LMFDB - Embedded newform 8004.2.a.i.1.10

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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} + \cdots - 208 $ x^16 - 5x^15 - 43x^14 + 234x^13 + 634x^12 - 4048x^11 - 3483x^10 + 32512x^9 - 137x^8 - 121665x^7 + 60168x^6 + 172218x^5 - 138024x^4 - 17844x^3 + 14352x^2 + 72*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.27509$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.27509 q^{5} +1.05879 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.27509 q^{5} +1.05879 q^{7} +1.00000 q^{9} +2.86975 q^{11} -0.442370 q^{13} -1.27509 q^{15} +7.11949 q^{17} +5.55811 q^{19} -1.05879 q^{21} +1.00000 q^{23} -3.37415 q^{25} -1.00000 q^{27} +1.00000 q^{29} +9.86978 q^{31} -2.86975 q^{33} +1.35005 q^{35} +3.13742 q^{37} +0.442370 q^{39} +0.570057 q^{41} +3.31512 q^{43} +1.27509 q^{45} +7.62112 q^{47} -5.87896 q^{49} -7.11949 q^{51} -0.0892956 q^{53} +3.65919 q^{55} -5.55811 q^{57} -7.70376 q^{59} +0.605364 q^{61} +1.05879 q^{63} -0.564061 q^{65} -2.08669 q^{67} -1.00000 q^{69} -9.62182 q^{71} +9.39503 q^{73} +3.37415 q^{75} +3.03847 q^{77} -4.40635 q^{79} +1.00000 q^{81} -11.8541 q^{83} +9.07799 q^{85} -1.00000 q^{87} -15.9142 q^{89} -0.468378 q^{91} -9.86978 q^{93} +7.08708 q^{95} +1.77186 q^{97} +2.86975 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9	$ 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9 + 5 * q^11 + 8 * q^13 - 5 * q^15 + 7 * q^17 + q^19 + 4 * q^21 + 16 * q^23 + 31 * q^25 - 16 * q^27 + 16 * q^29 - 2 * q^31 - 5 * q^33 + 5 * q^35 + 14 * q^37 - 8 * q^39 - q^41 - 13 * q^43 + 5 * q^45 - 4 * q^47 + 30 * q^49 - 7 * q^51 + 19 * q^53 - 37 * q^55 - q^57 + 12 * q^59 + 21 * q^61 - 4 * q^63 + 26 * q^65 - 11 * q^67 - 16 * q^69 + 7 * q^71 - 13 * q^73 - 31 * q^75 + 4 * q^77 - 18 * q^79 + 16 * q^81 + 25 * q^83 + 48 * q^85 - 16 * q^87 + 12 * q^89 - 11 * q^91 + 2 * q^93 - q^95 + 5 * 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
$5$	1.27509	0.570237	0.285119	−	0.958492i	$-0.407967\pi$
$5$	1.27509	0.570237	0.285119	+	0.958492i	$0.407967\pi$
$6$	0	0
$7$	1.05879	0.400186	0.200093	−	0.979777i	$-0.435876\pi$
$7$	1.05879	0.400186	0.200093	+	0.979777i	$0.435876\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.86975	0.865263	0.432631	−	0.901571i	$-0.357585\pi$
$11$	2.86975	0.865263	0.432631	+	0.901571i	$0.357585\pi$
$12$	0	0
$13$	−0.442370	−0.122691	−0.0613457	−	0.998117i	$-0.519539\pi$
$13$	−0.442370	−0.122691	−0.0613457	+	0.998117i	$0.519539\pi$
$14$	0	0
$15$	−1.27509	−0.329227
$16$	0	0
$17$	7.11949	1.72673	0.863366	−	0.504579i	$-0.168352\pi$
$17$	7.11949	1.72673	0.863366	+	0.504579i	$0.168352\pi$
$18$	0	0
$19$	5.55811	1.27512	0.637559	−	0.770402i	$-0.279944\pi$
$19$	5.55811	1.27512	0.637559	+	0.770402i	$0.279944\pi$
$20$	0	0
$21$	−1.05879	−0.231047
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.37415	−0.674830
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	9.86978	1.77266	0.886332	−	0.463050i	$-0.153245\pi$
$31$	9.86978	1.77266	0.886332	+	0.463050i	$0.153245\pi$
$32$	0	0
$33$	−2.86975	−0.499560
$34$	0	0
$35$	1.35005	0.228201
$36$	0	0
$37$	3.13742	0.515789	0.257894	−	0.966173i	$-0.416971\pi$
$37$	3.13742	0.515789	0.257894	+	0.966173i	$0.416971\pi$
$38$	0	0
$39$	0.442370	0.0708359
$40$	0	0
$41$	0.570057	0.0890280	0.0445140	−	0.999009i	$-0.485826\pi$
$41$	0.570057	0.0890280	0.0445140	+	0.999009i	$0.485826\pi$
$42$	0	0
$43$	3.31512	0.505551	0.252775	−	0.967525i	$-0.418657\pi$
$43$	3.31512	0.505551	0.252775	+	0.967525i	$0.418657\pi$
$44$	0	0
$45$	1.27509	0.190079
$46$	0	0
$47$	7.62112	1.11166	0.555828	−	0.831298i	$-0.312401\pi$
$47$	7.62112	1.11166	0.555828	+	0.831298i	$0.312401\pi$
$48$	0	0
$49$	−5.87896	−0.839851
$50$	0	0
$51$	−7.11949	−0.996929
$52$	0	0
$53$	−0.0892956	−0.0122657	−0.00613285	−	0.999981i	$-0.501952\pi$
$53$	−0.0892956	−0.0122657	−0.00613285	+	0.999981i	$0.501952\pi$
$54$	0	0
$55$	3.65919	0.493405
$56$	0	0
$57$	−5.55811	−0.736189
$58$	0	0
$59$	−7.70376	−1.00294	−0.501472	−	0.865174i	$-0.667208\pi$
$59$	−7.70376	−1.00294	−0.501472	+	0.865174i	$0.667208\pi$
$60$	0	0
$61$	0.605364	0.0775089	0.0387544	−	0.999249i	$-0.487661\pi$
$61$	0.605364	0.0775089	0.0387544	+	0.999249i	$0.487661\pi$
$62$	0	0
$63$	1.05879	0.133395
$64$	0	0
$65$	−0.564061	−0.0699632
$66$	0	0
$67$	−2.08669	−0.254930	−0.127465	−	0.991843i	$-0.540684\pi$
$67$	−2.08669	−0.254930	−0.127465	+	0.991843i	$0.540684\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−9.62182	−1.14190	−0.570950	−	0.820985i	$-0.693425\pi$
$71$	−9.62182	−1.14190	−0.570950	+	0.820985i	$0.693425\pi$
$72$	0	0
$73$	9.39503	1.09960	0.549802	−	0.835295i	$-0.314703\pi$
$73$	9.39503	1.09960	0.549802	+	0.835295i	$0.314703\pi$
$74$	0	0
$75$	3.37415	0.389613
$76$	0	0
$77$	3.03847	0.346266
$78$	0	0
$79$	−4.40635	−0.495754	−0.247877	−	0.968792i	$-0.579733\pi$
$79$	−4.40635	−0.495754	−0.247877	+	0.968792i	$0.579733\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.8541	−1.30116	−0.650580	−	0.759438i	$-0.725474\pi$
$83$	−11.8541	−1.30116	−0.650580	+	0.759438i	$0.725474\pi$
$84$	0	0
$85$	9.07799	0.984646
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−15.9142	−1.68690	−0.843449	−	0.537209i	$-0.819479\pi$
$89$	−15.9142	−1.68690	−0.843449	+	0.537209i	$0.819479\pi$
$90$	0	0
$91$	−0.468378	−0.0490993
$92$	0	0
$93$	−9.86978	−1.02345
$94$	0	0
$95$	7.08708	0.727119
$96$	0	0
$97$	1.77186	0.179905	0.0899527	−	0.995946i	$-0.471328\pi$
$97$	1.77186	0.179905	0.0899527	+	0.995946i	$0.471328\pi$
$98$	0	0
$99$	2.86975	0.288421
$100$	0	0
$101$	17.1221	1.70371	0.851856	−	0.523776i	$-0.175477\pi$
$101$	17.1221	1.70371	0.851856	+	0.523776i	$0.175477\pi$
$102$	0	0
$103$	17.4005	1.71452	0.857262	−	0.514880i	$-0.172164\pi$
$103$	17.4005	1.71452	0.857262	+	0.514880i	$0.172164\pi$
$104$	0	0
$105$	−1.35005	−0.131752
$106$	0	0
$107$	−0.113485	−0.0109710	−0.00548548	−	0.999985i	$-0.501746\pi$
$107$	−0.113485	−0.0109710	−0.00548548	+	0.999985i	$0.501746\pi$
$108$	0	0
$109$	3.63082	0.347769	0.173885	−	0.984766i	$-0.444368\pi$
$109$	3.63082	0.347769	0.173885	+	0.984766i	$0.444368\pi$
$110$	0	0
$111$	−3.13742	−0.297791
$112$	0	0
$113$	4.66039	0.438413	0.219206	−	0.975679i	$-0.429653\pi$
$113$	4.66039	0.438413	0.219206	+	0.975679i	$0.429653\pi$
$114$	0	0
$115$	1.27509	0.118903
$116$	0	0
$117$	−0.442370	−0.0408971
$118$	0	0
$119$	7.53806	0.691013
$120$	0	0
$121$	−2.76453	−0.251321
$122$	0	0
$123$	−0.570057	−0.0514003
$124$	0	0
$125$	−10.6778	−0.955050
$126$	0	0
$127$	3.38250	0.300149	0.150074	−	0.988675i	$-0.452049\pi$
$127$	3.38250	0.300149	0.150074	+	0.988675i	$0.452049\pi$
$128$	0	0
$129$	−3.31512	−0.291880
$130$	0	0
$131$	5.55867	0.485663	0.242832	−	0.970068i	$-0.421924\pi$
$131$	5.55867	0.485663	0.242832	+	0.970068i	$0.421924\pi$
$132$	0	0
$133$	5.88488	0.510284
$134$	0	0
$135$	−1.27509	−0.109742
$136$	0	0
$137$	−2.15471	−0.184090	−0.0920448	−	0.995755i	$-0.529340\pi$
$137$	−2.15471	−0.184090	−0.0920448	+	0.995755i	$0.529340\pi$
$138$	0	0
$139$	−15.7527	−1.33613	−0.668065	−	0.744103i	$-0.732877\pi$
$139$	−15.7527	−1.33613	−0.668065	+	0.744103i	$0.732877\pi$
$140$	0	0
$141$	−7.62112	−0.641814
$142$	0	0
$143$	−1.26949	−0.106160
$144$	0	0
$145$	1.27509	0.105890
$146$	0	0
$147$	5.87896	0.484888
$148$	0	0
$149$	24.0193	1.96774	0.983869	−	0.178890i	$-0.0572506\pi$
$149$	24.0193	1.96774	0.983869	+	0.178890i	$0.0572506\pi$
$150$	0	0
$151$	−22.3745	−1.82081	−0.910405	−	0.413719i	$-0.864230\pi$
$151$	−22.3745	−1.82081	−0.910405	+	0.413719i	$0.864230\pi$
$152$	0	0
$153$	7.11949	0.575577
$154$	0	0
$155$	12.5848	1.01084
$156$	0	0
$157$	−6.34045	−0.506023	−0.253011	−	0.967463i	$-0.581421\pi$
$157$	−6.34045	−0.506023	−0.253011	+	0.967463i	$0.581421\pi$
$158$	0	0
$159$	0.0892956	0.00708160
$160$	0	0
$161$	1.05879	0.0834445
$162$	0	0
$163$	−8.59993	−0.673599	−0.336799	−	0.941576i	$-0.609344\pi$
$163$	−8.59993	−0.673599	−0.336799	+	0.941576i	$0.609344\pi$
$164$	0	0
$165$	−3.65919	−0.284867
$166$	0	0
$167$	−16.5821	−1.28316	−0.641582	−	0.767055i	$-0.721721\pi$
$167$	−16.5821	−1.28316	−0.641582	+	0.767055i	$0.721721\pi$
$168$	0	0
$169$	−12.8043	−0.984947
$170$	0	0
$171$	5.55811	0.425039
$172$	0	0
$173$	16.7748	1.27537	0.637683	−	0.770299i	$-0.279893\pi$
$173$	16.7748	1.27537	0.637683	+	0.770299i	$0.279893\pi$
$174$	0	0
$175$	−3.57252	−0.270057
$176$	0	0
$177$	7.70376	0.579050
$178$	0	0
$179$	−20.3735	−1.52279	−0.761395	−	0.648288i	$-0.775485\pi$
$179$	−20.3735	−1.52279	−0.761395	+	0.648288i	$0.775485\pi$
$180$	0	0
$181$	10.7827	0.801471	0.400736	−	0.916194i	$-0.368755\pi$
$181$	10.7827	0.801471	0.400736	+	0.916194i	$0.368755\pi$
$182$	0	0
$183$	−0.605364	−0.0447498
$184$	0	0
$185$	4.00049	0.294122
$186$	0	0
$187$	20.4312	1.49408
$188$	0	0
$189$	−1.05879	−0.0770158
$190$	0	0
$191$	−14.5647	−1.05386	−0.526931	−	0.849908i	$-0.676657\pi$
$191$	−14.5647	−1.05386	−0.526931	+	0.849908i	$0.676657\pi$
$192$	0	0
$193$	−26.9558	−1.94032	−0.970160	−	0.242464i	$-0.922044\pi$
$193$	−26.9558	−1.94032	−0.970160	+	0.242464i	$0.922044\pi$
$194$	0	0
$195$	0.564061	0.0403933
$196$	0	0
$197$	7.37900	0.525732	0.262866	−	0.964832i	$-0.415332\pi$
$197$	7.37900	0.525732	0.262866	+	0.964832i	$0.415332\pi$
$198$	0	0
$199$	−4.10097	−0.290710	−0.145355	−	0.989380i	$-0.546432\pi$
$199$	−4.10097	−0.290710	−0.145355	+	0.989380i	$0.546432\pi$
$200$	0	0
$201$	2.08669	0.147184
$202$	0	0
$203$	1.05879	0.0743126
$204$	0	0
$205$	0.726873	0.0507670
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	15.9504	1.10331
$210$	0	0
$211$	17.3554	1.19479	0.597397	−	0.801946i	$-0.296202\pi$
$211$	17.3554	1.19479	0.597397	+	0.801946i	$0.296202\pi$
$212$	0	0
$213$	9.62182	0.659276
$214$	0	0
$215$	4.22707	0.288284
$216$	0	0
$217$	10.4500	0.709395
$218$	0	0
$219$	−9.39503	−0.634857
$220$	0	0
$221$	−3.14945	−0.211855
$222$	0	0
$223$	13.1742	0.882212	0.441106	−	0.897455i	$-0.354586\pi$
$223$	13.1742	0.882212	0.441106	+	0.897455i	$0.354586\pi$
$224$	0	0
$225$	−3.37415	−0.224943
$226$	0	0
$227$	−22.4629	−1.49091	−0.745457	−	0.666554i	$-0.767769\pi$
$227$	−22.4629	−1.49091	−0.745457	+	0.666554i	$0.767769\pi$
$228$	0	0
$229$	−1.02268	−0.0675808	−0.0337904	−	0.999429i	$-0.510758\pi$
$229$	−1.02268	−0.0675808	−0.0337904	+	0.999429i	$0.510758\pi$
$230$	0	0
$231$	−3.03847	−0.199917
$232$	0	0
$233$	8.36599	0.548074	0.274037	−	0.961719i	$-0.411641\pi$
$233$	8.36599	0.548074	0.274037	+	0.961719i	$0.411641\pi$
$234$	0	0
$235$	9.71761	0.633907
$236$	0	0
$237$	4.40635	0.286223
$238$	0	0
$239$	−25.0231	−1.61861	−0.809304	−	0.587390i	$-0.800156\pi$
$239$	−25.0231	−1.61861	−0.809304	+	0.587390i	$0.800156\pi$
$240$	0	0
$241$	7.76619	0.500264	0.250132	−	0.968212i	$-0.419526\pi$
$241$	7.76619	0.500264	0.250132	+	0.968212i	$0.419526\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−7.49620	−0.478914
$246$	0	0
$247$	−2.45874	−0.156446
$248$	0	0
$249$	11.8541	0.751225
$250$	0	0
$251$	19.1812	1.21071	0.605354	−	0.795957i	$-0.293032\pi$
$251$	19.1812	1.21071	0.605354	+	0.795957i	$0.293032\pi$
$252$	0	0
$253$	2.86975	0.180420
$254$	0	0
$255$	−9.07799	−0.568486
$256$	0	0
$257$	16.6896	1.04107	0.520533	−	0.853842i	$-0.325733\pi$
$257$	16.6896	1.04107	0.520533	+	0.853842i	$0.325733\pi$
$258$	0	0
$259$	3.32187	0.206411
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−21.2779	−1.31205	−0.656025	−	0.754739i	$-0.727763\pi$
$263$	−21.2779	−1.31205	−0.656025	+	0.754739i	$0.727763\pi$
$264$	0	0
$265$	−0.113860	−0.00699435
$266$	0	0
$267$	15.9142	0.973931
$268$	0	0
$269$	15.8508	0.966441	0.483221	−	0.875499i	$-0.339467\pi$
$269$	15.8508	0.966441	0.483221	+	0.875499i	$0.339467\pi$
$270$	0	0
$271$	2.60058	0.157974	0.0789869	−	0.996876i	$-0.474831\pi$
$271$	2.60058	0.157974	0.0789869	+	0.996876i	$0.474831\pi$
$272$	0	0
$273$	0.468378	0.0283475
$274$	0	0
$275$	−9.68297	−0.583905
$276$	0	0
$277$	−15.4758	−0.929852	−0.464926	−	0.885350i	$-0.653919\pi$
$277$	−15.4758	−0.929852	−0.464926	+	0.885350i	$0.653919\pi$
$278$	0	0
$279$	9.86978	0.590888
$280$	0	0
$281$	27.1129	1.61742	0.808710	−	0.588208i	$-0.200166\pi$
$281$	27.1129	1.61742	0.808710	+	0.588208i	$0.200166\pi$
$282$	0	0
$283$	−6.74351	−0.400860	−0.200430	−	0.979708i	$-0.564234\pi$
$283$	−6.74351	−0.400860	−0.200430	+	0.979708i	$0.564234\pi$
$284$	0	0
$285$	−7.08708	−0.419803
$286$	0	0
$287$	0.603572	0.0356277
$288$	0	0
$289$	33.6872	1.98160
$290$	0	0
$291$	−1.77186	−0.103868
$292$	0	0
$293$	0.678749	0.0396530	0.0198265	−	0.999803i	$-0.493689\pi$
$293$	0.678749	0.0396530	0.0198265	+	0.999803i	$0.493689\pi$
$294$	0	0
$295$	−9.82298	−0.571916
$296$	0	0
$297$	−2.86975	−0.166520
$298$	0	0
$299$	−0.442370	−0.0255829
$300$	0	0
$301$	3.51002	0.202314
$302$	0	0
$303$	−17.1221	−0.983639
$304$	0	0
$305$	0.771893	0.0441984
$306$	0	0
$307$	−19.8221	−1.13131	−0.565654	−	0.824643i	$-0.691376\pi$
$307$	−19.8221	−1.13131	−0.565654	+	0.824643i	$0.691376\pi$
$308$	0	0
$309$	−17.4005	−0.989881
$310$	0	0
$311$	27.9653	1.58577	0.792883	−	0.609374i	$-0.208579\pi$
$311$	27.9653	1.58577	0.792883	+	0.609374i	$0.208579\pi$
$312$	0	0
$313$	−4.12103	−0.232934	−0.116467	−	0.993195i	$-0.537157\pi$
$313$	−4.12103	−0.232934	−0.116467	+	0.993195i	$0.537157\pi$
$314$	0	0
$315$	1.35005	0.0760669
$316$	0	0
$317$	1.06358	0.0597366	0.0298683	−	0.999554i	$-0.490491\pi$
$317$	1.06358	0.0597366	0.0298683	+	0.999554i	$0.490491\pi$
$318$	0	0
$319$	2.86975	0.160675
$320$	0	0
$321$	0.113485	0.00633409
$322$	0	0
$323$	39.5709	2.20179
$324$	0	0
$325$	1.49262	0.0827958
$326$	0	0
$327$	−3.63082	−0.200785
$328$	0	0
$329$	8.06918	0.444869
$330$	0	0
$331$	12.6301	0.694211	0.347105	−	0.937826i	$-0.387165\pi$
$331$	12.6301	0.694211	0.347105	+	0.937826i	$0.387165\pi$
$332$	0	0
$333$	3.13742	0.171930
$334$	0	0
$335$	−2.66072	−0.145370
$336$	0	0
$337$	19.6462	1.07020	0.535098	−	0.844790i	$-0.320275\pi$
$337$	19.6462	1.07020	0.535098	+	0.844790i	$0.320275\pi$
$338$	0	0
$339$	−4.66039	−0.253118
$340$	0	0
$341$	28.3238	1.53382
$342$	0	0
$343$	−13.6361	−0.736282
$344$	0	0
$345$	−1.27509	−0.0686485
$346$	0	0
$347$	−16.5051	−0.886042	−0.443021	−	0.896511i	$-0.646093\pi$
$347$	−16.5051	−0.886042	−0.443021	+	0.896511i	$0.646093\pi$
$348$	0	0
$349$	−6.96572	−0.372867	−0.186433	−	0.982468i	$-0.559693\pi$
$349$	−6.96572	−0.372867	−0.186433	+	0.982468i	$0.559693\pi$
$350$	0	0
$351$	0.442370	0.0236120
$352$	0	0
$353$	−18.6023	−0.990100	−0.495050	−	0.868864i	$-0.664850\pi$
$353$	−18.6023	−0.990100	−0.495050	+	0.868864i	$0.664850\pi$
$354$	0	0
$355$	−12.2687	−0.651153
$356$	0	0
$357$	−7.53806	−0.398957
$358$	0	0
$359$	32.8220	1.73228	0.866138	−	0.499804i	$-0.166595\pi$
$359$	32.8220	1.73228	0.866138	+	0.499804i	$0.166595\pi$
$360$	0	0
$361$	11.8926	0.625925
$362$	0	0
$363$	2.76453	0.145100
$364$	0	0
$365$	11.9795	0.627035
$366$	0	0
$367$	−6.30126	−0.328923	−0.164462	−	0.986383i	$-0.552589\pi$
$367$	−6.30126	−0.328923	−0.164462	+	0.986383i	$0.552589\pi$
$368$	0	0
$369$	0.570057	0.0296760
$370$	0	0
$371$	−0.0945455	−0.00490856
$372$	0	0
$373$	8.07809	0.418268	0.209134	−	0.977887i	$-0.432936\pi$
$373$	8.07809	0.418268	0.209134	+	0.977887i	$0.432936\pi$
$374$	0	0
$375$	10.6778	0.551398
$376$	0	0
$377$	−0.442370	−0.0227832
$378$	0	0
$379$	3.83382	0.196930	0.0984651	−	0.995141i	$-0.468607\pi$
$379$	3.83382	0.196930	0.0984651	+	0.995141i	$0.468607\pi$
$380$	0	0
$381$	−3.38250	−0.173291
$382$	0	0
$383$	12.7371	0.650833	0.325416	−	0.945571i	$-0.394496\pi$
$383$	12.7371	0.650833	0.325416	+	0.945571i	$0.394496\pi$
$384$	0	0
$385$	3.87432	0.197454
$386$	0	0
$387$	3.31512	0.168517
$388$	0	0
$389$	−5.42536	−0.275077	−0.137538	−	0.990496i	$-0.543919\pi$
$389$	−5.42536	−0.275077	−0.137538	+	0.990496i	$0.543919\pi$
$390$	0	0
$391$	7.11949	0.360048
$392$	0	0
$393$	−5.55867	−0.280398
$394$	0	0
$395$	−5.61849	−0.282697
$396$	0	0
$397$	−36.9473	−1.85433	−0.927166	−	0.374651i	$-0.877763\pi$
$397$	−36.9473	−1.85433	−0.927166	+	0.374651i	$0.877763\pi$
$398$	0	0
$399$	−5.88488	−0.294613
$400$	0	0
$401$	31.3298	1.56454	0.782269	−	0.622941i	$-0.214062\pi$
$401$	31.3298	1.56454	0.782269	+	0.622941i	$0.214062\pi$
$402$	0	0
$403$	−4.36609	−0.217491
$404$	0	0
$405$	1.27509	0.0633597
$406$	0	0
$407$	9.00361	0.446293
$408$	0	0
$409$	3.22224	0.159329	0.0796646	−	0.996822i	$-0.474615\pi$
$409$	3.22224	0.159329	0.0796646	+	0.996822i	$0.474615\pi$
$410$	0	0
$411$	2.15471	0.106284
$412$	0	0
$413$	−8.15668	−0.401364
$414$	0	0
$415$	−15.1151	−0.741969
$416$	0	0
$417$	15.7527	0.771415
$418$	0	0
$419$	24.3527	1.18971	0.594854	−	0.803834i	$-0.297210\pi$
$419$	24.3527	1.18971	0.594854	+	0.803834i	$0.297210\pi$
$420$	0	0
$421$	−25.3720	−1.23655	−0.618277	−	0.785960i	$-0.712169\pi$
$421$	−25.3720	−1.23655	−0.618277	+	0.785960i	$0.712169\pi$
$422$	0	0
$423$	7.62112	0.370552
$424$	0	0
$425$	−24.0222	−1.16525
$426$	0	0
$427$	0.640954	0.0310179
$428$	0	0
$429$	1.26949	0.0612917
$430$	0	0
$431$	3.18144	0.153245	0.0766224	−	0.997060i	$-0.475586\pi$
$431$	3.18144	0.153245	0.0766224	+	0.997060i	$0.475586\pi$
$432$	0	0
$433$	37.0333	1.77971	0.889854	−	0.456246i	$-0.150806\pi$
$433$	37.0333	1.77971	0.889854	+	0.456246i	$0.150806\pi$
$434$	0	0
$435$	−1.27509	−0.0611358
$436$	0	0
$437$	5.55811	0.265880
$438$	0	0
$439$	27.5051	1.31275	0.656373	−	0.754436i	$-0.272090\pi$
$439$	27.5051	1.31275	0.656373	+	0.754436i	$0.272090\pi$
$440$	0	0
$441$	−5.87896	−0.279950
$442$	0	0
$443$	−8.27771	−0.393286	−0.196643	−	0.980475i	$-0.563004\pi$
$443$	−8.27771	−0.393286	−0.196643	+	0.980475i	$0.563004\pi$
$444$	0	0
$445$	−20.2920	−0.961932
$446$	0	0
$447$	−24.0193	−1.13607
$448$	0	0
$449$	−32.2605	−1.52247	−0.761234	−	0.648477i	$-0.775406\pi$
$449$	−32.2605	−1.52247	−0.761234	+	0.648477i	$0.775406\pi$
$450$	0	0
$451$	1.63592	0.0770326
$452$	0	0
$453$	22.3745	1.05124
$454$	0	0
$455$	−0.597223	−0.0279983
$456$	0	0
$457$	−34.4054	−1.60942	−0.804708	−	0.593671i	$-0.797678\pi$
$457$	−34.4054	−1.60942	−0.804708	+	0.593671i	$0.797678\pi$
$458$	0	0
$459$	−7.11949	−0.332310
$460$	0	0
$461$	−13.9731	−0.650793	−0.325396	−	0.945578i	$-0.605498\pi$
$461$	−13.9731	−0.650793	−0.325396	+	0.945578i	$0.605498\pi$
$462$	0	0
$463$	−21.7584	−1.01120	−0.505600	−	0.862768i	$-0.668729\pi$
$463$	−21.7584	−1.01120	−0.505600	+	0.862768i	$0.668729\pi$
$464$	0	0
$465$	−12.5848	−0.583608
$466$	0	0
$467$	33.0522	1.52948	0.764738	−	0.644342i	$-0.222869\pi$
$467$	33.0522	1.52948	0.764738	+	0.644342i	$0.222869\pi$
$468$	0	0
$469$	−2.20937	−0.102019
$470$	0	0
$471$	6.34045	0.292152
$472$	0	0
$473$	9.51356	0.437434
$474$	0	0
$475$	−18.7539	−0.860487
$476$	0	0
$477$	−0.0892956	−0.00408856
$478$	0	0
$479$	−26.2669	−1.20016	−0.600082	−	0.799939i	$-0.704865\pi$
$479$	−26.2669	−1.20016	−0.600082	+	0.799939i	$0.704865\pi$
$480$	0	0
$481$	−1.38790	−0.0632828
$482$	0	0
$483$	−1.05879	−0.0481767
$484$	0	0
$485$	2.25928	0.102589
$486$	0	0
$487$	−24.6659	−1.11772	−0.558858	−	0.829263i	$-0.688760\pi$
$487$	−24.6659	−1.11772	−0.558858	+	0.829263i	$0.688760\pi$
$488$	0	0
$489$	8.59993	0.388902
$490$	0	0
$491$	25.5713	1.15402	0.577008	−	0.816738i	$-0.304220\pi$
$491$	25.5713	1.15402	0.577008	+	0.816738i	$0.304220\pi$
$492$	0	0
$493$	7.11949	0.320646
$494$	0	0
$495$	3.65919	0.164468
$496$	0	0
$497$	−10.1875	−0.456972
$498$	0	0
$499$	−21.5606	−0.965185	−0.482593	−	0.875845i	$-0.660305\pi$
$499$	−21.5606	−0.965185	−0.482593	+	0.875845i	$0.660305\pi$
$500$	0	0
$501$	16.5821	0.740835
$502$	0	0
$503$	−5.45317	−0.243145	−0.121572	−	0.992583i	$-0.538794\pi$
$503$	−5.45317	−0.243145	−0.121572	+	0.992583i	$0.538794\pi$
$504$	0	0
$505$	21.8322	0.971520
$506$	0	0
$507$	12.8043	0.568659
$508$	0	0
$509$	−23.6991	−1.05045	−0.525223	−	0.850965i	$-0.676018\pi$
$509$	−23.6991	−1.05045	−0.525223	+	0.850965i	$0.676018\pi$
$510$	0	0
$511$	9.94738	0.440046
$512$	0	0
$513$	−5.55811	−0.245396
$514$	0	0
$515$	22.1872	0.977685
$516$	0	0
$517$	21.8707	0.961874
$518$	0	0
$519$	−16.7748	−0.736332
$520$	0	0
$521$	38.0359	1.66638	0.833191	−	0.552986i	$-0.186512\pi$
$521$	38.0359	1.66638	0.833191	+	0.552986i	$0.186512\pi$
$522$	0	0
$523$	34.1791	1.49455	0.747273	−	0.664517i	$-0.231363\pi$
$523$	34.1791	1.49455	0.747273	+	0.664517i	$0.231363\pi$
$524$	0	0
$525$	3.57252	0.155918
$526$	0	0
$527$	70.2678	3.06091
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−7.70376	−0.334315
$532$	0	0
$533$	−0.252176	−0.0109230
$534$	0	0
$535$	−0.144703	−0.00625605
$536$	0	0
$537$	20.3735	0.879183
$538$	0	0
$539$	−16.8712	−0.726692
$540$	0	0
$541$	7.81012	0.335783	0.167892	−	0.985805i	$-0.446304\pi$
$541$	7.81012	0.335783	0.167892	+	0.985805i	$0.446304\pi$
$542$	0	0
$543$	−10.7827	−0.462730
$544$	0	0
$545$	4.62961	0.198311
$546$	0	0
$547$	−14.6738	−0.627408	−0.313704	−	0.949521i	$-0.601570\pi$
$547$	−14.6738	−0.627408	−0.313704	+	0.949521i	$0.601570\pi$
$548$	0	0
$549$	0.605364	0.0258363
$550$	0	0
$551$	5.55811	0.236783
$552$	0	0
$553$	−4.66541	−0.198394
$554$	0	0
$555$	−4.00049	−0.169811
$556$	0	0
$557$	26.9549	1.14211	0.571057	−	0.820910i	$-0.306533\pi$
$557$	26.9549	1.14211	0.571057	+	0.820910i	$0.306533\pi$
$558$	0	0
$559$	−1.46651	−0.0620267
$560$	0	0
$561$	−20.4312	−0.862605
$562$	0	0
$563$	−5.18284	−0.218431	−0.109215	−	0.994018i	$-0.534834\pi$
$563$	−5.18284	−0.218431	−0.109215	+	0.994018i	$0.534834\pi$
$564$	0	0
$565$	5.94241	0.249999
$566$	0	0
$567$	1.05879	0.0444651
$568$	0	0
$569$	43.1031	1.80697	0.903487	−	0.428615i	$-0.140998\pi$
$569$	43.1031	1.80697	0.903487	+	0.428615i	$0.140998\pi$
$570$	0	0
$571$	22.6916	0.949615	0.474808	−	0.880090i	$-0.342518\pi$
$571$	22.6916	0.949615	0.474808	+	0.880090i	$0.342518\pi$
$572$	0	0
$573$	14.5647	0.608447
$574$	0	0
$575$	−3.37415	−0.140712
$576$	0	0
$577$	−40.6126	−1.69072	−0.845362	−	0.534194i	$-0.820615\pi$
$577$	−40.6126	−1.69072	−0.845362	+	0.534194i	$0.820615\pi$
$578$	0	0
$579$	26.9558	1.12024
$580$	0	0
$581$	−12.5511	−0.520705
$582$	0	0
$583$	−0.256256	−0.0106130
$584$	0	0
$585$	−0.564061	−0.0233211
$586$	0	0
$587$	−2.01872	−0.0833215	−0.0416608	−	0.999132i	$-0.513265\pi$
$587$	−2.01872	−0.0833215	−0.0416608	+	0.999132i	$0.513265\pi$
$588$	0	0
$589$	54.8573	2.26036
$590$	0	0
$591$	−7.37900	−0.303532
$592$	0	0
$593$	39.5678	1.62486	0.812428	−	0.583061i	$-0.198145\pi$
$593$	39.5678	1.62486	0.812428	+	0.583061i	$0.198145\pi$
$594$	0	0
$595$	9.61170	0.394041
$596$	0	0
$597$	4.10097	0.167841
$598$	0	0
$599$	−28.4345	−1.16180	−0.580902	−	0.813973i	$-0.697300\pi$
$599$	−28.4345	−1.16180	−0.580902	+	0.813973i	$0.697300\pi$
$600$	0	0
$601$	22.3696	0.912474	0.456237	−	0.889858i	$-0.349197\pi$
$601$	22.3696	0.912474	0.456237	+	0.889858i	$0.349197\pi$
$602$	0	0
$603$	−2.08669	−0.0849766
$604$	0	0
$605$	−3.52502	−0.143312
$606$	0	0
$607$	−13.9477	−0.566119	−0.283059	−	0.959102i	$-0.591349\pi$
$607$	−13.9477	−0.566119	−0.283059	+	0.959102i	$0.591349\pi$
$608$	0	0
$609$	−1.05879	−0.0429044
$610$	0	0
$611$	−3.37136	−0.136391
$612$	0	0
$613$	20.6578	0.834362	0.417181	−	0.908824i	$-0.363018\pi$
$613$	20.6578	0.834362	0.417181	+	0.908824i	$0.363018\pi$
$614$	0	0
$615$	−0.726873	−0.0293104
$616$	0	0
$617$	11.9068	0.479348	0.239674	−	0.970853i	$-0.422959\pi$
$617$	11.9068	0.479348	0.239674	+	0.970853i	$0.422959\pi$
$618$	0	0
$619$	−23.4110	−0.940969	−0.470485	−	0.882408i	$-0.655921\pi$
$619$	−23.4110	−0.940969	−0.470485	+	0.882408i	$0.655921\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−16.8498	−0.675073
$624$	0	0
$625$	3.25562	0.130225
$626$	0	0
$627$	−15.9504	−0.636997
$628$	0	0
$629$	22.3368	0.890628
$630$	0	0
$631$	7.21302	0.287146	0.143573	−	0.989640i	$-0.454141\pi$
$631$	7.21302	0.287146	0.143573	+	0.989640i	$0.454141\pi$
$632$	0	0
$633$	−17.3554	−0.689815
$634$	0	0
$635$	4.31299	0.171156
$636$	0	0
$637$	2.60068	0.103043
$638$	0	0
$639$	−9.62182	−0.380633
$640$	0	0
$641$	14.3952	0.568577	0.284289	−	0.958739i	$-0.408243\pi$
$641$	14.3952	0.568577	0.284289	+	0.958739i	$0.408243\pi$
$642$	0	0
$643$	19.1830	0.756503	0.378251	−	0.925703i	$-0.376526\pi$
$643$	19.1830	0.756503	0.378251	+	0.925703i	$0.376526\pi$
$644$	0	0
$645$	−4.22707	−0.166441
$646$	0	0
$647$	−10.3169	−0.405599	−0.202799	−	0.979220i	$-0.565004\pi$
$647$	−10.3169	−0.405599	−0.202799	+	0.979220i	$0.565004\pi$
$648$	0	0
$649$	−22.1079	−0.867810
$650$	0	0
$651$	−10.4500	−0.409569
$652$	0	0
$653$	−16.5857	−0.649049	−0.324525	−	0.945877i	$-0.605204\pi$
$653$	−16.5857	−0.649049	−0.324525	+	0.945877i	$0.605204\pi$
$654$	0	0
$655$	7.08780	0.276943
$656$	0	0
$657$	9.39503	0.366535
$658$	0	0
$659$	27.5776	1.07427	0.537135	−	0.843496i	$-0.319506\pi$
$659$	27.5776	1.07427	0.537135	+	0.843496i	$0.319506\pi$
$660$	0	0
$661$	14.0135	0.545060	0.272530	−	0.962147i	$-0.412140\pi$
$661$	14.0135	0.545060	0.272530	+	0.962147i	$0.412140\pi$
$662$	0	0
$663$	3.14945	0.122315
$664$	0	0
$665$	7.50375	0.290983
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−13.1742	−0.509345
$670$	0	0
$671$	1.73724	0.0670655
$672$	0	0
$673$	20.9827	0.808823	0.404411	−	0.914577i	$-0.367476\pi$
$673$	20.9827	0.808823	0.404411	+	0.914577i	$0.367476\pi$
$674$	0	0
$675$	3.37415	0.129871
$676$	0	0
$677$	7.83936	0.301291	0.150646	−	0.988588i	$-0.451865\pi$
$677$	7.83936	0.301291	0.150646	+	0.988588i	$0.451865\pi$
$678$	0	0
$679$	1.87603	0.0719955
$680$	0	0
$681$	22.4629	0.860779
$682$	0	0
$683$	−20.3927	−0.780304	−0.390152	−	0.920751i	$-0.627578\pi$
$683$	−20.3927	−0.780304	−0.390152	+	0.920751i	$0.627578\pi$
$684$	0	0
$685$	−2.74745	−0.104975
$686$	0	0
$687$	1.02268	0.0390178
$688$	0	0
$689$	0.0395017	0.00150490
$690$	0	0
$691$	−9.09234	−0.345889	−0.172944	−	0.984932i	$-0.555328\pi$
$691$	−9.09234	−0.345889	−0.172944	+	0.984932i	$0.555328\pi$
$692$	0	0
$693$	3.03847	0.115422
$694$	0	0
$695$	−20.0861	−0.761911
$696$	0	0
$697$	4.05852	0.153727
$698$	0	0
$699$	−8.36599	−0.316431
$700$	0	0
$701$	6.45943	0.243969	0.121985	−	0.992532i	$-0.461074\pi$
$701$	6.45943	0.243969	0.121985	+	0.992532i	$0.461074\pi$
$702$	0	0
$703$	17.4381	0.657691
$704$	0	0
$705$	−9.71761	−0.365986
$706$	0	0
$707$	18.1287	0.681801
$708$	0	0
$709$	−32.0419	−1.20336	−0.601680	−	0.798737i	$-0.705502\pi$
$709$	−32.0419	−1.20336	−0.601680	+	0.798737i	$0.705502\pi$
$710$	0	0
$711$	−4.40635	−0.165251
$712$	0	0
$713$	9.86978	0.369626
$714$	0	0
$715$	−1.61872	−0.0605365
$716$	0	0
$717$	25.0231	0.934504
$718$	0	0
$719$	−30.7996	−1.14863	−0.574315	−	0.818634i	$-0.694732\pi$
$719$	−30.7996	−1.14863	−0.574315	+	0.818634i	$0.694732\pi$
$720$	0	0
$721$	18.4235	0.686128
$722$	0	0
$723$	−7.76619	−0.288828
$724$	0	0
$725$	−3.37415	−0.125313
$726$	0	0
$727$	−33.6829	−1.24923	−0.624615	−	0.780933i	$-0.714744\pi$
$727$	−33.6829	−1.24923	−0.624615	+	0.780933i	$0.714744\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	23.6020	0.872950
$732$	0	0
$733$	34.2762	1.26602	0.633010	−	0.774144i	$-0.281819\pi$
$733$	34.2762	1.26602	0.633010	+	0.774144i	$0.281819\pi$
$734$	0	0
$735$	7.49620	0.276501
$736$	0	0
$737$	−5.98828	−0.220581
$738$	0	0
$739$	45.5235	1.67461	0.837305	−	0.546737i	$-0.184130\pi$
$739$	45.5235	1.67461	0.837305	+	0.546737i	$0.184130\pi$
$740$	0	0
$741$	2.45874	0.0903241
$742$	0	0
$743$	37.6596	1.38160	0.690799	−	0.723047i	$-0.257259\pi$
$743$	37.6596	1.38160	0.690799	+	0.723047i	$0.257259\pi$
$744$	0	0
$745$	30.6267	1.12208
$746$	0	0
$747$	−11.8541	−0.433720
$748$	0	0
$749$	−0.120157	−0.00439042
$750$	0	0
$751$	−15.9505	−0.582043	−0.291022	−	0.956716i	$-0.593995\pi$
$751$	−15.9505	−0.582043	−0.291022	+	0.956716i	$0.593995\pi$
$752$	0	0
$753$	−19.1812	−0.699002
$754$	0	0
$755$	−28.5295	−1.03829
$756$	0	0
$757$	−50.1992	−1.82452	−0.912261	−	0.409610i	$-0.865665\pi$
$757$	−50.1992	−1.82452	−0.912261	+	0.409610i	$0.865665\pi$
$758$	0	0
$759$	−2.86975	−0.104165
$760$	0	0
$761$	−35.9729	−1.30402	−0.652009	−	0.758211i	$-0.726074\pi$
$761$	−35.9729	−1.30402	−0.652009	+	0.758211i	$0.726074\pi$
$762$	0	0
$763$	3.84428	0.139172
$764$	0	0
$765$	9.07799	0.328215
$766$	0	0
$767$	3.40791	0.123053
$768$	0	0
$769$	36.1256	1.30272	0.651360	−	0.758768i	$-0.274199\pi$
$769$	36.1256	1.30272	0.651360	+	0.758768i	$0.274199\pi$
$770$	0	0
$771$	−16.6896	−0.601060
$772$	0	0
$773$	32.1947	1.15796	0.578982	−	0.815340i	$-0.303450\pi$
$773$	32.1947	1.15796	0.578982	+	0.815340i	$0.303450\pi$
$774$	0	0
$775$	−33.3021	−1.19625
$776$	0	0
$777$	−3.32187	−0.119172
$778$	0	0
$779$	3.16844	0.113521
$780$	0	0
$781$	−27.6122	−0.988043
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−8.08463	−0.288553
$786$	0	0
$787$	23.4543	0.836055	0.418027	−	0.908434i	$-0.362722\pi$
$787$	23.4543	0.836055	0.418027	+	0.908434i	$0.362722\pi$
$788$	0	0
$789$	21.2779	0.757512
$790$	0	0
$791$	4.93438	0.175447
$792$	0	0
$793$	−0.267795	−0.00950967
$794$	0	0
$795$	0.113860	0.00403819
$796$	0	0
$797$	−8.05138	−0.285194	−0.142597	−	0.989781i	$-0.545545\pi$
$797$	−8.05138	−0.285194	−0.142597	+	0.989781i	$0.545545\pi$
$798$	0	0
$799$	54.2586	1.91953
$800$	0	0
$801$	−15.9142	−0.562299
$802$	0	0
$803$	26.9614	0.951447
$804$	0	0
$805$	1.35005	0.0475831
$806$	0	0
$807$	−15.8508	−0.557975
$808$	0	0
$809$	31.7619	1.11669	0.558345	−	0.829609i	$-0.311437\pi$
$809$	31.7619	1.11669	0.558345	+	0.829609i	$0.311437\pi$
$810$	0	0
$811$	56.1925	1.97319	0.986593	−	0.163202i	$-0.0521823\pi$
$811$	56.1925	1.97319	0.986593	+	0.163202i	$0.0521823\pi$
$812$	0	0
$813$	−2.60058	−0.0912062
$814$	0	0
$815$	−10.9657	−0.384111
$816$	0	0
$817$	18.4258	0.644636
$818$	0	0
$819$	−0.468378	−0.0163664
$820$	0	0
$821$	30.5428	1.06595	0.532976	−	0.846131i	$-0.321074\pi$
$821$	30.5428	1.06595	0.532976	+	0.846131i	$0.321074\pi$
$822$	0	0
$823$	−22.9871	−0.801281	−0.400640	−	0.916235i	$-0.631212\pi$
$823$	−22.9871	−0.801281	−0.400640	+	0.916235i	$0.631212\pi$
$824$	0	0
$825$	9.68297	0.337118
$826$	0	0
$827$	9.10724	0.316690	0.158345	−	0.987384i	$-0.449384\pi$
$827$	9.10724	0.316690	0.158345	+	0.987384i	$0.449384\pi$
$828$	0	0
$829$	29.3903	1.02077	0.510384	−	0.859947i	$-0.329503\pi$
$829$	29.3903	1.02077	0.510384	+	0.859947i	$0.329503\pi$
$830$	0	0
$831$	15.4758	0.536850
$832$	0	0
$833$	−41.8552	−1.45020
$834$	0	0
$835$	−21.1437	−0.731707
$836$	0	0
$837$	−9.86978	−0.341149
$838$	0	0
$839$	57.1937	1.97455	0.987273	−	0.159036i	$-0.0508386\pi$
$839$	57.1937	1.97455	0.987273	+	0.159036i	$0.0508386\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−27.1129	−0.933818
$844$	0	0
$845$	−16.3266	−0.561653
$846$	0	0
$847$	−2.92706	−0.100575
$848$	0	0
$849$	6.74351	0.231437
$850$	0	0
$851$	3.13742	0.107549
$852$	0	0
$853$	−21.3277	−0.730248	−0.365124	−	0.930959i	$-0.618973\pi$
$853$	−21.3277	−0.730248	−0.365124	+	0.930959i	$0.618973\pi$
$854$	0	0
$855$	7.08708	0.242373
$856$	0	0
$857$	4.60456	0.157289	0.0786444	−	0.996903i	$-0.474941\pi$
$857$	4.60456	0.157289	0.0786444	+	0.996903i	$0.474941\pi$
$858$	0	0
$859$	−3.20234	−0.109262	−0.0546312	−	0.998507i	$-0.517398\pi$
$859$	−3.20234	−0.109262	−0.0546312	+	0.998507i	$0.517398\pi$
$860$	0	0
$861$	−0.603572	−0.0205697
$862$	0	0
$863$	−50.0387	−1.70334	−0.851669	−	0.524080i	$-0.824409\pi$
$863$	−50.0387	−1.70334	−0.851669	+	0.524080i	$0.824409\pi$
$864$	0	0
$865$	21.3894	0.727260
$866$	0	0
$867$	−33.6872	−1.14408
$868$	0	0
$869$	−12.6451	−0.428957
$870$	0	0
$871$	0.923090	0.0312777
$872$	0	0
$873$	1.77186	0.0599684
$874$	0	0
$875$	−11.3056	−0.382197
$876$	0	0
$877$	−53.0266	−1.79058	−0.895290	−	0.445483i	$-0.853032\pi$
$877$	−53.0266	−1.79058	−0.895290	+	0.445483i	$0.853032\pi$
$878$	0	0
$879$	−0.678749	−0.0228936
$880$	0	0
$881$	−43.3377	−1.46008	−0.730042	−	0.683403i	$-0.760499\pi$
$881$	−43.3377	−1.46008	−0.730042	+	0.683403i	$0.760499\pi$
$882$	0	0
$883$	−19.9719	−0.672107	−0.336054	−	0.941843i	$-0.609092\pi$
$883$	−19.9719	−0.672107	−0.336054	+	0.941843i	$0.609092\pi$
$884$	0	0
$885$	9.82298	0.330196
$886$	0	0
$887$	−8.94174	−0.300234	−0.150117	−	0.988668i	$-0.547965\pi$
$887$	−8.94174	−0.300234	−0.150117	+	0.988668i	$0.547965\pi$
$888$	0	0
$889$	3.58137	0.120115
$890$	0	0
$891$	2.86975	0.0961403
$892$	0	0
$893$	42.3590	1.41749
$894$	0	0
$895$	−25.9781	−0.868351
$896$	0	0
$897$	0.442370	0.0147703
$898$	0	0
$899$	9.86978	0.329176
$900$	0	0
$901$	−0.635740	−0.0211796
$902$	0	0
$903$	−3.51002	−0.116806
$904$	0	0
$905$	13.7489	0.457028
$906$	0	0
$907$	−14.9439	−0.496205	−0.248102	−	0.968734i	$-0.579807\pi$
$907$	−14.9439	−0.496205	−0.248102	+	0.968734i	$0.579807\pi$
$908$	0	0
$909$	17.1221	0.567904
$910$	0	0
$911$	−14.7265	−0.487910	−0.243955	−	0.969787i	$-0.578445\pi$
$911$	−14.7265	−0.487910	−0.243955	+	0.969787i	$0.578445\pi$
$912$	0	0
$913$	−34.0184	−1.12584
$914$	0	0
$915$	−0.771893	−0.0255180
$916$	0	0
$917$	5.88548	0.194356
$918$	0	0
$919$	−50.4312	−1.66357	−0.831786	−	0.555097i	$-0.812681\pi$
$919$	−50.4312	−1.66357	−0.831786	+	0.555097i	$0.812681\pi$
$920$	0	0
$921$	19.8221	0.653161
$922$	0	0
$923$	4.25640	0.140101
$924$	0	0
$925$	−10.5861	−0.348069
$926$	0	0
$927$	17.4005	0.571508
$928$	0	0
$929$	−47.0385	−1.54328	−0.771641	−	0.636058i	$-0.780564\pi$
$929$	−47.0385	−1.54328	−0.771641	+	0.636058i	$0.780564\pi$
$930$	0	0
$931$	−32.6759	−1.07091
$932$	0	0
$933$	−27.9653	−0.915542
$934$	0	0
$935$	26.0516	0.851977
$936$	0	0
$937$	28.9908	0.947089	0.473545	−	0.880770i	$-0.342974\pi$
$937$	28.9908	0.947089	0.473545	+	0.880770i	$0.342974\pi$
$938$	0	0
$939$	4.12103	0.134485
$940$	0	0
$941$	8.51175	0.277475	0.138738	−	0.990329i	$-0.455696\pi$
$941$	8.51175	0.277475	0.138738	+	0.990329i	$0.455696\pi$
$942$	0	0
$943$	0.570057	0.0185636
$944$	0	0
$945$	−1.35005	−0.0439173
$946$	0	0
$947$	2.39763	0.0779126	0.0389563	−	0.999241i	$-0.487597\pi$
$947$	2.39763	0.0779126	0.0389563	+	0.999241i	$0.487597\pi$
$948$	0	0
$949$	−4.15608	−0.134912
$950$	0	0
$951$	−1.06358	−0.0344890
$952$	0	0
$953$	−16.9655	−0.549566	−0.274783	−	0.961506i	$-0.588606\pi$
$953$	−16.9655	−0.549566	−0.274783	+	0.961506i	$0.588606\pi$
$954$	0	0
$955$	−18.5712	−0.600951
$956$	0	0
$957$	−2.86975	−0.0927659
$958$	0	0
$959$	−2.28139	−0.0736700
$960$	0	0
$961$	66.4125	2.14234
$962$	0	0
$963$	−0.113485	−0.00365699
$964$	0	0
$965$	−34.3710	−1.10644
$966$	0	0
$967$	−51.4075	−1.65315	−0.826577	−	0.562824i	$-0.809715\pi$
$967$	−51.4075	−1.65315	−0.826577	+	0.562824i	$0.809715\pi$
$968$	0	0
$969$	−39.5709	−1.27120
$970$	0	0
$971$	47.1585	1.51339	0.756694	−	0.653770i	$-0.226813\pi$
$971$	47.1585	1.51339	0.756694	+	0.653770i	$0.226813\pi$
$972$	0	0
$973$	−16.6789	−0.534700
$974$	0	0
$975$	−1.49262	−0.0478022
$976$	0	0
$977$	2.02506	0.0647874	0.0323937	−	0.999475i	$-0.489687\pi$
$977$	2.02506	0.0647874	0.0323937	+	0.999475i	$0.489687\pi$
$978$	0	0
$979$	−45.6697	−1.45961
$980$	0	0
$981$	3.63082	0.115923
$982$	0	0
$983$	23.1217	0.737467	0.368734	−	0.929535i	$-0.379791\pi$
$983$	23.1217	0.737467	0.368734	+	0.929535i	$0.379791\pi$
$984$	0	0
$985$	9.40888	0.299792
$986$	0	0
$987$	−8.06918	−0.256845
$988$	0	0
$989$	3.31512	0.105415
$990$	0	0
$991$	−21.7655	−0.691404	−0.345702	−	0.938344i	$-0.612359\pi$
$991$	−21.7655	−0.691404	−0.345702	+	0.938344i	$0.612359\pi$
$992$	0	0
$993$	−12.6301	−0.400803
$994$	0	0
$995$	−5.22910	−0.165774
$996$	0	0
$997$	16.6418	0.527050	0.263525	−	0.964653i	$-0.415115\pi$
$997$	16.6418	0.527050	0.263525	+	0.964653i	$0.415115\pi$
$998$	0	0
$999$	−3.13742	−0.0992636

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.i.1.10	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.i.1.10	✓	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.27509 q^{5} +1.05879 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.27509 q^{5} +1.05879 q^{7} +1.00000 q^{9} +2.86975 q^{11} -0.442370 q^{13} -1.27509 q^{15} +7.11949 q^{17} +5.55811 q^{19} -1.05879 q^{21} +1.00000 q^{23} -3.37415 q^{25} -1.00000 q^{27} +1.00000 q^{29} +9.86978 q^{31} -2.86975 q^{33} +1.35005 q^{35} +3.13742 q^{37} +0.442370 q^{39} +0.570057 q^{41} +3.31512 q^{43} +1.27509 q^{45} +7.62112 q^{47} -5.87896 q^{49} -7.11949 q^{51} -0.0892956 q^{53} +3.65919 q^{55} -5.55811 q^{57} -7.70376 q^{59} +0.605364 q^{61} +1.05879 q^{63} -0.564061 q^{65} -2.08669 q^{67} -1.00000 q^{69} -9.62182 q^{71} +9.39503 q^{73} +3.37415 q^{75} +3.03847 q^{77} -4.40635 q^{79} +1.00000 q^{81} -11.8541 q^{83} +9.07799 q^{85} -1.00000 q^{87} -15.9142 q^{89} -0.468378 q^{91} -9.86978 q^{93} +7.08708 q^{95} +1.77186 q^{97} +2.86975 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9 + 5 * q^11 + 8 * q^13 - 5 * q^15 + 7 * q^17 + q^19 + 4 * q^21 + 16 * q^23 + 31 * q^25 - 16 * q^27 + 16 * q^29 - 2 * q^31 - 5 * q^33 + 5 * q^35 + 14 * q^37 - 8 * q^39 - q^41 - 13 * q^43 + 5 * q^45 - 4 * q^47 + 30 * q^49 - 7 * q^51 + 19 * q^53 - 37 * q^55 - q^57 + 12 * q^59 + 21 * q^61 - 4 * q^63 + 26 * q^65 - 11 * q^67 - 16 * q^69 + 7 * q^71 - 13 * q^73 - 31 * q^75 + 4 * q^77 - 18 * q^79 + 16 * q^81 + 25 * q^83 + 48 * q^85 - 16 * q^87 + 12 * q^89 - 11 * q^91 + 2 * q^93 - q^95 + 5 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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