LMFDB - Embedded newform 3920.2.a.by.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(1,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.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:	$31.3013575923$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1960)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +O(q^{10})$	$q+2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +1.00000 q^{11} +0.414214 q^{13} +2.41421 q^{15} -2.41421 q^{17} +2.00000 q^{19} -6.24264 q^{23} +1.00000 q^{25} -0.414214 q^{27} +1.00000 q^{29} +10.2426 q^{31} +2.41421 q^{33} +11.8995 q^{37} +1.00000 q^{39} -4.58579 q^{41} +11.6569 q^{43} +2.82843 q^{45} +7.58579 q^{47} -5.82843 q^{51} +6.58579 q^{53} +1.00000 q^{55} +4.82843 q^{57} +1.75736 q^{59} +6.82843 q^{61} +0.414214 q^{65} +1.41421 q^{67} -15.0711 q^{69} +2.48528 q^{71} +10.8284 q^{73} +2.41421 q^{75} +3.34315 q^{79} -9.48528 q^{81} -11.3137 q^{83} -2.41421 q^{85} +2.41421 q^{87} -9.65685 q^{89} +24.7279 q^{93} +2.00000 q^{95} -14.0711 q^{97} +2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{11} - 2 q^{13} + 2 q^{15} - 2 q^{17} + 4 q^{19} - 4 q^{23} + 2 q^{25} + 2 q^{27} + 2 q^{29} + 12 q^{31} + 2 q^{33} + 4 q^{37} + 2 q^{39} - 12 q^{41} + 12 q^{43} + 18 q^{47} - 6 q^{51} + 16 q^{53} + 2 q^{55} + 4 q^{57} + 12 q^{59} + 8 q^{61} - 2 q^{65} - 16 q^{69} - 12 q^{71} + 16 q^{73} + 2 q^{75} + 18 q^{79} - 2 q^{81} - 2 q^{85} + 2 q^{87} - 8 q^{89} + 24 q^{93} + 4 q^{95} - 14 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^11 - 2 * q^13 + 2 * q^15 - 2 * q^17 + 4 * q^19 - 4 * q^23 + 2 * q^25 + 2 * q^27 + 2 * q^29 + 12 * q^31 + 2 * q^33 + 4 * q^37 + 2 * q^39 - 12 * q^41 + 12 * q^43 + 18 * q^47 - 6 * q^51 + 16 * q^53 + 2 * q^55 + 4 * q^57 + 12 * q^59 + 8 * q^61 - 2 * q^65 - 16 * q^69 - 12 * q^71 + 16 * q^73 + 2 * q^75 + 18 * q^79 - 2 * q^81 - 2 * q^85 + 2 * q^87 - 8 * q^89 + 24 * q^93 + 4 * q^95 - 14 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	0.414214	0.114882	0.0574411	−	0.998349i	$-0.481706\pi$
$13$	0.414214	0.114882	0.0574411	+	0.998349i	$0.481706\pi$
$14$	0	0
$15$	2.41421	0.623347
$16$	0	0
$17$	−2.41421	−0.585533	−0.292766	−	0.956184i	$-0.594576\pi$
$17$	−2.41421	−0.585533	−0.292766	+	0.956184i	$0.594576\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.24264	−1.30168	−0.650840	−	0.759215i	$-0.725583\pi$
$23$	−6.24264	−1.30168	−0.650840	+	0.759215i	$0.725583\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.414214	−0.0797154
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	10.2426	1.83963	0.919816	−	0.392349i	$-0.128338\pi$
$31$	10.2426	1.83963	0.919816	+	0.392349i	$0.128338\pi$
$32$	0	0
$33$	2.41421	0.420261
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.8995	1.95626	0.978132	−	0.207983i	$-0.0666898\pi$
$37$	11.8995	1.95626	0.978132	+	0.207983i	$0.0666898\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−4.58579	−0.716180	−0.358090	−	0.933687i	$-0.616572\pi$
$41$	−4.58579	−0.716180	−0.358090	+	0.933687i	$0.616572\pi$
$42$	0	0
$43$	11.6569	1.77765	0.888827	−	0.458243i	$-0.151521\pi$
$43$	11.6569	1.77765	0.888827	+	0.458243i	$0.151521\pi$
$44$	0	0
$45$	2.82843	0.421637
$46$	0	0
$47$	7.58579	1.10650	0.553250	−	0.833015i	$-0.313387\pi$
$47$	7.58579	1.10650	0.553250	+	0.833015i	$0.313387\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−5.82843	−0.816143
$52$	0	0
$53$	6.58579	0.904627	0.452314	−	0.891859i	$-0.350599\pi$
$53$	6.58579	0.904627	0.452314	+	0.891859i	$0.350599\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	4.82843	0.639541
$58$	0	0
$59$	1.75736	0.228789	0.114394	−	0.993435i	$-0.463507\pi$
$59$	1.75736	0.228789	0.114394	+	0.993435i	$0.463507\pi$
$60$	0	0
$61$	6.82843	0.874291	0.437145	−	0.899391i	$-0.355990\pi$
$61$	6.82843	0.874291	0.437145	+	0.899391i	$0.355990\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.414214	0.0513769
$66$	0	0
$67$	1.41421	0.172774	0.0863868	−	0.996262i	$-0.472468\pi$
$67$	1.41421	0.172774	0.0863868	+	0.996262i	$0.472468\pi$
$68$	0	0
$69$	−15.0711	−1.81434
$70$	0	0
$71$	2.48528	0.294949	0.147474	−	0.989066i	$-0.452886\pi$
$71$	2.48528	0.294949	0.147474	+	0.989066i	$0.452886\pi$
$72$	0	0
$73$	10.8284	1.26737	0.633686	−	0.773591i	$-0.281541\pi$
$73$	10.8284	1.26737	0.633686	+	0.773591i	$0.281541\pi$
$74$	0	0
$75$	2.41421	0.278769
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.34315	0.376133	0.188067	−	0.982156i	$-0.439778\pi$
$79$	3.34315	0.376133	0.188067	+	0.982156i	$0.439778\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	−11.3137	−1.24184	−0.620920	−	0.783874i	$-0.713241\pi$
$83$	−11.3137	−1.24184	−0.620920	+	0.783874i	$0.713241\pi$
$84$	0	0
$85$	−2.41421	−0.261858
$86$	0	0
$87$	2.41421	0.258831
$88$	0	0
$89$	−9.65685	−1.02362	−0.511812	−	0.859097i	$-0.671026\pi$
$89$	−9.65685	−1.02362	−0.511812	+	0.859097i	$0.671026\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	24.7279	2.56417
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−14.0711	−1.42870	−0.714350	−	0.699788i	$-0.753278\pi$
$97$	−14.0711	−1.42870	−0.714350	+	0.699788i	$0.753278\pi$
$98$	0	0
$99$	2.82843	0.284268
$100$	0	0
$101$	−10.4853	−1.04332	−0.521662	−	0.853152i	$-0.674688\pi$
$101$	−10.4853	−1.04332	−0.521662	+	0.853152i	$0.674688\pi$
$102$	0	0
$103$	6.75736	0.665822	0.332911	−	0.942958i	$-0.391969\pi$
$103$	6.75736	0.665822	0.332911	+	0.942958i	$0.391969\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.4853	1.40035	0.700173	−	0.713974i	$-0.253106\pi$
$107$	14.4853	1.40035	0.700173	+	0.713974i	$0.253106\pi$
$108$	0	0
$109$	−18.3137	−1.75414	−0.877068	−	0.480367i	$-0.840503\pi$
$109$	−18.3137	−1.75414	−0.877068	+	0.480367i	$0.840503\pi$
$110$	0	0
$111$	28.7279	2.72673
$112$	0	0
$113$	−9.07107	−0.853334	−0.426667	−	0.904409i	$-0.640312\pi$
$113$	−9.07107	−0.853334	−0.426667	+	0.904409i	$0.640312\pi$
$114$	0	0
$115$	−6.24264	−0.582129
$116$	0	0
$117$	1.17157	0.108312
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	−11.0711	−0.998245
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.75736	−0.333412	−0.166706	−	0.986007i	$-0.553313\pi$
$127$	−3.75736	−0.333412	−0.166706	+	0.986007i	$0.553313\pi$
$128$	0	0
$129$	28.1421	2.47778
$130$	0	0
$131$	20.2426	1.76861	0.884304	−	0.466912i	$-0.154633\pi$
$131$	20.2426	1.76861	0.884304	+	0.466912i	$0.154633\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.414214	−0.0356498
$136$	0	0
$137$	−19.3137	−1.65008	−0.825041	−	0.565073i	$-0.808848\pi$
$137$	−19.3137	−1.65008	−0.825041	+	0.565073i	$0.808848\pi$
$138$	0	0
$139$	1.41421	0.119952	0.0599760	−	0.998200i	$-0.480898\pi$
$139$	1.41421	0.119952	0.0599760	+	0.998200i	$0.480898\pi$
$140$	0	0
$141$	18.3137	1.54229
$142$	0	0
$143$	0.414214	0.0346383
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.17157	−0.751365	−0.375682	−	0.926749i	$-0.622592\pi$
$149$	−9.17157	−0.751365	−0.375682	+	0.926749i	$0.622592\pi$
$150$	0	0
$151$	6.65685	0.541727	0.270864	−	0.962618i	$-0.412691\pi$
$151$	6.65685	0.541727	0.270864	+	0.962618i	$0.412691\pi$
$152$	0	0
$153$	−6.82843	−0.552046
$154$	0	0
$155$	10.2426	0.822709
$156$	0	0
$157$	−10.4853	−0.836817	−0.418408	−	0.908259i	$-0.637412\pi$
$157$	−10.4853	−0.836817	−0.418408	+	0.908259i	$0.637412\pi$
$158$	0	0
$159$	15.8995	1.26091
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.92893	−0.229412	−0.114706	−	0.993400i	$-0.536593\pi$
$163$	−2.92893	−0.229412	−0.114706	+	0.993400i	$0.536593\pi$
$164$	0	0
$165$	2.41421	0.187946
$166$	0	0
$167$	7.58579	0.587006	0.293503	−	0.955958i	$-0.405179\pi$
$167$	7.58579	0.587006	0.293503	+	0.955958i	$0.405179\pi$
$168$	0	0
$169$	−12.8284	−0.986802
$170$	0	0
$171$	5.65685	0.432590
$172$	0	0
$173$	13.7279	1.04371	0.521857	−	0.853033i	$-0.325239\pi$
$173$	13.7279	1.04371	0.521857	+	0.853033i	$0.325239\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.24264	0.318896
$178$	0	0
$179$	−4.14214	−0.309598	−0.154799	−	0.987946i	$-0.549473\pi$
$179$	−4.14214	−0.309598	−0.154799	+	0.987946i	$0.549473\pi$
$180$	0	0
$181$	25.5563	1.89959	0.949794	−	0.312875i	$-0.101292\pi$
$181$	25.5563	1.89959	0.949794	+	0.312875i	$0.101292\pi$
$182$	0	0
$183$	16.4853	1.21863
$184$	0	0
$185$	11.8995	0.874868
$186$	0	0
$187$	−2.41421	−0.176545
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.8284	−1.00059	−0.500295	−	0.865855i	$-0.666775\pi$
$191$	−13.8284	−1.00059	−0.500295	+	0.865855i	$0.666775\pi$
$192$	0	0
$193$	0.686292	0.0494003	0.0247002	−	0.999695i	$-0.492137\pi$
$193$	0.686292	0.0494003	0.0247002	+	0.999695i	$0.492137\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	−12.2426	−0.872252	−0.436126	−	0.899886i	$-0.643650\pi$
$197$	−12.2426	−0.872252	−0.436126	+	0.899886i	$0.643650\pi$
$198$	0	0
$199$	−12.3848	−0.877934	−0.438967	−	0.898503i	$-0.644655\pi$
$199$	−12.3848	−0.877934	−0.438967	+	0.898503i	$0.644655\pi$
$200$	0	0
$201$	3.41421	0.240820
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4.58579	−0.320285
$206$	0	0
$207$	−17.6569	−1.22724
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	−17.1421	−1.18011	−0.590057	−	0.807362i	$-0.700895\pi$
$211$	−17.1421	−1.18011	−0.590057	+	0.807362i	$0.700895\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	11.6569	0.794991
$216$	0	0
$217$	0	0
$218$	0	0
$219$	26.1421	1.76652
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	0	0
$223$	−17.3848	−1.16417	−0.582085	−	0.813128i	$-0.697763\pi$
$223$	−17.3848	−1.16417	−0.582085	+	0.813128i	$0.697763\pi$
$224$	0	0
$225$	2.82843	0.188562
$226$	0	0
$227$	1.58579	0.105252	0.0526262	−	0.998614i	$-0.483241\pi$
$227$	1.58579	0.105252	0.0526262	+	0.998614i	$0.483241\pi$
$228$	0	0
$229$	4.10051	0.270969	0.135485	−	0.990779i	$-0.456741\pi$
$229$	4.10051	0.270969	0.135485	+	0.990779i	$0.456741\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.51472	0.230257	0.115128	−	0.993351i	$-0.463272\pi$
$233$	3.51472	0.230257	0.115128	+	0.993351i	$0.463272\pi$
$234$	0	0
$235$	7.58579	0.494842
$236$	0	0
$237$	8.07107	0.524272
$238$	0	0
$239$	4.65685	0.301227	0.150613	−	0.988593i	$-0.451875\pi$
$239$	4.65685	0.301227	0.150613	+	0.988593i	$0.451875\pi$
$240$	0	0
$241$	−10.3848	−0.668942	−0.334471	−	0.942406i	$-0.608558\pi$
$241$	−10.3848	−0.668942	−0.334471	+	0.942406i	$0.608558\pi$
$242$	0	0
$243$	−21.6569	−1.38929
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.828427	0.0527116
$248$	0	0
$249$	−27.3137	−1.73094
$250$	0	0
$251$	7.55635	0.476953	0.238476	−	0.971148i	$-0.423352\pi$
$251$	7.55635	0.476953	0.238476	+	0.971148i	$0.423352\pi$
$252$	0	0
$253$	−6.24264	−0.392471
$254$	0	0
$255$	−5.82843	−0.364990
$256$	0	0
$257$	−10.4853	−0.654054	−0.327027	−	0.945015i	$-0.606047\pi$
$257$	−10.4853	−0.654054	−0.327027	+	0.945015i	$0.606047\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.82843	0.175075
$262$	0	0
$263$	−18.9706	−1.16978	−0.584888	−	0.811114i	$-0.698861\pi$
$263$	−18.9706	−1.16978	−0.584888	+	0.811114i	$0.698861\pi$
$264$	0	0
$265$	6.58579	0.404562
$266$	0	0
$267$	−23.3137	−1.42678
$268$	0	0
$269$	−18.5858	−1.13320	−0.566598	−	0.823995i	$-0.691741\pi$
$269$	−18.5858	−1.13320	−0.566598	+	0.823995i	$0.691741\pi$
$270$	0	0
$271$	2.34315	0.142336	0.0711680	−	0.997464i	$-0.477327\pi$
$271$	2.34315	0.142336	0.0711680	+	0.997464i	$0.477327\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−13.0711	−0.785364	−0.392682	−	0.919674i	$-0.628453\pi$
$277$	−13.0711	−0.785364	−0.392682	+	0.919674i	$0.628453\pi$
$278$	0	0
$279$	28.9706	1.73442
$280$	0	0
$281$	18.3137	1.09250	0.546252	−	0.837621i	$-0.316054\pi$
$281$	18.3137	1.09250	0.546252	+	0.837621i	$0.316054\pi$
$282$	0	0
$283$	25.0416	1.48857	0.744285	−	0.667862i	$-0.232790\pi$
$283$	25.0416	1.48857	0.744285	+	0.667862i	$0.232790\pi$
$284$	0	0
$285$	4.82843	0.286011
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−11.1716	−0.657151
$290$	0	0
$291$	−33.9706	−1.99139
$292$	0	0
$293$	18.4142	1.07577	0.537885	−	0.843018i	$-0.319223\pi$
$293$	18.4142	1.07577	0.537885	+	0.843018i	$0.319223\pi$
$294$	0	0
$295$	1.75736	0.102317
$296$	0	0
$297$	−0.414214	−0.0240351
$298$	0	0
$299$	−2.58579	−0.149540
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−25.3137	−1.45423
$304$	0	0
$305$	6.82843	0.390995
$306$	0	0
$307$	−11.9289	−0.680820	−0.340410	−	0.940277i	$-0.610566\pi$
$307$	−11.9289	−0.680820	−0.340410	+	0.940277i	$0.610566\pi$
$308$	0	0
$309$	16.3137	0.928054
$310$	0	0
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	23.7279	1.34118	0.670591	−	0.741828i	$-0.266041\pi$
$313$	23.7279	1.34118	0.670591	+	0.741828i	$0.266041\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.3431	0.693260	0.346630	−	0.938002i	$-0.387326\pi$
$317$	12.3431	0.693260	0.346630	+	0.938002i	$0.387326\pi$
$318$	0	0
$319$	1.00000	0.0559893
$320$	0	0
$321$	34.9706	1.95187
$322$	0	0
$323$	−4.82843	−0.268661
$324$	0	0
$325$	0.414214	0.0229764
$326$	0	0
$327$	−44.2132	−2.44500
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−9.51472	−0.522976	−0.261488	−	0.965207i	$-0.584213\pi$
$331$	−9.51472	−0.522976	−0.261488	+	0.965207i	$0.584213\pi$
$332$	0	0
$333$	33.6569	1.84438
$334$	0	0
$335$	1.41421	0.0772667
$336$	0	0
$337$	−13.0711	−0.712026	−0.356013	−	0.934481i	$-0.615864\pi$
$337$	−13.0711	−0.712026	−0.356013	+	0.934481i	$0.615864\pi$
$338$	0	0
$339$	−21.8995	−1.18942
$340$	0	0
$341$	10.2426	0.554670
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−15.0711	−0.811399
$346$	0	0
$347$	−28.8701	−1.54983	−0.774913	−	0.632068i	$-0.782206\pi$
$347$	−28.8701	−1.54983	−0.774913	+	0.632068i	$0.782206\pi$
$348$	0	0
$349$	6.68629	0.357909	0.178954	−	0.983857i	$-0.442729\pi$
$349$	6.68629	0.357909	0.178954	+	0.983857i	$0.442729\pi$
$350$	0	0
$351$	−0.171573	−0.00915788
$352$	0	0
$353$	−30.2132	−1.60809	−0.804043	−	0.594571i	$-0.797322\pi$
$353$	−30.2132	−1.60809	−0.804043	+	0.594571i	$0.797322\pi$
$354$	0	0
$355$	2.48528	0.131905
$356$	0	0
$357$	0	0
$358$	0	0
$359$	29.6569	1.56523	0.782614	−	0.622507i	$-0.213886\pi$
$359$	29.6569	1.56523	0.782614	+	0.622507i	$0.213886\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−24.1421	−1.26713
$364$	0	0
$365$	10.8284	0.566786
$366$	0	0
$367$	11.2426	0.586861	0.293431	−	0.955980i	$-0.405203\pi$
$367$	11.2426	0.586861	0.293431	+	0.955980i	$0.405203\pi$
$368$	0	0
$369$	−12.9706	−0.675221
$370$	0	0
$371$	0	0
$372$	0	0
$373$	22.8284	1.18201	0.591006	−	0.806667i	$-0.298731\pi$
$373$	22.8284	1.18201	0.591006	+	0.806667i	$0.298731\pi$
$374$	0	0
$375$	2.41421	0.124669
$376$	0	0
$377$	0.414214	0.0213331
$378$	0	0
$379$	−28.6274	−1.47049	−0.735246	−	0.677801i	$-0.762933\pi$
$379$	−28.6274	−1.47049	−0.735246	+	0.677801i	$0.762933\pi$
$380$	0	0
$381$	−9.07107	−0.464725
$382$	0	0
$383$	22.8284	1.16648	0.583239	−	0.812301i	$-0.301785\pi$
$383$	22.8284	1.16648	0.583239	+	0.812301i	$0.301785\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	32.9706	1.67599
$388$	0	0
$389$	10.7990	0.547531	0.273765	−	0.961797i	$-0.411731\pi$
$389$	10.7990	0.547531	0.273765	+	0.961797i	$0.411731\pi$
$390$	0	0
$391$	15.0711	0.762177
$392$	0	0
$393$	48.8701	2.46517
$394$	0	0
$395$	3.34315	0.168212
$396$	0	0
$397$	5.58579	0.280343	0.140171	−	0.990127i	$-0.455235\pi$
$397$	5.58579	0.280343	0.140171	+	0.990127i	$0.455235\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.4558	−1.22127	−0.610633	−	0.791913i	$-0.709085\pi$
$401$	−24.4558	−1.22127	−0.610633	+	0.791913i	$0.709085\pi$
$402$	0	0
$403$	4.24264	0.211341
$404$	0	0
$405$	−9.48528	−0.471327
$406$	0	0
$407$	11.8995	0.589836
$408$	0	0
$409$	33.1127	1.63732	0.818659	−	0.574280i	$-0.194718\pi$
$409$	33.1127	1.63732	0.818659	+	0.574280i	$0.194718\pi$
$410$	0	0
$411$	−46.6274	−2.29996
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−11.3137	−0.555368
$416$	0	0
$417$	3.41421	0.167195
$418$	0	0
$419$	−28.5858	−1.39651	−0.698254	−	0.715851i	$-0.746039\pi$
$419$	−28.5858	−1.39651	−0.698254	+	0.715851i	$0.746039\pi$
$420$	0	0
$421$	−26.3137	−1.28245	−0.641226	−	0.767352i	$-0.721574\pi$
$421$	−26.3137	−1.28245	−0.641226	+	0.767352i	$0.721574\pi$
$422$	0	0
$423$	21.4558	1.04322
$424$	0	0
$425$	−2.41421	−0.117107
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−22.6569	−1.09134	−0.545671	−	0.837999i	$-0.683725\pi$
$431$	−22.6569	−1.09134	−0.545671	+	0.837999i	$0.683725\pi$
$432$	0	0
$433$	−6.97056	−0.334984	−0.167492	−	0.985873i	$-0.553567\pi$
$433$	−6.97056	−0.334984	−0.167492	+	0.985873i	$0.553567\pi$
$434$	0	0
$435$	2.41421	0.115753
$436$	0	0
$437$	−12.4853	−0.597252
$438$	0	0
$439$	−3.75736	−0.179329	−0.0896645	−	0.995972i	$-0.528579\pi$
$439$	−3.75736	−0.179329	−0.0896645	+	0.995972i	$0.528579\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	34.7696	1.65195	0.825976	−	0.563706i	$-0.190625\pi$
$443$	34.7696	1.65195	0.825976	+	0.563706i	$0.190625\pi$
$444$	0	0
$445$	−9.65685	−0.457779
$446$	0	0
$447$	−22.1421	−1.04729
$448$	0	0
$449$	−27.4853	−1.29711	−0.648555	−	0.761168i	$-0.724626\pi$
$449$	−27.4853	−1.29711	−0.648555	+	0.761168i	$0.724626\pi$
$450$	0	0
$451$	−4.58579	−0.215936
$452$	0	0
$453$	16.0711	0.755085
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.92893	0.230566	0.115283	−	0.993333i	$-0.463223\pi$
$457$	4.92893	0.230566	0.115283	+	0.993333i	$0.463223\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	32.2843	1.50363	0.751814	−	0.659375i	$-0.229179\pi$
$461$	32.2843	1.50363	0.751814	+	0.659375i	$0.229179\pi$
$462$	0	0
$463$	−25.4558	−1.18303	−0.591517	−	0.806293i	$-0.701471\pi$
$463$	−25.4558	−1.18303	−0.591517	+	0.806293i	$0.701471\pi$
$464$	0	0
$465$	24.7279	1.14673
$466$	0	0
$467$	8.41421	0.389363	0.194682	−	0.980866i	$-0.437633\pi$
$467$	8.41421	0.389363	0.194682	+	0.980866i	$0.437633\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−25.3137	−1.16639
$472$	0	0
$473$	11.6569	0.535983
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	18.6274	0.852891
$478$	0	0
$479$	−40.5269	−1.85172	−0.925861	−	0.377864i	$-0.876659\pi$
$479$	−40.5269	−1.85172	−0.925861	+	0.377864i	$0.876659\pi$
$480$	0	0
$481$	4.92893	0.224740
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.0711	−0.638934
$486$	0	0
$487$	−1.27208	−0.0576434	−0.0288217	−	0.999585i	$-0.509175\pi$
$487$	−1.27208	−0.0576434	−0.0288217	+	0.999585i	$0.509175\pi$
$488$	0	0
$489$	−7.07107	−0.319765
$490$	0	0
$491$	11.4853	0.518323	0.259162	−	0.965834i	$-0.416554\pi$
$491$	11.4853	0.518323	0.259162	+	0.965834i	$0.416554\pi$
$492$	0	0
$493$	−2.41421	−0.108731
$494$	0	0
$495$	2.82843	0.127128
$496$	0	0
$497$	0	0
$498$	0	0
$499$	26.7990	1.19969	0.599844	−	0.800117i	$-0.295229\pi$
$499$	26.7990	1.19969	0.599844	+	0.800117i	$0.295229\pi$
$500$	0	0
$501$	18.3137	0.818196
$502$	0	0
$503$	−17.0416	−0.759849	−0.379924	−	0.925018i	$-0.624050\pi$
$503$	−17.0416	−0.759849	−0.379924	+	0.925018i	$0.624050\pi$
$504$	0	0
$505$	−10.4853	−0.466589
$506$	0	0
$507$	−30.9706	−1.37545
$508$	0	0
$509$	−16.2426	−0.719942	−0.359971	−	0.932963i	$-0.617213\pi$
$509$	−16.2426	−0.719942	−0.359971	+	0.932963i	$0.617213\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−0.828427	−0.0365760
$514$	0	0
$515$	6.75736	0.297765
$516$	0	0
$517$	7.58579	0.333623
$518$	0	0
$519$	33.1421	1.45478
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−15.5147	−0.678411	−0.339206	−	0.940712i	$-0.610158\pi$
$523$	−15.5147	−0.678411	−0.339206	+	0.940712i	$0.610158\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24.7279	−1.07717
$528$	0	0
$529$	15.9706	0.694372
$530$	0	0
$531$	4.97056	0.215704
$532$	0	0
$533$	−1.89949	−0.0822763
$534$	0	0
$535$	14.4853	0.626253
$536$	0	0
$537$	−10.0000	−0.431532
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−12.3137	−0.529408	−0.264704	−	0.964330i	$-0.585274\pi$
$541$	−12.3137	−0.529408	−0.264704	+	0.964330i	$0.585274\pi$
$542$	0	0
$543$	61.6985	2.64774
$544$	0	0
$545$	−18.3137	−0.784473
$546$	0	0
$547$	−35.1127	−1.50131	−0.750655	−	0.660694i	$-0.770262\pi$
$547$	−35.1127	−1.50131	−0.750655	+	0.660694i	$0.770262\pi$
$548$	0	0
$549$	19.3137	0.824289
$550$	0	0
$551$	2.00000	0.0852029
$552$	0	0
$553$	0	0
$554$	0	0
$555$	28.7279	1.21943
$556$	0	0
$557$	2.14214	0.0907652	0.0453826	−	0.998970i	$-0.485549\pi$
$557$	2.14214	0.0907652	0.0453826	+	0.998970i	$0.485549\pi$
$558$	0	0
$559$	4.82843	0.204221
$560$	0	0
$561$	−5.82843	−0.246076
$562$	0	0
$563$	27.9411	1.17758	0.588789	−	0.808287i	$-0.299605\pi$
$563$	27.9411	1.17758	0.588789	+	0.808287i	$0.299605\pi$
$564$	0	0
$565$	−9.07107	−0.381623
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−34.1421	−1.43131	−0.715656	−	0.698453i	$-0.753872\pi$
$569$	−34.1421	−1.43131	−0.715656	+	0.698453i	$0.753872\pi$
$570$	0	0
$571$	23.4558	0.981597	0.490798	−	0.871273i	$-0.336705\pi$
$571$	23.4558	0.981597	0.490798	+	0.871273i	$0.336705\pi$
$572$	0	0
$573$	−33.3848	−1.39467
$574$	0	0
$575$	−6.24264	−0.260336
$576$	0	0
$577$	36.4142	1.51594	0.757972	−	0.652287i	$-0.226190\pi$
$577$	36.4142	1.51594	0.757972	+	0.652287i	$0.226190\pi$
$578$	0	0
$579$	1.65685	0.0688565
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.58579	0.272755
$584$	0	0
$585$	1.17157	0.0484386
$586$	0	0
$587$	−13.8579	−0.571975	−0.285988	−	0.958233i	$-0.592322\pi$
$587$	−13.8579	−0.571975	−0.285988	+	0.958233i	$0.592322\pi$
$588$	0	0
$589$	20.4853	0.844081
$590$	0	0
$591$	−29.5563	−1.21579
$592$	0	0
$593$	−9.44365	−0.387804	−0.193902	−	0.981021i	$-0.562114\pi$
$593$	−9.44365	−0.387804	−0.193902	+	0.981021i	$0.562114\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−29.8995	−1.22370
$598$	0	0
$599$	36.3137	1.48374	0.741869	−	0.670545i	$-0.233940\pi$
$599$	36.3137	1.48374	0.741869	+	0.670545i	$0.233940\pi$
$600$	0	0
$601$	18.2843	0.745831	0.372915	−	0.927865i	$-0.378358\pi$
$601$	18.2843	0.745831	0.372915	+	0.927865i	$0.378358\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	−10.0000	−0.406558
$606$	0	0
$607$	20.0711	0.814660	0.407330	−	0.913281i	$-0.366460\pi$
$607$	20.0711	0.814660	0.407330	+	0.913281i	$0.366460\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.14214	0.127117
$612$	0	0
$613$	29.3137	1.18397	0.591985	−	0.805949i	$-0.298345\pi$
$613$	29.3137	1.18397	0.591985	+	0.805949i	$0.298345\pi$
$614$	0	0
$615$	−11.0711	−0.446429
$616$	0	0
$617$	−11.4142	−0.459519	−0.229759	−	0.973247i	$-0.573794\pi$
$617$	−11.4142	−0.459519	−0.229759	+	0.973247i	$0.573794\pi$
$618$	0	0
$619$	−15.0711	−0.605757	−0.302879	−	0.953029i	$-0.597948\pi$
$619$	−15.0711	−0.605757	−0.302879	+	0.953029i	$0.597948\pi$
$620$	0	0
$621$	2.58579	0.103764
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.82843	0.192829
$628$	0	0
$629$	−28.7279	−1.14546
$630$	0	0
$631$	37.6274	1.49792	0.748962	−	0.662613i	$-0.230553\pi$
$631$	37.6274	1.49792	0.748962	+	0.662613i	$0.230553\pi$
$632$	0	0
$633$	−41.3848	−1.64490
$634$	0	0
$635$	−3.75736	−0.149106
$636$	0	0
$637$	0	0
$638$	0	0
$639$	7.02944	0.278080
$640$	0	0
$641$	32.2843	1.27515	0.637576	−	0.770387i	$-0.279937\pi$
$641$	32.2843	1.27515	0.637576	+	0.770387i	$0.279937\pi$
$642$	0	0
$643$	−30.2132	−1.19149	−0.595746	−	0.803173i	$-0.703144\pi$
$643$	−30.2132	−1.19149	−0.595746	+	0.803173i	$0.703144\pi$
$644$	0	0
$645$	28.1421	1.10810
$646$	0	0
$647$	18.1421	0.713241	0.356620	−	0.934249i	$-0.383929\pi$
$647$	18.1421	0.713241	0.356620	+	0.934249i	$0.383929\pi$
$648$	0	0
$649$	1.75736	0.0689824
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.2843	−0.872051	−0.436025	−	0.899934i	$-0.643614\pi$
$653$	−22.2843	−0.872051	−0.436025	+	0.899934i	$0.643614\pi$
$654$	0	0
$655$	20.2426	0.790945
$656$	0	0
$657$	30.6274	1.19489
$658$	0	0
$659$	−17.4853	−0.681130	−0.340565	−	0.940221i	$-0.610618\pi$
$659$	−17.4853	−0.681130	−0.340565	+	0.940221i	$0.610618\pi$
$660$	0	0
$661$	30.8284	1.19909	0.599543	−	0.800342i	$-0.295349\pi$
$661$	30.8284	1.19909	0.599543	+	0.800342i	$0.295349\pi$
$662$	0	0
$663$	−2.41421	−0.0937603
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.24264	−0.241716
$668$	0	0
$669$	−41.9706	−1.62268
$670$	0	0
$671$	6.82843	0.263609
$672$	0	0
$673$	50.8284	1.95929	0.979646	−	0.200733i	$-0.0643324\pi$
$673$	50.8284	1.95929	0.979646	+	0.200733i	$0.0643324\pi$
$674$	0	0
$675$	−0.414214	−0.0159431
$676$	0	0
$677$	−36.0711	−1.38632	−0.693162	−	0.720782i	$-0.743783\pi$
$677$	−36.0711	−1.38632	−0.693162	+	0.720782i	$0.743783\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	3.82843	0.146706
$682$	0	0
$683$	13.1716	0.503996	0.251998	−	0.967728i	$-0.418912\pi$
$683$	13.1716	0.503996	0.251998	+	0.967728i	$0.418912\pi$
$684$	0	0
$685$	−19.3137	−0.737939
$686$	0	0
$687$	9.89949	0.377689
$688$	0	0
$689$	2.72792	0.103926
$690$	0	0
$691$	10.1421	0.385825	0.192913	−	0.981216i	$-0.438207\pi$
$691$	10.1421	0.385825	0.192913	+	0.981216i	$0.438207\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.41421	0.0536442
$696$	0	0
$697$	11.0711	0.419347
$698$	0	0
$699$	8.48528	0.320943
$700$	0	0
$701$	−37.1421	−1.40284	−0.701420	−	0.712749i	$-0.747450\pi$
$701$	−37.1421	−1.40284	−0.701420	+	0.712749i	$0.747450\pi$
$702$	0	0
$703$	23.7990	0.897596
$704$	0	0
$705$	18.3137	0.689734
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−21.3431	−0.801559	−0.400779	−	0.916175i	$-0.631261\pi$
$709$	−21.3431	−0.801559	−0.400779	+	0.916175i	$0.631261\pi$
$710$	0	0
$711$	9.45584	0.354622
$712$	0	0
$713$	−63.9411	−2.39461
$714$	0	0
$715$	0.414214	0.0154907
$716$	0	0
$717$	11.2426	0.419864
$718$	0	0
$719$	49.6985	1.85344	0.926720	−	0.375752i	$-0.122615\pi$
$719$	49.6985	1.85344	0.926720	+	0.375752i	$0.122615\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−25.0711	−0.932403
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	41.3137	1.53224	0.766120	−	0.642697i	$-0.222185\pi$
$727$	41.3137	1.53224	0.766120	+	0.642697i	$0.222185\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	−28.1421	−1.04087
$732$	0	0
$733$	26.6985	0.986131	0.493066	−	0.869992i	$-0.335876\pi$
$733$	26.6985	0.986131	0.493066	+	0.869992i	$0.335876\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.41421	0.0520932
$738$	0	0
$739$	4.79899	0.176534	0.0882668	−	0.996097i	$-0.471867\pi$
$739$	4.79899	0.176534	0.0882668	+	0.996097i	$0.471867\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	5.89949	0.216431	0.108216	−	0.994127i	$-0.465486\pi$
$743$	5.89949	0.216431	0.108216	+	0.994127i	$0.465486\pi$
$744$	0	0
$745$	−9.17157	−0.336020
$746$	0	0
$747$	−32.0000	−1.17082
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−25.6274	−0.935158	−0.467579	−	0.883951i	$-0.654874\pi$
$751$	−25.6274	−0.935158	−0.467579	+	0.883951i	$0.654874\pi$
$752$	0	0
$753$	18.2426	0.664799
$754$	0	0
$755$	6.65685	0.242268
$756$	0	0
$757$	−36.7696	−1.33641	−0.668206	−	0.743976i	$-0.732938\pi$
$757$	−36.7696	−1.33641	−0.668206	+	0.743976i	$0.732938\pi$
$758$	0	0
$759$	−15.0711	−0.547045
$760$	0	0
$761$	−4.44365	−0.161082	−0.0805411	−	0.996751i	$-0.525665\pi$
$761$	−4.44365	−0.161082	−0.0805411	+	0.996751i	$0.525665\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−6.82843	−0.246882
$766$	0	0
$767$	0.727922	0.0262837
$768$	0	0
$769$	18.4853	0.666596	0.333298	−	0.942821i	$-0.391838\pi$
$769$	18.4853	0.666596	0.333298	+	0.942821i	$0.391838\pi$
$770$	0	0
$771$	−25.3137	−0.911651
$772$	0	0
$773$	21.3848	0.769157	0.384578	−	0.923092i	$-0.374347\pi$
$773$	21.3848	0.769157	0.384578	+	0.923092i	$0.374347\pi$
$774$	0	0
$775$	10.2426	0.367927
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.17157	−0.328606
$780$	0	0
$781$	2.48528	0.0889304
$782$	0	0
$783$	−0.414214	−0.0148028
$784$	0	0
$785$	−10.4853	−0.374236
$786$	0	0
$787$	−42.8406	−1.52710	−0.763552	−	0.645747i	$-0.776546\pi$
$787$	−42.8406	−1.52710	−0.763552	+	0.645747i	$0.776546\pi$
$788$	0	0
$789$	−45.7990	−1.63049
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.82843	0.100440
$794$	0	0
$795$	15.8995	0.563897
$796$	0	0
$797$	−35.9289	−1.27267	−0.636334	−	0.771414i	$-0.719550\pi$
$797$	−35.9289	−1.27267	−0.636334	+	0.771414i	$0.719550\pi$
$798$	0	0
$799$	−18.3137	−0.647892
$800$	0	0
$801$	−27.3137	−0.965082
$802$	0	0
$803$	10.8284	0.382127
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−44.8701	−1.57950
$808$	0	0
$809$	48.9411	1.72068	0.860339	−	0.509722i	$-0.170252\pi$
$809$	48.9411	1.72068	0.860339	+	0.509722i	$0.170252\pi$
$810$	0	0
$811$	−39.5563	−1.38901	−0.694506	−	0.719487i	$-0.744377\pi$
$811$	−39.5563	−1.38901	−0.694506	+	0.719487i	$0.744377\pi$
$812$	0	0
$813$	5.65685	0.198395
$814$	0	0
$815$	−2.92893	−0.102596
$816$	0	0
$817$	23.3137	0.815643
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.4853	0.749841	0.374921	−	0.927057i	$-0.377670\pi$
$821$	21.4853	0.749841	0.374921	+	0.927057i	$0.377670\pi$
$822$	0	0
$823$	36.0416	1.25633	0.628166	−	0.778079i	$-0.283806\pi$
$823$	36.0416	1.25633	0.628166	+	0.778079i	$0.283806\pi$
$824$	0	0
$825$	2.41421	0.0840521
$826$	0	0
$827$	54.0416	1.87921	0.939606	−	0.342259i	$-0.111192\pi$
$827$	54.0416	1.87921	0.939606	+	0.342259i	$0.111192\pi$
$828$	0	0
$829$	−7.41421	−0.257506	−0.128753	−	0.991677i	$-0.541097\pi$
$829$	−7.41421	−0.257506	−0.128753	+	0.991677i	$0.541097\pi$
$830$	0	0
$831$	−31.5563	−1.09468
$832$	0	0
$833$	0	0
$834$	0	0
$835$	7.58579	0.262517
$836$	0	0
$837$	−4.24264	−0.146647
$838$	0	0
$839$	7.75736	0.267814	0.133907	−	0.990994i	$-0.457248\pi$
$839$	7.75736	0.267814	0.133907	+	0.990994i	$0.457248\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	44.2132	1.52278
$844$	0	0
$845$	−12.8284	−0.441311
$846$	0	0
$847$	0	0
$848$	0	0
$849$	60.4558	2.07484
$850$	0	0
$851$	−74.2843	−2.54643
$852$	0	0
$853$	−9.31371	−0.318895	−0.159448	−	0.987206i	$-0.550971\pi$
$853$	−9.31371	−0.318895	−0.159448	+	0.987206i	$0.550971\pi$
$854$	0	0
$855$	5.65685	0.193460
$856$	0	0
$857$	−54.4853	−1.86118	−0.930591	−	0.366061i	$-0.880706\pi$
$857$	−54.4853	−1.86118	−0.930591	+	0.366061i	$0.880706\pi$
$858$	0	0
$859$	−53.4558	−1.82389	−0.911945	−	0.410313i	$-0.865420\pi$
$859$	−53.4558	−1.82389	−0.911945	+	0.410313i	$0.865420\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22.1838	−0.755144	−0.377572	−	0.925980i	$-0.623241\pi$
$863$	−22.1838	−0.755144	−0.377572	+	0.925980i	$0.623241\pi$
$864$	0	0
$865$	13.7279	0.466763
$866$	0	0
$867$	−26.9706	−0.915968
$868$	0	0
$869$	3.34315	0.113408
$870$	0	0
$871$	0.585786	0.0198486
$872$	0	0
$873$	−39.7990	−1.34699
$874$	0	0
$875$	0	0
$876$	0	0
$877$	26.9706	0.910731	0.455366	−	0.890305i	$-0.349509\pi$
$877$	26.9706	0.910731	0.455366	+	0.890305i	$0.349509\pi$
$878$	0	0
$879$	44.4558	1.49946
$880$	0	0
$881$	−17.6569	−0.594875	−0.297437	−	0.954741i	$-0.596132\pi$
$881$	−17.6569	−0.594875	−0.297437	+	0.954741i	$0.596132\pi$
$882$	0	0
$883$	−35.1127	−1.18164	−0.590818	−	0.806805i	$-0.701195\pi$
$883$	−35.1127	−1.18164	−0.590818	+	0.806805i	$0.701195\pi$
$884$	0	0
$885$	4.24264	0.142615
$886$	0	0
$887$	−18.2843	−0.613926	−0.306963	−	0.951721i	$-0.599313\pi$
$887$	−18.2843	−0.613926	−0.306963	+	0.951721i	$0.599313\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−9.48528	−0.317769
$892$	0	0
$893$	15.1716	0.507697
$894$	0	0
$895$	−4.14214	−0.138456
$896$	0	0
$897$	−6.24264	−0.208436
$898$	0	0
$899$	10.2426	0.341611
$900$	0	0
$901$	−15.8995	−0.529689
$902$	0	0
$903$	0	0
$904$	0	0
$905$	25.5563	0.849522
$906$	0	0
$907$	7.69848	0.255624	0.127812	−	0.991798i	$-0.459205\pi$
$907$	7.69848	0.255624	0.127812	+	0.991798i	$0.459205\pi$
$908$	0	0
$909$	−29.6569	−0.983656
$910$	0	0
$911$	−20.6863	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$911$	−20.6863	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$912$	0	0
$913$	−11.3137	−0.374429
$914$	0	0
$915$	16.4853	0.544987
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−28.5980	−0.943360	−0.471680	−	0.881770i	$-0.656352\pi$
$919$	−28.5980	−0.943360	−0.471680	+	0.881770i	$0.656352\pi$
$920$	0	0
$921$	−28.7990	−0.948959
$922$	0	0
$923$	1.02944	0.0338843
$924$	0	0
$925$	11.8995	0.391253
$926$	0	0
$927$	19.1127	0.627743
$928$	0	0
$929$	34.5269	1.13279	0.566396	−	0.824133i	$-0.308337\pi$
$929$	34.5269	1.13279	0.566396	+	0.824133i	$0.308337\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	24.1421	0.790378
$934$	0	0
$935$	−2.41421	−0.0789532
$936$	0	0
$937$	−14.2132	−0.464325	−0.232163	−	0.972677i	$-0.574580\pi$
$937$	−14.2132	−0.464325	−0.232163	+	0.972677i	$0.574580\pi$
$938$	0	0
$939$	57.2843	1.86940
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	0	0
$943$	28.6274	0.932237
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.89949	−0.321690	−0.160845	−	0.986980i	$-0.551422\pi$
$947$	−9.89949	−0.321690	−0.160845	+	0.986980i	$0.551422\pi$
$948$	0	0
$949$	4.48528	0.145598
$950$	0	0
$951$	29.7990	0.966298
$952$	0	0
$953$	58.3848	1.89127	0.945634	−	0.325232i	$-0.105442\pi$
$953$	58.3848	1.89127	0.945634	+	0.325232i	$0.105442\pi$
$954$	0	0
$955$	−13.8284	−0.447477
$956$	0	0
$957$	2.41421	0.0780404
$958$	0	0
$959$	0	0
$960$	0	0
$961$	73.9117	2.38425
$962$	0	0
$963$	40.9706	1.32026
$964$	0	0
$965$	0.686292	0.0220925
$966$	0	0
$967$	−28.8284	−0.927060	−0.463530	−	0.886081i	$-0.653417\pi$
$967$	−28.8284	−0.927060	−0.463530	+	0.886081i	$0.653417\pi$
$968$	0	0
$969$	−11.6569	−0.374472
$970$	0	0
$971$	−28.1838	−0.904460	−0.452230	−	0.891901i	$-0.649371\pi$
$971$	−28.1838	−0.904460	−0.452230	+	0.891901i	$0.649371\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	42.0833	1.34636	0.673181	−	0.739477i	$-0.264927\pi$
$977$	42.0833	1.34636	0.673181	+	0.739477i	$0.264927\pi$
$978$	0	0
$979$	−9.65685	−0.308634
$980$	0	0
$981$	−51.7990	−1.65381
$982$	0	0
$983$	33.3848	1.06481	0.532404	−	0.846490i	$-0.321289\pi$
$983$	33.3848	1.06481	0.532404	+	0.846490i	$0.321289\pi$
$984$	0	0
$985$	−12.2426	−0.390083
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−72.7696	−2.31394
$990$	0	0
$991$	5.31371	0.168796	0.0843978	−	0.996432i	$-0.473103\pi$
$991$	5.31371	0.168796	0.0843978	+	0.996432i	$0.473103\pi$
$992$	0	0
$993$	−22.9706	−0.728949
$994$	0	0
$995$	−12.3848	−0.392624
$996$	0	0
$997$	12.8995	0.408531	0.204266	−	0.978916i	$-0.434519\pi$
$997$	12.8995	0.408531	0.204266	+	0.978916i	$0.434519\pi$
$998$	0	0
$999$	−4.92893	−0.155945

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.a.by.1.2		2
4.3	odd	2		1960.2.a.q.1.1	✓	2
7.6	odd	2		3920.2.a.bn.1.1		2
20.19	odd	2		9800.2.a.ca.1.2		2
28.3	even	6		1960.2.q.p.961.1		4
28.11	odd	6		1960.2.q.v.961.2		4
28.19	even	6		1960.2.q.p.361.1		4
28.23	odd	6		1960.2.q.v.361.2		4
28.27	even	2		1960.2.a.u.1.2	yes	2
140.139	even	2		9800.2.a.bs.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1960.2.a.q.1.1	✓	2	4.3	odd	2
1960.2.a.u.1.2	yes	2	28.27	even	2
1960.2.q.p.361.1		4	28.19	even	6
1960.2.q.p.961.1		4	28.3	even	6
1960.2.q.v.361.2		4	28.23	odd	6
1960.2.q.v.961.2		4	28.11	odd	6
3920.2.a.bn.1.1		2	7.6	odd	2
3920.2.a.by.1.2		2	1.1	even	1	trivial
9800.2.a.bs.1.1		2	140.139	even	2
9800.2.a.ca.1.2		2	20.19	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:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +O(q^{10})\)	\(q+2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +1.00000 q^{11} +0.414214 q^{13} +2.41421 q^{15} -2.41421 q^{17} +2.00000 q^{19} -6.24264 q^{23} +1.00000 q^{25} -0.414214 q^{27} +1.00000 q^{29} +10.2426 q^{31} +2.41421 q^{33} +11.8995 q^{37} +1.00000 q^{39} -4.58579 q^{41} +11.6569 q^{43} +2.82843 q^{45} +7.58579 q^{47} -5.82843 q^{51} +6.58579 q^{53} +1.00000 q^{55} +4.82843 q^{57} +1.75736 q^{59} +6.82843 q^{61} +0.414214 q^{65} +1.41421 q^{67} -15.0711 q^{69} +2.48528 q^{71} +10.8284 q^{73} +2.41421 q^{75} +3.34315 q^{79} -9.48528 q^{81} -11.3137 q^{83} -2.41421 q^{85} +2.41421 q^{87} -9.65685 q^{89} +24.7279 q^{93} +2.00000 q^{95} -14.0711 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{11} - 2 q^{13} + 2 q^{15} - 2 q^{17} + 4 q^{19} - 4 q^{23} + 2 q^{25} + 2 q^{27} + 2 q^{29} + 12 q^{31} + 2 q^{33} + 4 q^{37} + 2 q^{39} - 12 q^{41} + 12 q^{43} + 18 q^{47} - 6 q^{51} + 16 q^{53} + 2 q^{55} + 4 q^{57} + 12 q^{59} + 8 q^{61} - 2 q^{65} - 16 q^{69} - 12 q^{71} + 16 q^{73} + 2 q^{75} + 18 q^{79} - 2 q^{81} - 2 q^{85} + 2 q^{87} - 8 q^{89} + 24 q^{93} + 4 q^{95} - 14 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^11 - 2 * q^13 + 2 * q^15 - 2 * q^17 + 4 * q^19 - 4 * q^23 + 2 * q^25 + 2 * q^27 + 2 * q^29 + 12 * q^31 + 2 * q^33 + 4 * q^37 + 2 * q^39 - 12 * q^41 + 12 * q^43 + 18 * q^47 - 6 * q^51 + 16 * q^53 + 2 * q^55 + 4 * q^57 + 12 * q^59 + 8 * q^61 - 2 * q^65 - 16 * q^69 - 12 * q^71 + 16 * q^73 + 2 * q^75 + 18 * q^79 - 2 * q^81 - 2 * q^85 + 2 * q^87 - 8 * q^89 + 24 * q^93 + 4 * q^95 - 14 * q^97

Introduction

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