LMFDB - Embedded newform 8008.2.a.v.1.1

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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 19 x^{9} + 33 x^{8} + 120 x^{7} - 178 x^{6} - 296 x^{5} + 380 x^{4} + 280 x^{3} + \cdots + 64 $ x^11 - 2x^10 - 19x^9 + 33x^8 + 120x^7 - 178x^6 - 296x^5 + 380x^4 + 280x^3 - 295x^2 - 87x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.33407$ 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)

$f(q)$	$=$	$q-3.33407 q^{3} -0.131360 q^{5} -1.00000 q^{7} +8.11603 q^{9} +O(q^{10})$	$q-3.33407 q^{3} -0.131360 q^{5} -1.00000 q^{7} +8.11603 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.437964 q^{15} +4.35509 q^{17} +0.765047 q^{19} +3.33407 q^{21} -2.35398 q^{23} -4.98274 q^{25} -17.0572 q^{27} +1.55007 q^{29} +3.44072 q^{31} -3.33407 q^{33} +0.131360 q^{35} -6.91967 q^{37} +3.33407 q^{39} +7.35180 q^{41} -9.25933 q^{43} -1.06612 q^{45} -2.35398 q^{47} +1.00000 q^{49} -14.5202 q^{51} +6.98712 q^{53} -0.131360 q^{55} -2.55072 q^{57} -0.151719 q^{59} -7.77746 q^{61} -8.11603 q^{63} +0.131360 q^{65} -7.77938 q^{67} +7.84834 q^{69} -0.493113 q^{71} -3.05167 q^{73} +16.6128 q^{75} -1.00000 q^{77} +1.38849 q^{79} +32.5219 q^{81} +4.49597 q^{83} -0.572085 q^{85} -5.16804 q^{87} +6.66657 q^{89} +1.00000 q^{91} -11.4716 q^{93} -0.100497 q^{95} +18.6373 q^{97} +8.11603 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) $ 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9	$ 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9} + 11 q^{11} - 11 q^{13} - 7 q^{15} - 4 q^{17} - 16 q^{19} + 2 q^{21} - 3 q^{23} + 11 q^{25} - 11 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} - 2 q^{35} - 8 q^{37} + 2 q^{39} + 4 q^{41} - 30 q^{43} + 13 q^{45} - 3 q^{47} + 11 q^{49} - 14 q^{51} - 5 q^{53} + 2 q^{55} - 22 q^{57} + 11 q^{59} + 15 q^{61} - 9 q^{63} - 2 q^{65} - 41 q^{67} + 12 q^{69} + q^{71} - 8 q^{73} - 24 q^{75} - 11 q^{77} - 26 q^{79} + 19 q^{81} - 31 q^{83} - 27 q^{85} - 25 q^{87} + 2 q^{89} + 11 q^{91} - 37 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) $ 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9 + 11 * q^11 - 11 * q^13 - 7 * q^15 - 4 * q^17 - 16 * q^19 + 2 * q^21 - 3 * q^23 + 11 * q^25 - 11 * q^27 + q^29 + 14 * q^31 - 2 * q^33 - 2 * q^35 - 8 * q^37 + 2 * q^39 + 4 * q^41 - 30 * q^43 + 13 * q^45 - 3 * q^47 + 11 * q^49 - 14 * q^51 - 5 * q^53 + 2 * q^55 - 22 * q^57 + 11 * q^59 + 15 * q^61 - 9 * q^63 - 2 * q^65 - 41 * q^67 + 12 * q^69 + q^71 - 8 * q^73 - 24 * q^75 - 11 * q^77 - 26 * q^79 + 19 * q^81 - 31 * q^83 - 27 * q^85 - 25 * q^87 + 2 * q^89 + 11 * q^91 - 37 * q^93 - 6 * q^97 + 9 * 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
$3$	−3.33407	−1.92493	−0.962463	−	0.271411i	$-0.912510\pi$
$3$	−3.33407	−1.92493	−0.962463	+	0.271411i	$0.912510\pi$
$4$	0	0
$5$	−0.131360	−0.0587461	−0.0293730	−	0.999569i	$-0.509351\pi$
$5$	−0.131360	−0.0587461	−0.0293730	+	0.999569i	$0.509351\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	8.11603	2.70534
$10$	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
$15$	0.437964	0.113082
$16$	0	0
$17$	4.35509	1.05626	0.528132	−	0.849162i	$-0.322893\pi$
$17$	4.35509	1.05626	0.528132	+	0.849162i	$0.322893\pi$
$18$	0	0
$19$	0.765047	0.175514	0.0877569	−	0.996142i	$-0.472030\pi$
$19$	0.765047	0.175514	0.0877569	+	0.996142i	$0.472030\pi$
$20$	0	0
$21$	3.33407	0.727554
$22$	0	0
$23$	−2.35398	−0.490839	−0.245419	−	0.969417i	$-0.578926\pi$
$23$	−2.35398	−0.490839	−0.245419	+	0.969417i	$0.578926\pi$
$24$	0	0
$25$	−4.98274	−0.996549
$26$	0	0
$27$	−17.0572	−3.28266
$28$	0	0
$29$	1.55007	0.287840	0.143920	−	0.989589i	$-0.454029\pi$
$29$	1.55007	0.287840	0.143920	+	0.989589i	$0.454029\pi$
$30$	0	0
$31$	3.44072	0.617971	0.308985	−	0.951067i	$-0.400011\pi$
$31$	3.44072	0.617971	0.308985	+	0.951067i	$0.400011\pi$
$32$	0	0
$33$	−3.33407	−0.580387
$34$	0	0
$35$	0.131360	0.0222039
$36$	0	0
$37$	−6.91967	−1.13759	−0.568793	−	0.822481i	$-0.692589\pi$
$37$	−6.91967	−1.13759	−0.568793	+	0.822481i	$0.692589\pi$
$38$	0	0
$39$	3.33407	0.533879
$40$	0	0
$41$	7.35180	1.14816	0.574079	−	0.818800i	$-0.305360\pi$
$41$	7.35180	1.14816	0.574079	+	0.818800i	$0.305360\pi$
$42$	0	0
$43$	−9.25933	−1.41203	−0.706017	−	0.708195i	$-0.749510\pi$
$43$	−9.25933	−1.41203	−0.706017	+	0.708195i	$0.749510\pi$
$44$	0	0
$45$	−1.06612	−0.158928
$46$	0	0
$47$	−2.35398	−0.343363	−0.171682	−	0.985152i	$-0.554920\pi$
$47$	−2.35398	−0.343363	−0.171682	+	0.985152i	$0.554920\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−14.5202	−2.03323
$52$	0	0
$53$	6.98712	0.959754	0.479877	−	0.877336i	$-0.340681\pi$
$53$	6.98712	0.959754	0.479877	+	0.877336i	$0.340681\pi$
$54$	0	0
$55$	−0.131360	−0.0177126
$56$	0	0
$57$	−2.55072	−0.337851
$58$	0	0
$59$	−0.151719	−0.0197521	−0.00987604	−	0.999951i	$-0.503144\pi$
$59$	−0.151719	−0.0197521	−0.00987604	+	0.999951i	$0.503144\pi$
$60$	0	0
$61$	−7.77746	−0.995802	−0.497901	−	0.867234i	$-0.665896\pi$
$61$	−7.77746	−0.995802	−0.497901	+	0.867234i	$0.665896\pi$
$62$	0	0
$63$	−8.11603	−1.02252
$64$	0	0
$65$	0.131360	0.0162932
$66$	0	0
$67$	−7.77938	−0.950402	−0.475201	−	0.879877i	$-0.657625\pi$
$67$	−7.77938	−0.950402	−0.475201	+	0.879877i	$0.657625\pi$
$68$	0	0
$69$	7.84834	0.944829
$70$	0	0
$71$	−0.493113	−0.0585217	−0.0292609	−	0.999572i	$-0.509315\pi$
$71$	−0.493113	−0.0585217	−0.0292609	+	0.999572i	$0.509315\pi$
$72$	0	0
$73$	−3.05167	−0.357170	−0.178585	−	0.983924i	$-0.557152\pi$
$73$	−3.05167	−0.357170	−0.178585	+	0.983924i	$0.557152\pi$
$74$	0	0
$75$	16.6128	1.91828
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	1.38849	0.156217	0.0781087	−	0.996945i	$-0.475112\pi$
$79$	1.38849	0.156217	0.0781087	+	0.996945i	$0.475112\pi$
$80$	0	0
$81$	32.5219	3.61354
$82$	0	0
$83$	4.49597	0.493497	0.246748	−	0.969080i	$-0.420638\pi$
$83$	4.49597	0.493497	0.246748	+	0.969080i	$0.420638\pi$
$84$	0	0
$85$	−0.572085	−0.0620513
$86$	0	0
$87$	−5.16804	−0.554072
$88$	0	0
$89$	6.66657	0.706655	0.353327	−	0.935500i	$-0.385050\pi$
$89$	6.66657	0.706655	0.353327	+	0.935500i	$0.385050\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−11.4716	−1.18955
$94$	0	0
$95$	−0.100497	−0.0103107
$96$	0	0
$97$	18.6373	1.89233	0.946166	−	0.323681i	$-0.104921\pi$
$97$	18.6373	1.89233	0.946166	+	0.323681i	$0.104921\pi$
$98$	0	0
$99$	8.11603	0.815692
$100$	0	0
$101$	15.4024	1.53259	0.766297	−	0.642486i	$-0.222097\pi$
$101$	15.4024	1.53259	0.766297	+	0.642486i	$0.222097\pi$
$102$	0	0
$103$	−3.82506	−0.376895	−0.188447	−	0.982083i	$-0.560345\pi$
$103$	−3.82506	−0.376895	−0.188447	+	0.982083i	$0.560345\pi$
$104$	0	0
$105$	−0.437964	−0.0427409
$106$	0	0
$107$	5.58414	0.539839	0.269920	−	0.962883i	$-0.413003\pi$
$107$	5.58414	0.539839	0.269920	+	0.962883i	$0.413003\pi$
$108$	0	0
$109$	−2.66796	−0.255544	−0.127772	−	0.991804i	$-0.540783\pi$
$109$	−2.66796	−0.255544	−0.127772	+	0.991804i	$0.540783\pi$
$110$	0	0
$111$	23.0707	2.18977
$112$	0	0
$113$	−14.9938	−1.41050	−0.705251	−	0.708958i	$-0.749166\pi$
$113$	−14.9938	−1.41050	−0.705251	+	0.708958i	$0.749166\pi$
$114$	0	0
$115$	0.309219	0.0288348
$116$	0	0
$117$	−8.11603	−0.750327
$118$	0	0
$119$	−4.35509	−0.399230
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−24.5114	−2.21012
$124$	0	0
$125$	1.31134	0.117289
$126$	0	0
$127$	2.20075	0.195285	0.0976424	−	0.995222i	$-0.468870\pi$
$127$	2.20075	0.195285	0.0976424	+	0.995222i	$0.468870\pi$
$128$	0	0
$129$	30.8713	2.71806
$130$	0	0
$131$	−1.36405	−0.119178	−0.0595888	−	0.998223i	$-0.518979\pi$
$131$	−1.36405	−0.119178	−0.0595888	+	0.998223i	$0.518979\pi$
$132$	0	0
$133$	−0.765047	−0.0663380
$134$	0	0
$135$	2.24064	0.192843
$136$	0	0
$137$	−6.02782	−0.514991	−0.257496	−	0.966279i	$-0.582897\pi$
$137$	−6.02782	−0.514991	−0.257496	+	0.966279i	$0.582897\pi$
$138$	0	0
$139$	−10.0216	−0.850019	−0.425009	−	0.905189i	$-0.639729\pi$
$139$	−10.0216	−0.850019	−0.425009	+	0.905189i	$0.639729\pi$
$140$	0	0
$141$	7.84834	0.660949
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−0.203617	−0.0169095
$146$	0	0
$147$	−3.33407	−0.274990
$148$	0	0
$149$	19.0745	1.56264	0.781322	−	0.624128i	$-0.214546\pi$
$149$	19.0745	1.56264	0.781322	+	0.624128i	$0.214546\pi$
$150$	0	0
$151$	−11.7413	−0.955498	−0.477749	−	0.878497i	$-0.658547\pi$
$151$	−11.7413	−0.955498	−0.477749	+	0.878497i	$0.658547\pi$
$152$	0	0
$153$	35.3460	2.85756
$154$	0	0
$155$	−0.451973	−0.0363034
$156$	0	0
$157$	−13.6860	−1.09226	−0.546129	−	0.837701i	$-0.683899\pi$
$157$	−13.6860	−1.09226	−0.546129	+	0.837701i	$0.683899\pi$
$158$	0	0
$159$	−23.2955	−1.84746
$160$	0	0
$161$	2.35398	0.185520
$162$	0	0
$163$	7.12800	0.558308	0.279154	−	0.960246i	$-0.409946\pi$
$163$	7.12800	0.558308	0.279154	+	0.960246i	$0.409946\pi$
$164$	0	0
$165$	0.437964	0.0340955
$166$	0	0
$167$	19.7445	1.52787	0.763937	−	0.645291i	$-0.223264\pi$
$167$	19.7445	1.52787	0.763937	+	0.645291i	$0.223264\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.20915	0.474825
$172$	0	0
$173$	6.64607	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$173$	6.64607	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$174$	0	0
$175$	4.98274	0.376660
$176$	0	0
$177$	0.505841	0.0380213
$178$	0	0
$179$	−14.7828	−1.10492	−0.552460	−	0.833539i	$-0.686311\pi$
$179$	−14.7828	−1.10492	−0.552460	+	0.833539i	$0.686311\pi$
$180$	0	0
$181$	9.96875	0.740972	0.370486	−	0.928838i	$-0.379191\pi$
$181$	9.96875	0.740972	0.370486	+	0.928838i	$0.379191\pi$
$182$	0	0
$183$	25.9306	1.91685
$184$	0	0
$185$	0.908968	0.0668287
$186$	0	0
$187$	4.35509	0.318476
$188$	0	0
$189$	17.0572	1.24073
$190$	0	0
$191$	−21.6792	−1.56865	−0.784324	−	0.620351i	$-0.786990\pi$
$191$	−21.6792	−1.56865	−0.784324	+	0.620351i	$0.786990\pi$
$192$	0	0
$193$	7.76049	0.558612	0.279306	−	0.960202i	$-0.409896\pi$
$193$	7.76049	0.558612	0.279306	+	0.960202i	$0.409896\pi$
$194$	0	0
$195$	−0.437964	−0.0313633
$196$	0	0
$197$	−10.4152	−0.742052	−0.371026	−	0.928622i	$-0.620994\pi$
$197$	−10.4152	−0.742052	−0.371026	+	0.928622i	$0.620994\pi$
$198$	0	0
$199$	10.6780	0.756940	0.378470	−	0.925614i	$-0.376450\pi$
$199$	10.6780	0.756940	0.378470	+	0.925614i	$0.376450\pi$
$200$	0	0
$201$	25.9370	1.82946
$202$	0	0
$203$	−1.55007	−0.108793
$204$	0	0
$205$	−0.965734	−0.0674498
$206$	0	0
$207$	−19.1050	−1.32789
$208$	0	0
$209$	0.765047	0.0529194
$210$	0	0
$211$	16.6665	1.14737	0.573683	−	0.819077i	$-0.305514\pi$
$211$	16.6665	1.14737	0.573683	+	0.819077i	$0.305514\pi$
$212$	0	0
$213$	1.64407	0.112650
$214$	0	0
$215$	1.21631	0.0829514
$216$	0	0
$217$	−3.44072	−0.233571
$218$	0	0
$219$	10.1745	0.687527
$220$	0	0
$221$	−4.35509	−0.292955
$222$	0	0
$223$	17.4843	1.17084	0.585418	−	0.810732i	$-0.300930\pi$
$223$	17.4843	1.17084	0.585418	+	0.810732i	$0.300930\pi$
$224$	0	0
$225$	−40.4401	−2.69601
$226$	0	0
$227$	9.07798	0.602527	0.301263	−	0.953541i	$-0.402592\pi$
$227$	9.07798	0.602527	0.301263	+	0.953541i	$0.402592\pi$
$228$	0	0
$229$	−5.14364	−0.339901	−0.169951	−	0.985453i	$-0.554361\pi$
$229$	−5.14364	−0.339901	−0.169951	+	0.985453i	$0.554361\pi$
$230$	0	0
$231$	3.33407	0.219366
$232$	0	0
$233$	8.89654	0.582832	0.291416	−	0.956596i	$-0.405874\pi$
$233$	8.89654	0.582832	0.291416	+	0.956596i	$0.405874\pi$
$234$	0	0
$235$	0.309219	0.0201712
$236$	0	0
$237$	−4.62933	−0.300707
$238$	0	0
$239$	5.98466	0.387116	0.193558	−	0.981089i	$-0.437997\pi$
$239$	5.98466	0.387116	0.193558	+	0.981089i	$0.437997\pi$
$240$	0	0
$241$	−20.7900	−1.33920	−0.669600	−	0.742722i	$-0.733534\pi$
$241$	−20.7900	−1.33920	−0.669600	+	0.742722i	$0.733534\pi$
$242$	0	0
$243$	−57.2586	−3.67314
$244$	0	0
$245$	−0.131360	−0.00839229
$246$	0	0
$247$	−0.765047	−0.0486788
$248$	0	0
$249$	−14.9899	−0.949945
$250$	0	0
$251$	−7.73079	−0.487963	−0.243982	−	0.969780i	$-0.578454\pi$
$251$	−7.73079	−0.487963	−0.243982	+	0.969780i	$0.578454\pi$
$252$	0	0
$253$	−2.35398	−0.147993
$254$	0	0
$255$	1.90737	0.119444
$256$	0	0
$257$	8.98032	0.560177	0.280089	−	0.959974i	$-0.409636\pi$
$257$	8.98032	0.560177	0.280089	+	0.959974i	$0.409636\pi$
$258$	0	0
$259$	6.91967	0.429967
$260$	0	0
$261$	12.5804	0.778707
$262$	0	0
$263$	−8.88531	−0.547892	−0.273946	−	0.961745i	$-0.588329\pi$
$263$	−8.88531	−0.547892	−0.273946	+	0.961745i	$0.588329\pi$
$264$	0	0
$265$	−0.917829	−0.0563818
$266$	0	0
$267$	−22.2268	−1.36026
$268$	0	0
$269$	−1.70873	−0.104183	−0.0520915	−	0.998642i	$-0.516589\pi$
$269$	−1.70873	−0.104183	−0.0520915	+	0.998642i	$0.516589\pi$
$270$	0	0
$271$	14.3865	0.873920	0.436960	−	0.899481i	$-0.356055\pi$
$271$	14.3865	0.873920	0.436960	+	0.899481i	$0.356055\pi$
$272$	0	0
$273$	−3.33407	−0.201787
$274$	0	0
$275$	−4.98274	−0.300471
$276$	0	0
$277$	−7.21972	−0.433791	−0.216895	−	0.976195i	$-0.569593\pi$
$277$	−7.21972	−0.433791	−0.216895	+	0.976195i	$0.569593\pi$
$278$	0	0
$279$	27.9250	1.67182
$280$	0	0
$281$	23.1518	1.38112	0.690560	−	0.723275i	$-0.257364\pi$
$281$	23.1518	1.38112	0.690560	+	0.723275i	$0.257364\pi$
$282$	0	0
$283$	−27.4350	−1.63084	−0.815421	−	0.578868i	$-0.803494\pi$
$283$	−27.4350	−1.63084	−0.815421	+	0.578868i	$0.803494\pi$
$284$	0	0
$285$	0.335063	0.0198474
$286$	0	0
$287$	−7.35180	−0.433963
$288$	0	0
$289$	1.96678	0.115693
$290$	0	0
$291$	−62.1381	−3.64260
$292$	0	0
$293$	−15.5213	−0.906765	−0.453382	−	0.891316i	$-0.649783\pi$
$293$	−15.5213	−0.906765	−0.453382	+	0.891316i	$0.649783\pi$
$294$	0	0
$295$	0.0199298	0.00116036
$296$	0	0
$297$	−17.0572	−0.989760
$298$	0	0
$299$	2.35398	0.136134
$300$	0	0
$301$	9.25933	0.533699
$302$	0	0
$303$	−51.3527	−2.95013
$304$	0	0
$305$	1.02165	0.0584995
$306$	0	0
$307$	−19.8814	−1.13469	−0.567346	−	0.823480i	$-0.692030\pi$
$307$	−19.8814	−1.13469	−0.567346	+	0.823480i	$0.692030\pi$
$308$	0	0
$309$	12.7530	0.725495
$310$	0	0
$311$	−16.2048	−0.918890	−0.459445	−	0.888206i	$-0.651952\pi$
$311$	−16.2048	−0.918890	−0.459445	+	0.888206i	$0.651952\pi$
$312$	0	0
$313$	−14.0158	−0.792220	−0.396110	−	0.918203i	$-0.629640\pi$
$313$	−14.0158	−0.792220	−0.396110	+	0.918203i	$0.629640\pi$
$314$	0	0
$315$	1.06612	0.0600692
$316$	0	0
$317$	−12.2844	−0.689959	−0.344979	−	0.938610i	$-0.612114\pi$
$317$	−12.2844	−0.689959	−0.344979	+	0.938610i	$0.612114\pi$
$318$	0	0
$319$	1.55007	0.0867871
$320$	0	0
$321$	−18.6179	−1.03915
$322$	0	0
$323$	3.33185	0.185389
$324$	0	0
$325$	4.98274	0.276393
$326$	0	0
$327$	8.89516	0.491904
$328$	0	0
$329$	2.35398	0.129779
$330$	0	0
$331$	−3.87931	−0.213226	−0.106613	−	0.994301i	$-0.534001\pi$
$331$	−3.87931	−0.213226	−0.106613	+	0.994301i	$0.534001\pi$
$332$	0	0
$333$	−56.1602	−3.07756
$334$	0	0
$335$	1.02190	0.0558324
$336$	0	0
$337$	15.0265	0.818544	0.409272	−	0.912412i	$-0.365783\pi$
$337$	15.0265	0.818544	0.409272	+	0.912412i	$0.365783\pi$
$338$	0	0
$339$	49.9905	2.71511
$340$	0	0
$341$	3.44072	0.186325
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.03096	−0.0555050
$346$	0	0
$347$	−34.2138	−1.83669	−0.918347	−	0.395777i	$-0.870475\pi$
$347$	−34.2138	−1.83669	−0.918347	+	0.395777i	$0.870475\pi$
$348$	0	0
$349$	34.3616	1.83933	0.919667	−	0.392700i	$-0.128459\pi$
$349$	34.3616	1.83933	0.919667	+	0.392700i	$0.128459\pi$
$350$	0	0
$351$	17.0572	0.910447
$352$	0	0
$353$	−34.8717	−1.85604	−0.928018	−	0.372536i	$-0.878488\pi$
$353$	−34.8717	−1.85604	−0.928018	+	0.372536i	$0.878488\pi$
$354$	0	0
$355$	0.0647754	0.00343792
$356$	0	0
$357$	14.5202	0.768489
$358$	0	0
$359$	−10.4610	−0.552108	−0.276054	−	0.961142i	$-0.589027\pi$
$359$	−10.4610	−0.552108	−0.276054	+	0.961142i	$0.589027\pi$
$360$	0	0
$361$	−18.4147	−0.969195
$362$	0	0
$363$	−3.33407	−0.174993
$364$	0	0
$365$	0.400867	0.0209824
$366$	0	0
$367$	30.9235	1.61419	0.807096	−	0.590420i	$-0.201038\pi$
$367$	30.9235	1.61419	0.807096	+	0.590420i	$0.201038\pi$
$368$	0	0
$369$	59.6674	3.10616
$370$	0	0
$371$	−6.98712	−0.362753
$372$	0	0
$373$	−19.5450	−1.01200	−0.506001	−	0.862533i	$-0.668877\pi$
$373$	−19.5450	−1.01200	−0.506001	+	0.862533i	$0.668877\pi$
$374$	0	0
$375$	−4.37208	−0.225773
$376$	0	0
$377$	−1.55007	−0.0798325
$378$	0	0
$379$	−11.8988	−0.611200	−0.305600	−	0.952160i	$-0.598857\pi$
$379$	−11.8988	−0.611200	−0.305600	+	0.952160i	$0.598857\pi$
$380$	0	0
$381$	−7.33745	−0.375909
$382$	0	0
$383$	8.14824	0.416356	0.208178	−	0.978091i	$-0.433247\pi$
$383$	8.14824	0.416356	0.208178	+	0.978091i	$0.433247\pi$
$384$	0	0
$385$	0.131360	0.00669473
$386$	0	0
$387$	−75.1490	−3.82004
$388$	0	0
$389$	39.0264	1.97872	0.989359	−	0.145493i	$-0.0464770\pi$
$389$	39.0264	1.97872	0.989359	+	0.145493i	$0.0464770\pi$
$390$	0	0
$391$	−10.2518	−0.518455
$392$	0	0
$393$	4.54784	0.229408
$394$	0	0
$395$	−0.182392	−0.00917716
$396$	0	0
$397$	−6.18126	−0.310228	−0.155114	−	0.987897i	$-0.549575\pi$
$397$	−6.18126	−0.310228	−0.155114	+	0.987897i	$0.549575\pi$
$398$	0	0
$399$	2.55072	0.127696
$400$	0	0
$401$	−33.0280	−1.64934	−0.824669	−	0.565616i	$-0.808638\pi$
$401$	−33.0280	−1.64934	−0.824669	+	0.565616i	$0.808638\pi$
$402$	0	0
$403$	−3.44072	−0.171394
$404$	0	0
$405$	−4.27208	−0.212281
$406$	0	0
$407$	−6.91967	−0.342995
$408$	0	0
$409$	−35.1053	−1.73585	−0.867923	−	0.496698i	$-0.834545\pi$
$409$	−35.1053	−1.73585	−0.867923	+	0.496698i	$0.834545\pi$
$410$	0	0
$411$	20.0972	0.991321
$412$	0	0
$413$	0.151719	0.00746558
$414$	0	0
$415$	−0.590591	−0.0289910
$416$	0	0
$417$	33.4126	1.63622
$418$	0	0
$419$	−3.69266	−0.180398	−0.0901991	−	0.995924i	$-0.528750\pi$
$419$	−3.69266	−0.180398	−0.0901991	+	0.995924i	$0.528750\pi$
$420$	0	0
$421$	−21.5864	−1.05206	−0.526029	−	0.850467i	$-0.676320\pi$
$421$	−21.5864	−1.05206	−0.526029	+	0.850467i	$0.676320\pi$
$422$	0	0
$423$	−19.1050	−0.928916
$424$	0	0
$425$	−21.7003	−1.05262
$426$	0	0
$427$	7.77746	0.376378
$428$	0	0
$429$	3.33407	0.160970
$430$	0	0
$431$	25.0783	1.20798	0.603990	−	0.796992i	$-0.293577\pi$
$431$	25.0783	1.20798	0.603990	+	0.796992i	$0.293577\pi$
$432$	0	0
$433$	4.64244	0.223102	0.111551	−	0.993759i	$-0.464418\pi$
$433$	4.64244	0.223102	0.111551	+	0.993759i	$0.464418\pi$
$434$	0	0
$435$	0.678874	0.0325495
$436$	0	0
$437$	−1.80091	−0.0861490
$438$	0	0
$439$	20.1516	0.961785	0.480893	−	0.876780i	$-0.340313\pi$
$439$	20.1516	0.961785	0.480893	+	0.876780i	$0.340313\pi$
$440$	0	0
$441$	8.11603	0.386478
$442$	0	0
$443$	16.0015	0.760256	0.380128	−	0.924934i	$-0.375880\pi$
$443$	16.0015	0.760256	0.380128	+	0.924934i	$0.375880\pi$
$444$	0	0
$445$	−0.875721	−0.0415132
$446$	0	0
$447$	−63.5958	−3.00798
$448$	0	0
$449$	−19.5565	−0.922929	−0.461465	−	0.887159i	$-0.652676\pi$
$449$	−19.5565	−0.922929	−0.461465	+	0.887159i	$0.652676\pi$
$450$	0	0
$451$	7.35180	0.346183
$452$	0	0
$453$	39.1465	1.83926
$454$	0	0
$455$	−0.131360	−0.00615826
$456$	0	0
$457$	−25.5393	−1.19468	−0.597338	−	0.801989i	$-0.703775\pi$
$457$	−25.5393	−1.19468	−0.597338	+	0.801989i	$0.703775\pi$
$458$	0	0
$459$	−74.2857	−3.46736
$460$	0	0
$461$	1.76652	0.0822750	0.0411375	−	0.999153i	$-0.486902\pi$
$461$	1.76652	0.0822750	0.0411375	+	0.999153i	$0.486902\pi$
$462$	0	0
$463$	26.4558	1.22951	0.614753	−	0.788720i	$-0.289256\pi$
$463$	26.4558	1.22951	0.614753	+	0.788720i	$0.289256\pi$
$464$	0	0
$465$	1.50691	0.0698813
$466$	0	0
$467$	7.49460	0.346809	0.173404	−	0.984851i	$-0.444523\pi$
$467$	7.49460	0.346809	0.173404	+	0.984851i	$0.444523\pi$
$468$	0	0
$469$	7.77938	0.359218
$470$	0	0
$471$	45.6299	2.10252
$472$	0	0
$473$	−9.25933	−0.425744
$474$	0	0
$475$	−3.81203	−0.174908
$476$	0	0
$477$	56.7076	2.59646
$478$	0	0
$479$	−38.1474	−1.74300	−0.871500	−	0.490395i	$-0.836852\pi$
$479$	−38.1474	−1.74300	−0.871500	+	0.490395i	$0.836852\pi$
$480$	0	0
$481$	6.91967	0.315510
$482$	0	0
$483$	−7.84834	−0.357112
$484$	0	0
$485$	−2.44820	−0.111167
$486$	0	0
$487$	−4.23729	−0.192010	−0.0960050	−	0.995381i	$-0.530606\pi$
$487$	−4.23729	−0.192010	−0.0960050	+	0.995381i	$0.530606\pi$
$488$	0	0
$489$	−23.7653	−1.07470
$490$	0	0
$491$	4.21926	0.190413	0.0952063	−	0.995458i	$-0.469649\pi$
$491$	4.21926	0.190413	0.0952063	+	0.995458i	$0.469649\pi$
$492$	0	0
$493$	6.75068	0.304035
$494$	0	0
$495$	−1.06612	−0.0479187
$496$	0	0
$497$	0.493113	0.0221191
$498$	0	0
$499$	−12.5672	−0.562584	−0.281292	−	0.959622i	$-0.590763\pi$
$499$	−12.5672	−0.562584	−0.281292	+	0.959622i	$0.590763\pi$
$500$	0	0
$501$	−65.8295	−2.94105
$502$	0	0
$503$	0.278377	0.0124122	0.00620610	−	0.999981i	$-0.498025\pi$
$503$	0.278377	0.0124122	0.00620610	+	0.999981i	$0.498025\pi$
$504$	0	0
$505$	−2.02326	−0.0900339
$506$	0	0
$507$	−3.33407	−0.148071
$508$	0	0
$509$	5.55657	0.246291	0.123145	−	0.992389i	$-0.460702\pi$
$509$	5.55657	0.246291	0.123145	+	0.992389i	$0.460702\pi$
$510$	0	0
$511$	3.05167	0.134998
$512$	0	0
$513$	−13.0496	−0.576153
$514$	0	0
$515$	0.502461	0.0221411
$516$	0	0
$517$	−2.35398	−0.103528
$518$	0	0
$519$	−22.1585	−0.972649
$520$	0	0
$521$	−39.3183	−1.72257	−0.861284	−	0.508124i	$-0.830339\pi$
$521$	−39.3183	−1.72257	−0.861284	+	0.508124i	$0.830339\pi$
$522$	0	0
$523$	−15.7210	−0.687433	−0.343717	−	0.939073i	$-0.611686\pi$
$523$	−15.7210	−0.687433	−0.343717	+	0.939073i	$0.611686\pi$
$524$	0	0
$525$	−16.6128	−0.725043
$526$	0	0
$527$	14.9846	0.652740
$528$	0	0
$529$	−17.4588	−0.759077
$530$	0	0
$531$	−1.23135	−0.0534362
$532$	0	0
$533$	−7.35180	−0.318442
$534$	0	0
$535$	−0.733534	−0.0317134
$536$	0	0
$537$	49.2870	2.12689
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−2.41251	−0.103722	−0.0518610	−	0.998654i	$-0.516515\pi$
$541$	−2.41251	−0.103722	−0.0518610	+	0.998654i	$0.516515\pi$
$542$	0	0
$543$	−33.2365	−1.42632
$544$	0	0
$545$	0.350463	0.0150122
$546$	0	0
$547$	−6.56886	−0.280864	−0.140432	−	0.990090i	$-0.544849\pi$
$547$	−6.56886	−0.280864	−0.140432	+	0.990090i	$0.544849\pi$
$548$	0	0
$549$	−63.1221	−2.69399
$550$	0	0
$551$	1.18587	0.0505200
$552$	0	0
$553$	−1.38849	−0.0590447
$554$	0	0
$555$	−3.03057	−0.128640
$556$	0	0
$557$	−4.40486	−0.186640	−0.0933200	−	0.995636i	$-0.529748\pi$
$557$	−4.40486	−0.186640	−0.0933200	+	0.995636i	$0.529748\pi$
$558$	0	0
$559$	9.25933	0.391628
$560$	0	0
$561$	−14.5202	−0.613042
$562$	0	0
$563$	−22.7506	−0.958825	−0.479412	−	0.877590i	$-0.659150\pi$
$563$	−22.7506	−0.958825	−0.479412	+	0.877590i	$0.659150\pi$
$564$	0	0
$565$	1.96959	0.0828614
$566$	0	0
$567$	−32.5219	−1.36579
$568$	0	0
$569$	−1.11234	−0.0466319	−0.0233159	−	0.999728i	$-0.507422\pi$
$569$	−1.11234	−0.0466319	−0.0233159	+	0.999728i	$0.507422\pi$
$570$	0	0
$571$	−24.2442	−1.01459	−0.507294	−	0.861773i	$-0.669354\pi$
$571$	−24.2442	−1.01459	−0.507294	+	0.861773i	$0.669354\pi$
$572$	0	0
$573$	72.2799	3.01953
$574$	0	0
$575$	11.7293	0.489145
$576$	0	0
$577$	39.2412	1.63363	0.816815	−	0.576899i	$-0.195737\pi$
$577$	39.2412	1.63363	0.816815	+	0.576899i	$0.195737\pi$
$578$	0	0
$579$	−25.8740	−1.07529
$580$	0	0
$581$	−4.49597	−0.186524
$582$	0	0
$583$	6.98712	0.289377
$584$	0	0
$585$	1.06612	0.0440788
$586$	0	0
$587$	−23.3803	−0.965007	−0.482504	−	0.875894i	$-0.660273\pi$
$587$	−23.3803	−0.965007	−0.482504	+	0.875894i	$0.660273\pi$
$588$	0	0
$589$	2.63231	0.108462
$590$	0	0
$591$	34.7250	1.42840
$592$	0	0
$593$	7.97296	0.327410	0.163705	−	0.986509i	$-0.447655\pi$
$593$	7.97296	0.327410	0.163705	+	0.986509i	$0.447655\pi$
$594$	0	0
$595$	0.572085	0.0234532
$596$	0	0
$597$	−35.6011	−1.45705
$598$	0	0
$599$	−19.8961	−0.812932	−0.406466	−	0.913666i	$-0.633239\pi$
$599$	−19.8961	−0.812932	−0.406466	+	0.913666i	$0.633239\pi$
$600$	0	0
$601$	18.5273	0.755743	0.377871	−	0.925858i	$-0.376656\pi$
$601$	18.5273	0.755743	0.377871	+	0.925858i	$0.376656\pi$
$602$	0	0
$603$	−63.1377	−2.57117
$604$	0	0
$605$	−0.131360	−0.00534055
$606$	0	0
$607$	−1.20417	−0.0488756	−0.0244378	−	0.999701i	$-0.507780\pi$
$607$	−1.20417	−0.0488756	−0.0244378	+	0.999701i	$0.507780\pi$
$608$	0	0
$609$	5.16804	0.209419
$610$	0	0
$611$	2.35398	0.0952318
$612$	0	0
$613$	−37.4757	−1.51363	−0.756814	−	0.653630i	$-0.773245\pi$
$613$	−37.4757	−1.51363	−0.756814	+	0.653630i	$0.773245\pi$
$614$	0	0
$615$	3.21983	0.129836
$616$	0	0
$617$	−7.47398	−0.300891	−0.150446	−	0.988618i	$-0.548071\pi$
$617$	−7.47398	−0.300891	−0.150446	+	0.988618i	$0.548071\pi$
$618$	0	0
$619$	21.3392	0.857696	0.428848	−	0.903377i	$-0.358920\pi$
$619$	21.3392	0.857696	0.428848	+	0.903377i	$0.358920\pi$
$620$	0	0
$621$	40.1523	1.61126
$622$	0	0
$623$	−6.66657	−0.267090
$624$	0	0
$625$	24.7415	0.989659
$626$	0	0
$627$	−2.55072	−0.101866
$628$	0	0
$629$	−30.1357	−1.20159
$630$	0	0
$631$	−26.5065	−1.05521	−0.527604	−	0.849490i	$-0.676910\pi$
$631$	−26.5065	−1.05521	−0.527604	+	0.849490i	$0.676910\pi$
$632$	0	0
$633$	−55.5671	−2.20860
$634$	0	0
$635$	−0.289090	−0.0114722
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−4.00212	−0.158321
$640$	0	0
$641$	−22.0146	−0.869526	−0.434763	−	0.900545i	$-0.643168\pi$
$641$	−22.0146	−0.869526	−0.434763	+	0.900545i	$0.643168\pi$
$642$	0	0
$643$	−40.6392	−1.60266	−0.801328	−	0.598226i	$-0.795873\pi$
$643$	−40.6392	−1.60266	−0.801328	+	0.598226i	$0.795873\pi$
$644$	0	0
$645$	−4.05525	−0.159675
$646$	0	0
$647$	46.9827	1.84708	0.923540	−	0.383503i	$-0.125282\pi$
$647$	46.9827	1.84708	0.923540	+	0.383503i	$0.125282\pi$
$648$	0	0
$649$	−0.151719	−0.00595548
$650$	0	0
$651$	11.4716	0.449607
$652$	0	0
$653$	−9.69116	−0.379245	−0.189622	−	0.981857i	$-0.560726\pi$
$653$	−9.69116	−0.379245	−0.189622	+	0.981857i	$0.560726\pi$
$654$	0	0
$655$	0.179182	0.00700121
$656$	0	0
$657$	−24.7674	−0.966269
$658$	0	0
$659$	27.5066	1.07151	0.535753	−	0.844375i	$-0.320028\pi$
$659$	27.5066	1.07151	0.535753	+	0.844375i	$0.320028\pi$
$660$	0	0
$661$	−21.0568	−0.819016	−0.409508	−	0.912307i	$-0.634300\pi$
$661$	−21.0568	−0.819016	−0.409508	+	0.912307i	$0.634300\pi$
$662$	0	0
$663$	14.5202	0.563917
$664$	0	0
$665$	0.100497	0.00389710
$666$	0	0
$667$	−3.64883	−0.141283
$668$	0	0
$669$	−58.2939	−2.25377
$670$	0	0
$671$	−7.77746	−0.300246
$672$	0	0
$673$	23.8236	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$673$	23.8236	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$674$	0	0
$675$	84.9917	3.27133
$676$	0	0
$677$	−38.2567	−1.47033	−0.735163	−	0.677891i	$-0.762894\pi$
$677$	−38.2567	−1.47033	−0.735163	+	0.677891i	$0.762894\pi$
$678$	0	0
$679$	−18.6373	−0.715234
$680$	0	0
$681$	−30.2666	−1.15982
$682$	0	0
$683$	48.5941	1.85940	0.929701	−	0.368315i	$-0.120065\pi$
$683$	48.5941	1.85940	0.929701	+	0.368315i	$0.120065\pi$
$684$	0	0
$685$	0.791815	0.0302537
$686$	0	0
$687$	17.1493	0.654285
$688$	0	0
$689$	−6.98712	−0.266188
$690$	0	0
$691$	−4.14380	−0.157638	−0.0788188	−	0.996889i	$-0.525115\pi$
$691$	−4.14380	−0.157638	−0.0788188	+	0.996889i	$0.525115\pi$
$692$	0	0
$693$	−8.11603	−0.308303
$694$	0	0
$695$	1.31644	0.0499352
$696$	0	0
$697$	32.0177	1.21276
$698$	0	0
$699$	−29.6617	−1.12191
$700$	0	0
$701$	8.94798	0.337961	0.168980	−	0.985619i	$-0.445953\pi$
$701$	8.94798	0.337961	0.168980	+	0.985619i	$0.445953\pi$
$702$	0	0
$703$	−5.29387	−0.199662
$704$	0	0
$705$	−1.03096	−0.0388282
$706$	0	0
$707$	−15.4024	−0.579266
$708$	0	0
$709$	−35.0954	−1.31803	−0.659017	−	0.752128i	$-0.729028\pi$
$709$	−35.0954	−1.31803	−0.659017	+	0.752128i	$0.729028\pi$
$710$	0	0
$711$	11.2690	0.422622
$712$	0	0
$713$	−8.09938	−0.303324
$714$	0	0
$715$	0.131360	0.00491259
$716$	0	0
$717$	−19.9533	−0.745169
$718$	0	0
$719$	18.8101	0.701500	0.350750	−	0.936469i	$-0.385927\pi$
$719$	18.8101	0.701500	0.350750	+	0.936469i	$0.385927\pi$
$720$	0	0
$721$	3.82506	0.142453
$722$	0	0
$723$	69.3152	2.57786
$724$	0	0
$725$	−7.72359	−0.286847
$726$	0	0
$727$	−30.4039	−1.12762	−0.563809	−	0.825905i	$-0.690664\pi$
$727$	−30.4039	−1.12762	−0.563809	+	0.825905i	$0.690664\pi$
$728$	0	0
$729$	93.3387	3.45699
$730$	0	0
$731$	−40.3252	−1.49148
$732$	0	0
$733$	−22.8265	−0.843114	−0.421557	−	0.906802i	$-0.638516\pi$
$733$	−22.8265	−0.843114	−0.421557	+	0.906802i	$0.638516\pi$
$734$	0	0
$735$	0.437964	0.0161546
$736$	0	0
$737$	−7.77938	−0.286557
$738$	0	0
$739$	10.4878	0.385798	0.192899	−	0.981219i	$-0.438211\pi$
$739$	10.4878	0.385798	0.192899	+	0.981219i	$0.438211\pi$
$740$	0	0
$741$	2.55072	0.0937031
$742$	0	0
$743$	−21.8544	−0.801759	−0.400880	−	0.916131i	$-0.631295\pi$
$743$	−21.8544	−0.801759	−0.400880	+	0.916131i	$0.631295\pi$
$744$	0	0
$745$	−2.50563	−0.0917992
$746$	0	0
$747$	36.4894	1.33508
$748$	0	0
$749$	−5.58414	−0.204040
$750$	0	0
$751$	−10.0271	−0.365895	−0.182947	−	0.983123i	$-0.558564\pi$
$751$	−10.0271	−0.365895	−0.182947	+	0.983123i	$0.558564\pi$
$752$	0	0
$753$	25.7750	0.939293
$754$	0	0
$755$	1.54235	0.0561317
$756$	0	0
$757$	−36.8082	−1.33781	−0.668907	−	0.743346i	$-0.733238\pi$
$757$	−36.8082	−1.33781	−0.668907	+	0.743346i	$0.733238\pi$
$758$	0	0
$759$	7.84834	0.284877
$760$	0	0
$761$	12.3389	0.447286	0.223643	−	0.974671i	$-0.428205\pi$
$761$	12.3389	0.447286	0.223643	+	0.974671i	$0.428205\pi$
$762$	0	0
$763$	2.66796	0.0965865
$764$	0	0
$765$	−4.64306	−0.167870
$766$	0	0
$767$	0.151719	0.00547824
$768$	0	0
$769$	27.4637	0.990365	0.495182	−	0.868789i	$-0.335101\pi$
$769$	27.4637	0.990365	0.495182	+	0.868789i	$0.335101\pi$
$770$	0	0
$771$	−29.9410	−1.07830
$772$	0	0
$773$	−21.2895	−0.765731	−0.382865	−	0.923804i	$-0.625063\pi$
$773$	−21.2895	−0.765731	−0.382865	+	0.923804i	$0.625063\pi$
$774$	0	0
$775$	−17.1442	−0.615838
$776$	0	0
$777$	−23.0707	−0.827655
$778$	0	0
$779$	5.62447	0.201518
$780$	0	0
$781$	−0.493113	−0.0176450
$782$	0	0
$783$	−26.4398	−0.944883
$784$	0	0
$785$	1.79779	0.0641658
$786$	0	0
$787$	−50.9382	−1.81575	−0.907875	−	0.419240i	$-0.862297\pi$
$787$	−50.9382	−1.81575	−0.907875	+	0.419240i	$0.862297\pi$
$788$	0	0
$789$	29.6242	1.05465
$790$	0	0
$791$	14.9938	0.533120
$792$	0	0
$793$	7.77746	0.276186
$794$	0	0
$795$	3.06011	0.108531
$796$	0	0
$797$	12.5689	0.445214	0.222607	−	0.974908i	$-0.428543\pi$
$797$	12.5689	0.445214	0.222607	+	0.974908i	$0.428543\pi$
$798$	0	0
$799$	−10.2518	−0.362682
$800$	0	0
$801$	54.1061	1.91174
$802$	0	0
$803$	−3.05167	−0.107691
$804$	0	0
$805$	−0.309219	−0.0108985
$806$	0	0
$807$	5.69702	0.200545
$808$	0	0
$809$	24.0565	0.845783	0.422891	−	0.906180i	$-0.361015\pi$
$809$	24.0565	0.845783	0.422891	+	0.906180i	$0.361015\pi$
$810$	0	0
$811$	−10.4503	−0.366961	−0.183480	−	0.983023i	$-0.558736\pi$
$811$	−10.4503	−0.366961	−0.183480	+	0.983023i	$0.558736\pi$
$812$	0	0
$813$	−47.9658	−1.68223
$814$	0	0
$815$	−0.936335	−0.0327984
$816$	0	0
$817$	−7.08382	−0.247832
$818$	0	0
$819$	8.11603	0.283597
$820$	0	0
$821$	14.4334	0.503729	0.251864	−	0.967763i	$-0.418956\pi$
$821$	14.4334	0.503729	0.251864	+	0.967763i	$0.418956\pi$
$822$	0	0
$823$	15.1010	0.526388	0.263194	−	0.964743i	$-0.415224\pi$
$823$	15.1010	0.526388	0.263194	+	0.964743i	$0.415224\pi$
$824$	0	0
$825$	16.6128	0.578384
$826$	0	0
$827$	−4.92095	−0.171118	−0.0855592	−	0.996333i	$-0.527268\pi$
$827$	−4.92095	−0.171118	−0.0855592	+	0.996333i	$0.527268\pi$
$828$	0	0
$829$	−2.48798	−0.0864111	−0.0432055	−	0.999066i	$-0.513757\pi$
$829$	−2.48798	−0.0864111	−0.0432055	+	0.999066i	$0.513757\pi$
$830$	0	0
$831$	24.0711	0.835016
$832$	0	0
$833$	4.35509	0.150895
$834$	0	0
$835$	−2.59364	−0.0897566
$836$	0	0
$837$	−58.6890	−2.02859
$838$	0	0
$839$	44.1545	1.52438	0.762192	−	0.647351i	$-0.224123\pi$
$839$	44.1545	1.52438	0.762192	+	0.647351i	$0.224123\pi$
$840$	0	0
$841$	−26.5973	−0.917148
$842$	0	0
$843$	−77.1897	−2.65855
$844$	0	0
$845$	−0.131360	−0.00451893
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	91.4703	3.13925
$850$	0	0
$851$	16.2888	0.558371
$852$	0	0
$853$	25.7989	0.883336	0.441668	−	0.897179i	$-0.354387\pi$
$853$	25.7989	0.883336	0.441668	+	0.897179i	$0.354387\pi$
$854$	0	0
$855$	−0.815635	−0.0278941
$856$	0	0
$857$	−8.87595	−0.303197	−0.151598	−	0.988442i	$-0.548442\pi$
$857$	−8.87595	−0.303197	−0.151598	+	0.988442i	$0.548442\pi$
$858$	0	0
$859$	25.8810	0.883049	0.441524	−	0.897249i	$-0.354438\pi$
$859$	25.8810	0.883049	0.441524	+	0.897249i	$0.354438\pi$
$860$	0	0
$861$	24.5114	0.835347
$862$	0	0
$863$	−21.7619	−0.740785	−0.370393	−	0.928875i	$-0.620777\pi$
$863$	−21.7619	−0.740785	−0.370393	+	0.928875i	$0.620777\pi$
$864$	0	0
$865$	−0.873029	−0.0296839
$866$	0	0
$867$	−6.55740	−0.222701
$868$	0	0
$869$	1.38849	0.0471013
$870$	0	0
$871$	7.77938	0.263594
$872$	0	0
$873$	151.261	5.11941
$874$	0	0
$875$	−1.31134	−0.0443312
$876$	0	0
$877$	−2.16341	−0.0730530	−0.0365265	−	0.999333i	$-0.511629\pi$
$877$	−2.16341	−0.0730530	−0.0365265	+	0.999333i	$0.511629\pi$
$878$	0	0
$879$	51.7492	1.74546
$880$	0	0
$881$	−1.39781	−0.0470935	−0.0235468	−	0.999723i	$-0.507496\pi$
$881$	−1.39781	−0.0470935	−0.0235468	+	0.999723i	$0.507496\pi$
$882$	0	0
$883$	19.5992	0.659565	0.329783	−	0.944057i	$-0.393025\pi$
$883$	19.5992	0.659565	0.329783	+	0.944057i	$0.393025\pi$
$884$	0	0
$885$	−0.0664473	−0.00223360
$886$	0	0
$887$	0.151305	0.00508033	0.00254016	−	0.999997i	$-0.499191\pi$
$887$	0.151305	0.00508033	0.00254016	+	0.999997i	$0.499191\pi$
$888$	0	0
$889$	−2.20075	−0.0738107
$890$	0	0
$891$	32.5219	1.08952
$892$	0	0
$893$	−1.80091	−0.0602650
$894$	0	0
$895$	1.94187	0.0649097
$896$	0	0
$897$	−7.84834	−0.262048
$898$	0	0
$899$	5.33334	0.177877
$900$	0	0
$901$	30.4295	1.01375
$902$	0	0
$903$	−30.8713	−1.02733
$904$	0	0
$905$	−1.30950	−0.0435292
$906$	0	0
$907$	51.7316	1.71772	0.858859	−	0.512211i	$-0.171174\pi$
$907$	51.7316	1.71772	0.858859	+	0.512211i	$0.171174\pi$
$908$	0	0
$909$	125.006	4.14620
$910$	0	0
$911$	7.05918	0.233881	0.116941	−	0.993139i	$-0.462691\pi$
$911$	7.05918	0.233881	0.116941	+	0.993139i	$0.462691\pi$
$912$	0	0
$913$	4.49597	0.148795
$914$	0	0
$915$	−3.40625	−0.112607
$916$	0	0
$917$	1.36405	0.0450449
$918$	0	0
$919$	−57.2866	−1.88971	−0.944855	−	0.327488i	$-0.893798\pi$
$919$	−57.2866	−1.88971	−0.944855	+	0.327488i	$0.893798\pi$
$920$	0	0
$921$	66.2860	2.18420
$922$	0	0
$923$	0.493113	0.0162310
$924$	0	0
$925$	34.4789	1.13366
$926$	0	0
$927$	−31.0443	−1.01963
$928$	0	0
$929$	35.9526	1.17957	0.589784	−	0.807561i	$-0.299213\pi$
$929$	35.9526	1.17957	0.589784	+	0.807561i	$0.299213\pi$
$930$	0	0
$931$	0.765047	0.0250734
$932$	0	0
$933$	54.0280	1.76880
$934$	0	0
$935$	−0.572085	−0.0187092
$936$	0	0
$937$	23.4837	0.767180	0.383590	−	0.923504i	$-0.374688\pi$
$937$	23.4837	0.767180	0.383590	+	0.923504i	$0.374688\pi$
$938$	0	0
$939$	46.7297	1.52497
$940$	0	0
$941$	−27.1625	−0.885471	−0.442736	−	0.896652i	$-0.645992\pi$
$941$	−27.1625	−0.885471	−0.442736	+	0.896652i	$0.645992\pi$
$942$	0	0
$943$	−17.3060	−0.563561
$944$	0	0
$945$	−2.24064	−0.0728880
$946$	0	0
$947$	−57.1940	−1.85855	−0.929277	−	0.369383i	$-0.879569\pi$
$947$	−57.1940	−1.85855	−0.929277	+	0.369383i	$0.879569\pi$
$948$	0	0
$949$	3.05167	0.0990612
$950$	0	0
$951$	40.9569	1.32812
$952$	0	0
$953$	−16.5759	−0.536947	−0.268474	−	0.963287i	$-0.586519\pi$
$953$	−16.5759	−0.536947	−0.268474	+	0.963287i	$0.586519\pi$
$954$	0	0
$955$	2.84778	0.0921519
$956$	0	0
$957$	−5.16804	−0.167059
$958$	0	0
$959$	6.02782	0.194648
$960$	0	0
$961$	−19.1615	−0.618112
$962$	0	0
$963$	45.3211	1.46045
$964$	0	0
$965$	−1.01942	−0.0328162
$966$	0	0
$967$	10.2590	0.329908	0.164954	−	0.986301i	$-0.447252\pi$
$967$	10.2590	0.329908	0.164954	+	0.986301i	$0.447252\pi$
$968$	0	0
$969$	−11.1086	−0.356860
$970$	0	0
$971$	−40.7338	−1.30721	−0.653605	−	0.756836i	$-0.726744\pi$
$971$	−40.7338	−1.30721	−0.653605	+	0.756836i	$0.726744\pi$
$972$	0	0
$973$	10.0216	0.321277
$974$	0	0
$975$	−16.6128	−0.532036
$976$	0	0
$977$	−2.37413	−0.0759552	−0.0379776	−	0.999279i	$-0.512092\pi$
$977$	−2.37413	−0.0759552	−0.0379776	+	0.999279i	$0.512092\pi$
$978$	0	0
$979$	6.66657	0.213064
$980$	0	0
$981$	−21.6532	−0.691334
$982$	0	0
$983$	36.8610	1.17568	0.587842	−	0.808976i	$-0.299978\pi$
$983$	36.8610	1.17568	0.587842	+	0.808976i	$0.299978\pi$
$984$	0	0
$985$	1.36814	0.0435926
$986$	0	0
$987$	−7.84834	−0.249815
$988$	0	0
$989$	21.7963	0.693081
$990$	0	0
$991$	−22.0726	−0.701159	−0.350580	−	0.936533i	$-0.614015\pi$
$991$	−22.0726	−0.701159	−0.350580	+	0.936533i	$0.614015\pi$
$992$	0	0
$993$	12.9339	0.410444
$994$	0	0
$995$	−1.40266	−0.0444672
$996$	0	0
$997$	48.3632	1.53168	0.765839	−	0.643032i	$-0.222324\pi$
$997$	48.3632	1.53168	0.765839	+	0.643032i	$0.222324\pi$
$998$	0	0
$999$	118.030	3.73431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.v.1.1	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.v.1.1	✓	11	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)

\(f(q)\)	\(=\)	\(q-3.33407 q^{3} -0.131360 q^{5} -1.00000 q^{7} +8.11603 q^{9} +O(q^{10})\)	\(q-3.33407 q^{3} -0.131360 q^{5} -1.00000 q^{7} +8.11603 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.437964 q^{15} +4.35509 q^{17} +0.765047 q^{19} +3.33407 q^{21} -2.35398 q^{23} -4.98274 q^{25} -17.0572 q^{27} +1.55007 q^{29} +3.44072 q^{31} -3.33407 q^{33} +0.131360 q^{35} -6.91967 q^{37} +3.33407 q^{39} +7.35180 q^{41} -9.25933 q^{43} -1.06612 q^{45} -2.35398 q^{47} +1.00000 q^{49} -14.5202 q^{51} +6.98712 q^{53} -0.131360 q^{55} -2.55072 q^{57} -0.151719 q^{59} -7.77746 q^{61} -8.11603 q^{63} +0.131360 q^{65} -7.77938 q^{67} +7.84834 q^{69} -0.493113 q^{71} -3.05167 q^{73} +16.6128 q^{75} -1.00000 q^{77} +1.38849 q^{79} +32.5219 q^{81} +4.49597 q^{83} -0.572085 q^{85} -5.16804 q^{87} +6.66657 q^{89} +1.00000 q^{91} -11.4716 q^{93} -0.100497 q^{95} +18.6373 q^{97} +8.11603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) \) 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9	\( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9} + 11 q^{11} - 11 q^{13} - 7 q^{15} - 4 q^{17} - 16 q^{19} + 2 q^{21} - 3 q^{23} + 11 q^{25} - 11 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} - 2 q^{35} - 8 q^{37} + 2 q^{39} + 4 q^{41} - 30 q^{43} + 13 q^{45} - 3 q^{47} + 11 q^{49} - 14 q^{51} - 5 q^{53} + 2 q^{55} - 22 q^{57} + 11 q^{59} + 15 q^{61} - 9 q^{63} - 2 q^{65} - 41 q^{67} + 12 q^{69} + q^{71} - 8 q^{73} - 24 q^{75} - 11 q^{77} - 26 q^{79} + 19 q^{81} - 31 q^{83} - 27 q^{85} - 25 q^{87} + 2 q^{89} + 11 q^{91} - 37 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) \) 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9 + 11 * q^11 - 11 * q^13 - 7 * q^15 - 4 * q^17 - 16 * q^19 + 2 * q^21 - 3 * q^23 + 11 * q^25 - 11 * q^27 + q^29 + 14 * q^31 - 2 * q^33 - 2 * q^35 - 8 * q^37 + 2 * q^39 + 4 * q^41 - 30 * q^43 + 13 * q^45 - 3 * q^47 + 11 * q^49 - 14 * q^51 - 5 * q^53 + 2 * q^55 - 22 * q^57 + 11 * q^59 + 15 * q^61 - 9 * q^63 - 2 * q^65 - 41 * q^67 + 12 * q^69 + q^71 - 8 * q^73 - 24 * q^75 - 11 * q^77 - 26 * q^79 + 19 * q^81 - 31 * q^83 - 27 * q^85 - 25 * q^87 + 2 * q^89 + 11 * q^91 - 37 * q^93 - 6 * q^97 + 9 * q^99

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