LMFDB - Embedded newform 2151.2.a.f.1.7

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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 10x^{5} + 8x^{4} + 22x^{3} - 5x^{2} - 7x + 2 $ x^7 - x^6 - 10x^5 + 8x^4 + 22x^3 - 5x^2 - 7*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.456608$ 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)

$f(q)$	$=$	$q+2.51244 q^{2} +4.31233 q^{4} -3.31233 q^{5} -0.339486 q^{7} +5.80958 q^{8} +O(q^{10})$	\(q+2.51244 q^{2} +4.31233 q^{4} -3.31233 q^{5} -0.339486 q^{7} +5.80958 q^{8} -8.32202 q^{10} -3.90863 q^{11} -2.59922 q^{13} -0.852938 q^{14} +5.97153 q^{16} -7.00969 q^{17} -0.785005 q^{19} -14.2839 q^{20} -9.82018 q^{22} -4.71311 q^{23} +5.97153 q^{25} -6.53037 q^{26} -1.46398 q^{28} +9.39317 q^{29} -2.58404 q^{31} +3.38393 q^{32} -17.6114 q^{34} +1.12449 q^{35} +3.01060 q^{37} -1.97227 q^{38} -19.2433 q^{40} +6.89866 q^{41} -1.04078 q^{43} -16.8553 q^{44} -11.8414 q^{46} +3.53094 q^{47} -6.88475 q^{49} +15.0031 q^{50} -11.2087 q^{52} -9.53300 q^{53} +12.9467 q^{55} -1.97227 q^{56} +23.5997 q^{58} +0.267120 q^{59} -14.4942 q^{61} -6.49222 q^{62} -3.44116 q^{64} +8.60948 q^{65} +15.6793 q^{67} -30.2281 q^{68} +2.82521 q^{70} +4.18091 q^{71} +8.50986 q^{73} +7.56394 q^{74} -3.38520 q^{76} +1.32693 q^{77} +10.0735 q^{79} -19.7797 q^{80} +17.3324 q^{82} +1.82018 q^{83} +23.2184 q^{85} -2.61488 q^{86} -22.7075 q^{88} -12.9931 q^{89} +0.882400 q^{91} -20.3245 q^{92} +8.87126 q^{94} +2.60020 q^{95} -4.21796 q^{97} -17.2975 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 10 q^{4} - 3 q^{5} + 3 q^{7}+O(q^{10}) $ 7 * q + 10 * q^4 - 3 * q^5 + 3 * q^7	$ 7 q + 10 q^{4} - 3 q^{5} + 3 q^{7} - 11 q^{11} - 5 q^{13} - 6 q^{14} - 4 q^{16} - 11 q^{17} - 8 q^{19} - 34 q^{20} - 19 q^{22} - 26 q^{23} - 4 q^{25} - 6 q^{26} - 2 q^{28} - 6 q^{29} - 2 q^{31} + 5 q^{32} - 40 q^{34} + 5 q^{35} + 12 q^{37} + 9 q^{38} - 5 q^{40} - 26 q^{41} + 10 q^{43} - 3 q^{44} + 6 q^{46} - 7 q^{47} + 2 q^{49} + 5 q^{50} - 22 q^{52} - 2 q^{53} - 8 q^{55} + 9 q^{56} - 6 q^{58} - 6 q^{59} - 8 q^{61} + 2 q^{62} - 18 q^{64} + 17 q^{65} + 24 q^{67} + 9 q^{68} - 3 q^{70} + 25 q^{71} - 16 q^{73} + 9 q^{74} - 32 q^{76} - 24 q^{77} + 19 q^{79} - 18 q^{80} + 39 q^{82} - 37 q^{83} - 20 q^{85} + q^{86} - 21 q^{88} - 29 q^{89} - 19 q^{91} - 52 q^{92} - 22 q^{94} + 24 q^{95} - 12 q^{97} - 23 q^{98}+O(q^{100}) $ 7 * q + 10 * q^4 - 3 * q^5 + 3 * q^7 - 11 * q^11 - 5 * q^13 - 6 * q^14 - 4 * q^16 - 11 * q^17 - 8 * q^19 - 34 * q^20 - 19 * q^22 - 26 * q^23 - 4 * q^25 - 6 * q^26 - 2 * q^28 - 6 * q^29 - 2 * q^31 + 5 * q^32 - 40 * q^34 + 5 * q^35 + 12 * q^37 + 9 * q^38 - 5 * q^40 - 26 * q^41 + 10 * q^43 - 3 * q^44 + 6 * q^46 - 7 * q^47 + 2 * q^49 + 5 * q^50 - 22 * q^52 - 2 * q^53 - 8 * q^55 + 9 * q^56 - 6 * q^58 - 6 * q^59 - 8 * q^61 + 2 * q^62 - 18 * q^64 + 17 * q^65 + 24 * q^67 + 9 * q^68 - 3 * q^70 + 25 * q^71 - 16 * q^73 + 9 * q^74 - 32 * q^76 - 24 * q^77 + 19 * q^79 - 18 * q^80 + 39 * q^82 - 37 * q^83 - 20 * q^85 + q^86 - 21 * q^88 - 29 * q^89 - 19 * q^91 - 52 * q^92 - 22 * q^94 + 24 * q^95 - 12 * q^97 - 23 * 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 $

$2$	2.51244	1.77656	0.888280	−	0.459302i	$-0.151901\pi$
$2$	2.51244	1.77656	0.888280	+	0.459302i	$0.151901\pi$
$3$	0	0
$3$	0	0
$4$	4.31233	2.15617
$5$	−3.31233	−1.48132	−0.740660	−	0.671880i	$-0.765487\pi$
$5$	−3.31233	−1.48132	−0.740660	+	0.671880i	$0.765487\pi$
$6$	0	0
$7$	−0.339486	−0.128314	−0.0641569	−	0.997940i	$-0.520436\pi$
$7$	−0.339486	−0.128314	−0.0641569	+	0.997940i	$0.520436\pi$
$8$	5.80958	2.05400
$9$	0	0
$10$	−8.32202	−2.63165
$11$	−3.90863	−1.17850	−0.589248	−	0.807952i	$-0.700576\pi$
$11$	−3.90863	−1.17850	−0.589248	+	0.807952i	$0.700576\pi$
$12$	0	0
$13$	−2.59922	−0.720894	−0.360447	−	0.932780i	$-0.617376\pi$
$13$	−2.59922	−0.720894	−0.360447	+	0.932780i	$0.617376\pi$
$14$	−0.852938	−0.227957
$15$	0	0
$16$	5.97153	1.49288
$17$	−7.00969	−1.70010	−0.850049	−	0.526703i	$-0.823428\pi$
$17$	−7.00969	−1.70010	−0.850049	+	0.526703i	$0.823428\pi$
$18$	0	0
$19$	−0.785005	−0.180092	−0.0900462	−	0.995938i	$-0.528701\pi$
$19$	−0.785005	−0.180092	−0.0900462	+	0.995938i	$0.528701\pi$
$20$	−14.2839	−3.19397
$21$	0	0
$22$	−9.82018	−2.09367
$23$	−4.71311	−0.982751	−0.491376	−	0.870948i	$-0.663506\pi$
$23$	−4.71311	−0.982751	−0.491376	+	0.870948i	$0.663506\pi$
$24$	0	0
$25$	5.97153	1.19431
$26$	−6.53037	−1.28071
$27$	0	0
$28$	−1.46398	−0.276666
$29$	9.39317	1.74427	0.872134	−	0.489267i	$-0.162736\pi$
$29$	9.39317	1.74427	0.872134	+	0.489267i	$0.162736\pi$
$30$	0	0
$31$	−2.58404	−0.464106	−0.232053	−	0.972703i	$-0.574544\pi$
$31$	−2.58404	−0.464106	−0.232053	+	0.972703i	$0.574544\pi$
$32$	3.38393	0.598200
$33$	0	0
$34$	−17.6114	−3.02033
$35$	1.12449	0.190074
$36$	0	0
$37$	3.01060	0.494940	0.247470	−	0.968896i	$-0.420401\pi$
$37$	3.01060	0.494940	0.247470	+	0.968896i	$0.420401\pi$
$38$	−1.97227	−0.319945
$39$	0	0
$40$	−19.2433	−3.04263
$41$	6.89866	1.07739	0.538695	−	0.842501i	$-0.318917\pi$
$41$	6.89866	1.07739	0.538695	+	0.842501i	$0.318917\pi$
$42$	0	0
$43$	−1.04078	−0.158717	−0.0793585	−	0.996846i	$-0.525287\pi$
$43$	−1.04078	−0.158717	−0.0793585	+	0.996846i	$0.525287\pi$
$44$	−16.8553	−2.54103
$45$	0	0
$46$	−11.8414	−1.74592
$47$	3.53094	0.515041	0.257520	−	0.966273i	$-0.417095\pi$
$47$	3.53094	0.515041	0.257520	+	0.966273i	$0.417095\pi$
$48$	0	0
$49$	−6.88475	−0.983536
$50$	15.0031	2.12176
$51$	0	0
$52$	−11.2087	−1.55437
$53$	−9.53300	−1.30946	−0.654729	−	0.755864i	$-0.727217\pi$
$53$	−9.53300	−1.30946	−0.654729	+	0.755864i	$0.727217\pi$
$54$	0	0
$55$	12.9467	1.74573
$56$	−1.97227	−0.263556
$57$	0	0
$58$	23.5997	3.09880
$59$	0.267120	0.0347761	0.0173880	−	0.999849i	$-0.494465\pi$
$59$	0.267120	0.0347761	0.0173880	+	0.999849i	$0.494465\pi$
$60$	0	0
$61$	−14.4942	−1.85580	−0.927898	−	0.372835i	$-0.878386\pi$
$61$	−14.4942	−1.85580	−0.927898	+	0.372835i	$0.878386\pi$
$62$	−6.49222	−0.824513
$63$	0	0
$64$	−3.44116	−0.430145
$65$	8.60948	1.06787
$66$	0	0
$67$	15.6793	1.91553	0.957766	−	0.287547i	$-0.0928398\pi$
$67$	15.6793	1.91553	0.957766	+	0.287547i	$0.0928398\pi$
$68$	−30.2281	−3.66569
$69$	0	0
$70$	2.82521	0.337677
$71$	4.18091	0.496182	0.248091	−	0.968737i	$-0.420197\pi$
$71$	4.18091	0.496182	0.248091	+	0.968737i	$0.420197\pi$
$72$	0	0
$73$	8.50986	0.996003	0.498002	−	0.867176i	$-0.334067\pi$
$73$	8.50986	0.996003	0.498002	+	0.867176i	$0.334067\pi$
$74$	7.56394	0.879290
$75$	0	0
$76$	−3.38520	−0.388309
$77$	1.32693	0.151217
$78$	0	0
$79$	10.0735	1.13335	0.566676	−	0.823941i	$-0.308229\pi$
$79$	10.0735	1.13335	0.566676	+	0.823941i	$0.308229\pi$
$80$	−19.7797	−2.21144
$81$	0	0
$82$	17.3324	1.91405
$83$	1.82018	0.199791	0.0998954	−	0.994998i	$-0.468149\pi$
$83$	1.82018	0.199791	0.0998954	+	0.994998i	$0.468149\pi$
$84$	0	0
$85$	23.2184	2.51839
$86$	−2.61488	−0.281970
$87$	0	0
$88$	−22.7075	−2.42063
$89$	−12.9931	−1.37727	−0.688635	−	0.725109i	$-0.741790\pi$
$89$	−12.9931	−1.37727	−0.688635	+	0.725109i	$0.741790\pi$
$90$	0	0
$91$	0.882400	0.0925006
$92$	−20.3245	−2.11897
$93$	0	0
$94$	8.87126	0.915001
$95$	2.60020	0.266774
$96$	0	0
$97$	−4.21796	−0.428269	−0.214135	−	0.976804i	$-0.568693\pi$
$97$	−4.21796	−0.428269	−0.214135	+	0.976804i	$0.568693\pi$
$98$	−17.2975	−1.74731
$99$	0	0
$100$	25.7512	2.57512
$101$	−1.50648	−0.149900	−0.0749500	−	0.997187i	$-0.523880\pi$
$101$	−1.50648	−0.149900	−0.0749500	+	0.997187i	$0.523880\pi$
$102$	0	0
$103$	−19.1609	−1.88798	−0.943988	−	0.329981i	$-0.892958\pi$
$103$	−19.1609	−1.88798	−0.943988	+	0.329981i	$0.892958\pi$
$104$	−15.1004	−1.48071
$105$	0	0
$106$	−23.9510	−2.32633
$107$	−11.3518	−1.09742	−0.548711	−	0.836012i	$-0.684881\pi$
$107$	−11.3518	−1.09742	−0.548711	+	0.836012i	$0.684881\pi$
$108$	0	0
$109$	0.457195	0.0437913	0.0218957	−	0.999760i	$-0.493030\pi$
$109$	0.457195	0.0437913	0.0218957	+	0.999760i	$0.493030\pi$
$110$	32.5277	3.10139
$111$	0	0
$112$	−2.02725	−0.191558
$113$	4.50041	0.423363	0.211682	−	0.977339i	$-0.432106\pi$
$113$	4.50041	0.423363	0.211682	+	0.977339i	$0.432106\pi$
$114$	0	0
$115$	15.6114	1.45577
$116$	40.5065	3.76093
$117$	0	0
$118$	0.671122	0.0617818
$119$	2.37969	0.218146
$120$	0	0
$121$	4.27739	0.388854
$122$	−36.4158	−3.29693
$123$	0	0
$124$	−11.1432	−1.00069
$125$	−3.21804	−0.287830
$126$	0	0
$127$	−9.12525	−0.809735	−0.404868	−	0.914375i	$-0.632682\pi$
$127$	−9.12525	−0.809735	−0.404868	+	0.914375i	$0.632682\pi$
$128$	−15.4136	−1.36238
$129$	0	0
$130$	21.6308	1.89714
$131$	−9.98155	−0.872092	−0.436046	−	0.899924i	$-0.643621\pi$
$131$	−9.98155	−0.872092	−0.436046	+	0.899924i	$0.643621\pi$
$132$	0	0
$133$	0.266498	0.0231083
$134$	39.3933	3.40306
$135$	0	0
$136$	−40.7233	−3.49200
$137$	7.09709	0.606345	0.303173	−	0.952936i	$-0.401954\pi$
$137$	7.09709	0.606345	0.303173	+	0.952936i	$0.401954\pi$
$138$	0	0
$139$	8.01634	0.679937	0.339969	−	0.940437i	$-0.389584\pi$
$139$	8.01634	0.679937	0.339969	+	0.940437i	$0.389584\pi$
$140$	4.84918	0.409830
$141$	0	0
$142$	10.5043	0.881498
$143$	10.1594	0.849571
$144$	0	0
$145$	−31.1133	−2.58382
$146$	21.3805	1.76946
$147$	0	0
$148$	12.9827	1.06717
$149$	5.25047	0.430135	0.215067	−	0.976599i	$-0.431003\pi$
$149$	5.25047	0.430135	0.215067	+	0.976599i	$0.431003\pi$
$150$	0	0
$151$	5.65388	0.460106	0.230053	−	0.973178i	$-0.426110\pi$
$151$	5.65388	0.460106	0.230053	+	0.973178i	$0.426110\pi$
$152$	−4.56055	−0.369909
$153$	0	0
$154$	3.33382	0.268647
$155$	8.55918	0.687490
$156$	0	0
$157$	6.47990	0.517152	0.258576	−	0.965991i	$-0.416747\pi$
$157$	6.47990	0.517152	0.258576	+	0.965991i	$0.416747\pi$
$158$	25.3089	2.01347
$159$	0	0
$160$	−11.2087	−0.886125
$161$	1.60004	0.126101
$162$	0	0
$163$	1.72542	0.135145	0.0675725	−	0.997714i	$-0.478475\pi$
$163$	1.72542	0.135145	0.0675725	+	0.997714i	$0.478475\pi$
$164$	29.7493	2.32303
$165$	0	0
$166$	4.57309	0.354940
$167$	−8.11926	−0.628287	−0.314144	−	0.949376i	$-0.601717\pi$
$167$	−8.11926	−0.628287	−0.314144	+	0.949376i	$0.601717\pi$
$168$	0	0
$169$	−6.24406	−0.480312
$170$	58.3347	4.47407
$171$	0	0
$172$	−4.48817	−0.342220
$173$	23.0251	1.75056	0.875281	−	0.483614i	$-0.160676\pi$
$173$	23.0251	1.75056	0.875281	+	0.483614i	$0.160676\pi$
$174$	0	0
$175$	−2.02725	−0.153246
$176$	−23.3405	−1.75936
$177$	0	0
$178$	−32.6444	−2.44680
$179$	−17.9424	−1.34108	−0.670540	−	0.741874i	$-0.733937\pi$
$179$	−17.9424	−1.34108	−0.670540	+	0.741874i	$0.733937\pi$
$180$	0	0
$181$	−16.9966	−1.26335	−0.631675	−	0.775233i	$-0.717632\pi$
$181$	−16.9966	−1.26335	−0.631675	+	0.775233i	$0.717632\pi$
$182$	2.21697	0.164333
$183$	0	0
$184$	−27.3812	−2.01857
$185$	−9.97210	−0.733164
$186$	0	0
$187$	27.3983	2.00356
$188$	15.2266	1.11051
$189$	0	0
$190$	6.53282	0.473941
$191$	−20.2579	−1.46581	−0.732907	−	0.680329i	$-0.761837\pi$
$191$	−20.2579	−1.46581	−0.732907	+	0.680329i	$0.761837\pi$
$192$	0	0
$193$	1.18824	0.0855317	0.0427658	−	0.999085i	$-0.486383\pi$
$193$	1.18824	0.0855317	0.0427658	+	0.999085i	$0.486383\pi$
$194$	−10.5974	−0.760846
$195$	0	0
$196$	−29.6893	−2.12067
$197$	12.7167	0.906029	0.453014	−	0.891503i	$-0.350349\pi$
$197$	12.7167	0.906029	0.453014	+	0.891503i	$0.350349\pi$
$198$	0	0
$199$	20.2823	1.43778	0.718888	−	0.695125i	$-0.244651\pi$
$199$	20.2823	1.43778	0.718888	+	0.695125i	$0.244651\pi$
$200$	34.6921	2.45310
$201$	0	0
$202$	−3.78492	−0.266306
$203$	−3.18885	−0.223814
$204$	0	0
$205$	−22.8507	−1.59596
$206$	−48.1404	−3.35410
$207$	0	0
$208$	−15.5213	−1.07621
$209$	3.06829	0.212238
$210$	0	0
$211$	−23.5461	−1.62098	−0.810491	−	0.585751i	$-0.800800\pi$
$211$	−23.5461	−1.62098	−0.810491	+	0.585751i	$0.800800\pi$
$212$	−41.1094	−2.82341
$213$	0	0
$214$	−28.5207	−1.94963
$215$	3.44740	0.235111
$216$	0	0
$217$	0.877245	0.0595512
$218$	1.14867	0.0777979
$219$	0	0
$220$	55.8304	3.76408
$221$	18.2197	1.22559
$222$	0	0
$223$	−7.46082	−0.499614	−0.249807	−	0.968296i	$-0.580367\pi$
$223$	−7.46082	−0.499614	−0.249807	+	0.968296i	$0.580367\pi$
$224$	−1.14880	−0.0767573
$225$	0	0
$226$	11.3070	0.752130
$227$	10.1488	0.673598	0.336799	−	0.941577i	$-0.390656\pi$
$227$	10.1488	0.673598	0.336799	+	0.941577i	$0.390656\pi$
$228$	0	0
$229$	10.9282	0.722157	0.361078	−	0.932535i	$-0.382409\pi$
$229$	10.9282	0.722157	0.361078	+	0.932535i	$0.382409\pi$
$230$	39.2226	2.58626
$231$	0	0
$232$	54.5704	3.58272
$233$	5.53317	0.362490	0.181245	−	0.983438i	$-0.441987\pi$
$233$	5.53317	0.362490	0.181245	+	0.983438i	$0.441987\pi$
$234$	0	0
$235$	−11.6956	−0.762940
$236$	1.15191	0.0749829
$237$	0	0
$238$	5.97882	0.387550
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−10.6252	−0.684427	−0.342213	−	0.939622i	$-0.611177\pi$
$241$	−10.6252	−0.684427	−0.342213	+	0.939622i	$0.611177\pi$
$242$	10.7467	0.690822
$243$	0	0
$244$	−62.5039	−4.00140
$245$	22.8046	1.45693
$246$	0	0
$247$	2.04040	0.129828
$248$	−15.0122	−0.953273
$249$	0	0
$250$	−8.08512	−0.511348
$251$	10.6061	0.669448	0.334724	−	0.942316i	$-0.391357\pi$
$251$	10.6061	0.669448	0.334724	+	0.942316i	$0.391357\pi$
$252$	0	0
$253$	18.4218	1.15817
$254$	−22.9266	−1.43854
$255$	0	0
$256$	−31.8432	−1.99020
$257$	6.49909	0.405402	0.202701	−	0.979241i	$-0.435028\pi$
$257$	6.49909	0.405402	0.202701	+	0.979241i	$0.435028\pi$
$258$	0	0
$259$	−1.02206	−0.0635076
$260$	37.1269	2.30251
$261$	0	0
$262$	−25.0780	−1.54932
$263$	13.1895	0.813301	0.406650	−	0.913584i	$-0.366697\pi$
$263$	13.1895	0.813301	0.406650	+	0.913584i	$0.366697\pi$
$264$	0	0
$265$	31.5764	1.93973
$266$	0.669560	0.0410534
$267$	0	0
$268$	67.6144	4.13021
$269$	22.5835	1.37694	0.688470	−	0.725265i	$-0.258283\pi$
$269$	22.5835	1.37694	0.688470	+	0.725265i	$0.258283\pi$
$270$	0	0
$271$	−20.2609	−1.23076	−0.615382	−	0.788229i	$-0.710998\pi$
$271$	−20.2609	−1.23076	−0.615382	+	0.788229i	$0.710998\pi$
$272$	−41.8586	−2.53805
$273$	0	0
$274$	17.8310	1.07721
$275$	−23.3405	−1.40749
$276$	0	0
$277$	32.5177	1.95380	0.976900	−	0.213698i	$-0.0685509\pi$
$277$	32.5177	1.95380	0.976900	+	0.213698i	$0.0685509\pi$
$278$	20.1405	1.20795
$279$	0	0
$280$	6.53282	0.390411
$281$	−21.4050	−1.27691	−0.638457	−	0.769657i	$-0.720427\pi$
$281$	−21.4050	−1.27691	−0.638457	+	0.769657i	$0.720427\pi$
$282$	0	0
$283$	13.0075	0.773216	0.386608	−	0.922244i	$-0.373647\pi$
$283$	13.0075	0.773216	0.386608	+	0.922244i	$0.373647\pi$
$284$	18.0294	1.06985
$285$	0	0
$286$	25.5248	1.50931
$287$	−2.34200	−0.138244
$288$	0	0
$289$	32.1357	1.89033
$290$	−78.1701	−4.59031
$291$	0	0
$292$	36.6973	2.14755
$293$	−20.7662	−1.21318	−0.606588	−	0.795016i	$-0.707462\pi$
$293$	−20.7662	−1.21318	−0.606588	+	0.795016i	$0.707462\pi$
$294$	0	0
$295$	−0.884790	−0.0515145
$296$	17.4903	1.01660
$297$	0	0
$298$	13.1915	0.764161
$299$	12.2504	0.708460
$300$	0	0
$301$	0.353330	0.0203656
$302$	14.2050	0.817406
$303$	0	0
$304$	−4.68768	−0.268857
$305$	48.0097	2.74903
$306$	0	0
$307$	−26.0185	−1.48496	−0.742478	−	0.669871i	$-0.766349\pi$
$307$	−26.0185	−1.48496	−0.742478	+	0.669871i	$0.766349\pi$
$308$	5.72215	0.326050
$309$	0	0
$310$	21.5044	1.22137
$311$	15.7782	0.894702	0.447351	−	0.894359i	$-0.352368\pi$
$311$	15.7782	0.894702	0.447351	+	0.894359i	$0.352368\pi$
$312$	0	0
$313$	−33.0258	−1.86673	−0.933364	−	0.358931i	$-0.883141\pi$
$313$	−33.0258	−1.86673	−0.933364	+	0.358931i	$0.883141\pi$
$314$	16.2803	0.918752
$315$	0	0
$316$	43.4401	2.44369
$317$	−11.0140	−0.618605	−0.309303	−	0.950964i	$-0.600096\pi$
$317$	−11.0140	−0.618605	−0.309303	+	0.950964i	$0.600096\pi$
$318$	0	0
$319$	−36.7144	−2.05561
$320$	11.3983	0.637182
$321$	0	0
$322$	4.01999	0.224025
$323$	5.50264	0.306175
$324$	0	0
$325$	−15.5213	−0.860969
$326$	4.33500	0.240093
$327$	0	0
$328$	40.0783	2.21296
$329$	−1.19871	−0.0660869
$330$	0	0
$331$	−11.0114	−0.605243	−0.302622	−	0.953111i	$-0.597862\pi$
$331$	−11.0114	−0.605243	−0.302622	+	0.953111i	$0.597862\pi$
$332$	7.84922	0.430782
$333$	0	0
$334$	−20.3991	−1.11619
$335$	−51.9351	−2.83752
$336$	0	0
$337$	21.3718	1.16419	0.582097	−	0.813119i	$-0.302232\pi$
$337$	21.3718	1.16419	0.582097	+	0.813119i	$0.302232\pi$
$338$	−15.6878	−0.853303
$339$	0	0
$340$	100.125	5.43006
$341$	10.1000	0.546948
$342$	0	0
$343$	4.71368	0.254515
$344$	−6.04648	−0.326004
$345$	0	0
$346$	57.8490	3.10998
$347$	13.0033	0.698054	0.349027	−	0.937113i	$-0.386512\pi$
$347$	13.0033	0.698054	0.349027	+	0.937113i	$0.386512\pi$
$348$	0	0
$349$	−22.9842	−1.23032	−0.615158	−	0.788404i	$-0.710908\pi$
$349$	−22.9842	−1.23032	−0.615158	+	0.788404i	$0.710908\pi$
$350$	−5.09335	−0.272251
$351$	0	0
$352$	−13.2265	−0.704977
$353$	28.6105	1.52278	0.761392	−	0.648291i	$-0.224516\pi$
$353$	28.6105	1.52278	0.761392	+	0.648291i	$0.224516\pi$
$354$	0	0
$355$	−13.8485	−0.735004
$356$	−56.0307	−2.96962
$357$	0	0
$358$	−45.0792	−2.38251
$359$	−32.3316	−1.70640	−0.853199	−	0.521586i	$-0.825341\pi$
$359$	−32.3316	−1.70640	−0.853199	+	0.521586i	$0.825341\pi$
$360$	0	0
$361$	−18.3838	−0.967567
$362$	−42.7030	−2.24442
$363$	0	0
$364$	3.80520	0.199447
$365$	−28.1875	−1.47540
$366$	0	0
$367$	1.85904	0.0970410	0.0485205	−	0.998822i	$-0.484549\pi$
$367$	1.85904	0.0970410	0.0485205	+	0.998822i	$0.484549\pi$
$368$	−28.1445	−1.46713
$369$	0	0
$370$	−25.0543	−1.30251
$371$	3.23632	0.168022
$372$	0	0
$373$	−24.6782	−1.27779	−0.638893	−	0.769295i	$-0.720607\pi$
$373$	−24.6782	−1.27779	−0.638893	+	0.769295i	$0.720607\pi$
$374$	68.8364	3.55944
$375$	0	0
$376$	20.5133	1.05789
$377$	−24.4149	−1.25743
$378$	0	0
$379$	1.69911	0.0872772	0.0436386	−	0.999047i	$-0.486105\pi$
$379$	1.69911	0.0872772	0.0436386	+	0.999047i	$0.486105\pi$
$380$	11.2129	0.575210
$381$	0	0
$382$	−50.8968	−2.60411
$383$	26.2060	1.33906	0.669532	−	0.742783i	$-0.266495\pi$
$383$	26.2060	1.33906	0.669532	+	0.742783i	$0.266495\pi$
$384$	0	0
$385$	−4.39522	−0.224001
$386$	2.98539	0.151952
$387$	0	0
$388$	−18.1893	−0.923419
$389$	8.40497	0.426149	0.213075	−	0.977036i	$-0.431652\pi$
$389$	8.40497	0.426149	0.213075	+	0.977036i	$0.431652\pi$
$390$	0	0
$391$	33.0374	1.67077
$392$	−39.9975	−2.02018
$393$	0	0
$394$	31.9499	1.60961
$395$	−33.3666	−1.67886
$396$	0	0
$397$	−7.01413	−0.352029	−0.176014	−	0.984388i	$-0.556321\pi$
$397$	−7.01413	−0.352029	−0.176014	+	0.984388i	$0.556321\pi$
$398$	50.9581	2.55430
$399$	0	0
$400$	35.6592	1.78296
$401$	11.0698	0.552802	0.276401	−	0.961042i	$-0.410858\pi$
$401$	11.0698	0.552802	0.276401	+	0.961042i	$0.410858\pi$
$402$	0	0
$403$	6.71648	0.334571
$404$	−6.49643	−0.323209
$405$	0	0
$406$	−8.01179	−0.397618
$407$	−11.7673	−0.583285
$408$	0	0
$409$	−32.6034	−1.61213	−0.806067	−	0.591824i	$-0.798408\pi$
$409$	−32.6034	−1.61213	−0.806067	+	0.591824i	$0.798408\pi$
$410$	−57.4108	−2.83532
$411$	0	0
$412$	−82.6279	−4.07079
$413$	−0.0906836	−0.00446225
$414$	0	0
$415$	−6.02904	−0.295954
$416$	−8.79558	−0.431239
$417$	0	0
$418$	7.70889	0.377054
$419$	−21.1624	−1.03385	−0.516925	−	0.856031i	$-0.672923\pi$
$419$	−21.1624	−1.03385	−0.516925	+	0.856031i	$0.672923\pi$
$420$	0	0
$421$	−1.74016	−0.0848100	−0.0424050	−	0.999101i	$-0.513502\pi$
$421$	−1.74016	−0.0848100	−0.0424050	+	0.999101i	$0.513502\pi$
$422$	−59.1581	−2.87977
$423$	0	0
$424$	−55.3827	−2.68962
$425$	−41.8586	−2.03044
$426$	0	0
$427$	4.92059	0.238124
$428$	−48.9528	−2.36622
$429$	0	0
$430$	8.66136	0.417688
$431$	−16.2927	−0.784793	−0.392397	−	0.919796i	$-0.628354\pi$
$431$	−16.2927	−0.784793	−0.392397	+	0.919796i	$0.628354\pi$
$432$	0	0
$433$	−31.3283	−1.50554	−0.752771	−	0.658283i	$-0.771283\pi$
$433$	−31.3283	−1.50554	−0.752771	+	0.658283i	$0.771283\pi$
$434$	2.20402	0.105796
$435$	0	0
$436$	1.97157	0.0944213
$437$	3.69981	0.176986
$438$	0	0
$439$	−6.09888	−0.291084	−0.145542	−	0.989352i	$-0.546493\pi$
$439$	−6.09888	−0.291084	−0.145542	+	0.989352i	$0.546493\pi$
$440$	75.2148	3.58572
$441$	0	0
$442$	45.7759	2.17734
$443$	23.5958	1.12107	0.560535	−	0.828131i	$-0.310596\pi$
$443$	23.5958	1.12107	0.560535	+	0.828131i	$0.310596\pi$
$444$	0	0
$445$	43.0375	2.04018
$446$	−18.7448	−0.887594
$447$	0	0
$448$	1.16823	0.0551936
$449$	1.88592	0.0890019	0.0445009	−	0.999009i	$-0.485830\pi$
$449$	1.88592	0.0890019	0.0445009	+	0.999009i	$0.485830\pi$
$450$	0	0
$451$	−26.9643	−1.26970
$452$	19.4073	0.912841
$453$	0	0
$454$	25.4981	1.19669
$455$	−2.92280	−0.137023
$456$	0	0
$457$	28.3489	1.32610	0.663052	−	0.748573i	$-0.269261\pi$
$457$	28.3489	1.32610	0.663052	+	0.748573i	$0.269261\pi$
$458$	27.4564	1.28296
$459$	0	0
$460$	67.3214	3.13888
$461$	−39.9847	−1.86227	−0.931137	−	0.364670i	$-0.881182\pi$
$461$	−39.9847	−1.86227	−0.931137	+	0.364670i	$0.881182\pi$
$462$	0	0
$463$	36.3265	1.68824	0.844118	−	0.536157i	$-0.180124\pi$
$463$	36.3265	1.68824	0.844118	+	0.536157i	$0.180124\pi$
$464$	56.0916	2.60399
$465$	0	0
$466$	13.9017	0.643986
$467$	−5.53348	−0.256059	−0.128029	−	0.991770i	$-0.540865\pi$
$467$	−5.53348	−0.256059	−0.128029	+	0.991770i	$0.540865\pi$
$468$	0	0
$469$	−5.32291	−0.245789
$470$	−29.3846	−1.35541
$471$	0	0
$472$	1.55186	0.0714299
$473$	4.06801	0.187047
$474$	0	0
$475$	−4.68768	−0.215086
$476$	10.2620	0.470359
$477$	0	0
$478$	2.51244	0.114916
$479$	0.518428	0.0236876	0.0118438	−	0.999930i	$-0.496230\pi$
$479$	0.518428	0.0236876	0.0118438	+	0.999930i	$0.496230\pi$
$480$	0	0
$481$	−7.82521	−0.356799
$482$	−26.6950	−1.21593
$483$	0	0
$484$	18.4455	0.838434
$485$	13.9713	0.634404
$486$	0	0
$487$	13.1010	0.593663	0.296831	−	0.954930i	$-0.404070\pi$
$487$	13.1010	0.593663	0.296831	+	0.954930i	$0.404070\pi$
$488$	−84.2054	−3.81180
$489$	0	0
$490$	57.2950	2.58832
$491$	43.1943	1.94933	0.974666	−	0.223666i	$-0.0718024\pi$
$491$	43.1943	1.94933	0.974666	+	0.223666i	$0.0718024\pi$
$492$	0	0
$493$	−65.8432	−2.96543
$494$	5.12637	0.230646
$495$	0	0
$496$	−15.4307	−0.692857
$497$	−1.41936	−0.0636670
$498$	0	0
$499$	−32.9794	−1.47636	−0.738181	−	0.674603i	$-0.764315\pi$
$499$	−32.9794	−1.47636	−0.738181	+	0.674603i	$0.764315\pi$
$500$	−13.8773	−0.620610
$501$	0	0
$502$	26.6470	1.18932
$503$	−9.78083	−0.436106	−0.218053	−	0.975937i	$-0.569971\pi$
$503$	−9.78083	−0.436106	−0.218053	+	0.975937i	$0.569971\pi$
$504$	0	0
$505$	4.98995	0.222050
$506$	46.2836	2.05756
$507$	0	0
$508$	−39.3511	−1.74592
$509$	4.87132	0.215918	0.107959	−	0.994155i	$-0.465569\pi$
$509$	4.87132	0.215918	0.107959	+	0.994155i	$0.465569\pi$
$510$	0	0
$511$	−2.88898	−0.127801
$512$	−49.1770	−2.17334
$513$	0	0
$514$	16.3285	0.720221
$515$	63.4671	2.79669
$516$	0	0
$517$	−13.8012	−0.606974
$518$	−2.56785	−0.112825
$519$	0	0
$520$	50.0174	2.19341
$521$	−28.6067	−1.25328	−0.626640	−	0.779309i	$-0.715570\pi$
$521$	−28.6067	−1.25328	−0.626640	+	0.779309i	$0.715570\pi$
$522$	0	0
$523$	−14.3356	−0.626850	−0.313425	−	0.949613i	$-0.601476\pi$
$523$	−14.3356	−0.626850	−0.313425	+	0.949613i	$0.601476\pi$
$524$	−43.0437	−1.88037
$525$	0	0
$526$	33.1378	1.44488
$527$	18.1133	0.789027
$528$	0	0
$529$	−0.786589	−0.0341995
$530$	79.3338	3.44604
$531$	0	0
$532$	1.14923	0.0498254
$533$	−17.9311	−0.776684
$534$	0	0
$535$	37.6010	1.62563
$536$	91.0902	3.93450
$537$	0	0
$538$	56.7396	2.44622
$539$	26.9099	1.15909
$540$	0	0
$541$	−30.6702	−1.31861	−0.659307	−	0.751873i	$-0.729150\pi$
$541$	−30.6702	−1.31861	−0.659307	+	0.751873i	$0.729150\pi$
$542$	−50.9043	−2.18653
$543$	0	0
$544$	−23.7203	−1.01700
$545$	−1.51438	−0.0648689
$546$	0	0
$547$	−16.1307	−0.689698	−0.344849	−	0.938658i	$-0.612070\pi$
$547$	−16.1307	−0.689698	−0.344849	+	0.938658i	$0.612070\pi$
$548$	30.6050	1.30738
$549$	0	0
$550$	−58.6415	−2.50048
$551$	−7.37368	−0.314129
$552$	0	0
$553$	−3.41980	−0.145425
$554$	81.6986	3.47104
$555$	0	0
$556$	34.5691	1.46606
$557$	27.4105	1.16142	0.580709	−	0.814111i	$-0.302775\pi$
$557$	27.4105	1.16142	0.580709	+	0.814111i	$0.302775\pi$
$558$	0	0
$559$	2.70521	0.114418
$560$	6.71494	0.283758
$561$	0	0
$562$	−53.7786	−2.26851
$563$	13.5265	0.570074	0.285037	−	0.958516i	$-0.407994\pi$
$563$	13.5265	0.570074	0.285037	+	0.958516i	$0.407994\pi$
$564$	0	0
$565$	−14.9069	−0.627136
$566$	32.6805	1.37366
$567$	0	0
$568$	24.2893	1.01916
$569$	−12.0812	−0.506472	−0.253236	−	0.967404i	$-0.581495\pi$
$569$	−12.0812	−0.506472	−0.253236	+	0.967404i	$0.581495\pi$
$570$	0	0
$571$	20.7080	0.866603	0.433302	−	0.901249i	$-0.357348\pi$
$571$	20.7080	0.866603	0.433302	+	0.901249i	$0.357348\pi$
$572$	43.8107	1.83182
$573$	0	0
$574$	−5.88413	−0.245599
$575$	−28.1445	−1.17371
$576$	0	0
$577$	14.9403	0.621974	0.310987	−	0.950414i	$-0.399341\pi$
$577$	14.9403	0.621974	0.310987	+	0.950414i	$0.399341\pi$
$578$	80.7388	3.35829
$579$	0	0
$580$	−134.171	−5.57114
$581$	−0.617927	−0.0256359
$582$	0	0
$583$	37.2610	1.54319
$584$	49.4387	2.04579
$585$	0	0
$586$	−52.1738	−2.15528
$587$	−30.3707	−1.25353	−0.626767	−	0.779206i	$-0.715622\pi$
$587$	−30.3707	−1.25353	−0.626767	+	0.779206i	$0.715622\pi$
$588$	0	0
$589$	2.02848	0.0835820
$590$	−2.22298	−0.0915185
$591$	0	0
$592$	17.9779	0.738887
$593$	−43.1538	−1.77211	−0.886057	−	0.463577i	$-0.846566\pi$
$593$	−43.1538	−1.77211	−0.886057	+	0.463577i	$0.846566\pi$
$594$	0	0
$595$	−7.88233	−0.323144
$596$	22.6417	0.927442
$597$	0	0
$598$	30.7784	1.25862
$599$	7.61255	0.311040	0.155520	−	0.987833i	$-0.450295\pi$
$599$	7.61255	0.311040	0.155520	+	0.987833i	$0.450295\pi$
$600$	0	0
$601$	−31.0315	−1.26580	−0.632901	−	0.774233i	$-0.718136\pi$
$601$	−31.0315	−1.26580	−0.632901	+	0.774233i	$0.718136\pi$
$602$	0.887718	0.0361807
$603$	0	0
$604$	24.3814	0.992065
$605$	−14.1681	−0.576017
$606$	0	0
$607$	−32.0721	−1.30177	−0.650883	−	0.759178i	$-0.725601\pi$
$607$	−32.0721	−1.30177	−0.650883	+	0.759178i	$0.725601\pi$
$608$	−2.65640	−0.107731
$609$	0	0
$610$	120.621	4.88381
$611$	−9.17770	−0.371290
$612$	0	0
$613$	−17.4164	−0.703440	−0.351720	−	0.936105i	$-0.614403\pi$
$613$	−17.4164	−0.703440	−0.351720	+	0.936105i	$0.614403\pi$
$614$	−65.3698	−2.63811
$615$	0	0
$616$	7.70889	0.310600
$617$	−19.4595	−0.783411	−0.391705	−	0.920091i	$-0.628115\pi$
$617$	−19.4595	−0.783411	−0.391705	+	0.920091i	$0.628115\pi$
$618$	0	0
$619$	36.8726	1.48204	0.741018	−	0.671485i	$-0.234343\pi$
$619$	36.8726	1.48204	0.741018	+	0.671485i	$0.234343\pi$
$620$	36.9100	1.48234
$621$	0	0
$622$	39.6418	1.58949
$623$	4.41099	0.176723
$624$	0	0
$625$	−19.1985	−0.767938
$626$	−82.9751	−3.31635
$627$	0	0
$628$	27.9435	1.11507
$629$	−21.1034	−0.841446
$630$	0	0
$631$	−30.6426	−1.21986	−0.609932	−	0.792454i	$-0.708803\pi$
$631$	−30.6426	−1.21986	−0.609932	+	0.792454i	$0.708803\pi$
$632$	58.5225	2.32790
$633$	0	0
$634$	−27.6718	−1.09899
$635$	30.2259	1.19948
$636$	0	0
$637$	17.8950	0.709025
$638$	−92.2426	−3.65192
$639$	0	0
$640$	51.0548	2.01812
$641$	14.3819	0.568053	0.284026	−	0.958816i	$-0.408330\pi$
$641$	14.3819	0.568053	0.284026	+	0.958816i	$0.408330\pi$
$642$	0	0
$643$	8.77952	0.346231	0.173115	−	0.984902i	$-0.444617\pi$
$643$	8.77952	0.346231	0.173115	+	0.984902i	$0.444617\pi$
$644$	6.89989	0.271894
$645$	0	0
$646$	13.8250	0.543938
$647$	15.2115	0.598025	0.299013	−	0.954249i	$-0.403343\pi$
$647$	15.2115	0.598025	0.299013	+	0.954249i	$0.403343\pi$
$648$	0	0
$649$	−1.04407	−0.0409835
$650$	−38.9963	−1.52956
$651$	0	0
$652$	7.44057	0.291395
$653$	−1.00930	−0.0394970	−0.0197485	−	0.999805i	$-0.506287\pi$
$653$	−1.00930	−0.0394970	−0.0197485	+	0.999805i	$0.506287\pi$
$654$	0	0
$655$	33.0622	1.29185
$656$	41.1956	1.60842
$657$	0	0
$658$	−3.01167	−0.117407
$659$	−13.1750	−0.513226	−0.256613	−	0.966514i	$-0.582607\pi$
$659$	−13.1750	−0.513226	−0.256613	+	0.966514i	$0.582607\pi$
$660$	0	0
$661$	10.3593	0.402932	0.201466	−	0.979496i	$-0.435430\pi$
$661$	10.3593	0.402932	0.201466	+	0.979496i	$0.435430\pi$
$662$	−27.6655	−1.07525
$663$	0	0
$664$	10.5745	0.410370
$665$	−0.882731	−0.0342308
$666$	0	0
$667$	−44.2711	−1.71418
$668$	−35.0129	−1.35469
$669$	0	0
$670$	−130.483	−5.04102
$671$	56.6526	2.18705
$672$	0	0
$673$	−20.9075	−0.805925	−0.402962	−	0.915217i	$-0.632019\pi$
$673$	−20.9075	−0.805925	−0.402962	+	0.915217i	$0.632019\pi$
$674$	53.6952	2.06826
$675$	0	0
$676$	−26.9264	−1.03563
$677$	21.9562	0.843845	0.421922	−	0.906632i	$-0.361355\pi$
$677$	21.9562	0.843845	0.421922	+	0.906632i	$0.361355\pi$
$678$	0	0
$679$	1.43194	0.0549529
$680$	134.889	5.17276
$681$	0	0
$682$	25.3757	0.971685
$683$	−50.3300	−1.92582	−0.962912	−	0.269816i	$-0.913037\pi$
$683$	−50.3300	−1.92582	−0.962912	+	0.269816i	$0.913037\pi$
$684$	0	0
$685$	−23.5079	−0.898191
$686$	11.8428	0.452161
$687$	0	0
$688$	−6.21504	−0.236946
$689$	24.7784	0.943980
$690$	0	0
$691$	43.0784	1.63878	0.819390	−	0.573236i	$-0.194312\pi$
$691$	43.0784	1.63878	0.819390	+	0.573236i	$0.194312\pi$
$692$	99.2917	3.77450
$693$	0	0
$694$	32.6699	1.24013
$695$	−26.5528	−1.00720
$696$	0	0
$697$	−48.3575	−1.83167
$698$	−57.7463	−2.18573
$699$	0	0
$700$	−8.74219	−0.330424
$701$	9.39461	0.354830	0.177415	−	0.984136i	$-0.443227\pi$
$701$	9.39461	0.354830	0.177415	+	0.984136i	$0.443227\pi$
$702$	0	0
$703$	−2.36334	−0.0891349
$704$	13.4502	0.506925
$705$	0	0
$706$	71.8821	2.70532
$707$	0.511428	0.0192342
$708$	0	0
$709$	−12.1188	−0.455129	−0.227565	−	0.973763i	$-0.573076\pi$
$709$	−12.1188	−0.455129	−0.227565	+	0.973763i	$0.573076\pi$
$710$	−34.7936	−1.30578
$711$	0	0
$712$	−75.4846	−2.82891
$713$	12.1788	0.456101
$714$	0	0
$715$	−33.6513	−1.25849
$716$	−77.3736	−2.89159
$717$	0	0
$718$	−81.2311	−3.03152
$719$	−42.4247	−1.58217	−0.791087	−	0.611704i	$-0.790484\pi$
$719$	−42.4247	−1.58217	−0.791087	+	0.611704i	$0.790484\pi$
$720$	0	0
$721$	6.50485	0.242253
$722$	−46.1880	−1.71894
$723$	0	0
$724$	−73.2952	−2.72399
$725$	56.0916	2.08319
$726$	0	0
$727$	35.5278	1.31765	0.658826	−	0.752295i	$-0.271053\pi$
$727$	35.5278	1.31765	0.658826	+	0.752295i	$0.271053\pi$
$728$	5.12637	0.189996
$729$	0	0
$730$	−70.8192	−2.62113
$731$	7.29552	0.269835
$732$	0	0
$733$	41.3676	1.52795	0.763973	−	0.645248i	$-0.223246\pi$
$733$	41.3676	1.52795	0.763973	+	0.645248i	$0.223246\pi$
$734$	4.67072	0.172399
$735$	0	0
$736$	−15.9488	−0.587882
$737$	−61.2846	−2.25745
$738$	0	0
$739$	34.9372	1.28519	0.642593	−	0.766208i	$-0.277859\pi$
$739$	34.9372	1.28519	0.642593	+	0.766208i	$0.277859\pi$
$740$	−43.0030	−1.58082
$741$	0	0
$742$	8.13105	0.298500
$743$	−6.87545	−0.252236	−0.126118	−	0.992015i	$-0.540252\pi$
$743$	−6.87545	−0.252236	−0.126118	+	0.992015i	$0.540252\pi$
$744$	0	0
$745$	−17.3913	−0.637167
$746$	−62.0023	−2.27006
$747$	0	0
$748$	118.150	4.32001
$749$	3.85379	0.140814
$750$	0	0
$751$	15.1306	0.552123	0.276061	−	0.961140i	$-0.410971\pi$
$751$	15.1306	0.552123	0.276061	+	0.961140i	$0.410971\pi$
$752$	21.0851	0.768896
$753$	0	0
$754$	−61.3409	−2.23390
$755$	−18.7275	−0.681564
$756$	0	0
$757$	−3.55855	−0.129338	−0.0646688	−	0.997907i	$-0.520599\pi$
$757$	−3.55855	−0.129338	−0.0646688	+	0.997907i	$0.520599\pi$
$758$	4.26889	0.155053
$759$	0	0
$760$	15.1060	0.547954
$761$	2.53251	0.0918034	0.0459017	−	0.998946i	$-0.485384\pi$
$761$	2.53251	0.0918034	0.0459017	+	0.998946i	$0.485384\pi$
$762$	0	0
$763$	−0.155211	−0.00561903
$764$	−87.3590	−3.16054
$765$	0	0
$766$	65.8408	2.37893
$767$	−0.694304	−0.0250699
$768$	0	0
$769$	24.7355	0.891984	0.445992	−	0.895037i	$-0.352851\pi$
$769$	24.7355	0.891984	0.445992	+	0.895037i	$0.352851\pi$
$770$	−11.0427	−0.397952
$771$	0	0
$772$	5.12410	0.184420
$773$	1.06309	0.0382368	0.0191184	−	0.999817i	$-0.493914\pi$
$773$	1.06309	0.0382368	0.0191184	+	0.999817i	$0.493914\pi$
$774$	0	0
$775$	−15.4307	−0.554285
$776$	−24.5046	−0.879664
$777$	0	0
$778$	21.1170	0.757079
$779$	−5.41548	−0.194030
$780$	0	0
$781$	−16.3416	−0.584749
$782$	83.0044	2.96823
$783$	0	0
$784$	−41.1125	−1.46830
$785$	−21.4636	−0.766067
$786$	0	0
$787$	3.13133	0.111620	0.0558099	−	0.998441i	$-0.482226\pi$
$787$	3.13133	0.111620	0.0558099	+	0.998441i	$0.482226\pi$
$788$	54.8387	1.95355
$789$	0	0
$790$	−83.8314	−2.98259
$791$	−1.52783	−0.0543234
$792$	0	0
$793$	37.6737	1.33783
$794$	−17.6225	−0.625400
$795$	0	0
$796$	87.4642	3.10008
$797$	−28.8529	−1.02202	−0.511011	−	0.859574i	$-0.670729\pi$
$797$	−28.8529	−1.02202	−0.511011	+	0.859574i	$0.670729\pi$
$798$	0	0
$799$	−24.7508	−0.875620
$800$	20.2073	0.714434
$801$	0	0
$802$	27.8123	0.982085
$803$	−33.2619	−1.17379
$804$	0	0
$805$	−5.29985	−0.186795
$806$	16.8747	0.594386
$807$	0	0
$808$	−8.75200	−0.307894
$809$	−40.9345	−1.43918	−0.719591	−	0.694398i	$-0.755671\pi$
$809$	−40.9345	−1.43918	−0.719591	+	0.694398i	$0.755671\pi$
$810$	0	0
$811$	41.8222	1.46858	0.734288	−	0.678838i	$-0.237516\pi$
$811$	41.8222	1.46858	0.734288	+	0.678838i	$0.237516\pi$
$812$	−13.7514	−0.482579
$813$	0	0
$814$	−29.5646	−1.03624
$815$	−5.71515	−0.200193
$816$	0	0
$817$	0.817015	0.0285837
$818$	−81.9140	−2.86405
$819$	0	0
$820$	−98.5396	−3.44115
$821$	28.7502	1.00339	0.501694	−	0.865045i	$-0.332710\pi$
$821$	28.7502	1.00339	0.501694	+	0.865045i	$0.332710\pi$
$822$	0	0
$823$	42.8358	1.49316	0.746580	−	0.665295i	$-0.231694\pi$
$823$	42.8358	1.49316	0.746580	+	0.665295i	$0.231694\pi$
$824$	−111.317	−3.87790
$825$	0	0
$826$	−0.227837	−0.00792745
$827$	−10.3083	−0.358454	−0.179227	−	0.983808i	$-0.557360\pi$
$827$	−10.3083	−0.358454	−0.179227	+	0.983808i	$0.557360\pi$
$828$	0	0
$829$	−0.839454	−0.0291554	−0.0145777	−	0.999894i	$-0.504640\pi$
$829$	−0.839454	−0.0291554	−0.0145777	+	0.999894i	$0.504640\pi$
$830$	−15.1476	−0.525780
$831$	0	0
$832$	8.94434	0.310089
$833$	48.2599	1.67211
$834$	0	0
$835$	26.8937	0.930694
$836$	13.2315	0.457621
$837$	0	0
$838$	−53.1691	−1.83670
$839$	28.7550	0.992732	0.496366	−	0.868113i	$-0.334667\pi$
$839$	28.7550	0.992732	0.496366	+	0.868113i	$0.334667\pi$
$840$	0	0
$841$	59.2317	2.04247
$842$	−4.37203	−0.150670
$843$	0	0
$844$	−101.539	−3.49511
$845$	20.6824	0.711495
$846$	0	0
$847$	−1.45212	−0.0498953
$848$	−56.9266	−1.95487
$849$	0	0
$850$	−105.167	−3.60720
$851$	−14.1893	−0.486403
$852$	0	0
$853$	39.8737	1.36525	0.682625	−	0.730769i	$-0.260838\pi$
$853$	39.8737	1.36525	0.682625	+	0.730769i	$0.260838\pi$
$854$	12.3627	0.423042
$855$	0	0
$856$	−65.9493	−2.25410
$857$	−43.2839	−1.47855	−0.739275	−	0.673404i	$-0.764831\pi$
$857$	−43.2839	−1.47855	−0.739275	+	0.673404i	$0.764831\pi$
$858$	0	0
$859$	22.4608	0.766353	0.383177	−	0.923675i	$-0.374830\pi$
$859$	22.4608	0.766353	0.383177	+	0.923675i	$0.374830\pi$
$860$	14.8663	0.506937
$861$	0	0
$862$	−40.9344	−1.39423
$863$	9.22650	0.314074	0.157037	−	0.987593i	$-0.449806\pi$
$863$	9.22650	0.314074	0.157037	+	0.987593i	$0.449806\pi$
$864$	0	0
$865$	−76.2666	−2.59314
$866$	−78.7103	−2.67468
$867$	0	0
$868$	3.78297	0.128402
$869$	−39.3734	−1.33565
$870$	0	0
$871$	−40.7540	−1.38090
$872$	2.65611	0.0899472
$873$	0	0
$874$	9.29554	0.314426
$875$	1.09248	0.0369326
$876$	0	0
$877$	−5.52941	−0.186715	−0.0933574	−	0.995633i	$-0.529760\pi$
$877$	−5.52941	−0.186715	−0.0933574	+	0.995633i	$0.529760\pi$
$878$	−15.3230	−0.517128
$879$	0	0
$880$	77.3115	2.60617
$881$	35.3261	1.19016	0.595082	−	0.803665i	$-0.297119\pi$
$881$	35.3261	1.19016	0.595082	+	0.803665i	$0.297119\pi$
$882$	0	0
$883$	34.0864	1.14710	0.573549	−	0.819171i	$-0.305566\pi$
$883$	34.0864	1.14710	0.573549	+	0.819171i	$0.305566\pi$
$884$	78.5694	2.64258
$885$	0	0
$886$	59.2829	1.99165
$887$	−49.0600	−1.64727	−0.823637	−	0.567118i	$-0.808058\pi$
$887$	−49.0600	−1.64727	−0.823637	+	0.567118i	$0.808058\pi$
$888$	0	0
$889$	3.09790	0.103900
$890$	108.129	3.62449
$891$	0	0
$892$	−32.1735	−1.07725
$893$	−2.77181	−0.0927550
$894$	0	0
$895$	59.4312	1.98657
$896$	5.23269	0.174812
$897$	0	0
$898$	4.73824	0.158117
$899$	−24.2723	−0.809526
$900$	0	0
$901$	66.8233	2.22621
$902$	−67.7461	−2.25570
$903$	0	0
$904$	26.1455	0.869587
$905$	56.2985	1.87143
$906$	0	0
$907$	−4.35662	−0.144659	−0.0723297	−	0.997381i	$-0.523043\pi$
$907$	−4.35662	−0.144659	−0.0723297	+	0.997381i	$0.523043\pi$
$908$	43.7649	1.45239
$909$	0	0
$910$	−7.34335	−0.243430
$911$	−2.13391	−0.0706997	−0.0353498	−	0.999375i	$-0.511255\pi$
$911$	−2.13391	−0.0706997	−0.0353498	+	0.999375i	$0.511255\pi$
$912$	0	0
$913$	−7.11442	−0.235453
$914$	71.2247	2.35590
$915$	0	0
$916$	47.1261	1.55709
$917$	3.38860	0.111901
$918$	0	0
$919$	−56.2342	−1.85499	−0.927497	−	0.373831i	$-0.878044\pi$
$919$	−56.2342	−1.85499	−0.927497	+	0.373831i	$0.878044\pi$
$920$	90.6956	2.99014
$921$	0	0
$922$	−100.459	−3.30844
$923$	−10.8671	−0.357695
$924$	0	0
$925$	17.9779	0.591110
$926$	91.2680	2.99925
$927$	0	0
$928$	31.7858	1.04342
$929$	−1.32924	−0.0436109	−0.0218054	−	0.999762i	$-0.506941\pi$
$929$	−1.32924	−0.0436109	−0.0218054	+	0.999762i	$0.506941\pi$
$930$	0	0
$931$	5.40456	0.177127
$932$	23.8609	0.781589
$933$	0	0
$934$	−13.9025	−0.454904
$935$	−90.7521	−2.96791
$936$	0	0
$937$	−26.7000	−0.872250	−0.436125	−	0.899886i	$-0.643649\pi$
$937$	−26.7000	−0.872250	−0.436125	+	0.899886i	$0.643649\pi$
$938$	−13.3735	−0.436659
$939$	0	0
$940$	−50.4355	−1.64502
$941$	−15.1933	−0.495287	−0.247643	−	0.968851i	$-0.579656\pi$
$941$	−15.1933	−0.495287	−0.247643	+	0.968851i	$0.579656\pi$
$942$	0	0
$943$	−32.5142	−1.05881
$944$	1.59512	0.0519166
$945$	0	0
$946$	10.2206	0.332301
$947$	−1.20963	−0.0393076	−0.0196538	−	0.999807i	$-0.506256\pi$
$947$	−1.20963	−0.0393076	−0.0196538	+	0.999807i	$0.506256\pi$
$948$	0	0
$949$	−22.1190	−0.718013
$950$	−11.7775	−0.382112
$951$	0	0
$952$	13.8250	0.448071
$953$	40.3475	1.30698	0.653491	−	0.756934i	$-0.273304\pi$
$953$	40.3475	1.30698	0.653491	+	0.756934i	$0.273304\pi$
$954$	0	0
$955$	67.1010	2.17134
$956$	4.31233	0.139471
$957$	0	0
$958$	1.30252	0.0420824
$959$	−2.40937	−0.0778025
$960$	0	0
$961$	−24.3228	−0.784605
$962$	−19.6603	−0.633875
$963$	0	0
$964$	−45.8192	−1.47574
$965$	−3.93586	−0.126700
$966$	0	0
$967$	−4.56739	−0.146877	−0.0734386	−	0.997300i	$-0.523397\pi$
$967$	−4.56739	−0.146877	−0.0734386	+	0.997300i	$0.523397\pi$
$968$	24.8499	0.798705
$969$	0	0
$970$	35.1020	1.12706
$971$	4.34649	0.139485	0.0697427	−	0.997565i	$-0.477782\pi$
$971$	4.34649	0.139485	0.0697427	+	0.997565i	$0.477782\pi$
$972$	0	0
$973$	−2.72144	−0.0872453
$974$	32.9154	1.05468
$975$	0	0
$976$	−86.5528	−2.77049
$977$	−38.9859	−1.24727	−0.623635	−	0.781716i	$-0.714345\pi$
$977$	−38.9859	−1.24727	−0.623635	+	0.781716i	$0.714345\pi$
$978$	0	0
$979$	50.7853	1.62311
$980$	98.3408	3.14138
$981$	0	0
$982$	108.523	3.46310
$983$	−38.0354	−1.21314	−0.606571	−	0.795029i	$-0.707455\pi$
$983$	−38.0354	−1.21314	−0.606571	+	0.795029i	$0.707455\pi$
$984$	0	0
$985$	−42.1220	−1.34212
$986$	−165.427	−5.26826
$987$	0	0
$988$	8.79888	0.279930
$989$	4.90530	0.155979
$990$	0	0
$991$	−48.0407	−1.52606	−0.763032	−	0.646360i	$-0.776290\pi$
$991$	−48.0407	−1.52606	−0.763032	+	0.646360i	$0.776290\pi$
$992$	−8.74419	−0.277628
$993$	0	0
$994$	−3.56605	−0.113108
$995$	−67.1818	−2.12981
$996$	0	0
$997$	−46.7500	−1.48059	−0.740293	−	0.672284i	$-0.765313\pi$
$997$	−46.7500	−1.48059	−0.740293	+	0.672284i	$0.765313\pi$
$998$	−82.8586	−2.62284
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.f.1.7		7
3.2	odd	2		717.2.a.e.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.e.1.1	✓	7	3.2	odd	2
2151.2.a.f.1.7		7	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)

\(f(q)\)	\(=\)	\(q+2.51244 q^{2} +4.31233 q^{4} -3.31233 q^{5} -0.339486 q^{7} +5.80958 q^{8} +O(q^{10})\)	\(q+2.51244 q^{2} +4.31233 q^{4} -3.31233 q^{5} -0.339486 q^{7} +5.80958 q^{8} -8.32202 q^{10} -3.90863 q^{11} -2.59922 q^{13} -0.852938 q^{14} +5.97153 q^{16} -7.00969 q^{17} -0.785005 q^{19} -14.2839 q^{20} -9.82018 q^{22} -4.71311 q^{23} +5.97153 q^{25} -6.53037 q^{26} -1.46398 q^{28} +9.39317 q^{29} -2.58404 q^{31} +3.38393 q^{32} -17.6114 q^{34} +1.12449 q^{35} +3.01060 q^{37} -1.97227 q^{38} -19.2433 q^{40} +6.89866 q^{41} -1.04078 q^{43} -16.8553 q^{44} -11.8414 q^{46} +3.53094 q^{47} -6.88475 q^{49} +15.0031 q^{50} -11.2087 q^{52} -9.53300 q^{53} +12.9467 q^{55} -1.97227 q^{56} +23.5997 q^{58} +0.267120 q^{59} -14.4942 q^{61} -6.49222 q^{62} -3.44116 q^{64} +8.60948 q^{65} +15.6793 q^{67} -30.2281 q^{68} +2.82521 q^{70} +4.18091 q^{71} +8.50986 q^{73} +7.56394 q^{74} -3.38520 q^{76} +1.32693 q^{77} +10.0735 q^{79} -19.7797 q^{80} +17.3324 q^{82} +1.82018 q^{83} +23.2184 q^{85} -2.61488 q^{86} -22.7075 q^{88} -12.9931 q^{89} +0.882400 q^{91} -20.3245 q^{92} +8.87126 q^{94} +2.60020 q^{95} -4.21796 q^{97} -17.2975 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 10 q^{4} - 3 q^{5} + 3 q^{7}+O(q^{10}) \) 7 * q + 10 * q^4 - 3 * q^5 + 3 * q^7	\( 7 q + 10 q^{4} - 3 q^{5} + 3 q^{7} - 11 q^{11} - 5 q^{13} - 6 q^{14} - 4 q^{16} - 11 q^{17} - 8 q^{19} - 34 q^{20} - 19 q^{22} - 26 q^{23} - 4 q^{25} - 6 q^{26} - 2 q^{28} - 6 q^{29} - 2 q^{31} + 5 q^{32} - 40 q^{34} + 5 q^{35} + 12 q^{37} + 9 q^{38} - 5 q^{40} - 26 q^{41} + 10 q^{43} - 3 q^{44} + 6 q^{46} - 7 q^{47} + 2 q^{49} + 5 q^{50} - 22 q^{52} - 2 q^{53} - 8 q^{55} + 9 q^{56} - 6 q^{58} - 6 q^{59} - 8 q^{61} + 2 q^{62} - 18 q^{64} + 17 q^{65} + 24 q^{67} + 9 q^{68} - 3 q^{70} + 25 q^{71} - 16 q^{73} + 9 q^{74} - 32 q^{76} - 24 q^{77} + 19 q^{79} - 18 q^{80} + 39 q^{82} - 37 q^{83} - 20 q^{85} + q^{86} - 21 q^{88} - 29 q^{89} - 19 q^{91} - 52 q^{92} - 22 q^{94} + 24 q^{95} - 12 q^{97} - 23 q^{98}+O(q^{100}) \) 7 * q + 10 * q^4 - 3 * q^5 + 3 * q^7 - 11 * q^11 - 5 * q^13 - 6 * q^14 - 4 * q^16 - 11 * q^17 - 8 * q^19 - 34 * q^20 - 19 * q^22 - 26 * q^23 - 4 * q^25 - 6 * q^26 - 2 * q^28 - 6 * q^29 - 2 * q^31 + 5 * q^32 - 40 * q^34 + 5 * q^35 + 12 * q^37 + 9 * q^38 - 5 * q^40 - 26 * q^41 + 10 * q^43 - 3 * q^44 + 6 * q^46 - 7 * q^47 + 2 * q^49 + 5 * q^50 - 22 * q^52 - 2 * q^53 - 8 * q^55 + 9 * q^56 - 6 * q^58 - 6 * q^59 - 8 * q^61 + 2 * q^62 - 18 * q^64 + 17 * q^65 + 24 * q^67 + 9 * q^68 - 3 * q^70 + 25 * q^71 - 16 * q^73 + 9 * q^74 - 32 * q^76 - 24 * q^77 + 19 * q^79 - 18 * q^80 + 39 * q^82 - 37 * q^83 - 20 * q^85 + q^86 - 21 * q^88 - 29 * q^89 - 19 * q^91 - 52 * q^92 - 22 * q^94 + 24 * q^95 - 12 * q^97 - 23 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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