LMFDB - Embedded newform 8012.2.a.b.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8012,2,Mod(1,8012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8012 = 2^{2} \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8012.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:	$63.9761420994$
Analytic rank:	$0$
Dimension:	$88$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56711 q^{3} -4.07865 q^{5} +0.492202 q^{7} +3.59005 q^{9} +O(q^{10})$	$q-2.56711 q^{3} -4.07865 q^{5} +0.492202 q^{7} +3.59005 q^{9} -0.646602 q^{11} -3.61633 q^{13} +10.4703 q^{15} +5.84303 q^{17} +0.227938 q^{19} -1.26354 q^{21} -1.86937 q^{23} +11.6354 q^{25} -1.51473 q^{27} -0.578110 q^{29} -0.432367 q^{31} +1.65990 q^{33} -2.00752 q^{35} +4.50346 q^{37} +9.28352 q^{39} -3.02576 q^{41} -6.66297 q^{43} -14.6426 q^{45} +9.00184 q^{47} -6.75774 q^{49} -14.9997 q^{51} -13.1786 q^{53} +2.63726 q^{55} -0.585142 q^{57} +6.61515 q^{59} -7.99481 q^{61} +1.76703 q^{63} +14.7497 q^{65} -6.65548 q^{67} +4.79888 q^{69} -14.6092 q^{71} -8.75250 q^{73} -29.8693 q^{75} -0.318259 q^{77} +5.79948 q^{79} -6.88167 q^{81} -0.273304 q^{83} -23.8316 q^{85} +1.48407 q^{87} +6.31041 q^{89} -1.77997 q^{91} +1.10993 q^{93} -0.929678 q^{95} +16.6865 q^{97} -2.32134 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9}+O(q^{10}) $ 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9	$ 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9} + 16 q^{11} - 3 q^{13} + 21 q^{15} + 32 q^{17} + 49 q^{19} + 7 q^{21} + 24 q^{23} + 114 q^{25} + 82 q^{27} + 2 q^{29} + 40 q^{31} + 31 q^{33} + 26 q^{35} + 37 q^{39} + 22 q^{41} + 98 q^{43} + 12 q^{45} + 48 q^{47} + 132 q^{49} + 55 q^{51} - q^{53} + 116 q^{55} + 62 q^{57} + 34 q^{59} + 132 q^{63} + 39 q^{65} + 75 q^{67} + 15 q^{69} + 24 q^{71} + 104 q^{73} + 87 q^{75} + 4 q^{77} + 111 q^{79} + 128 q^{81} + 64 q^{83} + 7 q^{85} + 115 q^{87} + 14 q^{89} + 73 q^{91} - 7 q^{93} + 51 q^{95} + 117 q^{97} + 72 q^{99}+O(q^{100}) $ 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9 + 16 * q^11 - 3 * q^13 + 21 * q^15 + 32 * q^17 + 49 * q^19 + 7 * q^21 + 24 * q^23 + 114 * q^25 + 82 * q^27 + 2 * q^29 + 40 * q^31 + 31 * q^33 + 26 * q^35 + 37 * q^39 + 22 * q^41 + 98 * q^43 + 12 * q^45 + 48 * q^47 + 132 * q^49 + 55 * q^51 - q^53 + 116 * q^55 + 62 * q^57 + 34 * q^59 + 132 * q^63 + 39 * q^65 + 75 * q^67 + 15 * q^69 + 24 * q^71 + 104 * q^73 + 87 * q^75 + 4 * q^77 + 111 * q^79 + 128 * q^81 + 64 * q^83 + 7 * q^85 + 115 * q^87 + 14 * q^89 + 73 * q^91 - 7 * q^93 + 51 * q^95 + 117 * q^97 + 72 * 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
$3$	−2.56711	−1.48212	−0.741061	−	0.671438i	$-0.765677\pi$
$3$	−2.56711	−1.48212	−0.741061	+	0.671438i	$0.765677\pi$
$4$	0	0
$5$	−4.07865	−1.82403	−0.912013	−	0.410161i	$-0.865473\pi$
$5$	−4.07865	−1.82403	−0.912013	+	0.410161i	$0.865473\pi$
$6$	0	0
$7$	0.492202	0.186035	0.0930175	−	0.995664i	$-0.470349\pi$
$7$	0.492202	0.186035	0.0930175	+	0.995664i	$0.470349\pi$
$8$	0	0
$9$	3.59005	1.19668
$10$	0	0
$11$	−0.646602	−0.194958	−0.0974789	−	0.995238i	$-0.531078\pi$
$11$	−0.646602	−0.194958	−0.0974789	+	0.995238i	$0.531078\pi$
$12$	0	0
$13$	−3.61633	−1.00299	−0.501495	−	0.865161i	$-0.667216\pi$
$13$	−3.61633	−1.00299	−0.501495	+	0.865161i	$0.667216\pi$
$14$	0	0
$15$	10.4703	2.70343
$16$	0	0
$17$	5.84303	1.41714	0.708571	−	0.705639i	$-0.249340\pi$
$17$	5.84303	1.41714	0.708571	+	0.705639i	$0.249340\pi$
$18$	0	0
$19$	0.227938	0.0522925	0.0261463	−	0.999658i	$-0.491676\pi$
$19$	0.227938	0.0522925	0.0261463	+	0.999658i	$0.491676\pi$
$20$	0	0
$21$	−1.26354	−0.275726
$22$	0	0
$23$	−1.86937	−0.389790	−0.194895	−	0.980824i	$-0.562437\pi$
$23$	−1.86937	−0.389790	−0.194895	+	0.980824i	$0.562437\pi$
$24$	0	0
$25$	11.6354	2.32707
$26$	0	0
$27$	−1.51473	−0.291511
$28$	0	0
$29$	−0.578110	−0.107352	−0.0536762	−	0.998558i	$-0.517094\pi$
$29$	−0.578110	−0.107352	−0.0536762	+	0.998558i	$0.517094\pi$
$30$	0	0
$31$	−0.432367	−0.0776555	−0.0388277	−	0.999246i	$-0.512362\pi$
$31$	−0.432367	−0.0776555	−0.0388277	+	0.999246i	$0.512362\pi$
$32$	0	0
$33$	1.65990	0.288951
$34$	0	0
$35$	−2.00752	−0.339333
$36$	0	0
$37$	4.50346	0.740365	0.370182	−	0.928959i	$-0.379295\pi$
$37$	4.50346	0.740365	0.370182	+	0.928959i	$0.379295\pi$
$38$	0	0
$39$	9.28352	1.48655
$40$	0	0
$41$	−3.02576	−0.472544	−0.236272	−	0.971687i	$-0.575926\pi$
$41$	−3.02576	−0.472544	−0.236272	+	0.971687i	$0.575926\pi$
$42$	0	0
$43$	−6.66297	−1.01609	−0.508047	−	0.861330i	$-0.669632\pi$
$43$	−6.66297	−1.01609	−0.508047	+	0.861330i	$0.669632\pi$
$44$	0	0
$45$	−14.6426	−2.18278
$46$	0	0
$47$	9.00184	1.31305	0.656527	−	0.754303i	$-0.272025\pi$
$47$	9.00184	1.31305	0.656527	+	0.754303i	$0.272025\pi$
$48$	0	0
$49$	−6.75774	−0.965391
$50$	0	0
$51$	−14.9997	−2.10038
$52$	0	0
$53$	−13.1786	−1.81022	−0.905112	−	0.425174i	$-0.860213\pi$
$53$	−13.1786	−1.81022	−0.905112	+	0.425174i	$0.860213\pi$
$54$	0	0
$55$	2.63726	0.355608
$56$	0	0
$57$	−0.585142	−0.0775039
$58$	0	0
$59$	6.61515	0.861219	0.430610	−	0.902538i	$-0.358299\pi$
$59$	6.61515	0.861219	0.430610	+	0.902538i	$0.358299\pi$
$60$	0	0
$61$	−7.99481	−1.02363	−0.511815	−	0.859096i	$-0.671027\pi$
$61$	−7.99481	−1.02363	−0.511815	+	0.859096i	$0.671027\pi$
$62$	0	0
$63$	1.76703	0.222625
$64$	0	0
$65$	14.7497	1.82948
$66$	0	0
$67$	−6.65548	−0.813096	−0.406548	−	0.913629i	$-0.633268\pi$
$67$	−6.65548	−0.813096	−0.406548	+	0.913629i	$0.633268\pi$
$68$	0	0
$69$	4.79888	0.577717
$70$	0	0
$71$	−14.6092	−1.73379	−0.866896	−	0.498490i	$-0.833888\pi$
$71$	−14.6092	−1.73379	−0.866896	+	0.498490i	$0.833888\pi$
$72$	0	0
$73$	−8.75250	−1.02440	−0.512202	−	0.858865i	$-0.671170\pi$
$73$	−8.75250	−1.02440	−0.512202	+	0.858865i	$0.671170\pi$
$74$	0	0
$75$	−29.8693	−3.44900
$76$	0	0
$77$	−0.318259	−0.0362689
$78$	0	0
$79$	5.79948	0.652493	0.326246	−	0.945285i	$-0.394216\pi$
$79$	5.79948	0.652493	0.326246	+	0.945285i	$0.394216\pi$
$80$	0	0
$81$	−6.88167	−0.764630
$82$	0	0
$83$	−0.273304	−0.0299990	−0.0149995	−	0.999888i	$-0.504775\pi$
$83$	−0.273304	−0.0299990	−0.0149995	+	0.999888i	$0.504775\pi$
$84$	0	0
$85$	−23.8316	−2.58490
$86$	0	0
$87$	1.48407	0.159109
$88$	0	0
$89$	6.31041	0.668902	0.334451	−	0.942413i	$-0.391449\pi$
$89$	6.31041	0.668902	0.334451	+	0.942413i	$0.391449\pi$
$90$	0	0
$91$	−1.77997	−0.186591
$92$	0	0
$93$	1.10993	0.115095
$94$	0	0
$95$	−0.929678	−0.0953830
$96$	0	0
$97$	16.6865	1.69426	0.847128	−	0.531390i	$-0.178330\pi$
$97$	16.6865	1.69426	0.847128	+	0.531390i	$0.178330\pi$
$98$	0	0
$99$	−2.32134	−0.233303
$100$	0	0
$101$	1.95861	0.194889	0.0974445	−	0.995241i	$-0.468933\pi$
$101$	1.95861	0.194889	0.0974445	+	0.995241i	$0.468933\pi$
$102$	0	0
$103$	−11.7651	−1.15925	−0.579627	−	0.814882i	$-0.696802\pi$
$103$	−11.7651	−1.15925	−0.579627	+	0.814882i	$0.696802\pi$
$104$	0	0
$105$	5.15352	0.502932
$106$	0	0
$107$	−10.0192	−0.968590	−0.484295	−	0.874905i	$-0.660924\pi$
$107$	−10.0192	−0.968590	−0.484295	+	0.874905i	$0.660924\pi$
$108$	0	0
$109$	4.87316	0.466764	0.233382	−	0.972385i	$-0.425021\pi$
$109$	4.87316	0.466764	0.233382	+	0.972385i	$0.425021\pi$
$110$	0	0
$111$	−11.5609	−1.09731
$112$	0	0
$113$	16.9960	1.59885	0.799423	−	0.600769i	$-0.205139\pi$
$113$	16.9960	1.59885	0.799423	+	0.600769i	$0.205139\pi$
$114$	0	0
$115$	7.62450	0.710988
$116$	0	0
$117$	−12.9828	−1.20026
$118$	0	0
$119$	2.87595	0.263638
$120$	0	0
$121$	−10.5819	−0.961991
$122$	0	0
$123$	7.76746	0.700368
$124$	0	0
$125$	−27.0633	−2.42061
$126$	0	0
$127$	8.21606	0.729057	0.364529	−	0.931192i	$-0.381230\pi$
$127$	8.21606	0.729057	0.364529	+	0.931192i	$0.381230\pi$
$128$	0	0
$129$	17.1046	1.50597
$130$	0	0
$131$	−21.3135	−1.86217	−0.931084	−	0.364805i	$-0.881136\pi$
$131$	−21.3135	−1.86217	−0.931084	+	0.364805i	$0.881136\pi$
$132$	0	0
$133$	0.112192	0.00972824
$134$	0	0
$135$	6.17807	0.531723
$136$	0	0
$137$	−21.4610	−1.83353	−0.916767	−	0.399423i	$-0.869210\pi$
$137$	−21.4610	−1.83353	−0.916767	+	0.399423i	$0.869210\pi$
$138$	0	0
$139$	6.55452	0.555947	0.277974	−	0.960589i	$-0.410337\pi$
$139$	6.55452	0.555947	0.277974	+	0.960589i	$0.410337\pi$
$140$	0	0
$141$	−23.1087	−1.94610
$142$	0	0
$143$	2.33833	0.195541
$144$	0	0
$145$	2.35791	0.195814
$146$	0	0
$147$	17.3479	1.43083
$148$	0	0
$149$	20.8686	1.70962	0.854809	−	0.518942i	$-0.173674\pi$
$149$	20.8686	1.70962	0.854809	+	0.518942i	$0.173674\pi$
$150$	0	0
$151$	−15.7062	−1.27815	−0.639076	−	0.769144i	$-0.720683\pi$
$151$	−15.7062	−1.27815	−0.639076	+	0.769144i	$0.720683\pi$
$152$	0	0
$153$	20.9768	1.69587
$154$	0	0
$155$	1.76347	0.141646
$156$	0	0
$157$	−0.838730	−0.0669379	−0.0334690	−	0.999440i	$-0.510655\pi$
$157$	−0.838730	−0.0669379	−0.0334690	+	0.999440i	$0.510655\pi$
$158$	0	0
$159$	33.8310	2.68297
$160$	0	0
$161$	−0.920107	−0.0725146
$162$	0	0
$163$	10.8162	0.847188	0.423594	−	0.905852i	$-0.360768\pi$
$163$	10.8162	0.847188	0.423594	+	0.905852i	$0.360768\pi$
$164$	0	0
$165$	−6.77014	−0.527054
$166$	0	0
$167$	−11.3952	−0.881787	−0.440894	−	0.897559i	$-0.645338\pi$
$167$	−11.3952	−0.881787	−0.440894	+	0.897559i	$0.645338\pi$
$168$	0	0
$169$	0.0778466	0.00598820
$170$	0	0
$171$	0.818309	0.0625777
$172$	0	0
$173$	−17.9372	−1.36374	−0.681871	−	0.731473i	$-0.738833\pi$
$173$	−17.9372	−1.36374	−0.681871	+	0.731473i	$0.738833\pi$
$174$	0	0
$175$	5.72695	0.432917
$176$	0	0
$177$	−16.9818	−1.27643
$178$	0	0
$179$	0.176541	0.0131953	0.00659766	−	0.999978i	$-0.497900\pi$
$179$	0.176541	0.0131953	0.00659766	+	0.999978i	$0.497900\pi$
$180$	0	0
$181$	12.6521	0.940423	0.470212	−	0.882554i	$-0.344178\pi$
$181$	12.6521	0.940423	0.470212	+	0.882554i	$0.344178\pi$
$182$	0	0
$183$	20.5235	1.51714
$184$	0	0
$185$	−18.3680	−1.35044
$186$	0	0
$187$	−3.77811	−0.276283
$188$	0	0
$189$	−0.745556	−0.0542312
$190$	0	0
$191$	7.07234	0.511736	0.255868	−	0.966712i	$-0.417639\pi$
$191$	7.07234	0.511736	0.255868	+	0.966712i	$0.417639\pi$
$192$	0	0
$193$	−5.27013	−0.379352	−0.189676	−	0.981847i	$-0.560744\pi$
$193$	−5.27013	−0.379352	−0.189676	+	0.981847i	$0.560744\pi$
$194$	0	0
$195$	−37.8642	−2.71151
$196$	0	0
$197$	−12.1534	−0.865893	−0.432947	−	0.901420i	$-0.642526\pi$
$197$	−12.1534	−0.865893	−0.432947	+	0.901420i	$0.642526\pi$
$198$	0	0
$199$	9.15910	0.649271	0.324636	−	0.945839i	$-0.394758\pi$
$199$	9.15910	0.649271	0.324636	+	0.945839i	$0.394758\pi$
$200$	0	0
$201$	17.0854	1.20511
$202$	0	0
$203$	−0.284547	−0.0199713
$204$	0	0
$205$	12.3410	0.861933
$206$	0	0
$207$	−6.71114	−0.466456
$208$	0	0
$209$	−0.147385	−0.0101948
$210$	0	0
$211$	5.69609	0.392135	0.196067	−	0.980590i	$-0.437183\pi$
$211$	5.69609	0.392135	0.196067	+	0.980590i	$0.437183\pi$
$212$	0	0
$213$	37.5034	2.56969
$214$	0	0
$215$	27.1759	1.85338
$216$	0	0
$217$	−0.212812	−0.0144466
$218$	0	0
$219$	22.4686	1.51829
$220$	0	0
$221$	−21.1303	−1.42138
$222$	0	0
$223$	12.9947	0.870192	0.435096	−	0.900384i	$-0.356714\pi$
$223$	12.9947	0.870192	0.435096	+	0.900384i	$0.356714\pi$
$224$	0	0
$225$	41.7716	2.78477
$226$	0	0
$227$	−6.62867	−0.439960	−0.219980	−	0.975504i	$-0.570599\pi$
$227$	−6.62867	−0.439960	−0.219980	+	0.975504i	$0.570599\pi$
$228$	0	0
$229$	−23.9848	−1.58496	−0.792481	−	0.609897i	$-0.791211\pi$
$229$	−23.9848	−1.58496	−0.792481	+	0.609897i	$0.791211\pi$
$230$	0	0
$231$	0.817005	0.0537550
$232$	0	0
$233$	−26.0482	−1.70647	−0.853236	−	0.521526i	$-0.825363\pi$
$233$	−26.0482	−1.70647	−0.853236	+	0.521526i	$0.825363\pi$
$234$	0	0
$235$	−36.7153	−2.39504
$236$	0	0
$237$	−14.8879	−0.967074
$238$	0	0
$239$	−17.5107	−1.13267	−0.566336	−	0.824174i	$-0.691640\pi$
$239$	−17.5107	−1.13267	−0.566336	+	0.824174i	$0.691640\pi$
$240$	0	0
$241$	29.4095	1.89443	0.947217	−	0.320594i	$-0.103883\pi$
$241$	29.4095	1.89443	0.947217	+	0.320594i	$0.103883\pi$
$242$	0	0
$243$	22.2102	1.42479
$244$	0	0
$245$	27.5624	1.76090
$246$	0	0
$247$	−0.824299	−0.0524489
$248$	0	0
$249$	0.701601	0.0444621
$250$	0	0
$251$	−3.92310	−0.247624	−0.123812	−	0.992306i	$-0.539512\pi$
$251$	−3.92310	−0.247624	−0.123812	+	0.992306i	$0.539512\pi$
$252$	0	0
$253$	1.20874	0.0759926
$254$	0	0
$255$	61.1785	3.83114
$256$	0	0
$257$	−27.1257	−1.69205	−0.846027	−	0.533140i	$-0.821012\pi$
$257$	−27.1257	−1.69205	−0.846027	+	0.533140i	$0.821012\pi$
$258$	0	0
$259$	2.21661	0.137734
$260$	0	0
$261$	−2.07545	−0.128467
$262$	0	0
$263$	5.19047	0.320058	0.160029	−	0.987112i	$-0.448841\pi$
$263$	5.19047	0.320058	0.160029	+	0.987112i	$0.448841\pi$
$264$	0	0
$265$	53.7510	3.30190
$266$	0	0
$267$	−16.1995	−0.991395
$268$	0	0
$269$	−12.1190	−0.738907	−0.369454	−	0.929249i	$-0.620455\pi$
$269$	−12.1190	−0.738907	−0.369454	+	0.929249i	$0.620455\pi$
$270$	0	0
$271$	25.7936	1.56685	0.783425	−	0.621486i	$-0.213471\pi$
$271$	25.7936	1.56685	0.783425	+	0.621486i	$0.213471\pi$
$272$	0	0
$273$	4.56937	0.276551
$274$	0	0
$275$	−7.52344	−0.453681
$276$	0	0
$277$	0.910611	0.0547133	0.0273567	−	0.999626i	$-0.491291\pi$
$277$	0.910611	0.0547133	0.0273567	+	0.999626i	$0.491291\pi$
$278$	0	0
$279$	−1.55222	−0.0929291
$280$	0	0
$281$	19.2040	1.14561	0.572806	−	0.819691i	$-0.305855\pi$
$281$	19.2040	1.14561	0.572806	+	0.819691i	$0.305855\pi$
$282$	0	0
$283$	−20.2627	−1.20450	−0.602248	−	0.798309i	$-0.705728\pi$
$283$	−20.2627	−1.20450	−0.602248	+	0.798309i	$0.705728\pi$
$284$	0	0
$285$	2.38659	0.141369
$286$	0	0
$287$	−1.48928	−0.0879097
$288$	0	0
$289$	17.1410	1.00829
$290$	0	0
$291$	−42.8360	−2.51109
$292$	0	0
$293$	−22.7443	−1.32874	−0.664369	−	0.747405i	$-0.731300\pi$
$293$	−22.7443	−1.32874	−0.664369	+	0.747405i	$0.731300\pi$
$294$	0	0
$295$	−26.9809	−1.57089
$296$	0	0
$297$	0.979430	0.0568323
$298$	0	0
$299$	6.76026	0.390956
$300$	0	0
$301$	−3.27953	−0.189029
$302$	0	0
$303$	−5.02797	−0.288849
$304$	0	0
$305$	32.6080	1.86713
$306$	0	0
$307$	32.6338	1.86251	0.931255	−	0.364368i	$-0.118715\pi$
$307$	32.6338	1.86251	0.931255	+	0.364368i	$0.118715\pi$
$308$	0	0
$309$	30.2024	1.71816
$310$	0	0
$311$	−3.12496	−0.177200	−0.0886001	−	0.996067i	$-0.528239\pi$
$311$	−3.12496	−0.177200	−0.0886001	+	0.996067i	$0.528239\pi$
$312$	0	0
$313$	−12.1016	−0.684022	−0.342011	−	0.939696i	$-0.611108\pi$
$313$	−12.1016	−0.684022	−0.342011	+	0.939696i	$0.611108\pi$
$314$	0	0
$315$	−7.20710	−0.406074
$316$	0	0
$317$	13.0728	0.734239	0.367120	−	0.930174i	$-0.380344\pi$
$317$	13.0728	0.734239	0.367120	+	0.930174i	$0.380344\pi$
$318$	0	0
$319$	0.373807	0.0209292
$320$	0	0
$321$	25.7203	1.43557
$322$	0	0
$323$	1.33185	0.0741060
$324$	0	0
$325$	−42.0773	−2.33403
$326$	0	0
$327$	−12.5099	−0.691801
$328$	0	0
$329$	4.43072	0.244274
$330$	0	0
$331$	9.82123	0.539824	0.269912	−	0.962885i	$-0.413005\pi$
$331$	9.82123	0.539824	0.269912	+	0.962885i	$0.413005\pi$
$332$	0	0
$333$	16.1677	0.885983
$334$	0	0
$335$	27.1454	1.48311
$336$	0	0
$337$	6.65351	0.362440	0.181220	−	0.983443i	$-0.441995\pi$
$337$	6.65351	0.362440	0.181220	+	0.983443i	$0.441995\pi$
$338$	0	0
$339$	−43.6305	−2.36968
$340$	0	0
$341$	0.279569	0.0151395
$342$	0	0
$343$	−6.77159	−0.365631
$344$	0	0
$345$	−19.5729	−1.05377
$346$	0	0
$347$	13.5225	0.725926	0.362963	−	0.931803i	$-0.381765\pi$
$347$	13.5225	0.725926	0.362963	+	0.931803i	$0.381765\pi$
$348$	0	0
$349$	18.6766	0.999736	0.499868	−	0.866102i	$-0.333382\pi$
$349$	18.6766	0.999736	0.499868	+	0.866102i	$0.333382\pi$
$350$	0	0
$351$	5.47778	0.292382
$352$	0	0
$353$	29.8256	1.58746	0.793728	−	0.608272i	$-0.208137\pi$
$353$	29.8256	1.58746	0.793728	+	0.608272i	$0.208137\pi$
$354$	0	0
$355$	59.5857	3.16248
$356$	0	0
$357$	−7.38288	−0.390744
$358$	0	0
$359$	−24.0380	−1.26868	−0.634338	−	0.773056i	$-0.718727\pi$
$359$	−24.0380	−1.26868	−0.634338	+	0.773056i	$0.718727\pi$
$360$	0	0
$361$	−18.9480	−0.997265
$362$	0	0
$363$	27.1649	1.42579
$364$	0	0
$365$	35.6984	1.86854
$366$	0	0
$367$	16.4940	0.860982	0.430491	−	0.902595i	$-0.358340\pi$
$367$	16.4940	0.860982	0.430491	+	0.902595i	$0.358340\pi$
$368$	0	0
$369$	−10.8626	−0.565486
$370$	0	0
$371$	−6.48655	−0.336765
$372$	0	0
$373$	28.0753	1.45368	0.726842	−	0.686805i	$-0.240987\pi$
$373$	28.0753	1.45368	0.726842	+	0.686805i	$0.240987\pi$
$374$	0	0
$375$	69.4744	3.58764
$376$	0	0
$377$	2.09064	0.107673
$378$	0	0
$379$	31.0051	1.59262	0.796312	−	0.604886i	$-0.206781\pi$
$379$	31.0051	1.59262	0.796312	+	0.604886i	$0.206781\pi$
$380$	0	0
$381$	−21.0915	−1.08055
$382$	0	0
$383$	32.0992	1.64019	0.820097	−	0.572224i	$-0.193919\pi$
$383$	32.0992	1.64019	0.820097	+	0.572224i	$0.193919\pi$
$384$	0	0
$385$	1.29806	0.0661555
$386$	0	0
$387$	−23.9204	−1.21594
$388$	0	0
$389$	−15.0441	−0.762767	−0.381384	−	0.924417i	$-0.624552\pi$
$389$	−15.0441	−0.762767	−0.381384	+	0.924417i	$0.624552\pi$
$390$	0	0
$391$	−10.9228	−0.552388
$392$	0	0
$393$	54.7141	2.75996
$394$	0	0
$395$	−23.6540	−1.19016
$396$	0	0
$397$	15.6032	0.783101	0.391551	−	0.920157i	$-0.371939\pi$
$397$	15.6032	0.783101	0.391551	+	0.920157i	$0.371939\pi$
$398$	0	0
$399$	−0.288008	−0.0144184
$400$	0	0
$401$	−10.9070	−0.544669	−0.272334	−	0.962203i	$-0.587796\pi$
$401$	−10.9070	−0.544669	−0.272334	+	0.962203i	$0.587796\pi$
$402$	0	0
$403$	1.56358	0.0778876
$404$	0	0
$405$	28.0679	1.39471
$406$	0	0
$407$	−2.91195	−0.144340
$408$	0	0
$409$	0.973555	0.0481392	0.0240696	−	0.999710i	$-0.492338\pi$
$409$	0.973555	0.0481392	0.0240696	+	0.999710i	$0.492338\pi$
$410$	0	0
$411$	55.0926	2.71752
$412$	0	0
$413$	3.25599	0.160217
$414$	0	0
$415$	1.11471	0.0547189
$416$	0	0
$417$	−16.8262	−0.823981
$418$	0	0
$419$	−1.30843	−0.0639211	−0.0319606	−	0.999489i	$-0.510175\pi$
$419$	−1.30843	−0.0639211	−0.0319606	+	0.999489i	$0.510175\pi$
$420$	0	0
$421$	−5.61052	−0.273440	−0.136720	−	0.990610i	$-0.543656\pi$
$421$	−5.61052	−0.273440	−0.136720	+	0.990610i	$0.543656\pi$
$422$	0	0
$423$	32.3171	1.57131
$424$	0	0
$425$	67.9857	3.29779
$426$	0	0
$427$	−3.93506	−0.190431
$428$	0	0
$429$	−6.00274	−0.289815
$430$	0	0
$431$	−24.4123	−1.17590	−0.587950	−	0.808898i	$-0.700065\pi$
$431$	−24.4123	−1.17590	−0.587950	+	0.808898i	$0.700065\pi$
$432$	0	0
$433$	4.55425	0.218863	0.109432	−	0.993994i	$-0.465097\pi$
$433$	4.55425	0.218863	0.109432	+	0.993994i	$0.465097\pi$
$434$	0	0
$435$	−6.05301	−0.290220
$436$	0	0
$437$	−0.426100	−0.0203831
$438$	0	0
$439$	−5.97806	−0.285317	−0.142659	−	0.989772i	$-0.545565\pi$
$439$	−5.97806	−0.285317	−0.142659	+	0.989772i	$0.545565\pi$
$440$	0	0
$441$	−24.2606	−1.15527
$442$	0	0
$443$	22.0314	1.04674	0.523371	−	0.852105i	$-0.324674\pi$
$443$	22.0314	1.04674	0.523371	+	0.852105i	$0.324674\pi$
$444$	0	0
$445$	−25.7379	−1.22010
$446$	0	0
$447$	−53.5719	−2.53386
$448$	0	0
$449$	−25.3817	−1.19784	−0.598919	−	0.800809i	$-0.704403\pi$
$449$	−25.3817	−1.19784	−0.598919	+	0.800809i	$0.704403\pi$
$450$	0	0
$451$	1.95646	0.0921261
$452$	0	0
$453$	40.3195	1.89438
$454$	0	0
$455$	7.25985	0.340347
$456$	0	0
$457$	14.8603	0.695136	0.347568	−	0.937655i	$-0.387008\pi$
$457$	14.8603	0.695136	0.347568	+	0.937655i	$0.387008\pi$
$458$	0	0
$459$	−8.85064	−0.413112
$460$	0	0
$461$	−20.2858	−0.944804	−0.472402	−	0.881383i	$-0.656613\pi$
$461$	−20.2858	−0.944804	−0.472402	+	0.881383i	$0.656613\pi$
$462$	0	0
$463$	−42.5505	−1.97749	−0.988744	−	0.149617i	$-0.952196\pi$
$463$	−42.5505	−1.97749	−0.988744	+	0.149617i	$0.952196\pi$
$464$	0	0
$465$	−4.52703	−0.209936
$466$	0	0
$467$	18.2001	0.842199	0.421099	−	0.907014i	$-0.361644\pi$
$467$	18.2001	0.842199	0.421099	+	0.907014i	$0.361644\pi$
$468$	0	0
$469$	−3.27584	−0.151264
$470$	0	0
$471$	2.15311	0.0992102
$472$	0	0
$473$	4.30829	0.198095
$474$	0	0
$475$	2.65214	0.121688
$476$	0	0
$477$	−47.3120	−2.16627
$478$	0	0
$479$	−14.0182	−0.640508	−0.320254	−	0.947332i	$-0.603768\pi$
$479$	−14.0182	−0.640508	−0.320254	+	0.947332i	$0.603768\pi$
$480$	0	0
$481$	−16.2860	−0.742578
$482$	0	0
$483$	2.36202	0.107475
$484$	0	0
$485$	−68.0582	−3.09037
$486$	0	0
$487$	36.8264	1.66876	0.834381	−	0.551188i	$-0.185825\pi$
$487$	36.8264	1.66876	0.834381	+	0.551188i	$0.185825\pi$
$488$	0	0
$489$	−27.7663	−1.25564
$490$	0	0
$491$	9.04194	0.408057	0.204028	−	0.978965i	$-0.434597\pi$
$491$	9.04194	0.408057	0.204028	+	0.978965i	$0.434597\pi$
$492$	0	0
$493$	−3.37791	−0.152134
$494$	0	0
$495$	9.46791	0.425551
$496$	0	0
$497$	−7.19067	−0.322546
$498$	0	0
$499$	−11.6172	−0.520055	−0.260028	−	0.965601i	$-0.583732\pi$
$499$	−11.6172	−0.520055	−0.260028	+	0.965601i	$0.583732\pi$
$500$	0	0
$501$	29.2527	1.30692
$502$	0	0
$503$	13.4010	0.597522	0.298761	−	0.954328i	$-0.403427\pi$
$503$	13.4010	0.597522	0.298761	+	0.954328i	$0.403427\pi$
$504$	0	0
$505$	−7.98848	−0.355483
$506$	0	0
$507$	−0.199841	−0.00887525
$508$	0	0
$509$	−2.42707	−0.107578	−0.0537890	−	0.998552i	$-0.517130\pi$
$509$	−2.42707	−0.107578	−0.0537890	+	0.998552i	$0.517130\pi$
$510$	0	0
$511$	−4.30800	−0.190575
$512$	0	0
$513$	−0.345265	−0.0152438
$514$	0	0
$515$	47.9859	2.11451
$516$	0	0
$517$	−5.82060	−0.255990
$518$	0	0
$519$	46.0468	2.02123
$520$	0	0
$521$	33.3100	1.45934	0.729669	−	0.683801i	$-0.239674\pi$
$521$	33.3100	1.45934	0.729669	+	0.683801i	$0.239674\pi$
$522$	0	0
$523$	−16.6749	−0.729141	−0.364570	−	0.931176i	$-0.618784\pi$
$523$	−16.6749	−0.729141	−0.364570	+	0.931176i	$0.618784\pi$
$524$	0	0
$525$	−14.7017	−0.641635
$526$	0	0
$527$	−2.52633	−0.110049
$528$	0	0
$529$	−19.5055	−0.848063
$530$	0	0
$531$	23.7488	1.03061
$532$	0	0
$533$	10.9421	0.473957
$534$	0	0
$535$	40.8647	1.76673
$536$	0	0
$537$	−0.453201	−0.0195571
$538$	0	0
$539$	4.36956	0.188210
$540$	0	0
$541$	10.8455	0.466283	0.233141	−	0.972443i	$-0.425099\pi$
$541$	10.8455	0.466283	0.233141	+	0.972443i	$0.425099\pi$
$542$	0	0
$543$	−32.4793	−1.39382
$544$	0	0
$545$	−19.8759	−0.851389
$546$	0	0
$547$	44.3383	1.89577	0.947884	−	0.318615i	$-0.103218\pi$
$547$	44.3383	1.89577	0.947884	+	0.318615i	$0.103218\pi$
$548$	0	0
$549$	−28.7018	−1.22496
$550$	0	0
$551$	−0.131773	−0.00561373
$552$	0	0
$553$	2.85452	0.121386
$554$	0	0
$555$	47.1528	2.00152
$556$	0	0
$557$	−17.4536	−0.739532	−0.369766	−	0.929125i	$-0.620562\pi$
$557$	−17.4536	−0.739532	−0.369766	+	0.929125i	$0.620562\pi$
$558$	0	0
$559$	24.0955	1.01913
$560$	0	0
$561$	9.69883	0.409485
$562$	0	0
$563$	28.3757	1.19589	0.597946	−	0.801537i	$-0.295984\pi$
$563$	28.3757	1.19589	0.597946	+	0.801537i	$0.295984\pi$
$564$	0	0
$565$	−69.3205	−2.91634
$566$	0	0
$567$	−3.38717	−0.142248
$568$	0	0
$569$	−17.6301	−0.739094	−0.369547	−	0.929212i	$-0.620487\pi$
$569$	−17.6301	−0.739094	−0.369547	+	0.929212i	$0.620487\pi$
$570$	0	0
$571$	2.00427	0.0838762	0.0419381	−	0.999120i	$-0.486647\pi$
$571$	2.00427	0.0838762	0.0419381	+	0.999120i	$0.486647\pi$
$572$	0	0
$573$	−18.1555	−0.758455
$574$	0	0
$575$	−21.7508	−0.907070
$576$	0	0
$577$	34.8857	1.45231	0.726155	−	0.687531i	$-0.241306\pi$
$577$	34.8857	1.45231	0.726155	+	0.687531i	$0.241306\pi$
$578$	0	0
$579$	13.5290	0.562246
$580$	0	0
$581$	−0.134521	−0.00558086
$582$	0	0
$583$	8.52132	0.352917
$584$	0	0
$585$	52.9524	2.18931
$586$	0	0
$587$	26.0493	1.07517	0.537586	−	0.843209i	$-0.319337\pi$
$587$	26.0493	1.07517	0.537586	+	0.843209i	$0.319337\pi$
$588$	0	0
$589$	−0.0985529	−0.00406080
$590$	0	0
$591$	31.1991	1.28336
$592$	0	0
$593$	−0.0480149	−0.00197173	−0.000985867	−	1.00000i	$-0.500314\pi$
$593$	−0.0480149	−0.00197173	−0.000985867		1.00000i	$0.500314\pi$
$594$	0	0
$595$	−11.7300	−0.480883
$596$	0	0
$597$	−23.5124	−0.962299
$598$	0	0
$599$	27.1090	1.10764	0.553822	−	0.832635i	$-0.313169\pi$
$599$	27.1090	1.10764	0.553822	+	0.832635i	$0.313169\pi$
$600$	0	0
$601$	−20.5316	−0.837502	−0.418751	−	0.908101i	$-0.637532\pi$
$601$	−20.5316	−0.837502	−0.418751	+	0.908101i	$0.637532\pi$
$602$	0	0
$603$	−23.8935	−0.973020
$604$	0	0
$605$	43.1599	1.75470
$606$	0	0
$607$	16.5459	0.671576	0.335788	−	0.941938i	$-0.390997\pi$
$607$	16.5459	0.671576	0.335788	+	0.941938i	$0.390997\pi$
$608$	0	0
$609$	0.730464	0.0295999
$610$	0	0
$611$	−32.5536	−1.31698
$612$	0	0
$613$	4.34374	0.175442	0.0877210	−	0.996145i	$-0.472042\pi$
$613$	4.34374	0.175442	0.0877210	+	0.996145i	$0.472042\pi$
$614$	0	0
$615$	−31.6807	−1.27749
$616$	0	0
$617$	44.4503	1.78950	0.894752	−	0.446564i	$-0.147353\pi$
$617$	44.4503	1.78950	0.894752	+	0.446564i	$0.147353\pi$
$618$	0	0
$619$	13.1844	0.529927	0.264963	−	0.964258i	$-0.414640\pi$
$619$	13.1844	0.529927	0.264963	+	0.964258i	$0.414640\pi$
$620$	0	0
$621$	2.83160	0.113628
$622$	0	0
$623$	3.10600	0.124439
$624$	0	0
$625$	52.2048	2.08819
$626$	0	0
$627$	0.378354	0.0151100
$628$	0	0
$629$	26.3139	1.04920
$630$	0	0
$631$	−29.0295	−1.15565	−0.577823	−	0.816162i	$-0.696098\pi$
$631$	−29.0295	−1.15565	−0.577823	+	0.816162i	$0.696098\pi$
$632$	0	0
$633$	−14.6225	−0.581191
$634$	0	0
$635$	−33.5104	−1.32982
$636$	0	0
$637$	24.4382	0.968277
$638$	0	0
$639$	−52.4478	−2.07480
$640$	0	0
$641$	31.9546	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$641$	31.9546	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$642$	0	0
$643$	46.0769	1.81709	0.908547	−	0.417782i	$-0.137192\pi$
$643$	46.0769	1.81709	0.908547	+	0.417782i	$0.137192\pi$
$644$	0	0
$645$	−69.7635	−2.74694
$646$	0	0
$647$	45.2134	1.77752	0.888760	−	0.458372i	$-0.151567\pi$
$647$	45.2134	1.77752	0.888760	+	0.458372i	$0.151567\pi$
$648$	0	0
$649$	−4.27737	−0.167901
$650$	0	0
$651$	0.546312	0.0214117
$652$	0	0
$653$	−29.4981	−1.15435	−0.577175	−	0.816621i	$-0.695845\pi$
$653$	−29.4981	−1.15435	−0.577175	+	0.816621i	$0.695845\pi$
$654$	0	0
$655$	86.9302	3.39664
$656$	0	0
$657$	−31.4220	−1.22589
$658$	0	0
$659$	−1.46787	−0.0571803	−0.0285901	−	0.999591i	$-0.509102\pi$
$659$	−1.46787	−0.0571803	−0.0285901	+	0.999591i	$0.509102\pi$
$660$	0	0
$661$	−48.8650	−1.90063	−0.950315	−	0.311291i	$-0.899239\pi$
$661$	−48.8650	−1.90063	−0.950315	+	0.311291i	$0.899239\pi$
$662$	0	0
$663$	54.2439	2.10666
$664$	0	0
$665$	−0.457590	−0.0177446
$666$	0	0
$667$	1.08070	0.0418449
$668$	0	0
$669$	−33.3589	−1.28973
$670$	0	0
$671$	5.16946	0.199565
$672$	0	0
$673$	−1.60144	−0.0617311	−0.0308656	−	0.999524i	$-0.509826\pi$
$673$	−1.60144	−0.0617311	−0.0308656	+	0.999524i	$0.509826\pi$
$674$	0	0
$675$	−17.6245	−0.678367
$676$	0	0
$677$	18.2030	0.699598	0.349799	−	0.936825i	$-0.386250\pi$
$677$	18.2030	0.699598	0.349799	+	0.936825i	$0.386250\pi$
$678$	0	0
$679$	8.21312	0.315191
$680$	0	0
$681$	17.0165	0.652075
$682$	0	0
$683$	9.89414	0.378589	0.189294	−	0.981920i	$-0.439380\pi$
$683$	9.89414	0.378589	0.189294	+	0.981920i	$0.439380\pi$
$684$	0	0
$685$	87.5317	3.34441
$686$	0	0
$687$	61.5717	2.34911
$688$	0	0
$689$	47.6583	1.81564
$690$	0	0
$691$	15.7539	0.599307	0.299654	−	0.954048i	$-0.403129\pi$
$691$	15.7539	0.599307	0.299654	+	0.954048i	$0.403129\pi$
$692$	0	0
$693$	−1.14257	−0.0434025
$694$	0	0
$695$	−26.7336	−1.01406
$696$	0	0
$697$	−17.6796	−0.669662
$698$	0	0
$699$	66.8685	2.52920
$700$	0	0
$701$	−28.7988	−1.08771	−0.543857	−	0.839178i	$-0.683037\pi$
$701$	−28.7988	−1.08771	−0.543857	+	0.839178i	$0.683037\pi$
$702$	0	0
$703$	1.02651	0.0387155
$704$	0	0
$705$	94.2523	3.54975
$706$	0	0
$707$	0.964032	0.0362562
$708$	0	0
$709$	−26.4836	−0.994614	−0.497307	−	0.867575i	$-0.665678\pi$
$709$	−26.4836	−0.994614	−0.497307	+	0.867575i	$0.665678\pi$
$710$	0	0
$711$	20.8205	0.780828
$712$	0	0
$713$	0.808254	0.0302694
$714$	0	0
$715$	−9.53720	−0.356671
$716$	0	0
$717$	44.9519	1.67876
$718$	0	0
$719$	14.1241	0.526739	0.263370	−	0.964695i	$-0.415166\pi$
$719$	14.1241	0.526739	0.263370	+	0.964695i	$0.415166\pi$
$720$	0	0
$721$	−5.79083	−0.215662
$722$	0	0
$723$	−75.4975	−2.80778
$724$	0	0
$725$	−6.72652	−0.249817
$726$	0	0
$727$	18.1019	0.671363	0.335682	−	0.941975i	$-0.391033\pi$
$727$	18.1019	0.671363	0.335682	+	0.941975i	$0.391033\pi$
$728$	0	0
$729$	−36.3711	−1.34708
$730$	0	0
$731$	−38.9319	−1.43995
$732$	0	0
$733$	−46.3363	−1.71147	−0.855735	−	0.517414i	$-0.826895\pi$
$733$	−46.3363	−1.71147	−0.855735	+	0.517414i	$0.826895\pi$
$734$	0	0
$735$	−70.7558	−2.60987
$736$	0	0
$737$	4.30344	0.158519
$738$	0	0
$739$	37.5493	1.38127	0.690636	−	0.723203i	$-0.257331\pi$
$739$	37.5493	1.38127	0.690636	+	0.723203i	$0.257331\pi$
$740$	0	0
$741$	2.11607	0.0777356
$742$	0	0
$743$	−27.2238	−0.998746	−0.499373	−	0.866387i	$-0.666436\pi$
$743$	−27.2238	−0.998746	−0.499373	+	0.866387i	$0.666436\pi$
$744$	0	0
$745$	−85.1155	−3.11839
$746$	0	0
$747$	−0.981175	−0.0358993
$748$	0	0
$749$	−4.93146	−0.180192
$750$	0	0
$751$	−12.6397	−0.461228	−0.230614	−	0.973045i	$-0.574073\pi$
$751$	−12.6397	−0.461228	−0.230614	+	0.973045i	$0.574073\pi$
$752$	0	0
$753$	10.0710	0.367009
$754$	0	0
$755$	64.0600	2.33138
$756$	0	0
$757$	−35.8851	−1.30427	−0.652134	−	0.758104i	$-0.726126\pi$
$757$	−35.8851	−1.30427	−0.652134	+	0.758104i	$0.726126\pi$
$758$	0	0
$759$	−3.10296	−0.112630
$760$	0	0
$761$	−25.2356	−0.914789	−0.457395	−	0.889264i	$-0.651217\pi$
$761$	−25.2356	−0.914789	−0.457395	+	0.889264i	$0.651217\pi$
$762$	0	0
$763$	2.39858	0.0868343
$764$	0	0
$765$	−85.5569	−3.09332
$766$	0	0
$767$	−23.9226	−0.863794
$768$	0	0
$769$	−52.2185	−1.88305	−0.941523	−	0.336948i	$-0.890605\pi$
$769$	−52.2185	−1.88305	−0.941523	+	0.336948i	$0.890605\pi$
$770$	0	0
$771$	69.6347	2.50783
$772$	0	0
$773$	−23.6388	−0.850229	−0.425114	−	0.905140i	$-0.639766\pi$
$773$	−23.6388	−0.850229	−0.425114	+	0.905140i	$0.639766\pi$
$774$	0	0
$775$	−5.03075	−0.180710
$776$	0	0
$777$	−5.69029	−0.204138
$778$	0	0
$779$	−0.689685	−0.0247105
$780$	0	0
$781$	9.44632	0.338016
$782$	0	0
$783$	0.875684	0.0312944
$784$	0	0
$785$	3.42088	0.122097
$786$	0	0
$787$	−9.89477	−0.352710	−0.176355	−	0.984327i	$-0.556431\pi$
$787$	−9.89477	−0.352710	−0.176355	+	0.984327i	$0.556431\pi$
$788$	0	0
$789$	−13.3245	−0.474365
$790$	0	0
$791$	8.36545	0.297441
$792$	0	0
$793$	28.9119	1.02669
$794$	0	0
$795$	−137.985	−4.89381
$796$	0	0
$797$	−41.1571	−1.45786	−0.728930	−	0.684589i	$-0.759982\pi$
$797$	−41.1571	−1.45786	−0.728930	+	0.684589i	$0.759982\pi$
$798$	0	0
$799$	52.5980	1.86078
$800$	0	0
$801$	22.6547	0.800465
$802$	0	0
$803$	5.65938	0.199715
$804$	0	0
$805$	3.75279	0.132269
$806$	0	0
$807$	31.1107	1.09515
$808$	0	0
$809$	−44.7363	−1.57285	−0.786423	−	0.617688i	$-0.788069\pi$
$809$	−44.7363	−1.57285	−0.786423	+	0.617688i	$0.788069\pi$
$810$	0	0
$811$	31.6526	1.11147	0.555737	−	0.831358i	$-0.312436\pi$
$811$	31.6526	1.11147	0.555737	+	0.831358i	$0.312436\pi$
$812$	0	0
$813$	−66.2150	−2.32226
$814$	0	0
$815$	−44.1153	−1.54529
$816$	0	0
$817$	−1.51874	−0.0531341
$818$	0	0
$819$	−6.39017	−0.223291
$820$	0	0
$821$	10.3019	0.359540	0.179770	−	0.983709i	$-0.442465\pi$
$821$	10.3019	0.359540	0.179770	+	0.983709i	$0.442465\pi$
$822$	0	0
$823$	51.2845	1.78767	0.893833	−	0.448400i	$-0.148006\pi$
$823$	51.2845	1.78767	0.893833	+	0.448400i	$0.148006\pi$
$824$	0	0
$825$	19.3135	0.672410
$826$	0	0
$827$	−0.971433	−0.0337800	−0.0168900	−	0.999857i	$-0.505377\pi$
$827$	−0.971433	−0.0337800	−0.0168900	+	0.999857i	$0.505377\pi$
$828$	0	0
$829$	16.6923	0.579747	0.289873	−	0.957065i	$-0.406387\pi$
$829$	16.6923	0.579747	0.289873	+	0.957065i	$0.406387\pi$
$830$	0	0
$831$	−2.33764	−0.0810918
$832$	0	0
$833$	−39.4856	−1.36810
$834$	0	0
$835$	46.4770	1.60840
$836$	0	0
$837$	0.654922	0.0226374
$838$	0	0
$839$	−49.0654	−1.69392	−0.846962	−	0.531653i	$-0.821571\pi$
$839$	−49.0654	−1.69392	−0.846962	+	0.531653i	$0.821571\pi$
$840$	0	0
$841$	−28.6658	−0.988475
$842$	0	0
$843$	−49.2987	−1.69794
$844$	0	0
$845$	−0.317509	−0.0109226
$846$	0	0
$847$	−5.20844	−0.178964
$848$	0	0
$849$	52.0167	1.78521
$850$	0	0
$851$	−8.41863	−0.288587
$852$	0	0
$853$	39.4889	1.35207	0.676037	−	0.736868i	$-0.263696\pi$
$853$	39.4889	1.35207	0.676037	+	0.736868i	$0.263696\pi$
$854$	0	0
$855$	−3.33759	−0.114143
$856$	0	0
$857$	9.61095	0.328304	0.164152	−	0.986435i	$-0.447511\pi$
$857$	9.61095	0.328304	0.164152	+	0.986435i	$0.447511\pi$
$858$	0	0
$859$	−15.5030	−0.528954	−0.264477	−	0.964392i	$-0.585199\pi$
$859$	−15.5030	−0.528954	−0.264477	+	0.964392i	$0.585199\pi$
$860$	0	0
$861$	3.82316	0.130293
$862$	0	0
$863$	0.135515	0.00461300	0.00230650	−	0.999997i	$-0.499266\pi$
$863$	0.135515	0.00461300	0.00230650	+	0.999997i	$0.499266\pi$
$864$	0	0
$865$	73.1596	2.48750
$866$	0	0
$867$	−44.0028	−1.49441
$868$	0	0
$869$	−3.74995	−0.127209
$870$	0	0
$871$	24.0684	0.815527
$872$	0	0
$873$	59.9054	2.02749
$874$	0	0
$875$	−13.3206	−0.450319
$876$	0	0
$877$	−41.3907	−1.39767	−0.698833	−	0.715285i	$-0.746297\pi$
$877$	−41.3907	−1.39767	−0.698833	+	0.715285i	$0.746297\pi$
$878$	0	0
$879$	58.3872	1.96935
$880$	0	0
$881$	−11.6497	−0.392488	−0.196244	−	0.980555i	$-0.562875\pi$
$881$	−11.6497	−0.392488	−0.196244	+	0.980555i	$0.562875\pi$
$882$	0	0
$883$	0.100812	0.00339260	0.00169630	−	0.999999i	$-0.499460\pi$
$883$	0.100812	0.00339260	0.00169630	+	0.999999i	$0.499460\pi$
$884$	0	0
$885$	69.2628	2.32825
$886$	0	0
$887$	−8.71399	−0.292587	−0.146294	−	0.989241i	$-0.546734\pi$
$887$	−8.71399	−0.292587	−0.146294	+	0.989241i	$0.546734\pi$
$888$	0	0
$889$	4.04396	0.135630
$890$	0	0
$891$	4.44970	0.149071
$892$	0	0
$893$	2.05186	0.0686629
$894$	0	0
$895$	−0.720050	−0.0240686
$896$	0	0
$897$	−17.3543	−0.579444
$898$	0	0
$899$	0.249956	0.00833650
$900$	0	0
$901$	−77.0031	−2.56534
$902$	0	0
$903$	8.41891	0.280164
$904$	0	0
$905$	−51.6035	−1.71536
$906$	0	0
$907$	−13.1227	−0.435733	−0.217867	−	0.975979i	$-0.569910\pi$
$907$	−13.1227	−0.435733	−0.217867	+	0.975979i	$0.569910\pi$
$908$	0	0
$909$	7.03152	0.233221
$910$	0	0
$911$	−53.7177	−1.77975	−0.889873	−	0.456208i	$-0.849207\pi$
$911$	−53.7177	−1.77975	−0.889873	+	0.456208i	$0.849207\pi$
$912$	0	0
$913$	0.176719	0.00584853
$914$	0	0
$915$	−83.7083	−2.76731
$916$	0	0
$917$	−10.4905	−0.346428
$918$	0	0
$919$	16.1895	0.534043	0.267021	−	0.963691i	$-0.413961\pi$
$919$	16.1895	0.534043	0.267021	+	0.963691i	$0.413961\pi$
$920$	0	0
$921$	−83.7746	−2.76047
$922$	0	0
$923$	52.8316	1.73897
$924$	0	0
$925$	52.3994	1.72288
$926$	0	0
$927$	−42.2375	−1.38726
$928$	0	0
$929$	−3.12780	−0.102620	−0.0513098	−	0.998683i	$-0.516340\pi$
$929$	−3.12780	−0.102620	−0.0513098	+	0.998683i	$0.516340\pi$
$930$	0	0
$931$	−1.54034	−0.0504827
$932$	0	0
$933$	8.02211	0.262632
$934$	0	0
$935$	15.4096	0.503947
$936$	0	0
$937$	2.12174	0.0693143	0.0346572	−	0.999399i	$-0.488966\pi$
$937$	2.12174	0.0693143	0.0346572	+	0.999399i	$0.488966\pi$
$938$	0	0
$939$	31.0661	1.01380
$940$	0	0
$941$	12.4260	0.405076	0.202538	−	0.979274i	$-0.435081\pi$
$941$	12.4260	0.405076	0.202538	+	0.979274i	$0.435081\pi$
$942$	0	0
$943$	5.65626	0.184193
$944$	0	0
$945$	3.04086	0.0989191
$946$	0	0
$947$	−22.9990	−0.747367	−0.373684	−	0.927556i	$-0.621905\pi$
$947$	−22.9990	−0.747367	−0.373684	+	0.927556i	$0.621905\pi$
$948$	0	0
$949$	31.6519	1.02747
$950$	0	0
$951$	−33.5592	−1.08823
$952$	0	0
$953$	26.4279	0.856085	0.428042	−	0.903759i	$-0.359203\pi$
$953$	26.4279	0.856085	0.428042	+	0.903759i	$0.359203\pi$
$954$	0	0
$955$	−28.8456	−0.933420
$956$	0	0
$957$	−0.959604	−0.0310196
$958$	0	0
$959$	−10.5631	−0.341101
$960$	0	0
$961$	−30.8131	−0.993970
$962$	0	0
$963$	−35.9694	−1.15910
$964$	0	0
$965$	21.4950	0.691948
$966$	0	0
$967$	35.2560	1.13376	0.566878	−	0.823802i	$-0.308151\pi$
$967$	35.2560	1.13376	0.566878	+	0.823802i	$0.308151\pi$
$968$	0	0
$969$	−3.41900	−0.109834
$970$	0	0
$971$	−20.5527	−0.659566	−0.329783	−	0.944057i	$-0.606976\pi$
$971$	−20.5527	−0.659566	−0.329783	+	0.944057i	$0.606976\pi$
$972$	0	0
$973$	3.22615	0.103426
$974$	0	0
$975$	108.017	3.45932
$976$	0	0
$977$	−8.53273	−0.272986	−0.136493	−	0.990641i	$-0.543583\pi$
$977$	−8.53273	−0.272986	−0.136493	+	0.990641i	$0.543583\pi$
$978$	0	0
$979$	−4.08032	−0.130408
$980$	0	0
$981$	17.4949	0.558569
$982$	0	0
$983$	−26.2839	−0.838325	−0.419163	−	0.907911i	$-0.637676\pi$
$983$	−26.2839	−0.838325	−0.419163	+	0.907911i	$0.637676\pi$
$984$	0	0
$985$	49.5694	1.57941
$986$	0	0
$987$	−11.3742	−0.362043
$988$	0	0
$989$	12.4555	0.396063
$990$	0	0
$991$	9.39177	0.298339	0.149170	−	0.988812i	$-0.452340\pi$
$991$	9.39177	0.298339	0.149170	+	0.988812i	$0.452340\pi$
$992$	0	0
$993$	−25.2122	−0.800084
$994$	0	0
$995$	−37.3567	−1.18429
$996$	0	0
$997$	38.1458	1.20809	0.604045	−	0.796950i	$-0.293555\pi$
$997$	38.1458	1.20809	0.604045	+	0.796950i	$0.293555\pi$
$998$	0	0
$999$	−6.82155	−0.215824

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8012.2.a.b.1.11	✓	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8012.2.a.b.1.11	✓	88	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8012 = 2^{2} \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8012.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56711 q^{3} -4.07865 q^{5} +0.492202 q^{7} +3.59005 q^{9} +O(q^{10})\)	\(q-2.56711 q^{3} -4.07865 q^{5} +0.492202 q^{7} +3.59005 q^{9} -0.646602 q^{11} -3.61633 q^{13} +10.4703 q^{15} +5.84303 q^{17} +0.227938 q^{19} -1.26354 q^{21} -1.86937 q^{23} +11.6354 q^{25} -1.51473 q^{27} -0.578110 q^{29} -0.432367 q^{31} +1.65990 q^{33} -2.00752 q^{35} +4.50346 q^{37} +9.28352 q^{39} -3.02576 q^{41} -6.66297 q^{43} -14.6426 q^{45} +9.00184 q^{47} -6.75774 q^{49} -14.9997 q^{51} -13.1786 q^{53} +2.63726 q^{55} -0.585142 q^{57} +6.61515 q^{59} -7.99481 q^{61} +1.76703 q^{63} +14.7497 q^{65} -6.65548 q^{67} +4.79888 q^{69} -14.6092 q^{71} -8.75250 q^{73} -29.8693 q^{75} -0.318259 q^{77} +5.79948 q^{79} -6.88167 q^{81} -0.273304 q^{83} -23.8316 q^{85} +1.48407 q^{87} +6.31041 q^{89} -1.77997 q^{91} +1.10993 q^{93} -0.929678 q^{95} +16.6865 q^{97} -2.32134 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9}+O(q^{10}) \) 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9	\( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9} + 16 q^{11} - 3 q^{13} + 21 q^{15} + 32 q^{17} + 49 q^{19} + 7 q^{21} + 24 q^{23} + 114 q^{25} + 82 q^{27} + 2 q^{29} + 40 q^{31} + 31 q^{33} + 26 q^{35} + 37 q^{39} + 22 q^{41} + 98 q^{43} + 12 q^{45} + 48 q^{47} + 132 q^{49} + 55 q^{51} - q^{53} + 116 q^{55} + 62 q^{57} + 34 q^{59} + 132 q^{63} + 39 q^{65} + 75 q^{67} + 15 q^{69} + 24 q^{71} + 104 q^{73} + 87 q^{75} + 4 q^{77} + 111 q^{79} + 128 q^{81} + 64 q^{83} + 7 q^{85} + 115 q^{87} + 14 q^{89} + 73 q^{91} - 7 q^{93} + 51 q^{95} + 117 q^{97} + 72 q^{99}+O(q^{100}) \) 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9 + 16 * q^11 - 3 * q^13 + 21 * q^15 + 32 * q^17 + 49 * q^19 + 7 * q^21 + 24 * q^23 + 114 * q^25 + 82 * q^27 + 2 * q^29 + 40 * q^31 + 31 * q^33 + 26 * q^35 + 37 * q^39 + 22 * q^41 + 98 * q^43 + 12 * q^45 + 48 * q^47 + 132 * q^49 + 55 * q^51 - q^53 + 116 * q^55 + 62 * q^57 + 34 * q^59 + 132 * q^63 + 39 * q^65 + 75 * q^67 + 15 * q^69 + 24 * q^71 + 104 * q^73 + 87 * q^75 + 4 * q^77 + 111 * q^79 + 128 * q^81 + 64 * q^83 + 7 * q^85 + 115 * q^87 + 14 * q^89 + 73 * q^91 - 7 * q^93 + 51 * q^95 + 117 * q^97 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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