LMFDB - Embedded newform 8008.2.a.u.1.2

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} - 2x^{9} - 16x^{8} + 26x^{7} + 93x^{6} - 113x^{5} - 230x^{4} + 197x^{3} + 201x^{2} - 115x + 8 $ x^10 - 2x^9 - 16x^8 + 26x^7 + 93x^6 - 113x^5 - 230x^4 + 197x^3 + 201x^2 - 115*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			1.2
Root			$-1.80136$ 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)

$f(q)$	$=$	$q-1.80136 q^{3} -2.97899 q^{5} -1.00000 q^{7} +0.244902 q^{9} +O(q^{10})$	$q-1.80136 q^{3} -2.97899 q^{5} -1.00000 q^{7} +0.244902 q^{9} -1.00000 q^{11} -1.00000 q^{13} +5.36624 q^{15} -3.36273 q^{17} -0.650951 q^{19} +1.80136 q^{21} -5.75145 q^{23} +3.87440 q^{25} +4.96293 q^{27} +3.84151 q^{29} +4.81258 q^{31} +1.80136 q^{33} +2.97899 q^{35} +5.38609 q^{37} +1.80136 q^{39} +0.331597 q^{41} -4.36013 q^{43} -0.729562 q^{45} +10.3033 q^{47} +1.00000 q^{49} +6.05749 q^{51} +2.85065 q^{53} +2.97899 q^{55} +1.17260 q^{57} +6.24631 q^{59} +8.41953 q^{61} -0.244902 q^{63} +2.97899 q^{65} -11.3964 q^{67} +10.3604 q^{69} -11.4219 q^{71} -4.43520 q^{73} -6.97920 q^{75} +1.00000 q^{77} +14.2508 q^{79} -9.67473 q^{81} -4.23541 q^{83} +10.0175 q^{85} -6.91995 q^{87} +7.54738 q^{89} +1.00000 q^{91} -8.66920 q^{93} +1.93918 q^{95} -7.46541 q^{97} -0.244902 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9	$ 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} - 3 q^{17} + 13 q^{19} - 2 q^{21} + 6 q^{25} + 14 q^{27} - 3 q^{29} - q^{31} - 2 q^{33} + 4 q^{35} - 5 q^{37} - 2 q^{39} + 4 q^{41} + 19 q^{43} - 11 q^{45} + q^{47} + 10 q^{49} + 15 q^{51} - 6 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 18 q^{61} - 6 q^{63} + 4 q^{65} + q^{67} - 11 q^{69} - 21 q^{71} - 10 q^{73} + 10 q^{77} + 3 q^{79} - 30 q^{81} + 6 q^{83} - 33 q^{85} - 9 q^{87} - 22 q^{89} + 10 q^{91} - 34 q^{93} - 3 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) $ 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 - 3 * q^17 + 13 * q^19 - 2 * q^21 + 6 * q^25 + 14 * q^27 - 3 * q^29 - q^31 - 2 * q^33 + 4 * q^35 - 5 * q^37 - 2 * q^39 + 4 * q^41 + 19 * q^43 - 11 * q^45 + q^47 + 10 * q^49 + 15 * q^51 - 6 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 18 * q^61 - 6 * q^63 + 4 * q^65 + q^67 - 11 * q^69 - 21 * q^71 - 10 * q^73 + 10 * q^77 + 3 * q^79 - 30 * q^81 + 6 * q^83 - 33 * q^85 - 9 * q^87 - 22 * q^89 + 10 * q^91 - 34 * q^93 - 3 * q^95 - 9 * q^97 - 6 * 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
$3$	−1.80136	−1.04002	−0.520008	−	0.854161i	$-0.674071\pi$
$3$	−1.80136	−1.04002	−0.520008	+	0.854161i	$0.674071\pi$
$4$	0	0
$5$	−2.97899	−1.33225	−0.666123	−	0.745842i	$-0.732047\pi$
$5$	−2.97899	−1.33225	−0.666123	+	0.745842i	$0.732047\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.244902	0.0816340
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	5.36624	1.38556
$16$	0	0
$17$	−3.36273	−0.815581	−0.407791	−	0.913076i	$-0.633701\pi$
$17$	−3.36273	−0.815581	−0.407791	+	0.913076i	$0.633701\pi$
$18$	0	0
$19$	−0.650951	−0.149338	−0.0746692	−	0.997208i	$-0.523790\pi$
$19$	−0.650951	−0.149338	−0.0746692	+	0.997208i	$0.523790\pi$
$20$	0	0
$21$	1.80136	0.393089
$22$	0	0
$23$	−5.75145	−1.19926	−0.599631	−	0.800277i	$-0.704686\pi$
$23$	−5.75145	−1.19926	−0.599631	+	0.800277i	$0.704686\pi$
$24$	0	0
$25$	3.87440	0.774880
$26$	0	0
$27$	4.96293	0.955116
$28$	0	0
$29$	3.84151	0.713351	0.356676	−	0.934228i	$-0.383910\pi$
$29$	3.84151	0.713351	0.356676	+	0.934228i	$0.383910\pi$
$30$	0	0
$31$	4.81258	0.864365	0.432183	−	0.901786i	$-0.357744\pi$
$31$	4.81258	0.864365	0.432183	+	0.901786i	$0.357744\pi$
$32$	0	0
$33$	1.80136	0.313577
$34$	0	0
$35$	2.97899	0.503542
$36$	0	0
$37$	5.38609	0.885467	0.442734	−	0.896653i	$-0.354009\pi$
$37$	5.38609	0.885467	0.442734	+	0.896653i	$0.354009\pi$
$38$	0	0
$39$	1.80136	0.288449
$40$	0	0
$41$	0.331597	0.0517868	0.0258934	−	0.999665i	$-0.491757\pi$
$41$	0.331597	0.0517868	0.0258934	+	0.999665i	$0.491757\pi$
$42$	0	0
$43$	−4.36013	−0.664914	−0.332457	−	0.943118i	$-0.607878\pi$
$43$	−4.36013	−0.664914	−0.332457	+	0.943118i	$0.607878\pi$
$44$	0	0
$45$	−0.729562	−0.108757
$46$	0	0
$47$	10.3033	1.50289	0.751447	−	0.659793i	$-0.229356\pi$
$47$	10.3033	1.50289	0.751447	+	0.659793i	$0.229356\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.05749	0.848218
$52$	0	0
$53$	2.85065	0.391566	0.195783	−	0.980647i	$-0.437275\pi$
$53$	2.85065	0.391566	0.195783	+	0.980647i	$0.437275\pi$
$54$	0	0
$55$	2.97899	0.401687
$56$	0	0
$57$	1.17260	0.155314
$58$	0	0
$59$	6.24631	0.813200	0.406600	−	0.913606i	$-0.366714\pi$
$59$	6.24631	0.813200	0.406600	+	0.913606i	$0.366714\pi$
$60$	0	0
$61$	8.41953	1.07801	0.539005	−	0.842302i	$-0.318800\pi$
$61$	8.41953	1.07801	0.539005	+	0.842302i	$0.318800\pi$
$62$	0	0
$63$	−0.244902	−0.0308548
$64$	0	0
$65$	2.97899	0.369499
$66$	0	0
$67$	−11.3964	−1.39229	−0.696145	−	0.717901i	$-0.745103\pi$
$67$	−11.3964	−1.39229	−0.696145	+	0.717901i	$0.745103\pi$
$68$	0	0
$69$	10.3604	1.24725
$70$	0	0
$71$	−11.4219	−1.35553	−0.677763	−	0.735281i	$-0.737050\pi$
$71$	−11.4219	−1.35553	−0.677763	+	0.735281i	$0.737050\pi$
$72$	0	0
$73$	−4.43520	−0.519101	−0.259551	−	0.965729i	$-0.583574\pi$
$73$	−4.43520	−0.519101	−0.259551	+	0.965729i	$0.583574\pi$
$74$	0	0
$75$	−6.97920	−0.805888
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	14.2508	1.60334	0.801671	−	0.597766i	$-0.203945\pi$
$79$	14.2508	1.60334	0.801671	+	0.597766i	$0.203945\pi$
$80$	0	0
$81$	−9.67473	−1.07497
$82$	0	0
$83$	−4.23541	−0.464896	−0.232448	−	0.972609i	$-0.574674\pi$
$83$	−4.23541	−0.464896	−0.232448	+	0.972609i	$0.574674\pi$
$84$	0	0
$85$	10.0175	1.08655
$86$	0	0
$87$	−6.91995	−0.741897
$88$	0	0
$89$	7.54738	0.800020	0.400010	−	0.916511i	$-0.369007\pi$
$89$	7.54738	0.800020	0.400010	+	0.916511i	$0.369007\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−8.66920	−0.898954
$94$	0	0
$95$	1.93918	0.198956
$96$	0	0
$97$	−7.46541	−0.757998	−0.378999	−	0.925397i	$-0.623732\pi$
$97$	−7.46541	−0.757998	−0.378999	+	0.925397i	$0.623732\pi$
$98$	0	0
$99$	−0.244902	−0.0246136
$100$	0	0
$101$	−12.5569	−1.24945	−0.624727	−	0.780843i	$-0.714790\pi$
$101$	−12.5569	−1.24945	−0.624727	+	0.780843i	$0.714790\pi$
$102$	0	0
$103$	13.6746	1.34739	0.673697	−	0.739008i	$-0.264705\pi$
$103$	13.6746	1.34739	0.673697	+	0.739008i	$0.264705\pi$
$104$	0	0
$105$	−5.36624	−0.523692
$106$	0	0
$107$	−4.07770	−0.394206	−0.197103	−	0.980383i	$-0.563153\pi$
$107$	−4.07770	−0.394206	−0.197103	+	0.980383i	$0.563153\pi$
$108$	0	0
$109$	0.162712	0.0155850	0.00779251	−	0.999970i	$-0.497520\pi$
$109$	0.162712	0.0155850	0.00779251	+	0.999970i	$0.497520\pi$
$110$	0	0
$111$	−9.70229	−0.920900
$112$	0	0
$113$	−11.3896	−1.07144	−0.535721	−	0.844395i	$-0.679960\pi$
$113$	−11.3896	−1.07144	−0.535721	+	0.844395i	$0.679960\pi$
$114$	0	0
$115$	17.1335	1.59771
$116$	0	0
$117$	−0.244902	−0.0226412
$118$	0	0
$119$	3.36273	0.308261
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.597327	−0.0538591
$124$	0	0
$125$	3.35315	0.299915
$126$	0	0
$127$	1.53467	0.136180	0.0680898	−	0.997679i	$-0.478310\pi$
$127$	1.53467	0.136180	0.0680898	+	0.997679i	$0.478310\pi$
$128$	0	0
$129$	7.85417	0.691521
$130$	0	0
$131$	20.0747	1.75393	0.876965	−	0.480554i	$-0.159564\pi$
$131$	20.0747	1.75393	0.876965	+	0.480554i	$0.159564\pi$
$132$	0	0
$133$	0.650951	0.0564446
$134$	0	0
$135$	−14.7845	−1.27245
$136$	0	0
$137$	6.81319	0.582090	0.291045	−	0.956709i	$-0.405997\pi$
$137$	6.81319	0.582090	0.291045	+	0.956709i	$0.405997\pi$
$138$	0	0
$139$	10.8278	0.918405	0.459203	−	0.888331i	$-0.348135\pi$
$139$	10.8278	0.918405	0.459203	+	0.888331i	$0.348135\pi$
$140$	0	0
$141$	−18.5600	−1.56303
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−11.4438	−0.950359
$146$	0	0
$147$	−1.80136	−0.148574
$148$	0	0
$149$	15.5704	1.27557	0.637787	−	0.770213i	$-0.279850\pi$
$149$	15.5704	1.27557	0.637787	+	0.770213i	$0.279850\pi$
$150$	0	0
$151$	−3.92435	−0.319359	−0.159680	−	0.987169i	$-0.551046\pi$
$151$	−3.92435	−0.319359	−0.159680	+	0.987169i	$0.551046\pi$
$152$	0	0
$153$	−0.823539	−0.0665792
$154$	0	0
$155$	−14.3367	−1.15155
$156$	0	0
$157$	4.52819	0.361389	0.180694	−	0.983539i	$-0.442165\pi$
$157$	4.52819	0.361389	0.180694	+	0.983539i	$0.442165\pi$
$158$	0	0
$159$	−5.13504	−0.407236
$160$	0	0
$161$	5.75145	0.453278
$162$	0	0
$163$	15.5743	1.21987	0.609937	−	0.792450i	$-0.291195\pi$
$163$	15.5743	1.21987	0.609937	+	0.792450i	$0.291195\pi$
$164$	0	0
$165$	−5.36624	−0.417761
$166$	0	0
$167$	2.96986	0.229815	0.114907	−	0.993376i	$-0.463343\pi$
$167$	2.96986	0.229815	0.114907	+	0.993376i	$0.463343\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.159419	−0.0121911
$172$	0	0
$173$	9.74884	0.741191	0.370595	−	0.928794i	$-0.379154\pi$
$173$	9.74884	0.741191	0.370595	+	0.928794i	$0.379154\pi$
$174$	0	0
$175$	−3.87440	−0.292877
$176$	0	0
$177$	−11.2519	−0.845741
$178$	0	0
$179$	10.3831	0.776069	0.388034	−	0.921645i	$-0.373154\pi$
$179$	10.3831	0.776069	0.388034	+	0.921645i	$0.373154\pi$
$180$	0	0
$181$	−15.6304	−1.16180	−0.580901	−	0.813974i	$-0.697300\pi$
$181$	−15.6304	−1.16180	−0.580901	+	0.813974i	$0.697300\pi$
$182$	0	0
$183$	−15.1666	−1.12115
$184$	0	0
$185$	−16.0451	−1.17966
$186$	0	0
$187$	3.36273	0.245907
$188$	0	0
$189$	−4.96293	−0.361000
$190$	0	0
$191$	−23.0063	−1.66468	−0.832340	−	0.554266i	$-0.812999\pi$
$191$	−23.0063	−1.66468	−0.832340	+	0.554266i	$0.812999\pi$
$192$	0	0
$193$	−1.80559	−0.129969	−0.0649847	−	0.997886i	$-0.520700\pi$
$193$	−1.80559	−0.129969	−0.0649847	+	0.997886i	$0.520700\pi$
$194$	0	0
$195$	−5.36624	−0.384285
$196$	0	0
$197$	−22.0515	−1.57110	−0.785551	−	0.618797i	$-0.787620\pi$
$197$	−22.0515	−1.57110	−0.785551	+	0.618797i	$0.787620\pi$
$198$	0	0
$199$	−1.13812	−0.0806792	−0.0403396	−	0.999186i	$-0.512844\pi$
$199$	−1.13812	−0.0806792	−0.0403396	+	0.999186i	$0.512844\pi$
$200$	0	0
$201$	20.5290	1.44800
$202$	0	0
$203$	−3.84151	−0.269621
$204$	0	0
$205$	−0.987826	−0.0689928
$206$	0	0
$207$	−1.40854	−0.0979006
$208$	0	0
$209$	0.650951	0.0450272
$210$	0	0
$211$	14.3439	0.987473	0.493737	−	0.869612i	$-0.335631\pi$
$211$	14.3439	0.987473	0.493737	+	0.869612i	$0.335631\pi$
$212$	0	0
$213$	20.5749	1.40977
$214$	0	0
$215$	12.9888	0.885829
$216$	0	0
$217$	−4.81258	−0.326699
$218$	0	0
$219$	7.98940	0.539874
$220$	0	0
$221$	3.36273	0.226201
$222$	0	0
$223$	17.4436	1.16811	0.584056	−	0.811713i	$-0.301465\pi$
$223$	17.4436	1.16811	0.584056	+	0.811713i	$0.301465\pi$
$224$	0	0
$225$	0.948849	0.0632566
$226$	0	0
$227$	−6.54401	−0.434341	−0.217170	−	0.976134i	$-0.569683\pi$
$227$	−6.54401	−0.434341	−0.217170	+	0.976134i	$0.569683\pi$
$228$	0	0
$229$	−6.77490	−0.447698	−0.223849	−	0.974624i	$-0.571862\pi$
$229$	−6.77490	−0.447698	−0.223849	+	0.974624i	$0.571862\pi$
$230$	0	0
$231$	−1.80136	−0.118521
$232$	0	0
$233$	10.4613	0.685341	0.342671	−	0.939456i	$-0.388669\pi$
$233$	10.4613	0.685341	0.342671	+	0.939456i	$0.388669\pi$
$234$	0	0
$235$	−30.6935	−2.00223
$236$	0	0
$237$	−25.6709	−1.66750
$238$	0	0
$239$	−5.22666	−0.338084	−0.169042	−	0.985609i	$-0.554067\pi$
$239$	−5.22666	−0.338084	−0.169042	+	0.985609i	$0.554067\pi$
$240$	0	0
$241$	−2.09179	−0.134744	−0.0673722	−	0.997728i	$-0.521461\pi$
$241$	−2.09179	−0.134744	−0.0673722	+	0.997728i	$0.521461\pi$
$242$	0	0
$243$	2.53890	0.162871
$244$	0	0
$245$	−2.97899	−0.190321
$246$	0	0
$247$	0.650951	0.0414190
$248$	0	0
$249$	7.62950	0.483500
$250$	0	0
$251$	8.24646	0.520512	0.260256	−	0.965540i	$-0.416193\pi$
$251$	8.24646	0.520512	0.260256	+	0.965540i	$0.416193\pi$
$252$	0	0
$253$	5.75145	0.361591
$254$	0	0
$255$	−18.0452	−1.13003
$256$	0	0
$257$	−9.41326	−0.587183	−0.293591	−	0.955931i	$-0.594851\pi$
$257$	−9.41326	−0.587183	−0.293591	+	0.955931i	$0.594851\pi$
$258$	0	0
$259$	−5.38609	−0.334675
$260$	0	0
$261$	0.940795	0.0582337
$262$	0	0
$263$	−15.2008	−0.937324	−0.468662	−	0.883378i	$-0.655264\pi$
$263$	−15.2008	−0.937324	−0.468662	+	0.883378i	$0.655264\pi$
$264$	0	0
$265$	−8.49206	−0.521663
$266$	0	0
$267$	−13.5956	−0.832034
$268$	0	0
$269$	−4.44135	−0.270794	−0.135397	−	0.990791i	$-0.543231\pi$
$269$	−4.44135	−0.270794	−0.135397	+	0.990791i	$0.543231\pi$
$270$	0	0
$271$	22.9282	1.39279	0.696393	−	0.717661i	$-0.254787\pi$
$271$	22.9282	1.39279	0.696393	+	0.717661i	$0.254787\pi$
$272$	0	0
$273$	−1.80136	−0.109023
$274$	0	0
$275$	−3.87440	−0.233635
$276$	0	0
$277$	−7.57984	−0.455429	−0.227714	−	0.973728i	$-0.573125\pi$
$277$	−7.57984	−0.455429	−0.227714	+	0.973728i	$0.573125\pi$
$278$	0	0
$279$	1.17861	0.0705617
$280$	0	0
$281$	−4.29252	−0.256070	−0.128035	−	0.991770i	$-0.540867\pi$
$281$	−4.29252	−0.256070	−0.128035	+	0.991770i	$0.540867\pi$
$282$	0	0
$283$	18.1474	1.07875	0.539375	−	0.842066i	$-0.318661\pi$
$283$	18.1474	1.07875	0.539375	+	0.842066i	$0.318661\pi$
$284$	0	0
$285$	−3.49316	−0.206917
$286$	0	0
$287$	−0.331597	−0.0195736
$288$	0	0
$289$	−5.69207	−0.334828
$290$	0	0
$291$	13.4479	0.788330
$292$	0	0
$293$	−22.3980	−1.30850	−0.654251	−	0.756277i	$-0.727016\pi$
$293$	−22.3980	−1.30850	−0.654251	+	0.756277i	$0.727016\pi$
$294$	0	0
$295$	−18.6077	−1.08338
$296$	0	0
$297$	−4.96293	−0.287978
$298$	0	0
$299$	5.75145	0.332615
$300$	0	0
$301$	4.36013	0.251314
$302$	0	0
$303$	22.6195	1.29945
$304$	0	0
$305$	−25.0817	−1.43618
$306$	0	0
$307$	23.2327	1.32596	0.662979	−	0.748638i	$-0.269292\pi$
$307$	23.2327	1.32596	0.662979	+	0.748638i	$0.269292\pi$
$308$	0	0
$309$	−24.6328	−1.40131
$310$	0	0
$311$	−14.9345	−0.846859	−0.423429	−	0.905929i	$-0.639174\pi$
$311$	−14.9345	−0.846859	−0.423429	+	0.905929i	$0.639174\pi$
$312$	0	0
$313$	−18.4203	−1.04118	−0.520590	−	0.853807i	$-0.674288\pi$
$313$	−18.4203	−1.04118	−0.520590	+	0.853807i	$0.674288\pi$
$314$	0	0
$315$	0.729562	0.0411062
$316$	0	0
$317$	−17.3295	−0.973323	−0.486661	−	0.873591i	$-0.661785\pi$
$317$	−17.3295	−0.973323	−0.486661	+	0.873591i	$0.661785\pi$
$318$	0	0
$319$	−3.84151	−0.215083
$320$	0	0
$321$	7.34542	0.409981
$322$	0	0
$323$	2.18897	0.121798
$324$	0	0
$325$	−3.87440	−0.214913
$326$	0	0
$327$	−0.293104	−0.0162087
$328$	0	0
$329$	−10.3033	−0.568041
$330$	0	0
$331$	21.8640	1.20175	0.600876	−	0.799342i	$-0.294818\pi$
$331$	21.8640	1.20175	0.600876	+	0.799342i	$0.294818\pi$
$332$	0	0
$333$	1.31906	0.0722843
$334$	0	0
$335$	33.9498	1.85487
$336$	0	0
$337$	−14.5332	−0.791674	−0.395837	−	0.918321i	$-0.629546\pi$
$337$	−14.5332	−0.791674	−0.395837	+	0.918321i	$0.629546\pi$
$338$	0	0
$339$	20.5167	1.11432
$340$	0	0
$341$	−4.81258	−0.260616
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−30.8637	−1.66165
$346$	0	0
$347$	−25.6184	−1.37527	−0.687634	−	0.726058i	$-0.741351\pi$
$347$	−25.6184	−1.37527	−0.687634	+	0.726058i	$0.741351\pi$
$348$	0	0
$349$	−1.20847	−0.0646880	−0.0323440	−	0.999477i	$-0.510297\pi$
$349$	−1.20847	−0.0646880	−0.0323440	+	0.999477i	$0.510297\pi$
$350$	0	0
$351$	−4.96293	−0.264901
$352$	0	0
$353$	0.692068	0.0368351	0.0184175	−	0.999830i	$-0.494137\pi$
$353$	0.692068	0.0368351	0.0184175	+	0.999830i	$0.494137\pi$
$354$	0	0
$355$	34.0256	1.80589
$356$	0	0
$357$	−6.05749	−0.320596
$358$	0	0
$359$	−30.5310	−1.61137	−0.805683	−	0.592347i	$-0.798202\pi$
$359$	−30.5310	−1.61137	−0.805683	+	0.592347i	$0.798202\pi$
$360$	0	0
$361$	−18.5763	−0.977698
$362$	0	0
$363$	−1.80136	−0.0945469
$364$	0	0
$365$	13.2124	0.691571
$366$	0	0
$367$	−27.3898	−1.42973	−0.714867	−	0.699260i	$-0.753513\pi$
$367$	−27.3898	−1.42973	−0.714867	+	0.699260i	$0.753513\pi$
$368$	0	0
$369$	0.0812089	0.00422757
$370$	0	0
$371$	−2.85065	−0.147998
$372$	0	0
$373$	12.5046	0.647464	0.323732	−	0.946149i	$-0.395062\pi$
$373$	12.5046	0.647464	0.323732	+	0.946149i	$0.395062\pi$
$374$	0	0
$375$	−6.04024	−0.311917
$376$	0	0
$377$	−3.84151	−0.197848
$378$	0	0
$379$	−14.9991	−0.770451	−0.385225	−	0.922822i	$-0.625876\pi$
$379$	−14.9991	−0.770451	−0.385225	+	0.922822i	$0.625876\pi$
$380$	0	0
$381$	−2.76449	−0.141629
$382$	0	0
$383$	−26.6330	−1.36088	−0.680442	−	0.732802i	$-0.738212\pi$
$383$	−26.6330	−1.36088	−0.680442	+	0.732802i	$0.738212\pi$
$384$	0	0
$385$	−2.97899	−0.151824
$386$	0	0
$387$	−1.06781	−0.0542796
$388$	0	0
$389$	−36.2672	−1.83882	−0.919410	−	0.393301i	$-0.871333\pi$
$389$	−36.2672	−1.83882	−0.919410	+	0.393301i	$0.871333\pi$
$390$	0	0
$391$	19.3406	0.978095
$392$	0	0
$393$	−36.1617	−1.82412
$394$	0	0
$395$	−42.4531	−2.13605
$396$	0	0
$397$	28.0713	1.40886	0.704429	−	0.709774i	$-0.251203\pi$
$397$	28.0713	1.40886	0.704429	+	0.709774i	$0.251203\pi$
$398$	0	0
$399$	−1.17260	−0.0587033
$400$	0	0
$401$	−4.24452	−0.211961	−0.105981	−	0.994368i	$-0.533798\pi$
$401$	−4.24452	−0.211961	−0.105981	+	0.994368i	$0.533798\pi$
$402$	0	0
$403$	−4.81258	−0.239732
$404$	0	0
$405$	28.8210	1.43212
$406$	0	0
$407$	−5.38609	−0.266978
$408$	0	0
$409$	−8.33469	−0.412124	−0.206062	−	0.978539i	$-0.566065\pi$
$409$	−8.33469	−0.412124	−0.206062	+	0.978539i	$0.566065\pi$
$410$	0	0
$411$	−12.2730	−0.605383
$412$	0	0
$413$	−6.24631	−0.307361
$414$	0	0
$415$	12.6173	0.619356
$416$	0	0
$417$	−19.5049	−0.955157
$418$	0	0
$419$	11.4213	0.557966	0.278983	−	0.960296i	$-0.410003\pi$
$419$	11.4213	0.557966	0.278983	+	0.960296i	$0.410003\pi$
$420$	0	0
$421$	23.8374	1.16176	0.580881	−	0.813988i	$-0.302708\pi$
$421$	23.8374	1.16176	0.580881	+	0.813988i	$0.302708\pi$
$422$	0	0
$423$	2.52331	0.122687
$424$	0	0
$425$	−13.0286	−0.631978
$426$	0	0
$427$	−8.41953	−0.407450
$428$	0	0
$429$	−1.80136	−0.0869705
$430$	0	0
$431$	−2.72332	−0.131178	−0.0655889	−	0.997847i	$-0.520893\pi$
$431$	−2.72332	−0.131178	−0.0655889	+	0.997847i	$0.520893\pi$
$432$	0	0
$433$	−29.8039	−1.43228	−0.716141	−	0.697956i	$-0.754093\pi$
$433$	−29.8039	−1.43228	−0.716141	+	0.697956i	$0.754093\pi$
$434$	0	0
$435$	20.6145	0.988389
$436$	0	0
$437$	3.74392	0.179096
$438$	0	0
$439$	−25.4516	−1.21474	−0.607370	−	0.794419i	$-0.707776\pi$
$439$	−25.4516	−1.21474	−0.607370	+	0.794419i	$0.707776\pi$
$440$	0	0
$441$	0.244902	0.0116620
$442$	0	0
$443$	25.7795	1.22482	0.612410	−	0.790540i	$-0.290200\pi$
$443$	25.7795	1.22482	0.612410	+	0.790540i	$0.290200\pi$
$444$	0	0
$445$	−22.4836	−1.06582
$446$	0	0
$447$	−28.0478	−1.32662
$448$	0	0
$449$	0.122520	0.00578207	0.00289103	−	0.999996i	$-0.499080\pi$
$449$	0.122520	0.00578207	0.00289103	+	0.999996i	$0.499080\pi$
$450$	0	0
$451$	−0.331597	−0.0156143
$452$	0	0
$453$	7.06918	0.332139
$454$	0	0
$455$	−2.97899	−0.139657
$456$	0	0
$457$	−20.7921	−0.972611	−0.486306	−	0.873789i	$-0.661656\pi$
$457$	−20.7921	−0.972611	−0.486306	+	0.873789i	$0.661656\pi$
$458$	0	0
$459$	−16.6890	−0.778974
$460$	0	0
$461$	−27.5042	−1.28100	−0.640500	−	0.767958i	$-0.721273\pi$
$461$	−27.5042	−1.28100	−0.640500	+	0.767958i	$0.721273\pi$
$462$	0	0
$463$	0.762117	0.0354186	0.0177093	−	0.999843i	$-0.494363\pi$
$463$	0.762117	0.0354186	0.0177093	+	0.999843i	$0.494363\pi$
$464$	0	0
$465$	25.8255	1.19763
$466$	0	0
$467$	20.3063	0.939662	0.469831	−	0.882756i	$-0.344315\pi$
$467$	20.3063	0.939662	0.469831	+	0.882756i	$0.344315\pi$
$468$	0	0
$469$	11.3964	0.526236
$470$	0	0
$471$	−8.15691	−0.375850
$472$	0	0
$473$	4.36013	0.200479
$474$	0	0
$475$	−2.52205	−0.115719
$476$	0	0
$477$	0.698130	0.0319652
$478$	0	0
$479$	8.54857	0.390594	0.195297	−	0.980744i	$-0.437433\pi$
$479$	8.54857	0.390594	0.195297	+	0.980744i	$0.437433\pi$
$480$	0	0
$481$	−5.38609	−0.245584
$482$	0	0
$483$	−10.3604	−0.471417
$484$	0	0
$485$	22.2394	1.00984
$486$	0	0
$487$	−4.60102	−0.208492	−0.104246	−	0.994552i	$-0.533243\pi$
$487$	−4.60102	−0.208492	−0.104246	+	0.994552i	$0.533243\pi$
$488$	0	0
$489$	−28.0549	−1.26869
$490$	0	0
$491$	0.990276	0.0446906	0.0223453	−	0.999750i	$-0.492887\pi$
$491$	0.990276	0.0446906	0.0223453	+	0.999750i	$0.492887\pi$
$492$	0	0
$493$	−12.9180	−0.581796
$494$	0	0
$495$	0.729562	0.0327914
$496$	0	0
$497$	11.4219	0.512341
$498$	0	0
$499$	−31.6376	−1.41629	−0.708147	−	0.706065i	$-0.750468\pi$
$499$	−31.6376	−1.41629	−0.708147	+	0.706065i	$0.750468\pi$
$500$	0	0
$501$	−5.34979	−0.239011
$502$	0	0
$503$	−1.31444	−0.0586078	−0.0293039	−	0.999571i	$-0.509329\pi$
$503$	−1.31444	−0.0586078	−0.0293039	+	0.999571i	$0.509329\pi$
$504$	0	0
$505$	37.4068	1.66458
$506$	0	0
$507$	−1.80136	−0.0800013
$508$	0	0
$509$	11.3782	0.504331	0.252165	−	0.967684i	$-0.418857\pi$
$509$	11.3782	0.504331	0.252165	+	0.967684i	$0.418857\pi$
$510$	0	0
$511$	4.43520	0.196202
$512$	0	0
$513$	−3.23062	−0.142635
$514$	0	0
$515$	−40.7364	−1.79506
$516$	0	0
$517$	−10.3033	−0.453140
$518$	0	0
$519$	−17.5612	−0.770850
$520$	0	0
$521$	8.20888	0.359638	0.179819	−	0.983700i	$-0.442449\pi$
$521$	8.20888	0.359638	0.179819	+	0.983700i	$0.442449\pi$
$522$	0	0
$523$	−35.3039	−1.54373	−0.771866	−	0.635785i	$-0.780676\pi$
$523$	−35.3039	−1.54373	−0.771866	+	0.635785i	$0.780676\pi$
$524$	0	0
$525$	6.97920	0.304597
$526$	0	0
$527$	−16.1834	−0.704960
$528$	0	0
$529$	10.0792	0.438228
$530$	0	0
$531$	1.52973	0.0663848
$532$	0	0
$533$	−0.331597	−0.0143631
$534$	0	0
$535$	12.1474	0.525180
$536$	0	0
$537$	−18.7037	−0.807124
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−0.679250	−0.0292032	−0.0146016	−	0.999893i	$-0.504648\pi$
$541$	−0.679250	−0.0292032	−0.0146016	+	0.999893i	$0.504648\pi$
$542$	0	0
$543$	28.1561	1.20829
$544$	0	0
$545$	−0.484719	−0.0207631
$546$	0	0
$547$	44.5901	1.90653	0.953267	−	0.302128i	$-0.0976972\pi$
$547$	44.5901	1.90653	0.953267	+	0.302128i	$0.0976972\pi$
$548$	0	0
$549$	2.06196	0.0880024
$550$	0	0
$551$	−2.50064	−0.106531
$552$	0	0
$553$	−14.2508	−0.606006
$554$	0	0
$555$	28.9030	1.22687
$556$	0	0
$557$	−41.0257	−1.73832	−0.869158	−	0.494535i	$-0.835339\pi$
$557$	−41.0257	−1.73832	−0.869158	+	0.494535i	$0.835339\pi$
$558$	0	0
$559$	4.36013	0.184414
$560$	0	0
$561$	−6.05749	−0.255747
$562$	0	0
$563$	30.8679	1.30093	0.650464	−	0.759537i	$-0.274575\pi$
$563$	30.8679	1.30093	0.650464	+	0.759537i	$0.274575\pi$
$564$	0	0
$565$	33.9295	1.42742
$566$	0	0
$567$	9.67473	0.406300
$568$	0	0
$569$	24.9698	1.04679	0.523394	−	0.852091i	$-0.324666\pi$
$569$	24.9698	1.04679	0.523394	+	0.852091i	$0.324666\pi$
$570$	0	0
$571$	36.9566	1.54659	0.773293	−	0.634049i	$-0.218608\pi$
$571$	36.9566	1.54659	0.773293	+	0.634049i	$0.218608\pi$
$572$	0	0
$573$	41.4427	1.73129
$574$	0	0
$575$	−22.2834	−0.929284
$576$	0	0
$577$	25.9924	1.08208	0.541038	−	0.840998i	$-0.318032\pi$
$577$	25.9924	1.08208	0.541038	+	0.840998i	$0.318032\pi$
$578$	0	0
$579$	3.25253	0.135170
$580$	0	0
$581$	4.23541	0.175714
$582$	0	0
$583$	−2.85065	−0.118062
$584$	0	0
$585$	0.729562	0.0301637
$586$	0	0
$587$	23.6105	0.974510	0.487255	−	0.873260i	$-0.337998\pi$
$587$	23.6105	0.974510	0.487255	+	0.873260i	$0.337998\pi$
$588$	0	0
$589$	−3.13276	−0.129083
$590$	0	0
$591$	39.7227	1.63397
$592$	0	0
$593$	−1.12660	−0.0462638	−0.0231319	−	0.999732i	$-0.507364\pi$
$593$	−1.12660	−0.0462638	−0.0231319	+	0.999732i	$0.507364\pi$
$594$	0	0
$595$	−10.0175	−0.410679
$596$	0	0
$597$	2.05016	0.0839076
$598$	0	0
$599$	28.4490	1.16239	0.581196	−	0.813763i	$-0.302585\pi$
$599$	28.4490	1.16239	0.581196	+	0.813763i	$0.302585\pi$
$600$	0	0
$601$	10.1591	0.414398	0.207199	−	0.978299i	$-0.433565\pi$
$601$	10.1591	0.414398	0.207199	+	0.978299i	$0.433565\pi$
$602$	0	0
$603$	−2.79100	−0.113658
$604$	0	0
$605$	−2.97899	−0.121113
$606$	0	0
$607$	−4.58121	−0.185945	−0.0929727	−	0.995669i	$-0.529637\pi$
$607$	−4.58121	−0.185945	−0.0929727	+	0.995669i	$0.529637\pi$
$608$	0	0
$609$	6.91995	0.280411
$610$	0	0
$611$	−10.3033	−0.416828
$612$	0	0
$613$	23.5802	0.952394	0.476197	−	0.879339i	$-0.342015\pi$
$613$	23.5802	0.952394	0.476197	+	0.879339i	$0.342015\pi$
$614$	0	0
$615$	1.77943	0.0717536
$616$	0	0
$617$	12.0743	0.486093	0.243047	−	0.970015i	$-0.421853\pi$
$617$	12.0743	0.486093	0.243047	+	0.970015i	$0.421853\pi$
$618$	0	0
$619$	3.53753	0.142185	0.0710926	−	0.997470i	$-0.477351\pi$
$619$	3.53753	0.142185	0.0710926	+	0.997470i	$0.477351\pi$
$620$	0	0
$621$	−28.5440	−1.14543
$622$	0	0
$623$	−7.54738	−0.302379
$624$	0	0
$625$	−29.3610	−1.17444
$626$	0	0
$627$	−1.17260	−0.0468291
$628$	0	0
$629$	−18.1119	−0.722170
$630$	0	0
$631$	−8.63310	−0.343678	−0.171839	−	0.985125i	$-0.554971\pi$
$631$	−8.63310	−0.343678	−0.171839	+	0.985125i	$0.554971\pi$
$632$	0	0
$633$	−25.8385	−1.02699
$634$	0	0
$635$	−4.57176	−0.181425
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−2.79724	−0.110657
$640$	0	0
$641$	−33.8877	−1.33848	−0.669242	−	0.743045i	$-0.733381\pi$
$641$	−33.8877	−1.33848	−0.669242	+	0.743045i	$0.733381\pi$
$642$	0	0
$643$	15.2758	0.602418	0.301209	−	0.953558i	$-0.402610\pi$
$643$	15.2758	0.602418	0.301209	+	0.953558i	$0.402610\pi$
$644$	0	0
$645$	−23.3975	−0.921277
$646$	0	0
$647$	−28.9361	−1.13760	−0.568799	−	0.822477i	$-0.692592\pi$
$647$	−28.9361	−1.13760	−0.568799	+	0.822477i	$0.692592\pi$
$648$	0	0
$649$	−6.24631	−0.245189
$650$	0	0
$651$	8.66920	0.339773
$652$	0	0
$653$	26.2086	1.02562	0.512811	−	0.858501i	$-0.328604\pi$
$653$	26.2086	1.02562	0.512811	+	0.858501i	$0.328604\pi$
$654$	0	0
$655$	−59.8023	−2.33667
$656$	0	0
$657$	−1.08619	−0.0423763
$658$	0	0
$659$	22.1501	0.862847	0.431424	−	0.902150i	$-0.358011\pi$
$659$	22.1501	0.862847	0.431424	+	0.902150i	$0.358011\pi$
$660$	0	0
$661$	−0.223564	−0.00869563	−0.00434781	−	0.999991i	$-0.501384\pi$
$661$	−0.223564	−0.00869563	−0.00434781	+	0.999991i	$0.501384\pi$
$662$	0	0
$663$	−6.05749	−0.235253
$664$	0	0
$665$	−1.93918	−0.0751982
$666$	0	0
$667$	−22.0943	−0.855494
$668$	0	0
$669$	−31.4223	−1.21486
$670$	0	0
$671$	−8.41953	−0.325032
$672$	0	0
$673$	−28.5052	−1.09879	−0.549396	−	0.835562i	$-0.685142\pi$
$673$	−28.5052	−1.09879	−0.549396	+	0.835562i	$0.685142\pi$
$674$	0	0
$675$	19.2284	0.740100
$676$	0	0
$677$	−9.20819	−0.353899	−0.176950	−	0.984220i	$-0.556623\pi$
$677$	−9.20819	−0.353899	−0.176950	+	0.984220i	$0.556623\pi$
$678$	0	0
$679$	7.46541	0.286496
$680$	0	0
$681$	11.7881	0.451722
$682$	0	0
$683$	14.5445	0.556528	0.278264	−	0.960505i	$-0.410241\pi$
$683$	14.5445	0.556528	0.278264	+	0.960505i	$0.410241\pi$
$684$	0	0
$685$	−20.2965	−0.775488
$686$	0	0
$687$	12.2040	0.465614
$688$	0	0
$689$	−2.85065	−0.108601
$690$	0	0
$691$	−9.08049	−0.345438	−0.172719	−	0.984971i	$-0.555255\pi$
$691$	−9.08049	−0.345438	−0.172719	+	0.984971i	$0.555255\pi$
$692$	0	0
$693$	0.244902	0.00930306
$694$	0	0
$695$	−32.2561	−1.22354
$696$	0	0
$697$	−1.11507	−0.0422363
$698$	0	0
$699$	−18.8445	−0.712766
$700$	0	0
$701$	−16.7898	−0.634141	−0.317071	−	0.948402i	$-0.602699\pi$
$701$	−16.7898	−0.634141	−0.317071	+	0.948402i	$0.602699\pi$
$702$	0	0
$703$	−3.50608	−0.132234
$704$	0	0
$705$	55.2901	2.08235
$706$	0	0
$707$	12.5569	0.472250
$708$	0	0
$709$	−15.3306	−0.575754	−0.287877	−	0.957667i	$-0.592949\pi$
$709$	−15.3306	−0.575754	−0.287877	+	0.957667i	$0.592949\pi$
$710$	0	0
$711$	3.49006	0.130887
$712$	0	0
$713$	−27.6794	−1.03660
$714$	0	0
$715$	−2.97899	−0.111408
$716$	0	0
$717$	9.41509	0.351613
$718$	0	0
$719$	−0.748354	−0.0279089	−0.0139544	−	0.999903i	$-0.504442\pi$
$719$	−0.748354	−0.0279089	−0.0139544	+	0.999903i	$0.504442\pi$
$720$	0	0
$721$	−13.6746	−0.509267
$722$	0	0
$723$	3.76808	0.140136
$724$	0	0
$725$	14.8836	0.552762
$726$	0	0
$727$	−17.7351	−0.657760	−0.328880	−	0.944372i	$-0.606671\pi$
$727$	−17.7351	−0.657760	−0.328880	+	0.944372i	$0.606671\pi$
$728$	0	0
$729$	24.4507	0.905582
$730$	0	0
$731$	14.6619	0.542291
$732$	0	0
$733$	−18.4312	−0.680771	−0.340386	−	0.940286i	$-0.610558\pi$
$733$	−18.4312	−0.680771	−0.340386	+	0.940286i	$0.610558\pi$
$734$	0	0
$735$	5.36624	0.197937
$736$	0	0
$737$	11.3964	0.419791
$738$	0	0
$739$	46.5589	1.71270	0.856348	−	0.516400i	$-0.172728\pi$
$739$	46.5589	1.71270	0.856348	+	0.516400i	$0.172728\pi$
$740$	0	0
$741$	−1.17260	−0.0430765
$742$	0	0
$743$	37.6360	1.38073	0.690366	−	0.723460i	$-0.257449\pi$
$743$	37.6360	1.38073	0.690366	+	0.723460i	$0.257449\pi$
$744$	0	0
$745$	−46.3840	−1.69938
$746$	0	0
$747$	−1.03726	−0.0379514
$748$	0	0
$749$	4.07770	0.148996
$750$	0	0
$751$	−5.32903	−0.194459	−0.0972296	−	0.995262i	$-0.530998\pi$
$751$	−5.32903	−0.194459	−0.0972296	+	0.995262i	$0.530998\pi$
$752$	0	0
$753$	−14.8548	−0.541341
$754$	0	0
$755$	11.6906	0.425465
$756$	0	0
$757$	−41.9962	−1.52638	−0.763190	−	0.646175i	$-0.776368\pi$
$757$	−41.9962	−1.52638	−0.763190	+	0.646175i	$0.776368\pi$
$758$	0	0
$759$	−10.3604	−0.376060
$760$	0	0
$761$	15.1395	0.548805	0.274402	−	0.961615i	$-0.411520\pi$
$761$	15.1395	0.548805	0.274402	+	0.961615i	$0.411520\pi$
$762$	0	0
$763$	−0.162712	−0.00589059
$764$	0	0
$765$	2.45332	0.0886999
$766$	0	0
$767$	−6.24631	−0.225541
$768$	0	0
$769$	1.54087	0.0555652	0.0277826	−	0.999614i	$-0.491155\pi$
$769$	1.54087	0.0555652	0.0277826	+	0.999614i	$0.491155\pi$
$770$	0	0
$771$	16.9567	0.610680
$772$	0	0
$773$	7.08798	0.254937	0.127469	−	0.991843i	$-0.459315\pi$
$773$	7.08798	0.254937	0.127469	+	0.991843i	$0.459315\pi$
$774$	0	0
$775$	18.6459	0.669780
$776$	0	0
$777$	9.70229	0.348068
$778$	0	0
$779$	−0.215854	−0.00773376
$780$	0	0
$781$	11.4219	0.408706
$782$	0	0
$783$	19.0651	0.681333
$784$	0	0
$785$	−13.4894	−0.481459
$786$	0	0
$787$	−27.6820	−0.986757	−0.493378	−	0.869815i	$-0.664238\pi$
$787$	−27.6820	−0.986757	−0.493378	+	0.869815i	$0.664238\pi$
$788$	0	0
$789$	27.3822	0.974832
$790$	0	0
$791$	11.3896	0.404967
$792$	0	0
$793$	−8.41953	−0.298986
$794$	0	0
$795$	15.2973	0.542538
$796$	0	0
$797$	53.6390	1.89999	0.949996	−	0.312263i	$-0.101087\pi$
$797$	53.6390	1.89999	0.949996	+	0.312263i	$0.101087\pi$
$798$	0	0
$799$	−34.6473	−1.22573
$800$	0	0
$801$	1.84837	0.0653089
$802$	0	0
$803$	4.43520	0.156515
$804$	0	0
$805$	−17.1335	−0.603878
$806$	0	0
$807$	8.00048	0.281630
$808$	0	0
$809$	3.62134	0.127319	0.0636597	−	0.997972i	$-0.479723\pi$
$809$	3.62134	0.127319	0.0636597	+	0.997972i	$0.479723\pi$
$810$	0	0
$811$	−9.07667	−0.318725	−0.159363	−	0.987220i	$-0.550944\pi$
$811$	−9.07667	−0.318725	−0.159363	+	0.987220i	$0.550944\pi$
$812$	0	0
$813$	−41.3019	−1.44852
$814$	0	0
$815$	−46.3957	−1.62517
$816$	0	0
$817$	2.83823	0.0992972
$818$	0	0
$819$	0.244902	0.00855757
$820$	0	0
$821$	−11.6994	−0.408313	−0.204157	−	0.978938i	$-0.565445\pi$
$821$	−11.6994	−0.408313	−0.204157	+	0.978938i	$0.565445\pi$
$822$	0	0
$823$	33.7856	1.17769	0.588845	−	0.808246i	$-0.299583\pi$
$823$	33.7856	1.17769	0.588845	+	0.808246i	$0.299583\pi$
$824$	0	0
$825$	6.97920	0.242984
$826$	0	0
$827$	8.94379	0.311006	0.155503	−	0.987835i	$-0.450300\pi$
$827$	8.94379	0.311006	0.155503	+	0.987835i	$0.450300\pi$
$828$	0	0
$829$	44.3182	1.53923	0.769617	−	0.638506i	$-0.220447\pi$
$829$	44.3182	1.53923	0.769617	+	0.638506i	$0.220447\pi$
$830$	0	0
$831$	13.6540	0.473653
$832$	0	0
$833$	−3.36273	−0.116512
$834$	0	0
$835$	−8.84719	−0.306170
$836$	0	0
$837$	23.8845	0.825569
$838$	0	0
$839$	−23.6729	−0.817279	−0.408640	−	0.912696i	$-0.633997\pi$
$839$	−23.6729	−0.817279	−0.408640	+	0.912696i	$0.633997\pi$
$840$	0	0
$841$	−14.2428	−0.491130
$842$	0	0
$843$	7.73238	0.266317
$844$	0	0
$845$	−2.97899	−0.102480
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−32.6900	−1.12192
$850$	0	0
$851$	−30.9778	−1.06191
$852$	0	0
$853$	−4.24737	−0.145427	−0.0727136	−	0.997353i	$-0.523166\pi$
$853$	−4.24737	−0.145427	−0.0727136	+	0.997353i	$0.523166\pi$
$854$	0	0
$855$	0.474909	0.0162416
$856$	0	0
$857$	−53.6007	−1.83096	−0.915482	−	0.402359i	$-0.868190\pi$
$857$	−53.6007	−1.83096	−0.915482	+	0.402359i	$0.868190\pi$
$858$	0	0
$859$	17.7557	0.605816	0.302908	−	0.953020i	$-0.402042\pi$
$859$	17.7557	0.605816	0.302908	+	0.953020i	$0.402042\pi$
$860$	0	0
$861$	0.597327	0.0203568
$862$	0	0
$863$	10.8866	0.370586	0.185293	−	0.982683i	$-0.440677\pi$
$863$	10.8866	0.370586	0.185293	+	0.982683i	$0.440677\pi$
$864$	0	0
$865$	−29.0417	−0.987448
$866$	0	0
$867$	10.2535	0.348226
$868$	0	0
$869$	−14.2508	−0.483426
$870$	0	0
$871$	11.3964	0.386152
$872$	0	0
$873$	−1.82830	−0.0618784
$874$	0	0
$875$	−3.35315	−0.113357
$876$	0	0
$877$	−5.43991	−0.183693	−0.0918463	−	0.995773i	$-0.529277\pi$
$877$	−5.43991	−0.183693	−0.0918463	+	0.995773i	$0.529277\pi$
$878$	0	0
$879$	40.3468	1.36086
$880$	0	0
$881$	19.3905	0.653282	0.326641	−	0.945148i	$-0.394083\pi$
$881$	19.3905	0.653282	0.326641	+	0.945148i	$0.394083\pi$
$882$	0	0
$883$	9.44681	0.317910	0.158955	−	0.987286i	$-0.449187\pi$
$883$	9.44681	0.317910	0.158955	+	0.987286i	$0.449187\pi$
$884$	0	0
$885$	33.5192	1.12674
$886$	0	0
$887$	−24.2956	−0.815767	−0.407883	−	0.913034i	$-0.633733\pi$
$887$	−24.2956	−0.815767	−0.407883	+	0.913034i	$0.633733\pi$
$888$	0	0
$889$	−1.53467	−0.0514711
$890$	0	0
$891$	9.67473	0.324116
$892$	0	0
$893$	−6.70696	−0.224440
$894$	0	0
$895$	−30.9312	−1.03391
$896$	0	0
$897$	−10.3604	−0.345925
$898$	0	0
$899$	18.4876	0.616596
$900$	0	0
$901$	−9.58595	−0.319354
$902$	0	0
$903$	−7.85417	−0.261370
$904$	0	0
$905$	46.5630	1.54781
$906$	0	0
$907$	43.4805	1.44375	0.721873	−	0.692026i	$-0.243282\pi$
$907$	43.4805	1.44375	0.721873	+	0.692026i	$0.243282\pi$
$908$	0	0
$909$	−3.07520	−0.101998
$910$	0	0
$911$	−12.1619	−0.402940	−0.201470	−	0.979495i	$-0.564572\pi$
$911$	−12.1619	−0.402940	−0.201470	+	0.979495i	$0.564572\pi$
$912$	0	0
$913$	4.23541	0.140172
$914$	0	0
$915$	45.1812	1.49365
$916$	0	0
$917$	−20.0747	−0.662923
$918$	0	0
$919$	−46.3321	−1.52836	−0.764178	−	0.645006i	$-0.776855\pi$
$919$	−46.3321	−1.52836	−0.764178	+	0.645006i	$0.776855\pi$
$920$	0	0
$921$	−41.8504	−1.37902
$922$	0	0
$923$	11.4219	0.375955
$924$	0	0
$925$	20.8679	0.686131
$926$	0	0
$927$	3.34893	0.109993
$928$	0	0
$929$	33.4516	1.09751	0.548756	−	0.835982i	$-0.315101\pi$
$929$	33.4516	1.09751	0.548756	+	0.835982i	$0.315101\pi$
$930$	0	0
$931$	−0.650951	−0.0213341
$932$	0	0
$933$	26.9025	0.880747
$934$	0	0
$935$	−10.0175	−0.327609
$936$	0	0
$937$	51.7909	1.69193	0.845967	−	0.533235i	$-0.179024\pi$
$937$	51.7909	1.69193	0.845967	+	0.533235i	$0.179024\pi$
$938$	0	0
$939$	33.1817	1.08284
$940$	0	0
$941$	−29.5429	−0.963071	−0.481535	−	0.876427i	$-0.659921\pi$
$941$	−29.5429	−0.963071	−0.481535	+	0.876427i	$0.659921\pi$
$942$	0	0
$943$	−1.90717	−0.0621059
$944$	0	0
$945$	14.7845	0.480941
$946$	0	0
$947$	−34.9610	−1.13608	−0.568040	−	0.823001i	$-0.692298\pi$
$947$	−34.9610	−1.13608	−0.568040	+	0.823001i	$0.692298\pi$
$948$	0	0
$949$	4.43520	0.143973
$950$	0	0
$951$	31.2167	1.01227
$952$	0	0
$953$	6.41068	0.207662	0.103831	−	0.994595i	$-0.466890\pi$
$953$	6.41068	0.207662	0.103831	+	0.994595i	$0.466890\pi$
$954$	0	0
$955$	68.5357	2.21776
$956$	0	0
$957$	6.91995	0.223690
$958$	0	0
$959$	−6.81319	−0.220009
$960$	0	0
$961$	−7.83904	−0.252872
$962$	0	0
$963$	−0.998638	−0.0321807
$964$	0	0
$965$	5.37885	0.173151
$966$	0	0
$967$	−59.4885	−1.91302	−0.956510	−	0.291699i	$-0.905780\pi$
$967$	−59.4885	−1.91302	−0.956510	+	0.291699i	$0.905780\pi$
$968$	0	0
$969$	−3.94313	−0.126672
$970$	0	0
$971$	34.9057	1.12018	0.560089	−	0.828432i	$-0.310767\pi$
$971$	34.9057	1.12018	0.560089	+	0.828432i	$0.310767\pi$
$972$	0	0
$973$	−10.8278	−0.347125
$974$	0	0
$975$	6.97920	0.223513
$976$	0	0
$977$	−8.31265	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$977$	−8.31265	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$978$	0	0
$979$	−7.54738	−0.241215
$980$	0	0
$981$	0.0398486	0.00127227
$982$	0	0
$983$	−47.2357	−1.50658	−0.753292	−	0.657686i	$-0.771535\pi$
$983$	−47.2357	−1.50658	−0.753292	+	0.657686i	$0.771535\pi$
$984$	0	0
$985$	65.6912	2.09309
$986$	0	0
$987$	18.5600	0.590772
$988$	0	0
$989$	25.0771	0.797405
$990$	0	0
$991$	46.3448	1.47219	0.736095	−	0.676878i	$-0.236667\pi$
$991$	46.3448	1.47219	0.736095	+	0.676878i	$0.236667\pi$
$992$	0	0
$993$	−39.3849	−1.24984
$994$	0	0
$995$	3.39045	0.107485
$996$	0	0
$997$	45.0438	1.42655	0.713275	−	0.700884i	$-0.247211\pi$
$997$	45.0438	1.42655	0.713275	+	0.700884i	$0.247211\pi$
$998$	0	0
$999$	26.7307	0.845723

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.u.1.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.u.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-1.80136 q^{3} -2.97899 q^{5} -1.00000 q^{7} +0.244902 q^{9} +O(q^{10})\)	\(q-1.80136 q^{3} -2.97899 q^{5} -1.00000 q^{7} +0.244902 q^{9} -1.00000 q^{11} -1.00000 q^{13} +5.36624 q^{15} -3.36273 q^{17} -0.650951 q^{19} +1.80136 q^{21} -5.75145 q^{23} +3.87440 q^{25} +4.96293 q^{27} +3.84151 q^{29} +4.81258 q^{31} +1.80136 q^{33} +2.97899 q^{35} +5.38609 q^{37} +1.80136 q^{39} +0.331597 q^{41} -4.36013 q^{43} -0.729562 q^{45} +10.3033 q^{47} +1.00000 q^{49} +6.05749 q^{51} +2.85065 q^{53} +2.97899 q^{55} +1.17260 q^{57} +6.24631 q^{59} +8.41953 q^{61} -0.244902 q^{63} +2.97899 q^{65} -11.3964 q^{67} +10.3604 q^{69} -11.4219 q^{71} -4.43520 q^{73} -6.97920 q^{75} +1.00000 q^{77} +14.2508 q^{79} -9.67473 q^{81} -4.23541 q^{83} +10.0175 q^{85} -6.91995 q^{87} +7.54738 q^{89} +1.00000 q^{91} -8.66920 q^{93} +1.93918 q^{95} -7.46541 q^{97} -0.244902 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} - 3 q^{17} + 13 q^{19} - 2 q^{21} + 6 q^{25} + 14 q^{27} - 3 q^{29} - q^{31} - 2 q^{33} + 4 q^{35} - 5 q^{37} - 2 q^{39} + 4 q^{41} + 19 q^{43} - 11 q^{45} + q^{47} + 10 q^{49} + 15 q^{51} - 6 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 18 q^{61} - 6 q^{63} + 4 q^{65} + q^{67} - 11 q^{69} - 21 q^{71} - 10 q^{73} + 10 q^{77} + 3 q^{79} - 30 q^{81} + 6 q^{83} - 33 q^{85} - 9 q^{87} - 22 q^{89} + 10 q^{91} - 34 q^{93} - 3 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) \) 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 - 3 * q^17 + 13 * q^19 - 2 * q^21 + 6 * q^25 + 14 * q^27 - 3 * q^29 - q^31 - 2 * q^33 + 4 * q^35 - 5 * q^37 - 2 * q^39 + 4 * q^41 + 19 * q^43 - 11 * q^45 + q^47 + 10 * q^49 + 15 * q^51 - 6 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 18 * q^61 - 6 * q^63 + 4 * q^65 + q^67 - 11 * q^69 - 21 * q^71 - 10 * q^73 + 10 * q^77 + 3 * q^79 - 30 * q^81 + 6 * q^83 - 33 * q^85 - 9 * q^87 - 22 * q^89 + 10 * q^91 - 34 * q^93 - 3 * q^95 - 9 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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