LMFDB - Embedded newform 8008.2.a.i.1.3

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:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.11491$ 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)

$f(q)$	$=$	$q+2.47283 q^{3} -0.114908 q^{5} -1.00000 q^{7} +3.11491 q^{9} +O(q^{10})$	$q+2.47283 q^{3} -0.114908 q^{5} -1.00000 q^{7} +3.11491 q^{9} -1.00000 q^{11} +1.00000 q^{13} -0.284147 q^{15} -5.83076 q^{17} +2.75698 q^{19} -2.47283 q^{21} +3.81756 q^{23} -4.98680 q^{25} +0.284147 q^{27} -5.17548 q^{29} -5.35793 q^{31} -2.47283 q^{33} +0.114908 q^{35} +2.64207 q^{37} +2.47283 q^{39} -8.79811 q^{41} +9.34472 q^{43} -0.357926 q^{45} -5.51396 q^{47} +1.00000 q^{49} -14.4185 q^{51} -13.0062 q^{53} +0.114908 q^{55} +6.81756 q^{57} +3.92622 q^{59} +5.11491 q^{61} -3.11491 q^{63} -0.114908 q^{65} -5.75698 q^{67} +9.44018 q^{69} -8.83076 q^{71} -14.4596 q^{73} -12.3315 q^{75} +1.00000 q^{77} +8.64832 q^{79} -8.64207 q^{81} +9.93246 q^{83} +0.669998 q^{85} -12.7981 q^{87} +11.9868 q^{89} -1.00000 q^{91} -13.2493 q^{93} -0.316798 q^{95} -1.73530 q^{97} -3.11491 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 6 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 6 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 2 q^{3} + 6 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + q^{15} - 13 q^{17} + q^{19} - 2 q^{21} - 13 q^{23} + 5 q^{25} - q^{27} + 8 q^{29} - 17 q^{31} - 2 q^{33} - 6 q^{35} + 7 q^{37} + 2 q^{39} - 10 q^{41} + 9 q^{43} - 2 q^{45} - 2 q^{47} + 3 q^{49} - 27 q^{51} - 11 q^{53} - 6 q^{55} - 4 q^{57} + 9 q^{59} + 9 q^{61} - 3 q^{63} + 6 q^{65} - 10 q^{67} + 11 q^{69} - 22 q^{71} - 18 q^{73} + 2 q^{75} + 3 q^{77} - 3 q^{79} - 25 q^{81} - q^{83} - 28 q^{85} - 22 q^{87} + 16 q^{89} - 3 q^{91} - 19 q^{93} - 11 q^{95} + q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 6 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + q^15 - 13 * q^17 + q^19 - 2 * q^21 - 13 * q^23 + 5 * q^25 - q^27 + 8 * q^29 - 17 * q^31 - 2 * q^33 - 6 * q^35 + 7 * q^37 + 2 * q^39 - 10 * q^41 + 9 * q^43 - 2 * q^45 - 2 * q^47 + 3 * q^49 - 27 * q^51 - 11 * q^53 - 6 * q^55 - 4 * q^57 + 9 * q^59 + 9 * q^61 - 3 * q^63 + 6 * q^65 - 10 * q^67 + 11 * q^69 - 22 * q^71 - 18 * q^73 + 2 * q^75 + 3 * q^77 - 3 * q^79 - 25 * q^81 - q^83 - 28 * q^85 - 22 * q^87 + 16 * q^89 - 3 * q^91 - 19 * q^93 - 11 * q^95 + q^97 - 3 * 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
$3$	2.47283	1.42769	0.713846	−	0.700303i	$-0.246952\pi$
$3$	2.47283	1.42769	0.713846	+	0.700303i	$0.246952\pi$
$4$	0	0
$5$	−0.114908	−0.0513882	−0.0256941	−	0.999670i	$-0.508180\pi$
$5$	−0.114908	−0.0513882	−0.0256941	+	0.999670i	$0.508180\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.11491	1.03830
$10$	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
$15$	−0.284147	−0.0733665
$16$	0	0
$17$	−5.83076	−1.41417	−0.707084	−	0.707130i	$-0.749990\pi$
$17$	−5.83076	−1.41417	−0.707084	+	0.707130i	$0.749990\pi$
$18$	0	0
$19$	2.75698	0.632495	0.316247	−	0.948677i	$-0.397577\pi$
$19$	2.75698	0.632495	0.316247	+	0.948677i	$0.397577\pi$
$20$	0	0
$21$	−2.47283	−0.539617
$22$	0	0
$23$	3.81756	0.796016	0.398008	−	0.917382i	$-0.369702\pi$
$23$	3.81756	0.796016	0.398008	+	0.917382i	$0.369702\pi$
$24$	0	0
$25$	−4.98680	−0.997359
$26$	0	0
$27$	0.284147	0.0546842
$28$	0	0
$29$	−5.17548	−0.961063	−0.480532	−	0.876977i	$-0.659556\pi$
$29$	−5.17548	−0.961063	−0.480532	+	0.876977i	$0.659556\pi$
$30$	0	0
$31$	−5.35793	−0.962312	−0.481156	−	0.876635i	$-0.659783\pi$
$31$	−5.35793	−0.962312	−0.481156	+	0.876635i	$0.659783\pi$
$32$	0	0
$33$	−2.47283	−0.430465
$34$	0	0
$35$	0.114908	0.0194229
$36$	0	0
$37$	2.64207	0.434354	0.217177	−	0.976132i	$-0.430315\pi$
$37$	2.64207	0.434354	0.217177	+	0.976132i	$0.430315\pi$
$38$	0	0
$39$	2.47283	0.395970
$40$	0	0
$41$	−8.79811	−1.37403	−0.687017	−	0.726641i	$-0.741080\pi$
$41$	−8.79811	−1.37403	−0.687017	+	0.726641i	$0.741080\pi$
$42$	0	0
$43$	9.34472	1.42506	0.712528	−	0.701643i	$-0.247550\pi$
$43$	9.34472	1.42506	0.712528	+	0.701643i	$0.247550\pi$
$44$	0	0
$45$	−0.357926	−0.0533565
$46$	0	0
$47$	−5.51396	−0.804294	−0.402147	−	0.915575i	$-0.631736\pi$
$47$	−5.51396	−0.804294	−0.402147	+	0.915575i	$0.631736\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−14.4185	−2.01899
$52$	0	0
$53$	−13.0062	−1.78654	−0.893272	−	0.449516i	$-0.851597\pi$
$53$	−13.0062	−1.78654	−0.893272	+	0.449516i	$0.851597\pi$
$54$	0	0
$55$	0.114908	0.0154941
$56$	0	0
$57$	6.81756	0.903007
$58$	0	0
$59$	3.92622	0.511150	0.255575	−	0.966789i	$-0.417735\pi$
$59$	3.92622	0.511150	0.255575	+	0.966789i	$0.417735\pi$
$60$	0	0
$61$	5.11491	0.654897	0.327448	−	0.944869i	$-0.393811\pi$
$61$	5.11491	0.654897	0.327448	+	0.944869i	$0.393811\pi$
$62$	0	0
$63$	−3.11491	−0.392441
$64$	0	0
$65$	−0.114908	−0.0142525
$66$	0	0
$67$	−5.75698	−0.703327	−0.351664	−	0.936126i	$-0.614384\pi$
$67$	−5.75698	−0.703327	−0.351664	+	0.936126i	$0.614384\pi$
$68$	0	0
$69$	9.44018	1.13646
$70$	0	0
$71$	−8.83076	−1.04802	−0.524009	−	0.851713i	$-0.675564\pi$
$71$	−8.83076	−1.04802	−0.524009	+	0.851713i	$0.675564\pi$
$72$	0	0
$73$	−14.4596	−1.69237	−0.846186	−	0.532888i	$-0.821107\pi$
$73$	−14.4596	−1.69237	−0.846186	+	0.532888i	$0.821107\pi$
$74$	0	0
$75$	−12.3315	−1.42392
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	8.64832	0.973012	0.486506	−	0.873677i	$-0.338271\pi$
$79$	8.64832	0.973012	0.486506	+	0.873677i	$0.338271\pi$
$80$	0	0
$81$	−8.64207	−0.960230
$82$	0	0
$83$	9.93246	1.09023	0.545115	−	0.838361i	$-0.316486\pi$
$83$	9.93246	1.09023	0.545115	+	0.838361i	$0.316486\pi$
$84$	0	0
$85$	0.669998	0.0726715
$86$	0	0
$87$	−12.7981	−1.37210
$88$	0	0
$89$	11.9868	1.27060	0.635299	−	0.772266i	$-0.280877\pi$
$89$	11.9868	1.27060	0.635299	+	0.772266i	$0.280877\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−13.2493	−1.37388
$94$	0	0
$95$	−0.316798	−0.0325028
$96$	0	0
$97$	−1.73530	−0.176193	−0.0880965	−	0.996112i	$-0.528078\pi$
$97$	−1.73530	−0.176193	−0.0880965	+	0.996112i	$0.528078\pi$
$98$	0	0
$99$	−3.11491	−0.313060
$100$	0	0
$101$	−4.52717	−0.450470	−0.225235	−	0.974304i	$-0.572315\pi$
$101$	−4.52717	−0.450470	−0.225235	+	0.974304i	$0.572315\pi$
$102$	0	0
$103$	−6.56829	−0.647193	−0.323597	−	0.946195i	$-0.604892\pi$
$103$	−6.56829	−0.647193	−0.323597	+	0.946195i	$0.604892\pi$
$104$	0	0
$105$	0.284147	0.0277299
$106$	0	0
$107$	−14.7500	−1.42594	−0.712969	−	0.701195i	$-0.752650\pi$
$107$	−14.7500	−1.42594	−0.712969	+	0.701195i	$0.752650\pi$
$108$	0	0
$109$	14.9868	1.43547	0.717737	−	0.696314i	$-0.245178\pi$
$109$	14.9868	1.43547	0.717737	+	0.696314i	$0.245178\pi$
$110$	0	0
$111$	6.53341	0.620124
$112$	0	0
$113$	−13.1692	−1.23886	−0.619429	−	0.785053i	$-0.712636\pi$
$113$	−13.1692	−1.23886	−0.619429	+	0.785053i	$0.712636\pi$
$114$	0	0
$115$	−0.438666	−0.0409058
$116$	0	0
$117$	3.11491	0.287973
$118$	0	0
$119$	5.83076	0.534505
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−21.7563	−1.96170
$124$	0	0
$125$	1.14756	0.102641
$126$	0	0
$127$	−11.8238	−1.04919	−0.524596	−	0.851351i	$-0.675784\pi$
$127$	−11.8238	−1.04919	−0.524596	+	0.851351i	$0.675784\pi$
$128$	0	0
$129$	23.1079	2.03454
$130$	0	0
$131$	8.60719	0.752014	0.376007	−	0.926617i	$-0.377297\pi$
$131$	8.60719	0.752014	0.376007	+	0.926617i	$0.377297\pi$
$132$	0	0
$133$	−2.75698	−0.239061
$134$	0	0
$135$	−0.0326507	−0.00281012
$136$	0	0
$137$	−4.78963	−0.409206	−0.204603	−	0.978845i	$-0.565590\pi$
$137$	−4.78963	−0.409206	−0.204603	+	0.978845i	$0.565590\pi$
$138$	0	0
$139$	2.71585	0.230356	0.115178	−	0.993345i	$-0.463256\pi$
$139$	2.71585	0.230356	0.115178	+	0.993345i	$0.463256\pi$
$140$	0	0
$141$	−13.6351	−1.14828
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	0.594702	0.0493873
$146$	0	0
$147$	2.47283	0.203956
$148$	0	0
$149$	13.9325	1.14139	0.570696	−	0.821161i	$-0.306673\pi$
$149$	13.9325	1.14139	0.570696	+	0.821161i	$0.306673\pi$
$150$	0	0
$151$	−0.338479	−0.0275451	−0.0137725	−	0.999905i	$-0.504384\pi$
$151$	−0.338479	−0.0275451	−0.0137725	+	0.999905i	$0.504384\pi$
$152$	0	0
$153$	−18.1623	−1.46833
$154$	0	0
$155$	0.615666	0.0494515
$156$	0	0
$157$	16.4247	1.31084	0.655419	−	0.755266i	$-0.272492\pi$
$157$	16.4247	1.31084	0.655419	+	0.755266i	$0.272492\pi$
$158$	0	0
$159$	−32.1623	−2.55063
$160$	0	0
$161$	−3.81756	−0.300866
$162$	0	0
$163$	−0.527166	−0.0412908	−0.0206454	−	0.999787i	$-0.506572\pi$
$163$	−0.527166	−0.0412908	−0.0206454	+	0.999787i	$0.506572\pi$
$164$	0	0
$165$	0.284147	0.0221208
$166$	0	0
$167$	−6.65528	−0.515001	−0.257500	−	0.966278i	$-0.582899\pi$
$167$	−6.65528	−0.515001	−0.257500	+	0.966278i	$0.582899\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	8.58774	0.656721
$172$	0	0
$173$	−20.1623	−1.53291	−0.766455	−	0.642298i	$-0.777981\pi$
$173$	−20.1623	−1.53291	−0.766455	+	0.642298i	$0.777981\pi$
$174$	0	0
$175$	4.98680	0.376966
$176$	0	0
$177$	9.70889	0.729765
$178$	0	0
$179$	−25.7089	−1.92157	−0.960786	−	0.277290i	$-0.910564\pi$
$179$	−25.7089	−1.92157	−0.960786	+	0.277290i	$0.910564\pi$
$180$	0	0
$181$	17.3315	1.28824	0.644121	−	0.764924i	$-0.277223\pi$
$181$	17.3315	1.28824	0.644121	+	0.764924i	$0.277223\pi$
$182$	0	0
$183$	12.6483	0.934990
$184$	0	0
$185$	−0.303594	−0.0223207
$186$	0	0
$187$	5.83076	0.426387
$188$	0	0
$189$	−0.284147	−0.0206687
$190$	0	0
$191$	5.10170	0.369146	0.184573	−	0.982819i	$-0.440910\pi$
$191$	5.10170	0.369146	0.184573	+	0.982819i	$0.440910\pi$
$192$	0	0
$193$	−8.96735	−0.645484	−0.322742	−	0.946487i	$-0.604605\pi$
$193$	−8.96735	−0.645484	−0.322742	+	0.946487i	$0.604605\pi$
$194$	0	0
$195$	−0.284147	−0.0203482
$196$	0	0
$197$	12.3726	0.881515	0.440757	−	0.897626i	$-0.354710\pi$
$197$	12.3726	0.881515	0.440757	+	0.897626i	$0.354710\pi$
$198$	0	0
$199$	−19.2555	−1.36499	−0.682493	−	0.730892i	$-0.739104\pi$
$199$	−19.2555	−1.36499	−0.682493	+	0.730892i	$0.739104\pi$
$200$	0	0
$201$	−14.2361	−1.00413
$202$	0	0
$203$	5.17548	0.363248
$204$	0	0
$205$	1.01097	0.0706091
$206$	0	0
$207$	11.8913	0.826505
$208$	0	0
$209$	−2.75698	−0.190704
$210$	0	0
$211$	15.5745	1.07220	0.536098	−	0.844156i	$-0.319898\pi$
$211$	15.5745	1.07220	0.536098	+	0.844156i	$0.319898\pi$
$212$	0	0
$213$	−21.8370	−1.49625
$214$	0	0
$215$	−1.07378	−0.0732311
$216$	0	0
$217$	5.35793	0.363720
$218$	0	0
$219$	−35.7563	−2.41618
$220$	0	0
$221$	−5.83076	−0.392219
$222$	0	0
$223$	−0.680967	−0.0456010	−0.0228005	−	0.999740i	$-0.507258\pi$
$223$	−0.680967	−0.0456010	−0.0228005	+	0.999740i	$0.507258\pi$
$224$	0	0
$225$	−15.5334	−1.03556
$226$	0	0
$227$	19.6134	1.30179	0.650895	−	0.759168i	$-0.274394\pi$
$227$	19.6134	1.30179	0.650895	+	0.759168i	$0.274394\pi$
$228$	0	0
$229$	−4.80435	−0.317481	−0.158740	−	0.987320i	$-0.550743\pi$
$229$	−4.80435	−0.317481	−0.158740	+	0.987320i	$0.550743\pi$
$230$	0	0
$231$	2.47283	0.162701
$232$	0	0
$233$	−14.0823	−0.922559	−0.461280	−	0.887255i	$-0.652610\pi$
$233$	−14.0823	−0.922559	−0.461280	+	0.887255i	$0.652610\pi$
$234$	0	0
$235$	0.633596	0.0413312
$236$	0	0
$237$	21.3859	1.38916
$238$	0	0
$239$	20.8370	1.34783	0.673917	−	0.738807i	$-0.264611\pi$
$239$	20.8370	1.34783	0.673917	+	0.738807i	$0.264611\pi$
$240$	0	0
$241$	−11.5140	−0.741680	−0.370840	−	0.928697i	$-0.620930\pi$
$241$	−11.5140	−0.741680	−0.370840	+	0.928697i	$0.620930\pi$
$242$	0	0
$243$	−22.2229	−1.42560
$244$	0	0
$245$	−0.114908	−0.00734117
$246$	0	0
$247$	2.75698	0.175423
$248$	0	0
$249$	24.5613	1.55651
$250$	0	0
$251$	−18.4270	−1.16310	−0.581550	−	0.813510i	$-0.697554\pi$
$251$	−18.4270	−1.16310	−0.581550	+	0.813510i	$0.697554\pi$
$252$	0	0
$253$	−3.81756	−0.240008
$254$	0	0
$255$	1.65679	0.103753
$256$	0	0
$257$	−11.2966	−0.704665	−0.352332	−	0.935875i	$-0.614611\pi$
$257$	−11.2966	−0.704665	−0.352332	+	0.935875i	$0.614611\pi$
$258$	0	0
$259$	−2.64207	−0.164170
$260$	0	0
$261$	−16.1212	−0.997874
$262$	0	0
$263$	14.3921	0.887455	0.443727	−	0.896162i	$-0.353656\pi$
$263$	14.3921	0.887455	0.443727	+	0.896162i	$0.353656\pi$
$264$	0	0
$265$	1.49452	0.0918074
$266$	0	0
$267$	29.6414	1.81402
$268$	0	0
$269$	20.5683	1.25407	0.627036	−	0.778991i	$-0.284268\pi$
$269$	20.5683	1.25407	0.627036	+	0.778991i	$0.284268\pi$
$270$	0	0
$271$	−0.694171	−0.0421679	−0.0210839	−	0.999778i	$-0.506712\pi$
$271$	−0.694171	−0.0421679	−0.0210839	+	0.999778i	$0.506712\pi$
$272$	0	0
$273$	−2.47283	−0.149663
$274$	0	0
$275$	4.98680	0.300715
$276$	0	0
$277$	−23.4053	−1.40629	−0.703144	−	0.711047i	$-0.748221\pi$
$277$	−23.4053	−1.40629	−0.703144	+	0.711047i	$0.748221\pi$
$278$	0	0
$279$	−16.6894	−0.999171
$280$	0	0
$281$	−23.9255	−1.42728	−0.713638	−	0.700515i	$-0.752954\pi$
$281$	−23.9255	−1.42728	−0.713638	+	0.700515i	$0.752954\pi$
$282$	0	0
$283$	13.5264	0.804064	0.402032	−	0.915626i	$-0.368304\pi$
$283$	13.5264	0.804064	0.402032	+	0.915626i	$0.368304\pi$
$284$	0	0
$285$	−0.783389	−0.0464039
$286$	0	0
$287$	8.79811	0.519336
$288$	0	0
$289$	16.9978	0.999869
$290$	0	0
$291$	−4.29111	−0.251549
$292$	0	0
$293$	−3.40530	−0.198940	−0.0994698	−	0.995041i	$-0.531715\pi$
$293$	−3.40530	−0.198940	−0.0994698	+	0.995041i	$0.531715\pi$
$294$	0	0
$295$	−0.451152	−0.0262671
$296$	0	0
$297$	−0.284147	−0.0164879
$298$	0	0
$299$	3.81756	0.220775
$300$	0	0
$301$	−9.34472	−0.538621
$302$	0	0
$303$	−11.1949	−0.643132
$304$	0	0
$305$	−0.587741	−0.0336540
$306$	0	0
$307$	−5.91302	−0.337474	−0.168737	−	0.985661i	$-0.553969\pi$
$307$	−5.91302	−0.337474	−0.168737	+	0.985661i	$0.553969\pi$
$308$	0	0
$309$	−16.2423	−0.923992
$310$	0	0
$311$	−26.0411	−1.47666	−0.738328	−	0.674441i	$-0.764385\pi$
$311$	−26.0411	−1.47666	−0.738328	+	0.674441i	$0.764385\pi$
$312$	0	0
$313$	−12.6810	−0.716771	−0.358385	−	0.933574i	$-0.616673\pi$
$313$	−12.6810	−0.716771	−0.358385	+	0.933574i	$0.616673\pi$
$314$	0	0
$315$	0.357926	0.0201669
$316$	0	0
$317$	−6.64207	−0.373056	−0.186528	−	0.982450i	$-0.559724\pi$
$317$	−6.64207	−0.373056	−0.186528	+	0.982450i	$0.559724\pi$
$318$	0	0
$319$	5.17548	0.289771
$320$	0	0
$321$	−36.4744	−2.03580
$322$	0	0
$323$	−16.0753	−0.894453
$324$	0	0
$325$	−4.98680	−0.276618
$326$	0	0
$327$	37.0599	2.04941
$328$	0	0
$329$	5.51396	0.303995
$330$	0	0
$331$	−3.88509	−0.213544	−0.106772	−	0.994284i	$-0.534051\pi$
$331$	−3.88509	−0.213544	−0.106772	+	0.994284i	$0.534051\pi$
$332$	0	0
$333$	8.22982	0.450991
$334$	0	0
$335$	0.661521	0.0361427
$336$	0	0
$337$	10.6506	0.580173	0.290086	−	0.957001i	$-0.406316\pi$
$337$	10.6506	0.580173	0.290086	+	0.957001i	$0.406316\pi$
$338$	0	0
$339$	−32.5653	−1.76871
$340$	0	0
$341$	5.35793	0.290148
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.08475	−0.0584009
$346$	0	0
$347$	3.01320	0.161757	0.0808786	−	0.996724i	$-0.474227\pi$
$347$	3.01320	0.161757	0.0808786	+	0.996724i	$0.474227\pi$
$348$	0	0
$349$	−7.16300	−0.383426	−0.191713	−	0.981451i	$-0.561404\pi$
$349$	−7.16300	−0.383426	−0.191713	+	0.981451i	$0.561404\pi$
$350$	0	0
$351$	0.284147	0.0151667
$352$	0	0
$353$	29.2229	1.55538	0.777688	−	0.628651i	$-0.216392\pi$
$353$	29.2229	1.55538	0.777688	+	0.628651i	$0.216392\pi$
$354$	0	0
$355$	1.01472	0.0538558
$356$	0	0
$357$	14.4185	0.763108
$358$	0	0
$359$	26.3246	1.38936	0.694679	−	0.719320i	$-0.255547\pi$
$359$	26.3246	1.38936	0.694679	+	0.719320i	$0.255547\pi$
$360$	0	0
$361$	−11.3991	−0.599950
$362$	0	0
$363$	2.47283	0.129790
$364$	0	0
$365$	1.66152	0.0869680
$366$	0	0
$367$	2.85021	0.148780	0.0743898	−	0.997229i	$-0.476299\pi$
$367$	2.85021	0.148780	0.0743898	+	0.997229i	$0.476299\pi$
$368$	0	0
$369$	−27.4053	−1.42666
$370$	0	0
$371$	13.0062	0.675251
$372$	0	0
$373$	31.1057	1.61059	0.805296	−	0.592872i	$-0.202006\pi$
$373$	31.1057	1.61059	0.805296	+	0.592872i	$0.202006\pi$
$374$	0	0
$375$	2.83772	0.146539
$376$	0	0
$377$	−5.17548	−0.266551
$378$	0	0
$379$	−15.9170	−0.817603	−0.408801	−	0.912623i	$-0.634053\pi$
$379$	−15.9170	−0.817603	−0.408801	+	0.912623i	$0.634053\pi$
$380$	0	0
$381$	−29.2383	−1.49792
$382$	0	0
$383$	16.3983	0.837916	0.418958	−	0.908006i	$-0.362395\pi$
$383$	16.3983	0.837916	0.418958	+	0.908006i	$0.362395\pi$
$384$	0	0
$385$	−0.114908	−0.00585623
$386$	0	0
$387$	29.1079	1.47964
$388$	0	0
$389$	−26.4270	−1.33990	−0.669951	−	0.742406i	$-0.733685\pi$
$389$	−26.4270	−1.33990	−0.669951	+	0.742406i	$0.733685\pi$
$390$	0	0
$391$	−22.2593	−1.12570
$392$	0	0
$393$	21.2841	1.07364
$394$	0	0
$395$	−0.993757	−0.0500013
$396$	0	0
$397$	22.5202	1.13026	0.565128	−	0.825003i	$-0.308827\pi$
$397$	22.5202	1.13026	0.565128	+	0.825003i	$0.308827\pi$
$398$	0	0
$399$	−6.81756	−0.341305
$400$	0	0
$401$	−15.0404	−0.751082	−0.375541	−	0.926806i	$-0.622543\pi$
$401$	−15.0404	−0.751082	−0.375541	+	0.926806i	$0.622543\pi$
$402$	0	0
$403$	−5.35793	−0.266897
$404$	0	0
$405$	0.993039	0.0493445
$406$	0	0
$407$	−2.64207	−0.130963
$408$	0	0
$409$	−21.3664	−1.05650	−0.528250	−	0.849089i	$-0.677152\pi$
$409$	−21.3664	−1.05650	−0.528250	+	0.849089i	$0.677152\pi$
$410$	0	0
$411$	−11.8440	−0.584220
$412$	0	0
$413$	−3.92622	−0.193197
$414$	0	0
$415$	−1.14132	−0.0560250
$416$	0	0
$417$	6.71585	0.328877
$418$	0	0
$419$	−5.69569	−0.278253	−0.139126	−	0.990275i	$-0.544429\pi$
$419$	−5.69569	−0.278253	−0.139126	+	0.990275i	$0.544429\pi$
$420$	0	0
$421$	−20.0558	−0.977462	−0.488731	−	0.872434i	$-0.662540\pi$
$421$	−20.0558	−0.977462	−0.488731	+	0.872434i	$0.662540\pi$
$422$	0	0
$423$	−17.1755	−0.835101
$424$	0	0
$425$	29.0768	1.41043
$426$	0	0
$427$	−5.11491	−0.247528
$428$	0	0
$429$	−2.47283	−0.119390
$430$	0	0
$431$	8.86341	0.426935	0.213468	−	0.976950i	$-0.431524\pi$
$431$	8.86341	0.426935	0.213468	+	0.976950i	$0.431524\pi$
$432$	0	0
$433$	28.2897	1.35951	0.679757	−	0.733437i	$-0.262085\pi$
$433$	28.2897	1.35951	0.679757	+	0.733437i	$0.262085\pi$
$434$	0	0
$435$	1.47060	0.0705098
$436$	0	0
$437$	10.5249	0.503476
$438$	0	0
$439$	−4.33848	−0.207064	−0.103532	−	0.994626i	$-0.533014\pi$
$439$	−4.33848	−0.207064	−0.103532	+	0.994626i	$0.533014\pi$
$440$	0	0
$441$	3.11491	0.148329
$442$	0	0
$443$	−8.57677	−0.407495	−0.203747	−	0.979023i	$-0.565312\pi$
$443$	−8.57677	−0.407495	−0.203747	+	0.979023i	$0.565312\pi$
$444$	0	0
$445$	−1.37737	−0.0652938
$446$	0	0
$447$	34.4527	1.62956
$448$	0	0
$449$	−34.3594	−1.62152	−0.810761	−	0.585377i	$-0.800946\pi$
$449$	−34.3594	−1.62152	−0.810761	+	0.585377i	$0.800946\pi$
$450$	0	0
$451$	8.79811	0.414287
$452$	0	0
$453$	−0.837003	−0.0393259
$454$	0	0
$455$	0.114908	0.00538695
$456$	0	0
$457$	−10.1234	−0.473552	−0.236776	−	0.971564i	$-0.576091\pi$
$457$	−10.1234	−0.473552	−0.236776	+	0.971564i	$0.576091\pi$
$458$	0	0
$459$	−1.65679	−0.0773325
$460$	0	0
$461$	33.2577	1.54897	0.774484	−	0.632594i	$-0.218010\pi$
$461$	33.2577	1.54897	0.774484	+	0.632594i	$0.218010\pi$
$462$	0	0
$463$	−31.1281	−1.44665	−0.723323	−	0.690510i	$-0.757386\pi$
$463$	−31.1281	−1.44665	−0.723323	+	0.690510i	$0.757386\pi$
$464$	0	0
$465$	1.52244	0.0706015
$466$	0	0
$467$	−8.36417	−0.387048	−0.193524	−	0.981096i	$-0.561992\pi$
$467$	−8.36417	−0.387048	−0.193524	+	0.981096i	$0.561992\pi$
$468$	0	0
$469$	5.75698	0.265833
$470$	0	0
$471$	40.6157	1.87147
$472$	0	0
$473$	−9.34472	−0.429671
$474$	0	0
$475$	−13.7485	−0.630825
$476$	0	0
$477$	−40.5132	−1.85497
$478$	0	0
$479$	12.9045	0.589623	0.294812	−	0.955555i	$-0.404743\pi$
$479$	12.9045	0.589623	0.294812	+	0.955555i	$0.404743\pi$
$480$	0	0
$481$	2.64207	0.120468
$482$	0	0
$483$	−9.44018	−0.429543
$484$	0	0
$485$	0.199399	0.00905424
$486$	0	0
$487$	−15.4402	−0.699661	−0.349831	−	0.936813i	$-0.613761\pi$
$487$	−15.4402	−0.699661	−0.349831	+	0.936813i	$0.613761\pi$
$488$	0	0
$489$	−1.30359	−0.0589506
$490$	0	0
$491$	22.7159	1.02515	0.512576	−	0.858642i	$-0.328691\pi$
$491$	22.7159	1.02515	0.512576	+	0.858642i	$0.328691\pi$
$492$	0	0
$493$	30.1770	1.35910
$494$	0	0
$495$	0.357926	0.0160876
$496$	0	0
$497$	8.83076	0.396114
$498$	0	0
$499$	35.7974	1.60251	0.801256	−	0.598322i	$-0.204166\pi$
$499$	35.7974	1.60251	0.801256	+	0.598322i	$0.204166\pi$
$500$	0	0
$501$	−16.4574	−0.735262
$502$	0	0
$503$	−11.5264	−0.513939	−0.256969	−	0.966420i	$-0.582724\pi$
$503$	−11.5264	−0.513939	−0.256969	+	0.966420i	$0.582724\pi$
$504$	0	0
$505$	0.520206	0.0231488
$506$	0	0
$507$	2.47283	0.109822
$508$	0	0
$509$	−0.373365	−0.0165491	−0.00827455	−	0.999966i	$-0.502634\pi$
$509$	−0.373365	−0.0165491	−0.00827455	+	0.999966i	$0.502634\pi$
$510$	0	0
$511$	14.4596	0.639656
$512$	0	0
$513$	0.783389	0.0345875
$514$	0	0
$515$	0.754747	0.0332581
$516$	0	0
$517$	5.51396	0.242504
$518$	0	0
$519$	−49.8580	−2.18852
$520$	0	0
$521$	27.8734	1.22116	0.610578	−	0.791956i	$-0.290937\pi$
$521$	27.8734	1.22116	0.610578	+	0.791956i	$0.290937\pi$
$522$	0	0
$523$	−36.6894	−1.60432	−0.802159	−	0.597111i	$-0.796315\pi$
$523$	−36.6894	−1.60432	−0.802159	+	0.597111i	$0.796315\pi$
$524$	0	0
$525$	12.3315	0.538192
$526$	0	0
$527$	31.2408	1.36087
$528$	0	0
$529$	−8.42626	−0.366359
$530$	0	0
$531$	12.2298	0.530729
$532$	0	0
$533$	−8.79811	−0.381088
$534$	0	0
$535$	1.69489	0.0732764
$536$	0	0
$537$	−63.5738	−2.74341
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	37.7827	1.62440	0.812202	−	0.583376i	$-0.198269\pi$
$541$	37.7827	1.62440	0.812202	+	0.583376i	$0.198269\pi$
$542$	0	0
$543$	42.8580	1.83921
$544$	0	0
$545$	−1.72210	−0.0737665
$546$	0	0
$547$	0.754747	0.0322706	0.0161353	−	0.999870i	$-0.494864\pi$
$547$	0.754747	0.0322706	0.0161353	+	0.999870i	$0.494864\pi$
$548$	0	0
$549$	15.9325	0.679981
$550$	0	0
$551$	−14.2687	−0.607867
$552$	0	0
$553$	−8.64832	−0.367764
$554$	0	0
$555$	−0.750738	−0.0318671
$556$	0	0
$557$	13.1079	0.555402	0.277701	−	0.960668i	$-0.410428\pi$
$557$	13.1079	0.555402	0.277701	+	0.960668i	$0.410428\pi$
$558$	0	0
$559$	9.34472	0.395240
$560$	0	0
$561$	14.4185	0.608750
$562$	0	0
$563$	−31.5918	−1.33143	−0.665717	−	0.746205i	$-0.731874\pi$
$563$	−31.5918	−1.33143	−0.665717	+	0.746205i	$0.731874\pi$
$564$	0	0
$565$	1.51324	0.0636627
$566$	0	0
$567$	8.64207	0.362933
$568$	0	0
$569$	−6.59470	−0.276464	−0.138232	−	0.990400i	$-0.544142\pi$
$569$	−6.59470	−0.276464	−0.138232	+	0.990400i	$0.544142\pi$
$570$	0	0
$571$	30.8781	1.29221	0.646105	−	0.763249i	$-0.276397\pi$
$571$	30.8781	1.29221	0.646105	+	0.763249i	$0.276397\pi$
$572$	0	0
$573$	12.6157	0.527027
$574$	0	0
$575$	−19.0374	−0.793913
$576$	0	0
$577$	3.28343	0.136691	0.0683455	−	0.997662i	$-0.478228\pi$
$577$	3.28343	0.136691	0.0683455	+	0.997662i	$0.478228\pi$
$578$	0	0
$579$	−22.1748	−0.921552
$580$	0	0
$581$	−9.93246	−0.412068
$582$	0	0
$583$	13.0062	0.538664
$584$	0	0
$585$	−0.357926	−0.0147984
$586$	0	0
$587$	4.12115	0.170098	0.0850490	−	0.996377i	$-0.472895\pi$
$587$	4.12115	0.170098	0.0850490	+	0.996377i	$0.472895\pi$
$588$	0	0
$589$	−14.7717	−0.608657
$590$	0	0
$591$	30.5955	1.25853
$592$	0	0
$593$	−46.1600	−1.89557	−0.947783	−	0.318916i	$-0.896681\pi$
$593$	−46.1600	−1.89557	−0.947783	+	0.318916i	$0.896681\pi$
$594$	0	0
$595$	−0.669998	−0.0274673
$596$	0	0
$597$	−47.6157	−1.94878
$598$	0	0
$599$	24.3635	0.995464	0.497732	−	0.867331i	$-0.334166\pi$
$599$	24.3635	0.995464	0.497732	+	0.867331i	$0.334166\pi$
$600$	0	0
$601$	−18.4115	−0.751022	−0.375511	−	0.926818i	$-0.622533\pi$
$601$	−18.4115	−0.751022	−0.375511	+	0.926818i	$0.622533\pi$
$602$	0	0
$603$	−17.9325	−0.730266
$604$	0	0
$605$	−0.114908	−0.00467166
$606$	0	0
$607$	−22.6630	−0.919864	−0.459932	−	0.887954i	$-0.652126\pi$
$607$	−22.6630	−0.919864	−0.459932	+	0.887954i	$0.652126\pi$
$608$	0	0
$609$	12.7981	0.518606
$610$	0	0
$611$	−5.51396	−0.223071
$612$	0	0
$613$	34.4068	1.38968	0.694839	−	0.719165i	$-0.255476\pi$
$613$	34.4068	1.38968	0.694839	+	0.719165i	$0.255476\pi$
$614$	0	0
$615$	2.49996	0.100808
$616$	0	0
$617$	−26.6630	−1.07341	−0.536707	−	0.843769i	$-0.680332\pi$
$617$	−26.6630	−1.07341	−0.536707	+	0.843769i	$0.680332\pi$
$618$	0	0
$619$	−34.2772	−1.37772	−0.688858	−	0.724896i	$-0.741888\pi$
$619$	−34.2772	−1.37772	−0.688858	+	0.724896i	$0.741888\pi$
$620$	0	0
$621$	1.08475	0.0435294
$622$	0	0
$623$	−11.9868	−0.480241
$624$	0	0
$625$	24.8021	0.992085
$626$	0	0
$627$	−6.81756	−0.272267
$628$	0	0
$629$	−15.4053	−0.614249
$630$	0	0
$631$	−11.9215	−0.474587	−0.237294	−	0.971438i	$-0.576260\pi$
$631$	−11.9215	−0.474587	−0.237294	+	0.971438i	$0.576260\pi$
$632$	0	0
$633$	38.5132	1.53076
$634$	0	0
$635$	1.35864	0.0539161
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−27.5070	−1.08816
$640$	0	0
$641$	43.9729	1.73682	0.868412	−	0.495843i	$-0.165141\pi$
$641$	43.9729	1.73682	0.868412	+	0.495843i	$0.165141\pi$
$642$	0	0
$643$	34.3983	1.35654	0.678269	−	0.734814i	$-0.262730\pi$
$643$	34.3983	1.35654	0.678269	+	0.734814i	$0.262730\pi$
$644$	0	0
$645$	−2.65528	−0.104551
$646$	0	0
$647$	−25.6964	−1.01023	−0.505115	−	0.863052i	$-0.668550\pi$
$647$	−25.6964	−1.01023	−0.505115	+	0.863052i	$0.668550\pi$
$648$	0	0
$649$	−3.92622	−0.154118
$650$	0	0
$651$	13.2493	0.519280
$652$	0	0
$653$	44.1857	1.72912	0.864561	−	0.502528i	$-0.167597\pi$
$653$	44.1857	1.72912	0.864561	+	0.502528i	$0.167597\pi$
$654$	0	0
$655$	−0.989031	−0.0386446
$656$	0	0
$657$	−45.0404	−1.75719
$658$	0	0
$659$	−41.7346	−1.62575	−0.812874	−	0.582439i	$-0.802098\pi$
$659$	−41.7346	−1.62575	−0.812874	+	0.582439i	$0.802098\pi$
$660$	0	0
$661$	34.1538	1.32843	0.664214	−	0.747542i	$-0.268766\pi$
$661$	34.1538	1.32843	0.664214	+	0.747542i	$0.268766\pi$
$662$	0	0
$663$	−14.4185	−0.559968
$664$	0	0
$665$	0.316798	0.0122849
$666$	0	0
$667$	−19.7577	−0.765021
$668$	0	0
$669$	−1.68392	−0.0651041
$670$	0	0
$671$	−5.11491	−0.197459
$672$	0	0
$673$	−6.41627	−0.247329	−0.123664	−	0.992324i	$-0.539465\pi$
$673$	−6.41627	−0.247329	−0.123664	+	0.992324i	$0.539465\pi$
$674$	0	0
$675$	−1.41698	−0.0545398
$676$	0	0
$677$	46.9798	1.80558	0.902791	−	0.430080i	$-0.141515\pi$
$677$	46.9798	1.80558	0.902791	+	0.430080i	$0.141515\pi$
$678$	0	0
$679$	1.73530	0.0665947
$680$	0	0
$681$	48.5008	1.85855
$682$	0	0
$683$	−9.31832	−0.356555	−0.178278	−	0.983980i	$-0.557053\pi$
$683$	−9.31832	−0.356555	−0.178278	+	0.983980i	$0.557053\pi$
$684$	0	0
$685$	0.550365	0.0210284
$686$	0	0
$687$	−11.8804	−0.453264
$688$	0	0
$689$	−13.0062	−0.495498
$690$	0	0
$691$	−3.93871	−0.149835	−0.0749177	−	0.997190i	$-0.523869\pi$
$691$	−3.93871	−0.149835	−0.0749177	+	0.997190i	$0.523869\pi$
$692$	0	0
$693$	3.11491	0.118326
$694$	0	0
$695$	−0.312072	−0.0118376
$696$	0	0
$697$	51.2997	1.94311
$698$	0	0
$699$	−34.8231	−1.31713
$700$	0	0
$701$	−13.3928	−0.505840	−0.252920	−	0.967487i	$-0.581391\pi$
$701$	−13.3928	−0.505840	−0.252920	+	0.967487i	$0.581391\pi$
$702$	0	0
$703$	7.28415	0.274727
$704$	0	0
$705$	1.56678	0.0590082
$706$	0	0
$707$	4.52717	0.170262
$708$	0	0
$709$	37.3176	1.40149	0.700746	−	0.713411i	$-0.252851\pi$
$709$	37.3176	1.40149	0.700746	+	0.713411i	$0.252851\pi$
$710$	0	0
$711$	26.9387	1.01028
$712$	0	0
$713$	−20.4542	−0.766015
$714$	0	0
$715$	0.114908	0.00429730
$716$	0	0
$717$	51.5264	1.92429
$718$	0	0
$719$	7.69168	0.286851	0.143426	−	0.989661i	$-0.454188\pi$
$719$	7.69168	0.286851	0.143426	+	0.989661i	$0.454188\pi$
$720$	0	0
$721$	6.56829	0.244616
$722$	0	0
$723$	−28.4721	−1.05889
$724$	0	0
$725$	25.8091	0.958525
$726$	0	0
$727$	45.3587	1.68226	0.841131	−	0.540831i	$-0.181890\pi$
$727$	45.3587	1.68226	0.841131	+	0.540831i	$0.181890\pi$
$728$	0	0
$729$	−29.0272	−1.07508
$730$	0	0
$731$	−54.4868	−2.01527
$732$	0	0
$733$	−3.75626	−0.138741	−0.0693704	−	0.997591i	$-0.522099\pi$
$733$	−3.75626	−0.138741	−0.0693704	+	0.997591i	$0.522099\pi$
$734$	0	0
$735$	−0.284147	−0.0104809
$736$	0	0
$737$	5.75698	0.212061
$738$	0	0
$739$	26.3246	0.968365	0.484182	−	0.874967i	$-0.339117\pi$
$739$	26.3246	0.968365	0.484182	+	0.874967i	$0.339117\pi$
$740$	0	0
$741$	6.81756	0.250449
$742$	0	0
$743$	14.9193	0.547335	0.273667	−	0.961824i	$-0.411763\pi$
$743$	14.9193	0.547335	0.273667	+	0.961824i	$0.411763\pi$
$744$	0	0
$745$	−1.60095	−0.0586541
$746$	0	0
$747$	30.9387	1.13199
$748$	0	0
$749$	14.7500	0.538954
$750$	0	0
$751$	−34.3425	−1.25318	−0.626588	−	0.779351i	$-0.715549\pi$
$751$	−34.3425	−1.25318	−0.626588	+	0.779351i	$0.715549\pi$
$752$	0	0
$753$	−45.5669	−1.66055
$754$	0	0
$755$	0.0388938	0.00141549
$756$	0	0
$757$	−10.2320	−0.371890	−0.185945	−	0.982560i	$-0.559535\pi$
$757$	−10.2320	−0.371890	−0.185945	+	0.982560i	$0.559535\pi$
$758$	0	0
$759$	−9.44018	−0.342657
$760$	0	0
$761$	26.9582	0.977232	0.488616	−	0.872499i	$-0.337502\pi$
$761$	26.9582	0.977232	0.488616	+	0.872499i	$0.337502\pi$
$762$	0	0
$763$	−14.9868	−0.542558
$764$	0	0
$765$	2.08698	0.0754550
$766$	0	0
$767$	3.92622	0.141768
$768$	0	0
$769$	49.0265	1.76794	0.883970	−	0.467543i	$-0.154861\pi$
$769$	49.0265	1.76794	0.883970	+	0.467543i	$0.154861\pi$
$770$	0	0
$771$	−27.9347	−1.00604
$772$	0	0
$773$	52.3587	1.88321	0.941606	−	0.336716i	$-0.109316\pi$
$773$	52.3587	1.88321	0.941606	+	0.336716i	$0.109316\pi$
$774$	0	0
$775$	26.7189	0.959771
$776$	0	0
$777$	−6.53341	−0.234385
$778$	0	0
$779$	−24.2562	−0.869069
$780$	0	0
$781$	8.83076	0.315989
$782$	0	0
$783$	−1.47060	−0.0525549
$784$	0	0
$785$	−1.88733	−0.0673616
$786$	0	0
$787$	−18.6414	−0.664493	−0.332246	−	0.943193i	$-0.607807\pi$
$787$	−18.6414	−0.664493	−0.332246	+	0.943193i	$0.607807\pi$
$788$	0	0
$789$	35.5893	1.26701
$790$	0	0
$791$	13.1692	0.468244
$792$	0	0
$793$	5.11491	0.181636
$794$	0	0
$795$	3.69569	0.131073
$796$	0	0
$797$	−16.1421	−0.571783	−0.285892	−	0.958262i	$-0.592290\pi$
$797$	−16.1421	−0.571783	−0.285892	+	0.958262i	$0.592290\pi$
$798$	0	0
$799$	32.1506	1.13741
$800$	0	0
$801$	37.3378	1.31926
$802$	0	0
$803$	14.4596	0.510269
$804$	0	0
$805$	0.438666	0.0154609
$806$	0	0
$807$	50.8620	1.79043
$808$	0	0
$809$	53.8121	1.89193	0.945967	−	0.324264i	$-0.105117\pi$
$809$	53.8121	1.89193	0.945967	+	0.324264i	$0.105117\pi$
$810$	0	0
$811$	28.7306	1.00887	0.504433	−	0.863451i	$-0.331701\pi$
$811$	28.7306	1.00887	0.504433	+	0.863451i	$0.331701\pi$
$812$	0	0
$813$	−1.71657	−0.0602027
$814$	0	0
$815$	0.0605754	0.00212186
$816$	0	0
$817$	25.7632	0.901341
$818$	0	0
$819$	−3.11491	−0.108844
$820$	0	0
$821$	−9.87412	−0.344609	−0.172305	−	0.985044i	$-0.555121\pi$
$821$	−9.87412	−0.344609	−0.172305	+	0.985044i	$0.555121\pi$
$822$	0	0
$823$	14.5459	0.507038	0.253519	−	0.967330i	$-0.418412\pi$
$823$	14.5459	0.507038	0.253519	+	0.967330i	$0.418412\pi$
$824$	0	0
$825$	12.3315	0.429328
$826$	0	0
$827$	12.6894	0.441255	0.220628	−	0.975358i	$-0.429189\pi$
$827$	12.6894	0.441255	0.220628	+	0.975358i	$0.429189\pi$
$828$	0	0
$829$	15.2104	0.528278	0.264139	−	0.964485i	$-0.414912\pi$
$829$	15.2104	0.528278	0.264139	+	0.964485i	$0.414912\pi$
$830$	0	0
$831$	−57.8774	−2.00775
$832$	0	0
$833$	−5.83076	−0.202024
$834$	0	0
$835$	0.764742	0.0264650
$836$	0	0
$837$	−1.52244	−0.0526232
$838$	0	0
$839$	−9.14060	−0.315568	−0.157784	−	0.987474i	$-0.550435\pi$
$839$	−9.14060	−0.315568	−0.157784	+	0.987474i	$0.550435\pi$
$840$	0	0
$841$	−2.21438	−0.0763578
$842$	0	0
$843$	−59.1638	−2.03771
$844$	0	0
$845$	−0.114908	−0.00395294
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	33.4487	1.14796
$850$	0	0
$851$	10.0863	0.345753
$852$	0	0
$853$	7.34000	0.251317	0.125658	−	0.992074i	$-0.459896\pi$
$853$	7.34000	0.251317	0.125658	+	0.992074i	$0.459896\pi$
$854$	0	0
$855$	−0.986796	−0.0337477
$856$	0	0
$857$	38.7820	1.32477	0.662383	−	0.749165i	$-0.269545\pi$
$857$	38.7820	1.32477	0.662383	+	0.749165i	$0.269545\pi$
$858$	0	0
$859$	−40.0474	−1.36640	−0.683199	−	0.730232i	$-0.739412\pi$
$859$	−40.0474	−1.36640	−0.683199	+	0.730232i	$0.739412\pi$
$860$	0	0
$861$	21.7563	0.741451
$862$	0	0
$863$	−11.7827	−0.401087	−0.200543	−	0.979685i	$-0.564271\pi$
$863$	−11.7827	−0.401087	−0.200543	+	0.979685i	$0.564271\pi$
$864$	0	0
$865$	2.31680	0.0787735
$866$	0	0
$867$	42.0327	1.42750
$868$	0	0
$869$	−8.64832	−0.293374
$870$	0	0
$871$	−5.75698	−0.195068
$872$	0	0
$873$	−5.40530	−0.182942
$874$	0	0
$875$	−1.14756	−0.0387945
$876$	0	0
$877$	−4.57302	−0.154420	−0.0772100	−	0.997015i	$-0.524601\pi$
$877$	−4.57302	−0.154420	−0.0772100	+	0.997015i	$0.524601\pi$
$878$	0	0
$879$	−8.42074	−0.284024
$880$	0	0
$881$	−7.31903	−0.246584	−0.123292	−	0.992370i	$-0.539345\pi$
$881$	−7.31903	−0.246584	−0.123292	+	0.992370i	$0.539345\pi$
$882$	0	0
$883$	38.0947	1.28199	0.640995	−	0.767545i	$-0.278522\pi$
$883$	38.0947	1.28199	0.640995	+	0.767545i	$0.278522\pi$
$884$	0	0
$885$	−1.11562	−0.0375013
$886$	0	0
$887$	−12.9721	−0.435560	−0.217780	−	0.975998i	$-0.569881\pi$
$887$	−12.9721	−0.435560	−0.217780	+	0.975998i	$0.569881\pi$
$888$	0	0
$889$	11.8238	0.396558
$890$	0	0
$891$	8.64207	0.289520
$892$	0	0
$893$	−15.2019	−0.508712
$894$	0	0
$895$	2.95415	0.0987462
$896$	0	0
$897$	9.44018	0.315199
$898$	0	0
$899$	27.7299	0.924842
$900$	0	0
$901$	75.8363	2.52647
$902$	0	0
$903$	−23.1079	−0.768984
$904$	0	0
$905$	−1.99152	−0.0662004
$906$	0	0
$907$	−41.3744	−1.37382	−0.686908	−	0.726745i	$-0.741032\pi$
$907$	−41.3744	−1.37382	−0.686908	+	0.726745i	$0.741032\pi$
$908$	0	0
$909$	−14.1017	−0.467724
$910$	0	0
$911$	2.47212	0.0819049	0.0409524	−	0.999161i	$-0.486961\pi$
$911$	2.47212	0.0819049	0.0409524	+	0.999161i	$0.486961\pi$
$912$	0	0
$913$	−9.93246	−0.328717
$914$	0	0
$915$	−1.45339	−0.0480475
$916$	0	0
$917$	−8.60719	−0.284234
$918$	0	0
$919$	15.5885	0.514216	0.257108	−	0.966383i	$-0.417230\pi$
$919$	15.5885	0.514216	0.257108	+	0.966383i	$0.417230\pi$
$920$	0	0
$921$	−14.6219	−0.481808
$922$	0	0
$923$	−8.83076	−0.290668
$924$	0	0
$925$	−13.1755	−0.433207
$926$	0	0
$927$	−20.4596	−0.671982
$928$	0	0
$929$	31.7391	1.04132	0.520662	−	0.853763i	$-0.325685\pi$
$929$	31.7391	1.04132	0.520662	+	0.853763i	$0.325685\pi$
$930$	0	0
$931$	2.75698	0.0903564
$932$	0	0
$933$	−64.3954	−2.10821
$934$	0	0
$935$	−0.669998	−0.0219113
$936$	0	0
$937$	−26.9993	−0.882028	−0.441014	−	0.897500i	$-0.645381\pi$
$937$	−26.9993	−0.882028	−0.441014	+	0.897500i	$0.645381\pi$
$938$	0	0
$939$	−31.3579	−1.02333
$940$	0	0
$941$	−7.40530	−0.241406	−0.120703	−	0.992689i	$-0.538515\pi$
$941$	−7.40530	−0.241406	−0.120703	+	0.992689i	$0.538515\pi$
$942$	0	0
$943$	−33.5873	−1.09375
$944$	0	0
$945$	0.0326507	0.00106213
$946$	0	0
$947$	−5.37514	−0.174669	−0.0873343	−	0.996179i	$-0.527835\pi$
$947$	−5.37514	−0.174669	−0.0873343	+	0.996179i	$0.527835\pi$
$948$	0	0
$949$	−14.4596	−0.469379
$950$	0	0
$951$	−16.4247	−0.532609
$952$	0	0
$953$	−53.2577	−1.72519	−0.862594	−	0.505897i	$-0.831161\pi$
$953$	−53.2577	−1.72519	−0.862594	+	0.505897i	$0.831161\pi$
$954$	0	0
$955$	−0.586224	−0.0189698
$956$	0	0
$957$	12.7981	0.413704
$958$	0	0
$959$	4.78963	0.154665
$960$	0	0
$961$	−2.29263	−0.0739556
$962$	0	0
$963$	−45.9450	−1.48056
$964$	0	0
$965$	1.03042	0.0331703
$966$	0	0
$967$	−10.7019	−0.344151	−0.172075	−	0.985084i	$-0.555047\pi$
$967$	−10.7019	−0.344151	−0.172075	+	0.985084i	$0.555047\pi$
$968$	0	0
$969$	−39.7515	−1.27700
$970$	0	0
$971$	−57.5452	−1.84671	−0.923356	−	0.383944i	$-0.874566\pi$
$971$	−57.5452	−1.84671	−0.923356	+	0.383944i	$0.874566\pi$
$972$	0	0
$973$	−2.71585	−0.0870662
$974$	0	0
$975$	−12.3315	−0.394925
$976$	0	0
$977$	33.7214	1.07884	0.539421	−	0.842036i	$-0.318643\pi$
$977$	33.7214	1.07884	0.539421	+	0.842036i	$0.318643\pi$
$978$	0	0
$979$	−11.9868	−0.383100
$980$	0	0
$981$	46.6825	1.49046
$982$	0	0
$983$	−14.3734	−0.458439	−0.229220	−	0.973375i	$-0.573617\pi$
$983$	−14.3734	−0.458439	−0.229220	+	0.973375i	$0.573617\pi$
$984$	0	0
$985$	−1.42171	−0.0452995
$986$	0	0
$987$	13.6351	0.434010
$988$	0	0
$989$	35.6740	1.13437
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	−9.60719	−0.304875
$994$	0	0
$995$	2.21260	0.0701442
$996$	0	0
$997$	28.6050	0.905928	0.452964	−	0.891529i	$-0.350367\pi$
$997$	28.6050	0.905928	0.452964	+	0.891529i	$0.350367\pi$
$998$	0	0
$999$	0.750738	0.0237523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.i.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.i.1.3	✓	3	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+2.47283 q^{3} -0.114908 q^{5} -1.00000 q^{7} +3.11491 q^{9} +O(q^{10})\)	\(q+2.47283 q^{3} -0.114908 q^{5} -1.00000 q^{7} +3.11491 q^{9} -1.00000 q^{11} +1.00000 q^{13} -0.284147 q^{15} -5.83076 q^{17} +2.75698 q^{19} -2.47283 q^{21} +3.81756 q^{23} -4.98680 q^{25} +0.284147 q^{27} -5.17548 q^{29} -5.35793 q^{31} -2.47283 q^{33} +0.114908 q^{35} +2.64207 q^{37} +2.47283 q^{39} -8.79811 q^{41} +9.34472 q^{43} -0.357926 q^{45} -5.51396 q^{47} +1.00000 q^{49} -14.4185 q^{51} -13.0062 q^{53} +0.114908 q^{55} +6.81756 q^{57} +3.92622 q^{59} +5.11491 q^{61} -3.11491 q^{63} -0.114908 q^{65} -5.75698 q^{67} +9.44018 q^{69} -8.83076 q^{71} -14.4596 q^{73} -12.3315 q^{75} +1.00000 q^{77} +8.64832 q^{79} -8.64207 q^{81} +9.93246 q^{83} +0.669998 q^{85} -12.7981 q^{87} +11.9868 q^{89} -1.00000 q^{91} -13.2493 q^{93} -0.316798 q^{95} -1.73530 q^{97} -3.11491 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 6 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 6 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 2 q^{3} + 6 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + q^{15} - 13 q^{17} + q^{19} - 2 q^{21} - 13 q^{23} + 5 q^{25} - q^{27} + 8 q^{29} - 17 q^{31} - 2 q^{33} - 6 q^{35} + 7 q^{37} + 2 q^{39} - 10 q^{41} + 9 q^{43} - 2 q^{45} - 2 q^{47} + 3 q^{49} - 27 q^{51} - 11 q^{53} - 6 q^{55} - 4 q^{57} + 9 q^{59} + 9 q^{61} - 3 q^{63} + 6 q^{65} - 10 q^{67} + 11 q^{69} - 22 q^{71} - 18 q^{73} + 2 q^{75} + 3 q^{77} - 3 q^{79} - 25 q^{81} - q^{83} - 28 q^{85} - 22 q^{87} + 16 q^{89} - 3 q^{91} - 19 q^{93} - 11 q^{95} + q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 6 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + q^15 - 13 * q^17 + q^19 - 2 * q^21 - 13 * q^23 + 5 * q^25 - q^27 + 8 * q^29 - 17 * q^31 - 2 * q^33 - 6 * q^35 + 7 * q^37 + 2 * q^39 - 10 * q^41 + 9 * q^43 - 2 * q^45 - 2 * q^47 + 3 * q^49 - 27 * q^51 - 11 * q^53 - 6 * q^55 - 4 * q^57 + 9 * q^59 + 9 * q^61 - 3 * q^63 + 6 * q^65 - 10 * q^67 + 11 * q^69 - 22 * q^71 - 18 * q^73 + 2 * q^75 + 3 * q^77 - 3 * q^79 - 25 * q^81 - q^83 - 28 * q^85 - 22 * q^87 + 16 * q^89 - 3 * q^91 - 19 * q^93 - 11 * q^95 + q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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