LMFDB - Embedded newform 9522.2.a.bt.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

sage:traces = [5,-5,0,5,7,0,-7,-5,0,-7,13,0,-4,7,0,5,9,0,-11,7,0,-13,0,0,-2, 4,0,-7,7] 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:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.284630$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.51334 q^{5} -2.59435 q^{7} -1.00000 q^{8} +1.51334 q^{10} +5.00714 q^{11} -4.82306 q^{13} +2.59435 q^{14} +1.00000 q^{16} +0.863693 q^{17} -7.00158 q^{19} -1.51334 q^{20} -5.00714 q^{22} -2.70981 q^{25} +4.82306 q^{26} -2.59435 q^{28} -3.11362 q^{29} -1.17428 q^{31} -1.00000 q^{32} -0.863693 q^{34} +3.92613 q^{35} -0.602123 q^{37} +7.00158 q^{38} +1.51334 q^{40} -5.29335 q^{41} +1.45925 q^{43} +5.00714 q^{44} +12.6797 q^{47} -0.269342 q^{49} +2.70981 q^{50} -4.82306 q^{52} -4.23281 q^{53} -7.57749 q^{55} +2.59435 q^{56} +3.11362 q^{58} -3.98142 q^{59} -7.26229 q^{61} +1.17428 q^{62} +1.00000 q^{64} +7.29891 q^{65} -16.1371 q^{67} +0.863693 q^{68} -3.92613 q^{70} -5.88463 q^{71} +14.3814 q^{73} +0.602123 q^{74} -7.00158 q^{76} -12.9903 q^{77} +10.3402 q^{79} -1.51334 q^{80} +5.29335 q^{82} -2.64101 q^{83} -1.30706 q^{85} -1.45925 q^{86} -5.00714 q^{88} +5.23754 q^{89} +12.5127 q^{91} -12.6797 q^{94} +10.5958 q^{95} +14.9518 q^{97} +0.269342 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} + 7 q^{5} - 7 q^{7} - 5 q^{8} - 7 q^{10} + 13 q^{11} - 4 q^{13} + 7 q^{14} + 5 q^{16} + 9 q^{17} - 11 q^{19} + 7 q^{20} - 13 q^{22} - 2 q^{25} + 4 q^{26} - 7 q^{28} + 7 q^{29} - 8 q^{31}+ \cdots + 12 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 + 7 * q^5 - 7 * q^7 - 5 * q^8 - 7 * q^10 + 13 * q^11 - 4 * q^13 + 7 * q^14 + 5 * q^16 + 9 * q^17 - 11 * q^19 + 7 * q^20 - 13 * q^22 - 2 * q^25 + 4 * q^26 - 7 * q^28 + 7 * q^29 - 8 * q^31 - 5 * q^32 - 9 * q^34 - q^35 - 12 * q^37 + 11 * q^38 - 7 * q^40 + 10 * q^41 - 4 * q^43 + 13 * q^44 + 24 * q^47 - 12 * q^49 + 2 * q^50 - 4 * q^52 + 9 * q^53 + 16 * q^55 + 7 * q^56 - 7 * q^58 + 14 * q^59 - 5 * q^61 + 8 * q^62 + 5 * q^64 + 12 * q^65 - 13 * q^67 + 9 * q^68 + q^70 + 19 * q^71 + 4 * q^73 + 12 * q^74 - 11 * q^76 - 5 * q^77 - 4 * q^79 + 7 * q^80 - 10 * q^82 + 24 * q^83 + 17 * q^85 + 4 * q^86 - 13 * q^88 + 4 * q^89 + 21 * q^91 - 24 * q^94 + 11 * q^95 + 9 * q^97 + 12 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.51334	−0.676785	−0.338392	−	0.941005i	$-0.609883\pi$
$5$	−1.51334	−0.676785	−0.338392	+	0.941005i	$0.609883\pi$
$6$	0	0
$7$	−2.59435	−0.980573	−0.490286	−	0.871561i	$-0.663108\pi$
$7$	−2.59435	−0.980573	−0.490286	+	0.871561i	$0.663108\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.51334	0.478559
$11$	5.00714	1.50971	0.754855	−	0.655892i	$-0.227707\pi$
$11$	5.00714	1.50971	0.754855	+	0.655892i	$0.227707\pi$
$12$	0	0
$13$	−4.82306	−1.33768	−0.668838	−	0.743408i	$-0.733208\pi$
$13$	−4.82306	−1.33768	−0.668838	+	0.743408i	$0.733208\pi$
$14$	2.59435	0.693370
$15$	0	0
$16$	1.00000	0.250000
$17$	0.863693	0.209476	0.104738	−	0.994500i	$-0.466600\pi$
$17$	0.863693	0.209476	0.104738	+	0.994500i	$0.466600\pi$
$18$	0	0
$19$	−7.00158	−1.60627	−0.803137	−	0.595795i	$-0.796837\pi$
$19$	−7.00158	−1.60627	−0.803137	+	0.595795i	$0.796837\pi$
$20$	−1.51334	−0.338392
$21$	0	0
$22$	−5.00714	−1.06753
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−2.70981	−0.541962
$26$	4.82306	0.945880
$27$	0	0
$28$	−2.59435	−0.490286
$29$	−3.11362	−0.578185	−0.289093	−	0.957301i	$-0.593354\pi$
$29$	−3.11362	−0.578185	−0.289093	+	0.957301i	$0.593354\pi$
$30$	0	0
$31$	−1.17428	−0.210907	−0.105453	−	0.994424i	$-0.533629\pi$
$31$	−1.17428	−0.210907	−0.105453	+	0.994424i	$0.533629\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.863693	−0.148122
$35$	3.92613	0.663637
$36$	0	0
$37$	−0.602123	−0.0989883	−0.0494942	−	0.998774i	$-0.515761\pi$
$37$	−0.602123	−0.0989883	−0.0494942	+	0.998774i	$0.515761\pi$
$38$	7.00158	1.13581
$39$	0	0
$40$	1.51334	0.239280
$41$	−5.29335	−0.826683	−0.413341	−	0.910576i	$-0.635638\pi$
$41$	−5.29335	−0.826683	−0.413341	+	0.910576i	$0.635638\pi$
$42$	0	0
$43$	1.45925	0.222534	0.111267	−	0.993791i	$-0.464509\pi$
$43$	1.45925	0.222534	0.111267	+	0.993791i	$0.464509\pi$
$44$	5.00714	0.754855
$45$	0	0
$46$	0	0
$47$	12.6797	1.84952	0.924760	−	0.380552i	$-0.124266\pi$
$47$	12.6797	1.84952	0.924760	+	0.380552i	$0.124266\pi$
$48$	0	0
$49$	−0.269342	−0.0384774
$50$	2.70981	0.383225
$51$	0	0
$52$	−4.82306	−0.668838
$53$	−4.23281	−0.581422	−0.290711	−	0.956811i	$-0.593892\pi$
$53$	−4.23281	−0.581422	−0.290711	+	0.956811i	$0.593892\pi$
$54$	0	0
$55$	−7.57749	−1.02175
$56$	2.59435	0.346685
$57$	0	0
$58$	3.11362	0.408839
$59$	−3.98142	−0.518337	−0.259168	−	0.965832i	$-0.583448\pi$
$59$	−3.98142	−0.518337	−0.259168	+	0.965832i	$0.583448\pi$
$60$	0	0
$61$	−7.26229	−0.929841	−0.464920	−	0.885352i	$-0.653917\pi$
$61$	−7.26229	−0.929841	−0.464920	+	0.885352i	$0.653917\pi$
$62$	1.17428	0.149134
$63$	0	0
$64$	1.00000	0.125000
$65$	7.29891	0.905319
$66$	0	0
$67$	−16.1371	−1.97146	−0.985729	−	0.168340i	$-0.946159\pi$
$67$	−16.1371	−1.97146	−0.985729	+	0.168340i	$0.946159\pi$
$68$	0.863693	0.104738
$69$	0	0
$70$	−3.92613	−0.469262
$71$	−5.88463	−0.698377	−0.349188	−	0.937053i	$-0.613543\pi$
$71$	−5.88463	−0.698377	−0.349188	+	0.937053i	$0.613543\pi$
$72$	0	0
$73$	14.3814	1.68322	0.841608	−	0.540089i	$-0.181609\pi$
$73$	14.3814	1.68322	0.841608	+	0.540089i	$0.181609\pi$
$74$	0.602123	0.0699953
$75$	0	0
$76$	−7.00158	−0.803137
$77$	−12.9903	−1.48038
$78$	0	0
$79$	10.3402	1.16336	0.581681	−	0.813417i	$-0.302395\pi$
$79$	10.3402	1.16336	0.581681	+	0.813417i	$0.302395\pi$
$80$	−1.51334	−0.169196
$81$	0	0
$82$	5.29335	0.584553
$83$	−2.64101	−0.289888	−0.144944	−	0.989440i	$-0.546300\pi$
$83$	−2.64101	−0.289888	−0.144944	+	0.989440i	$0.546300\pi$
$84$	0	0
$85$	−1.30706	−0.141770
$86$	−1.45925	−0.157355
$87$	0	0
$88$	−5.00714	−0.533763
$89$	5.23754	0.555178	0.277589	−	0.960700i	$-0.410465\pi$
$89$	5.23754	0.555178	0.277589	+	0.960700i	$0.410465\pi$
$90$	0	0
$91$	12.5127	1.31169
$92$	0	0
$93$	0	0
$94$	−12.6797	−1.30781
$95$	10.5958	1.08710
$96$	0	0
$97$	14.9518	1.51812	0.759062	−	0.651018i	$-0.225658\pi$
$97$	14.9518	1.51812	0.759062	+	0.651018i	$0.225658\pi$
$98$	0.269342	0.0272077
$99$	0	0
$100$	−2.70981	−0.270981
$101$	0.530283	0.0527652	0.0263826	−	0.999652i	$-0.491601\pi$
$101$	0.530283	0.0527652	0.0263826	+	0.999652i	$0.491601\pi$
$102$	0	0
$103$	0.919917	0.0906421	0.0453211	−	0.998972i	$-0.485569\pi$
$103$	0.919917	0.0906421	0.0453211	+	0.998972i	$0.485569\pi$
$104$	4.82306	0.472940
$105$	0	0
$106$	4.23281	0.411127
$107$	−13.4562	−1.30086	−0.650428	−	0.759568i	$-0.725410\pi$
$107$	−13.4562	−1.30086	−0.650428	+	0.759568i	$0.725410\pi$
$108$	0	0
$109$	−4.97597	−0.476611	−0.238306	−	0.971190i	$-0.576592\pi$
$109$	−4.97597	−0.476611	−0.238306	+	0.971190i	$0.576592\pi$
$110$	7.57749	0.722486
$111$	0	0
$112$	−2.59435	−0.245143
$113$	13.6038	1.27974	0.639869	−	0.768484i	$-0.278989\pi$
$113$	13.6038	1.27974	0.639869	+	0.768484i	$0.278989\pi$
$114$	0	0
$115$	0	0
$116$	−3.11362	−0.289093
$117$	0	0
$118$	3.98142	0.366519
$119$	−2.24072	−0.205407
$120$	0	0
$121$	14.0715	1.27922
$122$	7.26229	0.657497
$123$	0	0
$124$	−1.17428	−0.105453
$125$	11.6675	1.04358
$126$	0	0
$127$	6.73013	0.597203	0.298601	−	0.954378i	$-0.403480\pi$
$127$	6.73013	0.597203	0.298601	+	0.954378i	$0.403480\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−7.29891	−0.640157
$131$	−18.9969	−1.65976	−0.829882	−	0.557939i	$-0.811592\pi$
$131$	−18.9969	−1.65976	−0.829882	+	0.557939i	$0.811592\pi$
$132$	0	0
$133$	18.1646	1.57507
$134$	16.1371	1.39403
$135$	0	0
$136$	−0.863693	−0.0740611
$137$	2.62684	0.224426	0.112213	−	0.993684i	$-0.464206\pi$
$137$	2.62684	0.224426	0.112213	+	0.993684i	$0.464206\pi$
$138$	0	0
$139$	−18.4268	−1.56294	−0.781471	−	0.623941i	$-0.785530\pi$
$139$	−18.4268	−1.56294	−0.781471	+	0.623941i	$0.785530\pi$
$140$	3.92613	0.331818
$141$	0	0
$142$	5.88463	0.493827
$143$	−24.1497	−2.01950
$144$	0	0
$145$	4.71196	0.391307
$146$	−14.3814	−1.19021
$147$	0	0
$148$	−0.602123	−0.0494942
$149$	3.95602	0.324090	0.162045	−	0.986783i	$-0.448191\pi$
$149$	3.95602	0.324090	0.162045	+	0.986783i	$0.448191\pi$
$150$	0	0
$151$	−16.2471	−1.32217	−0.661085	−	0.750311i	$-0.729904\pi$
$151$	−16.2471	−1.32217	−0.661085	+	0.750311i	$0.729904\pi$
$152$	7.00158	0.567903
$153$	0	0
$154$	12.9903	1.04679
$155$	1.77708	0.142739
$156$	0	0
$157$	20.3422	1.62349	0.811743	−	0.584015i	$-0.198519\pi$
$157$	20.3422	1.62349	0.811743	+	0.584015i	$0.198519\pi$
$158$	−10.3402	−0.822621
$159$	0	0
$160$	1.51334	0.119640
$161$	0	0
$162$	0	0
$163$	−14.9527	−1.17119	−0.585593	−	0.810605i	$-0.699138\pi$
$163$	−14.9527	−1.17119	−0.585593	+	0.810605i	$0.699138\pi$
$164$	−5.29335	−0.413341
$165$	0	0
$166$	2.64101	0.204982
$167$	12.0503	0.932477	0.466238	−	0.884659i	$-0.345609\pi$
$167$	12.0503	0.932477	0.466238	+	0.884659i	$0.345609\pi$
$168$	0	0
$169$	10.2619	0.789376
$170$	1.30706	0.100247
$171$	0	0
$172$	1.45925	0.111267
$173$	−15.5014	−1.17855	−0.589273	−	0.807934i	$-0.700586\pi$
$173$	−15.5014	−1.17855	−0.589273	+	0.807934i	$0.700586\pi$
$174$	0	0
$175$	7.03020	0.531433
$176$	5.00714	0.377428
$177$	0	0
$178$	−5.23754	−0.392570
$179$	−7.21524	−0.539292	−0.269646	−	0.962960i	$-0.586907\pi$
$179$	−7.21524	−0.539292	−0.269646	+	0.962960i	$0.586907\pi$
$180$	0	0
$181$	3.34058	0.248304	0.124152	−	0.992263i	$-0.460379\pi$
$181$	3.34058	0.248304	0.124152	+	0.992263i	$0.460379\pi$
$182$	−12.5127	−0.927504
$183$	0	0
$184$	0	0
$185$	0.911214	0.0669938
$186$	0	0
$187$	4.32463	0.316249
$188$	12.6797	0.924760
$189$	0	0
$190$	−10.5958	−0.768697
$191$	17.0123	1.23097	0.615485	−	0.788149i	$-0.288960\pi$
$191$	17.0123	1.23097	0.615485	+	0.788149i	$0.288960\pi$
$192$	0	0
$193$	−22.7417	−1.63698	−0.818491	−	0.574519i	$-0.805189\pi$
$193$	−22.7417	−1.63698	−0.818491	+	0.574519i	$0.805189\pi$
$194$	−14.9518	−1.07348
$195$	0	0
$196$	−0.269342	−0.0192387
$197$	−10.0792	−0.718116	−0.359058	−	0.933315i	$-0.616902\pi$
$197$	−10.0792	−0.718116	−0.359058	+	0.933315i	$0.616902\pi$
$198$	0	0
$199$	−9.43241	−0.668646	−0.334323	−	0.942459i	$-0.608508\pi$
$199$	−9.43241	−0.668646	−0.334323	+	0.942459i	$0.608508\pi$
$200$	2.70981	0.191613
$201$	0	0
$202$	−0.530283	−0.0373106
$203$	8.07783	0.566952
$204$	0	0
$205$	8.01063	0.559487
$206$	−0.919917	−0.0640937
$207$	0	0
$208$	−4.82306	−0.334419
$209$	−35.0579	−2.42501
$210$	0	0
$211$	−3.37926	−0.232638	−0.116319	−	0.993212i	$-0.537109\pi$
$211$	−3.37926	−0.232638	−0.116319	+	0.993212i	$0.537109\pi$
$212$	−4.23281	−0.290711
$213$	0	0
$214$	13.4562	0.919844
$215$	−2.20834	−0.150608
$216$	0	0
$217$	3.04649	0.206809
$218$	4.97597	0.337015
$219$	0	0
$220$	−7.57749	−0.510875
$221$	−4.16564	−0.280211
$222$	0	0
$223$	−17.3295	−1.16047	−0.580236	−	0.814448i	$-0.697040\pi$
$223$	−17.3295	−1.16047	−0.580236	+	0.814448i	$0.697040\pi$
$224$	2.59435	0.173342
$225$	0	0
$226$	−13.6038	−0.904911
$227$	−11.9206	−0.791197	−0.395598	−	0.918424i	$-0.629463\pi$
$227$	−11.9206	−0.791197	−0.395598	+	0.918424i	$0.629463\pi$
$228$	0	0
$229$	16.9000	1.11678	0.558390	−	0.829578i	$-0.311419\pi$
$229$	16.9000	1.11678	0.558390	+	0.829578i	$0.311419\pi$
$230$	0	0
$231$	0	0
$232$	3.11362	0.204419
$233$	−28.4924	−1.86660	−0.933299	−	0.359099i	$-0.883084\pi$
$233$	−28.4924	−1.86660	−0.933299	+	0.359099i	$0.883084\pi$
$234$	0	0
$235$	−19.1886	−1.25173
$236$	−3.98142	−0.259168
$237$	0	0
$238$	2.24072	0.145245
$239$	22.0994	1.42949	0.714745	−	0.699385i	$-0.246543\pi$
$239$	22.0994	1.42949	0.714745	+	0.699385i	$0.246543\pi$
$240$	0	0
$241$	−1.59815	−0.102946	−0.0514728	−	0.998674i	$-0.516392\pi$
$241$	−1.59815	−0.102946	−0.0514728	+	0.998674i	$0.516392\pi$
$242$	−14.0715	−0.904548
$243$	0	0
$244$	−7.26229	−0.464920
$245$	0.407605	0.0260410
$246$	0	0
$247$	33.7690	2.14867
$248$	1.17428	0.0745668
$249$	0	0
$250$	−11.6675	−0.737920
$251$	17.5887	1.11019	0.555094	−	0.831787i	$-0.312682\pi$
$251$	17.5887	1.11019	0.555094	+	0.831787i	$0.312682\pi$
$252$	0	0
$253$	0	0
$254$	−6.73013	−0.422286
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.6964	1.29101	0.645504	−	0.763757i	$-0.276648\pi$
$257$	20.6964	1.29101	0.645504	+	0.763757i	$0.276648\pi$
$258$	0	0
$259$	1.56212	0.0970653
$260$	7.29891	0.452659
$261$	0	0
$262$	18.9969	1.17363
$263$	27.5247	1.69724	0.848622	−	0.529000i	$-0.177433\pi$
$263$	27.5247	1.69724	0.848622	+	0.529000i	$0.177433\pi$
$264$	0	0
$265$	6.40567	0.393497
$266$	−18.1646	−1.11374
$267$	0	0
$268$	−16.1371	−0.985729
$269$	13.2757	0.809434	0.404717	−	0.914442i	$-0.367370\pi$
$269$	13.2757	0.809434	0.404717	+	0.914442i	$0.367370\pi$
$270$	0	0
$271$	24.0786	1.46267	0.731335	−	0.682019i	$-0.238898\pi$
$271$	24.0786	1.46267	0.731335	+	0.682019i	$0.238898\pi$
$272$	0.863693	0.0523691
$273$	0	0
$274$	−2.62684	−0.158693
$275$	−13.5684	−0.818206
$276$	0	0
$277$	−18.3404	−1.10197	−0.550983	−	0.834516i	$-0.685747\pi$
$277$	−18.3404	−1.10197	−0.550983	+	0.834516i	$0.685747\pi$
$278$	18.4268	1.10517
$279$	0	0
$280$	−3.92613	−0.234631
$281$	0.0199733	0.00119151	0.000595754	−	1.00000i	$-0.499810\pi$
$281$	0.0199733	0.00119151	0.000595754		1.00000i	$0.499810\pi$
$282$	0	0
$283$	17.5537	1.04346	0.521729	−	0.853111i	$-0.325287\pi$
$283$	17.5537	1.04346	0.521729	+	0.853111i	$0.325287\pi$
$284$	−5.88463	−0.349188
$285$	0	0
$286$	24.1497	1.42800
$287$	13.7328	0.810623
$288$	0	0
$289$	−16.2540	−0.956120
$290$	−4.71196	−0.276696
$291$	0	0
$292$	14.3814	0.841608
$293$	2.90645	0.169797	0.0848983	−	0.996390i	$-0.472943\pi$
$293$	2.90645	0.169797	0.0848983	+	0.996390i	$0.472943\pi$
$294$	0	0
$295$	6.02523	0.350803
$296$	0.602123	0.0349977
$297$	0	0
$298$	−3.95602	−0.229166
$299$	0	0
$300$	0	0
$301$	−3.78581	−0.218211
$302$	16.2471	0.934915
$303$	0	0
$304$	−7.00158	−0.401568
$305$	10.9903	0.629302
$306$	0	0
$307$	−12.7120	−0.725511	−0.362755	−	0.931884i	$-0.618164\pi$
$307$	−12.7120	−0.725511	−0.362755	+	0.931884i	$0.618164\pi$
$308$	−12.9903	−0.740190
$309$	0	0
$310$	−1.77708	−0.100931
$311$	21.3006	1.20785	0.603924	−	0.797042i	$-0.293603\pi$
$311$	21.3006	1.20785	0.603924	+	0.797042i	$0.293603\pi$
$312$	0	0
$313$	1.89134	0.106905	0.0534524	−	0.998570i	$-0.482977\pi$
$313$	1.89134	0.106905	0.0534524	+	0.998570i	$0.482977\pi$
$314$	−20.3422	−1.14798
$315$	0	0
$316$	10.3402	0.581681
$317$	21.8115	1.22505	0.612527	−	0.790450i	$-0.290153\pi$
$317$	21.8115	1.22505	0.612527	+	0.790450i	$0.290153\pi$
$318$	0	0
$319$	−15.5903	−0.872892
$320$	−1.51334	−0.0845981
$321$	0	0
$322$	0	0
$323$	−6.04722	−0.336476
$324$	0	0
$325$	13.0696	0.724970
$326$	14.9527	0.828153
$327$	0	0
$328$	5.29335	0.292277
$329$	−32.8955	−1.81359
$330$	0	0
$331$	17.4183	0.957396	0.478698	−	0.877980i	$-0.341109\pi$
$331$	17.4183	0.957396	0.478698	+	0.877980i	$0.341109\pi$
$332$	−2.64101	−0.144944
$333$	0	0
$334$	−12.0503	−0.659361
$335$	24.4208	1.33425
$336$	0	0
$337$	−17.9480	−0.977689	−0.488844	−	0.872371i	$-0.662581\pi$
$337$	−17.9480	−0.977689	−0.488844	+	0.872371i	$0.662581\pi$
$338$	−10.2619	−0.558173
$339$	0	0
$340$	−1.30706	−0.0708852
$341$	−5.87978	−0.318408
$342$	0	0
$343$	18.8592	1.01830
$344$	−1.45925	−0.0786776
$345$	0	0
$346$	15.5014	0.833358
$347$	4.82518	0.259029	0.129515	−	0.991578i	$-0.458658\pi$
$347$	4.82518	0.259029	0.129515	+	0.991578i	$0.458658\pi$
$348$	0	0
$349$	10.3559	0.554337	0.277169	−	0.960821i	$-0.410604\pi$
$349$	10.3559	0.554337	0.277169	+	0.960821i	$0.410604\pi$
$350$	−7.03020	−0.375780
$351$	0	0
$352$	−5.00714	−0.266882
$353$	29.3716	1.56329	0.781646	−	0.623723i	$-0.214381\pi$
$353$	29.3716	1.56329	0.781646	+	0.623723i	$0.214381\pi$
$354$	0	0
$355$	8.90543	0.472651
$356$	5.23754	0.277589
$357$	0	0
$358$	7.21524	0.381337
$359$	9.99774	0.527660	0.263830	−	0.964569i	$-0.415014\pi$
$359$	9.99774	0.527660	0.263830	+	0.964569i	$0.415014\pi$
$360$	0	0
$361$	30.0222	1.58011
$362$	−3.34058	−0.175577
$363$	0	0
$364$	12.5127	0.655844
$365$	−21.7639	−1.13918
$366$	0	0
$367$	16.7049	0.871991	0.435995	−	0.899949i	$-0.356396\pi$
$367$	16.7049	0.871991	0.435995	+	0.899949i	$0.356396\pi$
$368$	0	0
$369$	0	0
$370$	−0.911214	−0.0473718
$371$	10.9814	0.570126
$372$	0	0
$373$	33.2235	1.72025	0.860125	−	0.510084i	$-0.170386\pi$
$373$	33.2235	1.72025	0.860125	+	0.510084i	$0.170386\pi$
$374$	−4.32463	−0.223622
$375$	0	0
$376$	−12.6797	−0.653904
$377$	15.0172	0.773424
$378$	0	0
$379$	8.45299	0.434201	0.217100	−	0.976149i	$-0.430340\pi$
$379$	8.45299	0.434201	0.217100	+	0.976149i	$0.430340\pi$
$380$	10.5958	0.543551
$381$	0	0
$382$	−17.0123	−0.870427
$383$	4.40327	0.224997	0.112498	−	0.993652i	$-0.464115\pi$
$383$	4.40327	0.224997	0.112498	+	0.993652i	$0.464115\pi$
$384$	0	0
$385$	19.6587	1.00190
$386$	22.7417	1.15752
$387$	0	0
$388$	14.9518	0.759062
$389$	−20.2882	−1.02865	−0.514325	−	0.857595i	$-0.671958\pi$
$389$	−20.2882	−1.02865	−0.514325	+	0.857595i	$0.671958\pi$
$390$	0	0
$391$	0	0
$392$	0.269342	0.0136038
$393$	0	0
$394$	10.0792	0.507784
$395$	−15.6482	−0.787346
$396$	0	0
$397$	−14.3569	−0.720555	−0.360277	−	0.932845i	$-0.617318\pi$
$397$	−14.3569	−0.720555	−0.360277	+	0.932845i	$0.617318\pi$
$398$	9.43241	0.472804
$399$	0	0
$400$	−2.70981	−0.135491
$401$	11.5320	0.575879	0.287940	−	0.957649i	$-0.407030\pi$
$401$	11.5320	0.575879	0.287940	+	0.957649i	$0.407030\pi$
$402$	0	0
$403$	5.66362	0.282125
$404$	0.530283	0.0263826
$405$	0	0
$406$	−8.07783	−0.400896
$407$	−3.01491	−0.149444
$408$	0	0
$409$	−23.6704	−1.17042	−0.585212	−	0.810880i	$-0.698989\pi$
$409$	−23.6704	−1.17042	−0.585212	+	0.810880i	$0.698989\pi$
$410$	−8.01063	−0.395617
$411$	0	0
$412$	0.919917	0.0453211
$413$	10.3292	0.508267
$414$	0	0
$415$	3.99674	0.196192
$416$	4.82306	0.236470
$417$	0	0
$418$	35.0579	1.71474
$419$	2.23033	0.108959	0.0544794	−	0.998515i	$-0.482650\pi$
$419$	2.23033	0.108959	0.0544794	+	0.998515i	$0.482650\pi$
$420$	0	0
$421$	−18.0322	−0.878834	−0.439417	−	0.898283i	$-0.644815\pi$
$421$	−18.0322	−0.878834	−0.439417	+	0.898283i	$0.644815\pi$
$422$	3.37926	0.164500
$423$	0	0
$424$	4.23281	0.205564
$425$	−2.34045	−0.113528
$426$	0	0
$427$	18.8409	0.911776
$428$	−13.4562	−0.650428
$429$	0	0
$430$	2.20834	0.106496
$431$	−22.3676	−1.07741	−0.538705	−	0.842495i	$-0.681086\pi$
$431$	−22.3676	−1.07741	−0.538705	+	0.842495i	$0.681086\pi$
$432$	0	0
$433$	17.5048	0.841227	0.420614	−	0.907240i	$-0.361815\pi$
$433$	17.5048	0.841227	0.420614	+	0.907240i	$0.361815\pi$
$434$	−3.04649	−0.146236
$435$	0	0
$436$	−4.97597	−0.238306
$437$	0	0
$438$	0	0
$439$	−15.1141	−0.721357	−0.360678	−	0.932690i	$-0.617455\pi$
$439$	−15.1141	−0.721357	−0.360678	+	0.932690i	$0.617455\pi$
$440$	7.57749	0.361243
$441$	0	0
$442$	4.16564	0.198139
$443$	11.8244	0.561794	0.280897	−	0.959738i	$-0.409368\pi$
$443$	11.8244	0.561794	0.280897	+	0.959738i	$0.409368\pi$
$444$	0	0
$445$	−7.92616	−0.375736
$446$	17.3295	0.820578
$447$	0	0
$448$	−2.59435	−0.122572
$449$	4.68716	0.221201	0.110600	−	0.993865i	$-0.464723\pi$
$449$	4.68716	0.221201	0.110600	+	0.993865i	$0.464723\pi$
$450$	0	0
$451$	−26.5046	−1.24805
$452$	13.6038	0.639869
$453$	0	0
$454$	11.9206	0.559461
$455$	−18.9359	−0.887731
$456$	0	0
$457$	−28.8624	−1.35013	−0.675064	−	0.737759i	$-0.735884\pi$
$457$	−28.8624	−1.35013	−0.675064	+	0.737759i	$0.735884\pi$
$458$	−16.9000	−0.789683
$459$	0	0
$460$	0	0
$461$	1.23414	0.0574794	0.0287397	−	0.999587i	$-0.490851\pi$
$461$	1.23414	0.0574794	0.0287397	+	0.999587i	$0.490851\pi$
$462$	0	0
$463$	0.545786	0.0253648	0.0126824	−	0.999920i	$-0.495963\pi$
$463$	0.545786	0.0253648	0.0126824	+	0.999920i	$0.495963\pi$
$464$	−3.11362	−0.144546
$465$	0	0
$466$	28.4924	1.31988
$467$	−15.9286	−0.737089	−0.368544	−	0.929610i	$-0.620144\pi$
$467$	−15.9286	−0.737089	−0.368544	+	0.929610i	$0.620144\pi$
$468$	0	0
$469$	41.8653	1.93316
$470$	19.1886	0.885104
$471$	0	0
$472$	3.98142	0.183260
$473$	7.30668	0.335962
$474$	0	0
$475$	18.9730	0.870539
$476$	−2.24072	−0.102703
$477$	0	0
$478$	−22.0994	−1.01080
$479$	0.255466	0.0116725	0.00583626	−	0.999983i	$-0.498142\pi$
$479$	0.255466	0.0116725	0.00583626	+	0.999983i	$0.498142\pi$
$480$	0	0
$481$	2.90407	0.132414
$482$	1.59815	0.0727935
$483$	0	0
$484$	14.0715	0.639612
$485$	−22.6271	−1.02744
$486$	0	0
$487$	25.5758	1.15895	0.579476	−	0.814989i	$-0.303257\pi$
$487$	25.5758	1.15895	0.579476	+	0.814989i	$0.303257\pi$
$488$	7.26229	0.328748
$489$	0	0
$490$	−0.407605	−0.0184137
$491$	7.52661	0.339671	0.169836	−	0.985472i	$-0.445676\pi$
$491$	7.52661	0.339671	0.169836	+	0.985472i	$0.445676\pi$
$492$	0	0
$493$	−2.68921	−0.121116
$494$	−33.7690	−1.51934
$495$	0	0
$496$	−1.17428	−0.0527267
$497$	15.2668	0.684809
$498$	0	0
$499$	−40.1412	−1.79697	−0.898483	−	0.439007i	$-0.855330\pi$
$499$	−40.1412	−1.79697	−0.898483	+	0.439007i	$0.855330\pi$
$500$	11.6675	0.521788
$501$	0	0
$502$	−17.5887	−0.785022
$503$	24.1482	1.07671	0.538357	−	0.842717i	$-0.319045\pi$
$503$	24.1482	1.07671	0.538357	+	0.842717i	$0.319045\pi$
$504$	0	0
$505$	−0.802497	−0.0357107
$506$	0	0
$507$	0	0
$508$	6.73013	0.298601
$509$	−0.639983	−0.0283667	−0.0141834	−	0.999899i	$-0.504515\pi$
$509$	−0.639983	−0.0283667	−0.0141834	+	0.999899i	$0.504515\pi$
$510$	0	0
$511$	−37.3104	−1.65052
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−20.6964	−0.912880
$515$	−1.39214	−0.0613452
$516$	0	0
$517$	63.4889	2.79224
$518$	−1.56212	−0.0686355
$519$	0	0
$520$	−7.29891	−0.320079
$521$	−22.7385	−0.996191	−0.498095	−	0.867122i	$-0.665967\pi$
$521$	−22.7385	−0.996191	−0.498095	+	0.867122i	$0.665967\pi$
$522$	0	0
$523$	0.840829	0.0367669	0.0183834	−	0.999831i	$-0.494148\pi$
$523$	0.840829	0.0367669	0.0183834	+	0.999831i	$0.494148\pi$
$524$	−18.9969	−0.829882
$525$	0	0
$526$	−27.5247	−1.20013
$527$	−1.01422	−0.0441800
$528$	0	0
$529$	0	0
$530$	−6.40567	−0.278245
$531$	0	0
$532$	18.1646	0.787534
$533$	25.5302	1.10583
$534$	0	0
$535$	20.3637	0.880399
$536$	16.1371	0.697016
$537$	0	0
$538$	−13.2757	−0.572356
$539$	−1.34863	−0.0580898
$540$	0	0
$541$	22.8587	0.982773	0.491386	−	0.870942i	$-0.336490\pi$
$541$	22.8587	0.982773	0.491386	+	0.870942i	$0.336490\pi$
$542$	−24.0786	−1.03426
$543$	0	0
$544$	−0.863693	−0.0370305
$545$	7.53032	0.322563
$546$	0	0
$547$	10.4614	0.447296	0.223648	−	0.974670i	$-0.428203\pi$
$547$	10.4614	0.447296	0.223648	+	0.974670i	$0.428203\pi$
$548$	2.62684	0.112213
$549$	0	0
$550$	13.5684	0.578559
$551$	21.8003	0.928723
$552$	0	0
$553$	−26.8261	−1.14076
$554$	18.3404	0.779208
$555$	0	0
$556$	−18.4268	−0.781471
$557$	26.2874	1.11383	0.556916	−	0.830569i	$-0.311984\pi$
$557$	26.2874	1.11383	0.556916	+	0.830569i	$0.311984\pi$
$558$	0	0
$559$	−7.03806	−0.297678
$560$	3.92613	0.165909
$561$	0	0
$562$	−0.0199733	−0.000842524 0
$563$	27.8694	1.17456	0.587278	−	0.809386i	$-0.300200\pi$
$563$	27.8694	1.17456	0.587278	+	0.809386i	$0.300200\pi$
$564$	0	0
$565$	−20.5871	−0.866107
$566$	−17.5537	−0.737836
$567$	0	0
$568$	5.88463	0.246914
$569$	32.2063	1.35016	0.675080	−	0.737745i	$-0.264109\pi$
$569$	32.2063	1.35016	0.675080	+	0.737745i	$0.264109\pi$
$570$	0	0
$571$	16.4966	0.690361	0.345181	−	0.938536i	$-0.387818\pi$
$571$	16.4966	0.690361	0.345181	+	0.938536i	$0.387818\pi$
$572$	−24.1497	−1.00975
$573$	0	0
$574$	−13.7328	−0.573197
$575$	0	0
$576$	0	0
$577$	−6.57419	−0.273687	−0.136844	−	0.990593i	$-0.543696\pi$
$577$	−6.57419	−0.273687	−0.136844	+	0.990593i	$0.543696\pi$
$578$	16.2540	0.676079
$579$	0	0
$580$	4.71196	0.195653
$581$	6.85170	0.284257
$582$	0	0
$583$	−21.1943	−0.877778
$584$	−14.3814	−0.595107
$585$	0	0
$586$	−2.90645	−0.120064
$587$	15.4230	0.636574	0.318287	−	0.947994i	$-0.396892\pi$
$587$	15.4230	0.636574	0.318287	+	0.947994i	$0.396892\pi$
$588$	0	0
$589$	8.22181	0.338774
$590$	−6.02523	−0.248055
$591$	0	0
$592$	−0.602123	−0.0247471
$593$	14.3695	0.590086	0.295043	−	0.955484i	$-0.404666\pi$
$593$	14.3695	0.590086	0.295043	+	0.955484i	$0.404666\pi$
$594$	0	0
$595$	3.39097	0.139016
$596$	3.95602	0.162045
$597$	0	0
$598$	0	0
$599$	−7.73316	−0.315968	−0.157984	−	0.987442i	$-0.550500\pi$
$599$	−7.73316	−0.315968	−0.157984	+	0.987442i	$0.550500\pi$
$600$	0	0
$601$	32.0245	1.30631	0.653153	−	0.757226i	$-0.273446\pi$
$601$	32.0245	1.30631	0.653153	+	0.757226i	$0.273446\pi$
$602$	3.78581	0.154298
$603$	0	0
$604$	−16.2471	−0.661085
$605$	−21.2949	−0.865760
$606$	0	0
$607$	−0.918396	−0.0372765	−0.0186383	−	0.999826i	$-0.505933\pi$
$607$	−0.918396	−0.0372765	−0.0186383	+	0.999826i	$0.505933\pi$
$608$	7.00158	0.283952
$609$	0	0
$610$	−10.9903	−0.444984
$611$	−61.1548	−2.47406
$612$	0	0
$613$	19.7621	0.798182	0.399091	−	0.916911i	$-0.369326\pi$
$613$	19.7621	0.798182	0.399091	+	0.916911i	$0.369326\pi$
$614$	12.7120	0.513013
$615$	0	0
$616$	12.9903	0.523393
$617$	−6.90005	−0.277786	−0.138893	−	0.990307i	$-0.544354\pi$
$617$	−6.90005	−0.277786	−0.138893	+	0.990307i	$0.544354\pi$
$618$	0	0
$619$	22.7324	0.913691	0.456846	−	0.889546i	$-0.348979\pi$
$619$	22.7324	0.913691	0.456846	+	0.889546i	$0.348979\pi$
$620$	1.77708	0.0713693
$621$	0	0
$622$	−21.3006	−0.854077
$623$	−13.5880	−0.544392
$624$	0	0
$625$	−4.10787	−0.164315
$626$	−1.89134	−0.0755931
$627$	0	0
$628$	20.3422	0.811743
$629$	−0.520049	−0.0207357
$630$	0	0
$631$	33.6909	1.34121	0.670607	−	0.741813i	$-0.266034\pi$
$631$	33.6909	1.34121	0.670607	+	0.741813i	$0.266034\pi$
$632$	−10.3402	−0.411311
$633$	0	0
$634$	−21.8115	−0.866244
$635$	−10.1850	−0.404178
$636$	0	0
$637$	1.29905	0.0514704
$638$	15.5903	0.617228
$639$	0	0
$640$	1.51334	0.0598199
$641$	4.23932	0.167443	0.0837216	−	0.996489i	$-0.473319\pi$
$641$	4.23932	0.167443	0.0837216	+	0.996489i	$0.473319\pi$
$642$	0	0
$643$	40.0914	1.58105	0.790525	−	0.612430i	$-0.209808\pi$
$643$	40.0914	1.58105	0.790525	+	0.612430i	$0.209808\pi$
$644$	0	0
$645$	0	0
$646$	6.04722	0.237925
$647$	5.69698	0.223971	0.111986	−	0.993710i	$-0.464279\pi$
$647$	5.69698	0.223971	0.111986	+	0.993710i	$0.464279\pi$
$648$	0	0
$649$	−19.9355	−0.782538
$650$	−13.0696	−0.512631
$651$	0	0
$652$	−14.9527	−0.585593
$653$	−32.2988	−1.26395	−0.631975	−	0.774989i	$-0.717755\pi$
$653$	−32.2988	−1.26395	−0.631975	+	0.774989i	$0.717755\pi$
$654$	0	0
$655$	28.7487	1.12330
$656$	−5.29335	−0.206671
$657$	0	0
$658$	32.8955	1.28240
$659$	−9.24211	−0.360021	−0.180011	−	0.983665i	$-0.557613\pi$
$659$	−9.24211	−0.360021	−0.180011	+	0.983665i	$0.557613\pi$
$660$	0	0
$661$	−32.3877	−1.25974	−0.629868	−	0.776702i	$-0.716891\pi$
$661$	−32.3877	−1.25974	−0.629868	+	0.776702i	$0.716891\pi$
$662$	−17.4183	−0.676981
$663$	0	0
$664$	2.64101	0.102491
$665$	−27.4891	−1.06598
$666$	0	0
$667$	0	0
$668$	12.0503	0.466238
$669$	0	0
$670$	−24.4208	−0.943459
$671$	−36.3633	−1.40379
$672$	0	0
$673$	38.7223	1.49264	0.746318	−	0.665590i	$-0.231820\pi$
$673$	38.7223	1.49264	0.746318	+	0.665590i	$0.231820\pi$
$674$	17.9480	0.691330
$675$	0	0
$676$	10.2619	0.394688
$677$	−29.2630	−1.12467	−0.562334	−	0.826910i	$-0.690097\pi$
$677$	−29.2630	−1.12467	−0.562334	+	0.826910i	$0.690097\pi$
$678$	0	0
$679$	−38.7902	−1.48863
$680$	1.30706	0.0501234
$681$	0	0
$682$	5.87978	0.225148
$683$	32.8985	1.25883	0.629413	−	0.777071i	$-0.283295\pi$
$683$	32.8985	1.25883	0.629413	+	0.777071i	$0.283295\pi$
$684$	0	0
$685$	−3.97529	−0.151888
$686$	−18.8592	−0.720049
$687$	0	0
$688$	1.45925	0.0556335
$689$	20.4151	0.777754
$690$	0	0
$691$	−12.3215	−0.468730	−0.234365	−	0.972149i	$-0.575301\pi$
$691$	−12.3215	−0.468730	−0.234365	+	0.972149i	$0.575301\pi$
$692$	−15.5014	−0.589273
$693$	0	0
$694$	−4.82518	−0.183161
$695$	27.8860	1.05778
$696$	0	0
$697$	−4.57183	−0.173171
$698$	−10.3559	−0.391976
$699$	0	0
$700$	7.03020	0.265717
$701$	27.3050	1.03129	0.515647	−	0.856801i	$-0.327552\pi$
$701$	27.3050	1.03129	0.515647	+	0.856801i	$0.327552\pi$
$702$	0	0
$703$	4.21581	0.159002
$704$	5.00714	0.188714
$705$	0	0
$706$	−29.3716	−1.10541
$707$	−1.37574	−0.0517401
$708$	0	0
$709$	11.1293	0.417969	0.208985	−	0.977919i	$-0.432984\pi$
$709$	11.1293	0.417969	0.208985	+	0.977919i	$0.432984\pi$
$710$	−8.90543	−0.334215
$711$	0	0
$712$	−5.23754	−0.196285
$713$	0	0
$714$	0	0
$715$	36.5467	1.36677
$716$	−7.21524	−0.269646
$717$	0	0
$718$	−9.99774	−0.373112
$719$	−25.7081	−0.958750	−0.479375	−	0.877610i	$-0.659137\pi$
$719$	−25.7081	−0.958750	−0.479375	+	0.877610i	$0.659137\pi$
$720$	0	0
$721$	−2.38659	−0.0888812
$722$	−30.0222	−1.11731
$723$	0	0
$724$	3.34058	0.124152
$725$	8.43733	0.313354
$726$	0	0
$727$	0.0480451	0.00178189	0.000890947	−	1.00000i	$-0.499716\pi$
$727$	0.0480451	0.00178189	0.000890947		1.00000i	$0.499716\pi$
$728$	−12.5127	−0.463752
$729$	0	0
$730$	21.7639	0.805518
$731$	1.26035	0.0466156
$732$	0	0
$733$	30.6005	1.13026	0.565128	−	0.825003i	$-0.308827\pi$
$733$	30.6005	1.13026	0.565128	+	0.825003i	$0.308827\pi$
$734$	−16.7049	−0.616591
$735$	0	0
$736$	0	0
$737$	−80.8007	−2.97633
$738$	0	0
$739$	−12.0286	−0.442481	−0.221240	−	0.975219i	$-0.571011\pi$
$739$	−12.0286	−0.442481	−0.221240	+	0.975219i	$0.571011\pi$
$740$	0.911214	0.0334969
$741$	0	0
$742$	−10.9814	−0.403140
$743$	10.1814	0.373521	0.186760	−	0.982405i	$-0.440201\pi$
$743$	10.1814	0.373521	0.186760	+	0.982405i	$0.440201\pi$
$744$	0	0
$745$	−5.98679	−0.219339
$746$	−33.2235	−1.21640
$747$	0	0
$748$	4.32463	0.158124
$749$	34.9100	1.27558
$750$	0	0
$751$	−9.06980	−0.330962	−0.165481	−	0.986213i	$-0.552918\pi$
$751$	−9.06980	−0.330962	−0.165481	+	0.986213i	$0.552918\pi$
$752$	12.6797	0.462380
$753$	0	0
$754$	−15.0172	−0.546893
$755$	24.5873	0.894824
$756$	0	0
$757$	17.5078	0.636333	0.318166	−	0.948035i	$-0.396933\pi$
$757$	17.5078	0.636333	0.318166	+	0.948035i	$0.396933\pi$
$758$	−8.45299	−0.307026
$759$	0	0
$760$	−10.5958	−0.384348
$761$	24.9741	0.905310	0.452655	−	0.891686i	$-0.350477\pi$
$761$	24.9741	0.905310	0.452655	+	0.891686i	$0.350477\pi$
$762$	0	0
$763$	12.9094	0.467352
$764$	17.0123	0.615485
$765$	0	0
$766$	−4.40327	−0.159097
$767$	19.2026	0.693367
$768$	0	0
$769$	−8.51279	−0.306979	−0.153490	−	0.988150i	$-0.549051\pi$
$769$	−8.51279	−0.306979	−0.153490	+	0.988150i	$0.549051\pi$
$770$	−19.6587	−0.708450
$771$	0	0
$772$	−22.7417	−0.818491
$773$	−49.0834	−1.76541	−0.882703	−	0.469931i	$-0.844279\pi$
$773$	−49.0834	−1.76541	−0.882703	+	0.469931i	$0.844279\pi$
$774$	0	0
$775$	3.18207	0.114303
$776$	−14.9518	−0.536738
$777$	0	0
$778$	20.2882	0.727366
$779$	37.0619	1.32788
$780$	0	0
$781$	−29.4652	−1.05435
$782$	0	0
$783$	0	0
$784$	−0.269342	−0.00961936
$785$	−30.7846	−1.09875
$786$	0	0
$787$	4.33510	0.154530	0.0772649	−	0.997011i	$-0.475381\pi$
$787$	4.33510	0.154530	0.0772649	+	0.997011i	$0.475381\pi$
$788$	−10.0792	−0.359058
$789$	0	0
$790$	15.6482	0.556738
$791$	−35.2930	−1.25488
$792$	0	0
$793$	35.0264	1.24383
$794$	14.3569	0.509509
$795$	0	0
$796$	−9.43241	−0.334323
$797$	4.48314	0.158801	0.0794005	−	0.996843i	$-0.474699\pi$
$797$	4.48314	0.158801	0.0794005	+	0.996843i	$0.474699\pi$
$798$	0	0
$799$	10.9513	0.387431
$800$	2.70981	0.0958063
$801$	0	0
$802$	−11.5320	−0.407208
$803$	72.0097	2.54117
$804$	0	0
$805$	0	0
$806$	−5.66362	−0.199492
$807$	0	0
$808$	−0.530283	−0.0186553
$809$	−28.1045	−0.988100	−0.494050	−	0.869433i	$-0.664484\pi$
$809$	−28.1045	−0.988100	−0.494050	+	0.869433i	$0.664484\pi$
$810$	0	0
$811$	43.7318	1.53563	0.767815	−	0.640672i	$-0.221344\pi$
$811$	43.7318	1.53563	0.767815	+	0.640672i	$0.221344\pi$
$812$	8.07783	0.283476
$813$	0	0
$814$	3.01491	0.105673
$815$	22.6285	0.792641
$816$	0	0
$817$	−10.2171	−0.357450
$818$	23.6704	0.827615
$819$	0	0
$820$	8.01063	0.279743
$821$	6.53023	0.227907	0.113953	−	0.993486i	$-0.463649\pi$
$821$	6.53023	0.227907	0.113953	+	0.993486i	$0.463649\pi$
$822$	0	0
$823$	−0.913788	−0.0318527	−0.0159263	−	0.999873i	$-0.505070\pi$
$823$	−0.913788	−0.0318527	−0.0159263	+	0.999873i	$0.505070\pi$
$824$	−0.919917	−0.0320468
$825$	0	0
$826$	−10.3292	−0.359399
$827$	19.0247	0.661554	0.330777	−	0.943709i	$-0.392689\pi$
$827$	19.0247	0.661554	0.330777	+	0.943709i	$0.392689\pi$
$828$	0	0
$829$	14.2601	0.495272	0.247636	−	0.968853i	$-0.420346\pi$
$829$	14.2601	0.495272	0.247636	+	0.968853i	$0.420346\pi$
$830$	−3.99674	−0.138729
$831$	0	0
$832$	−4.82306	−0.167209
$833$	−0.232629	−0.00806012
$834$	0	0
$835$	−18.2361	−0.631086
$836$	−35.0579	−1.21250
$837$	0	0
$838$	−2.23033	−0.0770456
$839$	−55.4731	−1.91514	−0.957571	−	0.288196i	$-0.906944\pi$
$839$	−55.4731	−1.91514	−0.957571	+	0.288196i	$0.906944\pi$
$840$	0	0
$841$	−19.3054	−0.665702
$842$	18.0322	0.621430
$843$	0	0
$844$	−3.37926	−0.116319
$845$	−15.5297	−0.534238
$846$	0	0
$847$	−36.5063	−1.25437
$848$	−4.23281	−0.145355
$849$	0	0
$850$	2.34045	0.0802766
$851$	0	0
$852$	0	0
$853$	−7.42049	−0.254073	−0.127036	−	0.991898i	$-0.540547\pi$
$853$	−7.42049	−0.254073	−0.127036	+	0.991898i	$0.540547\pi$
$854$	−18.8409	−0.644723
$855$	0	0
$856$	13.4562	0.459922
$857$	25.9013	0.884772	0.442386	−	0.896825i	$-0.354132\pi$
$857$	25.9013	0.884772	0.442386	+	0.896825i	$0.354132\pi$
$858$	0	0
$859$	−41.1780	−1.40498	−0.702488	−	0.711695i	$-0.747928\pi$
$859$	−41.1780	−1.40498	−0.702488	+	0.711695i	$0.747928\pi$
$860$	−2.20834	−0.0753038
$861$	0	0
$862$	22.3676	0.761843
$863$	−50.2342	−1.70999	−0.854995	−	0.518636i	$-0.826440\pi$
$863$	−50.2342	−1.70999	−0.854995	+	0.518636i	$0.826440\pi$
$864$	0	0
$865$	23.4588	0.797622
$866$	−17.5048	−0.594837
$867$	0	0
$868$	3.04649	0.103405
$869$	51.7748	1.75634
$870$	0	0
$871$	77.8301	2.63717
$872$	4.97597	0.168508
$873$	0	0
$874$	0	0
$875$	−30.2697	−1.02330
$876$	0	0
$877$	18.5028	0.624794	0.312397	−	0.949952i	$-0.398868\pi$
$877$	18.5028	0.624794	0.312397	+	0.949952i	$0.398868\pi$
$878$	15.1141	0.510076
$879$	0	0
$880$	−7.57749	−0.255437
$881$	−13.6757	−0.460746	−0.230373	−	0.973102i	$-0.573995\pi$
$881$	−13.6757	−0.460746	−0.230373	+	0.973102i	$0.573995\pi$
$882$	0	0
$883$	−1.23466	−0.0415497	−0.0207749	−	0.999784i	$-0.506613\pi$
$883$	−1.23466	−0.0415497	−0.0207749	+	0.999784i	$0.506613\pi$
$884$	−4.16564	−0.140106
$885$	0	0
$886$	−11.8244	−0.397248
$887$	13.0032	0.436606	0.218303	−	0.975881i	$-0.429948\pi$
$887$	13.0032	0.436606	0.218303	+	0.975881i	$0.429948\pi$
$888$	0	0
$889$	−17.4603	−0.585601
$890$	7.92616	0.265686
$891$	0	0
$892$	−17.3295	−0.580236
$893$	−88.7777	−2.97083
$894$	0	0
$895$	10.9191	0.364985
$896$	2.59435	0.0866712
$897$	0	0
$898$	−4.68716	−0.156412
$899$	3.65626	0.121943
$900$	0	0
$901$	−3.65585	−0.121794
$902$	26.5046	0.882506
$903$	0	0
$904$	−13.6038	−0.452455
$905$	−5.05543	−0.168048
$906$	0	0
$907$	−7.84547	−0.260505	−0.130252	−	0.991481i	$-0.541579\pi$
$907$	−7.84547	−0.260505	−0.130252	+	0.991481i	$0.541579\pi$
$908$	−11.9206	−0.395598
$909$	0	0
$910$	18.9359	0.627720
$911$	43.3450	1.43608	0.718042	−	0.696000i	$-0.245039\pi$
$911$	43.3450	1.43608	0.718042	+	0.696000i	$0.245039\pi$
$912$	0	0
$913$	−13.2239	−0.437647
$914$	28.8624	0.954685
$915$	0	0
$916$	16.9000	0.558390
$917$	49.2846	1.62752
$918$	0	0
$919$	−39.3327	−1.29747	−0.648733	−	0.761016i	$-0.724701\pi$
$919$	−39.3327	−1.29747	−0.648733	+	0.761016i	$0.724701\pi$
$920$	0	0
$921$	0	0
$922$	−1.23414	−0.0406441
$923$	28.3819	0.934202
$924$	0	0
$925$	1.63164	0.0536479
$926$	−0.545786	−0.0179357
$927$	0	0
$928$	3.11362	0.102210
$929$	−29.5966	−0.971033	−0.485516	−	0.874228i	$-0.661368\pi$
$929$	−29.5966	−0.971033	−0.485516	+	0.874228i	$0.661368\pi$
$930$	0	0
$931$	1.88582	0.0618053
$932$	−28.4924	−0.933299
$933$	0	0
$934$	15.9286	0.521200
$935$	−6.54463	−0.214032
$936$	0	0
$937$	−25.1762	−0.822470	−0.411235	−	0.911529i	$-0.634902\pi$
$937$	−25.1762	−0.822470	−0.411235	+	0.911529i	$0.634902\pi$
$938$	−41.8653	−1.36695
$939$	0	0
$940$	−19.1886	−0.625863
$941$	−16.8886	−0.550552	−0.275276	−	0.961365i	$-0.588769\pi$
$941$	−16.8886	−0.550552	−0.275276	+	0.961365i	$0.588769\pi$
$942$	0	0
$943$	0	0
$944$	−3.98142	−0.129584
$945$	0	0
$946$	−7.30668	−0.237561
$947$	15.4544	0.502200	0.251100	−	0.967961i	$-0.419208\pi$
$947$	15.4544	0.502200	0.251100	+	0.967961i	$0.419208\pi$
$948$	0	0
$949$	−69.3624	−2.25160
$950$	−18.9730	−0.615564
$951$	0	0
$952$	2.24072	0.0726223
$953$	−1.03906	−0.0336585	−0.0168292	−	0.999858i	$-0.505357\pi$
$953$	−1.03906	−0.0336585	−0.0168292	+	0.999858i	$0.505357\pi$
$954$	0	0
$955$	−25.7454	−0.833102
$956$	22.0994	0.714745
$957$	0	0
$958$	−0.255466	−0.00825372
$959$	−6.81495	−0.220066
$960$	0	0
$961$	−29.6211	−0.955518
$962$	−2.90407	−0.0936311
$963$	0	0
$964$	−1.59815	−0.0514728
$965$	34.4158	1.10789
$966$	0	0
$967$	23.1795	0.745404	0.372702	−	0.927951i	$-0.378431\pi$
$967$	23.1795	0.745404	0.372702	+	0.927951i	$0.378431\pi$
$968$	−14.0715	−0.452274
$969$	0	0
$970$	22.6271	0.726512
$971$	−6.18163	−0.198378	−0.0991890	−	0.995069i	$-0.531625\pi$
$971$	−6.18163	−0.198378	−0.0991890	+	0.995069i	$0.531625\pi$
$972$	0	0
$973$	47.8056	1.53258
$974$	−25.5758	−0.819502
$975$	0	0
$976$	−7.26229	−0.232460
$977$	30.4534	0.974292	0.487146	−	0.873321i	$-0.338038\pi$
$977$	30.4534	0.974292	0.487146	+	0.873321i	$0.338038\pi$
$978$	0	0
$979$	26.2251	0.838158
$980$	0.407605	0.0130205
$981$	0	0
$982$	−7.52661	−0.240184
$983$	−1.47151	−0.0469338	−0.0234669	−	0.999725i	$-0.507470\pi$
$983$	−1.47151	−0.0469338	−0.0234669	+	0.999725i	$0.507470\pi$
$984$	0	0
$985$	15.2533	0.486010
$986$	2.68921	0.0856420
$987$	0	0
$988$	33.7690	1.07434
$989$	0	0
$990$	0	0
$991$	−31.5601	−1.00254	−0.501269	−	0.865291i	$-0.667133\pi$
$991$	−31.5601	−1.00254	−0.501269	+	0.865291i	$0.667133\pi$
$992$	1.17428	0.0372834
$993$	0	0
$994$	−15.2668	−0.484233
$995$	14.2744	0.452529
$996$	0	0
$997$	15.3785	0.487042	0.243521	−	0.969896i	$-0.421698\pi$
$997$	15.3785	0.487042	0.243521	+	0.969896i	$0.421698\pi$
$998$	40.1412	1.27065
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.bt.1.1		5
3.2	odd	2		3174.2.a.bc.1.5		5
23.2	even	11		414.2.i.d.73.1		10
23.12	even	11		414.2.i.d.397.1		10
23.22	odd	2		9522.2.a.bq.1.5		5
69.2	odd	22		138.2.e.a.73.1	✓	10
69.35	odd	22		138.2.e.a.121.1	yes	10
69.68	even	2		3174.2.a.bd.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.a.73.1	✓	10	69.2	odd	22
138.2.e.a.121.1	yes	10	69.35	odd	22
414.2.i.d.73.1		10	23.2	even	11
414.2.i.d.397.1		10	23.12	even	11
3174.2.a.bc.1.5		5	3.2	odd	2
3174.2.a.bd.1.1		5	69.68	even	2
9522.2.a.bq.1.5		5	23.22	odd	2
9522.2.a.bt.1.1		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.51334 q^{5} -2.59435 q^{7} -1.00000 q^{8} +1.51334 q^{10} +5.00714 q^{11} -4.82306 q^{13} +2.59435 q^{14} +1.00000 q^{16} +0.863693 q^{17} -7.00158 q^{19} -1.51334 q^{20} -5.00714 q^{22} -2.70981 q^{25} +4.82306 q^{26} -2.59435 q^{28} -3.11362 q^{29} -1.17428 q^{31} -1.00000 q^{32} -0.863693 q^{34} +3.92613 q^{35} -0.602123 q^{37} +7.00158 q^{38} +1.51334 q^{40} -5.29335 q^{41} +1.45925 q^{43} +5.00714 q^{44} +12.6797 q^{47} -0.269342 q^{49} +2.70981 q^{50} -4.82306 q^{52} -4.23281 q^{53} -7.57749 q^{55} +2.59435 q^{56} +3.11362 q^{58} -3.98142 q^{59} -7.26229 q^{61} +1.17428 q^{62} +1.00000 q^{64} +7.29891 q^{65} -16.1371 q^{67} +0.863693 q^{68} -3.92613 q^{70} -5.88463 q^{71} +14.3814 q^{73} +0.602123 q^{74} -7.00158 q^{76} -12.9903 q^{77} +10.3402 q^{79} -1.51334 q^{80} +5.29335 q^{82} -2.64101 q^{83} -1.30706 q^{85} -1.45925 q^{86} -5.00714 q^{88} +5.23754 q^{89} +12.5127 q^{91} -12.6797 q^{94} +10.5958 q^{95} +14.9518 q^{97} +0.269342 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} + 7 q^{5} - 7 q^{7} - 5 q^{8} - 7 q^{10} + 13 q^{11} - 4 q^{13} + 7 q^{14} + 5 q^{16} + 9 q^{17} - 11 q^{19} + 7 q^{20} - 13 q^{22} - 2 q^{25} + 4 q^{26} - 7 q^{28} + 7 q^{29} - 8 q^{31}+ \cdots + 12 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 + 7 * q^5 - 7 * q^7 - 5 * q^8 - 7 * q^10 + 13 * q^11 - 4 * q^13 + 7 * q^14 + 5 * q^16 + 9 * q^17 - 11 * q^19 + 7 * q^20 - 13 * q^22 - 2 * q^25 + 4 * q^26 - 7 * q^28 + 7 * q^29 - 8 * q^31 - 5 * q^32 - 9 * q^34 - q^35 - 12 * q^37 + 11 * q^38 - 7 * q^40 + 10 * q^41 - 4 * q^43 + 13 * q^44 + 24 * q^47 - 12 * q^49 + 2 * q^50 - 4 * q^52 + 9 * q^53 + 16 * q^55 + 7 * q^56 - 7 * q^58 + 14 * q^59 - 5 * q^61 + 8 * q^62 + 5 * q^64 + 12 * q^65 - 13 * q^67 + 9 * q^68 + q^70 + 19 * q^71 + 4 * q^73 + 12 * q^74 - 11 * q^76 - 5 * q^77 - 4 * q^79 + 7 * q^80 - 10 * q^82 + 24 * q^83 + 17 * q^85 + 4 * q^86 - 13 * q^88 + 4 * q^89 + 21 * q^91 - 24 * q^94 + 11 * q^95 + 9 * q^97 + 12 * q^98

Introduction

L-functions

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