LMFDB - Embedded newform 1860.2.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1860,2,Mod(1,1860)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1860.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1860.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$14.8521747760$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.224148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} + 9x + 14 $ x^4 - x^3 - 11x^2 + 9x + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.829933$ of defining polynomial
Character	$\chi$	$=$	1860.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +1.61894 q^{7} +1.00000 q^{9} -1.65987 q^{11} +5.27881 q^{13} -1.00000 q^{15} +4.48128 q^{17} -1.61894 q^{21} -4.48128 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.76009 q^{29} +1.00000 q^{31} +1.65987 q^{33} +1.61894 q^{35} +3.61894 q^{37} -5.27881 q^{39} +3.65987 q^{41} +1.65987 q^{43} +1.00000 q^{45} -0.756606 q^{47} -4.37903 q^{49} -4.48128 q^{51} -2.75661 q^{53} -1.65987 q^{55} +11.4848 q^{59} +14.2823 q^{61} +1.61894 q^{63} +5.27881 q^{65} -5.00348 q^{67} +4.48128 q^{69} -0.862336 q^{71} -2.93867 q^{73} -1.00000 q^{75} -2.68723 q^{77} +8.41647 q^{79} +1.00000 q^{81} +0.481278 q^{83} +4.48128 q^{85} +3.76009 q^{87} +5.13766 q^{89} +8.54608 q^{91} -1.00000 q^{93} +9.30269 q^{97} -1.65987 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} + 4 q^{21} - 2 q^{23} + 4 q^{25} - 4 q^{27} + 20 q^{29} + 4 q^{31} - 2 q^{33} - 4 q^{35} + 4 q^{37} - 2 q^{39} + 6 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^15 + 2 * q^17 + 4 * q^21 - 2 * q^23 + 4 * q^25 - 4 * q^27 + 20 * q^29 + 4 * q^31 - 2 * q^33 - 4 * q^35 + 4 * q^37 - 2 * q^39 + 6 * q^41 - 2 * q^43 + 4 * q^45 + 2 * q^47 + 28 * q^49 - 2 * q^51 - 6 * q^53 + 2 * q^55 + 8 * q^61 - 4 * q^63 + 2 * q^65 + 10 * q^67 + 2 * q^69 + 2 * q^71 + 16 * q^73 - 4 * q^75 + 32 * q^77 + 20 * q^79 + 4 * q^81 - 14 * q^83 + 2 * q^85 - 20 * q^87 + 26 * q^89 + 16 * q^91 - 4 * q^93 + 14 * q^97 + 2 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.61894	0.611902	0.305951	−	0.952047i	$-0.401026\pi$
$7$	1.61894	0.611902	0.305951	+	0.952047i	$0.401026\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.65987	−0.500469	−0.250234	−	0.968185i	$-0.580508\pi$
$11$	−1.65987	−0.500469	−0.250234	+	0.968185i	$0.580508\pi$
$12$	0	0
$13$	5.27881	1.46408	0.732039	−	0.681263i	$-0.238569\pi$
$13$	5.27881	1.46408	0.732039	+	0.681263i	$0.238569\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	4.48128	1.08687	0.543435	−	0.839451i	$-0.317124\pi$
$17$	4.48128	1.08687	0.543435	+	0.839451i	$0.317124\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−1.61894	−0.353282
$22$	0	0
$23$	−4.48128	−0.934411	−0.467205	−	0.884149i	$-0.654739\pi$
$23$	−4.48128	−0.934411	−0.467205	+	0.884149i	$0.654739\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.76009	−0.698230	−0.349115	−	0.937080i	$-0.613518\pi$
$29$	−3.76009	−0.698230	−0.349115	+	0.937080i	$0.613518\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	1.65987	0.288946
$34$	0	0
$35$	1.61894	0.273651
$36$	0	0
$37$	3.61894	0.594950	0.297475	−	0.954730i	$-0.403855\pi$
$37$	3.61894	0.594950	0.297475	+	0.954730i	$0.403855\pi$
$38$	0	0
$39$	−5.27881	−0.845286
$40$	0	0
$41$	3.65987	0.571575	0.285788	−	0.958293i	$-0.407745\pi$
$41$	3.65987	0.571575	0.285788	+	0.958293i	$0.407745\pi$
$42$	0	0
$43$	1.65987	0.253127	0.126564	−	0.991958i	$-0.459605\pi$
$43$	1.65987	0.253127	0.126564	+	0.991958i	$0.459605\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−0.756606	−0.110362	−0.0551812	−	0.998476i	$-0.517574\pi$
$47$	−0.756606	−0.110362	−0.0551812	+	0.998476i	$0.517574\pi$
$48$	0	0
$49$	−4.37903	−0.625575
$50$	0	0
$51$	−4.48128	−0.627504
$52$	0	0
$53$	−2.75661	−0.378649	−0.189324	−	0.981915i	$-0.560630\pi$
$53$	−2.75661	−0.378649	−0.189324	+	0.981915i	$0.560630\pi$
$54$	0	0
$55$	−1.65987	−0.223816
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.4848	1.49519	0.747594	−	0.664156i	$-0.231209\pi$
$59$	11.4848	1.49519	0.747594	+	0.664156i	$0.231209\pi$
$60$	0	0
$61$	14.2823	1.82866	0.914330	−	0.404970i	$-0.132718\pi$
$61$	14.2823	1.82866	0.914330	+	0.404970i	$0.132718\pi$
$62$	0	0
$63$	1.61894	0.203967
$64$	0	0
$65$	5.27881	0.654756
$66$	0	0
$67$	−5.00348	−0.611272	−0.305636	−	0.952148i	$-0.598869\pi$
$67$	−5.00348	−0.611272	−0.305636	+	0.952148i	$0.598869\pi$
$68$	0	0
$69$	4.48128	0.539482
$70$	0	0
$71$	−0.862336	−0.102340	−0.0511702	−	0.998690i	$-0.516295\pi$
$71$	−0.862336	−0.102340	−0.0511702	+	0.998690i	$0.516295\pi$
$72$	0	0
$73$	−2.93867	−0.343946	−0.171973	−	0.985102i	$-0.555014\pi$
$73$	−2.93867	−0.343946	−0.171973	+	0.985102i	$0.555014\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−2.68723	−0.306238
$78$	0	0
$79$	8.41647	0.946927	0.473464	−	0.880813i	$-0.343004\pi$
$79$	8.41647	0.946927	0.473464	+	0.880813i	$0.343004\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.481278	0.0528271	0.0264135	−	0.999651i	$-0.491591\pi$
$83$	0.481278	0.0528271	0.0264135	+	0.999651i	$0.491591\pi$
$84$	0	0
$85$	4.48128	0.486063
$86$	0	0
$87$	3.76009	0.403124
$88$	0	0
$89$	5.13766	0.544591	0.272296	−	0.962214i	$-0.412217\pi$
$89$	5.13766	0.544591	0.272296	+	0.962214i	$0.412217\pi$
$90$	0	0
$91$	8.54608	0.895873
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.30269	0.944545	0.472272	−	0.881453i	$-0.343434\pi$
$97$	9.30269	0.944545	0.472272	+	0.881453i	$0.343434\pi$
$98$	0	0
$99$	−1.65987	−0.166823
$100$	0	0
$101$	−4.62242	−0.459948	−0.229974	−	0.973197i	$-0.573864\pi$
$101$	−4.62242	−0.459948	−0.229974	+	0.973197i	$0.573864\pi$
$102$	0	0
$103$	0.0409247	0.00403243	0.00201621	−	0.999998i	$-0.499358\pi$
$103$	0.0409247	0.00403243	0.00201621	+	0.999998i	$0.499358\pi$
$104$	0	0
$105$	−1.61894	−0.157993
$106$	0	0
$107$	−2.48128	−0.239874	−0.119937	−	0.992781i	$-0.538269\pi$
$107$	−2.48128	−0.239874	−0.119937	+	0.992781i	$0.538269\pi$
$108$	0	0
$109$	3.85886	0.369611	0.184806	−	0.982775i	$-0.440834\pi$
$109$	3.85886	0.369611	0.184806	+	0.982775i	$0.440834\pi$
$110$	0	0
$111$	−3.61894	−0.343495
$112$	0	0
$113$	−12.5576	−1.18132	−0.590661	−	0.806920i	$-0.701133\pi$
$113$	−12.5576	−1.18132	−0.590661	+	0.806920i	$0.701133\pi$
$114$	0	0
$115$	−4.48128	−0.417881
$116$	0	0
$117$	5.27881	0.488026
$118$	0	0
$119$	7.25493	0.665058
$120$	0	0
$121$	−8.24484	−0.749531
$122$	0	0
$123$	−3.65987	−0.329999
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.55762	−0.581894	−0.290947	−	0.956739i	$-0.593970\pi$
$127$	−6.55762	−0.581894	−0.290947	+	0.956739i	$0.593970\pi$
$128$	0	0
$129$	−1.65987	−0.146143
$130$	0	0
$131$	8.99797	0.786156	0.393078	−	0.919505i	$-0.371410\pi$
$131$	8.99797	0.786156	0.393078	+	0.919505i	$0.371410\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	20.7636	1.77395	0.886976	−	0.461816i	$-0.152802\pi$
$137$	20.7636	1.77395	0.886976	+	0.461816i	$0.152802\pi$
$138$	0	0
$139$	17.0444	1.44569	0.722844	−	0.691011i	$-0.242835\pi$
$139$	17.0444	1.44569	0.722844	+	0.691011i	$0.242835\pi$
$140$	0	0
$141$	0.756606	0.0637177
$142$	0	0
$143$	−8.76212	−0.732725
$144$	0	0
$145$	−3.76009	−0.312258
$146$	0	0
$147$	4.37903	0.361176
$148$	0	0
$149$	17.5850	1.44062	0.720309	−	0.693654i	$-0.244000\pi$
$149$	17.5850	1.44062	0.720309	+	0.693654i	$0.244000\pi$
$150$	0	0
$151$	10.1411	0.825275	0.412637	−	0.910895i	$-0.364608\pi$
$151$	10.1411	0.825275	0.412637	+	0.910895i	$0.364608\pi$
$152$	0	0
$153$	4.48128	0.362290
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	14.1356	1.12815	0.564073	−	0.825725i	$-0.309234\pi$
$157$	14.1356	1.12815	0.564073	+	0.825725i	$0.309234\pi$
$158$	0	0
$159$	2.75661	0.218613
$160$	0	0
$161$	−7.25493	−0.571768
$162$	0	0
$163$	3.27881	0.256816	0.128408	−	0.991721i	$-0.459013\pi$
$163$	3.27881	0.256816	0.128408	+	0.991721i	$0.459013\pi$
$164$	0	0
$165$	1.65987	0.129220
$166$	0	0
$167$	−3.91815	−0.303196	−0.151598	−	0.988442i	$-0.548442\pi$
$167$	−3.91815	−0.303196	−0.151598	+	0.988442i	$0.548442\pi$
$168$	0	0
$169$	14.8658	1.14352
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.23788	−0.398229	−0.199114	−	0.979976i	$-0.563807\pi$
$173$	−5.23788	−0.398229	−0.199114	+	0.979976i	$0.563807\pi$
$174$	0	0
$175$	1.61894	0.122380
$176$	0	0
$177$	−11.4848	−0.863247
$178$	0	0
$179$	−8.28229	−0.619047	−0.309524	−	0.950892i	$-0.600170\pi$
$179$	−8.28229	−0.619047	−0.309524	+	0.950892i	$0.600170\pi$
$180$	0	0
$181$	14.8329	1.10252	0.551262	−	0.834332i	$-0.314146\pi$
$181$	14.8329	1.10252	0.551262	+	0.834332i	$0.314146\pi$
$182$	0	0
$183$	−14.2823	−1.05578
$184$	0	0
$185$	3.61894	0.266070
$186$	0	0
$187$	−7.43832	−0.543944
$188$	0	0
$189$	−1.61894	−0.117761
$190$	0	0
$191$	5.07982	0.367563	0.183781	−	0.982967i	$-0.441166\pi$
$191$	5.07982	0.367563	0.183781	+	0.982967i	$0.441166\pi$
$192$	0	0
$193$	−17.4554	−1.25646	−0.628232	−	0.778026i	$-0.716221\pi$
$193$	−17.4554	−1.25646	−0.628232	+	0.778026i	$0.716221\pi$
$194$	0	0
$195$	−5.27881	−0.378023
$196$	0	0
$197$	5.16154	0.367745	0.183872	−	0.982950i	$-0.441137\pi$
$197$	5.16154	0.367745	0.183872	+	0.982950i	$0.441137\pi$
$198$	0	0
$199$	21.4609	1.52132	0.760661	−	0.649150i	$-0.224875\pi$
$199$	21.4609	1.52132	0.760661	+	0.649150i	$0.224875\pi$
$200$	0	0
$201$	5.00348	0.352918
$202$	0	0
$203$	−6.08736	−0.427249
$204$	0	0
$205$	3.65987	0.255616
$206$	0	0
$207$	−4.48128	−0.311470
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−21.9959	−1.51426	−0.757131	−	0.653263i	$-0.773400\pi$
$211$	−21.9959	−1.51426	−0.757131	+	0.653263i	$0.773400\pi$
$212$	0	0
$213$	0.862336	0.0590863
$214$	0	0
$215$	1.65987	0.113202
$216$	0	0
$217$	1.61894	0.109901
$218$	0	0
$219$	2.93867	0.198577
$220$	0	0
$221$	23.6558	1.59126
$222$	0	0
$223$	0.897750	0.0601178	0.0300589	−	0.999548i	$-0.490431\pi$
$223$	0.897750	0.0601178	0.0300589	+	0.999548i	$0.490431\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−3.99449	−0.265124	−0.132562	−	0.991175i	$-0.542320\pi$
$227$	−3.99449	−0.265124	−0.132562	+	0.991175i	$0.542320\pi$
$228$	0	0
$229$	18.0708	1.19415	0.597077	−	0.802184i	$-0.296329\pi$
$229$	18.0708	1.19415	0.597077	+	0.802184i	$0.296329\pi$
$230$	0	0
$231$	2.68723	0.176807
$232$	0	0
$233$	−26.0014	−1.70341	−0.851706	−	0.524020i	$-0.824432\pi$
$233$	−26.0014	−1.70341	−0.851706	+	0.524020i	$0.824432\pi$
$234$	0	0
$235$	−0.756606	−0.0493555
$236$	0	0
$237$	−8.41647	−0.546709
$238$	0	0
$239$	−21.8603	−1.41403	−0.707013	−	0.707201i	$-0.749958\pi$
$239$	−21.8603	−1.41403	−0.707013	+	0.707201i	$0.749958\pi$
$240$	0	0
$241$	11.8773	0.765087	0.382544	−	0.923937i	$-0.375048\pi$
$241$	11.8773	0.765087	0.382544	+	0.923937i	$0.375048\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.37903	−0.279766
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.481278	−0.0304997
$250$	0	0
$251$	−27.6020	−1.74222	−0.871112	−	0.491084i	$-0.836601\pi$
$251$	−27.6020	−1.74222	−0.871112	+	0.491084i	$0.836601\pi$
$252$	0	0
$253$	7.43832	0.467643
$254$	0	0
$255$	−4.48128	−0.280628
$256$	0	0
$257$	−5.71916	−0.356751	−0.178376	−	0.983962i	$-0.557084\pi$
$257$	−5.71916	−0.356751	−0.178376	+	0.983962i	$0.557084\pi$
$258$	0	0
$259$	5.85886	0.364052
$260$	0	0
$261$	−3.76009	−0.232743
$262$	0	0
$263$	−20.0708	−1.23762	−0.618810	−	0.785541i	$-0.712385\pi$
$263$	−20.0708	−1.23762	−0.618810	+	0.785541i	$0.712385\pi$
$264$	0	0
$265$	−2.75661	−0.169337
$266$	0	0
$267$	−5.13766	−0.314420
$268$	0	0
$269$	−4.60405	−0.280714	−0.140357	−	0.990101i	$-0.544825\pi$
$269$	−4.60405	−0.280714	−0.140357	+	0.990101i	$0.544825\pi$
$270$	0	0
$271$	−22.8399	−1.38743	−0.693713	−	0.720252i	$-0.744026\pi$
$271$	−22.8399	−1.38743	−0.693713	+	0.720252i	$0.744026\pi$
$272$	0	0
$273$	−8.54608	−0.517232
$274$	0	0
$275$	−1.65987	−0.100094
$276$	0	0
$277$	−13.4144	−0.805996	−0.402998	−	0.915201i	$-0.632032\pi$
$277$	−13.4144	−0.805996	−0.402998	+	0.915201i	$0.632032\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−2.08185	−0.124193	−0.0620964	−	0.998070i	$-0.519779\pi$
$281$	−2.08185	−0.124193	−0.0620964	+	0.998070i	$0.519779\pi$
$282$	0	0
$283$	−0.0238807	−0.00141956	−0.000709780	−	1.00000i	$-0.500226\pi$
$283$	−0.0238807	−0.00141956	−0.000709780		1.00000i	$0.500226\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.92511	0.349748
$288$	0	0
$289$	3.08185	0.181285
$290$	0	0
$291$	−9.30269	−0.545333
$292$	0	0
$293$	−7.92366	−0.462905	−0.231453	−	0.972846i	$-0.574348\pi$
$293$	−7.92366	−0.462905	−0.231453	+	0.972846i	$0.574348\pi$
$294$	0	0
$295$	11.4848	0.668668
$296$	0	0
$297$	1.65987	0.0963152
$298$	0	0
$299$	−23.6558	−1.36805
$300$	0	0
$301$	2.68723	0.154889
$302$	0	0
$303$	4.62242	0.265551
$304$	0	0
$305$	14.2823	0.817801
$306$	0	0
$307$	8.30617	0.474058	0.237029	−	0.971503i	$-0.423826\pi$
$307$	8.30617	0.474058	0.237029	+	0.971503i	$0.423826\pi$
$308$	0	0
$309$	−0.0409247	−0.00232812
$310$	0	0
$311$	−28.1002	−1.59342	−0.796709	−	0.604364i	$-0.793427\pi$
$311$	−28.1002	−1.59342	−0.796709	+	0.604364i	$0.793427\pi$
$312$	0	0
$313$	11.8364	0.669034	0.334517	−	0.942390i	$-0.391427\pi$
$313$	11.8364	0.669034	0.334517	+	0.942390i	$0.391427\pi$
$314$	0	0
$315$	1.61894	0.0912170
$316$	0	0
$317$	−3.23237	−0.181548	−0.0907741	−	0.995872i	$-0.528934\pi$
$317$	−3.23237	−0.181548	−0.0907741	+	0.995872i	$0.528934\pi$
$318$	0	0
$319$	6.24124	0.349442
$320$	0	0
$321$	2.48128	0.138491
$322$	0	0
$323$	0	0
$324$	0	0
$325$	5.27881	0.292816
$326$	0	0
$327$	−3.85886	−0.213395
$328$	0	0
$329$	−1.22490	−0.0675310
$330$	0	0
$331$	−14.3416	−0.788285	−0.394142	−	0.919049i	$-0.628958\pi$
$331$	−14.3416	−0.788285	−0.394142	+	0.919049i	$0.628958\pi$
$332$	0	0
$333$	3.61894	0.198317
$334$	0	0
$335$	−5.00348	−0.273369
$336$	0	0
$337$	13.4144	0.730731	0.365366	−	0.930864i	$-0.380944\pi$
$337$	13.4144	0.730731	0.365366	+	0.930864i	$0.380944\pi$
$338$	0	0
$339$	12.5576	0.682036
$340$	0	0
$341$	−1.65987	−0.0898868
$342$	0	0
$343$	−18.4220	−0.994694
$344$	0	0
$345$	4.48128	0.241264
$346$	0	0
$347$	−27.8025	−1.49251	−0.746257	−	0.665658i	$-0.768151\pi$
$347$	−27.8025	−1.49251	−0.746257	+	0.665658i	$0.768151\pi$
$348$	0	0
$349$	−16.4234	−0.879126	−0.439563	−	0.898212i	$-0.644867\pi$
$349$	−16.4234	−0.879126	−0.439563	+	0.898212i	$0.644867\pi$
$350$	0	0
$351$	−5.27881	−0.281762
$352$	0	0
$353$	−8.19899	−0.436388	−0.218194	−	0.975905i	$-0.570017\pi$
$353$	−8.19899	−0.436388	−0.218194	+	0.975905i	$0.570017\pi$
$354$	0	0
$355$	−0.862336	−0.0457680
$356$	0	0
$357$	−7.25493	−0.383971
$358$	0	0
$359$	10.7226	0.565919	0.282960	−	0.959132i	$-0.408684\pi$
$359$	10.7226	0.565919	0.282960	+	0.959132i	$0.408684\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	8.24484	0.432742
$364$	0	0
$365$	−2.93867	−0.153817
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	3.65987	0.190525
$370$	0	0
$371$	−4.46278	−0.231696
$372$	0	0
$373$	3.72467	0.192856	0.0964281	−	0.995340i	$-0.469258\pi$
$373$	3.72467	0.192856	0.0964281	+	0.995340i	$0.469258\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−19.8488	−1.02226
$378$	0	0
$379$	7.15603	0.367581	0.183790	−	0.982965i	$-0.441163\pi$
$379$	7.15603	0.367581	0.183790	+	0.982965i	$0.441163\pi$
$380$	0	0
$381$	6.55762	0.335957
$382$	0	0
$383$	2.95704	0.151098	0.0755490	−	0.997142i	$-0.475929\pi$
$383$	2.95704	0.151098	0.0755490	+	0.997142i	$0.475929\pi$
$384$	0	0
$385$	−2.68723	−0.136954
$386$	0	0
$387$	1.65987	0.0843758
$388$	0	0
$389$	−1.89978	−0.0963227	−0.0481613	−	0.998840i	$-0.515336\pi$
$389$	−1.89978	−0.0963227	−0.0481613	+	0.998840i	$0.515336\pi$
$390$	0	0
$391$	−20.0818	−1.01558
$392$	0	0
$393$	−8.99797	−0.453888
$394$	0	0
$395$	8.41647	0.423479
$396$	0	0
$397$	−12.3231	−0.618478	−0.309239	−	0.950984i	$-0.600074\pi$
$397$	−12.3231	−0.618478	−0.309239	+	0.950984i	$0.600074\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.13766	−0.256563	−0.128281	−	0.991738i	$-0.540946\pi$
$401$	−5.13766	−0.256563	−0.128281	+	0.991738i	$0.540946\pi$
$402$	0	0
$403$	5.27881	0.262956
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−6.00696	−0.297754
$408$	0	0
$409$	25.4383	1.25784	0.628922	−	0.777468i	$-0.283496\pi$
$409$	25.4383	1.25784	0.628922	+	0.777468i	$0.283496\pi$
$410$	0	0
$411$	−20.7636	−1.02419
$412$	0	0
$413$	18.5932	0.914909
$414$	0	0
$415$	0.481278	0.0236250
$416$	0	0
$417$	−17.0444	−0.834668
$418$	0	0
$419$	−8.78049	−0.428955	−0.214477	−	0.976729i	$-0.568805\pi$
$419$	−8.78049	−0.428955	−0.214477	+	0.976729i	$0.568805\pi$
$420$	0	0
$421$	−7.37903	−0.359632	−0.179816	−	0.983700i	$-0.557550\pi$
$421$	−7.37903	−0.359632	−0.179816	+	0.983700i	$0.557550\pi$
$422$	0	0
$423$	−0.756606	−0.0367874
$424$	0	0
$425$	4.48128	0.217374
$426$	0	0
$427$	23.1222	1.11896
$428$	0	0
$429$	8.76212	0.423039
$430$	0	0
$431$	22.9401	1.10499	0.552493	−	0.833517i	$-0.313676\pi$
$431$	22.9401	1.10499	0.552493	+	0.833517i	$0.313676\pi$
$432$	0	0
$433$	−21.5441	−1.03534	−0.517671	−	0.855580i	$-0.673201\pi$
$433$	−21.5441	−1.03534	−0.517671	+	0.855580i	$0.673201\pi$
$434$	0	0
$435$	3.76009	0.180282
$436$	0	0
$437$	0	0
$438$	0	0
$439$	17.9251	0.855519	0.427759	−	0.903893i	$-0.359303\pi$
$439$	17.9251	0.855519	0.427759	+	0.903893i	$0.359303\pi$
$440$	0	0
$441$	−4.37903	−0.208525
$442$	0	0
$443$	32.5591	1.54693	0.773464	−	0.633840i	$-0.218522\pi$
$443$	32.5591	1.54693	0.773464	+	0.633840i	$0.218522\pi$
$444$	0	0
$445$	5.13766	0.243549
$446$	0	0
$447$	−17.5850	−0.831741
$448$	0	0
$449$	−3.61343	−0.170528	−0.0852642	−	0.996358i	$-0.527173\pi$
$449$	−3.61343	−0.170528	−0.0852642	+	0.996358i	$0.527173\pi$
$450$	0	0
$451$	−6.07489	−0.286055
$452$	0	0
$453$	−10.1411	−0.476473
$454$	0	0
$455$	8.54608	0.400647
$456$	0	0
$457$	−34.5407	−1.61575	−0.807873	−	0.589357i	$-0.799381\pi$
$457$	−34.5407	−1.61575	−0.807873	+	0.589357i	$0.799381\pi$
$458$	0	0
$459$	−4.48128	−0.209168
$460$	0	0
$461$	−11.2803	−0.525374	−0.262687	−	0.964881i	$-0.584609\pi$
$461$	−11.2803	−0.525374	−0.262687	+	0.964881i	$0.584609\pi$
$462$	0	0
$463$	−19.8673	−0.923310	−0.461655	−	0.887060i	$-0.652744\pi$
$463$	−19.8673	−0.923310	−0.461655	+	0.887060i	$0.652744\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	24.4772	1.13267	0.566335	−	0.824175i	$-0.308361\pi$
$467$	24.4772	1.13267	0.566335	+	0.824175i	$0.308361\pi$
$468$	0	0
$469$	−8.10034	−0.374039
$470$	0	0
$471$	−14.1356	−0.651336
$472$	0	0
$473$	−2.75516	−0.126682
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.75661	−0.126216
$478$	0	0
$479$	−22.9401	−1.04816	−0.524081	−	0.851669i	$-0.675591\pi$
$479$	−22.9401	−1.04816	−0.524081	+	0.851669i	$0.675591\pi$
$480$	0	0
$481$	19.1037	0.871054
$482$	0	0
$483$	7.25493	0.330111
$484$	0	0
$485$	9.30269	0.422413
$486$	0	0
$487$	−13.7247	−0.621924	−0.310962	−	0.950422i	$-0.600651\pi$
$487$	−13.7247	−0.621924	−0.310962	+	0.950422i	$0.600651\pi$
$488$	0	0
$489$	−3.27881	−0.148273
$490$	0	0
$491$	−0.751095	−0.0338965	−0.0169482	−	0.999856i	$-0.505395\pi$
$491$	−0.751095	−0.0338965	−0.0169482	+	0.999856i	$0.505395\pi$
$492$	0	0
$493$	−16.8500	−0.758885
$494$	0	0
$495$	−1.65987	−0.0746054
$496$	0	0
$497$	−1.39607	−0.0626224
$498$	0	0
$499$	33.1152	1.48244	0.741221	−	0.671261i	$-0.234247\pi$
$499$	33.1152	1.48244	0.741221	+	0.671261i	$0.234247\pi$
$500$	0	0
$501$	3.91815	0.175050
$502$	0	0
$503$	−30.6817	−1.36803	−0.684015	−	0.729468i	$-0.739768\pi$
$503$	−30.6817	−1.36803	−0.684015	+	0.729468i	$0.739768\pi$
$504$	0	0
$505$	−4.62242	−0.205695
$506$	0	0
$507$	−14.8658	−0.660214
$508$	0	0
$509$	5.05581	0.224095	0.112048	−	0.993703i	$-0.464259\pi$
$509$	5.05581	0.224095	0.112048	+	0.993703i	$0.464259\pi$
$510$	0	0
$511$	−4.75754	−0.210461
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.0409247	0.00180336
$516$	0	0
$517$	1.25586	0.0552329
$518$	0	0
$519$	5.23788	0.229918
$520$	0	0
$521$	−9.45537	−0.414247	−0.207124	−	0.978315i	$-0.566410\pi$
$521$	−9.45537	−0.414247	−0.207124	+	0.978315i	$0.566410\pi$
$522$	0	0
$523$	1.16010	0.0507274	0.0253637	−	0.999678i	$-0.491926\pi$
$523$	1.16010	0.0507274	0.0253637	+	0.999678i	$0.491926\pi$
$524$	0	0
$525$	−1.61894	−0.0706564
$526$	0	0
$527$	4.48128	0.195208
$528$	0	0
$529$	−2.91815	−0.126876
$530$	0	0
$531$	11.4848	0.498396
$532$	0	0
$533$	19.3197	0.836831
$534$	0	0
$535$	−2.48128	−0.107275
$536$	0	0
$537$	8.28229	0.357407
$538$	0	0
$539$	7.26860	0.313081
$540$	0	0
$541$	−7.50864	−0.322822	−0.161411	−	0.986887i	$-0.551604\pi$
$541$	−7.50864	−0.322822	−0.161411	+	0.986887i	$0.551604\pi$
$542$	0	0
$543$	−14.8329	−0.636543
$544$	0	0
$545$	3.85886	0.165295
$546$	0	0
$547$	5.75458	0.246048	0.123024	−	0.992404i	$-0.460741\pi$
$547$	5.75458	0.246048	0.123024	+	0.992404i	$0.460741\pi$
$548$	0	0
$549$	14.2823	0.609553
$550$	0	0
$551$	0	0
$552$	0	0
$553$	13.6258	0.579427
$554$	0	0
$555$	−3.61894	−0.153616
$556$	0	0
$557$	20.7636	0.879781	0.439890	−	0.898052i	$-0.355017\pi$
$557$	20.7636	0.879781	0.439890	+	0.898052i	$0.355017\pi$
$558$	0	0
$559$	8.76212	0.370598
$560$	0	0
$561$	7.43832	0.314046
$562$	0	0
$563$	1.80101	0.0759035	0.0379518	−	0.999280i	$-0.487917\pi$
$563$	1.80101	0.0759035	0.0379518	+	0.999280i	$0.487917\pi$
$564$	0	0
$565$	−12.5576	−0.528303
$566$	0	0
$567$	1.61894	0.0679892
$568$	0	0
$569$	10.2469	0.429571	0.214786	−	0.976661i	$-0.431095\pi$
$569$	10.2469	0.429571	0.214786	+	0.976661i	$0.431095\pi$
$570$	0	0
$571$	7.24890	0.303357	0.151679	−	0.988430i	$-0.451532\pi$
$571$	7.24890	0.303357	0.151679	+	0.988430i	$0.451532\pi$
$572$	0	0
$573$	−5.07982	−0.212212
$574$	0	0
$575$	−4.48128	−0.186882
$576$	0	0
$577$	−22.6155	−0.941494	−0.470747	−	0.882268i	$-0.656016\pi$
$577$	−22.6155	−0.941494	−0.470747	+	0.882268i	$0.656016\pi$
$578$	0	0
$579$	17.4554	0.725420
$580$	0	0
$581$	0.779160	0.0323250
$582$	0	0
$583$	4.57560	0.189502
$584$	0	0
$585$	5.27881	0.218252
$586$	0	0
$587$	−3.52017	−0.145293	−0.0726465	−	0.997358i	$-0.523144\pi$
$587$	−3.52017	−0.145293	−0.0726465	+	0.997358i	$0.523144\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−5.16154	−0.212318
$592$	0	0
$593$	−10.8807	−0.446817	−0.223409	−	0.974725i	$-0.571718\pi$
$593$	−10.8807	−0.446817	−0.223409	+	0.974725i	$0.571718\pi$
$594$	0	0
$595$	7.25493	0.297423
$596$	0	0
$597$	−21.4609	−0.878335
$598$	0	0
$599$	−43.9735	−1.79671	−0.898354	−	0.439271i	$-0.855237\pi$
$599$	−43.9735	−1.79671	−0.898354	+	0.439271i	$0.855237\pi$
$600$	0	0
$601$	28.2782	1.15349	0.576746	−	0.816923i	$-0.304322\pi$
$601$	28.2782	1.15349	0.576746	+	0.816923i	$0.304322\pi$
$602$	0	0
$603$	−5.00348	−0.203757
$604$	0	0
$605$	−8.24484	−0.335201
$606$	0	0
$607$	4.72119	0.191627	0.0958136	−	0.995399i	$-0.469455\pi$
$607$	4.72119	0.191627	0.0958136	+	0.995399i	$0.469455\pi$
$608$	0	0
$609$	6.08736	0.246672
$610$	0	0
$611$	−3.99398	−0.161579
$612$	0	0
$613$	−19.8842	−0.803115	−0.401557	−	0.915834i	$-0.631531\pi$
$613$	−19.8842	−0.803115	−0.401557	+	0.915834i	$0.631531\pi$
$614$	0	0
$615$	−3.65987	−0.147580
$616$	0	0
$617$	34.4350	1.38630	0.693150	−	0.720794i	$-0.256222\pi$
$617$	34.4350	1.38630	0.693150	+	0.720794i	$0.256222\pi$
$618$	0	0
$619$	−28.4194	−1.14227	−0.571135	−	0.820856i	$-0.693497\pi$
$619$	−28.4194	−1.14227	−0.571135	+	0.820856i	$0.693497\pi$
$620$	0	0
$621$	4.48128	0.179827
$622$	0	0
$623$	8.31758	0.333237
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.2175	0.646633
$630$	0	0
$631$	27.5910	1.09838	0.549190	−	0.835697i	$-0.314936\pi$
$631$	27.5910	1.09838	0.549190	+	0.835697i	$0.314936\pi$
$632$	0	0
$633$	21.9959	0.874260
$634$	0	0
$635$	−6.55762	−0.260231
$636$	0	0
$637$	−23.1160	−0.915891
$638$	0	0
$639$	−0.862336	−0.0341135
$640$	0	0
$641$	−5.47374	−0.216200	−0.108100	−	0.994140i	$-0.534477\pi$
$641$	−5.47374	−0.216200	−0.108100	+	0.994140i	$0.534477\pi$
$642$	0	0
$643$	34.2892	1.35224	0.676118	−	0.736793i	$-0.263661\pi$
$643$	34.2892	1.35224	0.676118	+	0.736793i	$0.263661\pi$
$644$	0	0
$645$	−1.65987	−0.0653572
$646$	0	0
$647$	9.51321	0.374003	0.187001	−	0.982360i	$-0.440123\pi$
$647$	9.51321	0.374003	0.187001	+	0.982360i	$0.440123\pi$
$648$	0	0
$649$	−19.0632	−0.748295
$650$	0	0
$651$	−1.61894	−0.0634513
$652$	0	0
$653$	−45.7561	−1.79058	−0.895288	−	0.445487i	$-0.853030\pi$
$653$	−45.7561	−1.79058	−0.895288	+	0.445487i	$0.853030\pi$
$654$	0	0
$655$	8.99797	0.351580
$656$	0	0
$657$	−2.93867	−0.114649
$658$	0	0
$659$	10.8215	0.421547	0.210774	−	0.977535i	$-0.432402\pi$
$659$	10.8215	0.421547	0.210774	+	0.977535i	$0.432402\pi$
$660$	0	0
$661$	−12.4280	−0.483393	−0.241697	−	0.970352i	$-0.577704\pi$
$661$	−12.4280	−0.483393	−0.241697	+	0.970352i	$0.577704\pi$
$662$	0	0
$663$	−23.6558	−0.918715
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.8500	0.652434
$668$	0	0
$669$	−0.897750	−0.0347090
$670$	0	0
$671$	−23.7067	−0.915187
$672$	0	0
$673$	−43.1591	−1.66366	−0.831830	−	0.555031i	$-0.812706\pi$
$673$	−43.1591	−1.66366	−0.831830	+	0.555031i	$0.812706\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	47.7192	1.83400	0.916998	−	0.398891i	$-0.130605\pi$
$677$	47.7192	1.83400	0.916998	+	0.398891i	$0.130605\pi$
$678$	0	0
$679$	15.0605	0.577969
$680$	0	0
$681$	3.99449	0.153069
$682$	0	0
$683$	29.1097	1.11385	0.556926	−	0.830562i	$-0.311981\pi$
$683$	29.1097	1.11385	0.556926	+	0.830562i	$0.311981\pi$
$684$	0	0
$685$	20.7636	0.793335
$686$	0	0
$687$	−18.0708	−0.689445
$688$	0	0
$689$	−14.5516	−0.554372
$690$	0	0
$691$	20.8807	0.794339	0.397170	−	0.917745i	$-0.369992\pi$
$691$	20.8807	0.794339	0.397170	+	0.917745i	$0.369992\pi$
$692$	0	0
$693$	−2.68723	−0.102079
$694$	0	0
$695$	17.0444	0.646531
$696$	0	0
$697$	16.4009	0.621228
$698$	0	0
$699$	26.0014	0.983465
$700$	0	0
$701$	−14.2175	−0.536987	−0.268493	−	0.963282i	$-0.586526\pi$
$701$	−14.2175	−0.536987	−0.268493	+	0.963282i	$0.586526\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0.756606	0.0284954
$706$	0	0
$707$	−7.48343	−0.281443
$708$	0	0
$709$	47.6798	1.79065	0.895326	−	0.445411i	$-0.146942\pi$
$709$	47.6798	1.79065	0.895326	+	0.445411i	$0.146942\pi$
$710$	0	0
$711$	8.41647	0.315642
$712$	0	0
$713$	−4.48128	−0.167825
$714$	0	0
$715$	−8.76212	−0.327685
$716$	0	0
$717$	21.8603	0.816388
$718$	0	0
$719$	38.2244	1.42553	0.712766	−	0.701402i	$-0.247442\pi$
$719$	38.2244	1.42553	0.712766	+	0.701402i	$0.247442\pi$
$720$	0	0
$721$	0.0662547	0.00246745
$722$	0	0
$723$	−11.8773	−0.441723
$724$	0	0
$725$	−3.76009	−0.139646
$726$	0	0
$727$	15.0104	0.556706	0.278353	−	0.960479i	$-0.410211\pi$
$727$	15.0104	0.556706	0.278353	+	0.960479i	$0.410211\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.43832	0.275116
$732$	0	0
$733$	6.43497	0.237681	0.118840	−	0.992913i	$-0.462082\pi$
$733$	6.43497	0.237681	0.118840	+	0.992913i	$0.462082\pi$
$734$	0	0
$735$	4.37903	0.161523
$736$	0	0
$737$	8.30511	0.305923
$738$	0	0
$739$	12.3456	0.454142	0.227071	−	0.973878i	$-0.427085\pi$
$739$	12.3456	0.454142	0.227071	+	0.973878i	$0.427085\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−53.9959	−1.98092	−0.990459	−	0.137805i	$-0.955995\pi$
$743$	−53.9959	−1.98092	−0.990459	+	0.137805i	$0.955995\pi$
$744$	0	0
$745$	17.5850	0.644264
$746$	0	0
$747$	0.481278	0.0176090
$748$	0	0
$749$	−4.01704	−0.146780
$750$	0	0
$751$	−0.197540	−0.00720834	−0.00360417	−	0.999994i	$-0.501147\pi$
$751$	−0.197540	−0.00720834	−0.00360417	+	0.999994i	$0.501147\pi$
$752$	0	0
$753$	27.6020	1.00587
$754$	0	0
$755$	10.1411	0.369074
$756$	0	0
$757$	−23.2747	−0.845935	−0.422968	−	0.906145i	$-0.639012\pi$
$757$	−23.2747	−0.845935	−0.422968	+	0.906145i	$0.639012\pi$
$758$	0	0
$759$	−7.43832	−0.269994
$760$	0	0
$761$	−7.00899	−0.254076	−0.127038	−	0.991898i	$-0.540547\pi$
$761$	−7.00899	−0.254076	−0.127038	+	0.991898i	$0.540547\pi$
$762$	0	0
$763$	6.24726	0.226166
$764$	0	0
$765$	4.48128	0.162021
$766$	0	0
$767$	60.6258	2.18907
$768$	0	0
$769$	−39.1815	−1.41292	−0.706460	−	0.707753i	$-0.749709\pi$
$769$	−39.1815	−1.41292	−0.706460	+	0.707753i	$0.749709\pi$
$770$	0	0
$771$	5.71916	0.205971
$772$	0	0
$773$	−16.0833	−0.578476	−0.289238	−	0.957257i	$-0.593402\pi$
$773$	−16.0833	−0.578476	−0.289238	+	0.957257i	$0.593402\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−5.85886	−0.210185
$778$	0	0
$779$	0	0
$780$	0	0
$781$	1.43136	0.0512182
$782$	0	0
$783$	3.76009	0.134375
$784$	0	0
$785$	14.1356	0.504522
$786$	0	0
$787$	−48.3500	−1.72349	−0.861746	−	0.507341i	$-0.830629\pi$
$787$	−48.3500	−1.72349	−0.861746	+	0.507341i	$0.830629\pi$
$788$	0	0
$789$	20.0708	0.714540
$790$	0	0
$791$	−20.3300	−0.722853
$792$	0	0
$793$	75.3935	2.67730
$794$	0	0
$795$	2.75661	0.0977667
$796$	0	0
$797$	7.23933	0.256430	0.128215	−	0.991746i	$-0.459075\pi$
$797$	7.23933	0.256430	0.128215	+	0.991746i	$0.459075\pi$
$798$	0	0
$799$	−3.39056	−0.119949
$800$	0	0
$801$	5.13766	0.181530
$802$	0	0
$803$	4.87781	0.172134
$804$	0	0
$805$	−7.25493	−0.255703
$806$	0	0
$807$	4.60405	0.162070
$808$	0	0
$809$	−6.91010	−0.242946	−0.121473	−	0.992595i	$-0.538762\pi$
$809$	−6.91010	−0.242946	−0.121473	+	0.992595i	$0.538762\pi$
$810$	0	0
$811$	−28.4827	−1.00016	−0.500082	−	0.865978i	$-0.666697\pi$
$811$	−28.4827	−1.00016	−0.500082	+	0.865978i	$0.666697\pi$
$812$	0	0
$813$	22.8399	0.801030
$814$	0	0
$815$	3.27881	0.114852
$816$	0	0
$817$	0	0
$818$	0	0
$819$	8.54608	0.298624
$820$	0	0
$821$	44.1890	1.54221	0.771104	−	0.636709i	$-0.219705\pi$
$821$	44.1890	1.54221	0.771104	+	0.636709i	$0.219705\pi$
$822$	0	0
$823$	−40.2883	−1.40436	−0.702181	−	0.711998i	$-0.747790\pi$
$823$	−40.2883	−1.40436	−0.702181	+	0.711998i	$0.747790\pi$
$824$	0	0
$825$	1.65987	0.0577891
$826$	0	0
$827$	39.2504	1.36487	0.682434	−	0.730947i	$-0.260921\pi$
$827$	39.2504	1.36487	0.682434	+	0.730947i	$0.260921\pi$
$828$	0	0
$829$	−40.5466	−1.40824	−0.704121	−	0.710080i	$-0.748659\pi$
$829$	−40.5466	−1.40824	−0.704121	+	0.710080i	$0.748659\pi$
$830$	0	0
$831$	13.4144	0.465342
$832$	0	0
$833$	−19.6236	−0.679919
$834$	0	0
$835$	−3.91815	−0.135593
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−26.3585	−0.909997	−0.454998	−	0.890492i	$-0.650360\pi$
$839$	−26.3585	−0.909997	−0.454998	+	0.890492i	$0.650360\pi$
$840$	0	0
$841$	−14.8618	−0.512474
$842$	0	0
$843$	2.08185	0.0717027
$844$	0	0
$845$	14.8658	0.511400
$846$	0	0
$847$	−13.3479	−0.458640
$848$	0	0
$849$	0.0238807	0.000819583 0
$850$	0	0
$851$	−16.2175	−0.555928
$852$	0	0
$853$	−16.4220	−0.562278	−0.281139	−	0.959667i	$-0.590712\pi$
$853$	−16.4220	−0.562278	−0.281139	+	0.959667i	$0.590712\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.39392	0.286731	0.143365	−	0.989670i	$-0.454208\pi$
$857$	8.39392	0.286731	0.143365	+	0.989670i	$0.454208\pi$
$858$	0	0
$859$	23.4493	0.800081	0.400041	−	0.916497i	$-0.368996\pi$
$859$	23.4493	0.800081	0.400041	+	0.916497i	$0.368996\pi$
$860$	0	0
$861$	−5.92511	−0.201927
$862$	0	0
$863$	−8.43497	−0.287130	−0.143565	−	0.989641i	$-0.545857\pi$
$863$	−8.43497	−0.287130	−0.143565	+	0.989641i	$0.545857\pi$
$864$	0	0
$865$	−5.23788	−0.178093
$866$	0	0
$867$	−3.08185	−0.104665
$868$	0	0
$869$	−13.9702	−0.473907
$870$	0	0
$871$	−26.4124	−0.894950
$872$	0	0
$873$	9.30269	0.314848
$874$	0	0
$875$	1.61894	0.0547302
$876$	0	0
$877$	−27.0163	−0.912277	−0.456138	−	0.889909i	$-0.650768\pi$
$877$	−27.0163	−0.912277	−0.456138	+	0.889909i	$0.650768\pi$
$878$	0	0
$879$	7.92366	0.267258
$880$	0	0
$881$	21.9365	0.739060	0.369530	−	0.929219i	$-0.379519\pi$
$881$	21.9365	0.739060	0.369530	+	0.929219i	$0.379519\pi$
$882$	0	0
$883$	4.19442	0.141153	0.0705767	−	0.997506i	$-0.477516\pi$
$883$	4.19442	0.141153	0.0705767	+	0.997506i	$0.477516\pi$
$884$	0	0
$885$	−11.4848	−0.386056
$886$	0	0
$887$	−27.3851	−0.919500	−0.459750	−	0.888048i	$-0.652061\pi$
$887$	−27.3851	−0.919500	−0.459750	+	0.888048i	$0.652061\pi$
$888$	0	0
$889$	−10.6164	−0.356063
$890$	0	0
$891$	−1.65987	−0.0556076
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−8.28229	−0.276846
$896$	0	0
$897$	23.6558	0.789844
$898$	0	0
$899$	−3.76009	−0.125406
$900$	0	0
$901$	−12.3531	−0.411542
$902$	0	0
$903$	−2.68723	−0.0894253
$904$	0	0
$905$	14.8329	0.493064
$906$	0	0
$907$	21.9831	0.729936	0.364968	−	0.931020i	$-0.381080\pi$
$907$	21.9831	0.729936	0.364968	+	0.931020i	$0.381080\pi$
$908$	0	0
$909$	−4.62242	−0.153316
$910$	0	0
$911$	−31.4554	−1.04216	−0.521081	−	0.853507i	$-0.674471\pi$
$911$	−31.4554	−1.04216	−0.521081	+	0.853507i	$0.674471\pi$
$912$	0	0
$913$	−0.798857	−0.0264383
$914$	0	0
$915$	−14.2823	−0.472158
$916$	0	0
$917$	14.5672	0.481051
$918$	0	0
$919$	1.51321	0.0499162	0.0249581	−	0.999688i	$-0.492055\pi$
$919$	1.51321	0.0499162	0.0249581	+	0.999688i	$0.492055\pi$
$920$	0	0
$921$	−8.30617	−0.273698
$922$	0	0
$923$	−4.55211	−0.149834
$924$	0	0
$925$	3.61894	0.118990
$926$	0	0
$927$	0.0409247	0.00134414
$928$	0	0
$929$	33.6264	1.10325	0.551623	−	0.834093i	$-0.314009\pi$
$929$	33.6264	1.10325	0.551623	+	0.834093i	$0.314009\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	28.1002	0.919960
$934$	0	0
$935$	−7.43832	−0.243259
$936$	0	0
$937$	−37.3027	−1.21863	−0.609313	−	0.792930i	$-0.708555\pi$
$937$	−37.3027	−1.21863	−0.609313	+	0.792930i	$0.708555\pi$
$938$	0	0
$939$	−11.8364	−0.386267
$940$	0	0
$941$	21.7022	0.707473	0.353736	−	0.935345i	$-0.384911\pi$
$941$	21.7022	0.707473	0.353736	+	0.935345i	$0.384911\pi$
$942$	0	0
$943$	−16.4009	−0.534086
$944$	0	0
$945$	−1.61894	−0.0526642
$946$	0	0
$947$	−4.91624	−0.159756	−0.0798782	−	0.996805i	$-0.525453\pi$
$947$	−4.91624	−0.159756	−0.0798782	+	0.996805i	$0.525453\pi$
$948$	0	0
$949$	−15.5127	−0.503564
$950$	0	0
$951$	3.23237	0.104817
$952$	0	0
$953$	55.6853	1.80382	0.901912	−	0.431919i	$-0.142164\pi$
$953$	55.6853	1.80382	0.901912	+	0.431919i	$0.142164\pi$
$954$	0	0
$955$	5.07982	0.164379
$956$	0	0
$957$	−6.24124	−0.201751
$958$	0	0
$959$	33.6150	1.08549
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−2.48128	−0.0799581
$964$	0	0
$965$	−17.4554	−0.561908
$966$	0	0
$967$	22.1767	0.713154	0.356577	−	0.934266i	$-0.383944\pi$
$967$	22.1767	0.713154	0.356577	+	0.934266i	$0.383944\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−33.4977	−1.07499	−0.537497	−	0.843266i	$-0.680630\pi$
$971$	−33.4977	−1.07499	−0.537497	+	0.843266i	$0.680630\pi$
$972$	0	0
$973$	27.5939	0.884620
$974$	0	0
$975$	−5.27881	−0.169057
$976$	0	0
$977$	25.0403	0.801112	0.400556	−	0.916272i	$-0.368817\pi$
$977$	25.0403	0.801112	0.400556	+	0.916272i	$0.368817\pi$
$978$	0	0
$979$	−8.52784	−0.272551
$980$	0	0
$981$	3.85886	0.123204
$982$	0	0
$983$	14.7581	0.470709	0.235354	−	0.971910i	$-0.424375\pi$
$983$	14.7581	0.470709	0.235354	+	0.971910i	$0.424375\pi$
$984$	0	0
$985$	5.16154	0.164460
$986$	0	0
$987$	1.22490	0.0389890
$988$	0	0
$989$	−7.43832	−0.236525
$990$	0	0
$991$	6.83990	0.217277	0.108638	−	0.994081i	$-0.465351\pi$
$991$	6.83990	0.217277	0.108638	+	0.994081i	$0.465351\pi$
$992$	0	0
$993$	14.3416	0.455116
$994$	0	0
$995$	21.4609	0.680356
$996$	0	0
$997$	−55.8663	−1.76930	−0.884652	−	0.466252i	$-0.845604\pi$
$997$	−55.8663	−1.76930	−0.884652	+	0.466252i	$0.845604\pi$
$998$	0	0
$999$	−3.61894	−0.114498

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1860.2.a.i.1.3	✓	4
3.2	odd	2		5580.2.a.m.1.3		4
4.3	odd	2		7440.2.a.cb.1.2		4
5.2	odd	4		9300.2.g.s.3349.7		8
5.3	odd	4		9300.2.g.s.3349.2		8
5.4	even	2		9300.2.a.x.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.i.1.3	✓	4	1.1	even	1	trivial
5580.2.a.m.1.3		4	3.2	odd	2
7440.2.a.cb.1.2		4	4.3	odd	2
9300.2.a.x.1.2		4	5.4	even	2
9300.2.g.s.3349.2		8	5.3	odd	4
9300.2.g.s.3349.7		8	5.2	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1860.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +1.61894 q^{7} +1.00000 q^{9} -1.65987 q^{11} +5.27881 q^{13} -1.00000 q^{15} +4.48128 q^{17} -1.61894 q^{21} -4.48128 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.76009 q^{29} +1.00000 q^{31} +1.65987 q^{33} +1.61894 q^{35} +3.61894 q^{37} -5.27881 q^{39} +3.65987 q^{41} +1.65987 q^{43} +1.00000 q^{45} -0.756606 q^{47} -4.37903 q^{49} -4.48128 q^{51} -2.75661 q^{53} -1.65987 q^{55} +11.4848 q^{59} +14.2823 q^{61} +1.61894 q^{63} +5.27881 q^{65} -5.00348 q^{67} +4.48128 q^{69} -0.862336 q^{71} -2.93867 q^{73} -1.00000 q^{75} -2.68723 q^{77} +8.41647 q^{79} +1.00000 q^{81} +0.481278 q^{83} +4.48128 q^{85} +3.76009 q^{87} +5.13766 q^{89} +8.54608 q^{91} -1.00000 q^{93} +9.30269 q^{97} -1.65987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} + 4 q^{21} - 2 q^{23} + 4 q^{25} - 4 q^{27} + 20 q^{29} + 4 q^{31} - 2 q^{33} - 4 q^{35} + 4 q^{37} - 2 q^{39} + 6 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^15 + 2 * q^17 + 4 * q^21 - 2 * q^23 + 4 * q^25 - 4 * q^27 + 20 * q^29 + 4 * q^31 - 2 * q^33 - 4 * q^35 + 4 * q^37 - 2 * q^39 + 6 * q^41 - 2 * q^43 + 4 * q^45 + 2 * q^47 + 28 * q^49 - 2 * q^51 - 6 * q^53 + 2 * q^55 + 8 * q^61 - 4 * q^63 + 2 * q^65 + 10 * q^67 + 2 * q^69 + 2 * q^71 + 16 * q^73 - 4 * q^75 + 32 * q^77 + 20 * q^79 + 4 * q^81 - 14 * q^83 + 2 * q^85 - 20 * q^87 + 26 * q^89 + 16 * q^91 - 4 * q^93 + 14 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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