LMFDB - Embedded newform 8008.2.a.l.1.5

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 24x^{4} - 10x^{3} - 18x^{2} - 2x + 1 $ x^8 - 2x^7 - 9x^6 + 12x^5 + 24x^4 - 10x^3 - 18x^2 - 2*x + 1
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			1.5
Root			$2.95468$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.837498 q^{3} -2.01769 q^{5} +1.00000 q^{7} -2.29860 q^{9} +O(q^{10})$	$q-0.837498 q^{3} -2.01769 q^{5} +1.00000 q^{7} -2.29860 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.68982 q^{15} -4.49920 q^{17} +5.07513 q^{19} -0.837498 q^{21} -2.66698 q^{23} -0.928910 q^{25} +4.43757 q^{27} -1.39916 q^{29} +8.88513 q^{31} -0.837498 q^{33} -2.01769 q^{35} -5.57018 q^{37} -0.837498 q^{39} -1.42111 q^{41} +9.54597 q^{43} +4.63786 q^{45} -7.06818 q^{47} +1.00000 q^{49} +3.76807 q^{51} -2.80634 q^{53} -2.01769 q^{55} -4.25041 q^{57} +8.77426 q^{59} +9.79866 q^{61} -2.29860 q^{63} -2.01769 q^{65} -6.10795 q^{67} +2.23359 q^{69} +0.130592 q^{71} -1.60457 q^{73} +0.777961 q^{75} +1.00000 q^{77} +5.77603 q^{79} +3.17933 q^{81} -6.63075 q^{83} +9.07800 q^{85} +1.17180 q^{87} -4.56314 q^{89} +1.00000 q^{91} -7.44129 q^{93} -10.2401 q^{95} -5.32631 q^{97} -2.29860 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9	$ 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9} + 8 q^{11} + 8 q^{13} - 5 q^{15} + 3 q^{17} - 19 q^{19} - 5 q^{21} + q^{23} + 15 q^{25} - 11 q^{27} - 21 q^{29} + 2 q^{31} - 5 q^{33} - 7 q^{35} - 12 q^{37} - 5 q^{39} - 6 q^{41} - 19 q^{43} - 17 q^{45} - 7 q^{47} + 8 q^{49} - 19 q^{51} - 26 q^{53} - 7 q^{55} + 2 q^{57} - 17 q^{59} + 7 q^{63} - 7 q^{65} - 24 q^{67} + 20 q^{69} + 2 q^{71} + 2 q^{73} + 18 q^{75} + 8 q^{77} + 7 q^{79} - 4 q^{81} - 6 q^{83} + 19 q^{85} - 13 q^{87} + 13 q^{89} + 8 q^{91} + 13 q^{93} + 21 q^{95} + 18 q^{97} + 7 q^{99}+O(q^{100}) $ 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9 + 8 * q^11 + 8 * q^13 - 5 * q^15 + 3 * q^17 - 19 * q^19 - 5 * q^21 + q^23 + 15 * q^25 - 11 * q^27 - 21 * q^29 + 2 * q^31 - 5 * q^33 - 7 * q^35 - 12 * q^37 - 5 * q^39 - 6 * q^41 - 19 * q^43 - 17 * q^45 - 7 * q^47 + 8 * q^49 - 19 * q^51 - 26 * q^53 - 7 * q^55 + 2 * q^57 - 17 * q^59 + 7 * q^63 - 7 * q^65 - 24 * q^67 + 20 * q^69 + 2 * q^71 + 2 * q^73 + 18 * q^75 + 8 * q^77 + 7 * q^79 - 4 * q^81 - 6 * q^83 + 19 * q^85 - 13 * q^87 + 13 * q^89 + 8 * q^91 + 13 * q^93 + 21 * q^95 + 18 * q^97 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.837498	−0.483530	−0.241765	−	0.970335i	$-0.577726\pi$
$3$	−0.837498	−0.483530	−0.241765	+	0.970335i	$0.577726\pi$
$4$	0	0
$5$	−2.01769	−0.902340	−0.451170	−	0.892438i	$-0.648993\pi$
$5$	−2.01769	−0.902340	−0.451170	+	0.892438i	$0.648993\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.29860	−0.766199
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.68982	0.436309
$16$	0	0
$17$	−4.49920	−1.09122	−0.545608	−	0.838041i	$-0.683701\pi$
$17$	−4.49920	−1.09122	−0.545608	+	0.838041i	$0.683701\pi$
$18$	0	0
$19$	5.07513	1.16431	0.582157	−	0.813076i	$-0.302209\pi$
$19$	5.07513	1.16431	0.582157	+	0.813076i	$0.302209\pi$
$20$	0	0
$21$	−0.837498	−0.182757
$22$	0	0
$23$	−2.66698	−0.556104	−0.278052	−	0.960566i	$-0.589689\pi$
$23$	−2.66698	−0.556104	−0.278052	+	0.960566i	$0.589689\pi$
$24$	0	0
$25$	−0.928910	−0.185782
$26$	0	0
$27$	4.43757	0.854010
$28$	0	0
$29$	−1.39916	−0.259818	−0.129909	−	0.991526i	$-0.541469\pi$
$29$	−1.39916	−0.259818	−0.129909	+	0.991526i	$0.541469\pi$
$30$	0	0
$31$	8.88513	1.59582	0.797909	−	0.602778i	$-0.205940\pi$
$31$	8.88513	1.59582	0.797909	+	0.602778i	$0.205940\pi$
$32$	0	0
$33$	−0.837498	−0.145790
$34$	0	0
$35$	−2.01769	−0.341053
$36$	0	0
$37$	−5.57018	−0.915732	−0.457866	−	0.889021i	$-0.651386\pi$
$37$	−5.57018	−0.915732	−0.457866	+	0.889021i	$0.651386\pi$
$38$	0	0
$39$	−0.837498	−0.134107
$40$	0	0
$41$	−1.42111	−0.221940	−0.110970	−	0.993824i	$-0.535396\pi$
$41$	−1.42111	−0.221940	−0.110970	+	0.993824i	$0.535396\pi$
$42$	0	0
$43$	9.54597	1.45575	0.727874	−	0.685711i	$-0.240509\pi$
$43$	9.54597	1.45575	0.727874	+	0.685711i	$0.240509\pi$
$44$	0	0
$45$	4.63786	0.691372
$46$	0	0
$47$	−7.06818	−1.03100	−0.515500	−	0.856889i	$-0.672394\pi$
$47$	−7.06818	−1.03100	−0.515500	+	0.856889i	$0.672394\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.76807	0.527635
$52$	0	0
$53$	−2.80634	−0.385481	−0.192741	−	0.981250i	$-0.561738\pi$
$53$	−2.80634	−0.385481	−0.192741	+	0.981250i	$0.561738\pi$
$54$	0	0
$55$	−2.01769	−0.272066
$56$	0	0
$57$	−4.25041	−0.562981
$58$	0	0
$59$	8.77426	1.14231	0.571156	−	0.820842i	$-0.306495\pi$
$59$	8.77426	1.14231	0.571156	+	0.820842i	$0.306495\pi$
$60$	0	0
$61$	9.79866	1.25459	0.627295	−	0.778782i	$-0.284162\pi$
$61$	9.79866	1.25459	0.627295	+	0.778782i	$0.284162\pi$
$62$	0	0
$63$	−2.29860	−0.289596
$64$	0	0
$65$	−2.01769	−0.250264
$66$	0	0
$67$	−6.10795	−0.746205	−0.373103	−	0.927790i	$-0.621706\pi$
$67$	−6.10795	−0.746205	−0.373103	+	0.927790i	$0.621706\pi$
$68$	0	0
$69$	2.23359	0.268893
$70$	0	0
$71$	0.130592	0.0154984	0.00774922	−	0.999970i	$-0.497533\pi$
$71$	0.130592	0.0154984	0.00774922	+	0.999970i	$0.497533\pi$
$72$	0	0
$73$	−1.60457	−0.187801	−0.0939003	−	0.995582i	$-0.529933\pi$
$73$	−1.60457	−0.187801	−0.0939003	+	0.995582i	$0.529933\pi$
$74$	0	0
$75$	0.777961	0.0898312
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	5.77603	0.649854	0.324927	−	0.945739i	$-0.394660\pi$
$79$	5.77603	0.649854	0.324927	+	0.945739i	$0.394660\pi$
$80$	0	0
$81$	3.17933	0.353259
$82$	0	0
$83$	−6.63075	−0.727820	−0.363910	−	0.931434i	$-0.618558\pi$
$83$	−6.63075	−0.727820	−0.363910	+	0.931434i	$0.618558\pi$
$84$	0	0
$85$	9.07800	0.984648
$86$	0	0
$87$	1.17180	0.125630
$88$	0	0
$89$	−4.56314	−0.483691	−0.241846	−	0.970315i	$-0.577753\pi$
$89$	−4.56314	−0.483691	−0.241846	+	0.970315i	$0.577753\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−7.44129	−0.771625
$94$	0	0
$95$	−10.2401	−1.05061
$96$	0	0
$97$	−5.32631	−0.540805	−0.270402	−	0.962747i	$-0.587157\pi$
$97$	−5.32631	−0.540805	−0.270402	+	0.962747i	$0.587157\pi$
$98$	0	0
$99$	−2.29860	−0.231018
$100$	0	0
$101$	−2.35314	−0.234147	−0.117073	−	0.993123i	$-0.537351\pi$
$101$	−2.35314	−0.234147	−0.117073	+	0.993123i	$0.537351\pi$
$102$	0	0
$103$	−0.498813	−0.0491495	−0.0245747	−	0.999698i	$-0.507823\pi$
$103$	−0.498813	−0.0491495	−0.0245747	+	0.999698i	$0.507823\pi$
$104$	0	0
$105$	1.68982	0.164909
$106$	0	0
$107$	4.60037	0.444735	0.222367	−	0.974963i	$-0.428622\pi$
$107$	4.60037	0.444735	0.222367	+	0.974963i	$0.428622\pi$
$108$	0	0
$109$	4.68683	0.448917	0.224458	−	0.974484i	$-0.427939\pi$
$109$	4.68683	0.448917	0.224458	+	0.974484i	$0.427939\pi$
$110$	0	0
$111$	4.66502	0.442784
$112$	0	0
$113$	−1.30599	−0.122857	−0.0614284	−	0.998111i	$-0.519566\pi$
$113$	−1.30599	−0.122857	−0.0614284	+	0.998111i	$0.519566\pi$
$114$	0	0
$115$	5.38115	0.501795
$116$	0	0
$117$	−2.29860	−0.212505
$118$	0	0
$119$	−4.49920	−0.412441
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.19017	0.107314
$124$	0	0
$125$	11.9627	1.06998
$126$	0	0
$127$	8.49397	0.753718	0.376859	−	0.926271i	$-0.377004\pi$
$127$	8.49397	0.753718	0.376859	+	0.926271i	$0.377004\pi$
$128$	0	0
$129$	−7.99474	−0.703898
$130$	0	0
$131$	−19.0888	−1.66780	−0.833899	−	0.551917i	$-0.813896\pi$
$131$	−19.0888	−1.66780	−0.833899	+	0.551917i	$0.813896\pi$
$132$	0	0
$133$	5.07513	0.440069
$134$	0	0
$135$	−8.95365	−0.770608
$136$	0	0
$137$	3.66606	0.313213	0.156606	−	0.987661i	$-0.449945\pi$
$137$	3.66606	0.313213	0.156606	+	0.987661i	$0.449945\pi$
$138$	0	0
$139$	2.64809	0.224608	0.112304	−	0.993674i	$-0.464177\pi$
$139$	2.64809	0.224608	0.112304	+	0.993674i	$0.464177\pi$
$140$	0	0
$141$	5.91959	0.498520
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	2.82309	0.234445
$146$	0	0
$147$	−0.837498	−0.0690757
$148$	0	0
$149$	12.7653	1.04577	0.522887	−	0.852402i	$-0.324855\pi$
$149$	12.7653	1.04577	0.522887	+	0.852402i	$0.324855\pi$
$150$	0	0
$151$	−0.524149	−0.0426547	−0.0213273	−	0.999773i	$-0.506789\pi$
$151$	−0.524149	−0.0426547	−0.0213273	+	0.999773i	$0.506789\pi$
$152$	0	0
$153$	10.3418	0.836088
$154$	0	0
$155$	−17.9275	−1.43997
$156$	0	0
$157$	−1.74536	−0.139295	−0.0696476	−	0.997572i	$-0.522187\pi$
$157$	−1.74536	−0.139295	−0.0696476	+	0.997572i	$0.522187\pi$
$158$	0	0
$159$	2.35031	0.186392
$160$	0	0
$161$	−2.66698	−0.210188
$162$	0	0
$163$	−15.8878	−1.24443	−0.622214	−	0.782847i	$-0.713766\pi$
$163$	−15.8878	−1.24443	−0.622214	+	0.782847i	$0.713766\pi$
$164$	0	0
$165$	1.68982	0.131552
$166$	0	0
$167$	11.7847	0.911927	0.455964	−	0.889998i	$-0.349295\pi$
$167$	11.7847	0.911927	0.455964	+	0.889998i	$0.349295\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−11.6657	−0.892096
$172$	0	0
$173$	−23.6218	−1.79593	−0.897967	−	0.440063i	$-0.854956\pi$
$173$	−23.6218	−1.79593	−0.897967	+	0.440063i	$0.854956\pi$
$174$	0	0
$175$	−0.928910	−0.0702190
$176$	0	0
$177$	−7.34843	−0.552342
$178$	0	0
$179$	−10.2754	−0.768023	−0.384012	−	0.923328i	$-0.625458\pi$
$179$	−10.2754	−0.768023	−0.384012	+	0.923328i	$0.625458\pi$
$180$	0	0
$181$	−6.35884	−0.472649	−0.236324	−	0.971674i	$-0.575943\pi$
$181$	−6.35884	−0.472649	−0.236324	+	0.971674i	$0.575943\pi$
$182$	0	0
$183$	−8.20636	−0.606632
$184$	0	0
$185$	11.2389	0.826302
$186$	0	0
$187$	−4.49920	−0.329014
$188$	0	0
$189$	4.43757	0.322785
$190$	0	0
$191$	4.60768	0.333400	0.166700	−	0.986008i	$-0.446689\pi$
$191$	4.60768	0.333400	0.166700	+	0.986008i	$0.446689\pi$
$192$	0	0
$193$	−7.66135	−0.551476	−0.275738	−	0.961233i	$-0.588922\pi$
$193$	−7.66135	−0.551476	−0.275738	+	0.961233i	$0.588922\pi$
$194$	0	0
$195$	1.68982	0.121010
$196$	0	0
$197$	−15.7362	−1.12116	−0.560578	−	0.828101i	$-0.689421\pi$
$197$	−15.7362	−1.12116	−0.560578	+	0.828101i	$0.689421\pi$
$198$	0	0
$199$	5.90372	0.418504	0.209252	−	0.977862i	$-0.432897\pi$
$199$	5.90372	0.418504	0.209252	+	0.977862i	$0.432897\pi$
$200$	0	0
$201$	5.11540	0.360813
$202$	0	0
$203$	−1.39916	−0.0982021
$204$	0	0
$205$	2.86736	0.200265
$206$	0	0
$207$	6.13031	0.426086
$208$	0	0
$209$	5.07513	0.351054
$210$	0	0
$211$	16.8478	1.15985	0.579926	−	0.814669i	$-0.303081\pi$
$211$	16.8478	1.15985	0.579926	+	0.814669i	$0.303081\pi$
$212$	0	0
$213$	−0.109371	−0.00749396
$214$	0	0
$215$	−19.2609	−1.31358
$216$	0	0
$217$	8.88513	0.603162
$218$	0	0
$219$	1.34382	0.0908072
$220$	0	0
$221$	−4.49920	−0.302649
$222$	0	0
$223$	−17.9885	−1.20460	−0.602300	−	0.798270i	$-0.705749\pi$
$223$	−17.9885	−1.20460	−0.602300	+	0.798270i	$0.705749\pi$
$224$	0	0
$225$	2.13519	0.142346
$226$	0	0
$227$	−2.56821	−0.170458	−0.0852291	−	0.996361i	$-0.527162\pi$
$227$	−2.56821	−0.170458	−0.0852291	+	0.996361i	$0.527162\pi$
$228$	0	0
$229$	−17.3843	−1.14879	−0.574395	−	0.818579i	$-0.694762\pi$
$229$	−17.3843	−1.14879	−0.574395	+	0.818579i	$0.694762\pi$
$230$	0	0
$231$	−0.837498	−0.0551034
$232$	0	0
$233$	−3.87264	−0.253705	−0.126852	−	0.991922i	$-0.540487\pi$
$233$	−3.87264	−0.253705	−0.126852	+	0.991922i	$0.540487\pi$
$234$	0	0
$235$	14.2614	0.930313
$236$	0	0
$237$	−4.83742	−0.314224
$238$	0	0
$239$	−13.2816	−0.859117	−0.429559	−	0.903039i	$-0.641331\pi$
$239$	−13.2816	−0.859117	−0.429559	+	0.903039i	$0.641331\pi$
$240$	0	0
$241$	23.9292	1.54141	0.770706	−	0.637191i	$-0.219904\pi$
$241$	23.9292	1.54141	0.770706	+	0.637191i	$0.219904\pi$
$242$	0	0
$243$	−15.9754	−1.02482
$244$	0	0
$245$	−2.01769	−0.128906
$246$	0	0
$247$	5.07513	0.322923
$248$	0	0
$249$	5.55325	0.351923
$250$	0	0
$251$	18.4563	1.16495	0.582474	−	0.812849i	$-0.302085\pi$
$251$	18.4563	1.16495	0.582474	+	0.812849i	$0.302085\pi$
$252$	0	0
$253$	−2.66698	−0.167672
$254$	0	0
$255$	−7.60281	−0.476107
$256$	0	0
$257$	−21.1626	−1.32009	−0.660043	−	0.751228i	$-0.729462\pi$
$257$	−21.1626	−1.32009	−0.660043	+	0.751228i	$0.729462\pi$
$258$	0	0
$259$	−5.57018	−0.346114
$260$	0	0
$261$	3.21611	0.199072
$262$	0	0
$263$	22.9895	1.41759	0.708796	−	0.705413i	$-0.249239\pi$
$263$	22.9895	1.41759	0.708796	+	0.705413i	$0.249239\pi$
$264$	0	0
$265$	5.66235	0.347835
$266$	0	0
$267$	3.82162	0.233879
$268$	0	0
$269$	−13.9560	−0.850910	−0.425455	−	0.904980i	$-0.639886\pi$
$269$	−13.9560	−0.850910	−0.425455	+	0.904980i	$0.639886\pi$
$270$	0	0
$271$	−11.9513	−0.725988	−0.362994	−	0.931791i	$-0.618245\pi$
$271$	−11.9513	−0.725988	−0.362994	+	0.931791i	$0.618245\pi$
$272$	0	0
$273$	−0.837498	−0.0506877
$274$	0	0
$275$	−0.928910	−0.0560154
$276$	0	0
$277$	−12.3183	−0.740137	−0.370068	−	0.929005i	$-0.620666\pi$
$277$	−12.3183	−0.740137	−0.370068	+	0.929005i	$0.620666\pi$
$278$	0	0
$279$	−20.4233	−1.22271
$280$	0	0
$281$	17.0693	1.01827	0.509133	−	0.860688i	$-0.329966\pi$
$281$	17.0693	1.01827	0.509133	+	0.860688i	$0.329966\pi$
$282$	0	0
$283$	−5.14963	−0.306114	−0.153057	−	0.988217i	$-0.548912\pi$
$283$	−5.14963	−0.306114	−0.153057	+	0.988217i	$0.548912\pi$
$284$	0	0
$285$	8.57603	0.508000
$286$	0	0
$287$	−1.42111	−0.0838853
$288$	0	0
$289$	3.24277	0.190751
$290$	0	0
$291$	4.46077	0.261495
$292$	0	0
$293$	−19.6684	−1.14904	−0.574520	−	0.818491i	$-0.694811\pi$
$293$	−19.6684	−1.14904	−0.574520	+	0.818491i	$0.694811\pi$
$294$	0	0
$295$	−17.7038	−1.03075
$296$	0	0
$297$	4.43757	0.257494
$298$	0	0
$299$	−2.66698	−0.154236
$300$	0	0
$301$	9.54597	0.550221
$302$	0	0
$303$	1.97075	0.113217
$304$	0	0
$305$	−19.7707	−1.13207
$306$	0	0
$307$	−21.4206	−1.22254	−0.611268	−	0.791423i	$-0.709340\pi$
$307$	−21.4206	−1.22254	−0.611268	+	0.791423i	$0.709340\pi$
$308$	0	0
$309$	0.417755	0.0237652
$310$	0	0
$311$	−13.3419	−0.756550	−0.378275	−	0.925693i	$-0.623483\pi$
$311$	−13.3419	−0.756550	−0.378275	+	0.925693i	$0.623483\pi$
$312$	0	0
$313$	−6.57123	−0.371428	−0.185714	−	0.982604i	$-0.559460\pi$
$313$	−6.57123	−0.371428	−0.185714	+	0.982604i	$0.559460\pi$
$314$	0	0
$315$	4.63786	0.261314
$316$	0	0
$317$	14.0257	0.787764	0.393882	−	0.919161i	$-0.371132\pi$
$317$	14.0257	0.787764	0.393882	+	0.919161i	$0.371132\pi$
$318$	0	0
$319$	−1.39916	−0.0783382
$320$	0	0
$321$	−3.85280	−0.215043
$322$	0	0
$323$	−22.8340	−1.27052
$324$	0	0
$325$	−0.928910	−0.0515267
$326$	0	0
$327$	−3.92521	−0.217065
$328$	0	0
$329$	−7.06818	−0.389682
$330$	0	0
$331$	−7.38116	−0.405705	−0.202853	−	0.979209i	$-0.565021\pi$
$331$	−7.38116	−0.405705	−0.202853	+	0.979209i	$0.565021\pi$
$332$	0	0
$333$	12.8036	0.701633
$334$	0	0
$335$	12.3240	0.673331
$336$	0	0
$337$	29.2916	1.59561	0.797807	−	0.602913i	$-0.205993\pi$
$337$	29.2916	1.59561	0.797807	+	0.602913i	$0.205993\pi$
$338$	0	0
$339$	1.09376	0.0594050
$340$	0	0
$341$	8.88513	0.481157
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−4.50671	−0.242633
$346$	0	0
$347$	−23.9020	−1.28312	−0.641562	−	0.767071i	$-0.721713\pi$
$347$	−23.9020	−1.28312	−0.641562	+	0.767071i	$0.721713\pi$
$348$	0	0
$349$	−31.4932	−1.68579	−0.842897	−	0.538075i	$-0.819152\pi$
$349$	−31.4932	−1.68579	−0.842897	+	0.538075i	$0.819152\pi$
$350$	0	0
$351$	4.43757	0.236860
$352$	0	0
$353$	−3.67504	−0.195603	−0.0978014	−	0.995206i	$-0.531181\pi$
$353$	−3.67504	−0.195603	−0.0978014	+	0.995206i	$0.531181\pi$
$354$	0	0
$355$	−0.263495	−0.0139849
$356$	0	0
$357$	3.76807	0.199427
$358$	0	0
$359$	−10.9985	−0.580481	−0.290240	−	0.956954i	$-0.593735\pi$
$359$	−10.9985	−0.580481	−0.290240	+	0.956954i	$0.593735\pi$
$360$	0	0
$361$	6.75692	0.355628
$362$	0	0
$363$	−0.837498	−0.0439573
$364$	0	0
$365$	3.23753	0.169460
$366$	0	0
$367$	−13.8922	−0.725167	−0.362584	−	0.931951i	$-0.618105\pi$
$367$	−13.8922	−0.725167	−0.362584	+	0.931951i	$0.618105\pi$
$368$	0	0
$369$	3.26655	0.170050
$370$	0	0
$371$	−2.80634	−0.145698
$372$	0	0
$373$	23.9494	1.24005	0.620027	−	0.784580i	$-0.287122\pi$
$373$	23.9494	1.24005	0.620027	+	0.784580i	$0.287122\pi$
$374$	0	0
$375$	−10.0188	−0.517367
$376$	0	0
$377$	−1.39916	−0.0720606
$378$	0	0
$379$	−1.74244	−0.0895033	−0.0447517	−	0.998998i	$-0.514250\pi$
$379$	−1.74244	−0.0895033	−0.0447517	+	0.998998i	$0.514250\pi$
$380$	0	0
$381$	−7.11369	−0.364445
$382$	0	0
$383$	−14.9832	−0.765608	−0.382804	−	0.923830i	$-0.625042\pi$
$383$	−14.9832	−0.765608	−0.382804	+	0.923830i	$0.625042\pi$
$384$	0	0
$385$	−2.01769	−0.102831
$386$	0	0
$387$	−21.9423	−1.11539
$388$	0	0
$389$	15.4527	0.783483	0.391742	−	0.920075i	$-0.371873\pi$
$389$	15.4527	0.783483	0.391742	+	0.920075i	$0.371873\pi$
$390$	0	0
$391$	11.9993	0.606829
$392$	0	0
$393$	15.9869	0.806430
$394$	0	0
$395$	−11.6543	−0.586389
$396$	0	0
$397$	−24.2858	−1.21887	−0.609434	−	0.792837i	$-0.708603\pi$
$397$	−24.2858	−1.21887	−0.609434	+	0.792837i	$0.708603\pi$
$398$	0	0
$399$	−4.25041	−0.212787
$400$	0	0
$401$	−26.1948	−1.30810	−0.654052	−	0.756450i	$-0.726932\pi$
$401$	−26.1948	−1.30810	−0.654052	+	0.756450i	$0.726932\pi$
$402$	0	0
$403$	8.88513	0.442600
$404$	0	0
$405$	−6.41492	−0.318760
$406$	0	0
$407$	−5.57018	−0.276104
$408$	0	0
$409$	−26.5408	−1.31236	−0.656180	−	0.754604i	$-0.727829\pi$
$409$	−26.5408	−1.31236	−0.656180	+	0.754604i	$0.727829\pi$
$410$	0	0
$411$	−3.07032	−0.151448
$412$	0	0
$413$	8.77426	0.431753
$414$	0	0
$415$	13.3788	0.656741
$416$	0	0
$417$	−2.21777	−0.108605
$418$	0	0
$419$	11.9049	0.581593	0.290797	−	0.956785i	$-0.406080\pi$
$419$	11.9049	0.581593	0.290797	+	0.956785i	$0.406080\pi$
$420$	0	0
$421$	−2.08981	−0.101851	−0.0509257	−	0.998702i	$-0.516217\pi$
$421$	−2.08981	−0.101851	−0.0509257	+	0.998702i	$0.516217\pi$
$422$	0	0
$423$	16.2469	0.789951
$424$	0	0
$425$	4.17935	0.202728
$426$	0	0
$427$	9.79866	0.474190
$428$	0	0
$429$	−0.837498	−0.0404348
$430$	0	0
$431$	−0.160891	−0.00774986	−0.00387493	−	0.999992i	$-0.501233\pi$
$431$	−0.160891	−0.00774986	−0.00387493	+	0.999992i	$0.501233\pi$
$432$	0	0
$433$	−22.9479	−1.10281	−0.551403	−	0.834239i	$-0.685907\pi$
$433$	−22.9479	−1.10281	−0.551403	+	0.834239i	$0.685907\pi$
$434$	0	0
$435$	−2.36433	−0.113361
$436$	0	0
$437$	−13.5353	−0.647480
$438$	0	0
$439$	1.30870	0.0624610	0.0312305	−	0.999512i	$-0.490057\pi$
$439$	1.30870	0.0624610	0.0312305	+	0.999512i	$0.490057\pi$
$440$	0	0
$441$	−2.29860	−0.109457
$442$	0	0
$443$	−34.7947	−1.65315	−0.826574	−	0.562828i	$-0.809713\pi$
$443$	−34.7947	−1.65315	−0.826574	+	0.562828i	$0.809713\pi$
$444$	0	0
$445$	9.20701	0.436454
$446$	0	0
$447$	−10.6909	−0.505663
$448$	0	0
$449$	−7.26956	−0.343072	−0.171536	−	0.985178i	$-0.554873\pi$
$449$	−7.26956	−0.343072	−0.171536	+	0.985178i	$0.554873\pi$
$450$	0	0
$451$	−1.42111	−0.0669173
$452$	0	0
$453$	0.438974	0.0206248
$454$	0	0
$455$	−2.01769	−0.0945910
$456$	0	0
$457$	24.8019	1.16018	0.580092	−	0.814551i	$-0.303016\pi$
$457$	24.8019	1.16018	0.580092	+	0.814551i	$0.303016\pi$
$458$	0	0
$459$	−19.9655	−0.931909
$460$	0	0
$461$	6.15766	0.286791	0.143395	−	0.989665i	$-0.454198\pi$
$461$	6.15766	0.286791	0.143395	+	0.989665i	$0.454198\pi$
$462$	0	0
$463$	−30.5900	−1.42164	−0.710819	−	0.703375i	$-0.751675\pi$
$463$	−30.5900	−1.42164	−0.710819	+	0.703375i	$0.751675\pi$
$464$	0	0
$465$	15.0142	0.696269
$466$	0	0
$467$	0.310576	0.0143717	0.00718586	−	0.999974i	$-0.497713\pi$
$467$	0.310576	0.0143717	0.00718586	+	0.999974i	$0.497713\pi$
$468$	0	0
$469$	−6.10795	−0.282039
$470$	0	0
$471$	1.46174	0.0673534
$472$	0	0
$473$	9.54597	0.438924
$474$	0	0
$475$	−4.71434	−0.216309
$476$	0	0
$477$	6.45065	0.295355
$478$	0	0
$479$	24.8538	1.13560	0.567799	−	0.823167i	$-0.307795\pi$
$479$	24.8538	1.13560	0.567799	+	0.823167i	$0.307795\pi$
$480$	0	0
$481$	−5.57018	−0.253978
$482$	0	0
$483$	2.23359	0.101632
$484$	0	0
$485$	10.7469	0.487990
$486$	0	0
$487$	2.77781	0.125874	0.0629372	−	0.998017i	$-0.479953\pi$
$487$	2.77781	0.125874	0.0629372	+	0.998017i	$0.479953\pi$
$488$	0	0
$489$	13.3060	0.601718
$490$	0	0
$491$	20.7022	0.934277	0.467139	−	0.884184i	$-0.345285\pi$
$491$	20.7022	0.934277	0.467139	+	0.884184i	$0.345285\pi$
$492$	0	0
$493$	6.29512	0.283518
$494$	0	0
$495$	4.63786	0.208457
$496$	0	0
$497$	0.130592	0.00585786
$498$	0	0
$499$	−24.5401	−1.09857	−0.549283	−	0.835636i	$-0.685099\pi$
$499$	−24.5401	−1.09857	−0.549283	+	0.835636i	$0.685099\pi$
$500$	0	0
$501$	−9.86966	−0.440944
$502$	0	0
$503$	−27.6565	−1.23314	−0.616571	−	0.787299i	$-0.711479\pi$
$503$	−27.6565	−1.23314	−0.616571	+	0.787299i	$0.711479\pi$
$504$	0	0
$505$	4.74792	0.211280
$506$	0	0
$507$	−0.837498	−0.0371946
$508$	0	0
$509$	−10.8175	−0.479476	−0.239738	−	0.970838i	$-0.577061\pi$
$509$	−10.8175	−0.479476	−0.239738	+	0.970838i	$0.577061\pi$
$510$	0	0
$511$	−1.60457	−0.0709819
$512$	0	0
$513$	22.5212	0.994336
$514$	0	0
$515$	1.00645	0.0443495
$516$	0	0
$517$	−7.06818	−0.310858
$518$	0	0
$519$	19.7832	0.868388
$520$	0	0
$521$	−6.39448	−0.280147	−0.140074	−	0.990141i	$-0.544734\pi$
$521$	−6.39448	−0.280147	−0.140074	+	0.990141i	$0.544734\pi$
$522$	0	0
$523$	−11.2869	−0.493543	−0.246771	−	0.969074i	$-0.579370\pi$
$523$	−11.2869	−0.493543	−0.246771	+	0.969074i	$0.579370\pi$
$524$	0	0
$525$	0.777961	0.0339530
$526$	0	0
$527$	−39.9760	−1.74138
$528$	0	0
$529$	−15.8872	−0.690748
$530$	0	0
$531$	−20.1685	−0.875238
$532$	0	0
$533$	−1.42111	−0.0615550
$534$	0	0
$535$	−9.28214	−0.401302
$536$	0	0
$537$	8.60567	0.371362
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−12.5781	−0.540774	−0.270387	−	0.962752i	$-0.587152\pi$
$541$	−12.5781	−0.540774	−0.270387	+	0.962752i	$0.587152\pi$
$542$	0	0
$543$	5.32552	0.228540
$544$	0	0
$545$	−9.45659	−0.405076
$546$	0	0
$547$	−26.8532	−1.14816	−0.574080	−	0.818799i	$-0.694640\pi$
$547$	−26.8532	−1.14816	−0.574080	+	0.818799i	$0.694640\pi$
$548$	0	0
$549$	−22.5232	−0.961265
$550$	0	0
$551$	−7.10094	−0.302510
$552$	0	0
$553$	5.77603	0.245622
$554$	0	0
$555$	−9.41258	−0.399542
$556$	0	0
$557$	−4.64331	−0.196743	−0.0983716	−	0.995150i	$-0.531363\pi$
$557$	−4.64331	−0.196743	−0.0983716	+	0.995150i	$0.531363\pi$
$558$	0	0
$559$	9.54597	0.403752
$560$	0	0
$561$	3.76807	0.159088
$562$	0	0
$563$	−30.2654	−1.27553	−0.637767	−	0.770229i	$-0.720142\pi$
$563$	−30.2654	−1.27553	−0.637767	+	0.770229i	$0.720142\pi$
$564$	0	0
$565$	2.63508	0.110859
$566$	0	0
$567$	3.17933	0.133519
$568$	0	0
$569$	18.7290	0.785159	0.392580	−	0.919718i	$-0.371583\pi$
$569$	18.7290	0.785159	0.392580	+	0.919718i	$0.371583\pi$
$570$	0	0
$571$	37.4458	1.56706	0.783530	−	0.621354i	$-0.213417\pi$
$571$	37.4458	1.56706	0.783530	+	0.621354i	$0.213417\pi$
$572$	0	0
$573$	−3.85892	−0.161209
$574$	0	0
$575$	2.47739	0.103314
$576$	0	0
$577$	7.89511	0.328678	0.164339	−	0.986404i	$-0.447451\pi$
$577$	7.89511	0.328678	0.164339	+	0.986404i	$0.447451\pi$
$578$	0	0
$579$	6.41637	0.266655
$580$	0	0
$581$	−6.63075	−0.275090
$582$	0	0
$583$	−2.80634	−0.116227
$584$	0	0
$585$	4.63786	0.191752
$586$	0	0
$587$	−14.5163	−0.599153	−0.299576	−	0.954072i	$-0.596845\pi$
$587$	−14.5163	−0.599153	−0.299576	+	0.954072i	$0.596845\pi$
$588$	0	0
$589$	45.0932	1.85803
$590$	0	0
$591$	13.1790	0.542113
$592$	0	0
$593$	44.6710	1.83442	0.917210	−	0.398405i	$-0.130436\pi$
$593$	44.6710	1.83442	0.917210	+	0.398405i	$0.130436\pi$
$594$	0	0
$595$	9.07800	0.372162
$596$	0	0
$597$	−4.94436	−0.202359
$598$	0	0
$599$	3.51032	0.143428	0.0717140	−	0.997425i	$-0.477153\pi$
$599$	3.51032	0.143428	0.0717140	+	0.997425i	$0.477153\pi$
$600$	0	0
$601$	43.6651	1.78113	0.890567	−	0.454851i	$-0.150308\pi$
$601$	43.6651	1.78113	0.890567	+	0.454851i	$0.150308\pi$
$602$	0	0
$603$	14.0397	0.571742
$604$	0	0
$605$	−2.01769	−0.0820309
$606$	0	0
$607$	−16.4122	−0.666150	−0.333075	−	0.942900i	$-0.608086\pi$
$607$	−16.4122	−0.666150	−0.333075	+	0.942900i	$0.608086\pi$
$608$	0	0
$609$	1.17180	0.0474837
$610$	0	0
$611$	−7.06818	−0.285948
$612$	0	0
$613$	−9.57539	−0.386747	−0.193373	−	0.981125i	$-0.561943\pi$
$613$	−9.57539	−0.386747	−0.193373	+	0.981125i	$0.561943\pi$
$614$	0	0
$615$	−2.40141	−0.0968342
$616$	0	0
$617$	5.82026	0.234315	0.117157	−	0.993113i	$-0.462622\pi$
$617$	5.82026	0.234315	0.117157	+	0.993113i	$0.462622\pi$
$618$	0	0
$619$	−38.6673	−1.55417	−0.777085	−	0.629396i	$-0.783302\pi$
$619$	−38.6673	−1.55417	−0.777085	+	0.629396i	$0.783302\pi$
$620$	0	0
$621$	−11.8349	−0.474919
$622$	0	0
$623$	−4.56314	−0.182818
$624$	0	0
$625$	−19.4926	−0.779703
$626$	0	0
$627$	−4.25041	−0.169745
$628$	0	0
$629$	25.0613	0.999261
$630$	0	0
$631$	−9.74889	−0.388097	−0.194049	−	0.980992i	$-0.562162\pi$
$631$	−9.74889	−0.388097	−0.194049	+	0.980992i	$0.562162\pi$
$632$	0	0
$633$	−14.1100	−0.560823
$634$	0	0
$635$	−17.1382	−0.680110
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−0.300179	−0.0118749
$640$	0	0
$641$	−10.7306	−0.423833	−0.211917	−	0.977288i	$-0.567970\pi$
$641$	−10.7306	−0.423833	−0.211917	+	0.977288i	$0.567970\pi$
$642$	0	0
$643$	32.8094	1.29388	0.646939	−	0.762542i	$-0.276049\pi$
$643$	32.8094	1.29388	0.646939	+	0.762542i	$0.276049\pi$
$644$	0	0
$645$	16.1309	0.635155
$646$	0	0
$647$	39.0787	1.53634	0.768171	−	0.640245i	$-0.221167\pi$
$647$	39.0787	1.53634	0.768171	+	0.640245i	$0.221167\pi$
$648$	0	0
$649$	8.77426	0.344420
$650$	0	0
$651$	−7.44129	−0.291647
$652$	0	0
$653$	21.4486	0.839349	0.419675	−	0.907675i	$-0.362144\pi$
$653$	21.4486	0.839349	0.419675	+	0.907675i	$0.362144\pi$
$654$	0	0
$655$	38.5154	1.50492
$656$	0	0
$657$	3.68826	0.143893
$658$	0	0
$659$	29.8570	1.16307	0.581533	−	0.813523i	$-0.302453\pi$
$659$	29.8570	1.16307	0.581533	+	0.813523i	$0.302453\pi$
$660$	0	0
$661$	−32.6339	−1.26931	−0.634656	−	0.772795i	$-0.718858\pi$
$661$	−32.6339	−1.26931	−0.634656	+	0.772795i	$0.718858\pi$
$662$	0	0
$663$	3.76807	0.146340
$664$	0	0
$665$	−10.2401	−0.397092
$666$	0	0
$667$	3.73155	0.144486
$668$	0	0
$669$	15.0653	0.582460
$670$	0	0
$671$	9.79866	0.378273
$672$	0	0
$673$	5.73184	0.220946	0.110473	−	0.993879i	$-0.464763\pi$
$673$	5.73184	0.220946	0.110473	+	0.993879i	$0.464763\pi$
$674$	0	0
$675$	−4.12210	−0.158660
$676$	0	0
$677$	−37.5347	−1.44258	−0.721289	−	0.692634i	$-0.756450\pi$
$677$	−37.5347	−1.44258	−0.721289	+	0.692634i	$0.756450\pi$
$678$	0	0
$679$	−5.32631	−0.204405
$680$	0	0
$681$	2.15087	0.0824216
$682$	0	0
$683$	35.2032	1.34701	0.673507	−	0.739181i	$-0.264787\pi$
$683$	35.2032	1.34701	0.673507	+	0.739181i	$0.264787\pi$
$684$	0	0
$685$	−7.39699	−0.282625
$686$	0	0
$687$	14.5594	0.555474
$688$	0	0
$689$	−2.80634	−0.106913
$690$	0	0
$691$	−39.3720	−1.49778	−0.748891	−	0.662693i	$-0.769413\pi$
$691$	−39.3720	−1.49778	−0.748891	+	0.662693i	$0.769413\pi$
$692$	0	0
$693$	−2.29860	−0.0873165
$694$	0	0
$695$	−5.34303	−0.202673
$696$	0	0
$697$	6.39384	0.242184
$698$	0	0
$699$	3.24333	0.122674
$700$	0	0
$701$	1.19501	0.0451350	0.0225675	−	0.999745i	$-0.492816\pi$
$701$	1.19501	0.0451350	0.0225675	+	0.999745i	$0.492816\pi$
$702$	0	0
$703$	−28.2694	−1.06620
$704$	0	0
$705$	−11.9439	−0.449834
$706$	0	0
$707$	−2.35314	−0.0884991
$708$	0	0
$709$	−9.10231	−0.341844	−0.170922	−	0.985285i	$-0.554675\pi$
$709$	−9.10231	−0.341844	−0.170922	+	0.985285i	$0.554675\pi$
$710$	0	0
$711$	−13.2768	−0.497917
$712$	0	0
$713$	−23.6965	−0.887441
$714$	0	0
$715$	−2.01769	−0.0754575
$716$	0	0
$717$	11.1233	0.415409
$718$	0	0
$719$	−38.8761	−1.44983	−0.724917	−	0.688836i	$-0.758122\pi$
$719$	−38.8761	−1.44983	−0.724917	+	0.688836i	$0.758122\pi$
$720$	0	0
$721$	−0.498813	−0.0185768
$722$	0	0
$723$	−20.0406	−0.745319
$724$	0	0
$725$	1.29970	0.0482696
$726$	0	0
$727$	−9.48491	−0.351776	−0.175888	−	0.984410i	$-0.556280\pi$
$727$	−9.48491	−0.351776	−0.175888	+	0.984410i	$0.556280\pi$
$728$	0	0
$729$	3.84136	0.142273
$730$	0	0
$731$	−42.9492	−1.58853
$732$	0	0
$733$	46.3751	1.71290	0.856452	−	0.516226i	$-0.172664\pi$
$733$	46.3751	1.71290	0.856452	+	0.516226i	$0.172664\pi$
$734$	0	0
$735$	1.68982	0.0623298
$736$	0	0
$737$	−6.10795	−0.224989
$738$	0	0
$739$	−40.6254	−1.49443	−0.747215	−	0.664583i	$-0.768609\pi$
$739$	−40.6254	−1.49443	−0.747215	+	0.664583i	$0.768609\pi$
$740$	0	0
$741$	−4.25041	−0.156143
$742$	0	0
$743$	40.8861	1.49996	0.749982	−	0.661458i	$-0.230062\pi$
$743$	40.8861	1.49996	0.749982	+	0.661458i	$0.230062\pi$
$744$	0	0
$745$	−25.7565	−0.943645
$746$	0	0
$747$	15.2414	0.557655
$748$	0	0
$749$	4.60037	0.168094
$750$	0	0
$751$	−39.8723	−1.45496	−0.727480	−	0.686129i	$-0.759309\pi$
$751$	−39.8723	−1.45496	−0.727480	+	0.686129i	$0.759309\pi$
$752$	0	0
$753$	−15.4571	−0.563287
$754$	0	0
$755$	1.05757	0.0384890
$756$	0	0
$757$	9.06992	0.329652	0.164826	−	0.986323i	$-0.447294\pi$
$757$	9.06992	0.329652	0.164826	+	0.986323i	$0.447294\pi$
$758$	0	0
$759$	2.23359	0.0810743
$760$	0	0
$761$	8.52271	0.308948	0.154474	−	0.987997i	$-0.450632\pi$
$761$	8.52271	0.308948	0.154474	+	0.987997i	$0.450632\pi$
$762$	0	0
$763$	4.68683	0.169675
$764$	0	0
$765$	−20.8667	−0.754436
$766$	0	0
$767$	8.77426	0.316820
$768$	0	0
$769$	17.2812	0.623177	0.311589	−	0.950217i	$-0.399139\pi$
$769$	17.2812	0.623177	0.311589	+	0.950217i	$0.399139\pi$
$770$	0	0
$771$	17.7236	0.638301
$772$	0	0
$773$	28.5034	1.02520	0.512598	−	0.858629i	$-0.328683\pi$
$773$	28.5034	1.02520	0.512598	+	0.858629i	$0.328683\pi$
$774$	0	0
$775$	−8.25349	−0.296474
$776$	0	0
$777$	4.66502	0.167357
$778$	0	0
$779$	−7.21230	−0.258407
$780$	0	0
$781$	0.130592	0.00467295
$782$	0	0
$783$	−6.20889	−0.221887
$784$	0	0
$785$	3.52161	0.125692
$786$	0	0
$787$	9.55084	0.340451	0.170225	−	0.985405i	$-0.445550\pi$
$787$	9.55084	0.340451	0.170225	+	0.985405i	$0.445550\pi$
$788$	0	0
$789$	−19.2537	−0.685449
$790$	0	0
$791$	−1.30599	−0.0464355
$792$	0	0
$793$	9.79866	0.347961
$794$	0	0
$795$	−4.74221	−0.168189
$796$	0	0
$797$	−13.9603	−0.494498	−0.247249	−	0.968952i	$-0.579527\pi$
$797$	−13.9603	−0.494498	−0.247249	+	0.968952i	$0.579527\pi$
$798$	0	0
$799$	31.8011	1.12504
$800$	0	0
$801$	10.4888	0.370604
$802$	0	0
$803$	−1.60457	−0.0566240
$804$	0	0
$805$	5.38115	0.189661
$806$	0	0
$807$	11.6881	0.411441
$808$	0	0
$809$	−53.7630	−1.89021	−0.945104	−	0.326769i	$-0.894040\pi$
$809$	−53.7630	−1.89021	−0.945104	+	0.326769i	$0.894040\pi$
$810$	0	0
$811$	−56.0260	−1.96734	−0.983670	−	0.179982i	$-0.942396\pi$
$811$	−56.0260	−1.96734	−0.983670	+	0.179982i	$0.942396\pi$
$812$	0	0
$813$	10.0092	0.351037
$814$	0	0
$815$	32.0567	1.12290
$816$	0	0
$817$	48.4470	1.69495
$818$	0	0
$819$	−2.29860	−0.0803195
$820$	0	0
$821$	3.96258	0.138295	0.0691475	−	0.997606i	$-0.477972\pi$
$821$	3.96258	0.138295	0.0691475	+	0.997606i	$0.477972\pi$
$822$	0	0
$823$	20.8305	0.726105	0.363053	−	0.931769i	$-0.381735\pi$
$823$	20.8305	0.726105	0.363053	+	0.931769i	$0.381735\pi$
$824$	0	0
$825$	0.777961	0.0270851
$826$	0	0
$827$	−1.40418	−0.0488282	−0.0244141	−	0.999702i	$-0.507772\pi$
$827$	−1.40418	−0.0488282	−0.0244141	+	0.999702i	$0.507772\pi$
$828$	0	0
$829$	−1.36878	−0.0475397	−0.0237699	−	0.999717i	$-0.507567\pi$
$829$	−1.36878	−0.0475397	−0.0237699	+	0.999717i	$0.507567\pi$
$830$	0	0
$831$	10.3166	0.357878
$832$	0	0
$833$	−4.49920	−0.155888
$834$	0	0
$835$	−23.7779	−0.822869
$836$	0	0
$837$	39.4284	1.36284
$838$	0	0
$839$	12.4137	0.428570	0.214285	−	0.976771i	$-0.431258\pi$
$839$	12.4137	0.428570	0.214285	+	0.976771i	$0.431258\pi$
$840$	0	0
$841$	−27.0423	−0.932494
$842$	0	0
$843$	−14.2955	−0.492363
$844$	0	0
$845$	−2.01769	−0.0694108
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	4.31281	0.148015
$850$	0	0
$851$	14.8556	0.509242
$852$	0	0
$853$	0.985384	0.0337389	0.0168695	−	0.999858i	$-0.494630\pi$
$853$	0.985384	0.0337389	0.0168695	+	0.999858i	$0.494630\pi$
$854$	0	0
$855$	23.5378	0.804974
$856$	0	0
$857$	−37.7763	−1.29041	−0.645206	−	0.764009i	$-0.723228\pi$
$857$	−37.7763	−1.29041	−0.645206	+	0.764009i	$0.723228\pi$
$858$	0	0
$859$	−33.7261	−1.15072	−0.575359	−	0.817901i	$-0.695138\pi$
$859$	−33.7261	−1.15072	−0.575359	+	0.817901i	$0.695138\pi$
$860$	0	0
$861$	1.19017	0.0405611
$862$	0	0
$863$	−28.0287	−0.954108	−0.477054	−	0.878874i	$-0.658295\pi$
$863$	−28.0287	−0.954108	−0.477054	+	0.878874i	$0.658295\pi$
$864$	0	0
$865$	47.6616	1.62054
$866$	0	0
$867$	−2.71582	−0.0922339
$868$	0	0
$869$	5.77603	0.195938
$870$	0	0
$871$	−6.10795	−0.206960
$872$	0	0
$873$	12.2430	0.414364
$874$	0	0
$875$	11.9627	0.404414
$876$	0	0
$877$	−1.37499	−0.0464303	−0.0232151	−	0.999730i	$-0.507390\pi$
$877$	−1.37499	−0.0464303	−0.0232151	+	0.999730i	$0.507390\pi$
$878$	0	0
$879$	16.4723	0.555595
$880$	0	0
$881$	−11.2890	−0.380336	−0.190168	−	0.981752i	$-0.560903\pi$
$881$	−11.2890	−0.380336	−0.190168	+	0.981752i	$0.560903\pi$
$882$	0	0
$883$	29.1209	0.979997	0.489999	−	0.871723i	$-0.336997\pi$
$883$	29.1209	0.979997	0.489999	+	0.871723i	$0.336997\pi$
$884$	0	0
$885$	14.8269	0.498400
$886$	0	0
$887$	45.3579	1.52297	0.761485	−	0.648183i	$-0.224471\pi$
$887$	45.3579	1.52297	0.761485	+	0.648183i	$0.224471\pi$
$888$	0	0
$889$	8.49397	0.284879
$890$	0	0
$891$	3.17933	0.106512
$892$	0	0
$893$	−35.8719	−1.20041
$894$	0	0
$895$	20.7327	0.693018
$896$	0	0
$897$	2.23359	0.0745775
$898$	0	0
$899$	−12.4318	−0.414623
$900$	0	0
$901$	12.6263	0.420643
$902$	0	0
$903$	−7.99474	−0.266048
$904$	0	0
$905$	12.8302	0.426490
$906$	0	0
$907$	−52.3007	−1.73662	−0.868308	−	0.496026i	$-0.834792\pi$
$907$	−52.3007	−1.73662	−0.868308	+	0.496026i	$0.834792\pi$
$908$	0	0
$909$	5.40893	0.179403
$910$	0	0
$911$	−55.0509	−1.82392	−0.911959	−	0.410280i	$-0.865431\pi$
$911$	−55.0509	−1.82392	−0.911959	+	0.410280i	$0.865431\pi$
$912$	0	0
$913$	−6.63075	−0.219446
$914$	0	0
$915$	16.5579	0.547388
$916$	0	0
$917$	−19.0888	−0.630368
$918$	0	0
$919$	15.9109	0.524853	0.262426	−	0.964952i	$-0.415477\pi$
$919$	15.9109	0.524853	0.262426	+	0.964952i	$0.415477\pi$
$920$	0	0
$921$	17.9397	0.591133
$922$	0	0
$923$	0.130592	0.00429849
$924$	0	0
$925$	5.17420	0.170127
$926$	0	0
$927$	1.14657	0.0376583
$928$	0	0
$929$	−48.4828	−1.59067	−0.795334	−	0.606172i	$-0.792704\pi$
$929$	−48.4828	−1.59067	−0.795334	+	0.606172i	$0.792704\pi$
$930$	0	0
$931$	5.07513	0.166331
$932$	0	0
$933$	11.1738	0.365815
$934$	0	0
$935$	9.07800	0.296882
$936$	0	0
$937$	−23.0414	−0.752730	−0.376365	−	0.926472i	$-0.622826\pi$
$937$	−23.0414	−0.752730	−0.376365	+	0.926472i	$0.622826\pi$
$938$	0	0
$939$	5.50340	0.179597
$940$	0	0
$941$	24.2637	0.790975	0.395487	−	0.918471i	$-0.370576\pi$
$941$	24.2637	0.790975	0.395487	+	0.918471i	$0.370576\pi$
$942$	0	0
$943$	3.79007	0.123422
$944$	0	0
$945$	−8.95365	−0.291262
$946$	0	0
$947$	−46.5846	−1.51379	−0.756897	−	0.653534i	$-0.773286\pi$
$947$	−46.5846	−1.51379	−0.756897	+	0.653534i	$0.773286\pi$
$948$	0	0
$949$	−1.60457	−0.0520865
$950$	0	0
$951$	−11.7465	−0.380908
$952$	0	0
$953$	6.65715	0.215646	0.107823	−	0.994170i	$-0.465612\pi$
$953$	6.65715	0.215646	0.107823	+	0.994170i	$0.465612\pi$
$954$	0	0
$955$	−9.29688	−0.300840
$956$	0	0
$957$	1.17180	0.0378789
$958$	0	0
$959$	3.66606	0.118383
$960$	0	0
$961$	47.9456	1.54663
$962$	0	0
$963$	−10.5744	−0.340755
$964$	0	0
$965$	15.4583	0.497619
$966$	0	0
$967$	17.2543	0.554860	0.277430	−	0.960746i	$-0.410517\pi$
$967$	17.2543	0.554860	0.277430	+	0.960746i	$0.410517\pi$
$968$	0	0
$969$	19.1234	0.614333
$970$	0	0
$971$	−3.24158	−0.104027	−0.0520136	−	0.998646i	$-0.516564\pi$
$971$	−3.24158	−0.104027	−0.0520136	+	0.998646i	$0.516564\pi$
$972$	0	0
$973$	2.64809	0.0848938
$974$	0	0
$975$	0.777961	0.0249147
$976$	0	0
$977$	−16.2883	−0.521110	−0.260555	−	0.965459i	$-0.583906\pi$
$977$	−16.2883	−0.521110	−0.260555	+	0.965459i	$0.583906\pi$
$978$	0	0
$979$	−4.56314	−0.145838
$980$	0	0
$981$	−10.7731	−0.343960
$982$	0	0
$983$	44.4656	1.41823	0.709117	−	0.705091i	$-0.249094\pi$
$983$	44.4656	1.41823	0.709117	+	0.705091i	$0.249094\pi$
$984$	0	0
$985$	31.7508	1.01167
$986$	0	0
$987$	5.91959	0.188423
$988$	0	0
$989$	−25.4589	−0.809547
$990$	0	0
$991$	6.19915	0.196923	0.0984613	−	0.995141i	$-0.468608\pi$
$991$	6.19915	0.196923	0.0984613	+	0.995141i	$0.468608\pi$
$992$	0	0
$993$	6.18171	0.196171
$994$	0	0
$995$	−11.9119	−0.377633
$996$	0	0
$997$	35.6698	1.12967	0.564836	−	0.825203i	$-0.308939\pi$
$997$	35.6698	1.12967	0.564836	+	0.825203i	$0.308939\pi$
$998$	0	0
$999$	−24.7180	−0.782044

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.l.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.l.1.5	✓	8	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)

\(f(q)\)	\(=\)	\(q-0.837498 q^{3} -2.01769 q^{5} +1.00000 q^{7} -2.29860 q^{9} +O(q^{10})\)	\(q-0.837498 q^{3} -2.01769 q^{5} +1.00000 q^{7} -2.29860 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.68982 q^{15} -4.49920 q^{17} +5.07513 q^{19} -0.837498 q^{21} -2.66698 q^{23} -0.928910 q^{25} +4.43757 q^{27} -1.39916 q^{29} +8.88513 q^{31} -0.837498 q^{33} -2.01769 q^{35} -5.57018 q^{37} -0.837498 q^{39} -1.42111 q^{41} +9.54597 q^{43} +4.63786 q^{45} -7.06818 q^{47} +1.00000 q^{49} +3.76807 q^{51} -2.80634 q^{53} -2.01769 q^{55} -4.25041 q^{57} +8.77426 q^{59} +9.79866 q^{61} -2.29860 q^{63} -2.01769 q^{65} -6.10795 q^{67} +2.23359 q^{69} +0.130592 q^{71} -1.60457 q^{73} +0.777961 q^{75} +1.00000 q^{77} +5.77603 q^{79} +3.17933 q^{81} -6.63075 q^{83} +9.07800 q^{85} +1.17180 q^{87} -4.56314 q^{89} +1.00000 q^{91} -7.44129 q^{93} -10.2401 q^{95} -5.32631 q^{97} -2.29860 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9	\( 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9} + 8 q^{11} + 8 q^{13} - 5 q^{15} + 3 q^{17} - 19 q^{19} - 5 q^{21} + q^{23} + 15 q^{25} - 11 q^{27} - 21 q^{29} + 2 q^{31} - 5 q^{33} - 7 q^{35} - 12 q^{37} - 5 q^{39} - 6 q^{41} - 19 q^{43} - 17 q^{45} - 7 q^{47} + 8 q^{49} - 19 q^{51} - 26 q^{53} - 7 q^{55} + 2 q^{57} - 17 q^{59} + 7 q^{63} - 7 q^{65} - 24 q^{67} + 20 q^{69} + 2 q^{71} + 2 q^{73} + 18 q^{75} + 8 q^{77} + 7 q^{79} - 4 q^{81} - 6 q^{83} + 19 q^{85} - 13 q^{87} + 13 q^{89} + 8 q^{91} + 13 q^{93} + 21 q^{95} + 18 q^{97} + 7 q^{99}+O(q^{100}) \) 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9 + 8 * q^11 + 8 * q^13 - 5 * q^15 + 3 * q^17 - 19 * q^19 - 5 * q^21 + q^23 + 15 * q^25 - 11 * q^27 - 21 * q^29 + 2 * q^31 - 5 * q^33 - 7 * q^35 - 12 * q^37 - 5 * q^39 - 6 * q^41 - 19 * q^43 - 17 * q^45 - 7 * q^47 + 8 * q^49 - 19 * q^51 - 26 * q^53 - 7 * q^55 + 2 * q^57 - 17 * q^59 + 7 * q^63 - 7 * q^65 - 24 * q^67 + 20 * q^69 + 2 * q^71 + 2 * q^73 + 18 * q^75 + 8 * q^77 + 7 * q^79 - 4 * q^81 - 6 * q^83 + 19 * q^85 - 13 * q^87 + 13 * q^89 + 8 * q^91 + 13 * q^93 + 21 * q^95 + 18 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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