LMFDB - Embedded newform 8001.2.a.ba.1.30

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.30
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.50508 q^{2} +0.265269 q^{4} -2.12799 q^{5} +1.00000 q^{7} -2.61091 q^{8} +O(q^{10})$	\(q+1.50508 q^{2} +0.265269 q^{4} -2.12799 q^{5} +1.00000 q^{7} -2.61091 q^{8} -3.20280 q^{10} +1.63093 q^{11} +2.77338 q^{13} +1.50508 q^{14} -4.46017 q^{16} -4.81311 q^{17} +1.82102 q^{19} -0.564490 q^{20} +2.45468 q^{22} +4.21796 q^{23} -0.471655 q^{25} +4.17416 q^{26} +0.265269 q^{28} -2.17542 q^{29} -5.01640 q^{31} -1.49110 q^{32} -7.24411 q^{34} -2.12799 q^{35} -4.80716 q^{37} +2.74078 q^{38} +5.55599 q^{40} -0.803009 q^{41} +11.2895 q^{43} +0.432635 q^{44} +6.34837 q^{46} -7.13834 q^{47} +1.00000 q^{49} -0.709880 q^{50} +0.735692 q^{52} +7.35769 q^{53} -3.47061 q^{55} -2.61091 q^{56} -3.27418 q^{58} +1.99972 q^{59} +1.64713 q^{61} -7.55008 q^{62} +6.67612 q^{64} -5.90173 q^{65} +10.7832 q^{67} -1.27677 q^{68} -3.20280 q^{70} +4.48471 q^{71} +5.53294 q^{73} -7.23516 q^{74} +0.483060 q^{76} +1.63093 q^{77} -0.583282 q^{79} +9.49120 q^{80} -1.20859 q^{82} -4.31083 q^{83} +10.2422 q^{85} +16.9916 q^{86} -4.25822 q^{88} -0.157199 q^{89} +2.77338 q^{91} +1.11889 q^{92} -10.7438 q^{94} -3.87512 q^{95} +2.33005 q^{97} +1.50508 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) $ 40 * q + 54 * q^4 + 40 * q^7	$ 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) $ 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

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 $

$2$	1.50508	1.06425	0.532127	−	0.846665i	$-0.321393\pi$
$2$	1.50508	1.06425	0.532127	+	0.846665i	$0.321393\pi$
$3$	0	0
$3$	0	0
$4$	0.265269	0.132634
$5$	−2.12799	−0.951666	−0.475833	−	0.879536i	$-0.657853\pi$
$5$	−2.12799	−0.951666	−0.475833	+	0.879536i	$0.657853\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.61091	−0.923096
$9$	0	0
$10$	−3.20280	−1.01281
$11$	1.63093	0.491744	0.245872	−	0.969302i	$-0.420926\pi$
$11$	1.63093	0.491744	0.245872	+	0.969302i	$0.420926\pi$
$12$	0	0
$13$	2.77338	0.769198	0.384599	−	0.923084i	$-0.374340\pi$
$13$	2.77338	0.769198	0.384599	+	0.923084i	$0.374340\pi$
$14$	1.50508	0.402250
$15$	0	0
$16$	−4.46017	−1.11504
$17$	−4.81311	−1.16735	−0.583675	−	0.811988i	$-0.698386\pi$
$17$	−4.81311	−1.16735	−0.583675	+	0.811988i	$0.698386\pi$
$18$	0	0
$19$	1.82102	0.417771	0.208885	−	0.977940i	$-0.433016\pi$
$19$	1.82102	0.417771	0.208885	+	0.977940i	$0.433016\pi$
$20$	−0.564490	−0.126224
$21$	0	0
$22$	2.45468	0.523340
$23$	4.21796	0.879505	0.439752	−	0.898119i	$-0.355066\pi$
$23$	4.21796	0.879505	0.439752	+	0.898119i	$0.355066\pi$
$24$	0	0
$25$	−0.471655	−0.0943311
$26$	4.17416	0.818621
$27$	0	0
$28$	0.265269	0.0501311
$29$	−2.17542	−0.403965	−0.201983	−	0.979389i	$-0.564738\pi$
$29$	−2.17542	−0.403965	−0.201983	+	0.979389i	$0.564738\pi$
$30$	0	0
$31$	−5.01640	−0.900972	−0.450486	−	0.892784i	$-0.648749\pi$
$31$	−5.01640	−0.900972	−0.450486	+	0.892784i	$0.648749\pi$
$32$	−1.49110	−0.263591
$33$	0	0
$34$	−7.24411	−1.24236
$35$	−2.12799	−0.359696
$36$	0	0
$37$	−4.80716	−0.790292	−0.395146	−	0.918618i	$-0.629306\pi$
$37$	−4.80716	−0.790292	−0.395146	+	0.918618i	$0.629306\pi$
$38$	2.74078	0.444614
$39$	0	0
$40$	5.55599	0.878480
$41$	−0.803009	−0.125409	−0.0627044	−	0.998032i	$-0.519973\pi$
$41$	−0.803009	−0.125409	−0.0627044	+	0.998032i	$0.519973\pi$
$42$	0	0
$43$	11.2895	1.72164	0.860818	−	0.508913i	$-0.169953\pi$
$43$	11.2895	1.72164	0.860818	+	0.508913i	$0.169953\pi$
$44$	0.432635	0.0652223
$45$	0	0
$46$	6.34837	0.936016
$47$	−7.13834	−1.04123	−0.520617	−	0.853790i	$-0.674298\pi$
$47$	−7.13834	−1.04123	−0.520617	+	0.853790i	$0.674298\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.709880	−0.100392
$51$	0	0
$52$	0.735692	0.102022
$53$	7.35769	1.01066	0.505328	−	0.862927i	$-0.331371\pi$
$53$	7.35769	1.01066	0.505328	+	0.862927i	$0.331371\pi$
$54$	0	0
$55$	−3.47061	−0.467977
$56$	−2.61091	−0.348898
$57$	0	0
$58$	−3.27418	−0.429921
$59$	1.99972	0.260341	0.130170	−	0.991492i	$-0.458448\pi$
$59$	1.99972	0.260341	0.130170	+	0.991492i	$0.458448\pi$
$60$	0	0
$61$	1.64713	0.210893	0.105447	−	0.994425i	$-0.466373\pi$
$61$	1.64713	0.210893	0.105447	+	0.994425i	$0.466373\pi$
$62$	−7.55008	−0.958862
$63$	0	0
$64$	6.67612	0.834515
$65$	−5.90173	−0.732019
$66$	0	0
$67$	10.7832	1.31737	0.658687	−	0.752417i	$-0.271112\pi$
$67$	10.7832	1.31737	0.658687	+	0.752417i	$0.271112\pi$
$68$	−1.27677	−0.154831
$69$	0	0
$70$	−3.20280	−0.382808
$71$	4.48471	0.532238	0.266119	−	0.963940i	$-0.414259\pi$
$71$	4.48471	0.532238	0.266119	+	0.963940i	$0.414259\pi$
$72$	0	0
$73$	5.53294	0.647582	0.323791	−	0.946129i	$-0.395043\pi$
$73$	5.53294	0.647582	0.323791	+	0.946129i	$0.395043\pi$
$74$	−7.23516	−0.841070
$75$	0	0
$76$	0.483060	0.0554108
$77$	1.63093	0.185862
$78$	0	0
$79$	−0.583282	−0.0656244	−0.0328122	−	0.999462i	$-0.510446\pi$
$79$	−0.583282	−0.0656244	−0.0328122	+	0.999462i	$0.510446\pi$
$80$	9.49120	1.06115
$81$	0	0
$82$	−1.20859	−0.133467
$83$	−4.31083	−0.473175	−0.236588	−	0.971610i	$-0.576029\pi$
$83$	−4.31083	−0.473175	−0.236588	+	0.971610i	$0.576029\pi$
$84$	0	0
$85$	10.2422	1.11093
$86$	16.9916	1.83226
$87$	0	0
$88$	−4.25822	−0.453927
$89$	−0.157199	−0.0166630	−0.00833151	−	0.999965i	$-0.502652\pi$
$89$	−0.157199	−0.0166630	−0.00833151	+	0.999965i	$0.502652\pi$
$90$	0	0
$91$	2.77338	0.290729
$92$	1.11889	0.116653
$93$	0	0
$94$	−10.7438	−1.10814
$95$	−3.87512	−0.397579
$96$	0	0
$97$	2.33005	0.236581	0.118290	−	0.992979i	$-0.462259\pi$
$97$	2.33005	0.236581	0.118290	+	0.992979i	$0.462259\pi$
$98$	1.50508	0.152036
$99$	0	0
$100$	−0.125116	−0.0125116
$101$	5.25621	0.523013	0.261506	−	0.965202i	$-0.415781\pi$
$101$	5.25621	0.523013	0.261506	+	0.965202i	$0.415781\pi$
$102$	0	0
$103$	12.1853	1.20066	0.600328	−	0.799754i	$-0.295037\pi$
$103$	12.1853	1.20066	0.600328	+	0.799754i	$0.295037\pi$
$104$	−7.24105	−0.710044
$105$	0	0
$106$	11.0739	1.07559
$107$	−10.1443	−0.980690	−0.490345	−	0.871528i	$-0.663129\pi$
$107$	−10.1443	−0.980690	−0.490345	+	0.871528i	$0.663129\pi$
$108$	0	0
$109$	5.63005	0.539261	0.269630	−	0.962964i	$-0.413098\pi$
$109$	5.63005	0.539261	0.269630	+	0.962964i	$0.413098\pi$
$110$	−5.22355	−0.498046
$111$	0	0
$112$	−4.46017	−0.421446
$113$	−10.0500	−0.945425	−0.472713	−	0.881217i	$-0.656725\pi$
$113$	−10.0500	−0.945425	−0.472713	+	0.881217i	$0.656725\pi$
$114$	0	0
$115$	−8.97577	−0.836995
$116$	−0.577071	−0.0535797
$117$	0	0
$118$	3.00973	0.277069
$119$	−4.81311	−0.441217
$120$	0	0
$121$	−8.34006	−0.758187
$122$	2.47906	0.224444
$123$	0	0
$124$	−1.33069	−0.119500
$125$	11.6436	1.04144
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	13.0303	1.15173
$129$	0	0
$130$	−8.88258	−0.779054
$131$	−1.48299	−0.129570	−0.0647848	−	0.997899i	$-0.520636\pi$
$131$	−1.48299	−0.129570	−0.0647848	+	0.997899i	$0.520636\pi$
$132$	0	0
$133$	1.82102	0.157903
$134$	16.2296	1.40202
$135$	0	0
$136$	12.5666	1.07758
$137$	11.8958	1.01633	0.508164	−	0.861261i	$-0.330325\pi$
$137$	11.8958	1.01633	0.508164	+	0.861261i	$0.330325\pi$
$138$	0	0
$139$	−0.572818	−0.0485858	−0.0242929	−	0.999705i	$-0.507733\pi$
$139$	−0.572818	−0.0485858	−0.0242929	+	0.999705i	$0.507733\pi$
$140$	−0.564490	−0.0477081
$141$	0	0
$142$	6.74986	0.566435
$143$	4.52320	0.378249
$144$	0	0
$145$	4.62927	0.384440
$146$	8.32752	0.689191
$147$	0	0
$148$	−1.27519	−0.104820
$149$	1.43612	0.117651	0.0588256	−	0.998268i	$-0.481264\pi$
$149$	1.43612	0.117651	0.0588256	+	0.998268i	$0.481264\pi$
$150$	0	0
$151$	−14.2726	−1.16149	−0.580745	−	0.814085i	$-0.697239\pi$
$151$	−14.2726	−1.16149	−0.580745	+	0.814085i	$0.697239\pi$
$152$	−4.75452	−0.385643
$153$	0	0
$154$	2.45468	0.197804
$155$	10.6748	0.857424
$156$	0	0
$157$	22.4735	1.79358	0.896791	−	0.442453i	$-0.145892\pi$
$157$	22.4735	1.79358	0.896791	+	0.442453i	$0.145892\pi$
$158$	−0.877887	−0.0698409
$159$	0	0
$160$	3.17304	0.250851
$161$	4.21796	0.332422
$162$	0	0
$163$	10.6555	0.834604	0.417302	−	0.908768i	$-0.362976\pi$
$163$	10.6555	0.834604	0.417302	+	0.908768i	$0.362976\pi$
$164$	−0.213013	−0.0166335
$165$	0	0
$166$	−6.48815	−0.503578
$167$	−20.1342	−1.55803	−0.779015	−	0.627006i	$-0.784280\pi$
$167$	−20.1342	−1.55803	−0.779015	+	0.627006i	$0.784280\pi$
$168$	0	0
$169$	−5.30836	−0.408335
$170$	15.4154	1.18231
$171$	0	0
$172$	2.99476	0.228348
$173$	7.40946	0.563331	0.281665	−	0.959513i	$-0.409113\pi$
$173$	7.40946	0.563331	0.281665	+	0.959513i	$0.409113\pi$
$174$	0	0
$175$	−0.471655	−0.0356538
$176$	−7.27423	−0.548316
$177$	0	0
$178$	−0.236597	−0.0177337
$179$	13.2628	0.991306	0.495653	−	0.868521i	$-0.334929\pi$
$179$	13.2628	0.991306	0.495653	+	0.868521i	$0.334929\pi$
$180$	0	0
$181$	13.2421	0.984281	0.492140	−	0.870516i	$-0.336215\pi$
$181$	13.2421	0.984281	0.492140	+	0.870516i	$0.336215\pi$
$182$	4.17416	0.309410
$183$	0	0
$184$	−11.0127	−0.811868
$185$	10.2296	0.752094
$186$	0	0
$187$	−7.84985	−0.574038
$188$	−1.89358	−0.138103
$189$	0	0
$190$	−5.83236	−0.423124
$191$	−6.93155	−0.501549	−0.250775	−	0.968046i	$-0.580685\pi$
$191$	−6.93155	−0.501549	−0.250775	+	0.968046i	$0.580685\pi$
$192$	0	0
$193$	12.9826	0.934510	0.467255	−	0.884123i	$-0.345243\pi$
$193$	12.9826	0.934510	0.467255	+	0.884123i	$0.345243\pi$
$194$	3.50692	0.251782
$195$	0	0
$196$	0.265269	0.0189478
$197$	0.943824	0.0672446	0.0336223	−	0.999435i	$-0.489296\pi$
$197$	0.943824	0.0672446	0.0336223	+	0.999435i	$0.489296\pi$
$198$	0	0
$199$	15.1662	1.07511	0.537553	−	0.843230i	$-0.319349\pi$
$199$	15.1662	1.07511	0.537553	+	0.843230i	$0.319349\pi$
$200$	1.23145	0.0870767
$201$	0	0
$202$	7.91102	0.556618
$203$	−2.17542	−0.152685
$204$	0	0
$205$	1.70879	0.119347
$206$	18.3399	1.27780
$207$	0	0
$208$	−12.3698	−0.857688
$209$	2.96996	0.205437
$210$	0	0
$211$	20.1654	1.38824	0.694122	−	0.719857i	$-0.255793\pi$
$211$	20.1654	1.38824	0.694122	+	0.719857i	$0.255793\pi$
$212$	1.95177	0.134048
$213$	0	0
$214$	−15.2680	−1.04370
$215$	−24.0240	−1.63842
$216$	0	0
$217$	−5.01640	−0.340535
$218$	8.47368	0.573910
$219$	0	0
$220$	−0.920644	−0.0620698
$221$	−13.3486	−0.897923
$222$	0	0
$223$	25.2773	1.69269	0.846346	−	0.532633i	$-0.178798\pi$
$223$	25.2773	1.69269	0.846346	+	0.532633i	$0.178798\pi$
$224$	−1.49110	−0.0996280
$225$	0	0
$226$	−15.1261	−1.00617
$227$	5.94333	0.394473	0.197236	−	0.980356i	$-0.436803\pi$
$227$	5.94333	0.394473	0.197236	+	0.980356i	$0.436803\pi$
$228$	0	0
$229$	16.5825	1.09580	0.547902	−	0.836542i	$-0.315427\pi$
$229$	16.5825	1.09580	0.547902	+	0.836542i	$0.315427\pi$
$230$	−13.5093	−0.890775
$231$	0	0
$232$	5.67983	0.372899
$233$	26.7788	1.75434	0.877168	−	0.480183i	$-0.159430\pi$
$233$	26.7788	1.75434	0.877168	+	0.480183i	$0.159430\pi$
$234$	0	0
$235$	15.1903	0.990907
$236$	0.530463	0.0345302
$237$	0	0
$238$	−7.24411	−0.469566
$239$	7.15117	0.462571	0.231285	−	0.972886i	$-0.425707\pi$
$239$	7.15117	0.462571	0.231285	+	0.972886i	$0.425707\pi$
$240$	0	0
$241$	16.1647	1.04126	0.520630	−	0.853783i	$-0.325697\pi$
$241$	16.1647	1.04126	0.520630	+	0.853783i	$0.325697\pi$
$242$	−12.5525	−0.806903
$243$	0	0
$244$	0.436932	0.0279717
$245$	−2.12799	−0.135952
$246$	0	0
$247$	5.05039	0.321348
$248$	13.0974	0.831684
$249$	0	0
$250$	17.5246	1.10835
$251$	−10.5416	−0.665382	−0.332691	−	0.943036i	$-0.607957\pi$
$251$	−10.5416	−0.665382	−0.332691	+	0.943036i	$0.607957\pi$
$252$	0	0
$253$	6.87920	0.432492
$254$	−1.50508	−0.0944372
$255$	0	0
$256$	6.25941	0.391213
$257$	−8.14743	−0.508222	−0.254111	−	0.967175i	$-0.581783\pi$
$257$	−8.14743	−0.508222	−0.254111	+	0.967175i	$0.581783\pi$
$258$	0	0
$259$	−4.80716	−0.298702
$260$	−1.56555	−0.0970910
$261$	0	0
$262$	−2.23202	−0.137895
$263$	13.8490	0.853963	0.426982	−	0.904260i	$-0.359577\pi$
$263$	13.8490	0.853963	0.426982	+	0.904260i	$0.359577\pi$
$264$	0	0
$265$	−15.6571	−0.961808
$266$	2.74078	0.168048
$267$	0	0
$268$	2.86044	0.174729
$269$	8.14970	0.496896	0.248448	−	0.968645i	$-0.420079\pi$
$269$	8.14970	0.496896	0.248448	+	0.968645i	$0.420079\pi$
$270$	0	0
$271$	−12.7842	−0.776583	−0.388291	−	0.921537i	$-0.626935\pi$
$271$	−12.7842	−0.776583	−0.388291	+	0.921537i	$0.626935\pi$
$272$	21.4673	1.30164
$273$	0	0
$274$	17.9042	1.08163
$275$	−0.769238	−0.0463868
$276$	0	0
$277$	3.03321	0.182248	0.0911240	−	0.995840i	$-0.470954\pi$
$277$	3.03321	0.182248	0.0911240	+	0.995840i	$0.470954\pi$
$278$	−0.862138	−0.0517076
$279$	0	0
$280$	5.55599	0.332034
$281$	33.1775	1.97920	0.989602	−	0.143832i	$-0.0459426\pi$
$281$	33.1775	1.97920	0.989602	+	0.143832i	$0.0459426\pi$
$282$	0	0
$283$	−1.84353	−0.109587	−0.0547933	−	0.998498i	$-0.517450\pi$
$283$	−1.84353	−0.109587	−0.0547933	+	0.998498i	$0.517450\pi$
$284$	1.18966	0.0705930
$285$	0	0
$286$	6.80778	0.402552
$287$	−0.803009	−0.0474001
$288$	0	0
$289$	6.16599	0.362705
$290$	6.96743	0.409142
$291$	0	0
$292$	1.46772	0.0858917
$293$	19.4459	1.13604	0.568021	−	0.823014i	$-0.307709\pi$
$293$	19.4459	1.13604	0.568021	+	0.823014i	$0.307709\pi$
$294$	0	0
$295$	−4.25538	−0.247758
$296$	12.5511	0.729516
$297$	0	0
$298$	2.16147	0.125211
$299$	11.6980	0.676513
$300$	0	0
$301$	11.2895	0.650717
$302$	−21.4815	−1.23612
$303$	0	0
$304$	−8.12206	−0.465832
$305$	−3.50507	−0.200700
$306$	0	0
$307$	−0.861448	−0.0491654	−0.0245827	−	0.999698i	$-0.507826\pi$
$307$	−0.861448	−0.0491654	−0.0245827	+	0.999698i	$0.507826\pi$
$308$	0.432635	0.0246517
$309$	0	0
$310$	16.0665	0.912516
$311$	−5.78371	−0.327964	−0.163982	−	0.986463i	$-0.552434\pi$
$311$	−5.78371	−0.327964	−0.163982	+	0.986463i	$0.552434\pi$
$312$	0	0
$313$	−25.3373	−1.43215	−0.716074	−	0.698024i	$-0.754063\pi$
$313$	−25.3373	−1.43215	−0.716074	+	0.698024i	$0.754063\pi$
$314$	33.8245	1.90883
$315$	0	0
$316$	−0.154727	−0.00870405
$317$	12.8867	0.723791	0.361896	−	0.932219i	$-0.382130\pi$
$317$	12.8867	0.723791	0.361896	+	0.932219i	$0.382130\pi$
$318$	0	0
$319$	−3.54796	−0.198648
$320$	−14.2067	−0.794180
$321$	0	0
$322$	6.34837	0.353781
$323$	−8.76477	−0.487685
$324$	0	0
$325$	−1.30808	−0.0725592
$326$	16.0374	0.888230
$327$	0	0
$328$	2.09658	0.115764
$329$	−7.13834	−0.393549
$330$	0	0
$331$	−13.4461	−0.739062	−0.369531	−	0.929218i	$-0.620482\pi$
$331$	−13.4461	−0.739062	−0.369531	+	0.929218i	$0.620482\pi$
$332$	−1.14353	−0.0627594
$333$	0	0
$334$	−30.3036	−1.65814
$335$	−22.9465	−1.25370
$336$	0	0
$337$	7.39436	0.402796	0.201398	−	0.979509i	$-0.435451\pi$
$337$	7.39436	0.402796	0.201398	+	0.979509i	$0.435451\pi$
$338$	−7.98951	−0.434572
$339$	0	0
$340$	2.71695	0.147347
$341$	−8.18140	−0.443048
$342$	0	0
$343$	1.00000	0.0539949
$344$	−29.4759	−1.58924
$345$	0	0
$346$	11.1518	0.599527
$347$	5.77786	0.310172	0.155086	−	0.987901i	$-0.450435\pi$
$347$	5.77786	0.310172	0.155086	+	0.987901i	$0.450435\pi$
$348$	0	0
$349$	−17.3767	−0.930155	−0.465078	−	0.885270i	$-0.653974\pi$
$349$	−17.3767	−0.930155	−0.465078	+	0.885270i	$0.653974\pi$
$350$	−0.709880	−0.0379447
$351$	0	0
$352$	−2.43188	−0.129619
$353$	−20.2089	−1.07561	−0.537805	−	0.843069i	$-0.680746\pi$
$353$	−20.2089	−1.07561	−0.537805	+	0.843069i	$0.680746\pi$
$354$	0	0
$355$	−9.54343	−0.506513
$356$	−0.0416999	−0.00221009
$357$	0	0
$358$	19.9615	1.05500
$359$	23.5996	1.24554	0.622769	−	0.782406i	$-0.286008\pi$
$359$	23.5996	1.24554	0.622769	+	0.782406i	$0.286008\pi$
$360$	0	0
$361$	−15.6839	−0.825467
$362$	19.9305	1.04752
$363$	0	0
$364$	0.735692	0.0385607
$365$	−11.7740	−0.616282
$366$	0	0
$367$	10.3692	0.541266	0.270633	−	0.962683i	$-0.412767\pi$
$367$	10.3692	0.541266	0.270633	+	0.962683i	$0.412767\pi$
$368$	−18.8128	−0.980685
$369$	0	0
$370$	15.3964	0.800418
$371$	7.35769	0.381992
$372$	0	0
$373$	−25.7799	−1.33483	−0.667415	−	0.744686i	$-0.732599\pi$
$373$	−25.7799	−1.33483	−0.667415	+	0.744686i	$0.732599\pi$
$374$	−11.8147	−0.610921
$375$	0	0
$376$	18.6376	0.961159
$377$	−6.03327	−0.310729
$378$	0	0
$379$	30.5474	1.56912	0.784558	−	0.620056i	$-0.212890\pi$
$379$	30.5474	1.56912	0.784558	+	0.620056i	$0.212890\pi$
$380$	−1.02795	−0.0527326
$381$	0	0
$382$	−10.4325	−0.533775
$383$	−29.0438	−1.48407	−0.742034	−	0.670362i	$-0.766139\pi$
$383$	−29.0438	−1.48407	−0.742034	+	0.670362i	$0.766139\pi$
$384$	0	0
$385$	−3.47061	−0.176879
$386$	19.5399	0.994555
$387$	0	0
$388$	0.618090	0.0313788
$389$	25.0195	1.26854	0.634270	−	0.773111i	$-0.281301\pi$
$389$	25.0195	1.26854	0.634270	+	0.773111i	$0.281301\pi$
$390$	0	0
$391$	−20.3015	−1.02669
$392$	−2.61091	−0.131871
$393$	0	0
$394$	1.42053	0.0715653
$395$	1.24122	0.0624525
$396$	0	0
$397$	33.8034	1.69654	0.848272	−	0.529561i	$-0.177643\pi$
$397$	33.8034	1.69654	0.848272	+	0.529561i	$0.177643\pi$
$398$	22.8264	1.14418
$399$	0	0
$400$	2.10366	0.105183
$401$	16.5983	0.828877	0.414439	−	0.910077i	$-0.363978\pi$
$401$	16.5983	0.828877	0.414439	+	0.910077i	$0.363978\pi$
$402$	0	0
$403$	−13.9124	−0.693025
$404$	1.39431	0.0693695
$405$	0	0
$406$	−3.27418	−0.162495
$407$	−7.84015	−0.388622
$408$	0	0
$409$	−17.6138	−0.870944	−0.435472	−	0.900202i	$-0.643419\pi$
$409$	−17.6138	−0.870944	−0.435472	+	0.900202i	$0.643419\pi$
$410$	2.57187	0.127016
$411$	0	0
$412$	3.23239	0.159248
$413$	1.99972	0.0983996
$414$	0	0
$415$	9.17341	0.450305
$416$	−4.13538	−0.202754
$417$	0	0
$418$	4.47003	0.218636
$419$	−31.9337	−1.56007	−0.780033	−	0.625739i	$-0.784798\pi$
$419$	−31.9337	−1.56007	−0.780033	+	0.625739i	$0.784798\pi$
$420$	0	0
$421$	−26.7893	−1.30563	−0.652816	−	0.757517i	$-0.726412\pi$
$421$	−26.7893	−1.30563	−0.652816	+	0.757517i	$0.726412\pi$
$422$	30.3506	1.47744
$423$	0	0
$424$	−19.2103	−0.932933
$425$	2.27013	0.110117
$426$	0	0
$427$	1.64713	0.0797101
$428$	−2.69098	−0.130073
$429$	0	0
$430$	−36.1581	−1.74370
$431$	3.12430	0.150492	0.0752462	−	0.997165i	$-0.476026\pi$
$431$	3.12430	0.150492	0.0752462	+	0.997165i	$0.476026\pi$
$432$	0	0
$433$	16.7813	0.806456	0.403228	−	0.915100i	$-0.367888\pi$
$433$	16.7813	0.806456	0.403228	+	0.915100i	$0.367888\pi$
$434$	−7.55008	−0.362416
$435$	0	0
$436$	1.49348	0.0715245
$437$	7.68099	0.367432
$438$	0	0
$439$	−1.99581	−0.0952548	−0.0476274	−	0.998865i	$-0.515166\pi$
$439$	−1.99581	−0.0952548	−0.0476274	+	0.998865i	$0.515166\pi$
$440$	9.06145	0.431988
$441$	0	0
$442$	−20.0907	−0.955617
$443$	5.38599	0.255896	0.127948	−	0.991781i	$-0.459161\pi$
$443$	5.38599	0.255896	0.127948	+	0.991781i	$0.459161\pi$
$444$	0	0
$445$	0.334517	0.0158576
$446$	38.0444	1.80145
$447$	0	0
$448$	6.67612	0.315417
$449$	3.18246	0.150190	0.0750948	−	0.997176i	$-0.476074\pi$
$449$	3.18246	0.150190	0.0750948	+	0.997176i	$0.476074\pi$
$450$	0	0
$451$	−1.30965	−0.0616691
$452$	−2.66595	−0.125396
$453$	0	0
$454$	8.94519	0.419819
$455$	−5.90173	−0.276677
$456$	0	0
$457$	18.3721	0.859409	0.429704	−	0.902970i	$-0.358618\pi$
$457$	18.3721	0.859409	0.429704	+	0.902970i	$0.358618\pi$
$458$	24.9581	1.16621
$459$	0	0
$460$	−2.38099	−0.111014
$461$	−2.21772	−0.103289	−0.0516447	−	0.998666i	$-0.516446\pi$
$461$	−2.21772	−0.103289	−0.0516447	+	0.998666i	$0.516446\pi$
$462$	0	0
$463$	8.10883	0.376849	0.188425	−	0.982088i	$-0.439662\pi$
$463$	8.10883	0.376849	0.188425	+	0.982088i	$0.439662\pi$
$464$	9.70274	0.450438
$465$	0	0
$466$	40.3042	1.86706
$467$	15.3655	0.711029	0.355514	−	0.934671i	$-0.384306\pi$
$467$	15.3655	0.711029	0.355514	+	0.934671i	$0.384306\pi$
$468$	0	0
$469$	10.7832	0.497921
$470$	22.8627	1.05458
$471$	0	0
$472$	−5.22108	−0.240320
$473$	18.4124	0.846605
$474$	0	0
$475$	−0.858894	−0.0394088
$476$	−1.27677	−0.0585205
$477$	0	0
$478$	10.7631	0.492292
$479$	−39.2875	−1.79509	−0.897545	−	0.440922i	$-0.854652\pi$
$479$	−39.2875	−1.79509	−0.897545	+	0.440922i	$0.854652\pi$
$480$	0	0
$481$	−13.3321	−0.607891
$482$	24.3292	1.10816
$483$	0	0
$484$	−2.21236	−0.100562
$485$	−4.95833	−0.225146
$486$	0	0
$487$	13.7758	0.624243	0.312121	−	0.950042i	$-0.398960\pi$
$487$	13.7758	0.624243	0.312121	+	0.950042i	$0.398960\pi$
$488$	−4.30051	−0.194675
$489$	0	0
$490$	−3.20280	−0.144688
$491$	−4.77600	−0.215538	−0.107769	−	0.994176i	$-0.534371\pi$
$491$	−4.77600	−0.215538	−0.107769	+	0.994176i	$0.534371\pi$
$492$	0	0
$493$	10.4705	0.471569
$494$	7.60124	0.341996
$495$	0	0
$496$	22.3740	1.00462
$497$	4.48471	0.201167
$498$	0	0
$499$	19.9600	0.893533	0.446767	−	0.894651i	$-0.352575\pi$
$499$	19.9600	0.893533	0.446767	+	0.894651i	$0.352575\pi$
$500$	3.08869	0.138131
$501$	0	0
$502$	−15.8660	−0.708134
$503$	−18.5831	−0.828581	−0.414290	−	0.910145i	$-0.635970\pi$
$503$	−18.5831	−0.828581	−0.414290	+	0.910145i	$0.635970\pi$
$504$	0	0
$505$	−11.1852	−0.497733
$506$	10.3538	0.460280
$507$	0	0
$508$	−0.265269	−0.0117694
$509$	−38.2223	−1.69417	−0.847086	−	0.531455i	$-0.821645\pi$
$509$	−38.2223	−1.69417	−0.847086	+	0.531455i	$0.821645\pi$
$510$	0	0
$511$	5.53294	0.244763
$512$	−16.6397	−0.735377
$513$	0	0
$514$	−12.2625	−0.540877
$515$	−25.9303	−1.14262
$516$	0	0
$517$	−11.6421	−0.512021
$518$	−7.23516	−0.317895
$519$	0	0
$520$	15.4089	0.675725
$521$	−4.27005	−0.187074	−0.0935371	−	0.995616i	$-0.529817\pi$
$521$	−4.27005	−0.187074	−0.0935371	+	0.995616i	$0.529817\pi$
$522$	0	0
$523$	−2.89861	−0.126747	−0.0633736	−	0.997990i	$-0.520186\pi$
$523$	−2.89861	−0.126747	−0.0633736	+	0.997990i	$0.520186\pi$
$524$	−0.393392	−0.0171854
$525$	0	0
$526$	20.8438	0.908833
$527$	24.1444	1.05175
$528$	0	0
$529$	−5.20884	−0.226471
$530$	−23.5652	−1.02361
$531$	0	0
$532$	0.483060	0.0209433
$533$	−2.22705	−0.0964642
$534$	0	0
$535$	21.5871	0.933290
$536$	−28.1539	−1.21606
$537$	0	0
$538$	12.2660	0.528823
$539$	1.63093	0.0702492
$540$	0	0
$541$	18.8368	0.809856	0.404928	−	0.914349i	$-0.367297\pi$
$541$	18.8368	0.809856	0.404928	+	0.914349i	$0.367297\pi$
$542$	−19.2412	−0.826480
$543$	0	0
$544$	7.17680	0.307703
$545$	−11.9807	−0.513196
$546$	0	0
$547$	29.4065	1.25733	0.628665	−	0.777676i	$-0.283602\pi$
$547$	29.4065	1.25733	0.628665	+	0.777676i	$0.283602\pi$
$548$	3.15559	0.134800
$549$	0	0
$550$	−1.15777	−0.0493673
$551$	−3.96148	−0.168765
$552$	0	0
$553$	−0.583282	−0.0248037
$554$	4.56523	0.193958
$555$	0	0
$556$	−0.151951	−0.00644415
$557$	−45.9015	−1.94491	−0.972455	−	0.233090i	$-0.925116\pi$
$557$	−45.9015	−1.94491	−0.972455	+	0.233090i	$0.925116\pi$
$558$	0	0
$559$	31.3101	1.32428
$560$	9.49120	0.401076
$561$	0	0
$562$	49.9348	2.10637
$563$	8.87157	0.373892	0.186946	−	0.982370i	$-0.440141\pi$
$563$	8.87157	0.373892	0.186946	+	0.982370i	$0.440141\pi$
$564$	0	0
$565$	21.3863	0.899729
$566$	−2.77466	−0.116628
$567$	0	0
$568$	−11.7092	−0.491307
$569$	16.3396	0.684991	0.342495	−	0.939520i	$-0.388728\pi$
$569$	16.3396	0.684991	0.342495	+	0.939520i	$0.388728\pi$
$570$	0	0
$571$	17.9875	0.752754	0.376377	−	0.926467i	$-0.377170\pi$
$571$	17.9875	0.752754	0.376377	+	0.926467i	$0.377170\pi$
$572$	1.19986	0.0501688
$573$	0	0
$574$	−1.20859	−0.0504457
$575$	−1.98942	−0.0829646
$576$	0	0
$577$	−18.6836	−0.777808	−0.388904	−	0.921278i	$-0.627146\pi$
$577$	−18.6836	−0.777808	−0.388904	+	0.921278i	$0.627146\pi$
$578$	9.28031	0.386010
$579$	0	0
$580$	1.22800	0.0509900
$581$	−4.31083	−0.178844
$582$	0	0
$583$	11.9999	0.496985
$584$	−14.4460	−0.597780
$585$	0	0
$586$	29.2677	1.20904
$587$	−20.4150	−0.842617	−0.421309	−	0.906917i	$-0.638429\pi$
$587$	−20.4150	−0.842617	−0.421309	+	0.906917i	$0.638429\pi$
$588$	0	0
$589$	−9.13496	−0.376400
$590$	−6.40469	−0.263677
$591$	0	0
$592$	21.4407	0.881209
$593$	−38.0384	−1.56205	−0.781025	−	0.624500i	$-0.785303\pi$
$593$	−38.0384	−1.56205	−0.781025	+	0.624500i	$0.785303\pi$
$594$	0	0
$595$	10.2422	0.419891
$596$	0.380957	0.0156046
$597$	0	0
$598$	17.6064	0.719981
$599$	−35.1546	−1.43638	−0.718189	−	0.695848i	$-0.755029\pi$
$599$	−35.1546	−1.43638	−0.718189	+	0.695848i	$0.755029\pi$
$600$	0	0
$601$	−34.5758	−1.41038	−0.705189	−	0.709020i	$-0.749138\pi$
$601$	−34.5758	−1.41038	−0.705189	+	0.709020i	$0.749138\pi$
$602$	16.9916	0.692528
$603$	0	0
$604$	−3.78609	−0.154054
$605$	17.7476	0.721541
$606$	0	0
$607$	−0.0631932	−0.00256493	−0.00128247	−	0.999999i	$-0.500408\pi$
$607$	−0.0631932	−0.00256493	−0.00128247	+	0.999999i	$0.500408\pi$
$608$	−2.71532	−0.110121
$609$	0	0
$610$	−5.27542	−0.213596
$611$	−19.7973	−0.800914
$612$	0	0
$613$	24.0028	0.969466	0.484733	−	0.874662i	$-0.338917\pi$
$613$	24.0028	0.969466	0.484733	+	0.874662i	$0.338917\pi$
$614$	−1.29655	−0.0523245
$615$	0	0
$616$	−4.25822	−0.171568
$617$	8.66784	0.348954	0.174477	−	0.984661i	$-0.444177\pi$
$617$	8.66784	0.348954	0.174477	+	0.984661i	$0.444177\pi$
$618$	0	0
$619$	−25.2979	−1.01681	−0.508404	−	0.861119i	$-0.669764\pi$
$619$	−25.2979	−1.01681	−0.508404	+	0.861119i	$0.669764\pi$
$620$	2.83170	0.113724
$621$	0	0
$622$	−8.70496	−0.349037
$623$	−0.157199	−0.00629803
$624$	0	0
$625$	−22.4193	−0.896771
$626$	−38.1347	−1.52417
$627$	0	0
$628$	5.96153	0.237891
$629$	23.1374	0.922547
$630$	0	0
$631$	−3.28821	−0.130901	−0.0654507	−	0.997856i	$-0.520849\pi$
$631$	−3.28821	−0.130901	−0.0654507	+	0.997856i	$0.520849\pi$
$632$	1.52290	0.0605776
$633$	0	0
$634$	19.3956	0.770297
$635$	2.12799	0.0844467
$636$	0	0
$637$	2.77338	0.109885
$638$	−5.33997	−0.211411
$639$	0	0
$640$	−27.7283	−1.09606
$641$	31.2786	1.23543	0.617716	−	0.786401i	$-0.288058\pi$
$641$	31.2786	1.23543	0.617716	+	0.786401i	$0.288058\pi$
$642$	0	0
$643$	−27.4185	−1.08128	−0.540640	−	0.841254i	$-0.681818\pi$
$643$	−27.4185	−1.08128	−0.540640	+	0.841254i	$0.681818\pi$
$644$	1.11889	0.0440906
$645$	0	0
$646$	−13.1917	−0.519020
$647$	−35.8863	−1.41084	−0.705418	−	0.708791i	$-0.749241\pi$
$647$	−35.8863	−1.41084	−0.705418	+	0.708791i	$0.749241\pi$
$648$	0	0
$649$	3.26140	0.128021
$650$	−1.96877	−0.0772214
$651$	0	0
$652$	2.82658	0.110697
$653$	−20.9115	−0.818329	−0.409165	−	0.912461i	$-0.634180\pi$
$653$	−20.9115	−0.818329	−0.409165	+	0.912461i	$0.634180\pi$
$654$	0	0
$655$	3.15579	0.123307
$656$	3.58155	0.139836
$657$	0	0
$658$	−10.7438	−0.418836
$659$	4.11845	0.160432	0.0802160	−	0.996778i	$-0.474439\pi$
$659$	4.11845	0.160432	0.0802160	+	0.996778i	$0.474439\pi$
$660$	0	0
$661$	34.1674	1.32896	0.664479	−	0.747307i	$-0.268653\pi$
$661$	34.1674	1.32896	0.664479	+	0.747307i	$0.268653\pi$
$662$	−20.2374	−0.786549
$663$	0	0
$664$	11.2552	0.436787
$665$	−3.87512	−0.150271
$666$	0	0
$667$	−9.17583	−0.355289
$668$	−5.34097	−0.206648
$669$	0	0
$670$	−34.5363	−1.33426
$671$	2.68635	0.103706
$672$	0	0
$673$	−5.60403	−0.216020	−0.108010	−	0.994150i	$-0.534448\pi$
$673$	−5.60403	−0.216020	−0.108010	+	0.994150i	$0.534448\pi$
$674$	11.1291	0.428677
$675$	0	0
$676$	−1.40814	−0.0541593
$677$	7.27308	0.279527	0.139764	−	0.990185i	$-0.455366\pi$
$677$	7.27308	0.279527	0.139764	+	0.990185i	$0.455366\pi$
$678$	0	0
$679$	2.33005	0.0894192
$680$	−26.7416	−1.02549
$681$	0	0
$682$	−12.3137	−0.471515
$683$	−25.5922	−0.979259	−0.489630	−	0.871930i	$-0.662868\pi$
$683$	−25.5922	−0.979259	−0.489630	+	0.871930i	$0.662868\pi$
$684$	0	0
$685$	−25.3142	−0.967204
$686$	1.50508	0.0574643
$687$	0	0
$688$	−50.3532	−1.91970
$689$	20.4057	0.777395
$690$	0	0
$691$	−14.6598	−0.557686	−0.278843	−	0.960337i	$-0.589951\pi$
$691$	−14.6598	−0.557686	−0.278843	+	0.960337i	$0.589951\pi$
$692$	1.96550	0.0747171
$693$	0	0
$694$	8.69615	0.330102
$695$	1.21895	0.0462375
$696$	0	0
$697$	3.86497	0.146396
$698$	−26.1534	−0.989920
$699$	0	0
$700$	−0.125116	−0.00472892
$701$	−27.0381	−1.02122	−0.510608	−	0.859814i	$-0.670580\pi$
$701$	−27.0381	−1.02122	−0.510608	+	0.859814i	$0.670580\pi$
$702$	0	0
$703$	−8.75394	−0.330161
$704$	10.8883	0.410368
$705$	0	0
$706$	−30.4160	−1.14472
$707$	5.25621	0.197680
$708$	0	0
$709$	−46.6488	−1.75193	−0.875966	−	0.482372i	$-0.839775\pi$
$709$	−46.6488	−1.75193	−0.875966	+	0.482372i	$0.839775\pi$
$710$	−14.3636	−0.539058
$711$	0	0
$712$	0.410431	0.0153816
$713$	−21.1589	−0.792409
$714$	0	0
$715$	−9.62532	−0.359966
$716$	3.51820	0.131481
$717$	0	0
$718$	35.5193	1.32557
$719$	−33.5776	−1.25224	−0.626118	−	0.779729i	$-0.715357\pi$
$719$	−33.5776	−1.25224	−0.626118	+	0.779729i	$0.715357\pi$
$720$	0	0
$721$	12.1853	0.453805
$722$	−23.6055	−0.878506
$723$	0	0
$724$	3.51273	0.130550
$725$	1.02605	0.0381065
$726$	0	0
$727$	30.6696	1.13747	0.568737	−	0.822520i	$-0.307432\pi$
$727$	30.6696	1.13747	0.568737	+	0.822520i	$0.307432\pi$
$728$	−7.24105	−0.268371
$729$	0	0
$730$	−17.7209	−0.655880
$731$	−54.3376	−2.00975
$732$	0	0
$733$	26.0914	0.963706	0.481853	−	0.876252i	$-0.339964\pi$
$733$	26.0914	0.963706	0.481853	+	0.876252i	$0.339964\pi$
$734$	15.6064	0.576043
$735$	0	0
$736$	−6.28938	−0.231830
$737$	17.5866	0.647812
$738$	0	0
$739$	−14.6632	−0.539396	−0.269698	−	0.962945i	$-0.586924\pi$
$739$	−14.6632	−0.539396	−0.269698	+	0.962945i	$0.586924\pi$
$740$	2.71359	0.0997536
$741$	0	0
$742$	11.0739	0.406536
$743$	−7.35661	−0.269888	−0.134944	−	0.990853i	$-0.543085\pi$
$743$	−7.35661	−0.269888	−0.134944	+	0.990853i	$0.543085\pi$
$744$	0	0
$745$	−3.05604	−0.111965
$746$	−38.8008	−1.42060
$747$	0	0
$748$	−2.08232	−0.0761372
$749$	−10.1443	−0.370666
$750$	0	0
$751$	−25.6001	−0.934161	−0.467081	−	0.884215i	$-0.654694\pi$
$751$	−25.6001	−0.934161	−0.467081	+	0.884215i	$0.654694\pi$
$752$	31.8382	1.16102
$753$	0	0
$754$	−9.08056	−0.330694
$755$	30.3720	1.10535
$756$	0	0
$757$	18.8028	0.683401	0.341700	−	0.939809i	$-0.388997\pi$
$757$	18.8028	0.683401	0.341700	+	0.939809i	$0.388997\pi$
$758$	45.9763	1.66994
$759$	0	0
$760$	10.1176	0.367003
$761$	−45.6268	−1.65397	−0.826985	−	0.562224i	$-0.809946\pi$
$761$	−45.6268	−1.65397	−0.826985	+	0.562224i	$0.809946\pi$
$762$	0	0
$763$	5.63005	0.203821
$764$	−1.83872	−0.0665227
$765$	0	0
$766$	−43.7133	−1.57942
$767$	5.54598	0.200254
$768$	0	0
$769$	−1.98470	−0.0715699	−0.0357850	−	0.999360i	$-0.511393\pi$
$769$	−1.98470	−0.0715699	−0.0357850	+	0.999360i	$0.511393\pi$
$770$	−5.22355	−0.188244
$771$	0	0
$772$	3.44389	0.123948
$773$	−10.7260	−0.385789	−0.192894	−	0.981220i	$-0.561787\pi$
$773$	−10.7260	−0.385789	−0.192894	+	0.981220i	$0.561787\pi$
$774$	0	0
$775$	2.36601	0.0849896
$776$	−6.08356	−0.218387
$777$	0	0
$778$	37.6564	1.35005
$779$	−1.46230	−0.0523922
$780$	0	0
$781$	7.31426	0.261725
$782$	−30.5554	−1.09266
$783$	0	0
$784$	−4.46017	−0.159292
$785$	−47.8235	−1.70689
$786$	0	0
$787$	−36.4882	−1.30067	−0.650333	−	0.759650i	$-0.725370\pi$
$787$	−36.4882	−1.30067	−0.650333	+	0.759650i	$0.725370\pi$
$788$	0.250367	0.00891896
$789$	0	0
$790$	1.86814	0.0664653
$791$	−10.0500	−0.357337
$792$	0	0
$793$	4.56812	0.162219
$794$	50.8768	1.80555
$795$	0	0
$796$	4.02313	0.142596
$797$	−17.3175	−0.613416	−0.306708	−	0.951804i	$-0.599227\pi$
$797$	−17.3175	−0.613416	−0.306708	+	0.951804i	$0.599227\pi$
$798$	0	0
$799$	34.3576	1.21548
$800$	0.703283	0.0248648
$801$	0	0
$802$	24.9817	0.882135
$803$	9.02385	0.318445
$804$	0	0
$805$	−8.97577	−0.316354
$806$	−20.9393	−0.737554
$807$	0	0
$808$	−13.7235	−0.482791
$809$	34.3146	1.20644	0.603219	−	0.797575i	$-0.293884\pi$
$809$	34.3146	1.20644	0.603219	+	0.797575i	$0.293884\pi$
$810$	0	0
$811$	0.0816467	0.00286700	0.00143350	−	0.999999i	$-0.499544\pi$
$811$	0.0816467	0.00286700	0.00143350	+	0.999999i	$0.499544\pi$
$812$	−0.577071	−0.0202512
$813$	0	0
$814$	−11.8001	−0.413592
$815$	−22.6748	−0.794264
$816$	0	0
$817$	20.5585	0.719249
$818$	−26.5101	−0.926905
$819$	0	0
$820$	0.453290	0.0158296
$821$	43.5520	1.51997	0.759987	−	0.649938i	$-0.225205\pi$
$821$	43.5520	1.51997	0.759987	+	0.649938i	$0.225205\pi$
$822$	0	0
$823$	1.81861	0.0633928	0.0316964	−	0.999498i	$-0.489909\pi$
$823$	1.81861	0.0633928	0.0316964	+	0.999498i	$0.489909\pi$
$824$	−31.8148	−1.10832
$825$	0	0
$826$	3.00973	0.104722
$827$	−2.64469	−0.0919649	−0.0459824	−	0.998942i	$-0.514642\pi$
$827$	−2.64469	−0.0919649	−0.0459824	+	0.998942i	$0.514642\pi$
$828$	0	0
$829$	−36.9868	−1.28461	−0.642303	−	0.766451i	$-0.722021\pi$
$829$	−36.9868	−1.28461	−0.642303	+	0.766451i	$0.722021\pi$
$830$	13.8067	0.479239
$831$	0	0
$832$	18.5154	0.641907
$833$	−4.81311	−0.166764
$834$	0	0
$835$	42.8453	1.48272
$836$	0.787838	0.0272480
$837$	0	0
$838$	−48.0629	−1.66030
$839$	35.5013	1.22564	0.612820	−	0.790223i	$-0.290035\pi$
$839$	35.5013	1.22564	0.612820	+	0.790223i	$0.290035\pi$
$840$	0	0
$841$	−24.2676	−0.836812
$842$	−40.3201	−1.38952
$843$	0	0
$844$	5.34926	0.184129
$845$	11.2961	0.388599
$846$	0	0
$847$	−8.34006	−0.286568
$848$	−32.8166	−1.12693
$849$	0	0
$850$	3.41673	0.117193
$851$	−20.2764	−0.695065
$852$	0	0
$853$	43.0930	1.47548	0.737738	−	0.675087i	$-0.235894\pi$
$853$	43.0930	1.47548	0.737738	+	0.675087i	$0.235894\pi$
$854$	2.47906	0.0848318
$855$	0	0
$856$	26.4860	0.905271
$857$	38.6248	1.31940	0.659700	−	0.751529i	$-0.270683\pi$
$857$	38.6248	1.31940	0.659700	+	0.751529i	$0.270683\pi$
$858$	0	0
$859$	5.37477	0.183385	0.0916924	−	0.995787i	$-0.470772\pi$
$859$	5.37477	0.183385	0.0916924	+	0.995787i	$0.470772\pi$
$860$	−6.37282	−0.217311
$861$	0	0
$862$	4.70233	0.160162
$863$	−40.1594	−1.36704	−0.683520	−	0.729932i	$-0.739552\pi$
$863$	−40.1594	−1.36704	−0.683520	+	0.729932i	$0.739552\pi$
$864$	0	0
$865$	−15.7673	−0.536103
$866$	25.2571	0.858273
$867$	0	0
$868$	−1.33069	−0.0451667
$869$	−0.951293	−0.0322704
$870$	0	0
$871$	29.9059	1.01332
$872$	−14.6995	−0.497790
$873$	0	0
$874$	11.5605	0.391040
$875$	11.6436	0.393627
$876$	0	0
$877$	27.6508	0.933702	0.466851	−	0.884336i	$-0.345388\pi$
$877$	27.6508	0.933702	0.466851	+	0.884336i	$0.345388\pi$
$878$	−3.00385	−0.101375
$879$	0	0
$880$	15.4795	0.521814
$881$	−6.15905	−0.207504	−0.103752	−	0.994603i	$-0.533085\pi$
$881$	−6.15905	−0.207504	−0.103752	+	0.994603i	$0.533085\pi$
$882$	0	0
$883$	22.7955	0.767130	0.383565	−	0.923514i	$-0.374696\pi$
$883$	22.7955	0.767130	0.383565	+	0.923514i	$0.374696\pi$
$884$	−3.54096	−0.119095
$885$	0	0
$886$	8.10635	0.272338
$887$	10.9460	0.367531	0.183765	−	0.982970i	$-0.441171\pi$
$887$	10.9460	0.367531	0.183765	+	0.982970i	$0.441171\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	0.503475	0.0168765
$891$	0	0
$892$	6.70528	0.224509
$893$	−12.9991	−0.434997
$894$	0	0
$895$	−28.2231	−0.943393
$896$	13.0303	0.435312
$897$	0	0
$898$	4.78986	0.159840
$899$	10.9128	0.363961
$900$	0	0
$901$	−35.4133	−1.17979
$902$	−1.97113	−0.0656315
$903$	0	0
$904$	26.2397	0.872718
$905$	−28.1792	−0.936707
$906$	0	0
$907$	18.1632	0.603099	0.301550	−	0.953450i	$-0.402496\pi$
$907$	18.1632	0.603099	0.301550	+	0.953450i	$0.402496\pi$
$908$	1.57658	0.0523207
$909$	0	0
$910$	−8.88258	−0.294455
$911$	−2.71351	−0.0899024	−0.0449512	−	0.998989i	$-0.514313\pi$
$911$	−2.71351	−0.0899024	−0.0449512	+	0.998989i	$0.514313\pi$
$912$	0	0
$913$	−7.03068	−0.232681
$914$	27.6514	0.914628
$915$	0	0
$916$	4.39883	0.145341
$917$	−1.48299	−0.0489727
$918$	0	0
$919$	32.2240	1.06297	0.531486	−	0.847067i	$-0.321634\pi$
$919$	32.2240	1.06297	0.531486	+	0.847067i	$0.321634\pi$
$920$	23.4349	0.772627
$921$	0	0
$922$	−3.33784	−0.109926
$923$	12.4378	0.409396
$924$	0	0
$925$	2.26732	0.0745491
$926$	12.2044	0.401063
$927$	0	0
$928$	3.24376	0.106482
$929$	9.80980	0.321849	0.160925	−	0.986967i	$-0.448552\pi$
$929$	9.80980	0.321849	0.160925	+	0.986967i	$0.448552\pi$
$930$	0	0
$931$	1.82102	0.0596816
$932$	7.10358	0.232685
$933$	0	0
$934$	23.1263	0.756714
$935$	16.7044	0.546292
$936$	0	0
$937$	1.87596	0.0612850	0.0306425	−	0.999530i	$-0.490245\pi$
$937$	1.87596	0.0612850	0.0306425	+	0.999530i	$0.490245\pi$
$938$	16.2296	0.529914
$939$	0	0
$940$	4.02952	0.131428
$941$	−9.37897	−0.305746	−0.152873	−	0.988246i	$-0.548853\pi$
$941$	−9.37897	−0.305746	−0.152873	+	0.988246i	$0.548853\pi$
$942$	0	0
$943$	−3.38706	−0.110298
$944$	−8.91907	−0.290291
$945$	0	0
$946$	27.7122	0.901001
$947$	−13.5267	−0.439559	−0.219779	−	0.975550i	$-0.570534\pi$
$947$	−13.5267	−0.439559	−0.219779	+	0.975550i	$0.570534\pi$
$948$	0	0
$949$	15.3450	0.498118
$950$	−1.29271	−0.0419409
$951$	0	0
$952$	12.5666	0.407286
$953$	−38.8476	−1.25840	−0.629199	−	0.777244i	$-0.716617\pi$
$953$	−38.8476	−1.25840	−0.629199	+	0.777244i	$0.716617\pi$
$954$	0	0
$955$	14.7503	0.477307
$956$	1.89698	0.0613528
$957$	0	0
$958$	−59.1308	−1.91043
$959$	11.8958	0.384136
$960$	0	0
$961$	−5.83576	−0.188250
$962$	−20.0659	−0.646949
$963$	0	0
$964$	4.28799	0.138107
$965$	−27.6269	−0.889341
$966$	0	0
$967$	−46.4337	−1.49321	−0.746604	−	0.665268i	$-0.768317\pi$
$967$	−46.4337	−1.49321	−0.746604	+	0.665268i	$0.768317\pi$
$968$	21.7752	0.699880
$969$	0	0
$970$	−7.46269	−0.239612
$971$	−17.6091	−0.565104	−0.282552	−	0.959252i	$-0.591181\pi$
$971$	−17.6091	−0.565104	−0.282552	+	0.959252i	$0.591181\pi$
$972$	0	0
$973$	−0.572818	−0.0183637
$974$	20.7338	0.664352
$975$	0	0
$976$	−7.34647	−0.235155
$977$	7.43882	0.237989	0.118995	−	0.992895i	$-0.462033\pi$
$977$	7.43882	0.237989	0.118995	+	0.992895i	$0.462033\pi$
$978$	0	0
$979$	−0.256380	−0.00819394
$980$	−0.564490	−0.0180320
$981$	0	0
$982$	−7.18826	−0.229387
$983$	32.9210	1.05002	0.525008	−	0.851097i	$-0.324062\pi$
$983$	32.9210	1.05002	0.525008	+	0.851097i	$0.324062\pi$
$984$	0	0
$985$	−2.00845	−0.0639945
$986$	15.7590	0.501868
$987$	0	0
$988$	1.33971	0.0426219
$989$	47.6187	1.51419
$990$	0	0
$991$	38.9537	1.23741	0.618703	−	0.785625i	$-0.287658\pi$
$991$	38.9537	1.23741	0.618703	+	0.785625i	$0.287658\pi$
$992$	7.47993	0.237488
$993$	0	0
$994$	6.74986	0.214092
$995$	−32.2736	−1.02314
$996$	0	0
$997$	13.7715	0.436149	0.218075	−	0.975932i	$-0.430022\pi$
$997$	13.7715	0.436149	0.218075	+	0.975932i	$0.430022\pi$
$998$	30.0414	0.950945
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.30	yes	40
3.2	odd	2	inner	8001.2.a.ba.1.11	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.11	✓	40	3.2	odd	2	inner
8001.2.a.ba.1.30	yes	40	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+1.50508 q^{2} +0.265269 q^{4} -2.12799 q^{5} +1.00000 q^{7} -2.61091 q^{8} +O(q^{10})\)	\(q+1.50508 q^{2} +0.265269 q^{4} -2.12799 q^{5} +1.00000 q^{7} -2.61091 q^{8} -3.20280 q^{10} +1.63093 q^{11} +2.77338 q^{13} +1.50508 q^{14} -4.46017 q^{16} -4.81311 q^{17} +1.82102 q^{19} -0.564490 q^{20} +2.45468 q^{22} +4.21796 q^{23} -0.471655 q^{25} +4.17416 q^{26} +0.265269 q^{28} -2.17542 q^{29} -5.01640 q^{31} -1.49110 q^{32} -7.24411 q^{34} -2.12799 q^{35} -4.80716 q^{37} +2.74078 q^{38} +5.55599 q^{40} -0.803009 q^{41} +11.2895 q^{43} +0.432635 q^{44} +6.34837 q^{46} -7.13834 q^{47} +1.00000 q^{49} -0.709880 q^{50} +0.735692 q^{52} +7.35769 q^{53} -3.47061 q^{55} -2.61091 q^{56} -3.27418 q^{58} +1.99972 q^{59} +1.64713 q^{61} -7.55008 q^{62} +6.67612 q^{64} -5.90173 q^{65} +10.7832 q^{67} -1.27677 q^{68} -3.20280 q^{70} +4.48471 q^{71} +5.53294 q^{73} -7.23516 q^{74} +0.483060 q^{76} +1.63093 q^{77} -0.583282 q^{79} +9.49120 q^{80} -1.20859 q^{82} -4.31083 q^{83} +10.2422 q^{85} +16.9916 q^{86} -4.25822 q^{88} -0.157199 q^{89} +2.77338 q^{91} +1.11889 q^{92} -10.7438 q^{94} -3.87512 q^{95} +2.33005 q^{97} +1.50508 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 40 * q + 54 * q^4 + 40 * q^7	\( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) \) 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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