LMFDB - Embedded newform 2151.2.a.k.1.2

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			1.2
Root			$-2.12763$ 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)

$f(q)$	$=$	$q-2.12763 q^{2} +2.52683 q^{4} +2.10198 q^{5} +0.789580 q^{7} -1.12089 q^{8} +O(q^{10})$	\(q-2.12763 q^{2} +2.52683 q^{4} +2.10198 q^{5} +0.789580 q^{7} -1.12089 q^{8} -4.47224 q^{10} +4.48898 q^{11} -4.31061 q^{13} -1.67994 q^{14} -2.66880 q^{16} -6.64316 q^{17} +3.39928 q^{19} +5.31133 q^{20} -9.55090 q^{22} +9.17525 q^{23} -0.581692 q^{25} +9.17139 q^{26} +1.99513 q^{28} +0.384188 q^{29} +2.78887 q^{31} +7.92002 q^{32} +14.1342 q^{34} +1.65968 q^{35} -0.131728 q^{37} -7.23242 q^{38} -2.35609 q^{40} +8.64441 q^{41} -3.94739 q^{43} +11.3429 q^{44} -19.5216 q^{46} +13.3918 q^{47} -6.37656 q^{49} +1.23763 q^{50} -10.8921 q^{52} +0.625549 q^{53} +9.43573 q^{55} -0.885034 q^{56} -0.817411 q^{58} -7.17980 q^{59} +6.78363 q^{61} -5.93370 q^{62} -11.5133 q^{64} -9.06079 q^{65} +0.301693 q^{67} -16.7861 q^{68} -3.53119 q^{70} +2.81573 q^{71} -2.28817 q^{73} +0.280269 q^{74} +8.58938 q^{76} +3.54441 q^{77} +14.2938 q^{79} -5.60977 q^{80} -18.3921 q^{82} -14.1111 q^{83} -13.9638 q^{85} +8.39860 q^{86} -5.03166 q^{88} +12.1171 q^{89} -3.40357 q^{91} +23.1843 q^{92} -28.4927 q^{94} +7.14521 q^{95} -10.1442 q^{97} +13.5670 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.12763	−1.50446	−0.752232	−	0.658898i	$-0.771023\pi$
$2$	−2.12763	−1.50446	−0.752232	+	0.658898i	$0.771023\pi$
$3$	0	0
$3$	0	0
$4$	2.52683	1.26341
$5$	2.10198	0.940033	0.470016	−	0.882658i	$-0.344248\pi$
$5$	2.10198	0.940033	0.470016	+	0.882658i	$0.344248\pi$
$6$	0	0
$7$	0.789580	0.298433	0.149217	−	0.988805i	$-0.452325\pi$
$7$	0.789580	0.298433	0.149217	+	0.988805i	$0.452325\pi$
$8$	−1.12089	−0.396295
$9$	0	0
$10$	−4.47224	−1.41425
$11$	4.48898	1.35348	0.676739	−	0.736223i	$-0.263393\pi$
$11$	4.48898	1.35348	0.676739	+	0.736223i	$0.263393\pi$
$12$	0	0
$13$	−4.31061	−1.19555	−0.597773	−	0.801665i	$-0.703948\pi$
$13$	−4.31061	−1.19555	−0.597773	+	0.801665i	$0.703948\pi$
$14$	−1.67994	−0.448982
$15$	0	0
$16$	−2.66880	−0.667201
$17$	−6.64316	−1.61120	−0.805602	−	0.592458i	$-0.798158\pi$
$17$	−6.64316	−1.61120	−0.805602	+	0.592458i	$0.798158\pi$
$18$	0	0
$19$	3.39928	0.779848	0.389924	−	0.920847i	$-0.372501\pi$
$19$	3.39928	0.779848	0.389924	+	0.920847i	$0.372501\pi$
$20$	5.31133	1.18765
$21$	0	0
$22$	−9.55090	−2.03626
$23$	9.17525	1.91317	0.956586	−	0.291451i	$-0.0941379\pi$
$23$	9.17525	1.91317	0.956586	+	0.291451i	$0.0941379\pi$
$24$	0	0
$25$	−0.581692	−0.116338
$26$	9.17139	1.79866
$27$	0	0
$28$	1.99513	0.377045
$29$	0.384188	0.0713418	0.0356709	−	0.999364i	$-0.488643\pi$
$29$	0.384188	0.0713418	0.0356709	+	0.999364i	$0.488643\pi$
$30$	0	0
$31$	2.78887	0.500896	0.250448	−	0.968130i	$-0.419422\pi$
$31$	2.78887	0.500896	0.250448	+	0.968130i	$0.419422\pi$
$32$	7.92002	1.40008
$33$	0	0
$34$	14.1342	2.42400
$35$	1.65968	0.280537
$36$	0	0
$37$	−0.131728	−0.0216560	−0.0108280	−	0.999941i	$-0.503447\pi$
$37$	−0.131728	−0.0216560	−0.0108280	+	0.999941i	$0.503447\pi$
$38$	−7.23242	−1.17325
$39$	0	0
$40$	−2.35609	−0.372530
$41$	8.64441	1.35003	0.675015	−	0.737804i	$-0.264137\pi$
$41$	8.64441	1.35003	0.675015	+	0.737804i	$0.264137\pi$
$42$	0	0
$43$	−3.94739	−0.601971	−0.300986	−	0.953629i	$-0.597316\pi$
$43$	−3.94739	−0.601971	−0.300986	+	0.953629i	$0.597316\pi$
$44$	11.3429	1.71000
$45$	0	0
$46$	−19.5216	−2.87830
$47$	13.3918	1.95339	0.976694	−	0.214637i	$-0.0688569\pi$
$47$	13.3918	1.95339	0.976694	+	0.214637i	$0.0688569\pi$
$48$	0	0
$49$	−6.37656	−0.910938
$50$	1.23763	0.175027
$51$	0	0
$52$	−10.8921	−1.51047
$53$	0.625549	0.0859257	0.0429629	−	0.999077i	$-0.486320\pi$
$53$	0.625549	0.0859257	0.0429629	+	0.999077i	$0.486320\pi$
$54$	0	0
$55$	9.43573	1.27231
$56$	−0.885034	−0.118268
$57$	0	0
$58$	−0.817411	−0.107331
$59$	−7.17980	−0.934731	−0.467365	−	0.884064i	$-0.654797\pi$
$59$	−7.17980	−0.934731	−0.467365	+	0.884064i	$0.654797\pi$
$60$	0	0
$61$	6.78363	0.868555	0.434278	−	0.900779i	$-0.357004\pi$
$61$	6.78363	0.868555	0.434278	+	0.900779i	$0.357004\pi$
$62$	−5.93370	−0.753580
$63$	0	0
$64$	−11.5133	−1.43916
$65$	−9.06079	−1.12385
$66$	0	0
$67$	0.301693	0.0368576	0.0184288	−	0.999830i	$-0.494134\pi$
$67$	0.301693	0.0368576	0.0184288	+	0.999830i	$0.494134\pi$
$68$	−16.7861	−2.03561
$69$	0	0
$70$	−3.53119	−0.422058
$71$	2.81573	0.334166	0.167083	−	0.985943i	$-0.446565\pi$
$71$	2.81573	0.334166	0.167083	+	0.985943i	$0.446565\pi$
$72$	0	0
$73$	−2.28817	−0.267810	−0.133905	−	0.990994i	$-0.542752\pi$
$73$	−2.28817	−0.267810	−0.133905	+	0.990994i	$0.542752\pi$
$74$	0.280269	0.0325807
$75$	0	0
$76$	8.58938	0.985270
$77$	3.54441	0.403923
$78$	0	0
$79$	14.2938	1.60818	0.804090	−	0.594508i	$-0.202653\pi$
$79$	14.2938	1.60818	0.804090	+	0.594508i	$0.202653\pi$
$80$	−5.60977	−0.627191
$81$	0	0
$82$	−18.3921	−2.03107
$83$	−14.1111	−1.54890	−0.774450	−	0.632635i	$-0.781973\pi$
$83$	−14.1111	−1.54890	−0.774450	+	0.632635i	$0.781973\pi$
$84$	0	0
$85$	−13.9638	−1.51458
$86$	8.39860	0.905644
$87$	0	0
$88$	−5.03166	−0.536377
$89$	12.1171	1.28441	0.642207	−	0.766531i	$-0.278019\pi$
$89$	12.1171	1.28441	0.642207	+	0.766531i	$0.278019\pi$
$90$	0	0
$91$	−3.40357	−0.356791
$92$	23.1843	2.41713
$93$	0	0
$94$	−28.4927	−2.93880
$95$	7.14521	0.733083
$96$	0	0
$97$	−10.1442	−1.02999	−0.514993	−	0.857194i	$-0.672206\pi$
$97$	−10.1442	−1.02999	−0.514993	+	0.857194i	$0.672206\pi$
$98$	13.5670	1.37047
$99$	0	0
$100$	−1.46984	−0.146984
$101$	4.78785	0.476409	0.238205	−	0.971215i	$-0.423441\pi$
$101$	4.78785	0.476409	0.238205	+	0.971215i	$0.423441\pi$
$102$	0	0
$103$	15.2517	1.50280	0.751399	−	0.659848i	$-0.229379\pi$
$103$	15.2517	1.50280	0.751399	+	0.659848i	$0.229379\pi$
$104$	4.83172	0.473789
$105$	0	0
$106$	−1.33094	−0.129272
$107$	−13.0380	−1.26043	−0.630217	−	0.776419i	$-0.717034\pi$
$107$	−13.0380	−1.26043	−0.630217	+	0.776419i	$0.717034\pi$
$108$	0	0
$109$	6.33358	0.606647	0.303324	−	0.952888i	$-0.401904\pi$
$109$	6.33358	0.606647	0.303324	+	0.952888i	$0.401904\pi$
$110$	−20.0758	−1.91415
$111$	0	0
$112$	−2.10724	−0.199115
$113$	−14.7090	−1.38371	−0.691854	−	0.722038i	$-0.743206\pi$
$113$	−14.7090	−1.38371	−0.691854	+	0.722038i	$0.743206\pi$
$114$	0	0
$115$	19.2862	1.79844
$116$	0.970775	0.0901342
$117$	0	0
$118$	15.2760	1.40627
$119$	−5.24531	−0.480837
$120$	0	0
$121$	9.15094	0.831903
$122$	−14.4331	−1.30671
$123$	0	0
$124$	7.04699	0.632838
$125$	−11.7326	−1.04939
$126$	0	0
$127$	12.3025	1.09167	0.545837	−	0.837892i	$-0.316212\pi$
$127$	12.3025	1.09167	0.545837	+	0.837892i	$0.316212\pi$
$128$	8.65603	0.765093
$129$	0	0
$130$	19.2781	1.69080
$131$	18.2760	1.59678	0.798389	−	0.602142i	$-0.205686\pi$
$131$	18.2760	1.59678	0.798389	+	0.602142i	$0.205686\pi$
$132$	0	0
$133$	2.68400	0.232733
$134$	−0.641892	−0.0554510
$135$	0	0
$136$	7.44626	0.638512
$137$	13.4590	1.14988	0.574938	−	0.818197i	$-0.305026\pi$
$137$	13.4590	1.14988	0.574938	+	0.818197i	$0.305026\pi$
$138$	0	0
$139$	−14.8779	−1.26192	−0.630962	−	0.775814i	$-0.717340\pi$
$139$	−14.8779	−1.26192	−0.630962	+	0.775814i	$0.717340\pi$
$140$	4.19372	0.354434
$141$	0	0
$142$	−5.99084	−0.502740
$143$	−19.3502	−1.61815
$144$	0	0
$145$	0.807554	0.0670637
$146$	4.86838	0.402910
$147$	0	0
$148$	−0.332854	−0.0273604
$149$	−21.7502	−1.78185	−0.890923	−	0.454154i	$-0.849942\pi$
$149$	−21.7502	−1.78185	−0.890923	+	0.454154i	$0.849942\pi$
$150$	0	0
$151$	−9.40234	−0.765151	−0.382576	−	0.923924i	$-0.624963\pi$
$151$	−9.40234	−0.765151	−0.382576	+	0.923924i	$0.624963\pi$
$152$	−3.81022	−0.309050
$153$	0	0
$154$	−7.54121	−0.607688
$155$	5.86214	0.470859
$156$	0	0
$157$	−3.15258	−0.251603	−0.125802	−	0.992055i	$-0.540150\pi$
$157$	−3.15258	−0.251603	−0.125802	+	0.992055i	$0.540150\pi$
$158$	−30.4120	−2.41945
$159$	0	0
$160$	16.6477	1.31612
$161$	7.24460	0.570954
$162$	0	0
$163$	4.24712	0.332660	0.166330	−	0.986070i	$-0.446808\pi$
$163$	4.24712	0.332660	0.166330	+	0.986070i	$0.446808\pi$
$164$	21.8429	1.70564
$165$	0	0
$166$	30.0234	2.33026
$167$	5.46371	0.422794	0.211397	−	0.977400i	$-0.432199\pi$
$167$	5.46371	0.422794	0.211397	+	0.977400i	$0.432199\pi$
$168$	0	0
$169$	5.58132	0.429332
$170$	29.7098	2.27864
$171$	0	0
$172$	−9.97437	−0.760538
$173$	2.70195	0.205426	0.102713	−	0.994711i	$-0.467248\pi$
$173$	2.70195	0.205426	0.102713	+	0.994711i	$0.467248\pi$
$174$	0	0
$175$	−0.459293	−0.0347193
$176$	−11.9802	−0.903042
$177$	0	0
$178$	−25.7808	−1.93235
$179$	21.4365	1.60224	0.801119	−	0.598505i	$-0.204238\pi$
$179$	21.4365	1.60224	0.801119	+	0.598505i	$0.204238\pi$
$180$	0	0
$181$	25.4855	1.89432	0.947159	−	0.320763i	$-0.103939\pi$
$181$	25.4855	1.89432	0.947159	+	0.320763i	$0.103939\pi$
$182$	7.24155	0.536779
$183$	0	0
$184$	−10.2845	−0.758180
$185$	−0.276890	−0.0203573
$186$	0	0
$187$	−29.8210	−2.18073
$188$	33.8386	2.46793
$189$	0	0
$190$	−15.2024	−1.10290
$191$	20.1181	1.45570	0.727849	−	0.685738i	$-0.240520\pi$
$191$	20.1181	1.45570	0.727849	+	0.685738i	$0.240520\pi$
$192$	0	0
$193$	−11.9500	−0.860180	−0.430090	−	0.902786i	$-0.641518\pi$
$193$	−11.9500	−0.860180	−0.430090	+	0.902786i	$0.641518\pi$
$194$	21.5831	1.54958
$195$	0	0
$196$	−16.1125	−1.15089
$197$	−3.53147	−0.251607	−0.125803	−	0.992055i	$-0.540151\pi$
$197$	−3.53147	−0.251607	−0.125803	+	0.992055i	$0.540151\pi$
$198$	0	0
$199$	1.98449	0.140677	0.0703383	−	0.997523i	$-0.477592\pi$
$199$	1.98449	0.140677	0.0703383	+	0.997523i	$0.477592\pi$
$200$	0.652014	0.0461044
$201$	0	0
$202$	−10.1868	−0.716741
$203$	0.303347	0.0212908
$204$	0	0
$205$	18.1703	1.26907
$206$	−32.4501	−2.26091
$207$	0	0
$208$	11.5042	0.797670
$209$	15.2593	1.05551
$210$	0	0
$211$	−7.46169	−0.513684	−0.256842	−	0.966453i	$-0.582682\pi$
$211$	−7.46169	−0.513684	−0.256842	+	0.966453i	$0.582682\pi$
$212$	1.58065	0.108560
$213$	0	0
$214$	27.7402	1.89628
$215$	−8.29732	−0.565873
$216$	0	0
$217$	2.20204	0.149484
$218$	−13.4755	−0.912679
$219$	0	0
$220$	23.8424	1.60746
$221$	28.6360	1.92627
$222$	0	0
$223$	21.8418	1.46264	0.731318	−	0.682036i	$-0.238905\pi$
$223$	21.8418	1.46264	0.731318	+	0.682036i	$0.238905\pi$
$224$	6.25349	0.417829
$225$	0	0
$226$	31.2954	2.08174
$227$	24.6350	1.63508	0.817542	−	0.575869i	$-0.195336\pi$
$227$	24.6350	1.63508	0.817542	+	0.575869i	$0.195336\pi$
$228$	0	0
$229$	−11.4551	−0.756975	−0.378488	−	0.925606i	$-0.623556\pi$
$229$	−11.4551	−0.756975	−0.378488	+	0.925606i	$0.623556\pi$
$230$	−41.0339	−2.70569
$231$	0	0
$232$	−0.430633	−0.0282724
$233$	3.84650	0.251992	0.125996	−	0.992031i	$-0.459787\pi$
$233$	3.84650	0.251992	0.125996	+	0.992031i	$0.459787\pi$
$234$	0	0
$235$	28.1492	1.83625
$236$	−18.1421	−1.18095
$237$	0	0
$238$	11.1601	0.723402
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−18.2874	−1.17799	−0.588997	−	0.808135i	$-0.700477\pi$
$241$	−18.2874	−1.17799	−0.588997	+	0.808135i	$0.700477\pi$
$242$	−19.4698	−1.25157
$243$	0	0
$244$	17.1411	1.09734
$245$	−13.4034	−0.856311
$246$	0	0
$247$	−14.6529	−0.932345
$248$	−3.12602	−0.198503
$249$	0	0
$250$	24.9627	1.57878
$251$	4.28904	0.270722	0.135361	−	0.990796i	$-0.456781\pi$
$251$	4.28904	0.270722	0.135361	+	0.990796i	$0.456781\pi$
$252$	0	0
$253$	41.1875	2.58944
$254$	−26.1753	−1.64238
$255$	0	0
$256$	4.60972	0.288108
$257$	−18.0042	−1.12307	−0.561535	−	0.827453i	$-0.689789\pi$
$257$	−18.0042	−1.12307	−0.561535	+	0.827453i	$0.689789\pi$
$258$	0	0
$259$	−0.104010	−0.00646287
$260$	−22.8950	−1.41989
$261$	0	0
$262$	−38.8845	−2.40229
$263$	7.38769	0.455545	0.227772	−	0.973714i	$-0.426856\pi$
$263$	7.38769	0.455545	0.227772	+	0.973714i	$0.426856\pi$
$264$	0	0
$265$	1.31489	0.0807730
$266$	−5.71058	−0.350138
$267$	0	0
$268$	0.762325	0.0465664
$269$	15.8974	0.969283	0.484641	−	0.874713i	$-0.338950\pi$
$269$	15.8974	0.969283	0.484641	+	0.874713i	$0.338950\pi$
$270$	0	0
$271$	19.7386	1.19903	0.599516	−	0.800363i	$-0.295360\pi$
$271$	19.7386	1.19903	0.599516	+	0.800363i	$0.295360\pi$
$272$	17.7293	1.07500
$273$	0	0
$274$	−28.6357	−1.72995
$275$	−2.61121	−0.157462
$276$	0	0
$277$	29.9558	1.79987	0.899935	−	0.436024i	$-0.143614\pi$
$277$	29.9558	1.79987	0.899935	+	0.436024i	$0.143614\pi$
$278$	31.6547	1.89852
$279$	0	0
$280$	−1.86032	−0.111175
$281$	4.27216	0.254855	0.127428	−	0.991848i	$-0.459328\pi$
$281$	4.27216	0.254855	0.127428	+	0.991848i	$0.459328\pi$
$282$	0	0
$283$	−27.6755	−1.64514	−0.822570	−	0.568664i	$-0.807460\pi$
$283$	−27.6755	−1.64514	−0.822570	+	0.568664i	$0.807460\pi$
$284$	7.11486	0.422189
$285$	0	0
$286$	41.1702	2.43444
$287$	6.82546	0.402894
$288$	0	0
$289$	27.1316	1.59598
$290$	−1.71818	−0.100895
$291$	0	0
$292$	−5.78180	−0.338354
$293$	10.5839	0.618320	0.309160	−	0.951010i	$-0.399952\pi$
$293$	10.5839	0.618320	0.309160	+	0.951010i	$0.399952\pi$
$294$	0	0
$295$	−15.0918	−0.878677
$296$	0.147653	0.00858216
$297$	0	0
$298$	46.2765	2.68072
$299$	−39.5509	−2.28729
$300$	0	0
$301$	−3.11678	−0.179648
$302$	20.0047	1.15114
$303$	0	0
$304$	−9.07201	−0.520315
$305$	14.2590	0.816470
$306$	0	0
$307$	25.6078	1.46152	0.730758	−	0.682637i	$-0.239167\pi$
$307$	25.6078	1.46152	0.730758	+	0.682637i	$0.239167\pi$
$308$	8.95611	0.510322
$309$	0	0
$310$	−12.4725	−0.708390
$311$	−13.8656	−0.786246	−0.393123	−	0.919486i	$-0.628605\pi$
$311$	−13.8656	−0.786246	−0.393123	+	0.919486i	$0.628605\pi$
$312$	0	0
$313$	4.30573	0.243374	0.121687	−	0.992569i	$-0.461170\pi$
$313$	4.30573	0.243374	0.121687	+	0.992569i	$0.461170\pi$
$314$	6.70753	0.378528
$315$	0	0
$316$	36.1180	2.03179
$317$	2.12732	0.119482	0.0597410	−	0.998214i	$-0.480973\pi$
$317$	2.12732	0.119482	0.0597410	+	0.998214i	$0.480973\pi$
$318$	0	0
$319$	1.72461	0.0965596
$320$	−24.2007	−1.35286
$321$	0	0
$322$	−15.4139	−0.858980
$323$	−22.5820	−1.25649
$324$	0	0
$325$	2.50745	0.139088
$326$	−9.03631	−0.500475
$327$	0	0
$328$	−9.68944	−0.535010
$329$	10.5739	0.582956
$330$	0	0
$331$	−13.3771	−0.735269	−0.367635	−	0.929970i	$-0.619832\pi$
$331$	−13.3771	−0.735269	−0.367635	+	0.929970i	$0.619832\pi$
$332$	−35.6564	−1.95690
$333$	0	0
$334$	−11.6248	−0.636079
$335$	0.634151	0.0346474
$336$	0	0
$337$	12.2307	0.666247	0.333123	−	0.942883i	$-0.391897\pi$
$337$	12.2307	0.666247	0.333123	+	0.942883i	$0.391897\pi$
$338$	−11.8750	−0.645915
$339$	0	0
$340$	−35.2840	−1.91354
$341$	12.5192	0.677952
$342$	0	0
$343$	−10.5619	−0.570288
$344$	4.42460	0.238558
$345$	0	0
$346$	−5.74876	−0.309055
$347$	−16.4824	−0.884820	−0.442410	−	0.896813i	$-0.645876\pi$
$347$	−16.4824	−0.884820	−0.442410	+	0.896813i	$0.645876\pi$
$348$	0	0
$349$	−7.08592	−0.379301	−0.189650	−	0.981852i	$-0.560735\pi$
$349$	−7.08592	−0.379301	−0.189650	+	0.981852i	$0.560735\pi$
$350$	0.977207	0.0522339
$351$	0	0
$352$	35.5528	1.89497
$353$	33.6523	1.79113	0.895567	−	0.444927i	$-0.146770\pi$
$353$	33.6523	1.79113	0.895567	+	0.444927i	$0.146770\pi$
$354$	0	0
$355$	5.91860	0.314127
$356$	30.6179	1.62274
$357$	0	0
$358$	−45.6090	−2.41051
$359$	−11.2250	−0.592435	−0.296218	−	0.955120i	$-0.595725\pi$
$359$	−11.2250	−0.592435	−0.296218	+	0.955120i	$0.595725\pi$
$360$	0	0
$361$	−7.44491	−0.391837
$362$	−54.2237	−2.84994
$363$	0	0
$364$	−8.60023	−0.450774
$365$	−4.80968	−0.251750
$366$	0	0
$367$	9.75843	0.509386	0.254693	−	0.967022i	$-0.418026\pi$
$367$	9.75843	0.509386	0.254693	+	0.967022i	$0.418026\pi$
$368$	−24.4869	−1.27647
$369$	0	0
$370$	0.589120	0.0306269
$371$	0.493921	0.0256431
$372$	0	0
$373$	4.61889	0.239157	0.119579	−	0.992825i	$-0.461846\pi$
$373$	4.61889	0.239157	0.119579	+	0.992825i	$0.461846\pi$
$374$	63.4482	3.28083
$375$	0	0
$376$	−15.0107	−0.774118
$377$	−1.65608	−0.0852925
$378$	0	0
$379$	27.8720	1.43169	0.715845	−	0.698259i	$-0.246042\pi$
$379$	27.8720	1.43169	0.715845	+	0.698259i	$0.246042\pi$
$380$	18.0547	0.926186
$381$	0	0
$382$	−42.8040	−2.19005
$383$	−22.8580	−1.16799	−0.583994	−	0.811758i	$-0.698511\pi$
$383$	−22.8580	−1.16799	−0.583994	+	0.811758i	$0.698511\pi$
$384$	0	0
$385$	7.45027	0.379701
$386$	25.4252	1.29411
$387$	0	0
$388$	−25.6326	−1.30130
$389$	−29.2780	−1.48445	−0.742227	−	0.670149i	$-0.766230\pi$
$389$	−29.2780	−1.48445	−0.742227	+	0.670149i	$0.766230\pi$
$390$	0	0
$391$	−60.9527	−3.08251
$392$	7.14743	0.361000
$393$	0	0
$394$	7.51367	0.378533
$395$	30.0453	1.51174
$396$	0	0
$397$	−32.1482	−1.61347	−0.806735	−	0.590913i	$-0.798768\pi$
$397$	−32.1482	−1.61347	−0.806735	+	0.590913i	$0.798768\pi$
$398$	−4.22226	−0.211643
$399$	0	0
$400$	1.55242	0.0776212
$401$	−8.54317	−0.426626	−0.213313	−	0.976984i	$-0.568425\pi$
$401$	−8.54317	−0.426626	−0.213313	+	0.976984i	$0.568425\pi$
$402$	0	0
$403$	−12.0217	−0.598845
$404$	12.0981	0.601902
$405$	0	0
$406$	−0.645411	−0.0320312
$407$	−0.591325	−0.0293109
$408$	0	0
$409$	−4.37441	−0.216301	−0.108150	−	0.994135i	$-0.534493\pi$
$409$	−4.37441	−0.216301	−0.108150	+	0.994135i	$0.534493\pi$
$410$	−38.6598	−1.90927
$411$	0	0
$412$	38.5385	1.89865
$413$	−5.66903	−0.278955
$414$	0	0
$415$	−29.6613	−1.45602
$416$	−34.1401	−1.67386
$417$	0	0
$418$	−32.4662	−1.58797
$419$	−6.49460	−0.317282	−0.158641	−	0.987336i	$-0.550711\pi$
$419$	−6.49460	−0.317282	−0.158641	+	0.987336i	$0.550711\pi$
$420$	0	0
$421$	16.6912	0.813481	0.406741	−	0.913544i	$-0.366665\pi$
$421$	16.6912	0.813481	0.406741	+	0.913544i	$0.366665\pi$
$422$	15.8757	0.772819
$423$	0	0
$424$	−0.701172	−0.0340519
$425$	3.86428	0.187445
$426$	0	0
$427$	5.35623	0.259206
$428$	−32.9448	−1.59245
$429$	0	0
$430$	17.6537	0.851335
$431$	−13.7066	−0.660226	−0.330113	−	0.943941i	$-0.607087\pi$
$431$	−13.7066	−0.660226	−0.330113	+	0.943941i	$0.607087\pi$
$432$	0	0
$433$	−21.1110	−1.01453	−0.507264	−	0.861791i	$-0.669343\pi$
$433$	−21.1110	−1.01453	−0.507264	+	0.861791i	$0.669343\pi$
$434$	−4.68513	−0.224893
$435$	0	0
$436$	16.0039	0.766446
$437$	31.1892	1.49198
$438$	0	0
$439$	−36.5659	−1.74520	−0.872598	−	0.488438i	$-0.837567\pi$
$439$	−36.5659	−1.74520	−0.872598	+	0.488438i	$0.837567\pi$
$440$	−10.5764	−0.504212
$441$	0	0
$442$	−60.9270	−2.89800
$443$	2.61320	0.124157	0.0620785	−	0.998071i	$-0.480227\pi$
$443$	2.61320	0.124157	0.0620785	+	0.998071i	$0.480227\pi$
$444$	0	0
$445$	25.4699	1.20739
$446$	−46.4714	−2.20048
$447$	0	0
$448$	−9.09067	−0.429494
$449$	21.8573	1.03151	0.515756	−	0.856736i	$-0.327511\pi$
$449$	21.8573	1.03151	0.515756	+	0.856736i	$0.327511\pi$
$450$	0	0
$451$	38.8046	1.82724
$452$	−37.1671	−1.74819
$453$	0	0
$454$	−52.4143	−2.45993
$455$	−7.15423	−0.335395
$456$	0	0
$457$	−33.7368	−1.57814	−0.789070	−	0.614304i	$-0.789437\pi$
$457$	−33.7368	−1.57814	−0.789070	+	0.614304i	$0.789437\pi$
$458$	24.3723	1.13884
$459$	0	0
$460$	48.7328	2.27218
$461$	−12.2555	−0.570795	−0.285397	−	0.958409i	$-0.592126\pi$
$461$	−12.2555	−0.570795	−0.285397	+	0.958409i	$0.592126\pi$
$462$	0	0
$463$	−40.6275	−1.88812	−0.944061	−	0.329772i	$-0.893028\pi$
$463$	−40.6275	−1.88812	−0.944061	+	0.329772i	$0.893028\pi$
$464$	−1.02532	−0.0475994
$465$	0	0
$466$	−8.18394	−0.379113
$467$	10.4900	0.485419	0.242710	−	0.970099i	$-0.421964\pi$
$467$	10.4900	0.485419	0.242710	+	0.970099i	$0.421964\pi$
$468$	0	0
$469$	0.238211	0.0109995
$470$	−59.8911	−2.76257
$471$	0	0
$472$	8.04778	0.370429
$473$	−17.7198	−0.814755
$474$	0	0
$475$	−1.97733	−0.0907263
$476$	−13.2540	−0.607495
$477$	0	0
$478$	−2.12763	−0.0973157
$479$	−27.6348	−1.26267	−0.631333	−	0.775512i	$-0.717492\pi$
$479$	−27.6348	−1.26267	−0.631333	+	0.775512i	$0.717492\pi$
$480$	0	0
$481$	0.567828	0.0258907
$482$	38.9089	1.77225
$483$	0	0
$484$	23.1228	1.05104
$485$	−21.3229	−0.968221
$486$	0	0
$487$	−17.4271	−0.789696	−0.394848	−	0.918746i	$-0.629203\pi$
$487$	−17.4271	−0.789696	−0.394848	+	0.918746i	$0.629203\pi$
$488$	−7.60372	−0.344204
$489$	0	0
$490$	28.5175	1.28829
$491$	30.0841	1.35768	0.678838	−	0.734288i	$-0.262484\pi$
$491$	30.0841	1.35768	0.678838	+	0.734288i	$0.262484\pi$
$492$	0	0
$493$	−2.55222	−0.114946
$494$	31.1761	1.40268
$495$	0	0
$496$	−7.44295	−0.334198
$497$	2.22324	0.0997262
$498$	0	0
$499$	11.5301	0.516157	0.258079	−	0.966124i	$-0.416911\pi$
$499$	11.5301	0.516157	0.258079	+	0.966124i	$0.416911\pi$
$500$	−29.6462	−1.32582
$501$	0	0
$502$	−9.12552	−0.407292
$503$	7.70067	0.343356	0.171678	−	0.985153i	$-0.445081\pi$
$503$	7.70067	0.343356	0.171678	+	0.985153i	$0.445081\pi$
$504$	0	0
$505$	10.0640	0.447840
$506$	−87.6319	−3.89571
$507$	0	0
$508$	31.0864	1.37923
$509$	33.1324	1.46857	0.734284	−	0.678842i	$-0.237518\pi$
$509$	33.1324	1.46857	0.734284	+	0.678842i	$0.237518\pi$
$510$	0	0
$511$	−1.80669	−0.0799234
$512$	−27.1199	−1.19854
$513$	0	0
$514$	38.3063	1.68962
$515$	32.0588	1.41268
$516$	0	0
$517$	60.1153	2.64387
$518$	0.221295	0.00972315
$519$	0	0
$520$	10.1562	0.445377
$521$	36.2158	1.58664	0.793321	−	0.608804i	$-0.208350\pi$
$521$	36.2158	1.58664	0.793321	+	0.608804i	$0.208350\pi$
$522$	0	0
$523$	−13.3239	−0.582613	−0.291306	−	0.956630i	$-0.594090\pi$
$523$	−13.3239	−0.582613	−0.291306	+	0.956630i	$0.594090\pi$
$524$	46.1801	2.01739
$525$	0	0
$526$	−15.7183	−0.685351
$527$	−18.5269	−0.807045
$528$	0	0
$529$	61.1852	2.66023
$530$	−2.79760	−0.121520
$531$	0	0
$532$	6.78201	0.294037
$533$	−37.2626	−1.61402
$534$	0	0
$535$	−27.4056	−1.18485
$536$	−0.338165	−0.0146065
$537$	0	0
$538$	−33.8239	−1.45825
$539$	−28.6243	−1.23293
$540$	0	0
$541$	−15.2808	−0.656972	−0.328486	−	0.944509i	$-0.606538\pi$
$541$	−15.2808	−0.656972	−0.328486	+	0.944509i	$0.606538\pi$
$542$	−41.9964	−1.80390
$543$	0	0
$544$	−52.6140	−2.25581
$545$	13.3130	0.570268
$546$	0	0
$547$	−24.9953	−1.06872	−0.534362	−	0.845256i	$-0.679448\pi$
$547$	−24.9953	−1.06872	−0.534362	+	0.845256i	$0.679448\pi$
$548$	34.0084	1.45277
$549$	0	0
$550$	5.55569	0.236895
$551$	1.30596	0.0556358
$552$	0	0
$553$	11.2861	0.479935
$554$	−63.7350	−2.70784
$555$	0	0
$556$	−37.5938	−1.59433
$557$	12.6114	0.534360	0.267180	−	0.963647i	$-0.413908\pi$
$557$	12.6114	0.534360	0.267180	+	0.963647i	$0.413908\pi$
$558$	0	0
$559$	17.0156	0.719685
$560$	−4.42936	−0.187175
$561$	0	0
$562$	−9.08958	−0.383421
$563$	−0.586176	−0.0247044	−0.0123522	−	0.999924i	$-0.503932\pi$
$563$	−0.586176	−0.0247044	−0.0123522	+	0.999924i	$0.503932\pi$
$564$	0	0
$565$	−30.9180	−1.30073
$566$	58.8834	2.47505
$567$	0	0
$568$	−3.15613	−0.132428
$569$	41.0817	1.72224	0.861118	−	0.508406i	$-0.169765\pi$
$569$	41.0817	1.72224	0.861118	+	0.508406i	$0.169765\pi$
$570$	0	0
$571$	−7.37931	−0.308814	−0.154407	−	0.988007i	$-0.549347\pi$
$571$	−7.37931	−0.308814	−0.154407	+	0.988007i	$0.549347\pi$
$572$	−48.8946	−2.04439
$573$	0	0
$574$	−14.5221	−0.606140
$575$	−5.33717	−0.222576
$576$	0	0
$577$	−0.852487	−0.0354895	−0.0177447	−	0.999843i	$-0.505649\pi$
$577$	−0.852487	−0.0354895	−0.0177447	+	0.999843i	$0.505649\pi$
$578$	−57.7261	−2.40109
$579$	0	0
$580$	2.04055	0.0847291
$581$	−11.1419	−0.462243
$582$	0	0
$583$	2.80807	0.116299
$584$	2.56479	0.106132
$585$	0	0
$586$	−22.5187	−0.930240
$587$	24.7875	1.02309	0.511544	−	0.859257i	$-0.329074\pi$
$587$	24.7875	1.02309	0.511544	+	0.859257i	$0.329074\pi$
$588$	0	0
$589$	9.48015	0.390623
$590$	32.1098	1.32194
$591$	0	0
$592$	0.351557	0.0144489
$593$	40.5797	1.66641	0.833204	−	0.552966i	$-0.186504\pi$
$593$	40.5797	1.66641	0.833204	+	0.552966i	$0.186504\pi$
$594$	0	0
$595$	−11.0255	−0.452002
$596$	−54.9590	−2.25121
$597$	0	0
$598$	84.1498	3.44114
$599$	−21.9846	−0.898267	−0.449134	−	0.893465i	$-0.648267\pi$
$599$	−21.9846	−0.898267	−0.449134	+	0.893465i	$0.648267\pi$
$600$	0	0
$601$	11.3425	0.462669	0.231335	−	0.972874i	$-0.425691\pi$
$601$	11.3425	0.462669	0.231335	+	0.972874i	$0.425691\pi$
$602$	6.63137	0.270275
$603$	0	0
$604$	−23.7581	−0.966702
$605$	19.2351	0.782016
$606$	0	0
$607$	−2.12766	−0.0863590	−0.0431795	−	0.999067i	$-0.513749\pi$
$607$	−2.12766	−0.0863590	−0.0431795	+	0.999067i	$0.513749\pi$
$608$	26.9224	1.09185
$609$	0	0
$610$	−30.3380	−1.22835
$611$	−57.7266	−2.33537
$612$	0	0
$613$	−45.6639	−1.84435	−0.922175	−	0.386773i	$-0.873590\pi$
$613$	−45.6639	−1.84435	−0.922175	+	0.386773i	$0.873590\pi$
$614$	−54.4841	−2.19880
$615$	0	0
$616$	−3.97290	−0.160073
$617$	10.0287	0.403742	0.201871	−	0.979412i	$-0.435298\pi$
$617$	10.0287	0.403742	0.201871	+	0.979412i	$0.435298\pi$
$618$	0	0
$619$	−4.86175	−0.195410	−0.0977052	−	0.995215i	$-0.531150\pi$
$619$	−4.86175	−0.195410	−0.0977052	+	0.995215i	$0.531150\pi$
$620$	14.8126	0.594889
$621$	0	0
$622$	29.5009	1.18288
$623$	9.56745	0.383312
$624$	0	0
$625$	−21.7532	−0.870127
$626$	−9.16101	−0.366148
$627$	0	0
$628$	−7.96602	−0.317879
$629$	0.875092	0.0348922
$630$	0	0
$631$	−28.5095	−1.13495	−0.567473	−	0.823392i	$-0.692079\pi$
$631$	−28.5095	−1.13495	−0.567473	+	0.823392i	$0.692079\pi$
$632$	−16.0218	−0.637314
$633$	0	0
$634$	−4.52615	−0.179756
$635$	25.8596	1.02621
$636$	0	0
$637$	27.4868	1.08907
$638$	−3.66934	−0.145271
$639$	0	0
$640$	18.1948	0.719212
$641$	19.0522	0.752517	0.376258	−	0.926515i	$-0.377210\pi$
$641$	19.0522	0.752517	0.376258	+	0.926515i	$0.377210\pi$
$642$	0	0
$643$	16.7496	0.660539	0.330269	−	0.943887i	$-0.392860\pi$
$643$	16.7496	0.660539	0.330269	+	0.943887i	$0.392860\pi$
$644$	18.3058	0.721351
$645$	0	0
$646$	48.0461	1.89035
$647$	30.9651	1.21736	0.608682	−	0.793414i	$-0.291698\pi$
$647$	30.9651	1.21736	0.608682	+	0.793414i	$0.291698\pi$
$648$	0	0
$649$	−32.2300	−1.26514
$650$	−5.33493	−0.209253
$651$	0	0
$652$	10.7317	0.420287
$653$	−36.1207	−1.41351	−0.706756	−	0.707458i	$-0.749842\pi$
$653$	−36.1207	−1.41351	−0.706756	+	0.707458i	$0.749842\pi$
$654$	0	0
$655$	38.4156	1.50102
$656$	−23.0702	−0.900741
$657$	0	0
$658$	−22.4973	−0.877037
$659$	−23.2340	−0.905069	−0.452535	−	0.891747i	$-0.649480\pi$
$659$	−23.2340	−0.905069	−0.452535	+	0.891747i	$0.649480\pi$
$660$	0	0
$661$	46.5878	1.81206	0.906028	−	0.423218i	$-0.139100\pi$
$661$	46.5878	1.81206	0.906028	+	0.423218i	$0.139100\pi$
$662$	28.4615	1.10619
$663$	0	0
$664$	15.8171	0.613821
$665$	5.64171	0.218776
$666$	0	0
$667$	3.52502	0.136489
$668$	13.8058	0.534164
$669$	0	0
$670$	−1.34924	−0.0521257
$671$	30.4516	1.17557
$672$	0	0
$673$	−5.17261	−0.199390	−0.0996948	−	0.995018i	$-0.531787\pi$
$673$	−5.17261	−0.199390	−0.0996948	+	0.995018i	$0.531787\pi$
$674$	−26.0224	−1.00234
$675$	0	0
$676$	14.1030	0.542424
$677$	14.8722	0.571584	0.285792	−	0.958292i	$-0.407743\pi$
$677$	14.8722	0.571584	0.285792	+	0.958292i	$0.407743\pi$
$678$	0	0
$679$	−8.00966	−0.307382
$680$	15.6519	0.600222
$681$	0	0
$682$	−26.6362	−1.01995
$683$	−47.9804	−1.83592	−0.917960	−	0.396672i	$-0.870165\pi$
$683$	−47.9804	−1.83592	−0.917960	+	0.396672i	$0.870165\pi$
$684$	0	0
$685$	28.2904	1.08092
$686$	22.4718	0.857977
$687$	0	0
$688$	10.5348	0.401636
$689$	−2.69649	−0.102728
$690$	0	0
$691$	−10.4497	−0.397524	−0.198762	−	0.980048i	$-0.563692\pi$
$691$	−10.4497	−0.397524	−0.198762	+	0.980048i	$0.563692\pi$
$692$	6.82736	0.259537
$693$	0	0
$694$	35.0684	1.33118
$695$	−31.2729	−1.18625
$696$	0	0
$697$	−57.4262	−2.17517
$698$	15.0762	0.570644
$699$	0	0
$700$	−1.16055	−0.0438648
$701$	−22.4554	−0.848129	−0.424065	−	0.905632i	$-0.639397\pi$
$701$	−22.4554	−0.848129	−0.424065	+	0.905632i	$0.639397\pi$
$702$	0	0
$703$	−0.447781	−0.0168884
$704$	−51.6829	−1.94787
$705$	0	0
$706$	−71.5999	−2.69470
$707$	3.78040	0.142176
$708$	0	0
$709$	−2.79774	−0.105071	−0.0525356	−	0.998619i	$-0.516730\pi$
$709$	−2.79774	−0.105071	−0.0525356	+	0.998619i	$0.516730\pi$
$710$	−12.5926	−0.472592
$711$	0	0
$712$	−13.5820	−0.509007
$713$	25.5886	0.958300
$714$	0	0
$715$	−40.6737	−1.52111
$716$	54.1662	2.02429
$717$	0	0
$718$	23.8828	0.891298
$719$	−15.6650	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$719$	−15.6650	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$720$	0	0
$721$	12.0425	0.448485
$722$	15.8400	0.589505
$723$	0	0
$724$	64.3973	2.39331
$725$	−0.223479	−0.00829980
$726$	0	0
$727$	15.9906	0.593058	0.296529	−	0.955024i	$-0.404171\pi$
$727$	15.9906	0.593058	0.296529	+	0.955024i	$0.404171\pi$
$728$	3.81503	0.141395
$729$	0	0
$730$	10.2332	0.378749
$731$	26.2232	0.969898
$732$	0	0
$733$	−51.6646	−1.90828	−0.954138	−	0.299369i	$-0.903224\pi$
$733$	−51.6646	−1.90828	−0.954138	+	0.299369i	$0.903224\pi$
$734$	−20.7624	−0.766353
$735$	0	0
$736$	72.6682	2.67858
$737$	1.35429	0.0498860
$738$	0	0
$739$	−19.0256	−0.699866	−0.349933	−	0.936775i	$-0.613796\pi$
$739$	−19.0256	−0.699866	−0.349933	+	0.936775i	$0.613796\pi$
$740$	−0.699652	−0.0257197
$741$	0	0
$742$	−1.05088	−0.0385791
$743$	−12.4637	−0.457247	−0.228624	−	0.973515i	$-0.573422\pi$
$743$	−12.4637	−0.457247	−0.228624	+	0.973515i	$0.573422\pi$
$744$	0	0
$745$	−45.7184	−1.67499
$746$	−9.82731	−0.359803
$747$	0	0
$748$	−75.3525	−2.75516
$749$	−10.2946	−0.376156
$750$	0	0
$751$	10.9426	0.399303	0.199651	−	0.979867i	$-0.436019\pi$
$751$	10.9426	0.399303	0.199651	+	0.979867i	$0.436019\pi$
$752$	−35.7400	−1.30330
$753$	0	0
$754$	3.52353	0.128320
$755$	−19.7635	−0.719267
$756$	0	0
$757$	−4.14204	−0.150545	−0.0752725	−	0.997163i	$-0.523983\pi$
$757$	−4.14204	−0.150545	−0.0752725	+	0.997163i	$0.523983\pi$
$758$	−59.3015	−2.15393
$759$	0	0
$760$	−8.00900	−0.290517
$761$	24.2607	0.879449	0.439725	−	0.898133i	$-0.355076\pi$
$761$	24.2607	0.879449	0.439725	+	0.898133i	$0.355076\pi$
$762$	0	0
$763$	5.00087	0.181044
$764$	50.8350	1.83915
$765$	0	0
$766$	48.6334	1.75720
$767$	30.9493	1.11751
$768$	0	0
$769$	−4.60195	−0.165951	−0.0829753	−	0.996552i	$-0.526442\pi$
$769$	−4.60195	−0.165951	−0.0829753	+	0.996552i	$0.526442\pi$
$770$	−15.8514	−0.571246
$771$	0	0
$772$	−30.1956	−1.08676
$773$	−24.5840	−0.884227	−0.442113	−	0.896959i	$-0.645771\pi$
$773$	−24.5840	−0.884227	−0.442113	+	0.896959i	$0.645771\pi$
$774$	0	0
$775$	−1.62227	−0.0582735
$776$	11.3705	0.408179
$777$	0	0
$778$	62.2929	2.23331
$779$	29.3848	1.05282
$780$	0	0
$781$	12.6397	0.452286
$782$	129.685	4.63752
$783$	0	0
$784$	17.0178	0.607778
$785$	−6.62665	−0.236515
$786$	0	0
$787$	10.5232	0.375110	0.187555	−	0.982254i	$-0.439944\pi$
$787$	10.5232	0.375110	0.187555	+	0.982254i	$0.439944\pi$
$788$	−8.92340	−0.317883
$789$	0	0
$790$	−63.9253	−2.27436
$791$	−11.6140	−0.412945
$792$	0	0
$793$	−29.2416	−1.03840
$794$	68.3995	2.42741
$795$	0	0
$796$	5.01445	0.177733
$797$	−14.8664	−0.526594	−0.263297	−	0.964715i	$-0.584810\pi$
$797$	−14.8664	−0.526594	−0.263297	+	0.964715i	$0.584810\pi$
$798$	0	0
$799$	−88.9636	−3.14730
$800$	−4.60702	−0.162883
$801$	0	0
$802$	18.1767	0.641843
$803$	−10.2715	−0.362475
$804$	0	0
$805$	15.2280	0.536716
$806$	25.5778	0.900940
$807$	0	0
$808$	−5.36666	−0.188799
$809$	−37.7784	−1.32822	−0.664108	−	0.747636i	$-0.731189\pi$
$809$	−37.7784	−1.32822	−0.664108	+	0.747636i	$0.731189\pi$
$810$	0	0
$811$	17.9591	0.630629	0.315315	−	0.948987i	$-0.397890\pi$
$811$	17.9591	0.630629	0.315315	+	0.948987i	$0.397890\pi$
$812$	0.766505	0.0268991
$813$	0	0
$814$	1.25812	0.0440972
$815$	8.92735	0.312711
$816$	0	0
$817$	−13.4183	−0.469446
$818$	9.30714	0.325417
$819$	0	0
$820$	45.9133	1.60336
$821$	35.4571	1.23746	0.618730	−	0.785604i	$-0.287648\pi$
$821$	35.4571	1.23746	0.618730	+	0.785604i	$0.287648\pi$
$822$	0	0
$823$	−8.61910	−0.300443	−0.150221	−	0.988652i	$-0.547999\pi$
$823$	−8.61910	−0.300443	−0.150221	+	0.988652i	$0.547999\pi$
$824$	−17.0955	−0.595551
$825$	0	0
$826$	12.0616	0.419678
$827$	−17.3968	−0.604948	−0.302474	−	0.953158i	$-0.597812\pi$
$827$	−17.3968	−0.604948	−0.302474	+	0.953158i	$0.597812\pi$
$828$	0	0
$829$	−0.772910	−0.0268443	−0.0134221	−	0.999910i	$-0.504273\pi$
$829$	−0.772910	−0.0268443	−0.0134221	+	0.999910i	$0.504273\pi$
$830$	63.1084	2.19052
$831$	0	0
$832$	49.6293	1.72059
$833$	42.3605	1.46771
$834$	0	0
$835$	11.4846	0.397441
$836$	38.5576	1.33354
$837$	0	0
$838$	13.8181	0.477339
$839$	26.3653	0.910230	0.455115	−	0.890433i	$-0.349598\pi$
$839$	26.3653	0.910230	0.455115	+	0.890433i	$0.349598\pi$
$840$	0	0
$841$	−28.8524	−0.994910
$842$	−35.5129	−1.22385
$843$	0	0
$844$	−18.8544	−0.648995
$845$	11.7318	0.403586
$846$	0	0
$847$	7.22540	0.248268
$848$	−1.66947	−0.0573297
$849$	0	0
$850$	−8.22177	−0.282004
$851$	−1.20864	−0.0414316
$852$	0	0
$853$	−0.382069	−0.0130818	−0.00654089	−	0.999979i	$-0.502082\pi$
$853$	−0.382069	−0.0130818	−0.00654089	+	0.999979i	$0.502082\pi$
$854$	−11.3961	−0.389966
$855$	0	0
$856$	14.6142	0.499504
$857$	−40.4862	−1.38298	−0.691492	−	0.722384i	$-0.743046\pi$
$857$	−40.4862	−1.38298	−0.691492	+	0.722384i	$0.743046\pi$
$858$	0	0
$859$	−4.24423	−0.144811	−0.0724056	−	0.997375i	$-0.523068\pi$
$859$	−4.24423	−0.144811	−0.0724056	+	0.997375i	$0.523068\pi$
$860$	−20.9659	−0.714931
$861$	0	0
$862$	29.1627	0.993286
$863$	−1.91165	−0.0650733	−0.0325367	−	0.999471i	$-0.510359\pi$
$863$	−1.91165	−0.0650733	−0.0325367	+	0.999471i	$0.510359\pi$
$864$	0	0
$865$	5.67944	0.193107
$866$	44.9164	1.52632
$867$	0	0
$868$	5.56417	0.188860
$869$	64.1646	2.17664
$870$	0	0
$871$	−1.30048	−0.0440650
$872$	−7.09926	−0.240411
$873$	0	0
$874$	−66.3593	−2.24464
$875$	−9.26382	−0.313174
$876$	0	0
$877$	26.1285	0.882298	0.441149	−	0.897434i	$-0.354571\pi$
$877$	26.1285	0.882298	0.441149	+	0.897434i	$0.354571\pi$
$878$	77.7989	2.62559
$879$	0	0
$880$	−25.1821	−0.848889
$881$	24.0315	0.809642	0.404821	−	0.914396i	$-0.367334\pi$
$881$	24.0315	0.809642	0.404821	+	0.914396i	$0.367334\pi$
$882$	0	0
$883$	21.4066	0.720389	0.360194	−	0.932877i	$-0.382710\pi$
$883$	21.4066	0.720389	0.360194	+	0.932877i	$0.382710\pi$
$884$	72.3583	2.43367
$885$	0	0
$886$	−5.55993	−0.186790
$887$	−14.2082	−0.477066	−0.238533	−	0.971134i	$-0.576666\pi$
$887$	−14.2082	−0.477066	−0.238533	+	0.971134i	$0.576666\pi$
$888$	0	0
$889$	9.71384	0.325792
$890$	−54.1907	−1.81648
$891$	0	0
$892$	55.1905	1.84791
$893$	45.5223	1.52335
$894$	0	0
$895$	45.0590	1.50616
$896$	6.83464	0.228329
$897$	0	0
$898$	−46.5044	−1.55187
$899$	1.07145	0.0357348
$900$	0	0
$901$	−4.15562	−0.138444
$902$	−82.5619	−2.74901
$903$	0	0
$904$	16.4872	0.548356
$905$	53.5698	1.78072
$906$	0	0
$907$	19.2537	0.639310	0.319655	−	0.947534i	$-0.396433\pi$
$907$	19.2537	0.639310	0.319655	+	0.947534i	$0.396433\pi$
$908$	62.2484	2.06579
$909$	0	0
$910$	15.2216	0.504590
$911$	−44.9443	−1.48907	−0.744536	−	0.667583i	$-0.767329\pi$
$911$	−44.9443	−1.48907	−0.744536	+	0.667583i	$0.767329\pi$
$912$	0	0
$913$	−63.3447	−2.09640
$914$	71.7795	2.37425
$915$	0	0
$916$	−28.9451	−0.956372
$917$	14.4303	0.476532
$918$	0	0
$919$	43.4424	1.43303	0.716516	−	0.697571i	$-0.245736\pi$
$919$	43.4424	1.43303	0.716516	+	0.697571i	$0.245736\pi$
$920$	−21.6177	−0.712714
$921$	0	0
$922$	26.0752	0.858740
$923$	−12.1375	−0.399511
$924$	0	0
$925$	0.0766253	0.00251942
$926$	86.4405	2.84061
$927$	0	0
$928$	3.04277	0.0998839
$929$	−1.23801	−0.0406177	−0.0203089	−	0.999794i	$-0.506465\pi$
$929$	−1.23801	−0.0406177	−0.0203089	+	0.999794i	$0.506465\pi$
$930$	0	0
$931$	−21.6757	−0.710393
$932$	9.71942	0.318370
$933$	0	0
$934$	−22.3189	−0.730296
$935$	−62.6831	−2.04996
$936$	0	0
$937$	−42.3900	−1.38482	−0.692410	−	0.721504i	$-0.743451\pi$
$937$	−42.3900	−1.38482	−0.692410	+	0.721504i	$0.743451\pi$
$938$	−0.506825	−0.0165484
$939$	0	0
$940$	71.1280	2.31994
$941$	−8.79386	−0.286672	−0.143336	−	0.989674i	$-0.545783\pi$
$941$	−8.79386	−0.286672	−0.143336	+	0.989674i	$0.545783\pi$
$942$	0	0
$943$	79.3146	2.58284
$944$	19.1615	0.623653
$945$	0	0
$946$	37.7011	1.22577
$947$	−46.4504	−1.50943	−0.754717	−	0.656051i	$-0.772226\pi$
$947$	−46.4504	−1.50943	−0.754717	+	0.656051i	$0.772226\pi$
$948$	0	0
$949$	9.86339	0.320179
$950$	4.20704	0.136495
$951$	0	0
$952$	5.87942	0.190553
$953$	−23.4320	−0.759038	−0.379519	−	0.925184i	$-0.623911\pi$
$953$	−23.4320	−0.759038	−0.379519	+	0.925184i	$0.623911\pi$
$954$	0	0
$955$	42.2879	1.36840
$956$	2.52683	0.0817234
$957$	0	0
$958$	58.7967	1.89963
$959$	10.6269	0.343161
$960$	0	0
$961$	−23.2222	−0.749103
$962$	−1.20813	−0.0389517
$963$	0	0
$964$	−46.2091	−1.48829
$965$	−25.1186	−0.808597
$966$	0	0
$967$	−30.8007	−0.990483	−0.495242	−	0.868755i	$-0.664921\pi$
$967$	−30.8007	−0.990483	−0.495242	+	0.868755i	$0.664921\pi$
$968$	−10.2572	−0.329679
$969$	0	0
$970$	45.3672	1.45665
$971$	43.0309	1.38093	0.690464	−	0.723367i	$-0.257406\pi$
$971$	43.0309	1.38093	0.690464	+	0.723367i	$0.257406\pi$
$972$	0	0
$973$	−11.7473	−0.376600
$974$	37.0784	1.18807
$975$	0	0
$976$	−18.1042	−0.579501
$977$	−49.7231	−1.59078	−0.795392	−	0.606096i	$-0.792735\pi$
$977$	−49.7231	−1.59078	−0.795392	+	0.606096i	$0.792735\pi$
$978$	0	0
$979$	54.3936	1.73843
$980$	−33.8680	−1.08187
$981$	0	0
$982$	−64.0079	−2.04257
$983$	14.4551	0.461046	0.230523	−	0.973067i	$-0.425956\pi$
$983$	14.4551	0.461046	0.230523	+	0.973067i	$0.425956\pi$
$984$	0	0
$985$	−7.42306	−0.236518
$986$	5.43019	0.172932
$987$	0	0
$988$	−37.0254	−1.17794
$989$	−36.2183	−1.15167
$990$	0	0
$991$	13.1410	0.417438	0.208719	−	0.977976i	$-0.433071\pi$
$991$	13.1410	0.417438	0.208719	+	0.977976i	$0.433071\pi$
$992$	22.0879	0.701292
$993$	0	0
$994$	−4.73025	−0.150034
$995$	4.17135	0.132241
$996$	0	0
$997$	9.68247	0.306647	0.153324	−	0.988176i	$-0.451002\pi$
$997$	9.68247	0.306647	0.153324	+	0.988176i	$0.451002\pi$
$998$	−24.5318	−0.776540
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.k.1.2	yes	20
3.2	odd	2		2151.2.a.j.1.19	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.19	✓	20	3.2	odd	2
2151.2.a.k.1.2	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)

\(f(q)\)	\(=\)	\(q-2.12763 q^{2} +2.52683 q^{4} +2.10198 q^{5} +0.789580 q^{7} -1.12089 q^{8} +O(q^{10})\)	\(q-2.12763 q^{2} +2.52683 q^{4} +2.10198 q^{5} +0.789580 q^{7} -1.12089 q^{8} -4.47224 q^{10} +4.48898 q^{11} -4.31061 q^{13} -1.67994 q^{14} -2.66880 q^{16} -6.64316 q^{17} +3.39928 q^{19} +5.31133 q^{20} -9.55090 q^{22} +9.17525 q^{23} -0.581692 q^{25} +9.17139 q^{26} +1.99513 q^{28} +0.384188 q^{29} +2.78887 q^{31} +7.92002 q^{32} +14.1342 q^{34} +1.65968 q^{35} -0.131728 q^{37} -7.23242 q^{38} -2.35609 q^{40} +8.64441 q^{41} -3.94739 q^{43} +11.3429 q^{44} -19.5216 q^{46} +13.3918 q^{47} -6.37656 q^{49} +1.23763 q^{50} -10.8921 q^{52} +0.625549 q^{53} +9.43573 q^{55} -0.885034 q^{56} -0.817411 q^{58} -7.17980 q^{59} +6.78363 q^{61} -5.93370 q^{62} -11.5133 q^{64} -9.06079 q^{65} +0.301693 q^{67} -16.7861 q^{68} -3.53119 q^{70} +2.81573 q^{71} -2.28817 q^{73} +0.280269 q^{74} +8.58938 q^{76} +3.54441 q^{77} +14.2938 q^{79} -5.60977 q^{80} -18.3921 q^{82} -14.1111 q^{83} -13.9638 q^{85} +8.39860 q^{86} -5.03166 q^{88} +12.1171 q^{89} -3.40357 q^{91} +23.1843 q^{92} -28.4927 q^{94} +7.14521 q^{95} -10.1442 q^{97} +13.5670 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) 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

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Database

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