LMFDB - Embedded newform 9522.2.a.bt.1.5

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 = [5,-5,0,5,7,0,-7,-5,0,-7,13,0,-4,7,0,5,9,0,-11,7,0,-13,0,0,-2, 4,0,-7,7] 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:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.830830$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.20362 q^{5} +1.51334 q^{7} -1.00000 q^{8} -3.20362 q^{10} +6.53843 q^{11} +2.88612 q^{13} -1.51334 q^{14} +1.00000 q^{16} +1.19647 q^{17} +0.555554 q^{19} +3.20362 q^{20} -6.53843 q^{22} +5.26315 q^{25} -2.88612 q^{26} +1.51334 q^{28} +8.66918 q^{29} -6.73559 q^{31} -1.00000 q^{32} -1.19647 q^{34} +4.84815 q^{35} -1.45380 q^{37} -0.555554 q^{38} -3.20362 q^{40} +1.84796 q^{41} +2.17379 q^{43} +6.53843 q^{44} +11.4135 q^{47} -4.70981 q^{49} -5.26315 q^{50} +2.88612 q^{52} -12.0613 q^{53} +20.9466 q^{55} -1.51334 q^{56} -8.66918 q^{58} -3.96140 q^{59} +9.03450 q^{61} +6.73559 q^{62} +1.00000 q^{64} +9.24603 q^{65} -7.66794 q^{67} +1.19647 q^{68} -4.84815 q^{70} +9.45607 q^{71} -0.627598 q^{73} +1.45380 q^{74} +0.555554 q^{76} +9.89485 q^{77} -13.7721 q^{79} +3.20362 q^{80} -1.84796 q^{82} +15.5888 q^{83} +3.83304 q^{85} -2.17379 q^{86} -6.53843 q^{88} +5.61844 q^{89} +4.36768 q^{91} -11.4135 q^{94} +1.77978 q^{95} -7.24727 q^{97} +4.70981 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} + 7 q^{5} - 7 q^{7} - 5 q^{8} - 7 q^{10} + 13 q^{11} - 4 q^{13} + 7 q^{14} + 5 q^{16} + 9 q^{17} - 11 q^{19} + 7 q^{20} - 13 q^{22} - 2 q^{25} + 4 q^{26} - 7 q^{28} + 7 q^{29} - 8 q^{31}+ \cdots + 12 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 + 7 * q^5 - 7 * q^7 - 5 * q^8 - 7 * q^10 + 13 * q^11 - 4 * q^13 + 7 * q^14 + 5 * q^16 + 9 * q^17 - 11 * q^19 + 7 * q^20 - 13 * q^22 - 2 * q^25 + 4 * q^26 - 7 * q^28 + 7 * q^29 - 8 * q^31 - 5 * q^32 - 9 * q^34 - q^35 - 12 * q^37 + 11 * q^38 - 7 * q^40 + 10 * q^41 - 4 * q^43 + 13 * q^44 + 24 * q^47 - 12 * q^49 + 2 * q^50 - 4 * q^52 + 9 * q^53 + 16 * q^55 + 7 * q^56 - 7 * q^58 + 14 * q^59 - 5 * q^61 + 8 * q^62 + 5 * q^64 + 12 * q^65 - 13 * q^67 + 9 * q^68 + q^70 + 19 * q^71 + 4 * q^73 + 12 * q^74 - 11 * q^76 - 5 * q^77 - 4 * q^79 + 7 * q^80 - 10 * q^82 + 24 * q^83 + 17 * q^85 + 4 * q^86 - 13 * q^88 + 4 * q^89 + 21 * q^91 - 24 * q^94 + 11 * q^95 + 9 * q^97 + 12 * 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
$5$	3.20362	1.43270	0.716350	−	0.697741i	$-0.245811\pi$
$5$	3.20362	1.43270	0.716350	+	0.697741i	$0.245811\pi$
$6$	0	0
$7$	1.51334	0.571988	0.285994	−	0.958231i	$-0.407676\pi$
$7$	1.51334	0.571988	0.285994	+	0.958231i	$0.407676\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.20362	−1.01307
$11$	6.53843	1.97141	0.985705	−	0.168479i	$-0.0538855\pi$
$11$	6.53843	1.97141	0.985705	+	0.168479i	$0.0538855\pi$
$12$	0	0
$13$	2.88612	0.800466	0.400233	−	0.916413i	$-0.368929\pi$
$13$	2.88612	0.800466	0.400233	+	0.916413i	$0.368929\pi$
$14$	−1.51334	−0.404456
$15$	0	0
$16$	1.00000	0.250000
$17$	1.19647	0.290188	0.145094	−	0.989418i	$-0.453652\pi$
$17$	1.19647	0.290188	0.145094	+	0.989418i	$0.453652\pi$
$18$	0	0
$19$	0.555554	0.127453	0.0637264	−	0.997967i	$-0.479701\pi$
$19$	0.555554	0.127453	0.0637264	+	0.997967i	$0.479701\pi$
$20$	3.20362	0.716350
$21$	0	0
$22$	−6.53843	−1.39400
$23$	0	0
$23$	0	0
$24$	0	0
$25$	5.26315	1.05263
$26$	−2.88612	−0.566015
$27$	0	0
$28$	1.51334	0.285994
$29$	8.66918	1.60983	0.804913	−	0.593393i	$-0.202212\pi$
$29$	8.66918	1.60983	0.804913	+	0.593393i	$0.202212\pi$
$30$	0	0
$31$	−6.73559	−1.20975	−0.604874	−	0.796321i	$-0.706776\pi$
$31$	−6.73559	−1.20975	−0.604874	+	0.796321i	$0.706776\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.19647	−0.205194
$35$	4.84815	0.819487
$36$	0	0
$37$	−1.45380	−0.239003	−0.119502	−	0.992834i	$-0.538130\pi$
$37$	−1.45380	−0.239003	−0.119502	+	0.992834i	$0.538130\pi$
$38$	−0.555554	−0.0901228
$39$	0	0
$40$	−3.20362	−0.506536
$41$	1.84796	0.288602	0.144301	−	0.989534i	$-0.453907\pi$
$41$	1.84796	0.288602	0.144301	+	0.989534i	$0.453907\pi$
$42$	0	0
$43$	2.17379	0.331500	0.165750	−	0.986168i	$-0.446996\pi$
$43$	2.17379	0.331500	0.165750	+	0.986168i	$0.446996\pi$
$44$	6.53843	0.985705
$45$	0	0
$46$	0	0
$47$	11.4135	1.66483	0.832416	−	0.554152i	$-0.186957\pi$
$47$	11.4135	1.66483	0.832416	+	0.554152i	$0.186957\pi$
$48$	0	0
$49$	−4.70981	−0.672830
$50$	−5.26315	−0.744322
$51$	0	0
$52$	2.88612	0.400233
$53$	−12.0613	−1.65675	−0.828375	−	0.560175i	$-0.810734\pi$
$53$	−12.0613	−1.65675	−0.828375	+	0.560175i	$0.810734\pi$
$54$	0	0
$55$	20.9466	2.82444
$56$	−1.51334	−0.202228
$57$	0	0
$58$	−8.66918	−1.13832
$59$	−3.96140	−0.515730	−0.257865	−	0.966181i	$-0.583019\pi$
$59$	−3.96140	−0.515730	−0.257865	+	0.966181i	$0.583019\pi$
$60$	0	0
$61$	9.03450	1.15675	0.578375	−	0.815771i	$-0.303687\pi$
$61$	9.03450	1.15675	0.578375	+	0.815771i	$0.303687\pi$
$62$	6.73559	0.855420
$63$	0	0
$64$	1.00000	0.125000
$65$	9.24603	1.14683
$66$	0	0
$67$	−7.66794	−0.936788	−0.468394	−	0.883520i	$-0.655167\pi$
$67$	−7.66794	−0.936788	−0.468394	+	0.883520i	$0.655167\pi$
$68$	1.19647	0.145094
$69$	0	0
$70$	−4.84815	−0.579465
$71$	9.45607	1.12223	0.561114	−	0.827738i	$-0.310373\pi$
$71$	9.45607	1.12223	0.561114	+	0.827738i	$0.310373\pi$
$72$	0	0
$73$	−0.627598	−0.0734548	−0.0367274	−	0.999325i	$-0.511693\pi$
$73$	−0.627598	−0.0734548	−0.0367274	+	0.999325i	$0.511693\pi$
$74$	1.45380	0.169001
$75$	0	0
$76$	0.555554	0.0637264
$77$	9.89485	1.12762
$78$	0	0
$79$	−13.7721	−1.54949	−0.774743	−	0.632276i	$-0.782121\pi$
$79$	−13.7721	−1.54949	−0.774743	+	0.632276i	$0.782121\pi$
$80$	3.20362	0.358175
$81$	0	0
$82$	−1.84796	−0.204073
$83$	15.5888	1.71110	0.855549	−	0.517722i	$-0.173220\pi$
$83$	15.5888	1.71110	0.855549	+	0.517722i	$0.173220\pi$
$84$	0	0
$85$	3.83304	0.415752
$86$	−2.17379	−0.234406
$87$	0	0
$88$	−6.53843	−0.696999
$89$	5.61844	0.595553	0.297777	−	0.954636i	$-0.403755\pi$
$89$	5.61844	0.595553	0.297777	+	0.954636i	$0.403755\pi$
$90$	0	0
$91$	4.36768	0.457857
$92$	0	0
$93$	0	0
$94$	−11.4135	−1.17721
$95$	1.77978	0.182602
$96$	0	0
$97$	−7.24727	−0.735848	−0.367924	−	0.929856i	$-0.619931\pi$
$97$	−7.24727	−0.735848	−0.367924	+	0.929856i	$0.619931\pi$
$98$	4.70981	0.475763
$99$	0	0
$100$	5.26315	0.526315
$101$	1.02378	0.101870	0.0509348	−	0.998702i	$-0.483780\pi$
$101$	1.02378	0.101870	0.0509348	+	0.998702i	$0.483780\pi$
$102$	0	0
$103$	4.54925	0.448251	0.224125	−	0.974560i	$-0.428048\pi$
$103$	4.54925	0.448251	0.224125	+	0.974560i	$0.428048\pi$
$104$	−2.88612	−0.283008
$105$	0	0
$106$	12.0613	1.17150
$107$	0.602987	0.0582929	0.0291465	−	0.999575i	$-0.490721\pi$
$107$	0.602987	0.0582929	0.0291465	+	0.999575i	$0.490721\pi$
$108$	0	0
$109$	−5.87577	−0.562796	−0.281398	−	0.959591i	$-0.590798\pi$
$109$	−5.87577	−0.562796	−0.281398	+	0.959591i	$0.590798\pi$
$110$	−20.9466	−1.99718
$111$	0	0
$112$	1.51334	0.142997
$113$	−12.3076	−1.15781	−0.578903	−	0.815397i	$-0.696519\pi$
$113$	−12.3076	−1.15781	−0.578903	+	0.815397i	$0.696519\pi$
$114$	0	0
$115$	0	0
$116$	8.66918	0.804913
$117$	0	0
$118$	3.96140	0.364676
$119$	1.81067	0.165984
$120$	0	0
$121$	31.7511	2.88646
$122$	−9.03450	−0.817945
$123$	0	0
$124$	−6.73559	−0.604874
$125$	0.843041	0.0754039
$126$	0	0
$127$	8.86983	0.787070	0.393535	−	0.919310i	$-0.371252\pi$
$127$	8.86983	0.787070	0.393535	+	0.919310i	$0.371252\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−9.24603	−0.810930
$131$	−12.5385	−1.09549	−0.547745	−	0.836645i	$-0.684514\pi$
$131$	−12.5385	−1.09549	−0.547745	+	0.836645i	$0.684514\pi$
$132$	0	0
$133$	0.840741	0.0729015
$134$	7.66794	0.662409
$135$	0	0
$136$	−1.19647	−0.102597
$137$	13.6483	1.16606	0.583028	−	0.812452i	$-0.301868\pi$
$137$	13.6483	1.16606	0.583028	+	0.812452i	$0.301868\pi$
$138$	0	0
$139$	−11.6537	−0.988455	−0.494227	−	0.869333i	$-0.664549\pi$
$139$	−11.6537	−0.988455	−0.494227	+	0.869333i	$0.664549\pi$
$140$	4.84815	0.409743
$141$	0	0
$142$	−9.45607	−0.793536
$143$	18.8707	1.57805
$144$	0	0
$145$	27.7727	2.30640
$146$	0.627598	0.0519404
$147$	0	0
$148$	−1.45380	−0.119502
$149$	−17.3034	−1.41755	−0.708776	−	0.705434i	$-0.750752\pi$
$149$	−17.3034	−1.41755	−0.708776	+	0.705434i	$0.750752\pi$
$150$	0	0
$151$	−4.83249	−0.393262	−0.196631	−	0.980478i	$-0.563000\pi$
$151$	−4.83249	−0.393262	−0.196631	+	0.980478i	$0.563000\pi$
$152$	−0.555554	−0.0450614
$153$	0	0
$154$	−9.89485	−0.797349
$155$	−21.5782	−1.73321
$156$	0	0
$157$	−5.05031	−0.403059	−0.201529	−	0.979482i	$-0.564591\pi$
$157$	−5.05031	−0.403059	−0.201529	+	0.979482i	$0.564591\pi$
$158$	13.7721	1.09565
$159$	0	0
$160$	−3.20362	−0.253268
$161$	0	0
$162$	0	0
$163$	3.56743	0.279423	0.139711	−	0.990192i	$-0.455383\pi$
$163$	3.56743	0.279423	0.139711	+	0.990192i	$0.455383\pi$
$164$	1.84796	0.144301
$165$	0	0
$166$	−15.5888	−1.20993
$167$	−4.56038	−0.352893	−0.176446	−	0.984310i	$-0.556460\pi$
$167$	−4.56038	−0.352893	−0.176446	+	0.984310i	$0.556460\pi$
$168$	0	0
$169$	−4.67030	−0.359254
$170$	−3.83304	−0.293981
$171$	0	0
$172$	2.17379	0.165750
$173$	15.6132	1.18705	0.593525	−	0.804816i	$-0.297736\pi$
$173$	15.6132	1.18705	0.593525	+	0.804816i	$0.297736\pi$
$174$	0	0
$175$	7.96492	0.602092
$176$	6.53843	0.492853
$177$	0	0
$178$	−5.61844	−0.421120
$179$	−6.40521	−0.478748	−0.239374	−	0.970927i	$-0.576942\pi$
$179$	−6.40521	−0.478748	−0.239374	+	0.970927i	$0.576942\pi$
$180$	0	0
$181$	−22.1363	−1.64538	−0.822691	−	0.568489i	$-0.807528\pi$
$181$	−22.1363	−1.64538	−0.822691	+	0.568489i	$0.807528\pi$
$182$	−4.36768	−0.323754
$183$	0	0
$184$	0	0
$185$	−4.65742	−0.342420
$186$	0	0
$187$	7.82306	0.572079
$188$	11.4135	0.832416
$189$	0	0
$190$	−1.77978	−0.129119
$191$	2.14915	0.155507	0.0777534	−	0.996973i	$-0.475225\pi$
$191$	2.14915	0.155507	0.0777534	+	0.996973i	$0.475225\pi$
$192$	0	0
$193$	−11.2460	−0.809506	−0.404753	−	0.914426i	$-0.632643\pi$
$193$	−11.2460	−0.809506	−0.404753	+	0.914426i	$0.632643\pi$
$194$	7.24727	0.520323
$195$	0	0
$196$	−4.70981	−0.336415
$197$	−10.4125	−0.741860	−0.370930	−	0.928661i	$-0.620961\pi$
$197$	−10.4125	−0.741860	−0.370930	+	0.928661i	$0.620961\pi$
$198$	0	0
$199$	15.0998	1.07039	0.535197	−	0.844727i	$-0.320237\pi$
$199$	15.0998	1.07039	0.535197	+	0.844727i	$0.320237\pi$
$200$	−5.26315	−0.372161
$201$	0	0
$202$	−1.02378	−0.0720327
$203$	13.1194	0.920800
$204$	0	0
$205$	5.92014	0.413480
$206$	−4.54925	−0.316961
$207$	0	0
$208$	2.88612	0.200117
$209$	3.63245	0.251262
$210$	0	0
$211$	11.4828	0.790512	0.395256	−	0.918571i	$-0.370656\pi$
$211$	11.4828	0.790512	0.395256	+	0.918571i	$0.370656\pi$
$212$	−12.0613	−0.828375
$213$	0	0
$214$	−0.602987	−0.0412193
$215$	6.96398	0.474940
$216$	0	0
$217$	−10.1932	−0.691961
$218$	5.87577	0.397957
$219$	0	0
$220$	20.9466	1.41222
$221$	3.45317	0.232285
$222$	0	0
$223$	−15.8063	−1.05847	−0.529235	−	0.848475i	$-0.677521\pi$
$223$	−15.8063	−1.05847	−0.529235	+	0.848475i	$0.677521\pi$
$224$	−1.51334	−0.101114
$225$	0	0
$226$	12.3076	0.818692
$227$	2.59469	0.172216	0.0861079	−	0.996286i	$-0.472557\pi$
$227$	2.59469	0.172216	0.0861079	+	0.996286i	$0.472557\pi$
$228$	0	0
$229$	9.99374	0.660405	0.330202	−	0.943910i	$-0.392883\pi$
$229$	9.99374	0.660405	0.330202	+	0.943910i	$0.392883\pi$
$230$	0	0
$231$	0	0
$232$	−8.66918	−0.569159
$233$	27.2948	1.78814	0.894070	−	0.447927i	$-0.147838\pi$
$233$	27.2948	1.78814	0.894070	+	0.447927i	$0.147838\pi$
$234$	0	0
$235$	36.5645	2.38520
$236$	−3.96140	−0.257865
$237$	0	0
$238$	−1.81067	−0.117368
$239$	−14.3187	−0.926198	−0.463099	−	0.886306i	$-0.653263\pi$
$239$	−14.3187	−0.926198	−0.463099	+	0.886306i	$0.653263\pi$
$240$	0	0
$241$	14.1957	0.914428	0.457214	−	0.889357i	$-0.348847\pi$
$241$	14.1957	0.914428	0.457214	+	0.889357i	$0.348847\pi$
$242$	−31.7511	−2.04103
$243$	0	0
$244$	9.03450	0.578375
$245$	−15.0884	−0.963964
$246$	0	0
$247$	1.60340	0.102022
$248$	6.73559	0.427710
$249$	0	0
$250$	−0.843041	−0.0533186
$251$	−11.9645	−0.755194	−0.377597	−	0.925970i	$-0.623250\pi$
$251$	−11.9645	−0.755194	−0.377597	+	0.925970i	$0.623250\pi$
$252$	0	0
$253$	0	0
$254$	−8.86983	−0.556543
$255$	0	0
$256$	1.00000	0.0625000
$257$	−25.9967	−1.62163	−0.810815	−	0.585302i	$-0.800976\pi$
$257$	−25.9967	−1.62163	−0.810815	+	0.585302i	$0.800976\pi$
$258$	0	0
$259$	−2.20009	−0.136707
$260$	9.24603	0.573414
$261$	0	0
$262$	12.5385	0.774629
$263$	−5.74265	−0.354107	−0.177054	−	0.984201i	$-0.556657\pi$
$263$	−5.74265	−0.354107	−0.177054	+	0.984201i	$0.556657\pi$
$264$	0	0
$265$	−38.6398	−2.37363
$266$	−0.840741	−0.0515491
$267$	0	0
$268$	−7.66794	−0.468394
$269$	−8.17926	−0.498699	−0.249349	−	0.968414i	$-0.580217\pi$
$269$	−8.17926	−0.498699	−0.249349	+	0.968414i	$0.580217\pi$
$270$	0	0
$271$	−13.5053	−0.820386	−0.410193	−	0.911999i	$-0.634539\pi$
$271$	−13.5053	−0.820386	−0.410193	+	0.911999i	$0.634539\pi$
$272$	1.19647	0.0725469
$273$	0	0
$274$	−13.6483	−0.824526
$275$	34.4128	2.07517
$276$	0	0
$277$	14.9697	0.899440	0.449720	−	0.893169i	$-0.351524\pi$
$277$	14.9697	0.899440	0.449720	+	0.893169i	$0.351524\pi$
$278$	11.6537	0.698943
$279$	0	0
$280$	−4.84815	−0.289732
$281$	−26.9740	−1.60913	−0.804566	−	0.593863i	$-0.797602\pi$
$281$	−26.9740	−1.60913	−0.804566	+	0.593863i	$0.797602\pi$
$282$	0	0
$283$	13.7193	0.815528	0.407764	−	0.913087i	$-0.366309\pi$
$283$	13.7193	0.815528	0.407764	+	0.913087i	$0.366309\pi$
$284$	9.45607	0.561114
$285$	0	0
$286$	−18.8707	−1.11585
$287$	2.79658	0.165077
$288$	0	0
$289$	−15.5685	−0.915791
$290$	−27.7727	−1.63087
$291$	0	0
$292$	−0.627598	−0.0367274
$293$	−9.20499	−0.537761	−0.268881	−	0.963174i	$-0.586654\pi$
$293$	−9.20499	−0.537761	−0.268881	+	0.963174i	$0.586654\pi$
$294$	0	0
$295$	−12.6908	−0.738887
$296$	1.45380	0.0845004
$297$	0	0
$298$	17.3034	1.00236
$299$	0	0
$300$	0	0
$301$	3.28968	0.189614
$302$	4.83249	0.278078
$303$	0	0
$304$	0.555554	0.0318632
$305$	28.9431	1.65728
$306$	0	0
$307$	1.75676	0.100264	0.0501319	−	0.998743i	$-0.484036\pi$
$307$	1.75676	0.100264	0.0501319	+	0.998743i	$0.484036\pi$
$308$	9.89485	0.563811
$309$	0	0
$310$	21.5782	1.22556
$311$	−30.3629	−1.72172	−0.860860	−	0.508841i	$-0.830074\pi$
$311$	−30.3629	−1.72172	−0.860860	+	0.508841i	$0.830074\pi$
$312$	0	0
$313$	11.0192	0.622843	0.311421	−	0.950272i	$-0.399195\pi$
$313$	11.0192	0.622843	0.311421	+	0.950272i	$0.399195\pi$
$314$	5.05031	0.285006
$315$	0	0
$316$	−13.7721	−0.774743
$317$	10.1471	0.569920	0.284960	−	0.958539i	$-0.408020\pi$
$317$	10.1471	0.569920	0.284960	+	0.958539i	$0.408020\pi$
$318$	0	0
$319$	56.6828	3.17363
$320$	3.20362	0.179088
$321$	0	0
$322$	0	0
$323$	0.664706	0.0369852
$324$	0	0
$325$	15.1901	0.842595
$326$	−3.56743	−0.197582
$327$	0	0
$328$	−1.84796	−0.102036
$329$	17.2725	0.952263
$330$	0	0
$331$	29.7790	1.63680	0.818401	−	0.574648i	$-0.194861\pi$
$331$	29.7790	1.63680	0.818401	+	0.574648i	$0.194861\pi$
$332$	15.5888	0.855549
$333$	0	0
$334$	4.56038	0.249533
$335$	−24.5651	−1.34214
$336$	0	0
$337$	−19.0588	−1.03820	−0.519100	−	0.854714i	$-0.673733\pi$
$337$	−19.0588	−1.03820	−0.519100	+	0.854714i	$0.673733\pi$
$338$	4.67030	0.254031
$339$	0	0
$340$	3.83304	0.207876
$341$	−44.0402	−2.38491
$342$	0	0
$343$	−17.7209	−0.956838
$344$	−2.17379	−0.117203
$345$	0	0
$346$	−15.6132	−0.839371
$347$	12.1929	0.654550	0.327275	−	0.944929i	$-0.393870\pi$
$347$	12.1929	0.654550	0.327275	+	0.944929i	$0.393870\pi$
$348$	0	0
$349$	−15.4022	−0.824459	−0.412229	−	0.911080i	$-0.635250\pi$
$349$	−15.4022	−0.824459	−0.412229	+	0.911080i	$0.635250\pi$
$350$	−7.96492	−0.425743
$351$	0	0
$352$	−6.53843	−0.348499
$353$	−18.5308	−0.986295	−0.493147	−	0.869946i	$-0.664154\pi$
$353$	−18.5308	−0.986295	−0.493147	+	0.869946i	$0.664154\pi$
$354$	0	0
$355$	30.2936	1.60782
$356$	5.61844	0.297777
$357$	0	0
$358$	6.40521	0.338526
$359$	11.7601	0.620673	0.310337	−	0.950627i	$-0.399558\pi$
$359$	11.7601	0.620673	0.310337	+	0.950627i	$0.399558\pi$
$360$	0	0
$361$	−18.6914	−0.983756
$362$	22.1363	1.16346
$363$	0	0
$364$	4.36768	0.228928
$365$	−2.01058	−0.105239
$366$	0	0
$367$	−7.81907	−0.408152	−0.204076	−	0.978955i	$-0.565419\pi$
$367$	−7.81907	−0.408152	−0.204076	+	0.978955i	$0.565419\pi$
$368$	0	0
$369$	0	0
$370$	4.65742	0.242127
$371$	−18.2528	−0.947640
$372$	0	0
$373$	2.13817	0.110710	0.0553550	−	0.998467i	$-0.482371\pi$
$373$	2.13817	0.110710	0.0553550	+	0.998467i	$0.482371\pi$
$374$	−7.82306	−0.404521
$375$	0	0
$376$	−11.4135	−0.588607
$377$	25.0203	1.28861
$378$	0	0
$379$	−21.6725	−1.11324	−0.556621	−	0.830767i	$-0.687902\pi$
$379$	−21.6725	−1.11324	−0.556621	+	0.830767i	$0.687902\pi$
$380$	1.77978	0.0913009
$381$	0	0
$382$	−2.14915	−0.109960
$383$	−5.00975	−0.255986	−0.127993	−	0.991775i	$-0.540854\pi$
$383$	−5.00975	−0.255986	−0.127993	+	0.991775i	$0.540854\pi$
$384$	0	0
$385$	31.6993	1.61555
$386$	11.2460	0.572407
$387$	0	0
$388$	−7.24727	−0.367924
$389$	6.02177	0.305316	0.152658	−	0.988279i	$-0.451217\pi$
$389$	6.02177	0.305316	0.152658	+	0.988279i	$0.451217\pi$
$390$	0	0
$391$	0	0
$392$	4.70981	0.237881
$393$	0	0
$394$	10.4125	0.524574
$395$	−44.1206	−2.21995
$396$	0	0
$397$	24.0018	1.20461	0.602307	−	0.798265i	$-0.294248\pi$
$397$	24.0018	1.20461	0.602307	+	0.798265i	$0.294248\pi$
$398$	−15.0998	−0.756883
$399$	0	0
$400$	5.26315	0.263158
$401$	14.4766	0.722926	0.361463	−	0.932387i	$-0.382277\pi$
$401$	14.4766	0.722926	0.361463	+	0.932387i	$0.382277\pi$
$402$	0	0
$403$	−19.4397	−0.968362
$404$	1.02378	0.0509348
$405$	0	0
$406$	−13.1194	−0.651104
$407$	−9.50557	−0.471173
$408$	0	0
$409$	−32.6435	−1.61412	−0.807059	−	0.590471i	$-0.798942\pi$
$409$	−32.6435	−1.61412	−0.807059	+	0.590471i	$0.798942\pi$
$410$	−5.92014	−0.292375
$411$	0	0
$412$	4.54925	0.224125
$413$	−5.99493	−0.294991
$414$	0	0
$415$	49.9407	2.45149
$416$	−2.88612	−0.141504
$417$	0	0
$418$	−3.63245	−0.177669
$419$	36.2427	1.77057	0.885286	−	0.465046i	$-0.153962\pi$
$419$	36.2427	1.77057	0.885286	+	0.465046i	$0.153962\pi$
$420$	0	0
$421$	6.98570	0.340462	0.170231	−	0.985404i	$-0.445549\pi$
$421$	6.98570	0.340462	0.170231	+	0.985404i	$0.445549\pi$
$422$	−11.4828	−0.558976
$423$	0	0
$424$	12.0613	0.585749
$425$	6.29722	0.305460
$426$	0	0
$427$	13.6722	0.661646
$428$	0.602987	0.0291465
$429$	0	0
$430$	−6.96398	−0.335833
$431$	33.3054	1.60426	0.802132	−	0.597146i	$-0.203699\pi$
$431$	33.3054	1.60426	0.802132	+	0.597146i	$0.203699\pi$
$432$	0	0
$433$	−21.5646	−1.03633	−0.518165	−	0.855281i	$-0.673385\pi$
$433$	−21.5646	−1.03633	−0.518165	+	0.855281i	$0.673385\pi$
$434$	10.1932	0.489290
$435$	0	0
$436$	−5.87577	−0.281398
$437$	0	0
$438$	0	0
$439$	17.2393	0.822785	0.411393	−	0.911458i	$-0.365043\pi$
$439$	17.2393	0.822785	0.411393	+	0.911458i	$0.365043\pi$
$440$	−20.9466	−0.998591
$441$	0	0
$442$	−3.45317	−0.164251
$443$	−18.9166	−0.898754	−0.449377	−	0.893342i	$-0.648354\pi$
$443$	−18.9166	−0.898754	−0.449377	+	0.893342i	$0.648354\pi$
$444$	0	0
$445$	17.9993	0.853250
$446$	15.8063	0.748452
$447$	0	0
$448$	1.51334	0.0714985
$449$	−23.0511	−1.08785	−0.543924	−	0.839134i	$-0.683062\pi$
$449$	−23.0511	−1.08785	−0.543924	+	0.839134i	$0.683062\pi$
$450$	0	0
$451$	12.0827	0.568953
$452$	−12.3076	−0.578903
$453$	0	0
$454$	−2.59469	−0.121775
$455$	13.9924	0.655972
$456$	0	0
$457$	15.1904	0.710578	0.355289	−	0.934757i	$-0.384383\pi$
$457$	15.1904	0.710578	0.355289	+	0.934757i	$0.384383\pi$
$458$	−9.99374	−0.466977
$459$	0	0
$460$	0	0
$461$	−19.2440	−0.896281	−0.448141	−	0.893963i	$-0.647914\pi$
$461$	−19.2440	−0.896281	−0.448141	+	0.893963i	$0.647914\pi$
$462$	0	0
$463$	−3.03240	−0.140927	−0.0704637	−	0.997514i	$-0.522448\pi$
$463$	−3.03240	−0.140927	−0.0704637	+	0.997514i	$0.522448\pi$
$464$	8.66918	0.402456
$465$	0	0
$466$	−27.2948	−1.26441
$467$	−2.85012	−0.131888	−0.0659439	−	0.997823i	$-0.521006\pi$
$467$	−2.85012	−0.131888	−0.0659439	+	0.997823i	$0.521006\pi$
$468$	0	0
$469$	−11.6042	−0.535831
$470$	−36.5645	−1.68659
$471$	0	0
$472$	3.96140	0.182338
$473$	14.2132	0.653522
$474$	0	0
$475$	2.92397	0.134161
$476$	1.81067	0.0829918
$477$	0	0
$478$	14.3187	0.654921
$479$	−37.2607	−1.70249	−0.851243	−	0.524772i	$-0.824151\pi$
$479$	−37.2607	−1.70249	−0.851243	+	0.524772i	$0.824151\pi$
$480$	0	0
$481$	−4.19584	−0.191314
$482$	−14.1957	−0.646598
$483$	0	0
$484$	31.7511	1.44323
$485$	−23.2175	−1.05425
$486$	0	0
$487$	−30.7455	−1.39321	−0.696606	−	0.717454i	$-0.745307\pi$
$487$	−30.7455	−1.39321	−0.696606	+	0.717454i	$0.745307\pi$
$488$	−9.03450	−0.408973
$489$	0	0
$490$	15.0884	0.681625
$491$	−7.81494	−0.352683	−0.176342	−	0.984329i	$-0.556426\pi$
$491$	−7.81494	−0.352683	−0.176342	+	0.984329i	$0.556426\pi$
$492$	0	0
$493$	10.3724	0.467151
$494$	−1.60340	−0.0721403
$495$	0	0
$496$	−6.73559	−0.302437
$497$	14.3102	0.641901
$498$	0	0
$499$	12.9597	0.580154	0.290077	−	0.957003i	$-0.406319\pi$
$499$	12.9597	0.580154	0.290077	+	0.957003i	$0.406319\pi$
$500$	0.843041	0.0377020
$501$	0	0
$502$	11.9645	0.534003
$503$	−19.6304	−0.875276	−0.437638	−	0.899151i	$-0.644185\pi$
$503$	−19.6304	−0.875276	−0.437638	+	0.899151i	$0.644185\pi$
$504$	0	0
$505$	3.27979	0.145949
$506$	0	0
$507$	0	0
$508$	8.86983	0.393535
$509$	−6.25244	−0.277134	−0.138567	−	0.990353i	$-0.544250\pi$
$509$	−6.25244	−0.277134	−0.138567	+	0.990353i	$0.544250\pi$
$510$	0	0
$511$	−0.949767	−0.0420152
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	25.9967	1.14667
$515$	14.5740	0.642209
$516$	0	0
$517$	74.6264	3.28207
$518$	2.20009	0.0966664
$519$	0	0
$520$	−9.24603	−0.405465
$521$	−23.9904	−1.05104	−0.525518	−	0.850782i	$-0.676129\pi$
$521$	−23.9904	−1.05104	−0.525518	+	0.850782i	$0.676129\pi$
$522$	0	0
$523$	−7.42756	−0.324784	−0.162392	−	0.986726i	$-0.551921\pi$
$523$	−7.42756	−0.324784	−0.162392	+	0.986726i	$0.551921\pi$
$524$	−12.5385	−0.547745
$525$	0	0
$526$	5.74265	0.250392
$527$	−8.05895	−0.351054
$528$	0	0
$529$	0	0
$530$	38.6398	1.67841
$531$	0	0
$532$	0.840741	0.0364507
$533$	5.33343	0.231016
$534$	0	0
$535$	1.93174	0.0835163
$536$	7.66794	0.331204
$537$	0	0
$538$	8.17926	0.352633
$539$	−30.7948	−1.32642
$540$	0	0
$541$	20.4393	0.878752	0.439376	−	0.898303i	$-0.355200\pi$
$541$	20.4393	0.878752	0.439376	+	0.898303i	$0.355200\pi$
$542$	13.5053	0.580101
$543$	0	0
$544$	−1.19647	−0.0512984
$545$	−18.8237	−0.806318
$546$	0	0
$547$	−9.31027	−0.398078	−0.199039	−	0.979992i	$-0.563782\pi$
$547$	−9.31027	−0.398078	−0.199039	+	0.979992i	$0.563782\pi$
$548$	13.6483	0.583028
$549$	0	0
$550$	−34.4128	−1.46736
$551$	4.81620	0.205177
$552$	0	0
$553$	−20.8419	−0.886287
$554$	−14.9697	−0.636000
$555$	0	0
$556$	−11.6537	−0.494227
$557$	31.9209	1.35253	0.676265	−	0.736658i	$-0.263597\pi$
$557$	31.9209	1.35253	0.676265	+	0.736658i	$0.263597\pi$
$558$	0	0
$559$	6.27382	0.265354
$560$	4.84815	0.204872
$561$	0	0
$562$	26.9740	1.13783
$563$	16.8512	0.710195	0.355097	−	0.934829i	$-0.384448\pi$
$563$	16.8512	0.710195	0.355097	+	0.934829i	$0.384448\pi$
$564$	0	0
$565$	−39.4289	−1.65879
$566$	−13.7193	−0.576665
$567$	0	0
$568$	−9.45607	−0.396768
$569$	2.18185	0.0914680	0.0457340	−	0.998954i	$-0.485437\pi$
$569$	2.18185	0.0914680	0.0457340	+	0.998954i	$0.485437\pi$
$570$	0	0
$571$	−8.97740	−0.375693	−0.187846	−	0.982198i	$-0.560151\pi$
$571$	−8.97740	−0.375693	−0.187846	+	0.982198i	$0.560151\pi$
$572$	18.8707	0.789024
$573$	0	0
$574$	−2.79658	−0.116727
$575$	0	0
$576$	0	0
$577$	−41.1842	−1.71452	−0.857261	−	0.514882i	$-0.827836\pi$
$577$	−41.1842	−1.71452	−0.857261	+	0.514882i	$0.827836\pi$
$578$	15.5685	0.647562
$579$	0	0
$580$	27.7727	1.15320
$581$	23.5912	0.978727
$582$	0	0
$583$	−78.8620	−3.26613
$584$	0.627598	0.0259702
$585$	0	0
$586$	9.20499	0.380254
$587$	−9.61116	−0.396695	−0.198348	−	0.980132i	$-0.563557\pi$
$587$	−9.61116	−0.396695	−0.198348	+	0.980132i	$0.563557\pi$
$588$	0	0
$589$	−3.74198	−0.154186
$590$	12.6908	0.522472
$591$	0	0
$592$	−1.45380	−0.0597508
$593$	36.4575	1.49713	0.748564	−	0.663062i	$-0.230744\pi$
$593$	36.4575	1.49713	0.748564	+	0.663062i	$0.230744\pi$
$594$	0	0
$595$	5.80068	0.237805
$596$	−17.3034	−0.708776
$597$	0	0
$598$	0	0
$599$	6.93439	0.283331	0.141666	−	0.989915i	$-0.454754\pi$
$599$	6.93439	0.283331	0.141666	+	0.989915i	$0.454754\pi$
$600$	0	0
$601$	16.7707	0.684089	0.342045	−	0.939684i	$-0.388881\pi$
$601$	16.7707	0.684089	0.342045	+	0.939684i	$0.388881\pi$
$602$	−3.28968	−0.134077
$603$	0	0
$604$	−4.83249	−0.196631
$605$	101.718	4.13543
$606$	0	0
$607$	−10.8715	−0.441259	−0.220629	−	0.975358i	$-0.570811\pi$
$607$	−10.8715	−0.441259	−0.220629	+	0.975358i	$0.570811\pi$
$608$	−0.555554	−0.0225307
$609$	0	0
$610$	−28.9431	−1.17187
$611$	32.9408	1.33264
$612$	0	0
$613$	20.7458	0.837914	0.418957	−	0.908006i	$-0.362396\pi$
$613$	20.7458	0.837914	0.418957	+	0.908006i	$0.362396\pi$
$614$	−1.75676	−0.0708972
$615$	0	0
$616$	−9.89485	−0.398675
$617$	−3.09354	−0.124541	−0.0622706	−	0.998059i	$-0.519834\pi$
$617$	−3.09354	−0.124541	−0.0622706	+	0.998059i	$0.519834\pi$
$618$	0	0
$619$	−13.0130	−0.523038	−0.261519	−	0.965198i	$-0.584223\pi$
$619$	−13.0130	−0.523038	−0.261519	+	0.965198i	$0.584223\pi$
$620$	−21.5782	−0.866603
$621$	0	0
$622$	30.3629	1.21744
$623$	8.50259	0.340649
$624$	0	0
$625$	−23.6150	−0.944599
$626$	−11.0192	−0.440416
$627$	0	0
$628$	−5.05031	−0.201529
$629$	−1.73943	−0.0693557
$630$	0	0
$631$	9.82295	0.391045	0.195523	−	0.980699i	$-0.437360\pi$
$631$	9.82295	0.391045	0.195523	+	0.980699i	$0.437360\pi$
$632$	13.7721	0.547826
$633$	0	0
$634$	−10.1471	−0.402994
$635$	28.4155	1.12764
$636$	0	0
$637$	−13.5931	−0.538578
$638$	−56.6828	−2.24409
$639$	0	0
$640$	−3.20362	−0.126634
$641$	−13.6360	−0.538590	−0.269295	−	0.963058i	$-0.586791\pi$
$641$	−13.6360	−0.538590	−0.269295	+	0.963058i	$0.586791\pi$
$642$	0	0
$643$	3.57857	0.141125	0.0705626	−	0.997507i	$-0.477521\pi$
$643$	3.57857	0.141125	0.0705626	+	0.997507i	$0.477521\pi$
$644$	0	0
$645$	0	0
$646$	−0.664706	−0.0261525
$647$	26.2492	1.03196	0.515982	−	0.856599i	$-0.327427\pi$
$647$	26.2492	1.03196	0.515982	+	0.856599i	$0.327427\pi$
$648$	0	0
$649$	−25.9013	−1.01672
$650$	−15.1901	−0.595805
$651$	0	0
$652$	3.56743	0.139711
$653$	−25.7390	−1.00725	−0.503623	−	0.863923i	$-0.668000\pi$
$653$	−25.7390	−1.00725	−0.503623	+	0.863923i	$0.668000\pi$
$654$	0	0
$655$	−40.1684	−1.56951
$656$	1.84796	0.0721505
$657$	0	0
$658$	−17.2725	−0.673352
$659$	10.1996	0.397318	0.198659	−	0.980069i	$-0.436341\pi$
$659$	10.1996	0.397318	0.198659	+	0.980069i	$0.436341\pi$
$660$	0	0
$661$	−29.9545	−1.16509	−0.582547	−	0.812797i	$-0.697944\pi$
$661$	−29.9545	−1.16509	−0.582547	+	0.812797i	$0.697944\pi$
$662$	−29.7790	−1.15739
$663$	0	0
$664$	−15.5888	−0.604965
$665$	2.69341	0.104446
$666$	0	0
$667$	0	0
$668$	−4.56038	−0.176446
$669$	0	0
$670$	24.5651	0.949034
$671$	59.0714	2.28043
$672$	0	0
$673$	−2.02716	−0.0781413	−0.0390707	−	0.999236i	$-0.512440\pi$
$673$	−2.02716	−0.0781413	−0.0390707	+	0.999236i	$0.512440\pi$
$674$	19.0588	0.734118
$675$	0	0
$676$	−4.67030	−0.179627
$677$	−2.46315	−0.0946664	−0.0473332	−	0.998879i	$-0.515072\pi$
$677$	−2.46315	−0.0946664	−0.0473332	+	0.998879i	$0.515072\pi$
$678$	0	0
$679$	−10.9676	−0.420896
$680$	−3.83304	−0.146990
$681$	0	0
$682$	44.0402	1.68638
$683$	30.3002	1.15941	0.579703	−	0.814828i	$-0.303169\pi$
$683$	30.3002	1.15941	0.579703	+	0.814828i	$0.303169\pi$
$684$	0	0
$685$	43.7240	1.67061
$686$	17.7209	0.676587
$687$	0	0
$688$	2.17379	0.0828749
$689$	−34.8104	−1.32617
$690$	0	0
$691$	−1.71993	−0.0654294	−0.0327147	−	0.999465i	$-0.510415\pi$
$691$	−1.71993	−0.0654294	−0.0327147	+	0.999465i	$0.510415\pi$
$692$	15.6132	0.593525
$693$	0	0
$694$	−12.1929	−0.462837
$695$	−37.3340	−1.41616
$696$	0	0
$697$	2.21103	0.0837487
$698$	15.4022	0.582980
$699$	0	0
$700$	7.96492	0.301046
$701$	0.739062	0.0279140	0.0139570	−	0.999903i	$-0.495557\pi$
$701$	0.739062	0.0279140	0.0139570	+	0.999903i	$0.495557\pi$
$702$	0	0
$703$	−0.807665	−0.0304616
$704$	6.53843	0.246426
$705$	0	0
$706$	18.5308	0.697416
$707$	1.54932	0.0582682
$708$	0	0
$709$	22.5819	0.848081	0.424040	−	0.905643i	$-0.360611\pi$
$709$	22.5819	0.848081	0.424040	+	0.905643i	$0.360611\pi$
$710$	−30.2936	−1.13690
$711$	0	0
$712$	−5.61844	−0.210560
$713$	0	0
$714$	0	0
$715$	60.4545	2.26087
$716$	−6.40521	−0.239374
$717$	0	0
$718$	−11.7601	−0.438882
$719$	−10.9509	−0.408400	−0.204200	−	0.978929i	$-0.565459\pi$
$719$	−10.9509	−0.408400	−0.204200	+	0.978929i	$0.565459\pi$
$720$	0	0
$721$	6.88454	0.256394
$722$	18.6914	0.695620
$723$	0	0
$724$	−22.1363	−0.822691
$725$	45.6272	1.69455
$726$	0	0
$727$	37.2793	1.38261	0.691307	−	0.722562i	$-0.257035\pi$
$727$	37.2793	1.38261	0.691307	+	0.722562i	$0.257035\pi$
$728$	−4.36768	−0.161877
$729$	0	0
$730$	2.01058	0.0744150
$731$	2.60088	0.0961971
$732$	0	0
$733$	32.1008	1.18567	0.592835	−	0.805324i	$-0.298009\pi$
$733$	32.1008	1.18567	0.592835	+	0.805324i	$0.298009\pi$
$734$	7.81907	0.288607
$735$	0	0
$736$	0	0
$737$	−50.1363	−1.84679
$738$	0	0
$739$	37.5076	1.37974	0.689870	−	0.723934i	$-0.257668\pi$
$739$	37.5076	1.37974	0.689870	+	0.723934i	$0.257668\pi$
$740$	−4.65742	−0.171210
$741$	0	0
$742$	18.2528	0.670083
$743$	−31.8791	−1.16953	−0.584766	−	0.811202i	$-0.698814\pi$
$743$	−31.8791	−1.16953	−0.584766	+	0.811202i	$0.698814\pi$
$744$	0	0
$745$	−55.4335	−2.03093
$746$	−2.13817	−0.0782838
$747$	0	0
$748$	7.82306	0.286039
$749$	0.912522	0.0333428
$750$	0	0
$751$	−17.9494	−0.654983	−0.327491	−	0.944854i	$-0.606203\pi$
$751$	−17.9494	−0.654983	−0.327491	+	0.944854i	$0.606203\pi$
$752$	11.4135	0.416208
$753$	0	0
$754$	−25.0203	−0.911186
$755$	−15.4814	−0.563427
$756$	0	0
$757$	−10.9617	−0.398410	−0.199205	−	0.979958i	$-0.563836\pi$
$757$	−10.9617	−0.398410	−0.199205	+	0.979958i	$0.563836\pi$
$758$	21.6725	0.787180
$759$	0	0
$760$	−1.77978	−0.0645595
$761$	0.365855	0.0132622	0.00663111	−	0.999978i	$-0.497889\pi$
$761$	0.365855	0.0132622	0.00663111	+	0.999978i	$0.497889\pi$
$762$	0	0
$763$	−8.89201	−0.321912
$764$	2.14915	0.0777534
$765$	0	0
$766$	5.00975	0.181010
$767$	−11.4331	−0.412825
$768$	0	0
$769$	0.305975	0.0110337	0.00551686	−	0.999985i	$-0.498244\pi$
$769$	0.305975	0.0110337	0.00551686	+	0.999985i	$0.498244\pi$
$770$	−31.6993	−1.14236
$771$	0	0
$772$	−11.2460	−0.404753
$773$	3.53034	0.126977	0.0634887	−	0.997983i	$-0.479777\pi$
$773$	3.53034	0.126977	0.0634887	+	0.997983i	$0.479777\pi$
$774$	0	0
$775$	−35.4504	−1.27342
$776$	7.24727	0.260162
$777$	0	0
$778$	−6.02177	−0.215891
$779$	1.02664	0.0367832
$780$	0	0
$781$	61.8278	2.21237
$782$	0	0
$783$	0	0
$784$	−4.70981	−0.168208
$785$	−16.1793	−0.577462
$786$	0	0
$787$	−2.24943	−0.0801837	−0.0400918	−	0.999196i	$-0.512765\pi$
$787$	−2.24943	−0.0801837	−0.0400918	+	0.999196i	$0.512765\pi$
$788$	−10.4125	−0.370930
$789$	0	0
$790$	44.1206	1.56974
$791$	−18.6256	−0.662250
$792$	0	0
$793$	26.0747	0.925939
$794$	−24.0018	−0.851790
$795$	0	0
$796$	15.0998	0.535197
$797$	39.6694	1.40516	0.702581	−	0.711604i	$-0.252031\pi$
$797$	39.6694	1.40516	0.702581	+	0.711604i	$0.252031\pi$
$798$	0	0
$799$	13.6560	0.483113
$800$	−5.26315	−0.186081
$801$	0	0
$802$	−14.4766	−0.511186
$803$	−4.10350	−0.144810
$804$	0	0
$805$	0	0
$806$	19.4397	0.684735
$807$	0	0
$808$	−1.02378	−0.0360164
$809$	38.4898	1.35323	0.676614	−	0.736338i	$-0.263447\pi$
$809$	38.4898	1.35323	0.676614	+	0.736338i	$0.263447\pi$
$810$	0	0
$811$	−14.4390	−0.507021	−0.253511	−	0.967333i	$-0.581585\pi$
$811$	−14.4390	−0.507021	−0.253511	+	0.967333i	$0.581585\pi$
$812$	13.1194	0.460400
$813$	0	0
$814$	9.50557	0.333170
$815$	11.4287	0.400329
$816$	0	0
$817$	1.20766	0.0422506
$818$	32.6435	1.14135
$819$	0	0
$820$	5.92014	0.206740
$821$	−29.2724	−1.02161	−0.510806	−	0.859696i	$-0.670653\pi$
$821$	−29.2724	−1.02161	−0.510806	+	0.859696i	$0.670653\pi$
$822$	0	0
$823$	−16.6990	−0.582091	−0.291045	−	0.956709i	$-0.594003\pi$
$823$	−16.6990	−0.582091	−0.291045	+	0.956709i	$0.594003\pi$
$824$	−4.54925	−0.158481
$825$	0	0
$826$	5.99493	0.208590
$827$	20.3196	0.706582	0.353291	−	0.935514i	$-0.385063\pi$
$827$	20.3196	0.706582	0.353291	+	0.935514i	$0.385063\pi$
$828$	0	0
$829$	0.285271	0.00990788	0.00495394	−	0.999988i	$-0.498423\pi$
$829$	0.285271	0.00990788	0.00495394	+	0.999988i	$0.498423\pi$
$830$	−49.9407	−1.73347
$831$	0	0
$832$	2.88612	0.100058
$833$	−5.63517	−0.195247
$834$	0	0
$835$	−14.6097	−0.505589
$836$	3.63245	0.125631
$837$	0	0
$838$	−36.2427	−1.25198
$839$	24.4524	0.844192	0.422096	−	0.906551i	$-0.361295\pi$
$839$	24.4524	0.844192	0.422096	+	0.906551i	$0.361295\pi$
$840$	0	0
$841$	46.1546	1.59154
$842$	−6.98570	−0.240743
$843$	0	0
$844$	11.4828	0.395256
$845$	−14.9618	−0.514703
$846$	0	0
$847$	48.0500	1.65102
$848$	−12.0613	−0.414187
$849$	0	0
$850$	−6.29722	−0.215993
$851$	0	0
$852$	0	0
$853$	28.9592	0.991545	0.495773	−	0.868452i	$-0.334885\pi$
$853$	28.9592	0.991545	0.495773	+	0.868452i	$0.334885\pi$
$854$	−13.6722	−0.467855
$855$	0	0
$856$	−0.602987	−0.0206097
$857$	30.4523	1.04023	0.520116	−	0.854096i	$-0.325889\pi$
$857$	30.4523	1.04023	0.520116	+	0.854096i	$0.325889\pi$
$858$	0	0
$859$	42.3725	1.44573	0.722866	−	0.690988i	$-0.242824\pi$
$859$	42.3725	1.44573	0.722866	+	0.690988i	$0.242824\pi$
$860$	6.96398	0.237470
$861$	0	0
$862$	−33.3054	−1.13439
$863$	49.6854	1.69131	0.845655	−	0.533729i	$-0.179210\pi$
$863$	49.6854	1.69131	0.845655	+	0.533729i	$0.179210\pi$
$864$	0	0
$865$	50.0187	1.70069
$866$	21.5646	0.732796
$867$	0	0
$868$	−10.1932	−0.345980
$869$	−90.0482	−3.05467
$870$	0	0
$871$	−22.1306	−0.749867
$872$	5.87577	0.198979
$873$	0	0
$874$	0	0
$875$	1.27581	0.0431301
$876$	0	0
$877$	4.98560	0.168352	0.0841759	−	0.996451i	$-0.473174\pi$
$877$	4.98560	0.168352	0.0841759	+	0.996451i	$0.473174\pi$
$878$	−17.2393	−0.581797
$879$	0	0
$880$	20.9466	0.706110
$881$	−49.2503	−1.65928	−0.829642	−	0.558296i	$-0.811455\pi$
$881$	−49.2503	−1.65928	−0.829642	+	0.558296i	$0.811455\pi$
$882$	0	0
$883$	−29.1640	−0.981448	−0.490724	−	0.871315i	$-0.663268\pi$
$883$	−29.1640	−0.981448	−0.490724	+	0.871315i	$0.663268\pi$
$884$	3.45317	0.116143
$885$	0	0
$886$	18.9166	0.635515
$887$	−6.09320	−0.204590	−0.102295	−	0.994754i	$-0.532619\pi$
$887$	−6.09320	−0.204590	−0.102295	+	0.994754i	$0.532619\pi$
$888$	0	0
$889$	13.4230	0.450194
$890$	−17.9993	−0.603339
$891$	0	0
$892$	−15.8063	−0.529235
$893$	6.34082	0.212188
$894$	0	0
$895$	−20.5198	−0.685903
$896$	−1.51334	−0.0505570
$897$	0	0
$898$	23.0511	0.769225
$899$	−58.3920	−1.94748
$900$	0	0
$901$	−14.4310	−0.480768
$902$	−12.0827	−0.402311
$903$	0	0
$904$	12.3076	0.409346
$905$	−70.9163	−2.35734
$906$	0	0
$907$	−2.46052	−0.0817004	−0.0408502	−	0.999165i	$-0.513007\pi$
$907$	−2.46052	−0.0817004	−0.0408502	+	0.999165i	$0.513007\pi$
$908$	2.59469	0.0861079
$909$	0	0
$910$	−13.9924	−0.463842
$911$	52.4143	1.73656	0.868281	−	0.496073i	$-0.165225\pi$
$911$	52.4143	1.73656	0.868281	+	0.496073i	$0.165225\pi$
$912$	0	0
$913$	101.927	3.37328
$914$	−15.1904	−0.502454
$915$	0	0
$916$	9.99374	0.330202
$917$	−18.9749	−0.626607
$918$	0	0
$919$	−12.0132	−0.396280	−0.198140	−	0.980174i	$-0.563490\pi$
$919$	−12.0132	−0.396280	−0.198140	+	0.980174i	$0.563490\pi$
$920$	0	0
$921$	0	0
$922$	19.2440	0.633767
$923$	27.2914	0.898306
$924$	0	0
$925$	−7.65157	−0.251582
$926$	3.03240	0.0996507
$927$	0	0
$928$	−8.66918	−0.284580
$929$	17.2437	0.565747	0.282874	−	0.959157i	$-0.408712\pi$
$929$	17.2437	0.565747	0.282874	+	0.959157i	$0.408712\pi$
$930$	0	0
$931$	−2.61656	−0.0857541
$932$	27.2948	0.894070
$933$	0	0
$934$	2.85012	0.0932588
$935$	25.0621	0.819617
$936$	0	0
$937$	19.8875	0.649696	0.324848	−	0.945766i	$-0.394687\pi$
$937$	19.8875	0.649696	0.324848	+	0.945766i	$0.394687\pi$
$938$	11.6042	0.378890
$939$	0	0
$940$	36.5645	1.19260
$941$	14.7942	0.482278	0.241139	−	0.970491i	$-0.422479\pi$
$941$	14.7942	0.482278	0.241139	+	0.970491i	$0.422479\pi$
$942$	0	0
$943$	0	0
$944$	−3.96140	−0.128933
$945$	0	0
$946$	−14.2132	−0.462110
$947$	18.1501	0.589799	0.294900	−	0.955528i	$-0.404714\pi$
$947$	18.1501	0.589799	0.294900	+	0.955528i	$0.404714\pi$
$948$	0	0
$949$	−1.81132	−0.0587981
$950$	−2.92397	−0.0948660
$951$	0	0
$952$	−1.81067	−0.0586841
$953$	26.1661	0.847605	0.423802	−	0.905755i	$-0.360695\pi$
$953$	26.1661	0.847605	0.423802	+	0.905755i	$0.360695\pi$
$954$	0	0
$955$	6.88504	0.222795
$956$	−14.3187	−0.463099
$957$	0	0
$958$	37.2607	1.20384
$959$	20.6545	0.666969
$960$	0	0
$961$	14.3681	0.463488
$962$	4.19584	0.135279
$963$	0	0
$964$	14.1957	0.457214
$965$	−36.0279	−1.15978
$966$	0	0
$967$	−22.8164	−0.733726	−0.366863	−	0.930275i	$-0.619568\pi$
$967$	−22.8164	−0.733726	−0.366863	+	0.930275i	$0.619568\pi$
$968$	−31.7511	−1.02052
$969$	0	0
$970$	23.2175	0.745468
$971$	2.27238	0.0729240	0.0364620	−	0.999335i	$-0.488391\pi$
$971$	2.27238	0.0729240	0.0364620	+	0.999335i	$0.488391\pi$
$972$	0	0
$973$	−17.6360	−0.565384
$974$	30.7455	0.985150
$975$	0	0
$976$	9.03450	0.289187
$977$	37.7855	1.20887	0.604433	−	0.796656i	$-0.293400\pi$
$977$	37.7855	1.20887	0.604433	+	0.796656i	$0.293400\pi$
$978$	0	0
$979$	36.7358	1.17408
$980$	−15.0884	−0.481982
$981$	0	0
$982$	7.81494	0.249385
$983$	53.3063	1.70021	0.850104	−	0.526615i	$-0.176539\pi$
$983$	53.3063	1.70021	0.850104	+	0.526615i	$0.176539\pi$
$984$	0	0
$985$	−33.3577	−1.06286
$986$	−10.3724	−0.330326
$987$	0	0
$988$	1.60340	0.0510109
$989$	0	0
$990$	0	0
$991$	17.1670	0.545328	0.272664	−	0.962109i	$-0.412095\pi$
$991$	17.1670	0.545328	0.272664	+	0.962109i	$0.412095\pi$
$992$	6.73559	0.213855
$993$	0	0
$994$	−14.3102	−0.453893
$995$	48.3739	1.53356
$996$	0	0
$997$	−28.8767	−0.914535	−0.457268	−	0.889329i	$-0.651172\pi$
$997$	−28.8767	−0.914535	−0.457268	+	0.889329i	$0.651172\pi$
$998$	−12.9597	−0.410231
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.bt.1.5		5
3.2	odd	2		3174.2.a.bc.1.1		5
23.3	even	11		414.2.i.d.55.1		10
23.8	even	11		414.2.i.d.271.1		10
23.22	odd	2		9522.2.a.bq.1.1		5
69.8	odd	22		138.2.e.a.133.1	yes	10
69.26	odd	22		138.2.e.a.55.1	✓	10
69.68	even	2		3174.2.a.bd.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.a.55.1	✓	10	69.26	odd	22
138.2.e.a.133.1	yes	10	69.8	odd	22
414.2.i.d.55.1		10	23.3	even	11
414.2.i.d.271.1		10	23.8	even	11
3174.2.a.bc.1.1		5	3.2	odd	2
3174.2.a.bd.1.5		5	69.68	even	2
9522.2.a.bq.1.1		5	23.22	odd	2
9522.2.a.bt.1.5		5	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.20362 q^{5} +1.51334 q^{7} -1.00000 q^{8} -3.20362 q^{10} +6.53843 q^{11} +2.88612 q^{13} -1.51334 q^{14} +1.00000 q^{16} +1.19647 q^{17} +0.555554 q^{19} +3.20362 q^{20} -6.53843 q^{22} +5.26315 q^{25} -2.88612 q^{26} +1.51334 q^{28} +8.66918 q^{29} -6.73559 q^{31} -1.00000 q^{32} -1.19647 q^{34} +4.84815 q^{35} -1.45380 q^{37} -0.555554 q^{38} -3.20362 q^{40} +1.84796 q^{41} +2.17379 q^{43} +6.53843 q^{44} +11.4135 q^{47} -4.70981 q^{49} -5.26315 q^{50} +2.88612 q^{52} -12.0613 q^{53} +20.9466 q^{55} -1.51334 q^{56} -8.66918 q^{58} -3.96140 q^{59} +9.03450 q^{61} +6.73559 q^{62} +1.00000 q^{64} +9.24603 q^{65} -7.66794 q^{67} +1.19647 q^{68} -4.84815 q^{70} +9.45607 q^{71} -0.627598 q^{73} +1.45380 q^{74} +0.555554 q^{76} +9.89485 q^{77} -13.7721 q^{79} +3.20362 q^{80} -1.84796 q^{82} +15.5888 q^{83} +3.83304 q^{85} -2.17379 q^{86} -6.53843 q^{88} +5.61844 q^{89} +4.36768 q^{91} -11.4135 q^{94} +1.77978 q^{95} -7.24727 q^{97} +4.70981 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} + 7 q^{5} - 7 q^{7} - 5 q^{8} - 7 q^{10} + 13 q^{11} - 4 q^{13} + 7 q^{14} + 5 q^{16} + 9 q^{17} - 11 q^{19} + 7 q^{20} - 13 q^{22} - 2 q^{25} + 4 q^{26} - 7 q^{28} + 7 q^{29} - 8 q^{31}+ \cdots + 12 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 + 7 * q^5 - 7 * q^7 - 5 * q^8 - 7 * q^10 + 13 * q^11 - 4 * q^13 + 7 * q^14 + 5 * q^16 + 9 * q^17 - 11 * q^19 + 7 * q^20 - 13 * q^22 - 2 * q^25 + 4 * q^26 - 7 * q^28 + 7 * q^29 - 8 * q^31 - 5 * q^32 - 9 * q^34 - q^35 - 12 * q^37 + 11 * q^38 - 7 * q^40 + 10 * q^41 - 4 * q^43 + 13 * q^44 + 24 * q^47 - 12 * q^49 + 2 * q^50 - 4 * q^52 + 9 * q^53 + 16 * q^55 + 7 * q^56 - 7 * q^58 + 14 * q^59 - 5 * q^61 + 8 * q^62 + 5 * q^64 + 12 * q^65 - 13 * q^67 + 9 * q^68 + q^70 + 19 * q^71 + 4 * q^73 + 12 * q^74 - 11 * q^76 - 5 * q^77 - 4 * q^79 + 7 * q^80 - 10 * q^82 + 24 * q^83 + 17 * q^85 + 4 * q^86 - 13 * q^88 + 4 * q^89 + 21 * q^91 - 24 * q^94 + 11 * q^95 + 9 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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