LMFDB - Embedded newform 1521.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.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:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 169)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.554958 q^{2} -1.69202 q^{4} +2.80194 q^{5} -2.69202 q^{7} -2.04892 q^{8} +O(q^{10})$	$q+0.554958 q^{2} -1.69202 q^{4} +2.80194 q^{5} -2.69202 q^{7} -2.04892 q^{8} +1.55496 q^{10} +1.19806 q^{11} -1.49396 q^{14} +2.24698 q^{16} -1.13706 q^{17} +1.93900 q^{19} -4.74094 q^{20} +0.664874 q^{22} +4.60388 q^{23} +2.85086 q^{25} +4.55496 q^{28} +7.89977 q^{29} +5.89977 q^{31} +5.34481 q^{32} -0.631023 q^{34} -7.54288 q^{35} +0.951083 q^{37} +1.07606 q^{38} -5.74094 q^{40} -3.31767 q^{41} +7.15883 q^{43} -2.02715 q^{44} +2.55496 q^{46} +7.69202 q^{47} +0.246980 q^{49} +1.58211 q^{50} -5.87263 q^{53} +3.35690 q^{55} +5.51573 q^{56} +4.38404 q^{58} +0.0120816 q^{59} -8.03684 q^{61} +3.27413 q^{62} -1.52781 q^{64} -9.25667 q^{67} +1.92394 q^{68} -4.18598 q^{70} +13.7409 q^{71} +12.8170 q^{73} +0.527811 q^{74} -3.28083 q^{76} -3.22521 q^{77} +0.807315 q^{79} +6.29590 q^{80} -1.84117 q^{82} +16.3327 q^{83} -3.18598 q^{85} +3.97285 q^{86} -2.45473 q^{88} +14.7289 q^{89} -7.78986 q^{92} +4.26875 q^{94} +5.43296 q^{95} +3.13169 q^{97} +0.137063 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 4 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 4 * q^5 - 3 * q^7 + 3 * q^8	$ 3 q + 2 q^{2} + 4 q^{5} - 3 q^{7} + 3 q^{8} + 5 q^{10} + 8 q^{11} + 5 q^{14} + 2 q^{16} + 2 q^{17} - 4 q^{19} + 3 q^{22} + 5 q^{23} - 5 q^{25} + 14 q^{28} + q^{29} - 5 q^{31} - 7 q^{32} + 13 q^{34} - 4 q^{35} + 12 q^{37} - 12 q^{38} - 3 q^{40} + 7 q^{41} + 13 q^{43} + 8 q^{46} + 18 q^{47} - 4 q^{49} - q^{50} - q^{53} + 6 q^{55} + 4 q^{56} + 3 q^{58} + 19 q^{59} + 4 q^{61} - q^{62} - 11 q^{64} - q^{67} + 21 q^{68} + 2 q^{70} + 27 q^{71} + 9 q^{73} + 8 q^{74} - 21 q^{76} - 8 q^{77} - 5 q^{79} + 5 q^{80} - 14 q^{82} + 7 q^{83} + 5 q^{85} + 18 q^{86} + 15 q^{88} + 11 q^{89} + 5 q^{94} - 3 q^{95} + 7 q^{97} - 5 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 4 * q^5 - 3 * q^7 + 3 * q^8 + 5 * q^10 + 8 * q^11 + 5 * q^14 + 2 * q^16 + 2 * q^17 - 4 * q^19 + 3 * q^22 + 5 * q^23 - 5 * q^25 + 14 * q^28 + q^29 - 5 * q^31 - 7 * q^32 + 13 * q^34 - 4 * q^35 + 12 * q^37 - 12 * q^38 - 3 * q^40 + 7 * q^41 + 13 * q^43 + 8 * q^46 + 18 * q^47 - 4 * q^49 - q^50 - q^53 + 6 * q^55 + 4 * q^56 + 3 * q^58 + 19 * q^59 + 4 * q^61 - q^62 - 11 * q^64 - q^67 + 21 * q^68 + 2 * q^70 + 27 * q^71 + 9 * q^73 + 8 * q^74 - 21 * q^76 - 8 * q^77 - 5 * q^79 + 5 * q^80 - 14 * q^82 + 7 * q^83 + 5 * q^85 + 18 * q^86 + 15 * q^88 + 11 * q^89 + 5 * q^94 - 3 * q^95 + 7 * q^97 - 5 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	0.554958	0.392415	0.196207	−	0.980562i	$-0.437137\pi$
$2$	0.554958	0.392415	0.196207	+	0.980562i	$0.437137\pi$
$3$	0	0
$3$	0	0
$4$	−1.69202	−0.846011
$5$	2.80194	1.25306	0.626532	−	0.779395i	$-0.284474\pi$
$5$	2.80194	1.25306	0.626532	+	0.779395i	$0.284474\pi$
$6$	0	0
$7$	−2.69202	−1.01749	−0.508744	−	0.860918i	$-0.669890\pi$
$7$	−2.69202	−1.01749	−0.508744	+	0.860918i	$0.669890\pi$
$8$	−2.04892	−0.724402
$9$	0	0
$10$	1.55496	0.491721
$11$	1.19806	0.361229	0.180615	−	0.983554i	$-0.442191\pi$
$11$	1.19806	0.361229	0.180615	+	0.983554i	$0.442191\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−1.49396	−0.399277
$15$	0	0
$16$	2.24698	0.561745
$17$	−1.13706	−0.275778	−0.137889	−	0.990448i	$-0.544032\pi$
$17$	−1.13706	−0.275778	−0.137889	+	0.990448i	$0.544032\pi$
$18$	0	0
$19$	1.93900	0.444837	0.222419	−	0.974951i	$-0.428605\pi$
$19$	1.93900	0.444837	0.222419	+	0.974951i	$0.428605\pi$
$20$	−4.74094	−1.06011
$21$	0	0
$22$	0.664874	0.141752
$23$	4.60388	0.959974	0.479987	−	0.877275i	$-0.340641\pi$
$23$	4.60388	0.959974	0.479987	+	0.877275i	$0.340641\pi$
$24$	0	0
$25$	2.85086	0.570171
$26$	0	0
$27$	0	0
$28$	4.55496	0.860806
$29$	7.89977	1.46695	0.733475	−	0.679716i	$-0.237897\pi$
$29$	7.89977	1.46695	0.733475	+	0.679716i	$0.237897\pi$
$30$	0	0
$31$	5.89977	1.05963	0.529815	−	0.848113i	$-0.322261\pi$
$31$	5.89977	1.05963	0.529815	+	0.848113i	$0.322261\pi$
$32$	5.34481	0.944839
$33$	0	0
$34$	−0.631023	−0.108219
$35$	−7.54288	−1.27498
$36$	0	0
$37$	0.951083	0.156357	0.0781785	−	0.996939i	$-0.475090\pi$
$37$	0.951083	0.156357	0.0781785	+	0.996939i	$0.475090\pi$
$38$	1.07606	0.174561
$39$	0	0
$40$	−5.74094	−0.907722
$41$	−3.31767	−0.518133	−0.259066	−	0.965860i	$-0.583415\pi$
$41$	−3.31767	−0.518133	−0.259066	+	0.965860i	$0.583415\pi$
$42$	0	0
$43$	7.15883	1.09171	0.545856	−	0.837879i	$-0.316205\pi$
$43$	7.15883	1.09171	0.545856	+	0.837879i	$0.316205\pi$
$44$	−2.02715	−0.305604
$45$	0	0
$46$	2.55496	0.376708
$47$	7.69202	1.12200	0.560998	−	0.827817i	$-0.310417\pi$
$47$	7.69202	1.12200	0.560998	+	0.827817i	$0.310417\pi$
$48$	0	0
$49$	0.246980	0.0352828
$50$	1.58211	0.223743
$51$	0	0
$52$	0	0
$53$	−5.87263	−0.806667	−0.403334	−	0.915053i	$-0.632149\pi$
$53$	−5.87263	−0.806667	−0.403334	+	0.915053i	$0.632149\pi$
$54$	0	0
$55$	3.35690	0.452644
$56$	5.51573	0.737070
$57$	0	0
$58$	4.38404	0.575653
$59$	0.0120816	0.00157289	0.000786444	−	1.00000i	$-0.499750\pi$
$59$	0.0120816	0.00157289	0.000786444		1.00000i	$0.499750\pi$
$60$	0	0
$61$	−8.03684	−1.02901	−0.514506	−	0.857487i	$-0.672025\pi$
$61$	−8.03684	−1.02901	−0.514506	+	0.857487i	$0.672025\pi$
$62$	3.27413	0.415815
$63$	0	0
$64$	−1.52781	−0.190976
$65$	0	0
$66$	0	0
$67$	−9.25667	−1.13088	−0.565441	−	0.824789i	$-0.691294\pi$
$67$	−9.25667	−1.13088	−0.565441	+	0.824789i	$0.691294\pi$
$68$	1.92394	0.233311
$69$	0	0
$70$	−4.18598	−0.500320
$71$	13.7409	1.63075	0.815375	−	0.578934i	$-0.196531\pi$
$71$	13.7409	1.63075	0.815375	+	0.578934i	$0.196531\pi$
$72$	0	0
$73$	12.8170	1.50012	0.750058	−	0.661372i	$-0.230025\pi$
$73$	12.8170	1.50012	0.750058	+	0.661372i	$0.230025\pi$
$74$	0.527811	0.0613568
$75$	0	0
$76$	−3.28083	−0.376337
$77$	−3.22521	−0.367547
$78$	0	0
$79$	0.807315	0.0908300	0.0454150	−	0.998968i	$-0.485539\pi$
$79$	0.807315	0.0908300	0.0454150	+	0.998968i	$0.485539\pi$
$80$	6.29590	0.703903
$81$	0	0
$82$	−1.84117	−0.203323
$83$	16.3327	1.79275	0.896375	−	0.443296i	$-0.146191\pi$
$83$	16.3327	1.79275	0.896375	+	0.443296i	$0.146191\pi$
$84$	0	0
$85$	−3.18598	−0.345568
$86$	3.97285	0.428404
$87$	0	0
$88$	−2.45473	−0.261675
$89$	14.7289	1.56126	0.780628	−	0.624996i	$-0.214899\pi$
$89$	14.7289	1.56126	0.780628	+	0.624996i	$0.214899\pi$
$90$	0	0
$91$	0	0
$92$	−7.78986	−0.812149
$93$	0	0
$94$	4.26875	0.440288
$95$	5.43296	0.557410
$96$	0	0
$97$	3.13169	0.317975	0.158987	−	0.987281i	$-0.449177\pi$
$97$	3.13169	0.317975	0.158987	+	0.987281i	$0.449177\pi$
$98$	0.137063	0.0138455
$99$	0	0
$100$	−4.82371	−0.482371
$101$	−5.29052	−0.526426	−0.263213	−	0.964738i	$-0.584782\pi$
$101$	−5.29052	−0.526426	−0.263213	+	0.964738i	$0.584782\pi$
$102$	0	0
$103$	−13.5308	−1.33323	−0.666614	−	0.745403i	$-0.732257\pi$
$103$	−13.5308	−1.33323	−0.666614	+	0.745403i	$0.732257\pi$
$104$	0	0
$105$	0	0
$106$	−3.25906	−0.316548
$107$	−5.63102	−0.544371	−0.272186	−	0.962245i	$-0.587747\pi$
$107$	−5.63102	−0.544371	−0.272186	+	0.962245i	$0.587747\pi$
$108$	0	0
$109$	4.17629	0.400016	0.200008	−	0.979794i	$-0.435903\pi$
$109$	4.17629	0.400016	0.200008	+	0.979794i	$0.435903\pi$
$110$	1.86294	0.177624
$111$	0	0
$112$	−6.04892	−0.571569
$113$	−7.64310	−0.719003	−0.359501	−	0.933145i	$-0.617053\pi$
$113$	−7.64310	−0.719003	−0.359501	+	0.933145i	$0.617053\pi$
$114$	0	0
$115$	12.8998	1.20291
$116$	−13.3666	−1.24106
$117$	0	0
$118$	0.00670477	0.000617224 0
$119$	3.06100	0.280601
$120$	0	0
$121$	−9.56465	−0.869513
$122$	−4.46011	−0.403799
$123$	0	0
$124$	−9.98254	−0.896459
$125$	−6.02177	−0.538604
$126$	0	0
$127$	6.77777	0.601430	0.300715	−	0.953714i	$-0.402775\pi$
$127$	6.77777	0.601430	0.300715	+	0.953714i	$0.402775\pi$
$128$	−11.5375	−1.01978
$129$	0	0
$130$	0	0
$131$	13.6799	1.19522	0.597611	−	0.801786i	$-0.296117\pi$
$131$	13.6799	1.19522	0.597611	+	0.801786i	$0.296117\pi$
$132$	0	0
$133$	−5.21983	−0.452617
$134$	−5.13706	−0.443775
$135$	0	0
$136$	2.32975	0.199774
$137$	−12.9879	−1.10963	−0.554816	−	0.831973i	$-0.687211\pi$
$137$	−12.9879	−1.10963	−0.554816	+	0.831973i	$0.687211\pi$
$138$	0	0
$139$	12.0465	1.02177	0.510886	−	0.859648i	$-0.329317\pi$
$139$	12.0465	1.02177	0.510886	+	0.859648i	$0.329317\pi$
$140$	12.7627	1.07865
$141$	0	0
$142$	7.62565	0.639930
$143$	0	0
$144$	0	0
$145$	22.1347	1.83818
$146$	7.11290	0.588668
$147$	0	0
$148$	−1.60925	−0.132280
$149$	0.740939	0.0607001	0.0303500	−	0.999539i	$-0.490338\pi$
$149$	0.740939	0.0607001	0.0303500	+	0.999539i	$0.490338\pi$
$150$	0	0
$151$	−19.0737	−1.55219	−0.776097	−	0.630614i	$-0.782803\pi$
$151$	−19.0737	−1.55219	−0.776097	+	0.630614i	$0.782803\pi$
$152$	−3.97285	−0.322241
$153$	0	0
$154$	−1.78986	−0.144231
$155$	16.5308	1.32779
$156$	0	0
$157$	−4.02177	−0.320972	−0.160486	−	0.987038i	$-0.551306\pi$
$157$	−4.02177	−0.320972	−0.160486	+	0.987038i	$0.551306\pi$
$158$	0.448026	0.0356430
$159$	0	0
$160$	14.9758	1.18394
$161$	−12.3937	−0.976763
$162$	0	0
$163$	15.1371	1.18563	0.592813	−	0.805340i	$-0.298017\pi$
$163$	15.1371	1.18563	0.592813	+	0.805340i	$0.298017\pi$
$164$	5.61356	0.438346
$165$	0	0
$166$	9.06398	0.703502
$167$	6.26337	0.484674	0.242337	−	0.970192i	$-0.422086\pi$
$167$	6.26337	0.484674	0.242337	+	0.970192i	$0.422086\pi$
$168$	0	0
$169$	0	0
$170$	−1.76809	−0.135606
$171$	0	0
$172$	−12.1129	−0.923600
$173$	−16.3913	−1.24621	−0.623105	−	0.782138i	$-0.714129\pi$
$173$	−16.3913	−1.24621	−0.623105	+	0.782138i	$0.714129\pi$
$174$	0	0
$175$	−7.67456	−0.580142
$176$	2.69202	0.202919
$177$	0	0
$178$	8.17390	0.612660
$179$	2.45473	0.183475	0.0917376	−	0.995783i	$-0.470758\pi$
$179$	2.45473	0.183475	0.0917376	+	0.995783i	$0.470758\pi$
$180$	0	0
$181$	11.8073	0.877631	0.438815	−	0.898577i	$-0.355398\pi$
$181$	11.8073	0.877631	0.438815	+	0.898577i	$0.355398\pi$
$182$	0	0
$183$	0	0
$184$	−9.43296	−0.695407
$185$	2.66487	0.195925
$186$	0	0
$187$	−1.36227	−0.0996192
$188$	−13.0151	−0.949221
$189$	0	0
$190$	3.01507	0.218736
$191$	8.99330	0.650732	0.325366	−	0.945588i	$-0.394512\pi$
$191$	8.99330	0.650732	0.325366	+	0.945588i	$0.394512\pi$
$192$	0	0
$193$	13.5254	0.973581	0.486790	−	0.873519i	$-0.338168\pi$
$193$	13.5254	0.973581	0.486790	+	0.873519i	$0.338168\pi$
$194$	1.73795	0.124778
$195$	0	0
$196$	−0.417895	−0.0298496
$197$	−12.9758	−0.924490	−0.462245	−	0.886752i	$-0.652956\pi$
$197$	−12.9758	−0.924490	−0.462245	+	0.886752i	$0.652956\pi$
$198$	0	0
$199$	−13.5864	−0.963116	−0.481558	−	0.876414i	$-0.659929\pi$
$199$	−13.5864	−0.963116	−0.481558	+	0.876414i	$0.659929\pi$
$200$	−5.84117	−0.413033
$201$	0	0
$202$	−2.93602	−0.206577
$203$	−21.2664	−1.49261
$204$	0	0
$205$	−9.29590	−0.649254
$206$	−7.50902	−0.523179
$207$	0	0
$208$	0	0
$209$	2.32304	0.160688
$210$	0	0
$211$	10.4601	0.720103	0.360052	−	0.932932i	$-0.382759\pi$
$211$	10.4601	0.720103	0.360052	+	0.932932i	$0.382759\pi$
$212$	9.93661	0.682449
$213$	0	0
$214$	−3.12498	−0.213619
$215$	20.0586	1.36799
$216$	0	0
$217$	−15.8823	−1.07816
$218$	2.31767	0.156972
$219$	0	0
$220$	−5.67994	−0.382941
$221$	0	0
$222$	0	0
$223$	−11.4058	−0.763790	−0.381895	−	0.924206i	$-0.624728\pi$
$223$	−11.4058	−0.763790	−0.381895	+	0.924206i	$0.624728\pi$
$224$	−14.3884	−0.961362
$225$	0	0
$226$	−4.24160	−0.282147
$227$	−10.6407	−0.706249	−0.353124	−	0.935576i	$-0.614881\pi$
$227$	−10.6407	−0.706249	−0.353124	+	0.935576i	$0.614881\pi$
$228$	0	0
$229$	−1.13946	−0.0752974	−0.0376487	−	0.999291i	$-0.511987\pi$
$229$	−1.13946	−0.0752974	−0.0376487	+	0.999291i	$0.511987\pi$
$230$	7.15883	0.472040
$231$	0	0
$232$	−16.1860	−1.06266
$233$	10.8509	0.710863	0.355432	−	0.934702i	$-0.384334\pi$
$233$	10.8509	0.710863	0.355432	+	0.934702i	$0.384334\pi$
$234$	0	0
$235$	21.5526	1.40593
$236$	−0.0204423	−0.00133068
$237$	0	0
$238$	1.69873	0.110112
$239$	11.9293	0.771643	0.385822	−	0.922573i	$-0.373918\pi$
$239$	11.9293	0.771643	0.385822	+	0.922573i	$0.373918\pi$
$240$	0	0
$241$	−3.64848	−0.235019	−0.117510	−	0.993072i	$-0.537491\pi$
$241$	−3.64848	−0.235019	−0.117510	+	0.993072i	$0.537491\pi$
$242$	−5.30798	−0.341210
$243$	0	0
$244$	13.5985	0.870555
$245$	0.692021	0.0442116
$246$	0	0
$247$	0	0
$248$	−12.0881	−0.767598
$249$	0	0
$250$	−3.34183	−0.211356
$251$	−1.37329	−0.0866813	−0.0433406	−	0.999060i	$-0.513800\pi$
$251$	−1.37329	−0.0866813	−0.0433406	+	0.999060i	$0.513800\pi$
$252$	0	0
$253$	5.51573	0.346771
$254$	3.76138	0.236010
$255$	0	0
$256$	−3.34721	−0.209200
$257$	−29.4359	−1.83616	−0.918082	−	0.396391i	$-0.870263\pi$
$257$	−29.4359	−1.83616	−0.918082	+	0.396391i	$0.870263\pi$
$258$	0	0
$259$	−2.56033	−0.159091
$260$	0	0
$261$	0	0
$262$	7.59179	0.469023
$263$	−10.6963	−0.659564	−0.329782	−	0.944057i	$-0.606975\pi$
$263$	−10.6963	−0.659564	−0.329782	+	0.944057i	$0.606975\pi$
$264$	0	0
$265$	−16.4547	−1.01081
$266$	−2.89679	−0.177613
$267$	0	0
$268$	15.6625	0.956738
$269$	10.1860	0.621050	0.310525	−	0.950565i	$-0.399495\pi$
$269$	10.1860	0.621050	0.310525	+	0.950565i	$0.399495\pi$
$270$	0	0
$271$	−29.4523	−1.78910	−0.894551	−	0.446966i	$-0.852505\pi$
$271$	−29.4523	−1.78910	−0.894551	+	0.446966i	$0.852505\pi$
$272$	−2.55496	−0.154917
$273$	0	0
$274$	−7.20775	−0.435436
$275$	3.41550	0.205963
$276$	0	0
$277$	−10.2446	−0.615538	−0.307769	−	0.951461i	$-0.599582\pi$
$277$	−10.2446	−0.615538	−0.307769	+	0.951461i	$0.599582\pi$
$278$	6.68532	0.400959
$279$	0	0
$280$	15.4547	0.923597
$281$	11.5646	0.689889	0.344944	−	0.938623i	$-0.387898\pi$
$281$	11.5646	0.689889	0.344944	+	0.938623i	$0.387898\pi$
$282$	0	0
$283$	−30.7090	−1.82546	−0.912730	−	0.408562i	$-0.866030\pi$
$283$	−30.7090	−1.82546	−0.912730	+	0.408562i	$0.866030\pi$
$284$	−23.2500	−1.37963
$285$	0	0
$286$	0	0
$287$	8.93123	0.527194
$288$	0	0
$289$	−15.7071	−0.923946
$290$	12.2838	0.721330
$291$	0	0
$292$	−21.6866	−1.26911
$293$	18.6082	1.08710	0.543551	−	0.839376i	$-0.317079\pi$
$293$	18.6082	1.08710	0.543551	+	0.839376i	$0.317079\pi$
$294$	0	0
$295$	0.0338518	0.00197093
$296$	−1.94869	−0.113265
$297$	0	0
$298$	0.411190	0.0238196
$299$	0	0
$300$	0	0
$301$	−19.2717	−1.11080
$302$	−10.5851	−0.609103
$303$	0	0
$304$	4.35690	0.249885
$305$	−22.5187	−1.28942
$306$	0	0
$307$	8.94438	0.510483	0.255241	−	0.966877i	$-0.417845\pi$
$307$	8.94438	0.510483	0.255241	+	0.966877i	$0.417845\pi$
$308$	5.45712	0.310948
$309$	0	0
$310$	9.17390	0.521042
$311$	−21.0398	−1.19306	−0.596529	−	0.802591i	$-0.703454\pi$
$311$	−21.0398	−1.19306	−0.596529	+	0.802591i	$0.703454\pi$
$312$	0	0
$313$	−7.12737	−0.402863	−0.201432	−	0.979503i	$-0.564559\pi$
$313$	−7.12737	−0.402863	−0.201432	+	0.979503i	$0.564559\pi$
$314$	−2.23191	−0.125954
$315$	0	0
$316$	−1.36599	−0.0768431
$317$	−23.9651	−1.34601	−0.673007	−	0.739636i	$-0.734997\pi$
$317$	−23.9651	−1.34601	−0.673007	+	0.739636i	$0.734997\pi$
$318$	0	0
$319$	9.46442	0.529906
$320$	−4.28083	−0.239306
$321$	0	0
$322$	−6.87800	−0.383296
$323$	−2.20477	−0.122677
$324$	0	0
$325$	0	0
$326$	8.40044	0.465257
$327$	0	0
$328$	6.79763	0.375336
$329$	−20.7071	−1.14162
$330$	0	0
$331$	2.89546	0.159149	0.0795745	−	0.996829i	$-0.474644\pi$
$331$	2.89546	0.159149	0.0795745	+	0.996829i	$0.474644\pi$
$332$	−27.6353	−1.51669
$333$	0	0
$334$	3.47591	0.190193
$335$	−25.9366	−1.41707
$336$	0	0
$337$	−3.10560	−0.169173	−0.0845865	−	0.996416i	$-0.526957\pi$
$337$	−3.10560	−0.169173	−0.0845865	+	0.996416i	$0.526957\pi$
$338$	0	0
$339$	0	0
$340$	5.39075	0.292354
$341$	7.06829	0.382770
$342$	0	0
$343$	18.1793	0.981589
$344$	−14.6679	−0.790838
$345$	0	0
$346$	−9.09651	−0.489031
$347$	11.3787	0.610839	0.305419	−	0.952218i	$-0.401203\pi$
$347$	11.3787	0.610839	0.305419	+	0.952218i	$0.401203\pi$
$348$	0	0
$349$	−3.34721	−0.179172	−0.0895859	−	0.995979i	$-0.528554\pi$
$349$	−3.34721	−0.179172	−0.0895859	+	0.995979i	$0.528554\pi$
$350$	−4.25906	−0.227656
$351$	0	0
$352$	6.40342	0.341303
$353$	−0.637727	−0.0339428	−0.0169714	−	0.999856i	$-0.505402\pi$
$353$	−0.637727	−0.0339428	−0.0169714	+	0.999856i	$0.505402\pi$
$354$	0	0
$355$	38.5013	2.04343
$356$	−24.9215	−1.32084
$357$	0	0
$358$	1.36227	0.0719983
$359$	21.4590	1.13256	0.566282	−	0.824211i	$-0.308381\pi$
$359$	21.4590	1.13256	0.566282	+	0.824211i	$0.308381\pi$
$360$	0	0
$361$	−15.2403	−0.802120
$362$	6.55257	0.344395
$363$	0	0
$364$	0	0
$365$	35.9124	1.87974
$366$	0	0
$367$	−9.38703	−0.489999	−0.244999	−	0.969523i	$-0.578788\pi$
$367$	−9.38703	−0.489999	−0.244999	+	0.969523i	$0.578788\pi$
$368$	10.3448	0.539261
$369$	0	0
$370$	1.47889	0.0768840
$371$	15.8092	0.820775
$372$	0	0
$373$	27.7265	1.43562	0.717811	−	0.696238i	$-0.245144\pi$
$373$	27.7265	1.43562	0.717811	+	0.696238i	$0.245144\pi$
$374$	−0.756004	−0.0390921
$375$	0	0
$376$	−15.7603	−0.812776
$377$	0	0
$378$	0	0
$379$	35.8702	1.84253	0.921265	−	0.388935i	$-0.127157\pi$
$379$	35.8702	1.84253	0.921265	+	0.388935i	$0.127157\pi$
$380$	−9.19269	−0.471575
$381$	0	0
$382$	4.99090	0.255357
$383$	−4.85517	−0.248087	−0.124044	−	0.992277i	$-0.539586\pi$
$383$	−4.85517	−0.248087	−0.124044	+	0.992277i	$0.539586\pi$
$384$	0	0
$385$	−9.03684	−0.460560
$386$	7.50604	0.382047
$387$	0	0
$388$	−5.29888	−0.269010
$389$	−2.38537	−0.120943	−0.0604716	−	0.998170i	$-0.519260\pi$
$389$	−2.38537	−0.120943	−0.0604716	+	0.998170i	$0.519260\pi$
$390$	0	0
$391$	−5.23490	−0.264740
$392$	−0.506041	−0.0255589
$393$	0	0
$394$	−7.20105	−0.362783
$395$	2.26205	0.113816
$396$	0	0
$397$	−15.2664	−0.766196	−0.383098	−	0.923708i	$-0.625143\pi$
$397$	−15.2664	−0.766196	−0.383098	+	0.923708i	$0.625143\pi$
$398$	−7.53989	−0.377941
$399$	0	0
$400$	6.40581	0.320291
$401$	−12.7584	−0.637124	−0.318562	−	0.947902i	$-0.603200\pi$
$401$	−12.7584	−0.637124	−0.318562	+	0.947902i	$0.603200\pi$
$402$	0	0
$403$	0	0
$404$	8.95167	0.445362
$405$	0	0
$406$	−11.8019	−0.585720
$407$	1.13946	0.0564807
$408$	0	0
$409$	−25.3588	−1.25391	−0.626956	−	0.779054i	$-0.715700\pi$
$409$	−25.3588	−1.25391	−0.626956	+	0.779054i	$0.715700\pi$
$410$	−5.15883	−0.254777
$411$	0	0
$412$	22.8944	1.12793
$413$	−0.0325239	−0.00160040
$414$	0	0
$415$	45.7633	2.24643
$416$	0	0
$417$	0	0
$418$	1.28919	0.0630565
$419$	11.6673	0.569983	0.284992	−	0.958530i	$-0.408009\pi$
$419$	11.6673	0.569983	0.284992	+	0.958530i	$0.408009\pi$
$420$	0	0
$421$	−8.29291	−0.404172	−0.202086	−	0.979368i	$-0.564772\pi$
$421$	−8.29291	−0.404172	−0.202086	+	0.979368i	$0.564772\pi$
$422$	5.80492	0.282579
$423$	0	0
$424$	12.0325	0.584351
$425$	−3.24160	−0.157241
$426$	0	0
$427$	21.6353	1.04701
$428$	9.52781	0.460544
$429$	0	0
$430$	11.1317	0.536818
$431$	−0.932296	−0.0449071	−0.0224536	−	0.999748i	$-0.507148\pi$
$431$	−0.932296	−0.0449071	−0.0224536	+	0.999748i	$0.507148\pi$
$432$	0	0
$433$	−13.3502	−0.641569	−0.320785	−	0.947152i	$-0.603947\pi$
$433$	−13.3502	−0.641569	−0.320785	+	0.947152i	$0.603947\pi$
$434$	−8.81402	−0.423086
$435$	0	0
$436$	−7.06638	−0.338418
$437$	8.92692	0.427032
$438$	0	0
$439$	13.9922	0.667813	0.333906	−	0.942606i	$-0.391633\pi$
$439$	13.9922	0.667813	0.333906	+	0.942606i	$0.391633\pi$
$440$	−6.87800	−0.327896
$441$	0	0
$442$	0	0
$443$	−23.7017	−1.12610	−0.563051	−	0.826422i	$-0.690373\pi$
$443$	−23.7017	−1.12610	−0.563051	+	0.826422i	$0.690373\pi$
$444$	0	0
$445$	41.2693	1.95635
$446$	−6.32975	−0.299722
$447$	0	0
$448$	4.11290	0.194316
$449$	12.5864	0.593990	0.296995	−	0.954879i	$-0.404016\pi$
$449$	12.5864	0.593990	0.296995	+	0.954879i	$0.404016\pi$
$450$	0	0
$451$	−3.97477	−0.187165
$452$	12.9323	0.608284
$453$	0	0
$454$	−5.90515	−0.277142
$455$	0	0
$456$	0	0
$457$	−33.6383	−1.57353	−0.786767	−	0.617250i	$-0.788247\pi$
$457$	−33.6383	−1.57353	−0.786767	+	0.617250i	$0.788247\pi$
$458$	−0.632351	−0.0295478
$459$	0	0
$460$	−21.8267	−1.01767
$461$	1.40283	0.0653363	0.0326681	−	0.999466i	$-0.489600\pi$
$461$	1.40283	0.0653363	0.0326681	+	0.999466i	$0.489600\pi$
$462$	0	0
$463$	−15.2010	−0.706453	−0.353226	−	0.935538i	$-0.614915\pi$
$463$	−15.2010	−0.706453	−0.353226	+	0.935538i	$0.614915\pi$
$464$	17.7506	0.824052
$465$	0	0
$466$	6.02177	0.278953
$467$	39.3414	1.82050	0.910250	−	0.414058i	$-0.135889\pi$
$467$	39.3414	1.82050	0.910250	+	0.414058i	$0.135889\pi$
$468$	0	0
$469$	24.9191	1.15066
$470$	11.9608	0.551709
$471$	0	0
$472$	−0.0247542	−0.00113940
$473$	8.57673	0.394358
$474$	0	0
$475$	5.52781	0.253633
$476$	−5.17928	−0.237392
$477$	0	0
$478$	6.62027	0.302804
$479$	22.3690	1.02206	0.511032	−	0.859561i	$-0.329263\pi$
$479$	22.3690	1.02206	0.511032	+	0.859561i	$0.329263\pi$
$480$	0	0
$481$	0	0
$482$	−2.02475	−0.0922250
$483$	0	0
$484$	16.1836	0.735618
$485$	8.77479	0.398443
$486$	0	0
$487$	22.9205	1.03863	0.519313	−	0.854584i	$-0.326188\pi$
$487$	22.9205	1.03863	0.519313	+	0.854584i	$0.326188\pi$
$488$	16.4668	0.745418
$489$	0	0
$490$	0.384043	0.0173493
$491$	−1.84356	−0.0831987	−0.0415993	−	0.999134i	$-0.513245\pi$
$491$	−1.84356	−0.0831987	−0.0415993	+	0.999134i	$0.513245\pi$
$492$	0	0
$493$	−8.98254	−0.404553
$494$	0	0
$495$	0	0
$496$	13.2567	0.595242
$497$	−36.9909	−1.65927
$498$	0	0
$499$	−12.0344	−0.538736	−0.269368	−	0.963037i	$-0.586815\pi$
$499$	−12.0344	−0.538736	−0.269368	+	0.963037i	$0.586815\pi$
$500$	10.1890	0.455664
$501$	0	0
$502$	−0.762118	−0.0340150
$503$	30.5056	1.36018	0.680088	−	0.733130i	$-0.261942\pi$
$503$	30.5056	1.36018	0.680088	+	0.733130i	$0.261942\pi$
$504$	0	0
$505$	−14.8237	−0.659646
$506$	3.06100	0.136078
$507$	0	0
$508$	−11.4681	−0.508816
$509$	1.51142	0.0669924	0.0334962	−	0.999439i	$-0.489336\pi$
$509$	1.51142	0.0669924	0.0334962	+	0.999439i	$0.489336\pi$
$510$	0	0
$511$	−34.5036	−1.52635
$512$	21.2174	0.937687
$513$	0	0
$514$	−16.3357	−0.720538
$515$	−37.9124	−1.67062
$516$	0	0
$517$	9.21552	0.405298
$518$	−1.42088	−0.0624298
$519$	0	0
$520$	0	0
$521$	5.64012	0.247098	0.123549	−	0.992338i	$-0.460572\pi$
$521$	5.64012	0.247098	0.123549	+	0.992338i	$0.460572\pi$
$522$	0	0
$523$	−31.7506	−1.38836	−0.694179	−	0.719802i	$-0.744232\pi$
$523$	−31.7506	−1.38836	−0.694179	+	0.719802i	$0.744232\pi$
$524$	−23.1468	−1.01117
$525$	0	0
$526$	−5.93602	−0.258823
$527$	−6.70841	−0.292223
$528$	0	0
$529$	−1.80433	−0.0784492
$530$	−9.13169	−0.396655
$531$	0	0
$532$	8.83207	0.382919
$533$	0	0
$534$	0	0
$535$	−15.7778	−0.682133
$536$	18.9661	0.819213
$537$	0	0
$538$	5.65279	0.243709
$539$	0.295897	0.0127452
$540$	0	0
$541$	24.3297	1.04602	0.523009	−	0.852327i	$-0.324809\pi$
$541$	24.3297	1.04602	0.523009	+	0.852327i	$0.324809\pi$
$542$	−16.3448	−0.702070
$543$	0	0
$544$	−6.07739	−0.260566
$545$	11.7017	0.501246
$546$	0	0
$547$	−8.18896	−0.350135	−0.175067	−	0.984556i	$-0.556014\pi$
$547$	−8.18896	−0.350135	−0.175067	+	0.984556i	$0.556014\pi$
$548$	21.9758	0.938761
$549$	0	0
$550$	1.89546	0.0808227
$551$	15.3177	0.652555
$552$	0	0
$553$	−2.17331	−0.0924185
$554$	−5.68532	−0.241546
$555$	0	0
$556$	−20.3830	−0.864431
$557$	−25.3327	−1.07338	−0.536691	−	0.843779i	$-0.680326\pi$
$557$	−25.3327	−1.07338	−0.536691	+	0.843779i	$0.680326\pi$
$558$	0	0
$559$	0	0
$560$	−16.9487	−0.716213
$561$	0	0
$562$	6.41789	0.270723
$563$	25.3937	1.07022	0.535109	−	0.844783i	$-0.320270\pi$
$563$	25.3937	1.07022	0.535109	+	0.844783i	$0.320270\pi$
$564$	0	0
$565$	−21.4155	−0.900957
$566$	−17.0422	−0.716338
$567$	0	0
$568$	−28.1540	−1.18132
$569$	31.1347	1.30523	0.652617	−	0.757688i	$-0.273671\pi$
$569$	31.1347	1.30523	0.652617	+	0.757688i	$0.273671\pi$
$570$	0	0
$571$	−20.5090	−0.858276	−0.429138	−	0.903239i	$-0.641183\pi$
$571$	−20.5090	−0.858276	−0.429138	+	0.903239i	$0.641183\pi$
$572$	0	0
$573$	0	0
$574$	4.95646	0.206879
$575$	13.1250	0.547350
$576$	0	0
$577$	15.6890	0.653143	0.326572	−	0.945172i	$-0.394107\pi$
$577$	15.6890	0.653143	0.326572	+	0.945172i	$0.394107\pi$
$578$	−8.71678	−0.362570
$579$	0	0
$580$	−37.4523	−1.55512
$581$	−43.9681	−1.82410
$582$	0	0
$583$	−7.03577	−0.291392
$584$	−26.2610	−1.08669
$585$	0	0
$586$	10.3268	0.426595
$587$	−30.5687	−1.26171	−0.630853	−	0.775903i	$-0.717295\pi$
$587$	−30.5687	−1.26171	−0.630853	+	0.775903i	$0.717295\pi$
$588$	0	0
$589$	11.4397	0.471363
$590$	0.0187864	0.000773422 0
$591$	0	0
$592$	2.13706	0.0878328
$593$	29.6883	1.21915	0.609576	−	0.792727i	$-0.291340\pi$
$593$	29.6883	1.21915	0.609576	+	0.792727i	$0.291340\pi$
$594$	0	0
$595$	8.57673	0.351612
$596$	−1.25368	−0.0513529
$597$	0	0
$598$	0	0
$599$	−24.2325	−0.990113	−0.495057	−	0.868861i	$-0.664853\pi$
$599$	−24.2325	−0.990113	−0.495057	+	0.868861i	$0.664853\pi$
$600$	0	0
$601$	16.4819	0.672310	0.336155	−	0.941807i	$-0.390873\pi$
$601$	16.4819	0.672310	0.336155	+	0.941807i	$0.390873\pi$
$602$	−10.6950	−0.435896
$603$	0	0
$604$	32.2731	1.31317
$605$	−26.7995	−1.08956
$606$	0	0
$607$	1.43190	0.0581188	0.0290594	−	0.999578i	$-0.490749\pi$
$607$	1.43190	0.0581188	0.0290594	+	0.999578i	$0.490749\pi$
$608$	10.3636	0.420300
$609$	0	0
$610$	−12.4969	−0.505986
$611$	0	0
$612$	0	0
$613$	3.84846	0.155438	0.0777190	−	0.996975i	$-0.475236\pi$
$613$	3.84846	0.155438	0.0777190	+	0.996975i	$0.475236\pi$
$614$	4.96376	0.200321
$615$	0	0
$616$	6.60819	0.266251
$617$	15.0388	0.605437	0.302719	−	0.953080i	$-0.402106\pi$
$617$	15.0388	0.605437	0.302719	+	0.953080i	$0.402106\pi$
$618$	0	0
$619$	12.8170	0.515159	0.257579	−	0.966257i	$-0.417075\pi$
$619$	12.8170	0.515159	0.257579	+	0.966257i	$0.417075\pi$
$620$	−27.9705	−1.12332
$621$	0	0
$622$	−11.6762	−0.468174
$623$	−39.6504	−1.58856
$624$	0	0
$625$	−31.1269	−1.24508
$626$	−3.95539	−0.158089
$627$	0	0
$628$	6.80492	0.271546
$629$	−1.08144	−0.0431199
$630$	0	0
$631$	25.7517	1.02516	0.512579	−	0.858640i	$-0.328690\pi$
$631$	25.7517	1.02516	0.512579	+	0.858640i	$0.328690\pi$
$632$	−1.65412	−0.0657974
$633$	0	0
$634$	−13.2996	−0.528195
$635$	18.9909	0.753631
$636$	0	0
$637$	0	0
$638$	5.25236	0.207943
$639$	0	0
$640$	−32.3274	−1.27785
$641$	−24.4571	−0.965998	−0.482999	−	0.875621i	$-0.660453\pi$
$641$	−24.4571	−0.965998	−0.482999	+	0.875621i	$0.660453\pi$
$642$	0	0
$643$	−9.97344	−0.393314	−0.196657	−	0.980472i	$-0.563009\pi$
$643$	−9.97344	−0.393314	−0.196657	+	0.980472i	$0.563009\pi$
$644$	20.9705	0.826352
$645$	0	0
$646$	−1.22355	−0.0481401
$647$	−11.8431	−0.465600	−0.232800	−	0.972525i	$-0.574789\pi$
$647$	−11.8431	−0.465600	−0.232800	+	0.972525i	$0.574789\pi$
$648$	0	0
$649$	0.0144745	0.000568173 0
$650$	0	0
$651$	0	0
$652$	−25.6122	−1.00305
$653$	7.47411	0.292484	0.146242	−	0.989249i	$-0.453282\pi$
$653$	7.47411	0.292484	0.146242	+	0.989249i	$0.453282\pi$
$654$	0	0
$655$	38.3303	1.49769
$656$	−7.45473	−0.291058
$657$	0	0
$658$	−11.4916	−0.447988
$659$	−34.1739	−1.33123	−0.665613	−	0.746297i	$-0.731830\pi$
$659$	−34.1739	−1.33123	−0.665613	+	0.746297i	$0.731830\pi$
$660$	0	0
$661$	33.6088	1.30723	0.653615	−	0.756827i	$-0.273252\pi$
$661$	33.6088	1.30723	0.653615	+	0.756827i	$0.273252\pi$
$662$	1.60686	0.0624524
$663$	0	0
$664$	−33.4644	−1.29867
$665$	−14.6256	−0.567158
$666$	0	0
$667$	36.3696	1.40824
$668$	−10.5978	−0.410040
$669$	0	0
$670$	−14.3937	−0.556078
$671$	−9.62863	−0.371709
$672$	0	0
$673$	48.0320	1.85150	0.925750	−	0.378137i	$-0.123435\pi$
$673$	48.0320	1.85150	0.925750	+	0.378137i	$0.123435\pi$
$674$	−1.72348	−0.0663860
$675$	0	0
$676$	0	0
$677$	33.6582	1.29359	0.646794	−	0.762665i	$-0.276109\pi$
$677$	33.6582	1.29359	0.646794	+	0.762665i	$0.276109\pi$
$678$	0	0
$679$	−8.43057	−0.323535
$680$	6.52781	0.250330
$681$	0	0
$682$	3.92261	0.150204
$683$	−15.9041	−0.608553	−0.304276	−	0.952584i	$-0.598415\pi$
$683$	−15.9041	−0.608553	−0.304276	+	0.952584i	$0.598415\pi$
$684$	0	0
$685$	−36.3913	−1.39044
$686$	10.0887	0.385190
$687$	0	0
$688$	16.0858	0.613264
$689$	0	0
$690$	0	0
$691$	33.1903	1.26262	0.631309	−	0.775531i	$-0.282518\pi$
$691$	33.1903	1.26262	0.631309	+	0.775531i	$0.282518\pi$
$692$	27.7345	1.05431
$693$	0	0
$694$	6.31468	0.239702
$695$	33.7536	1.28035
$696$	0	0
$697$	3.77240	0.142890
$698$	−1.85756	−0.0703097
$699$	0	0
$700$	12.9855	0.490807
$701$	14.9129	0.563253	0.281627	−	0.959524i	$-0.409126\pi$
$701$	14.9129	0.563253	0.281627	+	0.959524i	$0.409126\pi$
$702$	0	0
$703$	1.84415	0.0695534
$704$	−1.83041	−0.0689863
$705$	0	0
$706$	−0.353912	−0.0133197
$707$	14.2422	0.535633
$708$	0	0
$709$	38.4312	1.44331	0.721656	−	0.692252i	$-0.243381\pi$
$709$	38.4312	1.44331	0.721656	+	0.692252i	$0.243381\pi$
$710$	21.3666	0.801874
$711$	0	0
$712$	−30.1782	−1.13098
$713$	27.1618	1.01722
$714$	0	0
$715$	0	0
$716$	−4.15346	−0.155222
$717$	0	0
$718$	11.9089	0.444435
$719$	11.4373	0.426538	0.213269	−	0.976993i	$-0.431589\pi$
$719$	11.4373	0.426538	0.213269	+	0.976993i	$0.431589\pi$
$720$	0	0
$721$	36.4252	1.35654
$722$	−8.45771	−0.314764
$723$	0	0
$724$	−19.9782	−0.742485
$725$	22.5211	0.836413
$726$	0	0
$727$	3.63640	0.134867	0.0674333	−	0.997724i	$-0.478519\pi$
$727$	3.63640	0.134867	0.0674333	+	0.997724i	$0.478519\pi$
$728$	0	0
$729$	0	0
$730$	19.9299	0.737639
$731$	−8.14005	−0.301071
$732$	0	0
$733$	−3.52217	−0.130094	−0.0650472	−	0.997882i	$-0.520720\pi$
$733$	−3.52217	−0.130094	−0.0650472	+	0.997882i	$0.520720\pi$
$734$	−5.20941	−0.192283
$735$	0	0
$736$	24.6069	0.907021
$737$	−11.0901	−0.408508
$738$	0	0
$739$	0.420288	0.0154605	0.00773027	−	0.999970i	$-0.497539\pi$
$739$	0.420288	0.0154605	0.00773027	+	0.999970i	$0.497539\pi$
$740$	−4.50902	−0.165755
$741$	0	0
$742$	8.77346	0.322084
$743$	25.3623	0.930452	0.465226	−	0.885192i	$-0.345973\pi$
$743$	25.3623	0.930452	0.465226	+	0.885192i	$0.345973\pi$
$744$	0	0
$745$	2.07606	0.0760611
$746$	15.3870	0.563359
$747$	0	0
$748$	2.30499	0.0842790
$749$	15.1588	0.553892
$750$	0	0
$751$	0.650874	0.0237507	0.0118754	−	0.999929i	$-0.496220\pi$
$751$	0.650874	0.0237507	0.0118754	+	0.999929i	$0.496220\pi$
$752$	17.2838	0.630276
$753$	0	0
$754$	0	0
$755$	−53.4432	−1.94500
$756$	0	0
$757$	−16.7909	−0.610276	−0.305138	−	0.952308i	$-0.598703\pi$
$757$	−16.7909	−0.610276	−0.305138	+	0.952308i	$0.598703\pi$
$758$	19.9065	0.723036
$759$	0	0
$760$	−11.1317	−0.403789
$761$	−30.9221	−1.12093	−0.560463	−	0.828179i	$-0.689377\pi$
$761$	−30.9221	−1.12093	−0.560463	+	0.828179i	$0.689377\pi$
$762$	0	0
$763$	−11.2427	−0.407012
$764$	−15.2168	−0.550526
$765$	0	0
$766$	−2.69441	−0.0973531
$767$	0	0
$768$	0	0
$769$	−43.7689	−1.57835	−0.789174	−	0.614169i	$-0.789491\pi$
$769$	−43.7689	−1.57835	−0.789174	+	0.614169i	$0.789491\pi$
$770$	−5.01507	−0.180730
$771$	0	0
$772$	−22.8853	−0.823660
$773$	42.4209	1.52577	0.762886	−	0.646532i	$-0.223782\pi$
$773$	42.4209	1.52577	0.762886	+	0.646532i	$0.223782\pi$
$774$	0	0
$775$	16.8194	0.604171
$776$	−6.41657	−0.230341
$777$	0	0
$778$	−1.32378	−0.0474598
$779$	−6.43296	−0.230485
$780$	0	0
$781$	16.4625	0.589075
$782$	−2.90515	−0.103888
$783$	0	0
$784$	0.554958	0.0198199
$785$	−11.2687	−0.402199
$786$	0	0
$787$	−36.0116	−1.28368	−0.641838	−	0.766841i	$-0.721828\pi$
$787$	−36.0116	−1.28368	−0.641838	+	0.766841i	$0.721828\pi$
$788$	21.9554	0.782129
$789$	0	0
$790$	1.25534	0.0446630
$791$	20.5754	0.731577
$792$	0	0
$793$	0	0
$794$	−8.47219	−0.300667
$795$	0	0
$796$	22.9885	0.814806
$797$	31.7101	1.12323	0.561614	−	0.827399i	$-0.310181\pi$
$797$	31.7101	1.12323	0.561614	+	0.827399i	$0.310181\pi$
$798$	0	0
$799$	−8.74632	−0.309422
$800$	15.2373	0.538720
$801$	0	0
$802$	−7.08038	−0.250017
$803$	15.3556	0.541886
$804$	0	0
$805$	−34.7265	−1.22395
$806$	0	0
$807$	0	0
$808$	10.8398	0.381344
$809$	45.2814	1.59201	0.796005	−	0.605290i	$-0.206943\pi$
$809$	45.2814	1.59201	0.796005	+	0.605290i	$0.206943\pi$
$810$	0	0
$811$	−42.8635	−1.50514	−0.752571	−	0.658511i	$-0.771187\pi$
$811$	−42.8635	−1.50514	−0.752571	+	0.658511i	$0.771187\pi$
$812$	35.9831	1.26276
$813$	0	0
$814$	0.632351	0.0221639
$815$	42.4131	1.48567
$816$	0	0
$817$	13.8810	0.485634
$818$	−14.0731	−0.492054
$819$	0	0
$820$	15.7289	0.549276
$821$	7.82776	0.273191	0.136595	−	0.990627i	$-0.456384\pi$
$821$	7.82776	0.273191	0.136595	+	0.990627i	$0.456384\pi$
$822$	0	0
$823$	−36.7754	−1.28191	−0.640955	−	0.767579i	$-0.721461\pi$
$823$	−36.7754	−1.28191	−0.640955	+	0.767579i	$0.721461\pi$
$824$	27.7235	0.965793
$825$	0	0
$826$	−0.0180494	−0.000628019 0
$827$	−47.3293	−1.64580	−0.822900	−	0.568186i	$-0.807645\pi$
$827$	−47.3293	−1.64580	−0.822900	+	0.568186i	$0.807645\pi$
$828$	0	0
$829$	25.2687	0.877620	0.438810	−	0.898580i	$-0.355400\pi$
$829$	25.2687	0.877620	0.438810	+	0.898580i	$0.355400\pi$
$830$	25.3967	0.881533
$831$	0	0
$832$	0	0
$833$	−0.280831	−0.00973023
$834$	0	0
$835$	17.5496	0.607328
$836$	−3.93064	−0.135944
$837$	0	0
$838$	6.47484	0.223670
$839$	37.6883	1.30114	0.650572	−	0.759444i	$-0.274529\pi$
$839$	37.6883	1.30114	0.650572	+	0.759444i	$0.274529\pi$
$840$	0	0
$841$	33.4064	1.15194
$842$	−4.60222	−0.158603
$843$	0	0
$844$	−17.6987	−0.609215
$845$	0	0
$846$	0	0
$847$	25.7482	0.884720
$848$	−13.1957	−0.453141
$849$	0	0
$850$	−1.79895	−0.0617036
$851$	4.37867	0.150099
$852$	0	0
$853$	−31.0121	−1.06183	−0.530917	−	0.847424i	$-0.678152\pi$
$853$	−31.0121	−1.06183	−0.530917	+	0.847424i	$0.678152\pi$
$854$	12.0067	0.410861
$855$	0	0
$856$	11.5375	0.394344
$857$	−12.4692	−0.425940	−0.212970	−	0.977059i	$-0.568314\pi$
$857$	−12.4692	−0.425940	−0.212970	+	0.977059i	$0.568314\pi$
$858$	0	0
$859$	−17.3163	−0.590826	−0.295413	−	0.955370i	$-0.595457\pi$
$859$	−17.3163	−0.590826	−0.295413	+	0.955370i	$0.595457\pi$
$860$	−33.9396	−1.15733
$861$	0	0
$862$	−0.517385	−0.0176222
$863$	3.46383	0.117910	0.0589550	−	0.998261i	$-0.481223\pi$
$863$	3.46383	0.117910	0.0589550	+	0.998261i	$0.481223\pi$
$864$	0	0
$865$	−45.9275	−1.56158
$866$	−7.40880	−0.251761
$867$	0	0
$868$	26.8732	0.912136
$869$	0.967213	0.0328105
$870$	0	0
$871$	0	0
$872$	−8.55688	−0.289772
$873$	0	0
$874$	4.95407	0.167574
$875$	16.2107	0.548023
$876$	0	0
$877$	57.2549	1.93336	0.966680	−	0.255989i	$-0.0824010\pi$
$877$	57.2549	1.93336	0.966680	+	0.255989i	$0.0824010\pi$
$878$	7.76510	0.262059
$879$	0	0
$880$	7.54288	0.254270
$881$	43.1782	1.45471	0.727355	−	0.686261i	$-0.240749\pi$
$881$	43.1782	1.45471	0.727355	+	0.686261i	$0.240749\pi$
$882$	0	0
$883$	49.9560	1.68115	0.840576	−	0.541693i	$-0.182216\pi$
$883$	49.9560	1.68115	0.840576	+	0.541693i	$0.182216\pi$
$884$	0	0
$885$	0	0
$886$	−13.1535	−0.441899
$887$	−17.6746	−0.593454	−0.296727	−	0.954962i	$-0.595895\pi$
$887$	−17.6746	−0.593454	−0.296727	+	0.954962i	$0.595895\pi$
$888$	0	0
$889$	−18.2459	−0.611948
$890$	22.9028	0.767702
$891$	0	0
$892$	19.2989	0.646174
$893$	14.9148	0.499106
$894$	0	0
$895$	6.87800	0.229906
$896$	31.0592	1.03761
$897$	0	0
$898$	6.98493	0.233090
$899$	46.6069	1.55443
$900$	0	0
$901$	6.67755	0.222461
$902$	−2.20583	−0.0734462
$903$	0	0
$904$	15.6601	0.520847
$905$	33.0834	1.09973
$906$	0	0
$907$	7.73423	0.256811	0.128406	−	0.991722i	$-0.459014\pi$
$907$	7.73423	0.256811	0.128406	+	0.991722i	$0.459014\pi$
$908$	18.0043	0.597494
$909$	0	0
$910$	0	0
$911$	−39.6179	−1.31260	−0.656299	−	0.754501i	$-0.727879\pi$
$911$	−39.6179	−1.31260	−0.656299	+	0.754501i	$0.727879\pi$
$912$	0	0
$913$	19.5676	0.647594
$914$	−18.6679	−0.617478
$915$	0	0
$916$	1.92798	0.0637024
$917$	−36.8267	−1.21612
$918$	0	0
$919$	14.6213	0.482313	0.241157	−	0.970486i	$-0.422473\pi$
$919$	14.6213	0.482313	0.241157	+	0.970486i	$0.422473\pi$
$920$	−26.4306	−0.871390
$921$	0	0
$922$	0.778512	0.0256389
$923$	0	0
$924$	0	0
$925$	2.71140	0.0891502
$926$	−8.43594	−0.277222
$927$	0	0
$928$	42.2228	1.38603
$929$	−3.55735	−0.116713	−0.0583565	−	0.998296i	$-0.518586\pi$
$929$	−3.55735	−0.116713	−0.0583565	+	0.998296i	$0.518586\pi$
$930$	0	0
$931$	0.478894	0.0156951
$932$	−18.3599	−0.601398
$933$	0	0
$934$	21.8328	0.714391
$935$	−3.81700	−0.124829
$936$	0	0
$937$	34.5526	1.12878	0.564392	−	0.825507i	$-0.309111\pi$
$937$	34.5526	1.12878	0.564392	+	0.825507i	$0.309111\pi$
$938$	13.8291	0.451536
$939$	0	0
$940$	−36.4674	−1.18944
$941$	−20.6233	−0.672299	−0.336149	−	0.941809i	$-0.609125\pi$
$941$	−20.6233	−0.672299	−0.336149	+	0.941809i	$0.609125\pi$
$942$	0	0
$943$	−15.2741	−0.497394
$944$	0.0271471	0.000883562 0
$945$	0	0
$946$	4.75973	0.154752
$947$	−29.4999	−0.958619	−0.479309	−	0.877646i	$-0.659113\pi$
$947$	−29.4999	−0.958619	−0.479309	+	0.877646i	$0.659113\pi$
$948$	0	0
$949$	0	0
$950$	3.06770	0.0995295
$951$	0	0
$952$	−6.27173	−0.203268
$953$	−26.2389	−0.849963	−0.424981	−	0.905202i	$-0.639719\pi$
$953$	−26.2389	−0.849963	−0.424981	+	0.905202i	$0.639719\pi$
$954$	0	0
$955$	25.1987	0.815409
$956$	−20.1847	−0.652818
$957$	0	0
$958$	12.4138	0.401073
$959$	34.9638	1.12904
$960$	0	0
$961$	3.80731	0.122817
$962$	0	0
$963$	0	0
$964$	6.17331	0.198829
$965$	37.8974	1.21996
$966$	0	0
$967$	17.5176	0.563330	0.281665	−	0.959513i	$-0.409113\pi$
$967$	17.5176	0.563330	0.281665	+	0.959513i	$0.409113\pi$
$968$	19.5972	0.629877
$969$	0	0
$970$	4.86964	0.156355
$971$	−20.5120	−0.658262	−0.329131	−	0.944284i	$-0.606756\pi$
$971$	−20.5120	−0.658262	−0.329131	+	0.944284i	$0.606756\pi$
$972$	0	0
$973$	−32.4295	−1.03964
$974$	12.7199	0.407572
$975$	0	0
$976$	−18.0586	−0.578042
$977$	−25.4450	−0.814059	−0.407030	−	0.913415i	$-0.633435\pi$
$977$	−25.4450	−0.814059	−0.407030	+	0.913415i	$0.633435\pi$
$978$	0	0
$979$	17.6461	0.563971
$980$	−1.17092	−0.0374035
$981$	0	0
$982$	−1.02310	−0.0326484
$983$	39.5244	1.26063	0.630316	−	0.776339i	$-0.282926\pi$
$983$	39.5244	1.26063	0.630316	+	0.776339i	$0.282926\pi$
$984$	0	0
$985$	−36.3575	−1.15845
$986$	−4.98493	−0.158753
$987$	0	0
$988$	0	0
$989$	32.9584	1.04802
$990$	0	0
$991$	−29.8377	−0.947826	−0.473913	−	0.880572i	$-0.657159\pi$
$991$	−29.8377	−0.947826	−0.473913	+	0.880572i	$0.657159\pi$
$992$	31.5332	1.00118
$993$	0	0
$994$	−20.5284	−0.651121
$995$	−38.0683	−1.20685
$996$	0	0
$997$	−4.93123	−0.156174	−0.0780868	−	0.996947i	$-0.524881\pi$
$997$	−4.93123	−0.156174	−0.0780868	+	0.996947i	$0.524881\pi$
$998$	−6.67861	−0.211408
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.r.1.2		3
3.2	odd	2		169.2.a.b.1.2	✓	3
12.11	even	2		2704.2.a.z.1.1		3
13.5	odd	4		1521.2.b.l.1351.3		6
13.8	odd	4		1521.2.b.l.1351.4		6
13.12	even	2		1521.2.a.o.1.2		3
15.14	odd	2		4225.2.a.bg.1.2		3
21.20	even	2		8281.2.a.bf.1.2		3
39.2	even	12		169.2.e.b.147.3		12
39.5	even	4		169.2.b.b.168.4		6
39.8	even	4		169.2.b.b.168.3		6
39.11	even	12		169.2.e.b.147.4		12
39.17	odd	6		169.2.c.b.146.2		6
39.20	even	12		169.2.e.b.23.3		12
39.23	odd	6		169.2.c.b.22.2		6
39.29	odd	6		169.2.c.c.22.2		6
39.32	even	12		169.2.e.b.23.4		12
39.35	odd	6		169.2.c.c.146.2		6
39.38	odd	2		169.2.a.c.1.2	yes	3
156.47	odd	4		2704.2.f.o.337.1		6
156.83	odd	4		2704.2.f.o.337.2		6
156.155	even	2		2704.2.a.ba.1.1		3
195.194	odd	2		4225.2.a.bb.1.2		3
273.272	even	2		8281.2.a.bj.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.2	✓	3	3.2	odd	2
169.2.a.c.1.2	yes	3	39.38	odd	2
169.2.b.b.168.3		6	39.8	even	4
169.2.b.b.168.4		6	39.5	even	4
169.2.c.b.22.2		6	39.23	odd	6
169.2.c.b.146.2		6	39.17	odd	6
169.2.c.c.22.2		6	39.29	odd	6
169.2.c.c.146.2		6	39.35	odd	6
169.2.e.b.23.3		12	39.20	even	12
169.2.e.b.23.4		12	39.32	even	12
169.2.e.b.147.3		12	39.2	even	12
169.2.e.b.147.4		12	39.11	even	12
1521.2.a.o.1.2		3	13.12	even	2
1521.2.a.r.1.2		3	1.1	even	1	trivial
1521.2.b.l.1351.3		6	13.5	odd	4
1521.2.b.l.1351.4		6	13.8	odd	4
2704.2.a.z.1.1		3	12.11	even	2
2704.2.a.ba.1.1		3	156.155	even	2
2704.2.f.o.337.1		6	156.47	odd	4
2704.2.f.o.337.2		6	156.83	odd	4
4225.2.a.bb.1.2		3	195.194	odd	2
4225.2.a.bg.1.2		3	15.14	odd	2
8281.2.a.bf.1.2		3	21.20	even	2
8281.2.a.bj.1.2		3	273.272	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+0.554958 q^{2} -1.69202 q^{4} +2.80194 q^{5} -2.69202 q^{7} -2.04892 q^{8} +O(q^{10})\)	\(q+0.554958 q^{2} -1.69202 q^{4} +2.80194 q^{5} -2.69202 q^{7} -2.04892 q^{8} +1.55496 q^{10} +1.19806 q^{11} -1.49396 q^{14} +2.24698 q^{16} -1.13706 q^{17} +1.93900 q^{19} -4.74094 q^{20} +0.664874 q^{22} +4.60388 q^{23} +2.85086 q^{25} +4.55496 q^{28} +7.89977 q^{29} +5.89977 q^{31} +5.34481 q^{32} -0.631023 q^{34} -7.54288 q^{35} +0.951083 q^{37} +1.07606 q^{38} -5.74094 q^{40} -3.31767 q^{41} +7.15883 q^{43} -2.02715 q^{44} +2.55496 q^{46} +7.69202 q^{47} +0.246980 q^{49} +1.58211 q^{50} -5.87263 q^{53} +3.35690 q^{55} +5.51573 q^{56} +4.38404 q^{58} +0.0120816 q^{59} -8.03684 q^{61} +3.27413 q^{62} -1.52781 q^{64} -9.25667 q^{67} +1.92394 q^{68} -4.18598 q^{70} +13.7409 q^{71} +12.8170 q^{73} +0.527811 q^{74} -3.28083 q^{76} -3.22521 q^{77} +0.807315 q^{79} +6.29590 q^{80} -1.84117 q^{82} +16.3327 q^{83} -3.18598 q^{85} +3.97285 q^{86} -2.45473 q^{88} +14.7289 q^{89} -7.78986 q^{92} +4.26875 q^{94} +5.43296 q^{95} +3.13169 q^{97} +0.137063 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 4 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 4 * q^5 - 3 * q^7 + 3 * q^8	\( 3 q + 2 q^{2} + 4 q^{5} - 3 q^{7} + 3 q^{8} + 5 q^{10} + 8 q^{11} + 5 q^{14} + 2 q^{16} + 2 q^{17} - 4 q^{19} + 3 q^{22} + 5 q^{23} - 5 q^{25} + 14 q^{28} + q^{29} - 5 q^{31} - 7 q^{32} + 13 q^{34} - 4 q^{35} + 12 q^{37} - 12 q^{38} - 3 q^{40} + 7 q^{41} + 13 q^{43} + 8 q^{46} + 18 q^{47} - 4 q^{49} - q^{50} - q^{53} + 6 q^{55} + 4 q^{56} + 3 q^{58} + 19 q^{59} + 4 q^{61} - q^{62} - 11 q^{64} - q^{67} + 21 q^{68} + 2 q^{70} + 27 q^{71} + 9 q^{73} + 8 q^{74} - 21 q^{76} - 8 q^{77} - 5 q^{79} + 5 q^{80} - 14 q^{82} + 7 q^{83} + 5 q^{85} + 18 q^{86} + 15 q^{88} + 11 q^{89} + 5 q^{94} - 3 q^{95} + 7 q^{97} - 5 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 4 * q^5 - 3 * q^7 + 3 * q^8 + 5 * q^10 + 8 * q^11 + 5 * q^14 + 2 * q^16 + 2 * q^17 - 4 * q^19 + 3 * q^22 + 5 * q^23 - 5 * q^25 + 14 * q^28 + q^29 - 5 * q^31 - 7 * q^32 + 13 * q^34 - 4 * q^35 + 12 * q^37 - 12 * q^38 - 3 * q^40 + 7 * q^41 + 13 * q^43 + 8 * q^46 + 18 * q^47 - 4 * q^49 - q^50 - q^53 + 6 * q^55 + 4 * q^56 + 3 * q^58 + 19 * q^59 + 4 * q^61 - q^62 - 11 * q^64 - q^67 + 21 * q^68 + 2 * q^70 + 27 * q^71 + 9 * q^73 + 8 * q^74 - 21 * q^76 - 8 * q^77 - 5 * q^79 + 5 * q^80 - 14 * q^82 + 7 * q^83 + 5 * q^85 + 18 * q^86 + 15 * q^88 + 11 * q^89 + 5 * q^94 - 3 * q^95 + 7 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

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Representations

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Database

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