LMFDB - Embedded newform 9522.2.a.ba.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9522,2,Mod(1,9522)] 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("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9522 = 2 \cdot 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9522.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,1,0,-3,-2,0,-1,-3,0,3,3,0,2,4,0,8,1,0,3,0,0,11,-3,0, -3,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3174)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	9522.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.70156 q^{5} -4.70156 q^{7} -1.00000 q^{8} -3.70156 q^{10} -4.70156 q^{11} -1.70156 q^{13} +4.70156 q^{14} +1.00000 q^{16} +2.00000 q^{17} +4.00000 q^{19} +3.70156 q^{20} +4.70156 q^{22} +8.70156 q^{25} +1.70156 q^{26} -4.70156 q^{28} +6.40312 q^{29} +0.701562 q^{31} -1.00000 q^{32} -2.00000 q^{34} -17.4031 q^{35} +3.40312 q^{37} -4.00000 q^{38} -3.70156 q^{40} -9.10469 q^{41} +1.40312 q^{43} -4.70156 q^{44} -4.00000 q^{47} +15.1047 q^{49} -8.70156 q^{50} -1.70156 q^{52} -6.40312 q^{53} -17.4031 q^{55} +4.70156 q^{56} -6.40312 q^{58} -7.29844 q^{59} +13.7016 q^{61} -0.701562 q^{62} +1.00000 q^{64} -6.29844 q^{65} -12.0000 q^{67} +2.00000 q^{68} +17.4031 q^{70} -2.59688 q^{71} -6.40312 q^{73} -3.40312 q^{74} +4.00000 q^{76} +22.1047 q^{77} +16.7016 q^{79} +3.70156 q^{80} +9.10469 q^{82} +12.7016 q^{83} +7.40312 q^{85} -1.40312 q^{86} +4.70156 q^{88} -4.29844 q^{89} +8.00000 q^{91} +4.00000 q^{94} +14.8062 q^{95} +7.80625 q^{97} -15.1047 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + q^{5} - 3 q^{7} - 2 q^{8} - q^{10} - 3 q^{11} + 3 q^{13} + 3 q^{14} + 2 q^{16} + 4 q^{17} + 8 q^{19} + q^{20} + 3 q^{22} + 11 q^{25} - 3 q^{26} - 3 q^{28} - 5 q^{31} - 2 q^{32}+ \cdots - 11 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + q^5 - 3 * q^7 - 2 * q^8 - q^10 - 3 * q^11 + 3 * q^13 + 3 * q^14 + 2 * q^16 + 4 * q^17 + 8 * q^19 + q^20 + 3 * q^22 + 11 * q^25 - 3 * q^26 - 3 * q^28 - 5 * q^31 - 2 * q^32 - 4 * q^34 - 22 * q^35 - 6 * q^37 - 8 * q^38 - q^40 + q^41 - 10 * q^43 - 3 * q^44 - 8 * q^47 + 11 * q^49 - 11 * q^50 + 3 * q^52 - 22 * q^55 + 3 * q^56 - 21 * q^59 + 21 * q^61 + 5 * q^62 + 2 * q^64 - 19 * q^65 - 24 * q^67 + 4 * q^68 + 22 * q^70 - 18 * q^71 + 6 * q^74 + 8 * q^76 + 25 * q^77 + 27 * q^79 + q^80 - q^82 + 19 * q^83 + 2 * q^85 + 10 * q^86 + 3 * q^88 - 15 * q^89 + 16 * q^91 + 8 * q^94 + 4 * q^95 - 10 * q^97 - 11 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.70156	1.65539	0.827694	−	0.561179i	$-0.189652\pi$
$5$	3.70156	1.65539	0.827694	+	0.561179i	$0.189652\pi$
$6$	0	0
$7$	−4.70156	−1.77702	−0.888512	−	0.458854i	$-0.848260\pi$
$7$	−4.70156	−1.77702	−0.888512	+	0.458854i	$0.848260\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.70156	−1.17054
$11$	−4.70156	−1.41757	−0.708787	−	0.705422i	$-0.750757\pi$
$11$	−4.70156	−1.41757	−0.708787	+	0.705422i	$0.750757\pi$
$12$	0	0
$13$	−1.70156	−0.471928	−0.235964	−	0.971762i	$-0.575825\pi$
$13$	−1.70156	−0.471928	−0.235964	+	0.971762i	$0.575825\pi$
$14$	4.70156	1.25655
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	3.70156	0.827694
$21$	0	0
$22$	4.70156	1.00238
$23$	0	0
$23$	0	0
$24$	0	0
$25$	8.70156	1.74031
$26$	1.70156	0.333704
$27$	0	0
$28$	−4.70156	−0.888512
$29$	6.40312	1.18903	0.594515	−	0.804084i	$-0.297344\pi$
$29$	6.40312	1.18903	0.594515	+	0.804084i	$0.297344\pi$
$30$	0	0
$31$	0.701562	0.126004	0.0630021	−	0.998013i	$-0.479933\pi$
$31$	0.701562	0.126004	0.0630021	+	0.998013i	$0.479933\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	−17.4031	−2.94166
$36$	0	0
$37$	3.40312	0.559470	0.279735	−	0.960077i	$-0.409753\pi$
$37$	3.40312	0.559470	0.279735	+	0.960077i	$0.409753\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−3.70156	−0.585268
$41$	−9.10469	−1.42191	−0.710957	−	0.703236i	$-0.751738\pi$
$41$	−9.10469	−1.42191	−0.710957	+	0.703236i	$0.751738\pi$
$42$	0	0
$43$	1.40312	0.213974	0.106987	−	0.994260i	$-0.465880\pi$
$43$	1.40312	0.213974	0.106987	+	0.994260i	$0.465880\pi$
$44$	−4.70156	−0.708787
$45$	0	0
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	15.1047	2.15781
$50$	−8.70156	−1.23059
$51$	0	0
$52$	−1.70156	−0.235964
$53$	−6.40312	−0.879537	−0.439768	−	0.898111i	$-0.644939\pi$
$53$	−6.40312	−0.879537	−0.439768	+	0.898111i	$0.644939\pi$
$54$	0	0
$55$	−17.4031	−2.34664
$56$	4.70156	0.628273
$57$	0	0
$58$	−6.40312	−0.840771
$59$	−7.29844	−0.950176	−0.475088	−	0.879938i	$-0.657584\pi$
$59$	−7.29844	−0.950176	−0.475088	+	0.879938i	$0.657584\pi$
$60$	0	0
$61$	13.7016	1.75431	0.877153	−	0.480212i	$-0.159440\pi$
$61$	13.7016	1.75431	0.877153	+	0.480212i	$0.159440\pi$
$62$	−0.701562	−0.0890985
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.29844	−0.781225
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	17.4031	2.08007
$71$	−2.59688	−0.308192	−0.154096	−	0.988056i	$-0.549247\pi$
$71$	−2.59688	−0.308192	−0.154096	+	0.988056i	$0.549247\pi$
$72$	0	0
$73$	−6.40312	−0.749429	−0.374715	−	0.927140i	$-0.622259\pi$
$73$	−6.40312	−0.749429	−0.374715	+	0.927140i	$0.622259\pi$
$74$	−3.40312	−0.395605
$75$	0	0
$76$	4.00000	0.458831
$77$	22.1047	2.51906
$78$	0	0
$79$	16.7016	1.87907	0.939536	−	0.342449i	$-0.111257\pi$
$79$	16.7016	1.87907	0.939536	+	0.342449i	$0.111257\pi$
$80$	3.70156	0.413847
$81$	0	0
$82$	9.10469	1.00544
$83$	12.7016	1.39418	0.697089	−	0.716985i	$-0.254478\pi$
$83$	12.7016	1.39418	0.697089	+	0.716985i	$0.254478\pi$
$84$	0	0
$85$	7.40312	0.802982
$86$	−1.40312	−0.151303
$87$	0	0
$88$	4.70156	0.501188
$89$	−4.29844	−0.455634	−0.227817	−	0.973704i	$-0.573159\pi$
$89$	−4.29844	−0.455634	−0.227817	+	0.973704i	$0.573159\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	0	0
$94$	4.00000	0.412568
$95$	14.8062	1.51909
$96$	0	0
$97$	7.80625	0.792604	0.396302	−	0.918120i	$-0.370293\pi$
$97$	7.80625	0.792604	0.396302	+	0.918120i	$0.370293\pi$
$98$	−15.1047	−1.52580
$99$	0	0
$100$	8.70156	0.870156
$101$	−0.403124	−0.0401124	−0.0200562	−	0.999799i	$-0.506385\pi$
$101$	−0.403124	−0.0401124	−0.0200562	+	0.999799i	$0.506385\pi$
$102$	0	0
$103$	−14.1047	−1.38978	−0.694888	−	0.719118i	$-0.744546\pi$
$103$	−14.1047	−1.38978	−0.694888	+	0.719118i	$0.744546\pi$
$104$	1.70156	0.166852
$105$	0	0
$106$	6.40312	0.621926
$107$	−18.8062	−1.81807	−0.909034	−	0.416721i	$-0.863179\pi$
$107$	−18.8062	−1.81807	−0.909034	+	0.416721i	$0.863179\pi$
$108$	0	0
$109$	9.70156	0.929241	0.464621	−	0.885510i	$-0.346191\pi$
$109$	9.70156	0.929241	0.464621	+	0.885510i	$0.346191\pi$
$110$	17.4031	1.65932
$111$	0	0
$112$	−4.70156	−0.444256
$113$	6.29844	0.592507	0.296254	−	0.955109i	$-0.404263\pi$
$113$	6.29844	0.592507	0.296254	+	0.955109i	$0.404263\pi$
$114$	0	0
$115$	0	0
$116$	6.40312	0.594515
$117$	0	0
$118$	7.29844	0.671876
$119$	−9.40312	−0.861983
$120$	0	0
$121$	11.1047	1.00952
$122$	−13.7016	−1.24048
$123$	0	0
$124$	0.701562	0.0630021
$125$	13.7016	1.22550
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	6.29844	0.552410
$131$	0.701562	0.0612958	0.0306479	−	0.999530i	$-0.490243\pi$
$131$	0.701562	0.0612958	0.0306479	+	0.999530i	$0.490243\pi$
$132$	0	0
$133$	−18.8062	−1.63071
$134$	12.0000	1.03664
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−1.70156	−0.145374	−0.0726871	−	0.997355i	$-0.523157\pi$
$137$	−1.70156	−0.145374	−0.0726871	+	0.997355i	$0.523157\pi$
$138$	0	0
$139$	−14.8062	−1.25585	−0.627925	−	0.778274i	$-0.716095\pi$
$139$	−14.8062	−1.25585	−0.627925	+	0.778274i	$0.716095\pi$
$140$	−17.4031	−1.47083
$141$	0	0
$142$	2.59688	0.217925
$143$	8.00000	0.668994
$144$	0	0
$145$	23.7016	1.96831
$146$	6.40312	0.529926
$147$	0	0
$148$	3.40312	0.279735
$149$	−19.1047	−1.56512	−0.782558	−	0.622577i	$-0.786086\pi$
$149$	−19.1047	−1.56512	−0.782558	+	0.622577i	$0.786086\pi$
$150$	0	0
$151$	0.701562	0.0570923	0.0285462	−	0.999592i	$-0.490912\pi$
$151$	0.701562	0.0570923	0.0285462	+	0.999592i	$0.490912\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	−22.1047	−1.78125
$155$	2.59688	0.208586
$156$	0	0
$157$	13.7016	1.09350	0.546752	−	0.837295i	$-0.315864\pi$
$157$	13.7016	1.09350	0.546752	+	0.837295i	$0.315864\pi$
$158$	−16.7016	−1.32870
$159$	0	0
$160$	−3.70156	−0.292634
$161$	0	0
$162$	0	0
$163$	−1.40312	−0.109901	−0.0549506	−	0.998489i	$-0.517500\pi$
$163$	−1.40312	−0.109901	−0.0549506	+	0.998489i	$0.517500\pi$
$164$	−9.10469	−0.710957
$165$	0	0
$166$	−12.7016	−0.985832
$167$	−14.8062	−1.14574	−0.572871	−	0.819646i	$-0.694170\pi$
$167$	−14.8062	−1.14574	−0.572871	+	0.819646i	$0.694170\pi$
$168$	0	0
$169$	−10.1047	−0.777284
$170$	−7.40312	−0.567794
$171$	0	0
$172$	1.40312	0.106987
$173$	3.10469	0.236045	0.118022	−	0.993011i	$-0.462345\pi$
$173$	3.10469	0.236045	0.118022	+	0.993011i	$0.462345\pi$
$174$	0	0
$175$	−40.9109	−3.09258
$176$	−4.70156	−0.354394
$177$	0	0
$178$	4.29844	0.322182
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−15.4031	−1.14491	−0.572453	−	0.819938i	$-0.694008\pi$
$181$	−15.4031	−1.14491	−0.572453	+	0.819938i	$0.694008\pi$
$182$	−8.00000	−0.592999
$183$	0	0
$184$	0	0
$185$	12.5969	0.926141
$186$	0	0
$187$	−9.40312	−0.687625
$188$	−4.00000	−0.291730
$189$	0	0
$190$	−14.8062	−1.07416
$191$	5.40312	0.390956	0.195478	−	0.980708i	$-0.437374\pi$
$191$	5.40312	0.390956	0.195478	+	0.980708i	$0.437374\pi$
$192$	0	0
$193$	−11.5969	−0.834761	−0.417381	−	0.908732i	$-0.637052\pi$
$193$	−11.5969	−0.834761	−0.417381	+	0.908732i	$0.637052\pi$
$194$	−7.80625	−0.560456
$195$	0	0
$196$	15.1047	1.07891
$197$	−5.59688	−0.398761	−0.199380	−	0.979922i	$-0.563893\pi$
$197$	−5.59688	−0.398761	−0.199380	+	0.979922i	$0.563893\pi$
$198$	0	0
$199$	−6.10469	−0.432750	−0.216375	−	0.976310i	$-0.569423\pi$
$199$	−6.10469	−0.432750	−0.216375	+	0.976310i	$0.569423\pi$
$200$	−8.70156	−0.615293
$201$	0	0
$202$	0.403124	0.0283637
$203$	−30.1047	−2.11293
$204$	0	0
$205$	−33.7016	−2.35382
$206$	14.1047	0.982720
$207$	0	0
$208$	−1.70156	−0.117982
$209$	−18.8062	−1.30086
$210$	0	0
$211$	24.2094	1.66664	0.833321	−	0.552789i	$-0.186437\pi$
$211$	24.2094	1.66664	0.833321	+	0.552789i	$0.186437\pi$
$212$	−6.40312	−0.439768
$213$	0	0
$214$	18.8062	1.28557
$215$	5.19375	0.354211
$216$	0	0
$217$	−3.29844	−0.223913
$218$	−9.70156	−0.657073
$219$	0	0
$220$	−17.4031	−1.17332
$221$	−3.40312	−0.228919
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	4.70156	0.314136
$225$	0	0
$226$	−6.29844	−0.418966
$227$	−20.7016	−1.37401	−0.687005	−	0.726652i	$-0.741075\pi$
$227$	−20.7016	−1.37401	−0.687005	+	0.726652i	$0.741075\pi$
$228$	0	0
$229$	−18.2094	−1.20331	−0.601655	−	0.798756i	$-0.705492\pi$
$229$	−18.2094	−1.20331	−0.601655	+	0.798756i	$0.705492\pi$
$230$	0	0
$231$	0	0
$232$	−6.40312	−0.420386
$233$	−22.5078	−1.47454	−0.737268	−	0.675601i	$-0.763884\pi$
$233$	−22.5078	−1.47454	−0.737268	+	0.675601i	$0.763884\pi$
$234$	0	0
$235$	−14.8062	−0.965853
$236$	−7.29844	−0.475088
$237$	0	0
$238$	9.40312	0.609514
$239$	14.8062	0.957737	0.478868	−	0.877887i	$-0.341047\pi$
$239$	14.8062	0.957737	0.478868	+	0.877887i	$0.341047\pi$
$240$	0	0
$241$	0.298438	0.0192241	0.00961204	−	0.999954i	$-0.496940\pi$
$241$	0.298438	0.0192241	0.00961204	+	0.999954i	$0.496940\pi$
$242$	−11.1047	−0.713836
$243$	0	0
$244$	13.7016	0.877153
$245$	55.9109	3.57202
$246$	0	0
$247$	−6.80625	−0.433071
$248$	−0.701562	−0.0445492
$249$	0	0
$250$	−13.7016	−0.866563
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	0	0
$254$	4.00000	0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	−7.70156	−0.480410	−0.240205	−	0.970722i	$-0.577215\pi$
$257$	−7.70156	−0.480410	−0.240205	+	0.970722i	$0.577215\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	−6.29844	−0.390613
$261$	0	0
$262$	−0.701562	−0.0433427
$263$	10.8062	0.666342	0.333171	−	0.942866i	$-0.391881\pi$
$263$	10.8062	0.666342	0.333171	+	0.942866i	$0.391881\pi$
$264$	0	0
$265$	−23.7016	−1.45598
$266$	18.8062	1.15309
$267$	0	0
$268$	−12.0000	−0.733017
$269$	5.29844	0.323051	0.161526	−	0.986869i	$-0.448359\pi$
$269$	5.29844	0.323051	0.161526	+	0.986869i	$0.448359\pi$
$270$	0	0
$271$	3.50781	0.213084	0.106542	−	0.994308i	$-0.466022\pi$
$271$	3.50781	0.213084	0.106542	+	0.994308i	$0.466022\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	1.70156	0.102795
$275$	−40.9109	−2.46702
$276$	0	0
$277$	8.80625	0.529116	0.264558	−	0.964370i	$-0.414774\pi$
$277$	8.80625	0.529116	0.264558	+	0.964370i	$0.414774\pi$
$278$	14.8062	0.888020
$279$	0	0
$280$	17.4031	1.04004
$281$	3.40312	0.203013	0.101507	−	0.994835i	$-0.467634\pi$
$281$	3.40312	0.203013	0.101507	+	0.994835i	$0.467634\pi$
$282$	0	0
$283$	−25.6125	−1.52250	−0.761252	−	0.648456i	$-0.775415\pi$
$283$	−25.6125	−1.52250	−0.761252	+	0.648456i	$0.775415\pi$
$284$	−2.59688	−0.154096
$285$	0	0
$286$	−8.00000	−0.473050
$287$	42.8062	2.52677
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−23.7016	−1.39180
$291$	0	0
$292$	−6.40312	−0.374715
$293$	23.2094	1.35591	0.677953	−	0.735105i	$-0.262867\pi$
$293$	23.2094	1.35591	0.677953	+	0.735105i	$0.262867\pi$
$294$	0	0
$295$	−27.0156	−1.57291
$296$	−3.40312	−0.197803
$297$	0	0
$298$	19.1047	1.10670
$299$	0	0
$300$	0	0
$301$	−6.59688	−0.380238
$302$	−0.701562	−0.0403704
$303$	0	0
$304$	4.00000	0.229416
$305$	50.7172	2.90406
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	22.1047	1.25953
$309$	0	0
$310$	−2.59688	−0.147493
$311$	−5.19375	−0.294511	−0.147255	−	0.989099i	$-0.547044\pi$
$311$	−5.19375	−0.294511	−0.147255	+	0.989099i	$0.547044\pi$
$312$	0	0
$313$	−27.7016	−1.56578	−0.782892	−	0.622157i	$-0.786257\pi$
$313$	−27.7016	−1.56578	−0.782892	+	0.622157i	$0.786257\pi$
$314$	−13.7016	−0.773224
$315$	0	0
$316$	16.7016	0.939536
$317$	−5.59688	−0.314352	−0.157176	−	0.987571i	$-0.550239\pi$
$317$	−5.59688	−0.314352	−0.157176	+	0.987571i	$0.550239\pi$
$318$	0	0
$319$	−30.1047	−1.68554
$320$	3.70156	0.206924
$321$	0	0
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	−14.8062	−0.821303
$326$	1.40312	0.0777119
$327$	0	0
$328$	9.10469	0.502722
$329$	18.8062	1.03682
$330$	0	0
$331$	−9.40312	−0.516842	−0.258421	−	0.966032i	$-0.583202\pi$
$331$	−9.40312	−0.516842	−0.258421	+	0.966032i	$0.583202\pi$
$332$	12.7016	0.697089
$333$	0	0
$334$	14.8062	0.810162
$335$	−44.4187	−2.42686
$336$	0	0
$337$	−11.7016	−0.637425	−0.318712	−	0.947851i	$-0.603250\pi$
$337$	−11.7016	−0.637425	−0.318712	+	0.947851i	$0.603250\pi$
$338$	10.1047	0.549622
$339$	0	0
$340$	7.40312	0.401491
$341$	−3.29844	−0.178620
$342$	0	0
$343$	−38.1047	−2.05746
$344$	−1.40312	−0.0756514
$345$	0	0
$346$	−3.10469	−0.166909
$347$	−3.29844	−0.177069	−0.0885347	−	0.996073i	$-0.528218\pi$
$347$	−3.29844	−0.177069	−0.0885347	+	0.996073i	$0.528218\pi$
$348$	0	0
$349$	−20.8062	−1.11373	−0.556866	−	0.830602i	$-0.687996\pi$
$349$	−20.8062	−1.11373	−0.556866	+	0.830602i	$0.687996\pi$
$350$	40.9109	2.18678
$351$	0	0
$352$	4.70156	0.250594
$353$	−19.9109	−1.05975	−0.529876	−	0.848075i	$-0.677762\pi$
$353$	−19.9109	−1.05975	−0.529876	+	0.848075i	$0.677762\pi$
$354$	0	0
$355$	−9.61250	−0.510178
$356$	−4.29844	−0.227817
$357$	0	0
$358$	0	0
$359$	−14.8062	−0.781444	−0.390722	−	0.920509i	$-0.627775\pi$
$359$	−14.8062	−0.781444	−0.390722	+	0.920509i	$0.627775\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	15.4031	0.809570
$363$	0	0
$364$	8.00000	0.419314
$365$	−23.7016	−1.24060
$366$	0	0
$367$	6.10469	0.318662	0.159331	−	0.987225i	$-0.449066\pi$
$367$	6.10469	0.318662	0.159331	+	0.987225i	$0.449066\pi$
$368$	0	0
$369$	0	0
$370$	−12.5969	−0.654880
$371$	30.1047	1.56296
$372$	0	0
$373$	7.19375	0.372478	0.186239	−	0.982504i	$-0.440370\pi$
$373$	7.19375	0.372478	0.186239	+	0.982504i	$0.440370\pi$
$374$	9.40312	0.486224
$375$	0	0
$376$	4.00000	0.206284
$377$	−10.8953	−0.561137
$378$	0	0
$379$	12.2094	0.627153	0.313577	−	0.949563i	$-0.398473\pi$
$379$	12.2094	0.627153	0.313577	+	0.949563i	$0.398473\pi$
$380$	14.8062	0.759545
$381$	0	0
$382$	−5.40312	−0.276448
$383$	13.4031	0.684868	0.342434	−	0.939542i	$-0.388749\pi$
$383$	13.4031	0.684868	0.342434	+	0.939542i	$0.388749\pi$
$384$	0	0
$385$	81.8219	4.17003
$386$	11.5969	0.590265
$387$	0	0
$388$	7.80625	0.396302
$389$	−19.8953	−1.00873	−0.504366	−	0.863490i	$-0.668274\pi$
$389$	−19.8953	−1.00873	−0.504366	+	0.863490i	$0.668274\pi$
$390$	0	0
$391$	0	0
$392$	−15.1047	−0.762902
$393$	0	0
$394$	5.59688	0.281967
$395$	61.8219	3.11060
$396$	0	0
$397$	−28.2984	−1.42026	−0.710129	−	0.704072i	$-0.751363\pi$
$397$	−28.2984	−1.42026	−0.710129	+	0.704072i	$0.751363\pi$
$398$	6.10469	0.306000
$399$	0	0
$400$	8.70156	0.435078
$401$	−8.50781	−0.424860	−0.212430	−	0.977176i	$-0.568138\pi$
$401$	−8.50781	−0.424860	−0.212430	+	0.977176i	$0.568138\pi$
$402$	0	0
$403$	−1.19375	−0.0594650
$404$	−0.403124	−0.0200562
$405$	0	0
$406$	30.1047	1.49407
$407$	−16.0000	−0.793091
$408$	0	0
$409$	−5.50781	−0.272344	−0.136172	−	0.990685i	$-0.543480\pi$
$409$	−5.50781	−0.272344	−0.136172	+	0.990685i	$0.543480\pi$
$410$	33.7016	1.66440
$411$	0	0
$412$	−14.1047	−0.694888
$413$	34.3141	1.68848
$414$	0	0
$415$	47.0156	2.30791
$416$	1.70156	0.0834259
$417$	0	0
$418$	18.8062	0.919844
$419$	−27.5078	−1.34384	−0.671922	−	0.740622i	$-0.734531\pi$
$419$	−27.5078	−1.34384	−0.671922	+	0.740622i	$0.734531\pi$
$420$	0	0
$421$	31.6125	1.54070	0.770349	−	0.637622i	$-0.220082\pi$
$421$	31.6125	1.54070	0.770349	+	0.637622i	$0.220082\pi$
$422$	−24.2094	−1.17849
$423$	0	0
$424$	6.40312	0.310963
$425$	17.4031	0.844176
$426$	0	0
$427$	−64.4187	−3.11744
$428$	−18.8062	−0.909034
$429$	0	0
$430$	−5.19375	−0.250465
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	19.1047	0.918113	0.459056	−	0.888407i	$-0.348188\pi$
$433$	19.1047	0.918113	0.459056	+	0.888407i	$0.348188\pi$
$434$	3.29844	0.158330
$435$	0	0
$436$	9.70156	0.464621
$437$	0	0
$438$	0	0
$439$	−12.7016	−0.606212	−0.303106	−	0.952957i	$-0.598024\pi$
$439$	−12.7016	−0.606212	−0.303106	+	0.952957i	$0.598024\pi$
$440$	17.4031	0.829661
$441$	0	0
$442$	3.40312	0.161870
$443$	−20.9109	−0.993508	−0.496754	−	0.867891i	$-0.665475\pi$
$443$	−20.9109	−0.993508	−0.496754	+	0.867891i	$0.665475\pi$
$444$	0	0
$445$	−15.9109	−0.754251
$446$	0	0
$447$	0	0
$448$	−4.70156	−0.222128
$449$	−34.7172	−1.63841	−0.819203	−	0.573504i	$-0.805584\pi$
$449$	−34.7172	−1.63841	−0.819203	+	0.573504i	$0.805584\pi$
$450$	0	0
$451$	42.8062	2.01567
$452$	6.29844	0.296254
$453$	0	0
$454$	20.7016	0.971572
$455$	29.6125	1.38826
$456$	0	0
$457$	38.6125	1.80622	0.903108	−	0.429413i	$-0.141280\pi$
$457$	38.6125	1.80622	0.903108	+	0.429413i	$0.141280\pi$
$458$	18.2094	0.850868
$459$	0	0
$460$	0	0
$461$	−7.00000	−0.326023	−0.163011	−	0.986624i	$-0.552121\pi$
$461$	−7.00000	−0.326023	−0.163011	+	0.986624i	$0.552121\pi$
$462$	0	0
$463$	−9.89531	−0.459874	−0.229937	−	0.973205i	$-0.573852\pi$
$463$	−9.89531	−0.459874	−0.229937	+	0.973205i	$0.573852\pi$
$464$	6.40312	0.297258
$465$	0	0
$466$	22.5078	1.04265
$467$	−4.91093	−0.227251	−0.113625	−	0.993524i	$-0.536246\pi$
$467$	−4.91093	−0.227251	−0.113625	+	0.993524i	$0.536246\pi$
$468$	0	0
$469$	56.4187	2.60518
$470$	14.8062	0.682961
$471$	0	0
$472$	7.29844	0.335938
$473$	−6.59688	−0.303325
$474$	0	0
$475$	34.8062	1.59702
$476$	−9.40312	−0.430991
$477$	0	0
$478$	−14.8062	−0.677222
$479$	1.40312	0.0641104	0.0320552	−	0.999486i	$-0.489795\pi$
$479$	1.40312	0.0641104	0.0320552	+	0.999486i	$0.489795\pi$
$480$	0	0
$481$	−5.79063	−0.264030
$482$	−0.298438	−0.0135935
$483$	0	0
$484$	11.1047	0.504758
$485$	28.8953	1.31207
$486$	0	0
$487$	23.2984	1.05575	0.527876	−	0.849321i	$-0.322988\pi$
$487$	23.2984	1.05575	0.527876	+	0.849321i	$0.322988\pi$
$488$	−13.7016	−0.620241
$489$	0	0
$490$	−55.9109	−2.52580
$491$	24.7016	1.11477	0.557383	−	0.830256i	$-0.311806\pi$
$491$	24.7016	1.11477	0.557383	+	0.830256i	$0.311806\pi$
$492$	0	0
$493$	12.8062	0.576764
$494$	6.80625	0.306228
$495$	0	0
$496$	0.701562	0.0315011
$497$	12.2094	0.547665
$498$	0	0
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	13.7016	0.612752
$501$	0	0
$502$	4.00000	0.178529
$503$	9.40312	0.419265	0.209632	−	0.977780i	$-0.432773\pi$
$503$	9.40312	0.419265	0.209632	+	0.977780i	$0.432773\pi$
$504$	0	0
$505$	−1.49219	−0.0664016
$506$	0	0
$507$	0	0
$508$	−4.00000	−0.177471
$509$	−35.2094	−1.56063	−0.780314	−	0.625388i	$-0.784941\pi$
$509$	−35.2094	−1.56063	−0.780314	+	0.625388i	$0.784941\pi$
$510$	0	0
$511$	30.1047	1.33175
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	7.70156	0.339701
$515$	−52.2094	−2.30062
$516$	0	0
$517$	18.8062	0.827098
$518$	16.0000	0.703000
$519$	0	0
$520$	6.29844	0.276205
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	−32.2094	−1.40842	−0.704209	−	0.709993i	$-0.748698\pi$
$523$	−32.2094	−1.40842	−0.704209	+	0.709993i	$0.748698\pi$
$524$	0.701562	0.0306479
$525$	0	0
$526$	−10.8062	−0.471175
$527$	1.40312	0.0611211
$528$	0	0
$529$	0	0
$530$	23.7016	1.02953
$531$	0	0
$532$	−18.8062	−0.815354
$533$	15.4922	0.671041
$534$	0	0
$535$	−69.6125	−3.00961
$536$	12.0000	0.518321
$537$	0	0
$538$	−5.29844	−0.228432
$539$	−71.0156	−3.05886
$540$	0	0
$541$	−36.5078	−1.56959	−0.784797	−	0.619753i	$-0.787233\pi$
$541$	−36.5078	−1.56959	−0.784797	+	0.619753i	$0.787233\pi$
$542$	−3.50781	−0.150673
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	35.9109	1.53826
$546$	0	0
$547$	29.6125	1.26614	0.633069	−	0.774095i	$-0.281795\pi$
$547$	29.6125	1.26614	0.633069	+	0.774095i	$0.281795\pi$
$548$	−1.70156	−0.0726871
$549$	0	0
$550$	40.9109	1.74445
$551$	25.6125	1.09113
$552$	0	0
$553$	−78.5234	−3.33916
$554$	−8.80625	−0.374142
$555$	0	0
$556$	−14.8062	−0.627925
$557$	−13.0000	−0.550828	−0.275414	−	0.961326i	$-0.588815\pi$
$557$	−13.0000	−0.550828	−0.275414	+	0.961326i	$0.588815\pi$
$558$	0	0
$559$	−2.38750	−0.100981
$560$	−17.4031	−0.735416
$561$	0	0
$562$	−3.40312	−0.143552
$563$	32.9109	1.38703	0.693515	−	0.720442i	$-0.256061\pi$
$563$	32.9109	1.38703	0.693515	+	0.720442i	$0.256061\pi$
$564$	0	0
$565$	23.3141	0.980830
$566$	25.6125	1.07657
$567$	0	0
$568$	2.59688	0.108962
$569$	−27.1047	−1.13629	−0.568144	−	0.822929i	$-0.692338\pi$
$569$	−27.1047	−1.13629	−0.568144	+	0.822929i	$0.692338\pi$
$570$	0	0
$571$	32.4187	1.35668	0.678341	−	0.734747i	$-0.262699\pi$
$571$	32.4187	1.35668	0.678341	+	0.734747i	$0.262699\pi$
$572$	8.00000	0.334497
$573$	0	0
$574$	−42.8062	−1.78670
$575$	0	0
$576$	0	0
$577$	−10.7016	−0.445512	−0.222756	−	0.974874i	$-0.571505\pi$
$577$	−10.7016	−0.445512	−0.222756	+	0.974874i	$0.571505\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	23.7016	0.984154
$581$	−59.7172	−2.47749
$582$	0	0
$583$	30.1047	1.24681
$584$	6.40312	0.264963
$585$	0	0
$586$	−23.2094	−0.958770
$587$	−1.19375	−0.0492714	−0.0246357	−	0.999696i	$-0.507843\pi$
$587$	−1.19375	−0.0492714	−0.0246357	+	0.999696i	$0.507843\pi$
$588$	0	0
$589$	2.80625	0.115629
$590$	27.0156	1.11222
$591$	0	0
$592$	3.40312	0.139868
$593$	−24.5969	−1.01007	−0.505036	−	0.863098i	$-0.668521\pi$
$593$	−24.5969	−1.01007	−0.505036	+	0.863098i	$0.668521\pi$
$594$	0	0
$595$	−34.8062	−1.42692
$596$	−19.1047	−0.782558
$597$	0	0
$598$	0	0
$599$	−21.1938	−0.865953	−0.432977	−	0.901405i	$-0.642537\pi$
$599$	−21.1938	−0.865953	−0.432977	+	0.901405i	$0.642537\pi$
$600$	0	0
$601$	−15.8062	−0.644750	−0.322375	−	0.946612i	$-0.604481\pi$
$601$	−15.8062	−0.644750	−0.322375	+	0.946612i	$0.604481\pi$
$602$	6.59688	0.268869
$603$	0	0
$604$	0.701562	0.0285462
$605$	41.1047	1.67114
$606$	0	0
$607$	−24.9109	−1.01110	−0.505552	−	0.862796i	$-0.668711\pi$
$607$	−24.9109	−1.01110	−0.505552	+	0.862796i	$0.668711\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	−50.7172	−2.05348
$611$	6.80625	0.275351
$612$	0	0
$613$	47.3141	1.91100	0.955498	−	0.294996i	$-0.0953184\pi$
$613$	47.3141	1.91100	0.955498	+	0.294996i	$0.0953184\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	−22.1047	−0.890623
$617$	0.596876	0.0240293	0.0120147	−	0.999928i	$-0.496176\pi$
$617$	0.596876	0.0240293	0.0120147	+	0.999928i	$0.496176\pi$
$618$	0	0
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	2.59688	0.104293
$621$	0	0
$622$	5.19375	0.208250
$623$	20.2094	0.809671
$624$	0	0
$625$	7.20937	0.288375
$626$	27.7016	1.10718
$627$	0	0
$628$	13.7016	0.546752
$629$	6.80625	0.271383
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−16.7016	−0.664352
$633$	0	0
$634$	5.59688	0.222280
$635$	−14.8062	−0.587568
$636$	0	0
$637$	−25.7016	−1.01833
$638$	30.1047	1.19186
$639$	0	0
$640$	−3.70156	−0.146317
$641$	45.1047	1.78153	0.890764	−	0.454466i	$-0.150170\pi$
$641$	45.1047	1.78153	0.890764	+	0.454466i	$0.150170\pi$
$642$	0	0
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	0	0
$645$	0	0
$646$	−8.00000	−0.314756
$647$	−39.0156	−1.53386	−0.766931	−	0.641729i	$-0.778217\pi$
$647$	−39.0156	−1.53386	−0.766931	+	0.641729i	$0.778217\pi$
$648$	0	0
$649$	34.3141	1.34694
$650$	14.8062	0.580749
$651$	0	0
$652$	−1.40312	−0.0549506
$653$	−11.0000	−0.430463	−0.215232	−	0.976563i	$-0.569051\pi$
$653$	−11.0000	−0.430463	−0.215232	+	0.976563i	$0.569051\pi$
$654$	0	0
$655$	2.59688	0.101468
$656$	−9.10469	−0.355478
$657$	0	0
$658$	−18.8062	−0.733144
$659$	−2.80625	−0.109316	−0.0546580	−	0.998505i	$-0.517407\pi$
$659$	−2.80625	−0.109316	−0.0546580	+	0.998505i	$0.517407\pi$
$660$	0	0
$661$	−40.1203	−1.56050	−0.780250	−	0.625468i	$-0.784908\pi$
$661$	−40.1203	−1.56050	−0.780250	+	0.625468i	$0.784908\pi$
$662$	9.40312	0.365463
$663$	0	0
$664$	−12.7016	−0.492916
$665$	−69.6125	−2.69946
$666$	0	0
$667$	0	0
$668$	−14.8062	−0.572871
$669$	0	0
$670$	44.4187	1.71605
$671$	−64.4187	−2.48686
$672$	0	0
$673$	17.5078	0.674877	0.337438	−	0.941348i	$-0.390440\pi$
$673$	17.5078	0.674877	0.337438	+	0.941348i	$0.390440\pi$
$674$	11.7016	0.450727
$675$	0	0
$676$	−10.1047	−0.388642
$677$	11.0000	0.422764	0.211382	−	0.977403i	$-0.432204\pi$
$677$	11.0000	0.422764	0.211382	+	0.977403i	$0.432204\pi$
$678$	0	0
$679$	−36.7016	−1.40848
$680$	−7.40312	−0.283897
$681$	0	0
$682$	3.29844	0.126304
$683$	−41.6125	−1.59226	−0.796129	−	0.605127i	$-0.793122\pi$
$683$	−41.6125	−1.59226	−0.796129	+	0.605127i	$0.793122\pi$
$684$	0	0
$685$	−6.29844	−0.240651
$686$	38.1047	1.45484
$687$	0	0
$688$	1.40312	0.0534936
$689$	10.8953	0.415078
$690$	0	0
$691$	24.0000	0.913003	0.456502	−	0.889723i	$-0.349102\pi$
$691$	24.0000	0.913003	0.456502	+	0.889723i	$0.349102\pi$
$692$	3.10469	0.118022
$693$	0	0
$694$	3.29844	0.125207
$695$	−54.8062	−2.07892
$696$	0	0
$697$	−18.2094	−0.689729
$698$	20.8062	0.787528
$699$	0	0
$700$	−40.9109	−1.54629
$701$	−24.1047	−0.910421	−0.455211	−	0.890384i	$-0.650436\pi$
$701$	−24.1047	−0.910421	−0.455211	+	0.890384i	$0.650436\pi$
$702$	0	0
$703$	13.6125	0.513405
$704$	−4.70156	−0.177197
$705$	0	0
$706$	19.9109	0.749358
$707$	1.89531	0.0712806
$708$	0	0
$709$	17.9109	0.672659	0.336330	−	0.941744i	$-0.390814\pi$
$709$	17.9109	0.672659	0.336330	+	0.941744i	$0.390814\pi$
$710$	9.61250	0.360751
$711$	0	0
$712$	4.29844	0.161091
$713$	0	0
$714$	0	0
$715$	29.6125	1.10744
$716$	0	0
$717$	0	0
$718$	14.8062	0.552564
$719$	9.40312	0.350677	0.175339	−	0.984508i	$-0.443898\pi$
$719$	9.40312	0.350677	0.175339	+	0.984508i	$0.443898\pi$
$720$	0	0
$721$	66.3141	2.46966
$722$	3.00000	0.111648
$723$	0	0
$724$	−15.4031	−0.572453
$725$	55.7172	2.06928
$726$	0	0
$727$	37.6125	1.39497	0.697485	−	0.716599i	$-0.254302\pi$
$727$	37.6125	1.39497	0.697485	+	0.716599i	$0.254302\pi$
$728$	−8.00000	−0.296500
$729$	0	0
$730$	23.7016	0.877234
$731$	2.80625	0.103793
$732$	0	0
$733$	−33.3141	−1.23048	−0.615241	−	0.788339i	$-0.710941\pi$
$733$	−33.3141	−1.23048	−0.615241	+	0.788339i	$0.710941\pi$
$734$	−6.10469	−0.225328
$735$	0	0
$736$	0	0
$737$	56.4187	2.07821
$738$	0	0
$739$	13.6125	0.500744	0.250372	−	0.968150i	$-0.419447\pi$
$739$	13.6125	0.500744	0.250372	+	0.968150i	$0.419447\pi$
$740$	12.5969	0.463070
$741$	0	0
$742$	−30.1047	−1.10518
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	−70.7172	−2.59088
$746$	−7.19375	−0.263382
$747$	0	0
$748$	−9.40312	−0.343812
$749$	88.4187	3.23075
$750$	0	0
$751$	−6.31406	−0.230403	−0.115202	−	0.993342i	$-0.536751\pi$
$751$	−6.31406	−0.230403	−0.115202	+	0.993342i	$0.536751\pi$
$752$	−4.00000	−0.145865
$753$	0	0
$754$	10.8953	0.396784
$755$	2.59688	0.0945100
$756$	0	0
$757$	24.2984	0.883142	0.441571	−	0.897226i	$-0.354421\pi$
$757$	24.2984	0.883142	0.441571	+	0.897226i	$0.354421\pi$
$758$	−12.2094	−0.443464
$759$	0	0
$760$	−14.8062	−0.537079
$761$	28.5078	1.03341	0.516704	−	0.856164i	$-0.327159\pi$
$761$	28.5078	1.03341	0.516704	+	0.856164i	$0.327159\pi$
$762$	0	0
$763$	−45.6125	−1.65128
$764$	5.40312	0.195478
$765$	0	0
$766$	−13.4031	−0.484275
$767$	12.4187	0.448415
$768$	0	0
$769$	6.91093	0.249215	0.124607	−	0.992206i	$-0.460233\pi$
$769$	6.91093	0.249215	0.124607	+	0.992206i	$0.460233\pi$
$770$	−81.8219	−2.94866
$771$	0	0
$772$	−11.5969	−0.417381
$773$	34.7016	1.24813	0.624064	−	0.781373i	$-0.285480\pi$
$773$	34.7016	1.24813	0.624064	+	0.781373i	$0.285480\pi$
$774$	0	0
$775$	6.10469	0.219287
$776$	−7.80625	−0.280228
$777$	0	0
$778$	19.8953	0.713282
$779$	−36.4187	−1.30484
$780$	0	0
$781$	12.2094	0.436886
$782$	0	0
$783$	0	0
$784$	15.1047	0.539453
$785$	50.7172	1.81017
$786$	0	0
$787$	−51.2250	−1.82597	−0.912987	−	0.407989i	$-0.866230\pi$
$787$	−51.2250	−1.82597	−0.912987	+	0.407989i	$0.866230\pi$
$788$	−5.59688	−0.199380
$789$	0	0
$790$	−61.8219	−2.19952
$791$	−29.6125	−1.05290
$792$	0	0
$793$	−23.3141	−0.827907
$794$	28.2984	1.00427
$795$	0	0
$796$	−6.10469	−0.216375
$797$	−4.80625	−0.170246	−0.0851230	−	0.996370i	$-0.527128\pi$
$797$	−4.80625	−0.170246	−0.0851230	+	0.996370i	$0.527128\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	−8.70156	−0.307647
$801$	0	0
$802$	8.50781	0.300421
$803$	30.1047	1.06237
$804$	0	0
$805$	0	0
$806$	1.19375	0.0420481
$807$	0	0
$808$	0.403124	0.0141819
$809$	47.4031	1.66661	0.833303	−	0.552817i	$-0.186447\pi$
$809$	47.4031	1.66661	0.833303	+	0.552817i	$0.186447\pi$
$810$	0	0
$811$	−19.0156	−0.667729	−0.333864	−	0.942621i	$-0.608353\pi$
$811$	−19.0156	−0.667729	−0.333864	+	0.942621i	$0.608353\pi$
$812$	−30.1047	−1.05647
$813$	0	0
$814$	16.0000	0.560800
$815$	−5.19375	−0.181929
$816$	0	0
$817$	5.61250	0.196356
$818$	5.50781	0.192576
$819$	0	0
$820$	−33.7016	−1.17691
$821$	34.4031	1.20068	0.600339	−	0.799746i	$-0.295032\pi$
$821$	34.4031	1.20068	0.600339	+	0.799746i	$0.295032\pi$
$822$	0	0
$823$	37.6125	1.31109	0.655545	−	0.755156i	$-0.272439\pi$
$823$	37.6125	1.31109	0.655545	+	0.755156i	$0.272439\pi$
$824$	14.1047	0.491360
$825$	0	0
$826$	−34.3141	−1.19394
$827$	−5.19375	−0.180604	−0.0903022	−	0.995914i	$-0.528783\pi$
$827$	−5.19375	−0.180604	−0.0903022	+	0.995914i	$0.528783\pi$
$828$	0	0
$829$	29.1047	1.01085	0.505424	−	0.862871i	$-0.331336\pi$
$829$	29.1047	1.01085	0.505424	+	0.862871i	$0.331336\pi$
$830$	−47.0156	−1.63194
$831$	0	0
$832$	−1.70156	−0.0589911
$833$	30.2094	1.04669
$834$	0	0
$835$	−54.8062	−1.89665
$836$	−18.8062	−0.650428
$837$	0	0
$838$	27.5078	0.950242
$839$	−41.6125	−1.43662	−0.718311	−	0.695722i	$-0.755085\pi$
$839$	−41.6125	−1.43662	−0.718311	+	0.695722i	$0.755085\pi$
$840$	0	0
$841$	12.0000	0.413793
$842$	−31.6125	−1.08944
$843$	0	0
$844$	24.2094	0.833321
$845$	−37.4031	−1.28671
$846$	0	0
$847$	−52.2094	−1.79394
$848$	−6.40312	−0.219884
$849$	0	0
$850$	−17.4031	−0.596922
$851$	0	0
$852$	0	0
$853$	−34.4187	−1.17848	−0.589238	−	0.807960i	$-0.700572\pi$
$853$	−34.4187	−1.17848	−0.589238	+	0.807960i	$0.700572\pi$
$854$	64.4187	2.20436
$855$	0	0
$856$	18.8062	0.642784
$857$	−49.1047	−1.67738	−0.838692	−	0.544606i	$-0.816679\pi$
$857$	−49.1047	−1.67738	−0.838692	+	0.544606i	$0.816679\pi$
$858$	0	0
$859$	20.2094	0.689535	0.344767	−	0.938688i	$-0.387958\pi$
$859$	20.2094	0.689535	0.344767	+	0.938688i	$0.387958\pi$
$860$	5.19375	0.177105
$861$	0	0
$862$	−8.00000	−0.272481
$863$	−16.2094	−0.551773	−0.275887	−	0.961190i	$-0.588971\pi$
$863$	−16.2094	−0.551773	−0.275887	+	0.961190i	$0.588971\pi$
$864$	0	0
$865$	11.4922	0.390746
$866$	−19.1047	−0.649204
$867$	0	0
$868$	−3.29844	−0.111956
$869$	−78.5234	−2.66372
$870$	0	0
$871$	20.4187	0.691863
$872$	−9.70156	−0.328536
$873$	0	0
$874$	0	0
$875$	−64.4187	−2.17775
$876$	0	0
$877$	26.2094	0.885028	0.442514	−	0.896762i	$-0.354087\pi$
$877$	26.2094	0.885028	0.442514	+	0.896762i	$0.354087\pi$
$878$	12.7016	0.428657
$879$	0	0
$880$	−17.4031	−0.586659
$881$	25.1047	0.845798	0.422899	−	0.906177i	$-0.361012\pi$
$881$	25.1047	0.845798	0.422899	+	0.906177i	$0.361012\pi$
$882$	0	0
$883$	−9.40312	−0.316440	−0.158220	−	0.987404i	$-0.550576\pi$
$883$	−9.40312	−0.316440	−0.158220	+	0.987404i	$0.550576\pi$
$884$	−3.40312	−0.114459
$885$	0	0
$886$	20.9109	0.702517
$887$	2.80625	0.0942246	0.0471123	−	0.998890i	$-0.484998\pi$
$887$	2.80625	0.0942246	0.0471123	+	0.998890i	$0.484998\pi$
$888$	0	0
$889$	18.8062	0.630741
$890$	15.9109	0.533336
$891$	0	0
$892$	0	0
$893$	−16.0000	−0.535420
$894$	0	0
$895$	0	0
$896$	4.70156	0.157068
$897$	0	0
$898$	34.7172	1.15853
$899$	4.49219	0.149823
$900$	0	0
$901$	−12.8062	−0.426638
$902$	−42.8062	−1.42529
$903$	0	0
$904$	−6.29844	−0.209483
$905$	−57.0156	−1.89526
$906$	0	0
$907$	6.59688	0.219046	0.109523	−	0.993984i	$-0.465068\pi$
$907$	6.59688	0.219046	0.109523	+	0.993984i	$0.465068\pi$
$908$	−20.7016	−0.687005
$909$	0	0
$910$	−29.6125	−0.981645
$911$	−19.7906	−0.655693	−0.327846	−	0.944731i	$-0.606323\pi$
$911$	−19.7906	−0.655693	−0.327846	+	0.944731i	$0.606323\pi$
$912$	0	0
$913$	−59.7172	−1.97635
$914$	−38.6125	−1.27719
$915$	0	0
$916$	−18.2094	−0.601655
$917$	−3.29844	−0.108924
$918$	0	0
$919$	−54.8062	−1.80789	−0.903946	−	0.427647i	$-0.859343\pi$
$919$	−54.8062	−1.80789	−0.903946	+	0.427647i	$0.859343\pi$
$920$	0	0
$921$	0	0
$922$	7.00000	0.230533
$923$	4.41875	0.145445
$924$	0	0
$925$	29.6125	0.973653
$926$	9.89531	0.325180
$927$	0	0
$928$	−6.40312	−0.210193
$929$	26.2094	0.859902	0.429951	−	0.902852i	$-0.358531\pi$
$929$	26.2094	0.859902	0.429951	+	0.902852i	$0.358531\pi$
$930$	0	0
$931$	60.4187	1.98014
$932$	−22.5078	−0.737268
$933$	0	0
$934$	4.91093	0.160691
$935$	−34.8062	−1.13829
$936$	0	0
$937$	15.8953	0.519277	0.259639	−	0.965706i	$-0.416397\pi$
$937$	15.8953	0.519277	0.259639	+	0.965706i	$0.416397\pi$
$938$	−56.4187	−1.84214
$939$	0	0
$940$	−14.8062	−0.482927
$941$	9.71718	0.316771	0.158386	−	0.987377i	$-0.449371\pi$
$941$	9.71718	0.316771	0.158386	+	0.987377i	$0.449371\pi$
$942$	0	0
$943$	0	0
$944$	−7.29844	−0.237544
$945$	0	0
$946$	6.59688	0.214483
$947$	−15.2984	−0.497132	−0.248566	−	0.968615i	$-0.579959\pi$
$947$	−15.2984	−0.497132	−0.248566	+	0.968615i	$0.579959\pi$
$948$	0	0
$949$	10.8953	0.353677
$950$	−34.8062	−1.12926
$951$	0	0
$952$	9.40312	0.304757
$953$	54.7172	1.77246	0.886232	−	0.463242i	$-0.153314\pi$
$953$	54.7172	1.77246	0.886232	+	0.463242i	$0.153314\pi$
$954$	0	0
$955$	20.0000	0.647185
$956$	14.8062	0.478868
$957$	0	0
$958$	−1.40312	−0.0453329
$959$	8.00000	0.258333
$960$	0	0
$961$	−30.5078	−0.984123
$962$	5.79063	0.186697
$963$	0	0
$964$	0.298438	0.00961204
$965$	−42.9266	−1.38185
$966$	0	0
$967$	26.3141	0.846203	0.423102	−	0.906082i	$-0.360941\pi$
$967$	26.3141	0.846203	0.423102	+	0.906082i	$0.360941\pi$
$968$	−11.1047	−0.356918
$969$	0	0
$970$	−28.8953	−0.927773
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	0	0
$973$	69.6125	2.23167
$974$	−23.2984	−0.746530
$975$	0	0
$976$	13.7016	0.438576
$977$	16.8953	0.540529	0.270264	−	0.962786i	$-0.412889\pi$
$977$	16.8953	0.540529	0.270264	+	0.962786i	$0.412889\pi$
$978$	0	0
$979$	20.2094	0.645894
$980$	55.9109	1.78601
$981$	0	0
$982$	−24.7016	−0.788259
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	0	0
$985$	−20.7172	−0.660104
$986$	−12.8062	−0.407834
$987$	0	0
$988$	−6.80625	−0.216536
$989$	0	0
$990$	0	0
$991$	−35.5078	−1.12794	−0.563971	−	0.825794i	$-0.690727\pi$
$991$	−35.5078	−1.12794	−0.563971	+	0.825794i	$0.690727\pi$
$992$	−0.701562	−0.0222746
$993$	0	0
$994$	−12.2094	−0.387258
$995$	−22.5969	−0.716369
$996$	0	0
$997$	54.9266	1.73954	0.869771	−	0.493456i	$-0.164267\pi$
$997$	54.9266	1.73954	0.869771	+	0.493456i	$0.164267\pi$
$998$	−8.00000	−0.253236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.ba.1.2		2
3.2	odd	2		3174.2.a.o.1.1	✓	2
23.22	odd	2		9522.2.a.r.1.1		2
69.68	even	2		3174.2.a.r.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3174.2.a.o.1.1	✓	2	3.2	odd	2
3174.2.a.r.1.2	yes	2	69.68	even	2
9522.2.a.r.1.1		2	23.22	odd	2
9522.2.a.ba.1.2		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.70156 q^{5} -4.70156 q^{7} -1.00000 q^{8} -3.70156 q^{10} -4.70156 q^{11} -1.70156 q^{13} +4.70156 q^{14} +1.00000 q^{16} +2.00000 q^{17} +4.00000 q^{19} +3.70156 q^{20} +4.70156 q^{22} +8.70156 q^{25} +1.70156 q^{26} -4.70156 q^{28} +6.40312 q^{29} +0.701562 q^{31} -1.00000 q^{32} -2.00000 q^{34} -17.4031 q^{35} +3.40312 q^{37} -4.00000 q^{38} -3.70156 q^{40} -9.10469 q^{41} +1.40312 q^{43} -4.70156 q^{44} -4.00000 q^{47} +15.1047 q^{49} -8.70156 q^{50} -1.70156 q^{52} -6.40312 q^{53} -17.4031 q^{55} +4.70156 q^{56} -6.40312 q^{58} -7.29844 q^{59} +13.7016 q^{61} -0.701562 q^{62} +1.00000 q^{64} -6.29844 q^{65} -12.0000 q^{67} +2.00000 q^{68} +17.4031 q^{70} -2.59688 q^{71} -6.40312 q^{73} -3.40312 q^{74} +4.00000 q^{76} +22.1047 q^{77} +16.7016 q^{79} +3.70156 q^{80} +9.10469 q^{82} +12.7016 q^{83} +7.40312 q^{85} -1.40312 q^{86} +4.70156 q^{88} -4.29844 q^{89} +8.00000 q^{91} +4.00000 q^{94} +14.8062 q^{95} +7.80625 q^{97} -15.1047 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + q^{5} - 3 q^{7} - 2 q^{8} - q^{10} - 3 q^{11} + 3 q^{13} + 3 q^{14} + 2 q^{16} + 4 q^{17} + 8 q^{19} + q^{20} + 3 q^{22} + 11 q^{25} - 3 q^{26} - 3 q^{28} - 5 q^{31} - 2 q^{32}+ \cdots - 11 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + q^5 - 3 * q^7 - 2 * q^8 - q^10 - 3 * q^11 + 3 * q^13 + 3 * q^14 + 2 * q^16 + 4 * q^17 + 8 * q^19 + q^20 + 3 * q^22 + 11 * q^25 - 3 * q^26 - 3 * q^28 - 5 * q^31 - 2 * q^32 - 4 * q^34 - 22 * q^35 - 6 * q^37 - 8 * q^38 - q^40 + q^41 - 10 * q^43 - 3 * q^44 - 8 * q^47 + 11 * q^49 - 11 * q^50 + 3 * q^52 - 22 * q^55 + 3 * q^56 - 21 * q^59 + 21 * q^61 + 5 * q^62 + 2 * q^64 - 19 * q^65 - 24 * q^67 + 4 * q^68 + 22 * q^70 - 18 * q^71 + 6 * q^74 + 8 * q^76 + 25 * q^77 + 27 * q^79 + q^80 - q^82 + 19 * q^83 + 2 * q^85 + 10 * q^86 + 3 * q^88 - 15 * q^89 + 16 * q^91 + 8 * q^94 + 4 * q^95 - 10 * q^97 - 11 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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