LMFDB - Embedded newform 1521.2.b.m.1351.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1351,1521)]

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, 1]))

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

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

chi := DirichletCharacter("1521.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 507)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.5
Root			$1.24698i$ of defining polynomial
Character	$\chi$	$=$	1521.1351
Dual form			1521.2.b.m.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.24698i q^{2} +0.445042 q^{4} -2.80194i q^{5} -4.80194i q^{7} +3.04892i q^{8} +O(q^{10})$	$q+1.24698i q^{2} +0.445042 q^{4} -2.80194i q^{5} -4.80194i q^{7} +3.04892i q^{8} +3.49396 q^{10} -1.46681i q^{11} +5.98792 q^{14} -2.91185 q^{16} -2.44504 q^{17} -2.54288i q^{19} -1.24698i q^{20} +1.82908 q^{22} -3.51573 q^{23} -2.85086 q^{25} -2.13706i q^{28} -1.85086 q^{29} +7.63102i q^{31} +2.46681i q^{32} -3.04892i q^{34} -13.4547 q^{35} -4.55496i q^{37} +3.17092 q^{38} +8.54288 q^{40} +1.24698i q^{41} -2.38404 q^{43} -0.652793i q^{44} -4.38404i q^{46} -12.8170i q^{47} -16.0586 q^{49} -3.55496i q^{50} +8.85086 q^{53} -4.10992 q^{55} +14.6407 q^{56} -2.30798i q^{58} -2.17629i q^{59} -7.82908 q^{61} -9.51573 q^{62} -8.89977 q^{64} +3.58211i q^{67} -1.08815 q^{68} -16.7778i q^{70} -8.83877i q^{71} -7.69202i q^{73} +5.67994 q^{74} -1.13169i q^{76} -7.04354 q^{77} -4.02177 q^{79} +8.15883i q^{80} -1.55496 q^{82} -0.652793i q^{83} +6.85086i q^{85} -2.97285i q^{86} +4.47219 q^{88} -6.29590i q^{89} -1.56465 q^{92} +15.9825 q^{94} -7.12498 q^{95} +10.0315i q^{97} -20.0248i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{4}+O(q^{10}) $ 6 * q + 2 * q^4	$ 6 q + 2 q^{4} + 2 q^{10} - 2 q^{14} - 10 q^{16} - 14 q^{17} - 10 q^{22} + 4 q^{23} + 10 q^{25} + 16 q^{29} - 36 q^{35} + 40 q^{38} + 14 q^{40} + 6 q^{43} - 34 q^{49} + 26 q^{53} - 26 q^{55} + 14 q^{56} - 26 q^{61} - 32 q^{62} - 8 q^{64} - 14 q^{68} - 14 q^{74} - 30 q^{77} - 18 q^{79} - 10 q^{82} + 14 q^{88} + 34 q^{92} + 64 q^{94} + 6 q^{95}+O(q^{100}) $ 6 * q + 2 * q^4 + 2 * q^10 - 2 * q^14 - 10 * q^16 - 14 * q^17 - 10 * q^22 + 4 * q^23 + 10 * q^25 + 16 * q^29 - 36 * q^35 + 40 * q^38 + 14 * q^40 + 6 * q^43 - 34 * q^49 + 26 * q^53 - 26 * q^55 + 14 * q^56 - 26 * q^61 - 32 * q^62 - 8 * q^64 - 14 * q^68 - 14 * q^74 - 30 * q^77 - 18 * q^79 - 10 * q^82 + 14 * q^88 + 34 * q^92 + 64 * q^94 + 6 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.24698i	0.881748i	0.897569	+	0.440874i	$0.145331\pi$
$2$	1.24698i	0.881748i	−0.897569	+	0.440874i	$0.854669\pi$
$3$	0	0
$3$	0	0
$4$	0.445042	0.222521
$5$	− 2.80194i	− 1.25306i	−0.779395	−	0.626532i	$-0.784474\pi$
$5$	− 2.80194i	− 1.25306i	0.779395	−	0.626532i	$-0.215526\pi$
$6$	0	0
$7$	− 4.80194i	− 1.81496i	−0.420093	−	0.907481i	$-0.638003\pi$
$7$	− 4.80194i	− 1.81496i	0.420093	−	0.907481i	$-0.361997\pi$
$8$	3.04892i	1.07796i
$9$	0	0
$10$	3.49396	1.10489
$11$	− 1.46681i	− 0.442260i	−0.975244	−	0.221130i	$-0.929025\pi$
$11$	− 1.46681i	− 0.442260i	0.975244	−	0.221130i	$-0.0709746\pi$
$12$	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	$-0.595821\pi$
$17$	−2.44504	−0.593010	−0.296505	+	0.955031i	$0.595821\pi$
$18$	0	0
$19$	− 2.54288i	− 0.583376i	−0.956514	−	0.291688i	$-0.905783\pi$
$19$	− 2.54288i	− 0.583376i	0.956514	−	0.291688i	$-0.0942169\pi$
$20$	− 1.24698i	− 0.278833i
$21$	0	0
$22$	1.82908	0.389962
$23$	−3.51573	−0.733080	−0.366540	−	0.930402i	$-0.619458\pi$
$23$	−3.51573	−0.733080	−0.366540	+	0.930402i	$0.619458\pi$
$24$	0	0
$25$	−2.85086	−0.570171
$26$	0	0
$27$	0	0
$28$	− 2.13706i	− 0.403867i
$29$	−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
$31$	7.63102i	1.37057i	0.728274	+	0.685286i	$0.240323\pi$
$31$	7.63102i	1.37057i	−0.728274	+	0.685286i	$0.759677\pi$
$32$	2.46681i	0.436075i
$33$	0	0
$34$	− 3.04892i	− 0.522885i
$35$	−13.4547	−2.27426
$36$	0	0
$37$	− 4.55496i	− 0.748831i	−0.927261	−	0.374415i	$-0.877843\pi$
$37$	− 4.55496i	− 0.748831i	0.927261	−	0.374415i	$-0.122157\pi$
$38$	3.17092	0.514390
$39$	0	0
$40$	8.54288	1.35075
$41$	1.24698i	0.194745i	0.995248	+	0.0973727i	$0.0310439\pi$
$41$	1.24698i	0.194745i	−0.995248	+	0.0973727i	$0.968956\pi$
$42$	0	0
$43$	−2.38404	−0.363563	−0.181782	−	0.983339i	$-0.558186\pi$
$43$	−2.38404	−0.363563	−0.181782	+	0.983339i	$0.558186\pi$
$44$	− 0.652793i	− 0.0984122i
$45$	0	0
$46$	− 4.38404i	− 0.646392i
$47$	− 12.8170i	− 1.86955i	−0.355238	−	0.934776i	$-0.615600\pi$
$47$	− 12.8170i	− 1.86955i	0.355238	−	0.934776i	$-0.384400\pi$
$48$	0	0
$49$	−16.0586	−2.29409
$50$	− 3.55496i	− 0.502747i
$51$	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
$58$	− 2.30798i	− 0.303052i
$59$	− 2.17629i	− 0.283329i	−0.989915	−	0.141665i	$-0.954755\pi$
$59$	− 2.17629i	− 0.283329i	0.989915	−	0.141665i	$-0.0452454\pi$
$60$	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
$67$	3.58211i	0.437624i	0.975767	+	0.218812i	$0.0702181\pi$
$67$	3.58211i	0.437624i	−0.975767	+	0.218812i	$0.929782\pi$
$68$	−1.08815	−0.131957
$69$	0	0
$70$	− 16.7778i	− 2.00533i
$71$	− 8.83877i	− 1.04897i	−0.851420	−	0.524485i	$-0.824258\pi$
$71$	− 8.83877i	− 1.04897i	0.851420	−	0.524485i	$-0.175742\pi$
$72$	0	0
$73$	− 7.69202i	− 0.900283i	−0.892957	−	0.450142i	$-0.851374\pi$
$73$	− 7.69202i	− 0.900283i	0.892957	−	0.450142i	$-0.148626\pi$
$74$	5.67994	0.660280
$75$	0	0
$76$	− 1.13169i	− 0.129813i
$77$	−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	$0.572644\pi$
$80$	8.15883i	0.912185i
$81$	0	0
$82$	−1.55496	−0.171716
$83$	− 0.652793i	− 0.0716533i	−0.999358	−	0.0358267i	$-0.988594\pi$
$83$	− 0.652793i	− 0.0716533i	0.999358	−	0.0358267i	$-0.0114064\pi$
$84$	0	0
$85$	6.85086i	0.743080i
$86$	− 2.97285i	− 0.320571i
$87$	0	0
$88$	4.47219	0.476737
$89$	− 6.29590i	− 0.667364i	−0.942686	−	0.333682i	$-0.891709\pi$
$89$	− 6.29590i	− 0.667364i	0.942686	−	0.333682i	$-0.108291\pi$
$90$	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
$97$	10.0315i	1.01854i	0.860607	+	0.509270i	$0.170085\pi$
$97$	10.0315i	1.01854i	−0.860607	+	0.509270i	$0.829915\pi$
$98$	− 20.0248i	− 2.02281i
$99$	0	0
$100$	−1.26875	−0.126875
$101$	13.8877	1.38188	0.690938	−	0.722914i	$-0.257198\pi$
$101$	13.8877	1.38188	0.690938	+	0.722914i	$0.257198\pi$
$102$	0	0
$103$	17.4034	1.71481	0.857405	−	0.514642i	$-0.172075\pi$
$103$	17.4034	1.71481	0.857405	+	0.514642i	$0.172075\pi$
$104$	0	0
$105$	0	0
$106$	11.0368i	1.07199i
$107$	−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
$109$	− 1.07069i	− 0.102553i	−0.998684	−	0.0512766i	$-0.983671\pi$
$109$	− 1.07069i	− 0.102553i	0.998684	−	0.0512766i	$-0.0163290\pi$
$110$	− 5.12498i	− 0.488648i
$111$	0	0
$112$	13.9825i	1.32123i
$113$	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
$115$	9.85086i	0.918597i
$116$	−0.823708	−0.0764794
$117$	0	0
$118$	2.71379	0.249825
$119$	11.7409i	1.07629i
$120$	0	0
$121$	8.84846	0.804406
$122$	− 9.76271i	− 0.883874i
$123$	0	0
$124$	3.39612i	0.304981i
$125$	− 6.02177i	− 0.538604i
$126$	0	0
$127$	9.53750	0.846316	0.423158	−	0.906056i	$-0.360921\pi$
$127$	9.53750	0.846316	0.423158	+	0.906056i	$0.360921\pi$
$128$	− 6.16421i	− 0.544844i
$129$	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
$136$	− 7.45473i	− 0.639238i
$137$	16.1836i	1.38266i	0.722540	+	0.691329i	$0.242974\pi$
$137$	16.1836i	1.38266i	−0.722540	+	0.691329i	$0.757026\pi$
$138$	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
$145$	5.18598i	0.430672i
$146$	9.59179	0.793823
$147$	0	0
$148$	− 2.02715i	− 0.166630i
$149$	14.3502i	1.17561i	0.809001	+	0.587807i	$0.200008\pi$
$149$	14.3502i	1.17561i	−0.809001	+	0.587807i	$0.799992\pi$
$150$	0	0
$151$	1.96615i	0.160003i	0.996795	+	0.0800014i	$0.0254925\pi$
$151$	1.96615i	0.160003i	−0.996795	+	0.0800014i	$0.974508\pi$
$152$	7.75302	0.628853
$153$	0	0
$154$	− 8.78315i	− 0.707767i
$155$	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	$0.359555\pi$
$158$	− 5.01507i	− 0.398977i
$159$	0	0
$160$	6.91185	0.546430
$161$	16.8823i	1.33051i
$162$	0	0
$163$	3.89977i	0.305454i	0.988268	+	0.152727i	$0.0488055\pi$
$163$	3.89977i	0.305454i	−0.988268	+	0.152727i	$0.951195\pi$
$164$	0.554958i	0.0433349i
$165$	0	0
$166$	0.814019	0.0631802
$167$	− 21.0194i	− 1.62653i	−0.581895	−	0.813264i	$-0.697688\pi$
$167$	− 21.0194i	− 1.62653i	0.581895	−	0.813264i	$-0.302312\pi$
$168$	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	$-0.332187\pi$
$173$	13.2349	1.00623	0.503115	+	0.864219i	$0.332187\pi$
$174$	0	0
$175$	13.6896i	1.03484i
$176$	4.27114i	0.321950i
$177$	0	0
$178$	7.85086	0.588446
$179$	8.52781	0.637399	0.318699	−	0.947856i	$-0.396754\pi$
$179$	8.52781	0.637399	0.318699	+	0.947856i	$0.396754\pi$
$180$	0	0
$181$	3.63640	0.270291	0.135146	−	0.990826i	$-0.456850\pi$
$181$	3.63640	0.270291	0.135146	+	0.990826i	$0.456850\pi$
$182$	0	0
$183$	0	0
$184$	− 10.7192i	− 0.790228i
$185$	−12.7627	−0.938333
$186$	0	0
$187$	3.58642i	0.262265i
$188$	− 5.70410i	− 0.416014i
$189$	0	0
$190$	− 8.88471i	− 0.644564i
$191$	−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
$193$	− 8.42758i	− 0.606631i	−0.952890	−	0.303315i	$-0.901906\pi$
$193$	− 8.42758i	− 0.606631i	0.952890	−	0.303315i	$-0.0980936\pi$
$194$	−12.5090	−0.898096
$195$	0	0
$196$	−7.14675	−0.510482
$197$	− 26.4765i	− 1.88637i	−0.332264	−	0.943186i	$-0.607813\pi$
$197$	− 26.4765i	− 1.88637i	0.332264	−	0.943186i	$-0.392187\pi$
$198$	0	0
$199$	14.2524	1.01032	0.505161	−	0.863025i	$-0.331433\pi$
$199$	14.2524	1.01032	0.505161	+	0.863025i	$0.331433\pi$
$200$	− 8.69202i	− 0.614619i
$201$	0	0
$202$	17.3177i	1.21847i
$203$	8.88769i	0.623794i
$204$	0	0
$205$	3.49396	0.244029
$206$	21.7017i	1.51203i
$207$	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
$214$	− 13.1588i	− 0.899519i
$215$	6.67994i	0.455568i
$216$	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
$223$	− 18.6504i	− 1.24892i	−0.781056	−	0.624462i	$-0.785318\pi$
$223$	− 18.6504i	− 1.24892i	0.781056	−	0.624462i	$-0.214682\pi$
$224$	11.8455	0.791459
$225$	0	0
$226$	20.6136i	1.37119i
$227$	9.75733i	0.647617i	0.946123	+	0.323808i	$0.104963\pi$
$227$	9.75733i	0.647617i	−0.946123	+	0.323808i	$0.895037\pi$
$228$	0	0
$229$	− 2.86294i	− 0.189188i	−0.995516	−	0.0945941i	$-0.969845\pi$
$229$	− 2.86294i	− 0.189188i	0.995516	−	0.0945941i	$-0.0301553\pi$
$230$	−12.2838	−0.809971
$231$	0	0
$232$	− 5.64310i	− 0.370488i
$233$	−5.78554	−0.379024	−0.189512	−	0.981878i	$-0.560691\pi$
$233$	−5.78554	−0.379024	−0.189512	+	0.981878i	$0.560691\pi$
$234$	0	0
$235$	−35.9124	−2.34267
$236$	− 0.968541i	− 0.0630467i
$237$	0	0
$238$	−14.6407	−0.949016
$239$	7.09246i	0.458773i	0.973335	+	0.229386i	$0.0736720\pi$
$239$	7.09246i	0.458773i	−0.973335	+	0.229386i	$0.926328\pi$
$240$	0	0
$241$	3.89977i	0.251206i	0.992081	+	0.125603i	$0.0400866\pi$
$241$	3.89977i	0.251206i	−0.992081	+	0.125603i	$0.959913\pi$
$242$	11.0339i	0.709283i
$243$	0	0
$244$	−3.48427	−0.223058
$245$	44.9952i	2.87464i
$246$	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	$-0.524587\pi$
$251$	−2.44504	−0.154330	−0.0771648	+	0.997018i	$0.524587\pi$
$252$	0	0
$253$	5.15691i	0.324212i
$254$	11.8931i	0.746237i
$255$	0	0
$256$	−10.1129	−0.632056
$257$	−14.1304	−0.881428	−0.440714	−	0.897648i	$-0.645275\pi$
$257$	−14.1304	−0.881428	−0.440714	+	0.897648i	$0.645275\pi$
$258$	0	0
$259$	−21.8726	−1.35910
$260$	0	0
$261$	0	0
$262$	6.86964i	0.424408i
$263$	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
$265$	− 24.7995i	− 1.52342i
$266$	− 15.2265i	− 0.933599i
$267$	0	0
$268$	1.59419i	0.0973805i
$269$	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
$271$	3.19029i	0.193796i	0.995294	+	0.0968982i	$0.0308921\pi$
$271$	3.19029i	0.193796i	−0.995294	+	0.0968982i	$0.969108\pi$
$272$	7.11960	0.431689
$273$	0	0
$274$	−20.1806	−1.21915
$275$	4.18167i	0.252164i
$276$	0	0
$277$	−21.9366	−1.31804	−0.659022	−	0.752124i	$-0.729029\pi$
$277$	−21.9366	−1.31804	−0.659022	+	0.752124i	$0.729029\pi$
$278$	− 13.1045i	− 0.785958i
$279$	0	0
$280$	− 41.0224i	− 2.45155i
$281$	11.9903i	0.715282i	0.933859	+	0.357641i	$0.116419\pi$
$281$	11.9903i	0.715282i	−0.933859	+	0.357641i	$0.883581\pi$
$282$	0	0
$283$	14.3666	0.854005	0.427002	−	0.904250i	$-0.359570\pi$
$283$	14.3666	0.854005	0.427002	+	0.904250i	$0.359570\pi$
$284$	− 3.93362i	− 0.233418i
$285$	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
$292$	− 3.42327i	− 0.200332i
$293$	18.7584i	1.09588i	0.836519	+	0.547939i	$0.184587\pi$
$293$	18.7584i	1.09588i	−0.836519	+	0.547939i	$0.815413\pi$
$294$	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
$301$	11.4480i	0.659853i
$302$	−2.45175	−0.141082
$303$	0	0
$304$	7.40449i	0.424676i
$305$	21.9366i	1.25609i
$306$	0	0
$307$	− 25.6262i	− 1.46257i	−0.682074	−	0.731283i	$-0.738922\pi$
$307$	− 25.6262i	− 1.46257i	0.682074	−	0.731283i	$-0.261078\pi$
$308$	−3.13467	−0.178614
$309$	0	0
$310$	26.6625i	1.51433i
$311$	11.3817	0.645394	0.322697	−	0.946502i	$-0.395410\pi$
$311$	11.3817	0.645394	0.322697	+	0.946502i	$0.395410\pi$
$312$	0	0
$313$	27.5743	1.55859	0.779297	−	0.626655i	$-0.215576\pi$
$313$	27.5743	1.55859	0.779297	+	0.626655i	$0.215576\pi$
$314$	13.3448i	0.753091i
$315$	0	0
$316$	−1.78986	−0.100687
$317$	11.2597i	0.632405i	0.948692	+	0.316203i	$0.102408\pi$
$317$	11.2597i	0.632405i	−0.948692	+	0.316203i	$0.897592\pi$
$318$	0	0
$319$	2.71486i	0.152003i
$320$	24.9366i	1.39400i
$321$	0	0
$322$	−21.0519	−1.17318
$323$	6.21744i	0.345948i
$324$	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
$331$	− 11.9065i	− 0.654439i	−0.944948	−	0.327220i	$-0.893888\pi$
$331$	− 11.9065i	− 0.654439i	0.944948	−	0.327220i	$-0.106112\pi$
$332$	− 0.290520i	− 0.0159444i
$333$	0	0
$334$	26.2107	1.43419
$335$	10.0368	0.548371
$336$	0	0
$337$	−17.1672	−0.935157	−0.467578	−	0.883952i	$-0.654873\pi$
$337$	−17.1672	−0.935157	−0.467578	+	0.883952i	$0.654873\pi$
$338$	0	0
$339$	0	0
$340$	3.04892i	0.165351i
$341$	11.1933	0.606150
$342$	0	0
$343$	43.4989i	2.34872i
$344$	− 7.26875i	− 0.391905i
$345$	0	0
$346$	16.5036i	0.887242i
$347$	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
$349$	4.57242i	0.244756i	0.992484	+	0.122378i	$0.0390520\pi$
$349$	4.57242i	0.244756i	−0.992484	+	0.122378i	$0.960948\pi$
$350$	−17.0707	−0.912467
$351$	0	0
$352$	3.61835	0.192859
$353$	6.07606i	0.323396i	0.986840	+	0.161698i	$0.0516971\pi$
$353$	6.07606i	0.323396i	−0.986840	+	0.161698i	$0.948303\pi$
$354$	0	0
$355$	−24.7657	−1.31443
$356$	− 2.80194i	− 0.148502i
$357$	0	0
$358$	10.6340i	0.562025i
$359$	− 14.9661i	− 0.789883i	−0.918706	−	0.394942i	$-0.870765\pi$
$359$	− 14.9661i	− 0.789883i	0.918706	−	0.394942i	$-0.129235\pi$
$360$	0	0
$361$	12.5338	0.659673
$362$	4.53452i	0.238329i
$363$	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
$370$	− 15.9148i	− 0.827373i
$371$	− 42.5013i	− 2.20656i
$372$	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
$379$	− 26.5851i	− 1.36558i	−0.730613	−	0.682792i	$-0.760765\pi$
$379$	− 26.5851i	− 1.36558i	0.730613	−	0.682792i	$-0.239235\pi$
$380$	−3.17092	−0.162665
$381$	0	0
$382$	− 26.6625i	− 1.36417i
$383$	14.3502i	0.733261i	0.930367	+	0.366630i	$0.119489\pi$
$383$	14.3502i	0.733261i	−0.930367	+	0.366630i	$0.880511\pi$
$384$	0	0
$385$	19.7356i	1.00582i
$386$	10.5090	0.534895
$387$	0	0
$388$	4.46442i	0.226647i
$389$	−22.6582	−1.14881	−0.574407	−	0.818570i	$-0.694767\pi$
$389$	−22.6582	−1.14881	−0.574407	+	0.818570i	$0.694767\pi$
$390$	0	0
$391$	8.59611	0.434724
$392$	− 48.9614i	− 2.47292i
$393$	0	0
$394$	33.0157	1.66330
$395$	11.2687i	0.566992i
$396$	0	0
$397$	− 7.90754i	− 0.396868i	−0.980114	−	0.198434i	$-0.936414\pi$
$397$	− 7.90754i	− 0.396868i	0.980114	−	0.198434i	$-0.0635856\pi$
$398$	17.7724i	0.890850i
$399$	0	0
$400$	8.30127	0.415064
$401$	2.93661i	0.146647i	0.997308	+	0.0733236i	$0.0233606\pi$
$401$	2.93661i	0.146647i	−0.997308	+	0.0733236i	$0.976639\pi$
$402$	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
$409$	11.7549i	0.581244i	0.956838	+	0.290622i	$0.0938623\pi$
$409$	11.7549i	0.581244i	−0.956838	+	0.290622i	$0.906138\pi$
$410$	4.35690i	0.215172i
$411$	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
$418$	− 4.65114i	− 0.227495i
$419$	−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
$421$	25.6963i	1.25236i	0.779677	+	0.626181i	$0.215383\pi$
$421$	25.6963i	1.25236i	−0.779677	+	0.626181i	$0.784617\pi$
$422$	2.30798i	0.112351i
$423$	0	0
$424$	26.9855i	1.31053i
$425$	6.97046	0.338117
$426$	0	0
$427$	37.5948i	1.81934i
$428$	−4.69633	−0.227006
$429$	0	0
$430$	−8.32975	−0.401696
$431$	− 8.94198i	− 0.430720i	−0.976535	−	0.215360i	$-0.930907\pi$
$431$	− 8.94198i	− 0.430720i	0.976535	−	0.215360i	$-0.0690925\pi$
$432$	0	0
$433$	−2.91484	−0.140078	−0.0700391	−	0.997544i	$-0.522312\pi$
$433$	−2.91484	−0.140078	−0.0700391	+	0.997544i	$0.522312\pi$
$434$	45.6939i	2.19338i
$435$	0	0
$436$	− 0.476501i	− 0.0228203i
$437$	8.94007i	0.427661i
$438$	0	0
$439$	−9.05861	−0.432344	−0.216172	−	0.976355i	$-0.569357\pi$
$439$	−9.05861	−0.432344	−0.216172	+	0.976355i	$0.569357\pi$
$440$	− 12.5308i	− 0.597382i
$441$	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
$448$	42.7362i	2.01909i
$449$	28.7579i	1.35717i	0.734522	+	0.678585i	$0.237407\pi$
$449$	28.7579i	1.35717i	−0.734522	+	0.678585i	$0.762593\pi$
$450$	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
$457$	− 19.0761i	− 0.892341i	−0.894948	−	0.446170i	$-0.852788\pi$
$457$	− 19.0761i	− 0.892341i	0.894948	−	0.446170i	$-0.147212\pi$
$458$	3.57002	0.166816
$459$	0	0
$460$	4.38404i	0.204407i
$461$	− 31.7332i	− 1.47796i	−0.673727	−	0.738981i	$-0.735308\pi$
$461$	− 31.7332i	− 1.47796i	0.673727	−	0.738981i	$-0.264692\pi$
$462$	0	0
$463$	36.4784i	1.69530i	0.530559	+	0.847648i	$0.321982\pi$
$463$	36.4784i	1.69530i	−0.530559	+	0.847648i	$0.678018\pi$
$464$	5.38942	0.250198
$465$	0	0
$466$	− 7.21446i	− 0.334203i
$467$	13.0000	0.601568	0.300784	−	0.953692i	$-0.402752\pi$
$467$	13.0000	0.601568	0.300784	+	0.953692i	$0.402752\pi$
$468$	0	0
$469$	17.2010	0.794271
$470$	− 44.7821i	− 2.06564i
$471$	0	0
$472$	6.63533	0.305416
$473$	3.49694i	0.160790i
$474$	0	0
$475$	7.24937i	0.332624i
$476$	5.22521i	0.239497i
$477$	0	0
$478$	−8.84415	−0.404522
$479$	5.61655i	0.256627i	0.991734	+	0.128313i	$0.0409563\pi$
$479$	5.61655i	0.256627i	−0.991734	+	0.128313i	$0.959044\pi$
$480$	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
$487$	− 9.75733i	− 0.442147i	−0.975257	−	0.221073i	$-0.929044\pi$
$487$	− 9.75733i	− 0.442147i	0.975257	−	0.221073i	$-0.0709561\pi$
$488$	− 23.8702i	− 1.08055i
$489$	0	0
$490$	−56.1081	−2.53471
$491$	−7.38835	−0.333432	−0.166716	−	0.986005i	$-0.553316\pi$
$491$	−7.38835	−0.333432	−0.166716	+	0.986005i	$0.553316\pi$
$492$	0	0
$493$	4.52542	0.203815
$494$	0	0
$495$	0	0
$496$	− 22.2204i	− 0.997726i
$497$	−42.4432	−1.90384
$498$	0	0
$499$	− 43.2814i	− 1.93754i	−0.247956	−	0.968771i	$-0.579759\pi$
$499$	− 43.2814i	− 1.93754i	0.247956	−	0.968771i	$-0.420241\pi$
$500$	− 2.67994i	− 0.119851i
$501$	0	0
$502$	− 3.04892i	− 0.136080i
$503$	−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
$505$	− 38.9124i	− 1.73158i
$506$	−6.43057	−0.285874
$507$	0	0
$508$	4.24459	0.188323
$509$	41.5448i	1.84144i	0.390223	+	0.920720i	$0.372398\pi$
$509$	41.5448i	1.84144i	−0.390223	+	0.920720i	$0.627602\pi$
$510$	0	0
$511$	−36.9366	−1.63398
$512$	− 24.9390i	− 1.10216i
$513$	0	0
$514$	− 17.6203i	− 0.777197i
$515$	− 48.7633i	− 2.14877i
$516$	0	0
$517$	−18.8001	−0.826829
$518$	− 27.2747i	− 1.19838i
$519$	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
$526$	29.5827i	1.28987i
$527$	− 18.6582i	− 0.812763i
$528$	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
$535$	29.5676i	1.27832i
$536$	−10.9215	−0.471739
$537$	0	0
$538$	7.37734i	0.318060i
$539$	23.5550i	1.01458i
$540$	0	0
$541$	− 31.3534i	− 1.34799i	−0.738736	−	0.673995i	$-0.764577\pi$
$541$	− 31.3534i	− 1.34799i	0.738736	−	0.673995i	$-0.235423\pi$
$542$	−3.97823	−0.170880
$543$	0	0
$544$	− 6.03146i	− 0.258597i
$545$	−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	$0.359865\pi$
$548$	7.20237i	0.307670i
$549$	0	0
$550$	−5.21446	−0.222345
$551$	4.70650i	0.200503i
$552$	0	0
$553$	19.3123i	0.821242i
$554$	− 27.3545i	− 1.16218i
$555$	0	0
$556$	−4.67696	−0.198347
$557$	33.2349i	1.40821i	0.710097	+	0.704104i	$0.248651\pi$
$557$	33.2349i	1.40821i	−0.710097	+	0.704104i	$0.751349\pi$
$558$	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	$-0.474004\pi$
$563$	3.87130	0.163156	0.0815779	+	0.996667i	$0.474004\pi$
$564$	0	0
$565$	− 46.3183i	− 1.94862i
$566$	17.9148i	0.753017i
$567$	0	0
$568$	26.9487	1.13074
$569$	20.1457	0.844551	0.422276	−	0.906468i	$-0.361231\pi$
$569$	20.1457	0.844551	0.422276	+	0.906468i	$0.361231\pi$
$570$	0	0
$571$	32.1269	1.34447	0.672234	−	0.740338i	$-0.265335\pi$
$571$	32.1269	1.34447	0.672234	+	0.740338i	$0.265335\pi$
$572$	0	0
$573$	0	0
$574$	7.46681i	0.311659i
$575$	10.0228	0.417981
$576$	0	0
$577$	− 16.7506i	− 0.697338i	−0.937246	−	0.348669i	$-0.886634\pi$
$577$	− 16.7506i	− 0.697338i	0.937246	−	0.348669i	$-0.113366\pi$
$578$	− 13.7439i	− 0.571672i
$579$	0	0
$580$	2.30798i	0.0958336i
$581$	−3.13467	−0.130048
$582$	0	0
$583$	− 12.9825i	− 0.537682i
$584$	23.4523	0.970465
$585$	0	0
$586$	−23.3913	−0.966287
$587$	6.73795i	0.278105i	0.990285	+	0.139053i	$0.0444057\pi$
$587$	6.73795i	0.278105i	−0.990285	+	0.139053i	$0.955594\pi$
$588$	0	0
$589$	19.4047	0.799559
$590$	− 7.60388i	− 0.313047i
$591$	0	0
$592$	13.2634i	0.545121i
$593$	18.1172i	0.743985i	0.928236	+	0.371992i	$0.121325\pi$
$593$	18.1172i	0.743985i	−0.928236	+	0.371992i	$0.878675\pi$
$594$	0	0
$595$	32.8974	1.34866
$596$	6.38644i	0.261599i
$597$	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
$604$	0.875018i	0.0356040i
$605$	− 24.7928i	− 1.00797i
$606$	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
$613$	48.1782i	1.94590i	0.231017	+	0.972950i	$0.425795\pi$
$613$	48.1782i	1.94590i	−0.231017	+	0.972950i	$0.574205\pi$
$614$	31.9554	1.28961
$615$	0	0
$616$	− 21.4752i	− 0.865259i
$617$	− 30.3043i	− 1.22000i	−0.792400	−	0.610002i	$-0.791169\pi$
$617$	− 30.3043i	− 1.22000i	0.792400	−	0.610002i	$-0.208831\pi$
$618$	0	0
$619$	10.9041i	0.438272i	0.975694	+	0.219136i	$0.0703239\pi$
$619$	10.9041i	0.438272i	−0.975694	+	0.219136i	$0.929676\pi$
$620$	9.51573	0.382161
$621$	0	0
$622$	14.1927i	0.569075i
$623$	−30.2325	−1.21124
$624$	0	0
$625$	−31.1269	−1.24508
$626$	34.3846i	1.37429i
$627$	0	0
$628$	4.76271	0.190053
$629$	11.1371i	0.444064i
$630$	0	0
$631$	10.4523i	0.416101i	0.978118	+	0.208050i	$0.0667118\pi$
$631$	10.4523i	0.416101i	−0.978118	+	0.208050i	$0.933288\pi$
$632$	− 12.2620i	− 0.487758i
$633$	0	0
$634$	−14.0406	−0.557622
$635$	− 26.7235i	− 1.06049i
$636$	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	$-0.612957\pi$
$641$	−17.5942	−0.694929	−0.347464	+	0.937693i	$0.612957\pi$
$642$	0	0
$643$	22.6058i	0.891486i	0.895161	+	0.445743i	$0.147060\pi$
$643$	22.6058i	0.891486i	−0.895161	+	0.445743i	$0.852940\pi$
$644$	7.51334i	0.296067i
$645$	0	0
$646$	−7.75302	−0.305039
$647$	−24.7918	−0.974665	−0.487333	−	0.873216i	$-0.662030\pi$
$647$	−24.7918	−0.974665	−0.487333	+	0.873216i	$0.662030\pi$
$648$	0	0
$649$	−3.19221	−0.125305
$650$	0	0
$651$	0	0
$652$	1.73556i	0.0679699i
$653$	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
$655$	− 15.4359i	− 0.603132i
$656$	− 3.63102i	− 0.141768i
$657$	0	0
$658$	− 76.7472i	− 2.99192i
$659$	−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
$661$	15.9541i	0.620541i	0.950648	+	0.310271i	$0.100420\pi$
$661$	15.9541i	0.620541i	−0.950648	+	0.310271i	$0.899580\pi$
$662$	14.8471	0.577050
$663$	0	0
$664$	1.99031	0.0772391
$665$	34.2137i	1.32675i
$666$	0	0
$667$	6.50711	0.251956
$668$	− 9.35450i	− 0.361937i
$669$	0	0
$670$	12.5157i	0.483525i
$671$	11.4838i	0.443327i
$672$	0	0
$673$	7.38835	0.284800	0.142400	−	0.989809i	$-0.454518\pi$
$673$	7.38835	0.284800	0.142400	+	0.989809i	$0.454518\pi$
$674$	− 21.4071i	− 0.824572i
$675$	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
$682$	13.9578i	0.534471i
$683$	9.10023i	0.348211i	0.984727	+	0.174105i	$0.0557033\pi$
$683$	9.10023i	0.348211i	−0.984727	+	0.174105i	$0.944297\pi$
$684$	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
$691$	13.7711i	0.523876i	0.965085	+	0.261938i	$0.0843616\pi$
$691$	13.7711i	0.523876i	−0.965085	+	0.261938i	$0.915638\pi$
$692$	5.89008	0.223907
$693$	0	0
$694$	30.2717i	1.14910i
$695$	29.4456i	1.11694i
$696$	0	0
$697$	− 3.04892i	− 0.115486i
$698$	−5.70171	−0.215813
$699$	0	0
$700$	6.09246i	0.230273i
$701$	46.5090	1.75662	0.878311	−	0.478090i	$-0.158671\pi$
$701$	46.5090	1.75662	0.878311	+	0.478090i	$0.158671\pi$
$702$	0	0
$703$	−11.5827	−0.436850
$704$	13.0543i	0.492002i
$705$	0	0
$706$	−7.57673	−0.285154
$707$	− 66.6878i	− 2.50805i
$708$	0	0
$709$	− 7.24565i	− 0.272116i	−0.990701	−	0.136058i	$-0.956557\pi$
$709$	− 7.24565i	− 0.272116i	0.990701	−	0.136058i	$-0.0434434\pi$
$710$	− 30.8823i	− 1.15899i
$711$	0	0
$712$	19.1957	0.719388
$713$	− 26.8286i	− 1.00474i
$714$	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	$-0.342170\pi$
$719$	25.5147	0.951536	0.475768	+	0.879571i	$0.342170\pi$
$720$	0	0
$721$	− 83.5701i	− 3.11231i
$722$	15.6294i	0.581665i
$723$	0	0
$724$	1.61835	0.0601455
$725$	5.27652	0.195965
$726$	0	0
$727$	14.4873	0.537303	0.268651	−	0.963238i	$-0.413422\pi$
$727$	14.4873	0.537303	0.268651	+	0.963238i	$0.413422\pi$
$728$	0	0
$729$	0	0
$730$	− 26.8756i	− 0.994711i
$731$	5.82908	0.215596
$732$	0	0
$733$	37.5036i	1.38523i	0.721308	+	0.692614i	$0.243541\pi$
$733$	37.5036i	1.38523i	−0.721308	+	0.692614i	$0.756459\pi$
$734$	46.2422i	1.70683i
$735$	0	0
$736$	− 8.67264i	− 0.319678i
$737$	5.25428	0.193544
$738$	0	0
$739$	− 43.1876i	− 1.58868i	−0.607472	−	0.794341i	$-0.707816\pi$
$739$	− 43.1876i	− 1.58868i	0.607472	−	0.794341i	$-0.292184\pi$
$740$	−5.67994	−0.208799
$741$	0	0
$742$	52.9982	1.94563
$743$	− 13.4765i	− 0.494405i	−0.968964	−	0.247202i	$-0.920489\pi$
$743$	− 13.4765i	− 0.494405i	0.968964	−	0.247202i	$-0.0795113\pi$
$744$	0	0
$745$	40.2083	1.47312
$746$	− 45.5260i	− 1.66683i
$747$	0	0
$748$	1.59611i	0.0583594i
$749$	50.6728i	1.85154i
$750$	0	0
$751$	−35.5894	−1.29868	−0.649338	−	0.760500i	$-0.724954\pi$
$751$	−35.5894	−1.29868	−0.649338	+	0.760500i	$0.724954\pi$
$752$	37.3212i	1.36097i
$753$	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
$760$	− 21.7235i	− 0.787993i
$761$	30.9071i	1.12038i	0.828364	+	0.560190i	$0.189272\pi$
$761$	30.9071i	1.12038i	−0.828364	+	0.560190i	$0.810728\pi$
$762$	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
$769$	11.9892i	0.432343i	0.976355	+	0.216172i	$0.0693571\pi$
$769$	11.9892i	0.432343i	−0.976355	+	0.216172i	$0.930643\pi$
$770$	−24.6098	−0.886877
$771$	0	0
$772$	− 3.75063i	− 0.134988i
$773$	− 43.9661i	− 1.58135i	−0.612235	−	0.790676i	$-0.709729\pi$
$773$	− 43.9661i	− 1.58135i	0.612235	−	0.790676i	$-0.290271\pi$
$774$	0	0
$775$	− 21.7549i	− 0.781460i
$776$	−30.5851	−1.09794
$777$	0	0
$778$	− 28.2543i	− 1.01296i
$779$	3.17092	0.113610
$780$	0	0
$781$	−12.9648	−0.463918
$782$	10.7192i	0.383317i
$783$	0	0
$784$	46.7603	1.67001
$785$	− 29.9855i	− 1.07023i
$786$	0	0
$787$	− 30.3763i	− 1.08280i	−0.840766	−	0.541399i	$-0.817895\pi$
$787$	− 30.3763i	− 1.08280i	0.840766	−	0.541399i	$-0.182105\pi$
$788$	− 11.7832i	− 0.419757i
$789$	0	0
$790$	−14.0519	−0.499944
$791$	− 79.3798i	− 2.82242i
$792$	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	$-0.118786\pi$
$797$	52.5763	1.86235	0.931173	+	0.364577i	$0.118786\pi$
$798$	0	0
$799$	31.3381i	1.10866i
$800$	− 7.03252i	− 0.248637i
$801$	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
$808$	42.3424i	1.48960i
$809$	−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
$811$	− 1.36526i	− 0.0479406i	−0.999713	−	0.0239703i	$-0.992369\pi$
$811$	− 1.36526i	− 0.0479406i	0.999713	−	0.0239703i	$-0.00763072\pi$
$812$	3.95539i	0.138807i
$813$	0	0
$814$	− 8.33140i	− 0.292016i
$815$	10.9269	0.382753
$816$	0	0
$817$	6.06233i	0.212094i
$818$	−14.6582	−0.512511
$819$	0	0
$820$	1.55496	0.0543015
$821$	− 0.665939i	− 0.0232414i	−0.999932	−	0.0116207i	$-0.996301\pi$
$821$	− 0.665939i	− 0.0232414i	0.999932	−	0.0116207i	$-0.00369907\pi$
$822$	0	0
$823$	10.0592	0.350642	0.175321	−	0.984511i	$-0.443904\pi$
$823$	10.0592	0.350642	0.175321	+	0.984511i	$0.443904\pi$
$824$	53.0616i	1.84849i
$825$	0	0
$826$	− 13.0315i	− 0.453422i
$827$	− 37.3038i	− 1.29718i	−0.761138	−	0.648590i	$-0.775359\pi$
$827$	− 37.3038i	− 1.29718i	0.761138	−	0.648590i	$-0.224641\pi$
$828$	0	0
$829$	42.6209	1.48028	0.740142	−	0.672451i	$-0.234758\pi$
$829$	42.6209	1.48028	0.740142	+	0.672451i	$0.234758\pi$
$830$	− 2.28083i	− 0.0791688i
$831$	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
$838$	− 9.15511i	− 0.316258i
$839$	4.14005i	0.142930i	0.997443	+	0.0714652i	$0.0227675\pi$
$839$	4.14005i	0.142930i	−0.997443	+	0.0714652i	$0.977233\pi$
$840$	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
$847$	− 42.4898i	− 1.45997i
$848$	−25.7724	−0.885028
$849$	0	0
$850$	8.69202i	0.298134i
$851$	16.0140i	0.548953i
$852$	0	0
$853$	− 17.3502i	− 0.594059i	−0.954868	−	0.297030i	$-0.904004\pi$
$853$	− 17.3502i	− 0.594059i	0.954868	−	0.297030i	$-0.0959960\pi$
$854$	−46.8799	−1.60420
$855$	0	0
$856$	− 32.1739i	− 1.09968i
$857$	41.0180	1.40115	0.700575	−	0.713579i	$-0.252927\pi$
$857$	41.0180	1.40115	0.700575	+	0.713579i	$0.252927\pi$
$858$	0	0
$859$	6.59286	0.224945	0.112473	−	0.993655i	$-0.464123\pi$
$859$	6.59286	0.224945	0.112473	+	0.993655i	$0.464123\pi$
$860$	2.97285i	0.101373i
$861$	0	0
$862$	11.1505	0.379787
$863$	16.6455i	0.566619i	0.959028	+	0.283310i	$0.0914324\pi$
$863$	16.6455i	0.566619i	−0.959028	+	0.283310i	$0.908568\pi$
$864$	0	0
$865$	− 37.0834i	− 1.26087i
$866$	− 3.63474i	− 0.123514i
$867$	0	0
$868$	16.3080	0.553529
$869$	5.89918i	0.200116i
$870$	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
$877$	54.4965i	1.84022i	0.391666	+	0.920108i	$0.371899\pi$
$877$	54.4965i	1.84022i	−0.391666	+	0.920108i	$0.628101\pi$
$878$	− 11.2959i	− 0.381218i
$879$	0	0
$880$	11.9675	0.403424
$881$	−9.00670	−0.303444	−0.151722	−	0.988423i	$-0.548482\pi$
$881$	−9.00670	−0.303444	−0.151722	+	0.988423i	$0.548482\pi$
$882$	0	0
$883$	−18.8907	−0.635722	−0.317861	−	0.948137i	$-0.602965\pi$
$883$	−18.8907	−0.635722	−0.317861	+	0.948137i	$0.602965\pi$
$884$	0	0
$885$	0	0
$886$	− 14.0067i	− 0.470564i
$887$	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
$889$	− 45.7985i	− 1.53603i
$890$	− 21.9976i	− 0.737361i
$891$	0	0
$892$	− 8.30021i	− 0.277912i
$893$	−32.5921	−1.09065
$894$	0	0
$895$	− 23.8944i	− 0.798702i
$896$	−29.6002	−0.988872
$897$	0	0
$898$	−35.8605	−1.19668
$899$	− 14.1239i	− 0.471059i
$900$	0	0
$901$	−21.6407	−0.720957
$902$	2.28083i	0.0759434i
$903$	0	0
$904$	50.4010i	1.67631i
$905$	− 10.1890i	− 0.338693i
$906$	0	0
$907$	35.3013	1.17216	0.586080	−	0.810253i	$-0.300671\pi$
$907$	35.3013	1.17216	0.586080	+	0.810253i	$0.300671\pi$
$908$	4.34242i	0.144108i
$909$	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
$916$	− 1.27413i	− 0.0420983i
$917$	− 26.4540i	− 0.873588i
$918$	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
$925$	12.9855i	0.426961i
$926$	−45.4878	−1.49482
$927$	0	0
$928$	− 4.56571i	− 0.149877i
$929$	25.6267i	0.840785i	0.907342	+	0.420393i	$0.138108\pi$
$929$	25.6267i	0.840785i	−0.907342	+	0.420393i	$0.861892\pi$
$930$	0	0
$931$	40.8351i	1.33831i
$932$	−2.57481	−0.0843407
$933$	0	0
$934$	16.2107i	0.530431i
$935$	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	$0.460648\pi$
$938$	21.4494i	0.700346i
$939$	0	0
$940$	−15.9825	−0.521293
$941$	12.6418i	0.412110i	0.978540	+	0.206055i	$0.0660626\pi$
$941$	12.6418i	0.412110i	−0.978540	+	0.206055i	$0.933937\pi$
$942$	0	0
$943$	− 4.38404i	− 0.142764i
$944$	6.33704i	0.206253i
$945$	0	0
$946$	−4.36062	−0.141776
$947$	27.9801i	0.909233i	0.890687	+	0.454616i	$0.150224\pi$
$947$	27.9801i	0.909233i	−0.890687	+	0.454616i	$0.849776\pi$
$948$	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	$-0.479351\pi$
$953$	4.00239	0.129650	0.0648251	+	0.997897i	$0.479351\pi$
$954$	0	0
$955$	59.9101i	1.93864i
$956$	3.15644i	0.102087i
$957$	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
$964$	1.73556i	0.0558987i
$965$	−23.6136	−0.760148
$966$	0	0
$967$	12.2239i	0.393094i	0.980494	+	0.196547i	$0.0629728\pi$
$967$	12.2239i	0.393094i	−0.980494	+	0.196547i	$0.937027\pi$
$968$	26.9782i	0.867113i
$969$	0	0
$970$	35.0495i	1.12537i
$971$	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
$973$	50.4637i	1.61779i
$974$	12.1672	0.389862
$975$	0	0
$976$	22.7972	0.729719
$977$	− 18.7313i	− 0.599266i	−0.954055	−	0.299633i	$-0.903136\pi$
$977$	− 18.7313i	− 0.599266i	0.954055	−	0.299633i	$-0.0968642\pi$
$978$	0	0
$979$	−9.23490	−0.295149
$980$	20.0248i	0.639667i
$981$	0	0
$982$	− 9.21313i	− 0.294003i
$983$	12.0954i	0.385785i	0.981220	+	0.192892i	$0.0617868\pi$
$983$	12.0954i	0.385785i	−0.981220	+	0.192892i	$0.938213\pi$
$984$	0	0
$985$	−74.1855	−2.36375
$986$	5.64310i	0.179713i
$987$	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
$994$	− 52.9259i	− 1.67871i
$995$	− 39.9342i	− 1.26600i
$996$	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.b.m.1351.5		6
3.2	odd	2		507.2.b.g.337.2		6
13.5	odd	4		1521.2.a.p.1.3		3
13.8	odd	4		1521.2.a.q.1.1		3
13.12	even	2	inner	1521.2.b.m.1351.2		6
39.2	even	12		507.2.e.j.22.3		6
39.5	even	4		507.2.a.k.1.1	yes	3
39.8	even	4		507.2.a.j.1.3	✓	3
39.11	even	12		507.2.e.k.22.1		6
39.17	odd	6		507.2.j.h.361.5		12
39.20	even	12		507.2.e.k.484.1		6
39.23	odd	6		507.2.j.h.316.2		12
39.29	odd	6		507.2.j.h.316.5		12
39.32	even	12		507.2.e.j.484.3		6
39.35	odd	6		507.2.j.h.361.2		12
39.38	odd	2		507.2.b.g.337.5		6
156.47	odd	4		8112.2.a.by.1.1		3
156.83	odd	4		8112.2.a.cf.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.j.1.3	✓	3	39.8	even	4
507.2.a.k.1.1	yes	3	39.5	even	4
507.2.b.g.337.2		6	3.2	odd	2
507.2.b.g.337.5		6	39.38	odd	2
507.2.e.j.22.3		6	39.2	even	12
507.2.e.j.484.3		6	39.32	even	12
507.2.e.k.22.1		6	39.11	even	12
507.2.e.k.484.1		6	39.20	even	12
507.2.j.h.316.2		12	39.23	odd	6
507.2.j.h.316.5		12	39.29	odd	6
507.2.j.h.361.2		12	39.35	odd	6
507.2.j.h.361.5		12	39.17	odd	6
1521.2.a.p.1.3		3	13.5	odd	4
1521.2.a.q.1.1		3	13.8	odd	4
1521.2.b.m.1351.2		6	13.12	even	2	inner
1521.2.b.m.1351.5		6	1.1	even	1	trivial
8112.2.a.by.1.1		3	156.47	odd	4
8112.2.a.cf.1.3		3	156.83	odd	4

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.24698i q^{2} +0.445042 q^{4} -2.80194i q^{5} -4.80194i q^{7} +3.04892i q^{8} +O(q^{10})\)	\(q+1.24698i q^{2} +0.445042 q^{4} -2.80194i q^{5} -4.80194i q^{7} +3.04892i q^{8} +3.49396 q^{10} -1.46681i q^{11} +5.98792 q^{14} -2.91185 q^{16} -2.44504 q^{17} -2.54288i q^{19} -1.24698i q^{20} +1.82908 q^{22} -3.51573 q^{23} -2.85086 q^{25} -2.13706i q^{28} -1.85086 q^{29} +7.63102i q^{31} +2.46681i q^{32} -3.04892i q^{34} -13.4547 q^{35} -4.55496i q^{37} +3.17092 q^{38} +8.54288 q^{40} +1.24698i q^{41} -2.38404 q^{43} -0.652793i q^{44} -4.38404i q^{46} -12.8170i q^{47} -16.0586 q^{49} -3.55496i q^{50} +8.85086 q^{53} -4.10992 q^{55} +14.6407 q^{56} -2.30798i q^{58} -2.17629i q^{59} -7.82908 q^{61} -9.51573 q^{62} -8.89977 q^{64} +3.58211i q^{67} -1.08815 q^{68} -16.7778i q^{70} -8.83877i q^{71} -7.69202i q^{73} +5.67994 q^{74} -1.13169i q^{76} -7.04354 q^{77} -4.02177 q^{79} +8.15883i q^{80} -1.55496 q^{82} -0.652793i q^{83} +6.85086i q^{85} -2.97285i q^{86} +4.47219 q^{88} -6.29590i q^{89} -1.56465 q^{92} +15.9825 q^{94} -7.12498 q^{95} +10.0315i q^{97} -20.0248i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{4}+O(q^{10}) \) 6 * q + 2 * q^4	\( 6 q + 2 q^{4} + 2 q^{10} - 2 q^{14} - 10 q^{16} - 14 q^{17} - 10 q^{22} + 4 q^{23} + 10 q^{25} + 16 q^{29} - 36 q^{35} + 40 q^{38} + 14 q^{40} + 6 q^{43} - 34 q^{49} + 26 q^{53} - 26 q^{55} + 14 q^{56} - 26 q^{61} - 32 q^{62} - 8 q^{64} - 14 q^{68} - 14 q^{74} - 30 q^{77} - 18 q^{79} - 10 q^{82} + 14 q^{88} + 34 q^{92} + 64 q^{94} + 6 q^{95}+O(q^{100}) \) 6 * q + 2 * q^4 + 2 * q^10 - 2 * q^14 - 10 * q^16 - 14 * q^17 - 10 * q^22 + 4 * q^23 + 10 * q^25 + 16 * q^29 - 36 * q^35 + 40 * q^38 + 14 * q^40 + 6 * q^43 - 34 * q^49 + 26 * q^53 - 26 * q^55 + 14 * q^56 - 26 * q^61 - 32 * q^62 - 8 * q^64 - 14 * q^68 - 14 * q^74 - 30 * q^77 - 18 * q^79 - 10 * q^82 + 14 * q^88 + 34 * q^92 + 64 * q^94 + 6 * q^95

Introduction

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