LMFDB - Embedded newform 8008.2.a.s.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 15x^{8} + 43x^{7} + 66x^{6} - 173x^{5} - 127x^{4} + 246x^{3} + 99x^{2} - 82x + 8 $ x^10 - 3x^9 - 15x^8 + 43x^7 + 66x^6 - 173x^5 - 127x^4 + 246x^3 + 99x^2 - 82*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.119595$ 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)

$f(q)$	$=$	$q-0.119595 q^{3} -2.15143 q^{5} +1.00000 q^{7} -2.98570 q^{9} +O(q^{10})$	$q-0.119595 q^{3} -2.15143 q^{5} +1.00000 q^{7} -2.98570 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.257301 q^{15} -0.163520 q^{17} -3.89504 q^{19} -0.119595 q^{21} +6.01792 q^{23} -0.371333 q^{25} +0.715859 q^{27} +2.38380 q^{29} +1.09603 q^{31} -0.119595 q^{33} -2.15143 q^{35} +8.82041 q^{37} +0.119595 q^{39} +1.81454 q^{41} +2.23845 q^{43} +6.42353 q^{45} -3.87753 q^{47} +1.00000 q^{49} +0.0195561 q^{51} +9.71172 q^{53} -2.15143 q^{55} +0.465827 q^{57} -9.50729 q^{59} +11.6906 q^{61} -2.98570 q^{63} +2.15143 q^{65} -1.70331 q^{67} -0.719713 q^{69} +5.97880 q^{71} -9.80405 q^{73} +0.0444096 q^{75} +1.00000 q^{77} -9.91892 q^{79} +8.87148 q^{81} -10.9092 q^{83} +0.351802 q^{85} -0.285091 q^{87} -12.5726 q^{89} -1.00000 q^{91} -0.131080 q^{93} +8.37993 q^{95} -3.55279 q^{97} -2.98570 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9}+O(q^{10}) $ 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9	$ 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9} + 10 q^{11} - 10 q^{13} - 5 q^{15} - 11 q^{17} + 2 q^{19} - 3 q^{21} - 8 q^{23} + 2 q^{25} - 15 q^{27} - 8 q^{29} - 23 q^{31} - 3 q^{33} - 4 q^{35} + 10 q^{37} + 3 q^{39} - 18 q^{41} + 12 q^{43} - 10 q^{45} - 36 q^{47} + 10 q^{49} + 9 q^{51} - 21 q^{53} - 4 q^{55} - 30 q^{57} - 13 q^{59} - 2 q^{61} + 9 q^{63} + 4 q^{65} - 4 q^{67} - 26 q^{69} - 24 q^{71} - 23 q^{73} - 28 q^{75} + 10 q^{77} + 14 q^{79} + 30 q^{81} - 9 q^{83} - 17 q^{85} + 7 q^{87} - 18 q^{89} - 10 q^{91} + q^{93} - 4 q^{95} - 9 q^{97} + 9 q^{99}+O(q^{100}) $ 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9 + 10 * q^11 - 10 * q^13 - 5 * q^15 - 11 * q^17 + 2 * q^19 - 3 * q^21 - 8 * q^23 + 2 * q^25 - 15 * q^27 - 8 * q^29 - 23 * q^31 - 3 * q^33 - 4 * q^35 + 10 * q^37 + 3 * q^39 - 18 * q^41 + 12 * q^43 - 10 * q^45 - 36 * q^47 + 10 * q^49 + 9 * q^51 - 21 * q^53 - 4 * q^55 - 30 * q^57 - 13 * q^59 - 2 * q^61 + 9 * q^63 + 4 * q^65 - 4 * q^67 - 26 * q^69 - 24 * q^71 - 23 * q^73 - 28 * q^75 + 10 * q^77 + 14 * q^79 + 30 * q^81 - 9 * q^83 - 17 * q^85 + 7 * q^87 - 18 * q^89 - 10 * q^91 + q^93 - 4 * q^95 - 9 * 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
$3$	−0.119595	−0.0690482	−0.0345241	−	0.999404i	$-0.510992\pi$
$3$	−0.119595	−0.0690482	−0.0345241	+	0.999404i	$0.510992\pi$
$4$	0	0
$5$	−2.15143	−0.962150	−0.481075	−	0.876679i	$-0.659754\pi$
$5$	−2.15143	−0.962150	−0.481075	+	0.876679i	$0.659754\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.98570	−0.995232
$10$	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
$15$	0.257301	0.0664347
$16$	0	0
$17$	−0.163520	−0.0396593	−0.0198297	−	0.999803i	$-0.506312\pi$
$17$	−0.163520	−0.0396593	−0.0198297	+	0.999803i	$0.506312\pi$
$18$	0	0
$19$	−3.89504	−0.893584	−0.446792	−	0.894638i	$-0.647434\pi$
$19$	−3.89504	−0.893584	−0.446792	+	0.894638i	$0.647434\pi$
$20$	0	0
$21$	−0.119595	−0.0260977
$22$	0	0
$23$	6.01792	1.25482	0.627412	−	0.778688i	$-0.284114\pi$
$23$	6.01792	1.25482	0.627412	+	0.778688i	$0.284114\pi$
$24$	0	0
$25$	−0.371333	−0.0742667
$26$	0	0
$27$	0.715859	0.137767
$28$	0	0
$29$	2.38380	0.442661	0.221330	−	0.975199i	$-0.428960\pi$
$29$	2.38380	0.442661	0.221330	+	0.975199i	$0.428960\pi$
$30$	0	0
$31$	1.09603	0.196853	0.0984267	−	0.995144i	$-0.468619\pi$
$31$	1.09603	0.196853	0.0984267	+	0.995144i	$0.468619\pi$
$32$	0	0
$33$	−0.119595	−0.0208188
$34$	0	0
$35$	−2.15143	−0.363659
$36$	0	0
$37$	8.82041	1.45007	0.725033	−	0.688714i	$-0.241824\pi$
$37$	8.82041	1.45007	0.725033	+	0.688714i	$0.241824\pi$
$38$	0	0
$39$	0.119595	0.0191505
$40$	0	0
$41$	1.81454	0.283383	0.141691	−	0.989911i	$-0.454746\pi$
$41$	1.81454	0.283383	0.141691	+	0.989911i	$0.454746\pi$
$42$	0	0
$43$	2.23845	0.341361	0.170680	−	0.985326i	$-0.445403\pi$
$43$	2.23845	0.341361	0.170680	+	0.985326i	$0.445403\pi$
$44$	0	0
$45$	6.42353	0.957563
$46$	0	0
$47$	−3.87753	−0.565596	−0.282798	−	0.959179i	$-0.591263\pi$
$47$	−3.87753	−0.565596	−0.282798	+	0.959179i	$0.591263\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.0195561	0.00273840
$52$	0	0
$53$	9.71172	1.33401	0.667003	−	0.745055i	$-0.267577\pi$
$53$	9.71172	1.33401	0.667003	+	0.745055i	$0.267577\pi$
$54$	0	0
$55$	−2.15143	−0.290099
$56$	0	0
$57$	0.465827	0.0617003
$58$	0	0
$59$	−9.50729	−1.23774	−0.618872	−	0.785492i	$-0.712410\pi$
$59$	−9.50729	−1.23774	−0.618872	+	0.785492i	$0.712410\pi$
$60$	0	0
$61$	11.6906	1.49682	0.748412	−	0.663235i	$-0.230817\pi$
$61$	11.6906	1.49682	0.748412	+	0.663235i	$0.230817\pi$
$62$	0	0
$63$	−2.98570	−0.376162
$64$	0	0
$65$	2.15143	0.266852
$66$	0	0
$67$	−1.70331	−0.208093	−0.104046	−	0.994572i	$-0.533179\pi$
$67$	−1.70331	−0.208093	−0.104046	+	0.994572i	$0.533179\pi$
$68$	0	0
$69$	−0.719713	−0.0866432
$70$	0	0
$71$	5.97880	0.709553	0.354776	−	0.934951i	$-0.384557\pi$
$71$	5.97880	0.709553	0.354776	+	0.934951i	$0.384557\pi$
$72$	0	0
$73$	−9.80405	−1.14748	−0.573739	−	0.819038i	$-0.694508\pi$
$73$	−9.80405	−1.14748	−0.573739	+	0.819038i	$0.694508\pi$
$74$	0	0
$75$	0.0444096	0.00512798
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−9.91892	−1.11597	−0.557983	−	0.829852i	$-0.688425\pi$
$79$	−9.91892	−1.11597	−0.557983	+	0.829852i	$0.688425\pi$
$80$	0	0
$81$	8.87148	0.985720
$82$	0	0
$83$	−10.9092	−1.19745	−0.598723	−	0.800956i	$-0.704325\pi$
$83$	−10.9092	−1.19745	−0.598723	+	0.800956i	$0.704325\pi$
$84$	0	0
$85$	0.351802	0.0381583
$86$	0	0
$87$	−0.285091	−0.0305649
$88$	0	0
$89$	−12.5726	−1.33270	−0.666348	−	0.745641i	$-0.732144\pi$
$89$	−12.5726	−1.33270	−0.666348	+	0.745641i	$0.732144\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−0.131080	−0.0135924
$94$	0	0
$95$	8.37993	0.859762
$96$	0	0
$97$	−3.55279	−0.360731	−0.180366	−	0.983600i	$-0.557728\pi$
$97$	−3.55279	−0.360731	−0.180366	+	0.983600i	$0.557728\pi$
$98$	0	0
$99$	−2.98570	−0.300074
$100$	0	0
$101$	−7.32627	−0.728991	−0.364496	−	0.931205i	$-0.618759\pi$
$101$	−7.32627	−0.728991	−0.364496	+	0.931205i	$0.618759\pi$
$102$	0	0
$103$	10.4024	1.02498	0.512488	−	0.858694i	$-0.328724\pi$
$103$	10.4024	1.02498	0.512488	+	0.858694i	$0.328724\pi$
$104$	0	0
$105$	0.257301	0.0251100
$106$	0	0
$107$	−11.6898	−1.13010	−0.565049	−	0.825057i	$-0.691143\pi$
$107$	−11.6898	−1.13010	−0.565049	+	0.825057i	$0.691143\pi$
$108$	0	0
$109$	−11.5540	−1.10667	−0.553337	−	0.832957i	$-0.686646\pi$
$109$	−11.5540	−1.10667	−0.553337	+	0.832957i	$0.686646\pi$
$110$	0	0
$111$	−1.05488	−0.100124
$112$	0	0
$113$	13.0117	1.22404	0.612020	−	0.790843i	$-0.290357\pi$
$113$	13.0117	1.22404	0.612020	+	0.790843i	$0.290357\pi$
$114$	0	0
$115$	−12.9472	−1.20733
$116$	0	0
$117$	2.98570	0.276028
$118$	0	0
$119$	−0.163520	−0.0149898
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.217009	−0.0195671
$124$	0	0
$125$	11.5561	1.03361
$126$	0	0
$127$	−3.52594	−0.312877	−0.156438	−	0.987688i	$-0.550001\pi$
$127$	−3.52594	−0.312877	−0.156438	+	0.987688i	$0.550001\pi$
$128$	0	0
$129$	−0.267708	−0.0235703
$130$	0	0
$131$	−0.868196	−0.0758546	−0.0379273	−	0.999281i	$-0.512076\pi$
$131$	−0.868196	−0.0758546	−0.0379273	+	0.999281i	$0.512076\pi$
$132$	0	0
$133$	−3.89504	−0.337743
$134$	0	0
$135$	−1.54012	−0.132553
$136$	0	0
$137$	−7.38909	−0.631293	−0.315646	−	0.948877i	$-0.602221\pi$
$137$	−7.38909	−0.631293	−0.315646	+	0.948877i	$0.602221\pi$
$138$	0	0
$139$	11.7454	0.996233	0.498116	−	0.867110i	$-0.334025\pi$
$139$	11.7454	0.996233	0.498116	+	0.867110i	$0.334025\pi$
$140$	0	0
$141$	0.463733	0.0390534
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−5.12859	−0.425906
$146$	0	0
$147$	−0.119595	−0.00986402
$148$	0	0
$149$	−20.6424	−1.69109	−0.845546	−	0.533902i	$-0.820725\pi$
$149$	−20.6424	−1.69109	−0.845546	+	0.533902i	$0.820725\pi$
$150$	0	0
$151$	−1.20254	−0.0978614	−0.0489307	−	0.998802i	$-0.515581\pi$
$151$	−1.20254	−0.0978614	−0.0489307	+	0.998802i	$0.515581\pi$
$152$	0	0
$153$	0.488220	0.0394703
$154$	0	0
$155$	−2.35804	−0.189403
$156$	0	0
$157$	−8.68020	−0.692756	−0.346378	−	0.938095i	$-0.612588\pi$
$157$	−8.68020	−0.692756	−0.346378	+	0.938095i	$0.612588\pi$
$158$	0	0
$159$	−1.16147	−0.0921107
$160$	0	0
$161$	6.01792	0.474279
$162$	0	0
$163$	9.98508	0.782092	0.391046	−	0.920371i	$-0.372113\pi$
$163$	9.98508	0.782092	0.391046	+	0.920371i	$0.372113\pi$
$164$	0	0
$165$	0.257301	0.0200308
$166$	0	0
$167$	5.24442	0.405825	0.202913	−	0.979197i	$-0.434959\pi$
$167$	5.24442	0.405825	0.202913	+	0.979197i	$0.434959\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	11.6294	0.889324
$172$	0	0
$173$	−0.338435	−0.0257307	−0.0128654	−	0.999917i	$-0.504095\pi$
$173$	−0.338435	−0.0257307	−0.0128654	+	0.999917i	$0.504095\pi$
$174$	0	0
$175$	−0.371333	−0.0280702
$176$	0	0
$177$	1.13702	0.0854639
$178$	0	0
$179$	4.43026	0.331133	0.165567	−	0.986199i	$-0.447055\pi$
$179$	4.43026	0.331133	0.165567	+	0.986199i	$0.447055\pi$
$180$	0	0
$181$	−20.5674	−1.52877	−0.764383	−	0.644763i	$-0.776956\pi$
$181$	−20.5674	−1.52877	−0.764383	+	0.644763i	$0.776956\pi$
$182$	0	0
$183$	−1.39813	−0.103353
$184$	0	0
$185$	−18.9765	−1.39518
$186$	0	0
$187$	−0.163520	−0.0119577
$188$	0	0
$189$	0.715859	0.0520711
$190$	0	0
$191$	−14.2351	−1.03002	−0.515008	−	0.857186i	$-0.672211\pi$
$191$	−14.2351	−1.03002	−0.515008	+	0.857186i	$0.672211\pi$
$192$	0	0
$193$	8.94871	0.644142	0.322071	−	0.946715i	$-0.395621\pi$
$193$	8.94871	0.644142	0.322071	+	0.946715i	$0.395621\pi$
$194$	0	0
$195$	−0.257301	−0.0184257
$196$	0	0
$197$	1.56430	0.111452	0.0557258	−	0.998446i	$-0.482253\pi$
$197$	1.56430	0.111452	0.0557258	+	0.998446i	$0.482253\pi$
$198$	0	0
$199$	−11.5413	−0.818144	−0.409072	−	0.912502i	$-0.634148\pi$
$199$	−11.5413	−0.818144	−0.409072	+	0.912502i	$0.634148\pi$
$200$	0	0
$201$	0.203708	0.0143684
$202$	0	0
$203$	2.38380	0.167310
$204$	0	0
$205$	−3.90385	−0.272657
$206$	0	0
$207$	−17.9677	−1.24884
$208$	0	0
$209$	−3.89504	−0.269426
$210$	0	0
$211$	6.07254	0.418051	0.209025	−	0.977910i	$-0.432971\pi$
$211$	6.07254	0.418051	0.209025	+	0.977910i	$0.432971\pi$
$212$	0	0
$213$	−0.715034	−0.0489933
$214$	0	0
$215$	−4.81588	−0.328441
$216$	0	0
$217$	1.09603	0.0744036
$218$	0	0
$219$	1.17251	0.0792312
$220$	0	0
$221$	0.163520	0.0109995
$222$	0	0
$223$	10.1423	0.679178	0.339589	−	0.940574i	$-0.389712\pi$
$223$	10.1423	0.679178	0.339589	+	0.940574i	$0.389712\pi$
$224$	0	0
$225$	1.10869	0.0739126
$226$	0	0
$227$	28.3909	1.88437	0.942185	−	0.335093i	$-0.108768\pi$
$227$	28.3909	1.88437	0.942185	+	0.335093i	$0.108768\pi$
$228$	0	0
$229$	−6.13291	−0.405274	−0.202637	−	0.979254i	$-0.564951\pi$
$229$	−6.13291	−0.405274	−0.202637	+	0.979254i	$0.564951\pi$
$230$	0	0
$231$	−0.119595	−0.00786877
$232$	0	0
$233$	−23.0219	−1.50821	−0.754107	−	0.656752i	$-0.771930\pi$
$233$	−23.0219	−1.50821	−0.754107	+	0.656752i	$0.771930\pi$
$234$	0	0
$235$	8.34225	0.544189
$236$	0	0
$237$	1.18625	0.0770554
$238$	0	0
$239$	−29.5790	−1.91330	−0.956652	−	0.291234i	$-0.905934\pi$
$239$	−29.5790	−1.91330	−0.956652	+	0.291234i	$0.905934\pi$
$240$	0	0
$241$	7.67003	0.494070	0.247035	−	0.969007i	$-0.420544\pi$
$241$	7.67003	0.494070	0.247035	+	0.969007i	$0.420544\pi$
$242$	0	0
$243$	−3.20856	−0.205829
$244$	0	0
$245$	−2.15143	−0.137450
$246$	0	0
$247$	3.89504	0.247836
$248$	0	0
$249$	1.30469	0.0826814
$250$	0	0
$251$	1.49165	0.0941522	0.0470761	−	0.998891i	$-0.485010\pi$
$251$	1.49165	0.0941522	0.0470761	+	0.998891i	$0.485010\pi$
$252$	0	0
$253$	6.01792	0.378344
$254$	0	0
$255$	−0.0420737	−0.00263476
$256$	0	0
$257$	−28.8249	−1.79805	−0.899024	−	0.437900i	$-0.855722\pi$
$257$	−28.8249	−1.79805	−0.899024	+	0.437900i	$0.855722\pi$
$258$	0	0
$259$	8.82041	0.548074
$260$	0	0
$261$	−7.11731	−0.440550
$262$	0	0
$263$	−7.41321	−0.457118	−0.228559	−	0.973530i	$-0.573401\pi$
$263$	−7.41321	−0.457118	−0.228559	+	0.973530i	$0.573401\pi$
$264$	0	0
$265$	−20.8941	−1.28352
$266$	0	0
$267$	1.50362	0.0920203
$268$	0	0
$269$	24.3012	1.48167	0.740836	−	0.671685i	$-0.234429\pi$
$269$	24.3012	1.48167	0.740836	+	0.671685i	$0.234429\pi$
$270$	0	0
$271$	−19.2935	−1.17200	−0.585998	−	0.810313i	$-0.699297\pi$
$271$	−19.2935	−1.17200	−0.585998	+	0.810313i	$0.699297\pi$
$272$	0	0
$273$	0.119595	0.00723821
$274$	0	0
$275$	−0.371333	−0.0223922
$276$	0	0
$277$	0.331256	0.0199032	0.00995161	−	0.999950i	$-0.496832\pi$
$277$	0.331256	0.0199032	0.00995161	+	0.999950i	$0.496832\pi$
$278$	0	0
$279$	−3.27242	−0.195915
$280$	0	0
$281$	29.9430	1.78625	0.893125	−	0.449809i	$-0.148508\pi$
$281$	29.9430	1.78625	0.893125	+	0.449809i	$0.148508\pi$
$282$	0	0
$283$	−31.7542	−1.88759	−0.943795	−	0.330532i	$-0.892772\pi$
$283$	−31.7542	−1.88759	−0.943795	+	0.330532i	$0.892772\pi$
$284$	0	0
$285$	−1.00220	−0.0593650
$286$	0	0
$287$	1.81454	0.107109
$288$	0	0
$289$	−16.9733	−0.998427
$290$	0	0
$291$	0.424896	0.0249078
$292$	0	0
$293$	30.3663	1.77402	0.887008	−	0.461753i	$-0.152779\pi$
$293$	30.3663	1.77402	0.887008	+	0.461753i	$0.152779\pi$
$294$	0	0
$295$	20.4543	1.19090
$296$	0	0
$297$	0.715859	0.0415383
$298$	0	0
$299$	−6.01792	−0.348025
$300$	0	0
$301$	2.23845	0.129022
$302$	0	0
$303$	0.876185	0.0503355
$304$	0	0
$305$	−25.1515	−1.44017
$306$	0	0
$307$	5.66163	0.323126	0.161563	−	0.986862i	$-0.448346\pi$
$307$	5.66163	0.323126	0.161563	+	0.986862i	$0.448346\pi$
$308$	0	0
$309$	−1.24407	−0.0707727
$310$	0	0
$311$	−18.6228	−1.05600	−0.528002	−	0.849243i	$-0.677059\pi$
$311$	−18.6228	−1.05600	−0.528002	+	0.849243i	$0.677059\pi$
$312$	0	0
$313$	−5.36033	−0.302984	−0.151492	−	0.988459i	$-0.548408\pi$
$313$	−5.36033	−0.302984	−0.151492	+	0.988459i	$0.548408\pi$
$314$	0	0
$315$	6.42353	0.361925
$316$	0	0
$317$	6.29687	0.353668	0.176834	−	0.984241i	$-0.443414\pi$
$317$	6.29687	0.353668	0.176834	+	0.984241i	$0.443414\pi$
$318$	0	0
$319$	2.38380	0.133467
$320$	0	0
$321$	1.39804	0.0780312
$322$	0	0
$323$	0.636916	0.0354390
$324$	0	0
$325$	0.371333	0.0205979
$326$	0	0
$327$	1.38180	0.0764138
$328$	0	0
$329$	−3.87753	−0.213775
$330$	0	0
$331$	12.9868	0.713817	0.356909	−	0.934139i	$-0.383831\pi$
$331$	12.9868	0.713817	0.356909	+	0.934139i	$0.383831\pi$
$332$	0	0
$333$	−26.3351	−1.44315
$334$	0	0
$335$	3.66457	0.200217
$336$	0	0
$337$	−19.7956	−1.07833	−0.539167	−	0.842199i	$-0.681261\pi$
$337$	−19.7956	−1.07833	−0.539167	+	0.842199i	$0.681261\pi$
$338$	0	0
$339$	−1.55614	−0.0845177
$340$	0	0
$341$	1.09603	0.0593535
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.54841	0.0833638
$346$	0	0
$347$	−9.56370	−0.513406	−0.256703	−	0.966490i	$-0.582636\pi$
$347$	−9.56370	−0.513406	−0.256703	+	0.966490i	$0.582636\pi$
$348$	0	0
$349$	−30.7001	−1.64334	−0.821669	−	0.569965i	$-0.806957\pi$
$349$	−30.7001	−1.64334	−0.821669	+	0.569965i	$0.806957\pi$
$350$	0	0
$351$	−0.715859	−0.0382097
$352$	0	0
$353$	0.578176	0.0307732	0.0153866	−	0.999882i	$-0.495102\pi$
$353$	0.578176	0.0307732	0.0153866	+	0.999882i	$0.495102\pi$
$354$	0	0
$355$	−12.8630	−0.682697
$356$	0	0
$357$	0.0195561	0.00103502
$358$	0	0
$359$	5.58955	0.295005	0.147502	−	0.989062i	$-0.452877\pi$
$359$	5.58955	0.295005	0.147502	+	0.989062i	$0.452877\pi$
$360$	0	0
$361$	−3.82864	−0.201507
$362$	0	0
$363$	−0.119595	−0.00627711
$364$	0	0
$365$	21.0928	1.10405
$366$	0	0
$367$	−14.2423	−0.743444	−0.371722	−	0.928344i	$-0.621233\pi$
$367$	−14.2423	−0.743444	−0.371722	+	0.928344i	$0.621233\pi$
$368$	0	0
$369$	−5.41765	−0.282032
$370$	0	0
$371$	9.71172	0.504207
$372$	0	0
$373$	6.51300	0.337230	0.168615	−	0.985682i	$-0.446071\pi$
$373$	6.51300	0.337230	0.168615	+	0.985682i	$0.446071\pi$
$374$	0	0
$375$	−1.38205	−0.0713686
$376$	0	0
$377$	−2.38380	−0.122772
$378$	0	0
$379$	7.87835	0.404684	0.202342	−	0.979315i	$-0.435145\pi$
$379$	7.87835	0.404684	0.202342	+	0.979315i	$0.435145\pi$
$380$	0	0
$381$	0.421685	0.0216036
$382$	0	0
$383$	−5.27789	−0.269688	−0.134844	−	0.990867i	$-0.543053\pi$
$383$	−5.27789	−0.269688	−0.134844	+	0.990867i	$0.543053\pi$
$384$	0	0
$385$	−2.15143	−0.109647
$386$	0	0
$387$	−6.68334	−0.339733
$388$	0	0
$389$	−19.4046	−0.983853	−0.491927	−	0.870637i	$-0.663707\pi$
$389$	−19.4046	−0.983853	−0.491927	+	0.870637i	$0.663707\pi$
$390$	0	0
$391$	−0.984049	−0.0497655
$392$	0	0
$393$	0.103832	0.00523762
$394$	0	0
$395$	21.3399	1.07373
$396$	0	0
$397$	−28.1409	−1.41235	−0.706177	−	0.708036i	$-0.749582\pi$
$397$	−28.1409	−1.41235	−0.706177	+	0.708036i	$0.749582\pi$
$398$	0	0
$399$	0.465827	0.0233205
$400$	0	0
$401$	−33.2391	−1.65988	−0.829941	−	0.557851i	$-0.811626\pi$
$401$	−33.2391	−1.65988	−0.829941	+	0.557851i	$0.811626\pi$
$402$	0	0
$403$	−1.09603	−0.0545973
$404$	0	0
$405$	−19.0864	−0.948411
$406$	0	0
$407$	8.82041	0.437212
$408$	0	0
$409$	−2.72127	−0.134558	−0.0672792	−	0.997734i	$-0.521432\pi$
$409$	−2.72127	−0.134558	−0.0672792	+	0.997734i	$0.521432\pi$
$410$	0	0
$411$	0.883698	0.0435896
$412$	0	0
$413$	−9.50729	−0.467823
$414$	0	0
$415$	23.4705	1.15212
$416$	0	0
$417$	−1.40469	−0.0687880
$418$	0	0
$419$	25.3853	1.24015	0.620077	−	0.784541i	$-0.287101\pi$
$419$	25.3853	1.24015	0.620077	+	0.784541i	$0.287101\pi$
$420$	0	0
$421$	6.28935	0.306524	0.153262	−	0.988186i	$-0.451022\pi$
$421$	6.28935	0.306524	0.153262	+	0.988186i	$0.451022\pi$
$422$	0	0
$423$	11.5771	0.562900
$424$	0	0
$425$	0.0607203	0.00294537
$426$	0	0
$427$	11.6906	0.565746
$428$	0	0
$429$	0.119595	0.00577410
$430$	0	0
$431$	−33.1281	−1.59572	−0.797862	−	0.602840i	$-0.794036\pi$
$431$	−33.1281	−1.59572	−0.797862	+	0.602840i	$0.794036\pi$
$432$	0	0
$433$	11.7776	0.565997	0.282998	−	0.959120i	$-0.408671\pi$
$433$	11.7776	0.565997	0.282998	+	0.959120i	$0.408671\pi$
$434$	0	0
$435$	0.613353	0.0294080
$436$	0	0
$437$	−23.4401	−1.12129
$438$	0	0
$439$	6.37075	0.304060	0.152030	−	0.988376i	$-0.451419\pi$
$439$	6.37075	0.304060	0.152030	+	0.988376i	$0.451419\pi$
$440$	0	0
$441$	−2.98570	−0.142176
$442$	0	0
$443$	29.6939	1.41080	0.705400	−	0.708809i	$-0.250767\pi$
$443$	29.6939	1.41080	0.705400	+	0.708809i	$0.250767\pi$
$444$	0	0
$445$	27.0492	1.28225
$446$	0	0
$447$	2.46873	0.116767
$448$	0	0
$449$	−17.8363	−0.841747	−0.420873	−	0.907119i	$-0.638276\pi$
$449$	−17.8363	−0.841747	−0.420873	+	0.907119i	$0.638276\pi$
$450$	0	0
$451$	1.81454	0.0854431
$452$	0	0
$453$	0.143818	0.00675715
$454$	0	0
$455$	2.15143	0.100861
$456$	0	0
$457$	−1.82471	−0.0853563	−0.0426781	−	0.999089i	$-0.513589\pi$
$457$	−1.82471	−0.0853563	−0.0426781	+	0.999089i	$0.513589\pi$
$458$	0	0
$459$	−0.117057	−0.00546375
$460$	0	0
$461$	−35.2993	−1.64405	−0.822027	−	0.569448i	$-0.807157\pi$
$461$	−35.2993	−1.64405	−0.822027	+	0.569448i	$0.807157\pi$
$462$	0	0
$463$	30.9929	1.44036	0.720181	−	0.693787i	$-0.244059\pi$
$463$	30.9929	1.44036	0.720181	+	0.693787i	$0.244059\pi$
$464$	0	0
$465$	0.282010	0.0130779
$466$	0	0
$467$	−35.2736	−1.63227	−0.816134	−	0.577863i	$-0.803887\pi$
$467$	−35.2736	−1.63227	−0.816134	+	0.577863i	$0.803887\pi$
$468$	0	0
$469$	−1.70331	−0.0786517
$470$	0	0
$471$	1.03811	0.0478335
$472$	0	0
$473$	2.23845	0.102924
$474$	0	0
$475$	1.44636	0.0663635
$476$	0	0
$477$	−28.9962	−1.32765
$478$	0	0
$479$	−11.0942	−0.506907	−0.253453	−	0.967348i	$-0.581566\pi$
$479$	−11.0942	−0.506907	−0.253453	+	0.967348i	$0.581566\pi$
$480$	0	0
$481$	−8.82041	−0.402176
$482$	0	0
$483$	−0.719713	−0.0327481
$484$	0	0
$485$	7.64359	0.347078
$486$	0	0
$487$	−25.9669	−1.17667	−0.588337	−	0.808616i	$-0.700217\pi$
$487$	−25.9669	−1.17667	−0.588337	+	0.808616i	$0.700217\pi$
$488$	0	0
$489$	−1.19416	−0.0540020
$490$	0	0
$491$	−24.6151	−1.11086	−0.555432	−	0.831562i	$-0.687447\pi$
$491$	−24.6151	−1.11086	−0.555432	+	0.831562i	$0.687447\pi$
$492$	0	0
$493$	−0.389798	−0.0175556
$494$	0	0
$495$	6.42353	0.288716
$496$	0	0
$497$	5.97880	0.268186
$498$	0	0
$499$	−6.90959	−0.309316	−0.154658	−	0.987968i	$-0.549428\pi$
$499$	−6.90959	−0.309316	−0.154658	+	0.987968i	$0.549428\pi$
$500$	0	0
$501$	−0.627206	−0.0280215
$502$	0	0
$503$	−34.2983	−1.52929	−0.764643	−	0.644454i	$-0.777085\pi$
$503$	−34.2983	−1.52929	−0.764643	+	0.644454i	$0.777085\pi$
$504$	0	0
$505$	15.7620	0.701399
$506$	0	0
$507$	−0.119595	−0.00531140
$508$	0	0
$509$	−4.03658	−0.178918	−0.0894591	−	0.995990i	$-0.528514\pi$
$509$	−4.03658	−0.178918	−0.0894591	+	0.995990i	$0.528514\pi$
$510$	0	0
$511$	−9.80405	−0.433706
$512$	0	0
$513$	−2.78830	−0.123107
$514$	0	0
$515$	−22.3800	−0.986181
$516$	0	0
$517$	−3.87753	−0.170534
$518$	0	0
$519$	0.0404751	0.00177666
$520$	0	0
$521$	23.8521	1.04498	0.522489	−	0.852646i	$-0.325004\pi$
$521$	23.8521	1.04498	0.522489	+	0.852646i	$0.325004\pi$
$522$	0	0
$523$	14.1881	0.620403	0.310202	−	0.950671i	$-0.399603\pi$
$523$	14.1881	0.620403	0.310202	+	0.950671i	$0.399603\pi$
$524$	0	0
$525$	0.0444096	0.00193819
$526$	0	0
$527$	−0.179223	−0.00780708
$528$	0	0
$529$	13.2154	0.574582
$530$	0	0
$531$	28.3859	1.23184
$532$	0	0
$533$	−1.81454	−0.0785962
$534$	0	0
$535$	25.1499	1.08732
$536$	0	0
$537$	−0.529837	−0.0228642
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−43.4502	−1.86807	−0.934035	−	0.357181i	$-0.883738\pi$
$541$	−43.4502	−1.86807	−0.934035	+	0.357181i	$0.883738\pi$
$542$	0	0
$543$	2.45976	0.105558
$544$	0	0
$545$	24.8577	1.06479
$546$	0	0
$547$	36.7951	1.57325	0.786623	−	0.617434i	$-0.211828\pi$
$547$	36.7951	1.57325	0.786623	+	0.617434i	$0.211828\pi$
$548$	0	0
$549$	−34.9045	−1.48969
$550$	0	0
$551$	−9.28501	−0.395555
$552$	0	0
$553$	−9.91892	−0.421796
$554$	0	0
$555$	2.26950	0.0963347
$556$	0	0
$557$	41.9050	1.77557	0.887786	−	0.460256i	$-0.152242\pi$
$557$	41.9050	1.77557	0.887786	+	0.460256i	$0.152242\pi$
$558$	0	0
$559$	−2.23845	−0.0946765
$560$	0	0
$561$	0.0195561	0.000825660 0
$562$	0	0
$563$	−5.19160	−0.218800	−0.109400	−	0.993998i	$-0.534893\pi$
$563$	−5.19160	−0.218800	−0.109400	+	0.993998i	$0.534893\pi$
$564$	0	0
$565$	−27.9938	−1.17771
$566$	0	0
$567$	8.87148	0.372567
$568$	0	0
$569$	−5.84472	−0.245024	−0.122512	−	0.992467i	$-0.539095\pi$
$569$	−5.84472	−0.245024	−0.122512	+	0.992467i	$0.539095\pi$
$570$	0	0
$571$	7.22873	0.302513	0.151257	−	0.988495i	$-0.451668\pi$
$571$	7.22873	0.302513	0.151257	+	0.988495i	$0.451668\pi$
$572$	0	0
$573$	1.70244	0.0711207
$574$	0	0
$575$	−2.23466	−0.0931916
$576$	0	0
$577$	30.1333	1.25447	0.627234	−	0.778831i	$-0.284187\pi$
$577$	30.1333	1.25447	0.627234	+	0.778831i	$0.284187\pi$
$578$	0	0
$579$	−1.07022	−0.0444768
$580$	0	0
$581$	−10.9092	−0.452592
$582$	0	0
$583$	9.71172	0.402218
$584$	0	0
$585$	−6.42353	−0.265580
$586$	0	0
$587$	8.67479	0.358047	0.179023	−	0.983845i	$-0.442706\pi$
$587$	8.67479	0.358047	0.179023	+	0.983845i	$0.442706\pi$
$588$	0	0
$589$	−4.26910	−0.175905
$590$	0	0
$591$	−0.187082	−0.00769553
$592$	0	0
$593$	−34.9197	−1.43398	−0.716991	−	0.697082i	$-0.754481\pi$
$593$	−34.9197	−1.43398	−0.716991	+	0.697082i	$0.754481\pi$
$594$	0	0
$595$	0.351802	0.0144225
$596$	0	0
$597$	1.38029	0.0564913
$598$	0	0
$599$	15.3618	0.627665	0.313832	−	0.949478i	$-0.398387\pi$
$599$	15.3618	0.627665	0.313832	+	0.949478i	$0.398387\pi$
$600$	0	0
$601$	−20.4421	−0.833852	−0.416926	−	0.908940i	$-0.636893\pi$
$601$	−20.4421	−0.833852	−0.416926	+	0.908940i	$0.636893\pi$
$602$	0	0
$603$	5.08558	0.207101
$604$	0	0
$605$	−2.15143	−0.0874682
$606$	0	0
$607$	32.9713	1.33826	0.669131	−	0.743144i	$-0.266666\pi$
$607$	32.9713	1.33826	0.669131	+	0.743144i	$0.266666\pi$
$608$	0	0
$609$	−0.285091	−0.0115525
$610$	0	0
$611$	3.87753	0.156868
$612$	0	0
$613$	10.6930	0.431885	0.215943	−	0.976406i	$-0.430718\pi$
$613$	10.6930	0.431885	0.215943	+	0.976406i	$0.430718\pi$
$614$	0	0
$615$	0.466881	0.0188265
$616$	0	0
$617$	−49.3733	−1.98769	−0.993846	−	0.110767i	$-0.964669\pi$
$617$	−49.3733	−1.98769	−0.993846	+	0.110767i	$0.964669\pi$
$618$	0	0
$619$	7.10230	0.285466	0.142733	−	0.989761i	$-0.454411\pi$
$619$	7.10230	0.285466	0.142733	+	0.989761i	$0.454411\pi$
$620$	0	0
$621$	4.30798	0.172873
$622$	0	0
$623$	−12.5726	−0.503712
$624$	0	0
$625$	−23.0054	−0.920218
$626$	0	0
$627$	0.465827	0.0186034
$628$	0	0
$629$	−1.44231	−0.0575087
$630$	0	0
$631$	−1.73628	−0.0691201	−0.0345601	−	0.999403i	$-0.511003\pi$
$631$	−1.73628	−0.0691201	−0.0345601	+	0.999403i	$0.511003\pi$
$632$	0	0
$633$	−0.726245	−0.0288656
$634$	0	0
$635$	7.58583	0.301035
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−17.8509	−0.706170
$640$	0	0
$641$	−11.5510	−0.456236	−0.228118	−	0.973634i	$-0.573257\pi$
$641$	−11.5510	−0.456236	−0.228118	+	0.973634i	$0.573257\pi$
$642$	0	0
$643$	−13.2685	−0.523257	−0.261628	−	0.965169i	$-0.584260\pi$
$643$	−13.2685	−0.523257	−0.261628	+	0.965169i	$0.584260\pi$
$644$	0	0
$645$	0.575955	0.0226782
$646$	0	0
$647$	12.8930	0.506876	0.253438	−	0.967352i	$-0.418439\pi$
$647$	12.8930	0.506876	0.253438	+	0.967352i	$0.418439\pi$
$648$	0	0
$649$	−9.50729	−0.373194
$650$	0	0
$651$	−0.131080	−0.00513743
$652$	0	0
$653$	−40.1699	−1.57197	−0.785985	−	0.618246i	$-0.787843\pi$
$653$	−40.1699	−1.57197	−0.785985	+	0.618246i	$0.787843\pi$
$654$	0	0
$655$	1.86787	0.0729836
$656$	0	0
$657$	29.2719	1.14201
$658$	0	0
$659$	31.1428	1.21315	0.606576	−	0.795026i	$-0.292543\pi$
$659$	31.1428	1.21315	0.606576	+	0.795026i	$0.292543\pi$
$660$	0	0
$661$	−20.6894	−0.804726	−0.402363	−	0.915480i	$-0.631811\pi$
$661$	−20.6894	−0.804726	−0.402363	+	0.915480i	$0.631811\pi$
$662$	0	0
$663$	−0.0195561	−0.000759497 0
$664$	0	0
$665$	8.37993	0.324960
$666$	0	0
$667$	14.3455	0.555461
$668$	0	0
$669$	−1.21297	−0.0468960
$670$	0	0
$671$	11.6906	0.451309
$672$	0	0
$673$	51.6793	1.99209	0.996046	−	0.0888396i	$-0.0283159\pi$
$673$	51.6793	1.99209	0.996046	+	0.0888396i	$0.0283159\pi$
$674$	0	0
$675$	−0.265822	−0.0102315
$676$	0	0
$677$	31.3116	1.20340	0.601702	−	0.798721i	$-0.294489\pi$
$677$	31.3116	1.20340	0.601702	+	0.798721i	$0.294489\pi$
$678$	0	0
$679$	−3.55279	−0.136344
$680$	0	0
$681$	−3.39541	−0.130112
$682$	0	0
$683$	−33.0714	−1.26544	−0.632722	−	0.774379i	$-0.718062\pi$
$683$	−33.0714	−1.26544	−0.632722	+	0.774379i	$0.718062\pi$
$684$	0	0
$685$	15.8971	0.607398
$686$	0	0
$687$	0.733465	0.0279834
$688$	0	0
$689$	−9.71172	−0.369987
$690$	0	0
$691$	−5.45643	−0.207573	−0.103786	−	0.994600i	$-0.533096\pi$
$691$	−5.45643	−0.207573	−0.103786	+	0.994600i	$0.533096\pi$
$692$	0	0
$693$	−2.98570	−0.113417
$694$	0	0
$695$	−25.2695	−0.958526
$696$	0	0
$697$	−0.296712	−0.0112388
$698$	0	0
$699$	2.75330	0.104139
$700$	0	0
$701$	−3.01735	−0.113964	−0.0569818	−	0.998375i	$-0.518148\pi$
$701$	−3.01735	−0.113964	−0.0569818	+	0.998375i	$0.518148\pi$
$702$	0	0
$703$	−34.3559	−1.29576
$704$	0	0
$705$	−0.997691	−0.0375752
$706$	0	0
$707$	−7.32627	−0.275533
$708$	0	0
$709$	−9.80405	−0.368199	−0.184099	−	0.982908i	$-0.558937\pi$
$709$	−9.80405	−0.368199	−0.184099	+	0.982908i	$0.558937\pi$
$710$	0	0
$711$	29.6149	1.11065
$712$	0	0
$713$	6.59584	0.247016
$714$	0	0
$715$	2.15143	0.0804591
$716$	0	0
$717$	3.53749	0.132110
$718$	0	0
$719$	7.18664	0.268016	0.134008	−	0.990980i	$-0.457215\pi$
$719$	7.18664	0.268016	0.134008	+	0.990980i	$0.457215\pi$
$720$	0	0
$721$	10.4024	0.387404
$722$	0	0
$723$	−0.917296	−0.0341146
$724$	0	0
$725$	−0.885185	−0.0328750
$726$	0	0
$727$	3.77771	0.140108	0.0700538	−	0.997543i	$-0.477683\pi$
$727$	3.77771	0.140108	0.0700538	+	0.997543i	$0.477683\pi$
$728$	0	0
$729$	−26.2307	−0.971508
$730$	0	0
$731$	−0.366031	−0.0135382
$732$	0	0
$733$	4.43791	0.163918	0.0819589	−	0.996636i	$-0.473882\pi$
$733$	4.43791	0.163918	0.0819589	+	0.996636i	$0.473882\pi$
$734$	0	0
$735$	0.257301	0.00949067
$736$	0	0
$737$	−1.70331	−0.0627424
$738$	0	0
$739$	20.7927	0.764870	0.382435	−	0.923982i	$-0.375086\pi$
$739$	20.7927	0.764870	0.382435	+	0.923982i	$0.375086\pi$
$740$	0	0
$741$	−0.465827	−0.0171126
$742$	0	0
$743$	23.2084	0.851432	0.425716	−	0.904857i	$-0.360022\pi$
$743$	23.2084	0.851432	0.425716	+	0.904857i	$0.360022\pi$
$744$	0	0
$745$	44.4108	1.62708
$746$	0	0
$747$	32.5717	1.19174
$748$	0	0
$749$	−11.6898	−0.427137
$750$	0	0
$751$	40.7473	1.48689	0.743445	−	0.668797i	$-0.233190\pi$
$751$	40.7473	1.48689	0.743445	+	0.668797i	$0.233190\pi$
$752$	0	0
$753$	−0.178394	−0.00650103
$754$	0	0
$755$	2.58719	0.0941573
$756$	0	0
$757$	28.6752	1.04222	0.521110	−	0.853490i	$-0.325518\pi$
$757$	28.6752	1.04222	0.521110	+	0.853490i	$0.325518\pi$
$758$	0	0
$759$	−0.719713	−0.0261239
$760$	0	0
$761$	7.91078	0.286766	0.143383	−	0.989667i	$-0.454202\pi$
$761$	7.91078	0.286766	0.143383	+	0.989667i	$0.454202\pi$
$762$	0	0
$763$	−11.5540	−0.418283
$764$	0	0
$765$	−1.05037	−0.0379763
$766$	0	0
$767$	9.50729	0.343288
$768$	0	0
$769$	21.2191	0.765179	0.382590	−	0.923918i	$-0.375032\pi$
$769$	21.2191	0.765179	0.382590	+	0.923918i	$0.375032\pi$
$770$	0	0
$771$	3.44731	0.124152
$772$	0	0
$773$	−26.0827	−0.938129	−0.469065	−	0.883164i	$-0.655409\pi$
$773$	−26.0827	−0.938129	−0.469065	+	0.883164i	$0.655409\pi$
$774$	0	0
$775$	−0.406994	−0.0146197
$776$	0	0
$777$	−1.05488	−0.0378435
$778$	0	0
$779$	−7.06769	−0.253226
$780$	0	0
$781$	5.97880	0.213938
$782$	0	0
$783$	1.70647	0.0609841
$784$	0	0
$785$	18.6749	0.666535
$786$	0	0
$787$	−4.49290	−0.160155	−0.0800773	−	0.996789i	$-0.525517\pi$
$787$	−4.49290	−0.160155	−0.0800773	+	0.996789i	$0.525517\pi$
$788$	0	0
$789$	0.886582	0.0315632
$790$	0	0
$791$	13.0117	0.462643
$792$	0	0
$793$	−11.6906	−0.415144
$794$	0	0
$795$	2.49883	0.0886244
$796$	0	0
$797$	−52.1048	−1.84565	−0.922823	−	0.385225i	$-0.874124\pi$
$797$	−52.1048	−1.84565	−0.922823	+	0.385225i	$0.874124\pi$
$798$	0	0
$799$	0.634053	0.0224312
$800$	0	0
$801$	37.5381	1.32634
$802$	0	0
$803$	−9.80405	−0.345977
$804$	0	0
$805$	−12.9472	−0.456327
$806$	0	0
$807$	−2.90631	−0.102307
$808$	0	0
$809$	−40.8545	−1.43637	−0.718183	−	0.695854i	$-0.755026\pi$
$809$	−40.8545	−1.43637	−0.718183	+	0.695854i	$0.755026\pi$
$810$	0	0
$811$	−26.2827	−0.922912	−0.461456	−	0.887163i	$-0.652673\pi$
$811$	−26.2827	−0.922912	−0.461456	+	0.887163i	$0.652673\pi$
$812$	0	0
$813$	2.30740	0.0809241
$814$	0	0
$815$	−21.4822	−0.752490
$816$	0	0
$817$	−8.71887	−0.305035
$818$	0	0
$819$	2.98570	0.104329
$820$	0	0
$821$	14.4438	0.504090	0.252045	−	0.967715i	$-0.418897\pi$
$821$	14.4438	0.504090	0.252045	+	0.967715i	$0.418897\pi$
$822$	0	0
$823$	40.6120	1.41564	0.707822	−	0.706391i	$-0.249678\pi$
$823$	40.6120	1.41564	0.707822	+	0.706391i	$0.249678\pi$
$824$	0	0
$825$	0.0444096	0.00154614
$826$	0	0
$827$	−12.6662	−0.440446	−0.220223	−	0.975450i	$-0.570679\pi$
$827$	−12.6662	−0.440446	−0.220223	+	0.975450i	$0.570679\pi$
$828$	0	0
$829$	38.0883	1.32286	0.661430	−	0.750007i	$-0.269950\pi$
$829$	38.0883	1.32286	0.661430	+	0.750007i	$0.269950\pi$
$830$	0	0
$831$	−0.0396165	−0.00137428
$832$	0	0
$833$	−0.163520	−0.00566562
$834$	0	0
$835$	−11.2830	−0.390465
$836$	0	0
$837$	0.784605	0.0271199
$838$	0	0
$839$	−52.4116	−1.80945	−0.904724	−	0.425998i	$-0.859923\pi$
$839$	−52.4116	−1.80945	−0.904724	+	0.425998i	$0.859923\pi$
$840$	0	0
$841$	−23.3175	−0.804051
$842$	0	0
$843$	−3.58103	−0.123337
$844$	0	0
$845$	−2.15143	−0.0740116
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	3.79764	0.130335
$850$	0	0
$851$	53.0805	1.81958
$852$	0	0
$853$	19.8226	0.678711	0.339356	−	0.940658i	$-0.389791\pi$
$853$	19.8226	0.678711	0.339356	+	0.940658i	$0.389791\pi$
$854$	0	0
$855$	−25.0199	−0.855663
$856$	0	0
$857$	−3.16240	−0.108026	−0.0540128	−	0.998540i	$-0.517201\pi$
$857$	−3.16240	−0.108026	−0.0540128	+	0.998540i	$0.517201\pi$
$858$	0	0
$859$	2.77991	0.0948495	0.0474247	−	0.998875i	$-0.484899\pi$
$859$	2.77991	0.0948495	0.0474247	+	0.998875i	$0.484899\pi$
$860$	0	0
$861$	−0.217009	−0.00739565
$862$	0	0
$863$	−46.5329	−1.58400	−0.791999	−	0.610522i	$-0.790960\pi$
$863$	−46.5329	−1.58400	−0.791999	+	0.610522i	$0.790960\pi$
$864$	0	0
$865$	0.728120	0.0247568
$866$	0	0
$867$	2.02992	0.0689396
$868$	0	0
$869$	−9.91892	−0.336476
$870$	0	0
$871$	1.70331	0.0577146
$872$	0	0
$873$	10.6076	0.359011
$874$	0	0
$875$	11.5561	0.390666
$876$	0	0
$877$	−11.9700	−0.404197	−0.202099	−	0.979365i	$-0.564776\pi$
$877$	−11.9700	−0.404197	−0.202099	+	0.979365i	$0.564776\pi$
$878$	0	0
$879$	−3.63165	−0.122493
$880$	0	0
$881$	−35.7418	−1.20417	−0.602086	−	0.798431i	$-0.705664\pi$
$881$	−35.7418	−1.20417	−0.602086	+	0.798431i	$0.705664\pi$
$882$	0	0
$883$	56.0664	1.88678	0.943392	−	0.331681i	$-0.107616\pi$
$883$	56.0664	1.88678	0.943392	+	0.331681i	$0.107616\pi$
$884$	0	0
$885$	−2.44623	−0.0822292
$886$	0	0
$887$	−36.4762	−1.22475	−0.612375	−	0.790568i	$-0.709786\pi$
$887$	−36.4762	−1.22475	−0.612375	+	0.790568i	$0.709786\pi$
$888$	0	0
$889$	−3.52594	−0.118256
$890$	0	0
$891$	8.87148	0.297206
$892$	0	0
$893$	15.1032	0.505408
$894$	0	0
$895$	−9.53142	−0.318600
$896$	0	0
$897$	0.719713	0.0240305
$898$	0	0
$899$	2.61273	0.0871393
$900$	0	0
$901$	−1.58806	−0.0529058
$902$	0	0
$903$	−0.267708	−0.00890875
$904$	0	0
$905$	44.2495	1.47090
$906$	0	0
$907$	−20.6454	−0.685520	−0.342760	−	0.939423i	$-0.611362\pi$
$907$	−20.6454	−0.685520	−0.342760	+	0.939423i	$0.611362\pi$
$908$	0	0
$909$	21.8740	0.725516
$910$	0	0
$911$	−18.8773	−0.625432	−0.312716	−	0.949847i	$-0.601239\pi$
$911$	−18.8773	−0.625432	−0.312716	+	0.949847i	$0.601239\pi$
$912$	0	0
$913$	−10.9092	−0.361043
$914$	0	0
$915$	3.00799	0.0994410
$916$	0	0
$917$	−0.868196	−0.0286704
$918$	0	0
$919$	29.7179	0.980302	0.490151	−	0.871637i	$-0.336942\pi$
$919$	29.7179	0.980302	0.490151	+	0.871637i	$0.336942\pi$
$920$	0	0
$921$	−0.677103	−0.0223113
$922$	0	0
$923$	−5.97880	−0.196795
$924$	0	0
$925$	−3.27531	−0.107692
$926$	0	0
$927$	−31.0583	−1.02009
$928$	0	0
$929$	−17.6450	−0.578913	−0.289457	−	0.957191i	$-0.593475\pi$
$929$	−17.6450	−0.578913	−0.289457	+	0.957191i	$0.593475\pi$
$930$	0	0
$931$	−3.89504	−0.127655
$932$	0	0
$933$	2.22720	0.0729152
$934$	0	0
$935$	0.351802	0.0115051
$936$	0	0
$937$	−10.9420	−0.357458	−0.178729	−	0.983898i	$-0.557199\pi$
$937$	−10.9420	−0.357458	−0.178729	+	0.983898i	$0.557199\pi$
$938$	0	0
$939$	0.641068	0.0209205
$940$	0	0
$941$	−27.2108	−0.887046	−0.443523	−	0.896263i	$-0.646272\pi$
$941$	−27.2108	−0.887046	−0.443523	+	0.896263i	$0.646272\pi$
$942$	0	0
$943$	10.9197	0.355595
$944$	0	0
$945$	−1.54012	−0.0501002
$946$	0	0
$947$	11.5707	0.375997	0.187999	−	0.982169i	$-0.439800\pi$
$947$	11.5707	0.375997	0.187999	+	0.982169i	$0.439800\pi$
$948$	0	0
$949$	9.80405	0.318253
$950$	0	0
$951$	−0.753074	−0.0244201
$952$	0	0
$953$	44.6890	1.44762	0.723809	−	0.690001i	$-0.242390\pi$
$953$	44.6890	1.44762	0.723809	+	0.690001i	$0.242390\pi$
$954$	0	0
$955$	30.6259	0.991030
$956$	0	0
$957$	−0.285091	−0.00921567
$958$	0	0
$959$	−7.38909	−0.238606
$960$	0	0
$961$	−29.7987	−0.961249
$962$	0	0
$963$	34.9023	1.12471
$964$	0	0
$965$	−19.2526	−0.619762
$966$	0	0
$967$	6.14376	0.197570	0.0987850	−	0.995109i	$-0.468504\pi$
$967$	6.14376	0.197570	0.0987850	+	0.995109i	$0.468504\pi$
$968$	0	0
$969$	−0.0761719	−0.00244700
$970$	0	0
$971$	−6.46574	−0.207496	−0.103748	−	0.994604i	$-0.533083\pi$
$971$	−6.46574	−0.207496	−0.103748	+	0.994604i	$0.533083\pi$
$972$	0	0
$973$	11.7454	0.376541
$974$	0	0
$975$	−0.0444096	−0.00142225
$976$	0	0
$977$	36.9554	1.18231	0.591153	−	0.806559i	$-0.298673\pi$
$977$	36.9554	1.18231	0.591153	+	0.806559i	$0.298673\pi$
$978$	0	0
$979$	−12.5726	−0.401823
$980$	0	0
$981$	34.4968	1.10140
$982$	0	0
$983$	−10.3963	−0.331592	−0.165796	−	0.986160i	$-0.553019\pi$
$983$	−10.3963	−0.331592	−0.165796	+	0.986160i	$0.553019\pi$
$984$	0	0
$985$	−3.36548	−0.107233
$986$	0	0
$987$	0.463733	0.0147608
$988$	0	0
$989$	13.4708	0.428348
$990$	0	0
$991$	25.9179	0.823308	0.411654	−	0.911340i	$-0.364951\pi$
$991$	25.9179	0.823308	0.411654	+	0.911340i	$0.364951\pi$
$992$	0	0
$993$	−1.55315	−0.0492878
$994$	0	0
$995$	24.8304	0.787178
$996$	0	0
$997$	−26.7969	−0.848666	−0.424333	−	0.905506i	$-0.639491\pi$
$997$	−26.7969	−0.848666	−0.424333	+	0.905506i	$0.639491\pi$
$998$	0	0
$999$	6.31417	0.199771

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.s.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.s.1.6	✓	10	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)

\(f(q)\)	\(=\)	\(q-0.119595 q^{3} -2.15143 q^{5} +1.00000 q^{7} -2.98570 q^{9} +O(q^{10})\)	\(q-0.119595 q^{3} -2.15143 q^{5} +1.00000 q^{7} -2.98570 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.257301 q^{15} -0.163520 q^{17} -3.89504 q^{19} -0.119595 q^{21} +6.01792 q^{23} -0.371333 q^{25} +0.715859 q^{27} +2.38380 q^{29} +1.09603 q^{31} -0.119595 q^{33} -2.15143 q^{35} +8.82041 q^{37} +0.119595 q^{39} +1.81454 q^{41} +2.23845 q^{43} +6.42353 q^{45} -3.87753 q^{47} +1.00000 q^{49} +0.0195561 q^{51} +9.71172 q^{53} -2.15143 q^{55} +0.465827 q^{57} -9.50729 q^{59} +11.6906 q^{61} -2.98570 q^{63} +2.15143 q^{65} -1.70331 q^{67} -0.719713 q^{69} +5.97880 q^{71} -9.80405 q^{73} +0.0444096 q^{75} +1.00000 q^{77} -9.91892 q^{79} +8.87148 q^{81} -10.9092 q^{83} +0.351802 q^{85} -0.285091 q^{87} -12.5726 q^{89} -1.00000 q^{91} -0.131080 q^{93} +8.37993 q^{95} -3.55279 q^{97} -2.98570 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9}+O(q^{10}) \) 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9	\( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9} + 10 q^{11} - 10 q^{13} - 5 q^{15} - 11 q^{17} + 2 q^{19} - 3 q^{21} - 8 q^{23} + 2 q^{25} - 15 q^{27} - 8 q^{29} - 23 q^{31} - 3 q^{33} - 4 q^{35} + 10 q^{37} + 3 q^{39} - 18 q^{41} + 12 q^{43} - 10 q^{45} - 36 q^{47} + 10 q^{49} + 9 q^{51} - 21 q^{53} - 4 q^{55} - 30 q^{57} - 13 q^{59} - 2 q^{61} + 9 q^{63} + 4 q^{65} - 4 q^{67} - 26 q^{69} - 24 q^{71} - 23 q^{73} - 28 q^{75} + 10 q^{77} + 14 q^{79} + 30 q^{81} - 9 q^{83} - 17 q^{85} + 7 q^{87} - 18 q^{89} - 10 q^{91} + q^{93} - 4 q^{95} - 9 q^{97} + 9 q^{99}+O(q^{100}) \) 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9 + 10 * q^11 - 10 * q^13 - 5 * q^15 - 11 * q^17 + 2 * q^19 - 3 * q^21 - 8 * q^23 + 2 * q^25 - 15 * q^27 - 8 * q^29 - 23 * q^31 - 3 * q^33 - 4 * q^35 + 10 * q^37 + 3 * q^39 - 18 * q^41 + 12 * q^43 - 10 * q^45 - 36 * q^47 + 10 * q^49 + 9 * q^51 - 21 * q^53 - 4 * q^55 - 30 * q^57 - 13 * q^59 - 2 * q^61 + 9 * q^63 + 4 * q^65 - 4 * q^67 - 26 * q^69 - 24 * q^71 - 23 * q^73 - 28 * q^75 + 10 * q^77 + 14 * q^79 + 30 * q^81 - 9 * q^83 - 17 * q^85 + 7 * q^87 - 18 * q^89 - 10 * q^91 + q^93 - 4 * q^95 - 9 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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