LMFDB - Embedded newform 8044.2.a.b.1.12

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:	$0$
Dimension:	$87$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-2.42698 q^{3} -3.68217 q^{5} -0.776134 q^{7} +2.89025 q^{9} +O(q^{10})$	$q-2.42698 q^{3} -3.68217 q^{5} -0.776134 q^{7} +2.89025 q^{9} -2.29225 q^{11} +6.47628 q^{13} +8.93656 q^{15} +2.63364 q^{17} +1.08226 q^{19} +1.88366 q^{21} +9.06346 q^{23} +8.55835 q^{25} +0.266359 q^{27} +7.44001 q^{29} +2.47843 q^{31} +5.56326 q^{33} +2.85785 q^{35} -7.19170 q^{37} -15.7178 q^{39} -5.54709 q^{41} -7.07339 q^{43} -10.6424 q^{45} +4.41288 q^{47} -6.39762 q^{49} -6.39181 q^{51} +6.36246 q^{53} +8.44045 q^{55} -2.62663 q^{57} +14.7487 q^{59} -10.7657 q^{61} -2.24322 q^{63} -23.8468 q^{65} -5.54943 q^{67} -21.9969 q^{69} -5.30688 q^{71} -2.58691 q^{73} -20.7710 q^{75} +1.77909 q^{77} +15.0114 q^{79} -9.31720 q^{81} -9.99003 q^{83} -9.69751 q^{85} -18.0568 q^{87} -2.20716 q^{89} -5.02646 q^{91} -6.01512 q^{93} -3.98506 q^{95} +17.6115 q^{97} -6.62518 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) $ 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9	$ 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9} + 36 q^{11} - q^{13} + 16 q^{15} + 31 q^{17} + 35 q^{19} - 3 q^{21} + 39 q^{23} + 93 q^{25} + 55 q^{27} - 5 q^{29} + 46 q^{31} + 25 q^{33} + 68 q^{35} - 11 q^{37} + 54 q^{39} + 83 q^{41} + 28 q^{43} - 14 q^{45} + 48 q^{47} + 103 q^{49} + 77 q^{51} + 3 q^{53} + 35 q^{55} + 14 q^{57} + 122 q^{59} - 13 q^{61} + 39 q^{63} + 41 q^{65} + 32 q^{67} - 10 q^{69} + 100 q^{71} + 34 q^{73} + 97 q^{75} + 4 q^{77} + 52 q^{79} + 131 q^{81} + 67 q^{83} - 2 q^{85} + 89 q^{87} + 68 q^{89} + 75 q^{91} + 138 q^{95} + 36 q^{97} + 107 q^{99}+O(q^{100}) $ 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9 + 36 * q^11 - q^13 + 16 * q^15 + 31 * q^17 + 35 * q^19 - 3 * q^21 + 39 * q^23 + 93 * q^25 + 55 * q^27 - 5 * q^29 + 46 * q^31 + 25 * q^33 + 68 * q^35 - 11 * q^37 + 54 * q^39 + 83 * q^41 + 28 * q^43 - 14 * q^45 + 48 * q^47 + 103 * q^49 + 77 * q^51 + 3 * q^53 + 35 * q^55 + 14 * q^57 + 122 * q^59 - 13 * q^61 + 39 * q^63 + 41 * q^65 + 32 * q^67 - 10 * q^69 + 100 * q^71 + 34 * q^73 + 97 * q^75 + 4 * q^77 + 52 * q^79 + 131 * q^81 + 67 * q^83 - 2 * q^85 + 89 * q^87 + 68 * q^89 + 75 * q^91 + 138 * q^95 + 36 * q^97 + 107 * 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
$3$	−2.42698	−1.40122	−0.700610	−	0.713545i	$-0.747089\pi$
$3$	−2.42698	−1.40122	−0.700610	+	0.713545i	$0.747089\pi$
$4$	0	0
$5$	−3.68217	−1.64671	−0.823357	−	0.567523i	$-0.807902\pi$
$5$	−3.68217	−1.64671	−0.823357	+	0.567523i	$0.807902\pi$
$6$	0	0
$7$	−0.776134	−0.293351	−0.146676	−	0.989185i	$-0.546857\pi$
$7$	−0.776134	−0.293351	−0.146676	+	0.989185i	$0.546857\pi$
$8$	0	0
$9$	2.89025	0.963417
$10$	0	0
$11$	−2.29225	−0.691140	−0.345570	−	0.938393i	$-0.612314\pi$
$11$	−2.29225	−0.691140	−0.345570	+	0.938393i	$0.612314\pi$
$12$	0	0
$13$	6.47628	1.79620	0.898099	−	0.439793i	$-0.144948\pi$
$13$	6.47628	1.79620	0.898099	+	0.439793i	$0.144948\pi$
$14$	0	0
$15$	8.93656	2.30741
$16$	0	0
$17$	2.63364	0.638752	0.319376	−	0.947628i	$-0.396527\pi$
$17$	2.63364	0.638752	0.319376	+	0.947628i	$0.396527\pi$
$18$	0	0
$19$	1.08226	0.248288	0.124144	−	0.992264i	$-0.460382\pi$
$19$	1.08226	0.248288	0.124144	+	0.992264i	$0.460382\pi$
$20$	0	0
$21$	1.88366	0.411049
$22$	0	0
$23$	9.06346	1.88986	0.944931	−	0.327271i	$-0.106129\pi$
$23$	9.06346	1.88986	0.944931	+	0.327271i	$0.106129\pi$
$24$	0	0
$25$	8.55835	1.71167
$26$	0	0
$27$	0.266359	0.0512609
$28$	0	0
$29$	7.44001	1.38157	0.690787	−	0.723058i	$-0.257264\pi$
$29$	7.44001	1.38157	0.690787	+	0.723058i	$0.257264\pi$
$30$	0	0
$31$	2.47843	0.445140	0.222570	−	0.974917i	$-0.428555\pi$
$31$	2.47843	0.445140	0.222570	+	0.974917i	$0.428555\pi$
$32$	0	0
$33$	5.56326	0.968439
$34$	0	0
$35$	2.85785	0.483065
$36$	0	0
$37$	−7.19170	−1.18231	−0.591154	−	0.806558i	$-0.701328\pi$
$37$	−7.19170	−1.18231	−0.591154	+	0.806558i	$0.701328\pi$
$38$	0	0
$39$	−15.7178	−2.51687
$40$	0	0
$41$	−5.54709	−0.866310	−0.433155	−	0.901319i	$-0.642600\pi$
$41$	−5.54709	−0.866310	−0.433155	+	0.901319i	$0.642600\pi$
$42$	0	0
$43$	−7.07339	−1.07868	−0.539341	−	0.842088i	$-0.681327\pi$
$43$	−7.07339	−1.07868	−0.539341	+	0.842088i	$0.681327\pi$
$44$	0	0
$45$	−10.6424	−1.58647
$46$	0	0
$47$	4.41288	0.643685	0.321842	−	0.946793i	$-0.395698\pi$
$47$	4.41288	0.643685	0.321842	+	0.946793i	$0.395698\pi$
$48$	0	0
$49$	−6.39762	−0.913945
$50$	0	0
$51$	−6.39181	−0.895032
$52$	0	0
$53$	6.36246	0.873951	0.436975	−	0.899473i	$-0.356050\pi$
$53$	6.36246	0.873951	0.436975	+	0.899473i	$0.356050\pi$
$54$	0	0
$55$	8.44045	1.13811
$56$	0	0
$57$	−2.62663	−0.347906
$58$	0	0
$59$	14.7487	1.92012	0.960059	−	0.279799i	$-0.0902678\pi$
$59$	14.7487	1.92012	0.960059	+	0.279799i	$0.0902678\pi$
$60$	0	0
$61$	−10.7657	−1.37841	−0.689203	−	0.724569i	$-0.742039\pi$
$61$	−10.7657	−1.37841	−0.689203	+	0.724569i	$0.742039\pi$
$62$	0	0
$63$	−2.24322	−0.282619
$64$	0	0
$65$	−23.8468	−2.95783
$66$	0	0
$67$	−5.54943	−0.677971	−0.338986	−	0.940792i	$-0.610084\pi$
$67$	−5.54943	−0.677971	−0.338986	+	0.940792i	$0.610084\pi$
$68$	0	0
$69$	−21.9969	−2.64811
$70$	0	0
$71$	−5.30688	−0.629811	−0.314906	−	0.949123i	$-0.601973\pi$
$71$	−5.30688	−0.629811	−0.314906	+	0.949123i	$0.601973\pi$
$72$	0	0
$73$	−2.58691	−0.302775	−0.151388	−	0.988474i	$-0.548374\pi$
$73$	−2.58691	−0.302775	−0.151388	+	0.988474i	$0.548374\pi$
$74$	0	0
$75$	−20.7710	−2.39843
$76$	0	0
$77$	1.77909	0.202747
$78$	0	0
$79$	15.0114	1.68892	0.844459	−	0.535620i	$-0.179922\pi$
$79$	15.0114	1.68892	0.844459	+	0.535620i	$0.179922\pi$
$80$	0	0
$81$	−9.31720	−1.03524
$82$	0	0
$83$	−9.99003	−1.09655	−0.548274	−	0.836299i	$-0.684715\pi$
$83$	−9.99003	−1.09655	−0.548274	+	0.836299i	$0.684715\pi$
$84$	0	0
$85$	−9.69751	−1.05184
$86$	0	0
$87$	−18.0568	−1.93589
$88$	0	0
$89$	−2.20716	−0.233958	−0.116979	−	0.993134i	$-0.537321\pi$
$89$	−2.20716	−0.233958	−0.116979	+	0.993134i	$0.537321\pi$
$90$	0	0
$91$	−5.02646	−0.526916
$92$	0	0
$93$	−6.01512	−0.623739
$94$	0	0
$95$	−3.98506	−0.408859
$96$	0	0
$97$	17.6115	1.78818	0.894090	−	0.447888i	$-0.147824\pi$
$97$	17.6115	1.78818	0.894090	+	0.447888i	$0.147824\pi$
$98$	0	0
$99$	−6.62518	−0.665856
$100$	0	0
$101$	−6.69763	−0.666439	−0.333220	−	0.942849i	$-0.608135\pi$
$101$	−6.69763	−0.666439	−0.333220	+	0.942849i	$0.608135\pi$
$102$	0	0
$103$	1.50282	0.148077	0.0740384	−	0.997255i	$-0.476411\pi$
$103$	1.50282	0.148077	0.0740384	+	0.997255i	$0.476411\pi$
$104$	0	0
$105$	−6.93597	−0.676881
$106$	0	0
$107$	15.6253	1.51055	0.755277	−	0.655405i	$-0.227502\pi$
$107$	15.6253	1.51055	0.755277	+	0.655405i	$0.227502\pi$
$108$	0	0
$109$	1.75333	0.167939	0.0839695	−	0.996468i	$-0.473240\pi$
$109$	1.75333	0.167939	0.0839695	+	0.996468i	$0.473240\pi$
$110$	0	0
$111$	17.4541	1.65667
$112$	0	0
$113$	−0.744793	−0.0700642	−0.0350321	−	0.999386i	$-0.511153\pi$
$113$	−0.744793	−0.0700642	−0.0350321	+	0.999386i	$0.511153\pi$
$114$	0	0
$115$	−33.3732	−3.11206
$116$	0	0
$117$	18.7181	1.73049
$118$	0	0
$119$	−2.04406	−0.187379
$120$	0	0
$121$	−5.74558	−0.522325
$122$	0	0
$123$	13.4627	1.21389
$124$	0	0
$125$	−13.1024	−1.17192
$126$	0	0
$127$	9.73102	0.863488	0.431744	−	0.901996i	$-0.357898\pi$
$127$	9.73102	0.863488	0.431744	+	0.901996i	$0.357898\pi$
$128$	0	0
$129$	17.1670	1.51147
$130$	0	0
$131$	−0.647637	−0.0565843	−0.0282922	−	0.999600i	$-0.509007\pi$
$131$	−0.647637	−0.0565843	−0.0282922	+	0.999600i	$0.509007\pi$
$132$	0	0
$133$	−0.839979	−0.0728354
$134$	0	0
$135$	−0.980779	−0.0844120
$136$	0	0
$137$	−12.8643	−1.09908	−0.549538	−	0.835469i	$-0.685196\pi$
$137$	−12.8643	−1.09908	−0.549538	+	0.835469i	$0.685196\pi$
$138$	0	0
$139$	2.35422	0.199682	0.0998411	−	0.995003i	$-0.468167\pi$
$139$	2.35422	0.199682	0.0998411	+	0.995003i	$0.468167\pi$
$140$	0	0
$141$	−10.7100	−0.901944
$142$	0	0
$143$	−14.8453	−1.24142
$144$	0	0
$145$	−27.3953	−2.27506
$146$	0	0
$147$	15.5269	1.28064
$148$	0	0
$149$	−17.3832	−1.42409	−0.712043	−	0.702136i	$-0.752230\pi$
$149$	−17.3832	−1.42409	−0.712043	+	0.702136i	$0.752230\pi$
$150$	0	0
$151$	14.2344	1.15838	0.579189	−	0.815193i	$-0.303369\pi$
$151$	14.2344	1.15838	0.579189	+	0.815193i	$0.303369\pi$
$152$	0	0
$153$	7.61189	0.615385
$154$	0	0
$155$	−9.12601	−0.733018
$156$	0	0
$157$	8.06692	0.643811	0.321905	−	0.946772i	$-0.395677\pi$
$157$	8.06692	0.643811	0.321905	+	0.946772i	$0.395677\pi$
$158$	0	0
$159$	−15.4416	−1.22460
$160$	0	0
$161$	−7.03445	−0.554393
$162$	0	0
$163$	21.0657	1.64999	0.824996	−	0.565139i	$-0.191177\pi$
$163$	21.0657	1.64999	0.824996	+	0.565139i	$0.191177\pi$
$164$	0	0
$165$	−20.4848	−1.59474
$166$	0	0
$167$	16.8536	1.30417	0.652086	−	0.758145i	$-0.273894\pi$
$167$	16.8536	1.30417	0.652086	+	0.758145i	$0.273894\pi$
$168$	0	0
$169$	28.9423	2.22633
$170$	0	0
$171$	3.12801	0.239205
$172$	0	0
$173$	3.35460	0.255045	0.127523	−	0.991836i	$-0.459297\pi$
$173$	3.35460	0.255045	0.127523	+	0.991836i	$0.459297\pi$
$174$	0	0
$175$	−6.64242	−0.502120
$176$	0	0
$177$	−35.7949	−2.69051
$178$	0	0
$179$	−16.9057	−1.26359	−0.631795	−	0.775136i	$-0.717682\pi$
$179$	−16.9057	−1.26359	−0.631795	+	0.775136i	$0.717682\pi$
$180$	0	0
$181$	7.80334	0.580018	0.290009	−	0.957024i	$-0.406342\pi$
$181$	7.80334	0.580018	0.290009	+	0.957024i	$0.406342\pi$
$182$	0	0
$183$	26.1282	1.93145
$184$	0	0
$185$	26.4810	1.94693
$186$	0	0
$187$	−6.03697	−0.441467
$188$	0	0
$189$	−0.206730	−0.0150374
$190$	0	0
$191$	25.7033	1.85983	0.929914	−	0.367778i	$-0.119881\pi$
$191$	25.7033	1.85983	0.929914	+	0.367778i	$0.119881\pi$
$192$	0	0
$193$	4.32239	0.311132	0.155566	−	0.987825i	$-0.450280\pi$
$193$	4.32239	0.311132	0.155566	+	0.987825i	$0.450280\pi$
$194$	0	0
$195$	57.8757	4.14456
$196$	0	0
$197$	9.78096	0.696865	0.348432	−	0.937334i	$-0.386714\pi$
$197$	9.78096	0.696865	0.348432	+	0.937334i	$0.386714\pi$
$198$	0	0
$199$	−12.9764	−0.919870	−0.459935	−	0.887952i	$-0.652127\pi$
$199$	−12.9764	−0.919870	−0.459935	+	0.887952i	$0.652127\pi$
$200$	0	0
$201$	13.4684	0.949987
$202$	0	0
$203$	−5.77444	−0.405286
$204$	0	0
$205$	20.4253	1.42657
$206$	0	0
$207$	26.1957	1.82072
$208$	0	0
$209$	−2.48081	−0.171602
$210$	0	0
$211$	−26.5951	−1.83088	−0.915442	−	0.402449i	$-0.868159\pi$
$211$	−26.5951	−1.83088	−0.915442	+	0.402449i	$0.868159\pi$
$212$	0	0
$213$	12.8797	0.882504
$214$	0	0
$215$	26.0454	1.77628
$216$	0	0
$217$	−1.92360	−0.130582
$218$	0	0
$219$	6.27840	0.424255
$220$	0	0
$221$	17.0562	1.14733
$222$	0	0
$223$	−14.8709	−0.995832	−0.497916	−	0.867225i	$-0.665901\pi$
$223$	−14.8709	−0.995832	−0.497916	+	0.867225i	$0.665901\pi$
$224$	0	0
$225$	24.7358	1.64905
$226$	0	0
$227$	−23.5851	−1.56540	−0.782699	−	0.622400i	$-0.786158\pi$
$227$	−23.5851	−1.56540	−0.782699	+	0.622400i	$0.786158\pi$
$228$	0	0
$229$	1.75377	0.115892	0.0579462	−	0.998320i	$-0.481545\pi$
$229$	1.75377	0.115892	0.0579462	+	0.998320i	$0.481545\pi$
$230$	0	0
$231$	−4.31783	−0.284093
$232$	0	0
$233$	17.0060	1.11410	0.557049	−	0.830480i	$-0.311934\pi$
$233$	17.0060	1.11410	0.557049	+	0.830480i	$0.311934\pi$
$234$	0	0
$235$	−16.2490	−1.05997
$236$	0	0
$237$	−36.4325	−2.36655
$238$	0	0
$239$	14.0203	0.906900	0.453450	−	0.891282i	$-0.350193\pi$
$239$	14.0203	0.906900	0.453450	+	0.891282i	$0.350193\pi$
$240$	0	0
$241$	−3.82747	−0.246549	−0.123275	−	0.992373i	$-0.539340\pi$
$241$	−3.82747	−0.246549	−0.123275	+	0.992373i	$0.539340\pi$
$242$	0	0
$243$	21.8136	1.39934
$244$	0	0
$245$	23.5571	1.50501
$246$	0	0
$247$	7.00903	0.445974
$248$	0	0
$249$	24.2456	1.53650
$250$	0	0
$251$	4.82305	0.304428	0.152214	−	0.988348i	$-0.451360\pi$
$251$	4.82305	0.304428	0.152214	+	0.988348i	$0.451360\pi$
$252$	0	0
$253$	−20.7757	−1.30616
$254$	0	0
$255$	23.5357	1.47386
$256$	0	0
$257$	18.2204	1.13655	0.568277	−	0.822837i	$-0.307610\pi$
$257$	18.2204	1.13655	0.568277	+	0.822837i	$0.307610\pi$
$258$	0	0
$259$	5.58172	0.346831
$260$	0	0
$261$	21.5035	1.33103
$262$	0	0
$263$	−7.64478	−0.471397	−0.235699	−	0.971826i	$-0.575738\pi$
$263$	−7.64478	−0.471397	−0.235699	+	0.971826i	$0.575738\pi$
$264$	0	0
$265$	−23.4276	−1.43915
$266$	0	0
$267$	5.35674	0.327827
$268$	0	0
$269$	3.49357	0.213006	0.106503	−	0.994312i	$-0.466035\pi$
$269$	3.49357	0.213006	0.106503	+	0.994312i	$0.466035\pi$
$270$	0	0
$271$	−27.1951	−1.65199	−0.825993	−	0.563681i	$-0.809385\pi$
$271$	−27.1951	−1.65199	−0.825993	+	0.563681i	$0.809385\pi$
$272$	0	0
$273$	12.1991	0.738326
$274$	0	0
$275$	−19.6179	−1.18300
$276$	0	0
$277$	−28.3283	−1.70208	−0.851041	−	0.525100i	$-0.824028\pi$
$277$	−28.3283	−1.70208	−0.851041	+	0.525100i	$0.824028\pi$
$278$	0	0
$279$	7.16330	0.428855
$280$	0	0
$281$	−21.1397	−1.26109	−0.630543	−	0.776154i	$-0.717168\pi$
$281$	−21.1397	−1.26109	−0.630543	+	0.776154i	$0.717168\pi$
$282$	0	0
$283$	−14.5263	−0.863497	−0.431749	−	0.901994i	$-0.642103\pi$
$283$	−14.5263	−0.863497	−0.431749	+	0.901994i	$0.642103\pi$
$284$	0	0
$285$	9.67169	0.572901
$286$	0	0
$287$	4.30528	0.254133
$288$	0	0
$289$	−10.0639	−0.591996
$290$	0	0
$291$	−42.7429	−2.50563
$292$	0	0
$293$	−6.03216	−0.352403	−0.176201	−	0.984354i	$-0.556381\pi$
$293$	−6.03216	−0.352403	−0.176201	+	0.984354i	$0.556381\pi$
$294$	0	0
$295$	−54.3072	−3.16189
$296$	0	0
$297$	−0.610563	−0.0354284
$298$	0	0
$299$	58.6975	3.39456
$300$	0	0
$301$	5.48990	0.316432
$302$	0	0
$303$	16.2550	0.933828
$304$	0	0
$305$	39.6411	2.26984
$306$	0	0
$307$	−4.27880	−0.244204	−0.122102	−	0.992518i	$-0.538963\pi$
$307$	−4.27880	−0.244204	−0.122102	+	0.992518i	$0.538963\pi$
$308$	0	0
$309$	−3.64731	−0.207488
$310$	0	0
$311$	19.0598	1.08078	0.540391	−	0.841414i	$-0.318276\pi$
$311$	19.0598	1.08078	0.540391	+	0.841414i	$0.318276\pi$
$312$	0	0
$313$	14.7379	0.833034	0.416517	−	0.909128i	$-0.363251\pi$
$313$	14.7379	0.833034	0.416517	+	0.909128i	$0.363251\pi$
$314$	0	0
$315$	8.25991	0.465393
$316$	0	0
$317$	−10.0483	−0.564367	−0.282184	−	0.959360i	$-0.591059\pi$
$317$	−10.0483	−0.564367	−0.282184	+	0.959360i	$0.591059\pi$
$318$	0	0
$319$	−17.0544	−0.954862
$320$	0	0
$321$	−37.9223	−2.11662
$322$	0	0
$323$	2.85029	0.158594
$324$	0	0
$325$	55.4263	3.07450
$326$	0	0
$327$	−4.25531	−0.235319
$328$	0	0
$329$	−3.42499	−0.188826
$330$	0	0
$331$	−21.6980	−1.19263	−0.596314	−	0.802751i	$-0.703369\pi$
$331$	−21.6980	−1.19263	−0.596314	+	0.802751i	$0.703369\pi$
$332$	0	0
$333$	−20.7858	−1.13906
$334$	0	0
$335$	20.4339	1.11643
$336$	0	0
$337$	8.84040	0.481567	0.240784	−	0.970579i	$-0.422596\pi$
$337$	8.84040	0.481567	0.240784	+	0.970579i	$0.422596\pi$
$338$	0	0
$339$	1.80760	0.0981754
$340$	0	0
$341$	−5.68120	−0.307654
$342$	0	0
$343$	10.3983	0.561458
$344$	0	0
$345$	80.9961	4.36068
$346$	0	0
$347$	−7.38632	−0.396518	−0.198259	−	0.980150i	$-0.563529\pi$
$347$	−7.38632	−0.396518	−0.198259	+	0.980150i	$0.563529\pi$
$348$	0	0
$349$	22.0024	1.17776	0.588881	−	0.808219i	$-0.299569\pi$
$349$	22.0024	1.17776	0.588881	+	0.808219i	$0.299569\pi$
$350$	0	0
$351$	1.72502	0.0920747
$352$	0	0
$353$	−11.3927	−0.606372	−0.303186	−	0.952931i	$-0.598050\pi$
$353$	−11.3927	−0.606372	−0.303186	+	0.952931i	$0.598050\pi$
$354$	0	0
$355$	19.5408	1.03712
$356$	0	0
$357$	4.96090	0.262559
$358$	0	0
$359$	−22.7932	−1.20298	−0.601490	−	0.798880i	$-0.705426\pi$
$359$	−22.7932	−1.20298	−0.601490	+	0.798880i	$0.705426\pi$
$360$	0	0
$361$	−17.8287	−0.938353
$362$	0	0
$363$	13.9444	0.731893
$364$	0	0
$365$	9.52545	0.498585
$366$	0	0
$367$	10.5101	0.548622	0.274311	−	0.961641i	$-0.411550\pi$
$367$	10.5101	0.548622	0.274311	+	0.961641i	$0.411550\pi$
$368$	0	0
$369$	−16.0325	−0.834618
$370$	0	0
$371$	−4.93812	−0.256374
$372$	0	0
$373$	−32.0295	−1.65843	−0.829213	−	0.558933i	$-0.811211\pi$
$373$	−32.0295	−1.65843	−0.829213	+	0.558933i	$0.811211\pi$
$374$	0	0
$375$	31.7994	1.64211
$376$	0	0
$377$	48.1836	2.48158
$378$	0	0
$379$	8.94630	0.459541	0.229770	−	0.973245i	$-0.426202\pi$
$379$	8.94630	0.459541	0.229770	+	0.973245i	$0.426202\pi$
$380$	0	0
$381$	−23.6170	−1.20994
$382$	0	0
$383$	10.8080	0.552262	0.276131	−	0.961120i	$-0.410948\pi$
$383$	10.8080	0.552262	0.276131	+	0.961120i	$0.410948\pi$
$384$	0	0
$385$	−6.55092	−0.333866
$386$	0	0
$387$	−20.4439	−1.03922
$388$	0	0
$389$	8.85585	0.449009	0.224505	−	0.974473i	$-0.427924\pi$
$389$	8.85585	0.449009	0.224505	+	0.974473i	$0.427924\pi$
$390$	0	0
$391$	23.8699	1.20715
$392$	0	0
$393$	1.57180	0.0792871
$394$	0	0
$395$	−55.2746	−2.78117
$396$	0	0
$397$	−1.53214	−0.0768960	−0.0384480	−	0.999261i	$-0.512241\pi$
$397$	−1.53214	−0.0768960	−0.0384480	+	0.999261i	$0.512241\pi$
$398$	0	0
$399$	2.03862	0.102058
$400$	0	0
$401$	16.7244	0.835176	0.417588	−	0.908636i	$-0.362876\pi$
$401$	16.7244	0.835176	0.417588	+	0.908636i	$0.362876\pi$
$402$	0	0
$403$	16.0510	0.799559
$404$	0	0
$405$	34.3075	1.70475
$406$	0	0
$407$	16.4852	0.817141
$408$	0	0
$409$	18.9349	0.936269	0.468135	−	0.883657i	$-0.344926\pi$
$409$	18.9349	0.936269	0.468135	+	0.883657i	$0.344926\pi$
$410$	0	0
$411$	31.2216	1.54005
$412$	0	0
$413$	−11.4470	−0.563268
$414$	0	0
$415$	36.7849	1.80570
$416$	0	0
$417$	−5.71365	−0.279799
$418$	0	0
$419$	37.2378	1.81918	0.909592	−	0.415502i	$-0.136394\pi$
$419$	37.2378	1.81918	0.909592	+	0.415502i	$0.136394\pi$
$420$	0	0
$421$	−9.95604	−0.485228	−0.242614	−	0.970123i	$-0.578005\pi$
$421$	−9.95604	−0.485228	−0.242614	+	0.970123i	$0.578005\pi$
$422$	0	0
$423$	12.7543	0.620137
$424$	0	0
$425$	22.5396	1.09333
$426$	0	0
$427$	8.35561	0.404357
$428$	0	0
$429$	36.0292	1.73951
$430$	0	0
$431$	17.6806	0.851643	0.425821	−	0.904807i	$-0.359985\pi$
$431$	17.6806	0.851643	0.425821	+	0.904807i	$0.359985\pi$
$432$	0	0
$433$	−35.6500	−1.71323	−0.856615	−	0.515955i	$-0.827437\pi$
$433$	−35.6500	−1.71323	−0.856615	+	0.515955i	$0.827437\pi$
$434$	0	0
$435$	66.4881	3.18786
$436$	0	0
$437$	9.80902	0.469229
$438$	0	0
$439$	8.13521	0.388272	0.194136	−	0.980975i	$-0.437810\pi$
$439$	8.13521	0.388272	0.194136	+	0.980975i	$0.437810\pi$
$440$	0	0
$441$	−18.4907	−0.880510
$442$	0	0
$443$	−15.8912	−0.755014	−0.377507	−	0.926007i	$-0.623219\pi$
$443$	−15.8912	−0.755014	−0.377507	+	0.926007i	$0.623219\pi$
$444$	0	0
$445$	8.12712	0.385263
$446$	0	0
$447$	42.1887	1.99546
$448$	0	0
$449$	15.4566	0.729442	0.364721	−	0.931117i	$-0.381164\pi$
$449$	15.4566	0.729442	0.364721	+	0.931117i	$0.381164\pi$
$450$	0	0
$451$	12.7153	0.598741
$452$	0	0
$453$	−34.5466	−1.62314
$454$	0	0
$455$	18.5083	0.867681
$456$	0	0
$457$	−36.5578	−1.71010	−0.855050	−	0.518545i	$-0.826474\pi$
$457$	−36.5578	−1.71010	−0.855050	+	0.518545i	$0.826474\pi$
$458$	0	0
$459$	0.701495	0.0327430
$460$	0	0
$461$	18.1136	0.843633	0.421816	−	0.906681i	$-0.361393\pi$
$461$	18.1136	0.843633	0.421816	+	0.906681i	$0.361393\pi$
$462$	0	0
$463$	17.5370	0.815012	0.407506	−	0.913202i	$-0.366398\pi$
$463$	17.5370	0.815012	0.407506	+	0.913202i	$0.366398\pi$
$464$	0	0
$465$	22.1487	1.02712
$466$	0	0
$467$	28.7489	1.33034	0.665170	−	0.746692i	$-0.268359\pi$
$467$	28.7489	1.33034	0.665170	+	0.746692i	$0.268359\pi$
$468$	0	0
$469$	4.30710	0.198884
$470$	0	0
$471$	−19.5783	−0.902120
$472$	0	0
$473$	16.2140	0.745520
$474$	0	0
$475$	9.26237	0.424987
$476$	0	0
$477$	18.3891	0.841979
$478$	0	0
$479$	12.6912	0.579874	0.289937	−	0.957046i	$-0.406366\pi$
$479$	12.6912	0.579874	0.289937	+	0.957046i	$0.406366\pi$
$480$	0	0
$481$	−46.5755	−2.12366
$482$	0	0
$483$	17.0725	0.776826
$484$	0	0
$485$	−64.8486	−2.94462
$486$	0	0
$487$	12.1986	0.552771	0.276385	−	0.961047i	$-0.410863\pi$
$487$	12.1986	0.552771	0.276385	+	0.961047i	$0.410863\pi$
$488$	0	0
$489$	−51.1261	−2.31200
$490$	0	0
$491$	−2.33897	−0.105556	−0.0527781	−	0.998606i	$-0.516808\pi$
$491$	−2.33897	−0.105556	−0.0527781	+	0.998606i	$0.516808\pi$
$492$	0	0
$493$	19.5943	0.882484
$494$	0	0
$495$	24.3950	1.09648
$496$	0	0
$497$	4.11885	0.184756
$498$	0	0
$499$	10.5290	0.471344	0.235672	−	0.971833i	$-0.424271\pi$
$499$	10.5290	0.471344	0.235672	+	0.971833i	$0.424271\pi$
$500$	0	0
$501$	−40.9034	−1.82743
$502$	0	0
$503$	32.6398	1.45534	0.727668	−	0.685929i	$-0.240604\pi$
$503$	32.6398	1.45534	0.727668	+	0.685929i	$0.240604\pi$
$504$	0	0
$505$	24.6618	1.09744
$506$	0	0
$507$	−70.2424	−3.11957
$508$	0	0
$509$	16.2323	0.719483	0.359742	−	0.933052i	$-0.382865\pi$
$509$	16.2323	0.719483	0.359742	+	0.933052i	$0.382865\pi$
$510$	0	0
$511$	2.00779	0.0888195
$512$	0	0
$513$	0.288270	0.0127274
$514$	0	0
$515$	−5.53362	−0.243840
$516$	0	0
$517$	−10.1154	−0.444876
$518$	0	0
$519$	−8.14156	−0.357375
$520$	0	0
$521$	−11.6261	−0.509351	−0.254675	−	0.967027i	$-0.581969\pi$
$521$	−11.6261	−0.509351	−0.254675	+	0.967027i	$0.581969\pi$
$522$	0	0
$523$	−30.8712	−1.34990	−0.674952	−	0.737862i	$-0.735836\pi$
$523$	−30.8712	−1.34990	−0.674952	+	0.737862i	$0.735836\pi$
$524$	0	0
$525$	16.1211	0.703581
$526$	0	0
$527$	6.52731	0.284334
$528$	0	0
$529$	59.1462	2.57158
$530$	0	0
$531$	42.6274	1.84987
$532$	0	0
$533$	−35.9245	−1.55606
$534$	0	0
$535$	−57.5349	−2.48745
$536$	0	0
$537$	41.0298	1.77057
$538$	0	0
$539$	14.6650	0.631664
$540$	0	0
$541$	33.0743	1.42197	0.710987	−	0.703205i	$-0.248248\pi$
$541$	33.0743	1.42197	0.710987	+	0.703205i	$0.248248\pi$
$542$	0	0
$543$	−18.9386	−0.812732
$544$	0	0
$545$	−6.45607	−0.276548
$546$	0	0
$547$	−6.42381	−0.274662	−0.137331	−	0.990525i	$-0.543852\pi$
$547$	−6.42381	−0.274662	−0.137331	+	0.990525i	$0.543852\pi$
$548$	0	0
$549$	−31.1155	−1.32798
$550$	0	0
$551$	8.05203	0.343028
$552$	0	0
$553$	−11.6509	−0.495446
$554$	0	0
$555$	−64.2691	−2.72807
$556$	0	0
$557$	−8.21465	−0.348066	−0.174033	−	0.984740i	$-0.555680\pi$
$557$	−8.21465	−0.348066	−0.174033	+	0.984740i	$0.555680\pi$
$558$	0	0
$559$	−45.8093	−1.93753
$560$	0	0
$561$	14.6516	0.618593
$562$	0	0
$563$	23.8941	1.00702	0.503508	−	0.863991i	$-0.332042\pi$
$563$	23.8941	1.00702	0.503508	+	0.863991i	$0.332042\pi$
$564$	0	0
$565$	2.74245	0.115376
$566$	0	0
$567$	7.23140	0.303690
$568$	0	0
$569$	−2.51286	−0.105345	−0.0526724	−	0.998612i	$-0.516774\pi$
$569$	−2.51286	−0.105345	−0.0526724	+	0.998612i	$0.516774\pi$
$570$	0	0
$571$	2.11324	0.0884363	0.0442182	−	0.999022i	$-0.485920\pi$
$571$	2.11324	0.0884363	0.0442182	+	0.999022i	$0.485920\pi$
$572$	0	0
$573$	−62.3815	−2.60603
$574$	0	0
$575$	77.5682	3.23482
$576$	0	0
$577$	0.131902	0.00549116	0.00274558	−	0.999996i	$-0.499126\pi$
$577$	0.131902	0.00549116	0.00274558	+	0.999996i	$0.499126\pi$
$578$	0	0
$579$	−10.4904	−0.435965
$580$	0	0
$581$	7.75360	0.321673
$582$	0	0
$583$	−14.5844	−0.604022
$584$	0	0
$585$	−68.9231	−2.84962
$586$	0	0
$587$	−15.6891	−0.647559	−0.323779	−	0.946133i	$-0.604954\pi$
$587$	−15.6891	−0.647559	−0.323779	+	0.946133i	$0.604954\pi$
$588$	0	0
$589$	2.68231	0.110523
$590$	0	0
$591$	−23.7382	−0.976460
$592$	0	0
$593$	−21.6442	−0.888822	−0.444411	−	0.895823i	$-0.646587\pi$
$593$	−21.6442	−0.888822	−0.444411	+	0.895823i	$0.646587\pi$
$594$	0	0
$595$	7.52657	0.308559
$596$	0	0
$597$	31.4934	1.28894
$598$	0	0
$599$	23.1435	0.945620	0.472810	−	0.881165i	$-0.343240\pi$
$599$	23.1435	0.945620	0.472810	+	0.881165i	$0.343240\pi$
$600$	0	0
$601$	−6.64157	−0.270915	−0.135458	−	0.990783i	$-0.543250\pi$
$601$	−6.64157	−0.270915	−0.135458	+	0.990783i	$0.543250\pi$
$602$	0	0
$603$	−16.0393	−0.653169
$604$	0	0
$605$	21.1562	0.860121
$606$	0	0
$607$	38.5947	1.56651	0.783255	−	0.621701i	$-0.213558\pi$
$607$	38.5947	1.56651	0.783255	+	0.621701i	$0.213558\pi$
$608$	0	0
$609$	14.0145	0.567895
$610$	0	0
$611$	28.5791	1.15619
$612$	0	0
$613$	−43.7390	−1.76660	−0.883301	−	0.468805i	$-0.844685\pi$
$613$	−43.7390	−1.76660	−0.883301	+	0.468805i	$0.844685\pi$
$614$	0	0
$615$	−49.5719	−1.99893
$616$	0	0
$617$	−6.17660	−0.248661	−0.124330	−	0.992241i	$-0.539678\pi$
$617$	−6.17660	−0.248661	−0.124330	+	0.992241i	$0.539678\pi$
$618$	0	0
$619$	−30.2469	−1.21572	−0.607862	−	0.794043i	$-0.707973\pi$
$619$	−30.2469	−1.21572	−0.607862	+	0.794043i	$0.707973\pi$
$620$	0	0
$621$	2.41414	0.0968759
$622$	0	0
$623$	1.71305	0.0686319
$624$	0	0
$625$	5.45360	0.218144
$626$	0	0
$627$	6.02090	0.240451
$628$	0	0
$629$	−18.9404	−0.755202
$630$	0	0
$631$	−24.6898	−0.982884	−0.491442	−	0.870910i	$-0.663530\pi$
$631$	−24.6898	−0.982884	−0.491442	+	0.870910i	$0.663530\pi$
$632$	0	0
$633$	64.5460	2.56547
$634$	0	0
$635$	−35.8312	−1.42192
$636$	0	0
$637$	−41.4328	−1.64163
$638$	0	0
$639$	−15.3382	−0.606771
$640$	0	0
$641$	−21.7601	−0.859472	−0.429736	−	0.902955i	$-0.641393\pi$
$641$	−21.7601	−0.859472	−0.429736	+	0.902955i	$0.641393\pi$
$642$	0	0
$643$	−9.20131	−0.362864	−0.181432	−	0.983403i	$-0.558073\pi$
$643$	−9.20131	−0.362864	−0.181432	+	0.983403i	$0.558073\pi$
$644$	0	0
$645$	−63.2117	−2.48896
$646$	0	0
$647$	22.9154	0.900898	0.450449	−	0.892802i	$-0.351264\pi$
$647$	22.9154	0.900898	0.450449	+	0.892802i	$0.351264\pi$
$648$	0	0
$649$	−33.8077	−1.32707
$650$	0	0
$651$	4.66854	0.182974
$652$	0	0
$653$	15.3994	0.602624	0.301312	−	0.953526i	$-0.402575\pi$
$653$	15.3994	0.602624	0.301312	+	0.953526i	$0.402575\pi$
$654$	0	0
$655$	2.38471	0.0931782
$656$	0	0
$657$	−7.47683	−0.291699
$658$	0	0
$659$	2.87231	0.111889	0.0559446	−	0.998434i	$-0.482183\pi$
$659$	2.87231	0.111889	0.0559446	+	0.998434i	$0.482183\pi$
$660$	0	0
$661$	−39.6824	−1.54347	−0.771733	−	0.635946i	$-0.780610\pi$
$661$	−39.6824	−1.54347	−0.771733	+	0.635946i	$0.780610\pi$
$662$	0	0
$663$	−41.3952	−1.60766
$664$	0	0
$665$	3.09294	0.119939
$666$	0	0
$667$	67.4322	2.61098
$668$	0	0
$669$	36.0915	1.39538
$670$	0	0
$671$	24.6777	0.952671
$672$	0	0
$673$	0.0123909	0.000477636 0	0.000238818	−	1.00000i	$-0.499924\pi$
$673$	0.0123909	0.000477636 0	0.000238818		1.00000i	$0.499924\pi$
$674$	0	0
$675$	2.27960	0.0877417
$676$	0	0
$677$	−32.2684	−1.24018	−0.620088	−	0.784532i	$-0.712903\pi$
$677$	−32.2684	−1.24018	−0.620088	+	0.784532i	$0.712903\pi$
$678$	0	0
$679$	−13.6689	−0.524564
$680$	0	0
$681$	57.2407	2.19347
$682$	0	0
$683$	45.2044	1.72970	0.864849	−	0.502033i	$-0.167414\pi$
$683$	45.2044	1.72970	0.864849	+	0.502033i	$0.167414\pi$
$684$	0	0
$685$	47.3687	1.80986
$686$	0	0
$687$	−4.25637	−0.162391
$688$	0	0
$689$	41.2051	1.56979
$690$	0	0
$691$	−23.1209	−0.879561	−0.439780	−	0.898105i	$-0.644944\pi$
$691$	−23.1209	−0.879561	−0.439780	+	0.898105i	$0.644944\pi$
$692$	0	0
$693$	5.14203	0.195330
$694$	0	0
$695$	−8.66862	−0.328820
$696$	0	0
$697$	−14.6091	−0.553357
$698$	0	0
$699$	−41.2732	−1.56110
$700$	0	0
$701$	37.3914	1.41225	0.706127	−	0.708085i	$-0.250441\pi$
$701$	37.3914	1.41225	0.706127	+	0.708085i	$0.250441\pi$
$702$	0	0
$703$	−7.78330	−0.293553
$704$	0	0
$705$	39.4360	1.48524
$706$	0	0
$707$	5.19826	0.195501
$708$	0	0
$709$	36.1971	1.35941	0.679705	−	0.733486i	$-0.262108\pi$
$709$	36.1971	1.35941	0.679705	+	0.733486i	$0.262108\pi$
$710$	0	0
$711$	43.3868	1.62713
$712$	0	0
$713$	22.4632	0.841253
$714$	0	0
$715$	54.6628	2.04427
$716$	0	0
$717$	−34.0271	−1.27077
$718$	0	0
$719$	27.0021	1.00701	0.503504	−	0.863993i	$-0.332044\pi$
$719$	27.0021	1.00701	0.503504	+	0.863993i	$0.332044\pi$
$720$	0	0
$721$	−1.16639	−0.0434385
$722$	0	0
$723$	9.28921	0.345469
$724$	0	0
$725$	63.6742	2.36480
$726$	0	0
$727$	22.8832	0.848692	0.424346	−	0.905500i	$-0.360504\pi$
$727$	22.8832	0.848692	0.424346	+	0.905500i	$0.360504\pi$
$728$	0	0
$729$	−24.9897	−0.925545
$730$	0	0
$731$	−18.6288	−0.689010
$732$	0	0
$733$	48.9960	1.80971	0.904854	−	0.425722i	$-0.139980\pi$
$733$	48.9960	1.80971	0.904854	+	0.425722i	$0.139980\pi$
$734$	0	0
$735$	−57.1727	−2.10885
$736$	0	0
$737$	12.7207	0.468573
$738$	0	0
$739$	−35.0032	−1.28761	−0.643807	−	0.765188i	$-0.722646\pi$
$739$	−35.0032	−1.28761	−0.643807	+	0.765188i	$0.722646\pi$
$740$	0	0
$741$	−17.0108	−0.624907
$742$	0	0
$743$	28.5897	1.04886	0.524428	−	0.851455i	$-0.324279\pi$
$743$	28.5897	1.04886	0.524428	+	0.851455i	$0.324279\pi$
$744$	0	0
$745$	64.0078	2.34506
$746$	0	0
$747$	−28.8737	−1.05643
$748$	0	0
$749$	−12.1273	−0.443123
$750$	0	0
$751$	−17.2903	−0.630934	−0.315467	−	0.948937i	$-0.602161\pi$
$751$	−17.2903	−0.630934	−0.315467	+	0.948937i	$0.602161\pi$
$752$	0	0
$753$	−11.7055	−0.426570
$754$	0	0
$755$	−52.4134	−1.90752
$756$	0	0
$757$	30.1311	1.09513	0.547567	−	0.836762i	$-0.315554\pi$
$757$	30.1311	1.09513	0.547567	+	0.836762i	$0.315554\pi$
$758$	0	0
$759$	50.4224	1.83022
$760$	0	0
$761$	37.4077	1.35603	0.678015	−	0.735048i	$-0.262840\pi$
$761$	37.4077	1.35603	0.678015	+	0.735048i	$0.262840\pi$
$762$	0	0
$763$	−1.36082	−0.0492651
$764$	0	0
$765$	−28.0282	−1.01336
$766$	0	0
$767$	95.5168	3.44891
$768$	0	0
$769$	8.98913	0.324156	0.162078	−	0.986778i	$-0.448180\pi$
$769$	8.98913	0.324156	0.162078	+	0.986778i	$0.448180\pi$
$770$	0	0
$771$	−44.2205	−1.59256
$772$	0	0
$773$	−25.5723	−0.919772	−0.459886	−	0.887978i	$-0.652110\pi$
$773$	−25.5723	−0.919772	−0.459886	+	0.887978i	$0.652110\pi$
$774$	0	0
$775$	21.2113	0.761932
$776$	0	0
$777$	−13.5468	−0.485987
$778$	0	0
$779$	−6.00340	−0.215094
$780$	0	0
$781$	12.1647	0.435288
$782$	0	0
$783$	1.98172	0.0708207
$784$	0	0
$785$	−29.7038	−1.06017
$786$	0	0
$787$	−40.5153	−1.44421	−0.722107	−	0.691782i	$-0.756826\pi$
$787$	−40.5153	−1.44421	−0.722107	+	0.691782i	$0.756826\pi$
$788$	0	0
$789$	18.5537	0.660531
$790$	0	0
$791$	0.578059	0.0205534
$792$	0	0
$793$	−69.7217	−2.47589
$794$	0	0
$795$	56.8585	2.01656
$796$	0	0
$797$	−33.2895	−1.17918	−0.589588	−	0.807704i	$-0.700710\pi$
$797$	−33.2895	−1.17918	−0.589588	+	0.807704i	$0.700710\pi$
$798$	0	0
$799$	11.6220	0.411155
$800$	0	0
$801$	−6.37924	−0.225399
$802$	0	0
$803$	5.92986	0.209260
$804$	0	0
$805$	25.9020	0.912927
$806$	0	0
$807$	−8.47883	−0.298469
$808$	0	0
$809$	−36.9713	−1.29984	−0.649921	−	0.760002i	$-0.725198\pi$
$809$	−36.9713	−1.29984	−0.649921	+	0.760002i	$0.725198\pi$
$810$	0	0
$811$	51.3187	1.80204	0.901022	−	0.433774i	$-0.142818\pi$
$811$	51.3187	1.80204	0.901022	+	0.433774i	$0.142818\pi$
$812$	0	0
$813$	66.0021	2.31479
$814$	0	0
$815$	−77.5674	−2.71707
$816$	0	0
$817$	−7.65525	−0.267823
$818$	0	0
$819$	−14.5277	−0.507640
$820$	0	0
$821$	−46.7017	−1.62990	−0.814950	−	0.579532i	$-0.803235\pi$
$821$	−46.7017	−1.62990	−0.814950	+	0.579532i	$0.803235\pi$
$822$	0	0
$823$	8.00778	0.279134	0.139567	−	0.990213i	$-0.455429\pi$
$823$	8.00778	0.279134	0.139567	+	0.990213i	$0.455429\pi$
$824$	0	0
$825$	47.6123	1.65765
$826$	0	0
$827$	−34.8800	−1.21290	−0.606449	−	0.795123i	$-0.707406\pi$
$827$	−34.8800	−1.21290	−0.606449	+	0.795123i	$0.707406\pi$
$828$	0	0
$829$	−28.7580	−0.998807	−0.499403	−	0.866370i	$-0.666447\pi$
$829$	−28.7580	−0.998807	−0.499403	+	0.866370i	$0.666447\pi$
$830$	0	0
$831$	68.7523	2.38499
$832$	0	0
$833$	−16.8490	−0.583785
$834$	0	0
$835$	−62.0578	−2.14760
$836$	0	0
$837$	0.660154	0.0228183
$838$	0	0
$839$	19.2941	0.666105	0.333053	−	0.942908i	$-0.391921\pi$
$839$	19.2941	0.666105	0.333053	+	0.942908i	$0.391921\pi$
$840$	0	0
$841$	26.3537	0.908749
$842$	0	0
$843$	51.3056	1.76706
$844$	0	0
$845$	−106.570	−3.66613
$846$	0	0
$847$	4.45934	0.153225
$848$	0	0
$849$	35.2550	1.20995
$850$	0	0
$851$	−65.1817	−2.23440
$852$	0	0
$853$	39.1952	1.34202	0.671009	−	0.741449i	$-0.265861\pi$
$853$	39.1952	1.34202	0.671009	+	0.741449i	$0.265861\pi$
$854$	0	0
$855$	−11.5178	−0.393902
$856$	0	0
$857$	44.5202	1.52078	0.760391	−	0.649466i	$-0.225007\pi$
$857$	44.5202	1.52078	0.760391	+	0.649466i	$0.225007\pi$
$858$	0	0
$859$	13.2603	0.452435	0.226217	−	0.974077i	$-0.427364\pi$
$859$	13.2603	0.452435	0.226217	+	0.974077i	$0.427364\pi$
$860$	0	0
$861$	−10.4489	−0.356096
$862$	0	0
$863$	−33.9850	−1.15686	−0.578432	−	0.815731i	$-0.696335\pi$
$863$	−33.9850	−1.15686	−0.578432	+	0.815731i	$0.696335\pi$
$864$	0	0
$865$	−12.3522	−0.419987
$866$	0	0
$867$	24.4250	0.829516
$868$	0	0
$869$	−34.4100	−1.16728
$870$	0	0
$871$	−35.9397	−1.21777
$872$	0	0
$873$	50.9017	1.72276
$874$	0	0
$875$	10.1692	0.343783
$876$	0	0
$877$	−3.52762	−0.119119	−0.0595597	−	0.998225i	$-0.518970\pi$
$877$	−3.52762	−0.119119	−0.0595597	+	0.998225i	$0.518970\pi$
$878$	0	0
$879$	14.6400	0.493794
$880$	0	0
$881$	46.8725	1.57917	0.789587	−	0.613638i	$-0.210295\pi$
$881$	46.8725	1.57917	0.789587	+	0.613638i	$0.210295\pi$
$882$	0	0
$883$	2.68994	0.0905237	0.0452618	−	0.998975i	$-0.485588\pi$
$883$	2.68994	0.0905237	0.0452618	+	0.998975i	$0.485588\pi$
$884$	0	0
$885$	131.803	4.43050
$886$	0	0
$887$	12.2955	0.412841	0.206421	−	0.978463i	$-0.433818\pi$
$887$	12.2955	0.412841	0.206421	+	0.978463i	$0.433818\pi$
$888$	0	0
$889$	−7.55257	−0.253305
$890$	0	0
$891$	21.3574	0.715499
$892$	0	0
$893$	4.77589	0.159819
$894$	0	0
$895$	62.2495	2.08077
$896$	0	0
$897$	−142.458	−4.75653
$898$	0	0
$899$	18.4396	0.614994
$900$	0	0
$901$	16.7564	0.558238
$902$	0	0
$903$	−13.3239	−0.443391
$904$	0	0
$905$	−28.7332	−0.955124
$906$	0	0
$907$	−6.81934	−0.226432	−0.113216	−	0.993570i	$-0.536115\pi$
$907$	−6.81934	−0.226432	−0.113216	+	0.993570i	$0.536115\pi$
$908$	0	0
$909$	−19.3578	−0.642059
$910$	0	0
$911$	15.0769	0.499520	0.249760	−	0.968308i	$-0.419648\pi$
$911$	15.0769	0.499520	0.249760	+	0.968308i	$0.419648\pi$
$912$	0	0
$913$	22.8997	0.757868
$914$	0	0
$915$	−96.2082	−3.18055
$916$	0	0
$917$	0.502653	0.0165991
$918$	0	0
$919$	−33.7058	−1.11185	−0.555925	−	0.831232i	$-0.687636\pi$
$919$	−33.7058	−1.11185	−0.555925	+	0.831232i	$0.687636\pi$
$920$	0	0
$921$	10.3846	0.342183
$922$	0	0
$923$	−34.3689	−1.13127
$924$	0	0
$925$	−61.5491	−2.02372
$926$	0	0
$927$	4.34351	0.142660
$928$	0	0
$929$	−2.06716	−0.0678215	−0.0339107	−	0.999425i	$-0.510796\pi$
$929$	−2.06716	−0.0678215	−0.0339107	+	0.999425i	$0.510796\pi$
$930$	0	0
$931$	−6.92389	−0.226921
$932$	0	0
$933$	−46.2578	−1.51441
$934$	0	0
$935$	22.2291	0.726971
$936$	0	0
$937$	−26.1520	−0.854347	−0.427174	−	0.904170i	$-0.640491\pi$
$937$	−26.1520	−0.854347	−0.427174	+	0.904170i	$0.640491\pi$
$938$	0	0
$939$	−35.7686	−1.16726
$940$	0	0
$941$	2.23332	0.0728041	0.0364021	−	0.999337i	$-0.488410\pi$
$941$	2.23332	0.0728041	0.0364021	+	0.999337i	$0.488410\pi$
$942$	0	0
$943$	−50.2758	−1.63721
$944$	0	0
$945$	0.761216	0.0247624
$946$	0	0
$947$	−31.7361	−1.03129	−0.515643	−	0.856804i	$-0.672447\pi$
$947$	−31.7361	−1.03129	−0.515643	+	0.856804i	$0.672447\pi$
$948$	0	0
$949$	−16.7536	−0.543845
$950$	0	0
$951$	24.3870	0.790803
$952$	0	0
$953$	36.6839	1.18831	0.594154	−	0.804351i	$-0.297487\pi$
$953$	36.6839	1.18831	0.594154	+	0.804351i	$0.297487\pi$
$954$	0	0
$955$	−94.6439	−3.06260
$956$	0	0
$957$	41.3907	1.33797
$958$	0	0
$959$	9.98445	0.322415
$960$	0	0
$961$	−24.8574	−0.801851
$962$	0	0
$963$	45.1610	1.45529
$964$	0	0
$965$	−15.9157	−0.512346
$966$	0	0
$967$	32.3030	1.03880	0.519398	−	0.854533i	$-0.326156\pi$
$967$	32.3030	1.03880	0.519398	+	0.854533i	$0.326156\pi$
$968$	0	0
$969$	−6.91760	−0.222225
$970$	0	0
$971$	6.19924	0.198943	0.0994715	−	0.995040i	$-0.468285\pi$
$971$	6.19924	0.198943	0.0994715	+	0.995040i	$0.468285\pi$
$972$	0	0
$973$	−1.82719	−0.0585770
$974$	0	0
$975$	−134.519	−4.30805
$976$	0	0
$977$	−42.3272	−1.35417	−0.677084	−	0.735906i	$-0.736757\pi$
$977$	−42.3272	−1.35417	−0.677084	+	0.735906i	$0.736757\pi$
$978$	0	0
$979$	5.05936	0.161698
$980$	0	0
$981$	5.06757	0.161795
$982$	0	0
$983$	−17.5267	−0.559016	−0.279508	−	0.960143i	$-0.590171\pi$
$983$	−17.5267	−0.559016	−0.279508	+	0.960143i	$0.590171\pi$
$984$	0	0
$985$	−36.0151	−1.14754
$986$	0	0
$987$	8.31239	0.264586
$988$	0	0
$989$	−64.1093	−2.03856
$990$	0	0
$991$	−23.8164	−0.756551	−0.378276	−	0.925693i	$-0.623483\pi$
$991$	−23.8164	−0.756551	−0.378276	+	0.925693i	$0.623483\pi$
$992$	0	0
$993$	52.6606	1.67113
$994$	0	0
$995$	47.7812	1.51476
$996$	0	0
$997$	−40.7744	−1.29134	−0.645669	−	0.763618i	$-0.723421\pi$
$997$	−40.7744	−1.29134	−0.645669	+	0.763618i	$0.723421\pi$
$998$	0	0
$999$	−1.91558	−0.0606062

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.b.1.12	✓	87

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.b.1.12	✓	87	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)

\(f(q)\)	\(=\)	\(q-2.42698 q^{3} -3.68217 q^{5} -0.776134 q^{7} +2.89025 q^{9} +O(q^{10})\)	\(q-2.42698 q^{3} -3.68217 q^{5} -0.776134 q^{7} +2.89025 q^{9} -2.29225 q^{11} +6.47628 q^{13} +8.93656 q^{15} +2.63364 q^{17} +1.08226 q^{19} +1.88366 q^{21} +9.06346 q^{23} +8.55835 q^{25} +0.266359 q^{27} +7.44001 q^{29} +2.47843 q^{31} +5.56326 q^{33} +2.85785 q^{35} -7.19170 q^{37} -15.7178 q^{39} -5.54709 q^{41} -7.07339 q^{43} -10.6424 q^{45} +4.41288 q^{47} -6.39762 q^{49} -6.39181 q^{51} +6.36246 q^{53} +8.44045 q^{55} -2.62663 q^{57} +14.7487 q^{59} -10.7657 q^{61} -2.24322 q^{63} -23.8468 q^{65} -5.54943 q^{67} -21.9969 q^{69} -5.30688 q^{71} -2.58691 q^{73} -20.7710 q^{75} +1.77909 q^{77} +15.0114 q^{79} -9.31720 q^{81} -9.99003 q^{83} -9.69751 q^{85} -18.0568 q^{87} -2.20716 q^{89} -5.02646 q^{91} -6.01512 q^{93} -3.98506 q^{95} +17.6115 q^{97} -6.62518 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) \) 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9} + 36 q^{11} - q^{13} + 16 q^{15} + 31 q^{17} + 35 q^{19} - 3 q^{21} + 39 q^{23} + 93 q^{25} + 55 q^{27} - 5 q^{29} + 46 q^{31} + 25 q^{33} + 68 q^{35} - 11 q^{37} + 54 q^{39} + 83 q^{41} + 28 q^{43} - 14 q^{45} + 48 q^{47} + 103 q^{49} + 77 q^{51} + 3 q^{53} + 35 q^{55} + 14 q^{57} + 122 q^{59} - 13 q^{61} + 39 q^{63} + 41 q^{65} + 32 q^{67} - 10 q^{69} + 100 q^{71} + 34 q^{73} + 97 q^{75} + 4 q^{77} + 52 q^{79} + 131 q^{81} + 67 q^{83} - 2 q^{85} + 89 q^{87} + 68 q^{89} + 75 q^{91} + 138 q^{95} + 36 q^{97} + 107 q^{99}+O(q^{100}) \) 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9 + 36 * q^11 - q^13 + 16 * q^15 + 31 * q^17 + 35 * q^19 - 3 * q^21 + 39 * q^23 + 93 * q^25 + 55 * q^27 - 5 * q^29 + 46 * q^31 + 25 * q^33 + 68 * q^35 - 11 * q^37 + 54 * q^39 + 83 * q^41 + 28 * q^43 - 14 * q^45 + 48 * q^47 + 103 * q^49 + 77 * q^51 + 3 * q^53 + 35 * q^55 + 14 * q^57 + 122 * q^59 - 13 * q^61 + 39 * q^63 + 41 * q^65 + 32 * q^67 - 10 * q^69 + 100 * q^71 + 34 * q^73 + 97 * q^75 + 4 * q^77 + 52 * q^79 + 131 * q^81 + 67 * q^83 - 2 * q^85 + 89 * q^87 + 68 * q^89 + 75 * q^91 + 138 * q^95 + 36 * q^97 + 107 * q^99

Introduction

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