LMFDB - Embedded newform 3762.2.a.bd.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3762,2,Mod(1,3762)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [3,-3,0,3,-5,0,1,-3,0,5,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$30.0397212404$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.3
Root			$2.39138$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +0.391382 q^{5} -1.71871 q^{7} -1.00000 q^{8} -0.391382 q^{10} +1.00000 q^{11} +3.71871 q^{13} +1.71871 q^{14} +1.00000 q^{16} -5.43742 q^{17} -1.00000 q^{19} +0.391382 q^{20} -1.00000 q^{22} +3.43742 q^{23} -4.84682 q^{25} -3.71871 q^{26} -1.71871 q^{28} +3.82880 q^{29} -3.06406 q^{31} -1.00000 q^{32} +5.43742 q^{34} -0.672673 q^{35} -4.65465 q^{37} +1.00000 q^{38} -0.391382 q^{40} -1.06406 q^{41} -3.73673 q^{43} +1.00000 q^{44} -3.43742 q^{46} +8.22018 q^{47} -4.04604 q^{49} +4.84682 q^{50} +3.71871 q^{52} +1.21724 q^{53} +0.391382 q^{55} +1.71871 q^{56} -3.82880 q^{58} -12.4404 q^{59} +2.00000 q^{61} +3.06406 q^{62} +1.00000 q^{64} +1.45544 q^{65} -1.71871 q^{67} -5.43742 q^{68} +0.672673 q^{70} -1.04604 q^{71} +13.6576 q^{73} +4.65465 q^{74} -1.00000 q^{76} -1.71871 q^{77} +13.0029 q^{79} +0.391382 q^{80} +1.06406 q^{82} -8.51949 q^{83} -2.12811 q^{85} +3.73673 q^{86} -1.00000 q^{88} -17.6936 q^{89} -6.39138 q^{91} +3.43742 q^{92} -8.22018 q^{94} -0.391382 q^{95} -4.65465 q^{97} +4.04604 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} + q^{7} - 3 q^{8} + 5 q^{10} + 3 q^{11} + 5 q^{13} - q^{14} + 3 q^{16} - 4 q^{17} - 3 q^{19} - 5 q^{20} - 3 q^{22} - 2 q^{23} + 4 q^{25} - 5 q^{26} + q^{28} - 7 q^{29}+ \cdots + 6 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 5 * q^5 + q^7 - 3 * q^8 + 5 * q^10 + 3 * q^11 + 5 * q^13 - q^14 + 3 * q^16 - 4 * q^17 - 3 * q^19 - 5 * q^20 - 3 * q^22 - 2 * q^23 + 4 * q^25 - 5 * q^26 + q^28 - 7 * q^29 - 3 * q^31 - 3 * q^32 + 4 * q^34 - 2 * q^35 - 14 * q^37 + 3 * q^38 + 5 * q^40 + 3 * q^41 - 5 * q^43 + 3 * q^44 + 2 * q^46 - 6 * q^49 - 4 * q^50 + 5 * q^52 + 16 * q^53 - 5 * q^55 - q^56 + 7 * q^58 + 12 * q^59 + 6 * q^61 + 3 * q^62 + 3 * q^64 - 8 * q^65 + q^67 - 4 * q^68 + 2 * q^70 + 3 * q^71 + 4 * q^73 + 14 * q^74 - 3 * q^76 + q^77 + 2 * q^79 - 5 * q^80 - 3 * q^82 - 7 * q^83 + 6 * q^85 + 5 * q^86 - 3 * q^88 - 16 * q^89 - 13 * q^91 - 2 * q^92 + 5 * q^95 - 14 * q^97 + 6 * 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
$5$	0.391382	0.175032	0.0875158	−	0.996163i	$-0.472107\pi$
$5$	0.391382	0.175032	0.0875158	+	0.996163i	$0.472107\pi$
$6$	0	0
$7$	−1.71871	−0.649611	−0.324806	−	0.945781i	$-0.605299\pi$
$7$	−1.71871	−0.649611	−0.324806	+	0.945781i	$0.605299\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.391382	−0.123766
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	3.71871	1.03138	0.515692	−	0.856774i	$-0.327535\pi$
$13$	3.71871	1.03138	0.515692	+	0.856774i	$0.327535\pi$
$14$	1.71871	0.459344
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.43742	−1.31877	−0.659384	−	0.751806i	$-0.729183\pi$
$17$	−5.43742	−1.31877	−0.659384	+	0.751806i	$0.729183\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.391382	0.0875158
$21$	0	0
$22$	−1.00000	−0.213201
$23$	3.43742	0.716751	0.358376	−	0.933577i	$-0.383331\pi$
$23$	3.43742	0.716751	0.358376	+	0.933577i	$0.383331\pi$
$24$	0	0
$25$	−4.84682	−0.969364
$26$	−3.71871	−0.729299
$27$	0	0
$28$	−1.71871	−0.324806
$29$	3.82880	0.710991	0.355495	−	0.934678i	$-0.384312\pi$
$29$	3.82880	0.710991	0.355495	+	0.934678i	$0.384312\pi$
$30$	0	0
$31$	−3.06406	−0.550321	−0.275160	−	0.961398i	$-0.588731\pi$
$31$	−3.06406	−0.550321	−0.275160	+	0.961398i	$0.588731\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.43742	0.932510
$35$	−0.672673	−0.113702
$36$	0	0
$37$	−4.65465	−0.765221	−0.382610	−	0.923910i	$-0.624975\pi$
$37$	−4.65465	−0.765221	−0.382610	+	0.923910i	$0.624975\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−0.391382	−0.0618830
$41$	−1.06406	−0.166177	−0.0830887	−	0.996542i	$-0.526478\pi$
$41$	−1.06406	−0.166177	−0.0830887	+	0.996542i	$0.526478\pi$
$42$	0	0
$43$	−3.73673	−0.569846	−0.284923	−	0.958550i	$-0.591968\pi$
$43$	−3.73673	−0.569846	−0.284923	+	0.958550i	$0.591968\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−3.43742	−0.506820
$47$	8.22018	1.19904	0.599519	−	0.800361i	$-0.295359\pi$
$47$	8.22018	1.19904	0.599519	+	0.800361i	$0.295359\pi$
$48$	0	0
$49$	−4.04604	−0.578005
$50$	4.84682	0.685444
$51$	0	0
$52$	3.71871	0.515692
$53$	1.21724	0.167200	0.0836001	−	0.996499i	$-0.473358\pi$
$53$	1.21724	0.167200	0.0836001	+	0.996499i	$0.473358\pi$
$54$	0	0
$55$	0.391382	0.0527740
$56$	1.71871	0.229672
$57$	0	0
$58$	−3.82880	−0.502746
$59$	−12.4404	−1.61960	−0.809799	−	0.586707i	$-0.800424\pi$
$59$	−12.4404	−1.61960	−0.809799	+	0.586707i	$0.800424\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	3.06406	0.389135
$63$	0	0
$64$	1.00000	0.125000
$65$	1.45544	0.180525
$66$	0	0
$67$	−1.71871	−0.209974	−0.104987	−	0.994474i	$-0.533480\pi$
$67$	−1.71871	−0.209974	−0.104987	+	0.994474i	$0.533480\pi$
$68$	−5.43742	−0.659384
$69$	0	0
$70$	0.672673	0.0803998
$71$	−1.04604	−0.124142	−0.0620709	−	0.998072i	$-0.519770\pi$
$71$	−1.04604	−0.124142	−0.0620709	+	0.998072i	$0.519770\pi$
$72$	0	0
$73$	13.6576	1.59850	0.799251	−	0.600998i	$-0.205230\pi$
$73$	13.6576	1.59850	0.799251	+	0.600998i	$0.205230\pi$
$74$	4.65465	0.541093
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−1.71871	−0.195865
$78$	0	0
$79$	13.0029	1.46295	0.731473	−	0.681870i	$-0.238833\pi$
$79$	13.0029	1.46295	0.731473	+	0.681870i	$0.238833\pi$
$80$	0.391382	0.0437579
$81$	0	0
$82$	1.06406	0.117505
$83$	−8.51949	−0.935136	−0.467568	−	0.883957i	$-0.654870\pi$
$83$	−8.51949	−0.935136	−0.467568	+	0.883957i	$0.654870\pi$
$84$	0	0
$85$	−2.12811	−0.230826
$86$	3.73673	0.402942
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−17.6936	−1.87552	−0.937761	−	0.347281i	$-0.887105\pi$
$89$	−17.6936	−1.87552	−0.937761	+	0.347281i	$0.887105\pi$
$90$	0	0
$91$	−6.39138	−0.669999
$92$	3.43742	0.358376
$93$	0	0
$94$	−8.22018	−0.847847
$95$	−0.391382	−0.0401550
$96$	0	0
$97$	−4.65465	−0.472609	−0.236304	−	0.971679i	$-0.575936\pi$
$97$	−4.65465	−0.472609	−0.236304	+	0.971679i	$0.575936\pi$
$98$	4.04604	0.408711
$99$	0	0
$100$	−4.84682	−0.484682
$101$	9.65760	0.960967	0.480484	−	0.877004i	$-0.340461\pi$
$101$	9.65760	0.960967	0.480484	+	0.877004i	$0.340461\pi$
$102$	0	0
$103$	−6.95396	−0.685194	−0.342597	−	0.939482i	$-0.611307\pi$
$103$	−6.95396	−0.685194	−0.342597	+	0.939482i	$0.611307\pi$
$104$	−3.71871	−0.364649
$105$	0	0
$106$	−1.21724	−0.118228
$107$	−2.65465	−0.256635	−0.128318	−	0.991733i	$-0.540958\pi$
$107$	−2.65465	−0.256635	−0.128318	+	0.991733i	$0.540958\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−0.391382	−0.0373168
$111$	0	0
$112$	−1.71871	−0.162403
$113$	−7.52949	−0.708315	−0.354158	−	0.935186i	$-0.615232\pi$
$113$	−7.52949	−0.708315	−0.354158	+	0.935186i	$0.615232\pi$
$114$	0	0
$115$	1.34535	0.125454
$116$	3.82880	0.355495
$117$	0	0
$118$	12.4404	1.14523
$119$	9.34535	0.856686
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.00000	−0.181071
$123$	0	0
$124$	−3.06406	−0.275160
$125$	−3.85387	−0.344701
$126$	0	0
$127$	−5.00295	−0.443940	−0.221970	−	0.975054i	$-0.571249\pi$
$127$	−5.00295	−0.443940	−0.221970	+	0.975054i	$0.571249\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.45544	−0.127650
$131$	6.50147	0.568036	0.284018	−	0.958819i	$-0.408332\pi$
$131$	6.50147	0.568036	0.284018	+	0.958819i	$0.408332\pi$
$132$	0	0
$133$	1.71871	0.149031
$134$	1.71871	0.148474
$135$	0	0
$136$	5.43742	0.466255
$137$	−15.7187	−1.34294	−0.671470	−	0.741032i	$-0.734337\pi$
$137$	−15.7187	−1.34294	−0.671470	+	0.741032i	$0.734337\pi$
$138$	0	0
$139$	−11.1381	−0.944722	−0.472361	−	0.881405i	$-0.656598\pi$
$139$	−11.1381	−0.944722	−0.472361	+	0.881405i	$0.656598\pi$
$140$	−0.672673	−0.0568512
$141$	0	0
$142$	1.04604	0.0877815
$143$	3.71871	0.310974
$144$	0	0
$145$	1.49853	0.124446
$146$	−13.6576	−1.13031
$147$	0	0
$148$	−4.65465	−0.382610
$149$	−11.3453	−0.929447	−0.464723	−	0.885456i	$-0.653846\pi$
$149$	−11.3453	−0.929447	−0.464723	+	0.885456i	$0.653846\pi$
$150$	0	0
$151$	−14.6547	−1.19258	−0.596289	−	0.802770i	$-0.703359\pi$
$151$	−14.6547	−1.19258	−0.596289	+	0.802770i	$0.703359\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	1.71871	0.138498
$155$	−1.19922	−0.0963234
$156$	0	0
$157$	−10.4835	−0.836671	−0.418335	−	0.908293i	$-0.637386\pi$
$157$	−10.4835	−0.836671	−0.418335	+	0.908293i	$0.637386\pi$
$158$	−13.0029	−1.03446
$159$	0	0
$160$	−0.391382	−0.0309415
$161$	−5.90793	−0.465610
$162$	0	0
$163$	−3.47346	−0.272062	−0.136031	−	0.990705i	$-0.543435\pi$
$163$	−3.47346	−0.272062	−0.136031	+	0.990705i	$0.543435\pi$
$164$	−1.06406	−0.0830887
$165$	0	0
$166$	8.51949	0.661241
$167$	−23.0950	−1.78715	−0.893573	−	0.448917i	$-0.851810\pi$
$167$	−23.0950	−1.78715	−0.893573	+	0.448917i	$0.851810\pi$
$168$	0	0
$169$	0.828802	0.0637540
$170$	2.12811	0.163219
$171$	0	0
$172$	−3.73673	−0.284923
$173$	18.1771	1.38198	0.690990	−	0.722865i	$-0.257175\pi$
$173$	18.1771	1.38198	0.690990	+	0.722865i	$0.257175\pi$
$174$	0	0
$175$	8.33028	0.629710
$176$	1.00000	0.0753778
$177$	0	0
$178$	17.6936	1.32619
$179$	−11.2842	−0.843424	−0.421712	−	0.906730i	$-0.638571\pi$
$179$	−11.2842	−0.843424	−0.421712	+	0.906730i	$0.638571\pi$
$180$	0	0
$181$	21.8778	1.62616	0.813082	−	0.582150i	$-0.197788\pi$
$181$	21.8778	1.62616	0.813082	+	0.582150i	$0.197788\pi$
$182$	6.39138	0.473761
$183$	0	0
$184$	−3.43742	−0.253410
$185$	−1.82175	−0.133938
$186$	0	0
$187$	−5.43742	−0.397623
$188$	8.22018	0.599519
$189$	0	0
$190$	0.391382	0.0283939
$191$	7.09502	0.513378	0.256689	−	0.966494i	$-0.417368\pi$
$191$	7.09502	0.513378	0.256689	+	0.966494i	$0.417368\pi$
$192$	0	0
$193$	12.6476	0.910394	0.455197	−	0.890391i	$-0.349569\pi$
$193$	12.6476	0.910394	0.455197	+	0.890391i	$0.349569\pi$
$194$	4.65465	0.334185
$195$	0	0
$196$	−4.04604	−0.289003
$197$	−8.43447	−0.600931	−0.300466	−	0.953793i	$-0.597142\pi$
$197$	−8.43447	−0.600931	−0.300466	+	0.953793i	$0.597142\pi$
$198$	0	0
$199$	−2.09207	−0.148303	−0.0741516	−	0.997247i	$-0.523625\pi$
$199$	−2.09207	−0.148303	−0.0741516	+	0.997247i	$0.523625\pi$
$200$	4.84682	0.342722
$201$	0	0
$202$	−9.65760	−0.679507
$203$	−6.58060	−0.461867
$204$	0	0
$205$	−0.416452	−0.0290863
$206$	6.95396	0.484506
$207$	0	0
$208$	3.71871	0.257846
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−6.09207	−0.419396	−0.209698	−	0.977766i	$-0.567248\pi$
$211$	−6.09207	−0.419396	−0.209698	+	0.977766i	$0.567248\pi$
$212$	1.21724	0.0836001
$213$	0	0
$214$	2.65465	0.181468
$215$	−1.46249	−0.0997409
$216$	0	0
$217$	5.26622	0.357494
$218$	−6.00000	−0.406371
$219$	0	0
$220$	0.391382	0.0263870
$221$	−20.2202	−1.36016
$222$	0	0
$223$	−18.8748	−1.26395	−0.631976	−	0.774988i	$-0.717756\pi$
$223$	−18.8748	−1.26395	−0.631976	+	0.774988i	$0.717756\pi$
$224$	1.71871	0.114836
$225$	0	0
$226$	7.52949	0.500854
$227$	−20.9669	−1.39162	−0.695811	−	0.718225i	$-0.744955\pi$
$227$	−20.9669	−1.39162	−0.695811	+	0.718225i	$0.744955\pi$
$228$	0	0
$229$	13.8108	0.912642	0.456321	−	0.889815i	$-0.349167\pi$
$229$	13.8108	0.912642	0.456321	+	0.889815i	$0.349167\pi$
$230$	−1.34535	−0.0887094
$231$	0	0
$232$	−3.82880	−0.251373
$233$	−4.91087	−0.321722	−0.160861	−	0.986977i	$-0.551427\pi$
$233$	−4.91087	−0.321722	−0.160861	+	0.986977i	$0.551427\pi$
$234$	0	0
$235$	3.21724	0.209869
$236$	−12.4404	−0.809799
$237$	0	0
$238$	−9.34535	−0.605769
$239$	14.0490	0.908753	0.454377	−	0.890810i	$-0.349862\pi$
$239$	14.0490	0.908753	0.454377	+	0.890810i	$0.349862\pi$
$240$	0	0
$241$	22.7397	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$241$	22.7397	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	2.00000	0.128037
$245$	−1.58355	−0.101169
$246$	0	0
$247$	−3.71871	−0.236616
$248$	3.06406	0.194568
$249$	0	0
$250$	3.85387	0.243740
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	3.43742	0.216109
$254$	5.00295	0.313913
$255$	0	0
$256$	1.00000	0.0625000
$257$	19.4433	1.21284	0.606420	−	0.795144i	$-0.292605\pi$
$257$	19.4433	1.21284	0.606420	+	0.795144i	$0.292605\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	1.45544	0.0902624
$261$	0	0
$262$	−6.50147	−0.401662
$263$	−14.6476	−0.903210	−0.451605	−	0.892218i	$-0.649148\pi$
$263$	−14.6476	−0.903210	−0.451605	+	0.892218i	$0.649148\pi$
$264$	0	0
$265$	0.476404	0.0292653
$266$	−1.71871	−0.105381
$267$	0	0
$268$	−1.71871	−0.104987
$269$	−8.34829	−0.509004	−0.254502	−	0.967072i	$-0.581912\pi$
$269$	−8.34829	−0.509004	−0.254502	+	0.967072i	$0.581912\pi$
$270$	0	0
$271$	−10.5375	−0.640108	−0.320054	−	0.947399i	$-0.603701\pi$
$271$	−10.5375	−0.640108	−0.320054	+	0.947399i	$0.603701\pi$
$272$	−5.43742	−0.329692
$273$	0	0
$274$	15.7187	0.949602
$275$	−4.84682	−0.292274
$276$	0	0
$277$	−22.8807	−1.37477	−0.687385	−	0.726293i	$-0.741242\pi$
$277$	−22.8807	−1.37477	−0.687385	+	0.726293i	$0.741242\pi$
$278$	11.1381	0.668020
$279$	0	0
$280$	0.672673	0.0401999
$281$	4.39138	0.261968	0.130984	−	0.991384i	$-0.458186\pi$
$281$	4.39138	0.261968	0.130984	+	0.991384i	$0.458186\pi$
$282$	0	0
$283$	−24.0670	−1.43063	−0.715317	−	0.698800i	$-0.753718\pi$
$283$	−24.0670	−1.43063	−0.715317	+	0.698800i	$0.753718\pi$
$284$	−1.04604	−0.0620709
$285$	0	0
$286$	−3.71871	−0.219892
$287$	1.82880	0.107951
$288$	0	0
$289$	12.5655	0.739149
$290$	−1.49853	−0.0879965
$291$	0	0
$292$	13.6576	0.799251
$293$	27.2901	1.59431	0.797153	−	0.603777i	$-0.206338\pi$
$293$	27.2901	1.59431	0.797153	+	0.603777i	$0.206338\pi$
$294$	0	0
$295$	−4.86894	−0.283481
$296$	4.65465	0.270546
$297$	0	0
$298$	11.3453	0.657218
$299$	12.7828	0.739246
$300$	0	0
$301$	6.42235	0.370178
$302$	14.6547	0.843281
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0.782765	0.0448210
$306$	0	0
$307$	−9.22313	−0.526392	−0.263196	−	0.964742i	$-0.584777\pi$
$307$	−9.22313	−0.526392	−0.263196	+	0.964742i	$0.584777\pi$
$308$	−1.71871	−0.0979326
$309$	0	0
$310$	1.19922	0.0681110
$311$	−6.34829	−0.359979	−0.179989	−	0.983669i	$-0.557606\pi$
$311$	−6.34829	−0.359979	−0.179989	+	0.983669i	$0.557606\pi$
$312$	0	0
$313$	13.3943	0.757092	0.378546	−	0.925582i	$-0.376424\pi$
$313$	13.3943	0.757092	0.378546	+	0.925582i	$0.376424\pi$
$314$	10.4835	0.591616
$315$	0	0
$316$	13.0029	0.731473
$317$	−31.0029	−1.74130	−0.870650	−	0.491904i	$-0.836301\pi$
$317$	−31.0029	−1.74130	−0.870650	+	0.491904i	$0.836301\pi$
$318$	0	0
$319$	3.82880	0.214372
$320$	0.391382	0.0218789
$321$	0	0
$322$	5.90793	0.329236
$323$	5.43742	0.302546
$324$	0	0
$325$	−18.0239	−0.999787
$326$	3.47346	0.192377
$327$	0	0
$328$	1.06406	0.0587526
$329$	−14.1281	−0.778908
$330$	0	0
$331$	−16.2993	−0.895891	−0.447946	−	0.894061i	$-0.647844\pi$
$331$	−16.2993	−0.895891	−0.447946	+	0.894061i	$0.647844\pi$
$332$	−8.51949	−0.467568
$333$	0	0
$334$	23.0950	1.26370
$335$	−0.672673	−0.0367520
$336$	0	0
$337$	−3.19217	−0.173888	−0.0869442	−	0.996213i	$-0.527710\pi$
$337$	−3.19217	−0.173888	−0.0869442	+	0.996213i	$0.527710\pi$
$338$	−0.828802	−0.0450809
$339$	0	0
$340$	−2.12811	−0.115413
$341$	−3.06406	−0.165928
$342$	0	0
$343$	18.9849	1.02509
$344$	3.73673	0.201471
$345$	0	0
$346$	−18.1771	−0.977207
$347$	−24.6966	−1.32578	−0.662891	−	0.748716i	$-0.730671\pi$
$347$	−24.6966	−1.32578	−0.662891	+	0.748716i	$0.730671\pi$
$348$	0	0
$349$	−4.43447	−0.237372	−0.118686	−	0.992932i	$-0.537868\pi$
$349$	−4.43447	−0.237372	−0.118686	+	0.992932i	$0.537868\pi$
$350$	−8.33028	−0.445272
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−25.3152	−1.34739	−0.673696	−	0.739008i	$-0.735294\pi$
$353$	−25.3152	−1.34739	−0.673696	+	0.739008i	$0.735294\pi$
$354$	0	0
$355$	−0.409400	−0.0217287
$356$	−17.6936	−0.937761
$357$	0	0
$358$	11.2842	0.596391
$359$	13.9749	0.737569	0.368784	−	0.929515i	$-0.379774\pi$
$359$	13.9749	0.737569	0.368784	+	0.929515i	$0.379774\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−21.8778	−1.14987
$363$	0	0
$364$	−6.39138	−0.334999
$365$	5.34535	0.279788
$366$	0	0
$367$	−25.3453	−1.32302	−0.661508	−	0.749938i	$-0.730083\pi$
$367$	−25.3453	−1.32302	−0.661508	+	0.749938i	$0.730083\pi$
$368$	3.43742	0.179188
$369$	0	0
$370$	1.82175	0.0947083
$371$	−2.09207	−0.108615
$372$	0	0
$373$	8.95396	0.463619	0.231809	−	0.972761i	$-0.425535\pi$
$373$	8.95396	0.463619	0.231809	+	0.972761i	$0.425535\pi$
$374$	5.43742	0.281162
$375$	0	0
$376$	−8.22018	−0.423924
$377$	14.2382	0.733305
$378$	0	0
$379$	−25.8527	−1.32796	−0.663982	−	0.747748i	$-0.731135\pi$
$379$	−25.8527	−1.32796	−0.663982	+	0.747748i	$0.731135\pi$
$380$	−0.391382	−0.0200775
$381$	0	0
$382$	−7.09502	−0.363013
$383$	20.6296	1.05412	0.527061	−	0.849827i	$-0.323294\pi$
$383$	20.6296	1.05412	0.527061	+	0.849827i	$0.323294\pi$
$384$	0	0
$385$	−0.672673	−0.0342826
$386$	−12.6476	−0.643746
$387$	0	0
$388$	−4.65465	−0.236304
$389$	−15.6086	−0.791388	−0.395694	−	0.918382i	$-0.629496\pi$
$389$	−15.6086	−0.791388	−0.395694	+	0.918382i	$0.629496\pi$
$390$	0	0
$391$	−18.6907	−0.945229
$392$	4.04604	0.204356
$393$	0	0
$394$	8.43447	0.424922
$395$	5.08913	0.256062
$396$	0	0
$397$	−34.2512	−1.71902	−0.859508	−	0.511122i	$-0.829230\pi$
$397$	−34.2512	−1.71902	−0.859508	+	0.511122i	$0.829230\pi$
$398$	2.09207	0.104866
$399$	0	0
$400$	−4.84682	−0.242341
$401$	−4.09207	−0.204348	−0.102174	−	0.994767i	$-0.532580\pi$
$401$	−4.09207	−0.204348	−0.102174	+	0.994767i	$0.532580\pi$
$402$	0	0
$403$	−11.3943	−0.567592
$404$	9.65760	0.480484
$405$	0	0
$406$	6.58060	0.326590
$407$	−4.65465	−0.230723
$408$	0	0
$409$	2.48346	0.122799	0.0613995	−	0.998113i	$-0.480444\pi$
$409$	2.48346	0.122799	0.0613995	+	0.998113i	$0.480444\pi$
$410$	0.416452	0.0205671
$411$	0	0
$412$	−6.95396	−0.342597
$413$	21.3814	1.05211
$414$	0	0
$415$	−3.33438	−0.163678
$416$	−3.71871	−0.182325
$417$	0	0
$418$	1.00000	0.0489116
$419$	−35.0950	−1.71450	−0.857252	−	0.514897i	$-0.827830\pi$
$419$	−35.0950	−1.71450	−0.857252	+	0.514897i	$0.827830\pi$
$420$	0	0
$421$	−12.3483	−0.601819	−0.300910	−	0.953653i	$-0.597290\pi$
$421$	−12.3483	−0.601819	−0.300910	+	0.953653i	$0.597290\pi$
$422$	6.09207	0.296558
$423$	0	0
$424$	−1.21724	−0.0591142
$425$	26.3542	1.27837
$426$	0	0
$427$	−3.43742	−0.166348
$428$	−2.65465	−0.128318
$429$	0	0
$430$	1.46249	0.0705275
$431$	6.94691	0.334621	0.167310	−	0.985904i	$-0.446492\pi$
$431$	6.94691	0.334621	0.167310	+	0.985904i	$0.446492\pi$
$432$	0	0
$433$	−13.2533	−0.636912	−0.318456	−	0.947938i	$-0.603164\pi$
$433$	−13.2533	−0.636912	−0.318456	+	0.947938i	$0.603164\pi$
$434$	−5.26622	−0.252787
$435$	0	0
$436$	6.00000	0.287348
$437$	−3.43742	−0.164434
$438$	0	0
$439$	15.3152	0.730955	0.365477	−	0.930820i	$-0.380906\pi$
$439$	15.3152	0.730955	0.365477	+	0.930820i	$0.380906\pi$
$440$	−0.391382	−0.0186584
$441$	0	0
$442$	20.2202	0.961776
$443$	20.8807	0.992074	0.496037	−	0.868301i	$-0.334788\pi$
$443$	20.8807	0.992074	0.496037	+	0.868301i	$0.334788\pi$
$444$	0	0
$445$	−6.92498	−0.328275
$446$	18.8748	0.893750
$447$	0	0
$448$	−1.71871	−0.0812014
$449$	28.7887	1.35862	0.679310	−	0.733851i	$-0.262279\pi$
$449$	28.7887	1.35862	0.679310	+	0.733851i	$0.262279\pi$
$450$	0	0
$451$	−1.06406	−0.0501044
$452$	−7.52949	−0.354158
$453$	0	0
$454$	20.9669	0.984026
$455$	−2.50147	−0.117271
$456$	0	0
$457$	−32.7526	−1.53210	−0.766052	−	0.642779i	$-0.777781\pi$
$457$	−32.7526	−1.53210	−0.766052	+	0.642779i	$0.777781\pi$
$458$	−13.8108	−0.645336
$459$	0	0
$460$	1.34535	0.0627271
$461$	−20.2261	−0.942023	−0.471011	−	0.882127i	$-0.656111\pi$
$461$	−20.2261	−0.942023	−0.471011	+	0.882127i	$0.656111\pi$
$462$	0	0
$463$	−18.6907	−0.868630	−0.434315	−	0.900761i	$-0.643010\pi$
$463$	−18.6907	−0.868630	−0.434315	+	0.900761i	$0.643010\pi$
$464$	3.82880	0.177748
$465$	0	0
$466$	4.91087	0.227492
$467$	−34.5685	−1.59964	−0.799819	−	0.600241i	$-0.795071\pi$
$467$	−34.5685	−1.59964	−0.799819	+	0.600241i	$0.795071\pi$
$468$	0	0
$469$	2.95396	0.136401
$470$	−3.21724	−0.148400
$471$	0	0
$472$	12.4404	0.572614
$473$	−3.73673	−0.171815
$474$	0	0
$475$	4.84682	0.222387
$476$	9.34535	0.428343
$477$	0	0
$478$	−14.0490	−0.642586
$479$	17.7187	0.809589	0.404794	−	0.914408i	$-0.367343\pi$
$479$	17.7187	0.809589	0.404794	+	0.914408i	$0.367343\pi$
$480$	0	0
$481$	−17.3093	−0.789237
$482$	−22.7397	−1.03576
$483$	0	0
$484$	1.00000	0.0454545
$485$	−1.82175	−0.0827214
$486$	0	0
$487$	37.7426	1.71028	0.855141	−	0.518396i	$-0.173471\pi$
$487$	37.7426	1.71028	0.855141	+	0.518396i	$0.173471\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	1.58355	0.0715374
$491$	−0.373364	−0.0168497	−0.00842485	−	0.999965i	$-0.502682\pi$
$491$	−0.373364	−0.0168497	−0.00842485	+	0.999965i	$0.502682\pi$
$492$	0	0
$493$	−20.8188	−0.937632
$494$	3.71871	0.167313
$495$	0	0
$496$	−3.06406	−0.137580
$497$	1.79783	0.0806439
$498$	0	0
$499$	−14.8388	−0.664276	−0.332138	−	0.943231i	$-0.607770\pi$
$499$	−14.8388	−0.664276	−0.332138	+	0.943231i	$0.607770\pi$
$500$	−3.85387	−0.172350
$501$	0	0
$502$	−12.0000	−0.535586
$503$	−33.2662	−1.48327	−0.741634	−	0.670805i	$-0.765949\pi$
$503$	−33.2662	−1.48327	−0.741634	+	0.670805i	$0.765949\pi$
$504$	0	0
$505$	3.77982	0.168200
$506$	−3.43742	−0.152812
$507$	0	0
$508$	−5.00295	−0.221970
$509$	8.47051	0.375449	0.187724	−	0.982222i	$-0.439889\pi$
$509$	8.47051	0.375449	0.187724	+	0.982222i	$0.439889\pi$
$510$	0	0
$511$	−23.4735	−1.03840
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−19.4433	−0.857608
$515$	−2.72166	−0.119931
$516$	0	0
$517$	8.22018	0.361523
$518$	−8.00000	−0.351500
$519$	0	0
$520$	−1.45544	−0.0638252
$521$	37.3152	1.63481	0.817404	−	0.576064i	$-0.195412\pi$
$521$	37.3152	1.63481	0.817404	+	0.576064i	$0.195412\pi$
$522$	0	0
$523$	−31.6576	−1.38429	−0.692145	−	0.721758i	$-0.743334\pi$
$523$	−31.6576	−1.38429	−0.692145	+	0.721758i	$0.743334\pi$
$524$	6.50147	0.284018
$525$	0	0
$526$	14.6476	0.638666
$527$	16.6606	0.725745
$528$	0	0
$529$	−11.1841	−0.486267
$530$	−0.476404	−0.0206937
$531$	0	0
$532$	1.71871	0.0745155
$533$	−3.95691	−0.171393
$534$	0	0
$535$	−1.03899	−0.0449192
$536$	1.71871	0.0742370
$537$	0	0
$538$	8.34829	0.359921
$539$	−4.04604	−0.174275
$540$	0	0
$541$	28.0921	1.20777	0.603886	−	0.797070i	$-0.293618\pi$
$541$	28.0921	1.20777	0.603886	+	0.797070i	$0.293618\pi$
$542$	10.5375	0.452625
$543$	0	0
$544$	5.43742	0.233127
$545$	2.34829	0.100590
$546$	0	0
$547$	24.1841	1.03404	0.517020	−	0.855973i	$-0.327041\pi$
$547$	24.1841	1.03404	0.517020	+	0.855973i	$0.327041\pi$
$548$	−15.7187	−0.671470
$549$	0	0
$550$	4.84682	0.206669
$551$	−3.82880	−0.163112
$552$	0	0
$553$	−22.3483	−0.950346
$554$	22.8807	0.972109
$555$	0	0
$556$	−11.1381	−0.472361
$557$	−39.9699	−1.69358	−0.846789	−	0.531929i	$-0.821467\pi$
$557$	−39.9699	−1.69358	−0.846789	+	0.531929i	$0.821467\pi$
$558$	0	0
$559$	−13.8958	−0.587730
$560$	−0.672673	−0.0284256
$561$	0	0
$562$	−4.39138	−0.185239
$563$	−16.4764	−0.694398	−0.347199	−	0.937792i	$-0.612867\pi$
$563$	−16.4764	−0.694398	−0.347199	+	0.937792i	$0.612867\pi$
$564$	0	0
$565$	−2.94691	−0.123977
$566$	24.0670	1.01161
$567$	0	0
$568$	1.04604	0.0438907
$569$	24.9418	1.04562	0.522808	−	0.852450i	$-0.324884\pi$
$569$	24.9418	1.04562	0.522808	+	0.852450i	$0.324884\pi$
$570$	0	0
$571$	−8.50737	−0.356022	−0.178011	−	0.984028i	$-0.556966\pi$
$571$	−8.50737	−0.356022	−0.178011	+	0.984028i	$0.556966\pi$
$572$	3.71871	0.155487
$573$	0	0
$574$	−1.82880	−0.0763327
$575$	−16.6606	−0.694793
$576$	0	0
$577$	31.8527	1.32605	0.663023	−	0.748599i	$-0.269273\pi$
$577$	31.8527	1.32605	0.663023	+	0.748599i	$0.269273\pi$
$578$	−12.5655	−0.522657
$579$	0	0
$580$	1.49853	0.0622229
$581$	14.6425	0.607475
$582$	0	0
$583$	1.21724	0.0504127
$584$	−13.6576	−0.565156
$585$	0	0
$586$	−27.2901	−1.12735
$587$	−2.43447	−0.100481	−0.0502407	−	0.998737i	$-0.515999\pi$
$587$	−2.43447	−0.100481	−0.0502407	+	0.998737i	$0.515999\pi$
$588$	0	0
$589$	3.06406	0.126252
$590$	4.86894	0.200451
$591$	0	0
$592$	−4.65465	−0.191305
$593$	18.1841	0.746733	0.373367	−	0.927684i	$-0.378203\pi$
$593$	18.1841	0.746733	0.373367	+	0.927684i	$0.378203\pi$
$594$	0	0
$595$	3.65760	0.149947
$596$	−11.3453	−0.464723
$597$	0	0
$598$	−12.7828	−0.522726
$599$	−4.33534	−0.177137	−0.0885687	−	0.996070i	$-0.528229\pi$
$599$	−4.33534	−0.177137	−0.0885687	+	0.996070i	$0.528229\pi$
$600$	0	0
$601$	16.2453	0.662658	0.331329	−	0.943515i	$-0.392503\pi$
$601$	16.2453	0.662658	0.331329	+	0.943515i	$0.392503\pi$
$602$	−6.42235	−0.261755
$603$	0	0
$604$	−14.6547	−0.596289
$605$	0.391382	0.0159120
$606$	0	0
$607$	41.6995	1.69253	0.846266	−	0.532761i	$-0.178845\pi$
$607$	41.6995	1.69253	0.846266	+	0.532761i	$0.178845\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−0.782765	−0.0316932
$611$	30.5685	1.23667
$612$	0	0
$613$	48.1900	1.94638	0.973189	−	0.230008i	$-0.0738751\pi$
$613$	48.1900	1.94638	0.973189	+	0.230008i	$0.0738751\pi$
$614$	9.22313	0.372215
$615$	0	0
$616$	1.71871	0.0692488
$617$	−15.7187	−0.632811	−0.316406	−	0.948624i	$-0.602476\pi$
$617$	−15.7187	−0.632811	−0.316406	+	0.948624i	$0.602476\pi$
$618$	0	0
$619$	44.6606	1.79506	0.897530	−	0.440954i	$-0.145360\pi$
$619$	44.6606	1.79506	0.897530	+	0.440954i	$0.145360\pi$
$620$	−1.19922	−0.0481617
$621$	0	0
$622$	6.34829	0.254543
$623$	30.4102	1.21836
$624$	0	0
$625$	22.7258	0.909030
$626$	−13.3943	−0.535345
$627$	0	0
$628$	−10.4835	−0.418335
$629$	25.3093	1.00915
$630$	0	0
$631$	2.38433	0.0949187	0.0474593	−	0.998873i	$-0.484888\pi$
$631$	2.38433	0.0949187	0.0474593	+	0.998873i	$0.484888\pi$
$632$	−13.0029	−0.517230
$633$	0	0
$634$	31.0029	1.23128
$635$	−1.95807	−0.0777035
$636$	0	0
$637$	−15.0460	−0.596146
$638$	−3.82880	−0.151584
$639$	0	0
$640$	−0.391382	−0.0154707
$641$	45.6936	1.80479	0.902395	−	0.430910i	$-0.141807\pi$
$641$	45.6936	1.80479	0.902395	+	0.430910i	$0.141807\pi$
$642$	0	0
$643$	42.2621	1.66666	0.833328	−	0.552779i	$-0.186433\pi$
$643$	42.2621	1.66666	0.833328	+	0.552779i	$0.186433\pi$
$644$	−5.90793	−0.232805
$645$	0	0
$646$	−5.43742	−0.213932
$647$	38.7166	1.52211	0.761053	−	0.648690i	$-0.224683\pi$
$647$	38.7166	1.52211	0.761053	+	0.648690i	$0.224683\pi$
$648$	0	0
$649$	−12.4404	−0.488327
$650$	18.0239	0.706956
$651$	0	0
$652$	−3.47346	−0.136031
$653$	−38.6915	−1.51412	−0.757058	−	0.653348i	$-0.773364\pi$
$653$	−38.6915	−1.51412	−0.757058	+	0.653348i	$0.773364\pi$
$654$	0	0
$655$	2.54456	0.0994243
$656$	−1.06406	−0.0415444
$657$	0	0
$658$	14.1281	0.550771
$659$	−12.7467	−0.496542	−0.248271	−	0.968691i	$-0.579862\pi$
$659$	−12.7467	−0.496542	−0.248271	+	0.968691i	$0.579862\pi$
$660$	0	0
$661$	−21.1671	−0.823305	−0.411652	−	0.911341i	$-0.635048\pi$
$661$	−21.1671	−0.823305	−0.411652	+	0.911341i	$0.635048\pi$
$662$	16.2993	0.633491
$663$	0	0
$664$	8.51949	0.330620
$665$	0.672673	0.0260851
$666$	0	0
$667$	13.1612	0.509604
$668$	−23.0950	−0.893573
$669$	0	0
$670$	0.672673	0.0259876
$671$	2.00000	0.0772091
$672$	0	0
$673$	2.25115	0.0867755	0.0433878	−	0.999058i	$-0.486185\pi$
$673$	2.25115	0.0867755	0.0433878	+	0.999058i	$0.486185\pi$
$674$	3.19217	0.122958
$675$	0	0
$676$	0.828802	0.0318770
$677$	9.57848	0.368131	0.184065	−	0.982914i	$-0.441074\pi$
$677$	9.57848	0.368131	0.184065	+	0.982914i	$0.441074\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	2.12811	0.0816093
$681$	0	0
$682$	3.06406	0.117329
$683$	32.0118	1.22490	0.612449	−	0.790510i	$-0.290185\pi$
$683$	32.0118	1.22490	0.612449	+	0.790510i	$0.290185\pi$
$684$	0	0
$685$	−6.15203	−0.235057
$686$	−18.9849	−0.724848
$687$	0	0
$688$	−3.73673	−0.142461
$689$	4.52654	0.172448
$690$	0	0
$691$	−25.9699	−0.987940	−0.493970	−	0.869479i	$-0.664455\pi$
$691$	−25.9699	−0.987940	−0.493970	+	0.869479i	$0.664455\pi$
$692$	18.1771	0.690990
$693$	0	0
$694$	24.6966	0.937470
$695$	−4.35926	−0.165356
$696$	0	0
$697$	5.78571	0.219150
$698$	4.43447	0.167847
$699$	0	0
$700$	8.33028	0.314855
$701$	21.0950	0.796748	0.398374	−	0.917223i	$-0.369575\pi$
$701$	21.0950	0.796748	0.398374	+	0.917223i	$0.369575\pi$
$702$	0	0
$703$	4.65465	0.175554
$704$	1.00000	0.0376889
$705$	0	0
$706$	25.3152	0.952750
$707$	−16.5986	−0.624255
$708$	0	0
$709$	29.1981	1.09656	0.548278	−	0.836296i	$-0.315284\pi$
$709$	29.1981	1.09656	0.548278	+	0.836296i	$0.315284\pi$
$710$	0.409400	0.0153645
$711$	0	0
$712$	17.6936	0.663097
$713$	−10.5324	−0.394443
$714$	0	0
$715$	1.45544	0.0544303
$716$	−11.2842	−0.421712
$717$	0	0
$718$	−13.9749	−0.521540
$719$	13.9079	0.518678	0.259339	−	0.965786i	$-0.416495\pi$
$719$	13.9079	0.518678	0.259339	+	0.965786i	$0.416495\pi$
$720$	0	0
$721$	11.9518	0.445110
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	21.8778	0.813082
$725$	−18.5575	−0.689209
$726$	0	0
$727$	−42.0059	−1.55791	−0.778956	−	0.627078i	$-0.784251\pi$
$727$	−42.0059	−1.55791	−0.778956	+	0.627078i	$0.784251\pi$
$728$	6.39138	0.236880
$729$	0	0
$730$	−5.34535	−0.197840
$731$	20.3182	0.751494
$732$	0	0
$733$	36.0780	1.33257	0.666285	−	0.745697i	$-0.267883\pi$
$733$	36.0780	1.33257	0.666285	+	0.745697i	$0.267883\pi$
$734$	25.3453	0.935514
$735$	0	0
$736$	−3.43742	−0.126705
$737$	−1.71871	−0.0633095
$738$	0	0
$739$	49.9988	1.83924	0.919619	−	0.392812i	$-0.128498\pi$
$739$	49.9988	1.83924	0.919619	+	0.392812i	$0.128498\pi$
$740$	−1.82175	−0.0669689
$741$	0	0
$742$	2.09207	0.0768025
$743$	−0.220184	−0.00807777	−0.00403889	−	0.999992i	$-0.501286\pi$
$743$	−0.220184	−0.00807777	−0.00403889	+	0.999992i	$0.501286\pi$
$744$	0	0
$745$	−4.44037	−0.162683
$746$	−8.95396	−0.327828
$747$	0	0
$748$	−5.43742	−0.198812
$749$	4.56258	0.166713
$750$	0	0
$751$	−23.1311	−0.844064	−0.422032	−	0.906581i	$-0.638683\pi$
$751$	−23.1311	−0.844064	−0.422032	+	0.906581i	$0.638683\pi$
$752$	8.22018	0.299759
$753$	0	0
$754$	−14.2382	−0.518525
$755$	−5.73557	−0.208739
$756$	0	0
$757$	37.4304	1.36043	0.680215	−	0.733013i	$-0.261886\pi$
$757$	37.4304	1.36043	0.680215	+	0.733013i	$0.261886\pi$
$758$	25.8527	0.939013
$759$	0	0
$760$	0.391382	0.0141969
$761$	33.9499	1.23068	0.615341	−	0.788261i	$-0.289018\pi$
$761$	33.9499	1.23068	0.615341	+	0.788261i	$0.289018\pi$
$762$	0	0
$763$	−10.3123	−0.373329
$764$	7.09502	0.256689
$765$	0	0
$766$	−20.6296	−0.745377
$767$	−46.2621	−1.67043
$768$	0	0
$769$	−44.7025	−1.61201	−0.806006	−	0.591907i	$-0.798375\pi$
$769$	−44.7025	−1.61201	−0.806006	+	0.591907i	$0.798375\pi$
$770$	0.672673	0.0242414
$771$	0	0
$772$	12.6476	0.455197
$773$	44.8748	1.61404	0.807018	−	0.590527i	$-0.201080\pi$
$773$	44.8748	1.61404	0.807018	+	0.590527i	$0.201080\pi$
$774$	0	0
$775$	14.8509	0.533461
$776$	4.65465	0.167092
$777$	0	0
$778$	15.6086	0.559596
$779$	1.06406	0.0381237
$780$	0	0
$781$	−1.04604	−0.0374301
$782$	18.6907	0.668378
$783$	0	0
$784$	−4.04604	−0.144501
$785$	−4.10304	−0.146444
$786$	0	0
$787$	11.8418	0.422113	0.211056	−	0.977474i	$-0.432310\pi$
$787$	11.8418	0.422113	0.211056	+	0.977474i	$0.432310\pi$
$788$	−8.43447	−0.300466
$789$	0	0
$790$	−5.08913	−0.181063
$791$	12.9410	0.460129
$792$	0	0
$793$	7.43742	0.264111
$794$	34.2512	1.21553
$795$	0	0
$796$	−2.09207	−0.0741516
$797$	−0.128110	−0.00453789	−0.00226895	−	0.999997i	$-0.500722\pi$
$797$	−0.128110	−0.00453789	−0.00226895	+	0.999997i	$0.500722\pi$
$798$	0	0
$799$	−44.6966	−1.58125
$800$	4.84682	0.171361
$801$	0	0
$802$	4.09207	0.144496
$803$	13.6576	0.481966
$804$	0	0
$805$	−2.31226	−0.0814964
$806$	11.3943	0.401348
$807$	0	0
$808$	−9.65760	−0.339753
$809$	−17.5354	−0.616512	−0.308256	−	0.951304i	$-0.599745\pi$
$809$	−17.5354	−0.616512	−0.308256	+	0.951304i	$0.599745\pi$
$810$	0	0
$811$	−39.3512	−1.38181	−0.690905	−	0.722946i	$-0.742788\pi$
$811$	−39.3512	−1.38181	−0.690905	+	0.722946i	$0.742788\pi$
$812$	−6.58060	−0.230934
$813$	0	0
$814$	4.65465	0.163146
$815$	−1.35945	−0.0476194
$816$	0	0
$817$	3.73673	0.130732
$818$	−2.48346	−0.0868320
$819$	0	0
$820$	−0.416452	−0.0145431
$821$	−14.0980	−0.492023	−0.246011	−	0.969267i	$-0.579120\pi$
$821$	−14.0980	−0.492023	−0.246011	+	0.969267i	$0.579120\pi$
$822$	0	0
$823$	−1.12516	−0.0392207	−0.0196103	−	0.999808i	$-0.506243\pi$
$823$	−1.12516	−0.0392207	−0.0196103	+	0.999808i	$0.506243\pi$
$824$	6.95396	0.242253
$825$	0	0
$826$	−21.3814	−0.743953
$827$	33.8217	1.17610	0.588049	−	0.808825i	$-0.299896\pi$
$827$	33.8217	1.17610	0.588049	+	0.808825i	$0.299896\pi$
$828$	0	0
$829$	−36.1900	−1.25693	−0.628466	−	0.777837i	$-0.716317\pi$
$829$	−36.1900	−1.25693	−0.628466	+	0.777837i	$0.716317\pi$
$830$	3.33438	0.115738
$831$	0	0
$832$	3.71871	0.128923
$833$	22.0000	0.762255
$834$	0	0
$835$	−9.03899	−0.312807
$836$	−1.00000	−0.0345857
$837$	0	0
$838$	35.0950	1.21234
$839$	31.8347	1.09906	0.549528	−	0.835475i	$-0.314808\pi$
$839$	31.8347	1.09906	0.549528	+	0.835475i	$0.314808\pi$
$840$	0	0
$841$	−14.3403	−0.494492
$842$	12.3483	0.425550
$843$	0	0
$844$	−6.09207	−0.209698
$845$	0.324378	0.0111590
$846$	0	0
$847$	−1.71871	−0.0590556
$848$	1.21724	0.0418000
$849$	0	0
$850$	−26.3542	−0.903941
$851$	−16.0000	−0.548473
$852$	0	0
$853$	−31.1009	−1.06488	−0.532438	−	0.846469i	$-0.678724\pi$
$853$	−31.1009	−1.06488	−0.532438	+	0.846469i	$0.678724\pi$
$854$	3.43742	0.117626
$855$	0	0
$856$	2.65465	0.0907342
$857$	35.4985	1.21261	0.606303	−	0.795234i	$-0.292652\pi$
$857$	35.4985	1.21261	0.606303	+	0.795234i	$0.292652\pi$
$858$	0	0
$859$	−5.63760	−0.192352	−0.0961762	−	0.995364i	$-0.530661\pi$
$859$	−5.63760	−0.192352	−0.0961762	+	0.995364i	$0.530661\pi$
$860$	−1.46249	−0.0498705
$861$	0	0
$862$	−6.94691	−0.236613
$863$	−31.9949	−1.08912	−0.544560	−	0.838722i	$-0.683303\pi$
$863$	−31.9949	−1.08912	−0.544560	+	0.838722i	$0.683303\pi$
$864$	0	0
$865$	7.11420	0.241890
$866$	13.2533	0.450364
$867$	0	0
$868$	5.26622	0.178747
$869$	13.0029	0.441095
$870$	0	0
$871$	−6.39138	−0.216564
$872$	−6.00000	−0.203186
$873$	0	0
$874$	3.43742	0.116272
$875$	6.62369	0.223921
$876$	0	0
$877$	−3.23019	−0.109076	−0.0545378	−	0.998512i	$-0.517369\pi$
$877$	−3.23019	−0.109076	−0.0545378	+	0.998512i	$0.517369\pi$
$878$	−15.3152	−0.516863
$879$	0	0
$880$	0.391382	0.0131935
$881$	24.7217	0.832894	0.416447	−	0.909160i	$-0.363275\pi$
$881$	24.7217	0.832894	0.416447	+	0.909160i	$0.363275\pi$
$882$	0	0
$883$	11.9138	0.400932	0.200466	−	0.979701i	$-0.435754\pi$
$883$	11.9138	0.400932	0.200466	+	0.979701i	$0.435754\pi$
$884$	−20.2202	−0.680078
$885$	0	0
$886$	−20.8807	−0.701502
$887$	12.1841	0.409104	0.204552	−	0.978856i	$-0.434426\pi$
$887$	12.1841	0.409104	0.204552	+	0.978856i	$0.434426\pi$
$888$	0	0
$889$	8.59862	0.288388
$890$	6.92498	0.232126
$891$	0	0
$892$	−18.8748	−0.631976
$893$	−8.22018	−0.275078
$894$	0	0
$895$	−4.41645	−0.147626
$896$	1.71871	0.0574181
$897$	0	0
$898$	−28.7887	−0.960690
$899$	−11.7317	−0.391273
$900$	0	0
$901$	−6.61862	−0.220498
$902$	1.06406	0.0354292
$903$	0	0
$904$	7.52949	0.250427
$905$	8.56258	0.284630
$906$	0	0
$907$	45.3211	1.50486	0.752431	−	0.658671i	$-0.228881\pi$
$907$	45.3211	1.50486	0.752431	+	0.658671i	$0.228881\pi$
$908$	−20.9669	−0.695811
$909$	0	0
$910$	2.50147	0.0829231
$911$	13.6376	0.451834	0.225917	−	0.974147i	$-0.427462\pi$
$911$	13.6376	0.451834	0.225917	+	0.974147i	$0.427462\pi$
$912$	0	0
$913$	−8.51949	−0.281954
$914$	32.7526	1.08336
$915$	0	0
$916$	13.8108	0.456321
$917$	−11.1741	−0.369003
$918$	0	0
$919$	39.5283	1.30392	0.651960	−	0.758254i	$-0.273947\pi$
$919$	39.5283	1.30392	0.651960	+	0.758254i	$0.273947\pi$
$920$	−1.34535	−0.0443547
$921$	0	0
$922$	20.2261	0.666111
$923$	−3.88991	−0.128038
$924$	0	0
$925$	22.5603	0.741777
$926$	18.6907	0.614214
$927$	0	0
$928$	−3.82880	−0.125687
$929$	−6.89991	−0.226379	−0.113189	−	0.993573i	$-0.536107\pi$
$929$	−6.89991	−0.226379	−0.113189	+	0.993573i	$0.536107\pi$
$930$	0	0
$931$	4.04604	0.132604
$932$	−4.91087	−0.160861
$933$	0	0
$934$	34.5685	1.13112
$935$	−2.12811	−0.0695966
$936$	0	0
$937$	−21.3873	−0.698692	−0.349346	−	0.936994i	$-0.613596\pi$
$937$	−21.3873	−0.698692	−0.349346	+	0.936994i	$0.613596\pi$
$938$	−2.95396	−0.0964503
$939$	0	0
$940$	3.21724	0.104935
$941$	27.5655	0.898611	0.449305	−	0.893378i	$-0.351672\pi$
$941$	27.5655	0.898611	0.449305	+	0.893378i	$0.351672\pi$
$942$	0	0
$943$	−3.65760	−0.119108
$944$	−12.4404	−0.404899
$945$	0	0
$946$	3.73673	0.121491
$947$	9.22313	0.299712	0.149856	−	0.988708i	$-0.452119\pi$
$947$	9.22313	0.299712	0.149856	+	0.988708i	$0.452119\pi$
$948$	0	0
$949$	50.7887	1.64867
$950$	−4.84682	−0.157252
$951$	0	0
$952$	−9.34535	−0.302884
$953$	−22.6966	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$953$	−22.6966	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$954$	0	0
$955$	2.77687	0.0898573
$956$	14.0490	0.454377
$957$	0	0
$958$	−17.7187	−0.572466
$959$	27.0159	0.872389
$960$	0	0
$961$	−21.6116	−0.697147
$962$	17.3093	0.558075
$963$	0	0
$964$	22.7397	0.732396
$965$	4.95005	0.159348
$966$	0	0
$967$	−48.6966	−1.56598	−0.782988	−	0.622036i	$-0.786306\pi$
$967$	−48.6966	−1.56598	−0.782988	+	0.622036i	$0.786306\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	1.82175	0.0584929
$971$	19.5165	0.626316	0.313158	−	0.949701i	$-0.398613\pi$
$971$	19.5165	0.626316	0.313158	+	0.949701i	$0.398613\pi$
$972$	0	0
$973$	19.1432	0.613702
$974$	−37.7426	−1.20935
$975$	0	0
$976$	2.00000	0.0640184
$977$	−37.3512	−1.19497	−0.597486	−	0.801879i	$-0.703834\pi$
$977$	−37.3512	−1.19497	−0.597486	+	0.801879i	$0.703834\pi$
$978$	0	0
$979$	−17.6936	−0.565491
$980$	−1.58355	−0.0505846
$981$	0	0
$982$	0.373364	0.0119145
$983$	45.8888	1.46362	0.731812	−	0.681507i	$-0.238675\pi$
$983$	45.8888	1.46362	0.731812	+	0.681507i	$0.238675\pi$
$984$	0	0
$985$	−3.30110	−0.105182
$986$	20.8188	0.663006
$987$	0	0
$988$	−3.71871	−0.118308
$989$	−12.8447	−0.408438
$990$	0	0
$991$	−30.9418	−0.982900	−0.491450	−	0.870906i	$-0.663533\pi$
$991$	−30.9418	−0.982900	−0.491450	+	0.870906i	$0.663533\pi$
$992$	3.06406	0.0972838
$993$	0	0
$994$	−1.79783	−0.0570238
$995$	−0.818801	−0.0259577
$996$	0	0
$997$	49.8418	1.57850	0.789252	−	0.614069i	$-0.210469\pi$
$997$	49.8418	1.57850	0.789252	+	0.614069i	$0.210469\pi$
$998$	14.8388	0.469714
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3762.2.a.bd.1.3		3
3.2	odd	2		418.2.a.h.1.3	✓	3
12.11	even	2		3344.2.a.p.1.1		3
33.32	even	2		4598.2.a.bm.1.3		3
57.56	even	2		7942.2.a.bc.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.h.1.3	✓	3	3.2	odd	2
3344.2.a.p.1.1		3	12.11	even	2
3762.2.a.bd.1.3		3	1.1	even	1	trivial
4598.2.a.bm.1.3		3	33.32	even	2
7942.2.a.bc.1.1		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.391382 q^{5} -1.71871 q^{7} -1.00000 q^{8} -0.391382 q^{10} +1.00000 q^{11} +3.71871 q^{13} +1.71871 q^{14} +1.00000 q^{16} -5.43742 q^{17} -1.00000 q^{19} +0.391382 q^{20} -1.00000 q^{22} +3.43742 q^{23} -4.84682 q^{25} -3.71871 q^{26} -1.71871 q^{28} +3.82880 q^{29} -3.06406 q^{31} -1.00000 q^{32} +5.43742 q^{34} -0.672673 q^{35} -4.65465 q^{37} +1.00000 q^{38} -0.391382 q^{40} -1.06406 q^{41} -3.73673 q^{43} +1.00000 q^{44} -3.43742 q^{46} +8.22018 q^{47} -4.04604 q^{49} +4.84682 q^{50} +3.71871 q^{52} +1.21724 q^{53} +0.391382 q^{55} +1.71871 q^{56} -3.82880 q^{58} -12.4404 q^{59} +2.00000 q^{61} +3.06406 q^{62} +1.00000 q^{64} +1.45544 q^{65} -1.71871 q^{67} -5.43742 q^{68} +0.672673 q^{70} -1.04604 q^{71} +13.6576 q^{73} +4.65465 q^{74} -1.00000 q^{76} -1.71871 q^{77} +13.0029 q^{79} +0.391382 q^{80} +1.06406 q^{82} -8.51949 q^{83} -2.12811 q^{85} +3.73673 q^{86} -1.00000 q^{88} -17.6936 q^{89} -6.39138 q^{91} +3.43742 q^{92} -8.22018 q^{94} -0.391382 q^{95} -4.65465 q^{97} +4.04604 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} + q^{7} - 3 q^{8} + 5 q^{10} + 3 q^{11} + 5 q^{13} - q^{14} + 3 q^{16} - 4 q^{17} - 3 q^{19} - 5 q^{20} - 3 q^{22} - 2 q^{23} + 4 q^{25} - 5 q^{26} + q^{28} - 7 q^{29}+ \cdots + 6 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 5 * q^5 + q^7 - 3 * q^8 + 5 * q^10 + 3 * q^11 + 5 * q^13 - q^14 + 3 * q^16 - 4 * q^17 - 3 * q^19 - 5 * q^20 - 3 * q^22 - 2 * q^23 + 4 * q^25 - 5 * q^26 + q^28 - 7 * q^29 - 3 * q^31 - 3 * q^32 + 4 * q^34 - 2 * q^35 - 14 * q^37 + 3 * q^38 + 5 * q^40 + 3 * q^41 - 5 * q^43 + 3 * q^44 + 2 * q^46 - 6 * q^49 - 4 * q^50 + 5 * q^52 + 16 * q^53 - 5 * q^55 - q^56 + 7 * q^58 + 12 * q^59 + 6 * q^61 + 3 * q^62 + 3 * q^64 - 8 * q^65 + q^67 - 4 * q^68 + 2 * q^70 + 3 * q^71 + 4 * q^73 + 14 * q^74 - 3 * q^76 + q^77 + 2 * q^79 - 5 * q^80 - 3 * q^82 - 7 * q^83 + 6 * q^85 + 5 * q^86 - 3 * q^88 - 16 * q^89 - 13 * q^91 - 2 * q^92 + 5 * q^95 - 14 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

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Database

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