LMFDB - Embedded newform 8032.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.26665 q^{3} -1.74507 q^{5} -2.44649 q^{7} +7.67103 q^{9} +O(q^{10})$	$q-3.26665 q^{3} -1.74507 q^{5} -2.44649 q^{7} +7.67103 q^{9} +4.91815 q^{11} -0.951412 q^{13} +5.70053 q^{15} -0.287060 q^{17} +3.32910 q^{19} +7.99183 q^{21} -8.64311 q^{23} -1.95474 q^{25} -15.2586 q^{27} -2.58254 q^{29} -3.10051 q^{31} -16.0659 q^{33} +4.26929 q^{35} +3.76104 q^{37} +3.10793 q^{39} +11.7169 q^{41} -2.03971 q^{43} -13.3865 q^{45} -6.06500 q^{47} -1.01470 q^{49} +0.937726 q^{51} +1.73480 q^{53} -8.58251 q^{55} -10.8750 q^{57} -6.43022 q^{59} +9.69981 q^{61} -18.7671 q^{63} +1.66028 q^{65} +5.73425 q^{67} +28.2340 q^{69} -15.7854 q^{71} +0.931761 q^{73} +6.38546 q^{75} -12.0322 q^{77} +9.72571 q^{79} +26.8316 q^{81} +10.2370 q^{83} +0.500939 q^{85} +8.43627 q^{87} +3.19539 q^{89} +2.32762 q^{91} +10.1283 q^{93} -5.80951 q^{95} +14.2427 q^{97} +37.7273 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9	$ 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9} - 13 q^{11} - 7 q^{13} - 28 q^{15} - 9 q^{17} + 17 q^{19} - 6 q^{21} - 43 q^{23} + 34 q^{25} - 12 q^{27} - q^{29} - 39 q^{31} - 17 q^{35} - q^{37} - 48 q^{39} - 3 q^{41} + 19 q^{43} - 66 q^{47} + 25 q^{49} + 14 q^{51} + 3 q^{53} - 50 q^{55} - 14 q^{57} - 27 q^{59} + 15 q^{61} - 75 q^{63} - 6 q^{65} - 8 q^{67} + 18 q^{69} - 64 q^{71} - 15 q^{73} - 9 q^{75} - 71 q^{79} + 6 q^{81} - 60 q^{83} + 15 q^{85} - 64 q^{87} - 32 q^{89} + 26 q^{91} - 4 q^{93} - 72 q^{95} - 4 q^{97} - 13 q^{99}+O(q^{100}) $ 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9 - 13 * q^11 - 7 * q^13 - 28 * q^15 - 9 * q^17 + 17 * q^19 - 6 * q^21 - 43 * q^23 + 34 * q^25 - 12 * q^27 - q^29 - 39 * q^31 - 17 * q^35 - q^37 - 48 * q^39 - 3 * q^41 + 19 * q^43 - 66 * q^47 + 25 * q^49 + 14 * q^51 + 3 * q^53 - 50 * q^55 - 14 * q^57 - 27 * q^59 + 15 * q^61 - 75 * q^63 - 6 * q^65 - 8 * q^67 + 18 * q^69 - 64 * q^71 - 15 * q^73 - 9 * q^75 - 71 * q^79 + 6 * q^81 - 60 * q^83 + 15 * q^85 - 64 * q^87 - 32 * q^89 + 26 * q^91 - 4 * q^93 - 72 * q^95 - 4 * q^97 - 13 * 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
$3$	−3.26665	−1.88600	−0.943002	−	0.332788i	$-0.892011\pi$
$3$	−3.26665	−1.88600	−0.943002	+	0.332788i	$0.892011\pi$
$4$	0	0
$5$	−1.74507	−0.780418	−0.390209	−	0.920726i	$-0.627597\pi$
$5$	−1.74507	−0.780418	−0.390209	+	0.920726i	$0.627597\pi$
$6$	0	0
$7$	−2.44649	−0.924686	−0.462343	−	0.886701i	$-0.652991\pi$
$7$	−2.44649	−0.924686	−0.462343	+	0.886701i	$0.652991\pi$
$8$	0	0
$9$	7.67103	2.55701
$10$	0	0
$11$	4.91815	1.48288	0.741440	−	0.671020i	$-0.234143\pi$
$11$	4.91815	1.48288	0.741440	+	0.671020i	$0.234143\pi$
$12$	0	0
$13$	−0.951412	−0.263874	−0.131937	−	0.991258i	$-0.542120\pi$
$13$	−0.951412	−0.263874	−0.131937	+	0.991258i	$0.542120\pi$
$14$	0	0
$15$	5.70053	1.47187
$16$	0	0
$17$	−0.287060	−0.0696223	−0.0348111	−	0.999394i	$-0.511083\pi$
$17$	−0.287060	−0.0696223	−0.0348111	+	0.999394i	$0.511083\pi$
$18$	0	0
$19$	3.32910	0.763749	0.381874	−	0.924214i	$-0.375279\pi$
$19$	3.32910	0.763749	0.381874	+	0.924214i	$0.375279\pi$
$20$	0	0
$21$	7.99183	1.74396
$22$	0	0
$23$	−8.64311	−1.80221	−0.901106	−	0.433599i	$-0.857244\pi$
$23$	−8.64311	−1.80221	−0.901106	+	0.433599i	$0.857244\pi$
$24$	0	0
$25$	−1.95474	−0.390948
$26$	0	0
$27$	−15.2586	−2.93652
$28$	0	0
$29$	−2.58254	−0.479566	−0.239783	−	0.970827i	$-0.577076\pi$
$29$	−2.58254	−0.479566	−0.239783	+	0.970827i	$0.577076\pi$
$30$	0	0
$31$	−3.10051	−0.556867	−0.278434	−	0.960455i	$-0.589815\pi$
$31$	−3.10051	−0.556867	−0.278434	+	0.960455i	$0.589815\pi$
$32$	0	0
$33$	−16.0659	−2.79671
$34$	0	0
$35$	4.26929	0.721641
$36$	0	0
$37$	3.76104	0.618311	0.309155	−	0.951012i	$-0.399954\pi$
$37$	3.76104	0.618311	0.309155	+	0.951012i	$0.399954\pi$
$38$	0	0
$39$	3.10793	0.497668
$40$	0	0
$41$	11.7169	1.82988	0.914938	−	0.403596i	$-0.132240\pi$
$41$	11.7169	1.82988	0.914938	+	0.403596i	$0.132240\pi$
$42$	0	0
$43$	−2.03971	−0.311053	−0.155526	−	0.987832i	$-0.549707\pi$
$43$	−2.03971	−0.311053	−0.155526	+	0.987832i	$0.549707\pi$
$44$	0	0
$45$	−13.3865	−1.99553
$46$	0	0
$47$	−6.06500	−0.884671	−0.442336	−	0.896850i	$-0.645850\pi$
$47$	−6.06500	−0.884671	−0.442336	+	0.896850i	$0.645850\pi$
$48$	0	0
$49$	−1.01470	−0.144956
$50$	0	0
$51$	0.937726	0.131308
$52$	0	0
$53$	1.73480	0.238293	0.119147	−	0.992877i	$-0.461984\pi$
$53$	1.73480	0.238293	0.119147	+	0.992877i	$0.461984\pi$
$54$	0	0
$55$	−8.58251	−1.15727
$56$	0	0
$57$	−10.8750	−1.44043
$58$	0	0
$59$	−6.43022	−0.837144	−0.418572	−	0.908184i	$-0.637469\pi$
$59$	−6.43022	−0.837144	−0.418572	+	0.908184i	$0.637469\pi$
$60$	0	0
$61$	9.69981	1.24193	0.620967	−	0.783837i	$-0.286740\pi$
$61$	9.69981	1.24193	0.620967	+	0.783837i	$0.286740\pi$
$62$	0	0
$63$	−18.7671	−2.36443
$64$	0	0
$65$	1.66028	0.205932
$66$	0	0
$67$	5.73425	0.700551	0.350275	−	0.936647i	$-0.386088\pi$
$67$	5.73425	0.700551	0.350275	+	0.936647i	$0.386088\pi$
$68$	0	0
$69$	28.2340	3.39898
$70$	0	0
$71$	−15.7854	−1.87338	−0.936692	−	0.350155i	$-0.886129\pi$
$71$	−15.7854	−1.87338	−0.936692	+	0.350155i	$0.886129\pi$
$72$	0	0
$73$	0.931761	0.109054	0.0545272	−	0.998512i	$-0.482635\pi$
$73$	0.931761	0.109054	0.0545272	+	0.998512i	$0.482635\pi$
$74$	0	0
$75$	6.38546	0.737329
$76$	0	0
$77$	−12.0322	−1.37120
$78$	0	0
$79$	9.72571	1.09423	0.547114	−	0.837058i	$-0.315726\pi$
$79$	9.72571	1.09423	0.547114	+	0.837058i	$0.315726\pi$
$80$	0	0
$81$	26.8316	2.98128
$82$	0	0
$83$	10.2370	1.12366	0.561831	−	0.827252i	$-0.310097\pi$
$83$	10.2370	1.12366	0.561831	+	0.827252i	$0.310097\pi$
$84$	0	0
$85$	0.500939	0.0543345
$86$	0	0
$87$	8.43627	0.904463
$88$	0	0
$89$	3.19539	0.338711	0.169355	−	0.985555i	$-0.445831\pi$
$89$	3.19539	0.338711	0.169355	+	0.985555i	$0.445831\pi$
$90$	0	0
$91$	2.32762	0.244001
$92$	0	0
$93$	10.1283	1.05025
$94$	0	0
$95$	−5.80951	−0.596043
$96$	0	0
$97$	14.2427	1.44612	0.723061	−	0.690784i	$-0.242734\pi$
$97$	14.2427	1.44612	0.723061	+	0.690784i	$0.242734\pi$
$98$	0	0
$99$	37.7273	3.79173
$100$	0	0
$101$	4.90129	0.487697	0.243848	−	0.969813i	$-0.421590\pi$
$101$	4.90129	0.487697	0.243848	+	0.969813i	$0.421590\pi$
$102$	0	0
$103$	−2.07603	−0.204558	−0.102279	−	0.994756i	$-0.532613\pi$
$103$	−2.07603	−0.204558	−0.102279	+	0.994756i	$0.532613\pi$
$104$	0	0
$105$	−13.9463	−1.36102
$106$	0	0
$107$	5.60475	0.541832	0.270916	−	0.962603i	$-0.412673\pi$
$107$	5.60475	0.541832	0.270916	+	0.962603i	$0.412673\pi$
$108$	0	0
$109$	−3.81314	−0.365232	−0.182616	−	0.983184i	$-0.558457\pi$
$109$	−3.81314	−0.365232	−0.182616	+	0.983184i	$0.558457\pi$
$110$	0	0
$111$	−12.2860	−1.16614
$112$	0	0
$113$	7.92510	0.745531	0.372766	−	0.927926i	$-0.378410\pi$
$113$	7.92510	0.745531	0.372766	+	0.927926i	$0.378410\pi$
$114$	0	0
$115$	15.0828	1.40648
$116$	0	0
$117$	−7.29831	−0.674729
$118$	0	0
$119$	0.702289	0.0643787
$120$	0	0
$121$	13.1882	1.19893
$122$	0	0
$123$	−38.2751	−3.45115
$124$	0	0
$125$	12.1365	1.08552
$126$	0	0
$127$	−4.73628	−0.420276	−0.210138	−	0.977672i	$-0.567391\pi$
$127$	−4.73628	−0.420276	−0.210138	+	0.977672i	$0.567391\pi$
$128$	0	0
$129$	6.66302	0.586646
$130$	0	0
$131$	15.6756	1.36959	0.684793	−	0.728738i	$-0.259893\pi$
$131$	15.6756	1.36959	0.684793	+	0.728738i	$0.259893\pi$
$132$	0	0
$133$	−8.14461	−0.706227
$134$	0	0
$135$	26.6273	2.29171
$136$	0	0
$137$	1.02007	0.0871503	0.0435752	−	0.999050i	$-0.486125\pi$
$137$	1.02007	0.0871503	0.0435752	+	0.999050i	$0.486125\pi$
$138$	0	0
$139$	−9.99810	−0.848027	−0.424014	−	0.905656i	$-0.639379\pi$
$139$	−9.99810	−0.848027	−0.424014	+	0.905656i	$0.639379\pi$
$140$	0	0
$141$	19.8123	1.66849
$142$	0	0
$143$	−4.67919	−0.391294
$144$	0	0
$145$	4.50671	0.374262
$146$	0	0
$147$	3.31466	0.273388
$148$	0	0
$149$	−19.0443	−1.56017	−0.780085	−	0.625674i	$-0.784824\pi$
$149$	−19.0443	−1.56017	−0.780085	+	0.625674i	$0.784824\pi$
$150$	0	0
$151$	−18.5778	−1.51184	−0.755921	−	0.654663i	$-0.772811\pi$
$151$	−18.5778	−1.51184	−0.755921	+	0.654663i	$0.772811\pi$
$152$	0	0
$153$	−2.20204	−0.178025
$154$	0	0
$155$	5.41059	0.434589
$156$	0	0
$157$	21.6862	1.73074	0.865372	−	0.501130i	$-0.167082\pi$
$157$	21.6862	1.73074	0.865372	+	0.501130i	$0.167082\pi$
$158$	0	0
$159$	−5.66699	−0.449421
$160$	0	0
$161$	21.1453	1.66648
$162$	0	0
$163$	−16.0080	−1.25384	−0.626922	−	0.779082i	$-0.715686\pi$
$163$	−16.0080	−1.25384	−0.626922	+	0.779082i	$0.715686\pi$
$164$	0	0
$165$	28.0361	2.18261
$166$	0	0
$167$	−6.00162	−0.464419	−0.232210	−	0.972666i	$-0.574596\pi$
$167$	−6.00162	−0.464419	−0.232210	+	0.972666i	$0.574596\pi$
$168$	0	0
$169$	−12.0948	−0.930370
$170$	0	0
$171$	25.5376	1.95291
$172$	0	0
$173$	6.21284	0.472353	0.236177	−	0.971710i	$-0.424106\pi$
$173$	6.21284	0.472353	0.236177	+	0.971710i	$0.424106\pi$
$174$	0	0
$175$	4.78225	0.361504
$176$	0	0
$177$	21.0053	1.57886
$178$	0	0
$179$	16.2855	1.21724	0.608618	−	0.793463i	$-0.291724\pi$
$179$	16.2855	1.21724	0.608618	+	0.793463i	$0.291724\pi$
$180$	0	0
$181$	10.9072	0.810729	0.405364	−	0.914155i	$-0.367145\pi$
$181$	10.9072	0.810729	0.405364	+	0.914155i	$0.367145\pi$
$182$	0	0
$183$	−31.6859	−2.34229
$184$	0	0
$185$	−6.56326	−0.482541
$186$	0	0
$187$	−1.41181	−0.103241
$188$	0	0
$189$	37.3300	2.71536
$190$	0	0
$191$	−0.155040	−0.0112183	−0.00560915	−	0.999984i	$-0.501785\pi$
$191$	−0.155040	−0.0112183	−0.00560915	+	0.999984i	$0.501785\pi$
$192$	0	0
$193$	−17.9201	−1.28992	−0.644958	−	0.764218i	$-0.723125\pi$
$193$	−17.9201	−1.28992	−0.644958	+	0.764218i	$0.723125\pi$
$194$	0	0
$195$	−5.42355	−0.388389
$196$	0	0
$197$	17.6957	1.26077	0.630385	−	0.776283i	$-0.282897\pi$
$197$	17.6957	1.26077	0.630385	+	0.776283i	$0.282897\pi$
$198$	0	0
$199$	20.9687	1.48643	0.743215	−	0.669052i	$-0.233300\pi$
$199$	20.9687	1.48643	0.743215	+	0.669052i	$0.233300\pi$
$200$	0	0
$201$	−18.7318	−1.32124
$202$	0	0
$203$	6.31816	0.443448
$204$	0	0
$205$	−20.4468	−1.42807
$206$	0	0
$207$	−66.3015	−4.60827
$208$	0	0
$209$	16.3730	1.13255
$210$	0	0
$211$	23.0958	1.58998	0.794990	−	0.606623i	$-0.207476\pi$
$211$	23.0958	1.58998	0.794990	+	0.606623i	$0.207476\pi$
$212$	0	0
$213$	51.5655	3.53321
$214$	0	0
$215$	3.55943	0.242751
$216$	0	0
$217$	7.58535	0.514927
$218$	0	0
$219$	−3.04374	−0.205677
$220$	0	0
$221$	0.273112	0.0183715
$222$	0	0
$223$	−8.13982	−0.545083	−0.272541	−	0.962144i	$-0.587864\pi$
$223$	−8.13982	−0.545083	−0.272541	+	0.962144i	$0.587864\pi$
$224$	0	0
$225$	−14.9949	−0.999658
$226$	0	0
$227$	−5.59973	−0.371667	−0.185834	−	0.982581i	$-0.559499\pi$
$227$	−5.59973	−0.371667	−0.185834	+	0.982581i	$0.559499\pi$
$228$	0	0
$229$	−15.7591	−1.04139	−0.520696	−	0.853742i	$-0.674327\pi$
$229$	−15.7591	−1.04139	−0.520696	+	0.853742i	$0.674327\pi$
$230$	0	0
$231$	39.3050	2.58608
$232$	0	0
$233$	−6.14544	−0.402601	−0.201300	−	0.979530i	$-0.564517\pi$
$233$	−6.14544	−0.402601	−0.201300	+	0.979530i	$0.564517\pi$
$234$	0	0
$235$	10.5838	0.690413
$236$	0	0
$237$	−31.7705	−2.06372
$238$	0	0
$239$	6.31840	0.408703	0.204352	−	0.978898i	$-0.434491\pi$
$239$	6.31840	0.408703	0.204352	+	0.978898i	$0.434491\pi$
$240$	0	0
$241$	−11.1648	−0.719189	−0.359594	−	0.933109i	$-0.617085\pi$
$241$	−11.1648	−0.719189	−0.359594	+	0.933109i	$0.617085\pi$
$242$	0	0
$243$	−41.8735	−2.68619
$244$	0	0
$245$	1.77071	0.113127
$246$	0	0
$247$	−3.16735	−0.201534
$248$	0	0
$249$	−33.4409	−2.11923
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−42.5081	−2.67246
$254$	0	0
$255$	−1.63639	−0.102475
$256$	0	0
$257$	−0.387607	−0.0241782	−0.0120891	−	0.999927i	$-0.503848\pi$
$257$	−0.387607	−0.0241782	−0.0120891	+	0.999927i	$0.503848\pi$
$258$	0	0
$259$	−9.20133	−0.571743
$260$	0	0
$261$	−19.8107	−1.22625
$262$	0	0
$263$	8.75286	0.539724	0.269862	−	0.962899i	$-0.413022\pi$
$263$	8.75286	0.539724	0.269862	+	0.962899i	$0.413022\pi$
$264$	0	0
$265$	−3.02734	−0.185968
$266$	0	0
$267$	−10.4382	−0.638810
$268$	0	0
$269$	−11.2602	−0.686545	−0.343273	−	0.939236i	$-0.611536\pi$
$269$	−11.2602	−0.686545	−0.343273	+	0.939236i	$0.611536\pi$
$270$	0	0
$271$	22.0106	1.33705	0.668523	−	0.743691i	$-0.266927\pi$
$271$	22.0106	1.33705	0.668523	+	0.743691i	$0.266927\pi$
$272$	0	0
$273$	−7.60353	−0.460186
$274$	0	0
$275$	−9.61371	−0.579729
$276$	0	0
$277$	−20.4216	−1.22701	−0.613507	−	0.789689i	$-0.710242\pi$
$277$	−20.4216	−1.22701	−0.613507	+	0.789689i	$0.710242\pi$
$278$	0	0
$279$	−23.7841	−1.42391
$280$	0	0
$281$	6.92654	0.413203	0.206601	−	0.978425i	$-0.433760\pi$
$281$	6.92654	0.413203	0.206601	+	0.978425i	$0.433760\pi$
$282$	0	0
$283$	24.8584	1.47768	0.738839	−	0.673882i	$-0.235374\pi$
$283$	24.8584	1.47768	0.738839	+	0.673882i	$0.235374\pi$
$284$	0	0
$285$	18.9777	1.12414
$286$	0	0
$287$	−28.6653	−1.69206
$288$	0	0
$289$	−16.9176	−0.995153
$290$	0	0
$291$	−46.5258	−2.72739
$292$	0	0
$293$	−4.22266	−0.246690	−0.123345	−	0.992364i	$-0.539362\pi$
$293$	−4.22266	−0.246690	−0.123345	+	0.992364i	$0.539362\pi$
$294$	0	0
$295$	11.2212	0.653322
$296$	0	0
$297$	−75.0442	−4.35451
$298$	0	0
$299$	8.22316	0.475557
$300$	0	0
$301$	4.99012	0.287626
$302$	0	0
$303$	−16.0108	−0.919797
$304$	0	0
$305$	−16.9268	−0.969227
$306$	0	0
$307$	−22.3878	−1.27774	−0.638868	−	0.769316i	$-0.720597\pi$
$307$	−22.3878	−1.27774	−0.638868	+	0.769316i	$0.720597\pi$
$308$	0	0
$309$	6.78168	0.385796
$310$	0	0
$311$	−13.7175	−0.777847	−0.388923	−	0.921270i	$-0.627153\pi$
$311$	−13.7175	−0.777847	−0.388923	+	0.921270i	$0.627153\pi$
$312$	0	0
$313$	−8.43072	−0.476532	−0.238266	−	0.971200i	$-0.576579\pi$
$313$	−8.43072	−0.476532	−0.238266	+	0.971200i	$0.576579\pi$
$314$	0	0
$315$	32.7498	1.84524
$316$	0	0
$317$	−6.50313	−0.365252	−0.182626	−	0.983182i	$-0.558460\pi$
$317$	−6.50313	−0.365252	−0.182626	+	0.983182i	$0.558460\pi$
$318$	0	0
$319$	−12.7013	−0.711138
$320$	0	0
$321$	−18.3088	−1.02190
$322$	0	0
$323$	−0.955652	−0.0531739
$324$	0	0
$325$	1.85976	0.103161
$326$	0	0
$327$	12.4562	0.688829
$328$	0	0
$329$	14.8380	0.818043
$330$	0	0
$331$	−1.45593	−0.0800252	−0.0400126	−	0.999199i	$-0.512740\pi$
$331$	−1.45593	−0.0800252	−0.0400126	+	0.999199i	$0.512740\pi$
$332$	0	0
$333$	28.8510	1.58103
$334$	0	0
$335$	−10.0067	−0.546722
$336$	0	0
$337$	−29.0591	−1.58295	−0.791474	−	0.611203i	$-0.790686\pi$
$337$	−29.0591	−1.58295	−0.791474	+	0.611203i	$0.790686\pi$
$338$	0	0
$339$	−25.8886	−1.40607
$340$	0	0
$341$	−15.2488	−0.825767
$342$	0	0
$343$	19.6079	1.05872
$344$	0	0
$345$	−49.2703	−2.65262
$346$	0	0
$347$	−34.6311	−1.85909	−0.929547	−	0.368704i	$-0.879802\pi$
$347$	−34.6311	−1.85909	−0.929547	+	0.368704i	$0.879802\pi$
$348$	0	0
$349$	17.7590	0.950615	0.475307	−	0.879820i	$-0.342337\pi$
$349$	17.7590	0.950615	0.475307	+	0.879820i	$0.342337\pi$
$350$	0	0
$351$	14.5172	0.774873
$352$	0	0
$353$	−18.2176	−0.969624	−0.484812	−	0.874618i	$-0.661112\pi$
$353$	−18.2176	−0.969624	−0.484812	+	0.874618i	$0.661112\pi$
$354$	0	0
$355$	27.5466	1.46202
$356$	0	0
$357$	−2.29413	−0.121418
$358$	0	0
$359$	9.81251	0.517885	0.258942	−	0.965893i	$-0.416626\pi$
$359$	9.81251	0.517885	0.258942	+	0.965893i	$0.416626\pi$
$360$	0	0
$361$	−7.91707	−0.416688
$362$	0	0
$363$	−43.0814	−2.26119
$364$	0	0
$365$	−1.62598	−0.0851079
$366$	0	0
$367$	31.2411	1.63077	0.815386	−	0.578918i	$-0.196525\pi$
$367$	31.2411	1.63077	0.815386	+	0.578918i	$0.196525\pi$
$368$	0	0
$369$	89.8808	4.67901
$370$	0	0
$371$	−4.24417	−0.220346
$372$	0	0
$373$	−9.68456	−0.501448	−0.250724	−	0.968059i	$-0.580669\pi$
$373$	−9.68456	−0.501448	−0.250724	+	0.968059i	$0.580669\pi$
$374$	0	0
$375$	−39.6457	−2.04730
$376$	0	0
$377$	2.45706	0.126545
$378$	0	0
$379$	−34.0227	−1.74763	−0.873813	−	0.486261i	$-0.838360\pi$
$379$	−34.0227	−1.74763	−0.873813	+	0.486261i	$0.838360\pi$
$380$	0	0
$381$	15.4718	0.792643
$382$	0	0
$383$	27.1812	1.38890	0.694448	−	0.719543i	$-0.255649\pi$
$383$	27.1812	1.38890	0.694448	+	0.719543i	$0.255649\pi$
$384$	0	0
$385$	20.9970	1.07011
$386$	0	0
$387$	−15.6467	−0.795364
$388$	0	0
$389$	1.20740	0.0612176	0.0306088	−	0.999531i	$-0.490255\pi$
$389$	1.20740	0.0612176	0.0306088	+	0.999531i	$0.490255\pi$
$390$	0	0
$391$	2.48109	0.125474
$392$	0	0
$393$	−51.2068	−2.58304
$394$	0	0
$395$	−16.9720	−0.853955
$396$	0	0
$397$	9.47528	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$397$	9.47528	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$398$	0	0
$399$	26.6056	1.33195
$400$	0	0
$401$	−19.8636	−0.991939	−0.495969	−	0.868340i	$-0.665187\pi$
$401$	−19.8636	−0.991939	−0.495969	+	0.868340i	$0.665187\pi$
$402$	0	0
$403$	2.94986	0.146943
$404$	0	0
$405$	−46.8229	−2.32665
$406$	0	0
$407$	18.4974	0.916880
$408$	0	0
$409$	27.2327	1.34657	0.673284	−	0.739384i	$-0.264883\pi$
$409$	27.2327	1.34657	0.673284	+	0.739384i	$0.264883\pi$
$410$	0	0
$411$	−3.33221	−0.164366
$412$	0	0
$413$	15.7315	0.774095
$414$	0	0
$415$	−17.8643	−0.876926
$416$	0	0
$417$	32.6603	1.59938
$418$	0	0
$419$	−7.70570	−0.376448	−0.188224	−	0.982126i	$-0.560273\pi$
$419$	−7.70570	−0.376448	−0.188224	+	0.982126i	$0.560273\pi$
$420$	0	0
$421$	−1.20612	−0.0587829	−0.0293915	−	0.999568i	$-0.509357\pi$
$421$	−1.20612	−0.0587829	−0.0293915	+	0.999568i	$0.509357\pi$
$422$	0	0
$423$	−46.5248	−2.26211
$424$	0	0
$425$	0.561128	0.0272187
$426$	0	0
$427$	−23.7305	−1.14840
$428$	0	0
$429$	15.2853	0.737981
$430$	0	0
$431$	3.53540	0.170294	0.0851471	−	0.996368i	$-0.472864\pi$
$431$	3.53540	0.170294	0.0851471	+	0.996368i	$0.472864\pi$
$432$	0	0
$433$	14.9773	0.719764	0.359882	−	0.932998i	$-0.382817\pi$
$433$	14.9773	0.719764	0.359882	+	0.932998i	$0.382817\pi$
$434$	0	0
$435$	−14.7219	−0.705859
$436$	0	0
$437$	−28.7738	−1.37644
$438$	0	0
$439$	6.09414	0.290857	0.145429	−	0.989369i	$-0.453544\pi$
$439$	6.09414	0.290857	0.145429	+	0.989369i	$0.453544\pi$
$440$	0	0
$441$	−7.78375	−0.370655
$442$	0	0
$443$	22.4404	1.06617	0.533087	−	0.846060i	$-0.321032\pi$
$443$	22.4404	1.06617	0.533087	+	0.846060i	$0.321032\pi$
$444$	0	0
$445$	−5.57617	−0.264336
$446$	0	0
$447$	62.2111	2.94249
$448$	0	0
$449$	15.0819	0.711760	0.355880	−	0.934532i	$-0.384181\pi$
$449$	15.0819	0.711760	0.355880	+	0.934532i	$0.384181\pi$
$450$	0	0
$451$	57.6256	2.71348
$452$	0	0
$453$	60.6873	2.85134
$454$	0	0
$455$	−4.06185	−0.190423
$456$	0	0
$457$	−35.8363	−1.67635	−0.838176	−	0.545399i	$-0.816378\pi$
$457$	−35.8363	−1.67635	−0.838176	+	0.545399i	$0.816378\pi$
$458$	0	0
$459$	4.38014	0.204447
$460$	0	0
$461$	7.07190	0.329371	0.164686	−	0.986346i	$-0.447339\pi$
$461$	7.07190	0.329371	0.164686	+	0.986346i	$0.447339\pi$
$462$	0	0
$463$	−18.4908	−0.859340	−0.429670	−	0.902986i	$-0.641370\pi$
$463$	−18.4908	−0.859340	−0.429670	+	0.902986i	$0.641370\pi$
$464$	0	0
$465$	−17.6745	−0.819637
$466$	0	0
$467$	−31.8202	−1.47246	−0.736231	−	0.676731i	$-0.763396\pi$
$467$	−31.8202	−1.47246	−0.736231	+	0.676731i	$0.763396\pi$
$468$	0	0
$469$	−14.0288	−0.647789
$470$	0	0
$471$	−70.8412	−3.26419
$472$	0	0
$473$	−10.0316	−0.461253
$474$	0	0
$475$	−6.50753	−0.298586
$476$	0	0
$477$	13.3077	0.609317
$478$	0	0
$479$	−25.0651	−1.14525	−0.572627	−	0.819816i	$-0.694076\pi$
$479$	−25.0651	−1.14525	−0.572627	+	0.819816i	$0.694076\pi$
$480$	0	0
$481$	−3.57830	−0.163156
$482$	0	0
$483$	−69.0742	−3.14299
$484$	0	0
$485$	−24.8544	−1.12858
$486$	0	0
$487$	25.8059	1.16938	0.584688	−	0.811258i	$-0.301217\pi$
$487$	25.8059	1.16938	0.584688	+	0.811258i	$0.301217\pi$
$488$	0	0
$489$	52.2926	2.36475
$490$	0	0
$491$	35.4957	1.60190	0.800949	−	0.598733i	$-0.204329\pi$
$491$	35.4957	1.60190	0.800949	+	0.598733i	$0.204329\pi$
$492$	0	0
$493$	0.741344	0.0333885
$494$	0	0
$495$	−65.8366	−2.95914
$496$	0	0
$497$	38.6188	1.73229
$498$	0	0
$499$	−7.24948	−0.324531	−0.162266	−	0.986747i	$-0.551880\pi$
$499$	−7.24948	−0.324531	−0.162266	+	0.986747i	$0.551880\pi$
$500$	0	0
$501$	19.6052	0.875896
$502$	0	0
$503$	−14.4334	−0.643556	−0.321778	−	0.946815i	$-0.604280\pi$
$503$	−14.4334	−0.643556	−0.321778	+	0.946815i	$0.604280\pi$
$504$	0	0
$505$	−8.55308	−0.380607
$506$	0	0
$507$	39.5096	1.75468
$508$	0	0
$509$	20.5364	0.910259	0.455129	−	0.890425i	$-0.349593\pi$
$509$	20.5364	0.910259	0.455129	+	0.890425i	$0.349593\pi$
$510$	0	0
$511$	−2.27954	−0.100841
$512$	0	0
$513$	−50.7975	−2.24277
$514$	0	0
$515$	3.62282	0.159640
$516$	0	0
$517$	−29.8286	−1.31186
$518$	0	0
$519$	−20.2952	−0.890860
$520$	0	0
$521$	17.0592	0.747377	0.373689	−	0.927554i	$-0.378093\pi$
$521$	17.0592	0.747377	0.373689	+	0.927554i	$0.378093\pi$
$522$	0	0
$523$	−22.9344	−1.00285	−0.501426	−	0.865200i	$-0.667191\pi$
$523$	−22.9344	−1.00285	−0.501426	+	0.865200i	$0.667191\pi$
$524$	0	0
$525$	−15.6220	−0.681798
$526$	0	0
$527$	0.890031	0.0387704
$528$	0	0
$529$	51.7033	2.24797
$530$	0	0
$531$	−49.3264	−2.14058
$532$	0	0
$533$	−11.1476	−0.482857
$534$	0	0
$535$	−9.78067	−0.422855
$536$	0	0
$537$	−53.1991	−2.29571
$538$	0	0
$539$	−4.99043	−0.214953
$540$	0	0
$541$	29.7711	1.27996	0.639980	−	0.768391i	$-0.278942\pi$
$541$	29.7711	1.27996	0.639980	+	0.768391i	$0.278942\pi$
$542$	0	0
$543$	−35.6302	−1.52904
$544$	0	0
$545$	6.65418	0.285034
$546$	0	0
$547$	36.3643	1.55483	0.777413	−	0.628990i	$-0.216531\pi$
$547$	36.3643	1.55483	0.777413	+	0.628990i	$0.216531\pi$
$548$	0	0
$549$	74.4075	3.17563
$550$	0	0
$551$	−8.59755	−0.366268
$552$	0	0
$553$	−23.7938	−1.01182
$554$	0	0
$555$	21.4399	0.910073
$556$	0	0
$557$	6.93370	0.293790	0.146895	−	0.989152i	$-0.453072\pi$
$557$	6.93370	0.293790	0.146895	+	0.989152i	$0.453072\pi$
$558$	0	0
$559$	1.94060	0.0820788
$560$	0	0
$561$	4.61188	0.194714
$562$	0	0
$563$	−24.8361	−1.04672	−0.523358	−	0.852113i	$-0.675321\pi$
$563$	−24.8361	−1.04672	−0.523358	+	0.852113i	$0.675321\pi$
$564$	0	0
$565$	−13.8298	−0.581826
$566$	0	0
$567$	−65.6431	−2.75675
$568$	0	0
$569$	−28.7230	−1.20413	−0.602066	−	0.798446i	$-0.705656\pi$
$569$	−28.7230	−1.20413	−0.602066	+	0.798446i	$0.705656\pi$
$570$	0	0
$571$	10.6850	0.447152	0.223576	−	0.974686i	$-0.428227\pi$
$571$	10.6850	0.447152	0.223576	+	0.974686i	$0.428227\pi$
$572$	0	0
$573$	0.506462	0.0211578
$574$	0	0
$575$	16.8950	0.704571
$576$	0	0
$577$	32.7816	1.36472	0.682358	−	0.731018i	$-0.260954\pi$
$577$	32.7816	1.36472	0.682358	+	0.731018i	$0.260954\pi$
$578$	0	0
$579$	58.5387	2.43279
$580$	0	0
$581$	−25.0448	−1.03903
$582$	0	0
$583$	8.53201	0.353360
$584$	0	0
$585$	12.7360	0.526570
$586$	0	0
$587$	−47.0046	−1.94009	−0.970043	−	0.242934i	$-0.921890\pi$
$587$	−47.0046	−1.94009	−0.970043	+	0.242934i	$0.921890\pi$
$588$	0	0
$589$	−10.3219	−0.425307
$590$	0	0
$591$	−57.8059	−2.37782
$592$	0	0
$593$	−5.69539	−0.233882	−0.116941	−	0.993139i	$-0.537309\pi$
$593$	−5.69539	−0.233882	−0.116941	+	0.993139i	$0.537309\pi$
$594$	0	0
$595$	−1.22554	−0.0502423
$596$	0	0
$597$	−68.4974	−2.80341
$598$	0	0
$599$	−43.6444	−1.78326	−0.891631	−	0.452762i	$-0.850439\pi$
$599$	−43.6444	−1.78326	−0.891631	+	0.452762i	$0.850439\pi$
$600$	0	0
$601$	−41.2531	−1.68275	−0.841375	−	0.540452i	$-0.818253\pi$
$601$	−41.2531	−1.68275	−0.841375	+	0.540452i	$0.818253\pi$
$602$	0	0
$603$	43.9876	1.79131
$604$	0	0
$605$	−23.0144	−0.935666
$606$	0	0
$607$	−2.10730	−0.0855327	−0.0427663	−	0.999085i	$-0.513617\pi$
$607$	−2.10730	−0.0855327	−0.0427663	+	0.999085i	$0.513617\pi$
$608$	0	0
$609$	−20.6392	−0.836344
$610$	0	0
$611$	5.77032	0.233442
$612$	0	0
$613$	−33.0176	−1.33357	−0.666784	−	0.745251i	$-0.732329\pi$
$613$	−33.0176	−1.33357	−0.666784	+	0.745251i	$0.732329\pi$
$614$	0	0
$615$	66.7926	2.69334
$616$	0	0
$617$	−10.4198	−0.419485	−0.209742	−	0.977757i	$-0.567263\pi$
$617$	−10.4198	−0.419485	−0.209742	+	0.977757i	$0.567263\pi$
$618$	0	0
$619$	8.07396	0.324520	0.162260	−	0.986748i	$-0.448122\pi$
$619$	8.07396	0.324520	0.162260	+	0.986748i	$0.448122\pi$
$620$	0	0
$621$	131.882	5.29224
$622$	0	0
$623$	−7.81749	−0.313201
$624$	0	0
$625$	−11.4053	−0.456211
$626$	0	0
$627$	−53.4851	−2.13599
$628$	0	0
$629$	−1.07964	−0.0430482
$630$	0	0
$631$	10.6988	0.425913	0.212956	−	0.977062i	$-0.431691\pi$
$631$	10.6988	0.425913	0.212956	+	0.977062i	$0.431691\pi$
$632$	0	0
$633$	−75.4459	−2.99871
$634$	0	0
$635$	8.26512	0.327991
$636$	0	0
$637$	0.965393	0.0382503
$638$	0	0
$639$	−121.090	−4.79026
$640$	0	0
$641$	26.4145	1.04331	0.521655	−	0.853157i	$-0.325315\pi$
$641$	26.4145	1.04331	0.521655	+	0.853157i	$0.325315\pi$
$642$	0	0
$643$	14.1368	0.557501	0.278750	−	0.960364i	$-0.410080\pi$
$643$	14.1368	0.557501	0.278750	+	0.960364i	$0.410080\pi$
$644$	0	0
$645$	−11.6274	−0.457829
$646$	0	0
$647$	9.78214	0.384576	0.192288	−	0.981339i	$-0.438409\pi$
$647$	9.78214	0.384576	0.192288	+	0.981339i	$0.438409\pi$
$648$	0	0
$649$	−31.6248	−1.24138
$650$	0	0
$651$	−24.7787	−0.971154
$652$	0	0
$653$	−30.4280	−1.19074	−0.595371	−	0.803451i	$-0.702995\pi$
$653$	−30.4280	−1.19074	−0.595371	+	0.803451i	$0.702995\pi$
$654$	0	0
$655$	−27.3550	−1.06885
$656$	0	0
$657$	7.14756	0.278853
$658$	0	0
$659$	−22.8622	−0.890584	−0.445292	−	0.895385i	$-0.646900\pi$
$659$	−22.8622	−0.890584	−0.445292	+	0.895385i	$0.646900\pi$
$660$	0	0
$661$	−32.5628	−1.26655	−0.633273	−	0.773929i	$-0.718289\pi$
$661$	−32.5628	−1.26655	−0.633273	+	0.773929i	$0.718289\pi$
$662$	0	0
$663$	−0.892164	−0.0346488
$664$	0	0
$665$	14.2129	0.551152
$666$	0	0
$667$	22.3212	0.864279
$668$	0	0
$669$	26.5900	1.02803
$670$	0	0
$671$	47.7052	1.84164
$672$	0	0
$673$	3.76363	0.145077	0.0725386	−	0.997366i	$-0.476890\pi$
$673$	3.76363	0.145077	0.0725386	+	0.997366i	$0.476890\pi$
$674$	0	0
$675$	29.8266	1.14803
$676$	0	0
$677$	26.5725	1.02127	0.510633	−	0.859799i	$-0.329411\pi$
$677$	26.5725	1.02127	0.510633	+	0.859799i	$0.329411\pi$
$678$	0	0
$679$	−34.8445	−1.33721
$680$	0	0
$681$	18.2924	0.700966
$682$	0	0
$683$	−23.3625	−0.893943	−0.446971	−	0.894548i	$-0.647497\pi$
$683$	−23.3625	−0.893943	−0.446971	+	0.894548i	$0.647497\pi$
$684$	0	0
$685$	−1.78009	−0.0680137
$686$	0	0
$687$	51.4796	1.96407
$688$	0	0
$689$	−1.65051	−0.0628794
$690$	0	0
$691$	16.2609	0.618594	0.309297	−	0.950965i	$-0.399906\pi$
$691$	16.2609	0.618594	0.309297	+	0.950965i	$0.399906\pi$
$692$	0	0
$693$	−92.2993	−3.50616
$694$	0	0
$695$	17.4474	0.661816
$696$	0	0
$697$	−3.36346	−0.127400
$698$	0	0
$699$	20.0750	0.759307
$700$	0	0
$701$	−1.07308	−0.0405297	−0.0202649	−	0.999795i	$-0.506451\pi$
$701$	−1.07308	−0.0405297	−0.0202649	+	0.999795i	$0.506451\pi$
$702$	0	0
$703$	12.5209	0.472234
$704$	0	0
$705$	−34.5737	−1.30212
$706$	0	0
$707$	−11.9909	−0.450966
$708$	0	0
$709$	−22.0806	−0.829254	−0.414627	−	0.909992i	$-0.636088\pi$
$709$	−22.0806	−0.829254	−0.414627	+	0.909992i	$0.636088\pi$
$710$	0	0
$711$	74.6062	2.79795
$712$	0	0
$713$	26.7980	1.00359
$714$	0	0
$715$	8.16550	0.305373
$716$	0	0
$717$	−20.6400	−0.770815
$718$	0	0
$719$	−47.9072	−1.78664	−0.893319	−	0.449422i	$-0.851630\pi$
$719$	−47.9072	−1.78664	−0.893319	+	0.449422i	$0.851630\pi$
$720$	0	0
$721$	5.07899	0.189152
$722$	0	0
$723$	36.4716	1.35639
$724$	0	0
$725$	5.04820	0.187485
$726$	0	0
$727$	44.4908	1.65007	0.825037	−	0.565079i	$-0.191154\pi$
$727$	44.4908	1.65007	0.825037	+	0.565079i	$0.191154\pi$
$728$	0	0
$729$	56.2916	2.08488
$730$	0	0
$731$	0.585519	0.0216562
$732$	0	0
$733$	10.8946	0.402403	0.201201	−	0.979550i	$-0.435515\pi$
$733$	10.8946	0.402403	0.201201	+	0.979550i	$0.435515\pi$
$734$	0	0
$735$	−5.78430	−0.213357
$736$	0	0
$737$	28.2019	1.03883
$738$	0	0
$739$	−11.5658	−0.425453	−0.212727	−	0.977112i	$-0.568234\pi$
$739$	−11.5658	−0.425453	−0.212727	+	0.977112i	$0.568234\pi$
$740$	0	0
$741$	10.3466	0.380093
$742$	0	0
$743$	22.2502	0.816281	0.408141	−	0.912919i	$-0.366177\pi$
$743$	22.2502	0.816281	0.408141	+	0.912919i	$0.366177\pi$
$744$	0	0
$745$	33.2336	1.21758
$746$	0	0
$747$	78.5287	2.87321
$748$	0	0
$749$	−13.7120	−0.501024
$750$	0	0
$751$	14.8602	0.542257	0.271129	−	0.962543i	$-0.412603\pi$
$751$	14.8602	0.542257	0.271129	+	0.962543i	$0.412603\pi$
$752$	0	0
$753$	3.26665	0.119043
$754$	0	0
$755$	32.4195	1.17987
$756$	0	0
$757$	−1.91370	−0.0695547	−0.0347774	−	0.999395i	$-0.511072\pi$
$757$	−1.91370	−0.0695547	−0.0347774	+	0.999395i	$0.511072\pi$
$758$	0	0
$759$	138.859	5.04027
$760$	0	0
$761$	−38.0636	−1.37980	−0.689902	−	0.723903i	$-0.742346\pi$
$761$	−38.0636	−1.37980	−0.689902	+	0.723903i	$0.742346\pi$
$762$	0	0
$763$	9.32879	0.337725
$764$	0	0
$765$	3.84272	0.138934
$766$	0	0
$767$	6.11779	0.220901
$768$	0	0
$769$	−23.6169	−0.851648	−0.425824	−	0.904806i	$-0.640016\pi$
$769$	−23.6169	−0.851648	−0.425824	+	0.904806i	$0.640016\pi$
$770$	0	0
$771$	1.26618	0.0456002
$772$	0	0
$773$	−21.8779	−0.786893	−0.393446	−	0.919348i	$-0.628717\pi$
$773$	−21.8779	−0.786893	−0.393446	+	0.919348i	$0.628717\pi$
$774$	0	0
$775$	6.06068	0.217706
$776$	0	0
$777$	30.0576	1.07831
$778$	0	0
$779$	39.0068	1.39756
$780$	0	0
$781$	−77.6351	−2.77800
$782$	0	0
$783$	39.4060	1.40826
$784$	0	0
$785$	−37.8438	−1.35070
$786$	0	0
$787$	0.900271	0.0320912	0.0160456	−	0.999871i	$-0.494892\pi$
$787$	0.900271	0.0320912	0.0160456	+	0.999871i	$0.494892\pi$
$788$	0	0
$789$	−28.5926	−1.01792
$790$	0	0
$791$	−19.3887	−0.689382
$792$	0	0
$793$	−9.22852	−0.327714
$794$	0	0
$795$	9.88928	0.350736
$796$	0	0
$797$	5.31131	0.188136	0.0940681	−	0.995566i	$-0.470013\pi$
$797$	5.31131	0.188136	0.0940681	+	0.995566i	$0.470013\pi$
$798$	0	0
$799$	1.74102	0.0615928
$800$	0	0
$801$	24.5119	0.866086
$802$	0	0
$803$	4.58254	0.161714
$804$	0	0
$805$	−36.8999	−1.30055
$806$	0	0
$807$	36.7831	1.29483
$808$	0	0
$809$	45.9316	1.61487	0.807434	−	0.589958i	$-0.200856\pi$
$809$	45.9316	1.61487	0.807434	+	0.589958i	$0.200856\pi$
$810$	0	0
$811$	−4.56332	−0.160240	−0.0801199	−	0.996785i	$-0.525530\pi$
$811$	−4.56332	−0.160240	−0.0801199	+	0.996785i	$0.525530\pi$
$812$	0	0
$813$	−71.9009	−2.52167
$814$	0	0
$815$	27.9351	0.978523
$816$	0	0
$817$	−6.79040	−0.237566
$818$	0	0
$819$	17.8552	0.623912
$820$	0	0
$821$	−7.13600	−0.249048	−0.124524	−	0.992217i	$-0.539740\pi$
$821$	−7.13600	−0.249048	−0.124524	+	0.992217i	$0.539740\pi$
$822$	0	0
$823$	−50.4114	−1.75723	−0.878616	−	0.477530i	$-0.841532\pi$
$823$	−50.4114	−1.75723	−0.878616	+	0.477530i	$0.841532\pi$
$824$	0	0
$825$	31.4047	1.09337
$826$	0	0
$827$	−33.8612	−1.17747	−0.588735	−	0.808326i	$-0.700374\pi$
$827$	−33.8612	−1.17747	−0.588735	+	0.808326i	$0.700374\pi$
$828$	0	0
$829$	−54.5515	−1.89465	−0.947326	−	0.320271i	$-0.896226\pi$
$829$	−54.5515	−1.89465	−0.947326	+	0.320271i	$0.896226\pi$
$830$	0	0
$831$	66.7102	2.31415
$832$	0	0
$833$	0.291278	0.0100922
$834$	0	0
$835$	10.4732	0.362441
$836$	0	0
$837$	47.3094	1.63525
$838$	0	0
$839$	−20.6065	−0.711416	−0.355708	−	0.934597i	$-0.615760\pi$
$839$	−20.6065	−0.711416	−0.355708	+	0.934597i	$0.615760\pi$
$840$	0	0
$841$	−22.3305	−0.770017
$842$	0	0
$843$	−22.6266	−0.779301
$844$	0	0
$845$	21.1063	0.726078
$846$	0	0
$847$	−32.2649	−1.10863
$848$	0	0
$849$	−81.2037	−2.78691
$850$	0	0
$851$	−32.5070	−1.11433
$852$	0	0
$853$	20.2163	0.692194	0.346097	−	0.938199i	$-0.387507\pi$
$853$	20.2163	0.692194	0.346097	+	0.938199i	$0.387507\pi$
$854$	0	0
$855$	−44.5649	−1.52409
$856$	0	0
$857$	24.7015	0.843788	0.421894	−	0.906645i	$-0.361365\pi$
$857$	24.7015	0.843788	0.421894	+	0.906645i	$0.361365\pi$
$858$	0	0
$859$	−6.14590	−0.209695	−0.104848	−	0.994488i	$-0.533436\pi$
$859$	−6.14590	−0.209695	−0.104848	+	0.994488i	$0.533436\pi$
$860$	0	0
$861$	93.6396	3.19123
$862$	0	0
$863$	45.1276	1.53616	0.768081	−	0.640353i	$-0.221212\pi$
$863$	45.1276	1.53616	0.768081	+	0.640353i	$0.221212\pi$
$864$	0	0
$865$	−10.8418	−0.368633
$866$	0	0
$867$	55.2639	1.87686
$868$	0	0
$869$	47.8325	1.62261
$870$	0	0
$871$	−5.45564	−0.184857
$872$	0	0
$873$	109.256	3.69775
$874$	0	0
$875$	−29.6918	−1.00377
$876$	0	0
$877$	44.4816	1.50204	0.751018	−	0.660281i	$-0.229563\pi$
$877$	44.4816	1.50204	0.751018	+	0.660281i	$0.229563\pi$
$878$	0	0
$879$	13.7940	0.465259
$880$	0	0
$881$	−8.09272	−0.272651	−0.136325	−	0.990664i	$-0.543529\pi$
$881$	−8.09272	−0.272651	−0.136325	+	0.990664i	$0.543529\pi$
$882$	0	0
$883$	36.0878	1.21445	0.607225	−	0.794530i	$-0.292283\pi$
$883$	36.0878	1.21445	0.607225	+	0.794530i	$0.292283\pi$
$884$	0	0
$885$	−36.6557	−1.23217
$886$	0	0
$887$	−23.0987	−0.775580	−0.387790	−	0.921748i	$-0.626761\pi$
$887$	−23.0987	−0.775580	−0.387790	+	0.921748i	$0.626761\pi$
$888$	0	0
$889$	11.5872	0.388624
$890$	0	0
$891$	131.962	4.42088
$892$	0	0
$893$	−20.1910	−0.675667
$894$	0	0
$895$	−28.4193	−0.949953
$896$	0	0
$897$	−26.8622	−0.896903
$898$	0	0
$899$	8.00718	0.267055
$900$	0	0
$901$	−0.497992	−0.0165905
$902$	0	0
$903$	−16.3010	−0.542463
$904$	0	0
$905$	−19.0339	−0.632707
$906$	0	0
$907$	−3.94419	−0.130965	−0.0654824	−	0.997854i	$-0.520859\pi$
$907$	−3.94419	−0.130965	−0.0654824	+	0.997854i	$0.520859\pi$
$908$	0	0
$909$	37.5979	1.24704
$910$	0	0
$911$	−45.1958	−1.49740	−0.748702	−	0.662907i	$-0.769322\pi$
$911$	−45.1958	−1.49740	−0.748702	+	0.662907i	$0.769322\pi$
$912$	0	0
$913$	50.3474	1.66625
$914$	0	0
$915$	55.2941	1.82797
$916$	0	0
$917$	−38.3502	−1.26644
$918$	0	0
$919$	47.7457	1.57498	0.787492	−	0.616324i	$-0.211379\pi$
$919$	47.7457	1.57498	0.787492	+	0.616324i	$0.211379\pi$
$920$	0	0
$921$	73.1330	2.40982
$922$	0	0
$923$	15.0184	0.494338
$924$	0	0
$925$	−7.35185	−0.241727
$926$	0	0
$927$	−15.9253	−0.523056
$928$	0	0
$929$	45.6456	1.49759	0.748793	−	0.662804i	$-0.230634\pi$
$929$	45.6456	1.49759	0.748793	+	0.662804i	$0.230634\pi$
$930$	0	0
$931$	−3.37802	−0.110710
$932$	0	0
$933$	44.8103	1.46702
$934$	0	0
$935$	2.46369	0.0805714
$936$	0	0
$937$	−32.6680	−1.06722	−0.533608	−	0.845732i	$-0.679164\pi$
$937$	−32.6680	−1.06722	−0.533608	+	0.845732i	$0.679164\pi$
$938$	0	0
$939$	27.5402	0.898741
$940$	0	0
$941$	−24.6353	−0.803087	−0.401543	−	0.915840i	$-0.631526\pi$
$941$	−24.6353	−0.803087	−0.401543	+	0.915840i	$0.631526\pi$
$942$	0	0
$943$	−101.271	−3.29782
$944$	0	0
$945$	−65.1434	−2.11912
$946$	0	0
$947$	−52.0922	−1.69277	−0.846385	−	0.532571i	$-0.821226\pi$
$947$	−52.0922	−1.69277	−0.846385	+	0.532571i	$0.821226\pi$
$948$	0	0
$949$	−0.886488	−0.0287766
$950$	0	0
$951$	21.2435	0.688867
$952$	0	0
$953$	42.4727	1.37583	0.687913	−	0.725793i	$-0.258527\pi$
$953$	42.4727	1.37583	0.687913	+	0.725793i	$0.258527\pi$
$954$	0	0
$955$	0.270555	0.00875497
$956$	0	0
$957$	41.4909	1.34121
$958$	0	0
$959$	−2.49559	−0.0805867
$960$	0	0
$961$	−21.3869	−0.689899
$962$	0	0
$963$	42.9942	1.38547
$964$	0	0
$965$	31.2718	1.00667
$966$	0	0
$967$	−35.2776	−1.13445	−0.567226	−	0.823562i	$-0.691984\pi$
$967$	−35.2776	−1.13445	−0.567226	+	0.823562i	$0.691984\pi$
$968$	0	0
$969$	3.12179	0.100286
$970$	0	0
$971$	−40.8393	−1.31059	−0.655297	−	0.755371i	$-0.727457\pi$
$971$	−40.8393	−1.31059	−0.655297	+	0.755371i	$0.727457\pi$
$972$	0	0
$973$	24.4602	0.784159
$974$	0	0
$975$	−6.07521	−0.194562
$976$	0	0
$977$	−28.2186	−0.902792	−0.451396	−	0.892324i	$-0.649074\pi$
$977$	−28.2186	−0.902792	−0.451396	+	0.892324i	$0.649074\pi$
$978$	0	0
$979$	15.7154	0.502267
$980$	0	0
$981$	−29.2507	−0.933902
$982$	0	0
$983$	14.5450	0.463913	0.231956	−	0.972726i	$-0.425487\pi$
$983$	14.5450	0.463913	0.231956	+	0.972726i	$0.425487\pi$
$984$	0	0
$985$	−30.8803	−0.983927
$986$	0	0
$987$	−48.4705	−1.54283
$988$	0	0
$989$	17.6294	0.560583
$990$	0	0
$991$	−7.52353	−0.238993	−0.119496	−	0.992835i	$-0.538128\pi$
$991$	−7.52353	−0.238993	−0.119496	+	0.992835i	$0.538128\pi$
$992$	0	0
$993$	4.75602	0.150928
$994$	0	0
$995$	−36.5918	−1.16004
$996$	0	0
$997$	−10.3616	−0.328156	−0.164078	−	0.986447i	$-0.552465\pi$
$997$	−10.3616	−0.328156	−0.164078	+	0.986447i	$0.552465\pi$
$998$	0	0
$999$	−57.3882	−1.81568

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.h.1.1	✓	30
4.3	odd	2		8032.2.a.i.1.30	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.h.1.1	✓	30	1.1	even	1	trivial
8032.2.a.i.1.30	yes	30	4.3	odd	2

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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-3.26665 q^{3} -1.74507 q^{5} -2.44649 q^{7} +7.67103 q^{9} +O(q^{10})\)	\(q-3.26665 q^{3} -1.74507 q^{5} -2.44649 q^{7} +7.67103 q^{9} +4.91815 q^{11} -0.951412 q^{13} +5.70053 q^{15} -0.287060 q^{17} +3.32910 q^{19} +7.99183 q^{21} -8.64311 q^{23} -1.95474 q^{25} -15.2586 q^{27} -2.58254 q^{29} -3.10051 q^{31} -16.0659 q^{33} +4.26929 q^{35} +3.76104 q^{37} +3.10793 q^{39} +11.7169 q^{41} -2.03971 q^{43} -13.3865 q^{45} -6.06500 q^{47} -1.01470 q^{49} +0.937726 q^{51} +1.73480 q^{53} -8.58251 q^{55} -10.8750 q^{57} -6.43022 q^{59} +9.69981 q^{61} -18.7671 q^{63} +1.66028 q^{65} +5.73425 q^{67} +28.2340 q^{69} -15.7854 q^{71} +0.931761 q^{73} +6.38546 q^{75} -12.0322 q^{77} +9.72571 q^{79} +26.8316 q^{81} +10.2370 q^{83} +0.500939 q^{85} +8.43627 q^{87} +3.19539 q^{89} +2.32762 q^{91} +10.1283 q^{93} -5.80951 q^{95} +14.2427 q^{97} +37.7273 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9	\( 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9} - 13 q^{11} - 7 q^{13} - 28 q^{15} - 9 q^{17} + 17 q^{19} - 6 q^{21} - 43 q^{23} + 34 q^{25} - 12 q^{27} - q^{29} - 39 q^{31} - 17 q^{35} - q^{37} - 48 q^{39} - 3 q^{41} + 19 q^{43} - 66 q^{47} + 25 q^{49} + 14 q^{51} + 3 q^{53} - 50 q^{55} - 14 q^{57} - 27 q^{59} + 15 q^{61} - 75 q^{63} - 6 q^{65} - 8 q^{67} + 18 q^{69} - 64 q^{71} - 15 q^{73} - 9 q^{75} - 71 q^{79} + 6 q^{81} - 60 q^{83} + 15 q^{85} - 64 q^{87} - 32 q^{89} + 26 q^{91} - 4 q^{93} - 72 q^{95} - 4 q^{97} - 13 q^{99}+O(q^{100}) \) 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9 - 13 * q^11 - 7 * q^13 - 28 * q^15 - 9 * q^17 + 17 * q^19 - 6 * q^21 - 43 * q^23 + 34 * q^25 - 12 * q^27 - q^29 - 39 * q^31 - 17 * q^35 - q^37 - 48 * q^39 - 3 * q^41 + 19 * q^43 - 66 * q^47 + 25 * q^49 + 14 * q^51 + 3 * q^53 - 50 * q^55 - 14 * q^57 - 27 * q^59 + 15 * q^61 - 75 * q^63 - 6 * q^65 - 8 * q^67 + 18 * q^69 - 64 * q^71 - 15 * q^73 - 9 * q^75 - 71 * q^79 + 6 * q^81 - 60 * q^83 + 15 * q^85 - 64 * q^87 - 32 * q^89 + 26 * q^91 - 4 * q^93 - 72 * q^95 - 4 * q^97 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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