LMFDB - Embedded newform 9522.2.a.ce.1.7

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 = [8,-8,0,8,0,0,0,-8,0,0,0,0,12,0,0,8,0,0,0,0,0,0,0,0,32,-12,0, 0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.819879542784.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 151x^{4} - 440x^{3} - 298x^{2} + 532x - 146 $ x^8 - 4x^7 - 22x^6 + 80x^5 + 151x^4 - 440x^3 - 298x^2 + 532*x - 146
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1058)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-3.30747$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.17513 q^{5} +3.35963 q^{7} -1.00000 q^{8} -3.17513 q^{10} +2.44949 q^{11} +5.10767 q^{13} -3.35963 q^{14} +1.00000 q^{16} +3.15163 q^{17} +3.69277 q^{19} +3.17513 q^{20} -2.44949 q^{22} +5.08146 q^{25} -5.10767 q^{26} +3.35963 q^{28} -9.59798 q^{29} +3.46410 q^{31} -1.00000 q^{32} -3.15163 q^{34} +10.6673 q^{35} -3.35963 q^{37} -3.69277 q^{38} -3.17513 q^{40} +1.82053 q^{41} +10.9161 q^{43} +2.44949 q^{44} -7.95442 q^{47} +4.28714 q^{49} -5.08146 q^{50} +5.10767 q^{52} +3.70634 q^{53} +7.77745 q^{55} -3.35963 q^{56} +9.59798 q^{58} -0.330219 q^{59} +14.6312 q^{61} -3.46410 q^{62} +1.00000 q^{64} +16.2175 q^{65} -1.58515 q^{67} +3.15163 q^{68} -10.6673 q^{70} -9.25560 q^{71} -7.66728 q^{73} +3.35963 q^{74} +3.69277 q^{76} +8.22939 q^{77} -10.9447 q^{79} +3.17513 q^{80} -1.82053 q^{82} -2.27856 q^{83} +10.0068 q^{85} -10.9161 q^{86} -2.44949 q^{88} +1.78805 q^{89} +17.1599 q^{91} +7.95442 q^{94} +11.7250 q^{95} +1.34491 q^{97} -4.28714 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 12 q^{13} + 8 q^{16} + 32 q^{25} - 12 q^{26} - 8 q^{32} + 12 q^{35} - 12 q^{41} + 12 q^{47} + 32 q^{49} - 32 q^{50} + 12 q^{52} + 12 q^{55} - 24 q^{59} + 8 q^{64} - 12 q^{70}+ \cdots - 32 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 12 * q^13 + 8 * q^16 + 32 * q^25 - 12 * q^26 - 8 * q^32 + 12 * q^35 - 12 * q^41 + 12 * q^47 + 32 * q^49 - 32 * q^50 + 12 * q^52 + 12 * q^55 - 24 * q^59 + 8 * q^64 - 12 * q^70 + 12 * q^71 + 12 * q^73 + 12 * q^82 - 36 * q^85 - 12 * q^94 + 84 * q^95 - 32 * 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
$5$	3.17513	1.41996	0.709981	−	0.704221i	$-0.248704\pi$
$5$	3.17513	1.41996	0.709981	+	0.704221i	$0.248704\pi$
$6$	0	0
$7$	3.35963	1.26982	0.634911	−	0.772585i	$-0.281037\pi$
$7$	3.35963	1.26982	0.634911	+	0.772585i	$0.281037\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.17513	−1.00406
$11$	2.44949	0.738549	0.369274	−	0.929320i	$-0.379606\pi$
$11$	2.44949	0.738549	0.369274	+	0.929320i	$0.379606\pi$
$12$	0	0
$13$	5.10767	1.41661	0.708306	−	0.705905i	$-0.249459\pi$
$13$	5.10767	1.41661	0.708306	+	0.705905i	$0.249459\pi$
$14$	−3.35963	−0.897900
$15$	0	0
$16$	1.00000	0.250000
$17$	3.15163	0.764383	0.382191	−	0.924083i	$-0.375170\pi$
$17$	3.15163	0.764383	0.382191	+	0.924083i	$0.375170\pi$
$18$	0	0
$19$	3.69277	0.847179	0.423590	−	0.905854i	$-0.360770\pi$
$19$	3.69277	0.847179	0.423590	+	0.905854i	$0.360770\pi$
$20$	3.17513	0.709981
$21$	0	0
$22$	−2.44949	−0.522233
$23$	0	0
$23$	0	0
$24$	0	0
$25$	5.08146	1.01629
$26$	−5.10767	−1.00170
$27$	0	0
$28$	3.35963	0.634911
$29$	−9.59798	−1.78230	−0.891150	−	0.453708i	$-0.850101\pi$
$29$	−9.59798	−1.78230	−0.891150	+	0.453708i	$0.850101\pi$
$30$	0	0
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.15163	−0.540500
$35$	10.6673	1.80310
$36$	0	0
$37$	−3.35963	−0.552320	−0.276160	−	0.961112i	$-0.589062\pi$
$37$	−3.35963	−0.552320	−0.276160	+	0.961112i	$0.589062\pi$
$38$	−3.69277	−0.599046
$39$	0	0
$40$	−3.17513	−0.502032
$41$	1.82053	0.284319	0.142160	−	0.989844i	$-0.454595\pi$
$41$	1.82053	0.284319	0.142160	+	0.989844i	$0.454595\pi$
$42$	0	0
$43$	10.9161	1.66469	0.832345	−	0.554257i	$-0.186998\pi$
$43$	10.9161	1.66469	0.832345	+	0.554257i	$0.186998\pi$
$44$	2.44949	0.369274
$45$	0	0
$46$	0	0
$47$	−7.95442	−1.16027	−0.580135	−	0.814520i	$-0.697000\pi$
$47$	−7.95442	−1.16027	−0.580135	+	0.814520i	$0.697000\pi$
$48$	0	0
$49$	4.28714	0.612448
$50$	−5.08146	−0.718627
$51$	0	0
$52$	5.10767	0.708306
$53$	3.70634	0.509105	0.254552	−	0.967059i	$-0.418072\pi$
$53$	3.70634	0.509105	0.254552	+	0.967059i	$0.418072\pi$
$54$	0	0
$55$	7.77745	1.04871
$56$	−3.35963	−0.448950
$57$	0	0
$58$	9.59798	1.26028
$59$	−0.330219	−0.0429909	−0.0214954	−	0.999769i	$-0.506843\pi$
$59$	−0.330219	−0.0429909	−0.0214954	+	0.999769i	$0.506843\pi$
$60$	0	0
$61$	14.6312	1.87333	0.936665	−	0.350226i	$-0.113895\pi$
$61$	14.6312	1.87333	0.936665	+	0.350226i	$0.113895\pi$
$62$	−3.46410	−0.439941
$63$	0	0
$64$	1.00000	0.125000
$65$	16.2175	2.01154
$66$	0	0
$67$	−1.58515	−0.193657	−0.0968283	−	0.995301i	$-0.530870\pi$
$67$	−1.58515	−0.193657	−0.0968283	+	0.995301i	$0.530870\pi$
$68$	3.15163	0.382191
$69$	0	0
$70$	−10.6673	−1.27498
$71$	−9.25560	−1.09844	−0.549219	−	0.835679i	$-0.685075\pi$
$71$	−9.25560	−1.09844	−0.549219	+	0.835679i	$0.685075\pi$
$72$	0	0
$73$	−7.66728	−0.897387	−0.448693	−	0.893686i	$-0.648110\pi$
$73$	−7.66728	−0.897387	−0.448693	+	0.893686i	$0.648110\pi$
$74$	3.35963	0.390549
$75$	0	0
$76$	3.69277	0.423590
$77$	8.22939	0.937826
$78$	0	0
$79$	−10.9447	−1.23138	−0.615688	−	0.787990i	$-0.711122\pi$
$79$	−10.9447	−1.23138	−0.615688	+	0.787990i	$0.711122\pi$
$80$	3.17513	0.354990
$81$	0	0
$82$	−1.82053	−0.201044
$83$	−2.27856	−0.250104	−0.125052	−	0.992150i	$-0.539910\pi$
$83$	−2.27856	−0.250104	−0.125052	+	0.992150i	$0.539910\pi$
$84$	0	0
$85$	10.0068	1.08539
$86$	−10.9161	−1.17711
$87$	0	0
$88$	−2.44949	−0.261116
$89$	1.78805	0.189533	0.0947667	−	0.995500i	$-0.469789\pi$
$89$	1.78805	0.189533	0.0947667	+	0.995500i	$0.469789\pi$
$90$	0	0
$91$	17.1599	1.79885
$92$	0	0
$93$	0	0
$94$	7.95442	0.820435
$95$	11.7250	1.20296
$96$	0	0
$97$	1.34491	0.136555	0.0682775	−	0.997666i	$-0.478250\pi$
$97$	1.34491	0.136555	0.0682775	+	0.997666i	$0.478250\pi$
$98$	−4.28714	−0.433066
$99$	0	0
$100$	5.08146	0.508146
$101$	−6.34238	−0.631091	−0.315545	−	0.948910i	$-0.602187\pi$
$101$	−6.34238	−0.631091	−0.315545	+	0.948910i	$0.602187\pi$
$102$	0	0
$103$	−0.785011	−0.0773495	−0.0386747	−	0.999252i	$-0.512314\pi$
$103$	−0.785011	−0.0773495	−0.0386747	+	0.999252i	$0.512314\pi$
$104$	−5.10767	−0.500848
$105$	0	0
$106$	−3.70634	−0.359991
$107$	12.3861	1.19741	0.598703	−	0.800971i	$-0.295683\pi$
$107$	12.3861	1.19741	0.598703	+	0.800971i	$0.295683\pi$
$108$	0	0
$109$	7.76957	0.744190	0.372095	−	0.928195i	$-0.378640\pi$
$109$	7.76957	0.744190	0.372095	+	0.928195i	$0.378640\pi$
$110$	−7.77745	−0.741551
$111$	0	0
$112$	3.35963	0.317456
$113$	−2.42599	−0.228218	−0.114109	−	0.993468i	$-0.536401\pi$
$113$	−2.42599	−0.228218	−0.114109	+	0.993468i	$0.536401\pi$
$114$	0	0
$115$	0	0
$116$	−9.59798	−0.891150
$117$	0	0
$118$	0.330219	0.0303991
$119$	10.5883	0.970630
$120$	0	0
$121$	−5.00000	−0.454545
$122$	−14.6312	−1.32464
$123$	0	0
$124$	3.46410	0.311086
$125$	0.258641	0.0231336
$126$	0	0
$127$	−4.76529	−0.422851	−0.211425	−	0.977394i	$-0.567811\pi$
$127$	−4.76529	−0.422851	−0.211425	+	0.977394i	$0.567811\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−16.2175	−1.42237
$131$	15.3108	1.33771	0.668857	−	0.743391i	$-0.266784\pi$
$131$	15.3108	1.33771	0.668857	+	0.743391i	$0.266784\pi$
$132$	0	0
$133$	12.4064	1.07577
$134$	1.58515	0.136936
$135$	0	0
$136$	−3.15163	−0.270250
$137$	1.42778	0.121984	0.0609918	−	0.998138i	$-0.480574\pi$
$137$	1.42778	0.121984	0.0609918	+	0.998138i	$0.480574\pi$
$138$	0	0
$139$	−11.7871	−0.999770	−0.499885	−	0.866092i	$-0.666624\pi$
$139$	−11.7871	−0.999770	−0.499885	+	0.866092i	$0.666624\pi$
$140$	10.6673	0.901549
$141$	0	0
$142$	9.25560	0.776713
$143$	12.5112	1.04624
$144$	0	0
$145$	−30.4749	−2.53080
$146$	7.66728	0.634548
$147$	0	0
$148$	−3.35963	−0.276160
$149$	−3.03279	−0.248456	−0.124228	−	0.992254i	$-0.539645\pi$
$149$	−3.03279	−0.248456	−0.124228	+	0.992254i	$0.539645\pi$
$150$	0	0
$151$	−8.54274	−0.695198	−0.347599	−	0.937643i	$-0.613003\pi$
$151$	−8.54274	−0.695198	−0.347599	+	0.937643i	$0.613003\pi$
$152$	−3.69277	−0.299523
$153$	0	0
$154$	−8.22939	−0.663143
$155$	10.9990	0.883459
$156$	0	0
$157$	9.24799	0.738070	0.369035	−	0.929415i	$-0.379688\pi$
$157$	9.24799	0.738070	0.369035	+	0.929415i	$0.379688\pi$
$158$	10.9447	0.870714
$159$	0	0
$160$	−3.17513	−0.251016
$161$	0	0
$162$	0	0
$163$	−18.9210	−1.48201	−0.741003	−	0.671501i	$-0.765650\pi$
$163$	−18.9210	−1.48201	−0.741003	+	0.671501i	$0.765650\pi$
$164$	1.82053	0.142160
$165$	0	0
$166$	2.27856	0.176850
$167$	−12.8177	−0.991864	−0.495932	−	0.868361i	$-0.665174\pi$
$167$	−12.8177	−0.991864	−0.495932	+	0.868361i	$0.665174\pi$
$168$	0	0
$169$	13.0883	1.00679
$170$	−10.0068	−0.767490
$171$	0	0
$172$	10.9161	0.832345
$173$	−15.5980	−1.18589	−0.592946	−	0.805242i	$-0.702035\pi$
$173$	−15.5980	−1.18589	−0.592946	+	0.805242i	$0.702035\pi$
$174$	0	0
$175$	17.0718	1.29051
$176$	2.44949	0.184637
$177$	0	0
$178$	−1.78805	−0.134020
$179$	7.13902	0.533596	0.266798	−	0.963753i	$-0.414034\pi$
$179$	7.13902	0.533596	0.266798	+	0.963753i	$0.414034\pi$
$180$	0	0
$181$	9.57129	0.711428	0.355714	−	0.934595i	$-0.384238\pi$
$181$	9.57129	0.711428	0.355714	+	0.934595i	$0.384238\pi$
$182$	−17.1599	−1.27198
$183$	0	0
$184$	0	0
$185$	−10.6673	−0.784274
$186$	0	0
$187$	7.71989	0.564534
$188$	−7.95442	−0.580135
$189$	0	0
$190$	−11.7250	−0.850623
$191$	−3.58630	−0.259496	−0.129748	−	0.991547i	$-0.541417\pi$
$191$	−3.58630	−0.259496	−0.129748	+	0.991547i	$0.541417\pi$
$192$	0	0
$193$	−20.2152	−1.45512	−0.727559	−	0.686045i	$-0.759345\pi$
$193$	−20.2152	−1.45512	−0.727559	+	0.686045i	$0.759345\pi$
$194$	−1.34491	−0.0965590
$195$	0	0
$196$	4.28714	0.306224
$197$	19.7890	1.40991	0.704954	−	0.709253i	$-0.250968\pi$
$197$	19.7890	1.40991	0.704954	+	0.709253i	$0.250968\pi$
$198$	0	0
$199$	−0.457068	−0.0324007	−0.0162003	−	0.999869i	$-0.505157\pi$
$199$	−0.457068	−0.0324007	−0.0162003	+	0.999869i	$0.505157\pi$
$200$	−5.08146	−0.359313
$201$	0	0
$202$	6.34238	0.446249
$203$	−32.2457	−2.26321
$204$	0	0
$205$	5.78043	0.403723
$206$	0.785011	0.0546943
$207$	0	0
$208$	5.10767	0.354153
$209$	9.04540	0.625683
$210$	0	0
$211$	0.904495	0.0622680	0.0311340	−	0.999515i	$-0.490088\pi$
$211$	0.904495	0.0622680	0.0311340	+	0.999515i	$0.490088\pi$
$212$	3.70634	0.254552
$213$	0	0
$214$	−12.3861	−0.846693
$215$	34.6601	2.36380
$216$	0	0
$217$	11.6381	0.790047
$218$	−7.76957	−0.526222
$219$	0	0
$220$	7.77745	0.524356
$221$	16.0975	1.08283
$222$	0	0
$223$	−11.6339	−0.779061	−0.389530	−	0.921014i	$-0.627363\pi$
$223$	−11.6339	−0.779061	−0.389530	+	0.921014i	$0.627363\pi$
$224$	−3.35963	−0.224475
$225$	0	0
$226$	2.42599	0.161374
$227$	−13.1524	−0.872955	−0.436478	−	0.899715i	$-0.643774\pi$
$227$	−13.1524	−0.872955	−0.436478	+	0.899715i	$0.643774\pi$
$228$	0	0
$229$	13.5935	0.898281	0.449141	−	0.893461i	$-0.351730\pi$
$229$	13.5935	0.898281	0.449141	+	0.893461i	$0.351730\pi$
$230$	0	0
$231$	0	0
$232$	9.59798	0.630139
$233$	−4.85641	−0.318154	−0.159077	−	0.987266i	$-0.550852\pi$
$233$	−4.85641	−0.318154	−0.159077	+	0.987266i	$0.550852\pi$
$234$	0	0
$235$	−25.2563	−1.64754
$236$	−0.330219	−0.0214954
$237$	0	0
$238$	−10.5883	−0.686339
$239$	17.3396	1.12160	0.560802	−	0.827950i	$-0.310493\pi$
$239$	17.3396	1.12160	0.560802	+	0.827950i	$0.310493\pi$
$240$	0	0
$241$	14.8021	0.953488	0.476744	−	0.879042i	$-0.341817\pi$
$241$	14.8021	0.953488	0.476744	+	0.879042i	$0.341817\pi$
$242$	5.00000	0.321412
$243$	0	0
$244$	14.6312	0.936665
$245$	13.6122	0.869653
$246$	0	0
$247$	18.8614	1.20013
$248$	−3.46410	−0.219971
$249$	0	0
$250$	−0.258641	−0.0163579
$251$	−0.0793307	−0.00500731	−0.00250366	−	0.999997i	$-0.500797\pi$
$251$	−0.0793307	−0.00500731	−0.00250366	+	0.999997i	$0.500797\pi$
$252$	0	0
$253$	0	0
$254$	4.76529	0.299001
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.4518	−0.901476	−0.450738	−	0.892656i	$-0.648839\pi$
$257$	−14.4518	−0.901476	−0.450738	+	0.892656i	$0.648839\pi$
$258$	0	0
$259$	−11.2871	−0.701349
$260$	16.2175	1.00577
$261$	0	0
$262$	−15.3108	−0.945907
$263$	−22.4007	−1.38129	−0.690645	−	0.723194i	$-0.742673\pi$
$263$	−22.4007	−1.38129	−0.690645	+	0.723194i	$0.742673\pi$
$264$	0	0
$265$	11.7681	0.722909
$266$	−12.4064	−0.760682
$267$	0	0
$268$	−1.58515	−0.0968283
$269$	−29.2100	−1.78097	−0.890483	−	0.455016i	$-0.849633\pi$
$269$	−29.2100	−1.78097	−0.890483	+	0.455016i	$0.849633\pi$
$270$	0	0
$271$	24.6145	1.49522	0.747612	−	0.664136i	$-0.231200\pi$
$271$	24.6145	1.49522	0.747612	+	0.664136i	$0.231200\pi$
$272$	3.15163	0.191096
$273$	0	0
$274$	−1.42778	−0.0862555
$275$	12.4470	0.750581
$276$	0	0
$277$	5.16260	0.310190	0.155095	−	0.987900i	$-0.450432\pi$
$277$	5.16260	0.310190	0.155095	+	0.987900i	$0.450432\pi$
$278$	11.7871	0.706944
$279$	0	0
$280$	−10.6673	−0.637492
$281$	−16.3662	−0.976323	−0.488162	−	0.872753i	$-0.662332\pi$
$281$	−16.3662	−0.976323	−0.488162	+	0.872753i	$0.662332\pi$
$282$	0	0
$283$	−18.1024	−1.07607	−0.538037	−	0.842921i	$-0.680834\pi$
$283$	−18.1024	−1.07607	−0.538037	+	0.842921i	$0.680834\pi$
$284$	−9.25560	−0.549219
$285$	0	0
$286$	−12.5112	−0.739802
$287$	6.11632	0.361035
$288$	0	0
$289$	−7.06723	−0.415719
$290$	30.4749	1.78955
$291$	0	0
$292$	−7.66728	−0.448693
$293$	13.8462	0.808902	0.404451	−	0.914560i	$-0.367463\pi$
$293$	13.8462	0.808902	0.404451	+	0.914560i	$0.367463\pi$
$294$	0	0
$295$	−1.04849	−0.0610454
$296$	3.35963	0.195275
$297$	0	0
$298$	3.03279	0.175685
$299$	0	0
$300$	0	0
$301$	36.6741	2.11386
$302$	8.54274	0.491579
$303$	0	0
$304$	3.69277	0.211795
$305$	46.4559	2.66006
$306$	0	0
$307$	31.2180	1.78170	0.890852	−	0.454293i	$-0.150108\pi$
$307$	31.2180	1.78170	0.890852	+	0.454293i	$0.150108\pi$
$308$	8.22939	0.468913
$309$	0	0
$310$	−10.9990	−0.624700
$311$	22.6673	1.28534	0.642672	−	0.766142i	$-0.277826\pi$
$311$	22.6673	1.28534	0.642672	+	0.766142i	$0.277826\pi$
$312$	0	0
$313$	−10.1260	−0.572355	−0.286178	−	0.958177i	$-0.592385\pi$
$313$	−10.1260	−0.572355	−0.286178	+	0.958177i	$0.592385\pi$
$314$	−9.24799	−0.521894
$315$	0	0
$316$	−10.9447	−0.615688
$317$	−31.1529	−1.74972	−0.874860	−	0.484375i	$-0.839047\pi$
$317$	−31.1529	−1.74972	−0.874860	+	0.484375i	$0.839047\pi$
$318$	0	0
$319$	−23.5102	−1.31632
$320$	3.17513	0.177495
$321$	0	0
$322$	0	0
$323$	11.6382	0.647569
$324$	0	0
$325$	25.9544	1.43969
$326$	18.9210	1.04794
$327$	0	0
$328$	−1.82053	−0.100522
$329$	−26.7239	−1.47334
$330$	0	0
$331$	−6.78216	−0.372781	−0.186390	−	0.982476i	$-0.559679\pi$
$331$	−6.78216	−0.372781	−0.186390	+	0.982476i	$0.559679\pi$
$332$	−2.27856	−0.125052
$333$	0	0
$334$	12.8177	0.701354
$335$	−5.03305	−0.274985
$336$	0	0
$337$	−32.0732	−1.74714	−0.873570	−	0.486699i	$-0.838201\pi$
$337$	−32.0732	−1.74714	−0.873570	+	0.486699i	$0.838201\pi$
$338$	−13.0883	−0.711910
$339$	0	0
$340$	10.0068	0.542697
$341$	8.48528	0.459504
$342$	0	0
$343$	−9.11422	−0.492122
$344$	−10.9161	−0.588557
$345$	0	0
$346$	15.5980	0.838553
$347$	10.5473	0.566206	0.283103	−	0.959089i	$-0.408636\pi$
$347$	10.5473	0.566206	0.283103	+	0.959089i	$0.408636\pi$
$348$	0	0
$349$	−1.69349	−0.0906504	−0.0453252	−	0.998972i	$-0.514432\pi$
$349$	−1.69349	−0.0906504	−0.0453252	+	0.998972i	$0.514432\pi$
$350$	−17.0718	−0.912528
$351$	0	0
$352$	−2.44949	−0.130558
$353$	−0.870319	−0.0463224	−0.0231612	−	0.999732i	$-0.507373\pi$
$353$	−0.870319	−0.0463224	−0.0231612	+	0.999732i	$0.507373\pi$
$354$	0	0
$355$	−29.3877	−1.55974
$356$	1.78805	0.0947667
$357$	0	0
$358$	−7.13902	−0.377309
$359$	−14.5546	−0.768163	−0.384081	−	0.923299i	$-0.625482\pi$
$359$	−14.5546	−0.768163	−0.384081	+	0.923299i	$0.625482\pi$
$360$	0	0
$361$	−5.36345	−0.282287
$362$	−9.57129	−0.503056
$363$	0	0
$364$	17.1599	0.899423
$365$	−24.3446	−1.27426
$366$	0	0
$367$	−37.8381	−1.97513	−0.987566	−	0.157203i	$-0.949752\pi$
$367$	−37.8381	−1.97513	−0.987566	+	0.157203i	$0.949752\pi$
$368$	0	0
$369$	0	0
$370$	10.6673	0.554565
$371$	12.4519	0.646472
$372$	0	0
$373$	15.2480	0.789512	0.394756	−	0.918786i	$-0.370829\pi$
$373$	15.2480	0.789512	0.394756	+	0.918786i	$0.370829\pi$
$374$	−7.71989	−0.399186
$375$	0	0
$376$	7.95442	0.410218
$377$	−49.0233	−2.52483
$378$	0	0
$379$	−12.4418	−0.639092	−0.319546	−	0.947571i	$-0.603530\pi$
$379$	−12.4418	−0.639092	−0.319546	+	0.947571i	$0.603530\pi$
$380$	11.7250	0.601481
$381$	0	0
$382$	3.58630	0.183491
$383$	5.62972	0.287665	0.143833	−	0.989602i	$-0.454057\pi$
$383$	5.62972	0.287665	0.143833	+	0.989602i	$0.454057\pi$
$384$	0	0
$385$	26.1294	1.33168
$386$	20.2152	1.02892
$387$	0	0
$388$	1.34491	0.0682775
$389$	−19.8769	−1.00780	−0.503898	−	0.863763i	$-0.668101\pi$
$389$	−19.8769	−1.00780	−0.503898	+	0.863763i	$0.668101\pi$
$390$	0	0
$391$	0	0
$392$	−4.28714	−0.216533
$393$	0	0
$394$	−19.7890	−0.996955
$395$	−34.7509	−1.74851
$396$	0	0
$397$	20.7871	1.04328	0.521638	−	0.853167i	$-0.325321\pi$
$397$	20.7871	1.04328	0.521638	+	0.853167i	$0.325321\pi$
$398$	0.457068	0.0229107
$399$	0	0
$400$	5.08146	0.254073
$401$	−29.8197	−1.48913	−0.744563	−	0.667552i	$-0.767342\pi$
$401$	−29.8197	−1.48913	−0.744563	+	0.667552i	$0.767342\pi$
$402$	0	0
$403$	17.6935	0.881375
$404$	−6.34238	−0.315545
$405$	0	0
$406$	32.2457	1.60033
$407$	−8.22939	−0.407916
$408$	0	0
$409$	−15.2913	−0.756106	−0.378053	−	0.925784i	$-0.623406\pi$
$409$	−15.2913	−0.756106	−0.378053	+	0.925784i	$0.623406\pi$
$410$	−5.78043	−0.285475
$411$	0	0
$412$	−0.785011	−0.0386747
$413$	−1.10942	−0.0545907
$414$	0	0
$415$	−7.23471	−0.355138
$416$	−5.10767	−0.250424
$417$	0	0
$418$	−9.04540	−0.442425
$419$	1.00960	0.0493222	0.0246611	−	0.999696i	$-0.492149\pi$
$419$	1.00960	0.0493222	0.0246611	+	0.999696i	$0.492149\pi$
$420$	0	0
$421$	−25.8310	−1.25892	−0.629462	−	0.777031i	$-0.716725\pi$
$421$	−25.8310	−1.25892	−0.629462	+	0.777031i	$0.716725\pi$
$422$	−0.904495	−0.0440301
$423$	0	0
$424$	−3.70634	−0.179996
$425$	16.0149	0.776836
$426$	0	0
$427$	49.1554	2.37880
$428$	12.3861	0.598703
$429$	0	0
$430$	−34.6601	−1.67146
$431$	30.4056	1.46458	0.732292	−	0.680991i	$-0.238451\pi$
$431$	30.4056	1.46458	0.732292	+	0.680991i	$0.238451\pi$
$432$	0	0
$433$	−18.5048	−0.889285	−0.444642	−	0.895708i	$-0.646669\pi$
$433$	−18.5048	−0.889285	−0.444642	+	0.895708i	$0.646669\pi$
$434$	−11.6381	−0.558647
$435$	0	0
$436$	7.76957	0.372095
$437$	0	0
$438$	0	0
$439$	−24.3202	−1.16074	−0.580370	−	0.814353i	$-0.697092\pi$
$439$	−24.3202	−1.16074	−0.580370	+	0.814353i	$0.697092\pi$
$440$	−7.77745	−0.370775
$441$	0	0
$442$	−16.0975	−0.765679
$443$	1.99548	0.0948081	0.0474041	−	0.998876i	$-0.484905\pi$
$443$	1.99548	0.0948081	0.0474041	+	0.998876i	$0.484905\pi$
$444$	0	0
$445$	5.67731	0.269130
$446$	11.6339	0.550879
$447$	0	0
$448$	3.35963	0.158728
$449$	−2.46617	−0.116386	−0.0581928	−	0.998305i	$-0.518534\pi$
$449$	−2.46617	−0.116386	−0.0581928	+	0.998305i	$0.518534\pi$
$450$	0	0
$451$	4.45938	0.209984
$452$	−2.42599	−0.114109
$453$	0	0
$454$	13.1524	0.617273
$455$	54.4849	2.55429
$456$	0	0
$457$	26.5566	1.24227	0.621133	−	0.783705i	$-0.286673\pi$
$457$	26.5566	1.24227	0.621133	+	0.783705i	$0.286673\pi$
$458$	−13.5935	−0.635181
$459$	0	0
$460$	0	0
$461$	6.13388	0.285683	0.142842	−	0.989746i	$-0.454376\pi$
$461$	6.13388	0.285683	0.142842	+	0.989746i	$0.454376\pi$
$462$	0	0
$463$	13.5549	0.629950	0.314975	−	0.949100i	$-0.398004\pi$
$463$	13.5549	0.629950	0.314975	+	0.949100i	$0.398004\pi$
$464$	−9.59798	−0.445575
$465$	0	0
$466$	4.85641	0.224969
$467$	−27.1783	−1.25766	−0.628831	−	0.777542i	$-0.716466\pi$
$467$	−27.1783	−1.25766	−0.628831	+	0.777542i	$0.716466\pi$
$468$	0	0
$469$	−5.32551	−0.245909
$470$	25.2563	1.16499
$471$	0	0
$472$	0.330219	0.0151996
$473$	26.7389	1.22946
$474$	0	0
$475$	18.7647	0.860981
$476$	10.5883	0.485315
$477$	0	0
$478$	−17.3396	−0.793093
$479$	34.8649	1.59302	0.796510	−	0.604626i	$-0.206677\pi$
$479$	34.8649	1.59302	0.796510	+	0.604626i	$0.206677\pi$
$480$	0	0
$481$	−17.1599	−0.782424
$482$	−14.8021	−0.674218
$483$	0	0
$484$	−5.00000	−0.227273
$485$	4.27027	0.193903
$486$	0	0
$487$	36.9107	1.67258	0.836292	−	0.548284i	$-0.184719\pi$
$487$	36.9107	1.67258	0.836292	+	0.548284i	$0.184719\pi$
$488$	−14.6312	−0.662322
$489$	0	0
$490$	−13.6122	−0.614938
$491$	−31.6407	−1.42792	−0.713962	−	0.700184i	$-0.753101\pi$
$491$	−31.6407	−1.42792	−0.713962	+	0.700184i	$0.753101\pi$
$492$	0	0
$493$	−30.2493	−1.36236
$494$	−18.8614	−0.848617
$495$	0	0
$496$	3.46410	0.155543
$497$	−31.0954	−1.39482
$498$	0	0
$499$	−32.9954	−1.47708	−0.738539	−	0.674211i	$-0.764484\pi$
$499$	−32.9954	−1.47708	−0.738539	+	0.674211i	$0.764484\pi$
$500$	0.258641	0.0115668
$501$	0	0
$502$	0.0793307	0.00354070
$503$	17.6948	0.788974	0.394487	−	0.918902i	$-0.370922\pi$
$503$	17.6948	0.788974	0.394487	+	0.918902i	$0.370922\pi$
$504$	0	0
$505$	−20.1379	−0.896125
$506$	0	0
$507$	0	0
$508$	−4.76529	−0.211425
$509$	13.0864	0.580045	0.290023	−	0.957020i	$-0.406337\pi$
$509$	13.0864	0.580045	0.290023	+	0.957020i	$0.406337\pi$
$510$	0	0
$511$	−25.7592	−1.13952
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	14.4518	0.637439
$515$	−2.49251	−0.109833
$516$	0	0
$517$	−19.4843	−0.856917
$518$	11.2871	0.495928
$519$	0	0
$520$	−16.2175	−0.711185
$521$	−28.8417	−1.26358	−0.631790	−	0.775140i	$-0.717679\pi$
$521$	−28.8417	−1.26358	−0.631790	+	0.775140i	$0.717679\pi$
$522$	0	0
$523$	34.1876	1.49492	0.747459	−	0.664307i	$-0.231273\pi$
$523$	34.1876	1.49492	0.747459	+	0.664307i	$0.231273\pi$
$524$	15.3108	0.668857
$525$	0	0
$526$	22.4007	0.976719
$527$	10.9176	0.475577
$528$	0	0
$529$	0	0
$530$	−11.7681	−0.511174
$531$	0	0
$532$	12.4064	0.537884
$533$	9.29868	0.402771
$534$	0	0
$535$	39.3273	1.70027
$536$	1.58515	0.0684679
$537$	0	0
$538$	29.2100	1.25933
$539$	10.5013	0.452323
$540$	0	0
$541$	−37.6314	−1.61790	−0.808949	−	0.587879i	$-0.799963\pi$
$541$	−37.6314	−1.61790	−0.808949	+	0.587879i	$0.799963\pi$
$542$	−24.6145	−1.05728
$543$	0	0
$544$	−3.15163	−0.135125
$545$	24.6694	1.05672
$546$	0	0
$547$	18.3783	0.785798	0.392899	−	0.919582i	$-0.371472\pi$
$547$	18.3783	0.785798	0.392899	+	0.919582i	$0.371472\pi$
$548$	1.42778	0.0609918
$549$	0	0
$550$	−12.4470	−0.530741
$551$	−35.4431	−1.50993
$552$	0	0
$553$	−36.7702	−1.56363
$554$	−5.16260	−0.219338
$555$	0	0
$556$	−11.7871	−0.499885
$557$	−6.39908	−0.271138	−0.135569	−	0.990768i	$-0.543286\pi$
$557$	−6.39908	−0.271138	−0.135569	+	0.990768i	$0.543286\pi$
$558$	0	0
$559$	55.7559	2.35822
$560$	10.6673	0.450775
$561$	0	0
$562$	16.3662	0.690365
$563$	0.485405	0.0204574	0.0102287	−	0.999948i	$-0.496744\pi$
$563$	0.485405	0.0204574	0.0102287	+	0.999948i	$0.496744\pi$
$564$	0	0
$565$	−7.70283	−0.324061
$566$	18.1024	0.760900
$567$	0	0
$568$	9.25560	0.388356
$569$	−3.48840	−0.146241	−0.0731207	−	0.997323i	$-0.523296\pi$
$569$	−3.48840	−0.146241	−0.0731207	+	0.997323i	$0.523296\pi$
$570$	0	0
$571$	34.3011	1.43546	0.717729	−	0.696323i	$-0.245182\pi$
$571$	34.3011	1.43546	0.717729	+	0.696323i	$0.245182\pi$
$572$	12.5112	0.523119
$573$	0	0
$574$	−6.11632	−0.255290
$575$	0	0
$576$	0	0
$577$	31.8779	1.32709	0.663547	−	0.748134i	$-0.269050\pi$
$577$	31.8779	1.32709	0.663547	+	0.748134i	$0.269050\pi$
$578$	7.06723	0.293958
$579$	0	0
$580$	−30.4749	−1.26540
$581$	−7.65511	−0.317588
$582$	0	0
$583$	9.07864	0.375999
$584$	7.66728	0.317274
$585$	0	0
$586$	−13.8462	−0.571980
$587$	−5.30401	−0.218920	−0.109460	−	0.993991i	$-0.534912\pi$
$587$	−5.30401	−0.218920	−0.109460	+	0.993991i	$0.534912\pi$
$588$	0	0
$589$	12.7921	0.527090
$590$	1.04849	0.0431656
$591$	0	0
$592$	−3.35963	−0.138080
$593$	−28.6262	−1.17554	−0.587769	−	0.809029i	$-0.699994\pi$
$593$	−28.6262	−1.17554	−0.587769	+	0.809029i	$0.699994\pi$
$594$	0	0
$595$	33.6193	1.37826
$596$	−3.03279	−0.124228
$597$	0	0
$598$	0	0
$599$	24.2993	0.992843	0.496421	−	0.868082i	$-0.334647\pi$
$599$	24.2993	0.992843	0.496421	+	0.868082i	$0.334647\pi$
$600$	0	0
$601$	−9.32985	−0.380573	−0.190286	−	0.981729i	$-0.560942\pi$
$601$	−9.32985	−0.380573	−0.190286	+	0.981729i	$0.560942\pi$
$602$	−36.6741	−1.49473
$603$	0	0
$604$	−8.54274	−0.347599
$605$	−15.8757	−0.645437
$606$	0	0
$607$	−4.14543	−0.168258	−0.0841288	−	0.996455i	$-0.526811\pi$
$607$	−4.14543	−0.168258	−0.0841288	+	0.996455i	$0.526811\pi$
$608$	−3.69277	−0.149762
$609$	0	0
$610$	−46.4559	−1.88094
$611$	−40.6285	−1.64365
$612$	0	0
$613$	−34.0907	−1.37691	−0.688455	−	0.725279i	$-0.741711\pi$
$613$	−34.0907	−1.37691	−0.688455	+	0.725279i	$0.741711\pi$
$614$	−31.2180	−1.25986
$615$	0	0
$616$	−8.22939	−0.331571
$617$	17.5304	0.705746	0.352873	−	0.935671i	$-0.385205\pi$
$617$	17.5304	0.705746	0.352873	+	0.935671i	$0.385205\pi$
$618$	0	0
$619$	−33.6769	−1.35359	−0.676794	−	0.736173i	$-0.736631\pi$
$619$	−33.6769	−1.35359	−0.676794	+	0.736173i	$0.736631\pi$
$620$	10.9990	0.441730
$621$	0	0
$622$	−22.6673	−0.908875
$623$	6.00721	0.240674
$624$	0	0
$625$	−24.5861	−0.983443
$626$	10.1260	0.404716
$627$	0	0
$628$	9.24799	0.369035
$629$	−10.5883	−0.422184
$630$	0	0
$631$	−31.0952	−1.23788	−0.618941	−	0.785438i	$-0.712438\pi$
$631$	−31.0952	−1.23788	−0.618941	+	0.785438i	$0.712438\pi$
$632$	10.9447	0.435357
$633$	0	0
$634$	31.1529	1.23724
$635$	−15.1304	−0.600432
$636$	0	0
$637$	21.8973	0.867602
$638$	23.5102	0.930776
$639$	0	0
$640$	−3.17513	−0.125508
$641$	2.67953	0.105835	0.0529176	−	0.998599i	$-0.483148\pi$
$641$	2.67953	0.105835	0.0529176	+	0.998599i	$0.483148\pi$
$642$	0	0
$643$	38.9484	1.53597	0.767987	−	0.640465i	$-0.221258\pi$
$643$	38.9484	1.53597	0.767987	+	0.640465i	$0.221258\pi$
$644$	0	0
$645$	0	0
$646$	−11.6382	−0.457901
$647$	−6.67756	−0.262522	−0.131261	−	0.991348i	$-0.541903\pi$
$647$	−6.67756	−0.262522	−0.131261	+	0.991348i	$0.541903\pi$
$648$	0	0
$649$	−0.808868	−0.0317509
$650$	−25.9544	−1.01802
$651$	0	0
$652$	−18.9210	−0.741003
$653$	5.28463	0.206804	0.103402	−	0.994640i	$-0.467027\pi$
$653$	5.28463	0.206804	0.103402	+	0.994640i	$0.467027\pi$
$654$	0	0
$655$	48.6139	1.89950
$656$	1.82053	0.0710799
$657$	0	0
$658$	26.7239	1.04181
$659$	3.38557	0.131883	0.0659415	−	0.997823i	$-0.478995\pi$
$659$	3.38557	0.131883	0.0659415	+	0.997823i	$0.478995\pi$
$660$	0	0
$661$	−17.5536	−0.682756	−0.341378	−	0.939926i	$-0.610894\pi$
$661$	−17.5536	−0.682756	−0.341378	+	0.939926i	$0.610894\pi$
$662$	6.78216	0.263596
$663$	0	0
$664$	2.27856	0.0884251
$665$	39.3918	1.52755
$666$	0	0
$667$	0	0
$668$	−12.8177	−0.495932
$669$	0	0
$670$	5.03305	0.194444
$671$	35.8389	1.38355
$672$	0	0
$673$	−20.6172	−0.794734	−0.397367	−	0.917660i	$-0.630076\pi$
$673$	−20.6172	−0.794734	−0.397367	+	0.917660i	$0.630076\pi$
$674$	32.0732	1.23541
$675$	0	0
$676$	13.0883	0.503396
$677$	−6.83911	−0.262848	−0.131424	−	0.991326i	$-0.541955\pi$
$677$	−6.83911	−0.262848	−0.131424	+	0.991326i	$0.541955\pi$
$678$	0	0
$679$	4.51841	0.173401
$680$	−10.0068	−0.383745
$681$	0	0
$682$	−8.48528	−0.324918
$683$	12.5861	0.481595	0.240798	−	0.970575i	$-0.422591\pi$
$683$	12.5861	0.481595	0.240798	+	0.970575i	$0.422591\pi$
$684$	0	0
$685$	4.53339	0.173212
$686$	9.11422	0.347983
$687$	0	0
$688$	10.9161	0.416173
$689$	18.9308	0.721204
$690$	0	0
$691$	18.3065	0.696412	0.348206	−	0.937418i	$-0.386791\pi$
$691$	18.3065	0.696412	0.348206	+	0.937418i	$0.386791\pi$
$692$	−15.5980	−0.592946
$693$	0	0
$694$	−10.5473	−0.400368
$695$	−37.4256	−1.41964
$696$	0	0
$697$	5.73765	0.217329
$698$	1.69349	0.0640995
$699$	0	0
$700$	17.0718	0.645255
$701$	42.7172	1.61341	0.806703	−	0.590957i	$-0.201250\pi$
$701$	42.7172	1.61341	0.806703	+	0.590957i	$0.201250\pi$
$702$	0	0
$703$	−12.4064	−0.467914
$704$	2.44949	0.0923186
$705$	0	0
$706$	0.870319	0.0327549
$707$	−21.3081	−0.801373
$708$	0	0
$709$	−15.4559	−0.580459	−0.290230	−	0.956957i	$-0.593732\pi$
$709$	−15.4559	−0.580459	−0.290230	+	0.956957i	$0.593732\pi$
$710$	29.3877	1.10290
$711$	0	0
$712$	−1.78805	−0.0670102
$713$	0	0
$714$	0	0
$715$	39.7247	1.48562
$716$	7.13902	0.266798
$717$	0	0
$718$	14.5546	0.543173
$719$	30.8651	1.15108	0.575538	−	0.817775i	$-0.304793\pi$
$719$	30.8651	1.15108	0.575538	+	0.817775i	$0.304793\pi$
$720$	0	0
$721$	−2.63735	−0.0982201
$722$	5.36345	0.199607
$723$	0	0
$724$	9.57129	0.355714
$725$	−48.7718	−1.81134
$726$	0	0
$727$	−2.43290	−0.0902314	−0.0451157	−	0.998982i	$-0.514366\pi$
$727$	−2.43290	−0.0902314	−0.0451157	+	0.998982i	$0.514366\pi$
$728$	−17.1599	−0.635988
$729$	0	0
$730$	24.3446	0.901035
$731$	34.4035	1.27246
$732$	0	0
$733$	4.34901	0.160634	0.0803172	−	0.996769i	$-0.474407\pi$
$733$	4.34901	0.160634	0.0803172	+	0.996769i	$0.474407\pi$
$734$	37.8381	1.39663
$735$	0	0
$736$	0	0
$737$	−3.88280	−0.143025
$738$	0	0
$739$	38.7250	1.42452	0.712261	−	0.701915i	$-0.247671\pi$
$739$	38.7250	1.42452	0.712261	+	0.701915i	$0.247671\pi$
$740$	−10.6673	−0.392137
$741$	0	0
$742$	−12.4519	−0.457125
$743$	−9.99749	−0.366772	−0.183386	−	0.983041i	$-0.558706\pi$
$743$	−9.99749	−0.366772	−0.183386	+	0.983041i	$0.558706\pi$
$744$	0	0
$745$	−9.62952	−0.352798
$746$	−15.2480	−0.558269
$747$	0	0
$748$	7.71989	0.282267
$749$	41.6126	1.52049
$750$	0	0
$751$	51.9604	1.89606	0.948030	−	0.318180i	$-0.103072\pi$
$751$	51.9604	1.89606	0.948030	+	0.318180i	$0.103072\pi$
$752$	−7.95442	−0.290068
$753$	0	0
$754$	49.0233	1.78532
$755$	−27.1243	−0.987155
$756$	0	0
$757$	10.9531	0.398096	0.199048	−	0.979990i	$-0.436215\pi$
$757$	10.9531	0.398096	0.199048	+	0.979990i	$0.436215\pi$
$758$	12.4418	0.451906
$759$	0	0
$760$	−11.7250	−0.425311
$761$	−1.53603	−0.0556812	−0.0278406	−	0.999612i	$-0.508863\pi$
$761$	−1.53603	−0.0556812	−0.0278406	+	0.999612i	$0.508863\pi$
$762$	0	0
$763$	26.1029	0.944989
$764$	−3.58630	−0.129748
$765$	0	0
$766$	−5.62972	−0.203410
$767$	−1.68665	−0.0609014
$768$	0	0
$769$	25.1481	0.906865	0.453432	−	0.891291i	$-0.350199\pi$
$769$	25.1481	0.906865	0.453432	+	0.891291i	$0.350199\pi$
$770$	−26.1294	−0.941638
$771$	0	0
$772$	−20.2152	−0.727559
$773$	−5.66039	−0.203590	−0.101795	−	0.994805i	$-0.532459\pi$
$773$	−5.66039	−0.203590	−0.101795	+	0.994805i	$0.532459\pi$
$774$	0	0
$775$	17.6027	0.632307
$776$	−1.34491	−0.0482795
$777$	0	0
$778$	19.8769	0.712620
$779$	6.72281	0.240870
$780$	0	0
$781$	−22.6715	−0.811250
$782$	0	0
$783$	0	0
$784$	4.28714	0.153112
$785$	29.3636	1.04803
$786$	0	0
$787$	28.6070	1.01973	0.509864	−	0.860255i	$-0.329696\pi$
$787$	28.6070	1.01973	0.509864	+	0.860255i	$0.329696\pi$
$788$	19.7890	0.704954
$789$	0	0
$790$	34.7509	1.23638
$791$	−8.15043	−0.289796
$792$	0	0
$793$	74.7312	2.65378
$794$	−20.7871	−0.737707
$795$	0	0
$796$	−0.457068	−0.0162003
$797$	−20.0053	−0.708623	−0.354312	−	0.935127i	$-0.615285\pi$
$797$	−20.0053	−0.708623	−0.354312	+	0.935127i	$0.615285\pi$
$798$	0	0
$799$	−25.0694	−0.886891
$800$	−5.08146	−0.179657
$801$	0	0
$802$	29.8197	1.05297
$803$	−18.7809	−0.662764
$804$	0	0
$805$	0	0
$806$	−17.6935	−0.623227
$807$	0	0
$808$	6.34238	0.223124
$809$	53.0334	1.86456	0.932278	−	0.361742i	$-0.117818\pi$
$809$	53.0334	1.86456	0.932278	+	0.361742i	$0.117818\pi$
$810$	0	0
$811$	−1.45444	−0.0510723	−0.0255361	−	0.999674i	$-0.508129\pi$
$811$	−1.45444	−0.0510723	−0.0255361	+	0.999674i	$0.508129\pi$
$812$	−32.2457	−1.13160
$813$	0	0
$814$	8.22939	0.288440
$815$	−60.0766	−2.10439
$816$	0	0
$817$	40.3107	1.41029
$818$	15.2913	0.534647
$819$	0	0
$820$	5.78043	0.201861
$821$	55.2983	1.92992	0.964962	−	0.262389i	$-0.0845103\pi$
$821$	55.2983	1.92992	0.964962	+	0.262389i	$0.0845103\pi$
$822$	0	0
$823$	−5.59548	−0.195046	−0.0975231	−	0.995233i	$-0.531092\pi$
$823$	−5.59548	−0.195046	−0.0975231	+	0.995233i	$0.531092\pi$
$824$	0.785011	0.0273472
$825$	0	0
$826$	1.10942	0.0386015
$827$	54.6141	1.89912	0.949560	−	0.313586i	$-0.101531\pi$
$827$	54.6141	1.89912	0.949560	+	0.313586i	$0.101531\pi$
$828$	0	0
$829$	−34.6310	−1.20279	−0.601393	−	0.798954i	$-0.705387\pi$
$829$	−34.6310	−1.20279	−0.601393	+	0.798954i	$0.705387\pi$
$830$	7.23471	0.251121
$831$	0	0
$832$	5.10767	0.177077
$833$	13.5115	0.468145
$834$	0	0
$835$	−40.6979	−1.40841
$836$	9.04540	0.312842
$837$	0	0
$838$	−1.00960	−0.0348761
$839$	12.7414	0.439882	0.219941	−	0.975513i	$-0.429414\pi$
$839$	12.7414	0.439882	0.219941	+	0.975513i	$0.429414\pi$
$840$	0	0
$841$	63.1213	2.17660
$842$	25.8310	0.890194
$843$	0	0
$844$	0.904495	0.0311340
$845$	41.5571	1.42961
$846$	0	0
$847$	−16.7982	−0.577192
$848$	3.70634	0.127276
$849$	0	0
$850$	−16.0149	−0.549306
$851$	0	0
$852$	0	0
$853$	−9.48342	−0.324706	−0.162353	−	0.986733i	$-0.551908\pi$
$853$	−9.48342	−0.324706	−0.162353	+	0.986733i	$0.551908\pi$
$854$	−49.1554	−1.68206
$855$	0	0
$856$	−12.3861	−0.423347
$857$	−15.9525	−0.544928	−0.272464	−	0.962166i	$-0.587839\pi$
$857$	−15.9525	−0.544928	−0.272464	+	0.962166i	$0.587839\pi$
$858$	0	0
$859$	−44.5572	−1.52027	−0.760136	−	0.649765i	$-0.774867\pi$
$859$	−44.5572	−1.52027	−0.760136	+	0.649765i	$0.774867\pi$
$860$	34.6601	1.18190
$861$	0	0
$862$	−30.4056	−1.03562
$863$	37.1835	1.26574	0.632870	−	0.774258i	$-0.281877\pi$
$863$	37.1835	1.26574	0.632870	+	0.774258i	$0.281877\pi$
$864$	0	0
$865$	−49.5256	−1.68392
$866$	18.5048	0.628819
$867$	0	0
$868$	11.6381	0.395023
$869$	−26.8089	−0.909431
$870$	0	0
$871$	−8.09641	−0.274336
$872$	−7.76957	−0.263111
$873$	0	0
$874$	0	0
$875$	0.868940	0.0293755
$876$	0	0
$877$	11.3152	0.382088	0.191044	−	0.981581i	$-0.438813\pi$
$877$	11.3152	0.382088	0.191044	+	0.981581i	$0.438813\pi$
$878$	24.3202	0.820767
$879$	0	0
$880$	7.77745	0.262178
$881$	52.0831	1.75472	0.877362	−	0.479830i	$-0.159302\pi$
$881$	52.0831	1.75472	0.877362	+	0.479830i	$0.159302\pi$
$882$	0	0
$883$	6.48749	0.218321	0.109161	−	0.994024i	$-0.465184\pi$
$883$	6.48749	0.218321	0.109161	+	0.994024i	$0.465184\pi$
$884$	16.0975	0.541417
$885$	0	0
$886$	−1.99548	−0.0670395
$887$	47.3448	1.58968	0.794842	−	0.606816i	$-0.207554\pi$
$887$	47.3448	1.58968	0.794842	+	0.606816i	$0.207554\pi$
$888$	0	0
$889$	−16.0096	−0.536945
$890$	−5.67731	−0.190304
$891$	0	0
$892$	−11.6339	−0.389530
$893$	−29.3738	−0.982957
$894$	0	0
$895$	22.6673	0.757686
$896$	−3.35963	−0.112237
$897$	0	0
$898$	2.46617	0.0822971
$899$	−33.2484	−1.10890
$900$	0	0
$901$	11.6810	0.389151
$902$	−4.45938	−0.148481
$903$	0	0
$904$	2.42599	0.0806872
$905$	30.3901	1.01020
$906$	0	0
$907$	0.566196	0.0188002	0.00940012	−	0.999956i	$-0.497008\pi$
$907$	0.566196	0.0188002	0.00940012	+	0.999956i	$0.497008\pi$
$908$	−13.1524	−0.436478
$909$	0	0
$910$	−54.4849	−1.80616
$911$	1.80334	0.0597475	0.0298737	−	0.999554i	$-0.490489\pi$
$911$	1.80334	0.0597475	0.0298737	+	0.999554i	$0.490489\pi$
$912$	0	0
$913$	−5.58130	−0.184714
$914$	−26.5566	−0.878415
$915$	0	0
$916$	13.5935	0.449141
$917$	51.4388	1.69866
$918$	0	0
$919$	7.57488	0.249872	0.124936	−	0.992165i	$-0.460127\pi$
$919$	7.57488	0.249872	0.124936	+	0.992165i	$0.460127\pi$
$920$	0	0
$921$	0	0
$922$	−6.13388	−0.202009
$923$	−47.2746	−1.55606
$924$	0	0
$925$	−17.0718	−0.561319
$926$	−13.5549	−0.445442
$927$	0	0
$928$	9.59798	0.315069
$929$	2.82336	0.0926313	0.0463156	−	0.998927i	$-0.485252\pi$
$929$	2.82336	0.0926313	0.0463156	+	0.998927i	$0.485252\pi$
$930$	0	0
$931$	15.8314	0.518854
$932$	−4.85641	−0.159077
$933$	0	0
$934$	27.1783	0.889302
$935$	24.5117	0.801617
$936$	0	0
$937$	−41.3956	−1.35234	−0.676168	−	0.736748i	$-0.736361\pi$
$937$	−41.3956	−1.35234	−0.676168	+	0.736748i	$0.736361\pi$
$938$	5.32551	0.173884
$939$	0	0
$940$	−25.2563	−0.823770
$941$	5.44038	0.177351	0.0886756	−	0.996061i	$-0.471737\pi$
$941$	5.44038	0.177351	0.0886756	+	0.996061i	$0.471737\pi$
$942$	0	0
$943$	0	0
$944$	−0.330219	−0.0107477
$945$	0	0
$946$	−26.7389	−0.869356
$947$	56.0285	1.82068	0.910341	−	0.413860i	$-0.135820\pi$
$947$	56.0285	1.82068	0.910341	+	0.413860i	$0.135820\pi$
$948$	0	0
$949$	−39.1619	−1.27125
$950$	−18.7647	−0.608806
$951$	0	0
$952$	−10.5883	−0.343170
$953$	43.6931	1.41536	0.707679	−	0.706534i	$-0.249742\pi$
$953$	43.6931	1.41536	0.707679	+	0.706534i	$0.249742\pi$
$954$	0	0
$955$	−11.3870	−0.368474
$956$	17.3396	0.560802
$957$	0	0
$958$	−34.8649	−1.12643
$959$	4.79682	0.154898
$960$	0	0
$961$	−19.0000	−0.612903
$962$	17.1599	0.553257
$963$	0	0
$964$	14.8021	0.476744
$965$	−64.1858	−2.06621
$966$	0	0
$967$	23.6586	0.760808	0.380404	−	0.924820i	$-0.375785\pi$
$967$	23.6586	0.760808	0.380404	+	0.924820i	$0.375785\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	−4.27027	−0.137110
$971$	56.8330	1.82386	0.911928	−	0.410350i	$-0.134593\pi$
$971$	56.8330	1.82386	0.911928	+	0.410350i	$0.134593\pi$
$972$	0	0
$973$	−39.6004	−1.26953
$974$	−36.9107	−1.18270
$975$	0	0
$976$	14.6312	0.468333
$977$	−27.2500	−0.871806	−0.435903	−	0.899994i	$-0.643571\pi$
$977$	−27.2500	−0.871806	−0.435903	+	0.899994i	$0.643571\pi$
$978$	0	0
$979$	4.37982	0.139980
$980$	13.6122	0.434827
$981$	0	0
$982$	31.6407	1.00970
$983$	−34.3668	−1.09613	−0.548065	−	0.836435i	$-0.684636\pi$
$983$	−34.3668	−1.09613	−0.548065	+	0.836435i	$0.684636\pi$
$984$	0	0
$985$	62.8327	2.00201
$986$	30.2493	0.963334
$987$	0	0
$988$	18.8614	0.600063
$989$	0	0
$990$	0	0
$991$	−40.7022	−1.29295	−0.646474	−	0.762936i	$-0.723757\pi$
$991$	−40.7022	−1.29295	−0.646474	+	0.762936i	$0.723757\pi$
$992$	−3.46410	−0.109985
$993$	0	0
$994$	31.0954	0.986287
$995$	−1.45125	−0.0460077
$996$	0	0
$997$	36.8183	1.16605	0.583023	−	0.812455i	$-0.301870\pi$
$997$	36.8183	1.16605	0.583023	+	0.812455i	$0.301870\pi$
$998$	32.9954	1.04445
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.ce.1.7		8
3.2	odd	2		1058.2.a.n.1.3	✓	8
12.11	even	2		8464.2.a.cb.1.5		8
23.22	odd	2	inner	9522.2.a.ce.1.2		8
69.68	even	2		1058.2.a.n.1.4	yes	8
276.275	odd	2		8464.2.a.cb.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1058.2.a.n.1.3	✓	8	3.2	odd	2
1058.2.a.n.1.4	yes	8	69.68	even	2
8464.2.a.cb.1.5		8	12.11	even	2
8464.2.a.cb.1.6		8	276.275	odd	2
9522.2.a.ce.1.2		8	23.22	odd	2	inner
9522.2.a.ce.1.7		8	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.17513 q^{5} +3.35963 q^{7} -1.00000 q^{8} -3.17513 q^{10} +2.44949 q^{11} +5.10767 q^{13} -3.35963 q^{14} +1.00000 q^{16} +3.15163 q^{17} +3.69277 q^{19} +3.17513 q^{20} -2.44949 q^{22} +5.08146 q^{25} -5.10767 q^{26} +3.35963 q^{28} -9.59798 q^{29} +3.46410 q^{31} -1.00000 q^{32} -3.15163 q^{34} +10.6673 q^{35} -3.35963 q^{37} -3.69277 q^{38} -3.17513 q^{40} +1.82053 q^{41} +10.9161 q^{43} +2.44949 q^{44} -7.95442 q^{47} +4.28714 q^{49} -5.08146 q^{50} +5.10767 q^{52} +3.70634 q^{53} +7.77745 q^{55} -3.35963 q^{56} +9.59798 q^{58} -0.330219 q^{59} +14.6312 q^{61} -3.46410 q^{62} +1.00000 q^{64} +16.2175 q^{65} -1.58515 q^{67} +3.15163 q^{68} -10.6673 q^{70} -9.25560 q^{71} -7.66728 q^{73} +3.35963 q^{74} +3.69277 q^{76} +8.22939 q^{77} -10.9447 q^{79} +3.17513 q^{80} -1.82053 q^{82} -2.27856 q^{83} +10.0068 q^{85} -10.9161 q^{86} -2.44949 q^{88} +1.78805 q^{89} +17.1599 q^{91} +7.95442 q^{94} +11.7250 q^{95} +1.34491 q^{97} -4.28714 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 12 q^{13} + 8 q^{16} + 32 q^{25} - 12 q^{26} - 8 q^{32} + 12 q^{35} - 12 q^{41} + 12 q^{47} + 32 q^{49} - 32 q^{50} + 12 q^{52} + 12 q^{55} - 24 q^{59} + 8 q^{64} - 12 q^{70}+ \cdots - 32 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 12 * q^13 + 8 * q^16 + 32 * q^25 - 12 * q^26 - 8 * q^32 + 12 * q^35 - 12 * q^41 + 12 * q^47 + 32 * q^49 - 32 * q^50 + 12 * q^52 + 12 * q^55 - 24 * q^59 + 8 * q^64 - 12 * q^70 + 12 * q^71 + 12 * q^73 + 12 * q^82 - 36 * q^85 - 12 * q^94 + 84 * q^95 - 32 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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