LMFDB - Embedded newform 3321.2.a.i.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3321,2,Mod(1,3321)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3321, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3321.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3321 = 3^{4} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3321.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:	$26.5183185113$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 20 x^{14} + 158 x^{12} - 2 x^{11} - 629 x^{10} + 25 x^{9} + 1329 x^{8} - 116 x^{7} - 1433 x^{6} + \cdots + 3 $ x^16 - 20x^14 + 158x^12 - 2x^11 - 629x^10 + 25x^9 + 1329x^8 - 116x^7 - 1433x^6 + 222x^5 + 691x^4 - 133x^3 - 104x^2 + 9*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 369)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$1.93363$ of defining polynomial
Character	$\chi$	$=$	3321.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.93363 q^{2} +1.73893 q^{4} -3.73785 q^{5} +0.466713 q^{7} -0.504807 q^{8} -7.22763 q^{10} +5.31685 q^{11} -1.11629 q^{13} +0.902452 q^{14} -4.45398 q^{16} +3.25843 q^{17} +0.0992768 q^{19} -6.49988 q^{20} +10.2808 q^{22} -3.59301 q^{23} +8.97154 q^{25} -2.15849 q^{26} +0.811584 q^{28} -4.76962 q^{29} -3.11403 q^{31} -7.60274 q^{32} +6.30061 q^{34} -1.74451 q^{35} -9.77389 q^{37} +0.191965 q^{38} +1.88689 q^{40} -1.00000 q^{41} -6.38245 q^{43} +9.24564 q^{44} -6.94756 q^{46} -3.33536 q^{47} -6.78218 q^{49} +17.3477 q^{50} -1.94115 q^{52} +6.33395 q^{53} -19.8736 q^{55} -0.235600 q^{56} -9.22268 q^{58} +4.41235 q^{59} -8.03663 q^{61} -6.02138 q^{62} -5.79295 q^{64} +4.17251 q^{65} -5.81691 q^{67} +5.66620 q^{68} -3.37323 q^{70} -10.1380 q^{71} +8.00974 q^{73} -18.8991 q^{74} +0.172636 q^{76} +2.48144 q^{77} -13.6652 q^{79} +16.6483 q^{80} -1.93363 q^{82} +7.52659 q^{83} -12.1795 q^{85} -12.3413 q^{86} -2.68398 q^{88} +4.21348 q^{89} -0.520986 q^{91} -6.24800 q^{92} -6.44936 q^{94} -0.371082 q^{95} -18.7285 q^{97} -13.1142 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{4} - q^{5} - 13 q^{7} - 8 q^{10} + 2 q^{11} - 5 q^{13} - q^{14} - 8 q^{16} + q^{17} - 19 q^{19} - 11 q^{20} - 14 q^{22} - 2 q^{23} - 3 q^{25} + 15 q^{26} - 24 q^{28} + 10 q^{29} - 39 q^{31}+ \cdots - 48 q^{98}+O(q^{100}) $ 16 * q + 8 * q^4 - q^5 - 13 * q^7 - 8 * q^10 + 2 * q^11 - 5 * q^13 - q^14 - 8 * q^16 + q^17 - 19 * q^19 - 11 * q^20 - 14 * q^22 - 2 * q^23 - 3 * q^25 + 15 * q^26 - 24 * q^28 + 10 * q^29 - 39 * q^31 + 10 * q^32 - 18 * q^34 + 2 * q^35 - 26 * q^37 + 14 * q^38 - 18 * q^40 - 16 * q^41 - 20 * q^43 - 23 * q^44 - 26 * q^46 - 5 * q^47 - 7 * q^49 - 9 * q^50 - 14 * q^52 + 7 * q^53 - 37 * q^55 + 2 * q^56 - 24 * q^58 - 6 * q^59 - 21 * q^61 - 5 * q^62 - 34 * q^64 - 2 * q^65 - 32 * q^67 + 8 * q^68 - 22 * q^70 - 20 * q^71 - 17 * q^73 - 28 * q^74 - 40 * q^76 + 22 * q^77 - 69 * q^79 + 39 * q^80 - 10 * q^83 - 42 * q^85 + 8 * q^86 - 42 * q^88 + 11 * q^89 - 51 * q^91 + 10 * q^92 - 13 * q^94 + 22 * q^95 - 25 * q^97 - 48 * 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 $

$2$	1.93363	1.36728	0.683642	−	0.729817i	$-0.260395\pi$
$2$	1.93363	1.36728	0.683642	+	0.729817i	$0.260395\pi$
$3$	0	0
$3$	0	0
$4$	1.73893	0.869467
$5$	−3.73785	−1.67162	−0.835809	−	0.549020i	$-0.815001\pi$
$5$	−3.73785	−1.67162	−0.835809	+	0.549020i	$0.815001\pi$
$6$	0	0
$7$	0.466713	0.176401	0.0882005	−	0.996103i	$-0.471888\pi$
$7$	0.466713	0.176401	0.0882005	+	0.996103i	$0.471888\pi$
$8$	−0.504807	−0.178476
$9$	0	0
$10$	−7.22763	−2.28558
$11$	5.31685	1.60309	0.801545	−	0.597934i	$-0.204012\pi$
$11$	5.31685	1.60309	0.801545	+	0.597934i	$0.204012\pi$
$12$	0	0
$13$	−1.11629	−0.309602	−0.154801	−	0.987946i	$-0.549474\pi$
$13$	−1.11629	−0.309602	−0.154801	+	0.987946i	$0.549474\pi$
$14$	0.902452	0.241190
$15$	0	0
$16$	−4.45398	−1.11349
$17$	3.25843	0.790286	0.395143	−	0.918620i	$-0.370695\pi$
$17$	3.25843	0.790286	0.395143	+	0.918620i	$0.370695\pi$
$18$	0	0
$19$	0.0992768	0.0227756	0.0113878	−	0.999935i	$-0.496375\pi$
$19$	0.0992768	0.0227756	0.0113878	+	0.999935i	$0.496375\pi$
$20$	−6.49988	−1.45342
$21$	0	0
$22$	10.2808	2.19188
$23$	−3.59301	−0.749194	−0.374597	−	0.927188i	$-0.622219\pi$
$23$	−3.59301	−0.749194	−0.374597	+	0.927188i	$0.622219\pi$
$24$	0	0
$25$	8.97154	1.79431
$26$	−2.15849	−0.423314
$27$	0	0
$28$	0.811584	0.153375
$29$	−4.76962	−0.885695	−0.442848	−	0.896597i	$-0.646032\pi$
$29$	−4.76962	−0.885695	−0.442848	+	0.896597i	$0.646032\pi$
$30$	0	0
$31$	−3.11403	−0.559296	−0.279648	−	0.960103i	$-0.590218\pi$
$31$	−3.11403	−0.559296	−0.279648	+	0.960103i	$0.590218\pi$
$32$	−7.60274	−1.34399
$33$	0	0
$34$	6.30061	1.08055
$35$	−1.74451	−0.294875
$36$	0	0
$37$	−9.77389	−1.60682	−0.803409	−	0.595427i	$-0.796983\pi$
$37$	−9.77389	−1.60682	−0.803409	+	0.595427i	$0.796983\pi$
$38$	0.191965	0.0311408
$39$	0	0
$40$	1.88689	0.298344
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−6.38245	−0.973314	−0.486657	−	0.873593i	$-0.661784\pi$
$43$	−6.38245	−0.973314	−0.486657	+	0.873593i	$0.661784\pi$
$44$	9.24564	1.39383
$45$	0	0
$46$	−6.94756	−1.02436
$47$	−3.33536	−0.486512	−0.243256	−	0.969962i	$-0.578216\pi$
$47$	−3.33536	−0.486512	−0.243256	+	0.969962i	$0.578216\pi$
$48$	0	0
$49$	−6.78218	−0.968883
$50$	17.3477	2.45333
$51$	0	0
$52$	−1.94115	−0.269189
$53$	6.33395	0.870035	0.435017	−	0.900422i	$-0.356742\pi$
$53$	6.33395	0.870035	0.435017	+	0.900422i	$0.356742\pi$
$54$	0	0
$55$	−19.8736	−2.67975
$56$	−0.235600	−0.0314834
$57$	0	0
$58$	−9.22268	−1.21100
$59$	4.41235	0.574438	0.287219	−	0.957865i	$-0.407269\pi$
$59$	4.41235	0.574438	0.287219	+	0.957865i	$0.407269\pi$
$60$	0	0
$61$	−8.03663	−1.02898	−0.514492	−	0.857495i	$-0.672020\pi$
$61$	−8.03663	−1.02898	−0.514492	+	0.857495i	$0.672020\pi$
$62$	−6.02138	−0.764716
$63$	0	0
$64$	−5.79295	−0.724119
$65$	4.17251	0.517537
$66$	0	0
$67$	−5.81691	−0.710649	−0.355324	−	0.934743i	$-0.615630\pi$
$67$	−5.81691	−0.710649	−0.355324	+	0.934743i	$0.615630\pi$
$68$	5.66620	0.687127
$69$	0	0
$70$	−3.37323	−0.403178
$71$	−10.1380	−1.20316	−0.601579	−	0.798814i	$-0.705461\pi$
$71$	−10.1380	−1.20316	−0.601579	+	0.798814i	$0.705461\pi$
$72$	0	0
$73$	8.00974	0.937469	0.468735	−	0.883339i	$-0.344710\pi$
$73$	8.00974	0.937469	0.468735	+	0.883339i	$0.344710\pi$
$74$	−18.8991	−2.19698
$75$	0	0
$76$	0.172636	0.0198027
$77$	2.48144	0.282787
$78$	0	0
$79$	−13.6652	−1.53745	−0.768725	−	0.639580i	$-0.779108\pi$
$79$	−13.6652	−1.53745	−0.768725	+	0.639580i	$0.779108\pi$
$80$	16.6483	1.86134
$81$	0	0
$82$	−1.93363	−0.213534
$83$	7.52659	0.826150	0.413075	−	0.910697i	$-0.364455\pi$
$83$	7.52659	0.826150	0.413075	+	0.910697i	$0.364455\pi$
$84$	0	0
$85$	−12.1795	−1.32106
$86$	−12.3413	−1.33080
$87$	0	0
$88$	−2.68398	−0.286113
$89$	4.21348	0.446628	0.223314	−	0.974747i	$-0.428312\pi$
$89$	4.21348	0.446628	0.223314	+	0.974747i	$0.428312\pi$
$90$	0	0
$91$	−0.520986	−0.0546142
$92$	−6.24800	−0.651399
$93$	0	0
$94$	−6.44936	−0.665201
$95$	−0.371082	−0.0380722
$96$	0	0
$97$	−18.7285	−1.90159	−0.950796	−	0.309818i	$-0.899732\pi$
$97$	−18.7285	−1.90159	−0.950796	+	0.309818i	$0.899732\pi$
$98$	−13.1142	−1.32474
$99$	0	0
$100$	15.6009	1.56009
$101$	−16.4375	−1.63559	−0.817795	−	0.575510i	$-0.804804\pi$
$101$	−16.4375	−1.63559	−0.817795	+	0.575510i	$0.804804\pi$
$102$	0	0
$103$	11.2296	1.10648	0.553241	−	0.833021i	$-0.313391\pi$
$103$	11.2296	1.10648	0.553241	+	0.833021i	$0.313391\pi$
$104$	0.563509	0.0552566
$105$	0	0
$106$	12.2475	1.18959
$107$	1.34625	0.130147	0.0650737	−	0.997880i	$-0.479272\pi$
$107$	1.34625	0.130147	0.0650737	+	0.997880i	$0.479272\pi$
$108$	0	0
$109$	13.4302	1.28638	0.643191	−	0.765706i	$-0.277610\pi$
$109$	13.4302	1.28638	0.643191	+	0.765706i	$0.277610\pi$
$110$	−38.4282	−3.66399
$111$	0	0
$112$	−2.07873	−0.196422
$113$	−15.1796	−1.42798	−0.713988	−	0.700158i	$-0.753113\pi$
$113$	−15.1796	−1.42798	−0.713988	+	0.700158i	$0.753113\pi$
$114$	0	0
$115$	13.4301	1.25237
$116$	−8.29404	−0.770083
$117$	0	0
$118$	8.53185	0.785421
$119$	1.52075	0.139407
$120$	0	0
$121$	17.2689	1.56990
$122$	−15.5399	−1.40691
$123$	0	0
$124$	−5.41508	−0.486289
$125$	−14.8450	−1.32778
$126$	0	0
$127$	−16.7423	−1.48564	−0.742821	−	0.669490i	$-0.766513\pi$
$127$	−16.7423	−1.48564	−0.742821	+	0.669490i	$0.766513\pi$
$128$	4.00405	0.353911
$129$	0	0
$130$	8.06811	0.707620
$131$	14.5900	1.27474	0.637368	−	0.770560i	$-0.280023\pi$
$131$	14.5900	1.27474	0.637368	+	0.770560i	$0.280023\pi$
$132$	0	0
$133$	0.0463338	0.00401765
$134$	−11.2478	−0.971659
$135$	0	0
$136$	−1.64488	−0.141047
$137$	17.8451	1.52461	0.762306	−	0.647216i	$-0.224067\pi$
$137$	17.8451	1.52461	0.762306	+	0.647216i	$0.224067\pi$
$138$	0	0
$139$	7.33393	0.622056	0.311028	−	0.950401i	$-0.399327\pi$
$139$	7.33393	0.622056	0.311028	+	0.950401i	$0.399327\pi$
$140$	−3.03358	−0.256384
$141$	0	0
$142$	−19.6031	−1.64506
$143$	−5.93513	−0.496320
$144$	0	0
$145$	17.8281	1.48054
$146$	15.4879	1.28179
$147$	0	0
$148$	−16.9961	−1.39707
$149$	1.66212	0.136166	0.0680832	−	0.997680i	$-0.478312\pi$
$149$	1.66212	0.136166	0.0680832	+	0.997680i	$0.478312\pi$
$150$	0	0
$151$	−3.34143	−0.271922	−0.135961	−	0.990714i	$-0.543412\pi$
$151$	−3.34143	−0.271922	−0.135961	+	0.990714i	$0.543412\pi$
$152$	−0.0501156	−0.00406491
$153$	0	0
$154$	4.79820	0.386650
$155$	11.6398	0.934929
$156$	0	0
$157$	−19.8228	−1.58203	−0.791015	−	0.611797i	$-0.790447\pi$
$157$	−19.8228	−1.58203	−0.791015	+	0.611797i	$0.790447\pi$
$158$	−26.4234	−2.10213
$159$	0	0
$160$	28.4179	2.24663
$161$	−1.67691	−0.132159
$162$	0	0
$163$	13.9568	1.09318	0.546592	−	0.837399i	$-0.315925\pi$
$163$	13.9568	1.09318	0.546592	+	0.837399i	$0.315925\pi$
$164$	−1.73893	−0.135788
$165$	0	0
$166$	14.5537	1.12958
$167$	24.0910	1.86422	0.932110	−	0.362175i	$-0.117966\pi$
$167$	24.0910	1.86422	0.932110	+	0.362175i	$0.117966\pi$
$168$	0	0
$169$	−11.7539	−0.904146
$170$	−23.5507	−1.80626
$171$	0	0
$172$	−11.0987	−0.846264
$173$	−6.43765	−0.489446	−0.244723	−	0.969593i	$-0.578697\pi$
$173$	−6.43765	−0.489446	−0.244723	+	0.969593i	$0.578697\pi$
$174$	0	0
$175$	4.18714	0.316518
$176$	−23.6811	−1.78503
$177$	0	0
$178$	8.14732	0.610667
$179$	−1.85372	−0.138553	−0.0692766	−	0.997597i	$-0.522069\pi$
$179$	−1.85372	−0.138553	−0.0692766	+	0.997597i	$0.522069\pi$
$180$	0	0
$181$	17.1429	1.27422	0.637111	−	0.770772i	$-0.280129\pi$
$181$	17.1429	1.27422	0.637111	+	0.770772i	$0.280129\pi$
$182$	−1.00740	−0.0746731
$183$	0	0
$184$	1.81378	0.133713
$185$	36.5334	2.68599
$186$	0	0
$187$	17.3246	1.26690
$188$	−5.79997	−0.423006
$189$	0	0
$190$	−0.717536	−0.0520555
$191$	6.07807	0.439793	0.219897	−	0.975523i	$-0.429428\pi$
$191$	6.07807	0.439793	0.219897	+	0.975523i	$0.429428\pi$
$192$	0	0
$193$	2.94619	0.212072	0.106036	−	0.994362i	$-0.466184\pi$
$193$	2.94619	0.212072	0.106036	+	0.994362i	$0.466184\pi$
$194$	−36.2140	−2.60002
$195$	0	0
$196$	−11.7938	−0.842411
$197$	−20.7418	−1.47779	−0.738895	−	0.673820i	$-0.764652\pi$
$197$	−20.7418	−1.47779	−0.738895	+	0.673820i	$0.764652\pi$
$198$	0	0
$199$	−26.3268	−1.86626	−0.933129	−	0.359543i	$-0.882933\pi$
$199$	−26.3268	−1.86626	−0.933129	+	0.359543i	$0.882933\pi$
$200$	−4.52889	−0.320241
$201$	0	0
$202$	−31.7840	−2.23632
$203$	−2.22604	−0.156238
$204$	0	0
$205$	3.73785	0.261063
$206$	21.7139	1.51288
$207$	0	0
$208$	4.97192	0.344740
$209$	0.527839	0.0365114
$210$	0	0
$211$	−13.6255	−0.938015	−0.469008	−	0.883194i	$-0.655388\pi$
$211$	−13.6255	−0.938015	−0.469008	+	0.883194i	$0.655388\pi$
$212$	11.0143	0.756466
$213$	0	0
$214$	2.60316	0.177948
$215$	23.8566	1.62701
$216$	0	0
$217$	−1.45336	−0.0986603
$218$	25.9691	1.75885
$219$	0	0
$220$	−34.5588	−2.32996
$221$	−3.63735	−0.244674
$222$	0	0
$223$	11.1100	0.743981	0.371991	−	0.928236i	$-0.378675\pi$
$223$	11.1100	0.743981	0.371991	+	0.928236i	$0.378675\pi$
$224$	−3.54830	−0.237081
$225$	0	0
$226$	−29.3517	−1.95245
$227$	−5.08420	−0.337450	−0.168725	−	0.985663i	$-0.553965\pi$
$227$	−5.08420	−0.337450	−0.168725	+	0.985663i	$0.553965\pi$
$228$	0	0
$229$	−1.63577	−0.108095	−0.0540475	−	0.998538i	$-0.517212\pi$
$229$	−1.63577	−0.108095	−0.0540475	+	0.998538i	$0.517212\pi$
$230$	25.9689	1.71234
$231$	0	0
$232$	2.40773	0.158076
$233$	−19.0351	−1.24703	−0.623515	−	0.781812i	$-0.714296\pi$
$233$	−19.0351	−1.24703	−0.623515	+	0.781812i	$0.714296\pi$
$234$	0	0
$235$	12.4671	0.813263
$236$	7.67277	0.499455
$237$	0	0
$238$	2.94058	0.190609
$239$	−1.29228	−0.0835909	−0.0417954	−	0.999126i	$-0.513308\pi$
$239$	−1.29228	−0.0835909	−0.0417954	+	0.999126i	$0.513308\pi$
$240$	0	0
$241$	1.29411	0.0833610	0.0416805	−	0.999131i	$-0.486729\pi$
$241$	1.29411	0.0833610	0.0416805	+	0.999131i	$0.486729\pi$
$242$	33.3916	2.14650
$243$	0	0
$244$	−13.9752	−0.894668
$245$	25.3508	1.61960
$246$	0	0
$247$	−0.110821	−0.00705139
$248$	1.57198	0.0998209
$249$	0	0
$250$	−28.7048	−1.81545
$251$	31.3179	1.97677	0.988383	−	0.151982i	$-0.0485656\pi$
$251$	31.3179	1.97677	0.988383	+	0.151982i	$0.0485656\pi$
$252$	0	0
$253$	−19.1035	−1.20103
$254$	−32.3735	−2.03129
$255$	0	0
$256$	19.3283	1.20802
$257$	−19.4095	−1.21073	−0.605367	−	0.795947i	$-0.706974\pi$
$257$	−19.4095	−1.21073	−0.605367	+	0.795947i	$0.706974\pi$
$258$	0	0
$259$	−4.56161	−0.283444
$260$	7.25572	0.449981
$261$	0	0
$262$	28.2117	1.74293
$263$	32.1771	1.98413	0.992064	−	0.125737i	$-0.0401297\pi$
$263$	32.1771	1.98413	0.992064	+	0.125737i	$0.0401297\pi$
$264$	0	0
$265$	−23.6754	−1.45437
$266$	0.0895925	0.00549327
$267$	0	0
$268$	−10.1152	−0.617886
$269$	−13.1228	−0.800114	−0.400057	−	0.916490i	$-0.631010\pi$
$269$	−13.1228	−0.800114	−0.400057	+	0.916490i	$0.631010\pi$
$270$	0	0
$271$	18.4975	1.12364	0.561821	−	0.827258i	$-0.310101\pi$
$271$	18.4975	1.12364	0.561821	+	0.827258i	$0.310101\pi$
$272$	−14.5130	−0.879979
$273$	0	0
$274$	34.5059	2.08458
$275$	47.7003	2.87644
$276$	0	0
$277$	25.6709	1.54242	0.771208	−	0.636583i	$-0.219653\pi$
$277$	25.6709	1.54242	0.771208	+	0.636583i	$0.219653\pi$
$278$	14.1811	0.850527
$279$	0	0
$280$	0.880638	0.0526282
$281$	1.74772	0.104260	0.0521301	−	0.998640i	$-0.483399\pi$
$281$	1.74772	0.104260	0.0521301	+	0.998640i	$0.483399\pi$
$282$	0	0
$283$	−2.77648	−0.165045	−0.0825223	−	0.996589i	$-0.526298\pi$
$283$	−2.77648	−0.165045	−0.0825223	+	0.996589i	$0.526298\pi$
$284$	−17.6293	−1.04611
$285$	0	0
$286$	−11.4764	−0.678611
$287$	−0.466713	−0.0275492
$288$	0	0
$289$	−6.38262	−0.375448
$290$	34.4730	2.02433
$291$	0	0
$292$	13.9284	0.815098
$293$	2.13045	0.124462	0.0622310	−	0.998062i	$-0.480178\pi$
$293$	2.13045	0.124462	0.0622310	+	0.998062i	$0.480178\pi$
$294$	0	0
$295$	−16.4927	−0.960242
$296$	4.93393	0.286779
$297$	0	0
$298$	3.21393	0.186178
$299$	4.01083	0.231952
$300$	0	0
$301$	−2.97877	−0.171694
$302$	−6.46110	−0.371795
$303$	0	0
$304$	−0.442176	−0.0253606
$305$	30.0397	1.72007
$306$	0	0
$307$	−5.11496	−0.291926	−0.145963	−	0.989290i	$-0.546628\pi$
$307$	−5.11496	−0.291926	−0.145963	+	0.989290i	$0.546628\pi$
$308$	4.31507	0.245874
$309$	0	0
$310$	22.5070	1.27831
$311$	−6.09037	−0.345353	−0.172677	−	0.984979i	$-0.555242\pi$
$311$	−6.09037	−0.345353	−0.172677	+	0.984979i	$0.555242\pi$
$312$	0	0
$313$	21.6034	1.22109	0.610547	−	0.791980i	$-0.290950\pi$
$313$	21.6034	1.22109	0.610547	+	0.791980i	$0.290950\pi$
$314$	−38.3300	−2.16308
$315$	0	0
$316$	−23.7628	−1.33676
$317$	14.5774	0.818746	0.409373	−	0.912367i	$-0.365748\pi$
$317$	14.5774	0.818746	0.409373	+	0.912367i	$0.365748\pi$
$318$	0	0
$319$	−25.3593	−1.41985
$320$	21.6532	1.21045
$321$	0	0
$322$	−3.24252	−0.180699
$323$	0.323487	0.0179993
$324$	0	0
$325$	−10.0148	−0.555522
$326$	26.9874	1.49469
$327$	0	0
$328$	0.504807	0.0278733
$329$	−1.55666	−0.0858213
$330$	0	0
$331$	−8.88484	−0.488355	−0.244178	−	0.969731i	$-0.578518\pi$
$331$	−8.88484	−0.488355	−0.244178	+	0.969731i	$0.578518\pi$
$332$	13.0882	0.718310
$333$	0	0
$334$	46.5832	2.54892
$335$	21.7428	1.18793
$336$	0	0
$337$	−12.7849	−0.696440	−0.348220	−	0.937413i	$-0.613214\pi$
$337$	−12.7849	−0.696440	−0.348220	+	0.937413i	$0.613214\pi$
$338$	−22.7277	−1.23623
$339$	0	0
$340$	−21.1794	−1.14861
$341$	−16.5568	−0.896601
$342$	0	0
$343$	−6.43233	−0.347313
$344$	3.22190	0.173713
$345$	0	0
$346$	−12.4480	−0.669211
$347$	−16.9803	−0.911551	−0.455775	−	0.890095i	$-0.650638\pi$
$347$	−16.9803	−0.911551	−0.455775	+	0.890095i	$0.650638\pi$
$348$	0	0
$349$	−25.7205	−1.37679	−0.688393	−	0.725337i	$-0.741684\pi$
$349$	−25.7205	−1.37679	−0.688393	+	0.725337i	$0.741684\pi$
$350$	8.09638	0.432770
$351$	0	0
$352$	−40.4226	−2.15453
$353$	17.9704	0.956469	0.478234	−	0.878232i	$-0.341277\pi$
$353$	17.9704	0.956469	0.478234	+	0.878232i	$0.341277\pi$
$354$	0	0
$355$	37.8943	2.01122
$356$	7.32696	0.388328
$357$	0	0
$358$	−3.58441	−0.189442
$359$	−12.2194	−0.644914	−0.322457	−	0.946584i	$-0.604509\pi$
$359$	−12.2194	−0.644914	−0.322457	+	0.946584i	$0.604509\pi$
$360$	0	0
$361$	−18.9901	−0.999481
$362$	33.1481	1.74222
$363$	0	0
$364$	−0.905960	−0.0474852
$365$	−29.9392	−1.56709
$366$	0	0
$367$	6.13582	0.320287	0.160144	−	0.987094i	$-0.448804\pi$
$367$	6.13582	0.320287	0.160144	+	0.987094i	$0.448804\pi$
$368$	16.0032	0.834224
$369$	0	0
$370$	70.6421	3.67251
$371$	2.95614	0.153475
$372$	0	0
$373$	30.2011	1.56375	0.781876	−	0.623434i	$-0.214263\pi$
$373$	30.2011	1.56375	0.781876	+	0.623434i	$0.214263\pi$
$374$	33.4994	1.73221
$375$	0	0
$376$	1.68371	0.0868309
$377$	5.32426	0.274213
$378$	0	0
$379$	−8.34193	−0.428496	−0.214248	−	0.976779i	$-0.568730\pi$
$379$	−8.34193	−0.428496	−0.214248	+	0.976779i	$0.568730\pi$
$380$	−0.645287	−0.0331025
$381$	0	0
$382$	11.7527	0.601323
$383$	16.2687	0.831290	0.415645	−	0.909527i	$-0.363556\pi$
$383$	16.2687	0.831290	0.415645	+	0.909527i	$0.363556\pi$
$384$	0	0
$385$	−9.27527	−0.472712
$386$	5.69686	0.289962
$387$	0	0
$388$	−32.5676	−1.65337
$389$	−20.6572	−1.04736	−0.523681	−	0.851915i	$-0.675441\pi$
$389$	−20.6572	−1.04736	−0.523681	+	0.851915i	$0.675441\pi$
$390$	0	0
$391$	−11.7076	−0.592078
$392$	3.42369	0.172922
$393$	0	0
$394$	−40.1070	−2.02056
$395$	51.0783	2.57003
$396$	0	0
$397$	18.1534	0.911091	0.455545	−	0.890213i	$-0.349444\pi$
$397$	18.1534	0.911091	0.455545	+	0.890213i	$0.349444\pi$
$398$	−50.9063	−2.55170
$399$	0	0
$400$	−39.9590	−1.99795
$401$	20.5366	1.02555	0.512774	−	0.858524i	$-0.328618\pi$
$401$	20.5366	1.02555	0.512774	+	0.858524i	$0.328618\pi$
$402$	0	0
$403$	3.47615	0.173159
$404$	−28.5837	−1.42209
$405$	0	0
$406$	−4.30435	−0.213621
$407$	−51.9663	−2.57587
$408$	0	0
$409$	0.796814	0.0393999	0.0197000	−	0.999806i	$-0.493729\pi$
$409$	0.796814	0.0393999	0.0197000	+	0.999806i	$0.493729\pi$
$410$	7.22763	0.356947
$411$	0	0
$412$	19.5275	0.962050
$413$	2.05930	0.101332
$414$	0	0
$415$	−28.1333	−1.38101
$416$	8.48684	0.416101
$417$	0	0
$418$	1.02065	0.0499215
$419$	33.1628	1.62011	0.810054	−	0.586355i	$-0.199438\pi$
$419$	33.1628	1.62011	0.810054	+	0.586355i	$0.199438\pi$
$420$	0	0
$421$	9.48072	0.462062	0.231031	−	0.972946i	$-0.425790\pi$
$421$	9.48072	0.462062	0.231031	+	0.972946i	$0.425790\pi$
$422$	−26.3466	−1.28253
$423$	0	0
$424$	−3.19742	−0.155280
$425$	29.2331	1.41802
$426$	0	0
$427$	−3.75080	−0.181514
$428$	2.34105	0.113159
$429$	0	0
$430$	46.1300	2.22459
$431$	11.6853	0.562861	0.281431	−	0.959582i	$-0.409191\pi$
$431$	11.6853	0.562861	0.281431	+	0.959582i	$0.409191\pi$
$432$	0	0
$433$	2.69870	0.129691	0.0648456	−	0.997895i	$-0.479345\pi$
$433$	2.69870	0.129691	0.0648456	+	0.997895i	$0.479345\pi$
$434$	−2.81026	−0.134897
$435$	0	0
$436$	23.3543	1.11847
$437$	−0.356702	−0.0170634
$438$	0	0
$439$	21.3726	1.02006	0.510029	−	0.860157i	$-0.329635\pi$
$439$	21.3726	1.02006	0.510029	+	0.860157i	$0.329635\pi$
$440$	10.0323	0.478272
$441$	0	0
$442$	−7.03329	−0.334539
$443$	−16.3610	−0.777335	−0.388668	−	0.921378i	$-0.627065\pi$
$443$	−16.3610	−0.777335	−0.388668	+	0.921378i	$0.627065\pi$
$444$	0	0
$445$	−15.7494	−0.746591
$446$	21.4827	1.01723
$447$	0	0
$448$	−2.70365	−0.127735
$449$	−7.88783	−0.372250	−0.186125	−	0.982526i	$-0.559593\pi$
$449$	−7.88783	−0.372250	−0.186125	+	0.982526i	$0.559593\pi$
$450$	0	0
$451$	−5.31685	−0.250361
$452$	−26.3963	−1.24158
$453$	0	0
$454$	−9.83097	−0.461390
$455$	1.94737	0.0912940
$456$	0	0
$457$	−18.9732	−0.887530	−0.443765	−	0.896143i	$-0.646358\pi$
$457$	−18.9732	−0.887530	−0.443765	+	0.896143i	$0.646358\pi$
$458$	−3.16299	−0.147797
$459$	0	0
$460$	23.3541	1.08889
$461$	−10.1269	−0.471658	−0.235829	−	0.971795i	$-0.575781\pi$
$461$	−10.1269	−0.471658	−0.235829	+	0.971795i	$0.575781\pi$
$462$	0	0
$463$	−14.9152	−0.693166	−0.346583	−	0.938019i	$-0.612658\pi$
$463$	−14.9152	−0.693166	−0.346583	+	0.938019i	$0.612658\pi$
$464$	21.2438	0.986217
$465$	0	0
$466$	−36.8068	−1.70504
$467$	−2.20817	−0.102182	−0.0510910	−	0.998694i	$-0.516270\pi$
$467$	−2.20817	−0.102182	−0.0510910	+	0.998694i	$0.516270\pi$
$468$	0	0
$469$	−2.71483	−0.125359
$470$	24.1068	1.11196
$471$	0	0
$472$	−2.22738	−0.102524
$473$	−33.9345	−1.56031
$474$	0	0
$475$	0.890665	0.0408665
$476$	2.64449	0.121210
$477$	0	0
$478$	−2.49880	−0.114292
$479$	−1.16968	−0.0534438	−0.0267219	−	0.999643i	$-0.508507\pi$
$479$	−1.16968	−0.0534438	−0.0267219	+	0.999643i	$0.508507\pi$
$480$	0	0
$481$	10.9105	0.497474
$482$	2.50233	0.113978
$483$	0	0
$484$	30.0294	1.36497
$485$	70.0044	3.17874
$486$	0	0
$487$	−20.3072	−0.920206	−0.460103	−	0.887866i	$-0.652188\pi$
$487$	−20.3072	−0.920206	−0.460103	+	0.887866i	$0.652188\pi$
$488$	4.05695	0.183649
$489$	0	0
$490$	49.0191	2.21446
$491$	15.0932	0.681145	0.340572	−	0.940218i	$-0.389379\pi$
$491$	15.0932	0.681145	0.340572	+	0.940218i	$0.389379\pi$
$492$	0	0
$493$	−15.5415	−0.699953
$494$	−0.214288	−0.00964126
$495$	0	0
$496$	13.8698	0.622772
$497$	−4.73153	−0.212238
$498$	0	0
$499$	−10.1937	−0.456333	−0.228167	−	0.973622i	$-0.573273\pi$
$499$	−10.1937	−0.456333	−0.228167	+	0.973622i	$0.573273\pi$
$500$	−25.8145	−1.15446
$501$	0	0
$502$	60.5572	2.70280
$503$	−2.51574	−0.112171	−0.0560857	−	0.998426i	$-0.517862\pi$
$503$	−2.51574	−0.112171	−0.0560857	+	0.998426i	$0.517862\pi$
$504$	0	0
$505$	61.4408	2.73408
$506$	−36.9391	−1.64214
$507$	0	0
$508$	−29.1138	−1.29172
$509$	6.45906	0.286293	0.143146	−	0.989702i	$-0.454278\pi$
$509$	6.45906	0.286293	0.143146	+	0.989702i	$0.454278\pi$
$510$	0	0
$511$	3.73825	0.165371
$512$	29.3656	1.29779
$513$	0	0
$514$	−37.5309	−1.65542
$515$	−41.9745	−1.84962
$516$	0	0
$517$	−17.7336	−0.779923
$518$	−8.82047	−0.387549
$519$	0	0
$520$	−2.10631	−0.0923680
$521$	−1.31900	−0.0577866	−0.0288933	−	0.999583i	$-0.509198\pi$
$521$	−1.31900	−0.0577866	−0.0288933	+	0.999583i	$0.509198\pi$
$522$	0	0
$523$	18.1005	0.791482	0.395741	−	0.918362i	$-0.370488\pi$
$523$	18.1005	0.791482	0.395741	+	0.918362i	$0.370488\pi$
$524$	25.3711	1.10834
$525$	0	0
$526$	62.2187	2.71287
$527$	−10.1468	−0.442003
$528$	0	0
$529$	−10.0903	−0.438708
$530$	−45.7794	−1.98853
$531$	0	0
$532$	0.0805714	0.00349321
$533$	1.11629	0.0483517
$534$	0	0
$535$	−5.03210	−0.217557
$536$	2.93642	0.126834
$537$	0	0
$538$	−25.3747	−1.09398
$539$	−36.0598	−1.55321
$540$	0	0
$541$	19.2348	0.826968	0.413484	−	0.910511i	$-0.364312\pi$
$541$	19.2348	0.826968	0.413484	+	0.910511i	$0.364312\pi$
$542$	35.7674	1.53634
$543$	0	0
$544$	−24.7730	−1.06213
$545$	−50.2002	−2.15034
$546$	0	0
$547$	−3.56821	−0.152566	−0.0762828	−	0.997086i	$-0.524305\pi$
$547$	−3.56821	−0.152566	−0.0762828	+	0.997086i	$0.524305\pi$
$548$	31.0315	1.32560
$549$	0	0
$550$	92.2348	3.93291
$551$	−0.473512	−0.0201723
$552$	0	0
$553$	−6.37771	−0.271208
$554$	49.6381	2.10892
$555$	0	0
$556$	12.7532	0.540857
$557$	14.5428	0.616198	0.308099	−	0.951354i	$-0.400307\pi$
$557$	14.5428	0.616198	0.308099	+	0.951354i	$0.400307\pi$
$558$	0	0
$559$	7.12464	0.301340
$560$	7.76999	0.328342
$561$	0	0
$562$	3.37945	0.142553
$563$	10.1406	0.427377	0.213688	−	0.976902i	$-0.431452\pi$
$563$	10.1406	0.427377	0.213688	+	0.976902i	$0.431452\pi$
$564$	0	0
$565$	56.7391	2.38703
$566$	−5.36869	−0.225663
$567$	0	0
$568$	5.11772	0.214735
$569$	−34.4882	−1.44582	−0.722910	−	0.690942i	$-0.757196\pi$
$569$	−34.4882	−1.44582	−0.722910	+	0.690942i	$0.757196\pi$
$570$	0	0
$571$	−37.0652	−1.55113	−0.775565	−	0.631267i	$-0.782535\pi$
$571$	−37.0652	−1.55113	−0.775565	+	0.631267i	$0.782535\pi$
$572$	−10.3208	−0.431534
$573$	0	0
$574$	−0.902452	−0.0376676
$575$	−32.2348	−1.34428
$576$	0	0
$577$	−44.8314	−1.86636	−0.933178	−	0.359415i	$-0.882976\pi$
$577$	−44.8314	−1.86636	−0.933178	+	0.359415i	$0.882976\pi$
$578$	−12.3416	−0.513344
$579$	0	0
$580$	31.0019	1.28728
$581$	3.51276	0.145734
$582$	0	0
$583$	33.6766	1.39474
$584$	−4.04337	−0.167316
$585$	0	0
$586$	4.11950	0.170175
$587$	−11.8592	−0.489480	−0.244740	−	0.969589i	$-0.578703\pi$
$587$	−11.8592	−0.489480	−0.244740	+	0.969589i	$0.578703\pi$
$588$	0	0
$589$	−0.309150	−0.0127383
$590$	−31.8908	−1.31292
$591$	0	0
$592$	43.5327	1.78918
$593$	−19.8984	−0.817128	−0.408564	−	0.912730i	$-0.633970\pi$
$593$	−19.8984	−0.817128	−0.408564	+	0.912730i	$0.633970\pi$
$594$	0	0
$595$	−5.68435	−0.233036
$596$	2.89032	0.118392
$597$	0	0
$598$	7.75547	0.317145
$599$	42.9203	1.75367	0.876837	−	0.480787i	$-0.159649\pi$
$599$	42.9203	1.75367	0.876837	+	0.480787i	$0.159649\pi$
$600$	0	0
$601$	−25.5492	−1.04217	−0.521087	−	0.853504i	$-0.674473\pi$
$601$	−25.5492	−1.04217	−0.521087	+	0.853504i	$0.674473\pi$
$602$	−5.75985	−0.234754
$603$	0	0
$604$	−5.81053	−0.236427
$605$	−64.5485	−2.62427
$606$	0	0
$607$	4.07435	0.165373	0.0826863	−	0.996576i	$-0.473650\pi$
$607$	4.07435	0.165373	0.0826863	+	0.996576i	$0.473650\pi$
$608$	−0.754775	−0.0306102
$609$	0	0
$610$	58.0858	2.35182
$611$	3.72322	0.150625
$612$	0	0
$613$	−1.39932	−0.0565182	−0.0282591	−	0.999601i	$-0.508996\pi$
$613$	−1.39932	−0.0565182	−0.0282591	+	0.999601i	$0.508996\pi$
$614$	−9.89045	−0.399146
$615$	0	0
$616$	−1.25265	−0.0504707
$617$	−31.3506	−1.26213	−0.631064	−	0.775731i	$-0.717381\pi$
$617$	−31.3506	−1.26213	−0.631064	+	0.775731i	$0.717381\pi$
$618$	0	0
$619$	33.4416	1.34413	0.672065	−	0.740492i	$-0.265408\pi$
$619$	33.4416	1.34413	0.672065	+	0.740492i	$0.265408\pi$
$620$	20.2408	0.812889
$621$	0	0
$622$	−11.7765	−0.472196
$623$	1.96649	0.0787857
$624$	0	0
$625$	10.6308	0.425231
$626$	41.7729	1.66958
$627$	0	0
$628$	−34.4705	−1.37552
$629$	−31.8476	−1.26985
$630$	0	0
$631$	−32.0308	−1.27513	−0.637563	−	0.770398i	$-0.720057\pi$
$631$	−32.0308	−1.27513	−0.637563	+	0.770398i	$0.720057\pi$
$632$	6.89826	0.274398
$633$	0	0
$634$	28.1872	1.11946
$635$	62.5804	2.48343
$636$	0	0
$637$	7.57086	0.299968
$638$	−49.0356	−1.94134
$639$	0	0
$640$	−14.9665	−0.591605
$641$	7.36684	0.290973	0.145486	−	0.989360i	$-0.453525\pi$
$641$	7.36684	0.290973	0.145486	+	0.989360i	$0.453525\pi$
$642$	0	0
$643$	−8.46317	−0.333755	−0.166877	−	0.985978i	$-0.553368\pi$
$643$	−8.46317	−0.333755	−0.166877	+	0.985978i	$0.553368\pi$
$644$	−2.91603	−0.114908
$645$	0	0
$646$	0.625504	0.0246101
$647$	20.6282	0.810980	0.405490	−	0.914100i	$-0.367101\pi$
$647$	20.6282	0.810980	0.405490	+	0.914100i	$0.367101\pi$
$648$	0	0
$649$	23.4598	0.920876
$650$	−19.3650	−0.759556
$651$	0	0
$652$	24.2700	0.950486
$653$	−2.40219	−0.0940048	−0.0470024	−	0.998895i	$-0.514967\pi$
$653$	−2.40219	−0.0940048	−0.0470024	+	0.998895i	$0.514967\pi$
$654$	0	0
$655$	−54.5353	−2.13087
$656$	4.45398	0.173899
$657$	0	0
$658$	−3.01000	−0.117342
$659$	39.9128	1.55478	0.777390	−	0.629018i	$-0.216543\pi$
$659$	39.9128	1.55478	0.777390	+	0.629018i	$0.216543\pi$
$660$	0	0
$661$	16.0322	0.623581	0.311791	−	0.950151i	$-0.399071\pi$
$661$	16.0322	0.623581	0.311791	+	0.950151i	$0.399071\pi$
$662$	−17.1800	−0.667721
$663$	0	0
$664$	−3.79947	−0.147448
$665$	−0.173189	−0.00671598
$666$	0	0
$667$	17.1373	0.663558
$668$	41.8927	1.62088
$669$	0	0
$670$	42.0425	1.62424
$671$	−42.7295	−1.64956
$672$	0	0
$673$	−22.3712	−0.862347	−0.431174	−	0.902269i	$-0.641900\pi$
$673$	−22.3712	−0.862347	−0.431174	+	0.902269i	$0.641900\pi$
$674$	−24.7214	−0.952232
$675$	0	0
$676$	−20.4393	−0.786125
$677$	−40.1056	−1.54138	−0.770692	−	0.637208i	$-0.780089\pi$
$677$	−40.1056	−1.54138	−0.770692	+	0.637208i	$0.780089\pi$
$678$	0	0
$679$	−8.74084	−0.335443
$680$	6.14831	0.235777
$681$	0	0
$682$	−32.0148	−1.22591
$683$	−4.42587	−0.169351	−0.0846755	−	0.996409i	$-0.526985\pi$
$683$	−4.42587	−0.169351	−0.0846755	+	0.996409i	$0.526985\pi$
$684$	0	0
$685$	−66.7025	−2.54857
$686$	−12.4378	−0.474876
$687$	0	0
$688$	28.4273	1.08378
$689$	−7.07050	−0.269365
$690$	0	0
$691$	−23.9837	−0.912382	−0.456191	−	0.889882i	$-0.650787\pi$
$691$	−23.9837	−0.912382	−0.456191	+	0.889882i	$0.650787\pi$
$692$	−11.1946	−0.425557
$693$	0	0
$694$	−32.8337	−1.24635
$695$	−27.4131	−1.03984
$696$	0	0
$697$	−3.25843	−0.123422
$698$	−49.7340	−1.88246
$699$	0	0
$700$	7.28115	0.275202
$701$	−5.86495	−0.221516	−0.110758	−	0.993847i	$-0.535328\pi$
$701$	−5.86495	−0.221516	−0.110758	+	0.993847i	$0.535328\pi$
$702$	0	0
$703$	−0.970320	−0.0365963
$704$	−30.8002	−1.16083
$705$	0	0
$706$	34.7482	1.30776
$707$	−7.67159	−0.288520
$708$	0	0
$709$	8.54455	0.320897	0.160449	−	0.987044i	$-0.448706\pi$
$709$	8.54455	0.320897	0.160449	+	0.987044i	$0.448706\pi$
$710$	73.2736	2.74991
$711$	0	0
$712$	−2.12699	−0.0797124
$713$	11.1887	0.419021
$714$	0	0
$715$	22.1846	0.829658
$716$	−3.22349	−0.120467
$717$	0	0
$718$	−23.6278	−0.881780
$719$	−40.3054	−1.50314	−0.751569	−	0.659654i	$-0.770703\pi$
$719$	−40.3054	−1.50314	−0.751569	+	0.659654i	$0.770703\pi$
$720$	0	0
$721$	5.24099	0.195185
$722$	−36.7200	−1.36658
$723$	0	0
$724$	29.8104	1.10789
$725$	−42.7908	−1.58921
$726$	0	0
$727$	9.89041	0.366815	0.183407	−	0.983037i	$-0.441287\pi$
$727$	9.89041	0.366815	0.183407	+	0.983037i	$0.441287\pi$
$728$	0.262997	0.00974733
$729$	0	0
$730$	−57.8915	−2.14266
$731$	−20.7968	−0.769197
$732$	0	0
$733$	−15.6316	−0.577368	−0.288684	−	0.957424i	$-0.593218\pi$
$733$	−15.6316	−0.577368	−0.288684	+	0.957424i	$0.593218\pi$
$734$	11.8644	0.437924
$735$	0	0
$736$	27.3167	1.00691
$737$	−30.9276	−1.13923
$738$	0	0
$739$	−7.50399	−0.276039	−0.138019	−	0.990430i	$-0.544074\pi$
$739$	−7.50399	−0.276039	−0.138019	+	0.990430i	$0.544074\pi$
$740$	63.5291	2.33538
$741$	0	0
$742$	5.71609	0.209844
$743$	−6.60676	−0.242379	−0.121189	−	0.992629i	$-0.538671\pi$
$743$	−6.60676	−0.242379	−0.121189	+	0.992629i	$0.538671\pi$
$744$	0	0
$745$	−6.21277	−0.227618
$746$	58.3977	2.13809
$747$	0	0
$748$	30.1263	1.10153
$749$	0.628315	0.0229581
$750$	0	0
$751$	−33.2748	−1.21421	−0.607107	−	0.794620i	$-0.707670\pi$
$751$	−33.2748	−1.21421	−0.607107	+	0.794620i	$0.707670\pi$
$752$	14.8556	0.541729
$753$	0	0
$754$	10.2952	0.374928
$755$	12.4898	0.454550
$756$	0	0
$757$	9.81356	0.356680	0.178340	−	0.983969i	$-0.442927\pi$
$757$	9.81356	0.356680	0.178340	+	0.983969i	$0.442927\pi$
$758$	−16.1302	−0.585876
$759$	0	0
$760$	0.187325	0.00679498
$761$	21.7418	0.788138	0.394069	−	0.919081i	$-0.371067\pi$
$761$	21.7418	0.788138	0.394069	+	0.919081i	$0.371067\pi$
$762$	0	0
$763$	6.26806	0.226919
$764$	10.5694	0.382386
$765$	0	0
$766$	31.4576	1.13661
$767$	−4.92544	−0.177847
$768$	0	0
$769$	12.2590	0.442071	0.221035	−	0.975266i	$-0.429056\pi$
$769$	12.2590	0.442071	0.221035	+	0.975266i	$0.429056\pi$
$770$	−17.9350	−0.646331
$771$	0	0
$772$	5.12323	0.184389
$773$	23.3957	0.841486	0.420743	−	0.907180i	$-0.361769\pi$
$773$	23.3957	0.841486	0.420743	+	0.907180i	$0.361769\pi$
$774$	0	0
$775$	−27.9376	−1.00355
$776$	9.45428	0.339389
$777$	0	0
$778$	−39.9434	−1.43204
$779$	−0.0992768	−0.00355696
$780$	0	0
$781$	−53.9021	−1.92877
$782$	−22.6382	−0.809539
$783$	0	0
$784$	30.2077	1.07885
$785$	74.0946	2.64455
$786$	0	0
$787$	36.2291	1.29143	0.645715	−	0.763579i	$-0.276560\pi$
$787$	36.2291	1.29143	0.645715	+	0.763579i	$0.276560\pi$
$788$	−36.0686	−1.28489
$789$	0	0
$790$	98.7667	3.51396
$791$	−7.08452	−0.251896
$792$	0	0
$793$	8.97118	0.318576
$794$	35.1019	1.24572
$795$	0	0
$796$	−45.7805	−1.62265
$797$	0.114523	0.00405662	0.00202831	−	0.999998i	$-0.499354\pi$
$797$	0.114523	0.00405662	0.00202831	+	0.999998i	$0.499354\pi$
$798$	0	0
$799$	−10.8680	−0.384484
$800$	−68.2083	−2.41153
$801$	0	0
$802$	39.7102	1.40222
$803$	42.5866	1.50285
$804$	0	0
$805$	6.26802	0.220919
$806$	6.72159	0.236758
$807$	0	0
$808$	8.29775	0.291914
$809$	20.5034	0.720861	0.360431	−	0.932786i	$-0.382630\pi$
$809$	20.5034	0.720861	0.360431	+	0.932786i	$0.382630\pi$
$810$	0	0
$811$	16.6957	0.586264	0.293132	−	0.956072i	$-0.405302\pi$
$811$	16.6957	0.586264	0.293132	+	0.956072i	$0.405302\pi$
$812$	−3.87094	−0.135843
$813$	0	0
$814$	−100.484	−3.52195
$815$	−52.1686	−1.82739
$816$	0	0
$817$	−0.633629	−0.0221679
$818$	1.54075	0.0538709
$819$	0	0
$820$	6.49988	0.226985
$821$	−19.3688	−0.675977	−0.337989	−	0.941150i	$-0.609747\pi$
$821$	−19.3688	−0.675977	−0.337989	+	0.941150i	$0.609747\pi$
$822$	0	0
$823$	34.3954	1.19895	0.599475	−	0.800394i	$-0.295376\pi$
$823$	34.3954	1.19895	0.599475	+	0.800394i	$0.295376\pi$
$824$	−5.66877	−0.197481
$825$	0	0
$826$	3.98193	0.138549
$827$	41.0753	1.42833	0.714163	−	0.699979i	$-0.246807\pi$
$827$	41.0753	1.42833	0.714163	+	0.699979i	$0.246807\pi$
$828$	0	0
$829$	−18.2219	−0.632874	−0.316437	−	0.948614i	$-0.602487\pi$
$829$	−18.2219	−0.632874	−0.316437	+	0.948614i	$0.602487\pi$
$830$	−54.3994	−1.88823
$831$	0	0
$832$	6.46659	0.224189
$833$	−22.0993	−0.765694
$834$	0	0
$835$	−90.0487	−3.11626
$836$	0.917878	0.0317455
$837$	0	0
$838$	64.1246	2.21515
$839$	−13.6864	−0.472507	−0.236254	−	0.971691i	$-0.575920\pi$
$839$	−13.6864	−0.472507	−0.236254	+	0.971691i	$0.575920\pi$
$840$	0	0
$841$	−6.25076	−0.215544
$842$	18.3322	0.631770
$843$	0	0
$844$	−23.6938	−0.815573
$845$	43.9344	1.51139
$846$	0	0
$847$	8.05961	0.276932
$848$	−28.2113	−0.968779
$849$	0	0
$850$	56.5262	1.93883
$851$	35.1177	1.20382
$852$	0	0
$853$	54.9535	1.88157	0.940786	−	0.339001i	$-0.110089\pi$
$853$	54.9535	1.88157	0.940786	+	0.339001i	$0.110089\pi$
$854$	−7.25267	−0.248181
$855$	0	0
$856$	−0.679599	−0.0232282
$857$	13.3533	0.456139	0.228069	−	0.973645i	$-0.426759\pi$
$857$	13.3533	0.456139	0.228069	+	0.973645i	$0.426759\pi$
$858$	0	0
$859$	−10.2230	−0.348805	−0.174402	−	0.984674i	$-0.555799\pi$
$859$	−10.2230	−0.348805	−0.174402	+	0.984674i	$0.555799\pi$
$860$	41.4851	1.41463
$861$	0	0
$862$	22.5951	0.769592
$863$	7.53999	0.256664	0.128332	−	0.991731i	$-0.459038\pi$
$863$	7.53999	0.256664	0.128332	+	0.991731i	$0.459038\pi$
$864$	0	0
$865$	24.0630	0.818166
$866$	5.21829	0.177325
$867$	0	0
$868$	−2.52729	−0.0857819
$869$	−72.6555	−2.46467
$870$	0	0
$871$	6.49334	0.220018
$872$	−6.77967	−0.229588
$873$	0	0
$874$	−0.689731	−0.0233305
$875$	−6.92837	−0.234222
$876$	0	0
$877$	58.1790	1.96457	0.982283	−	0.187404i	$-0.0600072\pi$
$877$	58.1790	1.96457	0.982283	+	0.187404i	$0.0600072\pi$
$878$	41.3267	1.39471
$879$	0	0
$880$	88.5165	2.98389
$881$	−35.8338	−1.20727	−0.603636	−	0.797260i	$-0.706282\pi$
$881$	−35.8338	−1.20727	−0.603636	+	0.797260i	$0.706282\pi$
$882$	0	0
$883$	29.7633	1.00161	0.500807	−	0.865559i	$-0.333037\pi$
$883$	29.7633	1.00161	0.500807	+	0.865559i	$0.333037\pi$
$884$	−6.32510	−0.212736
$885$	0	0
$886$	−31.6362	−1.06284
$887$	−27.4150	−0.920506	−0.460253	−	0.887788i	$-0.652241\pi$
$887$	−27.4150	−0.920506	−0.460253	+	0.887788i	$0.652241\pi$
$888$	0	0
$889$	−7.81387	−0.262069
$890$	−30.4535	−1.02080
$891$	0	0
$892$	19.3196	0.646867
$893$	−0.331124	−0.0110806
$894$	0	0
$895$	6.92892	0.231608
$896$	1.86874	0.0624303
$897$	0	0
$898$	−15.2522	−0.508971
$899$	14.8527	0.495366
$900$	0	0
$901$	20.6387	0.687576
$902$	−10.2808	−0.342314
$903$	0	0
$904$	7.66276	0.254860
$905$	−64.0776	−2.13001
$906$	0	0
$907$	5.49533	0.182469	0.0912346	−	0.995829i	$-0.470919\pi$
$907$	5.49533	0.182469	0.0912346	+	0.995829i	$0.470919\pi$
$908$	−8.84108	−0.293401
$909$	0	0
$910$	3.76549	0.124825
$911$	−30.0729	−0.996360	−0.498180	−	0.867074i	$-0.665998\pi$
$911$	−30.0729	−0.996360	−0.498180	+	0.867074i	$0.665998\pi$
$912$	0	0
$913$	40.0177	1.32439
$914$	−36.6872	−1.21351
$915$	0	0
$916$	−2.84450	−0.0939850
$917$	6.80936	0.224865
$918$	0	0
$919$	7.92860	0.261540	0.130770	−	0.991413i	$-0.458255\pi$
$919$	7.92860	0.261540	0.130770	+	0.991413i	$0.458255\pi$
$920$	−6.77962	−0.223518
$921$	0	0
$922$	−19.5817	−0.644891
$923$	11.3169	0.372500
$924$	0	0
$925$	−87.6868	−2.88313
$926$	−28.8404	−0.947755
$927$	0	0
$928$	36.2622	1.19036
$929$	45.7764	1.50188	0.750938	−	0.660373i	$-0.229602\pi$
$929$	45.7764	1.50188	0.750938	+	0.660373i	$0.229602\pi$
$930$	0	0
$931$	−0.673313	−0.0220669
$932$	−33.1007	−1.08425
$933$	0	0
$934$	−4.26979	−0.139712
$935$	−64.7568	−2.11777
$936$	0	0
$937$	−1.60857	−0.0525498	−0.0262749	−	0.999655i	$-0.508365\pi$
$937$	−1.60857	−0.0525498	−0.0262749	+	0.999655i	$0.508365\pi$
$938$	−5.24948	−0.171402
$939$	0	0
$940$	21.6794	0.707105
$941$	−27.7771	−0.905506	−0.452753	−	0.891636i	$-0.649558\pi$
$941$	−27.7771	−0.905506	−0.452753	+	0.891636i	$0.649558\pi$
$942$	0	0
$943$	3.59301	0.117004
$944$	−19.6525	−0.639634
$945$	0	0
$946$	−65.6169	−2.13339
$947$	−3.15789	−0.102618	−0.0513088	−	0.998683i	$-0.516339\pi$
$947$	−3.15789	−0.102618	−0.0513088	+	0.998683i	$0.516339\pi$
$948$	0	0
$949$	−8.94117	−0.290243
$950$	1.72222	0.0558761
$951$	0	0
$952$	−0.767687	−0.0248809
$953$	17.9816	0.582483	0.291241	−	0.956650i	$-0.405932\pi$
$953$	17.9816	0.582483	0.291241	+	0.956650i	$0.405932\pi$
$954$	0	0
$955$	−22.7189	−0.735167
$956$	−2.24719	−0.0726795
$957$	0	0
$958$	−2.26172	−0.0730729
$959$	8.32857	0.268943
$960$	0	0
$961$	−21.3028	−0.687188
$962$	21.0968	0.680189
$963$	0	0
$964$	2.25037	0.0724796
$965$	−11.0124	−0.354503
$966$	0	0
$967$	−19.4803	−0.626444	−0.313222	−	0.949680i	$-0.601408\pi$
$967$	−19.4803	−0.626444	−0.313222	+	0.949680i	$0.601408\pi$
$968$	−8.71745	−0.280189
$969$	0	0
$970$	135.363	4.34623
$971$	2.32619	0.0746509	0.0373255	−	0.999303i	$-0.488116\pi$
$971$	2.32619	0.0746509	0.0373255	+	0.999303i	$0.488116\pi$
$972$	0	0
$973$	3.42284	0.109731
$974$	−39.2666	−1.25818
$975$	0	0
$976$	35.7950	1.14577
$977$	55.3654	1.77130	0.885648	−	0.464357i	$-0.153714\pi$
$977$	55.3654	1.77130	0.885648	+	0.464357i	$0.153714\pi$
$978$	0	0
$979$	22.4024	0.715985
$980$	44.0833	1.40819
$981$	0	0
$982$	29.1846	0.931319
$983$	−23.0896	−0.736443	−0.368222	−	0.929738i	$-0.620033\pi$
$983$	−23.0896	−0.736443	−0.368222	+	0.929738i	$0.620033\pi$
$984$	0	0
$985$	77.5297	2.47030
$986$	−30.0515	−0.957034
$987$	0	0
$988$	−0.192711	−0.00613095
$989$	22.9322	0.729202
$990$	0	0
$991$	38.4258	1.22064	0.610318	−	0.792156i	$-0.291042\pi$
$991$	38.4258	1.22064	0.610318	+	0.792156i	$0.291042\pi$
$992$	23.6751	0.751686
$993$	0	0
$994$	−9.14904	−0.290190
$995$	98.4057	3.11967
$996$	0	0
$997$	11.2216	0.355393	0.177696	−	0.984085i	$-0.443135\pi$
$997$	11.2216	0.355393	0.177696	+	0.984085i	$0.443135\pi$
$998$	−19.7109	−0.623937
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3321.2.a.i.1.14		16
3.2	odd	2		3321.2.a.j.1.3		16
9.2	odd	6		1107.2.e.a.739.14		32
9.4	even	3		369.2.e.a.124.3	✓	32
9.5	odd	6		1107.2.e.a.370.14		32
9.7	even	3		369.2.e.a.247.3	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
369.2.e.a.124.3	✓	32	9.4	even	3
369.2.e.a.247.3	yes	32	9.7	even	3
1107.2.e.a.370.14		32	9.5	odd	6
1107.2.e.a.739.14		32	9.2	odd	6
3321.2.a.i.1.14		16	1.1	even	1	trivial
3321.2.a.j.1.3		16	3.2	odd	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:	\( N \)	\(=\)	\( 3321 = 3^{4} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3321.a (trivial)

\(f(q)\)	\(=\)	\(q+1.93363 q^{2} +1.73893 q^{4} -3.73785 q^{5} +0.466713 q^{7} -0.504807 q^{8} -7.22763 q^{10} +5.31685 q^{11} -1.11629 q^{13} +0.902452 q^{14} -4.45398 q^{16} +3.25843 q^{17} +0.0992768 q^{19} -6.49988 q^{20} +10.2808 q^{22} -3.59301 q^{23} +8.97154 q^{25} -2.15849 q^{26} +0.811584 q^{28} -4.76962 q^{29} -3.11403 q^{31} -7.60274 q^{32} +6.30061 q^{34} -1.74451 q^{35} -9.77389 q^{37} +0.191965 q^{38} +1.88689 q^{40} -1.00000 q^{41} -6.38245 q^{43} +9.24564 q^{44} -6.94756 q^{46} -3.33536 q^{47} -6.78218 q^{49} +17.3477 q^{50} -1.94115 q^{52} +6.33395 q^{53} -19.8736 q^{55} -0.235600 q^{56} -9.22268 q^{58} +4.41235 q^{59} -8.03663 q^{61} -6.02138 q^{62} -5.79295 q^{64} +4.17251 q^{65} -5.81691 q^{67} +5.66620 q^{68} -3.37323 q^{70} -10.1380 q^{71} +8.00974 q^{73} -18.8991 q^{74} +0.172636 q^{76} +2.48144 q^{77} -13.6652 q^{79} +16.6483 q^{80} -1.93363 q^{82} +7.52659 q^{83} -12.1795 q^{85} -12.3413 q^{86} -2.68398 q^{88} +4.21348 q^{89} -0.520986 q^{91} -6.24800 q^{92} -6.44936 q^{94} -0.371082 q^{95} -18.7285 q^{97} -13.1142 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4} - q^{5} - 13 q^{7} - 8 q^{10} + 2 q^{11} - 5 q^{13} - q^{14} - 8 q^{16} + q^{17} - 19 q^{19} - 11 q^{20} - 14 q^{22} - 2 q^{23} - 3 q^{25} + 15 q^{26} - 24 q^{28} + 10 q^{29} - 39 q^{31}+ \cdots - 48 q^{98}+O(q^{100}) \) 16 * q + 8 * q^4 - q^5 - 13 * q^7 - 8 * q^10 + 2 * q^11 - 5 * q^13 - q^14 - 8 * q^16 + q^17 - 19 * q^19 - 11 * q^20 - 14 * q^22 - 2 * q^23 - 3 * q^25 + 15 * q^26 - 24 * q^28 + 10 * q^29 - 39 * q^31 + 10 * q^32 - 18 * q^34 + 2 * q^35 - 26 * q^37 + 14 * q^38 - 18 * q^40 - 16 * q^41 - 20 * q^43 - 23 * q^44 - 26 * q^46 - 5 * q^47 - 7 * q^49 - 9 * q^50 - 14 * q^52 + 7 * q^53 - 37 * q^55 + 2 * q^56 - 24 * q^58 - 6 * q^59 - 21 * q^61 - 5 * q^62 - 34 * q^64 - 2 * q^65 - 32 * q^67 + 8 * q^68 - 22 * q^70 - 20 * q^71 - 17 * q^73 - 28 * q^74 - 40 * q^76 + 22 * q^77 - 69 * q^79 + 39 * q^80 - 10 * q^83 - 42 * q^85 + 8 * q^86 - 42 * q^88 + 11 * q^89 - 51 * q^91 + 10 * q^92 - 13 * q^94 + 22 * q^95 - 25 * q^97 - 48 * q^98

Introduction

L-functions

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