LMFDB - Embedded newform 8008.2.a.z.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 35 x^{13} + 32 x^{12} + 477 x^{11} - 392 x^{10} - 3236 x^{9} + 2330 x^{8} + \cdots + 2560 $ x^15 - x^14 - 35x^13 + 32x^12 + 477x^11 - 392x^10 - 3236x^9 + 2330x^8 + 11690x^7 - 7119x^6 - 22246x^5 + 11137x^4 + 20034x^3 - 8392x^2 - 6016*x + 2560
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-2.42593$ 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)

$f(q)$	$=$	$q+2.42593 q^{3} +3.43967 q^{5} -1.00000 q^{7} +2.88513 q^{9} +O(q^{10})$	$q+2.42593 q^{3} +3.43967 q^{5} -1.00000 q^{7} +2.88513 q^{9} -1.00000 q^{11} +1.00000 q^{13} +8.34439 q^{15} -4.39446 q^{17} +3.63761 q^{19} -2.42593 q^{21} -1.12036 q^{23} +6.83133 q^{25} -0.278666 q^{27} -3.89071 q^{29} +9.45333 q^{31} -2.42593 q^{33} -3.43967 q^{35} +2.37053 q^{37} +2.42593 q^{39} +4.58480 q^{41} +7.49006 q^{43} +9.92390 q^{45} +4.83430 q^{47} +1.00000 q^{49} -10.6606 q^{51} -1.84952 q^{53} -3.43967 q^{55} +8.82459 q^{57} +9.91141 q^{59} -0.650840 q^{61} -2.88513 q^{63} +3.43967 q^{65} -7.75105 q^{67} -2.71791 q^{69} +9.76069 q^{71} +2.15596 q^{73} +16.5723 q^{75} +1.00000 q^{77} +1.46419 q^{79} -9.33141 q^{81} -0.343435 q^{83} -15.1155 q^{85} -9.43859 q^{87} +11.6650 q^{89} -1.00000 q^{91} +22.9331 q^{93} +12.5122 q^{95} +19.2176 q^{97} -2.88513 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * 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
$3$	2.42593	1.40061	0.700305	−	0.713843i	$-0.253047\pi$
$3$	2.42593	1.40061	0.700305	+	0.713843i	$0.253047\pi$
$4$	0	0
$5$	3.43967	1.53827	0.769134	−	0.639088i	$-0.220688\pi$
$5$	3.43967	1.53827	0.769134	+	0.639088i	$0.220688\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.88513	0.961710
$10$	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
$15$	8.34439	2.15451
$16$	0	0
$17$	−4.39446	−1.06581	−0.532907	−	0.846174i	$-0.678900\pi$
$17$	−4.39446	−1.06581	−0.532907	+	0.846174i	$0.678900\pi$
$18$	0	0
$19$	3.63761	0.834525	0.417263	−	0.908786i	$-0.362990\pi$
$19$	3.63761	0.834525	0.417263	+	0.908786i	$0.362990\pi$
$20$	0	0
$21$	−2.42593	−0.529381
$22$	0	0
$23$	−1.12036	−0.233611	−0.116806	−	0.993155i	$-0.537265\pi$
$23$	−1.12036	−0.233611	−0.116806	+	0.993155i	$0.537265\pi$
$24$	0	0
$25$	6.83133	1.36627
$26$	0	0
$27$	−0.278666	−0.0536292
$28$	0	0
$29$	−3.89071	−0.722487	−0.361244	−	0.932472i	$-0.617648\pi$
$29$	−3.89071	−0.722487	−0.361244	+	0.932472i	$0.617648\pi$
$30$	0	0
$31$	9.45333	1.69787	0.848935	−	0.528498i	$-0.177245\pi$
$31$	9.45333	1.69787	0.848935	+	0.528498i	$0.177245\pi$
$32$	0	0
$33$	−2.42593	−0.422300
$34$	0	0
$35$	−3.43967	−0.581410
$36$	0	0
$37$	2.37053	0.389713	0.194856	−	0.980832i	$-0.437576\pi$
$37$	2.37053	0.389713	0.194856	+	0.980832i	$0.437576\pi$
$38$	0	0
$39$	2.42593	0.388459
$40$	0	0
$41$	4.58480	0.716026	0.358013	−	0.933717i	$-0.383454\pi$
$41$	4.58480	0.716026	0.358013	+	0.933717i	$0.383454\pi$
$42$	0	0
$43$	7.49006	1.14222	0.571112	−	0.820872i	$-0.306512\pi$
$43$	7.49006	1.14222	0.571112	+	0.820872i	$0.306512\pi$
$44$	0	0
$45$	9.92390	1.47937
$46$	0	0
$47$	4.83430	0.705155	0.352578	−	0.935783i	$-0.385305\pi$
$47$	4.83430	0.705155	0.352578	+	0.935783i	$0.385305\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−10.6606	−1.49279
$52$	0	0
$53$	−1.84952	−0.254051	−0.127025	−	0.991899i	$-0.540543\pi$
$53$	−1.84952	−0.254051	−0.127025	+	0.991899i	$0.540543\pi$
$54$	0	0
$55$	−3.43967	−0.463805
$56$	0	0
$57$	8.82459	1.16885
$58$	0	0
$59$	9.91141	1.29036	0.645178	−	0.764032i	$-0.276783\pi$
$59$	9.91141	1.29036	0.645178	+	0.764032i	$0.276783\pi$
$60$	0	0
$61$	−0.650840	−0.0833316	−0.0416658	−	0.999132i	$-0.513266\pi$
$61$	−0.650840	−0.0833316	−0.0416658	+	0.999132i	$0.513266\pi$
$62$	0	0
$63$	−2.88513	−0.363492
$64$	0	0
$65$	3.43967	0.426639
$66$	0	0
$67$	−7.75105	−0.946941	−0.473471	−	0.880810i	$-0.656999\pi$
$67$	−7.75105	−0.946941	−0.473471	+	0.880810i	$0.656999\pi$
$68$	0	0
$69$	−2.71791	−0.327198
$70$	0	0
$71$	9.76069	1.15838	0.579190	−	0.815192i	$-0.303369\pi$
$71$	9.76069	1.15838	0.579190	+	0.815192i	$0.303369\pi$
$72$	0	0
$73$	2.15596	0.252336	0.126168	−	0.992009i	$-0.459732\pi$
$73$	2.15596	0.252336	0.126168	+	0.992009i	$0.459732\pi$
$74$	0	0
$75$	16.5723	1.91361
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	1.46419	0.164734	0.0823670	−	0.996602i	$-0.473752\pi$
$79$	1.46419	0.164734	0.0823670	+	0.996602i	$0.473752\pi$
$80$	0	0
$81$	−9.33141	−1.03682
$82$	0	0
$83$	−0.343435	−0.0376969	−0.0188485	−	0.999822i	$-0.506000\pi$
$83$	−0.343435	−0.0376969	−0.0188485	+	0.999822i	$0.506000\pi$
$84$	0	0
$85$	−15.1155	−1.63951
$86$	0	0
$87$	−9.43859	−1.01192
$88$	0	0
$89$	11.6650	1.23649	0.618243	−	0.785987i	$-0.287845\pi$
$89$	11.6650	1.23649	0.618243	+	0.785987i	$0.287845\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	22.9331	2.37805
$94$	0	0
$95$	12.5122	1.28372
$96$	0	0
$97$	19.2176	1.95125	0.975624	−	0.219449i	$-0.0704260\pi$
$97$	19.2176	1.95125	0.975624	+	0.219449i	$0.0704260\pi$
$98$	0	0
$99$	−2.88513	−0.289967
$100$	0	0
$101$	12.0837	1.20238	0.601189	−	0.799107i	$-0.294694\pi$
$101$	12.0837	1.20238	0.601189	+	0.799107i	$0.294694\pi$
$102$	0	0
$103$	−12.3618	−1.21804	−0.609020	−	0.793155i	$-0.708437\pi$
$103$	−12.3618	−1.21804	−0.609020	+	0.793155i	$0.708437\pi$
$104$	0	0
$105$	−8.34439	−0.814330
$106$	0	0
$107$	4.62696	0.447305	0.223653	−	0.974669i	$-0.428202\pi$
$107$	4.62696	0.447305	0.223653	+	0.974669i	$0.428202\pi$
$108$	0	0
$109$	−7.56094	−0.724206	−0.362103	−	0.932138i	$-0.617941\pi$
$109$	−7.56094	−0.724206	−0.362103	+	0.932138i	$0.617941\pi$
$110$	0	0
$111$	5.75074	0.545836
$112$	0	0
$113$	13.1550	1.23752	0.618758	−	0.785581i	$-0.287636\pi$
$113$	13.1550	1.23752	0.618758	+	0.785581i	$0.287636\pi$
$114$	0	0
$115$	−3.85367	−0.359356
$116$	0	0
$117$	2.88513	0.266730
$118$	0	0
$119$	4.39446	0.402840
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	11.1224	1.00287
$124$	0	0
$125$	6.29917	0.563415
$126$	0	0
$127$	−1.71926	−0.152560	−0.0762798	−	0.997086i	$-0.524304\pi$
$127$	−1.71926	−0.152560	−0.0762798	+	0.997086i	$0.524304\pi$
$128$	0	0
$129$	18.1704	1.59981
$130$	0	0
$131$	−7.54999	−0.659646	−0.329823	−	0.944043i	$-0.606989\pi$
$131$	−7.54999	−0.659646	−0.329823	+	0.944043i	$0.606989\pi$
$132$	0	0
$133$	−3.63761	−0.315421
$134$	0	0
$135$	−0.958517	−0.0824960
$136$	0	0
$137$	12.8810	1.10050	0.550250	−	0.835000i	$-0.314532\pi$
$137$	12.8810	1.10050	0.550250	+	0.835000i	$0.314532\pi$
$138$	0	0
$139$	−18.2450	−1.54752	−0.773758	−	0.633481i	$-0.781625\pi$
$139$	−18.2450	−1.54752	−0.773758	+	0.633481i	$0.781625\pi$
$140$	0	0
$141$	11.7277	0.987648
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−13.3828	−1.11138
$146$	0	0
$147$	2.42593	0.200087
$148$	0	0
$149$	−6.65303	−0.545037	−0.272519	−	0.962151i	$-0.587857\pi$
$149$	−6.65303	−0.545037	−0.272519	+	0.962151i	$0.587857\pi$
$150$	0	0
$151$	8.54966	0.695762	0.347881	−	0.937539i	$-0.386901\pi$
$151$	8.54966	0.695762	0.347881	+	0.937539i	$0.386901\pi$
$152$	0	0
$153$	−12.6786	−1.02500
$154$	0	0
$155$	32.5164	2.61178
$156$	0	0
$157$	−15.9131	−1.27001	−0.635003	−	0.772510i	$-0.719001\pi$
$157$	−15.9131	−1.27001	−0.635003	+	0.772510i	$0.719001\pi$
$158$	0	0
$159$	−4.48680	−0.355826
$160$	0	0
$161$	1.12036	0.0882967
$162$	0	0
$163$	−19.0212	−1.48985	−0.744927	−	0.667146i	$-0.767516\pi$
$163$	−19.0212	−1.48985	−0.744927	+	0.667146i	$0.767516\pi$
$164$	0	0
$165$	−8.34439	−0.649610
$166$	0	0
$167$	−22.7915	−1.76366	−0.881830	−	0.471567i	$-0.843689\pi$
$167$	−22.7915	−1.76366	−0.881830	+	0.471567i	$0.843689\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	10.4950	0.802571
$172$	0	0
$173$	19.0035	1.44481	0.722406	−	0.691469i	$-0.243036\pi$
$173$	19.0035	1.44481	0.722406	+	0.691469i	$0.243036\pi$
$174$	0	0
$175$	−6.83133	−0.516400
$176$	0	0
$177$	24.0444	1.80729
$178$	0	0
$179$	−12.3235	−0.921102	−0.460551	−	0.887633i	$-0.652348\pi$
$179$	−12.3235	−0.921102	−0.460551	+	0.887633i	$0.652348\pi$
$180$	0	0
$181$	14.9713	1.11281	0.556404	−	0.830912i	$-0.312181\pi$
$181$	14.9713	1.11281	0.556404	+	0.830912i	$0.312181\pi$
$182$	0	0
$183$	−1.57889	−0.116715
$184$	0	0
$185$	8.15384	0.599482
$186$	0	0
$187$	4.39446	0.321355
$188$	0	0
$189$	0.278666	0.0202699
$190$	0	0
$191$	−10.8565	−0.785550	−0.392775	−	0.919635i	$-0.628485\pi$
$191$	−10.8565	−0.785550	−0.392775	+	0.919635i	$0.628485\pi$
$192$	0	0
$193$	2.42633	0.174651	0.0873257	−	0.996180i	$-0.472168\pi$
$193$	2.42633	0.174651	0.0873257	+	0.996180i	$0.472168\pi$
$194$	0	0
$195$	8.34439	0.597555
$196$	0	0
$197$	0.00290513	0.000206982 0	0.000103491	−	1.00000i	$-0.499967\pi$
$197$	0.00290513	0.000206982 0	0.000103491		1.00000i	$0.499967\pi$
$198$	0	0
$199$	2.31984	0.164449	0.0822246	−	0.996614i	$-0.473798\pi$
$199$	2.31984	0.164449	0.0822246	+	0.996614i	$0.473798\pi$
$200$	0	0
$201$	−18.8035	−1.32630
$202$	0	0
$203$	3.89071	0.273074
$204$	0	0
$205$	15.7702	1.10144
$206$	0	0
$207$	−3.23238	−0.224666
$208$	0	0
$209$	−3.63761	−0.251619
$210$	0	0
$211$	−26.9952	−1.85842	−0.929212	−	0.369546i	$-0.879513\pi$
$211$	−26.9952	−1.85842	−0.929212	+	0.369546i	$0.879513\pi$
$212$	0	0
$213$	23.6787	1.62244
$214$	0	0
$215$	25.7633	1.75705
$216$	0	0
$217$	−9.45333	−0.641734
$218$	0	0
$219$	5.23020	0.353424
$220$	0	0
$221$	−4.39446	−0.295603
$222$	0	0
$223$	−4.87673	−0.326570	−0.163285	−	0.986579i	$-0.552209\pi$
$223$	−4.87673	−0.326570	−0.163285	+	0.986579i	$0.552209\pi$
$224$	0	0
$225$	19.7093	1.31395
$226$	0	0
$227$	−3.57904	−0.237549	−0.118775	−	0.992921i	$-0.537897\pi$
$227$	−3.57904	−0.237549	−0.118775	+	0.992921i	$0.537897\pi$
$228$	0	0
$229$	3.43624	0.227073	0.113537	−	0.993534i	$-0.463782\pi$
$229$	3.43624	0.227073	0.113537	+	0.993534i	$0.463782\pi$
$230$	0	0
$231$	2.42593	0.159614
$232$	0	0
$233$	20.1985	1.32325	0.661623	−	0.749837i	$-0.269868\pi$
$233$	20.1985	1.32325	0.661623	+	0.749837i	$0.269868\pi$
$234$	0	0
$235$	16.6284	1.08472
$236$	0	0
$237$	3.55201	0.230728
$238$	0	0
$239$	−20.3332	−1.31524	−0.657622	−	0.753348i	$-0.728438\pi$
$239$	−20.3332	−1.31524	−0.657622	+	0.753348i	$0.728438\pi$
$240$	0	0
$241$	13.0403	0.839999	0.420000	−	0.907524i	$-0.362030\pi$
$241$	13.0403	0.839999	0.420000	+	0.907524i	$0.362030\pi$
$242$	0	0
$243$	−21.8013	−1.39856
$244$	0	0
$245$	3.43967	0.219752
$246$	0	0
$247$	3.63761	0.231456
$248$	0	0
$249$	−0.833150	−0.0527987
$250$	0	0
$251$	−6.35716	−0.401261	−0.200630	−	0.979667i	$-0.564299\pi$
$251$	−6.35716	−0.401261	−0.200630	+	0.979667i	$0.564299\pi$
$252$	0	0
$253$	1.12036	0.0704364
$254$	0	0
$255$	−36.6691	−2.29631
$256$	0	0
$257$	−12.9787	−0.809589	−0.404794	−	0.914408i	$-0.632657\pi$
$257$	−12.9787	−0.809589	−0.404794	+	0.914408i	$0.632657\pi$
$258$	0	0
$259$	−2.37053	−0.147298
$260$	0	0
$261$	−11.2252	−0.694823
$262$	0	0
$263$	24.6095	1.51749	0.758744	−	0.651389i	$-0.225814\pi$
$263$	24.6095	1.51749	0.758744	+	0.651389i	$0.225814\pi$
$264$	0	0
$265$	−6.36173	−0.390798
$266$	0	0
$267$	28.2984	1.73184
$268$	0	0
$269$	9.34666	0.569876	0.284938	−	0.958546i	$-0.408027\pi$
$269$	9.34666	0.569876	0.284938	+	0.958546i	$0.408027\pi$
$270$	0	0
$271$	−5.21743	−0.316936	−0.158468	−	0.987364i	$-0.550656\pi$
$271$	−5.21743	−0.316936	−0.158468	+	0.987364i	$0.550656\pi$
$272$	0	0
$273$	−2.42593	−0.146824
$274$	0	0
$275$	−6.83133	−0.411945
$276$	0	0
$277$	−4.04369	−0.242962	−0.121481	−	0.992594i	$-0.538764\pi$
$277$	−4.04369	−0.242962	−0.121481	+	0.992594i	$0.538764\pi$
$278$	0	0
$279$	27.2741	1.63286
$280$	0	0
$281$	−27.7220	−1.65376	−0.826879	−	0.562380i	$-0.809886\pi$
$281$	−27.7220	−1.65376	−0.826879	+	0.562380i	$0.809886\pi$
$282$	0	0
$283$	−0.542146	−0.0322272	−0.0161136	−	0.999870i	$-0.505129\pi$
$283$	−0.542146	−0.0322272	−0.0161136	+	0.999870i	$0.505129\pi$
$284$	0	0
$285$	30.3537	1.79800
$286$	0	0
$287$	−4.58480	−0.270632
$288$	0	0
$289$	2.31129	0.135958
$290$	0	0
$291$	46.6204	2.73294
$292$	0	0
$293$	−18.0888	−1.05676	−0.528380	−	0.849008i	$-0.677200\pi$
$293$	−18.0888	−1.05676	−0.528380	+	0.849008i	$0.677200\pi$
$294$	0	0
$295$	34.0920	1.98491
$296$	0	0
$297$	0.278666	0.0161698
$298$	0	0
$299$	−1.12036	−0.0647921
$300$	0	0
$301$	−7.49006	−0.431720
$302$	0	0
$303$	29.3143	1.68406
$304$	0	0
$305$	−2.23868	−0.128186
$306$	0	0
$307$	−9.22000	−0.526213	−0.263107	−	0.964767i	$-0.584747\pi$
$307$	−9.22000	−0.526213	−0.263107	+	0.964767i	$0.584747\pi$
$308$	0	0
$309$	−29.9887	−1.70600
$310$	0	0
$311$	14.4886	0.821572	0.410786	−	0.911732i	$-0.365254\pi$
$311$	14.4886	0.821572	0.410786	+	0.911732i	$0.365254\pi$
$312$	0	0
$313$	0.851282	0.0481173	0.0240586	−	0.999711i	$-0.492341\pi$
$313$	0.851282	0.0481173	0.0240586	+	0.999711i	$0.492341\pi$
$314$	0	0
$315$	−9.92390	−0.559148
$316$	0	0
$317$	2.07945	0.116794	0.0583969	−	0.998293i	$-0.481401\pi$
$317$	2.07945	0.116794	0.0583969	+	0.998293i	$0.481401\pi$
$318$	0	0
$319$	3.89071	0.217838
$320$	0	0
$321$	11.2247	0.626500
$322$	0	0
$323$	−15.9853	−0.889448
$324$	0	0
$325$	6.83133	0.378934
$326$	0	0
$327$	−18.3423	−1.01433
$328$	0	0
$329$	−4.83430	−0.266524
$330$	0	0
$331$	−13.1667	−0.723708	−0.361854	−	0.932235i	$-0.617856\pi$
$331$	−13.1667	−0.723708	−0.361854	+	0.932235i	$0.617856\pi$
$332$	0	0
$333$	6.83929	0.374791
$334$	0	0
$335$	−26.6610	−1.45665
$336$	0	0
$337$	−18.1014	−0.986048	−0.493024	−	0.870016i	$-0.664108\pi$
$337$	−18.1014	−0.986048	−0.493024	+	0.870016i	$0.664108\pi$
$338$	0	0
$339$	31.9131	1.73328
$340$	0	0
$341$	−9.45333	−0.511927
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−9.34872	−0.503318
$346$	0	0
$347$	−16.6464	−0.893625	−0.446812	−	0.894628i	$-0.647441\pi$
$347$	−16.6464	−0.893625	−0.446812	+	0.894628i	$0.647441\pi$
$348$	0	0
$349$	−0.136903	−0.00732825	−0.00366413	−	0.999993i	$-0.501166\pi$
$349$	−0.136903	−0.00732825	−0.00366413	+	0.999993i	$0.501166\pi$
$350$	0	0
$351$	−0.278666	−0.0148741
$352$	0	0
$353$	10.6570	0.567216	0.283608	−	0.958940i	$-0.408469\pi$
$353$	10.6570	0.567216	0.283608	+	0.958940i	$0.408469\pi$
$354$	0	0
$355$	33.5736	1.78190
$356$	0	0
$357$	10.6606	0.564221
$358$	0	0
$359$	−24.9224	−1.31535	−0.657677	−	0.753300i	$-0.728461\pi$
$359$	−24.9224	−1.31535	−0.657677	+	0.753300i	$0.728461\pi$
$360$	0	0
$361$	−5.76778	−0.303567
$362$	0	0
$363$	2.42593	0.127328
$364$	0	0
$365$	7.41578	0.388159
$366$	0	0
$367$	4.71677	0.246214	0.123107	−	0.992393i	$-0.460714\pi$
$367$	4.71677	0.246214	0.123107	+	0.992393i	$0.460714\pi$
$368$	0	0
$369$	13.2278	0.688610
$370$	0	0
$371$	1.84952	0.0960222
$372$	0	0
$373$	−6.53298	−0.338265	−0.169132	−	0.985593i	$-0.554097\pi$
$373$	−6.53298	−0.338265	−0.169132	+	0.985593i	$0.554097\pi$
$374$	0	0
$375$	15.2813	0.789125
$376$	0	0
$377$	−3.89071	−0.200382
$378$	0	0
$379$	−29.2490	−1.50242	−0.751210	−	0.660063i	$-0.770529\pi$
$379$	−29.2490	−1.50242	−0.751210	+	0.660063i	$0.770529\pi$
$380$	0	0
$381$	−4.17080	−0.213676
$382$	0	0
$383$	11.0743	0.565869	0.282934	−	0.959139i	$-0.408692\pi$
$383$	11.0743	0.565869	0.282934	+	0.959139i	$0.408692\pi$
$384$	0	0
$385$	3.43967	0.175302
$386$	0	0
$387$	21.6098	1.09849
$388$	0	0
$389$	−12.7247	−0.645168	−0.322584	−	0.946541i	$-0.604551\pi$
$389$	−12.7247	−0.645168	−0.322584	+	0.946541i	$0.604551\pi$
$390$	0	0
$391$	4.92338	0.248986
$392$	0	0
$393$	−18.3157	−0.923907
$394$	0	0
$395$	5.03632	0.253405
$396$	0	0
$397$	34.0119	1.70701	0.853505	−	0.521084i	$-0.174472\pi$
$397$	34.0119	1.70701	0.853505	+	0.521084i	$0.174472\pi$
$398$	0	0
$399$	−8.82459	−0.441782
$400$	0	0
$401$	−5.96380	−0.297818	−0.148909	−	0.988851i	$-0.547576\pi$
$401$	−5.96380	−0.297818	−0.148909	+	0.988851i	$0.547576\pi$
$402$	0	0
$403$	9.45333	0.470904
$404$	0	0
$405$	−32.0970	−1.59491
$406$	0	0
$407$	−2.37053	−0.117503
$408$	0	0
$409$	5.32269	0.263190	0.131595	−	0.991304i	$-0.457990\pi$
$409$	5.32269	0.263190	0.131595	+	0.991304i	$0.457990\pi$
$410$	0	0
$411$	31.2484	1.54137
$412$	0	0
$413$	−9.91141	−0.487709
$414$	0	0
$415$	−1.18130	−0.0579879
$416$	0	0
$417$	−44.2610	−2.16747
$418$	0	0
$419$	−13.7043	−0.669499	−0.334750	−	0.942307i	$-0.608652\pi$
$419$	−13.7043	−0.669499	−0.334750	+	0.942307i	$0.608652\pi$
$420$	0	0
$421$	2.63192	0.128272	0.0641359	−	0.997941i	$-0.479571\pi$
$421$	2.63192	0.128272	0.0641359	+	0.997941i	$0.479571\pi$
$422$	0	0
$423$	13.9476	0.678155
$424$	0	0
$425$	−30.0200	−1.45618
$426$	0	0
$427$	0.650840	0.0314964
$428$	0	0
$429$	−2.42593	−0.117125
$430$	0	0
$431$	−7.69378	−0.370596	−0.185298	−	0.982682i	$-0.559325\pi$
$431$	−7.69378	−0.370596	−0.185298	+	0.982682i	$0.559325\pi$
$432$	0	0
$433$	−32.6632	−1.56970	−0.784848	−	0.619689i	$-0.787259\pi$
$433$	−32.6632	−1.56970	−0.784848	+	0.619689i	$0.787259\pi$
$434$	0	0
$435$	−32.4656	−1.55661
$436$	0	0
$437$	−4.07543	−0.194954
$438$	0	0
$439$	−8.99271	−0.429199	−0.214599	−	0.976702i	$-0.568845\pi$
$439$	−8.99271	−0.429199	−0.214599	+	0.976702i	$0.568845\pi$
$440$	0	0
$441$	2.88513	0.137387
$442$	0	0
$443$	−34.5622	−1.64210	−0.821050	−	0.570856i	$-0.806612\pi$
$443$	−34.5622	−1.64210	−0.821050	+	0.570856i	$0.806612\pi$
$444$	0	0
$445$	40.1237	1.90205
$446$	0	0
$447$	−16.1398	−0.763385
$448$	0	0
$449$	7.75887	0.366164	0.183082	−	0.983098i	$-0.441393\pi$
$449$	7.75887	0.366164	0.183082	+	0.983098i	$0.441393\pi$
$450$	0	0
$451$	−4.58480	−0.215890
$452$	0	0
$453$	20.7409	0.974491
$454$	0	0
$455$	−3.43967	−0.161254
$456$	0	0
$457$	−17.9485	−0.839596	−0.419798	−	0.907617i	$-0.637899\pi$
$457$	−17.9485	−0.839596	−0.419798	+	0.907617i	$0.637899\pi$
$458$	0	0
$459$	1.22458	0.0571587
$460$	0	0
$461$	−8.07951	−0.376300	−0.188150	−	0.982140i	$-0.560249\pi$
$461$	−8.07951	−0.376300	−0.188150	+	0.982140i	$0.560249\pi$
$462$	0	0
$463$	27.5133	1.27865	0.639326	−	0.768936i	$-0.279213\pi$
$463$	27.5133	1.27865	0.639326	+	0.768936i	$0.279213\pi$
$464$	0	0
$465$	78.8824	3.65808
$466$	0	0
$467$	34.3596	1.58997	0.794986	−	0.606628i	$-0.207478\pi$
$467$	34.3596	1.58997	0.794986	+	0.606628i	$0.207478\pi$
$468$	0	0
$469$	7.75105	0.357910
$470$	0	0
$471$	−38.6041	−1.77878
$472$	0	0
$473$	−7.49006	−0.344393
$474$	0	0
$475$	24.8497	1.14018
$476$	0	0
$477$	−5.33610	−0.244323
$478$	0	0
$479$	−27.7006	−1.26567	−0.632836	−	0.774286i	$-0.718109\pi$
$479$	−27.7006	−1.26567	−0.632836	+	0.774286i	$0.718109\pi$
$480$	0	0
$481$	2.37053	0.108087
$482$	0	0
$483$	2.71791	0.123669
$484$	0	0
$485$	66.1021	3.00154
$486$	0	0
$487$	−24.9726	−1.13162	−0.565809	−	0.824536i	$-0.691436\pi$
$487$	−24.9726	−1.13162	−0.565809	+	0.824536i	$0.691436\pi$
$488$	0	0
$489$	−46.1441	−2.08671
$490$	0	0
$491$	4.47595	0.201997	0.100998	−	0.994887i	$-0.467796\pi$
$491$	4.47595	0.201997	0.100998	+	0.994887i	$0.467796\pi$
$492$	0	0
$493$	17.0976	0.770036
$494$	0	0
$495$	−9.92390	−0.446046
$496$	0	0
$497$	−9.76069	−0.437827
$498$	0	0
$499$	−23.5884	−1.05596	−0.527982	−	0.849256i	$-0.677051\pi$
$499$	−23.5884	−1.05596	−0.527982	+	0.849256i	$0.677051\pi$
$500$	0	0
$501$	−55.2906	−2.47020
$502$	0	0
$503$	19.8609	0.885553	0.442776	−	0.896632i	$-0.353994\pi$
$503$	19.8609	0.885553	0.442776	+	0.896632i	$0.353994\pi$
$504$	0	0
$505$	41.5641	1.84958
$506$	0	0
$507$	2.42593	0.107739
$508$	0	0
$509$	15.6950	0.695668	0.347834	−	0.937556i	$-0.386917\pi$
$509$	15.6950	0.695668	0.347834	+	0.937556i	$0.386917\pi$
$510$	0	0
$511$	−2.15596	−0.0953739
$512$	0	0
$513$	−1.01368	−0.0447549
$514$	0	0
$515$	−42.5204	−1.87367
$516$	0	0
$517$	−4.83430	−0.212612
$518$	0	0
$519$	46.1012	2.02362
$520$	0	0
$521$	−31.1916	−1.36653	−0.683265	−	0.730171i	$-0.739441\pi$
$521$	−31.1916	−1.36653	−0.683265	+	0.730171i	$0.739441\pi$
$522$	0	0
$523$	−7.13776	−0.312113	−0.156056	−	0.987748i	$-0.549878\pi$
$523$	−7.13776	−0.312113	−0.156056	+	0.987748i	$0.549878\pi$
$524$	0	0
$525$	−16.5723	−0.723275
$526$	0	0
$527$	−41.5423	−1.80961
$528$	0	0
$529$	−21.7448	−0.945426
$530$	0	0
$531$	28.5957	1.24095
$532$	0	0
$533$	4.58480	0.198590
$534$	0	0
$535$	15.9152	0.688075
$536$	0	0
$537$	−29.8959	−1.29011
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−33.5896	−1.44413	−0.722064	−	0.691826i	$-0.756807\pi$
$541$	−33.5896	−1.44413	−0.722064	+	0.691826i	$0.756807\pi$
$542$	0	0
$543$	36.3193	1.55861
$544$	0	0
$545$	−26.0071	−1.11402
$546$	0	0
$547$	24.4328	1.04467	0.522335	−	0.852741i	$-0.325061\pi$
$547$	24.4328	1.04467	0.522335	+	0.852741i	$0.325061\pi$
$548$	0	0
$549$	−1.87776	−0.0801408
$550$	0	0
$551$	−14.1529	−0.602934
$552$	0	0
$553$	−1.46419	−0.0622636
$554$	0	0
$555$	19.7806	0.839641
$556$	0	0
$557$	−29.5692	−1.25289	−0.626444	−	0.779466i	$-0.715490\pi$
$557$	−29.5692	−1.25289	−0.626444	+	0.779466i	$0.715490\pi$
$558$	0	0
$559$	7.49006	0.316796
$560$	0	0
$561$	10.6606	0.450093
$562$	0	0
$563$	30.0903	1.26815	0.634077	−	0.773270i	$-0.281380\pi$
$563$	30.0903	1.26815	0.634077	+	0.773270i	$0.281380\pi$
$564$	0	0
$565$	45.2488	1.90363
$566$	0	0
$567$	9.33141	0.391883
$568$	0	0
$569$	−23.9871	−1.00559	−0.502796	−	0.864405i	$-0.667695\pi$
$569$	−23.9871	−1.00559	−0.502796	+	0.864405i	$0.667695\pi$
$570$	0	0
$571$	−9.07260	−0.379676	−0.189838	−	0.981815i	$-0.560796\pi$
$571$	−9.07260	−0.379676	−0.189838	+	0.981815i	$0.560796\pi$
$572$	0	0
$573$	−26.3371	−1.10025
$574$	0	0
$575$	−7.65355	−0.319175
$576$	0	0
$577$	−16.2386	−0.676023	−0.338011	−	0.941142i	$-0.609754\pi$
$577$	−16.2386	−0.676023	−0.338011	+	0.941142i	$0.609754\pi$
$578$	0	0
$579$	5.88611	0.244618
$580$	0	0
$581$	0.343435	0.0142481
$582$	0	0
$583$	1.84952	0.0765992
$584$	0	0
$585$	9.92390	0.410303
$586$	0	0
$587$	−16.1644	−0.667178	−0.333589	−	0.942719i	$-0.608260\pi$
$587$	−16.1644	−0.667178	−0.333589	+	0.942719i	$0.608260\pi$
$588$	0	0
$589$	34.3876	1.41691
$590$	0	0
$591$	0.00704763	0.000289901 0
$592$	0	0
$593$	−6.65555	−0.273311	−0.136655	−	0.990619i	$-0.543635\pi$
$593$	−6.65555	−0.273311	−0.136655	+	0.990619i	$0.543635\pi$
$594$	0	0
$595$	15.1155	0.619675
$596$	0	0
$597$	5.62777	0.230329
$598$	0	0
$599$	−11.6931	−0.477768	−0.238884	−	0.971048i	$-0.576782\pi$
$599$	−11.6931	−0.477768	−0.238884	+	0.971048i	$0.576782\pi$
$600$	0	0
$601$	13.0615	0.532791	0.266396	−	0.963864i	$-0.414167\pi$
$601$	13.0615	0.532791	0.266396	+	0.963864i	$0.414167\pi$
$602$	0	0
$603$	−22.3628	−0.910683
$604$	0	0
$605$	3.43967	0.139842
$606$	0	0
$607$	11.4942	0.466535	0.233268	−	0.972413i	$-0.425058\pi$
$607$	11.4942	0.466535	0.233268	+	0.972413i	$0.425058\pi$
$608$	0	0
$609$	9.43859	0.382471
$610$	0	0
$611$	4.83430	0.195575
$612$	0	0
$613$	0.173262	0.00699801	0.00349900	−	0.999994i	$-0.498886\pi$
$613$	0.173262	0.00699801	0.00349900	+	0.999994i	$0.498886\pi$
$614$	0	0
$615$	38.2574	1.54269
$616$	0	0
$617$	34.0799	1.37200	0.686002	−	0.727600i	$-0.259364\pi$
$617$	34.0799	1.37200	0.686002	+	0.727600i	$0.259364\pi$
$618$	0	0
$619$	14.6506	0.588859	0.294429	−	0.955673i	$-0.404870\pi$
$619$	14.6506	0.588859	0.294429	+	0.955673i	$0.404870\pi$
$620$	0	0
$621$	0.312206	0.0125284
$622$	0	0
$623$	−11.6650	−0.467348
$624$	0	0
$625$	−12.4896	−0.499583
$626$	0	0
$627$	−8.82459	−0.352420
$628$	0	0
$629$	−10.4172	−0.415361
$630$	0	0
$631$	33.9339	1.35089	0.675443	−	0.737412i	$-0.263953\pi$
$631$	33.9339	1.35089	0.675443	+	0.737412i	$0.263953\pi$
$632$	0	0
$633$	−65.4884	−2.60293
$634$	0	0
$635$	−5.91368	−0.234677
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	28.1609	1.11403
$640$	0	0
$641$	37.5354	1.48256	0.741280	−	0.671196i	$-0.234219\pi$
$641$	37.5354	1.48256	0.741280	+	0.671196i	$0.234219\pi$
$642$	0	0
$643$	−8.50207	−0.335289	−0.167644	−	0.985848i	$-0.553616\pi$
$643$	−8.50207	−0.335289	−0.167644	+	0.985848i	$0.553616\pi$
$644$	0	0
$645$	62.5000	2.46094
$646$	0	0
$647$	9.99110	0.392791	0.196395	−	0.980525i	$-0.437076\pi$
$647$	9.99110	0.392791	0.196395	+	0.980525i	$0.437076\pi$
$648$	0	0
$649$	−9.91141	−0.389057
$650$	0	0
$651$	−22.9331	−0.898820
$652$	0	0
$653$	34.8978	1.36565	0.682827	−	0.730580i	$-0.260750\pi$
$653$	34.8978	1.36565	0.682827	+	0.730580i	$0.260750\pi$
$654$	0	0
$655$	−25.9695	−1.01471
$656$	0	0
$657$	6.22021	0.242674
$658$	0	0
$659$	8.98982	0.350194	0.175097	−	0.984551i	$-0.443976\pi$
$659$	8.98982	0.350194	0.175097	+	0.984551i	$0.443976\pi$
$660$	0	0
$661$	−38.0385	−1.47953	−0.739763	−	0.672867i	$-0.765063\pi$
$661$	−38.0385	−1.47953	−0.739763	+	0.672867i	$0.765063\pi$
$662$	0	0
$663$	−10.6606	−0.414025
$664$	0	0
$665$	−12.5122	−0.485202
$666$	0	0
$667$	4.35900	0.168781
$668$	0	0
$669$	−11.8306	−0.457397
$670$	0	0
$671$	0.650840	0.0251254
$672$	0	0
$673$	−19.0582	−0.734641	−0.367321	−	0.930094i	$-0.619725\pi$
$673$	−19.0582	−0.734641	−0.367321	+	0.930094i	$0.619725\pi$
$674$	0	0
$675$	−1.90366	−0.0732718
$676$	0	0
$677$	−7.32867	−0.281664	−0.140832	−	0.990034i	$-0.544978\pi$
$677$	−7.32867	−0.281664	−0.140832	+	0.990034i	$0.544978\pi$
$678$	0	0
$679$	−19.2176	−0.737502
$680$	0	0
$681$	−8.68249	−0.332714
$682$	0	0
$683$	−17.0488	−0.652356	−0.326178	−	0.945308i	$-0.605761\pi$
$683$	−17.0488	−0.652356	−0.326178	+	0.945308i	$0.605761\pi$
$684$	0	0
$685$	44.3065	1.69286
$686$	0	0
$687$	8.33608	0.318041
$688$	0	0
$689$	−1.84952	−0.0704610
$690$	0	0
$691$	−24.6681	−0.938419	−0.469209	−	0.883087i	$-0.655461\pi$
$691$	−24.6681	−0.938419	−0.469209	+	0.883087i	$0.655461\pi$
$692$	0	0
$693$	2.88513	0.109597
$694$	0	0
$695$	−62.7566	−2.38049
$696$	0	0
$697$	−20.1477	−0.763150
$698$	0	0
$699$	49.0000	1.85335
$700$	0	0
$701$	−39.0516	−1.47496	−0.737480	−	0.675369i	$-0.763984\pi$
$701$	−39.0516	−1.47496	−0.737480	+	0.675369i	$0.763984\pi$
$702$	0	0
$703$	8.62307	0.325225
$704$	0	0
$705$	40.3393	1.51927
$706$	0	0
$707$	−12.0837	−0.454456
$708$	0	0
$709$	−23.3790	−0.878017	−0.439008	−	0.898483i	$-0.644670\pi$
$709$	−23.3790	−0.878017	−0.439008	+	0.898483i	$0.644670\pi$
$710$	0	0
$711$	4.22437	0.158426
$712$	0	0
$713$	−10.5911	−0.396641
$714$	0	0
$715$	−3.43967	−0.128636
$716$	0	0
$717$	−49.3269	−1.84215
$718$	0	0
$719$	−17.0261	−0.634966	−0.317483	−	0.948264i	$-0.602838\pi$
$719$	−17.0261	−0.634966	−0.317483	+	0.948264i	$0.602838\pi$
$720$	0	0
$721$	12.3618	0.460376
$722$	0	0
$723$	31.6348	1.17651
$724$	0	0
$725$	−26.5787	−0.987110
$726$	0	0
$727$	−43.2062	−1.60243	−0.801214	−	0.598378i	$-0.795812\pi$
$727$	−43.2062	−1.60243	−0.801214	+	0.598378i	$0.795812\pi$
$728$	0	0
$729$	−24.8943	−0.922010
$730$	0	0
$731$	−32.9148	−1.21740
$732$	0	0
$733$	44.9053	1.65861	0.829307	−	0.558793i	$-0.188735\pi$
$733$	44.9053	1.65861	0.829307	+	0.558793i	$0.188735\pi$
$734$	0	0
$735$	8.34439	0.307788
$736$	0	0
$737$	7.75105	0.285513
$738$	0	0
$739$	−13.6940	−0.503743	−0.251871	−	0.967761i	$-0.581046\pi$
$739$	−13.6940	−0.503743	−0.251871	+	0.967761i	$0.581046\pi$
$740$	0	0
$741$	8.82459	0.324179
$742$	0	0
$743$	−13.0544	−0.478918	−0.239459	−	0.970906i	$-0.576970\pi$
$743$	−13.0544	−0.478918	−0.239459	+	0.970906i	$0.576970\pi$
$744$	0	0
$745$	−22.8842	−0.838413
$746$	0	0
$747$	−0.990856	−0.0362535
$748$	0	0
$749$	−4.62696	−0.169065
$750$	0	0
$751$	−10.5301	−0.384248	−0.192124	−	0.981371i	$-0.561538\pi$
$751$	−10.5301	−0.384248	−0.192124	+	0.981371i	$0.561538\pi$
$752$	0	0
$753$	−15.4220	−0.562010
$754$	0	0
$755$	29.4080	1.07027
$756$	0	0
$757$	32.4193	1.17830	0.589150	−	0.808023i	$-0.299463\pi$
$757$	32.4193	1.17830	0.589150	+	0.808023i	$0.299463\pi$
$758$	0	0
$759$	2.71791	0.0986540
$760$	0	0
$761$	37.5052	1.35956	0.679781	−	0.733415i	$-0.262075\pi$
$761$	37.5052	1.35956	0.679781	+	0.733415i	$0.262075\pi$
$762$	0	0
$763$	7.56094	0.273724
$764$	0	0
$765$	−43.6102	−1.57673
$766$	0	0
$767$	9.91141	0.357880
$768$	0	0
$769$	36.3325	1.31018	0.655091	−	0.755550i	$-0.272630\pi$
$769$	36.3325	1.31018	0.655091	+	0.755550i	$0.272630\pi$
$770$	0	0
$771$	−31.4854	−1.13392
$772$	0	0
$773$	37.0878	1.33396	0.666978	−	0.745078i	$-0.267588\pi$
$773$	37.0878	1.33396	0.666978	+	0.745078i	$0.267588\pi$
$774$	0	0
$775$	64.5788	2.31974
$776$	0	0
$777$	−5.75074	−0.206307
$778$	0	0
$779$	16.6777	0.597542
$780$	0	0
$781$	−9.76069	−0.349265
$782$	0	0
$783$	1.08421	0.0387464
$784$	0	0
$785$	−54.7359	−1.95361
$786$	0	0
$787$	22.3705	0.797421	0.398710	−	0.917077i	$-0.369458\pi$
$787$	22.3705	0.797421	0.398710	+	0.917077i	$0.369458\pi$
$788$	0	0
$789$	59.7009	2.12541
$790$	0	0
$791$	−13.1550	−0.467737
$792$	0	0
$793$	−0.650840	−0.0231120
$794$	0	0
$795$	−15.4331	−0.547356
$796$	0	0
$797$	9.42396	0.333814	0.166907	−	0.985973i	$-0.446622\pi$
$797$	9.42396	0.333814	0.166907	+	0.985973i	$0.446622\pi$
$798$	0	0
$799$	−21.2441	−0.751564
$800$	0	0
$801$	33.6550	1.18914
$802$	0	0
$803$	−2.15596	−0.0760820
$804$	0	0
$805$	3.85367	0.135824
$806$	0	0
$807$	22.6743	0.798174
$808$	0	0
$809$	48.0837	1.69053	0.845267	−	0.534345i	$-0.179442\pi$
$809$	48.0837	1.69053	0.845267	+	0.534345i	$0.179442\pi$
$810$	0	0
$811$	35.7446	1.25516	0.627581	−	0.778552i	$-0.284045\pi$
$811$	35.7446	1.25516	0.627581	+	0.778552i	$0.284045\pi$
$812$	0	0
$813$	−12.6571	−0.443904
$814$	0	0
$815$	−65.4266	−2.29179
$816$	0	0
$817$	27.2459	0.953215
$818$	0	0
$819$	−2.88513	−0.100815
$820$	0	0
$821$	−28.9272	−1.00957	−0.504783	−	0.863246i	$-0.668427\pi$
$821$	−28.9272	−1.00957	−0.504783	+	0.863246i	$0.668427\pi$
$822$	0	0
$823$	17.1149	0.596587	0.298293	−	0.954474i	$-0.403583\pi$
$823$	17.1149	0.596587	0.298293	+	0.954474i	$0.403583\pi$
$824$	0	0
$825$	−16.5723	−0.576974
$826$	0	0
$827$	24.9579	0.867870	0.433935	−	0.900944i	$-0.357125\pi$
$827$	24.9579	0.867870	0.433935	+	0.900944i	$0.357125\pi$
$828$	0	0
$829$	35.6164	1.23701	0.618505	−	0.785781i	$-0.287739\pi$
$829$	35.6164	1.23701	0.618505	+	0.785781i	$0.287739\pi$
$830$	0	0
$831$	−9.80972	−0.340295
$832$	0	0
$833$	−4.39446	−0.152259
$834$	0	0
$835$	−78.3953	−2.71298
$836$	0	0
$837$	−2.63432	−0.0910554
$838$	0	0
$839$	25.9135	0.894634	0.447317	−	0.894376i	$-0.352380\pi$
$839$	25.9135	0.894634	0.447317	+	0.894376i	$0.352380\pi$
$840$	0	0
$841$	−13.8624	−0.478012
$842$	0	0
$843$	−67.2517	−2.31627
$844$	0	0
$845$	3.43967	0.118328
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−1.31521	−0.0451378
$850$	0	0
$851$	−2.65585	−0.0910412
$852$	0	0
$853$	47.6882	1.63281	0.816406	−	0.577479i	$-0.195963\pi$
$853$	47.6882	1.63281	0.816406	+	0.577479i	$0.195963\pi$
$854$	0	0
$855$	36.0993	1.23457
$856$	0	0
$857$	6.18455	0.211260	0.105630	−	0.994405i	$-0.466314\pi$
$857$	6.18455	0.211260	0.105630	+	0.994405i	$0.466314\pi$
$858$	0	0
$859$	−9.04710	−0.308683	−0.154342	−	0.988018i	$-0.549326\pi$
$859$	−9.04710	−0.308683	−0.154342	+	0.988018i	$0.549326\pi$
$860$	0	0
$861$	−11.1224	−0.379051
$862$	0	0
$863$	46.4843	1.58234	0.791171	−	0.611595i	$-0.209472\pi$
$863$	46.4843	1.58234	0.791171	+	0.611595i	$0.209472\pi$
$864$	0	0
$865$	65.3659	2.22251
$866$	0	0
$867$	5.60702	0.190424
$868$	0	0
$869$	−1.46419	−0.0496691
$870$	0	0
$871$	−7.75105	−0.262634
$872$	0	0
$873$	55.4452	1.87653
$874$	0	0
$875$	−6.29917	−0.212951
$876$	0	0
$877$	21.1617	0.714581	0.357290	−	0.933993i	$-0.383701\pi$
$877$	21.1617	0.714581	0.357290	+	0.933993i	$0.383701\pi$
$878$	0	0
$879$	−43.8822	−1.48011
$880$	0	0
$881$	35.8315	1.20719	0.603597	−	0.797289i	$-0.293734\pi$
$881$	35.8315	1.20719	0.603597	+	0.797289i	$0.293734\pi$
$882$	0	0
$883$	20.5652	0.692074	0.346037	−	0.938221i	$-0.387527\pi$
$883$	20.5652	0.692074	0.346037	+	0.938221i	$0.387527\pi$
$884$	0	0
$885$	82.7047	2.78009
$886$	0	0
$887$	21.4208	0.719241	0.359621	−	0.933099i	$-0.382906\pi$
$887$	21.4208	0.719241	0.359621	+	0.933099i	$0.382906\pi$
$888$	0	0
$889$	1.71926	0.0576621
$890$	0	0
$891$	9.33141	0.312614
$892$	0	0
$893$	17.5853	0.588470
$894$	0	0
$895$	−42.3888	−1.41690
$896$	0	0
$897$	−2.71791	−0.0907485
$898$	0	0
$899$	−36.7802	−1.22669
$900$	0	0
$901$	8.12763	0.270771
$902$	0	0
$903$	−18.1704	−0.604672
$904$	0	0
$905$	51.4964	1.71180
$906$	0	0
$907$	13.3939	0.444739	0.222369	−	0.974963i	$-0.428621\pi$
$907$	13.3939	0.444739	0.222369	+	0.974963i	$0.428621\pi$
$908$	0	0
$909$	34.8632	1.15634
$910$	0	0
$911$	−48.6453	−1.61169	−0.805845	−	0.592127i	$-0.798288\pi$
$911$	−48.6453	−1.61169	−0.805845	+	0.592127i	$0.798288\pi$
$912$	0	0
$913$	0.343435	0.0113661
$914$	0	0
$915$	−5.43087	−0.179539
$916$	0	0
$917$	7.54999	0.249323
$918$	0	0
$919$	−39.5391	−1.30428	−0.652138	−	0.758100i	$-0.726128\pi$
$919$	−39.5391	−1.30428	−0.652138	+	0.758100i	$0.726128\pi$
$920$	0	0
$921$	−22.3671	−0.737020
$922$	0	0
$923$	9.76069	0.321277
$924$	0	0
$925$	16.1939	0.532451
$926$	0	0
$927$	−35.6653	−1.17140
$928$	0	0
$929$	3.60362	0.118231	0.0591154	−	0.998251i	$-0.481172\pi$
$929$	3.60362	0.118231	0.0591154	+	0.998251i	$0.481172\pi$
$930$	0	0
$931$	3.63761	0.119218
$932$	0	0
$933$	35.1483	1.15070
$934$	0	0
$935$	15.1155	0.494330
$936$	0	0
$937$	−6.43742	−0.210301	−0.105151	−	0.994456i	$-0.533532\pi$
$937$	−6.43742	−0.210301	−0.105151	+	0.994456i	$0.533532\pi$
$938$	0	0
$939$	2.06515	0.0673936
$940$	0	0
$941$	45.6353	1.48767	0.743833	−	0.668365i	$-0.233006\pi$
$941$	45.6353	1.48767	0.743833	+	0.668365i	$0.233006\pi$
$942$	0	0
$943$	−5.13663	−0.167272
$944$	0	0
$945$	0.958517	0.0311806
$946$	0	0
$947$	6.98768	0.227069	0.113535	−	0.993534i	$-0.463783\pi$
$947$	6.98768	0.227069	0.113535	+	0.993534i	$0.463783\pi$
$948$	0	0
$949$	2.15596	0.0699853
$950$	0	0
$951$	5.04461	0.163583
$952$	0	0
$953$	16.5062	0.534687	0.267343	−	0.963601i	$-0.413854\pi$
$953$	16.5062	0.534687	0.267343	+	0.963601i	$0.413854\pi$
$954$	0	0
$955$	−37.3428	−1.20839
$956$	0	0
$957$	9.43859	0.305106
$958$	0	0
$959$	−12.8810	−0.415950
$960$	0	0
$961$	58.3655	1.88276
$962$	0	0
$963$	13.3494	0.430178
$964$	0	0
$965$	8.34579	0.268660
$966$	0	0
$967$	11.3163	0.363908	0.181954	−	0.983307i	$-0.441758\pi$
$967$	11.3163	0.363908	0.181954	+	0.983307i	$0.441758\pi$
$968$	0	0
$969$	−38.7793	−1.24577
$970$	0	0
$971$	24.4783	0.785545	0.392772	−	0.919636i	$-0.371516\pi$
$971$	24.4783	0.785545	0.392772	+	0.919636i	$0.371516\pi$
$972$	0	0
$973$	18.2450	0.584906
$974$	0	0
$975$	16.5723	0.530739
$976$	0	0
$977$	−32.4595	−1.03847	−0.519236	−	0.854631i	$-0.673783\pi$
$977$	−32.4595	−1.03847	−0.519236	+	0.854631i	$0.673783\pi$
$978$	0	0
$979$	−11.6650	−0.372815
$980$	0	0
$981$	−21.8143	−0.696477
$982$	0	0
$983$	48.0393	1.53221	0.766107	−	0.642713i	$-0.222191\pi$
$983$	48.0393	1.53221	0.766107	+	0.642713i	$0.222191\pi$
$984$	0	0
$985$	0.00999267	0.000318393 0
$986$	0	0
$987$	−11.7277	−0.373296
$988$	0	0
$989$	−8.39156	−0.266836
$990$	0	0
$991$	−35.9924	−1.14334	−0.571668	−	0.820485i	$-0.693704\pi$
$991$	−35.9924	−1.14334	−0.571668	+	0.820485i	$0.693704\pi$
$992$	0	0
$993$	−31.9415	−1.01363
$994$	0	0
$995$	7.97949	0.252967
$996$	0	0
$997$	24.3025	0.769667	0.384834	−	0.922986i	$-0.374259\pi$
$997$	24.3025	0.769667	0.384834	+	0.922986i	$0.374259\pi$
$998$	0	0
$999$	−0.660585	−0.0209000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.z.1.13	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.z.1.13	✓	15	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+2.42593 q^{3} +3.43967 q^{5} -1.00000 q^{7} +2.88513 q^{9} +O(q^{10})\)	\(q+2.42593 q^{3} +3.43967 q^{5} -1.00000 q^{7} +2.88513 q^{9} -1.00000 q^{11} +1.00000 q^{13} +8.34439 q^{15} -4.39446 q^{17} +3.63761 q^{19} -2.42593 q^{21} -1.12036 q^{23} +6.83133 q^{25} -0.278666 q^{27} -3.89071 q^{29} +9.45333 q^{31} -2.42593 q^{33} -3.43967 q^{35} +2.37053 q^{37} +2.42593 q^{39} +4.58480 q^{41} +7.49006 q^{43} +9.92390 q^{45} +4.83430 q^{47} +1.00000 q^{49} -10.6606 q^{51} -1.84952 q^{53} -3.43967 q^{55} +8.82459 q^{57} +9.91141 q^{59} -0.650840 q^{61} -2.88513 q^{63} +3.43967 q^{65} -7.75105 q^{67} -2.71791 q^{69} +9.76069 q^{71} +2.15596 q^{73} +16.5723 q^{75} +1.00000 q^{77} +1.46419 q^{79} -9.33141 q^{81} -0.343435 q^{83} -15.1155 q^{85} -9.43859 q^{87} +11.6650 q^{89} -1.00000 q^{91} +22.9331 q^{93} +12.5122 q^{95} +19.2176 q^{97} -2.88513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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