LMFDB - Embedded newform 3720.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3720,2,Mod(1,3720)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$29.7043495519$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.78292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 8x + 18 $ x^4 - x^3 - 10x^2 + 8x + 18
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			1.2
Root			$-1.09502$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -1.09502 q^{7} +1.00000 q^{9} +5.25711 q^{11} +1.24976 q^{13} +1.00000 q^{15} -2.64620 q^{17} +3.25711 q^{19} +1.09502 q^{21} -6.99097 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.89596 q^{29} +1.00000 q^{31} -5.25711 q^{33} +1.09502 q^{35} -3.24976 q^{37} -1.24976 q^{39} +4.00000 q^{41} -4.34477 q^{43} -1.00000 q^{45} +7.14571 q^{47} -5.80094 q^{49} +2.64620 q^{51} +6.80094 q^{53} -5.25711 q^{55} -3.25711 q^{57} -8.80829 q^{59} +2.91234 q^{61} -1.09502 q^{63} -1.24976 q^{65} +1.43979 q^{67} +6.99097 q^{69} -4.24073 q^{71} -8.69689 q^{73} -1.00000 q^{75} -5.75662 q^{77} +12.0933 q^{79} +1.00000 q^{81} +0.0433434 q^{83} +2.64620 q^{85} -5.89596 q^{87} +4.63885 q^{89} -1.36850 q^{91} -1.00000 q^{93} -3.25711 q^{95} +0.309478 q^{97} +5.25711 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} + q^{11} - 2 q^{13} + 4 q^{15} + 4 q^{17} - 7 q^{19} - q^{21} - q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} + 2 q^{39}+ \cdots + q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 + q^11 - 2 * q^13 + 4 * q^15 + 4 * q^17 - 7 * q^19 - q^21 - q^23 + 4 * q^25 - 4 * q^27 + 2 * q^29 + 4 * q^31 - q^33 - q^35 - 6 * q^37 + 2 * q^39 + 16 * q^41 - 5 * q^43 - 4 * q^45 - 7 * q^49 - 4 * q^51 + 11 * q^53 - q^55 + 7 * q^57 - 6 * q^59 + 4 * q^61 + q^63 + 2 * q^65 - 12 * q^67 + q^69 + 17 * q^71 + 3 * q^73 - 4 * q^75 + 11 * q^77 + 3 * q^79 + 4 * q^81 - 10 * q^83 - 4 * q^85 - 2 * q^87 + 17 * q^89 + 6 * q^91 - 4 * q^93 + 7 * q^95 - 2 * q^97 + 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
$7$	−1.09502	−0.413877	−0.206939	−	0.978354i	$-0.566350\pi$
$7$	−1.09502	−0.413877	−0.206939	+	0.978354i	$0.566350\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.25711	1.58508	0.792539	−	0.609822i	$-0.208759\pi$
$11$	5.25711	1.58508	0.792539	+	0.609822i	$0.208759\pi$
$12$	0	0
$13$	1.24976	0.346620	0.173310	−	0.984867i	$-0.444554\pi$
$13$	1.24976	0.346620	0.173310	+	0.984867i	$0.444554\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−2.64620	−0.641798	−0.320899	−	0.947113i	$-0.603985\pi$
$17$	−2.64620	−0.641798	−0.320899	+	0.947113i	$0.603985\pi$
$18$	0	0
$19$	3.25711	0.747232	0.373616	−	0.927584i	$-0.378118\pi$
$19$	3.25711	0.747232	0.373616	+	0.927584i	$0.378118\pi$
$20$	0	0
$21$	1.09502	0.238952
$22$	0	0
$23$	−6.99097	−1.45772	−0.728859	−	0.684664i	$-0.759949\pi$
$23$	−6.99097	−1.45772	−0.728859	+	0.684664i	$0.759949\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.89596	1.09485	0.547426	−	0.836854i	$-0.315608\pi$
$29$	5.89596	1.09485	0.547426	+	0.836854i	$0.315608\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−5.25711	−0.915145
$34$	0	0
$35$	1.09502	0.185092
$36$	0	0
$37$	−3.24976	−0.534257	−0.267128	−	0.963661i	$-0.586075\pi$
$37$	−3.24976	−0.534257	−0.267128	+	0.963661i	$0.586075\pi$
$38$	0	0
$39$	−1.24976	−0.200121
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	−4.34477	−0.662571	−0.331286	−	0.943530i	$-0.607482\pi$
$43$	−4.34477	−0.662571	−0.331286	+	0.943530i	$0.607482\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	7.14571	1.04231	0.521155	−	0.853462i	$-0.325502\pi$
$47$	7.14571	1.04231	0.521155	+	0.853462i	$0.325502\pi$
$48$	0	0
$49$	−5.80094	−0.828706
$50$	0	0
$51$	2.64620	0.370542
$52$	0	0
$53$	6.80094	0.934181	0.467090	−	0.884210i	$-0.345302\pi$
$53$	6.80094	0.934181	0.467090	+	0.884210i	$0.345302\pi$
$54$	0	0
$55$	−5.25711	−0.708868
$56$	0	0
$57$	−3.25711	−0.431414
$58$	0	0
$59$	−8.80829	−1.14674	−0.573371	−	0.819296i	$-0.694365\pi$
$59$	−8.80829	−1.14674	−0.573371	+	0.819296i	$0.694365\pi$
$60$	0	0
$61$	2.91234	0.372886	0.186443	−	0.982466i	$-0.440304\pi$
$61$	2.91234	0.372886	0.186443	+	0.982466i	$0.440304\pi$
$62$	0	0
$63$	−1.09502	−0.137959
$64$	0	0
$65$	−1.24976	−0.155013
$66$	0	0
$67$	1.43979	0.175898	0.0879491	−	0.996125i	$-0.471969\pi$
$67$	1.43979	0.175898	0.0879491	+	0.996125i	$0.471969\pi$
$68$	0	0
$69$	6.99097	0.841614
$70$	0	0
$71$	−4.24073	−0.503282	−0.251641	−	0.967821i	$-0.580970\pi$
$71$	−4.24073	−0.503282	−0.251641	+	0.967821i	$0.580970\pi$
$72$	0	0
$73$	−8.69689	−1.01789	−0.508947	−	0.860798i	$-0.669965\pi$
$73$	−8.69689	−1.01789	−0.508947	+	0.860798i	$0.669965\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−5.75662	−0.656027
$78$	0	0
$79$	12.0933	1.36061	0.680304	−	0.732931i	$-0.261848\pi$
$79$	12.0933	1.36061	0.680304	+	0.732931i	$0.261848\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.0433434	0.00475755	0.00237878	−	0.999997i	$-0.499243\pi$
$83$	0.0433434	0.00475755	0.00237878	+	0.999997i	$0.499243\pi$
$84$	0	0
$85$	2.64620	0.287021
$86$	0	0
$87$	−5.89596	−0.632113
$88$	0	0
$89$	4.63885	0.491717	0.245858	−	0.969306i	$-0.420930\pi$
$89$	4.63885	0.491717	0.245858	+	0.969306i	$0.420930\pi$
$90$	0	0
$91$	−1.36850	−0.143458
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−3.25711	−0.334172
$96$	0	0
$97$	0.309478	0.0314228	0.0157114	−	0.999877i	$-0.494999\pi$
$97$	0.309478	0.0314228	0.0157114	+	0.999877i	$0.494999\pi$
$98$	0	0
$99$	5.25711	0.528359
$100$	0	0
$101$	16.4789	1.63971	0.819857	−	0.572568i	$-0.194053\pi$
$101$	16.4789	1.63971	0.819857	+	0.572568i	$0.194053\pi$
$102$	0	0
$103$	−15.0417	−1.48210	−0.741050	−	0.671450i	$-0.765672\pi$
$103$	−15.0417	−1.48210	−0.741050	+	0.671450i	$0.765672\pi$
$104$	0	0
$105$	−1.09502	−0.106863
$106$	0	0
$107$	−10.0786	−0.974339	−0.487169	−	0.873308i	$-0.661971\pi$
$107$	−10.0786	−0.974339	−0.487169	+	0.873308i	$0.661971\pi$
$108$	0	0
$109$	10.2481	0.981588	0.490794	−	0.871276i	$-0.336707\pi$
$109$	10.2481	0.981588	0.490794	+	0.871276i	$0.336707\pi$
$110$	0	0
$111$	3.24976	0.308453
$112$	0	0
$113$	10.1547	0.955278	0.477639	−	0.878556i	$-0.341493\pi$
$113$	10.1547	0.955278	0.477639	+	0.878556i	$0.341493\pi$
$114$	0	0
$115$	6.99097	0.651911
$116$	0	0
$117$	1.24976	0.115540
$118$	0	0
$119$	2.89763	0.265626
$120$	0	0
$121$	16.6372	1.51247
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	21.7033	1.92585	0.962927	−	0.269763i	$-0.0869453\pi$
$127$	21.7033	1.92585	0.962927	+	0.269763i	$0.0869453\pi$
$128$	0	0
$129$	4.34477	0.382536
$130$	0	0
$131$	−19.0997	−1.66875	−0.834375	−	0.551197i	$-0.814171\pi$
$131$	−19.0997	−1.66875	−0.834375	+	0.551197i	$0.814171\pi$
$132$	0	0
$133$	−3.56659	−0.309262
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	10.3367	0.883126	0.441563	−	0.897230i	$-0.354424\pi$
$137$	10.3367	0.883126	0.441563	+	0.897230i	$0.354424\pi$
$138$	0	0
$139$	−1.80997	−0.153519	−0.0767597	−	0.997050i	$-0.524457\pi$
$139$	−1.80997	−0.153519	−0.0767597	+	0.997050i	$0.524457\pi$
$140$	0	0
$141$	−7.14571	−0.601777
$142$	0	0
$143$	6.57010	0.549419
$144$	0	0
$145$	−5.89596	−0.489632
$146$	0	0
$147$	5.80094	0.478453
$148$	0	0
$149$	15.2571	1.24991	0.624955	−	0.780660i	$-0.285117\pi$
$149$	15.2571	1.24991	0.624955	+	0.780660i	$0.285117\pi$
$150$	0	0
$151$	6.26613	0.509931	0.254965	−	0.966950i	$-0.417936\pi$
$151$	6.26613	0.509931	0.254965	+	0.966950i	$0.417936\pi$
$152$	0	0
$153$	−2.64620	−0.213933
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	21.2391	1.69506	0.847530	−	0.530747i	$-0.178089\pi$
$157$	21.2391	1.69506	0.847530	+	0.530747i	$0.178089\pi$
$158$	0	0
$159$	−6.80094	−0.539350
$160$	0	0
$161$	7.65523	0.603317
$162$	0	0
$163$	15.3511	1.20239	0.601197	−	0.799101i	$-0.294691\pi$
$163$	15.3511	1.20239	0.601197	+	0.799101i	$0.294691\pi$
$164$	0	0
$165$	5.25711	0.409265
$166$	0	0
$167$	18.6690	1.44465	0.722323	−	0.691555i	$-0.243074\pi$
$167$	18.6690	1.44465	0.722323	+	0.691555i	$0.243074\pi$
$168$	0	0
$169$	−11.4381	−0.879855
$170$	0	0
$171$	3.25711	0.249077
$172$	0	0
$173$	13.4118	1.01968	0.509842	−	0.860268i	$-0.329704\pi$
$173$	13.4118	1.01968	0.509842	+	0.860268i	$0.329704\pi$
$174$	0	0
$175$	−1.09502	−0.0827755
$176$	0	0
$177$	8.80829	0.662072
$178$	0	0
$179$	−24.4962	−1.83093	−0.915464	−	0.402399i	$-0.868176\pi$
$179$	−24.4962	−1.83093	−0.915464	+	0.402399i	$0.868176\pi$
$180$	0	0
$181$	17.0343	1.26615	0.633075	−	0.774090i	$-0.281792\pi$
$181$	17.0343	1.26615	0.633075	+	0.774090i	$0.281792\pi$
$182$	0	0
$183$	−2.91234	−0.215286
$184$	0	0
$185$	3.24976	0.238927
$186$	0	0
$187$	−13.9114	−1.01730
$188$	0	0
$189$	1.09502	0.0796507
$190$	0	0
$191$	12.4988	0.904382	0.452191	−	0.891921i	$-0.350642\pi$
$191$	12.4988	0.904382	0.452191	+	0.891921i	$0.350642\pi$
$192$	0	0
$193$	−5.60188	−0.403232	−0.201616	−	0.979465i	$-0.564619\pi$
$193$	−5.60188	−0.403232	−0.201616	+	0.979465i	$0.564619\pi$
$194$	0	0
$195$	1.24976	0.0894968
$196$	0	0
$197$	24.2481	1.72760	0.863802	−	0.503831i	$-0.168077\pi$
$197$	24.2481	1.72760	0.863802	+	0.503831i	$0.168077\pi$
$198$	0	0
$199$	−4.61091	−0.326859	−0.163429	−	0.986555i	$-0.552256\pi$
$199$	−4.61091	−0.326859	−0.163429	+	0.986555i	$0.552256\pi$
$200$	0	0
$201$	−1.43979	−0.101555
$202$	0	0
$203$	−6.45617	−0.453134
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	−6.99097	−0.485906
$208$	0	0
$209$	17.1230	1.18442
$210$	0	0
$211$	−2.35947	−0.162433	−0.0812165	−	0.996696i	$-0.525881\pi$
$211$	−2.35947	−0.162433	−0.0812165	+	0.996696i	$0.525881\pi$
$212$	0	0
$213$	4.24073	0.290570
$214$	0	0
$215$	4.34477	0.296311
$216$	0	0
$217$	−1.09502	−0.0743346
$218$	0	0
$219$	8.69689	0.587682
$220$	0	0
$221$	−3.30710	−0.222460
$222$	0	0
$223$	0.792889	0.0530958	0.0265479	−	0.999648i	$-0.491549\pi$
$223$	0.792889	0.0530958	0.0265479	+	0.999648i	$0.491549\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	22.7976	1.51313	0.756564	−	0.653919i	$-0.226876\pi$
$227$	22.7976	1.51313	0.756564	+	0.653919i	$0.226876\pi$
$228$	0	0
$229$	−6.60539	−0.436497	−0.218248	−	0.975893i	$-0.570034\pi$
$229$	−6.60539	−0.436497	−0.218248	+	0.975893i	$0.570034\pi$
$230$	0	0
$231$	5.75662	0.378758
$232$	0	0
$233$	28.7417	1.88293	0.941466	−	0.337109i	$-0.109449\pi$
$233$	28.7417	1.88293	0.941466	+	0.337109i	$0.109449\pi$
$234$	0	0
$235$	−7.14571	−0.466135
$236$	0	0
$237$	−12.0933	−0.785547
$238$	0	0
$239$	−2.10139	−0.135928	−0.0679638	−	0.997688i	$-0.521650\pi$
$239$	−2.10139	−0.135928	−0.0679638	+	0.997688i	$0.521650\pi$
$240$	0	0
$241$	4.11945	0.265357	0.132678	−	0.991159i	$-0.457642\pi$
$241$	4.11945	0.265357	0.132678	+	0.991159i	$0.457642\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	5.80094	0.370608
$246$	0	0
$247$	4.07059	0.259005
$248$	0	0
$249$	−0.0433434	−0.00274677
$250$	0	0
$251$	24.1161	1.52219	0.761097	−	0.648638i	$-0.224661\pi$
$251$	24.1161	1.52219	0.761097	+	0.648638i	$0.224661\pi$
$252$	0	0
$253$	−36.7523	−2.31060
$254$	0	0
$255$	−2.64620	−0.165711
$256$	0	0
$257$	−4.09334	−0.255336	−0.127668	−	0.991817i	$-0.540749\pi$
$257$	−4.09334	−0.255336	−0.127668	+	0.991817i	$0.540749\pi$
$258$	0	0
$259$	3.55854	0.221117
$260$	0	0
$261$	5.89596	0.364950
$262$	0	0
$263$	10.5289	0.649241	0.324620	−	0.945844i	$-0.394763\pi$
$263$	10.5289	0.649241	0.324620	+	0.945844i	$0.394763\pi$
$264$	0	0
$265$	−6.80094	−0.417778
$266$	0	0
$267$	−4.63885	−0.283893
$268$	0	0
$269$	−5.61728	−0.342492	−0.171246	−	0.985228i	$-0.554779\pi$
$269$	−5.61728	−0.342492	−0.171246	+	0.985228i	$0.554779\pi$
$270$	0	0
$271$	−26.8556	−1.63136	−0.815682	−	0.578501i	$-0.803638\pi$
$271$	−26.8556	−1.63136	−0.815682	+	0.578501i	$0.803638\pi$
$272$	0	0
$273$	1.36850	0.0828255
$274$	0	0
$275$	5.25711	0.317015
$276$	0	0
$277$	−31.1578	−1.87209	−0.936044	−	0.351883i	$-0.885542\pi$
$277$	−31.1578	−1.87209	−0.936044	+	0.351883i	$0.885542\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	22.3948	1.33596	0.667980	−	0.744179i	$-0.267159\pi$
$281$	22.3948	1.33596	0.667980	+	0.744179i	$0.267159\pi$
$282$	0	0
$283$	−0.616096	−0.0366231	−0.0183116	−	0.999832i	$-0.505829\pi$
$283$	−0.616096	−0.0366231	−0.0183116	+	0.999832i	$0.505829\pi$
$284$	0	0
$285$	3.25711	0.192934
$286$	0	0
$287$	−4.38007	−0.258547
$288$	0	0
$289$	−9.99762	−0.588096
$290$	0	0
$291$	−0.309478	−0.0181419
$292$	0	0
$293$	−28.0400	−1.63811	−0.819057	−	0.573712i	$-0.805503\pi$
$293$	−28.0400	−1.63811	−0.819057	+	0.573712i	$0.805503\pi$
$294$	0	0
$295$	8.80829	0.512839
$296$	0	0
$297$	−5.25711	−0.305048
$298$	0	0
$299$	−8.73701	−0.505274
$300$	0	0
$301$	4.75760	0.274223
$302$	0	0
$303$	−16.4789	−0.946689
$304$	0	0
$305$	−2.91234	−0.166760
$306$	0	0
$307$	−19.6118	−1.11930	−0.559651	−	0.828728i	$-0.689065\pi$
$307$	−19.6118	−1.11930	−0.559651	+	0.828728i	$0.689065\pi$
$308$	0	0
$309$	15.0417	0.855691
$310$	0	0
$311$	31.2750	1.77344	0.886722	−	0.462302i	$-0.152976\pi$
$311$	31.2750	1.77344	0.886722	+	0.462302i	$0.152976\pi$
$312$	0	0
$313$	−0.630799	−0.0356549	−0.0178274	−	0.999841i	$-0.505675\pi$
$313$	−0.630799	−0.0356549	−0.0178274	+	0.999841i	$0.505675\pi$
$314$	0	0
$315$	1.09502	0.0616972
$316$	0	0
$317$	−3.88272	−0.218075	−0.109038	−	0.994038i	$-0.534777\pi$
$317$	−3.88272	−0.218075	−0.109038	+	0.994038i	$0.534777\pi$
$318$	0	0
$319$	30.9957	1.73542
$320$	0	0
$321$	10.0786	0.562535
$322$	0	0
$323$	−8.61896	−0.479572
$324$	0	0
$325$	1.24976	0.0693240
$326$	0	0
$327$	−10.2481	−0.566720
$328$	0	0
$329$	−7.82467	−0.431388
$330$	0	0
$331$	−11.3504	−0.623877	−0.311938	−	0.950102i	$-0.600978\pi$
$331$	−11.3504	−0.623877	−0.311938	+	0.950102i	$0.600978\pi$
$332$	0	0
$333$	−3.24976	−0.178086
$334$	0	0
$335$	−1.43979	−0.0786640
$336$	0	0
$337$	20.6435	1.12453	0.562263	−	0.826959i	$-0.309931\pi$
$337$	20.6435	1.12453	0.562263	+	0.826959i	$0.309931\pi$
$338$	0	0
$339$	−10.1547	−0.551530
$340$	0	0
$341$	5.25711	0.284688
$342$	0	0
$343$	14.0172	0.756860
$344$	0	0
$345$	−6.99097	−0.376381
$346$	0	0
$347$	−21.6166	−1.16044	−0.580219	−	0.814460i	$-0.697033\pi$
$347$	−21.6166	−1.16044	−0.580219	+	0.814460i	$0.697033\pi$
$348$	0	0
$349$	−4.42341	−0.236780	−0.118390	−	0.992967i	$-0.537773\pi$
$349$	−4.42341	−0.236780	−0.118390	+	0.992967i	$0.537773\pi$
$350$	0	0
$351$	−1.24976	−0.0667070
$352$	0	0
$353$	15.8680	0.844569	0.422285	−	0.906463i	$-0.361228\pi$
$353$	15.8680	0.844569	0.422285	+	0.906463i	$0.361228\pi$
$354$	0	0
$355$	4.24073	0.225074
$356$	0	0
$357$	−2.89763	−0.153359
$358$	0	0
$359$	8.59767	0.453768	0.226884	−	0.973922i	$-0.427146\pi$
$359$	8.59767	0.453768	0.226884	+	0.973922i	$0.427146\pi$
$360$	0	0
$361$	−8.39126	−0.441645
$362$	0	0
$363$	−16.6372	−0.873225
$364$	0	0
$365$	8.69689	0.455216
$366$	0	0
$367$	−12.3389	−0.644084	−0.322042	−	0.946725i	$-0.604369\pi$
$367$	−12.3389	−0.644084	−0.322042	+	0.946725i	$0.604369\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	0	0
$371$	−7.44714	−0.386636
$372$	0	0
$373$	28.2528	1.46287	0.731437	−	0.681909i	$-0.238850\pi$
$373$	28.2528	1.46287	0.731437	+	0.681909i	$0.238850\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	7.36850	0.379497
$378$	0	0
$379$	−32.0172	−1.64462	−0.822308	−	0.569043i	$-0.807314\pi$
$379$	−32.0172	−1.64462	−0.822308	+	0.569043i	$0.807314\pi$
$380$	0	0
$381$	−21.7033	−1.11189
$382$	0	0
$383$	11.0082	0.562493	0.281247	−	0.959636i	$-0.409252\pi$
$383$	11.0082	0.562493	0.281247	+	0.959636i	$0.409252\pi$
$384$	0	0
$385$	5.75662	0.293384
$386$	0	0
$387$	−4.34477	−0.220857
$388$	0	0
$389$	−4.60020	−0.233239	−0.116620	−	0.993177i	$-0.537206\pi$
$389$	−4.60020	−0.233239	−0.116620	+	0.993177i	$0.537206\pi$
$390$	0	0
$391$	18.4995	0.935560
$392$	0	0
$393$	19.0997	0.963453
$394$	0	0
$395$	−12.0933	−0.608482
$396$	0	0
$397$	−6.47892	−0.325168	−0.162584	−	0.986695i	$-0.551983\pi$
$397$	−6.47892	−0.325168	−0.162584	+	0.986695i	$0.551983\pi$
$398$	0	0
$399$	3.56659	0.178553
$400$	0	0
$401$	0.0506947	0.00253157	0.00126579	−	0.999999i	$-0.499597\pi$
$401$	0.0506947	0.00253157	0.00126579	+	0.999999i	$0.499597\pi$
$402$	0	0
$403$	1.24976	0.0622548
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−17.0843	−0.846838
$408$	0	0
$409$	18.1867	0.899273	0.449637	−	0.893212i	$-0.351553\pi$
$409$	18.1867	0.899273	0.449637	+	0.893212i	$0.351553\pi$
$410$	0	0
$411$	−10.3367	−0.509873
$412$	0	0
$413$	9.64522	0.474610
$414$	0	0
$415$	−0.0433434	−0.00212764
$416$	0	0
$417$	1.80997	0.0886345
$418$	0	0
$419$	13.4798	0.658530	0.329265	−	0.944238i	$-0.393199\pi$
$419$	13.4798	0.658530	0.329265	+	0.944238i	$0.393199\pi$
$420$	0	0
$421$	7.14571	0.348261	0.174130	−	0.984723i	$-0.444289\pi$
$421$	7.14571	0.348261	0.174130	+	0.984723i	$0.444289\pi$
$422$	0	0
$423$	7.14571	0.347436
$424$	0	0
$425$	−2.64620	−0.128360
$426$	0	0
$427$	−3.18905	−0.154329
$428$	0	0
$429$	−6.57010	−0.317207
$430$	0	0
$431$	39.6551	1.91012	0.955060	−	0.296414i	$-0.0957907\pi$
$431$	39.6551	1.91012	0.955060	+	0.296414i	$0.0957907\pi$
$432$	0	0
$433$	13.4011	0.644018	0.322009	−	0.946737i	$-0.395642\pi$
$433$	13.4011	0.644018	0.322009	+	0.946737i	$0.395642\pi$
$434$	0	0
$435$	5.89596	0.282689
$436$	0	0
$437$	−22.7703	−1.08925
$438$	0	0
$439$	−26.8943	−1.28359	−0.641797	−	0.766875i	$-0.721811\pi$
$439$	−26.8943	−1.28359	−0.641797	+	0.766875i	$0.721811\pi$
$440$	0	0
$441$	−5.80094	−0.276235
$442$	0	0
$443$	−17.7692	−0.844238	−0.422119	−	0.906540i	$-0.638714\pi$
$443$	−17.7692	−0.844238	−0.422119	+	0.906540i	$0.638714\pi$
$444$	0	0
$445$	−4.63885	−0.219903
$446$	0	0
$447$	−15.2571	−0.721636
$448$	0	0
$449$	16.8642	0.795870	0.397935	−	0.917414i	$-0.369727\pi$
$449$	16.8642	0.795870	0.397935	+	0.917414i	$0.369727\pi$
$450$	0	0
$451$	21.0284	0.990190
$452$	0	0
$453$	−6.26613	−0.294409
$454$	0	0
$455$	1.36850	0.0641564
$456$	0	0
$457$	9.67728	0.452684	0.226342	−	0.974048i	$-0.427323\pi$
$457$	9.67728	0.452684	0.226342	+	0.974048i	$0.427323\pi$
$458$	0	0
$459$	2.64620	0.123514
$460$	0	0
$461$	15.2442	0.709995	0.354998	−	0.934867i	$-0.384482\pi$
$461$	15.2442	0.709995	0.354998	+	0.934867i	$0.384482\pi$
$462$	0	0
$463$	−42.3028	−1.96598	−0.982988	−	0.183668i	$-0.941203\pi$
$463$	−42.3028	−1.96598	−0.982988	+	0.183668i	$0.941203\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	13.1277	0.607475	0.303738	−	0.952756i	$-0.401765\pi$
$467$	13.1277	0.607475	0.303738	+	0.952756i	$0.401765\pi$
$468$	0	0
$469$	−1.57659	−0.0728002
$470$	0	0
$471$	−21.2391	−0.978644
$472$	0	0
$473$	−22.8409	−1.05023
$474$	0	0
$475$	3.25711	0.149446
$476$	0	0
$477$	6.80094	0.311394
$478$	0	0
$479$	19.0791	0.871747	0.435874	−	0.900008i	$-0.356439\pi$
$479$	19.0791	0.871747	0.435874	+	0.900008i	$0.356439\pi$
$480$	0	0
$481$	−4.06140	−0.185184
$482$	0	0
$483$	−7.65523	−0.348325
$484$	0	0
$485$	−0.309478	−0.0140527
$486$	0	0
$487$	21.2365	0.962318	0.481159	−	0.876633i	$-0.340216\pi$
$487$	21.2365	0.962318	0.481159	+	0.876633i	$0.340216\pi$
$488$	0	0
$489$	−15.3511	−0.694203
$490$	0	0
$491$	−7.66797	−0.346051	−0.173025	−	0.984917i	$-0.555354\pi$
$491$	−7.66797	−0.346051	−0.173025	+	0.984917i	$0.555354\pi$
$492$	0	0
$493$	−15.6019	−0.702673
$494$	0	0
$495$	−5.25711	−0.236289
$496$	0	0
$497$	4.64367	0.208297
$498$	0	0
$499$	−38.7582	−1.73505	−0.867527	−	0.497390i	$-0.834292\pi$
$499$	−38.7582	−1.73505	−0.867527	+	0.497390i	$0.834292\pi$
$500$	0	0
$501$	−18.6690	−0.834067
$502$	0	0
$503$	−12.9249	−0.576292	−0.288146	−	0.957587i	$-0.593039\pi$
$503$	−12.9249	−0.576292	−0.288146	+	0.957587i	$0.593039\pi$
$504$	0	0
$505$	−16.4789	−0.733302
$506$	0	0
$507$	11.4381	0.507984
$508$	0	0
$509$	−3.05421	−0.135375	−0.0676877	−	0.997707i	$-0.521562\pi$
$509$	−3.05421	−0.135375	−0.0676877	+	0.997707i	$0.521562\pi$
$510$	0	0
$511$	9.52324	0.421283
$512$	0	0
$513$	−3.25711	−0.143805
$514$	0	0
$515$	15.0417	0.662815
$516$	0	0
$517$	37.5658	1.65214
$518$	0	0
$519$	−13.4118	−0.588715
$520$	0	0
$521$	−6.13415	−0.268742	−0.134371	−	0.990931i	$-0.542901\pi$
$521$	−6.13415	−0.268742	−0.134371	+	0.990931i	$0.542901\pi$
$522$	0	0
$523$	14.6984	0.642714	0.321357	−	0.946958i	$-0.395861\pi$
$523$	14.6984	0.642714	0.321357	+	0.946958i	$0.395861\pi$
$524$	0	0
$525$	1.09502	0.0477904
$526$	0	0
$527$	−2.64620	−0.115270
$528$	0	0
$529$	25.8737	1.12494
$530$	0	0
$531$	−8.80829	−0.382247
$532$	0	0
$533$	4.99902	0.216532
$534$	0	0
$535$	10.0786	0.435737
$536$	0	0
$537$	24.4962	1.05709
$538$	0	0
$539$	−30.4962	−1.31356
$540$	0	0
$541$	20.5542	0.883694	0.441847	−	0.897090i	$-0.354323\pi$
$541$	20.5542	0.883694	0.441847	+	0.897090i	$0.354323\pi$
$542$	0	0
$543$	−17.0343	−0.731012
$544$	0	0
$545$	−10.2481	−0.438979
$546$	0	0
$547$	7.08285	0.302841	0.151420	−	0.988469i	$-0.451615\pi$
$547$	7.08285	0.302841	0.151420	+	0.988469i	$0.451615\pi$
$548$	0	0
$549$	2.91234	0.124295
$550$	0	0
$551$	19.2038	0.818108
$552$	0	0
$553$	−13.2424	−0.563124
$554$	0	0
$555$	−3.24976	−0.137944
$556$	0	0
$557$	2.18338	0.0925128	0.0462564	−	0.998930i	$-0.485271\pi$
$557$	2.18338	0.0925128	0.0462564	+	0.998930i	$0.485271\pi$
$558$	0	0
$559$	−5.42990	−0.229660
$560$	0	0
$561$	13.9114	0.587338
$562$	0	0
$563$	−34.5164	−1.45469	−0.727346	−	0.686271i	$-0.759246\pi$
$563$	−34.5164	−1.45469	−0.727346	+	0.686271i	$0.759246\pi$
$564$	0	0
$565$	−10.1547	−0.427213
$566$	0	0
$567$	−1.09502	−0.0459864
$568$	0	0
$569$	5.65453	0.237050	0.118525	−	0.992951i	$-0.462183\pi$
$569$	5.65453	0.237050	0.118525	+	0.992951i	$0.462183\pi$
$570$	0	0
$571$	10.1900	0.426440	0.213220	−	0.977004i	$-0.431605\pi$
$571$	10.1900	0.426440	0.213220	+	0.977004i	$0.431605\pi$
$572$	0	0
$573$	−12.4988	−0.522145
$574$	0	0
$575$	−6.99097	−0.291544
$576$	0	0
$577$	1.70327	0.0709080	0.0354540	−	0.999371i	$-0.488712\pi$
$577$	1.70327	0.0709080	0.0354540	+	0.999371i	$0.488712\pi$
$578$	0	0
$579$	5.60188	0.232806
$580$	0	0
$581$	−0.0474617	−0.00196904
$582$	0	0
$583$	35.7533	1.48075
$584$	0	0
$585$	−1.24976	−0.0516710
$586$	0	0
$587$	12.7042	0.524360	0.262180	−	0.965019i	$-0.415559\pi$
$587$	12.7042	0.524360	0.262180	+	0.965019i	$0.415559\pi$
$588$	0	0
$589$	3.25711	0.134207
$590$	0	0
$591$	−24.2481	−0.997433
$592$	0	0
$593$	−28.7023	−1.17866	−0.589331	−	0.807892i	$-0.700609\pi$
$593$	−28.7023	−1.17866	−0.589331	+	0.807892i	$0.700609\pi$
$594$	0	0
$595$	−2.89763	−0.118791
$596$	0	0
$597$	4.61091	0.188712
$598$	0	0
$599$	12.8597	0.525432	0.262716	−	0.964873i	$-0.415382\pi$
$599$	12.8597	0.525432	0.262716	+	0.964873i	$0.415382\pi$
$600$	0	0
$601$	12.5289	0.511065	0.255533	−	0.966800i	$-0.417749\pi$
$601$	12.5289	0.511065	0.255533	+	0.966800i	$0.417749\pi$
$602$	0	0
$603$	1.43979	0.0586327
$604$	0	0
$605$	−16.6372	−0.676397
$606$	0	0
$607$	−5.22916	−0.212245	−0.106123	−	0.994353i	$-0.533844\pi$
$607$	−5.22916	−0.212245	−0.106123	+	0.994353i	$0.533844\pi$
$608$	0	0
$609$	6.45617	0.261617
$610$	0	0
$611$	8.93039	0.361285
$612$	0	0
$613$	−16.8697	−0.681360	−0.340680	−	0.940179i	$-0.610657\pi$
$613$	−16.8697	−0.681360	−0.340680	+	0.940179i	$0.610657\pi$
$614$	0	0
$615$	4.00000	0.161296
$616$	0	0
$617$	43.3551	1.74541	0.872706	−	0.488246i	$-0.162363\pi$
$617$	43.3551	1.74541	0.872706	+	0.488246i	$0.162363\pi$
$618$	0	0
$619$	−14.5448	−0.584605	−0.292303	−	0.956326i	$-0.594421\pi$
$619$	−14.5448	−0.584605	−0.292303	+	0.956326i	$0.594421\pi$
$620$	0	0
$621$	6.99097	0.280538
$622$	0	0
$623$	−5.07962	−0.203510
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−17.1230	−0.683825
$628$	0	0
$629$	8.59950	0.342885
$630$	0	0
$631$	10.4970	0.417878	0.208939	−	0.977929i	$-0.432999\pi$
$631$	10.4970	0.417878	0.208939	+	0.977929i	$0.432999\pi$
$632$	0	0
$633$	2.35947	0.0937807
$634$	0	0
$635$	−21.7033	−0.861268
$636$	0	0
$637$	−7.24976	−0.287246
$638$	0	0
$639$	−4.24073	−0.167761
$640$	0	0
$641$	−33.2864	−1.31473	−0.657367	−	0.753571i	$-0.728330\pi$
$641$	−33.2864	−1.31473	−0.657367	+	0.753571i	$0.728330\pi$
$642$	0	0
$643$	19.3585	0.763425	0.381712	−	0.924281i	$-0.375335\pi$
$643$	19.3585	0.763425	0.381712	+	0.924281i	$0.375335\pi$
$644$	0	0
$645$	−4.34477	−0.171075
$646$	0	0
$647$	−19.3732	−0.761639	−0.380820	−	0.924649i	$-0.624358\pi$
$647$	−19.3732	−0.761639	−0.380820	+	0.924649i	$0.624358\pi$
$648$	0	0
$649$	−46.3061	−1.81767
$650$	0	0
$651$	1.09502	0.0429171
$652$	0	0
$653$	0.456168	0.0178512	0.00892561	−	0.999960i	$-0.497159\pi$
$653$	0.456168	0.0178512	0.00892561	+	0.999960i	$0.497159\pi$
$654$	0	0
$655$	19.0997	0.746288
$656$	0	0
$657$	−8.69689	−0.339298
$658$	0	0
$659$	49.2158	1.91718	0.958588	−	0.284797i	$-0.0919263\pi$
$659$	49.2158	1.91718	0.958588	+	0.284797i	$0.0919263\pi$
$660$	0	0
$661$	−45.1984	−1.75802	−0.879008	−	0.476807i	$-0.841794\pi$
$661$	−45.1984	−1.75802	−0.879008	+	0.476807i	$0.841794\pi$
$662$	0	0
$663$	3.30710	0.128437
$664$	0	0
$665$	3.56659	0.138306
$666$	0	0
$667$	−41.2185	−1.59599
$668$	0	0
$669$	−0.792889	−0.0306549
$670$	0	0
$671$	15.3105	0.591054
$672$	0	0
$673$	15.1611	0.584418	0.292209	−	0.956354i	$-0.405610\pi$
$673$	15.1611	0.584418	0.292209	+	0.956354i	$0.405610\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−3.78386	−0.145426	−0.0727128	−	0.997353i	$-0.523166\pi$
$677$	−3.78386	−0.145426	−0.0727128	+	0.997353i	$0.523166\pi$
$678$	0	0
$679$	−0.338884	−0.0130052
$680$	0	0
$681$	−22.7976	−0.873605
$682$	0	0
$683$	−25.3479	−0.969911	−0.484955	−	0.874539i	$-0.661164\pi$
$683$	−25.3479	−0.969911	−0.484955	+	0.874539i	$0.661164\pi$
$684$	0	0
$685$	−10.3367	−0.394946
$686$	0	0
$687$	6.60539	0.252011
$688$	0	0
$689$	8.49951	0.323806
$690$	0	0
$691$	22.0500	0.838821	0.419411	−	0.907797i	$-0.362237\pi$
$691$	22.0500	0.838821	0.419411	+	0.907797i	$0.362237\pi$
$692$	0	0
$693$	−5.75662	−0.218676
$694$	0	0
$695$	1.80997	0.0686560
$696$	0	0
$697$	−10.5848	−0.400928
$698$	0	0
$699$	−28.7417	−1.08711
$700$	0	0
$701$	−4.72819	−0.178581	−0.0892906	−	0.996006i	$-0.528460\pi$
$701$	−4.72819	−0.178581	−0.0892906	+	0.996006i	$0.528460\pi$
$702$	0	0
$703$	−10.5848	−0.399213
$704$	0	0
$705$	7.14571	0.269123
$706$	0	0
$707$	−18.0447	−0.678640
$708$	0	0
$709$	−41.5305	−1.55971	−0.779855	−	0.625960i	$-0.784707\pi$
$709$	−41.5305	−1.55971	−0.779855	+	0.625960i	$0.784707\pi$
$710$	0	0
$711$	12.0933	0.453536
$712$	0	0
$713$	−6.99097	−0.261814
$714$	0	0
$715$	−6.57010	−0.245708
$716$	0	0
$717$	2.10139	0.0784778
$718$	0	0
$719$	−18.2028	−0.678849	−0.339425	−	0.940633i	$-0.610232\pi$
$719$	−18.2028	−0.678849	−0.339425	+	0.940633i	$0.610232\pi$
$720$	0	0
$721$	16.4709	0.613407
$722$	0	0
$723$	−4.11945	−0.153204
$724$	0	0
$725$	5.89596	0.218970
$726$	0	0
$727$	2.21209	0.0820418	0.0410209	−	0.999158i	$-0.486939\pi$
$727$	2.21209	0.0820418	0.0410209	+	0.999158i	$0.486939\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	11.4971	0.425237
$732$	0	0
$733$	−39.0578	−1.44263	−0.721317	−	0.692605i	$-0.756463\pi$
$733$	−39.0578	−1.44263	−0.721317	+	0.692605i	$0.756463\pi$
$734$	0	0
$735$	−5.80094	−0.213971
$736$	0	0
$737$	7.56912	0.278812
$738$	0	0
$739$	−1.11491	−0.0410126	−0.0205063	−	0.999790i	$-0.506528\pi$
$739$	−1.11491	−0.0410126	−0.0205063	+	0.999790i	$0.506528\pi$
$740$	0	0
$741$	−4.07059	−0.149537
$742$	0	0
$743$	−26.7556	−0.981569	−0.490785	−	0.871281i	$-0.663290\pi$
$743$	−26.7556	−0.981569	−0.490785	+	0.871281i	$0.663290\pi$
$744$	0	0
$745$	−15.2571	−0.558977
$746$	0	0
$747$	0.0433434	0.00158585
$748$	0	0
$749$	11.0363	0.403257
$750$	0	0
$751$	−14.0783	−0.513723	−0.256862	−	0.966448i	$-0.582688\pi$
$751$	−14.0783	−0.513723	−0.256862	+	0.966448i	$0.582688\pi$
$752$	0	0
$753$	−24.1161	−0.878839
$754$	0	0
$755$	−6.26613	−0.228048
$756$	0	0
$757$	2.16209	0.0785825	0.0392913	−	0.999228i	$-0.487490\pi$
$757$	2.16209	0.0785825	0.0392913	+	0.999228i	$0.487490\pi$
$758$	0	0
$759$	36.7523	1.33402
$760$	0	0
$761$	−3.46450	−0.125588	−0.0627940	−	0.998027i	$-0.520001\pi$
$761$	−3.46450	−0.125588	−0.0627940	+	0.998027i	$0.520001\pi$
$762$	0	0
$763$	−11.2218	−0.406257
$764$	0	0
$765$	2.64620	0.0956736
$766$	0	0
$767$	−11.0082	−0.397483
$768$	0	0
$769$	−14.0786	−0.507689	−0.253844	−	0.967245i	$-0.581695\pi$
$769$	−14.0786	−0.507689	−0.253844	+	0.967245i	$0.581695\pi$
$770$	0	0
$771$	4.09334	0.147418
$772$	0	0
$773$	−45.8327	−1.64849	−0.824244	−	0.566235i	$-0.808400\pi$
$773$	−45.8327	−1.64849	−0.824244	+	0.566235i	$0.808400\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−3.55854	−0.127662
$778$	0	0
$779$	13.0284	0.466792
$780$	0	0
$781$	−22.2940	−0.797740
$782$	0	0
$783$	−5.89596	−0.210704
$784$	0	0
$785$	−21.2391	−0.758054
$786$	0	0
$787$	12.6234	0.449977	0.224989	−	0.974361i	$-0.427765\pi$
$787$	12.6234	0.449977	0.224989	+	0.974361i	$0.427765\pi$
$788$	0	0
$789$	−10.5289	−0.374839
$790$	0	0
$791$	−11.1196	−0.395368
$792$	0	0
$793$	3.63971	0.129250
$794$	0	0
$795$	6.80094	0.241204
$796$	0	0
$797$	11.9841	0.424499	0.212249	−	0.977216i	$-0.431921\pi$
$797$	11.9841	0.424499	0.212249	+	0.977216i	$0.431921\pi$
$798$	0	0
$799$	−18.9090	−0.668952
$800$	0	0
$801$	4.63885	0.163906
$802$	0	0
$803$	−45.7205	−1.61344
$804$	0	0
$805$	−7.65523	−0.269811
$806$	0	0
$807$	5.61728	0.197738
$808$	0	0
$809$	14.9611	0.526003	0.263002	−	0.964795i	$-0.415288\pi$
$809$	14.9611	0.526003	0.263002	+	0.964795i	$0.415288\pi$
$810$	0	0
$811$	16.9604	0.595559	0.297780	−	0.954635i	$-0.403754\pi$
$811$	16.9604	0.595559	0.297780	+	0.954635i	$0.403754\pi$
$812$	0	0
$813$	26.8556	0.941868
$814$	0	0
$815$	−15.3511	−0.537727
$816$	0	0
$817$	−14.1514	−0.495094
$818$	0	0
$819$	−1.36850	−0.0478194
$820$	0	0
$821$	−10.8167	−0.377506	−0.188753	−	0.982025i	$-0.560445\pi$
$821$	−10.8167	−0.377506	−0.188753	+	0.982025i	$0.560445\pi$
$822$	0	0
$823$	−11.4085	−0.397675	−0.198838	−	0.980032i	$-0.563717\pi$
$823$	−11.4085	−0.397675	−0.198838	+	0.980032i	$0.563717\pi$
$824$	0	0
$825$	−5.25711	−0.183029
$826$	0	0
$827$	40.6556	1.41373	0.706867	−	0.707347i	$-0.250108\pi$
$827$	40.6556	1.41373	0.706867	+	0.707347i	$0.250108\pi$
$828$	0	0
$829$	13.0078	0.451781	0.225890	−	0.974153i	$-0.427471\pi$
$829$	13.0078	0.451781	0.225890	+	0.974153i	$0.427471\pi$
$830$	0	0
$831$	31.1578	1.08085
$832$	0	0
$833$	15.3504	0.531861
$834$	0	0
$835$	−18.6690	−0.646066
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−54.8677	−1.89424	−0.947121	−	0.320876i	$-0.896023\pi$
$839$	−54.8677	−1.89424	−0.947121	+	0.320876i	$0.896023\pi$
$840$	0	0
$841$	5.76229	0.198700
$842$	0	0
$843$	−22.3948	−0.771317
$844$	0	0
$845$	11.4381	0.393483
$846$	0	0
$847$	−18.2180	−0.625977
$848$	0	0
$849$	0.616096	0.0211444
$850$	0	0
$851$	22.7189	0.778796
$852$	0	0
$853$	−57.6338	−1.97334	−0.986672	−	0.162721i	$-0.947973\pi$
$853$	−57.6338	−1.97334	−0.986672	+	0.162721i	$0.947973\pi$
$854$	0	0
$855$	−3.25711	−0.111391
$856$	0	0
$857$	−8.34731	−0.285139	−0.142569	−	0.989785i	$-0.545536\pi$
$857$	−8.34731	−0.285139	−0.142569	+	0.989785i	$0.545536\pi$
$858$	0	0
$859$	−25.7886	−0.879894	−0.439947	−	0.898024i	$-0.645003\pi$
$859$	−25.7886	−0.879894	−0.439947	+	0.898024i	$0.645003\pi$
$860$	0	0
$861$	4.38007	0.149272
$862$	0	0
$863$	8.32672	0.283445	0.141722	−	0.989906i	$-0.454736\pi$
$863$	8.32672	0.283445	0.141722	+	0.989906i	$0.454736\pi$
$864$	0	0
$865$	−13.4118	−0.456016
$866$	0	0
$867$	9.99762	0.339537
$868$	0	0
$869$	63.5760	2.15667
$870$	0	0
$871$	1.79938	0.0609698
$872$	0	0
$873$	0.309478	0.0104743
$874$	0	0
$875$	1.09502	0.0370183
$876$	0	0
$877$	57.7377	1.94967	0.974833	−	0.222938i	$-0.0715649\pi$
$877$	57.7377	1.94967	0.974833	+	0.222938i	$0.0715649\pi$
$878$	0	0
$879$	28.0400	0.945765
$880$	0	0
$881$	−32.6002	−1.09833	−0.549164	−	0.835714i	$-0.685054\pi$
$881$	−32.6002	−1.09833	−0.549164	+	0.835714i	$0.685054\pi$
$882$	0	0
$883$	−28.0955	−0.945489	−0.472744	−	0.881200i	$-0.656737\pi$
$883$	−28.0955	−0.945489	−0.472744	+	0.881200i	$0.656737\pi$
$884$	0	0
$885$	−8.80829	−0.296087
$886$	0	0
$887$	−36.1741	−1.21461	−0.607304	−	0.794469i	$-0.707749\pi$
$887$	−36.1741	−1.21461	−0.607304	+	0.794469i	$0.707749\pi$
$888$	0	0
$889$	−23.7654	−0.797067
$890$	0	0
$891$	5.25711	0.176120
$892$	0	0
$893$	23.2743	0.778846
$894$	0	0
$895$	24.4962	0.818816
$896$	0	0
$897$	8.73701	0.291720
$898$	0	0
$899$	5.89596	0.196641
$900$	0	0
$901$	−17.9966	−0.599555
$902$	0	0
$903$	−4.75760	−0.158323
$904$	0	0
$905$	−17.0343	−0.566240
$906$	0	0
$907$	−14.3488	−0.476443	−0.238222	−	0.971211i	$-0.576564\pi$
$907$	−14.3488	−0.476443	−0.238222	+	0.971211i	$0.576564\pi$
$908$	0	0
$909$	16.4789	0.546571
$910$	0	0
$911$	7.48579	0.248015	0.124008	−	0.992281i	$-0.460425\pi$
$911$	7.48579	0.248015	0.124008	+	0.992281i	$0.460425\pi$
$912$	0	0
$913$	0.227861	0.00754109
$914$	0	0
$915$	2.91234	0.0962788
$916$	0	0
$917$	20.9145	0.690658
$918$	0	0
$919$	−9.84077	−0.324617	−0.162309	−	0.986740i	$-0.551894\pi$
$919$	−9.84077	−0.324617	−0.162309	+	0.986740i	$0.551894\pi$
$920$	0	0
$921$	19.6118	0.646230
$922$	0	0
$923$	−5.29987	−0.174447
$924$	0	0
$925$	−3.24976	−0.106851
$926$	0	0
$927$	−15.0417	−0.494033
$928$	0	0
$929$	34.2227	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$929$	34.2227	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$930$	0	0
$931$	−18.8943	−0.619235
$932$	0	0
$933$	−31.2750	−1.02390
$934$	0	0
$935$	13.9114	0.454950
$936$	0	0
$937$	−51.5838	−1.68517	−0.842585	−	0.538563i	$-0.818967\pi$
$937$	−51.5838	−1.68517	−0.842585	+	0.538563i	$0.818967\pi$
$938$	0	0
$939$	0.630799	0.0205853
$940$	0	0
$941$	6.44628	0.210143	0.105071	−	0.994465i	$-0.466493\pi$
$941$	6.44628	0.210143	0.105071	+	0.994465i	$0.466493\pi$
$942$	0	0
$943$	−27.9639	−0.910629
$944$	0	0
$945$	−1.09502	−0.0356209
$946$	0	0
$947$	29.2584	0.950772	0.475386	−	0.879777i	$-0.342308\pi$
$947$	29.2584	0.950772	0.475386	+	0.879777i	$0.342308\pi$
$948$	0	0
$949$	−10.8690	−0.352822
$950$	0	0
$951$	3.88272	0.125906
$952$	0	0
$953$	49.3471	1.59851	0.799255	−	0.600993i	$-0.205228\pi$
$953$	49.3471	1.59851	0.799255	+	0.600993i	$0.205228\pi$
$954$	0	0
$955$	−12.4988	−0.404452
$956$	0	0
$957$	−30.9957	−1.00195
$958$	0	0
$959$	−11.3189	−0.365506
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−10.0786	−0.324780
$964$	0	0
$965$	5.60188	0.180331
$966$	0	0
$967$	50.6842	1.62990	0.814948	−	0.579535i	$-0.196766\pi$
$967$	50.6842	1.62990	0.814948	+	0.579535i	$0.196766\pi$
$968$	0	0
$969$	8.61896	0.276881
$970$	0	0
$971$	6.76711	0.217167	0.108583	−	0.994087i	$-0.465369\pi$
$971$	6.76711	0.217167	0.108583	+	0.994087i	$0.465369\pi$
$972$	0	0
$973$	1.98194	0.0635382
$974$	0	0
$975$	−1.24976	−0.0400242
$976$	0	0
$977$	16.5848	0.530595	0.265297	−	0.964167i	$-0.414530\pi$
$977$	16.5848	0.530595	0.265297	+	0.964167i	$0.414530\pi$
$978$	0	0
$979$	24.3869	0.779409
$980$	0	0
$981$	10.2481	0.327196
$982$	0	0
$983$	15.1171	0.482160	0.241080	−	0.970505i	$-0.422498\pi$
$983$	15.1171	0.482160	0.241080	+	0.970505i	$0.422498\pi$
$984$	0	0
$985$	−24.2481	−0.772608
$986$	0	0
$987$	7.82467	0.249062
$988$	0	0
$989$	30.3742	0.965843
$990$	0	0
$991$	60.9948	1.93757	0.968783	−	0.247912i	$-0.0797443\pi$
$991$	60.9948	1.93757	0.968783	+	0.247912i	$0.0797443\pi$
$992$	0	0
$993$	11.3504	0.360196
$994$	0	0
$995$	4.61091	0.146176
$996$	0	0
$997$	40.2355	1.27427	0.637136	−	0.770751i	$-0.280119\pi$
$997$	40.2355	1.27427	0.637136	+	0.770751i	$0.280119\pi$
$998$	0	0
$999$	3.24976	0.102818

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3720.2.a.r.1.2	✓	4
4.3	odd	2		7440.2.a.bx.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.r.1.2	✓	4	1.1	even	1	trivial
7440.2.a.bx.1.3		4	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -1.09502 q^{7} +1.00000 q^{9} +5.25711 q^{11} +1.24976 q^{13} +1.00000 q^{15} -2.64620 q^{17} +3.25711 q^{19} +1.09502 q^{21} -6.99097 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.89596 q^{29} +1.00000 q^{31} -5.25711 q^{33} +1.09502 q^{35} -3.24976 q^{37} -1.24976 q^{39} +4.00000 q^{41} -4.34477 q^{43} -1.00000 q^{45} +7.14571 q^{47} -5.80094 q^{49} +2.64620 q^{51} +6.80094 q^{53} -5.25711 q^{55} -3.25711 q^{57} -8.80829 q^{59} +2.91234 q^{61} -1.09502 q^{63} -1.24976 q^{65} +1.43979 q^{67} +6.99097 q^{69} -4.24073 q^{71} -8.69689 q^{73} -1.00000 q^{75} -5.75662 q^{77} +12.0933 q^{79} +1.00000 q^{81} +0.0433434 q^{83} +2.64620 q^{85} -5.89596 q^{87} +4.63885 q^{89} -1.36850 q^{91} -1.00000 q^{93} -3.25711 q^{95} +0.309478 q^{97} +5.25711 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} + q^{11} - 2 q^{13} + 4 q^{15} + 4 q^{17} - 7 q^{19} - q^{21} - q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} + 2 q^{39}+ \cdots + q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 + q^11 - 2 * q^13 + 4 * q^15 + 4 * q^17 - 7 * q^19 - q^21 - q^23 + 4 * q^25 - 4 * q^27 + 2 * q^29 + 4 * q^31 - q^33 - q^35 - 6 * q^37 + 2 * q^39 + 16 * q^41 - 5 * q^43 - 4 * q^45 - 7 * q^49 - 4 * q^51 + 11 * q^53 - q^55 + 7 * q^57 - 6 * q^59 + 4 * q^61 + q^63 + 2 * q^65 - 12 * q^67 + q^69 + 17 * q^71 + 3 * q^73 - 4 * q^75 + 11 * q^77 + 3 * q^79 + 4 * q^81 - 10 * q^83 - 4 * q^85 - 2 * q^87 + 17 * q^89 + 6 * q^91 - 4 * q^93 + 7 * q^95 - 2 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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