LMFDB - Embedded newform 2151.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.2585660609.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 4x^{6} + 15x^{5} + x^{4} - 19x^{3} + 6x^{2} + 3x - 1 $ x^8 - 3x^7 - 4x^6 + 15x^5 + x^4 - 19x^3 + 6x^2 + 3x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.551277$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77982 q^{2} +1.16777 q^{4} +3.42938 q^{5} -4.73145 q^{7} +1.48122 q^{8} +O(q^{10})$	\(q-1.77982 q^{2} +1.16777 q^{4} +3.42938 q^{5} -4.73145 q^{7} +1.48122 q^{8} -6.10369 q^{10} +4.00716 q^{11} +5.24378 q^{13} +8.42114 q^{14} -4.97185 q^{16} +6.72530 q^{17} +5.62991 q^{19} +4.00473 q^{20} -7.13205 q^{22} -3.38512 q^{23} +6.76062 q^{25} -9.33300 q^{26} -5.52526 q^{28} -5.64920 q^{29} -4.15051 q^{31} +5.88659 q^{32} -11.9699 q^{34} -16.2259 q^{35} +0.459217 q^{37} -10.0203 q^{38} +5.07965 q^{40} +7.63479 q^{41} -11.1621 q^{43} +4.67946 q^{44} +6.02492 q^{46} +9.77015 q^{47} +15.3866 q^{49} -12.0327 q^{50} +6.12354 q^{52} -2.85055 q^{53} +13.7421 q^{55} -7.00830 q^{56} +10.0546 q^{58} -2.31331 q^{59} +6.70980 q^{61} +7.38717 q^{62} -0.533387 q^{64} +17.9829 q^{65} -4.26337 q^{67} +7.85363 q^{68} +28.8793 q^{70} +2.74894 q^{71} -5.07465 q^{73} -0.817325 q^{74} +6.57446 q^{76} -18.9597 q^{77} -13.2608 q^{79} -17.0503 q^{80} -13.5886 q^{82} -6.60862 q^{83} +23.0636 q^{85} +19.8666 q^{86} +5.93548 q^{88} +7.24765 q^{89} -24.8106 q^{91} -3.95306 q^{92} -17.3891 q^{94} +19.3071 q^{95} -6.52674 q^{97} -27.3854 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8}+O(q^{10}) $ 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8	$ 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8} + 4 q^{10} + 19 q^{11} - 3 q^{13} + 8 q^{14} + 9 q^{16} + 13 q^{17} - 6 q^{19} + 18 q^{20} + 3 q^{22} + 18 q^{23} + 7 q^{25} - 2 q^{28} + 10 q^{29} - 2 q^{31} + 20 q^{32} - 4 q^{34} + 7 q^{35} - 8 q^{37} - 9 q^{38} + 29 q^{40} + 22 q^{41} - 24 q^{43} + 7 q^{44} + 30 q^{46} + 17 q^{47} + 15 q^{49} - 24 q^{50} + 22 q^{52} + 32 q^{55} - 19 q^{56} + 18 q^{58} + 24 q^{59} + 10 q^{61} - 30 q^{62} + 33 q^{64} + 17 q^{65} - 48 q^{67} + 21 q^{68} + 31 q^{70} + 17 q^{71} + 2 q^{73} - 9 q^{74} + 10 q^{76} - 10 q^{77} + 17 q^{79} - 8 q^{80} - 17 q^{82} + 37 q^{83} + 28 q^{85} + q^{86} + 15 q^{88} + 41 q^{89} - 39 q^{91} + 38 q^{92} + 2 q^{94} - 16 q^{95} - 20 q^{97} - 46 q^{98}+O(q^{100}) $ 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8 + 4 * q^10 + 19 * q^11 - 3 * q^13 + 8 * q^14 + 9 * q^16 + 13 * q^17 - 6 * q^19 + 18 * q^20 + 3 * q^22 + 18 * q^23 + 7 * q^25 - 2 * q^28 + 10 * q^29 - 2 * q^31 + 20 * q^32 - 4 * q^34 + 7 * q^35 - 8 * q^37 - 9 * q^38 + 29 * q^40 + 22 * q^41 - 24 * q^43 + 7 * q^44 + 30 * q^46 + 17 * q^47 + 15 * q^49 - 24 * q^50 + 22 * q^52 + 32 * q^55 - 19 * q^56 + 18 * q^58 + 24 * q^59 + 10 * q^61 - 30 * q^62 + 33 * q^64 + 17 * q^65 - 48 * q^67 + 21 * q^68 + 31 * q^70 + 17 * q^71 + 2 * q^73 - 9 * q^74 + 10 * q^76 - 10 * q^77 + 17 * q^79 - 8 * q^80 - 17 * q^82 + 37 * q^83 + 28 * q^85 + q^86 + 15 * q^88 + 41 * q^89 - 39 * q^91 + 38 * q^92 + 2 * q^94 - 16 * q^95 - 20 * q^97 - 46 * 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 $

$2$	−1.77982	−1.25853	−0.629263	−	0.777193i	$-0.716643\pi$
$2$	−1.77982	−1.25853	−0.629263	+	0.777193i	$0.716643\pi$
$3$	0	0
$3$	0	0
$4$	1.16777	0.583887
$5$	3.42938	1.53366	0.766832	−	0.641848i	$-0.221832\pi$
$5$	3.42938	1.53366	0.766832	+	0.641848i	$0.221832\pi$
$6$	0	0
$7$	−4.73145	−1.78832	−0.894159	−	0.447749i	$-0.852226\pi$
$7$	−4.73145	−1.78832	−0.894159	+	0.447749i	$0.852226\pi$
$8$	1.48122	0.523689
$9$	0	0
$10$	−6.10369	−1.93015
$11$	4.00716	1.20821	0.604103	−	0.796907i	$-0.293532\pi$
$11$	4.00716	1.20821	0.604103	+	0.796907i	$0.293532\pi$
$12$	0	0
$13$	5.24378	1.45436	0.727181	−	0.686446i	$-0.240830\pi$
$13$	5.24378	1.45436	0.727181	+	0.686446i	$0.240830\pi$
$14$	8.42114	2.25064
$15$	0	0
$16$	−4.97185	−1.24296
$17$	6.72530	1.63113	0.815563	−	0.578669i	$-0.196428\pi$
$17$	6.72530	1.63113	0.815563	+	0.578669i	$0.196428\pi$
$18$	0	0
$19$	5.62991	1.29159	0.645795	−	0.763511i	$-0.276526\pi$
$19$	5.62991	1.29159	0.645795	+	0.763511i	$0.276526\pi$
$20$	4.00473	0.895486
$21$	0	0
$22$	−7.13205	−1.52056
$23$	−3.38512	−0.705847	−0.352924	−	0.935652i	$-0.614812\pi$
$23$	−3.38512	−0.705847	−0.352924	+	0.935652i	$0.614812\pi$
$24$	0	0
$25$	6.76062	1.35212
$26$	−9.33300	−1.83035
$27$	0	0
$28$	−5.52526	−1.04418
$29$	−5.64920	−1.04903	−0.524515	−	0.851401i	$-0.675753\pi$
$29$	−5.64920	−1.04903	−0.524515	+	0.851401i	$0.675753\pi$
$30$	0	0
$31$	−4.15051	−0.745453	−0.372727	−	0.927941i	$-0.621577\pi$
$31$	−4.15051	−0.745453	−0.372727	+	0.927941i	$0.621577\pi$
$32$	5.88659	1.04061
$33$	0	0
$34$	−11.9699	−2.05281
$35$	−16.2259	−2.74268
$36$	0	0
$37$	0.459217	0.0754948	0.0377474	−	0.999287i	$-0.487982\pi$
$37$	0.459217	0.0754948	0.0377474	+	0.999287i	$0.487982\pi$
$38$	−10.0203	−1.62550
$39$	0	0
$40$	5.07965	0.803163
$41$	7.63479	1.19235	0.596177	−	0.802853i	$-0.296686\pi$
$41$	7.63479	1.19235	0.596177	+	0.802853i	$0.296686\pi$
$42$	0	0
$43$	−11.1621	−1.70220	−0.851102	−	0.525001i	$-0.824065\pi$
$43$	−11.1621	−1.70220	−0.851102	+	0.525001i	$0.824065\pi$
$44$	4.67946	0.705455
$45$	0	0
$46$	6.02492	0.888327
$47$	9.77015	1.42512	0.712561	−	0.701610i	$-0.247535\pi$
$47$	9.77015	1.42512	0.712561	+	0.701610i	$0.247535\pi$
$48$	0	0
$49$	15.3866	2.19808
$50$	−12.0327	−1.70168
$51$	0	0
$52$	6.12354	0.849182
$53$	−2.85055	−0.391553	−0.195776	−	0.980649i	$-0.562723\pi$
$53$	−2.85055	−0.391553	−0.195776	+	0.980649i	$0.562723\pi$
$54$	0	0
$55$	13.7421	1.85298
$56$	−7.00830	−0.936523
$57$	0	0
$58$	10.0546	1.32023
$59$	−2.31331	−0.301168	−0.150584	−	0.988597i	$-0.548115\pi$
$59$	−2.31331	−0.301168	−0.150584	+	0.988597i	$0.548115\pi$
$60$	0	0
$61$	6.70980	0.859102	0.429551	−	0.903043i	$-0.358672\pi$
$61$	6.70980	0.859102	0.429551	+	0.903043i	$0.358672\pi$
$62$	7.38717	0.938172
$63$	0	0
$64$	−0.533387	−0.0666733
$65$	17.9829	2.23050
$66$	0	0
$67$	−4.26337	−0.520854	−0.260427	−	0.965494i	$-0.583863\pi$
$67$	−4.26337	−0.520854	−0.260427	+	0.965494i	$0.583863\pi$
$68$	7.85363	0.952392
$69$	0	0
$70$	28.8793	3.45173
$71$	2.74894	0.326240	0.163120	−	0.986606i	$-0.447844\pi$
$71$	2.74894	0.326240	0.163120	+	0.986606i	$0.447844\pi$
$72$	0	0
$73$	−5.07465	−0.593943	−0.296971	−	0.954886i	$-0.595977\pi$
$73$	−5.07465	−0.593943	−0.296971	+	0.954886i	$0.595977\pi$
$74$	−0.817325	−0.0950121
$75$	0	0
$76$	6.57446	0.754142
$77$	−18.9597	−2.16066
$78$	0	0
$79$	−13.2608	−1.49195	−0.745977	−	0.665971i	$-0.768017\pi$
$79$	−13.2608	−1.49195	−0.745977	+	0.665971i	$0.768017\pi$
$80$	−17.0503	−1.90629
$81$	0	0
$82$	−13.5886	−1.50061
$83$	−6.60862	−0.725390	−0.362695	−	0.931908i	$-0.618143\pi$
$83$	−6.60862	−0.725390	−0.362695	+	0.931908i	$0.618143\pi$
$84$	0	0
$85$	23.0636	2.50160
$86$	19.8666	2.14227
$87$	0	0
$88$	5.93548	0.632724
$89$	7.24765	0.768250	0.384125	−	0.923281i	$-0.374503\pi$
$89$	7.24765	0.768250	0.384125	+	0.923281i	$0.374503\pi$
$90$	0	0
$91$	−24.8106	−2.60086
$92$	−3.95306	−0.412135
$93$	0	0
$94$	−17.3891	−1.79355
$95$	19.3071	1.98086
$96$	0	0
$97$	−6.52674	−0.662690	−0.331345	−	0.943510i	$-0.607502\pi$
$97$	−6.52674	−0.662690	−0.331345	+	0.943510i	$0.607502\pi$
$98$	−27.3854	−2.76634
$99$	0	0
$100$	7.89487	0.789487
$101$	2.74004	0.272644	0.136322	−	0.990665i	$-0.456472\pi$
$101$	2.74004	0.272644	0.136322	+	0.990665i	$0.456472\pi$
$102$	0	0
$103$	7.97929	0.786223	0.393111	−	0.919491i	$-0.371399\pi$
$103$	7.97929	0.786223	0.393111	+	0.919491i	$0.371399\pi$
$104$	7.76717	0.761633
$105$	0	0
$106$	5.07347	0.492779
$107$	8.00547	0.773918	0.386959	−	0.922097i	$-0.373525\pi$
$107$	8.00547	0.773918	0.386959	+	0.922097i	$0.373525\pi$
$108$	0	0
$109$	−8.05542	−0.771569	−0.385785	−	0.922589i	$-0.626069\pi$
$109$	−8.05542	−0.771569	−0.385785	+	0.922589i	$0.626069\pi$
$110$	−24.4585	−2.33202
$111$	0	0
$112$	23.5241	2.22281
$113$	3.85280	0.362441	0.181221	−	0.983442i	$-0.441995\pi$
$113$	3.85280	0.362441	0.181221	+	0.983442i	$0.441995\pi$
$114$	0	0
$115$	−11.6089	−1.08253
$116$	−6.59699	−0.612515
$117$	0	0
$118$	4.11729	0.379027
$119$	−31.8204	−2.91697
$120$	0	0
$121$	5.05736	0.459760
$122$	−11.9423	−1.08120
$123$	0	0
$124$	−4.84685	−0.435260
$125$	6.03783	0.540040
$126$	0	0
$127$	13.9548	1.23829	0.619144	−	0.785277i	$-0.287480\pi$
$127$	13.9548	1.23829	0.619144	+	0.785277i	$0.287480\pi$
$128$	−10.8238	−0.956701
$129$	0	0
$130$	−32.0064	−2.80714
$131$	7.84323	0.685266	0.342633	−	0.939469i	$-0.388681\pi$
$131$	7.84323	0.685266	0.342633	+	0.939469i	$0.388681\pi$
$132$	0	0
$133$	−26.6376	−2.30977
$134$	7.58805	0.655508
$135$	0	0
$136$	9.96163	0.854203
$137$	11.5407	0.985989	0.492995	−	0.870032i	$-0.335902\pi$
$137$	11.5407	0.985989	0.492995	+	0.870032i	$0.335902\pi$
$138$	0	0
$139$	8.37720	0.710545	0.355272	−	0.934763i	$-0.384388\pi$
$139$	8.37720	0.710545	0.355272	+	0.934763i	$0.384388\pi$
$140$	−18.9482	−1.60141
$141$	0	0
$142$	−4.89264	−0.410581
$143$	21.0127	1.75717
$144$	0	0
$145$	−19.3732	−1.60886
$146$	9.03199	0.747492
$147$	0	0
$148$	0.536261	0.0440804
$149$	−5.22819	−0.428310	−0.214155	−	0.976800i	$-0.568700\pi$
$149$	−5.22819	−0.428310	−0.214155	+	0.976800i	$0.568700\pi$
$150$	0	0
$151$	14.2029	1.15581	0.577907	−	0.816103i	$-0.303870\pi$
$151$	14.2029	1.15581	0.577907	+	0.816103i	$0.303870\pi$
$152$	8.33912	0.676392
$153$	0	0
$154$	33.7449	2.71924
$155$	−14.2337	−1.14327
$156$	0	0
$157$	−5.45566	−0.435409	−0.217704	−	0.976015i	$-0.569857\pi$
$157$	−5.45566	−0.435409	−0.217704	+	0.976015i	$0.569857\pi$
$158$	23.6019	1.87766
$159$	0	0
$160$	20.1873	1.59595
$161$	16.0165	1.26228
$162$	0	0
$163$	−24.1335	−1.89028	−0.945142	−	0.326661i	$-0.894076\pi$
$163$	−24.1335	−1.89028	−0.945142	+	0.326661i	$0.894076\pi$
$164$	8.91571	0.696200
$165$	0	0
$166$	11.7622	0.912922
$167$	8.67535	0.671319	0.335659	−	0.941983i	$-0.391041\pi$
$167$	8.67535	0.671319	0.335659	+	0.941983i	$0.391041\pi$
$168$	0	0
$169$	14.4972	1.11517
$170$	−41.0491	−3.14832
$171$	0	0
$172$	−13.0348	−0.993894
$173$	−18.1422	−1.37932	−0.689662	−	0.724132i	$-0.742241\pi$
$173$	−18.1422	−1.37932	−0.689662	+	0.724132i	$0.742241\pi$
$174$	0	0
$175$	−31.9875	−2.41803
$176$	−19.9230	−1.50175
$177$	0	0
$178$	−12.8995	−0.966862
$179$	6.97740	0.521515	0.260758	−	0.965404i	$-0.416028\pi$
$179$	6.97740	0.521515	0.260758	+	0.965404i	$0.416028\pi$
$180$	0	0
$181$	25.0686	1.86334	0.931668	−	0.363310i	$-0.118353\pi$
$181$	25.0686	1.86334	0.931668	+	0.363310i	$0.118353\pi$
$182$	44.1586	3.27325
$183$	0	0
$184$	−5.01410	−0.369644
$185$	1.57483	0.115784
$186$	0	0
$187$	26.9494	1.97073
$188$	11.4093	0.832110
$189$	0	0
$190$	−34.3632	−2.49297
$191$	−0.232350	−0.0168123	−0.00840613	−	0.999965i	$-0.502676\pi$
$191$	−0.232350	−0.0168123	−0.00840613	+	0.999965i	$0.502676\pi$
$192$	0	0
$193$	6.60999	0.475798	0.237899	−	0.971290i	$-0.423541\pi$
$193$	6.60999	0.475798	0.237899	+	0.971290i	$0.423541\pi$
$194$	11.6164	0.834012
$195$	0	0
$196$	17.9680	1.28343
$197$	−5.87846	−0.418823	−0.209412	−	0.977828i	$-0.567155\pi$
$197$	−5.87846	−0.418823	−0.209412	+	0.977828i	$0.567155\pi$
$198$	0	0
$199$	−2.34692	−0.166369	−0.0831845	−	0.996534i	$-0.526509\pi$
$199$	−2.34692	−0.166369	−0.0831845	+	0.996534i	$0.526509\pi$
$200$	10.0139	0.708093
$201$	0	0
$202$	−4.87679	−0.343130
$203$	26.7289	1.87600
$204$	0	0
$205$	26.1826	1.82867
$206$	−14.2017	−0.989481
$207$	0	0
$208$	−26.0713	−1.80772
$209$	22.5600	1.56051
$210$	0	0
$211$	22.8761	1.57485	0.787426	−	0.616409i	$-0.211413\pi$
$211$	22.8761	1.57485	0.787426	+	0.616409i	$0.211413\pi$
$212$	−3.32879	−0.228622
$213$	0	0
$214$	−14.2483	−0.973996
$215$	−38.2790	−2.61061
$216$	0	0
$217$	19.6379	1.33311
$218$	14.3372	0.971040
$219$	0	0
$220$	16.0476	1.08193
$221$	35.2660	2.37225
$222$	0	0
$223$	7.43053	0.497585	0.248793	−	0.968557i	$-0.419966\pi$
$223$	7.43053	0.497585	0.248793	+	0.968557i	$0.419966\pi$
$224$	−27.8521	−1.86095
$225$	0	0
$226$	−6.85731	−0.456141
$227$	−16.2807	−1.08059	−0.540293	−	0.841477i	$-0.681687\pi$
$227$	−16.2807	−1.08059	−0.540293	+	0.841477i	$0.681687\pi$
$228$	0	0
$229$	7.56557	0.499947	0.249974	−	0.968253i	$-0.419578\pi$
$229$	7.56557	0.499947	0.249974	+	0.968253i	$0.419578\pi$
$230$	20.6617	1.36239
$231$	0	0
$232$	−8.36769	−0.549366
$233$	11.4082	0.747379	0.373690	−	0.927554i	$-0.378093\pi$
$233$	11.4082	0.747379	0.373690	+	0.927554i	$0.378093\pi$
$234$	0	0
$235$	33.5055	2.18566
$236$	−2.70143	−0.175848
$237$	0	0
$238$	56.6347	3.67108
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	6.23773	0.401808	0.200904	−	0.979611i	$-0.435612\pi$
$241$	6.23773	0.401808	0.200904	+	0.979611i	$0.435612\pi$
$242$	−9.00121	−0.578620
$243$	0	0
$244$	7.83553	0.501618
$245$	52.7664	3.37112
$246$	0	0
$247$	29.5220	1.87844
$248$	−6.14780	−0.390386
$249$	0	0
$250$	−10.7463	−0.679654
$251$	9.00060	0.568113	0.284056	−	0.958808i	$-0.408320\pi$
$251$	9.00060	0.568113	0.284056	+	0.958808i	$0.408320\pi$
$252$	0	0
$253$	−13.5647	−0.852808
$254$	−24.8371	−1.55842
$255$	0	0
$256$	20.3313	1.27071
$257$	−15.6150	−0.974037	−0.487018	−	0.873392i	$-0.661916\pi$
$257$	−15.6150	−0.974037	−0.487018	+	0.873392i	$0.661916\pi$
$258$	0	0
$259$	−2.17276	−0.135009
$260$	20.9999	1.30236
$261$	0	0
$262$	−13.9596	−0.862425
$263$	27.7226	1.70945	0.854723	−	0.519084i	$-0.173727\pi$
$263$	27.7226	1.70945	0.854723	+	0.519084i	$0.173727\pi$
$264$	0	0
$265$	−9.77559	−0.600510
$266$	47.4103	2.90691
$267$	0	0
$268$	−4.97865	−0.304119
$269$	3.41673	0.208322	0.104161	−	0.994560i	$-0.466784\pi$
$269$	3.41673	0.208322	0.104161	+	0.994560i	$0.466784\pi$
$270$	0	0
$271$	−22.6817	−1.37781	−0.688906	−	0.724851i	$-0.741909\pi$
$271$	−22.6817	−1.37781	−0.688906	+	0.724851i	$0.741909\pi$
$272$	−33.4372	−2.02743
$273$	0	0
$274$	−20.5404	−1.24089
$275$	27.0909	1.63364
$276$	0	0
$277$	−21.9387	−1.31817	−0.659083	−	0.752070i	$-0.729056\pi$
$277$	−21.9387	−1.31817	−0.659083	+	0.752070i	$0.729056\pi$
$278$	−14.9099	−0.894239
$279$	0	0
$280$	−24.0341	−1.43631
$281$	9.11756	0.543908	0.271954	−	0.962310i	$-0.412330\pi$
$281$	9.11756	0.543908	0.271954	+	0.962310i	$0.412330\pi$
$282$	0	0
$283$	6.85690	0.407600	0.203800	−	0.979012i	$-0.434671\pi$
$283$	6.85690	0.407600	0.203800	+	0.979012i	$0.434671\pi$
$284$	3.21014	0.190487
$285$	0	0
$286$	−37.3988	−2.21144
$287$	−36.1236	−2.13231
$288$	0	0
$289$	28.2297	1.66057
$290$	34.4810	2.02479
$291$	0	0
$292$	−5.92604	−0.346795
$293$	−23.8933	−1.39586	−0.697931	−	0.716165i	$-0.745896\pi$
$293$	−23.8933	−1.39586	−0.697931	+	0.716165i	$0.745896\pi$
$294$	0	0
$295$	−7.93322	−0.461890
$296$	0.680200	0.0395358
$297$	0	0
$298$	9.30525	0.539039
$299$	−17.7508	−1.02656
$300$	0	0
$301$	52.8128	3.04408
$302$	−25.2786	−1.45462
$303$	0	0
$304$	−27.9911	−1.60540
$305$	23.0104	1.31757
$306$	0	0
$307$	−10.9848	−0.626936	−0.313468	−	0.949599i	$-0.601491\pi$
$307$	−10.9848	−0.626936	−0.313468	+	0.949599i	$0.601491\pi$
$308$	−22.1406	−1.26158
$309$	0	0
$310$	25.3334	1.43884
$311$	−16.5644	−0.939282	−0.469641	−	0.882857i	$-0.655617\pi$
$311$	−16.5644	−0.939282	−0.469641	+	0.882857i	$0.655617\pi$
$312$	0	0
$313$	4.01041	0.226682	0.113341	−	0.993556i	$-0.463845\pi$
$313$	4.01041	0.226682	0.113341	+	0.993556i	$0.463845\pi$
$314$	9.71011	0.547973
$315$	0	0
$316$	−15.4856	−0.871133
$317$	4.09934	0.230242	0.115121	−	0.993351i	$-0.463274\pi$
$317$	4.09934	0.230242	0.115121	+	0.993351i	$0.463274\pi$
$318$	0	0
$319$	−22.6373	−1.26744
$320$	−1.82918	−0.102254
$321$	0	0
$322$	−28.5066	−1.58861
$323$	37.8629	2.10675
$324$	0	0
$325$	35.4512	1.96648
$326$	42.9534	2.37897
$327$	0	0
$328$	11.3088	0.624423
$329$	−46.2269	−2.54857
$330$	0	0
$331$	19.5113	1.07244	0.536218	−	0.844079i	$-0.319852\pi$
$331$	19.5113	1.07244	0.536218	+	0.844079i	$0.319852\pi$
$332$	−7.71737	−0.423546
$333$	0	0
$334$	−15.4406	−0.844872
$335$	−14.6207	−0.798814
$336$	0	0
$337$	−22.6668	−1.23474	−0.617370	−	0.786673i	$-0.711802\pi$
$337$	−22.6668	−1.23474	−0.617370	+	0.786673i	$0.711802\pi$
$338$	−25.8024	−1.40347
$339$	0	0
$340$	26.9330	1.46065
$341$	−16.6318	−0.900661
$342$	0	0
$343$	−39.6807	−2.14256
$344$	−16.5335	−0.891425
$345$	0	0
$346$	32.2899	1.73591
$347$	16.9676	0.910870	0.455435	−	0.890269i	$-0.349484\pi$
$347$	16.9676	0.910870	0.455435	+	0.890269i	$0.349484\pi$
$348$	0	0
$349$	−30.5651	−1.63611	−0.818057	−	0.575137i	$-0.804949\pi$
$349$	−30.5651	−1.63611	−0.818057	+	0.575137i	$0.804949\pi$
$350$	56.9321	3.04315
$351$	0	0
$352$	23.5885	1.25727
$353$	17.7900	0.946868	0.473434	−	0.880829i	$-0.343014\pi$
$353$	17.7900	0.946868	0.473434	+	0.880829i	$0.343014\pi$
$354$	0	0
$355$	9.42716	0.500342
$356$	8.46362	0.448571
$357$	0	0
$358$	−12.4185	−0.656340
$359$	12.8446	0.677915	0.338957	−	0.940802i	$-0.389926\pi$
$359$	12.8446	0.677915	0.338957	+	0.940802i	$0.389926\pi$
$360$	0	0
$361$	12.6959	0.668205
$362$	−44.6178	−2.34506
$363$	0	0
$364$	−28.9732	−1.51861
$365$	−17.4029	−0.910909
$366$	0	0
$367$	10.6810	0.557544	0.278772	−	0.960357i	$-0.410073\pi$
$367$	10.6810	0.557544	0.278772	+	0.960357i	$0.410073\pi$
$368$	16.8303	0.877342
$369$	0	0
$370$	−2.80292	−0.145717
$371$	13.4872	0.700221
$372$	0	0
$373$	−9.38807	−0.486096	−0.243048	−	0.970014i	$-0.578147\pi$
$373$	−9.38807	−0.486096	−0.243048	+	0.970014i	$0.578147\pi$
$374$	−47.9652	−2.48022
$375$	0	0
$376$	14.4717	0.746321
$377$	−29.6232	−1.52567
$378$	0	0
$379$	−1.85543	−0.0953070	−0.0476535	−	0.998864i	$-0.515174\pi$
$379$	−1.85543	−0.0953070	−0.0476535	+	0.998864i	$0.515174\pi$
$380$	22.5463	1.15660
$381$	0	0
$382$	0.413542	0.0211587
$383$	−8.56801	−0.437805	−0.218902	−	0.975747i	$-0.570248\pi$
$383$	−8.56801	−0.437805	−0.218902	+	0.975747i	$0.570248\pi$
$384$	0	0
$385$	−65.0199	−3.31372
$386$	−11.7646	−0.598803
$387$	0	0
$388$	−7.62175	−0.386936
$389$	22.5044	1.14102	0.570509	−	0.821291i	$-0.306746\pi$
$389$	22.5044	1.14102	0.570509	+	0.821291i	$0.306746\pi$
$390$	0	0
$391$	−22.7660	−1.15132
$392$	22.7909	1.15111
$393$	0	0
$394$	10.4626	0.527100
$395$	−45.4762	−2.28816
$396$	0	0
$397$	−25.5169	−1.28066	−0.640329	−	0.768101i	$-0.721202\pi$
$397$	−25.5169	−1.28066	−0.640329	+	0.768101i	$0.721202\pi$
$398$	4.17711	0.209380
$399$	0	0
$400$	−33.6128	−1.68064
$401$	−6.83902	−0.341524	−0.170762	−	0.985312i	$-0.554623\pi$
$401$	−6.83902	−0.341524	−0.170762	+	0.985312i	$0.554623\pi$
$402$	0	0
$403$	−21.7643	−1.08416
$404$	3.19975	0.159193
$405$	0	0
$406$	−47.5727	−2.36100
$407$	1.84016	0.0912132
$408$	0	0
$409$	10.7561	0.531856	0.265928	−	0.963993i	$-0.414322\pi$
$409$	10.7561	0.531856	0.265928	+	0.963993i	$0.414322\pi$
$410$	−46.6004	−2.30143
$411$	0	0
$412$	9.31800	0.459065
$413$	10.9453	0.538584
$414$	0	0
$415$	−22.6634	−1.11250
$416$	30.8679	1.51343
$417$	0	0
$418$	−40.1528	−1.96394
$419$	−17.3249	−0.846378	−0.423189	−	0.906042i	$-0.639089\pi$
$419$	−17.3249	−0.846378	−0.423189	+	0.906042i	$0.639089\pi$
$420$	0	0
$421$	11.5786	0.564305	0.282152	−	0.959370i	$-0.408952\pi$
$421$	11.5786	0.564305	0.282152	+	0.959370i	$0.408952\pi$
$422$	−40.7153	−1.98199
$423$	0	0
$424$	−4.22228	−0.205052
$425$	45.4672	2.20548
$426$	0	0
$427$	−31.7471	−1.53635
$428$	9.34858	0.451881
$429$	0	0
$430$	68.1299	3.28552
$431$	6.01184	0.289580	0.144790	−	0.989462i	$-0.453749\pi$
$431$	6.01184	0.289580	0.144790	+	0.989462i	$0.453749\pi$
$432$	0	0
$433$	24.6242	1.18336	0.591681	−	0.806172i	$-0.298465\pi$
$433$	24.6242	1.18336	0.591681	+	0.806172i	$0.298465\pi$
$434$	−34.9520	−1.67775
$435$	0	0
$436$	−9.40691	−0.450509
$437$	−19.0579	−0.911665
$438$	0	0
$439$	−1.61597	−0.0771258	−0.0385629	−	0.999256i	$-0.512278\pi$
$439$	−1.61597	−0.0771258	−0.0385629	+	0.999256i	$0.512278\pi$
$440$	20.3550	0.970386
$441$	0	0
$442$	−62.7672	−2.98553
$443$	30.2686	1.43810	0.719052	−	0.694957i	$-0.244576\pi$
$443$	30.2686	1.43810	0.719052	+	0.694957i	$0.244576\pi$
$444$	0	0
$445$	24.8549	1.17824
$446$	−13.2250	−0.626224
$447$	0	0
$448$	2.52369	0.119233
$449$	31.5649	1.48964	0.744819	−	0.667267i	$-0.232536\pi$
$449$	31.5649	1.48964	0.744819	+	0.667267i	$0.232536\pi$
$450$	0	0
$451$	30.5939	1.44061
$452$	4.49920	0.211625
$453$	0	0
$454$	28.9767	1.35994
$455$	−85.0850	−3.98885
$456$	0	0
$457$	27.2477	1.27460	0.637298	−	0.770618i	$-0.280052\pi$
$457$	27.2477	1.27460	0.637298	+	0.770618i	$0.280052\pi$
$458$	−13.4654	−0.629196
$459$	0	0
$460$	−13.5565	−0.632076
$461$	−4.32913	−0.201628	−0.100814	−	0.994905i	$-0.532145\pi$
$461$	−4.32913	−0.201628	−0.100814	+	0.994905i	$0.532145\pi$
$462$	0	0
$463$	−8.44105	−0.392289	−0.196145	−	0.980575i	$-0.562842\pi$
$463$	−8.44105	−0.392289	−0.196145	+	0.980575i	$0.562842\pi$
$464$	28.0870	1.30391
$465$	0	0
$466$	−20.3047	−0.940596
$467$	29.3336	1.35739	0.678697	−	0.734418i	$-0.262545\pi$
$467$	29.3336	1.35739	0.678697	+	0.734418i	$0.262545\pi$
$468$	0	0
$469$	20.1719	0.931452
$470$	−59.6339	−2.75071
$471$	0	0
$472$	−3.42652	−0.157718
$473$	−44.7283	−2.05661
$474$	0	0
$475$	38.0617	1.74639
$476$	−37.1590	−1.70318
$477$	0	0
$478$	1.77982	0.0814073
$479$	27.9268	1.27601	0.638004	−	0.770033i	$-0.279760\pi$
$479$	27.9268	1.27601	0.638004	+	0.770033i	$0.279760\pi$
$480$	0	0
$481$	2.40803	0.109797
$482$	−11.1021	−0.505685
$483$	0	0
$484$	5.90585	0.268448
$485$	−22.3826	−1.01634
$486$	0	0
$487$	−33.3345	−1.51053	−0.755265	−	0.655420i	$-0.772492\pi$
$487$	−33.3345	−1.51053	−0.755265	+	0.655420i	$0.772492\pi$
$488$	9.93867	0.449903
$489$	0	0
$490$	−93.9149	−4.24264
$491$	38.1236	1.72050	0.860248	−	0.509876i	$-0.170309\pi$
$491$	38.1236	1.72050	0.860248	+	0.509876i	$0.170309\pi$
$492$	0	0
$493$	−37.9926	−1.71110
$494$	−52.5439	−2.36406
$495$	0	0
$496$	20.6357	0.926571
$497$	−13.0065	−0.583420
$498$	0	0
$499$	−32.1099	−1.43744	−0.718719	−	0.695301i	$-0.755271\pi$
$499$	−32.1099	−1.43744	−0.718719	+	0.695301i	$0.755271\pi$
$500$	7.05081	0.315322
$501$	0	0
$502$	−16.0195	−0.714985
$503$	−41.0835	−1.83182	−0.915910	−	0.401383i	$-0.868530\pi$
$503$	−41.0835	−1.83182	−0.915910	+	0.401383i	$0.868530\pi$
$504$	0	0
$505$	9.39664	0.418145
$506$	24.1429	1.07328
$507$	0	0
$508$	16.2960	0.723020
$509$	−21.6072	−0.957724	−0.478862	−	0.877890i	$-0.658951\pi$
$509$	−21.6072	−0.957724	−0.478862	+	0.877890i	$0.658951\pi$
$510$	0	0
$511$	24.0104	1.06216
$512$	−14.5385	−0.642515
$513$	0	0
$514$	27.7919	1.22585
$515$	27.3640	1.20580
$516$	0	0
$517$	39.1506	1.72184
$518$	3.86713	0.169912
$519$	0	0
$520$	26.6365	1.16809
$521$	39.6888	1.73880	0.869398	−	0.494112i	$-0.164507\pi$
$521$	39.6888	1.73880	0.869398	+	0.494112i	$0.164507\pi$
$522$	0	0
$523$	−20.5276	−0.897610	−0.448805	−	0.893630i	$-0.648150\pi$
$523$	−20.5276	−0.897610	−0.448805	+	0.893630i	$0.648150\pi$
$524$	9.15912	0.400118
$525$	0	0
$526$	−49.3413	−2.15138
$527$	−27.9134	−1.21593
$528$	0	0
$529$	−11.5409	−0.501780
$530$	17.3988	0.755757
$531$	0	0
$532$	−31.1067	−1.34865
$533$	40.0351	1.73411
$534$	0	0
$535$	27.4538	1.18693
$536$	−6.31498	−0.272765
$537$	0	0
$538$	−6.08118	−0.262178
$539$	61.6566	2.65574
$540$	0	0
$541$	27.9331	1.20094	0.600468	−	0.799649i	$-0.294981\pi$
$541$	27.9331	1.20094	0.600468	+	0.799649i	$0.294981\pi$
$542$	40.3693	1.73401
$543$	0	0
$544$	39.5891	1.69737
$545$	−27.6251	−1.18333
$546$	0	0
$547$	−37.2253	−1.59164	−0.795820	−	0.605533i	$-0.792960\pi$
$547$	−37.2253	−1.59164	−0.795820	+	0.605533i	$0.792960\pi$
$548$	13.4769	0.575706
$549$	0	0
$550$	−48.2170	−2.05598
$551$	−31.8045	−1.35492
$552$	0	0
$553$	62.7427	2.66809
$554$	39.0470	1.65895
$555$	0	0
$556$	9.78267	0.414878
$557$	−12.2692	−0.519864	−0.259932	−	0.965627i	$-0.583700\pi$
$557$	−12.2692	−0.519864	−0.259932	+	0.965627i	$0.583700\pi$
$558$	0	0
$559$	−58.5315	−2.47562
$560$	80.6728	3.40905
$561$	0	0
$562$	−16.2276	−0.684522
$563$	−36.2109	−1.52611	−0.763054	−	0.646335i	$-0.776301\pi$
$563$	−36.2109	−1.52611	−0.763054	+	0.646335i	$0.776301\pi$
$564$	0	0
$565$	13.2127	0.555863
$566$	−12.2041	−0.512976
$567$	0	0
$568$	4.07178	0.170848
$569$	−37.9372	−1.59041	−0.795206	−	0.606339i	$-0.792637\pi$
$569$	−37.9372	−1.59041	−0.795206	+	0.606339i	$0.792637\pi$
$570$	0	0
$571$	−32.9880	−1.38050	−0.690252	−	0.723569i	$-0.742501\pi$
$571$	−32.9880	−1.38050	−0.690252	+	0.723569i	$0.742501\pi$
$572$	24.5380	1.02599
$573$	0	0
$574$	64.2937	2.68357
$575$	−22.8855	−0.954393
$576$	0	0
$577$	11.4000	0.474587	0.237293	−	0.971438i	$-0.423740\pi$
$577$	11.4000	0.474587	0.237293	+	0.971438i	$0.423740\pi$
$578$	−50.2439	−2.08987
$579$	0	0
$580$	−22.6236	−0.939392
$581$	31.2683	1.29723
$582$	0	0
$583$	−11.4226	−0.473076
$584$	−7.51666	−0.311042
$585$	0	0
$586$	42.5259	1.75673
$587$	10.6481	0.439493	0.219746	−	0.975557i	$-0.429477\pi$
$587$	10.6481	0.439493	0.219746	+	0.975557i	$0.429477\pi$
$588$	0	0
$589$	−23.3670	−0.962820
$590$	14.1197	0.581300
$591$	0	0
$592$	−2.28316	−0.0938373
$593$	−27.2403	−1.11862	−0.559312	−	0.828957i	$-0.688935\pi$
$593$	−27.2403	−1.11862	−0.559312	+	0.828957i	$0.688935\pi$
$594$	0	0
$595$	−109.124	−4.47365
$596$	−6.10534	−0.250084
$597$	0	0
$598$	31.5933	1.29195
$599$	11.9855	0.489714	0.244857	−	0.969559i	$-0.421259\pi$
$599$	11.9855	0.489714	0.244857	+	0.969559i	$0.421259\pi$
$600$	0	0
$601$	−0.792780	−0.0323382	−0.0161691	−	0.999869i	$-0.505147\pi$
$601$	−0.792780	−0.0323382	−0.0161691	+	0.999869i	$0.505147\pi$
$602$	−93.9976	−3.83106
$603$	0	0
$604$	16.5857	0.674864
$605$	17.3436	0.705117
$606$	0	0
$607$	3.75115	0.152255	0.0761273	−	0.997098i	$-0.475744\pi$
$607$	3.75115	0.152255	0.0761273	+	0.997098i	$0.475744\pi$
$608$	33.1410	1.34404
$609$	0	0
$610$	−40.9545	−1.65820
$611$	51.2325	2.07264
$612$	0	0
$613$	−14.4116	−0.582078	−0.291039	−	0.956711i	$-0.594001\pi$
$613$	−14.4116	−0.582078	−0.291039	+	0.956711i	$0.594001\pi$
$614$	19.5510	0.789015
$615$	0	0
$616$	−28.0834	−1.13151
$617$	−31.1233	−1.25298	−0.626489	−	0.779431i	$-0.715508\pi$
$617$	−31.1233	−1.25298	−0.626489	+	0.779431i	$0.715508\pi$
$618$	0	0
$619$	−15.8065	−0.635316	−0.317658	−	0.948205i	$-0.602896\pi$
$619$	−15.8065	−0.635316	−0.317658	+	0.948205i	$0.602896\pi$
$620$	−16.6217	−0.667543
$621$	0	0
$622$	29.4818	1.18211
$623$	−34.2919	−1.37388
$624$	0	0
$625$	−13.0971	−0.523885
$626$	−7.13783	−0.285285
$627$	0	0
$628$	−6.37097	−0.254229
$629$	3.08837	0.123141
$630$	0	0
$631$	−23.6849	−0.942879	−0.471440	−	0.881898i	$-0.656265\pi$
$631$	−23.6849	−0.942879	−0.471440	+	0.881898i	$0.656265\pi$
$632$	−19.6421	−0.781321
$633$	0	0
$634$	−7.29611	−0.289766
$635$	47.8563	1.89912
$636$	0	0
$637$	80.6838	3.19681
$638$	40.2904	1.59511
$639$	0	0
$640$	−37.1190	−1.46726
$641$	−9.37034	−0.370106	−0.185053	−	0.982729i	$-0.559246\pi$
$641$	−9.37034	−0.370106	−0.185053	+	0.982729i	$0.559246\pi$
$642$	0	0
$643$	−37.4178	−1.47561	−0.737806	−	0.675012i	$-0.764138\pi$
$643$	−37.4178	−1.47561	−0.737806	+	0.675012i	$0.764138\pi$
$644$	18.7037	0.737028
$645$	0	0
$646$	−67.3892	−2.65139
$647$	0.801402	0.0315064	0.0157532	−	0.999876i	$-0.494985\pi$
$647$	0.801402	0.0315064	0.0157532	+	0.999876i	$0.494985\pi$
$648$	0	0
$649$	−9.26982	−0.363872
$650$	−63.0968	−2.47486
$651$	0	0
$652$	−28.1825	−1.10371
$653$	−7.97090	−0.311925	−0.155963	−	0.987763i	$-0.549848\pi$
$653$	−7.97090	−0.311925	−0.155963	+	0.987763i	$0.549848\pi$
$654$	0	0
$655$	26.8974	1.05097
$656$	−37.9591	−1.48205
$657$	0	0
$658$	82.2758	3.20744
$659$	−0.758492	−0.0295466	−0.0147733	−	0.999891i	$-0.504703\pi$
$659$	−0.758492	−0.0295466	−0.0147733	+	0.999891i	$0.504703\pi$
$660$	0	0
$661$	−1.38017	−0.0536823	−0.0268412	−	0.999640i	$-0.508545\pi$
$661$	−1.38017	−0.0536823	−0.0268412	+	0.999640i	$0.508545\pi$
$662$	−34.7266	−1.34969
$663$	0	0
$664$	−9.78880	−0.379879
$665$	−91.3504	−3.54242
$666$	0	0
$667$	19.1232	0.740455
$668$	10.1308	0.391974
$669$	0	0
$670$	26.0223	1.00533
$671$	26.8873	1.03797
$672$	0	0
$673$	−31.0254	−1.19594	−0.597970	−	0.801519i	$-0.704026\pi$
$673$	−31.0254	−1.19594	−0.597970	+	0.801519i	$0.704026\pi$
$674$	40.3430	1.55395
$675$	0	0
$676$	16.9294	0.651132
$677$	0.0415111	0.00159540	0.000797700	−	1.00000i	$-0.499746\pi$
$677$	0.0415111	0.00159540	0.000797700		1.00000i	$0.499746\pi$
$678$	0	0
$679$	30.8809	1.18510
$680$	34.1622	1.31006
$681$	0	0
$682$	29.6016	1.13350
$683$	8.72694	0.333927	0.166963	−	0.985963i	$-0.446604\pi$
$683$	8.72694	0.333927	0.166963	+	0.985963i	$0.446604\pi$
$684$	0	0
$685$	39.5774	1.51218
$686$	70.6246	2.69646
$687$	0	0
$688$	55.4963	2.11578
$689$	−14.9476	−0.569459
$690$	0	0
$691$	33.9570	1.29179	0.645893	−	0.763428i	$-0.276485\pi$
$691$	33.9570	1.29179	0.645893	+	0.763428i	$0.276485\pi$
$692$	−21.1859	−0.805369
$693$	0	0
$694$	−30.1994	−1.14635
$695$	28.7286	1.08974
$696$	0	0
$697$	51.3463	1.94488
$698$	54.4006	2.05909
$699$	0	0
$700$	−37.3542	−1.41185
$701$	−51.6303	−1.95005	−0.975025	−	0.222097i	$-0.928710\pi$
$701$	−51.6303	−1.95005	−0.975025	+	0.222097i	$0.928710\pi$
$702$	0	0
$703$	2.58535	0.0975084
$704$	−2.13737	−0.0805551
$705$	0	0
$706$	−31.6631	−1.19166
$707$	−12.9644	−0.487575
$708$	0	0
$709$	42.9923	1.61461	0.807305	−	0.590134i	$-0.200925\pi$
$709$	42.9923	1.61461	0.807305	+	0.590134i	$0.200925\pi$
$710$	−16.7787	−0.629693
$711$	0	0
$712$	10.7353	0.402324
$713$	14.0500	0.526176
$714$	0	0
$715$	72.0603	2.69490
$716$	8.14802	0.304506
$717$	0	0
$718$	−22.8612	−0.853173
$719$	−34.9968	−1.30516	−0.652580	−	0.757720i	$-0.726313\pi$
$719$	−34.9968	−1.30516	−0.652580	+	0.757720i	$0.726313\pi$
$720$	0	0
$721$	−37.7536	−1.40602
$722$	−22.5965	−0.840953
$723$	0	0
$724$	29.2745	1.08798
$725$	−38.1921	−1.41842
$726$	0	0
$727$	3.22142	0.119476	0.0597380	−	0.998214i	$-0.480973\pi$
$727$	3.22142	0.119476	0.0597380	+	0.998214i	$0.480973\pi$
$728$	−36.7499	−1.36204
$729$	0	0
$730$	30.9741	1.14640
$731$	−75.0684	−2.77651
$732$	0	0
$733$	2.46216	0.0909421	0.0454710	−	0.998966i	$-0.485521\pi$
$733$	2.46216	0.0909421	0.0454710	+	0.998966i	$0.485521\pi$
$734$	−19.0103	−0.701683
$735$	0	0
$736$	−19.9268	−0.734513
$737$	−17.0840	−0.629298
$738$	0	0
$739$	26.0766	0.959245	0.479622	−	0.877475i	$-0.340774\pi$
$739$	26.0766	0.959245	0.479622	+	0.877475i	$0.340774\pi$
$740$	1.83904	0.0676045
$741$	0	0
$742$	−24.0048	−0.881246
$743$	−19.0129	−0.697517	−0.348758	−	0.937213i	$-0.613397\pi$
$743$	−19.0129	−0.697517	−0.348758	+	0.937213i	$0.613397\pi$
$744$	0	0
$745$	−17.9294	−0.656883
$746$	16.7091	0.611764
$747$	0	0
$748$	31.4708	1.15069
$749$	−37.8775	−1.38401
$750$	0	0
$751$	−31.5500	−1.15127	−0.575637	−	0.817705i	$-0.695246\pi$
$751$	−31.5500	−1.15127	−0.575637	+	0.817705i	$0.695246\pi$
$752$	−48.5757	−1.77137
$753$	0	0
$754$	52.7240	1.92009
$755$	48.7070	1.77263
$756$	0	0
$757$	−16.4479	−0.597808	−0.298904	−	0.954283i	$-0.596621\pi$
$757$	−16.4479	−0.597808	−0.298904	+	0.954283i	$0.596621\pi$
$758$	3.30234	0.119946
$759$	0	0
$760$	28.5980	1.03736
$761$	−41.2041	−1.49365	−0.746823	−	0.665022i	$-0.768422\pi$
$761$	−41.2041	−1.49365	−0.746823	+	0.665022i	$0.768422\pi$
$762$	0	0
$763$	38.1138	1.37981
$764$	−0.271332	−0.00981645
$765$	0	0
$766$	15.2496	0.550989
$767$	−12.1305	−0.438007
$768$	0	0
$769$	19.8311	0.715129	0.357564	−	0.933889i	$-0.383607\pi$
$769$	19.8311	0.715129	0.357564	+	0.933889i	$0.383607\pi$
$770$	115.724	4.17040
$771$	0	0
$772$	7.71897	0.277812
$773$	21.3987	0.769659	0.384829	−	0.922988i	$-0.374260\pi$
$773$	21.3987	0.769659	0.384829	+	0.922988i	$0.374260\pi$
$774$	0	0
$775$	−28.0600	−1.00795
$776$	−9.66751	−0.347044
$777$	0	0
$778$	−40.0539	−1.43600
$779$	42.9832	1.54003
$780$	0	0
$781$	11.0155	0.394164
$782$	40.5194	1.44897
$783$	0	0
$784$	−76.4998	−2.73214
$785$	−18.7095	−0.667771
$786$	0	0
$787$	22.5971	0.805501	0.402751	−	0.915310i	$-0.368054\pi$
$787$	22.5971	0.805501	0.402751	+	0.915310i	$0.368054\pi$
$788$	−6.86471	−0.244545
$789$	0	0
$790$	80.9397	2.87970
$791$	−18.2293	−0.648160
$792$	0	0
$793$	35.1847	1.24945
$794$	45.4156	1.61174
$795$	0	0
$796$	−2.74067	−0.0971406
$797$	4.16632	0.147579	0.0737893	−	0.997274i	$-0.476491\pi$
$797$	4.16632	0.147579	0.0737893	+	0.997274i	$0.476491\pi$
$798$	0	0
$799$	65.7072	2.32455
$800$	39.7970	1.40704
$801$	0	0
$802$	12.1723	0.429817
$803$	−20.3350	−0.717605
$804$	0	0
$805$	54.9267	1.93591
$806$	38.7367	1.36444
$807$	0	0
$808$	4.05860	0.142781
$809$	−9.82492	−0.345426	−0.172713	−	0.984972i	$-0.555253\pi$
$809$	−9.82492	−0.345426	−0.172713	+	0.984972i	$0.555253\pi$
$810$	0	0
$811$	−20.9928	−0.737158	−0.368579	−	0.929596i	$-0.620156\pi$
$811$	−20.9928	−0.737158	−0.368579	+	0.929596i	$0.620156\pi$
$812$	31.2133	1.09537
$813$	0	0
$814$	−3.27516	−0.114794
$815$	−82.7629	−2.89906
$816$	0	0
$817$	−62.8416	−2.19855
$818$	−19.1440	−0.669354
$819$	0	0
$820$	30.5753	1.06774
$821$	7.00931	0.244627	0.122313	−	0.992492i	$-0.460969\pi$
$821$	7.00931	0.244627	0.122313	+	0.992492i	$0.460969\pi$
$822$	0	0
$823$	26.6502	0.928966	0.464483	−	0.885582i	$-0.346240\pi$
$823$	26.6502	0.928966	0.464483	+	0.885582i	$0.346240\pi$
$824$	11.8191	0.411736
$825$	0	0
$826$	−19.4807	−0.677822
$827$	−21.7044	−0.754737	−0.377369	−	0.926063i	$-0.623171\pi$
$827$	−21.7044	−0.754737	−0.377369	+	0.926063i	$0.623171\pi$
$828$	0	0
$829$	39.5208	1.37261	0.686307	−	0.727312i	$-0.259231\pi$
$829$	39.5208	1.37261	0.686307	+	0.727312i	$0.259231\pi$
$830$	40.3369	1.40012
$831$	0	0
$832$	−2.79696	−0.0969671
$833$	103.479	3.58535
$834$	0	0
$835$	29.7510	1.02958
$836$	26.3449	0.911159
$837$	0	0
$838$	30.8353	1.06519
$839$	−10.8212	−0.373588	−0.186794	−	0.982399i	$-0.559810\pi$
$839$	−10.8212	−0.373588	−0.186794	+	0.982399i	$0.559810\pi$
$840$	0	0
$841$	2.91349	0.100465
$842$	−20.6078	−0.710192
$843$	0	0
$844$	26.7140	0.919535
$845$	49.7163	1.71029
$846$	0	0
$847$	−23.9286	−0.822198
$848$	14.1725	0.486685
$849$	0	0
$850$	−80.9236	−2.77566
$851$	−1.55451	−0.0532878
$852$	0	0
$853$	−32.0566	−1.09760	−0.548799	−	0.835954i	$-0.684915\pi$
$853$	−32.0566	−1.09760	−0.548799	+	0.835954i	$0.684915\pi$
$854$	56.5042	1.93353
$855$	0	0
$856$	11.8578	0.405293
$857$	21.5957	0.737695	0.368847	−	0.929490i	$-0.379752\pi$
$857$	21.5957	0.737695	0.368847	+	0.929490i	$0.379752\pi$
$858$	0	0
$859$	32.2907	1.10174	0.550872	−	0.834590i	$-0.314295\pi$
$859$	32.2907	1.10174	0.550872	+	0.834590i	$0.314295\pi$
$860$	−44.7012	−1.52430
$861$	0	0
$862$	−10.7000	−0.364444
$863$	−52.6342	−1.79169	−0.895845	−	0.444366i	$-0.853429\pi$
$863$	−52.6342	−1.79169	−0.895845	+	0.444366i	$0.853429\pi$
$864$	0	0
$865$	−62.2163	−2.11542
$866$	−43.8267	−1.48929
$867$	0	0
$868$	22.9326	0.778384
$869$	−53.1381	−1.80259
$870$	0	0
$871$	−22.3562	−0.757509
$872$	−11.9318	−0.404063
$873$	0	0
$874$	33.9198	1.14735
$875$	−28.5676	−0.965763
$876$	0	0
$877$	49.8039	1.68176	0.840879	−	0.541224i	$-0.182039\pi$
$877$	49.8039	1.68176	0.840879	+	0.541224i	$0.182039\pi$
$878$	2.87614	0.0970649
$879$	0	0
$880$	−68.3235	−2.30319
$881$	57.1487	1.92539	0.962694	−	0.270592i	$-0.0872194\pi$
$881$	57.1487	1.92539	0.962694	+	0.270592i	$0.0872194\pi$
$882$	0	0
$883$	−43.6202	−1.46794	−0.733969	−	0.679183i	$-0.762334\pi$
$883$	−43.6202	−1.46794	−0.733969	+	0.679183i	$0.762334\pi$
$884$	41.1827	1.38512
$885$	0	0
$886$	−53.8727	−1.80989
$887$	11.6036	0.389612	0.194806	−	0.980842i	$-0.437592\pi$
$887$	11.6036	0.389612	0.194806	+	0.980842i	$0.437592\pi$
$888$	0	0
$889$	−66.0264	−2.21445
$890$	−44.2374	−1.48284
$891$	0	0
$892$	8.67718	0.290533
$893$	55.0051	1.84067
$894$	0	0
$895$	23.9281	0.799829
$896$	51.2124	1.71089
$897$	0	0
$898$	−56.1799	−1.87475
$899$	23.4471	0.782003
$900$	0	0
$901$	−19.1708	−0.638671
$902$	−54.4517	−1.81304
$903$	0	0
$904$	5.70683	0.189806
$905$	85.9698	2.85773
$906$	0	0
$907$	31.6917	1.05231	0.526153	−	0.850390i	$-0.323634\pi$
$907$	31.6917	1.05231	0.526153	+	0.850390i	$0.323634\pi$
$908$	−19.0121	−0.630940
$909$	0	0
$910$	151.436	5.02007
$911$	20.5202	0.679864	0.339932	−	0.940450i	$-0.389596\pi$
$911$	20.5202	0.679864	0.339932	+	0.940450i	$0.389596\pi$
$912$	0	0
$913$	−26.4818	−0.876420
$914$	−48.4962	−1.60411
$915$	0	0
$916$	8.83488	0.291913
$917$	−37.1098	−1.22547
$918$	0	0
$919$	−39.0439	−1.28794	−0.643970	−	0.765051i	$-0.722714\pi$
$919$	−39.0439	−1.28794	−0.643970	+	0.765051i	$0.722714\pi$
$920$	−17.1952	−0.566910
$921$	0	0
$922$	7.70509	0.253754
$923$	14.4148	0.474470
$924$	0	0
$925$	3.10459	0.102078
$926$	15.0236	0.493706
$927$	0	0
$928$	−33.2545	−1.09163
$929$	−25.8989	−0.849716	−0.424858	−	0.905260i	$-0.639676\pi$
$929$	−25.8989	−0.849716	−0.424858	+	0.905260i	$0.639676\pi$
$930$	0	0
$931$	86.6251	2.83902
$932$	13.3223	0.436385
$933$	0	0
$934$	−52.2086	−1.70832
$935$	92.4196	3.02244
$936$	0	0
$937$	−13.6459	−0.445791	−0.222896	−	0.974842i	$-0.571551\pi$
$937$	−13.6459	−0.445791	−0.222896	+	0.974842i	$0.571551\pi$
$938$	−35.9024	−1.17226
$939$	0	0
$940$	39.1268	1.27618
$941$	−6.90137	−0.224978	−0.112489	−	0.993653i	$-0.535882\pi$
$941$	−6.90137	−0.224978	−0.112489	+	0.993653i	$0.535882\pi$
$942$	0	0
$943$	−25.8447	−0.841620
$944$	11.5015	0.374340
$945$	0	0
$946$	79.6086	2.58830
$947$	39.6899	1.28975	0.644875	−	0.764288i	$-0.276910\pi$
$947$	39.6899	1.28975	0.644875	+	0.764288i	$0.276910\pi$
$948$	0	0
$949$	−26.6103	−0.863808
$950$	−67.7431	−2.19788
$951$	0	0
$952$	−47.1329	−1.52759
$953$	−36.6847	−1.18833	−0.594166	−	0.804342i	$-0.702518\pi$
$953$	−36.6847	−1.18833	−0.594166	+	0.804342i	$0.702518\pi$
$954$	0	0
$955$	−0.796815	−0.0257843
$956$	−1.16777	−0.0377685
$957$	0	0
$958$	−49.7048	−1.60589
$959$	−54.6042	−1.76326
$960$	0	0
$961$	−13.7733	−0.444299
$962$	−4.28587	−0.138182
$963$	0	0
$964$	7.28426	0.234610
$965$	22.6681	0.729713
$966$	0	0
$967$	32.5258	1.04596	0.522980	−	0.852345i	$-0.324820\pi$
$967$	32.5258	1.04596	0.522980	+	0.852345i	$0.324820\pi$
$968$	7.49105	0.240771
$969$	0	0
$970$	39.8372	1.27909
$971$	10.0062	0.321114	0.160557	−	0.987027i	$-0.448671\pi$
$971$	10.0062	0.321114	0.160557	+	0.987027i	$0.448671\pi$
$972$	0	0
$973$	−39.6363	−1.27068
$974$	59.3295	1.90104
$975$	0	0
$976$	−33.3602	−1.06783
$977$	−18.7495	−0.599851	−0.299925	−	0.953963i	$-0.596962\pi$
$977$	−18.7495	−0.599851	−0.299925	+	0.953963i	$0.596962\pi$
$978$	0	0
$979$	29.0425	0.928203
$980$	61.6192	1.96835
$981$	0	0
$982$	−67.8534	−2.16529
$983$	10.3615	0.330480	0.165240	−	0.986253i	$-0.447160\pi$
$983$	10.3615	0.330480	0.165240	+	0.986253i	$0.447160\pi$
$984$	0	0
$985$	−20.1595	−0.642334
$986$	67.6201	2.15346
$987$	0	0
$988$	34.4750	1.09680
$989$	37.7851	1.20150
$990$	0	0
$991$	−16.7232	−0.531231	−0.265616	−	0.964079i	$-0.585575\pi$
$991$	−16.7232	−0.531231	−0.265616	+	0.964079i	$0.585575\pi$
$992$	−24.4323	−0.775727
$993$	0	0
$994$	23.1492	0.734249
$995$	−8.04848	−0.255154
$996$	0	0
$997$	−29.1968	−0.924672	−0.462336	−	0.886705i	$-0.652989\pi$
$997$	−29.1968	−0.924672	−0.462336	+	0.886705i	$0.652989\pi$
$998$	57.1500	1.80905
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.g.1.1		8
3.2	odd	2		717.2.a.f.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.f.1.8	✓	8	3.2	odd	2
2151.2.a.g.1.1		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-1.77982 q^{2} +1.16777 q^{4} +3.42938 q^{5} -4.73145 q^{7} +1.48122 q^{8} +O(q^{10})\)	\(q-1.77982 q^{2} +1.16777 q^{4} +3.42938 q^{5} -4.73145 q^{7} +1.48122 q^{8} -6.10369 q^{10} +4.00716 q^{11} +5.24378 q^{13} +8.42114 q^{14} -4.97185 q^{16} +6.72530 q^{17} +5.62991 q^{19} +4.00473 q^{20} -7.13205 q^{22} -3.38512 q^{23} +6.76062 q^{25} -9.33300 q^{26} -5.52526 q^{28} -5.64920 q^{29} -4.15051 q^{31} +5.88659 q^{32} -11.9699 q^{34} -16.2259 q^{35} +0.459217 q^{37} -10.0203 q^{38} +5.07965 q^{40} +7.63479 q^{41} -11.1621 q^{43} +4.67946 q^{44} +6.02492 q^{46} +9.77015 q^{47} +15.3866 q^{49} -12.0327 q^{50} +6.12354 q^{52} -2.85055 q^{53} +13.7421 q^{55} -7.00830 q^{56} +10.0546 q^{58} -2.31331 q^{59} +6.70980 q^{61} +7.38717 q^{62} -0.533387 q^{64} +17.9829 q^{65} -4.26337 q^{67} +7.85363 q^{68} +28.8793 q^{70} +2.74894 q^{71} -5.07465 q^{73} -0.817325 q^{74} +6.57446 q^{76} -18.9597 q^{77} -13.2608 q^{79} -17.0503 q^{80} -13.5886 q^{82} -6.60862 q^{83} +23.0636 q^{85} +19.8666 q^{86} +5.93548 q^{88} +7.24765 q^{89} -24.8106 q^{91} -3.95306 q^{92} -17.3891 q^{94} +19.3071 q^{95} -6.52674 q^{97} -27.3854 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8}+O(q^{10}) \) 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8	\( 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8} + 4 q^{10} + 19 q^{11} - 3 q^{13} + 8 q^{14} + 9 q^{16} + 13 q^{17} - 6 q^{19} + 18 q^{20} + 3 q^{22} + 18 q^{23} + 7 q^{25} - 2 q^{28} + 10 q^{29} - 2 q^{31} + 20 q^{32} - 4 q^{34} + 7 q^{35} - 8 q^{37} - 9 q^{38} + 29 q^{40} + 22 q^{41} - 24 q^{43} + 7 q^{44} + 30 q^{46} + 17 q^{47} + 15 q^{49} - 24 q^{50} + 22 q^{52} + 32 q^{55} - 19 q^{56} + 18 q^{58} + 24 q^{59} + 10 q^{61} - 30 q^{62} + 33 q^{64} + 17 q^{65} - 48 q^{67} + 21 q^{68} + 31 q^{70} + 17 q^{71} + 2 q^{73} - 9 q^{74} + 10 q^{76} - 10 q^{77} + 17 q^{79} - 8 q^{80} - 17 q^{82} + 37 q^{83} + 28 q^{85} + q^{86} + 15 q^{88} + 41 q^{89} - 39 q^{91} + 38 q^{92} + 2 q^{94} - 16 q^{95} - 20 q^{97} - 46 q^{98}+O(q^{100}) \) 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8 + 4 * q^10 + 19 * q^11 - 3 * q^13 + 8 * q^14 + 9 * q^16 + 13 * q^17 - 6 * q^19 + 18 * q^20 + 3 * q^22 + 18 * q^23 + 7 * q^25 - 2 * q^28 + 10 * q^29 - 2 * q^31 + 20 * q^32 - 4 * q^34 + 7 * q^35 - 8 * q^37 - 9 * q^38 + 29 * q^40 + 22 * q^41 - 24 * q^43 + 7 * q^44 + 30 * q^46 + 17 * q^47 + 15 * q^49 - 24 * q^50 + 22 * q^52 + 32 * q^55 - 19 * q^56 + 18 * q^58 + 24 * q^59 + 10 * q^61 - 30 * q^62 + 33 * q^64 + 17 * q^65 - 48 * q^67 + 21 * q^68 + 31 * q^70 + 17 * q^71 + 2 * q^73 - 9 * q^74 + 10 * q^76 - 10 * q^77 + 17 * q^79 - 8 * q^80 - 17 * q^82 + 37 * q^83 + 28 * q^85 + q^86 + 15 * q^88 + 41 * q^89 - 39 * q^91 + 38 * q^92 + 2 * q^94 - 16 * q^95 - 20 * q^97 - 46 * q^98

Introduction

L-functions

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