LMFDB - Embedded newform 936.2.ba.b.593.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [936,2,Mod(161,936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(936, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("936.161");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	936.ba (of order $4$, degree $2$, 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:	$7.47399762919$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	12.0.125772815663104.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 27x^{8} + 107x^{4} + 1 $ x^12 + 27x^8 + 107x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.1
Root			$1.04736 - 1.04736i$ of defining polynomial
Character	$\chi$	$=$	936.593
Dual form			936.2.ba.b.161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.93897 + 2.93897i) q^{5} +(-1.67513 + 1.67513i) q^{7} +(-0.569973 - 0.569973i) q^{11} +(1.48119 - 3.28726i) q^{13} -1.41421 q^{17} +(2.86907 + 2.86907i) q^{19} -5.32940 q^{23} -12.2750i q^{25} -0.274268i q^{29} +(-5.83146 - 5.83146i) q^{31} -9.84630i q^{35} +(2.61213 - 2.61213i) q^{37} +(3.21323 - 3.21323i) q^{41} -8.96239i q^{43} +(4.16801 + 4.16801i) q^{47} +1.38787i q^{49} -3.32377i q^{53} +3.35026 q^{55} +(-9.95684 - 9.95684i) q^{59} -5.66291 q^{61} +(5.30796 + 14.0143i) q^{65} +(-1.28726 - 1.28726i) q^{67} +(-11.8664 + 11.8664i) q^{71} +(-0.581810 + 0.581810i) q^{73} +1.90956 q^{77} -12.3127 q^{79} +(1.02941 - 1.02941i) q^{83} +(4.15633 - 4.15633i) q^{85} +(8.26836 + 8.26836i) q^{89} +(3.02539 + 7.98778i) q^{91} -16.8642 q^{95} +(12.1187 + 12.1187i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{13} + 16 q^{19} - 8 q^{31} + 28 q^{37} + 56 q^{61} + 8 q^{67} - 12 q^{73} - 64 q^{79} + 8 q^{85} - 24 q^{91} + 60 q^{97}+O(q^{100}) $ 12 * q - 4 * q^13 + 16 * q^19 - 8 * q^31 + 28 * q^37 + 56 * q^61 + 8 * q^67 - 12 * q^73 - 64 * q^79 + 8 * q^85 - 24 * q^91 + 60 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.93897	+	2.93897i	−1.31435	+	1.31435i	−0.396167	+	0.918179i	$0.629660\pi$
$5$	−2.93897	+	2.93897i	−1.31435	+	1.31435i	−0.918179	+	0.396167i	$0.870340\pi$
$6$		0			0
$7$	−1.67513	+	1.67513i	−0.633140	+	0.633140i	−0.948854	−	0.315714i	$-0.897756\pi$
$7$	−1.67513	+	1.67513i	−0.633140	+	0.633140i	0.315714	+	0.948854i	$0.397756\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.569973	−	0.569973i	−0.171853	−	0.171853i	0.615940	−	0.787793i	$-0.288776\pi$
$11$	−0.569973	−	0.569973i	−0.171853	−	0.171853i	−0.787793	+	0.615940i	$0.788776\pi$
$12$		0			0
$13$	1.48119	−	3.28726i	0.410809	−	0.911721i
$13$	1.48119	−	3.28726i	0.410809	−	0.911721i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.41421			−0.342997			−0.171499	−	0.985184i	$-0.554861\pi$
$17$	−1.41421			−0.342997			−0.171499	+	0.985184i	$0.554861\pi$
$18$		0			0
$19$	2.86907	+	2.86907i	0.658209	+	0.658209i	0.954956	−	0.296747i	$-0.0959018\pi$
$19$	2.86907	+	2.86907i	0.658209	+	0.658209i	−0.296747	+	0.954956i	$0.595902\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.32940			−1.11126			−0.555628	−	0.831431i	$-0.687522\pi$
$23$	−5.32940			−1.11126			−0.555628	+	0.831431i	$0.687522\pi$
$24$		0			0
$25$		−	12.2750i		−	2.45501i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	0.274268i		−	0.0509302i	−0.999676	−	0.0254651i	$-0.991893\pi$
$29$		−	0.274268i		−	0.0509302i	0.999676	−	0.0254651i	$-0.00810668\pi$
$30$		0			0
$31$	−5.83146	−	5.83146i	−1.04736	−	1.04736i	−0.998821	−	0.0485391i	$-0.984543\pi$
$31$	−5.83146	−	5.83146i	−1.04736	−	1.04736i	−0.0485391	−	0.998821i	$-0.515457\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	9.84630i		−	1.66433i
$36$		0			0
$37$	2.61213	−	2.61213i	0.429431	−	0.429431i	−0.459003	−	0.888434i	$-0.651793\pi$
$37$	2.61213	−	2.61213i	0.429431	−	0.429431i	0.888434	+	0.459003i	$0.151793\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.21323	−	3.21323i	0.501823	−	0.501823i	−0.410181	−	0.912004i	$-0.634535\pi$
$41$	3.21323	−	3.21323i	0.501823	−	0.501823i	0.912004	+	0.410181i	$0.134535\pi$
$42$		0			0
$43$		−	8.96239i		−	1.36675i	−0.730067	−	0.683376i	$-0.760511\pi$
$43$		−	8.96239i		−	1.36675i	0.730067	−	0.683376i	$-0.239489\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.16801	+	4.16801i	0.607967	+	0.607967i	0.942414	−	0.334447i	$-0.108550\pi$
$47$	4.16801	+	4.16801i	0.607967	+	0.607967i	−0.334447	+	0.942414i	$0.608550\pi$
$48$		0			0
$49$			1.38787i			0.198268i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	3.32377i		−	0.456555i	−0.973596	−	0.228278i	$-0.926691\pi$
$53$		−	3.32377i		−	0.456555i	0.973596	−	0.228278i	$-0.0733094\pi$
$54$		0			0
$55$	3.35026			0.451749
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.95684	−	9.95684i	−1.29627	−	1.29627i	−0.930838	−	0.365433i	$-0.880921\pi$
$59$	−9.95684	−	9.95684i	−1.29627	−	1.29627i	−0.365433	−	0.930838i	$-0.619079\pi$
$60$		0			0
$61$	−5.66291			−0.725062			−0.362531	−	0.931972i	$-0.618087\pi$
$61$	−5.66291			−0.725062			−0.362531	+	0.931972i	$0.618087\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.30796	+	14.0143i	0.658371	+	1.73826i
$66$		0			0
$67$	−1.28726	−	1.28726i	−0.157264	−	0.157264i	0.624089	−	0.781353i	$-0.285470\pi$
$67$	−1.28726	−	1.28726i	−0.157264	−	0.157264i	−0.781353	+	0.624089i	$0.785470\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−11.8664	+	11.8664i	−1.40828	+	1.40828i	−0.639450	+	0.768832i	$0.720838\pi$
$71$	−11.8664	+	11.8664i	−1.40828	+	1.40828i	−0.768832	+	0.639450i	$0.779162\pi$
$72$		0			0
$73$	−0.581810	+	0.581810i	−0.0680957	+	0.0680957i	−0.740334	−	0.672239i	$-0.765333\pi$
$73$	−0.581810	+	0.581810i	−0.0680957	+	0.0680957i	0.672239	+	0.740334i	$0.265333\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.90956			0.217614
$78$		0			0
$79$	−12.3127			−1.38528			−0.692641	−	0.721283i	$-0.743553\pi$
$79$	−12.3127			−1.38528			−0.692641	+	0.721283i	$0.743553\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.02941	−	1.02941i	0.112992	−	0.112992i	−0.648350	−	0.761342i	$-0.724541\pi$
$83$	1.02941	−	1.02941i	0.112992	−	0.112992i	0.761342	+	0.648350i	$0.224541\pi$
$84$		0			0
$85$	4.15633	−	4.15633i	0.450817	−	0.450817i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.26836	+	8.26836i	0.876445	+	0.876445i	0.993165	−	0.116720i	$-0.0372381\pi$
$89$	8.26836	+	8.26836i	0.876445	+	0.876445i	−0.116720	+	0.993165i	$0.537238\pi$
$90$		0			0
$91$	3.02539	+	7.98778i	0.317147	+	0.837347i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−16.8642			−1.73023
$96$		0			0
$97$	12.1187	+	12.1187i	1.23047	+	1.23047i	0.963784	+	0.266685i	$0.0859284\pi$
$97$	12.1187	+	12.1187i	1.23047	+	1.23047i	0.266685	+	0.963784i	$0.414072\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.5787			−1.25162			−0.625812	−	0.779974i	$-0.715232\pi$
$101$	−12.5787			−1.25162			−0.625812	+	0.779974i	$0.715232\pi$
$102$		0			0
$103$			2.57452i			0.253675i	0.991924	+	0.126837i	$0.0404826\pi$
$103$			2.57452i			0.253675i	−0.991924	+	0.126837i	$0.959517\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	16.1664i		−	1.56286i	−0.623990	−	0.781432i	$-0.714490\pi$
$107$		−	16.1664i		−	1.56286i	0.623990	−	0.781432i	$-0.285510\pi$
$108$		0			0
$109$	12.5066	+	12.5066i	1.19791	+	1.19791i	0.974790	+	0.223124i	$0.0716254\pi$
$109$	12.5066	+	12.5066i	1.19791	+	1.19791i	0.223124	+	0.974790i	$0.428375\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2.55416i		−	0.240275i	−0.992757	−	0.120138i	$-0.961666\pi$
$113$		−	2.55416i		−	0.240275i	0.992757	−	0.120138i	$-0.0383336\pi$
$114$		0			0
$115$	15.6629	−	15.6629i	1.46057	−	1.46057i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.36899	−	2.36899i	0.217165	−	0.217165i
$120$		0			0
$121$		−	10.3503i		−	0.940933i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	21.3811	+	21.3811i	1.91238	+	1.91238i
$126$		0			0
$127$			8.18664i			0.726447i	0.931702	+	0.363224i	$0.118324\pi$
$127$			8.18664i			0.726447i	−0.931702	+	0.363224i	$0.881676\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			6.42647i			0.561483i	0.959783	+	0.280742i	$0.0905804\pi$
$131$			6.42647i			0.561483i	−0.959783	+	0.280742i	$0.909420\pi$
$132$		0			0
$133$	−9.61213			−0.833477
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.6892	−	12.6892i	−1.08411	−	1.08411i	−0.996121	−	0.0879906i	$-0.971955\pi$
$137$	−12.6892	−	12.6892i	−1.08411	−	1.08411i	−0.0879906	−	0.996121i	$-0.528045\pi$
$138$		0			0
$139$	−14.2374			−1.20760			−0.603801	−	0.797135i	$-0.706348\pi$
$139$	−14.2374			−1.20760			−0.603801	+	0.797135i	$0.706348\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.71789	+	1.02941i	−0.227281	+	0.0860834i
$144$		0			0
$145$	0.806063	+	0.806063i	0.0669399	+	0.0669399i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.85221	−	1.85221i	0.151739	−	0.151739i	−0.627155	−	0.778894i	$-0.715781\pi$
$149$	1.85221	−	1.85221i	0.151739	−	0.151739i	0.778894	+	0.627155i	$0.215781\pi$
$150$		0			0
$151$	−14.4812	+	14.4812i	−1.17846	+	1.17846i	−0.198327	+	0.980136i	$0.563551\pi$
$151$	−14.4812	+	14.4812i	−1.17846	+	1.17846i	−0.980136	+	0.198327i	$0.936449\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	34.2769			2.75319
$156$		0			0
$157$	−0.463096			−0.0369591			−0.0184795	−	0.999829i	$-0.505883\pi$
$157$	−0.463096			−0.0369591			−0.0184795	+	0.999829i	$0.505883\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.92744	−	8.92744i	0.703581	−	0.703581i
$162$		0			0
$163$	−10.4060	+	10.4060i	−0.815059	+	0.815059i	−0.985387	−	0.170329i	$-0.945517\pi$
$163$	−10.4060	+	10.4060i	−0.815059	+	0.815059i	0.170329	+	0.985387i	$0.445517\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.36677	−	7.36677i	−0.570058	−	0.570058i	0.362086	−	0.932145i	$-0.382064\pi$
$167$	−7.36677	−	7.36677i	−0.570058	−	0.570058i	−0.932145	+	0.362086i	$0.882064\pi$
$168$		0			0
$169$	−8.61213	−	9.73813i	−0.662471	−	0.749087i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.18576			0.546324			0.273162	−	0.961968i	$-0.411931\pi$
$173$	7.18576			0.546324			0.273162	+	0.961968i	$0.411931\pi$
$174$		0			0
$175$	20.5623	+	20.5623i	1.55436	+	1.55436i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	19.1441			1.43090			0.715448	−	0.698666i	$-0.246223\pi$
$179$	19.1441			1.43090			0.715448	+	0.698666i	$0.246223\pi$
$180$		0			0
$181$		−	21.7137i		−	1.61397i	−0.590574	−	0.806983i	$-0.701099\pi$
$181$		−	21.7137i		−	1.61397i	0.590574	−	0.806983i	$-0.298901\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			15.3539i			1.12884i
$186$		0			0
$187$	0.806063	+	0.806063i	0.0589452	+	0.0589452i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	22.6992i		−	1.64246i	−0.570598	−	0.821230i	$-0.693288\pi$
$191$		−	22.6992i		−	1.64246i	0.570598	−	0.821230i	$-0.306712\pi$
$192$		0			0
$193$	−12.8496	+	12.8496i	−0.924931	+	0.924931i	−0.997373	−	0.0724415i	$-0.976921\pi$
$193$	−12.8496	+	12.8496i	−0.924931	+	0.924931i	0.0724415	+	0.997373i	$0.476921\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.2367	−	12.2367i	0.871831	−	0.871831i	−0.120841	−	0.992672i	$-0.538559\pi$
$197$	12.2367	−	12.2367i	0.871831	−	0.871831i	0.992672	+	0.120841i	$0.0385590\pi$
$198$		0			0
$199$			7.19982i			0.510381i	0.966891	+	0.255191i	$0.0821383\pi$
$199$			7.19982i			0.510381i	−0.966891	+	0.255191i	$0.917862\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.459434	+	0.459434i	0.0322460	+	0.0322460i
$204$		0			0
$205$			18.8872i			1.31914i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	3.27058i		−	0.226231i
$210$		0			0
$211$	−10.7005			−0.736654			−0.368327	−	0.929696i	$-0.620069\pi$
$211$	−10.7005			−0.736654			−0.368327	+	0.929696i	$0.620069\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	26.3402	+	26.3402i	1.79638	+	1.79638i
$216$		0			0
$217$	19.5369			1.32625
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.09473	+	4.64888i	−0.140906	+	0.312718i
$222$		0			0
$223$	8.09332	+	8.09332i	0.541969	+	0.541969i	0.924106	−	0.382137i	$-0.124812\pi$
$223$	8.09332	+	8.09332i	0.541969	+	0.541969i	−0.382137	+	0.924106i	$0.624812\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.03931	−	7.03931i	0.467216	−	0.467216i	−0.433796	−	0.901011i	$-0.642826\pi$
$227$	7.03931	−	7.03931i	0.467216	−	0.467216i	0.901011	+	0.433796i	$0.142826\pi$
$228$		0			0
$229$	−1.68006	+	1.68006i	−0.111021	+	0.111021i	−0.760435	−	0.649414i	$-0.775014\pi$
$229$	−1.68006	+	1.68006i	−0.111021	+	0.111021i	0.649414	+	0.760435i	$0.275014\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−18.7841			−1.23058			−0.615292	−	0.788299i	$-0.710962\pi$
$233$	−18.7841			−1.23058			−0.615292	+	0.788299i	$0.710962\pi$
$234$		0			0
$235$	−24.4993			−1.59816
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.62938	+	9.62938i	−0.622873	+	0.622873i	−0.946265	−	0.323392i	$-0.895177\pi$
$239$	−9.62938	+	9.62938i	−0.622873	+	0.622873i	0.323392	+	0.946265i	$0.395177\pi$
$240$		0			0
$241$	−12.1187	+	12.1187i	−0.780635	+	0.780635i	−0.979938	−	0.199303i	$-0.936132\pi$
$241$	−12.1187	+	12.1187i	−0.780635	+	0.780635i	0.199303	+	0.979938i	$0.436132\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.07891	−	4.07891i	−0.260592	−	0.260592i
$246$		0			0
$247$	13.6810	−	5.18172i	0.870502	−	0.329705i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.33409			−0.210446			−0.105223	−	0.994449i	$-0.533556\pi$
$251$	−3.33409			−0.210446			−0.105223	+	0.994449i	$0.533556\pi$
$252$		0			0
$253$	3.03761	+	3.03761i	0.190973	+	0.190973i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.72767			−0.294904			−0.147452	−	0.989069i	$-0.547107\pi$
$257$	−4.72767			−0.294904			−0.147452	+	0.989069i	$0.547107\pi$
$258$		0			0
$259$			8.75131i			0.543780i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			18.3745i			1.13302i	0.824056	+	0.566509i	$0.191706\pi$
$263$			18.3745i			1.13302i	−0.824056	+	0.566509i	$0.808294\pi$
$264$		0			0
$265$	9.76845	+	9.76845i	0.600071	+	0.600071i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	5.01225i		−	0.305602i	−0.988257	−	0.152801i	$-0.951171\pi$
$269$		−	5.01225i		−	0.305602i	0.988257	−	0.152801i	$-0.0488294\pi$
$270$		0			0
$271$	12.3757	−	12.3757i	0.751767	−	0.751767i	−0.223041	−	0.974809i	$-0.571599\pi$
$271$	12.3757	−	12.3757i	0.751767	−	0.751767i	0.974809	+	0.223041i	$0.0715985\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.99644	+	6.99644i	−0.421901	+	0.421901i
$276$		0			0
$277$			0.237428i			0.0142656i	0.999975	+	0.00713282i	$0.00227047\pi$
$277$			0.237428i			0.0142656i	−0.999975	+	0.00713282i	$0.997730\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.05097	+	5.05097i	0.301316	+	0.301316i	0.841528	−	0.540213i	$-0.181656\pi$
$281$	5.05097	+	5.05097i	0.301316	+	0.301316i	−0.540213	+	0.841528i	$0.681656\pi$
$282$		0			0
$283$			8.12601i			0.483041i	0.970396	+	0.241521i	$0.0776461\pi$
$283$			8.12601i			0.483041i	−0.970396	+	0.241521i	$0.922354\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			10.7652i			0.635448i
$288$		0			0
$289$	−15.0000			−0.882353
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−11.7414	−	11.7414i	−0.685939	−	0.685939i	0.275393	−	0.961332i	$-0.411192\pi$
$293$	−11.7414	−	11.7414i	−0.685939	−	0.685939i	−0.961332	+	0.275393i	$0.911192\pi$
$294$		0			0
$295$	58.5256			3.40749
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.89387	+	17.5191i	−0.456514	+	1.01316i
$300$		0			0
$301$	15.0132	+	15.0132i	0.865345	+	0.865345i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	16.6431	−	16.6431i	0.952981	−	0.952981i
$306$		0			0
$307$	−7.36248	+	7.36248i	−0.420199	+	0.420199i	−0.885272	−	0.465073i	$-0.846028\pi$
$307$	−7.36248	+	7.36248i	−0.420199	+	0.420199i	0.465073	+	0.885272i	$0.346028\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−13.2661			−0.752254			−0.376127	−	0.926568i	$-0.622744\pi$
$311$	−13.2661			−0.752254			−0.376127	+	0.926568i	$0.622744\pi$
$312$		0			0
$313$	2.82653			0.159765			0.0798825	−	0.996804i	$-0.474545\pi$
$313$	2.82653			0.159765			0.0798825	+	0.996804i	$0.474545\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.48750	+	3.48750i	−0.195878	+	0.195878i	−0.798230	−	0.602353i	$-0.794230\pi$
$317$	−3.48750	+	3.48750i	−0.195878	+	0.195878i	0.602353	+	0.798230i	$0.294230\pi$
$318$		0			0
$319$	−0.156325	+	0.156325i	−0.00875253	+	0.00875253i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.05747	−	4.05747i	−0.225764	−	0.225764i
$324$		0			0
$325$	−40.3512	−	18.1817i	−2.23828	−	1.00854i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−13.9639			−0.769857
$330$		0			0
$331$	−5.26282	−	5.26282i	−0.289271	−	0.289271i	0.547521	−	0.836792i	$-0.315572\pi$
$331$	−5.26282	−	5.26282i	−0.289271	−	0.289271i	−0.836792	+	0.547521i	$0.815572\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.56641			0.413397
$336$		0			0
$337$			34.6761i			1.88893i	0.328617	+	0.944463i	$0.393418\pi$
$337$			34.6761i			1.88893i	−0.328617	+	0.944463i	$0.606582\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			6.64754i			0.359985i
$342$		0			0
$343$	−14.0508	−	14.0508i	−0.758671	−	0.758671i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	6.13357i		−	0.329267i	−0.986355	−	0.164634i	$-0.947356\pi$
$347$		−	6.13357i		−	0.329267i	0.986355	−	0.164634i	$-0.0526442\pi$
$348$		0			0
$349$	−4.66291	+	4.66291i	−0.249600	+	0.249600i	−0.820806	−	0.571206i	$-0.806476\pi$
$349$	−4.66291	+	4.66291i	−0.249600	+	0.249600i	0.571206	+	0.820806i	$0.306476\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.73945	−	6.73945i	0.358705	−	0.358705i	−0.504631	−	0.863335i	$-0.668371\pi$
$353$	6.73945	−	6.73945i	0.358705	−	0.358705i	0.863335	+	0.504631i	$0.168371\pi$
$354$		0			0
$355$		−	69.7499i		−	3.70194i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.23346	−	9.23346i	−0.487323	−	0.487323i	0.420137	−	0.907461i	$-0.361982\pi$
$359$	−9.23346	−	9.23346i	−0.487323	−	0.487323i	−0.907461	+	0.420137i	$0.861982\pi$
$360$		0			0
$361$		−	2.53690i		−	0.133521i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	3.41984i		−	0.179002i
$366$		0			0
$367$	18.7612			0.979324			0.489662	−	0.871912i	$-0.337120\pi$
$367$	18.7612			0.979324			0.489662	+	0.871912i	$0.337120\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.56775	+	5.56775i	0.289063	+	0.289063i
$372$		0			0
$373$	10.2012			0.528200			0.264100	−	0.964495i	$-0.414925\pi$
$373$	10.2012			0.528200			0.264100	+	0.964495i	$0.414925\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.901589	−	0.406244i	−0.0464342	−	0.0209226i
$378$		0			0
$379$	−25.4191	−	25.4191i	−1.30569	−	1.30569i	−0.924492	−	0.381202i	$-0.875510\pi$
$379$	−25.4191	−	25.4191i	−1.30569	−	1.30569i	−0.381202	−	0.924492i	$-0.624490\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10.9219	−	10.9219i	0.558085	−	0.558085i	−0.370677	−	0.928762i	$-0.620874\pi$
$383$	10.9219	−	10.9219i	0.558085	−	0.558085i	0.928762	+	0.370677i	$0.120874\pi$
$384$		0			0
$385$	−5.61213	+	5.61213i	−0.286020	+	0.286020i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.3622			−0.677491			−0.338745	−	0.940878i	$-0.610003\pi$
$389$	−13.3622			−0.677491			−0.338745	+	0.940878i	$0.610003\pi$
$390$		0			0
$391$	7.53690			0.381158
$392$		0			0
$393$		0			0
$394$		0			0
$395$	36.1865	−	36.1865i	1.82074	−	1.82074i
$396$		0			0
$397$	16.8496	−	16.8496i	0.845655	−	0.845655i	−0.143933	−	0.989587i	$-0.545975\pi$
$397$	16.8496	−	16.8496i	0.845655	−	0.845655i	0.989587	+	0.143933i	$0.0459749\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−14.2955	−	14.2955i	−0.713886	−	0.713886i	0.253460	−	0.967346i	$-0.418431\pi$
$401$	−14.2955	−	14.2955i	−0.713886	−	0.713886i	−0.967346	+	0.253460i	$0.918431\pi$
$402$		0			0
$403$	−27.8070	+	10.5320i	−1.38517	+	0.524635i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.97768			−0.147598
$408$		0			0
$409$	−8.81924	−	8.81924i	−0.436083	−	0.436083i	0.454608	−	0.890691i	$-0.349779\pi$
$409$	−8.81924	−	8.81924i	−0.436083	−	0.436083i	−0.890691	+	0.454608i	$0.849779\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	33.3580			1.64144
$414$		0			0
$415$			6.05079i			0.297021i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			23.3970i			1.14302i	0.820595	+	0.571510i	$0.193642\pi$
$419$			23.3970i			1.14302i	−0.820595	+	0.571510i	$0.806358\pi$
$420$		0			0
$421$	19.5950	+	19.5950i	0.955001	+	0.955001i	0.999030	−	0.0440292i	$-0.0140195\pi$
$421$	19.5950	+	19.5950i	0.955001	+	0.955001i	−0.0440292	+	0.999030i	$0.514019\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			17.3595i			0.842061i
$426$		0			0
$427$	9.48612	−	9.48612i	0.459065	−	0.459065i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15.5246	+	15.5246i	−0.747793	+	0.747793i	−0.974064	−	0.226271i	$-0.927347\pi$
$431$	−15.5246	+	15.5246i	−0.747793	+	0.747793i	0.226271	+	0.974064i	$0.427347\pi$
$432$		0			0
$433$		−	18.2619i		−	0.877609i	−0.898583	−	0.438805i	$-0.855402\pi$
$433$		−	18.2619i		−	0.877609i	0.898583	−	0.438805i	$-0.144598\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−15.2904	−	15.2904i	−0.731439	−	0.731439i
$438$		0			0
$439$		−	24.0362i		−	1.14719i	−0.819141	−	0.573593i	$-0.805549\pi$
$439$		−	24.0362i		−	1.14719i	0.819141	−	0.573593i	$-0.194451\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			33.1370i			1.57438i	0.616708	+	0.787192i	$0.288466\pi$
$443$			33.1370i			1.57438i	−0.616708	+	0.787192i	$0.711534\pi$
$444$		0			0
$445$	−48.6009			−2.30390
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5.61814	+	5.61814i	0.265136	+	0.265136i	0.827137	−	0.562001i	$-0.189968\pi$
$449$	5.61814	+	5.61814i	0.265136	+	0.265136i	−0.562001	+	0.827137i	$0.689968\pi$
$450$		0			0
$451$	−3.66291			−0.172480
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−32.3673	−	14.5843i	−1.51740	−	0.683722i
$456$		0			0
$457$	−18.1695	−	18.1695i	−0.849933	−	0.849933i	0.140191	−	0.990124i	$-0.455228\pi$
$457$	−18.1695	−	18.1695i	−0.849933	−	0.849933i	−0.990124	+	0.140191i	$0.955228\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	22.9590	−	22.9590i	1.06931	−	1.06931i	0.0718967	−	0.997412i	$-0.477095\pi$
$461$	22.9590	−	22.9590i	1.06931	−	1.06931i	0.997412	−	0.0718967i	$-0.0229052\pi$
$462$		0			0
$463$	−12.4001	+	12.4001i	−0.576281	+	0.576281i	−0.933877	−	0.357595i	$-0.883597\pi$
$463$	−12.4001	+	12.4001i	−0.576281	+	0.576281i	0.357595	+	0.933877i	$0.383597\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.3904			0.943555			0.471778	−	0.881718i	$-0.343613\pi$
$467$	20.3904			0.943555			0.471778	+	0.881718i	$0.343613\pi$
$468$		0			0
$469$	4.31265			0.199140
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−5.10832	+	5.10832i	−0.234881	+	0.234881i
$474$		0			0
$475$	35.2179	−	35.2179i	1.61591	−	1.61591i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−13.6613	−	13.6613i	−0.624199	−	0.624199i	0.322403	−	0.946602i	$-0.395509\pi$
$479$	−13.6613	−	13.6613i	−0.624199	−	0.624199i	−0.946602	+	0.322403i	$0.895509\pi$
$480$		0			0
$481$	−4.71767	−	12.4558i	−0.215107	−	0.567936i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−71.2330			−3.23452
$486$		0			0
$487$	−7.10650	−	7.10650i	−0.322026	−	0.322026i	0.527518	−	0.849544i	$-0.323123\pi$
$487$	−7.10650	−	7.10650i	−0.322026	−	0.322026i	−0.849544	+	0.527518i	$0.823123\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	19.6497			0.886780			0.443390	−	0.896329i	$-0.353776\pi$
$491$	19.6497			0.886780			0.443390	+	0.896329i	$0.353776\pi$
$492$		0			0
$493$			0.387873i			0.0174689i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	39.7556i		−	1.78328i
$498$		0			0
$499$	6.96476	+	6.96476i	0.311785	+	0.311785i	0.845601	−	0.533816i	$-0.179242\pi$
$499$	6.96476	+	6.96476i	0.311785	+	0.311785i	−0.533816	+	0.845601i	$0.679242\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			11.0581i			0.493055i	0.969136	+	0.246528i	$0.0792896\pi$
$503$			11.0581i			0.493055i	−0.969136	+	0.246528i	$0.920710\pi$
$504$		0			0
$505$	36.9683	−	36.9683i	1.64507	−	1.64507i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−28.4031	+	28.4031i	−1.25895	+	1.25895i	−0.307350	+	0.951597i	$0.599442\pi$
$509$	−28.4031	+	28.4031i	−1.25895	+	1.25895i	−0.951597	+	0.307350i	$0.900558\pi$
$510$		0			0
$511$		−	1.94921i		−	0.0862282i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7.56641	−	7.56641i	−0.333416	−	0.333416i
$516$		0			0
$517$		−	4.75131i		−	0.208962i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			13.3193i			0.583531i	0.956490	+	0.291765i	$0.0942426\pi$
$521$			13.3193i			0.583531i	−0.956490	+	0.291765i	$0.905757\pi$
$522$		0			0
$523$	24.8773			1.08781			0.543905	−	0.839147i	$-0.316945\pi$
$523$	24.8773			1.08781			0.543905	+	0.839147i	$0.316945\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.24692	+	8.24692i	0.359242	+	0.359242i
$528$		0			0
$529$	5.40246			0.234890
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−5.80330	−	15.3221i	−0.251369	−	0.663676i
$534$		0			0
$535$	47.5125	+	47.5125i	2.05414	+	2.05414i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.791050	−	0.791050i	0.0340729	−	0.0340729i
$540$		0			0
$541$	−7.67021	+	7.67021i	−0.329768	+	0.329768i	−0.852498	−	0.522730i	$-0.824914\pi$
$541$	−7.67021	+	7.67021i	−0.329768	+	0.329768i	0.522730	+	0.852498i	$0.324914\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−73.5129			−3.14894
$546$		0			0
$547$	−4.71511			−0.201604			−0.100802	−	0.994907i	$-0.532141\pi$
$547$	−4.71511			−0.201604			−0.100802	+	0.994907i	$0.532141\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.786893	−	0.786893i	0.0335228	−	0.0335228i
$552$		0			0
$553$	20.6253	−	20.6253i	0.877077	−	0.877077i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−23.0840	−	23.0840i	−0.978102	−	0.978102i	0.0216630	−	0.999765i	$-0.493104\pi$
$557$	−23.0840	−	23.0840i	−0.978102	−	0.978102i	−0.999765	+	0.0216630i	$0.993104\pi$
$558$		0			0
$559$	−29.4617	−	13.2750i	−1.24610	−	0.561474i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−16.6860			−0.703230			−0.351615	−	0.936145i	$-0.614367\pi$
$563$	−16.6860			−0.703230			−0.351615	+	0.936145i	$0.614367\pi$
$564$		0			0
$565$	7.50659	+	7.50659i	0.315804	+	0.315804i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.610041			0.0255742			0.0127871	−	0.999918i	$-0.495930\pi$
$569$	0.610041			0.0255742			0.0127871	+	0.999918i	$0.495930\pi$
$570$		0			0
$571$		−	5.63656i		−	0.235883i	−0.993021	−	0.117941i	$-0.962370\pi$
$571$		−	5.63656i		−	0.235883i	0.993021	−	0.117941i	$-0.0376295\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			65.4185i			2.72814i
$576$		0			0
$577$	17.2619	+	17.2619i	0.718621	+	0.718621i	0.968323	−	0.249702i	$-0.0803326\pi$
$577$	17.2619	+	17.2619i	0.718621	+	0.718621i	−0.249702	+	0.968323i	$0.580333\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			3.44878i			0.143080i
$582$		0			0
$583$	−1.89446	+	1.89446i	−0.0784605	+	0.0784605i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.76076	−	2.76076i	0.113949	−	0.113949i	−0.647833	−	0.761782i	$-0.724325\pi$
$587$	2.76076	−	2.76076i	0.113949	−	0.113949i	0.761782	+	0.647833i	$0.224325\pi$
$588$		0			0
$589$		−	33.4617i		−	1.37876i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	19.0970	+	19.0970i	0.784221	+	0.784221i	0.980540	−	0.196319i	$-0.0628987\pi$
$593$	19.0970	+	19.0970i	0.784221	+	0.784221i	−0.196319	+	0.980540i	$0.562899\pi$
$594$		0			0
$595$			13.9248i			0.570860i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			26.9316i			1.10039i	0.835035	+	0.550197i	$0.185447\pi$
$599$			26.9316i			1.10039i	−0.835035	+	0.550197i	$0.814553\pi$
$600$		0			0
$601$	−13.4255			−0.547637			−0.273818	−	0.961781i	$-0.588287\pi$
$601$	−13.4255			−0.547637			−0.273818	+	0.961781i	$0.588287\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	30.4191	+	30.4191i	1.23671	+	1.23671i
$606$		0			0
$607$	34.6107			1.40481			0.702403	−	0.711780i	$-0.252111\pi$
$607$	34.6107			1.40481			0.702403	+	0.711780i	$0.252111\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	19.8750	−	7.52770i	0.804055	−	0.304538i
$612$		0			0
$613$	−12.1490	−	12.1490i	−0.490695	−	0.490695i	0.417830	−	0.908525i	$-0.362791\pi$
$613$	−12.1490	−	12.1490i	−0.490695	−	0.490695i	−0.908525	+	0.417830i	$0.862791\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	22.6848	−	22.6848i	0.913254	−	0.913254i	−0.0832726	−	0.996527i	$-0.526537\pi$
$617$	22.6848	−	22.6848i	0.913254	−	0.913254i	0.996527	+	0.0832726i	$0.0265372\pi$
$618$		0			0
$619$	−13.2873	+	13.2873i	−0.534060	+	0.534060i	−0.921778	−	0.387718i	$-0.873263\pi$
$619$	−13.2873	+	13.2873i	−0.534060	+	0.534060i	0.387718	+	0.921778i	$0.373263\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−27.7012			−1.10982
$624$		0			0
$625$	−64.3014			−2.57206
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.69411	+	3.69411i	−0.147294	+	0.147294i
$630$		0			0
$631$	−21.0313	+	21.0313i	−0.837242	+	0.837242i	−0.988495	−	0.151253i	$-0.951669\pi$
$631$	−21.0313	+	21.0313i	−0.837242	+	0.837242i	0.151253	+	0.988495i	$0.451669\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.0603	−	24.0603i	−0.954802	−	0.954802i
$636$		0			0
$637$	4.56230	+	2.05571i	0.180765	+	0.0814502i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12.6934			0.501358			0.250679	−	0.968070i	$-0.419346\pi$
$641$	12.6934			0.501358			0.250679	+	0.968070i	$0.419346\pi$
$642$		0			0
$643$	13.1006	+	13.1006i	0.516638	+	0.516638i	0.916552	−	0.399915i	$-0.130960\pi$
$643$	13.1006	+	13.1006i	0.516638	+	0.516638i	−0.399915	+	0.916552i	$0.630960\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−2.24533			−0.0882731			−0.0441365	−	0.999026i	$-0.514054\pi$
$647$	−2.24533			−0.0882731			−0.0441365	+	0.999026i	$0.514054\pi$
$648$		0			0
$649$			11.3503i			0.445537i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	32.6277i		−	1.27682i	−0.769697	−	0.638410i	$-0.779592\pi$
$653$		−	32.6277i		−	1.27682i	0.769697	−	0.638410i	$-0.220408\pi$
$654$		0			0
$655$	−18.8872	−	18.8872i	−0.737983	−	0.737983i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	0.476714i		−	0.0185702i	−0.999957	−	0.00928508i	$-0.997044\pi$
$659$		−	0.476714i		−	0.0185702i	0.999957	−	0.00928508i	$-0.00295557\pi$
$660$		0			0
$661$	−5.49929	+	5.49929i	−0.213898	+	0.213898i	−0.805921	−	0.592023i	$-0.798329\pi$
$661$	−5.49929	+	5.49929i	−0.213898	+	0.213898i	0.592023	+	0.805921i	$0.298329\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	28.2497	−	28.2497i	1.09548	−	1.09548i
$666$		0			0
$667$			1.46168i			0.0565965i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3.22771	+	3.22771i	0.124604	+	0.124604i
$672$		0			0
$673$		−	3.08840i		−	0.119049i	−0.998227	−	0.0595245i	$-0.981042\pi$
$673$		−	3.08840i		−	0.119049i	0.998227	−	0.0595245i	$-0.0189584\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			43.9843i			1.69045i	0.534409	+	0.845226i	$0.320534\pi$
$677$			43.9843i			1.69045i	−0.534409	+	0.845226i	$0.679466\pi$
$678$		0			0
$679$	−40.6009			−1.55812
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.77536	−	6.77536i	−0.259252	−	0.259252i	0.565498	−	0.824750i	$-0.308684\pi$
$683$	−6.77536	−	6.77536i	−0.259252	−	0.259252i	−0.824750	+	0.565498i	$0.808684\pi$
$684$		0			0
$685$	74.5863			2.84980
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−10.9261	−	4.92315i	−0.416251	−	0.187557i
$690$		0			0
$691$	10.0630	+	10.0630i	0.382815	+	0.382815i	0.872115	−	0.489301i	$-0.162748\pi$
$691$	10.0630	+	10.0630i	0.382815	+	0.382815i	−0.489301	+	0.872115i	$0.662748\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	41.8433	−	41.8433i	1.58721	−	1.58721i
$696$		0			0
$697$	−4.54420	+	4.54420i	−0.172124	+	0.172124i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−4.89756			−0.184978			−0.0924891	−	0.995714i	$-0.529482\pi$
$701$	−4.89756			−0.184978			−0.0924891	+	0.995714i	$0.529482\pi$
$702$		0			0
$703$	14.9887			0.565311
$704$		0			0
$705$		0			0
$706$		0			0
$707$	21.0709	−	21.0709i	0.792453	−	0.792453i
$708$		0			0
$709$	−34.5428	+	34.5428i	−1.29728	+	1.29728i	−0.367100	+	0.930181i	$0.619649\pi$
$709$	−34.5428	+	34.5428i	−1.29728	+	1.29728i	−0.930181	+	0.367100i	$0.880351\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	31.0781	+	31.0781i	1.16389	+	1.16389i
$714$		0			0
$715$	4.96239	−	11.0132i	0.185583	−	0.411869i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	50.3369			1.87725			0.938625	−	0.344939i	$-0.112101\pi$
$719$	50.3369			1.87725			0.938625	+	0.344939i	$0.112101\pi$
$720$		0			0
$721$	−4.31265	−	4.31265i	−0.160611	−	0.160611i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.36665			−0.125034
$726$		0			0
$727$		−	52.7024i		−	1.95463i	−0.211802	−	0.977313i	$-0.567933\pi$
$727$		−	52.7024i		−	1.95463i	0.211802	−	0.977313i	$-0.432067\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			12.6747i			0.468792i
$732$		0			0
$733$	−34.5329	−	34.5329i	−1.27550	−	1.27550i	−0.943158	−	0.332345i	$-0.892160\pi$
$733$	−34.5329	−	34.5329i	−1.27550	−	1.27550i	−0.332345	−	0.943158i	$-0.607840\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1.46740i			0.0540525i
$738$		0			0
$739$	−12.3611	+	12.3611i	−0.454709	+	0.454709i	−0.896914	−	0.442205i	$-0.854196\pi$
$739$	−12.3611	+	12.3611i	−0.454709	+	0.454709i	0.442205	+	0.896914i	$0.354196\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12.3431	+	12.3431i	−0.452825	+	0.452825i	−0.896291	−	0.443466i	$-0.853749\pi$
$743$	−12.3431	+	12.3431i	−0.452825	+	0.452825i	0.443466	+	0.896291i	$0.353749\pi$
$744$		0			0
$745$			10.8872i			0.398875i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	27.0808	+	27.0808i	0.989512	+	0.989512i
$750$		0			0
$751$		−	26.3028i		−	0.959803i	−0.877322	−	0.479901i	$-0.840672\pi$
$751$		−	26.3028i		−	0.959803i	0.877322	−	0.479901i	$-0.159328\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	85.1195i		−	3.09781i
$756$		0			0
$757$	31.3620			1.13987			0.569936	−	0.821689i	$-0.306968\pi$
$757$	31.3620			1.13987			0.569936	+	0.821689i	$0.306968\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5.45025	−	5.45025i	−0.197571	−	0.197571i	0.601387	−	0.798958i	$-0.294615\pi$
$761$	−5.45025	−	5.45025i	−0.197571	−	0.197571i	−0.798958	+	0.601387i	$0.794615\pi$
$762$		0			0
$763$	−41.9003			−1.51689
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−47.4787	+	17.9827i	−1.71436	+	0.649317i
$768$		0			0
$769$	32.7499	+	32.7499i	1.18099	+	1.18099i	0.979488	+	0.201504i	$0.0645828\pi$
$769$	32.7499	+	32.7499i	1.18099	+	1.18099i	0.201504	+	0.979488i	$0.435417\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19.8992	−	19.8992i	0.715725	−	0.715725i	−0.252002	−	0.967727i	$-0.581089\pi$
$773$	19.8992	−	19.8992i	0.715725	−	0.715725i	0.967727	+	0.252002i	$0.0810889\pi$
$774$		0			0
$775$	−71.5814	+	71.5814i	−2.57128	+	2.57128i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	18.4380			0.660609
$780$		0			0
$781$	13.5271			0.484036
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.36102	−	1.36102i	0.0485770	−	0.0485770i
$786$		0			0
$787$	35.6263	−	35.6263i	1.26994	−	1.26994i	0.323820	−	0.946119i	$-0.395033\pi$
$787$	35.6263	−	35.6263i	1.26994	−	1.26994i	0.946119	−	0.323820i	$-0.104967\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4.27855	+	4.27855i	0.152128	+	0.152128i
$792$		0			0
$793$	−8.38787	+	18.6155i	−0.297862	+	0.661054i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11.9666			0.423880			0.211940	−	0.977283i	$-0.432022\pi$
$797$	11.9666			0.423880			0.211940	+	0.977283i	$0.432022\pi$
$798$		0			0
$799$	−5.89446	−	5.89446i	−0.208531	−	0.208531i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.663232			0.0234049
$804$		0			0
$805$			52.4749i			1.84950i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			42.0969i			1.48005i	0.672579	+	0.740025i	$0.265186\pi$
$809$			42.0969i			1.48005i	−0.672579	+	0.740025i	$0.734814\pi$
$810$		0			0
$811$	35.8432	+	35.8432i	1.25863	+	1.25863i	0.951749	+	0.306876i	$0.0992837\pi$
$811$	35.8432	+	35.8432i	1.25863	+	1.25863i	0.306876	+	0.951749i	$0.400716\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	61.1656i		−	2.14254i
$816$		0			0
$817$	25.7137	−	25.7137i	0.899608	−	0.899608i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	26.0617	−	26.0617i	0.909561	−	0.909561i	−0.0866760	−	0.996237i	$-0.527624\pi$
$821$	26.0617	−	26.0617i	0.909561	−	0.909561i	0.996237	+	0.0866760i	$0.0276245\pi$
$822$		0			0
$823$		−	27.2896i		−	0.951257i	−0.879646	−	0.475628i	$-0.842221\pi$
$823$		−	27.2896i		−	0.951257i	0.879646	−	0.475628i	$-0.157779\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15.9668	−	15.9668i	−0.555218	−	0.555218i	0.372724	−	0.927942i	$-0.378424\pi$
$827$	−15.9668	−	15.9668i	−0.555218	−	0.555218i	−0.927942	+	0.372724i	$0.878424\pi$
$828$		0			0
$829$			11.4255i			0.396823i	0.980119	+	0.198412i	$0.0635783\pi$
$829$			11.4255i			0.396823i	−0.980119	+	0.198412i	$0.936422\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	1.96275i		−	0.0680052i
$834$		0			0
$835$	43.3014			1.49851
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.83358	+	1.83358i	0.0633022	+	0.0633022i	0.738049	−	0.674747i	$-0.235747\pi$
$839$	1.83358	+	1.83358i	0.0633022	+	0.0633022i	−0.674747	+	0.738049i	$0.735747\pi$
$840$		0			0
$841$	28.9248			0.997406
$842$		0			0
$843$		0			0
$844$		0			0
$845$	53.9308	+	3.30930i	1.85528	+	0.113843i
$846$		0			0
$847$	17.3380	+	17.3380i	0.595742	+	0.595742i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−13.9211	+	13.9211i	−0.477208	+	0.477208i
$852$		0			0
$853$	−13.8872	+	13.8872i	−0.475488	+	0.475488i	−0.903685	−	0.428198i	$-0.859149\pi$
$853$	−13.8872	+	13.8872i	−0.475488	+	0.475488i	0.428198	+	0.903685i	$0.359149\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	30.3411			1.03643			0.518216	−	0.855250i	$-0.326596\pi$
$857$	30.3411			1.03643			0.518216	+	0.855250i	$0.326596\pi$
$858$		0			0
$859$	−26.3587			−0.899347			−0.449674	−	0.893193i	$-0.648460\pi$
$859$	−26.3587			−0.899347			−0.449674	+	0.893193i	$0.648460\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−32.8599	+	32.8599i	−1.11856	+	1.11856i	−0.126611	+	0.991952i	$0.540410\pi$
$863$	−32.8599	+	32.8599i	−1.11856	+	1.11856i	−0.991952	+	0.126611i	$0.959590\pi$
$864$		0			0
$865$	−21.1187	+	21.1187i	−0.718058	+	0.718058i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.01788	+	7.01788i	0.238065	+	0.238065i
$870$		0			0
$871$	−6.13823	+	2.32487i	−0.207986	+	0.0787752i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−71.6322			−2.42161
$876$		0			0
$877$	19.3127	+	19.3127i	0.652142	+	0.652142i	0.953508	−	0.301367i	$-0.0974428\pi$
$877$	19.3127	+	19.3127i	0.652142	+	0.652142i	−0.301367	+	0.953508i	$0.597443\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.1179			−0.475644			−0.237822	−	0.971309i	$-0.576433\pi$
$881$	−14.1179			−0.475644			−0.237822	+	0.971309i	$0.576433\pi$
$882$		0			0
$883$		−	24.3780i		−	0.820386i	−0.911999	−	0.410193i	$-0.865461\pi$
$883$		−	24.3780i		−	0.820386i	0.911999	−	0.410193i	$-0.134539\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			5.32108i			0.178664i	0.996002	+	0.0893322i	$0.0284733\pi$
$887$			5.32108i			0.178664i	−0.996002	+	0.0893322i	$0.971527\pi$
$888$		0			0
$889$	−13.7137	−	13.7137i	−0.459943	−	0.459943i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			23.9166i			0.800339i
$894$		0			0
$895$	−56.2638	+	56.2638i	−1.88069	+	1.88069i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.59938	+	1.59938i	−0.0533423	+	0.0533423i
$900$		0			0
$901$			4.70052i			0.156597i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	63.8158	+	63.8158i	2.12131	+	2.12131i
$906$		0			0
$907$			43.5125i			1.44481i	0.691471	+	0.722404i	$0.256963\pi$
$907$			43.5125i			1.44481i	−0.691471	+	0.722404i	$0.743037\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			12.9593i			0.429361i	0.976684	+	0.214681i	$0.0688711\pi$
$911$			12.9593i			0.429361i	−0.976684	+	0.214681i	$0.931129\pi$
$912$		0			0
$913$	−1.17347			−0.0388361
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.7652	−	10.7652i	−0.355497	−	0.355497i
$918$		0			0
$919$	−1.73340			−0.0571794			−0.0285897	−	0.999591i	$-0.509102\pi$
$919$	−1.73340			−0.0571794			−0.0285897	+	0.999591i	$0.509102\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	21.4315	+	56.5844i	0.705426	+	1.86250i
$924$		0			0
$925$	−32.0640	−	32.0640i	−1.05426	−	1.05426i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−21.1539	+	21.1539i	−0.694035	+	0.694035i	−0.963117	−	0.269082i	$-0.913280\pi$
$929$	−21.1539	+	21.1539i	−0.694035	+	0.694035i	0.269082	+	0.963117i	$0.413280\pi$
$930$		0			0
$931$	−3.98190	+	3.98190i	−0.130502	+	0.130502i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.73799			−0.154949
$936$		0			0
$937$	35.7090			1.16656			0.583280	−	0.812271i	$-0.301769\pi$
$937$	35.7090			1.16656			0.583280	+	0.812271i	$0.301769\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−26.3892	+	26.3892i	−0.860263	+	0.860263i	−0.991368	−	0.131106i	$-0.958147\pi$
$941$	−26.3892	+	26.3892i	−0.860263	+	0.860263i	0.131106	+	0.991368i	$0.458147\pi$
$942$		0			0
$943$	−17.1246	+	17.1246i	−0.557653	+	0.557653i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.94560	+	5.94560i	0.193206	+	0.193206i	0.797080	−	0.603874i	$-0.206377\pi$
$947$	5.94560	+	5.94560i	0.193206	+	0.193206i	−0.603874	+	0.797080i	$0.706377\pi$
$948$		0			0
$949$	1.05079	+	2.77433i	0.0341099	+	0.0900586i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−31.8152			−1.03060			−0.515298	−	0.857011i	$-0.672319\pi$
$953$	−31.8152			−1.03060			−0.515298	+	0.857011i	$0.672319\pi$
$954$		0			0
$955$	66.7123	+	66.7123i	2.15876	+	2.15876i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	42.5122			1.37279
$960$		0			0
$961$			37.0118i			1.19393i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	75.5288i		−	2.43136i
$966$		0			0
$967$	11.5042	+	11.5042i	0.369951	+	0.369951i	0.867459	−	0.497508i	$-0.165752\pi$
$967$	11.5042	+	11.5042i	0.369951	+	0.369951i	−0.497508	+	0.867459i	$0.665752\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			8.78649i			0.281972i	0.990012	+	0.140986i	$0.0450273\pi$
$971$			8.78649i			0.281972i	−0.990012	+	0.140986i	$0.954973\pi$
$972$		0			0
$973$	23.8496	−	23.8496i	0.764582	−	0.764582i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.26173	−	5.26173i	0.168338	−	0.168338i	−0.617911	−	0.786248i	$-0.712021\pi$
$977$	5.26173	−	5.26173i	0.168338	−	0.168338i	0.786248	+	0.617911i	$0.212021\pi$
$978$		0			0
$979$		−	9.42548i		−	0.301240i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−28.7223	−	28.7223i	−0.916098	−	0.916098i	0.0806448	−	0.996743i	$-0.474302\pi$
$983$	−28.7223	−	28.7223i	−0.916098	−	0.916098i	−0.996743	+	0.0806448i	$0.974302\pi$
$984$		0			0
$985$			71.9267i			2.29177i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			47.7641i			1.51881i
$990$		0			0
$991$	−10.8218			−0.343766			−0.171883	−	0.985117i	$-0.554985\pi$
$991$	−10.8218			−0.343766			−0.171883	+	0.985117i	$0.554985\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−21.1600	−	21.1600i	−0.670818	−	0.670818i
$996$		0			0
$997$	−20.4288			−0.646987			−0.323493	−	0.946230i	$-0.604857\pi$
$997$	−20.4288			−0.646987			−0.323493	+	0.946230i	$0.604857\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	936.2.ba.b.593.1	yes	12
3.2	odd	2	inner	936.2.ba.b.593.6	yes	12
4.3	odd	2		1872.2.bi.e.593.1		12
12.11	even	2		1872.2.bi.e.593.6		12
13.5	odd	4	inner	936.2.ba.b.161.6	yes	12
39.5	even	4	inner	936.2.ba.b.161.1	✓	12
52.31	even	4		1872.2.bi.e.161.6		12
156.83	odd	4		1872.2.bi.e.161.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.ba.b.161.1	✓	12	39.5	even	4	inner
936.2.ba.b.161.6	yes	12	13.5	odd	4	inner
936.2.ba.b.593.1	yes	12	1.1	even	1	trivial
936.2.ba.b.593.6	yes	12	3.2	odd	2	inner
1872.2.bi.e.161.1		12	156.83	odd	4
1872.2.bi.e.161.6		12	52.31	even	4
1872.2.bi.e.593.1		12	4.3	odd	2
1872.2.bi.e.593.6		12	12.11	even	2

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:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	936.ba (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.93897 + 2.93897i) q^{5} +(-1.67513 + 1.67513i) q^{7} +(-0.569973 - 0.569973i) q^{11} +(1.48119 - 3.28726i) q^{13} -1.41421 q^{17} +(2.86907 + 2.86907i) q^{19} -5.32940 q^{23} -12.2750i q^{25} -0.274268i q^{29} +(-5.83146 - 5.83146i) q^{31} -9.84630i q^{35} +(2.61213 - 2.61213i) q^{37} +(3.21323 - 3.21323i) q^{41} -8.96239i q^{43} +(4.16801 + 4.16801i) q^{47} +1.38787i q^{49} -3.32377i q^{53} +3.35026 q^{55} +(-9.95684 - 9.95684i) q^{59} -5.66291 q^{61} +(5.30796 + 14.0143i) q^{65} +(-1.28726 - 1.28726i) q^{67} +(-11.8664 + 11.8664i) q^{71} +(-0.581810 + 0.581810i) q^{73} +1.90956 q^{77} -12.3127 q^{79} +(1.02941 - 1.02941i) q^{83} +(4.15633 - 4.15633i) q^{85} +(8.26836 + 8.26836i) q^{89} +(3.02539 + 7.98778i) q^{91} -16.8642 q^{95} +(12.1187 + 12.1187i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{13} + 16 q^{19} - 8 q^{31} + 28 q^{37} + 56 q^{61} + 8 q^{67} - 12 q^{73} - 64 q^{79} + 8 q^{85} - 24 q^{91} + 60 q^{97}+O(q^{100}) \) 12 * q - 4 * q^13 + 16 * q^19 - 8 * q^31 + 28 * q^37 + 56 * q^61 + 8 * q^67 - 12 * q^73 - 64 * q^79 + 8 * q^85 - 24 * q^91 + 60 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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