LMFDB - Embedded newform 8004.2.a.d.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:	$8$
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			1.1
Root			$-1.85152$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.41642 q^{5} +2.05098 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.41642 q^{5} +2.05098 q^{7} +1.00000 q^{9} +1.14688 q^{11} -4.51492 q^{13} -3.41642 q^{15} +0.446852 q^{17} -2.11134 q^{19} +2.05098 q^{21} +1.00000 q^{23} +6.67190 q^{25} +1.00000 q^{27} +1.00000 q^{29} +6.57178 q^{31} +1.14688 q^{33} -7.00700 q^{35} -4.89567 q^{37} -4.51492 q^{39} +3.92571 q^{41} +3.16338 q^{43} -3.41642 q^{45} -0.451968 q^{47} -2.79348 q^{49} +0.446852 q^{51} +4.11866 q^{53} -3.91820 q^{55} -2.11134 q^{57} -6.31529 q^{59} -8.75827 q^{61} +2.05098 q^{63} +15.4248 q^{65} -10.6292 q^{67} +1.00000 q^{69} +13.3216 q^{71} -12.6838 q^{73} +6.67190 q^{75} +2.35222 q^{77} +2.30778 q^{79} +1.00000 q^{81} -4.59898 q^{83} -1.52663 q^{85} +1.00000 q^{87} +14.8381 q^{89} -9.26001 q^{91} +6.57178 q^{93} +7.21320 q^{95} +8.14524 q^{97} +1.14688 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.41642	−1.52787	−0.763934	−	0.645294i	$-0.776735\pi$
$5$	−3.41642	−1.52787	−0.763934	+	0.645294i	$0.776735\pi$
$6$	0	0
$7$	2.05098	0.775198	0.387599	−	0.921828i	$-0.373305\pi$
$7$	2.05098	0.775198	0.387599	+	0.921828i	$0.373305\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.14688	0.345796	0.172898	−	0.984940i	$-0.444687\pi$
$11$	1.14688	0.345796	0.172898	+	0.984940i	$0.444687\pi$
$12$	0	0
$13$	−4.51492	−1.25221	−0.626106	−	0.779738i	$-0.715352\pi$
$13$	−4.51492	−1.25221	−0.626106	+	0.779738i	$0.715352\pi$
$14$	0	0
$15$	−3.41642	−0.882115
$16$	0	0
$17$	0.446852	0.108377	0.0541887	−	0.998531i	$-0.482743\pi$
$17$	0.446852	0.108377	0.0541887	+	0.998531i	$0.482743\pi$
$18$	0	0
$19$	−2.11134	−0.484374	−0.242187	−	0.970230i	$-0.577865\pi$
$19$	−2.11134	−0.484374	−0.242187	+	0.970230i	$0.577865\pi$
$20$	0	0
$21$	2.05098	0.447561
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	6.67190	1.33438
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	6.57178	1.18033	0.590163	−	0.807284i	$-0.299063\pi$
$31$	6.57178	1.18033	0.590163	+	0.807284i	$0.299063\pi$
$32$	0	0
$33$	1.14688	0.199645
$34$	0	0
$35$	−7.00700	−1.18440
$36$	0	0
$37$	−4.89567	−0.804843	−0.402421	−	0.915455i	$-0.631831\pi$
$37$	−4.89567	−0.804843	−0.402421	+	0.915455i	$0.631831\pi$
$38$	0	0
$39$	−4.51492	−0.722965
$40$	0	0
$41$	3.92571	0.613093	0.306547	−	0.951856i	$-0.400826\pi$
$41$	3.92571	0.613093	0.306547	+	0.951856i	$0.400826\pi$
$42$	0	0
$43$	3.16338	0.482410	0.241205	−	0.970474i	$-0.422457\pi$
$43$	3.16338	0.482410	0.241205	+	0.970474i	$0.422457\pi$
$44$	0	0
$45$	−3.41642	−0.509289
$46$	0	0
$47$	−0.451968	−0.0659262	−0.0329631	−	0.999457i	$-0.510494\pi$
$47$	−0.451968	−0.0659262	−0.0329631	+	0.999457i	$0.510494\pi$
$48$	0	0
$49$	−2.79348	−0.399068
$50$	0	0
$51$	0.446852	0.0625717
$52$	0	0
$53$	4.11866	0.565742	0.282871	−	0.959158i	$-0.408713\pi$
$53$	4.11866	0.565742	0.282871	+	0.959158i	$0.408713\pi$
$54$	0	0
$55$	−3.91820	−0.528330
$56$	0	0
$57$	−2.11134	−0.279653
$58$	0	0
$59$	−6.31529	−0.822180	−0.411090	−	0.911595i	$-0.634852\pi$
$59$	−6.31529	−0.822180	−0.411090	+	0.911595i	$0.634852\pi$
$60$	0	0
$61$	−8.75827	−1.12138	−0.560690	−	0.828025i	$-0.689464\pi$
$61$	−8.75827	−1.12138	−0.560690	+	0.828025i	$0.689464\pi$
$62$	0	0
$63$	2.05098	0.258399
$64$	0	0
$65$	15.4248	1.91322
$66$	0	0
$67$	−10.6292	−1.29856	−0.649280	−	0.760549i	$-0.724930\pi$
$67$	−10.6292	−1.29856	−0.649280	+	0.760549i	$0.724930\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	13.3216	1.58099	0.790494	−	0.612469i	$-0.209824\pi$
$71$	13.3216	1.58099	0.790494	+	0.612469i	$0.209824\pi$
$72$	0	0
$73$	−12.6838	−1.48452	−0.742262	−	0.670110i	$-0.766247\pi$
$73$	−12.6838	−1.48452	−0.742262	+	0.670110i	$0.766247\pi$
$74$	0	0
$75$	6.67190	0.770405
$76$	0	0
$77$	2.35222	0.268060
$78$	0	0
$79$	2.30778	0.259646	0.129823	−	0.991537i	$-0.458559\pi$
$79$	2.30778	0.259646	0.129823	+	0.991537i	$0.458559\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.59898	−0.504804	−0.252402	−	0.967622i	$-0.581221\pi$
$83$	−4.59898	−0.504804	−0.252402	+	0.967622i	$0.581221\pi$
$84$	0	0
$85$	−1.52663	−0.165586
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	14.8381	1.57284	0.786418	−	0.617695i	$-0.211933\pi$
$89$	14.8381	1.57284	0.786418	+	0.617695i	$0.211933\pi$
$90$	0	0
$91$	−9.26001	−0.970713
$92$	0	0
$93$	6.57178	0.681462
$94$	0	0
$95$	7.21320	0.740059
$96$	0	0
$97$	8.14524	0.827024	0.413512	−	0.910499i	$-0.364302\pi$
$97$	8.14524	0.827024	0.413512	+	0.910499i	$0.364302\pi$
$98$	0	0
$99$	1.14688	0.115265
$100$	0	0
$101$	−10.8719	−1.08180	−0.540899	−	0.841088i	$-0.681916\pi$
$101$	−10.8719	−1.08180	−0.540899	+	0.841088i	$0.681916\pi$
$102$	0	0
$103$	14.4916	1.42790	0.713950	−	0.700196i	$-0.246904\pi$
$103$	14.4916	1.42790	0.713950	+	0.700196i	$0.246904\pi$
$104$	0	0
$105$	−7.00700	−0.683814
$106$	0	0
$107$	−11.0590	−1.06912	−0.534558	−	0.845132i	$-0.679522\pi$
$107$	−11.0590	−1.06912	−0.534558	+	0.845132i	$0.679522\pi$
$108$	0	0
$109$	−5.38049	−0.515358	−0.257679	−	0.966231i	$-0.582958\pi$
$109$	−5.38049	−0.515358	−0.257679	+	0.966231i	$0.582958\pi$
$110$	0	0
$111$	−4.89567	−0.464676
$112$	0	0
$113$	−2.54011	−0.238954	−0.119477	−	0.992837i	$-0.538122\pi$
$113$	−2.54011	−0.238954	−0.119477	+	0.992837i	$0.538122\pi$
$114$	0	0
$115$	−3.41642	−0.318582
$116$	0	0
$117$	−4.51492	−0.417404
$118$	0	0
$119$	0.916484	0.0840139
$120$	0	0
$121$	−9.68468	−0.880425
$122$	0	0
$123$	3.92571	0.353969
$124$	0	0
$125$	−5.71192	−0.510889
$126$	0	0
$127$	−7.41710	−0.658161	−0.329080	−	0.944302i	$-0.606739\pi$
$127$	−7.41710	−0.658161	−0.329080	+	0.944302i	$0.606739\pi$
$128$	0	0
$129$	3.16338	0.278520
$130$	0	0
$131$	−6.83180	−0.596897	−0.298448	−	0.954426i	$-0.596469\pi$
$131$	−6.83180	−0.596897	−0.298448	+	0.954426i	$0.596469\pi$
$132$	0	0
$133$	−4.33031	−0.375486
$134$	0	0
$135$	−3.41642	−0.294038
$136$	0	0
$137$	−14.7282	−1.25832	−0.629159	−	0.777276i	$-0.716601\pi$
$137$	−14.7282	−1.25832	−0.629159	+	0.777276i	$0.716601\pi$
$138$	0	0
$139$	−20.3372	−1.72497	−0.862487	−	0.506079i	$-0.831095\pi$
$139$	−20.3372	−1.72497	−0.862487	+	0.506079i	$0.831095\pi$
$140$	0	0
$141$	−0.451968	−0.0380625
$142$	0	0
$143$	−5.17805	−0.433010
$144$	0	0
$145$	−3.41642	−0.283718
$146$	0	0
$147$	−2.79348	−0.230402
$148$	0	0
$149$	1.98828	0.162887	0.0814433	−	0.996678i	$-0.474047\pi$
$149$	1.98828	0.162887	0.0814433	+	0.996678i	$0.474047\pi$
$150$	0	0
$151$	−0.688066	−0.0559940	−0.0279970	−	0.999608i	$-0.508913\pi$
$151$	−0.688066	−0.0559940	−0.0279970	+	0.999608i	$0.508913\pi$
$152$	0	0
$153$	0.446852	0.0361258
$154$	0	0
$155$	−22.4519	−1.80338
$156$	0	0
$157$	7.26724	0.579989	0.289994	−	0.957028i	$-0.406347\pi$
$157$	7.26724	0.579989	0.289994	+	0.957028i	$0.406347\pi$
$158$	0	0
$159$	4.11866	0.326631
$160$	0	0
$161$	2.05098	0.161640
$162$	0	0
$163$	−21.3903	−1.67542	−0.837711	−	0.546114i	$-0.816106\pi$
$163$	−21.3903	−1.67542	−0.837711	+	0.546114i	$0.816106\pi$
$164$	0	0
$165$	−3.91820	−0.305032
$166$	0	0
$167$	3.78952	0.293242	0.146621	−	0.989193i	$-0.453160\pi$
$167$	3.78952	0.293242	0.146621	+	0.989193i	$0.453160\pi$
$168$	0	0
$169$	7.38447	0.568036
$170$	0	0
$171$	−2.11134	−0.161458
$172$	0	0
$173$	21.3538	1.62350	0.811750	−	0.584005i	$-0.198515\pi$
$173$	21.3538	1.62350	0.811750	+	0.584005i	$0.198515\pi$
$174$	0	0
$175$	13.6839	1.03441
$176$	0	0
$177$	−6.31529	−0.474686
$178$	0	0
$179$	−19.0982	−1.42746	−0.713732	−	0.700418i	$-0.752997\pi$
$179$	−19.0982	−1.42746	−0.713732	+	0.700418i	$0.752997\pi$
$180$	0	0
$181$	10.7729	0.800747	0.400374	−	0.916352i	$-0.368880\pi$
$181$	10.7729	0.800747	0.400374	+	0.916352i	$0.368880\pi$
$182$	0	0
$183$	−8.75827	−0.647430
$184$	0	0
$185$	16.7256	1.22969
$186$	0	0
$187$	0.512483	0.0374765
$188$	0	0
$189$	2.05098	0.149187
$190$	0	0
$191$	−13.8685	−1.00349	−0.501746	−	0.865015i	$-0.667309\pi$
$191$	−13.8685	−1.00349	−0.501746	+	0.865015i	$0.667309\pi$
$192$	0	0
$193$	24.4103	1.75709	0.878545	−	0.477659i	$-0.158515\pi$
$193$	24.4103	1.75709	0.878545	+	0.477659i	$0.158515\pi$
$194$	0	0
$195$	15.4248	1.10460
$196$	0	0
$197$	−15.9348	−1.13531	−0.567655	−	0.823267i	$-0.692149\pi$
$197$	−15.9348	−1.13531	−0.567655	+	0.823267i	$0.692149\pi$
$198$	0	0
$199$	−22.3201	−1.58223	−0.791115	−	0.611667i	$-0.790499\pi$
$199$	−22.3201	−1.58223	−0.791115	+	0.611667i	$0.790499\pi$
$200$	0	0
$201$	−10.6292	−0.749724
$202$	0	0
$203$	2.05098	0.143951
$204$	0	0
$205$	−13.4119	−0.936725
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−2.42144	−0.167494
$210$	0	0
$211$	−3.36302	−0.231520	−0.115760	−	0.993277i	$-0.536930\pi$
$211$	−3.36302	−0.231520	−0.115760	+	0.993277i	$0.536930\pi$
$212$	0	0
$213$	13.3216	0.912784
$214$	0	0
$215$	−10.8074	−0.737060
$216$	0	0
$217$	13.4786	0.914987
$218$	0	0
$219$	−12.6838	−0.857090
$220$	0	0
$221$	−2.01750	−0.135712
$222$	0	0
$223$	−16.0725	−1.07629	−0.538147	−	0.842851i	$-0.680875\pi$
$223$	−16.0725	−1.07629	−0.538147	+	0.842851i	$0.680875\pi$
$224$	0	0
$225$	6.67190	0.444794
$226$	0	0
$227$	18.1471	1.20447	0.602233	−	0.798320i	$-0.294278\pi$
$227$	18.1471	1.20447	0.602233	+	0.798320i	$0.294278\pi$
$228$	0	0
$229$	−7.11132	−0.469929	−0.234965	−	0.972004i	$-0.575497\pi$
$229$	−7.11132	−0.469929	−0.234965	+	0.972004i	$0.575497\pi$
$230$	0	0
$231$	2.35222	0.154765
$232$	0	0
$233$	3.07312	0.201327	0.100663	−	0.994921i	$-0.467903\pi$
$233$	3.07312	0.201327	0.100663	+	0.994921i	$0.467903\pi$
$234$	0	0
$235$	1.54411	0.100727
$236$	0	0
$237$	2.30778	0.149906
$238$	0	0
$239$	7.61297	0.492442	0.246221	−	0.969214i	$-0.420811\pi$
$239$	7.61297	0.492442	0.246221	+	0.969214i	$0.420811\pi$
$240$	0	0
$241$	−20.4791	−1.31917	−0.659587	−	0.751629i	$-0.729269\pi$
$241$	−20.4791	−1.31917	−0.659587	+	0.751629i	$0.729269\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	9.54368	0.609724
$246$	0	0
$247$	9.53251	0.606539
$248$	0	0
$249$	−4.59898	−0.291449
$250$	0	0
$251$	10.7623	0.679313	0.339656	−	0.940550i	$-0.389689\pi$
$251$	10.7623	0.679313	0.339656	+	0.940550i	$0.389689\pi$
$252$	0	0
$253$	1.14688	0.0721034
$254$	0	0
$255$	−1.52663	−0.0956014
$256$	0	0
$257$	−19.2212	−1.19899	−0.599493	−	0.800380i	$-0.704631\pi$
$257$	−19.2212	−1.19899	−0.599493	+	0.800380i	$0.704631\pi$
$258$	0	0
$259$	−10.0409	−0.623913
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	8.85390	0.545955	0.272978	−	0.962020i	$-0.411992\pi$
$263$	8.85390	0.545955	0.272978	+	0.962020i	$0.411992\pi$
$264$	0	0
$265$	−14.0711	−0.864379
$266$	0	0
$267$	14.8381	0.908077
$268$	0	0
$269$	9.97519	0.608198	0.304099	−	0.952640i	$-0.401645\pi$
$269$	9.97519	0.608198	0.304099	+	0.952640i	$0.401645\pi$
$270$	0	0
$271$	−14.7260	−0.894541	−0.447270	−	0.894399i	$-0.647604\pi$
$271$	−14.7260	−0.894541	−0.447270	+	0.894399i	$0.647604\pi$
$272$	0	0
$273$	−9.26001	−0.560441
$274$	0	0
$275$	7.65184	0.461423
$276$	0	0
$277$	0.848598	0.0509873	0.0254937	−	0.999675i	$-0.491884\pi$
$277$	0.848598	0.0509873	0.0254937	+	0.999675i	$0.491884\pi$
$278$	0	0
$279$	6.57178	0.393442
$280$	0	0
$281$	−22.8691	−1.36426	−0.682128	−	0.731233i	$-0.738945\pi$
$281$	−22.8691	−1.36426	−0.682128	+	0.731233i	$0.738945\pi$
$282$	0	0
$283$	−8.10218	−0.481624	−0.240812	−	0.970572i	$-0.577414\pi$
$283$	−8.10218	−0.481624	−0.240812	+	0.970572i	$0.577414\pi$
$284$	0	0
$285$	7.21320	0.427273
$286$	0	0
$287$	8.05156	0.475268
$288$	0	0
$289$	−16.8003	−0.988254
$290$	0	0
$291$	8.14524	0.477482
$292$	0	0
$293$	24.6949	1.44269	0.721344	−	0.692577i	$-0.243525\pi$
$293$	24.6949	1.44269	0.721344	+	0.692577i	$0.243525\pi$
$294$	0	0
$295$	21.5757	1.25618
$296$	0	0
$297$	1.14688	0.0665485
$298$	0	0
$299$	−4.51492	−0.261104
$300$	0	0
$301$	6.48803	0.373964
$302$	0	0
$303$	−10.8719	−0.624576
$304$	0	0
$305$	29.9219	1.71332
$306$	0	0
$307$	−7.75138	−0.442395	−0.221197	−	0.975229i	$-0.570997\pi$
$307$	−7.75138	−0.442395	−0.221197	+	0.975229i	$0.570997\pi$
$308$	0	0
$309$	14.4916	0.824399
$310$	0	0
$311$	11.2388	0.637292	0.318646	−	0.947874i	$-0.396772\pi$
$311$	11.2388	0.637292	0.318646	+	0.947874i	$0.396772\pi$
$312$	0	0
$313$	12.3629	0.698794	0.349397	−	0.936975i	$-0.386386\pi$
$313$	12.3629	0.698794	0.349397	+	0.936975i	$0.386386\pi$
$314$	0	0
$315$	−7.00700	−0.394800
$316$	0	0
$317$	−10.5402	−0.591994	−0.295997	−	0.955189i	$-0.595652\pi$
$317$	−10.5402	−0.591994	−0.295997	+	0.955189i	$0.595652\pi$
$318$	0	0
$319$	1.14688	0.0642127
$320$	0	0
$321$	−11.0590	−0.617254
$322$	0	0
$323$	−0.943454	−0.0524952
$324$	0	0
$325$	−30.1231	−1.67093
$326$	0	0
$327$	−5.38049	−0.297542
$328$	0	0
$329$	−0.926977	−0.0511059
$330$	0	0
$331$	−26.4647	−1.45463	−0.727316	−	0.686303i	$-0.759233\pi$
$331$	−26.4647	−1.45463	−0.727316	+	0.686303i	$0.759233\pi$
$332$	0	0
$333$	−4.89567	−0.268281
$334$	0	0
$335$	36.3137	1.98403
$336$	0	0
$337$	−24.0080	−1.30780	−0.653901	−	0.756580i	$-0.726869\pi$
$337$	−24.0080	−1.30780	−0.653901	+	0.756580i	$0.726869\pi$
$338$	0	0
$339$	−2.54011	−0.137960
$340$	0	0
$341$	7.53701	0.408152
$342$	0	0
$343$	−20.0862	−1.08455
$344$	0	0
$345$	−3.41642	−0.183934
$346$	0	0
$347$	−14.2863	−0.766929	−0.383465	−	0.923556i	$-0.625269\pi$
$347$	−14.2863	−0.766929	−0.383465	+	0.923556i	$0.625269\pi$
$348$	0	0
$349$	31.7682	1.70051	0.850257	−	0.526368i	$-0.176446\pi$
$349$	31.7682	1.70051	0.850257	+	0.526368i	$0.176446\pi$
$350$	0	0
$351$	−4.51492	−0.240988
$352$	0	0
$353$	19.4300	1.03416	0.517078	−	0.855938i	$-0.327020\pi$
$353$	19.4300	1.03416	0.517078	+	0.855938i	$0.327020\pi$
$354$	0	0
$355$	−45.5123	−2.41554
$356$	0	0
$357$	0.916484	0.0485055
$358$	0	0
$359$	13.6319	0.719465	0.359733	−	0.933055i	$-0.382868\pi$
$359$	13.6319	0.719465	0.359733	+	0.933055i	$0.382868\pi$
$360$	0	0
$361$	−14.5423	−0.765382
$362$	0	0
$363$	−9.68468	−0.508314
$364$	0	0
$365$	43.3331	2.26816
$366$	0	0
$367$	−27.1439	−1.41690	−0.708450	−	0.705761i	$-0.750605\pi$
$367$	−27.1439	−1.41690	−0.708450	+	0.705761i	$0.750605\pi$
$368$	0	0
$369$	3.92571	0.204364
$370$	0	0
$371$	8.44730	0.438562
$372$	0	0
$373$	38.3540	1.98589	0.992947	−	0.118562i	$-0.0378284\pi$
$373$	38.3540	1.98589	0.992947	+	0.118562i	$0.0378284\pi$
$374$	0	0
$375$	−5.71192	−0.294962
$376$	0	0
$377$	−4.51492	−0.232530
$378$	0	0
$379$	−35.1885	−1.80751	−0.903757	−	0.428046i	$-0.859202\pi$
$379$	−35.1885	−1.80751	−0.903757	+	0.428046i	$0.859202\pi$
$380$	0	0
$381$	−7.41710	−0.379989
$382$	0	0
$383$	−11.7678	−0.601307	−0.300653	−	0.953733i	$-0.597205\pi$
$383$	−11.7678	−0.601307	−0.300653	+	0.953733i	$0.597205\pi$
$384$	0	0
$385$	−8.03616	−0.409561
$386$	0	0
$387$	3.16338	0.160803
$388$	0	0
$389$	−16.4904	−0.836096	−0.418048	−	0.908425i	$-0.637286\pi$
$389$	−16.4904	−0.836096	−0.418048	+	0.908425i	$0.637286\pi$
$390$	0	0
$391$	0.446852	0.0225983
$392$	0	0
$393$	−6.83180	−0.344618
$394$	0	0
$395$	−7.88434	−0.396704
$396$	0	0
$397$	7.18614	0.360662	0.180331	−	0.983606i	$-0.442283\pi$
$397$	7.18614	0.360662	0.180331	+	0.983606i	$0.442283\pi$
$398$	0	0
$399$	−4.33031	−0.216787
$400$	0	0
$401$	−0.904789	−0.0451830	−0.0225915	−	0.999745i	$-0.507192\pi$
$401$	−0.904789	−0.0451830	−0.0225915	+	0.999745i	$0.507192\pi$
$402$	0	0
$403$	−29.6710	−1.47802
$404$	0	0
$405$	−3.41642	−0.169763
$406$	0	0
$407$	−5.61472	−0.278311
$408$	0	0
$409$	35.7504	1.76774	0.883872	−	0.467730i	$-0.154928\pi$
$409$	35.7504	1.76774	0.883872	+	0.467730i	$0.154928\pi$
$410$	0	0
$411$	−14.7282	−0.726491
$412$	0	0
$413$	−12.9525	−0.637353
$414$	0	0
$415$	15.7120	0.771274
$416$	0	0
$417$	−20.3372	−0.995915
$418$	0	0
$419$	38.9086	1.90081	0.950404	−	0.311018i	$-0.100670\pi$
$419$	38.9086	1.90081	0.950404	+	0.311018i	$0.100670\pi$
$420$	0	0
$421$	−28.3864	−1.38347	−0.691735	−	0.722151i	$-0.743154\pi$
$421$	−28.3864	−1.38347	−0.691735	+	0.722151i	$0.743154\pi$
$422$	0	0
$423$	−0.451968	−0.0219754
$424$	0	0
$425$	2.98135	0.144617
$426$	0	0
$427$	−17.9630	−0.869292
$428$	0	0
$429$	−5.17805	−0.249998
$430$	0	0
$431$	−0.922354	−0.0444282	−0.0222141	−	0.999753i	$-0.507072\pi$
$431$	−0.922354	−0.0444282	−0.0222141	+	0.999753i	$0.507072\pi$
$432$	0	0
$433$	11.8030	0.567217	0.283608	−	0.958940i	$-0.408468\pi$
$433$	11.8030	0.567217	0.283608	+	0.958940i	$0.408468\pi$
$434$	0	0
$435$	−3.41642	−0.163805
$436$	0	0
$437$	−2.11134	−0.100999
$438$	0	0
$439$	21.5722	1.02958	0.514791	−	0.857315i	$-0.327869\pi$
$439$	21.5722	1.02958	0.514791	+	0.857315i	$0.327869\pi$
$440$	0	0
$441$	−2.79348	−0.133023
$442$	0	0
$443$	8.00039	0.380110	0.190055	−	0.981773i	$-0.439133\pi$
$443$	8.00039	0.380110	0.190055	+	0.981773i	$0.439133\pi$
$444$	0	0
$445$	−50.6931	−2.40309
$446$	0	0
$447$	1.98828	0.0940426
$448$	0	0
$449$	−15.4319	−0.728277	−0.364138	−	0.931345i	$-0.618637\pi$
$449$	−15.4319	−0.728277	−0.364138	+	0.931345i	$0.618637\pi$
$450$	0	0
$451$	4.50230	0.212005
$452$	0	0
$453$	−0.688066	−0.0323281
$454$	0	0
$455$	31.6360	1.48312
$456$	0	0
$457$	−35.0563	−1.63987	−0.819933	−	0.572460i	$-0.805989\pi$
$457$	−35.0563	−1.63987	−0.819933	+	0.572460i	$0.805989\pi$
$458$	0	0
$459$	0.446852	0.0208572
$460$	0	0
$461$	−34.9727	−1.62884	−0.814420	−	0.580276i	$-0.802945\pi$
$461$	−34.9727	−1.62884	−0.814420	+	0.580276i	$0.802945\pi$
$462$	0	0
$463$	−17.9578	−0.834569	−0.417284	−	0.908776i	$-0.637018\pi$
$463$	−17.9578	−0.834569	−0.417284	+	0.908776i	$0.637018\pi$
$464$	0	0
$465$	−22.4519	−1.04118
$466$	0	0
$467$	−22.6365	−1.04749	−0.523747	−	0.851874i	$-0.675466\pi$
$467$	−22.6365	−1.04749	−0.523747	+	0.851874i	$0.675466\pi$
$468$	0	0
$469$	−21.8002	−1.00664
$470$	0	0
$471$	7.26724	0.334857
$472$	0	0
$473$	3.62800	0.166816
$474$	0	0
$475$	−14.0866	−0.646339
$476$	0	0
$477$	4.11866	0.188581
$478$	0	0
$479$	3.26212	0.149050	0.0745250	−	0.997219i	$-0.476256\pi$
$479$	3.26212	0.149050	0.0745250	+	0.997219i	$0.476256\pi$
$480$	0	0
$481$	22.1035	1.00783
$482$	0	0
$483$	2.05098	0.0933229
$484$	0	0
$485$	−27.8275	−1.26358
$486$	0	0
$487$	−36.0283	−1.63260	−0.816300	−	0.577628i	$-0.803978\pi$
$487$	−36.0283	−1.63260	−0.816300	+	0.577628i	$0.803978\pi$
$488$	0	0
$489$	−21.3903	−0.967305
$490$	0	0
$491$	−31.4507	−1.41935	−0.709674	−	0.704530i	$-0.751158\pi$
$491$	−31.4507	−1.41935	−0.709674	+	0.704530i	$0.751158\pi$
$492$	0	0
$493$	0.446852	0.0201252
$494$	0	0
$495$	−3.91820	−0.176110
$496$	0	0
$497$	27.3224	1.22558
$498$	0	0
$499$	16.5170	0.739402	0.369701	−	0.929151i	$-0.379460\pi$
$499$	16.5170	0.739402	0.369701	+	0.929151i	$0.379460\pi$
$500$	0	0
$501$	3.78952	0.169303
$502$	0	0
$503$	−30.9673	−1.38076	−0.690381	−	0.723446i	$-0.742557\pi$
$503$	−30.9673	−1.38076	−0.690381	+	0.723446i	$0.742557\pi$
$504$	0	0
$505$	37.1430	1.65284
$506$	0	0
$507$	7.38447	0.327956
$508$	0	0
$509$	−14.5132	−0.643287	−0.321644	−	0.946861i	$-0.604235\pi$
$509$	−14.5132	−0.643287	−0.321644	+	0.946861i	$0.604235\pi$
$510$	0	0
$511$	−26.0142	−1.15080
$512$	0	0
$513$	−2.11134	−0.0932178
$514$	0	0
$515$	−49.5094	−2.18164
$516$	0	0
$517$	−0.518350	−0.0227970
$518$	0	0
$519$	21.3538	0.937329
$520$	0	0
$521$	−4.08813	−0.179104	−0.0895522	−	0.995982i	$-0.528544\pi$
$521$	−4.08813	−0.179104	−0.0895522	+	0.995982i	$0.528544\pi$
$522$	0	0
$523$	21.1992	0.926976	0.463488	−	0.886103i	$-0.346598\pi$
$523$	21.1992	0.926976	0.463488	+	0.886103i	$0.346598\pi$
$524$	0	0
$525$	13.6839	0.597216
$526$	0	0
$527$	2.93661	0.127921
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.31529	−0.274060
$532$	0	0
$533$	−17.7243	−0.767723
$534$	0	0
$535$	37.7822	1.63347
$536$	0	0
$537$	−19.0982	−0.824147
$538$	0	0
$539$	−3.20377	−0.137996
$540$	0	0
$541$	12.4030	0.533245	0.266622	−	0.963801i	$-0.414092\pi$
$541$	12.4030	0.533245	0.266622	+	0.963801i	$0.414092\pi$
$542$	0	0
$543$	10.7729	0.462312
$544$	0	0
$545$	18.3820	0.787399
$546$	0	0
$547$	13.8751	0.593255	0.296627	−	0.954993i	$-0.404138\pi$
$547$	13.8751	0.593255	0.296627	+	0.954993i	$0.404138\pi$
$548$	0	0
$549$	−8.75827	−0.373794
$550$	0	0
$551$	−2.11134	−0.0899460
$552$	0	0
$553$	4.73321	0.201277
$554$	0	0
$555$	16.7256	0.709964
$556$	0	0
$557$	9.49572	0.402347	0.201173	−	0.979556i	$-0.435525\pi$
$557$	9.49572	0.402347	0.201173	+	0.979556i	$0.435525\pi$
$558$	0	0
$559$	−14.2824	−0.604080
$560$	0	0
$561$	0.512483	0.0216370
$562$	0	0
$563$	6.34046	0.267219	0.133609	−	0.991034i	$-0.457343\pi$
$563$	6.34046	0.267219	0.133609	+	0.991034i	$0.457343\pi$
$564$	0	0
$565$	8.67809	0.365090
$566$	0	0
$567$	2.05098	0.0861331
$568$	0	0
$569$	−11.6377	−0.487876	−0.243938	−	0.969791i	$-0.578439\pi$
$569$	−11.6377	−0.487876	−0.243938	+	0.969791i	$0.578439\pi$
$570$	0	0
$571$	−25.7525	−1.07771	−0.538854	−	0.842399i	$-0.681143\pi$
$571$	−25.7525	−1.07771	−0.538854	+	0.842399i	$0.681143\pi$
$572$	0	0
$573$	−13.8685	−0.579367
$574$	0	0
$575$	6.67190	0.278238
$576$	0	0
$577$	−0.196347	−0.00817405	−0.00408702	−	0.999992i	$-0.501301\pi$
$577$	−0.196347	−0.00817405	−0.00408702	+	0.999992i	$0.501301\pi$
$578$	0	0
$579$	24.4103	1.01446
$580$	0	0
$581$	−9.43243	−0.391323
$582$	0	0
$583$	4.72359	0.195631
$584$	0	0
$585$	15.4248	0.637739
$586$	0	0
$587$	3.75537	0.155001	0.0775003	−	0.996992i	$-0.475306\pi$
$587$	3.75537	0.155001	0.0775003	+	0.996992i	$0.475306\pi$
$588$	0	0
$589$	−13.8752	−0.571719
$590$	0	0
$591$	−15.9348	−0.655471
$592$	0	0
$593$	−22.7741	−0.935219	−0.467609	−	0.883935i	$-0.654885\pi$
$593$	−22.7741	−0.935219	−0.467609	+	0.883935i	$0.654885\pi$
$594$	0	0
$595$	−3.13109	−0.128362
$596$	0	0
$597$	−22.3201	−0.913501
$598$	0	0
$599$	0.514422	0.0210187	0.0105093	−	0.999945i	$-0.496655\pi$
$599$	0.514422	0.0210187	0.0105093	+	0.999945i	$0.496655\pi$
$600$	0	0
$601$	−18.2236	−0.743358	−0.371679	−	0.928361i	$-0.621218\pi$
$601$	−18.2236	−0.743358	−0.371679	+	0.928361i	$0.621218\pi$
$602$	0	0
$603$	−10.6292	−0.432854
$604$	0	0
$605$	33.0869	1.34517
$606$	0	0
$607$	−12.4283	−0.504449	−0.252224	−	0.967669i	$-0.581162\pi$
$607$	−12.4283	−0.504449	−0.252224	+	0.967669i	$0.581162\pi$
$608$	0	0
$609$	2.05098	0.0831099
$610$	0	0
$611$	2.04060	0.0825537
$612$	0	0
$613$	−22.1624	−0.895131	−0.447566	−	0.894251i	$-0.647709\pi$
$613$	−22.1624	−0.895131	−0.447566	+	0.894251i	$0.647709\pi$
$614$	0	0
$615$	−13.4119	−0.540819
$616$	0	0
$617$	−12.4373	−0.500705	−0.250352	−	0.968155i	$-0.580547\pi$
$617$	−12.4373	−0.500705	−0.250352	+	0.968155i	$0.580547\pi$
$618$	0	0
$619$	31.1769	1.25311	0.626553	−	0.779379i	$-0.284465\pi$
$619$	31.1769	1.25311	0.626553	+	0.779379i	$0.284465\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	30.4327	1.21926
$624$	0	0
$625$	−13.8452	−0.553809
$626$	0	0
$627$	−2.42144	−0.0967030
$628$	0	0
$629$	−2.18764	−0.0872268
$630$	0	0
$631$	−23.1401	−0.921191	−0.460596	−	0.887610i	$-0.652364\pi$
$631$	−23.1401	−0.921191	−0.460596	+	0.887610i	$0.652364\pi$
$632$	0	0
$633$	−3.36302	−0.133668
$634$	0	0
$635$	25.3399	1.00558
$636$	0	0
$637$	12.6123	0.499718
$638$	0	0
$639$	13.3216	0.526996
$640$	0	0
$641$	26.5718	1.04952	0.524761	−	0.851250i	$-0.324155\pi$
$641$	26.5718	1.04952	0.524761	+	0.851250i	$0.324155\pi$
$642$	0	0
$643$	14.2684	0.562691	0.281345	−	0.959607i	$-0.409219\pi$
$643$	14.2684	0.562691	0.281345	+	0.959607i	$0.409219\pi$
$644$	0	0
$645$	−10.8074	−0.425542
$646$	0	0
$647$	21.4489	0.843242	0.421621	−	0.906772i	$-0.361461\pi$
$647$	21.4489	0.843242	0.421621	+	0.906772i	$0.361461\pi$
$648$	0	0
$649$	−7.24285	−0.284307
$650$	0	0
$651$	13.4786	0.528268
$652$	0	0
$653$	−26.4337	−1.03443	−0.517216	−	0.855855i	$-0.673032\pi$
$653$	−26.4337	−1.03443	−0.517216	+	0.855855i	$0.673032\pi$
$654$	0	0
$655$	23.3403	0.911979
$656$	0	0
$657$	−12.6838	−0.494841
$658$	0	0
$659$	26.2186	1.02133	0.510666	−	0.859779i	$-0.329399\pi$
$659$	26.2186	1.02133	0.510666	+	0.859779i	$0.329399\pi$
$660$	0	0
$661$	29.9343	1.16431	0.582154	−	0.813078i	$-0.302210\pi$
$661$	29.9343	1.16431	0.582154	+	0.813078i	$0.302210\pi$
$662$	0	0
$663$	−2.01750	−0.0783531
$664$	0	0
$665$	14.7941	0.573692
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−16.0725	−0.621398
$670$	0	0
$671$	−10.0446	−0.387769
$672$	0	0
$673$	−10.9988	−0.423971	−0.211986	−	0.977273i	$-0.567993\pi$
$673$	−10.9988	−0.423971	−0.211986	+	0.977273i	$0.567993\pi$
$674$	0	0
$675$	6.67190	0.256802
$676$	0	0
$677$	24.7102	0.949688	0.474844	−	0.880070i	$-0.342504\pi$
$677$	24.7102	0.949688	0.474844	+	0.880070i	$0.342504\pi$
$678$	0	0
$679$	16.7057	0.641107
$680$	0	0
$681$	18.1471	0.695399
$682$	0	0
$683$	8.25274	0.315782	0.157891	−	0.987457i	$-0.449530\pi$
$683$	8.25274	0.315782	0.157891	+	0.987457i	$0.449530\pi$
$684$	0	0
$685$	50.3178	1.92254
$686$	0	0
$687$	−7.11132	−0.271314
$688$	0	0
$689$	−18.5954	−0.708429
$690$	0	0
$691$	−10.9738	−0.417463	−0.208732	−	0.977973i	$-0.566934\pi$
$691$	−10.9738	−0.417463	−0.208732	+	0.977973i	$0.566934\pi$
$692$	0	0
$693$	2.35222	0.0893534
$694$	0	0
$695$	69.4802	2.63553
$696$	0	0
$697$	1.75421	0.0664455
$698$	0	0
$699$	3.07312	0.116236
$700$	0	0
$701$	−43.2229	−1.63251	−0.816254	−	0.577694i	$-0.803953\pi$
$701$	−43.2229	−1.63251	−0.816254	+	0.577694i	$0.803953\pi$
$702$	0	0
$703$	10.3364	0.389845
$704$	0	0
$705$	1.54411	0.0581545
$706$	0	0
$707$	−22.2981	−0.838607
$708$	0	0
$709$	7.16787	0.269195	0.134598	−	0.990900i	$-0.457026\pi$
$709$	7.16787	0.269195	0.134598	+	0.990900i	$0.457026\pi$
$710$	0	0
$711$	2.30778	0.0865485
$712$	0	0
$713$	6.57178	0.246115
$714$	0	0
$715$	17.6904	0.661582
$716$	0	0
$717$	7.61297	0.284312
$718$	0	0
$719$	8.57336	0.319732	0.159866	−	0.987139i	$-0.448894\pi$
$719$	8.57336	0.319732	0.159866	+	0.987139i	$0.448894\pi$
$720$	0	0
$721$	29.7220	1.10691
$722$	0	0
$723$	−20.4791	−0.761625
$724$	0	0
$725$	6.67190	0.247788
$726$	0	0
$727$	−6.72086	−0.249263	−0.124631	−	0.992203i	$-0.539775\pi$
$727$	−6.72086	−0.249263	−0.124631	+	0.992203i	$0.539775\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.41356	0.0522824
$732$	0	0
$733$	24.5994	0.908601	0.454301	−	0.890848i	$-0.349889\pi$
$733$	24.5994	0.908601	0.454301	+	0.890848i	$0.349889\pi$
$734$	0	0
$735$	9.54368	0.352024
$736$	0	0
$737$	−12.1903	−0.449037
$738$	0	0
$739$	22.8595	0.840901	0.420450	−	0.907316i	$-0.361872\pi$
$739$	22.8595	0.840901	0.420450	+	0.907316i	$0.361872\pi$
$740$	0	0
$741$	9.53251	0.350185
$742$	0	0
$743$	−25.5707	−0.938099	−0.469050	−	0.883172i	$-0.655403\pi$
$743$	−25.5707	−0.938099	−0.469050	+	0.883172i	$0.655403\pi$
$744$	0	0
$745$	−6.79280	−0.248869
$746$	0	0
$747$	−4.59898	−0.168268
$748$	0	0
$749$	−22.6818	−0.828776
$750$	0	0
$751$	−7.27611	−0.265509	−0.132754	−	0.991149i	$-0.542382\pi$
$751$	−7.27611	−0.265509	−0.132754	+	0.991149i	$0.542382\pi$
$752$	0	0
$753$	10.7623	0.392201
$754$	0	0
$755$	2.35072	0.0855514
$756$	0	0
$757$	18.9497	0.688738	0.344369	−	0.938834i	$-0.388093\pi$
$757$	18.9497	0.688738	0.344369	+	0.938834i	$0.388093\pi$
$758$	0	0
$759$	1.14688	0.0416289
$760$	0	0
$761$	11.2163	0.406591	0.203295	−	0.979117i	$-0.434835\pi$
$761$	11.2163	0.406591	0.203295	+	0.979117i	$0.434835\pi$
$762$	0	0
$763$	−11.0353	−0.399504
$764$	0	0
$765$	−1.52663	−0.0551955
$766$	0	0
$767$	28.5130	1.02954
$768$	0	0
$769$	−37.2889	−1.34467	−0.672337	−	0.740245i	$-0.734709\pi$
$769$	−37.2889	−1.34467	−0.672337	+	0.740245i	$0.734709\pi$
$770$	0	0
$771$	−19.2212	−0.692234
$772$	0	0
$773$	54.5495	1.96201	0.981005	−	0.193983i	$-0.0621408\pi$
$773$	54.5495	1.96201	0.981005	+	0.193983i	$0.0621408\pi$
$774$	0	0
$775$	43.8463	1.57501
$776$	0	0
$777$	−10.0409	−0.360216
$778$	0	0
$779$	−8.28850	−0.296966
$780$	0	0
$781$	15.2783	0.546699
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−24.8279	−0.886146
$786$	0	0
$787$	27.0718	0.965004	0.482502	−	0.875895i	$-0.339728\pi$
$787$	27.0718	0.965004	0.482502	+	0.875895i	$0.339728\pi$
$788$	0	0
$789$	8.85390	0.315207
$790$	0	0
$791$	−5.20973	−0.185236
$792$	0	0
$793$	39.5428	1.40421
$794$	0	0
$795$	−14.0711	−0.499049
$796$	0	0
$797$	4.74869	0.168207	0.0841036	−	0.996457i	$-0.473197\pi$
$797$	4.74869	0.168207	0.0841036	+	0.996457i	$0.473197\pi$
$798$	0	0
$799$	−0.201962	−0.00714492
$800$	0	0
$801$	14.8381	0.524279
$802$	0	0
$803$	−14.5467	−0.513342
$804$	0	0
$805$	−7.00700	−0.246964
$806$	0	0
$807$	9.97519	0.351143
$808$	0	0
$809$	28.8430	1.01407	0.507033	−	0.861927i	$-0.330742\pi$
$809$	28.8430	1.01407	0.507033	+	0.861927i	$0.330742\pi$
$810$	0	0
$811$	3.47894	0.122162	0.0610810	−	0.998133i	$-0.480545\pi$
$811$	3.47894	0.122162	0.0610810	+	0.998133i	$0.480545\pi$
$812$	0	0
$813$	−14.7260	−0.516463
$814$	0	0
$815$	73.0783	2.55982
$816$	0	0
$817$	−6.67895	−0.233667
$818$	0	0
$819$	−9.26001	−0.323571
$820$	0	0
$821$	−34.1095	−1.19043	−0.595216	−	0.803566i	$-0.702933\pi$
$821$	−34.1095	−1.19043	−0.595216	+	0.803566i	$0.702933\pi$
$822$	0	0
$823$	44.6176	1.55527	0.777636	−	0.628715i	$-0.216419\pi$
$823$	44.6176	1.55527	0.777636	+	0.628715i	$0.216419\pi$
$824$	0	0
$825$	7.65184	0.266403
$826$	0	0
$827$	33.0174	1.14813	0.574064	−	0.818810i	$-0.305366\pi$
$827$	33.0174	1.14813	0.574064	+	0.818810i	$0.305366\pi$
$828$	0	0
$829$	−40.4901	−1.40628	−0.703139	−	0.711052i	$-0.748219\pi$
$829$	−40.4901	−1.40628	−0.703139	+	0.711052i	$0.748219\pi$
$830$	0	0
$831$	0.848598	0.0294376
$832$	0	0
$833$	−1.24827	−0.0432500
$834$	0	0
$835$	−12.9466	−0.448035
$836$	0	0
$837$	6.57178	0.227154
$838$	0	0
$839$	−41.3358	−1.42707	−0.713536	−	0.700619i	$-0.752907\pi$
$839$	−41.3358	−1.42707	−0.713536	+	0.700619i	$0.752907\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−22.8691	−0.787654
$844$	0	0
$845$	−25.2284	−0.867885
$846$	0	0
$847$	−19.8631	−0.682504
$848$	0	0
$849$	−8.10218	−0.278066
$850$	0	0
$851$	−4.89567	−0.167821
$852$	0	0
$853$	−13.0312	−0.446179	−0.223090	−	0.974798i	$-0.571614\pi$
$853$	−13.0312	−0.446179	−0.223090	+	0.974798i	$0.571614\pi$
$854$	0	0
$855$	7.21320	0.246686
$856$	0	0
$857$	−5.49480	−0.187699	−0.0938494	−	0.995586i	$-0.529917\pi$
$857$	−5.49480	−0.187699	−0.0938494	+	0.995586i	$0.529917\pi$
$858$	0	0
$859$	−21.8403	−0.745180	−0.372590	−	0.927996i	$-0.621530\pi$
$859$	−21.8403	−0.745180	−0.372590	+	0.927996i	$0.621530\pi$
$860$	0	0
$861$	8.05156	0.274396
$862$	0	0
$863$	7.67863	0.261384	0.130692	−	0.991423i	$-0.458280\pi$
$863$	7.67863	0.261384	0.130692	+	0.991423i	$0.458280\pi$
$864$	0	0
$865$	−72.9535	−2.48049
$866$	0	0
$867$	−16.8003	−0.570569
$868$	0	0
$869$	2.64674	0.0897844
$870$	0	0
$871$	47.9899	1.62607
$872$	0	0
$873$	8.14524	0.275675
$874$	0	0
$875$	−11.7150	−0.396040
$876$	0	0
$877$	32.3068	1.09092	0.545461	−	0.838136i	$-0.316355\pi$
$877$	32.3068	1.09092	0.545461	+	0.838136i	$0.316355\pi$
$878$	0	0
$879$	24.6949	0.832937
$880$	0	0
$881$	−24.7489	−0.833812	−0.416906	−	0.908950i	$-0.636886\pi$
$881$	−24.7489	−0.833812	−0.416906	+	0.908950i	$0.636886\pi$
$882$	0	0
$883$	−23.2435	−0.782207	−0.391103	−	0.920347i	$-0.627907\pi$
$883$	−23.2435	−0.782207	−0.391103	+	0.920347i	$0.627907\pi$
$884$	0	0
$885$	21.5757	0.725258
$886$	0	0
$887$	26.2844	0.882544	0.441272	−	0.897374i	$-0.354527\pi$
$887$	26.2844	0.882544	0.441272	+	0.897374i	$0.354527\pi$
$888$	0	0
$889$	−15.2123	−0.510205
$890$	0	0
$891$	1.14688	0.0384218
$892$	0	0
$893$	0.954255	0.0319329
$894$	0	0
$895$	65.2473	2.18098
$896$	0	0
$897$	−4.51492	−0.150749
$898$	0	0
$899$	6.57178	0.219181
$900$	0	0
$901$	1.84043	0.0613136
$902$	0	0
$903$	6.48803	0.215908
$904$	0	0
$905$	−36.8049	−1.22344
$906$	0	0
$907$	54.1109	1.79672	0.898361	−	0.439258i	$-0.144759\pi$
$907$	54.1109	1.79672	0.898361	+	0.439258i	$0.144759\pi$
$908$	0	0
$909$	−10.8719	−0.360599
$910$	0	0
$911$	−14.6064	−0.483932	−0.241966	−	0.970285i	$-0.577792\pi$
$911$	−14.6064	−0.483932	−0.241966	+	0.970285i	$0.577792\pi$
$912$	0	0
$913$	−5.27446	−0.174559
$914$	0	0
$915$	29.9219	0.989187
$916$	0	0
$917$	−14.0119	−0.462713
$918$	0	0
$919$	−3.49205	−0.115192	−0.0575961	−	0.998340i	$-0.518344\pi$
$919$	−3.49205	−0.115192	−0.0575961	+	0.998340i	$0.518344\pi$
$920$	0	0
$921$	−7.75138	−0.255417
$922$	0	0
$923$	−60.1461	−1.97973
$924$	0	0
$925$	−32.6634	−1.07397
$926$	0	0
$927$	14.4916	0.475967
$928$	0	0
$929$	−12.6091	−0.413690	−0.206845	−	0.978374i	$-0.566320\pi$
$929$	−12.6091	−0.413690	−0.206845	+	0.978374i	$0.566320\pi$
$930$	0	0
$931$	5.89797	0.193298
$932$	0	0
$933$	11.2388	0.367941
$934$	0	0
$935$	−1.75086	−0.0572591
$936$	0	0
$937$	21.6562	0.707477	0.353739	−	0.935344i	$-0.384910\pi$
$937$	21.6562	0.707477	0.353739	+	0.935344i	$0.384910\pi$
$938$	0	0
$939$	12.3629	0.403449
$940$	0	0
$941$	44.3084	1.44441	0.722206	−	0.691678i	$-0.243128\pi$
$941$	44.3084	1.44441	0.722206	+	0.691678i	$0.243128\pi$
$942$	0	0
$943$	3.92571	0.127839
$944$	0	0
$945$	−7.00700	−0.227938
$946$	0	0
$947$	57.6519	1.87344	0.936718	−	0.350085i	$-0.113847\pi$
$947$	57.6519	1.87344	0.936718	+	0.350085i	$0.113847\pi$
$948$	0	0
$949$	57.2662	1.85894
$950$	0	0
$951$	−10.5402	−0.341788
$952$	0	0
$953$	14.3695	0.465472	0.232736	−	0.972540i	$-0.425232\pi$
$953$	14.3695	0.465472	0.232736	+	0.972540i	$0.425232\pi$
$954$	0	0
$955$	47.3807	1.53320
$956$	0	0
$957$	1.14688	0.0370732
$958$	0	0
$959$	−30.2073	−0.975446
$960$	0	0
$961$	12.1883	0.393172
$962$	0	0
$963$	−11.0590	−0.356372
$964$	0	0
$965$	−83.3957	−2.68460
$966$	0	0
$967$	10.7620	0.346082	0.173041	−	0.984915i	$-0.444641\pi$
$967$	10.7620	0.346082	0.173041	+	0.984915i	$0.444641\pi$
$968$	0	0
$969$	−0.943454	−0.0303081
$970$	0	0
$971$	−30.9320	−0.992656	−0.496328	−	0.868135i	$-0.665319\pi$
$971$	−30.9320	−0.992656	−0.496328	+	0.868135i	$0.665319\pi$
$972$	0	0
$973$	−41.7111	−1.33720
$974$	0	0
$975$	−30.1231	−0.964711
$976$	0	0
$977$	27.4813	0.879204	0.439602	−	0.898193i	$-0.355120\pi$
$977$	27.4813	0.879204	0.439602	+	0.898193i	$0.355120\pi$
$978$	0	0
$979$	17.0175	0.543880
$980$	0	0
$981$	−5.38049	−0.171786
$982$	0	0
$983$	14.6830	0.468316	0.234158	−	0.972199i	$-0.424767\pi$
$983$	14.6830	0.468316	0.234158	+	0.972199i	$0.424767\pi$
$984$	0	0
$985$	54.4400	1.73460
$986$	0	0
$987$	−0.926977	−0.0295060
$988$	0	0
$989$	3.16338	0.100590
$990$	0	0
$991$	8.32406	0.264422	0.132211	−	0.991222i	$-0.457792\pi$
$991$	8.32406	0.264422	0.132211	+	0.991222i	$0.457792\pi$
$992$	0	0
$993$	−26.4647	−0.839832
$994$	0	0
$995$	76.2548	2.41744
$996$	0	0
$997$	20.9269	0.662762	0.331381	−	0.943497i	$-0.392486\pi$
$997$	20.9269	0.662762	0.331381	+	0.943497i	$0.392486\pi$
$998$	0	0
$999$	−4.89567	−0.154892

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.d.1.1	✓	8	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.41642 q^{5} +2.05098 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.41642 q^{5} +2.05098 q^{7} +1.00000 q^{9} +1.14688 q^{11} -4.51492 q^{13} -3.41642 q^{15} +0.446852 q^{17} -2.11134 q^{19} +2.05098 q^{21} +1.00000 q^{23} +6.67190 q^{25} +1.00000 q^{27} +1.00000 q^{29} +6.57178 q^{31} +1.14688 q^{33} -7.00700 q^{35} -4.89567 q^{37} -4.51492 q^{39} +3.92571 q^{41} +3.16338 q^{43} -3.41642 q^{45} -0.451968 q^{47} -2.79348 q^{49} +0.446852 q^{51} +4.11866 q^{53} -3.91820 q^{55} -2.11134 q^{57} -6.31529 q^{59} -8.75827 q^{61} +2.05098 q^{63} +15.4248 q^{65} -10.6292 q^{67} +1.00000 q^{69} +13.3216 q^{71} -12.6838 q^{73} +6.67190 q^{75} +2.35222 q^{77} +2.30778 q^{79} +1.00000 q^{81} -4.59898 q^{83} -1.52663 q^{85} +1.00000 q^{87} +14.8381 q^{89} -9.26001 q^{91} +6.57178 q^{93} +7.21320 q^{95} +8.14524 q^{97} +1.14688 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) 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

Groups

Database

Properties

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