LMFDB - Embedded newform 8008.2.a.x.1.7

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 17 x^{10} + 79 x^{9} + 80 x^{8} - 536 x^{7} - 4 x^{6} + 1484 x^{5} - 682 x^{4} + \cdots - 64 $ x^12 - 4x^11 - 17x^10 + 79x^9 + 80x^8 - 536x^7 - 4x^6 + 1484x^5 - 682x^4 - 1431x^3 + 1069x^2 - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.818045$ 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)

$f(q)$	$=$	$q+0.818045 q^{3} -0.570018 q^{5} +1.00000 q^{7} -2.33080 q^{9} +O(q^{10})$	$q+0.818045 q^{3} -0.570018 q^{5} +1.00000 q^{7} -2.33080 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.466300 q^{15} -6.67104 q^{17} +1.44676 q^{19} +0.818045 q^{21} +5.53644 q^{23} -4.67508 q^{25} -4.36084 q^{27} -5.42048 q^{29} -1.22873 q^{31} +0.818045 q^{33} -0.570018 q^{35} -4.75597 q^{37} +0.818045 q^{39} +3.95779 q^{41} +8.80275 q^{43} +1.32860 q^{45} +7.01984 q^{47} +1.00000 q^{49} -5.45721 q^{51} +0.903831 q^{53} -0.570018 q^{55} +1.18351 q^{57} +7.67516 q^{59} +4.80619 q^{61} -2.33080 q^{63} -0.570018 q^{65} +12.1752 q^{67} +4.52905 q^{69} +2.78201 q^{71} +0.712954 q^{73} -3.82442 q^{75} +1.00000 q^{77} +7.71952 q^{79} +3.42505 q^{81} +8.30702 q^{83} +3.80261 q^{85} -4.43420 q^{87} +11.1244 q^{89} +1.00000 q^{91} -1.00516 q^{93} -0.824678 q^{95} -1.86301 q^{97} -2.33080 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) $ 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9	$ 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) $ 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9 + 12 * q^11 + 12 * q^13 - 3 * q^15 + 16 * q^17 - 2 * q^19 + 4 * q^21 + 9 * q^23 + 14 * q^25 + 7 * q^27 + 15 * q^29 + 10 * q^31 + 4 * q^33 + 6 * q^35 + 18 * q^37 + 4 * q^39 + 24 * q^41 + 15 * q^45 + 5 * q^47 + 12 * q^49 + 4 * q^51 + 15 * q^53 + 6 * q^55 - 4 * q^57 + 15 * q^59 + 17 * q^61 + 14 * q^63 + 6 * q^65 - 7 * q^67 + 9 * q^71 + 32 * q^73 - 8 * q^75 + 12 * q^77 + 20 * q^79 - 4 * q^81 - 5 * q^83 + 25 * q^85 + 19 * q^87 + 16 * q^89 + 12 * q^91 + 21 * q^93 + 8 * q^95 + 10 * 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
$3$	0.818045	0.472298	0.236149	−	0.971717i	$-0.424115\pi$
$3$	0.818045	0.472298	0.236149	+	0.971717i	$0.424115\pi$
$4$	0	0
$5$	−0.570018	−0.254920	−0.127460	−	0.991844i	$-0.540682\pi$
$5$	−0.570018	−0.254920	−0.127460	+	0.991844i	$0.540682\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.33080	−0.776934
$10$	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
$15$	−0.466300	−0.120398
$16$	0	0
$17$	−6.67104	−1.61797	−0.808983	−	0.587832i	$-0.799981\pi$
$17$	−6.67104	−1.61797	−0.808983	+	0.587832i	$0.799981\pi$
$18$	0	0
$19$	1.44676	0.331909	0.165955	−	0.986133i	$-0.446929\pi$
$19$	1.44676	0.331909	0.165955	+	0.986133i	$0.446929\pi$
$20$	0	0
$21$	0.818045	0.178512
$22$	0	0
$23$	5.53644	1.15443	0.577214	−	0.816593i	$-0.304140\pi$
$23$	5.53644	1.15443	0.577214	+	0.816593i	$0.304140\pi$
$24$	0	0
$25$	−4.67508	−0.935016
$26$	0	0
$27$	−4.36084	−0.839243
$28$	0	0
$29$	−5.42048	−1.00656	−0.503279	−	0.864124i	$-0.667873\pi$
$29$	−5.42048	−1.00656	−0.503279	+	0.864124i	$0.667873\pi$
$30$	0	0
$31$	−1.22873	−0.220687	−0.110343	−	0.993894i	$-0.535195\pi$
$31$	−1.22873	−0.220687	−0.110343	+	0.993894i	$0.535195\pi$
$32$	0	0
$33$	0.818045	0.142403
$34$	0	0
$35$	−0.570018	−0.0963506
$36$	0	0
$37$	−4.75597	−0.781877	−0.390939	−	0.920417i	$-0.627850\pi$
$37$	−4.75597	−0.781877	−0.390939	+	0.920417i	$0.627850\pi$
$38$	0	0
$39$	0.818045	0.130992
$40$	0	0
$41$	3.95779	0.618102	0.309051	−	0.951045i	$-0.399989\pi$
$41$	3.95779	0.618102	0.309051	+	0.951045i	$0.399989\pi$
$42$	0	0
$43$	8.80275	1.34241	0.671203	−	0.741273i	$-0.265778\pi$
$43$	8.80275	1.34241	0.671203	+	0.741273i	$0.265778\pi$
$44$	0	0
$45$	1.32860	0.198056
$46$	0	0
$47$	7.01984	1.02395	0.511975	−	0.859000i	$-0.328914\pi$
$47$	7.01984	1.02395	0.511975	+	0.859000i	$0.328914\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.45721	−0.764162
$52$	0	0
$53$	0.903831	0.124151	0.0620754	−	0.998071i	$-0.480228\pi$
$53$	0.903831	0.124151	0.0620754	+	0.998071i	$0.480228\pi$
$54$	0	0
$55$	−0.570018	−0.0768612
$56$	0	0
$57$	1.18351	0.156760
$58$	0	0
$59$	7.67516	0.999220	0.499610	−	0.866250i	$-0.333477\pi$
$59$	7.67516	0.999220	0.499610	+	0.866250i	$0.333477\pi$
$60$	0	0
$61$	4.80619	0.615369	0.307685	−	0.951488i	$-0.400446\pi$
$61$	4.80619	0.615369	0.307685	+	0.951488i	$0.400446\pi$
$62$	0	0
$63$	−2.33080	−0.293654
$64$	0	0
$65$	−0.570018	−0.0707020
$66$	0	0
$67$	12.1752	1.48743	0.743717	−	0.668495i	$-0.233061\pi$
$67$	12.1752	1.48743	0.743717	+	0.668495i	$0.233061\pi$
$68$	0	0
$69$	4.52905	0.545234
$70$	0	0
$71$	2.78201	0.330163	0.165082	−	0.986280i	$-0.447211\pi$
$71$	2.78201	0.330163	0.165082	+	0.986280i	$0.447211\pi$
$72$	0	0
$73$	0.712954	0.0834449	0.0417225	−	0.999129i	$-0.486715\pi$
$73$	0.712954	0.0834449	0.0417225	+	0.999129i	$0.486715\pi$
$74$	0	0
$75$	−3.82442	−0.441606
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	7.71952	0.868514	0.434257	−	0.900789i	$-0.357011\pi$
$79$	7.71952	0.868514	0.434257	+	0.900789i	$0.357011\pi$
$80$	0	0
$81$	3.42505	0.380561
$82$	0	0
$83$	8.30702	0.911814	0.455907	−	0.890027i	$-0.349315\pi$
$83$	8.30702	0.911814	0.455907	+	0.890027i	$0.349315\pi$
$84$	0	0
$85$	3.80261	0.412451
$86$	0	0
$87$	−4.43420	−0.475396
$88$	0	0
$89$	11.1244	1.17919	0.589595	−	0.807699i	$-0.299288\pi$
$89$	11.1244	1.17919	0.589595	+	0.807699i	$0.299288\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−1.00516	−0.104230
$94$	0	0
$95$	−0.824678	−0.0846102
$96$	0	0
$97$	−1.86301	−0.189160	−0.0945800	−	0.995517i	$-0.530151\pi$
$97$	−1.86301	−0.189160	−0.0945800	+	0.995517i	$0.530151\pi$
$98$	0	0
$99$	−2.33080	−0.234254
$100$	0	0
$101$	−5.55238	−0.552482	−0.276241	−	0.961088i	$-0.589089\pi$
$101$	−5.55238	−0.552482	−0.276241	+	0.961088i	$0.589089\pi$
$102$	0	0
$103$	−11.4535	−1.12855	−0.564273	−	0.825589i	$-0.690843\pi$
$103$	−11.4535	−1.12855	−0.564273	+	0.825589i	$0.690843\pi$
$104$	0	0
$105$	−0.466300	−0.0455062
$106$	0	0
$107$	7.61855	0.736513	0.368257	−	0.929724i	$-0.379955\pi$
$107$	7.61855	0.736513	0.368257	+	0.929724i	$0.379955\pi$
$108$	0	0
$109$	−0.854120	−0.0818098	−0.0409049	−	0.999163i	$-0.513024\pi$
$109$	−0.854120	−0.0818098	−0.0409049	+	0.999163i	$0.513024\pi$
$110$	0	0
$111$	−3.89060	−0.369279
$112$	0	0
$113$	0.343634	0.0323264	0.0161632	−	0.999869i	$-0.494855\pi$
$113$	0.343634	0.0323264	0.0161632	+	0.999869i	$0.494855\pi$
$114$	0	0
$115$	−3.15587	−0.294286
$116$	0	0
$117$	−2.33080	−0.215483
$118$	0	0
$119$	−6.67104	−0.611533
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.23765	0.291929
$124$	0	0
$125$	5.51497	0.493274
$126$	0	0
$127$	−1.32571	−0.117637	−0.0588187	−	0.998269i	$-0.518733\pi$
$127$	−1.32571	−0.117637	−0.0588187	+	0.998269i	$0.518733\pi$
$128$	0	0
$129$	7.20104	0.634017
$130$	0	0
$131$	7.70498	0.673187	0.336594	−	0.941650i	$-0.390725\pi$
$131$	7.70498	0.673187	0.336594	+	0.941650i	$0.390725\pi$
$132$	0	0
$133$	1.44676	0.125450
$134$	0	0
$135$	2.48575	0.213940
$136$	0	0
$137$	−9.95919	−0.850871	−0.425436	−	0.904989i	$-0.639879\pi$
$137$	−9.95919	−0.850871	−0.425436	+	0.904989i	$0.639879\pi$
$138$	0	0
$139$	−5.88077	−0.498800	−0.249400	−	0.968401i	$-0.580233\pi$
$139$	−5.88077	−0.498800	−0.249400	+	0.968401i	$0.580233\pi$
$140$	0	0
$141$	5.74255	0.483610
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	3.08977	0.256592
$146$	0	0
$147$	0.818045	0.0674712
$148$	0	0
$149$	−2.85567	−0.233945	−0.116973	−	0.993135i	$-0.537319\pi$
$149$	−2.85567	−0.233945	−0.116973	+	0.993135i	$0.537319\pi$
$150$	0	0
$151$	2.66954	0.217244	0.108622	−	0.994083i	$-0.465356\pi$
$151$	2.66954	0.217244	0.108622	+	0.994083i	$0.465356\pi$
$152$	0	0
$153$	15.5489	1.25705
$154$	0	0
$155$	0.700399	0.0562574
$156$	0	0
$157$	−12.0457	−0.961354	−0.480677	−	0.876898i	$-0.659609\pi$
$157$	−12.0457	−0.961354	−0.480677	+	0.876898i	$0.659609\pi$
$158$	0	0
$159$	0.739374	0.0586362
$160$	0	0
$161$	5.53644	0.436332
$162$	0	0
$163$	9.44765	0.739997	0.369999	−	0.929032i	$-0.379358\pi$
$163$	9.44765	0.739997	0.369999	+	0.929032i	$0.379358\pi$
$164$	0	0
$165$	−0.466300	−0.0363014
$166$	0	0
$167$	9.86884	0.763674	0.381837	−	0.924230i	$-0.375292\pi$
$167$	9.86884	0.763674	0.381837	+	0.924230i	$0.375292\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.37211	−0.257872
$172$	0	0
$173$	8.56202	0.650958	0.325479	−	0.945549i	$-0.394474\pi$
$173$	8.56202	0.650958	0.325479	+	0.945549i	$0.394474\pi$
$174$	0	0
$175$	−4.67508	−0.353403
$176$	0	0
$177$	6.27862	0.471930
$178$	0	0
$179$	6.46535	0.483243	0.241622	−	0.970371i	$-0.422321\pi$
$179$	6.46535	0.483243	0.241622	+	0.970371i	$0.422321\pi$
$180$	0	0
$181$	−10.5160	−0.781648	−0.390824	−	0.920465i	$-0.627810\pi$
$181$	−10.5160	−0.781648	−0.390824	+	0.920465i	$0.627810\pi$
$182$	0	0
$183$	3.93168	0.290638
$184$	0	0
$185$	2.71099	0.199316
$186$	0	0
$187$	−6.67104	−0.487835
$188$	0	0
$189$	−4.36084	−0.317204
$190$	0	0
$191$	−9.10318	−0.658683	−0.329341	−	0.944211i	$-0.606827\pi$
$191$	−9.10318	−0.658683	−0.329341	+	0.944211i	$0.606827\pi$
$192$	0	0
$193$	10.1246	0.728785	0.364393	−	0.931245i	$-0.381277\pi$
$193$	10.1246	0.728785	0.364393	+	0.931245i	$0.381277\pi$
$194$	0	0
$195$	−0.466300	−0.0333924
$196$	0	0
$197$	9.06904	0.646142	0.323071	−	0.946375i	$-0.395285\pi$
$197$	9.06904	0.646142	0.323071	+	0.946375i	$0.395285\pi$
$198$	0	0
$199$	11.7962	0.836209	0.418104	−	0.908399i	$-0.362695\pi$
$199$	11.7962	0.836209	0.418104	+	0.908399i	$0.362695\pi$
$200$	0	0
$201$	9.95984	0.702513
$202$	0	0
$203$	−5.42048	−0.380443
$204$	0	0
$205$	−2.25601	−0.157566
$206$	0	0
$207$	−12.9043	−0.896914
$208$	0	0
$209$	1.44676	0.100074
$210$	0	0
$211$	−20.1385	−1.38639	−0.693197	−	0.720748i	$-0.743798\pi$
$211$	−20.1385	−1.38639	−0.693197	+	0.720748i	$0.743798\pi$
$212$	0	0
$213$	2.27580	0.155936
$214$	0	0
$215$	−5.01772	−0.342206
$216$	0	0
$217$	−1.22873	−0.0834118
$218$	0	0
$219$	0.583228	0.0394109
$220$	0	0
$221$	−6.67104	−0.448743
$222$	0	0
$223$	5.64728	0.378170	0.189085	−	0.981961i	$-0.439448\pi$
$223$	5.64728	0.378170	0.189085	+	0.981961i	$0.439448\pi$
$224$	0	0
$225$	10.8967	0.726446
$226$	0	0
$227$	17.2027	1.14178	0.570892	−	0.821025i	$-0.306598\pi$
$227$	17.2027	1.14178	0.570892	+	0.821025i	$0.306598\pi$
$228$	0	0
$229$	22.3106	1.47433	0.737164	−	0.675714i	$-0.236165\pi$
$229$	22.3106	1.47433	0.737164	+	0.675714i	$0.236165\pi$
$230$	0	0
$231$	0.818045	0.0538234
$232$	0	0
$233$	−15.0144	−0.983623	−0.491811	−	0.870702i	$-0.663665\pi$
$233$	−15.0144	−0.983623	−0.491811	+	0.870702i	$0.663665\pi$
$234$	0	0
$235$	−4.00144	−0.261025
$236$	0	0
$237$	6.31491	0.410198
$238$	0	0
$239$	14.4122	0.932251	0.466125	−	0.884719i	$-0.345650\pi$
$239$	14.4122	0.932251	0.466125	+	0.884719i	$0.345650\pi$
$240$	0	0
$241$	−10.2983	−0.663373	−0.331686	−	0.943390i	$-0.607618\pi$
$241$	−10.2983	−0.663373	−0.331686	+	0.943390i	$0.607618\pi$
$242$	0	0
$243$	15.8843	1.01898
$244$	0	0
$245$	−0.570018	−0.0364171
$246$	0	0
$247$	1.44676	0.0920550
$248$	0	0
$249$	6.79551	0.430648
$250$	0	0
$251$	1.02767	0.0648657	0.0324329	−	0.999474i	$-0.489674\pi$
$251$	1.02767	0.0648657	0.0324329	+	0.999474i	$0.489674\pi$
$252$	0	0
$253$	5.53644	0.348073
$254$	0	0
$255$	3.11071	0.194800
$256$	0	0
$257$	−7.01161	−0.437372	−0.218686	−	0.975795i	$-0.570177\pi$
$257$	−7.01161	−0.437372	−0.218686	+	0.975795i	$0.570177\pi$
$258$	0	0
$259$	−4.75597	−0.295522
$260$	0	0
$261$	12.6341	0.782030
$262$	0	0
$263$	−15.5656	−0.959818	−0.479909	−	0.877318i	$-0.659330\pi$
$263$	−15.5656	−0.959818	−0.479909	+	0.877318i	$0.659330\pi$
$264$	0	0
$265$	−0.515200	−0.0316485
$266$	0	0
$267$	9.10030	0.556929
$268$	0	0
$269$	21.9731	1.33972	0.669861	−	0.742487i	$-0.266354\pi$
$269$	21.9731	1.33972	0.669861	+	0.742487i	$0.266354\pi$
$270$	0	0
$271$	−6.59902	−0.400862	−0.200431	−	0.979708i	$-0.564234\pi$
$271$	−6.59902	−0.400862	−0.200431	+	0.979708i	$0.564234\pi$
$272$	0	0
$273$	0.818045	0.0495103
$274$	0	0
$275$	−4.67508	−0.281918
$276$	0	0
$277$	−2.77936	−0.166995	−0.0834977	−	0.996508i	$-0.526609\pi$
$277$	−2.77936	−0.166995	−0.0834977	+	0.996508i	$0.526609\pi$
$278$	0	0
$279$	2.86393	0.171459
$280$	0	0
$281$	7.89313	0.470865	0.235432	−	0.971891i	$-0.424349\pi$
$281$	7.89313	0.470865	0.235432	+	0.971891i	$0.424349\pi$
$282$	0	0
$283$	12.9998	0.772760	0.386380	−	0.922340i	$-0.373725\pi$
$283$	12.9998	0.772760	0.386380	+	0.922340i	$0.373725\pi$
$284$	0	0
$285$	−0.674624	−0.0399612
$286$	0	0
$287$	3.95779	0.233621
$288$	0	0
$289$	27.5028	1.61781
$290$	0	0
$291$	−1.52403	−0.0893400
$292$	0	0
$293$	1.18902	0.0694635	0.0347318	−	0.999397i	$-0.488942\pi$
$293$	1.18902	0.0694635	0.0347318	+	0.999397i	$0.488942\pi$
$294$	0	0
$295$	−4.37498	−0.254721
$296$	0	0
$297$	−4.36084	−0.253041
$298$	0	0
$299$	5.53644	0.320180
$300$	0	0
$301$	8.80275	0.507382
$302$	0	0
$303$	−4.54210	−0.260937
$304$	0	0
$305$	−2.73961	−0.156870
$306$	0	0
$307$	−5.12065	−0.292251	−0.146126	−	0.989266i	$-0.546680\pi$
$307$	−5.12065	−0.292251	−0.146126	+	0.989266i	$0.546680\pi$
$308$	0	0
$309$	−9.36946	−0.533010
$310$	0	0
$311$	14.4114	0.817197	0.408598	−	0.912714i	$-0.366018\pi$
$311$	14.4114	0.817197	0.408598	+	0.912714i	$0.366018\pi$
$312$	0	0
$313$	4.93964	0.279205	0.139602	−	0.990208i	$-0.455418\pi$
$313$	4.93964	0.279205	0.139602	+	0.990208i	$0.455418\pi$
$314$	0	0
$315$	1.32860	0.0748581
$316$	0	0
$317$	−0.0737361	−0.00414143	−0.00207072	−	0.999998i	$-0.500659\pi$
$317$	−0.0737361	−0.00414143	−0.00207072	+	0.999998i	$0.500659\pi$
$318$	0	0
$319$	−5.42048	−0.303489
$320$	0	0
$321$	6.23231	0.347854
$322$	0	0
$323$	−9.65139	−0.537018
$324$	0	0
$325$	−4.67508	−0.259327
$326$	0	0
$327$	−0.698708	−0.0386386
$328$	0	0
$329$	7.01984	0.387017
$330$	0	0
$331$	28.8113	1.58361	0.791806	−	0.610772i	$-0.209141\pi$
$331$	28.8113	1.58361	0.791806	+	0.610772i	$0.209141\pi$
$332$	0	0
$333$	11.0852	0.607467
$334$	0	0
$335$	−6.94006	−0.379176
$336$	0	0
$337$	−27.0283	−1.47232	−0.736162	−	0.676806i	$-0.763364\pi$
$337$	−27.0283	−1.47232	−0.736162	+	0.676806i	$0.763364\pi$
$338$	0	0
$339$	0.281108	0.0152677
$340$	0	0
$341$	−1.22873	−0.0665396
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.58164	−0.138991
$346$	0	0
$347$	32.8824	1.76522	0.882611	−	0.470105i	$-0.155784\pi$
$347$	32.8824	1.76522	0.882611	+	0.470105i	$0.155784\pi$
$348$	0	0
$349$	11.8839	0.636130	0.318065	−	0.948069i	$-0.396967\pi$
$349$	11.8839	0.636130	0.318065	+	0.948069i	$0.396967\pi$
$350$	0	0
$351$	−4.36084	−0.232764
$352$	0	0
$353$	−11.3768	−0.605525	−0.302763	−	0.953066i	$-0.597909\pi$
$353$	−11.3768	−0.605525	−0.302763	+	0.953066i	$0.597909\pi$
$354$	0	0
$355$	−1.58579	−0.0841651
$356$	0	0
$357$	−5.45721	−0.288826
$358$	0	0
$359$	−6.28006	−0.331449	−0.165724	−	0.986172i	$-0.552996\pi$
$359$	−6.28006	−0.331449	−0.165724	+	0.986172i	$0.552996\pi$
$360$	0	0
$361$	−16.9069	−0.889836
$362$	0	0
$363$	0.818045	0.0429362
$364$	0	0
$365$	−0.406396	−0.0212718
$366$	0	0
$367$	2.22792	0.116296	0.0581482	−	0.998308i	$-0.481480\pi$
$367$	2.22792	0.116296	0.0581482	+	0.998308i	$0.481480\pi$
$368$	0	0
$369$	−9.22482	−0.480225
$370$	0	0
$371$	0.903831	0.0469246
$372$	0	0
$373$	15.4407	0.799492	0.399746	−	0.916626i	$-0.369098\pi$
$373$	15.4407	0.799492	0.399746	+	0.916626i	$0.369098\pi$
$374$	0	0
$375$	4.51149	0.232972
$376$	0	0
$377$	−5.42048	−0.279169
$378$	0	0
$379$	−32.3370	−1.66104	−0.830520	−	0.556989i	$-0.811957\pi$
$379$	−32.3370	−1.66104	−0.830520	+	0.556989i	$0.811957\pi$
$380$	0	0
$381$	−1.08449	−0.0555600
$382$	0	0
$383$	15.6693	0.800663	0.400332	−	0.916370i	$-0.368895\pi$
$383$	15.6693	0.800663	0.400332	+	0.916370i	$0.368895\pi$
$384$	0	0
$385$	−0.570018	−0.0290508
$386$	0	0
$387$	−20.5175	−1.04296
$388$	0	0
$389$	22.7035	1.15111	0.575556	−	0.817762i	$-0.304786\pi$
$389$	22.7035	1.15111	0.575556	+	0.817762i	$0.304786\pi$
$390$	0	0
$391$	−36.9338	−1.86782
$392$	0	0
$393$	6.30302	0.317945
$394$	0	0
$395$	−4.40026	−0.221401
$396$	0	0
$397$	16.9704	0.851722	0.425861	−	0.904789i	$-0.359971\pi$
$397$	16.9704	0.851722	0.425861	+	0.904789i	$0.359971\pi$
$398$	0	0
$399$	1.18351	0.0592498
$400$	0	0
$401$	8.12649	0.405818	0.202909	−	0.979198i	$-0.434960\pi$
$401$	8.12649	0.405818	0.202909	+	0.979198i	$0.434960\pi$
$402$	0	0
$403$	−1.22873	−0.0612075
$404$	0	0
$405$	−1.95234	−0.0970125
$406$	0	0
$407$	−4.75597	−0.235745
$408$	0	0
$409$	−22.3440	−1.10484	−0.552419	−	0.833566i	$-0.686295\pi$
$409$	−22.3440	−1.10484	−0.552419	+	0.833566i	$0.686295\pi$
$410$	0	0
$411$	−8.14706	−0.401865
$412$	0	0
$413$	7.67516	0.377670
$414$	0	0
$415$	−4.73515	−0.232439
$416$	0	0
$417$	−4.81073	−0.235583
$418$	0	0
$419$	−17.4069	−0.850382	−0.425191	−	0.905104i	$-0.639793\pi$
$419$	−17.4069	−0.850382	−0.425191	+	0.905104i	$0.639793\pi$
$420$	0	0
$421$	−17.7927	−0.867164	−0.433582	−	0.901114i	$-0.642751\pi$
$421$	−17.7927	−0.867164	−0.433582	+	0.901114i	$0.642751\pi$
$422$	0	0
$423$	−16.3619	−0.795542
$424$	0	0
$425$	31.1877	1.51282
$426$	0	0
$427$	4.80619	0.232588
$428$	0	0
$429$	0.818045	0.0394956
$430$	0	0
$431$	30.5912	1.47353	0.736763	−	0.676151i	$-0.236353\pi$
$431$	30.5912	1.47353	0.736763	+	0.676151i	$0.236353\pi$
$432$	0	0
$433$	23.7185	1.13984	0.569919	−	0.821701i	$-0.306975\pi$
$433$	23.7185	1.13984	0.569919	+	0.821701i	$0.306975\pi$
$434$	0	0
$435$	2.52757	0.121188
$436$	0	0
$437$	8.00989	0.383165
$438$	0	0
$439$	−7.16522	−0.341977	−0.170989	−	0.985273i	$-0.554696\pi$
$439$	−7.16522	−0.341977	−0.170989	+	0.985273i	$0.554696\pi$
$440$	0	0
$441$	−2.33080	−0.110991
$442$	0	0
$443$	−16.1890	−0.769164	−0.384582	−	0.923091i	$-0.625654\pi$
$443$	−16.1890	−0.769164	−0.384582	+	0.923091i	$0.625654\pi$
$444$	0	0
$445$	−6.34113	−0.300599
$446$	0	0
$447$	−2.33606	−0.110492
$448$	0	0
$449$	4.19349	0.197903	0.0989516	−	0.995092i	$-0.468451\pi$
$449$	4.19349	0.197903	0.0989516	+	0.995092i	$0.468451\pi$
$450$	0	0
$451$	3.95779	0.186365
$452$	0	0
$453$	2.18380	0.102604
$454$	0	0
$455$	−0.570018	−0.0267228
$456$	0	0
$457$	−9.92075	−0.464073	−0.232037	−	0.972707i	$-0.574539\pi$
$457$	−9.92075	−0.464073	−0.232037	+	0.972707i	$0.574539\pi$
$458$	0	0
$459$	29.0913	1.35787
$460$	0	0
$461$	−33.0489	−1.53924	−0.769621	−	0.638501i	$-0.779555\pi$
$461$	−33.0489	−1.53924	−0.769621	+	0.638501i	$0.779555\pi$
$462$	0	0
$463$	9.22639	0.428787	0.214393	−	0.976747i	$-0.431223\pi$
$463$	9.22639	0.428787	0.214393	+	0.976747i	$0.431223\pi$
$464$	0	0
$465$	0.572958	0.0265703
$466$	0	0
$467$	19.7127	0.912195	0.456097	−	0.889930i	$-0.349247\pi$
$467$	19.7127	0.912195	0.456097	+	0.889930i	$0.349247\pi$
$468$	0	0
$469$	12.1752	0.562197
$470$	0	0
$471$	−9.85395	−0.454046
$472$	0	0
$473$	8.80275	0.404751
$474$	0	0
$475$	−6.76371	−0.310340
$476$	0	0
$477$	−2.10665	−0.0964570
$478$	0	0
$479$	24.1089	1.10157	0.550783	−	0.834649i	$-0.314329\pi$
$479$	24.1089	1.10157	0.550783	+	0.834649i	$0.314329\pi$
$480$	0	0
$481$	−4.75597	−0.216854
$482$	0	0
$483$	4.52905	0.206079
$484$	0	0
$485$	1.06195	0.0482206
$486$	0	0
$487$	−18.3059	−0.829519	−0.414759	−	0.909931i	$-0.636134\pi$
$487$	−18.3059	−0.829519	−0.414759	+	0.909931i	$0.636134\pi$
$488$	0	0
$489$	7.72860	0.349499
$490$	0	0
$491$	−19.4472	−0.877641	−0.438820	−	0.898575i	$-0.644604\pi$
$491$	−19.4472	−0.877641	−0.438820	+	0.898575i	$0.644604\pi$
$492$	0	0
$493$	36.1603	1.62858
$494$	0	0
$495$	1.32860	0.0597161
$496$	0	0
$497$	2.78201	0.124790
$498$	0	0
$499$	34.7804	1.55699	0.778493	−	0.627653i	$-0.215985\pi$
$499$	34.7804	1.55699	0.778493	+	0.627653i	$0.215985\pi$
$500$	0	0
$501$	8.07315	0.360682
$502$	0	0
$503$	22.2752	0.993201	0.496600	−	0.867979i	$-0.334581\pi$
$503$	22.2752	0.993201	0.496600	+	0.867979i	$0.334581\pi$
$504$	0	0
$505$	3.16496	0.140839
$506$	0	0
$507$	0.818045	0.0363306
$508$	0	0
$509$	−19.3234	−0.856496	−0.428248	−	0.903661i	$-0.640869\pi$
$509$	−19.3234	−0.856496	−0.428248	+	0.903661i	$0.640869\pi$
$510$	0	0
$511$	0.712954	0.0315392
$512$	0	0
$513$	−6.30908	−0.278552
$514$	0	0
$515$	6.52869	0.287688
$516$	0	0
$517$	7.01984	0.308732
$518$	0	0
$519$	7.00411	0.307447
$520$	0	0
$521$	−1.64735	−0.0721716	−0.0360858	−	0.999349i	$-0.511489\pi$
$521$	−1.64735	−0.0721716	−0.0360858	+	0.999349i	$0.511489\pi$
$522$	0	0
$523$	−13.1680	−0.575798	−0.287899	−	0.957661i	$-0.592957\pi$
$523$	−13.1680	−0.575798	−0.287899	+	0.957661i	$0.592957\pi$
$524$	0	0
$525$	−3.82442	−0.166912
$526$	0	0
$527$	8.19692	0.357064
$528$	0	0
$529$	7.65214	0.332702
$530$	0	0
$531$	−17.8893	−0.776328
$532$	0	0
$533$	3.95779	0.171431
$534$	0	0
$535$	−4.34271	−0.187752
$536$	0	0
$537$	5.28895	0.228235
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	38.7216	1.66477	0.832386	−	0.554197i	$-0.186974\pi$
$541$	38.7216	1.66477	0.832386	+	0.554197i	$0.186974\pi$
$542$	0	0
$543$	−8.60256	−0.369171
$544$	0	0
$545$	0.486863	0.0208549
$546$	0	0
$547$	−29.2156	−1.24917	−0.624585	−	0.780957i	$-0.714732\pi$
$547$	−29.2156	−1.24917	−0.624585	+	0.780957i	$0.714732\pi$
$548$	0	0
$549$	−11.2023	−0.478102
$550$	0	0
$551$	−7.84213	−0.334086
$552$	0	0
$553$	7.71952	0.328267
$554$	0	0
$555$	2.21771	0.0941366
$556$	0	0
$557$	11.3763	0.482027	0.241014	−	0.970522i	$-0.422520\pi$
$557$	11.3763	0.482027	0.241014	+	0.970522i	$0.422520\pi$
$558$	0	0
$559$	8.80275	0.372317
$560$	0	0
$561$	−5.45721	−0.230404
$562$	0	0
$563$	14.1216	0.595154	0.297577	−	0.954698i	$-0.403822\pi$
$563$	14.1216	0.595154	0.297577	+	0.954698i	$0.403822\pi$
$564$	0	0
$565$	−0.195878	−0.00824063
$566$	0	0
$567$	3.42505	0.143839
$568$	0	0
$569$	43.4765	1.82263	0.911316	−	0.411708i	$-0.135068\pi$
$569$	43.4765	1.82263	0.911316	+	0.411708i	$0.135068\pi$
$570$	0	0
$571$	−20.3405	−0.851225	−0.425613	−	0.904906i	$-0.639941\pi$
$571$	−20.3405	−0.851225	−0.425613	+	0.904906i	$0.639941\pi$
$572$	0	0
$573$	−7.44681	−0.311095
$574$	0	0
$575$	−25.8833	−1.07941
$576$	0	0
$577$	−30.8614	−1.28478	−0.642388	−	0.766379i	$-0.722056\pi$
$577$	−30.8614	−1.28478	−0.642388	+	0.766379i	$0.722056\pi$
$578$	0	0
$579$	8.28238	0.344204
$580$	0	0
$581$	8.30702	0.344633
$582$	0	0
$583$	0.903831	0.0374329
$584$	0	0
$585$	1.32860	0.0549308
$586$	0	0
$587$	−30.9757	−1.27850	−0.639252	−	0.768997i	$-0.720756\pi$
$587$	−30.9757	−1.27850	−0.639252	+	0.768997i	$0.720756\pi$
$588$	0	0
$589$	−1.77768	−0.0732480
$590$	0	0
$591$	7.41888	0.305172
$592$	0	0
$593$	25.5979	1.05118	0.525590	−	0.850738i	$-0.323845\pi$
$593$	25.5979	1.05118	0.525590	+	0.850738i	$0.323845\pi$
$594$	0	0
$595$	3.80261	0.155892
$596$	0	0
$597$	9.64980	0.394940
$598$	0	0
$599$	30.0093	1.22615	0.613074	−	0.790025i	$-0.289933\pi$
$599$	30.0093	1.22615	0.613074	+	0.790025i	$0.289933\pi$
$600$	0	0
$601$	13.6581	0.557126	0.278563	−	0.960418i	$-0.410142\pi$
$601$	13.6581	0.557126	0.278563	+	0.960418i	$0.410142\pi$
$602$	0	0
$603$	−28.3779	−1.15564
$604$	0	0
$605$	−0.570018	−0.0231745
$606$	0	0
$607$	13.8543	0.562328	0.281164	−	0.959660i	$-0.409279\pi$
$607$	13.8543	0.562328	0.281164	+	0.959660i	$0.409279\pi$
$608$	0	0
$609$	−4.43420	−0.179683
$610$	0	0
$611$	7.01984	0.283993
$612$	0	0
$613$	44.7600	1.80784	0.903919	−	0.427704i	$-0.140677\pi$
$613$	44.7600	1.80784	0.903919	+	0.427704i	$0.140677\pi$
$614$	0	0
$615$	−1.84552	−0.0744184
$616$	0	0
$617$	−36.4361	−1.46686	−0.733431	−	0.679764i	$-0.762082\pi$
$617$	−36.4361	−1.46686	−0.733431	+	0.679764i	$0.762082\pi$
$618$	0	0
$619$	−26.9891	−1.08478	−0.542392	−	0.840125i	$-0.682481\pi$
$619$	−26.9891	−1.08478	−0.542392	+	0.840125i	$0.682481\pi$
$620$	0	0
$621$	−24.1435	−0.968845
$622$	0	0
$623$	11.1244	0.445692
$624$	0	0
$625$	20.2318	0.809271
$626$	0	0
$627$	1.18351	0.0472650
$628$	0	0
$629$	31.7273	1.26505
$630$	0	0
$631$	22.4321	0.893007	0.446503	−	0.894782i	$-0.352669\pi$
$631$	22.4321	0.893007	0.446503	+	0.894782i	$0.352669\pi$
$632$	0	0
$633$	−16.4742	−0.654792
$634$	0	0
$635$	0.755676	0.0299881
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−6.48431	−0.256515
$640$	0	0
$641$	26.6682	1.05333	0.526666	−	0.850072i	$-0.323442\pi$
$641$	26.6682	1.05333	0.526666	+	0.850072i	$0.323442\pi$
$642$	0	0
$643$	7.14083	0.281607	0.140803	−	0.990038i	$-0.455031\pi$
$643$	7.14083	0.281607	0.140803	+	0.990038i	$0.455031\pi$
$644$	0	0
$645$	−4.10472	−0.161623
$646$	0	0
$647$	4.78481	0.188110	0.0940552	−	0.995567i	$-0.470017\pi$
$647$	4.78481	0.188110	0.0940552	+	0.995567i	$0.470017\pi$
$648$	0	0
$649$	7.67516	0.301276
$650$	0	0
$651$	−1.00516	−0.0393952
$652$	0	0
$653$	−6.66276	−0.260734	−0.130367	−	0.991466i	$-0.541616\pi$
$653$	−6.66276	−0.260734	−0.130367	+	0.991466i	$0.541616\pi$
$654$	0	0
$655$	−4.39198	−0.171609
$656$	0	0
$657$	−1.66175	−0.0648312
$658$	0	0
$659$	−20.2129	−0.787383	−0.393691	−	0.919243i	$-0.628802\pi$
$659$	−20.2129	−0.787383	−0.393691	+	0.919243i	$0.628802\pi$
$660$	0	0
$661$	−16.1754	−0.629151	−0.314575	−	0.949233i	$-0.601862\pi$
$661$	−16.1754	−0.629151	−0.314575	+	0.949233i	$0.601862\pi$
$662$	0	0
$663$	−5.45721	−0.211941
$664$	0	0
$665$	−0.824678	−0.0319796
$666$	0	0
$667$	−30.0102	−1.16200
$668$	0	0
$669$	4.61972	0.178609
$670$	0	0
$671$	4.80619	0.185541
$672$	0	0
$673$	−48.5079	−1.86984	−0.934922	−	0.354854i	$-0.884531\pi$
$673$	−48.5079	−1.86984	−0.934922	+	0.354854i	$0.884531\pi$
$674$	0	0
$675$	20.3873	0.784706
$676$	0	0
$677$	−22.0267	−0.846555	−0.423278	−	0.906000i	$-0.639120\pi$
$677$	−22.0267	−0.846555	−0.423278	+	0.906000i	$0.639120\pi$
$678$	0	0
$679$	−1.86301	−0.0714958
$680$	0	0
$681$	14.0726	0.539263
$682$	0	0
$683$	4.33737	0.165965	0.0829824	−	0.996551i	$-0.473555\pi$
$683$	4.33737	0.165965	0.0829824	+	0.996551i	$0.473555\pi$
$684$	0	0
$685$	5.67692	0.216904
$686$	0	0
$687$	18.2511	0.696323
$688$	0	0
$689$	0.903831	0.0344332
$690$	0	0
$691$	12.7644	0.485581	0.242790	−	0.970079i	$-0.421937\pi$
$691$	12.7644	0.485581	0.242790	+	0.970079i	$0.421937\pi$
$692$	0	0
$693$	−2.33080	−0.0885399
$694$	0	0
$695$	3.35214	0.127154
$696$	0	0
$697$	−26.4026	−1.00007
$698$	0	0
$699$	−12.2824	−0.464563
$700$	0	0
$701$	12.9683	0.489807	0.244903	−	0.969548i	$-0.421244\pi$
$701$	12.9683	0.489807	0.244903	+	0.969548i	$0.421244\pi$
$702$	0	0
$703$	−6.88075	−0.259512
$704$	0	0
$705$	−3.27335	−0.123282
$706$	0	0
$707$	−5.55238	−0.208819
$708$	0	0
$709$	37.6924	1.41557	0.707784	−	0.706429i	$-0.249695\pi$
$709$	37.6924	1.41557	0.707784	+	0.706429i	$0.249695\pi$
$710$	0	0
$711$	−17.9927	−0.674778
$712$	0	0
$713$	−6.80280	−0.254767
$714$	0	0
$715$	−0.570018	−0.0213175
$716$	0	0
$717$	11.7899	0.440301
$718$	0	0
$719$	6.89254	0.257049	0.128524	−	0.991706i	$-0.458976\pi$
$719$	6.89254	0.257049	0.128524	+	0.991706i	$0.458976\pi$
$720$	0	0
$721$	−11.4535	−0.426550
$722$	0	0
$723$	−8.42448	−0.313310
$724$	0	0
$725$	25.3412	0.941148
$726$	0	0
$727$	−20.1915	−0.748859	−0.374430	−	0.927255i	$-0.622162\pi$
$727$	−20.1915	−0.748859	−0.374430	+	0.927255i	$0.622162\pi$
$728$	0	0
$729$	2.71896	0.100702
$730$	0	0
$731$	−58.7235	−2.17197
$732$	0	0
$733$	−42.3381	−1.56379	−0.781897	−	0.623408i	$-0.785748\pi$
$733$	−42.3381	−1.56379	−0.781897	+	0.623408i	$0.785748\pi$
$734$	0	0
$735$	−0.466300	−0.0171997
$736$	0	0
$737$	12.1752	0.448478
$738$	0	0
$739$	−10.9440	−0.402581	−0.201291	−	0.979532i	$-0.564514\pi$
$739$	−10.9440	−0.402581	−0.201291	+	0.979532i	$0.564514\pi$
$740$	0	0
$741$	1.18351	0.0434774
$742$	0	0
$743$	4.34348	0.159347	0.0796734	−	0.996821i	$-0.474612\pi$
$743$	4.34348	0.159347	0.0796734	+	0.996821i	$0.474612\pi$
$744$	0	0
$745$	1.62778	0.0596373
$746$	0	0
$747$	−19.3620	−0.708420
$748$	0	0
$749$	7.61855	0.278376
$750$	0	0
$751$	−26.6706	−0.973224	−0.486612	−	0.873618i	$-0.661767\pi$
$751$	−26.6706	−0.973224	−0.486612	+	0.873618i	$0.661767\pi$
$752$	0	0
$753$	0.840677	0.0306360
$754$	0	0
$755$	−1.52169	−0.0553798
$756$	0	0
$757$	−12.2682	−0.445894	−0.222947	−	0.974831i	$-0.571568\pi$
$757$	−12.2682	−0.445894	−0.222947	+	0.974831i	$0.571568\pi$
$758$	0	0
$759$	4.52905	0.164394
$760$	0	0
$761$	−13.8134	−0.500736	−0.250368	−	0.968151i	$-0.580552\pi$
$761$	−13.8134	−0.500736	−0.250368	+	0.968151i	$0.580552\pi$
$762$	0	0
$763$	−0.854120	−0.0309212
$764$	0	0
$765$	−8.86314	−0.320448
$766$	0	0
$767$	7.67516	0.277134
$768$	0	0
$769$	52.3780	1.88880	0.944400	−	0.328799i	$-0.106644\pi$
$769$	52.3780	1.88880	0.944400	+	0.328799i	$0.106644\pi$
$770$	0	0
$771$	−5.73581	−0.206570
$772$	0	0
$773$	−11.8966	−0.427891	−0.213945	−	0.976846i	$-0.568631\pi$
$773$	−11.8966	−0.427891	−0.213945	+	0.976846i	$0.568631\pi$
$774$	0	0
$775$	5.74442	0.206346
$776$	0	0
$777$	−3.89060	−0.139574
$778$	0	0
$779$	5.72596	0.205154
$780$	0	0
$781$	2.78201	0.0995480
$782$	0	0
$783$	23.6378	0.844747
$784$	0	0
$785$	6.86628	0.245068
$786$	0	0
$787$	14.6010	0.520470	0.260235	−	0.965545i	$-0.416200\pi$
$787$	14.6010	0.520470	0.260235	+	0.965545i	$0.416200\pi$
$788$	0	0
$789$	−12.7334	−0.453320
$790$	0	0
$791$	0.343634	0.0122182
$792$	0	0
$793$	4.80619	0.170673
$794$	0	0
$795$	−0.421457	−0.0149475
$796$	0	0
$797$	−27.1575	−0.961968	−0.480984	−	0.876729i	$-0.659721\pi$
$797$	−27.1575	−0.961968	−0.480984	+	0.876729i	$0.659721\pi$
$798$	0	0
$799$	−46.8297	−1.65671
$800$	0	0
$801$	−25.9289	−0.916152
$802$	0	0
$803$	0.712954	0.0251596
$804$	0	0
$805$	−3.15587	−0.111230
$806$	0	0
$807$	17.9750	0.632748
$808$	0	0
$809$	0.582552	0.0204814	0.0102407	−	0.999948i	$-0.496740\pi$
$809$	0.582552	0.0204814	0.0102407	+	0.999948i	$0.496740\pi$
$810$	0	0
$811$	−48.1539	−1.69091	−0.845456	−	0.534045i	$-0.820671\pi$
$811$	−48.1539	−1.69091	−0.845456	+	0.534045i	$0.820671\pi$
$812$	0	0
$813$	−5.39829	−0.189326
$814$	0	0
$815$	−5.38533	−0.188640
$816$	0	0
$817$	12.7355	0.445557
$818$	0	0
$819$	−2.33080	−0.0814448
$820$	0	0
$821$	−20.3958	−0.711817	−0.355909	−	0.934521i	$-0.615829\pi$
$821$	−20.3958	−0.711817	−0.355909	+	0.934521i	$0.615829\pi$
$822$	0	0
$823$	−49.5107	−1.72584	−0.862918	−	0.505344i	$-0.831365\pi$
$823$	−49.5107	−1.72584	−0.862918	+	0.505344i	$0.831365\pi$
$824$	0	0
$825$	−3.82442	−0.133149
$826$	0	0
$827$	−27.2695	−0.948255	−0.474127	−	0.880456i	$-0.657236\pi$
$827$	−27.2695	−0.948255	−0.474127	+	0.880456i	$0.657236\pi$
$828$	0	0
$829$	48.3449	1.67909	0.839544	−	0.543291i	$-0.182822\pi$
$829$	48.3449	1.67909	0.839544	+	0.543291i	$0.182822\pi$
$830$	0	0
$831$	−2.27364	−0.0788717
$832$	0	0
$833$	−6.67104	−0.231138
$834$	0	0
$835$	−5.62541	−0.194675
$836$	0	0
$837$	5.35830	0.185210
$838$	0	0
$839$	4.30293	0.148554	0.0742769	−	0.997238i	$-0.476335\pi$
$839$	4.30293	0.148554	0.0742769	+	0.997238i	$0.476335\pi$
$840$	0	0
$841$	0.381624	0.0131594
$842$	0	0
$843$	6.45694	0.222389
$844$	0	0
$845$	−0.570018	−0.0196092
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	10.6344	0.364973
$850$	0	0
$851$	−26.3312	−0.902620
$852$	0	0
$853$	28.2136	0.966016	0.483008	−	0.875616i	$-0.339544\pi$
$853$	28.2136	0.966016	0.483008	+	0.875616i	$0.339544\pi$
$854$	0	0
$855$	1.92216	0.0657366
$856$	0	0
$857$	25.9068	0.884959	0.442480	−	0.896779i	$-0.354099\pi$
$857$	25.9068	0.884959	0.442480	+	0.896779i	$0.354099\pi$
$858$	0	0
$859$	−19.2972	−0.658411	−0.329206	−	0.944258i	$-0.606781\pi$
$859$	−19.2972	−0.658411	−0.329206	+	0.944258i	$0.606781\pi$
$860$	0	0
$861$	3.23765	0.110339
$862$	0	0
$863$	20.8662	0.710294	0.355147	−	0.934810i	$-0.384431\pi$
$863$	20.8662	0.710294	0.355147	+	0.934810i	$0.384431\pi$
$864$	0	0
$865$	−4.88050	−0.165942
$866$	0	0
$867$	22.4985	0.764090
$868$	0	0
$869$	7.71952	0.261867
$870$	0	0
$871$	12.1752	0.412540
$872$	0	0
$873$	4.34231	0.146965
$874$	0	0
$875$	5.51497	0.186440
$876$	0	0
$877$	11.5027	0.388419	0.194209	−	0.980960i	$-0.437786\pi$
$877$	11.5027	0.388419	0.194209	+	0.980960i	$0.437786\pi$
$878$	0	0
$879$	0.972675	0.0328075
$880$	0	0
$881$	−23.7693	−0.800807	−0.400404	−	0.916339i	$-0.631130\pi$
$881$	−23.7693	−0.800807	−0.400404	+	0.916339i	$0.631130\pi$
$882$	0	0
$883$	−47.0419	−1.58309	−0.791543	−	0.611113i	$-0.790722\pi$
$883$	−47.0419	−1.58309	−0.791543	+	0.611113i	$0.790722\pi$
$884$	0	0
$885$	−3.57893	−0.120304
$886$	0	0
$887$	4.76222	0.159900	0.0799499	−	0.996799i	$-0.474524\pi$
$887$	4.76222	0.159900	0.0799499	+	0.996799i	$0.474524\pi$
$888$	0	0
$889$	−1.32571	−0.0444628
$890$	0	0
$891$	3.42505	0.114744
$892$	0	0
$893$	10.1560	0.339858
$894$	0	0
$895$	−3.68537	−0.123188
$896$	0	0
$897$	4.52905	0.151221
$898$	0	0
$899$	6.66032	0.222134
$900$	0	0
$901$	−6.02950	−0.200872
$902$	0	0
$903$	7.20104	0.239636
$904$	0	0
$905$	5.99431	0.199257
$906$	0	0
$907$	27.8847	0.925896	0.462948	−	0.886386i	$-0.346792\pi$
$907$	27.8847	0.925896	0.462948	+	0.886386i	$0.346792\pi$
$908$	0	0
$909$	12.9415	0.429243
$910$	0	0
$911$	2.65070	0.0878218	0.0439109	−	0.999035i	$-0.486018\pi$
$911$	2.65070	0.0878218	0.0439109	+	0.999035i	$0.486018\pi$
$912$	0	0
$913$	8.30702	0.274922
$914$	0	0
$915$	−2.24113	−0.0740893
$916$	0	0
$917$	7.70498	0.254441
$918$	0	0
$919$	20.5771	0.678776	0.339388	−	0.940646i	$-0.389780\pi$
$919$	20.5771	0.678776	0.339388	+	0.940646i	$0.389780\pi$
$920$	0	0
$921$	−4.18892	−0.138030
$922$	0	0
$923$	2.78201	0.0915708
$924$	0	0
$925$	22.2346	0.731068
$926$	0	0
$927$	26.6958	0.876805
$928$	0	0
$929$	−30.1233	−0.988314	−0.494157	−	0.869373i	$-0.664523\pi$
$929$	−30.1233	−0.988314	−0.494157	+	0.869373i	$0.664523\pi$
$930$	0	0
$931$	1.44676	0.0474156
$932$	0	0
$933$	11.7892	0.385961
$934$	0	0
$935$	3.80261	0.124359
$936$	0	0
$937$	49.0880	1.60363	0.801817	−	0.597569i	$-0.203867\pi$
$937$	49.0880	1.60363	0.801817	+	0.597569i	$0.203867\pi$
$938$	0	0
$939$	4.04084	0.131868
$940$	0	0
$941$	−36.3851	−1.18612	−0.593061	−	0.805158i	$-0.702081\pi$
$941$	−36.3851	−1.18612	−0.593061	+	0.805158i	$0.702081\pi$
$942$	0	0
$943$	21.9120	0.713554
$944$	0	0
$945$	2.48575	0.0808616
$946$	0	0
$947$	−26.3682	−0.856851	−0.428426	−	0.903577i	$-0.640932\pi$
$947$	−26.3682	−0.856851	−0.428426	+	0.903577i	$0.640932\pi$
$948$	0	0
$949$	0.712954	0.0231435
$950$	0	0
$951$	−0.0603194	−0.00195599
$952$	0	0
$953$	58.4482	1.89332	0.946662	−	0.322228i	$-0.104432\pi$
$953$	58.4482	1.89332	0.946662	+	0.322228i	$0.104432\pi$
$954$	0	0
$955$	5.18897	0.167911
$956$	0	0
$957$	−4.43420	−0.143337
$958$	0	0
$959$	−9.95919	−0.321599
$960$	0	0
$961$	−29.4902	−0.951297
$962$	0	0
$963$	−17.7573	−0.572222
$964$	0	0
$965$	−5.77121	−0.185782
$966$	0	0
$967$	47.7140	1.53438	0.767189	−	0.641421i	$-0.221655\pi$
$967$	47.7140	1.53438	0.767189	+	0.641421i	$0.221655\pi$
$968$	0	0
$969$	−7.89527	−0.253632
$970$	0	0
$971$	6.43239	0.206425	0.103213	−	0.994659i	$-0.467088\pi$
$971$	6.43239	0.206425	0.103213	+	0.994659i	$0.467088\pi$
$972$	0	0
$973$	−5.88077	−0.188529
$974$	0	0
$975$	−3.82442	−0.122480
$976$	0	0
$977$	−3.87238	−0.123888	−0.0619442	−	0.998080i	$-0.519730\pi$
$977$	−3.87238	−0.123888	−0.0619442	+	0.998080i	$0.519730\pi$
$978$	0	0
$979$	11.1244	0.355539
$980$	0	0
$981$	1.99078	0.0635609
$982$	0	0
$983$	18.0250	0.574907	0.287453	−	0.957795i	$-0.407191\pi$
$983$	18.0250	0.574907	0.287453	+	0.957795i	$0.407191\pi$
$984$	0	0
$985$	−5.16951	−0.164714
$986$	0	0
$987$	5.74255	0.182787
$988$	0	0
$989$	48.7359	1.54971
$990$	0	0
$991$	18.5366	0.588835	0.294418	−	0.955677i	$-0.404874\pi$
$991$	18.5366	0.588835	0.294418	+	0.955677i	$0.404874\pi$
$992$	0	0
$993$	23.5689	0.747938
$994$	0	0
$995$	−6.72403	−0.213166
$996$	0	0
$997$	10.0345	0.317797	0.158899	−	0.987295i	$-0.449206\pi$
$997$	10.0345	0.317797	0.158899	+	0.987295i	$0.449206\pi$
$998$	0	0
$999$	20.7400	0.656185

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.x.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.x.1.7	✓	12	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)

\(f(q)\)	\(=\)	\(q+0.818045 q^{3} -0.570018 q^{5} +1.00000 q^{7} -2.33080 q^{9} +O(q^{10})\)	\(q+0.818045 q^{3} -0.570018 q^{5} +1.00000 q^{7} -2.33080 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.466300 q^{15} -6.67104 q^{17} +1.44676 q^{19} +0.818045 q^{21} +5.53644 q^{23} -4.67508 q^{25} -4.36084 q^{27} -5.42048 q^{29} -1.22873 q^{31} +0.818045 q^{33} -0.570018 q^{35} -4.75597 q^{37} +0.818045 q^{39} +3.95779 q^{41} +8.80275 q^{43} +1.32860 q^{45} +7.01984 q^{47} +1.00000 q^{49} -5.45721 q^{51} +0.903831 q^{53} -0.570018 q^{55} +1.18351 q^{57} +7.67516 q^{59} +4.80619 q^{61} -2.33080 q^{63} -0.570018 q^{65} +12.1752 q^{67} +4.52905 q^{69} +2.78201 q^{71} +0.712954 q^{73} -3.82442 q^{75} +1.00000 q^{77} +7.71952 q^{79} +3.42505 q^{81} +8.30702 q^{83} +3.80261 q^{85} -4.43420 q^{87} +11.1244 q^{89} +1.00000 q^{91} -1.00516 q^{93} -0.824678 q^{95} -1.86301 q^{97} -2.33080 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) \) 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) \) 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9 + 12 * q^11 + 12 * q^13 - 3 * q^15 + 16 * q^17 - 2 * q^19 + 4 * q^21 + 9 * q^23 + 14 * q^25 + 7 * q^27 + 15 * q^29 + 10 * q^31 + 4 * q^33 + 6 * q^35 + 18 * q^37 + 4 * q^39 + 24 * q^41 + 15 * q^45 + 5 * q^47 + 12 * q^49 + 4 * q^51 + 15 * q^53 + 6 * q^55 - 4 * q^57 + 15 * q^59 + 17 * q^61 + 14 * q^63 + 6 * q^65 - 7 * q^67 + 9 * q^71 + 32 * q^73 - 8 * q^75 + 12 * q^77 + 20 * q^79 - 4 * q^81 - 5 * q^83 + 25 * q^85 + 19 * q^87 + 16 * q^89 + 12 * q^91 + 21 * q^93 + 8 * q^95 + 10 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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