LMFDB - Embedded newform 8032.2.a.i.1.9

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:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-1.69973 q^{3} +2.39251 q^{5} -2.42748 q^{7} -0.110913 q^{9} +O(q^{10})$	$q-1.69973 q^{3} +2.39251 q^{5} -2.42748 q^{7} -0.110913 q^{9} +6.20535 q^{11} -2.05323 q^{13} -4.06663 q^{15} +3.12693 q^{17} +8.56198 q^{19} +4.12606 q^{21} +5.19359 q^{23} +0.724125 q^{25} +5.28772 q^{27} -0.325590 q^{29} -1.08445 q^{31} -10.5474 q^{33} -5.80777 q^{35} +6.84140 q^{37} +3.48994 q^{39} -4.00470 q^{41} +2.67606 q^{43} -0.265360 q^{45} +5.81569 q^{47} -1.10736 q^{49} -5.31494 q^{51} +10.8809 q^{53} +14.8464 q^{55} -14.5531 q^{57} -7.22001 q^{59} +1.37579 q^{61} +0.269238 q^{63} -4.91238 q^{65} -9.94262 q^{67} -8.82770 q^{69} -1.18331 q^{71} -15.2452 q^{73} -1.23082 q^{75} -15.0633 q^{77} -11.4910 q^{79} -8.65496 q^{81} +10.7485 q^{83} +7.48122 q^{85} +0.553415 q^{87} -7.55730 q^{89} +4.98416 q^{91} +1.84327 q^{93} +20.4847 q^{95} -9.70061 q^{97} -0.688252 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.69973	−0.981340	−0.490670	−	0.871345i	$-0.663248\pi$
$3$	−1.69973	−0.981340	−0.490670	+	0.871345i	$0.663248\pi$
$4$	0	0
$5$	2.39251	1.06996	0.534982	−	0.844863i	$-0.320318\pi$
$5$	2.39251	1.06996	0.534982	+	0.844863i	$0.320318\pi$
$6$	0	0
$7$	−2.42748	−0.917500	−0.458750	−	0.888565i	$-0.651703\pi$
$7$	−2.42748	−0.917500	−0.458750	+	0.888565i	$0.651703\pi$
$8$	0	0
$9$	−0.110913	−0.0369709
$10$	0	0
$11$	6.20535	1.87098	0.935492	−	0.353349i	$-0.114957\pi$
$11$	6.20535	1.87098	0.935492	+	0.353349i	$0.114957\pi$
$12$	0	0
$13$	−2.05323	−0.569463	−0.284731	−	0.958607i	$-0.591904\pi$
$13$	−2.05323	−0.569463	−0.284731	+	0.958607i	$0.591904\pi$
$14$	0	0
$15$	−4.06663	−1.05000
$16$	0	0
$17$	3.12693	0.758391	0.379196	−	0.925317i	$-0.376201\pi$
$17$	3.12693	0.758391	0.379196	+	0.925317i	$0.376201\pi$
$18$	0	0
$19$	8.56198	1.96425	0.982127	−	0.188219i	$-0.0602716\pi$
$19$	8.56198	1.96425	0.982127	+	0.188219i	$0.0602716\pi$
$20$	0	0
$21$	4.12606	0.900380
$22$	0	0
$23$	5.19359	1.08294	0.541469	−	0.840721i	$-0.317868\pi$
$23$	5.19359	1.08294	0.541469	+	0.840721i	$0.317868\pi$
$24$	0	0
$25$	0.724125	0.144825
$26$	0	0
$27$	5.28772	1.01762
$28$	0	0
$29$	−0.325590	−0.0604605	−0.0302303	−	0.999543i	$-0.509624\pi$
$29$	−0.325590	−0.0604605	−0.0302303	+	0.999543i	$0.509624\pi$
$30$	0	0
$31$	−1.08445	−0.194772	−0.0973861	−	0.995247i	$-0.531048\pi$
$31$	−1.08445	−0.194772	−0.0973861	+	0.995247i	$0.531048\pi$
$32$	0	0
$33$	−10.5474	−1.83607
$34$	0	0
$35$	−5.80777	−0.981692
$36$	0	0
$37$	6.84140	1.12472	0.562360	−	0.826893i	$-0.309894\pi$
$37$	6.84140	1.12472	0.562360	+	0.826893i	$0.309894\pi$
$38$	0	0
$39$	3.48994	0.558837
$40$	0	0
$41$	−4.00470	−0.625430	−0.312715	−	0.949847i	$-0.601238\pi$
$41$	−4.00470	−0.625430	−0.312715	+	0.949847i	$0.601238\pi$
$42$	0	0
$43$	2.67606	0.408096	0.204048	−	0.978961i	$-0.434590\pi$
$43$	2.67606	0.408096	0.204048	+	0.978961i	$0.434590\pi$
$44$	0	0
$45$	−0.265360	−0.0395576
$46$	0	0
$47$	5.81569	0.848306	0.424153	−	0.905591i	$-0.360572\pi$
$47$	5.81569	0.848306	0.424153	+	0.905591i	$0.360572\pi$
$48$	0	0
$49$	−1.10736	−0.158194
$50$	0	0
$51$	−5.31494	−0.744240
$52$	0	0
$53$	10.8809	1.49460	0.747301	−	0.664486i	$-0.231349\pi$
$53$	10.8809	1.49460	0.747301	+	0.664486i	$0.231349\pi$
$54$	0	0
$55$	14.8464	2.00189
$56$	0	0
$57$	−14.5531	−1.92760
$58$	0	0
$59$	−7.22001	−0.939965	−0.469983	−	0.882676i	$-0.655740\pi$
$59$	−7.22001	−0.939965	−0.469983	+	0.882676i	$0.655740\pi$
$60$	0	0
$61$	1.37579	0.176152	0.0880758	−	0.996114i	$-0.471928\pi$
$61$	1.37579	0.176152	0.0880758	+	0.996114i	$0.471928\pi$
$62$	0	0
$63$	0.269238	0.0339208
$64$	0	0
$65$	−4.91238	−0.609305
$66$	0	0
$67$	−9.94262	−1.21468	−0.607342	−	0.794440i	$-0.707764\pi$
$67$	−9.94262	−1.21468	−0.607342	+	0.794440i	$0.707764\pi$
$68$	0	0
$69$	−8.82770	−1.06273
$70$	0	0
$71$	−1.18331	−0.140433	−0.0702167	−	0.997532i	$-0.522369\pi$
$71$	−1.18331	−0.140433	−0.0702167	+	0.997532i	$0.522369\pi$
$72$	0	0
$73$	−15.2452	−1.78432	−0.892159	−	0.451722i	$-0.850810\pi$
$73$	−15.2452	−1.78432	−0.892159	+	0.451722i	$0.850810\pi$
$74$	0	0
$75$	−1.23082	−0.142123
$76$	0	0
$77$	−15.0633	−1.71663
$78$	0	0
$79$	−11.4910	−1.29284	−0.646422	−	0.762980i	$-0.723735\pi$
$79$	−11.4910	−1.29284	−0.646422	+	0.762980i	$0.723735\pi$
$80$	0	0
$81$	−8.65496	−0.961662
$82$	0	0
$83$	10.7485	1.17980	0.589900	−	0.807477i	$-0.299167\pi$
$83$	10.7485	1.17980	0.589900	+	0.807477i	$0.299167\pi$
$84$	0	0
$85$	7.48122	0.811452
$86$	0	0
$87$	0.553415	0.0593324
$88$	0	0
$89$	−7.55730	−0.801072	−0.400536	−	0.916281i	$-0.631176\pi$
$89$	−7.55730	−0.801072	−0.400536	+	0.916281i	$0.631176\pi$
$90$	0	0
$91$	4.98416	0.522482
$92$	0	0
$93$	1.84327	0.191138
$94$	0	0
$95$	20.4847	2.10168
$96$	0	0
$97$	−9.70061	−0.984948	−0.492474	−	0.870327i	$-0.663907\pi$
$97$	−9.70061	−0.984948	−0.492474	+	0.870327i	$0.663907\pi$
$98$	0	0
$99$	−0.688252	−0.0691720
$100$	0	0
$101$	−2.42253	−0.241051	−0.120525	−	0.992710i	$-0.538458\pi$
$101$	−2.42253	−0.241051	−0.120525	+	0.992710i	$0.538458\pi$
$102$	0	0
$103$	18.0815	1.78163	0.890813	−	0.454371i	$-0.150136\pi$
$103$	18.0815	1.78163	0.890813	+	0.454371i	$0.150136\pi$
$104$	0	0
$105$	9.87165	0.963375
$106$	0	0
$107$	17.3994	1.68206	0.841032	−	0.540986i	$-0.181949\pi$
$107$	17.3994	1.68206	0.841032	+	0.540986i	$0.181949\pi$
$108$	0	0
$109$	7.79723	0.746839	0.373420	−	0.927663i	$-0.378185\pi$
$109$	7.79723	0.746839	0.373420	+	0.927663i	$0.378185\pi$
$110$	0	0
$111$	−11.6285	−1.10373
$112$	0	0
$113$	0.766145	0.0720728	0.0360364	−	0.999350i	$-0.488527\pi$
$113$	0.766145	0.0720728	0.0360364	+	0.999350i	$0.488527\pi$
$114$	0	0
$115$	12.4257	1.15871
$116$	0	0
$117$	0.227729	0.0210536
$118$	0	0
$119$	−7.59054	−0.695824
$120$	0	0
$121$	27.5064	2.50058
$122$	0	0
$123$	6.80692	0.613759
$124$	0	0
$125$	−10.2301	−0.915007
$126$	0	0
$127$	7.15927	0.635283	0.317641	−	0.948211i	$-0.397109\pi$
$127$	7.15927	0.635283	0.317641	+	0.948211i	$0.397109\pi$
$128$	0	0
$129$	−4.54859	−0.400481
$130$	0	0
$131$	−15.3202	−1.33854	−0.669268	−	0.743021i	$-0.733392\pi$
$131$	−15.3202	−1.33854	−0.669268	+	0.743021i	$0.733392\pi$
$132$	0	0
$133$	−20.7840	−1.80220
$134$	0	0
$135$	12.6509	1.08882
$136$	0	0
$137$	1.34576	0.114976	0.0574881	−	0.998346i	$-0.481691\pi$
$137$	1.34576	0.114976	0.0574881	+	0.998346i	$0.481691\pi$
$138$	0	0
$139$	−16.1448	−1.36939	−0.684693	−	0.728832i	$-0.740064\pi$
$139$	−16.1448	−1.36939	−0.684693	+	0.728832i	$0.740064\pi$
$140$	0	0
$141$	−9.88512	−0.832477
$142$	0	0
$143$	−12.7410	−1.06546
$144$	0	0
$145$	−0.778979	−0.0646907
$146$	0	0
$147$	1.88222	0.155243
$148$	0	0
$149$	13.7554	1.12688	0.563442	−	0.826156i	$-0.309477\pi$
$149$	13.7554	1.12688	0.563442	+	0.826156i	$0.309477\pi$
$150$	0	0
$151$	15.0437	1.22424	0.612119	−	0.790765i	$-0.290317\pi$
$151$	15.0437	1.22424	0.612119	+	0.790765i	$0.290317\pi$
$152$	0	0
$153$	−0.346816	−0.0280384
$154$	0	0
$155$	−2.59455	−0.208399
$156$	0	0
$157$	−5.16506	−0.412216	−0.206108	−	0.978529i	$-0.566080\pi$
$157$	−5.16506	−0.412216	−0.206108	+	0.978529i	$0.566080\pi$
$158$	0	0
$159$	−18.4946	−1.46671
$160$	0	0
$161$	−12.6073	−0.993595
$162$	0	0
$163$	12.6565	0.991332	0.495666	−	0.868513i	$-0.334924\pi$
$163$	12.6565	0.991332	0.495666	+	0.868513i	$0.334924\pi$
$164$	0	0
$165$	−25.2349	−1.96453
$166$	0	0
$167$	−15.5702	−1.20486	−0.602428	−	0.798173i	$-0.705800\pi$
$167$	−15.5702	−1.20486	−0.602428	+	0.798173i	$0.705800\pi$
$168$	0	0
$169$	−8.78426	−0.675712
$170$	0	0
$171$	−0.949633	−0.0726203
$172$	0	0
$173$	−24.8774	−1.89139	−0.945695	−	0.325055i	$-0.894617\pi$
$173$	−24.8774	−1.89139	−0.945695	+	0.325055i	$0.894617\pi$
$174$	0	0
$175$	−1.75780	−0.132877
$176$	0	0
$177$	12.2721	0.922426
$178$	0	0
$179$	12.4745	0.932387	0.466194	−	0.884683i	$-0.345625\pi$
$179$	12.4745	0.932387	0.466194	+	0.884683i	$0.345625\pi$
$180$	0	0
$181$	19.6397	1.45981	0.729903	−	0.683551i	$-0.239565\pi$
$181$	19.6397	1.45981	0.729903	+	0.683551i	$0.239565\pi$
$182$	0	0
$183$	−2.33847	−0.172865
$184$	0	0
$185$	16.3682	1.20341
$186$	0	0
$187$	19.4037	1.41894
$188$	0	0
$189$	−12.8358	−0.933667
$190$	0	0
$191$	10.1217	0.732379	0.366189	−	0.930540i	$-0.380662\pi$
$191$	10.1217	0.732379	0.366189	+	0.930540i	$0.380662\pi$
$192$	0	0
$193$	8.15307	0.586871	0.293436	−	0.955979i	$-0.405201\pi$
$193$	8.15307	0.586871	0.293436	+	0.955979i	$0.405201\pi$
$194$	0	0
$195$	8.34972	0.597936
$196$	0	0
$197$	−11.5141	−0.820344	−0.410172	−	0.912008i	$-0.634531\pi$
$197$	−11.5141	−0.820344	−0.410172	+	0.912008i	$0.634531\pi$
$198$	0	0
$199$	18.2537	1.29397	0.646986	−	0.762502i	$-0.276029\pi$
$199$	18.2537	1.29397	0.646986	+	0.762502i	$0.276029\pi$
$200$	0	0
$201$	16.8998	1.19202
$202$	0	0
$203$	0.790362	0.0554725
$204$	0	0
$205$	−9.58131	−0.669188
$206$	0	0
$207$	−0.576035	−0.0400372
$208$	0	0
$209$	53.1301	3.67509
$210$	0	0
$211$	−3.40373	−0.234323	−0.117161	−	0.993113i	$-0.537379\pi$
$211$	−3.40373	−0.234323	−0.117161	+	0.993113i	$0.537379\pi$
$212$	0	0
$213$	2.01132	0.137813
$214$	0	0
$215$	6.40252	0.436648
$216$	0	0
$217$	2.63247	0.178703
$218$	0	0
$219$	25.9128	1.75102
$220$	0	0
$221$	−6.42029	−0.431876
$222$	0	0
$223$	10.8043	0.723507	0.361754	−	0.932274i	$-0.382178\pi$
$223$	10.8043	0.723507	0.361754	+	0.932274i	$0.382178\pi$
$224$	0	0
$225$	−0.0803147	−0.00535431
$226$	0	0
$227$	−13.2506	−0.879475	−0.439738	−	0.898126i	$-0.644929\pi$
$227$	−13.2506	−0.879475	−0.439738	+	0.898126i	$0.644929\pi$
$228$	0	0
$229$	2.60567	0.172187	0.0860937	−	0.996287i	$-0.472562\pi$
$229$	2.60567	0.172187	0.0860937	+	0.996287i	$0.472562\pi$
$230$	0	0
$231$	25.6036	1.68460
$232$	0	0
$233$	−0.0949309	−0.00621913	−0.00310957	−	0.999995i	$-0.500990\pi$
$233$	−0.0949309	−0.00621913	−0.00310957	+	0.999995i	$0.500990\pi$
$234$	0	0
$235$	13.9141	0.907658
$236$	0	0
$237$	19.5317	1.26872
$238$	0	0
$239$	21.1932	1.37088	0.685438	−	0.728131i	$-0.259611\pi$
$239$	21.1932	1.37088	0.685438	+	0.728131i	$0.259611\pi$
$240$	0	0
$241$	−20.3905	−1.31347	−0.656735	−	0.754121i	$-0.728063\pi$
$241$	−20.3905	−1.31347	−0.656735	+	0.754121i	$0.728063\pi$
$242$	0	0
$243$	−1.15204	−0.0739035
$244$	0	0
$245$	−2.64938	−0.169262
$246$	0	0
$247$	−17.5797	−1.11857
$248$	0	0
$249$	−18.2695	−1.15778
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	32.2280	2.02616
$254$	0	0
$255$	−12.7161	−0.796311
$256$	0	0
$257$	21.6917	1.35309	0.676547	−	0.736400i	$-0.263476\pi$
$257$	21.6917	1.35309	0.676547	+	0.736400i	$0.263476\pi$
$258$	0	0
$259$	−16.6073	−1.03193
$260$	0	0
$261$	0.0361121	0.00223528
$262$	0	0
$263$	−16.9047	−1.04239	−0.521196	−	0.853437i	$-0.674514\pi$
$263$	−16.9047	−1.04239	−0.521196	+	0.853437i	$0.674514\pi$
$264$	0	0
$265$	26.0326	1.59917
$266$	0	0
$267$	12.8454	0.786124
$268$	0	0
$269$	18.7724	1.14458	0.572288	−	0.820053i	$-0.306056\pi$
$269$	18.7724	1.14458	0.572288	+	0.820053i	$0.306056\pi$
$270$	0	0
$271$	12.3019	0.747287	0.373643	−	0.927572i	$-0.378108\pi$
$271$	12.3019	0.747287	0.373643	+	0.927572i	$0.378108\pi$
$272$	0	0
$273$	−8.47173	−0.512733
$274$	0	0
$275$	4.49345	0.270965
$276$	0	0
$277$	3.00443	0.180518	0.0902592	−	0.995918i	$-0.471230\pi$
$277$	3.00443	0.180518	0.0902592	+	0.995918i	$0.471230\pi$
$278$	0	0
$279$	0.120279	0.00720091
$280$	0	0
$281$	−21.0024	−1.25290	−0.626449	−	0.779463i	$-0.715492\pi$
$281$	−21.0024	−1.25290	−0.626449	+	0.779463i	$0.715492\pi$
$282$	0	0
$283$	−7.26601	−0.431919	−0.215960	−	0.976402i	$-0.569288\pi$
$283$	−7.26601	−0.431919	−0.215960	+	0.976402i	$0.569288\pi$
$284$	0	0
$285$	−34.8184	−2.06247
$286$	0	0
$287$	9.72132	0.573831
$288$	0	0
$289$	−7.22232	−0.424843
$290$	0	0
$291$	16.4884	0.966570
$292$	0	0
$293$	4.10451	0.239788	0.119894	−	0.992787i	$-0.461745\pi$
$293$	4.10451	0.239788	0.119894	+	0.992787i	$0.461745\pi$
$294$	0	0
$295$	−17.2740	−1.00573
$296$	0	0
$297$	32.8121	1.90395
$298$	0	0
$299$	−10.6636	−0.616693
$300$	0	0
$301$	−6.49608	−0.374428
$302$	0	0
$303$	4.11765	0.236553
$304$	0	0
$305$	3.29159	0.188476
$306$	0	0
$307$	−29.1665	−1.66462	−0.832309	−	0.554312i	$-0.812981\pi$
$307$	−29.1665	−1.66462	−0.832309	+	0.554312i	$0.812981\pi$
$308$	0	0
$309$	−30.7337	−1.74838
$310$	0	0
$311$	−2.59370	−0.147075	−0.0735375	−	0.997292i	$-0.523429\pi$
$311$	−2.59370	−0.147075	−0.0735375	+	0.997292i	$0.523429\pi$
$312$	0	0
$313$	13.2214	0.747319	0.373659	−	0.927566i	$-0.378103\pi$
$313$	13.2214	0.747319	0.373659	+	0.927566i	$0.378103\pi$
$314$	0	0
$315$	0.644156	0.0362941
$316$	0	0
$317$	−19.4426	−1.09201	−0.546003	−	0.837783i	$-0.683851\pi$
$317$	−19.4426	−1.09201	−0.546003	+	0.837783i	$0.683851\pi$
$318$	0	0
$319$	−2.02040	−0.113121
$320$	0	0
$321$	−29.5743	−1.65068
$322$	0	0
$323$	26.7727	1.48967
$324$	0	0
$325$	−1.48679	−0.0824725
$326$	0	0
$327$	−13.2532	−0.732904
$328$	0	0
$329$	−14.1175	−0.778320
$330$	0	0
$331$	18.3431	1.00823	0.504114	−	0.863637i	$-0.331819\pi$
$331$	18.3431	1.00823	0.504114	+	0.863637i	$0.331819\pi$
$332$	0	0
$333$	−0.758799	−0.0415819
$334$	0	0
$335$	−23.7879	−1.29967
$336$	0	0
$337$	−2.84191	−0.154809	−0.0774044	−	0.997000i	$-0.524663\pi$
$337$	−2.84191	−0.154809	−0.0774044	+	0.997000i	$0.524663\pi$
$338$	0	0
$339$	−1.30224	−0.0707280
$340$	0	0
$341$	−6.72937	−0.364416
$342$	0	0
$343$	19.6804	1.06264
$344$	0	0
$345$	−21.1204	−1.13708
$346$	0	0
$347$	−23.9457	−1.28547	−0.642735	−	0.766088i	$-0.722201\pi$
$347$	−23.9457	−1.28547	−0.642735	+	0.766088i	$0.722201\pi$
$348$	0	0
$349$	11.3385	0.606938	0.303469	−	0.952841i	$-0.401855\pi$
$349$	11.3385	0.606938	0.303469	+	0.952841i	$0.401855\pi$
$350$	0	0
$351$	−10.8569	−0.579498
$352$	0	0
$353$	27.9663	1.48850	0.744248	−	0.667903i	$-0.232808\pi$
$353$	27.9663	1.48850	0.744248	+	0.667903i	$0.232808\pi$
$354$	0	0
$355$	−2.83109	−0.150259
$356$	0	0
$357$	12.9019	0.682840
$358$	0	0
$359$	6.44355	0.340078	0.170039	−	0.985437i	$-0.445611\pi$
$359$	6.44355	0.340078	0.170039	+	0.985437i	$0.445611\pi$
$360$	0	0
$361$	54.3076	2.85829
$362$	0	0
$363$	−46.7534	−2.45392
$364$	0	0
$365$	−36.4744	−1.90916
$366$	0	0
$367$	−15.4902	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$367$	−15.4902	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$368$	0	0
$369$	0.444173	0.0231227
$370$	0	0
$371$	−26.4130	−1.37130
$372$	0	0
$373$	26.4832	1.37125	0.685625	−	0.727955i	$-0.259529\pi$
$373$	26.4832	1.37125	0.685625	+	0.727955i	$0.259529\pi$
$374$	0	0
$375$	17.3884	0.897934
$376$	0	0
$377$	0.668510	0.0344300
$378$	0	0
$379$	−16.8138	−0.863669	−0.431835	−	0.901953i	$-0.642134\pi$
$379$	−16.8138	−0.863669	−0.431835	+	0.901953i	$0.642134\pi$
$380$	0	0
$381$	−12.1688	−0.623429
$382$	0	0
$383$	16.2662	0.831164	0.415582	−	0.909556i	$-0.363578\pi$
$383$	16.2662	0.831164	0.415582	+	0.909556i	$0.363578\pi$
$384$	0	0
$385$	−36.0393	−1.83673
$386$	0	0
$387$	−0.296810	−0.0150877
$388$	0	0
$389$	−36.1661	−1.83369	−0.916847	−	0.399239i	$-0.869274\pi$
$389$	−36.1661	−1.83369	−0.916847	+	0.399239i	$0.869274\pi$
$390$	0	0
$391$	16.2400	0.821291
$392$	0	0
$393$	26.0403	1.31356
$394$	0	0
$395$	−27.4925	−1.38330
$396$	0	0
$397$	27.7195	1.39120	0.695601	−	0.718428i	$-0.255138\pi$
$397$	27.7195	1.39120	0.695601	+	0.718428i	$0.255138\pi$
$398$	0	0
$399$	35.3272	1.76857
$400$	0	0
$401$	−21.8713	−1.09220	−0.546101	−	0.837719i	$-0.683888\pi$
$401$	−21.8713	−1.09220	−0.546101	+	0.837719i	$0.683888\pi$
$402$	0	0
$403$	2.22661	0.110916
$404$	0	0
$405$	−20.7071	−1.02894
$406$	0	0
$407$	42.4533	2.10433
$408$	0	0
$409$	27.7330	1.37131	0.685655	−	0.727927i	$-0.259516\pi$
$409$	27.7330	1.37131	0.685655	+	0.727927i	$0.259516\pi$
$410$	0	0
$411$	−2.28743	−0.112831
$412$	0	0
$413$	17.5264	0.862418
$414$	0	0
$415$	25.7159	1.26234
$416$	0	0
$417$	27.4419	1.34383
$418$	0	0
$419$	15.1183	0.738576	0.369288	−	0.929315i	$-0.379602\pi$
$419$	15.1183	0.738576	0.369288	+	0.929315i	$0.379602\pi$
$420$	0	0
$421$	8.40491	0.409630	0.204815	−	0.978801i	$-0.434341\pi$
$421$	8.40491	0.409630	0.204815	+	0.978801i	$0.434341\pi$
$422$	0	0
$423$	−0.645035	−0.0313627
$424$	0	0
$425$	2.26429	0.109834
$426$	0	0
$427$	−3.33969	−0.161619
$428$	0	0
$429$	21.6563	1.04557
$430$	0	0
$431$	−39.5058	−1.90293	−0.951464	−	0.307759i	$-0.900421\pi$
$431$	−39.5058	−1.90293	−0.951464	+	0.307759i	$0.900421\pi$
$432$	0	0
$433$	23.6471	1.13641	0.568203	−	0.822888i	$-0.307639\pi$
$433$	23.6471	1.13641	0.568203	+	0.822888i	$0.307639\pi$
$434$	0	0
$435$	1.32405	0.0634836
$436$	0	0
$437$	44.4674	2.12717
$438$	0	0
$439$	11.8604	0.566064	0.283032	−	0.959110i	$-0.408660\pi$
$439$	11.8604	0.566064	0.283032	+	0.959110i	$0.408660\pi$
$440$	0	0
$441$	0.122820	0.00584859
$442$	0	0
$443$	6.42239	0.305137	0.152569	−	0.988293i	$-0.451246\pi$
$443$	6.42239	0.305137	0.152569	+	0.988293i	$0.451246\pi$
$444$	0	0
$445$	−18.0809	−0.857119
$446$	0	0
$447$	−23.3804	−1.10586
$448$	0	0
$449$	−35.6494	−1.68240	−0.841200	−	0.540724i	$-0.818150\pi$
$449$	−35.6494	−1.68240	−0.841200	+	0.540724i	$0.818150\pi$
$450$	0	0
$451$	−24.8506	−1.17017
$452$	0	0
$453$	−25.5703	−1.20140
$454$	0	0
$455$	11.9247	0.559037
$456$	0	0
$457$	41.9896	1.96419	0.982095	−	0.188384i	$-0.0603250\pi$
$457$	41.9896	1.96419	0.982095	+	0.188384i	$0.0603250\pi$
$458$	0	0
$459$	16.5343	0.771755
$460$	0	0
$461$	20.8908	0.972983	0.486492	−	0.873685i	$-0.338276\pi$
$461$	20.8908	0.972983	0.486492	+	0.873685i	$0.338276\pi$
$462$	0	0
$463$	19.0446	0.885079	0.442540	−	0.896749i	$-0.354078\pi$
$463$	19.0446	0.885079	0.442540	+	0.896749i	$0.354078\pi$
$464$	0	0
$465$	4.41004	0.204511
$466$	0	0
$467$	4.02510	0.186259	0.0931297	−	0.995654i	$-0.470313\pi$
$467$	4.02510	0.186259	0.0931297	+	0.995654i	$0.470313\pi$
$468$	0	0
$469$	24.1355	1.11447
$470$	0	0
$471$	8.77921	0.404525
$472$	0	0
$473$	16.6059	0.763541
$474$	0	0
$475$	6.19995	0.284473
$476$	0	0
$477$	−1.20683	−0.0552568
$478$	0	0
$479$	−36.7400	−1.67869	−0.839347	−	0.543596i	$-0.817062\pi$
$479$	−36.7400	−1.67869	−0.839347	+	0.543596i	$0.817062\pi$
$480$	0	0
$481$	−14.0470	−0.640486
$482$	0	0
$483$	21.4290	0.975055
$484$	0	0
$485$	−23.2089	−1.05386
$486$	0	0
$487$	−36.6390	−1.66027	−0.830135	−	0.557562i	$-0.811737\pi$
$487$	−36.6390	−1.66027	−0.830135	+	0.557562i	$0.811737\pi$
$488$	0	0
$489$	−21.5126	−0.972834
$490$	0	0
$491$	−31.9656	−1.44259	−0.721294	−	0.692629i	$-0.756453\pi$
$491$	−31.9656	−1.44259	−0.721294	+	0.692629i	$0.756453\pi$
$492$	0	0
$493$	−1.01810	−0.0458527
$494$	0	0
$495$	−1.64665	−0.0740116
$496$	0	0
$497$	2.87247	0.128848
$498$	0	0
$499$	0.843767	0.0377722	0.0188861	−	0.999822i	$-0.493988\pi$
$499$	0.843767	0.0377722	0.0188861	+	0.999822i	$0.493988\pi$
$500$	0	0
$501$	26.4651	1.18237
$502$	0	0
$503$	21.3296	0.951039	0.475519	−	0.879705i	$-0.342260\pi$
$503$	21.3296	0.951039	0.475519	+	0.879705i	$0.342260\pi$
$504$	0	0
$505$	−5.79594	−0.257916
$506$	0	0
$507$	14.9309	0.663104
$508$	0	0
$509$	−2.75750	−0.122224	−0.0611120	−	0.998131i	$-0.519465\pi$
$509$	−2.75750	−0.122224	−0.0611120	+	0.998131i	$0.519465\pi$
$510$	0	0
$511$	37.0074	1.63711
$512$	0	0
$513$	45.2733	1.99887
$514$	0	0
$515$	43.2603	1.90628
$516$	0	0
$517$	36.0884	1.58717
$518$	0	0
$519$	42.2848	1.85610
$520$	0	0
$521$	−26.3855	−1.15597	−0.577985	−	0.816047i	$-0.696161\pi$
$521$	−26.3855	−1.15597	−0.577985	+	0.816047i	$0.696161\pi$
$522$	0	0
$523$	0.653108	0.0285584	0.0142792	−	0.999898i	$-0.495455\pi$
$523$	0.653108	0.0285584	0.0142792	+	0.999898i	$0.495455\pi$
$524$	0	0
$525$	2.98778	0.130397
$526$	0	0
$527$	−3.39098	−0.147714
$528$	0	0
$529$	3.97335	0.172754
$530$	0	0
$531$	0.800791	0.0347514
$532$	0	0
$533$	8.22257	0.356159
$534$	0	0
$535$	41.6283	1.79975
$536$	0	0
$537$	−21.2033	−0.914989
$538$	0	0
$539$	−6.87156	−0.295979
$540$	0	0
$541$	9.31263	0.400381	0.200191	−	0.979757i	$-0.435844\pi$
$541$	9.31263	0.400381	0.200191	+	0.979757i	$0.435844\pi$
$542$	0	0
$543$	−33.3822	−1.43257
$544$	0	0
$545$	18.6550	0.799092
$546$	0	0
$547$	11.0974	0.474491	0.237245	−	0.971450i	$-0.423756\pi$
$547$	11.0974	0.474491	0.237245	+	0.971450i	$0.423756\pi$
$548$	0	0
$549$	−0.152593	−0.00651249
$550$	0	0
$551$	−2.78770	−0.118760
$552$	0	0
$553$	27.8942	1.18618
$554$	0	0
$555$	−27.8215	−1.18096
$556$	0	0
$557$	−35.2491	−1.49355	−0.746777	−	0.665075i	$-0.768400\pi$
$557$	−35.2491	−1.49355	−0.746777	+	0.665075i	$0.768400\pi$
$558$	0	0
$559$	−5.49457	−0.232396
$560$	0	0
$561$	−32.9810	−1.39246
$562$	0	0
$563$	−7.81017	−0.329159	−0.164580	−	0.986364i	$-0.552627\pi$
$563$	−7.81017	−0.329159	−0.164580	+	0.986364i	$0.552627\pi$
$564$	0	0
$565$	1.83301	0.0771154
$566$	0	0
$567$	21.0097	0.882325
$568$	0	0
$569$	43.0003	1.80267	0.901333	−	0.433128i	$-0.142590\pi$
$569$	43.0003	1.80267	0.901333	+	0.433128i	$0.142590\pi$
$570$	0	0
$571$	−19.7859	−0.828016	−0.414008	−	0.910273i	$-0.635871\pi$
$571$	−19.7859	−0.828016	−0.414008	+	0.910273i	$0.635871\pi$
$572$	0	0
$573$	−17.2041	−0.718713
$574$	0	0
$575$	3.76081	0.156836
$576$	0	0
$577$	26.3408	1.09658	0.548291	−	0.836287i	$-0.315278\pi$
$577$	26.3408	1.09658	0.548291	+	0.836287i	$0.315278\pi$
$578$	0	0
$579$	−13.8580	−0.575920
$580$	0	0
$581$	−26.0917	−1.08247
$582$	0	0
$583$	67.5196	2.79638
$584$	0	0
$585$	0.544845	0.0225266
$586$	0	0
$587$	36.1305	1.49127	0.745633	−	0.666357i	$-0.232147\pi$
$587$	36.1305	1.49127	0.745633	+	0.666357i	$0.232147\pi$
$588$	0	0
$589$	−9.28501	−0.382582
$590$	0	0
$591$	19.5708	0.805037
$592$	0	0
$593$	−19.0228	−0.781172	−0.390586	−	0.920566i	$-0.627728\pi$
$593$	−19.0228	−0.781172	−0.390586	+	0.920566i	$0.627728\pi$
$594$	0	0
$595$	−18.1605	−0.744507
$596$	0	0
$597$	−31.0264	−1.26983
$598$	0	0
$599$	19.7192	0.805703	0.402851	−	0.915265i	$-0.368019\pi$
$599$	19.7192	0.805703	0.402851	+	0.915265i	$0.368019\pi$
$600$	0	0
$601$	−30.6683	−1.25099	−0.625493	−	0.780230i	$-0.715102\pi$
$601$	−30.6683	−1.25099	−0.625493	+	0.780230i	$0.715102\pi$
$602$	0	0
$603$	1.10276	0.0449080
$604$	0	0
$605$	65.8094	2.67553
$606$	0	0
$607$	26.1378	1.06090	0.530451	−	0.847716i	$-0.322023\pi$
$607$	26.1378	1.06090	0.530451	+	0.847716i	$0.322023\pi$
$608$	0	0
$609$	−1.34340	−0.0544374
$610$	0	0
$611$	−11.9409	−0.483079
$612$	0	0
$613$	28.3076	1.14333	0.571667	−	0.820486i	$-0.306297\pi$
$613$	28.3076	1.14333	0.571667	+	0.820486i	$0.306297\pi$
$614$	0	0
$615$	16.2857	0.656701
$616$	0	0
$617$	13.5957	0.547342	0.273671	−	0.961823i	$-0.411762\pi$
$617$	13.5957	0.547342	0.273671	+	0.961823i	$0.411762\pi$
$618$	0	0
$619$	−25.4158	−1.02155	−0.510773	−	0.859716i	$-0.670641\pi$
$619$	−25.4158	−1.02155	−0.510773	+	0.859716i	$0.670641\pi$
$620$	0	0
$621$	27.4622	1.10202
$622$	0	0
$623$	18.3452	0.734983
$624$	0	0
$625$	−28.0963	−1.12385
$626$	0	0
$627$	−90.3069	−3.60651
$628$	0	0
$629$	21.3926	0.852978
$630$	0	0
$631$	8.13001	0.323651	0.161825	−	0.986819i	$-0.448262\pi$
$631$	8.13001	0.323651	0.161825	+	0.986819i	$0.448262\pi$
$632$	0	0
$633$	5.78543	0.229950
$634$	0	0
$635$	17.1287	0.679730
$636$	0	0
$637$	2.27366	0.0900859
$638$	0	0
$639$	0.131245	0.00519195
$640$	0	0
$641$	−33.7164	−1.33172	−0.665860	−	0.746077i	$-0.731935\pi$
$641$	−33.7164	−1.33172	−0.665860	+	0.746077i	$0.731935\pi$
$642$	0	0
$643$	36.0193	1.42046	0.710232	−	0.703968i	$-0.248590\pi$
$643$	36.0193	1.42046	0.710232	+	0.703968i	$0.248590\pi$
$644$	0	0
$645$	−10.8826	−0.428501
$646$	0	0
$647$	−5.35730	−0.210617	−0.105309	−	0.994440i	$-0.533583\pi$
$647$	−5.35730	−0.210617	−0.105309	+	0.994440i	$0.533583\pi$
$648$	0	0
$649$	−44.8027	−1.75866
$650$	0	0
$651$	−4.47449	−0.175369
$652$	0	0
$653$	48.3770	1.89314	0.946570	−	0.322499i	$-0.104523\pi$
$653$	48.3770	1.89314	0.946570	+	0.322499i	$0.104523\pi$
$654$	0	0
$655$	−36.6539	−1.43219
$656$	0	0
$657$	1.69089	0.0659678
$658$	0	0
$659$	3.36691	0.131156	0.0655781	−	0.997847i	$-0.479111\pi$
$659$	3.36691	0.131156	0.0655781	+	0.997847i	$0.479111\pi$
$660$	0	0
$661$	−24.6756	−0.959771	−0.479885	−	0.877331i	$-0.659322\pi$
$661$	−24.6756	−0.959771	−0.479885	+	0.877331i	$0.659322\pi$
$662$	0	0
$663$	10.9128	0.423817
$664$	0	0
$665$	−49.7260	−1.92829
$666$	0	0
$667$	−1.69098	−0.0654750
$668$	0	0
$669$	−18.3644	−0.710007
$670$	0	0
$671$	8.53725	0.329577
$672$	0	0
$673$	−6.67636	−0.257355	−0.128677	−	0.991687i	$-0.541073\pi$
$673$	−6.67636	−0.257355	−0.128677	+	0.991687i	$0.541073\pi$
$674$	0	0
$675$	3.82897	0.147377
$676$	0	0
$677$	27.8785	1.07146	0.535729	−	0.844390i	$-0.320037\pi$
$677$	27.8785	1.07146	0.535729	+	0.844390i	$0.320037\pi$
$678$	0	0
$679$	23.5480	0.903690
$680$	0	0
$681$	22.5225	0.863065
$682$	0	0
$683$	32.3355	1.23728	0.618642	−	0.785673i	$-0.287683\pi$
$683$	32.3355	1.23728	0.618642	+	0.785673i	$0.287683\pi$
$684$	0	0
$685$	3.21975	0.123020
$686$	0	0
$687$	−4.42893	−0.168974
$688$	0	0
$689$	−22.3409	−0.851121
$690$	0	0
$691$	7.99236	0.304044	0.152022	−	0.988377i	$-0.451422\pi$
$691$	7.99236	0.304044	0.152022	+	0.988377i	$0.451422\pi$
$692$	0	0
$693$	1.67072	0.0634653
$694$	0	0
$695$	−38.6267	−1.46520
$696$	0	0
$697$	−12.5224	−0.474320
$698$	0	0
$699$	0.161357	0.00610309
$700$	0	0
$701$	−0.175662	−0.00663467	−0.00331733	−	0.999994i	$-0.501056\pi$
$701$	−0.175662	−0.00663467	−0.00331733	+	0.999994i	$0.501056\pi$
$702$	0	0
$703$	58.5760	2.20924
$704$	0	0
$705$	−23.6503	−0.890721
$706$	0	0
$707$	5.88063	0.221164
$708$	0	0
$709$	2.79083	0.104812	0.0524058	−	0.998626i	$-0.483311\pi$
$709$	2.79083	0.104812	0.0524058	+	0.998626i	$0.483311\pi$
$710$	0	0
$711$	1.27450	0.0477976
$712$	0	0
$713$	−5.63216	−0.210926
$714$	0	0
$715$	−30.4830	−1.14000
$716$	0	0
$717$	−36.0228	−1.34530
$718$	0	0
$719$	34.7514	1.29601	0.648004	−	0.761637i	$-0.275604\pi$
$719$	34.7514	1.29601	0.648004	+	0.761637i	$0.275604\pi$
$720$	0	0
$721$	−43.8925	−1.63464
$722$	0	0
$723$	34.6584	1.28896
$724$	0	0
$725$	−0.235768	−0.00875620
$726$	0	0
$727$	−18.2904	−0.678354	−0.339177	−	0.940723i	$-0.610149\pi$
$727$	−18.2904	−0.678354	−0.339177	+	0.940723i	$0.610149\pi$
$728$	0	0
$729$	27.9230	1.03419
$730$	0	0
$731$	8.36786	0.309496
$732$	0	0
$733$	−37.1412	−1.37184	−0.685921	−	0.727676i	$-0.740601\pi$
$733$	−37.1412	−1.37184	−0.685921	+	0.727676i	$0.740601\pi$
$734$	0	0
$735$	4.50323	0.166104
$736$	0	0
$737$	−61.6974	−2.27265
$738$	0	0
$739$	8.46048	0.311224	0.155612	−	0.987818i	$-0.450265\pi$
$739$	8.46048	0.311224	0.155612	+	0.987818i	$0.450265\pi$
$740$	0	0
$741$	29.8808	1.09770
$742$	0	0
$743$	50.4024	1.84909	0.924543	−	0.381079i	$-0.124447\pi$
$743$	50.4024	1.84909	0.924543	+	0.381079i	$0.124447\pi$
$744$	0	0
$745$	32.9099	1.20573
$746$	0	0
$747$	−1.19214	−0.0436183
$748$	0	0
$749$	−42.2366	−1.54329
$750$	0	0
$751$	17.9518	0.655070	0.327535	−	0.944839i	$-0.393782\pi$
$751$	17.9518	0.655070	0.327535	+	0.944839i	$0.393782\pi$
$752$	0	0
$753$	−1.69973	−0.0619417
$754$	0	0
$755$	35.9923	1.30989
$756$	0	0
$757$	3.40161	0.123634	0.0618169	−	0.998088i	$-0.480311\pi$
$757$	3.40161	0.123634	0.0618169	+	0.998088i	$0.480311\pi$
$758$	0	0
$759$	−54.7790	−1.98835
$760$	0	0
$761$	43.8162	1.58834	0.794169	−	0.607698i	$-0.207907\pi$
$761$	43.8162	1.58834	0.794169	+	0.607698i	$0.207907\pi$
$762$	0	0
$763$	−18.9276	−0.685225
$764$	0	0
$765$	−0.829763	−0.0300001
$766$	0	0
$767$	14.8243	0.535275
$768$	0	0
$769$	13.6043	0.490584	0.245292	−	0.969449i	$-0.421116\pi$
$769$	13.6043	0.490584	0.245292	+	0.969449i	$0.421116\pi$
$770$	0	0
$771$	−36.8701	−1.32785
$772$	0	0
$773$	−18.4983	−0.665338	−0.332669	−	0.943044i	$-0.607949\pi$
$773$	−18.4983	−0.665338	−0.332669	+	0.943044i	$0.607949\pi$
$774$	0	0
$775$	−0.785274	−0.0282079
$776$	0	0
$777$	28.2280	1.01267
$778$	0	0
$779$	−34.2882	−1.22850
$780$	0	0
$781$	−7.34288	−0.262749
$782$	0	0
$783$	−1.72163	−0.0615259
$784$	0	0
$785$	−12.3575	−0.441057
$786$	0	0
$787$	47.9986	1.71097	0.855483	−	0.517831i	$-0.173261\pi$
$787$	47.9986	1.71097	0.855483	+	0.517831i	$0.173261\pi$
$788$	0	0
$789$	28.7335	1.02294
$790$	0	0
$791$	−1.85980	−0.0661268
$792$	0	0
$793$	−2.82481	−0.100312
$794$	0	0
$795$	−44.2485	−1.56933
$796$	0	0
$797$	14.9036	0.527911	0.263955	−	0.964535i	$-0.414973\pi$
$797$	14.9036	0.527911	0.263955	+	0.964535i	$0.414973\pi$
$798$	0	0
$799$	18.1853	0.643348
$800$	0	0
$801$	0.838200	0.0296164
$802$	0	0
$803$	−94.6019	−3.33843
$804$	0	0
$805$	−30.1632	−1.06311
$806$	0	0
$807$	−31.9081	−1.12322
$808$	0	0
$809$	8.47760	0.298057	0.149028	−	0.988833i	$-0.452385\pi$
$809$	8.47760	0.298057	0.149028	+	0.988833i	$0.452385\pi$
$810$	0	0
$811$	31.1733	1.09464	0.547322	−	0.836922i	$-0.315647\pi$
$811$	31.1733	1.09464	0.547322	+	0.836922i	$0.315647\pi$
$812$	0	0
$813$	−20.9099	−0.733343
$814$	0	0
$815$	30.2808	1.06069
$816$	0	0
$817$	22.9124	0.801604
$818$	0	0
$819$	−0.552807	−0.0193166
$820$	0	0
$821$	39.1999	1.36808	0.684042	−	0.729442i	$-0.260220\pi$
$821$	39.1999	1.36808	0.684042	+	0.729442i	$0.260220\pi$
$822$	0	0
$823$	−1.14105	−0.0397745	−0.0198872	−	0.999802i	$-0.506331\pi$
$823$	−1.14105	−0.0397745	−0.0198872	+	0.999802i	$0.506331\pi$
$824$	0	0
$825$	−7.63766	−0.265909
$826$	0	0
$827$	28.4216	0.988317	0.494159	−	0.869372i	$-0.335476\pi$
$827$	28.4216	0.988317	0.494159	+	0.869372i	$0.335476\pi$
$828$	0	0
$829$	−16.1809	−0.561986	−0.280993	−	0.959710i	$-0.590664\pi$
$829$	−16.1809	−0.561986	−0.280993	+	0.959710i	$0.590664\pi$
$830$	0	0
$831$	−5.10672	−0.177150
$832$	0	0
$833$	−3.46264	−0.119973
$834$	0	0
$835$	−37.2519	−1.28915
$836$	0	0
$837$	−5.73424	−0.198204
$838$	0	0
$839$	−55.8554	−1.92834	−0.964172	−	0.265279i	$-0.914536\pi$
$839$	−55.8554	−1.92834	−0.964172	+	0.265279i	$0.914536\pi$
$840$	0	0
$841$	−28.8940	−0.996345
$842$	0	0
$843$	35.6984	1.22952
$844$	0	0
$845$	−21.0165	−0.722988
$846$	0	0
$847$	−66.7710	−2.29428
$848$	0	0
$849$	12.3503	0.423860
$850$	0	0
$851$	35.5314	1.21800
$852$	0	0
$853$	−37.4538	−1.28240	−0.641198	−	0.767376i	$-0.721562\pi$
$853$	−37.4538	−1.28240	−0.641198	+	0.767376i	$0.721562\pi$
$854$	0	0
$855$	−2.27201	−0.0777011
$856$	0	0
$857$	1.30226	0.0444843	0.0222421	−	0.999753i	$-0.492920\pi$
$857$	1.30226	0.0444843	0.0222421	+	0.999753i	$0.492920\pi$
$858$	0	0
$859$	14.1917	0.484214	0.242107	−	0.970250i	$-0.422161\pi$
$859$	14.1917	0.484214	0.242107	+	0.970250i	$0.422161\pi$
$860$	0	0
$861$	−16.5236	−0.563124
$862$	0	0
$863$	6.30149	0.214505	0.107253	−	0.994232i	$-0.465795\pi$
$863$	6.30149	0.214505	0.107253	+	0.994232i	$0.465795\pi$
$864$	0	0
$865$	−59.5194	−2.02372
$866$	0	0
$867$	12.2760	0.416915
$868$	0	0
$869$	−71.3059	−2.41889
$870$	0	0
$871$	20.4145	0.691718
$872$	0	0
$873$	1.07592	0.0364144
$874$	0	0
$875$	24.8333	0.839519
$876$	0	0
$877$	−18.5850	−0.627571	−0.313785	−	0.949494i	$-0.601597\pi$
$877$	−18.5850	−0.627571	−0.313785	+	0.949494i	$0.601597\pi$
$878$	0	0
$879$	−6.97657	−0.235314
$880$	0	0
$881$	−17.0648	−0.574928	−0.287464	−	0.957791i	$-0.592812\pi$
$881$	−17.0648	−0.574928	−0.287464	+	0.957791i	$0.592812\pi$
$882$	0	0
$883$	−9.61132	−0.323447	−0.161723	−	0.986836i	$-0.551705\pi$
$883$	−9.61132	−0.323447	−0.161723	+	0.986836i	$0.551705\pi$
$884$	0	0
$885$	29.3611	0.986963
$886$	0	0
$887$	26.0768	0.875573	0.437787	−	0.899079i	$-0.355763\pi$
$887$	26.0768	0.875573	0.437787	+	0.899079i	$0.355763\pi$
$888$	0	0
$889$	−17.3790	−0.582872
$890$	0	0
$891$	−53.7071	−1.79925
$892$	0	0
$893$	49.7939	1.66629
$894$	0	0
$895$	29.8454	0.997622
$896$	0	0
$897$	18.1253	0.605186
$898$	0	0
$899$	0.353085	0.0117760
$900$	0	0
$901$	34.0237	1.13349
$902$	0	0
$903$	11.0416	0.367441
$904$	0	0
$905$	46.9882	1.56194
$906$	0	0
$907$	−22.6626	−0.752499	−0.376250	−	0.926518i	$-0.622786\pi$
$907$	−22.6626	−0.752499	−0.376250	+	0.926518i	$0.622786\pi$
$908$	0	0
$909$	0.268690	0.00891187
$910$	0	0
$911$	−42.3348	−1.40261	−0.701307	−	0.712860i	$-0.747400\pi$
$911$	−42.3348	−1.40261	−0.701307	+	0.712860i	$0.747400\pi$
$912$	0	0
$913$	66.6981	2.20738
$914$	0	0
$915$	−5.59483	−0.184959
$916$	0	0
$917$	37.1895	1.22811
$918$	0	0
$919$	2.08030	0.0686226	0.0343113	−	0.999411i	$-0.489076\pi$
$919$	2.08030	0.0686226	0.0343113	+	0.999411i	$0.489076\pi$
$920$	0	0
$921$	49.5752	1.63356
$922$	0	0
$923$	2.42961	0.0799717
$924$	0	0
$925$	4.95403	0.162888
$926$	0	0
$927$	−2.00547	−0.0658683
$928$	0	0
$929$	−5.92001	−0.194229	−0.0971146	−	0.995273i	$-0.530961\pi$
$929$	−5.92001	−0.194229	−0.0971146	+	0.995273i	$0.530961\pi$
$930$	0	0
$931$	−9.48121	−0.310734
$932$	0	0
$933$	4.40859	0.144331
$934$	0	0
$935$	46.4236	1.51821
$936$	0	0
$937$	29.8764	0.976020	0.488010	−	0.872838i	$-0.337723\pi$
$937$	29.8764	0.976020	0.488010	+	0.872838i	$0.337723\pi$
$938$	0	0
$939$	−22.4729	−0.733374
$940$	0	0
$941$	8.22997	0.268289	0.134145	−	0.990962i	$-0.457171\pi$
$941$	8.22997	0.268289	0.134145	+	0.990962i	$0.457171\pi$
$942$	0	0
$943$	−20.7988	−0.677301
$944$	0	0
$945$	−30.7098	−0.998991
$946$	0	0
$947$	−36.8599	−1.19779	−0.598893	−	0.800829i	$-0.704392\pi$
$947$	−36.8599	−1.19779	−0.598893	+	0.800829i	$0.704392\pi$
$948$	0	0
$949$	31.3019	1.01610
$950$	0	0
$951$	33.0472	1.07163
$952$	0	0
$953$	−2.04281	−0.0661731	−0.0330866	−	0.999452i	$-0.510534\pi$
$953$	−2.04281	−0.0661731	−0.0330866	+	0.999452i	$0.510534\pi$
$954$	0	0
$955$	24.2163	0.783620
$956$	0	0
$957$	3.43414	0.111010
$958$	0	0
$959$	−3.26680	−0.105491
$960$	0	0
$961$	−29.8240	−0.962064
$962$	0	0
$963$	−1.92981	−0.0621874
$964$	0	0
$965$	19.5063	0.627932
$966$	0	0
$967$	−1.36293	−0.0438290	−0.0219145	−	0.999760i	$-0.506976\pi$
$967$	−1.36293	−0.0438290	−0.0219145	+	0.999760i	$0.506976\pi$
$968$	0	0
$969$	−45.5064	−1.46188
$970$	0	0
$971$	41.2914	1.32511	0.662553	−	0.749015i	$-0.269473\pi$
$971$	41.2914	1.32511	0.662553	+	0.749015i	$0.269473\pi$
$972$	0	0
$973$	39.1912	1.25641
$974$	0	0
$975$	2.52715	0.0809336
$976$	0	0
$977$	−8.02253	−0.256664	−0.128332	−	0.991731i	$-0.540962\pi$
$977$	−8.02253	−0.256664	−0.128332	+	0.991731i	$0.540962\pi$
$978$	0	0
$979$	−46.8957	−1.49879
$980$	0	0
$981$	−0.864812	−0.0276113
$982$	0	0
$983$	3.19812	0.102004	0.0510021	−	0.998699i	$-0.483758\pi$
$983$	3.19812	0.102004	0.0510021	+	0.998699i	$0.483758\pi$
$984$	0	0
$985$	−27.5476	−0.877740
$986$	0	0
$987$	23.9959	0.763797
$988$	0	0
$989$	13.8984	0.441943
$990$	0	0
$991$	20.2194	0.642291	0.321145	−	0.947030i	$-0.395932\pi$
$991$	20.2194	0.642291	0.321145	+	0.947030i	$0.395932\pi$
$992$	0	0
$993$	−31.1783	−0.989414
$994$	0	0
$995$	43.6723	1.38451
$996$	0	0
$997$	−45.1730	−1.43064	−0.715321	−	0.698796i	$-0.753720\pi$
$997$	−45.1730	−1.43064	−0.715321	+	0.698796i	$0.753720\pi$
$998$	0	0
$999$	36.1754	1.14454

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.i.1.9	yes	30
4.3	odd	2		8032.2.a.h.1.22	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.h.1.22	✓	30	4.3	odd	2
8032.2.a.i.1.9	yes	30	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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-1.69973 q^{3} +2.39251 q^{5} -2.42748 q^{7} -0.110913 q^{9} +O(q^{10})\)	\(q-1.69973 q^{3} +2.39251 q^{5} -2.42748 q^{7} -0.110913 q^{9} +6.20535 q^{11} -2.05323 q^{13} -4.06663 q^{15} +3.12693 q^{17} +8.56198 q^{19} +4.12606 q^{21} +5.19359 q^{23} +0.724125 q^{25} +5.28772 q^{27} -0.325590 q^{29} -1.08445 q^{31} -10.5474 q^{33} -5.80777 q^{35} +6.84140 q^{37} +3.48994 q^{39} -4.00470 q^{41} +2.67606 q^{43} -0.265360 q^{45} +5.81569 q^{47} -1.10736 q^{49} -5.31494 q^{51} +10.8809 q^{53} +14.8464 q^{55} -14.5531 q^{57} -7.22001 q^{59} +1.37579 q^{61} +0.269238 q^{63} -4.91238 q^{65} -9.94262 q^{67} -8.82770 q^{69} -1.18331 q^{71} -15.2452 q^{73} -1.23082 q^{75} -15.0633 q^{77} -11.4910 q^{79} -8.65496 q^{81} +10.7485 q^{83} +7.48122 q^{85} +0.553415 q^{87} -7.55730 q^{89} +4.98416 q^{91} +1.84327 q^{93} +20.4847 q^{95} -9.70061 q^{97} -0.688252 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) \) 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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