LMFDB - Embedded newform 3762.2.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3762,2,Mod(1,3762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3762, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3762.a (trivial)

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:	yes
Analytic conductor:	$30.0397212404$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.523976$ of defining polynomial
Character	$\chi$	$=$	3762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} -2.72545 q^{5} -4.67750 q^{7} +1.00000 q^{8} -2.72545 q^{10} +1.00000 q^{11} -2.67750 q^{13} -4.67750 q^{14} +1.00000 q^{16} -0.201472 q^{17} -1.00000 q^{19} -2.72545 q^{20} +1.00000 q^{22} +1.79853 q^{23} +2.42807 q^{25} -2.67750 q^{26} -4.67750 q^{28} +4.92692 q^{29} -2.57193 q^{31} +1.00000 q^{32} -0.201472 q^{34} +12.7483 q^{35} +4.40294 q^{37} -1.00000 q^{38} -2.72545 q^{40} -4.97487 q^{41} -11.7734 q^{43} +1.00000 q^{44} +1.79853 q^{46} +12.4989 q^{47} +14.8790 q^{49} +2.42807 q^{50} -2.67750 q^{52} +4.10557 q^{53} -2.72545 q^{55} -4.67750 q^{56} +4.92692 q^{58} +5.24943 q^{59} +1.59706 q^{61} -2.57193 q^{62} +1.00000 q^{64} +7.29738 q^{65} +9.08044 q^{67} -0.201472 q^{68} +12.7483 q^{70} -12.8214 q^{71} +2.20147 q^{73} +4.40294 q^{74} -1.00000 q^{76} -4.67750 q^{77} +11.8538 q^{79} -2.72545 q^{80} -4.97487 q^{82} +13.6775 q^{83} +0.549103 q^{85} -11.7734 q^{86} +1.00000 q^{88} +5.45090 q^{89} +12.5240 q^{91} +1.79853 q^{92} +12.4989 q^{94} +2.72545 q^{95} +16.8059 q^{97} +14.8790 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{10} + 3 q^{11} - 6 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 15 q^{23} + 12 q^{25} - 6 q^{28} - 6 q^{29} - 3 q^{31} + 3 q^{32}+ \cdots + 27 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^7 + 3 * q^8 + 3 * q^10 + 3 * q^11 - 6 * q^14 + 3 * q^16 + 9 * q^17 - 3 * q^19 + 3 * q^20 + 3 * q^22 + 15 * q^23 + 12 * q^25 - 6 * q^28 - 6 * q^29 - 3 * q^31 + 3 * q^32 + 9 * q^34 - 6 * q^37 - 3 * q^38 + 3 * q^40 + 9 * q^41 - 21 * q^43 + 3 * q^44 + 15 * q^46 + 12 * q^47 + 27 * q^49 + 12 * q^50 + 9 * q^53 + 3 * q^55 - 6 * q^56 - 6 * q^58 + 3 * q^59 + 24 * q^61 - 3 * q^62 + 3 * q^64 + 6 * q^65 + 9 * q^68 - 21 * q^71 - 3 * q^73 - 6 * q^74 - 3 * q^76 - 6 * q^77 - 6 * q^79 + 3 * q^80 + 9 * q^82 + 33 * q^83 + 24 * q^85 - 21 * q^86 + 3 * q^88 - 6 * q^89 + 36 * q^91 + 15 * q^92 + 12 * q^94 - 3 * q^95 + 12 * q^97 + 27 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.72545	−1.21886	−0.609429	−	0.792841i	$-0.708601\pi$
$5$	−2.72545	−1.21886	−0.609429	+	0.792841i	$0.708601\pi$
$6$	0	0
$7$	−4.67750	−1.76793	−0.883964	−	0.467556i	$-0.845135\pi$
$7$	−4.67750	−1.76793	−0.883964	+	0.467556i	$0.845135\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.72545	−0.861863
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.67750	−0.742604	−0.371302	−	0.928512i	$-0.621089\pi$
$13$	−2.67750	−0.742604	−0.371302	+	0.928512i	$0.621089\pi$
$14$	−4.67750	−1.25011
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.201472	−0.0488642	−0.0244321	−	0.999701i	$-0.507778\pi$
$17$	−0.201472	−0.0488642	−0.0244321	+	0.999701i	$0.507778\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.72545	−0.609429
$21$	0	0
$22$	1.00000	0.213201
$23$	1.79853	0.375019	0.187509	−	0.982263i	$-0.439958\pi$
$23$	1.79853	0.375019	0.187509	+	0.982263i	$0.439958\pi$
$24$	0	0
$25$	2.42807	0.485614
$26$	−2.67750	−0.525100
$27$	0	0
$28$	−4.67750	−0.883964
$29$	4.92692	0.914906	0.457453	−	0.889234i	$-0.348762\pi$
$29$	4.92692	0.914906	0.457453	+	0.889234i	$0.348762\pi$
$30$	0	0
$31$	−2.57193	−0.461932	−0.230966	−	0.972962i	$-0.574189\pi$
$31$	−2.57193	−0.461932	−0.230966	+	0.972962i	$0.574189\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.201472	−0.0345522
$35$	12.7483	2.15485
$36$	0	0
$37$	4.40294	0.723840	0.361920	−	0.932209i	$-0.382121\pi$
$37$	4.40294	0.723840	0.361920	+	0.932209i	$0.382121\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−2.72545	−0.430931
$41$	−4.97487	−0.776945	−0.388472	−	0.921460i	$-0.626997\pi$
$41$	−4.97487	−0.776945	−0.388472	+	0.921460i	$0.626997\pi$
$42$	0	0
$43$	−11.7734	−1.79543	−0.897713	−	0.440580i	$-0.854773\pi$
$43$	−11.7734	−1.79543	−0.897713	+	0.440580i	$0.854773\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	1.79853	0.265178
$47$	12.4989	1.82314	0.911572	−	0.411140i	$-0.134869\pi$
$47$	12.4989	1.82314	0.911572	+	0.411140i	$0.134869\pi$
$48$	0	0
$49$	14.8790	2.12557
$50$	2.42807	0.343381
$51$	0	0
$52$	−2.67750	−0.371302
$53$	4.10557	0.563943	0.281971	−	0.959423i	$-0.409012\pi$
$53$	4.10557	0.563943	0.281971	+	0.959423i	$0.409012\pi$
$54$	0	0
$55$	−2.72545	−0.367499
$56$	−4.67750	−0.625057
$57$	0	0
$58$	4.92692	0.646936
$59$	5.24943	0.683417	0.341708	−	0.939806i	$-0.388994\pi$
$59$	5.24943	0.683417	0.341708	+	0.939806i	$0.388994\pi$
$60$	0	0
$61$	1.59706	0.204482	0.102241	−	0.994760i	$-0.467399\pi$
$61$	1.59706	0.204482	0.102241	+	0.994760i	$0.467399\pi$
$62$	−2.57193	−0.326635
$63$	0	0
$64$	1.00000	0.125000
$65$	7.29738	0.905128
$66$	0	0
$67$	9.08044	1.10935	0.554676	−	0.832066i	$-0.312842\pi$
$67$	9.08044	1.10935	0.554676	+	0.832066i	$0.312842\pi$
$68$	−0.201472	−0.0244321
$69$	0	0
$70$	12.7483	1.52371
$71$	−12.8214	−1.52161	−0.760807	−	0.648978i	$-0.775197\pi$
$71$	−12.8214	−1.52161	−0.760807	+	0.648978i	$0.775197\pi$
$72$	0	0
$73$	2.20147	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$73$	2.20147	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$74$	4.40294	0.511832
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−4.67750	−0.533050
$78$	0	0
$79$	11.8538	1.33366	0.666831	−	0.745209i	$-0.267650\pi$
$79$	11.8538	1.33366	0.666831	+	0.745209i	$0.267650\pi$
$80$	−2.72545	−0.304714
$81$	0	0
$82$	−4.97487	−0.549383
$83$	13.6775	1.50130	0.750650	−	0.660700i	$-0.229740\pi$
$83$	13.6775	1.50130	0.750650	+	0.660700i	$0.229740\pi$
$84$	0	0
$85$	0.549103	0.0595585
$86$	−11.7734	−1.26956
$87$	0	0
$88$	1.00000	0.106600
$89$	5.45090	0.577794	0.288897	−	0.957360i	$-0.406711\pi$
$89$	5.45090	0.577794	0.288897	+	0.957360i	$0.406711\pi$
$90$	0	0
$91$	12.5240	1.31287
$92$	1.79853	0.187509
$93$	0	0
$94$	12.4989	1.28916
$95$	2.72545	0.279625
$96$	0	0
$97$	16.8059	1.70638	0.853190	−	0.521601i	$-0.174665\pi$
$97$	16.8059	1.70638	0.853190	+	0.521601i	$0.174665\pi$
$98$	14.8790	1.50300
$99$	0	0
$100$	2.42807	0.242807
$101$	−7.45090	−0.741392	−0.370696	−	0.928754i	$-0.620881\pi$
$101$	−7.45090	−0.741392	−0.370696	+	0.928754i	$0.620881\pi$
$102$	0	0
$103$	−14.4834	−1.42709	−0.713545	−	0.700609i	$-0.752912\pi$
$103$	−14.4834	−1.42709	−0.713545	+	0.700609i	$0.752912\pi$
$104$	−2.67750	−0.262550
$105$	0	0
$106$	4.10557	0.398768
$107$	−1.15352	−0.111515	−0.0557575	−	0.998444i	$-0.517757\pi$
$107$	−1.15352	−0.111515	−0.0557575	+	0.998444i	$0.517757\pi$
$108$	0	0
$109$	−9.55646	−0.915343	−0.457672	−	0.889121i	$-0.651316\pi$
$109$	−9.55646	−0.915343	−0.457672	+	0.889121i	$0.651316\pi$
$110$	−2.72545	−0.259861
$111$	0	0
$112$	−4.67750	−0.441982
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	−4.90179	−0.457095
$116$	4.92692	0.457453
$117$	0	0
$118$	5.24943	0.483249
$119$	0.942386	0.0863884
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.59706	0.144591
$123$	0	0
$124$	−2.57193	−0.230966
$125$	7.00966	0.626963
$126$	0	0
$127$	5.75794	0.510934	0.255467	−	0.966818i	$-0.417771\pi$
$127$	5.75794	0.510934	0.255467	+	0.966818i	$0.417771\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	7.29738	0.640022
$131$	−1.42807	−0.124771	−0.0623856	−	0.998052i	$-0.519871\pi$
$131$	−1.42807	−0.124771	−0.0623856	+	0.998052i	$0.519871\pi$
$132$	0	0
$133$	4.67750	0.405590
$134$	9.08044	0.784431
$135$	0	0
$136$	−0.201472	−0.0172761
$137$	14.5313	1.24150	0.620748	−	0.784010i	$-0.286829\pi$
$137$	14.5313	1.24150	0.620748	+	0.784010i	$0.286829\pi$
$138$	0	0
$139$	16.1763	1.37206	0.686030	−	0.727573i	$-0.259352\pi$
$139$	16.1763	1.37206	0.686030	+	0.727573i	$0.259352\pi$
$140$	12.7483	1.07743
$141$	0	0
$142$	−12.8214	−1.07594
$143$	−2.67750	−0.223903
$144$	0	0
$145$	−13.4281	−1.11514
$146$	2.20147	0.182195
$147$	0	0
$148$	4.40294	0.361920
$149$	−6.30704	−0.516693	−0.258346	−	0.966052i	$-0.583178\pi$
$149$	−6.30704	−0.516693	−0.258346	+	0.966052i	$0.583178\pi$
$150$	0	0
$151$	0.498850	0.0405959	0.0202979	−	0.999794i	$-0.493539\pi$
$151$	0.498850	0.0405959	0.0202979	+	0.999794i	$0.493539\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−4.67750	−0.376923
$155$	7.00966	0.563030
$156$	0	0
$157$	−13.6272	−1.08757	−0.543786	−	0.839224i	$-0.683010\pi$
$157$	−13.6272	−1.08757	−0.543786	+	0.839224i	$0.683010\pi$
$158$	11.8538	0.943041
$159$	0	0
$160$	−2.72545	−0.215466
$161$	−8.41261	−0.663006
$162$	0	0
$163$	15.9497	1.24928	0.624640	−	0.780913i	$-0.285246\pi$
$163$	15.9497	1.24928	0.624640	+	0.780913i	$0.285246\pi$
$164$	−4.97487	−0.388472
$165$	0	0
$166$	13.6775	1.06158
$167$	−18.8059	−1.45524	−0.727622	−	0.685979i	$-0.759374\pi$
$167$	−18.8059	−1.45524	−0.727622	+	0.685979i	$0.759374\pi$
$168$	0	0
$169$	−5.83102	−0.448540
$170$	0.549103	0.0421142
$171$	0	0
$172$	−11.7734	−0.897713
$173$	3.22430	0.245139	0.122569	−	0.992460i	$-0.460887\pi$
$173$	3.22430	0.245139	0.122569	+	0.992460i	$0.460887\pi$
$174$	0	0
$175$	−11.3573	−0.858531
$176$	1.00000	0.0753778
$177$	0	0
$178$	5.45090	0.408562
$179$	−13.3778	−0.999905	−0.499953	−	0.866053i	$-0.666649\pi$
$179$	−13.3778	−0.999905	−0.499953	+	0.866053i	$0.666649\pi$
$180$	0	0
$181$	15.9497	1.18554	0.592768	−	0.805373i	$-0.298035\pi$
$181$	15.9497	1.18554	0.592768	+	0.805373i	$0.298035\pi$
$182$	12.5240	0.928339
$183$	0	0
$184$	1.79853	0.132589
$185$	−12.0000	−0.882258
$186$	0	0
$187$	−0.201472	−0.0147331
$188$	12.4989	0.911572
$189$	0	0
$190$	2.72545	0.197725
$191$	24.0553	1.74058	0.870291	−	0.492538i	$-0.163931\pi$
$191$	24.0553	1.74058	0.870291	+	0.492538i	$0.163931\pi$
$192$	0	0
$193$	13.0325	0.938099	0.469050	−	0.883172i	$-0.344597\pi$
$193$	13.0325	0.938099	0.469050	+	0.883172i	$0.344597\pi$
$194$	16.8059	1.20659
$195$	0	0
$196$	14.8790	1.06278
$197$	5.59706	0.398774	0.199387	−	0.979921i	$-0.436105\pi$
$197$	5.59706	0.398774	0.199387	+	0.979921i	$0.436105\pi$
$198$	0	0
$199$	−7.39558	−0.524259	−0.262129	−	0.965033i	$-0.584425\pi$
$199$	−7.39558	−0.524259	−0.262129	+	0.965033i	$0.584425\pi$
$200$	2.42807	0.171691
$201$	0	0
$202$	−7.45090	−0.524243
$203$	−23.0457	−1.61749
$204$	0	0
$205$	13.5588	0.946985
$206$	−14.4834	−1.00911
$207$	0	0
$208$	−2.67750	−0.185651
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−14.2974	−0.984272	−0.492136	−	0.870518i	$-0.663784\pi$
$211$	−14.2974	−0.984272	−0.492136	+	0.870518i	$0.663784\pi$
$212$	4.10557	0.281971
$213$	0	0
$214$	−1.15352	−0.0788530
$215$	32.0878	2.18837
$216$	0	0
$217$	12.0302	0.816662
$218$	−9.55646	−0.647245
$219$	0	0
$220$	−2.72545	−0.183750
$221$	0.539441	0.0362868
$222$	0	0
$223$	−0.402945	−0.0269832	−0.0134916	−	0.999909i	$-0.504295\pi$
$223$	−0.402945	−0.0269832	−0.0134916	+	0.999909i	$0.504295\pi$
$224$	−4.67750	−0.312528
$225$	0	0
$226$	8.00000	0.532152
$227$	21.0074	1.39431	0.697154	−	0.716922i	$-0.254449\pi$
$227$	21.0074	1.39431	0.697154	+	0.716922i	$0.254449\pi$
$228$	0	0
$229$	−14.3299	−0.946944	−0.473472	−	0.880809i	$-0.657000\pi$
$229$	−14.3299	−0.946944	−0.473472	+	0.880809i	$0.657000\pi$
$230$	−4.90179	−0.323215
$231$	0	0
$232$	4.92692	0.323468
$233$	24.4029	1.59869	0.799345	−	0.600872i	$-0.205180\pi$
$233$	24.4029	1.59869	0.799345	+	0.600872i	$0.205180\pi$
$234$	0	0
$235$	−34.0650	−2.22215
$236$	5.24943	0.341708
$237$	0	0
$238$	0.942386	0.0610858
$239$	12.6849	0.820515	0.410258	−	0.911970i	$-0.365439\pi$
$239$	12.6849	0.820515	0.410258	+	0.911970i	$0.365439\pi$
$240$	0	0
$241$	−23.9343	−1.54174	−0.770871	−	0.636991i	$-0.780179\pi$
$241$	−23.9343	−1.54174	−0.770871	+	0.636991i	$0.780179\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	1.59706	0.102241
$245$	−40.5519	−2.59076
$246$	0	0
$247$	2.67750	0.170365
$248$	−2.57193	−0.163318
$249$	0	0
$250$	7.00966	0.443330
$251$	−18.4989	−1.16764	−0.583819	−	0.811884i	$-0.698442\pi$
$251$	−18.4989	−1.16764	−0.583819	+	0.811884i	$0.698442\pi$
$252$	0	0
$253$	1.79853	0.113072
$254$	5.75794	0.361285
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.7579	−1.73149	−0.865746	−	0.500483i	$-0.833156\pi$
$257$	−27.7579	−1.73149	−0.865746	+	0.500483i	$0.833156\pi$
$258$	0	0
$259$	−20.5948	−1.27970
$260$	7.29738	0.452564
$261$	0	0
$262$	−1.42807	−0.0882265
$263$	−20.2723	−1.25004	−0.625020	−	0.780608i	$-0.714909\pi$
$263$	−20.2723	−1.25004	−0.625020	+	0.780608i	$0.714909\pi$
$264$	0	0
$265$	−11.1895	−0.687366
$266$	4.67750	0.286796
$267$	0	0
$268$	9.08044	0.554676
$269$	15.6427	0.953753	0.476876	−	0.878970i	$-0.341769\pi$
$269$	15.6427	0.953753	0.476876	+	0.878970i	$0.341769\pi$
$270$	0	0
$271$	−12.3395	−0.749573	−0.374786	−	0.927111i	$-0.622284\pi$
$271$	−12.3395	−0.749573	−0.374786	+	0.927111i	$0.622284\pi$
$272$	−0.201472	−0.0122161
$273$	0	0
$274$	14.5313	0.877870
$275$	2.42807	0.146418
$276$	0	0
$277$	8.49885	0.510646	0.255323	−	0.966856i	$-0.417818\pi$
$277$	8.49885	0.510646	0.255323	+	0.966856i	$0.417818\pi$
$278$	16.1763	0.970193
$279$	0	0
$280$	12.7483	0.761855
$281$	7.12839	0.425244	0.212622	−	0.977134i	$-0.431800\pi$
$281$	7.12839	0.425244	0.212622	+	0.977134i	$0.431800\pi$
$282$	0	0
$283$	11.5240	0.685029	0.342515	−	0.939512i	$-0.388721\pi$
$283$	11.5240	0.685029	0.342515	+	0.939512i	$0.388721\pi$
$284$	−12.8214	−0.760807
$285$	0	0
$286$	−2.67750	−0.158324
$287$	23.2700	1.37358
$288$	0	0
$289$	−16.9594	−0.997612
$290$	−13.4281	−0.788523
$291$	0	0
$292$	2.20147	0.128831
$293$	−10.6775	−0.623786	−0.311893	−	0.950117i	$-0.600963\pi$
$293$	−10.6775	−0.623786	−0.311893	+	0.950117i	$0.600963\pi$
$294$	0	0
$295$	−14.3070	−0.832988
$296$	4.40294	0.255916
$297$	0	0
$298$	−6.30704	−0.365357
$299$	−4.81555	−0.278490
$300$	0	0
$301$	55.0700	3.17418
$302$	0.498850	0.0287056
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	−4.35269	−0.249234
$306$	0	0
$307$	−30.2568	−1.72685	−0.863423	−	0.504481i	$-0.831684\pi$
$307$	−30.2568	−1.72685	−0.863423	+	0.504481i	$0.831684\pi$
$308$	−4.67750	−0.266525
$309$	0	0
$310$	7.00966	0.398122
$311$	−8.79623	−0.498788	−0.249394	−	0.968402i	$-0.580231\pi$
$311$	−8.79623	−0.498788	−0.249394	+	0.968402i	$0.580231\pi$
$312$	0	0
$313$	10.1860	0.575747	0.287874	−	0.957668i	$-0.407052\pi$
$313$	10.1860	0.575747	0.287874	+	0.957668i	$0.407052\pi$
$314$	−13.6272	−0.769030
$315$	0	0
$316$	11.8538	0.666831
$317$	12.0097	0.674530	0.337265	−	0.941410i	$-0.390498\pi$
$317$	12.0097	0.674530	0.337265	+	0.941410i	$0.390498\pi$
$318$	0	0
$319$	4.92692	0.275855
$320$	−2.72545	−0.152357
$321$	0	0
$322$	−8.41261	−0.468816
$323$	0.201472	0.0112102
$324$	0	0
$325$	−6.50115	−0.360619
$326$	15.9497	0.883375
$327$	0	0
$328$	−4.97487	−0.274691
$329$	−58.4633	−3.22319
$330$	0	0
$331$	26.1667	1.43825	0.719126	−	0.694880i	$-0.244543\pi$
$331$	26.1667	1.43825	0.719126	+	0.694880i	$0.244543\pi$
$332$	13.6775	0.750650
$333$	0	0
$334$	−18.8059	−1.02901
$335$	−24.7483	−1.35214
$336$	0	0
$337$	17.3778	0.946630	0.473315	−	0.880893i	$-0.343057\pi$
$337$	17.3778	0.946630	0.473315	+	0.880893i	$0.343057\pi$
$338$	−5.83102	−0.317165
$339$	0	0
$340$	0.549103	0.0297793
$341$	−2.57193	−0.139278
$342$	0	0
$343$	−36.8538	−1.98992
$344$	−11.7734	−0.634779
$345$	0	0
$346$	3.22430	0.173339
$347$	12.4029	0.665825	0.332912	−	0.942958i	$-0.391969\pi$
$347$	12.4029	0.665825	0.332912	+	0.942958i	$0.391969\pi$
$348$	0	0
$349$	−23.8036	−1.27418	−0.637088	−	0.770791i	$-0.719861\pi$
$349$	−23.8036	−1.27418	−0.637088	+	0.770791i	$0.719861\pi$
$350$	−11.3573	−0.607073
$351$	0	0
$352$	1.00000	0.0533002
$353$	24.2664	1.29157	0.645786	−	0.763518i	$-0.276530\pi$
$353$	24.2664	1.29157	0.645786	+	0.763518i	$0.276530\pi$
$354$	0	0
$355$	34.9439	1.85463
$356$	5.45090	0.288897
$357$	0	0
$358$	−13.3778	−0.707040
$359$	−20.3395	−1.07348	−0.536740	−	0.843748i	$-0.680344\pi$
$359$	−20.3395	−1.07348	−0.536740	+	0.843748i	$0.680344\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	15.9497	0.838300
$363$	0	0
$364$	12.5240	0.656435
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−23.0170	−1.20148	−0.600739	−	0.799445i	$-0.705127\pi$
$367$	−23.0170	−1.20148	−0.600739	+	0.799445i	$0.705127\pi$
$368$	1.79853	0.0937547
$369$	0	0
$370$	−12.0000	−0.623850
$371$	−19.2038	−0.997010
$372$	0	0
$373$	−18.5890	−0.962499	−0.481250	−	0.876584i	$-0.659817\pi$
$373$	−18.5890	−0.962499	−0.481250	+	0.876584i	$0.659817\pi$
$374$	−0.201472	−0.0104179
$375$	0	0
$376$	12.4989	0.644579
$377$	−13.1918	−0.679413
$378$	0	0
$379$	−10.5816	−0.543540	−0.271770	−	0.962362i	$-0.587609\pi$
$379$	−10.5816	−0.543540	−0.271770	+	0.962362i	$0.587609\pi$
$380$	2.72545	0.139813
$381$	0	0
$382$	24.0553	1.23078
$383$	26.0878	1.33302	0.666512	−	0.745494i	$-0.267786\pi$
$383$	26.0878	1.33302	0.666512	+	0.745494i	$0.267786\pi$
$384$	0	0
$385$	12.7483	0.649712
$386$	13.0325	0.663336
$387$	0	0
$388$	16.8059	0.853190
$389$	9.46636	0.479964	0.239982	−	0.970777i	$-0.422859\pi$
$389$	9.46636	0.479964	0.239982	+	0.970777i	$0.422859\pi$
$390$	0	0
$391$	−0.362354	−0.0183250
$392$	14.8790	0.751501
$393$	0	0
$394$	5.59706	0.281976
$395$	−32.3070	−1.62554
$396$	0	0
$397$	−25.1667	−1.26308	−0.631540	−	0.775343i	$-0.717577\pi$
$397$	−25.1667	−1.26308	−0.631540	+	0.775343i	$0.717577\pi$
$398$	−7.39558	−0.370707
$399$	0	0
$400$	2.42807	0.121404
$401$	16.0650	0.802247	0.401123	−	0.916024i	$-0.368620\pi$
$401$	16.0650	0.802247	0.401123	+	0.916024i	$0.368620\pi$
$402$	0	0
$403$	6.88633	0.343033
$404$	−7.45090	−0.370696
$405$	0	0
$406$	−23.0457	−1.14374
$407$	4.40294	0.218246
$408$	0	0
$409$	−19.5313	−0.965763	−0.482881	−	0.875686i	$-0.660410\pi$
$409$	−19.5313	−0.965763	−0.482881	+	0.875686i	$0.660410\pi$
$410$	13.5588	0.669620
$411$	0	0
$412$	−14.4834	−0.713545
$413$	−24.5542	−1.20823
$414$	0	0
$415$	−37.2773	−1.82987
$416$	−2.67750	−0.131275
$417$	0	0
$418$	−1.00000	−0.0489116
$419$	−24.9018	−1.21653	−0.608266	−	0.793733i	$-0.708135\pi$
$419$	−24.9018	−1.21653	−0.608266	+	0.793733i	$0.708135\pi$
$420$	0	0
$421$	26.6044	1.29662	0.648310	−	0.761377i	$-0.275476\pi$
$421$	26.6044	1.29662	0.648310	+	0.761377i	$0.275476\pi$
$422$	−14.2974	−0.695985
$423$	0	0
$424$	4.10557	0.199384
$425$	−0.489189	−0.0237292
$426$	0	0
$427$	−7.47022	−0.361509
$428$	−1.15352	−0.0557575
$429$	0	0
$430$	32.0878	1.54741
$431$	32.5948	1.57003	0.785017	−	0.619474i	$-0.212654\pi$
$431$	32.5948	1.57003	0.785017	+	0.619474i	$0.212654\pi$
$432$	0	0
$433$	19.7386	0.948577	0.474289	−	0.880369i	$-0.342705\pi$
$433$	19.7386	0.948577	0.474289	+	0.880369i	$0.342705\pi$
$434$	12.0302	0.577468
$435$	0	0
$436$	−9.55646	−0.457672
$437$	−1.79853	−0.0860352
$438$	0	0
$439$	−24.4029	−1.16469	−0.582345	−	0.812942i	$-0.697865\pi$
$439$	−24.4029	−1.16469	−0.582345	+	0.812942i	$0.697865\pi$
$440$	−2.72545	−0.129931
$441$	0	0
$442$	0.539441	0.0256586
$443$	16.5948	0.788441	0.394220	−	0.919016i	$-0.371015\pi$
$443$	16.5948	0.788441	0.394220	+	0.919016i	$0.371015\pi$
$444$	0	0
$445$	−14.8561	−0.704249
$446$	−0.402945	−0.0190800
$447$	0	0
$448$	−4.67750	−0.220991
$449$	20.9520	0.988788	0.494394	−	0.869238i	$-0.335390\pi$
$449$	20.9520	0.988788	0.494394	+	0.869238i	$0.335390\pi$
$450$	0	0
$451$	−4.97487	−0.234258
$452$	8.00000	0.376288
$453$	0	0
$454$	21.0074	0.985924
$455$	−34.1335	−1.60020
$456$	0	0
$457$	23.3144	1.09060	0.545301	−	0.838240i	$-0.316415\pi$
$457$	23.3144	1.09060	0.545301	+	0.838240i	$0.316415\pi$
$458$	−14.3299	−0.669591
$459$	0	0
$460$	−4.90179	−0.228547
$461$	−20.8059	−0.969027	−0.484513	−	0.874784i	$-0.661003\pi$
$461$	−20.8059	−0.969027	−0.484513	+	0.874784i	$0.661003\pi$
$462$	0	0
$463$	−9.29002	−0.431744	−0.215872	−	0.976422i	$-0.569259\pi$
$463$	−9.29002	−0.431744	−0.215872	+	0.976422i	$0.569259\pi$
$464$	4.92692	0.228727
$465$	0	0
$466$	24.4029	1.13044
$467$	−12.0457	−0.557406	−0.278703	−	0.960377i	$-0.589905\pi$
$467$	−12.0457	−0.557406	−0.278703	+	0.960377i	$0.589905\pi$
$468$	0	0
$469$	−42.4737	−1.96125
$470$	−34.0650	−1.57130
$471$	0	0
$472$	5.24943	0.241624
$473$	−11.7734	−0.541342
$474$	0	0
$475$	−2.42807	−0.111408
$476$	0.942386	0.0431942
$477$	0	0
$478$	12.6849	0.580192
$479$	−0.188307	−0.00860396	−0.00430198	−	0.999991i	$-0.501369\pi$
$479$	−0.188307	−0.00860396	−0.00430198	+	0.999991i	$0.501369\pi$
$480$	0	0
$481$	−11.7889	−0.537526
$482$	−23.9343	−1.09018
$483$	0	0
$484$	1.00000	0.0454545
$485$	−45.8036	−2.07983
$486$	0	0
$487$	−33.3852	−1.51283	−0.756413	−	0.654094i	$-0.773050\pi$
$487$	−33.3852	−1.51283	−0.756413	+	0.654094i	$0.773050\pi$
$488$	1.59706	0.0722953
$489$	0	0
$490$	−40.5519	−1.83195
$491$	−12.5217	−0.565095	−0.282548	−	0.959253i	$-0.591180\pi$
$491$	−12.5217	−0.565095	−0.282548	+	0.959253i	$0.591180\pi$
$492$	0	0
$493$	−0.992638	−0.0447062
$494$	2.67750	0.120466
$495$	0	0
$496$	−2.57193	−0.115483
$497$	59.9718	2.69010
$498$	0	0
$499$	16.3070	0.730003	0.365002	−	0.931007i	$-0.381068\pi$
$499$	16.3070	0.730003	0.365002	+	0.931007i	$0.381068\pi$
$500$	7.00966	0.313482
$501$	0	0
$502$	−18.4989	−0.825644
$503$	18.9726	0.845945	0.422973	−	0.906142i	$-0.360987\pi$
$503$	18.9726	0.845945	0.422973	+	0.906142i	$0.360987\pi$
$504$	0	0
$505$	20.3070	0.903651
$506$	1.79853	0.0799543
$507$	0	0
$508$	5.75794	0.255467
$509$	30.6907	1.36034	0.680170	−	0.733055i	$-0.261906\pi$
$509$	30.6907	1.36034	0.680170	+	0.733055i	$0.261906\pi$
$510$	0	0
$511$	−10.2974	−0.455529
$512$	1.00000	0.0441942
$513$	0	0
$514$	−27.7579	−1.22435
$515$	39.4737	1.73942
$516$	0	0
$517$	12.4989	0.549699
$518$	−20.5948	−0.904882
$519$	0	0
$520$	7.29738	0.320011
$521$	42.1106	1.84490	0.922450	−	0.386116i	$-0.126184\pi$
$521$	42.1106	1.84490	0.922450	+	0.386116i	$0.126184\pi$
$522$	0	0
$523$	37.8133	1.65346	0.826729	−	0.562600i	$-0.190199\pi$
$523$	37.8133	1.65346	0.826729	+	0.562600i	$0.190199\pi$
$524$	−1.42807	−0.0623856
$525$	0	0
$526$	−20.2723	−0.883912
$527$	0.518173	0.0225720
$528$	0	0
$529$	−19.7653	−0.859361
$530$	−11.1895	−0.486041
$531$	0	0
$532$	4.67750	0.202795
$533$	13.3202	0.576962
$534$	0	0
$535$	3.14386	0.135921
$536$	9.08044	0.392215
$537$	0	0
$538$	15.6427	0.674405
$539$	14.8790	0.640883
$540$	0	0
$541$	13.5661	0.583253	0.291627	−	0.956532i	$-0.405804\pi$
$541$	13.5661	0.583253	0.291627	+	0.956532i	$0.405804\pi$
$542$	−12.3395	−0.530028
$543$	0	0
$544$	−0.201472	−0.00863806
$545$	26.0457	1.11567
$546$	0	0
$547$	46.4177	1.98468	0.992338	−	0.123552i	$-0.0394286\pi$
$547$	46.4177	1.98468	0.992338	+	0.123552i	$0.0394286\pi$
$548$	14.5313	0.620748
$549$	0	0
$550$	2.42807	0.103533
$551$	−4.92692	−0.209894
$552$	0	0
$553$	−55.4463	−2.35782
$554$	8.49885	0.361082
$555$	0	0
$556$	16.1763	0.686030
$557$	38.2065	1.61886	0.809431	−	0.587214i	$-0.199775\pi$
$557$	38.2065	1.61886	0.809431	+	0.587214i	$0.199775\pi$
$558$	0	0
$559$	31.5232	1.33329
$560$	12.7483	0.538713
$561$	0	0
$562$	7.12839	0.300693
$563$	7.01702	0.295732	0.147866	−	0.989007i	$-0.452760\pi$
$563$	7.01702	0.295732	0.147866	+	0.989007i	$0.452760\pi$
$564$	0	0
$565$	−21.8036	−0.917284
$566$	11.5240	0.484389
$567$	0	0
$568$	−12.8214	−0.537972
$569$	41.4235	1.73656	0.868281	−	0.496072i	$-0.165225\pi$
$569$	41.4235	1.73656	0.868281	+	0.496072i	$0.165225\pi$
$570$	0	0
$571$	12.1381	0.507962	0.253981	−	0.967209i	$-0.418260\pi$
$571$	12.1381	0.507962	0.253981	+	0.967209i	$0.418260\pi$
$572$	−2.67750	−0.111952
$573$	0	0
$574$	23.2700	0.971269
$575$	4.36695	0.182115
$576$	0	0
$577$	12.8384	0.534469	0.267234	−	0.963632i	$-0.413890\pi$
$577$	12.8384	0.534469	0.267234	+	0.963632i	$0.413890\pi$
$578$	−16.9594	−0.705418
$579$	0	0
$580$	−13.4281	−0.557570
$581$	−63.9764	−2.65419
$582$	0	0
$583$	4.10557	0.170035
$584$	2.20147	0.0910976
$585$	0	0
$586$	−10.6775	−0.441083
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	2.57193	0.105974
$590$	−14.3070	−0.589011
$591$	0	0
$592$	4.40294	0.180960
$593$	−40.4177	−1.65975	−0.829877	−	0.557946i	$-0.811590\pi$
$593$	−40.4177	−1.65975	−0.829877	+	0.557946i	$0.811590\pi$
$594$	0	0
$595$	−2.56842	−0.105295
$596$	−6.30704	−0.258346
$597$	0	0
$598$	−4.81555	−0.196923
$599$	−0.679795	−0.0277757	−0.0138878	−	0.999904i	$-0.504421\pi$
$599$	−0.679795	−0.0277757	−0.0138878	+	0.999904i	$0.504421\pi$
$600$	0	0
$601$	21.5240	0.877981	0.438991	−	0.898492i	$-0.355336\pi$
$601$	21.5240	0.877981	0.438991	+	0.898492i	$0.355336\pi$
$602$	55.0700	2.24449
$603$	0	0
$604$	0.498850	0.0202979
$605$	−2.72545	−0.110805
$606$	0	0
$607$	13.3550	0.542062	0.271031	−	0.962571i	$-0.412635\pi$
$607$	13.3550	0.542062	0.271031	+	0.962571i	$0.412635\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	−4.35269	−0.176235
$611$	−33.4656	−1.35387
$612$	0	0
$613$	9.11293	0.368068	0.184034	−	0.982920i	$-0.441084\pi$
$613$	9.11293	0.368068	0.184034	+	0.982920i	$0.441084\pi$
$614$	−30.2568	−1.22106
$615$	0	0
$616$	−4.67750	−0.188462
$617$	−0.329866	−0.0132799	−0.00663995	−	0.999978i	$-0.502114\pi$
$617$	−0.329866	−0.0132799	−0.00663995	+	0.999978i	$0.502114\pi$
$618$	0	0
$619$	8.92112	0.358570	0.179285	−	0.983797i	$-0.442622\pi$
$619$	8.92112	0.358570	0.179285	+	0.983797i	$0.442622\pi$
$620$	7.00966	0.281515
$621$	0	0
$622$	−8.79623	−0.352697
$623$	−25.4966	−1.02150
$624$	0	0
$625$	−31.2448	−1.24979
$626$	10.1860	0.407115
$627$	0	0
$628$	−13.6272	−0.543786
$629$	−0.887072	−0.0353699
$630$	0	0
$631$	44.6597	1.77788	0.888938	−	0.458028i	$-0.151444\pi$
$631$	44.6597	1.77788	0.888938	+	0.458028i	$0.151444\pi$
$632$	11.8538	0.471521
$633$	0	0
$634$	12.0097	0.476965
$635$	−15.6930	−0.622756
$636$	0	0
$637$	−39.8384	−1.57845
$638$	4.92692	0.195059
$639$	0	0
$640$	−2.72545	−0.107733
$641$	6.93272	0.273826	0.136913	−	0.990583i	$-0.456282\pi$
$641$	6.93272	0.273826	0.136913	+	0.990583i	$0.456282\pi$
$642$	0	0
$643$	23.4966	0.926614	0.463307	−	0.886198i	$-0.346663\pi$
$643$	23.4966	0.926614	0.463307	+	0.886198i	$0.346663\pi$
$644$	−8.41261	−0.331503
$645$	0	0
$646$	0.201472	0.00792682
$647$	−20.9880	−0.825125	−0.412562	−	0.910929i	$-0.635366\pi$
$647$	−20.9880	−0.825125	−0.412562	+	0.910929i	$0.635366\pi$
$648$	0	0
$649$	5.24943	0.206058
$650$	−6.50115	−0.254996
$651$	0	0
$652$	15.9497	0.624640
$653$	44.1335	1.72708	0.863538	−	0.504284i	$-0.168244\pi$
$653$	44.1335	1.72708	0.863538	+	0.504284i	$0.168244\pi$
$654$	0	0
$655$	3.89213	0.152078
$656$	−4.97487	−0.194236
$657$	0	0
$658$	−58.4633	−2.27914
$659$	−24.0862	−0.938267	−0.469133	−	0.883127i	$-0.655434\pi$
$659$	−24.0862	−0.938267	−0.469133	+	0.883127i	$0.655434\pi$
$660$	0	0
$661$	−9.65237	−0.375434	−0.187717	−	0.982223i	$-0.560109\pi$
$661$	−9.65237	−0.375434	−0.187717	+	0.982223i	$0.560109\pi$
$662$	26.1667	1.01700
$663$	0	0
$664$	13.6775	0.530790
$665$	−12.7483	−0.494357
$666$	0	0
$667$	8.86120	0.343107
$668$	−18.8059	−0.727622
$669$	0	0
$670$	−24.7483	−0.956109
$671$	1.59706	0.0616536
$672$	0	0
$673$	24.6176	0.948938	0.474469	−	0.880272i	$-0.342640\pi$
$673$	24.6176	0.948938	0.474469	+	0.880272i	$0.342640\pi$
$674$	17.3778	0.669369
$675$	0	0
$676$	−5.83102	−0.224270
$677$	−21.5719	−0.829077	−0.414538	−	0.910032i	$-0.636057\pi$
$677$	−21.5719	−0.829077	−0.414538	+	0.910032i	$0.636057\pi$
$678$	0	0
$679$	−78.6095	−3.01675
$680$	0.549103	0.0210571
$681$	0	0
$682$	−2.57193	−0.0984843
$683$	0.997701	0.0381759	0.0190880	−	0.999818i	$-0.493924\pi$
$683$	0.997701	0.0381759	0.0190880	+	0.999818i	$0.493924\pi$
$684$	0	0
$685$	−39.6044	−1.51321
$686$	−36.8538	−1.40709
$687$	0	0
$688$	−11.7734	−0.448857
$689$	−10.9926	−0.418786
$690$	0	0
$691$	−9.00230	−0.342464	−0.171232	−	0.985231i	$-0.554775\pi$
$691$	−9.00230	−0.342464	−0.171232	+	0.985231i	$0.554775\pi$
$692$	3.22430	0.122569
$693$	0	0
$694$	12.4029	0.470809
$695$	−44.0878	−1.67235
$696$	0	0
$697$	1.00230	0.0379648
$698$	−23.8036	−0.900979
$699$	0	0
$700$	−11.3573	−0.429265
$701$	15.0936	0.570078	0.285039	−	0.958516i	$-0.407994\pi$
$701$	15.0936	0.570078	0.285039	+	0.958516i	$0.407994\pi$
$702$	0	0
$703$	−4.40294	−0.166060
$704$	1.00000	0.0376889
$705$	0	0
$706$	24.2664	0.913280
$707$	34.8515	1.31073
$708$	0	0
$709$	−22.9246	−0.860952	−0.430476	−	0.902602i	$-0.641654\pi$
$709$	−22.9246	−0.860952	−0.430476	+	0.902602i	$0.641654\pi$
$710$	34.9439	1.31142
$711$	0	0
$712$	5.45090	0.204281
$713$	−4.62569	−0.173233
$714$	0	0
$715$	7.29738	0.272906
$716$	−13.3778	−0.499953
$717$	0	0
$718$	−20.3395	−0.759064
$719$	−6.70032	−0.249880	−0.124940	−	0.992164i	$-0.539874\pi$
$719$	−6.70032	−0.249880	−0.124940	+	0.992164i	$0.539874\pi$
$720$	0	0
$721$	67.7460	2.52299
$722$	1.00000	0.0372161
$723$	0	0
$724$	15.9497	0.592768
$725$	11.9629	0.444291
$726$	0	0
$727$	−18.4583	−0.684579	−0.342289	−	0.939595i	$-0.611202\pi$
$727$	−18.4583	−0.684579	−0.342289	+	0.939595i	$0.611202\pi$
$728$	12.5240	0.464169
$729$	0	0
$730$	−6.00000	−0.222070
$731$	2.37201	0.0877321
$732$	0	0
$733$	41.8995	1.54759	0.773797	−	0.633434i	$-0.218355\pi$
$733$	41.8995	1.54759	0.773797	+	0.633434i	$0.218355\pi$
$734$	−23.0170	−0.849574
$735$	0	0
$736$	1.79853	0.0662946
$737$	9.08044	0.334482
$738$	0	0
$739$	−34.8361	−1.28147	−0.640733	−	0.767764i	$-0.721369\pi$
$739$	−34.8361	−1.28147	−0.640733	+	0.767764i	$0.721369\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	−19.2038	−0.704993
$743$	25.7889	0.946102	0.473051	−	0.881035i	$-0.343153\pi$
$743$	25.7889	0.946102	0.473051	+	0.881035i	$0.343153\pi$
$744$	0	0
$745$	17.1895	0.629775
$746$	−18.5890	−0.680590
$747$	0	0
$748$	−0.201472	−0.00736656
$749$	5.39558	0.197150
$750$	0	0
$751$	−33.5011	−1.22247	−0.611237	−	0.791447i	$-0.709328\pi$
$751$	−33.5011	−1.22247	−0.611237	+	0.791447i	$0.709328\pi$
$752$	12.4989	0.455786
$753$	0	0
$754$	−13.1918	−0.480417
$755$	−1.35959	−0.0494806
$756$	0	0
$757$	7.53134	0.273731	0.136866	−	0.990590i	$-0.456297\pi$
$757$	7.53134	0.273731	0.136866	+	0.990590i	$0.456297\pi$
$758$	−10.5816	−0.384341
$759$	0	0
$760$	2.72545	0.0988624
$761$	−38.1969	−1.38464	−0.692318	−	0.721593i	$-0.743410\pi$
$761$	−38.1969	−1.38464	−0.692318	+	0.721593i	$0.743410\pi$
$762$	0	0
$763$	44.7003	1.61826
$764$	24.0553	0.870291
$765$	0	0
$766$	26.0878	0.942591
$767$	−14.0553	−0.507508
$768$	0	0
$769$	16.6501	0.600417	0.300208	−	0.953874i	$-0.402944\pi$
$769$	16.6501	0.600417	0.300208	+	0.953874i	$0.402944\pi$
$770$	12.7483	0.459416
$771$	0	0
$772$	13.0325	0.469050
$773$	23.6330	0.850022	0.425011	−	0.905188i	$-0.360270\pi$
$773$	23.6330	0.850022	0.425011	+	0.905188i	$0.360270\pi$
$774$	0	0
$775$	−6.24483	−0.224321
$776$	16.8059	0.603296
$777$	0	0
$778$	9.46636	0.339386
$779$	4.97487	0.178243
$780$	0	0
$781$	−12.8214	−0.458784
$782$	−0.362354	−0.0129577
$783$	0	0
$784$	14.8790	0.531392
$785$	37.1404	1.32560
$786$	0	0
$787$	−20.3933	−0.726942	−0.363471	−	0.931606i	$-0.618408\pi$
$787$	−20.3933	−0.726942	−0.363471	+	0.931606i	$0.618408\pi$
$788$	5.59706	0.199387
$789$	0	0
$790$	−32.3070	−1.14943
$791$	−37.4200	−1.33050
$792$	0	0
$793$	−4.27611	−0.151849
$794$	−25.1667	−0.893132
$795$	0	0
$796$	−7.39558	−0.262129
$797$	9.70262	0.343685	0.171842	−	0.985124i	$-0.445028\pi$
$797$	9.70262	0.343685	0.171842	+	0.985124i	$0.445028\pi$
$798$	0	0
$799$	−2.51817	−0.0890865
$800$	2.42807	0.0858453
$801$	0	0
$802$	16.0650	0.567274
$803$	2.20147	0.0776883
$804$	0	0
$805$	22.9281	0.808110
$806$	6.88633	0.242561
$807$	0	0
$808$	−7.45090	−0.262122
$809$	27.0723	0.951813	0.475906	−	0.879496i	$-0.342120\pi$
$809$	27.0723	0.951813	0.475906	+	0.879496i	$0.342120\pi$
$810$	0	0
$811$	51.9548	1.82438	0.912190	−	0.409767i	$-0.134390\pi$
$811$	51.9548	1.82438	0.912190	+	0.409767i	$0.134390\pi$
$812$	−23.0457	−0.808744
$813$	0	0
$814$	4.40294	0.154323
$815$	−43.4702	−1.52270
$816$	0	0
$817$	11.7734	0.411899
$818$	−19.5313	−0.682897
$819$	0	0
$820$	13.5588	0.473493
$821$	25.6574	0.895451	0.447725	−	0.894171i	$-0.352234\pi$
$821$	25.6574	0.895451	0.447725	+	0.894171i	$0.352234\pi$
$822$	0	0
$823$	−41.8635	−1.45927	−0.729635	−	0.683837i	$-0.760310\pi$
$823$	−41.8635	−1.45927	−0.729635	+	0.683837i	$0.760310\pi$
$824$	−14.4834	−0.504553
$825$	0	0
$826$	−24.5542	−0.854349
$827$	−6.15582	−0.214059	−0.107029	−	0.994256i	$-0.534134\pi$
$827$	−6.15582	−0.214059	−0.107029	+	0.994256i	$0.534134\pi$
$828$	0	0
$829$	−0.247126	−0.00858303	−0.00429151	−	0.999991i	$-0.501366\pi$
$829$	−0.247126	−0.00858303	−0.00429151	+	0.999991i	$0.501366\pi$
$830$	−37.2773	−1.29391
$831$	0	0
$832$	−2.67750	−0.0928255
$833$	−2.99770	−0.103864
$834$	0	0
$835$	51.2545	1.77373
$836$	−1.00000	−0.0345857
$837$	0	0
$838$	−24.9018	−0.860218
$839$	−16.9320	−0.584557	−0.292278	−	0.956333i	$-0.594413\pi$
$839$	−16.9320	−0.584557	−0.292278	+	0.956333i	$0.594413\pi$
$840$	0	0
$841$	−4.72545	−0.162947
$842$	26.6044	0.916849
$843$	0	0
$844$	−14.2974	−0.492136
$845$	15.8921	0.546706
$846$	0	0
$847$	−4.67750	−0.160721
$848$	4.10557	0.140986
$849$	0	0
$850$	−0.489189	−0.0167790
$851$	7.91882	0.271454
$852$	0	0
$853$	21.6930	0.742753	0.371376	−	0.928482i	$-0.378886\pi$
$853$	21.6930	0.742753	0.371376	+	0.928482i	$0.378886\pi$
$854$	−7.47022	−0.255626
$855$	0	0
$856$	−1.15352	−0.0394265
$857$	−34.7328	−1.18645	−0.593225	−	0.805037i	$-0.702146\pi$
$857$	−34.7328	−1.18645	−0.593225	+	0.805037i	$0.702146\pi$
$858$	0	0
$859$	−15.3047	−0.522191	−0.261095	−	0.965313i	$-0.584084\pi$
$859$	−15.3047	−0.522191	−0.261095	+	0.965313i	$0.584084\pi$
$860$	32.0878	1.09418
$861$	0	0
$862$	32.5948	1.11018
$863$	−35.8457	−1.22020	−0.610102	−	0.792323i	$-0.708871\pi$
$863$	−35.8457	−1.22020	−0.610102	+	0.792323i	$0.708871\pi$
$864$	0	0
$865$	−8.78766	−0.298789
$866$	19.7386	0.670745
$867$	0	0
$868$	12.0302	0.408331
$869$	11.8538	0.402114
$870$	0	0
$871$	−24.3128	−0.823809
$872$	−9.55646	−0.323623
$873$	0	0
$874$	−1.79853	−0.0608361
$875$	−32.7877	−1.10843
$876$	0	0
$877$	−45.4907	−1.53611	−0.768057	−	0.640382i	$-0.778776\pi$
$877$	−45.4907	−1.53611	−0.768057	+	0.640382i	$0.778776\pi$
$878$	−24.4029	−0.823559
$879$	0	0
$880$	−2.72545	−0.0918749
$881$	−31.3322	−1.05561	−0.527804	−	0.849366i	$-0.676984\pi$
$881$	−31.3322	−1.05561	−0.527804	+	0.849366i	$0.676984\pi$
$882$	0	0
$883$	35.3550	1.18979	0.594895	−	0.803803i	$-0.297194\pi$
$883$	35.3550	1.18979	0.594895	+	0.803803i	$0.297194\pi$
$884$	0.539441	0.0181434
$885$	0	0
$886$	16.5948	0.557512
$887$	16.6141	0.557846	0.278923	−	0.960313i	$-0.410023\pi$
$887$	16.6141	0.557846	0.278923	+	0.960313i	$0.410023\pi$
$888$	0	0
$889$	−26.9327	−0.903295
$890$	−14.8561	−0.497979
$891$	0	0
$892$	−0.402945	−0.0134916
$893$	−12.4989	−0.418258
$894$	0	0
$895$	36.4606	1.21874
$896$	−4.67750	−0.156264
$897$	0	0
$898$	20.9520	0.699179
$899$	−12.6717	−0.422625
$900$	0	0
$901$	−0.827158	−0.0275566
$902$	−4.97487	−0.165645
$903$	0	0
$904$	8.00000	0.266076
$905$	−43.4702	−1.44500
$906$	0	0
$907$	54.8612	1.82164	0.910818	−	0.412808i	$-0.135452\pi$
$907$	54.8612	1.82164	0.910818	+	0.412808i	$0.135452\pi$
$908$	21.0074	0.697154
$909$	0	0
$910$	−34.1335	−1.13151
$911$	−2.98298	−0.0988304	−0.0494152	−	0.998778i	$-0.515736\pi$
$911$	−2.98298	−0.0988304	−0.0494152	+	0.998778i	$0.515736\pi$
$912$	0	0
$913$	13.6775	0.452659
$914$	23.3144	0.771172
$915$	0	0
$916$	−14.3299	−0.473472
$917$	6.67980	0.220586
$918$	0	0
$919$	−26.6701	−0.879767	−0.439883	−	0.898055i	$-0.644980\pi$
$919$	−26.6701	−0.879767	−0.439883	+	0.898055i	$0.644980\pi$
$920$	−4.90179	−0.161607
$921$	0	0
$922$	−20.8059	−0.685205
$923$	34.3291	1.12996
$924$	0	0
$925$	10.6907	0.351507
$926$	−9.29002	−0.305289
$927$	0	0
$928$	4.92692	0.161734
$929$	−43.7595	−1.43570	−0.717851	−	0.696197i	$-0.754874\pi$
$929$	−43.7595	−1.43570	−0.717851	+	0.696197i	$0.754874\pi$
$930$	0	0
$931$	−14.8790	−0.487638
$932$	24.4029	0.799345
$933$	0	0
$934$	−12.0457	−0.394146
$935$	0.549103	0.0179576
$936$	0	0
$937$	47.7823	1.56098	0.780490	−	0.625168i	$-0.214970\pi$
$937$	47.7823	1.56098	0.780490	+	0.625168i	$0.214970\pi$
$938$	−42.4737	−1.38682
$939$	0	0
$940$	−34.0650	−1.11108
$941$	−36.2664	−1.18225	−0.591126	−	0.806579i	$-0.701316\pi$
$941$	−36.2664	−1.18225	−0.591126	+	0.806579i	$0.701316\pi$
$942$	0	0
$943$	−8.94745	−0.291369
$944$	5.24943	0.170854
$945$	0	0
$946$	−11.7734	−0.382786
$947$	−35.5468	−1.15512	−0.577558	−	0.816350i	$-0.695994\pi$
$947$	−35.5468	−1.15512	−0.577558	+	0.816350i	$0.695994\pi$
$948$	0	0
$949$	−5.89443	−0.191341
$950$	−2.42807	−0.0787770
$951$	0	0
$952$	0.942386	0.0305429
$953$	−23.2088	−0.751808	−0.375904	−	0.926659i	$-0.622668\pi$
$953$	−23.2088	−0.751808	−0.375904	+	0.926659i	$0.622668\pi$
$954$	0	0
$955$	−65.5615	−2.12152
$956$	12.6849	0.410258
$957$	0	0
$958$	−0.188307	−0.00608392
$959$	−67.9703	−2.19487
$960$	0	0
$961$	−24.3852	−0.786619
$962$	−11.7889	−0.380088
$963$	0	0
$964$	−23.9343	−0.770871
$965$	−35.5194	−1.14341
$966$	0	0
$967$	−6.09591	−0.196031	−0.0980156	−	0.995185i	$-0.531249\pi$
$967$	−6.09591	−0.196031	−0.0980156	+	0.995185i	$0.531249\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−45.8036	−1.47066
$971$	−34.9822	−1.12263	−0.561317	−	0.827601i	$-0.689705\pi$
$971$	−34.9822	−1.12263	−0.561317	+	0.827601i	$0.689705\pi$
$972$	0	0
$973$	−75.6648	−2.42570
$974$	−33.3852	−1.06973
$975$	0	0
$976$	1.59706	0.0511205
$977$	10.9018	0.348779	0.174390	−	0.984677i	$-0.444205\pi$
$977$	10.9018	0.348779	0.174390	+	0.984677i	$0.444205\pi$
$978$	0	0
$979$	5.45090	0.174211
$980$	−40.5519	−1.29538
$981$	0	0
$982$	−12.5217	−0.399583
$983$	25.4281	0.811030	0.405515	−	0.914089i	$-0.367092\pi$
$983$	25.4281	0.811030	0.405515	+	0.914089i	$0.367092\pi$
$984$	0	0
$985$	−15.2545	−0.486048
$986$	−0.992638	−0.0316120
$987$	0	0
$988$	2.67750	0.0851825
$989$	−21.1748	−0.673319
$990$	0	0
$991$	49.3925	1.56901	0.784503	−	0.620125i	$-0.212918\pi$
$991$	49.3925	1.56901	0.784503	+	0.620125i	$0.212918\pi$
$992$	−2.57193	−0.0816588
$993$	0	0
$994$	59.9718	1.90219
$995$	20.1563	0.638997
$996$	0	0
$997$	20.7409	0.656871	0.328436	−	0.944526i	$-0.393479\pi$
$997$	20.7409	0.656871	0.328436	+	0.944526i	$0.393479\pi$
$998$	16.3070	0.516190
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3762.2.a.bg.1.1		3
3.2	odd	2		418.2.a.g.1.2	✓	3
12.11	even	2		3344.2.a.q.1.2		3
33.32	even	2		4598.2.a.bo.1.2		3
57.56	even	2		7942.2.a.bi.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.g.1.2	✓	3	3.2	odd	2
3344.2.a.q.1.2		3	12.11	even	2
3762.2.a.bg.1.1		3	1.1	even	1	trivial
4598.2.a.bo.1.2		3	33.32	even	2
7942.2.a.bi.1.2		3	57.56	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:	\( N \)	\(=\)	\( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3762.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.72545 q^{5} -4.67750 q^{7} +1.00000 q^{8} -2.72545 q^{10} +1.00000 q^{11} -2.67750 q^{13} -4.67750 q^{14} +1.00000 q^{16} -0.201472 q^{17} -1.00000 q^{19} -2.72545 q^{20} +1.00000 q^{22} +1.79853 q^{23} +2.42807 q^{25} -2.67750 q^{26} -4.67750 q^{28} +4.92692 q^{29} -2.57193 q^{31} +1.00000 q^{32} -0.201472 q^{34} +12.7483 q^{35} +4.40294 q^{37} -1.00000 q^{38} -2.72545 q^{40} -4.97487 q^{41} -11.7734 q^{43} +1.00000 q^{44} +1.79853 q^{46} +12.4989 q^{47} +14.8790 q^{49} +2.42807 q^{50} -2.67750 q^{52} +4.10557 q^{53} -2.72545 q^{55} -4.67750 q^{56} +4.92692 q^{58} +5.24943 q^{59} +1.59706 q^{61} -2.57193 q^{62} +1.00000 q^{64} +7.29738 q^{65} +9.08044 q^{67} -0.201472 q^{68} +12.7483 q^{70} -12.8214 q^{71} +2.20147 q^{73} +4.40294 q^{74} -1.00000 q^{76} -4.67750 q^{77} +11.8538 q^{79} -2.72545 q^{80} -4.97487 q^{82} +13.6775 q^{83} +0.549103 q^{85} -11.7734 q^{86} +1.00000 q^{88} +5.45090 q^{89} +12.5240 q^{91} +1.79853 q^{92} +12.4989 q^{94} +2.72545 q^{95} +16.8059 q^{97} +14.8790 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{10} + 3 q^{11} - 6 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 15 q^{23} + 12 q^{25} - 6 q^{28} - 6 q^{29} - 3 q^{31} + 3 q^{32}+ \cdots + 27 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^7 + 3 * q^8 + 3 * q^10 + 3 * q^11 - 6 * q^14 + 3 * q^16 + 9 * q^17 - 3 * q^19 + 3 * q^20 + 3 * q^22 + 15 * q^23 + 12 * q^25 - 6 * q^28 - 6 * q^29 - 3 * q^31 + 3 * q^32 + 9 * q^34 - 6 * q^37 - 3 * q^38 + 3 * q^40 + 9 * q^41 - 21 * q^43 + 3 * q^44 + 15 * q^46 + 12 * q^47 + 27 * q^49 + 12 * q^50 + 9 * q^53 + 3 * q^55 - 6 * q^56 - 6 * q^58 + 3 * q^59 + 24 * q^61 - 3 * q^62 + 3 * q^64 + 6 * q^65 + 9 * q^68 - 21 * q^71 - 3 * q^73 - 6 * q^74 - 3 * q^76 - 6 * q^77 - 6 * q^79 + 3 * q^80 + 9 * q^82 + 33 * q^83 + 24 * q^85 - 21 * q^86 + 3 * q^88 - 6 * q^89 + 36 * q^91 + 15 * q^92 + 12 * q^94 - 3 * q^95 + 12 * q^97 + 27 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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