LMFDB - Embedded newform 4008.2.a.m.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + \cdots + 12144 $ x^13 - 2x^12 - 49x^11 + 99x^10 + 901x^9 - 1879x^8 - 7582x^7 + 16968x^6 + 26911x^5 - 72240x^4 - 14532x^3 + 112850x^2 - 72184x + 12144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$3.38131$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.38131 q^{5} +2.18014 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.38131 q^{5} +2.18014 q^{7} +1.00000 q^{9} +1.42337 q^{11} +5.00143 q^{13} +3.38131 q^{15} +5.50797 q^{17} -7.82733 q^{19} +2.18014 q^{21} +7.75758 q^{23} +6.43326 q^{25} +1.00000 q^{27} +6.29374 q^{29} -9.08688 q^{31} +1.42337 q^{33} +7.37172 q^{35} +4.72415 q^{37} +5.00143 q^{39} -11.8364 q^{41} -7.11615 q^{43} +3.38131 q^{45} -8.21355 q^{47} -2.24700 q^{49} +5.50797 q^{51} -10.8063 q^{53} +4.81285 q^{55} -7.82733 q^{57} -6.83956 q^{59} -1.73720 q^{61} +2.18014 q^{63} +16.9114 q^{65} -1.61452 q^{67} +7.75758 q^{69} +16.0154 q^{71} +4.43455 q^{73} +6.43326 q^{75} +3.10314 q^{77} +4.26342 q^{79} +1.00000 q^{81} +4.41440 q^{83} +18.6241 q^{85} +6.29374 q^{87} -8.56376 q^{89} +10.9038 q^{91} -9.08688 q^{93} -26.4666 q^{95} -10.2977 q^{97} +1.42337 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9	$ 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9} + 11 q^{11} + 12 q^{13} + 2 q^{15} + 15 q^{17} + 14 q^{19} + q^{21} + 9 q^{23} + 37 q^{25} + 13 q^{27} - 3 q^{29} - 17 q^{31} + 11 q^{33} + 15 q^{35} + 16 q^{37} + 12 q^{39} + 12 q^{41} + 20 q^{43} + 2 q^{45} - 6 q^{47} + 26 q^{49} + 15 q^{51} - 12 q^{53} + 7 q^{55} + 14 q^{57} + 14 q^{59} + 24 q^{61} + q^{63} + 8 q^{65} + 3 q^{67} + 9 q^{69} + 17 q^{71} + 34 q^{73} + 37 q^{75} + 30 q^{77} + 10 q^{79} + 13 q^{81} + 44 q^{83} + 25 q^{85} - 3 q^{87} + 25 q^{89} + 29 q^{91} - 17 q^{93} - 15 q^{95} + 38 q^{97} + 11 q^{99}+O(q^{100}) $ 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9 + 11 * q^11 + 12 * q^13 + 2 * q^15 + 15 * q^17 + 14 * q^19 + q^21 + 9 * q^23 + 37 * q^25 + 13 * q^27 - 3 * q^29 - 17 * q^31 + 11 * q^33 + 15 * q^35 + 16 * q^37 + 12 * q^39 + 12 * q^41 + 20 * q^43 + 2 * q^45 - 6 * q^47 + 26 * q^49 + 15 * q^51 - 12 * q^53 + 7 * q^55 + 14 * q^57 + 14 * q^59 + 24 * q^61 + q^63 + 8 * q^65 + 3 * q^67 + 9 * q^69 + 17 * q^71 + 34 * q^73 + 37 * q^75 + 30 * q^77 + 10 * q^79 + 13 * q^81 + 44 * q^83 + 25 * q^85 - 3 * q^87 + 25 * q^89 + 29 * q^91 - 17 * q^93 - 15 * q^95 + 38 * q^97 + 11 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.38131	1.51217	0.756084	−	0.654475i	$-0.227110\pi$
$5$	3.38131	1.51217	0.756084	+	0.654475i	$0.227110\pi$
$6$	0	0
$7$	2.18014	0.824014	0.412007	−	0.911181i	$-0.364828\pi$
$7$	2.18014	0.824014	0.412007	+	0.911181i	$0.364828\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.42337	0.429162	0.214581	−	0.976706i	$-0.431161\pi$
$11$	1.42337	0.429162	0.214581	+	0.976706i	$0.431161\pi$
$12$	0	0
$13$	5.00143	1.38715	0.693574	−	0.720386i	$-0.256035\pi$
$13$	5.00143	1.38715	0.693574	+	0.720386i	$0.256035\pi$
$14$	0	0
$15$	3.38131	0.873051
$16$	0	0
$17$	5.50797	1.33588	0.667939	−	0.744216i	$-0.267177\pi$
$17$	5.50797	1.33588	0.667939	+	0.744216i	$0.267177\pi$
$18$	0	0
$19$	−7.82733	−1.79571	−0.897856	−	0.440289i	$-0.854876\pi$
$19$	−7.82733	−1.79571	−0.897856	+	0.440289i	$0.854876\pi$
$20$	0	0
$21$	2.18014	0.475745
$22$	0	0
$23$	7.75758	1.61757	0.808784	−	0.588106i	$-0.200126\pi$
$23$	7.75758	1.61757	0.808784	+	0.588106i	$0.200126\pi$
$24$	0	0
$25$	6.43326	1.28665
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.29374	1.16872	0.584359	−	0.811495i	$-0.301346\pi$
$29$	6.29374	1.16872	0.584359	+	0.811495i	$0.301346\pi$
$30$	0	0
$31$	−9.08688	−1.63205	−0.816026	−	0.578015i	$-0.803828\pi$
$31$	−9.08688	−1.63205	−0.816026	+	0.578015i	$0.803828\pi$
$32$	0	0
$33$	1.42337	0.247777
$34$	0	0
$35$	7.37172	1.24605
$36$	0	0
$37$	4.72415	0.776646	0.388323	−	0.921523i	$-0.373055\pi$
$37$	4.72415	0.776646	0.388323	+	0.921523i	$0.373055\pi$
$38$	0	0
$39$	5.00143	0.800870
$40$	0	0
$41$	−11.8364	−1.84853	−0.924267	−	0.381748i	$-0.875322\pi$
$41$	−11.8364	−1.84853	−0.924267	+	0.381748i	$0.875322\pi$
$42$	0	0
$43$	−7.11615	−1.08520	−0.542601	−	0.839991i	$-0.682560\pi$
$43$	−7.11615	−1.08520	−0.542601	+	0.839991i	$0.682560\pi$
$44$	0	0
$45$	3.38131	0.504056
$46$	0	0
$47$	−8.21355	−1.19807	−0.599035	−	0.800723i	$-0.704449\pi$
$47$	−8.21355	−1.19807	−0.599035	+	0.800723i	$0.704449\pi$
$48$	0	0
$49$	−2.24700	−0.321000
$50$	0	0
$51$	5.50797	0.771270
$52$	0	0
$53$	−10.8063	−1.48436	−0.742179	−	0.670201i	$-0.766208\pi$
$53$	−10.8063	−1.48436	−0.742179	+	0.670201i	$0.766208\pi$
$54$	0	0
$55$	4.81285	0.648964
$56$	0	0
$57$	−7.82733	−1.03676
$58$	0	0
$59$	−6.83956	−0.890435	−0.445218	−	0.895422i	$-0.646874\pi$
$59$	−6.83956	−0.890435	−0.445218	+	0.895422i	$0.646874\pi$
$60$	0	0
$61$	−1.73720	−0.222426	−0.111213	−	0.993797i	$-0.535474\pi$
$61$	−1.73720	−0.222426	−0.111213	+	0.993797i	$0.535474\pi$
$62$	0	0
$63$	2.18014	0.274671
$64$	0	0
$65$	16.9114	2.09760
$66$	0	0
$67$	−1.61452	−0.197245	−0.0986225	−	0.995125i	$-0.531444\pi$
$67$	−1.61452	−0.197245	−0.0986225	+	0.995125i	$0.531444\pi$
$68$	0	0
$69$	7.75758	0.933903
$70$	0	0
$71$	16.0154	1.90068	0.950340	−	0.311214i	$-0.100736\pi$
$71$	16.0154	1.90068	0.950340	+	0.311214i	$0.100736\pi$
$72$	0	0
$73$	4.43455	0.519025	0.259513	−	0.965740i	$-0.416438\pi$
$73$	4.43455	0.519025	0.259513	+	0.965740i	$0.416438\pi$
$74$	0	0
$75$	6.43326	0.742849
$76$	0	0
$77$	3.10314	0.353635
$78$	0	0
$79$	4.26342	0.479673	0.239836	−	0.970813i	$-0.422906\pi$
$79$	4.26342	0.479673	0.239836	+	0.970813i	$0.422906\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.41440	0.484544	0.242272	−	0.970208i	$-0.422107\pi$
$83$	4.41440	0.484544	0.242272	+	0.970208i	$0.422107\pi$
$84$	0	0
$85$	18.6241	2.02007
$86$	0	0
$87$	6.29374	0.674759
$88$	0	0
$89$	−8.56376	−0.907757	−0.453879	−	0.891064i	$-0.649960\pi$
$89$	−8.56376	−0.907757	−0.453879	+	0.891064i	$0.649960\pi$
$90$	0	0
$91$	10.9038	1.14303
$92$	0	0
$93$	−9.08688	−0.942266
$94$	0	0
$95$	−26.4666	−2.71542
$96$	0	0
$97$	−10.2977	−1.04557	−0.522786	−	0.852464i	$-0.675107\pi$
$97$	−10.2977	−1.04557	−0.522786	+	0.852464i	$0.675107\pi$
$98$	0	0
$99$	1.42337	0.143054
$100$	0	0
$101$	5.31154	0.528518	0.264259	−	0.964452i	$-0.414873\pi$
$101$	5.31154	0.528518	0.264259	+	0.964452i	$0.414873\pi$
$102$	0	0
$103$	−0.971769	−0.0957513	−0.0478756	−	0.998853i	$-0.515245\pi$
$103$	−0.971769	−0.0957513	−0.0478756	+	0.998853i	$0.515245\pi$
$104$	0	0
$105$	7.37172	0.719406
$106$	0	0
$107$	−13.7609	−1.33031	−0.665156	−	0.746704i	$-0.731635\pi$
$107$	−13.7609	−1.33031	−0.665156	+	0.746704i	$0.731635\pi$
$108$	0	0
$109$	−5.15480	−0.493741	−0.246870	−	0.969049i	$-0.579402\pi$
$109$	−5.15480	−0.493741	−0.246870	+	0.969049i	$0.579402\pi$
$110$	0	0
$111$	4.72415	0.448397
$112$	0	0
$113$	−17.9885	−1.69221	−0.846106	−	0.533015i	$-0.821059\pi$
$113$	−17.9885	−1.69221	−0.846106	+	0.533015i	$0.821059\pi$
$114$	0	0
$115$	26.2308	2.44603
$116$	0	0
$117$	5.00143	0.462383
$118$	0	0
$119$	12.0081	1.10078
$120$	0	0
$121$	−8.97402	−0.815820
$122$	0	0
$123$	−11.8364	−1.06725
$124$	0	0
$125$	4.84630	0.433466
$126$	0	0
$127$	−16.3783	−1.45334	−0.726668	−	0.686989i	$-0.758932\pi$
$127$	−16.3783	−1.45334	−0.726668	+	0.686989i	$0.758932\pi$
$128$	0	0
$129$	−7.11615	−0.626542
$130$	0	0
$131$	−12.0066	−1.04902	−0.524512	−	0.851403i	$-0.675752\pi$
$131$	−12.0066	−1.04902	−0.524512	+	0.851403i	$0.675752\pi$
$132$	0	0
$133$	−17.0647	−1.47969
$134$	0	0
$135$	3.38131	0.291017
$136$	0	0
$137$	7.24650	0.619110	0.309555	−	0.950881i	$-0.399820\pi$
$137$	7.24650	0.619110	0.309555	+	0.950881i	$0.399820\pi$
$138$	0	0
$139$	−7.39592	−0.627314	−0.313657	−	0.949536i	$-0.601554\pi$
$139$	−7.39592	−0.627314	−0.313657	+	0.949536i	$0.601554\pi$
$140$	0	0
$141$	−8.21355	−0.691706
$142$	0	0
$143$	7.11888	0.595310
$144$	0	0
$145$	21.2811	1.76730
$146$	0	0
$147$	−2.24700	−0.185330
$148$	0	0
$149$	23.8445	1.95341	0.976707	−	0.214577i	$-0.0688373\pi$
$149$	23.8445	1.95341	0.976707	+	0.214577i	$0.0688373\pi$
$150$	0	0
$151$	1.42298	0.115800	0.0579002	−	0.998322i	$-0.481559\pi$
$151$	1.42298	0.115800	0.0579002	+	0.998322i	$0.481559\pi$
$152$	0	0
$153$	5.50797	0.445293
$154$	0	0
$155$	−30.7256	−2.46794
$156$	0	0
$157$	19.8822	1.58677	0.793386	−	0.608718i	$-0.208316\pi$
$157$	19.8822	1.58677	0.793386	+	0.608718i	$0.208316\pi$
$158$	0	0
$159$	−10.8063	−0.856995
$160$	0	0
$161$	16.9126	1.33290
$162$	0	0
$163$	14.3419	1.12335	0.561674	−	0.827359i	$-0.310158\pi$
$163$	14.3419	1.12335	0.561674	+	0.827359i	$0.310158\pi$
$164$	0	0
$165$	4.81285	0.374680
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	12.0143	0.924179
$170$	0	0
$171$	−7.82733	−0.598571
$172$	0	0
$173$	−4.94096	−0.375654	−0.187827	−	0.982202i	$-0.560144\pi$
$173$	−4.94096	−0.375654	−0.187827	+	0.982202i	$0.560144\pi$
$174$	0	0
$175$	14.0254	1.06022
$176$	0	0
$177$	−6.83956	−0.514093
$178$	0	0
$179$	1.94308	0.145233	0.0726163	−	0.997360i	$-0.476865\pi$
$179$	1.94308	0.145233	0.0726163	+	0.997360i	$0.476865\pi$
$180$	0	0
$181$	11.2531	0.836439	0.418220	−	0.908346i	$-0.362654\pi$
$181$	11.2531	0.836439	0.418220	+	0.908346i	$0.362654\pi$
$182$	0	0
$183$	−1.73720	−0.128418
$184$	0	0
$185$	15.9738	1.17442
$186$	0	0
$187$	7.83986	0.573308
$188$	0	0
$189$	2.18014	0.158582
$190$	0	0
$191$	−8.97683	−0.649541	−0.324770	−	0.945793i	$-0.605287\pi$
$191$	−8.97683	−0.649541	−0.324770	+	0.945793i	$0.605287\pi$
$192$	0	0
$193$	−10.0704	−0.724884	−0.362442	−	0.932006i	$-0.618057\pi$
$193$	−10.0704	−0.724884	−0.362442	+	0.932006i	$0.618057\pi$
$194$	0	0
$195$	16.9114	1.21105
$196$	0	0
$197$	−19.4156	−1.38330	−0.691652	−	0.722231i	$-0.743117\pi$
$197$	−19.4156	−1.38330	−0.691652	+	0.722231i	$0.743117\pi$
$198$	0	0
$199$	5.98916	0.424560	0.212280	−	0.977209i	$-0.431911\pi$
$199$	5.98916	0.424560	0.212280	+	0.977209i	$0.431911\pi$
$200$	0	0
$201$	−1.61452	−0.113880
$202$	0	0
$203$	13.7212	0.963040
$204$	0	0
$205$	−40.0225	−2.79529
$206$	0	0
$207$	7.75758	0.539189
$208$	0	0
$209$	−11.1412	−0.770651
$210$	0	0
$211$	−7.53399	−0.518661	−0.259331	−	0.965789i	$-0.583502\pi$
$211$	−7.53399	−0.518661	−0.259331	+	0.965789i	$0.583502\pi$
$212$	0	0
$213$	16.0154	1.09736
$214$	0	0
$215$	−24.0619	−1.64101
$216$	0	0
$217$	−19.8107	−1.34484
$218$	0	0
$219$	4.43455	0.299659
$220$	0	0
$221$	27.5477	1.85306
$222$	0	0
$223$	1.15074	0.0770591	0.0385295	−	0.999257i	$-0.487733\pi$
$223$	1.15074	0.0770591	0.0385295	+	0.999257i	$0.487733\pi$
$224$	0	0
$225$	6.43326	0.428884
$226$	0	0
$227$	−25.0325	−1.66147	−0.830733	−	0.556671i	$-0.812078\pi$
$227$	−25.0325	−1.66147	−0.830733	+	0.556671i	$0.812078\pi$
$228$	0	0
$229$	7.40955	0.489637	0.244818	−	0.969569i	$-0.421272\pi$
$229$	7.40955	0.489637	0.244818	+	0.969569i	$0.421272\pi$
$230$	0	0
$231$	3.10314	0.204171
$232$	0	0
$233$	−2.93657	−0.192381	−0.0961906	−	0.995363i	$-0.530666\pi$
$233$	−2.93657	−0.192381	−0.0961906	+	0.995363i	$0.530666\pi$
$234$	0	0
$235$	−27.7726	−1.81168
$236$	0	0
$237$	4.26342	0.276939
$238$	0	0
$239$	26.6134	1.72148	0.860738	−	0.509048i	$-0.170002\pi$
$239$	26.6134	1.72148	0.860738	+	0.509048i	$0.170002\pi$
$240$	0	0
$241$	16.8419	1.08488	0.542441	−	0.840094i	$-0.317500\pi$
$241$	16.8419	1.08488	0.542441	+	0.840094i	$0.317500\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−7.59781	−0.485406
$246$	0	0
$247$	−39.1479	−2.49092
$248$	0	0
$249$	4.41440	0.279751
$250$	0	0
$251$	5.78068	0.364873	0.182437	−	0.983218i	$-0.441602\pi$
$251$	5.78068	0.364873	0.182437	+	0.983218i	$0.441602\pi$
$252$	0	0
$253$	11.0419	0.694198
$254$	0	0
$255$	18.6241	1.16629
$256$	0	0
$257$	28.5244	1.77930	0.889651	−	0.456641i	$-0.150948\pi$
$257$	28.5244	1.77930	0.889651	+	0.456641i	$0.150948\pi$
$258$	0	0
$259$	10.2993	0.639967
$260$	0	0
$261$	6.29374	0.389573
$262$	0	0
$263$	4.27282	0.263473	0.131737	−	0.991285i	$-0.457945\pi$
$263$	4.27282	0.263473	0.131737	+	0.991285i	$0.457945\pi$
$264$	0	0
$265$	−36.5394	−2.24460
$266$	0	0
$267$	−8.56376	−0.524094
$268$	0	0
$269$	7.38718	0.450404	0.225202	−	0.974312i	$-0.427696\pi$
$269$	7.38718	0.450404	0.225202	+	0.974312i	$0.427696\pi$
$270$	0	0
$271$	14.7734	0.897420	0.448710	−	0.893677i	$-0.351884\pi$
$271$	14.7734	0.897420	0.448710	+	0.893677i	$0.351884\pi$
$272$	0	0
$273$	10.9038	0.659929
$274$	0	0
$275$	9.15690	0.552182
$276$	0	0
$277$	−2.05325	−0.123368	−0.0616839	−	0.998096i	$-0.519647\pi$
$277$	−2.05325	−0.123368	−0.0616839	+	0.998096i	$0.519647\pi$
$278$	0	0
$279$	−9.08688	−0.544018
$280$	0	0
$281$	25.3998	1.51523	0.757613	−	0.652704i	$-0.226365\pi$
$281$	25.3998	1.51523	0.757613	+	0.652704i	$0.226365\pi$
$282$	0	0
$283$	2.37163	0.140979	0.0704894	−	0.997513i	$-0.477544\pi$
$283$	2.37163	0.140979	0.0704894	+	0.997513i	$0.477544\pi$
$284$	0	0
$285$	−26.4666	−1.56775
$286$	0	0
$287$	−25.8050	−1.52322
$288$	0	0
$289$	13.3377	0.784571
$290$	0	0
$291$	−10.2977	−0.603661
$292$	0	0
$293$	8.38097	0.489621	0.244811	−	0.969571i	$-0.421274\pi$
$293$	8.38097	0.489621	0.244811	+	0.969571i	$0.421274\pi$
$294$	0	0
$295$	−23.1267	−1.34649
$296$	0	0
$297$	1.42337	0.0825922
$298$	0	0
$299$	38.7990	2.24381
$300$	0	0
$301$	−15.5142	−0.894222
$302$	0	0
$303$	5.31154	0.305140
$304$	0	0
$305$	−5.87403	−0.336346
$306$	0	0
$307$	23.1839	1.32318	0.661588	−	0.749868i	$-0.269883\pi$
$307$	23.1839	1.32318	0.661588	+	0.749868i	$0.269883\pi$
$308$	0	0
$309$	−0.971769	−0.0552820
$310$	0	0
$311$	22.1324	1.25501	0.627506	−	0.778612i	$-0.284076\pi$
$311$	22.1324	1.25501	0.627506	+	0.778612i	$0.284076\pi$
$312$	0	0
$313$	−25.6297	−1.44868	−0.724338	−	0.689445i	$-0.757854\pi$
$313$	−25.6297	−1.44868	−0.724338	+	0.689445i	$0.757854\pi$
$314$	0	0
$315$	7.37172	0.415349
$316$	0	0
$317$	−34.5554	−1.94082	−0.970412	−	0.241454i	$-0.922376\pi$
$317$	−34.5554	−1.94082	−0.970412	+	0.241454i	$0.922376\pi$
$318$	0	0
$319$	8.95830	0.501569
$320$	0	0
$321$	−13.7609	−0.768056
$322$	0	0
$323$	−43.1127	−2.39885
$324$	0	0
$325$	32.1755	1.78478
$326$	0	0
$327$	−5.15480	−0.285061
$328$	0	0
$329$	−17.9067	−0.987227
$330$	0	0
$331$	−9.70159	−0.533247	−0.266624	−	0.963801i	$-0.585908\pi$
$331$	−9.70159	−0.533247	−0.266624	+	0.963801i	$0.585908\pi$
$332$	0	0
$333$	4.72415	0.258882
$334$	0	0
$335$	−5.45920	−0.298268
$336$	0	0
$337$	7.56357	0.412014	0.206007	−	0.978551i	$-0.433953\pi$
$337$	7.56357	0.412014	0.206007	+	0.978551i	$0.433953\pi$
$338$	0	0
$339$	−17.9885	−0.976999
$340$	0	0
$341$	−12.9340	−0.700414
$342$	0	0
$343$	−20.1597	−1.08852
$344$	0	0
$345$	26.2308	1.41222
$346$	0	0
$347$	−7.21673	−0.387415	−0.193707	−	0.981059i	$-0.562051\pi$
$347$	−7.21673	−0.387415	−0.193707	+	0.981059i	$0.562051\pi$
$348$	0	0
$349$	13.2805	0.710889	0.355444	−	0.934697i	$-0.384330\pi$
$349$	13.2805	0.710889	0.355444	+	0.934697i	$0.384330\pi$
$350$	0	0
$351$	5.00143	0.266957
$352$	0	0
$353$	−16.4226	−0.874087	−0.437043	−	0.899440i	$-0.643974\pi$
$353$	−16.4226	−0.874087	−0.437043	+	0.899440i	$0.643974\pi$
$354$	0	0
$355$	54.1531	2.87415
$356$	0	0
$357$	12.0081	0.635537
$358$	0	0
$359$	0.150138	0.00792397	0.00396198	−	0.999992i	$-0.498739\pi$
$359$	0.150138	0.00792397	0.00396198	+	0.999992i	$0.498739\pi$
$360$	0	0
$361$	42.2671	2.22458
$362$	0	0
$363$	−8.97402	−0.471014
$364$	0	0
$365$	14.9946	0.784853
$366$	0	0
$367$	21.6880	1.13211	0.566053	−	0.824369i	$-0.308470\pi$
$367$	21.6880	1.13211	0.566053	+	0.824369i	$0.308470\pi$
$368$	0	0
$369$	−11.8364	−0.616178
$370$	0	0
$371$	−23.5592	−1.22313
$372$	0	0
$373$	17.2389	0.892595	0.446298	−	0.894885i	$-0.352742\pi$
$373$	17.2389	0.892595	0.446298	+	0.894885i	$0.352742\pi$
$374$	0	0
$375$	4.84630	0.250262
$376$	0	0
$377$	31.4777	1.62118
$378$	0	0
$379$	−5.48673	−0.281834	−0.140917	−	0.990021i	$-0.545005\pi$
$379$	−5.48673	−0.281834	−0.140917	+	0.990021i	$0.545005\pi$
$380$	0	0
$381$	−16.3783	−0.839084
$382$	0	0
$383$	−15.0351	−0.768256	−0.384128	−	0.923280i	$-0.625498\pi$
$383$	−15.0351	−0.768256	−0.384128	+	0.923280i	$0.625498\pi$
$384$	0	0
$385$	10.4927	0.534756
$386$	0	0
$387$	−7.11615	−0.361734
$388$	0	0
$389$	−22.5552	−1.14359	−0.571797	−	0.820395i	$-0.693753\pi$
$389$	−22.5552	−1.14359	−0.571797	+	0.820395i	$0.693753\pi$
$390$	0	0
$391$	42.7285	2.16087
$392$	0	0
$393$	−12.0066	−0.605655
$394$	0	0
$395$	14.4160	0.725346
$396$	0	0
$397$	14.6044	0.732975	0.366488	−	0.930423i	$-0.380560\pi$
$397$	14.6044	0.732975	0.366488	+	0.930423i	$0.380560\pi$
$398$	0	0
$399$	−17.0647	−0.854301
$400$	0	0
$401$	−8.53046	−0.425991	−0.212995	−	0.977053i	$-0.568322\pi$
$401$	−8.53046	−0.425991	−0.212995	+	0.977053i	$0.568322\pi$
$402$	0	0
$403$	−45.4474	−2.26390
$404$	0	0
$405$	3.38131	0.168019
$406$	0	0
$407$	6.72421	0.333307
$408$	0	0
$409$	32.9598	1.62976	0.814879	−	0.579631i	$-0.196803\pi$
$409$	32.9598	1.62976	0.814879	+	0.579631i	$0.196803\pi$
$410$	0	0
$411$	7.24650	0.357444
$412$	0	0
$413$	−14.9112	−0.733731
$414$	0	0
$415$	14.9265	0.732711
$416$	0	0
$417$	−7.39592	−0.362180
$418$	0	0
$419$	22.4830	1.09837	0.549183	−	0.835702i	$-0.314939\pi$
$419$	22.4830	1.09837	0.549183	+	0.835702i	$0.314939\pi$
$420$	0	0
$421$	−2.13889	−0.104243	−0.0521215	−	0.998641i	$-0.516598\pi$
$421$	−2.13889	−0.104243	−0.0521215	+	0.998641i	$0.516598\pi$
$422$	0	0
$423$	−8.21355	−0.399356
$424$	0	0
$425$	35.4342	1.71881
$426$	0	0
$427$	−3.78734	−0.183282
$428$	0	0
$429$	7.11888	0.343703
$430$	0	0
$431$	−37.6297	−1.81256	−0.906278	−	0.422681i	$-0.861089\pi$
$431$	−37.6297	−1.81256	−0.906278	+	0.422681i	$0.861089\pi$
$432$	0	0
$433$	4.34024	0.208579	0.104289	−	0.994547i	$-0.466743\pi$
$433$	4.34024	0.208579	0.104289	+	0.994547i	$0.466743\pi$
$434$	0	0
$435$	21.2811	1.02035
$436$	0	0
$437$	−60.7212	−2.90469
$438$	0	0
$439$	−3.88893	−0.185609	−0.0928044	−	0.995684i	$-0.529583\pi$
$439$	−3.88893	−0.185609	−0.0928044	+	0.995684i	$0.529583\pi$
$440$	0	0
$441$	−2.24700	−0.107000
$442$	0	0
$443$	−4.35696	−0.207005	−0.103503	−	0.994629i	$-0.533005\pi$
$443$	−4.35696	−0.207005	−0.103503	+	0.994629i	$0.533005\pi$
$444$	0	0
$445$	−28.9567	−1.37268
$446$	0	0
$447$	23.8445	1.12780
$448$	0	0
$449$	18.8021	0.887328	0.443664	−	0.896193i	$-0.353678\pi$
$449$	18.8021	0.887328	0.443664	+	0.896193i	$0.353678\pi$
$450$	0	0
$451$	−16.8475	−0.793319
$452$	0	0
$453$	1.42298	0.0668573
$454$	0	0
$455$	36.8692	1.72845
$456$	0	0
$457$	26.5750	1.24313	0.621563	−	0.783364i	$-0.286498\pi$
$457$	26.5750	1.24313	0.621563	+	0.783364i	$0.286498\pi$
$458$	0	0
$459$	5.50797	0.257090
$460$	0	0
$461$	20.0895	0.935662	0.467831	−	0.883818i	$-0.345036\pi$
$461$	20.0895	0.935662	0.467831	+	0.883818i	$0.345036\pi$
$462$	0	0
$463$	29.0599	1.35053	0.675264	−	0.737576i	$-0.264030\pi$
$463$	29.0599	1.35053	0.675264	+	0.737576i	$0.264030\pi$
$464$	0	0
$465$	−30.7256	−1.42486
$466$	0	0
$467$	25.4129	1.17597	0.587984	−	0.808872i	$-0.299922\pi$
$467$	25.4129	1.17597	0.587984	+	0.808872i	$0.299922\pi$
$468$	0	0
$469$	−3.51988	−0.162533
$470$	0	0
$471$	19.8822	0.916124
$472$	0	0
$473$	−10.1289	−0.465727
$474$	0	0
$475$	−50.3553	−2.31046
$476$	0	0
$477$	−10.8063	−0.494786
$478$	0	0
$479$	2.72548	0.124530	0.0622651	−	0.998060i	$-0.480168\pi$
$479$	2.72548	0.124530	0.0622651	+	0.998060i	$0.480168\pi$
$480$	0	0
$481$	23.6275	1.07732
$482$	0	0
$483$	16.9126	0.769550
$484$	0	0
$485$	−34.8197	−1.58108
$486$	0	0
$487$	23.1754	1.05018	0.525089	−	0.851047i	$-0.324032\pi$
$487$	23.1754	1.05018	0.525089	+	0.851047i	$0.324032\pi$
$488$	0	0
$489$	14.3419	0.648565
$490$	0	0
$491$	−22.3994	−1.01087	−0.505434	−	0.862865i	$-0.668668\pi$
$491$	−22.3994	−1.01087	−0.505434	+	0.862865i	$0.668668\pi$
$492$	0	0
$493$	34.6657	1.56126
$494$	0	0
$495$	4.81285	0.216321
$496$	0	0
$497$	34.9158	1.56619
$498$	0	0
$499$	−22.4678	−1.00580	−0.502898	−	0.864346i	$-0.667733\pi$
$499$	−22.4678	−1.00580	−0.502898	+	0.864346i	$0.667733\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	34.4958	1.53809	0.769046	−	0.639193i	$-0.220731\pi$
$503$	34.4958	1.53809	0.769046	+	0.639193i	$0.220731\pi$
$504$	0	0
$505$	17.9600	0.799208
$506$	0	0
$507$	12.0143	0.533575
$508$	0	0
$509$	36.0445	1.59765	0.798823	−	0.601567i	$-0.205457\pi$
$509$	36.0445	1.59765	0.798823	+	0.601567i	$0.205457\pi$
$510$	0	0
$511$	9.66794	0.427684
$512$	0	0
$513$	−7.82733	−0.345585
$514$	0	0
$515$	−3.28585	−0.144792
$516$	0	0
$517$	−11.6909	−0.514165
$518$	0	0
$519$	−4.94096	−0.216884
$520$	0	0
$521$	−15.3650	−0.673154	−0.336577	−	0.941656i	$-0.609269\pi$
$521$	−15.3650	−0.673154	−0.336577	+	0.941656i	$0.609269\pi$
$522$	0	0
$523$	11.7306	0.512941	0.256471	−	0.966552i	$-0.417440\pi$
$523$	11.7306	0.512941	0.256471	+	0.966552i	$0.417440\pi$
$524$	0	0
$525$	14.0254	0.612118
$526$	0	0
$527$	−50.0503	−2.18022
$528$	0	0
$529$	37.1801	1.61653
$530$	0	0
$531$	−6.83956	−0.296812
$532$	0	0
$533$	−59.1989	−2.56419
$534$	0	0
$535$	−46.5297	−2.01166
$536$	0	0
$537$	1.94308	0.0838501
$538$	0	0
$539$	−3.19831	−0.137761
$540$	0	0
$541$	−26.9718	−1.15961	−0.579804	−	0.814756i	$-0.696871\pi$
$541$	−26.9718	−1.15961	−0.579804	+	0.814756i	$0.696871\pi$
$542$	0	0
$543$	11.2531	0.482919
$544$	0	0
$545$	−17.4300	−0.746619
$546$	0	0
$547$	38.4592	1.64440	0.822198	−	0.569202i	$-0.192748\pi$
$547$	38.4592	1.64440	0.822198	+	0.569202i	$0.192748\pi$
$548$	0	0
$549$	−1.73720	−0.0741420
$550$	0	0
$551$	−49.2632	−2.09868
$552$	0	0
$553$	9.29485	0.395257
$554$	0	0
$555$	15.9738	0.678051
$556$	0	0
$557$	−15.1906	−0.643648	−0.321824	−	0.946800i	$-0.604296\pi$
$557$	−15.1906	−0.643648	−0.321824	+	0.946800i	$0.604296\pi$
$558$	0	0
$559$	−35.5909	−1.50534
$560$	0	0
$561$	7.83986	0.330999
$562$	0	0
$563$	13.3816	0.563969	0.281985	−	0.959419i	$-0.409007\pi$
$563$	13.3816	0.563969	0.281985	+	0.959419i	$0.409007\pi$
$564$	0	0
$565$	−60.8246	−2.55891
$566$	0	0
$567$	2.18014	0.0915572
$568$	0	0
$569$	−19.6459	−0.823600	−0.411800	−	0.911274i	$-0.635100\pi$
$569$	−19.6459	−0.823600	−0.411800	+	0.911274i	$0.635100\pi$
$570$	0	0
$571$	−15.2035	−0.636249	−0.318124	−	0.948049i	$-0.603053\pi$
$571$	−15.2035	−0.636249	−0.318124	+	0.948049i	$0.603053\pi$
$572$	0	0
$573$	−8.97683	−0.375013
$574$	0	0
$575$	49.9066	2.08125
$576$	0	0
$577$	0.699702	0.0291290	0.0145645	−	0.999894i	$-0.495364\pi$
$577$	0.699702	0.0291290	0.0145645	+	0.999894i	$0.495364\pi$
$578$	0	0
$579$	−10.0704	−0.418512
$580$	0	0
$581$	9.62400	0.399271
$582$	0	0
$583$	−15.3813	−0.637030
$584$	0	0
$585$	16.9114	0.699200
$586$	0	0
$587$	−12.4446	−0.513645	−0.256822	−	0.966459i	$-0.582676\pi$
$587$	−12.4446	−0.513645	−0.256822	+	0.966459i	$0.582676\pi$
$588$	0	0
$589$	71.1260	2.93070
$590$	0	0
$591$	−19.4156	−0.798651
$592$	0	0
$593$	17.1187	0.702982	0.351491	−	0.936191i	$-0.385675\pi$
$593$	17.1187	0.702982	0.351491	+	0.936191i	$0.385675\pi$
$594$	0	0
$595$	40.6032	1.66457
$596$	0	0
$597$	5.98916	0.245120
$598$	0	0
$599$	−16.8211	−0.687289	−0.343645	−	0.939100i	$-0.611662\pi$
$599$	−16.8211	−0.687289	−0.343645	+	0.939100i	$0.611662\pi$
$600$	0	0
$601$	−22.5830	−0.921178	−0.460589	−	0.887614i	$-0.652362\pi$
$601$	−22.5830	−0.921178	−0.460589	+	0.887614i	$0.652362\pi$
$602$	0	0
$603$	−1.61452	−0.0657484
$604$	0	0
$605$	−30.3440	−1.23366
$606$	0	0
$607$	−9.60797	−0.389976	−0.194988	−	0.980806i	$-0.562467\pi$
$607$	−9.60797	−0.389976	−0.194988	+	0.980806i	$0.562467\pi$
$608$	0	0
$609$	13.7212	0.556012
$610$	0	0
$611$	−41.0795	−1.66190
$612$	0	0
$613$	31.4590	1.27062	0.635308	−	0.772259i	$-0.280873\pi$
$613$	31.4590	1.27062	0.635308	+	0.772259i	$0.280873\pi$
$614$	0	0
$615$	−40.0225	−1.61386
$616$	0	0
$617$	−15.2764	−0.615005	−0.307503	−	0.951547i	$-0.599493\pi$
$617$	−15.2764	−0.615005	−0.307503	+	0.951547i	$0.599493\pi$
$618$	0	0
$619$	34.1164	1.37126	0.685628	−	0.727952i	$-0.259528\pi$
$619$	34.1164	1.37126	0.685628	+	0.727952i	$0.259528\pi$
$620$	0	0
$621$	7.75758	0.311301
$622$	0	0
$623$	−18.6702	−0.748005
$624$	0	0
$625$	−15.7795	−0.631178
$626$	0	0
$627$	−11.1412	−0.444935
$628$	0	0
$629$	26.0205	1.03750
$630$	0	0
$631$	−29.8395	−1.18789	−0.593946	−	0.804505i	$-0.702431\pi$
$631$	−29.8395	−1.18789	−0.593946	+	0.804505i	$0.702431\pi$
$632$	0	0
$633$	−7.53399	−0.299449
$634$	0	0
$635$	−55.3800	−2.19769
$636$	0	0
$637$	−11.2382	−0.445275
$638$	0	0
$639$	16.0154	0.633560
$640$	0	0
$641$	1.98414	0.0783690	0.0391845	−	0.999232i	$-0.487524\pi$
$641$	1.98414	0.0783690	0.0391845	+	0.999232i	$0.487524\pi$
$642$	0	0
$643$	15.6832	0.618484	0.309242	−	0.950983i	$-0.399925\pi$
$643$	15.6832	0.618484	0.309242	+	0.950983i	$0.399925\pi$
$644$	0	0
$645$	−24.0619	−0.947436
$646$	0	0
$647$	−9.16451	−0.360294	−0.180147	−	0.983640i	$-0.557657\pi$
$647$	−9.16451	−0.360294	−0.180147	+	0.983640i	$0.557657\pi$
$648$	0	0
$649$	−9.73521	−0.382141
$650$	0	0
$651$	−19.8107	−0.776441
$652$	0	0
$653$	12.0883	0.473050	0.236525	−	0.971625i	$-0.423991\pi$
$653$	12.0883	0.473050	0.236525	+	0.971625i	$0.423991\pi$
$654$	0	0
$655$	−40.5982	−1.58630
$656$	0	0
$657$	4.43455	0.173008
$658$	0	0
$659$	−42.2128	−1.64438	−0.822188	−	0.569217i	$-0.807247\pi$
$659$	−42.2128	−1.64438	−0.822188	+	0.569217i	$0.807247\pi$
$660$	0	0
$661$	−10.0116	−0.389404	−0.194702	−	0.980862i	$-0.562374\pi$
$661$	−10.0116	−0.389404	−0.194702	+	0.980862i	$0.562374\pi$
$662$	0	0
$663$	27.5477	1.06987
$664$	0	0
$665$	−57.7009	−2.23754
$666$	0	0
$667$	48.8242	1.89048
$668$	0	0
$669$	1.15074	0.0444901
$670$	0	0
$671$	−2.47268	−0.0954567
$672$	0	0
$673$	15.8885	0.612457	0.306229	−	0.951958i	$-0.400933\pi$
$673$	15.8885	0.612457	0.306229	+	0.951958i	$0.400933\pi$
$674$	0	0
$675$	6.43326	0.247616
$676$	0	0
$677$	31.1756	1.19817	0.599087	−	0.800684i	$-0.295530\pi$
$677$	31.1756	1.19817	0.599087	+	0.800684i	$0.295530\pi$
$678$	0	0
$679$	−22.4504	−0.861566
$680$	0	0
$681$	−25.0325	−0.959248
$682$	0	0
$683$	19.7517	0.755780	0.377890	−	0.925851i	$-0.376650\pi$
$683$	19.7517	0.755780	0.377890	+	0.925851i	$0.376650\pi$
$684$	0	0
$685$	24.5027	0.936199
$686$	0	0
$687$	7.40955	0.282692
$688$	0	0
$689$	−54.0469	−2.05902
$690$	0	0
$691$	−35.7668	−1.36063	−0.680316	−	0.732919i	$-0.738158\pi$
$691$	−35.7668	−1.36063	−0.680316	+	0.732919i	$0.738158\pi$
$692$	0	0
$693$	3.10314	0.117878
$694$	0	0
$695$	−25.0079	−0.948604
$696$	0	0
$697$	−65.1944	−2.46942
$698$	0	0
$699$	−2.93657	−0.111071
$700$	0	0
$701$	−23.2240	−0.877160	−0.438580	−	0.898692i	$-0.644518\pi$
$701$	−23.2240	−0.877160	−0.438580	+	0.898692i	$0.644518\pi$
$702$	0	0
$703$	−36.9775	−1.39463
$704$	0	0
$705$	−27.7726	−1.04598
$706$	0	0
$707$	11.5799	0.435506
$708$	0	0
$709$	19.8679	0.746156	0.373078	−	0.927800i	$-0.378302\pi$
$709$	19.8679	0.746156	0.373078	+	0.927800i	$0.378302\pi$
$710$	0	0
$711$	4.26342	0.159891
$712$	0	0
$713$	−70.4923	−2.63996
$714$	0	0
$715$	24.0711	0.900209
$716$	0	0
$717$	26.6134	0.993895
$718$	0	0
$719$	5.79193	0.216003	0.108001	−	0.994151i	$-0.465555\pi$
$719$	5.79193	0.216003	0.108001	+	0.994151i	$0.465555\pi$
$720$	0	0
$721$	−2.11859	−0.0789004
$722$	0	0
$723$	16.8419	0.626357
$724$	0	0
$725$	40.4893	1.50373
$726$	0	0
$727$	−22.2264	−0.824331	−0.412165	−	0.911109i	$-0.635227\pi$
$727$	−22.2264	−0.824331	−0.412165	+	0.911109i	$0.635227\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−39.1955	−1.44970
$732$	0	0
$733$	−2.28416	−0.0843672	−0.0421836	−	0.999110i	$-0.513431\pi$
$733$	−2.28416	−0.0843672	−0.0421836	+	0.999110i	$0.513431\pi$
$734$	0	0
$735$	−7.59781	−0.280249
$736$	0	0
$737$	−2.29806	−0.0846500
$738$	0	0
$739$	1.67059	0.0614538	0.0307269	−	0.999528i	$-0.490218\pi$
$739$	1.67059	0.0614538	0.0307269	+	0.999528i	$0.490218\pi$
$740$	0	0
$741$	−39.1479	−1.43813
$742$	0	0
$743$	−49.4665	−1.81475	−0.907375	−	0.420323i	$-0.861917\pi$
$743$	−49.4665	−1.81475	−0.907375	+	0.420323i	$0.861917\pi$
$744$	0	0
$745$	80.6255	2.95389
$746$	0	0
$747$	4.41440	0.161515
$748$	0	0
$749$	−30.0006	−1.09620
$750$	0	0
$751$	38.2975	1.39749	0.698747	−	0.715369i	$-0.253741\pi$
$751$	38.2975	1.39749	0.698747	+	0.715369i	$0.253741\pi$
$752$	0	0
$753$	5.78068	0.210660
$754$	0	0
$755$	4.81153	0.175110
$756$	0	0
$757$	30.3444	1.10288	0.551442	−	0.834213i	$-0.314078\pi$
$757$	30.3444	1.10288	0.551442	+	0.834213i	$0.314078\pi$
$758$	0	0
$759$	11.0419	0.400795
$760$	0	0
$761$	13.5231	0.490214	0.245107	−	0.969496i	$-0.421177\pi$
$761$	13.5231	0.490214	0.245107	+	0.969496i	$0.421177\pi$
$762$	0	0
$763$	−11.2382	−0.406849
$764$	0	0
$765$	18.6241	0.673358
$766$	0	0
$767$	−34.2076	−1.23517
$768$	0	0
$769$	12.8744	0.464261	0.232131	−	0.972685i	$-0.425430\pi$
$769$	12.8744	0.464261	0.232131	+	0.972685i	$0.425430\pi$
$770$	0	0
$771$	28.5244	1.02728
$772$	0	0
$773$	−5.91163	−0.212627	−0.106313	−	0.994333i	$-0.533905\pi$
$773$	−5.91163	−0.212627	−0.106313	+	0.994333i	$0.533905\pi$
$774$	0	0
$775$	−58.4583	−2.09988
$776$	0	0
$777$	10.2993	0.369485
$778$	0	0
$779$	92.6473	3.31943
$780$	0	0
$781$	22.7958	0.815699
$782$	0	0
$783$	6.29374	0.224920
$784$	0	0
$785$	67.2279	2.39947
$786$	0	0
$787$	26.3311	0.938601	0.469301	−	0.883039i	$-0.344506\pi$
$787$	26.3311	0.938601	0.469301	+	0.883039i	$0.344506\pi$
$788$	0	0
$789$	4.27282	0.152116
$790$	0	0
$791$	−39.2173	−1.39441
$792$	0	0
$793$	−8.68851	−0.308538
$794$	0	0
$795$	−36.5394	−1.29592
$796$	0	0
$797$	−50.7848	−1.79889	−0.899445	−	0.437035i	$-0.856029\pi$
$797$	−50.7848	−1.79889	−0.899445	+	0.437035i	$0.856029\pi$
$798$	0	0
$799$	−45.2400	−1.60047
$800$	0	0
$801$	−8.56376	−0.302586
$802$	0	0
$803$	6.31200	0.222746
$804$	0	0
$805$	57.1867	2.01557
$806$	0	0
$807$	7.38718	0.260041
$808$	0	0
$809$	22.0939	0.776781	0.388391	−	0.921495i	$-0.373031\pi$
$809$	22.0939	0.776781	0.388391	+	0.921495i	$0.373031\pi$
$810$	0	0
$811$	−10.9802	−0.385567	−0.192783	−	0.981241i	$-0.561751\pi$
$811$	−10.9802	−0.385567	−0.192783	+	0.981241i	$0.561751\pi$
$812$	0	0
$813$	14.7734	0.518126
$814$	0	0
$815$	48.4946	1.69869
$816$	0	0
$817$	55.7004	1.94871
$818$	0	0
$819$	10.9038	0.381010
$820$	0	0
$821$	22.8208	0.796451	0.398226	−	0.917287i	$-0.369626\pi$
$821$	22.8208	0.796451	0.398226	+	0.917287i	$0.369626\pi$
$822$	0	0
$823$	4.73811	0.165160	0.0825802	−	0.996584i	$-0.473684\pi$
$823$	4.73811	0.165160	0.0825802	+	0.996584i	$0.473684\pi$
$824$	0	0
$825$	9.15690	0.318802
$826$	0	0
$827$	17.6761	0.614658	0.307329	−	0.951603i	$-0.400565\pi$
$827$	17.6761	0.614658	0.307329	+	0.951603i	$0.400565\pi$
$828$	0	0
$829$	−6.25537	−0.217258	−0.108629	−	0.994082i	$-0.534646\pi$
$829$	−6.25537	−0.217258	−0.108629	+	0.994082i	$0.534646\pi$
$830$	0	0
$831$	−2.05325	−0.0712264
$832$	0	0
$833$	−12.3764	−0.428817
$834$	0	0
$835$	−3.38131	−0.117015
$836$	0	0
$837$	−9.08688	−0.314089
$838$	0	0
$839$	31.0898	1.07334	0.536669	−	0.843793i	$-0.319682\pi$
$839$	31.0898	1.07334	0.536669	+	0.843793i	$0.319682\pi$
$840$	0	0
$841$	10.6111	0.365901
$842$	0	0
$843$	25.3998	0.874816
$844$	0	0
$845$	40.6242	1.39751
$846$	0	0
$847$	−19.5646	−0.672248
$848$	0	0
$849$	2.37163	0.0813941
$850$	0	0
$851$	36.6480	1.25628
$852$	0	0
$853$	−3.71143	−0.127077	−0.0635385	−	0.997979i	$-0.520239\pi$
$853$	−3.71143	−0.127077	−0.0635385	+	0.997979i	$0.520239\pi$
$854$	0	0
$855$	−26.4666	−0.905140
$856$	0	0
$857$	−26.9041	−0.919026	−0.459513	−	0.888171i	$-0.651976\pi$
$857$	−26.9041	−0.919026	−0.459513	+	0.888171i	$0.651976\pi$
$858$	0	0
$859$	−15.3548	−0.523901	−0.261950	−	0.965081i	$-0.584366\pi$
$859$	−15.3548	−0.523901	−0.261950	+	0.965081i	$0.584366\pi$
$860$	0	0
$861$	−25.8050	−0.879430
$862$	0	0
$863$	−52.1838	−1.77636	−0.888179	−	0.459498i	$-0.848029\pi$
$863$	−52.1838	−1.77636	−0.888179	+	0.459498i	$0.848029\pi$
$864$	0	0
$865$	−16.7069	−0.568052
$866$	0	0
$867$	13.3377	0.452972
$868$	0	0
$869$	6.06842	0.205857
$870$	0	0
$871$	−8.07492	−0.273608
$872$	0	0
$873$	−10.2977	−0.348524
$874$	0	0
$875$	10.5656	0.357183
$876$	0	0
$877$	48.6674	1.64338	0.821691	−	0.569934i	$-0.193031\pi$
$877$	48.6674	1.64338	0.821691	+	0.569934i	$0.193031\pi$
$878$	0	0
$879$	8.38097	0.282683
$880$	0	0
$881$	−0.713672	−0.0240442	−0.0120221	−	0.999928i	$-0.503827\pi$
$881$	−0.713672	−0.0240442	−0.0120221	+	0.999928i	$0.503827\pi$
$882$	0	0
$883$	−8.62220	−0.290160	−0.145080	−	0.989420i	$-0.546344\pi$
$883$	−8.62220	−0.290160	−0.145080	+	0.989420i	$0.546344\pi$
$884$	0	0
$885$	−23.1267	−0.777395
$886$	0	0
$887$	8.51090	0.285768	0.142884	−	0.989739i	$-0.454362\pi$
$887$	8.51090	0.285768	0.142884	+	0.989739i	$0.454362\pi$
$888$	0	0
$889$	−35.7069	−1.19757
$890$	0	0
$891$	1.42337	0.0476846
$892$	0	0
$893$	64.2902	2.15139
$894$	0	0
$895$	6.57016	0.219616
$896$	0	0
$897$	38.7990	1.29546
$898$	0	0
$899$	−57.1905	−1.90741
$900$	0	0
$901$	−59.5207	−1.98292
$902$	0	0
$903$	−15.5142	−0.516279
$904$	0	0
$905$	38.0504	1.26484
$906$	0	0
$907$	−29.3979	−0.976143	−0.488071	−	0.872804i	$-0.662299\pi$
$907$	−29.3979	−0.976143	−0.488071	+	0.872804i	$0.662299\pi$
$908$	0	0
$909$	5.31154	0.176173
$910$	0	0
$911$	0.694625	0.0230140	0.0115070	−	0.999934i	$-0.496337\pi$
$911$	0.694625	0.0230140	0.0115070	+	0.999934i	$0.496337\pi$
$912$	0	0
$913$	6.28332	0.207947
$914$	0	0
$915$	−5.87403	−0.194189
$916$	0	0
$917$	−26.1761	−0.864412
$918$	0	0
$919$	−45.2759	−1.49351	−0.746757	−	0.665097i	$-0.768390\pi$
$919$	−45.2759	−1.49351	−0.746757	+	0.665097i	$0.768390\pi$
$920$	0	0
$921$	23.1839	0.763936
$922$	0	0
$923$	80.1000	2.63652
$924$	0	0
$925$	30.3917	0.999273
$926$	0	0
$927$	−0.971769	−0.0319171
$928$	0	0
$929$	−50.0209	−1.64113	−0.820566	−	0.571551i	$-0.806342\pi$
$929$	−50.0209	−1.64113	−0.820566	+	0.571551i	$0.806342\pi$
$930$	0	0
$931$	17.5880	0.576424
$932$	0	0
$933$	22.1324	0.724581
$934$	0	0
$935$	26.5090	0.866938
$936$	0	0
$937$	−52.3630	−1.71062	−0.855312	−	0.518114i	$-0.826634\pi$
$937$	−52.3630	−1.71062	−0.855312	+	0.518114i	$0.826634\pi$
$938$	0	0
$939$	−25.6297	−0.836393
$940$	0	0
$941$	−49.6749	−1.61935	−0.809677	−	0.586875i	$-0.800358\pi$
$941$	−49.6749	−1.61935	−0.809677	+	0.586875i	$0.800358\pi$
$942$	0	0
$943$	−91.8218	−2.99013
$944$	0	0
$945$	7.37172	0.239802
$946$	0	0
$947$	−31.6023	−1.02694	−0.513468	−	0.858109i	$-0.671640\pi$
$947$	−31.6023	−1.02694	−0.513468	+	0.858109i	$0.671640\pi$
$948$	0	0
$949$	22.1791	0.719965
$950$	0	0
$951$	−34.5554	−1.12054
$952$	0	0
$953$	−44.2157	−1.43229	−0.716143	−	0.697953i	$-0.754094\pi$
$953$	−44.2157	−1.43229	−0.716143	+	0.697953i	$0.754094\pi$
$954$	0	0
$955$	−30.3535	−0.982215
$956$	0	0
$957$	8.95830	0.289581
$958$	0	0
$959$	15.7984	0.510156
$960$	0	0
$961$	51.5715	1.66360
$962$	0	0
$963$	−13.7609	−0.443437
$964$	0	0
$965$	−34.0512	−1.09615
$966$	0	0
$967$	−31.8623	−1.02462	−0.512311	−	0.858800i	$-0.671210\pi$
$967$	−31.8623	−1.02462	−0.512311	+	0.858800i	$0.671210\pi$
$968$	0	0
$969$	−43.1127	−1.38498
$970$	0	0
$971$	24.6768	0.791917	0.395958	−	0.918268i	$-0.370412\pi$
$971$	24.6768	0.791917	0.395958	+	0.918268i	$0.370412\pi$
$972$	0	0
$973$	−16.1241	−0.516916
$974$	0	0
$975$	32.1755	1.03044
$976$	0	0
$977$	−27.3621	−0.875392	−0.437696	−	0.899123i	$-0.644205\pi$
$977$	−27.3621	−0.875392	−0.437696	+	0.899123i	$0.644205\pi$
$978$	0	0
$979$	−12.1894	−0.389574
$980$	0	0
$981$	−5.15480	−0.164580
$982$	0	0
$983$	−10.2808	−0.327906	−0.163953	−	0.986468i	$-0.552425\pi$
$983$	−10.2808	−0.327906	−0.163953	+	0.986468i	$0.552425\pi$
$984$	0	0
$985$	−65.6502	−2.09179
$986$	0	0
$987$	−17.9067	−0.569975
$988$	0	0
$989$	−55.2041	−1.75539
$990$	0	0
$991$	6.86212	0.217982	0.108991	−	0.994043i	$-0.465238\pi$
$991$	6.86212	0.217982	0.108991	+	0.994043i	$0.465238\pi$
$992$	0	0
$993$	−9.70159	−0.307871
$994$	0	0
$995$	20.2512	0.642006
$996$	0	0
$997$	41.7713	1.32291	0.661456	−	0.749984i	$-0.269939\pi$
$997$	41.7713	1.32291	0.661456	+	0.749984i	$0.269939\pi$
$998$	0	0
$999$	4.72415	0.149466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.m.1.12	✓	13
4.3	odd	2		8016.2.a.bg.1.12		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.m.1.12	✓	13	1.1	even	1	trivial
8016.2.a.bg.1.12		13	4.3	odd	2

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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.38131 q^{5} +2.18014 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.38131 q^{5} +2.18014 q^{7} +1.00000 q^{9} +1.42337 q^{11} +5.00143 q^{13} +3.38131 q^{15} +5.50797 q^{17} -7.82733 q^{19} +2.18014 q^{21} +7.75758 q^{23} +6.43326 q^{25} +1.00000 q^{27} +6.29374 q^{29} -9.08688 q^{31} +1.42337 q^{33} +7.37172 q^{35} +4.72415 q^{37} +5.00143 q^{39} -11.8364 q^{41} -7.11615 q^{43} +3.38131 q^{45} -8.21355 q^{47} -2.24700 q^{49} +5.50797 q^{51} -10.8063 q^{53} +4.81285 q^{55} -7.82733 q^{57} -6.83956 q^{59} -1.73720 q^{61} +2.18014 q^{63} +16.9114 q^{65} -1.61452 q^{67} +7.75758 q^{69} +16.0154 q^{71} +4.43455 q^{73} +6.43326 q^{75} +3.10314 q^{77} +4.26342 q^{79} +1.00000 q^{81} +4.41440 q^{83} +18.6241 q^{85} +6.29374 q^{87} -8.56376 q^{89} +10.9038 q^{91} -9.08688 q^{93} -26.4666 q^{95} -10.2977 q^{97} +1.42337 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9	\( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9} + 11 q^{11} + 12 q^{13} + 2 q^{15} + 15 q^{17} + 14 q^{19} + q^{21} + 9 q^{23} + 37 q^{25} + 13 q^{27} - 3 q^{29} - 17 q^{31} + 11 q^{33} + 15 q^{35} + 16 q^{37} + 12 q^{39} + 12 q^{41} + 20 q^{43} + 2 q^{45} - 6 q^{47} + 26 q^{49} + 15 q^{51} - 12 q^{53} + 7 q^{55} + 14 q^{57} + 14 q^{59} + 24 q^{61} + q^{63} + 8 q^{65} + 3 q^{67} + 9 q^{69} + 17 q^{71} + 34 q^{73} + 37 q^{75} + 30 q^{77} + 10 q^{79} + 13 q^{81} + 44 q^{83} + 25 q^{85} - 3 q^{87} + 25 q^{89} + 29 q^{91} - 17 q^{93} - 15 q^{95} + 38 q^{97} + 11 q^{99}+O(q^{100}) \) 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9 + 11 * q^11 + 12 * q^13 + 2 * q^15 + 15 * q^17 + 14 * q^19 + q^21 + 9 * q^23 + 37 * q^25 + 13 * q^27 - 3 * q^29 - 17 * q^31 + 11 * q^33 + 15 * q^35 + 16 * q^37 + 12 * q^39 + 12 * q^41 + 20 * q^43 + 2 * q^45 - 6 * q^47 + 26 * q^49 + 15 * q^51 - 12 * q^53 + 7 * q^55 + 14 * q^57 + 14 * q^59 + 24 * q^61 + q^63 + 8 * q^65 + 3 * q^67 + 9 * q^69 + 17 * q^71 + 34 * q^73 + 37 * q^75 + 30 * q^77 + 10 * q^79 + 13 * q^81 + 44 * q^83 + 25 * q^85 - 3 * q^87 + 25 * q^89 + 29 * q^91 - 17 * q^93 - 15 * q^95 + 38 * q^97 + 11 * q^99

Introduction

L-functions

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Varieties

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Database

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