LMFDB - Embedded newform 1521.2.a.p.1.3

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 507)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ 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)

$f(q)$	$=$	$q+1.24698 q^{2} -0.445042 q^{4} -2.80194 q^{5} +4.80194 q^{7} -3.04892 q^{8} +O(q^{10})$	$q+1.24698 q^{2} -0.445042 q^{4} -2.80194 q^{5} +4.80194 q^{7} -3.04892 q^{8} -3.49396 q^{10} +1.46681 q^{11} +5.98792 q^{14} -2.91185 q^{16} +2.44504 q^{17} -2.54288 q^{19} +1.24698 q^{20} +1.82908 q^{22} +3.51573 q^{23} +2.85086 q^{25} -2.13706 q^{28} -1.85086 q^{29} +7.63102 q^{31} +2.46681 q^{32} +3.04892 q^{34} -13.4547 q^{35} +4.55496 q^{37} -3.17092 q^{38} +8.54288 q^{40} +1.24698 q^{41} +2.38404 q^{43} -0.652793 q^{44} +4.38404 q^{46} +12.8170 q^{47} +16.0586 q^{49} +3.55496 q^{50} +8.85086 q^{53} -4.10992 q^{55} -14.6407 q^{56} -2.30798 q^{58} +2.17629 q^{59} -7.82908 q^{61} +9.51573 q^{62} +8.89977 q^{64} +3.58211 q^{67} -1.08815 q^{68} -16.7778 q^{70} -8.83877 q^{71} +7.69202 q^{73} +5.67994 q^{74} +1.13169 q^{76} +7.04354 q^{77} -4.02177 q^{79} +8.15883 q^{80} +1.55496 q^{82} -0.652793 q^{83} -6.85086 q^{85} +2.97285 q^{86} -4.47219 q^{88} +6.29590 q^{89} -1.56465 q^{92} +15.9825 q^{94} +7.12498 q^{95} +10.0315 q^{97} +20.0248 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - q^{4} - 4 q^{5} + 10 q^{7}+O(q^{10}) $ 3 * q - q^2 - q^4 - 4 * q^5 + 10 * q^7	$ 3 q - q^{2} - q^{4} - 4 q^{5} + 10 q^{7} - q^{10} + q^{11} - q^{14} - 5 q^{16} + 7 q^{17} + 11 q^{19} - q^{20} - 5 q^{22} - 2 q^{23} - 5 q^{25} - q^{28} + 8 q^{29} + 8 q^{31} + 4 q^{32} - 18 q^{35} + 14 q^{37} - 20 q^{38} + 7 q^{40} - q^{41} - 3 q^{43} + 16 q^{44} + 3 q^{46} + 9 q^{47} + 17 q^{49} + 11 q^{50} + 13 q^{53} - 13 q^{55} - 7 q^{56} - 12 q^{58} + 14 q^{59} - 13 q^{61} + 16 q^{62} + 4 q^{64} + 5 q^{67} - 7 q^{68} - 8 q^{70} + 6 q^{71} + 18 q^{73} - 7 q^{74} + q^{76} + 15 q^{77} - 9 q^{79} + 16 q^{80} + 5 q^{82} + 16 q^{83} - 7 q^{85} + 15 q^{86} - 7 q^{88} + 5 q^{89} + 17 q^{92} + 32 q^{94} - 3 q^{95} + 5 q^{97} + 13 q^{98}+O(q^{100}) $ 3 * q - q^2 - q^4 - 4 * q^5 + 10 * q^7 - q^10 + q^11 - q^14 - 5 * q^16 + 7 * q^17 + 11 * q^19 - q^20 - 5 * q^22 - 2 * q^23 - 5 * q^25 - q^28 + 8 * q^29 + 8 * q^31 + 4 * q^32 - 18 * q^35 + 14 * q^37 - 20 * q^38 + 7 * q^40 - q^41 - 3 * q^43 + 16 * q^44 + 3 * q^46 + 9 * q^47 + 17 * q^49 + 11 * q^50 + 13 * q^53 - 13 * q^55 - 7 * q^56 - 12 * q^58 + 14 * q^59 - 13 * q^61 + 16 * q^62 + 4 * q^64 + 5 * q^67 - 7 * q^68 - 8 * q^70 + 6 * q^71 + 18 * q^73 - 7 * q^74 + q^76 + 15 * q^77 - 9 * q^79 + 16 * q^80 + 5 * q^82 + 16 * q^83 - 7 * q^85 + 15 * q^86 - 7 * q^88 + 5 * q^89 + 17 * q^92 + 32 * q^94 - 3 * q^95 + 5 * q^97 + 13 * 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 $

$2$	1.24698	0.881748	0.440874	−	0.897569i	$-0.354669\pi$
$2$	1.24698	0.881748	0.440874	+	0.897569i	$0.354669\pi$
$3$	0	0
$3$	0	0
$4$	−0.445042	−0.222521
$5$	−2.80194	−1.25306	−0.626532	−	0.779395i	$-0.715526\pi$
$5$	−2.80194	−1.25306	−0.626532	+	0.779395i	$0.715526\pi$
$6$	0	0
$7$	4.80194	1.81496	0.907481	−	0.420093i	$-0.138003\pi$
$7$	4.80194	1.81496	0.907481	+	0.420093i	$0.138003\pi$
$8$	−3.04892	−1.07796
$9$	0	0
$10$	−3.49396	−1.10489
$11$	1.46681	0.442260	0.221130	−	0.975244i	$-0.429025\pi$
$11$	1.46681	0.442260	0.221130	+	0.975244i	$0.429025\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	5.98792	1.60034
$15$	0	0
$16$	−2.91185	−0.727963
$17$	2.44504	0.593010	0.296505	−	0.955031i	$-0.404179\pi$
$17$	2.44504	0.593010	0.296505	+	0.955031i	$0.404179\pi$
$18$	0	0
$19$	−2.54288	−0.583376	−0.291688	−	0.956514i	$-0.594217\pi$
$19$	−2.54288	−0.583376	−0.291688	+	0.956514i	$0.594217\pi$
$20$	1.24698	0.278833
$21$	0	0
$22$	1.82908	0.389962
$23$	3.51573	0.733080	0.366540	−	0.930402i	$-0.380542\pi$
$23$	3.51573	0.733080	0.366540	+	0.930402i	$0.380542\pi$
$24$	0	0
$25$	2.85086	0.570171
$26$	0	0
$27$	0	0
$28$	−2.13706	−0.403867
$29$	−1.85086	−0.343695	−0.171848	−	0.985124i	$-0.554974\pi$
$29$	−1.85086	−0.343695	−0.171848	+	0.985124i	$0.554974\pi$
$30$	0	0
$31$	7.63102	1.37057	0.685286	−	0.728274i	$-0.259677\pi$
$31$	7.63102	1.37057	0.685286	+	0.728274i	$0.259677\pi$
$32$	2.46681	0.436075
$33$	0	0
$34$	3.04892	0.522885
$35$	−13.4547	−2.27426
$36$	0	0
$37$	4.55496	0.748831	0.374415	−	0.927261i	$-0.377843\pi$
$37$	4.55496	0.748831	0.374415	+	0.927261i	$0.377843\pi$
$38$	−3.17092	−0.514390
$39$	0	0
$40$	8.54288	1.35075
$41$	1.24698	0.194745	0.0973727	−	0.995248i	$-0.468956\pi$
$41$	1.24698	0.194745	0.0973727	+	0.995248i	$0.468956\pi$
$42$	0	0
$43$	2.38404	0.363563	0.181782	−	0.983339i	$-0.441814\pi$
$43$	2.38404	0.363563	0.181782	+	0.983339i	$0.441814\pi$
$44$	−0.652793	−0.0984122
$45$	0	0
$46$	4.38404	0.646392
$47$	12.8170	1.86955	0.934776	−	0.355238i	$-0.115600\pi$
$47$	12.8170	1.86955	0.934776	+	0.355238i	$0.115600\pi$
$48$	0	0
$49$	16.0586	2.29409
$50$	3.55496	0.502747
$51$	0	0
$52$	0	0
$53$	8.85086	1.21576	0.607879	−	0.794030i	$-0.292020\pi$
$53$	8.85086	1.21576	0.607879	+	0.794030i	$0.292020\pi$
$54$	0	0
$55$	−4.10992	−0.554181
$56$	−14.6407	−1.95645
$57$	0	0
$58$	−2.30798	−0.303052
$59$	2.17629	0.283329	0.141665	−	0.989915i	$-0.454755\pi$
$59$	2.17629	0.283329	0.141665	+	0.989915i	$0.454755\pi$
$60$	0	0
$61$	−7.82908	−1.00241	−0.501206	−	0.865328i	$-0.667110\pi$
$61$	−7.82908	−1.00241	−0.501206	+	0.865328i	$0.667110\pi$
$62$	9.51573	1.20850
$63$	0	0
$64$	8.89977	1.11247
$65$	0	0
$66$	0	0
$67$	3.58211	0.437624	0.218812	−	0.975767i	$-0.429782\pi$
$67$	3.58211	0.437624	0.218812	+	0.975767i	$0.429782\pi$
$68$	−1.08815	−0.131957
$69$	0	0
$70$	−16.7778	−2.00533
$71$	−8.83877	−1.04897	−0.524485	−	0.851420i	$-0.675742\pi$
$71$	−8.83877	−1.04897	−0.524485	+	0.851420i	$0.675742\pi$
$72$	0	0
$73$	7.69202	0.900283	0.450142	−	0.892957i	$-0.351374\pi$
$73$	7.69202	0.900283	0.450142	+	0.892957i	$0.351374\pi$
$74$	5.67994	0.660280
$75$	0	0
$76$	1.13169	0.129813
$77$	7.04354	0.802686
$78$	0	0
$79$	−4.02177	−0.452485	−0.226242	−	0.974071i	$-0.572644\pi$
$79$	−4.02177	−0.452485	−0.226242	+	0.974071i	$0.572644\pi$
$80$	8.15883	0.912185
$81$	0	0
$82$	1.55496	0.171716
$83$	−0.652793	−0.0716533	−0.0358267	−	0.999358i	$-0.511406\pi$
$83$	−0.652793	−0.0716533	−0.0358267	+	0.999358i	$0.511406\pi$
$84$	0	0
$85$	−6.85086	−0.743080
$86$	2.97285	0.320571
$87$	0	0
$88$	−4.47219	−0.476737
$89$	6.29590	0.667364	0.333682	−	0.942686i	$-0.391709\pi$
$89$	6.29590	0.667364	0.333682	+	0.942686i	$0.391709\pi$
$90$	0	0
$91$	0	0
$92$	−1.56465	−0.163126
$93$	0	0
$94$	15.9825	1.64847
$95$	7.12498	0.731008
$96$	0	0
$97$	10.0315	1.01854	0.509270	−	0.860607i	$-0.329915\pi$
$97$	10.0315	1.01854	0.509270	+	0.860607i	$0.329915\pi$
$98$	20.0248	2.02281
$99$	0	0
$100$	−1.26875	−0.126875
$101$	−13.8877	−1.38188	−0.690938	−	0.722914i	$-0.742802\pi$
$101$	−13.8877	−1.38188	−0.690938	+	0.722914i	$0.742802\pi$
$102$	0	0
$103$	−17.4034	−1.71481	−0.857405	−	0.514642i	$-0.827925\pi$
$103$	−17.4034	−1.71481	−0.857405	+	0.514642i	$0.827925\pi$
$104$	0	0
$105$	0	0
$106$	11.0368	1.07199
$107$	−10.5526	−1.02015	−0.510077	−	0.860128i	$-0.670383\pi$
$107$	−10.5526	−1.02015	−0.510077	+	0.860128i	$0.670383\pi$
$108$	0	0
$109$	−1.07069	−0.102553	−0.0512766	−	0.998684i	$-0.516329\pi$
$109$	−1.07069	−0.102553	−0.0512766	+	0.998684i	$0.516329\pi$
$110$	−5.12498	−0.488648
$111$	0	0
$112$	−13.9825	−1.32123
$113$	16.5308	1.55509	0.777543	−	0.628830i	$-0.216466\pi$
$113$	16.5308	1.55509	0.777543	+	0.628830i	$0.216466\pi$
$114$	0	0
$115$	−9.85086	−0.918597
$116$	0.823708	0.0764794
$117$	0	0
$118$	2.71379	0.249825
$119$	11.7409	1.07629
$120$	0	0
$121$	−8.84846	−0.804406
$122$	−9.76271	−0.883874
$123$	0	0
$124$	−3.39612	−0.304981
$125$	6.02177	0.538604
$126$	0	0
$127$	−9.53750	−0.846316	−0.423158	−	0.906056i	$-0.639079\pi$
$127$	−9.53750	−0.846316	−0.423158	+	0.906056i	$0.639079\pi$
$128$	6.16421	0.544844
$129$	0	0
$130$	0	0
$131$	5.50902	0.481326	0.240663	−	0.970609i	$-0.422635\pi$
$131$	5.50902	0.481326	0.240663	+	0.970609i	$0.422635\pi$
$132$	0	0
$133$	−12.2107	−1.05880
$134$	4.46681	0.385874
$135$	0	0
$136$	−7.45473	−0.639238
$137$	−16.1836	−1.38266	−0.691329	−	0.722540i	$-0.742974\pi$
$137$	−16.1836	−1.38266	−0.691329	+	0.722540i	$0.742974\pi$
$138$	0	0
$139$	−10.5090	−0.891364	−0.445682	−	0.895191i	$-0.647039\pi$
$139$	−10.5090	−0.891364	−0.445682	+	0.895191i	$0.647039\pi$
$140$	5.98792	0.506071
$141$	0	0
$142$	−11.0218	−0.924926
$143$	0	0
$144$	0	0
$145$	5.18598	0.430672
$146$	9.59179	0.793823
$147$	0	0
$148$	−2.02715	−0.166630
$149$	14.3502	1.17561	0.587807	−	0.809001i	$-0.299992\pi$
$149$	14.3502	1.17561	0.587807	+	0.809001i	$0.299992\pi$
$150$	0	0
$151$	−1.96615	−0.160003	−0.0800014	−	0.996795i	$-0.525492\pi$
$151$	−1.96615	−0.160003	−0.0800014	+	0.996795i	$0.525492\pi$
$152$	7.75302	0.628853
$153$	0	0
$154$	8.78315	0.707767
$155$	−21.3817	−1.71742
$156$	0	0
$157$	10.7017	0.854089	0.427045	−	0.904231i	$-0.359555\pi$
$157$	10.7017	0.854089	0.427045	+	0.904231i	$0.359555\pi$
$158$	−5.01507	−0.398977
$159$	0	0
$160$	−6.91185	−0.546430
$161$	16.8823	1.33051
$162$	0	0
$163$	−3.89977	−0.305454	−0.152727	−	0.988268i	$-0.548805\pi$
$163$	−3.89977	−0.305454	−0.152727	+	0.988268i	$0.548805\pi$
$164$	−0.554958	−0.0433349
$165$	0	0
$166$	−0.814019	−0.0631802
$167$	21.0194	1.62653	0.813264	−	0.581895i	$-0.197688\pi$
$167$	21.0194	1.62653	0.813264	+	0.581895i	$0.197688\pi$
$168$	0	0
$169$	0	0
$170$	−8.54288	−0.655209
$171$	0	0
$172$	−1.06100	−0.0809004
$173$	−13.2349	−1.00623	−0.503115	−	0.864219i	$-0.667813\pi$
$173$	−13.2349	−1.00623	−0.503115	+	0.864219i	$0.667813\pi$
$174$	0	0
$175$	13.6896	1.03484
$176$	−4.27114	−0.321950
$177$	0	0
$178$	7.85086	0.588446
$179$	−8.52781	−0.637399	−0.318699	−	0.947856i	$-0.603246\pi$
$179$	−8.52781	−0.637399	−0.318699	+	0.947856i	$0.603246\pi$
$180$	0	0
$181$	−3.63640	−0.270291	−0.135146	−	0.990826i	$-0.543150\pi$
$181$	−3.63640	−0.270291	−0.135146	+	0.990826i	$0.543150\pi$
$182$	0	0
$183$	0	0
$184$	−10.7192	−0.790228
$185$	−12.7627	−0.938333
$186$	0	0
$187$	3.58642	0.262265
$188$	−5.70410	−0.416014
$189$	0	0
$190$	8.88471	0.644564
$191$	−21.3817	−1.54712	−0.773561	−	0.633722i	$-0.781526\pi$
$191$	−21.3817	−1.54712	−0.773561	+	0.633722i	$0.781526\pi$
$192$	0	0
$193$	8.42758	0.606631	0.303315	−	0.952890i	$-0.401906\pi$
$193$	8.42758	0.606631	0.303315	+	0.952890i	$0.401906\pi$
$194$	12.5090	0.898096
$195$	0	0
$196$	−7.14675	−0.510482
$197$	−26.4765	−1.88637	−0.943186	−	0.332264i	$-0.892187\pi$
$197$	−26.4765	−1.88637	−0.943186	+	0.332264i	$0.892187\pi$
$198$	0	0
$199$	−14.2524	−1.01032	−0.505161	−	0.863025i	$-0.668567\pi$
$199$	−14.2524	−1.01032	−0.505161	+	0.863025i	$0.668567\pi$
$200$	−8.69202	−0.614619
$201$	0	0
$202$	−17.3177	−1.21847
$203$	−8.88769	−0.623794
$204$	0	0
$205$	−3.49396	−0.244029
$206$	−21.7017	−1.51203
$207$	0	0
$208$	0	0
$209$	−3.72992	−0.258004
$210$	0	0
$211$	1.85086	0.127418	0.0637091	−	0.997969i	$-0.479707\pi$
$211$	1.85086	0.127418	0.0637091	+	0.997969i	$0.479707\pi$
$212$	−3.93900	−0.270532
$213$	0	0
$214$	−13.1588	−0.899519
$215$	−6.67994	−0.455568
$216$	0	0
$217$	36.6437	2.48754
$218$	−1.33513	−0.0904261
$219$	0	0
$220$	1.82908	0.123317
$221$	0	0
$222$	0	0
$223$	−18.6504	−1.24892	−0.624462	−	0.781056i	$-0.714682\pi$
$223$	−18.6504	−1.24892	−0.624462	+	0.781056i	$0.714682\pi$
$224$	11.8455	0.791459
$225$	0	0
$226$	20.6136	1.37119
$227$	9.75733	0.647617	0.323808	−	0.946123i	$-0.395037\pi$
$227$	9.75733	0.647617	0.323808	+	0.946123i	$0.395037\pi$
$228$	0	0
$229$	2.86294	0.189188	0.0945941	−	0.995516i	$-0.469845\pi$
$229$	2.86294	0.189188	0.0945941	+	0.995516i	$0.469845\pi$
$230$	−12.2838	−0.809971
$231$	0	0
$232$	5.64310	0.370488
$233$	5.78554	0.379024	0.189512	−	0.981878i	$-0.439309\pi$
$233$	5.78554	0.379024	0.189512	+	0.981878i	$0.439309\pi$
$234$	0	0
$235$	−35.9124	−2.34267
$236$	−0.968541	−0.0630467
$237$	0	0
$238$	14.6407	0.949016
$239$	7.09246	0.458773	0.229386	−	0.973335i	$-0.426328\pi$
$239$	7.09246	0.458773	0.229386	+	0.973335i	$0.426328\pi$
$240$	0	0
$241$	−3.89977	−0.251206	−0.125603	−	0.992081i	$-0.540087\pi$
$241$	−3.89977	−0.251206	−0.125603	+	0.992081i	$0.540087\pi$
$242$	−11.0339	−0.709283
$243$	0	0
$244$	3.48427	0.223058
$245$	−44.9952	−2.87464
$246$	0	0
$247$	0	0
$248$	−23.2664	−1.47742
$249$	0	0
$250$	7.50902	0.474912
$251$	2.44504	0.154330	0.0771648	−	0.997018i	$-0.475413\pi$
$251$	2.44504	0.154330	0.0771648	+	0.997018i	$0.475413\pi$
$252$	0	0
$253$	5.15691	0.324212
$254$	−11.8931	−0.746237
$255$	0	0
$256$	−10.1129	−0.632056
$257$	14.1304	0.881428	0.440714	−	0.897648i	$-0.354725\pi$
$257$	14.1304	0.881428	0.440714	+	0.897648i	$0.354725\pi$
$258$	0	0
$259$	21.8726	1.35910
$260$	0	0
$261$	0	0
$262$	6.86964	0.424408
$263$	23.7235	1.46285	0.731426	−	0.681921i	$-0.238855\pi$
$263$	23.7235	1.46285	0.731426	+	0.681921i	$0.238855\pi$
$264$	0	0
$265$	−24.7995	−1.52342
$266$	−15.2265	−0.933599
$267$	0	0
$268$	−1.59419	−0.0973805
$269$	5.91617	0.360715	0.180357	−	0.983601i	$-0.442275\pi$
$269$	5.91617	0.360715	0.180357	+	0.983601i	$0.442275\pi$
$270$	0	0
$271$	−3.19029	−0.193796	−0.0968982	−	0.995294i	$-0.530892\pi$
$271$	−3.19029	−0.193796	−0.0968982	+	0.995294i	$0.530892\pi$
$272$	−7.11960	−0.431689
$273$	0	0
$274$	−20.1806	−1.21915
$275$	4.18167	0.252164
$276$	0	0
$277$	21.9366	1.31804	0.659022	−	0.752124i	$-0.270971\pi$
$277$	21.9366	1.31804	0.659022	+	0.752124i	$0.270971\pi$
$278$	−13.1045	−0.785958
$279$	0	0
$280$	41.0224	2.45155
$281$	−11.9903	−0.715282	−0.357641	−	0.933859i	$-0.616419\pi$
$281$	−11.9903	−0.715282	−0.357641	+	0.933859i	$0.616419\pi$
$282$	0	0
$283$	−14.3666	−0.854005	−0.427002	−	0.904250i	$-0.640430\pi$
$283$	−14.3666	−0.854005	−0.427002	+	0.904250i	$0.640430\pi$
$284$	3.93362	0.233418
$285$	0	0
$286$	0	0
$287$	5.98792	0.353456
$288$	0	0
$289$	−11.0218	−0.648339
$290$	6.46681	0.379744
$291$	0	0
$292$	−3.42327	−0.200332
$293$	−18.7584	−1.09588	−0.547939	−	0.836519i	$-0.684587\pi$
$293$	−18.7584	−1.09588	−0.547939	+	0.836519i	$0.684587\pi$
$294$	0	0
$295$	−6.09783	−0.355030
$296$	−13.8877	−0.807206
$297$	0	0
$298$	17.8944	1.03659
$299$	0	0
$300$	0	0
$301$	11.4480	0.659853
$302$	−2.45175	−0.141082
$303$	0	0
$304$	7.40449	0.424676
$305$	21.9366	1.25609
$306$	0	0
$307$	25.6262	1.46257	0.731283	−	0.682074i	$-0.238922\pi$
$307$	25.6262	1.46257	0.731283	+	0.682074i	$0.238922\pi$
$308$	−3.13467	−0.178614
$309$	0	0
$310$	−26.6625	−1.51433
$311$	−11.3817	−0.645394	−0.322697	−	0.946502i	$-0.604590\pi$
$311$	−11.3817	−0.645394	−0.322697	+	0.946502i	$0.604590\pi$
$312$	0	0
$313$	27.5743	1.55859	0.779297	−	0.626655i	$-0.215576\pi$
$313$	27.5743	1.55859	0.779297	+	0.626655i	$0.215576\pi$
$314$	13.3448	0.753091
$315$	0	0
$316$	1.78986	0.100687
$317$	11.2597	0.632405	0.316203	−	0.948692i	$-0.397592\pi$
$317$	11.2597	0.632405	0.316203	+	0.948692i	$0.397592\pi$
$318$	0	0
$319$	−2.71486	−0.152003
$320$	−24.9366	−1.39400
$321$	0	0
$322$	21.0519	1.17318
$323$	−6.21744	−0.345948
$324$	0	0
$325$	0	0
$326$	−4.86294	−0.269333
$327$	0	0
$328$	−3.80194	−0.209927
$329$	61.5465	3.39317
$330$	0	0
$331$	−11.9065	−0.654439	−0.327220	−	0.944948i	$-0.606112\pi$
$331$	−11.9065	−0.654439	−0.327220	+	0.944948i	$0.606112\pi$
$332$	0.290520	0.0159444
$333$	0	0
$334$	26.2107	1.43419
$335$	−10.0368	−0.548371
$336$	0	0
$337$	17.1672	0.935157	0.467578	−	0.883952i	$-0.345127\pi$
$337$	17.1672	0.935157	0.467578	+	0.883952i	$0.345127\pi$
$338$	0	0
$339$	0	0
$340$	3.04892	0.165351
$341$	11.1933	0.606150
$342$	0	0
$343$	43.4989	2.34872
$344$	−7.26875	−0.391905
$345$	0	0
$346$	−16.5036	−0.887242
$347$	24.2760	1.30321	0.651603	−	0.758560i	$-0.274097\pi$
$347$	24.2760	1.30321	0.651603	+	0.758560i	$0.274097\pi$
$348$	0	0
$349$	−4.57242	−0.244756	−0.122378	−	0.992484i	$-0.539052\pi$
$349$	−4.57242	−0.244756	−0.122378	+	0.992484i	$0.539052\pi$
$350$	17.0707	0.912467
$351$	0	0
$352$	3.61835	0.192859
$353$	6.07606	0.323396	0.161698	−	0.986840i	$-0.448303\pi$
$353$	6.07606	0.323396	0.161698	+	0.986840i	$0.448303\pi$
$354$	0	0
$355$	24.7657	1.31443
$356$	−2.80194	−0.148502
$357$	0	0
$358$	−10.6340	−0.562025
$359$	14.9661	0.789883	0.394942	−	0.918706i	$-0.370765\pi$
$359$	14.9661	0.789883	0.394942	+	0.918706i	$0.370765\pi$
$360$	0	0
$361$	−12.5338	−0.659673
$362$	−4.53452	−0.238329
$363$	0	0
$364$	0	0
$365$	−21.5526	−1.12811
$366$	0	0
$367$	37.0834	1.93574	0.967868	−	0.251459i	$-0.0809105\pi$
$367$	37.0834	1.93574	0.967868	+	0.251459i	$0.0809105\pi$
$368$	−10.2373	−0.533656
$369$	0	0
$370$	−15.9148	−0.827373
$371$	42.5013	2.20656
$372$	0	0
$373$	−36.5090	−1.89037	−0.945183	−	0.326542i	$-0.894117\pi$
$373$	−36.5090	−1.89037	−0.945183	+	0.326542i	$0.894117\pi$
$374$	4.47219	0.231251
$375$	0	0
$376$	−39.0780	−2.01529
$377$	0	0
$378$	0	0
$379$	−26.5851	−1.36558	−0.682792	−	0.730613i	$-0.739235\pi$
$379$	−26.5851	−1.36558	−0.682792	+	0.730613i	$0.739235\pi$
$380$	−3.17092	−0.162665
$381$	0	0
$382$	−26.6625	−1.36417
$383$	14.3502	0.733261	0.366630	−	0.930367i	$-0.380511\pi$
$383$	14.3502	0.733261	0.366630	+	0.930367i	$0.380511\pi$
$384$	0	0
$385$	−19.7356	−1.00582
$386$	10.5090	0.534895
$387$	0	0
$388$	−4.46442	−0.226647
$389$	22.6582	1.14881	0.574407	−	0.818570i	$-0.305233\pi$
$389$	22.6582	1.14881	0.574407	+	0.818570i	$0.305233\pi$
$390$	0	0
$391$	8.59611	0.434724
$392$	−48.9614	−2.47292
$393$	0	0
$394$	−33.0157	−1.66330
$395$	11.2687	0.566992
$396$	0	0
$397$	7.90754	0.396868	0.198434	−	0.980114i	$-0.436414\pi$
$397$	7.90754	0.396868	0.198434	+	0.980114i	$0.436414\pi$
$398$	−17.7724	−0.890850
$399$	0	0
$400$	−8.30127	−0.415064
$401$	−2.93661	−0.146647	−0.0733236	−	0.997308i	$-0.523361\pi$
$401$	−2.93661	−0.146647	−0.0733236	+	0.997308i	$0.523361\pi$
$402$	0	0
$403$	0	0
$404$	6.18060	0.307497
$405$	0	0
$406$	−11.0828	−0.550029
$407$	6.68127	0.331178
$408$	0	0
$409$	11.7549	0.581244	0.290622	−	0.956838i	$-0.406138\pi$
$409$	11.7549	0.581244	0.290622	+	0.956838i	$0.406138\pi$
$410$	−4.35690	−0.215172
$411$	0	0
$412$	7.74525	0.381581
$413$	10.4504	0.514231
$414$	0	0
$415$	1.82908	0.0897862
$416$	0	0
$417$	0	0
$418$	−4.65114	−0.227495
$419$	−7.34183	−0.358672	−0.179336	−	0.983788i	$-0.557395\pi$
$419$	−7.34183	−0.358672	−0.179336	+	0.983788i	$0.557395\pi$
$420$	0	0
$421$	25.6963	1.25236	0.626181	−	0.779677i	$-0.284617\pi$
$421$	25.6963	1.25236	0.626181	+	0.779677i	$0.284617\pi$
$422$	2.30798	0.112351
$423$	0	0
$424$	−26.9855	−1.31053
$425$	6.97046	0.338117
$426$	0	0
$427$	−37.5948	−1.81934
$428$	4.69633	0.227006
$429$	0	0
$430$	−8.32975	−0.401696
$431$	−8.94198	−0.430720	−0.215360	−	0.976535i	$-0.569093\pi$
$431$	−8.94198	−0.430720	−0.215360	+	0.976535i	$0.569093\pi$
$432$	0	0
$433$	2.91484	0.140078	0.0700391	−	0.997544i	$-0.477688\pi$
$433$	2.91484	0.140078	0.0700391	+	0.997544i	$0.477688\pi$
$434$	45.6939	2.19338
$435$	0	0
$436$	0.476501	0.0228203
$437$	−8.94007	−0.427661
$438$	0	0
$439$	9.05861	0.432344	0.216172	−	0.976355i	$-0.430643\pi$
$439$	9.05861	0.432344	0.216172	+	0.976355i	$0.430643\pi$
$440$	12.5308	0.597382
$441$	0	0
$442$	0	0
$443$	−11.2325	−0.533672	−0.266836	−	0.963742i	$-0.585978\pi$
$443$	−11.2325	−0.533672	−0.266836	+	0.963742i	$0.585978\pi$
$444$	0	0
$445$	−17.6407	−0.836250
$446$	−23.2567	−1.10124
$447$	0	0
$448$	42.7362	2.01909
$449$	−28.7579	−1.35717	−0.678585	−	0.734522i	$-0.737407\pi$
$449$	−28.7579	−1.35717	−0.678585	+	0.734522i	$0.737407\pi$
$450$	0	0
$451$	1.82908	0.0861282
$452$	−7.35690	−0.346039
$453$	0	0
$454$	12.1672	0.571035
$455$	0	0
$456$	0	0
$457$	−19.0761	−0.892341	−0.446170	−	0.894948i	$-0.647212\pi$
$457$	−19.0761	−0.892341	−0.446170	+	0.894948i	$0.647212\pi$
$458$	3.57002	0.166816
$459$	0	0
$460$	4.38404	0.204407
$461$	−31.7332	−1.47796	−0.738981	−	0.673727i	$-0.764692\pi$
$461$	−31.7332	−1.47796	−0.738981	+	0.673727i	$0.764692\pi$
$462$	0	0
$463$	−36.4784	−1.69530	−0.847648	−	0.530559i	$-0.821982\pi$
$463$	−36.4784	−1.69530	−0.847648	+	0.530559i	$0.821982\pi$
$464$	5.38942	0.250198
$465$	0	0
$466$	7.21446	0.334203
$467$	−13.0000	−0.601568	−0.300784	−	0.953692i	$-0.597248\pi$
$467$	−13.0000	−0.601568	−0.300784	+	0.953692i	$0.597248\pi$
$468$	0	0
$469$	17.2010	0.794271
$470$	−44.7821	−2.06564
$471$	0	0
$472$	−6.63533	−0.305416
$473$	3.49694	0.160790
$474$	0	0
$475$	−7.24937	−0.332624
$476$	−5.22521	−0.239497
$477$	0	0
$478$	8.84415	0.404522
$479$	−5.61655	−0.256627	−0.128313	−	0.991734i	$-0.540956\pi$
$479$	−5.61655	−0.256627	−0.128313	+	0.991734i	$0.540956\pi$
$480$	0	0
$481$	0	0
$482$	−4.86294	−0.221501
$483$	0	0
$484$	3.93794	0.178997
$485$	−28.1075	−1.27630
$486$	0	0
$487$	−9.75733	−0.442147	−0.221073	−	0.975257i	$-0.570956\pi$
$487$	−9.75733	−0.442147	−0.221073	+	0.975257i	$0.570956\pi$
$488$	23.8702	1.08055
$489$	0	0
$490$	−56.1081	−2.53471
$491$	7.38835	0.333432	0.166716	−	0.986005i	$-0.446684\pi$
$491$	7.38835	0.333432	0.166716	+	0.986005i	$0.446684\pi$
$492$	0	0
$493$	−4.52542	−0.203815
$494$	0	0
$495$	0	0
$496$	−22.2204	−0.997726
$497$	−42.4432	−1.90384
$498$	0	0
$499$	−43.2814	−1.93754	−0.968771	−	0.247956i	$-0.920241\pi$
$499$	−43.2814	−1.93754	−0.968771	+	0.247956i	$0.920241\pi$
$500$	−2.67994	−0.119851
$501$	0	0
$502$	3.04892	0.136080
$503$	−10.7670	−0.480078	−0.240039	−	0.970763i	$-0.577160\pi$
$503$	−10.7670	−0.480078	−0.240039	+	0.970763i	$0.577160\pi$
$504$	0	0
$505$	38.9124	1.73158
$506$	6.43057	0.285874
$507$	0	0
$508$	4.24459	0.188323
$509$	41.5448	1.84144	0.920720	−	0.390223i	$-0.127602\pi$
$509$	41.5448	1.84144	0.920720	+	0.390223i	$0.127602\pi$
$510$	0	0
$511$	36.9366	1.63398
$512$	−24.9390	−1.10216
$513$	0	0
$514$	17.6203	0.777197
$515$	48.7633	2.14877
$516$	0	0
$517$	18.8001	0.826829
$518$	27.2747	1.19838
$519$	0	0
$520$	0	0
$521$	−25.7198	−1.12680	−0.563402	−	0.826183i	$-0.690508\pi$
$521$	−25.7198	−1.12680	−0.563402	+	0.826183i	$0.690508\pi$
$522$	0	0
$523$	8.59286	0.375739	0.187870	−	0.982194i	$-0.439842\pi$
$523$	8.59286	0.375739	0.187870	+	0.982194i	$0.439842\pi$
$524$	−2.45175	−0.107105
$525$	0	0
$526$	29.5827	1.28987
$527$	18.6582	0.812763
$528$	0	0
$529$	−10.6396	−0.462593
$530$	−30.9245	−1.34328
$531$	0	0
$532$	5.43429	0.235606
$533$	0	0
$534$	0	0
$535$	29.5676	1.27832
$536$	−10.9215	−0.471739
$537$	0	0
$538$	7.37734	0.318060
$539$	23.5550	1.01458
$540$	0	0
$541$	31.3534	1.34799	0.673995	−	0.738736i	$-0.264577\pi$
$541$	31.3534	1.34799	0.673995	+	0.738736i	$0.264577\pi$
$542$	−3.97823	−0.170880
$543$	0	0
$544$	6.03146	0.258597
$545$	3.00000	0.128506
$546$	0	0
$547$	19.9342	0.852325	0.426163	−	0.904647i	$-0.359865\pi$
$547$	19.9342	0.852325	0.426163	+	0.904647i	$0.359865\pi$
$548$	7.20237	0.307670
$549$	0	0
$550$	5.21446	0.222345
$551$	4.70650	0.200503
$552$	0	0
$553$	−19.3123	−0.821242
$554$	27.3545	1.16218
$555$	0	0
$556$	4.67696	0.198347
$557$	−33.2349	−1.40821	−0.704104	−	0.710097i	$-0.748651\pi$
$557$	−33.2349	−1.40821	−0.704104	+	0.710097i	$0.748651\pi$
$558$	0	0
$559$	0	0
$560$	39.1782	1.65558
$561$	0	0
$562$	−14.9517	−0.630698
$563$	−3.87130	−0.163156	−0.0815779	−	0.996667i	$-0.525996\pi$
$563$	−3.87130	−0.163156	−0.0815779	+	0.996667i	$0.525996\pi$
$564$	0	0
$565$	−46.3183	−1.94862
$566$	−17.9148	−0.753017
$567$	0	0
$568$	26.9487	1.13074
$569$	−20.1457	−0.844551	−0.422276	−	0.906468i	$-0.638769\pi$
$569$	−20.1457	−0.844551	−0.422276	+	0.906468i	$0.638769\pi$
$570$	0	0
$571$	−32.1269	−1.34447	−0.672234	−	0.740338i	$-0.734665\pi$
$571$	−32.1269	−1.34447	−0.672234	+	0.740338i	$0.734665\pi$
$572$	0	0
$573$	0	0
$574$	7.46681	0.311659
$575$	10.0228	0.417981
$576$	0	0
$577$	−16.7506	−0.697338	−0.348669	−	0.937246i	$-0.613366\pi$
$577$	−16.7506	−0.697338	−0.348669	+	0.937246i	$0.613366\pi$
$578$	−13.7439	−0.571672
$579$	0	0
$580$	−2.30798	−0.0958336
$581$	−3.13467	−0.130048
$582$	0	0
$583$	12.9825	0.537682
$584$	−23.4523	−0.970465
$585$	0	0
$586$	−23.3913	−0.966287
$587$	6.73795	0.278105	0.139053	−	0.990285i	$-0.455594\pi$
$587$	6.73795	0.278105	0.139053	+	0.990285i	$0.455594\pi$
$588$	0	0
$589$	−19.4047	−0.799559
$590$	−7.60388	−0.313047
$591$	0	0
$592$	−13.2634	−0.545121
$593$	−18.1172	−0.743985	−0.371992	−	0.928236i	$-0.621325\pi$
$593$	−18.1172	−0.743985	−0.371992	+	0.928236i	$0.621325\pi$
$594$	0	0
$595$	−32.8974	−1.34866
$596$	−6.38644	−0.261599
$597$	0	0
$598$	0	0
$599$	26.7851	1.09441	0.547204	−	0.836999i	$-0.315692\pi$
$599$	26.7851	1.09441	0.547204	+	0.836999i	$0.315692\pi$
$600$	0	0
$601$	−4.70171	−0.191787	−0.0958934	−	0.995392i	$-0.530571\pi$
$601$	−4.70171	−0.191787	−0.0958934	+	0.995392i	$0.530571\pi$
$602$	14.2755	0.581824
$603$	0	0
$604$	0.875018	0.0356040
$605$	24.7928	1.00797
$606$	0	0
$607$	27.6396	1.12186	0.560929	−	0.827864i	$-0.310444\pi$
$607$	27.6396	1.12186	0.560929	+	0.827864i	$0.310444\pi$
$608$	−6.27280	−0.254396
$609$	0	0
$610$	27.3545	1.10755
$611$	0	0
$612$	0	0
$613$	48.1782	1.94590	0.972950	−	0.231017i	$-0.0742052\pi$
$613$	48.1782	1.94590	0.972950	+	0.231017i	$0.0742052\pi$
$614$	31.9554	1.28961
$615$	0	0
$616$	−21.4752	−0.865259
$617$	−30.3043	−1.22000	−0.610002	−	0.792400i	$-0.708831\pi$
$617$	−30.3043	−1.22000	−0.610002	+	0.792400i	$0.708831\pi$
$618$	0	0
$619$	−10.9041	−0.438272	−0.219136	−	0.975694i	$-0.570324\pi$
$619$	−10.9041	−0.438272	−0.219136	+	0.975694i	$0.570324\pi$
$620$	9.51573	0.382161
$621$	0	0
$622$	−14.1927	−0.569075
$623$	30.2325	1.21124
$624$	0	0
$625$	−31.1269	−1.24508
$626$	34.3846	1.37429
$627$	0	0
$628$	−4.76271	−0.190053
$629$	11.1371	0.444064
$630$	0	0
$631$	−10.4523	−0.416101	−0.208050	−	0.978118i	$-0.566712\pi$
$631$	−10.4523	−0.416101	−0.208050	+	0.978118i	$0.566712\pi$
$632$	12.2620	0.487758
$633$	0	0
$634$	14.0406	0.557622
$635$	26.7235	1.06049
$636$	0	0
$637$	0	0
$638$	−3.38537	−0.134028
$639$	0	0
$640$	−17.2717	−0.682725
$641$	17.5942	0.694929	0.347464	−	0.937693i	$-0.387043\pi$
$641$	17.5942	0.694929	0.347464	+	0.937693i	$0.387043\pi$
$642$	0	0
$643$	22.6058	0.891486	0.445743	−	0.895161i	$-0.352940\pi$
$643$	22.6058	0.891486	0.445743	+	0.895161i	$0.352940\pi$
$644$	−7.51334	−0.296067
$645$	0	0
$646$	−7.75302	−0.305039
$647$	24.7918	0.974665	0.487333	−	0.873216i	$-0.337970\pi$
$647$	24.7918	0.974665	0.487333	+	0.873216i	$0.337970\pi$
$648$	0	0
$649$	3.19221	0.125305
$650$	0	0
$651$	0	0
$652$	1.73556	0.0679699
$653$	21.8106	0.853513	0.426757	−	0.904367i	$-0.359656\pi$
$653$	21.8106	0.853513	0.426757	+	0.904367i	$0.359656\pi$
$654$	0	0
$655$	−15.4359	−0.603132
$656$	−3.63102	−0.141768
$657$	0	0
$658$	76.7472	2.99192
$659$	−16.5526	−0.644796	−0.322398	−	0.946604i	$-0.604489\pi$
$659$	−16.5526	−0.644796	−0.322398	+	0.946604i	$0.604489\pi$
$660$	0	0
$661$	−15.9541	−0.620541	−0.310271	−	0.950648i	$-0.600420\pi$
$661$	−15.9541	−0.620541	−0.310271	+	0.950648i	$0.600420\pi$
$662$	−14.8471	−0.577050
$663$	0	0
$664$	1.99031	0.0772391
$665$	34.2137	1.32675
$666$	0	0
$667$	−6.50711	−0.251956
$668$	−9.35450	−0.361937
$669$	0	0
$670$	−12.5157	−0.483525
$671$	−11.4838	−0.443327
$672$	0	0
$673$	−7.38835	−0.284800	−0.142400	−	0.989809i	$-0.545482\pi$
$673$	−7.38835	−0.284800	−0.142400	+	0.989809i	$0.545482\pi$
$674$	21.4071	0.824572
$675$	0	0
$676$	0	0
$677$	22.1454	0.851118	0.425559	−	0.904931i	$-0.360078\pi$
$677$	22.1454	0.851118	0.425559	+	0.904931i	$0.360078\pi$
$678$	0	0
$679$	48.1704	1.84861
$680$	20.8877	0.801006
$681$	0	0
$682$	13.9578	0.534471
$683$	−9.10023	−0.348211	−0.174105	−	0.984727i	$-0.555703\pi$
$683$	−9.10023	−0.348211	−0.174105	+	0.984727i	$0.555703\pi$
$684$	0	0
$685$	45.3454	1.73256
$686$	54.2422	2.07098
$687$	0	0
$688$	−6.94198	−0.264661
$689$	0	0
$690$	0	0
$691$	13.7711	0.523876	0.261938	−	0.965085i	$-0.415638\pi$
$691$	13.7711	0.523876	0.261938	+	0.965085i	$0.415638\pi$
$692$	5.89008	0.223907
$693$	0	0
$694$	30.2717	1.14910
$695$	29.4456	1.11694
$696$	0	0
$697$	3.04892	0.115486
$698$	−5.70171	−0.215813
$699$	0	0
$700$	−6.09246	−0.230273
$701$	−46.5090	−1.75662	−0.878311	−	0.478090i	$-0.841329\pi$
$701$	−46.5090	−1.75662	−0.878311	+	0.478090i	$0.841329\pi$
$702$	0	0
$703$	−11.5827	−0.436850
$704$	13.0543	0.492002
$705$	0	0
$706$	7.57673	0.285154
$707$	−66.6878	−2.50805
$708$	0	0
$709$	7.24565	0.272116	0.136058	−	0.990701i	$-0.456557\pi$
$709$	7.24565	0.272116	0.136058	+	0.990701i	$0.456557\pi$
$710$	30.8823	1.15899
$711$	0	0
$712$	−19.1957	−0.719388
$713$	26.8286	1.00474
$714$	0	0
$715$	0	0
$716$	3.79523	0.141835
$717$	0	0
$718$	18.6625	0.696478
$719$	−25.5147	−0.951536	−0.475768	−	0.879571i	$-0.657830\pi$
$719$	−25.5147	−0.951536	−0.475768	+	0.879571i	$0.657830\pi$
$720$	0	0
$721$	−83.5701	−3.11231
$722$	−15.6294	−0.581665
$723$	0	0
$724$	1.61835	0.0601455
$725$	−5.27652	−0.195965
$726$	0	0
$727$	−14.4873	−0.537303	−0.268651	−	0.963238i	$-0.586578\pi$
$727$	−14.4873	−0.537303	−0.268651	+	0.963238i	$0.586578\pi$
$728$	0	0
$729$	0	0
$730$	−26.8756	−0.994711
$731$	5.82908	0.215596
$732$	0	0
$733$	37.5036	1.38523	0.692614	−	0.721308i	$-0.256459\pi$
$733$	37.5036	1.38523	0.692614	+	0.721308i	$0.256459\pi$
$734$	46.2422	1.70683
$735$	0	0
$736$	8.67264	0.319678
$737$	5.25428	0.193544
$738$	0	0
$739$	43.1876	1.58868	0.794341	−	0.607472i	$-0.207816\pi$
$739$	43.1876	1.58868	0.794341	+	0.607472i	$0.207816\pi$
$740$	5.67994	0.208799
$741$	0	0
$742$	52.9982	1.94563
$743$	−13.4765	−0.494405	−0.247202	−	0.968964i	$-0.579511\pi$
$743$	−13.4765	−0.494405	−0.247202	+	0.968964i	$0.579511\pi$
$744$	0	0
$745$	−40.2083	−1.47312
$746$	−45.5260	−1.66683
$747$	0	0
$748$	−1.59611	−0.0583594
$749$	−50.6728	−1.85154
$750$	0	0
$751$	35.5894	1.29868	0.649338	−	0.760500i	$-0.275046\pi$
$751$	35.5894	1.29868	0.649338	+	0.760500i	$0.275046\pi$
$752$	−37.3212	−1.36097
$753$	0	0
$754$	0	0
$755$	5.50902	0.200494
$756$	0	0
$757$	−12.2107	−0.443807	−0.221903	−	0.975069i	$-0.571227\pi$
$757$	−12.2107	−0.443807	−0.221903	+	0.975069i	$0.571227\pi$
$758$	−33.1511	−1.20410
$759$	0	0
$760$	−21.7235	−0.787993
$761$	−30.9071	−1.12038	−0.560190	−	0.828364i	$-0.689272\pi$
$761$	−30.9071	−1.12038	−0.560190	+	0.828364i	$0.689272\pi$
$762$	0	0
$763$	−5.14138	−0.186130
$764$	9.51573	0.344267
$765$	0	0
$766$	17.8944	0.646551
$767$	0	0
$768$	0	0
$769$	11.9892	0.432343	0.216172	−	0.976355i	$-0.430643\pi$
$769$	11.9892	0.432343	0.216172	+	0.976355i	$0.430643\pi$
$770$	−24.6098	−0.886877
$771$	0	0
$772$	−3.75063	−0.134988
$773$	−43.9661	−1.58135	−0.790676	−	0.612235i	$-0.790271\pi$
$773$	−43.9661	−1.58135	−0.790676	+	0.612235i	$0.790271\pi$
$774$	0	0
$775$	21.7549	0.781460
$776$	−30.5851	−1.09794
$777$	0	0
$778$	28.2543	1.01296
$779$	−3.17092	−0.113610
$780$	0	0
$781$	−12.9648	−0.463918
$782$	10.7192	0.383317
$783$	0	0
$784$	−46.7603	−1.67001
$785$	−29.9855	−1.07023
$786$	0	0
$787$	30.3763	1.08280	0.541399	−	0.840766i	$-0.317895\pi$
$787$	30.3763	1.08280	0.541399	+	0.840766i	$0.317895\pi$
$788$	11.7832	0.419757
$789$	0	0
$790$	14.0519	0.499944
$791$	79.3798	2.82242
$792$	0	0
$793$	0	0
$794$	9.86054	0.349938
$795$	0	0
$796$	6.34290	0.224818
$797$	−52.5763	−1.86235	−0.931173	−	0.364577i	$-0.881214\pi$
$797$	−52.5763	−1.86235	−0.931173	+	0.364577i	$0.881214\pi$
$798$	0	0
$799$	31.3381	1.10866
$800$	7.03252	0.248637
$801$	0	0
$802$	−3.66189	−0.129306
$803$	11.2828	0.398160
$804$	0	0
$805$	−47.3032	−1.66722
$806$	0	0
$807$	0	0
$808$	42.3424	1.48960
$809$	−49.4215	−1.73757	−0.868783	−	0.495193i	$-0.835097\pi$
$809$	−49.4215	−1.73757	−0.868783	+	0.495193i	$0.835097\pi$
$810$	0	0
$811$	−1.36526	−0.0479406	−0.0239703	−	0.999713i	$-0.507631\pi$
$811$	−1.36526	−0.0479406	−0.0239703	+	0.999713i	$0.507631\pi$
$812$	3.95539	0.138807
$813$	0	0
$814$	8.33140	0.292016
$815$	10.9269	0.382753
$816$	0	0
$817$	−6.06233	−0.212094
$818$	14.6582	0.512511
$819$	0	0
$820$	1.55496	0.0543015
$821$	−0.665939	−0.0232414	−0.0116207	−	0.999932i	$-0.503699\pi$
$821$	−0.665939	−0.0232414	−0.0116207	+	0.999932i	$0.503699\pi$
$822$	0	0
$823$	−10.0592	−0.350642	−0.175321	−	0.984511i	$-0.556096\pi$
$823$	−10.0592	−0.350642	−0.175321	+	0.984511i	$0.556096\pi$
$824$	53.0616	1.84849
$825$	0	0
$826$	13.0315	0.453422
$827$	37.3038	1.29718	0.648590	−	0.761138i	$-0.275359\pi$
$827$	37.3038	1.29718	0.648590	+	0.761138i	$0.275359\pi$
$828$	0	0
$829$	−42.6209	−1.48028	−0.740142	−	0.672451i	$-0.765242\pi$
$829$	−42.6209	−1.48028	−0.740142	+	0.672451i	$0.765242\pi$
$830$	2.28083	0.0791688
$831$	0	0
$832$	0	0
$833$	39.2640	1.36042
$834$	0	0
$835$	−58.8950	−2.03815
$836$	1.65997	0.0574113
$837$	0	0
$838$	−9.15511	−0.316258
$839$	−4.14005	−0.142930	−0.0714652	−	0.997443i	$-0.522767\pi$
$839$	−4.14005	−0.142930	−0.0714652	+	0.997443i	$0.522767\pi$
$840$	0	0
$841$	−25.5743	−0.881874
$842$	32.0428	1.10427
$843$	0	0
$844$	−0.823708	−0.0283532
$845$	0	0
$846$	0	0
$847$	−42.4898	−1.45997
$848$	−25.7724	−0.885028
$849$	0	0
$850$	8.69202	0.298134
$851$	16.0140	0.548953
$852$	0	0
$853$	17.3502	0.594059	0.297030	−	0.954868i	$-0.404004\pi$
$853$	17.3502	0.594059	0.297030	+	0.954868i	$0.404004\pi$
$854$	−46.8799	−1.60420
$855$	0	0
$856$	32.1739	1.09968
$857$	−41.0180	−1.40115	−0.700575	−	0.713579i	$-0.747073\pi$
$857$	−41.0180	−1.40115	−0.700575	+	0.713579i	$0.747073\pi$
$858$	0	0
$859$	6.59286	0.224945	0.112473	−	0.993655i	$-0.464123\pi$
$859$	6.59286	0.224945	0.112473	+	0.993655i	$0.464123\pi$
$860$	2.97285	0.101373
$861$	0	0
$862$	−11.1505	−0.379787
$863$	16.6455	0.566619	0.283310	−	0.959028i	$-0.408568\pi$
$863$	16.6455	0.566619	0.283310	+	0.959028i	$0.408568\pi$
$864$	0	0
$865$	37.0834	1.26087
$866$	3.63474	0.123514
$867$	0	0
$868$	−16.3080	−0.553529
$869$	−5.89918	−0.200116
$870$	0	0
$871$	0	0
$872$	3.26444	0.110548
$873$	0	0
$874$	−11.1481	−0.377089
$875$	28.9162	0.977545
$876$	0	0
$877$	54.4965	1.84022	0.920108	−	0.391666i	$-0.128101\pi$
$877$	54.4965	1.84022	0.920108	+	0.391666i	$0.128101\pi$
$878$	11.2959	0.381218
$879$	0	0
$880$	11.9675	0.403424
$881$	9.00670	0.303444	0.151722	−	0.988423i	$-0.451518\pi$
$881$	9.00670	0.303444	0.151722	+	0.988423i	$0.451518\pi$
$882$	0	0
$883$	18.8907	0.635722	0.317861	−	0.948137i	$-0.397035\pi$
$883$	18.8907	0.635722	0.317861	+	0.948137i	$0.397035\pi$
$884$	0	0
$885$	0	0
$886$	−14.0067	−0.470564
$887$	46.9124	1.57517	0.787583	−	0.616209i	$-0.211332\pi$
$887$	46.9124	1.57517	0.787583	+	0.616209i	$0.211332\pi$
$888$	0	0
$889$	−45.7985	−1.53603
$890$	−21.9976	−0.737361
$891$	0	0
$892$	8.30021	0.277912
$893$	−32.5921	−1.09065
$894$	0	0
$895$	23.8944	0.798702
$896$	29.6002	0.988872
$897$	0	0
$898$	−35.8605	−1.19668
$899$	−14.1239	−0.471059
$900$	0	0
$901$	21.6407	0.720957
$902$	2.28083	0.0759434
$903$	0	0
$904$	−50.4010	−1.67631
$905$	10.1890	0.338693
$906$	0	0
$907$	−35.3013	−1.17216	−0.586080	−	0.810253i	$-0.699329\pi$
$907$	−35.3013	−1.17216	−0.586080	+	0.810253i	$0.699329\pi$
$908$	−4.34242	−0.144108
$909$	0	0
$910$	0	0
$911$	9.80731	0.324931	0.162465	−	0.986714i	$-0.448055\pi$
$911$	9.80731	0.324931	0.162465	+	0.986714i	$0.448055\pi$
$912$	0	0
$913$	−0.957524	−0.0316894
$914$	−23.7875	−0.786819
$915$	0	0
$916$	−1.27413	−0.0420983
$917$	26.4540	0.873588
$918$	0	0
$919$	18.4655	0.609120	0.304560	−	0.952493i	$-0.401491\pi$
$919$	18.4655	0.609120	0.304560	+	0.952493i	$0.401491\pi$
$920$	30.0344	0.990206
$921$	0	0
$922$	−39.5706	−1.30319
$923$	0	0
$924$	0	0
$925$	12.9855	0.426961
$926$	−45.4878	−1.49482
$927$	0	0
$928$	−4.56571	−0.149877
$929$	25.6267	0.840785	0.420393	−	0.907342i	$-0.361892\pi$
$929$	25.6267	0.840785	0.420393	+	0.907342i	$0.361892\pi$
$930$	0	0
$931$	−40.8351	−1.33831
$932$	−2.57481	−0.0843407
$933$	0	0
$934$	−16.2107	−0.530431
$935$	−10.0489	−0.328635
$936$	0	0
$937$	7.54932	0.246625	0.123313	−	0.992368i	$-0.460648\pi$
$937$	7.54932	0.246625	0.123313	+	0.992368i	$0.460648\pi$
$938$	21.4494	0.700346
$939$	0	0
$940$	15.9825	0.521293
$941$	12.6418	0.412110	0.206055	−	0.978540i	$-0.433937\pi$
$941$	12.6418	0.412110	0.206055	+	0.978540i	$0.433937\pi$
$942$	0	0
$943$	4.38404	0.142764
$944$	−6.33704	−0.206253
$945$	0	0
$946$	4.36062	0.141776
$947$	−27.9801	−0.909233	−0.454616	−	0.890687i	$-0.650224\pi$
$947$	−27.9801	−0.909233	−0.454616	+	0.890687i	$0.650224\pi$
$948$	0	0
$949$	0	0
$950$	−9.03982	−0.293290
$951$	0	0
$952$	−35.7972	−1.16019
$953$	−4.00239	−0.129650	−0.0648251	−	0.997897i	$-0.520649\pi$
$953$	−4.00239	−0.129650	−0.0648251	+	0.997897i	$0.520649\pi$
$954$	0	0
$955$	59.9101	1.93864
$956$	−3.15644	−0.102087
$957$	0	0
$958$	−7.00372	−0.226280
$959$	−77.7126	−2.50947
$960$	0	0
$961$	27.2325	0.878468
$962$	0	0
$963$	0	0
$964$	1.73556	0.0558987
$965$	−23.6136	−0.760148
$966$	0	0
$967$	12.2239	0.393094	0.196547	−	0.980494i	$-0.437027\pi$
$967$	12.2239	0.393094	0.196547	+	0.980494i	$0.437027\pi$
$968$	26.9782	0.867113
$969$	0	0
$970$	−35.0495	−1.12537
$971$	23.6401	0.758648	0.379324	−	0.925264i	$-0.376157\pi$
$971$	23.6401	0.758648	0.379324	+	0.925264i	$0.376157\pi$
$972$	0	0
$973$	−50.4637	−1.61779
$974$	−12.1672	−0.389862
$975$	0	0
$976$	22.7972	0.729719
$977$	−18.7313	−0.599266	−0.299633	−	0.954055i	$-0.596864\pi$
$977$	−18.7313	−0.599266	−0.299633	+	0.954055i	$0.596864\pi$
$978$	0	0
$979$	9.23490	0.295149
$980$	20.0248	0.639667
$981$	0	0
$982$	9.21313	0.294003
$983$	−12.0954	−0.385785	−0.192892	−	0.981220i	$-0.561787\pi$
$983$	−12.0954	−0.385785	−0.192892	+	0.981220i	$0.561787\pi$
$984$	0	0
$985$	74.1855	2.36375
$986$	−5.64310	−0.179713
$987$	0	0
$988$	0	0
$989$	8.38165	0.266521
$990$	0	0
$991$	−28.5526	−0.907002	−0.453501	−	0.891256i	$-0.649825\pi$
$991$	−28.5526	−0.907002	−0.453501	+	0.891256i	$0.649825\pi$
$992$	18.8243	0.597672
$993$	0	0
$994$	−52.9259	−1.67871
$995$	39.9342	1.26600
$996$	0	0
$997$	−23.9347	−0.758019	−0.379010	−	0.925393i	$-0.623735\pi$
$997$	−23.9347	−0.758019	−0.379010	+	0.925393i	$0.623735\pi$
$998$	−53.9711	−1.70842
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.p.1.3		3
3.2	odd	2		507.2.a.k.1.1	yes	3
12.11	even	2		8112.2.a.cf.1.3		3
13.5	odd	4		1521.2.b.m.1351.2		6
13.8	odd	4		1521.2.b.m.1351.5		6
13.12	even	2		1521.2.a.q.1.1		3
39.2	even	12		507.2.j.h.316.2		12
39.5	even	4		507.2.b.g.337.5		6
39.8	even	4		507.2.b.g.337.2		6
39.11	even	12		507.2.j.h.316.5		12
39.17	odd	6		507.2.e.k.484.1		6
39.20	even	12		507.2.j.h.361.2		12
39.23	odd	6		507.2.e.k.22.1		6
39.29	odd	6		507.2.e.j.22.3		6
39.32	even	12		507.2.j.h.361.5		12
39.35	odd	6		507.2.e.j.484.3		6
39.38	odd	2		507.2.a.j.1.3	✓	3
156.155	even	2		8112.2.a.by.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.j.1.3	✓	3	39.38	odd	2
507.2.a.k.1.1	yes	3	3.2	odd	2
507.2.b.g.337.2		6	39.8	even	4
507.2.b.g.337.5		6	39.5	even	4
507.2.e.j.22.3		6	39.29	odd	6
507.2.e.j.484.3		6	39.35	odd	6
507.2.e.k.22.1		6	39.23	odd	6
507.2.e.k.484.1		6	39.17	odd	6
507.2.j.h.316.2		12	39.2	even	12
507.2.j.h.316.5		12	39.11	even	12
507.2.j.h.361.2		12	39.20	even	12
507.2.j.h.361.5		12	39.32	even	12
1521.2.a.p.1.3		3	1.1	even	1	trivial
1521.2.a.q.1.1		3	13.12	even	2
1521.2.b.m.1351.2		6	13.5	odd	4
1521.2.b.m.1351.5		6	13.8	odd	4
8112.2.a.by.1.1		3	156.155	even	2
8112.2.a.cf.1.3		3	12.11	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)

\(f(q)\)	\(=\)	\(q+1.24698 q^{2} -0.445042 q^{4} -2.80194 q^{5} +4.80194 q^{7} -3.04892 q^{8} +O(q^{10})\)	\(q+1.24698 q^{2} -0.445042 q^{4} -2.80194 q^{5} +4.80194 q^{7} -3.04892 q^{8} -3.49396 q^{10} +1.46681 q^{11} +5.98792 q^{14} -2.91185 q^{16} +2.44504 q^{17} -2.54288 q^{19} +1.24698 q^{20} +1.82908 q^{22} +3.51573 q^{23} +2.85086 q^{25} -2.13706 q^{28} -1.85086 q^{29} +7.63102 q^{31} +2.46681 q^{32} +3.04892 q^{34} -13.4547 q^{35} +4.55496 q^{37} -3.17092 q^{38} +8.54288 q^{40} +1.24698 q^{41} +2.38404 q^{43} -0.652793 q^{44} +4.38404 q^{46} +12.8170 q^{47} +16.0586 q^{49} +3.55496 q^{50} +8.85086 q^{53} -4.10992 q^{55} -14.6407 q^{56} -2.30798 q^{58} +2.17629 q^{59} -7.82908 q^{61} +9.51573 q^{62} +8.89977 q^{64} +3.58211 q^{67} -1.08815 q^{68} -16.7778 q^{70} -8.83877 q^{71} +7.69202 q^{73} +5.67994 q^{74} +1.13169 q^{76} +7.04354 q^{77} -4.02177 q^{79} +8.15883 q^{80} +1.55496 q^{82} -0.652793 q^{83} -6.85086 q^{85} +2.97285 q^{86} -4.47219 q^{88} +6.29590 q^{89} -1.56465 q^{92} +15.9825 q^{94} +7.12498 q^{95} +10.0315 q^{97} +20.0248 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - q^{4} - 4 q^{5} + 10 q^{7}+O(q^{10}) \) 3 * q - q^2 - q^4 - 4 * q^5 + 10 * q^7	\( 3 q - q^{2} - q^{4} - 4 q^{5} + 10 q^{7} - q^{10} + q^{11} - q^{14} - 5 q^{16} + 7 q^{17} + 11 q^{19} - q^{20} - 5 q^{22} - 2 q^{23} - 5 q^{25} - q^{28} + 8 q^{29} + 8 q^{31} + 4 q^{32} - 18 q^{35} + 14 q^{37} - 20 q^{38} + 7 q^{40} - q^{41} - 3 q^{43} + 16 q^{44} + 3 q^{46} + 9 q^{47} + 17 q^{49} + 11 q^{50} + 13 q^{53} - 13 q^{55} - 7 q^{56} - 12 q^{58} + 14 q^{59} - 13 q^{61} + 16 q^{62} + 4 q^{64} + 5 q^{67} - 7 q^{68} - 8 q^{70} + 6 q^{71} + 18 q^{73} - 7 q^{74} + q^{76} + 15 q^{77} - 9 q^{79} + 16 q^{80} + 5 q^{82} + 16 q^{83} - 7 q^{85} + 15 q^{86} - 7 q^{88} + 5 q^{89} + 17 q^{92} + 32 q^{94} - 3 q^{95} + 5 q^{97} + 13 q^{98}+O(q^{100}) \) 3 * q - q^2 - q^4 - 4 * q^5 + 10 * q^7 - q^10 + q^11 - q^14 - 5 * q^16 + 7 * q^17 + 11 * q^19 - q^20 - 5 * q^22 - 2 * q^23 - 5 * q^25 - q^28 + 8 * q^29 + 8 * q^31 + 4 * q^32 - 18 * q^35 + 14 * q^37 - 20 * q^38 + 7 * q^40 - q^41 - 3 * q^43 + 16 * q^44 + 3 * q^46 + 9 * q^47 + 17 * q^49 + 11 * q^50 + 13 * q^53 - 13 * q^55 - 7 * q^56 - 12 * q^58 + 14 * q^59 - 13 * q^61 + 16 * q^62 + 4 * q^64 + 5 * q^67 - 7 * q^68 - 8 * q^70 + 6 * q^71 + 18 * q^73 - 7 * q^74 + q^76 + 15 * q^77 - 9 * q^79 + 16 * q^80 + 5 * q^82 + 16 * q^83 - 7 * q^85 + 15 * q^86 - 7 * q^88 + 5 * q^89 + 17 * q^92 + 32 * q^94 - 3 * q^95 + 5 * q^97 + 13 * q^98

Introduction

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