LMFDB - Embedded newform 1860.2.a.i.1.1

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			1.1
Root			$-3.06316$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -4.64963 q^{7} +1.00000 q^{9} -6.12631 q^{11} +3.47668 q^{13} -1.00000 q^{15} -6.44607 q^{17} +4.64963 q^{21} +6.44607 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.96939 q^{29} +1.00000 q^{31} +6.12631 q^{33} -4.64963 q^{35} -2.64963 q^{37} -3.47668 q^{39} +8.12631 q^{41} +6.12631 q^{43} +1.00000 q^{45} +0.853176 q^{47} +14.6190 q^{49} +6.44607 q^{51} -1.14682 q^{53} -6.12631 q^{55} -10.5623 q^{59} +1.36047 q^{61} -4.64963 q^{63} +3.47668 q^{65} +6.11621 q^{67} -6.44607 q^{69} +3.79645 q^{71} -5.60300 q^{73} -1.00000 q^{75} +28.4850 q^{77} +11.2731 q^{79} +1.00000 q^{81} -10.4461 q^{83} -6.44607 q^{85} -8.96939 q^{87} +9.79645 q^{89} -16.1653 q^{91} -1.00000 q^{93} -17.0185 q^{97} -6.12631 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−4.64963	−1.75739	−0.878697	−	0.477381i	$-0.841586\pi$
$7$	−4.64963	−1.75739	−0.878697	+	0.477381i	$0.841586\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.12631	−1.84715	−0.923576	−	0.383415i	$-0.874748\pi$
$11$	−6.12631	−1.84715	−0.923576	+	0.383415i	$0.874748\pi$
$12$	0	0
$13$	3.47668	0.964259	0.482129	−	0.876100i	$-0.339863\pi$
$13$	3.47668	0.964259	0.482129	+	0.876100i	$0.339863\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−6.44607	−1.56340	−0.781701	−	0.623653i	$-0.785648\pi$
$17$	−6.44607	−1.56340	−0.781701	+	0.623653i	$0.785648\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	4.64963	1.01463
$22$	0	0
$23$	6.44607	1.34410	0.672050	−	0.740506i	$-0.265414\pi$
$23$	6.44607	1.34410	0.672050	+	0.740506i	$0.265414\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.96939	1.66557	0.832787	−	0.553594i	$-0.186744\pi$
$29$	8.96939	1.66557	0.832787	+	0.553594i	$0.186744\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	6.12631	1.06645
$34$	0	0
$35$	−4.64963	−0.785930
$36$	0	0
$37$	−2.64963	−0.435596	−0.217798	−	0.975994i	$-0.569887\pi$
$37$	−2.64963	−0.435596	−0.217798	+	0.975994i	$0.569887\pi$
$38$	0	0
$39$	−3.47668	−0.556715
$40$	0	0
$41$	8.12631	1.26912	0.634558	−	0.772875i	$-0.281182\pi$
$41$	8.12631	1.26912	0.634558	+	0.772875i	$0.281182\pi$
$42$	0	0
$43$	6.12631	0.934254	0.467127	−	0.884190i	$-0.345289\pi$
$43$	6.12631	0.934254	0.467127	+	0.884190i	$0.345289\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	0.853176	0.124449	0.0622243	−	0.998062i	$-0.480181\pi$
$47$	0.853176	0.124449	0.0622243	+	0.998062i	$0.480181\pi$
$48$	0	0
$49$	14.6190	2.08843
$50$	0	0
$51$	6.44607	0.902631
$52$	0	0
$53$	−1.14682	−0.157528	−0.0787642	−	0.996893i	$-0.525097\pi$
$53$	−1.14682	−0.157528	−0.0787642	+	0.996893i	$0.525097\pi$
$54$	0	0
$55$	−6.12631	−0.826071
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.5623	−1.37509	−0.687546	−	0.726141i	$-0.741312\pi$
$59$	−10.5623	−1.37509	−0.687546	+	0.726141i	$0.741312\pi$
$60$	0	0
$61$	1.36047	0.174191	0.0870953	−	0.996200i	$-0.472242\pi$
$61$	1.36047	0.174191	0.0870953	+	0.996200i	$0.472242\pi$
$62$	0	0
$63$	−4.64963	−0.585798
$64$	0	0
$65$	3.47668	0.431230
$66$	0	0
$67$	6.11621	0.747214	0.373607	−	0.927587i	$-0.378121\pi$
$67$	6.11621	0.747214	0.373607	+	0.927587i	$0.378121\pi$
$68$	0	0
$69$	−6.44607	−0.776016
$70$	0	0
$71$	3.79645	0.450556	0.225278	−	0.974295i	$-0.427671\pi$
$71$	3.79645	0.450556	0.225278	+	0.974295i	$0.427671\pi$
$72$	0	0
$73$	−5.60300	−0.655781	−0.327890	−	0.944716i	$-0.606338\pi$
$73$	−5.60300	−0.655781	−0.327890	+	0.944716i	$0.606338\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	28.4850	3.24617
$78$	0	0
$79$	11.2731	1.26833	0.634163	−	0.773199i	$-0.281345\pi$
$79$	11.2731	1.26833	0.634163	+	0.773199i	$0.281345\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.4461	−1.14661	−0.573303	−	0.819344i	$-0.694338\pi$
$83$	−10.4461	−1.14661	−0.573303	+	0.819344i	$0.694338\pi$
$84$	0	0
$85$	−6.44607	−0.699175
$86$	0	0
$87$	−8.96939	−0.961619
$88$	0	0
$89$	9.79645	1.03842	0.519211	−	0.854646i	$-0.326226\pi$
$89$	9.79645	1.03842	0.519211	+	0.854646i	$0.326226\pi$
$90$	0	0
$91$	−16.1653	−1.69458
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−17.0185	−1.72796	−0.863981	−	0.503524i	$-0.832037\pi$
$97$	−17.0185	−1.72796	−0.863981	+	0.503524i	$0.832037\pi$
$98$	0	0
$99$	−6.12631	−0.615717
$100$	0	0
$101$	12.7658	1.27025	0.635124	−	0.772410i	$-0.280949\pi$
$101$	12.7658	1.27025	0.635124	+	0.772410i	$0.280949\pi$
$102$	0	0
$103$	10.7759	1.06178	0.530892	−	0.847439i	$-0.321857\pi$
$103$	10.7759	1.06178	0.530892	+	0.847439i	$0.321857\pi$
$104$	0	0
$105$	4.64963	0.453757
$106$	0	0
$107$	8.44607	0.816513	0.408256	−	0.912867i	$-0.366137\pi$
$107$	8.44607	0.816513	0.408256	+	0.912867i	$0.366137\pi$
$108$	0	0
$109$	10.3198	0.988454	0.494227	−	0.869333i	$-0.335451\pi$
$109$	10.3198	0.988454	0.494227	+	0.869333i	$0.335451\pi$
$110$	0	0
$111$	2.64963	0.251491
$112$	0	0
$113$	−8.95337	−0.842262	−0.421131	−	0.907000i	$-0.638367\pi$
$113$	−8.95337	−0.842262	−0.421131	+	0.907000i	$0.638367\pi$
$114$	0	0
$115$	6.44607	0.601100
$116$	0	0
$117$	3.47668	0.321420
$118$	0	0
$119$	29.9718	2.74751
$120$	0	0
$121$	26.5317	2.41197
$122$	0	0
$123$	−8.12631	−0.732725
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.95337	−0.262069	−0.131035	−	0.991378i	$-0.541830\pi$
$127$	−2.95337	−0.262069	−0.131035	+	0.991378i	$0.541830\pi$
$128$	0	0
$129$	−6.12631	−0.539392
$130$	0	0
$131$	−16.2686	−1.42140	−0.710699	−	0.703496i	$-0.751621\pi$
$131$	−16.2686	−1.42140	−0.710699	+	0.703496i	$0.751621\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−3.08560	−0.263621	−0.131810	−	0.991275i	$-0.542079\pi$
$137$	−3.08560	−0.263621	−0.131810	+	0.991275i	$0.542079\pi$
$138$	0	0
$139$	16.6597	1.41306	0.706530	−	0.707683i	$-0.250260\pi$
$139$	16.6597	1.41306	0.706530	+	0.707683i	$0.250260\pi$
$140$	0	0
$141$	−0.853176	−0.0718504
$142$	0	0
$143$	−21.2993	−1.78113
$144$	0	0
$145$	8.96939	0.744867
$146$	0	0
$147$	−14.6190	−1.20576
$148$	0	0
$149$	−21.6580	−1.77429	−0.887146	−	0.461489i	$-0.847315\pi$
$149$	−21.6580	−1.77429	−0.887146	+	0.461489i	$0.847315\pi$
$150$	0	0
$151$	3.68024	0.299493	0.149747	−	0.988724i	$-0.452154\pi$
$151$	3.68024	0.299493	0.149747	+	0.988724i	$0.452154\pi$
$152$	0	0
$153$	−6.44607	−0.521134
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−6.47219	−0.516537	−0.258269	−	0.966073i	$-0.583152\pi$
$157$	−6.47219	−0.516537	−0.258269	+	0.966073i	$0.583152\pi$
$158$	0	0
$159$	1.14682	0.0909490
$160$	0	0
$161$	−29.9718	−2.36211
$162$	0	0
$163$	1.47668	0.115663	0.0578314	−	0.998326i	$-0.481581\pi$
$163$	1.47668	0.115663	0.0578314	+	0.998326i	$0.481581\pi$
$164$	0	0
$165$	6.12631	0.476933
$166$	0	0
$167$	17.5519	1.35820	0.679102	−	0.734044i	$-0.262369\pi$
$167$	17.5519	1.35820	0.679102	+	0.734044i	$0.262369\pi$
$168$	0	0
$169$	−0.912662	−0.0702047
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.29925	0.554952	0.277476	−	0.960733i	$-0.410502\pi$
$173$	7.29925	0.554952	0.277476	+	0.960733i	$0.410502\pi$
$174$	0	0
$175$	−4.64963	−0.351479
$176$	0	0
$177$	10.5623	0.793910
$178$	0	0
$179$	4.63953	0.346775	0.173387	−	0.984854i	$-0.444529\pi$
$179$	4.63953	0.346775	0.173387	+	0.984854i	$0.444529\pi$
$180$	0	0
$181$	20.5463	1.52719	0.763596	−	0.645694i	$-0.223432\pi$
$181$	20.5463	1.52719	0.763596	+	0.645694i	$0.223432\pi$
$182$	0	0
$183$	−1.36047	−0.100569
$184$	0	0
$185$	−2.64963	−0.194804
$186$	0	0
$187$	39.4906	2.88784
$188$	0	0
$189$	4.64963	0.338210
$190$	0	0
$191$	1.28323	0.0928514	0.0464257	−	0.998922i	$-0.485217\pi$
$191$	1.28323	0.0928514	0.0464257	+	0.998922i	$0.485217\pi$
$192$	0	0
$193$	−5.78043	−0.416084	−0.208042	−	0.978120i	$-0.566709\pi$
$193$	−5.78043	−0.416084	−0.208042	+	0.978120i	$0.566709\pi$
$194$	0	0
$195$	−3.47668	−0.248971
$196$	0	0
$197$	−14.6987	−1.04724	−0.523619	−	0.851952i	$-0.675419\pi$
$197$	−14.6987	−1.04724	−0.523619	+	0.851952i	$0.675419\pi$
$198$	0	0
$199$	23.9329	1.69656	0.848278	−	0.529552i	$-0.177640\pi$
$199$	23.9329	1.69656	0.848278	+	0.529552i	$0.177640\pi$
$200$	0	0
$201$	−6.11621	−0.431404
$202$	0	0
$203$	−41.7043	−2.92707
$204$	0	0
$205$	8.12631	0.567566
$206$	0	0
$207$	6.44607	0.448033
$208$	0	0
$209$	0	0
$210$	0	0
$211$	28.5373	1.96459	0.982294	−	0.187348i	$-0.0599891\pi$
$211$	28.5373	1.96459	0.982294	+	0.187348i	$0.0599891\pi$
$212$	0	0
$213$	−3.79645	−0.260128
$214$	0	0
$215$	6.12631	0.417811
$216$	0	0
$217$	−4.64963	−0.315637
$218$	0	0
$219$	5.60300	0.378615
$220$	0	0
$221$	−22.4110	−1.50752
$222$	0	0
$223$	−7.17294	−0.480336	−0.240168	−	0.970731i	$-0.577202\pi$
$223$	−7.17294	−0.480336	−0.240168	+	0.970731i	$0.577202\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	10.1524	0.673840	0.336920	−	0.941533i	$-0.390615\pi$
$227$	10.1524	0.673840	0.336920	+	0.941533i	$0.390615\pi$
$228$	0	0
$229$	11.2470	0.743224	0.371612	−	0.928388i	$-0.378805\pi$
$229$	11.2470	0.743224	0.371612	+	0.928388i	$0.378805\pi$
$230$	0	0
$231$	−28.4850	−1.87418
$232$	0	0
$233$	10.3849	0.680334	0.340167	−	0.940365i	$-0.389516\pi$
$233$	10.3849	0.680334	0.340167	+	0.940365i	$0.389516\pi$
$234$	0	0
$235$	0.853176	0.0556551
$236$	0	0
$237$	−11.2731	−0.732269
$238$	0	0
$239$	8.06509	0.521687	0.260844	−	0.965381i	$-0.415999\pi$
$239$	8.06509	0.521687	0.260844	+	0.965381i	$0.415999\pi$
$240$	0	0
$241$	17.2060	1.10834	0.554168	−	0.832405i	$-0.313037\pi$
$241$	17.2060	1.10834	0.554168	+	0.832405i	$0.313037\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	14.6190	0.933975
$246$	0	0
$247$	0	0
$248$	0	0
$249$	10.4461	0.661993
$250$	0	0
$251$	−23.6131	−1.49045	−0.745223	−	0.666816i	$-0.767657\pi$
$251$	−23.6131	−1.49045	−0.745223	+	0.666816i	$0.767657\pi$
$252$	0	0
$253$	−39.4906	−2.48276
$254$	0	0
$255$	6.44607	0.403669
$256$	0	0
$257$	17.7453	1.10692	0.553461	−	0.832875i	$-0.313307\pi$
$257$	17.7453	1.10692	0.553461	+	0.832875i	$0.313307\pi$
$258$	0	0
$259$	12.3198	0.765513
$260$	0	0
$261$	8.96939	0.555191
$262$	0	0
$263$	−13.2470	−0.816846	−0.408423	−	0.912793i	$-0.633921\pi$
$263$	−13.2470	−0.816846	−0.408423	+	0.912793i	$0.633921\pi$
$264$	0	0
$265$	−1.14682	−0.0704488
$266$	0	0
$267$	−9.79645	−0.599533
$268$	0	0
$269$	−25.8817	−1.57804	−0.789019	−	0.614369i	$-0.789410\pi$
$269$	−25.8817	−1.57804	−0.789019	+	0.614369i	$0.789410\pi$
$270$	0	0
$271$	−6.31384	−0.383539	−0.191769	−	0.981440i	$-0.561423\pi$
$271$	−6.31384	−0.383539	−0.191769	+	0.981440i	$0.561423\pi$
$272$	0	0
$273$	16.1653	0.978367
$274$	0	0
$275$	−6.12631	−0.369430
$276$	0	0
$277$	8.99551	0.540488	0.270244	−	0.962792i	$-0.412896\pi$
$277$	8.99551	0.540488	0.270244	+	0.962792i	$0.412896\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−23.5519	−1.40499	−0.702493	−	0.711690i	$-0.747930\pi$
$281$	−23.5519	−1.40499	−0.702493	+	0.711690i	$0.747930\pi$
$282$	0	0
$283$	24.4951	1.45609	0.728043	−	0.685532i	$-0.240430\pi$
$283$	24.4951	1.45609	0.728043	+	0.685532i	$0.240430\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−37.7843	−2.23034
$288$	0	0
$289$	24.5519	1.44423
$290$	0	0
$291$	17.0185	0.997640
$292$	0	0
$293$	−0.600556	−0.0350849	−0.0175424	−	0.999846i	$-0.505584\pi$
$293$	−0.600556	−0.0350849	−0.0175424	+	0.999846i	$0.505584\pi$
$294$	0	0
$295$	−10.5623	−0.614960
$296$	0	0
$297$	6.12631	0.355485
$298$	0	0
$299$	22.4110	1.29606
$300$	0	0
$301$	−28.4850	−1.64185
$302$	0	0
$303$	−12.7658	−0.733378
$304$	0	0
$305$	1.36047	0.0779004
$306$	0	0
$307$	−29.1347	−1.66280	−0.831402	−	0.555672i	$-0.812461\pi$
$307$	−29.1347	−1.66280	−0.831402	+	0.555672i	$0.812461\pi$
$308$	0	0
$309$	−10.7759	−0.613022
$310$	0	0
$311$	−10.9043	−0.618326	−0.309163	−	0.951009i	$-0.600049\pi$
$311$	−10.9043	−0.618326	−0.309163	+	0.951009i	$0.600049\pi$
$312$	0	0
$313$	6.43005	0.363448	0.181724	−	0.983350i	$-0.441832\pi$
$313$	6.43005	0.363448	0.181724	+	0.983350i	$0.441832\pi$
$314$	0	0
$315$	−4.64963	−0.261977
$316$	0	0
$317$	23.4517	1.31718	0.658589	−	0.752503i	$-0.271154\pi$
$317$	23.4517	1.31718	0.658589	+	0.752503i	$0.271154\pi$
$318$	0	0
$319$	−54.9493	−3.07657
$320$	0	0
$321$	−8.44607	−0.471414
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.47668	0.192852
$326$	0	0
$327$	−10.3198	−0.570684
$328$	0	0
$329$	−3.96695	−0.218705
$330$	0	0
$331$	26.5112	1.45719	0.728593	−	0.684947i	$-0.240175\pi$
$331$	26.5112	1.45719	0.728593	+	0.684947i	$0.240175\pi$
$332$	0	0
$333$	−2.64963	−0.145199
$334$	0	0
$335$	6.11621	0.334164
$336$	0	0
$337$	−8.99551	−0.490016	−0.245008	−	0.969521i	$-0.578791\pi$
$337$	−8.99551	−0.490016	−0.245008	+	0.969521i	$0.578791\pi$
$338$	0	0
$339$	8.95337	0.486280
$340$	0	0
$341$	−6.12631	−0.331758
$342$	0	0
$343$	−35.4256	−1.91280
$344$	0	0
$345$	−6.44607	−0.347045
$346$	0	0
$347$	10.5783	0.567873	0.283937	−	0.958843i	$-0.408359\pi$
$347$	10.5783	0.567873	0.283937	+	0.958843i	$0.408359\pi$
$348$	0	0
$349$	2.95929	0.158407	0.0792036	−	0.996858i	$-0.474762\pi$
$349$	2.95929	0.158407	0.0792036	+	0.996858i	$0.474762\pi$
$350$	0	0
$351$	−3.47668	−0.185572
$352$	0	0
$353$	−10.1935	−0.542543	−0.271271	−	0.962503i	$-0.587444\pi$
$353$	−10.1935	−0.542543	−0.271271	+	0.962503i	$0.587444\pi$
$354$	0	0
$355$	3.79645	0.201495
$356$	0	0
$357$	−29.9718	−1.58628
$358$	0	0
$359$	−23.8615	−1.25936	−0.629682	−	0.776853i	$-0.716815\pi$
$359$	−23.8615	−1.25936	−0.629682	+	0.776853i	$0.716815\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−26.5317	−1.39255
$364$	0	0
$365$	−5.60300	−0.293274
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	8.12631	0.423039
$370$	0	0
$371$	5.33230	0.276839
$372$	0	0
$373$	−5.59290	−0.289589	−0.144795	−	0.989462i	$-0.546252\pi$
$373$	−5.59290	−0.289589	−0.144795	+	0.989462i	$0.546252\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	31.1837	1.60604
$378$	0	0
$379$	−26.8511	−1.37925	−0.689625	−	0.724167i	$-0.742224\pi$
$379$	−26.8511	−1.37925	−0.689625	+	0.724167i	$0.742224\pi$
$380$	0	0
$381$	2.95337	0.151306
$382$	0	0
$383$	−33.0446	−1.68850	−0.844249	−	0.535950i	$-0.819953\pi$
$383$	−33.0446	−1.68850	−0.844249	+	0.535950i	$0.819953\pi$
$384$	0	0
$385$	28.4850	1.45173
$386$	0	0
$387$	6.12631	0.311418
$388$	0	0
$389$	−19.0957	−0.968190	−0.484095	−	0.875015i	$-0.660851\pi$
$389$	−19.0957	−0.968190	−0.484095	+	0.875015i	$0.660851\pi$
$390$	0	0
$391$	−41.5519	−2.10137
$392$	0	0
$393$	16.2686	0.820644
$394$	0	0
$395$	11.2731	0.567213
$396$	0	0
$397$	27.3974	1.37504	0.687518	−	0.726168i	$-0.258700\pi$
$397$	27.3974	1.37504	0.687518	+	0.726168i	$0.258700\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.79645	−0.489211	−0.244606	−	0.969623i	$-0.578659\pi$
$401$	−9.79645	−0.489211	−0.244606	+	0.969623i	$0.578659\pi$
$402$	0	0
$403$	3.47668	0.173186
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	16.2324	0.804611
$408$	0	0
$409$	−21.4906	−1.06264	−0.531322	−	0.847170i	$-0.678305\pi$
$409$	−21.4906	−1.06264	−0.531322	+	0.847170i	$0.678305\pi$
$410$	0	0
$411$	3.08560	0.152202
$412$	0	0
$413$	49.1107	2.41658
$414$	0	0
$415$	−10.4461	−0.512778
$416$	0	0
$417$	−16.6597	−0.815830
$418$	0	0
$419$	17.3483	0.847521	0.423760	−	0.905774i	$-0.360710\pi$
$419$	17.3483	0.847521	0.423760	+	0.905774i	$0.360710\pi$
$420$	0	0
$421$	11.6190	0.566276	0.283138	−	0.959079i	$-0.408625\pi$
$421$	11.6190	0.566276	0.283138	+	0.959079i	$0.408625\pi$
$422$	0	0
$423$	0.853176	0.0414829
$424$	0	0
$425$	−6.44607	−0.312681
$426$	0	0
$427$	−6.32569	−0.306121
$428$	0	0
$429$	21.2993	1.02834
$430$	0	0
$431$	−10.7819	−0.519344	−0.259672	−	0.965697i	$-0.583614\pi$
$431$	−10.7819	−0.519344	−0.259672	+	0.965697i	$0.583614\pi$
$432$	0	0
$433$	28.4339	1.36645	0.683224	−	0.730209i	$-0.260577\pi$
$433$	28.4339	1.36645	0.683224	+	0.730209i	$0.260577\pi$
$434$	0	0
$435$	−8.96939	−0.430049
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−25.7843	−1.23062	−0.615309	−	0.788286i	$-0.710969\pi$
$439$	−25.7843	−1.23062	−0.615309	+	0.788286i	$0.710969\pi$
$440$	0	0
$441$	14.6190	0.696144
$442$	0	0
$443$	−7.43148	−0.353080	−0.176540	−	0.984293i	$-0.556491\pi$
$443$	−7.43148	−0.353080	−0.176540	+	0.984293i	$0.556491\pi$
$444$	0	0
$445$	9.79645	0.464396
$446$	0	0
$447$	21.6580	1.02439
$448$	0	0
$449$	16.8021	0.792938	0.396469	−	0.918048i	$-0.370235\pi$
$449$	16.8021	0.792938	0.396469	+	0.918048i	$0.370235\pi$
$450$	0	0
$451$	−49.7843	−2.34425
$452$	0	0
$453$	−3.68024	−0.172913
$454$	0	0
$455$	−16.1653	−0.757840
$456$	0	0
$457$	−33.2161	−1.55378	−0.776892	−	0.629634i	$-0.783205\pi$
$457$	−33.2161	−1.55378	−0.776892	+	0.629634i	$0.783205\pi$
$458$	0	0
$459$	6.44607	0.300877
$460$	0	0
$461$	26.9082	1.25324	0.626619	−	0.779326i	$-0.284438\pi$
$461$	26.9082	1.25324	0.626619	+	0.779326i	$0.284438\pi$
$462$	0	0
$463$	32.2975	1.50099	0.750496	−	0.660875i	$-0.229814\pi$
$463$	32.2975	1.50099	0.750496	+	0.660875i	$0.229814\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	−36.9834	−1.71139	−0.855693	−	0.517484i	$-0.826869\pi$
$467$	−36.9834	−1.71139	−0.855693	+	0.517484i	$0.826869\pi$
$468$	0	0
$469$	−28.4381	−1.31315
$470$	0	0
$471$	6.47219	0.298223
$472$	0	0
$473$	−37.5317	−1.72571
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.14682	−0.0525095
$478$	0	0
$479$	10.7819	0.492636	0.246318	−	0.969189i	$-0.420779\pi$
$479$	10.7819	0.492636	0.246318	+	0.969189i	$0.420779\pi$
$480$	0	0
$481$	−9.21191	−0.420027
$482$	0	0
$483$	29.9718	1.36377
$484$	0	0
$485$	−17.0185	−0.772768
$486$	0	0
$487$	−4.40710	−0.199705	−0.0998524	−	0.995002i	$-0.531837\pi$
$487$	−4.40710	−0.199705	−0.0998524	+	0.995002i	$0.531837\pi$
$488$	0	0
$489$	−1.47668	−0.0667780
$490$	0	0
$491$	15.0056	0.677193	0.338597	−	0.940932i	$-0.390048\pi$
$491$	15.0056	0.677193	0.338597	+	0.940932i	$0.390048\pi$
$492$	0	0
$493$	−57.8173	−2.60396
$494$	0	0
$495$	−6.12631	−0.275357
$496$	0	0
$497$	−17.6521	−0.791803
$498$	0	0
$499$	25.9067	1.15974	0.579872	−	0.814707i	$-0.303102\pi$
$499$	25.9067	1.15974	0.579872	+	0.814707i	$0.303102\pi$
$500$	0	0
$501$	−17.5519	−0.784160
$502$	0	0
$503$	14.6375	0.652653	0.326326	−	0.945257i	$-0.394189\pi$
$503$	14.6375	0.652653	0.326326	+	0.945257i	$0.394189\pi$
$504$	0	0
$505$	12.7658	0.568072
$506$	0	0
$507$	0.912662	0.0405327
$508$	0	0
$509$	−11.7554	−0.521050	−0.260525	−	0.965467i	$-0.583896\pi$
$509$	−11.7554	−0.521050	−0.260525	+	0.965467i	$0.583896\pi$
$510$	0	0
$511$	26.0518	1.15246
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.7759	0.474844
$516$	0	0
$517$	−5.22682	−0.229875
$518$	0	0
$519$	−7.29925	−0.320402
$520$	0	0
$521$	2.21957	0.0972411	0.0486206	−	0.998817i	$-0.484517\pi$
$521$	2.21957	0.0972411	0.0486206	+	0.998817i	$0.484517\pi$
$522$	0	0
$523$	17.6862	0.773362	0.386681	−	0.922214i	$-0.373622\pi$
$523$	17.6862	0.773362	0.386681	+	0.922214i	$0.373622\pi$
$524$	0	0
$525$	4.64963	0.202926
$526$	0	0
$527$	−6.44607	−0.280795
$528$	0	0
$529$	18.5519	0.806603
$530$	0	0
$531$	−10.5623	−0.458364
$532$	0	0
$533$	28.2526	1.22376
$534$	0	0
$535$	8.44607	0.365156
$536$	0	0
$537$	−4.63953	−0.200210
$538$	0	0
$539$	−89.5606	−3.85765
$540$	0	0
$541$	39.0574	1.67921	0.839605	−	0.543197i	$-0.182786\pi$
$541$	39.0574	1.67921	0.839605	+	0.543197i	$0.182786\pi$
$542$	0	0
$543$	−20.5463	−0.881725
$544$	0	0
$545$	10.3198	0.442050
$546$	0	0
$547$	−21.1218	−0.903104	−0.451552	−	0.892245i	$-0.649129\pi$
$547$	−21.1218	−0.903104	−0.451552	+	0.892245i	$0.649129\pi$
$548$	0	0
$549$	1.36047	0.0580635
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−52.4158	−2.22895
$554$	0	0
$555$	2.64963	0.112470
$556$	0	0
$557$	−3.08560	−0.130741	−0.0653706	−	0.997861i	$-0.520823\pi$
$557$	−3.08560	−0.130741	−0.0653706	+	0.997861i	$0.520823\pi$
$558$	0	0
$559$	21.2993	0.900862
$560$	0	0
$561$	−39.4906	−1.66730
$562$	0	0
$563$	−0.193454	−0.00815310	−0.00407655	−	0.999992i	$-0.501298\pi$
$563$	−0.193454	−0.00815310	−0.00407655	+	0.999992i	$0.501298\pi$
$564$	0	0
$565$	−8.95337	−0.376671
$566$	0	0
$567$	−4.64963	−0.195266
$568$	0	0
$569$	0.736964	0.0308951	0.0154476	−	0.999881i	$-0.495083\pi$
$569$	0.736964	0.0308951	0.0154476	+	0.999881i	$0.495083\pi$
$570$	0	0
$571$	23.0056	0.962755	0.481377	−	0.876514i	$-0.340137\pi$
$571$	23.0056	0.962755	0.481377	+	0.876514i	$0.340137\pi$
$572$	0	0
$573$	−1.28323	−0.0536078
$574$	0	0
$575$	6.44607	0.268820
$576$	0	0
$577$	−27.4666	−1.14345	−0.571725	−	0.820446i	$-0.693725\pi$
$577$	−27.4666	−1.14345	−0.571725	+	0.820446i	$0.693725\pi$
$578$	0	0
$579$	5.78043	0.240226
$580$	0	0
$581$	48.5703	2.01504
$582$	0	0
$583$	7.02580	0.290979
$584$	0	0
$585$	3.47668	0.143743
$586$	0	0
$587$	21.9388	0.905510	0.452755	−	0.891635i	$-0.350441\pi$
$587$	21.9388	0.905510	0.452755	+	0.891635i	$0.350441\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	14.6987	0.604623
$592$	0	0
$593$	32.4440	1.33232	0.666158	−	0.745811i	$-0.267938\pi$
$593$	32.4440	1.33232	0.666158	+	0.745811i	$0.267938\pi$
$594$	0	0
$595$	29.9718	1.22873
$596$	0	0
$597$	−23.9329	−0.979507
$598$	0	0
$599$	18.4270	0.752906	0.376453	−	0.926436i	$-0.377144\pi$
$599$	18.4270	0.752906	0.376453	+	0.926436i	$0.377144\pi$
$600$	0	0
$601$	−35.1768	−1.43489	−0.717446	−	0.696614i	$-0.754689\pi$
$601$	−35.1768	−1.43489	−0.717446	+	0.696614i	$0.754689\pi$
$602$	0	0
$603$	6.11621	0.249071
$604$	0	0
$605$	26.5317	1.07867
$606$	0	0
$607$	6.52332	0.264773	0.132387	−	0.991198i	$-0.457736\pi$
$607$	6.52332	0.264773	0.132387	+	0.991198i	$0.457736\pi$
$608$	0	0
$609$	41.7043	1.68994
$610$	0	0
$611$	2.96623	0.120001
$612$	0	0
$613$	34.5602	1.39587	0.697937	−	0.716159i	$-0.254101\pi$
$613$	34.5602	1.39587	0.697937	+	0.716159i	$0.254101\pi$
$614$	0	0
$615$	−8.12631	−0.327684
$616$	0	0
$617$	36.1594	1.45572	0.727861	−	0.685725i	$-0.240515\pi$
$617$	36.1594	1.45572	0.727861	+	0.685725i	$0.240515\pi$
$618$	0	0
$619$	41.4966	1.66789	0.833944	−	0.551849i	$-0.186078\pi$
$619$	41.4966	1.66789	0.833944	+	0.551849i	$0.186078\pi$
$620$	0	0
$621$	−6.44607	−0.258672
$622$	0	0
$623$	−45.5498	−1.82491
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	17.0797	0.681011
$630$	0	0
$631$	−4.69176	−0.186776	−0.0933880	−	0.995630i	$-0.529770\pi$
$631$	−4.69176	−0.186776	−0.0933880	+	0.995630i	$0.529770\pi$
$632$	0	0
$633$	−28.5373	−1.13426
$634$	0	0
$635$	−2.95337	−0.117201
$636$	0	0
$637$	50.8257	2.01379
$638$	0	0
$639$	3.79645	0.150185
$640$	0	0
$641$	44.8671	1.77215	0.886073	−	0.463546i	$-0.153423\pi$
$641$	44.8671	1.77215	0.886073	+	0.463546i	$0.153423\pi$
$642$	0	0
$643$	−0.871953	−0.0343865	−0.0171932	−	0.999852i	$-0.505473\pi$
$643$	−0.871953	−0.0343865	−0.0171932	+	0.999852i	$0.505473\pi$
$644$	0	0
$645$	−6.12631	−0.241223
$646$	0	0
$647$	6.29365	0.247429	0.123714	−	0.992318i	$-0.460519\pi$
$647$	6.29365	0.247429	0.123714	+	0.992318i	$0.460519\pi$
$648$	0	0
$649$	64.7078	2.54001
$650$	0	0
$651$	4.64963	0.182233
$652$	0	0
$653$	−20.0271	−0.783722	−0.391861	−	0.920024i	$-0.628169\pi$
$653$	−20.0271	−0.783722	−0.391861	+	0.920024i	$0.628169\pi$
$654$	0	0
$655$	−16.2686	−0.635668
$656$	0	0
$657$	−5.60300	−0.218594
$658$	0	0
$659$	32.9614	1.28399	0.641997	−	0.766707i	$-0.278106\pi$
$659$	32.9614	1.28399	0.641997	+	0.766707i	$0.278106\pi$
$660$	0	0
$661$	−36.3918	−1.41548	−0.707738	−	0.706475i	$-0.750284\pi$
$661$	−36.3918	−1.41548	−0.707738	+	0.706475i	$0.750284\pi$
$662$	0	0
$663$	22.4110	0.870370
$664$	0	0
$665$	0	0
$666$	0	0
$667$	57.8173	2.23870
$668$	0	0
$669$	7.17294	0.277322
$670$	0	0
$671$	−8.33467	−0.321757
$672$	0	0
$673$	26.0870	1.00558	0.502791	−	0.864408i	$-0.332307\pi$
$673$	26.0870	1.00558	0.502791	+	0.864408i	$0.332307\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	24.2547	0.932183	0.466091	−	0.884737i	$-0.345662\pi$
$677$	24.2547	0.932183	0.466091	+	0.884737i	$0.345662\pi$
$678$	0	0
$679$	79.1295	3.03671
$680$	0	0
$681$	−10.1524	−0.389042
$682$	0	0
$683$	7.75431	0.296711	0.148355	−	0.988934i	$-0.452602\pi$
$683$	7.75431	0.296711	0.148355	+	0.988934i	$0.452602\pi$
$684$	0	0
$685$	−3.08560	−0.117895
$686$	0	0
$687$	−11.2470	−0.429100
$688$	0	0
$689$	−3.98714	−0.151898
$690$	0	0
$691$	−22.4440	−0.853811	−0.426905	−	0.904296i	$-0.640396\pi$
$691$	−22.4440	−0.853811	−0.426905	+	0.904296i	$0.640396\pi$
$692$	0	0
$693$	28.4850	1.08206
$694$	0	0
$695$	16.6597	0.631939
$696$	0	0
$697$	−52.3828	−1.98414
$698$	0	0
$699$	−10.3849	−0.392791
$700$	0	0
$701$	−15.0797	−0.569552	−0.284776	−	0.958594i	$-0.591919\pi$
$701$	−15.0797	−0.569552	−0.284776	+	0.958594i	$0.591919\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−0.853176	−0.0321325
$706$	0	0
$707$	−59.3564	−2.23233
$708$	0	0
$709$	14.6277	0.549354	0.274677	−	0.961537i	$-0.411429\pi$
$709$	14.6277	0.549354	0.274677	+	0.961537i	$0.411429\pi$
$710$	0	0
$711$	11.2731	0.422776
$712$	0	0
$713$	6.44607	0.241407
$714$	0	0
$715$	−21.2993	−0.796547
$716$	0	0
$717$	−8.06509	−0.301196
$718$	0	0
$719$	16.8473	0.628297	0.314148	−	0.949374i	$-0.398281\pi$
$719$	16.8473	0.628297	0.314148	+	0.949374i	$0.398281\pi$
$720$	0	0
$721$	−50.1041	−1.86597
$722$	0	0
$723$	−17.2060	−0.639898
$724$	0	0
$725$	8.96939	0.333115
$726$	0	0
$727$	−18.3486	−0.680513	−0.340257	−	0.940333i	$-0.610514\pi$
$727$	−18.3486	−0.680513	−0.340257	+	0.940333i	$0.610514\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−39.4906	−1.46061
$732$	0	0
$733$	8.15936	0.301373	0.150686	−	0.988582i	$-0.451852\pi$
$733$	8.15936	0.301373	0.150686	+	0.988582i	$0.451852\pi$
$734$	0	0
$735$	−14.6190	−0.539230
$736$	0	0
$737$	−37.4698	−1.38022
$738$	0	0
$739$	22.0261	0.810244	0.405122	−	0.914263i	$-0.367229\pi$
$739$	22.0261	0.810244	0.405122	+	0.914263i	$0.367229\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3.46272	−0.127035	−0.0635175	−	0.997981i	$-0.520232\pi$
$743$	−3.46272	−0.127035	−0.0635175	+	0.997981i	$0.520232\pi$
$744$	0	0
$745$	−21.6580	−0.793487
$746$	0	0
$747$	−10.4461	−0.382202
$748$	0	0
$749$	−39.2711	−1.43493
$750$	0	0
$751$	−38.5783	−1.40774	−0.703871	−	0.710328i	$-0.748547\pi$
$751$	−38.5783	−1.40774	−0.703871	+	0.710328i	$0.748547\pi$
$752$	0	0
$753$	23.6131	0.860509
$754$	0	0
$755$	3.68024	0.133938
$756$	0	0
$757$	29.0606	1.05623	0.528113	−	0.849174i	$-0.322900\pi$
$757$	29.0606	1.05623	0.528113	+	0.849174i	$0.322900\pi$
$758$	0	0
$759$	39.4906	1.43342
$760$	0	0
$761$	−10.0362	−0.363812	−0.181906	−	0.983316i	$-0.558227\pi$
$761$	−10.0362	−0.363812	−0.181906	+	0.983316i	$0.558227\pi$
$762$	0	0
$763$	−47.9830	−1.73710
$764$	0	0
$765$	−6.44607	−0.233058
$766$	0	0
$767$	−36.7217	−1.32595
$768$	0	0
$769$	18.1973	0.656212	0.328106	−	0.944641i	$-0.393590\pi$
$769$	18.1973	0.656212	0.328106	+	0.944641i	$0.393590\pi$
$770$	0	0
$771$	−17.7453	−0.639082
$772$	0	0
$773$	−1.16702	−0.0419747	−0.0209874	−	0.999780i	$-0.506681\pi$
$773$	−1.16702	−0.0419747	−0.0209874	+	0.999780i	$0.506681\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−12.3198	−0.441969
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−23.2582	−0.832245
$782$	0	0
$783$	−8.96939	−0.320540
$784$	0	0
$785$	−6.47219	−0.231002
$786$	0	0
$787$	51.1284	1.82253	0.911266	−	0.411818i	$-0.135106\pi$
$787$	51.1284	1.82253	0.911266	+	0.411818i	$0.135106\pi$
$788$	0	0
$789$	13.2470	0.471606
$790$	0	0
$791$	41.6298	1.48019
$792$	0	0
$793$	4.72993	0.167965
$794$	0	0
$795$	1.14682	0.0406736
$796$	0	0
$797$	−41.6841	−1.47653	−0.738263	−	0.674513i	$-0.764354\pi$
$797$	−41.6841	−1.47653	−0.738263	+	0.674513i	$0.764354\pi$
$798$	0	0
$799$	−5.49964	−0.194563
$800$	0	0
$801$	9.79645	0.346141
$802$	0	0
$803$	34.3257	1.21133
$804$	0	0
$805$	−29.9718	−1.05637
$806$	0	0
$807$	25.8817	0.911080
$808$	0	0
$809$	46.7867	1.64493	0.822467	−	0.568813i	$-0.192597\pi$
$809$	46.7867	1.64493	0.822467	+	0.568813i	$0.192597\pi$
$810$	0	0
$811$	18.8309	0.661243	0.330622	−	0.943763i	$-0.392742\pi$
$811$	18.8309	0.661243	0.330622	+	0.943763i	$0.392742\pi$
$812$	0	0
$813$	6.31384	0.221436
$814$	0	0
$815$	1.47668	0.0517260
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−16.1653	−0.564861
$820$	0	0
$821$	26.2237	0.915215	0.457608	−	0.889154i	$-0.348706\pi$
$821$	26.2237	0.915215	0.457608	+	0.889154i	$0.348706\pi$
$822$	0	0
$823$	−34.3267	−1.19655	−0.598277	−	0.801290i	$-0.704148\pi$
$823$	−34.3267	−1.19655	−0.598277	+	0.801290i	$0.704148\pi$
$824$	0	0
$825$	6.12631	0.213291
$826$	0	0
$827$	18.6208	0.647507	0.323753	−	0.946141i	$-0.395055\pi$
$827$	18.6208	0.647507	0.323753	+	0.946141i	$0.395055\pi$
$828$	0	0
$829$	−8.64852	−0.300375	−0.150188	−	0.988658i	$-0.547988\pi$
$829$	−8.64852	−0.300375	−0.150188	+	0.988658i	$0.547988\pi$
$830$	0	0
$831$	−8.99551	−0.312051
$832$	0	0
$833$	−94.2352	−3.26506
$834$	0	0
$835$	17.5519	0.607408
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	16.7739	0.579099	0.289549	−	0.957163i	$-0.406495\pi$
$839$	16.7739	0.579099	0.289549	+	0.957163i	$0.406495\pi$
$840$	0	0
$841$	51.4499	1.77414
$842$	0	0
$843$	23.5519	0.811170
$844$	0	0
$845$	−0.912662	−0.0313965
$846$	0	0
$847$	−123.362	−4.23878
$848$	0	0
$849$	−24.4951	−0.840671
$850$	0	0
$851$	−17.0797	−0.585484
$852$	0	0
$853$	−33.4256	−1.14447	−0.572235	−	0.820090i	$-0.693923\pi$
$853$	−33.4256	−1.14447	−0.572235	+	0.820090i	$0.693923\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38.1504	−1.30319	−0.651596	−	0.758566i	$-0.725900\pi$
$857$	−38.1504	−1.30319	−0.651596	+	0.758566i	$0.725900\pi$
$858$	0	0
$859$	4.81420	0.164259	0.0821293	−	0.996622i	$-0.473828\pi$
$859$	4.81420	0.164259	0.0821293	+	0.996622i	$0.473828\pi$
$860$	0	0
$861$	37.7843	1.28769
$862$	0	0
$863$	−10.1594	−0.345829	−0.172914	−	0.984937i	$-0.555318\pi$
$863$	−10.1594	−0.345829	−0.172914	+	0.984937i	$0.555318\pi$
$864$	0	0
$865$	7.29925	0.248182
$866$	0	0
$867$	−24.5519	−0.833825
$868$	0	0
$869$	−69.0627	−2.34279
$870$	0	0
$871$	21.2641	0.720508
$872$	0	0
$873$	−17.0185	−0.575988
$874$	0	0
$875$	−4.64963	−0.157186
$876$	0	0
$877$	36.9162	1.24657	0.623286	−	0.781994i	$-0.285797\pi$
$877$	36.9162	1.24657	0.623286	+	0.781994i	$0.285797\pi$
$878$	0	0
$879$	0.600556	0.0202563
$880$	0	0
$881$	−38.1994	−1.28697	−0.643486	−	0.765458i	$-0.722512\pi$
$881$	−38.1994	−1.28697	−0.643486	+	0.765458i	$0.722512\pi$
$882$	0	0
$883$	−37.1576	−1.25045	−0.625227	−	0.780443i	$-0.714994\pi$
$883$	−37.1576	−1.25045	−0.625227	+	0.780443i	$0.714994\pi$
$884$	0	0
$885$	10.5623	0.355047
$886$	0	0
$887$	−15.3472	−0.515309	−0.257654	−	0.966237i	$-0.582950\pi$
$887$	−15.3472	−0.515309	−0.257654	+	0.966237i	$0.582950\pi$
$888$	0	0
$889$	13.7321	0.460559
$890$	0	0
$891$	−6.12631	−0.205239
$892$	0	0
$893$	0	0
$894$	0	0
$895$	4.63953	0.155082
$896$	0	0
$897$	−22.4110	−0.748280
$898$	0	0
$899$	8.96939	0.299146
$900$	0	0
$901$	7.39251	0.246280
$902$	0	0
$903$	28.4850	0.947923
$904$	0	0
$905$	20.5463	0.682981
$906$	0	0
$907$	24.2627	0.805630	0.402815	−	0.915281i	$-0.368032\pi$
$907$	24.2627	0.805630	0.402815	+	0.915281i	$0.368032\pi$
$908$	0	0
$909$	12.7658	0.423416
$910$	0	0
$911$	−19.7804	−0.655355	−0.327677	−	0.944790i	$-0.606266\pi$
$911$	−19.7804	−0.655355	−0.327677	+	0.944790i	$0.606266\pi$
$912$	0	0
$913$	63.9959	2.11795
$914$	0	0
$915$	−1.36047	−0.0449758
$916$	0	0
$917$	75.6431	2.49795
$918$	0	0
$919$	−1.70635	−0.0562874	−0.0281437	−	0.999604i	$-0.508960\pi$
$919$	−1.70635	−0.0562874	−0.0281437	+	0.999604i	$0.508960\pi$
$920$	0	0
$921$	29.1347	0.960020
$922$	0	0
$923$	13.1991	0.434452
$924$	0	0
$925$	−2.64963	−0.0871191
$926$	0	0
$927$	10.7759	0.353928
$928$	0	0
$929$	−2.06825	−0.0678572	−0.0339286	−	0.999424i	$-0.510802\pi$
$929$	−2.06825	−0.0678572	−0.0339286	+	0.999424i	$0.510802\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	10.9043	0.356991
$934$	0	0
$935$	39.4906	1.29148
$936$	0	0
$937$	−10.9815	−0.358751	−0.179376	−	0.983781i	$-0.557408\pi$
$937$	−10.9815	−0.358751	−0.179376	+	0.983781i	$0.557408\pi$
$938$	0	0
$939$	−6.43005	−0.209837
$940$	0	0
$941$	0.517393	0.0168665	0.00843327	−	0.999964i	$-0.497316\pi$
$941$	0.517393	0.0168665	0.00843327	+	0.999964i	$0.497316\pi$
$942$	0	0
$943$	52.3828	1.70582
$944$	0	0
$945$	4.64963	0.151252
$946$	0	0
$947$	4.28671	0.139299	0.0696497	−	0.997572i	$-0.477812\pi$
$947$	4.28671	0.139299	0.0696497	+	0.997572i	$0.477812\pi$
$948$	0	0
$949$	−19.4798	−0.632343
$950$	0	0
$951$	−23.4517	−0.760473
$952$	0	0
$953$	36.7801	1.19142	0.595712	−	0.803198i	$-0.296870\pi$
$953$	36.7801	1.19142	0.595712	+	0.803198i	$0.296870\pi$
$954$	0	0
$955$	1.28323	0.0415244
$956$	0	0
$957$	54.9493	1.77626
$958$	0	0
$959$	14.3469	0.463285
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	8.44607	0.272171
$964$	0	0
$965$	−5.78043	−0.186079
$966$	0	0
$967$	49.8375	1.60267	0.801334	−	0.598218i	$-0.204124\pi$
$967$	49.8375	1.60267	0.801334	+	0.598218i	$0.204124\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.82849	0.122862	0.0614310	−	0.998111i	$-0.480434\pi$
$971$	3.82849	0.122862	0.0614310	+	0.998111i	$0.480434\pi$
$972$	0	0
$973$	−77.4615	−2.48330
$974$	0	0
$975$	−3.47668	−0.111343
$976$	0	0
$977$	−25.8776	−0.827896	−0.413948	−	0.910300i	$-0.635851\pi$
$977$	−25.8776	−0.827896	−0.413948	+	0.910300i	$0.635851\pi$
$978$	0	0
$979$	−60.0161	−1.91812
$980$	0	0
$981$	10.3198	0.329485
$982$	0	0
$983$	−23.2380	−0.741178	−0.370589	−	0.928797i	$-0.620844\pi$
$983$	−23.2380	−0.741178	−0.370589	+	0.928797i	$0.620844\pi$
$984$	0	0
$985$	−14.6987	−0.468339
$986$	0	0
$987$	3.96695	0.126269
$988$	0	0
$989$	39.4906	1.25573
$990$	0	0
$991$	−9.68616	−0.307691	−0.153845	−	0.988095i	$-0.549166\pi$
$991$	−9.68616	−0.307691	−0.153845	+	0.988095i	$0.549166\pi$
$992$	0	0
$993$	−26.5112	−0.841306
$994$	0	0
$995$	23.9329	0.758723
$996$	0	0
$997$	−32.9011	−1.04199	−0.520995	−	0.853560i	$-0.674439\pi$
$997$	−32.9011	−1.04199	−0.520995	+	0.853560i	$0.674439\pi$
$998$	0	0
$999$	2.64963	0.0838304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1860.2.a.i.1.1	✓	4
3.2	odd	2		5580.2.a.m.1.1		4
4.3	odd	2		7440.2.a.cb.1.4		4
5.2	odd	4		9300.2.g.s.3349.5		8
5.3	odd	4		9300.2.g.s.3349.4		8
5.4	even	2		9300.2.a.x.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.i.1.1	✓	4	1.1	even	1	trivial
5580.2.a.m.1.1		4	3.2	odd	2
7440.2.a.cb.1.4		4	4.3	odd	2
9300.2.a.x.1.4		4	5.4	even	2
9300.2.g.s.3349.4		8	5.3	odd	4
9300.2.g.s.3349.5		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -4.64963 q^{7} +1.00000 q^{9} -6.12631 q^{11} +3.47668 q^{13} -1.00000 q^{15} -6.44607 q^{17} +4.64963 q^{21} +6.44607 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.96939 q^{29} +1.00000 q^{31} +6.12631 q^{33} -4.64963 q^{35} -2.64963 q^{37} -3.47668 q^{39} +8.12631 q^{41} +6.12631 q^{43} +1.00000 q^{45} +0.853176 q^{47} +14.6190 q^{49} +6.44607 q^{51} -1.14682 q^{53} -6.12631 q^{55} -10.5623 q^{59} +1.36047 q^{61} -4.64963 q^{63} +3.47668 q^{65} +6.11621 q^{67} -6.44607 q^{69} +3.79645 q^{71} -5.60300 q^{73} -1.00000 q^{75} +28.4850 q^{77} +11.2731 q^{79} +1.00000 q^{81} -10.4461 q^{83} -6.44607 q^{85} -8.96939 q^{87} +9.79645 q^{89} -16.1653 q^{91} -1.00000 q^{93} -17.0185 q^{97} -6.12631 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} + 4 q^{21} - 2 q^{23} + 4 q^{25} - 4 q^{27} + 20 q^{29} + 4 q^{31} - 2 q^{33} - 4 q^{35} + 4 q^{37} - 2 q^{39} + 6 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) \) 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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