LMFDB - Embedded newform 8008.2.a.o.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 12x^{7} + 20x^{6} + 47x^{5} - 55x^{4} - 68x^{3} + 37x^{2} + 21x - 9 $ x^9 - 2x^8 - 12x^7 + 20x^6 + 47x^5 - 55x^4 - 68x^3 + 37x^2 + 21x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.706153$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.706153 q^{3} -2.65916 q^{5} +1.00000 q^{7} -2.50135 q^{9} +O(q^{10})$	$q+0.706153 q^{3} -2.65916 q^{5} +1.00000 q^{7} -2.50135 q^{9} -1.00000 q^{11} +1.00000 q^{13} -1.87777 q^{15} +2.74683 q^{17} +5.12167 q^{19} +0.706153 q^{21} +0.289721 q^{23} +2.07111 q^{25} -3.88479 q^{27} -2.97566 q^{29} -9.58049 q^{31} -0.706153 q^{33} -2.65916 q^{35} +1.75878 q^{37} +0.706153 q^{39} +4.26528 q^{41} +9.87772 q^{43} +6.65147 q^{45} +7.79171 q^{47} +1.00000 q^{49} +1.93968 q^{51} -2.99533 q^{53} +2.65916 q^{55} +3.61668 q^{57} +9.18318 q^{59} -4.30829 q^{61} -2.50135 q^{63} -2.65916 q^{65} -7.56458 q^{67} +0.204588 q^{69} -12.2875 q^{71} -2.16017 q^{73} +1.46252 q^{75} -1.00000 q^{77} +0.418699 q^{79} +4.76079 q^{81} -14.2595 q^{83} -7.30423 q^{85} -2.10127 q^{87} -15.3638 q^{89} +1.00000 q^{91} -6.76529 q^{93} -13.6193 q^{95} +16.5392 q^{97} +2.50135 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) $ 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9	$ 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9} - 9 q^{11} + 9 q^{13} - 4 q^{15} - 9 q^{17} + 9 q^{19} - 2 q^{21} - 10 q^{23} - q^{25} - 8 q^{27} - 13 q^{29} + 7 q^{31} + 2 q^{33} - 4 q^{35} - 15 q^{37} - 2 q^{39} - 18 q^{41} + 7 q^{43} + 9 q^{45} - 11 q^{47} + 9 q^{49} + q^{51} - 16 q^{53} + 4 q^{55} - 26 q^{57} - 2 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 6 q^{73} - 4 q^{75} - 9 q^{77} - 21 q^{79} - 23 q^{81} - 30 q^{83} + q^{85} + q^{87} - 36 q^{89} + 9 q^{91} - 22 q^{93} - 23 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) $ 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9 - 9 * q^11 + 9 * q^13 - 4 * q^15 - 9 * q^17 + 9 * q^19 - 2 * q^21 - 10 * q^23 - q^25 - 8 * q^27 - 13 * q^29 + 7 * q^31 + 2 * q^33 - 4 * q^35 - 15 * q^37 - 2 * q^39 - 18 * q^41 + 7 * q^43 + 9 * q^45 - 11 * q^47 + 9 * q^49 + q^51 - 16 * q^53 + 4 * q^55 - 26 * q^57 - 2 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + 13 * q^67 - 3 * q^69 - q^71 - 6 * q^73 - 4 * q^75 - 9 * q^77 - 21 * q^79 - 23 * q^81 - 30 * q^83 + q^85 + q^87 - 36 * q^89 + 9 * q^91 - 22 * q^93 - 23 * q^95 - 5 * q^97 - 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
$3$	0.706153	0.407697	0.203849	−	0.979002i	$-0.434655\pi$
$3$	0.706153	0.407697	0.203849	+	0.979002i	$0.434655\pi$
$4$	0	0
$5$	−2.65916	−1.18921	−0.594605	−	0.804018i	$-0.702692\pi$
$5$	−2.65916	−1.18921	−0.594605	+	0.804018i	$0.702692\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.50135	−0.833783
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.87777	−0.484838
$16$	0	0
$17$	2.74683	0.666203	0.333101	−	0.942891i	$-0.391905\pi$
$17$	2.74683	0.666203	0.333101	+	0.942891i	$0.391905\pi$
$18$	0	0
$19$	5.12167	1.17499	0.587496	−	0.809227i	$-0.300114\pi$
$19$	5.12167	1.17499	0.587496	+	0.809227i	$0.300114\pi$
$20$	0	0
$21$	0.706153	0.154095
$22$	0	0
$23$	0.289721	0.0604111	0.0302055	−	0.999544i	$-0.490384\pi$
$23$	0.289721	0.0604111	0.0302055	+	0.999544i	$0.490384\pi$
$24$	0	0
$25$	2.07111	0.414221
$26$	0	0
$27$	−3.88479	−0.747629
$28$	0	0
$29$	−2.97566	−0.552567	−0.276283	−	0.961076i	$-0.589103\pi$
$29$	−2.97566	−0.552567	−0.276283	+	0.961076i	$0.589103\pi$
$30$	0	0
$31$	−9.58049	−1.72071	−0.860354	−	0.509697i	$-0.829757\pi$
$31$	−9.58049	−1.72071	−0.860354	+	0.509697i	$0.829757\pi$
$32$	0	0
$33$	−0.706153	−0.122925
$34$	0	0
$35$	−2.65916	−0.449479
$36$	0	0
$37$	1.75878	0.289141	0.144571	−	0.989495i	$-0.453820\pi$
$37$	1.75878	0.289141	0.144571	+	0.989495i	$0.453820\pi$
$38$	0	0
$39$	0.706153	0.113075
$40$	0	0
$41$	4.26528	0.666125	0.333063	−	0.942905i	$-0.391918\pi$
$41$	4.26528	0.666125	0.333063	+	0.942905i	$0.391918\pi$
$42$	0	0
$43$	9.87772	1.50634	0.753169	−	0.657827i	$-0.228524\pi$
$43$	9.87772	1.50634	0.753169	+	0.657827i	$0.228524\pi$
$44$	0	0
$45$	6.65147	0.991543
$46$	0	0
$47$	7.79171	1.13654	0.568269	−	0.822843i	$-0.307613\pi$
$47$	7.79171	1.13654	0.568269	+	0.822843i	$0.307613\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.93968	0.271609
$52$	0	0
$53$	−2.99533	−0.411440	−0.205720	−	0.978611i	$-0.565954\pi$
$53$	−2.99533	−0.411440	−0.205720	+	0.978611i	$0.565954\pi$
$54$	0	0
$55$	2.65916	0.358560
$56$	0	0
$57$	3.61668	0.479041
$58$	0	0
$59$	9.18318	1.19555	0.597774	−	0.801665i	$-0.296052\pi$
$59$	9.18318	1.19555	0.597774	+	0.801665i	$0.296052\pi$
$60$	0	0
$61$	−4.30829	−0.551620	−0.275810	−	0.961212i	$-0.588946\pi$
$61$	−4.30829	−0.551620	−0.275810	+	0.961212i	$0.588946\pi$
$62$	0	0
$63$	−2.50135	−0.315140
$64$	0	0
$65$	−2.65916	−0.329828
$66$	0	0
$67$	−7.56458	−0.924161	−0.462080	−	0.886838i	$-0.652897\pi$
$67$	−7.56458	−0.924161	−0.462080	+	0.886838i	$0.652897\pi$
$68$	0	0
$69$	0.204588	0.0246294
$70$	0	0
$71$	−12.2875	−1.45825	−0.729127	−	0.684379i	$-0.760074\pi$
$71$	−12.2875	−1.45825	−0.729127	+	0.684379i	$0.760074\pi$
$72$	0	0
$73$	−2.16017	−0.252829	−0.126415	−	0.991977i	$-0.540347\pi$
$73$	−2.16017	−0.252829	−0.126415	+	0.991977i	$0.540347\pi$
$74$	0	0
$75$	1.46252	0.168877
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	0.418699	0.0471074	0.0235537	−	0.999723i	$-0.492502\pi$
$79$	0.418699	0.0471074	0.0235537	+	0.999723i	$0.492502\pi$
$80$	0	0
$81$	4.76079	0.528977
$82$	0	0
$83$	−14.2595	−1.56518	−0.782592	−	0.622534i	$-0.786103\pi$
$83$	−14.2595	−1.56518	−0.782592	+	0.622534i	$0.786103\pi$
$84$	0	0
$85$	−7.30423	−0.792255
$86$	0	0
$87$	−2.10127	−0.225280
$88$	0	0
$89$	−15.3638	−1.62856	−0.814281	−	0.580471i	$-0.802868\pi$
$89$	−15.3638	−1.62856	−0.814281	+	0.580471i	$0.802868\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−6.76529	−0.701528
$94$	0	0
$95$	−13.6193	−1.39731
$96$	0	0
$97$	16.5392	1.67931	0.839653	−	0.543124i	$-0.182759\pi$
$97$	16.5392	1.67931	0.839653	+	0.543124i	$0.182759\pi$
$98$	0	0
$99$	2.50135	0.251395
$100$	0	0
$101$	−1.17822	−0.117237	−0.0586185	−	0.998280i	$-0.518670\pi$
$101$	−1.17822	−0.117237	−0.0586185	+	0.998280i	$0.518670\pi$
$102$	0	0
$103$	10.0011	0.985437	0.492719	−	0.870189i	$-0.336003\pi$
$103$	10.0011	0.985437	0.492719	+	0.870189i	$0.336003\pi$
$104$	0	0
$105$	−1.87777	−0.183252
$106$	0	0
$107$	−6.24662	−0.603884	−0.301942	−	0.953326i	$-0.597635\pi$
$107$	−6.24662	−0.603884	−0.301942	+	0.953326i	$0.597635\pi$
$108$	0	0
$109$	4.50840	0.431826	0.215913	−	0.976413i	$-0.430727\pi$
$109$	4.50840	0.431826	0.215913	+	0.976413i	$0.430727\pi$
$110$	0	0
$111$	1.24196	0.117882
$112$	0	0
$113$	9.42187	0.886335	0.443167	−	0.896439i	$-0.353855\pi$
$113$	9.42187	0.886335	0.443167	+	0.896439i	$0.353855\pi$
$114$	0	0
$115$	−0.770414	−0.0718415
$116$	0	0
$117$	−2.50135	−0.231250
$118$	0	0
$119$	2.74683	0.251801
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.01194	0.271578
$124$	0	0
$125$	7.78838	0.696614
$126$	0	0
$127$	−18.1566	−1.61114	−0.805569	−	0.592502i	$-0.798140\pi$
$127$	−18.1566	−1.61114	−0.805569	+	0.592502i	$0.798140\pi$
$128$	0	0
$129$	6.97518	0.614130
$130$	0	0
$131$	−10.2455	−0.895153	−0.447577	−	0.894246i	$-0.647713\pi$
$131$	−10.2455	−0.895153	−0.447577	+	0.894246i	$0.647713\pi$
$132$	0	0
$133$	5.12167	0.444105
$134$	0	0
$135$	10.3303	0.889088
$136$	0	0
$137$	13.2220	1.12963	0.564817	−	0.825216i	$-0.308947\pi$
$137$	13.2220	1.12963	0.564817	+	0.825216i	$0.308947\pi$
$138$	0	0
$139$	−21.2360	−1.80121	−0.900606	−	0.434637i	$-0.856877\pi$
$139$	−21.2360	−1.80121	−0.900606	+	0.434637i	$0.856877\pi$
$140$	0	0
$141$	5.50214	0.463364
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	7.91275	0.657118
$146$	0	0
$147$	0.706153	0.0582425
$148$	0	0
$149$	−4.75850	−0.389832	−0.194916	−	0.980820i	$-0.562443\pi$
$149$	−4.75850	−0.389832	−0.194916	+	0.980820i	$0.562443\pi$
$150$	0	0
$151$	−5.69873	−0.463756	−0.231878	−	0.972745i	$-0.574487\pi$
$151$	−5.69873	−0.463756	−0.231878	+	0.972745i	$0.574487\pi$
$152$	0	0
$153$	−6.87077	−0.555469
$154$	0	0
$155$	25.4760	2.04628
$156$	0	0
$157$	−3.71574	−0.296548	−0.148274	−	0.988946i	$-0.547372\pi$
$157$	−3.71574	−0.296548	−0.148274	+	0.988946i	$0.547372\pi$
$158$	0	0
$159$	−2.11516	−0.167743
$160$	0	0
$161$	0.289721	0.0228332
$162$	0	0
$163$	13.6708	1.07078	0.535388	−	0.844606i	$-0.320165\pi$
$163$	13.6708	1.07078	0.535388	+	0.844606i	$0.320165\pi$
$164$	0	0
$165$	1.87777	0.146184
$166$	0	0
$167$	8.52736	0.659867	0.329934	−	0.944004i	$-0.392974\pi$
$167$	8.52736	0.659867	0.329934	+	0.944004i	$0.392974\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−12.8111	−0.979688
$172$	0	0
$173$	6.95872	0.529062	0.264531	−	0.964377i	$-0.414783\pi$
$173$	6.95872	0.529062	0.264531	+	0.964377i	$0.414783\pi$
$174$	0	0
$175$	2.07111	0.156561
$176$	0	0
$177$	6.48473	0.487422
$178$	0	0
$179$	−11.1176	−0.830971	−0.415485	−	0.909600i	$-0.636388\pi$
$179$	−11.1176	−0.830971	−0.415485	+	0.909600i	$0.636388\pi$
$180$	0	0
$181$	−0.934991	−0.0694973	−0.0347487	−	0.999396i	$-0.511063\pi$
$181$	−0.934991	−0.0694973	−0.0347487	+	0.999396i	$0.511063\pi$
$182$	0	0
$183$	−3.04231	−0.224894
$184$	0	0
$185$	−4.67686	−0.343850
$186$	0	0
$187$	−2.74683	−0.200868
$188$	0	0
$189$	−3.88479	−0.282577
$190$	0	0
$191$	−20.5619	−1.48781	−0.743905	−	0.668286i	$-0.767028\pi$
$191$	−20.5619	−1.48781	−0.743905	+	0.668286i	$0.767028\pi$
$192$	0	0
$193$	−15.5263	−1.11761	−0.558803	−	0.829300i	$-0.688739\pi$
$193$	−15.5263	−1.11761	−0.558803	+	0.829300i	$0.688739\pi$
$194$	0	0
$195$	−1.87777	−0.134470
$196$	0	0
$197$	−9.85207	−0.701931	−0.350965	−	0.936388i	$-0.614146\pi$
$197$	−9.85207	−0.701931	−0.350965	+	0.936388i	$0.614146\pi$
$198$	0	0
$199$	21.9645	1.55702	0.778511	−	0.627631i	$-0.215975\pi$
$199$	21.9645	1.55702	0.778511	+	0.627631i	$0.215975\pi$
$200$	0	0
$201$	−5.34175	−0.376778
$202$	0	0
$203$	−2.97566	−0.208851
$204$	0	0
$205$	−11.3421	−0.792163
$206$	0	0
$207$	−0.724694	−0.0503697
$208$	0	0
$209$	−5.12167	−0.354274
$210$	0	0
$211$	5.61872	0.386809	0.193404	−	0.981119i	$-0.438047\pi$
$211$	5.61872	0.386809	0.193404	+	0.981119i	$0.438047\pi$
$212$	0	0
$213$	−8.67682	−0.594526
$214$	0	0
$215$	−26.2664	−1.79135
$216$	0	0
$217$	−9.58049	−0.650366
$218$	0	0
$219$	−1.52541	−0.103078
$220$	0	0
$221$	2.74683	0.184771
$222$	0	0
$223$	−28.5031	−1.90871	−0.954353	−	0.298680i	$-0.903454\pi$
$223$	−28.5031	−1.90871	−0.954353	+	0.298680i	$0.903454\pi$
$224$	0	0
$225$	−5.18056	−0.345371
$226$	0	0
$227$	−20.2735	−1.34560	−0.672801	−	0.739823i	$-0.734909\pi$
$227$	−20.2735	−1.34560	−0.672801	+	0.739823i	$0.734909\pi$
$228$	0	0
$229$	−6.17959	−0.408359	−0.204179	−	0.978934i	$-0.565453\pi$
$229$	−6.17959	−0.408359	−0.204179	+	0.978934i	$0.565453\pi$
$230$	0	0
$231$	−0.706153	−0.0464614
$232$	0	0
$233$	−19.9970	−1.31005	−0.655025	−	0.755608i	$-0.727342\pi$
$233$	−19.9970	−1.31005	−0.655025	+	0.755608i	$0.727342\pi$
$234$	0	0
$235$	−20.7194	−1.35158
$236$	0	0
$237$	0.295666	0.0192055
$238$	0	0
$239$	−2.99793	−0.193920	−0.0969601	−	0.995288i	$-0.530912\pi$
$239$	−2.99793	−0.193920	−0.0969601	+	0.995288i	$0.530912\pi$
$240$	0	0
$241$	−12.0993	−0.779386	−0.389693	−	0.920945i	$-0.627419\pi$
$241$	−12.0993	−0.779386	−0.389693	+	0.920945i	$0.627419\pi$
$242$	0	0
$243$	15.0162	0.963291
$244$	0	0
$245$	−2.65916	−0.169887
$246$	0	0
$247$	5.12167	0.325884
$248$	0	0
$249$	−10.0694	−0.638122
$250$	0	0
$251$	−5.21684	−0.329284	−0.164642	−	0.986353i	$-0.552647\pi$
$251$	−5.21684	−0.329284	−0.164642	+	0.986353i	$0.552647\pi$
$252$	0	0
$253$	−0.289721	−0.0182146
$254$	0	0
$255$	−5.15790	−0.323001
$256$	0	0
$257$	−16.2029	−1.01071	−0.505354	−	0.862912i	$-0.668638\pi$
$257$	−16.2029	−1.01071	−0.505354	+	0.862912i	$0.668638\pi$
$258$	0	0
$259$	1.75878	0.109285
$260$	0	0
$261$	7.44317	0.460721
$262$	0	0
$263$	−9.58463	−0.591014	−0.295507	−	0.955341i	$-0.595489\pi$
$263$	−9.58463	−0.591014	−0.295507	+	0.955341i	$0.595489\pi$
$264$	0	0
$265$	7.96504	0.489288
$266$	0	0
$267$	−10.8492	−0.663960
$268$	0	0
$269$	−25.9944	−1.58491	−0.792454	−	0.609932i	$-0.791197\pi$
$269$	−25.9944	−1.58491	−0.792454	+	0.609932i	$0.791197\pi$
$270$	0	0
$271$	−11.3113	−0.687112	−0.343556	−	0.939132i	$-0.611632\pi$
$271$	−11.3113	−0.687112	−0.343556	+	0.939132i	$0.611632\pi$
$272$	0	0
$273$	0.706153	0.0427383
$274$	0	0
$275$	−2.07111	−0.124892
$276$	0	0
$277$	−15.2836	−0.918300	−0.459150	−	0.888359i	$-0.651846\pi$
$277$	−15.2836	−0.918300	−0.459150	+	0.888359i	$0.651846\pi$
$278$	0	0
$279$	23.9642	1.43470
$280$	0	0
$281$	−8.04823	−0.480117	−0.240059	−	0.970758i	$-0.577167\pi$
$281$	−8.04823	−0.480117	−0.240059	+	0.970758i	$0.577167\pi$
$282$	0	0
$283$	11.1503	0.662816	0.331408	−	0.943487i	$-0.392476\pi$
$283$	11.1503	0.662816	0.331408	+	0.943487i	$0.392476\pi$
$284$	0	0
$285$	−9.61732	−0.569681
$286$	0	0
$287$	4.26528	0.251772
$288$	0	0
$289$	−9.45495	−0.556174
$290$	0	0
$291$	11.6792	0.684648
$292$	0	0
$293$	21.7602	1.27124	0.635621	−	0.772001i	$-0.280744\pi$
$293$	21.7602	1.27124	0.635621	+	0.772001i	$0.280744\pi$
$294$	0	0
$295$	−24.4195	−1.42176
$296$	0	0
$297$	3.88479	0.225418
$298$	0	0
$299$	0.289721	0.0167550
$300$	0	0
$301$	9.87772	0.569342
$302$	0	0
$303$	−0.832001	−0.0477972
$304$	0	0
$305$	11.4564	0.655992
$306$	0	0
$307$	29.8947	1.70618	0.853090	−	0.521764i	$-0.174726\pi$
$307$	29.8947	1.70618	0.853090	+	0.521764i	$0.174726\pi$
$308$	0	0
$309$	7.06230	0.401760
$310$	0	0
$311$	−26.2272	−1.48721	−0.743603	−	0.668621i	$-0.766885\pi$
$311$	−26.2272	−1.48721	−0.743603	+	0.668621i	$0.766885\pi$
$312$	0	0
$313$	−20.0586	−1.13378	−0.566891	−	0.823793i	$-0.691854\pi$
$313$	−20.0586	−1.13378	−0.566891	+	0.823793i	$0.691854\pi$
$314$	0	0
$315$	6.65147	0.374768
$316$	0	0
$317$	−30.6189	−1.71973	−0.859866	−	0.510520i	$-0.829453\pi$
$317$	−30.6189	−1.71973	−0.859866	+	0.510520i	$0.829453\pi$
$318$	0	0
$319$	2.97566	0.166605
$320$	0	0
$321$	−4.41107	−0.246202
$322$	0	0
$323$	14.0683	0.782783
$324$	0	0
$325$	2.07111	0.114884
$326$	0	0
$327$	3.18362	0.176055
$328$	0	0
$329$	7.79171	0.429571
$330$	0	0
$331$	−31.8109	−1.74849	−0.874244	−	0.485487i	$-0.838642\pi$
$331$	−31.8109	−1.74849	−0.874244	+	0.485487i	$0.838642\pi$
$332$	0	0
$333$	−4.39931	−0.241081
$334$	0	0
$335$	20.1154	1.09902
$336$	0	0
$337$	28.0390	1.52738	0.763691	−	0.645582i	$-0.223385\pi$
$337$	28.0390	1.52738	0.763691	+	0.645582i	$0.223385\pi$
$338$	0	0
$339$	6.65328	0.361356
$340$	0	0
$341$	9.58049	0.518813
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.544030	−0.0292896
$346$	0	0
$347$	11.7499	0.630770	0.315385	−	0.948964i	$-0.397866\pi$
$347$	11.7499	0.630770	0.315385	+	0.948964i	$0.397866\pi$
$348$	0	0
$349$	−25.1848	−1.34811	−0.674055	−	0.738681i	$-0.735449\pi$
$349$	−25.1848	−1.34811	−0.674055	+	0.738681i	$0.735449\pi$
$350$	0	0
$351$	−3.88479	−0.207355
$352$	0	0
$353$	−31.2024	−1.66074	−0.830369	−	0.557215i	$-0.811870\pi$
$353$	−31.2024	−1.66074	−0.830369	+	0.557215i	$0.811870\pi$
$354$	0	0
$355$	32.6743	1.73417
$356$	0	0
$357$	1.93968	0.102659
$358$	0	0
$359$	−9.77327	−0.515813	−0.257907	−	0.966170i	$-0.583033\pi$
$359$	−9.77327	−0.515813	−0.257907	+	0.966170i	$0.583033\pi$
$360$	0	0
$361$	7.23154	0.380607
$362$	0	0
$363$	0.706153	0.0370634
$364$	0	0
$365$	5.74424	0.300667
$366$	0	0
$367$	23.9252	1.24888	0.624442	−	0.781071i	$-0.285327\pi$
$367$	23.9252	1.24888	0.624442	+	0.781071i	$0.285327\pi$
$368$	0	0
$369$	−10.6690	−0.555404
$370$	0	0
$371$	−2.99533	−0.155510
$372$	0	0
$373$	6.76029	0.350034	0.175017	−	0.984565i	$-0.444002\pi$
$373$	6.76029	0.350034	0.175017	+	0.984565i	$0.444002\pi$
$374$	0	0
$375$	5.49979	0.284008
$376$	0	0
$377$	−2.97566	−0.153254
$378$	0	0
$379$	30.7821	1.58117	0.790585	−	0.612353i	$-0.209777\pi$
$379$	30.7821	1.58117	0.790585	+	0.612353i	$0.209777\pi$
$380$	0	0
$381$	−12.8213	−0.656857
$382$	0	0
$383$	24.2700	1.24014	0.620069	−	0.784547i	$-0.287105\pi$
$383$	24.2700	1.24014	0.620069	+	0.784547i	$0.287105\pi$
$384$	0	0
$385$	2.65916	0.135523
$386$	0	0
$387$	−24.7076	−1.25596
$388$	0	0
$389$	−21.4727	−1.08871	−0.544355	−	0.838855i	$-0.683226\pi$
$389$	−21.4727	−1.08871	−0.544355	+	0.838855i	$0.683226\pi$
$390$	0	0
$391$	0.795814	0.0402460
$392$	0	0
$393$	−7.23488	−0.364952
$394$	0	0
$395$	−1.11339	−0.0560206
$396$	0	0
$397$	30.4094	1.52621	0.763103	−	0.646277i	$-0.223675\pi$
$397$	30.4094	1.52621	0.763103	+	0.646277i	$0.223675\pi$
$398$	0	0
$399$	3.61668	0.181061
$400$	0	0
$401$	−15.2545	−0.761773	−0.380887	−	0.924622i	$-0.624381\pi$
$401$	−15.2545	−0.761773	−0.380887	+	0.924622i	$0.624381\pi$
$402$	0	0
$403$	−9.58049	−0.477238
$404$	0	0
$405$	−12.6597	−0.629064
$406$	0	0
$407$	−1.75878	−0.0871793
$408$	0	0
$409$	39.8730	1.97160	0.985798	−	0.167937i	$-0.0537106\pi$
$409$	39.8730	1.97160	0.985798	+	0.167937i	$0.0537106\pi$
$410$	0	0
$411$	9.33677	0.460549
$412$	0	0
$413$	9.18318	0.451875
$414$	0	0
$415$	37.9183	1.86133
$416$	0	0
$417$	−14.9958	−0.734349
$418$	0	0
$419$	−19.0320	−0.929775	−0.464888	−	0.885370i	$-0.653905\pi$
$419$	−19.0320	−0.929775	−0.464888	+	0.885370i	$0.653905\pi$
$420$	0	0
$421$	11.1230	0.542102	0.271051	−	0.962565i	$-0.412629\pi$
$421$	11.1230	0.542102	0.271051	+	0.962565i	$0.412629\pi$
$422$	0	0
$423$	−19.4898	−0.947626
$424$	0	0
$425$	5.68897	0.275956
$426$	0	0
$427$	−4.30829	−0.208493
$428$	0	0
$429$	−0.706153	−0.0340934
$430$	0	0
$431$	9.54242	0.459642	0.229821	−	0.973233i	$-0.426186\pi$
$431$	9.54242	0.459642	0.229821	+	0.973233i	$0.426186\pi$
$432$	0	0
$433$	10.0604	0.483471	0.241736	−	0.970342i	$-0.422283\pi$
$433$	10.0604	0.483471	0.241736	+	0.970342i	$0.422283\pi$
$434$	0	0
$435$	5.58761	0.267905
$436$	0	0
$437$	1.48386	0.0709826
$438$	0	0
$439$	−20.6282	−0.984531	−0.492265	−	0.870445i	$-0.663831\pi$
$439$	−20.6282	−0.984531	−0.492265	+	0.870445i	$0.663831\pi$
$440$	0	0
$441$	−2.50135	−0.119112
$442$	0	0
$443$	21.8609	1.03864	0.519322	−	0.854579i	$-0.326185\pi$
$443$	21.8609	1.03864	0.519322	+	0.854579i	$0.326185\pi$
$444$	0	0
$445$	40.8548	1.93670
$446$	0	0
$447$	−3.36023	−0.158933
$448$	0	0
$449$	6.24690	0.294810	0.147405	−	0.989076i	$-0.452908\pi$
$449$	6.24690	0.294810	0.147405	+	0.989076i	$0.452908\pi$
$450$	0	0
$451$	−4.26528	−0.200844
$452$	0	0
$453$	−4.02418	−0.189072
$454$	0	0
$455$	−2.65916	−0.124663
$456$	0	0
$457$	−27.3174	−1.27786	−0.638928	−	0.769267i	$-0.720622\pi$
$457$	−27.3174	−1.27786	−0.638928	+	0.769267i	$0.720622\pi$
$458$	0	0
$459$	−10.6708	−0.498072
$460$	0	0
$461$	15.2281	0.709244	0.354622	−	0.935010i	$-0.384610\pi$
$461$	15.2281	0.709244	0.354622	+	0.935010i	$0.384610\pi$
$462$	0	0
$463$	9.41809	0.437696	0.218848	−	0.975759i	$-0.429770\pi$
$463$	9.41809	0.437696	0.218848	+	0.975759i	$0.429770\pi$
$464$	0	0
$465$	17.9900	0.834264
$466$	0	0
$467$	−22.1161	−1.02341	−0.511705	−	0.859161i	$-0.670986\pi$
$467$	−22.1161	−1.02341	−0.511705	+	0.859161i	$0.670986\pi$
$468$	0	0
$469$	−7.56458	−0.349300
$470$	0	0
$471$	−2.62388	−0.120902
$472$	0	0
$473$	−9.87772	−0.454178
$474$	0	0
$475$	10.6075	0.486707
$476$	0	0
$477$	7.49236	0.343051
$478$	0	0
$479$	−26.0779	−1.19153	−0.595765	−	0.803159i	$-0.703151\pi$
$479$	−26.0779	−1.19153	−0.595765	+	0.803159i	$0.703151\pi$
$480$	0	0
$481$	1.75878	0.0801933
$482$	0	0
$483$	0.204588	0.00930905
$484$	0	0
$485$	−43.9804	−1.99705
$486$	0	0
$487$	−4.71700	−0.213748	−0.106874	−	0.994273i	$-0.534084\pi$
$487$	−4.71700	−0.213748	−0.106874	+	0.994273i	$0.534084\pi$
$488$	0	0
$489$	9.65364	0.436553
$490$	0	0
$491$	30.0245	1.35499	0.677493	−	0.735530i	$-0.263067\pi$
$491$	30.0245	1.35499	0.677493	+	0.735530i	$0.263067\pi$
$492$	0	0
$493$	−8.17363	−0.368122
$494$	0	0
$495$	−6.65147	−0.298962
$496$	0	0
$497$	−12.2875	−0.551168
$498$	0	0
$499$	10.5683	0.473104	0.236552	−	0.971619i	$-0.423983\pi$
$499$	10.5683	0.473104	0.236552	+	0.971619i	$0.423983\pi$
$500$	0	0
$501$	6.02162	0.269026
$502$	0	0
$503$	−13.8927	−0.619443	−0.309721	−	0.950827i	$-0.600236\pi$
$503$	−13.8927	−0.619443	−0.309721	+	0.950827i	$0.600236\pi$
$504$	0	0
$505$	3.13306	0.139419
$506$	0	0
$507$	0.706153	0.0313613
$508$	0	0
$509$	−3.89035	−0.172437	−0.0862184	−	0.996276i	$-0.527478\pi$
$509$	−3.89035	−0.172437	−0.0862184	+	0.996276i	$0.527478\pi$
$510$	0	0
$511$	−2.16017	−0.0955604
$512$	0	0
$513$	−19.8966	−0.878458
$514$	0	0
$515$	−26.5945	−1.17189
$516$	0	0
$517$	−7.79171	−0.342679
$518$	0	0
$519$	4.91392	0.215697
$520$	0	0
$521$	−17.5391	−0.768401	−0.384201	−	0.923250i	$-0.625523\pi$
$521$	−17.5391	−0.768401	−0.384201	+	0.923250i	$0.625523\pi$
$522$	0	0
$523$	17.5562	0.767677	0.383839	−	0.923400i	$-0.374602\pi$
$523$	17.5562	0.767677	0.383839	+	0.923400i	$0.374602\pi$
$524$	0	0
$525$	1.46252	0.0638295
$526$	0	0
$527$	−26.3159	−1.14634
$528$	0	0
$529$	−22.9161	−0.996351
$530$	0	0
$531$	−22.9703	−0.996827
$532$	0	0
$533$	4.26528	0.184750
$534$	0	0
$535$	16.6107	0.718145
$536$	0	0
$537$	−7.85075	−0.338785
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−39.4616	−1.69659	−0.848293	−	0.529527i	$-0.822369\pi$
$541$	−39.4616	−1.69659	−0.848293	+	0.529527i	$0.822369\pi$
$542$	0	0
$543$	−0.660247	−0.0283339
$544$	0	0
$545$	−11.9885	−0.513532
$546$	0	0
$547$	29.5509	1.26350	0.631752	−	0.775170i	$-0.282336\pi$
$547$	29.5509	1.26350	0.631752	+	0.775170i	$0.282336\pi$
$548$	0	0
$549$	10.7765	0.459931
$550$	0	0
$551$	−15.2404	−0.649262
$552$	0	0
$553$	0.418699	0.0178049
$554$	0	0
$555$	−3.30258	−0.140187
$556$	0	0
$557$	14.1731	0.600535	0.300268	−	0.953855i	$-0.402924\pi$
$557$	14.1731	0.600535	0.300268	+	0.953855i	$0.402924\pi$
$558$	0	0
$559$	9.87772	0.417783
$560$	0	0
$561$	−1.93968	−0.0818933
$562$	0	0
$563$	−42.5452	−1.79307	−0.896534	−	0.442975i	$-0.853923\pi$
$563$	−42.5452	−1.79307	−0.896534	+	0.442975i	$0.853923\pi$
$564$	0	0
$565$	−25.0542	−1.05404
$566$	0	0
$567$	4.76079	0.199934
$568$	0	0
$569$	−8.63010	−0.361793	−0.180896	−	0.983502i	$-0.557900\pi$
$569$	−8.63010	−0.361793	−0.180896	+	0.983502i	$0.557900\pi$
$570$	0	0
$571$	25.0377	1.04780	0.523898	−	0.851781i	$-0.324477\pi$
$571$	25.0377	1.04780	0.523898	+	0.851781i	$0.324477\pi$
$572$	0	0
$573$	−14.5199	−0.606576
$574$	0	0
$575$	0.600044	0.0250236
$576$	0	0
$577$	16.7634	0.697871	0.348935	−	0.937147i	$-0.386543\pi$
$577$	16.7634	0.697871	0.348935	+	0.937147i	$0.386543\pi$
$578$	0	0
$579$	−10.9639	−0.455645
$580$	0	0
$581$	−14.2595	−0.591584
$582$	0	0
$583$	2.99533	0.124054
$584$	0	0
$585$	6.65147	0.275005
$586$	0	0
$587$	21.4409	0.884961	0.442481	−	0.896778i	$-0.354099\pi$
$587$	21.4409	0.884961	0.442481	+	0.896778i	$0.354099\pi$
$588$	0	0
$589$	−49.0682	−2.02182
$590$	0	0
$591$	−6.95706	−0.286175
$592$	0	0
$593$	−29.9934	−1.23168	−0.615840	−	0.787871i	$-0.711183\pi$
$593$	−29.9934	−1.23168	−0.615840	+	0.787871i	$0.711183\pi$
$594$	0	0
$595$	−7.30423	−0.299444
$596$	0	0
$597$	15.5103	0.634794
$598$	0	0
$599$	26.2863	1.07403	0.537015	−	0.843572i	$-0.319552\pi$
$599$	26.2863	1.07403	0.537015	+	0.843572i	$0.319552\pi$
$600$	0	0
$601$	−15.3877	−0.627677	−0.313838	−	0.949476i	$-0.601615\pi$
$601$	−15.3877	−0.627677	−0.313838	+	0.949476i	$0.601615\pi$
$602$	0	0
$603$	18.9216	0.770549
$604$	0	0
$605$	−2.65916	−0.108110
$606$	0	0
$607$	−11.9368	−0.484499	−0.242249	−	0.970214i	$-0.577885\pi$
$607$	−11.9368	−0.484499	−0.242249	+	0.970214i	$0.577885\pi$
$608$	0	0
$609$	−2.10127	−0.0851479
$610$	0	0
$611$	7.79171	0.315219
$612$	0	0
$613$	1.53507	0.0620008	0.0310004	−	0.999519i	$-0.490131\pi$
$613$	1.53507	0.0620008	0.0310004	+	0.999519i	$0.490131\pi$
$614$	0	0
$615$	−8.00922	−0.322963
$616$	0	0
$617$	17.5187	0.705274	0.352637	−	0.935760i	$-0.385285\pi$
$617$	17.5187	0.705274	0.352637	+	0.935760i	$0.385285\pi$
$618$	0	0
$619$	−11.2743	−0.453153	−0.226577	−	0.973993i	$-0.572753\pi$
$619$	−11.2743	−0.453153	−0.226577	+	0.973993i	$0.572753\pi$
$620$	0	0
$621$	−1.12551	−0.0451650
$622$	0	0
$623$	−15.3638	−0.615538
$624$	0	0
$625$	−31.0661	−1.24264
$626$	0	0
$627$	−3.61668	−0.144436
$628$	0	0
$629$	4.83105	0.192627
$630$	0	0
$631$	21.0674	0.838682	0.419341	−	0.907829i	$-0.362261\pi$
$631$	21.0674	0.838682	0.419341	+	0.907829i	$0.362261\pi$
$632$	0	0
$633$	3.96768	0.157701
$634$	0	0
$635$	48.2812	1.91598
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	30.7352	1.21587
$640$	0	0
$641$	22.7175	0.897287	0.448644	−	0.893711i	$-0.351907\pi$
$641$	22.7175	0.897287	0.448644	+	0.893711i	$0.351907\pi$
$642$	0	0
$643$	24.4238	0.963180	0.481590	−	0.876397i	$-0.340059\pi$
$643$	24.4238	0.963180	0.481590	+	0.876397i	$0.340059\pi$
$644$	0	0
$645$	−18.5481	−0.730330
$646$	0	0
$647$	−44.7633	−1.75983	−0.879913	−	0.475134i	$-0.842400\pi$
$647$	−44.7633	−1.75983	−0.879913	+	0.475134i	$0.842400\pi$
$648$	0	0
$649$	−9.18318	−0.360471
$650$	0	0
$651$	−6.76529	−0.265153
$652$	0	0
$653$	−13.8766	−0.543034	−0.271517	−	0.962434i	$-0.587525\pi$
$653$	−13.8766	−0.543034	−0.271517	+	0.962434i	$0.587525\pi$
$654$	0	0
$655$	27.2444	1.06453
$656$	0	0
$657$	5.40335	0.210805
$658$	0	0
$659$	26.4825	1.03161	0.515805	−	0.856706i	$-0.327493\pi$
$659$	26.4825	1.03161	0.515805	+	0.856706i	$0.327493\pi$
$660$	0	0
$661$	8.87976	0.345383	0.172691	−	0.984976i	$-0.444754\pi$
$661$	8.87976	0.345383	0.172691	+	0.984976i	$0.444754\pi$
$662$	0	0
$663$	1.93968	0.0753308
$664$	0	0
$665$	−13.6193	−0.528135
$666$	0	0
$667$	−0.862113	−0.0333812
$668$	0	0
$669$	−20.1275	−0.778175
$670$	0	0
$671$	4.30829	0.166320
$672$	0	0
$673$	−1.99486	−0.0768963	−0.0384481	−	0.999261i	$-0.512241\pi$
$673$	−1.99486	−0.0768963	−0.0384481	+	0.999261i	$0.512241\pi$
$674$	0	0
$675$	−8.04582	−0.309684
$676$	0	0
$677$	−7.61834	−0.292797	−0.146398	−	0.989226i	$-0.546768\pi$
$677$	−7.61834	−0.292797	−0.146398	+	0.989226i	$0.546768\pi$
$678$	0	0
$679$	16.5392	0.634718
$680$	0	0
$681$	−14.3162	−0.548599
$682$	0	0
$683$	42.0366	1.60849	0.804244	−	0.594300i	$-0.202571\pi$
$683$	42.0366	1.60849	0.804244	+	0.594300i	$0.202571\pi$
$684$	0	0
$685$	−35.1594	−1.34337
$686$	0	0
$687$	−4.36373	−0.166487
$688$	0	0
$689$	−2.99533	−0.114113
$690$	0	0
$691$	18.3474	0.697969	0.348984	−	0.937129i	$-0.386527\pi$
$691$	18.3474	0.697969	0.348984	+	0.937129i	$0.386527\pi$
$692$	0	0
$693$	2.50135	0.0950184
$694$	0	0
$695$	56.4697	2.14202
$696$	0	0
$697$	11.7160	0.443775
$698$	0	0
$699$	−14.1210	−0.534104
$700$	0	0
$701$	31.0483	1.17268	0.586340	−	0.810065i	$-0.300568\pi$
$701$	31.0483	1.17268	0.586340	+	0.810065i	$0.300568\pi$
$702$	0	0
$703$	9.00788	0.339738
$704$	0	0
$705$	−14.6310	−0.551037
$706$	0	0
$707$	−1.17822	−0.0443114
$708$	0	0
$709$	−13.5056	−0.507213	−0.253607	−	0.967307i	$-0.581617\pi$
$709$	−13.5056	−0.507213	−0.253607	+	0.967307i	$0.581617\pi$
$710$	0	0
$711$	−1.04731	−0.0392773
$712$	0	0
$713$	−2.77567	−0.103950
$714$	0	0
$715$	2.65916	0.0994468
$716$	0	0
$717$	−2.11700	−0.0790607
$718$	0	0
$719$	−19.7168	−0.735311	−0.367656	−	0.929962i	$-0.619840\pi$
$719$	−19.7168	−0.735311	−0.367656	+	0.929962i	$0.619840\pi$
$720$	0	0
$721$	10.0011	0.372460
$722$	0	0
$723$	−8.54397	−0.317754
$724$	0	0
$725$	−6.16292	−0.228885
$726$	0	0
$727$	−29.4321	−1.09158	−0.545788	−	0.837923i	$-0.683770\pi$
$727$	−29.4321	−1.09158	−0.545788	+	0.837923i	$0.683770\pi$
$728$	0	0
$729$	−3.67862	−0.136245
$730$	0	0
$731$	27.1324	1.00353
$732$	0	0
$733$	−13.2060	−0.487774	−0.243887	−	0.969804i	$-0.578423\pi$
$733$	−13.2060	−0.487774	−0.243887	+	0.969804i	$0.578423\pi$
$734$	0	0
$735$	−1.87777	−0.0692626
$736$	0	0
$737$	7.56458	0.278645
$738$	0	0
$739$	−42.6436	−1.56867	−0.784335	−	0.620338i	$-0.786996\pi$
$739$	−42.6436	−1.56867	−0.784335	+	0.620338i	$0.786996\pi$
$740$	0	0
$741$	3.61668	0.132862
$742$	0	0
$743$	49.1686	1.80382	0.901911	−	0.431922i	$-0.142164\pi$
$743$	49.1686	1.80382	0.901911	+	0.431922i	$0.142164\pi$
$744$	0	0
$745$	12.6536	0.463592
$746$	0	0
$747$	35.6680	1.30502
$748$	0	0
$749$	−6.24662	−0.228247
$750$	0	0
$751$	−29.3278	−1.07019	−0.535094	−	0.844793i	$-0.679724\pi$
$751$	−29.3278	−1.07019	−0.535094	+	0.844793i	$0.679724\pi$
$752$	0	0
$753$	−3.68389	−0.134248
$754$	0	0
$755$	15.1538	0.551504
$756$	0	0
$757$	39.1192	1.42181	0.710906	−	0.703287i	$-0.248285\pi$
$757$	39.1192	1.42181	0.710906	+	0.703287i	$0.248285\pi$
$758$	0	0
$759$	−0.204588	−0.00742606
$760$	0	0
$761$	−42.2199	−1.53047	−0.765235	−	0.643752i	$-0.777377\pi$
$761$	−42.2199	−1.53047	−0.765235	+	0.643752i	$0.777377\pi$
$762$	0	0
$763$	4.50840	0.163215
$764$	0	0
$765$	18.2704	0.660569
$766$	0	0
$767$	9.18318	0.331585
$768$	0	0
$769$	−22.5388	−0.812772	−0.406386	−	0.913702i	$-0.633211\pi$
$769$	−22.5388	−0.812772	−0.406386	+	0.913702i	$0.633211\pi$
$770$	0	0
$771$	−11.4417	−0.412063
$772$	0	0
$773$	41.7046	1.50001	0.750005	−	0.661433i	$-0.230051\pi$
$773$	41.7046	1.50001	0.750005	+	0.661433i	$0.230051\pi$
$774$	0	0
$775$	−19.8422	−0.712754
$776$	0	0
$777$	1.24196	0.0445552
$778$	0	0
$779$	21.8454	0.782692
$780$	0	0
$781$	12.2875	0.439680
$782$	0	0
$783$	11.5598	0.413115
$784$	0	0
$785$	9.88073	0.352658
$786$	0	0
$787$	5.63391	0.200827	0.100414	−	0.994946i	$-0.467983\pi$
$787$	5.63391	0.200827	0.100414	+	0.994946i	$0.467983\pi$
$788$	0	0
$789$	−6.76821	−0.240955
$790$	0	0
$791$	9.42187	0.335003
$792$	0	0
$793$	−4.30829	−0.152992
$794$	0	0
$795$	5.62453	0.199482
$796$	0	0
$797$	5.06533	0.179423	0.0897116	−	0.995968i	$-0.471405\pi$
$797$	5.06533	0.179423	0.0897116	+	0.995968i	$0.471405\pi$
$798$	0	0
$799$	21.4025	0.757165
$800$	0	0
$801$	38.4303	1.35787
$802$	0	0
$803$	2.16017	0.0762308
$804$	0	0
$805$	−0.770414	−0.0271535
$806$	0	0
$807$	−18.3560	−0.646163
$808$	0	0
$809$	−51.2102	−1.80045	−0.900227	−	0.435421i	$-0.856600\pi$
$809$	−51.2102	−1.80045	−0.900227	+	0.435421i	$0.856600\pi$
$810$	0	0
$811$	40.3267	1.41606	0.708031	−	0.706181i	$-0.249584\pi$
$811$	40.3267	1.41606	0.708031	+	0.706181i	$0.249584\pi$
$812$	0	0
$813$	−7.98750	−0.280134
$814$	0	0
$815$	−36.3527	−1.27338
$816$	0	0
$817$	50.5904	1.76994
$818$	0	0
$819$	−2.50135	−0.0874042
$820$	0	0
$821$	−9.26748	−0.323437	−0.161719	−	0.986837i	$-0.551704\pi$
$821$	−9.26748	−0.323437	−0.161719	+	0.986837i	$0.551704\pi$
$822$	0	0
$823$	16.8924	0.588831	0.294416	−	0.955677i	$-0.404875\pi$
$823$	16.8924	0.588831	0.294416	+	0.955677i	$0.404875\pi$
$824$	0	0
$825$	−1.46252	−0.0509183
$826$	0	0
$827$	26.7857	0.931432	0.465716	−	0.884934i	$-0.345797\pi$
$827$	26.7857	0.931432	0.465716	+	0.884934i	$0.345797\pi$
$828$	0	0
$829$	15.6299	0.542850	0.271425	−	0.962460i	$-0.412505\pi$
$829$	15.6299	0.542850	0.271425	+	0.962460i	$0.412505\pi$
$830$	0	0
$831$	−10.7925	−0.374389
$832$	0	0
$833$	2.74683	0.0951718
$834$	0	0
$835$	−22.6756	−0.784721
$836$	0	0
$837$	37.2182	1.28645
$838$	0	0
$839$	2.34750	0.0810449	0.0405224	−	0.999179i	$-0.487098\pi$
$839$	2.34750	0.0810449	0.0405224	+	0.999179i	$0.487098\pi$
$840$	0	0
$841$	−20.1454	−0.694670
$842$	0	0
$843$	−5.68328	−0.195743
$844$	0	0
$845$	−2.65916	−0.0914777
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	7.87381	0.270229
$850$	0	0
$851$	0.509555	0.0174673
$852$	0	0
$853$	−21.3451	−0.730842	−0.365421	−	0.930842i	$-0.619075\pi$
$853$	−21.3451	−0.730842	−0.365421	+	0.930842i	$0.619075\pi$
$854$	0	0
$855$	34.0667	1.16506
$856$	0	0
$857$	−21.9362	−0.749327	−0.374663	−	0.927161i	$-0.622242\pi$
$857$	−21.9362	−0.749327	−0.374663	+	0.927161i	$0.622242\pi$
$858$	0	0
$859$	−52.2894	−1.78409	−0.892046	−	0.451946i	$-0.850730\pi$
$859$	−52.2894	−1.78409	−0.892046	+	0.451946i	$0.850730\pi$
$860$	0	0
$861$	3.01194	0.102647
$862$	0	0
$863$	38.8440	1.32227	0.661133	−	0.750269i	$-0.270076\pi$
$863$	38.8440	1.32227	0.661133	+	0.750269i	$0.270076\pi$
$864$	0	0
$865$	−18.5043	−0.629166
$866$	0	0
$867$	−6.67664	−0.226751
$868$	0	0
$869$	−0.418699	−0.0142034
$870$	0	0
$871$	−7.56458	−0.256316
$872$	0	0
$873$	−41.3704	−1.40018
$874$	0	0
$875$	7.78838	0.263295
$876$	0	0
$877$	−19.9946	−0.675170	−0.337585	−	0.941295i	$-0.609610\pi$
$877$	−19.9946	−0.675170	−0.337585	+	0.941295i	$0.609610\pi$
$878$	0	0
$879$	15.3660	0.518283
$880$	0	0
$881$	34.9925	1.17893	0.589463	−	0.807795i	$-0.299339\pi$
$881$	34.9925	1.17893	0.589463	+	0.807795i	$0.299339\pi$
$882$	0	0
$883$	26.5127	0.892223	0.446112	−	0.894977i	$-0.352808\pi$
$883$	26.5127	0.892223	0.446112	+	0.894977i	$0.352808\pi$
$884$	0	0
$885$	−17.2439	−0.579647
$886$	0	0
$887$	−56.9918	−1.91360	−0.956798	−	0.290753i	$-0.906094\pi$
$887$	−56.9918	−1.91360	−0.956798	+	0.290753i	$0.906094\pi$
$888$	0	0
$889$	−18.1566	−0.608953
$890$	0	0
$891$	−4.76079	−0.159492
$892$	0	0
$893$	39.9066	1.33542
$894$	0	0
$895$	29.5635	0.988199
$896$	0	0
$897$	0.204588	0.00683098
$898$	0	0
$899$	28.5083	0.950806
$900$	0	0
$901$	−8.22764	−0.274102
$902$	0	0
$903$	6.97518	0.232119
$904$	0	0
$905$	2.48629	0.0826470
$906$	0	0
$907$	40.0445	1.32966	0.664828	−	0.746996i	$-0.268505\pi$
$907$	40.0445	1.32966	0.664828	+	0.746996i	$0.268505\pi$
$908$	0	0
$909$	2.94713	0.0977502
$910$	0	0
$911$	−18.3819	−0.609020	−0.304510	−	0.952509i	$-0.598493\pi$
$911$	−18.3819	−0.609020	−0.304510	+	0.952509i	$0.598493\pi$
$912$	0	0
$913$	14.2595	0.471921
$914$	0	0
$915$	8.08997	0.267446
$916$	0	0
$917$	−10.2455	−0.338336
$918$	0	0
$919$	−8.87782	−0.292852	−0.146426	−	0.989222i	$-0.546777\pi$
$919$	−8.87782	−0.292852	−0.146426	+	0.989222i	$0.546777\pi$
$920$	0	0
$921$	21.1102	0.695605
$922$	0	0
$923$	−12.2875	−0.404447
$924$	0	0
$925$	3.64261	0.119768
$926$	0	0
$927$	−25.0162	−0.821641
$928$	0	0
$929$	18.0112	0.590929	0.295465	−	0.955354i	$-0.404526\pi$
$929$	18.0112	0.590929	0.295465	+	0.955354i	$0.404526\pi$
$930$	0	0
$931$	5.12167	0.167856
$932$	0	0
$933$	−18.5204	−0.606330
$934$	0	0
$935$	7.30423	0.238874
$936$	0	0
$937$	21.6103	0.705979	0.352989	−	0.935627i	$-0.385165\pi$
$937$	21.6103	0.705979	0.352989	+	0.935627i	$0.385165\pi$
$938$	0	0
$939$	−14.1645	−0.462240
$940$	0	0
$941$	−32.2688	−1.05193	−0.525966	−	0.850506i	$-0.676296\pi$
$941$	−32.2688	−1.05193	−0.525966	+	0.850506i	$0.676296\pi$
$942$	0	0
$943$	1.23574	0.0402414
$944$	0	0
$945$	10.3303	0.336044
$946$	0	0
$947$	36.8907	1.19879	0.599393	−	0.800455i	$-0.295409\pi$
$947$	36.8907	1.19879	0.599393	+	0.800455i	$0.295409\pi$
$948$	0	0
$949$	−2.16017	−0.0701222
$950$	0	0
$951$	−21.6217	−0.701130
$952$	0	0
$953$	−30.8988	−1.00091	−0.500455	−	0.865763i	$-0.666834\pi$
$953$	−30.8988	−1.00091	−0.500455	+	0.865763i	$0.666834\pi$
$954$	0	0
$955$	54.6774	1.76932
$956$	0	0
$957$	2.10127	0.0679245
$958$	0	0
$959$	13.2220	0.426962
$960$	0	0
$961$	60.7859	1.96083
$962$	0	0
$963$	15.6250	0.503508
$964$	0	0
$965$	41.2868	1.32907
$966$	0	0
$967$	25.0053	0.804117	0.402059	−	0.915614i	$-0.368295\pi$
$967$	25.0053	0.804117	0.402059	+	0.915614i	$0.368295\pi$
$968$	0	0
$969$	9.93440	0.319139
$970$	0	0
$971$	−30.7707	−0.987478	−0.493739	−	0.869610i	$-0.664370\pi$
$971$	−30.7707	−0.987478	−0.493739	+	0.869610i	$0.664370\pi$
$972$	0	0
$973$	−21.2360	−0.680794
$974$	0	0
$975$	1.46252	0.0468381
$976$	0	0
$977$	−44.6789	−1.42940	−0.714702	−	0.699429i	$-0.753438\pi$
$977$	−44.6789	−1.42940	−0.714702	+	0.699429i	$0.753438\pi$
$978$	0	0
$979$	15.3638	0.491030
$980$	0	0
$981$	−11.2771	−0.360049
$982$	0	0
$983$	−37.7153	−1.20293	−0.601466	−	0.798899i	$-0.705416\pi$
$983$	−37.7153	−1.20293	−0.601466	+	0.798899i	$0.705416\pi$
$984$	0	0
$985$	26.1982	0.834743
$986$	0	0
$987$	5.50214	0.175135
$988$	0	0
$989$	2.86179	0.0909995
$990$	0	0
$991$	31.6757	1.00621	0.503105	−	0.864225i	$-0.332191\pi$
$991$	31.6757	1.00621	0.503105	+	0.864225i	$0.332191\pi$
$992$	0	0
$993$	−22.4634	−0.712854
$994$	0	0
$995$	−58.4070	−1.85163
$996$	0	0
$997$	−19.9744	−0.632594	−0.316297	−	0.948660i	$-0.602440\pi$
$997$	−19.9744	−0.632594	−0.316297	+	0.948660i	$0.602440\pi$
$998$	0	0
$999$	−6.83248	−0.216170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.o.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.o.1.6	✓	9	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.706153 q^{3} -2.65916 q^{5} +1.00000 q^{7} -2.50135 q^{9} +O(q^{10})\)	\(q+0.706153 q^{3} -2.65916 q^{5} +1.00000 q^{7} -2.50135 q^{9} -1.00000 q^{11} +1.00000 q^{13} -1.87777 q^{15} +2.74683 q^{17} +5.12167 q^{19} +0.706153 q^{21} +0.289721 q^{23} +2.07111 q^{25} -3.88479 q^{27} -2.97566 q^{29} -9.58049 q^{31} -0.706153 q^{33} -2.65916 q^{35} +1.75878 q^{37} +0.706153 q^{39} +4.26528 q^{41} +9.87772 q^{43} +6.65147 q^{45} +7.79171 q^{47} +1.00000 q^{49} +1.93968 q^{51} -2.99533 q^{53} +2.65916 q^{55} +3.61668 q^{57} +9.18318 q^{59} -4.30829 q^{61} -2.50135 q^{63} -2.65916 q^{65} -7.56458 q^{67} +0.204588 q^{69} -12.2875 q^{71} -2.16017 q^{73} +1.46252 q^{75} -1.00000 q^{77} +0.418699 q^{79} +4.76079 q^{81} -14.2595 q^{83} -7.30423 q^{85} -2.10127 q^{87} -15.3638 q^{89} +1.00000 q^{91} -6.76529 q^{93} -13.6193 q^{95} +16.5392 q^{97} +2.50135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) \) 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9	\( 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9} - 9 q^{11} + 9 q^{13} - 4 q^{15} - 9 q^{17} + 9 q^{19} - 2 q^{21} - 10 q^{23} - q^{25} - 8 q^{27} - 13 q^{29} + 7 q^{31} + 2 q^{33} - 4 q^{35} - 15 q^{37} - 2 q^{39} - 18 q^{41} + 7 q^{43} + 9 q^{45} - 11 q^{47} + 9 q^{49} + q^{51} - 16 q^{53} + 4 q^{55} - 26 q^{57} - 2 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 6 q^{73} - 4 q^{75} - 9 q^{77} - 21 q^{79} - 23 q^{81} - 30 q^{83} + q^{85} + q^{87} - 36 q^{89} + 9 q^{91} - 22 q^{93} - 23 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) \) 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9 - 9 * q^11 + 9 * q^13 - 4 * q^15 - 9 * q^17 + 9 * q^19 - 2 * q^21 - 10 * q^23 - q^25 - 8 * q^27 - 13 * q^29 + 7 * q^31 + 2 * q^33 - 4 * q^35 - 15 * q^37 - 2 * q^39 - 18 * q^41 + 7 * q^43 + 9 * q^45 - 11 * q^47 + 9 * q^49 + q^51 - 16 * q^53 + 4 * q^55 - 26 * q^57 - 2 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + 13 * q^67 - 3 * q^69 - q^71 - 6 * q^73 - 4 * q^75 - 9 * q^77 - 21 * q^79 - 23 * q^81 - 30 * q^83 + q^85 + q^87 - 36 * q^89 + 9 * q^91 - 22 * q^93 - 23 * q^95 - 5 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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