LMFDB - Embedded newform 2151.2.a.k.1.3

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + \cdots + 1 $ x^20 - 4x^19 - 21x^18 + 96x^17 + 164x^16 - 936x^15 - 540x^14 + 4804x^13 + 229x^12 - 14020x^11 + 3356x^10 + 23404x^9 - 9429x^8 - 21252x^7 + 10479x^6 + 9108x^5 - 4844x^4 - 1184x^3 + 640x^2 - 56*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.02621$ 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)

$f(q)$	$=$	$q-2.02621 q^{2} +2.10552 q^{4} +3.48073 q^{5} +1.37004 q^{7} -0.213806 q^{8} +O(q^{10})$	\(q-2.02621 q^{2} +2.10552 q^{4} +3.48073 q^{5} +1.37004 q^{7} -0.213806 q^{8} -7.05267 q^{10} +0.191564 q^{11} +4.24334 q^{13} -2.77599 q^{14} -3.77782 q^{16} +4.07863 q^{17} -5.16823 q^{19} +7.32874 q^{20} -0.388149 q^{22} +0.978985 q^{23} +7.11545 q^{25} -8.59789 q^{26} +2.88465 q^{28} +10.0111 q^{29} -4.04813 q^{31} +8.08227 q^{32} -8.26416 q^{34} +4.76874 q^{35} +2.15163 q^{37} +10.4719 q^{38} -0.744200 q^{40} +0.0317692 q^{41} +2.77809 q^{43} +0.403342 q^{44} -1.98363 q^{46} +0.680962 q^{47} -5.12298 q^{49} -14.4174 q^{50} +8.93444 q^{52} +4.91306 q^{53} +0.666782 q^{55} -0.292924 q^{56} -20.2846 q^{58} +3.94941 q^{59} +3.63175 q^{61} +8.20235 q^{62} -8.82072 q^{64} +14.7699 q^{65} +5.37197 q^{67} +8.58765 q^{68} -9.66247 q^{70} -8.95527 q^{71} -7.76915 q^{73} -4.35966 q^{74} -10.8818 q^{76} +0.262451 q^{77} -12.9221 q^{79} -13.1496 q^{80} -0.0643711 q^{82} +2.22931 q^{83} +14.1966 q^{85} -5.62898 q^{86} -0.0409576 q^{88} -9.94855 q^{89} +5.81356 q^{91} +2.06127 q^{92} -1.37977 q^{94} -17.9892 q^{95} -4.84284 q^{97} +10.3802 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) $ 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8	$ 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) $ 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8 + 4 * q^10 + 12 * q^11 - 4 * q^13 + 20 * q^14 + 22 * q^16 + 24 * q^17 - 4 * q^19 + 40 * q^20 - 6 * q^22 + 12 * q^23 + 22 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 - 4 * q^31 + 28 * q^32 + 8 * q^34 + 20 * q^35 - 10 * q^37 + 26 * q^38 + 6 * q^40 + 66 * q^41 + 8 * q^43 + 36 * q^44 - 12 * q^46 + 28 * q^47 + 18 * q^49 + 28 * q^50 - 18 * q^52 + 28 * q^53 - 4 * q^55 + 60 * q^56 + 54 * q^59 - 4 * q^61 + 20 * q^62 + 22 * q^64 + 42 * q^65 + 12 * q^67 + 12 * q^68 + 20 * q^70 + 36 * q^71 + 14 * q^73 - 50 * q^76 + 8 * q^77 - 12 * q^79 + 88 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 + 18 * q^86 - 10 * q^88 + 130 * q^89 - 6 * q^91 - 46 * q^92 - 26 * q^94 - 2 * q^97 + 12 * 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 $

$2$	−2.02621	−1.43275	−0.716373	−	0.697718i	$-0.754199\pi$
$2$	−2.02621	−1.43275	−0.716373	+	0.697718i	$0.754199\pi$
$3$	0	0
$3$	0	0
$4$	2.10552	1.05276
$5$	3.48073	1.55663	0.778314	−	0.627875i	$-0.216075\pi$
$5$	3.48073	1.55663	0.778314	+	0.627875i	$0.216075\pi$
$6$	0	0
$7$	1.37004	0.517828	0.258914	−	0.965900i	$-0.416635\pi$
$7$	1.37004	0.517828	0.258914	+	0.965900i	$0.416635\pi$
$8$	−0.213806	−0.0755918
$9$	0	0
$10$	−7.05267	−2.23025
$11$	0.191564	0.0577588	0.0288794	−	0.999583i	$-0.490806\pi$
$11$	0.191564	0.0577588	0.0288794	+	0.999583i	$0.490806\pi$
$12$	0	0
$13$	4.24334	1.17689	0.588445	−	0.808537i	$-0.299740\pi$
$13$	4.24334	1.17689	0.588445	+	0.808537i	$0.299740\pi$
$14$	−2.77599	−0.741915
$15$	0	0
$16$	−3.77782	−0.944456
$17$	4.07863	0.989214	0.494607	−	0.869117i	$-0.335312\pi$
$17$	4.07863	0.989214	0.494607	+	0.869117i	$0.335312\pi$
$18$	0	0
$19$	−5.16823	−1.18567	−0.592837	−	0.805323i	$-0.701992\pi$
$19$	−5.16823	−1.18567	−0.592837	+	0.805323i	$0.701992\pi$
$20$	7.32874	1.63876
$21$	0	0
$22$	−0.388149	−0.0827536
$23$	0.978985	0.204132	0.102066	−	0.994778i	$-0.467455\pi$
$23$	0.978985	0.204132	0.102066	+	0.994778i	$0.467455\pi$
$24$	0	0
$25$	7.11545	1.42309
$26$	−8.59789	−1.68619
$27$	0	0
$28$	2.88465	0.545148
$29$	10.0111	1.85901	0.929507	−	0.368806i	$-0.120233\pi$
$29$	10.0111	1.85901	0.929507	+	0.368806i	$0.120233\pi$
$30$	0	0
$31$	−4.04813	−0.727065	−0.363533	−	0.931582i	$-0.618429\pi$
$31$	−4.04813	−0.727065	−0.363533	+	0.931582i	$0.618429\pi$
$32$	8.08227	1.42876
$33$	0	0
$34$	−8.26416	−1.41729
$35$	4.76874	0.806065
$36$	0	0
$37$	2.15163	0.353726	0.176863	−	0.984235i	$-0.443405\pi$
$37$	2.15163	0.353726	0.176863	+	0.984235i	$0.443405\pi$
$38$	10.4719	1.69877
$39$	0	0
$40$	−0.744200	−0.117668
$41$	0.0317692	0.00496152	0.00248076	−	0.999997i	$-0.499210\pi$
$41$	0.0317692	0.00496152	0.00248076	+	0.999997i	$0.499210\pi$
$42$	0	0
$43$	2.77809	0.423654	0.211827	−	0.977307i	$-0.432059\pi$
$43$	2.77809	0.423654	0.211827	+	0.977307i	$0.432059\pi$
$44$	0.403342	0.0608061
$45$	0	0
$46$	−1.98363	−0.292470
$47$	0.680962	0.0993286	0.0496643	−	0.998766i	$-0.484185\pi$
$47$	0.680962	0.0993286	0.0496643	+	0.998766i	$0.484185\pi$
$48$	0	0
$49$	−5.12298	−0.731854
$50$	−14.4174	−2.03893
$51$	0	0
$52$	8.93444	1.23898
$53$	4.91306	0.674861	0.337431	−	0.941350i	$-0.390442\pi$
$53$	4.91306	0.674861	0.337431	+	0.941350i	$0.390442\pi$
$54$	0	0
$55$	0.666782	0.0899089
$56$	−0.292924	−0.0391436
$57$	0	0
$58$	−20.2846	−2.66349
$59$	3.94941	0.514169	0.257085	−	0.966389i	$-0.417238\pi$
$59$	3.94941	0.514169	0.257085	+	0.966389i	$0.417238\pi$
$60$	0	0
$61$	3.63175	0.464997	0.232499	−	0.972597i	$-0.425310\pi$
$61$	3.63175	0.464997	0.232499	+	0.972597i	$0.425310\pi$
$62$	8.20235	1.04170
$63$	0	0
$64$	−8.82072	−1.10259
$65$	14.7699	1.83198
$66$	0	0
$67$	5.37197	0.656290	0.328145	−	0.944627i	$-0.393576\pi$
$67$	5.37197	0.656290	0.328145	+	0.944627i	$0.393576\pi$
$68$	8.58765	1.04141
$69$	0	0
$70$	−9.66247	−1.15489
$71$	−8.95527	−1.06279	−0.531397	−	0.847123i	$-0.678333\pi$
$71$	−8.95527	−1.06279	−0.531397	+	0.847123i	$0.678333\pi$
$72$	0	0
$73$	−7.76915	−0.909310	−0.454655	−	0.890668i	$-0.650237\pi$
$73$	−7.76915	−0.909310	−0.454655	+	0.890668i	$0.650237\pi$
$74$	−4.35966	−0.506800
$75$	0	0
$76$	−10.8818	−1.24823
$77$	0.262451	0.0299091
$78$	0	0
$79$	−12.9221	−1.45385	−0.726924	−	0.686718i	$-0.759051\pi$
$79$	−12.9221	−1.45385	−0.726924	+	0.686718i	$0.759051\pi$
$80$	−13.1496	−1.47017
$81$	0	0
$82$	−0.0643711	−0.00710860
$83$	2.22931	0.244698	0.122349	−	0.992487i	$-0.460957\pi$
$83$	2.22931	0.244698	0.122349	+	0.992487i	$0.460957\pi$
$84$	0	0
$85$	14.1966	1.53984
$86$	−5.62898	−0.606989
$87$	0	0
$88$	−0.0409576	−0.00436609
$89$	−9.94855	−1.05454	−0.527272	−	0.849696i	$-0.676785\pi$
$89$	−9.94855	−1.05454	−0.527272	+	0.849696i	$0.676785\pi$
$90$	0	0
$91$	5.81356	0.609427
$92$	2.06127	0.214903
$93$	0	0
$94$	−1.37977	−0.142313
$95$	−17.9892	−1.84565
$96$	0	0
$97$	−4.84284	−0.491715	−0.245858	−	0.969306i	$-0.579070\pi$
$97$	−4.84284	−0.491715	−0.245858	+	0.969306i	$0.579070\pi$
$98$	10.3802	1.04856
$99$	0	0
$100$	14.9817	1.49817
$101$	12.8427	1.27790	0.638950	−	0.769249i	$-0.279369\pi$
$101$	12.8427	1.27790	0.638950	+	0.769249i	$0.279369\pi$
$102$	0	0
$103$	−9.83159	−0.968736	−0.484368	−	0.874864i	$-0.660950\pi$
$103$	−9.83159	−0.968736	−0.484368	+	0.874864i	$0.660950\pi$
$104$	−0.907252	−0.0889634
$105$	0	0
$106$	−9.95489	−0.966904
$107$	6.72004	0.649650	0.324825	−	0.945774i	$-0.394695\pi$
$107$	6.72004	0.649650	0.324825	+	0.945774i	$0.394695\pi$
$108$	0	0
$109$	17.0804	1.63601	0.818005	−	0.575211i	$-0.195080\pi$
$109$	17.0804	1.63601	0.818005	+	0.575211i	$0.195080\pi$
$110$	−1.35104	−0.128817
$111$	0	0
$112$	−5.17578	−0.489066
$113$	−12.2581	−1.15315	−0.576574	−	0.817045i	$-0.695611\pi$
$113$	−12.2581	−1.15315	−0.576574	+	0.817045i	$0.695611\pi$
$114$	0	0
$115$	3.40758	0.317758
$116$	21.0786	1.95709
$117$	0	0
$118$	−8.00233	−0.736674
$119$	5.58791	0.512243
$120$	0	0
$121$	−10.9633	−0.996664
$122$	−7.35867	−0.666223
$123$	0	0
$124$	−8.52341	−0.765425
$125$	7.36330	0.658593
$126$	0	0
$127$	−13.4140	−1.19030	−0.595151	−	0.803614i	$-0.702908\pi$
$127$	−13.4140	−1.19030	−0.595151	+	0.803614i	$0.702908\pi$
$128$	1.70807	0.150973
$129$	0	0
$130$	−29.9269	−2.62476
$131$	−15.1828	−1.32652	−0.663262	−	0.748387i	$-0.730829\pi$
$131$	−15.1828	−1.32652	−0.663262	+	0.748387i	$0.730829\pi$
$132$	0	0
$133$	−7.08070	−0.613974
$134$	−10.8847	−0.940297
$135$	0	0
$136$	−0.872037	−0.0747765
$137$	6.50914	0.556113	0.278057	−	0.960565i	$-0.410310\pi$
$137$	6.50914	0.556113	0.278057	+	0.960565i	$0.410310\pi$
$138$	0	0
$139$	−5.06206	−0.429359	−0.214679	−	0.976685i	$-0.568871\pi$
$139$	−5.06206	−0.429359	−0.214679	+	0.976685i	$0.568871\pi$
$140$	10.0407	0.848593
$141$	0	0
$142$	18.1452	1.52271
$143$	0.812872	0.0679758
$144$	0	0
$145$	34.8459	2.89379
$146$	15.7419	1.30281
$147$	0	0
$148$	4.53031	0.372389
$149$	−10.6906	−0.875805	−0.437903	−	0.899022i	$-0.644279\pi$
$149$	−10.6906	−0.875805	−0.437903	+	0.899022i	$0.644279\pi$
$150$	0	0
$151$	15.4018	1.25338	0.626691	−	0.779268i	$-0.284409\pi$
$151$	15.4018	1.25338	0.626691	+	0.779268i	$0.284409\pi$
$152$	1.10500	0.0896272
$153$	0	0
$154$	−0.531781	−0.0428521
$155$	−14.0904	−1.13177
$156$	0	0
$157$	−3.81548	−0.304509	−0.152254	−	0.988341i	$-0.548653\pi$
$157$	−3.81548	−0.304509	−0.152254	+	0.988341i	$0.548653\pi$
$158$	26.1828	2.08299
$159$	0	0
$160$	28.1322	2.22404
$161$	1.34125	0.105705
$162$	0	0
$163$	10.4217	0.816292	0.408146	−	0.912917i	$-0.366175\pi$
$163$	10.4217	0.816292	0.408146	+	0.912917i	$0.366175\pi$
$164$	0.0668908	0.00522329
$165$	0	0
$166$	−4.51705	−0.350591
$167$	1.09246	0.0845374	0.0422687	−	0.999106i	$-0.486541\pi$
$167$	1.09246	0.0845374	0.0422687	+	0.999106i	$0.486541\pi$
$168$	0	0
$169$	5.00594	0.385072
$170$	−28.7653	−2.20620
$171$	0	0
$172$	5.84932	0.446006
$173$	10.4717	0.796148	0.398074	−	0.917353i	$-0.369679\pi$
$173$	10.4717	0.796148	0.398074	+	0.917353i	$0.369679\pi$
$174$	0	0
$175$	9.74847	0.736915
$176$	−0.723696	−0.0545506
$177$	0	0
$178$	20.1578	1.51089
$179$	−21.7287	−1.62408	−0.812041	−	0.583600i	$-0.801643\pi$
$179$	−21.7287	−1.62408	−0.812041	+	0.583600i	$0.801643\pi$
$180$	0	0
$181$	12.6108	0.937351	0.468676	−	0.883370i	$-0.344731\pi$
$181$	12.6108	0.937351	0.468676	+	0.883370i	$0.344731\pi$
$182$	−11.7795	−0.873153
$183$	0	0
$184$	−0.209313	−0.0154308
$185$	7.48924	0.550620
$186$	0	0
$187$	0.781320	0.0571358
$188$	1.43378	0.104569
$189$	0	0
$190$	36.4498	2.64435
$191$	5.93669	0.429564	0.214782	−	0.976662i	$-0.431096\pi$
$191$	5.93669	0.429564	0.214782	+	0.976662i	$0.431096\pi$
$192$	0	0
$193$	20.5726	1.48085	0.740426	−	0.672138i	$-0.234624\pi$
$193$	20.5726	1.48085	0.740426	+	0.672138i	$0.234624\pi$
$194$	9.81259	0.704503
$195$	0	0
$196$	−10.7865	−0.770467
$197$	3.06174	0.218140	0.109070	−	0.994034i	$-0.465213\pi$
$197$	3.06174	0.218140	0.109070	+	0.994034i	$0.465213\pi$
$198$	0	0
$199$	23.1755	1.64286	0.821432	−	0.570306i	$-0.193175\pi$
$199$	23.1755	1.64286	0.821432	+	0.570306i	$0.193175\pi$
$200$	−1.52133	−0.107574
$201$	0	0
$202$	−26.0220	−1.83090
$203$	13.7156	0.962649
$204$	0	0
$205$	0.110580	0.00772324
$206$	19.9209	1.38795
$207$	0	0
$208$	−16.0306	−1.11152
$209$	−0.990047	−0.0684830
$210$	0	0
$211$	3.22027	0.221693	0.110846	−	0.993838i	$-0.464644\pi$
$211$	3.22027	0.221693	0.110846	+	0.993838i	$0.464644\pi$
$212$	10.3446	0.710467
$213$	0	0
$214$	−13.6162	−0.930784
$215$	9.66976	0.659472
$216$	0	0
$217$	−5.54611	−0.376494
$218$	−34.6085	−2.34399
$219$	0	0
$220$	1.40392	0.0946525
$221$	17.3070	1.16420
$222$	0	0
$223$	2.78961	0.186806	0.0934032	−	0.995628i	$-0.470225\pi$
$223$	2.78961	0.186806	0.0934032	+	0.995628i	$0.470225\pi$
$224$	11.0731	0.739850
$225$	0	0
$226$	24.8375	1.65217
$227$	15.2465	1.01195	0.505975	−	0.862548i	$-0.331133\pi$
$227$	15.2465	1.01195	0.505975	+	0.862548i	$0.331133\pi$
$228$	0	0
$229$	17.2752	1.14158	0.570788	−	0.821097i	$-0.306638\pi$
$229$	17.2752	1.14158	0.570788	+	0.821097i	$0.306638\pi$
$230$	−6.90446	−0.455267
$231$	0	0
$232$	−2.14043	−0.140526
$233$	−12.5824	−0.824298	−0.412149	−	0.911116i	$-0.635222\pi$
$233$	−12.5824	−0.824298	−0.412149	+	0.911116i	$0.635222\pi$
$234$	0	0
$235$	2.37024	0.154618
$236$	8.31556	0.541297
$237$	0	0
$238$	−11.3223	−0.733913
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−0.658194	−0.0423980	−0.0211990	−	0.999775i	$-0.506748\pi$
$241$	−0.658194	−0.0423980	−0.0211990	+	0.999775i	$0.506748\pi$
$242$	22.2139	1.42797
$243$	0	0
$244$	7.64671	0.489531
$245$	−17.8317	−1.13922
$246$	0	0
$247$	−21.9306	−1.39541
$248$	0.865514	0.0549602
$249$	0	0
$250$	−14.9196	−0.943597
$251$	18.5453	1.17057	0.585283	−	0.810829i	$-0.300983\pi$
$251$	18.5453	1.17057	0.585283	+	0.810829i	$0.300983\pi$
$252$	0	0
$253$	0.187538	0.0117904
$254$	27.1796	1.70540
$255$	0	0
$256$	14.1805	0.886283
$257$	23.0065	1.43511	0.717555	−	0.696502i	$-0.245261\pi$
$257$	23.0065	1.43511	0.717555	+	0.696502i	$0.245261\pi$
$258$	0	0
$259$	2.94783	0.183169
$260$	31.0983	1.92864
$261$	0	0
$262$	30.7634	1.90057
$263$	3.78505	0.233396	0.116698	−	0.993167i	$-0.462769\pi$
$263$	3.78505	0.233396	0.116698	+	0.993167i	$0.462769\pi$
$264$	0	0
$265$	17.1010	1.05051
$266$	14.3470	0.879669
$267$	0	0
$268$	11.3108	0.690916
$269$	−2.62526	−0.160065	−0.0800325	−	0.996792i	$-0.525502\pi$
$269$	−2.62526	−0.160065	−0.0800325	+	0.996792i	$0.525502\pi$
$270$	0	0
$271$	−11.6427	−0.707242	−0.353621	−	0.935389i	$-0.615050\pi$
$271$	−11.6427	−0.707242	−0.353621	+	0.935389i	$0.615050\pi$
$272$	−15.4084	−0.934270
$273$	0	0
$274$	−13.1889	−0.796769
$275$	1.36306	0.0821959
$276$	0	0
$277$	4.72065	0.283636	0.141818	−	0.989893i	$-0.454705\pi$
$277$	4.72065	0.283636	0.141818	+	0.989893i	$0.454705\pi$
$278$	10.2568	0.615162
$279$	0	0
$280$	−1.01959	−0.0609319
$281$	−25.0192	−1.49252	−0.746260	−	0.665654i	$-0.768153\pi$
$281$	−25.0192	−1.49252	−0.746260	+	0.665654i	$0.768153\pi$
$282$	0	0
$283$	−9.85106	−0.585585	−0.292792	−	0.956176i	$-0.594584\pi$
$283$	−9.85106	−0.585585	−0.292792	+	0.956176i	$0.594584\pi$
$284$	−18.8555	−1.11887
$285$	0	0
$286$	−1.64705	−0.0973920
$287$	0.0435252	0.00256921
$288$	0	0
$289$	−0.364740	−0.0214553
$290$	−70.6050	−4.14607
$291$	0	0
$292$	−16.3581	−0.957285
$293$	−21.2915	−1.24386	−0.621932	−	0.783071i	$-0.713652\pi$
$293$	−21.2915	−1.24386	−0.621932	+	0.783071i	$0.713652\pi$
$294$	0	0
$295$	13.7468	0.800370
$296$	−0.460032	−0.0267388
$297$	0	0
$298$	21.6613	1.25481
$299$	4.15417	0.240242
$300$	0	0
$301$	3.80610	0.219380
$302$	−31.2073	−1.79578
$303$	0	0
$304$	19.5247	1.11982
$305$	12.6411	0.723828
$306$	0	0
$307$	−17.0413	−0.972597	−0.486299	−	0.873793i	$-0.661653\pi$
$307$	−17.0413	−0.972597	−0.486299	+	0.873793i	$0.661653\pi$
$308$	0.552596	0.0314871
$309$	0	0
$310$	28.5501	1.62154
$311$	22.1896	1.25826	0.629130	−	0.777300i	$-0.283411\pi$
$311$	22.1896	1.25826	0.629130	+	0.777300i	$0.283411\pi$
$312$	0	0
$313$	−4.79466	−0.271010	−0.135505	−	0.990777i	$-0.543266\pi$
$313$	−4.79466	−0.271010	−0.135505	+	0.990777i	$0.543266\pi$
$314$	7.73096	0.436283
$315$	0	0
$316$	−27.2077	−1.53055
$317$	−10.5786	−0.594152	−0.297076	−	0.954854i	$-0.596012\pi$
$317$	−10.5786	−0.594152	−0.297076	+	0.954854i	$0.596012\pi$
$318$	0	0
$319$	1.91777	0.107374
$320$	−30.7025	−1.71632
$321$	0	0
$322$	−2.71766	−0.151449
$323$	−21.0793	−1.17288
$324$	0	0
$325$	30.1933	1.67482
$326$	−21.1166	−1.16954
$327$	0	0
$328$	−0.00679245	−0.000375050 0
$329$	0.932948	0.0514351
$330$	0	0
$331$	33.0018	1.81394	0.906972	−	0.421192i	$-0.138388\pi$
$331$	33.0018	1.81394	0.906972	+	0.421192i	$0.138388\pi$
$332$	4.69386	0.257609
$333$	0	0
$334$	−2.21356	−0.121121
$335$	18.6983	1.02160
$336$	0	0
$337$	−31.0856	−1.69334	−0.846671	−	0.532116i	$-0.821397\pi$
$337$	−31.0856	−1.69334	−0.846671	+	0.532116i	$0.821397\pi$
$338$	−10.1431	−0.551711
$339$	0	0
$340$	29.8912	1.62108
$341$	−0.775476	−0.0419944
$342$	0	0
$343$	−16.6090	−0.896802
$344$	−0.593972	−0.0320248
$345$	0	0
$346$	−21.2178	−1.14068
$347$	−22.9937	−1.23436	−0.617182	−	0.786820i	$-0.711726\pi$
$347$	−22.9937	−1.23436	−0.617182	+	0.786820i	$0.711726\pi$
$348$	0	0
$349$	17.7700	0.951208	0.475604	−	0.879659i	$-0.342230\pi$
$349$	17.7700	0.951208	0.475604	+	0.879659i	$0.342230\pi$
$350$	−19.7524	−1.05581
$351$	0	0
$352$	1.54827	0.0825233
$353$	−13.9632	−0.743188	−0.371594	−	0.928395i	$-0.621189\pi$
$353$	−13.9632	−0.743188	−0.371594	+	0.928395i	$0.621189\pi$
$354$	0	0
$355$	−31.1708	−1.65438
$356$	−20.9469	−1.11018
$357$	0	0
$358$	44.0269	2.32690
$359$	21.7844	1.14974	0.574869	−	0.818245i	$-0.305053\pi$
$359$	21.7844	1.14974	0.574869	+	0.818245i	$0.305053\pi$
$360$	0	0
$361$	7.71060	0.405821
$362$	−25.5521	−1.34299
$363$	0	0
$364$	12.2406	0.641580
$365$	−27.0423	−1.41546
$366$	0	0
$367$	5.31876	0.277637	0.138818	−	0.990318i	$-0.455670\pi$
$367$	5.31876	0.277637	0.138818	+	0.990318i	$0.455670\pi$
$368$	−3.69843	−0.192794
$369$	0	0
$370$	−15.1748	−0.788898
$371$	6.73111	0.349462
$372$	0	0
$373$	−23.1559	−1.19897	−0.599484	−	0.800387i	$-0.704628\pi$
$373$	−23.1559	−1.19897	−0.599484	+	0.800387i	$0.704628\pi$
$374$	−1.58312	−0.0818610
$375$	0	0
$376$	−0.145594	−0.00750843
$377$	42.4805	2.18786
$378$	0	0
$379$	−5.92833	−0.304518	−0.152259	−	0.988341i	$-0.548655\pi$
$379$	−5.92833	−0.304518	−0.152259	+	0.988341i	$0.548655\pi$
$380$	−37.8766	−1.94303
$381$	0	0
$382$	−12.0290	−0.615456
$383$	34.8364	1.78006	0.890029	−	0.455903i	$-0.150684\pi$
$383$	34.8364	1.78006	0.890029	+	0.455903i	$0.150684\pi$
$384$	0	0
$385$	0.913520	0.0465573
$386$	−41.6845	−2.12168
$387$	0	0
$388$	−10.1967	−0.517658
$389$	1.77761	0.0901285	0.0450643	−	0.998984i	$-0.485651\pi$
$389$	1.77761	0.0901285	0.0450643	+	0.998984i	$0.485651\pi$
$390$	0	0
$391$	3.99292	0.201931
$392$	1.09532	0.0553222
$393$	0	0
$394$	−6.20373	−0.312540
$395$	−44.9782	−2.26310
$396$	0	0
$397$	−12.8987	−0.647365	−0.323682	−	0.946166i	$-0.604921\pi$
$397$	−12.8987	−0.647365	−0.323682	+	0.946166i	$0.604921\pi$
$398$	−46.9583	−2.35381
$399$	0	0
$400$	−26.8809	−1.34405
$401$	25.4445	1.27064	0.635319	−	0.772250i	$-0.280869\pi$
$401$	25.4445	1.27064	0.635319	+	0.772250i	$0.280869\pi$
$402$	0	0
$403$	−17.1776	−0.855676
$404$	27.0406	1.34532
$405$	0	0
$406$	−27.7907	−1.37923
$407$	0.412176	0.0204308
$408$	0	0
$409$	−33.1140	−1.63738	−0.818690	−	0.574235i	$-0.805299\pi$
$409$	−33.1140	−1.63738	−0.818690	+	0.574235i	$0.805299\pi$
$410$	−0.224058	−0.0110654
$411$	0	0
$412$	−20.7006	−1.01985
$413$	5.41086	0.266251
$414$	0	0
$415$	7.75961	0.380904
$416$	34.2958	1.68149
$417$	0	0
$418$	2.00604	0.0981187
$419$	33.1423	1.61911	0.809553	−	0.587047i	$-0.199710\pi$
$419$	33.1423	1.61911	0.809553	+	0.587047i	$0.199710\pi$
$420$	0	0
$421$	−14.3129	−0.697570	−0.348785	−	0.937203i	$-0.613406\pi$
$421$	−14.3129	−0.697570	−0.348785	+	0.937203i	$0.613406\pi$
$422$	−6.52495	−0.317629
$423$	0	0
$424$	−1.05044	−0.0510140
$425$	29.0213	1.40774
$426$	0	0
$427$	4.97565	0.240789
$428$	14.1492	0.683926
$429$	0	0
$430$	−19.5929	−0.944856
$431$	24.1655	1.16401	0.582006	−	0.813185i	$-0.302268\pi$
$431$	24.1655	1.16401	0.582006	+	0.813185i	$0.302268\pi$
$432$	0	0
$433$	−4.95559	−0.238150	−0.119075	−	0.992885i	$-0.537993\pi$
$433$	−4.95559	−0.238150	−0.119075	+	0.992885i	$0.537993\pi$
$434$	11.2376	0.539421
$435$	0	0
$436$	35.9632	1.72233
$437$	−5.05962	−0.242034
$438$	0	0
$439$	−20.9535	−1.00005	−0.500027	−	0.866010i	$-0.666677\pi$
$439$	−20.9535	−1.00005	−0.500027	+	0.866010i	$0.666677\pi$
$440$	−0.142562	−0.00679638
$441$	0	0
$442$	−35.0677	−1.66800
$443$	3.09007	0.146814	0.0734069	−	0.997302i	$-0.476613\pi$
$443$	3.09007	0.146814	0.0734069	+	0.997302i	$0.476613\pi$
$444$	0	0
$445$	−34.6282	−1.64153
$446$	−5.65234	−0.267646
$447$	0	0
$448$	−12.0848	−0.570952
$449$	7.59148	0.358264	0.179132	−	0.983825i	$-0.442671\pi$
$449$	7.59148	0.358264	0.179132	+	0.983825i	$0.442671\pi$
$450$	0	0
$451$	0.00608585	0.000286571 0
$452$	−25.8097	−1.21399
$453$	0	0
$454$	−30.8927	−1.44987
$455$	20.2354	0.948650
$456$	0	0
$457$	−24.8554	−1.16269	−0.581344	−	0.813658i	$-0.697473\pi$
$457$	−24.8554	−1.16269	−0.581344	+	0.813658i	$0.697473\pi$
$458$	−35.0031	−1.63559
$459$	0	0
$460$	7.17472	0.334523
$461$	−6.55243	−0.305177	−0.152589	−	0.988290i	$-0.548761\pi$
$461$	−6.55243	−0.305177	−0.152589	+	0.988290i	$0.548761\pi$
$462$	0	0
$463$	6.94066	0.322560	0.161280	−	0.986909i	$-0.448438\pi$
$463$	6.94066	0.322560	0.161280	+	0.986909i	$0.448438\pi$
$464$	−37.8202	−1.75576
$465$	0	0
$466$	25.4945	1.18101
$467$	39.0558	1.80729	0.903644	−	0.428285i	$-0.140882\pi$
$467$	39.0558	1.80729	0.903644	+	0.428285i	$0.140882\pi$
$468$	0	0
$469$	7.35983	0.339845
$470$	−4.80261	−0.221528
$471$	0	0
$472$	−0.844407	−0.0388670
$473$	0.532182	0.0244697
$474$	0	0
$475$	−36.7743	−1.68732
$476$	11.7654	0.539268
$477$	0	0
$478$	−2.02621	−0.0926766
$479$	−18.4393	−0.842513	−0.421257	−	0.906942i	$-0.638411\pi$
$479$	−18.4393	−0.842513	−0.421257	+	0.906942i	$0.638411\pi$
$480$	0	0
$481$	9.13011	0.416297
$482$	1.33364	0.0607456
$483$	0	0
$484$	−23.0835	−1.04925
$485$	−16.8566	−0.765418
$486$	0	0
$487$	37.3485	1.69242	0.846212	−	0.532846i	$-0.178878\pi$
$487$	37.3485	1.69242	0.846212	+	0.532846i	$0.178878\pi$
$488$	−0.776489	−0.0351500
$489$	0	0
$490$	36.1307	1.63222
$491$	−33.4616	−1.51010	−0.755050	−	0.655668i	$-0.772387\pi$
$491$	−33.4616	−1.51010	−0.755050	+	0.655668i	$0.772387\pi$
$492$	0	0
$493$	40.8316	1.83896
$494$	44.4359	1.99926
$495$	0	0
$496$	15.2931	0.686681
$497$	−12.2691	−0.550345
$498$	0	0
$499$	−32.5621	−1.45768	−0.728841	−	0.684684i	$-0.759940\pi$
$499$	−32.5621	−1.45768	−0.728841	+	0.684684i	$0.759940\pi$
$500$	15.5036	0.693341
$501$	0	0
$502$	−37.5766	−1.67712
$503$	4.28613	0.191109	0.0955545	−	0.995424i	$-0.469538\pi$
$503$	4.28613	0.191109	0.0955545	+	0.995424i	$0.469538\pi$
$504$	0	0
$505$	44.7020	1.98921
$506$	−0.379992	−0.0168927
$507$	0	0
$508$	−28.2435	−1.25310
$509$	28.3767	1.25777	0.628887	−	0.777497i	$-0.283511\pi$
$509$	28.3767	1.25777	0.628887	+	0.777497i	$0.283511\pi$
$510$	0	0
$511$	−10.6441	−0.470866
$512$	−32.1489	−1.42079
$513$	0	0
$514$	−46.6161	−2.05615
$515$	−34.2211	−1.50796
$516$	0	0
$517$	0.130448	0.00573710
$518$	−5.97292	−0.262435
$519$	0	0
$520$	−3.15789	−0.138483
$521$	−33.0835	−1.44941	−0.724706	−	0.689058i	$-0.758024\pi$
$521$	−33.0835	−1.44941	−0.724706	+	0.689058i	$0.758024\pi$
$522$	0	0
$523$	36.8785	1.61258	0.806292	−	0.591517i	$-0.201471\pi$
$523$	36.8785	1.61258	0.806292	+	0.591517i	$0.201471\pi$
$524$	−31.9676	−1.39651
$525$	0	0
$526$	−7.66931	−0.334398
$527$	−16.5108	−0.719223
$528$	0	0
$529$	−22.0416	−0.958330
$530$	−34.6502	−1.50511
$531$	0	0
$532$	−14.9086	−0.646368
$533$	0.134808	0.00583917
$534$	0	0
$535$	23.3906	1.01126
$536$	−1.14856	−0.0496102
$537$	0	0
$538$	5.31932	0.229332
$539$	−0.981380	−0.0422710
$540$	0	0
$541$	27.7829	1.19448	0.597241	−	0.802062i	$-0.296264\pi$
$541$	27.7829	1.19448	0.597241	+	0.802062i	$0.296264\pi$
$542$	23.5905	1.01330
$543$	0	0
$544$	32.9646	1.41335
$545$	59.4523	2.54666
$546$	0	0
$547$	−23.6920	−1.01300	−0.506498	−	0.862241i	$-0.669060\pi$
$547$	−23.6920	−1.01300	−0.506498	+	0.862241i	$0.669060\pi$
$548$	13.7051	0.585454
$549$	0	0
$550$	−2.76185	−0.117766
$551$	−51.7396	−2.20418
$552$	0	0
$553$	−17.7038	−0.752843
$554$	−9.56502	−0.406379
$555$	0	0
$556$	−10.6583	−0.452012
$557$	16.2888	0.690178	0.345089	−	0.938570i	$-0.387849\pi$
$557$	16.2888	0.690178	0.345089	+	0.938570i	$0.387849\pi$
$558$	0	0
$559$	11.7884	0.498595
$560$	−18.0155	−0.761293
$561$	0	0
$562$	50.6941	2.13840
$563$	−0.216197	−0.00911160	−0.00455580	−	0.999990i	$-0.501450\pi$
$563$	−0.216197	−0.00911160	−0.00455580	+	0.999990i	$0.501450\pi$
$564$	0	0
$565$	−42.6672	−1.79502
$566$	19.9603	0.838994
$567$	0	0
$568$	1.91469	0.0803386
$569$	13.6370	0.571691	0.285845	−	0.958276i	$-0.407726\pi$
$569$	13.6370	0.571691	0.285845	+	0.958276i	$0.407726\pi$
$570$	0	0
$571$	−17.2641	−0.722482	−0.361241	−	0.932473i	$-0.617647\pi$
$571$	−17.2641	−0.722482	−0.361241	+	0.932473i	$0.617647\pi$
$572$	1.71152	0.0715622
$573$	0	0
$574$	−0.0881912	−0.00368103
$575$	6.96592	0.290499
$576$	0	0
$577$	−45.0478	−1.87537	−0.937683	−	0.347493i	$-0.887033\pi$
$577$	−45.0478	−1.87537	−0.937683	+	0.347493i	$0.887033\pi$
$578$	0.739039	0.0307400
$579$	0	0
$580$	73.3687	3.04647
$581$	3.05425	0.126712
$582$	0	0
$583$	0.941167	0.0389791
$584$	1.66109	0.0687364
$585$	0	0
$586$	43.1411	1.78214
$587$	6.87236	0.283653	0.141826	−	0.989892i	$-0.454703\pi$
$587$	6.87236	0.283653	0.141826	+	0.989892i	$0.454703\pi$
$588$	0	0
$589$	20.9216	0.862062
$590$	−27.8539	−1.14673
$591$	0	0
$592$	−8.12849	−0.334079
$593$	−11.7147	−0.481067	−0.240534	−	0.970641i	$-0.577322\pi$
$593$	−11.7147	−0.481067	−0.240534	+	0.970641i	$0.577322\pi$
$594$	0	0
$595$	19.4500	0.797371
$596$	−22.5092	−0.922013
$597$	0	0
$598$	−8.41721	−0.344205
$599$	29.8786	1.22081	0.610403	−	0.792091i	$-0.291008\pi$
$599$	29.8786	1.22081	0.610403	+	0.792091i	$0.291008\pi$
$600$	0	0
$601$	21.7040	0.885324	0.442662	−	0.896688i	$-0.354034\pi$
$601$	21.7040	0.885324	0.442662	+	0.896688i	$0.354034\pi$
$602$	−7.71195	−0.314316
$603$	0	0
$604$	32.4288	1.31951
$605$	−38.1602	−1.55143
$606$	0	0
$607$	−12.5826	−0.510713	−0.255357	−	0.966847i	$-0.582193\pi$
$607$	−12.5826	−0.510713	−0.255357	+	0.966847i	$0.582193\pi$
$608$	−41.7710	−1.69404
$609$	0	0
$610$	−25.6135	−1.03706
$611$	2.88956	0.116899
$612$	0	0
$613$	−12.3210	−0.497642	−0.248821	−	0.968549i	$-0.580043\pi$
$613$	−12.3210	−0.497642	−0.248821	+	0.968549i	$0.580043\pi$
$614$	34.5292	1.39348
$615$	0	0
$616$	−0.0561136	−0.00226088
$617$	−48.6690	−1.95934	−0.979670	−	0.200617i	$-0.935705\pi$
$617$	−48.6690	−1.95934	−0.979670	+	0.200617i	$0.935705\pi$
$618$	0	0
$619$	−40.9465	−1.64578	−0.822890	−	0.568201i	$-0.807640\pi$
$619$	−40.9465	−1.64578	−0.822890	+	0.568201i	$0.807640\pi$
$620$	−29.6677	−1.19148
$621$	0	0
$622$	−44.9608	−1.80277
$623$	−13.6299	−0.546072
$624$	0	0
$625$	−9.94763	−0.397905
$626$	9.71497	0.388288
$627$	0	0
$628$	−8.03357	−0.320574
$629$	8.77572	0.349911
$630$	0	0
$631$	−22.4459	−0.893558	−0.446779	−	0.894644i	$-0.647429\pi$
$631$	−22.4459	−0.893558	−0.446779	+	0.894644i	$0.647429\pi$
$632$	2.76282	0.109899
$633$	0	0
$634$	21.4344	0.851269
$635$	−46.6905	−1.85286
$636$	0	0
$637$	−21.7386	−0.861313
$638$	−3.88579	−0.153840
$639$	0	0
$640$	5.94531	0.235009
$641$	11.9975	0.473872	0.236936	−	0.971525i	$-0.423857\pi$
$641$	11.9975	0.473872	0.236936	+	0.971525i	$0.423857\pi$
$642$	0	0
$643$	−40.5278	−1.59826	−0.799130	−	0.601158i	$-0.794706\pi$
$643$	−40.5278	−1.59826	−0.799130	+	0.601158i	$0.794706\pi$
$644$	2.82403	0.111282
$645$	0	0
$646$	42.7111	1.68045
$647$	−10.4001	−0.408871	−0.204435	−	0.978880i	$-0.565536\pi$
$647$	−10.4001	−0.408871	−0.204435	+	0.978880i	$0.565536\pi$
$648$	0	0
$649$	0.756565	0.0296978
$650$	−61.1779	−2.39959
$651$	0	0
$652$	21.9431	0.859359
$653$	2.80015	0.109578	0.0547891	−	0.998498i	$-0.482551\pi$
$653$	2.80015	0.109578	0.0547891	+	0.998498i	$0.482551\pi$
$654$	0	0
$655$	−52.8470	−2.06490
$656$	−0.120019	−0.00468594
$657$	0	0
$658$	−1.89035	−0.0736934
$659$	−28.1624	−1.09705	−0.548525	−	0.836134i	$-0.684811\pi$
$659$	−28.1624	−1.09705	−0.548525	+	0.836134i	$0.684811\pi$
$660$	0	0
$661$	15.9663	0.621016	0.310508	−	0.950571i	$-0.399501\pi$
$661$	15.9663	0.621016	0.310508	+	0.950571i	$0.399501\pi$
$662$	−66.8685	−2.59892
$663$	0	0
$664$	−0.476640	−0.0184972
$665$	−24.6460	−0.955730
$666$	0	0
$667$	9.80071	0.379485
$668$	2.30020	0.0889976
$669$	0	0
$670$	−37.8867	−1.46369
$671$	0.695712	0.0268577
$672$	0	0
$673$	20.0409	0.772519	0.386260	−	0.922390i	$-0.373767\pi$
$673$	20.0409	0.772519	0.386260	+	0.922390i	$0.373767\pi$
$674$	62.9860	2.42613
$675$	0	0
$676$	10.5401	0.405389
$677$	36.8324	1.41558	0.707791	−	0.706422i	$-0.249692\pi$
$677$	36.8324	1.41558	0.707791	+	0.706422i	$0.249692\pi$
$678$	0	0
$679$	−6.63489	−0.254624
$680$	−3.03532	−0.116399
$681$	0	0
$682$	1.57128	0.0601673
$683$	−25.9817	−0.994161	−0.497080	−	0.867705i	$-0.665595\pi$
$683$	−25.9817	−0.994161	−0.497080	+	0.867705i	$0.665595\pi$
$684$	0	0
$685$	22.6565	0.865661
$686$	33.6533	1.28489
$687$	0	0
$688$	−10.4951	−0.400123
$689$	20.8478	0.794238
$690$	0	0
$691$	−7.84083	−0.298279	−0.149140	−	0.988816i	$-0.547650\pi$
$691$	−7.84083	−0.298279	−0.149140	+	0.988816i	$0.547650\pi$
$692$	22.0483	0.838152
$693$	0	0
$694$	46.5900	1.76853
$695$	−17.6196	−0.668351
$696$	0	0
$697$	0.129575	0.00490801
$698$	−36.0058	−1.36284
$699$	0	0
$700$	20.5256	0.775795
$701$	28.2197	1.06584	0.532921	−	0.846165i	$-0.321094\pi$
$701$	28.2197	1.06584	0.532921	+	0.846165i	$0.321094\pi$
$702$	0	0
$703$	−11.1201	−0.419404
$704$	−1.68973	−0.0636842
$705$	0	0
$706$	28.2924	1.06480
$707$	17.5951	0.661732
$708$	0	0
$709$	21.9221	0.823303	0.411652	−	0.911341i	$-0.364952\pi$
$709$	21.9221	0.823303	0.411652	+	0.911341i	$0.364952\pi$
$710$	63.1586	2.37030
$711$	0	0
$712$	2.12706	0.0797150
$713$	−3.96306	−0.148418
$714$	0	0
$715$	2.82938	0.105813
$716$	−45.7503	−1.70977
$717$	0	0
$718$	−44.1398	−1.64728
$719$	−51.4814	−1.91993	−0.959966	−	0.280118i	$-0.909626\pi$
$719$	−51.4814	−1.91993	−0.959966	+	0.280118i	$0.909626\pi$
$720$	0	0
$721$	−13.4697	−0.501638
$722$	−15.6233	−0.581438
$723$	0	0
$724$	26.5522	0.986806
$725$	71.2334	2.64554
$726$	0	0
$727$	−4.61082	−0.171006	−0.0855030	−	0.996338i	$-0.527250\pi$
$727$	−4.61082	−0.171006	−0.0855030	+	0.996338i	$0.527250\pi$
$728$	−1.24297	−0.0460677
$729$	0	0
$730$	54.7933	2.02799
$731$	11.3308	0.419085
$732$	0	0
$733$	−3.26928	−0.120754	−0.0603768	−	0.998176i	$-0.519230\pi$
$733$	−3.26928	−0.120754	−0.0603768	+	0.998176i	$0.519230\pi$
$734$	−10.7769	−0.397783
$735$	0	0
$736$	7.91242	0.291656
$737$	1.02908	0.0379065
$738$	0	0
$739$	35.3375	1.29991	0.649956	−	0.759972i	$-0.274787\pi$
$739$	35.3375	1.29991	0.649956	+	0.759972i	$0.274787\pi$
$740$	15.7688	0.579671
$741$	0	0
$742$	−13.6386	−0.500690
$743$	−45.5076	−1.66951	−0.834757	−	0.550619i	$-0.814392\pi$
$743$	−45.5076	−1.66951	−0.834757	+	0.550619i	$0.814392\pi$
$744$	0	0
$745$	−37.2109	−1.36330
$746$	46.9187	1.71782
$747$	0	0
$748$	1.64509	0.0601503
$749$	9.20674	0.336407
$750$	0	0
$751$	38.6081	1.40883	0.704415	−	0.709789i	$-0.251210\pi$
$751$	38.6081	1.40883	0.704415	+	0.709789i	$0.251210\pi$
$752$	−2.57256	−0.0938115
$753$	0	0
$754$	−86.0743	−3.13464
$755$	53.6095	1.95105
$756$	0	0
$757$	12.4991	0.454288	0.227144	−	0.973861i	$-0.427061\pi$
$757$	12.4991	0.454288	0.227144	+	0.973861i	$0.427061\pi$
$758$	12.0120	0.436297
$759$	0	0
$760$	3.84620	0.139516
$761$	42.9165	1.55572	0.777862	−	0.628435i	$-0.216304\pi$
$761$	42.9165	1.55572	0.777862	+	0.628435i	$0.216304\pi$
$762$	0	0
$763$	23.4010	0.847171
$764$	12.4998	0.452228
$765$	0	0
$766$	−70.5859	−2.55037
$767$	16.7587	0.605121
$768$	0	0
$769$	−41.9328	−1.51214	−0.756068	−	0.654493i	$-0.772882\pi$
$769$	−41.9328	−1.51214	−0.756068	+	0.654493i	$0.772882\pi$
$770$	−1.85098	−0.0667048
$771$	0	0
$772$	43.3161	1.55898
$773$	17.1599	0.617200	0.308600	−	0.951192i	$-0.400140\pi$
$773$	17.1599	0.617200	0.308600	+	0.951192i	$0.400140\pi$
$774$	0	0
$775$	−28.8042	−1.03468
$776$	1.03543	0.0371697
$777$	0	0
$778$	−3.60181	−0.129131
$779$	−0.164191	−0.00588274
$780$	0	0
$781$	−1.71551	−0.0613857
$782$	−8.09049	−0.289315
$783$	0	0
$784$	19.3537	0.691205
$785$	−13.2806	−0.474006
$786$	0	0
$787$	8.35629	0.297870	0.148935	−	0.988847i	$-0.452416\pi$
$787$	8.35629	0.297870	0.148935	+	0.988847i	$0.452416\pi$
$788$	6.44657	0.229649
$789$	0	0
$790$	91.1353	3.24245
$791$	−16.7942	−0.597132
$792$	0	0
$793$	15.4107	0.547251
$794$	26.1354	0.927509
$795$	0	0
$796$	48.7964	1.72954
$797$	−50.4354	−1.78651	−0.893257	−	0.449547i	$-0.851585\pi$
$797$	−50.4354	−1.78651	−0.893257	+	0.449547i	$0.851585\pi$
$798$	0	0
$799$	2.77740	0.0982572
$800$	57.5090	2.03325
$801$	0	0
$802$	−51.5559	−1.82050
$803$	−1.48829	−0.0525206
$804$	0	0
$805$	4.66853	0.164544
$806$	34.8054	1.22597
$807$	0	0
$808$	−2.74585	−0.0965988
$809$	−37.4116	−1.31532	−0.657661	−	0.753314i	$-0.728454\pi$
$809$	−37.4116	−1.31532	−0.657661	+	0.753314i	$0.728454\pi$
$810$	0	0
$811$	34.9470	1.22716	0.613578	−	0.789634i	$-0.289730\pi$
$811$	34.9470	1.22716	0.613578	+	0.789634i	$0.289730\pi$
$812$	28.8785	1.01344
$813$	0	0
$814$	−0.835154	−0.0292721
$815$	36.2751	1.27066
$816$	0	0
$817$	−14.3578	−0.502316
$818$	67.0958	2.34595
$819$	0	0
$820$	0.232828	0.00813072
$821$	51.6588	1.80291	0.901453	−	0.432877i	$-0.142501\pi$
$821$	51.6588	1.80291	0.901453	+	0.432877i	$0.142501\pi$
$822$	0	0
$823$	−23.3085	−0.812484	−0.406242	−	0.913765i	$-0.633161\pi$
$823$	−23.3085	−0.812484	−0.406242	+	0.913765i	$0.633161\pi$
$824$	2.10205	0.0732285
$825$	0	0
$826$	−10.9635	−0.381470
$827$	0.139400	0.00484741	0.00242370	−	0.999997i	$-0.499229\pi$
$827$	0.139400	0.00484741	0.00242370	+	0.999997i	$0.499229\pi$
$828$	0	0
$829$	−8.55617	−0.297168	−0.148584	−	0.988900i	$-0.547472\pi$
$829$	−8.55617	−0.297168	−0.148584	+	0.988900i	$0.547472\pi$
$830$	−15.7226	−0.545739
$831$	0	0
$832$	−37.4293	−1.29763
$833$	−20.8948	−0.723961
$834$	0	0
$835$	3.80257	0.131593
$836$	−2.08456	−0.0720962
$837$	0	0
$838$	−67.1532	−2.31977
$839$	7.49954	0.258913	0.129456	−	0.991585i	$-0.458677\pi$
$839$	7.49954	0.258913	0.129456	+	0.991585i	$0.458677\pi$
$840$	0	0
$841$	71.2220	2.45593
$842$	29.0010	0.999440
$843$	0	0
$844$	6.78035	0.233389
$845$	17.4243	0.599414
$846$	0	0
$847$	−15.0202	−0.516100
$848$	−18.5607	−0.637377
$849$	0	0
$850$	−58.8032	−2.01693
$851$	2.10642	0.0722070
$852$	0	0
$853$	7.83193	0.268160	0.134080	−	0.990971i	$-0.457192\pi$
$853$	7.83193	0.268160	0.134080	+	0.990971i	$0.457192\pi$
$854$	−10.0817	−0.344989
$855$	0	0
$856$	−1.43678	−0.0491083
$857$	−9.69988	−0.331342	−0.165671	−	0.986181i	$-0.552979\pi$
$857$	−9.69988	−0.331342	−0.165671	+	0.986181i	$0.552979\pi$
$858$	0	0
$859$	−3.41295	−0.116448	−0.0582242	−	0.998304i	$-0.518544\pi$
$859$	−3.41295	−0.116448	−0.0582242	+	0.998304i	$0.518544\pi$
$860$	20.3599	0.694266
$861$	0	0
$862$	−48.9643	−1.66773
$863$	37.9282	1.29109	0.645545	−	0.763722i	$-0.276630\pi$
$863$	37.9282	1.29109	0.645545	+	0.763722i	$0.276630\pi$
$864$	0	0
$865$	36.4491	1.23931
$866$	10.0411	0.341209
$867$	0	0
$868$	−11.6774	−0.396358
$869$	−2.47541	−0.0839725
$870$	0	0
$871$	22.7951	0.772382
$872$	−3.65190	−0.123669
$873$	0	0
$874$	10.2518	0.346774
$875$	10.0880	0.341038
$876$	0	0
$877$	44.6999	1.50941	0.754704	−	0.656066i	$-0.227781\pi$
$877$	44.6999	1.50941	0.754704	+	0.656066i	$0.227781\pi$
$878$	42.4561	1.43282
$879$	0	0
$880$	−2.51899	−0.0849150
$881$	32.1877	1.08443	0.542216	−	0.840239i	$-0.317586\pi$
$881$	32.1877	1.08443	0.542216	+	0.840239i	$0.317586\pi$
$882$	0	0
$883$	54.4253	1.83156	0.915778	−	0.401685i	$-0.131575\pi$
$883$	54.4253	1.83156	0.915778	+	0.401685i	$0.131575\pi$
$884$	36.4403	1.22562
$885$	0	0
$886$	−6.26113	−0.210347
$887$	−32.7726	−1.10040	−0.550198	−	0.835034i	$-0.685448\pi$
$887$	−32.7726	−1.10040	−0.550198	+	0.835034i	$0.685448\pi$
$888$	0	0
$889$	−18.3778	−0.616371
$890$	70.1639	2.35190
$891$	0	0
$892$	5.87359	0.196662
$893$	−3.51937	−0.117771
$894$	0	0
$895$	−75.6318	−2.52809
$896$	2.34013	0.0781781
$897$	0	0
$898$	−15.3819	−0.513301
$899$	−40.5262	−1.35162
$900$	0	0
$901$	20.0386	0.667582
$902$	−0.0123312	−0.000410584 0
$903$	0	0
$904$	2.62086	0.0871685
$905$	43.8946	1.45911
$906$	0	0
$907$	19.5122	0.647890	0.323945	−	0.946076i	$-0.394991\pi$
$907$	19.5122	0.647890	0.323945	+	0.946076i	$0.394991\pi$
$908$	32.1019	1.06534
$909$	0	0
$910$	−41.0011	−1.35917
$911$	−12.5747	−0.416618	−0.208309	−	0.978063i	$-0.566796\pi$
$911$	−12.5747	−0.416618	−0.208309	+	0.978063i	$0.566796\pi$
$912$	0	0
$913$	0.427056	0.0141335
$914$	50.3623	1.66584
$915$	0	0
$916$	36.3733	1.20181
$917$	−20.8010	−0.686911
$918$	0	0
$919$	−25.2431	−0.832693	−0.416347	−	0.909206i	$-0.636690\pi$
$919$	−25.2431	−0.832693	−0.416347	+	0.909206i	$0.636690\pi$
$920$	−0.728561	−0.0240199
$921$	0	0
$922$	13.2766	0.437241
$923$	−38.0003	−1.25079
$924$	0	0
$925$	15.3098	0.503384
$926$	−14.0632	−0.462146
$927$	0	0
$928$	80.9124	2.65608
$929$	−14.4960	−0.475599	−0.237799	−	0.971314i	$-0.576426\pi$
$929$	−14.4960	−0.475599	−0.237799	+	0.971314i	$0.576426\pi$
$930$	0	0
$931$	26.4767	0.867740
$932$	−26.4924	−0.867789
$933$	0	0
$934$	−79.1352	−2.58938
$935$	2.71956	0.0889391
$936$	0	0
$937$	−13.7602	−0.449525	−0.224762	−	0.974414i	$-0.572161\pi$
$937$	−13.7602	−0.449525	−0.224762	+	0.974414i	$0.572161\pi$
$938$	−14.9125	−0.486912
$939$	0	0
$940$	4.99059	0.162775
$941$	34.3696	1.12042	0.560209	−	0.828351i	$-0.310721\pi$
$941$	34.3696	1.12042	0.560209	+	0.828351i	$0.310721\pi$
$942$	0	0
$943$	0.0311016	0.00101281
$944$	−14.9202	−0.485610
$945$	0	0
$946$	−1.07831	−0.0350589
$947$	−5.24589	−0.170469	−0.0852343	−	0.996361i	$-0.527164\pi$
$947$	−5.24589	−0.170469	−0.0852343	+	0.996361i	$0.527164\pi$
$948$	0	0
$949$	−32.9671	−1.07016
$950$	74.5123	2.41750
$951$	0	0
$952$	−1.19473	−0.0387214
$953$	−28.2817	−0.916135	−0.458068	−	0.888917i	$-0.651458\pi$
$953$	−28.2817	−0.916135	−0.458068	+	0.888917i	$0.651458\pi$
$954$	0	0
$955$	20.6640	0.668671
$956$	2.10552	0.0680974
$957$	0	0
$958$	37.3619	1.20711
$959$	8.91780	0.287971
$960$	0	0
$961$	−14.6127	−0.471376
$962$	−18.4995	−0.596448
$963$	0	0
$964$	−1.38584	−0.0446349
$965$	71.6077	2.30513
$966$	0	0
$967$	3.52394	0.113322	0.0566611	−	0.998393i	$-0.481955\pi$
$967$	3.52394	0.113322	0.0566611	+	0.998393i	$0.481955\pi$
$968$	2.34402	0.0753397
$969$	0	0
$970$	34.1549	1.09665
$971$	−30.9946	−0.994665	−0.497333	−	0.867560i	$-0.665687\pi$
$971$	−30.9946	−0.994665	−0.497333	+	0.867560i	$0.665687\pi$
$972$	0	0
$973$	−6.93525	−0.222334
$974$	−75.6759	−2.42481
$975$	0	0
$976$	−13.7201	−0.439170
$977$	44.4189	1.42109	0.710543	−	0.703653i	$-0.248449\pi$
$977$	44.4189	1.42109	0.710543	+	0.703653i	$0.248449\pi$
$978$	0	0
$979$	−1.90579	−0.0609092
$980$	−37.5450	−1.19933
$981$	0	0
$982$	67.8001	2.16359
$983$	−28.7370	−0.916568	−0.458284	−	0.888806i	$-0.651536\pi$
$983$	−28.7370	−0.916568	−0.458284	+	0.888806i	$0.651536\pi$
$984$	0	0
$985$	10.6571	0.339563
$986$	−82.7333	−2.63477
$987$	0	0
$988$	−46.1752	−1.46903
$989$	2.71971	0.0864816
$990$	0	0
$991$	18.3353	0.582441	0.291220	−	0.956656i	$-0.405939\pi$
$991$	18.3353	0.582441	0.291220	+	0.956656i	$0.405939\pi$
$992$	−32.7181	−1.03880
$993$	0	0
$994$	24.8598	0.788504
$995$	80.6674	2.55733
$996$	0	0
$997$	18.7618	0.594192	0.297096	−	0.954848i	$-0.403982\pi$
$997$	18.7618	0.594192	0.297096	+	0.954848i	$0.403982\pi$
$998$	65.9777	2.08849
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.k.1.3	yes	20
3.2	odd	2		2151.2.a.j.1.18	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.18	✓	20	3.2	odd	2
2151.2.a.k.1.3	yes	20	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)

\(f(q)\)	\(=\)	\(q-2.02621 q^{2} +2.10552 q^{4} +3.48073 q^{5} +1.37004 q^{7} -0.213806 q^{8} +O(q^{10})\)	\(q-2.02621 q^{2} +2.10552 q^{4} +3.48073 q^{5} +1.37004 q^{7} -0.213806 q^{8} -7.05267 q^{10} +0.191564 q^{11} +4.24334 q^{13} -2.77599 q^{14} -3.77782 q^{16} +4.07863 q^{17} -5.16823 q^{19} +7.32874 q^{20} -0.388149 q^{22} +0.978985 q^{23} +7.11545 q^{25} -8.59789 q^{26} +2.88465 q^{28} +10.0111 q^{29} -4.04813 q^{31} +8.08227 q^{32} -8.26416 q^{34} +4.76874 q^{35} +2.15163 q^{37} +10.4719 q^{38} -0.744200 q^{40} +0.0317692 q^{41} +2.77809 q^{43} +0.403342 q^{44} -1.98363 q^{46} +0.680962 q^{47} -5.12298 q^{49} -14.4174 q^{50} +8.93444 q^{52} +4.91306 q^{53} +0.666782 q^{55} -0.292924 q^{56} -20.2846 q^{58} +3.94941 q^{59} +3.63175 q^{61} +8.20235 q^{62} -8.82072 q^{64} +14.7699 q^{65} +5.37197 q^{67} +8.58765 q^{68} -9.66247 q^{70} -8.95527 q^{71} -7.76915 q^{73} -4.35966 q^{74} -10.8818 q^{76} +0.262451 q^{77} -12.9221 q^{79} -13.1496 q^{80} -0.0643711 q^{82} +2.22931 q^{83} +14.1966 q^{85} -5.62898 q^{86} -0.0409576 q^{88} -9.94855 q^{89} +5.81356 q^{91} +2.06127 q^{92} -1.37977 q^{94} -17.9892 q^{95} -4.84284 q^{97} +10.3802 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) \) 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8 + 4 * q^10 + 12 * q^11 - 4 * q^13 + 20 * q^14 + 22 * q^16 + 24 * q^17 - 4 * q^19 + 40 * q^20 - 6 * q^22 + 12 * q^23 + 22 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 - 4 * q^31 + 28 * q^32 + 8 * q^34 + 20 * q^35 - 10 * q^37 + 26 * q^38 + 6 * q^40 + 66 * q^41 + 8 * q^43 + 36 * q^44 - 12 * q^46 + 28 * q^47 + 18 * q^49 + 28 * q^50 - 18 * q^52 + 28 * q^53 - 4 * q^55 + 60 * q^56 + 54 * q^59 - 4 * q^61 + 20 * q^62 + 22 * q^64 + 42 * q^65 + 12 * q^67 + 12 * q^68 + 20 * q^70 + 36 * q^71 + 14 * q^73 - 50 * q^76 + 8 * q^77 - 12 * q^79 + 88 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 + 18 * q^86 - 10 * q^88 + 130 * q^89 - 6 * q^91 - 46 * q^92 - 26 * q^94 - 2 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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