LMFDB - Embedded newform 8008.2.a.w.1.11

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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 3 x^{10} - 19 x^{9} + 55 x^{8} + 128 x^{7} - 361 x^{6} - 343 x^{5} + 1012 x^{4} + 215 x^{3} + \cdots + 160 $ x^11 - 3x^10 - 19x^9 + 55x^8 + 128x^7 - 361x^6 - 343x^5 + 1012x^4 + 215x^3 - 1090x^2 + 240x + 160
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$3.32763$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.32763 q^{3} -2.89815 q^{5} -1.00000 q^{7} +8.07314 q^{9} +O(q^{10})$	$q+3.32763 q^{3} -2.89815 q^{5} -1.00000 q^{7} +8.07314 q^{9} +1.00000 q^{11} +1.00000 q^{13} -9.64398 q^{15} +4.91732 q^{17} -1.97392 q^{19} -3.32763 q^{21} -0.822583 q^{23} +3.39928 q^{25} +16.8815 q^{27} +0.862485 q^{29} -1.11851 q^{31} +3.32763 q^{33} +2.89815 q^{35} -0.0683687 q^{37} +3.32763 q^{39} -0.550227 q^{41} +5.47505 q^{43} -23.3972 q^{45} +1.87682 q^{47} +1.00000 q^{49} +16.3630 q^{51} +1.87689 q^{53} -2.89815 q^{55} -6.56848 q^{57} +4.58138 q^{59} +3.40233 q^{61} -8.07314 q^{63} -2.89815 q^{65} -3.82803 q^{67} -2.73725 q^{69} -2.76928 q^{71} -4.48346 q^{73} +11.3116 q^{75} -1.00000 q^{77} +9.58848 q^{79} +31.9561 q^{81} +0.996079 q^{83} -14.2511 q^{85} +2.87003 q^{87} -3.92287 q^{89} -1.00000 q^{91} -3.72199 q^{93} +5.72072 q^{95} -1.45718 q^{97} +8.07314 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9}+O(q^{10}) $ 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9	$ 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9} + 11 q^{11} + 11 q^{13} + 7 q^{15} + 9 q^{17} + 20 q^{19} - 3 q^{21} + 12 q^{23} + 13 q^{25} + 15 q^{27} + 8 q^{29} + 7 q^{31} + 3 q^{33} + 2 q^{35} - 10 q^{37} + 3 q^{39} - 2 q^{41} + 24 q^{43} - 6 q^{45} + 2 q^{47} + 11 q^{49} + 17 q^{51} + 3 q^{53} - 2 q^{55} - 16 q^{57} + q^{59} - 22 q^{61} - 14 q^{63} - 2 q^{65} + 14 q^{67} - 22 q^{69} + 6 q^{71} + 3 q^{73} - 11 q^{77} + 8 q^{79} - 9 q^{81} + 29 q^{83} - 9 q^{85} + 19 q^{87} + 20 q^{89} - 11 q^{91} - q^{93} + 18 q^{95} - 25 q^{97} + 14 q^{99}+O(q^{100}) $ 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9 + 11 * q^11 + 11 * q^13 + 7 * q^15 + 9 * q^17 + 20 * q^19 - 3 * q^21 + 12 * q^23 + 13 * q^25 + 15 * q^27 + 8 * q^29 + 7 * q^31 + 3 * q^33 + 2 * q^35 - 10 * q^37 + 3 * q^39 - 2 * q^41 + 24 * q^43 - 6 * q^45 + 2 * q^47 + 11 * q^49 + 17 * q^51 + 3 * q^53 - 2 * q^55 - 16 * q^57 + q^59 - 22 * q^61 - 14 * q^63 - 2 * q^65 + 14 * q^67 - 22 * q^69 + 6 * q^71 + 3 * q^73 - 11 * q^77 + 8 * q^79 - 9 * q^81 + 29 * q^83 - 9 * q^85 + 19 * q^87 + 20 * q^89 - 11 * q^91 - q^93 + 18 * q^95 - 25 * q^97 + 14 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.32763	1.92121	0.960605	−	0.277918i	$-0.0896445\pi$
$3$	3.32763	1.92121	0.960605	+	0.277918i	$0.0896445\pi$
$4$	0	0
$5$	−2.89815	−1.29609	−0.648046	−	0.761601i	$-0.724414\pi$
$5$	−2.89815	−1.29609	−0.648046	+	0.761601i	$0.724414\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	8.07314	2.69105
$10$	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
$15$	−9.64398	−2.49007
$16$	0	0
$17$	4.91732	1.19263	0.596313	−	0.802752i	$-0.296632\pi$
$17$	4.91732	1.19263	0.596313	+	0.802752i	$0.296632\pi$
$18$	0	0
$19$	−1.97392	−0.452848	−0.226424	−	0.974029i	$-0.572704\pi$
$19$	−1.97392	−0.452848	−0.226424	+	0.974029i	$0.572704\pi$
$20$	0	0
$21$	−3.32763	−0.726149
$22$	0	0
$23$	−0.822583	−0.171520	−0.0857602	−	0.996316i	$-0.527332\pi$
$23$	−0.822583	−0.171520	−0.0857602	+	0.996316i	$0.527332\pi$
$24$	0	0
$25$	3.39928	0.679856
$26$	0	0
$27$	16.8815	3.24885
$28$	0	0
$29$	0.862485	0.160159	0.0800797	−	0.996788i	$-0.474483\pi$
$29$	0.862485	0.160159	0.0800797	+	0.996788i	$0.474483\pi$
$30$	0	0
$31$	−1.11851	−0.200890	−0.100445	−	0.994943i	$-0.532027\pi$
$31$	−1.11851	−0.200890	−0.100445	+	0.994943i	$0.532027\pi$
$32$	0	0
$33$	3.32763	0.579266
$34$	0	0
$35$	2.89815	0.489877
$36$	0	0
$37$	−0.0683687	−0.0112397	−0.00561987	−	0.999984i	$-0.501789\pi$
$37$	−0.0683687	−0.0112397	−0.00561987	+	0.999984i	$0.501789\pi$
$38$	0	0
$39$	3.32763	0.532848
$40$	0	0
$41$	−0.550227	−0.0859310	−0.0429655	−	0.999077i	$-0.513681\pi$
$41$	−0.550227	−0.0859310	−0.0429655	+	0.999077i	$0.513681\pi$
$42$	0	0
$43$	5.47505	0.834937	0.417468	−	0.908691i	$-0.362917\pi$
$43$	5.47505	0.834937	0.417468	+	0.908691i	$0.362917\pi$
$44$	0	0
$45$	−23.3972	−3.48784
$46$	0	0
$47$	1.87682	0.273763	0.136882	−	0.990587i	$-0.456292\pi$
$47$	1.87682	0.273763	0.136882	+	0.990587i	$0.456292\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	16.3630	2.29128
$52$	0	0
$53$	1.87689	0.257810	0.128905	−	0.991657i	$-0.458854\pi$
$53$	1.87689	0.257810	0.128905	+	0.991657i	$0.458854\pi$
$54$	0	0
$55$	−2.89815	−0.390787
$56$	0	0
$57$	−6.56848	−0.870016
$58$	0	0
$59$	4.58138	0.596445	0.298222	−	0.954496i	$-0.403606\pi$
$59$	4.58138	0.596445	0.298222	+	0.954496i	$0.403606\pi$
$60$	0	0
$61$	3.40233	0.435624	0.217812	−	0.975991i	$-0.430108\pi$
$61$	3.40233	0.435624	0.217812	+	0.975991i	$0.430108\pi$
$62$	0	0
$63$	−8.07314	−1.01712
$64$	0	0
$65$	−2.89815	−0.359471
$66$	0	0
$67$	−3.82803	−0.467668	−0.233834	−	0.972276i	$-0.575127\pi$
$67$	−3.82803	−0.467668	−0.233834	+	0.972276i	$0.575127\pi$
$68$	0	0
$69$	−2.73725	−0.329527
$70$	0	0
$71$	−2.76928	−0.328653	−0.164326	−	0.986406i	$-0.552545\pi$
$71$	−2.76928	−0.328653	−0.164326	+	0.986406i	$0.552545\pi$
$72$	0	0
$73$	−4.48346	−0.524749	−0.262374	−	0.964966i	$-0.584506\pi$
$73$	−4.48346	−0.524749	−0.262374	+	0.964966i	$0.584506\pi$
$74$	0	0
$75$	11.3116	1.30615
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	9.58848	1.07879	0.539394	−	0.842054i	$-0.318653\pi$
$79$	9.58848	1.07879	0.539394	+	0.842054i	$0.318653\pi$
$80$	0	0
$81$	31.9561	3.55068
$82$	0	0
$83$	0.996079	0.109334	0.0546669	−	0.998505i	$-0.482590\pi$
$83$	0.996079	0.109334	0.0546669	+	0.998505i	$0.482590\pi$
$84$	0	0
$85$	−14.2511	−1.54575
$86$	0	0
$87$	2.87003	0.307700
$88$	0	0
$89$	−3.92287	−0.415824	−0.207912	−	0.978148i	$-0.566667\pi$
$89$	−3.92287	−0.415824	−0.207912	+	0.978148i	$0.566667\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−3.72199	−0.385952
$94$	0	0
$95$	5.72072	0.586933
$96$	0	0
$97$	−1.45718	−0.147955	−0.0739773	−	0.997260i	$-0.523569\pi$
$97$	−1.45718	−0.147955	−0.0739773	+	0.997260i	$0.523569\pi$
$98$	0	0
$99$	8.07314	0.811381
$100$	0	0
$101$	−16.9047	−1.68208	−0.841038	−	0.540976i	$-0.818055\pi$
$101$	−16.9047	−1.68208	−0.841038	+	0.540976i	$0.818055\pi$
$102$	0	0
$103$	16.6904	1.64455	0.822275	−	0.569091i	$-0.192705\pi$
$103$	16.6904	1.64455	0.822275	+	0.569091i	$0.192705\pi$
$104$	0	0
$105$	9.64398	0.941156
$106$	0	0
$107$	−1.49677	−0.144698	−0.0723489	−	0.997379i	$-0.523050\pi$
$107$	−1.49677	−0.144698	−0.0723489	+	0.997379i	$0.523050\pi$
$108$	0	0
$109$	8.23268	0.788548	0.394274	−	0.918993i	$-0.370996\pi$
$109$	8.23268	0.788548	0.394274	+	0.918993i	$0.370996\pi$
$110$	0	0
$111$	−0.227506	−0.0215939
$112$	0	0
$113$	4.54322	0.427390	0.213695	−	0.976900i	$-0.431450\pi$
$113$	4.54322	0.427390	0.213695	+	0.976900i	$0.431450\pi$
$114$	0	0
$115$	2.38397	0.222306
$116$	0	0
$117$	8.07314	0.746362
$118$	0	0
$119$	−4.91732	−0.450770
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.83095	−0.165091
$124$	0	0
$125$	4.63913	0.414936
$126$	0	0
$127$	−5.00477	−0.444101	−0.222051	−	0.975035i	$-0.571275\pi$
$127$	−5.00477	−0.444101	−0.222051	+	0.975035i	$0.571275\pi$
$128$	0	0
$129$	18.2189	1.60409
$130$	0	0
$131$	15.5944	1.36249	0.681245	−	0.732056i	$-0.261439\pi$
$131$	15.5944	1.36249	0.681245	+	0.732056i	$0.261439\pi$
$132$	0	0
$133$	1.97392	0.171161
$134$	0	0
$135$	−48.9252	−4.21081
$136$	0	0
$137$	19.2848	1.64761	0.823804	−	0.566874i	$-0.191848\pi$
$137$	19.2848	1.64761	0.823804	+	0.566874i	$0.191848\pi$
$138$	0	0
$139$	16.7403	1.41990	0.709948	−	0.704255i	$-0.248719\pi$
$139$	16.7403	1.41990	0.709948	+	0.704255i	$0.248719\pi$
$140$	0	0
$141$	6.24538	0.525956
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−2.49961	−0.207582
$146$	0	0
$147$	3.32763	0.274458
$148$	0	0
$149$	−3.30677	−0.270901	−0.135451	−	0.990784i	$-0.543248\pi$
$149$	−3.30677	−0.270901	−0.135451	+	0.990784i	$0.543248\pi$
$150$	0	0
$151$	14.8537	1.20878	0.604388	−	0.796690i	$-0.293418\pi$
$151$	14.8537	1.20878	0.604388	+	0.796690i	$0.293418\pi$
$152$	0	0
$153$	39.6982	3.20941
$154$	0	0
$155$	3.24161	0.260372
$156$	0	0
$157$	−11.1753	−0.891885	−0.445942	−	0.895062i	$-0.647131\pi$
$157$	−11.1753	−0.891885	−0.445942	+	0.895062i	$0.647131\pi$
$158$	0	0
$159$	6.24560	0.495308
$160$	0	0
$161$	0.822583	0.0648286
$162$	0	0
$163$	8.24374	0.645700	0.322850	−	0.946450i	$-0.395359\pi$
$163$	8.24374	0.645700	0.322850	+	0.946450i	$0.395359\pi$
$164$	0	0
$165$	−9.64398	−0.750783
$166$	0	0
$167$	6.20682	0.480298	0.240149	−	0.970736i	$-0.422804\pi$
$167$	6.20682	0.480298	0.240149	+	0.970736i	$0.422804\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−15.9357	−1.21864
$172$	0	0
$173$	10.4828	0.796992	0.398496	−	0.917170i	$-0.369532\pi$
$173$	10.4828	0.796992	0.398496	+	0.917170i	$0.369532\pi$
$174$	0	0
$175$	−3.39928	−0.256961
$176$	0	0
$177$	15.2451	1.14590
$178$	0	0
$179$	−8.50142	−0.635426	−0.317713	−	0.948187i	$-0.602915\pi$
$179$	−8.50142	−0.635426	−0.317713	+	0.948187i	$0.602915\pi$
$180$	0	0
$181$	13.1364	0.976425	0.488212	−	0.872725i	$-0.337649\pi$
$181$	13.1364	0.976425	0.488212	+	0.872725i	$0.337649\pi$
$182$	0	0
$183$	11.3217	0.836924
$184$	0	0
$185$	0.198143	0.0145678
$186$	0	0
$187$	4.91732	0.359590
$188$	0	0
$189$	−16.8815	−1.22795
$190$	0	0
$191$	−23.8273	−1.72408	−0.862042	−	0.506836i	$-0.830815\pi$
$191$	−23.8273	−1.72408	−0.862042	+	0.506836i	$0.830815\pi$
$192$	0	0
$193$	−2.09464	−0.150776	−0.0753879	−	0.997154i	$-0.524020\pi$
$193$	−2.09464	−0.150776	−0.0753879	+	0.997154i	$0.524020\pi$
$194$	0	0
$195$	−9.64398	−0.690620
$196$	0	0
$197$	23.4207	1.66866	0.834330	−	0.551266i	$-0.185855\pi$
$197$	23.4207	1.66866	0.834330	+	0.551266i	$0.185855\pi$
$198$	0	0
$199$	−14.5711	−1.03292	−0.516459	−	0.856312i	$-0.672750\pi$
$199$	−14.5711	−1.03292	−0.516459	+	0.856312i	$0.672750\pi$
$200$	0	0
$201$	−12.7383	−0.898489
$202$	0	0
$203$	−0.862485	−0.0605346
$204$	0	0
$205$	1.59464	0.111375
$206$	0	0
$207$	−6.64082	−0.461569
$208$	0	0
$209$	−1.97392	−0.136539
$210$	0	0
$211$	−7.99205	−0.550195	−0.275098	−	0.961416i	$-0.588710\pi$
$211$	−7.99205	−0.550195	−0.275098	+	0.961416i	$0.588710\pi$
$212$	0	0
$213$	−9.21513	−0.631410
$214$	0	0
$215$	−15.8675	−1.08216
$216$	0	0
$217$	1.11851	0.0759293
$218$	0	0
$219$	−14.9193	−1.00815
$220$	0	0
$221$	4.91732	0.330775
$222$	0	0
$223$	−18.7698	−1.25692	−0.628458	−	0.777843i	$-0.716314\pi$
$223$	−18.7698	−1.25692	−0.628458	+	0.777843i	$0.716314\pi$
$224$	0	0
$225$	27.4428	1.82952
$226$	0	0
$227$	−1.13287	−0.0751912	−0.0375956	−	0.999293i	$-0.511970\pi$
$227$	−1.13287	−0.0751912	−0.0375956	+	0.999293i	$0.511970\pi$
$228$	0	0
$229$	−2.83456	−0.187313	−0.0936566	−	0.995605i	$-0.529856\pi$
$229$	−2.83456	−0.187313	−0.0936566	+	0.995605i	$0.529856\pi$
$230$	0	0
$231$	−3.32763	−0.218942
$232$	0	0
$233$	7.41351	0.485675	0.242837	−	0.970067i	$-0.421922\pi$
$233$	7.41351	0.485675	0.242837	+	0.970067i	$0.421922\pi$
$234$	0	0
$235$	−5.43932	−0.354822
$236$	0	0
$237$	31.9069	2.07258
$238$	0	0
$239$	−28.0781	−1.81622	−0.908110	−	0.418731i	$-0.862475\pi$
$239$	−28.0781	−1.81622	−0.908110	+	0.418731i	$0.862475\pi$
$240$	0	0
$241$	12.3519	0.795653	0.397826	−	0.917461i	$-0.369765\pi$
$241$	12.3519	0.795653	0.397826	+	0.917461i	$0.369765\pi$
$242$	0	0
$243$	55.6936	3.57275
$244$	0	0
$245$	−2.89815	−0.185156
$246$	0	0
$247$	−1.97392	−0.125597
$248$	0	0
$249$	3.31458	0.210053
$250$	0	0
$251$	9.16331	0.578383	0.289192	−	0.957271i	$-0.406614\pi$
$251$	9.16331	0.578383	0.289192	+	0.957271i	$0.406614\pi$
$252$	0	0
$253$	−0.822583	−0.0517153
$254$	0	0
$255$	−47.4225	−2.96972
$256$	0	0
$257$	20.8302	1.29936	0.649678	−	0.760210i	$-0.274904\pi$
$257$	20.8302	1.29936	0.649678	+	0.760210i	$0.274904\pi$
$258$	0	0
$259$	0.0683687	0.00424822
$260$	0	0
$261$	6.96296	0.430997
$262$	0	0
$263$	0.00437489	0.000269767 0	0.000134884	−	1.00000i	$-0.499957\pi$
$263$	0.00437489	0.000269767 0	0.000134884		1.00000i	$0.499957\pi$
$264$	0	0
$265$	−5.43951	−0.334146
$266$	0	0
$267$	−13.0539	−0.798884
$268$	0	0
$269$	−16.8988	−1.03034	−0.515170	−	0.857088i	$-0.672271\pi$
$269$	−16.8988	−1.03034	−0.515170	+	0.857088i	$0.672271\pi$
$270$	0	0
$271$	29.5668	1.79606	0.898029	−	0.439937i	$-0.144999\pi$
$271$	29.5668	1.79606	0.898029	+	0.439937i	$0.144999\pi$
$272$	0	0
$273$	−3.32763	−0.201397
$274$	0	0
$275$	3.39928	0.204984
$276$	0	0
$277$	−23.1287	−1.38967	−0.694835	−	0.719169i	$-0.744523\pi$
$277$	−23.1287	−1.38967	−0.694835	+	0.719169i	$0.744523\pi$
$278$	0	0
$279$	−9.02987	−0.540604
$280$	0	0
$281$	12.4053	0.740036	0.370018	−	0.929025i	$-0.379351\pi$
$281$	12.4053	0.740036	0.370018	+	0.929025i	$0.379351\pi$
$282$	0	0
$283$	−25.3205	−1.50515	−0.752574	−	0.658507i	$-0.771188\pi$
$283$	−25.3205	−1.50515	−0.752574	+	0.658507i	$0.771188\pi$
$284$	0	0
$285$	19.0364	1.12762
$286$	0	0
$287$	0.550227	0.0324789
$288$	0	0
$289$	7.18004	0.422356
$290$	0	0
$291$	−4.84897	−0.284252
$292$	0	0
$293$	−12.4009	−0.724470	−0.362235	−	0.932087i	$-0.617986\pi$
$293$	−12.4009	−0.724470	−0.362235	+	0.932087i	$0.617986\pi$
$294$	0	0
$295$	−13.2775	−0.773047
$296$	0	0
$297$	16.8815	0.979566
$298$	0	0
$299$	−0.822583	−0.0475712
$300$	0	0
$301$	−5.47505	−0.315577
$302$	0	0
$303$	−56.2525	−3.23162
$304$	0	0
$305$	−9.86047	−0.564609
$306$	0	0
$307$	6.09758	0.348007	0.174004	−	0.984745i	$-0.444330\pi$
$307$	6.09758	0.348007	0.174004	+	0.984745i	$0.444330\pi$
$308$	0	0
$309$	55.5394	3.15952
$310$	0	0
$311$	5.17787	0.293610	0.146805	−	0.989165i	$-0.453101\pi$
$311$	5.17787	0.293610	0.146805	+	0.989165i	$0.453101\pi$
$312$	0	0
$313$	1.94734	0.110070	0.0550351	−	0.998484i	$-0.482473\pi$
$313$	1.94734	0.110070	0.0550351	+	0.998484i	$0.482473\pi$
$314$	0	0
$315$	23.3972	1.31828
$316$	0	0
$317$	−0.124805	−0.00700976	−0.00350488	−	0.999994i	$-0.501116\pi$
$317$	−0.124805	−0.00700976	−0.00350488	+	0.999994i	$0.501116\pi$
$318$	0	0
$319$	0.862485	0.0482899
$320$	0	0
$321$	−4.98069	−0.277995
$322$	0	0
$323$	−9.70640	−0.540078
$324$	0	0
$325$	3.39928	0.188558
$326$	0	0
$327$	27.3953	1.51497
$328$	0	0
$329$	−1.87682	−0.103473
$330$	0	0
$331$	23.3852	1.28537	0.642684	−	0.766132i	$-0.277821\pi$
$331$	23.3852	1.28537	0.642684	+	0.766132i	$0.277821\pi$
$332$	0	0
$333$	−0.551950	−0.0302467
$334$	0	0
$335$	11.0942	0.606141
$336$	0	0
$337$	−23.9226	−1.30314	−0.651572	−	0.758586i	$-0.725890\pi$
$337$	−23.9226	−1.30314	−0.651572	+	0.758586i	$0.725890\pi$
$338$	0	0
$339$	15.1182	0.821106
$340$	0	0
$341$	−1.11851	−0.0605706
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	7.93297	0.427097
$346$	0	0
$347$	13.8239	0.742106	0.371053	−	0.928612i	$-0.378997\pi$
$347$	13.8239	0.742106	0.371053	+	0.928612i	$0.378997\pi$
$348$	0	0
$349$	15.9107	0.851682	0.425841	−	0.904798i	$-0.359978\pi$
$349$	15.9107	0.851682	0.425841	+	0.904798i	$0.359978\pi$
$350$	0	0
$351$	16.8815	0.901070
$352$	0	0
$353$	−5.23510	−0.278636	−0.139318	−	0.990248i	$-0.544491\pi$
$353$	−5.23510	−0.278636	−0.139318	+	0.990248i	$0.544491\pi$
$354$	0	0
$355$	8.02578	0.425964
$356$	0	0
$357$	−16.3630	−0.866024
$358$	0	0
$359$	36.0901	1.90476	0.952381	−	0.304909i	$-0.0986262\pi$
$359$	36.0901	1.90476	0.952381	+	0.304909i	$0.0986262\pi$
$360$	0	0
$361$	−15.1036	−0.794929
$362$	0	0
$363$	3.32763	0.174655
$364$	0	0
$365$	12.9937	0.680123
$366$	0	0
$367$	−22.7304	−1.18652	−0.593258	−	0.805012i	$-0.702159\pi$
$367$	−22.7304	−1.18652	−0.593258	+	0.805012i	$0.702159\pi$
$368$	0	0
$369$	−4.44206	−0.231244
$370$	0	0
$371$	−1.87689	−0.0974432
$372$	0	0
$373$	24.9066	1.28961	0.644807	−	0.764345i	$-0.276938\pi$
$373$	24.9066	1.28961	0.644807	+	0.764345i	$0.276938\pi$
$374$	0	0
$375$	15.4373	0.797180
$376$	0	0
$377$	0.862485	0.0444203
$378$	0	0
$379$	6.10321	0.313501	0.156751	−	0.987638i	$-0.449898\pi$
$379$	6.10321	0.313501	0.156751	+	0.987638i	$0.449898\pi$
$380$	0	0
$381$	−16.6540	−0.853212
$382$	0	0
$383$	1.57956	0.0807118	0.0403559	−	0.999185i	$-0.487151\pi$
$383$	1.57956	0.0807118	0.0403559	+	0.999185i	$0.487151\pi$
$384$	0	0
$385$	2.89815	0.147703
$386$	0	0
$387$	44.2008	2.24685
$388$	0	0
$389$	7.50847	0.380694	0.190347	−	0.981717i	$-0.439039\pi$
$389$	7.50847	0.380694	0.190347	+	0.981717i	$0.439039\pi$
$390$	0	0
$391$	−4.04490	−0.204560
$392$	0	0
$393$	51.8925	2.61763
$394$	0	0
$395$	−27.7889	−1.39821
$396$	0	0
$397$	−17.9702	−0.901897	−0.450948	−	0.892550i	$-0.648914\pi$
$397$	−17.9702	−0.901897	−0.450948	+	0.892550i	$0.648914\pi$
$398$	0	0
$399$	6.56848	0.328835
$400$	0	0
$401$	29.8900	1.49263	0.746317	−	0.665590i	$-0.231820\pi$
$401$	29.8900	1.49263	0.746317	+	0.665590i	$0.231820\pi$
$402$	0	0
$403$	−1.11851	−0.0557169
$404$	0	0
$405$	−92.6137	−4.60201
$406$	0	0
$407$	−0.0683687	−0.00338891
$408$	0	0
$409$	−30.2361	−1.49508	−0.747540	−	0.664217i	$-0.768765\pi$
$409$	−30.2361	−1.49508	−0.747540	+	0.664217i	$0.768765\pi$
$410$	0	0
$411$	64.1726	3.16540
$412$	0	0
$413$	−4.58138	−0.225435
$414$	0	0
$415$	−2.88679	−0.141707
$416$	0	0
$417$	55.7056	2.72792
$418$	0	0
$419$	−27.8139	−1.35880	−0.679398	−	0.733770i	$-0.737759\pi$
$419$	−27.8139	−1.35880	−0.679398	+	0.733770i	$0.737759\pi$
$420$	0	0
$421$	36.5304	1.78038	0.890191	−	0.455587i	$-0.150571\pi$
$421$	36.5304	1.78038	0.890191	+	0.455587i	$0.150571\pi$
$422$	0	0
$423$	15.1519	0.736709
$424$	0	0
$425$	16.7153	0.810813
$426$	0	0
$427$	−3.40233	−0.164650
$428$	0	0
$429$	3.32763	0.160660
$430$	0	0
$431$	15.3411	0.738956	0.369478	−	0.929239i	$-0.379536\pi$
$431$	15.3411	0.738956	0.369478	+	0.929239i	$0.379536\pi$
$432$	0	0
$433$	−28.5851	−1.37371	−0.686857	−	0.726793i	$-0.741010\pi$
$433$	−28.5851	−1.37371	−0.686857	+	0.726793i	$0.741010\pi$
$434$	0	0
$435$	−8.31779	−0.398808
$436$	0	0
$437$	1.62371	0.0776727
$438$	0	0
$439$	15.7445	0.751445	0.375722	−	0.926732i	$-0.377395\pi$
$439$	15.7445	0.751445	0.375722	+	0.926732i	$0.377395\pi$
$440$	0	0
$441$	8.07314	0.384435
$442$	0	0
$443$	−13.1939	−0.626860	−0.313430	−	0.949611i	$-0.601478\pi$
$443$	−13.1939	−0.626860	−0.313430	+	0.949611i	$0.601478\pi$
$444$	0	0
$445$	11.3691	0.538946
$446$	0	0
$447$	−11.0037	−0.520458
$448$	0	0
$449$	0.672072	0.0317170	0.0158585	−	0.999874i	$-0.494952\pi$
$449$	0.672072	0.0317170	0.0158585	+	0.999874i	$0.494952\pi$
$450$	0	0
$451$	−0.550227	−0.0259092
$452$	0	0
$453$	49.4276	2.32231
$454$	0	0
$455$	2.89815	0.135867
$456$	0	0
$457$	−22.4837	−1.05174	−0.525872	−	0.850564i	$-0.676261\pi$
$457$	−22.4837	−1.05174	−0.525872	+	0.850564i	$0.676261\pi$
$458$	0	0
$459$	83.0119	3.87466
$460$	0	0
$461$	28.4970	1.32724	0.663620	−	0.748070i	$-0.269019\pi$
$461$	28.4970	1.32724	0.663620	+	0.748070i	$0.269019\pi$
$462$	0	0
$463$	−17.7953	−0.827017	−0.413509	−	0.910500i	$-0.635697\pi$
$463$	−17.7953	−0.827017	−0.413509	+	0.910500i	$0.635697\pi$
$464$	0	0
$465$	10.7869	0.500229
$466$	0	0
$467$	29.6611	1.37255	0.686276	−	0.727341i	$-0.259244\pi$
$467$	29.6611	1.37255	0.686276	+	0.727341i	$0.259244\pi$
$468$	0	0
$469$	3.82803	0.176762
$470$	0	0
$471$	−37.1872	−1.71350
$472$	0	0
$473$	5.47505	0.251743
$474$	0	0
$475$	−6.70990	−0.307872
$476$	0	0
$477$	15.1524	0.693780
$478$	0	0
$479$	−29.1148	−1.33029	−0.665144	−	0.746715i	$-0.731630\pi$
$479$	−29.1148	−1.33029	−0.665144	+	0.746715i	$0.731630\pi$
$480$	0	0
$481$	−0.0683687	−0.00311734
$482$	0	0
$483$	2.73725	0.124549
$484$	0	0
$485$	4.22314	0.191763
$486$	0	0
$487$	−9.92422	−0.449709	−0.224855	−	0.974392i	$-0.572191\pi$
$487$	−9.92422	−0.449709	−0.224855	+	0.974392i	$0.572191\pi$
$488$	0	0
$489$	27.4321	1.24052
$490$	0	0
$491$	26.8171	1.21024	0.605118	−	0.796136i	$-0.293126\pi$
$491$	26.8171	1.21024	0.605118	+	0.796136i	$0.293126\pi$
$492$	0	0
$493$	4.24112	0.191010
$494$	0	0
$495$	−23.3972	−1.05162
$496$	0	0
$497$	2.76928	0.124219
$498$	0	0
$499$	−25.4328	−1.13853	−0.569264	−	0.822155i	$-0.692772\pi$
$499$	−25.4328	−1.13853	−0.569264	+	0.822155i	$0.692772\pi$
$500$	0	0
$501$	20.6540	0.922753
$502$	0	0
$503$	29.5784	1.31884	0.659418	−	0.751777i	$-0.270803\pi$
$503$	29.5784	1.31884	0.659418	+	0.751777i	$0.270803\pi$
$504$	0	0
$505$	48.9922	2.18013
$506$	0	0
$507$	3.32763	0.147785
$508$	0	0
$509$	−22.6565	−1.00423	−0.502116	−	0.864801i	$-0.667445\pi$
$509$	−22.6565	−1.00423	−0.502116	+	0.864801i	$0.667445\pi$
$510$	0	0
$511$	4.48346	0.198336
$512$	0	0
$513$	−33.3228	−1.47124
$514$	0	0
$515$	−48.3712	−2.13149
$516$	0	0
$517$	1.87682	0.0825427
$518$	0	0
$519$	34.8829	1.53119
$520$	0	0
$521$	−40.7840	−1.78678	−0.893389	−	0.449285i	$-0.851679\pi$
$521$	−40.7840	−1.78678	−0.893389	+	0.449285i	$0.851679\pi$
$522$	0	0
$523$	6.71808	0.293761	0.146881	−	0.989154i	$-0.453077\pi$
$523$	6.71808	0.293761	0.146881	+	0.989154i	$0.453077\pi$
$524$	0	0
$525$	−11.3116	−0.493677
$526$	0	0
$527$	−5.50007	−0.239587
$528$	0	0
$529$	−22.3234	−0.970581
$530$	0	0
$531$	36.9861	1.60506
$532$	0	0
$533$	−0.550227	−0.0238330
$534$	0	0
$535$	4.33785	0.187542
$536$	0	0
$537$	−28.2896	−1.22079
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−12.0271	−0.517086	−0.258543	−	0.966000i	$-0.583242\pi$
$541$	−12.0271	−0.517086	−0.258543	+	0.966000i	$0.583242\pi$
$542$	0	0
$543$	43.7133	1.87592
$544$	0	0
$545$	−23.8596	−1.02203
$546$	0	0
$547$	−35.7675	−1.52931	−0.764654	−	0.644441i	$-0.777090\pi$
$547$	−35.7675	−1.52931	−0.764654	+	0.644441i	$0.777090\pi$
$548$	0	0
$549$	27.4675	1.17228
$550$	0	0
$551$	−1.70248	−0.0725279
$552$	0	0
$553$	−9.58848	−0.407744
$554$	0	0
$555$	0.659347	0.0279877
$556$	0	0
$557$	4.17259	0.176798	0.0883992	−	0.996085i	$-0.471825\pi$
$557$	4.17259	0.176798	0.0883992	+	0.996085i	$0.471825\pi$
$558$	0	0
$559$	5.47505	0.231570
$560$	0	0
$561$	16.3630	0.690848
$562$	0	0
$563$	−2.68836	−0.113301	−0.0566505	−	0.998394i	$-0.518042\pi$
$563$	−2.68836	−0.113301	−0.0566505	+	0.998394i	$0.518042\pi$
$564$	0	0
$565$	−13.1669	−0.553937
$566$	0	0
$567$	−31.9561	−1.34203
$568$	0	0
$569$	27.8668	1.16824	0.584119	−	0.811668i	$-0.301440\pi$
$569$	27.8668	1.16824	0.584119	+	0.811668i	$0.301440\pi$
$570$	0	0
$571$	19.0827	0.798586	0.399293	−	0.916823i	$-0.369256\pi$
$571$	19.0827	0.798586	0.399293	+	0.916823i	$0.369256\pi$
$572$	0	0
$573$	−79.2886	−3.31233
$574$	0	0
$575$	−2.79619	−0.116609
$576$	0	0
$577$	13.9146	0.579274	0.289637	−	0.957137i	$-0.406465\pi$
$577$	13.9146	0.579274	0.289637	+	0.957137i	$0.406465\pi$
$578$	0	0
$579$	−6.97021	−0.289672
$580$	0	0
$581$	−0.996079	−0.0413243
$582$	0	0
$583$	1.87689	0.0777328
$584$	0	0
$585$	−23.3972	−0.967354
$586$	0	0
$587$	7.44256	0.307187	0.153594	−	0.988134i	$-0.450915\pi$
$587$	7.44256	0.307187	0.153594	+	0.988134i	$0.450915\pi$
$588$	0	0
$589$	2.20785	0.0909727
$590$	0	0
$591$	77.9356	3.20584
$592$	0	0
$593$	−46.6843	−1.91709	−0.958547	−	0.284933i	$-0.908029\pi$
$593$	−46.6843	−1.91709	−0.958547	+	0.284933i	$0.908029\pi$
$594$	0	0
$595$	14.2511	0.584240
$596$	0	0
$597$	−48.4873	−1.98445
$598$	0	0
$599$	−34.7874	−1.42137	−0.710687	−	0.703508i	$-0.751616\pi$
$599$	−34.7874	−1.42137	−0.710687	+	0.703508i	$0.751616\pi$
$600$	0	0
$601$	−26.6448	−1.08686	−0.543432	−	0.839453i	$-0.682875\pi$
$601$	−26.6448	−1.08686	−0.543432	+	0.839453i	$0.682875\pi$
$602$	0	0
$603$	−30.9042	−1.25852
$604$	0	0
$605$	−2.89815	−0.117827
$606$	0	0
$607$	−15.9669	−0.648075	−0.324038	−	0.946044i	$-0.605040\pi$
$607$	−15.9669	−0.648075	−0.324038	+	0.946044i	$0.605040\pi$
$608$	0	0
$609$	−2.87003	−0.116300
$610$	0	0
$611$	1.87682	0.0759282
$612$	0	0
$613$	−41.2325	−1.66537	−0.832683	−	0.553750i	$-0.813196\pi$
$613$	−41.2325	−1.66537	−0.832683	+	0.553750i	$0.813196\pi$
$614$	0	0
$615$	5.30638	0.213974
$616$	0	0
$617$	−13.7917	−0.555234	−0.277617	−	0.960692i	$-0.589545\pi$
$617$	−13.7917	−0.555234	−0.277617	+	0.960692i	$0.589545\pi$
$618$	0	0
$619$	4.80112	0.192973	0.0964867	−	0.995334i	$-0.469239\pi$
$619$	4.80112	0.192973	0.0964867	+	0.995334i	$0.469239\pi$
$620$	0	0
$621$	−13.8865	−0.557244
$622$	0	0
$623$	3.92287	0.157167
$624$	0	0
$625$	−30.4413	−1.21765
$626$	0	0
$627$	−6.56848	−0.262320
$628$	0	0
$629$	−0.336191	−0.0134048
$630$	0	0
$631$	17.6347	0.702028	0.351014	−	0.936370i	$-0.385837\pi$
$631$	17.6347	0.702028	0.351014	+	0.936370i	$0.385837\pi$
$632$	0	0
$633$	−26.5946	−1.05704
$634$	0	0
$635$	14.5046	0.575596
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−22.3567	−0.884419
$640$	0	0
$641$	4.82610	0.190619	0.0953097	−	0.995448i	$-0.469616\pi$
$641$	4.82610	0.190619	0.0953097	+	0.995448i	$0.469616\pi$
$642$	0	0
$643$	−41.3855	−1.63208	−0.816042	−	0.577992i	$-0.803836\pi$
$643$	−41.3855	−1.63208	−0.816042	+	0.577992i	$0.803836\pi$
$644$	0	0
$645$	−52.8013	−2.07905
$646$	0	0
$647$	−5.48152	−0.215501	−0.107750	−	0.994178i	$-0.534365\pi$
$647$	−5.48152	−0.215501	−0.107750	+	0.994178i	$0.534365\pi$
$648$	0	0
$649$	4.58138	0.179835
$650$	0	0
$651$	3.72199	0.145876
$652$	0	0
$653$	33.7967	1.32257	0.661284	−	0.750136i	$-0.270012\pi$
$653$	33.7967	1.32257	0.661284	+	0.750136i	$0.270012\pi$
$654$	0	0
$655$	−45.1950	−1.76591
$656$	0	0
$657$	−36.1956	−1.41212
$658$	0	0
$659$	26.0174	1.01350	0.506748	−	0.862095i	$-0.330848\pi$
$659$	26.0174	1.01350	0.506748	+	0.862095i	$0.330848\pi$
$660$	0	0
$661$	−41.7458	−1.62372	−0.811861	−	0.583851i	$-0.801545\pi$
$661$	−41.7458	−1.62372	−0.811861	+	0.583851i	$0.801545\pi$
$662$	0	0
$663$	16.3630	0.635488
$664$	0	0
$665$	−5.72072	−0.221840
$666$	0	0
$667$	−0.709466	−0.0274706
$668$	0	0
$669$	−62.4589	−2.41480
$670$	0	0
$671$	3.40233	0.131345
$672$	0	0
$673$	39.9625	1.54044	0.770221	−	0.637778i	$-0.220146\pi$
$673$	39.9625	1.54044	0.770221	+	0.637778i	$0.220146\pi$
$674$	0	0
$675$	57.3851	2.20875
$676$	0	0
$677$	12.3204	0.473510	0.236755	−	0.971569i	$-0.423916\pi$
$677$	12.3204	0.473510	0.236755	+	0.971569i	$0.423916\pi$
$678$	0	0
$679$	1.45718	0.0559216
$680$	0	0
$681$	−3.76977	−0.144458
$682$	0	0
$683$	−12.4273	−0.475516	−0.237758	−	0.971324i	$-0.576413\pi$
$683$	−12.4273	−0.475516	−0.237758	+	0.971324i	$0.576413\pi$
$684$	0	0
$685$	−55.8902	−2.13545
$686$	0	0
$687$	−9.43238	−0.359868
$688$	0	0
$689$	1.87689	0.0715038
$690$	0	0
$691$	−37.8793	−1.44100	−0.720498	−	0.693457i	$-0.756087\pi$
$691$	−37.8793	−1.44100	−0.720498	+	0.693457i	$0.756087\pi$
$692$	0	0
$693$	−8.07314	−0.306673
$694$	0	0
$695$	−48.5160	−1.84032
$696$	0	0
$697$	−2.70564	−0.102484
$698$	0	0
$699$	24.6694	0.933083
$700$	0	0
$701$	−12.4903	−0.471754	−0.235877	−	0.971783i	$-0.575796\pi$
$701$	−12.4903	−0.471754	−0.235877	+	0.971783i	$0.575796\pi$
$702$	0	0
$703$	0.134954	0.00508990
$704$	0	0
$705$	−18.1001	−0.681688
$706$	0	0
$707$	16.9047	0.635765
$708$	0	0
$709$	16.7793	0.630161	0.315081	−	0.949065i	$-0.397968\pi$
$709$	16.7793	0.630161	0.315081	+	0.949065i	$0.397968\pi$
$710$	0	0
$711$	77.4091	2.90307
$712$	0	0
$713$	0.920066	0.0344567
$714$	0	0
$715$	−2.89815	−0.108385
$716$	0	0
$717$	−93.4335	−3.48934
$718$	0	0
$719$	−44.0471	−1.64268	−0.821340	−	0.570439i	$-0.806773\pi$
$719$	−44.0471	−1.64268	−0.821340	+	0.570439i	$0.806773\pi$
$720$	0	0
$721$	−16.6904	−0.621581
$722$	0	0
$723$	41.1024	1.52862
$724$	0	0
$725$	2.93183	0.108885
$726$	0	0
$727$	3.65427	0.135529	0.0677646	−	0.997701i	$-0.478413\pi$
$727$	3.65427	0.135529	0.0677646	+	0.997701i	$0.478413\pi$
$728$	0	0
$729$	89.4596	3.31332
$730$	0	0
$731$	26.9226	0.995767
$732$	0	0
$733$	9.06609	0.334864	0.167432	−	0.985884i	$-0.446453\pi$
$733$	9.06609	0.334864	0.167432	+	0.985884i	$0.446453\pi$
$734$	0	0
$735$	−9.64398	−0.355724
$736$	0	0
$737$	−3.82803	−0.141007
$738$	0	0
$739$	43.5197	1.60090	0.800449	−	0.599401i	$-0.204594\pi$
$739$	43.5197	1.60090	0.800449	+	0.599401i	$0.204594\pi$
$740$	0	0
$741$	−6.56848	−0.241299
$742$	0	0
$743$	9.33707	0.342544	0.171272	−	0.985224i	$-0.445212\pi$
$743$	9.33707	0.342544	0.171272	+	0.985224i	$0.445212\pi$
$744$	0	0
$745$	9.58353	0.351113
$746$	0	0
$747$	8.04148	0.294222
$748$	0	0
$749$	1.49677	0.0546906
$750$	0	0
$751$	32.5557	1.18798	0.593988	−	0.804474i	$-0.297553\pi$
$751$	32.5557	1.18798	0.593988	+	0.804474i	$0.297553\pi$
$752$	0	0
$753$	30.4921	1.11120
$754$	0	0
$755$	−43.0483	−1.56669
$756$	0	0
$757$	−41.9790	−1.52575	−0.762876	−	0.646545i	$-0.776213\pi$
$757$	−41.9790	−1.52575	−0.762876	+	0.646545i	$0.776213\pi$
$758$	0	0
$759$	−2.73725	−0.0993560
$760$	0	0
$761$	−37.2571	−1.35057	−0.675285	−	0.737557i	$-0.735979\pi$
$761$	−37.2571	−1.35057	−0.675285	+	0.737557i	$0.735979\pi$
$762$	0	0
$763$	−8.23268	−0.298043
$764$	0	0
$765$	−115.051	−4.15969
$766$	0	0
$767$	4.58138	0.165424
$768$	0	0
$769$	−27.0933	−0.977010	−0.488505	−	0.872561i	$-0.662458\pi$
$769$	−27.0933	−0.977010	−0.488505	+	0.872561i	$0.662458\pi$
$770$	0	0
$771$	69.3154	2.49633
$772$	0	0
$773$	34.3710	1.23624	0.618120	−	0.786084i	$-0.287895\pi$
$773$	34.3710	1.23624	0.618120	+	0.786084i	$0.287895\pi$
$774$	0	0
$775$	−3.80212	−0.136576
$776$	0	0
$777$	0.227506	0.00816173
$778$	0	0
$779$	1.08610	0.0389137
$780$	0	0
$781$	−2.76928	−0.0990925
$782$	0	0
$783$	14.5601	0.520335
$784$	0	0
$785$	32.3877	1.15597
$786$	0	0
$787$	22.6476	0.807301	0.403651	−	0.914913i	$-0.367741\pi$
$787$	22.6476	0.807301	0.403651	+	0.914913i	$0.367741\pi$
$788$	0	0
$789$	0.0145580	0.000518280 0
$790$	0	0
$791$	−4.54322	−0.161538
$792$	0	0
$793$	3.40233	0.120820
$794$	0	0
$795$	−18.1007	−0.641965
$796$	0	0
$797$	5.03028	0.178182	0.0890909	−	0.996024i	$-0.471604\pi$
$797$	5.03028	0.178182	0.0890909	+	0.996024i	$0.471604\pi$
$798$	0	0
$799$	9.22895	0.326497
$800$	0	0
$801$	−31.6699	−1.11900
$802$	0	0
$803$	−4.48346	−0.158218
$804$	0	0
$805$	−2.38397	−0.0840239
$806$	0	0
$807$	−56.2331	−1.97950
$808$	0	0
$809$	−4.75585	−0.167207	−0.0836034	−	0.996499i	$-0.526643\pi$
$809$	−4.75585	−0.167207	−0.0836034	+	0.996499i	$0.526643\pi$
$810$	0	0
$811$	−10.9491	−0.384474	−0.192237	−	0.981349i	$-0.561574\pi$
$811$	−10.9491	−0.384474	−0.192237	+	0.981349i	$0.561574\pi$
$812$	0	0
$813$	98.3876	3.45060
$814$	0	0
$815$	−23.8916	−0.836886
$816$	0	0
$817$	−10.8073	−0.378100
$818$	0	0
$819$	−8.07314	−0.282098
$820$	0	0
$821$	4.78109	0.166861	0.0834306	−	0.996514i	$-0.473412\pi$
$821$	4.78109	0.166861	0.0834306	+	0.996514i	$0.473412\pi$
$822$	0	0
$823$	1.62528	0.0566538	0.0283269	−	0.999599i	$-0.490982\pi$
$823$	1.62528	0.0566538	0.0283269	+	0.999599i	$0.490982\pi$
$824$	0	0
$825$	11.3116	0.393818
$826$	0	0
$827$	−43.5928	−1.51587	−0.757936	−	0.652329i	$-0.773792\pi$
$827$	−43.5928	−1.51587	−0.757936	+	0.652329i	$0.773792\pi$
$828$	0	0
$829$	−26.5188	−0.921035	−0.460517	−	0.887651i	$-0.652336\pi$
$829$	−26.5188	−0.921035	−0.460517	+	0.887651i	$0.652336\pi$
$830$	0	0
$831$	−76.9639	−2.66985
$832$	0	0
$833$	4.91732	0.170375
$834$	0	0
$835$	−17.9883	−0.622511
$836$	0	0
$837$	−18.8821	−0.652662
$838$	0	0
$839$	−0.307806	−0.0106266	−0.00531332	−	0.999986i	$-0.501691\pi$
$839$	−0.307806	−0.0106266	−0.00531332	+	0.999986i	$0.501691\pi$
$840$	0	0
$841$	−28.2561	−0.974349
$842$	0	0
$843$	41.2802	1.42176
$844$	0	0
$845$	−2.89815	−0.0996994
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−84.2574	−2.89171
$850$	0	0
$851$	0.0562389	0.00192785
$852$	0	0
$853$	36.1987	1.23942	0.619711	−	0.784830i	$-0.287250\pi$
$853$	36.1987	1.23942	0.619711	+	0.784830i	$0.287250\pi$
$854$	0	0
$855$	46.1841	1.57946
$856$	0	0
$857$	−19.8306	−0.677400	−0.338700	−	0.940894i	$-0.609987\pi$
$857$	−19.8306	−0.677400	−0.338700	+	0.940894i	$0.609987\pi$
$858$	0	0
$859$	−48.8814	−1.66781	−0.833906	−	0.551907i	$-0.813900\pi$
$859$	−48.8814	−1.66781	−0.833906	+	0.551907i	$0.813900\pi$
$860$	0	0
$861$	1.83095	0.0623987
$862$	0	0
$863$	−11.5763	−0.394063	−0.197031	−	0.980397i	$-0.563130\pi$
$863$	−11.5763	−0.394063	−0.197031	+	0.980397i	$0.563130\pi$
$864$	0	0
$865$	−30.3807	−1.03298
$866$	0	0
$867$	23.8925	0.811433
$868$	0	0
$869$	9.58848	0.325267
$870$	0	0
$871$	−3.82803	−0.129708
$872$	0	0
$873$	−11.7640	−0.398152
$874$	0	0
$875$	−4.63913	−0.156831
$876$	0	0
$877$	−18.4937	−0.624488	−0.312244	−	0.950002i	$-0.601081\pi$
$877$	−18.4937	−0.624488	−0.312244	+	0.950002i	$0.601081\pi$
$878$	0	0
$879$	−41.2657	−1.39186
$880$	0	0
$881$	41.2247	1.38890	0.694448	−	0.719543i	$-0.255649\pi$
$881$	41.2247	1.38890	0.694448	+	0.719543i	$0.255649\pi$
$882$	0	0
$883$	−53.3548	−1.79553	−0.897766	−	0.440474i	$-0.854811\pi$
$883$	−53.3548	−1.79553	−0.897766	+	0.440474i	$0.854811\pi$
$884$	0	0
$885$	−44.1827	−1.48519
$886$	0	0
$887$	37.3740	1.25489	0.627447	−	0.778659i	$-0.284100\pi$
$887$	37.3740	1.25489	0.627447	+	0.778659i	$0.284100\pi$
$888$	0	0
$889$	5.00477	0.167855
$890$	0	0
$891$	31.9561	1.07057
$892$	0	0
$893$	−3.70470	−0.123973
$894$	0	0
$895$	24.6384	0.823571
$896$	0	0
$897$	−2.73725	−0.0913942
$898$	0	0
$899$	−0.964697	−0.0321745
$900$	0	0
$901$	9.22926	0.307471
$902$	0	0
$903$	−18.2189	−0.606289
$904$	0	0
$905$	−38.0714	−1.26554
$906$	0	0
$907$	−37.8160	−1.25566	−0.627831	−	0.778350i	$-0.716057\pi$
$907$	−37.8160	−1.25566	−0.627831	+	0.778350i	$0.716057\pi$
$908$	0	0
$909$	−136.474	−4.52654
$910$	0	0
$911$	11.4163	0.378238	0.189119	−	0.981954i	$-0.439437\pi$
$911$	11.4163	0.378238	0.189119	+	0.981954i	$0.439437\pi$
$912$	0	0
$913$	0.996079	0.0329654
$914$	0	0
$915$	−32.8120	−1.08473
$916$	0	0
$917$	−15.5944	−0.514973
$918$	0	0
$919$	−19.5554	−0.645072	−0.322536	−	0.946557i	$-0.604535\pi$
$919$	−19.5554	−0.645072	−0.322536	+	0.946557i	$0.604535\pi$
$920$	0	0
$921$	20.2905	0.668595
$922$	0	0
$923$	−2.76928	−0.0911518
$924$	0	0
$925$	−0.232404	−0.00764141
$926$	0	0
$927$	134.744	4.42556
$928$	0	0
$929$	−36.0563	−1.18297	−0.591484	−	0.806317i	$-0.701458\pi$
$929$	−36.0563	−1.18297	−0.591484	+	0.806317i	$0.701458\pi$
$930$	0	0
$931$	−1.97392	−0.0646926
$932$	0	0
$933$	17.2300	0.564086
$934$	0	0
$935$	−14.2511	−0.466062
$936$	0	0
$937$	18.0076	0.588284	0.294142	−	0.955762i	$-0.404966\pi$
$937$	18.0076	0.588284	0.294142	+	0.955762i	$0.404966\pi$
$938$	0	0
$939$	6.48004	0.211468
$940$	0	0
$941$	−3.11960	−0.101696	−0.0508480	−	0.998706i	$-0.516192\pi$
$941$	−3.11960	−0.101696	−0.0508480	+	0.998706i	$0.516192\pi$
$942$	0	0
$943$	0.452607	0.0147389
$944$	0	0
$945$	48.9252	1.59154
$946$	0	0
$947$	15.5970	0.506835	0.253417	−	0.967357i	$-0.418445\pi$
$947$	15.5970	0.506835	0.253417	+	0.967357i	$0.418445\pi$
$948$	0	0
$949$	−4.48346	−0.145539
$950$	0	0
$951$	−0.415306	−0.0134672
$952$	0	0
$953$	26.0193	0.842847	0.421424	−	0.906864i	$-0.361531\pi$
$953$	26.0193	0.842847	0.421424	+	0.906864i	$0.361531\pi$
$954$	0	0
$955$	69.0552	2.23457
$956$	0	0
$957$	2.87003	0.0927750
$958$	0	0
$959$	−19.2848	−0.622737
$960$	0	0
$961$	−29.7489	−0.959643
$962$	0	0
$963$	−12.0836	−0.389388
$964$	0	0
$965$	6.07060	0.195419
$966$	0	0
$967$	−0.0621427	−0.00199837	−0.000999187	−	1.00000i	$-0.500318\pi$
$967$	−0.0621427	−0.00199837	−0.000999187		1.00000i	$0.500318\pi$
$968$	0	0
$969$	−32.2993	−1.03760
$970$	0	0
$971$	19.0652	0.611831	0.305916	−	0.952059i	$-0.401037\pi$
$971$	19.0652	0.611831	0.305916	+	0.952059i	$0.401037\pi$
$972$	0	0
$973$	−16.7403	−0.536670
$974$	0	0
$975$	11.3116	0.362260
$976$	0	0
$977$	−52.3576	−1.67507	−0.837534	−	0.546386i	$-0.816003\pi$
$977$	−52.3576	−1.67507	−0.837534	+	0.546386i	$0.816003\pi$
$978$	0	0
$979$	−3.92287	−0.125376
$980$	0	0
$981$	66.4636	2.12202
$982$	0	0
$983$	−15.8504	−0.505548	−0.252774	−	0.967525i	$-0.581343\pi$
$983$	−15.8504	−0.505548	−0.252774	+	0.967525i	$0.581343\pi$
$984$	0	0
$985$	−67.8769	−2.16274
$986$	0	0
$987$	−6.24538	−0.198793
$988$	0	0
$989$	−4.50368	−0.143209
$990$	0	0
$991$	−48.0163	−1.52529	−0.762644	−	0.646818i	$-0.776099\pi$
$991$	−48.0163	−1.52529	−0.762644	+	0.646818i	$0.776099\pi$
$992$	0	0
$993$	77.8174	2.46946
$994$	0	0
$995$	42.2293	1.33876
$996$	0	0
$997$	4.89349	0.154978	0.0774892	−	0.996993i	$-0.475310\pi$
$997$	4.89349	0.154978	0.0774892	+	0.996993i	$0.475310\pi$
$998$	0	0
$999$	−1.15417	−0.0365163

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.w.1.11	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.w.1.11	✓	11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+3.32763 q^{3} -2.89815 q^{5} -1.00000 q^{7} +8.07314 q^{9} +O(q^{10})\)	\(q+3.32763 q^{3} -2.89815 q^{5} -1.00000 q^{7} +8.07314 q^{9} +1.00000 q^{11} +1.00000 q^{13} -9.64398 q^{15} +4.91732 q^{17} -1.97392 q^{19} -3.32763 q^{21} -0.822583 q^{23} +3.39928 q^{25} +16.8815 q^{27} +0.862485 q^{29} -1.11851 q^{31} +3.32763 q^{33} +2.89815 q^{35} -0.0683687 q^{37} +3.32763 q^{39} -0.550227 q^{41} +5.47505 q^{43} -23.3972 q^{45} +1.87682 q^{47} +1.00000 q^{49} +16.3630 q^{51} +1.87689 q^{53} -2.89815 q^{55} -6.56848 q^{57} +4.58138 q^{59} +3.40233 q^{61} -8.07314 q^{63} -2.89815 q^{65} -3.82803 q^{67} -2.73725 q^{69} -2.76928 q^{71} -4.48346 q^{73} +11.3116 q^{75} -1.00000 q^{77} +9.58848 q^{79} +31.9561 q^{81} +0.996079 q^{83} -14.2511 q^{85} +2.87003 q^{87} -3.92287 q^{89} -1.00000 q^{91} -3.72199 q^{93} +5.72072 q^{95} -1.45718 q^{97} +8.07314 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9}+O(q^{10}) \) 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9	\( 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9} + 11 q^{11} + 11 q^{13} + 7 q^{15} + 9 q^{17} + 20 q^{19} - 3 q^{21} + 12 q^{23} + 13 q^{25} + 15 q^{27} + 8 q^{29} + 7 q^{31} + 3 q^{33} + 2 q^{35} - 10 q^{37} + 3 q^{39} - 2 q^{41} + 24 q^{43} - 6 q^{45} + 2 q^{47} + 11 q^{49} + 17 q^{51} + 3 q^{53} - 2 q^{55} - 16 q^{57} + q^{59} - 22 q^{61} - 14 q^{63} - 2 q^{65} + 14 q^{67} - 22 q^{69} + 6 q^{71} + 3 q^{73} - 11 q^{77} + 8 q^{79} - 9 q^{81} + 29 q^{83} - 9 q^{85} + 19 q^{87} + 20 q^{89} - 11 q^{91} - q^{93} + 18 q^{95} - 25 q^{97} + 14 q^{99}+O(q^{100}) \) 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9 + 11 * q^11 + 11 * q^13 + 7 * q^15 + 9 * q^17 + 20 * q^19 - 3 * q^21 + 12 * q^23 + 13 * q^25 + 15 * q^27 + 8 * q^29 + 7 * q^31 + 3 * q^33 + 2 * q^35 - 10 * q^37 + 3 * q^39 - 2 * q^41 + 24 * q^43 - 6 * q^45 + 2 * q^47 + 11 * q^49 + 17 * q^51 + 3 * q^53 - 2 * q^55 - 16 * q^57 + q^59 - 22 * q^61 - 14 * q^63 - 2 * q^65 + 14 * q^67 - 22 * q^69 + 6 * q^71 + 3 * q^73 - 11 * q^77 + 8 * q^79 - 9 * q^81 + 29 * q^83 - 9 * q^85 + 19 * q^87 + 20 * q^89 - 11 * q^91 - q^93 + 18 * q^95 - 25 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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