LMFDB - Embedded newform 9522.2.a.cd.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 = [8,-8,0,8,0,0,0,-8,0,0,0,0,12,0,0,8,0,0,0,0,0,0,0,0,8,-12,0,0, 12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.546984493056.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 20x^{6} + 129x^{4} - 320x^{2} + 256 $ x^8 - 20x^6 + 129x^4 - 320*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.75786$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +0.343645 q^{5} -2.27550 q^{7} -1.00000 q^{8} -0.343645 q^{10} -3.94128 q^{11} +5.39592 q^{13} +2.27550 q^{14} +1.00000 q^{16} +7.63099 q^{17} -4.55099 q^{19} +0.343645 q^{20} +3.94128 q^{22} -4.88191 q^{25} -5.39592 q^{26} -2.27550 q^{28} -2.21804 q^{29} +2.04015 q^{31} -1.00000 q^{32} -7.63099 q^{34} -0.781962 q^{35} +7.47584 q^{37} +4.55099 q^{38} -0.343645 q^{40} -1.09007 q^{41} +2.22570 q^{43} -3.94128 q^{44} -13.2840 q^{47} -1.82212 q^{49} +4.88191 q^{50} +5.39592 q^{52} +5.62855 q^{53} -1.35440 q^{55} +2.27550 q^{56} +2.21804 q^{58} -8.39592 q^{59} +4.48986 q^{61} -2.04015 q^{62} +1.00000 q^{64} +1.85428 q^{65} +3.17641 q^{67} +7.63099 q^{68} +0.781962 q^{70} -3.50787 q^{71} -12.0977 q^{73} -7.47584 q^{74} -4.55099 q^{76} +8.96836 q^{77} +7.52246 q^{79} +0.343645 q^{80} +1.09007 q^{82} -5.31585 q^{83} +2.62235 q^{85} -2.22570 q^{86} +3.94128 q^{88} -12.5911 q^{89} -12.2784 q^{91} +13.2840 q^{94} -1.56392 q^{95} +13.1182 q^{97} +1.82212 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 12 q^{13} + 8 q^{16} + 8 q^{25} - 12 q^{26} + 12 q^{29} - 12 q^{31} - 8 q^{32} - 36 q^{35} - 24 q^{41} - 48 q^{47} - 16 q^{49} - 8 q^{50} + 12 q^{52} + 12 q^{55} - 12 q^{58}+ \cdots + 16 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 12 * q^13 + 8 * q^16 + 8 * q^25 - 12 * q^26 + 12 * q^29 - 12 * q^31 - 8 * q^32 - 36 * q^35 - 24 * q^41 - 48 * q^47 - 16 * q^49 - 8 * q^50 + 12 * q^52 + 12 * q^55 - 12 * q^58 - 36 * q^59 + 12 * q^62 + 8 * q^64 + 36 * q^70 - 24 * q^71 + 12 * q^73 - 12 * q^77 + 24 * q^82 + 12 * q^85 + 48 * q^94 - 72 * q^95 + 16 * 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
$5$	0.343645	0.153683	0.0768413	−	0.997043i	$-0.475517\pi$
$5$	0.343645	0.153683	0.0768413	+	0.997043i	$0.475517\pi$
$6$	0	0
$7$	−2.27550	−0.860057	−0.430028	−	0.902815i	$-0.641496\pi$
$7$	−2.27550	−0.860057	−0.430028	+	0.902815i	$0.641496\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.343645	−0.108670
$11$	−3.94128	−1.18834	−0.594170	−	0.804340i	$-0.702519\pi$
$11$	−3.94128	−1.18834	−0.594170	+	0.804340i	$0.702519\pi$
$12$	0	0
$13$	5.39592	1.49656	0.748280	−	0.663383i	$-0.230880\pi$
$13$	5.39592	1.49656	0.748280	+	0.663383i	$0.230880\pi$
$14$	2.27550	0.608152
$15$	0	0
$16$	1.00000	0.250000
$17$	7.63099	1.85079	0.925393	−	0.379009i	$-0.123735\pi$
$17$	7.63099	1.85079	0.925393	+	0.379009i	$0.123735\pi$
$18$	0	0
$19$	−4.55099	−1.04407	−0.522035	−	0.852924i	$-0.674827\pi$
$19$	−4.55099	−1.04407	−0.522035	+	0.852924i	$0.674827\pi$
$20$	0.343645	0.0768413
$21$	0	0
$22$	3.94128	0.840283
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.88191	−0.976382
$26$	−5.39592	−1.05823
$27$	0	0
$28$	−2.27550	−0.430028
$29$	−2.21804	−0.411879	−0.205940	−	0.978565i	$-0.566025\pi$
$29$	−2.21804	−0.411879	−0.205940	+	0.978565i	$0.566025\pi$
$30$	0	0
$31$	2.04015	0.366423	0.183211	−	0.983074i	$-0.441351\pi$
$31$	2.04015	0.366423	0.183211	+	0.983074i	$0.441351\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−7.63099	−1.30870
$35$	−0.781962	−0.132176
$36$	0	0
$37$	7.47584	1.22902	0.614510	−	0.788909i	$-0.289354\pi$
$37$	7.47584	1.22902	0.614510	+	0.788909i	$0.289354\pi$
$38$	4.55099	0.738268
$39$	0	0
$40$	−0.343645	−0.0543350
$41$	−1.09007	−0.170240	−0.0851198	−	0.996371i	$-0.527127\pi$
$41$	−1.09007	−0.170240	−0.0851198	+	0.996371i	$0.527127\pi$
$42$	0	0
$43$	2.22570	0.339416	0.169708	−	0.985494i	$-0.445718\pi$
$43$	2.22570	0.339416	0.169708	+	0.985494i	$0.445718\pi$
$44$	−3.94128	−0.594170
$45$	0	0
$46$	0	0
$47$	−13.2840	−1.93767	−0.968833	−	0.247714i	$-0.920321\pi$
$47$	−13.2840	−1.93767	−0.968833	+	0.247714i	$0.920321\pi$
$48$	0	0
$49$	−1.82212	−0.260302
$50$	4.88191	0.690406
$51$	0	0
$52$	5.39592	0.748280
$53$	5.62855	0.773141	0.386570	−	0.922260i	$-0.373660\pi$
$53$	5.62855	0.773141	0.386570	+	0.922260i	$0.373660\pi$
$54$	0	0
$55$	−1.35440	−0.182627
$56$	2.27550	0.304076
$57$	0	0
$58$	2.21804	0.291243
$59$	−8.39592	−1.09306	−0.546528	−	0.837441i	$-0.684051\pi$
$59$	−8.39592	−1.09306	−0.546528	+	0.837441i	$0.684051\pi$
$60$	0	0
$61$	4.48986	0.574868	0.287434	−	0.957800i	$-0.407198\pi$
$61$	4.48986	0.574868	0.287434	+	0.957800i	$0.407198\pi$
$62$	−2.04015	−0.259100
$63$	0	0
$64$	1.00000	0.125000
$65$	1.85428	0.229995
$66$	0	0
$67$	3.17641	0.388061	0.194030	−	0.980996i	$-0.437844\pi$
$67$	3.17641	0.388061	0.194030	+	0.980996i	$0.437844\pi$
$68$	7.63099	0.925393
$69$	0	0
$70$	0.781962	0.0934624
$71$	−3.50787	−0.416308	−0.208154	−	0.978096i	$-0.566745\pi$
$71$	−3.50787	−0.416308	−0.208154	+	0.978096i	$0.566745\pi$
$72$	0	0
$73$	−12.0977	−1.41593	−0.707964	−	0.706248i	$-0.750386\pi$
$73$	−12.0977	−1.41593	−0.707964	+	0.706248i	$0.750386\pi$
$74$	−7.47584	−0.869049
$75$	0	0
$76$	−4.55099	−0.522035
$77$	8.96836	1.02204
$78$	0	0
$79$	7.52246	0.846343	0.423172	−	0.906050i	$-0.360917\pi$
$79$	7.52246	0.846343	0.423172	+	0.906050i	$0.360917\pi$
$80$	0.343645	0.0384206
$81$	0	0
$82$	1.09007	0.120378
$83$	−5.31585	−0.583491	−0.291745	−	0.956496i	$-0.594236\pi$
$83$	−5.31585	−0.583491	−0.291745	+	0.956496i	$0.594236\pi$
$84$	0	0
$85$	2.62235	0.284434
$86$	−2.22570	−0.240003
$87$	0	0
$88$	3.94128	0.420141
$89$	−12.5911	−1.33465	−0.667327	−	0.744765i	$-0.732561\pi$
$89$	−12.5911	−1.33465	−0.667327	+	0.744765i	$0.732561\pi$
$90$	0	0
$91$	−12.2784	−1.28713
$92$	0	0
$93$	0	0
$94$	13.2840	1.37014
$95$	−1.56392	−0.160455
$96$	0	0
$97$	13.1182	1.33195	0.665975	−	0.745974i	$-0.268016\pi$
$97$	13.1182	1.33195	0.665975	+	0.745974i	$0.268016\pi$
$98$	1.82212	0.184062
$99$	0	0
$100$	−4.88191	−0.488191
$101$	−12.1681	−1.21077	−0.605387	−	0.795931i	$-0.706982\pi$
$101$	−12.1681	−1.21077	−0.605387	+	0.795931i	$0.706982\pi$
$102$	0	0
$103$	0.764861	0.0753640	0.0376820	−	0.999290i	$-0.488003\pi$
$103$	0.764861	0.0753640	0.0376820	+	0.999290i	$0.488003\pi$
$104$	−5.39592	−0.529114
$105$	0	0
$106$	−5.62855	−0.546693
$107$	9.39319	0.908074	0.454037	−	0.890983i	$-0.349983\pi$
$107$	9.39319	0.908074	0.454037	+	0.890983i	$0.349983\pi$
$108$	0	0
$109$	5.11843	0.490257	0.245128	−	0.969491i	$-0.421170\pi$
$109$	5.11843	0.490257	0.245128	+	0.969491i	$0.421170\pi$
$110$	1.35440	0.129137
$111$	0	0
$112$	−2.27550	−0.215014
$113$	15.8262	1.48881	0.744403	−	0.667730i	$-0.232734\pi$
$113$	15.8262	1.48881	0.744403	+	0.667730i	$0.232734\pi$
$114$	0	0
$115$	0	0
$116$	−2.21804	−0.205940
$117$	0	0
$118$	8.39592	0.772907
$119$	−17.3643	−1.59178
$120$	0	0
$121$	4.53365	0.412150
$122$	−4.48986	−0.406493
$123$	0	0
$124$	2.04015	0.183211
$125$	−3.39587	−0.303735
$126$	0	0
$127$	22.1561	1.96604	0.983019	−	0.183504i	$-0.0587441\pi$
$127$	22.1561	1.96604	0.983019	+	0.183504i	$0.0587441\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.85428	−0.162631
$131$	0.685756	0.0599148	0.0299574	−	0.999551i	$-0.490463\pi$
$131$	0.685756	0.0599148	0.0299574	+	0.999551i	$0.490463\pi$
$132$	0	0
$133$	10.3558	0.897959
$134$	−3.17641	−0.274400
$135$	0	0
$136$	−7.63099	−0.654352
$137$	17.2833	1.47661	0.738304	−	0.674468i	$-0.235627\pi$
$137$	17.2833	1.47661	0.738304	+	0.674468i	$0.235627\pi$
$138$	0	0
$139$	−11.2279	−0.952340	−0.476170	−	0.879353i	$-0.657975\pi$
$139$	−11.2279	−0.952340	−0.476170	+	0.879353i	$0.657975\pi$
$140$	−0.781962	−0.0660879
$141$	0	0
$142$	3.50787	0.294374
$143$	−21.2668	−1.77842
$144$	0	0
$145$	−0.762217	−0.0632987
$146$	12.0977	1.00121
$147$	0	0
$148$	7.47584	0.614510
$149$	2.38836	0.195662	0.0978311	−	0.995203i	$-0.468810\pi$
$149$	2.38836	0.195662	0.0978311	+	0.995203i	$0.468810\pi$
$150$	0	0
$151$	−14.7616	−1.20128	−0.600640	−	0.799520i	$-0.705087\pi$
$151$	−14.7616	−1.20128	−0.600640	+	0.799520i	$0.705087\pi$
$152$	4.55099	0.369134
$153$	0	0
$154$	−8.96836	−0.722691
$155$	0.701088	0.0563128
$156$	0	0
$157$	14.0363	1.12022	0.560108	−	0.828420i	$-0.310760\pi$
$157$	14.0363	1.12022	0.560108	+	0.828420i	$0.310760\pi$
$158$	−7.52246	−0.598455
$159$	0	0
$160$	−0.343645	−0.0271675
$161$	0	0
$162$	0	0
$163$	10.4361	0.817417	0.408708	−	0.912665i	$-0.365979\pi$
$163$	10.4361	0.817417	0.408708	+	0.912665i	$0.365979\pi$
$164$	−1.09007	−0.0851198
$165$	0	0
$166$	5.31585	0.412590
$167$	21.5837	1.67020	0.835098	−	0.550101i	$-0.185411\pi$
$167$	21.5837	1.67020	0.835098	+	0.550101i	$0.185411\pi$
$168$	0	0
$169$	16.1160	1.23969
$170$	−2.62235	−0.201125
$171$	0	0
$172$	2.22570	0.169708
$173$	−10.6844	−0.812319	−0.406159	−	0.913802i	$-0.633132\pi$
$173$	−10.6844	−0.812319	−0.406159	+	0.913802i	$0.633132\pi$
$174$	0	0
$175$	11.1088	0.839744
$176$	−3.94128	−0.297085
$177$	0	0
$178$	12.5911	0.943742
$179$	10.1561	0.759104	0.379552	−	0.925170i	$-0.376078\pi$
$179$	10.1561	0.759104	0.379552	+	0.925170i	$0.376078\pi$
$180$	0	0
$181$	−14.0457	−1.04401	−0.522005	−	0.852943i	$-0.674816\pi$
$181$	−14.0457	−1.04401	−0.522005	+	0.852943i	$0.674816\pi$
$182$	12.2784	0.910136
$183$	0	0
$184$	0	0
$185$	2.56903	0.188879
$186$	0	0
$187$	−30.0758	−2.19936
$188$	−13.2840	−0.968833
$189$	0	0
$190$	1.56392	0.113459
$191$	−15.3239	−1.10880	−0.554398	−	0.832252i	$-0.687052\pi$
$191$	−15.3239	−1.10880	−0.554398	+	0.832252i	$0.687052\pi$
$192$	0	0
$193$	1.52976	0.110114	0.0550572	−	0.998483i	$-0.482466\pi$
$193$	1.52976	0.110114	0.0550572	+	0.998483i	$0.482466\pi$
$194$	−13.1182	−0.941831
$195$	0	0
$196$	−1.82212	−0.130151
$197$	−21.5724	−1.53697	−0.768486	−	0.639866i	$-0.778990\pi$
$197$	−21.5724	−1.53697	−0.768486	+	0.639866i	$0.778990\pi$
$198$	0	0
$199$	−22.1817	−1.57242	−0.786210	−	0.617960i	$-0.787959\pi$
$199$	−22.1817	−1.57242	−0.786210	+	0.617960i	$0.787959\pi$
$200$	4.88191	0.345203
$201$	0	0
$202$	12.1681	0.856146
$203$	5.04714	0.354240
$204$	0	0
$205$	−0.374595	−0.0261629
$206$	−0.764861	−0.0532904
$207$	0	0
$208$	5.39592	0.374140
$209$	17.9367	1.24071
$210$	0	0
$211$	−2.13636	−0.147073	−0.0735366	−	0.997293i	$-0.523429\pi$
$211$	−2.13636	−0.147073	−0.0735366	+	0.997293i	$0.523429\pi$
$212$	5.62855	0.386570
$213$	0	0
$214$	−9.39319	−0.642105
$215$	0.764849	0.0521622
$216$	0	0
$217$	−4.64236	−0.315144
$218$	−5.11843	−0.346664
$219$	0	0
$220$	−1.35440	−0.0913135
$221$	41.1762	2.76981
$222$	0	0
$223$	14.1561	0.947964	0.473982	−	0.880535i	$-0.342816\pi$
$223$	14.1561	0.947964	0.473982	+	0.880535i	$0.342816\pi$
$224$	2.27550	0.152038
$225$	0	0
$226$	−15.8262	−1.05275
$227$	−20.9207	−1.38856	−0.694278	−	0.719707i	$-0.744276\pi$
$227$	−20.9207	−1.38856	−0.694278	+	0.719707i	$0.744276\pi$
$228$	0	0
$229$	−0.599552	−0.0396195	−0.0198098	−	0.999804i	$-0.506306\pi$
$229$	−0.599552	−0.0396195	−0.0198098	+	0.999804i	$0.506306\pi$
$230$	0	0
$231$	0	0
$232$	2.21804	0.145621
$233$	−18.4981	−1.21185	−0.605926	−	0.795521i	$-0.707197\pi$
$233$	−18.4981	−1.21185	−0.605926	+	0.795521i	$0.707197\pi$
$234$	0	0
$235$	−4.56497	−0.297786
$236$	−8.39592	−0.546528
$237$	0	0
$238$	17.3643	1.12556
$239$	14.5164	0.938987	0.469493	−	0.882936i	$-0.344437\pi$
$239$	14.5164	0.938987	0.469493	+	0.882936i	$0.344437\pi$
$240$	0	0
$241$	−12.1157	−0.780443	−0.390222	−	0.920721i	$-0.627602\pi$
$241$	−12.1157	−0.780443	−0.390222	+	0.920721i	$0.627602\pi$
$242$	−4.53365	−0.291434
$243$	0	0
$244$	4.48986	0.287434
$245$	−0.626161	−0.0400039
$246$	0	0
$247$	−24.5568	−1.56251
$248$	−2.04015	−0.129550
$249$	0	0
$250$	3.39587	0.214773
$251$	−16.7349	−1.05630	−0.528149	−	0.849152i	$-0.677114\pi$
$251$	−16.7349	−1.05630	−0.528149	+	0.849152i	$0.677114\pi$
$252$	0	0
$253$	0	0
$254$	−22.1561	−1.39020
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.2377	−0.763365	−0.381683	−	0.924293i	$-0.624655\pi$
$257$	−12.2377	−0.763365	−0.381683	+	0.924293i	$0.624655\pi$
$258$	0	0
$259$	−17.0112	−1.05703
$260$	1.85428	0.114998
$261$	0	0
$262$	−0.685756	−0.0423661
$263$	−27.3336	−1.68546	−0.842730	−	0.538337i	$-0.819053\pi$
$263$	−27.3336	−1.68546	−0.842730	+	0.538337i	$0.819053\pi$
$264$	0	0
$265$	1.93422	0.118818
$266$	−10.3558	−0.634953
$267$	0	0
$268$	3.17641	0.194030
$269$	−25.4560	−1.55208	−0.776040	−	0.630684i	$-0.782774\pi$
$269$	−25.4560	−1.55208	−0.776040	+	0.630684i	$0.782774\pi$
$270$	0	0
$271$	16.1061	0.978375	0.489188	−	0.872179i	$-0.337293\pi$
$271$	16.1061	0.978375	0.489188	+	0.872179i	$0.337293\pi$
$272$	7.63099	0.462696
$273$	0	0
$274$	−17.2833	−1.04412
$275$	19.2409	1.16027
$276$	0	0
$277$	8.29972	0.498682	0.249341	−	0.968416i	$-0.419786\pi$
$277$	8.29972	0.498682	0.249341	+	0.968416i	$0.419786\pi$
$278$	11.2279	0.673406
$279$	0	0
$280$	0.781962	0.0467312
$281$	−29.7692	−1.77588	−0.887942	−	0.459955i	$-0.847865\pi$
$281$	−29.7692	−1.77588	−0.887942	+	0.459955i	$0.847865\pi$
$282$	0	0
$283$	−19.0411	−1.13188	−0.565938	−	0.824448i	$-0.691486\pi$
$283$	−19.0411	−1.13188	−0.565938	+	0.824448i	$0.691486\pi$
$284$	−3.50787	−0.208154
$285$	0	0
$286$	21.2668	1.25753
$287$	2.48044	0.146416
$288$	0	0
$289$	41.2319	2.42541
$290$	0.762217	0.0447589
$291$	0	0
$292$	−12.0977	−0.707964
$293$	−0.667667	−0.0390056	−0.0195028	−	0.999810i	$-0.506208\pi$
$293$	−0.667667	−0.0390056	−0.0195028	+	0.999810i	$0.506208\pi$
$294$	0	0
$295$	−2.88521	−0.167984
$296$	−7.47584	−0.434524
$297$	0	0
$298$	−2.38836	−0.138354
$299$	0	0
$300$	0	0
$301$	−5.06456	−0.291917
$302$	14.7616	0.849433
$303$	0	0
$304$	−4.55099	−0.261017
$305$	1.54292	0.0883472
$306$	0	0
$307$	−31.3082	−1.78685	−0.893427	−	0.449208i	$-0.851706\pi$
$307$	−31.3082	−1.78685	−0.893427	+	0.449208i	$0.851706\pi$
$308$	8.96836	0.511020
$309$	0	0
$310$	−0.701088	−0.0398191
$311$	−10.4361	−0.591776	−0.295888	−	0.955223i	$-0.595615\pi$
$311$	−10.4361	−0.591776	−0.295888	+	0.955223i	$0.595615\pi$
$312$	0	0
$313$	0.0889007	0.00502496	0.00251248	−	0.999997i	$-0.499200\pi$
$313$	0.0889007	0.00502496	0.00251248	+	0.999997i	$0.499200\pi$
$314$	−14.0363	−0.792112
$315$	0	0
$316$	7.52246	0.423172
$317$	−5.50924	−0.309430	−0.154715	−	0.987959i	$-0.549446\pi$
$317$	−5.50924	−0.309430	−0.154715	+	0.987959i	$0.549446\pi$
$318$	0	0
$319$	8.74190	0.489452
$320$	0.343645	0.0192103
$321$	0	0
$322$	0	0
$323$	−34.7286	−1.93235
$324$	0	0
$325$	−26.3424	−1.46121
$326$	−10.4361	−0.578001
$327$	0	0
$328$	1.09007	0.0601888
$329$	30.2276	1.66650
$330$	0	0
$331$	−13.6640	−0.751041	−0.375521	−	0.926814i	$-0.622536\pi$
$331$	−13.6640	−0.751041	−0.375521	+	0.926814i	$0.622536\pi$
$332$	−5.31585	−0.291745
$333$	0	0
$334$	−21.5837	−1.18101
$335$	1.09156	0.0596382
$336$	0	0
$337$	5.71992	0.311584	0.155792	−	0.987790i	$-0.450207\pi$
$337$	5.71992	0.311584	0.155792	+	0.987790i	$0.450207\pi$
$338$	−16.1160	−0.876593
$339$	0	0
$340$	2.62235	0.142217
$341$	−8.04081	−0.435434
$342$	0	0
$343$	20.0747	1.08393
$344$	−2.22570	−0.120002
$345$	0	0
$346$	10.6844	0.574396
$347$	−34.1303	−1.83221	−0.916106	−	0.400935i	$-0.868685\pi$
$347$	−34.1303	−1.83221	−0.916106	+	0.400935i	$0.868685\pi$
$348$	0	0
$349$	7.36702	0.394347	0.197174	−	0.980369i	$-0.436824\pi$
$349$	7.36702	0.394347	0.197174	+	0.980369i	$0.436824\pi$
$350$	−11.1088	−0.593788
$351$	0	0
$352$	3.94128	0.210071
$353$	−6.03164	−0.321032	−0.160516	−	0.987033i	$-0.551316\pi$
$353$	−6.03164	−0.321032	−0.160516	+	0.987033i	$0.551316\pi$
$354$	0	0
$355$	−1.20546	−0.0639793
$356$	−12.5911	−0.667327
$357$	0	0
$358$	−10.1561	−0.536768
$359$	33.6448	1.77571	0.887853	−	0.460128i	$-0.152196\pi$
$359$	33.6448	1.77571	0.887853	+	0.460128i	$0.152196\pi$
$360$	0	0
$361$	1.71153	0.0900808
$362$	14.0457	0.738226
$363$	0	0
$364$	−12.2784	−0.643563
$365$	−4.15731	−0.217604
$366$	0	0
$367$	4.35116	0.227129	0.113564	−	0.993531i	$-0.463773\pi$
$367$	4.35116	0.227129	0.113564	+	0.993531i	$0.463773\pi$
$368$	0	0
$369$	0	0
$370$	−2.56903	−0.133558
$371$	−12.8077	−0.664945
$372$	0	0
$373$	27.8123	1.44006	0.720032	−	0.693941i	$-0.244127\pi$
$373$	27.8123	1.44006	0.720032	+	0.693941i	$0.244127\pi$
$374$	30.0758	1.55518
$375$	0	0
$376$	13.2840	0.685068
$377$	−11.9684	−0.616402
$378$	0	0
$379$	−38.2513	−1.96484	−0.982420	−	0.186685i	$-0.940226\pi$
$379$	−38.2513	−1.96484	−0.982420	+	0.186685i	$0.940226\pi$
$380$	−1.56392	−0.0802276
$381$	0	0
$382$	15.3239	0.784037
$383$	−28.1431	−1.43805	−0.719023	−	0.694987i	$-0.755410\pi$
$383$	−28.1431	−1.43805	−0.719023	+	0.694987i	$0.755410\pi$
$384$	0	0
$385$	3.08193	0.157070
$386$	−1.52976	−0.0778626
$387$	0	0
$388$	13.1182	0.665975
$389$	13.1813	0.668316	0.334158	−	0.942517i	$-0.391548\pi$
$389$	13.1813	0.668316	0.334158	+	0.942517i	$0.391548\pi$
$390$	0	0
$391$	0	0
$392$	1.82212	0.0920308
$393$	0	0
$394$	21.5724	1.08680
$395$	2.58505	0.130068
$396$	0	0
$397$	26.3484	1.32239	0.661194	−	0.750215i	$-0.270050\pi$
$397$	26.3484	1.32239	0.661194	+	0.750215i	$0.270050\pi$
$398$	22.1817	1.11187
$399$	0	0
$400$	−4.88191	−0.244095
$401$	−22.0600	−1.10162	−0.550811	−	0.834630i	$-0.685682\pi$
$401$	−22.0600	−1.10162	−0.550811	+	0.834630i	$0.685682\pi$
$402$	0	0
$403$	11.0085	0.548373
$404$	−12.1681	−0.605387
$405$	0	0
$406$	−5.04714	−0.250485
$407$	−29.4643	−1.46049
$408$	0	0
$409$	−3.95535	−0.195579	−0.0977897	−	0.995207i	$-0.531177\pi$
$409$	−3.95535	−0.195579	−0.0977897	+	0.995207i	$0.531177\pi$
$410$	0.374595	0.0184999
$411$	0	0
$412$	0.764861	0.0376820
$413$	19.1049	0.940090
$414$	0	0
$415$	−1.82676	−0.0896724
$416$	−5.39592	−0.264557
$417$	0	0
$418$	−17.9367	−0.877313
$419$	−17.0204	−0.831499	−0.415750	−	0.909479i	$-0.636481\pi$
$419$	−17.0204	−0.831499	−0.415750	+	0.909479i	$0.636481\pi$
$420$	0	0
$421$	−0.0964214	−0.00469929	−0.00234965	−	0.999997i	$-0.500748\pi$
$421$	−0.0964214	−0.00469929	−0.00234965	+	0.999997i	$0.500748\pi$
$422$	2.13636	0.103996
$423$	0	0
$424$	−5.62855	−0.273346
$425$	−37.2538	−1.80707
$426$	0	0
$427$	−10.2167	−0.494419
$428$	9.39319	0.454037
$429$	0	0
$430$	−0.764849	−0.0368843
$431$	−11.1725	−0.538162	−0.269081	−	0.963118i	$-0.586720\pi$
$431$	−11.1725	−0.538162	−0.269081	+	0.963118i	$0.586720\pi$
$432$	0	0
$433$	10.2293	0.491591	0.245795	−	0.969322i	$-0.420951\pi$
$433$	10.2293	0.491591	0.245795	+	0.969322i	$0.420951\pi$
$434$	4.64236	0.222841
$435$	0	0
$436$	5.11843	0.245128
$437$	0	0
$438$	0	0
$439$	−15.8163	−0.754869	−0.377434	−	0.926036i	$-0.623194\pi$
$439$	−15.8163	−0.754869	−0.377434	+	0.926036i	$0.623194\pi$
$440$	1.35440	0.0645684
$441$	0	0
$442$	−41.1762	−1.95855
$443$	29.1805	1.38641	0.693204	−	0.720741i	$-0.256198\pi$
$443$	29.1805	1.38641	0.693204	+	0.720741i	$0.256198\pi$
$444$	0	0
$445$	−4.32686	−0.205113
$446$	−14.1561	−0.670312
$447$	0	0
$448$	−2.27550	−0.107507
$449$	−13.0463	−0.615693	−0.307846	−	0.951436i	$-0.599608\pi$
$449$	−13.0463	−0.615693	−0.307846	+	0.951436i	$0.599608\pi$
$450$	0	0
$451$	4.29625	0.202302
$452$	15.8262	0.744403
$453$	0	0
$454$	20.9207	0.981857
$455$	−4.21941	−0.197809
$456$	0	0
$457$	−27.2862	−1.27639	−0.638196	−	0.769874i	$-0.720319\pi$
$457$	−27.2862	−1.27639	−0.638196	+	0.769874i	$0.720319\pi$
$458$	0.599552	0.0280152
$459$	0	0
$460$	0	0
$461$	−24.1404	−1.12433	−0.562165	−	0.827025i	$-0.690031\pi$
$461$	−24.1404	−1.12433	−0.562165	+	0.827025i	$0.690031\pi$
$462$	0	0
$463$	23.2636	1.08115	0.540575	−	0.841296i	$-0.318207\pi$
$463$	23.2636	1.08115	0.540575	+	0.841296i	$0.318207\pi$
$464$	−2.21804	−0.102970
$465$	0	0
$466$	18.4981	0.856909
$467$	6.99900	0.323875	0.161938	−	0.986801i	$-0.448226\pi$
$467$	6.99900	0.323875	0.161938	+	0.986801i	$0.448226\pi$
$468$	0	0
$469$	−7.22792	−0.333754
$470$	4.56497	0.210566
$471$	0	0
$472$	8.39592	0.386454
$473$	−8.77208	−0.403341
$474$	0	0
$475$	22.2175	1.01941
$476$	−17.3643	−0.795890
$477$	0	0
$478$	−14.5164	−0.663964
$479$	0.0996001	0.00455084	0.00227542	−	0.999997i	$-0.499276\pi$
$479$	0.0996001	0.00455084	0.00227542	+	0.999997i	$0.499276\pi$
$480$	0	0
$481$	40.3391	1.83930
$482$	12.1157	0.551857
$483$	0	0
$484$	4.53365	0.206075
$485$	4.50800	0.204698
$486$	0	0
$487$	−11.1120	−0.503531	−0.251765	−	0.967788i	$-0.581011\pi$
$487$	−11.1120	−0.503531	−0.251765	+	0.967788i	$0.581011\pi$
$488$	−4.48986	−0.203247
$489$	0	0
$490$	0.626161	0.0282871
$491$	42.3941	1.91322	0.956609	−	0.291375i	$-0.0941128\pi$
$491$	42.3941	1.91322	0.956609	+	0.291375i	$0.0941128\pi$
$492$	0	0
$493$	−16.9258	−0.762300
$494$	24.5568	1.10486
$495$	0	0
$496$	2.04015	0.0916056
$497$	7.98215	0.358048
$498$	0	0
$499$	12.4849	0.558901	0.279450	−	0.960160i	$-0.409848\pi$
$499$	12.4849	0.558901	0.279450	+	0.960160i	$0.409848\pi$
$500$	−3.39587	−0.151868
$501$	0	0
$502$	16.7349	0.746915
$503$	−28.8530	−1.28649	−0.643247	−	0.765659i	$-0.722413\pi$
$503$	−28.8530	−1.28649	−0.643247	+	0.765659i	$0.722413\pi$
$504$	0	0
$505$	−4.18151	−0.186075
$506$	0	0
$507$	0	0
$508$	22.1561	0.983019
$509$	−4.30108	−0.190642	−0.0953211	−	0.995447i	$-0.530388\pi$
$509$	−4.30108	−0.190642	−0.0953211	+	0.995447i	$0.530388\pi$
$510$	0	0
$511$	27.5283	1.21778
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	12.2377	0.539781
$515$	0.262841	0.0115821
$516$	0	0
$517$	52.3558	2.30260
$518$	17.0112	0.747431
$519$	0	0
$520$	−1.85428	−0.0813155
$521$	−4.27910	−0.187471	−0.0937353	−	0.995597i	$-0.529881\pi$
$521$	−4.27910	−0.187471	−0.0937353	+	0.995597i	$0.529881\pi$
$522$	0	0
$523$	12.9986	0.568388	0.284194	−	0.958767i	$-0.408274\pi$
$523$	12.9986	0.568388	0.284194	+	0.958767i	$0.408274\pi$
$524$	0.685756	0.0299574
$525$	0	0
$526$	27.3336	1.19180
$527$	15.5684	0.678170
$528$	0	0
$529$	0	0
$530$	−1.93422	−0.0840172
$531$	0	0
$532$	10.3558	0.448979
$533$	−5.88191	−0.254774
$534$	0	0
$535$	3.22792	0.139555
$536$	−3.17641	−0.137200
$537$	0	0
$538$	25.4560	1.09749
$539$	7.18146	0.309327
$540$	0	0
$541$	28.6710	1.23266	0.616332	−	0.787487i	$-0.288618\pi$
$541$	28.6710	1.23266	0.616332	+	0.787487i	$0.288618\pi$
$542$	−16.1061	−0.691816
$543$	0	0
$544$	−7.63099	−0.327176
$545$	1.75892	0.0753439
$546$	0	0
$547$	−33.9637	−1.45218	−0.726092	−	0.687598i	$-0.758665\pi$
$547$	−33.9637	−1.45218	−0.726092	+	0.687598i	$0.758665\pi$
$548$	17.2833	0.738304
$549$	0	0
$550$	−19.2409	−0.820437
$551$	10.0943	0.430031
$552$	0	0
$553$	−17.1173	−0.727903
$554$	−8.29972	−0.352621
$555$	0	0
$556$	−11.2279	−0.476170
$557$	18.3963	0.779475	0.389738	−	0.920926i	$-0.372566\pi$
$557$	18.3963	0.779475	0.389738	+	0.920926i	$0.372566\pi$
$558$	0	0
$559$	12.0097	0.507955
$560$	−0.781962	−0.0330439
$561$	0	0
$562$	29.7692	1.25574
$563$	−25.7820	−1.08658	−0.543290	−	0.839545i	$-0.682822\pi$
$563$	−25.7820	−1.08658	−0.543290	+	0.839545i	$0.682822\pi$
$564$	0	0
$565$	5.43860	0.228804
$566$	19.0411	0.800358
$567$	0	0
$568$	3.50787	0.147187
$569$	−5.78221	−0.242403	−0.121201	−	0.992628i	$-0.538675\pi$
$569$	−5.78221	−0.242403	−0.121201	+	0.992628i	$0.538675\pi$
$570$	0	0
$571$	20.2745	0.848462	0.424231	−	0.905554i	$-0.360544\pi$
$571$	20.2745	0.848462	0.424231	+	0.905554i	$0.360544\pi$
$572$	−21.2668	−0.889210
$573$	0	0
$574$	−2.48044	−0.103532
$575$	0	0
$576$	0	0
$577$	−26.2424	−1.09249	−0.546244	−	0.837626i	$-0.683943\pi$
$577$	−26.2424	−1.09249	−0.546244	+	0.837626i	$0.683943\pi$
$578$	−41.2319	−1.71502
$579$	0	0
$580$	−0.762217	−0.0316493
$581$	12.0962	0.501835
$582$	0	0
$583$	−22.1837	−0.918753
$584$	12.0977	0.500606
$585$	0	0
$586$	0.667667	0.0275811
$587$	−32.9237	−1.35891	−0.679453	−	0.733719i	$-0.737783\pi$
$587$	−32.9237	−1.35891	−0.679453	+	0.733719i	$0.737783\pi$
$588$	0	0
$589$	−9.28473	−0.382571
$590$	2.88521	0.118782
$591$	0	0
$592$	7.47584	0.307255
$593$	34.5877	1.42035	0.710173	−	0.704027i	$-0.248617\pi$
$593$	34.5877	1.42035	0.710173	+	0.704027i	$0.248617\pi$
$594$	0	0
$595$	−5.96714	−0.244629
$596$	2.38836	0.0978311
$597$	0	0
$598$	0	0
$599$	−1.28397	−0.0524616	−0.0262308	−	0.999656i	$-0.508350\pi$
$599$	−1.28397	−0.0524616	−0.0262308	+	0.999656i	$0.508350\pi$
$600$	0	0
$601$	12.4241	0.506788	0.253394	−	0.967363i	$-0.418453\pi$
$601$	12.4241	0.506788	0.253394	+	0.967363i	$0.418453\pi$
$602$	5.06456	0.206416
$603$	0	0
$604$	−14.7616	−0.600640
$605$	1.55796	0.0633403
$606$	0	0
$607$	−13.0903	−0.531321	−0.265660	−	0.964067i	$-0.585590\pi$
$607$	−13.0903	−0.531321	−0.265660	+	0.964067i	$0.585590\pi$
$608$	4.55099	0.184567
$609$	0	0
$610$	−1.54292	−0.0624709
$611$	−71.6793	−2.89983
$612$	0	0
$613$	−8.47582	−0.342335	−0.171168	−	0.985242i	$-0.554754\pi$
$613$	−8.47582	−0.342335	−0.171168	+	0.985242i	$0.554754\pi$
$614$	31.3082	1.26350
$615$	0	0
$616$	−8.96836	−0.361345
$617$	−18.4352	−0.742173	−0.371087	−	0.928598i	$-0.621015\pi$
$617$	−18.4352	−0.742173	−0.371087	+	0.928598i	$0.621015\pi$
$618$	0	0
$619$	−2.23967	−0.0900199	−0.0450100	−	0.998987i	$-0.514332\pi$
$619$	−2.23967	−0.0900199	−0.0450100	+	0.998987i	$0.514332\pi$
$620$	0.701088	0.0281564
$621$	0	0
$622$	10.4361	0.418449
$623$	28.6510	1.14788
$624$	0	0
$625$	23.2426	0.929703
$626$	−0.0889007	−0.00355318
$627$	0	0
$628$	14.0363	0.560108
$629$	57.0480	2.27465
$630$	0	0
$631$	−18.6363	−0.741901	−0.370951	−	0.928653i	$-0.620968\pi$
$631$	−18.6363	−0.741901	−0.370951	+	0.928653i	$0.620968\pi$
$632$	−7.52246	−0.299227
$633$	0	0
$634$	5.50924	0.218800
$635$	7.61383	0.302146
$636$	0	0
$637$	−9.83200	−0.389558
$638$	−8.74190	−0.346095
$639$	0	0
$640$	−0.343645	−0.0135837
$641$	−10.2777	−0.405943	−0.202972	−	0.979185i	$-0.565060\pi$
$641$	−10.2777	−0.405943	−0.202972	+	0.979185i	$0.565060\pi$
$642$	0	0
$643$	−13.9391	−0.549703	−0.274852	−	0.961487i	$-0.588629\pi$
$643$	−13.9391	−0.549703	−0.274852	+	0.961487i	$0.588629\pi$
$644$	0	0
$645$	0	0
$646$	34.7286	1.36638
$647$	22.0153	0.865509	0.432755	−	0.901512i	$-0.357542\pi$
$647$	22.0153	0.865509	0.432755	+	0.901512i	$0.357542\pi$
$648$	0	0
$649$	33.0906	1.29892
$650$	26.3424	1.03323
$651$	0	0
$652$	10.4361	0.408708
$653$	33.6547	1.31701	0.658506	−	0.752576i	$-0.271189\pi$
$653$	33.6547	1.31701	0.658506	+	0.752576i	$0.271189\pi$
$654$	0	0
$655$	0.235656	0.00920786
$656$	−1.09007	−0.0425599
$657$	0	0
$658$	−30.2276	−1.17840
$659$	18.3540	0.714970	0.357485	−	0.933919i	$-0.383634\pi$
$659$	18.3540	0.714970	0.357485	+	0.933919i	$0.383634\pi$
$660$	0	0
$661$	−22.3013	−0.867421	−0.433710	−	0.901052i	$-0.642796\pi$
$661$	−22.3013	−0.867421	−0.433710	+	0.901052i	$0.642796\pi$
$662$	13.6640	0.531066
$663$	0	0
$664$	5.31585	0.206295
$665$	3.55870	0.138001
$666$	0	0
$667$	0	0
$668$	21.5837	0.835098
$669$	0	0
$670$	−1.09156	−0.0421706
$671$	−17.6958	−0.683138
$672$	0	0
$673$	−29.4847	−1.13655	−0.568277	−	0.822838i	$-0.692390\pi$
$673$	−29.4847	−1.13655	−0.568277	+	0.822838i	$0.692390\pi$
$674$	−5.71992	−0.220323
$675$	0	0
$676$	16.1160	0.619845
$677$	−19.8135	−0.761495	−0.380747	−	0.924679i	$-0.624333\pi$
$677$	−19.8135	−0.761495	−0.380747	+	0.924679i	$0.624333\pi$
$678$	0	0
$679$	−29.8504	−1.14555
$680$	−2.62235	−0.100562
$681$	0	0
$682$	8.04081	0.307899
$683$	46.3953	1.77527	0.887633	−	0.460551i	$-0.152348\pi$
$683$	46.3953	1.77527	0.887633	+	0.460551i	$0.152348\pi$
$684$	0	0
$685$	5.93930	0.226929
$686$	−20.0747	−0.766455
$687$	0	0
$688$	2.22570	0.0848539
$689$	30.3712	1.15705
$690$	0	0
$691$	−20.2997	−0.772238	−0.386119	−	0.922449i	$-0.626185\pi$
$691$	−20.2997	−0.772238	−0.386119	+	0.922449i	$0.626185\pi$
$692$	−10.6844	−0.406159
$693$	0	0
$694$	34.1303	1.29557
$695$	−3.85841	−0.146358
$696$	0	0
$697$	−8.31827	−0.315077
$698$	−7.36702	−0.278846
$699$	0	0
$700$	11.1088	0.419872
$701$	18.8156	0.710656	0.355328	−	0.934742i	$-0.384369\pi$
$701$	18.8156	0.710656	0.355328	+	0.934742i	$0.384369\pi$
$702$	0	0
$703$	−34.0225	−1.28318
$704$	−3.94128	−0.148542
$705$	0	0
$706$	6.03164	0.227004
$707$	27.6885	1.04133
$708$	0	0
$709$	−30.2990	−1.13790	−0.568951	−	0.822371i	$-0.692651\pi$
$709$	−30.2990	−1.13790	−0.568951	+	0.822371i	$0.692651\pi$
$710$	1.20546	0.0452402
$711$	0	0
$712$	12.5911	0.471871
$713$	0	0
$714$	0	0
$715$	−7.30823	−0.273312
$716$	10.1561	0.379552
$717$	0	0
$718$	−33.6448	−1.25561
$719$	−15.8004	−0.589254	−0.294627	−	0.955612i	$-0.595195\pi$
$719$	−15.8004	−0.589254	−0.294627	+	0.955612i	$0.595195\pi$
$720$	0	0
$721$	−1.74044	−0.0648173
$722$	−1.71153	−0.0636967
$723$	0	0
$724$	−14.0457	−0.522005
$725$	10.8283	0.402151
$726$	0	0
$727$	32.6309	1.21021	0.605107	−	0.796144i	$-0.293130\pi$
$727$	32.6309	1.21021	0.605107	+	0.796144i	$0.293130\pi$
$728$	12.2784	0.455068
$729$	0	0
$730$	4.15731	0.153869
$731$	16.9843	0.628185
$732$	0	0
$733$	−51.0266	−1.88471	−0.942355	−	0.334614i	$-0.891394\pi$
$733$	−51.0266	−1.88471	−0.942355	+	0.334614i	$0.891394\pi$
$734$	−4.35116	−0.160604
$735$	0	0
$736$	0	0
$737$	−12.5191	−0.461148
$738$	0	0
$739$	−24.9237	−0.916833	−0.458417	−	0.888737i	$-0.651583\pi$
$739$	−24.9237	−0.916833	−0.458417	+	0.888737i	$0.651583\pi$
$740$	2.56903	0.0944395
$741$	0	0
$742$	12.8077	0.470187
$743$	−0.465432	−0.0170750	−0.00853752	−	0.999964i	$-0.502718\pi$
$743$	−0.465432	−0.0170750	−0.00853752	+	0.999964i	$0.502718\pi$
$744$	0	0
$745$	0.820748	0.0300699
$746$	−27.8123	−1.01828
$747$	0	0
$748$	−30.0758	−1.09968
$749$	−21.3742	−0.780995
$750$	0	0
$751$	28.2433	1.03061	0.515307	−	0.857006i	$-0.327678\pi$
$751$	28.2433	1.03061	0.515307	+	0.857006i	$0.327678\pi$
$752$	−13.2840	−0.484417
$753$	0	0
$754$	11.9684	0.435862
$755$	−5.07273	−0.184616
$756$	0	0
$757$	−38.2042	−1.38855	−0.694277	−	0.719707i	$-0.744276\pi$
$757$	−38.2042	−1.38855	−0.694277	+	0.719707i	$0.744276\pi$
$758$	38.2513	1.38935
$759$	0	0
$760$	1.56392	0.0567295
$761$	−51.7670	−1.87655	−0.938275	−	0.345889i	$-0.887577\pi$
$761$	−51.7670	−1.87655	−0.938275	+	0.345889i	$0.887577\pi$
$762$	0	0
$763$	−11.6470	−0.421649
$764$	−15.3239	−0.554398
$765$	0	0
$766$	28.1431	1.01685
$767$	−45.3037	−1.63582
$768$	0	0
$769$	−40.6546	−1.46604	−0.733021	−	0.680206i	$-0.761890\pi$
$769$	−40.6546	−1.46604	−0.733021	+	0.680206i	$0.761890\pi$
$770$	−3.08193	−0.111065
$771$	0	0
$772$	1.52976	0.0550572
$773$	−18.3489	−0.659966	−0.329983	−	0.943987i	$-0.607043\pi$
$773$	−18.3489	−0.659966	−0.329983	+	0.943987i	$0.607043\pi$
$774$	0	0
$775$	−9.95985	−0.357768
$776$	−13.1182	−0.470916
$777$	0	0
$778$	−13.1813	−0.472571
$779$	4.96088	0.177742
$780$	0	0
$781$	13.8255	0.494715
$782$	0	0
$783$	0	0
$784$	−1.82212	−0.0650756
$785$	4.82349	0.172158
$786$	0	0
$787$	−31.5743	−1.12550	−0.562750	−	0.826627i	$-0.690257\pi$
$787$	−31.5743	−1.12550	−0.562750	+	0.826627i	$0.690257\pi$
$788$	−21.5724	−0.768486
$789$	0	0
$790$	−2.58505	−0.0919721
$791$	−36.0125	−1.28046
$792$	0	0
$793$	24.2270	0.860324
$794$	−26.3484	−0.935069
$795$	0	0
$796$	−22.1817	−0.786210
$797$	28.1590	0.997443	0.498722	−	0.866762i	$-0.333803\pi$
$797$	28.1590	0.997443	0.498722	+	0.866762i	$0.333803\pi$
$798$	0	0
$799$	−101.370	−3.58621
$800$	4.88191	0.172602
$801$	0	0
$802$	22.0600	0.778965
$803$	47.6804	1.68260
$804$	0	0
$805$	0	0
$806$	−11.0085	−0.387758
$807$	0	0
$808$	12.1681	0.428073
$809$	−28.2952	−0.994807	−0.497403	−	0.867519i	$-0.665713\pi$
$809$	−28.2952	−0.994807	−0.497403	+	0.867519i	$0.665713\pi$
$810$	0	0
$811$	−33.5321	−1.17747	−0.588736	−	0.808325i	$-0.700374\pi$
$811$	−33.5321	−1.17747	−0.588736	+	0.808325i	$0.700374\pi$
$812$	5.04714	0.177120
$813$	0	0
$814$	29.4643	1.03272
$815$	3.58630	0.125623
$816$	0	0
$817$	−10.1291	−0.354373
$818$	3.95535	0.138296
$819$	0	0
$820$	−0.374595	−0.0130814
$821$	6.52389	0.227685	0.113843	−	0.993499i	$-0.463684\pi$
$821$	6.52389	0.227685	0.113843	+	0.993499i	$0.463684\pi$
$822$	0	0
$823$	11.4401	0.398777	0.199388	−	0.979921i	$-0.436105\pi$
$823$	11.4401	0.398777	0.199388	+	0.979921i	$0.436105\pi$
$824$	−0.764861	−0.0266452
$825$	0	0
$826$	−19.1049	−0.664744
$827$	−0.603974	−0.0210022	−0.0105011	−	0.999945i	$-0.503343\pi$
$827$	−0.603974	−0.0210022	−0.0105011	+	0.999945i	$0.503343\pi$
$828$	0	0
$829$	−30.3422	−1.05383	−0.526915	−	0.849918i	$-0.676651\pi$
$829$	−30.3422	−1.05383	−0.526915	+	0.849918i	$0.676651\pi$
$830$	1.82676	0.0634079
$831$	0	0
$832$	5.39592	0.187070
$833$	−13.9045	−0.481764
$834$	0	0
$835$	7.41712	0.256680
$836$	17.9367	0.620354
$837$	0	0
$838$	17.0204	0.587959
$839$	4.46537	0.154162	0.0770808	−	0.997025i	$-0.475440\pi$
$839$	4.46537	0.154162	0.0770808	+	0.997025i	$0.475440\pi$
$840$	0	0
$841$	−24.0803	−0.830355
$842$	0.0964214	0.00332290
$843$	0	0
$844$	−2.13636	−0.0735366
$845$	5.53817	0.190519
$846$	0	0
$847$	−10.3163	−0.354472
$848$	5.62855	0.193285
$849$	0	0
$850$	37.2538	1.27779
$851$	0	0
$852$	0	0
$853$	−16.7088	−0.572098	−0.286049	−	0.958215i	$-0.592342\pi$
$853$	−16.7088	−0.572098	−0.286049	+	0.958215i	$0.592342\pi$
$854$	10.2167	0.349607
$855$	0	0
$856$	−9.39319	−0.321052
$857$	34.1138	1.16531	0.582653	−	0.812721i	$-0.302015\pi$
$857$	34.1138	1.16531	0.582653	+	0.812721i	$0.302015\pi$
$858$	0	0
$859$	−7.33279	−0.250192	−0.125096	−	0.992145i	$-0.539924\pi$
$859$	−7.33279	−0.250192	−0.125096	+	0.992145i	$0.539924\pi$
$860$	0.764849	0.0260811
$861$	0	0
$862$	11.1725	0.380538
$863$	−4.93094	−0.167851	−0.0839256	−	0.996472i	$-0.526746\pi$
$863$	−4.93094	−0.167851	−0.0839256	+	0.996472i	$0.526746\pi$
$864$	0	0
$865$	−3.67163	−0.124839
$866$	−10.2293	−0.347607
$867$	0	0
$868$	−4.64236	−0.157572
$869$	−29.6481	−1.00574
$870$	0	0
$871$	17.1397	0.580756
$872$	−5.11843	−0.173332
$873$	0	0
$874$	0	0
$875$	7.72728	0.261230
$876$	0	0
$877$	−30.5434	−1.03138	−0.515688	−	0.856776i	$-0.672464\pi$
$877$	−30.5434	−1.03138	−0.515688	+	0.856776i	$0.672464\pi$
$878$	15.8163	0.533773
$879$	0	0
$880$	−1.35440	−0.0456568
$881$	−4.62398	−0.155786	−0.0778930	−	0.996962i	$-0.524819\pi$
$881$	−4.62398	−0.155786	−0.0778930	+	0.996962i	$0.524819\pi$
$882$	0	0
$883$	−58.8677	−1.98105	−0.990527	−	0.137317i	$-0.956152\pi$
$883$	−58.8677	−1.98105	−0.990527	+	0.137317i	$0.956152\pi$
$884$	41.1762	1.38491
$885$	0	0
$886$	−29.1805	−0.980339
$887$	9.77610	0.328249	0.164125	−	0.986440i	$-0.447520\pi$
$887$	9.77610	0.328249	0.164125	+	0.986440i	$0.447520\pi$
$888$	0	0
$889$	−50.4162	−1.69090
$890$	4.32686	0.145037
$891$	0	0
$892$	14.1561	0.473982
$893$	60.4553	2.02306
$894$	0	0
$895$	3.49010	0.116661
$896$	2.27550	0.0760190
$897$	0	0
$898$	13.0463	0.435360
$899$	−4.52514	−0.150922
$900$	0	0
$901$	42.9514	1.43092
$902$	−4.29625	−0.143049
$903$	0	0
$904$	−15.8262	−0.526373
$905$	−4.82674	−0.160446
$906$	0	0
$907$	−43.2122	−1.43484	−0.717419	−	0.696642i	$-0.754677\pi$
$907$	−43.2122	−1.43484	−0.717419	+	0.696642i	$0.754677\pi$
$908$	−20.9207	−0.694278
$909$	0	0
$910$	4.21941	0.139872
$911$	22.5418	0.746843	0.373421	−	0.927662i	$-0.378185\pi$
$911$	22.5418	0.746843	0.373421	+	0.927662i	$0.378185\pi$
$912$	0	0
$913$	20.9512	0.693385
$914$	27.2862	0.902546
$915$	0	0
$916$	−0.599552	−0.0198098
$917$	−1.56044	−0.0515301
$918$	0	0
$919$	−42.8829	−1.41457	−0.707287	−	0.706926i	$-0.750081\pi$
$919$	−42.8829	−1.41457	−0.707287	+	0.706926i	$0.750081\pi$
$920$	0	0
$921$	0	0
$922$	24.1404	0.795021
$923$	−18.9282	−0.623029
$924$	0	0
$925$	−36.4964	−1.19999
$926$	−23.2636	−0.764489
$927$	0	0
$928$	2.21804	0.0728107
$929$	24.8632	0.815734	0.407867	−	0.913041i	$-0.366273\pi$
$929$	24.8632	0.815734	0.407867	+	0.913041i	$0.366273\pi$
$930$	0	0
$931$	8.29244	0.271774
$932$	−18.4981	−0.605926
$933$	0	0
$934$	−6.99900	−0.229014
$935$	−10.3354	−0.338004
$936$	0	0
$937$	36.7797	1.20154	0.600771	−	0.799422i	$-0.294861\pi$
$937$	36.7797	1.20154	0.600771	+	0.799422i	$0.294861\pi$
$938$	7.22792	0.236000
$939$	0	0
$940$	−4.56497	−0.148893
$941$	−33.6741	−1.09774	−0.548872	−	0.835907i	$-0.684942\pi$
$941$	−33.6741	−1.09774	−0.548872	+	0.835907i	$0.684942\pi$
$942$	0	0
$943$	0	0
$944$	−8.39592	−0.273264
$945$	0	0
$946$	8.77208	0.285205
$947$	45.6580	1.48368	0.741842	−	0.670574i	$-0.233952\pi$
$947$	45.6580	1.48368	0.741842	+	0.670574i	$0.233952\pi$
$948$	0	0
$949$	−65.2782	−2.11902
$950$	−22.2175	−0.720832
$951$	0	0
$952$	17.3643	0.562780
$953$	16.3410	0.529337	0.264669	−	0.964339i	$-0.414737\pi$
$953$	16.3410	0.529337	0.264669	+	0.964339i	$0.414737\pi$
$954$	0	0
$955$	−5.26597	−0.170403
$956$	14.5164	0.469493
$957$	0	0
$958$	−0.0996001	−0.00321793
$959$	−39.3280	−1.26997
$960$	0	0
$961$	−26.8378	−0.865735
$962$	−40.3391	−1.30058
$963$	0	0
$964$	−12.1157	−0.390222
$965$	0.525693	0.0169227
$966$	0	0
$967$	35.2563	1.13377	0.566884	−	0.823798i	$-0.308149\pi$
$967$	35.2563	1.13377	0.566884	+	0.823798i	$0.308149\pi$
$968$	−4.53365	−0.145717
$969$	0	0
$970$	−4.50800	−0.144743
$971$	21.9679	0.704983	0.352492	−	0.935815i	$-0.385334\pi$
$971$	21.9679	0.704983	0.352492	+	0.935815i	$0.385334\pi$
$972$	0	0
$973$	25.5491	0.819066
$974$	11.1120	0.356050
$975$	0	0
$976$	4.48986	0.143717
$977$	24.5459	0.785294	0.392647	−	0.919689i	$-0.371559\pi$
$977$	24.5459	0.785294	0.392647	+	0.919689i	$0.371559\pi$
$978$	0	0
$979$	49.6250	1.58602
$980$	−0.626161	−0.0200020
$981$	0	0
$982$	−42.3941	−1.35285
$983$	−12.7196	−0.405693	−0.202847	−	0.979211i	$-0.565019\pi$
$983$	−12.7196	−0.405693	−0.202847	+	0.979211i	$0.565019\pi$
$984$	0	0
$985$	−7.41325	−0.236206
$986$	16.9258	0.539028
$987$	0	0
$988$	−24.5568	−0.781256
$989$	0	0
$990$	0	0
$991$	−53.5606	−1.70141	−0.850704	−	0.525646i	$-0.823824\pi$
$991$	−53.5606	−1.70141	−0.850704	+	0.525646i	$0.823824\pi$
$992$	−2.04015	−0.0647750
$993$	0	0
$994$	−7.98215	−0.253178
$995$	−7.62262	−0.241653
$996$	0	0
$997$	−26.1013	−0.826637	−0.413318	−	0.910587i	$-0.635630\pi$
$997$	−26.1013	−0.826637	−0.413318	+	0.910587i	$0.635630\pi$
$998$	−12.4849	−0.395202
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.cd.1.5	yes	8
3.2	odd	2		9522.2.a.cf.1.4	yes	8
23.22	odd	2	inner	9522.2.a.cd.1.4	✓	8
69.68	even	2		9522.2.a.cf.1.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9522.2.a.cd.1.4	✓	8	23.22	odd	2	inner
9522.2.a.cd.1.5	yes	8	1.1	even	1	trivial
9522.2.a.cf.1.4	yes	8	3.2	odd	2
9522.2.a.cf.1.5	yes	8	69.68	even	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.343645 q^{5} -2.27550 q^{7} -1.00000 q^{8} -0.343645 q^{10} -3.94128 q^{11} +5.39592 q^{13} +2.27550 q^{14} +1.00000 q^{16} +7.63099 q^{17} -4.55099 q^{19} +0.343645 q^{20} +3.94128 q^{22} -4.88191 q^{25} -5.39592 q^{26} -2.27550 q^{28} -2.21804 q^{29} +2.04015 q^{31} -1.00000 q^{32} -7.63099 q^{34} -0.781962 q^{35} +7.47584 q^{37} +4.55099 q^{38} -0.343645 q^{40} -1.09007 q^{41} +2.22570 q^{43} -3.94128 q^{44} -13.2840 q^{47} -1.82212 q^{49} +4.88191 q^{50} +5.39592 q^{52} +5.62855 q^{53} -1.35440 q^{55} +2.27550 q^{56} +2.21804 q^{58} -8.39592 q^{59} +4.48986 q^{61} -2.04015 q^{62} +1.00000 q^{64} +1.85428 q^{65} +3.17641 q^{67} +7.63099 q^{68} +0.781962 q^{70} -3.50787 q^{71} -12.0977 q^{73} -7.47584 q^{74} -4.55099 q^{76} +8.96836 q^{77} +7.52246 q^{79} +0.343645 q^{80} +1.09007 q^{82} -5.31585 q^{83} +2.62235 q^{85} -2.22570 q^{86} +3.94128 q^{88} -12.5911 q^{89} -12.2784 q^{91} +13.2840 q^{94} -1.56392 q^{95} +13.1182 q^{97} +1.82212 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 12 q^{13} + 8 q^{16} + 8 q^{25} - 12 q^{26} + 12 q^{29} - 12 q^{31} - 8 q^{32} - 36 q^{35} - 24 q^{41} - 48 q^{47} - 16 q^{49} - 8 q^{50} + 12 q^{52} + 12 q^{55} - 12 q^{58}+ \cdots + 16 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 12 * q^13 + 8 * q^16 + 8 * q^25 - 12 * q^26 + 12 * q^29 - 12 * q^31 - 8 * q^32 - 36 * q^35 - 24 * q^41 - 48 * q^47 - 16 * q^49 - 8 * q^50 + 12 * q^52 + 12 * q^55 - 12 * q^58 - 36 * q^59 + 12 * q^62 + 8 * q^64 + 36 * q^70 - 24 * q^71 + 12 * q^73 - 12 * q^77 + 24 * q^82 + 12 * q^85 + 48 * q^94 - 72 * q^95 + 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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