LMFDB - Embedded newform 9522.2.a.cc.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9522,2,Mod(1,9522)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);

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

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

Newform invariants

comment:select newform

sage:traces = [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			1.5
Root			$-3.55932$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.03363 q^{5} +3.88423 q^{7} +1.00000 q^{8} +3.03363 q^{10} +2.26481 q^{11} -4.29021 q^{13} +3.88423 q^{14} +1.00000 q^{16} +5.29845 q^{17} -1.61942 q^{19} +3.03363 q^{20} +2.26481 q^{22} +4.20293 q^{25} -4.29021 q^{26} +3.88423 q^{28} -5.78334 q^{29} -3.20293 q^{31} +1.00000 q^{32} +5.29845 q^{34} +11.7833 q^{35} +9.74629 q^{37} -1.61942 q^{38} +3.03363 q^{40} -7.08727 q^{41} +10.5151 q^{43} +2.26481 q^{44} -1.49314 q^{47} +8.08727 q^{49} +4.20293 q^{50} -4.29021 q^{52} -13.6305 q^{53} +6.87062 q^{55} +3.88423 q^{56} -5.78334 q^{58} +14.9863 q^{59} +12.9851 q^{61} -3.20293 q^{62} +1.00000 q^{64} -13.0149 q^{65} -12.9331 q^{67} +5.29845 q^{68} +11.7833 q^{70} -1.49314 q^{71} +1.70979 q^{73} +9.74629 q^{74} -1.61942 q^{76} +8.79707 q^{77} -0.645395 q^{79} +3.03363 q^{80} -7.08727 q^{82} -0.153200 q^{83} +16.0735 q^{85} +10.5151 q^{86} +2.26481 q^{88} -0.768819 q^{89} -16.6642 q^{91} -1.49314 q^{94} -4.91273 q^{95} -9.18268 q^{97} +8.08727 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.03363	1.35668	0.678341	−	0.734747i	$-0.262699\pi$
$5$	3.03363	1.35668	0.678341	+	0.734747i	$0.262699\pi$
$6$	0	0
$7$	3.88423	1.46810	0.734051	−	0.679094i	$-0.237627\pi$
$7$	3.88423	1.46810	0.734051	+	0.679094i	$0.237627\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.03363	0.959319
$11$	2.26481	0.682867	0.341434	−	0.939906i	$-0.389088\pi$
$11$	2.26481	0.682867	0.341434	+	0.939906i	$0.389088\pi$
$12$	0	0
$13$	−4.29021	−1.18989	−0.594944	−	0.803767i	$-0.702826\pi$
$13$	−4.29021	−1.18989	−0.594944	+	0.803767i	$0.702826\pi$
$14$	3.88423	1.03811
$15$	0	0
$16$	1.00000	0.250000
$17$	5.29845	1.28506	0.642531	−	0.766260i	$-0.277884\pi$
$17$	5.29845	1.28506	0.642531	+	0.766260i	$0.277884\pi$
$18$	0	0
$19$	−1.61942	−0.371520	−0.185760	−	0.982595i	$-0.559475\pi$
$19$	−1.61942	−0.371520	−0.185760	+	0.982595i	$0.559475\pi$
$20$	3.03363	0.678341
$21$	0	0
$22$	2.26481	0.482860
$23$	0	0
$23$	0	0
$24$	0	0
$25$	4.20293	0.840586
$26$	−4.29021	−0.841378
$27$	0	0
$28$	3.88423	0.734051
$29$	−5.78334	−1.07394	−0.536970	−	0.843601i	$-0.680431\pi$
$29$	−5.78334	−1.07394	−0.536970	+	0.843601i	$0.680431\pi$
$30$	0	0
$31$	−3.20293	−0.575263	−0.287632	−	0.957741i	$-0.592868\pi$
$31$	−3.20293	−0.575263	−0.287632	+	0.957741i	$0.592868\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.29845	0.908676
$35$	11.7833	1.99175
$36$	0	0
$37$	9.74629	1.60228	0.801140	−	0.598476i	$-0.204227\pi$
$37$	9.74629	1.60228	0.801140	+	0.598476i	$0.204227\pi$
$38$	−1.61942	−0.262705
$39$	0	0
$40$	3.03363	0.479660
$41$	−7.08727	−1.10685	−0.553423	−	0.832900i	$-0.686679\pi$
$41$	−7.08727	−1.10685	−0.553423	+	0.832900i	$0.686679\pi$
$42$	0	0
$43$	10.5151	1.60354	0.801770	−	0.597633i	$-0.203892\pi$
$43$	10.5151	1.60354	0.801770	+	0.597633i	$0.203892\pi$
$44$	2.26481	0.341434
$45$	0	0
$46$	0	0
$47$	−1.49314	−0.217796	−0.108898	−	0.994053i	$-0.534732\pi$
$47$	−1.49314	−0.217796	−0.108898	+	0.994053i	$0.534732\pi$
$48$	0	0
$49$	8.08727	1.15532
$50$	4.20293	0.594384
$51$	0	0
$52$	−4.29021	−0.594944
$53$	−13.6305	−1.87230	−0.936149	−	0.351605i	$-0.885636\pi$
$53$	−13.6305	−1.87230	−0.936149	+	0.351605i	$0.885636\pi$
$54$	0	0
$55$	6.87062	0.926434
$56$	3.88423	0.519053
$57$	0	0
$58$	−5.78334	−0.759390
$59$	14.9863	1.95105	0.975523	−	0.219896i	$-0.0705719\pi$
$59$	14.9863	1.95105	0.975523	+	0.219896i	$0.0705719\pi$
$60$	0	0
$61$	12.9851	1.66258	0.831288	−	0.555842i	$-0.187604\pi$
$61$	12.9851	1.66258	0.831288	+	0.555842i	$0.187604\pi$
$62$	−3.20293	−0.406773
$63$	0	0
$64$	1.00000	0.125000
$65$	−13.0149	−1.61430
$66$	0	0
$67$	−12.9331	−1.58003	−0.790017	−	0.613086i	$-0.789928\pi$
$67$	−12.9331	−1.58003	−0.790017	+	0.613086i	$0.789928\pi$
$68$	5.29845	0.642531
$69$	0	0
$70$	11.7833	1.40838
$71$	−1.49314	−0.177203	−0.0886014	−	0.996067i	$-0.528240\pi$
$71$	−1.49314	−0.177203	−0.0886014	+	0.996067i	$0.528240\pi$
$72$	0	0
$73$	1.70979	0.200116	0.100058	−	0.994982i	$-0.468097\pi$
$73$	1.70979	0.200116	0.100058	+	0.994982i	$0.468097\pi$
$74$	9.74629	1.13298
$75$	0	0
$76$	−1.61942	−0.185760
$77$	8.79707	1.00252
$78$	0	0
$79$	−0.645395	−0.0726126	−0.0363063	−	0.999341i	$-0.511559\pi$
$79$	−0.645395	−0.0726126	−0.0363063	+	0.999341i	$0.511559\pi$
$80$	3.03363	0.339171
$81$	0	0
$82$	−7.08727	−0.782659
$83$	−0.153200	−0.0168159	−0.00840797	−	0.999965i	$-0.502676\pi$
$83$	−0.153200	−0.0168159	−0.00840797	+	0.999965i	$0.502676\pi$
$84$	0	0
$85$	16.0735	1.74342
$86$	10.5151	1.13387
$87$	0	0
$88$	2.26481	0.241430
$89$	−0.768819	−0.0814947	−0.0407473	−	0.999169i	$-0.512974\pi$
$89$	−0.768819	−0.0814947	−0.0407473	+	0.999169i	$0.512974\pi$
$90$	0	0
$91$	−16.6642	−1.74688
$92$	0	0
$93$	0	0
$94$	−1.49314	−0.154005
$95$	−4.91273	−0.504035
$96$	0	0
$97$	−9.18268	−0.932360	−0.466180	−	0.884690i	$-0.654370\pi$
$97$	−9.18268	−0.932360	−0.466180	+	0.884690i	$0.654370\pi$
$98$	8.08727	0.816938
$99$	0	0
$100$	4.20293	0.420293
$101$	1.70979	0.170131	0.0850655	−	0.996375i	$-0.472890\pi$
$101$	1.70979	0.170131	0.0850655	+	0.996375i	$0.472890\pi$
$102$	0	0
$103$	10.6683	1.05118	0.525590	−	0.850738i	$-0.323845\pi$
$103$	10.6683	1.05118	0.525590	+	0.850738i	$0.323845\pi$
$104$	−4.29021	−0.420689
$105$	0	0
$106$	−13.6305	−1.32391
$107$	12.2877	1.18790	0.593950	−	0.804502i	$-0.297568\pi$
$107$	12.2877	1.18790	0.593950	+	0.804502i	$0.297568\pi$
$108$	0	0
$109$	7.63468	0.731270	0.365635	−	0.930758i	$-0.380852\pi$
$109$	7.63468	0.731270	0.365635	+	0.930758i	$0.380852\pi$
$110$	6.87062	0.655088
$111$	0	0
$112$	3.88423	0.367026
$113$	11.3657	1.06920	0.534598	−	0.845106i	$-0.320463\pi$
$113$	11.3657	1.06920	0.534598	+	0.845106i	$0.320463\pi$
$114$	0	0
$115$	0	0
$116$	−5.78334	−0.536970
$117$	0	0
$118$	14.9863	1.37960
$119$	20.5804	1.88660
$120$	0	0
$121$	−5.87062	−0.533692
$122$	12.9851	1.17562
$123$	0	0
$124$	−3.20293	−0.287632
$125$	−2.41801	−0.216274
$126$	0	0
$127$	16.4794	1.46231	0.731156	−	0.682211i	$-0.238981\pi$
$127$	16.4794	1.46231	0.731156	+	0.682211i	$0.238981\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−13.0149	−1.14148
$131$	3.20293	0.279841	0.139921	−	0.990163i	$-0.455315\pi$
$131$	3.20293	0.279841	0.139921	+	0.990163i	$0.455315\pi$
$132$	0	0
$133$	−6.29021	−0.545430
$134$	−12.9331	−1.11725
$135$	0	0
$136$	5.29845	0.454338
$137$	3.18683	0.272270	0.136135	−	0.990690i	$-0.456532\pi$
$137$	3.18683	0.272270	0.136135	+	0.990690i	$0.456532\pi$
$138$	0	0
$139$	−2.40586	−0.204063	−0.102031	−	0.994781i	$-0.532534\pi$
$139$	−2.40586	−0.204063	−0.102031	+	0.994781i	$0.532534\pi$
$140$	11.7833	0.995874
$141$	0	0
$142$	−1.49314	−0.125301
$143$	−9.71652	−0.812536
$144$	0	0
$145$	−17.5445	−1.45699
$146$	1.70979	0.141504
$147$	0	0
$148$	9.74629	0.801140
$149$	22.1158	1.81180	0.905899	−	0.423494i	$-0.139197\pi$
$149$	22.1158	1.81180	0.905899	+	0.423494i	$0.139197\pi$
$150$	0	0
$151$	−21.2765	−1.73146	−0.865728	−	0.500515i	$-0.833144\pi$
$151$	−21.2765	−1.73146	−0.865728	+	0.500515i	$0.833144\pi$
$152$	−1.61942	−0.131352
$153$	0	0
$154$	8.79707	0.708888
$155$	−9.71652	−0.780450
$156$	0	0
$157$	−9.43989	−0.753386	−0.376693	−	0.926338i	$-0.622939\pi$
$157$	−9.43989	−0.753386	−0.376693	+	0.926338i	$0.622939\pi$
$158$	−0.645395	−0.0513448
$159$	0	0
$160$	3.03363	0.239830
$161$	0	0
$162$	0	0
$163$	−4.69607	−0.367824	−0.183912	−	0.982943i	$-0.558876\pi$
$163$	−4.69607	−0.367824	−0.183912	+	0.982943i	$0.558876\pi$
$164$	−7.08727	−0.553423
$165$	0	0
$166$	−0.153200	−0.0118907
$167$	25.4931	1.97272	0.986359	−	0.164608i	$-0.0526359\pi$
$167$	25.4931	1.97272	0.986359	+	0.164608i	$0.0526359\pi$
$168$	0	0
$169$	5.40586	0.415836
$170$	16.0735	1.23278
$171$	0	0
$172$	10.5151	0.801770
$173$	−5.78334	−0.439699	−0.219850	−	0.975534i	$-0.570557\pi$
$173$	−5.78334	−0.439699	−0.219850	+	0.975534i	$0.570557\pi$
$174$	0	0
$175$	16.3252	1.23407
$176$	2.26481	0.170717
$177$	0	0
$178$	−0.768819	−0.0576254
$179$	−15.2029	−1.13632	−0.568160	−	0.822918i	$-0.692345\pi$
$179$	−15.2029	−1.13632	−0.568160	+	0.822918i	$0.692345\pi$
$180$	0	0
$181$	−5.79064	−0.430415	−0.215208	−	0.976568i	$-0.569043\pi$
$181$	−5.79064	−0.430415	−0.215208	+	0.976568i	$0.569043\pi$
$182$	−16.6642	−1.23523
$183$	0	0
$184$	0	0
$185$	29.5667	2.17379
$186$	0	0
$187$	12.0000	0.877527
$188$	−1.49314	−0.108898
$189$	0	0
$190$	−4.91273	−0.356407
$191$	−12.4409	−0.900194	−0.450097	−	0.892980i	$-0.648611\pi$
$191$	−12.4409	−0.900194	−0.450097	+	0.892980i	$0.648611\pi$
$192$	0	0
$193$	−11.6677	−0.839858	−0.419929	−	0.907557i	$-0.637945\pi$
$193$	−11.6677	−0.839858	−0.419929	+	0.907557i	$0.637945\pi$
$194$	−9.18268	−0.659278
$195$	0	0
$196$	8.08727	0.577662
$197$	−15.3921	−1.09664	−0.548322	−	0.836267i	$-0.684733\pi$
$197$	−15.3921	−1.09664	−0.548322	+	0.836267i	$0.684733\pi$
$198$	0	0
$199$	−1.77262	−0.125658	−0.0628289	−	0.998024i	$-0.520012\pi$
$199$	−1.77262	−0.125658	−0.0628289	+	0.998024i	$0.520012\pi$
$200$	4.20293	0.297192
$201$	0	0
$202$	1.70979	0.120301
$203$	−22.4639	−1.57665
$204$	0	0
$205$	−21.5002	−1.50164
$206$	10.6683	0.743297
$207$	0	0
$208$	−4.29021	−0.297472
$209$	−3.66769	−0.253699
$210$	0	0
$211$	16.6961	1.14940	0.574702	−	0.818363i	$-0.305118\pi$
$211$	16.6961	1.14940	0.574702	+	0.818363i	$0.305118\pi$
$212$	−13.6305	−0.936149
$213$	0	0
$214$	12.2877	0.839972
$215$	31.8990	2.17549
$216$	0	0
$217$	−12.4409	−0.844546
$218$	7.63468	0.517086
$219$	0	0
$220$	6.87062	0.463217
$221$	−22.7314	−1.52908
$222$	0	0
$223$	−21.2765	−1.42478	−0.712389	−	0.701785i	$-0.752387\pi$
$223$	−21.2765	−1.42478	−0.712389	+	0.701785i	$0.752387\pi$
$224$	3.88423	0.259526
$225$	0	0
$226$	11.3657	0.756036
$227$	−28.9519	−1.92160	−0.960802	−	0.277234i	$-0.910582\pi$
$227$	−28.9519	−1.92160	−0.960802	+	0.277234i	$0.910582\pi$
$228$	0	0
$229$	−12.1643	−0.803840	−0.401920	−	0.915675i	$-0.631657\pi$
$229$	−12.1643	−0.803840	−0.401920	+	0.915675i	$0.631657\pi$
$230$	0	0
$231$	0	0
$232$	−5.78334	−0.379695
$233$	20.1471	1.31988	0.659940	−	0.751318i	$-0.270582\pi$
$233$	20.1471	1.31988	0.659940	+	0.751318i	$0.270582\pi$
$234$	0	0
$235$	−4.52963	−0.295480
$236$	14.9863	0.975523
$237$	0	0
$238$	20.5804	1.33403
$239$	−2.98627	−0.193166	−0.0965830	−	0.995325i	$-0.530791\pi$
$239$	−2.98627	−0.193166	−0.0965830	+	0.995325i	$0.530791\pi$
$240$	0	0
$241$	−5.63744	−0.363140	−0.181570	−	0.983378i	$-0.558118\pi$
$241$	−5.63744	−0.363140	−0.181570	+	0.983378i	$0.558118\pi$
$242$	−5.87062	−0.377377
$243$	0	0
$244$	12.9851	0.831288
$245$	24.5338	1.56741
$246$	0	0
$247$	6.94764	0.442068
$248$	−3.20293	−0.203386
$249$	0	0
$250$	−2.41801	−0.152929
$251$	−6.79444	−0.428861	−0.214431	−	0.976739i	$-0.568790\pi$
$251$	−6.79444	−0.428861	−0.214431	+	0.976739i	$0.568790\pi$
$252$	0	0
$253$	0	0
$254$	16.4794	1.03401
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.5069	0.655400	0.327700	−	0.944782i	$-0.393726\pi$
$257$	10.5069	0.655400	0.327700	+	0.944782i	$0.393726\pi$
$258$	0	0
$259$	37.8569	2.35231
$260$	−13.0149	−0.807150
$261$	0	0
$262$	3.20293	0.197878
$263$	−24.2691	−1.49649	−0.748247	−	0.663420i	$-0.769104\pi$
$263$	−24.2691	−1.49649	−0.748247	+	0.663420i	$0.769104\pi$
$264$	0	0
$265$	−41.3500	−2.54011
$266$	−6.29021	−0.385677
$267$	0	0
$268$	−12.9331	−0.790017
$269$	−8.98627	−0.547903	−0.273951	−	0.961744i	$-0.588331\pi$
$269$	−8.98627	−0.547903	−0.273951	+	0.961744i	$0.588331\pi$
$270$	0	0
$271$	−16.9127	−1.02737	−0.513687	−	0.857977i	$-0.671721\pi$
$271$	−16.9127	−1.02737	−0.513687	+	0.857977i	$0.671721\pi$
$272$	5.29845	0.321266
$273$	0	0
$274$	3.18683	0.192524
$275$	9.51886	0.574009
$276$	0	0
$277$	−8.79707	−0.528565	−0.264282	−	0.964445i	$-0.585135\pi$
$277$	−8.79707	−0.528565	−0.264282	+	0.964445i	$0.585135\pi$
$278$	−2.40586	−0.144294
$279$	0	0
$280$	11.7833	0.704189
$281$	29.8739	1.78213	0.891064	−	0.453877i	$-0.149960\pi$
$281$	29.8739	1.78213	0.891064	+	0.453877i	$0.149960\pi$
$282$	0	0
$283$	−7.11271	−0.422807	−0.211403	−	0.977399i	$-0.567803\pi$
$283$	−7.11271	−0.422807	−0.211403	+	0.977399i	$0.567803\pi$
$284$	−1.49314	−0.0886014
$285$	0	0
$286$	−9.71652	−0.574550
$287$	−27.5286	−1.62496
$288$	0	0
$289$	11.0735	0.651385
$290$	−17.5445	−1.03025
$291$	0	0
$292$	1.70979	0.100058
$293$	−28.7571	−1.68000	−0.840002	−	0.542583i	$-0.817447\pi$
$293$	−28.7571	−1.68000	−0.840002	+	0.542583i	$0.817447\pi$
$294$	0	0
$295$	45.4629	2.64695
$296$	9.74629	0.566492
$297$	0	0
$298$	22.1158	1.28113
$299$	0	0
$300$	0	0
$301$	40.8432	2.35416
$302$	−21.2765	−1.22432
$303$	0	0
$304$	−1.61942	−0.0928801
$305$	39.3921	2.25559
$306$	0	0
$307$	−25.2765	−1.44260	−0.721302	−	0.692620i	$-0.756456\pi$
$307$	−25.2765	−1.44260	−0.721302	+	0.692620i	$0.756456\pi$
$308$	8.79707	0.501260
$309$	0	0
$310$	−9.71652	−0.551861
$311$	14.1745	0.803765	0.401882	−	0.915691i	$-0.368356\pi$
$311$	14.1745	0.803765	0.401882	+	0.915691i	$0.368356\pi$
$312$	0	0
$313$	4.40620	0.249053	0.124527	−	0.992216i	$-0.460259\pi$
$313$	4.40620	0.249053	0.124527	+	0.992216i	$0.460259\pi$
$314$	−9.43989	−0.532724
$315$	0	0
$316$	−0.645395	−0.0363063
$317$	−17.7833	−0.998812	−0.499406	−	0.866368i	$-0.666448\pi$
$317$	−17.7833	−0.998812	−0.499406	+	0.866368i	$0.666448\pi$
$318$	0	0
$319$	−13.0982	−0.733358
$320$	3.03363	0.169585
$321$	0	0
$322$	0	0
$323$	−8.58041	−0.477427
$324$	0	0
$325$	−18.0314	−1.00020
$326$	−4.69607	−0.260091
$327$	0	0
$328$	−7.08727	−0.391329
$329$	−5.79969	−0.319747
$330$	0	0
$331$	−7.30393	−0.401460	−0.200730	−	0.979647i	$-0.564331\pi$
$331$	−7.30393	−0.401460	−0.200730	+	0.979647i	$0.564331\pi$
$332$	−0.153200	−0.00840797
$333$	0	0
$334$	25.4931	1.39492
$335$	−39.2344	−2.14360
$336$	0	0
$337$	−21.0108	−1.14453	−0.572266	−	0.820068i	$-0.693936\pi$
$337$	−21.0108	−1.14453	−0.572266	+	0.820068i	$0.693936\pi$
$338$	5.40586	0.294040
$339$	0	0
$340$	16.0735	0.871710
$341$	−7.25404	−0.392829
$342$	0	0
$343$	4.22323	0.228033
$344$	10.5151	0.566937
$345$	0	0
$346$	−5.78334	−0.310914
$347$	31.9304	1.71412	0.857058	−	0.515220i	$-0.172290\pi$
$347$	31.9304	1.71412	0.857058	+	0.515220i	$0.172290\pi$
$348$	0	0
$349$	13.2618	0.709889	0.354945	−	0.934887i	$-0.384500\pi$
$349$	13.2618	0.709889	0.354945	+	0.934887i	$0.384500\pi$
$350$	16.3252	0.872617
$351$	0	0
$352$	2.26481	0.120715
$353$	21.6677	1.15325	0.576627	−	0.817007i	$-0.304369\pi$
$353$	21.6677	1.15325	0.576627	+	0.817007i	$0.304369\pi$
$354$	0	0
$355$	−4.52963	−0.240408
$356$	−0.768819	−0.0407473
$357$	0	0
$358$	−15.2029	−0.803500
$359$	−6.64124	−0.350511	−0.175256	−	0.984523i	$-0.556075\pi$
$359$	−6.64124	−0.350511	−0.175256	+	0.984523i	$0.556075\pi$
$360$	0	0
$361$	−16.3775	−0.861973
$362$	−5.79064	−0.304349
$363$	0	0
$364$	−16.6642	−0.873439
$365$	5.18689	0.271494
$366$	0	0
$367$	17.0032	0.887557	0.443779	−	0.896136i	$-0.353638\pi$
$367$	17.0032	0.887557	0.443779	+	0.896136i	$0.353638\pi$
$368$	0	0
$369$	0	0
$370$	29.5667	1.53710
$371$	−52.9442	−2.74872
$372$	0	0
$373$	15.9771	0.827264	0.413632	−	0.910444i	$-0.364260\pi$
$373$	15.9771	0.827264	0.413632	+	0.910444i	$0.364260\pi$
$374$	12.0000	0.620505
$375$	0	0
$376$	−1.49314	−0.0770026
$377$	24.8117	1.27787
$378$	0	0
$379$	15.2083	0.781198	0.390599	−	0.920561i	$-0.372268\pi$
$379$	15.2083	0.781198	0.390599	+	0.920561i	$0.372268\pi$
$380$	−4.91273	−0.252018
$381$	0	0
$382$	−12.4409	−0.636534
$383$	19.0822	0.975054	0.487527	−	0.873108i	$-0.337899\pi$
$383$	19.0822	0.975054	0.487527	+	0.873108i	$0.337899\pi$
$384$	0	0
$385$	26.6871	1.36010
$386$	−11.6677	−0.593870
$387$	0	0
$388$	−9.18268	−0.466180
$389$	−13.0566	−0.661994	−0.330997	−	0.943632i	$-0.607385\pi$
$389$	−13.0566	−0.661994	−0.330997	+	0.943632i	$0.607385\pi$
$390$	0	0
$391$	0	0
$392$	8.08727	0.408469
$393$	0	0
$394$	−15.3921	−0.775445
$395$	−1.95789	−0.0985122
$396$	0	0
$397$	−10.8117	−0.542625	−0.271312	−	0.962491i	$-0.587458\pi$
$397$	−10.8117	−0.542625	−0.271312	+	0.962491i	$0.587458\pi$
$398$	−1.77262	−0.0888534
$399$	0	0
$400$	4.20293	0.210147
$401$	−3.76081	−0.187806	−0.0939029	−	0.995581i	$-0.529934\pi$
$401$	−3.76081	−0.187806	−0.0939029	+	0.995581i	$0.529934\pi$
$402$	0	0
$403$	13.7412	0.684500
$404$	1.70979	0.0850655
$405$	0	0
$406$	−22.4639	−1.11486
$407$	22.0735	1.09415
$408$	0	0
$409$	24.4794	1.21043	0.605214	−	0.796062i	$-0.293087\pi$
$409$	24.4794	1.21043	0.605214	+	0.796062i	$0.293087\pi$
$410$	−21.5002	−1.06182
$411$	0	0
$412$	10.6683	0.525590
$413$	58.2102	2.86434
$414$	0	0
$415$	−0.464754	−0.0228139
$416$	−4.29021	−0.210345
$417$	0	0
$418$	−3.66769	−0.179392
$419$	25.8766	1.26416	0.632078	−	0.774905i	$-0.282202\pi$
$419$	25.8766	1.26416	0.632078	+	0.774905i	$0.282202\pi$
$420$	0	0
$421$	7.49184	0.365130	0.182565	−	0.983194i	$-0.441560\pi$
$421$	7.49184	0.365130	0.182565	+	0.983194i	$0.441560\pi$
$422$	16.6961	0.812752
$423$	0	0
$424$	−13.6305	−0.661957
$425$	22.2690	1.08021
$426$	0	0
$427$	50.4373	2.44083
$428$	12.2877	0.593950
$429$	0	0
$430$	31.8990	1.53831
$431$	11.8281	0.569741	0.284871	−	0.958566i	$-0.408049\pi$
$431$	11.8281	0.569741	0.284871	+	0.958566i	$0.408049\pi$
$432$	0	0
$433$	11.0267	0.529910	0.264955	−	0.964261i	$-0.414643\pi$
$433$	11.0267	0.529910	0.264955	+	0.964261i	$0.414643\pi$
$434$	−12.4409	−0.597184
$435$	0	0
$436$	7.63468	0.365635
$437$	0	0
$438$	0	0
$439$	−36.1471	−1.72521	−0.862603	−	0.505881i	$-0.831168\pi$
$439$	−36.1471	−1.72521	−0.862603	+	0.505881i	$0.831168\pi$
$440$	6.87062	0.327544
$441$	0	0
$442$	−22.7314	−1.08122
$443$	−2.98627	−0.141882	−0.0709411	−	0.997481i	$-0.522600\pi$
$443$	−2.98627	−0.141882	−0.0709411	+	0.997481i	$0.522600\pi$
$444$	0	0
$445$	−2.33231	−0.110562
$446$	−21.2765	−1.00747
$447$	0	0
$448$	3.88423	0.183513
$449$	−16.4794	−0.777711	−0.388856	−	0.921299i	$-0.627130\pi$
$449$	−16.4794	−0.777711	−0.388856	+	0.921299i	$0.627130\pi$
$450$	0	0
$451$	−16.0514	−0.755829
$452$	11.3657	0.534598
$453$	0	0
$454$	−28.9519	−1.35878
$455$	−50.5530	−2.36996
$456$	0	0
$457$	−17.3616	−0.812140	−0.406070	−	0.913842i	$-0.633101\pi$
$457$	−17.3616	−0.812140	−0.406070	+	0.913842i	$0.633101\pi$
$458$	−12.1643	−0.568401
$459$	0	0
$460$	0	0
$461$	14.5804	0.679077	0.339539	−	0.940592i	$-0.389729\pi$
$461$	14.5804	0.679077	0.339539	+	0.940592i	$0.389729\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−5.78334	−0.268485
$465$	0	0
$466$	20.1471	0.933296
$467$	−33.4815	−1.54934	−0.774670	−	0.632366i	$-0.782084\pi$
$467$	−33.4815	−1.54934	−0.774670	+	0.632366i	$0.782084\pi$
$468$	0	0
$469$	−50.2353	−2.31965
$470$	−4.52963	−0.208936
$471$	0	0
$472$	14.9863	0.689799
$473$	23.8148	1.09500
$474$	0	0
$475$	−6.80631	−0.312295
$476$	20.5804	0.943302
$477$	0	0
$478$	−2.98627	−0.136589
$479$	2.11161	0.0964821	0.0482411	−	0.998836i	$-0.484638\pi$
$479$	2.11161	0.0964821	0.0482411	+	0.998836i	$0.484638\pi$
$480$	0	0
$481$	−41.8136	−1.90654
$482$	−5.63744	−0.256778
$483$	0	0
$484$	−5.87062	−0.266846
$485$	−27.8569	−1.26492
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	12.9851	0.587809
$489$	0	0
$490$	24.5338	1.10833
$491$	−11.3500	−0.512219	−0.256110	−	0.966648i	$-0.582441\pi$
$491$	−11.3500	−0.512219	−0.256110	+	0.966648i	$0.582441\pi$
$492$	0	0
$493$	−30.6427	−1.38008
$494$	6.94764	0.312589
$495$	0	0
$496$	−3.20293	−0.143816
$497$	−5.79969	−0.260152
$498$	0	0
$499$	−12.4648	−0.557999	−0.279000	−	0.960291i	$-0.590003\pi$
$499$	−12.4648	−0.557999	−0.279000	+	0.960291i	$0.590003\pi$
$500$	−2.41801	−0.108137
$501$	0	0
$502$	−6.79444	−0.303251
$503$	−2.41801	−0.107814	−0.0539070	−	0.998546i	$-0.517167\pi$
$503$	−2.41801	−0.107814	−0.0539070	+	0.998546i	$0.517167\pi$
$504$	0	0
$505$	5.18689	0.230814
$506$	0	0
$507$	0	0
$508$	16.4794	0.731156
$509$	0.189204	0.00838632	0.00419316	−	0.999991i	$-0.498665\pi$
$509$	0.189204	0.00838632	0.00419316	+	0.999991i	$0.498665\pi$
$510$	0	0
$511$	6.64124	0.293791
$512$	1.00000	0.0441942
$513$	0	0
$514$	10.5069	0.463438
$515$	32.3638	1.42612
$516$	0	0
$517$	−3.38168	−0.148726
$518$	37.8569	1.66334
$519$	0	0
$520$	−13.0149	−0.570742
$521$	−1.03639	−0.0454052	−0.0227026	−	0.999742i	$-0.507227\pi$
$521$	−1.03639	−0.0454052	−0.0227026	+	0.999742i	$0.507227\pi$
$522$	0	0
$523$	−17.4628	−0.763593	−0.381797	−	0.924246i	$-0.624694\pi$
$523$	−17.4628	−0.763593	−0.381797	+	0.924246i	$0.624694\pi$
$524$	3.20293	0.139921
$525$	0	0
$526$	−24.2691	−1.05818
$527$	−16.9706	−0.739249
$528$	0	0
$529$	0	0
$530$	−41.3500	−1.79613
$531$	0	0
$532$	−6.29021	−0.272715
$533$	30.4059	1.31702
$534$	0	0
$535$	37.2765	1.61160
$536$	−12.9331	−0.558626
$537$	0	0
$538$	−8.98627	−0.387426
$539$	18.3162	0.788933
$540$	0	0
$541$	−24.4794	−1.05245	−0.526226	−	0.850345i	$-0.676393\pi$
$541$	−24.4794	−1.05245	−0.526226	+	0.850345i	$0.676393\pi$
$542$	−16.9127	−0.726464
$543$	0	0
$544$	5.29845	0.227169
$545$	23.1608	0.992101
$546$	0	0
$547$	−33.8569	−1.44762	−0.723808	−	0.690001i	$-0.757610\pi$
$547$	−33.8569	−1.44762	−0.723808	+	0.690001i	$0.757610\pi$
$548$	3.18683	0.136135
$549$	0	0
$550$	9.51886	0.405885
$551$	9.36566	0.398990
$552$	0	0
$553$	−2.50686	−0.106603
$554$	−8.79707	−0.373752
$555$	0	0
$556$	−2.40586	−0.102031
$557$	−4.18159	−0.177179	−0.0885897	−	0.996068i	$-0.528236\pi$
$557$	−4.18159	−0.177179	−0.0885897	+	0.996068i	$0.528236\pi$
$558$	0	0
$559$	−45.1120	−1.90803
$560$	11.7833	0.497937
$561$	0	0
$562$	29.8739	1.26016
$563$	32.5179	1.37046	0.685232	−	0.728325i	$-0.259701\pi$
$563$	32.5179	1.37046	0.685232	+	0.728325i	$0.259701\pi$
$564$	0	0
$565$	34.4794	1.45056
$566$	−7.11271	−0.298970
$567$	0	0
$568$	−1.49314	−0.0626506
$569$	34.9775	1.46633	0.733167	−	0.680049i	$-0.238041\pi$
$569$	34.9775	1.46633	0.733167	+	0.680049i	$0.238041\pi$
$570$	0	0
$571$	−7.41911	−0.310480	−0.155240	−	0.987877i	$-0.549615\pi$
$571$	−7.41911	−0.310480	−0.155240	+	0.987877i	$0.549615\pi$
$572$	−9.71652	−0.406268
$573$	0	0
$574$	−27.5286	−1.14902
$575$	0	0
$576$	0	0
$577$	−19.3500	−0.805552	−0.402776	−	0.915299i	$-0.631955\pi$
$577$	−19.3500	−0.805552	−0.402776	+	0.915299i	$0.631955\pi$
$578$	11.0735	0.460599
$579$	0	0
$580$	−17.5445	−0.728497
$581$	−0.595066	−0.0246875
$582$	0	0
$583$	−30.8706	−1.27853
$584$	1.70979	0.0707518
$585$	0	0
$586$	−28.7571	−1.18794
$587$	36.1618	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$587$	36.1618	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$588$	0	0
$589$	5.18689	0.213722
$590$	45.4629	1.87168
$591$	0	0
$592$	9.74629	0.400570
$593$	40.0461	1.64450	0.822248	−	0.569129i	$-0.192720\pi$
$593$	40.0461	1.64450	0.822248	+	0.569129i	$0.192720\pi$
$594$	0	0
$595$	62.4334	2.55952
$596$	22.1158	0.905899
$597$	0	0
$598$	0	0
$599$	−10.7550	−0.439436	−0.219718	−	0.975563i	$-0.570514\pi$
$599$	−10.7550	−0.439436	−0.219718	+	0.975563i	$0.570514\pi$
$600$	0	0
$601$	33.7559	1.37693	0.688466	−	0.725269i	$-0.258285\pi$
$601$	33.7559	1.37693	0.688466	+	0.725269i	$0.258285\pi$
$602$	40.8432	1.66464
$603$	0	0
$604$	−21.2765	−0.865728
$605$	−17.8093	−0.724051
$606$	0	0
$607$	−16.4648	−0.668284	−0.334142	−	0.942523i	$-0.608446\pi$
$607$	−16.4648	−0.668284	−0.334142	+	0.942523i	$0.608446\pi$
$608$	−1.61942	−0.0656761
$609$	0	0
$610$	39.3921	1.59494
$611$	6.40586	0.259153
$612$	0	0
$613$	40.8590	1.65028	0.825140	−	0.564929i	$-0.191096\pi$
$613$	40.8590	1.65028	0.825140	+	0.564929i	$0.191096\pi$
$614$	−25.2765	−1.02008
$615$	0	0
$616$	8.79707	0.354444
$617$	−28.9103	−1.16388	−0.581941	−	0.813231i	$-0.697707\pi$
$617$	−28.9103	−1.16388	−0.581941	+	0.813231i	$0.697707\pi$
$618$	0	0
$619$	28.9845	1.16498	0.582492	−	0.812836i	$-0.302078\pi$
$619$	28.9845	1.16498	0.582492	+	0.812836i	$0.302078\pi$
$620$	−9.71652	−0.390225
$621$	0	0
$622$	14.1745	0.568348
$623$	−2.98627	−0.119643
$624$	0	0
$625$	−28.3500	−1.13400
$626$	4.40620	0.176107
$627$	0	0
$628$	−9.43989	−0.376693
$629$	51.6402	2.05903
$630$	0	0
$631$	−22.6600	−0.902081	−0.451040	−	0.892504i	$-0.648947\pi$
$631$	−22.6600	−0.902081	−0.451040	+	0.892504i	$0.648947\pi$
$632$	−0.645395	−0.0256724
$633$	0	0
$634$	−17.7833	−0.706267
$635$	49.9925	1.98389
$636$	0	0
$637$	−34.6961	−1.37471
$638$	−13.0982	−0.518563
$639$	0	0
$640$	3.03363	0.119915
$641$	−20.1574	−0.796169	−0.398085	−	0.917349i	$-0.630325\pi$
$641$	−20.1574	−0.796169	−0.398085	+	0.917349i	$0.630325\pi$
$642$	0	0
$643$	−1.45586	−0.0574134	−0.0287067	−	0.999588i	$-0.509139\pi$
$643$	−1.45586	−0.0574134	−0.0287067	+	0.999588i	$0.509139\pi$
$644$	0	0
$645$	0	0
$646$	−8.58041	−0.337592
$647$	−37.3079	−1.46673	−0.733363	−	0.679838i	$-0.762050\pi$
$647$	−37.3079	−1.46673	−0.733363	+	0.679838i	$0.762050\pi$
$648$	0	0
$649$	33.9411	1.33231
$650$	−18.0314	−0.707251
$651$	0	0
$652$	−4.69607	−0.183912
$653$	−11.1019	−0.434452	−0.217226	−	0.976121i	$-0.569701\pi$
$653$	−11.1019	−0.434452	−0.217226	+	0.976121i	$0.569701\pi$
$654$	0	0
$655$	9.71652	0.379656
$656$	−7.08727	−0.276712
$657$	0	0
$658$	−5.79969	−0.226096
$659$	17.1238	0.667047	0.333524	−	0.942742i	$-0.391762\pi$
$659$	17.1238	0.667047	0.333524	+	0.942742i	$0.391762\pi$
$660$	0	0
$661$	−21.5744	−0.839148	−0.419574	−	0.907721i	$-0.637821\pi$
$661$	−21.5744	−0.839148	−0.419574	+	0.907721i	$0.637821\pi$
$662$	−7.30393	−0.283875
$663$	0	0
$664$	−0.153200	−0.00594533
$665$	−19.0822	−0.739975
$666$	0	0
$667$	0	0
$668$	25.4931	0.986359
$669$	0	0
$670$	−39.2344	−1.51576
$671$	29.4089	1.13532
$672$	0	0
$673$	−12.1892	−0.469859	−0.234930	−	0.972012i	$-0.575486\pi$
$673$	−12.1892	−0.469859	−0.234930	+	0.972012i	$0.575486\pi$
$674$	−21.0108	−0.809306
$675$	0	0
$676$	5.40586	0.207918
$677$	6.98929	0.268620	0.134310	−	0.990939i	$-0.457118\pi$
$677$	6.98929	0.268620	0.134310	+	0.990939i	$0.457118\pi$
$678$	0	0
$679$	−35.6677	−1.36880
$680$	16.0735	0.616392
$681$	0	0
$682$	−7.25404	−0.277772
$683$	−2.76961	−0.105976	−0.0529882	−	0.998595i	$-0.516875\pi$
$683$	−2.76961	−0.105976	−0.0529882	+	0.998595i	$0.516875\pi$
$684$	0	0
$685$	9.66769	0.369383
$686$	4.22323	0.161244
$687$	0	0
$688$	10.5151	0.400885
$689$	58.4778	2.22783
$690$	0	0
$691$	13.2765	0.505061	0.252531	−	0.967589i	$-0.418737\pi$
$691$	13.2765	0.505061	0.252531	+	0.967589i	$0.418737\pi$
$692$	−5.78334	−0.219850
$693$	0	0
$694$	31.9304	1.21206
$695$	−7.29850	−0.276848
$696$	0	0
$697$	−37.5516	−1.42237
$698$	13.2618	0.501967
$699$	0	0
$700$	16.3252	0.617033
$701$	33.3700	1.26037	0.630183	−	0.776447i	$-0.282980\pi$
$701$	33.3700	1.26037	0.630183	+	0.776447i	$0.282980\pi$
$702$	0	0
$703$	−15.7833	−0.595280
$704$	2.26481	0.0853584
$705$	0	0
$706$	21.6677	0.815474
$707$	6.64124	0.249770
$708$	0	0
$709$	−1.73098	−0.0650082	−0.0325041	−	0.999472i	$-0.510348\pi$
$709$	−1.73098	−0.0650082	−0.0325041	+	0.999472i	$0.510348\pi$
$710$	−4.52963	−0.169994
$711$	0	0
$712$	−0.768819	−0.0288127
$713$	0	0
$714$	0	0
$715$	−29.4764	−1.10235
$716$	−15.2029	−0.568160
$717$	0	0
$718$	−6.64124	−0.247849
$719$	34.7550	1.29614	0.648071	−	0.761580i	$-0.275576\pi$
$719$	34.7550	1.29614	0.648071	+	0.761580i	$0.275576\pi$
$720$	0	0
$721$	41.4382	1.54324
$722$	−16.3775	−0.609507
$723$	0	0
$724$	−5.79064	−0.215208
$725$	−24.3070	−0.902739
$726$	0	0
$727$	18.4368	0.683782	0.341891	−	0.939740i	$-0.388933\pi$
$727$	18.4368	0.683782	0.341891	+	0.939740i	$0.388933\pi$
$728$	−16.6642	−0.617615
$729$	0	0
$730$	5.18689	0.191975
$731$	55.7138	2.06065
$732$	0	0
$733$	−39.4254	−1.45621	−0.728105	−	0.685466i	$-0.759598\pi$
$733$	−39.4254	−1.45621	−0.728105	+	0.685466i	$0.759598\pi$
$734$	17.0032	0.627598
$735$	0	0
$736$	0	0
$737$	−29.2911	−1.07895
$738$	0	0
$739$	4.14709	0.152553	0.0762767	−	0.997087i	$-0.475697\pi$
$739$	4.14709	0.152553	0.0762767	+	0.997087i	$0.475697\pi$
$740$	29.5667	1.08689
$741$	0	0
$742$	−52.9442	−1.94364
$743$	36.4036	1.33552	0.667759	−	0.744377i	$-0.267254\pi$
$743$	36.4036	1.33552	0.667759	+	0.744377i	$0.267254\pi$
$744$	0	0
$745$	67.0913	2.45803
$746$	15.9771	0.584964
$747$	0	0
$748$	12.0000	0.438763
$749$	47.7284	1.74396
$750$	0	0
$751$	32.8258	1.19783	0.598915	−	0.800813i	$-0.295599\pi$
$751$	32.8258	1.19783	0.598915	+	0.800813i	$0.295599\pi$
$752$	−1.49314	−0.0544491
$753$	0	0
$754$	24.8117	0.903590
$755$	−64.5450	−2.34903
$756$	0	0
$757$	9.17232	0.333374	0.166687	−	0.986010i	$-0.446693\pi$
$757$	9.17232	0.333374	0.166687	+	0.986010i	$0.446693\pi$
$758$	15.2083	0.552391
$759$	0	0
$760$	−4.91273	−0.178203
$761$	−13.0873	−0.474413	−0.237207	−	0.971459i	$-0.576232\pi$
$761$	−13.0873	−0.474413	−0.237207	+	0.971459i	$0.576232\pi$
$762$	0	0
$763$	29.6549	1.07358
$764$	−12.4409	−0.450097
$765$	0	0
$766$	19.0822	0.689467
$767$	−64.2942	−2.32153
$768$	0	0
$769$	45.0330	1.62393	0.811966	−	0.583704i	$-0.198397\pi$
$769$	45.0330	1.62393	0.811966	+	0.583704i	$0.198397\pi$
$770$	26.6871	0.961736
$771$	0	0
$772$	−11.6677	−0.419929
$773$	4.57127	0.164417	0.0822086	−	0.996615i	$-0.473803\pi$
$773$	4.57127	0.164417	0.0822086	+	0.996615i	$0.473803\pi$
$774$	0	0
$775$	−13.4617	−0.483558
$776$	−9.18268	−0.329639
$777$	0	0
$778$	−13.0566	−0.468100
$779$	11.4773	0.411216
$780$	0	0
$781$	−3.38168	−0.121006
$782$	0	0
$783$	0	0
$784$	8.08727	0.288831
$785$	−28.6372	−1.02210
$786$	0	0
$787$	−19.8808	−0.708673	−0.354336	−	0.935118i	$-0.615293\pi$
$787$	−19.8808	−0.708673	−0.354336	+	0.935118i	$0.615293\pi$
$788$	−15.3921	−0.548322
$789$	0	0
$790$	−1.95789	−0.0696586
$791$	44.1471	1.56969
$792$	0	0
$793$	−55.7089	−1.97828
$794$	−10.8117	−0.383694
$795$	0	0
$796$	−1.77262	−0.0628289
$797$	−35.2140	−1.24734	−0.623672	−	0.781686i	$-0.714360\pi$
$797$	−35.2140	−1.24734	−0.623672	+	0.781686i	$0.714360\pi$
$798$	0	0
$799$	−7.91131	−0.279882
$800$	4.20293	0.148596
$801$	0	0
$802$	−3.76081	−0.132799
$803$	3.87237	0.136653
$804$	0	0
$805$	0	0
$806$	13.7412	0.484014
$807$	0	0
$808$	1.70979	0.0601504
$809$	−28.0735	−0.987013	−0.493507	−	0.869742i	$-0.664285\pi$
$809$	−28.0735	−0.987013	−0.493507	+	0.869742i	$0.664285\pi$
$810$	0	0
$811$	5.94111	0.208621	0.104310	−	0.994545i	$-0.466737\pi$
$811$	5.94111	0.208621	0.104310	+	0.994545i	$0.466737\pi$
$812$	−22.4639	−0.788327
$813$	0	0
$814$	22.0735	0.773677
$815$	−14.2461	−0.499021
$816$	0	0
$817$	−17.0284	−0.595748
$818$	24.4794	0.855902
$819$	0	0
$820$	−21.5002	−0.750819
$821$	−36.0314	−1.25751	−0.628753	−	0.777605i	$-0.716434\pi$
$821$	−36.0314	−1.25751	−0.628753	+	0.777605i	$0.716434\pi$
$822$	0	0
$823$	−6.62252	−0.230847	−0.115423	−	0.993316i	$-0.536822\pi$
$823$	−6.62252	−0.230847	−0.115423	+	0.993316i	$0.536822\pi$
$824$	10.6683	0.371648
$825$	0	0
$826$	58.2102	2.02539
$827$	−27.6818	−0.962592	−0.481296	−	0.876558i	$-0.659834\pi$
$827$	−27.6818	−0.962592	−0.481296	+	0.876558i	$0.659834\pi$
$828$	0	0
$829$	−4.33231	−0.150468	−0.0752338	−	0.997166i	$-0.523970\pi$
$829$	−4.33231	−0.150468	−0.0752338	+	0.997166i	$0.523970\pi$
$830$	−0.464754	−0.0161318
$831$	0	0
$832$	−4.29021	−0.148736
$833$	42.8500	1.48466
$834$	0	0
$835$	77.3368	2.67635
$836$	−3.66769	−0.126850
$837$	0	0
$838$	25.8766	0.893893
$839$	30.2530	1.04445	0.522226	−	0.852807i	$-0.325102\pi$
$839$	30.2530	1.04445	0.522226	+	0.852807i	$0.325102\pi$
$840$	0	0
$841$	4.44704	0.153346
$842$	7.49184	0.258186
$843$	0	0
$844$	16.6961	0.574702
$845$	16.3994	0.564157
$846$	0	0
$847$	−22.8028	−0.783515
$848$	−13.6305	−0.468074
$849$	0	0
$850$	22.2690	0.763821
$851$	0	0
$852$	0	0
$853$	5.21759	0.178647	0.0893234	−	0.996003i	$-0.471530\pi$
$853$	5.21759	0.178647	0.0893234	+	0.996003i	$0.471530\pi$
$854$	50.4373	1.72593
$855$	0	0
$856$	12.2877	0.419986
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	28.1471	0.960366	0.480183	−	0.877168i	$-0.340570\pi$
$859$	28.1471	0.960366	0.480183	+	0.877168i	$0.340570\pi$
$860$	31.8990	1.08775
$861$	0	0
$862$	11.8281	0.402868
$863$	28.3491	0.965014	0.482507	−	0.875892i	$-0.339726\pi$
$863$	28.3491	0.965014	0.482507	+	0.875892i	$0.339726\pi$
$864$	0	0
$865$	−17.5445	−0.596532
$866$	11.0267	0.374703
$867$	0	0
$868$	−12.4409	−0.422273
$869$	−1.46170	−0.0495847
$870$	0	0
$871$	55.4858	1.88006
$872$	7.63468	0.258543
$873$	0	0
$874$	0	0
$875$	−9.39214	−0.317512
$876$	0	0
$877$	43.1187	1.45602	0.728008	−	0.685569i	$-0.240446\pi$
$877$	43.1187	1.45602	0.728008	+	0.685569i	$0.240446\pi$
$878$	−36.1471	−1.21991
$879$	0	0
$880$	6.87062	0.231608
$881$	3.49323	0.117690	0.0588450	−	0.998267i	$-0.481258\pi$
$881$	3.49323	0.117690	0.0588450	+	0.998267i	$0.481258\pi$
$882$	0	0
$883$	40.2628	1.35495	0.677475	−	0.735546i	$-0.263074\pi$
$883$	40.2628	1.35495	0.677475	+	0.735546i	$0.263074\pi$
$884$	−22.7314	−0.764541
$885$	0	0
$886$	−2.98627	−0.100326
$887$	−42.2206	−1.41763	−0.708815	−	0.705394i	$-0.750770\pi$
$887$	−42.2206	−1.41763	−0.708815	+	0.705394i	$0.750770\pi$
$888$	0	0
$889$	64.0099	2.14682
$890$	−2.33231	−0.0781794
$891$	0	0
$892$	−21.2765	−0.712389
$893$	2.41801	0.0809158
$894$	0	0
$895$	−46.1201	−1.54163
$896$	3.88423	0.129763
$897$	0	0
$898$	−16.4794	−0.549925
$899$	18.5236	0.617798
$900$	0	0
$901$	−72.2206	−2.40602
$902$	−16.0514	−0.534452
$903$	0	0
$904$	11.3657	0.378018
$905$	−17.5667	−0.583936
$906$	0	0
$907$	13.0967	0.434869	0.217434	−	0.976075i	$-0.430231\pi$
$907$	13.0967	0.434869	0.217434	+	0.976075i	$0.430231\pi$
$908$	−28.9519	−0.960802
$909$	0	0
$910$	−50.5530	−1.67581
$911$	−50.6053	−1.67663	−0.838314	−	0.545188i	$-0.816458\pi$
$911$	−50.6053	−1.67663	−0.838314	+	0.545188i	$0.816458\pi$
$912$	0	0
$913$	−0.346971	−0.0114830
$914$	−17.3616	−0.574269
$915$	0	0
$916$	−12.1643	−0.401920
$917$	12.4409	0.410836
$918$	0	0
$919$	4.84790	0.159917	0.0799587	−	0.996798i	$-0.474521\pi$
$919$	4.84790	0.159917	0.0799587	+	0.996798i	$0.474521\pi$
$920$	0	0
$921$	0	0
$922$	14.5804	0.480180
$923$	6.40586	0.210852
$924$	0	0
$925$	40.9630	1.34686
$926$	−8.00000	−0.262896
$927$	0	0
$928$	−5.78334	−0.189847
$929$	−19.8990	−0.652865	−0.326432	−	0.945221i	$-0.605847\pi$
$929$	−19.8990	−0.652865	−0.326432	+	0.945221i	$0.605847\pi$
$930$	0	0
$931$	−13.0967	−0.429227
$932$	20.1471	0.659940
$933$	0	0
$934$	−33.4815	−1.09555
$935$	36.4036	1.19053
$936$	0	0
$937$	28.8388	0.942124	0.471062	−	0.882100i	$-0.343871\pi$
$937$	28.8388	0.942124	0.471062	+	0.882100i	$0.343871\pi$
$938$	−50.2353	−1.64024
$939$	0	0
$940$	−4.52963	−0.147740
$941$	34.2503	1.11653	0.558265	−	0.829663i	$-0.311467\pi$
$941$	34.2503	1.11653	0.558265	+	0.829663i	$0.311467\pi$
$942$	0	0
$943$	0	0
$944$	14.9863	0.487762
$945$	0	0
$946$	23.8148	0.774285
$947$	18.6225	0.605151	0.302575	−	0.953125i	$-0.402154\pi$
$947$	18.6225	0.605151	0.302575	+	0.953125i	$0.402154\pi$
$948$	0	0
$949$	−7.33537	−0.238116
$950$	−6.80631	−0.220826
$951$	0	0
$952$	20.5804	0.667015
$953$	−35.9412	−1.16425	−0.582125	−	0.813100i	$-0.697778\pi$
$953$	−35.9412	−1.16425	−0.582125	+	0.813100i	$0.697778\pi$
$954$	0	0
$955$	−37.7412	−1.22128
$956$	−2.98627	−0.0965830
$957$	0	0
$958$	2.11161	0.0682231
$959$	12.3784	0.399720
$960$	0	0
$961$	−20.7412	−0.669072
$962$	−41.8136	−1.34812
$963$	0	0
$964$	−5.63744	−0.181570
$965$	−35.3955	−1.13942
$966$	0	0
$967$	17.3775	0.558822	0.279411	−	0.960172i	$-0.409861\pi$
$967$	17.3775	0.558822	0.279411	+	0.960172i	$0.409861\pi$
$968$	−5.87062	−0.188689
$969$	0	0
$970$	−27.8569	−0.894431
$971$	−33.4815	−1.07447	−0.537237	−	0.843432i	$-0.680532\pi$
$971$	−33.4815	−1.07447	−0.537237	+	0.843432i	$0.680532\pi$
$972$	0	0
$973$	−9.34493	−0.299585
$974$	−16.0000	−0.512673
$975$	0	0
$976$	12.9851	0.415644
$977$	4.72447	0.151149	0.0755746	−	0.997140i	$-0.475921\pi$
$977$	4.72447	0.151149	0.0755746	+	0.997140i	$0.475921\pi$
$978$	0	0
$979$	−1.74123	−0.0556500
$980$	24.5338	0.783704
$981$	0	0
$982$	−11.3500	−0.362194
$983$	10.3293	0.329454	0.164727	−	0.986339i	$-0.447326\pi$
$983$	10.3293	0.329454	0.164727	+	0.986339i	$0.447326\pi$
$984$	0	0
$985$	−46.6941	−1.48780
$986$	−30.6427	−0.975863
$987$	0	0
$988$	6.94764	0.221034
$989$	0	0
$990$	0	0
$991$	−55.1187	−1.75090	−0.875452	−	0.483305i	$-0.839436\pi$
$991$	−55.1187	−1.75090	−0.875452	+	0.483305i	$0.839436\pi$
$992$	−3.20293	−0.101693
$993$	0	0
$994$	−5.79969	−0.183955
$995$	−5.37748	−0.170478
$996$	0	0
$997$	−59.0324	−1.86957	−0.934787	−	0.355209i	$-0.884410\pi$
$997$	−59.0324	−1.86957	−0.934787	+	0.355209i	$0.884410\pi$
$998$	−12.4648	−0.394565
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.cc.1.5	yes	6
3.2	odd	2		9522.2.a.cb.1.2	✓	6
23.22	odd	2	inner	9522.2.a.cc.1.2	yes	6
69.68	even	2		9522.2.a.cb.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9522.2.a.cb.1.2	✓	6	3.2	odd	2
9522.2.a.cb.1.5	yes	6	69.68	even	2
9522.2.a.cc.1.2	yes	6	23.22	odd	2	inner
9522.2.a.cc.1.5	yes	6	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.03363 q^{5} +3.88423 q^{7} +1.00000 q^{8} +3.03363 q^{10} +2.26481 q^{11} -4.29021 q^{13} +3.88423 q^{14} +1.00000 q^{16} +5.29845 q^{17} -1.61942 q^{19} +3.03363 q^{20} +2.26481 q^{22} +4.20293 q^{25} -4.29021 q^{26} +3.88423 q^{28} -5.78334 q^{29} -3.20293 q^{31} +1.00000 q^{32} +5.29845 q^{34} +11.7833 q^{35} +9.74629 q^{37} -1.61942 q^{38} +3.03363 q^{40} -7.08727 q^{41} +10.5151 q^{43} +2.26481 q^{44} -1.49314 q^{47} +8.08727 q^{49} +4.20293 q^{50} -4.29021 q^{52} -13.6305 q^{53} +6.87062 q^{55} +3.88423 q^{56} -5.78334 q^{58} +14.9863 q^{59} +12.9851 q^{61} -3.20293 q^{62} +1.00000 q^{64} -13.0149 q^{65} -12.9331 q^{67} +5.29845 q^{68} +11.7833 q^{70} -1.49314 q^{71} +1.70979 q^{73} +9.74629 q^{74} -1.61942 q^{76} +8.79707 q^{77} -0.645395 q^{79} +3.03363 q^{80} -7.08727 q^{82} -0.153200 q^{83} +16.0735 q^{85} +10.5151 q^{86} +2.26481 q^{88} -0.768819 q^{89} -16.6642 q^{91} -1.49314 q^{94} -4.91273 q^{95} -9.18268 q^{97} +8.08727 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 6 q^{16} + 18 q^{25} + 24 q^{29} - 12 q^{31} + 6 q^{32} + 12 q^{35} - 24 q^{41} + 24 q^{47} + 30 q^{49} + 18 q^{50} - 36 q^{55} + 24 q^{58} + 24 q^{59} - 12 q^{62} + 6 q^{64}+ \cdots + 30 q^{98}+O(q^{100}) \) 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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