LMFDB - Embedded newform 4032.2.v.d.1583.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1583,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("4032.1583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.v (of order $4$, 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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$18$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1583.5
Character	$\chi$	$=$	4032.1583
Dual form			4032.2.v.d.3599.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.68827 - 1.68827i) q^{5} +1.00000 q^{7} +O(q^{10})$	$q+(-1.68827 - 1.68827i) q^{5} +1.00000 q^{7} +(-1.15241 + 1.15241i) q^{11} +(-1.23570 - 1.23570i) q^{13} +0.707775i q^{17} +(5.82650 - 5.82650i) q^{19} +4.09895i q^{23} +0.700525i q^{25} +(-1.57312 + 1.57312i) q^{29} -6.27573i q^{31} +(-1.68827 - 1.68827i) q^{35} +(-4.88854 + 4.88854i) q^{37} +3.51037 q^{41} +(-4.39450 - 4.39450i) q^{43} +1.52691 q^{47} +1.00000 q^{49} +(2.98685 + 2.98685i) q^{53} +3.89116 q^{55} +(-0.0546910 + 0.0546910i) q^{59} +(-8.07633 - 8.07633i) q^{61} +4.17239i q^{65} +(9.27446 - 9.27446i) q^{67} +8.21700i q^{71} +0.995009i q^{73} +(-1.15241 + 1.15241i) q^{77} +6.85821i q^{79} +(-7.92662 - 7.92662i) q^{83} +(1.19492 - 1.19492i) q^{85} -14.3481 q^{89} +(-1.23570 - 1.23570i) q^{91} -19.6734 q^{95} -7.84344 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 36 q^{7}+O(q^{10}) $ 36 * q + 36 * q^7	$ 36 q + 36 q^{7} - 16 q^{13} + 16 q^{19} + 20 q^{37} - 36 q^{43} + 36 q^{49} - 32 q^{55} + 112 q^{61} + 36 q^{67} - 96 q^{85} - 16 q^{91} + 56 q^{97}+O(q^{100}) $ 36 * q + 36 * q^7 - 16 * q^13 + 16 * q^19 + 20 * q^37 - 36 * q^43 + 36 * q^49 - 32 * q^55 + 112 * q^61 + 36 * q^67 - 96 * q^85 - 16 * q^91 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.68827	−	1.68827i	−0.755018	−	0.755018i	0.220393	−	0.975411i	$-0.429266\pi$
$5$	−1.68827	−	1.68827i	−0.755018	−	0.755018i	−0.975411	+	0.220393i	$0.929266\pi$
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.15241	+	1.15241i	−0.347465	+	0.347465i	−0.859164	−	0.511700i	$-0.829016\pi$
$11$	−1.15241	+	1.15241i	−0.347465	+	0.347465i	0.511700	+	0.859164i	$0.329016\pi$
$12$		0			0
$13$	−1.23570	−	1.23570i	−0.342721	−	0.342721i	0.514668	−	0.857390i	$-0.327915\pi$
$13$	−1.23570	−	1.23570i	−0.342721	−	0.342721i	−0.857390	+	0.514668i	$0.827915\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.707775i			0.171661i	0.996310	+	0.0858303i	$0.0273543\pi$
$17$			0.707775i			0.171661i	−0.996310	+	0.0858303i	$0.972646\pi$
$18$		0			0
$19$	5.82650	−	5.82650i	1.33669	−	1.33669i	0.437446	−	0.899245i	$-0.355883\pi$
$19$	5.82650	−	5.82650i	1.33669	−	1.33669i	0.899245	−	0.437446i	$-0.144117\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.09895i			0.854689i	0.904089	+	0.427345i	$0.140551\pi$
$23$			4.09895i			0.854689i	−0.904089	+	0.427345i	$0.859449\pi$
$24$		0			0
$25$			0.700525i			0.140105i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.57312	+	1.57312i	−0.292122	+	0.292122i	−0.837918	−	0.545796i	$-0.816227\pi$
$29$	−1.57312	+	1.57312i	−0.292122	+	0.292122i	0.545796	+	0.837918i	$0.316227\pi$
$30$		0			0
$31$		−	6.27573i		−	1.12716i	−0.826063	−	0.563578i	$-0.809425\pi$
$31$		−	6.27573i		−	1.12716i	0.826063	−	0.563578i	$-0.190575\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.68827	−	1.68827i	−0.285370	−	0.285370i
$36$		0			0
$37$	−4.88854	+	4.88854i	−0.803672	+	0.803672i	−0.983667	−	0.179996i	$-0.942392\pi$
$37$	−4.88854	+	4.88854i	−0.803672	+	0.803672i	0.179996	+	0.983667i	$0.442392\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.51037			0.548227			0.274114	−	0.961697i	$-0.411616\pi$
$41$	3.51037			0.548227			0.274114	+	0.961697i	$0.411616\pi$
$42$		0			0
$43$	−4.39450	−	4.39450i	−0.670155	−	0.670155i	0.287597	−	0.957752i	$-0.407144\pi$
$43$	−4.39450	−	4.39450i	−0.670155	−	0.670155i	−0.957752	+	0.287597i	$0.907144\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.52691			0.222722			0.111361	−	0.993780i	$-0.464479\pi$
$47$	1.52691			0.222722			0.111361	+	0.993780i	$0.464479\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.98685	+	2.98685i	0.410275	+	0.410275i	0.881834	−	0.471559i	$-0.156309\pi$
$53$	2.98685	+	2.98685i	0.410275	+	0.410275i	−0.471559	+	0.881834i	$0.656309\pi$
$54$		0			0
$55$	3.89116			0.524684
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.0546910	+	0.0546910i	−0.00712016	+	0.00712016i	−0.710658	−	0.703538i	$-0.751603\pi$
$59$	−0.0546910	+	0.0546910i	−0.00712016	+	0.00712016i	0.703538	+	0.710658i	$0.251603\pi$
$60$		0			0
$61$	−8.07633	−	8.07633i	−1.03407	−	1.03407i	−0.999399	−	0.0346697i	$-0.988962\pi$
$61$	−8.07633	−	8.07633i	−1.03407	−	1.03407i	−0.0346697	−	0.999399i	$-0.511038\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			4.17239i			0.517522i
$66$		0			0
$67$	9.27446	−	9.27446i	1.13306	−	1.13306i	0.143390	−	0.989666i	$-0.454200\pi$
$67$	9.27446	−	9.27446i	1.13306	−	1.13306i	0.989666	−	0.143390i	$-0.0458003\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			8.21700i			0.975179i	0.873073	+	0.487589i	$0.162124\pi$
$71$			8.21700i			0.975179i	−0.873073	+	0.487589i	$0.837876\pi$
$72$		0			0
$73$			0.995009i			0.116457i	0.998303	+	0.0582285i	$0.0185452\pi$
$73$			0.995009i			0.116457i	−0.998303	+	0.0582285i	$0.981455\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.15241	+	1.15241i	−0.131329	+	0.131329i
$78$		0			0
$79$			6.85821i			0.771609i	0.922581	+	0.385805i	$0.126076\pi$
$79$			6.85821i			0.771609i	−0.922581	+	0.385805i	$0.873924\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.92662	−	7.92662i	−0.870059	−	0.870059i	0.122419	−	0.992479i	$-0.460935\pi$
$83$	−7.92662	−	7.92662i	−0.870059	−	0.870059i	−0.992479	+	0.122419i	$0.960935\pi$
$84$		0			0
$85$	1.19492	−	1.19492i	0.129607	−	0.129607i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−14.3481			−1.52089			−0.760447	−	0.649400i	$-0.775020\pi$
$89$	−14.3481			−1.52089			−0.760447	+	0.649400i	$0.775020\pi$
$90$		0			0
$91$	−1.23570	−	1.23570i	−0.129537	−	0.129537i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−19.6734			−2.01845
$96$		0			0
$97$	−7.84344			−0.796381			−0.398190	−	0.917303i	$-0.630362\pi$
$97$	−7.84344			−0.796381			−0.398190	+	0.917303i	$0.630362\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.20143	−	7.20143i	−0.716569	−	0.716569i	0.251332	−	0.967901i	$-0.419131\pi$
$101$	−7.20143	−	7.20143i	−0.716569	−	0.716569i	−0.967901	+	0.251332i	$0.919131\pi$
$102$		0			0
$103$	6.70608			0.660770			0.330385	−	0.943846i	$-0.392821\pi$
$103$	6.70608			0.660770			0.330385	+	0.943846i	$0.392821\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.64323	−	4.64323i	0.448878	−	0.448878i	−0.446103	−	0.894981i	$-0.647189\pi$
$107$	4.64323	−	4.64323i	0.448878	−	0.448878i	0.894981	+	0.446103i	$0.147189\pi$
$108$		0			0
$109$	−13.6438	−	13.6438i	−1.30684	−	1.30684i	−0.923682	−	0.383160i	$-0.874836\pi$
$109$	−13.6438	−	13.6438i	−1.30684	−	1.30684i	−0.383160	−	0.923682i	$-0.625164\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	10.1474i		−	0.954588i	−0.878744	−	0.477294i	$-0.841618\pi$
$113$		−	10.1474i		−	0.954588i	0.878744	−	0.477294i	$-0.158382\pi$
$114$		0			0
$115$	6.92013	−	6.92013i	0.645306	−	0.645306i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			0.707775i			0.0648816i
$120$		0			0
$121$			8.34390i			0.758537i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.25868	+	7.25868i	−0.649236	+	0.649236i
$126$		0			0
$127$			9.74658i			0.864869i	0.901665	+	0.432435i	$0.142345\pi$
$127$			9.74658i			0.864869i	−0.901665	+	0.432435i	$0.857655\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.0665	+	10.0665i	0.879516	+	0.879516i	0.993484	−	0.113968i	$-0.0363563\pi$
$131$	10.0665	+	10.0665i	0.879516	+	0.879516i	−0.113968	+	0.993484i	$0.536356\pi$
$132$		0			0
$133$	5.82650	−	5.82650i	0.505222	−	0.505222i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.71854			−0.659439			−0.329720	−	0.944079i	$-0.606954\pi$
$137$	−7.71854			−0.659439			−0.329720	+	0.944079i	$0.606954\pi$
$138$		0			0
$139$	−5.49781	−	5.49781i	−0.466318	−	0.466318i	0.434401	−	0.900719i	$-0.356960\pi$
$139$	−5.49781	−	5.49781i	−0.466318	−	0.466318i	−0.900719	+	0.434401i	$0.856960\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.84806			0.238167
$144$		0			0
$145$	5.31172			0.441114
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.99496	−	8.99496i	−0.736896	−	0.736896i	0.235080	−	0.971976i	$-0.424465\pi$
$149$	−8.99496	−	8.99496i	−0.736896	−	0.736896i	−0.971976	+	0.235080i	$0.924465\pi$
$150$		0			0
$151$	−20.8307			−1.69518			−0.847588	−	0.530654i	$-0.821946\pi$
$151$	−20.8307			−1.69518			−0.847588	+	0.530654i	$0.821946\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.5951	+	10.5951i	−0.851023	+	0.851023i
$156$		0			0
$157$	5.72797	+	5.72797i	0.457142	+	0.457142i	0.897716	−	0.440574i	$-0.145225\pi$
$157$	5.72797	+	5.72797i	0.457142	+	0.457142i	−0.440574	+	0.897716i	$0.645225\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			4.09895i			0.323042i
$162$		0			0
$163$	−6.36478	+	6.36478i	−0.498528	+	0.498528i	−0.910979	−	0.412452i	$-0.864673\pi$
$163$	−6.36478	+	6.36478i	−0.498528	+	0.498528i	0.412452	+	0.910979i	$0.364673\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	12.6570i		−	0.979428i	−0.871883	−	0.489714i	$-0.837101\pi$
$167$		−	12.6570i		−	0.979428i	0.871883	−	0.489714i	$-0.162899\pi$
$168$		0			0
$169$		−	9.94609i		−	0.765084i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.64804	−	1.64804i	0.125298	−	0.125298i	−0.641677	−	0.766975i	$-0.721761\pi$
$173$	1.64804	−	1.64804i	0.125298	−	0.125298i	0.766975	+	0.641677i	$0.221761\pi$
$174$		0			0
$175$			0.700525i			0.0529547i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.672637	+	0.672637i	0.0502752	+	0.0502752i	0.731797	−	0.681522i	$-0.238682\pi$
$179$	0.672637	+	0.672637i	0.0502752	+	0.0502752i	−0.681522	+	0.731797i	$0.738682\pi$
$180$		0			0
$181$	−14.4202	+	14.4202i	−1.07185	+	1.07185i	−0.0746373	+	0.997211i	$0.523780\pi$
$181$	−14.4202	+	14.4202i	−1.07185	+	1.07185i	−0.997211	+	0.0746373i	$0.976220\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	16.5064			1.21357
$186$		0			0
$187$	−0.815646	−	0.815646i	−0.0596460	−	0.0596460i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−15.0872			−1.09167			−0.545837	−	0.837891i	$-0.683788\pi$
$191$	−15.0872			−1.09167			−0.545837	+	0.837891i	$0.683788\pi$
$192$		0			0
$193$	26.8080			1.92968			0.964841	−	0.262833i	$-0.0846567\pi$
$193$	26.8080			1.92968			0.964841	+	0.262833i	$0.0846567\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−16.6175	−	16.6175i	−1.18395	−	1.18395i	−0.978713	−	0.205234i	$-0.934204\pi$
$197$	−16.6175	−	16.6175i	−1.18395	−	1.18395i	−0.205234	−	0.978713i	$-0.565796\pi$
$198$		0			0
$199$	8.38510			0.594404			0.297202	−	0.954815i	$-0.403947\pi$
$199$	8.38510			0.594404			0.297202	+	0.954815i	$0.403947\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.57312	+	1.57312i	−0.110412	+	0.110412i
$204$		0			0
$205$	−5.92645	−	5.92645i	−0.413922	−	0.413922i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			13.4290i			0.928905i
$210$		0			0
$211$	3.06931	−	3.06931i	0.211300	−	0.211300i	−0.593520	−	0.804819i	$-0.702262\pi$
$211$	3.06931	−	3.06931i	0.211300	−	0.211300i	0.804819	+	0.593520i	$0.202262\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			14.8382i			1.01196i
$216$		0			0
$217$		−	6.27573i		−	0.426025i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.874597	−	0.874597i	0.0588317	−	0.0588317i
$222$		0			0
$223$		−	0.496003i		−	0.0332148i	−0.999862	−	0.0166074i	$-0.994713\pi$
$223$		−	0.496003i		−	0.0332148i	0.999862	−	0.0166074i	$-0.00528655\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.5149	+	12.5149i	0.830642	+	0.830642i	0.987605	−	0.156962i	$-0.0501701\pi$
$227$	12.5149	+	12.5149i	0.830642	+	0.830642i	−0.156962	+	0.987605i	$0.550170\pi$
$228$		0			0
$229$	−11.2771	+	11.2771i	−0.745212	+	0.745212i	−0.973576	−	0.228364i	$-0.926663\pi$
$229$	−11.2771	+	11.2771i	−0.745212	+	0.745212i	0.228364	+	0.973576i	$0.426663\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−28.6384			−1.87616			−0.938082	−	0.346413i	$-0.887400\pi$
$233$	−28.6384			−1.87616			−0.938082	+	0.346413i	$0.887400\pi$
$234$		0			0
$235$	−2.57784	−	2.57784i	−0.168159	−	0.168159i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.6541			−0.818523			−0.409262	−	0.912417i	$-0.634214\pi$
$239$	−12.6541			−0.818523			−0.409262	+	0.912417i	$0.634214\pi$
$240$		0			0
$241$	−0.492768			−0.0317420			−0.0158710	−	0.999874i	$-0.505052\pi$
$241$	−0.492768			−0.0317420			−0.0158710	+	0.999874i	$0.505052\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.68827	−	1.68827i	−0.107860	−	0.107860i
$246$		0			0
$247$	−14.3996			−0.916225
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−13.2736	+	13.2736i	−0.837821	+	0.837821i	−0.988572	−	0.150750i	$-0.951831\pi$
$251$	−13.2736	+	13.2736i	−0.837821	+	0.837821i	0.150750	+	0.988572i	$0.451831\pi$
$252$		0			0
$253$	−4.72366	−	4.72366i	−0.296974	−	0.296974i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	5.22405i		−	0.325867i	−0.986637	−	0.162934i	$-0.947904\pi$
$257$		−	5.22405i		−	0.325867i	0.986637	−	0.162934i	$-0.0520956\pi$
$258$		0			0
$259$	−4.88854	+	4.88854i	−0.303759	+	0.303759i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	4.94773i		−	0.305090i	−0.988297	−	0.152545i	$-0.951253\pi$
$263$		−	4.94773i		−	0.305090i	0.988297	−	0.152545i	$-0.0487469\pi$
$264$		0			0
$265$		−	10.0852i		−	0.619531i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.70507	−	7.70507i	0.469786	−	0.469786i	−0.432059	−	0.901845i	$-0.642213\pi$
$269$	7.70507	−	7.70507i	0.469786	−	0.469786i	0.901845	+	0.432059i	$0.142213\pi$
$270$		0			0
$271$		−	3.41528i		−	0.207464i	−0.994605	−	0.103732i	$-0.966922\pi$
$271$		−	3.41528i		−	0.207464i	0.994605	−	0.103732i	$-0.0330784\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.807291	−	0.807291i	−0.0486815	−	0.0486815i
$276$		0			0
$277$	0.378237	−	0.378237i	0.0227260	−	0.0227260i	−0.695652	−	0.718378i	$-0.744885\pi$
$277$	0.378237	−	0.378237i	0.0227260	−	0.0227260i	0.718378	+	0.695652i	$0.244885\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−24.4196			−1.45675			−0.728377	−	0.685177i	$-0.759725\pi$
$281$	−24.4196			−1.45675			−0.728377	+	0.685177i	$0.759725\pi$
$282$		0			0
$283$	−12.5144	−	12.5144i	−0.743903	−	0.743903i	0.229423	−	0.973327i	$-0.426316\pi$
$283$	−12.5144	−	12.5144i	−0.743903	−	0.743903i	−0.973327	+	0.229423i	$0.926316\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.51037			0.207210
$288$		0			0
$289$	16.4991			0.970533
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.2457	−	16.2457i	−0.949084	−	0.949084i	0.0496807	−	0.998765i	$-0.484180\pi$
$293$	−16.2457	−	16.2457i	−0.949084	−	0.949084i	−0.998765	+	0.0496807i	$0.984180\pi$
$294$		0			0
$295$	0.184667			0.0107517
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.06506	−	5.06506i	0.292920	−	0.292920i
$300$		0			0
$301$	−4.39450	−	4.39450i	−0.253295	−	0.253295i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			27.2701i			1.56148i
$306$		0			0
$307$	18.1599	−	18.1599i	1.03644	−	1.03644i	0.0371306	−	0.999310i	$-0.488178\pi$
$307$	18.1599	−	18.1599i	1.03644	−	1.03644i	0.999310	−	0.0371306i	$-0.0118217\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	6.41299i		−	0.363647i	−0.983331	−	0.181824i	$-0.941800\pi$
$311$		−	6.41299i		−	0.363647i	0.983331	−	0.181824i	$-0.0582000\pi$
$312$		0			0
$313$			6.67820i			0.377474i	0.982028	+	0.188737i	$0.0604394\pi$
$313$			6.67820i			0.377474i	−0.982028	+	0.188737i	$0.939561\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.54976	−	6.54976i	0.367871	−	0.367871i	−0.498829	−	0.866700i	$-0.666236\pi$
$317$	6.54976	−	6.54976i	0.367871	−	0.367871i	0.866700	+	0.498829i	$0.166236\pi$
$318$		0			0
$319$		−	3.62576i		−	0.203004i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.12385	+	4.12385i	0.229457	+	0.229457i
$324$		0			0
$325$	0.865638	−	0.865638i	0.0480170	−	0.0480170i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.52691			0.0841812
$330$		0			0
$331$	−4.65194	−	4.65194i	−0.255694	−	0.255694i	0.567606	−	0.823300i	$-0.307870\pi$
$331$	−4.65194	−	4.65194i	−0.255694	−	0.255694i	−0.823300	+	0.567606i	$0.807870\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−31.3156			−1.71096
$336$		0			0
$337$	2.10465			0.114648			0.0573239	−	0.998356i	$-0.481743\pi$
$337$	2.10465			0.114648			0.0573239	+	0.998356i	$0.481743\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7.23222	+	7.23222i	0.391646	+	0.391646i
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	18.2837	−	18.2837i	0.981519	−	0.981519i	−0.0183129	−	0.999832i	$-0.505830\pi$
$347$	18.2837	−	18.2837i	0.981519	−	0.981519i	0.999832	+	0.0183129i	$0.00582951\pi$
$348$		0			0
$349$	−5.43242	−	5.43242i	−0.290791	−	0.290791i	0.546602	−	0.837393i	$-0.315921\pi$
$349$	−5.43242	−	5.43242i	−0.290791	−	0.290791i	−0.837393	+	0.546602i	$0.815921\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			32.1488i			1.71111i	0.517715	+	0.855553i	$0.326783\pi$
$353$			32.1488i			1.71111i	−0.517715	+	0.855553i	$0.673217\pi$
$354$		0			0
$355$	13.8725	−	13.8725i	0.736278	−	0.736278i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	10.2571i		−	0.541351i	−0.962671	−	0.270676i	$-0.912753\pi$
$359$		−	10.2571i		−	0.541351i	0.962671	−	0.270676i	$-0.0872471\pi$
$360$		0			0
$361$		−	48.8962i		−	2.57348i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.67985	−	1.67985i	0.0879272	−	0.0879272i
$366$		0			0
$367$			33.7666i			1.76260i	0.472555	+	0.881301i	$0.343332\pi$
$367$			33.7666i			1.76260i	−0.472555	+	0.881301i	$0.656668\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.98685	+	2.98685i	0.155069	+	0.155069i
$372$		0			0
$373$	17.6905	−	17.6905i	0.915978	−	0.915978i	−0.0807556	−	0.996734i	$-0.525733\pi$
$373$	17.6905	−	17.6905i	0.915978	−	0.915978i	0.996734	+	0.0807556i	$0.0257333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.88782			0.200233
$378$		0			0
$379$	−15.6915	−	15.6915i	−0.806016	−	0.806016i	0.178012	−	0.984028i	$-0.443033\pi$
$379$	−15.6915	−	15.6915i	−0.806016	−	0.806016i	−0.984028	+	0.178012i	$0.943033\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.74730			−0.242576			−0.121288	−	0.992617i	$-0.538702\pi$
$383$	−4.74730			−0.242576			−0.121288	+	0.992617i	$0.538702\pi$
$384$		0			0
$385$	3.89116			0.198312
$386$		0			0
$387$		0			0
$388$		0			0
$389$	15.8807	+	15.8807i	0.805182	+	0.805182i	0.983900	−	0.178718i	$-0.0571950\pi$
$389$	15.8807	+	15.8807i	0.805182	+	0.805182i	−0.178718	+	0.983900i	$0.557195\pi$
$390$		0			0
$391$	−2.90113			−0.146716
$392$		0			0
$393$		0			0
$394$		0			0
$395$	11.5785	−	11.5785i	0.582579	−	0.582579i
$396$		0			0
$397$	−7.12849	−	7.12849i	−0.357768	−	0.357768i	0.505221	−	0.862990i	$-0.331411\pi$
$397$	−7.12849	−	7.12849i	−0.357768	−	0.357768i	−0.862990	+	0.505221i	$0.831411\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	15.8573i		−	0.791875i	−0.918278	−	0.395937i	$-0.870420\pi$
$401$		−	15.8573i		−	0.791875i	0.918278	−	0.395937i	$-0.129580\pi$
$402$		0			0
$403$	−7.75492	+	7.75492i	−0.386300	+	0.386300i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	11.2672i		−	0.558495i
$408$		0			0
$409$			2.66835i			0.131942i	0.997822	+	0.0659708i	$0.0210144\pi$
$409$			2.66835i			0.131942i	−0.997822	+	0.0659708i	$0.978986\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.0546910	+	0.0546910i	−0.00269117	+	0.00269117i
$414$		0			0
$415$			26.7646i			1.31382i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.87910	−	2.87910i	−0.140653	−	0.140653i	0.633274	−	0.773927i	$-0.281710\pi$
$419$	−2.87910	−	2.87910i	−0.140653	−	0.140653i	−0.773927	+	0.633274i	$0.781710\pi$
$420$		0			0
$421$	4.47446	−	4.47446i	0.218072	−	0.218072i	−0.589614	−	0.807685i	$-0.700720\pi$
$421$	4.47446	−	4.47446i	0.218072	−	0.218072i	0.807685	+	0.589614i	$0.200720\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.495814			−0.0240505
$426$		0			0
$427$	−8.07633	−	8.07633i	−0.390841	−	0.390841i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−37.1722			−1.79052			−0.895260	−	0.445544i	$-0.853010\pi$
$431$	−37.1722			−1.79052			−0.895260	+	0.445544i	$0.853010\pi$
$432$		0			0
$433$	4.58851			0.220510			0.110255	−	0.993903i	$-0.464833\pi$
$433$	4.58851			0.220510			0.110255	+	0.993903i	$0.464833\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	23.8825	+	23.8825i	1.14246	+	1.14246i
$438$		0			0
$439$	26.4553			1.26264			0.631321	−	0.775521i	$-0.282513\pi$
$439$	26.4553			1.26264			0.631321	+	0.775521i	$0.282513\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.5727	−	16.5727i	0.787391	−	0.787391i	−0.193675	−	0.981066i	$-0.562041\pi$
$443$	16.5727	−	16.5727i	0.787391	−	0.787391i	0.981066	+	0.193675i	$0.0620406\pi$
$444$		0			0
$445$	24.2235	+	24.2235i	1.14830	+	1.14830i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			35.0202i			1.65271i	0.563153	+	0.826353i	$0.309588\pi$
$449$			35.0202i			1.65271i	−0.563153	+	0.826353i	$0.690412\pi$
$450$		0			0
$451$	−4.04538	+	4.04538i	−0.190490	+	0.190490i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			4.17239i			0.195605i
$456$		0			0
$457$			6.33352i			0.296269i	0.988967	+	0.148135i	$0.0473269\pi$
$457$			6.33352i			0.296269i	−0.988967	+	0.148135i	$0.952673\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.8206	−	15.8206i	0.736840	−	0.736840i	−0.235125	−	0.971965i	$-0.575550\pi$
$461$	15.8206	−	15.8206i	0.736840	−	0.736840i	0.971965	+	0.235125i	$0.0755499\pi$
$462$		0			0
$463$		−	5.57848i		−	0.259254i	−0.991563	−	0.129627i	$-0.958622\pi$
$463$		−	5.57848i		−	0.259254i	0.991563	−	0.129627i	$-0.0413780\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	13.4450	+	13.4450i	0.622159	+	0.622159i	0.946083	−	0.323924i	$-0.105002\pi$
$467$	13.4450	+	13.4450i	0.622159	+	0.622159i	−0.323924	+	0.946083i	$0.605002\pi$
$468$		0			0
$469$	9.27446	−	9.27446i	0.428255	−	0.428255i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.1285			0.465710
$474$		0			0
$475$	4.08161	+	4.08161i	0.187277	+	0.187277i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	25.7923			1.17848			0.589239	−	0.807959i	$-0.299428\pi$
$479$	25.7923			1.17848			0.589239	+	0.807959i	$0.299428\pi$
$480$		0			0
$481$	12.0815			0.550871
$482$		0			0
$483$		0			0
$484$		0			0
$485$	13.2419	+	13.2419i	0.601282	+	0.601282i
$486$		0			0
$487$	−11.6304			−0.527022			−0.263511	−	0.964656i	$-0.584881\pi$
$487$	−11.6304			−0.527022			−0.263511	+	0.964656i	$0.584881\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3.79791	+	3.79791i	−0.171397	+	0.171397i	−0.787593	−	0.616196i	$-0.788673\pi$
$491$	−3.79791	+	3.79791i	−0.171397	+	0.171397i	0.616196	+	0.787593i	$0.288673\pi$
$492$		0			0
$493$	−1.11342	−	1.11342i	−0.0501458	−	0.0501458i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			8.21700i			0.368583i
$498$		0			0
$499$	−24.9510	+	24.9510i	−1.11696	+	1.11696i	−0.124773	+	0.992185i	$0.539820\pi$
$499$	−24.9510	+	24.9510i	−1.11696	+	1.11696i	−0.992185	+	0.124773i	$0.960180\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	33.2186i		−	1.48114i	−0.671978	−	0.740571i	$-0.734555\pi$
$503$		−	33.2186i		−	1.48114i	0.671978	−	0.740571i	$-0.265445\pi$
$504$		0			0
$505$			24.3159i			1.08205i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	25.8128	−	25.8128i	1.14413	−	1.14413i	0.156445	−	0.987687i	$-0.449997\pi$
$509$	25.8128	−	25.8128i	1.14413	−	1.14413i	0.987687	−	0.156445i	$-0.0500034\pi$
$510$		0			0
$511$			0.995009i			0.0440166i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−11.3217	−	11.3217i	−0.498893	−	0.498893i
$516$		0			0
$517$	−1.75962	+	1.75962i	−0.0773882	+	0.0773882i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	38.7445			1.69743			0.848714	−	0.528852i	$-0.177377\pi$
$521$	38.7445			1.69743			0.848714	+	0.528852i	$0.177377\pi$
$522$		0			0
$523$	−10.9299	−	10.9299i	−0.477929	−	0.477929i	0.426540	−	0.904469i	$-0.359732\pi$
$523$	−10.9299	−	10.9299i	−0.477929	−	0.477929i	−0.904469	+	0.426540i	$0.859732\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.44180			0.193488
$528$		0			0
$529$	6.19865			0.269506
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.33776	−	4.33776i	−0.187889	−	0.187889i
$534$		0			0
$535$	−15.6781			−0.677822
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.15241	+	1.15241i	−0.0496378	+	0.0496378i
$540$		0			0
$541$	1.33220	+	1.33220i	0.0572756	+	0.0572756i	0.735164	−	0.677889i	$-0.237105\pi$
$541$	1.33220	+	1.33220i	0.0572756	+	0.0572756i	−0.677889	+	0.735164i	$0.737105\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			46.0690i			1.97338i
$546$		0			0
$547$	9.64149	−	9.64149i	0.412241	−	0.412241i	−0.470278	−	0.882518i	$-0.655846\pi$
$547$	9.64149	−	9.64149i	0.412241	−	0.412241i	0.882518	+	0.470278i	$0.155846\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			18.3316i			0.780952i
$552$		0			0
$553$			6.85821i			0.291641i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−28.7636	+	28.7636i	−1.21875	+	1.21875i	−0.250683	+	0.968069i	$0.580655\pi$
$557$	−28.7636	+	28.7636i	−1.21875	+	1.21875i	−0.968069	+	0.250683i	$0.919345\pi$
$558$		0			0
$559$			10.8606i			0.459353i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−8.12416	−	8.12416i	−0.342393	−	0.342393i	0.514874	−	0.857266i	$-0.327839\pi$
$563$	−8.12416	−	8.12416i	−0.342393	−	0.342393i	−0.857266	+	0.514874i	$0.827839\pi$
$564$		0			0
$565$	−17.1316	+	17.1316i	−0.720731	+	0.720731i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	29.0826			1.21920			0.609602	−	0.792707i	$-0.291329\pi$
$569$	29.0826			1.21920			0.609602	+	0.792707i	$0.291329\pi$
$570$		0			0
$571$	19.0450	+	19.0450i	0.797011	+	0.797011i	0.982623	−	0.185613i	$-0.0594269\pi$
$571$	19.0450	+	19.0450i	0.797011	+	0.797011i	−0.185613	+	0.982623i	$0.559427\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.87141			−0.119746
$576$		0			0
$577$	39.6045			1.64876			0.824378	−	0.566040i	$-0.191525\pi$
$577$	39.6045			1.64876			0.824378	+	0.566040i	$0.191525\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.92662	−	7.92662i	−0.328852	−	0.328852i
$582$		0			0
$583$	−6.88415			−0.285112
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−28.6807	+	28.6807i	−1.18378	+	1.18378i	−0.205020	+	0.978758i	$0.565726\pi$
$587$	−28.6807	+	28.6807i	−1.18378	+	1.18378i	−0.978758	+	0.205020i	$0.934274\pi$
$588$		0			0
$589$	−36.5656	−	36.5656i	−1.50666	−	1.50666i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	36.7230i		−	1.50803i	−0.656856	−	0.754016i	$-0.728114\pi$
$593$		−	36.7230i		−	1.50803i	0.656856	−	0.754016i	$-0.271886\pi$
$594$		0			0
$595$	1.19492	−	1.19492i	0.0489868	−	0.0489868i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			9.20943i			0.376287i	0.982142	+	0.188144i	$0.0602470\pi$
$599$			9.20943i			0.376287i	−0.982142	+	0.188144i	$0.939753\pi$
$600$		0			0
$601$			21.2489i			0.866761i	0.901211	+	0.433381i	$0.142679\pi$
$601$			21.2489i			0.866761i	−0.901211	+	0.433381i	$0.857321\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14.0868	−	14.0868i	0.572709	−	0.572709i
$606$		0			0
$607$		−	5.64124i		−	0.228971i	−0.993425	−	0.114485i	$-0.963478\pi$
$607$		−	5.64124i		−	0.228971i	0.993425	−	0.114485i	$-0.0365219\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.88680	−	1.88680i	−0.0763317	−	0.0763317i
$612$		0			0
$613$	−19.7439	+	19.7439i	−0.797447	+	0.797447i	−0.982692	−	0.185245i	$-0.940692\pi$
$613$	−19.7439	+	19.7439i	−0.797447	+	0.797447i	0.185245	+	0.982692i	$0.440692\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14.0703			−0.566451			−0.283225	−	0.959053i	$-0.591404\pi$
$617$	−14.0703			−0.566451			−0.283225	+	0.959053i	$0.591404\pi$
$618$		0			0
$619$	21.7632	+	21.7632i	0.874739	+	0.874739i	0.992984	−	0.118246i	$-0.0377270\pi$
$619$	21.7632	+	21.7632i	0.874739	+	0.874739i	−0.118246	+	0.992984i	$0.537727\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−14.3481			−0.574844
$624$		0			0
$625$	28.0119			1.12048
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.45999	−	3.45999i	−0.137959	−	0.137959i
$630$		0			0
$631$	33.0663			1.31635			0.658174	−	0.752866i	$-0.271329\pi$
$631$	33.0663			1.31635			0.658174	+	0.752866i	$0.271329\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	16.4549	−	16.4549i	0.652992	−	0.652992i
$636$		0			0
$637$	−1.23570	−	1.23570i	−0.0489602	−	0.0489602i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	40.6644i		−	1.60615i	−0.595879	−	0.803074i	$-0.703196\pi$
$641$		−	40.6644i		−	1.60615i	0.595879	−	0.803074i	$-0.296804\pi$
$642$		0			0
$643$	28.9151	−	28.9151i	1.14030	−	1.14030i	0.151904	−	0.988395i	$-0.451459\pi$
$643$	28.9151	−	28.9151i	1.14030	−	1.14030i	0.988395	−	0.151904i	$-0.0485406\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	43.9618i		−	1.72832i	−0.503219	−	0.864159i	$-0.667851\pi$
$647$		−	43.9618i		−	1.72832i	0.503219	−	0.864159i	$-0.332149\pi$
$648$		0			0
$649$		−	0.126053i		−	0.00494801i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13.8332	−	13.8332i	0.541335	−	0.541335i	−0.382585	−	0.923920i	$-0.624966\pi$
$653$	13.8332	−	13.8332i	0.541335	−	0.541335i	0.923920	+	0.382585i	$0.124966\pi$
$654$		0			0
$655$		−	33.9901i		−	1.32810i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.31948	+	3.31948i	0.129309	+	0.129309i	0.768799	−	0.639490i	$-0.220855\pi$
$659$	3.31948	+	3.31948i	0.129309	+	0.129309i	−0.639490	+	0.768799i	$0.720855\pi$
$660$		0			0
$661$	0.0259580	−	0.0259580i	0.00100965	−	0.00100965i	−0.706602	−	0.707611i	$-0.749773\pi$
$661$	0.0259580	−	0.0259580i	0.00100965	−	0.00100965i	0.707611	+	0.706602i	$0.249773\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−19.6734			−0.762903
$666$		0			0
$667$	−6.44815	−	6.44815i	−0.249673	−	0.249673i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	18.6145			0.718604
$672$		0			0
$673$	−30.4544			−1.17393			−0.586965	−	0.809613i	$-0.699677\pi$
$673$	−30.4544			−1.17393			−0.586965	+	0.809613i	$0.699677\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.9412	−	12.9412i	−0.497369	−	0.497369i	0.413249	−	0.910618i	$-0.364394\pi$
$677$	−12.9412	−	12.9412i	−0.497369	−	0.497369i	−0.910618	+	0.413249i	$0.864394\pi$
$678$		0			0
$679$	−7.84344			−0.301004
$680$		0			0
$681$		0			0
$682$		0			0
$683$	5.08157	−	5.08157i	0.194441	−	0.194441i	−0.603171	−	0.797612i	$-0.706096\pi$
$683$	5.08157	−	5.08157i	0.194441	−	0.194441i	0.797612	+	0.603171i	$0.206096\pi$
$684$		0			0
$685$	13.0310	+	13.0310i	0.497889	+	0.497889i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	7.38170i		−	0.281220i
$690$		0			0
$691$	−11.7924	+	11.7924i	−0.448606	+	0.448606i	−0.894891	−	0.446285i	$-0.852747\pi$
$691$	−11.7924	+	11.7924i	−0.448606	+	0.448606i	0.446285	+	0.894891i	$0.352747\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			18.5636i			0.704157i
$696$		0			0
$697$			2.48455i			0.0941090i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.76041	+	1.76041i	−0.0664897	+	0.0664897i	−0.739570	−	0.673080i	$-0.764971\pi$
$701$	−1.76041	+	1.76041i	−0.0664897	+	0.0664897i	0.673080	+	0.739570i	$0.264971\pi$
$702$		0			0
$703$			56.9662i			2.14852i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.20143	−	7.20143i	−0.270838	−	0.270838i
$708$		0			0
$709$	−13.3717	+	13.3717i	−0.502185	+	0.502185i	−0.912116	−	0.409931i	$-0.865553\pi$
$709$	−13.3717	+	13.3717i	−0.502185	+	0.502185i	0.409931	+	0.912116i	$0.365553\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	25.7239			0.963367
$714$		0			0
$715$	−4.80831	−	4.80831i	−0.179820	−	0.179820i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	29.9516			1.11700			0.558502	−	0.829503i	$-0.311376\pi$
$719$	29.9516			1.11700			0.558502	+	0.829503i	$0.311376\pi$
$720$		0			0
$721$	6.70608			0.249747
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.10201	−	1.10201i	−0.0409277	−	0.0409277i
$726$		0			0
$727$	−52.2974			−1.93960			−0.969801	−	0.243897i	$-0.921574\pi$
$727$	−52.2974			−1.93960			−0.969801	+	0.243897i	$0.921574\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3.11032	−	3.11032i	0.115039	−	0.115039i
$732$		0			0
$733$	14.9261	+	14.9261i	0.551308	+	0.551308i	0.926818	−	0.375511i	$-0.122533\pi$
$733$	14.9261	+	14.9261i	0.551308	+	0.551308i	−0.375511	+	0.926818i	$0.622533\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			21.3760i			0.787394i
$738$		0			0
$739$	−0.113962	+	0.113962i	−0.00419215	+	0.00419215i	−0.709200	−	0.705008i	$-0.750944\pi$
$739$	−0.113962	+	0.113962i	−0.00419215	+	0.00419215i	0.705008	+	0.709200i	$0.250944\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			30.4230i			1.11611i	0.829803	+	0.558056i	$0.188453\pi$
$743$			30.4230i			1.11611i	−0.829803	+	0.558056i	$0.811547\pi$
$744$		0			0
$745$			30.3719i			1.11274i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.64323	−	4.64323i	0.169660	−	0.169660i
$750$		0			0
$751$			33.5953i			1.22591i	0.790117	+	0.612956i	$0.210020\pi$
$751$			33.5953i			1.22591i	−0.790117	+	0.612956i	$0.789980\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	35.1679	+	35.1679i	1.27989	+	1.27989i
$756$		0			0
$757$	−6.89248	+	6.89248i	−0.250511	+	0.250511i	−0.821180	−	0.570669i	$-0.806684\pi$
$757$	−6.89248	+	6.89248i	−0.250511	+	0.250511i	0.570669	+	0.821180i	$0.306684\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	14.4591			0.524142			0.262071	−	0.965049i	$-0.415595\pi$
$761$	14.4591			0.524142			0.262071	+	0.965049i	$0.415595\pi$
$762$		0			0
$763$	−13.6438	−	13.6438i	−0.493940	−	0.493940i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.135163			0.00488046
$768$		0			0
$769$	−2.08889			−0.0753274			−0.0376637	−	0.999290i	$-0.511992\pi$
$769$	−2.08889			−0.0753274			−0.0376637	+	0.999290i	$0.511992\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−14.8285	−	14.8285i	−0.533345	−	0.533345i	0.388221	−	0.921566i	$-0.373090\pi$
$773$	−14.8285	−	14.8285i	−0.533345	−	0.533345i	−0.921566	+	0.388221i	$0.873090\pi$
$774$		0			0
$775$	4.39631			0.157920
$776$		0			0
$777$		0			0
$778$		0			0
$779$	20.4532	−	20.4532i	0.732810	−	0.732810i
$780$		0			0
$781$	−9.46935	−	9.46935i	−0.338840	−	0.338840i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	19.3407i		−	0.690301i
$786$		0			0
$787$	−5.49280	+	5.49280i	−0.195797	+	0.195797i	−0.798195	−	0.602398i	$-0.794212\pi$
$787$	−5.49280	+	5.49280i	−0.195797	+	0.195797i	0.602398	+	0.798195i	$0.294212\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	10.1474i		−	0.360800i
$792$		0			0
$793$			19.9598i			0.708795i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−32.4998	+	32.4998i	−1.15120	+	1.15120i	−0.164890	+	0.986312i	$0.552727\pi$
$797$	−32.4998	+	32.4998i	−1.15120	+	1.15120i	−0.986312	+	0.164890i	$0.947273\pi$
$798$		0			0
$799$			1.08071i			0.0382327i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.14666	−	1.14666i	−0.0404647	−	0.0404647i
$804$		0			0
$805$	6.92013	−	6.92013i	0.243903	−	0.243903i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−39.5019			−1.38881			−0.694407	−	0.719582i	$-0.744333\pi$
$809$	−39.5019			−1.38881			−0.694407	+	0.719582i	$0.744333\pi$
$810$		0			0
$811$	2.13298	+	2.13298i	0.0748991	+	0.0748991i	0.743564	−	0.668665i	$-0.233134\pi$
$811$	2.13298	+	2.13298i	0.0748991	+	0.0748991i	−0.668665	+	0.743564i	$0.733134\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	21.4909			0.752795
$816$		0			0
$817$	−51.2091			−1.79158
$818$		0			0
$819$		0			0
$820$		0			0
$821$	39.0080	+	39.0080i	1.36139	+	1.36139i	0.872148	+	0.489242i	$0.162726\pi$
$821$	39.0080	+	39.0080i	1.36139	+	1.36139i	0.489242	+	0.872148i	$0.337274\pi$
$822$		0			0
$823$	18.1502			0.632676			0.316338	−	0.948647i	$-0.397547\pi$
$823$	18.1502			0.632676			0.316338	+	0.948647i	$0.397547\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	33.2007	−	33.2007i	1.15450	−	1.15450i	0.168861	−	0.985640i	$-0.445991\pi$
$827$	33.2007	−	33.2007i	1.15450	−	1.15450i	0.985640	−	0.168861i	$-0.0540088\pi$
$828$		0			0
$829$	0.760686	+	0.760686i	0.0264197	+	0.0264197i	0.720193	−	0.693774i	$-0.244053\pi$
$829$	0.760686	+	0.760686i	0.0264197	+	0.0264197i	−0.693774	+	0.720193i	$0.744053\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			0.707775i			0.0245229i
$834$		0			0
$835$	−21.3685	+	21.3685i	−0.739486	+	0.739486i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			8.57102i			0.295905i	0.988995	+	0.147952i	$0.0472682\pi$
$839$			8.57102i			0.295905i	−0.988995	+	0.147952i	$0.952732\pi$
$840$		0			0
$841$			24.0506i			0.829330i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−16.7917	+	16.7917i	−0.577652	+	0.577652i
$846$		0			0
$847$			8.34390i			0.286700i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−20.0379	−	20.0379i	−0.686890	−	0.686890i
$852$		0			0
$853$	−18.5253	+	18.5253i	−0.634294	+	0.634294i	−0.949142	−	0.314848i	$-0.898047\pi$
$853$	−18.5253	+	18.5253i	−0.634294	+	0.634294i	0.314848	+	0.949142i	$0.398047\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	26.0459			0.889709			0.444855	−	0.895603i	$-0.353255\pi$
$857$	26.0459			0.889709			0.444855	+	0.895603i	$0.353255\pi$
$858$		0			0
$859$	33.2875	+	33.2875i	1.13575	+	1.13575i	0.989203	+	0.146550i	$0.0468170\pi$
$859$	33.2875	+	33.2875i	1.13575	+	1.13575i	0.146550	+	0.989203i	$0.453183\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	38.9833			1.32701			0.663503	−	0.748174i	$-0.269069\pi$
$863$	38.9833			1.32701			0.663503	+	0.748174i	$0.269069\pi$
$864$		0			0
$865$	−5.56468			−0.189205
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.90347	−	7.90347i	−0.268107	−	0.268107i
$870$		0			0
$871$	−22.9209			−0.776645
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−7.25868	+	7.25868i	−0.245388	+	0.245388i
$876$		0			0
$877$	5.19908	+	5.19908i	0.175560	+	0.175560i	0.789417	−	0.613857i	$-0.210383\pi$
$877$	5.19908	+	5.19908i	0.175560	+	0.175560i	−0.613857	+	0.789417i	$0.710383\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	56.4632i		−	1.90229i	−0.308741	−	0.951146i	$-0.599908\pi$
$881$		−	56.4632i		−	1.90229i	0.308741	−	0.951146i	$-0.400092\pi$
$882$		0			0
$883$	12.0242	−	12.0242i	0.404646	−	0.404646i	−0.475221	−	0.879867i	$-0.657632\pi$
$883$	12.0242	−	12.0242i	0.404646	−	0.404646i	0.879867	+	0.475221i	$0.157632\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			50.7848i			1.70519i	0.522576	+	0.852593i	$0.324971\pi$
$887$			50.7848i			1.70519i	−0.522576	+	0.852593i	$0.675029\pi$
$888$		0			0
$889$			9.74658i			0.326890i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8.89653	−	8.89653i	0.297711	−	0.297711i
$894$		0			0
$895$		−	2.27119i		−	0.0759174i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9.87250	+	9.87250i	0.329266	+	0.329266i
$900$		0			0
$901$	−2.11402	+	2.11402i	−0.0704281	+	0.0704281i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	48.6906			1.61853
$906$		0			0
$907$	−13.4804	−	13.4804i	−0.447609	−	0.447609i	0.446950	−	0.894559i	$-0.352510\pi$
$907$	−13.4804	−	13.4804i	−0.447609	−	0.447609i	−0.894559	+	0.446950i	$0.852510\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	8.03561			0.266231			0.133116	−	0.991100i	$-0.457502\pi$
$911$	8.03561			0.266231			0.133116	+	0.991100i	$0.457502\pi$
$912$		0			0
$913$	18.2694			0.604630
$914$		0			0
$915$		0			0
$916$		0			0
$917$	10.0665	+	10.0665i	0.332426	+	0.332426i
$918$		0			0
$919$	43.6527			1.43997			0.719986	−	0.693989i	$-0.244148\pi$
$919$	43.6527			1.43997			0.719986	+	0.693989i	$0.244148\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.1537	−	10.1537i	0.334215	−	0.334215i
$924$		0			0
$925$	−3.42455	−	3.42455i	−0.112598	−	0.112598i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	18.1698i		−	0.596131i	−0.954545	−	0.298066i	$-0.903659\pi$
$929$		−	18.1698i		−	0.596131i	0.954545	−	0.298066i	$-0.0963414\pi$
$930$		0			0
$931$	5.82650	−	5.82650i	0.190956	−	0.190956i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			2.75407i			0.0900676i
$936$		0			0
$937$		−	43.3589i		−	1.41647i	−0.705975	−	0.708236i	$-0.749491\pi$
$937$		−	43.3589i		−	1.41647i	0.705975	−	0.708236i	$-0.250509\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	18.1988	−	18.1988i	0.593265	−	0.593265i	−0.345247	−	0.938512i	$-0.612205\pi$
$941$	18.1988	−	18.1988i	0.593265	−	0.593265i	0.938512	+	0.345247i	$0.112205\pi$
$942$		0			0
$943$			14.3888i			0.468564i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	9.85230	+	9.85230i	0.320157	+	0.320157i	0.848827	−	0.528670i	$-0.177309\pi$
$947$	9.85230	+	9.85230i	0.320157	+	0.320157i	−0.528670	+	0.848827i	$0.677309\pi$
$948$		0			0
$949$	1.22953	−	1.22953i	0.0399123	−	0.0399123i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−24.8972			−0.806500			−0.403250	−	0.915090i	$-0.632119\pi$
$953$	−24.8972			−0.806500			−0.403250	+	0.915090i	$0.632119\pi$
$954$		0			0
$955$	25.4714	+	25.4714i	0.824234	+	0.824234i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7.71854			−0.249245
$960$		0			0
$961$	−8.38484			−0.270479
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−45.2592	−	45.2592i	−1.45695	−	1.45695i
$966$		0			0
$967$	34.4058			1.10642			0.553208	−	0.833043i	$-0.313404\pi$
$967$	34.4058			1.10642			0.553208	+	0.833043i	$0.313404\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	10.4399	−	10.4399i	0.335033	−	0.335033i	−0.519461	−	0.854494i	$-0.673867\pi$
$971$	10.4399	−	10.4399i	0.335033	−	0.335033i	0.854494	+	0.519461i	$0.173867\pi$
$972$		0			0
$973$	−5.49781	−	5.49781i	−0.176252	−	0.176252i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	0.188385i		−	0.00602698i	−0.999995	−	0.00301349i	$-0.999041\pi$
$977$		−	0.188385i		−	0.00602698i	0.999995	−	0.00301349i	$-0.000959225\pi$
$978$		0			0
$979$	16.5349	−	16.5349i	0.528457	−	0.528457i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	33.6193i		−	1.07229i	−0.844126	−	0.536144i	$-0.819880\pi$
$983$		−	33.6193i		−	1.07229i	0.844126	−	0.536144i	$-0.180120\pi$
$984$		0			0
$985$			56.1097i			1.78780i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	18.0128	−	18.0128i	0.572774	−	0.572774i
$990$		0			0
$991$			13.4425i			0.427015i	0.976941	+	0.213507i	$0.0684887\pi$
$991$			13.4425i			0.427015i	−0.976941	+	0.213507i	$0.931511\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−14.1563	−	14.1563i	−0.448786	−	0.448786i
$996$		0			0
$997$	−9.59749	+	9.59749i	−0.303956	+	0.303956i	−0.842559	−	0.538604i	$-0.818952\pi$
$997$	−9.59749	+	9.59749i	−0.303956	+	0.303956i	0.538604	+	0.842559i	$0.318952\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.v.d.1583.5		36
3.2	odd	2	inner	4032.2.v.d.1583.14		36
4.3	odd	2		1008.2.v.d.323.12	yes	36
12.11	even	2		1008.2.v.d.323.7	✓	36
16.5	even	4		1008.2.v.d.827.7	yes	36
16.11	odd	4	inner	4032.2.v.d.3599.14		36
48.5	odd	4		1008.2.v.d.827.12	yes	36
48.11	even	4	inner	4032.2.v.d.3599.5		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.d.323.7	✓	36	12.11	even	2
1008.2.v.d.323.12	yes	36	4.3	odd	2
1008.2.v.d.827.7	yes	36	16.5	even	4
1008.2.v.d.827.12	yes	36	48.5	odd	4
4032.2.v.d.1583.5		36	1.1	even	1	trivial
4032.2.v.d.1583.14		36	3.2	odd	2	inner
4032.2.v.d.3599.5		36	48.11	even	4	inner
4032.2.v.d.3599.14		36	16.11	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.68827 - 1.68827i) q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+(-1.68827 - 1.68827i) q^{5} +1.00000 q^{7} +(-1.15241 + 1.15241i) q^{11} +(-1.23570 - 1.23570i) q^{13} +0.707775i q^{17} +(5.82650 - 5.82650i) q^{19} +4.09895i q^{23} +0.700525i q^{25} +(-1.57312 + 1.57312i) q^{29} -6.27573i q^{31} +(-1.68827 - 1.68827i) q^{35} +(-4.88854 + 4.88854i) q^{37} +3.51037 q^{41} +(-4.39450 - 4.39450i) q^{43} +1.52691 q^{47} +1.00000 q^{49} +(2.98685 + 2.98685i) q^{53} +3.89116 q^{55} +(-0.0546910 + 0.0546910i) q^{59} +(-8.07633 - 8.07633i) q^{61} +4.17239i q^{65} +(9.27446 - 9.27446i) q^{67} +8.21700i q^{71} +0.995009i q^{73} +(-1.15241 + 1.15241i) q^{77} +6.85821i q^{79} +(-7.92662 - 7.92662i) q^{83} +(1.19492 - 1.19492i) q^{85} -14.3481 q^{89} +(-1.23570 - 1.23570i) q^{91} -19.6734 q^{95} -7.84344 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 36 q^{7}+O(q^{10}) \) 36 * q + 36 * q^7	\( 36 q + 36 q^{7} - 16 q^{13} + 16 q^{19} + 20 q^{37} - 36 q^{43} + 36 q^{49} - 32 q^{55} + 112 q^{61} + 36 q^{67} - 96 q^{85} - 16 q^{91} + 56 q^{97}+O(q^{100}) \) 36 * q + 36 * q^7 - 16 * q^13 + 16 * q^19 + 20 * q^37 - 36 * q^43 + 36 * q^49 - 32 * q^55 + 112 * q^61 + 36 * q^67 - 96 * q^85 - 16 * q^91 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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