LMFDB - Embedded newform 9522.2.a.bu.1.4

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,-5,0,5,8,0,-7,-5,0,-8,5,0,-7,7,0,5,13,0,-12,8,0,-5,0,0,1,7, 0,-7,4] 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:	$5$
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 46)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.30972$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.54620 q^{5} -2.08816 q^{7} -1.00000 q^{8} -2.54620 q^{10} +3.81881 q^{11} -6.19584 q^{13} +2.08816 q^{14} +1.00000 q^{16} +1.30258 q^{17} +2.48047 q^{19} +2.54620 q^{20} -3.81881 q^{22} +1.48314 q^{25} +6.19584 q^{26} -2.08816 q^{28} -7.16723 q^{29} -4.02157 q^{31} -1.00000 q^{32} -1.30258 q^{34} -5.31686 q^{35} +11.4282 q^{37} -2.48047 q^{38} -2.54620 q^{40} +4.29298 q^{41} +2.77780 q^{43} +3.81881 q^{44} -9.34150 q^{47} -2.63960 q^{49} -1.48314 q^{50} -6.19584 q^{52} -2.36584 q^{53} +9.72346 q^{55} +2.08816 q^{56} +7.16723 q^{58} -0.611476 q^{59} -4.03969 q^{61} +4.02157 q^{62} +1.00000 q^{64} -15.7759 q^{65} -12.8258 q^{67} +1.30258 q^{68} +5.31686 q^{70} +6.61122 q^{71} -2.92492 q^{73} -11.4282 q^{74} +2.48047 q^{76} -7.97428 q^{77} +1.88675 q^{79} +2.54620 q^{80} -4.29298 q^{82} +10.4404 q^{83} +3.31663 q^{85} -2.77780 q^{86} -3.81881 q^{88} +5.01225 q^{89} +12.9379 q^{91} +9.34150 q^{94} +6.31578 q^{95} -12.5459 q^{97} +2.63960 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} + 8 q^{5} - 7 q^{7} - 5 q^{8} - 8 q^{10} + 5 q^{11} - 7 q^{13} + 7 q^{14} + 5 q^{16} + 13 q^{17} - 12 q^{19} + 8 q^{20} - 5 q^{22} + q^{25} + 7 q^{26} - 7 q^{28} + 4 q^{29} + 6 q^{31}+ \cdots + 12 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 + 8 * q^5 - 7 * q^7 - 5 * q^8 - 8 * q^10 + 5 * q^11 - 7 * q^13 + 7 * q^14 + 5 * q^16 + 13 * q^17 - 12 * q^19 + 8 * q^20 - 5 * q^22 + q^25 + 7 * q^26 - 7 * q^28 + 4 * q^29 + 6 * q^31 - 5 * q^32 - 13 * q^34 - 9 * q^35 - 14 * q^37 + 12 * q^38 - 8 * q^40 - q^41 - 11 * q^43 + 5 * q^44 - 9 * q^47 - 12 * q^49 - q^50 - 7 * q^52 + 6 * q^53 - 14 * q^55 + 7 * q^56 - 4 * q^58 - 3 * q^59 - 13 * q^61 - 6 * q^62 + 5 * q^64 - 9 * q^65 - 10 * q^67 + 13 * q^68 + 9 * q^70 - 8 * q^71 - 4 * q^73 + 14 * q^74 - 12 * q^76 - 18 * q^77 + 18 * q^79 + 8 * q^80 + q^82 + 2 * q^83 + 23 * q^85 + 11 * q^86 - 5 * q^88 + 10 * q^89 + 12 * q^91 + 9 * q^94 - 28 * q^95 - 17 * 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
$5$	2.54620	1.13870	0.569348	−	0.822097i	$-0.307196\pi$
$5$	2.54620	1.13870	0.569348	+	0.822097i	$0.307196\pi$
$6$	0	0
$7$	−2.08816	−0.789249	−0.394624	−	0.918843i	$-0.629125\pi$
$7$	−2.08816	−0.789249	−0.394624	+	0.918843i	$0.629125\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.54620	−0.805179
$11$	3.81881	1.15142	0.575708	−	0.817655i	$-0.304727\pi$
$11$	3.81881	1.15142	0.575708	+	0.817655i	$0.304727\pi$
$12$	0	0
$13$	−6.19584	−1.71842	−0.859209	−	0.511625i	$-0.829044\pi$
$13$	−6.19584	−1.71842	−0.859209	+	0.511625i	$0.829044\pi$
$14$	2.08816	0.558083
$15$	0	0
$16$	1.00000	0.250000
$17$	1.30258	0.315922	0.157961	−	0.987445i	$-0.449508\pi$
$17$	1.30258	0.315922	0.157961	+	0.987445i	$0.449508\pi$
$18$	0	0
$19$	2.48047	0.569060	0.284530	−	0.958667i	$-0.408163\pi$
$19$	2.48047	0.569060	0.284530	+	0.958667i	$0.408163\pi$
$20$	2.54620	0.569348
$21$	0	0
$22$	−3.81881	−0.814174
$23$	0	0
$23$	0	0
$24$	0	0
$25$	1.48314	0.296627
$26$	6.19584	1.21511
$27$	0	0
$28$	−2.08816	−0.394624
$29$	−7.16723	−1.33092	−0.665460	−	0.746433i	$-0.731765\pi$
$29$	−7.16723	−1.33092	−0.665460	+	0.746433i	$0.731765\pi$
$30$	0	0
$31$	−4.02157	−0.722294	−0.361147	−	0.932509i	$-0.617615\pi$
$31$	−4.02157	−0.722294	−0.361147	+	0.932509i	$0.617615\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.30258	−0.223391
$35$	−5.31686	−0.898714
$36$	0	0
$37$	11.4282	1.87878	0.939390	−	0.342852i	$-0.111393\pi$
$37$	11.4282	1.87878	0.939390	+	0.342852i	$0.111393\pi$
$38$	−2.48047	−0.402386
$39$	0	0
$40$	−2.54620	−0.402590
$41$	4.29298	0.670451	0.335225	−	0.942138i	$-0.391188\pi$
$41$	4.29298	0.670451	0.335225	+	0.942138i	$0.391188\pi$
$42$	0	0
$43$	2.77780	0.423611	0.211806	−	0.977312i	$-0.432066\pi$
$43$	2.77780	0.423611	0.211806	+	0.977312i	$0.432066\pi$
$44$	3.81881	0.575708
$45$	0	0
$46$	0	0
$47$	−9.34150	−1.36260	−0.681299	−	0.732005i	$-0.738585\pi$
$47$	−9.34150	−1.36260	−0.681299	+	0.732005i	$0.738585\pi$
$48$	0	0
$49$	−2.63960	−0.377086
$50$	−1.48314	−0.209747
$51$	0	0
$52$	−6.19584	−0.859209
$53$	−2.36584	−0.324973	−0.162486	−	0.986711i	$-0.551951\pi$
$53$	−2.36584	−0.324973	−0.162486	+	0.986711i	$0.551951\pi$
$54$	0	0
$55$	9.72346	1.31111
$56$	2.08816	0.279042
$57$	0	0
$58$	7.16723	0.941103
$59$	−0.611476	−0.0796075	−0.0398037	−	0.999208i	$-0.512673\pi$
$59$	−0.611476	−0.0796075	−0.0398037	+	0.999208i	$0.512673\pi$
$60$	0	0
$61$	−4.03969	−0.517230	−0.258615	−	0.965980i	$-0.583266\pi$
$61$	−4.03969	−0.517230	−0.258615	+	0.965980i	$0.583266\pi$
$62$	4.02157	0.510739
$63$	0	0
$64$	1.00000	0.125000
$65$	−15.7759	−1.95675
$66$	0	0
$67$	−12.8258	−1.56693	−0.783463	−	0.621438i	$-0.786549\pi$
$67$	−12.8258	−1.56693	−0.783463	+	0.621438i	$0.786549\pi$
$68$	1.30258	0.157961
$69$	0	0
$70$	5.31686	0.635487
$71$	6.61122	0.784608	0.392304	−	0.919836i	$-0.371678\pi$
$71$	6.61122	0.784608	0.392304	+	0.919836i	$0.371678\pi$
$72$	0	0
$73$	−2.92492	−0.342336	−0.171168	−	0.985242i	$-0.554754\pi$
$73$	−2.92492	−0.342336	−0.171168	+	0.985242i	$0.554754\pi$
$74$	−11.4282	−1.32850
$75$	0	0
$76$	2.48047	0.284530
$77$	−7.97428	−0.908753
$78$	0	0
$79$	1.88675	0.212276	0.106138	−	0.994351i	$-0.466151\pi$
$79$	1.88675	0.212276	0.106138	+	0.994351i	$0.466151\pi$
$80$	2.54620	0.284674
$81$	0	0
$82$	−4.29298	−0.474080
$83$	10.4404	1.14598	0.572989	−	0.819563i	$-0.305784\pi$
$83$	10.4404	1.14598	0.572989	+	0.819563i	$0.305784\pi$
$84$	0	0
$85$	3.31663	0.359739
$86$	−2.77780	−0.299538
$87$	0	0
$88$	−3.81881	−0.407087
$89$	5.01225	0.531298	0.265649	−	0.964070i	$-0.414414\pi$
$89$	5.01225	0.531298	0.265649	+	0.964070i	$0.414414\pi$
$90$	0	0
$91$	12.9379	1.35626
$92$	0	0
$93$	0	0
$94$	9.34150	0.963503
$95$	6.31578	0.647986
$96$	0	0
$97$	−12.5459	−1.27385	−0.636924	−	0.770927i	$-0.719793\pi$
$97$	−12.5459	−1.27385	−0.636924	+	0.770927i	$0.719793\pi$
$98$	2.63960	0.266640
$99$	0	0
$100$	1.48314	0.148314
$101$	−3.52251	−0.350503	−0.175252	−	0.984524i	$-0.556074\pi$
$101$	−3.52251	−0.350503	−0.175252	+	0.984524i	$0.556074\pi$
$102$	0	0
$103$	−1.56622	−0.154325	−0.0771623	−	0.997019i	$-0.524586\pi$
$103$	−1.56622	−0.154325	−0.0771623	+	0.997019i	$0.524586\pi$
$104$	6.19584	0.607553
$105$	0	0
$106$	2.36584	0.229791
$107$	3.40318	0.328998	0.164499	−	0.986377i	$-0.447399\pi$
$107$	3.40318	0.328998	0.164499	+	0.986377i	$0.447399\pi$
$108$	0	0
$109$	8.62324	0.825956	0.412978	−	0.910741i	$-0.364489\pi$
$109$	8.62324	0.825956	0.412978	+	0.910741i	$0.364489\pi$
$110$	−9.72346	−0.927096
$111$	0	0
$112$	−2.08816	−0.197312
$113$	−3.56622	−0.335482	−0.167741	−	0.985831i	$-0.553647\pi$
$113$	−3.56622	−0.335482	−0.167741	+	0.985831i	$0.553647\pi$
$114$	0	0
$115$	0	0
$116$	−7.16723	−0.665460
$117$	0	0
$118$	0.611476	0.0562910
$119$	−2.71999	−0.249341
$120$	0	0
$121$	3.58334	0.325758
$122$	4.03969	0.365737
$123$	0	0
$124$	−4.02157	−0.361147
$125$	−8.95464	−0.800927
$126$	0	0
$127$	−5.62165	−0.498841	−0.249421	−	0.968395i	$-0.580240\pi$
$127$	−5.62165	−0.498841	−0.249421	+	0.968395i	$0.580240\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	15.7759	1.38363
$131$	−7.97296	−0.696601	−0.348301	−	0.937383i	$-0.613241\pi$
$131$	−7.97296	−0.696601	−0.348301	+	0.937383i	$0.613241\pi$
$132$	0	0
$133$	−5.17962	−0.449130
$134$	12.8258	1.10798
$135$	0	0
$136$	−1.30258	−0.111695
$137$	2.06934	0.176796	0.0883979	−	0.996085i	$-0.471825\pi$
$137$	2.06934	0.176796	0.0883979	+	0.996085i	$0.471825\pi$
$138$	0	0
$139$	−13.0620	−1.10790	−0.553950	−	0.832550i	$-0.686880\pi$
$139$	−13.0620	−1.10790	−0.553950	+	0.832550i	$0.686880\pi$
$140$	−5.31686	−0.449357
$141$	0	0
$142$	−6.61122	−0.554801
$143$	−23.6608	−1.97861
$144$	0	0
$145$	−18.2492	−1.51551
$146$	2.92492	0.242068
$147$	0	0
$148$	11.4282	0.939390
$149$	14.8452	1.21617	0.608083	−	0.793873i	$-0.291939\pi$
$149$	14.8452	1.21617	0.608083	+	0.793873i	$0.291939\pi$
$150$	0	0
$151$	3.85308	0.313559	0.156780	−	0.987634i	$-0.449889\pi$
$151$	3.85308	0.313559	0.156780	+	0.987634i	$0.449889\pi$
$152$	−2.48047	−0.201193
$153$	0	0
$154$	7.97428	0.642586
$155$	−10.2397	−0.822473
$156$	0	0
$157$	−22.6511	−1.80776	−0.903878	−	0.427790i	$-0.859292\pi$
$157$	−22.6511	−1.80776	−0.903878	+	0.427790i	$0.859292\pi$
$158$	−1.88675	−0.150102
$159$	0	0
$160$	−2.54620	−0.201295
$161$	0	0
$162$	0	0
$163$	−18.9635	−1.48533	−0.742667	−	0.669660i	$-0.766440\pi$
$163$	−18.9635	−1.48533	−0.742667	+	0.669660i	$0.766440\pi$
$164$	4.29298	0.335225
$165$	0	0
$166$	−10.4404	−0.810329
$167$	0.979634	0.0758063	0.0379032	−	0.999281i	$-0.487932\pi$
$167$	0.979634	0.0758063	0.0379032	+	0.999281i	$0.487932\pi$
$168$	0	0
$169$	25.3885	1.95296
$170$	−3.31663	−0.254374
$171$	0	0
$172$	2.77780	0.211806
$173$	8.92203	0.678330	0.339165	−	0.940727i	$-0.389856\pi$
$173$	8.92203	0.678330	0.339165	+	0.940727i	$0.389856\pi$
$174$	0	0
$175$	−3.09702	−0.234113
$176$	3.81881	0.287854
$177$	0	0
$178$	−5.01225	−0.375684
$179$	8.49152	0.634686	0.317343	−	0.948311i	$-0.397209\pi$
$179$	8.49152	0.634686	0.317343	+	0.948311i	$0.397209\pi$
$180$	0	0
$181$	0.537425	0.0399465	0.0199732	−	0.999801i	$-0.493642\pi$
$181$	0.537425	0.0399465	0.0199732	+	0.999801i	$0.493642\pi$
$182$	−12.9379	−0.959020
$183$	0	0
$184$	0	0
$185$	29.0984	2.13936
$186$	0	0
$187$	4.97431	0.363757
$188$	−9.34150	−0.681299
$189$	0	0
$190$	−6.31578	−0.458195
$191$	−26.8320	−1.94149	−0.970747	−	0.240105i	$-0.922818\pi$
$191$	−26.8320	−1.94149	−0.970747	+	0.240105i	$0.922818\pi$
$192$	0	0
$193$	−1.40434	−0.101087	−0.0505434	−	0.998722i	$-0.516095\pi$
$193$	−1.40434	−0.101087	−0.0505434	+	0.998722i	$0.516095\pi$
$194$	12.5459	0.900746
$195$	0	0
$196$	−2.63960	−0.188543
$197$	8.24471	0.587411	0.293706	−	0.955896i	$-0.405111\pi$
$197$	8.24471	0.587411	0.293706	+	0.955896i	$0.405111\pi$
$198$	0	0
$199$	3.11512	0.220825	0.110412	−	0.993886i	$-0.464783\pi$
$199$	3.11512	0.220825	0.110412	+	0.993886i	$0.464783\pi$
$200$	−1.48314	−0.104874
$201$	0	0
$202$	3.52251	0.247843
$203$	14.9663	1.05043
$204$	0	0
$205$	10.9308	0.763439
$206$	1.56622	0.109124
$207$	0	0
$208$	−6.19584	−0.429604
$209$	9.47247	0.655224
$210$	0	0
$211$	−22.0077	−1.51508	−0.757538	−	0.652791i	$-0.773598\pi$
$211$	−22.0077	−1.51508	−0.757538	+	0.652791i	$0.773598\pi$
$212$	−2.36584	−0.162486
$213$	0	0
$214$	−3.40318	−0.232637
$215$	7.07285	0.482364
$216$	0	0
$217$	8.39765	0.570070
$218$	−8.62324	−0.584039
$219$	0	0
$220$	9.72346	0.655556
$221$	−8.07058	−0.542886
$222$	0	0
$223$	12.3097	0.824321	0.412161	−	0.911111i	$-0.364774\pi$
$223$	12.3097	0.824321	0.412161	+	0.911111i	$0.364774\pi$
$224$	2.08816	0.139521
$225$	0	0
$226$	3.56622	0.237222
$227$	5.33249	0.353930	0.176965	−	0.984217i	$-0.443372\pi$
$227$	5.33249	0.353930	0.176965	+	0.984217i	$0.443372\pi$
$228$	0	0
$229$	−14.1070	−0.932216	−0.466108	−	0.884728i	$-0.654344\pi$
$229$	−14.1070	−0.932216	−0.466108	+	0.884728i	$0.654344\pi$
$230$	0	0
$231$	0	0
$232$	7.16723	0.470551
$233$	−17.1828	−1.12568	−0.562841	−	0.826565i	$-0.690292\pi$
$233$	−17.1828	−1.12568	−0.562841	+	0.826565i	$0.690292\pi$
$234$	0	0
$235$	−23.7853	−1.55158
$236$	−0.611476	−0.0398037
$237$	0	0
$238$	2.71999	0.176311
$239$	5.64042	0.364848	0.182424	−	0.983220i	$-0.441606\pi$
$239$	5.64042	0.364848	0.182424	+	0.983220i	$0.441606\pi$
$240$	0	0
$241$	5.03768	0.324505	0.162253	−	0.986749i	$-0.448124\pi$
$241$	5.03768	0.324505	0.162253	+	0.986749i	$0.448124\pi$
$242$	−3.58334	−0.230346
$243$	0	0
$244$	−4.03969	−0.258615
$245$	−6.72096	−0.429387
$246$	0	0
$247$	−15.3686	−0.977882
$248$	4.02157	0.255370
$249$	0	0
$250$	8.95464	0.566341
$251$	−17.6811	−1.11602	−0.558012	−	0.829833i	$-0.688436\pi$
$251$	−17.6811	−1.11602	−0.558012	+	0.829833i	$0.688436\pi$
$252$	0	0
$253$	0	0
$254$	5.62165	0.352734
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.0654	−1.37640	−0.688202	−	0.725520i	$-0.741600\pi$
$257$	−22.0654	−1.37640	−0.688202	+	0.725520i	$0.741600\pi$
$258$	0	0
$259$	−23.8638	−1.48282
$260$	−15.7759	−0.978377
$261$	0	0
$262$	7.97296	0.492571
$263$	−7.75842	−0.478405	−0.239202	−	0.970970i	$-0.576886\pi$
$263$	−7.75842	−0.478405	−0.239202	+	0.970970i	$0.576886\pi$
$264$	0	0
$265$	−6.02390	−0.370045
$266$	5.17962	0.317583
$267$	0	0
$268$	−12.8258	−0.783463
$269$	−17.0193	−1.03768	−0.518842	−	0.854870i	$-0.673637\pi$
$269$	−17.0193	−1.03768	−0.518842	+	0.854870i	$0.673637\pi$
$270$	0	0
$271$	11.6475	0.707533	0.353767	−	0.935334i	$-0.384901\pi$
$271$	11.6475	0.707533	0.353767	+	0.935334i	$0.384901\pi$
$272$	1.30258	0.0789805
$273$	0	0
$274$	−2.06934	−0.125014
$275$	5.66382	0.341541
$276$	0	0
$277$	16.8299	1.01121	0.505604	−	0.862766i	$-0.331270\pi$
$277$	16.8299	1.01121	0.505604	+	0.862766i	$0.331270\pi$
$278$	13.0620	0.783404
$279$	0	0
$280$	5.31686	0.317743
$281$	−5.29050	−0.315605	−0.157802	−	0.987471i	$-0.550441\pi$
$281$	−5.29050	−0.315605	−0.157802	+	0.987471i	$0.550441\pi$
$282$	0	0
$283$	15.3696	0.913627	0.456814	−	0.889562i	$-0.348991\pi$
$283$	15.3696	0.913627	0.456814	+	0.889562i	$0.348991\pi$
$284$	6.61122	0.392304
$285$	0	0
$286$	23.6608	1.39909
$287$	−8.96441	−0.529152
$288$	0	0
$289$	−15.3033	−0.900193
$290$	18.2492	1.07163
$291$	0	0
$292$	−2.92492	−0.171168
$293$	−10.7552	−0.628323	−0.314161	−	0.949370i	$-0.601723\pi$
$293$	−10.7552	−0.628323	−0.314161	+	0.949370i	$0.601723\pi$
$294$	0	0
$295$	−1.55694	−0.0906486
$296$	−11.4282	−0.664249
$297$	0	0
$298$	−14.8452	−0.859959
$299$	0	0
$300$	0	0
$301$	−5.80049	−0.334335
$302$	−3.85308	−0.221720
$303$	0	0
$304$	2.48047	0.142265
$305$	−10.2859	−0.588967
$306$	0	0
$307$	−25.7849	−1.47162	−0.735810	−	0.677188i	$-0.763199\pi$
$307$	−25.7849	−1.47162	−0.735810	+	0.677188i	$0.763199\pi$
$308$	−7.97428	−0.454377
$309$	0	0
$310$	10.2397	0.581576
$311$	−16.3169	−0.925244	−0.462622	−	0.886556i	$-0.653091\pi$
$311$	−16.3169	−0.925244	−0.462622	+	0.886556i	$0.653091\pi$
$312$	0	0
$313$	−12.5034	−0.706731	−0.353366	−	0.935485i	$-0.614963\pi$
$313$	−12.5034	−0.706731	−0.353366	+	0.935485i	$0.614963\pi$
$314$	22.6511	1.27828
$315$	0	0
$316$	1.88675	0.106138
$317$	13.5387	0.760410	0.380205	−	0.924902i	$-0.375853\pi$
$317$	13.5387	0.760410	0.380205	+	0.924902i	$0.375853\pi$
$318$	0	0
$319$	−27.3703	−1.53244
$320$	2.54620	0.142337
$321$	0	0
$322$	0	0
$323$	3.23101	0.179778
$324$	0	0
$325$	−9.18928	−0.509730
$326$	18.9635	1.05029
$327$	0	0
$328$	−4.29298	−0.237040
$329$	19.5065	1.07543
$330$	0	0
$331$	28.3332	1.55734	0.778668	−	0.627437i	$-0.215896\pi$
$331$	28.3332	1.55734	0.778668	+	0.627437i	$0.215896\pi$
$332$	10.4404	0.572989
$333$	0	0
$334$	−0.979634	−0.0536032
$335$	−32.6572	−1.78425
$336$	0	0
$337$	14.4856	0.789082	0.394541	−	0.918878i	$-0.370904\pi$
$337$	14.4856	0.789082	0.394541	+	0.918878i	$0.370904\pi$
$338$	−25.3885	−1.38095
$339$	0	0
$340$	3.31663	0.179869
$341$	−15.3576	−0.831661
$342$	0	0
$343$	20.1290	1.08686
$344$	−2.77780	−0.149769
$345$	0	0
$346$	−8.92203	−0.479651
$347$	−32.9050	−1.76643	−0.883217	−	0.468964i	$-0.844627\pi$
$347$	−32.9050	−1.76643	−0.883217	+	0.468964i	$0.844627\pi$
$348$	0	0
$349$	−11.3601	−0.608090	−0.304045	−	0.952658i	$-0.598337\pi$
$349$	−11.3601	−0.608090	−0.304045	+	0.952658i	$0.598337\pi$
$350$	3.09702	0.165543
$351$	0	0
$352$	−3.81881	−0.203543
$353$	−1.50436	−0.0800689	−0.0400344	−	0.999198i	$-0.512747\pi$
$353$	−1.50436	−0.0800689	−0.0400344	+	0.999198i	$0.512747\pi$
$354$	0	0
$355$	16.8335	0.893429
$356$	5.01225	0.265649
$357$	0	0
$358$	−8.49152	−0.448791
$359$	20.3544	1.07426	0.537131	−	0.843499i	$-0.319508\pi$
$359$	20.3544	1.07426	0.537131	+	0.843499i	$0.319508\pi$
$360$	0	0
$361$	−12.8472	−0.676171
$362$	−0.537425	−0.0282464
$363$	0	0
$364$	12.9379	0.678130
$365$	−7.44743	−0.389816
$366$	0	0
$367$	12.2425	0.639051	0.319525	−	0.947578i	$-0.396477\pi$
$367$	12.2425	0.639051	0.319525	+	0.947578i	$0.396477\pi$
$368$	0	0
$369$	0	0
$370$	−29.0984	−1.51275
$371$	4.94024	0.256484
$372$	0	0
$373$	−16.4239	−0.850398	−0.425199	−	0.905100i	$-0.639796\pi$
$373$	−16.4239	−0.850398	−0.425199	+	0.905100i	$0.639796\pi$
$374$	−4.97431	−0.257215
$375$	0	0
$376$	9.34150	0.481751
$377$	44.4070	2.28708
$378$	0	0
$379$	25.7579	1.32309	0.661547	−	0.749904i	$-0.269900\pi$
$379$	25.7579	1.32309	0.661547	+	0.749904i	$0.269900\pi$
$380$	6.31578	0.323993
$381$	0	0
$382$	26.8320	1.37284
$383$	−15.0901	−0.771070	−0.385535	−	0.922693i	$-0.625983\pi$
$383$	−15.0901	−0.771070	−0.385535	+	0.922693i	$0.625983\pi$
$384$	0	0
$385$	−20.3041	−1.03479
$386$	1.40434	0.0714792
$387$	0	0
$388$	−12.5459	−0.636924
$389$	22.9846	1.16537	0.582684	−	0.812699i	$-0.302003\pi$
$389$	22.9846	1.16537	0.582684	+	0.812699i	$0.302003\pi$
$390$	0	0
$391$	0	0
$392$	2.63960	0.133320
$393$	0	0
$394$	−8.24471	−0.415363
$395$	4.80405	0.241718
$396$	0	0
$397$	−29.6670	−1.48895	−0.744473	−	0.667653i	$-0.767299\pi$
$397$	−29.6670	−1.48895	−0.744473	+	0.667653i	$0.767299\pi$
$398$	−3.11512	−0.156147
$399$	0	0
$400$	1.48314	0.0741568
$401$	29.2330	1.45983	0.729913	−	0.683540i	$-0.239561\pi$
$401$	29.2330	1.45983	0.729913	+	0.683540i	$0.239561\pi$
$402$	0	0
$403$	24.9170	1.24120
$404$	−3.52251	−0.175252
$405$	0	0
$406$	−14.9663	−0.742764
$407$	43.6420	2.16326
$408$	0	0
$409$	33.6998	1.66635	0.833175	−	0.553009i	$-0.186521\pi$
$409$	33.6998	1.66635	0.833175	+	0.553009i	$0.186521\pi$
$410$	−10.9308	−0.539833
$411$	0	0
$412$	−1.56622	−0.0771623
$413$	1.27686	0.0628301
$414$	0	0
$415$	26.5833	1.30492
$416$	6.19584	0.303776
$417$	0	0
$418$	−9.47247	−0.463314
$419$	8.82612	0.431184	0.215592	−	0.976484i	$-0.430832\pi$
$419$	8.82612	0.431184	0.215592	+	0.976484i	$0.430832\pi$
$420$	0	0
$421$	−27.3740	−1.33413	−0.667063	−	0.745001i	$-0.732449\pi$
$421$	−27.3740	−1.33413	−0.667063	+	0.745001i	$0.732449\pi$
$422$	22.0077	1.07132
$423$	0	0
$424$	2.36584	0.114895
$425$	1.93190	0.0937111
$426$	0	0
$427$	8.43551	0.408223
$428$	3.40318	0.164499
$429$	0	0
$430$	−7.07285	−0.341083
$431$	−29.8688	−1.43873	−0.719365	−	0.694632i	$-0.755567\pi$
$431$	−29.8688	−1.43873	−0.719365	+	0.694632i	$0.755567\pi$
$432$	0	0
$433$	−27.3484	−1.31428	−0.657139	−	0.753769i	$-0.728234\pi$
$433$	−27.3484	−1.31428	−0.657139	+	0.753769i	$0.728234\pi$
$434$	−8.39765	−0.403100
$435$	0	0
$436$	8.62324	0.412978
$437$	0	0
$438$	0	0
$439$	−13.1196	−0.626164	−0.313082	−	0.949726i	$-0.601361\pi$
$439$	−13.1196	−0.626164	−0.313082	+	0.949726i	$0.601361\pi$
$440$	−9.72346	−0.463548
$441$	0	0
$442$	8.07058	0.383878
$443$	10.7702	0.511706	0.255853	−	0.966716i	$-0.417644\pi$
$443$	10.7702	0.511706	0.255853	+	0.966716i	$0.417644\pi$
$444$	0	0
$445$	12.7622	0.604986
$446$	−12.3097	−0.582883
$447$	0	0
$448$	−2.08816	−0.0986561
$449$	−4.69291	−0.221472	−0.110736	−	0.993850i	$-0.535321\pi$
$449$	−4.69291	−0.221472	−0.110736	+	0.993850i	$0.535321\pi$
$450$	0	0
$451$	16.3941	0.771968
$452$	−3.56622	−0.167741
$453$	0	0
$454$	−5.33249	−0.250266
$455$	32.9425	1.54437
$456$	0	0
$457$	−6.93568	−0.324437	−0.162219	−	0.986755i	$-0.551865\pi$
$457$	−6.93568	−0.324437	−0.162219	+	0.986755i	$0.551865\pi$
$458$	14.1070	0.659176
$459$	0	0
$460$	0	0
$461$	21.3273	0.993312	0.496656	−	0.867947i	$-0.334561\pi$
$461$	21.3273	0.993312	0.496656	+	0.867947i	$0.334561\pi$
$462$	0	0
$463$	−6.41490	−0.298126	−0.149063	−	0.988828i	$-0.547626\pi$
$463$	−6.41490	−0.298126	−0.149063	+	0.988828i	$0.547626\pi$
$464$	−7.16723	−0.332730
$465$	0	0
$466$	17.1828	0.795977
$467$	14.4723	0.669696	0.334848	−	0.942272i	$-0.391315\pi$
$467$	14.4723	0.669696	0.334848	+	0.942272i	$0.391315\pi$
$468$	0	0
$469$	26.7824	1.23670
$470$	23.7853	1.09714
$471$	0	0
$472$	0.611476	0.0281455
$473$	10.6079	0.487753
$474$	0	0
$475$	3.67888	0.168799
$476$	−2.71999	−0.124671
$477$	0	0
$478$	−5.64042	−0.257987
$479$	−5.37524	−0.245601	−0.122801	−	0.992431i	$-0.539188\pi$
$479$	−5.37524	−0.245601	−0.122801	+	0.992431i	$0.539188\pi$
$480$	0	0
$481$	−70.8071	−3.22853
$482$	−5.03768	−0.229460
$483$	0	0
$484$	3.58334	0.162879
$485$	−31.9445	−1.45052
$486$	0	0
$487$	3.21931	0.145881	0.0729405	−	0.997336i	$-0.476762\pi$
$487$	3.21931	0.145881	0.0729405	+	0.997336i	$0.476762\pi$
$488$	4.03969	0.182868
$489$	0	0
$490$	6.72096	0.303622
$491$	−30.0754	−1.35728	−0.678642	−	0.734470i	$-0.737431\pi$
$491$	−30.0754	−1.35728	−0.678642	+	0.734470i	$0.737431\pi$
$492$	0	0
$493$	−9.33588	−0.420467
$494$	15.3686	0.691467
$495$	0	0
$496$	−4.02157	−0.180574
$497$	−13.8053	−0.619251
$498$	0	0
$499$	17.7485	0.794533	0.397266	−	0.917703i	$-0.369959\pi$
$499$	17.7485	0.794533	0.397266	+	0.917703i	$0.369959\pi$
$500$	−8.95464	−0.400464
$501$	0	0
$502$	17.6811	0.789148
$503$	24.7648	1.10421	0.552105	−	0.833775i	$-0.313825\pi$
$503$	24.7648	1.10421	0.552105	+	0.833775i	$0.313825\pi$
$504$	0	0
$505$	−8.96902	−0.399116
$506$	0	0
$507$	0	0
$508$	−5.62165	−0.249421
$509$	31.0419	1.37591	0.687953	−	0.725755i	$-0.258509\pi$
$509$	31.0419	1.37591	0.687953	+	0.725755i	$0.258509\pi$
$510$	0	0
$511$	6.10769	0.270188
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	22.0654	0.973264
$515$	−3.98792	−0.175729
$516$	0	0
$517$	−35.6735	−1.56892
$518$	23.8638	1.04851
$519$	0	0
$520$	15.7759	0.691817
$521$	33.3116	1.45941	0.729703	−	0.683764i	$-0.239658\pi$
$521$	33.3116	1.45941	0.729703	+	0.683764i	$0.239658\pi$
$522$	0	0
$523$	−31.0406	−1.35731	−0.678655	−	0.734457i	$-0.737437\pi$
$523$	−31.0406	−1.35731	−0.678655	+	0.734457i	$0.737437\pi$
$524$	−7.97296	−0.348301
$525$	0	0
$526$	7.75842	0.338283
$527$	−5.23841	−0.228189
$528$	0	0
$529$	0	0
$530$	6.02390	0.261661
$531$	0	0
$532$	−5.17962	−0.224565
$533$	−26.5986	−1.15211
$534$	0	0
$535$	8.66518	0.374628
$536$	12.8258	0.553992
$537$	0	0
$538$	17.0193	0.733754
$539$	−10.0802	−0.434183
$540$	0	0
$541$	−5.14103	−0.221030	−0.110515	−	0.993874i	$-0.535250\pi$
$541$	−5.14103	−0.221030	−0.110515	+	0.993874i	$0.535250\pi$
$542$	−11.6475	−0.500301
$543$	0	0
$544$	−1.30258	−0.0558476
$545$	21.9565	0.940513
$546$	0	0
$547$	19.0128	0.812930	0.406465	−	0.913666i	$-0.366761\pi$
$547$	19.0128	0.812930	0.406465	+	0.913666i	$0.366761\pi$
$548$	2.06934	0.0883979
$549$	0	0
$550$	−5.66382	−0.241506
$551$	−17.7781	−0.757373
$552$	0	0
$553$	−3.93983	−0.167539
$554$	−16.8299	−0.715032
$555$	0	0
$556$	−13.0620	−0.553950
$557$	12.0728	0.511541	0.255770	−	0.966738i	$-0.417671\pi$
$557$	12.0728	0.511541	0.255770	+	0.966738i	$0.417671\pi$
$558$	0	0
$559$	−17.2108	−0.727941
$560$	−5.31686	−0.224678
$561$	0	0
$562$	5.29050	0.223166
$563$	−21.2359	−0.894988	−0.447494	−	0.894287i	$-0.647683\pi$
$563$	−21.2359	−0.894988	−0.447494	+	0.894287i	$0.647683\pi$
$564$	0	0
$565$	−9.08032	−0.382012
$566$	−15.3696	−0.646032
$567$	0	0
$568$	−6.61122	−0.277401
$569$	11.7217	0.491401	0.245700	−	0.969346i	$-0.420982\pi$
$569$	11.7217	0.491401	0.245700	+	0.969346i	$0.420982\pi$
$570$	0	0
$571$	44.3277	1.85506	0.927529	−	0.373751i	$-0.121929\pi$
$571$	44.3277	1.85506	0.927529	+	0.373751i	$0.121929\pi$
$572$	−23.6608	−0.989307
$573$	0	0
$574$	8.96441	0.374167
$575$	0	0
$576$	0	0
$577$	29.9213	1.24564	0.622819	−	0.782366i	$-0.285987\pi$
$577$	29.9213	1.24564	0.622819	+	0.782366i	$0.285987\pi$
$578$	15.3033	0.636533
$579$	0	0
$580$	−18.2492	−0.757756
$581$	−21.8011	−0.904462
$582$	0	0
$583$	−9.03470	−0.374179
$584$	2.92492	0.121034
$585$	0	0
$586$	10.7552	0.444291
$587$	−10.5653	−0.436077	−0.218038	−	0.975940i	$-0.569966\pi$
$587$	−10.5653	−0.436077	−0.218038	+	0.975940i	$0.569966\pi$
$588$	0	0
$589$	−9.97539	−0.411029
$590$	1.55694	0.0640983
$591$	0	0
$592$	11.4282	0.469695
$593$	−8.64892	−0.355169	−0.177584	−	0.984106i	$-0.556828\pi$
$593$	−8.64892	−0.355169	−0.177584	+	0.984106i	$0.556828\pi$
$594$	0	0
$595$	−6.92564	−0.283923
$596$	14.8452	0.608083
$597$	0	0
$598$	0	0
$599$	−20.5800	−0.840877	−0.420439	−	0.907321i	$-0.638124\pi$
$599$	−20.5800	−0.840877	−0.420439	+	0.907321i	$0.638124\pi$
$600$	0	0
$601$	17.6734	0.720914	0.360457	−	0.932776i	$-0.382621\pi$
$601$	17.6734	0.720914	0.360457	+	0.932776i	$0.382621\pi$
$602$	5.80049	0.236410
$603$	0	0
$604$	3.85308	0.156780
$605$	9.12390	0.370939
$606$	0	0
$607$	−41.9751	−1.70372	−0.851859	−	0.523771i	$-0.824525\pi$
$607$	−41.9751	−1.70372	−0.851859	+	0.523771i	$0.824525\pi$
$608$	−2.48047	−0.100596
$609$	0	0
$610$	10.2859	0.416463
$611$	57.8785	2.34151
$612$	0	0
$613$	−17.9340	−0.724348	−0.362174	−	0.932110i	$-0.617965\pi$
$613$	−17.9340	−0.724348	−0.362174	+	0.932110i	$0.617965\pi$
$614$	25.7849	1.04059
$615$	0	0
$616$	7.97428	0.321293
$617$	2.95899	0.119124	0.0595622	−	0.998225i	$-0.481030\pi$
$617$	2.95899	0.119124	0.0595622	+	0.998225i	$0.481030\pi$
$618$	0	0
$619$	−28.0180	−1.12614	−0.563070	−	0.826409i	$-0.690380\pi$
$619$	−28.0180	−1.12614	−0.563070	+	0.826409i	$0.690380\pi$
$620$	−10.2397	−0.411237
$621$	0	0
$622$	16.3169	0.654247
$623$	−10.4664	−0.419326
$624$	0	0
$625$	−30.2160	−1.20864
$626$	12.5034	0.499735
$627$	0	0
$628$	−22.6511	−0.903878
$629$	14.8861	0.593548
$630$	0	0
$631$	2.43747	0.0970340	0.0485170	−	0.998822i	$-0.484550\pi$
$631$	2.43747	0.0970340	0.0485170	+	0.998822i	$0.484550\pi$
$632$	−1.88675	−0.0750510
$633$	0	0
$634$	−13.5387	−0.537691
$635$	−14.3139	−0.568028
$636$	0	0
$637$	16.3546	0.647992
$638$	27.3703	1.08360
$639$	0	0
$640$	−2.54620	−0.100647
$641$	3.20307	0.126514	0.0632569	−	0.997997i	$-0.479851\pi$
$641$	3.20307	0.126514	0.0632569	+	0.997997i	$0.479851\pi$
$642$	0	0
$643$	−35.8609	−1.41422	−0.707108	−	0.707105i	$-0.750001\pi$
$643$	−35.8609	−1.41422	−0.707108	+	0.707105i	$0.750001\pi$
$644$	0	0
$645$	0	0
$646$	−3.23101	−0.127123
$647$	−9.86585	−0.387866	−0.193933	−	0.981015i	$-0.562125\pi$
$647$	−9.86585	−0.387866	−0.193933	+	0.981015i	$0.562125\pi$
$648$	0	0
$649$	−2.33511	−0.0916613
$650$	9.18928	0.360433
$651$	0	0
$652$	−18.9635	−0.742667
$653$	12.7863	0.500367	0.250183	−	0.968198i	$-0.419509\pi$
$653$	12.7863	0.500367	0.250183	+	0.968198i	$0.419509\pi$
$654$	0	0
$655$	−20.3008	−0.793216
$656$	4.29298	0.167613
$657$	0	0
$658$	−19.5065	−0.760443
$659$	−5.10798	−0.198979	−0.0994893	−	0.995039i	$-0.531721\pi$
$659$	−5.10798	−0.198979	−0.0994893	+	0.995039i	$0.531721\pi$
$660$	0	0
$661$	26.9524	1.04833	0.524163	−	0.851618i	$-0.324378\pi$
$661$	26.9524	1.04833	0.524163	+	0.851618i	$0.324378\pi$
$662$	−28.3332	−1.10120
$663$	0	0
$664$	−10.4404	−0.405165
$665$	−13.1883	−0.511422
$666$	0	0
$667$	0	0
$668$	0.979634	0.0379032
$669$	0	0
$670$	32.6572	1.26166
$671$	−15.4268	−0.595547
$672$	0	0
$673$	27.1196	1.04538	0.522691	−	0.852522i	$-0.324928\pi$
$673$	27.1196	1.04538	0.522691	+	0.852522i	$0.324928\pi$
$674$	−14.4856	−0.557965
$675$	0	0
$676$	25.3885	0.976480
$677$	28.2117	1.08426	0.542132	−	0.840294i	$-0.317617\pi$
$677$	28.2117	1.08426	0.542132	+	0.840294i	$0.317617\pi$
$678$	0	0
$679$	26.1979	1.00538
$680$	−3.31663	−0.127187
$681$	0	0
$682$	15.3576	0.588073
$683$	2.26699	0.0867439	0.0433720	−	0.999059i	$-0.486190\pi$
$683$	2.26699	0.0867439	0.0433720	+	0.999059i	$0.486190\pi$
$684$	0	0
$685$	5.26896	0.201317
$686$	−20.1290	−0.768529
$687$	0	0
$688$	2.77780	0.105903
$689$	14.6584	0.558439
$690$	0	0
$691$	2.62969	0.100038	0.0500191	−	0.998748i	$-0.484072\pi$
$691$	2.62969	0.100038	0.0500191	+	0.998748i	$0.484072\pi$
$692$	8.92203	0.339165
$693$	0	0
$694$	32.9050	1.24906
$695$	−33.2584	−1.26156
$696$	0	0
$697$	5.59195	0.211810
$698$	11.3601	0.429985
$699$	0	0
$700$	−3.09702	−0.117056
$701$	−17.2208	−0.650422	−0.325211	−	0.945641i	$-0.605435\pi$
$701$	−17.2208	−0.650422	−0.325211	+	0.945641i	$0.605435\pi$
$702$	0	0
$703$	28.3473	1.06914
$704$	3.81881	0.143927
$705$	0	0
$706$	1.50436	0.0566172
$707$	7.35555	0.276634
$708$	0	0
$709$	−36.1353	−1.35709	−0.678544	−	0.734559i	$-0.737389\pi$
$709$	−36.1353	−1.35709	−0.678544	+	0.734559i	$0.737389\pi$
$710$	−16.8335	−0.631750
$711$	0	0
$712$	−5.01225	−0.187842
$713$	0	0
$714$	0	0
$715$	−60.2451	−2.25304
$716$	8.49152	0.317343
$717$	0	0
$718$	−20.3544	−0.759618
$719$	−28.0142	−1.04476	−0.522378	−	0.852714i	$-0.674955\pi$
$719$	−28.0142	−1.04476	−0.522378	+	0.852714i	$0.674955\pi$
$720$	0	0
$721$	3.27052	0.121800
$722$	12.8472	0.478125
$723$	0	0
$724$	0.537425	0.0199732
$725$	−10.6300	−0.394787
$726$	0	0
$727$	−48.6681	−1.80500	−0.902501	−	0.430688i	$-0.858271\pi$
$727$	−48.6681	−1.80500	−0.902501	+	0.430688i	$0.858271\pi$
$728$	−12.9379	−0.479510
$729$	0	0
$730$	7.44743	0.275642
$731$	3.61831	0.133828
$732$	0	0
$733$	3.42952	0.126672	0.0633362	−	0.997992i	$-0.479826\pi$
$733$	3.42952	0.126672	0.0633362	+	0.997992i	$0.479826\pi$
$734$	−12.2425	−0.451877
$735$	0	0
$736$	0	0
$737$	−48.9795	−1.80418
$738$	0	0
$739$	−12.1486	−0.446894	−0.223447	−	0.974716i	$-0.571731\pi$
$739$	−12.1486	−0.446894	−0.223447	+	0.974716i	$0.571731\pi$
$740$	29.0984	1.06968
$741$	0	0
$742$	−4.94024	−0.181362
$743$	−44.9522	−1.64914	−0.824568	−	0.565763i	$-0.808582\pi$
$743$	−44.9522	−1.64914	−0.824568	+	0.565763i	$0.808582\pi$
$744$	0	0
$745$	37.7989	1.38484
$746$	16.4239	0.601322
$747$	0	0
$748$	4.97431	0.181879
$749$	−7.10637	−0.259661
$750$	0	0
$751$	−6.17883	−0.225469	−0.112734	−	0.993625i	$-0.535961\pi$
$751$	−6.17883	−0.225469	−0.112734	+	0.993625i	$0.535961\pi$
$752$	−9.34150	−0.340650
$753$	0	0
$754$	−44.4070	−1.61721
$755$	9.81071	0.357048
$756$	0	0
$757$	−13.7458	−0.499599	−0.249799	−	0.968298i	$-0.580365\pi$
$757$	−13.7458	−0.499599	−0.249799	+	0.968298i	$0.580365\pi$
$758$	−25.7579	−0.935569
$759$	0	0
$760$	−6.31578	−0.229098
$761$	51.1112	1.85278	0.926390	−	0.376566i	$-0.122895\pi$
$761$	51.1112	1.85278	0.926390	+	0.376566i	$0.122895\pi$
$762$	0	0
$763$	−18.0067	−0.651885
$764$	−26.8320	−0.970747
$765$	0	0
$766$	15.0901	0.545229
$767$	3.78861	0.136799
$768$	0	0
$769$	−15.6086	−0.562861	−0.281430	−	0.959582i	$-0.590809\pi$
$769$	−15.6086	−0.562861	−0.281430	+	0.959582i	$0.590809\pi$
$770$	20.3041	0.731709
$771$	0	0
$772$	−1.40434	−0.0505434
$773$	−2.63936	−0.0949310	−0.0474655	−	0.998873i	$-0.515114\pi$
$773$	−2.63936	−0.0949310	−0.0474655	+	0.998873i	$0.515114\pi$
$774$	0	0
$775$	−5.96453	−0.214252
$776$	12.5459	0.450373
$777$	0	0
$778$	−22.9846	−0.824040
$779$	10.6486	0.381527
$780$	0	0
$781$	25.2470	0.903409
$782$	0	0
$783$	0	0
$784$	−2.63960	−0.0942716
$785$	−57.6743	−2.05848
$786$	0	0
$787$	−36.5824	−1.30402	−0.652011	−	0.758210i	$-0.726074\pi$
$787$	−36.5824	−1.30402	−0.652011	+	0.758210i	$0.726074\pi$
$788$	8.24471	0.293706
$789$	0	0
$790$	−4.80405	−0.170920
$791$	7.44683	0.264779
$792$	0	0
$793$	25.0293	0.888817
$794$	29.6670	1.05284
$795$	0	0
$796$	3.11512	0.110412
$797$	−19.6922	−0.697534	−0.348767	−	0.937210i	$-0.613400\pi$
$797$	−19.6922	−0.697534	−0.348767	+	0.937210i	$0.613400\pi$
$798$	0	0
$799$	−12.1681	−0.430475
$800$	−1.48314	−0.0524368
$801$	0	0
$802$	−29.2330	−1.03225
$803$	−11.1697	−0.394171
$804$	0	0
$805$	0	0
$806$	−24.9170	−0.877664
$807$	0	0
$808$	3.52251	0.123922
$809$	13.5968	0.478038	0.239019	−	0.971015i	$-0.423174\pi$
$809$	13.5968	0.478038	0.239019	+	0.971015i	$0.423174\pi$
$810$	0	0
$811$	−22.6677	−0.795970	−0.397985	−	0.917392i	$-0.630290\pi$
$811$	−22.6677	−0.795970	−0.397985	+	0.917392i	$0.630290\pi$
$812$	14.9663	0.525214
$813$	0	0
$814$	−43.6420	−1.52965
$815$	−48.2848	−1.69134
$816$	0	0
$817$	6.89027	0.241060
$818$	−33.6998	−1.17829
$819$	0	0
$820$	10.9308	0.381720
$821$	16.4984	0.575799	0.287900	−	0.957661i	$-0.407043\pi$
$821$	16.4984	0.575799	0.287900	+	0.957661i	$0.407043\pi$
$822$	0	0
$823$	−30.5690	−1.06557	−0.532785	−	0.846251i	$-0.678854\pi$
$823$	−30.5690	−1.06557	−0.532785	+	0.846251i	$0.678854\pi$
$824$	1.56622	0.0545620
$825$	0	0
$826$	−1.27686	−0.0444276
$827$	24.1171	0.838633	0.419316	−	0.907840i	$-0.362270\pi$
$827$	24.1171	0.838633	0.419316	+	0.907840i	$0.362270\pi$
$828$	0	0
$829$	−11.3407	−0.393879	−0.196939	−	0.980416i	$-0.563100\pi$
$829$	−11.3407	−0.393879	−0.196939	+	0.980416i	$0.563100\pi$
$830$	−26.5833	−0.922718
$831$	0	0
$832$	−6.19584	−0.214802
$833$	−3.43830	−0.119130
$834$	0	0
$835$	2.49434	0.0863203
$836$	9.47247	0.327612
$837$	0	0
$838$	−8.82612	−0.304893
$839$	−18.8321	−0.650157	−0.325079	−	0.945687i	$-0.605391\pi$
$839$	−18.8321	−0.650157	−0.325079	+	0.945687i	$0.605391\pi$
$840$	0	0
$841$	22.3691	0.771349
$842$	27.3740	0.943370
$843$	0	0
$844$	−22.0077	−0.757538
$845$	64.6442	2.22383
$846$	0	0
$847$	−7.48257	−0.257104
$848$	−2.36584	−0.0812432
$849$	0	0
$850$	−1.93190	−0.0662637
$851$	0	0
$852$	0	0
$853$	2.46467	0.0843887	0.0421943	−	0.999109i	$-0.486565\pi$
$853$	2.46467	0.0843887	0.0421943	+	0.999109i	$0.486565\pi$
$854$	−8.43551	−0.288657
$855$	0	0
$856$	−3.40318	−0.116318
$857$	15.2700	0.521614	0.260807	−	0.965391i	$-0.416011\pi$
$857$	15.2700	0.521614	0.260807	+	0.965391i	$0.416011\pi$
$858$	0	0
$859$	−3.96389	−0.135246	−0.0676231	−	0.997711i	$-0.521542\pi$
$859$	−3.96389	−0.135246	−0.0676231	+	0.997711i	$0.521542\pi$
$860$	7.07285	0.241182
$861$	0	0
$862$	29.8688	1.01734
$863$	−7.20518	−0.245267	−0.122634	−	0.992452i	$-0.539134\pi$
$863$	−7.20518	−0.245267	−0.122634	+	0.992452i	$0.539134\pi$
$864$	0	0
$865$	22.7173	0.772411
$866$	27.3484	0.929335
$867$	0	0
$868$	8.39765	0.285035
$869$	7.20516	0.244418
$870$	0	0
$871$	79.4670	2.69264
$872$	−8.62324	−0.292020
$873$	0	0
$874$	0	0
$875$	18.6987	0.632131
$876$	0	0
$877$	−46.6196	−1.57423	−0.787116	−	0.616805i	$-0.788427\pi$
$877$	−46.6196	−1.57423	−0.787116	+	0.616805i	$0.788427\pi$
$878$	13.1196	0.442765
$879$	0	0
$880$	9.72346	0.327778
$881$	1.12202	0.0378019	0.0189009	−	0.999821i	$-0.493983\pi$
$881$	1.12202	0.0378019	0.0189009	+	0.999821i	$0.493983\pi$
$882$	0	0
$883$	−2.12015	−0.0713488	−0.0356744	−	0.999363i	$-0.511358\pi$
$883$	−2.12015	−0.0713488	−0.0356744	+	0.999363i	$0.511358\pi$
$884$	−8.07058	−0.271443
$885$	0	0
$886$	−10.7702	−0.361831
$887$	−15.6021	−0.523869	−0.261934	−	0.965086i	$-0.584360\pi$
$887$	−15.6021	−0.523869	−0.261934	+	0.965086i	$0.584360\pi$
$888$	0	0
$889$	11.7389	0.393710
$890$	−12.7622	−0.427790
$891$	0	0
$892$	12.3097	0.412161
$893$	−23.1714	−0.775400
$894$	0	0
$895$	21.6211	0.722714
$896$	2.08816	0.0697604
$897$	0	0
$898$	4.69291	0.156605
$899$	28.8235	0.961316
$900$	0	0
$901$	−3.08169	−0.102666
$902$	−16.3941	−0.545863
$903$	0	0
$904$	3.56622	0.118611
$905$	1.36839	0.0454869
$906$	0	0
$907$	30.4920	1.01247	0.506235	−	0.862395i	$-0.331037\pi$
$907$	30.4920	1.01247	0.506235	+	0.862395i	$0.331037\pi$
$908$	5.33249	0.176965
$909$	0	0
$910$	−32.9425	−1.09203
$911$	−35.8245	−1.18692	−0.593460	−	0.804864i	$-0.702238\pi$
$911$	−35.8245	−1.18692	−0.593460	+	0.804864i	$0.702238\pi$
$912$	0	0
$913$	39.8698	1.31950
$914$	6.93568	0.229412
$915$	0	0
$916$	−14.1070	−0.466108
$917$	16.6488	0.549792
$918$	0	0
$919$	−29.8477	−0.984585	−0.492292	−	0.870430i	$-0.663841\pi$
$919$	−29.8477	−0.984585	−0.492292	+	0.870430i	$0.663841\pi$
$920$	0	0
$921$	0	0
$922$	−21.3273	−0.702378
$923$	−40.9621	−1.34828
$924$	0	0
$925$	16.9495	0.557297
$926$	6.41490	0.210807
$927$	0	0
$928$	7.16723	0.235276
$929$	39.7085	1.30279	0.651397	−	0.758737i	$-0.274183\pi$
$929$	39.7085	1.30279	0.651397	+	0.758737i	$0.274183\pi$
$930$	0	0
$931$	−6.54747	−0.214585
$932$	−17.1828	−0.562841
$933$	0	0
$934$	−14.4723	−0.473547
$935$	12.6656	0.414209
$936$	0	0
$937$	−20.8680	−0.681728	−0.340864	−	0.940113i	$-0.610720\pi$
$937$	−20.8680	−0.681728	−0.340864	+	0.940113i	$0.610720\pi$
$938$	−26.7824	−0.874475
$939$	0	0
$940$	−23.7853	−0.775792
$941$	−35.1554	−1.14603	−0.573017	−	0.819543i	$-0.694227\pi$
$941$	−35.1554	−1.14603	−0.573017	+	0.819543i	$0.694227\pi$
$942$	0	0
$943$	0	0
$944$	−0.611476	−0.0199019
$945$	0	0
$946$	−10.6079	−0.344893
$947$	−28.6519	−0.931062	−0.465531	−	0.885032i	$-0.654137\pi$
$947$	−28.6519	−0.931062	−0.465531	+	0.885032i	$0.654137\pi$
$948$	0	0
$949$	18.1223	0.588276
$950$	−3.67888	−0.119359
$951$	0	0
$952$	2.71999	0.0881554
$953$	−19.5950	−0.634745	−0.317373	−	0.948301i	$-0.602801\pi$
$953$	−19.5950	−0.634745	−0.317373	+	0.948301i	$0.602801\pi$
$954$	0	0
$955$	−68.3196	−2.21077
$956$	5.64042	0.182424
$957$	0	0
$958$	5.37524	0.173666
$959$	−4.32111	−0.139536
$960$	0	0
$961$	−14.8270	−0.478291
$962$	70.8071	2.28291
$963$	0	0
$964$	5.03768	0.162253
$965$	−3.57574	−0.115107
$966$	0	0
$967$	−45.4032	−1.46007	−0.730035	−	0.683410i	$-0.760496\pi$
$967$	−45.4032	−1.46007	−0.730035	+	0.683410i	$0.760496\pi$
$968$	−3.58334	−0.115173
$969$	0	0
$970$	31.9445	1.02568
$971$	38.4219	1.23302	0.616509	−	0.787348i	$-0.288547\pi$
$971$	38.4219	1.23302	0.616509	+	0.787348i	$0.288547\pi$
$972$	0	0
$973$	27.2754	0.874409
$974$	−3.21931	−0.103153
$975$	0	0
$976$	−4.03969	−0.129307
$977$	−1.16775	−0.0373596	−0.0186798	−	0.999826i	$-0.505946\pi$
$977$	−1.16775	−0.0373596	−0.0186798	+	0.999826i	$0.505946\pi$
$978$	0	0
$979$	19.1409	0.611744
$980$	−6.72096	−0.214693
$981$	0	0
$982$	30.0754	0.959744
$983$	−55.6335	−1.77443	−0.887216	−	0.461355i	$-0.847364\pi$
$983$	−55.6335	−1.77443	−0.887216	+	0.461355i	$0.847364\pi$
$984$	0	0
$985$	20.9927	0.668883
$986$	9.33588	0.297315
$987$	0	0
$988$	−15.3686	−0.488941
$989$	0	0
$990$	0	0
$991$	29.3329	0.931792	0.465896	−	0.884840i	$-0.345732\pi$
$991$	29.3329	0.931792	0.465896	+	0.884840i	$0.345732\pi$
$992$	4.02157	0.127685
$993$	0	0
$994$	13.8053	0.437876
$995$	7.93171	0.251452
$996$	0	0
$997$	22.8243	0.722852	0.361426	−	0.932401i	$-0.382290\pi$
$997$	22.8243	0.722852	0.361426	+	0.932401i	$0.382290\pi$
$998$	−17.7485	−0.561819
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.bu.1.4		5
3.2	odd	2		1058.2.a.l.1.5		5
12.11	even	2		8464.2.a.bw.1.1		5
23.7	odd	22		414.2.i.f.325.1		10
23.10	odd	22		414.2.i.f.307.1		10
23.22	odd	2		9522.2.a.bp.1.2		5
69.53	even	22		46.2.c.a.3.1	✓	10
69.56	even	22		46.2.c.a.31.1	yes	10
69.68	even	2		1058.2.a.m.1.5		5
276.191	odd	22		368.2.m.b.49.1		10
276.263	odd	22		368.2.m.b.353.1		10
276.275	odd	2		8464.2.a.bx.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.2.c.a.3.1	✓	10	69.53	even	22
46.2.c.a.31.1	yes	10	69.56	even	22
368.2.m.b.49.1		10	276.191	odd	22
368.2.m.b.353.1		10	276.263	odd	22
414.2.i.f.307.1		10	23.10	odd	22
414.2.i.f.325.1		10	23.7	odd	22
1058.2.a.l.1.5		5	3.2	odd	2
1058.2.a.m.1.5		5	69.68	even	2
8464.2.a.bw.1.1		5	12.11	even	2
8464.2.a.bx.1.1		5	276.275	odd	2
9522.2.a.bp.1.2		5	23.22	odd	2
9522.2.a.bu.1.4		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.54620 q^{5} -2.08816 q^{7} -1.00000 q^{8} -2.54620 q^{10} +3.81881 q^{11} -6.19584 q^{13} +2.08816 q^{14} +1.00000 q^{16} +1.30258 q^{17} +2.48047 q^{19} +2.54620 q^{20} -3.81881 q^{22} +1.48314 q^{25} +6.19584 q^{26} -2.08816 q^{28} -7.16723 q^{29} -4.02157 q^{31} -1.00000 q^{32} -1.30258 q^{34} -5.31686 q^{35} +11.4282 q^{37} -2.48047 q^{38} -2.54620 q^{40} +4.29298 q^{41} +2.77780 q^{43} +3.81881 q^{44} -9.34150 q^{47} -2.63960 q^{49} -1.48314 q^{50} -6.19584 q^{52} -2.36584 q^{53} +9.72346 q^{55} +2.08816 q^{56} +7.16723 q^{58} -0.611476 q^{59} -4.03969 q^{61} +4.02157 q^{62} +1.00000 q^{64} -15.7759 q^{65} -12.8258 q^{67} +1.30258 q^{68} +5.31686 q^{70} +6.61122 q^{71} -2.92492 q^{73} -11.4282 q^{74} +2.48047 q^{76} -7.97428 q^{77} +1.88675 q^{79} +2.54620 q^{80} -4.29298 q^{82} +10.4404 q^{83} +3.31663 q^{85} -2.77780 q^{86} -3.81881 q^{88} +5.01225 q^{89} +12.9379 q^{91} +9.34150 q^{94} +6.31578 q^{95} -12.5459 q^{97} +2.63960 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} + 8 q^{5} - 7 q^{7} - 5 q^{8} - 8 q^{10} + 5 q^{11} - 7 q^{13} + 7 q^{14} + 5 q^{16} + 13 q^{17} - 12 q^{19} + 8 q^{20} - 5 q^{22} + q^{25} + 7 q^{26} - 7 q^{28} + 4 q^{29} + 6 q^{31}+ \cdots + 12 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 + 8 * q^5 - 7 * q^7 - 5 * q^8 - 8 * q^10 + 5 * q^11 - 7 * q^13 + 7 * q^14 + 5 * q^16 + 13 * q^17 - 12 * q^19 + 8 * q^20 - 5 * q^22 + q^25 + 7 * q^26 - 7 * q^28 + 4 * q^29 + 6 * q^31 - 5 * q^32 - 13 * q^34 - 9 * q^35 - 14 * q^37 + 12 * q^38 - 8 * q^40 - q^41 - 11 * q^43 + 5 * q^44 - 9 * q^47 - 12 * q^49 - q^50 - 7 * q^52 + 6 * q^53 - 14 * q^55 + 7 * q^56 - 4 * q^58 - 3 * q^59 - 13 * q^61 - 6 * q^62 + 5 * q^64 - 9 * q^65 - 10 * q^67 + 13 * q^68 + 9 * q^70 - 8 * q^71 - 4 * q^73 + 14 * q^74 - 12 * q^76 - 18 * q^77 + 18 * q^79 + 8 * q^80 + q^82 + 2 * q^83 + 23 * q^85 + 11 * q^86 - 5 * q^88 + 10 * q^89 + 12 * q^91 + 9 * q^94 - 28 * q^95 - 17 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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