LMFDB - Embedded newform 8044.2.a.a.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8044,2,Mod(1,8044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8044, 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("8044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8044 = 2^{2} \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8044.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:	$64.2316633859$
Analytic rank:	$1$
Dimension:	$80$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.88063 q^{3} +4.02746 q^{5} +2.18923 q^{7} +5.29800 q^{9} +O(q^{10})$	$q-2.88063 q^{3} +4.02746 q^{5} +2.18923 q^{7} +5.29800 q^{9} -2.81439 q^{11} +2.88746 q^{13} -11.6016 q^{15} +3.25726 q^{17} -4.41551 q^{19} -6.30634 q^{21} +0.689969 q^{23} +11.2205 q^{25} -6.61968 q^{27} -3.82147 q^{29} -3.66502 q^{31} +8.10719 q^{33} +8.81703 q^{35} -10.9330 q^{37} -8.31768 q^{39} -2.86122 q^{41} -2.80191 q^{43} +21.3375 q^{45} -9.38720 q^{47} -2.20729 q^{49} -9.38296 q^{51} -5.15319 q^{53} -11.3348 q^{55} +12.7194 q^{57} +14.9848 q^{59} +4.99398 q^{61} +11.5985 q^{63} +11.6291 q^{65} -12.4932 q^{67} -1.98754 q^{69} -14.1409 q^{71} -9.97005 q^{73} -32.3219 q^{75} -6.16133 q^{77} -11.6789 q^{79} +3.17482 q^{81} -5.38123 q^{83} +13.1185 q^{85} +11.0082 q^{87} -5.97089 q^{89} +6.32129 q^{91} +10.5575 q^{93} -17.7833 q^{95} -3.78569 q^{97} -14.9106 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * 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
$3$	−2.88063	−1.66313	−0.831565	−	0.555428i	$-0.812555\pi$
$3$	−2.88063	−1.66313	−0.831565	+	0.555428i	$0.812555\pi$
$4$	0	0
$5$	4.02746	1.80114	0.900568	−	0.434715i	$-0.143151\pi$
$5$	4.02746	1.80114	0.900568	+	0.434715i	$0.143151\pi$
$6$	0	0
$7$	2.18923	0.827450	0.413725	−	0.910402i	$-0.364228\pi$
$7$	2.18923	0.827450	0.413725	+	0.910402i	$0.364228\pi$
$8$	0	0
$9$	5.29800	1.76600
$10$	0	0
$11$	−2.81439	−0.848570	−0.424285	−	0.905529i	$-0.639474\pi$
$11$	−2.81439	−0.848570	−0.424285	+	0.905529i	$0.639474\pi$
$12$	0	0
$13$	2.88746	0.800836	0.400418	−	0.916333i	$-0.368865\pi$
$13$	2.88746	0.800836	0.400418	+	0.916333i	$0.368865\pi$
$14$	0	0
$15$	−11.6016	−2.99552
$16$	0	0
$17$	3.25726	0.790003	0.395001	−	0.918681i	$-0.370744\pi$
$17$	3.25726	0.790003	0.395001	+	0.918681i	$0.370744\pi$
$18$	0	0
$19$	−4.41551	−1.01299	−0.506494	−	0.862243i	$-0.669059\pi$
$19$	−4.41551	−1.01299	−0.506494	+	0.862243i	$0.669059\pi$
$20$	0	0
$21$	−6.30634	−1.37616
$22$	0	0
$23$	0.689969	0.143868	0.0719342	−	0.997409i	$-0.477083\pi$
$23$	0.689969	0.143868	0.0719342	+	0.997409i	$0.477083\pi$
$24$	0	0
$25$	11.2205	2.24409
$26$	0	0
$27$	−6.61968	−1.27396
$28$	0	0
$29$	−3.82147	−0.709629	−0.354814	−	0.934937i	$-0.615456\pi$
$29$	−3.82147	−0.709629	−0.354814	+	0.934937i	$0.615456\pi$
$30$	0	0
$31$	−3.66502	−0.658256	−0.329128	−	0.944285i	$-0.606755\pi$
$31$	−3.66502	−0.658256	−0.329128	+	0.944285i	$0.606755\pi$
$32$	0	0
$33$	8.10719	1.41128
$34$	0	0
$35$	8.81703	1.49035
$36$	0	0
$37$	−10.9330	−1.79737	−0.898683	−	0.438598i	$-0.855475\pi$
$37$	−10.9330	−1.79737	−0.898683	+	0.438598i	$0.855475\pi$
$38$	0	0
$39$	−8.31768	−1.33189
$40$	0	0
$41$	−2.86122	−0.446847	−0.223423	−	0.974721i	$-0.571723\pi$
$41$	−2.86122	−0.446847	−0.223423	+	0.974721i	$0.571723\pi$
$42$	0	0
$43$	−2.80191	−0.427287	−0.213644	−	0.976912i	$-0.568533\pi$
$43$	−2.80191	−0.427287	−0.213644	+	0.976912i	$0.568533\pi$
$44$	0	0
$45$	21.3375	3.18081
$46$	0	0
$47$	−9.38720	−1.36926	−0.684632	−	0.728889i	$-0.740037\pi$
$47$	−9.38720	−1.36926	−0.684632	+	0.728889i	$0.740037\pi$
$48$	0	0
$49$	−2.20729	−0.315327
$50$	0	0
$51$	−9.38296	−1.31388
$52$	0	0
$53$	−5.15319	−0.707845	−0.353922	−	0.935275i	$-0.615152\pi$
$53$	−5.15319	−0.707845	−0.353922	+	0.935275i	$0.615152\pi$
$54$	0	0
$55$	−11.3348	−1.52839
$56$	0	0
$57$	12.7194	1.68473
$58$	0	0
$59$	14.9848	1.95085	0.975427	−	0.220322i	$-0.0707107\pi$
$59$	14.9848	1.95085	0.975427	+	0.220322i	$0.0707107\pi$
$60$	0	0
$61$	4.99398	0.639413	0.319706	−	0.947517i	$-0.396416\pi$
$61$	4.99398	0.639413	0.319706	+	0.947517i	$0.396416\pi$
$62$	0	0
$63$	11.5985	1.46128
$64$	0	0
$65$	11.6291	1.44242
$66$	0	0
$67$	−12.4932	−1.52629	−0.763143	−	0.646230i	$-0.776345\pi$
$67$	−12.4932	−1.52629	−0.763143	+	0.646230i	$0.776345\pi$
$68$	0	0
$69$	−1.98754	−0.239272
$70$	0	0
$71$	−14.1409	−1.67822	−0.839111	−	0.543961i	$-0.816924\pi$
$71$	−14.1409	−1.67822	−0.839111	+	0.543961i	$0.816924\pi$
$72$	0	0
$73$	−9.97005	−1.16691	−0.583453	−	0.812147i	$-0.698299\pi$
$73$	−9.97005	−1.16691	−0.583453	+	0.812147i	$0.698299\pi$
$74$	0	0
$75$	−32.3219	−3.73222
$76$	0	0
$77$	−6.16133	−0.702148
$78$	0	0
$79$	−11.6789	−1.31397	−0.656987	−	0.753902i	$-0.728169\pi$
$79$	−11.6789	−1.31397	−0.656987	+	0.753902i	$0.728169\pi$
$80$	0	0
$81$	3.17482	0.352758
$82$	0	0
$83$	−5.38123	−0.590667	−0.295334	−	0.955394i	$-0.595431\pi$
$83$	−5.38123	−0.590667	−0.295334	+	0.955394i	$0.595431\pi$
$84$	0	0
$85$	13.1185	1.42290
$86$	0	0
$87$	11.0082	1.18020
$88$	0	0
$89$	−5.97089	−0.632913	−0.316457	−	0.948607i	$-0.602493\pi$
$89$	−5.97089	−0.632913	−0.316457	+	0.948607i	$0.602493\pi$
$90$	0	0
$91$	6.32129	0.662652
$92$	0	0
$93$	10.5575	1.09477
$94$	0	0
$95$	−17.7833	−1.82453
$96$	0	0
$97$	−3.78569	−0.384379	−0.192189	−	0.981358i	$-0.561559\pi$
$97$	−3.78569	−0.384379	−0.192189	+	0.981358i	$0.561559\pi$
$98$	0	0
$99$	−14.9106	−1.49857
$100$	0	0
$101$	−0.742353	−0.0738669	−0.0369334	−	0.999318i	$-0.511759\pi$
$101$	−0.742353	−0.0738669	−0.0369334	+	0.999318i	$0.511759\pi$
$102$	0	0
$103$	−9.33922	−0.920221	−0.460110	−	0.887862i	$-0.652190\pi$
$103$	−9.33922	−0.920221	−0.460110	+	0.887862i	$0.652190\pi$
$104$	0	0
$105$	−25.3985	−2.47864
$106$	0	0
$107$	−2.22599	−0.215195	−0.107597	−	0.994195i	$-0.534316\pi$
$107$	−2.22599	−0.215195	−0.107597	+	0.994195i	$0.534316\pi$
$108$	0	0
$109$	−10.4969	−1.00542	−0.502709	−	0.864456i	$-0.667663\pi$
$109$	−10.4969	−1.00542	−0.502709	+	0.864456i	$0.667663\pi$
$110$	0	0
$111$	31.4937	2.98925
$112$	0	0
$113$	−9.93364	−0.934478	−0.467239	−	0.884131i	$-0.654751\pi$
$113$	−9.93364	−0.934478	−0.467239	+	0.884131i	$0.654751\pi$
$114$	0	0
$115$	2.77883	0.259127
$116$	0	0
$117$	15.2977	1.41428
$118$	0	0
$119$	7.13089	0.653687
$120$	0	0
$121$	−3.07923	−0.279930
$122$	0	0
$123$	8.24209	0.743164
$124$	0	0
$125$	25.0527	2.24078
$126$	0	0
$127$	−10.6331	−0.943533	−0.471767	−	0.881723i	$-0.656384\pi$
$127$	−10.6331	−0.943533	−0.471767	+	0.881723i	$0.656384\pi$
$128$	0	0
$129$	8.07125	0.710634
$130$	0	0
$131$	2.21504	0.193529	0.0967643	−	0.995307i	$-0.469151\pi$
$131$	2.21504	0.193529	0.0967643	+	0.995307i	$0.469151\pi$
$132$	0	0
$133$	−9.66656	−0.838197
$134$	0	0
$135$	−26.6605	−2.29457
$136$	0	0
$137$	−4.53067	−0.387082	−0.193541	−	0.981092i	$-0.561997\pi$
$137$	−4.53067	−0.387082	−0.193541	+	0.981092i	$0.561997\pi$
$138$	0	0
$139$	13.4565	1.14137	0.570683	−	0.821170i	$-0.306678\pi$
$139$	13.4565	1.14137	0.570683	+	0.821170i	$0.306678\pi$
$140$	0	0
$141$	27.0410	2.27726
$142$	0	0
$143$	−8.12642	−0.679565
$144$	0	0
$145$	−15.3908	−1.27814
$146$	0	0
$147$	6.35838	0.524430
$148$	0	0
$149$	3.90700	0.320074	0.160037	−	0.987111i	$-0.448839\pi$
$149$	3.90700	0.320074	0.160037	+	0.987111i	$0.448839\pi$
$150$	0	0
$151$	14.7696	1.20193	0.600965	−	0.799275i	$-0.294783\pi$
$151$	14.7696	1.20193	0.600965	+	0.799275i	$0.294783\pi$
$152$	0	0
$153$	17.2570	1.39515
$154$	0	0
$155$	−14.7607	−1.18561
$156$	0	0
$157$	3.89008	0.310462	0.155231	−	0.987878i	$-0.450388\pi$
$157$	3.89008	0.310462	0.155231	+	0.987878i	$0.450388\pi$
$158$	0	0
$159$	14.8444	1.17724
$160$	0	0
$161$	1.51050	0.119044
$162$	0	0
$163$	4.37660	0.342802	0.171401	−	0.985201i	$-0.445171\pi$
$163$	4.37660	0.342802	0.171401	+	0.985201i	$0.445171\pi$
$164$	0	0
$165$	32.6514	2.54191
$166$	0	0
$167$	17.2986	1.33860	0.669301	−	0.742991i	$-0.266594\pi$
$167$	17.2986	1.33860	0.669301	+	0.742991i	$0.266594\pi$
$168$	0	0
$169$	−4.66260	−0.358661
$170$	0	0
$171$	−23.3934	−1.78894
$172$	0	0
$173$	7.01438	0.533293	0.266647	−	0.963794i	$-0.414084\pi$
$173$	7.01438	0.533293	0.266647	+	0.963794i	$0.414084\pi$
$174$	0	0
$175$	24.5641	1.85687
$176$	0	0
$177$	−43.1656	−3.24452
$178$	0	0
$179$	−21.8965	−1.63662	−0.818311	−	0.574776i	$-0.805089\pi$
$179$	−21.8965	−1.63662	−0.818311	+	0.574776i	$0.805089\pi$
$180$	0	0
$181$	13.7968	1.02551	0.512753	−	0.858536i	$-0.328626\pi$
$181$	13.7968	1.02551	0.512753	+	0.858536i	$0.328626\pi$
$182$	0	0
$183$	−14.3858	−1.06343
$184$	0	0
$185$	−44.0321	−3.23730
$186$	0	0
$187$	−9.16720	−0.670372
$188$	0	0
$189$	−14.4920	−1.05414
$190$	0	0
$191$	−11.5104	−0.832861	−0.416430	−	0.909168i	$-0.636719\pi$
$191$	−11.5104	−0.832861	−0.416430	+	0.909168i	$0.636719\pi$
$192$	0	0
$193$	9.84065	0.708346	0.354173	−	0.935180i	$-0.384762\pi$
$193$	9.84065	0.708346	0.354173	+	0.935180i	$0.384762\pi$
$194$	0	0
$195$	−33.4991	−2.39892
$196$	0	0
$197$	14.5789	1.03871	0.519353	−	0.854560i	$-0.326173\pi$
$197$	14.5789	1.03871	0.519353	+	0.854560i	$0.326173\pi$
$198$	0	0
$199$	12.4151	0.880081	0.440041	−	0.897978i	$-0.354964\pi$
$199$	12.4151	0.880081	0.440041	+	0.897978i	$0.354964\pi$
$200$	0	0
$201$	35.9882	2.53841
$202$	0	0
$203$	−8.36605	−0.587182
$204$	0	0
$205$	−11.5234	−0.804832
$206$	0	0
$207$	3.65546	0.254072
$208$	0	0
$209$	12.4270	0.859591
$210$	0	0
$211$	7.63380	0.525532	0.262766	−	0.964860i	$-0.415365\pi$
$211$	7.63380	0.525532	0.262766	+	0.964860i	$0.415365\pi$
$212$	0	0
$213$	40.7348	2.79110
$214$	0	0
$215$	−11.2846	−0.769602
$216$	0	0
$217$	−8.02355	−0.544674
$218$	0	0
$219$	28.7200	1.94072
$220$	0	0
$221$	9.40521	0.632663
$222$	0	0
$223$	−21.9379	−1.46907	−0.734535	−	0.678571i	$-0.762600\pi$
$223$	−21.9379	−1.46907	−0.734535	+	0.678571i	$0.762600\pi$
$224$	0	0
$225$	59.4460	3.96307
$226$	0	0
$227$	28.5318	1.89372	0.946860	−	0.321647i	$-0.104237\pi$
$227$	28.5318	1.89372	0.946860	+	0.321647i	$0.104237\pi$
$228$	0	0
$229$	15.8590	1.04799	0.523996	−	0.851721i	$-0.324441\pi$
$229$	15.8590	1.04799	0.523996	+	0.851721i	$0.324441\pi$
$230$	0	0
$231$	17.7485	1.16776
$232$	0	0
$233$	20.3382	1.33240	0.666200	−	0.745773i	$-0.267920\pi$
$233$	20.3382	1.33240	0.666200	+	0.745773i	$0.267920\pi$
$234$	0	0
$235$	−37.8066	−2.46623
$236$	0	0
$237$	33.6424	2.18531
$238$	0	0
$239$	7.88479	0.510024	0.255012	−	0.966938i	$-0.417921\pi$
$239$	7.88479	0.510024	0.255012	+	0.966938i	$0.417921\pi$
$240$	0	0
$241$	−1.19444	−0.0769409	−0.0384704	−	0.999260i	$-0.512249\pi$
$241$	−1.19444	−0.0769409	−0.0384704	+	0.999260i	$0.512249\pi$
$242$	0	0
$243$	10.7136	0.687277
$244$	0	0
$245$	−8.88978	−0.567947
$246$	0	0
$247$	−12.7496	−0.811238
$248$	0	0
$249$	15.5013	0.982356
$250$	0	0
$251$	5.49534	0.346862	0.173431	−	0.984846i	$-0.444515\pi$
$251$	5.49534	0.346862	0.173431	+	0.984846i	$0.444515\pi$
$252$	0	0
$253$	−1.94184	−0.122082
$254$	0	0
$255$	−37.7895	−2.36647
$256$	0	0
$257$	−17.7025	−1.10425	−0.552125	−	0.833761i	$-0.686183\pi$
$257$	−17.7025	−1.10425	−0.552125	+	0.833761i	$0.686183\pi$
$258$	0	0
$259$	−23.9347	−1.48723
$260$	0	0
$261$	−20.2461	−1.25320
$262$	0	0
$263$	−24.6126	−1.51768	−0.758838	−	0.651280i	$-0.774232\pi$
$263$	−24.6126	−1.51768	−0.758838	+	0.651280i	$0.774232\pi$
$264$	0	0
$265$	−20.7543	−1.27492
$266$	0	0
$267$	17.1999	1.05262
$268$	0	0
$269$	−13.6349	−0.831335	−0.415668	−	0.909517i	$-0.636452\pi$
$269$	−13.6349	−0.831335	−0.415668	+	0.909517i	$0.636452\pi$
$270$	0	0
$271$	16.1052	0.978320	0.489160	−	0.872194i	$-0.337303\pi$
$271$	16.1052	0.978320	0.489160	+	0.872194i	$0.337303\pi$
$272$	0	0
$273$	−18.2093	−1.10208
$274$	0	0
$275$	−31.5787	−1.90427
$276$	0	0
$277$	19.3838	1.16466	0.582329	−	0.812954i	$-0.302142\pi$
$277$	19.3838	1.16466	0.582329	+	0.812954i	$0.302142\pi$
$278$	0	0
$279$	−19.4173	−1.16248
$280$	0	0
$281$	−5.93293	−0.353929	−0.176965	−	0.984217i	$-0.556628\pi$
$281$	−5.93293	−0.353929	−0.176965	+	0.984217i	$0.556628\pi$
$282$	0	0
$283$	24.4726	1.45474	0.727372	−	0.686243i	$-0.240741\pi$
$283$	24.4726	1.45474	0.727372	+	0.686243i	$0.240741\pi$
$284$	0	0
$285$	51.2271	3.03443
$286$	0	0
$287$	−6.26385	−0.369743
$288$	0	0
$289$	−6.39023	−0.375896
$290$	0	0
$291$	10.9052	0.639272
$292$	0	0
$293$	3.27294	0.191207	0.0956036	−	0.995419i	$-0.469522\pi$
$293$	3.27294	0.191207	0.0956036	+	0.995419i	$0.469522\pi$
$294$	0	0
$295$	60.3507	3.51376
$296$	0	0
$297$	18.6303	1.08104
$298$	0	0
$299$	1.99226	0.115215
$300$	0	0
$301$	−6.13401	−0.353558
$302$	0	0
$303$	2.13844	0.122850
$304$	0	0
$305$	20.1131	1.15167
$306$	0	0
$307$	−9.01264	−0.514379	−0.257189	−	0.966361i	$-0.582796\pi$
$307$	−9.01264	−0.514379	−0.257189	+	0.966361i	$0.582796\pi$
$308$	0	0
$309$	26.9028	1.53045
$310$	0	0
$311$	17.8784	1.01379	0.506896	−	0.862007i	$-0.330793\pi$
$311$	17.8784	1.01379	0.506896	+	0.862007i	$0.330793\pi$
$312$	0	0
$313$	−10.1676	−0.574709	−0.287355	−	0.957824i	$-0.592776\pi$
$313$	−10.1676	−0.574709	−0.287355	+	0.957824i	$0.592776\pi$
$314$	0	0
$315$	46.7126	2.63196
$316$	0	0
$317$	23.7938	1.33639	0.668196	−	0.743985i	$-0.267067\pi$
$317$	23.7938	1.33639	0.668196	+	0.743985i	$0.267067\pi$
$318$	0	0
$319$	10.7551	0.602169
$320$	0	0
$321$	6.41225	0.357897
$322$	0	0
$323$	−14.3825	−0.800263
$324$	0	0
$325$	32.3986	1.79715
$326$	0	0
$327$	30.2375	1.67214
$328$	0	0
$329$	−20.5507	−1.13300
$330$	0	0
$331$	27.9231	1.53479	0.767395	−	0.641174i	$-0.221552\pi$
$331$	27.9231	1.53479	0.767395	+	0.641174i	$0.221552\pi$
$332$	0	0
$333$	−57.9228	−3.17415
$334$	0	0
$335$	−50.3158	−2.74905
$336$	0	0
$337$	13.1817	0.718055	0.359028	−	0.933327i	$-0.383108\pi$
$337$	13.1817	0.718055	0.359028	+	0.933327i	$0.383108\pi$
$338$	0	0
$339$	28.6151	1.55416
$340$	0	0
$341$	10.3148	0.558576
$342$	0	0
$343$	−20.1568	−1.08837
$344$	0	0
$345$	−8.00475	−0.430961
$346$	0	0
$347$	16.2251	0.871007	0.435503	−	0.900187i	$-0.356570\pi$
$347$	16.2251	0.871007	0.435503	+	0.900187i	$0.356570\pi$
$348$	0	0
$349$	11.2122	0.600176	0.300088	−	0.953912i	$-0.402984\pi$
$349$	11.2122	0.600176	0.300088	+	0.953912i	$0.402984\pi$
$350$	0	0
$351$	−19.1140	−1.02023
$352$	0	0
$353$	−26.1704	−1.39291	−0.696456	−	0.717600i	$-0.745241\pi$
$353$	−26.1704	−1.39291	−0.696456	+	0.717600i	$0.745241\pi$
$354$	0	0
$355$	−56.9521	−3.02270
$356$	0	0
$357$	−20.5414	−1.08717
$358$	0	0
$359$	6.09697	0.321786	0.160893	−	0.986972i	$-0.448563\pi$
$359$	6.09697	0.321786	0.160893	+	0.986972i	$0.448563\pi$
$360$	0	0
$361$	0.496758	0.0261452
$362$	0	0
$363$	8.87010	0.465559
$364$	0	0
$365$	−40.1540	−2.10176
$366$	0	0
$367$	1.41993	0.0741196	0.0370598	−	0.999313i	$-0.488201\pi$
$367$	1.41993	0.0741196	0.0370598	+	0.999313i	$0.488201\pi$
$368$	0	0
$369$	−15.1587	−0.789132
$370$	0	0
$371$	−11.2815	−0.585706
$372$	0	0
$373$	2.02630	0.104918	0.0524590	−	0.998623i	$-0.483294\pi$
$373$	2.02630	0.104918	0.0524590	+	0.998623i	$0.483294\pi$
$374$	0	0
$375$	−72.1674	−3.72671
$376$	0	0
$377$	−11.0343	−0.568296
$378$	0	0
$379$	−28.0338	−1.44000	−0.719999	−	0.693975i	$-0.755858\pi$
$379$	−28.0338	−1.44000	−0.719999	+	0.693975i	$0.755858\pi$
$380$	0	0
$381$	30.6299	1.56922
$382$	0	0
$383$	8.45631	0.432097	0.216049	−	0.976383i	$-0.430683\pi$
$383$	8.45631	0.432097	0.216049	+	0.976383i	$0.430683\pi$
$384$	0	0
$385$	−24.8145	−1.26467
$386$	0	0
$387$	−14.8445	−0.754589
$388$	0	0
$389$	−9.97459	−0.505732	−0.252866	−	0.967501i	$-0.581373\pi$
$389$	−9.97459	−0.505732	−0.252866	+	0.967501i	$0.581373\pi$
$390$	0	0
$391$	2.24741	0.113656
$392$	0	0
$393$	−6.38069	−0.321863
$394$	0	0
$395$	−47.0362	−2.36665
$396$	0	0
$397$	−29.6900	−1.49010	−0.745050	−	0.667009i	$-0.767574\pi$
$397$	−29.6900	−1.49010	−0.745050	+	0.667009i	$0.767574\pi$
$398$	0	0
$399$	27.8457	1.39403
$400$	0	0
$401$	5.99761	0.299507	0.149753	−	0.988723i	$-0.452152\pi$
$401$	5.99761	0.299507	0.149753	+	0.988723i	$0.452152\pi$
$402$	0	0
$403$	−10.5826	−0.527155
$404$	0	0
$405$	12.7865	0.635364
$406$	0	0
$407$	30.7696	1.52519
$408$	0	0
$409$	1.76904	0.0874734	0.0437367	−	0.999043i	$-0.486074\pi$
$409$	1.76904	0.0874734	0.0437367	+	0.999043i	$0.486074\pi$
$410$	0	0
$411$	13.0512	0.643767
$412$	0	0
$413$	32.8051	1.61423
$414$	0	0
$415$	−21.6727	−1.06387
$416$	0	0
$417$	−38.7632	−1.89824
$418$	0	0
$419$	−23.5304	−1.14954	−0.574768	−	0.818317i	$-0.694908\pi$
$419$	−23.5304	−1.14954	−0.574768	+	0.818317i	$0.694908\pi$
$420$	0	0
$421$	−16.8726	−0.822322	−0.411161	−	0.911563i	$-0.634877\pi$
$421$	−16.8726	−0.822322	−0.411161	+	0.911563i	$0.634877\pi$
$422$	0	0
$423$	−49.7334	−2.41812
$424$	0	0
$425$	36.5480	1.77284
$426$	0	0
$427$	10.9329	0.529082
$428$	0	0
$429$	23.4092	1.13021
$430$	0	0
$431$	−29.7006	−1.43063	−0.715314	−	0.698803i	$-0.753716\pi$
$431$	−29.7006	−1.43063	−0.715314	+	0.698803i	$0.753716\pi$
$432$	0	0
$433$	1.97169	0.0947536	0.0473768	−	0.998877i	$-0.484914\pi$
$433$	1.97169	0.0947536	0.0473768	+	0.998877i	$0.484914\pi$
$434$	0	0
$435$	44.3352	2.12571
$436$	0	0
$437$	−3.04657	−0.145737
$438$	0	0
$439$	29.8492	1.42462	0.712311	−	0.701864i	$-0.247648\pi$
$439$	29.8492	1.42462	0.712311	+	0.701864i	$0.247648\pi$
$440$	0	0
$441$	−11.6942	−0.556868
$442$	0	0
$443$	19.0344	0.904350	0.452175	−	0.891929i	$-0.350648\pi$
$443$	19.0344	0.904350	0.452175	+	0.891929i	$0.350648\pi$
$444$	0	0
$445$	−24.0475	−1.13996
$446$	0	0
$447$	−11.2546	−0.532325
$448$	0	0
$449$	−27.8222	−1.31301	−0.656504	−	0.754322i	$-0.727966\pi$
$449$	−27.8222	−1.31301	−0.656504	+	0.754322i	$0.727966\pi$
$450$	0	0
$451$	8.05257	0.379181
$452$	0	0
$453$	−42.5456	−1.99897
$454$	0	0
$455$	25.4588	1.19353
$456$	0	0
$457$	−1.03433	−0.0483839	−0.0241919	−	0.999707i	$-0.507701\pi$
$457$	−1.03433	−0.0483839	−0.0241919	+	0.999707i	$0.507701\pi$
$458$	0	0
$459$	−21.5621	−1.00643
$460$	0	0
$461$	−2.01510	−0.0938528	−0.0469264	−	0.998898i	$-0.514943\pi$
$461$	−2.01510	−0.0938528	−0.0469264	+	0.998898i	$0.514943\pi$
$462$	0	0
$463$	27.4114	1.27392	0.636958	−	0.770899i	$-0.280193\pi$
$463$	27.4114	1.27392	0.636958	+	0.770899i	$0.280193\pi$
$464$	0	0
$465$	42.5201	1.97182
$466$	0	0
$467$	−40.5519	−1.87652	−0.938259	−	0.345934i	$-0.887562\pi$
$467$	−40.5519	−1.87652	−0.938259	+	0.345934i	$0.887562\pi$
$468$	0	0
$469$	−27.3504	−1.26292
$470$	0	0
$471$	−11.2059	−0.516339
$472$	0	0
$473$	7.88566	0.362583
$474$	0	0
$475$	−49.5441	−2.27324
$476$	0	0
$477$	−27.3016	−1.25005
$478$	0	0
$479$	−28.6003	−1.30678	−0.653390	−	0.757022i	$-0.726654\pi$
$479$	−28.6003	−1.30678	−0.653390	+	0.757022i	$0.726654\pi$
$480$	0	0
$481$	−31.5684	−1.43940
$482$	0	0
$483$	−4.35118	−0.197985
$484$	0	0
$485$	−15.2467	−0.692319
$486$	0	0
$487$	2.75337	0.124767	0.0623836	−	0.998052i	$-0.480130\pi$
$487$	2.75337	0.124767	0.0623836	+	0.998052i	$0.480130\pi$
$488$	0	0
$489$	−12.6074	−0.570124
$490$	0	0
$491$	43.3598	1.95680	0.978400	−	0.206720i	$-0.0662789\pi$
$491$	43.3598	1.95680	0.978400	+	0.206720i	$0.0662789\pi$
$492$	0	0
$493$	−12.4475	−0.560609
$494$	0	0
$495$	−60.0520	−2.69914
$496$	0	0
$497$	−30.9577	−1.38864
$498$	0	0
$499$	41.8056	1.87148	0.935738	−	0.352695i	$-0.114735\pi$
$499$	41.8056	1.87148	0.935738	+	0.352695i	$0.114735\pi$
$500$	0	0
$501$	−49.8306	−2.22627
$502$	0	0
$503$	1.51368	0.0674916	0.0337458	−	0.999430i	$-0.489256\pi$
$503$	1.51368	0.0674916	0.0337458	+	0.999430i	$0.489256\pi$
$504$	0	0
$505$	−2.98980	−0.133044
$506$	0	0
$507$	13.4312	0.596500
$508$	0	0
$509$	1.94261	0.0861047	0.0430523	−	0.999073i	$-0.486292\pi$
$509$	1.94261	0.0861047	0.0430523	+	0.999073i	$0.486292\pi$
$510$	0	0
$511$	−21.8267	−0.965556
$512$	0	0
$513$	29.2293	1.29050
$514$	0	0
$515$	−37.6134	−1.65744
$516$	0	0
$517$	26.4192	1.16192
$518$	0	0
$519$	−20.2058	−0.886936
$520$	0	0
$521$	18.4113	0.806614	0.403307	−	0.915065i	$-0.367861\pi$
$521$	18.4113	0.806614	0.403307	+	0.915065i	$0.367861\pi$
$522$	0	0
$523$	13.3274	0.582768	0.291384	−	0.956606i	$-0.405884\pi$
$523$	13.3274	0.582768	0.291384	+	0.956606i	$0.405884\pi$
$524$	0	0
$525$	−70.7600	−3.08822
$526$	0	0
$527$	−11.9379	−0.520024
$528$	0	0
$529$	−22.5239	−0.979302
$530$	0	0
$531$	79.3895	3.44521
$532$	0	0
$533$	−8.26163	−0.357851
$534$	0	0
$535$	−8.96511	−0.387595
$536$	0	0
$537$	63.0756	2.72191
$538$	0	0
$539$	6.21217	0.267577
$540$	0	0
$541$	−29.6613	−1.27524	−0.637619	−	0.770352i	$-0.720081\pi$
$541$	−29.6613	−1.27524	−0.637619	+	0.770352i	$0.720081\pi$
$542$	0	0
$543$	−39.7433	−1.70555
$544$	0	0
$545$	−42.2757	−1.81089
$546$	0	0
$547$	−17.2208	−0.736308	−0.368154	−	0.929765i	$-0.620010\pi$
$547$	−17.2208	−0.736308	−0.368154	+	0.929765i	$0.620010\pi$
$548$	0	0
$549$	26.4581	1.12920
$550$	0	0
$551$	16.8737	0.718846
$552$	0	0
$553$	−25.5677	−1.08725
$554$	0	0
$555$	126.840	5.38405
$556$	0	0
$557$	28.2886	1.19863	0.599314	−	0.800514i	$-0.295440\pi$
$557$	28.2886	1.19863	0.599314	+	0.800514i	$0.295440\pi$
$558$	0	0
$559$	−8.09039	−0.342187
$560$	0	0
$561$	26.4073	1.11492
$562$	0	0
$563$	−27.8785	−1.17494	−0.587469	−	0.809247i	$-0.699875\pi$
$563$	−27.8785	−1.17494	−0.587469	+	0.809247i	$0.699875\pi$
$564$	0	0
$565$	−40.0074	−1.68312
$566$	0	0
$567$	6.95039	0.291889
$568$	0	0
$569$	16.7995	0.704270	0.352135	−	0.935949i	$-0.385456\pi$
$569$	16.7995	0.704270	0.352135	+	0.935949i	$0.385456\pi$
$570$	0	0
$571$	−23.8780	−0.999263	−0.499632	−	0.866238i	$-0.666531\pi$
$571$	−23.8780	−0.999263	−0.499632	+	0.866238i	$0.666531\pi$
$572$	0	0
$573$	33.1571	1.38516
$574$	0	0
$575$	7.74177	0.322854
$576$	0	0
$577$	−1.91836	−0.0798622	−0.0399311	−	0.999202i	$-0.512714\pi$
$577$	−1.91836	−0.0798622	−0.0399311	+	0.999202i	$0.512714\pi$
$578$	0	0
$579$	−28.3472	−1.17807
$580$	0	0
$581$	−11.7807	−0.488747
$582$	0	0
$583$	14.5031	0.600655
$584$	0	0
$585$	61.6111	2.54731
$586$	0	0
$587$	29.7367	1.22736	0.613682	−	0.789553i	$-0.289688\pi$
$587$	29.7367	1.22736	0.613682	+	0.789553i	$0.289688\pi$
$588$	0	0
$589$	16.1829	0.666806
$590$	0	0
$591$	−41.9965	−1.72750
$592$	0	0
$593$	2.05925	0.0845634	0.0422817	−	0.999106i	$-0.486537\pi$
$593$	2.05925	0.0845634	0.0422817	+	0.999106i	$0.486537\pi$
$594$	0	0
$595$	28.7194	1.17738
$596$	0	0
$597$	−35.7632	−1.46369
$598$	0	0
$599$	−27.3144	−1.11604	−0.558019	−	0.829828i	$-0.688438\pi$
$599$	−27.3144	−1.11604	−0.558019	+	0.829828i	$0.688438\pi$
$600$	0	0
$601$	12.5811	0.513192	0.256596	−	0.966519i	$-0.417399\pi$
$601$	12.5811	0.513192	0.256596	+	0.966519i	$0.417399\pi$
$602$	0	0
$603$	−66.1889	−2.69542
$604$	0	0
$605$	−12.4015	−0.504192
$606$	0	0
$607$	14.3640	0.583018	0.291509	−	0.956568i	$-0.405843\pi$
$607$	14.3640	0.583018	0.291509	+	0.956568i	$0.405843\pi$
$608$	0	0
$609$	24.0995	0.976560
$610$	0	0
$611$	−27.1051	−1.09656
$612$	0	0
$613$	−41.1018	−1.66008	−0.830042	−	0.557700i	$-0.811684\pi$
$613$	−41.1018	−1.66008	−0.830042	+	0.557700i	$0.811684\pi$
$614$	0	0
$615$	33.1947	1.33854
$616$	0	0
$617$	34.6348	1.39434	0.697172	−	0.716904i	$-0.254441\pi$
$617$	34.6348	1.39434	0.697172	+	0.716904i	$0.254441\pi$
$618$	0	0
$619$	−29.4686	−1.18444	−0.592221	−	0.805776i	$-0.701749\pi$
$619$	−29.4686	−1.18444	−0.592221	+	0.805776i	$0.701749\pi$
$620$	0	0
$621$	−4.56738	−0.183282
$622$	0	0
$623$	−13.0716	−0.523704
$624$	0	0
$625$	44.7965	1.79186
$626$	0	0
$627$	−35.7974	−1.42961
$628$	0	0
$629$	−35.6115	−1.41992
$630$	0	0
$631$	24.4432	0.973067	0.486533	−	0.873662i	$-0.338261\pi$
$631$	24.4432	0.973067	0.486533	+	0.873662i	$0.338261\pi$
$632$	0	0
$633$	−21.9901	−0.874028
$634$	0	0
$635$	−42.8243	−1.69943
$636$	0	0
$637$	−6.37346	−0.252525
$638$	0	0
$639$	−74.9187	−2.96374
$640$	0	0
$641$	−8.30973	−0.328214	−0.164107	−	0.986443i	$-0.552474\pi$
$641$	−8.30973	−0.328214	−0.164107	+	0.986443i	$0.552474\pi$
$642$	0	0
$643$	45.8173	1.80686	0.903429	−	0.428738i	$-0.141042\pi$
$643$	45.8173	1.80686	0.903429	+	0.428738i	$0.141042\pi$
$644$	0	0
$645$	32.5067	1.27995
$646$	0	0
$647$	−30.8809	−1.21405	−0.607027	−	0.794681i	$-0.707638\pi$
$647$	−30.8809	−1.21405	−0.607027	+	0.794681i	$0.707638\pi$
$648$	0	0
$649$	−42.1730	−1.65544
$650$	0	0
$651$	23.1128	0.905863
$652$	0	0
$653$	−31.0073	−1.21341	−0.606705	−	0.794927i	$-0.707509\pi$
$653$	−31.0073	−1.21341	−0.606705	+	0.794927i	$0.707509\pi$
$654$	0	0
$655$	8.92097	0.348571
$656$	0	0
$657$	−52.8214	−2.06076
$658$	0	0
$659$	15.4937	0.603549	0.301774	−	0.953379i	$-0.402421\pi$
$659$	15.4937	0.603549	0.301774	+	0.953379i	$0.402421\pi$
$660$	0	0
$661$	6.28068	0.244290	0.122145	−	0.992512i	$-0.461023\pi$
$661$	6.28068	0.244290	0.122145	+	0.992512i	$0.461023\pi$
$662$	0	0
$663$	−27.0929	−1.05220
$664$	0	0
$665$	−38.9317	−1.50971
$666$	0	0
$667$	−2.63669	−0.102093
$668$	0	0
$669$	63.1949	2.44325
$670$	0	0
$671$	−14.0550	−0.542586
$672$	0	0
$673$	−11.7391	−0.452509	−0.226255	−	0.974068i	$-0.572648\pi$
$673$	−11.7391	−0.452509	−0.226255	+	0.974068i	$0.572648\pi$
$674$	0	0
$675$	−74.2759	−2.85888
$676$	0	0
$677$	27.2012	1.04543	0.522713	−	0.852509i	$-0.324920\pi$
$677$	27.2012	1.04543	0.522713	+	0.852509i	$0.324920\pi$
$678$	0	0
$679$	−8.28774	−0.318054
$680$	0	0
$681$	−82.1893	−3.14950
$682$	0	0
$683$	−23.7803	−0.909927	−0.454963	−	0.890510i	$-0.650348\pi$
$683$	−23.7803	−0.909927	−0.454963	+	0.890510i	$0.650348\pi$
$684$	0	0
$685$	−18.2471	−0.697187
$686$	0	0
$687$	−45.6838	−1.74295
$688$	0	0
$689$	−14.8796	−0.566868
$690$	0	0
$691$	−11.5877	−0.440815	−0.220408	−	0.975408i	$-0.570739\pi$
$691$	−11.5877	−0.440815	−0.220408	+	0.975408i	$0.570739\pi$
$692$	0	0
$693$	−32.6427	−1.23999
$694$	0	0
$695$	54.1956	2.05576
$696$	0	0
$697$	−9.31973	−0.353010
$698$	0	0
$699$	−58.5867	−2.21595
$700$	0	0
$701$	23.0421	0.870288	0.435144	−	0.900361i	$-0.356697\pi$
$701$	23.0421	0.870288	0.435144	+	0.900361i	$0.356697\pi$
$702$	0	0
$703$	48.2746	1.82071
$704$	0	0
$705$	108.907	4.10166
$706$	0	0
$707$	−1.62518	−0.0611211
$708$	0	0
$709$	9.87677	0.370930	0.185465	−	0.982651i	$-0.440621\pi$
$709$	9.87677	0.370930	0.185465	+	0.982651i	$0.440621\pi$
$710$	0	0
$711$	−61.8746	−2.32048
$712$	0	0
$713$	−2.52875	−0.0947023
$714$	0	0
$715$	−32.7289	−1.22399
$716$	0	0
$717$	−22.7131	−0.848237
$718$	0	0
$719$	18.9798	0.707826	0.353913	−	0.935278i	$-0.384851\pi$
$719$	18.9798	0.707826	0.353913	+	0.935278i	$0.384851\pi$
$720$	0	0
$721$	−20.4457	−0.761436
$722$	0	0
$723$	3.44074	0.127963
$724$	0	0
$725$	−42.8786	−1.59247
$726$	0	0
$727$	−26.7037	−0.990385	−0.495193	−	0.868783i	$-0.664903\pi$
$727$	−26.7037	−0.990385	−0.495193	+	0.868783i	$0.664903\pi$
$728$	0	0
$729$	−40.3863	−1.49579
$730$	0	0
$731$	−9.12656	−0.337558
$732$	0	0
$733$	50.0068	1.84704	0.923521	−	0.383548i	$-0.125298\pi$
$733$	50.0068	1.84704	0.923521	+	0.383548i	$0.125298\pi$
$734$	0	0
$735$	25.6081	0.944570
$736$	0	0
$737$	35.1606	1.29516
$738$	0	0
$739$	−29.6690	−1.09139	−0.545696	−	0.837983i	$-0.683735\pi$
$739$	−29.6690	−1.09139	−0.545696	+	0.837983i	$0.683735\pi$
$740$	0	0
$741$	36.7268	1.34919
$742$	0	0
$743$	52.0775	1.91054	0.955269	−	0.295739i	$-0.0955659\pi$
$743$	52.0775	1.91054	0.955269	+	0.295739i	$0.0955659\pi$
$744$	0	0
$745$	15.7353	0.576497
$746$	0	0
$747$	−28.5098	−1.04312
$748$	0	0
$749$	−4.87320	−0.178063
$750$	0	0
$751$	−12.3957	−0.452327	−0.226164	−	0.974089i	$-0.572618\pi$
$751$	−12.3957	−0.452327	−0.226164	+	0.974089i	$0.572618\pi$
$752$	0	0
$753$	−15.8300	−0.576877
$754$	0	0
$755$	59.4839	2.16484
$756$	0	0
$757$	−36.7010	−1.33392	−0.666961	−	0.745093i	$-0.732405\pi$
$757$	−36.7010	−1.33392	−0.666961	+	0.745093i	$0.732405\pi$
$758$	0	0
$759$	5.59371	0.203039
$760$	0	0
$761$	−38.3328	−1.38956	−0.694781	−	0.719222i	$-0.744499\pi$
$761$	−38.3328	−1.38956	−0.694781	+	0.719222i	$0.744499\pi$
$762$	0	0
$763$	−22.9800	−0.831932
$764$	0	0
$765$	69.5019	2.51285
$766$	0	0
$767$	43.2679	1.56231
$768$	0	0
$769$	30.5522	1.10174	0.550871	−	0.834590i	$-0.314296\pi$
$769$	30.5522	1.10174	0.550871	+	0.834590i	$0.314296\pi$
$770$	0	0
$771$	50.9942	1.83651
$772$	0	0
$773$	−14.3474	−0.516040	−0.258020	−	0.966140i	$-0.583070\pi$
$773$	−14.3474	−0.516040	−0.258020	+	0.966140i	$0.583070\pi$
$774$	0	0
$775$	−41.1232	−1.47719
$776$	0	0
$777$	68.9469	2.47346
$778$	0	0
$779$	12.6337	0.452651
$780$	0	0
$781$	39.7981	1.42409
$782$	0	0
$783$	25.2969	0.904037
$784$	0	0
$785$	15.6672	0.559185
$786$	0	0
$787$	−38.1073	−1.35838	−0.679189	−	0.733964i	$-0.737668\pi$
$787$	−38.1073	−1.35838	−0.679189	+	0.733964i	$0.737668\pi$
$788$	0	0
$789$	70.8995	2.52409
$790$	0	0
$791$	−21.7470	−0.773233
$792$	0	0
$793$	14.4199	0.512065
$794$	0	0
$795$	59.7853	2.12037
$796$	0	0
$797$	19.5254	0.691625	0.345813	−	0.938304i	$-0.387603\pi$
$797$	19.5254	0.691625	0.345813	+	0.938304i	$0.387603\pi$
$798$	0	0
$799$	−30.5766	−1.08172
$800$	0	0
$801$	−31.6338	−1.11773
$802$	0	0
$803$	28.0596	0.990201
$804$	0	0
$805$	6.08348	0.214414
$806$	0	0
$807$	39.2771	1.38262
$808$	0	0
$809$	−17.6321	−0.619910	−0.309955	−	0.950751i	$-0.600314\pi$
$809$	−17.6321	−0.619910	−0.309955	+	0.950751i	$0.600314\pi$
$810$	0	0
$811$	4.35861	0.153052	0.0765258	−	0.997068i	$-0.475617\pi$
$811$	4.35861	0.153052	0.0765258	+	0.997068i	$0.475617\pi$
$812$	0	0
$813$	−46.3930	−1.62707
$814$	0	0
$815$	17.6266	0.617433
$816$	0	0
$817$	12.3719	0.432837
$818$	0	0
$819$	33.4902	1.17024
$820$	0	0
$821$	−5.14538	−0.179575	−0.0897875	−	0.995961i	$-0.528619\pi$
$821$	−5.14538	−0.179575	−0.0897875	+	0.995961i	$0.528619\pi$
$822$	0	0
$823$	55.6339	1.93928	0.969638	−	0.244544i	$-0.0786382\pi$
$823$	55.6339	1.93928	0.969638	+	0.244544i	$0.0786382\pi$
$824$	0	0
$825$	90.9665	3.16705
$826$	0	0
$827$	31.8159	1.10635	0.553173	−	0.833066i	$-0.313417\pi$
$827$	31.8159	1.10635	0.553173	+	0.833066i	$0.313417\pi$
$828$	0	0
$829$	43.8243	1.52208	0.761041	−	0.648704i	$-0.224689\pi$
$829$	43.8243	1.52208	0.761041	+	0.648704i	$0.224689\pi$
$830$	0	0
$831$	−55.8373	−1.93698
$832$	0	0
$833$	−7.18973	−0.249109
$834$	0	0
$835$	69.6693	2.41100
$836$	0	0
$837$	24.2612	0.838591
$838$	0	0
$839$	18.8342	0.650230	0.325115	−	0.945675i	$-0.394597\pi$
$839$	18.8342	0.650230	0.325115	+	0.945675i	$0.394597\pi$
$840$	0	0
$841$	−14.3964	−0.496427
$842$	0	0
$843$	17.0906	0.588630
$844$	0	0
$845$	−18.7784	−0.645998
$846$	0	0
$847$	−6.74112	−0.231628
$848$	0	0
$849$	−70.4964	−2.41943
$850$	0	0
$851$	−7.54340	−0.258584
$852$	0	0
$853$	2.32013	0.0794397	0.0397199	−	0.999211i	$-0.487353\pi$
$853$	2.32013	0.0794397	0.0397199	+	0.999211i	$0.487353\pi$
$854$	0	0
$855$	−94.2161	−3.22212
$856$	0	0
$857$	−16.1104	−0.550320	−0.275160	−	0.961398i	$-0.588731\pi$
$857$	−16.1104	−0.550320	−0.275160	+	0.961398i	$0.588731\pi$
$858$	0	0
$859$	42.3512	1.44500	0.722502	−	0.691369i	$-0.242992\pi$
$859$	42.3512	1.44500	0.722502	+	0.691369i	$0.242992\pi$
$860$	0	0
$861$	18.0438	0.614931
$862$	0	0
$863$	10.2014	0.347259	0.173629	−	0.984811i	$-0.444451\pi$
$863$	10.2014	0.347259	0.173629	+	0.984811i	$0.444451\pi$
$864$	0	0
$865$	28.2501	0.960534
$866$	0	0
$867$	18.4079	0.625163
$868$	0	0
$869$	32.8688	1.11500
$870$	0	0
$871$	−36.0735	−1.22230
$872$	0	0
$873$	−20.0566	−0.678813
$874$	0	0
$875$	54.8460	1.85413
$876$	0	0
$877$	−30.3354	−1.02435	−0.512177	−	0.858880i	$-0.671161\pi$
$877$	−30.3354	−1.02435	−0.512177	+	0.858880i	$0.671161\pi$
$878$	0	0
$879$	−9.42811	−0.318002
$880$	0	0
$881$	34.4805	1.16168	0.580839	−	0.814018i	$-0.302724\pi$
$881$	34.4805	1.16168	0.580839	+	0.814018i	$0.302724\pi$
$882$	0	0
$883$	−21.2115	−0.713824	−0.356912	−	0.934138i	$-0.616170\pi$
$883$	−21.2115	−0.713824	−0.356912	+	0.934138i	$0.616170\pi$
$884$	0	0
$885$	−173.848	−5.84383
$886$	0	0
$887$	42.0321	1.41130	0.705650	−	0.708560i	$-0.250655\pi$
$887$	42.0321	1.41130	0.705650	+	0.708560i	$0.250655\pi$
$888$	0	0
$889$	−23.2782	−0.780726
$890$	0	0
$891$	−8.93517	−0.299339
$892$	0	0
$893$	41.4493	1.38705
$894$	0	0
$895$	−88.1874	−2.94778
$896$	0	0
$897$	−5.73894	−0.191618
$898$	0	0
$899$	14.0057	0.467118
$900$	0	0
$901$	−16.7853	−0.559199
$902$	0	0
$903$	17.6698	0.588014
$904$	0	0
$905$	55.5660	1.84708
$906$	0	0
$907$	−16.9833	−0.563922	−0.281961	−	0.959426i	$-0.590985\pi$
$907$	−16.9833	−0.563922	−0.281961	+	0.959426i	$0.590985\pi$
$908$	0	0
$909$	−3.93299	−0.130449
$910$	0	0
$911$	−21.0992	−0.699048	−0.349524	−	0.936927i	$-0.613657\pi$
$911$	−21.0992	−0.699048	−0.349524	+	0.936927i	$0.613657\pi$
$912$	0	0
$913$	15.1449	0.501222
$914$	0	0
$915$	−57.9382	−1.91538
$916$	0	0
$917$	4.84921	0.160135
$918$	0	0
$919$	18.8808	0.622819	0.311409	−	0.950276i	$-0.399199\pi$
$919$	18.8808	0.622819	0.311409	+	0.950276i	$0.399199\pi$
$920$	0	0
$921$	25.9620	0.855478
$922$	0	0
$923$	−40.8313	−1.34398
$924$	0	0
$925$	−122.673	−4.03346
$926$	0	0
$927$	−49.4792	−1.62511
$928$	0	0
$929$	10.1420	0.332749	0.166375	−	0.986063i	$-0.446794\pi$
$929$	10.1420	0.332749	0.166375	+	0.986063i	$0.446794\pi$
$930$	0	0
$931$	9.74632	0.319423
$932$	0	0
$933$	−51.5010	−1.68607
$934$	0	0
$935$	−36.9206	−1.20743
$936$	0	0
$937$	53.5444	1.74922	0.874610	−	0.484828i	$-0.161118\pi$
$937$	53.5444	1.74922	0.874610	+	0.484828i	$0.161118\pi$
$938$	0	0
$939$	29.2892	0.955816
$940$	0	0
$941$	−38.6888	−1.26122	−0.630610	−	0.776100i	$-0.717195\pi$
$941$	−38.6888	−1.26122	−0.630610	+	0.776100i	$0.717195\pi$
$942$	0	0
$943$	−1.97415	−0.0642872
$944$	0	0
$945$	−58.3659	−1.89864
$946$	0	0
$947$	16.4543	0.534694	0.267347	−	0.963600i	$-0.413853\pi$
$947$	16.4543	0.534694	0.267347	+	0.963600i	$0.413853\pi$
$948$	0	0
$949$	−28.7881	−0.934501
$950$	0	0
$951$	−68.5410	−2.22259
$952$	0	0
$953$	5.87484	0.190305	0.0951523	−	0.995463i	$-0.469666\pi$
$953$	5.87484	0.190305	0.0951523	+	0.995463i	$0.469666\pi$
$954$	0	0
$955$	−46.3576	−1.50010
$956$	0	0
$957$	−30.9814	−1.00149
$958$	0	0
$959$	−9.91867	−0.320291
$960$	0	0
$961$	−17.5677	−0.566699
$962$	0	0
$963$	−11.7933	−0.380034
$964$	0	0
$965$	39.6329	1.27583
$966$	0	0
$967$	4.31660	0.138812	0.0694062	−	0.997588i	$-0.477890\pi$
$967$	4.31660	0.138812	0.0694062	+	0.997588i	$0.477890\pi$
$968$	0	0
$969$	41.4306	1.33094
$970$	0	0
$971$	17.7700	0.570266	0.285133	−	0.958488i	$-0.407962\pi$
$971$	17.7700	0.570266	0.285133	+	0.958488i	$0.407962\pi$
$972$	0	0
$973$	29.4594	0.944424
$974$	0	0
$975$	−93.3282	−2.98889
$976$	0	0
$977$	50.3810	1.61183	0.805916	−	0.592030i	$-0.201673\pi$
$977$	50.3810	1.61183	0.805916	+	0.592030i	$0.201673\pi$
$978$	0	0
$979$	16.8044	0.537071
$980$	0	0
$981$	−55.6124	−1.77557
$982$	0	0
$983$	−39.7248	−1.26702	−0.633512	−	0.773733i	$-0.718387\pi$
$983$	−39.7248	−1.26702	−0.633512	+	0.773733i	$0.718387\pi$
$984$	0	0
$985$	58.7161	1.87085
$986$	0	0
$987$	59.1988	1.88432
$988$	0	0
$989$	−1.93323	−0.0614731
$990$	0	0
$991$	−24.4204	−0.775740	−0.387870	−	0.921714i	$-0.626789\pi$
$991$	−24.4204	−0.775740	−0.387870	+	0.921714i	$0.626789\pi$
$992$	0	0
$993$	−80.4359	−2.55256
$994$	0	0
$995$	50.0013	1.58515
$996$	0	0
$997$	37.3298	1.18225	0.591123	−	0.806582i	$-0.298685\pi$
$997$	37.3298	1.18225	0.591123	+	0.806582i	$0.298685\pi$
$998$	0	0
$999$	72.3727	2.28977

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.a.1.7	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.a.1.7	✓	80	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:	\( N \)	\(=\)	\( 8044 = 2^{2} \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.88063 q^{3} +4.02746 q^{5} +2.18923 q^{7} +5.29800 q^{9} +O(q^{10})\)	\(q-2.88063 q^{3} +4.02746 q^{5} +2.18923 q^{7} +5.29800 q^{9} -2.81439 q^{11} +2.88746 q^{13} -11.6016 q^{15} +3.25726 q^{17} -4.41551 q^{19} -6.30634 q^{21} +0.689969 q^{23} +11.2205 q^{25} -6.61968 q^{27} -3.82147 q^{29} -3.66502 q^{31} +8.10719 q^{33} +8.81703 q^{35} -10.9330 q^{37} -8.31768 q^{39} -2.86122 q^{41} -2.80191 q^{43} +21.3375 q^{45} -9.38720 q^{47} -2.20729 q^{49} -9.38296 q^{51} -5.15319 q^{53} -11.3348 q^{55} +12.7194 q^{57} +14.9848 q^{59} +4.99398 q^{61} +11.5985 q^{63} +11.6291 q^{65} -12.4932 q^{67} -1.98754 q^{69} -14.1409 q^{71} -9.97005 q^{73} -32.3219 q^{75} -6.16133 q^{77} -11.6789 q^{79} +3.17482 q^{81} -5.38123 q^{83} +13.1185 q^{85} +11.0082 q^{87} -5.97089 q^{89} +6.32129 q^{91} +10.5575 q^{93} -17.7833 q^{95} -3.78569 q^{97} -14.9106 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * q^99

Introduction

L-functions

Modular forms

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Database

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