LMFDB - Embedded newform 2960.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, 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("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2960 = 2^{4} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2960.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:	$23.6357189983$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 740)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	2960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{3} +1.00000 q^{5} -4.73205 q^{7} -2.46410 q^{9} -1.46410 q^{11} +4.00000 q^{13} -0.732051 q^{15} -4.00000 q^{17} +4.19615 q^{19} +3.46410 q^{21} -4.92820 q^{23} +1.00000 q^{25} +4.00000 q^{27} +0.535898 q^{29} +0.196152 q^{31} +1.07180 q^{33} -4.73205 q^{35} +1.00000 q^{37} -2.92820 q^{39} -8.92820 q^{41} -6.00000 q^{43} -2.46410 q^{45} -4.73205 q^{47} +15.3923 q^{49} +2.92820 q^{51} -0.928203 q^{53} -1.46410 q^{55} -3.07180 q^{57} +13.6603 q^{59} +14.0000 q^{61} +11.6603 q^{63} +4.00000 q^{65} +8.73205 q^{67} +3.60770 q^{69} -8.00000 q^{71} +10.3923 q^{73} -0.732051 q^{75} +6.92820 q^{77} +4.19615 q^{79} +4.46410 q^{81} +15.6603 q^{83} -4.00000 q^{85} -0.392305 q^{87} -2.00000 q^{89} -18.9282 q^{91} -0.143594 q^{93} +4.19615 q^{95} -14.0000 q^{97} +3.60770 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} + 4 q^{11} + 8 q^{13} + 2 q^{15} - 8 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 8 q^{27} + 8 q^{29} - 10 q^{31} + 16 q^{33} - 6 q^{35} + 2 q^{37} + 8 q^{39} - 4 q^{41}+ \cdots + 28 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 + 4 * q^11 + 8 * q^13 + 2 * q^15 - 8 * q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 8 * q^27 + 8 * q^29 - 10 * q^31 + 16 * q^33 - 6 * q^35 + 2 * q^37 + 8 * q^39 - 4 * q^41 - 12 * q^43 + 2 * q^45 - 6 * q^47 + 10 * q^49 - 8 * q^51 + 12 * q^53 + 4 * q^55 - 20 * q^57 + 10 * q^59 + 28 * q^61 + 6 * q^63 + 8 * q^65 + 14 * q^67 + 28 * q^69 - 16 * q^71 + 2 * q^75 - 2 * q^79 + 2 * q^81 + 14 * q^83 - 8 * q^85 + 20 * q^87 - 4 * q^89 - 24 * q^91 - 28 * q^93 - 2 * q^95 - 28 * q^97 + 28 * 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
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.73205	−1.78855	−0.894274	−	0.447521i	$-0.852307\pi$
$7$	−4.73205	−1.78855	−0.894274	+	0.447521i	$0.852307\pi$
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−1.46410	−0.441443	−0.220722	−	0.975337i	$-0.570841\pi$
$11$	−1.46410	−0.441443	−0.220722	+	0.975337i	$0.570841\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−0.732051	−0.189015
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	4.19615	0.962663	0.481332	−	0.876539i	$-0.340153\pi$
$19$	4.19615	0.962663	0.481332	+	0.876539i	$0.340153\pi$
$20$	0	0
$21$	3.46410	0.755929
$22$	0	0
$23$	−4.92820	−1.02760	−0.513801	−	0.857910i	$-0.671763\pi$
$23$	−4.92820	−1.02760	−0.513801	+	0.857910i	$0.671763\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	0.535898	0.0995138	0.0497569	−	0.998761i	$-0.484155\pi$
$29$	0.535898	0.0995138	0.0497569	+	0.998761i	$0.484155\pi$
$30$	0	0
$31$	0.196152	0.0352300	0.0176150	−	0.999845i	$-0.494393\pi$
$31$	0.196152	0.0352300	0.0176150	+	0.999845i	$0.494393\pi$
$32$	0	0
$33$	1.07180	0.186576
$34$	0	0
$35$	−4.73205	−0.799863
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−2.92820	−0.468888
$40$	0	0
$41$	−8.92820	−1.39435	−0.697176	−	0.716900i	$-0.745560\pi$
$41$	−8.92820	−1.39435	−0.697176	+	0.716900i	$0.745560\pi$
$42$	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
$45$	−2.46410	−0.367327
$46$	0	0
$47$	−4.73205	−0.690241	−0.345120	−	0.938558i	$-0.612162\pi$
$47$	−4.73205	−0.690241	−0.345120	+	0.938558i	$0.612162\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	0	0
$51$	2.92820	0.410030
$52$	0	0
$53$	−0.928203	−0.127499	−0.0637493	−	0.997966i	$-0.520306\pi$
$53$	−0.928203	−0.127499	−0.0637493	+	0.997966i	$0.520306\pi$
$54$	0	0
$55$	−1.46410	−0.197419
$56$	0	0
$57$	−3.07180	−0.406869
$58$	0	0
$59$	13.6603	1.77841	0.889207	−	0.457505i	$-0.151257\pi$
$59$	13.6603	1.77841	0.889207	+	0.457505i	$0.151257\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	11.6603	1.46905
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	8.73205	1.06679	0.533395	−	0.845866i	$-0.320916\pi$
$67$	8.73205	1.06679	0.533395	+	0.845866i	$0.320916\pi$
$68$	0	0
$69$	3.60770	0.434315
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	10.3923	1.21633	0.608164	−	0.793812i	$-0.291906\pi$
$73$	10.3923	1.21633	0.608164	+	0.793812i	$0.291906\pi$
$74$	0	0
$75$	−0.732051	−0.0845299
$76$	0	0
$77$	6.92820	0.789542
$78$	0	0
$79$	4.19615	0.472104	0.236052	−	0.971740i	$-0.424146\pi$
$79$	4.19615	0.472104	0.236052	+	0.971740i	$0.424146\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	15.6603	1.71894	0.859468	−	0.511189i	$-0.170795\pi$
$83$	15.6603	1.71894	0.859468	+	0.511189i	$0.170795\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	−0.392305	−0.0420595
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−18.9282	−1.98421
$92$	0	0
$93$	−0.143594	−0.0148900
$94$	0	0
$95$	4.19615	0.430516
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	3.60770	0.362587
$100$	0	0
$101$	9.46410	0.941713	0.470857	−	0.882210i	$-0.343945\pi$
$101$	9.46410	0.941713	0.470857	+	0.882210i	$0.343945\pi$
$102$	0	0
$103$	14.3923	1.41812	0.709058	−	0.705150i	$-0.249120\pi$
$103$	14.3923	1.41812	0.709058	+	0.705150i	$0.249120\pi$
$104$	0	0
$105$	3.46410	0.338062
$106$	0	0
$107$	−10.1962	−0.985699	−0.492850	−	0.870114i	$-0.664045\pi$
$107$	−10.1962	−0.985699	−0.492850	+	0.870114i	$0.664045\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−0.732051	−0.0694832
$112$	0	0
$113$	−9.85641	−0.927213	−0.463606	−	0.886041i	$-0.653445\pi$
$113$	−9.85641	−0.927213	−0.463606	+	0.886041i	$0.653445\pi$
$114$	0	0
$115$	−4.92820	−0.459557
$116$	0	0
$117$	−9.85641	−0.911225
$118$	0	0
$119$	18.9282	1.73515
$120$	0	0
$121$	−8.85641	−0.805128
$122$	0	0
$123$	6.53590	0.589322
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.6603	1.74457	0.872283	−	0.489002i	$-0.162639\pi$
$127$	19.6603	1.74457	0.872283	+	0.489002i	$0.162639\pi$
$128$	0	0
$129$	4.39230	0.386721
$130$	0	0
$131$	13.2679	1.15923	0.579613	−	0.814892i	$-0.303204\pi$
$131$	13.2679	1.15923	0.579613	+	0.814892i	$0.303204\pi$
$132$	0	0
$133$	−19.8564	−1.72177
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	13.3205	1.13805	0.569024	−	0.822321i	$-0.307321\pi$
$137$	13.3205	1.13805	0.569024	+	0.822321i	$0.307321\pi$
$138$	0	0
$139$	5.07180	0.430184	0.215092	−	0.976594i	$-0.430995\pi$
$139$	5.07180	0.430184	0.215092	+	0.976594i	$0.430995\pi$
$140$	0	0
$141$	3.46410	0.291730
$142$	0	0
$143$	−5.85641	−0.489737
$144$	0	0
$145$	0.535898	0.0445039
$146$	0	0
$147$	−11.2679	−0.929365
$148$	0	0
$149$	2.53590	0.207749	0.103874	−	0.994590i	$-0.466876\pi$
$149$	2.53590	0.207749	0.103874	+	0.994590i	$0.466876\pi$
$150$	0	0
$151$	−8.39230	−0.682956	−0.341478	−	0.939890i	$-0.610927\pi$
$151$	−8.39230	−0.682956	−0.341478	+	0.939890i	$0.610927\pi$
$152$	0	0
$153$	9.85641	0.796843
$154$	0	0
$155$	0.196152	0.0157553
$156$	0	0
$157$	0.535898	0.0427693	0.0213847	−	0.999771i	$-0.493193\pi$
$157$	0.535898	0.0427693	0.0213847	+	0.999771i	$0.493193\pi$
$158$	0	0
$159$	0.679492	0.0538872
$160$	0	0
$161$	23.3205	1.83791
$162$	0	0
$163$	−8.53590	−0.668583	−0.334292	−	0.942470i	$-0.608497\pi$
$163$	−8.53590	−0.668583	−0.334292	+	0.942470i	$0.608497\pi$
$164$	0	0
$165$	1.07180	0.0834393
$166$	0	0
$167$	14.3923	1.11371	0.556855	−	0.830610i	$-0.312008\pi$
$167$	14.3923	1.11371	0.556855	+	0.830610i	$0.312008\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−10.3397	−0.790700
$172$	0	0
$173$	−11.4641	−0.871600	−0.435800	−	0.900044i	$-0.643534\pi$
$173$	−11.4641	−0.871600	−0.435800	+	0.900044i	$0.643534\pi$
$174$	0	0
$175$	−4.73205	−0.357709
$176$	0	0
$177$	−10.0000	−0.751646
$178$	0	0
$179$	−1.26795	−0.0947710	−0.0473855	−	0.998877i	$-0.515089\pi$
$179$	−1.26795	−0.0947710	−0.0473855	+	0.998877i	$0.515089\pi$
$180$	0	0
$181$	19.3205	1.43608	0.718041	−	0.696001i	$-0.245039\pi$
$181$	19.3205	1.43608	0.718041	+	0.696001i	$0.245039\pi$
$182$	0	0
$183$	−10.2487	−0.757607
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	5.85641	0.428263
$188$	0	0
$189$	−18.9282	−1.37682
$190$	0	0
$191$	16.5885	1.20030	0.600149	−	0.799888i	$-0.295108\pi$
$191$	16.5885	1.20030	0.600149	+	0.799888i	$0.295108\pi$
$192$	0	0
$193$	−4.92820	−0.354740	−0.177370	−	0.984144i	$-0.556759\pi$
$193$	−4.92820	−0.354740	−0.177370	+	0.984144i	$0.556759\pi$
$194$	0	0
$195$	−2.92820	−0.209693
$196$	0	0
$197$	−25.3205	−1.80401	−0.902006	−	0.431724i	$-0.857905\pi$
$197$	−25.3205	−1.80401	−0.902006	+	0.431724i	$0.857905\pi$
$198$	0	0
$199$	11.1244	0.788585	0.394292	−	0.918985i	$-0.370990\pi$
$199$	11.1244	0.788585	0.394292	+	0.918985i	$0.370990\pi$
$200$	0	0
$201$	−6.39230	−0.450878
$202$	0	0
$203$	−2.53590	−0.177985
$204$	0	0
$205$	−8.92820	−0.623573
$206$	0	0
$207$	12.1436	0.844038
$208$	0	0
$209$	−6.14359	−0.424961
$210$	0	0
$211$	−2.92820	−0.201586	−0.100793	−	0.994907i	$-0.532138\pi$
$211$	−2.92820	−0.201586	−0.100793	+	0.994907i	$0.532138\pi$
$212$	0	0
$213$	5.85641	0.401274
$214$	0	0
$215$	−6.00000	−0.409197
$216$	0	0
$217$	−0.928203	−0.0630105
$218$	0	0
$219$	−7.60770	−0.514080
$220$	0	0
$221$	−16.0000	−1.07628
$222$	0	0
$223$	2.19615	0.147065	0.0735326	−	0.997293i	$-0.476573\pi$
$223$	2.19615	0.147065	0.0735326	+	0.997293i	$0.476573\pi$
$224$	0	0
$225$	−2.46410	−0.164273
$226$	0	0
$227$	7.46410	0.495410	0.247705	−	0.968836i	$-0.420324\pi$
$227$	7.46410	0.495410	0.247705	+	0.968836i	$0.420324\pi$
$228$	0	0
$229$	20.9282	1.38297	0.691487	−	0.722389i	$-0.256956\pi$
$229$	20.9282	1.38297	0.691487	+	0.722389i	$0.256956\pi$
$230$	0	0
$231$	−5.07180	−0.333700
$232$	0	0
$233$	−11.4641	−0.751038	−0.375519	−	0.926815i	$-0.622536\pi$
$233$	−11.4641	−0.751038	−0.375519	+	0.926815i	$0.622536\pi$
$234$	0	0
$235$	−4.73205	−0.308685
$236$	0	0
$237$	−3.07180	−0.199535
$238$	0	0
$239$	30.4449	1.96931	0.984657	−	0.174500i	$-0.0558307\pi$
$239$	30.4449	1.96931	0.984657	+	0.174500i	$0.0558307\pi$
$240$	0	0
$241$	−23.4641	−1.51146	−0.755728	−	0.654886i	$-0.772717\pi$
$241$	−23.4641	−1.51146	−0.755728	+	0.654886i	$0.772717\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	0	0
$245$	15.3923	0.983378
$246$	0	0
$247$	16.7846	1.06798
$248$	0	0
$249$	−11.4641	−0.726508
$250$	0	0
$251$	13.2679	0.837466	0.418733	−	0.908110i	$-0.362474\pi$
$251$	13.2679	0.837466	0.418733	+	0.908110i	$0.362474\pi$
$252$	0	0
$253$	7.21539	0.453628
$254$	0	0
$255$	2.92820	0.183371
$256$	0	0
$257$	14.9282	0.931196	0.465598	−	0.884996i	$-0.345839\pi$
$257$	14.9282	0.931196	0.465598	+	0.884996i	$0.345839\pi$
$258$	0	0
$259$	−4.73205	−0.294035
$260$	0	0
$261$	−1.32051	−0.0817374
$262$	0	0
$263$	−17.1244	−1.05593	−0.527967	−	0.849265i	$-0.677045\pi$
$263$	−17.1244	−1.05593	−0.527967	+	0.849265i	$0.677045\pi$
$264$	0	0
$265$	−0.928203	−0.0570191
$266$	0	0
$267$	1.46410	0.0896016
$268$	0	0
$269$	8.39230	0.511688	0.255844	−	0.966718i	$-0.417647\pi$
$269$	8.39230	0.511688	0.255844	+	0.966718i	$0.417647\pi$
$270$	0	0
$271$	−21.8564	−1.32768	−0.663841	−	0.747874i	$-0.731075\pi$
$271$	−21.8564	−1.32768	−0.663841	+	0.747874i	$0.731075\pi$
$272$	0	0
$273$	13.8564	0.838628
$274$	0	0
$275$	−1.46410	−0.0882886
$276$	0	0
$277$	12.7846	0.768153	0.384076	−	0.923301i	$-0.374520\pi$
$277$	12.7846	0.768153	0.384076	+	0.923301i	$0.374520\pi$
$278$	0	0
$279$	−0.483340	−0.0289368
$280$	0	0
$281$	14.3923	0.858573	0.429286	−	0.903168i	$-0.358765\pi$
$281$	14.3923	0.858573	0.429286	+	0.903168i	$0.358765\pi$
$282$	0	0
$283$	−15.4641	−0.919245	−0.459623	−	0.888114i	$-0.652015\pi$
$283$	−15.4641	−0.919245	−0.459623	+	0.888114i	$0.652015\pi$
$284$	0	0
$285$	−3.07180	−0.181958
$286$	0	0
$287$	42.2487	2.49386
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	10.2487	0.600790
$292$	0	0
$293$	−12.9282	−0.755274	−0.377637	−	0.925954i	$-0.623263\pi$
$293$	−12.9282	−0.755274	−0.377637	+	0.925954i	$0.623263\pi$
$294$	0	0
$295$	13.6603	0.795331
$296$	0	0
$297$	−5.85641	−0.339823
$298$	0	0
$299$	−19.7128	−1.14002
$300$	0	0
$301$	28.3923	1.63651
$302$	0	0
$303$	−6.92820	−0.398015
$304$	0	0
$305$	14.0000	0.801638
$306$	0	0
$307$	−4.73205	−0.270072	−0.135036	−	0.990841i	$-0.543115\pi$
$307$	−4.73205	−0.270072	−0.135036	+	0.990841i	$0.543115\pi$
$308$	0	0
$309$	−10.5359	−0.599366
$310$	0	0
$311$	−15.8038	−0.896154	−0.448077	−	0.893995i	$-0.647891\pi$
$311$	−15.8038	−0.896154	−0.448077	+	0.893995i	$0.647891\pi$
$312$	0	0
$313$	15.8564	0.896257	0.448129	−	0.893969i	$-0.352091\pi$
$313$	15.8564	0.896257	0.448129	+	0.893969i	$0.352091\pi$
$314$	0	0
$315$	11.6603	0.656981
$316$	0	0
$317$	8.53590	0.479424	0.239712	−	0.970844i	$-0.422947\pi$
$317$	8.53590	0.479424	0.239712	+	0.970844i	$0.422947\pi$
$318$	0	0
$319$	−0.784610	−0.0439297
$320$	0	0
$321$	7.46410	0.416606
$322$	0	0
$323$	−16.7846	−0.933921
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−1.46410	−0.0809650
$328$	0	0
$329$	22.3923	1.23453
$330$	0	0
$331$	5.66025	0.311116	0.155558	−	0.987827i	$-0.450282\pi$
$331$	5.66025	0.311116	0.155558	+	0.987827i	$0.450282\pi$
$332$	0	0
$333$	−2.46410	−0.135032
$334$	0	0
$335$	8.73205	0.477083
$336$	0	0
$337$	4.53590	0.247086	0.123543	−	0.992339i	$-0.460574\pi$
$337$	4.53590	0.247086	0.123543	+	0.992339i	$0.460574\pi$
$338$	0	0
$339$	7.21539	0.391886
$340$	0	0
$341$	−0.287187	−0.0155521
$342$	0	0
$343$	−39.7128	−2.14429
$344$	0	0
$345$	3.60770	0.194232
$346$	0	0
$347$	−14.7846	−0.793679	−0.396840	−	0.917888i	$-0.629893\pi$
$347$	−14.7846	−0.793679	−0.396840	+	0.917888i	$0.629893\pi$
$348$	0	0
$349$	26.2487	1.40506	0.702531	−	0.711653i	$-0.252053\pi$
$349$	26.2487	1.40506	0.702531	+	0.711653i	$0.252053\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	−13.8564	−0.733359
$358$	0	0
$359$	−1.46410	−0.0772723	−0.0386362	−	0.999253i	$-0.512301\pi$
$359$	−1.46410	−0.0772723	−0.0386362	+	0.999253i	$0.512301\pi$
$360$	0	0
$361$	−1.39230	−0.0732792
$362$	0	0
$363$	6.48334	0.340287
$364$	0	0
$365$	10.3923	0.543958
$366$	0	0
$367$	−24.0526	−1.25553	−0.627767	−	0.778402i	$-0.716031\pi$
$367$	−24.0526	−1.25553	−0.627767	+	0.778402i	$0.716031\pi$
$368$	0	0
$369$	22.0000	1.14527
$370$	0	0
$371$	4.39230	0.228037
$372$	0	0
$373$	−32.2487	−1.66977	−0.834887	−	0.550421i	$-0.814467\pi$
$373$	−32.2487	−1.66977	−0.834887	+	0.550421i	$0.814467\pi$
$374$	0	0
$375$	−0.732051	−0.0378029
$376$	0	0
$377$	2.14359	0.110401
$378$	0	0
$379$	−20.3923	−1.04748	−0.523741	−	0.851877i	$-0.675464\pi$
$379$	−20.3923	−1.04748	−0.523741	+	0.851877i	$0.675464\pi$
$380$	0	0
$381$	−14.3923	−0.737340
$382$	0	0
$383$	15.8564	0.810225	0.405112	−	0.914267i	$-0.367232\pi$
$383$	15.8564	0.810225	0.405112	+	0.914267i	$0.367232\pi$
$384$	0	0
$385$	6.92820	0.353094
$386$	0	0
$387$	14.7846	0.751544
$388$	0	0
$389$	7.07180	0.358554	0.179277	−	0.983799i	$-0.442624\pi$
$389$	7.07180	0.358554	0.179277	+	0.983799i	$0.442624\pi$
$390$	0	0
$391$	19.7128	0.996920
$392$	0	0
$393$	−9.71281	−0.489947
$394$	0	0
$395$	4.19615	0.211131
$396$	0	0
$397$	29.7128	1.49124	0.745622	−	0.666370i	$-0.232153\pi$
$397$	29.7128	1.49124	0.745622	+	0.666370i	$0.232153\pi$
$398$	0	0
$399$	14.5359	0.727705
$400$	0	0
$401$	23.8564	1.19133	0.595666	−	0.803232i	$-0.296888\pi$
$401$	23.8564	1.19133	0.595666	+	0.803232i	$0.296888\pi$
$402$	0	0
$403$	0.784610	0.0390842
$404$	0	0
$405$	4.46410	0.221823
$406$	0	0
$407$	−1.46410	−0.0725728
$408$	0	0
$409$	7.46410	0.369076	0.184538	−	0.982825i	$-0.440921\pi$
$409$	7.46410	0.369076	0.184538	+	0.982825i	$0.440921\pi$
$410$	0	0
$411$	−9.75129	−0.480996
$412$	0	0
$413$	−64.6410	−3.18078
$414$	0	0
$415$	15.6603	0.768732
$416$	0	0
$417$	−3.71281	−0.181817
$418$	0	0
$419$	13.4641	0.657764	0.328882	−	0.944371i	$-0.393328\pi$
$419$	13.4641	0.657764	0.328882	+	0.944371i	$0.393328\pi$
$420$	0	0
$421$	18.7846	0.915506	0.457753	−	0.889079i	$-0.348654\pi$
$421$	18.7846	0.915506	0.457753	+	0.889079i	$0.348654\pi$
$422$	0	0
$423$	11.6603	0.566941
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	−66.2487	−3.20600
$428$	0	0
$429$	4.28719	0.206987
$430$	0	0
$431$	−19.5167	−0.940084	−0.470042	−	0.882644i	$-0.655761\pi$
$431$	−19.5167	−0.940084	−0.470042	+	0.882644i	$0.655761\pi$
$432$	0	0
$433$	25.3205	1.21683	0.608413	−	0.793621i	$-0.291806\pi$
$433$	25.3205	1.21683	0.608413	+	0.793621i	$0.291806\pi$
$434$	0	0
$435$	−0.392305	−0.0188096
$436$	0	0
$437$	−20.6795	−0.989234
$438$	0	0
$439$	27.1244	1.29457	0.647287	−	0.762246i	$-0.275903\pi$
$439$	27.1244	1.29457	0.647287	+	0.762246i	$0.275903\pi$
$440$	0	0
$441$	−37.9282	−1.80610
$442$	0	0
$443$	−14.1962	−0.674480	−0.337240	−	0.941419i	$-0.609493\pi$
$443$	−14.1962	−0.674480	−0.337240	+	0.941419i	$0.609493\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	0	0
$447$	−1.85641	−0.0878050
$448$	0	0
$449$	15.4641	0.729796	0.364898	−	0.931047i	$-0.381104\pi$
$449$	15.4641	0.729796	0.364898	+	0.931047i	$0.381104\pi$
$450$	0	0
$451$	13.0718	0.615527
$452$	0	0
$453$	6.14359	0.288651
$454$	0	0
$455$	−18.9282	−0.887368
$456$	0	0
$457$	−24.9282	−1.16609	−0.583046	−	0.812439i	$-0.698139\pi$
$457$	−24.9282	−1.16609	−0.583046	+	0.812439i	$0.698139\pi$
$458$	0	0
$459$	−16.0000	−0.746816
$460$	0	0
$461$	39.4641	1.83803	0.919013	−	0.394227i	$-0.128988\pi$
$461$	39.4641	1.83803	0.919013	+	0.394227i	$0.128988\pi$
$462$	0	0
$463$	−11.0718	−0.514550	−0.257275	−	0.966338i	$-0.582825\pi$
$463$	−11.0718	−0.514550	−0.257275	+	0.966338i	$0.582825\pi$
$464$	0	0
$465$	−0.143594	−0.00665899
$466$	0	0
$467$	−13.6077	−0.629689	−0.314845	−	0.949143i	$-0.601952\pi$
$467$	−13.6077	−0.629689	−0.314845	+	0.949143i	$0.601952\pi$
$468$	0	0
$469$	−41.3205	−1.90800
$470$	0	0
$471$	−0.392305	−0.0180765
$472$	0	0
$473$	8.78461	0.403917
$474$	0	0
$475$	4.19615	0.192533
$476$	0	0
$477$	2.28719	0.104723
$478$	0	0
$479$	−27.1244	−1.23934	−0.619672	−	0.784861i	$-0.712734\pi$
$479$	−27.1244	−1.23934	−0.619672	+	0.784861i	$0.712734\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	−17.0718	−0.776794
$484$	0	0
$485$	−14.0000	−0.635707
$486$	0	0
$487$	−8.53590	−0.386798	−0.193399	−	0.981120i	$-0.561951\pi$
$487$	−8.53590	−0.386798	−0.193399	+	0.981120i	$0.561951\pi$
$488$	0	0
$489$	6.24871	0.282576
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	−2.14359	−0.0965426
$494$	0	0
$495$	3.60770	0.162154
$496$	0	0
$497$	37.8564	1.69809
$498$	0	0
$499$	2.33975	0.104741	0.0523707	−	0.998628i	$-0.483322\pi$
$499$	2.33975	0.104741	0.0523707	+	0.998628i	$0.483322\pi$
$500$	0	0
$501$	−10.5359	−0.470709
$502$	0	0
$503$	31.8564	1.42041	0.710203	−	0.703996i	$-0.248603\pi$
$503$	31.8564	1.42041	0.710203	+	0.703996i	$0.248603\pi$
$504$	0	0
$505$	9.46410	0.421147
$506$	0	0
$507$	−2.19615	−0.0975346
$508$	0	0
$509$	−15.8564	−0.702823	−0.351411	−	0.936221i	$-0.614298\pi$
$509$	−15.8564	−0.702823	−0.351411	+	0.936221i	$0.614298\pi$
$510$	0	0
$511$	−49.1769	−2.17546
$512$	0	0
$513$	16.7846	0.741059
$514$	0	0
$515$	14.3923	0.634201
$516$	0	0
$517$	6.92820	0.304702
$518$	0	0
$519$	8.39230	0.368381
$520$	0	0
$521$	−38.5359	−1.68829	−0.844144	−	0.536116i	$-0.819891\pi$
$521$	−38.5359	−1.68829	−0.844144	+	0.536116i	$0.819891\pi$
$522$	0	0
$523$	10.7846	0.471578	0.235789	−	0.971804i	$-0.424233\pi$
$523$	10.7846	0.471578	0.235789	+	0.971804i	$0.424233\pi$
$524$	0	0
$525$	3.46410	0.151186
$526$	0	0
$527$	−0.784610	−0.0341781
$528$	0	0
$529$	1.28719	0.0559647
$530$	0	0
$531$	−33.6603	−1.46073
$532$	0	0
$533$	−35.7128	−1.54689
$534$	0	0
$535$	−10.1962	−0.440818
$536$	0	0
$537$	0.928203	0.0400549
$538$	0	0
$539$	−22.5359	−0.970690
$540$	0	0
$541$	−10.3923	−0.446800	−0.223400	−	0.974727i	$-0.571716\pi$
$541$	−10.3923	−0.446800	−0.223400	+	0.974727i	$0.571716\pi$
$542$	0	0
$543$	−14.1436	−0.606960
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	4.92820	0.210715	0.105357	−	0.994434i	$-0.466401\pi$
$547$	4.92820	0.210715	0.105357	+	0.994434i	$0.466401\pi$
$548$	0	0
$549$	−34.4974	−1.47231
$550$	0	0
$551$	2.24871	0.0957983
$552$	0	0
$553$	−19.8564	−0.844380
$554$	0	0
$555$	−0.732051	−0.0310738
$556$	0	0
$557$	25.8564	1.09557	0.547786	−	0.836619i	$-0.315471\pi$
$557$	25.8564	1.09557	0.547786	+	0.836619i	$0.315471\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	−4.28719	−0.181005
$562$	0	0
$563$	−11.0718	−0.466621	−0.233310	−	0.972402i	$-0.574956\pi$
$563$	−11.0718	−0.466621	−0.233310	+	0.972402i	$0.574956\pi$
$564$	0	0
$565$	−9.85641	−0.414662
$566$	0	0
$567$	−21.1244	−0.887140
$568$	0	0
$569$	−21.3205	−0.893802	−0.446901	−	0.894583i	$-0.647472\pi$
$569$	−21.3205	−0.893802	−0.446901	+	0.894583i	$0.647472\pi$
$570$	0	0
$571$	−39.7128	−1.66193	−0.830965	−	0.556325i	$-0.812211\pi$
$571$	−39.7128	−1.66193	−0.830965	+	0.556325i	$0.812211\pi$
$572$	0	0
$573$	−12.1436	−0.507306
$574$	0	0
$575$	−4.92820	−0.205520
$576$	0	0
$577$	−42.6410	−1.77517	−0.887584	−	0.460645i	$-0.847618\pi$
$577$	−42.6410	−1.77517	−0.887584	+	0.460645i	$0.847618\pi$
$578$	0	0
$579$	3.60770	0.149931
$580$	0	0
$581$	−74.1051	−3.07440
$582$	0	0
$583$	1.35898	0.0562834
$584$	0	0
$585$	−9.85641	−0.407512
$586$	0	0
$587$	12.9282	0.533604	0.266802	−	0.963751i	$-0.414033\pi$
$587$	12.9282	0.533604	0.266802	+	0.963751i	$0.414033\pi$
$588$	0	0
$589$	0.823085	0.0339146
$590$	0	0
$591$	18.5359	0.762465
$592$	0	0
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	0	0
$595$	18.9282	0.775981
$596$	0	0
$597$	−8.14359	−0.333295
$598$	0	0
$599$	−14.5359	−0.593921	−0.296960	−	0.954890i	$-0.595973\pi$
$599$	−14.5359	−0.593921	−0.296960	+	0.954890i	$0.595973\pi$
$600$	0	0
$601$	31.3205	1.27759	0.638795	−	0.769377i	$-0.279433\pi$
$601$	31.3205	1.27759	0.638795	+	0.769377i	$0.279433\pi$
$602$	0	0
$603$	−21.5167	−0.876226
$604$	0	0
$605$	−8.85641	−0.360064
$606$	0	0
$607$	32.9282	1.33651	0.668257	−	0.743930i	$-0.267041\pi$
$607$	32.9282	1.33651	0.668257	+	0.743930i	$0.267041\pi$
$608$	0	0
$609$	1.85641	0.0752254
$610$	0	0
$611$	−18.9282	−0.765753
$612$	0	0
$613$	44.6410	1.80303	0.901517	−	0.432744i	$-0.142455\pi$
$613$	44.6410	1.80303	0.901517	+	0.432744i	$0.142455\pi$
$614$	0	0
$615$	6.53590	0.263553
$616$	0	0
$617$	−38.7846	−1.56141	−0.780705	−	0.624900i	$-0.785140\pi$
$617$	−38.7846	−1.56141	−0.780705	+	0.624900i	$0.785140\pi$
$618$	0	0
$619$	9.07180	0.364626	0.182313	−	0.983241i	$-0.441642\pi$
$619$	9.07180	0.364626	0.182313	+	0.983241i	$0.441642\pi$
$620$	0	0
$621$	−19.7128	−0.791048
$622$	0	0
$623$	9.46410	0.379171
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.49742	0.179610
$628$	0	0
$629$	−4.00000	−0.159490
$630$	0	0
$631$	47.5167	1.89161	0.945804	−	0.324737i	$-0.105276\pi$
$631$	47.5167	1.89161	0.945804	+	0.324737i	$0.105276\pi$
$632$	0	0
$633$	2.14359	0.0852002
$634$	0	0
$635$	19.6603	0.780193
$636$	0	0
$637$	61.5692	2.43946
$638$	0	0
$639$	19.7128	0.779827
$640$	0	0
$641$	−20.3923	−0.805448	−0.402724	−	0.915322i	$-0.631936\pi$
$641$	−20.3923	−0.805448	−0.402724	+	0.915322i	$0.631936\pi$
$642$	0	0
$643$	26.3923	1.04081	0.520405	−	0.853919i	$-0.325781\pi$
$643$	26.3923	1.04081	0.520405	+	0.853919i	$0.325781\pi$
$644$	0	0
$645$	4.39230	0.172947
$646$	0	0
$647$	2.00000	0.0786281	0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000	0.0786281	0.0393141	+	0.999227i	$0.487483\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	0.679492	0.0266314
$652$	0	0
$653$	0.928203	0.0363234	0.0181617	−	0.999835i	$-0.494219\pi$
$653$	0.928203	0.0363234	0.0181617	+	0.999835i	$0.494219\pi$
$654$	0	0
$655$	13.2679	0.518422
$656$	0	0
$657$	−25.6077	−0.999051
$658$	0	0
$659$	−29.0718	−1.13248	−0.566238	−	0.824242i	$-0.691602\pi$
$659$	−29.0718	−1.13248	−0.566238	+	0.824242i	$0.691602\pi$
$660$	0	0
$661$	−48.6410	−1.89192	−0.945958	−	0.324289i	$-0.894875\pi$
$661$	−48.6410	−1.89192	−0.945958	+	0.324289i	$0.894875\pi$
$662$	0	0
$663$	11.7128	0.454888
$664$	0	0
$665$	−19.8564	−0.769998
$666$	0	0
$667$	−2.64102	−0.102261
$668$	0	0
$669$	−1.60770	−0.0621571
$670$	0	0
$671$	−20.4974	−0.791294
$672$	0	0
$673$	13.7128	0.528590	0.264295	−	0.964442i	$-0.414861\pi$
$673$	13.7128	0.528590	0.264295	+	0.964442i	$0.414861\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	45.0333	1.73077	0.865386	−	0.501107i	$-0.167074\pi$
$677$	45.0333	1.73077	0.865386	+	0.501107i	$0.167074\pi$
$678$	0	0
$679$	66.2487	2.54239
$680$	0	0
$681$	−5.46410	−0.209385
$682$	0	0
$683$	−28.9282	−1.10691	−0.553453	−	0.832880i	$-0.686690\pi$
$683$	−28.9282	−1.10691	−0.553453	+	0.832880i	$0.686690\pi$
$684$	0	0
$685$	13.3205	0.508950
$686$	0	0
$687$	−15.3205	−0.584514
$688$	0	0
$689$	−3.71281	−0.141447
$690$	0	0
$691$	−11.6077	−0.441578	−0.220789	−	0.975322i	$-0.570863\pi$
$691$	−11.6077	−0.441578	−0.220789	+	0.975322i	$0.570863\pi$
$692$	0	0
$693$	−17.0718	−0.648504
$694$	0	0
$695$	5.07180	0.192384
$696$	0	0
$697$	35.7128	1.35272
$698$	0	0
$699$	8.39230	0.317426
$700$	0	0
$701$	−11.8564	−0.447810	−0.223905	−	0.974611i	$-0.571881\pi$
$701$	−11.8564	−0.447810	−0.223905	+	0.974611i	$0.571881\pi$
$702$	0	0
$703$	4.19615	0.158261
$704$	0	0
$705$	3.46410	0.130466
$706$	0	0
$707$	−44.7846	−1.68430
$708$	0	0
$709$	24.2487	0.910679	0.455340	−	0.890318i	$-0.349518\pi$
$709$	24.2487	0.910679	0.455340	+	0.890318i	$0.349518\pi$
$710$	0	0
$711$	−10.3397	−0.387771
$712$	0	0
$713$	−0.966679	−0.0362024
$714$	0	0
$715$	−5.85641	−0.219017
$716$	0	0
$717$	−22.2872	−0.832330
$718$	0	0
$719$	−22.2487	−0.829737	−0.414868	−	0.909881i	$-0.636172\pi$
$719$	−22.2487	−0.829737	−0.414868	+	0.909881i	$0.636172\pi$
$720$	0	0
$721$	−68.1051	−2.53637
$722$	0	0
$723$	17.1769	0.638816
$724$	0	0
$725$	0.535898	0.0199028
$726$	0	0
$727$	−6.00000	−0.222528	−0.111264	−	0.993791i	$-0.535490\pi$
$727$	−6.00000	−0.222528	−0.111264	+	0.993791i	$0.535490\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	−11.2679	−0.415625
$736$	0	0
$737$	−12.7846	−0.470927
$738$	0	0
$739$	−3.21539	−0.118280	−0.0591400	−	0.998250i	$-0.518836\pi$
$739$	−3.21539	−0.118280	−0.0591400	+	0.998250i	$0.518836\pi$
$740$	0	0
$741$	−12.2872	−0.451381
$742$	0	0
$743$	−10.8756	−0.398989	−0.199494	−	0.979899i	$-0.563930\pi$
$743$	−10.8756	−0.398989	−0.199494	+	0.979899i	$0.563930\pi$
$744$	0	0
$745$	2.53590	0.0929081
$746$	0	0
$747$	−38.5885	−1.41188
$748$	0	0
$749$	48.2487	1.76297
$750$	0	0
$751$	16.6795	0.608643	0.304322	−	0.952569i	$-0.401570\pi$
$751$	16.6795	0.608643	0.304322	+	0.952569i	$0.401570\pi$
$752$	0	0
$753$	−9.71281	−0.353955
$754$	0	0
$755$	−8.39230	−0.305427
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	−5.28203	−0.191726
$760$	0	0
$761$	−41.7128	−1.51209	−0.756044	−	0.654521i	$-0.772870\pi$
$761$	−41.7128	−1.51209	−0.756044	+	0.654521i	$0.772870\pi$
$762$	0	0
$763$	−9.46410	−0.342623
$764$	0	0
$765$	9.85641	0.356359
$766$	0	0
$767$	54.6410	1.97297
$768$	0	0
$769$	39.8564	1.43726	0.718629	−	0.695393i	$-0.244770\pi$
$769$	39.8564	1.43726	0.718629	+	0.695393i	$0.244770\pi$
$770$	0	0
$771$	−10.9282	−0.393570
$772$	0	0
$773$	−13.6077	−0.489435	−0.244717	−	0.969594i	$-0.578695\pi$
$773$	−13.6077	−0.489435	−0.244717	+	0.969594i	$0.578695\pi$
$774$	0	0
$775$	0.196152	0.00704600
$776$	0	0
$777$	3.46410	0.124274
$778$	0	0
$779$	−37.4641	−1.34229
$780$	0	0
$781$	11.7128	0.419117
$782$	0	0
$783$	2.14359	0.0766058
$784$	0	0
$785$	0.535898	0.0191270
$786$	0	0
$787$	−34.1962	−1.21896	−0.609481	−	0.792801i	$-0.708622\pi$
$787$	−34.1962	−1.21896	−0.609481	+	0.792801i	$0.708622\pi$
$788$	0	0
$789$	12.5359	0.446290
$790$	0	0
$791$	46.6410	1.65836
$792$	0	0
$793$	56.0000	1.98862
$794$	0	0
$795$	0.679492	0.0240991
$796$	0	0
$797$	33.8564	1.19926	0.599628	−	0.800279i	$-0.295315\pi$
$797$	33.8564	1.19926	0.599628	+	0.800279i	$0.295315\pi$
$798$	0	0
$799$	18.9282	0.669632
$800$	0	0
$801$	4.92820	0.174129
$802$	0	0
$803$	−15.2154	−0.536939
$804$	0	0
$805$	23.3205	0.821940
$806$	0	0
$807$	−6.14359	−0.216265
$808$	0	0
$809$	−3.07180	−0.107999	−0.0539993	−	0.998541i	$-0.517197\pi$
$809$	−3.07180	−0.107999	−0.0539993	+	0.998541i	$0.517197\pi$
$810$	0	0
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	0	0
$815$	−8.53590	−0.298999
$816$	0	0
$817$	−25.1769	−0.880829
$818$	0	0
$819$	46.6410	1.62977
$820$	0	0
$821$	31.8564	1.11180	0.555898	−	0.831250i	$-0.312374\pi$
$821$	31.8564	1.11180	0.555898	+	0.831250i	$0.312374\pi$
$822$	0	0
$823$	−30.1962	−1.05257	−0.526286	−	0.850308i	$-0.676416\pi$
$823$	−30.1962	−1.05257	−0.526286	+	0.850308i	$0.676416\pi$
$824$	0	0
$825$	1.07180	0.0373152
$826$	0	0
$827$	34.3923	1.19594	0.597969	−	0.801519i	$-0.295975\pi$
$827$	34.3923	1.19594	0.597969	+	0.801519i	$0.295975\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	−9.35898	−0.324660
$832$	0	0
$833$	−61.5692	−2.13325
$834$	0	0
$835$	14.3923	0.498066
$836$	0	0
$837$	0.784610	0.0271201
$838$	0	0
$839$	5.07180	0.175098	0.0875489	−	0.996160i	$-0.472097\pi$
$839$	5.07180	0.175098	0.0875489	+	0.996160i	$0.472097\pi$
$840$	0	0
$841$	−28.7128	−0.990097
$842$	0	0
$843$	−10.5359	−0.362876
$844$	0	0
$845$	3.00000	0.103203
$846$	0	0
$847$	41.9090	1.44001
$848$	0	0
$849$	11.3205	0.388519
$850$	0	0
$851$	−4.92820	−0.168937
$852$	0	0
$853$	−9.71281	−0.332560	−0.166280	−	0.986079i	$-0.553176\pi$
$853$	−9.71281	−0.332560	−0.166280	+	0.986079i	$0.553176\pi$
$854$	0	0
$855$	−10.3397	−0.353612
$856$	0	0
$857$	31.8564	1.08819	0.544097	−	0.839022i	$-0.316872\pi$
$857$	31.8564	1.08819	0.544097	+	0.839022i	$0.316872\pi$
$858$	0	0
$859$	48.5885	1.65782	0.828908	−	0.559384i	$-0.188962\pi$
$859$	48.5885	1.65782	0.828908	+	0.559384i	$0.188962\pi$
$860$	0	0
$861$	−30.9282	−1.05403
$862$	0	0
$863$	−39.3731	−1.34027	−0.670137	−	0.742237i	$-0.733765\pi$
$863$	−39.3731	−1.34027	−0.670137	+	0.742237i	$0.733765\pi$
$864$	0	0
$865$	−11.4641	−0.389791
$866$	0	0
$867$	0.732051	0.0248617
$868$	0	0
$869$	−6.14359	−0.208407
$870$	0	0
$871$	34.9282	1.18350
$872$	0	0
$873$	34.4974	1.16756
$874$	0	0
$875$	−4.73205	−0.159973
$876$	0	0
$877$	27.4641	0.927397	0.463698	−	0.885993i	$-0.346522\pi$
$877$	27.4641	0.927397	0.463698	+	0.885993i	$0.346522\pi$
$878$	0	0
$879$	9.46410	0.319216
$880$	0	0
$881$	−12.6795	−0.427183	−0.213591	−	0.976923i	$-0.568516\pi$
$881$	−12.6795	−0.427183	−0.213591	+	0.976923i	$0.568516\pi$
$882$	0	0
$883$	23.1769	0.779965	0.389983	−	0.920822i	$-0.372481\pi$
$883$	23.1769	0.779965	0.389983	+	0.920822i	$0.372481\pi$
$884$	0	0
$885$	−10.0000	−0.336146
$886$	0	0
$887$	−53.9090	−1.81009	−0.905043	−	0.425320i	$-0.860161\pi$
$887$	−53.9090	−1.81009	−0.905043	+	0.425320i	$0.860161\pi$
$888$	0	0
$889$	−93.0333	−3.12024
$890$	0	0
$891$	−6.53590	−0.218961
$892$	0	0
$893$	−19.8564	−0.664469
$894$	0	0
$895$	−1.26795	−0.0423829
$896$	0	0
$897$	14.4308	0.481830
$898$	0	0
$899$	0.105118	0.00350587
$900$	0	0
$901$	3.71281	0.123692
$902$	0	0
$903$	−20.7846	−0.691669
$904$	0	0
$905$	19.3205	0.642235
$906$	0	0
$907$	5.32051	0.176665	0.0883323	−	0.996091i	$-0.471846\pi$
$907$	5.32051	0.176665	0.0883323	+	0.996091i	$0.471846\pi$
$908$	0	0
$909$	−23.3205	−0.773492
$910$	0	0
$911$	−11.1244	−0.368566	−0.184283	−	0.982873i	$-0.558996\pi$
$911$	−11.1244	−0.368566	−0.184283	+	0.982873i	$0.558996\pi$
$912$	0	0
$913$	−22.9282	−0.758813
$914$	0	0
$915$	−10.2487	−0.338812
$916$	0	0
$917$	−62.7846	−2.07333
$918$	0	0
$919$	8.87564	0.292780	0.146390	−	0.989227i	$-0.453234\pi$
$919$	8.87564	0.292780	0.146390	+	0.989227i	$0.453234\pi$
$920$	0	0
$921$	3.46410	0.114146
$922$	0	0
$923$	−32.0000	−1.05329
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	−35.4641	−1.16479
$928$	0	0
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	64.5885	2.11680
$932$	0	0
$933$	11.5692	0.378759
$934$	0	0
$935$	5.85641	0.191525
$936$	0	0
$937$	−21.3205	−0.696511	−0.348255	−	0.937400i	$-0.613226\pi$
$937$	−21.3205	−0.696511	−0.348255	+	0.937400i	$0.613226\pi$
$938$	0	0
$939$	−11.6077	−0.378803
$940$	0	0
$941$	31.6077	1.03038	0.515191	−	0.857076i	$-0.327721\pi$
$941$	31.6077	1.03038	0.515191	+	0.857076i	$0.327721\pi$
$942$	0	0
$943$	44.0000	1.43284
$944$	0	0
$945$	−18.9282	−0.615734
$946$	0	0
$947$	−18.1051	−0.588337	−0.294169	−	0.955754i	$-0.595043\pi$
$947$	−18.1051	−0.588337	−0.294169	+	0.955754i	$0.595043\pi$
$948$	0	0
$949$	41.5692	1.34939
$950$	0	0
$951$	−6.24871	−0.202628
$952$	0	0
$953$	−16.9282	−0.548358	−0.274179	−	0.961679i	$-0.588406\pi$
$953$	−16.9282	−0.548358	−0.274179	+	0.961679i	$0.588406\pi$
$954$	0	0
$955$	16.5885	0.536790
$956$	0	0
$957$	0.574374	0.0185669
$958$	0	0
$959$	−63.0333	−2.03545
$960$	0	0
$961$	−30.9615	−0.998759
$962$	0	0
$963$	25.1244	0.809621
$964$	0	0
$965$	−4.92820	−0.158644
$966$	0	0
$967$	17.6077	0.566225	0.283113	−	0.959087i	$-0.408633\pi$
$967$	17.6077	0.566225	0.283113	+	0.959087i	$0.408633\pi$
$968$	0	0
$969$	12.2872	0.394721
$970$	0	0
$971$	16.0000	0.513464	0.256732	−	0.966483i	$-0.417354\pi$
$971$	16.0000	0.513464	0.256732	+	0.966483i	$0.417354\pi$
$972$	0	0
$973$	−24.0000	−0.769405
$974$	0	0
$975$	−2.92820	−0.0937776
$976$	0	0
$977$	−6.78461	−0.217059	−0.108529	−	0.994093i	$-0.534614\pi$
$977$	−6.78461	−0.217059	−0.108529	+	0.994093i	$0.534614\pi$
$978$	0	0
$979$	2.92820	0.0935858
$980$	0	0
$981$	−4.92820	−0.157345
$982$	0	0
$983$	27.3731	0.873065	0.436533	−	0.899688i	$-0.356206\pi$
$983$	27.3731	0.873065	0.436533	+	0.899688i	$0.356206\pi$
$984$	0	0
$985$	−25.3205	−0.806778
$986$	0	0
$987$	−16.3923	−0.521773
$988$	0	0
$989$	29.5692	0.940246
$990$	0	0
$991$	−39.9090	−1.26775	−0.633875	−	0.773435i	$-0.718537\pi$
$991$	−39.9090	−1.26775	−0.633875	+	0.773435i	$0.718537\pi$
$992$	0	0
$993$	−4.14359	−0.131493
$994$	0	0
$995$	11.1244	0.352666
$996$	0	0
$997$	33.7128	1.06770	0.533848	−	0.845581i	$-0.320746\pi$
$997$	33.7128	1.06770	0.533848	+	0.845581i	$0.320746\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.p.1.1		2
4.3	odd	2		740.2.a.d.1.2	✓	2
12.11	even	2		6660.2.a.i.1.2		2
20.3	even	4		3700.2.d.g.149.3		4
20.7	even	4		3700.2.d.g.149.2		4
20.19	odd	2		3700.2.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.a.d.1.2	✓	2	4.3	odd	2
2960.2.a.p.1.1		2	1.1	even	1	trivial
3700.2.a.h.1.1		2	20.19	odd	2
3700.2.d.g.149.2		4	20.7	even	4
3700.2.d.g.149.3		4	20.3	even	4
6660.2.a.i.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{3} +1.00000 q^{5} -4.73205 q^{7} -2.46410 q^{9} -1.46410 q^{11} +4.00000 q^{13} -0.732051 q^{15} -4.00000 q^{17} +4.19615 q^{19} +3.46410 q^{21} -4.92820 q^{23} +1.00000 q^{25} +4.00000 q^{27} +0.535898 q^{29} +0.196152 q^{31} +1.07180 q^{33} -4.73205 q^{35} +1.00000 q^{37} -2.92820 q^{39} -8.92820 q^{41} -6.00000 q^{43} -2.46410 q^{45} -4.73205 q^{47} +15.3923 q^{49} +2.92820 q^{51} -0.928203 q^{53} -1.46410 q^{55} -3.07180 q^{57} +13.6603 q^{59} +14.0000 q^{61} +11.6603 q^{63} +4.00000 q^{65} +8.73205 q^{67} +3.60770 q^{69} -8.00000 q^{71} +10.3923 q^{73} -0.732051 q^{75} +6.92820 q^{77} +4.19615 q^{79} +4.46410 q^{81} +15.6603 q^{83} -4.00000 q^{85} -0.392305 q^{87} -2.00000 q^{89} -18.9282 q^{91} -0.143594 q^{93} +4.19615 q^{95} -14.0000 q^{97} +3.60770 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} + 4 q^{11} + 8 q^{13} + 2 q^{15} - 8 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 8 q^{27} + 8 q^{29} - 10 q^{31} + 16 q^{33} - 6 q^{35} + 2 q^{37} + 8 q^{39} - 4 q^{41}+ \cdots + 28 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 + 4 * q^11 + 8 * q^13 + 2 * q^15 - 8 * q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 8 * q^27 + 8 * q^29 - 10 * q^31 + 16 * q^33 - 6 * q^35 + 2 * q^37 + 8 * q^39 - 4 * q^41 - 12 * q^43 + 2 * q^45 - 6 * q^47 + 10 * q^49 - 8 * q^51 + 12 * q^53 + 4 * q^55 - 20 * q^57 + 10 * q^59 + 28 * q^61 + 6 * q^63 + 8 * q^65 + 14 * q^67 + 28 * q^69 - 16 * q^71 + 2 * q^75 - 2 * q^79 + 2 * q^81 + 14 * q^83 - 8 * q^85 + 20 * q^87 - 4 * q^89 - 24 * q^91 - 28 * q^93 - 2 * q^95 - 28 * q^97 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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