LMFDB - Embedded newform 8008.2.a.x.1.12

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 17 x^{10} + 79 x^{9} + 80 x^{8} - 536 x^{7} - 4 x^{6} + 1484 x^{5} - 682 x^{4} + \cdots - 64 $ x^12 - 4x^11 - 17x^10 + 79x^9 + 80x^8 - 536x^7 - 4x^6 + 1484x^5 - 682x^4 - 1431x^3 + 1069x^2 - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$3.00474$ 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)

$f(q)$	$=$	$q+3.00474 q^{3} -1.92256 q^{5} +1.00000 q^{7} +6.02845 q^{9} +O(q^{10})$	$q+3.00474 q^{3} -1.92256 q^{5} +1.00000 q^{7} +6.02845 q^{9} +1.00000 q^{11} +1.00000 q^{13} -5.77678 q^{15} -1.45347 q^{17} +2.52325 q^{19} +3.00474 q^{21} +1.34974 q^{23} -1.30378 q^{25} +9.09969 q^{27} +8.20197 q^{29} -2.65044 q^{31} +3.00474 q^{33} -1.92256 q^{35} +6.81400 q^{37} +3.00474 q^{39} -8.51767 q^{41} +0.841170 q^{43} -11.5900 q^{45} +10.9957 q^{47} +1.00000 q^{49} -4.36728 q^{51} +4.76646 q^{53} -1.92256 q^{55} +7.58172 q^{57} +0.793568 q^{59} +0.0741609 q^{61} +6.02845 q^{63} -1.92256 q^{65} -4.41870 q^{67} +4.05561 q^{69} -3.15343 q^{71} +8.22895 q^{73} -3.91751 q^{75} +1.00000 q^{77} -12.1669 q^{79} +9.25684 q^{81} +0.0518681 q^{83} +2.79437 q^{85} +24.6448 q^{87} -13.6860 q^{89} +1.00000 q^{91} -7.96386 q^{93} -4.85110 q^{95} +4.86013 q^{97} +6.02845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) $ 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9	$ 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) $ 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9 + 12 * q^11 + 12 * q^13 - 3 * q^15 + 16 * q^17 - 2 * q^19 + 4 * q^21 + 9 * q^23 + 14 * q^25 + 7 * q^27 + 15 * q^29 + 10 * q^31 + 4 * q^33 + 6 * q^35 + 18 * q^37 + 4 * q^39 + 24 * q^41 + 15 * q^45 + 5 * q^47 + 12 * q^49 + 4 * q^51 + 15 * q^53 + 6 * q^55 - 4 * q^57 + 15 * q^59 + 17 * q^61 + 14 * q^63 + 6 * q^65 - 7 * q^67 + 9 * q^71 + 32 * q^73 - 8 * q^75 + 12 * q^77 + 20 * q^79 - 4 * q^81 - 5 * q^83 + 25 * q^85 + 19 * q^87 + 16 * q^89 + 12 * q^91 + 21 * q^93 + 8 * q^95 + 10 * q^97 + 14 * 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
$3$	3.00474	1.73479	0.867393	−	0.497624i	$-0.165794\pi$
$3$	3.00474	1.73479	0.867393	+	0.497624i	$0.165794\pi$
$4$	0	0
$5$	−1.92256	−0.859793	−0.429897	−	0.902878i	$-0.641450\pi$
$5$	−1.92256	−0.859793	−0.429897	+	0.902878i	$0.641450\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	6.02845	2.00948
$10$	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
$15$	−5.77678	−1.49156
$16$	0	0
$17$	−1.45347	−0.352517	−0.176259	−	0.984344i	$-0.556399\pi$
$17$	−1.45347	−0.352517	−0.176259	+	0.984344i	$0.556399\pi$
$18$	0	0
$19$	2.52325	0.578874	0.289437	−	0.957197i	$-0.406532\pi$
$19$	2.52325	0.578874	0.289437	+	0.957197i	$0.406532\pi$
$20$	0	0
$21$	3.00474	0.655688
$22$	0	0
$23$	1.34974	0.281440	0.140720	−	0.990049i	$-0.455058\pi$
$23$	1.34974	0.281440	0.140720	+	0.990049i	$0.455058\pi$
$24$	0	0
$25$	−1.30378	−0.260755
$26$	0	0
$27$	9.09969	1.75124
$28$	0	0
$29$	8.20197	1.52307	0.761533	−	0.648126i	$-0.224447\pi$
$29$	8.20197	1.52307	0.761533	+	0.648126i	$0.224447\pi$
$30$	0	0
$31$	−2.65044	−0.476032	−0.238016	−	0.971261i	$-0.576497\pi$
$31$	−2.65044	−0.476032	−0.238016	+	0.971261i	$0.576497\pi$
$32$	0	0
$33$	3.00474	0.523058
$34$	0	0
$35$	−1.92256	−0.324971
$36$	0	0
$37$	6.81400	1.12022	0.560108	−	0.828420i	$-0.310760\pi$
$37$	6.81400	1.12022	0.560108	+	0.828420i	$0.310760\pi$
$38$	0	0
$39$	3.00474	0.481143
$40$	0	0
$41$	−8.51767	−1.33024	−0.665118	−	0.746738i	$-0.731619\pi$
$41$	−8.51767	−1.33024	−0.665118	+	0.746738i	$0.731619\pi$
$42$	0	0
$43$	0.841170	0.128277	0.0641386	−	0.997941i	$-0.479570\pi$
$43$	0.841170	0.128277	0.0641386	+	0.997941i	$0.479570\pi$
$44$	0	0
$45$	−11.5900	−1.72774
$46$	0	0
$47$	10.9957	1.60389	0.801944	−	0.597399i	$-0.203799\pi$
$47$	10.9957	1.60389	0.801944	+	0.597399i	$0.203799\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.36728	−0.611542
$52$	0	0
$53$	4.76646	0.654724	0.327362	−	0.944899i	$-0.393840\pi$
$53$	4.76646	0.654724	0.327362	+	0.944899i	$0.393840\pi$
$54$	0	0
$55$	−1.92256	−0.259237
$56$	0	0
$57$	7.58172	1.00422
$58$	0	0
$59$	0.793568	0.103314	0.0516569	−	0.998665i	$-0.483550\pi$
$59$	0.793568	0.103314	0.0516569	+	0.998665i	$0.483550\pi$
$60$	0	0
$61$	0.0741609	0.00949533	0.00474767	−	0.999989i	$-0.498489\pi$
$61$	0.0741609	0.00949533	0.00474767	+	0.999989i	$0.498489\pi$
$62$	0	0
$63$	6.02845	0.759513
$64$	0	0
$65$	−1.92256	−0.238464
$66$	0	0
$67$	−4.41870	−0.539830	−0.269915	−	0.962884i	$-0.586996\pi$
$67$	−4.41870	−0.539830	−0.269915	+	0.962884i	$0.586996\pi$
$68$	0	0
$69$	4.05561	0.488237
$70$	0	0
$71$	−3.15343	−0.374243	−0.187121	−	0.982337i	$-0.559916\pi$
$71$	−3.15343	−0.374243	−0.187121	+	0.982337i	$0.559916\pi$
$72$	0	0
$73$	8.22895	0.963126	0.481563	−	0.876412i	$-0.340069\pi$
$73$	8.22895	0.963126	0.481563	+	0.876412i	$0.340069\pi$
$74$	0	0
$75$	−3.91751	−0.452355
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−12.1669	−1.36888	−0.684442	−	0.729067i	$-0.739954\pi$
$79$	−12.1669	−1.36888	−0.684442	+	0.729067i	$0.739954\pi$
$80$	0	0
$81$	9.25684	1.02854
$82$	0	0
$83$	0.0518681	0.00569326	0.00284663	−	0.999996i	$-0.499094\pi$
$83$	0.0518681	0.00569326	0.00284663	+	0.999996i	$0.499094\pi$
$84$	0	0
$85$	2.79437	0.303092
$86$	0	0
$87$	24.6448	2.64219
$88$	0	0
$89$	−13.6860	−1.45071	−0.725355	−	0.688375i	$-0.758324\pi$
$89$	−13.6860	−1.45071	−0.725355	+	0.688375i	$0.758324\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−7.96386	−0.825814
$94$	0	0
$95$	−4.85110	−0.497712
$96$	0	0
$97$	4.86013	0.493472	0.246736	−	0.969083i	$-0.420642\pi$
$97$	4.86013	0.493472	0.246736	+	0.969083i	$0.420642\pi$
$98$	0	0
$99$	6.02845	0.605882
$100$	0	0
$101$	16.5054	1.64234	0.821172	−	0.570680i	$-0.193320\pi$
$101$	16.5054	1.64234	0.821172	+	0.570680i	$0.193320\pi$
$102$	0	0
$103$	4.31072	0.424748	0.212374	−	0.977188i	$-0.431880\pi$
$103$	4.31072	0.424748	0.212374	+	0.977188i	$0.431880\pi$
$104$	0	0
$105$	−5.77678	−0.563756
$106$	0	0
$107$	8.61158	0.832513	0.416256	−	0.909247i	$-0.363342\pi$
$107$	8.61158	0.832513	0.416256	+	0.909247i	$0.363342\pi$
$108$	0	0
$109$	−9.98620	−0.956504	−0.478252	−	0.878223i	$-0.658729\pi$
$109$	−9.98620	−0.956504	−0.478252	+	0.878223i	$0.658729\pi$
$110$	0	0
$111$	20.4743	1.94333
$112$	0	0
$113$	−2.11108	−0.198593	−0.0992967	−	0.995058i	$-0.531659\pi$
$113$	−2.11108	−0.198593	−0.0992967	+	0.995058i	$0.531659\pi$
$114$	0	0
$115$	−2.59495	−0.241980
$116$	0	0
$117$	6.02845	0.557330
$118$	0	0
$119$	−1.45347	−0.133239
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−25.5934	−2.30768
$124$	0	0
$125$	12.1194	1.08399
$126$	0	0
$127$	10.4612	0.928283	0.464142	−	0.885761i	$-0.346363\pi$
$127$	10.4612	0.928283	0.464142	+	0.885761i	$0.346363\pi$
$128$	0	0
$129$	2.52749	0.222533
$130$	0	0
$131$	17.8166	1.55664	0.778322	−	0.627865i	$-0.216071\pi$
$131$	17.8166	1.55664	0.778322	+	0.627865i	$0.216071\pi$
$132$	0	0
$133$	2.52325	0.218794
$134$	0	0
$135$	−17.4947	−1.50570
$136$	0	0
$137$	−4.16566	−0.355896	−0.177948	−	0.984040i	$-0.556946\pi$
$137$	−4.16566	−0.355896	−0.177948	+	0.984040i	$0.556946\pi$
$138$	0	0
$139$	7.00111	0.593826	0.296913	−	0.954904i	$-0.404043\pi$
$139$	7.00111	0.593826	0.296913	+	0.954904i	$0.404043\pi$
$140$	0	0
$141$	33.0392	2.78240
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−15.7687	−1.30952
$146$	0	0
$147$	3.00474	0.247827
$148$	0	0
$149$	11.5659	0.947517	0.473759	−	0.880655i	$-0.342897\pi$
$149$	11.5659	0.947517	0.473759	+	0.880655i	$0.342897\pi$
$150$	0	0
$151$	9.17948	0.747015	0.373508	−	0.927627i	$-0.378155\pi$
$151$	9.17948	0.747015	0.373508	+	0.927627i	$0.378155\pi$
$152$	0	0
$153$	−8.76214	−0.708377
$154$	0	0
$155$	5.09561	0.409289
$156$	0	0
$157$	−4.69168	−0.374437	−0.187218	−	0.982318i	$-0.559947\pi$
$157$	−4.69168	−0.374437	−0.187218	+	0.982318i	$0.559947\pi$
$158$	0	0
$159$	14.3220	1.13581
$160$	0	0
$161$	1.34974	0.106374
$162$	0	0
$163$	−3.01818	−0.236402	−0.118201	−	0.992990i	$-0.537713\pi$
$163$	−3.01818	−0.236402	−0.118201	+	0.992990i	$0.537713\pi$
$164$	0	0
$165$	−5.77678	−0.449722
$166$	0	0
$167$	−8.58119	−0.664032	−0.332016	−	0.943274i	$-0.607729\pi$
$167$	−8.58119	−0.664032	−0.332016	+	0.943274i	$0.607729\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	15.2113	1.16324
$172$	0	0
$173$	−8.17915	−0.621849	−0.310925	−	0.950435i	$-0.600639\pi$
$173$	−8.17915	−0.621849	−0.310925	+	0.950435i	$0.600639\pi$
$174$	0	0
$175$	−1.30378	−0.0985563
$176$	0	0
$177$	2.38446	0.179227
$178$	0	0
$179$	10.7584	0.804117	0.402059	−	0.915614i	$-0.368295\pi$
$179$	10.7584	0.804117	0.402059	+	0.915614i	$0.368295\pi$
$180$	0	0
$181$	18.9737	1.41031	0.705153	−	0.709055i	$-0.250878\pi$
$181$	18.9737	1.41031	0.705153	+	0.709055i	$0.250878\pi$
$182$	0	0
$183$	0.222834	0.0164724
$184$	0	0
$185$	−13.1003	−0.963154
$186$	0	0
$187$	−1.45347	−0.106288
$188$	0	0
$189$	9.09969	0.661905
$190$	0	0
$191$	8.38477	0.606701	0.303350	−	0.952879i	$-0.401895\pi$
$191$	8.38477	0.606701	0.303350	+	0.952879i	$0.401895\pi$
$192$	0	0
$193$	−16.5390	−1.19050	−0.595252	−	0.803539i	$-0.702948\pi$
$193$	−16.5390	−1.19050	−0.595252	+	0.803539i	$0.702948\pi$
$194$	0	0
$195$	−5.77678	−0.413684
$196$	0	0
$197$	−2.15248	−0.153358	−0.0766790	−	0.997056i	$-0.524432\pi$
$197$	−2.15248	−0.153358	−0.0766790	+	0.997056i	$0.524432\pi$
$198$	0	0
$199$	3.73433	0.264719	0.132360	−	0.991202i	$-0.457745\pi$
$199$	3.73433	0.264719	0.132360	+	0.991202i	$0.457745\pi$
$200$	0	0
$201$	−13.2770	−0.936489
$202$	0	0
$203$	8.20197	0.575665
$204$	0	0
$205$	16.3757	1.14373
$206$	0	0
$207$	8.13682	0.565548
$208$	0	0
$209$	2.52325	0.174537
$210$	0	0
$211$	−7.60286	−0.523402	−0.261701	−	0.965149i	$-0.584283\pi$
$211$	−7.60286	−0.523402	−0.261701	+	0.965149i	$0.584283\pi$
$212$	0	0
$213$	−9.47522	−0.649231
$214$	0	0
$215$	−1.61720	−0.110292
$216$	0	0
$217$	−2.65044	−0.179923
$218$	0	0
$219$	24.7258	1.67082
$220$	0	0
$221$	−1.45347	−0.0977706
$222$	0	0
$223$	1.21259	0.0812012	0.0406006	−	0.999175i	$-0.487073\pi$
$223$	1.21259	0.0812012	0.0406006	+	0.999175i	$0.487073\pi$
$224$	0	0
$225$	−7.85975	−0.523983
$226$	0	0
$227$	16.9189	1.12295	0.561474	−	0.827494i	$-0.310234\pi$
$227$	16.9189	1.12295	0.561474	+	0.827494i	$0.310234\pi$
$228$	0	0
$229$	−5.78657	−0.382388	−0.191194	−	0.981552i	$-0.561236\pi$
$229$	−5.78657	−0.382388	−0.191194	+	0.981552i	$0.561236\pi$
$230$	0	0
$231$	3.00474	0.197697
$232$	0	0
$233$	−8.57544	−0.561796	−0.280898	−	0.959738i	$-0.590632\pi$
$233$	−8.57544	−0.561796	−0.280898	+	0.959738i	$0.590632\pi$
$234$	0	0
$235$	−21.1399	−1.37901
$236$	0	0
$237$	−36.5584	−2.37472
$238$	0	0
$239$	−1.04957	−0.0678908	−0.0339454	−	0.999424i	$-0.510807\pi$
$239$	−1.04957	−0.0678908	−0.0339454	+	0.999424i	$0.510807\pi$
$240$	0	0
$241$	8.29942	0.534612	0.267306	−	0.963612i	$-0.413866\pi$
$241$	8.29942	0.534612	0.267306	+	0.963612i	$0.413866\pi$
$242$	0	0
$243$	0.515307	0.0330570
$244$	0	0
$245$	−1.92256	−0.122828
$246$	0	0
$247$	2.52325	0.160551
$248$	0	0
$249$	0.155850	0.00987659
$250$	0	0
$251$	9.53263	0.601694	0.300847	−	0.953672i	$-0.402731\pi$
$251$	9.53263	0.601694	0.300847	+	0.953672i	$0.402731\pi$
$252$	0	0
$253$	1.34974	0.0848572
$254$	0	0
$255$	8.39634	0.525800
$256$	0	0
$257$	−29.0900	−1.81458	−0.907291	−	0.420503i	$-0.861854\pi$
$257$	−29.0900	−1.81458	−0.907291	+	0.420503i	$0.861854\pi$
$258$	0	0
$259$	6.81400	0.423402
$260$	0	0
$261$	49.4451	3.06058
$262$	0	0
$263$	13.0430	0.804265	0.402133	−	0.915581i	$-0.368269\pi$
$263$	13.0430	0.804265	0.402133	+	0.915581i	$0.368269\pi$
$264$	0	0
$265$	−9.16379	−0.562927
$266$	0	0
$267$	−41.1228	−2.51667
$268$	0	0
$269$	13.4712	0.821352	0.410676	−	0.911781i	$-0.365293\pi$
$269$	13.4712	0.821352	0.410676	+	0.911781i	$0.365293\pi$
$270$	0	0
$271$	6.62745	0.402589	0.201294	−	0.979531i	$-0.435485\pi$
$271$	6.62745	0.402589	0.201294	+	0.979531i	$0.435485\pi$
$272$	0	0
$273$	3.00474	0.181855
$274$	0	0
$275$	−1.30378	−0.0786207
$276$	0	0
$277$	−26.5284	−1.59394	−0.796968	−	0.604022i	$-0.793564\pi$
$277$	−26.5284	−1.59394	−0.796968	+	0.604022i	$0.793564\pi$
$278$	0	0
$279$	−15.9780	−0.956579
$280$	0	0
$281$	0.170464	0.0101690	0.00508451	−	0.999987i	$-0.498382\pi$
$281$	0.170464	0.0101690	0.00508451	+	0.999987i	$0.498382\pi$
$282$	0	0
$283$	14.1869	0.843323	0.421661	−	0.906753i	$-0.361447\pi$
$283$	14.1869	0.843323	0.421661	+	0.906753i	$0.361447\pi$
$284$	0	0
$285$	−14.5763	−0.863424
$286$	0	0
$287$	−8.51767	−0.502782
$288$	0	0
$289$	−14.8874	−0.875732
$290$	0	0
$291$	14.6034	0.856068
$292$	0	0
$293$	−12.0263	−0.702584	−0.351292	−	0.936266i	$-0.614258\pi$
$293$	−12.0263	−0.702584	−0.351292	+	0.936266i	$0.614258\pi$
$294$	0	0
$295$	−1.52568	−0.0888284
$296$	0	0
$297$	9.09969	0.528018
$298$	0	0
$299$	1.34974	0.0780573
$300$	0	0
$301$	0.841170	0.0484842
$302$	0	0
$303$	49.5943	2.84912
$304$	0	0
$305$	−0.142579	−0.00816403
$306$	0	0
$307$	−31.4128	−1.79282	−0.896412	−	0.443223i	$-0.853835\pi$
$307$	−31.4128	−1.79282	−0.896412	+	0.443223i	$0.853835\pi$
$308$	0	0
$309$	12.9526	0.736847
$310$	0	0
$311$	−1.05864	−0.0600297	−0.0300149	−	0.999549i	$-0.509555\pi$
$311$	−1.05864	−0.0600297	−0.0300149	+	0.999549i	$0.509555\pi$
$312$	0	0
$313$	−18.9131	−1.06903	−0.534516	−	0.845158i	$-0.679506\pi$
$313$	−18.9131	−1.06903	−0.534516	+	0.845158i	$0.679506\pi$
$314$	0	0
$315$	−11.5900	−0.653024
$316$	0	0
$317$	−12.6816	−0.712268	−0.356134	−	0.934435i	$-0.615905\pi$
$317$	−12.6816	−0.712268	−0.356134	+	0.934435i	$0.615905\pi$
$318$	0	0
$319$	8.20197	0.459222
$320$	0	0
$321$	25.8755	1.44423
$322$	0	0
$323$	−3.66746	−0.204063
$324$	0	0
$325$	−1.30378	−0.0723205
$326$	0	0
$327$	−30.0059	−1.65933
$328$	0	0
$329$	10.9957	0.606213
$330$	0	0
$331$	−18.3915	−1.01089	−0.505443	−	0.862860i	$-0.668671\pi$
$331$	−18.3915	−1.01089	−0.505443	+	0.862860i	$0.668671\pi$
$332$	0	0
$333$	41.0779	2.25105
$334$	0	0
$335$	8.49519	0.464142
$336$	0	0
$337$	12.2475	0.667163	0.333582	−	0.942721i	$-0.391743\pi$
$337$	12.2475	0.667163	0.333582	+	0.942721i	$0.391743\pi$
$338$	0	0
$339$	−6.34323	−0.344517
$340$	0	0
$341$	−2.65044	−0.143529
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−7.79713	−0.419783
$346$	0	0
$347$	−12.5687	−0.674721	−0.337361	−	0.941375i	$-0.609534\pi$
$347$	−12.5687	−0.674721	−0.337361	+	0.941375i	$0.609534\pi$
$348$	0	0
$349$	12.9671	0.694112	0.347056	−	0.937844i	$-0.387181\pi$
$349$	12.9671	0.694112	0.347056	+	0.937844i	$0.387181\pi$
$350$	0	0
$351$	9.09969	0.485706
$352$	0	0
$353$	32.6565	1.73813	0.869066	−	0.494697i	$-0.164721\pi$
$353$	32.6565	1.73813	0.869066	+	0.494697i	$0.164721\pi$
$354$	0	0
$355$	6.06264	0.321771
$356$	0	0
$357$	−4.36728	−0.231141
$358$	0	0
$359$	−7.04593	−0.371870	−0.185935	−	0.982562i	$-0.559531\pi$
$359$	−7.04593	−0.371870	−0.185935	+	0.982562i	$0.559531\pi$
$360$	0	0
$361$	−12.6332	−0.664905
$362$	0	0
$363$	3.00474	0.157708
$364$	0	0
$365$	−15.8206	−0.828089
$366$	0	0
$367$	30.9711	1.61668	0.808339	−	0.588717i	$-0.200367\pi$
$367$	30.9711	1.61668	0.808339	+	0.588717i	$0.200367\pi$
$368$	0	0
$369$	−51.3483	−2.67309
$370$	0	0
$371$	4.76646	0.247462
$372$	0	0
$373$	31.8068	1.64689	0.823446	−	0.567395i	$-0.192049\pi$
$373$	31.8068	1.64689	0.823446	+	0.567395i	$0.192049\pi$
$374$	0	0
$375$	36.4155	1.88049
$376$	0	0
$377$	8.20197	0.422423
$378$	0	0
$379$	−33.2869	−1.70984	−0.854918	−	0.518763i	$-0.826393\pi$
$379$	−33.2869	−1.70984	−0.854918	+	0.518763i	$0.826393\pi$
$380$	0	0
$381$	31.4332	1.61037
$382$	0	0
$383$	−14.8874	−0.760711	−0.380355	−	0.924840i	$-0.624198\pi$
$383$	−14.8874	−0.760711	−0.380355	+	0.924840i	$0.624198\pi$
$384$	0	0
$385$	−1.92256	−0.0979825
$386$	0	0
$387$	5.07095	0.257771
$388$	0	0
$389$	−0.0841795	−0.00426807	−0.00213403	−	0.999998i	$-0.500679\pi$
$389$	−0.0841795	−0.00426807	−0.00213403	+	0.999998i	$0.500679\pi$
$390$	0	0
$391$	−1.96180	−0.0992123
$392$	0	0
$393$	53.5342	2.70044
$394$	0	0
$395$	23.3916	1.17696
$396$	0	0
$397$	3.87414	0.194438	0.0972188	−	0.995263i	$-0.469005\pi$
$397$	3.87414	0.194438	0.0972188	+	0.995263i	$0.469005\pi$
$398$	0	0
$399$	7.58172	0.379561
$400$	0	0
$401$	32.2430	1.61014	0.805071	−	0.593179i	$-0.202127\pi$
$401$	32.2430	1.61014	0.805071	+	0.593179i	$0.202127\pi$
$402$	0	0
$403$	−2.65044	−0.132028
$404$	0	0
$405$	−17.7968	−0.884330
$406$	0	0
$407$	6.81400	0.337758
$408$	0	0
$409$	−3.70773	−0.183335	−0.0916677	−	0.995790i	$-0.529220\pi$
$409$	−3.70773	−0.183335	−0.0916677	+	0.995790i	$0.529220\pi$
$410$	0	0
$411$	−12.5167	−0.617403
$412$	0	0
$413$	0.793568	0.0390489
$414$	0	0
$415$	−0.0997193	−0.00489503
$416$	0	0
$417$	21.0365	1.03016
$418$	0	0
$419$	11.9889	0.585697	0.292849	−	0.956159i	$-0.405397\pi$
$419$	11.9889	0.585697	0.292849	+	0.956159i	$0.405397\pi$
$420$	0	0
$421$	−21.9651	−1.07051	−0.535256	−	0.844690i	$-0.679785\pi$
$421$	−21.9651	−1.07051	−0.535256	+	0.844690i	$0.679785\pi$
$422$	0	0
$423$	66.2870	3.22299
$424$	0	0
$425$	1.89499	0.0919207
$426$	0	0
$427$	0.0741609	0.00358890
$428$	0	0
$429$	3.00474	0.145070
$430$	0	0
$431$	5.04774	0.243141	0.121571	−	0.992583i	$-0.461207\pi$
$431$	5.04774	0.243141	0.121571	+	0.992583i	$0.461207\pi$
$432$	0	0
$433$	−12.7715	−0.613761	−0.306880	−	0.951748i	$-0.599285\pi$
$433$	−12.7715	−0.613761	−0.306880	+	0.951748i	$0.599285\pi$
$434$	0	0
$435$	−47.3809	−2.27174
$436$	0	0
$437$	3.40573	0.162918
$438$	0	0
$439$	38.1059	1.81870	0.909348	−	0.416036i	$-0.136581\pi$
$439$	38.1059	1.81870	0.909348	+	0.416036i	$0.136581\pi$
$440$	0	0
$441$	6.02845	0.287069
$442$	0	0
$443$	23.2291	1.10365	0.551825	−	0.833960i	$-0.313932\pi$
$443$	23.2291	1.10365	0.551825	+	0.833960i	$0.313932\pi$
$444$	0	0
$445$	26.3121	1.24731
$446$	0	0
$447$	34.7526	1.64374
$448$	0	0
$449$	−29.3469	−1.38497	−0.692483	−	0.721434i	$-0.743483\pi$
$449$	−29.3469	−1.38497	−0.692483	+	0.721434i	$0.743483\pi$
$450$	0	0
$451$	−8.51767	−0.401081
$452$	0	0
$453$	27.5819	1.29591
$454$	0	0
$455$	−1.92256	−0.0901308
$456$	0	0
$457$	−20.2322	−0.946424	−0.473212	−	0.880949i	$-0.656905\pi$
$457$	−20.2322	−0.946424	−0.473212	+	0.880949i	$0.656905\pi$
$458$	0	0
$459$	−13.2261	−0.617341
$460$	0	0
$461$	−12.4363	−0.579217	−0.289608	−	0.957145i	$-0.593525\pi$
$461$	−12.4363	−0.579217	−0.289608	+	0.957145i	$0.593525\pi$
$462$	0	0
$463$	0.553881	0.0257410	0.0128705	−	0.999917i	$-0.495903\pi$
$463$	0.553881	0.0257410	0.0128705	+	0.999917i	$0.495903\pi$
$464$	0	0
$465$	15.3110	0.710030
$466$	0	0
$467$	15.9111	0.736278	0.368139	−	0.929771i	$-0.379995\pi$
$467$	15.9111	0.736278	0.368139	+	0.929771i	$0.379995\pi$
$468$	0	0
$469$	−4.41870	−0.204036
$470$	0	0
$471$	−14.0973	−0.649567
$472$	0	0
$473$	0.841170	0.0386770
$474$	0	0
$475$	−3.28976	−0.150945
$476$	0	0
$477$	28.7344	1.31566
$478$	0	0
$479$	9.55384	0.436526	0.218263	−	0.975890i	$-0.429961\pi$
$479$	9.55384	0.436526	0.218263	+	0.975890i	$0.429961\pi$
$480$	0	0
$481$	6.81400	0.310692
$482$	0	0
$483$	4.05561	0.184536
$484$	0	0
$485$	−9.34388	−0.424284
$486$	0	0
$487$	9.37205	0.424688	0.212344	−	0.977195i	$-0.431890\pi$
$487$	9.37205	0.424688	0.212344	+	0.977195i	$0.431890\pi$
$488$	0	0
$489$	−9.06883	−0.410107
$490$	0	0
$491$	−4.53410	−0.204621	−0.102310	−	0.994753i	$-0.532624\pi$
$491$	−4.53410	−0.204621	−0.102310	+	0.994753i	$0.532624\pi$
$492$	0	0
$493$	−11.9213	−0.536907
$494$	0	0
$495$	−11.5900	−0.520933
$496$	0	0
$497$	−3.15343	−0.141450
$498$	0	0
$499$	−4.60750	−0.206260	−0.103130	−	0.994668i	$-0.532886\pi$
$499$	−4.60750	−0.206260	−0.103130	+	0.994668i	$0.532886\pi$
$500$	0	0
$501$	−25.7842	−1.15195
$502$	0	0
$503$	20.5054	0.914289	0.457144	−	0.889393i	$-0.348872\pi$
$503$	20.5054	0.914289	0.457144	+	0.889393i	$0.348872\pi$
$504$	0	0
$505$	−31.7325	−1.41208
$506$	0	0
$507$	3.00474	0.133445
$508$	0	0
$509$	14.7334	0.653046	0.326523	−	0.945189i	$-0.394123\pi$
$509$	14.7334	0.653046	0.326523	+	0.945189i	$0.394123\pi$
$510$	0	0
$511$	8.22895	0.364027
$512$	0	0
$513$	22.9608	1.01375
$514$	0	0
$515$	−8.28761	−0.365196
$516$	0	0
$517$	10.9957	0.483591
$518$	0	0
$519$	−24.5762	−1.07878
$520$	0	0
$521$	−4.88741	−0.214121	−0.107061	−	0.994252i	$-0.534144\pi$
$521$	−4.88741	−0.214121	−0.107061	+	0.994252i	$0.534144\pi$
$522$	0	0
$523$	4.85075	0.212108	0.106054	−	0.994360i	$-0.466178\pi$
$523$	4.85075	0.212108	0.106054	+	0.994360i	$0.466178\pi$
$524$	0	0
$525$	−3.91751	−0.170974
$526$	0	0
$527$	3.85232	0.167810
$528$	0	0
$529$	−21.1782	−0.920792
$530$	0	0
$531$	4.78398	0.207607
$532$	0	0
$533$	−8.51767	−0.368941
$534$	0	0
$535$	−16.5562	−0.715789
$536$	0	0
$537$	32.3260	1.39497
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−42.2562	−1.81674	−0.908369	−	0.418171i	$-0.862671\pi$
$541$	−42.2562	−1.81674	−0.908369	+	0.418171i	$0.862671\pi$
$542$	0	0
$543$	57.0111	2.44658
$544$	0	0
$545$	19.1990	0.822396
$546$	0	0
$547$	−3.14577	−0.134504	−0.0672518	−	0.997736i	$-0.521423\pi$
$547$	−3.14577	−0.134504	−0.0672518	+	0.997736i	$0.521423\pi$
$548$	0	0
$549$	0.447075	0.0190807
$550$	0	0
$551$	20.6956	0.881664
$552$	0	0
$553$	−12.1669	−0.517389
$554$	0	0
$555$	−39.3630	−1.67087
$556$	0	0
$557$	28.5433	1.20942	0.604710	−	0.796446i	$-0.293289\pi$
$557$	28.5433	1.20942	0.604710	+	0.796446i	$0.293289\pi$
$558$	0	0
$559$	0.841170	0.0355777
$560$	0	0
$561$	−4.36728	−0.184387
$562$	0	0
$563$	−31.9252	−1.34549	−0.672744	−	0.739875i	$-0.734884\pi$
$563$	−31.9252	−1.34549	−0.672744	+	0.739875i	$0.734884\pi$
$564$	0	0
$565$	4.05866	0.170749
$566$	0	0
$567$	9.25684	0.388751
$568$	0	0
$569$	12.1499	0.509351	0.254675	−	0.967027i	$-0.418031\pi$
$569$	12.1499	0.509351	0.254675	+	0.967027i	$0.418031\pi$
$570$	0	0
$571$	13.8045	0.577700	0.288850	−	0.957374i	$-0.406727\pi$
$571$	13.8045	0.577700	0.288850	+	0.957374i	$0.406727\pi$
$572$	0	0
$573$	25.1940	1.05250
$574$	0	0
$575$	−1.75976	−0.0733869
$576$	0	0
$577$	−19.2234	−0.800279	−0.400139	−	0.916454i	$-0.631038\pi$
$577$	−19.2234	−0.800279	−0.400139	+	0.916454i	$0.631038\pi$
$578$	0	0
$579$	−49.6953	−2.06527
$580$	0	0
$581$	0.0518681	0.00215185
$582$	0	0
$583$	4.76646	0.197407
$584$	0	0
$585$	−11.5900	−0.479189
$586$	0	0
$587$	−15.2620	−0.629932	−0.314966	−	0.949103i	$-0.601993\pi$
$587$	−15.2620	−0.629932	−0.314966	+	0.949103i	$0.601993\pi$
$588$	0	0
$589$	−6.68772	−0.275563
$590$	0	0
$591$	−6.46765	−0.266043
$592$	0	0
$593$	12.4258	0.510267	0.255134	−	0.966906i	$-0.417881\pi$
$593$	12.4258	0.510267	0.255134	+	0.966906i	$0.417881\pi$
$594$	0	0
$595$	2.79437	0.114558
$596$	0	0
$597$	11.2207	0.459231
$598$	0	0
$599$	−44.2951	−1.80985	−0.904925	−	0.425571i	$-0.860073\pi$
$599$	−44.2951	−1.80985	−0.904925	+	0.425571i	$0.860073\pi$
$600$	0	0
$601$	36.7223	1.49793	0.748967	−	0.662608i	$-0.230550\pi$
$601$	36.7223	1.49793	0.748967	+	0.662608i	$0.230550\pi$
$602$	0	0
$603$	−26.6379	−1.08478
$604$	0	0
$605$	−1.92256	−0.0781630
$606$	0	0
$607$	−43.0488	−1.74730	−0.873649	−	0.486557i	$-0.838253\pi$
$607$	−43.0488	−1.74730	−0.873649	+	0.486557i	$0.838253\pi$
$608$	0	0
$609$	24.6448	0.998656
$610$	0	0
$611$	10.9957	0.444839
$612$	0	0
$613$	−6.35832	−0.256810	−0.128405	−	0.991722i	$-0.540986\pi$
$613$	−6.35832	−0.256810	−0.128405	+	0.991722i	$0.540986\pi$
$614$	0	0
$615$	49.2047	1.98412
$616$	0	0
$617$	1.61972	0.0652074	0.0326037	−	0.999468i	$-0.489620\pi$
$617$	1.61972	0.0652074	0.0326037	+	0.999468i	$0.489620\pi$
$618$	0	0
$619$	−23.6672	−0.951266	−0.475633	−	0.879644i	$-0.657781\pi$
$619$	−23.6672	−0.951266	−0.475633	+	0.879644i	$0.657781\pi$
$620$	0	0
$621$	12.2822	0.492867
$622$	0	0
$623$	−13.6860	−0.548317
$624$	0	0
$625$	−16.7813	−0.671251
$626$	0	0
$627$	7.58172	0.302785
$628$	0	0
$629$	−9.90392	−0.394895
$630$	0	0
$631$	−23.1311	−0.920835	−0.460418	−	0.887702i	$-0.652300\pi$
$631$	−23.1311	−0.920835	−0.460418	+	0.887702i	$0.652300\pi$
$632$	0	0
$633$	−22.8446	−0.907991
$634$	0	0
$635$	−20.1123	−0.798132
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−19.0103	−0.752034
$640$	0	0
$641$	27.8353	1.09943	0.549714	−	0.835353i	$-0.314737\pi$
$641$	27.8353	1.09943	0.549714	+	0.835353i	$0.314737\pi$
$642$	0	0
$643$	−30.5873	−1.20625	−0.603123	−	0.797648i	$-0.706077\pi$
$643$	−30.5873	−1.20625	−0.603123	+	0.797648i	$0.706077\pi$
$644$	0	0
$645$	−4.85925	−0.191333
$646$	0	0
$647$	−16.8726	−0.663332	−0.331666	−	0.943397i	$-0.607611\pi$
$647$	−16.8726	−0.663332	−0.331666	+	0.943397i	$0.607611\pi$
$648$	0	0
$649$	0.793568	0.0311503
$650$	0	0
$651$	−7.96386	−0.312128
$652$	0	0
$653$	6.17904	0.241805	0.120902	−	0.992664i	$-0.461421\pi$
$653$	6.17904	0.241805	0.120902	+	0.992664i	$0.461421\pi$
$654$	0	0
$655$	−34.2534	−1.33839
$656$	0	0
$657$	49.6078	1.93538
$658$	0	0
$659$	−14.2317	−0.554389	−0.277194	−	0.960814i	$-0.589405\pi$
$659$	−14.2317	−0.554389	−0.277194	+	0.960814i	$0.589405\pi$
$660$	0	0
$661$	26.7598	1.04084	0.520418	−	0.853912i	$-0.325776\pi$
$661$	26.7598	1.04084	0.520418	+	0.853912i	$0.325776\pi$
$662$	0	0
$663$	−4.36728	−0.169611
$664$	0	0
$665$	−4.85110	−0.188118
$666$	0	0
$667$	11.0705	0.428651
$668$	0	0
$669$	3.64352	0.140867
$670$	0	0
$671$	0.0741609	0.00286295
$672$	0	0
$673$	−8.83865	−0.340705	−0.170353	−	0.985383i	$-0.554491\pi$
$673$	−8.83865	−0.340705	−0.170353	+	0.985383i	$0.554491\pi$
$674$	0	0
$675$	−11.8640	−0.456644
$676$	0	0
$677$	−19.3231	−0.742648	−0.371324	−	0.928503i	$-0.621096\pi$
$677$	−19.3231	−0.742648	−0.371324	+	0.928503i	$0.621096\pi$
$678$	0	0
$679$	4.86013	0.186515
$680$	0	0
$681$	50.8369	1.94808
$682$	0	0
$683$	−39.9350	−1.52807	−0.764036	−	0.645174i	$-0.776785\pi$
$683$	−39.9350	−1.52807	−0.764036	+	0.645174i	$0.776785\pi$
$684$	0	0
$685$	8.00871	0.305997
$686$	0	0
$687$	−17.3871	−0.663361
$688$	0	0
$689$	4.76646	0.181588
$690$	0	0
$691$	−10.8333	−0.412118	−0.206059	−	0.978540i	$-0.566064\pi$
$691$	−10.8333	−0.412118	−0.206059	+	0.978540i	$0.566064\pi$
$692$	0	0
$693$	6.02845	0.229002
$694$	0	0
$695$	−13.4600	−0.510568
$696$	0	0
$697$	12.3801	0.468931
$698$	0	0
$699$	−25.7670	−0.974596
$700$	0	0
$701$	41.8049	1.57895	0.789474	−	0.613784i	$-0.210354\pi$
$701$	41.8049	1.57895	0.789474	+	0.613784i	$0.210354\pi$
$702$	0	0
$703$	17.1935	0.648464
$704$	0	0
$705$	−63.5197	−2.39229
$706$	0	0
$707$	16.5054	0.620748
$708$	0	0
$709$	−10.2172	−0.383716	−0.191858	−	0.981423i	$-0.561451\pi$
$709$	−10.2172	−0.383716	−0.191858	+	0.981423i	$0.561451\pi$
$710$	0	0
$711$	−73.3476	−2.75075
$712$	0	0
$713$	−3.57739	−0.133974
$714$	0	0
$715$	−1.92256	−0.0718995
$716$	0	0
$717$	−3.15367	−0.117776
$718$	0	0
$719$	26.7456	0.997444	0.498722	−	0.866762i	$-0.333803\pi$
$719$	26.7456	0.997444	0.498722	+	0.866762i	$0.333803\pi$
$720$	0	0
$721$	4.31072	0.160540
$722$	0	0
$723$	24.9376	0.927438
$724$	0	0
$725$	−10.6935	−0.397148
$726$	0	0
$727$	−23.0499	−0.854872	−0.427436	−	0.904046i	$-0.640583\pi$
$727$	−23.0499	−0.854872	−0.427436	+	0.904046i	$0.640583\pi$
$728$	0	0
$729$	−26.2222	−0.971191
$730$	0	0
$731$	−1.22261	−0.0452199
$732$	0	0
$733$	−4.90002	−0.180986	−0.0904932	−	0.995897i	$-0.528844\pi$
$733$	−4.90002	−0.180986	−0.0904932	+	0.995897i	$0.528844\pi$
$734$	0	0
$735$	−5.77678	−0.213080
$736$	0	0
$737$	−4.41870	−0.162765
$738$	0	0
$739$	7.95438	0.292607	0.146303	−	0.989240i	$-0.453262\pi$
$739$	7.95438	0.292607	0.146303	+	0.989240i	$0.453262\pi$
$740$	0	0
$741$	7.58172	0.278521
$742$	0	0
$743$	−23.2929	−0.854532	−0.427266	−	0.904126i	$-0.640523\pi$
$743$	−23.2929	−0.854532	−0.427266	+	0.904126i	$0.640523\pi$
$744$	0	0
$745$	−22.2361	−0.814669
$746$	0	0
$747$	0.312684	0.0114405
$748$	0	0
$749$	8.61158	0.314660
$750$	0	0
$751$	−25.4764	−0.929645	−0.464823	−	0.885404i	$-0.653882\pi$
$751$	−25.4764	−0.929645	−0.464823	+	0.885404i	$0.653882\pi$
$752$	0	0
$753$	28.6430	1.04381
$754$	0	0
$755$	−17.6481	−0.642279
$756$	0	0
$757$	−11.1082	−0.403735	−0.201867	−	0.979413i	$-0.564701\pi$
$757$	−11.1082	−0.403735	−0.201867	+	0.979413i	$0.564701\pi$
$758$	0	0
$759$	4.05561	0.147209
$760$	0	0
$761$	32.7977	1.18892	0.594458	−	0.804127i	$-0.297367\pi$
$761$	32.7977	1.18892	0.594458	+	0.804127i	$0.297367\pi$
$762$	0	0
$763$	−9.98620	−0.361525
$764$	0	0
$765$	16.8457	0.609058
$766$	0	0
$767$	0.793568	0.0286541
$768$	0	0
$769$	0.921791	0.0332406	0.0166203	−	0.999862i	$-0.494709\pi$
$769$	0.921791	0.0332406	0.0166203	+	0.999862i	$0.494709\pi$
$770$	0	0
$771$	−87.4077	−3.14791
$772$	0	0
$773$	−48.4330	−1.74201	−0.871006	−	0.491272i	$-0.836532\pi$
$773$	−48.4330	−1.74201	−0.871006	+	0.491272i	$0.836532\pi$
$774$	0	0
$775$	3.45558	0.124128
$776$	0	0
$777$	20.4743	0.734511
$778$	0	0
$779$	−21.4922	−0.770039
$780$	0	0
$781$	−3.15343	−0.112838
$782$	0	0
$783$	74.6354	2.66725
$784$	0	0
$785$	9.02001	0.321938
$786$	0	0
$787$	8.11485	0.289263	0.144632	−	0.989486i	$-0.453800\pi$
$787$	8.11485	0.289263	0.144632	+	0.989486i	$0.453800\pi$
$788$	0	0
$789$	39.1908	1.39523
$790$	0	0
$791$	−2.11108	−0.0750612
$792$	0	0
$793$	0.0741609	0.00263353
$794$	0	0
$795$	−27.5348	−0.976558
$796$	0	0
$797$	−46.3606	−1.64218	−0.821088	−	0.570801i	$-0.806633\pi$
$797$	−46.3606	−1.64218	−0.821088	+	0.570801i	$0.806633\pi$
$798$	0	0
$799$	−15.9819	−0.565398
$800$	0	0
$801$	−82.5052	−2.91518
$802$	0	0
$803$	8.22895	0.290393
$804$	0	0
$805$	−2.59495	−0.0914598
$806$	0	0
$807$	40.4774	1.42487
$808$	0	0
$809$	33.4868	1.17733	0.588667	−	0.808376i	$-0.299653\pi$
$809$	33.4868	1.17733	0.588667	+	0.808376i	$0.299653\pi$
$810$	0	0
$811$	−21.2185	−0.745081	−0.372541	−	0.928016i	$-0.621513\pi$
$811$	−21.2185	−0.745081	−0.372541	+	0.928016i	$0.621513\pi$
$812$	0	0
$813$	19.9137	0.698405
$814$	0	0
$815$	5.80262	0.203257
$816$	0	0
$817$	2.12249	0.0742564
$818$	0	0
$819$	6.02845	0.210651
$820$	0	0
$821$	4.09238	0.142825	0.0714125	−	0.997447i	$-0.477249\pi$
$821$	4.09238	0.142825	0.0714125	+	0.997447i	$0.477249\pi$
$822$	0	0
$823$	−11.5806	−0.403673	−0.201837	−	0.979419i	$-0.564691\pi$
$823$	−11.5806	−0.403673	−0.201837	+	0.979419i	$0.564691\pi$
$824$	0	0
$825$	−3.91751	−0.136390
$826$	0	0
$827$	−22.4410	−0.780350	−0.390175	−	0.920741i	$-0.627586\pi$
$827$	−22.4410	−0.780350	−0.390175	+	0.920741i	$0.627586\pi$
$828$	0	0
$829$	42.3128	1.46958	0.734792	−	0.678293i	$-0.237280\pi$
$829$	42.3128	1.46958	0.734792	+	0.678293i	$0.237280\pi$
$830$	0	0
$831$	−79.7108	−2.76514
$832$	0	0
$833$	−1.45347	−0.0503596
$834$	0	0
$835$	16.4978	0.570930
$836$	0	0
$837$	−24.1182	−0.833645
$838$	0	0
$839$	−16.7302	−0.577591	−0.288796	−	0.957391i	$-0.593255\pi$
$839$	−16.7302	−0.577591	−0.288796	+	0.957391i	$0.593255\pi$
$840$	0	0
$841$	38.2722	1.31973
$842$	0	0
$843$	0.512199	0.0176411
$844$	0	0
$845$	−1.92256	−0.0661379
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	42.6279	1.46298
$850$	0	0
$851$	9.19711	0.315273
$852$	0	0
$853$	−53.2223	−1.82230	−0.911148	−	0.412080i	$-0.864802\pi$
$853$	−53.2223	−1.82230	−0.911148	+	0.412080i	$0.864802\pi$
$854$	0	0
$855$	−29.2446	−1.00014
$856$	0	0
$857$	31.2259	1.06666	0.533329	−	0.845908i	$-0.320941\pi$
$857$	31.2259	1.06666	0.533329	+	0.845908i	$0.320941\pi$
$858$	0	0
$859$	−41.5663	−1.41823	−0.709113	−	0.705095i	$-0.750904\pi$
$859$	−41.5663	−1.41823	−0.709113	+	0.705095i	$0.750904\pi$
$860$	0	0
$861$	−25.5934	−0.872219
$862$	0	0
$863$	28.0445	0.954645	0.477323	−	0.878728i	$-0.341607\pi$
$863$	28.0445	0.954645	0.477323	+	0.878728i	$0.341607\pi$
$864$	0	0
$865$	15.7249	0.534662
$866$	0	0
$867$	−44.7328	−1.51921
$868$	0	0
$869$	−12.1669	−0.412734
$870$	0	0
$871$	−4.41870	−0.149722
$872$	0	0
$873$	29.2991	0.991623
$874$	0	0
$875$	12.1194	0.409709
$876$	0	0
$877$	−1.13200	−0.0382248	−0.0191124	−	0.999817i	$-0.506084\pi$
$877$	−1.13200	−0.0382248	−0.0191124	+	0.999817i	$0.506084\pi$
$878$	0	0
$879$	−36.1359	−1.21883
$880$	0	0
$881$	−24.2457	−0.816858	−0.408429	−	0.912790i	$-0.633923\pi$
$881$	−24.2457	−0.816858	−0.408429	+	0.912790i	$0.633923\pi$
$882$	0	0
$883$	−33.3262	−1.12152	−0.560758	−	0.827980i	$-0.689490\pi$
$883$	−33.3262	−1.12152	−0.560758	+	0.827980i	$0.689490\pi$
$884$	0	0
$885$	−4.58426	−0.154098
$886$	0	0
$887$	−18.2400	−0.612440	−0.306220	−	0.951961i	$-0.599064\pi$
$887$	−18.2400	−0.612440	−0.306220	+	0.951961i	$0.599064\pi$
$888$	0	0
$889$	10.4612	0.350858
$890$	0	0
$891$	9.25684	0.310116
$892$	0	0
$893$	27.7450	0.928450
$894$	0	0
$895$	−20.6835	−0.691375
$896$	0	0
$897$	4.05561	0.135413
$898$	0	0
$899$	−21.7388	−0.725029
$900$	0	0
$901$	−6.92789	−0.230801
$902$	0	0
$903$	2.52749	0.0841098
$904$	0	0
$905$	−36.4781	−1.21257
$906$	0	0
$907$	23.3918	0.776711	0.388355	−	0.921510i	$-0.373043\pi$
$907$	23.3918	0.776711	0.388355	+	0.921510i	$0.373043\pi$
$908$	0	0
$909$	99.5017	3.30026
$910$	0	0
$911$	−50.7503	−1.68143	−0.840717	−	0.541475i	$-0.817866\pi$
$911$	−50.7503	−1.68143	−0.840717	+	0.541475i	$0.817866\pi$
$912$	0	0
$913$	0.0518681	0.00171658
$914$	0	0
$915$	−0.428411	−0.0141628
$916$	0	0
$917$	17.8166	0.588356
$918$	0	0
$919$	−14.8112	−0.488577	−0.244288	−	0.969703i	$-0.578554\pi$
$919$	−14.8112	−0.488577	−0.244288	+	0.969703i	$0.578554\pi$
$920$	0	0
$921$	−94.3872	−3.11016
$922$	0	0
$923$	−3.15343	−0.103796
$924$	0	0
$925$	−8.88394	−0.292102
$926$	0	0
$927$	25.9870	0.853524
$928$	0	0
$929$	−0.644648	−0.0211502	−0.0105751	−	0.999944i	$-0.503366\pi$
$929$	−0.644648	−0.0211502	−0.0105751	+	0.999944i	$0.503366\pi$
$930$	0	0
$931$	2.52325	0.0826963
$932$	0	0
$933$	−3.18092	−0.104139
$934$	0	0
$935$	2.79437	0.0913856
$936$	0	0
$937$	21.5767	0.704878	0.352439	−	0.935835i	$-0.385352\pi$
$937$	21.5767	0.704878	0.352439	+	0.935835i	$0.385352\pi$
$938$	0	0
$939$	−56.8289	−1.85454
$940$	0	0
$941$	−43.0185	−1.40236	−0.701182	−	0.712982i	$-0.747344\pi$
$941$	−43.0185	−1.40236	−0.701182	+	0.712982i	$0.747344\pi$
$942$	0	0
$943$	−11.4966	−0.374381
$944$	0	0
$945$	−17.4947	−0.569102
$946$	0	0
$947$	39.6657	1.28896	0.644481	−	0.764620i	$-0.277073\pi$
$947$	39.6657	1.28896	0.644481	+	0.764620i	$0.277073\pi$
$948$	0	0
$949$	8.22895	0.267123
$950$	0	0
$951$	−38.1048	−1.23563
$952$	0	0
$953$	−24.8238	−0.804120	−0.402060	−	0.915613i	$-0.631706\pi$
$953$	−24.8238	−0.804120	−0.402060	+	0.915613i	$0.631706\pi$
$954$	0	0
$955$	−16.1202	−0.521637
$956$	0	0
$957$	24.6448	0.796652
$958$	0	0
$959$	−4.16566	−0.134516
$960$	0	0
$961$	−23.9752	−0.773393
$962$	0	0
$963$	51.9144	1.67292
$964$	0	0
$965$	31.7972	1.02359
$966$	0	0
$967$	−7.33577	−0.235902	−0.117951	−	0.993019i	$-0.537633\pi$
$967$	−7.33577	−0.235902	−0.117951	+	0.993019i	$0.537633\pi$
$968$	0	0
$969$	−11.0198	−0.354006
$970$	0	0
$971$	−6.72574	−0.215839	−0.107920	−	0.994160i	$-0.534419\pi$
$971$	−6.72574	−0.215839	−0.107920	+	0.994160i	$0.534419\pi$
$972$	0	0
$973$	7.00111	0.224445
$974$	0	0
$975$	−3.91751	−0.125461
$976$	0	0
$977$	19.1462	0.612542	0.306271	−	0.951944i	$-0.400919\pi$
$977$	19.1462	0.612542	0.306271	+	0.951944i	$0.400919\pi$
$978$	0	0
$979$	−13.6860	−0.437406
$980$	0	0
$981$	−60.2013	−1.92208
$982$	0	0
$983$	49.6162	1.58251	0.791256	−	0.611486i	$-0.209428\pi$
$983$	49.6162	1.58251	0.791256	+	0.611486i	$0.209428\pi$
$984$	0	0
$985$	4.13827	0.131856
$986$	0	0
$987$	33.0392	1.05165
$988$	0	0
$989$	1.13536	0.0361023
$990$	0	0
$991$	−6.17713	−0.196223	−0.0981115	−	0.995175i	$-0.531280\pi$
$991$	−6.17713	−0.196223	−0.0981115	+	0.995175i	$0.531280\pi$
$992$	0	0
$993$	−55.2615	−1.75367
$994$	0	0
$995$	−7.17945	−0.227604
$996$	0	0
$997$	52.4898	1.66237	0.831185	−	0.555996i	$-0.187663\pi$
$997$	52.4898	1.66237	0.831185	+	0.555996i	$0.187663\pi$
$998$	0	0
$999$	62.0053	1.96176

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.x.1.12	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.x.1.12	✓	12	1.1	even	1	trivial

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00474 q^{3} -1.92256 q^{5} +1.00000 q^{7} +6.02845 q^{9} +O(q^{10})\)	\(q+3.00474 q^{3} -1.92256 q^{5} +1.00000 q^{7} +6.02845 q^{9} +1.00000 q^{11} +1.00000 q^{13} -5.77678 q^{15} -1.45347 q^{17} +2.52325 q^{19} +3.00474 q^{21} +1.34974 q^{23} -1.30378 q^{25} +9.09969 q^{27} +8.20197 q^{29} -2.65044 q^{31} +3.00474 q^{33} -1.92256 q^{35} +6.81400 q^{37} +3.00474 q^{39} -8.51767 q^{41} +0.841170 q^{43} -11.5900 q^{45} +10.9957 q^{47} +1.00000 q^{49} -4.36728 q^{51} +4.76646 q^{53} -1.92256 q^{55} +7.58172 q^{57} +0.793568 q^{59} +0.0741609 q^{61} +6.02845 q^{63} -1.92256 q^{65} -4.41870 q^{67} +4.05561 q^{69} -3.15343 q^{71} +8.22895 q^{73} -3.91751 q^{75} +1.00000 q^{77} -12.1669 q^{79} +9.25684 q^{81} +0.0518681 q^{83} +2.79437 q^{85} +24.6448 q^{87} -13.6860 q^{89} +1.00000 q^{91} -7.96386 q^{93} -4.85110 q^{95} +4.86013 q^{97} +6.02845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) \) 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) \) 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9 + 12 * q^11 + 12 * q^13 - 3 * q^15 + 16 * q^17 - 2 * q^19 + 4 * q^21 + 9 * q^23 + 14 * q^25 + 7 * q^27 + 15 * q^29 + 10 * q^31 + 4 * q^33 + 6 * q^35 + 18 * q^37 + 4 * q^39 + 24 * q^41 + 15 * q^45 + 5 * q^47 + 12 * q^49 + 4 * q^51 + 15 * q^53 + 6 * q^55 - 4 * q^57 + 15 * q^59 + 17 * q^61 + 14 * q^63 + 6 * q^65 - 7 * q^67 + 9 * q^71 + 32 * q^73 - 8 * q^75 + 12 * q^77 + 20 * q^79 - 4 * q^81 - 5 * q^83 + 25 * q^85 + 19 * q^87 + 16 * q^89 + 12 * q^91 + 21 * q^93 + 8 * q^95 + 10 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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