LMFDB - Embedded newform 8004.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} + \cdots + 16 $ x^12 - 3x^11 - 31x^10 + 80x^9 + 347x^8 - 697x^7 - 1714x^6 + 2146x^5 + 3304x^4 - 1156x^3 - 616x^2 + 136*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.15588$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.15588 q^{5} -1.11441 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.15588 q^{5} -1.11441 q^{7} +1.00000 q^{9} -5.32243 q^{11} +0.315594 q^{13} +4.15588 q^{15} -1.36640 q^{17} +5.63415 q^{19} +1.11441 q^{21} +1.00000 q^{23} +12.2713 q^{25} -1.00000 q^{27} -1.00000 q^{29} -7.05741 q^{31} +5.32243 q^{33} +4.63136 q^{35} -5.05783 q^{37} -0.315594 q^{39} +2.34509 q^{41} +4.48164 q^{43} -4.15588 q^{45} +3.25596 q^{47} -5.75809 q^{49} +1.36640 q^{51} +7.64907 q^{53} +22.1194 q^{55} -5.63415 q^{57} -5.11563 q^{59} +1.44172 q^{61} -1.11441 q^{63} -1.31157 q^{65} +1.32653 q^{67} -1.00000 q^{69} +5.18601 q^{71} +2.17074 q^{73} -12.2713 q^{75} +5.93137 q^{77} +16.7829 q^{79} +1.00000 q^{81} +10.5447 q^{83} +5.67859 q^{85} +1.00000 q^{87} +7.27517 q^{89} -0.351701 q^{91} +7.05741 q^{93} -23.4149 q^{95} +17.8471 q^{97} -5.32243 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−4.15588	−1.85857	−0.929283	−	0.369368i	$-0.879574\pi$
$5$	−4.15588	−1.85857	−0.929283	+	0.369368i	$0.879574\pi$
$6$	0	0
$7$	−1.11441	−0.421208	−0.210604	−	0.977572i	$-0.567543\pi$
$7$	−1.11441	−0.421208	−0.210604	+	0.977572i	$0.567543\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.32243	−1.60477	−0.802386	−	0.596805i	$-0.796437\pi$
$11$	−5.32243	−1.60477	−0.802386	+	0.596805i	$0.796437\pi$
$12$	0	0
$13$	0.315594	0.0875301	0.0437650	−	0.999042i	$-0.486065\pi$
$13$	0.315594	0.0875301	0.0437650	+	0.999042i	$0.486065\pi$
$14$	0	0
$15$	4.15588	1.07304
$16$	0	0
$17$	−1.36640	−0.331400	−0.165700	−	0.986176i	$-0.552988\pi$
$17$	−1.36640	−0.331400	−0.165700	+	0.986176i	$0.552988\pi$
$18$	0	0
$19$	5.63415	1.29256	0.646282	−	0.763099i	$-0.276323\pi$
$19$	5.63415	1.29256	0.646282	+	0.763099i	$0.276323\pi$
$20$	0	0
$21$	1.11441	0.243184
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	12.2713	2.45427
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−7.05741	−1.26755	−0.633774	−	0.773518i	$-0.718495\pi$
$31$	−7.05741	−1.26755	−0.633774	+	0.773518i	$0.718495\pi$
$32$	0	0
$33$	5.32243	0.926516
$34$	0	0
$35$	4.63136	0.782842
$36$	0	0
$37$	−5.05783	−0.831502	−0.415751	−	0.909478i	$-0.636481\pi$
$37$	−5.05783	−0.831502	−0.415751	+	0.909478i	$0.636481\pi$
$38$	0	0
$39$	−0.315594	−0.0505355
$40$	0	0
$41$	2.34509	0.366241	0.183120	−	0.983090i	$-0.441380\pi$
$41$	2.34509	0.366241	0.183120	+	0.983090i	$0.441380\pi$
$42$	0	0
$43$	4.48164	0.683444	0.341722	−	0.939801i	$-0.388990\pi$
$43$	4.48164	0.683444	0.341722	+	0.939801i	$0.388990\pi$
$44$	0	0
$45$	−4.15588	−0.619522
$46$	0	0
$47$	3.25596	0.474930	0.237465	−	0.971396i	$-0.423683\pi$
$47$	3.25596	0.474930	0.237465	+	0.971396i	$0.423683\pi$
$48$	0	0
$49$	−5.75809	−0.822584
$50$	0	0
$51$	1.36640	0.191334
$52$	0	0
$53$	7.64907	1.05068	0.525341	−	0.850892i	$-0.323938\pi$
$53$	7.64907	1.05068	0.525341	+	0.850892i	$0.323938\pi$
$54$	0	0
$55$	22.1194	2.98258
$56$	0	0
$57$	−5.63415	−0.746262
$58$	0	0
$59$	−5.11563	−0.665998	−0.332999	−	0.942927i	$-0.608061\pi$
$59$	−5.11563	−0.665998	−0.332999	+	0.942927i	$0.608061\pi$
$60$	0	0
$61$	1.44172	0.184593	0.0922965	−	0.995732i	$-0.470579\pi$
$61$	1.44172	0.184593	0.0922965	+	0.995732i	$0.470579\pi$
$62$	0	0
$63$	−1.11441	−0.140403
$64$	0	0
$65$	−1.31157	−0.162680
$66$	0	0
$67$	1.32653	0.162062	0.0810310	−	0.996712i	$-0.474179\pi$
$67$	1.32653	0.162062	0.0810310	+	0.996712i	$0.474179\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	5.18601	0.615467	0.307733	−	0.951473i	$-0.400430\pi$
$71$	5.18601	0.615467	0.307733	+	0.951473i	$0.400430\pi$
$72$	0	0
$73$	2.17074	0.254066	0.127033	−	0.991898i	$-0.459455\pi$
$73$	2.17074	0.254066	0.127033	+	0.991898i	$0.459455\pi$
$74$	0	0
$75$	−12.2713	−1.41697
$76$	0	0
$77$	5.93137	0.675942
$78$	0	0
$79$	16.7829	1.88822	0.944111	−	0.329629i	$-0.106924\pi$
$79$	16.7829	1.88822	0.944111	+	0.329629i	$0.106924\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.5447	1.15743	0.578714	−	0.815531i	$-0.303555\pi$
$83$	10.5447	1.15743	0.578714	+	0.815531i	$0.303555\pi$
$84$	0	0
$85$	5.67859	0.615930
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	7.27517	0.771166	0.385583	−	0.922673i	$-0.374000\pi$
$89$	7.27517	0.771166	0.385583	+	0.922673i	$0.374000\pi$
$90$	0	0
$91$	−0.351701	−0.0368683
$92$	0	0
$93$	7.05741	0.731819
$94$	0	0
$95$	−23.4149	−2.40231
$96$	0	0
$97$	17.8471	1.81210	0.906051	−	0.423168i	$-0.139082\pi$
$97$	17.8471	1.81210	0.906051	+	0.423168i	$0.139082\pi$
$98$	0	0
$99$	−5.32243	−0.534924
$100$	0	0
$101$	2.52471	0.251218	0.125609	−	0.992080i	$-0.459912\pi$
$101$	2.52471	0.251218	0.125609	+	0.992080i	$0.459912\pi$
$102$	0	0
$103$	7.55977	0.744886	0.372443	−	0.928055i	$-0.378520\pi$
$103$	7.55977	0.744886	0.372443	+	0.928055i	$0.378520\pi$
$104$	0	0
$105$	−4.63136	−0.451974
$106$	0	0
$107$	−6.96602	−0.673431	−0.336715	−	0.941606i	$-0.609316\pi$
$107$	−6.96602	−0.673431	−0.336715	+	0.941606i	$0.609316\pi$
$108$	0	0
$109$	15.0178	1.43844	0.719221	−	0.694781i	$-0.244499\pi$
$109$	15.0178	1.43844	0.719221	+	0.694781i	$0.244499\pi$
$110$	0	0
$111$	5.05783	0.480068
$112$	0	0
$113$	−16.7693	−1.57752	−0.788759	−	0.614702i	$-0.789276\pi$
$113$	−16.7693	−1.57752	−0.788759	+	0.614702i	$0.789276\pi$
$114$	0	0
$115$	−4.15588	−0.387538
$116$	0	0
$117$	0.315594	0.0291767
$118$	0	0
$119$	1.52273	0.139588
$120$	0	0
$121$	17.3282	1.57530
$122$	0	0
$123$	−2.34509	−0.211449
$124$	0	0
$125$	−30.2189	−2.70286
$126$	0	0
$127$	10.4769	0.929676	0.464838	−	0.885396i	$-0.346113\pi$
$127$	10.4769	0.929676	0.464838	+	0.885396i	$0.346113\pi$
$128$	0	0
$129$	−4.48164	−0.394587
$130$	0	0
$131$	1.45472	0.127099	0.0635496	−	0.997979i	$-0.479758\pi$
$131$	1.45472	0.127099	0.0635496	+	0.997979i	$0.479758\pi$
$132$	0	0
$133$	−6.27876	−0.544437
$134$	0	0
$135$	4.15588	0.357681
$136$	0	0
$137$	−17.7948	−1.52031	−0.760155	−	0.649742i	$-0.774877\pi$
$137$	−17.7948	−1.52031	−0.760155	+	0.649742i	$0.774877\pi$
$138$	0	0
$139$	0.750874	0.0636883	0.0318442	−	0.999493i	$-0.489862\pi$
$139$	0.750874	0.0636883	0.0318442	+	0.999493i	$0.489862\pi$
$140$	0	0
$141$	−3.25596	−0.274201
$142$	0	0
$143$	−1.67973	−0.140466
$144$	0	0
$145$	4.15588	0.345127
$146$	0	0
$147$	5.75809	0.474919
$148$	0	0
$149$	−3.25094	−0.266327	−0.133164	−	0.991094i	$-0.542514\pi$
$149$	−3.25094	−0.266327	−0.133164	+	0.991094i	$0.542514\pi$
$150$	0	0
$151$	10.2670	0.835513	0.417756	−	0.908559i	$-0.362817\pi$
$151$	10.2670	0.835513	0.417756	+	0.908559i	$0.362817\pi$
$152$	0	0
$153$	−1.36640	−0.110467
$154$	0	0
$155$	29.3298	2.35582
$156$	0	0
$157$	−5.19176	−0.414347	−0.207174	−	0.978304i	$-0.566426\pi$
$157$	−5.19176	−0.414347	−0.207174	+	0.978304i	$0.566426\pi$
$158$	0	0
$159$	−7.64907	−0.606611
$160$	0	0
$161$	−1.11441	−0.0878278
$162$	0	0
$163$	−21.8048	−1.70788	−0.853942	−	0.520368i	$-0.825795\pi$
$163$	−21.8048	−1.70788	−0.853942	+	0.520368i	$0.825795\pi$
$164$	0	0
$165$	−22.1194	−1.72199
$166$	0	0
$167$	−13.6574	−1.05684	−0.528420	−	0.848983i	$-0.677215\pi$
$167$	−13.6574	−1.05684	−0.528420	+	0.848983i	$0.677215\pi$
$168$	0	0
$169$	−12.9004	−0.992338
$170$	0	0
$171$	5.63415	0.430854
$172$	0	0
$173$	−16.9846	−1.29131	−0.645657	−	0.763627i	$-0.723417\pi$
$173$	−16.9846	−1.29131	−0.645657	+	0.763627i	$0.723417\pi$
$174$	0	0
$175$	−13.6753	−1.03376
$176$	0	0
$177$	5.11563	0.384514
$178$	0	0
$179$	−24.7924	−1.85307	−0.926537	−	0.376203i	$-0.877229\pi$
$179$	−24.7924	−1.85307	−0.926537	+	0.376203i	$0.877229\pi$
$180$	0	0
$181$	1.46156	0.108637	0.0543184	−	0.998524i	$-0.482701\pi$
$181$	1.46156	0.108637	0.0543184	+	0.998524i	$0.482701\pi$
$182$	0	0
$183$	−1.44172	−0.106575
$184$	0	0
$185$	21.0197	1.54540
$186$	0	0
$187$	7.27256	0.531822
$188$	0	0
$189$	1.11441	0.0810614
$190$	0	0
$191$	23.7095	1.71556	0.857780	−	0.514018i	$-0.171843\pi$
$191$	23.7095	1.71556	0.857780	+	0.514018i	$0.171843\pi$
$192$	0	0
$193$	−10.7989	−0.777325	−0.388662	−	0.921380i	$-0.627063\pi$
$193$	−10.7989	−0.777325	−0.388662	+	0.921380i	$0.627063\pi$
$194$	0	0
$195$	1.31157	0.0939236
$196$	0	0
$197$	4.91211	0.349973	0.174987	−	0.984571i	$-0.444012\pi$
$197$	4.91211	0.349973	0.174987	+	0.984571i	$0.444012\pi$
$198$	0	0
$199$	−15.6593	−1.11006	−0.555030	−	0.831830i	$-0.687293\pi$
$199$	−15.6593	−1.11006	−0.555030	+	0.831830i	$0.687293\pi$
$200$	0	0
$201$	−1.32653	−0.0935665
$202$	0	0
$203$	1.11441	0.0782163
$204$	0	0
$205$	−9.74590	−0.680683
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−29.9874	−2.07427
$210$	0	0
$211$	−16.1130	−1.10927	−0.554633	−	0.832095i	$-0.687141\pi$
$211$	−16.1130	−1.10927	−0.554633	+	0.832095i	$0.687141\pi$
$212$	0	0
$213$	−5.18601	−0.355340
$214$	0	0
$215$	−18.6252	−1.27023
$216$	0	0
$217$	7.86485	0.533901
$218$	0	0
$219$	−2.17074	−0.146685
$220$	0	0
$221$	−0.431227	−0.0290075
$222$	0	0
$223$	15.7109	1.05208	0.526038	−	0.850461i	$-0.323677\pi$
$223$	15.7109	1.05208	0.526038	+	0.850461i	$0.323677\pi$
$224$	0	0
$225$	12.2713	0.818090
$226$	0	0
$227$	6.06788	0.402739	0.201370	−	0.979515i	$-0.435461\pi$
$227$	6.06788	0.402739	0.201370	+	0.979515i	$0.435461\pi$
$228$	0	0
$229$	−17.2636	−1.14081	−0.570405	−	0.821364i	$-0.693214\pi$
$229$	−17.2636	−1.14081	−0.570405	+	0.821364i	$0.693214\pi$
$230$	0	0
$231$	−5.93137	−0.390255
$232$	0	0
$233$	−13.0603	−0.855608	−0.427804	−	0.903871i	$-0.640713\pi$
$233$	−13.0603	−0.855608	−0.427804	+	0.903871i	$0.640713\pi$
$234$	0	0
$235$	−13.5314	−0.882689
$236$	0	0
$237$	−16.7829	−1.09017
$238$	0	0
$239$	5.18526	0.335406	0.167703	−	0.985838i	$-0.446365\pi$
$239$	5.18526	0.335406	0.167703	+	0.985838i	$0.446365\pi$
$240$	0	0
$241$	1.56824	0.101019	0.0505097	−	0.998724i	$-0.483915\pi$
$241$	1.56824	0.101019	0.0505097	+	0.998724i	$0.483915\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	23.9299	1.52883
$246$	0	0
$247$	1.77811	0.113138
$248$	0	0
$249$	−10.5447	−0.668241
$250$	0	0
$251$	−10.7456	−0.678257	−0.339128	−	0.940740i	$-0.610132\pi$
$251$	−10.7456	−0.678257	−0.339128	+	0.940740i	$0.610132\pi$
$252$	0	0
$253$	−5.32243	−0.334618
$254$	0	0
$255$	−5.67859	−0.355607
$256$	0	0
$257$	15.6578	0.976705	0.488352	−	0.872647i	$-0.337598\pi$
$257$	15.6578	0.976705	0.488352	+	0.872647i	$0.337598\pi$
$258$	0	0
$259$	5.63650	0.350235
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−26.4212	−1.62920	−0.814602	−	0.580020i	$-0.803045\pi$
$263$	−26.4212	−1.62920	−0.814602	+	0.580020i	$0.803045\pi$
$264$	0	0
$265$	−31.7886	−1.95276
$266$	0	0
$267$	−7.27517	−0.445233
$268$	0	0
$269$	−4.75677	−0.290025	−0.145013	−	0.989430i	$-0.546322\pi$
$269$	−4.75677	−0.290025	−0.145013	+	0.989430i	$0.546322\pi$
$270$	0	0
$271$	28.0807	1.70578	0.852889	−	0.522092i	$-0.174848\pi$
$271$	28.0807	1.70578	0.852889	+	0.522092i	$0.174848\pi$
$272$	0	0
$273$	0.351701	0.0212859
$274$	0	0
$275$	−65.3134	−3.93855
$276$	0	0
$277$	−4.76867	−0.286522	−0.143261	−	0.989685i	$-0.545759\pi$
$277$	−4.76867	−0.286522	−0.143261	+	0.989685i	$0.545759\pi$
$278$	0	0
$279$	−7.05741	−0.422516
$280$	0	0
$281$	9.24418	0.551462	0.275731	−	0.961235i	$-0.411080\pi$
$281$	9.24418	0.551462	0.275731	+	0.961235i	$0.411080\pi$
$282$	0	0
$283$	1.38881	0.0825564	0.0412782	−	0.999148i	$-0.486857\pi$
$283$	1.38881	0.0825564	0.0412782	+	0.999148i	$0.486857\pi$
$284$	0	0
$285$	23.4149	1.38698
$286$	0	0
$287$	−2.61339	−0.154263
$288$	0	0
$289$	−15.1330	−0.890174
$290$	0	0
$291$	−17.8471	−1.04622
$292$	0	0
$293$	16.1908	0.945874	0.472937	−	0.881096i	$-0.343194\pi$
$293$	16.1908	0.945874	0.472937	+	0.881096i	$0.343194\pi$
$294$	0	0
$295$	21.2600	1.23780
$296$	0	0
$297$	5.32243	0.308839
$298$	0	0
$299$	0.315594	0.0182513
$300$	0	0
$301$	−4.99439	−0.287872
$302$	0	0
$303$	−2.52471	−0.145041
$304$	0	0
$305$	−5.99161	−0.343078
$306$	0	0
$307$	−27.3469	−1.56077	−0.780386	−	0.625298i	$-0.784977\pi$
$307$	−27.3469	−1.56077	−0.780386	+	0.625298i	$0.784977\pi$
$308$	0	0
$309$	−7.55977	−0.430060
$310$	0	0
$311$	3.23554	0.183471	0.0917353	−	0.995783i	$-0.470759\pi$
$311$	3.23554	0.183471	0.0917353	+	0.995783i	$0.470759\pi$
$312$	0	0
$313$	15.5299	0.877804	0.438902	−	0.898535i	$-0.355368\pi$
$313$	15.5299	0.877804	0.438902	+	0.898535i	$0.355368\pi$
$314$	0	0
$315$	4.63136	0.260947
$316$	0	0
$317$	19.2374	1.08048	0.540241	−	0.841510i	$-0.318333\pi$
$317$	19.2374	1.08048	0.540241	+	0.841510i	$0.318333\pi$
$318$	0	0
$319$	5.32243	0.297999
$320$	0	0
$321$	6.96602	0.388805
$322$	0	0
$323$	−7.69850	−0.428356
$324$	0	0
$325$	3.87277	0.214822
$326$	0	0
$327$	−15.0178	−0.830485
$328$	0	0
$329$	−3.62847	−0.200044
$330$	0	0
$331$	11.7753	0.647228	0.323614	−	0.946189i	$-0.395102\pi$
$331$	11.7753	0.647228	0.323614	+	0.946189i	$0.395102\pi$
$332$	0	0
$333$	−5.05783	−0.277167
$334$	0	0
$335$	−5.51292	−0.301203
$336$	0	0
$337$	−8.90018	−0.484824	−0.242412	−	0.970173i	$-0.577939\pi$
$337$	−8.90018	−0.484824	−0.242412	+	0.970173i	$0.577939\pi$
$338$	0	0
$339$	16.7693	0.910781
$340$	0	0
$341$	37.5626	2.03413
$342$	0	0
$343$	14.2177	0.767686
$344$	0	0
$345$	4.15588	0.223745
$346$	0	0
$347$	−21.1007	−1.13274	−0.566371	−	0.824150i	$-0.691653\pi$
$347$	−21.1007	−1.13274	−0.566371	+	0.824150i	$0.691653\pi$
$348$	0	0
$349$	0.752308	0.0402701	0.0201351	−	0.999797i	$-0.493590\pi$
$349$	0.752308	0.0402701	0.0201351	+	0.999797i	$0.493590\pi$
$350$	0	0
$351$	−0.315594	−0.0168452
$352$	0	0
$353$	−1.23784	−0.0658834	−0.0329417	−	0.999457i	$-0.510488\pi$
$353$	−1.23784	−0.0658834	−0.0329417	+	0.999457i	$0.510488\pi$
$354$	0	0
$355$	−21.5525	−1.14389
$356$	0	0
$357$	−1.52273	−0.0805913
$358$	0	0
$359$	−10.5446	−0.556525	−0.278263	−	0.960505i	$-0.589759\pi$
$359$	−10.5446	−0.556525	−0.278263	+	0.960505i	$0.589759\pi$
$360$	0	0
$361$	12.7437	0.670719
$362$	0	0
$363$	−17.3282	−0.909497
$364$	0	0
$365$	−9.02135	−0.472199
$366$	0	0
$367$	−16.2585	−0.848685	−0.424343	−	0.905502i	$-0.639495\pi$
$367$	−16.2585	−0.848685	−0.424343	+	0.905502i	$0.639495\pi$
$368$	0	0
$369$	2.34509	0.122080
$370$	0	0
$371$	−8.52421	−0.442555
$372$	0	0
$373$	11.3203	0.586144	0.293072	−	0.956090i	$-0.405322\pi$
$373$	11.3203	0.586144	0.293072	+	0.956090i	$0.405322\pi$
$374$	0	0
$375$	30.2189	1.56050
$376$	0	0
$377$	−0.315594	−0.0162539
$378$	0	0
$379$	32.7924	1.68443	0.842217	−	0.539138i	$-0.181250\pi$
$379$	32.7924	1.68443	0.842217	+	0.539138i	$0.181250\pi$
$380$	0	0
$381$	−10.4769	−0.536749
$382$	0	0
$383$	25.3286	1.29423	0.647114	−	0.762393i	$-0.275976\pi$
$383$	25.3286	1.29423	0.647114	+	0.762393i	$0.275976\pi$
$384$	0	0
$385$	−24.6501	−1.25628
$386$	0	0
$387$	4.48164	0.227815
$388$	0	0
$389$	3.40417	0.172598	0.0862991	−	0.996269i	$-0.472496\pi$
$389$	3.40417	0.172598	0.0862991	+	0.996269i	$0.472496\pi$
$390$	0	0
$391$	−1.36640	−0.0691017
$392$	0	0
$393$	−1.45472	−0.0733807
$394$	0	0
$395$	−69.7476	−3.50939
$396$	0	0
$397$	14.9864	0.752145	0.376073	−	0.926590i	$-0.377274\pi$
$397$	14.9864	0.752145	0.376073	+	0.926590i	$0.377274\pi$
$398$	0	0
$399$	6.27876	0.314331
$400$	0	0
$401$	30.3023	1.51323	0.756613	−	0.653863i	$-0.226853\pi$
$401$	30.3023	1.51323	0.756613	+	0.653863i	$0.226853\pi$
$402$	0	0
$403$	−2.22728	−0.110949
$404$	0	0
$405$	−4.15588	−0.206507
$406$	0	0
$407$	26.9199	1.33437
$408$	0	0
$409$	−16.9216	−0.836717	−0.418359	−	0.908282i	$-0.637394\pi$
$409$	−16.9216	−0.836717	−0.418359	+	0.908282i	$0.637394\pi$
$410$	0	0
$411$	17.7948	0.877752
$412$	0	0
$413$	5.70091	0.280524
$414$	0	0
$415$	−43.8224	−2.15116
$416$	0	0
$417$	−0.750874	−0.0367705
$418$	0	0
$419$	−18.5534	−0.906394	−0.453197	−	0.891410i	$-0.649717\pi$
$419$	−18.5534	−0.906394	−0.453197	+	0.891410i	$0.649717\pi$
$420$	0	0
$421$	−30.7341	−1.49789	−0.748944	−	0.662634i	$-0.769439\pi$
$421$	−30.7341	−1.49789	−0.748944	+	0.662634i	$0.769439\pi$
$422$	0	0
$423$	3.25596	0.158310
$424$	0	0
$425$	−16.7676	−0.813346
$426$	0	0
$427$	−1.60666	−0.0777519
$428$	0	0
$429$	1.67973	0.0810980
$430$	0	0
$431$	35.0770	1.68960	0.844799	−	0.535083i	$-0.179720\pi$
$431$	35.0770	1.68960	0.844799	+	0.535083i	$0.179720\pi$
$432$	0	0
$433$	−27.1107	−1.30286	−0.651428	−	0.758711i	$-0.725830\pi$
$433$	−27.1107	−1.30286	−0.651428	+	0.758711i	$0.725830\pi$
$434$	0	0
$435$	−4.15588	−0.199259
$436$	0	0
$437$	5.63415	0.269518
$438$	0	0
$439$	−17.4657	−0.833594	−0.416797	−	0.909000i	$-0.636847\pi$
$439$	−17.4657	−0.833594	−0.416797	+	0.909000i	$0.636847\pi$
$440$	0	0
$441$	−5.75809	−0.274195
$442$	0	0
$443$	−7.96682	−0.378515	−0.189258	−	0.981927i	$-0.560608\pi$
$443$	−7.96682	−0.378515	−0.189258	+	0.981927i	$0.560608\pi$
$444$	0	0
$445$	−30.2347	−1.43326
$446$	0	0
$447$	3.25094	0.153764
$448$	0	0
$449$	3.54966	0.167519	0.0837594	−	0.996486i	$-0.473307\pi$
$449$	3.54966	0.167519	0.0837594	+	0.996486i	$0.473307\pi$
$450$	0	0
$451$	−12.4816	−0.587733
$452$	0	0
$453$	−10.2670	−0.482383
$454$	0	0
$455$	1.46163	0.0685222
$456$	0	0
$457$	0.905773	0.0423703	0.0211851	−	0.999776i	$-0.493256\pi$
$457$	0.905773	0.0423703	0.0211851	+	0.999776i	$0.493256\pi$
$458$	0	0
$459$	1.36640	0.0637780
$460$	0	0
$461$	−7.28459	−0.339277	−0.169639	−	0.985506i	$-0.554260\pi$
$461$	−7.28459	−0.339277	−0.169639	+	0.985506i	$0.554260\pi$
$462$	0	0
$463$	−5.99214	−0.278478	−0.139239	−	0.990259i	$-0.544466\pi$
$463$	−5.99214	−0.278478	−0.139239	+	0.990259i	$0.544466\pi$
$464$	0	0
$465$	−29.3298	−1.36014
$466$	0	0
$467$	−28.8181	−1.33354	−0.666772	−	0.745262i	$-0.732324\pi$
$467$	−28.8181	−1.33354	−0.666772	+	0.745262i	$0.732324\pi$
$468$	0	0
$469$	−1.47830	−0.0682617
$470$	0	0
$471$	5.19176	0.239223
$472$	0	0
$473$	−23.8532	−1.09677
$474$	0	0
$475$	69.1386	3.17230
$476$	0	0
$477$	7.64907	0.350227
$478$	0	0
$479$	−1.47774	−0.0675197	−0.0337598	−	0.999430i	$-0.510748\pi$
$479$	−1.47774	−0.0675197	−0.0337598	+	0.999430i	$0.510748\pi$
$480$	0	0
$481$	−1.59622	−0.0727814
$482$	0	0
$483$	1.11441	0.0507074
$484$	0	0
$485$	−74.1706	−3.36791
$486$	0	0
$487$	31.2807	1.41746	0.708732	−	0.705478i	$-0.249268\pi$
$487$	31.2807	1.41746	0.708732	+	0.705478i	$0.249268\pi$
$488$	0	0
$489$	21.8048	0.986047
$490$	0	0
$491$	16.5707	0.747827	0.373913	−	0.927464i	$-0.378016\pi$
$491$	16.5707	0.747827	0.373913	+	0.927464i	$0.378016\pi$
$492$	0	0
$493$	1.36640	0.0615395
$494$	0	0
$495$	22.1194	0.994192
$496$	0	0
$497$	−5.77935	−0.259239
$498$	0	0
$499$	−36.5634	−1.63680	−0.818401	−	0.574648i	$-0.805139\pi$
$499$	−36.5634	−1.63680	−0.818401	+	0.574648i	$0.805139\pi$
$500$	0	0
$501$	13.6574	0.610167
$502$	0	0
$503$	−11.0931	−0.494618	−0.247309	−	0.968937i	$-0.579546\pi$
$503$	−11.0931	−0.494618	−0.247309	+	0.968937i	$0.579546\pi$
$504$	0	0
$505$	−10.4924	−0.466906
$506$	0	0
$507$	12.9004	0.572927
$508$	0	0
$509$	17.2458	0.764406	0.382203	−	0.924078i	$-0.375166\pi$
$509$	17.2458	0.764406	0.382203	+	0.924078i	$0.375166\pi$
$510$	0	0
$511$	−2.41910	−0.107015
$512$	0	0
$513$	−5.63415	−0.248754
$514$	0	0
$515$	−31.4175	−1.38442
$516$	0	0
$517$	−17.3296	−0.762155
$518$	0	0
$519$	16.9846	0.745541
$520$	0	0
$521$	−34.7972	−1.52449	−0.762247	−	0.647286i	$-0.775904\pi$
$521$	−34.7972	−1.52449	−0.762247	+	0.647286i	$0.775904\pi$
$522$	0	0
$523$	35.5395	1.55404	0.777018	−	0.629479i	$-0.216732\pi$
$523$	35.5395	1.55404	0.777018	+	0.629479i	$0.216732\pi$
$524$	0	0
$525$	13.6753	0.596840
$526$	0	0
$527$	9.64324	0.420066
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.11563	−0.221999
$532$	0	0
$533$	0.740095	0.0320571
$534$	0	0
$535$	28.9500	1.25162
$536$	0	0
$537$	24.7924	1.06987
$538$	0	0
$539$	30.6470	1.32006
$540$	0	0
$541$	−37.3517	−1.60587	−0.802936	−	0.596065i	$-0.796730\pi$
$541$	−37.3517	−1.60587	−0.802936	+	0.596065i	$0.796730\pi$
$542$	0	0
$543$	−1.46156	−0.0627215
$544$	0	0
$545$	−62.4121	−2.67344
$546$	0	0
$547$	−37.1011	−1.58633	−0.793165	−	0.609007i	$-0.791568\pi$
$547$	−37.1011	−1.58633	−0.793165	+	0.609007i	$0.791568\pi$
$548$	0	0
$549$	1.44172	0.0615310
$550$	0	0
$551$	−5.63415	−0.240023
$552$	0	0
$553$	−18.7030	−0.795333
$554$	0	0
$555$	−21.0197	−0.892238
$556$	0	0
$557$	−37.2436	−1.57806	−0.789030	−	0.614355i	$-0.789416\pi$
$557$	−37.2436	−1.57806	−0.789030	+	0.614355i	$0.789416\pi$
$558$	0	0
$559$	1.41438	0.0598219
$560$	0	0
$561$	−7.27256	−0.307048
$562$	0	0
$563$	21.7473	0.916540	0.458270	−	0.888813i	$-0.348469\pi$
$563$	21.7473	0.916540	0.458270	+	0.888813i	$0.348469\pi$
$564$	0	0
$565$	69.6910	2.93192
$566$	0	0
$567$	−1.11441	−0.0468008
$568$	0	0
$569$	2.87553	0.120548	0.0602741	−	0.998182i	$-0.480803\pi$
$569$	2.87553	0.120548	0.0602741	+	0.998182i	$0.480803\pi$
$570$	0	0
$571$	11.8603	0.496339	0.248170	−	0.968717i	$-0.420171\pi$
$571$	11.8603	0.496339	0.248170	+	0.968717i	$0.420171\pi$
$572$	0	0
$573$	−23.7095	−0.990479
$574$	0	0
$575$	12.2713	0.511751
$576$	0	0
$577$	13.2791	0.552818	0.276409	−	0.961040i	$-0.410856\pi$
$577$	13.2791	0.552818	0.276409	+	0.961040i	$0.410856\pi$
$578$	0	0
$579$	10.7989	0.448789
$580$	0	0
$581$	−11.7511	−0.487517
$582$	0	0
$583$	−40.7116	−1.68610
$584$	0	0
$585$	−1.31157	−0.0542268
$586$	0	0
$587$	−13.3679	−0.551753	−0.275877	−	0.961193i	$-0.588968\pi$
$587$	−13.3679	−0.551753	−0.275877	+	0.961193i	$0.588968\pi$
$588$	0	0
$589$	−39.7625	−1.63839
$590$	0	0
$591$	−4.91211	−0.202057
$592$	0	0
$593$	4.50293	0.184913	0.0924566	−	0.995717i	$-0.470528\pi$
$593$	4.50293	0.184913	0.0924566	+	0.995717i	$0.470528\pi$
$594$	0	0
$595$	−6.32828	−0.259434
$596$	0	0
$597$	15.6593	0.640893
$598$	0	0
$599$	7.84271	0.320445	0.160222	−	0.987081i	$-0.448779\pi$
$599$	7.84271	0.320445	0.160222	+	0.987081i	$0.448779\pi$
$600$	0	0
$601$	−8.22447	−0.335483	−0.167742	−	0.985831i	$-0.553647\pi$
$601$	−8.22447	−0.335483	−0.167742	+	0.985831i	$0.553647\pi$
$602$	0	0
$603$	1.32653	0.0540207
$604$	0	0
$605$	−72.0141	−2.92779
$606$	0	0
$607$	38.0418	1.54407	0.772035	−	0.635580i	$-0.219239\pi$
$607$	38.0418	1.54407	0.772035	+	0.635580i	$0.219239\pi$
$608$	0	0
$609$	−1.11441	−0.0451582
$610$	0	0
$611$	1.02756	0.0415707
$612$	0	0
$613$	−9.57247	−0.386628	−0.193314	−	0.981137i	$-0.561924\pi$
$613$	−9.57247	−0.386628	−0.193314	+	0.981137i	$0.561924\pi$
$614$	0	0
$615$	9.74590	0.392993
$616$	0	0
$617$	21.5698	0.868366	0.434183	−	0.900825i	$-0.357037\pi$
$617$	21.5698	0.868366	0.434183	+	0.900825i	$0.357037\pi$
$618$	0	0
$619$	30.3002	1.21787	0.608934	−	0.793221i	$-0.291598\pi$
$619$	30.3002	1.21787	0.608934	+	0.793221i	$0.291598\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−8.10752	−0.324821
$624$	0	0
$625$	64.2293	2.56917
$626$	0	0
$627$	29.9874	1.19758
$628$	0	0
$629$	6.91101	0.275560
$630$	0	0
$631$	−1.11720	−0.0444749	−0.0222374	−	0.999753i	$-0.507079\pi$
$631$	−1.11720	−0.0444749	−0.0222374	+	0.999753i	$0.507079\pi$
$632$	0	0
$633$	16.1130	0.640435
$634$	0	0
$635$	−43.5408	−1.72786
$636$	0	0
$637$	−1.81722	−0.0720009
$638$	0	0
$639$	5.18601	0.205156
$640$	0	0
$641$	−7.79220	−0.307774	−0.153887	−	0.988088i	$-0.549179\pi$
$641$	−7.79220	−0.307774	−0.153887	+	0.988088i	$0.549179\pi$
$642$	0	0
$643$	35.8453	1.41360	0.706800	−	0.707413i	$-0.250138\pi$
$643$	35.8453	1.41360	0.706800	+	0.707413i	$0.250138\pi$
$644$	0	0
$645$	18.6252	0.733365
$646$	0	0
$647$	13.9579	0.548742	0.274371	−	0.961624i	$-0.411530\pi$
$647$	13.9579	0.548742	0.274371	+	0.961624i	$0.411530\pi$
$648$	0	0
$649$	27.2276	1.06878
$650$	0	0
$651$	−7.86485	−0.308248
$652$	0	0
$653$	−9.11600	−0.356737	−0.178368	−	0.983964i	$-0.557082\pi$
$653$	−9.11600	−0.356737	−0.178368	+	0.983964i	$0.557082\pi$
$654$	0	0
$655$	−6.04563	−0.236222
$656$	0	0
$657$	2.17074	0.0846887
$658$	0	0
$659$	−33.7781	−1.31581	−0.657904	−	0.753101i	$-0.728557\pi$
$659$	−33.7781	−1.31581	−0.657904	+	0.753101i	$0.728557\pi$
$660$	0	0
$661$	−28.6418	−1.11404	−0.557018	−	0.830500i	$-0.688055\pi$
$661$	−28.6418	−1.11404	−0.557018	+	0.830500i	$0.688055\pi$
$662$	0	0
$663$	0.431227	0.0167475
$664$	0	0
$665$	26.0938	1.01187
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−15.7109	−0.607417
$670$	0	0
$671$	−7.67344	−0.296230
$672$	0	0
$673$	20.2837	0.781879	0.390939	−	0.920416i	$-0.372150\pi$
$673$	20.2837	0.781879	0.390939	+	0.920416i	$0.372150\pi$
$674$	0	0
$675$	−12.2713	−0.472324
$676$	0	0
$677$	2.54900	0.0979659	0.0489830	−	0.998800i	$-0.484402\pi$
$677$	2.54900	0.0979659	0.0489830	+	0.998800i	$0.484402\pi$
$678$	0	0
$679$	−19.8890	−0.763271
$680$	0	0
$681$	−6.06788	−0.232522
$682$	0	0
$683$	9.01416	0.344917	0.172459	−	0.985017i	$-0.444829\pi$
$683$	9.01416	0.344917	0.172459	+	0.985017i	$0.444829\pi$
$684$	0	0
$685$	73.9530	2.82560
$686$	0	0
$687$	17.2636	0.658646
$688$	0	0
$689$	2.41400	0.0919662
$690$	0	0
$691$	−4.03353	−0.153443	−0.0767214	−	0.997053i	$-0.524445\pi$
$691$	−4.03353	−0.153443	−0.0767214	+	0.997053i	$0.524445\pi$
$692$	0	0
$693$	5.93137	0.225314
$694$	0	0
$695$	−3.12054	−0.118369
$696$	0	0
$697$	−3.20432	−0.121372
$698$	0	0
$699$	13.0603	0.493986
$700$	0	0
$701$	−25.9923	−0.981717	−0.490859	−	0.871239i	$-0.663317\pi$
$701$	−25.9923	−0.981717	−0.490859	+	0.871239i	$0.663317\pi$
$702$	0	0
$703$	−28.4966	−1.07477
$704$	0	0
$705$	13.5314	0.509621
$706$	0	0
$707$	−2.81356	−0.105815
$708$	0	0
$709$	−18.6909	−0.701951	−0.350975	−	0.936385i	$-0.614150\pi$
$709$	−18.6909	−0.701951	−0.350975	+	0.936385i	$0.614150\pi$
$710$	0	0
$711$	16.7829	0.629407
$712$	0	0
$713$	−7.05741	−0.264302
$714$	0	0
$715$	6.98075	0.261065
$716$	0	0
$717$	−5.18526	−0.193647
$718$	0	0
$719$	−44.0786	−1.64386	−0.821928	−	0.569591i	$-0.807102\pi$
$719$	−44.0786	−1.64386	−0.821928	+	0.569591i	$0.807102\pi$
$720$	0	0
$721$	−8.42468	−0.313752
$722$	0	0
$723$	−1.56824	−0.0583236
$724$	0	0
$725$	−12.2713	−0.455746
$726$	0	0
$727$	−27.0431	−1.00297	−0.501487	−	0.865165i	$-0.667213\pi$
$727$	−27.0431	−1.00297	−0.501487	+	0.865165i	$0.667213\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.12371	−0.226494
$732$	0	0
$733$	−11.0106	−0.406685	−0.203343	−	0.979108i	$-0.565181\pi$
$733$	−11.0106	−0.406685	−0.203343	+	0.979108i	$0.565181\pi$
$734$	0	0
$735$	−23.9299	−0.882669
$736$	0	0
$737$	−7.06039	−0.260073
$738$	0	0
$739$	−4.77272	−0.175567	−0.0877836	−	0.996140i	$-0.527978\pi$
$739$	−4.77272	−0.175567	−0.0877836	+	0.996140i	$0.527978\pi$
$740$	0	0
$741$	−1.77811	−0.0653203
$742$	0	0
$743$	25.1948	0.924307	0.462153	−	0.886800i	$-0.347077\pi$
$743$	25.1948	0.924307	0.462153	+	0.886800i	$0.347077\pi$
$744$	0	0
$745$	13.5105	0.494987
$746$	0	0
$747$	10.5447	0.385809
$748$	0	0
$749$	7.76301	0.283654
$750$	0	0
$751$	20.4836	0.747457	0.373729	−	0.927538i	$-0.378079\pi$
$751$	20.4836	0.747457	0.373729	+	0.927538i	$0.378079\pi$
$752$	0	0
$753$	10.7456	0.391592
$754$	0	0
$755$	−42.6682	−1.55286
$756$	0	0
$757$	−28.9761	−1.05315	−0.526577	−	0.850127i	$-0.676525\pi$
$757$	−28.9761	−1.05315	−0.526577	+	0.850127i	$0.676525\pi$
$758$	0	0
$759$	5.32243	0.193192
$760$	0	0
$761$	−34.4284	−1.24803	−0.624015	−	0.781413i	$-0.714499\pi$
$761$	−34.4284	−1.24803	−0.624015	+	0.781413i	$0.714499\pi$
$762$	0	0
$763$	−16.7360	−0.605883
$764$	0	0
$765$	5.67859	0.205310
$766$	0	0
$767$	−1.61446	−0.0582949
$768$	0	0
$769$	−14.7722	−0.532699	−0.266350	−	0.963876i	$-0.585818\pi$
$769$	−14.7722	−0.532699	−0.266350	+	0.963876i	$0.585818\pi$
$770$	0	0
$771$	−15.6578	−0.563901
$772$	0	0
$773$	32.7055	1.17634	0.588168	−	0.808739i	$-0.299849\pi$
$773$	32.7055	1.17634	0.588168	+	0.808739i	$0.299849\pi$
$774$	0	0
$775$	−86.6040	−3.11091
$776$	0	0
$777$	−5.63650	−0.202208
$778$	0	0
$779$	13.2126	0.473389
$780$	0	0
$781$	−27.6022	−0.987684
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	21.5763	0.770092
$786$	0	0
$787$	26.9355	0.960147	0.480074	−	0.877228i	$-0.340610\pi$
$787$	26.9355	0.960147	0.480074	+	0.877228i	$0.340610\pi$
$788$	0	0
$789$	26.4212	0.940621
$790$	0	0
$791$	18.6878	0.664463
$792$	0	0
$793$	0.454998	0.0161574
$794$	0	0
$795$	31.7886	1.12743
$796$	0	0
$797$	24.6641	0.873648	0.436824	−	0.899547i	$-0.356103\pi$
$797$	24.6641	0.873648	0.436824	+	0.899547i	$0.356103\pi$
$798$	0	0
$799$	−4.44894	−0.157392
$800$	0	0
$801$	7.27517	0.257055
$802$	0	0
$803$	−11.5536	−0.407718
$804$	0	0
$805$	4.63136	0.163234
$806$	0	0
$807$	4.75677	0.167446
$808$	0	0
$809$	−1.47151	−0.0517354	−0.0258677	−	0.999665i	$-0.508235\pi$
$809$	−1.47151	−0.0517354	−0.0258677	+	0.999665i	$0.508235\pi$
$810$	0	0
$811$	−1.29574	−0.0454997	−0.0227499	−	0.999741i	$-0.507242\pi$
$811$	−1.29574	−0.0454997	−0.0227499	+	0.999741i	$0.507242\pi$
$812$	0	0
$813$	−28.0807	−0.984832
$814$	0	0
$815$	90.6182	3.17422
$816$	0	0
$817$	25.2502	0.883394
$818$	0	0
$819$	−0.351701	−0.0122894
$820$	0	0
$821$	21.4366	0.748142	0.374071	−	0.927400i	$-0.377962\pi$
$821$	21.4366	0.748142	0.374071	+	0.927400i	$0.377962\pi$
$822$	0	0
$823$	19.9990	0.697120	0.348560	−	0.937286i	$-0.386671\pi$
$823$	19.9990	0.697120	0.348560	+	0.937286i	$0.386671\pi$
$824$	0	0
$825$	65.3134	2.27392
$826$	0	0
$827$	22.0401	0.766409	0.383205	−	0.923663i	$-0.374820\pi$
$827$	22.0401	0.766409	0.383205	+	0.923663i	$0.374820\pi$
$828$	0	0
$829$	−38.6535	−1.34249	−0.671245	−	0.741235i	$-0.734240\pi$
$829$	−38.6535	−1.34249	−0.671245	+	0.741235i	$0.734240\pi$
$830$	0	0
$831$	4.76867	0.165423
$832$	0	0
$833$	7.86784	0.272605
$834$	0	0
$835$	56.7585	1.96421
$836$	0	0
$837$	7.05741	0.243940
$838$	0	0
$839$	−4.43195	−0.153008	−0.0765039	−	0.997069i	$-0.524376\pi$
$839$	−4.43195	−0.153008	−0.0765039	+	0.997069i	$0.524376\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−9.24418	−0.318387
$844$	0	0
$845$	53.6125	1.84433
$846$	0	0
$847$	−19.3108	−0.663526
$848$	0	0
$849$	−1.38881	−0.0476640
$850$	0	0
$851$	−5.05783	−0.173380
$852$	0	0
$853$	−16.3932	−0.561291	−0.280646	−	0.959811i	$-0.590549\pi$
$853$	−16.3932	−0.561291	−0.280646	+	0.959811i	$0.590549\pi$
$854$	0	0
$855$	−23.4149	−0.800772
$856$	0	0
$857$	4.15179	0.141822	0.0709112	−	0.997483i	$-0.477409\pi$
$857$	4.15179	0.141822	0.0709112	+	0.997483i	$0.477409\pi$
$858$	0	0
$859$	30.0383	1.02489	0.512447	−	0.858719i	$-0.328739\pi$
$859$	30.0383	1.02489	0.512447	+	0.858719i	$0.328739\pi$
$860$	0	0
$861$	2.61339	0.0890640
$862$	0	0
$863$	24.1245	0.821208	0.410604	−	0.911814i	$-0.365318\pi$
$863$	24.1245	0.821208	0.410604	+	0.911814i	$0.365318\pi$
$864$	0	0
$865$	70.5860	2.39999
$866$	0	0
$867$	15.1330	0.513942
$868$	0	0
$869$	−89.3257	−3.03017
$870$	0	0
$871$	0.418647	0.0141853
$872$	0	0
$873$	17.8471	0.604034
$874$	0	0
$875$	33.6762	1.13846
$876$	0	0
$877$	43.4529	1.46730	0.733650	−	0.679528i	$-0.237815\pi$
$877$	43.4529	1.46730	0.733650	+	0.679528i	$0.237815\pi$
$878$	0	0
$879$	−16.1908	−0.546100
$880$	0	0
$881$	10.3229	0.347786	0.173893	−	0.984765i	$-0.444365\pi$
$881$	10.3229	0.347786	0.173893	+	0.984765i	$0.444365\pi$
$882$	0	0
$883$	30.1615	1.01502	0.507508	−	0.861647i	$-0.330567\pi$
$883$	30.1615	1.01502	0.507508	+	0.861647i	$0.330567\pi$
$884$	0	0
$885$	−21.2600	−0.714646
$886$	0	0
$887$	−35.0652	−1.17737	−0.588687	−	0.808361i	$-0.700355\pi$
$887$	−35.0652	−1.17737	−0.588687	+	0.808361i	$0.700355\pi$
$888$	0	0
$889$	−11.6756	−0.391587
$890$	0	0
$891$	−5.32243	−0.178308
$892$	0	0
$893$	18.3446	0.613877
$894$	0	0
$895$	103.034	3.44406
$896$	0	0
$897$	−0.315594	−0.0105374
$898$	0	0
$899$	7.05741	0.235378
$900$	0	0
$901$	−10.4517	−0.348196
$902$	0	0
$903$	4.99439	0.166203
$904$	0	0
$905$	−6.07406	−0.201909
$906$	0	0
$907$	18.5729	0.616704	0.308352	−	0.951272i	$-0.400223\pi$
$907$	18.5729	0.616704	0.308352	+	0.951272i	$0.400223\pi$
$908$	0	0
$909$	2.52471	0.0837394
$910$	0	0
$911$	25.8275	0.855704	0.427852	−	0.903849i	$-0.359270\pi$
$911$	25.8275	0.855704	0.427852	+	0.903849i	$0.359270\pi$
$912$	0	0
$913$	−56.1232	−1.85741
$914$	0	0
$915$	5.99161	0.198076
$916$	0	0
$917$	−1.62115	−0.0535351
$918$	0	0
$919$	−25.4838	−0.840632	−0.420316	−	0.907378i	$-0.638081\pi$
$919$	−25.4838	−0.840632	−0.420316	+	0.907378i	$0.638081\pi$
$920$	0	0
$921$	27.3469	0.901112
$922$	0	0
$923$	1.63668	0.0538718
$924$	0	0
$925$	−62.0664	−2.04073
$926$	0	0
$927$	7.55977	0.248295
$928$	0	0
$929$	9.01630	0.295815	0.147908	−	0.989001i	$-0.452746\pi$
$929$	9.01630	0.295815	0.147908	+	0.989001i	$0.452746\pi$
$930$	0	0
$931$	−32.4419	−1.06324
$932$	0	0
$933$	−3.23554	−0.105927
$934$	0	0
$935$	−30.2239	−0.988427
$936$	0	0
$937$	30.2582	0.988491	0.494245	−	0.869322i	$-0.335444\pi$
$937$	30.2582	0.988491	0.494245	+	0.869322i	$0.335444\pi$
$938$	0	0
$939$	−15.5299	−0.506800
$940$	0	0
$941$	−22.4808	−0.732852	−0.366426	−	0.930447i	$-0.619419\pi$
$941$	−22.4808	−0.732852	−0.366426	+	0.930447i	$0.619419\pi$
$942$	0	0
$943$	2.34509	0.0763665
$944$	0	0
$945$	−4.63136	−0.150658
$946$	0	0
$947$	−22.8549	−0.742684	−0.371342	−	0.928496i	$-0.621102\pi$
$947$	−22.8549	−0.742684	−0.371342	+	0.928496i	$0.621102\pi$
$948$	0	0
$949$	0.685073	0.0222384
$950$	0	0
$951$	−19.2374	−0.623817
$952$	0	0
$953$	−27.3766	−0.886814	−0.443407	−	0.896320i	$-0.646230\pi$
$953$	−27.3766	−0.886814	−0.443407	+	0.896320i	$0.646230\pi$
$954$	0	0
$955$	−98.5339	−3.18848
$956$	0	0
$957$	−5.32243	−0.172050
$958$	0	0
$959$	19.8307	0.640366
$960$	0	0
$961$	18.8071	0.606679
$962$	0	0
$963$	−6.96602	−0.224477
$964$	0	0
$965$	44.8791	1.44471
$966$	0	0
$967$	51.4022	1.65298	0.826492	−	0.562949i	$-0.190333\pi$
$967$	51.4022	1.65298	0.826492	+	0.562949i	$0.190333\pi$
$968$	0	0
$969$	7.69850	0.247311
$970$	0	0
$971$	16.1002	0.516679	0.258340	−	0.966054i	$-0.416825\pi$
$971$	16.1002	0.516679	0.258340	+	0.966054i	$0.416825\pi$
$972$	0	0
$973$	−0.836782	−0.0268260
$974$	0	0
$975$	−3.87277	−0.124028
$976$	0	0
$977$	23.1830	0.741689	0.370845	−	0.928695i	$-0.379068\pi$
$977$	23.1830	0.741689	0.370845	+	0.928695i	$0.379068\pi$
$978$	0	0
$979$	−38.7216	−1.23755
$980$	0	0
$981$	15.0178	0.479481
$982$	0	0
$983$	20.7740	0.662588	0.331294	−	0.943528i	$-0.392515\pi$
$983$	20.7740	0.662588	0.331294	+	0.943528i	$0.392515\pi$
$984$	0	0
$985$	−20.4142	−0.650449
$986$	0	0
$987$	3.62847	0.115496
$988$	0	0
$989$	4.48164	0.142508
$990$	0	0
$991$	−53.1822	−1.68939	−0.844694	−	0.535250i	$-0.820217\pi$
$991$	−53.1822	−1.68939	−0.844694	+	0.535250i	$0.820217\pi$
$992$	0	0
$993$	−11.7753	−0.373677
$994$	0	0
$995$	65.0783	2.06312
$996$	0	0
$997$	39.0853	1.23785	0.618923	−	0.785452i	$-0.287569\pi$
$997$	39.0853	1.23785	0.618923	+	0.785452i	$0.287569\pi$
$998$	0	0
$999$	5.05783	0.160023

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.g.1.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.g.1.1	✓	12	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.15588 q^{5} -1.11441 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.15588 q^{5} -1.11441 q^{7} +1.00000 q^{9} -5.32243 q^{11} +0.315594 q^{13} +4.15588 q^{15} -1.36640 q^{17} +5.63415 q^{19} +1.11441 q^{21} +1.00000 q^{23} +12.2713 q^{25} -1.00000 q^{27} -1.00000 q^{29} -7.05741 q^{31} +5.32243 q^{33} +4.63136 q^{35} -5.05783 q^{37} -0.315594 q^{39} +2.34509 q^{41} +4.48164 q^{43} -4.15588 q^{45} +3.25596 q^{47} -5.75809 q^{49} +1.36640 q^{51} +7.64907 q^{53} +22.1194 q^{55} -5.63415 q^{57} -5.11563 q^{59} +1.44172 q^{61} -1.11441 q^{63} -1.31157 q^{65} +1.32653 q^{67} -1.00000 q^{69} +5.18601 q^{71} +2.17074 q^{73} -12.2713 q^{75} +5.93137 q^{77} +16.7829 q^{79} +1.00000 q^{81} +10.5447 q^{83} +5.67859 q^{85} +1.00000 q^{87} +7.27517 q^{89} -0.351701 q^{91} +7.05741 q^{93} -23.4149 q^{95} +17.8471 q^{97} -5.32243 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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