LMFDB - Embedded newform 8004.2.a.d.1.8

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 19x^{5} + 19x^{4} - 35x^{3} - 10x^{2} + 18x - 4 $ x^8 - 3x^7 - 8x^6 + 19x^5 + 19x^4 - 35x^3 - 10x^2 + 18*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.87214$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.74094 q^{5} -3.17108 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.74094 q^{5} -3.17108 q^{7} +1.00000 q^{9} -0.319748 q^{11} -0.778524 q^{13} +2.74094 q^{15} -1.46825 q^{17} -0.913678 q^{19} -3.17108 q^{21} +1.00000 q^{23} +2.51273 q^{25} +1.00000 q^{27} +1.00000 q^{29} -0.759923 q^{31} -0.319748 q^{33} -8.69172 q^{35} -7.77804 q^{37} -0.778524 q^{39} -10.1559 q^{41} -4.11821 q^{43} +2.74094 q^{45} -0.0564046 q^{47} +3.05572 q^{49} -1.46825 q^{51} -11.1853 q^{53} -0.876409 q^{55} -0.913678 q^{57} +11.6998 q^{59} +10.6382 q^{61} -3.17108 q^{63} -2.13388 q^{65} +0.276436 q^{67} +1.00000 q^{69} -8.70814 q^{71} -11.0869 q^{73} +2.51273 q^{75} +1.01394 q^{77} -4.66744 q^{79} +1.00000 q^{81} +1.81955 q^{83} -4.02437 q^{85} +1.00000 q^{87} -15.6397 q^{89} +2.46876 q^{91} -0.759923 q^{93} -2.50433 q^{95} +9.80700 q^{97} -0.319748 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9	$ 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9} - 5 q^{11} - 4 q^{13} - 5 q^{15} - 3 q^{17} - 5 q^{19} - 4 q^{21} + 8 q^{23} - 5 q^{25} + 8 q^{27} + 8 q^{29} - 2 q^{31} - 5 q^{33} - 15 q^{35} - 10 q^{37} - 4 q^{39} - 11 q^{41} - 7 q^{43} - 5 q^{45} - 14 q^{47} - 18 q^{49} - 3 q^{51} - 15 q^{53} - 17 q^{55} - 5 q^{57} + 4 q^{59} + q^{61} - 4 q^{63} - 5 q^{67} + 8 q^{69} - q^{71} - 21 q^{73} - 5 q^{75} - 8 q^{79} + 8 q^{81} + 3 q^{83} + 8 q^{87} - 20 q^{89} - 7 q^{91} - 2 q^{93} - 3 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9 - 5 * q^11 - 4 * q^13 - 5 * q^15 - 3 * q^17 - 5 * q^19 - 4 * q^21 + 8 * q^23 - 5 * q^25 + 8 * q^27 + 8 * q^29 - 2 * q^31 - 5 * q^33 - 15 * q^35 - 10 * q^37 - 4 * q^39 - 11 * q^41 - 7 * q^43 - 5 * q^45 - 14 * q^47 - 18 * q^49 - 3 * q^51 - 15 * q^53 - 17 * q^55 - 5 * q^57 + 4 * q^59 + q^61 - 4 * q^63 - 5 * q^67 + 8 * q^69 - q^71 - 21 * q^73 - 5 * q^75 - 8 * q^79 + 8 * q^81 + 3 * q^83 + 8 * q^87 - 20 * q^89 - 7 * q^91 - 2 * q^93 - 3 * q^95 - 7 * 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
$5$	2.74094	1.22578	0.612892	−	0.790167i	$-0.290006\pi$
$5$	2.74094	1.22578	0.612892	+	0.790167i	$0.290006\pi$
$6$	0	0
$7$	−3.17108	−1.19855	−0.599277	−	0.800542i	$-0.704545\pi$
$7$	−3.17108	−1.19855	−0.599277	+	0.800542i	$0.704545\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.319748	−0.0964076	−0.0482038	−	0.998838i	$-0.515350\pi$
$11$	−0.319748	−0.0964076	−0.0482038	+	0.998838i	$0.515350\pi$
$12$	0	0
$13$	−0.778524	−0.215924	−0.107962	−	0.994155i	$-0.534432\pi$
$13$	−0.778524	−0.215924	−0.107962	+	0.994155i	$0.534432\pi$
$14$	0	0
$15$	2.74094	0.707707
$16$	0	0
$17$	−1.46825	−0.356102	−0.178051	−	0.984021i	$-0.556979\pi$
$17$	−1.46825	−0.356102	−0.178051	+	0.984021i	$0.556979\pi$
$18$	0	0
$19$	−0.913678	−0.209612	−0.104806	−	0.994493i	$-0.533422\pi$
$19$	−0.913678	−0.209612	−0.104806	+	0.994493i	$0.533422\pi$
$20$	0	0
$21$	−3.17108	−0.691985
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	2.51273	0.502546
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−0.759923	−0.136486	−0.0682431	−	0.997669i	$-0.521739\pi$
$31$	−0.759923	−0.136486	−0.0682431	+	0.997669i	$0.521739\pi$
$32$	0	0
$33$	−0.319748	−0.0556610
$34$	0	0
$35$	−8.69172	−1.46917
$36$	0	0
$37$	−7.77804	−1.27870	−0.639351	−	0.768915i	$-0.720797\pi$
$37$	−7.77804	−1.27870	−0.639351	+	0.768915i	$0.720797\pi$
$38$	0	0
$39$	−0.778524	−0.124664
$40$	0	0
$41$	−10.1559	−1.58609	−0.793043	−	0.609165i	$-0.791505\pi$
$41$	−10.1559	−1.58609	−0.793043	+	0.609165i	$0.791505\pi$
$42$	0	0
$43$	−4.11821	−0.628021	−0.314011	−	0.949419i	$-0.601673\pi$
$43$	−4.11821	−0.628021	−0.314011	+	0.949419i	$0.601673\pi$
$44$	0	0
$45$	2.74094	0.408595
$46$	0	0
$47$	−0.0564046	−0.00822745	−0.00411373	−	0.999992i	$-0.501309\pi$
$47$	−0.0564046	−0.00822745	−0.00411373	+	0.999992i	$0.501309\pi$
$48$	0	0
$49$	3.05572	0.436532
$50$	0	0
$51$	−1.46825	−0.205595
$52$	0	0
$53$	−11.1853	−1.53641	−0.768207	−	0.640201i	$-0.778851\pi$
$53$	−11.1853	−1.53641	−0.768207	+	0.640201i	$0.778851\pi$
$54$	0	0
$55$	−0.876409	−0.118175
$56$	0	0
$57$	−0.913678	−0.121020
$58$	0	0
$59$	11.6998	1.52318	0.761589	−	0.648060i	$-0.224419\pi$
$59$	11.6998	1.52318	0.761589	+	0.648060i	$0.224419\pi$
$60$	0	0
$61$	10.6382	1.36208	0.681042	−	0.732244i	$-0.261527\pi$
$61$	10.6382	1.36208	0.681042	+	0.732244i	$0.261527\pi$
$62$	0	0
$63$	−3.17108	−0.399518
$64$	0	0
$65$	−2.13388	−0.264676
$66$	0	0
$67$	0.276436	0.0337720	0.0168860	−	0.999857i	$-0.494625\pi$
$67$	0.276436	0.0337720	0.0168860	+	0.999857i	$0.494625\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−8.70814	−1.03347	−0.516733	−	0.856147i	$-0.672852\pi$
$71$	−8.70814	−1.03347	−0.516733	+	0.856147i	$0.672852\pi$
$72$	0	0
$73$	−11.0869	−1.29762	−0.648811	−	0.760950i	$-0.724733\pi$
$73$	−11.0869	−1.29762	−0.648811	+	0.760950i	$0.724733\pi$
$74$	0	0
$75$	2.51273	0.290145
$76$	0	0
$77$	1.01394	0.115550
$78$	0	0
$79$	−4.66744	−0.525128	−0.262564	−	0.964915i	$-0.584568\pi$
$79$	−4.66744	−0.525128	−0.262564	+	0.964915i	$0.584568\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.81955	0.199722	0.0998610	−	0.995001i	$-0.468160\pi$
$83$	1.81955	0.199722	0.0998610	+	0.995001i	$0.468160\pi$
$84$	0	0
$85$	−4.02437	−0.436504
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−15.6397	−1.65780	−0.828900	−	0.559397i	$-0.811033\pi$
$89$	−15.6397	−1.65780	−0.828900	+	0.559397i	$0.811033\pi$
$90$	0	0
$91$	2.46876	0.258796
$92$	0	0
$93$	−0.759923	−0.0788004
$94$	0	0
$95$	−2.50433	−0.256939
$96$	0	0
$97$	9.80700	0.995750	0.497875	−	0.867249i	$-0.334114\pi$
$97$	9.80700	0.995750	0.497875	+	0.867249i	$0.334114\pi$
$98$	0	0
$99$	−0.319748	−0.0321359
$100$	0	0
$101$	−0.127442	−0.0126810	−0.00634048	−	0.999980i	$-0.502018\pi$
$101$	−0.127442	−0.0126810	−0.00634048	+	0.999980i	$0.502018\pi$
$102$	0	0
$103$	2.60543	0.256721	0.128361	−	0.991728i	$-0.459029\pi$
$103$	2.60543	0.256721	0.128361	+	0.991728i	$0.459029\pi$
$104$	0	0
$105$	−8.69172	−0.848225
$106$	0	0
$107$	7.32175	0.707821	0.353910	−	0.935279i	$-0.384852\pi$
$107$	7.32175	0.707821	0.353910	+	0.935279i	$0.384852\pi$
$108$	0	0
$109$	8.76847	0.839867	0.419934	−	0.907555i	$-0.362053\pi$
$109$	8.76847	0.839867	0.419934	+	0.907555i	$0.362053\pi$
$110$	0	0
$111$	−7.77804	−0.738259
$112$	0	0
$113$	−7.11868	−0.669669	−0.334835	−	0.942277i	$-0.608680\pi$
$113$	−7.11868	−0.669669	−0.334835	+	0.942277i	$0.608680\pi$
$114$	0	0
$115$	2.74094	0.255594
$116$	0	0
$117$	−0.778524	−0.0719745
$118$	0	0
$119$	4.65592	0.426807
$120$	0	0
$121$	−10.8978	−0.990706
$122$	0	0
$123$	−10.1559	−0.915727
$124$	0	0
$125$	−6.81744	−0.609771
$126$	0	0
$127$	13.5344	1.20099	0.600494	−	0.799629i	$-0.294971\pi$
$127$	13.5344	1.20099	0.600494	+	0.799629i	$0.294971\pi$
$128$	0	0
$129$	−4.11821	−0.362588
$130$	0	0
$131$	−2.98124	−0.260472	−0.130236	−	0.991483i	$-0.541574\pi$
$131$	−2.98124	−0.260472	−0.130236	+	0.991483i	$0.541574\pi$
$132$	0	0
$133$	2.89734	0.251232
$134$	0	0
$135$	2.74094	0.235902
$136$	0	0
$137$	15.4718	1.32185	0.660923	−	0.750453i	$-0.270165\pi$
$137$	15.4718	1.32185	0.660923	+	0.750453i	$0.270165\pi$
$138$	0	0
$139$	6.12909	0.519863	0.259931	−	0.965627i	$-0.416300\pi$
$139$	6.12909	0.519863	0.259931	+	0.965627i	$0.416300\pi$
$140$	0	0
$141$	−0.0564046	−0.00475012
$142$	0	0
$143$	0.248931	0.0208167
$144$	0	0
$145$	2.74094	0.227622
$146$	0	0
$147$	3.05572	0.252032
$148$	0	0
$149$	−10.8122	−0.885773	−0.442887	−	0.896578i	$-0.646046\pi$
$149$	−10.8122	−0.885773	−0.442887	+	0.896578i	$0.646046\pi$
$150$	0	0
$151$	0.0371443	0.00302276	0.00151138	−	0.999999i	$-0.499519\pi$
$151$	0.0371443	0.00302276	0.00151138	+	0.999999i	$0.499519\pi$
$152$	0	0
$153$	−1.46825	−0.118701
$154$	0	0
$155$	−2.08290	−0.167303
$156$	0	0
$157$	−0.229620	−0.0183257	−0.00916283	−	0.999958i	$-0.502917\pi$
$157$	−0.229620	−0.0183257	−0.00916283	+	0.999958i	$0.502917\pi$
$158$	0	0
$159$	−11.1853	−0.887050
$160$	0	0
$161$	−3.17108	−0.249916
$162$	0	0
$163$	−13.6219	−1.06695	−0.533475	−	0.845816i	$-0.679114\pi$
$163$	−13.6219	−1.06695	−0.533475	+	0.845816i	$0.679114\pi$
$164$	0	0
$165$	−0.876409	−0.0682283
$166$	0	0
$167$	−11.0692	−0.856561	−0.428280	−	0.903646i	$-0.640880\pi$
$167$	−11.0692	−0.856561	−0.428280	+	0.903646i	$0.640880\pi$
$168$	0	0
$169$	−12.3939	−0.953377
$170$	0	0
$171$	−0.913678	−0.0698707
$172$	0	0
$173$	−10.3603	−0.787676	−0.393838	−	0.919180i	$-0.628853\pi$
$173$	−10.3603	−0.787676	−0.393838	+	0.919180i	$0.628853\pi$
$174$	0	0
$175$	−7.96806	−0.602329
$176$	0	0
$177$	11.6998	0.879408
$178$	0	0
$179$	−11.9142	−0.890506	−0.445253	−	0.895405i	$-0.646886\pi$
$179$	−11.9142	−0.890506	−0.445253	+	0.895405i	$0.646886\pi$
$180$	0	0
$181$	−10.8691	−0.807896	−0.403948	−	0.914782i	$-0.632362\pi$
$181$	−10.8691	−0.807896	−0.403948	+	0.914782i	$0.632362\pi$
$182$	0	0
$183$	10.6382	0.786400
$184$	0	0
$185$	−21.3191	−1.56741
$186$	0	0
$187$	0.469468	0.0343309
$188$	0	0
$189$	−3.17108	−0.230662
$190$	0	0
$191$	10.8586	0.785700	0.392850	−	0.919603i	$-0.371489\pi$
$191$	10.8586	0.785700	0.392850	+	0.919603i	$0.371489\pi$
$192$	0	0
$193$	6.28047	0.452078	0.226039	−	0.974118i	$-0.427422\pi$
$193$	6.28047	0.452078	0.226039	+	0.974118i	$0.427422\pi$
$194$	0	0
$195$	−2.13388	−0.152811
$196$	0	0
$197$	−22.1081	−1.57513	−0.787567	−	0.616228i	$-0.788660\pi$
$197$	−22.1081	−1.57513	−0.787567	+	0.616228i	$0.788660\pi$
$198$	0	0
$199$	−9.76637	−0.692320	−0.346160	−	0.938176i	$-0.612515\pi$
$199$	−9.76637	−0.692320	−0.346160	+	0.938176i	$0.612515\pi$
$200$	0	0
$201$	0.276436	0.0194983
$202$	0	0
$203$	−3.17108	−0.222566
$204$	0	0
$205$	−27.8367	−1.94420
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	0.292147	0.0202082
$210$	0	0
$211$	13.2681	0.913415	0.456708	−	0.889617i	$-0.349029\pi$
$211$	13.2681	0.913415	0.456708	+	0.889617i	$0.349029\pi$
$212$	0	0
$213$	−8.70814	−0.596672
$214$	0	0
$215$	−11.2878	−0.769819
$216$	0	0
$217$	2.40977	0.163586
$218$	0	0
$219$	−11.0869	−0.749182
$220$	0	0
$221$	1.14306	0.0768908
$222$	0	0
$223$	12.5412	0.839820	0.419910	−	0.907566i	$-0.362062\pi$
$223$	12.5412	0.839820	0.419910	+	0.907566i	$0.362062\pi$
$224$	0	0
$225$	2.51273	0.167515
$226$	0	0
$227$	11.4671	0.761097	0.380549	−	0.924761i	$-0.375735\pi$
$227$	11.4671	0.761097	0.380549	+	0.924761i	$0.375735\pi$
$228$	0	0
$229$	11.5362	0.762332	0.381166	−	0.924507i	$-0.375523\pi$
$229$	11.5362	0.762332	0.381166	+	0.924507i	$0.375523\pi$
$230$	0	0
$231$	1.01394	0.0667127
$232$	0	0
$233$	25.9119	1.69755	0.848773	−	0.528757i	$-0.177342\pi$
$233$	25.9119	1.69755	0.848773	+	0.528757i	$0.177342\pi$
$234$	0	0
$235$	−0.154601	−0.0100851
$236$	0	0
$237$	−4.66744	−0.303183
$238$	0	0
$239$	−18.4313	−1.19222	−0.596111	−	0.802902i	$-0.703288\pi$
$239$	−18.4313	−1.19222	−0.596111	+	0.802902i	$0.703288\pi$
$240$	0	0
$241$	−8.44255	−0.543832	−0.271916	−	0.962321i	$-0.587657\pi$
$241$	−8.44255	−0.543832	−0.271916	+	0.962321i	$0.587657\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	8.37554	0.535094
$246$	0	0
$247$	0.711320	0.0452602
$248$	0	0
$249$	1.81955	0.115310
$250$	0	0
$251$	−16.0222	−1.01131	−0.505657	−	0.862734i	$-0.668750\pi$
$251$	−16.0222	−1.01131	−0.505657	+	0.862734i	$0.668750\pi$
$252$	0	0
$253$	−0.319748	−0.0201024
$254$	0	0
$255$	−4.02437	−0.252016
$256$	0	0
$257$	−12.0434	−0.751249	−0.375624	−	0.926772i	$-0.622572\pi$
$257$	−12.0434	−0.751249	−0.375624	+	0.926772i	$0.622572\pi$
$258$	0	0
$259$	24.6648	1.53259
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	25.9976	1.60308	0.801540	−	0.597942i	$-0.204015\pi$
$263$	25.9976	1.60308	0.801540	+	0.597942i	$0.204015\pi$
$264$	0	0
$265$	−30.6581	−1.88331
$266$	0	0
$267$	−15.6397	−0.957131
$268$	0	0
$269$	−18.1511	−1.10669	−0.553346	−	0.832952i	$-0.686649\pi$
$269$	−18.1511	−1.10669	−0.553346	+	0.832952i	$0.686649\pi$
$270$	0	0
$271$	−13.2450	−0.804578	−0.402289	−	0.915513i	$-0.631785\pi$
$271$	−13.2450	−0.804578	−0.402289	+	0.915513i	$0.631785\pi$
$272$	0	0
$273$	2.46876	0.149416
$274$	0	0
$275$	−0.803441	−0.0484493
$276$	0	0
$277$	4.86516	0.292319	0.146160	−	0.989261i	$-0.453309\pi$
$277$	4.86516	0.292319	0.146160	+	0.989261i	$0.453309\pi$
$278$	0	0
$279$	−0.759923	−0.0454954
$280$	0	0
$281$	−25.5871	−1.52640	−0.763198	−	0.646164i	$-0.776372\pi$
$281$	−25.5871	−1.52640	−0.763198	+	0.646164i	$0.776372\pi$
$282$	0	0
$283$	−10.1261	−0.601936	−0.300968	−	0.953634i	$-0.597310\pi$
$283$	−10.1261	−0.601936	−0.300968	+	0.953634i	$0.597310\pi$
$284$	0	0
$285$	−2.50433	−0.148344
$286$	0	0
$287$	32.2052	1.90101
$288$	0	0
$289$	−14.8443	−0.873191
$290$	0	0
$291$	9.80700	0.574897
$292$	0	0
$293$	27.6514	1.61541	0.807705	−	0.589587i	$-0.200709\pi$
$293$	27.6514	1.61541	0.807705	+	0.589587i	$0.200709\pi$
$294$	0	0
$295$	32.0683	1.86709
$296$	0	0
$297$	−0.319748	−0.0185537
$298$	0	0
$299$	−0.778524	−0.0450232
$300$	0	0
$301$	13.0592	0.752718
$302$	0	0
$303$	−0.127442	−0.00732135
$304$	0	0
$305$	29.1587	1.66962
$306$	0	0
$307$	28.8959	1.64918	0.824589	−	0.565732i	$-0.191406\pi$
$307$	28.8959	1.64918	0.824589	+	0.565732i	$0.191406\pi$
$308$	0	0
$309$	2.60543	0.148218
$310$	0	0
$311$	−0.879658	−0.0498808	−0.0249404	−	0.999689i	$-0.507940\pi$
$311$	−0.879658	−0.0498808	−0.0249404	+	0.999689i	$0.507940\pi$
$312$	0	0
$313$	−23.7373	−1.34171	−0.670855	−	0.741589i	$-0.734073\pi$
$313$	−23.7373	−1.34171	−0.670855	+	0.741589i	$0.734073\pi$
$314$	0	0
$315$	−8.69172	−0.489723
$316$	0	0
$317$	−4.87172	−0.273623	−0.136811	−	0.990597i	$-0.543685\pi$
$317$	−4.87172	−0.273623	−0.136811	+	0.990597i	$0.543685\pi$
$318$	0	0
$319$	−0.319748	−0.0179024
$320$	0	0
$321$	7.32175	0.408660
$322$	0	0
$323$	1.34150	0.0746433
$324$	0	0
$325$	−1.95622	−0.108512
$326$	0	0
$327$	8.76847	0.484898
$328$	0	0
$329$	0.178863	0.00986104
$330$	0	0
$331$	−2.02968	−0.111561	−0.0557807	−	0.998443i	$-0.517765\pi$
$331$	−2.02968	−0.111561	−0.0557807	+	0.998443i	$0.517765\pi$
$332$	0	0
$333$	−7.77804	−0.426234
$334$	0	0
$335$	0.757692	0.0413971
$336$	0	0
$337$	22.2073	1.20971	0.604854	−	0.796336i	$-0.293231\pi$
$337$	22.2073	1.20971	0.604854	+	0.796336i	$0.293231\pi$
$338$	0	0
$339$	−7.11868	−0.386634
$340$	0	0
$341$	0.242984	0.0131583
$342$	0	0
$343$	12.5076	0.675347
$344$	0	0
$345$	2.74094	0.147567
$346$	0	0
$347$	−3.65654	−0.196293	−0.0981466	−	0.995172i	$-0.531291\pi$
$347$	−3.65654	−0.196293	−0.0981466	+	0.995172i	$0.531291\pi$
$348$	0	0
$349$	−7.99437	−0.427929	−0.213964	−	0.976841i	$-0.568638\pi$
$349$	−7.99437	−0.427929	−0.213964	+	0.976841i	$0.568638\pi$
$350$	0	0
$351$	−0.778524	−0.0415545
$352$	0	0
$353$	15.0602	0.801571	0.400786	−	0.916172i	$-0.368737\pi$
$353$	15.0602	0.801571	0.400786	+	0.916172i	$0.368737\pi$
$354$	0	0
$355$	−23.8684	−1.26681
$356$	0	0
$357$	4.65592	0.246417
$358$	0	0
$359$	19.0451	1.00516	0.502581	−	0.864530i	$-0.332384\pi$
$359$	19.0451	1.00516	0.502581	+	0.864530i	$0.332384\pi$
$360$	0	0
$361$	−18.1652	−0.956063
$362$	0	0
$363$	−10.8978	−0.571984
$364$	0	0
$365$	−30.3884	−1.59060
$366$	0	0
$367$	−25.0628	−1.30827	−0.654135	−	0.756378i	$-0.726967\pi$
$367$	−25.0628	−1.30827	−0.654135	+	0.756378i	$0.726967\pi$
$368$	0	0
$369$	−10.1559	−0.528695
$370$	0	0
$371$	35.4693	1.84148
$372$	0	0
$373$	13.6205	0.705241	0.352621	−	0.935766i	$-0.385291\pi$
$373$	13.6205	0.705241	0.352621	+	0.935766i	$0.385291\pi$
$374$	0	0
$375$	−6.81744	−0.352051
$376$	0	0
$377$	−0.778524	−0.0400960
$378$	0	0
$379$	−3.03758	−0.156030	−0.0780149	−	0.996952i	$-0.524858\pi$
$379$	−3.03758	−0.156030	−0.0780149	+	0.996952i	$0.524858\pi$
$380$	0	0
$381$	13.5344	0.693390
$382$	0	0
$383$	−3.08959	−0.157871	−0.0789353	−	0.996880i	$-0.525152\pi$
$383$	−3.08959	−0.157871	−0.0789353	+	0.996880i	$0.525152\pi$
$384$	0	0
$385$	2.77916	0.141639
$386$	0	0
$387$	−4.11821	−0.209340
$388$	0	0
$389$	−38.9192	−1.97328	−0.986640	−	0.162914i	$-0.947911\pi$
$389$	−38.9192	−1.97328	−0.986640	+	0.162914i	$0.947911\pi$
$390$	0	0
$391$	−1.46825	−0.0742524
$392$	0	0
$393$	−2.98124	−0.150384
$394$	0	0
$395$	−12.7932	−0.643694
$396$	0	0
$397$	−16.5420	−0.830222	−0.415111	−	0.909771i	$-0.636257\pi$
$397$	−16.5420	−0.830222	−0.415111	+	0.909771i	$0.636257\pi$
$398$	0	0
$399$	2.89734	0.145049
$400$	0	0
$401$	11.7079	0.584665	0.292333	−	0.956317i	$-0.405569\pi$
$401$	11.7079	0.584665	0.292333	+	0.956317i	$0.405569\pi$
$402$	0	0
$403$	0.591618	0.0294706
$404$	0	0
$405$	2.74094	0.136198
$406$	0	0
$407$	2.48701	0.123277
$408$	0	0
$409$	−10.8671	−0.537343	−0.268671	−	0.963232i	$-0.586585\pi$
$409$	−10.8671	−0.537343	−0.268671	+	0.963232i	$0.586585\pi$
$410$	0	0
$411$	15.4718	0.763169
$412$	0	0
$413$	−37.1008	−1.82561
$414$	0	0
$415$	4.98728	0.244816
$416$	0	0
$417$	6.12909	0.300143
$418$	0	0
$419$	−4.60704	−0.225068	−0.112534	−	0.993648i	$-0.535897\pi$
$419$	−4.60704	−0.225068	−0.112534	+	0.993648i	$0.535897\pi$
$420$	0	0
$421$	21.1985	1.03315	0.516577	−	0.856241i	$-0.327206\pi$
$421$	21.1985	1.03315	0.516577	+	0.856241i	$0.327206\pi$
$422$	0	0
$423$	−0.0564046	−0.00274248
$424$	0	0
$425$	−3.68931	−0.178958
$426$	0	0
$427$	−33.7346	−1.63253
$428$	0	0
$429$	0.248931	0.0120185
$430$	0	0
$431$	35.7258	1.72085	0.860425	−	0.509577i	$-0.170198\pi$
$431$	35.7258	1.72085	0.860425	+	0.509577i	$0.170198\pi$
$432$	0	0
$433$	3.59244	0.172642	0.0863209	−	0.996267i	$-0.472489\pi$
$433$	3.59244	0.172642	0.0863209	+	0.996267i	$0.472489\pi$
$434$	0	0
$435$	2.74094	0.131418
$436$	0	0
$437$	−0.913678	−0.0437072
$438$	0	0
$439$	11.7197	0.559349	0.279674	−	0.960095i	$-0.409773\pi$
$439$	11.7197	0.559349	0.279674	+	0.960095i	$0.409773\pi$
$440$	0	0
$441$	3.05572	0.145511
$442$	0	0
$443$	−23.0195	−1.09369	−0.546845	−	0.837234i	$-0.684171\pi$
$443$	−23.0195	−1.09369	−0.546845	+	0.837234i	$0.684171\pi$
$444$	0	0
$445$	−42.8673	−2.03210
$446$	0	0
$447$	−10.8122	−0.511401
$448$	0	0
$449$	−23.3026	−1.09972	−0.549858	−	0.835258i	$-0.685318\pi$
$449$	−23.3026	−1.09972	−0.549858	+	0.835258i	$0.685318\pi$
$450$	0	0
$451$	3.24733	0.152911
$452$	0	0
$453$	0.0371443	0.00174519
$454$	0	0
$455$	6.76671	0.317228
$456$	0	0
$457$	−10.1758	−0.476004	−0.238002	−	0.971265i	$-0.576493\pi$
$457$	−10.1758	−0.476004	−0.238002	+	0.971265i	$0.576493\pi$
$458$	0	0
$459$	−1.46825	−0.0685318
$460$	0	0
$461$	28.6775	1.33565	0.667823	−	0.744320i	$-0.267226\pi$
$461$	28.6775	1.33565	0.667823	+	0.744320i	$0.267226\pi$
$462$	0	0
$463$	−1.26235	−0.0586666	−0.0293333	−	0.999570i	$-0.509338\pi$
$463$	−1.26235	−0.0586666	−0.0293333	+	0.999570i	$0.509338\pi$
$464$	0	0
$465$	−2.08290	−0.0965922
$466$	0	0
$467$	−15.4777	−0.716221	−0.358111	−	0.933679i	$-0.616579\pi$
$467$	−15.4777	−0.716221	−0.358111	+	0.933679i	$0.616579\pi$
$468$	0	0
$469$	−0.876598	−0.0404775
$470$	0	0
$471$	−0.229620	−0.0105803
$472$	0	0
$473$	1.31679	0.0605460
$474$	0	0
$475$	−2.29583	−0.105340
$476$	0	0
$477$	−11.1853	−0.512138
$478$	0	0
$479$	−23.1503	−1.05776	−0.528882	−	0.848695i	$-0.677389\pi$
$479$	−23.1503	−1.05776	−0.528882	+	0.848695i	$0.677389\pi$
$480$	0	0
$481$	6.05539	0.276102
$482$	0	0
$483$	−3.17108	−0.144289
$484$	0	0
$485$	26.8804	1.22057
$486$	0	0
$487$	15.5288	0.703677	0.351839	−	0.936061i	$-0.385557\pi$
$487$	15.5288	0.703677	0.351839	+	0.936061i	$0.385557\pi$
$488$	0	0
$489$	−13.6219	−0.616004
$490$	0	0
$491$	31.1097	1.40396	0.701979	−	0.712197i	$-0.252300\pi$
$491$	31.1097	1.40396	0.701979	+	0.712197i	$0.252300\pi$
$492$	0	0
$493$	−1.46825	−0.0661265
$494$	0	0
$495$	−0.876409	−0.0393916
$496$	0	0
$497$	27.6142	1.23866
$498$	0	0
$499$	17.5446	0.785404	0.392702	−	0.919666i	$-0.371540\pi$
$499$	17.5446	0.785404	0.392702	+	0.919666i	$0.371540\pi$
$500$	0	0
$501$	−11.0692	−0.494536
$502$	0	0
$503$	−3.56113	−0.158783	−0.0793915	−	0.996844i	$-0.525298\pi$
$503$	−3.56113	−0.158783	−0.0793915	+	0.996844i	$0.525298\pi$
$504$	0	0
$505$	−0.349311	−0.0155441
$506$	0	0
$507$	−12.3939	−0.550432
$508$	0	0
$509$	−4.37498	−0.193918	−0.0969588	−	0.995288i	$-0.530912\pi$
$509$	−4.37498	−0.193918	−0.0969588	+	0.995288i	$0.530912\pi$
$510$	0	0
$511$	35.1574	1.55527
$512$	0	0
$513$	−0.913678	−0.0403399
$514$	0	0
$515$	7.14133	0.314685
$516$	0	0
$517$	0.0180352	0.000793189 0
$518$	0	0
$519$	−10.3603	−0.454765
$520$	0	0
$521$	12.2906	0.538463	0.269231	−	0.963076i	$-0.413230\pi$
$521$	12.2906	0.538463	0.269231	+	0.963076i	$0.413230\pi$
$522$	0	0
$523$	10.9510	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$523$	10.9510	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$524$	0	0
$525$	−7.96806	−0.347755
$526$	0	0
$527$	1.11575	0.0486030
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	11.6998	0.507726
$532$	0	0
$533$	7.90662	0.342474
$534$	0	0
$535$	20.0685	0.867635
$536$	0	0
$537$	−11.9142	−0.514134
$538$	0	0
$539$	−0.977061	−0.0420850
$540$	0	0
$541$	20.8769	0.897570	0.448785	−	0.893640i	$-0.351857\pi$
$541$	20.8769	0.897570	0.448785	+	0.893640i	$0.351857\pi$
$542$	0	0
$543$	−10.8691	−0.466439
$544$	0	0
$545$	24.0338	1.02950
$546$	0	0
$547$	−6.70207	−0.286560	−0.143280	−	0.989682i	$-0.545765\pi$
$547$	−6.70207	−0.286560	−0.143280	+	0.989682i	$0.545765\pi$
$548$	0	0
$549$	10.6382	0.454028
$550$	0	0
$551$	−0.913678	−0.0389240
$552$	0	0
$553$	14.8008	0.629394
$554$	0	0
$555$	−21.3191	−0.904946
$556$	0	0
$557$	41.6614	1.76525	0.882625	−	0.470078i	$-0.155774\pi$
$557$	41.6614	1.76525	0.882625	+	0.470078i	$0.155774\pi$
$558$	0	0
$559$	3.20613	0.135605
$560$	0	0
$561$	0.469468	0.0198210
$562$	0	0
$563$	29.6449	1.24938	0.624691	−	0.780872i	$-0.285225\pi$
$563$	29.6449	1.24938	0.624691	+	0.780872i	$0.285225\pi$
$564$	0	0
$565$	−19.5119	−0.820870
$566$	0	0
$567$	−3.17108	−0.133173
$568$	0	0
$569$	2.80536	0.117607	0.0588034	−	0.998270i	$-0.481272\pi$
$569$	2.80536	0.117607	0.0588034	+	0.998270i	$0.481272\pi$
$570$	0	0
$571$	−36.5749	−1.53061	−0.765306	−	0.643667i	$-0.777412\pi$
$571$	−36.5749	−1.53061	−0.765306	+	0.643667i	$0.777412\pi$
$572$	0	0
$573$	10.8586	0.453624
$574$	0	0
$575$	2.51273	0.104788
$576$	0	0
$577$	−14.9213	−0.621180	−0.310590	−	0.950544i	$-0.600527\pi$
$577$	−14.9213	−0.621180	−0.310590	+	0.950544i	$0.600527\pi$
$578$	0	0
$579$	6.28047	0.261007
$580$	0	0
$581$	−5.76995	−0.239378
$582$	0	0
$583$	3.57647	0.148122
$584$	0	0
$585$	−2.13388	−0.0882252
$586$	0	0
$587$	29.4095	1.21386	0.606930	−	0.794755i	$-0.292401\pi$
$587$	29.4095	1.21386	0.606930	+	0.794755i	$0.292401\pi$
$588$	0	0
$589$	0.694325	0.0286092
$590$	0	0
$591$	−22.1081	−0.909405
$592$	0	0
$593$	−15.3090	−0.628664	−0.314332	−	0.949313i	$-0.601780\pi$
$593$	−15.3090	−0.628664	−0.314332	+	0.949313i	$0.601780\pi$
$594$	0	0
$595$	12.7616	0.523174
$596$	0	0
$597$	−9.76637	−0.399711
$598$	0	0
$599$	−21.3689	−0.873109	−0.436555	−	0.899678i	$-0.643801\pi$
$599$	−21.3689	−0.873109	−0.436555	+	0.899678i	$0.643801\pi$
$600$	0	0
$601$	−37.6068	−1.53401	−0.767006	−	0.641640i	$-0.778254\pi$
$601$	−37.6068	−1.53401	−0.767006	+	0.641640i	$0.778254\pi$
$602$	0	0
$603$	0.276436	0.0112573
$604$	0	0
$605$	−29.8701	−1.21439
$606$	0	0
$607$	−8.27447	−0.335850	−0.167925	−	0.985800i	$-0.553707\pi$
$607$	−8.27447	−0.335850	−0.167925	+	0.985800i	$0.553707\pi$
$608$	0	0
$609$	−3.17108	−0.128498
$610$	0	0
$611$	0.0439123	0.00177650
$612$	0	0
$613$	29.1148	1.17594	0.587969	−	0.808884i	$-0.299928\pi$
$613$	29.1148	1.17594	0.587969	+	0.808884i	$0.299928\pi$
$614$	0	0
$615$	−27.8367	−1.12248
$616$	0	0
$617$	−18.3528	−0.738856	−0.369428	−	0.929259i	$-0.620446\pi$
$617$	−18.3528	−0.738856	−0.369428	+	0.929259i	$0.620446\pi$
$618$	0	0
$619$	37.7224	1.51619	0.758096	−	0.652143i	$-0.226130\pi$
$619$	37.7224	1.51619	0.758096	+	0.652143i	$0.226130\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	49.5945	1.98696
$624$	0	0
$625$	−31.2498	−1.24999
$626$	0	0
$627$	0.292147	0.0116672
$628$	0	0
$629$	11.4201	0.455348
$630$	0	0
$631$	−21.1508	−0.842002	−0.421001	−	0.907060i	$-0.638321\pi$
$631$	−21.1508	−0.842002	−0.421001	+	0.907060i	$0.638321\pi$
$632$	0	0
$633$	13.2681	0.527361
$634$	0	0
$635$	37.0970	1.47215
$636$	0	0
$637$	−2.37895	−0.0942575
$638$	0	0
$639$	−8.70814	−0.344489
$640$	0	0
$641$	−3.50898	−0.138596	−0.0692982	−	0.997596i	$-0.522076\pi$
$641$	−3.50898	−0.138596	−0.0692982	+	0.997596i	$0.522076\pi$
$642$	0	0
$643$	6.23550	0.245904	0.122952	−	0.992413i	$-0.460764\pi$
$643$	6.23550	0.245904	0.122952	+	0.992413i	$0.460764\pi$
$644$	0	0
$645$	−11.2878	−0.444455
$646$	0	0
$647$	−8.42133	−0.331077	−0.165538	−	0.986203i	$-0.552936\pi$
$647$	−8.42133	−0.331077	−0.165538	+	0.986203i	$0.552936\pi$
$648$	0	0
$649$	−3.74097	−0.146846
$650$	0	0
$651$	2.40977	0.0944465
$652$	0	0
$653$	38.9460	1.52408	0.762038	−	0.647532i	$-0.224199\pi$
$653$	38.9460	1.52408	0.762038	+	0.647532i	$0.224199\pi$
$654$	0	0
$655$	−8.17139	−0.319283
$656$	0	0
$657$	−11.0869	−0.432541
$658$	0	0
$659$	27.4963	1.07110	0.535552	−	0.844503i	$-0.320104\pi$
$659$	27.4963	1.07110	0.535552	+	0.844503i	$0.320104\pi$
$660$	0	0
$661$	2.68466	0.104421	0.0522106	−	0.998636i	$-0.483373\pi$
$661$	2.68466	0.104421	0.0522106	+	0.998636i	$0.483373\pi$
$662$	0	0
$663$	1.14306	0.0443929
$664$	0	0
$665$	7.94143	0.307956
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	12.5412	0.484870
$670$	0	0
$671$	−3.40155	−0.131315
$672$	0	0
$673$	−10.7614	−0.414820	−0.207410	−	0.978254i	$-0.566503\pi$
$673$	−10.7614	−0.414820	−0.207410	+	0.978254i	$0.566503\pi$
$674$	0	0
$675$	2.51273	0.0967151
$676$	0	0
$677$	−3.71712	−0.142860	−0.0714302	−	0.997446i	$-0.522756\pi$
$677$	−3.71712	−0.142860	−0.0714302	+	0.997446i	$0.522756\pi$
$678$	0	0
$679$	−31.0988	−1.19346
$680$	0	0
$681$	11.4671	0.439420
$682$	0	0
$683$	42.4607	1.62471	0.812357	−	0.583161i	$-0.198184\pi$
$683$	42.4607	1.62471	0.812357	+	0.583161i	$0.198184\pi$
$684$	0	0
$685$	42.4073	1.62030
$686$	0	0
$687$	11.5362	0.440132
$688$	0	0
$689$	8.70800	0.331748
$690$	0	0
$691$	6.56195	0.249628	0.124814	−	0.992180i	$-0.460167\pi$
$691$	6.56195	0.249628	0.124814	+	0.992180i	$0.460167\pi$
$692$	0	0
$693$	1.01394	0.0385166
$694$	0	0
$695$	16.7994	0.637239
$696$	0	0
$697$	14.9114	0.564808
$698$	0	0
$699$	25.9119	0.980079
$700$	0	0
$701$	−3.96758	−0.149854	−0.0749268	−	0.997189i	$-0.523872\pi$
$701$	−3.96758	−0.149854	−0.0749268	+	0.997189i	$0.523872\pi$
$702$	0	0
$703$	7.10663	0.268031
$704$	0	0
$705$	−0.154601	−0.00582262
$706$	0	0
$707$	0.404128	0.0151988
$708$	0	0
$709$	−34.0976	−1.28056	−0.640282	−	0.768140i	$-0.721182\pi$
$709$	−34.0976	−1.28056	−0.640282	+	0.768140i	$0.721182\pi$
$710$	0	0
$711$	−4.66744	−0.175043
$712$	0	0
$713$	−0.759923	−0.0284593
$714$	0	0
$715$	0.682305	0.0255168
$716$	0	0
$717$	−18.4313	−0.688330
$718$	0	0
$719$	−6.94170	−0.258882	−0.129441	−	0.991587i	$-0.541318\pi$
$719$	−6.94170	−0.258882	−0.129441	+	0.991587i	$0.541318\pi$
$720$	0	0
$721$	−8.26203	−0.307694
$722$	0	0
$723$	−8.44255	−0.313982
$724$	0	0
$725$	2.51273	0.0933205
$726$	0	0
$727$	−36.7564	−1.36322	−0.681611	−	0.731715i	$-0.738720\pi$
$727$	−36.7564	−1.36322	−0.681611	+	0.731715i	$0.738720\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.04655	0.223640
$732$	0	0
$733$	50.2866	1.85738	0.928690	−	0.370857i	$-0.120936\pi$
$733$	50.2866	1.85738	0.928690	+	0.370857i	$0.120936\pi$
$734$	0	0
$735$	8.37554	0.308936
$736$	0	0
$737$	−0.0883897	−0.00325588
$738$	0	0
$739$	−6.94487	−0.255471	−0.127736	−	0.991808i	$-0.540771\pi$
$739$	−6.94487	−0.255471	−0.127736	+	0.991808i	$0.540771\pi$
$740$	0	0
$741$	0.711320	0.0261310
$742$	0	0
$743$	0.278485	0.0102166	0.00510831	−	0.999987i	$-0.498374\pi$
$743$	0.278485	0.0102166	0.00510831	+	0.999987i	$0.498374\pi$
$744$	0	0
$745$	−29.6357	−1.08577
$746$	0	0
$747$	1.81955	0.0665740
$748$	0	0
$749$	−23.2178	−0.848361
$750$	0	0
$751$	−40.6746	−1.48424	−0.742118	−	0.670269i	$-0.766179\pi$
$751$	−40.6746	−1.48424	−0.742118	+	0.670269i	$0.766179\pi$
$752$	0	0
$753$	−16.0222	−0.583883
$754$	0	0
$755$	0.101810	0.00370525
$756$	0	0
$757$	−29.3657	−1.06732	−0.533658	−	0.845700i	$-0.679183\pi$
$757$	−29.3657	−1.06732	−0.533658	+	0.845700i	$0.679183\pi$
$758$	0	0
$759$	−0.319748	−0.0116061
$760$	0	0
$761$	−12.1509	−0.440471	−0.220236	−	0.975447i	$-0.570683\pi$
$761$	−12.1509	−0.440471	−0.220236	+	0.975447i	$0.570683\pi$
$762$	0	0
$763$	−27.8055	−1.00663
$764$	0	0
$765$	−4.02437	−0.145501
$766$	0	0
$767$	−9.10854	−0.328890
$768$	0	0
$769$	−2.20534	−0.0795266	−0.0397633	−	0.999209i	$-0.512660\pi$
$769$	−2.20534	−0.0795266	−0.0397633	+	0.999209i	$0.512660\pi$
$770$	0	0
$771$	−12.0434	−0.433734
$772$	0	0
$773$	−13.6949	−0.492570	−0.246285	−	0.969198i	$-0.579210\pi$
$773$	−13.6949	−0.492570	−0.246285	+	0.969198i	$0.579210\pi$
$774$	0	0
$775$	−1.90948	−0.0685907
$776$	0	0
$777$	24.6648	0.884843
$778$	0	0
$779$	9.27923	0.332463
$780$	0	0
$781$	2.78441	0.0996340
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−0.629374	−0.0224633
$786$	0	0
$787$	−34.4934	−1.22956	−0.614779	−	0.788699i	$-0.710755\pi$
$787$	−34.4934	−1.22956	−0.614779	+	0.788699i	$0.710755\pi$
$788$	0	0
$789$	25.9976	0.925538
$790$	0	0
$791$	22.5739	0.802635
$792$	0	0
$793$	−8.28211	−0.294106
$794$	0	0
$795$	−30.6581	−1.08733
$796$	0	0
$797$	44.3779	1.57195	0.785973	−	0.618261i	$-0.212163\pi$
$797$	44.3779	1.57195	0.785973	+	0.618261i	$0.212163\pi$
$798$	0	0
$799$	0.0828157	0.00292981
$800$	0	0
$801$	−15.6397	−0.552600
$802$	0	0
$803$	3.54501	0.125101
$804$	0	0
$805$	−8.69172	−0.306343
$806$	0	0
$807$	−18.1511	−0.638949
$808$	0	0
$809$	34.8304	1.22457	0.612286	−	0.790637i	$-0.290250\pi$
$809$	34.8304	1.22457	0.612286	+	0.790637i	$0.290250\pi$
$810$	0	0
$811$	−26.8425	−0.942567	−0.471283	−	0.881982i	$-0.656209\pi$
$811$	−26.8425	−0.942567	−0.471283	+	0.881982i	$0.656209\pi$
$812$	0	0
$813$	−13.2450	−0.464523
$814$	0	0
$815$	−37.3368	−1.30785
$816$	0	0
$817$	3.76272	0.131641
$818$	0	0
$819$	2.46876	0.0862654
$820$	0	0
$821$	23.0110	0.803091	0.401545	−	0.915839i	$-0.368473\pi$
$821$	23.0110	0.803091	0.401545	+	0.915839i	$0.368473\pi$
$822$	0	0
$823$	−26.5219	−0.924496	−0.462248	−	0.886751i	$-0.652957\pi$
$823$	−26.5219	−0.924496	−0.462248	+	0.886751i	$0.652957\pi$
$824$	0	0
$825$	−0.803441	−0.0279722
$826$	0	0
$827$	54.9516	1.91086	0.955428	−	0.295224i	$-0.0953944\pi$
$827$	54.9516	1.91086	0.955428	+	0.295224i	$0.0953944\pi$
$828$	0	0
$829$	42.6804	1.48235	0.741176	−	0.671310i	$-0.234268\pi$
$829$	42.6804	1.48235	0.741176	+	0.671310i	$0.234268\pi$
$830$	0	0
$831$	4.86516	0.168771
$832$	0	0
$833$	−4.48655	−0.155450
$834$	0	0
$835$	−30.3400	−1.04996
$836$	0	0
$837$	−0.759923	−0.0262668
$838$	0	0
$839$	−20.0404	−0.691871	−0.345936	−	0.938258i	$-0.612438\pi$
$839$	−20.0404	−0.691871	−0.345936	+	0.938258i	$0.612438\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−25.5871	−0.881266
$844$	0	0
$845$	−33.9709	−1.16863
$846$	0	0
$847$	34.5576	1.18741
$848$	0	0
$849$	−10.1261	−0.347528
$850$	0	0
$851$	−7.77804	−0.266628
$852$	0	0
$853$	−11.1868	−0.383028	−0.191514	−	0.981490i	$-0.561340\pi$
$853$	−11.1868	−0.383028	−0.191514	+	0.981490i	$0.561340\pi$
$854$	0	0
$855$	−2.50433	−0.0856464
$856$	0	0
$857$	−52.6242	−1.79761	−0.898805	−	0.438349i	$-0.855563\pi$
$857$	−52.6242	−1.79761	−0.898805	+	0.438349i	$0.855563\pi$
$858$	0	0
$859$	−6.92802	−0.236381	−0.118191	−	0.992991i	$-0.537709\pi$
$859$	−6.92802	−0.236381	−0.118191	+	0.992991i	$0.537709\pi$
$860$	0	0
$861$	32.2052	1.09755
$862$	0	0
$863$	−49.1189	−1.67203	−0.836013	−	0.548710i	$-0.815119\pi$
$863$	−49.1189	−1.67203	−0.836013	+	0.548710i	$0.815119\pi$
$864$	0	0
$865$	−28.3968	−0.965520
$866$	0	0
$867$	−14.8443	−0.504137
$868$	0	0
$869$	1.49240	0.0506264
$870$	0	0
$871$	−0.215212	−0.00729217
$872$	0	0
$873$	9.80700	0.331917
$874$	0	0
$875$	21.6186	0.730843
$876$	0	0
$877$	15.9647	0.539090	0.269545	−	0.962988i	$-0.413127\pi$
$877$	15.9647	0.539090	0.269545	+	0.962988i	$0.413127\pi$
$878$	0	0
$879$	27.6514	0.932658
$880$	0	0
$881$	36.7371	1.23770	0.618852	−	0.785507i	$-0.287598\pi$
$881$	36.7371	1.23770	0.618852	+	0.785507i	$0.287598\pi$
$882$	0	0
$883$	−39.0226	−1.31321	−0.656607	−	0.754233i	$-0.728009\pi$
$883$	−39.0226	−1.31321	−0.656607	+	0.754233i	$0.728009\pi$
$884$	0	0
$885$	32.0683	1.07796
$886$	0	0
$887$	14.0804	0.472772	0.236386	−	0.971659i	$-0.424037\pi$
$887$	14.0804	0.472772	0.236386	+	0.971659i	$0.424037\pi$
$888$	0	0
$889$	−42.9187	−1.43945
$890$	0	0
$891$	−0.319748	−0.0107120
$892$	0	0
$893$	0.0515356	0.00172457
$894$	0	0
$895$	−32.6559	−1.09157
$896$	0	0
$897$	−0.778524	−0.0259942
$898$	0	0
$899$	−0.759923	−0.0253449
$900$	0	0
$901$	16.4227	0.547120
$902$	0	0
$903$	13.0592	0.434582
$904$	0	0
$905$	−29.7916	−0.990307
$906$	0	0
$907$	−48.8506	−1.62206	−0.811028	−	0.585007i	$-0.801092\pi$
$907$	−48.8506	−1.62206	−0.811028	+	0.585007i	$0.801092\pi$
$908$	0	0
$909$	−0.127442	−0.00422699
$910$	0	0
$911$	2.02211	0.0669954	0.0334977	−	0.999439i	$-0.489335\pi$
$911$	2.02211	0.0669954	0.0334977	+	0.999439i	$0.489335\pi$
$912$	0	0
$913$	−0.581799	−0.0192547
$914$	0	0
$915$	29.1587	0.963956
$916$	0	0
$917$	9.45374	0.312190
$918$	0	0
$919$	39.3164	1.29693	0.648465	−	0.761245i	$-0.275411\pi$
$919$	39.3164	1.29693	0.648465	+	0.761245i	$0.275411\pi$
$920$	0	0
$921$	28.8959	0.952153
$922$	0	0
$923$	6.77949	0.223150
$924$	0	0
$925$	−19.5441	−0.642607
$926$	0	0
$927$	2.60543	0.0855737
$928$	0	0
$929$	−26.6454	−0.874206	−0.437103	−	0.899412i	$-0.643995\pi$
$929$	−26.6454	−0.874206	−0.437103	+	0.899412i	$0.643995\pi$
$930$	0	0
$931$	−2.79195	−0.0915024
$932$	0	0
$933$	−0.879658	−0.0287987
$934$	0	0
$935$	1.28678	0.0420823
$936$	0	0
$937$	−0.670073	−0.0218903	−0.0109452	−	0.999940i	$-0.503484\pi$
$937$	−0.670073	−0.0218903	−0.0109452	+	0.999940i	$0.503484\pi$
$938$	0	0
$939$	−23.7373	−0.774636
$940$	0	0
$941$	11.8283	0.385593	0.192797	−	0.981239i	$-0.438244\pi$
$941$	11.8283	0.385593	0.192797	+	0.981239i	$0.438244\pi$
$942$	0	0
$943$	−10.1559	−0.330722
$944$	0	0
$945$	−8.69172	−0.282742
$946$	0	0
$947$	−34.0658	−1.10699	−0.553494	−	0.832853i	$-0.686706\pi$
$947$	−34.0658	−1.10699	−0.553494	+	0.832853i	$0.686706\pi$
$948$	0	0
$949$	8.63140	0.280187
$950$	0	0
$951$	−4.87172	−0.157976
$952$	0	0
$953$	24.8288	0.804285	0.402142	−	0.915577i	$-0.368266\pi$
$953$	24.8288	0.804285	0.402142	+	0.915577i	$0.368266\pi$
$954$	0	0
$955$	29.7627	0.963099
$956$	0	0
$957$	−0.319748	−0.0103360
$958$	0	0
$959$	−49.0623	−1.58430
$960$	0	0
$961$	−30.4225	−0.981372
$962$	0	0
$963$	7.32175	0.235940
$964$	0	0
$965$	17.2144	0.554150
$966$	0	0
$967$	1.04696	0.0336681	0.0168340	−	0.999858i	$-0.494641\pi$
$967$	1.04696	0.0336681	0.0168340	+	0.999858i	$0.494641\pi$
$968$	0	0
$969$	1.34150	0.0430953
$970$	0	0
$971$	−27.3776	−0.878590	−0.439295	−	0.898343i	$-0.644772\pi$
$971$	−27.3776	−0.878590	−0.439295	+	0.898343i	$0.644772\pi$
$972$	0	0
$973$	−19.4358	−0.623083
$974$	0	0
$975$	−1.95622	−0.0626492
$976$	0	0
$977$	−12.0324	−0.384949	−0.192475	−	0.981302i	$-0.561651\pi$
$977$	−12.0324	−0.384949	−0.192475	+	0.981302i	$0.561651\pi$
$978$	0	0
$979$	5.00075	0.159825
$980$	0	0
$981$	8.76847	0.279956
$982$	0	0
$983$	39.1661	1.24921	0.624603	−	0.780943i	$-0.285261\pi$
$983$	39.1661	1.24921	0.624603	+	0.780943i	$0.285261\pi$
$984$	0	0
$985$	−60.5968	−1.93078
$986$	0	0
$987$	0.178863	0.00569328
$988$	0	0
$989$	−4.11821	−0.130952
$990$	0	0
$991$	30.0370	0.954156	0.477078	−	0.878861i	$-0.341696\pi$
$991$	30.0370	0.954156	0.477078	+	0.878861i	$0.341696\pi$
$992$	0	0
$993$	−2.02968	−0.0644100
$994$	0	0
$995$	−26.7690	−0.848634
$996$	0	0
$997$	3.90829	0.123777	0.0618884	−	0.998083i	$-0.480288\pi$
$997$	3.90829	0.123777	0.0618884	+	0.998083i	$0.480288\pi$
$998$	0	0
$999$	−7.77804	−0.246086

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.d.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.d.1.8	✓	8	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.74094 q^{5} -3.17108 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.74094 q^{5} -3.17108 q^{7} +1.00000 q^{9} -0.319748 q^{11} -0.778524 q^{13} +2.74094 q^{15} -1.46825 q^{17} -0.913678 q^{19} -3.17108 q^{21} +1.00000 q^{23} +2.51273 q^{25} +1.00000 q^{27} +1.00000 q^{29} -0.759923 q^{31} -0.319748 q^{33} -8.69172 q^{35} -7.77804 q^{37} -0.778524 q^{39} -10.1559 q^{41} -4.11821 q^{43} +2.74094 q^{45} -0.0564046 q^{47} +3.05572 q^{49} -1.46825 q^{51} -11.1853 q^{53} -0.876409 q^{55} -0.913678 q^{57} +11.6998 q^{59} +10.6382 q^{61} -3.17108 q^{63} -2.13388 q^{65} +0.276436 q^{67} +1.00000 q^{69} -8.70814 q^{71} -11.0869 q^{73} +2.51273 q^{75} +1.01394 q^{77} -4.66744 q^{79} +1.00000 q^{81} +1.81955 q^{83} -4.02437 q^{85} +1.00000 q^{87} -15.6397 q^{89} +2.46876 q^{91} -0.759923 q^{93} -2.50433 q^{95} +9.80700 q^{97} -0.319748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9	\( 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9} - 5 q^{11} - 4 q^{13} - 5 q^{15} - 3 q^{17} - 5 q^{19} - 4 q^{21} + 8 q^{23} - 5 q^{25} + 8 q^{27} + 8 q^{29} - 2 q^{31} - 5 q^{33} - 15 q^{35} - 10 q^{37} - 4 q^{39} - 11 q^{41} - 7 q^{43} - 5 q^{45} - 14 q^{47} - 18 q^{49} - 3 q^{51} - 15 q^{53} - 17 q^{55} - 5 q^{57} + 4 q^{59} + q^{61} - 4 q^{63} - 5 q^{67} + 8 q^{69} - q^{71} - 21 q^{73} - 5 q^{75} - 8 q^{79} + 8 q^{81} + 3 q^{83} + 8 q^{87} - 20 q^{89} - 7 q^{91} - 2 q^{93} - 3 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9 - 5 * q^11 - 4 * q^13 - 5 * q^15 - 3 * q^17 - 5 * q^19 - 4 * q^21 + 8 * q^23 - 5 * q^25 + 8 * q^27 + 8 * q^29 - 2 * q^31 - 5 * q^33 - 15 * q^35 - 10 * q^37 - 4 * q^39 - 11 * q^41 - 7 * q^43 - 5 * q^45 - 14 * q^47 - 18 * q^49 - 3 * q^51 - 15 * q^53 - 17 * q^55 - 5 * q^57 + 4 * q^59 + q^61 - 4 * q^63 - 5 * q^67 + 8 * q^69 - q^71 - 21 * q^73 - 5 * q^75 - 8 * q^79 + 8 * q^81 + 3 * q^83 + 8 * q^87 - 20 * q^89 - 7 * q^91 - 2 * q^93 - 3 * q^95 - 7 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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