LMFDB - Embedded newform 9522.2.a.bu.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,-5,0,5,8,0,-7,-5,0,-8,5,0,-7,7,0,5,13,0,-12,8,0,-5,0,0,1,7, 0,-7,4] 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:	$5$
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 46)
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.22871 q^{5} -0.627214 q^{7} -1.00000 q^{8} +1.22871 q^{10} +5.89037 q^{11} -2.35194 q^{13} +0.627214 q^{14} +1.00000 q^{16} +5.33992 q^{17} +0.270925 q^{19} -1.22871 q^{20} -5.89037 q^{22} -3.49028 q^{25} +2.35194 q^{26} -0.627214 q^{28} +1.98287 q^{29} +6.60353 q^{31} -1.00000 q^{32} -5.33992 q^{34} +0.770663 q^{35} -10.0704 q^{37} -0.270925 q^{38} +1.22871 q^{40} -10.8126 q^{41} -4.59725 q^{43} +5.89037 q^{44} -6.97259 q^{47} -6.60660 q^{49} +3.49028 q^{50} -2.35194 q^{52} -2.63694 q^{53} -7.23754 q^{55} +0.627214 q^{56} -1.98287 q^{58} -1.64927 q^{59} -9.13485 q^{61} -6.60353 q^{62} +1.00000 q^{64} +2.88984 q^{65} +6.19401 q^{67} +5.33992 q^{68} -0.770663 q^{70} +9.06233 q^{71} -5.93852 q^{73} +10.0704 q^{74} +0.270925 q^{76} -3.69452 q^{77} +6.00714 q^{79} -1.22871 q^{80} +10.8126 q^{82} -1.95305 q^{83} -6.56120 q^{85} +4.59725 q^{86} -5.89037 q^{88} +4.59045 q^{89} +1.47517 q^{91} +6.97259 q^{94} -0.332887 q^{95} -10.1844 q^{97} +6.60660 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} + 8 q^{5} - 7 q^{7} - 5 q^{8} - 8 q^{10} + 5 q^{11} - 7 q^{13} + 7 q^{14} + 5 q^{16} + 13 q^{17} - 12 q^{19} + 8 q^{20} - 5 q^{22} + q^{25} + 7 q^{26} - 7 q^{28} + 4 q^{29} + 6 q^{31}+ \cdots + 12 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 + 8 * q^5 - 7 * q^7 - 5 * q^8 - 8 * q^10 + 5 * q^11 - 7 * q^13 + 7 * q^14 + 5 * q^16 + 13 * q^17 - 12 * q^19 + 8 * q^20 - 5 * q^22 + q^25 + 7 * q^26 - 7 * q^28 + 4 * q^29 + 6 * q^31 - 5 * q^32 - 13 * q^34 - 9 * q^35 - 14 * q^37 + 12 * q^38 - 8 * q^40 - q^41 - 11 * q^43 + 5 * q^44 - 9 * q^47 - 12 * q^49 - q^50 - 7 * q^52 + 6 * q^53 - 14 * q^55 + 7 * q^56 - 4 * q^58 - 3 * q^59 - 13 * q^61 - 6 * q^62 + 5 * q^64 - 9 * q^65 - 10 * q^67 + 13 * q^68 + 9 * q^70 - 8 * q^71 - 4 * q^73 + 14 * q^74 - 12 * q^76 - 18 * q^77 + 18 * q^79 + 8 * q^80 + q^82 + 2 * q^83 + 23 * q^85 + 11 * q^86 - 5 * q^88 + 10 * q^89 + 12 * q^91 + 9 * q^94 - 28 * q^95 - 17 * 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
$5$	−1.22871	−0.549495	−0.274747	−	0.961516i	$-0.588594\pi$
$5$	−1.22871	−0.549495	−0.274747	+	0.961516i	$0.588594\pi$
$6$	0	0
$7$	−0.627214	−0.237065	−0.118532	−	0.992950i	$-0.537819\pi$
$7$	−0.627214	−0.237065	−0.118532	+	0.992950i	$0.537819\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.22871	0.388551
$11$	5.89037	1.77601	0.888006	−	0.459831i	$-0.152090\pi$
$11$	5.89037	1.77601	0.888006	+	0.459831i	$0.152090\pi$
$12$	0	0
$13$	−2.35194	−0.652310	−0.326155	−	0.945316i	$-0.605753\pi$
$13$	−2.35194	−0.652310	−0.326155	+	0.945316i	$0.605753\pi$
$14$	0.627214	0.167630
$15$	0	0
$16$	1.00000	0.250000
$17$	5.33992	1.29512	0.647561	−	0.762014i	$-0.275789\pi$
$17$	5.33992	1.29512	0.647561	+	0.762014i	$0.275789\pi$
$18$	0	0
$19$	0.270925	0.0621544	0.0310772	−	0.999517i	$-0.490106\pi$
$19$	0.270925	0.0621544	0.0310772	+	0.999517i	$0.490106\pi$
$20$	−1.22871	−0.274747
$21$	0	0
$22$	−5.89037	−1.25583
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.49028	−0.698056
$26$	2.35194	0.461253
$27$	0	0
$28$	−0.627214	−0.118532
$29$	1.98287	0.368211	0.184105	−	0.982907i	$-0.441061\pi$
$29$	1.98287	0.368211	0.184105	+	0.982907i	$0.441061\pi$
$30$	0	0
$31$	6.60353	1.18603	0.593014	−	0.805192i	$-0.297938\pi$
$31$	6.60353	1.18603	0.593014	+	0.805192i	$0.297938\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.33992	−0.915789
$35$	0.770663	0.130266
$36$	0	0
$37$	−10.0704	−1.65556	−0.827782	−	0.561050i	$-0.810398\pi$
$37$	−10.0704	−1.65556	−0.827782	+	0.561050i	$0.810398\pi$
$38$	−0.270925	−0.0439498
$39$	0	0
$40$	1.22871	0.194276
$41$	−10.8126	−1.68865	−0.844324	−	0.535833i	$-0.819998\pi$
$41$	−10.8126	−1.68865	−0.844324	+	0.535833i	$0.819998\pi$
$42$	0	0
$43$	−4.59725	−0.701074	−0.350537	−	0.936549i	$-0.614001\pi$
$43$	−4.59725	−0.701074	−0.350537	+	0.936549i	$0.614001\pi$
$44$	5.89037	0.888006
$45$	0	0
$46$	0	0
$47$	−6.97259	−1.01706	−0.508528	−	0.861045i	$-0.669810\pi$
$47$	−6.97259	−1.01706	−0.508528	+	0.861045i	$0.669810\pi$
$48$	0	0
$49$	−6.60660	−0.943800
$50$	3.49028	0.493600
$51$	0	0
$52$	−2.35194	−0.326155
$53$	−2.63694	−0.362212	−0.181106	−	0.983464i	$-0.557968\pi$
$53$	−2.63694	−0.362212	−0.181106	+	0.983464i	$0.557968\pi$
$54$	0	0
$55$	−7.23754	−0.975909
$56$	0.627214	0.0838151
$57$	0	0
$58$	−1.98287	−0.260364
$59$	−1.64927	−0.214717	−0.107358	−	0.994220i	$-0.534239\pi$
$59$	−1.64927	−0.214717	−0.107358	+	0.994220i	$0.534239\pi$
$60$	0	0
$61$	−9.13485	−1.16960	−0.584799	−	0.811178i	$-0.698827\pi$
$61$	−9.13485	−1.16960	−0.584799	+	0.811178i	$0.698827\pi$
$62$	−6.60353	−0.838649
$63$	0	0
$64$	1.00000	0.125000
$65$	2.88984	0.358441
$66$	0	0
$67$	6.19401	0.756718	0.378359	−	0.925659i	$-0.376488\pi$
$67$	6.19401	0.756718	0.378359	+	0.925659i	$0.376488\pi$
$68$	5.33992	0.647561
$69$	0	0
$70$	−0.770663	−0.0921118
$71$	9.06233	1.07550	0.537750	−	0.843104i	$-0.319274\pi$
$71$	9.06233	1.07550	0.537750	+	0.843104i	$0.319274\pi$
$72$	0	0
$73$	−5.93852	−0.695051	−0.347525	−	0.937671i	$-0.612978\pi$
$73$	−5.93852	−0.695051	−0.347525	+	0.937671i	$0.612978\pi$
$74$	10.0704	1.17066
$75$	0	0
$76$	0.270925	0.0310772
$77$	−3.69452	−0.421030
$78$	0	0
$79$	6.00714	0.675856	0.337928	−	0.941172i	$-0.390274\pi$
$79$	6.00714	0.675856	0.337928	+	0.941172i	$0.390274\pi$
$80$	−1.22871	−0.137374
$81$	0	0
$82$	10.8126	1.19405
$83$	−1.95305	−0.214375	−0.107187	−	0.994239i	$-0.534184\pi$
$83$	−1.95305	−0.214375	−0.107187	+	0.994239i	$0.534184\pi$
$84$	0	0
$85$	−6.56120	−0.711662
$86$	4.59725	0.495734
$87$	0	0
$88$	−5.89037	−0.627915
$89$	4.59045	0.486587	0.243293	−	0.969953i	$-0.421772\pi$
$89$	4.59045	0.486587	0.243293	+	0.969953i	$0.421772\pi$
$90$	0	0
$91$	1.47517	0.154640
$92$	0	0
$93$	0	0
$94$	6.97259	0.719168
$95$	−0.332887	−0.0341535
$96$	0	0
$97$	−10.1844	−1.03406	−0.517032	−	0.855966i	$-0.672963\pi$
$97$	−10.1844	−1.03406	−0.517032	+	0.855966i	$0.672963\pi$
$98$	6.60660	0.667368
$99$	0	0
$100$	−3.49028	−0.349028
$101$	14.2352	1.41646	0.708229	−	0.705983i	$-0.249495\pi$
$101$	14.2352	1.41646	0.708229	+	0.705983i	$0.249495\pi$
$102$	0	0
$103$	0.753717	0.0742659	0.0371330	−	0.999310i	$-0.488177\pi$
$103$	0.753717	0.0742659	0.0371330	+	0.999310i	$0.488177\pi$
$104$	2.35194	0.230627
$105$	0	0
$106$	2.63694	0.256122
$107$	0.593036	0.0573310	0.0286655	−	0.999589i	$-0.490874\pi$
$107$	0.593036	0.0573310	0.0286655	+	0.999589i	$0.490874\pi$
$108$	0	0
$109$	6.86224	0.657283	0.328642	−	0.944455i	$-0.393409\pi$
$109$	6.86224	0.657283	0.328642	+	0.944455i	$0.393409\pi$
$110$	7.23754	0.690072
$111$	0	0
$112$	−0.627214	−0.0592662
$113$	−1.24628	−0.117240	−0.0586202	−	0.998280i	$-0.518670\pi$
$113$	−1.24628	−0.117240	−0.0586202	+	0.998280i	$0.518670\pi$
$114$	0	0
$115$	0	0
$116$	1.98287	0.184105
$117$	0	0
$118$	1.64927	0.151828
$119$	−3.34928	−0.307028
$120$	0	0
$121$	23.6964	2.15422
$122$	9.13485	0.827031
$123$	0	0
$124$	6.60353	0.593014
$125$	10.4321	0.933072
$126$	0	0
$127$	−7.18567	−0.637625	−0.318812	−	0.947818i	$-0.603284\pi$
$127$	−7.18567	−0.637625	−0.318812	+	0.947818i	$0.603284\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.88984	−0.253456
$131$	0.658240	0.0575107	0.0287553	−	0.999586i	$-0.490846\pi$
$131$	0.658240	0.0575107	0.0287553	+	0.999586i	$0.490846\pi$
$132$	0	0
$133$	−0.169928	−0.0147346
$134$	−6.19401	−0.535081
$135$	0	0
$136$	−5.33992	−0.457895
$137$	−12.5241	−1.07001	−0.535003	−	0.844850i	$-0.679689\pi$
$137$	−12.5241	−1.07001	−0.535003	+	0.844850i	$0.679689\pi$
$138$	0	0
$139$	17.0602	1.44703	0.723515	−	0.690309i	$-0.242525\pi$
$139$	17.0602	1.44703	0.723515	+	0.690309i	$0.242525\pi$
$140$	0.770663	0.0651329
$141$	0	0
$142$	−9.06233	−0.760494
$143$	−13.8538	−1.15851
$144$	0	0
$145$	−2.43637	−0.202330
$146$	5.93852	0.491475
$147$	0	0
$148$	−10.0704	−0.827782
$149$	−18.4139	−1.50853	−0.754264	−	0.656571i	$-0.772006\pi$
$149$	−18.4139	−1.50853	−0.754264	+	0.656571i	$0.772006\pi$
$150$	0	0
$151$	−11.5003	−0.935883	−0.467941	−	0.883760i	$-0.655004\pi$
$151$	−11.5003	−0.935883	−0.467941	+	0.883760i	$0.655004\pi$
$152$	−0.270925	−0.0219749
$153$	0	0
$154$	3.69452	0.297713
$155$	−8.11380	−0.651716
$156$	0	0
$157$	−3.22814	−0.257633	−0.128817	−	0.991668i	$-0.541118\pi$
$157$	−3.22814	−0.257633	−0.128817	+	0.991668i	$0.541118\pi$
$158$	−6.00714	−0.477903
$159$	0	0
$160$	1.22871	0.0971379
$161$	0	0
$162$	0	0
$163$	11.0637	0.866576	0.433288	−	0.901256i	$-0.357353\pi$
$163$	11.0637	0.866576	0.433288	+	0.901256i	$0.357353\pi$
$164$	−10.8126	−0.844324
$165$	0	0
$166$	1.95305	0.151586
$167$	21.9317	1.69713	0.848563	−	0.529094i	$-0.177468\pi$
$167$	21.9317	1.69713	0.848563	+	0.529094i	$0.177468\pi$
$168$	0	0
$169$	−7.46838	−0.574491
$170$	6.56120	0.503221
$171$	0	0
$172$	−4.59725	−0.350537
$173$	−11.0953	−0.843561	−0.421780	−	0.906698i	$-0.638595\pi$
$173$	−11.0953	−0.843561	−0.421780	+	0.906698i	$0.638595\pi$
$174$	0	0
$175$	2.18915	0.165484
$176$	5.89037	0.444003
$177$	0	0
$178$	−4.59045	−0.344069
$179$	17.5377	1.31083	0.655414	−	0.755270i	$-0.272494\pi$
$179$	17.5377	1.31083	0.655414	+	0.755270i	$0.272494\pi$
$180$	0	0
$181$	−20.4060	−1.51676	−0.758382	−	0.651810i	$-0.774010\pi$
$181$	−20.4060	−1.51676	−0.758382	+	0.651810i	$0.774010\pi$
$182$	−1.47517	−0.109347
$183$	0	0
$184$	0	0
$185$	12.3736	0.909723
$186$	0	0
$187$	31.4541	2.30015
$188$	−6.97259	−0.508528
$189$	0	0
$190$	0.332887	0.0241502
$191$	−9.46247	−0.684680	−0.342340	−	0.939576i	$-0.611219\pi$
$191$	−9.46247	−0.684680	−0.342340	+	0.939576i	$0.611219\pi$
$192$	0	0
$193$	25.5609	1.83991	0.919956	−	0.392022i	$-0.128224\pi$
$193$	25.5609	1.83991	0.919956	+	0.392022i	$0.128224\pi$
$194$	10.1844	0.731194
$195$	0	0
$196$	−6.60660	−0.471900
$197$	−5.26954	−0.375439	−0.187719	−	0.982223i	$-0.560110\pi$
$197$	−5.26954	−0.375439	−0.187719	+	0.982223i	$0.560110\pi$
$198$	0	0
$199$	25.0160	1.77333	0.886667	−	0.462409i	$-0.153015\pi$
$199$	25.0160	1.77333	0.886667	+	0.462409i	$0.153015\pi$
$200$	3.49028	0.246800
$201$	0	0
$202$	−14.2352	−1.00159
$203$	−1.24369	−0.0872898
$204$	0	0
$205$	13.2856	0.927903
$206$	−0.753717	−0.0525139
$207$	0	0
$208$	−2.35194	−0.163078
$209$	1.59585	0.110387
$210$	0	0
$211$	−8.86632	−0.610383	−0.305191	−	0.952291i	$-0.598720\pi$
$211$	−8.86632	−0.610383	−0.305191	+	0.952291i	$0.598720\pi$
$212$	−2.63694	−0.181106
$213$	0	0
$214$	−0.593036	−0.0405391
$215$	5.64867	0.385236
$216$	0	0
$217$	−4.14183	−0.281166
$218$	−6.86224	−0.464769
$219$	0	0
$220$	−7.23754	−0.487955
$221$	−12.5592	−0.844821
$222$	0	0
$223$	−16.9830	−1.13727	−0.568633	−	0.822592i	$-0.692527\pi$
$223$	−16.9830	−1.13727	−0.568633	+	0.822592i	$0.692527\pi$
$224$	0.627214	0.0419075
$225$	0	0
$226$	1.24628	0.0829015
$227$	−17.9952	−1.19438	−0.597192	−	0.802098i	$-0.703717\pi$
$227$	−17.9952	−1.19438	−0.597192	+	0.802098i	$0.703717\pi$
$228$	0	0
$229$	−7.49653	−0.495385	−0.247692	−	0.968839i	$-0.579672\pi$
$229$	−7.49653	−0.495385	−0.247692	+	0.968839i	$0.579672\pi$
$230$	0	0
$231$	0	0
$232$	−1.98287	−0.130182
$233$	−5.52307	−0.361829	−0.180914	−	0.983499i	$-0.557906\pi$
$233$	−5.52307	−0.361829	−0.180914	+	0.983499i	$0.557906\pi$
$234$	0	0
$235$	8.56727	0.558867
$236$	−1.64927	−0.107358
$237$	0	0
$238$	3.34928	0.217101
$239$	−5.21901	−0.337590	−0.168795	−	0.985651i	$-0.553988\pi$
$239$	−5.21901	−0.337590	−0.168795	+	0.985651i	$0.553988\pi$
$240$	0	0
$241$	1.55390	0.100095	0.0500476	−	0.998747i	$-0.484063\pi$
$241$	1.55390	0.100095	0.0500476	+	0.998747i	$0.484063\pi$
$242$	−23.6964	−1.52326
$243$	0	0
$244$	−9.13485	−0.584799
$245$	8.11758	0.518613
$246$	0	0
$247$	−0.637198	−0.0405439
$248$	−6.60353	−0.419324
$249$	0	0
$250$	−10.4321	−0.659782
$251$	−8.92657	−0.563440	−0.281720	−	0.959497i	$-0.590905\pi$
$251$	−8.92657	−0.563440	−0.281720	+	0.959497i	$0.590905\pi$
$252$	0	0
$253$	0	0
$254$	7.18567	0.450869
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.64256	0.227216	0.113608	−	0.993526i	$-0.463759\pi$
$257$	3.64256	0.227216	0.113608	+	0.993526i	$0.463759\pi$
$258$	0	0
$259$	6.31630	0.392476
$260$	2.88984	0.179221
$261$	0	0
$262$	−0.658240	−0.0406662
$263$	16.1369	0.995044	0.497522	−	0.867451i	$-0.334243\pi$
$263$	16.1369	0.995044	0.497522	+	0.867451i	$0.334243\pi$
$264$	0	0
$265$	3.24003	0.199034
$266$	0.169928	0.0104189
$267$	0	0
$268$	6.19401	0.378359
$269$	4.49392	0.273999	0.137000	−	0.990571i	$-0.456254\pi$
$269$	4.49392	0.273999	0.137000	+	0.990571i	$0.456254\pi$
$270$	0	0
$271$	−3.50249	−0.212761	−0.106381	−	0.994325i	$-0.533926\pi$
$271$	−3.50249	−0.212761	−0.106381	+	0.994325i	$0.533926\pi$
$272$	5.33992	0.323780
$273$	0	0
$274$	12.5241	0.756608
$275$	−20.5590	−1.23976
$276$	0	0
$277$	−12.0692	−0.725166	−0.362583	−	0.931951i	$-0.618105\pi$
$277$	−12.0692	−0.725166	−0.362583	+	0.931951i	$0.618105\pi$
$278$	−17.0602	−1.02320
$279$	0	0
$280$	−0.770663	−0.0460559
$281$	−6.26602	−0.373799	−0.186900	−	0.982379i	$-0.559844\pi$
$281$	−6.26602	−0.373799	−0.186900	+	0.982379i	$0.559844\pi$
$282$	0	0
$283$	−31.6306	−1.88024	−0.940121	−	0.340842i	$-0.889288\pi$
$283$	−31.6306	−1.88024	−0.940121	+	0.340842i	$0.889288\pi$
$284$	9.06233	0.537750
$285$	0	0
$286$	13.8538	0.819191
$287$	6.78183	0.400319
$288$	0	0
$289$	11.5148	0.677339
$290$	2.43637	0.143069
$291$	0	0
$292$	−5.93852	−0.347525
$293$	6.77777	0.395961	0.197981	−	0.980206i	$-0.436562\pi$
$293$	6.77777	0.395961	0.197981	+	0.980206i	$0.436562\pi$
$294$	0	0
$295$	2.02647	0.117986
$296$	10.0704	0.585330
$297$	0	0
$298$	18.4139	1.06669
$299$	0	0
$300$	0	0
$301$	2.88346	0.166200
$302$	11.5003	0.661769
$303$	0	0
$304$	0.270925	0.0155386
$305$	11.2241	0.642688
$306$	0	0
$307$	−6.07347	−0.346631	−0.173316	−	0.984866i	$-0.555448\pi$
$307$	−6.07347	−0.346631	−0.173316	+	0.984866i	$0.555448\pi$
$308$	−3.69452	−0.210515
$309$	0	0
$310$	8.11380	0.460833
$311$	−10.2293	−0.580052	−0.290026	−	0.957019i	$-0.593664\pi$
$311$	−10.2293	−0.580052	−0.290026	+	0.957019i	$0.593664\pi$
$312$	0	0
$313$	−4.02016	−0.227233	−0.113616	−	0.993525i	$-0.536244\pi$
$313$	−4.02016	−0.227233	−0.113616	+	0.993525i	$0.536244\pi$
$314$	3.22814	0.182174
$315$	0	0
$316$	6.00714	0.337928
$317$	−4.99304	−0.280437	−0.140218	−	0.990121i	$-0.544780\pi$
$317$	−4.99304	−0.280437	−0.140218	+	0.990121i	$0.544780\pi$
$318$	0	0
$319$	11.6799	0.653947
$320$	−1.22871	−0.0686868
$321$	0	0
$322$	0	0
$323$	1.44672	0.0804974
$324$	0	0
$325$	8.20892	0.455349
$326$	−11.0637	−0.612762
$327$	0	0
$328$	10.8126	0.597027
$329$	4.37331	0.241108
$330$	0	0
$331$	4.90967	0.269860	0.134930	−	0.990855i	$-0.456919\pi$
$331$	4.90967	0.269860	0.134930	+	0.990855i	$0.456919\pi$
$332$	−1.95305	−0.107187
$333$	0	0
$334$	−21.9317	−1.20005
$335$	−7.61062	−0.415813
$336$	0	0
$337$	−29.6329	−1.61421	−0.807105	−	0.590408i	$-0.798967\pi$
$337$	−29.6329	−1.61421	−0.807105	+	0.590408i	$0.798967\pi$
$338$	7.46838	0.406227
$339$	0	0
$340$	−6.56120	−0.355831
$341$	38.8972	2.10640
$342$	0	0
$343$	8.53426	0.460807
$344$	4.59725	0.247867
$345$	0	0
$346$	11.0953	0.596488
$347$	12.8579	0.690248	0.345124	−	0.938557i	$-0.387837\pi$
$347$	12.8579	0.690248	0.345124	+	0.938557i	$0.387837\pi$
$348$	0	0
$349$	−23.5370	−1.25990	−0.629952	−	0.776634i	$-0.716926\pi$
$349$	−23.5370	−1.25990	−0.629952	+	0.776634i	$0.716926\pi$
$350$	−2.18915	−0.117015
$351$	0	0
$352$	−5.89037	−0.313958
$353$	−8.31289	−0.442450	−0.221225	−	0.975223i	$-0.571006\pi$
$353$	−8.31289	−0.442450	−0.221225	+	0.975223i	$0.571006\pi$
$354$	0	0
$355$	−11.1349	−0.590982
$356$	4.59045	0.243293
$357$	0	0
$358$	−17.5377	−0.926896
$359$	−19.3351	−1.02047	−0.510234	−	0.860036i	$-0.670441\pi$
$359$	−19.3351	−1.02047	−0.510234	+	0.860036i	$0.670441\pi$
$360$	0	0
$361$	−18.9266	−0.996137
$362$	20.4060	1.07251
$363$	0	0
$364$	1.47517	0.0773199
$365$	7.29670	0.381927
$366$	0	0
$367$	−30.4419	−1.58906	−0.794528	−	0.607228i	$-0.792282\pi$
$367$	−30.4419	−1.58906	−0.794528	+	0.607228i	$0.792282\pi$
$368$	0	0
$369$	0	0
$370$	−12.3736	−0.643272
$371$	1.65393	0.0858677
$372$	0	0
$373$	−19.2962	−0.999120	−0.499560	−	0.866279i	$-0.666505\pi$
$373$	−19.2962	−0.999120	−0.499560	+	0.866279i	$0.666505\pi$
$374$	−31.4541	−1.62645
$375$	0	0
$376$	6.97259	0.359584
$377$	−4.66360	−0.240188
$378$	0	0
$379$	−32.3335	−1.66086	−0.830430	−	0.557123i	$-0.811905\pi$
$379$	−32.3335	−1.66086	−0.830430	+	0.557123i	$0.811905\pi$
$380$	−0.332887	−0.0170767
$381$	0	0
$382$	9.46247	0.484142
$383$	−7.18995	−0.367389	−0.183695	−	0.982983i	$-0.558806\pi$
$383$	−7.18995	−0.367389	−0.183695	+	0.982983i	$0.558806\pi$
$384$	0	0
$385$	4.53949	0.231354
$386$	−25.5609	−1.30101
$387$	0	0
$388$	−10.1844	−0.517032
$389$	26.2852	1.33271	0.666357	−	0.745633i	$-0.267853\pi$
$389$	26.2852	1.33271	0.666357	+	0.745633i	$0.267853\pi$
$390$	0	0
$391$	0	0
$392$	6.60660	0.333684
$393$	0	0
$394$	5.26954	0.265475
$395$	−7.38102	−0.371379
$396$	0	0
$397$	−9.27172	−0.465334	−0.232667	−	0.972556i	$-0.574745\pi$
$397$	−9.27172	−0.465334	−0.232667	+	0.972556i	$0.574745\pi$
$398$	−25.0160	−1.25394
$399$	0	0
$400$	−3.49028	−0.174514
$401$	−28.0321	−1.39986	−0.699929	−	0.714213i	$-0.746785\pi$
$401$	−28.0321	−1.39986	−0.699929	+	0.714213i	$0.746785\pi$
$402$	0	0
$403$	−15.5311	−0.773659
$404$	14.2352	0.708229
$405$	0	0
$406$	1.24369	0.0617232
$407$	−59.3184	−2.94030
$408$	0	0
$409$	0.0443103	0.00219100	0.00109550	−	0.999999i	$-0.499651\pi$
$409$	0.0443103	0.00219100	0.00109550	+	0.999999i	$0.499651\pi$
$410$	−13.2856	−0.656127
$411$	0	0
$412$	0.753717	0.0371330
$413$	1.03445	0.0509017
$414$	0	0
$415$	2.39972	0.117798
$416$	2.35194	0.115313
$417$	0	0
$418$	−1.59585	−0.0780553
$419$	12.3373	0.602714	0.301357	−	0.953511i	$-0.402560\pi$
$419$	12.3373	0.602714	0.301357	+	0.953511i	$0.402560\pi$
$420$	0	0
$421$	−2.79149	−0.136049	−0.0680244	−	0.997684i	$-0.521670\pi$
$421$	−2.79149	−0.136049	−0.0680244	+	0.997684i	$0.521670\pi$
$422$	8.86632	0.431606
$423$	0	0
$424$	2.63694	0.128061
$425$	−18.6378	−0.904067
$426$	0	0
$427$	5.72951	0.277270
$428$	0.593036	0.0286655
$429$	0	0
$430$	−5.64867	−0.272403
$431$	3.49668	0.168429	0.0842146	−	0.996448i	$-0.473162\pi$
$431$	3.49668	0.168429	0.0842146	+	0.996448i	$0.473162\pi$
$432$	0	0
$433$	9.42923	0.453140	0.226570	−	0.973995i	$-0.427249\pi$
$433$	9.42923	0.453140	0.226570	+	0.973995i	$0.427249\pi$
$434$	4.14183	0.198814
$435$	0	0
$436$	6.86224	0.328642
$437$	0	0
$438$	0	0
$439$	22.5828	1.07782	0.538910	−	0.842363i	$-0.318836\pi$
$439$	22.5828	1.07782	0.538910	+	0.842363i	$0.318836\pi$
$440$	7.23754	0.345036
$441$	0	0
$442$	12.5592	0.597379
$443$	6.08056	0.288896	0.144448	−	0.989512i	$-0.453859\pi$
$443$	6.08056	0.288896	0.144448	+	0.989512i	$0.453859\pi$
$444$	0	0
$445$	−5.64032	−0.267377
$446$	16.9830	0.804168
$447$	0	0
$448$	−0.627214	−0.0296331
$449$	12.0641	0.569338	0.284669	−	0.958626i	$-0.408116\pi$
$449$	12.0641	0.569338	0.284669	+	0.958626i	$0.408116\pi$
$450$	0	0
$451$	−63.6903	−2.99906
$452$	−1.24628	−0.0586202
$453$	0	0
$454$	17.9952	0.844557
$455$	−1.81255	−0.0849737
$456$	0	0
$457$	20.1601	0.943050	0.471525	−	0.881853i	$-0.343704\pi$
$457$	20.1601	0.943050	0.471525	+	0.881853i	$0.343704\pi$
$458$	7.49653	0.350290
$459$	0	0
$460$	0	0
$461$	−34.1368	−1.58991	−0.794954	−	0.606670i	$-0.792505\pi$
$461$	−34.1368	−1.58991	−0.794954	+	0.606670i	$0.792505\pi$
$462$	0	0
$463$	11.6716	0.542424	0.271212	−	0.962520i	$-0.412575\pi$
$463$	11.6716	0.542424	0.271212	+	0.962520i	$0.412575\pi$
$464$	1.98287	0.0920526
$465$	0	0
$466$	5.52307	0.255851
$467$	4.74513	0.219579	0.109789	−	0.993955i	$-0.464982\pi$
$467$	4.74513	0.219579	0.109789	+	0.993955i	$0.464982\pi$
$468$	0	0
$469$	−3.88497	−0.179391
$470$	−8.56727	−0.395179
$471$	0	0
$472$	1.64927	0.0759138
$473$	−27.0795	−1.24512
$474$	0	0
$475$	−0.945602	−0.0433872
$476$	−3.34928	−0.153514
$477$	0	0
$478$	5.21901	0.238712
$479$	−29.7540	−1.35950	−0.679748	−	0.733445i	$-0.737911\pi$
$479$	−29.7540	−1.35950	−0.679748	+	0.733445i	$0.737911\pi$
$480$	0	0
$481$	23.6850	1.07994
$482$	−1.55390	−0.0707780
$483$	0	0
$484$	23.6964	1.07711
$485$	12.5136	0.568213
$486$	0	0
$487$	−8.71408	−0.394873	−0.197436	−	0.980316i	$-0.563262\pi$
$487$	−8.71408	−0.394873	−0.197436	+	0.980316i	$0.563262\pi$
$488$	9.13485	0.413515
$489$	0	0
$490$	−8.11758	−0.366715
$491$	−15.0620	−0.679738	−0.339869	−	0.940473i	$-0.610383\pi$
$491$	−15.0620	−0.679738	−0.339869	+	0.940473i	$0.610383\pi$
$492$	0	0
$493$	10.5884	0.476877
$494$	0.637198	0.0286689
$495$	0	0
$496$	6.60353	0.296507
$497$	−5.68402	−0.254963
$498$	0	0
$499$	31.5993	1.41458	0.707290	−	0.706923i	$-0.249917\pi$
$499$	31.5993	1.41458	0.707290	+	0.706923i	$0.249917\pi$
$500$	10.4321	0.466536
$501$	0	0
$502$	8.92657	0.398413
$503$	−33.4242	−1.49031	−0.745157	−	0.666890i	$-0.767625\pi$
$503$	−33.4242	−1.49031	−0.745157	+	0.666890i	$0.767625\pi$
$504$	0	0
$505$	−17.4909	−0.778336
$506$	0	0
$507$	0	0
$508$	−7.18567	−0.318812
$509$	17.9178	0.794191	0.397096	−	0.917777i	$-0.370018\pi$
$509$	17.9178	0.794191	0.397096	+	0.917777i	$0.370018\pi$
$510$	0	0
$511$	3.72472	0.164772
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−3.64256	−0.160666
$515$	−0.926098	−0.0408087
$516$	0	0
$517$	−41.0711	−1.80631
$518$	−6.31630	−0.277522
$519$	0	0
$520$	−2.88984	−0.126728
$521$	8.20662	0.359538	0.179769	−	0.983709i	$-0.442465\pi$
$521$	8.20662	0.359538	0.179769	+	0.983709i	$0.442465\pi$
$522$	0	0
$523$	6.49965	0.284210	0.142105	−	0.989852i	$-0.454613\pi$
$523$	6.49965	0.284210	0.142105	+	0.989852i	$0.454613\pi$
$524$	0.658240	0.0287553
$525$	0	0
$526$	−16.1369	−0.703602
$527$	35.2623	1.53605
$528$	0	0
$529$	0	0
$530$	−3.24003	−0.140738
$531$	0	0
$532$	−0.169928	−0.00736730
$533$	25.4306	1.10152
$534$	0	0
$535$	−0.728668	−0.0315031
$536$	−6.19401	−0.267540
$537$	0	0
$538$	−4.49392	−0.193747
$539$	−38.9153	−1.67620
$540$	0	0
$541$	−26.3569	−1.13317	−0.566585	−	0.824003i	$-0.691736\pi$
$541$	−26.3569	−1.13317	−0.566585	+	0.824003i	$0.691736\pi$
$542$	3.50249	0.150445
$543$	0	0
$544$	−5.33992	−0.228947
$545$	−8.43168	−0.361174
$546$	0	0
$547$	−20.2475	−0.865721	−0.432861	−	0.901461i	$-0.642496\pi$
$547$	−20.2475	−0.865721	−0.432861	+	0.901461i	$0.642496\pi$
$548$	−12.5241	−0.535003
$549$	0	0
$550$	20.5590	0.876640
$551$	0.537209	0.0228859
$552$	0	0
$553$	−3.76777	−0.160222
$554$	12.0692	0.512770
$555$	0	0
$556$	17.0602	0.723515
$557$	8.15671	0.345611	0.172806	−	0.984956i	$-0.444717\pi$
$557$	8.15671	0.345611	0.172806	+	0.984956i	$0.444717\pi$
$558$	0	0
$559$	10.8124	0.457318
$560$	0.770663	0.0325665
$561$	0	0
$562$	6.26602	0.264316
$563$	31.4513	1.32551	0.662757	−	0.748835i	$-0.269386\pi$
$563$	31.4513	1.32551	0.662757	+	0.748835i	$0.269386\pi$
$564$	0	0
$565$	1.53132	0.0644230
$566$	31.6306	1.32953
$567$	0	0
$568$	−9.06233	−0.380247
$569$	−26.6983	−1.11925	−0.559625	−	0.828746i	$-0.689055\pi$
$569$	−26.6983	−1.11925	−0.559625	+	0.828746i	$0.689055\pi$
$570$	0	0
$571$	28.8560	1.20759	0.603793	−	0.797141i	$-0.293655\pi$
$571$	28.8560	1.20759	0.603793	+	0.797141i	$0.293655\pi$
$572$	−13.8538	−0.579256
$573$	0	0
$574$	−6.78183	−0.283068
$575$	0	0
$576$	0	0
$577$	−28.0824	−1.16909	−0.584544	−	0.811362i	$-0.698726\pi$
$577$	−28.0824	−1.16909	−0.584544	+	0.811362i	$0.698726\pi$
$578$	−11.5148	−0.478951
$579$	0	0
$580$	−2.43637	−0.101165
$581$	1.22498	0.0508207
$582$	0	0
$583$	−15.5326	−0.643293
$584$	5.93852	0.245738
$585$	0	0
$586$	−6.77777	−0.279987
$587$	28.0401	1.15734	0.578669	−	0.815563i	$-0.303572\pi$
$587$	28.0401	1.15734	0.578669	+	0.815563i	$0.303572\pi$
$588$	0	0
$589$	1.78906	0.0737168
$590$	−2.02647	−0.0834284
$591$	0	0
$592$	−10.0704	−0.413891
$593$	7.57300	0.310986	0.155493	−	0.987837i	$-0.450303\pi$
$593$	7.57300	0.310986	0.155493	+	0.987837i	$0.450303\pi$
$594$	0	0
$595$	4.11528	0.168710
$596$	−18.4139	−0.754264
$597$	0	0
$598$	0	0
$599$	12.5238	0.511708	0.255854	−	0.966715i	$-0.417643\pi$
$599$	12.5238	0.511708	0.255854	+	0.966715i	$0.417643\pi$
$600$	0	0
$601$	−18.2262	−0.743463	−0.371732	−	0.928340i	$-0.621236\pi$
$601$	−18.2262	−0.743463	−0.371732	+	0.928340i	$0.621236\pi$
$602$	−2.88346	−0.117521
$603$	0	0
$604$	−11.5003	−0.467941
$605$	−29.1160	−1.18373
$606$	0	0
$607$	−8.86155	−0.359679	−0.179840	−	0.983696i	$-0.557558\pi$
$607$	−8.86155	−0.359679	−0.179840	+	0.983696i	$0.557558\pi$
$608$	−0.270925	−0.0109874
$609$	0	0
$610$	−11.2241	−0.454449
$611$	16.3991	0.663437
$612$	0	0
$613$	21.3136	0.860849	0.430425	−	0.902627i	$-0.358364\pi$
$613$	21.3136	0.860849	0.430425	+	0.902627i	$0.358364\pi$
$614$	6.07347	0.245105
$615$	0	0
$616$	3.69452	0.148857
$617$	35.7387	1.43879	0.719393	−	0.694603i	$-0.244420\pi$
$617$	35.7387	1.43879	0.719393	+	0.694603i	$0.244420\pi$
$618$	0	0
$619$	7.37359	0.296370	0.148185	−	0.988960i	$-0.452657\pi$
$619$	7.37359	0.296370	0.148185	+	0.988960i	$0.452657\pi$
$620$	−8.11380	−0.325858
$621$	0	0
$622$	10.2293	0.410159
$623$	−2.87920	−0.115353
$624$	0	0
$625$	4.63343	0.185337
$626$	4.02016	0.160678
$627$	0	0
$628$	−3.22814	−0.128817
$629$	−53.7752	−2.14416
$630$	0	0
$631$	19.2251	0.765338	0.382669	−	0.923886i	$-0.375005\pi$
$631$	19.2251	0.765338	0.382669	+	0.923886i	$0.375005\pi$
$632$	−6.00714	−0.238951
$633$	0	0
$634$	4.99304	0.198299
$635$	8.82908	0.350371
$636$	0	0
$637$	15.5383	0.615651
$638$	−11.6799	−0.462410
$639$	0	0
$640$	1.22871	0.0485689
$641$	15.5293	0.613371	0.306686	−	0.951811i	$-0.400780\pi$
$641$	15.5293	0.613371	0.306686	+	0.951811i	$0.400780\pi$
$642$	0	0
$643$	−12.8573	−0.507043	−0.253522	−	0.967330i	$-0.581589\pi$
$643$	−12.8573	−0.507043	−0.253522	+	0.967330i	$0.581589\pi$
$644$	0	0
$645$	0	0
$646$	−1.44672	−0.0569203
$647$	35.2209	1.38468	0.692339	−	0.721572i	$-0.256580\pi$
$647$	35.2209	1.38468	0.692339	+	0.721572i	$0.256580\pi$
$648$	0	0
$649$	−9.71480	−0.381339
$650$	−8.20892	−0.321980
$651$	0	0
$652$	11.0637	0.433288
$653$	−28.7710	−1.12590	−0.562948	−	0.826492i	$-0.690333\pi$
$653$	−28.7710	−1.12590	−0.562948	+	0.826492i	$0.690333\pi$
$654$	0	0
$655$	−0.808784	−0.0316018
$656$	−10.8126	−0.422162
$657$	0	0
$658$	−4.37331	−0.170489
$659$	3.01164	0.117317	0.0586584	−	0.998278i	$-0.481318\pi$
$659$	3.01164	0.117317	0.0586584	+	0.998278i	$0.481318\pi$
$660$	0	0
$661$	−9.58560	−0.372837	−0.186418	−	0.982470i	$-0.559688\pi$
$661$	−9.58560	−0.372837	−0.186418	+	0.982470i	$0.559688\pi$
$662$	−4.90967	−0.190820
$663$	0	0
$664$	1.95305	0.0757930
$665$	0.208792	0.00809659
$666$	0	0
$667$	0	0
$668$	21.9317	0.848563
$669$	0	0
$670$	7.61062	0.294024
$671$	−53.8076	−2.07722
$672$	0	0
$673$	15.9623	0.615301	0.307650	−	0.951499i	$-0.400457\pi$
$673$	15.9623	0.615301	0.307650	+	0.951499i	$0.400457\pi$
$674$	29.6329	1.14142
$675$	0	0
$676$	−7.46838	−0.287246
$677$	26.1072	1.00338	0.501690	−	0.865048i	$-0.332712\pi$
$677$	26.1072	1.00338	0.501690	+	0.865048i	$0.332712\pi$
$678$	0	0
$679$	6.38777	0.245140
$680$	6.56120	0.251611
$681$	0	0
$682$	−38.8972	−1.48945
$683$	41.3823	1.58345	0.791726	−	0.610877i	$-0.209183\pi$
$683$	41.3823	1.58345	0.791726	+	0.610877i	$0.209183\pi$
$684$	0	0
$685$	15.3884	0.587962
$686$	−8.53426	−0.325839
$687$	0	0
$688$	−4.59725	−0.175268
$689$	6.20193	0.236275
$690$	0	0
$691$	40.6203	1.54527	0.772635	−	0.634850i	$-0.218938\pi$
$691$	40.6203	1.54527	0.772635	+	0.634850i	$0.218938\pi$
$692$	−11.0953	−0.421780
$693$	0	0
$694$	−12.8579	−0.488079
$695$	−20.9620	−0.795135
$696$	0	0
$697$	−57.7386	−2.18700
$698$	23.5370	0.890887
$699$	0	0
$700$	2.18915	0.0827422
$701$	1.58900	0.0600157	0.0300078	−	0.999550i	$-0.490447\pi$
$701$	1.58900	0.0600157	0.0300078	+	0.999550i	$0.490447\pi$
$702$	0	0
$703$	−2.72832	−0.102900
$704$	5.89037	0.222002
$705$	0	0
$706$	8.31289	0.312860
$707$	−8.92853	−0.335792
$708$	0	0
$709$	4.32980	0.162609	0.0813046	−	0.996689i	$-0.474091\pi$
$709$	4.32980	0.162609	0.0813046	+	0.996689i	$0.474091\pi$
$710$	11.1349	0.417887
$711$	0	0
$712$	−4.59045	−0.172034
$713$	0	0
$714$	0	0
$715$	17.0222	0.636596
$716$	17.5377	0.655414
$717$	0	0
$718$	19.3351	0.721580
$719$	25.9486	0.967718	0.483859	−	0.875146i	$-0.339235\pi$
$719$	25.9486	0.967718	0.483859	+	0.875146i	$0.339235\pi$
$720$	0	0
$721$	−0.472742	−0.0176058
$722$	18.9266	0.704375
$723$	0	0
$724$	−20.4060	−0.758382
$725$	−6.92078	−0.257031
$726$	0	0
$727$	−17.3672	−0.644114	−0.322057	−	0.946720i	$-0.604374\pi$
$727$	−17.3672	−0.644114	−0.322057	+	0.946720i	$0.604374\pi$
$728$	−1.47517	−0.0546734
$729$	0	0
$730$	−7.29670	−0.270063
$731$	−24.5489	−0.907976
$732$	0	0
$733$	−47.1889	−1.74296	−0.871480	−	0.490431i	$-0.836839\pi$
$733$	−47.1889	−1.74296	−0.871480	+	0.490431i	$0.836839\pi$
$734$	30.4419	1.12363
$735$	0	0
$736$	0	0
$737$	36.4850	1.34394
$738$	0	0
$739$	15.0987	0.555416	0.277708	−	0.960666i	$-0.410425\pi$
$739$	15.0987	0.555416	0.277708	+	0.960666i	$0.410425\pi$
$740$	12.3736	0.454862
$741$	0	0
$742$	−1.65393	−0.0607176
$743$	−11.8979	−0.436490	−0.218245	−	0.975894i	$-0.570033\pi$
$743$	−11.8979	−0.436490	−0.218245	+	0.975894i	$0.570033\pi$
$744$	0	0
$745$	22.6253	0.828928
$746$	19.2962	0.706485
$747$	0	0
$748$	31.4541	1.15008
$749$	−0.371961	−0.0135912
$750$	0	0
$751$	−7.94974	−0.290090	−0.145045	−	0.989425i	$-0.546333\pi$
$751$	−7.94974	−0.290090	−0.145045	+	0.989425i	$0.546333\pi$
$752$	−6.97259	−0.254264
$753$	0	0
$754$	4.66360	0.169838
$755$	14.1305	0.514262
$756$	0	0
$757$	−22.2062	−0.807100	−0.403550	−	0.914958i	$-0.632224\pi$
$757$	−22.2062	−0.807100	−0.403550	+	0.914958i	$0.632224\pi$
$758$	32.3335	1.17441
$759$	0	0
$760$	0.332887	0.0120751
$761$	−3.39284	−0.122991	−0.0614953	−	0.998107i	$-0.519587\pi$
$761$	−3.39284	−0.122991	−0.0614953	+	0.998107i	$0.519587\pi$
$762$	0	0
$763$	−4.30410	−0.155819
$764$	−9.46247	−0.342340
$765$	0	0
$766$	7.18995	0.259783
$767$	3.87898	0.140062
$768$	0	0
$769$	−4.15320	−0.149768	−0.0748842	−	0.997192i	$-0.523859\pi$
$769$	−4.15320	−0.149768	−0.0748842	+	0.997192i	$0.523859\pi$
$770$	−4.53949	−0.163592
$771$	0	0
$772$	25.5609	0.919956
$773$	−28.2415	−1.01578	−0.507888	−	0.861423i	$-0.669574\pi$
$773$	−28.2415	−1.01578	−0.507888	+	0.861423i	$0.669574\pi$
$774$	0	0
$775$	−23.0481	−0.827914
$776$	10.1844	0.365597
$777$	0	0
$778$	−26.2852	−0.942371
$779$	−2.92941	−0.104957
$780$	0	0
$781$	53.3804	1.91010
$782$	0	0
$783$	0	0
$784$	−6.60660	−0.235950
$785$	3.96644	0.141568
$786$	0	0
$787$	−21.8912	−0.780336	−0.390168	−	0.920744i	$-0.627583\pi$
$787$	−21.8912	−0.780336	−0.390168	+	0.920744i	$0.627583\pi$
$788$	−5.26954	−0.187719
$789$	0	0
$790$	7.38102	0.262605
$791$	0.781687	0.0277936
$792$	0	0
$793$	21.4846	0.762941
$794$	9.27172	0.329041
$795$	0	0
$796$	25.0160	0.886667
$797$	−10.7266	−0.379956	−0.189978	−	0.981788i	$-0.560842\pi$
$797$	−10.7266	−0.379956	−0.189978	+	0.981788i	$0.560842\pi$
$798$	0	0
$799$	−37.2331	−1.31721
$800$	3.49028	0.123400
$801$	0	0
$802$	28.0321	0.989849
$803$	−34.9801	−1.23442
$804$	0	0
$805$	0	0
$806$	15.5311	0.547059
$807$	0	0
$808$	−14.2352	−0.500793
$809$	−33.6941	−1.18462	−0.592312	−	0.805709i	$-0.701785\pi$
$809$	−33.6941	−1.18462	−0.592312	+	0.805709i	$0.701785\pi$
$810$	0	0
$811$	34.7974	1.22190	0.610952	−	0.791668i	$-0.290787\pi$
$811$	34.7974	1.22190	0.610952	+	0.791668i	$0.290787\pi$
$812$	−1.24369	−0.0436449
$813$	0	0
$814$	59.3184	2.07911
$815$	−13.5940	−0.476179
$816$	0	0
$817$	−1.24551	−0.0435748
$818$	−0.0443103	−0.00154927
$819$	0	0
$820$	13.2856	0.463952
$821$	23.8353	0.831857	0.415929	−	0.909397i	$-0.363457\pi$
$821$	23.8353	0.831857	0.415929	+	0.909397i	$0.363457\pi$
$822$	0	0
$823$	−23.7843	−0.829069	−0.414534	−	0.910034i	$-0.636056\pi$
$823$	−23.7843	−0.829069	−0.414534	+	0.910034i	$0.636056\pi$
$824$	−0.753717	−0.0262570
$825$	0	0
$826$	−1.03445	−0.0359930
$827$	30.8236	1.07184	0.535921	−	0.844268i	$-0.319965\pi$
$827$	30.8236	1.07184	0.535921	+	0.844268i	$0.319965\pi$
$828$	0	0
$829$	−30.9416	−1.07465	−0.537324	−	0.843376i	$-0.680565\pi$
$829$	−30.9416	−1.07465	−0.537324	+	0.843376i	$0.680565\pi$
$830$	−2.39972	−0.0832957
$831$	0	0
$832$	−2.35194	−0.0815388
$833$	−35.2787	−1.22234
$834$	0	0
$835$	−26.9476	−0.932562
$836$	1.59585	0.0551935
$837$	0	0
$838$	−12.3373	−0.426183
$839$	−39.3250	−1.35765	−0.678825	−	0.734300i	$-0.737511\pi$
$839$	−39.3250	−1.35765	−0.678825	+	0.734300i	$0.737511\pi$
$840$	0	0
$841$	−25.0682	−0.864421
$842$	2.79149	0.0962010
$843$	0	0
$844$	−8.86632	−0.305191
$845$	9.17646	0.315680
$846$	0	0
$847$	−14.8627	−0.510690
$848$	−2.63694	−0.0905530
$849$	0	0
$850$	18.6378	0.639272
$851$	0	0
$852$	0	0
$853$	−5.80760	−0.198848	−0.0994242	−	0.995045i	$-0.531700\pi$
$853$	−5.80760	−0.198848	−0.0994242	+	0.995045i	$0.531700\pi$
$854$	−5.72951	−0.196060
$855$	0	0
$856$	−0.593036	−0.0202696
$857$	21.1686	0.723107	0.361553	−	0.932351i	$-0.382247\pi$
$857$	21.1686	0.723107	0.361553	+	0.932351i	$0.382247\pi$
$858$	0	0
$859$	2.69259	0.0918698	0.0459349	−	0.998944i	$-0.485373\pi$
$859$	2.69259	0.0918698	0.0459349	+	0.998944i	$0.485373\pi$
$860$	5.64867	0.192618
$861$	0	0
$862$	−3.49668	−0.119097
$863$	−20.1950	−0.687447	−0.343724	−	0.939071i	$-0.611688\pi$
$863$	−20.1950	−0.687447	−0.343724	+	0.939071i	$0.611688\pi$
$864$	0	0
$865$	13.6329	0.463532
$866$	−9.42923	−0.320418
$867$	0	0
$868$	−4.14183	−0.140583
$869$	35.3843	1.20033
$870$	0	0
$871$	−14.5679	−0.493615
$872$	−6.86224	−0.232385
$873$	0	0
$874$	0	0
$875$	−6.54314	−0.221199
$876$	0	0
$877$	21.0342	0.710273	0.355137	−	0.934814i	$-0.384434\pi$
$877$	21.0342	0.710273	0.355137	+	0.934814i	$0.384434\pi$
$878$	−22.5828	−0.762134
$879$	0	0
$880$	−7.23754	−0.243977
$881$	−24.6353	−0.829983	−0.414992	−	0.909825i	$-0.636215\pi$
$881$	−24.6353	−0.829983	−0.414992	+	0.909825i	$0.636215\pi$
$882$	0	0
$883$	−1.16589	−0.0392352	−0.0196176	−	0.999808i	$-0.506245\pi$
$883$	−1.16589	−0.0392352	−0.0196176	+	0.999808i	$0.506245\pi$
$884$	−12.5592	−0.422411
$885$	0	0
$886$	−6.08056	−0.204281
$887$	10.1795	0.341793	0.170897	−	0.985289i	$-0.445334\pi$
$887$	10.1795	0.341793	0.170897	+	0.985289i	$0.445334\pi$
$888$	0	0
$889$	4.50695	0.151158
$890$	5.64032	0.189064
$891$	0	0
$892$	−16.9830	−0.568633
$893$	−1.88905	−0.0632145
$894$	0	0
$895$	−21.5487	−0.720293
$896$	0.627214	0.0209538
$897$	0	0
$898$	−12.0641	−0.402583
$899$	13.0940	0.436708
$900$	0	0
$901$	−14.0811	−0.469108
$902$	63.6903	2.12066
$903$	0	0
$904$	1.24628	0.0414508
$905$	25.0730	0.833454
$906$	0	0
$907$	−19.6527	−0.652556	−0.326278	−	0.945274i	$-0.605795\pi$
$907$	−19.6527	−0.652556	−0.326278	+	0.945274i	$0.605795\pi$
$908$	−17.9952	−0.597192
$909$	0	0
$910$	1.81255	0.0600855
$911$	−47.8643	−1.58581	−0.792907	−	0.609343i	$-0.791433\pi$
$911$	−47.8643	−1.58581	−0.792907	+	0.609343i	$0.791433\pi$
$912$	0	0
$913$	−11.5042	−0.380732
$914$	−20.1601	−0.666837
$915$	0	0
$916$	−7.49653	−0.247692
$917$	−0.412858	−0.0136338
$918$	0	0
$919$	35.7413	1.17900	0.589499	−	0.807769i	$-0.299325\pi$
$919$	35.7413	1.17900	0.589499	+	0.807769i	$0.299325\pi$
$920$	0	0
$921$	0	0
$922$	34.1368	1.12423
$923$	−21.3140	−0.701560
$924$	0	0
$925$	35.1485	1.15568
$926$	−11.6716	−0.383552
$927$	0	0
$928$	−1.98287	−0.0650911
$929$	22.9339	0.752435	0.376218	−	0.926531i	$-0.377224\pi$
$929$	22.9339	0.752435	0.376218	+	0.926531i	$0.377224\pi$
$930$	0	0
$931$	−1.78989	−0.0586613
$932$	−5.52307	−0.180914
$933$	0	0
$934$	−4.74513	−0.155265
$935$	−38.6479	−1.26392
$936$	0	0
$937$	19.6756	0.642773	0.321386	−	0.946948i	$-0.395851\pi$
$937$	19.6756	0.642773	0.321386	+	0.946948i	$0.395851\pi$
$938$	3.88497	0.126849
$939$	0	0
$940$	8.56727	0.279434
$941$	−52.2897	−1.70460	−0.852299	−	0.523056i	$-0.824792\pi$
$941$	−52.2897	−1.70460	−0.852299	+	0.523056i	$0.824792\pi$
$942$	0	0
$943$	0	0
$944$	−1.64927	−0.0536792
$945$	0	0
$946$	27.0795	0.880430
$947$	26.3714	0.856956	0.428478	−	0.903552i	$-0.359050\pi$
$947$	26.3714	0.856956	0.428478	+	0.903552i	$0.359050\pi$
$948$	0	0
$949$	13.9670	0.453389
$950$	0.945602	0.0306794
$951$	0	0
$952$	3.34928	0.108551
$953$	41.4107	1.34142	0.670711	−	0.741718i	$-0.265989\pi$
$953$	41.4107	1.34142	0.670711	+	0.741718i	$0.265989\pi$
$954$	0	0
$955$	11.6266	0.376228
$956$	−5.21901	−0.168795
$957$	0	0
$958$	29.7540	0.961309
$959$	7.85529	0.253661
$960$	0	0
$961$	12.6066	0.406663
$962$	−23.6850	−0.763634
$963$	0	0
$964$	1.55390	0.0500476
$965$	−31.4068	−1.01102
$966$	0	0
$967$	−24.8632	−0.799546	−0.399773	−	0.916614i	$-0.630911\pi$
$967$	−24.8632	−0.799546	−0.399773	+	0.916614i	$0.630911\pi$
$968$	−23.6964	−0.761632
$969$	0	0
$970$	−12.5136	−0.401787
$971$	−52.7969	−1.69433	−0.847166	−	0.531328i	$-0.821693\pi$
$971$	−52.7969	−1.69433	−0.847166	+	0.531328i	$0.821693\pi$
$972$	0	0
$973$	−10.7004	−0.343040
$974$	8.71408	0.279217
$975$	0	0
$976$	−9.13485	−0.292400
$977$	−13.1353	−0.420236	−0.210118	−	0.977676i	$-0.567385\pi$
$977$	−13.1353	−0.420236	−0.210118	+	0.977676i	$0.567385\pi$
$978$	0	0
$979$	27.0394	0.864184
$980$	8.11758	0.259307
$981$	0	0
$982$	15.0620	0.480647
$983$	−53.7978	−1.71588	−0.857942	−	0.513746i	$-0.828257\pi$
$983$	−53.7978	−1.71588	−0.857942	+	0.513746i	$0.828257\pi$
$984$	0	0
$985$	6.47472	0.206302
$986$	−10.5884	−0.337203
$987$	0	0
$988$	−0.637198	−0.0202720
$989$	0	0
$990$	0	0
$991$	−43.6587	−1.38686	−0.693432	−	0.720522i	$-0.743902\pi$
$991$	−43.6587	−1.38686	−0.693432	+	0.720522i	$0.743902\pi$
$992$	−6.60353	−0.209662
$993$	0	0
$994$	5.68402	0.180286
$995$	−30.7373	−0.974438
$996$	0	0
$997$	−53.3789	−1.69053	−0.845263	−	0.534350i	$-0.820556\pi$
$997$	−53.3789	−1.69053	−0.845263	+	0.534350i	$0.820556\pi$
$998$	−31.5993	−1.00026
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.bu.1.1		5
3.2	odd	2		1058.2.a.l.1.2		5
12.11	even	2		8464.2.a.bw.1.4		5
23.15	odd	22		414.2.i.f.271.1		10
23.20	odd	22		414.2.i.f.55.1		10
23.22	odd	2		9522.2.a.bp.1.5		5
69.20	even	22		46.2.c.a.9.1	✓	10
69.38	even	22		46.2.c.a.41.1	yes	10
69.68	even	2		1058.2.a.m.1.2		5
276.107	odd	22		368.2.m.b.225.1		10
276.227	odd	22		368.2.m.b.193.1		10
276.275	odd	2		8464.2.a.bx.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.2.c.a.9.1	✓	10	69.20	even	22
46.2.c.a.41.1	yes	10	69.38	even	22
368.2.m.b.193.1		10	276.227	odd	22
368.2.m.b.225.1		10	276.107	odd	22
414.2.i.f.55.1		10	23.20	odd	22
414.2.i.f.271.1		10	23.15	odd	22
1058.2.a.l.1.2		5	3.2	odd	2
1058.2.a.m.1.2		5	69.68	even	2
8464.2.a.bw.1.4		5	12.11	even	2
8464.2.a.bx.1.4		5	276.275	odd	2
9522.2.a.bp.1.5		5	23.22	odd	2
9522.2.a.bu.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.22871 q^{5} -0.627214 q^{7} -1.00000 q^{8} +1.22871 q^{10} +5.89037 q^{11} -2.35194 q^{13} +0.627214 q^{14} +1.00000 q^{16} +5.33992 q^{17} +0.270925 q^{19} -1.22871 q^{20} -5.89037 q^{22} -3.49028 q^{25} +2.35194 q^{26} -0.627214 q^{28} +1.98287 q^{29} +6.60353 q^{31} -1.00000 q^{32} -5.33992 q^{34} +0.770663 q^{35} -10.0704 q^{37} -0.270925 q^{38} +1.22871 q^{40} -10.8126 q^{41} -4.59725 q^{43} +5.89037 q^{44} -6.97259 q^{47} -6.60660 q^{49} +3.49028 q^{50} -2.35194 q^{52} -2.63694 q^{53} -7.23754 q^{55} +0.627214 q^{56} -1.98287 q^{58} -1.64927 q^{59} -9.13485 q^{61} -6.60353 q^{62} +1.00000 q^{64} +2.88984 q^{65} +6.19401 q^{67} +5.33992 q^{68} -0.770663 q^{70} +9.06233 q^{71} -5.93852 q^{73} +10.0704 q^{74} +0.270925 q^{76} -3.69452 q^{77} +6.00714 q^{79} -1.22871 q^{80} +10.8126 q^{82} -1.95305 q^{83} -6.56120 q^{85} +4.59725 q^{86} -5.89037 q^{88} +4.59045 q^{89} +1.47517 q^{91} +6.97259 q^{94} -0.332887 q^{95} -10.1844 q^{97} +6.60660 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} + 8 q^{5} - 7 q^{7} - 5 q^{8} - 8 q^{10} + 5 q^{11} - 7 q^{13} + 7 q^{14} + 5 q^{16} + 13 q^{17} - 12 q^{19} + 8 q^{20} - 5 q^{22} + q^{25} + 7 q^{26} - 7 q^{28} + 4 q^{29} + 6 q^{31}+ \cdots + 12 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 + 8 * q^5 - 7 * q^7 - 5 * q^8 - 8 * q^10 + 5 * q^11 - 7 * q^13 + 7 * q^14 + 5 * q^16 + 13 * q^17 - 12 * q^19 + 8 * q^20 - 5 * q^22 + q^25 + 7 * q^26 - 7 * q^28 + 4 * q^29 + 6 * q^31 - 5 * q^32 - 13 * q^34 - 9 * q^35 - 14 * q^37 + 12 * q^38 - 8 * q^40 - q^41 - 11 * q^43 + 5 * q^44 - 9 * q^47 - 12 * q^49 - q^50 - 7 * q^52 + 6 * q^53 - 14 * q^55 + 7 * q^56 - 4 * q^58 - 3 * q^59 - 13 * q^61 - 6 * q^62 + 5 * q^64 - 9 * q^65 - 10 * q^67 + 13 * q^68 + 9 * q^70 - 8 * q^71 - 4 * q^73 + 14 * q^74 - 12 * q^76 - 18 * q^77 + 18 * q^79 + 8 * q^80 + q^82 + 2 * q^83 + 23 * q^85 + 11 * q^86 - 5 * q^88 + 10 * q^89 + 12 * q^91 + 9 * q^94 - 28 * q^95 - 17 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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