LMFDB - Embedded newform 8044.2.a.b.1.17

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			1.17
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)

$f(q)$	$=$	$q-2.04177 q^{3} -2.46672 q^{5} +0.440869 q^{7} +1.16884 q^{9} +O(q^{10})$	$q-2.04177 q^{3} -2.46672 q^{5} +0.440869 q^{7} +1.16884 q^{9} -2.38981 q^{11} +2.82754 q^{13} +5.03648 q^{15} -6.71766 q^{17} -6.76409 q^{19} -0.900155 q^{21} -2.02110 q^{23} +1.08469 q^{25} +3.73881 q^{27} -3.01178 q^{29} -6.41052 q^{31} +4.87945 q^{33} -1.08750 q^{35} -2.22423 q^{37} -5.77320 q^{39} +2.81107 q^{41} +3.17964 q^{43} -2.88320 q^{45} +0.353706 q^{47} -6.80563 q^{49} +13.7159 q^{51} -6.86586 q^{53} +5.89499 q^{55} +13.8107 q^{57} +5.46287 q^{59} -2.95810 q^{61} +0.515306 q^{63} -6.97475 q^{65} -11.2730 q^{67} +4.12663 q^{69} +10.5875 q^{71} -10.9415 q^{73} -2.21470 q^{75} -1.05359 q^{77} -17.3841 q^{79} -11.1403 q^{81} -15.4637 q^{83} +16.5706 q^{85} +6.14937 q^{87} -0.524282 q^{89} +1.24658 q^{91} +13.0888 q^{93} +16.6851 q^{95} +3.97230 q^{97} -2.79331 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.04177	−1.17882	−0.589409	−	0.807835i	$-0.700639\pi$
$3$	−2.04177	−1.17882	−0.589409	+	0.807835i	$0.700639\pi$
$4$	0	0
$5$	−2.46672	−1.10315	−0.551575	−	0.834126i	$-0.685973\pi$
$5$	−2.46672	−1.10315	−0.551575	+	0.834126i	$0.685973\pi$
$6$	0	0
$7$	0.440869	0.166633	0.0833164	−	0.996523i	$-0.473449\pi$
$7$	0.440869	0.166633	0.0833164	+	0.996523i	$0.473449\pi$
$8$	0	0
$9$	1.16884	0.389613
$10$	0	0
$11$	−2.38981	−0.720555	−0.360278	−	0.932845i	$-0.617318\pi$
$11$	−2.38981	−0.720555	−0.360278	+	0.932845i	$0.617318\pi$
$12$	0	0
$13$	2.82754	0.784219	0.392110	−	0.919919i	$-0.371745\pi$
$13$	2.82754	0.784219	0.392110	+	0.919919i	$0.371745\pi$
$14$	0	0
$15$	5.03648	1.30041
$16$	0	0
$17$	−6.71766	−1.62927	−0.814635	−	0.579973i	$-0.803063\pi$
$17$	−6.71766	−1.62927	−0.814635	+	0.579973i	$0.803063\pi$
$18$	0	0
$19$	−6.76409	−1.55179	−0.775894	−	0.630863i	$-0.782701\pi$
$19$	−6.76409	−1.55179	−0.775894	+	0.630863i	$0.782701\pi$
$20$	0	0
$21$	−0.900155	−0.196430
$22$	0	0
$23$	−2.02110	−0.421429	−0.210714	−	0.977548i	$-0.567579\pi$
$23$	−2.02110	−0.421429	−0.210714	+	0.977548i	$0.567579\pi$
$24$	0	0
$25$	1.08469	0.216938
$26$	0	0
$27$	3.73881	0.719535
$28$	0	0
$29$	−3.01178	−0.559273	−0.279637	−	0.960106i	$-0.590214\pi$
$29$	−3.01178	−0.559273	−0.279637	+	0.960106i	$0.590214\pi$
$30$	0	0
$31$	−6.41052	−1.15136	−0.575682	−	0.817674i	$-0.695263\pi$
$31$	−6.41052	−1.15136	−0.575682	+	0.817674i	$0.695263\pi$
$32$	0	0
$33$	4.87945	0.849404
$34$	0	0
$35$	−1.08750	−0.183821
$36$	0	0
$37$	−2.22423	−0.365661	−0.182831	−	0.983144i	$-0.558526\pi$
$37$	−2.22423	−0.365661	−0.182831	+	0.983144i	$0.558526\pi$
$38$	0	0
$39$	−5.77320	−0.924452
$40$	0	0
$41$	2.81107	0.439016	0.219508	−	0.975611i	$-0.429555\pi$
$41$	2.81107	0.439016	0.219508	+	0.975611i	$0.429555\pi$
$42$	0	0
$43$	3.17964	0.484890	0.242445	−	0.970165i	$-0.422051\pi$
$43$	3.17964	0.484890	0.242445	+	0.970165i	$0.422051\pi$
$44$	0	0
$45$	−2.88320	−0.429802
$46$	0	0
$47$	0.353706	0.0515934	0.0257967	−	0.999667i	$-0.491788\pi$
$47$	0.353706	0.0515934	0.0257967	+	0.999667i	$0.491788\pi$
$48$	0	0
$49$	−6.80563	−0.972233
$50$	0	0
$51$	13.7159	1.92061
$52$	0	0
$53$	−6.86586	−0.943099	−0.471549	−	0.881840i	$-0.656305\pi$
$53$	−6.86586	−0.943099	−0.471549	+	0.881840i	$0.656305\pi$
$54$	0	0
$55$	5.89499	0.794880
$56$	0	0
$57$	13.8107	1.82928
$58$	0	0
$59$	5.46287	0.711205	0.355602	−	0.934637i	$-0.384276\pi$
$59$	5.46287	0.711205	0.355602	+	0.934637i	$0.384276\pi$
$60$	0	0
$61$	−2.95810	−0.378746	−0.189373	−	0.981905i	$-0.560646\pi$
$61$	−2.95810	−0.378746	−0.189373	+	0.981905i	$0.560646\pi$
$62$	0	0
$63$	0.515306	0.0649224
$64$	0	0
$65$	−6.97475	−0.865111
$66$	0	0
$67$	−11.2730	−1.37722	−0.688608	−	0.725134i	$-0.741778\pi$
$67$	−11.2730	−1.37722	−0.688608	+	0.725134i	$0.741778\pi$
$68$	0	0
$69$	4.12663	0.496788
$70$	0	0
$71$	10.5875	1.25651	0.628254	−	0.778008i	$-0.283770\pi$
$71$	10.5875	1.25651	0.628254	+	0.778008i	$0.283770\pi$
$72$	0	0
$73$	−10.9415	−1.28061	−0.640304	−	0.768122i	$-0.721192\pi$
$73$	−10.9415	−1.28061	−0.640304	+	0.768122i	$0.721192\pi$
$74$	0	0
$75$	−2.21470	−0.255731
$76$	0	0
$77$	−1.05359	−0.120068
$78$	0	0
$79$	−17.3841	−1.95587	−0.977935	−	0.208910i	$-0.933008\pi$
$79$	−17.3841	−1.95587	−0.977935	+	0.208910i	$0.933008\pi$
$80$	0	0
$81$	−11.1403	−1.23781
$82$	0	0
$83$	−15.4637	−1.69736	−0.848679	−	0.528908i	$-0.822602\pi$
$83$	−15.4637	−1.69736	−0.848679	+	0.528908i	$0.822602\pi$
$84$	0	0
$85$	16.5706	1.79733
$86$	0	0
$87$	6.14937	0.659281
$88$	0	0
$89$	−0.524282	−0.0555737	−0.0277869	−	0.999614i	$-0.508846\pi$
$89$	−0.524282	−0.0555737	−0.0277869	+	0.999614i	$0.508846\pi$
$90$	0	0
$91$	1.24658	0.130677
$92$	0	0
$93$	13.0888	1.35725
$94$	0	0
$95$	16.6851	1.71185
$96$	0	0
$97$	3.97230	0.403326	0.201663	−	0.979455i	$-0.435365\pi$
$97$	3.97230	0.403326	0.201663	+	0.979455i	$0.435365\pi$
$98$	0	0
$99$	−2.79331	−0.280738
$100$	0	0
$101$	8.36420	0.832269	0.416134	−	0.909303i	$-0.363385\pi$
$101$	8.36420	0.832269	0.416134	+	0.909303i	$0.363385\pi$
$102$	0	0
$103$	1.83990	0.181291	0.0906453	−	0.995883i	$-0.471107\pi$
$103$	1.83990	0.181291	0.0906453	+	0.995883i	$0.471107\pi$
$104$	0	0
$105$	2.22043	0.216692
$106$	0	0
$107$	−3.10619	−0.300287	−0.150143	−	0.988664i	$-0.547974\pi$
$107$	−3.10619	−0.300287	−0.150143	+	0.988664i	$0.547974\pi$
$108$	0	0
$109$	−12.8398	−1.22983	−0.614915	−	0.788593i	$-0.710810\pi$
$109$	−12.8398	−1.22983	−0.614915	+	0.788593i	$0.710810\pi$
$110$	0	0
$111$	4.54138	0.431048
$112$	0	0
$113$	1.30360	0.122633	0.0613164	−	0.998118i	$-0.480470\pi$
$113$	1.30360	0.122633	0.0613164	+	0.998118i	$0.480470\pi$
$114$	0	0
$115$	4.98549	0.464899
$116$	0	0
$117$	3.30494	0.305542
$118$	0	0
$119$	−2.96161	−0.271490
$120$	0	0
$121$	−5.28880	−0.480800
$122$	0	0
$123$	−5.73957	−0.517520
$124$	0	0
$125$	9.65796	0.863834
$126$	0	0
$127$	17.6278	1.56422	0.782109	−	0.623142i	$-0.214144\pi$
$127$	17.6278	1.56422	0.782109	+	0.623142i	$0.214144\pi$
$128$	0	0
$129$	−6.49210	−0.571598
$130$	0	0
$131$	−7.88150	−0.688610	−0.344305	−	0.938858i	$-0.611885\pi$
$131$	−7.88150	−0.688610	−0.344305	+	0.938858i	$0.611885\pi$
$132$	0	0
$133$	−2.98208	−0.258579
$134$	0	0
$135$	−9.22260	−0.793755
$136$	0	0
$137$	−10.9914	−0.939057	−0.469528	−	0.882917i	$-0.655576\pi$
$137$	−10.9914	−0.939057	−0.469528	+	0.882917i	$0.655576\pi$
$138$	0	0
$139$	−13.4938	−1.14453	−0.572264	−	0.820069i	$-0.693935\pi$
$139$	−13.4938	−1.14453	−0.572264	+	0.820069i	$0.693935\pi$
$140$	0	0
$141$	−0.722188	−0.0608192
$142$	0	0
$143$	−6.75729	−0.565073
$144$	0	0
$145$	7.42920	0.616962
$146$	0	0
$147$	13.8956	1.14609
$148$	0	0
$149$	−10.6702	−0.874133	−0.437067	−	0.899429i	$-0.643983\pi$
$149$	−10.6702	−0.874133	−0.437067	+	0.899429i	$0.643983\pi$
$150$	0	0
$151$	−14.7557	−1.20081	−0.600403	−	0.799698i	$-0.704993\pi$
$151$	−14.7557	−1.20081	−0.600403	+	0.799698i	$0.704993\pi$
$152$	0	0
$153$	−7.85186	−0.634786
$154$	0	0
$155$	15.8129	1.27013
$156$	0	0
$157$	−16.2474	−1.29668	−0.648342	−	0.761349i	$-0.724537\pi$
$157$	−16.2474	−1.29668	−0.648342	+	0.761349i	$0.724537\pi$
$158$	0	0
$159$	14.0185	1.11174
$160$	0	0
$161$	−0.891042	−0.0702239
$162$	0	0
$163$	−24.8728	−1.94819	−0.974096	−	0.226136i	$-0.927391\pi$
$163$	−24.8728	−1.94819	−0.974096	+	0.226136i	$0.927391\pi$
$164$	0	0
$165$	−12.0362	−0.937019
$166$	0	0
$167$	12.1555	0.940624	0.470312	−	0.882500i	$-0.344141\pi$
$167$	12.1555	0.940624	0.470312	+	0.882500i	$0.344141\pi$
$168$	0	0
$169$	−5.00501	−0.385000
$170$	0	0
$171$	−7.90614	−0.604597
$172$	0	0
$173$	3.22518	0.245206	0.122603	−	0.992456i	$-0.460876\pi$
$173$	3.22518	0.245206	0.122603	+	0.992456i	$0.460876\pi$
$174$	0	0
$175$	0.478207	0.0361491
$176$	0	0
$177$	−11.1539	−0.838381
$178$	0	0
$179$	5.40100	0.403689	0.201845	−	0.979418i	$-0.435306\pi$
$179$	5.40100	0.403689	0.201845	+	0.979418i	$0.435306\pi$
$180$	0	0
$181$	−14.0822	−1.04672	−0.523359	−	0.852112i	$-0.675321\pi$
$181$	−14.0822	−1.04672	−0.523359	+	0.852112i	$0.675321\pi$
$182$	0	0
$183$	6.03978	0.446473
$184$	0	0
$185$	5.48655	0.403379
$186$	0	0
$187$	16.0539	1.17398
$188$	0	0
$189$	1.64833	0.119898
$190$	0	0
$191$	17.5615	1.27070	0.635351	−	0.772223i	$-0.280855\pi$
$191$	17.5615	1.27070	0.635351	+	0.772223i	$0.280855\pi$
$192$	0	0
$193$	−18.6108	−1.33963	−0.669817	−	0.742527i	$-0.733627\pi$
$193$	−18.6108	−1.33963	−0.669817	+	0.742527i	$0.733627\pi$
$194$	0	0
$195$	14.2409	1.01981
$196$	0	0
$197$	−6.06718	−0.432269	−0.216134	−	0.976364i	$-0.569345\pi$
$197$	−6.06718	−0.432269	−0.216134	+	0.976364i	$0.569345\pi$
$198$	0	0
$199$	−14.9216	−1.05777	−0.528883	−	0.848695i	$-0.677389\pi$
$199$	−14.9216	−1.05777	−0.528883	+	0.848695i	$0.677389\pi$
$200$	0	0
$201$	23.0169	1.62349
$202$	0	0
$203$	−1.32780	−0.0931933
$204$	0	0
$205$	−6.93412	−0.484300
$206$	0	0
$207$	−2.36234	−0.164194
$208$	0	0
$209$	16.1649	1.11815
$210$	0	0
$211$	23.5787	1.62323	0.811613	−	0.584196i	$-0.198590\pi$
$211$	23.5787	1.62323	0.811613	+	0.584196i	$0.198590\pi$
$212$	0	0
$213$	−21.6173	−1.48120
$214$	0	0
$215$	−7.84327	−0.534906
$216$	0	0
$217$	−2.82620	−0.191855
$218$	0	0
$219$	22.3401	1.50960
$220$	0	0
$221$	−18.9945	−1.27771
$222$	0	0
$223$	17.6346	1.18090	0.590450	−	0.807074i	$-0.298950\pi$
$223$	17.6346	1.18090	0.590450	+	0.807074i	$0.298950\pi$
$224$	0	0
$225$	1.26783	0.0845221
$226$	0	0
$227$	−5.23754	−0.347628	−0.173814	−	0.984779i	$-0.555609\pi$
$227$	−5.23754	−0.347628	−0.173814	+	0.984779i	$0.555609\pi$
$228$	0	0
$229$	−2.92504	−0.193292	−0.0966460	−	0.995319i	$-0.530811\pi$
$229$	−2.92504	−0.193292	−0.0966460	+	0.995319i	$0.530811\pi$
$230$	0	0
$231$	2.15120	0.141539
$232$	0	0
$233$	26.8022	1.75587	0.877935	−	0.478779i	$-0.158921\pi$
$233$	26.8022	1.75587	0.877935	+	0.478779i	$0.158921\pi$
$234$	0	0
$235$	−0.872493	−0.0569152
$236$	0	0
$237$	35.4945	2.30562
$238$	0	0
$239$	−12.0646	−0.780393	−0.390197	−	0.920732i	$-0.627593\pi$
$239$	−12.0646	−0.780393	−0.390197	+	0.920732i	$0.627593\pi$
$240$	0	0
$241$	5.04653	0.325076	0.162538	−	0.986702i	$-0.448032\pi$
$241$	5.04653	0.325076	0.162538	+	0.986702i	$0.448032\pi$
$242$	0	0
$243$	11.5296	0.739624
$244$	0	0
$245$	16.7876	1.07252
$246$	0	0
$247$	−19.1257	−1.21694
$248$	0	0
$249$	31.5733	2.00088
$250$	0	0
$251$	−6.73640	−0.425198	−0.212599	−	0.977140i	$-0.568193\pi$
$251$	−6.73640	−0.425198	−0.212599	+	0.977140i	$0.568193\pi$
$252$	0	0
$253$	4.83005	0.303663
$254$	0	0
$255$	−33.8333	−2.11872
$256$	0	0
$257$	18.8098	1.17333	0.586663	−	0.809831i	$-0.300441\pi$
$257$	18.8098	1.17333	0.586663	+	0.809831i	$0.300441\pi$
$258$	0	0
$259$	−0.980595	−0.0609312
$260$	0	0
$261$	−3.52029	−0.217900
$262$	0	0
$263$	16.4762	1.01596	0.507982	−	0.861368i	$-0.330392\pi$
$263$	16.4762	1.01596	0.507982	+	0.861368i	$0.330392\pi$
$264$	0	0
$265$	16.9361	1.04038
$266$	0	0
$267$	1.07046	0.0655114
$268$	0	0
$269$	0.336644	0.0205256	0.0102628	−	0.999947i	$-0.496733\pi$
$269$	0.336644	0.0205256	0.0102628	+	0.999947i	$0.496733\pi$
$270$	0	0
$271$	10.2961	0.625445	0.312722	−	0.949845i	$-0.398759\pi$
$271$	10.2961	0.625445	0.312722	+	0.949845i	$0.398759\pi$
$272$	0	0
$273$	−2.54523	−0.154044
$274$	0	0
$275$	−2.59221	−0.156316
$276$	0	0
$277$	28.4357	1.70854	0.854269	−	0.519832i	$-0.174005\pi$
$277$	28.4357	1.70854	0.854269	+	0.519832i	$0.174005\pi$
$278$	0	0
$279$	−7.49287	−0.448586
$280$	0	0
$281$	−26.9922	−1.61022	−0.805110	−	0.593125i	$-0.797894\pi$
$281$	−26.9922	−1.61022	−0.805110	+	0.593125i	$0.797894\pi$
$282$	0	0
$283$	−8.69075	−0.516612	−0.258306	−	0.966063i	$-0.583164\pi$
$283$	−8.69075	−0.516612	−0.258306	+	0.966063i	$0.583164\pi$
$284$	0	0
$285$	−34.0672	−2.01797
$286$	0	0
$287$	1.23931	0.0731544
$288$	0	0
$289$	28.1269	1.65452
$290$	0	0
$291$	−8.11054	−0.475448
$292$	0	0
$293$	2.36481	0.138154	0.0690769	−	0.997611i	$-0.477995\pi$
$293$	2.36481	0.138154	0.0690769	+	0.997611i	$0.477995\pi$
$294$	0	0
$295$	−13.4753	−0.784565
$296$	0	0
$297$	−8.93506	−0.518465
$298$	0	0
$299$	−5.71475	−0.330493
$300$	0	0
$301$	1.40180	0.0807987
$302$	0	0
$303$	−17.0778	−0.981094
$304$	0	0
$305$	7.29680	0.417814
$306$	0	0
$307$	−12.1223	−0.691859	−0.345929	−	0.938261i	$-0.612436\pi$
$307$	−12.1223	−0.691859	−0.345929	+	0.938261i	$0.612436\pi$
$308$	0	0
$309$	−3.75666	−0.213709
$310$	0	0
$311$	6.90389	0.391484	0.195742	−	0.980655i	$-0.437289\pi$
$311$	6.90389	0.391484	0.195742	+	0.980655i	$0.437289\pi$
$312$	0	0
$313$	1.19256	0.0674075	0.0337037	−	0.999432i	$-0.489270\pi$
$313$	1.19256	0.0674075	0.0337037	+	0.999432i	$0.489270\pi$
$314$	0	0
$315$	−1.27111	−0.0716191
$316$	0	0
$317$	−11.1650	−0.627088	−0.313544	−	0.949574i	$-0.601516\pi$
$317$	−11.1650	−0.627088	−0.313544	+	0.949574i	$0.601516\pi$
$318$	0	0
$319$	7.19758	0.402987
$320$	0	0
$321$	6.34214	0.353984
$322$	0	0
$323$	45.4388	2.52828
$324$	0	0
$325$	3.06701	0.170127
$326$	0	0
$327$	26.2160	1.44975
$328$	0	0
$329$	0.155938	0.00859715
$330$	0	0
$331$	−22.1526	−1.21762	−0.608809	−	0.793317i	$-0.708352\pi$
$331$	−22.1526	−1.21762	−0.608809	+	0.793317i	$0.708352\pi$
$332$	0	0
$333$	−2.59977	−0.142466
$334$	0	0
$335$	27.8073	1.51928
$336$	0	0
$337$	−2.42826	−0.132276	−0.0661379	−	0.997810i	$-0.521068\pi$
$337$	−2.42826	−0.132276	−0.0661379	+	0.997810i	$0.521068\pi$
$338$	0	0
$339$	−2.66166	−0.144562
$340$	0	0
$341$	15.3199	0.829621
$342$	0	0
$343$	−6.08648	−0.328639
$344$	0	0
$345$	−10.1792	−0.548032
$346$	0	0
$347$	33.0394	1.77365	0.886824	−	0.462108i	$-0.152907\pi$
$347$	33.0394	1.77365	0.886824	+	0.462108i	$0.152907\pi$
$348$	0	0
$349$	−16.9103	−0.905186	−0.452593	−	0.891717i	$-0.649501\pi$
$349$	−16.9103	−0.905186	−0.452593	+	0.891717i	$0.649501\pi$
$350$	0	0
$351$	10.5717	0.564273
$352$	0	0
$353$	3.63351	0.193392	0.0966960	−	0.995314i	$-0.469173\pi$
$353$	3.63351	0.193392	0.0966960	+	0.995314i	$0.469173\pi$
$354$	0	0
$355$	−26.1164	−1.38612
$356$	0	0
$357$	6.04693	0.320038
$358$	0	0
$359$	30.3141	1.59992	0.799960	−	0.600054i	$-0.204854\pi$
$359$	30.3141	1.59992	0.799960	+	0.600054i	$0.204854\pi$
$360$	0	0
$361$	26.7529	1.40805
$362$	0	0
$363$	10.7985	0.566776
$364$	0	0
$365$	26.9896	1.41270
$366$	0	0
$367$	7.16993	0.374267	0.187134	−	0.982334i	$-0.440080\pi$
$367$	7.16993	0.374267	0.187134	+	0.982334i	$0.440080\pi$
$368$	0	0
$369$	3.28569	0.171046
$370$	0	0
$371$	−3.02695	−0.157151
$372$	0	0
$373$	−14.7874	−0.765664	−0.382832	−	0.923818i	$-0.625051\pi$
$373$	−14.7874	−0.765664	−0.382832	+	0.923818i	$0.625051\pi$
$374$	0	0
$375$	−19.7194	−1.01830
$376$	0	0
$377$	−8.51593	−0.438593
$378$	0	0
$379$	−6.55118	−0.336512	−0.168256	−	0.985743i	$-0.553813\pi$
$379$	−6.55118	−0.336512	−0.168256	+	0.985743i	$0.553813\pi$
$380$	0	0
$381$	−35.9920	−1.84393
$382$	0	0
$383$	16.7835	0.857599	0.428799	−	0.903400i	$-0.358937\pi$
$383$	16.7835	0.857599	0.428799	+	0.903400i	$0.358937\pi$
$384$	0	0
$385$	2.59892	0.132453
$386$	0	0
$387$	3.71649	0.188920
$388$	0	0
$389$	−4.48716	−0.227508	−0.113754	−	0.993509i	$-0.536288\pi$
$389$	−4.48716	−0.227508	−0.113754	+	0.993509i	$0.536288\pi$
$390$	0	0
$391$	13.5771	0.686622
$392$	0	0
$393$	16.0922	0.811746
$394$	0	0
$395$	42.8818	2.15762
$396$	0	0
$397$	8.92943	0.448155	0.224078	−	0.974571i	$-0.428063\pi$
$397$	8.92943	0.448155	0.224078	+	0.974571i	$0.428063\pi$
$398$	0	0
$399$	6.08873	0.304818
$400$	0	0
$401$	−19.1152	−0.954567	−0.477283	−	0.878749i	$-0.658378\pi$
$401$	−19.1152	−0.954567	−0.477283	+	0.878749i	$0.658378\pi$
$402$	0	0
$403$	−18.1260	−0.902921
$404$	0	0
$405$	27.4800	1.36549
$406$	0	0
$407$	5.31549	0.263479
$408$	0	0
$409$	17.6999	0.875204	0.437602	−	0.899169i	$-0.355828\pi$
$409$	17.6999	0.875204	0.437602	+	0.899169i	$0.355828\pi$
$410$	0	0
$411$	22.4419	1.10698
$412$	0	0
$413$	2.40841	0.118510
$414$	0	0
$415$	38.1445	1.87244
$416$	0	0
$417$	27.5513	1.34919
$418$	0	0
$419$	36.0123	1.75932	0.879658	−	0.475606i	$-0.157771\pi$
$419$	36.0123	1.75932	0.879658	+	0.475606i	$0.157771\pi$
$420$	0	0
$421$	13.7470	0.669986	0.334993	−	0.942221i	$-0.391266\pi$
$421$	13.7470	0.669986	0.334993	+	0.942221i	$0.391266\pi$
$422$	0	0
$423$	0.413426	0.0201015
$424$	0	0
$425$	−7.28659	−0.353451
$426$	0	0
$427$	−1.30414	−0.0631116
$428$	0	0
$429$	13.7969	0.666119
$430$	0	0
$431$	4.52507	0.217965	0.108983	−	0.994044i	$-0.465241\pi$
$431$	4.52507	0.217965	0.108983	+	0.994044i	$0.465241\pi$
$432$	0	0
$433$	16.2965	0.783160	0.391580	−	0.920144i	$-0.371929\pi$
$433$	16.2965	0.783160	0.391580	+	0.920144i	$0.371929\pi$
$434$	0	0
$435$	−15.1688	−0.727286
$436$	0	0
$437$	13.6709	0.653968
$438$	0	0
$439$	30.6871	1.46462	0.732308	−	0.680974i	$-0.238443\pi$
$439$	30.6871	1.46462	0.732308	+	0.680974i	$0.238443\pi$
$440$	0	0
$441$	−7.95470	−0.378795
$442$	0	0
$443$	28.4194	1.35025	0.675123	−	0.737705i	$-0.264091\pi$
$443$	28.4194	1.35025	0.675123	+	0.737705i	$0.264091\pi$
$444$	0	0
$445$	1.29325	0.0613061
$446$	0	0
$447$	21.7860	1.03044
$448$	0	0
$449$	20.5427	0.969468	0.484734	−	0.874662i	$-0.338916\pi$
$449$	20.5427	0.969468	0.484734	+	0.874662i	$0.338916\pi$
$450$	0	0
$451$	−6.71793	−0.316335
$452$	0	0
$453$	30.1279	1.41553
$454$	0	0
$455$	−3.07495	−0.144156
$456$	0	0
$457$	−4.66152	−0.218056	−0.109028	−	0.994039i	$-0.534774\pi$
$457$	−4.66152	−0.218056	−0.109028	+	0.994039i	$0.534774\pi$
$458$	0	0
$459$	−25.1161	−1.17232
$460$	0	0
$461$	−23.0250	−1.07238	−0.536190	−	0.844097i	$-0.680137\pi$
$461$	−23.0250	−1.07238	−0.536190	+	0.844097i	$0.680137\pi$
$462$	0	0
$463$	13.9512	0.648367	0.324183	−	0.945994i	$-0.394911\pi$
$463$	13.9512	0.648367	0.324183	+	0.945994i	$0.394911\pi$
$464$	0	0
$465$	−32.2864	−1.49725
$466$	0	0
$467$	−9.52613	−0.440817	−0.220408	−	0.975408i	$-0.570739\pi$
$467$	−9.52613	−0.440817	−0.220408	+	0.975408i	$0.570739\pi$
$468$	0	0
$469$	−4.96992	−0.229490
$470$	0	0
$471$	33.1735	1.52856
$472$	0	0
$473$	−7.59874	−0.349390
$474$	0	0
$475$	−7.33695	−0.336643
$476$	0	0
$477$	−8.02510	−0.367444
$478$	0	0
$479$	−0.405774	−0.0185403	−0.00927015	−	0.999957i	$-0.502951\pi$
$479$	−0.405774	−0.0185403	−0.00927015	+	0.999957i	$0.502951\pi$
$480$	0	0
$481$	−6.28910	−0.286758
$482$	0	0
$483$	1.81931	0.0827813
$484$	0	0
$485$	−9.79854	−0.444929
$486$	0	0
$487$	−25.1162	−1.13812	−0.569062	−	0.822295i	$-0.692694\pi$
$487$	−25.1162	−1.13812	−0.569062	+	0.822295i	$0.692694\pi$
$488$	0	0
$489$	50.7847	2.29656
$490$	0	0
$491$	4.49446	0.202832	0.101416	−	0.994844i	$-0.467663\pi$
$491$	4.49446	0.202832	0.101416	+	0.994844i	$0.467663\pi$
$492$	0	0
$493$	20.2321	0.911207
$494$	0	0
$495$	6.89030	0.309696
$496$	0	0
$497$	4.66771	0.209376
$498$	0	0
$499$	−24.8036	−1.11036	−0.555180	−	0.831730i	$-0.687351\pi$
$499$	−24.8036	−1.11036	−0.555180	+	0.831730i	$0.687351\pi$
$500$	0	0
$501$	−24.8189	−1.10882
$502$	0	0
$503$	−10.9310	−0.487389	−0.243694	−	0.969852i	$-0.578359\pi$
$503$	−10.9310	−0.487389	−0.243694	+	0.969852i	$0.578359\pi$
$504$	0	0
$505$	−20.6321	−0.918117
$506$	0	0
$507$	10.2191	0.453846
$508$	0	0
$509$	−11.4525	−0.507623	−0.253811	−	0.967254i	$-0.581684\pi$
$509$	−11.4525	−0.507623	−0.253811	+	0.967254i	$0.581684\pi$
$510$	0	0
$511$	−4.82378	−0.213391
$512$	0	0
$513$	−25.2897	−1.11657
$514$	0	0
$515$	−4.53851	−0.199991
$516$	0	0
$517$	−0.845291	−0.0371759
$518$	0	0
$519$	−6.58508	−0.289053
$520$	0	0
$521$	28.5556	1.25104	0.625522	−	0.780206i	$-0.284886\pi$
$521$	28.5556	1.25104	0.625522	+	0.780206i	$0.284886\pi$
$522$	0	0
$523$	−23.0436	−1.00762	−0.503812	−	0.863813i	$-0.668070\pi$
$523$	−23.0436	−1.00762	−0.503812	+	0.863813i	$0.668070\pi$
$524$	0	0
$525$	−0.976391	−0.0426132
$526$	0	0
$527$	43.0637	1.87588
$528$	0	0
$529$	−18.9151	−0.822398
$530$	0	0
$531$	6.38522	0.277095
$532$	0	0
$533$	7.94842	0.344284
$534$	0	0
$535$	7.66209	0.331261
$536$	0	0
$537$	−11.0276	−0.475877
$538$	0	0
$539$	16.2642	0.700548
$540$	0	0
$541$	8.51911	0.366265	0.183133	−	0.983088i	$-0.441376\pi$
$541$	8.51911	0.366265	0.183133	+	0.983088i	$0.441376\pi$
$542$	0	0
$543$	28.7526	1.23389
$544$	0	0
$545$	31.6722	1.35669
$546$	0	0
$547$	−46.5984	−1.99240	−0.996202	−	0.0870667i	$-0.972251\pi$
$547$	−46.5984	−1.99240	−0.996202	+	0.0870667i	$0.972251\pi$
$548$	0	0
$549$	−3.45755	−0.147565
$550$	0	0
$551$	20.3719	0.867873
$552$	0	0
$553$	−7.66414	−0.325912
$554$	0	0
$555$	−11.2023	−0.475511
$556$	0	0
$557$	−33.6021	−1.42377	−0.711883	−	0.702298i	$-0.752157\pi$
$557$	−33.6021	−1.42377	−0.711883	+	0.702298i	$0.752157\pi$
$558$	0	0
$559$	8.99056	0.380260
$560$	0	0
$561$	−32.7785	−1.38391
$562$	0	0
$563$	−25.4490	−1.07255	−0.536274	−	0.844044i	$-0.680169\pi$
$563$	−25.4490	−1.07255	−0.536274	+	0.844044i	$0.680169\pi$
$564$	0	0
$565$	−3.21562	−0.135282
$566$	0	0
$567$	−4.91143	−0.206261
$568$	0	0
$569$	−36.2232	−1.51856	−0.759279	−	0.650766i	$-0.774448\pi$
$569$	−36.2232	−1.51856	−0.759279	+	0.650766i	$0.774448\pi$
$570$	0	0
$571$	8.73077	0.365371	0.182686	−	0.983171i	$-0.441521\pi$
$571$	8.73077	0.365371	0.182686	+	0.983171i	$0.441521\pi$
$572$	0	0
$573$	−35.8565	−1.49793
$574$	0	0
$575$	−2.19227	−0.0914241
$576$	0	0
$577$	−29.1459	−1.21336	−0.606679	−	0.794947i	$-0.707499\pi$
$577$	−29.1459	−1.21336	−0.606679	+	0.794947i	$0.707499\pi$
$578$	0	0
$579$	37.9990	1.57918
$580$	0	0
$581$	−6.81745	−0.282836
$582$	0	0
$583$	16.4081	0.679555
$584$	0	0
$585$	−8.15236	−0.337059
$586$	0	0
$587$	−23.5308	−0.971221	−0.485610	−	0.874175i	$-0.661403\pi$
$587$	−23.5308	−0.971221	−0.485610	+	0.874175i	$0.661403\pi$
$588$	0	0
$589$	43.3613	1.78667
$590$	0	0
$591$	12.3878	0.509566
$592$	0	0
$593$	−27.0675	−1.11153	−0.555765	−	0.831339i	$-0.687575\pi$
$593$	−27.0675	−1.11153	−0.555765	+	0.831339i	$0.687575\pi$
$594$	0	0
$595$	7.30545	0.299494
$596$	0	0
$597$	30.4666	1.24691
$598$	0	0
$599$	9.67890	0.395469	0.197735	−	0.980256i	$-0.436642\pi$
$599$	9.67890	0.395469	0.197735	+	0.980256i	$0.436642\pi$
$600$	0	0
$601$	43.3406	1.76790	0.883951	−	0.467580i	$-0.154874\pi$
$601$	43.3406	1.76790	0.883951	+	0.467580i	$0.154874\pi$
$602$	0	0
$603$	−13.1763	−0.536582
$604$	0	0
$605$	13.0460	0.530395
$606$	0	0
$607$	38.2167	1.55117	0.775585	−	0.631243i	$-0.217455\pi$
$607$	38.2167	1.55117	0.775585	+	0.631243i	$0.217455\pi$
$608$	0	0
$609$	2.71107	0.109858
$610$	0	0
$611$	1.00012	0.0404605
$612$	0	0
$613$	0.467312	0.0188745	0.00943727	−	0.999955i	$-0.496996\pi$
$613$	0.467312	0.0188745	0.00943727	+	0.999955i	$0.496996\pi$
$614$	0	0
$615$	14.1579	0.570902
$616$	0	0
$617$	−19.9134	−0.801685	−0.400843	−	0.916147i	$-0.631283\pi$
$617$	−19.9134	−0.801685	−0.400843	+	0.916147i	$0.631283\pi$
$618$	0	0
$619$	42.0121	1.68861	0.844305	−	0.535864i	$-0.180014\pi$
$619$	42.0121	1.68861	0.844305	+	0.535864i	$0.180014\pi$
$620$	0	0
$621$	−7.55653	−0.303233
$622$	0	0
$623$	−0.231140	−0.00926041
$624$	0	0
$625$	−29.2469	−1.16988
$626$	0	0
$627$	−33.0051	−1.31809
$628$	0	0
$629$	14.9416	0.595761
$630$	0	0
$631$	37.6802	1.50003	0.750013	−	0.661423i	$-0.230047\pi$
$631$	37.6802	1.50003	0.750013	+	0.661423i	$0.230047\pi$
$632$	0	0
$633$	−48.1424	−1.91349
$634$	0	0
$635$	−43.4829	−1.72556
$636$	0	0
$637$	−19.2432	−0.762444
$638$	0	0
$639$	12.3751	0.489552
$640$	0	0
$641$	−46.3653	−1.83132	−0.915659	−	0.401956i	$-0.868330\pi$
$641$	−46.3653	−1.83132	−0.915659	+	0.401956i	$0.868330\pi$
$642$	0	0
$643$	14.9334	0.588917	0.294458	−	0.955664i	$-0.404861\pi$
$643$	14.9334	0.588917	0.294458	+	0.955664i	$0.404861\pi$
$644$	0	0
$645$	16.0142	0.630558
$646$	0	0
$647$	11.6992	0.459944	0.229972	−	0.973197i	$-0.426137\pi$
$647$	11.6992	0.459944	0.229972	+	0.973197i	$0.426137\pi$
$648$	0	0
$649$	−13.0552	−0.512462
$650$	0	0
$651$	5.77046	0.226162
$652$	0	0
$653$	48.0788	1.88147	0.940735	−	0.339143i	$-0.110137\pi$
$653$	48.0788	1.88147	0.940735	+	0.339143i	$0.110137\pi$
$654$	0	0
$655$	19.4414	0.759639
$656$	0	0
$657$	−12.7889	−0.498942
$658$	0	0
$659$	−28.0791	−1.09381	−0.546904	−	0.837196i	$-0.684194\pi$
$659$	−28.0791	−1.09381	−0.546904	+	0.837196i	$0.684194\pi$
$660$	0	0
$661$	−1.07946	−0.0419863	−0.0209931	−	0.999780i	$-0.506683\pi$
$661$	−1.07946	−0.0419863	−0.0209931	+	0.999780i	$0.506683\pi$
$662$	0	0
$663$	38.7824	1.50618
$664$	0	0
$665$	7.35594	0.285251
$666$	0	0
$667$	6.08711	0.235694
$668$	0	0
$669$	−36.0059	−1.39207
$670$	0	0
$671$	7.06931	0.272908
$672$	0	0
$673$	−13.1331	−0.506243	−0.253121	−	0.967435i	$-0.581457\pi$
$673$	−13.1331	−0.506243	−0.253121	+	0.967435i	$0.581457\pi$
$674$	0	0
$675$	4.05546	0.156095
$676$	0	0
$677$	44.1544	1.69699	0.848495	−	0.529203i	$-0.177509\pi$
$677$	44.1544	1.69699	0.848495	+	0.529203i	$0.177509\pi$
$678$	0	0
$679$	1.75126	0.0672074
$680$	0	0
$681$	10.6939	0.409790
$682$	0	0
$683$	30.4803	1.16630	0.583148	−	0.812366i	$-0.301821\pi$
$683$	30.4803	1.16630	0.583148	+	0.812366i	$0.301821\pi$
$684$	0	0
$685$	27.1126	1.03592
$686$	0	0
$687$	5.97227	0.227856
$688$	0	0
$689$	−19.4135	−0.739596
$690$	0	0
$691$	−23.3914	−0.889850	−0.444925	−	0.895568i	$-0.646770\pi$
$691$	−23.3914	−0.889850	−0.444925	+	0.895568i	$0.646770\pi$
$692$	0	0
$693$	−1.23148	−0.0467802
$694$	0	0
$695$	33.2854	1.26259
$696$	0	0
$697$	−18.8838	−0.715275
$698$	0	0
$699$	−54.7240	−2.06985
$700$	0	0
$701$	−44.9604	−1.69813	−0.849065	−	0.528288i	$-0.822834\pi$
$701$	−44.9604	−1.69813	−0.849065	+	0.528288i	$0.822834\pi$
$702$	0	0
$703$	15.0449	0.567429
$704$	0	0
$705$	1.78143	0.0670927
$706$	0	0
$707$	3.68752	0.138683
$708$	0	0
$709$	−10.9229	−0.410217	−0.205108	−	0.978739i	$-0.565755\pi$
$709$	−10.9229	−0.410217	−0.205108	+	0.978739i	$0.565755\pi$
$710$	0	0
$711$	−20.3193	−0.762033
$712$	0	0
$713$	12.9563	0.485218
$714$	0	0
$715$	16.6683	0.623360
$716$	0	0
$717$	24.6332	0.919942
$718$	0	0
$719$	−8.91225	−0.332371	−0.166185	−	0.986095i	$-0.553145\pi$
$719$	−8.91225	−0.332371	−0.166185	+	0.986095i	$0.553145\pi$
$720$	0	0
$721$	0.811155	0.0302090
$722$	0	0
$723$	−10.3039	−0.383205
$724$	0	0
$725$	−3.26685	−0.121328
$726$	0	0
$727$	27.8731	1.03376	0.516878	−	0.856059i	$-0.327094\pi$
$727$	27.8731	1.03376	0.516878	+	0.856059i	$0.327094\pi$
$728$	0	0
$729$	9.88017	0.365932
$730$	0	0
$731$	−21.3597	−0.790018
$732$	0	0
$733$	−23.7776	−0.878245	−0.439122	−	0.898427i	$-0.644711\pi$
$733$	−23.7776	−0.878245	−0.439122	+	0.898427i	$0.644711\pi$
$734$	0	0
$735$	−34.2764	−1.26431
$736$	0	0
$737$	26.9403	0.992360
$738$	0	0
$739$	11.9926	0.441156	0.220578	−	0.975369i	$-0.429206\pi$
$739$	11.9926	0.441156	0.220578	+	0.975369i	$0.429206\pi$
$740$	0	0
$741$	39.0504	1.43455
$742$	0	0
$743$	−38.3843	−1.40819	−0.704093	−	0.710108i	$-0.748646\pi$
$743$	−38.3843	−1.40819	−0.704093	+	0.710108i	$0.748646\pi$
$744$	0	0
$745$	26.3203	0.964300
$746$	0	0
$747$	−18.0746	−0.661313
$748$	0	0
$749$	−1.36942	−0.0500376
$750$	0	0
$751$	−22.1974	−0.809994	−0.404997	−	0.914318i	$-0.632727\pi$
$751$	−22.1974	−0.809994	−0.404997	+	0.914318i	$0.632727\pi$
$752$	0	0
$753$	13.7542	0.501231
$754$	0	0
$755$	36.3982	1.32467
$756$	0	0
$757$	−41.7068	−1.51586	−0.757930	−	0.652335i	$-0.773789\pi$
$757$	−41.7068	−1.51586	−0.757930	+	0.652335i	$0.773789\pi$
$758$	0	0
$759$	−9.86187	−0.357963
$760$	0	0
$761$	3.98662	0.144515	0.0722573	−	0.997386i	$-0.476980\pi$
$761$	3.98662	0.144515	0.0722573	+	0.997386i	$0.476980\pi$
$762$	0	0
$763$	−5.66068	−0.204930
$764$	0	0
$765$	19.3683	0.700263
$766$	0	0
$767$	15.4465	0.557740
$768$	0	0
$769$	13.1305	0.473497	0.236748	−	0.971571i	$-0.423918\pi$
$769$	13.1305	0.473497	0.236748	+	0.971571i	$0.423918\pi$
$770$	0	0
$771$	−38.4054	−1.38314
$772$	0	0
$773$	1.06771	0.0384029	0.0192014	−	0.999816i	$-0.493888\pi$
$773$	1.06771	0.0384029	0.0192014	+	0.999816i	$0.493888\pi$
$774$	0	0
$775$	−6.95344	−0.249775
$776$	0	0
$777$	2.00215	0.0718268
$778$	0	0
$779$	−19.0143	−0.681259
$780$	0	0
$781$	−25.3022	−0.905384
$782$	0	0
$783$	−11.2605	−0.402417
$784$	0	0
$785$	40.0778	1.43044
$786$	0	0
$787$	10.8230	0.385797	0.192898	−	0.981219i	$-0.438211\pi$
$787$	10.8230	0.385797	0.192898	+	0.981219i	$0.438211\pi$
$788$	0	0
$789$	−33.6406	−1.19764
$790$	0	0
$791$	0.574719	0.0204347
$792$	0	0
$793$	−8.36416	−0.297020
$794$	0	0
$795$	−34.5798	−1.22642
$796$	0	0
$797$	9.65800	0.342104	0.171052	−	0.985262i	$-0.445283\pi$
$797$	9.65800	0.342104	0.171052	+	0.985262i	$0.445283\pi$
$798$	0	0
$799$	−2.37608	−0.0840596
$800$	0	0
$801$	−0.612801	−0.0216523
$802$	0	0
$803$	26.1482	0.922748
$804$	0	0
$805$	2.19795	0.0774675
$806$	0	0
$807$	−0.687352	−0.0241959
$808$	0	0
$809$	22.0407	0.774910	0.387455	−	0.921889i	$-0.373354\pi$
$809$	22.0407	0.774910	0.387455	+	0.921889i	$0.373354\pi$
$810$	0	0
$811$	19.8375	0.696588	0.348294	−	0.937385i	$-0.386761\pi$
$811$	19.8375	0.696588	0.348294	+	0.937385i	$0.386761\pi$
$812$	0	0
$813$	−21.0223	−0.737286
$814$	0	0
$815$	61.3543	2.14915
$816$	0	0
$817$	−21.5074	−0.752447
$818$	0	0
$819$	1.45705	0.0509134
$820$	0	0
$821$	−28.8992	−1.00859	−0.504294	−	0.863532i	$-0.668247\pi$
$821$	−28.8992	−1.00859	−0.504294	+	0.863532i	$0.668247\pi$
$822$	0	0
$823$	−15.8729	−0.553296	−0.276648	−	0.960971i	$-0.589224\pi$
$823$	−15.8729	−0.553296	−0.276648	+	0.960971i	$0.589224\pi$
$824$	0	0
$825$	5.29270	0.184268
$826$	0	0
$827$	24.9879	0.868916	0.434458	−	0.900692i	$-0.356940\pi$
$827$	24.9879	0.868916	0.434458	+	0.900692i	$0.356940\pi$
$828$	0	0
$829$	13.1305	0.456042	0.228021	−	0.973656i	$-0.426774\pi$
$829$	13.1305	0.456042	0.228021	+	0.973656i	$0.426774\pi$
$830$	0	0
$831$	−58.0593	−2.01406
$832$	0	0
$833$	45.7179	1.58403
$834$	0	0
$835$	−29.9843	−1.03765
$836$	0	0
$837$	−23.9677	−0.828446
$838$	0	0
$839$	−15.9324	−0.550048	−0.275024	−	0.961437i	$-0.588686\pi$
$839$	−15.9324	−0.550048	−0.275024	+	0.961437i	$0.588686\pi$
$840$	0	0
$841$	−19.9292	−0.687214
$842$	0	0
$843$	55.1120	1.89816
$844$	0	0
$845$	12.3459	0.424713
$846$	0	0
$847$	−2.33167	−0.0801171
$848$	0	0
$849$	17.7446	0.608991
$850$	0	0
$851$	4.49540	0.154100
$852$	0	0
$853$	11.3968	0.390218	0.195109	−	0.980782i	$-0.437494\pi$
$853$	11.3968	0.390218	0.195109	+	0.980782i	$0.437494\pi$
$854$	0	0
$855$	19.5022	0.666961
$856$	0	0
$857$	−23.5609	−0.804823	−0.402412	−	0.915459i	$-0.631828\pi$
$857$	−23.5609	−0.804823	−0.402412	+	0.915459i	$0.631828\pi$
$858$	0	0
$859$	−54.4389	−1.85743	−0.928716	−	0.370792i	$-0.879086\pi$
$859$	−54.4389	−1.85743	−0.928716	+	0.370792i	$0.879086\pi$
$860$	0	0
$861$	−2.53040	−0.0862358
$862$	0	0
$863$	38.9976	1.32749	0.663747	−	0.747957i	$-0.268965\pi$
$863$	38.9976	1.32749	0.663747	+	0.747957i	$0.268965\pi$
$864$	0	0
$865$	−7.95560	−0.270498
$866$	0	0
$867$	−57.4287	−1.95038
$868$	0	0
$869$	41.5448	1.40931
$870$	0	0
$871$	−31.8749	−1.08004
$872$	0	0
$873$	4.64298	0.157141
$874$	0	0
$875$	4.25789	0.143943
$876$	0	0
$877$	−18.9247	−0.639042	−0.319521	−	0.947579i	$-0.603522\pi$
$877$	−18.9247	−0.639042	−0.319521	+	0.947579i	$0.603522\pi$
$878$	0	0
$879$	−4.82841	−0.162858
$880$	0	0
$881$	6.23754	0.210148	0.105074	−	0.994464i	$-0.466492\pi$
$881$	6.23754	0.210148	0.105074	+	0.994464i	$0.466492\pi$
$882$	0	0
$883$	−23.0507	−0.775719	−0.387859	−	0.921719i	$-0.626785\pi$
$883$	−23.0507	−0.775719	−0.387859	+	0.921719i	$0.626785\pi$
$884$	0	0
$885$	27.5136	0.924860
$886$	0	0
$887$	11.1343	0.373853	0.186927	−	0.982374i	$-0.440147\pi$
$887$	11.1343	0.373853	0.186927	+	0.982374i	$0.440147\pi$
$888$	0	0
$889$	7.77157	0.260650
$890$	0	0
$891$	26.6233	0.891914
$892$	0	0
$893$	−2.39250	−0.0800620
$894$	0	0
$895$	−13.3227	−0.445330
$896$	0	0
$897$	11.6682	0.389591
$898$	0	0
$899$	19.3071	0.643926
$900$	0	0
$901$	46.1225	1.53656
$902$	0	0
$903$	−2.86217	−0.0952470
$904$	0	0
$905$	34.7367	1.15469
$906$	0	0
$907$	−0.137603	−0.00456904	−0.00228452	−	0.999997i	$-0.500727\pi$
$907$	−0.137603	−0.00456904	−0.00228452	+	0.999997i	$0.500727\pi$
$908$	0	0
$909$	9.77641	0.324263
$910$	0	0
$911$	21.7209	0.719644	0.359822	−	0.933021i	$-0.382837\pi$
$911$	21.7209	0.719644	0.359822	+	0.933021i	$0.382837\pi$
$912$	0	0
$913$	36.9552	1.22304
$914$	0	0
$915$	−14.8984	−0.492527
$916$	0	0
$917$	−3.47471	−0.114745
$918$	0	0
$919$	58.7543	1.93813	0.969064	−	0.246811i	$-0.0793828\pi$
$919$	58.7543	1.93813	0.969064	+	0.246811i	$0.0793828\pi$
$920$	0	0
$921$	24.7511	0.815576
$922$	0	0
$923$	29.9367	0.985378
$924$	0	0
$925$	−2.41261	−0.0793260
$926$	0	0
$927$	2.15055	0.0706332
$928$	0	0
$929$	−50.1488	−1.64533	−0.822665	−	0.568527i	$-0.807514\pi$
$929$	−50.1488	−1.64533	−0.822665	+	0.568527i	$0.807514\pi$
$930$	0	0
$931$	46.0339	1.50870
$932$	0	0
$933$	−14.0962	−0.461488
$934$	0	0
$935$	−39.6005	−1.29507
$936$	0	0
$937$	2.08032	0.0679611	0.0339806	−	0.999422i	$-0.489182\pi$
$937$	2.08032	0.0679611	0.0339806	+	0.999422i	$0.489182\pi$
$938$	0	0
$939$	−2.43494	−0.0794612
$940$	0	0
$941$	10.6957	0.348670	0.174335	−	0.984686i	$-0.444222\pi$
$941$	10.6957	0.348670	0.174335	+	0.984686i	$0.444222\pi$
$942$	0	0
$943$	−5.68146	−0.185014
$944$	0	0
$945$	−4.06596	−0.132266
$946$	0	0
$947$	−7.81569	−0.253976	−0.126988	−	0.991904i	$-0.540531\pi$
$947$	−7.81569	−0.253976	−0.126988	+	0.991904i	$0.540531\pi$
$948$	0	0
$949$	−30.9376	−1.00428
$950$	0	0
$951$	22.7964	0.739223
$952$	0	0
$953$	−41.8455	−1.35551	−0.677754	−	0.735289i	$-0.737046\pi$
$953$	−41.8455	−1.35551	−0.677754	+	0.735289i	$0.737046\pi$
$954$	0	0
$955$	−43.3191	−1.40177
$956$	0	0
$957$	−14.6958	−0.475049
$958$	0	0
$959$	−4.84576	−0.156478
$960$	0	0
$961$	10.0947	0.325637
$962$	0	0
$963$	−3.63064	−0.116996
$964$	0	0
$965$	45.9075	1.47782
$966$	0	0
$967$	35.3805	1.13776	0.568880	−	0.822420i	$-0.307377\pi$
$967$	35.3805	1.13776	0.568880	+	0.822420i	$0.307377\pi$
$968$	0	0
$969$	−92.7758	−2.98039
$970$	0	0
$971$	25.0243	0.803070	0.401535	−	0.915844i	$-0.368477\pi$
$971$	25.0243	0.803070	0.401535	+	0.915844i	$0.368477\pi$
$972$	0	0
$973$	−5.94900	−0.190716
$974$	0	0
$975$	−6.26215	−0.200549
$976$	0	0
$977$	−38.5046	−1.23187	−0.615935	−	0.787797i	$-0.711222\pi$
$977$	−38.5046	−1.23187	−0.615935	+	0.787797i	$0.711222\pi$
$978$	0	0
$979$	1.25293	0.0400439
$980$	0	0
$981$	−15.0077	−0.479158
$982$	0	0
$983$	−1.89450	−0.0604250	−0.0302125	−	0.999543i	$-0.509618\pi$
$983$	−1.89450	−0.0604250	−0.0302125	+	0.999543i	$0.509618\pi$
$984$	0	0
$985$	14.9660	0.476857
$986$	0	0
$987$	−0.318391	−0.0101345
$988$	0	0
$989$	−6.42637	−0.204347
$990$	0	0
$991$	18.3200	0.581954	0.290977	−	0.956730i	$-0.406020\pi$
$991$	18.3200	0.581954	0.290977	+	0.956730i	$0.406020\pi$
$992$	0	0
$993$	45.2306	1.43535
$994$	0	0
$995$	36.8074	1.16687
$996$	0	0
$997$	30.9885	0.981416	0.490708	−	0.871324i	$-0.336738\pi$
$997$	30.9885	0.981416	0.490708	+	0.871324i	$0.336738\pi$
$998$	0	0
$999$	−8.31598	−0.263106

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.b.1.17	✓	87

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.b.1.17	✓	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)

\(f(q)\)	\(=\)	\(q-2.04177 q^{3} -2.46672 q^{5} +0.440869 q^{7} +1.16884 q^{9} +O(q^{10})\)	\(q-2.04177 q^{3} -2.46672 q^{5} +0.440869 q^{7} +1.16884 q^{9} -2.38981 q^{11} +2.82754 q^{13} +5.03648 q^{15} -6.71766 q^{17} -6.76409 q^{19} -0.900155 q^{21} -2.02110 q^{23} +1.08469 q^{25} +3.73881 q^{27} -3.01178 q^{29} -6.41052 q^{31} +4.87945 q^{33} -1.08750 q^{35} -2.22423 q^{37} -5.77320 q^{39} +2.81107 q^{41} +3.17964 q^{43} -2.88320 q^{45} +0.353706 q^{47} -6.80563 q^{49} +13.7159 q^{51} -6.86586 q^{53} +5.89499 q^{55} +13.8107 q^{57} +5.46287 q^{59} -2.95810 q^{61} +0.515306 q^{63} -6.97475 q^{65} -11.2730 q^{67} +4.12663 q^{69} +10.5875 q^{71} -10.9415 q^{73} -2.21470 q^{75} -1.05359 q^{77} -17.3841 q^{79} -11.1403 q^{81} -15.4637 q^{83} +16.5706 q^{85} +6.14937 q^{87} -0.524282 q^{89} +1.24658 q^{91} +13.0888 q^{93} +16.6851 q^{95} +3.97230 q^{97} -2.79331 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

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