0.930150\pi$
$198$		0			0
$199$	−0.385220	+	0.222407i	−0.0273075	+	0.0157660i	−0.513592	−	0.858035i	$-0.671685\pi$
$199$	−0.385220	+	0.222407i	−0.0273075	+	0.0157660i	0.486284	+	0.873801i	$0.338352\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.57185	−	5.62688i	0.110322	−	0.394930i
$204$		0			0
$205$	−16.8553			−1.17723
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.42827	+	4.20588i	0.167967	+	0.290927i
$210$		0			0
$211$	−4.23912	+	7.34236i	−0.291833	+	0.505469i	−0.974243	−	0.225500i	$-0.927598\pi$
$211$	−4.23912	+	7.34236i	−0.291833	+	0.505469i	0.682410	+	0.730969i	$0.260932\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.639693	−	1.10798i	0.0436267	−	0.0755636i
$216$		0			0
$217$	−7.83710	−	8.00509i	−0.532017	−	0.543421i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			9.21402i			0.619802i
$222$		0			0
$223$	7.25738	−	4.19005i	0.485990	−	0.280587i	−0.236919	−	0.971529i	$-0.576138\pi$
$223$	7.25738	−	4.19005i	0.485990	−	0.280587i	0.722909	+	0.690943i	$0.242804\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.11075	−	1.92388i	0.0737233	−	0.127693i	−0.826807	−	0.562486i	$-0.809845\pi$
$227$	1.11075	−	1.92388i	0.0737233	−	0.127693i	0.900530	+	0.434793i	$0.143178\pi$
$228$		0			0
$229$	−20.4417	+	11.8020i	−1.35082	+	0.779898i	−0.988365	−	0.152102i	$-0.951396\pi$
$229$	−20.4417	+	11.8020i	−1.35082	+	0.779898i	−0.362458	+	0.932000i	$0.618062\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.69275	−	3.86406i	−0.438457	−	0.253143i	0.264486	−	0.964390i	$-0.414798\pi$
$233$	−6.69275	−	3.86406i	−0.438457	−	0.253143i	−0.702943	+	0.711246i	$0.748131\pi$
$234$		0			0
$235$	2.06703	+	3.58020i	0.134838	+	0.233546i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−16.2873	−	9.40346i	−1.05354	−	0.608259i	−0.129899	−	0.991527i	$-0.541465\pi$
$239$	−16.2873	−	9.40346i	−1.05354	−	0.608259i	−0.923637	+	0.383268i	$0.874799\pi$
$240$		0			0
$241$	21.5749	+	12.4563i	1.38976	+	0.802379i	0.993288	−	0.115668i	$-0.0369008\pi$
$241$	21.5749	+	12.4563i	1.38976	+	0.802379i	0.396473	+	0.918047i	$0.370234\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−25.2447	+	13.8697i	−1.61282	+	0.886105i
$246$		0			0
$247$	1.21503	+	2.10449i	0.0773104	+	0.133906i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.39294			−0.214161			−0.107080	−	0.994250i	$-0.534150\pi$
$251$	−3.39294			−0.214161			−0.107080	+	0.994250i	$0.534150\pi$
$252$		0			0
$253$	41.3177			2.59762
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.47711	−	4.29047i	−0.154518	−	0.267632i	0.778366	−	0.627811i	$-0.216049\pi$
$257$	−2.47711	−	4.29047i	−0.154518	−	0.267632i	−0.932883	+	0.360179i	$0.882716\pi$
$258$		0			0
$259$	−2.40046	−	9.35428i	−0.149157	−	0.581247i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.52396	−	1.45721i	−0.155634	−	0.0898553i	0.420160	−	0.907450i	$-0.361974\pi$
$263$	−2.52396	−	1.45721i	−0.155634	−	0.0898553i	−0.575794	+	0.817595i	$0.695307\pi$
$264$		0			0
$265$	8.01576	+	4.62790i	0.492404	+	0.284290i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.41429	−	16.3060i	−0.573999	−	0.994196i	−0.996150	−	0.0876702i	$-0.972058\pi$
$269$	−9.41429	−	16.3060i	−0.573999	−	0.994196i	0.422150	−	0.906526i	$-0.361276\pi$
$270$		0			0
$271$	−13.6959	−	7.90734i	−0.831967	−	0.480336i	0.0225585	−	0.999746i	$-0.492819\pi$
$271$	−13.6959	−	7.90734i	−0.831967	−	0.480336i	−0.854526	+	0.519409i	$0.826152\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−60.1543	+	34.7301i	−3.62744	+	2.09430i
$276$		0			0
$277$	2.45231	−	4.24753i	0.147345	−	0.255209i	−0.782900	−	0.622147i	$-0.786261\pi$
$277$	2.45231	−	4.24753i	0.147345	−	0.255209i	0.930245	+	0.366938i	$0.119594\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.77186	+	1.60034i	−0.165355	+	0.0954680i	−0.580394	−	0.814336i	$-0.697101\pi$
$281$	−2.77186	+	1.60034i	−0.165355	+	0.0954680i	0.415039	+	0.909804i	$0.363768\pi$
$282$		0			0
$283$		−	5.06740i		−	0.301226i	−0.988593	−	0.150613i	$-0.951875\pi$
$283$		−	5.06740i		−	0.301226i	0.988593	−	0.150613i	$-0.0481247\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.69382	−	10.4975i	−0.159011	−	0.619646i
$288$		0			0
$289$	3.49694	−	6.05687i	0.205702	−	0.356287i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.3751	+	21.4343i	−0.722961	+	1.25221i	0.236847	+	0.971547i	$0.423886\pi$
$293$	−12.3751	+	21.4343i	−0.722961	+	1.25221i	−0.959808	+	0.280658i	$0.909447\pi$
$294$		0			0
$295$	10.3590	+	17.9424i	0.603126	+	1.04464i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	20.6741			1.19561
$300$		0			0
$301$	0.792285	+	0.221322i	0.0456665	+	0.0127568i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−16.4048	+	9.47130i	−0.939335	+	0.542325i
$306$		0			0
$307$		−	19.8011i		−	1.13011i	−0.825054	−	0.565055i	$-0.808855\pi$
$307$		−	19.8011i		−	1.13011i	0.825054	−	0.565055i	$-0.191145\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0490			0.683235			0.341617	−	0.939839i	$-0.389025\pi$
$311$	12.0490			0.683235			0.341617	+	0.939839i	$0.389025\pi$
$312$		0			0
$313$			30.0873i			1.70064i	0.526269	+	0.850318i	$0.323590\pi$
$313$			30.0873i			1.70064i	−0.526269	+	0.850318i	$0.676410\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	12.6887i		−	0.712669i	−0.934358	−	0.356335i	$-0.884026\pi$
$317$		−	12.6887i		−	0.712669i	0.934358	−	0.356335i	$-0.115974\pi$
$318$		0			0
$319$	12.8547			0.719724
$320$		0			0
$321$		0			0
$322$		0			0
$323$			2.63896i			0.146836i
$324$		0			0
$325$	−30.0993	+	17.3778i	−1.66961	+	0.963950i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.89939	+	1.85953i	−0.104717	+	0.102519i
$330$		0			0
$331$	−5.90816			−0.324742			−0.162371	−	0.986730i	$-0.551914\pi$
$331$	−5.90816			−0.324742			−0.162371	+	0.986730i	$0.551914\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	20.5629	+	35.6161i	1.12347	+	1.94591i
$336$		0			0
$337$	−10.5675	+	18.3034i	−0.575647	+	0.997051i	0.420323	+	0.907374i	$0.361917\pi$
$337$	−10.5675	+	18.3034i	−0.575647	+	0.997051i	−0.995971	+	0.0896763i	$0.971417\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	12.3246	−	21.3468i	0.667414	−	1.15599i
$342$		0			0
$343$	−12.6727	−	13.5057i	−0.684259	−	0.729239i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	29.5975i		−	1.58888i	−0.607344	−	0.794439i	$-0.707765\pi$
$347$		−	29.5975i		−	1.58888i	0.607344	−	0.794439i	$-0.292235\pi$
$348$		0			0
$349$	6.60232	−	3.81185i	0.353414	−	0.204044i	−0.312774	−	0.949828i	$-0.601258\pi$
$349$	6.60232	−	3.81185i	0.353414	−	0.204044i	0.666188	+	0.745784i	$0.267925\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.24727	+	9.08854i	−0.279284	+	0.483734i	−0.971207	−	0.238237i	$-0.923430\pi$
$353$	−5.24727	+	9.08854i	−0.279284	+	0.483734i	0.691923	+	0.721971i	$0.256764\pi$
$354$		0			0
$355$	40.6909	−	23.4929i	2.15965	−	1.24687i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.65028	+	2.68484i	0.245432	+	0.141700i	0.617671	−	0.786437i	$-0.288076\pi$
$359$	4.65028	+	2.68484i	0.245432	+	0.141700i	−0.372239	+	0.928137i	$0.621410\pi$
$360$		0			0
$361$	−9.15201	−	15.8517i	−0.481685	−	0.834302i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.5498	+	7.24566i	0.656889	+	0.379255i
$366$		0			0
$367$	9.61757	+	5.55271i	0.502033	+	0.289849i	0.729553	−	0.683925i	$-0.239728\pi$
$367$	9.61757	+	5.55271i	0.502033	+	0.289849i	−0.227520	+	0.973773i	$0.573062\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.60117	+	5.73184i	−0.0831286	+	0.297582i
$372$		0			0
$373$	17.7021	+	30.6610i	0.916580	+	1.58756i	0.804571	+	0.593857i	$0.202395\pi$
$373$	17.7021	+	30.6610i	0.916580	+	1.58756i	0.112010	+	0.993707i	$0.464271\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.43208			0.331269
$378$		0			0
$379$	−26.5906			−1.36587			−0.682933	−	0.730481i	$-0.739296\pi$
$379$	−26.5906			−1.36587			−0.682933	+	0.730481i	$0.739296\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.2956	+	21.2967i	0.628277	+	1.08821i	0.987897	+	0.155110i	$0.0495731\pi$
$383$	12.2956	+	21.2967i	0.628277	+	1.08821i	−0.359620	+	0.933099i	$0.617094\pi$
$384$		0			0
$385$	−44.3363	−	45.2866i	−2.25958	−	2.30802i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−16.5400	−	9.54936i	−0.838611	−	0.484172i	0.0181812	−	0.999835i	$-0.494212\pi$
$389$	−16.5400	−	9.54936i	−0.838611	−	0.484172i	−0.856792	+	0.515663i	$0.827546\pi$
$390$		0			0
$391$	19.4435	+	11.2257i	0.983298	+	0.567708i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.38394	+	4.12910i	0.119949	+	0.207757i
$396$		0			0
$397$	17.8857	+	10.3263i	0.897655	+	0.518261i	0.876439	−	0.481513i	$-0.159913\pi$
$397$	17.8857	+	10.3263i	0.897655	+	0.518261i	0.0212165	+	0.999775i	$0.493246\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13.3677	−	7.71786i	0.667553	−	0.385412i	−0.127596	−	0.991826i	$-0.540726\pi$
$401$	13.3677	−	7.71786i	0.667553	−	0.385412i	0.795149	+	0.606414i	$0.207393\pi$
$402$		0			0
$403$	6.16683	−	10.6813i	0.307192	−	0.532072i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.4021	−	10.6244i	0.912157	−	0.526634i
$408$		0			0
$409$			18.6613i			0.922741i	0.887208	+	0.461370i	$0.152642\pi$
$409$			18.6613i			0.922741i	−0.887208	+	0.461370i	$0.847358\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.51889	+	9.31914i	−0.468394	+	0.458565i
$414$		0			0
$415$	−31.1769	+	53.9999i	−1.53041	+	2.65075i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.3576	−	24.8681i	0.701414	−	1.21488i	−0.266557	−	0.963819i	$-0.585886\pi$
$419$	14.3576	−	24.8681i	0.701414	−	1.21488i	0.967970	−	0.251065i	$-0.0807807\pi$
$420$		0			0
$421$	12.8338	+	22.2287i	0.625479	+	1.08336i	0.988448	+	0.151560i	$0.0484296\pi$
$421$	12.8338	+	22.2287i	0.625479	+	1.08336i	−0.362969	+	0.931801i	$0.618237\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−37.7436			−1.83083
$426$		0			0
$427$	−8.52053	−	8.70317i	−0.412337	−	0.421176i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−17.0243	+	9.82896i	−0.820030	+	0.473444i	−0.850427	−	0.526094i	$-0.823656\pi$
$431$	−17.0243	+	9.82896i	−0.820030	+	0.473444i	0.0303970	+	0.999538i	$0.490323\pi$
$432$		0			0
$433$		−	7.07732i		−	0.340114i	−0.985434	−	0.170057i	$-0.945605\pi$
$433$		−	7.07732i		−	0.340114i	0.985434	−	0.170057i	$-0.0543952\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.92120			0.283250
$438$		0			0
$439$			19.4414i			0.927889i	0.885864	+	0.463945i	$0.153566\pi$
$439$			19.4414i			0.927889i	−0.885864	+	0.463945i	$0.846434\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			34.5199i			1.64009i	0.572299	+	0.820045i	$0.306052\pi$
$443$			34.5199i			1.64009i	−0.572299	+	0.820045i	$0.693948\pi$
$444$		0			0
$445$	−39.6889			−1.88144
$446$		0			0
$447$		0			0
$448$		0			0
$449$			10.9491i			0.516720i	0.966049	+	0.258360i	$0.0831821\pi$
$449$			10.9491i			0.516720i	−0.966049	+	0.258360i	$0.916818\pi$
$450$		0			0
$451$	20.6510	−	11.9228i	0.972417	−	0.561425i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−22.1845	−	22.6600i	−1.04002	−	1.06232i
$456$		0			0
$457$	−19.1923			−0.897776			−0.448888	−	0.893588i	$-0.648180\pi$
$457$	−19.1923			−0.897776			−0.448888	+	0.893588i	$0.648180\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.2492	+	28.1444i	0.756799	+	1.31081i	0.944475	+	0.328584i	$0.106571\pi$
$461$	16.2492	+	28.1444i	0.756799	+	1.31081i	−0.187676	+	0.982231i	$0.560095\pi$
$462$		0			0
$463$	13.9136	−	24.0991i	0.646621	−	1.11998i	−0.337303	−	0.941396i	$-0.609515\pi$
$463$	13.9136	−	24.0991i	0.646621	−	1.11998i	0.983924	−	0.178585i	$-0.0571519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−0.0869153	+	0.150542i	−0.00402196	+	0.00696624i	−0.868029	−	0.496513i	$-0.834614\pi$
$467$	−0.0869153	+	0.150542i	−0.00402196	+	0.00696624i	0.864007	+	0.503479i	$0.167947\pi$
$468$		0			0
$469$	−18.8953	+	18.4987i	−0.872502	+	0.854192i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			1.80998i			0.0832232i
$474$		0			0
$475$	−8.62066	+	4.97714i	−0.395543	+	0.228367i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.46598	−	2.53915i	0.0669822	−	0.116017i	−0.830589	−	0.556885i	$-0.811996\pi$
$479$	1.46598	−	2.53915i	0.0669822	−	0.116017i	0.897572	+	0.440869i	$0.145330\pi$
$480$		0			0
$481$	9.20782	−	5.31614i	0.419840	−	0.242395i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	20.8610	+	12.0441i	0.947247	+	0.546893i
$486$		0			0
$487$	18.7014	+	32.3917i	0.847440	+	1.46781i	0.883486	+	0.468458i	$0.155190\pi$
$487$	18.7014	+	32.3917i	0.847440	+	1.46781i	−0.0360460	+	0.999350i	$0.511476\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.27844	+	4.20221i	0.328472	+	0.189643i	0.655162	−	0.755488i	$-0.272600\pi$
$491$	7.27844	+	4.20221i	0.328472	+	0.189643i	−0.326691	+	0.945131i	$0.605934\pi$
$492$		0			0
$493$	6.04921	+	3.49251i	0.272443	+	0.157295i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	21.1346	+	21.5876i	0.948015	+	0.968336i
$498$		0			0
$499$	0.0176239	+	0.0305256i	0.000788956	+	0.00136651i	0.866420	−	0.499317i	$-0.166416\pi$
$499$	0.0176239	+	0.0305256i	0.000788956	+	0.00136651i	−0.865631	+	0.500683i	$0.833082\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14.9085			−0.664736			−0.332368	−	0.943150i	$-0.607848\pi$
$503$	−14.9085			−0.664736			−0.332368	+	0.943150i	$0.607848\pi$
$504$		0			0
$505$	24.0827			1.07166
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.20640	+	12.4818i	0.319418	+	0.553248i	0.980367	−	0.197183i	$-0.0631793\pi$
$509$	7.20640	+	12.4818i	0.319418	+	0.553248i	−0.660949	+	0.750431i	$0.729846\pi$
$510$		0			0
$511$	−2.50687	+	8.97403i	−0.110897	+	0.396988i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.78053	+	1.60534i	0.122525	+	0.0707397i
$516$		0			0
$517$	−5.06501	−	2.92429i	−0.222759	−	0.128610i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10.4159	−	18.0409i	−0.456330	−	0.790386i	0.542434	−	0.840098i	$-0.317503\pi$
$521$	−10.4159	−	18.0409i	−0.456330	−	0.790386i	−0.998764	+	0.0497124i	$0.984170\pi$
$522$		0			0
$523$	12.6684	+	7.31408i	0.553949	+	0.319822i	0.750713	−	0.660628i	$-0.229710\pi$
$523$	12.6684	+	7.31408i	0.553949	+	0.319822i	−0.196764	+	0.980451i	$0.563043\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	11.5995	−	6.69698i	0.505282	−	0.291725i
$528$		0			0
$529$	13.6878	−	23.7079i	0.595121	−	1.03078i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10.3331	−	5.96582i	0.447576	−	0.258408i
$534$		0			0
$535$			33.9981i			1.46987i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	21.1186	−	34.8503i	0.909643	−	1.50111i
$540$		0			0
$541$	3.60713	−	6.24773i	0.155083	−	0.268611i	−0.778006	−	0.628256i	$-0.783769\pi$
$541$	3.60713	−	6.24773i	0.155083	−	0.268611i	0.933089	+	0.359645i	$0.117102\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.9609	−	25.9130i	0.640854	−	1.10999i
$546$		0			0
$547$	3.81987	+	6.61620i	0.163326	+	0.282888i	0.936059	−	0.351842i	$-0.114445\pi$
$547$	3.81987	+	6.61620i	0.163326	+	0.282888i	−0.772734	+	0.634730i	$0.781111\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.84219			0.0784801
$552$		0			0
$553$	−2.19060	+	2.14462i	−0.0931536	+	0.0911987i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.13965	+	4.69943i	−0.344888	+	0.199121i	−0.662432	−	0.749122i	$-0.730475\pi$
$557$	−8.13965	+	4.69943i	−0.344888	+	0.199121i	0.317543	+	0.948244i	$0.397142\pi$
$558$		0			0
$559$			0.905659i			0.0383053i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.12705			−0.173934			−0.0869671	−	0.996211i	$-0.527718\pi$
$563$	−4.12705			−0.173934			−0.0869671	+	0.996211i	$0.527718\pi$
$564$		0			0
$565$		−	76.4104i		−	3.21461i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	18.2143i		−	0.763583i	−0.924248	−	0.381792i	$-0.875307\pi$
$569$		−	18.2143i		−	0.763583i	0.924248	−	0.381792i	$-0.124693\pi$
$570$		0			0
$571$	−45.1826			−1.89083			−0.945417	−	0.325864i	$-0.894345\pi$
$571$	−45.1826			−1.89083			−0.945417	+	0.325864i	$0.894345\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			84.6876i			3.53172i
$576$		0			0
$577$	7.04322	−	4.06641i	0.293213	−	0.169287i	−0.346177	−	0.938169i	$-0.612520\pi$
$577$	7.04322	−	4.06641i	0.293213	−	0.169287i	0.639390	+	0.768883i	$0.279187\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−38.6138	−	10.7866i	−1.60197	−	0.447505i
$582$		0			0
$583$	−13.0945			−0.542317
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.4379	+	21.5430i	0.513366	+	0.889175i	0.999880	+	0.0155026i	$0.00493481\pi$
$587$	12.4379	+	21.5430i	0.513366	+	0.889175i	−0.486514	+	0.873673i	$0.661732\pi$
$588$		0			0
$589$	1.76622	−	3.05919i	0.0727760	−	0.126052i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−15.6229	+	27.0596i	−0.641554	+	1.11120i	0.343532	+	0.939141i	$0.388377\pi$
$593$	−15.6229	+	27.0596i	−0.641554	+	1.11120i	−0.985086	+	0.172063i	$0.944957\pi$
$594$		0			0
$595$	−8.55993	−	33.3570i	−0.350923	−	1.36750i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			6.39849i			0.261435i	0.991420	+	0.130718i	$0.0417281\pi$
$599$			6.39849i			0.261435i	−0.991420	+	0.130718i	$0.958272\pi$
$600$		0			0
$601$	−36.0541	+	20.8158i	−1.47068	+	0.849095i	−0.999458	−	0.0329215i	$-0.989519\pi$
$601$	−36.0541	+	20.8158i	−1.47068	+	0.849095i	−0.471218	+	0.882017i	$0.656186\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	47.0913	−	81.5645i	1.91453	−	3.31607i
$606$		0			0
$607$	−21.7167	+	12.5381i	−0.881452	+	0.508907i	−0.871137	−	0.491040i	$-0.836617\pi$
$607$	−21.7167	+	12.5381i	−0.881452	+	0.508907i	−0.0103153	+	0.999947i	$0.503284\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.53437	−	1.46322i	−0.102530	−	0.0591956i
$612$		0			0
$613$	15.3779	+	26.6352i	0.621106	+	1.07579i	0.989280	+	0.146030i	$0.0466497\pi$
$613$	15.3779	+	26.6352i	0.621106	+	1.07579i	−0.368174	+	0.929757i	$0.620017\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	10.1910	+	5.88375i	0.410272	+	0.236871i	0.690907	−	0.722944i	$-0.257212\pi$
$617$	10.1910	+	5.88375i	0.410272	+	0.236871i	−0.280634	+	0.959815i	$0.590545\pi$
$618$		0			0
$619$	6.14617	+	3.54849i	0.247035	+	0.142626i	0.618406	−	0.785859i	$-0.287779\pi$
$619$	6.14617	+	3.54849i	0.247035	+	0.142626i	−0.371371	+	0.928485i	$0.621112\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.34309	−	24.7182i	−0.254131	−	0.990315i
$624$		0			0
$625$	−28.8554	−	49.9791i	−1.15422	−	1.99916i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	11.5463			0.460381
$630$		0			0
$631$	13.0686			0.520252			0.260126	−	0.965575i	$-0.416236\pi$
$631$	13.0686			0.520252			0.260126	+	0.965575i	$0.416236\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.01897	−	15.6213i	−0.357907	−	0.619913i
$636$		0			0
$637$	10.5671	−	17.4380i	0.418683	−	0.690918i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10.9219	−	6.30573i	−0.431387	−	0.249062i	0.268550	−	0.963266i	$-0.413455\pi$
$641$	−10.9219	−	6.30573i	−0.431387	−	0.249062i	−0.699937	+	0.714204i	$0.746789\pi$
$642$		0			0
$643$	−12.3449	−	7.12736i	−0.486837	−	0.281076i	0.236424	−	0.971650i	$-0.424025\pi$
$643$	−12.3449	−	7.12736i	−0.486837	−	0.281076i	−0.723261	+	0.690574i	$0.757358\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−20.0060	−	34.6515i	−0.786519	−	1.36229i	−0.928088	−	0.372362i	$-0.878548\pi$
$647$	−20.0060	−	34.6515i	−0.786519	−	1.36229i	0.141569	−	0.989928i	$-0.454785\pi$
$648$		0			0
$649$	−25.3836	−	14.6552i	−0.996392	−	0.575267i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	12.6726	−	7.31656i	0.495919	−	0.286319i	−0.231108	−	0.972928i	$-0.574235\pi$
$653$	12.6726	−	7.31656i	0.495919	−	0.286319i	0.727027	+	0.686609i	$0.240902\pi$
$654$		0			0
$655$	32.3187	−	55.9777i	1.26280	−	2.18723i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	23.7553	−	13.7151i	0.925374	−	0.534265i	0.0400285	−	0.999199i	$-0.487255\pi$
$659$	23.7553	−	13.7151i	0.925374	−	0.534265i	0.885346	+	0.464934i	$0.153922\pi$
$660$		0			0
$661$		−	35.6097i		−	1.38506i	−0.721391	−	0.692528i	$-0.756497\pi$
$661$		−	35.6097i		−	1.38506i	0.721391	−	0.692528i	$-0.243503\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.35379	−	6.48998i	−0.246389	−	0.251671i
$666$		0			0
$667$	7.83637	−	13.5730i	0.303426	−	0.525548i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13.3993	−	23.2083i	0.517275	−	0.895947i
$672$		0			0
$673$	6.35937	+	11.0147i	0.245136	+	0.424587i	0.962170	−	0.272451i	$-0.0878342\pi$
$673$	6.35937	+	11.0147i	0.245136	+	0.424587i	−0.717034	+	0.697038i	$0.754501\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−28.0556			−1.07826			−0.539131	−	0.842222i	$-0.681247\pi$
$677$	−28.0556			−1.07826			−0.539131	+	0.842222i	$0.681247\pi$
$678$		0			0
$679$	−4.16703	+	14.9171i	−0.159916	+	0.572464i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−14.4564	+	8.34638i	−0.553157	+	0.319365i	−0.750394	−	0.660990i	$-0.770136\pi$
$683$	−14.4564	+	8.34638i	−0.553157	+	0.319365i	0.197237	+	0.980356i	$0.436803\pi$
$684$		0			0
$685$		−	75.9756i		−	2.90288i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.55205			−0.249613
$690$		0			0
$691$			30.4437i			1.15813i	0.815281	+	0.579066i	$0.196583\pi$
$691$			30.4437i			1.15813i	−0.815281	+	0.579066i	$0.803417\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			18.0802i			0.685820i
$696$		0			0
$697$	12.9574			0.490795
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	6.16940i		−	0.233015i	−0.993190	−	0.116507i	$-0.962830\pi$
$701$		−	6.16940i		−	0.233015i	0.993190	−	0.116507i	$-0.0371699\pi$
$702$		0			0
$703$	2.63719	−	1.52258i	0.0994633	−	0.0574252i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.84890	+	14.9987i	0.144753	+	0.564082i
$708$		0			0
$709$	25.4973			0.957572			0.478786	−	0.877932i	$-0.341077\pi$
$709$	25.4973			0.957572			0.478786	+	0.877932i	$0.341077\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−15.0264	−	26.0265i	−0.562744	−	0.974702i
$714$		0			0
$715$	34.8871	−	60.4263i	1.30471	−	2.25982i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12.1263	−	21.0034i	0.452236	−	0.783296i	−0.546289	−	0.837597i	$-0.683960\pi$
$719$	12.1263	−	21.0034i	0.452236	−	0.783296i	0.998525	+	0.0543012i	$0.0172931\pi$
$720$		0			0
$721$	−0.555418	+	1.98827i	−0.0206849	+	0.0740472i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			26.3478i			0.978534i
$726$		0			0
$727$	13.5251	−	7.80870i	0.501617	−	0.289609i	−0.227764	−	0.973716i	$-0.573141\pi$
$727$	13.5251	−	7.80870i	0.501617	−	0.289609i	0.729381	+	0.684108i	$0.239808\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.491758	+	0.851750i	−0.0181883	+	0.0315031i
$732$		0			0
$733$	30.9202	−	17.8518i	1.14206	−	0.659371i	0.195123	−	0.980779i	$-0.437489\pi$
$733$	30.9202	−	17.8518i	1.14206	−	0.659371i	0.946941	+	0.321408i	$0.104156\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−50.3871	−	29.0910i	−1.85603	−	1.07158i
$738$		0			0
$739$	7.27883	+	12.6073i	0.267756	+	0.463767i	0.968282	−	0.249860i	$-0.0803846\pi$
$739$	7.27883	+	12.6073i	0.267756	+	0.463767i	−0.700526	+	0.713627i	$0.747051\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−0.0287226	−	0.0165830i	−0.00105373	−	0.000608372i	0.499473	−	0.866329i	$-0.333527\pi$
$743$	−0.0287226	−	0.0165830i	−0.00105373	−	0.000608372i	−0.500527	+	0.865721i	$0.666860\pi$
$744$		0			0
$745$	73.0711	+	42.1876i	2.67712	+	1.54564i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−21.1740	+	5.43359i	−0.773681	+	0.198539i
$750$		0			0
$751$	−11.8949	−	20.6027i	−0.434053	−	0.751801i	0.563165	−	0.826344i	$-0.309584\pi$
$751$	−11.8949	−	20.6027i	−0.434053	−	0.751801i	−0.997218	+	0.0745430i	$0.976250\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	22.2840			0.810998
$756$		0			0
$757$	−33.2060			−1.20689			−0.603446	−	0.797404i	$-0.706206\pi$
$757$	−33.2060			−1.20689			−0.603446	+	0.797404i	$0.706206\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	21.2837	+	36.8645i	0.771535	+	1.33634i	0.936722	+	0.350075i	$0.113844\pi$
$761$	21.2837	+	36.8645i	0.771535	+	1.33634i	−0.165187	+	0.986262i	$0.552823\pi$
$762$		0			0
$763$	18.5296	+	5.17620i	0.670818	+	0.187391i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12.7011	−	7.33301i	−0.458612	−	0.264780i
$768$		0			0
$769$	−27.1091	−	15.6514i	−0.977578	−	0.564405i	−0.0760402	−	0.997105i	$-0.524228\pi$
$769$	−27.1091	−	15.6514i	−0.977578	−	0.564405i	−0.901538	+	0.432700i	$0.857561\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.165130	+	0.286014i	0.00593932	+	0.0102872i	0.868980	−	0.494848i	$-0.164776\pi$
$773$	0.165130	+	0.286014i	0.00593932	+	0.0102872i	−0.863040	+	0.505135i	$0.831443\pi$
$774$		0			0
$775$	43.7539	+	25.2613i	1.57168	+	0.907413i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.95947	−	1.70865i	0.106034	−	0.0612188i
$780$		0			0
$781$	−33.2361	+	57.5666i	−1.18928	+	2.05990i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−40.1031	+	23.1535i	−1.43134	+	0.826384i
$786$		0			0
$787$			7.50934i			0.267679i	0.991003	+	0.133840i	$0.0427307\pi$
$787$			7.50934i			0.267679i	−0.991003	+	0.133840i	$0.957269\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	47.5883	−	12.2119i	1.69205	−	0.434206i
$792$		0			0
$793$	6.70460	−	11.6127i	0.238087	−	0.412379i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.58136	−	6.20310i	0.126858	−	0.219725i	−0.795600	−	0.605823i	$-0.792844\pi$
$797$	3.58136	−	6.20310i	0.126858	−	0.219725i	0.922458	+	0.386098i	$0.126177\pi$
$798$		0			0
$799$	−1.58901	−	2.75225i	−0.0562151	−	0.0973675i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−20.5013			−0.723475
$804$		0			0
$805$	−74.8451	+	19.2065i	−2.63794	+	0.676939i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	39.9767	−	23.0806i	1.40551	−	0.811469i	0.410556	−	0.911836i	$-0.365335\pi$
$809$	39.9767	−	23.0806i	1.40551	−	0.811469i	0.994951	+	0.100366i	$0.0320014\pi$
$810$		0			0
$811$			22.8560i			0.802581i	0.915951	+	0.401290i	$0.131438\pi$
$811$			22.8560i			0.802581i	−0.915951	+	0.401290i	$0.868562\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.3545			0.362701
$816$		0			0
$817$			0.259387i			0.00907481i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	1.91410i		−	0.0668026i	−0.999442	−	0.0334013i	$-0.989366\pi$
$821$		−	1.91410i		−	0.0668026i	0.999442	−	0.0334013i	$-0.0106339\pi$
$822$		0			0
$823$	−15.0947			−0.526168			−0.263084	−	0.964773i	$-0.584740\pi$
$823$	−15.0947			−0.526168			−0.263084	+	0.964773i	$0.584740\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	14.4077i		−	0.501004i	−0.968116	−	0.250502i	$-0.919404\pi$
$827$		−	14.4077i		−	0.501004i	0.968116	−	0.250502i	$-0.0805956\pi$
$828$		0			0
$829$	−37.1450	+	21.4457i	−1.29010	+	0.744840i	−0.978672	−	0.205430i	$-0.934141\pi$
$829$	−37.1450	+	21.4457i	−1.29010	+	0.744840i	−0.311429	+	0.950270i	$0.600807\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	19.4066	−	10.6622i	0.672400	−	0.369424i
$834$		0			0
$835$	−77.2157			−2.67216
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.5327	−	21.7073i	−0.432678	−	0.749421i	0.564425	−	0.825485i	$-0.309098\pi$
$839$	−12.5327	−	21.7073i	−0.432678	−	0.749421i	−0.997103	+	0.0760639i	$0.975765\pi$
$840$		0			0
$841$	−12.0620	+	20.8919i	−0.415930	+	0.720412i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.29003	+	16.0908i	−0.319587	+	0.553541i
$846$		0			0
$847$	58.3244	+	16.2927i	2.00405	+	0.559825i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	25.9072i		−	0.888086i
$852$		0			0
$853$	14.9013	−	8.60328i	0.510211	−	0.294571i	−0.222709	−	0.974885i	$-0.571490\pi$
$853$	14.9013	−	8.60328i	0.510211	−	0.294571i	0.732920	+	0.680314i	$0.238157\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−24.3484	+	42.1727i	−0.831726	+	1.44059i	0.0649433	+	0.997889i	$0.479313\pi$
$857$	−24.3484	+	42.1727i	−0.831726	+	1.44059i	−0.896669	+	0.442702i	$0.854020\pi$
$858$		0			0
$859$	−2.68282	+	1.54893i	−0.0915366	+	0.0528487i	−0.545070	−	0.838391i	$-0.683497\pi$
$859$	−2.68282	+	1.54893i	−0.0915366	+	0.0528487i	0.453533	+	0.891239i	$0.350163\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	22.2443	+	12.8427i	0.757203	+	0.437172i	0.828291	−	0.560299i	$-0.189314\pi$
$863$	22.2443	+	12.8427i	0.757203	+	0.437172i	−0.0710874	+	0.997470i	$0.522647\pi$
$864$		0			0
$865$	19.9923	+	34.6276i	0.679757	+	1.17737i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5.84156	−	3.37262i	−0.198161	−	0.114408i
$870$		0			0
$871$	−25.2121	−	14.5562i	−0.854280	−	0.493219i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	53.9257	−	52.7941i	1.82302	−	1.78477i
$876$		0			0
$877$	−0.626626	−	1.08535i	−0.0211597	−	0.0366496i	0.855252	−	0.518213i	$-0.173403\pi$
$877$	−0.626626	−	1.08535i	−0.0211597	−	0.0366496i	−0.876411	+	0.481563i	$0.840069\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.7429			0.631464			0.315732	−	0.948848i	$-0.397750\pi$
$881$	18.7429			0.631464			0.315732	+	0.948848i	$0.397750\pi$
$882$		0			0
$883$	39.3701			1.32491			0.662454	−	0.749103i	$-0.269515\pi$
$883$	39.3701			1.32491			0.662454	+	0.749103i	$0.269515\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18.5954	+	32.2083i	0.624374	+	1.08145i	0.988662	+	0.150161i	$0.0479792\pi$
$887$	18.5954	+	32.2083i	0.624374	+	1.08145i	−0.364288	+	0.931286i	$0.618687\pi$
$888$		0			0
$889$	8.28752	−	8.11360i	0.277955	−	0.272122i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.725863	−	0.419077i	−0.0242901	−	0.0140239i
$894$		0			0
$895$	38.0445	+	21.9650i	1.27169	+	0.734210i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.67499	−	8.09732i	−0.155920	−	0.270061i
$900$		0			0
$901$	−6.16204	−	3.55766i	−0.205287	−	0.118523i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−15.4979	+	8.94774i	−0.515169	+	0.297433i
$906$		0			0
$907$	15.4295	−	26.7247i	0.512328	−	0.887378i	−0.487570	−	0.873084i	$-0.662117\pi$
$907$	15.4295	−	26.7247i	0.512328	−	0.887378i	0.999898	−	0.0142940i	$-0.00455008\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	14.7486	−	8.51511i	0.488643	−	0.282118i	−0.235368	−	0.971906i	$-0.575630\pi$
$911$	14.7486	−	8.51511i	0.488643	−	0.282118i	0.724011	+	0.689788i	$0.242296\pi$
$912$		0			0
$913$		−	88.2136i		−	2.91945i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	40.0280	+	11.1817i	1.32184	+	0.369252i
$918$		0			0
$919$	−8.44635	+	14.6295i	−0.278619	+	0.482583i	−0.971042	−	0.238909i	$-0.923210\pi$
$919$	−8.44635	+	14.6295i	−0.278619	+	0.482583i	0.692422	+	0.721492i	$0.256543\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−16.6303	+	28.8045i	−0.547393	+	0.948112i
$924$		0			0
$925$	21.7766	+	37.7182i	0.716010	+	1.24017i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	6.79356			0.222889			0.111445	−	0.993771i	$-0.464452\pi$
$929$	6.79356			0.222889			0.111445	+	0.993771i	$0.464452\pi$
$930$		0			0
$931$	3.02649	−	4.99436i	0.0991892	−	0.163684i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	65.6210	−	37.8863i	2.14604	−	1.23901i
$936$		0			0
$937$			5.14535i			0.168091i	0.996462	+	0.0840456i	$0.0267841\pi$
$937$			5.14535i			0.168091i	−0.996462	+	0.0840456i	$0.973216\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.920907			−0.0300207			−0.0150104	−	0.999887i	$-0.504778\pi$
$941$	−0.920907			−0.0300207			−0.0150104	+	0.999887i	$0.504778\pi$
$942$		0			0
$943$		−	29.0733i		−	0.946756i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			24.4835i			0.795606i	0.917471	+	0.397803i	$0.130227\pi$
$947$			24.4835i			0.795606i	−0.917471	+	0.397803i	$0.869773\pi$
$948$		0			0
$949$	−10.2582			−0.332995
$950$		0			0
$951$		0			0
$952$		0			0
$953$			9.80916i			0.317750i	0.987299	+	0.158875i	$0.0507866\pi$
$953$			9.80916i			0.317750i	−0.987299	+	0.158875i	$0.949213\pi$
$954$		0			0
$955$	−88.3471	+	51.0072i	−2.85884	+	1.65055i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	47.3175	−	12.1424i	1.52796	−	0.392099i
$960$		0			0
$961$	13.0712			0.421651
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−54.8327	−	94.9731i	−1.76513	−	3.05729i
$966$		0			0
$967$	−18.1376	+	31.4152i	−0.583264	+	1.01024i	0.411825	+	0.911263i	$0.364892\pi$
$967$	−18.1376	+	31.4152i	−0.583264	+	1.01024i	−0.995089	+	0.0989807i	$0.968442\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.09772	+	1.90130i	−0.0352274	+	0.0610157i	−0.883102	−	0.469182i	$-0.844549\pi$
$971$	−1.09772	+	1.90130i	−0.0352274	+	0.0610157i	0.847874	+	0.530198i	$0.177882\pi$
$972$		0			0
$973$	−11.2603	+	2.88958i	−0.360989	+	0.0926356i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			42.7181i			1.36667i	0.730103	+	0.683337i	$0.239472\pi$
$977$			42.7181i			1.36667i	−0.730103	+	0.683337i	$0.760528\pi$
$978$		0			0
$979$	48.6266	−	28.0746i	1.55411	−	0.897267i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−22.8781	+	39.6260i	−0.729698	+	1.26387i	0.227313	+	0.973822i	$0.427006\pi$
$983$	−22.8781	+	39.6260i	−0.729698	+	1.26387i	−0.957011	+	0.290052i	$0.906327\pi$
$984$		0			0
$985$	21.7757	−	12.5722i	0.693832	−	0.400584i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.91113	+	1.10339i	0.0607703	+	0.0350857i
$990$		0			0
$991$	−8.30690	−	14.3880i	−0.263877	−	0.457049i	0.703392	−	0.710803i	$-0.251668\pi$
$991$	−8.30690	−	14.3880i	−0.263877	−	0.457049i	−0.967269	+	0.253754i	$0.918335\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.58512	+	0.915169i	0.0502517	+	0.0290128i
$996$		0			0
$997$	−18.6045	−	10.7413i	−0.589210	−	0.340181i	0.175575	−	0.984466i	$-0.443822\pi$
$997$	−18.6045	−	10.7413i	−0.589210	−	0.340181i	−0.764785	+	0.644285i	$0.777155\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.bs.a.1097.1		48
3.2	odd	2		504.2.bs.a.257.23	✓	48
4.3	odd	2		3024.2.ca.e.2609.1		48
7.3	odd	6		1512.2.cx.a.17.1		48
9.2	odd	6		1512.2.cx.a.89.1		48
9.7	even	3		504.2.cx.a.425.17	yes	48
12.11	even	2		1008.2.ca.e.257.2		48
21.17	even	6		504.2.cx.a.185.17	yes	48
28.3	even	6		3024.2.df.e.17.1		48
36.7	odd	6		1008.2.df.e.929.8		48
36.11	even	6		3024.2.df.e.1601.1		48
63.38	even	6	inner	1512.2.bs.a.521.1		48
63.52	odd	6		504.2.bs.a.353.23	yes	48
84.59	odd	6		1008.2.df.e.689.8		48
252.115	even	6		1008.2.ca.e.353.2		48
252.227	odd	6		3024.2.ca.e.2033.1		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.bs.a.257.23	✓	48	3.2	odd	2
504.2.bs.a.353.23	yes	48	63.52	odd	6
504.2.cx.a.185.17	yes	48	21.17	even	6
504.2.cx.a.425.17	yes	48	9.7	even	3
1008.2.ca.e.257.2		48	12.11	even	2
1008.2.ca.e.353.2		48	252.115	even	6
1008.2.df.e.689.8		48	84.59	odd	6
1008.2.df.e.929.8		48	36.7	odd	6
1512.2.bs.a.521.1		48	63.38	even	6	inner
1512.2.bs.a.1097.1		48	1.1	even	1	trivial
1512.2.cx.a.17.1		48	7.3	odd	6
1512.2.cx.a.89.1		48	9.2	odd	6
3024.2.ca.e.2033.1		48	252.227	odd	6
3024.2.ca.e.2609.1		48	4.3	odd	2
3024.2.df.e.17.1		48	28.3	even	6
3024.2.df.e.1601.1		48	36.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.bs (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.05742 - 3.56356i) q^{5} +(1.89056 - 1.85089i) q^{7} +(5.04147 + 2.91069i) q^{11} +(2.52259 + 1.45642i) q^{13} +(1.58162 + 2.73945i) q^{17} +(0.722488 + 0.417129i) q^{19} +(6.14668 - 3.54879i) q^{23} +(-5.96595 + 10.3333i) q^{25} +(1.91234 - 1.10409i) q^{29} -4.23424i q^{31} +(-10.4854 - 2.92907i) q^{35} +(1.82507 - 3.16112i) q^{37} +(2.04811 - 3.54743i) q^{41} +(0.155460 + 0.269265i) q^{43} -1.00467 q^{47} +(0.148439 - 6.99843i) q^{49} +(-1.94801 + 1.12469i) q^{53} -23.9541i q^{55} -5.03496 q^{59} -4.60348i q^{61} -11.9859i q^{65} -9.99453 q^{67} +11.4186i q^{71} +(-3.04990 + 1.76086i) q^{73} +(14.9186 - 3.82834i) q^{77} -1.15870 q^{79} +(-7.57669 - 13.1232i) q^{83} +(6.50813 - 11.2724i) q^{85} +(4.82266 - 8.35309i) q^{89} +(7.46478 - 1.91558i) q^{91} -3.43284i q^{95} +(-5.06969 + 2.92699i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 12 q^{23} - 24 q^{25} - 18 q^{29} - 6 q^{41} - 6 q^{43} - 36 q^{47} + 6 q^{49} - 12 q^{53} + 36 q^{77} - 12 q^{79} - 18 q^{89} + 6 q^{91}+O(q^{100}) \) 48 * q + 12 * q^23 - 24 * q^25 - 18 * q^29 - 6 * q^41 - 6 * q^43 - 36 * q^47 + 6 * q^49 - 12 * q^53 + 36 * q^77 - 12 * q^79 - 18 * q^89 + 6 * q^91

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