LMFDB - Embedded newform 6040.2.a.s.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.225898 q^{3} +1.00000 q^{5} +2.54851 q^{7} -2.94897 q^{9} +O(q^{10})$	$q+0.225898 q^{3} +1.00000 q^{5} +2.54851 q^{7} -2.94897 q^{9} +6.42893 q^{11} +6.27964 q^{13} +0.225898 q^{15} +1.96728 q^{17} -4.69315 q^{19} +0.575703 q^{21} +5.48783 q^{23} +1.00000 q^{25} -1.34386 q^{27} -2.70227 q^{29} +2.37580 q^{31} +1.45228 q^{33} +2.54851 q^{35} +1.20453 q^{37} +1.41856 q^{39} -6.71782 q^{41} +6.37145 q^{43} -2.94897 q^{45} +8.80605 q^{47} -0.505112 q^{49} +0.444404 q^{51} +13.8904 q^{53} +6.42893 q^{55} -1.06017 q^{57} -12.9355 q^{59} +0.393893 q^{61} -7.51547 q^{63} +6.27964 q^{65} +7.61550 q^{67} +1.23969 q^{69} -2.49070 q^{71} -10.3667 q^{73} +0.225898 q^{75} +16.3842 q^{77} -8.76868 q^{79} +8.54333 q^{81} -4.47154 q^{83} +1.96728 q^{85} -0.610437 q^{87} -9.45198 q^{89} +16.0037 q^{91} +0.536687 q^{93} -4.69315 q^{95} -10.3255 q^{97} -18.9587 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) $ 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9	$ 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) $ 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9 + 17 * q^11 + 16 * q^13 + 2 * q^15 + 22 * q^17 + 16 * q^19 - q^21 + 7 * q^23 + 24 * q^25 - 4 * q^27 + 25 * q^29 + 28 * q^31 + 11 * q^33 + 3 * q^35 + 26 * q^37 + 13 * q^39 + 38 * q^41 - 13 * q^43 + 40 * q^45 + 12 * q^47 + 61 * q^49 + 53 * q^53 + 17 * q^55 + 30 * q^57 + 35 * q^59 + 44 * q^61 - 9 * q^63 + 16 * q^65 - 15 * q^67 + 9 * q^69 + 22 * q^71 + 31 * q^73 + 2 * q^75 + 26 * q^77 + 20 * q^79 + 88 * q^81 - 14 * q^83 + 22 * q^85 - 18 * q^87 + 37 * q^89 - 26 * q^91 + 13 * q^93 + 16 * q^95 + 21 * q^97 - 12 * 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
$3$	0.225898	0.130422	0.0652111	−	0.997871i	$-0.479228\pi$
$3$	0.225898	0.130422	0.0652111	+	0.997871i	$0.479228\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.54851	0.963245	0.481623	−	0.876379i	$-0.340048\pi$
$7$	2.54851	0.963245	0.481623	+	0.876379i	$0.340048\pi$
$8$	0	0
$9$	−2.94897	−0.982990
$10$	0	0
$11$	6.42893	1.93839	0.969197	−	0.246286i	$-0.0792103\pi$
$11$	6.42893	1.93839	0.969197	+	0.246286i	$0.0792103\pi$
$12$	0	0
$13$	6.27964	1.74166	0.870830	−	0.491584i	$-0.163582\pi$
$13$	6.27964	1.74166	0.870830	+	0.491584i	$0.163582\pi$
$14$	0	0
$15$	0.225898	0.0583266
$16$	0	0
$17$	1.96728	0.477135	0.238567	−	0.971126i	$-0.423322\pi$
$17$	1.96728	0.477135	0.238567	+	0.971126i	$0.423322\pi$
$18$	0	0
$19$	−4.69315	−1.07668	−0.538342	−	0.842727i	$-0.680949\pi$
$19$	−4.69315	−1.07668	−0.538342	+	0.842727i	$0.680949\pi$
$20$	0	0
$21$	0.575703	0.125629
$22$	0	0
$23$	5.48783	1.14429	0.572146	−	0.820152i	$-0.306111\pi$
$23$	5.48783	1.14429	0.572146	+	0.820152i	$0.306111\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.34386	−0.258626
$28$	0	0
$29$	−2.70227	−0.501798	−0.250899	−	0.968013i	$-0.580726\pi$
$29$	−2.70227	−0.501798	−0.250899	+	0.968013i	$0.580726\pi$
$30$	0	0
$31$	2.37580	0.426705	0.213353	−	0.976975i	$-0.431562\pi$
$31$	2.37580	0.426705	0.213353	+	0.976975i	$0.431562\pi$
$32$	0	0
$33$	1.45228	0.252810
$34$	0	0
$35$	2.54851	0.430776
$36$	0	0
$37$	1.20453	0.198023	0.0990115	−	0.995086i	$-0.468432\pi$
$37$	1.20453	0.198023	0.0990115	+	0.995086i	$0.468432\pi$
$38$	0	0
$39$	1.41856	0.227151
$40$	0	0
$41$	−6.71782	−1.04915	−0.524574	−	0.851365i	$-0.675775\pi$
$41$	−6.71782	−1.04915	−0.524574	+	0.851365i	$0.675775\pi$
$42$	0	0
$43$	6.37145	0.971637	0.485819	−	0.874060i	$-0.338522\pi$
$43$	6.37145	0.971637	0.485819	+	0.874060i	$0.338522\pi$
$44$	0	0
$45$	−2.94897	−0.439607
$46$	0	0
$47$	8.80605	1.28450	0.642248	−	0.766497i	$-0.278002\pi$
$47$	8.80605	1.28450	0.642248	+	0.766497i	$0.278002\pi$
$48$	0	0
$49$	−0.505112	−0.0721589
$50$	0	0
$51$	0.444404	0.0622290
$52$	0	0
$53$	13.8904	1.90799	0.953995	−	0.299821i	$-0.0969269\pi$
$53$	13.8904	1.90799	0.953995	+	0.299821i	$0.0969269\pi$
$54$	0	0
$55$	6.42893	0.866876
$56$	0	0
$57$	−1.06017	−0.140423
$58$	0	0
$59$	−12.9355	−1.68406	−0.842030	−	0.539430i	$-0.818640\pi$
$59$	−12.9355	−1.68406	−0.842030	+	0.539430i	$0.818640\pi$
$60$	0	0
$61$	0.393893	0.0504328	0.0252164	−	0.999682i	$-0.491973\pi$
$61$	0.393893	0.0504328	0.0252164	+	0.999682i	$0.491973\pi$
$62$	0	0
$63$	−7.51547	−0.946860
$64$	0	0
$65$	6.27964	0.778894
$66$	0	0
$67$	7.61550	0.930381	0.465190	−	0.885211i	$-0.345986\pi$
$67$	7.61550	0.930381	0.465190	+	0.885211i	$0.345986\pi$
$68$	0	0
$69$	1.23969	0.149241
$70$	0	0
$71$	−2.49070	−0.295592	−0.147796	−	0.989018i	$-0.547218\pi$
$71$	−2.49070	−0.295592	−0.147796	+	0.989018i	$0.547218\pi$
$72$	0	0
$73$	−10.3667	−1.21333	−0.606663	−	0.794959i	$-0.707492\pi$
$73$	−10.3667	−1.21333	−0.606663	+	0.794959i	$0.707492\pi$
$74$	0	0
$75$	0.225898	0.0260845
$76$	0	0
$77$	16.3842	1.86715
$78$	0	0
$79$	−8.76868	−0.986554	−0.493277	−	0.869872i	$-0.664201\pi$
$79$	−8.76868	−0.986554	−0.493277	+	0.869872i	$0.664201\pi$
$80$	0	0
$81$	8.54333	0.949259
$82$	0	0
$83$	−4.47154	−0.490816	−0.245408	−	0.969420i	$-0.578922\pi$
$83$	−4.47154	−0.490816	−0.245408	+	0.969420i	$0.578922\pi$
$84$	0	0
$85$	1.96728	0.213381
$86$	0	0
$87$	−0.610437	−0.0654457
$88$	0	0
$89$	−9.45198	−1.00191	−0.500954	−	0.865474i	$-0.667017\pi$
$89$	−9.45198	−1.00191	−0.500954	+	0.865474i	$0.667017\pi$
$90$	0	0
$91$	16.0037	1.67765
$92$	0	0
$93$	0.536687	0.0556519
$94$	0	0
$95$	−4.69315	−0.481507
$96$	0	0
$97$	−10.3255	−1.04840	−0.524199	−	0.851596i	$-0.675635\pi$
$97$	−10.3255	−1.04840	−0.524199	+	0.851596i	$0.675635\pi$
$98$	0	0
$99$	−18.9587	−1.90542
$100$	0	0
$101$	−4.02196	−0.400200	−0.200100	−	0.979776i	$-0.564127\pi$
$101$	−4.02196	−0.400200	−0.200100	+	0.979776i	$0.564127\pi$
$102$	0	0
$103$	0.518092	0.0510491	0.0255246	−	0.999674i	$-0.491874\pi$
$103$	0.518092	0.0510491	0.0255246	+	0.999674i	$0.491874\pi$
$104$	0	0
$105$	0.575703	0.0561828
$106$	0	0
$107$	3.72824	0.360423	0.180211	−	0.983628i	$-0.442322\pi$
$107$	3.72824	0.360423	0.180211	+	0.983628i	$0.442322\pi$
$108$	0	0
$109$	14.0034	1.34128	0.670640	−	0.741783i	$-0.266020\pi$
$109$	14.0034	1.34128	0.670640	+	0.741783i	$0.266020\pi$
$110$	0	0
$111$	0.272100	0.0258266
$112$	0	0
$113$	−13.2086	−1.24256	−0.621279	−	0.783590i	$-0.713387\pi$
$113$	−13.2086	−1.24256	−0.621279	+	0.783590i	$0.713387\pi$
$114$	0	0
$115$	5.48783	0.511743
$116$	0	0
$117$	−18.5185	−1.71203
$118$	0	0
$119$	5.01362	0.459598
$120$	0	0
$121$	30.3311	2.75737
$122$	0	0
$123$	−1.51754	−0.136832
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−0.151769	−0.0134673	−0.00673366	−	0.999977i	$-0.502143\pi$
$127$	−0.151769	−0.0134673	−0.00673366	+	0.999977i	$0.502143\pi$
$128$	0	0
$129$	1.43930	0.126723
$130$	0	0
$131$	−10.6355	−0.929230	−0.464615	−	0.885513i	$-0.653807\pi$
$131$	−10.6355	−0.929230	−0.464615	+	0.885513i	$0.653807\pi$
$132$	0	0
$133$	−11.9605	−1.03711
$134$	0	0
$135$	−1.34386	−0.115661
$136$	0	0
$137$	−15.4353	−1.31873	−0.659364	−	0.751823i	$-0.729175\pi$
$137$	−15.4353	−1.31873	−0.659364	+	0.751823i	$0.729175\pi$
$138$	0	0
$139$	−2.28870	−0.194125	−0.0970626	−	0.995278i	$-0.530945\pi$
$139$	−2.28870	−0.194125	−0.0970626	+	0.995278i	$0.530945\pi$
$140$	0	0
$141$	1.98927	0.167527
$142$	0	0
$143$	40.3714	3.37602
$144$	0	0
$145$	−2.70227	−0.224411
$146$	0	0
$147$	−0.114104	−0.00941112
$148$	0	0
$149$	−18.1498	−1.48689	−0.743443	−	0.668799i	$-0.766809\pi$
$149$	−18.1498	−1.48689	−0.743443	+	0.668799i	$0.766809\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−5.80144	−0.469019
$154$	0	0
$155$	2.37580	0.190828
$156$	0	0
$157$	−12.2409	−0.976928	−0.488464	−	0.872584i	$-0.662443\pi$
$157$	−12.2409	−0.976928	−0.488464	+	0.872584i	$0.662443\pi$
$158$	0	0
$159$	3.13781	0.248845
$160$	0	0
$161$	13.9858	1.10223
$162$	0	0
$163$	−9.25298	−0.724749	−0.362375	−	0.932033i	$-0.618034\pi$
$163$	−9.25298	−0.724749	−0.362375	+	0.932033i	$0.618034\pi$
$164$	0	0
$165$	1.45228	0.113060
$166$	0	0
$167$	−18.9403	−1.46564	−0.732822	−	0.680421i	$-0.761797\pi$
$167$	−18.9403	−1.46564	−0.732822	+	0.680421i	$0.761797\pi$
$168$	0	0
$169$	26.4339	2.03338
$170$	0	0
$171$	13.8400	1.05837
$172$	0	0
$173$	19.4586	1.47941	0.739703	−	0.672933i	$-0.234966\pi$
$173$	19.4586	1.47941	0.739703	+	0.672933i	$0.234966\pi$
$174$	0	0
$175$	2.54851	0.192649
$176$	0	0
$177$	−2.92211	−0.219639
$178$	0	0
$179$	6.84917	0.511931	0.255966	−	0.966686i	$-0.417607\pi$
$179$	6.84917	0.511931	0.255966	+	0.966686i	$0.417607\pi$
$180$	0	0
$181$	−16.1827	−1.20285	−0.601424	−	0.798930i	$-0.705400\pi$
$181$	−16.1827	−1.20285	−0.601424	+	0.798930i	$0.705400\pi$
$182$	0	0
$183$	0.0889795	0.00657756
$184$	0	0
$185$	1.20453	0.0885586
$186$	0	0
$187$	12.6475	0.924875
$188$	0	0
$189$	−3.42484	−0.249120
$190$	0	0
$191$	16.3026	1.17961	0.589806	−	0.807545i	$-0.299204\pi$
$191$	16.3026	1.17961	0.589806	+	0.807545i	$0.299204\pi$
$192$	0	0
$193$	23.4350	1.68689	0.843443	−	0.537218i	$-0.180525\pi$
$193$	23.4350	1.68689	0.843443	+	0.537218i	$0.180525\pi$
$194$	0	0
$195$	1.41856	0.101585
$196$	0	0
$197$	23.3651	1.66469	0.832346	−	0.554257i	$-0.186997\pi$
$197$	23.3651	1.66469	0.832346	+	0.554257i	$0.186997\pi$
$198$	0	0
$199$	−27.9721	−1.98289	−0.991446	−	0.130521i	$-0.958335\pi$
$199$	−27.9721	−1.98289	−0.991446	+	0.130521i	$0.958335\pi$
$200$	0	0
$201$	1.72033	0.121342
$202$	0	0
$203$	−6.88675	−0.483355
$204$	0	0
$205$	−6.71782	−0.469193
$206$	0	0
$207$	−16.1834	−1.12483
$208$	0	0
$209$	−30.1719	−2.08704
$210$	0	0
$211$	−7.91941	−0.545195	−0.272597	−	0.962128i	$-0.587883\pi$
$211$	−7.91941	−0.545195	−0.272597	+	0.962128i	$0.587883\pi$
$212$	0	0
$213$	−0.562645	−0.0385518
$214$	0	0
$215$	6.37145	0.434529
$216$	0	0
$217$	6.05473	0.411022
$218$	0	0
$219$	−2.34181	−0.158245
$220$	0	0
$221$	12.3538	0.831006
$222$	0	0
$223$	21.7218	1.45460	0.727299	−	0.686321i	$-0.240776\pi$
$223$	21.7218	1.45460	0.727299	+	0.686321i	$0.240776\pi$
$224$	0	0
$225$	−2.94897	−0.196598
$226$	0	0
$227$	2.20564	0.146394	0.0731969	−	0.997318i	$-0.476680\pi$
$227$	2.20564	0.146394	0.0731969	+	0.997318i	$0.476680\pi$
$228$	0	0
$229$	27.6054	1.82422	0.912109	−	0.409947i	$-0.134453\pi$
$229$	27.6054	1.82422	0.912109	+	0.409947i	$0.134453\pi$
$230$	0	0
$231$	3.70115	0.243518
$232$	0	0
$233$	17.7960	1.16585	0.582927	−	0.812525i	$-0.301907\pi$
$233$	17.7960	1.16585	0.582927	+	0.812525i	$0.301907\pi$
$234$	0	0
$235$	8.80605	0.574444
$236$	0	0
$237$	−1.98083	−0.128669
$238$	0	0
$239$	8.02414	0.519038	0.259519	−	0.965738i	$-0.416436\pi$
$239$	8.02414	0.519038	0.259519	+	0.965738i	$0.416436\pi$
$240$	0	0
$241$	−3.77681	−0.243285	−0.121643	−	0.992574i	$-0.538816\pi$
$241$	−3.77681	−0.243285	−0.121643	+	0.992574i	$0.538816\pi$
$242$	0	0
$243$	5.96150	0.382431
$244$	0	0
$245$	−0.505112	−0.0322704
$246$	0	0
$247$	−29.4713	−1.87522
$248$	0	0
$249$	−1.01011	−0.0640133
$250$	0	0
$251$	−10.9118	−0.688746	−0.344373	−	0.938833i	$-0.611909\pi$
$251$	−10.9118	−0.688746	−0.344373	+	0.938833i	$0.611909\pi$
$252$	0	0
$253$	35.2808	2.21809
$254$	0	0
$255$	0.444404	0.0278296
$256$	0	0
$257$	4.74353	0.295894	0.147947	−	0.988995i	$-0.452734\pi$
$257$	4.74353	0.295894	0.147947	+	0.988995i	$0.452734\pi$
$258$	0	0
$259$	3.06974	0.190745
$260$	0	0
$261$	7.96890	0.493263
$262$	0	0
$263$	6.19057	0.381727	0.190864	−	0.981617i	$-0.438871\pi$
$263$	6.19057	0.381727	0.190864	+	0.981617i	$0.438871\pi$
$264$	0	0
$265$	13.8904	0.853280
$266$	0	0
$267$	−2.13518	−0.130671
$268$	0	0
$269$	20.4490	1.24680	0.623399	−	0.781904i	$-0.285751\pi$
$269$	20.4490	1.24680	0.623399	+	0.781904i	$0.285751\pi$
$270$	0	0
$271$	−8.39144	−0.509744	−0.254872	−	0.966975i	$-0.582033\pi$
$271$	−8.39144	−0.509744	−0.254872	+	0.966975i	$0.582033\pi$
$272$	0	0
$273$	3.61521	0.218802
$274$	0	0
$275$	6.42893	0.387679
$276$	0	0
$277$	20.1280	1.20937	0.604687	−	0.796463i	$-0.293298\pi$
$277$	20.1280	1.20937	0.604687	+	0.796463i	$0.293298\pi$
$278$	0	0
$279$	−7.00615	−0.419447
$280$	0	0
$281$	−5.82695	−0.347607	−0.173803	−	0.984780i	$-0.555606\pi$
$281$	−5.82695	−0.347607	−0.173803	+	0.984780i	$0.555606\pi$
$282$	0	0
$283$	−26.1371	−1.55369	−0.776845	−	0.629692i	$-0.783181\pi$
$283$	−26.1371	−1.55369	−0.776845	+	0.629692i	$0.783181\pi$
$284$	0	0
$285$	−1.06017	−0.0627993
$286$	0	0
$287$	−17.1204	−1.01059
$288$	0	0
$289$	−13.1298	−0.772343
$290$	0	0
$291$	−2.33251	−0.136734
$292$	0	0
$293$	−16.2876	−0.951532	−0.475766	−	0.879572i	$-0.657829\pi$
$293$	−16.2876	−0.951532	−0.475766	+	0.879572i	$0.657829\pi$
$294$	0	0
$295$	−12.9355	−0.753135
$296$	0	0
$297$	−8.63958	−0.501319
$298$	0	0
$299$	34.4616	1.99297
$300$	0	0
$301$	16.2377	0.935925
$302$	0	0
$303$	−0.908552	−0.0521949
$304$	0	0
$305$	0.393893	0.0225542
$306$	0	0
$307$	−2.75513	−0.157243	−0.0786217	−	0.996905i	$-0.525052\pi$
$307$	−2.75513	−0.157243	−0.0786217	+	0.996905i	$0.525052\pi$
$308$	0	0
$309$	0.117036	0.00665794
$310$	0	0
$311$	−10.5286	−0.597021	−0.298510	−	0.954406i	$-0.596490\pi$
$311$	−10.5286	−0.597021	−0.298510	+	0.954406i	$0.596490\pi$
$312$	0	0
$313$	−5.87056	−0.331824	−0.165912	−	0.986141i	$-0.553057\pi$
$313$	−5.87056	−0.331824	−0.165912	+	0.986141i	$0.553057\pi$
$314$	0	0
$315$	−7.51547	−0.423449
$316$	0	0
$317$	34.5318	1.93950	0.969751	−	0.244098i	$-0.0784917\pi$
$317$	34.5318	1.93950	0.969751	+	0.244098i	$0.0784917\pi$
$318$	0	0
$319$	−17.3727	−0.972683
$320$	0	0
$321$	0.842202	0.0470071
$322$	0	0
$323$	−9.23273	−0.513723
$324$	0	0
$325$	6.27964	0.348332
$326$	0	0
$327$	3.16333	0.174933
$328$	0	0
$329$	22.4423	1.23728
$330$	0	0
$331$	−14.2308	−0.782194	−0.391097	−	0.920349i	$-0.627904\pi$
$331$	−14.2308	−0.782194	−0.391097	+	0.920349i	$0.627904\pi$
$332$	0	0
$333$	−3.55211	−0.194655
$334$	0	0
$335$	7.61550	0.416079
$336$	0	0
$337$	19.4326	1.05856	0.529281	−	0.848447i	$-0.322462\pi$
$337$	19.4326	1.05856	0.529281	+	0.848447i	$0.322462\pi$
$338$	0	0
$339$	−2.98379	−0.162057
$340$	0	0
$341$	15.2738	0.827124
$342$	0	0
$343$	−19.1268	−1.03275
$344$	0	0
$345$	1.23969	0.0667426
$346$	0	0
$347$	24.9078	1.33712	0.668559	−	0.743659i	$-0.266911\pi$
$347$	24.9078	1.33712	0.668559	+	0.743659i	$0.266911\pi$
$348$	0	0
$349$	35.1685	1.88253	0.941263	−	0.337674i	$-0.109640\pi$
$349$	35.1685	1.88253	0.941263	+	0.337674i	$0.109640\pi$
$350$	0	0
$351$	−8.43897	−0.450439
$352$	0	0
$353$	−3.34302	−0.177931	−0.0889656	−	0.996035i	$-0.528356\pi$
$353$	−3.34302	−0.177931	−0.0889656	+	0.996035i	$0.528356\pi$
$354$	0	0
$355$	−2.49070	−0.132193
$356$	0	0
$357$	1.13257	0.0599418
$358$	0	0
$359$	−11.5885	−0.611616	−0.305808	−	0.952093i	$-0.598927\pi$
$359$	−11.5885	−0.611616	−0.305808	+	0.952093i	$0.598927\pi$
$360$	0	0
$361$	3.02570	0.159247
$362$	0	0
$363$	6.85173	0.359623
$364$	0	0
$365$	−10.3667	−0.542616
$366$	0	0
$367$	1.02682	0.0535996	0.0267998	−	0.999641i	$-0.491468\pi$
$367$	1.02682	0.0535996	0.0267998	+	0.999641i	$0.491468\pi$
$368$	0	0
$369$	19.8107	1.03130
$370$	0	0
$371$	35.3997	1.83786
$372$	0	0
$373$	5.85542	0.303182	0.151591	−	0.988443i	$-0.451560\pi$
$373$	5.85542	0.303182	0.151591	+	0.988443i	$0.451560\pi$
$374$	0	0
$375$	0.225898	0.0116653
$376$	0	0
$377$	−16.9693	−0.873962
$378$	0	0
$379$	6.95861	0.357440	0.178720	−	0.983900i	$-0.442804\pi$
$379$	6.95861	0.357440	0.178720	+	0.983900i	$0.442804\pi$
$380$	0	0
$381$	−0.0342843	−0.00175644
$382$	0	0
$383$	1.19235	0.0609264	0.0304632	−	0.999536i	$-0.490302\pi$
$383$	1.19235	0.0609264	0.0304632	+	0.999536i	$0.490302\pi$
$384$	0	0
$385$	16.3842	0.835014
$386$	0	0
$387$	−18.7892	−0.955110
$388$	0	0
$389$	−12.9025	−0.654183	−0.327092	−	0.944993i	$-0.606069\pi$
$389$	−12.9025	−0.654183	−0.327092	+	0.944993i	$0.606069\pi$
$390$	0	0
$391$	10.7961	0.545981
$392$	0	0
$393$	−2.40254	−0.121192
$394$	0	0
$395$	−8.76868	−0.441200
$396$	0	0
$397$	10.3359	0.518742	0.259371	−	0.965778i	$-0.416485\pi$
$397$	10.3359	0.518742	0.259371	+	0.965778i	$0.416485\pi$
$398$	0	0
$399$	−2.70186	−0.135262
$400$	0	0
$401$	−29.7060	−1.48345	−0.741723	−	0.670707i	$-0.765991\pi$
$401$	−29.7060	−1.48345	−0.741723	+	0.670707i	$0.765991\pi$
$402$	0	0
$403$	14.9192	0.743176
$404$	0	0
$405$	8.54333	0.424522
$406$	0	0
$407$	7.74381	0.383847
$408$	0	0
$409$	−8.19018	−0.404978	−0.202489	−	0.979285i	$-0.564903\pi$
$409$	−8.19018	−0.404978	−0.202489	+	0.979285i	$0.564903\pi$
$410$	0	0
$411$	−3.48681	−0.171992
$412$	0	0
$413$	−32.9663	−1.62216
$414$	0	0
$415$	−4.47154	−0.219500
$416$	0	0
$417$	−0.517013	−0.0253183
$418$	0	0
$419$	−4.88199	−0.238501	−0.119250	−	0.992864i	$-0.538049\pi$
$419$	−4.88199	−0.238501	−0.119250	+	0.992864i	$0.538049\pi$
$420$	0	0
$421$	2.54796	0.124180	0.0620901	−	0.998071i	$-0.480223\pi$
$421$	2.54796	0.124180	0.0620901	+	0.998071i	$0.480223\pi$
$422$	0	0
$423$	−25.9688	−1.26265
$424$	0	0
$425$	1.96728	0.0954269
$426$	0	0
$427$	1.00384	0.0485791
$428$	0	0
$429$	9.11981	0.440309
$430$	0	0
$431$	24.2881	1.16992	0.584958	−	0.811063i	$-0.301111\pi$
$431$	24.2881	1.16992	0.584958	+	0.811063i	$0.301111\pi$
$432$	0	0
$433$	−5.70538	−0.274183	−0.137092	−	0.990558i	$-0.543775\pi$
$433$	−5.70538	−0.274183	−0.137092	+	0.990558i	$0.543775\pi$
$434$	0	0
$435$	−0.610437	−0.0292682
$436$	0	0
$437$	−25.7552	−1.23204
$438$	0	0
$439$	−11.7091	−0.558845	−0.279422	−	0.960168i	$-0.590143\pi$
$439$	−11.7091	−0.558845	−0.279422	+	0.960168i	$0.590143\pi$
$440$	0	0
$441$	1.48956	0.0709315
$442$	0	0
$443$	−7.12773	−0.338649	−0.169324	−	0.985560i	$-0.554159\pi$
$443$	−7.12773	−0.338649	−0.169324	+	0.985560i	$0.554159\pi$
$444$	0	0
$445$	−9.45198	−0.448067
$446$	0	0
$447$	−4.09999	−0.193923
$448$	0	0
$449$	20.3429	0.960039	0.480019	−	0.877258i	$-0.340630\pi$
$449$	20.3429	0.960039	0.480019	+	0.877258i	$0.340630\pi$
$450$	0	0
$451$	−43.1884	−2.03366
$452$	0	0
$453$	0.225898	0.0106136
$454$	0	0
$455$	16.0037	0.750266
$456$	0	0
$457$	−21.9848	−1.02841	−0.514204	−	0.857668i	$-0.671912\pi$
$457$	−21.9848	−1.02841	−0.514204	+	0.857668i	$0.671912\pi$
$458$	0	0
$459$	−2.64375	−0.123399
$460$	0	0
$461$	−8.65999	−0.403336	−0.201668	−	0.979454i	$-0.564636\pi$
$461$	−8.65999	−0.403336	−0.201668	+	0.979454i	$0.564636\pi$
$462$	0	0
$463$	35.9256	1.66960	0.834802	−	0.550550i	$-0.185582\pi$
$463$	35.9256	1.66960	0.834802	+	0.550550i	$0.185582\pi$
$464$	0	0
$465$	0.536687	0.0248883
$466$	0	0
$467$	−0.286275	−0.0132472	−0.00662362	−	0.999978i	$-0.502108\pi$
$467$	−0.286275	−0.0132472	−0.00662362	+	0.999978i	$0.502108\pi$
$468$	0	0
$469$	19.4081	0.896185
$470$	0	0
$471$	−2.76519	−0.127413
$472$	0	0
$473$	40.9616	1.88342
$474$	0	0
$475$	−4.69315	−0.215337
$476$	0	0
$477$	−40.9623	−1.87554
$478$	0	0
$479$	−24.0064	−1.09688	−0.548440	−	0.836190i	$-0.684778\pi$
$479$	−24.0064	−1.09688	−0.548440	+	0.836190i	$0.684778\pi$
$480$	0	0
$481$	7.56400	0.344889
$482$	0	0
$483$	3.15936	0.143756
$484$	0	0
$485$	−10.3255	−0.468858
$486$	0	0
$487$	−25.3192	−1.14732	−0.573661	−	0.819093i	$-0.694477\pi$
$487$	−25.3192	−1.14732	−0.573661	+	0.819093i	$0.694477\pi$
$488$	0	0
$489$	−2.09023	−0.0945234
$490$	0	0
$491$	7.97819	0.360051	0.180025	−	0.983662i	$-0.442382\pi$
$491$	7.97819	0.360051	0.180025	+	0.983662i	$0.442382\pi$
$492$	0	0
$493$	−5.31611	−0.239425
$494$	0	0
$495$	−18.9587	−0.852131
$496$	0	0
$497$	−6.34757	−0.284728
$498$	0	0
$499$	−15.4114	−0.689911	−0.344955	−	0.938619i	$-0.612106\pi$
$499$	−15.4114	−0.689911	−0.344955	+	0.938619i	$0.612106\pi$
$500$	0	0
$501$	−4.27857	−0.191153
$502$	0	0
$503$	−21.2439	−0.947219	−0.473610	−	0.880735i	$-0.657049\pi$
$503$	−21.2439	−0.947219	−0.473610	+	0.880735i	$0.657049\pi$
$504$	0	0
$505$	−4.02196	−0.178975
$506$	0	0
$507$	5.97137	0.265198
$508$	0	0
$509$	31.3295	1.38865	0.694327	−	0.719660i	$-0.255702\pi$
$509$	31.3295	1.38865	0.694327	+	0.719660i	$0.255702\pi$
$510$	0	0
$511$	−26.4195	−1.16873
$512$	0	0
$513$	6.30694	0.278458
$514$	0	0
$515$	0.518092	0.0228299
$516$	0	0
$517$	56.6135	2.48986
$518$	0	0
$519$	4.39565	0.192948
$520$	0	0
$521$	14.3209	0.627411	0.313705	−	0.949520i	$-0.398430\pi$
$521$	14.3209	0.627411	0.313705	+	0.949520i	$0.398430\pi$
$522$	0	0
$523$	0.235702	0.0103065	0.00515326	−	0.999987i	$-0.498360\pi$
$523$	0.235702	0.0103065	0.00515326	+	0.999987i	$0.498360\pi$
$524$	0	0
$525$	0.575703	0.0251257
$526$	0	0
$527$	4.67385	0.203596
$528$	0	0
$529$	7.11625	0.309402
$530$	0	0
$531$	38.1465	1.65542
$532$	0	0
$533$	−42.1855	−1.82726
$534$	0	0
$535$	3.72824	0.161186
$536$	0	0
$537$	1.54721	0.0667672
$538$	0	0
$539$	−3.24733	−0.139872
$540$	0	0
$541$	6.45227	0.277405	0.138702	−	0.990334i	$-0.455707\pi$
$541$	6.45227	0.277405	0.138702	+	0.990334i	$0.455707\pi$
$542$	0	0
$543$	−3.65563	−0.156878
$544$	0	0
$545$	14.0034	0.599838
$546$	0	0
$547$	−46.4073	−1.98423	−0.992116	−	0.125322i	$-0.960003\pi$
$547$	−46.4073	−1.98423	−0.992116	+	0.125322i	$0.960003\pi$
$548$	0	0
$549$	−1.16158	−0.0495749
$550$	0	0
$551$	12.6822	0.540278
$552$	0	0
$553$	−22.3471	−0.950293
$554$	0	0
$555$	0.272100	0.0115500
$556$	0	0
$557$	−19.2407	−0.815256	−0.407628	−	0.913148i	$-0.633644\pi$
$557$	−19.2407	−0.815256	−0.407628	+	0.913148i	$0.633644\pi$
$558$	0	0
$559$	40.0105	1.69226
$560$	0	0
$561$	2.85704	0.120624
$562$	0	0
$563$	−12.7424	−0.537027	−0.268514	−	0.963276i	$-0.586532\pi$
$563$	−12.7424	−0.537027	−0.268514	+	0.963276i	$0.586532\pi$
$564$	0	0
$565$	−13.2086	−0.555689
$566$	0	0
$567$	21.7727	0.914370
$568$	0	0
$569$	−4.82202	−0.202150	−0.101075	−	0.994879i	$-0.532228\pi$
$569$	−4.82202	−0.202150	−0.101075	+	0.994879i	$0.532228\pi$
$570$	0	0
$571$	2.06222	0.0863011	0.0431506	−	0.999069i	$-0.486260\pi$
$571$	2.06222	0.0863011	0.0431506	+	0.999069i	$0.486260\pi$
$572$	0	0
$573$	3.68272	0.153848
$574$	0	0
$575$	5.48783	0.228858
$576$	0	0
$577$	39.3151	1.63671	0.818354	−	0.574715i	$-0.194887\pi$
$577$	39.3151	1.63671	0.818354	+	0.574715i	$0.194887\pi$
$578$	0	0
$579$	5.29391	0.220008
$580$	0	0
$581$	−11.3958	−0.472776
$582$	0	0
$583$	89.3003	3.69844
$584$	0	0
$585$	−18.5185	−0.765645
$586$	0	0
$587$	−42.5516	−1.75629	−0.878145	−	0.478394i	$-0.841219\pi$
$587$	−42.5516	−1.75629	−0.878145	+	0.478394i	$0.841219\pi$
$588$	0	0
$589$	−11.1500	−0.459427
$590$	0	0
$591$	5.27812	0.217113
$592$	0	0
$593$	−0.637618	−0.0261838	−0.0130919	−	0.999914i	$-0.504167\pi$
$593$	−0.637618	−0.0261838	−0.0130919	+	0.999914i	$0.504167\pi$
$594$	0	0
$595$	5.01362	0.205538
$596$	0	0
$597$	−6.31885	−0.258613
$598$	0	0
$599$	40.8816	1.67038	0.835188	−	0.549964i	$-0.185359\pi$
$599$	40.8816	1.67038	0.835188	+	0.549964i	$0.185359\pi$
$600$	0	0
$601$	32.8920	1.34169	0.670846	−	0.741597i	$-0.265931\pi$
$601$	32.8920	1.34169	0.670846	+	0.741597i	$0.265931\pi$
$602$	0	0
$603$	−22.4579	−0.914555
$604$	0	0
$605$	30.3311	1.23313
$606$	0	0
$607$	5.82385	0.236383	0.118191	−	0.992991i	$-0.462290\pi$
$607$	5.82385	0.236383	0.118191	+	0.992991i	$0.462290\pi$
$608$	0	0
$609$	−1.55570	−0.0630402
$610$	0	0
$611$	55.2989	2.23715
$612$	0	0
$613$	3.04859	0.123131	0.0615656	−	0.998103i	$-0.480391\pi$
$613$	3.04859	0.123131	0.0615656	+	0.998103i	$0.480391\pi$
$614$	0	0
$615$	−1.51754	−0.0611932
$616$	0	0
$617$	23.2175	0.934703	0.467351	−	0.884072i	$-0.345208\pi$
$617$	23.2175	0.934703	0.467351	+	0.884072i	$0.345208\pi$
$618$	0	0
$619$	−9.70997	−0.390277	−0.195138	−	0.980776i	$-0.562516\pi$
$619$	−9.70997	−0.390277	−0.195138	+	0.980776i	$0.562516\pi$
$620$	0	0
$621$	−7.37487	−0.295943
$622$	0	0
$623$	−24.0884	−0.965083
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.81578	−0.272196
$628$	0	0
$629$	2.36964	0.0944836
$630$	0	0
$631$	−33.4095	−1.33001	−0.665005	−	0.746839i	$-0.731571\pi$
$631$	−33.4095	−1.33001	−0.665005	+	0.746839i	$0.731571\pi$
$632$	0	0
$633$	−1.78898	−0.0711056
$634$	0	0
$635$	−0.151769	−0.00602277
$636$	0	0
$637$	−3.17192	−0.125676
$638$	0	0
$639$	7.34501	0.290564
$640$	0	0
$641$	−12.7878	−0.505087	−0.252543	−	0.967586i	$-0.581267\pi$
$641$	−12.7878	−0.505087	−0.252543	+	0.967586i	$0.581267\pi$
$642$	0	0
$643$	−27.0200	−1.06556	−0.532782	−	0.846253i	$-0.678853\pi$
$643$	−27.0200	−1.06556	−0.532782	+	0.846253i	$0.678853\pi$
$644$	0	0
$645$	1.43930	0.0566723
$646$	0	0
$647$	0.406091	0.0159651	0.00798254	−	0.999968i	$-0.497459\pi$
$647$	0.406091	0.0159651	0.00798254	+	0.999968i	$0.497459\pi$
$648$	0	0
$649$	−83.1615	−3.26437
$650$	0	0
$651$	1.36775	0.0536064
$652$	0	0
$653$	30.0521	1.17603	0.588016	−	0.808850i	$-0.299909\pi$
$653$	30.0521	1.17603	0.588016	+	0.808850i	$0.299909\pi$
$654$	0	0
$655$	−10.6355	−0.415564
$656$	0	0
$657$	30.5710	1.19269
$658$	0	0
$659$	33.8083	1.31698	0.658491	−	0.752588i	$-0.271195\pi$
$659$	33.8083	1.31698	0.658491	+	0.752588i	$0.271195\pi$
$660$	0	0
$661$	−36.6213	−1.42440	−0.712202	−	0.701974i	$-0.752302\pi$
$661$	−36.6213	−1.42440	−0.712202	+	0.701974i	$0.752302\pi$
$662$	0	0
$663$	2.79070	0.108382
$664$	0	0
$665$	−11.9605	−0.463810
$666$	0	0
$667$	−14.8296	−0.574203
$668$	0	0
$669$	4.90690	0.189712
$670$	0	0
$671$	2.53231	0.0977586
$672$	0	0
$673$	−5.13627	−0.197989	−0.0989944	−	0.995088i	$-0.531563\pi$
$673$	−5.13627	−0.197989	−0.0989944	+	0.995088i	$0.531563\pi$
$674$	0	0
$675$	−1.34386	−0.0517252
$676$	0	0
$677$	14.5833	0.560481	0.280241	−	0.959930i	$-0.409586\pi$
$677$	14.5833	0.560481	0.280241	+	0.959930i	$0.409586\pi$
$678$	0	0
$679$	−26.3147	−1.00986
$680$	0	0
$681$	0.498251	0.0190930
$682$	0	0
$683$	50.0086	1.91353	0.956764	−	0.290866i	$-0.0939435\pi$
$683$	50.0086	1.91353	0.956764	+	0.290866i	$0.0939435\pi$
$684$	0	0
$685$	−15.4353	−0.589753
$686$	0	0
$687$	6.23601	0.237919
$688$	0	0
$689$	87.2267	3.32307
$690$	0	0
$691$	10.4496	0.397523	0.198761	−	0.980048i	$-0.436308\pi$
$691$	10.4496	0.397523	0.198761	+	0.980048i	$0.436308\pi$
$692$	0	0
$693$	−48.3164	−1.83539
$694$	0	0
$695$	−2.28870	−0.0868154
$696$	0	0
$697$	−13.2158	−0.500585
$698$	0	0
$699$	4.02008	0.152053
$700$	0	0
$701$	23.4512	0.885740	0.442870	−	0.896586i	$-0.353960\pi$
$701$	23.4512	0.885740	0.442870	+	0.896586i	$0.353960\pi$
$702$	0	0
$703$	−5.65303	−0.213208
$704$	0	0
$705$	1.98927	0.0749202
$706$	0	0
$707$	−10.2500	−0.385490
$708$	0	0
$709$	−42.2275	−1.58589	−0.792943	−	0.609296i	$-0.791452\pi$
$709$	−42.2275	−1.58589	−0.792943	+	0.609296i	$0.791452\pi$
$710$	0	0
$711$	25.8586	0.969773
$712$	0	0
$713$	13.0380	0.488275
$714$	0	0
$715$	40.3714	1.50980
$716$	0	0
$717$	1.81264	0.0676942
$718$	0	0
$719$	3.34602	0.124786	0.0623928	−	0.998052i	$-0.480127\pi$
$719$	3.34602	0.124786	0.0623928	+	0.998052i	$0.480127\pi$
$720$	0	0
$721$	1.32036	0.0491728
$722$	0	0
$723$	−0.853173	−0.0317298
$724$	0	0
$725$	−2.70227	−0.100360
$726$	0	0
$727$	−27.7567	−1.02944	−0.514719	−	0.857359i	$-0.672104\pi$
$727$	−27.7567	−1.02944	−0.514719	+	0.857359i	$0.672104\pi$
$728$	0	0
$729$	−24.2833	−0.899382
$730$	0	0
$731$	12.5344	0.463602
$732$	0	0
$733$	16.6782	0.616024	0.308012	−	0.951382i	$-0.400336\pi$
$733$	16.6782	0.616024	0.308012	+	0.951382i	$0.400336\pi$
$734$	0	0
$735$	−0.114104	−0.00420878
$736$	0	0
$737$	48.9595	1.80345
$738$	0	0
$739$	18.3654	0.675582	0.337791	−	0.941221i	$-0.390320\pi$
$739$	18.3654	0.675582	0.337791	+	0.941221i	$0.390320\pi$
$740$	0	0
$741$	−6.65752	−0.244570
$742$	0	0
$743$	25.1861	0.923989	0.461995	−	0.886883i	$-0.347134\pi$
$743$	25.1861	0.923989	0.461995	+	0.886883i	$0.347134\pi$
$744$	0	0
$745$	−18.1498	−0.664956
$746$	0	0
$747$	13.1865	0.482467
$748$	0	0
$749$	9.50145	0.347175
$750$	0	0
$751$	20.5196	0.748770	0.374385	−	0.927273i	$-0.377854\pi$
$751$	20.5196	0.748770	0.374385	+	0.927273i	$0.377854\pi$
$752$	0	0
$753$	−2.46495	−0.0898279
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	2.40151	0.0872844	0.0436422	−	0.999047i	$-0.486104\pi$
$757$	2.40151	0.0872844	0.0436422	+	0.999047i	$0.486104\pi$
$758$	0	0
$759$	7.96987	0.289288
$760$	0	0
$761$	42.3083	1.53368	0.766838	−	0.641840i	$-0.221829\pi$
$761$	42.3083	1.53368	0.766838	+	0.641840i	$0.221829\pi$
$762$	0	0
$763$	35.6877	1.29198
$764$	0	0
$765$	−5.80144	−0.209751
$766$	0	0
$767$	−81.2305	−2.93306
$768$	0	0
$769$	−17.9950	−0.648917	−0.324458	−	0.945900i	$-0.605182\pi$
$769$	−17.9950	−0.648917	−0.324458	+	0.945900i	$0.605182\pi$
$770$	0	0
$771$	1.07155	0.0385911
$772$	0	0
$773$	4.28042	0.153956	0.0769781	−	0.997033i	$-0.475473\pi$
$773$	4.28042	0.153956	0.0769781	+	0.997033i	$0.475473\pi$
$774$	0	0
$775$	2.37580	0.0853411
$776$	0	0
$777$	0.693449	0.0248774
$778$	0	0
$779$	31.5278	1.12960
$780$	0	0
$781$	−16.0125	−0.572974
$782$	0	0
$783$	3.63147	0.129778
$784$	0	0
$785$	−12.2409	−0.436896
$786$	0	0
$787$	−17.2732	−0.615724	−0.307862	−	0.951431i	$-0.599613\pi$
$787$	−17.2732	−0.615724	−0.307862	+	0.951431i	$0.599613\pi$
$788$	0	0
$789$	1.39844	0.0497857
$790$	0	0
$791$	−33.6621	−1.19689
$792$	0	0
$793$	2.47351	0.0878368
$794$	0	0
$795$	3.13781	0.111287
$796$	0	0
$797$	−12.1084	−0.428903	−0.214451	−	0.976735i	$-0.568796\pi$
$797$	−12.1084	−0.428903	−0.214451	+	0.976735i	$0.568796\pi$
$798$	0	0
$799$	17.3239	0.612877
$800$	0	0
$801$	27.8736	0.984866
$802$	0	0
$803$	−66.6465	−2.35190
$804$	0	0
$805$	13.9858	0.492934
$806$	0	0
$807$	4.61939	0.162610
$808$	0	0
$809$	3.28452	0.115478	0.0577388	−	0.998332i	$-0.481611\pi$
$809$	3.28452	0.115478	0.0577388	+	0.998332i	$0.481611\pi$
$810$	0	0
$811$	−9.05343	−0.317909	−0.158955	−	0.987286i	$-0.550812\pi$
$811$	−9.05343	−0.317909	−0.158955	+	0.987286i	$0.550812\pi$
$812$	0	0
$813$	−1.89561	−0.0664819
$814$	0	0
$815$	−9.25298	−0.324118
$816$	0	0
$817$	−29.9022	−1.04615
$818$	0	0
$819$	−47.1945	−1.64911
$820$	0	0
$821$	−28.9824	−1.01149	−0.505746	−	0.862683i	$-0.668783\pi$
$821$	−28.9824	−1.01149	−0.505746	+	0.862683i	$0.668783\pi$
$822$	0	0
$823$	−1.75812	−0.0612842	−0.0306421	−	0.999530i	$-0.509755\pi$
$823$	−1.75812	−0.0612842	−0.0306421	+	0.999530i	$0.509755\pi$
$824$	0	0
$825$	1.45228	0.0505620
$826$	0	0
$827$	37.4975	1.30392	0.651958	−	0.758255i	$-0.273948\pi$
$827$	37.4975	1.30392	0.651958	+	0.758255i	$0.273948\pi$
$828$	0	0
$829$	−43.9828	−1.52759	−0.763793	−	0.645461i	$-0.776665\pi$
$829$	−43.9828	−1.52759	−0.763793	+	0.645461i	$0.776665\pi$
$830$	0	0
$831$	4.54688	0.157729
$832$	0	0
$833$	−0.993695	−0.0344295
$834$	0	0
$835$	−18.9403	−0.655456
$836$	0	0
$837$	−3.19274	−0.110357
$838$	0	0
$839$	32.8734	1.13491	0.567457	−	0.823403i	$-0.307927\pi$
$839$	32.8734	1.13491	0.567457	+	0.823403i	$0.307927\pi$
$840$	0	0
$841$	−21.6978	−0.748198
$842$	0	0
$843$	−1.31630	−0.0453356
$844$	0	0
$845$	26.4339	0.909355
$846$	0	0
$847$	77.2990	2.65603
$848$	0	0
$849$	−5.90432	−0.202636
$850$	0	0
$851$	6.61023	0.226596
$852$	0	0
$853$	−41.8772	−1.43385	−0.716923	−	0.697152i	$-0.754450\pi$
$853$	−41.8772	−1.43385	−0.716923	+	0.697152i	$0.754450\pi$
$854$	0	0
$855$	13.8400	0.473317
$856$	0	0
$857$	6.37025	0.217604	0.108802	−	0.994063i	$-0.465299\pi$
$857$	6.37025	0.217604	0.108802	+	0.994063i	$0.465299\pi$
$858$	0	0
$859$	36.2751	1.23769	0.618846	−	0.785512i	$-0.287601\pi$
$859$	36.2751	1.23769	0.618846	+	0.785512i	$0.287601\pi$
$860$	0	0
$861$	−3.86747	−0.131803
$862$	0	0
$863$	−50.1150	−1.70593	−0.852967	−	0.521964i	$-0.825199\pi$
$863$	−50.1150	−1.70593	−0.852967	+	0.521964i	$0.825199\pi$
$864$	0	0
$865$	19.4586	0.661611
$866$	0	0
$867$	−2.96600	−0.100731
$868$	0	0
$869$	−56.3732	−1.91233
$870$	0	0
$871$	47.8226	1.62041
$872$	0	0
$873$	30.4497	1.03056
$874$	0	0
$875$	2.54851	0.0861553
$876$	0	0
$877$	38.1685	1.28886	0.644429	−	0.764664i	$-0.277095\pi$
$877$	38.1685	1.28886	0.644429	+	0.764664i	$0.277095\pi$
$878$	0	0
$879$	−3.67934	−0.124101
$880$	0	0
$881$	26.7317	0.900613	0.450306	−	0.892874i	$-0.351315\pi$
$881$	26.7317	0.900613	0.450306	+	0.892874i	$0.351315\pi$
$882$	0	0
$883$	−23.8503	−0.802626	−0.401313	−	0.915941i	$-0.631446\pi$
$883$	−23.8503	−0.802626	−0.401313	+	0.915941i	$0.631446\pi$
$884$	0	0
$885$	−2.92211	−0.0982256
$886$	0	0
$887$	35.6248	1.19616	0.598082	−	0.801435i	$-0.295930\pi$
$887$	35.6248	1.19616	0.598082	+	0.801435i	$0.295930\pi$
$888$	0	0
$889$	−0.386784	−0.0129723
$890$	0	0
$891$	54.9245	1.84004
$892$	0	0
$893$	−41.3282	−1.38299
$894$	0	0
$895$	6.84917	0.228943
$896$	0	0
$897$	7.78481	0.259927
$898$	0	0
$899$	−6.42003	−0.214120
$900$	0	0
$901$	27.3262	0.910369
$902$	0	0
$903$	3.66806	0.122065
$904$	0	0
$905$	−16.1827	−0.537930
$906$	0	0
$907$	−37.6364	−1.24970	−0.624848	−	0.780747i	$-0.714839\pi$
$907$	−37.6364	−1.24970	−0.624848	+	0.780747i	$0.714839\pi$
$908$	0	0
$909$	11.8606	0.393392
$910$	0	0
$911$	−55.3805	−1.83484	−0.917419	−	0.397923i	$-0.869731\pi$
$911$	−55.3805	−1.83484	−0.917419	+	0.397923i	$0.869731\pi$
$912$	0	0
$913$	−28.7472	−0.951395
$914$	0	0
$915$	0.0889795	0.00294157
$916$	0	0
$917$	−27.1047	−0.895076
$918$	0	0
$919$	−25.8038	−0.851189	−0.425594	−	0.904914i	$-0.639935\pi$
$919$	−25.8038	−0.851189	−0.425594	+	0.904914i	$0.639935\pi$
$920$	0	0
$921$	−0.622378	−0.0205080
$922$	0	0
$923$	−15.6407	−0.514821
$924$	0	0
$925$	1.20453	0.0396046
$926$	0	0
$927$	−1.52784	−0.0501808
$928$	0	0
$929$	4.99357	0.163834	0.0819168	−	0.996639i	$-0.473896\pi$
$929$	4.99357	0.163834	0.0819168	+	0.996639i	$0.473896\pi$
$930$	0	0
$931$	2.37057	0.0776923
$932$	0	0
$933$	−2.37838	−0.0778648
$934$	0	0
$935$	12.6475	0.413617
$936$	0	0
$937$	31.0935	1.01578	0.507891	−	0.861421i	$-0.330425\pi$
$937$	31.0935	1.01578	0.507891	+	0.861421i	$0.330425\pi$
$938$	0	0
$939$	−1.32615	−0.0432772
$940$	0	0
$941$	−25.5664	−0.833442	−0.416721	−	0.909034i	$-0.636821\pi$
$941$	−25.5664	−0.833442	−0.416721	+	0.909034i	$0.636821\pi$
$942$	0	0
$943$	−36.8663	−1.20053
$944$	0	0
$945$	−3.42484	−0.111410
$946$	0	0
$947$	29.1634	0.947685	0.473842	−	0.880610i	$-0.342867\pi$
$947$	29.1634	0.947685	0.473842	+	0.880610i	$0.342867\pi$
$948$	0	0
$949$	−65.0990	−2.11320
$950$	0	0
$951$	7.80067	0.252954
$952$	0	0
$953$	−21.9153	−0.709906	−0.354953	−	0.934884i	$-0.615503\pi$
$953$	−21.9153	−0.709906	−0.354953	+	0.934884i	$0.615503\pi$
$954$	0	0
$955$	16.3026	0.527539
$956$	0	0
$957$	−3.92445	−0.126860
$958$	0	0
$959$	−39.3370	−1.27026
$960$	0	0
$961$	−25.3556	−0.817922
$962$	0	0
$963$	−10.9945	−0.354292
$964$	0	0
$965$	23.4350	0.754399
$966$	0	0
$967$	−40.5707	−1.30467	−0.652333	−	0.757933i	$-0.726209\pi$
$967$	−40.5707	−1.30467	−0.652333	+	0.757933i	$0.726209\pi$
$968$	0	0
$969$	−2.08566	−0.0670009
$970$	0	0
$971$	−39.7809	−1.27663	−0.638315	−	0.769775i	$-0.720368\pi$
$971$	−39.7809	−1.27663	−0.638315	+	0.769775i	$0.720368\pi$
$972$	0	0
$973$	−5.83277	−0.186990
$974$	0	0
$975$	1.41856	0.0454303
$976$	0	0
$977$	−3.83667	−0.122746	−0.0613730	−	0.998115i	$-0.519548\pi$
$977$	−3.83667	−0.122746	−0.0613730	+	0.998115i	$0.519548\pi$
$978$	0	0
$979$	−60.7661	−1.94209
$980$	0	0
$981$	−41.2955	−1.31846
$982$	0	0
$983$	25.8052	0.823059	0.411530	−	0.911396i	$-0.364995\pi$
$983$	25.8052	0.823059	0.411530	+	0.911396i	$0.364995\pi$
$984$	0	0
$985$	23.3651	0.744473
$986$	0	0
$987$	5.06967	0.161369
$988$	0	0
$989$	34.9654	1.11184
$990$	0	0
$991$	−3.21775	−0.102215	−0.0511076	−	0.998693i	$-0.516275\pi$
$991$	−3.21775	−0.102215	−0.0511076	+	0.998693i	$0.516275\pi$
$992$	0	0
$993$	−3.21470	−0.102016
$994$	0	0
$995$	−27.9721	−0.886776
$996$	0	0
$997$	−36.7029	−1.16239	−0.581195	−	0.813764i	$-0.697415\pi$
$997$	−36.7029	−1.16239	−0.581195	+	0.813764i	$0.697415\pi$
$998$	0	0
$999$	−1.61872	−0.0512139

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.s.1.13	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.s.1.13	✓	24	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q+0.225898 q^{3} +1.00000 q^{5} +2.54851 q^{7} -2.94897 q^{9} +O(q^{10})\)	\(q+0.225898 q^{3} +1.00000 q^{5} +2.54851 q^{7} -2.94897 q^{9} +6.42893 q^{11} +6.27964 q^{13} +0.225898 q^{15} +1.96728 q^{17} -4.69315 q^{19} +0.575703 q^{21} +5.48783 q^{23} +1.00000 q^{25} -1.34386 q^{27} -2.70227 q^{29} +2.37580 q^{31} +1.45228 q^{33} +2.54851 q^{35} +1.20453 q^{37} +1.41856 q^{39} -6.71782 q^{41} +6.37145 q^{43} -2.94897 q^{45} +8.80605 q^{47} -0.505112 q^{49} +0.444404 q^{51} +13.8904 q^{53} +6.42893 q^{55} -1.06017 q^{57} -12.9355 q^{59} +0.393893 q^{61} -7.51547 q^{63} +6.27964 q^{65} +7.61550 q^{67} +1.23969 q^{69} -2.49070 q^{71} -10.3667 q^{73} +0.225898 q^{75} +16.3842 q^{77} -8.76868 q^{79} +8.54333 q^{81} -4.47154 q^{83} +1.96728 q^{85} -0.610437 q^{87} -9.45198 q^{89} +16.0037 q^{91} +0.536687 q^{93} -4.69315 q^{95} -10.3255 q^{97} -18.9587 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) \) 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9	\( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) \) 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9 + 17 * q^11 + 16 * q^13 + 2 * q^15 + 22 * q^17 + 16 * q^19 - q^21 + 7 * q^23 + 24 * q^25 - 4 * q^27 + 25 * q^29 + 28 * q^31 + 11 * q^33 + 3 * q^35 + 26 * q^37 + 13 * q^39 + 38 * q^41 - 13 * q^43 + 40 * q^45 + 12 * q^47 + 61 * q^49 + 53 * q^53 + 17 * q^55 + 30 * q^57 + 35 * q^59 + 44 * q^61 - 9 * q^63 + 16 * q^65 - 15 * q^67 + 9 * q^69 + 22 * q^71 + 31 * q^73 + 2 * q^75 + 26 * q^77 + 20 * q^79 + 88 * q^81 - 14 * q^83 + 22 * q^85 - 18 * q^87 + 37 * q^89 - 26 * q^91 + 13 * q^93 + 16 * q^95 + 21 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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