LMFDB - Embedded newform 7569.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7569,2,Mod(1,7569)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7569, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7569.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7569 = 3^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7569.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$60.4387692899$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +2.82843 q^{7} -1.58579 q^{8} +0.414214 q^{10} +2.41421 q^{11} +1.82843 q^{13} +1.17157 q^{14} +3.00000 q^{16} -4.82843 q^{17} -6.00000 q^{19} -1.82843 q^{20} +1.00000 q^{22} +7.65685 q^{23} -4.00000 q^{25} +0.757359 q^{26} -5.17157 q^{28} +4.07107 q^{31} +4.41421 q^{32} -2.00000 q^{34} +2.82843 q^{35} +4.00000 q^{37} -2.48528 q^{38} -1.58579 q^{40} +12.4853 q^{41} -6.41421 q^{43} -4.41421 q^{44} +3.17157 q^{46} +5.24264 q^{47} +1.00000 q^{49} -1.65685 q^{50} -3.34315 q^{52} +7.48528 q^{53} +2.41421 q^{55} -4.48528 q^{56} -7.65685 q^{59} -0.828427 q^{61} +1.68629 q^{62} -4.17157 q^{64} +1.82843 q^{65} -5.65685 q^{67} +8.82843 q^{68} +1.17157 q^{70} +3.17157 q^{71} -4.00000 q^{73} +1.65685 q^{74} +10.9706 q^{76} +6.82843 q^{77} -0.414214 q^{79} +3.00000 q^{80} +5.17157 q^{82} +3.65685 q^{83} -4.82843 q^{85} -2.65685 q^{86} -3.82843 q^{88} +4.48528 q^{89} +5.17157 q^{91} -14.0000 q^{92} +2.17157 q^{94} -6.00000 q^{95} +12.4853 q^{97} +0.414214 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8} - 2 q^{10} + 2 q^{11} - 2 q^{13} + 8 q^{14} + 6 q^{16} - 4 q^{17} - 12 q^{19} + 2 q^{20} + 2 q^{22} + 4 q^{23} - 8 q^{25} + 10 q^{26} - 16 q^{28} - 6 q^{31} + 6 q^{32}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 6 * q^8 - 2 * q^10 + 2 * q^11 - 2 * q^13 + 8 * q^14 + 6 * q^16 - 4 * q^17 - 12 * q^19 + 2 * q^20 + 2 * q^22 + 4 * q^23 - 8 * q^25 + 10 * q^26 - 16 * q^28 - 6 * q^31 + 6 * q^32 - 4 * q^34 + 8 * q^37 + 12 * q^38 - 6 * q^40 + 8 * q^41 - 10 * q^43 - 6 * q^44 + 12 * q^46 + 2 * q^47 + 2 * q^49 + 8 * q^50 - 18 * q^52 - 2 * q^53 + 2 * q^55 + 8 * q^56 - 4 * q^59 + 4 * q^61 + 26 * q^62 - 14 * q^64 - 2 * q^65 + 12 * q^68 + 8 * q^70 + 12 * q^71 - 8 * q^73 - 8 * q^74 - 12 * q^76 + 8 * q^77 + 2 * q^79 + 6 * q^80 + 16 * q^82 - 4 * q^83 - 4 * q^85 + 6 * q^86 - 2 * q^88 - 8 * q^89 + 16 * q^91 - 28 * q^92 + 10 * q^94 - 12 * q^95 + 8 * q^97 - 2 * 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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	−1.58579	−0.560660
$9$	0	0
$10$	0.414214	0.130986
$11$	2.41421	0.727913	0.363956	−	0.931416i	$-0.381426\pi$
$11$	2.41421	0.727913	0.363956	+	0.931416i	$0.381426\pi$
$12$	0	0
$13$	1.82843	0.507114	0.253557	−	0.967320i	$-0.418399\pi$
$13$	1.82843	0.507114	0.253557	+	0.967320i	$0.418399\pi$
$14$	1.17157	0.313116
$15$	0	0
$16$	3.00000	0.750000
$17$	−4.82843	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$17$	−4.82843	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−1.82843	−0.408849
$21$	0	0
$22$	1.00000	0.213201
$23$	7.65685	1.59656	0.798282	−	0.602284i	$-0.205742\pi$
$23$	7.65685	1.59656	0.798282	+	0.602284i	$0.205742\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0.757359	0.148530
$27$	0	0
$28$	−5.17157	−0.977335
$29$	0	0
$29$	0	0
$30$	0	0
$31$	4.07107	0.731185	0.365593	−	0.930775i	$-0.380866\pi$
$31$	4.07107	0.731185	0.365593	+	0.930775i	$0.380866\pi$
$32$	4.41421	0.780330
$33$	0	0
$34$	−2.00000	−0.342997
$35$	2.82843	0.478091
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	−2.48528	−0.403166
$39$	0	0
$40$	−1.58579	−0.250735
$41$	12.4853	1.94987	0.974937	−	0.222483i	$-0.0714160\pi$
$41$	12.4853	1.94987	0.974937	+	0.222483i	$0.0714160\pi$
$42$	0	0
$43$	−6.41421	−0.978158	−0.489079	−	0.872239i	$-0.662667\pi$
$43$	−6.41421	−0.978158	−0.489079	+	0.872239i	$0.662667\pi$
$44$	−4.41421	−0.665468
$45$	0	0
$46$	3.17157	0.467623
$47$	5.24264	0.764718	0.382359	−	0.924014i	$-0.375112\pi$
$47$	5.24264	0.764718	0.382359	+	0.924014i	$0.375112\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.65685	−0.234315
$51$	0	0
$52$	−3.34315	−0.463611
$53$	7.48528	1.02818	0.514091	−	0.857736i	$-0.328129\pi$
$53$	7.48528	1.02818	0.514091	+	0.857736i	$0.328129\pi$
$54$	0	0
$55$	2.41421	0.325532
$56$	−4.48528	−0.599371
$57$	0	0
$58$	0	0
$59$	−7.65685	−0.996838	−0.498419	−	0.866936i	$-0.666086\pi$
$59$	−7.65685	−0.996838	−0.498419	+	0.866936i	$0.666086\pi$
$60$	0	0
$61$	−0.828427	−0.106069	−0.0530346	−	0.998593i	$-0.516889\pi$
$61$	−0.828427	−0.106069	−0.0530346	+	0.998593i	$0.516889\pi$
$62$	1.68629	0.214159
$63$	0	0
$64$	−4.17157	−0.521447
$65$	1.82843	0.226788
$66$	0	0
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	8.82843	1.07060
$69$	0	0
$70$	1.17157	0.140030
$71$	3.17157	0.376396	0.188198	−	0.982131i	$-0.439735\pi$
$71$	3.17157	0.376396	0.188198	+	0.982131i	$0.439735\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	1.65685	0.192605
$75$	0	0
$76$	10.9706	1.25841
$77$	6.82843	0.778171
$78$	0	0
$79$	−0.414214	−0.0466027	−0.0233013	−	0.999728i	$-0.507418\pi$
$79$	−0.414214	−0.0466027	−0.0233013	+	0.999728i	$0.507418\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	5.17157	0.571105
$83$	3.65685	0.401392	0.200696	−	0.979654i	$-0.435680\pi$
$83$	3.65685	0.401392	0.200696	+	0.979654i	$0.435680\pi$
$84$	0	0
$85$	−4.82843	−0.523716
$86$	−2.65685	−0.286496
$87$	0	0
$88$	−3.82843	−0.408112
$89$	4.48528	0.475439	0.237719	−	0.971334i	$-0.423600\pi$
$89$	4.48528	0.475439	0.237719	+	0.971334i	$0.423600\pi$
$90$	0	0
$91$	5.17157	0.542128
$92$	−14.0000	−1.45960
$93$	0	0
$94$	2.17157	0.223981
$95$	−6.00000	−0.615587
$96$	0	0
$97$	12.4853	1.26769	0.633844	−	0.773461i	$-0.281476\pi$
$97$	12.4853	1.26769	0.633844	+	0.773461i	$0.281476\pi$
$98$	0.414214	0.0418419
$99$	0	0
$100$	7.31371	0.731371
$101$	−13.6569	−1.35891	−0.679454	−	0.733718i	$-0.737783\pi$
$101$	−13.6569	−1.35891	−0.679454	+	0.733718i	$0.737783\pi$
$102$	0	0
$103$	0.828427	0.0816274	0.0408137	−	0.999167i	$-0.487005\pi$
$103$	0.828427	0.0816274	0.0408137	+	0.999167i	$0.487005\pi$
$104$	−2.89949	−0.284319
$105$	0	0
$106$	3.10051	0.301148
$107$	9.17157	0.886649	0.443325	−	0.896361i	$-0.353799\pi$
$107$	9.17157	0.886649	0.443325	+	0.896361i	$0.353799\pi$
$108$	0	0
$109$	1.34315	0.128650	0.0643250	−	0.997929i	$-0.479511\pi$
$109$	1.34315	0.128650	0.0643250	+	0.997929i	$0.479511\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	8.48528	0.801784
$113$	9.31371	0.876160	0.438080	−	0.898936i	$-0.355659\pi$
$113$	9.31371	0.876160	0.438080	+	0.898936i	$0.355659\pi$
$114$	0	0
$115$	7.65685	0.714005
$116$	0	0
$117$	0	0
$118$	−3.17157	−0.291967
$119$	−13.6569	−1.25192
$120$	0	0
$121$	−5.17157	−0.470143
$122$	−0.343146	−0.0310670
$123$	0	0
$124$	−7.44365	−0.668460
$125$	−9.00000	−0.804984
$126$	0	0
$127$	15.6569	1.38932	0.694661	−	0.719338i	$-0.255555\pi$
$127$	15.6569	1.38932	0.694661	+	0.719338i	$0.255555\pi$
$128$	−10.5563	−0.933058
$129$	0	0
$130$	0.757359	0.0664248
$131$	−1.31371	−0.114779	−0.0573896	−	0.998352i	$-0.518278\pi$
$131$	−1.31371	−0.114779	−0.0573896	+	0.998352i	$0.518278\pi$
$132$	0	0
$133$	−16.9706	−1.47153
$134$	−2.34315	−0.202417
$135$	0	0
$136$	7.65685	0.656570
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	−5.17157	−0.437078
$141$	0	0
$142$	1.31371	0.110244
$143$	4.41421	0.369135
$144$	0	0
$145$	0	0
$146$	−1.65685	−0.137122
$147$	0	0
$148$	−7.31371	−0.601183
$149$	2.17157	0.177902	0.0889511	−	0.996036i	$-0.471649\pi$
$149$	2.17157	0.177902	0.0889511	+	0.996036i	$0.471649\pi$
$150$	0	0
$151$	14.1421	1.15087	0.575435	−	0.817847i	$-0.304833\pi$
$151$	14.1421	1.15087	0.575435	+	0.817847i	$0.304833\pi$
$152$	9.51472	0.771746
$153$	0	0
$154$	2.82843	0.227921
$155$	4.07107	0.326996
$156$	0	0
$157$	8.48528	0.677199	0.338600	−	0.940931i	$-0.390047\pi$
$157$	8.48528	0.677199	0.338600	+	0.940931i	$0.390047\pi$
$158$	−0.171573	−0.0136496
$159$	0	0
$160$	4.41421	0.348974
$161$	21.6569	1.70680
$162$	0	0
$163$	−18.0711	−1.41544	−0.707718	−	0.706495i	$-0.750275\pi$
$163$	−18.0711	−1.41544	−0.707718	+	0.706495i	$0.750275\pi$
$164$	−22.8284	−1.78260
$165$	0	0
$166$	1.51472	0.117565
$167$	8.82843	0.683164	0.341582	−	0.939852i	$-0.389037\pi$
$167$	8.82843	0.683164	0.341582	+	0.939852i	$0.389037\pi$
$168$	0	0
$169$	−9.65685	−0.742835
$170$	−2.00000	−0.153393
$171$	0	0
$172$	11.7279	0.894246
$173$	−23.6569	−1.79860	−0.899299	−	0.437335i	$-0.855922\pi$
$173$	−23.6569	−1.79860	−0.899299	+	0.437335i	$0.855922\pi$
$174$	0	0
$175$	−11.3137	−0.855236
$176$	7.24264	0.545935
$177$	0	0
$178$	1.85786	0.139253
$179$	−10.4853	−0.783707	−0.391853	−	0.920028i	$-0.628166\pi$
$179$	−10.4853	−0.783707	−0.391853	+	0.920028i	$0.628166\pi$
$180$	0	0
$181$	−14.3137	−1.06393	−0.531965	−	0.846766i	$-0.678546\pi$
$181$	−14.3137	−1.06393	−0.531965	+	0.846766i	$0.678546\pi$
$182$	2.14214	0.158786
$183$	0	0
$184$	−12.1421	−0.895130
$185$	4.00000	0.294086
$186$	0	0
$187$	−11.6569	−0.852434
$188$	−9.58579	−0.699115
$189$	0	0
$190$	−2.48528	−0.180301
$191$	2.68629	0.194373	0.0971866	−	0.995266i	$-0.469016\pi$
$191$	2.68629	0.194373	0.0971866	+	0.995266i	$0.469016\pi$
$192$	0	0
$193$	10.8284	0.779447	0.389724	−	0.920932i	$-0.372571\pi$
$193$	10.8284	0.779447	0.389724	+	0.920932i	$0.372571\pi$
$194$	5.17157	0.371297
$195$	0	0
$196$	−1.82843	−0.130602
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	16.4853	1.16861	0.584305	−	0.811534i	$-0.301367\pi$
$199$	16.4853	1.16861	0.584305	+	0.811534i	$0.301367\pi$
$200$	6.34315	0.448528
$201$	0	0
$202$	−5.65685	−0.398015
$203$	0	0
$204$	0	0
$205$	12.4853	0.872010
$206$	0.343146	0.0239081
$207$	0	0
$208$	5.48528	0.380336
$209$	−14.4853	−1.00197
$210$	0	0
$211$	−17.3848	−1.19682	−0.598409	−	0.801191i	$-0.704200\pi$
$211$	−17.3848	−1.19682	−0.598409	+	0.801191i	$0.704200\pi$
$212$	−13.6863	−0.939978
$213$	0	0
$214$	3.79899	0.259694
$215$	−6.41421	−0.437446
$216$	0	0
$217$	11.5147	0.781670
$218$	0.556349	0.0376807
$219$	0	0
$220$	−4.41421	−0.297606
$221$	−8.82843	−0.593864
$222$	0	0
$223$	−8.82843	−0.591195	−0.295598	−	0.955313i	$-0.595519\pi$
$223$	−8.82843	−0.591195	−0.295598	+	0.955313i	$0.595519\pi$
$224$	12.4853	0.834208
$225$	0	0
$226$	3.85786	0.256621
$227$	−20.1421	−1.33688	−0.668440	−	0.743766i	$-0.733038\pi$
$227$	−20.1421	−1.33688	−0.668440	+	0.743766i	$0.733038\pi$
$228$	0	0
$229$	20.4853	1.35371	0.676853	−	0.736118i	$-0.263343\pi$
$229$	20.4853	1.35371	0.676853	+	0.736118i	$0.263343\pi$
$230$	3.17157	0.209127
$231$	0	0
$232$	0	0
$233$	4.31371	0.282600	0.141300	−	0.989967i	$-0.454872\pi$
$233$	4.31371	0.282600	0.141300	+	0.989967i	$0.454872\pi$
$234$	0	0
$235$	5.24264	0.341992
$236$	14.0000	0.911322
$237$	0	0
$238$	−5.65685	−0.366679
$239$	8.34315	0.539673	0.269837	−	0.962906i	$-0.413030\pi$
$239$	8.34315	0.539673	0.269837	+	0.962906i	$0.413030\pi$
$240$	0	0
$241$	4.31371	0.277870	0.138935	−	0.990301i	$-0.455632\pi$
$241$	4.31371	0.277870	0.138935	+	0.990301i	$0.455632\pi$
$242$	−2.14214	−0.137702
$243$	0	0
$244$	1.51472	0.0969699
$245$	1.00000	0.0638877
$246$	0	0
$247$	−10.9706	−0.698040
$248$	−6.45584	−0.409947
$249$	0	0
$250$	−3.72792	−0.235774
$251$	5.92893	0.374231	0.187115	−	0.982338i	$-0.440086\pi$
$251$	5.92893	0.374231	0.187115	+	0.982338i	$0.440086\pi$
$252$	0	0
$253$	18.4853	1.16216
$254$	6.48528	0.406923
$255$	0	0
$256$	3.97056	0.248160
$257$	23.8284	1.48638	0.743188	−	0.669082i	$-0.233313\pi$
$257$	23.8284	1.48638	0.743188	+	0.669082i	$0.233313\pi$
$258$	0	0
$259$	11.3137	0.703000
$260$	−3.34315	−0.207333
$261$	0	0
$262$	−0.544156	−0.0336181
$263$	11.2426	0.693251	0.346625	−	0.938004i	$-0.387327\pi$
$263$	11.2426	0.693251	0.346625	+	0.938004i	$0.387327\pi$
$264$	0	0
$265$	7.48528	0.459817
$266$	−7.02944	−0.431002
$267$	0	0
$268$	10.3431	0.631808
$269$	−19.4558	−1.18624	−0.593122	−	0.805113i	$-0.702105\pi$
$269$	−19.4558	−1.18624	−0.593122	+	0.805113i	$0.702105\pi$
$270$	0	0
$271$	14.5563	0.884235	0.442118	−	0.896957i	$-0.354227\pi$
$271$	14.5563	0.884235	0.442118	+	0.896957i	$0.354227\pi$
$272$	−14.4853	−0.878299
$273$	0	0
$274$	4.97056	0.300283
$275$	−9.65685	−0.582330
$276$	0	0
$277$	5.31371	0.319270	0.159635	−	0.987176i	$-0.448968\pi$
$277$	5.31371	0.319270	0.159635	+	0.987176i	$0.448968\pi$
$278$	5.79899	0.347800
$279$	0	0
$280$	−4.48528	−0.268047
$281$	1.97056	0.117554	0.0587770	−	0.998271i	$-0.481280\pi$
$281$	1.97056	0.117554	0.0587770	+	0.998271i	$0.481280\pi$
$282$	0	0
$283$	0.343146	0.0203979	0.0101989	−	0.999948i	$-0.496754\pi$
$283$	0.343146	0.0203979	0.0101989	+	0.999948i	$0.496754\pi$
$284$	−5.79899	−0.344107
$285$	0	0
$286$	1.82843	0.108117
$287$	35.3137	2.08450
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	0	0
$292$	7.31371	0.428002
$293$	−3.65685	−0.213636	−0.106818	−	0.994279i	$-0.534066\pi$
$293$	−3.65685	−0.213636	−0.106818	+	0.994279i	$0.534066\pi$
$294$	0	0
$295$	−7.65685	−0.445799
$296$	−6.34315	−0.368688
$297$	0	0
$298$	0.899495	0.0521063
$299$	14.0000	0.809641
$300$	0	0
$301$	−18.1421	−1.04570
$302$	5.85786	0.337082
$303$	0	0
$304$	−18.0000	−1.03237
$305$	−0.828427	−0.0474356
$306$	0	0
$307$	16.8995	0.964505	0.482253	−	0.876032i	$-0.339819\pi$
$307$	16.8995	0.964505	0.482253	+	0.876032i	$0.339819\pi$
$308$	−12.4853	−0.711415
$309$	0	0
$310$	1.68629	0.0957749
$311$	25.3137	1.43541	0.717704	−	0.696348i	$-0.245193\pi$
$311$	25.3137	1.43541	0.717704	+	0.696348i	$0.245193\pi$
$312$	0	0
$313$	4.17157	0.235791	0.117896	−	0.993026i	$-0.462385\pi$
$313$	4.17157	0.235791	0.117896	+	0.993026i	$0.462385\pi$
$314$	3.51472	0.198347
$315$	0	0
$316$	0.757359	0.0426048
$317$	19.4558	1.09275	0.546375	−	0.837541i	$-0.316008\pi$
$317$	19.4558	1.09275	0.546375	+	0.837541i	$0.316008\pi$
$318$	0	0
$319$	0	0
$320$	−4.17157	−0.233198
$321$	0	0
$322$	8.97056	0.499910
$323$	28.9706	1.61197
$324$	0	0
$325$	−7.31371	−0.405692
$326$	−7.48528	−0.414571
$327$	0	0
$328$	−19.7990	−1.09322
$329$	14.8284	0.817518
$330$	0	0
$331$	−0.414214	−0.0227672	−0.0113836	−	0.999935i	$-0.503624\pi$
$331$	−0.414214	−0.0227672	−0.0113836	+	0.999935i	$0.503624\pi$
$332$	−6.68629	−0.366958
$333$	0	0
$334$	3.65685	0.200094
$335$	−5.65685	−0.309067
$336$	0	0
$337$	17.7990	0.969573	0.484786	−	0.874633i	$-0.338897\pi$
$337$	17.7990	0.969573	0.484786	+	0.874633i	$0.338897\pi$
$338$	−4.00000	−0.217571
$339$	0	0
$340$	8.82843	0.478789
$341$	9.82843	0.532239
$342$	0	0
$343$	−16.9706	−0.916324
$344$	10.1716	0.548414
$345$	0	0
$346$	−9.79899	−0.526797
$347$	14.4853	0.777611	0.388805	−	0.921320i	$-0.372888\pi$
$347$	14.4853	0.777611	0.388805	+	0.921320i	$0.372888\pi$
$348$	0	0
$349$	23.1421	1.23877	0.619385	−	0.785087i	$-0.287382\pi$
$349$	23.1421	1.23877	0.619385	+	0.785087i	$0.287382\pi$
$350$	−4.68629	−0.250493
$351$	0	0
$352$	10.6569	0.568012
$353$	6.97056	0.371006	0.185503	−	0.982644i	$-0.440609\pi$
$353$	6.97056	0.371006	0.185503	+	0.982644i	$0.440609\pi$
$354$	0	0
$355$	3.17157	0.168330
$356$	−8.20101	−0.434653
$357$	0	0
$358$	−4.34315	−0.229542
$359$	18.0711	0.953754	0.476877	−	0.878970i	$-0.341769\pi$
$359$	18.0711	0.953754	0.476877	+	0.878970i	$0.341769\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−5.92893	−0.311618
$363$	0	0
$364$	−9.45584	−0.495621
$365$	−4.00000	−0.209370
$366$	0	0
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	22.9706	1.19742
$369$	0	0
$370$	1.65685	0.0861358
$371$	21.1716	1.09917
$372$	0	0
$373$	−3.68629	−0.190869	−0.0954345	−	0.995436i	$-0.530424\pi$
$373$	−3.68629	−0.190869	−0.0954345	+	0.995436i	$0.530424\pi$
$374$	−4.82843	−0.249672
$375$	0	0
$376$	−8.31371	−0.428747
$377$	0	0
$378$	0	0
$379$	−26.9706	−1.38538	−0.692692	−	0.721233i	$-0.743576\pi$
$379$	−26.9706	−1.38538	−0.692692	+	0.721233i	$0.743576\pi$
$380$	10.9706	0.562778
$381$	0	0
$382$	1.11270	0.0569306
$383$	20.4853	1.04675	0.523374	−	0.852103i	$-0.324673\pi$
$383$	20.4853	1.04675	0.523374	+	0.852103i	$0.324673\pi$
$384$	0	0
$385$	6.82843	0.348009
$386$	4.48528	0.228295
$387$	0	0
$388$	−22.8284	−1.15894
$389$	36.9706	1.87448	0.937241	−	0.348682i	$-0.113371\pi$
$389$	36.9706	1.87448	0.937241	+	0.348682i	$0.113371\pi$
$390$	0	0
$391$	−36.9706	−1.86968
$392$	−1.58579	−0.0800943
$393$	0	0
$394$	−0.828427	−0.0417356
$395$	−0.414214	−0.0208413
$396$	0	0
$397$	30.6569	1.53862	0.769312	−	0.638874i	$-0.220599\pi$
$397$	30.6569	1.53862	0.769312	+	0.638874i	$0.220599\pi$
$398$	6.82843	0.342278
$399$	0	0
$400$	−12.0000	−0.600000
$401$	7.34315	0.366699	0.183350	−	0.983048i	$-0.441306\pi$
$401$	7.34315	0.366699	0.183350	+	0.983048i	$0.441306\pi$
$402$	0	0
$403$	7.44365	0.370795
$404$	24.9706	1.24233
$405$	0	0
$406$	0	0
$407$	9.65685	0.478672
$408$	0	0
$409$	−14.9706	−0.740247	−0.370123	−	0.928983i	$-0.620685\pi$
$409$	−14.9706	−0.740247	−0.370123	+	0.928983i	$0.620685\pi$
$410$	5.17157	0.255406
$411$	0	0
$412$	−1.51472	−0.0746248
$413$	−21.6569	−1.06566
$414$	0	0
$415$	3.65685	0.179508
$416$	8.07107	0.395717
$417$	0	0
$418$	−6.00000	−0.293470
$419$	26.4853	1.29389	0.646945	−	0.762536i	$-0.276046\pi$
$419$	26.4853	1.29389	0.646945	+	0.762536i	$0.276046\pi$
$420$	0	0
$421$	25.1127	1.22392	0.611959	−	0.790889i	$-0.290382\pi$
$421$	25.1127	1.22392	0.611959	+	0.790889i	$0.290382\pi$
$422$	−7.20101	−0.350540
$423$	0	0
$424$	−11.8701	−0.576461
$425$	19.3137	0.936852
$426$	0	0
$427$	−2.34315	−0.113393
$428$	−16.7696	−0.810587
$429$	0	0
$430$	−2.65685	−0.128125
$431$	−8.34315	−0.401875	−0.200938	−	0.979604i	$-0.564399\pi$
$431$	−8.34315	−0.401875	−0.200938	+	0.979604i	$0.564399\pi$
$432$	0	0
$433$	14.6274	0.702949	0.351474	−	0.936197i	$-0.385680\pi$
$433$	14.6274	0.702949	0.351474	+	0.936197i	$0.385680\pi$
$434$	4.76955	0.228946
$435$	0	0
$436$	−2.45584	−0.117614
$437$	−45.9411	−2.19766
$438$	0	0
$439$	−11.6569	−0.556351	−0.278176	−	0.960530i	$-0.589730\pi$
$439$	−11.6569	−0.556351	−0.278176	+	0.960530i	$0.589730\pi$
$440$	−3.82843	−0.182513
$441$	0	0
$442$	−3.65685	−0.173939
$443$	−35.6569	−1.69411	−0.847054	−	0.531507i	$-0.821626\pi$
$443$	−35.6569	−1.69411	−0.847054	+	0.531507i	$0.821626\pi$
$444$	0	0
$445$	4.48528	0.212623
$446$	−3.65685	−0.173157
$447$	0	0
$448$	−11.7990	−0.557450
$449$	−1.02944	−0.0485821	−0.0242911	−	0.999705i	$-0.507733\pi$
$449$	−1.02944	−0.0485821	−0.0242911	+	0.999705i	$0.507733\pi$
$450$	0	0
$451$	30.1421	1.41934
$452$	−17.0294	−0.800997
$453$	0	0
$454$	−8.34315	−0.391563
$455$	5.17157	0.242447
$456$	0	0
$457$	34.9706	1.63585	0.817927	−	0.575322i	$-0.195123\pi$
$457$	34.9706	1.63585	0.817927	+	0.575322i	$0.195123\pi$
$458$	8.48528	0.396491
$459$	0	0
$460$	−14.0000	−0.652753
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	0	0
$465$	0	0
$466$	1.78680	0.0827718
$467$	32.3553	1.49723	0.748613	−	0.663007i	$-0.230720\pi$
$467$	32.3553	1.49723	0.748613	+	0.663007i	$0.230720\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	2.17157	0.100167
$471$	0	0
$472$	12.1421	0.558887
$473$	−15.4853	−0.712014
$474$	0	0
$475$	24.0000	1.10120
$476$	24.9706	1.14452
$477$	0	0
$478$	3.45584	0.158067
$479$	−12.8995	−0.589393	−0.294696	−	0.955591i	$-0.595219\pi$
$479$	−12.8995	−0.589393	−0.294696	+	0.955591i	$0.595219\pi$
$480$	0	0
$481$	7.31371	0.333476
$482$	1.78680	0.0813864
$483$	0	0
$484$	9.45584	0.429811
$485$	12.4853	0.566927
$486$	0	0
$487$	−28.4853	−1.29079	−0.645396	−	0.763848i	$-0.723307\pi$
$487$	−28.4853	−1.29079	−0.645396	+	0.763848i	$0.723307\pi$
$488$	1.31371	0.0594688
$489$	0	0
$490$	0.414214	0.0187123
$491$	−12.7574	−0.575732	−0.287866	−	0.957671i	$-0.592946\pi$
$491$	−12.7574	−0.575732	−0.287866	+	0.957671i	$0.592946\pi$
$492$	0	0
$493$	0	0
$494$	−4.54416	−0.204451
$495$	0	0
$496$	12.2132	0.548389
$497$	8.97056	0.402385
$498$	0	0
$499$	−14.9706	−0.670174	−0.335087	−	0.942187i	$-0.608766\pi$
$499$	−14.9706	−0.670174	−0.335087	+	0.942187i	$0.608766\pi$
$500$	16.4558	0.735928
$501$	0	0
$502$	2.45584	0.109610
$503$	25.7279	1.14715	0.573576	−	0.819153i	$-0.305556\pi$
$503$	25.7279	1.14715	0.573576	+	0.819153i	$0.305556\pi$
$504$	0	0
$505$	−13.6569	−0.607722
$506$	7.65685	0.340389
$507$	0	0
$508$	−28.6274	−1.27014
$509$	27.4853	1.21826	0.609132	−	0.793069i	$-0.291518\pi$
$509$	27.4853	1.21826	0.609132	+	0.793069i	$0.291518\pi$
$510$	0	0
$511$	−11.3137	−0.500489
$512$	22.7574	1.00574
$513$	0	0
$514$	9.87006	0.435350
$515$	0.828427	0.0365049
$516$	0	0
$517$	12.6569	0.556648
$518$	4.68629	0.205904
$519$	0	0
$520$	−2.89949	−0.127151
$521$	0.857864	0.0375837	0.0187919	−	0.999823i	$-0.494018\pi$
$521$	0.857864	0.0375837	0.0187919	+	0.999823i	$0.494018\pi$
$522$	0	0
$523$	27.3137	1.19435	0.597173	−	0.802113i	$-0.296291\pi$
$523$	27.3137	1.19435	0.597173	+	0.802113i	$0.296291\pi$
$524$	2.40202	0.104933
$525$	0	0
$526$	4.65685	0.203048
$527$	−19.6569	−0.856266
$528$	0	0
$529$	35.6274	1.54902
$530$	3.10051	0.134677
$531$	0	0
$532$	31.0294	1.34530
$533$	22.8284	0.988809
$534$	0	0
$535$	9.17157	0.396522
$536$	8.97056	0.387469
$537$	0	0
$538$	−8.05887	−0.347443
$539$	2.41421	0.103988
$540$	0	0
$541$	21.6569	0.931101	0.465550	−	0.885021i	$-0.345856\pi$
$541$	21.6569	0.931101	0.465550	+	0.885021i	$0.345856\pi$
$542$	6.02944	0.258987
$543$	0	0
$544$	−21.3137	−0.913818
$545$	1.34315	0.0575340
$546$	0	0
$547$	−3.79899	−0.162433	−0.0812165	−	0.996696i	$-0.525881\pi$
$547$	−3.79899	−0.162433	−0.0812165	+	0.996696i	$0.525881\pi$
$548$	−21.9411	−0.937278
$549$	0	0
$550$	−4.00000	−0.170561
$551$	0	0
$552$	0	0
$553$	−1.17157	−0.0498203
$554$	2.20101	0.0935120
$555$	0	0
$556$	−25.5980	−1.08560
$557$	−5.31371	−0.225149	−0.112575	−	0.993643i	$-0.535910\pi$
$557$	−5.31371	−0.225149	−0.112575	+	0.993643i	$0.535910\pi$
$558$	0	0
$559$	−11.7279	−0.496038
$560$	8.48528	0.358569
$561$	0	0
$562$	0.816234	0.0344307
$563$	−9.24264	−0.389531	−0.194765	−	0.980850i	$-0.562395\pi$
$563$	−9.24264	−0.389531	−0.194765	+	0.980850i	$0.562395\pi$
$564$	0	0
$565$	9.31371	0.391831
$566$	0.142136	0.00597441
$567$	0	0
$568$	−5.02944	−0.211030
$569$	−28.3431	−1.18821	−0.594103	−	0.804389i	$-0.702493\pi$
$569$	−28.3431	−1.18821	−0.594103	+	0.804389i	$0.702493\pi$
$570$	0	0
$571$	−30.6274	−1.28172	−0.640859	−	0.767659i	$-0.721422\pi$
$571$	−30.6274	−1.28172	−0.640859	+	0.767659i	$0.721422\pi$
$572$	−8.07107	−0.337468
$573$	0	0
$574$	14.6274	0.610537
$575$	−30.6274	−1.27725
$576$	0	0
$577$	−9.79899	−0.407937	−0.203969	−	0.978977i	$-0.565384\pi$
$577$	−9.79899	−0.407937	−0.203969	+	0.978977i	$0.565384\pi$
$578$	2.61522	0.108779
$579$	0	0
$580$	0	0
$581$	10.3431	0.429106
$582$	0	0
$583$	18.0711	0.748427
$584$	6.34315	0.262481
$585$	0	0
$586$	−1.51472	−0.0625724
$587$	3.65685	0.150935	0.0754673	−	0.997148i	$-0.475955\pi$
$587$	3.65685	0.150935	0.0754673	+	0.997148i	$0.475955\pi$
$588$	0	0
$589$	−24.4264	−1.00647
$590$	−3.17157	−0.130572
$591$	0	0
$592$	12.0000	0.493197
$593$	2.51472	0.103267	0.0516336	−	0.998666i	$-0.483557\pi$
$593$	2.51472	0.103267	0.0516336	+	0.998666i	$0.483557\pi$
$594$	0	0
$595$	−13.6569	−0.559876
$596$	−3.97056	−0.162641
$597$	0	0
$598$	5.79899	0.237138
$599$	−43.8701	−1.79248	−0.896241	−	0.443567i	$-0.853713\pi$
$599$	−43.8701	−1.79248	−0.896241	+	0.443567i	$0.853713\pi$
$600$	0	0
$601$	22.8284	0.931191	0.465595	−	0.884998i	$-0.345840\pi$
$601$	22.8284	0.931191	0.465595	+	0.884998i	$0.345840\pi$
$602$	−7.51472	−0.306277
$603$	0	0
$604$	−25.8579	−1.05214
$605$	−5.17157	−0.210254
$606$	0	0
$607$	−17.7279	−0.719554	−0.359777	−	0.933038i	$-0.617147\pi$
$607$	−17.7279	−0.719554	−0.359777	+	0.933038i	$0.617147\pi$
$608$	−26.4853	−1.07412
$609$	0	0
$610$	−0.343146	−0.0138936
$611$	9.58579	0.387799
$612$	0	0
$613$	−9.00000	−0.363507	−0.181753	−	0.983344i	$-0.558177\pi$
$613$	−9.00000	−0.363507	−0.181753	+	0.983344i	$0.558177\pi$
$614$	7.00000	0.282497
$615$	0	0
$616$	−10.8284	−0.436290
$617$	23.3137	0.938575	0.469287	−	0.883046i	$-0.344511\pi$
$617$	23.3137	0.938575	0.469287	+	0.883046i	$0.344511\pi$
$618$	0	0
$619$	−36.4142	−1.46361	−0.731805	−	0.681514i	$-0.761322\pi$
$619$	−36.4142	−1.46361	−0.731805	+	0.681514i	$0.761322\pi$
$620$	−7.44365	−0.298944
$621$	0	0
$622$	10.4853	0.420421
$623$	12.6863	0.508266
$624$	0	0
$625$	11.0000	0.440000
$626$	1.72792	0.0690617
$627$	0	0
$628$	−15.5147	−0.619105
$629$	−19.3137	−0.770088
$630$	0	0
$631$	−31.1716	−1.24092	−0.620460	−	0.784238i	$-0.713054\pi$
$631$	−31.1716	−1.24092	−0.620460	+	0.784238i	$0.713054\pi$
$632$	0.656854	0.0261283
$633$	0	0
$634$	8.05887	0.320059
$635$	15.6569	0.621323
$636$	0	0
$637$	1.82843	0.0724449
$638$	0	0
$639$	0	0
$640$	−10.5563	−0.417276
$641$	−21.7990	−0.861008	−0.430504	−	0.902589i	$-0.641664\pi$
$641$	−21.7990	−0.861008	−0.430504	+	0.902589i	$0.641664\pi$
$642$	0	0
$643$	15.5147	0.611841	0.305920	−	0.952057i	$-0.401036\pi$
$643$	15.5147	0.611841	0.305920	+	0.952057i	$0.401036\pi$
$644$	−39.5980	−1.56038
$645$	0	0
$646$	12.0000	0.472134
$647$	−28.3431	−1.11428	−0.557142	−	0.830417i	$-0.688102\pi$
$647$	−28.3431	−1.11428	−0.557142	+	0.830417i	$0.688102\pi$
$648$	0	0
$649$	−18.4853	−0.725611
$650$	−3.02944	−0.118824
$651$	0	0
$652$	33.0416	1.29401
$653$	−1.85786	−0.0727039	−0.0363519	−	0.999339i	$-0.511574\pi$
$653$	−1.85786	−0.0727039	−0.0363519	+	0.999339i	$0.511574\pi$
$654$	0	0
$655$	−1.31371	−0.0513308
$656$	37.4558	1.46241
$657$	0	0
$658$	6.14214	0.239445
$659$	11.5858	0.451318	0.225659	−	0.974206i	$-0.427546\pi$
$659$	11.5858	0.451318	0.225659	+	0.974206i	$0.427546\pi$
$660$	0	0
$661$	10.6863	0.415649	0.207824	−	0.978166i	$-0.433362\pi$
$661$	10.6863	0.415649	0.207824	+	0.978166i	$0.433362\pi$
$662$	−0.171573	−0.00666837
$663$	0	0
$664$	−5.79899	−0.225044
$665$	−16.9706	−0.658090
$666$	0	0
$667$	0	0
$668$	−16.1421	−0.624558
$669$	0	0
$670$	−2.34315	−0.0905236
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	23.6274	0.910770	0.455385	−	0.890295i	$-0.349502\pi$
$673$	23.6274	0.910770	0.455385	+	0.890295i	$0.349502\pi$
$674$	7.37258	0.283981
$675$	0	0
$676$	17.6569	0.679110
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	0	0
$679$	35.3137	1.35522
$680$	7.65685	0.293627
$681$	0	0
$682$	4.07107	0.155889
$683$	12.9706	0.496305	0.248152	−	0.968721i	$-0.420177\pi$
$683$	12.9706	0.496305	0.248152	+	0.968721i	$0.420177\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	−7.02944	−0.268385
$687$	0	0
$688$	−19.2426	−0.733619
$689$	13.6863	0.521406
$690$	0	0
$691$	48.0000	1.82601	0.913003	−	0.407953i	$-0.133757\pi$
$691$	48.0000	1.82601	0.913003	+	0.407953i	$0.133757\pi$
$692$	43.2548	1.64430
$693$	0	0
$694$	6.00000	0.227757
$695$	14.0000	0.531050
$696$	0	0
$697$	−60.2843	−2.28343
$698$	9.58579	0.362827
$699$	0	0
$700$	20.6863	0.781868
$701$	−22.1127	−0.835185	−0.417593	−	0.908634i	$-0.637126\pi$
$701$	−22.1127	−0.835185	−0.417593	+	0.908634i	$0.637126\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	−10.0711	−0.379568
$705$	0	0
$706$	2.88730	0.108665
$707$	−38.6274	−1.45273
$708$	0	0
$709$	0.857864	0.0322178	0.0161089	−	0.999870i	$-0.494872\pi$
$709$	0.857864	0.0322178	0.0161089	+	0.999870i	$0.494872\pi$
$710$	1.31371	0.0493026
$711$	0	0
$712$	−7.11270	−0.266560
$713$	31.1716	1.16738
$714$	0	0
$715$	4.41421	0.165082
$716$	19.1716	0.716475
$717$	0	0
$718$	7.48528	0.279348
$719$	−8.14214	−0.303650	−0.151825	−	0.988407i	$-0.548515\pi$
$719$	−8.14214	−0.303650	−0.151825	+	0.988407i	$0.548515\pi$
$720$	0	0
$721$	2.34315	0.0872633
$722$	7.04163	0.262062
$723$	0	0
$724$	26.1716	0.972659
$725$	0	0
$726$	0	0
$727$	21.3137	0.790482	0.395241	−	0.918578i	$-0.370661\pi$
$727$	21.3137	0.790482	0.395241	+	0.918578i	$0.370661\pi$
$728$	−8.20101	−0.303950
$729$	0	0
$730$	−1.65685	−0.0613229
$731$	30.9706	1.14549
$732$	0	0
$733$	−49.2548	−1.81927	−0.909634	−	0.415410i	$-0.863638\pi$
$733$	−49.2548	−1.81927	−0.909634	+	0.415410i	$0.863638\pi$
$734$	−7.45584	−0.275200
$735$	0	0
$736$	33.7990	1.24585
$737$	−13.6569	−0.503057
$738$	0	0
$739$	10.0711	0.370470	0.185235	−	0.982694i	$-0.440695\pi$
$739$	10.0711	0.370470	0.185235	+	0.982694i	$0.440695\pi$
$740$	−7.31371	−0.268857
$741$	0	0
$742$	8.76955	0.321940
$743$	12.3431	0.452826	0.226413	−	0.974031i	$-0.427300\pi$
$743$	12.3431	0.452826	0.226413	+	0.974031i	$0.427300\pi$
$744$	0	0
$745$	2.17157	0.0795603
$746$	−1.52691	−0.0559042
$747$	0	0
$748$	21.3137	0.779306
$749$	25.9411	0.947868
$750$	0	0
$751$	−2.68629	−0.0980242	−0.0490121	−	0.998798i	$-0.515607\pi$
$751$	−2.68629	−0.0980242	−0.0490121	+	0.998798i	$0.515607\pi$
$752$	15.7279	0.573538
$753$	0	0
$754$	0	0
$755$	14.1421	0.514685
$756$	0	0
$757$	−42.4853	−1.54415	−0.772077	−	0.635529i	$-0.780782\pi$
$757$	−42.4853	−1.54415	−0.772077	+	0.635529i	$0.780782\pi$
$758$	−11.1716	−0.405770
$759$	0	0
$760$	9.51472	0.345135
$761$	33.5980	1.21793	0.608963	−	0.793199i	$-0.291586\pi$
$761$	33.5980	1.21793	0.608963	+	0.793199i	$0.291586\pi$
$762$	0	0
$763$	3.79899	0.137533
$764$	−4.91169	−0.177699
$765$	0	0
$766$	8.48528	0.306586
$767$	−14.0000	−0.505511
$768$	0	0
$769$	−13.1127	−0.472856	−0.236428	−	0.971649i	$-0.575977\pi$
$769$	−13.1127	−0.472856	−0.236428	+	0.971649i	$0.575977\pi$
$770$	2.82843	0.101929
$771$	0	0
$772$	−19.7990	−0.712581
$773$	−36.4853	−1.31228	−0.656142	−	0.754637i	$-0.727813\pi$
$773$	−36.4853	−1.31228	−0.656142	+	0.754637i	$0.727813\pi$
$774$	0	0
$775$	−16.2843	−0.584948
$776$	−19.7990	−0.710742
$777$	0	0
$778$	15.3137	0.549023
$779$	−74.9117	−2.68399
$780$	0	0
$781$	7.65685	0.273984
$782$	−15.3137	−0.547617
$783$	0	0
$784$	3.00000	0.107143
$785$	8.48528	0.302853
$786$	0	0
$787$	42.0833	1.50011	0.750053	−	0.661378i	$-0.230028\pi$
$787$	42.0833	1.50011	0.750053	+	0.661378i	$0.230028\pi$
$788$	3.65685	0.130270
$789$	0	0
$790$	−0.171573	−0.00610429
$791$	26.3431	0.936654
$792$	0	0
$793$	−1.51472	−0.0537892
$794$	12.6985	0.450652
$795$	0	0
$796$	−30.1421	−1.06836
$797$	−55.7401	−1.97442	−0.987208	−	0.159437i	$-0.949032\pi$
$797$	−55.7401	−1.97442	−0.987208	+	0.159437i	$0.949032\pi$
$798$	0	0
$799$	−25.3137	−0.895535
$800$	−17.6569	−0.624264
$801$	0	0
$802$	3.04163	0.107404
$803$	−9.65685	−0.340783
$804$	0	0
$805$	21.6569	0.763304
$806$	3.08326	0.108603
$807$	0	0
$808$	21.6569	0.761885
$809$	−20.2843	−0.713157	−0.356578	−	0.934265i	$-0.616057\pi$
$809$	−20.2843	−0.713157	−0.356578	+	0.934265i	$0.616057\pi$
$810$	0	0
$811$	5.17157	0.181598	0.0907992	−	0.995869i	$-0.471058\pi$
$811$	5.17157	0.181598	0.0907992	+	0.995869i	$0.471058\pi$
$812$	0	0
$813$	0	0
$814$	4.00000	0.140200
$815$	−18.0711	−0.633002
$816$	0	0
$817$	38.4853	1.34643
$818$	−6.20101	−0.216813
$819$	0	0
$820$	−22.8284	−0.797203
$821$	−15.4853	−0.540440	−0.270220	−	0.962799i	$-0.587096\pi$
$821$	−15.4853	−0.540440	−0.270220	+	0.962799i	$0.587096\pi$
$822$	0	0
$823$	−2.28427	−0.0796247	−0.0398123	−	0.999207i	$-0.512676\pi$
$823$	−2.28427	−0.0796247	−0.0398123	+	0.999207i	$0.512676\pi$
$824$	−1.31371	−0.0457652
$825$	0	0
$826$	−8.97056	−0.312126
$827$	13.1005	0.455549	0.227775	−	0.973714i	$-0.426855\pi$
$827$	13.1005	0.455549	0.227775	+	0.973714i	$0.426855\pi$
$828$	0	0
$829$	−9.79899	−0.340333	−0.170166	−	0.985415i	$-0.554431\pi$
$829$	−9.79899	−0.340333	−0.170166	+	0.985415i	$0.554431\pi$
$830$	1.51472	0.0525767
$831$	0	0
$832$	−7.62742	−0.264433
$833$	−4.82843	−0.167295
$834$	0	0
$835$	8.82843	0.305520
$836$	26.4853	0.916013
$837$	0	0
$838$	10.9706	0.378972
$839$	−22.0711	−0.761978	−0.380989	−	0.924580i	$-0.624416\pi$
$839$	−22.0711	−0.761978	−0.380989	+	0.924580i	$0.624416\pi$
$840$	0	0
$841$	0	0
$842$	10.4020	0.358477
$843$	0	0
$844$	31.7868	1.09415
$845$	−9.65685	−0.332206
$846$	0	0
$847$	−14.6274	−0.502604
$848$	22.4558	0.771137
$849$	0	0
$850$	8.00000	0.274398
$851$	30.6274	1.04989
$852$	0	0
$853$	−10.9706	−0.375625	−0.187812	−	0.982205i	$-0.560140\pi$
$853$	−10.9706	−0.375625	−0.187812	+	0.982205i	$0.560140\pi$
$854$	−0.970563	−0.0332120
$855$	0	0
$856$	−14.5442	−0.497109
$857$	11.8284	0.404051	0.202026	−	0.979380i	$-0.435248\pi$
$857$	11.8284	0.404051	0.202026	+	0.979380i	$0.435248\pi$
$858$	0	0
$859$	5.72792	0.195434	0.0977171	−	0.995214i	$-0.468846\pi$
$859$	5.72792	0.195434	0.0977171	+	0.995214i	$0.468846\pi$
$860$	11.7279	0.399919
$861$	0	0
$862$	−3.45584	−0.117707
$863$	45.1127	1.53565	0.767827	−	0.640657i	$-0.221338\pi$
$863$	45.1127	1.53565	0.767827	+	0.640657i	$0.221338\pi$
$864$	0	0
$865$	−23.6569	−0.804357
$866$	6.05887	0.205889
$867$	0	0
$868$	−21.0538	−0.714613
$869$	−1.00000	−0.0339227
$870$	0	0
$871$	−10.3431	−0.350464
$872$	−2.12994	−0.0721289
$873$	0	0
$874$	−19.0294	−0.643680
$875$	−25.4558	−0.860565
$876$	0	0
$877$	−8.85786	−0.299109	−0.149554	−	0.988753i	$-0.547784\pi$
$877$	−8.85786	−0.299109	−0.149554	+	0.988753i	$0.547784\pi$
$878$	−4.82843	−0.162952
$879$	0	0
$880$	7.24264	0.244149
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	−46.4264	−1.56237	−0.781186	−	0.624298i	$-0.785385\pi$
$883$	−46.4264	−1.56237	−0.781186	+	0.624298i	$0.785385\pi$
$884$	16.1421	0.542919
$885$	0	0
$886$	−14.7696	−0.496193
$887$	36.8995	1.23896	0.619482	−	0.785011i	$-0.287343\pi$
$887$	36.8995	1.23896	0.619482	+	0.785011i	$0.287343\pi$
$888$	0	0
$889$	44.2843	1.48525
$890$	1.85786	0.0622758
$891$	0	0
$892$	16.1421	0.540479
$893$	−31.4558	−1.05263
$894$	0	0
$895$	−10.4853	−0.350484
$896$	−29.8579	−0.997481
$897$	0	0
$898$	−0.426407	−0.0142294
$899$	0	0
$900$	0	0
$901$	−36.1421	−1.20407
$902$	12.4853	0.415714
$903$	0	0
$904$	−14.7696	−0.491228
$905$	−14.3137	−0.475804
$906$	0	0
$907$	34.2843	1.13839	0.569195	−	0.822202i	$-0.307255\pi$
$907$	34.2843	1.13839	0.569195	+	0.822202i	$0.307255\pi$
$908$	36.8284	1.22219
$909$	0	0
$910$	2.14214	0.0710111
$911$	−46.5563	−1.54248	−0.771240	−	0.636544i	$-0.780363\pi$
$911$	−46.5563	−1.54248	−0.771240	+	0.636544i	$0.780363\pi$
$912$	0	0
$913$	8.82843	0.292178
$914$	14.4853	0.479131
$915$	0	0
$916$	−37.4558	−1.23758
$917$	−3.71573	−0.122704
$918$	0	0
$919$	−20.1421	−0.664428	−0.332214	−	0.943204i	$-0.607796\pi$
$919$	−20.1421	−0.664428	−0.332214	+	0.943204i	$0.607796\pi$
$920$	−12.1421	−0.400314
$921$	0	0
$922$	5.79899	0.190980
$923$	5.79899	0.190876
$924$	0	0
$925$	−16.0000	−0.526077
$926$	−10.7696	−0.353909
$927$	0	0
$928$	0	0
$929$	−41.3137	−1.35546	−0.677729	−	0.735311i	$-0.737036\pi$
$929$	−41.3137	−1.35546	−0.677729	+	0.735311i	$0.737036\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	−7.88730	−0.258357
$933$	0	0
$934$	13.4020	0.438527
$935$	−11.6569	−0.381220
$936$	0	0
$937$	28.6274	0.935217	0.467608	−	0.883936i	$-0.345116\pi$
$937$	28.6274	0.935217	0.467608	+	0.883936i	$0.345116\pi$
$938$	−6.62742	−0.216393
$939$	0	0
$940$	−9.58579	−0.312654
$941$	−22.5980	−0.736673	−0.368337	−	0.929693i	$-0.620073\pi$
$941$	−22.5980	−0.736673	−0.368337	+	0.929693i	$0.620073\pi$
$942$	0	0
$943$	95.5980	3.11310
$944$	−22.9706	−0.747628
$945$	0	0
$946$	−6.41421	−0.208544
$947$	39.3848	1.27983	0.639917	−	0.768444i	$-0.278969\pi$
$947$	39.3848	1.27983	0.639917	+	0.768444i	$0.278969\pi$
$948$	0	0
$949$	−7.31371	−0.237413
$950$	9.94113	0.322533
$951$	0	0
$952$	21.6569	0.701903
$953$	−9.62742	−0.311863	−0.155931	−	0.987768i	$-0.549838\pi$
$953$	−9.62742	−0.311863	−0.155931	+	0.987768i	$0.549838\pi$
$954$	0	0
$955$	2.68629	0.0869264
$956$	−15.2548	−0.493377
$957$	0	0
$958$	−5.34315	−0.172629
$959$	33.9411	1.09602
$960$	0	0
$961$	−14.4264	−0.465368
$962$	3.02944	0.0976730
$963$	0	0
$964$	−7.88730	−0.254033
$965$	10.8284	0.348579
$966$	0	0
$967$	26.7574	0.860459	0.430229	−	0.902720i	$-0.358433\pi$
$967$	26.7574	0.860459	0.430229	+	0.902720i	$0.358433\pi$
$968$	8.20101	0.263590
$969$	0	0
$970$	5.17157	0.166049
$971$	−4.34315	−0.139378	−0.0696891	−	0.997569i	$-0.522201\pi$
$971$	−4.34315	−0.139378	−0.0696891	+	0.997569i	$0.522201\pi$
$972$	0	0
$973$	39.5980	1.26945
$974$	−11.7990	−0.378064
$975$	0	0
$976$	−2.48528	−0.0795519
$977$	−41.8284	−1.33821	−0.669105	−	0.743168i	$-0.733322\pi$
$977$	−41.8284	−1.33821	−0.669105	+	0.743168i	$0.733322\pi$
$978$	0	0
$979$	10.8284	0.346078
$980$	−1.82843	−0.0584070
$981$	0	0
$982$	−5.28427	−0.168628
$983$	31.8701	1.01650	0.508248	−	0.861210i	$-0.330293\pi$
$983$	31.8701	1.01650	0.508248	+	0.861210i	$0.330293\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	0	0
$988$	20.0589	0.638158
$989$	−49.1127	−1.56169
$990$	0	0
$991$	−7.17157	−0.227813	−0.113906	−	0.993492i	$-0.536336\pi$
$991$	−7.17157	−0.227813	−0.113906	+	0.993492i	$0.536336\pi$
$992$	17.9706	0.570566
$993$	0	0
$994$	3.71573	0.117856
$995$	16.4853	0.522619
$996$	0	0
$997$	28.2843	0.895772	0.447886	−	0.894091i	$-0.352177\pi$
$997$	28.2843	0.895772	0.447886	+	0.894091i	$0.352177\pi$
$998$	−6.20101	−0.196290
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.c.1.2		2
3.2	odd	2		841.2.a.d.1.1		2
29.28	even	2		261.2.a.d.1.1		2
87.2	even	28		841.2.e.k.236.2		24
87.5	odd	14		841.2.d.j.605.1		12
87.8	even	28		841.2.e.k.267.2		24
87.11	even	28		841.2.e.k.63.2		24
87.14	even	28		841.2.e.k.196.3		24
87.17	even	4		841.2.b.a.840.3		4
87.20	odd	14		841.2.d.f.574.1		12
87.23	odd	14		841.2.d.f.645.2		12
87.26	even	28		841.2.e.k.270.3		24
87.32	even	28		841.2.e.k.270.2		24
87.35	odd	14		841.2.d.j.645.1		12
87.38	odd	14		841.2.d.j.574.2		12
87.41	even	4		841.2.b.a.840.2		4
87.44	even	28		841.2.e.k.196.2		24
87.47	even	28		841.2.e.k.63.3		24
87.50	even	28		841.2.e.k.267.3		24
87.53	odd	14		841.2.d.f.605.2		12
87.56	even	28		841.2.e.k.236.3		24
87.62	odd	14		841.2.d.j.190.1		12
87.65	odd	14		841.2.d.f.571.2		12
87.68	even	28		841.2.e.k.651.2		24
87.71	odd	14		841.2.d.j.778.2		12
87.74	odd	14		841.2.d.f.778.1		12
87.77	even	28		841.2.e.k.651.3		24
87.80	odd	14		841.2.d.j.571.1		12
87.83	odd	14		841.2.d.f.190.2		12
87.86	odd	2		29.2.a.a.1.2	✓	2
116.115	odd	2		4176.2.a.bq.1.1		2
145.144	even	2		6525.2.a.o.1.2		2
348.347	even	2		464.2.a.h.1.2		2
435.173	even	4		725.2.b.b.349.2		4
435.347	even	4		725.2.b.b.349.3		4
435.434	odd	2		725.2.a.b.1.1		2
609.608	even	2		1421.2.a.j.1.2		2
696.173	odd	2		1856.2.a.r.1.2		2
696.347	even	2		1856.2.a.w.1.1		2
957.956	even	2		3509.2.a.j.1.1		2
1131.1130	odd	2		4901.2.a.g.1.1		2
1479.1478	odd	2		8381.2.a.e.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.2.a.a.1.2	✓	2	87.86	odd	2
261.2.a.d.1.1		2	29.28	even	2
464.2.a.h.1.2		2	348.347	even	2
725.2.a.b.1.1		2	435.434	odd	2
725.2.b.b.349.2		4	435.173	even	4
725.2.b.b.349.3		4	435.347	even	4
841.2.a.d.1.1		2	3.2	odd	2
841.2.b.a.840.2		4	87.41	even	4
841.2.b.a.840.3		4	87.17	even	4
841.2.d.f.190.2		12	87.83	odd	14
841.2.d.f.571.2		12	87.65	odd	14
841.2.d.f.574.1		12	87.20	odd	14
841.2.d.f.605.2		12	87.53	odd	14
841.2.d.f.645.2		12	87.23	odd	14
841.2.d.f.778.1		12	87.74	odd	14
841.2.d.j.190.1		12	87.62	odd	14
841.2.d.j.571.1		12	87.80	odd	14
841.2.d.j.574.2		12	87.38	odd	14
841.2.d.j.605.1		12	87.5	odd	14
841.2.d.j.645.1		12	87.35	odd	14
841.2.d.j.778.2		12	87.71	odd	14
841.2.e.k.63.2		24	87.11	even	28
841.2.e.k.63.3		24	87.47	even	28
841.2.e.k.196.2		24	87.44	even	28
841.2.e.k.196.3		24	87.14	even	28
841.2.e.k.236.2		24	87.2	even	28
841.2.e.k.236.3		24	87.56	even	28
841.2.e.k.267.2		24	87.8	even	28
841.2.e.k.267.3		24	87.50	even	28
841.2.e.k.270.2		24	87.32	even	28
841.2.e.k.270.3		24	87.26	even	28
841.2.e.k.651.2		24	87.68	even	28
841.2.e.k.651.3		24	87.77	even	28
1421.2.a.j.1.2		2	609.608	even	2
1856.2.a.r.1.2		2	696.173	odd	2
1856.2.a.w.1.1		2	696.347	even	2
3509.2.a.j.1.1		2	957.956	even	2
4176.2.a.bq.1.1		2	116.115	odd	2
4901.2.a.g.1.1		2	1131.1130	odd	2
6525.2.a.o.1.2		2	145.144	even	2
7569.2.a.c.1.2		2	1.1	even	1	trivial
8381.2.a.e.1.2		2	1479.1478	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +2.82843 q^{7} -1.58579 q^{8} +0.414214 q^{10} +2.41421 q^{11} +1.82843 q^{13} +1.17157 q^{14} +3.00000 q^{16} -4.82843 q^{17} -6.00000 q^{19} -1.82843 q^{20} +1.00000 q^{22} +7.65685 q^{23} -4.00000 q^{25} +0.757359 q^{26} -5.17157 q^{28} +4.07107 q^{31} +4.41421 q^{32} -2.00000 q^{34} +2.82843 q^{35} +4.00000 q^{37} -2.48528 q^{38} -1.58579 q^{40} +12.4853 q^{41} -6.41421 q^{43} -4.41421 q^{44} +3.17157 q^{46} +5.24264 q^{47} +1.00000 q^{49} -1.65685 q^{50} -3.34315 q^{52} +7.48528 q^{53} +2.41421 q^{55} -4.48528 q^{56} -7.65685 q^{59} -0.828427 q^{61} +1.68629 q^{62} -4.17157 q^{64} +1.82843 q^{65} -5.65685 q^{67} +8.82843 q^{68} +1.17157 q^{70} +3.17157 q^{71} -4.00000 q^{73} +1.65685 q^{74} +10.9706 q^{76} +6.82843 q^{77} -0.414214 q^{79} +3.00000 q^{80} +5.17157 q^{82} +3.65685 q^{83} -4.82843 q^{85} -2.65685 q^{86} -3.82843 q^{88} +4.48528 q^{89} +5.17157 q^{91} -14.0000 q^{92} +2.17157 q^{94} -6.00000 q^{95} +12.4853 q^{97} +0.414214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8} - 2 q^{10} + 2 q^{11} - 2 q^{13} + 8 q^{14} + 6 q^{16} - 4 q^{17} - 12 q^{19} + 2 q^{20} + 2 q^{22} + 4 q^{23} - 8 q^{25} + 10 q^{26} - 16 q^{28} - 6 q^{31} + 6 q^{32}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 6 * q^8 - 2 * q^10 + 2 * q^11 - 2 * q^13 + 8 * q^14 + 6 * q^16 - 4 * q^17 - 12 * q^19 + 2 * q^20 + 2 * q^22 + 4 * q^23 - 8 * q^25 + 10 * q^26 - 16 * q^28 - 6 * q^31 + 6 * q^32 - 4 * q^34 + 8 * q^37 + 12 * q^38 - 6 * q^40 + 8 * q^41 - 10 * q^43 - 6 * q^44 + 12 * q^46 + 2 * q^47 + 2 * q^49 + 8 * q^50 - 18 * q^52 - 2 * q^53 + 2 * q^55 + 8 * q^56 - 4 * q^59 + 4 * q^61 + 26 * q^62 - 14 * q^64 - 2 * q^65 + 12 * q^68 + 8 * q^70 + 12 * q^71 - 8 * q^73 - 8 * q^74 - 12 * q^76 + 8 * q^77 + 2 * q^79 + 6 * q^80 + 16 * q^82 - 4 * q^83 - 4 * q^85 + 6 * q^86 - 2 * q^88 - 8 * q^89 + 16 * q^91 - 28 * q^92 + 10 * q^94 - 12 * q^95 + 8 * q^97 - 2 * q^98

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