LMFDB - Embedded newform 3762.2.a.y.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
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			$-1.56155$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -0.561553 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -0.561553 q^{7} +1.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} +0.561553 q^{13} -0.561553 q^{14} +1.00000 q^{16} +0.561553 q^{17} -1.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} -1.43845 q^{23} -1.00000 q^{25} +0.561553 q^{26} -0.561553 q^{28} +5.68466 q^{29} +2.00000 q^{31} +1.00000 q^{32} +0.561553 q^{34} +1.12311 q^{35} -5.12311 q^{37} -1.00000 q^{38} -2.00000 q^{40} -2.00000 q^{41} -1.00000 q^{44} -1.43845 q^{46} -8.00000 q^{47} -6.68466 q^{49} -1.00000 q^{50} +0.561553 q^{52} -12.8078 q^{53} +2.00000 q^{55} -0.561553 q^{56} +5.68466 q^{58} +7.68466 q^{59} +6.24621 q^{61} +2.00000 q^{62} +1.00000 q^{64} -1.12311 q^{65} -7.68466 q^{67} +0.561553 q^{68} +1.12311 q^{70} -6.00000 q^{71} -9.68466 q^{73} -5.12311 q^{74} -1.00000 q^{76} +0.561553 q^{77} -4.00000 q^{79} -2.00000 q^{80} -2.00000 q^{82} -14.2462 q^{83} -1.12311 q^{85} -1.00000 q^{88} +0.876894 q^{89} -0.315342 q^{91} -1.43845 q^{92} -8.00000 q^{94} +2.00000 q^{95} -7.12311 q^{97} -6.68466 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 3 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8} - 4 q^{10} - 2 q^{11} - 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} - 2 q^{19} - 4 q^{20} - 2 q^{22} - 7 q^{23} - 2 q^{25} - 3 q^{26} + 3 q^{28} - q^{29} + 4 q^{31} + 2 q^{32} - 3 q^{34} - 6 q^{35} - 2 q^{37} - 2 q^{38} - 4 q^{40} - 4 q^{41} - 2 q^{44} - 7 q^{46} - 16 q^{47} - q^{49} - 2 q^{50} - 3 q^{52} - 5 q^{53} + 4 q^{55} + 3 q^{56} - q^{58} + 3 q^{59} - 4 q^{61} + 4 q^{62} + 2 q^{64} + 6 q^{65} - 3 q^{67} - 3 q^{68} - 6 q^{70} - 12 q^{71} - 7 q^{73} - 2 q^{74} - 2 q^{76} - 3 q^{77} - 8 q^{79} - 4 q^{80} - 4 q^{82} - 12 q^{83} + 6 q^{85} - 2 q^{88} + 10 q^{89} - 13 q^{91} - 7 q^{92} - 16 q^{94} + 4 q^{95} - 6 q^{97} - q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 3 * q^7 + 2 * q^8 - 4 * q^10 - 2 * q^11 - 3 * q^13 + 3 * q^14 + 2 * q^16 - 3 * q^17 - 2 * q^19 - 4 * q^20 - 2 * q^22 - 7 * q^23 - 2 * q^25 - 3 * q^26 + 3 * q^28 - q^29 + 4 * q^31 + 2 * q^32 - 3 * q^34 - 6 * q^35 - 2 * q^37 - 2 * q^38 - 4 * q^40 - 4 * q^41 - 2 * q^44 - 7 * q^46 - 16 * q^47 - q^49 - 2 * q^50 - 3 * q^52 - 5 * q^53 + 4 * q^55 + 3 * q^56 - q^58 + 3 * q^59 - 4 * q^61 + 4 * q^62 + 2 * q^64 + 6 * q^65 - 3 * q^67 - 3 * q^68 - 6 * q^70 - 12 * q^71 - 7 * q^73 - 2 * q^74 - 2 * q^76 - 3 * q^77 - 8 * q^79 - 4 * q^80 - 4 * q^82 - 12 * q^83 + 6 * q^85 - 2 * q^88 + 10 * q^89 - 13 * q^91 - 7 * q^92 - 16 * q^94 + 4 * q^95 - 6 * q^97 - q^98

Display 100 coefficients 10 coefficients

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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−0.561553	−0.212247	−0.106124	−	0.994353i	$-0.533844\pi$
$7$	−0.561553	−0.212247	−0.106124	+	0.994353i	$0.533844\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.00000	−0.632456
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.561553	0.155747	0.0778734	−	0.996963i	$-0.475187\pi$
$13$	0.561553	0.155747	0.0778734	+	0.996963i	$0.475187\pi$
$14$	−0.561553	−0.150081
$15$	0	0
$16$	1.00000	0.250000
$17$	0.561553	0.136197	0.0680983	−	0.997679i	$-0.478307\pi$
$17$	0.561553	0.136197	0.0680983	+	0.997679i	$0.478307\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−1.43845	−0.299937	−0.149968	−	0.988691i	$-0.547917\pi$
$23$	−1.43845	−0.299937	−0.149968	+	0.988691i	$0.547917\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0.561553	0.110130
$27$	0	0
$28$	−0.561553	−0.106124
$29$	5.68466	1.05561	0.527807	−	0.849364i	$-0.323014\pi$
$29$	5.68466	1.05561	0.527807	+	0.849364i	$0.323014\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.561553	0.0963055
$35$	1.12311	0.189839
$36$	0	0
$37$	−5.12311	−0.842233	−0.421117	−	0.907006i	$-0.638362\pi$
$37$	−5.12311	−0.842233	−0.421117	+	0.907006i	$0.638362\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−2.00000	−0.316228
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−1.43845	−0.212087
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−6.68466	−0.954951
$50$	−1.00000	−0.141421
$51$	0	0
$52$	0.561553	0.0778734
$53$	−12.8078	−1.75928	−0.879641	−	0.475638i	$-0.842217\pi$
$53$	−12.8078	−1.75928	−0.879641	+	0.475638i	$0.842217\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	−0.561553	−0.0750407
$57$	0	0
$58$	5.68466	0.746432
$59$	7.68466	1.00046	0.500229	−	0.865893i	$-0.333249\pi$
$59$	7.68466	1.00046	0.500229	+	0.865893i	$0.333249\pi$
$60$	0	0
$61$	6.24621	0.799745	0.399873	−	0.916571i	$-0.369054\pi$
$61$	6.24621	0.799745	0.399873	+	0.916571i	$0.369054\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.12311	−0.139304
$66$	0	0
$67$	−7.68466	−0.938830	−0.469415	−	0.882978i	$-0.655535\pi$
$67$	−7.68466	−0.938830	−0.469415	+	0.882978i	$0.655535\pi$
$68$	0.561553	0.0680983
$69$	0	0
$70$	1.12311	0.134237
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−9.68466	−1.13350	−0.566752	−	0.823889i	$-0.691800\pi$
$73$	−9.68466	−1.13350	−0.566752	+	0.823889i	$0.691800\pi$
$74$	−5.12311	−0.595549
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0.561553	0.0639949
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	−2.00000	−0.220863
$83$	−14.2462	−1.56372	−0.781862	−	0.623451i	$-0.785730\pi$
$83$	−14.2462	−1.56372	−0.781862	+	0.623451i	$0.785730\pi$
$84$	0	0
$85$	−1.12311	−0.121818
$86$	0	0
$87$	0	0
$88$	−1.00000	−0.106600
$89$	0.876894	0.0929506	0.0464753	−	0.998919i	$-0.485201\pi$
$89$	0.876894	0.0929506	0.0464753	+	0.998919i	$0.485201\pi$
$90$	0	0
$91$	−0.315342	−0.0330568
$92$	−1.43845	−0.149968
$93$	0	0
$94$	−8.00000	−0.825137
$95$	2.00000	0.205196
$96$	0	0
$97$	−7.12311	−0.723242	−0.361621	−	0.932325i	$-0.617777\pi$
$97$	−7.12311	−0.723242	−0.361621	+	0.932325i	$0.617777\pi$
$98$	−6.68466	−0.675252
$99$	0	0
$100$	−1.00000	−0.100000
$101$	6.87689	0.684277	0.342138	−	0.939650i	$-0.388849\pi$
$101$	6.87689	0.684277	0.342138	+	0.939650i	$0.388849\pi$
$102$	0	0
$103$	−13.3693	−1.31732	−0.658659	−	0.752442i	$-0.728876\pi$
$103$	−13.3693	−1.31732	−0.658659	+	0.752442i	$0.728876\pi$
$104$	0.561553	0.0550648
$105$	0	0
$106$	−12.8078	−1.24400
$107$	9.93087	0.960053	0.480027	−	0.877254i	$-0.340627\pi$
$107$	9.93087	0.960053	0.480027	+	0.877254i	$0.340627\pi$
$108$	0	0
$109$	−6.31534	−0.604900	−0.302450	−	0.953165i	$-0.597805\pi$
$109$	−6.31534	−0.604900	−0.302450	+	0.953165i	$0.597805\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	−0.561553	−0.0530618
$113$	3.12311	0.293797	0.146899	−	0.989152i	$-0.453071\pi$
$113$	3.12311	0.293797	0.146899	+	0.989152i	$0.453071\pi$
$114$	0	0
$115$	2.87689	0.268272
$116$	5.68466	0.527807
$117$	0	0
$118$	7.68466	0.707430
$119$	−0.315342	−0.0289073
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.24621	0.565505
$123$	0	0
$124$	2.00000	0.179605
$125$	12.0000	1.07331
$126$	0	0
$127$	2.24621	0.199319	0.0996595	−	0.995022i	$-0.468225\pi$
$127$	2.24621	0.199319	0.0996595	+	0.995022i	$0.468225\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.12311	−0.0985029
$131$	−1.12311	−0.0981262	−0.0490631	−	0.998796i	$-0.515624\pi$
$131$	−1.12311	−0.0981262	−0.0490631	+	0.998796i	$0.515624\pi$
$132$	0	0
$133$	0.561553	0.0486928
$134$	−7.68466	−0.663853
$135$	0	0
$136$	0.561553	0.0481528
$137$	−12.5616	−1.07321	−0.536603	−	0.843835i	$-0.680293\pi$
$137$	−12.5616	−1.07321	−0.536603	+	0.843835i	$0.680293\pi$
$138$	0	0
$139$	7.36932	0.625057	0.312529	−	0.949908i	$-0.398824\pi$
$139$	7.36932	0.625057	0.312529	+	0.949908i	$0.398824\pi$
$140$	1.12311	0.0949197
$141$	0	0
$142$	−6.00000	−0.503509
$143$	−0.561553	−0.0469594
$144$	0	0
$145$	−11.3693	−0.944170
$146$	−9.68466	−0.801508
$147$	0	0
$148$	−5.12311	−0.421117
$149$	6.87689	0.563377	0.281689	−	0.959506i	$-0.409106\pi$
$149$	6.87689	0.563377	0.281689	+	0.959506i	$0.409106\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	0.561553	0.0452512
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−14.4924	−1.15662	−0.578311	−	0.815817i	$-0.696288\pi$
$157$	−14.4924	−1.15662	−0.578311	+	0.815817i	$0.696288\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	−2.00000	−0.158114
$161$	0.807764	0.0636607
$162$	0	0
$163$	11.3693	0.890514	0.445257	−	0.895403i	$-0.353112\pi$
$163$	11.3693	0.890514	0.445257	+	0.895403i	$0.353112\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−14.2462	−1.10572
$167$	0.630683	0.0488037	0.0244019	−	0.999702i	$-0.492232\pi$
$167$	0.630683	0.0488037	0.0244019	+	0.999702i	$0.492232\pi$
$168$	0	0
$169$	−12.6847	−0.975743
$170$	−1.12311	−0.0861383
$171$	0	0
$172$	0	0
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	0.561553	0.0424494
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	0.876894	0.0657260
$179$	−5.75379	−0.430058	−0.215029	−	0.976608i	$-0.568985\pi$
$179$	−5.75379	−0.430058	−0.215029	+	0.976608i	$0.568985\pi$
$180$	0	0
$181$	−10.8769	−0.808473	−0.404237	−	0.914654i	$-0.632463\pi$
$181$	−10.8769	−0.808473	−0.404237	+	0.914654i	$0.632463\pi$
$182$	−0.315342	−0.0233747
$183$	0	0
$184$	−1.43845	−0.106044
$185$	10.2462	0.753316
$186$	0	0
$187$	−0.561553	−0.0410648
$188$	−8.00000	−0.583460
$189$	0	0
$190$	2.00000	0.145095
$191$	8.31534	0.601677	0.300838	−	0.953675i	$-0.402733\pi$
$191$	8.31534	0.601677	0.300838	+	0.953675i	$0.402733\pi$
$192$	0	0
$193$	7.12311	0.512732	0.256366	−	0.966580i	$-0.417475\pi$
$193$	7.12311	0.512732	0.256366	+	0.966580i	$0.417475\pi$
$194$	−7.12311	−0.511409
$195$	0	0
$196$	−6.68466	−0.477476
$197$	−18.2462	−1.29999	−0.649994	−	0.759939i	$-0.725229\pi$
$197$	−18.2462	−1.29999	−0.649994	+	0.759939i	$0.725229\pi$
$198$	0	0
$199$	11.6847	0.828303	0.414152	−	0.910208i	$-0.364078\pi$
$199$	11.6847	0.828303	0.414152	+	0.910208i	$0.364078\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	6.87689	0.483857
$203$	−3.19224	−0.224051
$204$	0	0
$205$	4.00000	0.279372
$206$	−13.3693	−0.931484
$207$	0	0
$208$	0.561553	0.0389367
$209$	1.00000	0.0691714
$210$	0	0
$211$	−0.315342	−0.0217090	−0.0108545	−	0.999941i	$-0.503455\pi$
$211$	−0.315342	−0.0217090	−0.0108545	+	0.999941i	$0.503455\pi$
$212$	−12.8078	−0.879641
$213$	0	0
$214$	9.93087	0.678860
$215$	0	0
$216$	0	0
$217$	−1.12311	−0.0762414
$218$	−6.31534	−0.427729
$219$	0	0
$220$	2.00000	0.134840
$221$	0.315342	0.0212122
$222$	0	0
$223$	24.7386	1.65662	0.828311	−	0.560269i	$-0.189302\pi$
$223$	24.7386	1.65662	0.828311	+	0.560269i	$0.189302\pi$
$224$	−0.561553	−0.0375203
$225$	0	0
$226$	3.12311	0.207746
$227$	−28.1771	−1.87018	−0.935089	−	0.354412i	$-0.884681\pi$
$227$	−28.1771	−1.87018	−0.935089	+	0.354412i	$0.884681\pi$
$228$	0	0
$229$	12.8769	0.850929	0.425465	−	0.904975i	$-0.360111\pi$
$229$	12.8769	0.850929	0.425465	+	0.904975i	$0.360111\pi$
$230$	2.87689	0.189697
$231$	0	0
$232$	5.68466	0.373216
$233$	28.7386	1.88273	0.941365	−	0.337389i	$-0.109544\pi$
$233$	28.7386	1.88273	0.941365	+	0.337389i	$0.109544\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	7.68466	0.500229
$237$	0	0
$238$	−0.315342	−0.0204406
$239$	−23.9309	−1.54796	−0.773980	−	0.633210i	$-0.781737\pi$
$239$	−23.9309	−1.54796	−0.773980	+	0.633210i	$0.781737\pi$
$240$	0	0
$241$	−19.6155	−1.26355	−0.631774	−	0.775153i	$-0.717673\pi$
$241$	−19.6155	−1.26355	−0.631774	+	0.775153i	$0.717673\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	6.24621	0.399873
$245$	13.3693	0.854134
$246$	0	0
$247$	−0.561553	−0.0357307
$248$	2.00000	0.127000
$249$	0	0
$250$	12.0000	0.758947
$251$	−17.1231	−1.08080	−0.540400	−	0.841408i	$-0.681727\pi$
$251$	−17.1231	−1.08080	−0.540400	+	0.841408i	$0.681727\pi$
$252$	0	0
$253$	1.43845	0.0904344
$254$	2.24621	0.140940
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	2.87689	0.178762
$260$	−1.12311	−0.0696521
$261$	0	0
$262$	−1.12311	−0.0693857
$263$	−8.87689	−0.547373	−0.273686	−	0.961819i	$-0.588243\pi$
$263$	−8.87689	−0.547373	−0.273686	+	0.961819i	$0.588243\pi$
$264$	0	0
$265$	25.6155	1.57355
$266$	0.561553	0.0344310
$267$	0	0
$268$	−7.68466	−0.469415
$269$	−2.87689	−0.175407	−0.0877037	−	0.996147i	$-0.527953\pi$
$269$	−2.87689	−0.175407	−0.0877037	+	0.996147i	$0.527953\pi$
$270$	0	0
$271$	25.6847	1.56023	0.780116	−	0.625635i	$-0.215160\pi$
$271$	25.6847	1.56023	0.780116	+	0.625635i	$0.215160\pi$
$272$	0.561553	0.0340491
$273$	0	0
$274$	−12.5616	−0.758871
$275$	1.00000	0.0603023
$276$	0	0
$277$	8.49242	0.510260	0.255130	−	0.966907i	$-0.417882\pi$
$277$	8.49242	0.510260	0.255130	+	0.966907i	$0.417882\pi$
$278$	7.36932	0.441982
$279$	0	0
$280$	1.12311	0.0671184
$281$	25.3693	1.51341	0.756703	−	0.653758i	$-0.226809\pi$
$281$	25.3693	1.51341	0.756703	+	0.653758i	$0.226809\pi$
$282$	0	0
$283$	11.3693	0.675836	0.337918	−	0.941176i	$-0.390277\pi$
$283$	11.3693	0.675836	0.337918	+	0.941176i	$0.390277\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−0.561553	−0.0332053
$287$	1.12311	0.0662948
$288$	0	0
$289$	−16.6847	−0.981450
$290$	−11.3693	−0.667629
$291$	0	0
$292$	−9.68466	−0.566752
$293$	−3.93087	−0.229644	−0.114822	−	0.993386i	$-0.536630\pi$
$293$	−3.93087	−0.229644	−0.114822	+	0.993386i	$0.536630\pi$
$294$	0	0
$295$	−15.3693	−0.894836
$296$	−5.12311	−0.297774
$297$	0	0
$298$	6.87689	0.398368
$299$	−0.807764	−0.0467142
$300$	0	0
$301$	0	0
$302$	4.00000	0.230174
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	−12.4924	−0.715314
$306$	0	0
$307$	−22.2462	−1.26966	−0.634829	−	0.772653i	$-0.718930\pi$
$307$	−22.2462	−1.26966	−0.634829	+	0.772653i	$0.718930\pi$
$308$	0.561553	0.0319974
$309$	0	0
$310$	−4.00000	−0.227185
$311$	4.80776	0.272623	0.136312	−	0.990666i	$-0.456475\pi$
$311$	4.80776	0.272623	0.136312	+	0.990666i	$0.456475\pi$
$312$	0	0
$313$	−0.561553	−0.0317408	−0.0158704	−	0.999874i	$-0.505052\pi$
$313$	−0.561553	−0.0317408	−0.0158704	+	0.999874i	$0.505052\pi$
$314$	−14.4924	−0.817855
$315$	0	0
$316$	−4.00000	−0.225018
$317$	7.05398	0.396191	0.198095	−	0.980183i	$-0.436524\pi$
$317$	7.05398	0.396191	0.198095	+	0.980183i	$0.436524\pi$
$318$	0	0
$319$	−5.68466	−0.318280
$320$	−2.00000	−0.111803
$321$	0	0
$322$	0.807764	0.0450149
$323$	−0.561553	−0.0312456
$324$	0	0
$325$	−0.561553	−0.0311493
$326$	11.3693	0.629688
$327$	0	0
$328$	−2.00000	−0.110432
$329$	4.49242	0.247675
$330$	0	0
$331$	−14.5616	−0.800375	−0.400188	−	0.916433i	$-0.631055\pi$
$331$	−14.5616	−0.800375	−0.400188	+	0.916433i	$0.631055\pi$
$332$	−14.2462	−0.781862
$333$	0	0
$334$	0.630683	0.0345094
$335$	15.3693	0.839715
$336$	0	0
$337$	11.1231	0.605914	0.302957	−	0.953004i	$-0.402026\pi$
$337$	11.1231	0.605914	0.302957	+	0.953004i	$0.402026\pi$
$338$	−12.6847	−0.689954
$339$	0	0
$340$	−1.12311	−0.0609090
$341$	−2.00000	−0.108306
$342$	0	0
$343$	7.68466	0.414933
$344$	0	0
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−9.61553	−0.516189	−0.258094	−	0.966120i	$-0.583095\pi$
$347$	−9.61553	−0.516189	−0.258094	+	0.966120i	$0.583095\pi$
$348$	0	0
$349$	21.6155	1.15705	0.578526	−	0.815664i	$-0.303628\pi$
$349$	21.6155	1.15705	0.578526	+	0.815664i	$0.303628\pi$
$350$	0.561553	0.0300163
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−22.1771	−1.18037	−0.590183	−	0.807269i	$-0.700945\pi$
$353$	−22.1771	−1.18037	−0.590183	+	0.807269i	$0.700945\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	0.876894	0.0464753
$357$	0	0
$358$	−5.75379	−0.304097
$359$	15.9309	0.840799	0.420400	−	0.907339i	$-0.361890\pi$
$359$	15.9309	0.840799	0.420400	+	0.907339i	$0.361890\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−10.8769	−0.571677
$363$	0	0
$364$	−0.315342	−0.0165284
$365$	19.3693	1.01384
$366$	0	0
$367$	30.2462	1.57884	0.789420	−	0.613854i	$-0.210382\pi$
$367$	30.2462	1.57884	0.789420	+	0.613854i	$0.210382\pi$
$368$	−1.43845	−0.0749842
$369$	0	0
$370$	10.2462	0.532675
$371$	7.19224	0.373402
$372$	0	0
$373$	3.43845	0.178036	0.0890180	−	0.996030i	$-0.471627\pi$
$373$	3.43845	0.178036	0.0890180	+	0.996030i	$0.471627\pi$
$374$	−0.561553	−0.0290372
$375$	0	0
$376$	−8.00000	−0.412568
$377$	3.19224	0.164409
$378$	0	0
$379$	6.56155	0.337044	0.168522	−	0.985698i	$-0.446101\pi$
$379$	6.56155	0.337044	0.168522	+	0.985698i	$0.446101\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	8.31534	0.425450
$383$	2.00000	0.102195	0.0510976	−	0.998694i	$-0.483728\pi$
$383$	2.00000	0.102195	0.0510976	+	0.998694i	$0.483728\pi$
$384$	0	0
$385$	−1.12311	−0.0572388
$386$	7.12311	0.362557
$387$	0	0
$388$	−7.12311	−0.361621
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−0.807764	−0.0408504
$392$	−6.68466	−0.337626
$393$	0	0
$394$	−18.2462	−0.919231
$395$	8.00000	0.402524
$396$	0	0
$397$	38.9848	1.95659	0.978297	−	0.207209i	$-0.0664381\pi$
$397$	38.9848	1.95659	0.978297	+	0.207209i	$0.0664381\pi$
$398$	11.6847	0.585699
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	7.75379	0.387206	0.193603	−	0.981080i	$-0.437983\pi$
$401$	7.75379	0.387206	0.193603	+	0.981080i	$0.437983\pi$
$402$	0	0
$403$	1.12311	0.0559459
$404$	6.87689	0.342138
$405$	0	0
$406$	−3.19224	−0.158428
$407$	5.12311	0.253943
$408$	0	0
$409$	16.2462	0.803323	0.401662	−	0.915788i	$-0.368433\pi$
$409$	16.2462	0.803323	0.401662	+	0.915788i	$0.368433\pi$
$410$	4.00000	0.197546
$411$	0	0
$412$	−13.3693	−0.658659
$413$	−4.31534	−0.212344
$414$	0	0
$415$	28.4924	1.39864
$416$	0.561553	0.0275324
$417$	0	0
$418$	1.00000	0.0489116
$419$	14.8769	0.726784	0.363392	−	0.931636i	$-0.381619\pi$
$419$	14.8769	0.726784	0.363392	+	0.931636i	$0.381619\pi$
$420$	0	0
$421$	−29.9309	−1.45874	−0.729371	−	0.684119i	$-0.760187\pi$
$421$	−29.9309	−1.45874	−0.729371	+	0.684119i	$0.760187\pi$
$422$	−0.315342	−0.0153506
$423$	0	0
$424$	−12.8078	−0.622000
$425$	−0.561553	−0.0272393
$426$	0	0
$427$	−3.50758	−0.169744
$428$	9.93087	0.480027
$429$	0	0
$430$	0	0
$431$	−31.8617	−1.53473	−0.767363	−	0.641213i	$-0.778432\pi$
$431$	−31.8617	−1.53473	−0.767363	+	0.641213i	$0.778432\pi$
$432$	0	0
$433$	6.63068	0.318650	0.159325	−	0.987226i	$-0.449068\pi$
$433$	6.63068	0.318650	0.159325	+	0.987226i	$0.449068\pi$
$434$	−1.12311	−0.0539108
$435$	0	0
$436$	−6.31534	−0.302450
$437$	1.43845	0.0688103
$438$	0	0
$439$	−9.61553	−0.458924	−0.229462	−	0.973318i	$-0.573697\pi$
$439$	−9.61553	−0.458924	−0.229462	+	0.973318i	$0.573697\pi$
$440$	2.00000	0.0953463
$441$	0	0
$442$	0.315342	0.0149993
$443$	−39.8617	−1.89389	−0.946944	−	0.321398i	$-0.895847\pi$
$443$	−39.8617	−1.89389	−0.946944	+	0.321398i	$0.895847\pi$
$444$	0	0
$445$	−1.75379	−0.0831376
$446$	24.7386	1.17141
$447$	0	0
$448$	−0.561553	−0.0265309
$449$	−23.6155	−1.11449	−0.557243	−	0.830350i	$-0.688141\pi$
$449$	−23.6155	−1.11449	−0.557243	+	0.830350i	$0.688141\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	3.12311	0.146899
$453$	0	0
$454$	−28.1771	−1.32242
$455$	0.630683	0.0295669
$456$	0	0
$457$	36.5616	1.71028	0.855139	−	0.518399i	$-0.173472\pi$
$457$	36.5616	1.71028	0.855139	+	0.518399i	$0.173472\pi$
$458$	12.8769	0.601698
$459$	0	0
$460$	2.87689	0.134136
$461$	−26.7386	−1.24534	−0.622671	−	0.782484i	$-0.713953\pi$
$461$	−26.7386	−1.24534	−0.622671	+	0.782484i	$0.713953\pi$
$462$	0	0
$463$	3.50758	0.163011	0.0815055	−	0.996673i	$-0.474027\pi$
$463$	3.50758	0.163011	0.0815055	+	0.996673i	$0.474027\pi$
$464$	5.68466	0.263904
$465$	0	0
$466$	28.7386	1.33129
$467$	−9.75379	−0.451352	−0.225676	−	0.974202i	$-0.572459\pi$
$467$	−9.75379	−0.451352	−0.225676	+	0.974202i	$0.572459\pi$
$468$	0	0
$469$	4.31534	0.199264
$470$	16.0000	0.738025
$471$	0	0
$472$	7.68466	0.353715
$473$	0	0
$474$	0	0
$475$	1.00000	0.0458831
$476$	−0.315342	−0.0144537
$477$	0	0
$478$	−23.9309	−1.09457
$479$	13.3693	0.610860	0.305430	−	0.952215i	$-0.401200\pi$
$479$	13.3693	0.610860	0.305430	+	0.952215i	$0.401200\pi$
$480$	0	0
$481$	−2.87689	−0.131175
$482$	−19.6155	−0.893463
$483$	0	0
$484$	1.00000	0.0454545
$485$	14.2462	0.646887
$486$	0	0
$487$	31.1231	1.41032	0.705161	−	0.709047i	$-0.250875\pi$
$487$	31.1231	1.41032	0.705161	+	0.709047i	$0.250875\pi$
$488$	6.24621	0.282753
$489$	0	0
$490$	13.3693	0.603964
$491$	39.3693	1.77671	0.888356	−	0.459155i	$-0.151848\pi$
$491$	39.3693	1.77671	0.888356	+	0.459155i	$0.151848\pi$
$492$	0	0
$493$	3.19224	0.143771
$494$	−0.561553	−0.0252655
$495$	0	0
$496$	2.00000	0.0898027
$497$	3.36932	0.151135
$498$	0	0
$499$	9.75379	0.436640	0.218320	−	0.975877i	$-0.429942\pi$
$499$	9.75379	0.436640	0.218320	+	0.975877i	$0.429942\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	−17.1231	−0.764242
$503$	−21.6847	−0.966871	−0.483436	−	0.875380i	$-0.660611\pi$
$503$	−21.6847	−0.966871	−0.483436	+	0.875380i	$0.660611\pi$
$504$	0	0
$505$	−13.7538	−0.612036
$506$	1.43845	0.0639468
$507$	0	0
$508$	2.24621	0.0996595
$509$	31.3693	1.39042	0.695210	−	0.718806i	$-0.255311\pi$
$509$	31.3693	1.39042	0.695210	+	0.718806i	$0.255311\pi$
$510$	0	0
$511$	5.43845	0.240583
$512$	1.00000	0.0441942
$513$	0	0
$514$	22.0000	0.970378
$515$	26.7386	1.17824
$516$	0	0
$517$	8.00000	0.351840
$518$	2.87689	0.126403
$519$	0	0
$520$	−1.12311	−0.0492514
$521$	7.12311	0.312069	0.156034	−	0.987752i	$-0.450129\pi$
$521$	7.12311	0.312069	0.156034	+	0.987752i	$0.450129\pi$
$522$	0	0
$523$	31.0540	1.35790	0.678948	−	0.734187i	$-0.262436\pi$
$523$	31.0540	1.35790	0.678948	+	0.734187i	$0.262436\pi$
$524$	−1.12311	−0.0490631
$525$	0	0
$526$	−8.87689	−0.387051
$527$	1.12311	0.0489232
$528$	0	0
$529$	−20.9309	−0.910038
$530$	25.6155	1.11267
$531$	0	0
$532$	0.561553	0.0243464
$533$	−1.12311	−0.0486471
$534$	0	0
$535$	−19.8617	−0.858698
$536$	−7.68466	−0.331927
$537$	0	0
$538$	−2.87689	−0.124032
$539$	6.68466	0.287929
$540$	0	0
$541$	−39.2311	−1.68667	−0.843337	−	0.537384i	$-0.819412\pi$
$541$	−39.2311	−1.68667	−0.843337	+	0.537384i	$0.819412\pi$
$542$	25.6847	1.10325
$543$	0	0
$544$	0.561553	0.0240764
$545$	12.6307	0.541039
$546$	0	0
$547$	−42.7386	−1.82737	−0.913686	−	0.406421i	$-0.866777\pi$
$547$	−42.7386	−1.82737	−0.913686	+	0.406421i	$0.866777\pi$
$548$	−12.5616	−0.536603
$549$	0	0
$550$	1.00000	0.0426401
$551$	−5.68466	−0.242175
$552$	0	0
$553$	2.24621	0.0955186
$554$	8.49242	0.360808
$555$	0	0
$556$	7.36932	0.312529
$557$	−17.1231	−0.725529	−0.362765	−	0.931881i	$-0.618167\pi$
$557$	−17.1231	−0.725529	−0.362765	+	0.931881i	$0.618167\pi$
$558$	0	0
$559$	0	0
$560$	1.12311	0.0474599
$561$	0	0
$562$	25.3693	1.07014
$563$	42.7386	1.80122	0.900609	−	0.434630i	$-0.143121\pi$
$563$	42.7386	1.80122	0.900609	+	0.434630i	$0.143121\pi$
$564$	0	0
$565$	−6.24621	−0.262780
$566$	11.3693	0.477888
$567$	0	0
$568$	−6.00000	−0.251754
$569$	21.3693	0.895848	0.447924	−	0.894072i	$-0.352163\pi$
$569$	21.3693	0.895848	0.447924	+	0.894072i	$0.352163\pi$
$570$	0	0
$571$	−5.61553	−0.235003	−0.117501	−	0.993073i	$-0.537488\pi$
$571$	−5.61553	−0.235003	−0.117501	+	0.993073i	$0.537488\pi$
$572$	−0.561553	−0.0234797
$573$	0	0
$574$	1.12311	0.0468775
$575$	1.43845	0.0599874
$576$	0	0
$577$	41.0540	1.70910	0.854550	−	0.519370i	$-0.173833\pi$
$577$	41.0540	1.70910	0.854550	+	0.519370i	$0.173833\pi$
$578$	−16.6847	−0.693990
$579$	0	0
$580$	−11.3693	−0.472085
$581$	8.00000	0.331896
$582$	0	0
$583$	12.8078	0.530443
$584$	−9.68466	−0.400754
$585$	0	0
$586$	−3.93087	−0.162383
$587$	40.4924	1.67130	0.835651	−	0.549261i	$-0.185091\pi$
$587$	40.4924	1.67130	0.835651	+	0.549261i	$0.185091\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	−15.3693	−0.632745
$591$	0	0
$592$	−5.12311	−0.210558
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	0.630683	0.0258555
$596$	6.87689	0.281689
$597$	0	0
$598$	−0.807764	−0.0330319
$599$	17.8617	0.729811	0.364905	−	0.931045i	$-0.381101\pi$
$599$	17.8617	0.729811	0.364905	+	0.931045i	$0.381101\pi$
$600$	0	0
$601$	21.3693	0.871673	0.435836	−	0.900026i	$-0.356453\pi$
$601$	21.3693	0.871673	0.435836	+	0.900026i	$0.356453\pi$
$602$	0	0
$603$	0	0
$604$	4.00000	0.162758
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	−13.6155	−0.552637	−0.276319	−	0.961066i	$-0.589115\pi$
$607$	−13.6155	−0.552637	−0.276319	+	0.961066i	$0.589115\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	−12.4924	−0.505803
$611$	−4.49242	−0.181744
$612$	0	0
$613$	18.2462	0.736958	0.368479	−	0.929636i	$-0.379879\pi$
$613$	18.2462	0.736958	0.368479	+	0.929636i	$0.379879\pi$
$614$	−22.2462	−0.897784
$615$	0	0
$616$	0.561553	0.0226256
$617$	−44.7386	−1.80111	−0.900555	−	0.434743i	$-0.856839\pi$
$617$	−44.7386	−1.80111	−0.900555	+	0.434743i	$0.856839\pi$
$618$	0	0
$619$	−9.75379	−0.392038	−0.196019	−	0.980600i	$-0.562801\pi$
$619$	−9.75379	−0.392038	−0.196019	+	0.980600i	$0.562801\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	4.80776	0.192774
$623$	−0.492423	−0.0197285
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−0.561553	−0.0224442
$627$	0	0
$628$	−14.4924	−0.578311
$629$	−2.87689	−0.114709
$630$	0	0
$631$	3.50758	0.139634	0.0698172	−	0.997560i	$-0.477758\pi$
$631$	3.50758	0.139634	0.0698172	+	0.997560i	$0.477758\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	7.05398	0.280149
$635$	−4.49242	−0.178276
$636$	0	0
$637$	−3.75379	−0.148731
$638$	−5.68466	−0.225058
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	−31.6155	−1.24874	−0.624369	−	0.781129i	$-0.714644\pi$
$641$	−31.6155	−1.24874	−0.624369	+	0.781129i	$0.714644\pi$
$642$	0	0
$643$	−12.6307	−0.498106	−0.249053	−	0.968490i	$-0.580119\pi$
$643$	−12.6307	−0.498106	−0.249053	+	0.968490i	$0.580119\pi$
$644$	0.807764	0.0318304
$645$	0	0
$646$	−0.561553	−0.0220940
$647$	47.5464	1.86924	0.934621	−	0.355646i	$-0.115739\pi$
$647$	47.5464	1.86924	0.934621	+	0.355646i	$0.115739\pi$
$648$	0	0
$649$	−7.68466	−0.301649
$650$	−0.561553	−0.0220259
$651$	0	0
$652$	11.3693	0.445257
$653$	15.6155	0.611083	0.305541	−	0.952179i	$-0.401163\pi$
$653$	15.6155	0.611083	0.305541	+	0.952179i	$0.401163\pi$
$654$	0	0
$655$	2.24621	0.0877667
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	4.49242	0.175133
$659$	14.4233	0.561852	0.280926	−	0.959729i	$-0.409359\pi$
$659$	14.4233	0.561852	0.280926	+	0.959729i	$0.409359\pi$
$660$	0	0
$661$	−26.5616	−1.03312	−0.516562	−	0.856250i	$-0.672789\pi$
$661$	−26.5616	−1.03312	−0.516562	+	0.856250i	$0.672789\pi$
$662$	−14.5616	−0.565951
$663$	0	0
$664$	−14.2462	−0.552860
$665$	−1.12311	−0.0435522
$666$	0	0
$667$	−8.17708	−0.316618
$668$	0.630683	0.0244019
$669$	0	0
$670$	15.3693	0.593769
$671$	−6.24621	−0.241132
$672$	0	0
$673$	−27.1231	−1.04552	−0.522759	−	0.852480i	$-0.675097\pi$
$673$	−27.1231	−1.04552	−0.522759	+	0.852480i	$0.675097\pi$
$674$	11.1231	0.428446
$675$	0	0
$676$	−12.6847	−0.487871
$677$	−6.17708	−0.237405	−0.118702	−	0.992930i	$-0.537873\pi$
$677$	−6.17708	−0.237405	−0.118702	+	0.992930i	$0.537873\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	−1.12311	−0.0430691
$681$	0	0
$682$	−2.00000	−0.0765840
$683$	−28.4924	−1.09023	−0.545116	−	0.838361i	$-0.683514\pi$
$683$	−28.4924	−1.09023	−0.545116	+	0.838361i	$0.683514\pi$
$684$	0	0
$685$	25.1231	0.959905
$686$	7.68466	0.293402
$687$	0	0
$688$	0	0
$689$	−7.19224	−0.274002
$690$	0	0
$691$	−2.38447	−0.0907096	−0.0453548	−	0.998971i	$-0.514442\pi$
$691$	−2.38447	−0.0907096	−0.0453548	+	0.998971i	$0.514442\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	−9.61553	−0.365000
$695$	−14.7386	−0.559068
$696$	0	0
$697$	−1.12311	−0.0425407
$698$	21.6155	0.818160
$699$	0	0
$700$	0.561553	0.0212247
$701$	40.9848	1.54798	0.773988	−	0.633200i	$-0.218259\pi$
$701$	40.9848	1.54798	0.773988	+	0.633200i	$0.218259\pi$
$702$	0	0
$703$	5.12311	0.193222
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−22.1771	−0.834645
$707$	−3.86174	−0.145236
$708$	0	0
$709$	−42.4924	−1.59584	−0.797918	−	0.602766i	$-0.794065\pi$
$709$	−42.4924	−1.59584	−0.797918	+	0.602766i	$0.794065\pi$
$710$	12.0000	0.450352
$711$	0	0
$712$	0.876894	0.0328630
$713$	−2.87689	−0.107741
$714$	0	0
$715$	1.12311	0.0420018
$716$	−5.75379	−0.215029
$717$	0	0
$718$	15.9309	0.594535
$719$	10.5616	0.393879	0.196940	−	0.980416i	$-0.436900\pi$
$719$	10.5616	0.393879	0.196940	+	0.980416i	$0.436900\pi$
$720$	0	0
$721$	7.50758	0.279597
$722$	1.00000	0.0372161
$723$	0	0
$724$	−10.8769	−0.404237
$725$	−5.68466	−0.211123
$726$	0	0
$727$	22.4233	0.831634	0.415817	−	0.909448i	$-0.363496\pi$
$727$	22.4233	0.831634	0.415817	+	0.909448i	$0.363496\pi$
$728$	−0.315342	−0.0116873
$729$	0	0
$730$	19.3693	0.716891
$731$	0	0
$732$	0	0
$733$	26.1080	0.964319	0.482160	−	0.876083i	$-0.339853\pi$
$733$	26.1080	0.964319	0.482160	+	0.876083i	$0.339853\pi$
$734$	30.2462	1.11641
$735$	0	0
$736$	−1.43845	−0.0530219
$737$	7.68466	0.283068
$738$	0	0
$739$	−44.9848	−1.65479	−0.827397	−	0.561617i	$-0.810179\pi$
$739$	−44.9848	−1.65479	−0.827397	+	0.561617i	$0.810179\pi$
$740$	10.2462	0.376658
$741$	0	0
$742$	7.19224	0.264035
$743$	42.2462	1.54986	0.774932	−	0.632045i	$-0.217784\pi$
$743$	42.2462	1.54986	0.774932	+	0.632045i	$0.217784\pi$
$744$	0	0
$745$	−13.7538	−0.503900
$746$	3.43845	0.125890
$747$	0	0
$748$	−0.561553	−0.0205324
$749$	−5.57671	−0.203768
$750$	0	0
$751$	41.2311	1.50454	0.752271	−	0.658853i	$-0.228958\pi$
$751$	41.2311	1.50454	0.752271	+	0.658853i	$0.228958\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	3.19224	0.116254
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−42.4924	−1.54441	−0.772207	−	0.635371i	$-0.780847\pi$
$757$	−42.4924	−1.54441	−0.772207	+	0.635371i	$0.780847\pi$
$758$	6.56155	0.238326
$759$	0	0
$760$	2.00000	0.0725476
$761$	42.8078	1.55178	0.775890	−	0.630868i	$-0.217301\pi$
$761$	42.8078	1.55178	0.775890	+	0.630868i	$0.217301\pi$
$762$	0	0
$763$	3.54640	0.128388
$764$	8.31534	0.300838
$765$	0	0
$766$	2.00000	0.0722629
$767$	4.31534	0.155818
$768$	0	0
$769$	−13.6847	−0.493481	−0.246741	−	0.969082i	$-0.579360\pi$
$769$	−13.6847	−0.493481	−0.246741	+	0.969082i	$0.579360\pi$
$770$	−1.12311	−0.0404739
$771$	0	0
$772$	7.12311	0.256366
$773$	−37.9309	−1.36428	−0.682139	−	0.731222i	$-0.738950\pi$
$773$	−37.9309	−1.36428	−0.682139	+	0.731222i	$0.738950\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	−7.12311	−0.255705
$777$	0	0
$778$	−18.0000	−0.645331
$779$	2.00000	0.0716574
$780$	0	0
$781$	6.00000	0.214697
$782$	−0.807764	−0.0288856
$783$	0	0
$784$	−6.68466	−0.238738
$785$	28.9848	1.03451
$786$	0	0
$787$	15.6847	0.559098	0.279549	−	0.960131i	$-0.409815\pi$
$787$	15.6847	0.559098	0.279549	+	0.960131i	$0.409815\pi$
$788$	−18.2462	−0.649994
$789$	0	0
$790$	8.00000	0.284627
$791$	−1.75379	−0.0623575
$792$	0	0
$793$	3.50758	0.124558
$794$	38.9848	1.38352
$795$	0	0
$796$	11.6847	0.414152
$797$	15.1922	0.538137	0.269068	−	0.963121i	$-0.413284\pi$
$797$	15.1922	0.538137	0.269068	+	0.963121i	$0.413284\pi$
$798$	0	0
$799$	−4.49242	−0.158930
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	7.75379	0.273796
$803$	9.68466	0.341764
$804$	0	0
$805$	−1.61553	−0.0569399
$806$	1.12311	0.0395597
$807$	0	0
$808$	6.87689	0.241928
$809$	−8.06913	−0.283696	−0.141848	−	0.989888i	$-0.545304\pi$
$809$	−8.06913	−0.283696	−0.141848	+	0.989888i	$0.545304\pi$
$810$	0	0
$811$	8.31534	0.291991	0.145996	−	0.989285i	$-0.453361\pi$
$811$	8.31534	0.291991	0.145996	+	0.989285i	$0.453361\pi$
$812$	−3.19224	−0.112026
$813$	0	0
$814$	5.12311	0.179565
$815$	−22.7386	−0.796500
$816$	0	0
$817$	0	0
$818$	16.2462	0.568035
$819$	0	0
$820$	4.00000	0.139686
$821$	20.9848	0.732376	0.366188	−	0.930541i	$-0.380663\pi$
$821$	20.9848	0.732376	0.366188	+	0.930541i	$0.380663\pi$
$822$	0	0
$823$	−13.9309	−0.485600	−0.242800	−	0.970076i	$-0.578066\pi$
$823$	−13.9309	−0.485600	−0.242800	+	0.970076i	$0.578066\pi$
$824$	−13.3693	−0.465742
$825$	0	0
$826$	−4.31534	−0.150150
$827$	−49.9309	−1.73627	−0.868133	−	0.496331i	$-0.834680\pi$
$827$	−49.9309	−1.73627	−0.868133	+	0.496331i	$0.834680\pi$
$828$	0	0
$829$	12.1771	0.422928	0.211464	−	0.977386i	$-0.432177\pi$
$829$	12.1771	0.422928	0.211464	+	0.977386i	$0.432177\pi$
$830$	28.4924	0.988986
$831$	0	0
$832$	0.561553	0.0194683
$833$	−3.75379	−0.130061
$834$	0	0
$835$	−1.26137	−0.0436514
$836$	1.00000	0.0345857
$837$	0	0
$838$	14.8769	0.513914
$839$	7.12311	0.245917	0.122958	−	0.992412i	$-0.460762\pi$
$839$	7.12311	0.245917	0.122958	+	0.992412i	$0.460762\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	−29.9309	−1.03149
$843$	0	0
$844$	−0.315342	−0.0108545
$845$	25.3693	0.872731
$846$	0	0
$847$	−0.561553	−0.0192952
$848$	−12.8078	−0.439820
$849$	0	0
$850$	−0.561553	−0.0192611
$851$	7.36932	0.252617
$852$	0	0
$853$	−52.3542	−1.79257	−0.896286	−	0.443476i	$-0.853745\pi$
$853$	−52.3542	−1.79257	−0.896286	+	0.443476i	$0.853745\pi$
$854$	−3.50758	−0.120027
$855$	0	0
$856$	9.93087	0.339430
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	15.8617	0.541196	0.270598	−	0.962692i	$-0.412779\pi$
$859$	15.8617	0.541196	0.270598	+	0.962692i	$0.412779\pi$
$860$	0	0
$861$	0	0
$862$	−31.8617	−1.08522
$863$	−0.246211	−0.00838113	−0.00419056	−	0.999991i	$-0.501334\pi$
$863$	−0.246211	−0.00838113	−0.00419056	+	0.999991i	$0.501334\pi$
$864$	0	0
$865$	36.0000	1.22404
$866$	6.63068	0.225320
$867$	0	0
$868$	−1.12311	−0.0381207
$869$	4.00000	0.135691
$870$	0	0
$871$	−4.31534	−0.146220
$872$	−6.31534	−0.213864
$873$	0	0
$874$	1.43845	0.0486562
$875$	−6.73863	−0.227807
$876$	0	0
$877$	−44.5616	−1.50474	−0.752368	−	0.658743i	$-0.771089\pi$
$877$	−44.5616	−1.50474	−0.752368	+	0.658743i	$0.771089\pi$
$878$	−9.61553	−0.324508
$879$	0	0
$880$	2.00000	0.0674200
$881$	−20.7386	−0.698702	−0.349351	−	0.936992i	$-0.613598\pi$
$881$	−20.7386	−0.698702	−0.349351	+	0.936992i	$0.613598\pi$
$882$	0	0
$883$	−5.26137	−0.177059	−0.0885295	−	0.996074i	$-0.528217\pi$
$883$	−5.26137	−0.177059	−0.0885295	+	0.996074i	$0.528217\pi$
$884$	0.315342	0.0106061
$885$	0	0
$886$	−39.8617	−1.33918
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	−1.26137	−0.0423049
$890$	−1.75379	−0.0587871
$891$	0	0
$892$	24.7386	0.828311
$893$	8.00000	0.267710
$894$	0	0
$895$	11.5076	0.384656
$896$	−0.561553	−0.0187602
$897$	0	0
$898$	−23.6155	−0.788060
$899$	11.3693	0.379188
$900$	0	0
$901$	−7.19224	−0.239608
$902$	2.00000	0.0665927
$903$	0	0
$904$	3.12311	0.103873
$905$	21.7538	0.723120
$906$	0	0
$907$	−29.7926	−0.989247	−0.494624	−	0.869107i	$-0.664694\pi$
$907$	−29.7926	−0.989247	−0.494624	+	0.869107i	$0.664694\pi$
$908$	−28.1771	−0.935089
$909$	0	0
$910$	0.630683	0.0209069
$911$	−18.9848	−0.628996	−0.314498	−	0.949258i	$-0.601836\pi$
$911$	−18.9848	−0.628996	−0.314498	+	0.949258i	$0.601836\pi$
$912$	0	0
$913$	14.2462	0.471481
$914$	36.5616	1.20935
$915$	0	0
$916$	12.8769	0.425465
$917$	0.630683	0.0208270
$918$	0	0
$919$	32.5616	1.07411	0.537053	−	0.843548i	$-0.319537\pi$
$919$	32.5616	1.07411	0.537053	+	0.843548i	$0.319537\pi$
$920$	2.87689	0.0948484
$921$	0	0
$922$	−26.7386	−0.880590
$923$	−3.36932	−0.110902
$924$	0	0
$925$	5.12311	0.168447
$926$	3.50758	0.115266
$927$	0	0
$928$	5.68466	0.186608
$929$	−11.9309	−0.391439	−0.195720	−	0.980660i	$-0.562704\pi$
$929$	−11.9309	−0.391439	−0.195720	+	0.980660i	$0.562704\pi$
$930$	0	0
$931$	6.68466	0.219081
$932$	28.7386	0.941365
$933$	0	0
$934$	−9.75379	−0.319154
$935$	1.12311	0.0367295
$936$	0	0
$937$	−0.699813	−0.0228619	−0.0114310	−	0.999935i	$-0.503639\pi$
$937$	−0.699813	−0.0228619	−0.0114310	+	0.999935i	$0.503639\pi$
$938$	4.31534	0.140901
$939$	0	0
$940$	16.0000	0.521862
$941$	−24.5616	−0.800684	−0.400342	−	0.916366i	$-0.631109\pi$
$941$	−24.5616	−0.800684	−0.400342	+	0.916366i	$0.631109\pi$
$942$	0	0
$943$	2.87689	0.0936846
$944$	7.68466	0.250114
$945$	0	0
$946$	0	0
$947$	36.9848	1.20185	0.600923	−	0.799307i	$-0.294800\pi$
$947$	36.9848	1.20185	0.600923	+	0.799307i	$0.294800\pi$
$948$	0	0
$949$	−5.43845	−0.176539
$950$	1.00000	0.0324443
$951$	0	0
$952$	−0.315342	−0.0102203
$953$	0.876894	0.0284054	0.0142027	−	0.999899i	$-0.495479\pi$
$953$	0.876894	0.0284054	0.0142027	+	0.999899i	$0.495479\pi$
$954$	0	0
$955$	−16.6307	−0.538156
$956$	−23.9309	−0.773980
$957$	0	0
$958$	13.3693	0.431943
$959$	7.05398	0.227785
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−2.87689	−0.0927548
$963$	0	0
$964$	−19.6155	−0.631774
$965$	−14.2462	−0.458602
$966$	0	0
$967$	53.8617	1.73208	0.866038	−	0.499978i	$-0.166658\pi$
$967$	53.8617	1.73208	0.866038	+	0.499978i	$0.166658\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	14.2462	0.457418
$971$	−2.24621	−0.0720843	−0.0360422	−	0.999350i	$-0.511475\pi$
$971$	−2.24621	−0.0720843	−0.0360422	+	0.999350i	$0.511475\pi$
$972$	0	0
$973$	−4.13826	−0.132667
$974$	31.1231	0.997249
$975$	0	0
$976$	6.24621	0.199936
$977$	−30.6307	−0.979962	−0.489981	−	0.871733i	$-0.662996\pi$
$977$	−30.6307	−0.979962	−0.489981	+	0.871733i	$0.662996\pi$
$978$	0	0
$979$	−0.876894	−0.0280257
$980$	13.3693	0.427067
$981$	0	0
$982$	39.3693	1.25633
$983$	31.7538	1.01279	0.506394	−	0.862302i	$-0.330978\pi$
$983$	31.7538	1.01279	0.506394	+	0.862302i	$0.330978\pi$
$984$	0	0
$985$	36.4924	1.16275
$986$	3.19224	0.101662
$987$	0	0
$988$	−0.561553	−0.0178654
$989$	0	0
$990$	0	0
$991$	12.8769	0.409048	0.204524	−	0.978862i	$-0.434435\pi$
$991$	12.8769	0.409048	0.204524	+	0.978862i	$0.434435\pi$
$992$	2.00000	0.0635001
$993$	0	0
$994$	3.36932	0.106868
$995$	−23.3693	−0.740857
$996$	0	0
$997$	−28.6307	−0.906743	−0.453371	−	0.891322i	$-0.649779\pi$
$997$	−28.6307	−0.906743	−0.453371	+	0.891322i	$0.649779\pi$
$998$	9.75379	0.308751
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3762.2.a.y.1.1		2
3.2	odd	2		418.2.a.e.1.2	✓	2
12.11	even	2		3344.2.a.k.1.1		2
33.32	even	2		4598.2.a.bj.1.2		2
57.56	even	2		7942.2.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.e.1.2	✓	2	3.2	odd	2
3344.2.a.k.1.1		2	12.11	even	2
3762.2.a.y.1.1		2	1.1	even	1	trivial
4598.2.a.bj.1.2		2	33.32	even	2
7942.2.a.x.1.1		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -0.561553 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -0.561553 q^{7} +1.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} +0.561553 q^{13} -0.561553 q^{14} +1.00000 q^{16} +0.561553 q^{17} -1.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} -1.43845 q^{23} -1.00000 q^{25} +0.561553 q^{26} -0.561553 q^{28} +5.68466 q^{29} +2.00000 q^{31} +1.00000 q^{32} +0.561553 q^{34} +1.12311 q^{35} -5.12311 q^{37} -1.00000 q^{38} -2.00000 q^{40} -2.00000 q^{41} -1.00000 q^{44} -1.43845 q^{46} -8.00000 q^{47} -6.68466 q^{49} -1.00000 q^{50} +0.561553 q^{52} -12.8078 q^{53} +2.00000 q^{55} -0.561553 q^{56} +5.68466 q^{58} +7.68466 q^{59} +6.24621 q^{61} +2.00000 q^{62} +1.00000 q^{64} -1.12311 q^{65} -7.68466 q^{67} +0.561553 q^{68} +1.12311 q^{70} -6.00000 q^{71} -9.68466 q^{73} -5.12311 q^{74} -1.00000 q^{76} +0.561553 q^{77} -4.00000 q^{79} -2.00000 q^{80} -2.00000 q^{82} -14.2462 q^{83} -1.12311 q^{85} -1.00000 q^{88} +0.876894 q^{89} -0.315342 q^{91} -1.43845 q^{92} -8.00000 q^{94} +2.00000 q^{95} -7.12311 q^{97} -6.68466 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 3 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8} - 4 q^{10} - 2 q^{11} - 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} - 2 q^{19} - 4 q^{20} - 2 q^{22} - 7 q^{23} - 2 q^{25} - 3 q^{26} + 3 q^{28} - q^{29} + 4 q^{31} + 2 q^{32} - 3 q^{34} - 6 q^{35} - 2 q^{37} - 2 q^{38} - 4 q^{40} - 4 q^{41} - 2 q^{44} - 7 q^{46} - 16 q^{47} - q^{49} - 2 q^{50} - 3 q^{52} - 5 q^{53} + 4 q^{55} + 3 q^{56} - q^{58} + 3 q^{59} - 4 q^{61} + 4 q^{62} + 2 q^{64} + 6 q^{65} - 3 q^{67} - 3 q^{68} - 6 q^{70} - 12 q^{71} - 7 q^{73} - 2 q^{74} - 2 q^{76} - 3 q^{77} - 8 q^{79} - 4 q^{80} - 4 q^{82} - 12 q^{83} + 6 q^{85} - 2 q^{88} + 10 q^{89} - 13 q^{91} - 7 q^{92} - 16 q^{94} + 4 q^{95} - 6 q^{97} - q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 3 * q^7 + 2 * q^8 - 4 * q^10 - 2 * q^11 - 3 * q^13 + 3 * q^14 + 2 * q^16 - 3 * q^17 - 2 * q^19 - 4 * q^20 - 2 * q^22 - 7 * q^23 - 2 * q^25 - 3 * q^26 + 3 * q^28 - q^29 + 4 * q^31 + 2 * q^32 - 3 * q^34 - 6 * q^35 - 2 * q^37 - 2 * q^38 - 4 * q^40 - 4 * q^41 - 2 * q^44 - 7 * q^46 - 16 * q^47 - q^49 - 2 * q^50 - 3 * q^52 - 5 * q^53 + 4 * q^55 + 3 * q^56 - q^58 + 3 * q^59 - 4 * q^61 + 4 * q^62 + 2 * q^64 + 6 * q^65 - 3 * q^67 - 3 * q^68 - 6 * q^70 - 12 * q^71 - 7 * q^73 - 2 * q^74 - 2 * q^76 - 3 * q^77 - 8 * q^79 - 4 * q^80 - 4 * q^82 - 12 * q^83 + 6 * q^85 - 2 * q^88 + 10 * q^89 - 13 * q^91 - 7 * q^92 - 16 * q^94 + 4 * q^95 - 6 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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