LMFDB - Embedded newform 8008.2.a.r.1.3

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 9x^{7} + 36x^{6} + 23x^{5} - 89x^{4} - 20x^{3} + 51x^{2} + 18x + 1 $ x^9 - 4x^8 - 9x^7 + 36x^6 + 23x^5 - 89x^4 - 20x^3 + 51x^2 + 18x + 1
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			1.3
Root			$1.66280$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.662800 q^{3} +3.68243 q^{5} -1.00000 q^{7} -2.56070 q^{9} +O(q^{10})$	$q-0.662800 q^{3} +3.68243 q^{5} -1.00000 q^{7} -2.56070 q^{9} -1.00000 q^{11} -1.00000 q^{13} -2.44072 q^{15} -3.51112 q^{17} +0.825213 q^{19} +0.662800 q^{21} +6.76276 q^{23} +8.56030 q^{25} +3.68563 q^{27} -1.78380 q^{29} -4.39988 q^{31} +0.662800 q^{33} -3.68243 q^{35} +0.723998 q^{37} +0.662800 q^{39} +0.255274 q^{41} +12.3779 q^{43} -9.42959 q^{45} -6.35690 q^{47} +1.00000 q^{49} +2.32717 q^{51} -6.27697 q^{53} -3.68243 q^{55} -0.546951 q^{57} +12.1837 q^{59} -10.9339 q^{61} +2.56070 q^{63} -3.68243 q^{65} -0.932939 q^{67} -4.48235 q^{69} +13.4110 q^{71} +9.71515 q^{73} -5.67377 q^{75} +1.00000 q^{77} -2.89800 q^{79} +5.23925 q^{81} -8.22637 q^{83} -12.9295 q^{85} +1.18230 q^{87} +2.87874 q^{89} +1.00000 q^{91} +2.91624 q^{93} +3.03879 q^{95} -6.52337 q^{97} +2.56070 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9}+O(q^{10}) $ 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9	$ 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9} - 9 q^{11} - 9 q^{13} - 3 q^{15} - 7 q^{17} + 13 q^{19} - 5 q^{21} + 9 q^{23} - 2 q^{25} + 5 q^{27} + q^{29} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 14 q^{37} - 5 q^{39} - 2 q^{41} + 5 q^{43} - 15 q^{45} + 13 q^{47} + 9 q^{49} - 3 q^{51} + 22 q^{53} - 3 q^{55} + 16 q^{57} + 43 q^{59} - 10 q^{61} - 8 q^{63} - 3 q^{65} + 26 q^{67} - 30 q^{69} + 18 q^{71} - 8 q^{73} + 28 q^{75} + 9 q^{77} - 9 q^{79} + 33 q^{81} + 4 q^{83} + 5 q^{85} + 33 q^{87} + 7 q^{89} + 9 q^{91} + 13 q^{93} + 7 q^{95} + 2 q^{97} - 8 q^{99}+O(q^{100}) $ 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9 - 9 * q^11 - 9 * q^13 - 3 * q^15 - 7 * q^17 + 13 * q^19 - 5 * q^21 + 9 * q^23 - 2 * q^25 + 5 * q^27 + q^29 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 14 * q^37 - 5 * q^39 - 2 * q^41 + 5 * q^43 - 15 * q^45 + 13 * q^47 + 9 * q^49 - 3 * q^51 + 22 * q^53 - 3 * q^55 + 16 * q^57 + 43 * q^59 - 10 * q^61 - 8 * q^63 - 3 * q^65 + 26 * q^67 - 30 * q^69 + 18 * q^71 - 8 * q^73 + 28 * q^75 + 9 * q^77 - 9 * q^79 + 33 * q^81 + 4 * q^83 + 5 * q^85 + 33 * q^87 + 7 * q^89 + 9 * q^91 + 13 * q^93 + 7 * q^95 + 2 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.662800	−0.382668	−0.191334	−	0.981525i	$-0.561281\pi$
$3$	−0.662800	−0.382668	−0.191334	+	0.981525i	$0.561281\pi$
$4$	0	0
$5$	3.68243	1.64683	0.823417	−	0.567437i	$-0.192065\pi$
$5$	3.68243	1.64683	0.823417	+	0.567437i	$0.192065\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.56070	−0.853565
$10$	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
$15$	−2.44072	−0.630190
$16$	0	0
$17$	−3.51112	−0.851572	−0.425786	−	0.904824i	$-0.640002\pi$
$17$	−3.51112	−0.851572	−0.425786	+	0.904824i	$0.640002\pi$
$18$	0	0
$19$	0.825213	0.189317	0.0946584	−	0.995510i	$-0.469824\pi$
$19$	0.825213	0.189317	0.0946584	+	0.995510i	$0.469824\pi$
$20$	0	0
$21$	0.662800	0.144635
$22$	0	0
$23$	6.76276	1.41013	0.705066	−	0.709142i	$-0.250917\pi$
$23$	6.76276	1.41013	0.705066	+	0.709142i	$0.250917\pi$
$24$	0	0
$25$	8.56030	1.71206
$26$	0	0
$27$	3.68563	0.709300
$28$	0	0
$29$	−1.78380	−0.331244	−0.165622	−	0.986189i	$-0.552963\pi$
$29$	−1.78380	−0.331244	−0.165622	+	0.986189i	$0.552963\pi$
$30$	0	0
$31$	−4.39988	−0.790242	−0.395121	−	0.918629i	$-0.629297\pi$
$31$	−4.39988	−0.790242	−0.395121	+	0.918629i	$0.629297\pi$
$32$	0	0
$33$	0.662800	0.115379
$34$	0	0
$35$	−3.68243	−0.622445
$36$	0	0
$37$	0.723998	0.119025	0.0595123	−	0.998228i	$-0.481045\pi$
$37$	0.723998	0.119025	0.0595123	+	0.998228i	$0.481045\pi$
$38$	0	0
$39$	0.662800	0.106133
$40$	0	0
$41$	0.255274	0.0398672	0.0199336	−	0.999801i	$-0.493655\pi$
$41$	0.255274	0.0398672	0.0199336	+	0.999801i	$0.493655\pi$
$42$	0	0
$43$	12.3779	1.88762	0.943810	−	0.330490i	$-0.107214\pi$
$43$	12.3779	1.88762	0.943810	+	0.330490i	$0.107214\pi$
$44$	0	0
$45$	−9.42959	−1.40568
$46$	0	0
$47$	−6.35690	−0.927249	−0.463625	−	0.886032i	$-0.653451\pi$
$47$	−6.35690	−0.927249	−0.463625	+	0.886032i	$0.653451\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.32717	0.325869
$52$	0	0
$53$	−6.27697	−0.862208	−0.431104	−	0.902302i	$-0.641876\pi$
$53$	−6.27697	−0.862208	−0.431104	+	0.902302i	$0.641876\pi$
$54$	0	0
$55$	−3.68243	−0.496539
$56$	0	0
$57$	−0.546951	−0.0724454
$58$	0	0
$59$	12.1837	1.58618	0.793090	−	0.609104i	$-0.208471\pi$
$59$	12.1837	1.58618	0.793090	+	0.609104i	$0.208471\pi$
$60$	0	0
$61$	−10.9339	−1.39994	−0.699972	−	0.714171i	$-0.746804\pi$
$61$	−10.9339	−1.39994	−0.699972	+	0.714171i	$0.746804\pi$
$62$	0	0
$63$	2.56070	0.322617
$64$	0	0
$65$	−3.68243	−0.456749
$66$	0	0
$67$	−0.932939	−0.113977	−0.0569883	−	0.998375i	$-0.518150\pi$
$67$	−0.932939	−0.113977	−0.0569883	+	0.998375i	$0.518150\pi$
$68$	0	0
$69$	−4.48235	−0.539612
$70$	0	0
$71$	13.4110	1.59159	0.795797	−	0.605564i	$-0.207052\pi$
$71$	13.4110	1.59159	0.795797	+	0.605564i	$0.207052\pi$
$72$	0	0
$73$	9.71515	1.13707	0.568536	−	0.822658i	$-0.307510\pi$
$73$	9.71515	1.13707	0.568536	+	0.822658i	$0.307510\pi$
$74$	0	0
$75$	−5.67377	−0.655150
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−2.89800	−0.326051	−0.163025	−	0.986622i	$-0.552125\pi$
$79$	−2.89800	−0.326051	−0.163025	+	0.986622i	$0.552125\pi$
$80$	0	0
$81$	5.23925	0.582139
$82$	0	0
$83$	−8.22637	−0.902962	−0.451481	−	0.892281i	$-0.649104\pi$
$83$	−8.22637	−0.902962	−0.451481	+	0.892281i	$0.649104\pi$
$84$	0	0
$85$	−12.9295	−1.40240
$86$	0	0
$87$	1.18230	0.126756
$88$	0	0
$89$	2.87874	0.305146	0.152573	−	0.988292i	$-0.451244\pi$
$89$	2.87874	0.305146	0.152573	+	0.988292i	$0.451244\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	2.91624	0.302400
$94$	0	0
$95$	3.03879	0.311773
$96$	0	0
$97$	−6.52337	−0.662348	−0.331174	−	0.943570i	$-0.607445\pi$
$97$	−6.52337	−0.662348	−0.331174	+	0.943570i	$0.607445\pi$
$98$	0	0
$99$	2.56070	0.257360
$100$	0	0
$101$	17.9821	1.78929	0.894644	−	0.446780i	$-0.147429\pi$
$101$	17.9821	1.78929	0.894644	+	0.446780i	$0.147429\pi$
$102$	0	0
$103$	4.18340	0.412202	0.206101	−	0.978531i	$-0.433922\pi$
$103$	4.18340	0.412202	0.206101	+	0.978531i	$0.433922\pi$
$104$	0	0
$105$	2.44072	0.238189
$106$	0	0
$107$	16.8532	1.62926	0.814630	−	0.579981i	$-0.196940\pi$
$107$	16.8532	1.62926	0.814630	+	0.579981i	$0.196940\pi$
$108$	0	0
$109$	−3.61941	−0.346676	−0.173338	−	0.984862i	$-0.555455\pi$
$109$	−3.61941	−0.346676	−0.173338	+	0.984862i	$0.555455\pi$
$110$	0	0
$111$	−0.479866	−0.0455469
$112$	0	0
$113$	−13.0712	−1.22963	−0.614817	−	0.788670i	$-0.710770\pi$
$113$	−13.0712	−1.22963	−0.614817	+	0.788670i	$0.710770\pi$
$114$	0	0
$115$	24.9034	2.32225
$116$	0	0
$117$	2.56070	0.236736
$118$	0	0
$119$	3.51112	0.321864
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.169196	−0.0152559
$124$	0	0
$125$	13.1106	1.17265
$126$	0	0
$127$	−4.91778	−0.436383	−0.218191	−	0.975906i	$-0.570016\pi$
$127$	−4.91778	−0.436383	−0.218191	+	0.975906i	$0.570016\pi$
$128$	0	0
$129$	−8.20410	−0.722331
$130$	0	0
$131$	−14.2514	−1.24515	−0.622577	−	0.782559i	$-0.713914\pi$
$131$	−14.2514	−1.24515	−0.622577	+	0.782559i	$0.713914\pi$
$132$	0	0
$133$	−0.825213	−0.0715550
$134$	0	0
$135$	13.5721	1.16810
$136$	0	0
$137$	10.9380	0.934500	0.467250	−	0.884125i	$-0.345245\pi$
$137$	10.9380	0.934500	0.467250	+	0.884125i	$0.345245\pi$
$138$	0	0
$139$	11.2530	0.954471	0.477236	−	0.878775i	$-0.341639\pi$
$139$	11.2530	0.954471	0.477236	+	0.878775i	$0.341639\pi$
$140$	0	0
$141$	4.21335	0.354828
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−6.56873	−0.545503
$146$	0	0
$147$	−0.662800	−0.0546668
$148$	0	0
$149$	19.5431	1.60103	0.800517	−	0.599310i	$-0.204558\pi$
$149$	19.5431	1.60103	0.800517	+	0.599310i	$0.204558\pi$
$150$	0	0
$151$	7.30428	0.594414	0.297207	−	0.954813i	$-0.403945\pi$
$151$	7.30428	0.594414	0.297207	+	0.954813i	$0.403945\pi$
$152$	0	0
$153$	8.99091	0.726872
$154$	0	0
$155$	−16.2023	−1.30140
$156$	0	0
$157$	7.05499	0.563050	0.281525	−	0.959554i	$-0.409160\pi$
$157$	7.05499	0.563050	0.281525	+	0.959554i	$0.409160\pi$
$158$	0	0
$159$	4.16038	0.329939
$160$	0	0
$161$	−6.76276	−0.532980
$162$	0	0
$163$	21.6290	1.69411	0.847056	−	0.531504i	$-0.178373\pi$
$163$	21.6290	1.69411	0.847056	+	0.531504i	$0.178373\pi$
$164$	0	0
$165$	2.44072	0.190009
$166$	0	0
$167$	7.08517	0.548267	0.274133	−	0.961692i	$-0.411609\pi$
$167$	7.08517	0.548267	0.274133	+	0.961692i	$0.411609\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.11312	−0.161594
$172$	0	0
$173$	−9.91485	−0.753812	−0.376906	−	0.926251i	$-0.623012\pi$
$173$	−9.91485	−0.753812	−0.376906	+	0.926251i	$0.623012\pi$
$174$	0	0
$175$	−8.56030	−0.647098
$176$	0	0
$177$	−8.07534	−0.606980
$178$	0	0
$179$	17.9158	1.33909	0.669545	−	0.742772i	$-0.266489\pi$
$179$	17.9158	1.33909	0.669545	+	0.742772i	$0.266489\pi$
$180$	0	0
$181$	5.88586	0.437492	0.218746	−	0.975782i	$-0.429803\pi$
$181$	5.88586	0.437492	0.218746	+	0.975782i	$0.429803\pi$
$182$	0	0
$183$	7.24699	0.535713
$184$	0	0
$185$	2.66607	0.196014
$186$	0	0
$187$	3.51112	0.256759
$188$	0	0
$189$	−3.68563	−0.268090
$190$	0	0
$191$	7.46752	0.540331	0.270165	−	0.962814i	$-0.412922\pi$
$191$	7.46752	0.540331	0.270165	+	0.962814i	$0.412922\pi$
$192$	0	0
$193$	14.5230	1.04538	0.522692	−	0.852521i	$-0.324928\pi$
$193$	14.5230	1.04538	0.522692	+	0.852521i	$0.324928\pi$
$194$	0	0
$195$	2.44072	0.174783
$196$	0	0
$197$	−11.9352	−0.850347	−0.425174	−	0.905112i	$-0.639787\pi$
$197$	−11.9352	−0.850347	−0.425174	+	0.905112i	$0.639787\pi$
$198$	0	0
$199$	18.8640	1.33724	0.668618	−	0.743606i	$-0.266886\pi$
$199$	18.8640	1.33724	0.668618	+	0.743606i	$0.266886\pi$
$200$	0	0
$201$	0.618352	0.0436152
$202$	0	0
$203$	1.78380	0.125198
$204$	0	0
$205$	0.940031	0.0656546
$206$	0	0
$207$	−17.3174	−1.20364
$208$	0	0
$209$	−0.825213	−0.0570811
$210$	0	0
$211$	−12.9835	−0.893821	−0.446910	−	0.894579i	$-0.647476\pi$
$211$	−12.9835	−0.893821	−0.446910	+	0.894579i	$0.647476\pi$
$212$	0	0
$213$	−8.88881	−0.609051
$214$	0	0
$215$	45.5809	3.10859
$216$	0	0
$217$	4.39988	0.298683
$218$	0	0
$219$	−6.43920	−0.435121
$220$	0	0
$221$	3.51112	0.236183
$222$	0	0
$223$	23.1567	1.55069	0.775345	−	0.631537i	$-0.217576\pi$
$223$	23.1567	1.55069	0.775345	+	0.631537i	$0.217576\pi$
$224$	0	0
$225$	−21.9203	−1.46136
$226$	0	0
$227$	2.89083	0.191871	0.0959355	−	0.995388i	$-0.469416\pi$
$227$	2.89083	0.191871	0.0959355	+	0.995388i	$0.469416\pi$
$228$	0	0
$229$	−19.9010	−1.31509	−0.657547	−	0.753414i	$-0.728406\pi$
$229$	−19.9010	−1.31509	−0.657547	+	0.753414i	$0.728406\pi$
$230$	0	0
$231$	−0.662800	−0.0436090
$232$	0	0
$233$	9.36944	0.613812	0.306906	−	0.951740i	$-0.400706\pi$
$233$	9.36944	0.613812	0.306906	+	0.951740i	$0.400706\pi$
$234$	0	0
$235$	−23.4089	−1.52703
$236$	0	0
$237$	1.92079	0.124769
$238$	0	0
$239$	16.0141	1.03586	0.517932	−	0.855422i	$-0.326702\pi$
$239$	16.0141	1.03586	0.517932	+	0.855422i	$0.326702\pi$
$240$	0	0
$241$	2.92306	0.188291	0.0941453	−	0.995558i	$-0.469988\pi$
$241$	2.92306	0.188291	0.0941453	+	0.995558i	$0.469988\pi$
$242$	0	0
$243$	−14.5295	−0.932066
$244$	0	0
$245$	3.68243	0.235262
$246$	0	0
$247$	−0.825213	−0.0525070
$248$	0	0
$249$	5.45244	0.345534
$250$	0	0
$251$	−23.5902	−1.48900	−0.744499	−	0.667624i	$-0.767312\pi$
$251$	−23.5902	−1.48900	−0.744499	+	0.667624i	$0.767312\pi$
$252$	0	0
$253$	−6.76276	−0.425171
$254$	0	0
$255$	8.56964	0.536652
$256$	0	0
$257$	27.6857	1.72699	0.863494	−	0.504358i	$-0.168271\pi$
$257$	27.6857	1.72699	0.863494	+	0.504358i	$0.168271\pi$
$258$	0	0
$259$	−0.723998	−0.0449871
$260$	0	0
$261$	4.56777	0.282738
$262$	0	0
$263$	−9.44668	−0.582507	−0.291254	−	0.956646i	$-0.594072\pi$
$263$	−9.44668	−0.582507	−0.291254	+	0.956646i	$0.594072\pi$
$264$	0	0
$265$	−23.1145	−1.41991
$266$	0	0
$267$	−1.90803	−0.116769
$268$	0	0
$269$	−5.14278	−0.313561	−0.156780	−	0.987633i	$-0.550112\pi$
$269$	−5.14278	−0.313561	−0.156780	+	0.987633i	$0.550112\pi$
$270$	0	0
$271$	8.46153	0.514001	0.257001	−	0.966411i	$-0.417266\pi$
$271$	8.46153	0.514001	0.257001	+	0.966411i	$0.417266\pi$
$272$	0	0
$273$	−0.662800	−0.0401145
$274$	0	0
$275$	−8.56030	−0.516206
$276$	0	0
$277$	−26.7048	−1.60453	−0.802267	−	0.596966i	$-0.796373\pi$
$277$	−26.7048	−1.60453	−0.802267	+	0.596966i	$0.796373\pi$
$278$	0	0
$279$	11.2668	0.674523
$280$	0	0
$281$	−28.4976	−1.70003	−0.850013	−	0.526761i	$-0.823406\pi$
$281$	−28.4976	−1.70003	−0.850013	+	0.526761i	$0.823406\pi$
$282$	0	0
$283$	9.22988	0.548659	0.274330	−	0.961636i	$-0.411544\pi$
$283$	9.22988	0.548659	0.274330	+	0.961636i	$0.411544\pi$
$284$	0	0
$285$	−2.01411	−0.119306
$286$	0	0
$287$	−0.255274	−0.0150684
$288$	0	0
$289$	−4.67204	−0.274826
$290$	0	0
$291$	4.32369	0.253459
$292$	0	0
$293$	−15.3245	−0.895269	−0.447634	−	0.894217i	$-0.647733\pi$
$293$	−15.3245	−0.895269	−0.447634	+	0.894217i	$0.647733\pi$
$294$	0	0
$295$	44.8656	2.61218
$296$	0	0
$297$	−3.68563	−0.213862
$298$	0	0
$299$	−6.76276	−0.391100
$300$	0	0
$301$	−12.3779	−0.713453
$302$	0	0
$303$	−11.9185	−0.684703
$304$	0	0
$305$	−40.2634	−2.30547
$306$	0	0
$307$	−28.9151	−1.65027	−0.825135	−	0.564936i	$-0.808901\pi$
$307$	−28.9151	−1.65027	−0.825135	+	0.564936i	$0.808901\pi$
$308$	0	0
$309$	−2.77275	−0.157736
$310$	0	0
$311$	31.0580	1.76114	0.880569	−	0.473918i	$-0.157161\pi$
$311$	31.0580	1.76114	0.880569	+	0.473918i	$0.157161\pi$
$312$	0	0
$313$	8.55044	0.483299	0.241650	−	0.970364i	$-0.422312\pi$
$313$	8.55044	0.483299	0.241650	+	0.970364i	$0.422312\pi$
$314$	0	0
$315$	9.42959	0.531297
$316$	0	0
$317$	10.2041	0.573117	0.286559	−	0.958063i	$-0.407489\pi$
$317$	10.2041	0.573117	0.286559	+	0.958063i	$0.407489\pi$
$318$	0	0
$319$	1.78380	0.0998737
$320$	0	0
$321$	−11.1703	−0.623465
$322$	0	0
$323$	−2.89742	−0.161217
$324$	0	0
$325$	−8.56030	−0.474840
$326$	0	0
$327$	2.39894	0.132662
$328$	0	0
$329$	6.35690	0.350467
$330$	0	0
$331$	3.59291	0.197484	0.0987421	−	0.995113i	$-0.468518\pi$
$331$	3.59291	0.197484	0.0987421	+	0.995113i	$0.468518\pi$
$332$	0	0
$333$	−1.85394	−0.101595
$334$	0	0
$335$	−3.43548	−0.187701
$336$	0	0
$337$	−20.5734	−1.12071	−0.560353	−	0.828254i	$-0.689334\pi$
$337$	−20.5734	−1.12071	−0.560353	+	0.828254i	$0.689334\pi$
$338$	0	0
$339$	8.66358	0.470541
$340$	0	0
$341$	4.39988	0.238267
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−16.5060	−0.888651
$346$	0	0
$347$	−34.1391	−1.83268	−0.916341	−	0.400399i	$-0.868872\pi$
$347$	−34.1391	−1.83268	−0.916341	+	0.400399i	$0.868872\pi$
$348$	0	0
$349$	9.95658	0.532963	0.266482	−	0.963840i	$-0.414139\pi$
$349$	9.95658	0.532963	0.266482	+	0.963840i	$0.414139\pi$
$350$	0	0
$351$	−3.68563	−0.196724
$352$	0	0
$353$	−13.0132	−0.692622	−0.346311	−	0.938120i	$-0.612566\pi$
$353$	−13.0132	−0.692622	−0.346311	+	0.938120i	$0.612566\pi$
$354$	0	0
$355$	49.3851	2.62109
$356$	0	0
$357$	−2.32717	−0.123167
$358$	0	0
$359$	−27.4596	−1.44926	−0.724630	−	0.689138i	$-0.757989\pi$
$359$	−27.4596	−1.44926	−0.724630	+	0.689138i	$0.757989\pi$
$360$	0	0
$361$	−18.3190	−0.964159
$362$	0	0
$363$	−0.662800	−0.0347880
$364$	0	0
$365$	35.7754	1.87257
$366$	0	0
$367$	30.7618	1.60575	0.802877	−	0.596145i	$-0.203302\pi$
$367$	30.7618	1.60575	0.802877	+	0.596145i	$0.203302\pi$
$368$	0	0
$369$	−0.653680	−0.0340292
$370$	0	0
$371$	6.27697	0.325884
$372$	0	0
$373$	33.2285	1.72050	0.860252	−	0.509869i	$-0.170306\pi$
$373$	33.2285	1.72050	0.860252	+	0.509869i	$0.170306\pi$
$374$	0	0
$375$	−8.68969	−0.448733
$376$	0	0
$377$	1.78380	0.0918705
$378$	0	0
$379$	10.8583	0.557756	0.278878	−	0.960327i	$-0.410038\pi$
$379$	10.8583	0.557756	0.278878	+	0.960327i	$0.410038\pi$
$380$	0	0
$381$	3.25951	0.166990
$382$	0	0
$383$	−24.6592	−1.26003	−0.630013	−	0.776585i	$-0.716950\pi$
$383$	−24.6592	−1.26003	−0.630013	+	0.776585i	$0.716950\pi$
$384$	0	0
$385$	3.68243	0.187674
$386$	0	0
$387$	−31.6962	−1.61121
$388$	0	0
$389$	−8.01827	−0.406542	−0.203271	−	0.979122i	$-0.565157\pi$
$389$	−8.01827	−0.406542	−0.203271	+	0.979122i	$0.565157\pi$
$390$	0	0
$391$	−23.7448	−1.20083
$392$	0	0
$393$	9.44585	0.476480
$394$	0	0
$395$	−10.6717	−0.536951
$396$	0	0
$397$	3.19643	0.160424	0.0802121	−	0.996778i	$-0.474440\pi$
$397$	3.19643	0.160424	0.0802121	+	0.996778i	$0.474440\pi$
$398$	0	0
$399$	0.546951	0.0273818
$400$	0	0
$401$	26.1405	1.30540	0.652698	−	0.757618i	$-0.273637\pi$
$401$	26.1405	1.30540	0.652698	+	0.757618i	$0.273637\pi$
$402$	0	0
$403$	4.39988	0.219174
$404$	0	0
$405$	19.2932	0.958687
$406$	0	0
$407$	−0.723998	−0.0358873
$408$	0	0
$409$	26.9572	1.33295	0.666473	−	0.745529i	$-0.267803\pi$
$409$	26.9572	1.33295	0.666473	+	0.745529i	$0.267803\pi$
$410$	0	0
$411$	−7.24974	−0.357603
$412$	0	0
$413$	−12.1837	−0.599520
$414$	0	0
$415$	−30.2931	−1.48703
$416$	0	0
$417$	−7.45852	−0.365245
$418$	0	0
$419$	−7.26784	−0.355057	−0.177529	−	0.984116i	$-0.556810\pi$
$419$	−7.26784	−0.355057	−0.177529	+	0.984116i	$0.556810\pi$
$420$	0	0
$421$	3.27812	0.159766	0.0798829	−	0.996804i	$-0.474545\pi$
$421$	3.27812	0.159766	0.0798829	+	0.996804i	$0.474545\pi$
$422$	0	0
$423$	16.2781	0.791468
$424$	0	0
$425$	−30.0563	−1.45794
$426$	0	0
$427$	10.9339	0.529129
$428$	0	0
$429$	−0.662800	−0.0320003
$430$	0	0
$431$	22.1739	1.06808	0.534039	−	0.845460i	$-0.320673\pi$
$431$	22.1739	1.06808	0.534039	+	0.845460i	$0.320673\pi$
$432$	0	0
$433$	38.4996	1.85017	0.925087	−	0.379756i	$-0.123992\pi$
$433$	38.4996	1.85017	0.925087	+	0.379756i	$0.123992\pi$
$434$	0	0
$435$	4.35375	0.208746
$436$	0	0
$437$	5.58071	0.266962
$438$	0	0
$439$	1.33225	0.0635848	0.0317924	−	0.999494i	$-0.489878\pi$
$439$	1.33225	0.0635848	0.0317924	+	0.999494i	$0.489878\pi$
$440$	0	0
$441$	−2.56070	−0.121938
$442$	0	0
$443$	8.57345	0.407337	0.203668	−	0.979040i	$-0.434714\pi$
$443$	8.57345	0.407337	0.203668	+	0.979040i	$0.434714\pi$
$444$	0	0
$445$	10.6008	0.502524
$446$	0	0
$447$	−12.9532	−0.612664
$448$	0	0
$449$	−27.8362	−1.31367	−0.656835	−	0.754035i	$-0.728105\pi$
$449$	−27.8362	−1.31367	−0.656835	+	0.754035i	$0.728105\pi$
$450$	0	0
$451$	−0.255274	−0.0120204
$452$	0	0
$453$	−4.84127	−0.227463
$454$	0	0
$455$	3.68243	0.172635
$456$	0	0
$457$	−8.49868	−0.397551	−0.198776	−	0.980045i	$-0.563697\pi$
$457$	−8.49868	−0.397551	−0.198776	+	0.980045i	$0.563697\pi$
$458$	0	0
$459$	−12.9407	−0.604019
$460$	0	0
$461$	9.32931	0.434509	0.217255	−	0.976115i	$-0.430290\pi$
$461$	9.32931	0.434509	0.217255	+	0.976115i	$0.430290\pi$
$462$	0	0
$463$	0.167613	0.00778965	0.00389482	−	0.999992i	$-0.498760\pi$
$463$	0.167613	0.00778965	0.00389482	+	0.999992i	$0.498760\pi$
$464$	0	0
$465$	10.7389	0.498003
$466$	0	0
$467$	−13.9264	−0.644437	−0.322218	−	0.946665i	$-0.604429\pi$
$467$	−13.9264	−0.644437	−0.322218	+	0.946665i	$0.604429\pi$
$468$	0	0
$469$	0.932939	0.0430791
$470$	0	0
$471$	−4.67605	−0.215461
$472$	0	0
$473$	−12.3779	−0.569139
$474$	0	0
$475$	7.06407	0.324122
$476$	0	0
$477$	16.0734	0.735951
$478$	0	0
$479$	0.357858	0.0163509	0.00817547	−	0.999967i	$-0.497398\pi$
$479$	0.357858	0.0163509	0.00817547	+	0.999967i	$0.497398\pi$
$480$	0	0
$481$	−0.723998	−0.0330115
$482$	0	0
$483$	4.48235	0.203954
$484$	0	0
$485$	−24.0219	−1.09078
$486$	0	0
$487$	−14.8774	−0.674158	−0.337079	−	0.941476i	$-0.609439\pi$
$487$	−14.8774	−0.674158	−0.337079	+	0.941476i	$0.609439\pi$
$488$	0	0
$489$	−14.3357	−0.648282
$490$	0	0
$491$	17.0522	0.769555	0.384777	−	0.923009i	$-0.374278\pi$
$491$	17.0522	0.769555	0.384777	+	0.923009i	$0.374278\pi$
$492$	0	0
$493$	6.26314	0.282078
$494$	0	0
$495$	9.42959	0.423829
$496$	0	0
$497$	−13.4110	−0.601566
$498$	0	0
$499$	27.5133	1.23166	0.615831	−	0.787878i	$-0.288820\pi$
$499$	27.5133	1.23166	0.615831	+	0.787878i	$0.288820\pi$
$500$	0	0
$501$	−4.69605	−0.209804
$502$	0	0
$503$	−22.3614	−0.997044	−0.498522	−	0.866877i	$-0.666124\pi$
$503$	−22.3614	−0.997044	−0.498522	+	0.866877i	$0.666124\pi$
$504$	0	0
$505$	66.2179	2.94666
$506$	0	0
$507$	−0.662800	−0.0294360
$508$	0	0
$509$	14.2826	0.633065	0.316532	−	0.948582i	$-0.397481\pi$
$509$	14.2826	0.633065	0.316532	+	0.948582i	$0.397481\pi$
$510$	0	0
$511$	−9.71515	−0.429773
$512$	0	0
$513$	3.04143	0.134282
$514$	0	0
$515$	15.4051	0.678829
$516$	0	0
$517$	6.35690	0.279576
$518$	0	0
$519$	6.57156	0.288460
$520$	0	0
$521$	20.2850	0.888703	0.444351	−	0.895853i	$-0.353434\pi$
$521$	20.2850	0.888703	0.444351	+	0.895853i	$0.353434\pi$
$522$	0	0
$523$	26.9676	1.17921	0.589605	−	0.807692i	$-0.299283\pi$
$523$	26.9676	1.17921	0.589605	+	0.807692i	$0.299283\pi$
$524$	0	0
$525$	5.67377	0.247624
$526$	0	0
$527$	15.4485	0.672948
$528$	0	0
$529$	22.7349	0.988473
$530$	0	0
$531$	−31.1987	−1.35391
$532$	0	0
$533$	−0.255274	−0.0110572
$534$	0	0
$535$	62.0607	2.68312
$536$	0	0
$537$	−11.8746	−0.512426
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−26.3337	−1.13218	−0.566088	−	0.824345i	$-0.691544\pi$
$541$	−26.3337	−1.13218	−0.566088	+	0.824345i	$0.691544\pi$
$542$	0	0
$543$	−3.90115	−0.167414
$544$	0	0
$545$	−13.3282	−0.570918
$546$	0	0
$547$	33.7841	1.44450	0.722252	−	0.691630i	$-0.243107\pi$
$547$	33.7841	1.44450	0.722252	+	0.691630i	$0.243107\pi$
$548$	0	0
$549$	27.9984	1.19494
$550$	0	0
$551$	−1.47202	−0.0627100
$552$	0	0
$553$	2.89800	0.123236
$554$	0	0
$555$	−1.76707	−0.0750081
$556$	0	0
$557$	30.2337	1.28104	0.640521	−	0.767941i	$-0.278719\pi$
$557$	30.2337	1.28104	0.640521	+	0.767941i	$0.278719\pi$
$558$	0	0
$559$	−12.3779	−0.523531
$560$	0	0
$561$	−2.32717	−0.0982532
$562$	0	0
$563$	−32.2072	−1.35737	−0.678687	−	0.734428i	$-0.737451\pi$
$563$	−32.2072	−1.35737	−0.678687	+	0.734428i	$0.737451\pi$
$564$	0	0
$565$	−48.1338	−2.02500
$566$	0	0
$567$	−5.23925	−0.220028
$568$	0	0
$569$	10.1836	0.426917	0.213459	−	0.976952i	$-0.431527\pi$
$569$	10.1836	0.426917	0.213459	+	0.976952i	$0.431527\pi$
$570$	0	0
$571$	20.9947	0.878600	0.439300	−	0.898340i	$-0.355227\pi$
$571$	20.9947	0.878600	0.439300	+	0.898340i	$0.355227\pi$
$572$	0	0
$573$	−4.94947	−0.206767
$574$	0	0
$575$	57.8912	2.41423
$576$	0	0
$577$	4.53677	0.188868	0.0944342	−	0.995531i	$-0.469896\pi$
$577$	4.53677	0.188868	0.0944342	+	0.995531i	$0.469896\pi$
$578$	0	0
$579$	−9.62581	−0.400035
$580$	0	0
$581$	8.22637	0.341288
$582$	0	0
$583$	6.27697	0.259966
$584$	0	0
$585$	9.42959	0.389866
$586$	0	0
$587$	32.6967	1.34954	0.674769	−	0.738029i	$-0.264243\pi$
$587$	32.6967	1.34954	0.674769	+	0.738029i	$0.264243\pi$
$588$	0	0
$589$	−3.63084	−0.149606
$590$	0	0
$591$	7.91065	0.325400
$592$	0	0
$593$	20.2147	0.830116	0.415058	−	0.909795i	$-0.363761\pi$
$593$	20.2147	0.830116	0.415058	+	0.909795i	$0.363761\pi$
$594$	0	0
$595$	12.9295	0.530056
$596$	0	0
$597$	−12.5031	−0.511717
$598$	0	0
$599$	39.5185	1.61468	0.807342	−	0.590084i	$-0.200905\pi$
$599$	39.5185	1.61468	0.807342	+	0.590084i	$0.200905\pi$
$600$	0	0
$601$	39.4259	1.60822	0.804108	−	0.594483i	$-0.202643\pi$
$601$	39.4259	1.60822	0.804108	+	0.594483i	$0.202643\pi$
$602$	0	0
$603$	2.38897	0.0972865
$604$	0	0
$605$	3.68243	0.149712
$606$	0	0
$607$	−6.27885	−0.254851	−0.127425	−	0.991848i	$-0.540671\pi$
$607$	−6.27885	−0.254851	−0.127425	+	0.991848i	$0.540671\pi$
$608$	0	0
$609$	−1.18230	−0.0479094
$610$	0	0
$611$	6.35690	0.257173
$612$	0	0
$613$	10.8511	0.438271	0.219136	−	0.975694i	$-0.429676\pi$
$613$	10.8511	0.438271	0.219136	+	0.975694i	$0.429676\pi$
$614$	0	0
$615$	−0.623052	−0.0251239
$616$	0	0
$617$	−36.6787	−1.47663	−0.738315	−	0.674456i	$-0.764378\pi$
$617$	−36.6787	−1.47663	−0.738315	+	0.674456i	$0.764378\pi$
$618$	0	0
$619$	−7.67982	−0.308678	−0.154339	−	0.988018i	$-0.549325\pi$
$619$	−7.67982	−0.308678	−0.154339	+	0.988018i	$0.549325\pi$
$620$	0	0
$621$	24.9250	1.00021
$622$	0	0
$623$	−2.87874	−0.115334
$624$	0	0
$625$	5.47728	0.219091
$626$	0	0
$627$	0.546951	0.0218431
$628$	0	0
$629$	−2.54204	−0.101358
$630$	0	0
$631$	−10.9426	−0.435618	−0.217809	−	0.975991i	$-0.569891\pi$
$631$	−10.9426	−0.435618	−0.217809	+	0.975991i	$0.569891\pi$
$632$	0	0
$633$	8.60546	0.342036
$634$	0	0
$635$	−18.1094	−0.718650
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−34.3415	−1.35853
$640$	0	0
$641$	22.0519	0.870997	0.435498	−	0.900190i	$-0.356572\pi$
$641$	22.0519	0.870997	0.435498	+	0.900190i	$0.356572\pi$
$642$	0	0
$643$	−29.5260	−1.16439	−0.582197	−	0.813048i	$-0.697807\pi$
$643$	−29.5260	−1.16439	−0.582197	+	0.813048i	$0.697807\pi$
$644$	0	0
$645$	−30.2110	−1.18956
$646$	0	0
$647$	4.74394	0.186503	0.0932517	−	0.995643i	$-0.470274\pi$
$647$	4.74394	0.186503	0.0932517	+	0.995643i	$0.470274\pi$
$648$	0	0
$649$	−12.1837	−0.478251
$650$	0	0
$651$	−2.91624	−0.114297
$652$	0	0
$653$	−35.3254	−1.38239	−0.691196	−	0.722668i	$-0.742916\pi$
$653$	−35.3254	−1.38239	−0.691196	+	0.722668i	$0.742916\pi$
$654$	0	0
$655$	−52.4799	−2.05056
$656$	0	0
$657$	−24.8775	−0.970565
$658$	0	0
$659$	−11.8680	−0.462313	−0.231156	−	0.972917i	$-0.574251\pi$
$659$	−11.8680	−0.462313	−0.231156	+	0.972917i	$0.574251\pi$
$660$	0	0
$661$	−6.93627	−0.269790	−0.134895	−	0.990860i	$-0.543070\pi$
$661$	−6.93627	−0.269790	−0.134895	+	0.990860i	$0.543070\pi$
$662$	0	0
$663$	−2.32717	−0.0903798
$664$	0	0
$665$	−3.03879	−0.117839
$666$	0	0
$667$	−12.0634	−0.467097
$668$	0	0
$669$	−15.3483	−0.593399
$670$	0	0
$671$	10.9339	0.422099
$672$	0	0
$673$	−0.852648	−0.0328672	−0.0164336	−	0.999865i	$-0.505231\pi$
$673$	−0.852648	−0.0328672	−0.0164336	+	0.999865i	$0.505231\pi$
$674$	0	0
$675$	31.5501	1.21436
$676$	0	0
$677$	−16.3212	−0.627274	−0.313637	−	0.949543i	$-0.601548\pi$
$677$	−16.3212	−0.627274	−0.313637	+	0.949543i	$0.601548\pi$
$678$	0	0
$679$	6.52337	0.250344
$680$	0	0
$681$	−1.91604	−0.0734228
$682$	0	0
$683$	41.8232	1.60032	0.800161	−	0.599786i	$-0.204747\pi$
$683$	41.8232	1.60032	0.800161	+	0.599786i	$0.204747\pi$
$684$	0	0
$685$	40.2786	1.53897
$686$	0	0
$687$	13.1904	0.503244
$688$	0	0
$689$	6.27697	0.239134
$690$	0	0
$691$	−28.4336	−1.08166	−0.540832	−	0.841131i	$-0.681891\pi$
$691$	−28.4336	−1.08166	−0.540832	+	0.841131i	$0.681891\pi$
$692$	0	0
$693$	−2.56070	−0.0972728
$694$	0	0
$695$	41.4386	1.57185
$696$	0	0
$697$	−0.896299	−0.0339498
$698$	0	0
$699$	−6.21006	−0.234886
$700$	0	0
$701$	−3.91079	−0.147708	−0.0738542	−	0.997269i	$-0.523530\pi$
$701$	−3.91079	−0.147708	−0.0738542	+	0.997269i	$0.523530\pi$
$702$	0	0
$703$	0.597453	0.0225333
$704$	0	0
$705$	15.5154	0.584343
$706$	0	0
$707$	−17.9821	−0.676287
$708$	0	0
$709$	19.7117	0.740288	0.370144	−	0.928974i	$-0.379308\pi$
$709$	19.7117	0.740288	0.370144	+	0.928974i	$0.379308\pi$
$710$	0	0
$711$	7.42090	0.278306
$712$	0	0
$713$	−29.7553	−1.11435
$714$	0	0
$715$	3.68243	0.137715
$716$	0	0
$717$	−10.6141	−0.396391
$718$	0	0
$719$	24.3379	0.907652	0.453826	−	0.891090i	$-0.350059\pi$
$719$	24.3379	0.907652	0.453826	+	0.891090i	$0.350059\pi$
$720$	0	0
$721$	−4.18340	−0.155798
$722$	0	0
$723$	−1.93740	−0.0720527
$724$	0	0
$725$	−15.2699	−0.567109
$726$	0	0
$727$	−16.7356	−0.620689	−0.310345	−	0.950624i	$-0.600444\pi$
$727$	−16.7356	−0.620689	−0.310345	+	0.950624i	$0.600444\pi$
$728$	0	0
$729$	−6.08764	−0.225468
$730$	0	0
$731$	−43.4605	−1.60744
$732$	0	0
$733$	−37.1377	−1.37171	−0.685855	−	0.727738i	$-0.740572\pi$
$733$	−37.1377	−1.37171	−0.685855	+	0.727738i	$0.740572\pi$
$734$	0	0
$735$	−2.44072	−0.0900271
$736$	0	0
$737$	0.932939	0.0343653
$738$	0	0
$739$	−15.5146	−0.570714	−0.285357	−	0.958421i	$-0.592112\pi$
$739$	−15.5146	−0.570714	−0.285357	+	0.958421i	$0.592112\pi$
$740$	0	0
$741$	0.546951	0.0200927
$742$	0	0
$743$	−29.3119	−1.07535	−0.537675	−	0.843152i	$-0.680697\pi$
$743$	−29.3119	−1.07535	−0.537675	+	0.843152i	$0.680697\pi$
$744$	0	0
$745$	71.9661	2.63664
$746$	0	0
$747$	21.0652	0.770737
$748$	0	0
$749$	−16.8532	−0.615802
$750$	0	0
$751$	−49.5788	−1.80915	−0.904577	−	0.426310i	$-0.859813\pi$
$751$	−49.5788	−1.80915	−0.904577	+	0.426310i	$0.859813\pi$
$752$	0	0
$753$	15.6356	0.569791
$754$	0	0
$755$	26.8975	0.978900
$756$	0	0
$757$	36.0248	1.30934	0.654671	−	0.755914i	$-0.272807\pi$
$757$	36.0248	1.30934	0.654671	+	0.755914i	$0.272807\pi$
$758$	0	0
$759$	4.48235	0.162699
$760$	0	0
$761$	0.600640	0.0217732	0.0108866	−	0.999941i	$-0.496535\pi$
$761$	0.600640	0.0217732	0.0108866	+	0.999941i	$0.496535\pi$
$762$	0	0
$763$	3.61941	0.131031
$764$	0	0
$765$	33.1084	1.19704
$766$	0	0
$767$	−12.1837	−0.439927
$768$	0	0
$769$	−45.2113	−1.63036	−0.815181	−	0.579206i	$-0.803363\pi$
$769$	−45.2113	−1.63036	−0.815181	+	0.579206i	$0.803363\pi$
$770$	0	0
$771$	−18.3501	−0.660863
$772$	0	0
$773$	28.7567	1.03431	0.517154	−	0.855892i	$-0.326991\pi$
$773$	28.7567	1.03431	0.517154	+	0.855892i	$0.326991\pi$
$774$	0	0
$775$	−37.6643	−1.35294
$776$	0	0
$777$	0.479866	0.0172151
$778$	0	0
$779$	0.210656	0.00754752
$780$	0	0
$781$	−13.4110	−0.479884
$782$	0	0
$783$	−6.57443	−0.234951
$784$	0	0
$785$	25.9795	0.927249
$786$	0	0
$787$	−20.8237	−0.742284	−0.371142	−	0.928576i	$-0.621034\pi$
$787$	−20.8237	−0.742284	−0.371142	+	0.928576i	$0.621034\pi$
$788$	0	0
$789$	6.26126	0.222907
$790$	0	0
$791$	13.0712	0.464758
$792$	0	0
$793$	10.9339	0.388274
$794$	0	0
$795$	15.3203	0.543355
$796$	0	0
$797$	15.0191	0.532003	0.266001	−	0.963973i	$-0.414297\pi$
$797$	15.0191	0.532003	0.266001	+	0.963973i	$0.414297\pi$
$798$	0	0
$799$	22.3198	0.789619
$800$	0	0
$801$	−7.37158	−0.260462
$802$	0	0
$803$	−9.71515	−0.342840
$804$	0	0
$805$	−24.9034	−0.877729
$806$	0	0
$807$	3.40863	0.119990
$808$	0	0
$809$	18.3714	0.645905	0.322952	−	0.946415i	$-0.395325\pi$
$809$	18.3714	0.645905	0.322952	+	0.946415i	$0.395325\pi$
$810$	0	0
$811$	3.83483	0.134659	0.0673295	−	0.997731i	$-0.478552\pi$
$811$	3.83483	0.134659	0.0673295	+	0.997731i	$0.478552\pi$
$812$	0	0
$813$	−5.60830	−0.196692
$814$	0	0
$815$	79.6472	2.78992
$816$	0	0
$817$	10.2144	0.357358
$818$	0	0
$819$	−2.56070	−0.0894780
$820$	0	0
$821$	−31.9142	−1.11381	−0.556906	−	0.830576i	$-0.688012\pi$
$821$	−31.9142	−1.11381	−0.556906	+	0.830576i	$0.688012\pi$
$822$	0	0
$823$	2.51185	0.0875577	0.0437789	−	0.999041i	$-0.486060\pi$
$823$	2.51185	0.0875577	0.0437789	+	0.999041i	$0.486060\pi$
$824$	0	0
$825$	5.67377	0.197535
$826$	0	0
$827$	12.8351	0.446320	0.223160	−	0.974782i	$-0.428363\pi$
$827$	12.8351	0.446320	0.223160	+	0.974782i	$0.428363\pi$
$828$	0	0
$829$	40.4480	1.40482	0.702409	−	0.711774i	$-0.252108\pi$
$829$	40.4480	1.40482	0.702409	+	0.711774i	$0.252108\pi$
$830$	0	0
$831$	17.6999	0.614003
$832$	0	0
$833$	−3.51112	−0.121653
$834$	0	0
$835$	26.0907	0.902904
$836$	0	0
$837$	−16.2163	−0.560518
$838$	0	0
$839$	−53.5362	−1.84827	−0.924137	−	0.382061i	$-0.875214\pi$
$839$	−53.5362	−1.84827	−0.924137	+	0.382061i	$0.875214\pi$
$840$	0	0
$841$	−25.8181	−0.890278
$842$	0	0
$843$	18.8882	0.650545
$844$	0	0
$845$	3.68243	0.126680
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−6.11756	−0.209954
$850$	0	0
$851$	4.89622	0.167840
$852$	0	0
$853$	−31.1122	−1.06526	−0.532631	−	0.846348i	$-0.678796\pi$
$853$	−31.1122	−1.06526	−0.532631	+	0.846348i	$0.678796\pi$
$854$	0	0
$855$	−7.78142	−0.266119
$856$	0	0
$857$	6.27950	0.214504	0.107252	−	0.994232i	$-0.465795\pi$
$857$	6.27950	0.214504	0.107252	+	0.994232i	$0.465795\pi$
$858$	0	0
$859$	41.2039	1.40586	0.702930	−	0.711259i	$-0.251875\pi$
$859$	41.2039	1.40586	0.702930	+	0.711259i	$0.251875\pi$
$860$	0	0
$861$	0.169196	0.00576618
$862$	0	0
$863$	51.0244	1.73689	0.868446	−	0.495785i	$-0.165119\pi$
$863$	51.0244	1.73689	0.868446	+	0.495785i	$0.165119\pi$
$864$	0	0
$865$	−36.5108	−1.24140
$866$	0	0
$867$	3.09663	0.105167
$868$	0	0
$869$	2.89800	0.0983080
$870$	0	0
$871$	0.932939	0.0316114
$872$	0	0
$873$	16.7044	0.565357
$874$	0	0
$875$	−13.1106	−0.443218
$876$	0	0
$877$	39.6082	1.33748	0.668738	−	0.743498i	$-0.266835\pi$
$877$	39.6082	1.33748	0.668738	+	0.743498i	$0.266835\pi$
$878$	0	0
$879$	10.1571	0.342590
$880$	0	0
$881$	23.3541	0.786819	0.393409	−	0.919363i	$-0.371296\pi$
$881$	23.3541	0.786819	0.393409	+	0.919363i	$0.371296\pi$
$882$	0	0
$883$	−42.2984	−1.42345	−0.711727	−	0.702456i	$-0.752087\pi$
$883$	−42.2984	−1.42345	−0.711727	+	0.702456i	$0.752087\pi$
$884$	0	0
$885$	−29.7369	−0.999595
$886$	0	0
$887$	−43.7457	−1.46884	−0.734419	−	0.678696i	$-0.762545\pi$
$887$	−43.7457	−1.46884	−0.734419	+	0.678696i	$0.762545\pi$
$888$	0	0
$889$	4.91778	0.164937
$890$	0	0
$891$	−5.23925	−0.175522
$892$	0	0
$893$	−5.24579	−0.175544
$894$	0	0
$895$	65.9737	2.20526
$896$	0	0
$897$	4.48235	0.149661
$898$	0	0
$899$	7.84852	0.261763
$900$	0	0
$901$	22.0392	0.734232
$902$	0	0
$903$	8.20410	0.273015
$904$	0	0
$905$	21.6743	0.720477
$906$	0	0
$907$	29.8364	0.990700	0.495350	−	0.868694i	$-0.335040\pi$
$907$	29.8364	0.990700	0.495350	+	0.868694i	$0.335040\pi$
$908$	0	0
$909$	−46.0467	−1.52727
$910$	0	0
$911$	−22.6755	−0.751272	−0.375636	−	0.926767i	$-0.622576\pi$
$911$	−22.6755	−0.751272	−0.375636	+	0.926767i	$0.622576\pi$
$912$	0	0
$913$	8.22637	0.272253
$914$	0	0
$915$	26.6866	0.882230
$916$	0	0
$917$	14.2514	0.470624
$918$	0	0
$919$	52.4830	1.73126	0.865628	−	0.500689i	$-0.166920\pi$
$919$	52.4830	1.73126	0.865628	+	0.500689i	$0.166920\pi$
$920$	0	0
$921$	19.1649	0.631505
$922$	0	0
$923$	−13.4110	−0.441429
$924$	0	0
$925$	6.19765	0.203777
$926$	0	0
$927$	−10.7124	−0.351842
$928$	0	0
$929$	−60.7848	−1.99429	−0.997143	−	0.0755400i	$-0.975932\pi$
$929$	−60.7848	−1.99429	−0.997143	+	0.0755400i	$0.975932\pi$
$930$	0	0
$931$	0.825213	0.0270453
$932$	0	0
$933$	−20.5852	−0.673931
$934$	0	0
$935$	12.9295	0.422839
$936$	0	0
$937$	53.8170	1.75813	0.879063	−	0.476706i	$-0.158169\pi$
$937$	53.8170	1.75813	0.879063	+	0.476706i	$0.158169\pi$
$938$	0	0
$939$	−5.66723	−0.184943
$940$	0	0
$941$	−51.9893	−1.69480	−0.847402	−	0.530952i	$-0.821834\pi$
$941$	−51.9893	−1.69480	−0.847402	+	0.530952i	$0.821834\pi$
$942$	0	0
$943$	1.72636	0.0562180
$944$	0	0
$945$	−13.5721	−0.441500
$946$	0	0
$947$	23.6454	0.768371	0.384186	−	0.923256i	$-0.374482\pi$
$947$	23.6454	0.768371	0.384186	+	0.923256i	$0.374482\pi$
$948$	0	0
$949$	−9.71515	−0.315367
$950$	0	0
$951$	−6.76325	−0.219314
$952$	0	0
$953$	51.5005	1.66826	0.834132	−	0.551565i	$-0.185969\pi$
$953$	51.5005	1.66826	0.834132	+	0.551565i	$0.185969\pi$
$954$	0	0
$955$	27.4986	0.889835
$956$	0	0
$957$	−1.18230	−0.0382184
$958$	0	0
$959$	−10.9380	−0.353208
$960$	0	0
$961$	−11.6410	−0.375517
$962$	0	0
$963$	−43.1559	−1.39068
$964$	0	0
$965$	53.4798	1.72157
$966$	0	0
$967$	25.7416	0.827794	0.413897	−	0.910324i	$-0.364167\pi$
$967$	25.7416	0.827794	0.413897	+	0.910324i	$0.364167\pi$
$968$	0	0
$969$	1.92041	0.0616925
$970$	0	0
$971$	6.42293	0.206122	0.103061	−	0.994675i	$-0.467136\pi$
$971$	6.42293	0.206122	0.103061	+	0.994675i	$0.467136\pi$
$972$	0	0
$973$	−11.2530	−0.360756
$974$	0	0
$975$	5.67377	0.181706
$976$	0	0
$977$	32.6086	1.04324	0.521620	−	0.853178i	$-0.325328\pi$
$977$	32.6086	1.04324	0.521620	+	0.853178i	$0.325328\pi$
$978$	0	0
$979$	−2.87874	−0.0920049
$980$	0	0
$981$	9.26820	0.295911
$982$	0	0
$983$	−42.3069	−1.34938	−0.674690	−	0.738101i	$-0.735723\pi$
$983$	−42.3069	−1.34938	−0.674690	+	0.738101i	$0.735723\pi$
$984$	0	0
$985$	−43.9505	−1.40038
$986$	0	0
$987$	−4.21335	−0.134113
$988$	0	0
$989$	83.7090	2.66179
$990$	0	0
$991$	−8.88513	−0.282246	−0.141123	−	0.989992i	$-0.545071\pi$
$991$	−8.88513	−0.282246	−0.141123	+	0.989992i	$0.545071\pi$
$992$	0	0
$993$	−2.38138	−0.0755708
$994$	0	0
$995$	69.4656	2.20221
$996$	0	0
$997$	37.8447	1.19855	0.599277	−	0.800542i	$-0.295455\pi$
$997$	37.8447	1.19855	0.599277	+	0.800542i	$0.295455\pi$
$998$	0	0
$999$	2.66839	0.0844241

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.r.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.r.1.3	✓	9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.662800 q^{3} +3.68243 q^{5} -1.00000 q^{7} -2.56070 q^{9} +O(q^{10})\)	\(q-0.662800 q^{3} +3.68243 q^{5} -1.00000 q^{7} -2.56070 q^{9} -1.00000 q^{11} -1.00000 q^{13} -2.44072 q^{15} -3.51112 q^{17} +0.825213 q^{19} +0.662800 q^{21} +6.76276 q^{23} +8.56030 q^{25} +3.68563 q^{27} -1.78380 q^{29} -4.39988 q^{31} +0.662800 q^{33} -3.68243 q^{35} +0.723998 q^{37} +0.662800 q^{39} +0.255274 q^{41} +12.3779 q^{43} -9.42959 q^{45} -6.35690 q^{47} +1.00000 q^{49} +2.32717 q^{51} -6.27697 q^{53} -3.68243 q^{55} -0.546951 q^{57} +12.1837 q^{59} -10.9339 q^{61} +2.56070 q^{63} -3.68243 q^{65} -0.932939 q^{67} -4.48235 q^{69} +13.4110 q^{71} +9.71515 q^{73} -5.67377 q^{75} +1.00000 q^{77} -2.89800 q^{79} +5.23925 q^{81} -8.22637 q^{83} -12.9295 q^{85} +1.18230 q^{87} +2.87874 q^{89} +1.00000 q^{91} +2.91624 q^{93} +3.03879 q^{95} -6.52337 q^{97} +2.56070 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9}+O(q^{10}) \) 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9	\( 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9} - 9 q^{11} - 9 q^{13} - 3 q^{15} - 7 q^{17} + 13 q^{19} - 5 q^{21} + 9 q^{23} - 2 q^{25} + 5 q^{27} + q^{29} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 14 q^{37} - 5 q^{39} - 2 q^{41} + 5 q^{43} - 15 q^{45} + 13 q^{47} + 9 q^{49} - 3 q^{51} + 22 q^{53} - 3 q^{55} + 16 q^{57} + 43 q^{59} - 10 q^{61} - 8 q^{63} - 3 q^{65} + 26 q^{67} - 30 q^{69} + 18 q^{71} - 8 q^{73} + 28 q^{75} + 9 q^{77} - 9 q^{79} + 33 q^{81} + 4 q^{83} + 5 q^{85} + 33 q^{87} + 7 q^{89} + 9 q^{91} + 13 q^{93} + 7 q^{95} + 2 q^{97} - 8 q^{99}+O(q^{100}) \) 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9 - 9 * q^11 - 9 * q^13 - 3 * q^15 - 7 * q^17 + 13 * q^19 - 5 * q^21 + 9 * q^23 - 2 * q^25 + 5 * q^27 + q^29 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 14 * q^37 - 5 * q^39 - 2 * q^41 + 5 * q^43 - 15 * q^45 + 13 * q^47 + 9 * q^49 - 3 * q^51 + 22 * q^53 - 3 * q^55 + 16 * q^57 + 43 * q^59 - 10 * q^61 - 8 * q^63 - 3 * q^65 + 26 * q^67 - 30 * q^69 + 18 * q^71 - 8 * q^73 + 28 * q^75 + 9 * q^77 - 9 * q^79 + 33 * q^81 + 4 * q^83 + 5 * q^85 + 33 * q^87 + 7 * q^89 + 9 * q^91 + 13 * q^93 + 7 * q^95 + 2 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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