LMFDB - Embedded newform 9522.2.a.cc.1.3

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 = [6,6,0,6,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,18,0,0,0,24] 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:	$6$
Coefficient field:	6.6.197448192.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 18x^{4} - 6x^{3} + 48x^{2} - 23 $ x^6 - 18x^4 - 6x^3 + 48*x^2 - 23
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.890237$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.741015 q^{5} +4.52757 q^{7} +1.00000 q^{8} -0.741015 q^{10} +3.85437 q^{11} -1.04795 q^{13} +4.52757 q^{14} +1.00000 q^{16} +3.11335 q^{17} -0.673199 q^{19} -0.741015 q^{20} +3.85437 q^{22} -4.45090 q^{25} -1.04795 q^{26} +4.52757 q^{28} +9.35499 q^{29} +5.45090 q^{31} +1.00000 q^{32} +3.11335 q^{34} -3.35499 q^{35} +0.958124 q^{37} -0.673199 q^{38} -0.741015 q^{40} -12.4989 q^{41} -3.63726 q^{43} +3.85437 q^{44} +10.4029 q^{47} +13.4989 q^{49} -4.45090 q^{50} -1.04795 q^{52} -5.48569 q^{53} -2.85614 q^{55} +4.52757 q^{56} +9.35499 q^{58} -8.80589 q^{59} +2.30452 q^{61} +5.45090 q^{62} +1.00000 q^{64} +0.776548 q^{65} +10.6405 q^{67} +3.11335 q^{68} -3.35499 q^{70} +10.4029 q^{71} +4.95205 q^{73} +0.958124 q^{74} -0.673199 q^{76} +17.4509 q^{77} -3.18117 q^{79} -0.741015 q^{80} -12.4989 q^{82} +10.8576 q^{83} -2.30704 q^{85} -3.63726 q^{86} +3.85437 q^{88} +4.59538 q^{89} -4.74468 q^{91} +10.4029 q^{94} +0.498850 q^{95} -7.64092 q^{97} +13.4989 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 6 q^{16} + 18 q^{25} + 24 q^{29} - 12 q^{31} + 6 q^{32} + 12 q^{35} - 24 q^{41} + 24 q^{47} + 30 q^{49} + 18 q^{50} - 36 q^{55} + 24 q^{58} + 24 q^{59} - 12 q^{62} + 6 q^{64}+ \cdots + 30 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 6 * q^16 + 18 * q^25 + 24 * q^29 - 12 * q^31 + 6 * q^32 + 12 * q^35 - 24 * q^41 + 24 * q^47 + 30 * q^49 + 18 * q^50 - 36 * q^55 + 24 * q^58 + 24 * q^59 - 12 * q^62 + 6 * q^64 + 12 * q^70 + 24 * q^71 + 36 * q^73 + 60 * q^77 - 24 * q^82 + 12 * q^85 + 24 * q^94 - 48 * q^95 + 30 * 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
$5$	−0.741015	−0.331392	−0.165696	−	0.986177i	$-0.552987\pi$
$5$	−0.741015	−0.331392	−0.165696	+	0.986177i	$0.552987\pi$
$6$	0	0
$7$	4.52757	1.71126	0.855629	−	0.517589i	$-0.173170\pi$
$7$	4.52757	1.71126	0.855629	+	0.517589i	$0.173170\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.741015	−0.234329
$11$	3.85437	1.16214	0.581068	−	0.813855i	$-0.302635\pi$
$11$	3.85437	1.16214	0.581068	+	0.813855i	$0.302635\pi$
$12$	0	0
$13$	−1.04795	−0.290650	−0.145325	−	0.989384i	$-0.546423\pi$
$13$	−1.04795	−0.290650	−0.145325	+	0.989384i	$0.546423\pi$
$14$	4.52757	1.21004
$15$	0	0
$16$	1.00000	0.250000
$17$	3.11335	0.755099	0.377549	−	0.925989i	$-0.376767\pi$
$17$	3.11335	0.755099	0.377549	+	0.925989i	$0.376767\pi$
$18$	0	0
$19$	−0.673199	−0.154442	−0.0772212	−	0.997014i	$-0.524605\pi$
$19$	−0.673199	−0.154442	−0.0772212	+	0.997014i	$0.524605\pi$
$20$	−0.741015	−0.165696
$21$	0	0
$22$	3.85437	0.821754
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.45090	−0.890179
$26$	−1.04795	−0.205520
$27$	0	0
$28$	4.52757	0.855629
$29$	9.35499	1.73718	0.868589	−	0.495533i	$-0.165027\pi$
$29$	9.35499	1.73718	0.868589	+	0.495533i	$0.165027\pi$
$30$	0	0
$31$	5.45090	0.979010	0.489505	−	0.872000i	$-0.337177\pi$
$31$	5.45090	0.979010	0.489505	+	0.872000i	$0.337177\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.11335	0.533935
$35$	−3.35499	−0.567097
$36$	0	0
$37$	0.958124	0.157515	0.0787573	−	0.996894i	$-0.474905\pi$
$37$	0.958124	0.157515	0.0787573	+	0.996894i	$0.474905\pi$
$38$	−0.673199	−0.109207
$39$	0	0
$40$	−0.741015	−0.117165
$41$	−12.4989	−1.95199	−0.975996	−	0.217787i	$-0.930116\pi$
$41$	−12.4989	−1.95199	−0.975996	+	0.217787i	$0.930116\pi$
$42$	0	0
$43$	−3.63726	−0.554677	−0.277338	−	0.960772i	$-0.589452\pi$
$43$	−3.63726	−0.554677	−0.277338	+	0.960772i	$0.589452\pi$
$44$	3.85437	0.581068
$45$	0	0
$46$	0	0
$47$	10.4029	1.51743	0.758713	−	0.651425i	$-0.225829\pi$
$47$	10.4029	1.51743	0.758713	+	0.651425i	$0.225829\pi$
$48$	0	0
$49$	13.4989	1.92841
$50$	−4.45090	−0.629452
$51$	0	0
$52$	−1.04795	−0.145325
$53$	−5.48569	−0.753517	−0.376759	−	0.926311i	$-0.622961\pi$
$53$	−5.48569	−0.753517	−0.376759	+	0.926311i	$0.622961\pi$
$54$	0	0
$55$	−2.85614	−0.385122
$56$	4.52757	0.605021
$57$	0	0
$58$	9.35499	1.22837
$59$	−8.80589	−1.14643	−0.573215	−	0.819405i	$-0.694304\pi$
$59$	−8.80589	−1.14643	−0.573215	+	0.819405i	$0.694304\pi$
$60$	0	0
$61$	2.30452	0.295064	0.147532	−	0.989057i	$-0.452867\pi$
$61$	2.30452	0.295064	0.147532	+	0.989057i	$0.452867\pi$
$62$	5.45090	0.692265
$63$	0	0
$64$	1.00000	0.125000
$65$	0.776548	0.0963190
$66$	0	0
$67$	10.6405	1.29995	0.649973	−	0.759958i	$-0.274780\pi$
$67$	10.6405	1.29995	0.649973	+	0.759958i	$0.274780\pi$
$68$	3.11335	0.377549
$69$	0	0
$70$	−3.35499	−0.400998
$71$	10.4029	1.23460	0.617301	−	0.786727i	$-0.288226\pi$
$71$	10.4029	1.23460	0.617301	+	0.786727i	$0.288226\pi$
$72$	0	0
$73$	4.95205	0.579593	0.289797	−	0.957088i	$-0.406412\pi$
$73$	4.95205	0.579593	0.289797	+	0.957088i	$0.406412\pi$
$74$	0.958124	0.111380
$75$	0	0
$76$	−0.673199	−0.0772212
$77$	17.4509	1.98871
$78$	0	0
$79$	−3.18117	−0.357909	−0.178955	−	0.983857i	$-0.557272\pi$
$79$	−3.18117	−0.357909	−0.178955	+	0.983857i	$0.557272\pi$
$80$	−0.741015	−0.0828479
$81$	0	0
$82$	−12.4989	−1.38027
$83$	10.8576	1.19178	0.595889	−	0.803067i	$-0.296800\pi$
$83$	10.8576	1.19178	0.595889	+	0.803067i	$0.296800\pi$
$84$	0	0
$85$	−2.30704	−0.250234
$86$	−3.63726	−0.392216
$87$	0	0
$88$	3.85437	0.410877
$89$	4.59538	0.487109	0.243555	−	0.969887i	$-0.421686\pi$
$89$	4.59538	0.487109	0.243555	+	0.969887i	$0.421686\pi$
$90$	0	0
$91$	−4.74468	−0.497377
$92$	0	0
$93$	0	0
$94$	10.4029	1.07298
$95$	0.498850	0.0511810
$96$	0	0
$97$	−7.64092	−0.775818	−0.387909	−	0.921698i	$-0.626802\pi$
$97$	−7.64092	−0.775818	−0.387909	+	0.921698i	$0.626802\pi$
$98$	13.4989	1.36359
$99$	0	0
$100$	−4.45090	−0.445090
$101$	4.95205	0.492747	0.246374	−	0.969175i	$-0.420761\pi$
$101$	4.95205	0.492747	0.246374	+	0.969175i	$0.420761\pi$
$102$	0	0
$103$	−14.4949	−1.42822	−0.714111	−	0.700032i	$-0.753169\pi$
$103$	−14.4949	−1.42822	−0.714111	+	0.700032i	$0.753169\pi$
$104$	−1.04795	−0.102760
$105$	0	0
$106$	−5.48569	−0.532817
$107$	−13.8217	−1.33619	−0.668096	−	0.744075i	$-0.732890\pi$
$107$	−13.8217	−1.33619	−0.668096	+	0.744075i	$0.732890\pi$
$108$	0	0
$109$	−13.7539	−1.31738	−0.658691	−	0.752414i	$-0.728889\pi$
$109$	−13.7539	−1.31738	−0.658691	+	0.752414i	$0.728889\pi$
$110$	−2.85614	−0.272322
$111$	0	0
$112$	4.52757	0.427815
$113$	1.63132	0.153462	0.0767310	−	0.997052i	$-0.475552\pi$
$113$	1.63132	0.153462	0.0767310	+	0.997052i	$0.475552\pi$
$114$	0	0
$115$	0	0
$116$	9.35499	0.868589
$117$	0	0
$118$	−8.80589	−0.810648
$119$	14.0959	1.29217
$120$	0	0
$121$	3.85614	0.350558
$122$	2.30452	0.208642
$123$	0	0
$124$	5.45090	0.489505
$125$	7.00325	0.626390
$126$	0	0
$127$	−19.2088	−1.70451	−0.852254	−	0.523128i	$-0.824765\pi$
$127$	−19.2088	−1.70451	−0.852254	+	0.523128i	$0.824765\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0.776548	0.0681078
$131$	−5.45090	−0.476247	−0.238124	−	0.971235i	$-0.576532\pi$
$131$	−5.45090	−0.476247	−0.238124	+	0.971235i	$0.576532\pi$
$132$	0	0
$133$	−3.04795	−0.264291
$134$	10.6405	0.919200
$135$	0	0
$136$	3.11335	0.266968
$137$	−11.5986	−0.990938	−0.495469	−	0.868626i	$-0.665004\pi$
$137$	−11.5986	−0.990938	−0.495469	+	0.868626i	$0.665004\pi$
$138$	0	0
$139$	14.9018	1.26395	0.631977	−	0.774987i	$-0.282244\pi$
$139$	14.9018	1.26395	0.631977	+	0.774987i	$0.282244\pi$
$140$	−3.35499	−0.283549
$141$	0	0
$142$	10.4029	0.872996
$143$	−4.03919	−0.337774
$144$	0	0
$145$	−6.93218	−0.575687
$146$	4.95205	0.409834
$147$	0	0
$148$	0.958124	0.0787573
$149$	−2.99959	−0.245736	−0.122868	−	0.992423i	$-0.539209\pi$
$149$	−2.99959	−0.245736	−0.122868	+	0.992423i	$0.539209\pi$
$150$	0	0
$151$	5.75794	0.468574	0.234287	−	0.972167i	$-0.424724\pi$
$151$	5.75794	0.468574	0.234287	+	0.972167i	$0.424724\pi$
$152$	−0.673199	−0.0546037
$153$	0	0
$154$	17.4509	1.40623
$155$	−4.03919	−0.324436
$156$	0	0
$157$	−22.6734	−1.80953	−0.904766	−	0.425910i	$-0.859954\pi$
$157$	−22.6734	−1.80953	−0.904766	+	0.425910i	$0.859954\pi$
$158$	−3.18117	−0.253080
$159$	0	0
$160$	−0.741015	−0.0585823
$161$	0	0
$162$	0	0
$163$	15.8538	1.24177	0.620884	−	0.783902i	$-0.286774\pi$
$163$	15.8538	1.24177	0.620884	+	0.783902i	$0.286774\pi$
$164$	−12.4989	−0.975996
$165$	0	0
$166$	10.8576	0.842715
$167$	13.5971	1.05217	0.526086	−	0.850431i	$-0.323659\pi$
$167$	13.5971	1.05217	0.526086	+	0.850431i	$0.323659\pi$
$168$	0	0
$169$	−11.9018	−0.915523
$170$	−2.30704	−0.176942
$171$	0	0
$172$	−3.63726	−0.277338
$173$	9.35499	0.711247	0.355623	−	0.934629i	$-0.384269\pi$
$173$	9.35499	0.711247	0.355623	+	0.934629i	$0.384269\pi$
$174$	0	0
$175$	−20.1517	−1.52333
$176$	3.85437	0.290534
$177$	0	0
$178$	4.59538	0.344438
$179$	−6.54910	−0.489503	−0.244751	−	0.969586i	$-0.578706\pi$
$179$	−6.54910	−0.489503	−0.244751	+	0.969586i	$0.578706\pi$
$180$	0	0
$181$	−17.1521	−1.27491	−0.637454	−	0.770488i	$-0.720012\pi$
$181$	−17.1521	−1.27491	−0.637454	+	0.770488i	$0.720012\pi$
$182$	−4.74468	−0.351699
$183$	0	0
$184$	0	0
$185$	−0.709984	−0.0521990
$186$	0	0
$187$	12.0000	0.877527
$188$	10.4029	0.758713
$189$	0	0
$190$	0.498850	0.0361904
$191$	24.6793	1.78573	0.892866	−	0.450323i	$-0.148691\pi$
$191$	24.6793	1.78573	0.892866	+	0.450323i	$0.148691\pi$
$192$	0	0
$193$	−10.5948	−0.762627	−0.381314	−	0.924446i	$-0.624528\pi$
$193$	−10.5948	−0.762627	−0.381314	+	0.924446i	$0.624528\pi$
$194$	−7.64092	−0.548586
$195$	0	0
$196$	13.4989	0.964204
$197$	25.7077	1.83160	0.915798	−	0.401638i	$-0.131559\pi$
$197$	25.7077	1.83160	0.915798	+	0.401638i	$0.131559\pi$
$198$	0	0
$199$	10.1844	0.721954	0.360977	−	0.932575i	$-0.382443\pi$
$199$	10.1844	0.721954	0.360977	+	0.932575i	$0.382443\pi$
$200$	−4.45090	−0.314726
$201$	0	0
$202$	4.95205	0.348425
$203$	42.3553	2.97276
$204$	0	0
$205$	9.26183	0.646874
$206$	−14.4949	−1.00991
$207$	0	0
$208$	−1.04795	−0.0726625
$209$	−2.59476	−0.179483
$210$	0	0
$211$	−3.85384	−0.265309	−0.132655	−	0.991162i	$-0.542350\pi$
$211$	−3.85384	−0.265309	−0.132655	+	0.991162i	$0.542350\pi$
$212$	−5.48569	−0.376759
$213$	0	0
$214$	−13.8217	−0.944830
$215$	2.69526	0.183815
$216$	0	0
$217$	24.6793	1.67534
$218$	−13.7539	−0.931529
$219$	0	0
$220$	−2.85614	−0.192561
$221$	−3.26265	−0.219469
$222$	0	0
$223$	5.75794	0.385580	0.192790	−	0.981240i	$-0.438246\pi$
$223$	5.75794	0.385580	0.192790	+	0.981240i	$0.438246\pi$
$224$	4.52757	0.302511
$225$	0	0
$226$	1.63132	0.108514
$227$	9.07700	0.602462	0.301231	−	0.953551i	$-0.402603\pi$
$227$	9.07700	0.602462	0.301231	+	0.953551i	$0.402603\pi$
$228$	0	0
$229$	6.04513	0.399473	0.199737	−	0.979850i	$-0.435991\pi$
$229$	6.04513	0.399473	0.199737	+	0.979850i	$0.435991\pi$
$230$	0	0
$231$	0	0
$232$	9.35499	0.614185
$233$	−16.6141	−1.08842	−0.544212	−	0.838948i	$-0.683171\pi$
$233$	−16.6141	−1.08842	−0.544212	+	0.838948i	$0.683171\pi$
$234$	0	0
$235$	−7.70873	−0.502862
$236$	−8.80589	−0.573215
$237$	0	0
$238$	14.0959	0.913702
$239$	20.8059	1.34582	0.672911	−	0.739724i	$-0.265044\pi$
$239$	20.8059	1.34582	0.672911	+	0.739724i	$0.265044\pi$
$240$	0	0
$241$	−28.0098	−1.80427	−0.902134	−	0.431457i	$-0.858000\pi$
$241$	−28.0098	−1.80427	−0.902134	+	0.431457i	$0.858000\pi$
$242$	3.85614	0.247882
$243$	0	0
$244$	2.30452	0.147532
$245$	−10.0028	−0.639058
$246$	0	0
$247$	0.705481	0.0448887
$248$	5.45090	0.346132
$249$	0	0
$250$	7.00325	0.442925
$251$	−11.5631	−0.729856	−0.364928	−	0.931036i	$-0.618906\pi$
$251$	−11.5631	−0.729856	−0.364928	+	0.931036i	$0.618906\pi$
$252$	0	0
$253$	0	0
$254$	−19.2088	−1.20527
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.4029	1.39746	0.698729	−	0.715387i	$-0.253749\pi$
$257$	22.4029	1.39746	0.698729	+	0.715387i	$0.253749\pi$
$258$	0	0
$259$	4.33797	0.269548
$260$	0.776548	0.0481595
$261$	0	0
$262$	−5.45090	−0.336758
$263$	5.92812	0.365543	0.182772	−	0.983155i	$-0.441493\pi$
$263$	5.92812	0.365543	0.182772	+	0.983155i	$0.441493\pi$
$264$	0	0
$265$	4.06498	0.249709
$266$	−3.04795	−0.186882
$267$	0	0
$268$	10.6405	0.649973
$269$	14.8059	0.902731	0.451366	−	0.892339i	$-0.350937\pi$
$269$	14.8059	0.902731	0.451366	+	0.892339i	$0.350937\pi$
$270$	0	0
$271$	−11.5011	−0.698645	−0.349323	−	0.937003i	$-0.613588\pi$
$271$	−11.5011	−0.698645	−0.349323	+	0.937003i	$0.613588\pi$
$272$	3.11335	0.188775
$273$	0	0
$274$	−11.5986	−0.700699
$275$	−17.1554	−1.03451
$276$	0	0
$277$	−17.4509	−1.04852	−0.524261	−	0.851557i	$-0.675659\pi$
$277$	−17.4509	−1.04852	−0.524261	+	0.851557i	$0.675659\pi$
$278$	14.9018	0.893750
$279$	0	0
$280$	−3.35499	−0.200499
$281$	−24.5300	−1.46334	−0.731669	−	0.681660i	$-0.761258\pi$
$281$	−24.5300	−1.46334	−0.731669	+	0.681660i	$0.761258\pi$
$282$	0	0
$283$	24.7116	1.46895	0.734475	−	0.678635i	$-0.237428\pi$
$283$	24.7116	1.46895	0.734475	+	0.678635i	$0.237428\pi$
$284$	10.4029	0.617301
$285$	0	0
$286$	−4.03919	−0.238843
$287$	−56.5894	−3.34036
$288$	0	0
$289$	−7.30704	−0.429826
$290$	−6.93218	−0.407072
$291$	0	0
$292$	4.95205	0.289797
$293$	−19.4211	−1.13459	−0.567297	−	0.823513i	$-0.692011\pi$
$293$	−19.4211	−1.13459	−0.567297	+	0.823513i	$0.692011\pi$
$294$	0	0
$295$	6.52529	0.379917
$296$	0.958124	0.0556898
$297$	0	0
$298$	−2.99959	−0.173762
$299$	0	0
$300$	0	0
$301$	−16.4679	−0.949195
$302$	5.75794	0.331332
$303$	0	0
$304$	−0.673199	−0.0386106
$305$	−1.70768	−0.0977817
$306$	0	0
$307$	1.75794	0.100331	0.0501654	−	0.998741i	$-0.484025\pi$
$307$	1.75794	0.100331	0.0501654	+	0.998741i	$0.484025\pi$
$308$	17.4509	0.994357
$309$	0	0
$310$	−4.03919	−0.229411
$311$	24.9977	1.41749	0.708745	−	0.705465i	$-0.249262\pi$
$311$	24.9977	1.41749	0.708745	+	0.705465i	$0.249262\pi$
$312$	0	0
$313$	15.4853	0.875280	0.437640	−	0.899150i	$-0.355814\pi$
$313$	15.4853	0.875280	0.437640	+	0.899150i	$0.355814\pi$
$314$	−22.6734	−1.27953
$315$	0	0
$316$	−3.18117	−0.178955
$317$	−2.64501	−0.148558	−0.0742792	−	0.997237i	$-0.523666\pi$
$317$	−2.64501	−0.148558	−0.0742792	+	0.997237i	$0.523666\pi$
$318$	0	0
$319$	36.0576	2.01884
$320$	−0.741015	−0.0414240
$321$	0	0
$322$	0	0
$323$	−2.09591	−0.116619
$324$	0	0
$325$	4.66433	0.258731
$326$	15.8538	0.878063
$327$	0	0
$328$	−12.4989	−0.690134
$329$	47.1000	2.59671
$330$	0	0
$331$	−27.8538	−1.53099	−0.765493	−	0.643444i	$-0.777505\pi$
$331$	−27.8538	−1.53099	−0.765493	+	0.643444i	$0.777505\pi$
$332$	10.8576	0.595889
$333$	0	0
$334$	13.5971	0.743998
$335$	−7.88477	−0.430791
$336$	0	0
$337$	−26.3921	−1.43767	−0.718835	−	0.695181i	$-0.755324\pi$
$337$	−26.3921	−1.43767	−0.718835	+	0.695181i	$0.755324\pi$
$338$	−11.9018	−0.647372
$339$	0	0
$340$	−2.30704	−0.125117
$341$	21.0098	1.13774
$342$	0	0
$343$	29.4240	1.58875
$344$	−3.63726	−0.196108
$345$	0	0
$346$	9.35499	0.502927
$347$	−19.9691	−1.07200	−0.535998	−	0.844219i	$-0.680065\pi$
$347$	−19.9691	−1.07200	−0.535998	+	0.844219i	$0.680065\pi$
$348$	0	0
$349$	29.4966	1.57891	0.789457	−	0.613806i	$-0.210362\pi$
$349$	29.4966	1.57891	0.789457	+	0.613806i	$0.210362\pi$
$350$	−20.1517	−1.07716
$351$	0	0
$352$	3.85437	0.205438
$353$	20.5948	1.09615	0.548074	−	0.836430i	$-0.315361\pi$
$353$	20.5948	1.09615	0.548074	+	0.836430i	$0.315361\pi$
$354$	0	0
$355$	−7.70873	−0.409137
$356$	4.59538	0.243555
$357$	0	0
$358$	−6.54910	−0.346131
$359$	−22.4207	−1.18332	−0.591660	−	0.806188i	$-0.701527\pi$
$359$	−22.4207	−1.18332	−0.591660	+	0.806188i	$0.701527\pi$
$360$	0	0
$361$	−18.5468	−0.976148
$362$	−17.1521	−0.901496
$363$	0	0
$364$	−4.74468	−0.248689
$365$	−3.66954	−0.192072
$366$	0	0
$367$	29.6411	1.54725	0.773626	−	0.633643i	$-0.218441\pi$
$367$	29.6411	1.54725	0.773626	+	0.633643i	$0.218441\pi$
$368$	0	0
$369$	0	0
$370$	−0.709984	−0.0369103
$371$	−24.8368	−1.28946
$372$	0	0
$373$	19.2040	0.994346	0.497173	−	0.867651i	$-0.334371\pi$
$373$	19.2040	0.994346	0.497173	+	0.867651i	$0.334371\pi$
$374$	12.0000	0.620505
$375$	0	0
$376$	10.4029	0.536491
$377$	−9.80359	−0.504911
$378$	0	0
$379$	23.7994	1.22249	0.611246	−	0.791440i	$-0.290668\pi$
$379$	23.7994	1.22249	0.611246	+	0.791440i	$0.290668\pi$
$380$	0.498850	0.0255905
$381$	0	0
$382$	24.6793	1.26270
$383$	−2.25858	−0.115408	−0.0577040	−	0.998334i	$-0.518378\pi$
$383$	−2.25858	−0.115408	−0.0577040	+	0.998334i	$0.518378\pi$
$384$	0	0
$385$	−12.9314	−0.659044
$386$	−10.5948	−0.539259
$387$	0	0
$388$	−7.64092	−0.387909
$389$	18.4171	0.933782	0.466891	−	0.884315i	$-0.345374\pi$
$389$	18.4171	0.933782	0.466891	+	0.884315i	$0.345374\pi$
$390$	0	0
$391$	0	0
$392$	13.4989	0.681795
$393$	0	0
$394$	25.7077	1.29513
$395$	2.35729	0.118608
$396$	0	0
$397$	23.8036	1.19467	0.597334	−	0.801993i	$-0.296227\pi$
$397$	23.8036	1.19467	0.597334	+	0.801993i	$0.296227\pi$
$398$	10.1844	0.510499
$399$	0	0
$400$	−4.45090	−0.222545
$401$	−12.3041	−0.614438	−0.307219	−	0.951639i	$-0.599398\pi$
$401$	−12.3041	−0.614438	−0.307219	+	0.951639i	$0.599398\pi$
$402$	0	0
$403$	−5.71228	−0.284549
$404$	4.95205	0.246374
$405$	0	0
$406$	42.3553	2.10206
$407$	3.69296	0.183053
$408$	0	0
$409$	−11.2088	−0.554241	−0.277121	−	0.960835i	$-0.589380\pi$
$409$	−11.2088	−0.554241	−0.277121	+	0.960835i	$0.589380\pi$
$410$	9.26183	0.457409
$411$	0	0
$412$	−14.4949	−0.714111
$413$	−39.8692	−1.96184
$414$	0	0
$415$	−8.04565	−0.394946
$416$	−1.04795	−0.0513801
$417$	0	0
$418$	−2.59476	−0.126914
$419$	9.30452	0.454556	0.227278	−	0.973830i	$-0.427017\pi$
$419$	9.30452	0.454556	0.227278	+	0.973830i	$0.427017\pi$
$420$	0	0
$421$	27.6893	1.34949	0.674747	−	0.738049i	$-0.264253\pi$
$421$	27.6893	1.34949	0.674747	+	0.738049i	$0.264253\pi$
$422$	−3.85384	−0.187602
$423$	0	0
$424$	−5.48569	−0.266409
$425$	−13.8572	−0.672173
$426$	0	0
$427$	10.4339	0.504931
$428$	−13.8217	−0.668096
$429$	0	0
$430$	2.69526	0.129977
$431$	18.7512	0.903213	0.451606	−	0.892217i	$-0.350851\pi$
$431$	18.7512	0.903213	0.451606	+	0.892217i	$0.350851\pi$
$432$	0	0
$433$	−23.2651	−1.11805	−0.559024	−	0.829151i	$-0.688824\pi$
$433$	−23.2651	−1.11805	−0.559024	+	0.829151i	$0.688824\pi$
$434$	24.6793	1.18464
$435$	0	0
$436$	−13.7539	−0.658691
$437$	0	0
$438$	0	0
$439$	0.614078	0.0293084	0.0146542	−	0.999893i	$-0.495335\pi$
$439$	0.614078	0.0293084	0.0146542	+	0.999893i	$0.495335\pi$
$440$	−2.85614	−0.136161
$441$	0	0
$442$	−3.26265	−0.155188
$443$	20.8059	0.988518	0.494259	−	0.869315i	$-0.335440\pi$
$443$	20.8059	0.988518	0.494259	+	0.869315i	$0.335440\pi$
$444$	0	0
$445$	−3.40524	−0.161424
$446$	5.75794	0.272646
$447$	0	0
$448$	4.52757	0.213907
$449$	19.2088	0.906521	0.453260	−	0.891378i	$-0.350261\pi$
$449$	19.2088	0.906521	0.453260	+	0.891378i	$0.350261\pi$
$450$	0	0
$451$	−48.1752	−2.26848
$452$	1.63132	0.0767310
$453$	0	0
$454$	9.07700	0.426005
$455$	3.51587	0.164827
$456$	0	0
$457$	−20.8709	−0.976298	−0.488149	−	0.872760i	$-0.662328\pi$
$457$	−20.8709	−0.976298	−0.488149	+	0.872760i	$0.662328\pi$
$458$	6.04513	0.282470
$459$	0	0
$460$	0	0
$461$	8.09591	0.377064	0.188532	−	0.982067i	$-0.439627\pi$
$461$	8.09591	0.377064	0.188532	+	0.982067i	$0.439627\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	9.35499	0.434295
$465$	0	0
$466$	−16.6141	−0.769632
$467$	1.36827	0.0633159	0.0316580	−	0.999499i	$-0.489921\pi$
$467$	1.36827	0.0633159	0.0316580	+	0.999499i	$0.489921\pi$
$468$	0	0
$469$	48.1756	2.22454
$470$	−7.70873	−0.355577
$471$	0	0
$472$	−8.80589	−0.405324
$473$	−14.0193	−0.644609
$474$	0	0
$475$	2.99634	0.137482
$476$	14.0959	0.646085
$477$	0	0
$478$	20.8059	0.951639
$479$	14.7120	0.672208	0.336104	−	0.941825i	$-0.390891\pi$
$479$	14.7120	0.672208	0.336104	+	0.941825i	$0.390891\pi$
$480$	0	0
$481$	−1.00407	−0.0457816
$482$	−28.0098	−1.27581
$483$	0	0
$484$	3.85614	0.175279
$485$	5.66203	0.257100
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	2.30452	0.104321
$489$	0	0
$490$	−10.0028	−0.451882
$491$	34.0650	1.53733	0.768665	−	0.639651i	$-0.220921\pi$
$491$	34.0650	1.53733	0.768665	+	0.639651i	$0.220921\pi$
$492$	0	0
$493$	29.1254	1.31174
$494$	0.705481	0.0317411
$495$	0	0
$496$	5.45090	0.244753
$497$	47.1000	2.11272
$498$	0	0
$499$	−20.0457	−0.897367	−0.448683	−	0.893691i	$-0.648107\pi$
$499$	−20.0457	−0.897367	−0.448683	+	0.893691i	$0.648107\pi$
$500$	7.00325	0.313195
$501$	0	0
$502$	−11.5631	−0.516086
$503$	7.00325	0.312260	0.156130	−	0.987737i	$-0.450098\pi$
$503$	7.00325	0.312260	0.156130	+	0.987737i	$0.450098\pi$
$504$	0	0
$505$	−3.66954	−0.163292
$506$	0	0
$507$	0	0
$508$	−19.2088	−0.852254
$509$	−32.2568	−1.42976	−0.714878	−	0.699249i	$-0.753518\pi$
$509$	−32.2568	−1.42976	−0.714878	+	0.699249i	$0.753518\pi$
$510$	0	0
$511$	22.4207	0.991834
$512$	1.00000	0.0441942
$513$	0	0
$514$	22.4029	0.988152
$515$	10.7409	0.473301
$516$	0	0
$517$	40.0968	1.76345
$518$	4.33797	0.190599
$519$	0	0
$520$	0.776548	0.0340539
$521$	−41.0226	−1.79723	−0.898616	−	0.438735i	$-0.855427\pi$
$521$	−41.0226	−1.79723	−0.898616	+	0.438735i	$0.855427\pi$
$522$	0	0
$523$	2.93178	0.128198	0.0640988	−	0.997944i	$-0.479583\pi$
$523$	2.93178	0.128198	0.0640988	+	0.997944i	$0.479583\pi$
$524$	−5.45090	−0.238124
$525$	0	0
$526$	5.92812	0.258478
$527$	16.9706	0.739249
$528$	0	0
$529$	0	0
$530$	4.06498	0.176571
$531$	0	0
$532$	−3.04795	−0.132146
$533$	13.0982	0.567346
$534$	0	0
$535$	10.2421	0.442803
$536$	10.6405	0.459600
$537$	0	0
$538$	14.8059	0.638327
$539$	52.0295	2.24107
$540$	0	0
$541$	11.2088	0.481905	0.240953	−	0.970537i	$-0.422540\pi$
$541$	11.2088	0.481905	0.240953	+	0.970537i	$0.422540\pi$
$542$	−11.5011	−0.494017
$543$	0	0
$544$	3.11335	0.133484
$545$	10.1918	0.436569
$546$	0	0
$547$	−0.337969	−0.0144505	−0.00722526	−	0.999974i	$-0.502300\pi$
$547$	−0.337969	−0.0144505	−0.00722526	+	0.999974i	$0.502300\pi$
$548$	−11.5986	−0.495469
$549$	0	0
$550$	−17.1554	−0.731508
$551$	−6.29777	−0.268294
$552$	0	0
$553$	−14.4029	−0.612476
$554$	−17.4509	−0.741418
$555$	0	0
$556$	14.9018	0.631977
$557$	−47.0645	−1.99419	−0.997093	−	0.0761938i	$-0.975723\pi$
$557$	−47.0645	−1.99419	−0.997093	+	0.0761938i	$0.975723\pi$
$558$	0	0
$559$	3.81167	0.161217
$560$	−3.35499	−0.141774
$561$	0	0
$562$	−24.5300	−1.03474
$563$	31.7252	1.33706	0.668530	−	0.743685i	$-0.266924\pi$
$563$	31.7252	1.33706	0.668530	+	0.743685i	$0.266924\pi$
$564$	0	0
$565$	−1.20883	−0.0508560
$566$	24.7116	1.03871
$567$	0	0
$568$	10.4029	0.436498
$569$	7.08148	0.296871	0.148436	−	0.988922i	$-0.452576\pi$
$569$	7.08148	0.296871	0.148436	+	0.988922i	$0.452576\pi$
$570$	0	0
$571$	46.4268	1.94290	0.971451	−	0.237241i	$-0.0762430\pi$
$571$	46.4268	1.94290	0.971451	+	0.237241i	$0.0762430\pi$
$572$	−4.03919	−0.168887
$573$	0	0
$574$	−56.5894	−2.36199
$575$	0	0
$576$	0	0
$577$	26.0650	1.08510	0.542550	−	0.840024i	$-0.317459\pi$
$577$	26.0650	1.08510	0.542550	+	0.840024i	$0.317459\pi$
$578$	−7.30704	−0.303933
$579$	0	0
$580$	−6.93218	−0.287843
$581$	49.1586	2.03944
$582$	0	0
$583$	−21.1439	−0.875689
$584$	4.95205	0.204917
$585$	0	0
$586$	−19.4211	−0.802279
$587$	−43.8686	−1.81065	−0.905325	−	0.424720i	$-0.860373\pi$
$587$	−43.8686	−1.81065	−0.905325	+	0.424720i	$0.860373\pi$
$588$	0	0
$589$	−3.66954	−0.151201
$590$	6.52529	0.268642
$591$	0	0
$592$	0.958124	0.0393787
$593$	−25.9188	−1.06436	−0.532179	−	0.846632i	$-0.678627\pi$
$593$	−25.9188	−1.06436	−0.532179	+	0.846632i	$0.678627\pi$
$594$	0	0
$595$	−10.4453	−0.428214
$596$	−2.99959	−0.122868
$597$	0	0
$598$	0	0
$599$	−15.0936	−0.616708	−0.308354	−	0.951272i	$-0.599778\pi$
$599$	−15.0936	−0.616708	−0.308354	+	0.951272i	$0.599778\pi$
$600$	0	0
$601$	−28.9668	−1.18158	−0.590790	−	0.806826i	$-0.701184\pi$
$601$	−28.9668	−1.18158	−0.590790	+	0.806826i	$0.701184\pi$
$602$	−16.4679	−0.671182
$603$	0	0
$604$	5.75794	0.234287
$605$	−2.85746	−0.116172
$606$	0	0
$607$	−24.0457	−0.975983	−0.487991	−	0.872848i	$-0.662270\pi$
$607$	−24.0457	−0.975983	−0.487991	+	0.872848i	$0.662270\pi$
$608$	−0.673199	−0.0273018
$609$	0	0
$610$	−1.70768	−0.0691421
$611$	−10.9018	−0.441039
$612$	0	0
$613$	−30.1546	−1.21793	−0.608966	−	0.793196i	$-0.708415\pi$
$613$	−30.1546	−1.21793	−0.608966	+	0.793196i	$0.708415\pi$
$614$	1.75794	0.0709445
$615$	0	0
$616$	17.4509	0.703117
$617$	−8.56351	−0.344754	−0.172377	−	0.985031i	$-0.555145\pi$
$617$	−8.56351	−0.344754	−0.172377	+	0.985031i	$0.555145\pi$
$618$	0	0
$619$	37.5346	1.50864	0.754322	−	0.656504i	$-0.227966\pi$
$619$	37.5346	1.50864	0.754322	+	0.656504i	$0.227966\pi$
$620$	−4.03919	−0.162218
$621$	0	0
$622$	24.9977	1.00232
$623$	20.8059	0.833570
$624$	0	0
$625$	17.0650	0.682599
$626$	15.4853	0.618916
$627$	0	0
$628$	−22.6734	−0.904766
$629$	2.98298	0.118939
$630$	0	0
$631$	−23.9842	−0.954797	−0.477398	−	0.878687i	$-0.658420\pi$
$631$	−23.9842	−0.954797	−0.477398	+	0.878687i	$0.658420\pi$
$632$	−3.18117	−0.126540
$633$	0	0
$634$	−2.64501	−0.105047
$635$	14.2340	0.564860
$636$	0	0
$637$	−14.1462	−0.560491
$638$	36.0576	1.42753
$639$	0	0
$640$	−0.741015	−0.0292912
$641$	28.5692	1.12842	0.564208	−	0.825633i	$-0.309182\pi$
$641$	28.5692	1.12842	0.564208	+	0.825633i	$0.309182\pi$
$642$	0	0
$643$	19.0547	0.751445	0.375722	−	0.926732i	$-0.377395\pi$
$643$	19.0547	0.751445	0.375722	+	0.926732i	$0.377395\pi$
$644$	0	0
$645$	0	0
$646$	−2.09591	−0.0824623
$647$	12.4223	0.488370	0.244185	−	0.969729i	$-0.421480\pi$
$647$	12.4223	0.488370	0.244185	+	0.969729i	$0.421480\pi$
$648$	0	0
$649$	−33.9411	−1.33231
$650$	4.66433	0.182950
$651$	0	0
$652$	15.8538	0.620884
$653$	26.7556	1.04703	0.523514	−	0.852017i	$-0.324621\pi$
$653$	26.7556	1.04703	0.523514	+	0.852017i	$0.324621\pi$
$654$	0	0
$655$	4.03919	0.157824
$656$	−12.4989	−0.487998
$657$	0	0
$658$	47.1000	1.83615
$659$	−27.8282	−1.08403	−0.542016	−	0.840368i	$-0.682339\pi$
$659$	−27.8282	−1.08403	−0.542016	+	0.840368i	$0.682339\pi$
$660$	0	0
$661$	−19.7093	−0.766603	−0.383302	−	0.923623i	$-0.625213\pi$
$661$	−19.7093	−0.766603	−0.383302	+	0.923623i	$0.625213\pi$
$662$	−27.8538	−1.08257
$663$	0	0
$664$	10.8576	0.421357
$665$	2.25858	0.0875839
$666$	0	0
$667$	0	0
$668$	13.5971	0.526086
$669$	0	0
$670$	−7.88477	−0.304615
$671$	8.88247	0.342904
$672$	0	0
$673$	20.2568	0.780842	0.390421	−	0.920636i	$-0.372330\pi$
$673$	20.2568	0.780842	0.390421	+	0.920636i	$0.372330\pi$
$674$	−26.3921	−1.01659
$675$	0	0
$676$	−11.9018	−0.457761
$677$	−16.9350	−0.650866	−0.325433	−	0.945565i	$-0.605510\pi$
$677$	−16.9350	−0.650866	−0.325433	+	0.945565i	$0.605510\pi$
$678$	0	0
$679$	−34.5948	−1.32762
$680$	−2.30704	−0.0884709
$681$	0	0
$682$	21.0098	0.804505
$683$	36.1609	1.38366	0.691829	−	0.722062i	$-0.256805\pi$
$683$	36.1609	1.38366	0.691829	+	0.722062i	$0.256805\pi$
$684$	0	0
$685$	8.59476	0.328389
$686$	29.4240	1.12341
$687$	0	0
$688$	−3.63726	−0.138669
$689$	5.74874	0.219010
$690$	0	0
$691$	−13.7579	−0.523377	−0.261688	−	0.965152i	$-0.584279\pi$
$691$	−13.7579	−0.523377	−0.261688	+	0.965152i	$0.584279\pi$
$692$	9.35499	0.355623
$693$	0	0
$694$	−19.9691	−0.758016
$695$	−11.0424	−0.418864
$696$	0	0
$697$	−38.9133	−1.47395
$698$	29.4966	1.11646
$699$	0	0
$700$	−20.1517	−0.761664
$701$	−8.15116	−0.307865	−0.153933	−	0.988081i	$-0.549194\pi$
$701$	−8.15116	−0.307865	−0.153933	+	0.988081i	$0.549194\pi$
$702$	0	0
$703$	−0.645008	−0.0243269
$704$	3.85437	0.145267
$705$	0	0
$706$	20.5948	0.775094
$707$	22.4207	0.843218
$708$	0	0
$709$	−7.45609	−0.280019	−0.140010	−	0.990150i	$-0.544713\pi$
$709$	−7.45609	−0.280019	−0.140010	+	0.990150i	$0.544713\pi$
$710$	−7.70873	−0.289304
$711$	0	0
$712$	4.59538	0.172219
$713$	0	0
$714$	0	0
$715$	2.99310	0.111936
$716$	−6.54910	−0.244751
$717$	0	0
$718$	−22.4207	−0.836734
$719$	39.0936	1.45795	0.728973	−	0.684543i	$-0.239998\pi$
$719$	39.0936	1.45795	0.728973	+	0.684543i	$0.239998\pi$
$720$	0	0
$721$	−65.6265	−2.44406
$722$	−18.5468	−0.690241
$723$	0	0
$724$	−17.1521	−0.637454
$725$	−41.6381	−1.54640
$726$	0	0
$727$	−5.43974	−0.201749	−0.100874	−	0.994899i	$-0.532164\pi$
$727$	−5.43974	−0.201749	−0.100874	+	0.994899i	$0.532164\pi$
$728$	−4.74468	−0.175849
$729$	0	0
$730$	−3.66954	−0.135816
$731$	−11.3241	−0.418836
$732$	0	0
$733$	−4.92625	−0.181955	−0.0909776	−	0.995853i	$-0.528999\pi$
$733$	−4.92625	−0.181955	−0.0909776	+	0.995853i	$0.528999\pi$
$734$	29.6411	1.09407
$735$	0	0
$736$	0	0
$737$	41.0124	1.51071
$738$	0	0
$739$	−32.6141	−1.19973	−0.599864	−	0.800102i	$-0.704779\pi$
$739$	−32.6141	−1.19973	−0.599864	+	0.800102i	$0.704779\pi$
$740$	−0.709984	−0.0260995
$741$	0	0
$742$	−24.8368	−0.911788
$743$	−8.89217	−0.326222	−0.163111	−	0.986608i	$-0.552153\pi$
$743$	−8.89217	−0.326222	−0.163111	+	0.986608i	$0.552153\pi$
$744$	0	0
$745$	2.22274	0.0814349
$746$	19.2040	0.703109
$747$	0	0
$748$	12.0000	0.438763
$749$	−62.5785	−2.28657
$750$	0	0
$751$	−35.1350	−1.28209	−0.641047	−	0.767502i	$-0.721500\pi$
$751$	−35.1350	−1.28209	−0.641047	+	0.767502i	$0.721500\pi$
$752$	10.4029	0.379356
$753$	0	0
$754$	−9.80359	−0.357026
$755$	−4.26671	−0.155282
$756$	0	0
$757$	−22.9446	−0.833937	−0.416968	−	0.908921i	$-0.636907\pi$
$757$	−22.9446	−0.833937	−0.416968	+	0.908921i	$0.636907\pi$
$758$	23.7994	0.864433
$759$	0	0
$760$	0.498850	0.0180952
$761$	−18.4989	−0.670583	−0.335291	−	0.942114i	$-0.608835\pi$
$761$	−18.4989	−0.670583	−0.335291	+	0.942114i	$0.608835\pi$
$762$	0	0
$763$	−62.2715	−2.25438
$764$	24.6793	0.892866
$765$	0	0
$766$	−2.25858	−0.0816057
$767$	9.22816	0.333209
$768$	0	0
$769$	36.0171	1.29881	0.649405	−	0.760443i	$-0.275018\pi$
$769$	36.0171	1.29881	0.649405	+	0.760443i	$0.275018\pi$
$770$	−12.9314	−0.466014
$771$	0	0
$772$	−10.5948	−0.381314
$773$	−9.93178	−0.357221	−0.178611	−	0.983920i	$-0.557160\pi$
$773$	−9.93178	−0.357221	−0.178611	+	0.983920i	$0.557160\pi$
$774$	0	0
$775$	−24.2614	−0.871495
$776$	−7.64092	−0.274293
$777$	0	0
$778$	18.4171	0.660284
$779$	8.41421	0.301471
$780$	0	0
$781$	40.0968	1.43477
$782$	0	0
$783$	0	0
$784$	13.4989	0.482102
$785$	16.8013	0.599664
$786$	0	0
$787$	9.93503	0.354146	0.177073	−	0.984198i	$-0.443337\pi$
$787$	9.93503	0.354146	0.177073	+	0.984198i	$0.443337\pi$
$788$	25.7077	0.915798
$789$	0	0
$790$	2.35729	0.0838687
$791$	7.38592	0.262613
$792$	0	0
$793$	−2.41503	−0.0857602
$794$	23.8036	0.844758
$795$	0	0
$796$	10.1844	0.360977
$797$	39.0572	1.38348	0.691738	−	0.722149i	$-0.256845\pi$
$797$	39.0572	1.38348	0.691738	+	0.722149i	$0.256845\pi$
$798$	0	0
$799$	32.3880	1.14581
$800$	−4.45090	−0.157363
$801$	0	0
$802$	−12.3041	−0.434473
$803$	19.0870	0.673566
$804$	0	0
$805$	0	0
$806$	−5.71228	−0.201207
$807$	0	0
$808$	4.95205	0.174212
$809$	−9.69296	−0.340786	−0.170393	−	0.985376i	$-0.554504\pi$
$809$	−9.69296	−0.340786	−0.170393	+	0.985376i	$0.554504\pi$
$810$	0	0
$811$	−18.9474	−0.665335	−0.332667	−	0.943044i	$-0.607949\pi$
$811$	−18.9474	−0.665335	−0.332667	+	0.943044i	$0.607949\pi$
$812$	42.3553	1.48638
$813$	0	0
$814$	3.69296	0.129438
$815$	−11.7479	−0.411512
$816$	0	0
$817$	2.44860	0.0856656
$818$	−11.2088	−0.391908
$819$	0	0
$820$	9.26183	0.323437
$821$	−13.3357	−0.465418	−0.232709	−	0.972546i	$-0.574759\pi$
$821$	−13.3357	−0.465418	−0.232709	+	0.972546i	$0.574759\pi$
$822$	0	0
$823$	−4.45320	−0.155229	−0.0776143	−	0.996983i	$-0.524730\pi$
$823$	−4.45320	−0.155229	−0.0776143	+	0.996983i	$0.524730\pi$
$824$	−14.4949	−0.504953
$825$	0	0
$826$	−39.8692	−1.38723
$827$	−45.7317	−1.59025	−0.795124	−	0.606446i	$-0.792594\pi$
$827$	−45.7317	−1.59025	−0.795124	+	0.606446i	$0.792594\pi$
$828$	0	0
$829$	−5.40524	−0.187732	−0.0938659	−	0.995585i	$-0.529923\pi$
$829$	−5.40524	−0.187732	−0.0938659	+	0.995585i	$0.529923\pi$
$830$	−8.04565	−0.279269
$831$	0	0
$832$	−1.04795	−0.0363312
$833$	42.0267	1.45614
$834$	0	0
$835$	−10.0756	−0.348681
$836$	−2.59476	−0.0897415
$837$	0	0
$838$	9.30452	0.321419
$839$	27.8709	0.962209	0.481105	−	0.876663i	$-0.340236\pi$
$839$	27.8709	0.962209	0.481105	+	0.876663i	$0.340236\pi$
$840$	0	0
$841$	58.5159	2.01779
$842$	27.6893	0.954236
$843$	0	0
$844$	−3.85384	−0.132655
$845$	8.81940	0.303397
$846$	0	0
$847$	17.4589	0.599896
$848$	−5.48569	−0.188379
$849$	0	0
$850$	−13.8572	−0.475298
$851$	0	0
$852$	0	0
$853$	−46.7054	−1.59916	−0.799581	−	0.600558i	$-0.794945\pi$
$853$	−46.7054	−1.59916	−0.799581	+	0.600558i	$0.794945\pi$
$854$	10.4339	0.357040
$855$	0	0
$856$	−13.8217	−0.472415
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−8.61408	−0.293909	−0.146954	−	0.989143i	$-0.546947\pi$
$859$	−8.61408	−0.293909	−0.146954	+	0.989143i	$0.546947\pi$
$860$	2.69526	0.0919076
$861$	0	0
$862$	18.7512	0.638668
$863$	49.9954	1.70186	0.850932	−	0.525276i	$-0.176038\pi$
$863$	49.9954	1.70186	0.850932	+	0.525276i	$0.176038\pi$
$864$	0	0
$865$	−6.93218	−0.235701
$866$	−23.2651	−0.790580
$867$	0	0
$868$	24.6793	0.837670
$869$	−12.2614	−0.415939
$870$	0	0
$871$	−11.1508	−0.377829
$872$	−13.7539	−0.465765
$873$	0	0
$874$	0	0
$875$	31.7077	1.07192
$876$	0	0
$877$	25.8345	0.872370	0.436185	−	0.899857i	$-0.356329\pi$
$877$	25.8345	0.872370	0.436185	+	0.899857i	$0.356329\pi$
$878$	0.614078	0.0207241
$879$	0	0
$880$	−2.85614	−0.0962805
$881$	−33.3139	−1.12237	−0.561186	−	0.827689i	$-0.689655\pi$
$881$	−33.3139	−1.12237	−0.561186	+	0.827689i	$0.689655\pi$
$882$	0	0
$883$	−10.5638	−0.355501	−0.177751	−	0.984076i	$-0.556882\pi$
$883$	−10.5638	−0.355501	−0.177751	+	0.984076i	$0.556882\pi$
$884$	−3.26265	−0.109735
$885$	0	0
$886$	20.8059	0.698988
$887$	12.9211	0.433849	0.216924	−	0.976188i	$-0.430398\pi$
$887$	12.9211	0.433849	0.216924	+	0.976188i	$0.430398\pi$
$888$	0	0
$889$	−86.9693	−2.91686
$890$	−3.40524	−0.114144
$891$	0	0
$892$	5.75794	0.192790
$893$	−7.00325	−0.234355
$894$	0	0
$895$	4.85298	0.162217
$896$	4.52757	0.151255
$897$	0	0
$898$	19.2088	0.641007
$899$	50.9931	1.70072
$900$	0	0
$901$	−17.0789	−0.568980
$902$	−48.1752	−1.60406
$903$	0	0
$904$	1.63132	0.0542570
$905$	12.7100	0.422494
$906$	0	0
$907$	9.08741	0.301743	0.150871	−	0.988553i	$-0.451792\pi$
$907$	9.08741	0.301743	0.150871	+	0.988553i	$0.451792\pi$
$908$	9.07700	0.301231
$909$	0	0
$910$	3.51587	0.116550
$911$	29.1965	0.967322	0.483661	−	0.875256i	$-0.339307\pi$
$911$	29.1965	0.967322	0.483661	+	0.875256i	$0.339307\pi$
$912$	0	0
$913$	41.8492	1.38501
$914$	−20.8709	−0.690347
$915$	0	0
$916$	6.04513	0.199737
$917$	−24.6793	−0.814982
$918$	0	0
$919$	−28.5659	−0.942304	−0.471152	−	0.882052i	$-0.656162\pi$
$919$	−28.5659	−0.942304	−0.471152	+	0.882052i	$0.656162\pi$
$920$	0	0
$921$	0	0
$922$	8.09591	0.266624
$923$	−10.9018	−0.358837
$924$	0	0
$925$	−4.26451	−0.140216
$926$	−8.00000	−0.262896
$927$	0	0
$928$	9.35499	0.307093
$929$	9.30474	0.305279	0.152639	−	0.988282i	$-0.451223\pi$
$929$	9.30474	0.305279	0.152639	+	0.988282i	$0.451223\pi$
$930$	0	0
$931$	−9.08741	−0.297828
$932$	−16.6141	−0.544212
$933$	0	0
$934$	1.36827	0.0447711
$935$	−8.89217	−0.290805
$936$	0	0
$937$	29.2851	0.956702	0.478351	−	0.878169i	$-0.341235\pi$
$937$	29.2851	0.956702	0.478351	+	0.878169i	$0.341235\pi$
$938$	48.1756	1.57299
$939$	0	0
$940$	−7.70873	−0.251431
$941$	−5.96365	−0.194409	−0.0972047	−	0.995264i	$-0.530990\pi$
$941$	−5.96365	−0.194409	−0.0972047	+	0.995264i	$0.530990\pi$
$942$	0	0
$943$	0	0
$944$	−8.80589	−0.286607
$945$	0	0
$946$	−14.0193	−0.455807
$947$	16.4532	0.534657	0.267329	−	0.963605i	$-0.413859\pi$
$947$	16.4532	0.534657	0.267329	+	0.963605i	$0.413859\pi$
$948$	0	0
$949$	−5.18951	−0.168459
$950$	2.99634	0.0972141
$951$	0	0
$952$	14.0959	0.456851
$953$	26.0120	0.842612	0.421306	−	0.906918i	$-0.361572\pi$
$953$	26.0120	0.842612	0.421306	+	0.906918i	$0.361572\pi$
$954$	0	0
$955$	−18.2877	−0.591777
$956$	20.8059	0.672911
$957$	0	0
$958$	14.7120	0.475323
$959$	−52.5136	−1.69575
$960$	0	0
$961$	−1.28772	−0.0415393
$962$	−1.00407	−0.0323725
$963$	0	0
$964$	−28.0098	−0.902134
$965$	7.85087	0.252728
$966$	0	0
$967$	19.5468	0.628583	0.314291	−	0.949327i	$-0.398233\pi$
$967$	19.5468	0.628583	0.314291	+	0.949327i	$0.398233\pi$
$968$	3.85614	0.123941
$969$	0	0
$970$	5.66203	0.181797
$971$	1.36827	0.0439098	0.0219549	−	0.999759i	$-0.493011\pi$
$971$	1.36827	0.0439098	0.0219549	+	0.999759i	$0.493011\pi$
$972$	0	0
$973$	67.4689	2.16295
$974$	−16.0000	−0.512673
$975$	0	0
$976$	2.30452	0.0737660
$977$	−20.7894	−0.665112	−0.332556	−	0.943084i	$-0.607911\pi$
$977$	−20.7894	−0.665112	−0.332556	+	0.943084i	$0.607911\pi$
$978$	0	0
$979$	17.7123	0.566087
$980$	−10.0028	−0.319529
$981$	0	0
$982$	34.0650	1.08706
$983$	−39.3913	−1.25639	−0.628193	−	0.778057i	$-0.716205\pi$
$983$	−39.3913	−1.25639	−0.628193	+	0.778057i	$0.716205\pi$
$984$	0	0
$985$	−19.0498	−0.606976
$986$	29.1254	0.927541
$987$	0	0
$988$	0.705481	0.0224443
$989$	0	0
$990$	0	0
$991$	−37.8345	−1.20185	−0.600927	−	0.799304i	$-0.705202\pi$
$991$	−37.8345	−1.20185	−0.600927	+	0.799304i	$0.705202\pi$
$992$	5.45090	0.173066
$993$	0	0
$994$	47.1000	1.49392
$995$	−7.54680	−0.239250
$996$	0	0
$997$	30.7247	0.973061	0.486531	−	0.873664i	$-0.338262\pi$
$997$	30.7247	0.973061	0.486531	+	0.873664i	$0.338262\pi$
$998$	−20.0457	−0.634534
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.cc.1.3	yes	6
3.2	odd	2		9522.2.a.cb.1.4	yes	6
23.22	odd	2	inner	9522.2.a.cc.1.4	yes	6
69.68	even	2		9522.2.a.cb.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9522.2.a.cb.1.3	✓	6	69.68	even	2
9522.2.a.cb.1.4	yes	6	3.2	odd	2
9522.2.a.cc.1.3	yes	6	1.1	even	1	trivial
9522.2.a.cc.1.4	yes	6	23.22	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.741015 q^{5} +4.52757 q^{7} +1.00000 q^{8} -0.741015 q^{10} +3.85437 q^{11} -1.04795 q^{13} +4.52757 q^{14} +1.00000 q^{16} +3.11335 q^{17} -0.673199 q^{19} -0.741015 q^{20} +3.85437 q^{22} -4.45090 q^{25} -1.04795 q^{26} +4.52757 q^{28} +9.35499 q^{29} +5.45090 q^{31} +1.00000 q^{32} +3.11335 q^{34} -3.35499 q^{35} +0.958124 q^{37} -0.673199 q^{38} -0.741015 q^{40} -12.4989 q^{41} -3.63726 q^{43} +3.85437 q^{44} +10.4029 q^{47} +13.4989 q^{49} -4.45090 q^{50} -1.04795 q^{52} -5.48569 q^{53} -2.85614 q^{55} +4.52757 q^{56} +9.35499 q^{58} -8.80589 q^{59} +2.30452 q^{61} +5.45090 q^{62} +1.00000 q^{64} +0.776548 q^{65} +10.6405 q^{67} +3.11335 q^{68} -3.35499 q^{70} +10.4029 q^{71} +4.95205 q^{73} +0.958124 q^{74} -0.673199 q^{76} +17.4509 q^{77} -3.18117 q^{79} -0.741015 q^{80} -12.4989 q^{82} +10.8576 q^{83} -2.30704 q^{85} -3.63726 q^{86} +3.85437 q^{88} +4.59538 q^{89} -4.74468 q^{91} +10.4029 q^{94} +0.498850 q^{95} -7.64092 q^{97} +13.4989 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 6 q^{16} + 18 q^{25} + 24 q^{29} - 12 q^{31} + 6 q^{32} + 12 q^{35} - 24 q^{41} + 24 q^{47} + 30 q^{49} + 18 q^{50} - 36 q^{55} + 24 q^{58} + 24 q^{59} - 12 q^{62} + 6 q^{64}+ \cdots + 30 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 6 * q^16 + 18 * q^25 + 24 * q^29 - 12 * q^31 + 6 * q^32 + 12 * q^35 - 24 * q^41 + 24 * q^47 + 30 * q^49 + 18 * q^50 - 36 * q^55 + 24 * q^58 + 24 * q^59 - 12 * q^62 + 6 * q^64 + 12 * q^70 + 24 * q^71 + 36 * q^73 + 60 * q^77 - 24 * q^82 + 12 * q^85 + 24 * q^94 - 48 * q^95 + 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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