LMFDB - Embedded newform 8008.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-0.618034 q^{3} +1.61803 q^{5} +1.00000 q^{7} -2.61803 q^{9} +O(q^{10})$	$q-0.618034 q^{3} +1.61803 q^{5} +1.00000 q^{7} -2.61803 q^{9} +1.00000 q^{11} +1.00000 q^{13} -1.00000 q^{15} -7.85410 q^{17} +0.381966 q^{19} -0.618034 q^{21} +5.70820 q^{23} -2.38197 q^{25} +3.47214 q^{27} -0.472136 q^{29} -2.76393 q^{31} -0.618034 q^{33} +1.61803 q^{35} +11.7082 q^{37} -0.618034 q^{39} -5.23607 q^{41} -2.61803 q^{43} -4.23607 q^{45} -4.76393 q^{47} +1.00000 q^{49} +4.85410 q^{51} +2.61803 q^{53} +1.61803 q^{55} -0.236068 q^{57} +1.23607 q^{59} -6.14590 q^{61} -2.61803 q^{63} +1.61803 q^{65} +13.0902 q^{67} -3.52786 q^{69} -12.5623 q^{71} +4.47214 q^{73} +1.47214 q^{75} +1.00000 q^{77} -15.0344 q^{79} +5.70820 q^{81} +0.381966 q^{83} -12.7082 q^{85} +0.291796 q^{87} -18.6180 q^{89} +1.00000 q^{91} +1.70820 q^{93} +0.618034 q^{95} -2.00000 q^{97} -2.61803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + q^5 + 2 * q^7 - 3 * q^9	$ 2 q + q^{3} + q^{5} + 2 q^{7} - 3 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} - 9 q^{17} + 3 q^{19} + q^{21} - 2 q^{23} - 7 q^{25} - 2 q^{27} + 8 q^{29} - 10 q^{31} + q^{33} + q^{35} + 10 q^{37} + q^{39} - 6 q^{41} - 3 q^{43} - 4 q^{45} - 14 q^{47} + 2 q^{49} + 3 q^{51} + 3 q^{53} + q^{55} + 4 q^{57} - 2 q^{59} - 19 q^{61} - 3 q^{63} + q^{65} + 15 q^{67} - 16 q^{69} - 5 q^{71} - 6 q^{75} + 2 q^{77} - q^{79} - 2 q^{81} + 3 q^{83} - 12 q^{85} + 14 q^{87} - 35 q^{89} + 2 q^{91} - 10 q^{93} - q^{95} - 4 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q + q^3 + q^5 + 2 * q^7 - 3 * q^9 + 2 * q^11 + 2 * q^13 - 2 * q^15 - 9 * q^17 + 3 * q^19 + q^21 - 2 * q^23 - 7 * q^25 - 2 * q^27 + 8 * q^29 - 10 * q^31 + q^33 + q^35 + 10 * q^37 + q^39 - 6 * q^41 - 3 * q^43 - 4 * q^45 - 14 * q^47 + 2 * q^49 + 3 * q^51 + 3 * q^53 + q^55 + 4 * q^57 - 2 * q^59 - 19 * q^61 - 3 * q^63 + q^65 + 15 * q^67 - 16 * q^69 - 5 * q^71 - 6 * q^75 + 2 * q^77 - q^79 - 2 * q^81 + 3 * q^83 - 12 * q^85 + 14 * q^87 - 35 * q^89 + 2 * q^91 - 10 * q^93 - q^95 - 4 * q^97 - 3 * 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
$3$	−0.618034	−0.356822	−0.178411	−	0.983956i	$-0.557096\pi$
$3$	−0.618034	−0.356822	−0.178411	+	0.983956i	$0.557096\pi$
$4$	0	0
$5$	1.61803	0.723607	0.361803	−	0.932254i	$-0.382161\pi$
$5$	1.61803	0.723607	0.361803	+	0.932254i	$0.382161\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.61803	−0.872678
$10$	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
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−7.85410	−1.90490	−0.952450	−	0.304696i	$-0.901445\pi$
$17$	−7.85410	−1.90490	−0.952450	+	0.304696i	$0.901445\pi$
$18$	0	0
$19$	0.381966	0.0876290	0.0438145	−	0.999040i	$-0.486049\pi$
$19$	0.381966	0.0876290	0.0438145	+	0.999040i	$0.486049\pi$
$20$	0	0
$21$	−0.618034	−0.134866
$22$	0	0
$23$	5.70820	1.19024	0.595121	−	0.803636i	$-0.297104\pi$
$23$	5.70820	1.19024	0.595121	+	0.803636i	$0.297104\pi$
$24$	0	0
$25$	−2.38197	−0.476393
$26$	0	0
$27$	3.47214	0.668213
$28$	0	0
$29$	−0.472136	−0.0876734	−0.0438367	−	0.999039i	$-0.513958\pi$
$29$	−0.472136	−0.0876734	−0.0438367	+	0.999039i	$0.513958\pi$
$30$	0	0
$31$	−2.76393	−0.496417	−0.248208	−	0.968707i	$-0.579842\pi$
$31$	−2.76393	−0.496417	−0.248208	+	0.968707i	$0.579842\pi$
$32$	0	0
$33$	−0.618034	−0.107586
$34$	0	0
$35$	1.61803	0.273498
$36$	0	0
$37$	11.7082	1.92482	0.962408	−	0.271606i	$-0.0875549\pi$
$37$	11.7082	1.92482	0.962408	+	0.271606i	$0.0875549\pi$
$38$	0	0
$39$	−0.618034	−0.0989646
$40$	0	0
$41$	−5.23607	−0.817736	−0.408868	−	0.912593i	$-0.634076\pi$
$41$	−5.23607	−0.817736	−0.408868	+	0.912593i	$0.634076\pi$
$42$	0	0
$43$	−2.61803	−0.399246	−0.199623	−	0.979873i	$-0.563972\pi$
$43$	−2.61803	−0.399246	−0.199623	+	0.979873i	$0.563972\pi$
$44$	0	0
$45$	−4.23607	−0.631476
$46$	0	0
$47$	−4.76393	−0.694891	−0.347445	−	0.937700i	$-0.612951\pi$
$47$	−4.76393	−0.694891	−0.347445	+	0.937700i	$0.612951\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.85410	0.679710
$52$	0	0
$53$	2.61803	0.359615	0.179807	−	0.983702i	$-0.442453\pi$
$53$	2.61803	0.359615	0.179807	+	0.983702i	$0.442453\pi$
$54$	0	0
$55$	1.61803	0.218176
$56$	0	0
$57$	−0.236068	−0.0312680
$58$	0	0
$59$	1.23607	0.160922	0.0804612	−	0.996758i	$-0.474361\pi$
$59$	1.23607	0.160922	0.0804612	+	0.996758i	$0.474361\pi$
$60$	0	0
$61$	−6.14590	−0.786902	−0.393451	−	0.919346i	$-0.628719\pi$
$61$	−6.14590	−0.786902	−0.393451	+	0.919346i	$0.628719\pi$
$62$	0	0
$63$	−2.61803	−0.329841
$64$	0	0
$65$	1.61803	0.200692
$66$	0	0
$67$	13.0902	1.59922	0.799609	−	0.600520i	$-0.205040\pi$
$67$	13.0902	1.59922	0.799609	+	0.600520i	$0.205040\pi$
$68$	0	0
$69$	−3.52786	−0.424705
$70$	0	0
$71$	−12.5623	−1.49087	−0.745436	−	0.666578i	$-0.767759\pi$
$71$	−12.5623	−1.49087	−0.745436	+	0.666578i	$0.767759\pi$
$72$	0	0
$73$	4.47214	0.523424	0.261712	−	0.965146i	$-0.415713\pi$
$73$	4.47214	0.523424	0.261712	+	0.965146i	$0.415713\pi$
$74$	0	0
$75$	1.47214	0.169988
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−15.0344	−1.69151	−0.845753	−	0.533574i	$-0.820849\pi$
$79$	−15.0344	−1.69151	−0.845753	+	0.533574i	$0.820849\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	0.381966	0.0419262	0.0209631	−	0.999780i	$-0.493327\pi$
$83$	0.381966	0.0419262	0.0209631	+	0.999780i	$0.493327\pi$
$84$	0	0
$85$	−12.7082	−1.37840
$86$	0	0
$87$	0.291796	0.0312838
$88$	0	0
$89$	−18.6180	−1.97351	−0.986754	−	0.162225i	$-0.948133\pi$
$89$	−18.6180	−1.97351	−0.986754	+	0.162225i	$0.948133\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	1.70820	0.177132
$94$	0	0
$95$	0.618034	0.0634089
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−2.61803	−0.263122
$100$	0	0
$101$	5.61803	0.559015	0.279508	−	0.960143i	$-0.409829\pi$
$101$	5.61803	0.559015	0.279508	+	0.960143i	$0.409829\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	−10.6180	−1.02648	−0.513242	−	0.858244i	$-0.671556\pi$
$107$	−10.6180	−1.02648	−0.513242	+	0.858244i	$0.671556\pi$
$108$	0	0
$109$	13.3262	1.27642	0.638211	−	0.769861i	$-0.279675\pi$
$109$	13.3262	1.27642	0.638211	+	0.769861i	$0.279675\pi$
$110$	0	0
$111$	−7.23607	−0.686817
$112$	0	0
$113$	−3.14590	−0.295941	−0.147971	−	0.988992i	$-0.547274\pi$
$113$	−3.14590	−0.295941	−0.147971	+	0.988992i	$0.547274\pi$
$114$	0	0
$115$	9.23607	0.861268
$116$	0	0
$117$	−2.61803	−0.242037
$118$	0	0
$119$	−7.85410	−0.719984
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.23607	0.291786
$124$	0	0
$125$	−11.9443	−1.06833
$126$	0	0
$127$	−6.85410	−0.608203	−0.304102	−	0.952640i	$-0.598356\pi$
$127$	−6.85410	−0.608203	−0.304102	+	0.952640i	$0.598356\pi$
$128$	0	0
$129$	1.61803	0.142460
$130$	0	0
$131$	9.52786	0.832453	0.416227	−	0.909261i	$-0.363352\pi$
$131$	9.52786	0.832453	0.416227	+	0.909261i	$0.363352\pi$
$132$	0	0
$133$	0.381966	0.0331207
$134$	0	0
$135$	5.61803	0.483523
$136$	0	0
$137$	10.6525	0.910102	0.455051	−	0.890465i	$-0.349621\pi$
$137$	10.6525	0.910102	0.455051	+	0.890465i	$0.349621\pi$
$138$	0	0
$139$	−23.4164	−1.98615	−0.993077	−	0.117466i	$-0.962523\pi$
$139$	−23.4164	−1.98615	−0.993077	+	0.117466i	$0.962523\pi$
$140$	0	0
$141$	2.94427	0.247952
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−0.763932	−0.0634411
$146$	0	0
$147$	−0.618034	−0.0509746
$148$	0	0
$149$	−14.2705	−1.16909	−0.584543	−	0.811363i	$-0.698726\pi$
$149$	−14.2705	−1.16909	−0.584543	+	0.811363i	$0.698726\pi$
$150$	0	0
$151$	−13.2361	−1.07714	−0.538568	−	0.842582i	$-0.681034\pi$
$151$	−13.2361	−1.07714	−0.538568	+	0.842582i	$0.681034\pi$
$152$	0	0
$153$	20.5623	1.66236
$154$	0	0
$155$	−4.47214	−0.359211
$156$	0	0
$157$	4.76393	0.380203	0.190102	−	0.981764i	$-0.439118\pi$
$157$	4.76393	0.380203	0.190102	+	0.981764i	$0.439118\pi$
$158$	0	0
$159$	−1.61803	−0.128318
$160$	0	0
$161$	5.70820	0.449869
$162$	0	0
$163$	0.854102	0.0668984	0.0334492	−	0.999440i	$-0.489351\pi$
$163$	0.854102	0.0668984	0.0334492	+	0.999440i	$0.489351\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	−25.0902	−1.94154	−0.970768	−	0.240020i	$-0.922846\pi$
$167$	−25.0902	−1.94154	−0.970768	+	0.240020i	$0.922846\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−13.7984	−1.04907	−0.524535	−	0.851389i	$-0.675761\pi$
$173$	−13.7984	−1.04907	−0.524535	+	0.851389i	$0.675761\pi$
$174$	0	0
$175$	−2.38197	−0.180060
$176$	0	0
$177$	−0.763932	−0.0574206
$178$	0	0
$179$	0.763932	0.0570990	0.0285495	−	0.999592i	$-0.490911\pi$
$179$	0.763932	0.0570990	0.0285495	+	0.999592i	$0.490911\pi$
$180$	0	0
$181$	10.4721	0.778388	0.389194	−	0.921156i	$-0.372754\pi$
$181$	10.4721	0.778388	0.389194	+	0.921156i	$0.372754\pi$
$182$	0	0
$183$	3.79837	0.280784
$184$	0	0
$185$	18.9443	1.39281
$186$	0	0
$187$	−7.85410	−0.574349
$188$	0	0
$189$	3.47214	0.252561
$190$	0	0
$191$	6.65248	0.481356	0.240678	−	0.970605i	$-0.422630\pi$
$191$	6.65248	0.481356	0.240678	+	0.970605i	$0.422630\pi$
$192$	0	0
$193$	−4.90983	−0.353417	−0.176709	−	0.984263i	$-0.556545\pi$
$193$	−4.90983	−0.353417	−0.176709	+	0.984263i	$0.556545\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	8.14590	0.580371	0.290186	−	0.956970i	$-0.406283\pi$
$197$	8.14590	0.580371	0.290186	+	0.956970i	$0.406283\pi$
$198$	0	0
$199$	−9.79837	−0.694588	−0.347294	−	0.937756i	$-0.612899\pi$
$199$	−9.79837	−0.694588	−0.347294	+	0.937756i	$0.612899\pi$
$200$	0	0
$201$	−8.09017	−0.570637
$202$	0	0
$203$	−0.472136	−0.0331374
$204$	0	0
$205$	−8.47214	−0.591720
$206$	0	0
$207$	−14.9443	−1.03870
$208$	0	0
$209$	0.381966	0.0264211
$210$	0	0
$211$	−10.3262	−0.710888	−0.355444	−	0.934698i	$-0.615670\pi$
$211$	−10.3262	−0.710888	−0.355444	+	0.934698i	$0.615670\pi$
$212$	0	0
$213$	7.76393	0.531976
$214$	0	0
$215$	−4.23607	−0.288897
$216$	0	0
$217$	−2.76393	−0.187628
$218$	0	0
$219$	−2.76393	−0.186769
$220$	0	0
$221$	−7.85410	−0.528324
$222$	0	0
$223$	−5.23607	−0.350633	−0.175317	−	0.984512i	$-0.556095\pi$
$223$	−5.23607	−0.350633	−0.175317	+	0.984512i	$0.556095\pi$
$224$	0	0
$225$	6.23607	0.415738
$226$	0	0
$227$	25.5623	1.69663	0.848315	−	0.529492i	$-0.177617\pi$
$227$	25.5623	1.69663	0.848315	+	0.529492i	$0.177617\pi$
$228$	0	0
$229$	−10.5623	−0.697977	−0.348988	−	0.937127i	$-0.613475\pi$
$229$	−10.5623	−0.697977	−0.348988	+	0.937127i	$0.613475\pi$
$230$	0	0
$231$	−0.618034	−0.0406637
$232$	0	0
$233$	19.7082	1.29113	0.645564	−	0.763706i	$-0.276622\pi$
$233$	19.7082	1.29113	0.645564	+	0.763706i	$0.276622\pi$
$234$	0	0
$235$	−7.70820	−0.502828
$236$	0	0
$237$	9.29180	0.603567
$238$	0	0
$239$	6.65248	0.430313	0.215156	−	0.976580i	$-0.430974\pi$
$239$	6.65248	0.430313	0.215156	+	0.976580i	$0.430974\pi$
$240$	0	0
$241$	6.18034	0.398111	0.199055	−	0.979988i	$-0.436213\pi$
$241$	6.18034	0.398111	0.199055	+	0.979988i	$0.436213\pi$
$242$	0	0
$243$	−13.9443	−0.894525
$244$	0	0
$245$	1.61803	0.103372
$246$	0	0
$247$	0.381966	0.0243039
$248$	0	0
$249$	−0.236068	−0.0149602
$250$	0	0
$251$	−1.14590	−0.0723284	−0.0361642	−	0.999346i	$-0.511514\pi$
$251$	−1.14590	−0.0723284	−0.0361642	+	0.999346i	$0.511514\pi$
$252$	0	0
$253$	5.70820	0.358872
$254$	0	0
$255$	7.85410	0.491843
$256$	0	0
$257$	19.1246	1.19296	0.596480	−	0.802628i	$-0.296565\pi$
$257$	19.1246	1.19296	0.596480	+	0.802628i	$0.296565\pi$
$258$	0	0
$259$	11.7082	0.727512
$260$	0	0
$261$	1.23607	0.0765107
$262$	0	0
$263$	−25.1459	−1.55056	−0.775281	−	0.631616i	$-0.782392\pi$
$263$	−25.1459	−1.55056	−0.775281	+	0.631616i	$0.782392\pi$
$264$	0	0
$265$	4.23607	0.260220
$266$	0	0
$267$	11.5066	0.704191
$268$	0	0
$269$	−3.52786	−0.215098	−0.107549	−	0.994200i	$-0.534300\pi$
$269$	−3.52786	−0.215098	−0.107549	+	0.994200i	$0.534300\pi$
$270$	0	0
$271$	−14.0344	−0.852532	−0.426266	−	0.904598i	$-0.640171\pi$
$271$	−14.0344	−0.852532	−0.426266	+	0.904598i	$0.640171\pi$
$272$	0	0
$273$	−0.618034	−0.0374051
$274$	0	0
$275$	−2.38197	−0.143638
$276$	0	0
$277$	5.70820	0.342973	0.171486	−	0.985186i	$-0.445143\pi$
$277$	5.70820	0.342973	0.171486	+	0.985186i	$0.445143\pi$
$278$	0	0
$279$	7.23607	0.433212
$280$	0	0
$281$	−28.6869	−1.71132	−0.855659	−	0.517540i	$-0.826848\pi$
$281$	−28.6869	−1.71132	−0.855659	+	0.517540i	$0.826848\pi$
$282$	0	0
$283$	5.88854	0.350038	0.175019	−	0.984565i	$-0.444001\pi$
$283$	5.88854	0.350038	0.175019	+	0.984565i	$0.444001\pi$
$284$	0	0
$285$	−0.381966	−0.0226257
$286$	0	0
$287$	−5.23607	−0.309075
$288$	0	0
$289$	44.6869	2.62864
$290$	0	0
$291$	1.23607	0.0724596
$292$	0	0
$293$	21.1246	1.23411	0.617056	−	0.786919i	$-0.288325\pi$
$293$	21.1246	1.23411	0.617056	+	0.786919i	$0.288325\pi$
$294$	0	0
$295$	2.00000	0.116445
$296$	0	0
$297$	3.47214	0.201474
$298$	0	0
$299$	5.70820	0.330114
$300$	0	0
$301$	−2.61803	−0.150901
$302$	0	0
$303$	−3.47214	−0.199469
$304$	0	0
$305$	−9.94427	−0.569407
$306$	0	0
$307$	−12.3262	−0.703496	−0.351748	−	0.936095i	$-0.614413\pi$
$307$	−12.3262	−0.703496	−0.351748	+	0.936095i	$0.614413\pi$
$308$	0	0
$309$	2.47214	0.140635
$310$	0	0
$311$	−4.85410	−0.275251	−0.137625	−	0.990484i	$-0.543947\pi$
$311$	−4.85410	−0.275251	−0.137625	+	0.990484i	$0.543947\pi$
$312$	0	0
$313$	18.7639	1.06060	0.530300	−	0.847810i	$-0.322079\pi$
$313$	18.7639	1.06060	0.530300	+	0.847810i	$0.322079\pi$
$314$	0	0
$315$	−4.23607	−0.238675
$316$	0	0
$317$	−21.2361	−1.19274	−0.596368	−	0.802711i	$-0.703390\pi$
$317$	−21.2361	−1.19274	−0.596368	+	0.802711i	$0.703390\pi$
$318$	0	0
$319$	−0.472136	−0.0264345
$320$	0	0
$321$	6.56231	0.366272
$322$	0	0
$323$	−3.00000	−0.166924
$324$	0	0
$325$	−2.38197	−0.132128
$326$	0	0
$327$	−8.23607	−0.455456
$328$	0	0
$329$	−4.76393	−0.262644
$330$	0	0
$331$	−7.32624	−0.402686	−0.201343	−	0.979521i	$-0.564531\pi$
$331$	−7.32624	−0.402686	−0.201343	+	0.979521i	$0.564531\pi$
$332$	0	0
$333$	−30.6525	−1.67975
$334$	0	0
$335$	21.1803	1.15721
$336$	0	0
$337$	−8.18034	−0.445612	−0.222806	−	0.974863i	$-0.571522\pi$
$337$	−8.18034	−0.445612	−0.222806	+	0.974863i	$0.571522\pi$
$338$	0	0
$339$	1.94427	0.105598
$340$	0	0
$341$	−2.76393	−0.149675
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−5.70820	−0.307319
$346$	0	0
$347$	−9.03444	−0.484994	−0.242497	−	0.970152i	$-0.577966\pi$
$347$	−9.03444	−0.484994	−0.242497	+	0.970152i	$0.577966\pi$
$348$	0	0
$349$	29.7082	1.59024	0.795122	−	0.606450i	$-0.207407\pi$
$349$	29.7082	1.59024	0.795122	+	0.606450i	$0.207407\pi$
$350$	0	0
$351$	3.47214	0.185329
$352$	0	0
$353$	−34.9443	−1.85990	−0.929948	−	0.367691i	$-0.880148\pi$
$353$	−34.9443	−1.85990	−0.929948	+	0.367691i	$0.880148\pi$
$354$	0	0
$355$	−20.3262	−1.07880
$356$	0	0
$357$	4.85410	0.256906
$358$	0	0
$359$	−25.2361	−1.33191	−0.665954	−	0.745992i	$-0.731975\pi$
$359$	−25.2361	−1.33191	−0.665954	+	0.745992i	$0.731975\pi$
$360$	0	0
$361$	−18.8541	−0.992321
$362$	0	0
$363$	−0.618034	−0.0324384
$364$	0	0
$365$	7.23607	0.378753
$366$	0	0
$367$	33.4508	1.74612	0.873060	−	0.487613i	$-0.162132\pi$
$367$	33.4508	1.74612	0.873060	+	0.487613i	$0.162132\pi$
$368$	0	0
$369$	13.7082	0.713621
$370$	0	0
$371$	2.61803	0.135922
$372$	0	0
$373$	−0.583592	−0.0302173	−0.0151086	−	0.999886i	$-0.504809\pi$
$373$	−0.583592	−0.0302173	−0.0151086	+	0.999886i	$0.504809\pi$
$374$	0	0
$375$	7.38197	0.381203
$376$	0	0
$377$	−0.472136	−0.0243162
$378$	0	0
$379$	26.3262	1.35229	0.676144	−	0.736769i	$-0.263650\pi$
$379$	26.3262	1.35229	0.676144	+	0.736769i	$0.263650\pi$
$380$	0	0
$381$	4.23607	0.217020
$382$	0	0
$383$	−15.8885	−0.811867	−0.405933	−	0.913903i	$-0.633053\pi$
$383$	−15.8885	−0.811867	−0.405933	+	0.913903i	$0.633053\pi$
$384$	0	0
$385$	1.61803	0.0824626
$386$	0	0
$387$	6.85410	0.348414
$388$	0	0
$389$	−14.8541	−0.753133	−0.376566	−	0.926390i	$-0.622895\pi$
$389$	−14.8541	−0.753133	−0.376566	+	0.926390i	$0.622895\pi$
$390$	0	0
$391$	−44.8328	−2.26729
$392$	0	0
$393$	−5.88854	−0.297038
$394$	0	0
$395$	−24.3262	−1.22399
$396$	0	0
$397$	−34.2148	−1.71719	−0.858595	−	0.512654i	$-0.828662\pi$
$397$	−34.2148	−1.71719	−0.858595	+	0.512654i	$0.828662\pi$
$398$	0	0
$399$	−0.236068	−0.0118182
$400$	0	0
$401$	−13.7082	−0.684555	−0.342278	−	0.939599i	$-0.611198\pi$
$401$	−13.7082	−0.684555	−0.342278	+	0.939599i	$0.611198\pi$
$402$	0	0
$403$	−2.76393	−0.137681
$404$	0	0
$405$	9.23607	0.458944
$406$	0	0
$407$	11.7082	0.580354
$408$	0	0
$409$	−22.6525	−1.12009	−0.560046	−	0.828461i	$-0.689217\pi$
$409$	−22.6525	−1.12009	−0.560046	+	0.828461i	$0.689217\pi$
$410$	0	0
$411$	−6.58359	−0.324745
$412$	0	0
$413$	1.23607	0.0608229
$414$	0	0
$415$	0.618034	0.0303381
$416$	0	0
$417$	14.4721	0.708704
$418$	0	0
$419$	5.56231	0.271736	0.135868	−	0.990727i	$-0.456618\pi$
$419$	5.56231	0.271736	0.135868	+	0.990727i	$0.456618\pi$
$420$	0	0
$421$	32.4721	1.58260	0.791298	−	0.611431i	$-0.209406\pi$
$421$	32.4721	1.58260	0.791298	+	0.611431i	$0.209406\pi$
$422$	0	0
$423$	12.4721	0.606416
$424$	0	0
$425$	18.7082	0.907481
$426$	0	0
$427$	−6.14590	−0.297421
$428$	0	0
$429$	−0.618034	−0.0298390
$430$	0	0
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	0	0
$433$	−8.94427	−0.429834	−0.214917	−	0.976632i	$-0.568948\pi$
$433$	−8.94427	−0.429834	−0.214917	+	0.976632i	$0.568948\pi$
$434$	0	0
$435$	0.472136	0.0226372
$436$	0	0
$437$	2.18034	0.104300
$438$	0	0
$439$	−21.5967	−1.03076	−0.515378	−	0.856963i	$-0.672349\pi$
$439$	−21.5967	−1.03076	−0.515378	+	0.856963i	$0.672349\pi$
$440$	0	0
$441$	−2.61803	−0.124668
$442$	0	0
$443$	25.3050	1.20227	0.601137	−	0.799146i	$-0.294714\pi$
$443$	25.3050	1.20227	0.601137	+	0.799146i	$0.294714\pi$
$444$	0	0
$445$	−30.1246	−1.42804
$446$	0	0
$447$	8.81966	0.417156
$448$	0	0
$449$	−14.3607	−0.677722	−0.338861	−	0.940836i	$-0.610042\pi$
$449$	−14.3607	−0.677722	−0.338861	+	0.940836i	$0.610042\pi$
$450$	0	0
$451$	−5.23607	−0.246557
$452$	0	0
$453$	8.18034	0.384346
$454$	0	0
$455$	1.61803	0.0758546
$456$	0	0
$457$	−22.7426	−1.06386	−0.531928	−	0.846790i	$-0.678532\pi$
$457$	−22.7426	−1.06386	−0.531928	+	0.846790i	$0.678532\pi$
$458$	0	0
$459$	−27.2705	−1.27288
$460$	0	0
$461$	2.76393	0.128729	0.0643646	−	0.997926i	$-0.479498\pi$
$461$	2.76393	0.128729	0.0643646	+	0.997926i	$0.479498\pi$
$462$	0	0
$463$	−16.9443	−0.787467	−0.393734	−	0.919225i	$-0.628817\pi$
$463$	−16.9443	−0.787467	−0.393734	+	0.919225i	$0.628817\pi$
$464$	0	0
$465$	2.76393	0.128174
$466$	0	0
$467$	−16.9098	−0.782494	−0.391247	−	0.920286i	$-0.627956\pi$
$467$	−16.9098	−0.782494	−0.391247	+	0.920286i	$0.627956\pi$
$468$	0	0
$469$	13.0902	0.604448
$470$	0	0
$471$	−2.94427	−0.135665
$472$	0	0
$473$	−2.61803	−0.120377
$474$	0	0
$475$	−0.909830	−0.0417459
$476$	0	0
$477$	−6.85410	−0.313828
$478$	0	0
$479$	30.1591	1.37800	0.689001	−	0.724760i	$-0.258049\pi$
$479$	30.1591	1.37800	0.689001	+	0.724760i	$0.258049\pi$
$480$	0	0
$481$	11.7082	0.533848
$482$	0	0
$483$	−3.52786	−0.160523
$484$	0	0
$485$	−3.23607	−0.146942
$486$	0	0
$487$	12.9443	0.586561	0.293280	−	0.956026i	$-0.405253\pi$
$487$	12.9443	0.586561	0.293280	+	0.956026i	$0.405253\pi$
$488$	0	0
$489$	−0.527864	−0.0238708
$490$	0	0
$491$	9.52786	0.429986	0.214993	−	0.976616i	$-0.431027\pi$
$491$	9.52786	0.429986	0.214993	+	0.976616i	$0.431027\pi$
$492$	0	0
$493$	3.70820	0.167009
$494$	0	0
$495$	−4.23607	−0.190397
$496$	0	0
$497$	−12.5623	−0.563496
$498$	0	0
$499$	−35.6869	−1.59757	−0.798783	−	0.601619i	$-0.794522\pi$
$499$	−35.6869	−1.59757	−0.798783	+	0.601619i	$0.794522\pi$
$500$	0	0
$501$	15.5066	0.692783
$502$	0	0
$503$	−13.0557	−0.582126	−0.291063	−	0.956704i	$-0.594009\pi$
$503$	−13.0557	−0.582126	−0.291063	+	0.956704i	$0.594009\pi$
$504$	0	0
$505$	9.09017	0.404507
$506$	0	0
$507$	−0.618034	−0.0274479
$508$	0	0
$509$	−22.3607	−0.991120	−0.495560	−	0.868574i	$-0.665037\pi$
$509$	−22.3607	−0.991120	−0.495560	+	0.868574i	$0.665037\pi$
$510$	0	0
$511$	4.47214	0.197836
$512$	0	0
$513$	1.32624	0.0585548
$514$	0	0
$515$	−6.47214	−0.285196
$516$	0	0
$517$	−4.76393	−0.209517
$518$	0	0
$519$	8.52786	0.374332
$520$	0	0
$521$	11.1246	0.487378	0.243689	−	0.969853i	$-0.421642\pi$
$521$	11.1246	0.487378	0.243689	+	0.969853i	$0.421642\pi$
$522$	0	0
$523$	15.2361	0.666227	0.333113	−	0.942887i	$-0.391901\pi$
$523$	15.2361	0.666227	0.333113	+	0.942887i	$0.391901\pi$
$524$	0	0
$525$	1.47214	0.0642493
$526$	0	0
$527$	21.7082	0.945624
$528$	0	0
$529$	9.58359	0.416678
$530$	0	0
$531$	−3.23607	−0.140433
$532$	0	0
$533$	−5.23607	−0.226799
$534$	0	0
$535$	−17.1803	−0.742771
$536$	0	0
$537$	−0.472136	−0.0203742
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	0	0
$543$	−6.47214	−0.277746
$544$	0	0
$545$	21.5623	0.923628
$546$	0	0
$547$	−17.5279	−0.749437	−0.374719	−	0.927139i	$-0.622261\pi$
$547$	−17.5279	−0.749437	−0.374719	+	0.927139i	$0.622261\pi$
$548$	0	0
$549$	16.0902	0.686712
$550$	0	0
$551$	−0.180340	−0.00768274
$552$	0	0
$553$	−15.0344	−0.639329
$554$	0	0
$555$	−11.7082	−0.496986
$556$	0	0
$557$	−6.27051	−0.265690	−0.132845	−	0.991137i	$-0.542411\pi$
$557$	−6.27051	−0.265690	−0.132845	+	0.991137i	$0.542411\pi$
$558$	0	0
$559$	−2.61803	−0.110731
$560$	0	0
$561$	4.85410	0.204940
$562$	0	0
$563$	17.8197	0.751009	0.375505	−	0.926821i	$-0.377469\pi$
$563$	17.8197	0.751009	0.375505	+	0.926821i	$0.377469\pi$
$564$	0	0
$565$	−5.09017	−0.214145
$566$	0	0
$567$	5.70820	0.239722
$568$	0	0
$569$	−0.875388	−0.0366982	−0.0183491	−	0.999832i	$-0.505841\pi$
$569$	−0.875388	−0.0366982	−0.0183491	+	0.999832i	$0.505841\pi$
$570$	0	0
$571$	−9.43769	−0.394955	−0.197478	−	0.980307i	$-0.563275\pi$
$571$	−9.43769	−0.394955	−0.197478	+	0.980307i	$0.563275\pi$
$572$	0	0
$573$	−4.11146	−0.171759
$574$	0	0
$575$	−13.5967	−0.567024
$576$	0	0
$577$	−16.4377	−0.684310	−0.342155	−	0.939643i	$-0.611157\pi$
$577$	−16.4377	−0.684310	−0.342155	+	0.939643i	$0.611157\pi$
$578$	0	0
$579$	3.03444	0.126107
$580$	0	0
$581$	0.381966	0.0158466
$582$	0	0
$583$	2.61803	0.108428
$584$	0	0
$585$	−4.23607	−0.175140
$586$	0	0
$587$	−21.5967	−0.891393	−0.445697	−	0.895184i	$-0.647044\pi$
$587$	−21.5967	−0.891393	−0.445697	+	0.895184i	$0.647044\pi$
$588$	0	0
$589$	−1.05573	−0.0435005
$590$	0	0
$591$	−5.03444	−0.207089
$592$	0	0
$593$	−13.8197	−0.567505	−0.283753	−	0.958897i	$-0.591579\pi$
$593$	−13.8197	−0.567505	−0.283753	+	0.958897i	$0.591579\pi$
$594$	0	0
$595$	−12.7082	−0.520986
$596$	0	0
$597$	6.05573	0.247844
$598$	0	0
$599$	−22.5410	−0.921001	−0.460501	−	0.887659i	$-0.652330\pi$
$599$	−22.5410	−0.921001	−0.460501	+	0.887659i	$0.652330\pi$
$600$	0	0
$601$	−2.61803	−0.106792	−0.0533959	−	0.998573i	$-0.517005\pi$
$601$	−2.61803	−0.106792	−0.0533959	+	0.998573i	$0.517005\pi$
$602$	0	0
$603$	−34.2705	−1.39560
$604$	0	0
$605$	1.61803	0.0657824
$606$	0	0
$607$	18.4721	0.749761	0.374880	−	0.927073i	$-0.377684\pi$
$607$	18.4721	0.749761	0.374880	+	0.927073i	$0.377684\pi$
$608$	0	0
$609$	0.291796	0.0118242
$610$	0	0
$611$	−4.76393	−0.192728
$612$	0	0
$613$	−8.83282	−0.356754	−0.178377	−	0.983962i	$-0.557085\pi$
$613$	−8.83282	−0.356754	−0.178377	+	0.983962i	$0.557085\pi$
$614$	0	0
$615$	5.23607	0.211139
$616$	0	0
$617$	24.3607	0.980724	0.490362	−	0.871519i	$-0.336865\pi$
$617$	24.3607	0.980724	0.490362	+	0.871519i	$0.336865\pi$
$618$	0	0
$619$	−29.2361	−1.17510	−0.587548	−	0.809189i	$-0.699907\pi$
$619$	−29.2361	−1.17510	−0.587548	+	0.809189i	$0.699907\pi$
$620$	0	0
$621$	19.8197	0.795336
$622$	0	0
$623$	−18.6180	−0.745916
$624$	0	0
$625$	−7.41641	−0.296656
$626$	0	0
$627$	−0.236068	−0.00942765
$628$	0	0
$629$	−91.9574	−3.66658
$630$	0	0
$631$	36.0344	1.43451	0.717254	−	0.696812i	$-0.245399\pi$
$631$	36.0344	1.43451	0.717254	+	0.696812i	$0.245399\pi$
$632$	0	0
$633$	6.38197	0.253660
$634$	0	0
$635$	−11.0902	−0.440100
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	32.8885	1.30105
$640$	0	0
$641$	22.7984	0.900482	0.450241	−	0.892907i	$-0.351338\pi$
$641$	22.7984	0.900482	0.450241	+	0.892907i	$0.351338\pi$
$642$	0	0
$643$	−27.4164	−1.08120	−0.540599	−	0.841281i	$-0.681802\pi$
$643$	−27.4164	−1.08120	−0.540599	+	0.841281i	$0.681802\pi$
$644$	0	0
$645$	2.61803	0.103085
$646$	0	0
$647$	−5.88854	−0.231503	−0.115751	−	0.993278i	$-0.536928\pi$
$647$	−5.88854	−0.231503	−0.115751	+	0.993278i	$0.536928\pi$
$648$	0	0
$649$	1.23607	0.0485199
$650$	0	0
$651$	1.70820	0.0669498
$652$	0	0
$653$	40.8673	1.59926	0.799630	−	0.600493i	$-0.205029\pi$
$653$	40.8673	1.59926	0.799630	+	0.600493i	$0.205029\pi$
$654$	0	0
$655$	15.4164	0.602369
$656$	0	0
$657$	−11.7082	−0.456781
$658$	0	0
$659$	20.5066	0.798823	0.399411	−	0.916772i	$-0.369215\pi$
$659$	20.5066	0.798823	0.399411	+	0.916772i	$0.369215\pi$
$660$	0	0
$661$	42.1591	1.63980	0.819899	−	0.572509i	$-0.194030\pi$
$661$	42.1591	1.63980	0.819899	+	0.572509i	$0.194030\pi$
$662$	0	0
$663$	4.85410	0.188518
$664$	0	0
$665$	0.618034	0.0239663
$666$	0	0
$667$	−2.69505	−0.104353
$668$	0	0
$669$	3.23607	0.125114
$670$	0	0
$671$	−6.14590	−0.237260
$672$	0	0
$673$	24.1803	0.932084	0.466042	−	0.884763i	$-0.345680\pi$
$673$	24.1803	0.932084	0.466042	+	0.884763i	$0.345680\pi$
$674$	0	0
$675$	−8.27051	−0.318332
$676$	0	0
$677$	20.0344	0.769986	0.384993	−	0.922919i	$-0.374204\pi$
$677$	20.0344	0.769986	0.384993	+	0.922919i	$0.374204\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	−15.7984	−0.605395
$682$	0	0
$683$	−33.6738	−1.28849	−0.644245	−	0.764819i	$-0.722828\pi$
$683$	−33.6738	−1.28849	−0.644245	+	0.764819i	$0.722828\pi$
$684$	0	0
$685$	17.2361	0.658556
$686$	0	0
$687$	6.52786	0.249054
$688$	0	0
$689$	2.61803	0.0997392
$690$	0	0
$691$	12.8328	0.488183	0.244092	−	0.969752i	$-0.421510\pi$
$691$	12.8328	0.488183	0.244092	+	0.969752i	$0.421510\pi$
$692$	0	0
$693$	−2.61803	−0.0994509
$694$	0	0
$695$	−37.8885	−1.43719
$696$	0	0
$697$	41.1246	1.55771
$698$	0	0
$699$	−12.1803	−0.460703
$700$	0	0
$701$	12.4721	0.471066	0.235533	−	0.971866i	$-0.424316\pi$
$701$	12.4721	0.471066	0.235533	+	0.971866i	$0.424316\pi$
$702$	0	0
$703$	4.47214	0.168670
$704$	0	0
$705$	4.76393	0.179420
$706$	0	0
$707$	5.61803	0.211288
$708$	0	0
$709$	−36.2492	−1.36137	−0.680684	−	0.732577i	$-0.738317\pi$
$709$	−36.2492	−1.36137	−0.680684	+	0.732577i	$0.738317\pi$
$710$	0	0
$711$	39.3607	1.47614
$712$	0	0
$713$	−15.7771	−0.590857
$714$	0	0
$715$	1.61803	0.0605110
$716$	0	0
$717$	−4.11146	−0.153545
$718$	0	0
$719$	−10.3262	−0.385104	−0.192552	−	0.981287i	$-0.561676\pi$
$719$	−10.3262	−0.385104	−0.192552	+	0.981287i	$0.561676\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	−3.81966	−0.142055
$724$	0	0
$725$	1.12461	0.0417670
$726$	0	0
$727$	−37.0344	−1.37353	−0.686766	−	0.726879i	$-0.740970\pi$
$727$	−37.0344	−1.37353	−0.686766	+	0.726879i	$0.740970\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	20.5623	0.760524
$732$	0	0
$733$	−23.1246	−0.854127	−0.427064	−	0.904222i	$-0.640452\pi$
$733$	−23.1246	−0.854127	−0.427064	+	0.904222i	$0.640452\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	13.0902	0.482183
$738$	0	0
$739$	−5.23607	−0.192612	−0.0963059	−	0.995352i	$-0.530703\pi$
$739$	−5.23607	−0.192612	−0.0963059	+	0.995352i	$0.530703\pi$
$740$	0	0
$741$	−0.236068	−0.00867217
$742$	0	0
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	0	0
$745$	−23.0902	−0.845958
$746$	0	0
$747$	−1.00000	−0.0365881
$748$	0	0
$749$	−10.6180	−0.387975
$750$	0	0
$751$	−22.3607	−0.815953	−0.407976	−	0.912992i	$-0.633765\pi$
$751$	−22.3607	−0.815953	−0.407976	+	0.912992i	$0.633765\pi$
$752$	0	0
$753$	0.708204	0.0258084
$754$	0	0
$755$	−21.4164	−0.779423
$756$	0	0
$757$	5.72949	0.208242	0.104121	−	0.994565i	$-0.466797\pi$
$757$	5.72949	0.208242	0.104121	+	0.994565i	$0.466797\pi$
$758$	0	0
$759$	−3.52786	−0.128053
$760$	0	0
$761$	−20.3607	−0.738074	−0.369037	−	0.929415i	$-0.620312\pi$
$761$	−20.3607	−0.738074	−0.369037	+	0.929415i	$0.620312\pi$
$762$	0	0
$763$	13.3262	0.482442
$764$	0	0
$765$	33.2705	1.20290
$766$	0	0
$767$	1.23607	0.0446318
$768$	0	0
$769$	50.0689	1.80553	0.902765	−	0.430134i	$-0.141534\pi$
$769$	50.0689	1.80553	0.902765	+	0.430134i	$0.141534\pi$
$770$	0	0
$771$	−11.8197	−0.425675
$772$	0	0
$773$	−14.2148	−0.511270	−0.255635	−	0.966773i	$-0.582285\pi$
$773$	−14.2148	−0.511270	−0.255635	+	0.966773i	$0.582285\pi$
$774$	0	0
$775$	6.58359	0.236490
$776$	0	0
$777$	−7.23607	−0.259592
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	−12.5623	−0.449515
$782$	0	0
$783$	−1.63932	−0.0585845
$784$	0	0
$785$	7.70820	0.275118
$786$	0	0
$787$	−32.9787	−1.17556	−0.587782	−	0.809019i	$-0.699999\pi$
$787$	−32.9787	−1.17556	−0.587782	+	0.809019i	$0.699999\pi$
$788$	0	0
$789$	15.5410	0.553275
$790$	0	0
$791$	−3.14590	−0.111855
$792$	0	0
$793$	−6.14590	−0.218247
$794$	0	0
$795$	−2.61803	−0.0928521
$796$	0	0
$797$	−15.4164	−0.546077	−0.273039	−	0.962003i	$-0.588029\pi$
$797$	−15.4164	−0.546077	−0.273039	+	0.962003i	$0.588029\pi$
$798$	0	0
$799$	37.4164	1.32370
$800$	0	0
$801$	48.7426	1.72224
$802$	0	0
$803$	4.47214	0.157818
$804$	0	0
$805$	9.23607	0.325529
$806$	0	0
$807$	2.18034	0.0767516
$808$	0	0
$809$	−19.1246	−0.672386	−0.336193	−	0.941793i	$-0.609139\pi$
$809$	−19.1246	−0.672386	−0.336193	+	0.941793i	$0.609139\pi$
$810$	0	0
$811$	29.5066	1.03612	0.518058	−	0.855345i	$-0.326655\pi$
$811$	29.5066	1.03612	0.518058	+	0.855345i	$0.326655\pi$
$812$	0	0
$813$	8.67376	0.304202
$814$	0	0
$815$	1.38197	0.0484082
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	0	0
$819$	−2.61803	−0.0914815
$820$	0	0
$821$	−16.5066	−0.576084	−0.288042	−	0.957618i	$-0.593004\pi$
$821$	−16.5066	−0.576084	−0.288042	+	0.957618i	$0.593004\pi$
$822$	0	0
$823$	8.36068	0.291435	0.145717	−	0.989326i	$-0.453451\pi$
$823$	8.36068	0.291435	0.145717	+	0.989326i	$0.453451\pi$
$824$	0	0
$825$	1.47214	0.0512532
$826$	0	0
$827$	20.7639	0.722033	0.361016	−	0.932559i	$-0.382430\pi$
$827$	20.7639	0.722033	0.361016	+	0.932559i	$0.382430\pi$
$828$	0	0
$829$	−23.0132	−0.799280	−0.399640	−	0.916672i	$-0.630865\pi$
$829$	−23.0132	−0.799280	−0.399640	+	0.916672i	$0.630865\pi$
$830$	0	0
$831$	−3.52786	−0.122380
$832$	0	0
$833$	−7.85410	−0.272129
$834$	0	0
$835$	−40.5967	−1.40491
$836$	0	0
$837$	−9.59675	−0.331712
$838$	0	0
$839$	2.76393	0.0954215	0.0477108	−	0.998861i	$-0.484807\pi$
$839$	2.76393	0.0954215	0.0477108	+	0.998861i	$0.484807\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	0	0
$843$	17.7295	0.610636
$844$	0	0
$845$	1.61803	0.0556621
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−3.63932	−0.124901
$850$	0	0
$851$	66.8328	2.29100
$852$	0	0
$853$	−7.41641	−0.253933	−0.126966	−	0.991907i	$-0.540524\pi$
$853$	−7.41641	−0.253933	−0.126966	+	0.991907i	$0.540524\pi$
$854$	0	0
$855$	−1.61803	−0.0553356
$856$	0	0
$857$	−4.96556	−0.169620	−0.0848101	−	0.996397i	$-0.527028\pi$
$857$	−4.96556	−0.169620	−0.0848101	+	0.996397i	$0.527028\pi$
$858$	0	0
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	0	0
$861$	3.23607	0.110285
$862$	0	0
$863$	−28.9443	−0.985274	−0.492637	−	0.870235i	$-0.663967\pi$
$863$	−28.9443	−0.985274	−0.492637	+	0.870235i	$0.663967\pi$
$864$	0	0
$865$	−22.3262	−0.759115
$866$	0	0
$867$	−27.6180	−0.937958
$868$	0	0
$869$	−15.0344	−0.510009
$870$	0	0
$871$	13.0902	0.443543
$872$	0	0
$873$	5.23607	0.177214
$874$	0	0
$875$	−11.9443	−0.403790
$876$	0	0
$877$	−5.09017	−0.171883	−0.0859414	−	0.996300i	$-0.527390\pi$
$877$	−5.09017	−0.171883	−0.0859414	+	0.996300i	$0.527390\pi$
$878$	0	0
$879$	−13.0557	−0.440359
$880$	0	0
$881$	30.7639	1.03646	0.518232	−	0.855240i	$-0.326591\pi$
$881$	30.7639	1.03646	0.518232	+	0.855240i	$0.326591\pi$
$882$	0	0
$883$	46.9443	1.57980	0.789900	−	0.613235i	$-0.210132\pi$
$883$	46.9443	1.57980	0.789900	+	0.613235i	$0.210132\pi$
$884$	0	0
$885$	−1.23607	−0.0415500
$886$	0	0
$887$	10.4721	0.351620	0.175810	−	0.984424i	$-0.443746\pi$
$887$	10.4721	0.351620	0.175810	+	0.984424i	$0.443746\pi$
$888$	0	0
$889$	−6.85410	−0.229879
$890$	0	0
$891$	5.70820	0.191232
$892$	0	0
$893$	−1.81966	−0.0608926
$894$	0	0
$895$	1.23607	0.0413172
$896$	0	0
$897$	−3.52786	−0.117792
$898$	0	0
$899$	1.30495	0.0435226
$900$	0	0
$901$	−20.5623	−0.685030
$902$	0	0
$903$	1.61803	0.0538448
$904$	0	0
$905$	16.9443	0.563247
$906$	0	0
$907$	36.5410	1.21332	0.606662	−	0.794960i	$-0.292508\pi$
$907$	36.5410	1.21332	0.606662	+	0.794960i	$0.292508\pi$
$908$	0	0
$909$	−14.7082	−0.487840
$910$	0	0
$911$	−14.5410	−0.481765	−0.240883	−	0.970554i	$-0.577437\pi$
$911$	−14.5410	−0.481765	−0.240883	+	0.970554i	$0.577437\pi$
$912$	0	0
$913$	0.381966	0.0126412
$914$	0	0
$915$	6.14590	0.203177
$916$	0	0
$917$	9.52786	0.314638
$918$	0	0
$919$	3.67376	0.121186	0.0605931	−	0.998163i	$-0.480701\pi$
$919$	3.67376	0.121186	0.0605931	+	0.998163i	$0.480701\pi$
$920$	0	0
$921$	7.61803	0.251023
$922$	0	0
$923$	−12.5623	−0.413493
$924$	0	0
$925$	−27.8885	−0.916970
$926$	0	0
$927$	10.4721	0.343950
$928$	0	0
$929$	15.2705	0.501009	0.250505	−	0.968115i	$-0.419403\pi$
$929$	15.2705	0.501009	0.250505	+	0.968115i	$0.419403\pi$
$930$	0	0
$931$	0.381966	0.0125184
$932$	0	0
$933$	3.00000	0.0982156
$934$	0	0
$935$	−12.7082	−0.415603
$936$	0	0
$937$	−33.1459	−1.08283	−0.541415	−	0.840756i	$-0.682111\pi$
$937$	−33.1459	−1.08283	−0.541415	+	0.840756i	$0.682111\pi$
$938$	0	0
$939$	−11.5967	−0.378446
$940$	0	0
$941$	19.8197	0.646102	0.323051	−	0.946381i	$-0.395291\pi$
$941$	19.8197	0.646102	0.323051	+	0.946381i	$0.395291\pi$
$942$	0	0
$943$	−29.8885	−0.973305
$944$	0	0
$945$	5.61803	0.182755
$946$	0	0
$947$	12.5066	0.406409	0.203205	−	0.979136i	$-0.434864\pi$
$947$	12.5066	0.406409	0.203205	+	0.979136i	$0.434864\pi$
$948$	0	0
$949$	4.47214	0.145172
$950$	0	0
$951$	13.1246	0.425595
$952$	0	0
$953$	40.2918	1.30518	0.652590	−	0.757712i	$-0.273683\pi$
$953$	40.2918	1.30518	0.652590	+	0.757712i	$0.273683\pi$
$954$	0	0
$955$	10.7639	0.348313
$956$	0	0
$957$	0.291796	0.00943243
$958$	0	0
$959$	10.6525	0.343986
$960$	0	0
$961$	−23.3607	−0.753570
$962$	0	0
$963$	27.7984	0.895790
$964$	0	0
$965$	−7.94427	−0.255735
$966$	0	0
$967$	−13.4164	−0.431443	−0.215721	−	0.976455i	$-0.569210\pi$
$967$	−13.4164	−0.431443	−0.215721	+	0.976455i	$0.569210\pi$
$968$	0	0
$969$	1.85410	0.0595623
$970$	0	0
$971$	−17.9098	−0.574754	−0.287377	−	0.957818i	$-0.592783\pi$
$971$	−17.9098	−0.574754	−0.287377	+	0.957818i	$0.592783\pi$
$972$	0	0
$973$	−23.4164	−0.750696
$974$	0	0
$975$	1.47214	0.0471461
$976$	0	0
$977$	−54.2492	−1.73559	−0.867793	−	0.496925i	$-0.834462\pi$
$977$	−54.2492	−1.73559	−0.867793	+	0.496925i	$0.834462\pi$
$978$	0	0
$979$	−18.6180	−0.595035
$980$	0	0
$981$	−34.8885	−1.11391
$982$	0	0
$983$	−50.2492	−1.60270	−0.801351	−	0.598195i	$-0.795885\pi$
$983$	−50.2492	−1.60270	−0.801351	+	0.598195i	$0.795885\pi$
$984$	0	0
$985$	13.1803	0.419961
$986$	0	0
$987$	2.94427	0.0937172
$988$	0	0
$989$	−14.9443	−0.475200
$990$	0	0
$991$	60.3607	1.91742	0.958710	−	0.284385i	$-0.0917895\pi$
$991$	60.3607	1.91742	0.958710	+	0.284385i	$0.0917895\pi$
$992$	0	0
$993$	4.52786	0.143687
$994$	0	0
$995$	−15.8541	−0.502609
$996$	0	0
$997$	−14.5623	−0.461193	−0.230596	−	0.973049i	$-0.574068\pi$
$997$	−14.5623	−0.461193	−0.230596	+	0.973049i	$0.574068\pi$
$998$	0	0
$999$	40.6525	1.28619

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.g.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.g.1.1	✓	2	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)

\(f(q)\)	\(=\)	\(q-0.618034 q^{3} +1.61803 q^{5} +1.00000 q^{7} -2.61803 q^{9} +O(q^{10})\)	\(q-0.618034 q^{3} +1.61803 q^{5} +1.00000 q^{7} -2.61803 q^{9} +1.00000 q^{11} +1.00000 q^{13} -1.00000 q^{15} -7.85410 q^{17} +0.381966 q^{19} -0.618034 q^{21} +5.70820 q^{23} -2.38197 q^{25} +3.47214 q^{27} -0.472136 q^{29} -2.76393 q^{31} -0.618034 q^{33} +1.61803 q^{35} +11.7082 q^{37} -0.618034 q^{39} -5.23607 q^{41} -2.61803 q^{43} -4.23607 q^{45} -4.76393 q^{47} +1.00000 q^{49} +4.85410 q^{51} +2.61803 q^{53} +1.61803 q^{55} -0.236068 q^{57} +1.23607 q^{59} -6.14590 q^{61} -2.61803 q^{63} +1.61803 q^{65} +13.0902 q^{67} -3.52786 q^{69} -12.5623 q^{71} +4.47214 q^{73} +1.47214 q^{75} +1.00000 q^{77} -15.0344 q^{79} +5.70820 q^{81} +0.381966 q^{83} -12.7082 q^{85} +0.291796 q^{87} -18.6180 q^{89} +1.00000 q^{91} +1.70820 q^{93} +0.618034 q^{95} -2.00000 q^{97} -2.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + q^5 + 2 * q^7 - 3 * q^9	\( 2 q + q^{3} + q^{5} + 2 q^{7} - 3 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} - 9 q^{17} + 3 q^{19} + q^{21} - 2 q^{23} - 7 q^{25} - 2 q^{27} + 8 q^{29} - 10 q^{31} + q^{33} + q^{35} + 10 q^{37} + q^{39} - 6 q^{41} - 3 q^{43} - 4 q^{45} - 14 q^{47} + 2 q^{49} + 3 q^{51} + 3 q^{53} + q^{55} + 4 q^{57} - 2 q^{59} - 19 q^{61} - 3 q^{63} + q^{65} + 15 q^{67} - 16 q^{69} - 5 q^{71} - 6 q^{75} + 2 q^{77} - q^{79} - 2 q^{81} + 3 q^{83} - 12 q^{85} + 14 q^{87} - 35 q^{89} + 2 q^{91} - 10 q^{93} - q^{95} - 4 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q + q^3 + q^5 + 2 * q^7 - 3 * q^9 + 2 * q^11 + 2 * q^13 - 2 * q^15 - 9 * q^17 + 3 * q^19 + q^21 - 2 * q^23 - 7 * q^25 - 2 * q^27 + 8 * q^29 - 10 * q^31 + q^33 + q^35 + 10 * q^37 + q^39 - 6 * q^41 - 3 * q^43 - 4 * q^45 - 14 * q^47 + 2 * q^49 + 3 * q^51 + 3 * q^53 + q^55 + 4 * q^57 - 2 * q^59 - 19 * q^61 - 3 * q^63 + q^65 + 15 * q^67 - 16 * q^69 - 5 * q^71 - 6 * q^75 + 2 * q^77 - q^79 - 2 * q^81 + 3 * q^83 - 12 * q^85 + 14 * q^87 - 35 * q^89 + 2 * q^91 - 10 * q^93 - q^95 - 4 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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