LMFDB - Embedded newform 3720.2.a.o.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3720,2,Mod(1,3720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3720.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3720.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,-3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.7043495519$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	3720.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +1.17009 q^{7} +1.00000 q^{9} -5.41855 q^{11} -1.17009 q^{13} -1.00000 q^{15} +7.12783 q^{17} -4.00000 q^{19} +1.17009 q^{21} -6.97107 q^{23} +1.00000 q^{25} +1.00000 q^{27} +9.21953 q^{29} -1.00000 q^{31} -5.41855 q^{33} -1.17009 q^{35} -2.09171 q^{37} -1.17009 q^{39} -5.26180 q^{41} -6.00000 q^{43} -1.00000 q^{45} -1.52586 q^{47} -5.63090 q^{49} +7.12783 q^{51} +0.474142 q^{53} +5.41855 q^{55} -4.00000 q^{57} +14.6381 q^{59} -11.9421 q^{61} +1.17009 q^{63} +1.17009 q^{65} -13.3268 q^{67} -6.97107 q^{69} -8.14116 q^{71} -8.74539 q^{73} +1.00000 q^{75} -6.34017 q^{77} +9.46800 q^{79} +1.00000 q^{81} -3.86603 q^{83} -7.12783 q^{85} +9.21953 q^{87} -2.04226 q^{89} -1.36910 q^{91} -1.00000 q^{93} +4.00000 q^{95} -2.73820 q^{97} -5.41855 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 3 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 3 q^{31} - 2 q^{33} + 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 - 3 * q^15 - 12 * q^19 - 2 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 - 3 * q^31 - 2 * q^33 + 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 - 18 * q^43 - 3 * q^45 - 2 * q^47 - 13 * q^49 + 4 * q^53 + 2 * q^55 - 12 * q^57 + 6 * q^59 - 6 * q^61 - 2 * q^63 - 2 * q^65 - 28 * q^67 - 6 * q^69 - 4 * q^71 + 3 * q^75 - 8 * q^77 - 4 * q^79 + 3 * q^81 + 2 * q^83 + 4 * q^87 - 22 * q^89 - 8 * q^91 - 3 * q^93 + 12 * q^95 - 16 * q^97 - 2 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.17009	0.442251	0.221126	−	0.975245i	$-0.429027\pi$
$7$	1.17009	0.442251	0.221126	+	0.975245i	$0.429027\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.41855	−1.63375	−0.816877	−	0.576812i	$-0.804297\pi$
$11$	−5.41855	−1.63375	−0.816877	+	0.576812i	$0.804297\pi$
$12$	0	0
$13$	−1.17009	−0.324524	−0.162262	−	0.986748i	$-0.551879\pi$
$13$	−1.17009	−0.324524	−0.162262	+	0.986748i	$0.551879\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	7.12783	1.72875	0.864376	−	0.502846i	$-0.167714\pi$
$17$	7.12783	1.72875	0.864376	+	0.502846i	$0.167714\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	1.17009	0.255334
$22$	0	0
$23$	−6.97107	−1.45357	−0.726784	−	0.686866i	$-0.758986\pi$
$23$	−6.97107	−1.45357	−0.726784	+	0.686866i	$0.758986\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	9.21953	1.71202	0.856012	−	0.516955i	$-0.172935\pi$
$29$	9.21953	1.71202	0.856012	+	0.516955i	$0.172935\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−5.41855	−0.943249
$34$	0	0
$35$	−1.17009	−0.197781
$36$	0	0
$37$	−2.09171	−0.343875	−0.171937	−	0.985108i	$-0.555003\pi$
$37$	−2.09171	−0.343875	−0.171937	+	0.985108i	$0.555003\pi$
$38$	0	0
$39$	−1.17009	−0.187364
$40$	0	0
$41$	−5.26180	−0.821754	−0.410877	−	0.911691i	$-0.634777\pi$
$41$	−5.26180	−0.821754	−0.410877	+	0.911691i	$0.634777\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−1.52586	−0.222569	−0.111285	−	0.993789i	$-0.535497\pi$
$47$	−1.52586	−0.222569	−0.111285	+	0.993789i	$0.535497\pi$
$48$	0	0
$49$	−5.63090	−0.804414
$50$	0	0
$51$	7.12783	0.998095
$52$	0	0
$53$	0.474142	0.0651284	0.0325642	−	0.999470i	$-0.489633\pi$
$53$	0.474142	0.0651284	0.0325642	+	0.999470i	$0.489633\pi$
$54$	0	0
$55$	5.41855	0.730637
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	14.6381	1.90572	0.952858	−	0.303416i	$-0.0981272\pi$
$59$	14.6381	1.90572	0.952858	+	0.303416i	$0.0981272\pi$
$60$	0	0
$61$	−11.9421	−1.52903	−0.764517	−	0.644603i	$-0.777023\pi$
$61$	−11.9421	−1.52903	−0.764517	+	0.644603i	$0.777023\pi$
$62$	0	0
$63$	1.17009	0.147417
$64$	0	0
$65$	1.17009	0.145131
$66$	0	0
$67$	−13.3268	−1.62813	−0.814066	−	0.580772i	$-0.802751\pi$
$67$	−13.3268	−1.62813	−0.814066	+	0.580772i	$0.802751\pi$
$68$	0	0
$69$	−6.97107	−0.839218
$70$	0	0
$71$	−8.14116	−0.966178	−0.483089	−	0.875571i	$-0.660485\pi$
$71$	−8.14116	−0.966178	−0.483089	+	0.875571i	$0.660485\pi$
$72$	0	0
$73$	−8.74539	−1.02357	−0.511785	−	0.859113i	$-0.671016\pi$
$73$	−8.74539	−1.02357	−0.511785	+	0.859113i	$0.671016\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−6.34017	−0.722530
$78$	0	0
$79$	9.46800	1.06523	0.532617	−	0.846357i	$-0.321209\pi$
$79$	9.46800	1.06523	0.532617	+	0.846357i	$0.321209\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.86603	−0.424352	−0.212176	−	0.977231i	$-0.568055\pi$
$83$	−3.86603	−0.424352	−0.212176	+	0.977231i	$0.568055\pi$
$84$	0	0
$85$	−7.12783	−0.773121
$86$	0	0
$87$	9.21953	0.988438
$88$	0	0
$89$	−2.04226	−0.216479	−0.108240	−	0.994125i	$-0.534521\pi$
$89$	−2.04226	−0.216479	−0.108240	+	0.994125i	$0.534521\pi$
$90$	0	0
$91$	−1.36910	−0.143521
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−2.73820	−0.278023	−0.139011	−	0.990291i	$-0.544392\pi$
$97$	−2.73820	−0.278023	−0.139011	+	0.990291i	$0.544392\pi$
$98$	0	0
$99$	−5.41855	−0.544585
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−8.92881	−0.879782	−0.439891	−	0.898051i	$-0.644983\pi$
$103$	−8.92881	−0.879782	−0.439891	+	0.898051i	$0.644983\pi$
$104$	0	0
$105$	−1.17009	−0.114189
$106$	0	0
$107$	−15.6514	−1.51308	−0.756540	−	0.653948i	$-0.773112\pi$
$107$	−15.6514	−1.51308	−0.756540	+	0.653948i	$0.773112\pi$
$108$	0	0
$109$	−14.0228	−1.34314	−0.671570	−	0.740941i	$-0.734380\pi$
$109$	−14.0228	−1.34314	−0.671570	+	0.740941i	$0.734380\pi$
$110$	0	0
$111$	−2.09171	−0.198536
$112$	0	0
$113$	5.23513	0.492480	0.246240	−	0.969209i	$-0.420805\pi$
$113$	5.23513	0.492480	0.246240	+	0.969209i	$0.420805\pi$
$114$	0	0
$115$	6.97107	0.650056
$116$	0	0
$117$	−1.17009	−0.108175
$118$	0	0
$119$	8.34017	0.764542
$120$	0	0
$121$	18.3607	1.66915
$122$	0	0
$123$	−5.26180	−0.474440
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.18342	−0.726161	−0.363080	−	0.931758i	$-0.618275\pi$
$127$	−8.18342	−0.726161	−0.363080	+	0.931758i	$0.618275\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	7.21953	0.630774	0.315387	−	0.948963i	$-0.397866\pi$
$131$	7.21953	0.630774	0.315387	+	0.948963i	$0.397866\pi$
$132$	0	0
$133$	−4.68035	−0.405837
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	8.38962	0.716774	0.358387	−	0.933573i	$-0.383327\pi$
$137$	8.38962	0.716774	0.358387	+	0.933573i	$0.383327\pi$
$138$	0	0
$139$	−18.3402	−1.55559	−0.777797	−	0.628516i	$-0.783663\pi$
$139$	−18.3402	−1.55559	−0.777797	+	0.628516i	$0.783663\pi$
$140$	0	0
$141$	−1.52586	−0.128500
$142$	0	0
$143$	6.34017	0.530192
$144$	0	0
$145$	−9.21953	−0.765641
$146$	0	0
$147$	−5.63090	−0.464429
$148$	0	0
$149$	12.0989	0.991180	0.495590	−	0.868557i	$-0.334952\pi$
$149$	12.0989	0.991180	0.495590	+	0.868557i	$0.334952\pi$
$150$	0	0
$151$	22.3318	1.81733	0.908667	−	0.417523i	$-0.137102\pi$
$151$	22.3318	1.81733	0.908667	+	0.417523i	$0.137102\pi$
$152$	0	0
$153$	7.12783	0.576251
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	14.8371	1.18413	0.592065	−	0.805890i	$-0.298313\pi$
$157$	14.8371	1.18413	0.592065	+	0.805890i	$0.298313\pi$
$158$	0	0
$159$	0.474142	0.0376019
$160$	0	0
$161$	−8.15676	−0.642842
$162$	0	0
$163$	−1.59478	−0.124913	−0.0624564	−	0.998048i	$-0.519893\pi$
$163$	−1.59478	−0.124913	−0.0624564	+	0.998048i	$0.519893\pi$
$164$	0	0
$165$	5.41855	0.421834
$166$	0	0
$167$	3.91548	0.302989	0.151494	−	0.988458i	$-0.451591\pi$
$167$	3.91548	0.302989	0.151494	+	0.988458i	$0.451591\pi$
$168$	0	0
$169$	−11.6309	−0.894684
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−1.23513	−0.0939054	−0.0469527	−	0.998897i	$-0.514951\pi$
$173$	−1.23513	−0.0939054	−0.0469527	+	0.998897i	$0.514951\pi$
$174$	0	0
$175$	1.17009	0.0884502
$176$	0	0
$177$	14.6381	1.10027
$178$	0	0
$179$	−15.4186	−1.15244	−0.576218	−	0.817296i	$-0.695472\pi$
$179$	−15.4186	−1.15244	−0.576218	+	0.817296i	$0.695472\pi$
$180$	0	0
$181$	−7.26180	−0.539765	−0.269882	−	0.962893i	$-0.586985\pi$
$181$	−7.26180	−0.539765	−0.269882	+	0.962893i	$0.586985\pi$
$182$	0	0
$183$	−11.9421	−0.882788
$184$	0	0
$185$	2.09171	0.153785
$186$	0	0
$187$	−38.6225	−2.82436
$188$	0	0
$189$	1.17009	0.0851113
$190$	0	0
$191$	17.8732	1.29326	0.646630	−	0.762803i	$-0.276178\pi$
$191$	17.8732	1.29326	0.646630	+	0.762803i	$0.276178\pi$
$192$	0	0
$193$	21.6742	1.56014	0.780072	−	0.625690i	$-0.215183\pi$
$193$	21.6742	1.56014	0.780072	+	0.625690i	$0.215183\pi$
$194$	0	0
$195$	1.17009	0.0837916
$196$	0	0
$197$	−22.0638	−1.57198	−0.785991	−	0.618238i	$-0.787847\pi$
$197$	−22.0638	−1.57198	−0.785991	+	0.618238i	$0.787847\pi$
$198$	0	0
$199$	−16.3896	−1.16183	−0.580915	−	0.813964i	$-0.697305\pi$
$199$	−16.3896	−1.16183	−0.580915	+	0.813964i	$0.697305\pi$
$200$	0	0
$201$	−13.3268	−0.940003
$202$	0	0
$203$	10.7877	0.757145
$204$	0	0
$205$	5.26180	0.367500
$206$	0	0
$207$	−6.97107	−0.484523
$208$	0	0
$209$	21.6742	1.49924
$210$	0	0
$211$	−11.6020	−0.798712	−0.399356	−	0.916796i	$-0.630766\pi$
$211$	−11.6020	−0.798712	−0.399356	+	0.916796i	$0.630766\pi$
$212$	0	0
$213$	−8.14116	−0.557823
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	−1.17009	−0.0794306
$218$	0	0
$219$	−8.74539	−0.590959
$220$	0	0
$221$	−8.34017	−0.561021
$222$	0	0
$223$	11.5441	0.773051	0.386525	−	0.922279i	$-0.373675\pi$
$223$	11.5441	0.773051	0.386525	+	0.922279i	$0.373675\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−14.7031	−0.975881	−0.487941	−	0.872877i	$-0.662252\pi$
$227$	−14.7031	−0.975881	−0.487941	+	0.872877i	$0.662252\pi$
$228$	0	0
$229$	−9.60197	−0.634516	−0.317258	−	0.948339i	$-0.602762\pi$
$229$	−9.60197	−0.634516	−0.317258	+	0.948339i	$0.602762\pi$
$230$	0	0
$231$	−6.34017	−0.417153
$232$	0	0
$233$	−10.4163	−0.682393	−0.341197	−	0.939992i	$-0.610832\pi$
$233$	−10.4163	−0.682393	−0.341197	+	0.939992i	$0.610832\pi$
$234$	0	0
$235$	1.52586	0.0995360
$236$	0	0
$237$	9.46800	0.615013
$238$	0	0
$239$	−9.02052	−0.583489	−0.291744	−	0.956496i	$-0.594236\pi$
$239$	−9.02052	−0.583489	−0.291744	+	0.956496i	$0.594236\pi$
$240$	0	0
$241$	3.97334	0.255945	0.127973	−	0.991778i	$-0.459153\pi$
$241$	3.97334	0.255945	0.127973	+	0.991778i	$0.459153\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	5.63090	0.359745
$246$	0	0
$247$	4.68035	0.297803
$248$	0	0
$249$	−3.86603	−0.245000
$250$	0	0
$251$	5.94214	0.375065	0.187532	−	0.982258i	$-0.439951\pi$
$251$	5.94214	0.375065	0.187532	+	0.982258i	$0.439951\pi$
$252$	0	0
$253$	37.7731	2.37477
$254$	0	0
$255$	−7.12783	−0.446362
$256$	0	0
$257$	13.7359	0.856824	0.428412	−	0.903583i	$-0.359073\pi$
$257$	13.7359	0.856824	0.428412	+	0.903583i	$0.359073\pi$
$258$	0	0
$259$	−2.44748	−0.152079
$260$	0	0
$261$	9.21953	0.570675
$262$	0	0
$263$	16.8638	1.03986	0.519932	−	0.854208i	$-0.325957\pi$
$263$	16.8638	1.03986	0.519932	+	0.854208i	$0.325957\pi$
$264$	0	0
$265$	−0.474142	−0.0291263
$266$	0	0
$267$	−2.04226	−0.124984
$268$	0	0
$269$	−17.5018	−1.06711	−0.533553	−	0.845766i	$-0.679144\pi$
$269$	−17.5018	−1.06711	−0.533553	+	0.845766i	$0.679144\pi$
$270$	0	0
$271$	6.02666	0.366094	0.183047	−	0.983104i	$-0.441404\pi$
$271$	6.02666	0.366094	0.183047	+	0.983104i	$0.441404\pi$
$272$	0	0
$273$	−1.36910	−0.0828618
$274$	0	0
$275$	−5.41855	−0.326751
$276$	0	0
$277$	16.2485	0.976276	0.488138	−	0.872767i	$-0.337676\pi$
$277$	16.2485	0.976276	0.488138	+	0.872767i	$0.337676\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−20.1711	−1.20331	−0.601654	−	0.798757i	$-0.705492\pi$
$281$	−20.1711	−1.20331	−0.601654	+	0.798757i	$0.705492\pi$
$282$	0	0
$283$	16.9021	1.00473	0.502364	−	0.864656i	$-0.332464\pi$
$283$	16.9021	1.00473	0.502364	+	0.864656i	$0.332464\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	0	0
$287$	−6.15676	−0.363422
$288$	0	0
$289$	33.8059	1.98858
$290$	0	0
$291$	−2.73820	−0.160516
$292$	0	0
$293$	13.3691	0.781031	0.390516	−	0.920596i	$-0.372297\pi$
$293$	13.3691	0.781031	0.390516	+	0.920596i	$0.372297\pi$
$294$	0	0
$295$	−14.6381	−0.852262
$296$	0	0
$297$	−5.41855	−0.314416
$298$	0	0
$299$	8.15676	0.471717
$300$	0	0
$301$	−7.02052	−0.404656
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.9421	0.683805
$306$	0	0
$307$	−9.48360	−0.541257	−0.270629	−	0.962684i	$-0.587232\pi$
$307$	−9.48360	−0.541257	−0.270629	+	0.962684i	$0.587232\pi$
$308$	0	0
$309$	−8.92881	−0.507942
$310$	0	0
$311$	−12.6381	−0.716640	−0.358320	−	0.933599i	$-0.616650\pi$
$311$	−12.6381	−0.716640	−0.358320	+	0.933599i	$0.616650\pi$
$312$	0	0
$313$	−5.30018	−0.299584	−0.149792	−	0.988718i	$-0.547860\pi$
$313$	−5.30018	−0.299584	−0.149792	+	0.988718i	$0.547860\pi$
$314$	0	0
$315$	−1.17009	−0.0659269
$316$	0	0
$317$	−8.25953	−0.463901	−0.231951	−	0.972728i	$-0.574511\pi$
$317$	−8.25953	−0.463901	−0.231951	+	0.972728i	$0.574511\pi$
$318$	0	0
$319$	−49.9565	−2.79703
$320$	0	0
$321$	−15.6514	−0.873577
$322$	0	0
$323$	−28.5113	−1.58641
$324$	0	0
$325$	−1.17009	−0.0649047
$326$	0	0
$327$	−14.0228	−0.775462
$328$	0	0
$329$	−1.78539	−0.0984315
$330$	0	0
$331$	−27.3112	−1.50116	−0.750581	−	0.660779i	$-0.770226\pi$
$331$	−27.3112	−1.50116	−0.750581	+	0.660779i	$0.770226\pi$
$332$	0	0
$333$	−2.09171	−0.114625
$334$	0	0
$335$	13.3268	0.728123
$336$	0	0
$337$	−6.77205	−0.368897	−0.184449	−	0.982842i	$-0.559050\pi$
$337$	−6.77205	−0.368897	−0.184449	+	0.982842i	$0.559050\pi$
$338$	0	0
$339$	5.23513	0.284333
$340$	0	0
$341$	5.41855	0.293431
$342$	0	0
$343$	−14.7792	−0.798004
$344$	0	0
$345$	6.97107	0.375310
$346$	0	0
$347$	−3.71769	−0.199576	−0.0997879	−	0.995009i	$-0.531816\pi$
$347$	−3.71769	−0.199576	−0.0997879	+	0.995009i	$0.531816\pi$
$348$	0	0
$349$	−6.60424	−0.353517	−0.176758	−	0.984254i	$-0.556561\pi$
$349$	−6.60424	−0.353517	−0.176758	+	0.984254i	$0.556561\pi$
$350$	0	0
$351$	−1.17009	−0.0624546
$352$	0	0
$353$	−32.2206	−1.71493	−0.857464	−	0.514544i	$-0.827961\pi$
$353$	−32.2206	−1.71493	−0.857464	+	0.514544i	$0.827961\pi$
$354$	0	0
$355$	8.14116	0.432088
$356$	0	0
$357$	8.34017	0.441409
$358$	0	0
$359$	2.93722	0.155021	0.0775103	−	0.996992i	$-0.475303\pi$
$359$	2.93722	0.155021	0.0775103	+	0.996992i	$0.475303\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	18.3607	0.963686
$364$	0	0
$365$	8.74539	0.457755
$366$	0	0
$367$	17.3074	0.903437	0.451719	−	0.892160i	$-0.350811\pi$
$367$	17.3074	0.903437	0.451719	+	0.892160i	$0.350811\pi$
$368$	0	0
$369$	−5.26180	−0.273918
$370$	0	0
$371$	0.554787	0.0288031
$372$	0	0
$373$	30.5958	1.58419	0.792096	−	0.610397i	$-0.208990\pi$
$373$	30.5958	1.58419	0.792096	+	0.610397i	$0.208990\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−10.7877	−0.555592
$378$	0	0
$379$	33.5585	1.72378	0.861892	−	0.507092i	$-0.169280\pi$
$379$	33.5585	1.72378	0.861892	+	0.507092i	$0.169280\pi$
$380$	0	0
$381$	−8.18342	−0.419249
$382$	0	0
$383$	36.2784	1.85374	0.926871	−	0.375380i	$-0.122488\pi$
$383$	36.2784	1.85374	0.926871	+	0.375380i	$0.122488\pi$
$384$	0	0
$385$	6.34017	0.323125
$386$	0	0
$387$	−6.00000	−0.304997
$388$	0	0
$389$	9.54533	0.483968	0.241984	−	0.970280i	$-0.422202\pi$
$389$	9.54533	0.483968	0.241984	+	0.970280i	$0.422202\pi$
$390$	0	0
$391$	−49.6886	−2.51286
$392$	0	0
$393$	7.21953	0.364177
$394$	0	0
$395$	−9.46800	−0.476387
$396$	0	0
$397$	4.05786	0.203658	0.101829	−	0.994802i	$-0.467531\pi$
$397$	4.05786	0.203658	0.101829	+	0.994802i	$0.467531\pi$
$398$	0	0
$399$	−4.68035	−0.234310
$400$	0	0
$401$	32.0833	1.60216	0.801082	−	0.598555i	$-0.204258\pi$
$401$	32.0833	1.60216	0.801082	+	0.598555i	$0.204258\pi$
$402$	0	0
$403$	1.17009	0.0582862
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	11.3340	0.561807
$408$	0	0
$409$	2.89496	0.143147	0.0715733	−	0.997435i	$-0.477198\pi$
$409$	2.89496	0.143147	0.0715733	+	0.997435i	$0.477198\pi$
$410$	0	0
$411$	8.38962	0.413830
$412$	0	0
$413$	17.1278	0.842805
$414$	0	0
$415$	3.86603	0.189776
$416$	0	0
$417$	−18.3402	−0.898122
$418$	0	0
$419$	−36.6381	−1.78989	−0.894944	−	0.446179i	$-0.852785\pi$
$419$	−36.6381	−1.78989	−0.894944	+	0.446179i	$0.852785\pi$
$420$	0	0
$421$	−8.26406	−0.402766	−0.201383	−	0.979513i	$-0.564544\pi$
$421$	−8.26406	−0.402766	−0.201383	+	0.979513i	$0.564544\pi$
$422$	0	0
$423$	−1.52586	−0.0741897
$424$	0	0
$425$	7.12783	0.345750
$426$	0	0
$427$	−13.9733	−0.676217
$428$	0	0
$429$	6.34017	0.306106
$430$	0	0
$431$	6.19902	0.298596	0.149298	−	0.988792i	$-0.452299\pi$
$431$	6.19902	0.298596	0.149298	+	0.988792i	$0.452299\pi$
$432$	0	0
$433$	−10.1061	−0.485667	−0.242834	−	0.970068i	$-0.578077\pi$
$433$	−10.1061	−0.485667	−0.242834	+	0.970068i	$0.578077\pi$
$434$	0	0
$435$	−9.21953	−0.442043
$436$	0	0
$437$	27.8843	1.33389
$438$	0	0
$439$	7.20394	0.343825	0.171913	−	0.985112i	$-0.445005\pi$
$439$	7.20394	0.343825	0.171913	+	0.985112i	$0.445005\pi$
$440$	0	0
$441$	−5.63090	−0.268138
$442$	0	0
$443$	22.7031	1.07866	0.539329	−	0.842095i	$-0.318678\pi$
$443$	22.7031	1.07866	0.539329	+	0.842095i	$0.318678\pi$
$444$	0	0
$445$	2.04226	0.0968124
$446$	0	0
$447$	12.0989	0.572258
$448$	0	0
$449$	4.56585	0.215476	0.107738	−	0.994179i	$-0.465639\pi$
$449$	4.56585	0.215476	0.107738	+	0.994179i	$0.465639\pi$
$450$	0	0
$451$	28.5113	1.34254
$452$	0	0
$453$	22.3318	1.04924
$454$	0	0
$455$	1.36910	0.0641845
$456$	0	0
$457$	22.8710	1.06986	0.534929	−	0.844897i	$-0.320338\pi$
$457$	22.8710	1.06986	0.534929	+	0.844897i	$0.320338\pi$
$458$	0	0
$459$	7.12783	0.332698
$460$	0	0
$461$	−32.0566	−1.49303	−0.746513	−	0.665371i	$-0.768274\pi$
$461$	−32.0566	−1.49303	−0.746513	+	0.665371i	$0.768274\pi$
$462$	0	0
$463$	10.0989	0.469336	0.234668	−	0.972076i	$-0.424600\pi$
$463$	10.0989	0.469336	0.234668	+	0.972076i	$0.424600\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	−0.630898	−0.0291945	−0.0145972	−	0.999893i	$-0.504647\pi$
$467$	−0.630898	−0.0291945	−0.0145972	+	0.999893i	$0.504647\pi$
$468$	0	0
$469$	−15.5936	−0.720044
$470$	0	0
$471$	14.8371	0.683658
$472$	0	0
$473$	32.5113	1.49487
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0.474142	0.0217095
$478$	0	0
$479$	10.9639	0.500953	0.250476	−	0.968123i	$-0.419413\pi$
$479$	10.9639	0.500953	0.250476	+	0.968123i	$0.419413\pi$
$480$	0	0
$481$	2.44748	0.111595
$482$	0	0
$483$	−8.15676	−0.371145
$484$	0	0
$485$	2.73820	0.124335
$486$	0	0
$487$	−37.0784	−1.68018	−0.840091	−	0.542446i	$-0.817498\pi$
$487$	−37.0784	−1.68018	−0.840091	+	0.542446i	$0.817498\pi$
$488$	0	0
$489$	−1.59478	−0.0721185
$490$	0	0
$491$	25.6430	1.15725	0.578626	−	0.815593i	$-0.303589\pi$
$491$	25.6430	1.15725	0.578626	+	0.815593i	$0.303589\pi$
$492$	0	0
$493$	65.7152	2.95967
$494$	0	0
$495$	5.41855	0.243546
$496$	0	0
$497$	−9.52586	−0.427293
$498$	0	0
$499$	−13.1629	−0.589252	−0.294626	−	0.955613i	$-0.595195\pi$
$499$	−13.1629	−0.589252	−0.294626	+	0.955613i	$0.595195\pi$
$500$	0	0
$501$	3.91548	0.174931
$502$	0	0
$503$	−0.362959	−0.0161836	−0.00809178	−	0.999967i	$-0.502576\pi$
$503$	−0.362959	−0.0161836	−0.00809178	+	0.999967i	$0.502576\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−11.6309	−0.516546
$508$	0	0
$509$	35.8154	1.58749	0.793744	−	0.608252i	$-0.208129\pi$
$509$	35.8154	1.58749	0.793744	+	0.608252i	$0.208129\pi$
$510$	0	0
$511$	−10.2329	−0.452675
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	8.92881	0.393450
$516$	0	0
$517$	8.26794	0.363624
$518$	0	0
$519$	−1.23513	−0.0542163
$520$	0	0
$521$	23.7854	1.04206	0.521028	−	0.853539i	$-0.325549\pi$
$521$	23.7854	1.04206	0.521028	+	0.853539i	$0.325549\pi$
$522$	0	0
$523$	6.24128	0.272912	0.136456	−	0.990646i	$-0.456429\pi$
$523$	6.24128	0.272912	0.136456	+	0.990646i	$0.456429\pi$
$524$	0	0
$525$	1.17009	0.0510668
$526$	0	0
$527$	−7.12783	−0.310493
$528$	0	0
$529$	25.5958	1.11286
$530$	0	0
$531$	14.6381	0.635239
$532$	0	0
$533$	6.15676	0.266679
$534$	0	0
$535$	15.6514	0.676670
$536$	0	0
$537$	−15.4186	−0.665360
$538$	0	0
$539$	30.5113	1.31421
$540$	0	0
$541$	1.04331	0.0448552	0.0224276	−	0.999748i	$-0.492860\pi$
$541$	1.04331	0.0448552	0.0224276	+	0.999748i	$0.492860\pi$
$542$	0	0
$543$	−7.26180	−0.311633
$544$	0	0
$545$	14.0228	0.600670
$546$	0	0
$547$	−35.6358	−1.52368	−0.761839	−	0.647767i	$-0.775703\pi$
$547$	−35.6358	−1.52368	−0.761839	+	0.647767i	$0.775703\pi$
$548$	0	0
$549$	−11.9421	−0.509678
$550$	0	0
$551$	−36.8781	−1.57106
$552$	0	0
$553$	11.0784	0.471101
$554$	0	0
$555$	2.09171	0.0887881
$556$	0	0
$557$	−6.64527	−0.281569	−0.140785	−	0.990040i	$-0.544963\pi$
$557$	−6.64527	−0.281569	−0.140785	+	0.990040i	$0.544963\pi$
$558$	0	0
$559$	7.02052	0.296936
$560$	0	0
$561$	−38.6225	−1.63064
$562$	0	0
$563$	−21.2800	−0.896847	−0.448424	−	0.893821i	$-0.648014\pi$
$563$	−21.2800	−0.896847	−0.448424	+	0.893821i	$0.648014\pi$
$564$	0	0
$565$	−5.23513	−0.220244
$566$	0	0
$567$	1.17009	0.0491390
$568$	0	0
$569$	32.7226	1.37180	0.685902	−	0.727694i	$-0.259408\pi$
$569$	32.7226	1.37180	0.685902	+	0.727694i	$0.259408\pi$
$570$	0	0
$571$	−13.1773	−0.551452	−0.275726	−	0.961236i	$-0.588918\pi$
$571$	−13.1773	−0.551452	−0.275726	+	0.961236i	$0.588918\pi$
$572$	0	0
$573$	17.8732	0.746664
$574$	0	0
$575$	−6.97107	−0.290714
$576$	0	0
$577$	−10.8059	−0.449856	−0.224928	−	0.974375i	$-0.572215\pi$
$577$	−10.8059	−0.449856	−0.224928	+	0.974375i	$0.572215\pi$
$578$	0	0
$579$	21.6742	0.900749
$580$	0	0
$581$	−4.52359	−0.187670
$582$	0	0
$583$	−2.56916	−0.106404
$584$	0	0
$585$	1.17009	0.0483771
$586$	0	0
$587$	37.6886	1.55557	0.777787	−	0.628528i	$-0.216342\pi$
$587$	37.6886	1.55557	0.777787	+	0.628528i	$0.216342\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	−22.0638	−0.907584
$592$	0	0
$593$	−19.8576	−0.815455	−0.407727	−	0.913104i	$-0.633679\pi$
$593$	−19.8576	−0.815455	−0.407727	+	0.913104i	$0.633679\pi$
$594$	0	0
$595$	−8.34017	−0.341914
$596$	0	0
$597$	−16.3896	−0.670783
$598$	0	0
$599$	41.1350	1.68073	0.840366	−	0.542020i	$-0.182340\pi$
$599$	41.1350	1.68073	0.840366	+	0.542020i	$0.182340\pi$
$600$	0	0
$601$	17.3874	0.709245	0.354622	−	0.935010i	$-0.384609\pi$
$601$	17.3874	0.709245	0.354622	+	0.935010i	$0.384609\pi$
$602$	0	0
$603$	−13.3268	−0.542711
$604$	0	0
$605$	−18.3607	−0.746468
$606$	0	0
$607$	−21.5558	−0.874924	−0.437462	−	0.899237i	$-0.644123\pi$
$607$	−21.5558	−0.874924	−0.437462	+	0.899237i	$0.644123\pi$
$608$	0	0
$609$	10.7877	0.437138
$610$	0	0
$611$	1.78539	0.0722290
$612$	0	0
$613$	33.0544	1.33505	0.667527	−	0.744586i	$-0.267353\pi$
$613$	33.0544	1.33505	0.667527	+	0.744586i	$0.267353\pi$
$614$	0	0
$615$	5.26180	0.212176
$616$	0	0
$617$	−38.9315	−1.56732	−0.783661	−	0.621189i	$-0.786650\pi$
$617$	−38.9315	−1.56732	−0.783661	+	0.621189i	$0.786650\pi$
$618$	0	0
$619$	32.6719	1.31320	0.656598	−	0.754241i	$-0.271995\pi$
$619$	32.6719	1.31320	0.656598	+	0.754241i	$0.271995\pi$
$620$	0	0
$621$	−6.97107	−0.279739
$622$	0	0
$623$	−2.38962	−0.0957382
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	21.6742	0.865584
$628$	0	0
$629$	−14.9093	−0.594474
$630$	0	0
$631$	−4.68035	−0.186322	−0.0931608	−	0.995651i	$-0.529697\pi$
$631$	−4.68035	−0.186322	−0.0931608	+	0.995651i	$0.529697\pi$
$632$	0	0
$633$	−11.6020	−0.461137
$634$	0	0
$635$	8.18342	0.324749
$636$	0	0
$637$	6.58864	0.261051
$638$	0	0
$639$	−8.14116	−0.322059
$640$	0	0
$641$	−16.0111	−0.632399	−0.316199	−	0.948693i	$-0.602407\pi$
$641$	−16.0111	−0.632399	−0.316199	+	0.948693i	$0.602407\pi$
$642$	0	0
$643$	6.24128	0.246132	0.123066	−	0.992398i	$-0.460727\pi$
$643$	6.24128	0.246132	0.123066	+	0.992398i	$0.460727\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	0	0
$647$	7.73206	0.303979	0.151989	−	0.988382i	$-0.451432\pi$
$647$	7.73206	0.303979	0.151989	+	0.988382i	$0.451432\pi$
$648$	0	0
$649$	−79.3172	−3.11347
$650$	0	0
$651$	−1.17009	−0.0458593
$652$	0	0
$653$	8.16063	0.319350	0.159675	−	0.987170i	$-0.448955\pi$
$653$	8.16063	0.319350	0.159675	+	0.987170i	$0.448955\pi$
$654$	0	0
$655$	−7.21953	−0.282091
$656$	0	0
$657$	−8.74539	−0.341190
$658$	0	0
$659$	−40.2544	−1.56809	−0.784045	−	0.620704i	$-0.786847\pi$
$659$	−40.2544	−1.56809	−0.784045	+	0.620704i	$0.786847\pi$
$660$	0	0
$661$	−49.7464	−1.93491	−0.967456	−	0.253039i	$-0.918570\pi$
$661$	−49.7464	−1.93491	−0.967456	+	0.253039i	$0.918570\pi$
$662$	0	0
$663$	−8.34017	−0.323905
$664$	0	0
$665$	4.68035	0.181496
$666$	0	0
$667$	−64.2700	−2.48855
$668$	0	0
$669$	11.5441	0.446321
$670$	0	0
$671$	64.7091	2.49807
$672$	0	0
$673$	−46.4606	−1.79093	−0.895463	−	0.445136i	$-0.853155\pi$
$673$	−46.4606	−1.79093	−0.895463	+	0.445136i	$0.853155\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	19.6248	0.754241	0.377120	−	0.926164i	$-0.376914\pi$
$677$	19.6248	0.754241	0.377120	+	0.926164i	$0.376914\pi$
$678$	0	0
$679$	−3.20394	−0.122956
$680$	0	0
$681$	−14.7031	−0.563425
$682$	0	0
$683$	−6.23740	−0.238668	−0.119334	−	0.992854i	$-0.538076\pi$
$683$	−6.23740	−0.238668	−0.119334	+	0.992854i	$0.538076\pi$
$684$	0	0
$685$	−8.38962	−0.320551
$686$	0	0
$687$	−9.60197	−0.366338
$688$	0	0
$689$	−0.554787	−0.0211357
$690$	0	0
$691$	−40.2823	−1.53241	−0.766206	−	0.642595i	$-0.777858\pi$
$691$	−40.2823	−1.53241	−0.766206	+	0.642595i	$0.777858\pi$
$692$	0	0
$693$	−6.34017	−0.240843
$694$	0	0
$695$	18.3402	0.695682
$696$	0	0
$697$	−37.5052	−1.42061
$698$	0	0
$699$	−10.4163	−0.393980
$700$	0	0
$701$	31.7998	1.20106	0.600530	−	0.799602i	$-0.294956\pi$
$701$	31.7998	1.20106	0.600530	+	0.799602i	$0.294956\pi$
$702$	0	0
$703$	8.36683	0.315561
$704$	0	0
$705$	1.52586	0.0574671
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−5.68649	−0.213561	−0.106780	−	0.994283i	$-0.534054\pi$
$709$	−5.68649	−0.213561	−0.106780	+	0.994283i	$0.534054\pi$
$710$	0	0
$711$	9.46800	0.355078
$712$	0	0
$713$	6.97107	0.261069
$714$	0	0
$715$	−6.34017	−0.237109
$716$	0	0
$717$	−9.02052	−0.336877
$718$	0	0
$719$	19.6286	0.732024	0.366012	−	0.930610i	$-0.380723\pi$
$719$	19.6286	0.732024	0.366012	+	0.930610i	$0.380723\pi$
$720$	0	0
$721$	−10.4475	−0.389084
$722$	0	0
$723$	3.97334	0.147770
$724$	0	0
$725$	9.21953	0.342405
$726$	0	0
$727$	28.1783	1.04508	0.522538	−	0.852616i	$-0.324985\pi$
$727$	28.1783	1.04508	0.522538	+	0.852616i	$0.324985\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−42.7670	−1.58179
$732$	0	0
$733$	4.63931	0.171357	0.0856784	−	0.996323i	$-0.472694\pi$
$733$	4.63931	0.171357	0.0856784	+	0.996323i	$0.472694\pi$
$734$	0	0
$735$	5.63090	0.207699
$736$	0	0
$737$	72.2122	2.65997
$738$	0	0
$739$	−32.0761	−1.17994	−0.589969	−	0.807426i	$-0.700860\pi$
$739$	−32.0761	−1.17994	−0.589969	+	0.807426i	$0.700860\pi$
$740$	0	0
$741$	4.68035	0.171937
$742$	0	0
$743$	12.5503	0.460424	0.230212	−	0.973140i	$-0.426058\pi$
$743$	12.5503	0.460424	0.230212	+	0.973140i	$0.426058\pi$
$744$	0	0
$745$	−12.0989	−0.443269
$746$	0	0
$747$	−3.86603	−0.141451
$748$	0	0
$749$	−18.3135	−0.669161
$750$	0	0
$751$	45.4908	1.65998	0.829991	−	0.557777i	$-0.188345\pi$
$751$	45.4908	1.65998	0.829991	+	0.557777i	$0.188345\pi$
$752$	0	0
$753$	5.94214	0.216544
$754$	0	0
$755$	−22.3318	−0.812736
$756$	0	0
$757$	1.93495	0.0703271	0.0351635	−	0.999382i	$-0.488805\pi$
$757$	1.93495	0.0703271	0.0351635	+	0.999382i	$0.488805\pi$
$758$	0	0
$759$	37.7731	1.37108
$760$	0	0
$761$	−38.2134	−1.38523	−0.692617	−	0.721305i	$-0.743542\pi$
$761$	−38.2134	−1.38523	−0.692617	+	0.721305i	$0.743542\pi$
$762$	0	0
$763$	−16.4079	−0.594005
$764$	0	0
$765$	−7.12783	−0.257707
$766$	0	0
$767$	−17.1278	−0.618450
$768$	0	0
$769$	30.6042	1.10362	0.551808	−	0.833971i	$-0.313938\pi$
$769$	30.6042	1.10362	0.551808	+	0.833971i	$0.313938\pi$
$770$	0	0
$771$	13.7359	0.494688
$772$	0	0
$773$	33.7959	1.21555	0.607777	−	0.794108i	$-0.292062\pi$
$773$	33.7959	1.21555	0.607777	+	0.794108i	$0.292062\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−2.44748	−0.0878029
$778$	0	0
$779$	21.0472	0.754094
$780$	0	0
$781$	44.1133	1.57850
$782$	0	0
$783$	9.21953	0.329479
$784$	0	0
$785$	−14.8371	−0.529559
$786$	0	0
$787$	−15.5897	−0.555712	−0.277856	−	0.960623i	$-0.589624\pi$
$787$	−15.5897	−0.555712	−0.277856	+	0.960623i	$0.589624\pi$
$788$	0	0
$789$	16.8638	0.600366
$790$	0	0
$791$	6.12556	0.217800
$792$	0	0
$793$	13.9733	0.496208
$794$	0	0
$795$	−0.474142	−0.0168161
$796$	0	0
$797$	43.4245	1.53818	0.769088	−	0.639143i	$-0.220711\pi$
$797$	43.4245	1.53818	0.769088	+	0.639143i	$0.220711\pi$
$798$	0	0
$799$	−10.8760	−0.384767
$800$	0	0
$801$	−2.04226	−0.0721597
$802$	0	0
$803$	47.3874	1.67226
$804$	0	0
$805$	8.15676	0.287488
$806$	0	0
$807$	−17.5018	−0.616094
$808$	0	0
$809$	−48.9471	−1.72089	−0.860444	−	0.509546i	$-0.829813\pi$
$809$	−48.9471	−1.72089	−0.860444	+	0.509546i	$0.829813\pi$
$810$	0	0
$811$	−33.1773	−1.16501	−0.582506	−	0.812827i	$-0.697928\pi$
$811$	−33.1773	−1.16501	−0.582506	+	0.812827i	$0.697928\pi$
$812$	0	0
$813$	6.02666	0.211364
$814$	0	0
$815$	1.59478	0.0558627
$816$	0	0
$817$	24.0000	0.839654
$818$	0	0
$819$	−1.36910	−0.0478403
$820$	0	0
$821$	−1.62985	−0.0568823	−0.0284411	−	0.999595i	$-0.509054\pi$
$821$	−1.62985	−0.0568823	−0.0284411	+	0.999595i	$0.509054\pi$
$822$	0	0
$823$	34.2434	1.19365	0.596824	−	0.802372i	$-0.296429\pi$
$823$	34.2434	1.19365	0.596824	+	0.802372i	$0.296429\pi$
$824$	0	0
$825$	−5.41855	−0.188650
$826$	0	0
$827$	49.5234	1.72210	0.861049	−	0.508522i	$-0.169808\pi$
$827$	49.5234	1.72210	0.861049	+	0.508522i	$0.169808\pi$
$828$	0	0
$829$	35.4740	1.23206	0.616031	−	0.787722i	$-0.288740\pi$
$829$	35.4740	1.23206	0.616031	+	0.787722i	$0.288740\pi$
$830$	0	0
$831$	16.2485	0.563653
$832$	0	0
$833$	−40.1361	−1.39063
$834$	0	0
$835$	−3.91548	−0.135501
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−26.2544	−0.906404	−0.453202	−	0.891408i	$-0.649718\pi$
$839$	−26.2544	−0.906404	−0.453202	+	0.891408i	$0.649718\pi$
$840$	0	0
$841$	55.9998	1.93103
$842$	0	0
$843$	−20.1711	−0.694731
$844$	0	0
$845$	11.6309	0.400115
$846$	0	0
$847$	21.4836	0.738185
$848$	0	0
$849$	16.9021	0.580080
$850$	0	0
$851$	14.5814	0.499846
$852$	0	0
$853$	−45.5729	−1.56039	−0.780193	−	0.625539i	$-0.784879\pi$
$853$	−45.5729	−1.56039	−0.780193	+	0.625539i	$0.784879\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	3.24742	0.110930	0.0554649	−	0.998461i	$-0.482336\pi$
$857$	3.24742	0.110930	0.0554649	+	0.998461i	$0.482336\pi$
$858$	0	0
$859$	−28.1978	−0.962096	−0.481048	−	0.876694i	$-0.659744\pi$
$859$	−28.1978	−0.962096	−0.481048	+	0.876694i	$0.659744\pi$
$860$	0	0
$861$	−6.15676	−0.209822
$862$	0	0
$863$	−34.4079	−1.17126	−0.585629	−	0.810579i	$-0.699152\pi$
$863$	−34.4079	−1.17126	−0.585629	+	0.810579i	$0.699152\pi$
$864$	0	0
$865$	1.23513	0.0419958
$866$	0	0
$867$	33.8059	1.14811
$868$	0	0
$869$	−51.3028	−1.74033
$870$	0	0
$871$	15.5936	0.528368
$872$	0	0
$873$	−2.73820	−0.0926742
$874$	0	0
$875$	−1.17009	−0.0395561
$876$	0	0
$877$	45.5897	1.53945	0.769727	−	0.638373i	$-0.220392\pi$
$877$	45.5897	1.53945	0.769727	+	0.638373i	$0.220392\pi$
$878$	0	0
$879$	13.3691	0.450929
$880$	0	0
$881$	23.1662	0.780489	0.390245	−	0.920711i	$-0.372390\pi$
$881$	23.1662	0.780489	0.390245	+	0.920711i	$0.372390\pi$
$882$	0	0
$883$	17.4863	0.588459	0.294230	−	0.955735i	$-0.404937\pi$
$883$	17.4863	0.588459	0.294230	+	0.955735i	$0.404937\pi$
$884$	0	0
$885$	−14.6381	−0.492054
$886$	0	0
$887$	−11.1012	−0.372741	−0.186370	−	0.982480i	$-0.559672\pi$
$887$	−11.1012	−0.372741	−0.186370	+	0.982480i	$0.559672\pi$
$888$	0	0
$889$	−9.57531	−0.321145
$890$	0	0
$891$	−5.41855	−0.181528
$892$	0	0
$893$	6.10343	0.204244
$894$	0	0
$895$	15.4186	0.515385
$896$	0	0
$897$	8.15676	0.272346
$898$	0	0
$899$	−9.21953	−0.307489
$900$	0	0
$901$	3.37960	0.112591
$902$	0	0
$903$	−7.02052	−0.233628
$904$	0	0
$905$	7.26180	0.241390
$906$	0	0
$907$	50.4463	1.67504	0.837520	−	0.546406i	$-0.184005\pi$
$907$	50.4463	1.67504	0.837520	+	0.546406i	$0.184005\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	43.4596	1.43988	0.719940	−	0.694036i	$-0.244169\pi$
$911$	43.4596	1.43988	0.719940	+	0.694036i	$0.244169\pi$
$912$	0	0
$913$	20.9483	0.693287
$914$	0	0
$915$	11.9421	0.394795
$916$	0	0
$917$	8.44748	0.278960
$918$	0	0
$919$	−17.4596	−0.575939	−0.287969	−	0.957640i	$-0.592980\pi$
$919$	−17.4596	−0.575939	−0.287969	+	0.957640i	$0.592980\pi$
$920$	0	0
$921$	−9.48360	−0.312495
$922$	0	0
$923$	9.52586	0.313547
$924$	0	0
$925$	−2.09171	−0.0687750
$926$	0	0
$927$	−8.92881	−0.293261
$928$	0	0
$929$	−9.13501	−0.299710	−0.149855	−	0.988708i	$-0.547881\pi$
$929$	−9.13501	−0.299710	−0.149855	+	0.988708i	$0.547881\pi$
$930$	0	0
$931$	22.5236	0.738181
$932$	0	0
$933$	−12.6381	−0.413752
$934$	0	0
$935$	38.6225	1.26309
$936$	0	0
$937$	18.9893	0.620354	0.310177	−	0.950679i	$-0.399612\pi$
$937$	18.9893	0.620354	0.310177	+	0.950679i	$0.399612\pi$
$938$	0	0
$939$	−5.30018	−0.172965
$940$	0	0
$941$	3.12064	0.101730	0.0508649	−	0.998706i	$-0.483802\pi$
$941$	3.12064	0.101730	0.0508649	+	0.998706i	$0.483802\pi$
$942$	0	0
$943$	36.6803	1.19448
$944$	0	0
$945$	−1.17009	−0.0380629
$946$	0	0
$947$	−6.60424	−0.214609	−0.107304	−	0.994226i	$-0.534222\pi$
$947$	−6.60424	−0.214609	−0.107304	+	0.994226i	$0.534222\pi$
$948$	0	0
$949$	10.2329	0.332173
$950$	0	0
$951$	−8.25953	−0.267834
$952$	0	0
$953$	12.8554	0.416426	0.208213	−	0.978084i	$-0.433235\pi$
$953$	12.8554	0.416426	0.208213	+	0.978084i	$0.433235\pi$
$954$	0	0
$955$	−17.8732	−0.578364
$956$	0	0
$957$	−49.9565	−1.61486
$958$	0	0
$959$	9.81658	0.316994
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−15.6514	−0.504360
$964$	0	0
$965$	−21.6742	−0.697717
$966$	0	0
$967$	36.7982	1.18335	0.591674	−	0.806177i	$-0.298467\pi$
$967$	36.7982	1.18335	0.591674	+	0.806177i	$0.298467\pi$
$968$	0	0
$969$	−28.5113	−0.915915
$970$	0	0
$971$	−37.4707	−1.20249	−0.601245	−	0.799065i	$-0.705329\pi$
$971$	−37.4707	−1.20249	−0.601245	+	0.799065i	$0.705329\pi$
$972$	0	0
$973$	−21.4596	−0.687963
$974$	0	0
$975$	−1.17009	−0.0374728
$976$	0	0
$977$	42.5113	1.36006	0.680029	−	0.733186i	$-0.261967\pi$
$977$	42.5113	1.36006	0.680029	+	0.733186i	$0.261967\pi$
$978$	0	0
$979$	11.0661	0.353674
$980$	0	0
$981$	−14.0228	−0.447713
$982$	0	0
$983$	12.8950	0.411285	0.205643	−	0.978627i	$-0.434072\pi$
$983$	12.8950	0.411285	0.205643	+	0.978627i	$0.434072\pi$
$984$	0	0
$985$	22.0638	0.703012
$986$	0	0
$987$	−1.78539	−0.0568295
$988$	0	0
$989$	41.8264	1.33000
$990$	0	0
$991$	−8.55025	−0.271608	−0.135804	−	0.990736i	$-0.543362\pi$
$991$	−8.55025	−0.271608	−0.135804	+	0.990736i	$0.543362\pi$
$992$	0	0
$993$	−27.3112	−0.866696
$994$	0	0
$995$	16.3896	0.519586
$996$	0	0
$997$	23.5897	0.747093	0.373546	−	0.927612i	$-0.378142\pi$
$997$	23.5897	0.747093	0.373546	+	0.927612i	$0.378142\pi$
$998$	0	0
$999$	−2.09171	−0.0661787

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3720.2.a.o.1.3	✓	3
4.3	odd	2		7440.2.a.bo.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.o.1.3	✓	3	1.1	even	1	trivial
7440.2.a.bo.1.1		3	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3720.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.17009 q^{7} +1.00000 q^{9} -5.41855 q^{11} -1.17009 q^{13} -1.00000 q^{15} +7.12783 q^{17} -4.00000 q^{19} +1.17009 q^{21} -6.97107 q^{23} +1.00000 q^{25} +1.00000 q^{27} +9.21953 q^{29} -1.00000 q^{31} -5.41855 q^{33} -1.17009 q^{35} -2.09171 q^{37} -1.17009 q^{39} -5.26180 q^{41} -6.00000 q^{43} -1.00000 q^{45} -1.52586 q^{47} -5.63090 q^{49} +7.12783 q^{51} +0.474142 q^{53} +5.41855 q^{55} -4.00000 q^{57} +14.6381 q^{59} -11.9421 q^{61} +1.17009 q^{63} +1.17009 q^{65} -13.3268 q^{67} -6.97107 q^{69} -8.14116 q^{71} -8.74539 q^{73} +1.00000 q^{75} -6.34017 q^{77} +9.46800 q^{79} +1.00000 q^{81} -3.86603 q^{83} -7.12783 q^{85} +9.21953 q^{87} -2.04226 q^{89} -1.36910 q^{91} -1.00000 q^{93} +4.00000 q^{95} -2.73820 q^{97} -5.41855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 3 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 3 q^{31} - 2 q^{33} + 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 - 3 * q^15 - 12 * q^19 - 2 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 - 3 * q^31 - 2 * q^33 + 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 - 18 * q^43 - 3 * q^45 - 2 * q^47 - 13 * q^49 + 4 * q^53 + 2 * q^55 - 12 * q^57 + 6 * q^59 - 6 * q^61 - 2 * q^63 - 2 * q^65 - 28 * q^67 - 6 * q^69 - 4 * q^71 + 3 * q^75 - 8 * q^77 - 4 * q^79 + 3 * q^81 + 2 * q^83 + 4 * q^87 - 22 * q^89 - 8 * q^91 - 3 * q^93 + 12 * q^95 - 16 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more