LMFDB - Embedded newform 2151.2.a.i.1.9

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:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 28 x^{15} - x^{14} + 319 x^{13} + 17 x^{12} - 1903 x^{11} - 91 x^{10} + 6377 x^{9} + 125 x^{8} + \cdots - 49 $ x^17 - 28x^15 - x^14 + 319x^13 + 17x^12 - 1903x^11 - 91x^10 + 6377x^9 + 125x^8 - 11967x^7 + 233x^6 + 11733x^5 - 503x^4 - 5015x^3 + 94x^2 + 609x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.0842488$ 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)

$f(q)$	$=$	$q+0.0842488 q^{2} -1.99290 q^{4} -2.54226 q^{5} -1.62505 q^{7} -0.336397 q^{8} +O(q^{10})$	\(q+0.0842488 q^{2} -1.99290 q^{4} -2.54226 q^{5} -1.62505 q^{7} -0.336397 q^{8} -0.214182 q^{10} -5.58262 q^{11} +1.97958 q^{13} -0.136909 q^{14} +3.95746 q^{16} -6.03619 q^{17} -3.46632 q^{19} +5.06647 q^{20} -0.470329 q^{22} +0.922884 q^{23} +1.46308 q^{25} +0.166778 q^{26} +3.23857 q^{28} -6.23503 q^{29} +0.733449 q^{31} +1.00621 q^{32} -0.508542 q^{34} +4.13131 q^{35} -5.69889 q^{37} -0.292033 q^{38} +0.855209 q^{40} -7.79470 q^{41} +5.65220 q^{43} +11.1256 q^{44} +0.0777518 q^{46} +13.2175 q^{47} -4.35920 q^{49} +0.123263 q^{50} -3.94512 q^{52} +9.19034 q^{53} +14.1925 q^{55} +0.546664 q^{56} -0.525294 q^{58} -9.53520 q^{59} +8.48893 q^{61} +0.0617922 q^{62} -7.83015 q^{64} -5.03262 q^{65} +8.45471 q^{67} +12.0295 q^{68} +0.348058 q^{70} -5.78462 q^{71} +14.8321 q^{73} -0.480124 q^{74} +6.90803 q^{76} +9.07205 q^{77} -0.759801 q^{79} -10.0609 q^{80} -0.656694 q^{82} +10.2404 q^{83} +15.3456 q^{85} +0.476191 q^{86} +1.87798 q^{88} -8.41501 q^{89} -3.21693 q^{91} -1.83922 q^{92} +1.11356 q^{94} +8.81228 q^{95} -6.97946 q^{97} -0.367257 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) $ 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8	$ 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8} + 5 q^{10} + q^{11} + 15 q^{13} + 3 q^{14} + 24 q^{16} - 4 q^{17} + 24 q^{19} - 4 q^{20} - 10 q^{22} + 9 q^{23} + 39 q^{25} + 12 q^{26} - 7 q^{28} + 2 q^{29} + 28 q^{31} + 31 q^{32} + 29 q^{34} + 24 q^{35} + 11 q^{37} + 19 q^{38} - 18 q^{40} - 20 q^{41} - 9 q^{43} + 43 q^{44} - 18 q^{46} + 18 q^{47} + 60 q^{49} + 61 q^{50} - q^{52} + 12 q^{53} - 10 q^{55} + 60 q^{56} - 38 q^{58} - q^{59} + 24 q^{61} + 33 q^{62} + 21 q^{64} - 2 q^{65} + 16 q^{67} + 10 q^{68} + 7 q^{70} - 12 q^{71} + 30 q^{73} + 21 q^{74} + 75 q^{76} + 15 q^{77} - 10 q^{79} - 32 q^{80} + 50 q^{82} + 16 q^{83} - 18 q^{85} + 3 q^{86} - 28 q^{88} - 65 q^{89} + 47 q^{91} - 24 q^{92} + 32 q^{94} + 37 q^{95} + 87 q^{97} + 9 q^{98}+O(q^{100}) $ 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8 + 5 * q^10 + q^11 + 15 * q^13 + 3 * q^14 + 24 * q^16 - 4 * q^17 + 24 * q^19 - 4 * q^20 - 10 * q^22 + 9 * q^23 + 39 * q^25 + 12 * q^26 - 7 * q^28 + 2 * q^29 + 28 * q^31 + 31 * q^32 + 29 * q^34 + 24 * q^35 + 11 * q^37 + 19 * q^38 - 18 * q^40 - 20 * q^41 - 9 * q^43 + 43 * q^44 - 18 * q^46 + 18 * q^47 + 60 * q^49 + 61 * q^50 - q^52 + 12 * q^53 - 10 * q^55 + 60 * q^56 - 38 * q^58 - q^59 + 24 * q^61 + 33 * q^62 + 21 * q^64 - 2 * q^65 + 16 * q^67 + 10 * q^68 + 7 * q^70 - 12 * q^71 + 30 * q^73 + 21 * q^74 + 75 * q^76 + 15 * q^77 - 10 * q^79 - 32 * q^80 + 50 * q^82 + 16 * q^83 - 18 * q^85 + 3 * q^86 - 28 * q^88 - 65 * q^89 + 47 * q^91 - 24 * q^92 + 32 * q^94 + 37 * q^95 + 87 * q^97 + 9 * 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 $

$2$	0.0842488	0.0595729	0.0297864	−	0.999556i	$-0.490517\pi$
$2$	0.0842488	0.0595729	0.0297864	+	0.999556i	$0.490517\pi$
$3$	0	0
$3$	0	0
$4$	−1.99290	−0.996451
$5$	−2.54226	−1.13693	−0.568466	−	0.822706i	$-0.692463\pi$
$5$	−2.54226	−1.13693	−0.568466	+	0.822706i	$0.692463\pi$
$6$	0	0
$7$	−1.62505	−0.614213	−0.307106	−	0.951675i	$-0.599361\pi$
$7$	−1.62505	−0.614213	−0.307106	+	0.951675i	$0.599361\pi$
$8$	−0.336397	−0.118934
$9$	0	0
$10$	−0.214182	−0.0677304
$11$	−5.58262	−1.68322	−0.841611	−	0.540084i	$-0.818392\pi$
$11$	−5.58262	−1.68322	−0.841611	+	0.540084i	$0.818392\pi$
$12$	0	0
$13$	1.97958	0.549038	0.274519	−	0.961582i	$-0.411481\pi$
$13$	1.97958	0.549038	0.274519	+	0.961582i	$0.411481\pi$
$14$	−0.136909	−0.0365904
$15$	0	0
$16$	3.95746	0.989366
$17$	−6.03619	−1.46399	−0.731996	−	0.681309i	$-0.761411\pi$
$17$	−6.03619	−1.46399	−0.731996	+	0.681309i	$0.761411\pi$
$18$	0	0
$19$	−3.46632	−0.795228	−0.397614	−	0.917553i	$-0.630162\pi$
$19$	−3.46632	−0.795228	−0.397614	+	0.917553i	$0.630162\pi$
$20$	5.06647	1.13290
$21$	0	0
$22$	−0.470329	−0.100274
$23$	0.922884	0.192435	0.0962173	−	0.995360i	$-0.469326\pi$
$23$	0.922884	0.192435	0.0962173	+	0.995360i	$0.469326\pi$
$24$	0	0
$25$	1.46308	0.292616
$26$	0.166778	0.0327078
$27$	0	0
$28$	3.23857	0.612033
$29$	−6.23503	−1.15782	−0.578908	−	0.815393i	$-0.696521\pi$
$29$	−6.23503	−1.15782	−0.578908	+	0.815393i	$0.696521\pi$
$30$	0	0
$31$	0.733449	0.131731	0.0658656	−	0.997829i	$-0.479019\pi$
$31$	0.733449	0.131731	0.0658656	+	0.997829i	$0.479019\pi$
$32$	1.00621	0.177874
$33$	0	0
$34$	−0.508542	−0.0872142
$35$	4.13131	0.698319
$36$	0	0
$37$	−5.69889	−0.936891	−0.468446	−	0.883492i	$-0.655186\pi$
$37$	−5.69889	−0.936891	−0.468446	+	0.883492i	$0.655186\pi$
$38$	−0.292033	−0.0473740
$39$	0	0
$40$	0.855209	0.135220
$41$	−7.79470	−1.21733	−0.608664	−	0.793428i	$-0.708294\pi$
$41$	−7.79470	−1.21733	−0.608664	+	0.793428i	$0.708294\pi$
$42$	0	0
$43$	5.65220	0.861953	0.430976	−	0.902363i	$-0.358169\pi$
$43$	5.65220	0.861953	0.430976	+	0.902363i	$0.358169\pi$
$44$	11.1256	1.67725
$45$	0	0
$46$	0.0777518	0.0114639
$47$	13.2175	1.92797	0.963987	−	0.265949i	$-0.0856853\pi$
$47$	13.2175	1.92797	0.963987	+	0.265949i	$0.0856853\pi$
$48$	0	0
$49$	−4.35920	−0.622743
$50$	0.123263	0.0174320
$51$	0	0
$52$	−3.94512	−0.547090
$53$	9.19034	1.26239	0.631195	−	0.775624i	$-0.282565\pi$
$53$	9.19034	1.26239	0.631195	+	0.775624i	$0.282565\pi$
$54$	0	0
$55$	14.1925	1.91371
$56$	0.546664	0.0730510
$57$	0	0
$58$	−0.525294	−0.0689745
$59$	−9.53520	−1.24138	−0.620688	−	0.784057i	$-0.713147\pi$
$59$	−9.53520	−1.24138	−0.620688	+	0.784057i	$0.713147\pi$
$60$	0	0
$61$	8.48893	1.08690	0.543448	−	0.839443i	$-0.317118\pi$
$61$	8.48893	1.08690	0.543448	+	0.839443i	$0.317118\pi$
$62$	0.0617922	0.00784761
$63$	0	0
$64$	−7.83015	−0.978769
$65$	−5.03262	−0.624219
$66$	0	0
$67$	8.45471	1.03291	0.516454	−	0.856315i	$-0.327252\pi$
$67$	8.45471	1.03291	0.516454	+	0.856315i	$0.327252\pi$
$68$	12.0295	1.45880
$69$	0	0
$70$	0.348058	0.0416009
$71$	−5.78462	−0.686508	−0.343254	−	0.939243i	$-0.611529\pi$
$71$	−5.78462	−0.686508	−0.343254	+	0.939243i	$0.611529\pi$
$72$	0	0
$73$	14.8321	1.73596	0.867982	−	0.496596i	$-0.165417\pi$
$73$	14.8321	1.73596	0.867982	+	0.496596i	$0.165417\pi$
$74$	−0.480124	−0.0558133
$75$	0	0
$76$	6.90803	0.792406
$77$	9.07205	1.03386
$78$	0	0
$79$	−0.759801	−0.0854843	−0.0427421	−	0.999086i	$-0.513609\pi$
$79$	−0.759801	−0.0854843	−0.0427421	+	0.999086i	$0.513609\pi$
$80$	−10.0609	−1.12484
$81$	0	0
$82$	−0.656694	−0.0725197
$83$	10.2404	1.12403	0.562017	−	0.827126i	$-0.310026\pi$
$83$	10.2404	1.12403	0.562017	+	0.827126i	$0.310026\pi$
$84$	0	0
$85$	15.3456	1.66446
$86$	0.476191	0.0513490
$87$	0	0
$88$	1.87798	0.200193
$89$	−8.41501	−0.891990	−0.445995	−	0.895036i	$-0.647150\pi$
$89$	−8.41501	−0.891990	−0.445995	+	0.895036i	$0.647150\pi$
$90$	0	0
$91$	−3.21693	−0.337226
$92$	−1.83922	−0.191752
$93$	0	0
$94$	1.11356	0.114855
$95$	8.81228	0.904121
$96$	0	0
$97$	−6.97946	−0.708657	−0.354328	−	0.935121i	$-0.615290\pi$
$97$	−6.97946	−0.708657	−0.354328	+	0.935121i	$0.615290\pi$
$98$	−0.367257	−0.0370986
$99$	0	0
$100$	−2.91578	−0.291578
$101$	4.61542	0.459251	0.229626	−	0.973279i	$-0.426250\pi$
$101$	4.61542	0.459251	0.229626	+	0.973279i	$0.426250\pi$
$102$	0	0
$103$	−7.96465	−0.784780	−0.392390	−	0.919799i	$-0.628352\pi$
$103$	−7.96465	−0.784780	−0.392390	+	0.919799i	$0.628352\pi$
$104$	−0.665927	−0.0652995
$105$	0	0
$106$	0.774275	0.0752043
$107$	−1.79986	−0.173999	−0.0869993	−	0.996208i	$-0.527728\pi$
$107$	−1.79986	−0.173999	−0.0869993	+	0.996208i	$0.527728\pi$
$108$	0	0
$109$	6.07012	0.581412	0.290706	−	0.956812i	$-0.406110\pi$
$109$	6.07012	0.581412	0.290706	+	0.956812i	$0.406110\pi$
$110$	1.19570	0.114005
$111$	0	0
$112$	−6.43109	−0.607681
$113$	−6.59861	−0.620745	−0.310372	−	0.950615i	$-0.600454\pi$
$113$	−6.59861	−0.620745	−0.310372	+	0.950615i	$0.600454\pi$
$114$	0	0
$115$	−2.34621	−0.218785
$116$	12.4258	1.15371
$117$	0	0
$118$	−0.803329	−0.0739524
$119$	9.80913	0.899202
$120$	0	0
$121$	20.1656	1.83324
$122$	0.715182	0.0647496
$123$	0	0
$124$	−1.46169	−0.131264
$125$	8.99176	0.804248
$126$	0	0
$127$	8.37945	0.743556	0.371778	−	0.928322i	$-0.378748\pi$
$127$	8.37945	0.743556	0.371778	+	0.928322i	$0.378748\pi$
$128$	−2.67209	−0.236182
$129$	0	0
$130$	−0.423992	−0.0371866
$131$	1.47397	0.128781	0.0643905	−	0.997925i	$-0.479490\pi$
$131$	1.47397	0.128781	0.0643905	+	0.997925i	$0.479490\pi$
$132$	0	0
$133$	5.63296	0.488439
$134$	0.712299	0.0615333
$135$	0	0
$136$	2.03056	0.174119
$137$	6.72307	0.574391	0.287195	−	0.957872i	$-0.407277\pi$
$137$	6.72307	0.574391	0.287195	+	0.957872i	$0.407277\pi$
$138$	0	0
$139$	−11.4955	−0.975035	−0.487517	−	0.873113i	$-0.662097\pi$
$139$	−11.4955	−0.975035	−0.487517	+	0.873113i	$0.662097\pi$
$140$	−8.23329	−0.695840
$141$	0	0
$142$	−0.487348	−0.0408973
$143$	−11.0513	−0.924153
$144$	0	0
$145$	15.8511	1.31636
$146$	1.24959	0.103416
$147$	0	0
$148$	11.3573	0.933566
$149$	19.8668	1.62755	0.813774	−	0.581181i	$-0.197409\pi$
$149$	19.8668	1.62755	0.813774	+	0.581181i	$0.197409\pi$
$150$	0	0
$151$	1.15574	0.0940525	0.0470262	−	0.998894i	$-0.485026\pi$
$151$	1.15574	0.0940525	0.0470262	+	0.998894i	$0.485026\pi$
$152$	1.16606	0.0945800
$153$	0	0
$154$	0.764310	0.0615898
$155$	−1.86462	−0.149770
$156$	0	0
$157$	−14.8367	−1.18410	−0.592048	−	0.805903i	$-0.701680\pi$
$157$	−14.8367	−1.18410	−0.592048	+	0.805903i	$0.701680\pi$
$158$	−0.0640123	−0.00509255
$159$	0	0
$160$	−2.55804	−0.202231
$161$	−1.49974	−0.118196
$162$	0	0
$163$	−21.1599	−1.65737	−0.828685	−	0.559716i	$-0.810910\pi$
$163$	−21.1599	−1.65737	−0.828685	+	0.559716i	$0.810910\pi$
$164$	15.5341	1.21301
$165$	0	0
$166$	0.862744	0.0669619
$167$	−12.7106	−0.983576	−0.491788	−	0.870715i	$-0.663657\pi$
$167$	−12.7106	−0.983576	−0.491788	+	0.870715i	$0.663657\pi$
$168$	0	0
$169$	−9.08124	−0.698557
$170$	1.29284	0.0991567
$171$	0	0
$172$	−11.2643	−0.858894
$173$	11.6580	0.886340	0.443170	−	0.896438i	$-0.353854\pi$
$173$	11.6580	0.886340	0.443170	+	0.896438i	$0.353854\pi$
$174$	0	0
$175$	−2.37759	−0.179729
$176$	−22.0930	−1.66532
$177$	0	0
$178$	−0.708955	−0.0531384
$179$	−6.65698	−0.497566	−0.248783	−	0.968559i	$-0.580031\pi$
$179$	−6.65698	−0.497566	−0.248783	+	0.968559i	$0.580031\pi$
$180$	0	0
$181$	−2.45448	−0.182440	−0.0912201	−	0.995831i	$-0.529077\pi$
$181$	−2.45448	−0.182440	−0.0912201	+	0.995831i	$0.529077\pi$
$182$	−0.271023	−0.0200895
$183$	0	0
$184$	−0.310455	−0.0228871
$185$	14.4881	1.06518
$186$	0	0
$187$	33.6977	2.46422
$188$	−26.3412	−1.92113
$189$	0	0
$190$	0.742424	0.0538611
$191$	5.20229	0.376424	0.188212	−	0.982128i	$-0.439731\pi$
$191$	5.20229	0.376424	0.188212	+	0.982128i	$0.439731\pi$
$192$	0	0
$193$	−18.3800	−1.32302	−0.661511	−	0.749936i	$-0.730084\pi$
$193$	−18.3800	−1.32302	−0.661511	+	0.749936i	$0.730084\pi$
$194$	−0.588011	−0.0422167
$195$	0	0
$196$	8.68746	0.620533
$197$	0.167128	0.0119074	0.00595369	−	0.999982i	$-0.498105\pi$
$197$	0.167128	0.0119074	0.00595369	+	0.999982i	$0.498105\pi$
$198$	0	0
$199$	0.894513	0.0634104	0.0317052	−	0.999497i	$-0.489906\pi$
$199$	0.894513	0.0634104	0.0317052	+	0.999497i	$0.489906\pi$
$200$	−0.492177	−0.0348021
$201$	0	0
$202$	0.388843	0.0273589
$203$	10.1323	0.711146
$204$	0	0
$205$	19.8161	1.38402
$206$	−0.671012	−0.0467516
$207$	0	0
$208$	7.83413	0.543199
$209$	19.3511	1.33855
$210$	0	0
$211$	−17.6853	−1.21751	−0.608754	−	0.793359i	$-0.708330\pi$
$211$	−17.6853	−1.21751	−0.608754	+	0.793359i	$0.708330\pi$
$212$	−18.3155	−1.25791
$213$	0	0
$214$	−0.151636	−0.0103656
$215$	−14.3694	−0.979982
$216$	0	0
$217$	−1.19189	−0.0809110
$218$	0.511400	0.0346364
$219$	0	0
$220$	−28.2842	−1.90692
$221$	−11.9492	−0.803787
$222$	0	0
$223$	15.0963	1.01092	0.505461	−	0.862849i	$-0.331322\pi$
$223$	15.0963	1.01092	0.505461	+	0.862849i	$0.331322\pi$
$224$	−1.63514	−0.109252
$225$	0	0
$226$	−0.555925	−0.0369796
$227$	−6.13326	−0.407079	−0.203539	−	0.979067i	$-0.565244\pi$
$227$	−6.13326	−0.407079	−0.203539	+	0.979067i	$0.565244\pi$
$228$	0	0
$229$	4.70479	0.310901	0.155451	−	0.987844i	$-0.450317\pi$
$229$	4.70479	0.310901	0.155451	+	0.987844i	$0.450317\pi$
$230$	−0.197665	−0.0130337
$231$	0	0
$232$	2.09745	0.137704
$233$	21.3977	1.40181	0.700905	−	0.713255i	$-0.252780\pi$
$233$	21.3977	1.40181	0.700905	+	0.713255i	$0.252780\pi$
$234$	0	0
$235$	−33.6024	−2.19198
$236$	19.0027	1.23697
$237$	0	0
$238$	0.826408	0.0535681
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	25.1034	1.61705	0.808526	−	0.588460i	$-0.200266\pi$
$241$	25.1034	1.61705	0.808526	+	0.588460i	$0.200266\pi$
$242$	1.69893	0.109211
$243$	0	0
$244$	−16.9176	−1.08304
$245$	11.0822	0.708017
$246$	0	0
$247$	−6.86187	−0.436611
$248$	−0.246730	−0.0156674
$249$	0	0
$250$	0.757545	0.0479114
$251$	−15.3440	−0.968504	−0.484252	−	0.874928i	$-0.660908\pi$
$251$	−15.3440	−0.968504	−0.484252	+	0.874928i	$0.660908\pi$
$252$	0	0
$253$	−5.15211	−0.323910
$254$	0.705959	0.0442958
$255$	0	0
$256$	15.4352	0.964699
$257$	−8.56433	−0.534228	−0.267114	−	0.963665i	$-0.586070\pi$
$257$	−8.56433	−0.534228	−0.267114	+	0.963665i	$0.586070\pi$
$258$	0	0
$259$	9.26100	0.575451
$260$	10.0295	0.622004
$261$	0	0
$262$	0.124180	0.00767186
$263$	−1.18083	−0.0728128	−0.0364064	−	0.999337i	$-0.511591\pi$
$263$	−1.18083	−0.0728128	−0.0364064	+	0.999337i	$0.511591\pi$
$264$	0	0
$265$	−23.3642	−1.43525
$266$	0.474570	0.0290977
$267$	0	0
$268$	−16.8494	−1.02924
$269$	−27.7393	−1.69129	−0.845647	−	0.533743i	$-0.820785\pi$
$269$	−27.7393	−1.69129	−0.845647	+	0.533743i	$0.820785\pi$
$270$	0	0
$271$	2.13029	0.129406	0.0647029	−	0.997905i	$-0.479390\pi$
$271$	2.13029	0.129406	0.0647029	+	0.997905i	$0.479390\pi$
$272$	−23.8880	−1.44842
$273$	0	0
$274$	0.566411	0.0342181
$275$	−8.16782	−0.492538
$276$	0	0
$277$	−30.1940	−1.81418	−0.907092	−	0.420933i	$-0.861703\pi$
$277$	−30.1940	−1.81418	−0.907092	+	0.420933i	$0.861703\pi$
$278$	−0.968481	−0.0580857
$279$	0	0
$280$	−1.38976	−0.0830541
$281$	31.3913	1.87265	0.936323	−	0.351141i	$-0.114206\pi$
$281$	31.3913	1.87265	0.936323	+	0.351141i	$0.114206\pi$
$282$	0	0
$283$	−15.6160	−0.928274	−0.464137	−	0.885763i	$-0.653636\pi$
$283$	−15.6160	−0.928274	−0.464137	+	0.885763i	$0.653636\pi$
$284$	11.5282	0.684072
$285$	0	0
$286$	−0.931056	−0.0550545
$287$	12.6668	0.747698
$288$	0	0
$289$	19.4356	1.14327
$290$	1.33543	0.0784194
$291$	0	0
$292$	−29.5589	−1.72980
$293$	−12.2717	−0.716922	−0.358461	−	0.933545i	$-0.616698\pi$
$293$	−12.2717	−0.716922	−0.358461	+	0.933545i	$0.616698\pi$
$294$	0	0
$295$	24.2409	1.41136
$296$	1.91709	0.111429
$297$	0	0
$298$	1.67375	0.0969577
$299$	1.82693	0.105654
$300$	0	0
$301$	−9.18513	−0.529422
$302$	0.0973694	0.00560298
$303$	0	0
$304$	−13.7178	−0.786772
$305$	−21.5811	−1.23573
$306$	0	0
$307$	−3.04981	−0.174062	−0.0870310	−	0.996206i	$-0.527738\pi$
$307$	−3.04981	−0.174062	−0.0870310	+	0.996206i	$0.527738\pi$
$308$	−18.0797	−1.03019
$309$	0	0
$310$	−0.157092	−0.00892221
$311$	−29.6460	−1.68107	−0.840535	−	0.541758i	$-0.817759\pi$
$311$	−29.6460	−1.68107	−0.840535	+	0.541758i	$0.817759\pi$
$312$	0	0
$313$	31.6240	1.78749	0.893747	−	0.448572i	$-0.148067\pi$
$313$	31.6240	1.78749	0.893747	+	0.448572i	$0.148067\pi$
$314$	−1.24997	−0.0705400
$315$	0	0
$316$	1.51421	0.0851809
$317$	15.3053	0.859629	0.429815	−	0.902917i	$-0.358579\pi$
$317$	15.3053	0.859629	0.429815	+	0.902917i	$0.358579\pi$
$318$	0	0
$319$	34.8078	1.94886
$320$	19.9063	1.11280
$321$	0	0
$322$	−0.126351	−0.00704126
$323$	20.9234	1.16421
$324$	0	0
$325$	2.89629	0.160657
$326$	−1.78269	−0.0987343
$327$	0	0
$328$	2.62211	0.144782
$329$	−21.4792	−1.18419
$330$	0	0
$331$	−32.4513	−1.78368	−0.891842	−	0.452346i	$-0.850587\pi$
$331$	−32.4513	−1.78368	−0.891842	+	0.452346i	$0.850587\pi$
$332$	−20.4082	−1.12004
$333$	0	0
$334$	−1.07085	−0.0585945
$335$	−21.4941	−1.17435
$336$	0	0
$337$	10.4316	0.568247	0.284123	−	0.958788i	$-0.408297\pi$
$337$	10.4316	0.568247	0.284123	+	0.958788i	$0.408297\pi$
$338$	−0.765084	−0.0416151
$339$	0	0
$340$	−30.5822	−1.65855
$341$	−4.09456	−0.221733
$342$	0	0
$343$	18.4593	0.996709
$344$	−1.90138	−0.102516
$345$	0	0
$346$	0.982171	0.0528019
$347$	13.1108	0.703823	0.351911	−	0.936033i	$-0.385532\pi$
$347$	13.1108	0.703823	0.351911	+	0.936033i	$0.385532\pi$
$348$	0	0
$349$	−16.5930	−0.888203	−0.444101	−	0.895977i	$-0.646477\pi$
$349$	−16.5930	−0.888203	−0.444101	+	0.895977i	$0.646477\pi$
$350$	−0.200309	−0.0107070
$351$	0	0
$352$	−5.61726	−0.299401
$353$	21.9787	1.16981	0.584903	−	0.811103i	$-0.301132\pi$
$353$	21.9787	1.16981	0.584903	+	0.811103i	$0.301132\pi$
$354$	0	0
$355$	14.7060	0.780514
$356$	16.7703	0.888824
$357$	0	0
$358$	−0.560843	−0.0296415
$359$	−31.2750	−1.65063	−0.825316	−	0.564672i	$-0.809003\pi$
$359$	−31.2750	−1.65063	−0.825316	+	0.564672i	$0.809003\pi$
$360$	0	0
$361$	−6.98463	−0.367612
$362$	−0.206787	−0.0108685
$363$	0	0
$364$	6.41103	0.336029
$365$	−37.7070	−1.97367
$366$	0	0
$367$	7.56961	0.395130	0.197565	−	0.980290i	$-0.436697\pi$
$367$	7.56961	0.395130	0.197565	+	0.980290i	$0.436697\pi$
$368$	3.65228	0.190388
$369$	0	0
$370$	1.22060	0.0634560
$371$	−14.9348	−0.775376
$372$	0	0
$373$	−0.475242	−0.0246071	−0.0123035	−	0.999924i	$-0.503916\pi$
$373$	−0.475242	−0.0246071	−0.0123035	+	0.999924i	$0.503916\pi$
$374$	2.83899	0.146801
$375$	0	0
$376$	−4.44634	−0.229302
$377$	−12.3428	−0.635685
$378$	0	0
$379$	21.7614	1.11781	0.558905	−	0.829232i	$-0.311222\pi$
$379$	21.7614	1.11781	0.558905	+	0.829232i	$0.311222\pi$
$380$	−17.5620	−0.900912
$381$	0	0
$382$	0.438286	0.0224247
$383$	19.7618	1.00978	0.504890	−	0.863184i	$-0.331533\pi$
$383$	19.7618	1.00978	0.504890	+	0.863184i	$0.331533\pi$
$384$	0	0
$385$	−23.0635	−1.17543
$386$	−1.54849	−0.0788162
$387$	0	0
$388$	13.9094	0.706142
$389$	2.42618	0.123012	0.0615061	−	0.998107i	$-0.480410\pi$
$389$	2.42618	0.123012	0.0615061	+	0.998107i	$0.480410\pi$
$390$	0	0
$391$	−5.57070	−0.281722
$392$	1.46642	0.0740655
$393$	0	0
$394$	0.0140803	0.000709358 0
$395$	1.93161	0.0971899
$396$	0	0
$397$	20.4499	1.02635	0.513176	−	0.858283i	$-0.328469\pi$
$397$	20.4499	1.02635	0.513176	+	0.858283i	$0.328469\pi$
$398$	0.0753617	0.00377754
$399$	0	0
$400$	5.79009	0.289505
$401$	13.6616	0.682226	0.341113	−	0.940022i	$-0.389196\pi$
$401$	13.6616	0.682226	0.341113	+	0.940022i	$0.389196\pi$
$402$	0	0
$403$	1.45192	0.0723255
$404$	−9.19807	−0.457621
$405$	0	0
$406$	0.853631	0.0423650
$407$	31.8147	1.57700
$408$	0	0
$409$	25.3514	1.25355	0.626773	−	0.779202i	$-0.284375\pi$
$409$	25.3514	1.25355	0.626773	+	0.779202i	$0.284375\pi$
$410$	1.66949	0.0824500
$411$	0	0
$412$	15.8728	0.781995
$413$	15.4952	0.762469
$414$	0	0
$415$	−26.0338	−1.27795
$416$	1.99187	0.0976595
$417$	0	0
$418$	1.63031	0.0797410
$419$	−22.0380	−1.07663	−0.538313	−	0.842745i	$-0.680938\pi$
$419$	−22.0380	−1.07663	−0.538313	+	0.842745i	$0.680938\pi$
$420$	0	0
$421$	−32.1366	−1.56624	−0.783120	−	0.621870i	$-0.786373\pi$
$421$	−32.1366	−1.56624	−0.783120	+	0.621870i	$0.786373\pi$
$422$	−1.48997	−0.0725305
$423$	0	0
$424$	−3.09161	−0.150142
$425$	−8.83144	−0.428388
$426$	0	0
$427$	−13.7950	−0.667585
$428$	3.58694	0.173381
$429$	0	0
$430$	−1.21060	−0.0583804
$431$	7.03594	0.338909	0.169455	−	0.985538i	$-0.445799\pi$
$431$	7.03594	0.338909	0.169455	+	0.985538i	$0.445799\pi$
$432$	0	0
$433$	−17.2012	−0.826638	−0.413319	−	0.910586i	$-0.635631\pi$
$433$	−17.2012	−0.826638	−0.413319	+	0.910586i	$0.635631\pi$
$434$	−0.100416	−0.00482010
$435$	0	0
$436$	−12.0971	−0.579348
$437$	−3.19901	−0.153029
$438$	0	0
$439$	−8.91600	−0.425537	−0.212769	−	0.977103i	$-0.568248\pi$
$439$	−8.91600	−0.425537	−0.212769	+	0.977103i	$0.568248\pi$
$440$	−4.77430	−0.227606
$441$	0	0
$442$	−1.00670	−0.0478839
$443$	13.7282	0.652246	0.326123	−	0.945327i	$-0.394258\pi$
$443$	13.7282	0.652246	0.326123	+	0.945327i	$0.394258\pi$
$444$	0	0
$445$	21.3931	1.01413
$446$	1.27185	0.0602236
$447$	0	0
$448$	12.7244	0.601172
$449$	−18.9547	−0.894527	−0.447263	−	0.894402i	$-0.647601\pi$
$449$	−18.9547	−0.894527	−0.447263	+	0.894402i	$0.647601\pi$
$450$	0	0
$451$	43.5148	2.04903
$452$	13.1504	0.618542
$453$	0	0
$454$	−0.516720	−0.0242509
$455$	8.17827	0.383403
$456$	0	0
$457$	31.2559	1.46209	0.731044	−	0.682330i	$-0.239033\pi$
$457$	31.2559	1.46209	0.731044	+	0.682330i	$0.239033\pi$
$458$	0.396373	0.0185213
$459$	0	0
$460$	4.67577	0.218009
$461$	20.8151	0.969456	0.484728	−	0.874665i	$-0.338919\pi$
$461$	20.8151	0.969456	0.484728	+	0.874665i	$0.338919\pi$
$462$	0	0
$463$	−10.1467	−0.471556	−0.235778	−	0.971807i	$-0.575764\pi$
$463$	−10.1467	−0.471556	−0.235778	+	0.971807i	$0.575764\pi$
$464$	−24.6749	−1.14550
$465$	0	0
$466$	1.80273	0.0835099
$467$	38.3603	1.77510	0.887551	−	0.460710i	$-0.152405\pi$
$467$	38.3603	1.77510	0.887551	+	0.460710i	$0.152405\pi$
$468$	0	0
$469$	−13.7394	−0.634425
$470$	−2.83096	−0.130582
$471$	0	0
$472$	3.20761	0.147642
$473$	−31.5541	−1.45086
$474$	0	0
$475$	−5.07151	−0.232697
$476$	−19.5486	−0.896011
$477$	0	0
$478$	−0.0842488	−0.00385345
$479$	43.0994	1.96926	0.984630	−	0.174651i	$-0.0558798\pi$
$479$	43.0994	1.96926	0.984630	+	0.174651i	$0.0558798\pi$
$480$	0	0
$481$	−11.2814	−0.514389
$482$	2.11493	0.0963325
$483$	0	0
$484$	−40.1881	−1.82673
$485$	17.7436	0.805695
$486$	0	0
$487$	24.0018	1.08762	0.543812	−	0.839207i	$-0.316980\pi$
$487$	24.0018	1.08762	0.543812	+	0.839207i	$0.316980\pi$
$488$	−2.85565	−0.129269
$489$	0	0
$490$	0.933663	0.0421786
$491$	−15.4064	−0.695281	−0.347641	−	0.937628i	$-0.613017\pi$
$491$	−15.4064	−0.695281	−0.347641	+	0.937628i	$0.613017\pi$
$492$	0	0
$493$	37.6358	1.69503
$494$	−0.578105	−0.0260102
$495$	0	0
$496$	2.90260	0.130330
$497$	9.40032	0.421662
$498$	0	0
$499$	33.8813	1.51674	0.758368	−	0.651827i	$-0.225997\pi$
$499$	33.8813	1.51674	0.758368	+	0.651827i	$0.225997\pi$
$500$	−17.9197	−0.801394
$501$	0	0
$502$	−1.29271	−0.0576966
$503$	−10.2100	−0.455243	−0.227621	−	0.973750i	$-0.573095\pi$
$503$	−10.2100	−0.455243	−0.227621	+	0.973750i	$0.573095\pi$
$504$	0	0
$505$	−11.7336	−0.522138
$506$	−0.434059	−0.0192963
$507$	0	0
$508$	−16.6994	−0.740917
$509$	39.2832	1.74120	0.870598	−	0.491996i	$-0.163732\pi$
$509$	39.2832	1.74120	0.870598	+	0.491996i	$0.163732\pi$
$510$	0	0
$511$	−24.1029	−1.06625
$512$	6.64458	0.293652
$513$	0	0
$514$	−0.721534	−0.0318255
$515$	20.2482	0.892242
$516$	0	0
$517$	−73.7884	−3.24521
$518$	0.780228	0.0342813
$519$	0	0
$520$	1.69296	0.0742411
$521$	1.82108	0.0797828	0.0398914	−	0.999204i	$-0.487299\pi$
$521$	1.82108	0.0797828	0.0398914	+	0.999204i	$0.487299\pi$
$522$	0	0
$523$	34.7796	1.52081	0.760404	−	0.649451i	$-0.225001\pi$
$523$	34.7796	1.52081	0.760404	+	0.649451i	$0.225001\pi$
$524$	−2.93747	−0.128324
$525$	0	0
$526$	−0.0994831	−0.00433767
$527$	−4.42724	−0.192853
$528$	0	0
$529$	−22.1483	−0.962969
$530$	−1.96841	−0.0855022
$531$	0	0
$532$	−11.2259	−0.486706
$533$	−15.4303	−0.668359
$534$	0	0
$535$	4.57570	0.197825
$536$	−2.84414	−0.122848
$537$	0	0
$538$	−2.33700	−0.100755
$539$	24.3357	1.04821
$540$	0	0
$541$	−31.9524	−1.37374	−0.686871	−	0.726779i	$-0.741016\pi$
$541$	−31.9524	−1.37374	−0.686871	+	0.726779i	$0.741016\pi$
$542$	0.179474	0.00770908
$543$	0	0
$544$	−6.07365	−0.260406
$545$	−15.4318	−0.661026
$546$	0	0
$547$	−17.2935	−0.739417	−0.369709	−	0.929148i	$-0.620543\pi$
$547$	−17.2935	−0.739417	−0.369709	+	0.929148i	$0.620543\pi$
$548$	−13.3984	−0.572352
$549$	0	0
$550$	−0.688129	−0.0293419
$551$	21.6126	0.920728
$552$	0	0
$553$	1.23472	0.0525055
$554$	−2.54381	−0.108076
$555$	0	0
$556$	22.9094	0.971575
$557$	−19.4625	−0.824653	−0.412327	−	0.911036i	$-0.635284\pi$
$557$	−19.4625	−0.824653	−0.412327	+	0.911036i	$0.635284\pi$
$558$	0	0
$559$	11.1890	0.473245
$560$	16.3495	0.690892
$561$	0	0
$562$	2.64468	0.111559
$563$	−13.9465	−0.587776	−0.293888	−	0.955840i	$-0.594949\pi$
$563$	−13.9465	−0.587776	−0.293888	+	0.955840i	$0.594949\pi$
$564$	0	0
$565$	16.7754	0.705745
$566$	−1.31563	−0.0553000
$567$	0	0
$568$	1.94593	0.0816495
$569$	3.72562	0.156186	0.0780931	−	0.996946i	$-0.475117\pi$
$569$	3.72562	0.156186	0.0780931	+	0.996946i	$0.475117\pi$
$570$	0	0
$571$	−28.0899	−1.17552	−0.587762	−	0.809034i	$-0.699991\pi$
$571$	−28.0899	−1.17552	−0.587762	+	0.809034i	$0.699991\pi$
$572$	22.0241	0.920873
$573$	0	0
$574$	1.06716	0.0445425
$575$	1.35025	0.0563095
$576$	0	0
$577$	17.3238	0.721201	0.360601	−	0.932720i	$-0.382572\pi$
$577$	17.3238	0.721201	0.360601	+	0.932720i	$0.382572\pi$
$578$	1.63742	0.0681079
$579$	0	0
$580$	−31.5896	−1.31169
$581$	−16.6413	−0.690395
$582$	0	0
$583$	−51.3062	−2.12488
$584$	−4.98947	−0.206466
$585$	0	0
$586$	−1.03388	−0.0427091
$587$	−28.8356	−1.19017	−0.595087	−	0.803662i	$-0.702882\pi$
$587$	−28.8356	−1.19017	−0.595087	+	0.803662i	$0.702882\pi$
$588$	0	0
$589$	−2.54237	−0.104756
$590$	2.04227	0.0840789
$591$	0	0
$592$	−22.5531	−0.926928
$593$	−12.9789	−0.532980	−0.266490	−	0.963838i	$-0.585864\pi$
$593$	−12.9789	−0.532980	−0.266490	+	0.963838i	$0.585864\pi$
$594$	0	0
$595$	−24.9374	−1.02233
$596$	−39.5925	−1.62177
$597$	0	0
$598$	0.153916	0.00629411
$599$	31.2951	1.27868	0.639341	−	0.768923i	$-0.279207\pi$
$599$	31.2951	1.27868	0.639341	+	0.768923i	$0.279207\pi$
$600$	0	0
$601$	13.9855	0.570482	0.285241	−	0.958456i	$-0.407926\pi$
$601$	13.9855	0.570482	0.285241	+	0.958456i	$0.407926\pi$
$602$	−0.773836	−0.0315392
$603$	0	0
$604$	−2.30327	−0.0937187
$605$	−51.2662	−2.08427
$606$	0	0
$607$	−27.2289	−1.10519	−0.552594	−	0.833450i	$-0.686362\pi$
$607$	−27.2289	−1.10519	−0.552594	+	0.833450i	$0.686362\pi$
$608$	−3.48783	−0.141450
$609$	0	0
$610$	−1.81818	−0.0736159
$611$	26.1652	1.05853
$612$	0	0
$613$	−30.9650	−1.25066	−0.625332	−	0.780359i	$-0.715037\pi$
$613$	−30.9650	−1.25066	−0.625332	+	0.780359i	$0.715037\pi$
$614$	−0.256943	−0.0103694
$615$	0	0
$616$	−3.05181	−0.122961
$617$	0.460357	0.0185333	0.00926664	−	0.999957i	$-0.497050\pi$
$617$	0.460357	0.0185333	0.00926664	+	0.999957i	$0.497050\pi$
$618$	0	0
$619$	−45.0133	−1.80924	−0.904619	−	0.426222i	$-0.859844\pi$
$619$	−45.0133	−1.80924	−0.904619	+	0.426222i	$0.859844\pi$
$620$	3.71600	0.149238
$621$	0	0
$622$	−2.49764	−0.100146
$623$	13.6749	0.547871
$624$	0	0
$625$	−30.1748	−1.20699
$626$	2.66428	0.106486
$627$	0	0
$628$	29.5680	1.17989
$629$	34.3996	1.37160
$630$	0	0
$631$	21.6168	0.860550	0.430275	−	0.902698i	$-0.358417\pi$
$631$	21.6168	0.860550	0.430275	+	0.902698i	$0.358417\pi$
$632$	0.255595	0.0101670
$633$	0	0
$634$	1.28945	0.0512106
$635$	−21.3027	−0.845374
$636$	0	0
$637$	−8.62941	−0.341910
$638$	2.93252	0.116099
$639$	0	0
$640$	6.79315	0.268523
$641$	19.6899	0.777703	0.388851	−	0.921301i	$-0.372872\pi$
$641$	19.6899	0.777703	0.388851	+	0.921301i	$0.372872\pi$
$642$	0	0
$643$	−2.20865	−0.0871008	−0.0435504	−	0.999051i	$-0.513867\pi$
$643$	−2.20865	−0.0871008	−0.0435504	+	0.999051i	$0.513867\pi$
$644$	2.98883	0.117776
$645$	0	0
$646$	1.76277	0.0693552
$647$	−11.6065	−0.456300	−0.228150	−	0.973626i	$-0.573268\pi$
$647$	−11.6065	−0.456300	−0.228150	+	0.973626i	$0.573268\pi$
$648$	0	0
$649$	53.2314	2.08951
$650$	0.244009	0.00957083
$651$	0	0
$652$	42.1696	1.65149
$653$	−7.28826	−0.285212	−0.142606	−	0.989780i	$-0.545548\pi$
$653$	−7.28826	−0.285212	−0.142606	+	0.989780i	$0.545548\pi$
$654$	0	0
$655$	−3.74720	−0.146415
$656$	−30.8472	−1.20438
$657$	0	0
$658$	−1.80960	−0.0705454
$659$	−6.81106	−0.265321	−0.132661	−	0.991162i	$-0.542352\pi$
$659$	−6.81106	−0.265321	−0.132661	+	0.991162i	$0.542352\pi$
$660$	0	0
$661$	8.21412	0.319492	0.159746	−	0.987158i	$-0.448932\pi$
$661$	8.21412	0.319492	0.159746	+	0.987158i	$0.448932\pi$
$662$	−2.73398	−0.106259
$663$	0	0
$664$	−3.44485	−0.133686
$665$	−14.3204	−0.555323
$666$	0	0
$667$	−5.75421	−0.222804
$668$	25.3310	0.980086
$669$	0	0
$670$	−1.81085	−0.0699592
$671$	−47.3905	−1.82949
$672$	0	0
$673$	34.8841	1.34468	0.672341	−	0.740242i	$-0.265289\pi$
$673$	34.8841	1.34468	0.672341	+	0.740242i	$0.265289\pi$
$674$	0.878852	0.0338521
$675$	0	0
$676$	18.0980	0.696078
$677$	−33.2258	−1.27697	−0.638485	−	0.769634i	$-0.720439\pi$
$677$	−33.2258	−1.27697	−0.638485	+	0.769634i	$0.720439\pi$
$678$	0	0
$679$	11.3420	0.435266
$680$	−5.16220	−0.197961
$681$	0	0
$682$	−0.344962	−0.0132093
$683$	−11.3681	−0.434989	−0.217495	−	0.976062i	$-0.569788\pi$
$683$	−11.3681	−0.434989	−0.217495	+	0.976062i	$0.569788\pi$
$684$	0	0
$685$	−17.0918	−0.653044
$686$	1.55517	0.0593769
$687$	0	0
$688$	22.3684	0.852786
$689$	18.1931	0.693101
$690$	0	0
$691$	1.77743	0.0676168	0.0338084	−	0.999428i	$-0.489236\pi$
$691$	1.77743	0.0676168	0.0338084	+	0.999428i	$0.489236\pi$
$692$	−23.2332	−0.883195
$693$	0	0
$694$	1.10457	0.0419288
$695$	29.2245	1.10855
$696$	0	0
$697$	47.0503	1.78216
$698$	−1.39794	−0.0529128
$699$	0	0
$700$	4.73830	0.179091
$701$	20.9286	0.790463	0.395231	−	0.918582i	$-0.370664\pi$
$701$	20.9286	0.790463	0.395231	+	0.918582i	$0.370664\pi$
$702$	0	0
$703$	19.7542	0.745042
$704$	43.7128	1.64749
$705$	0	0
$706$	1.85168	0.0696888
$707$	−7.50030	−0.282078
$708$	0	0
$709$	−6.68625	−0.251107	−0.125554	−	0.992087i	$-0.540071\pi$
$709$	−6.68625	−0.251107	−0.125554	+	0.992087i	$0.540071\pi$
$710$	1.23896	0.0464975
$711$	0	0
$712$	2.83079	0.106088
$713$	0.676888	0.0253496
$714$	0	0
$715$	28.0952	1.05070
$716$	13.2667	0.495800
$717$	0	0
$718$	−2.63488	−0.0983329
$719$	4.81432	0.179544	0.0897719	−	0.995962i	$-0.471386\pi$
$719$	4.81432	0.179544	0.0897719	+	0.995962i	$0.471386\pi$
$720$	0	0
$721$	12.9430	0.482022
$722$	−0.588447	−0.0218997
$723$	0	0
$724$	4.89154	0.181793
$725$	−9.12236	−0.338796
$726$	0	0
$727$	−23.4424	−0.869430	−0.434715	−	0.900568i	$-0.643151\pi$
$727$	−23.4424	−0.869430	−0.434715	+	0.900568i	$0.643151\pi$
$728$	1.08217	0.0401078
$729$	0	0
$730$	−3.17677	−0.117578
$731$	−34.1178	−1.26189
$732$	0	0
$733$	31.2409	1.15391	0.576954	−	0.816777i	$-0.304241\pi$
$733$	31.2409	1.15391	0.576954	+	0.816777i	$0.304241\pi$
$734$	0.637730	0.0235391
$735$	0	0
$736$	0.928611	0.0342291
$737$	−47.1994	−1.73861
$738$	0	0
$739$	−10.7053	−0.393802	−0.196901	−	0.980423i	$-0.563088\pi$
$739$	−10.7053	−0.393802	−0.196901	+	0.980423i	$0.563088\pi$
$740$	−28.8733	−1.06140
$741$	0	0
$742$	−1.25824	−0.0461914
$743$	−41.6683	−1.52866	−0.764330	−	0.644825i	$-0.776930\pi$
$743$	−41.6683	−1.52866	−0.764330	+	0.644825i	$0.776930\pi$
$744$	0	0
$745$	−50.5064	−1.85041
$746$	−0.0400386	−0.00146592
$747$	0	0
$748$	−67.1563	−2.45548
$749$	2.92486	0.106872
$750$	0	0
$751$	−24.2164	−0.883670	−0.441835	−	0.897096i	$-0.645672\pi$
$751$	−24.2164	−0.883670	−0.441835	+	0.897096i	$0.645672\pi$
$752$	52.3079	1.90747
$753$	0	0
$754$	−1.03986	−0.0378696
$755$	−2.93818	−0.106931
$756$	0	0
$757$	30.7717	1.11842	0.559208	−	0.829027i	$-0.311105\pi$
$757$	30.7717	1.11842	0.559208	+	0.829027i	$0.311105\pi$
$758$	1.83337	0.0665911
$759$	0	0
$760$	−2.96443	−0.107531
$761$	28.3794	1.02875	0.514376	−	0.857565i	$-0.328024\pi$
$761$	28.3794	1.02875	0.514376	+	0.857565i	$0.328024\pi$
$762$	0	0
$763$	−9.86427	−0.357110
$764$	−10.3676	−0.375088
$765$	0	0
$766$	1.66491	0.0601555
$767$	−18.8757	−0.681563
$768$	0	0
$769$	7.32105	0.264004	0.132002	−	0.991249i	$-0.457860\pi$
$769$	7.32105	0.264004	0.132002	+	0.991249i	$0.457860\pi$
$770$	−1.94307	−0.0700235
$771$	0	0
$772$	36.6296	1.31833
$773$	20.8330	0.749312	0.374656	−	0.927164i	$-0.377761\pi$
$773$	20.8330	0.749312	0.374656	+	0.927164i	$0.377761\pi$
$774$	0	0
$775$	1.07310	0.0385467
$776$	2.34787	0.0842836
$777$	0	0
$778$	0.204403	0.00732820
$779$	27.0189	0.968053
$780$	0	0
$781$	32.2933	1.15555
$782$	−0.469325	−0.0167830
$783$	0	0
$784$	−17.2514	−0.616120
$785$	37.7187	1.34624
$786$	0	0
$787$	−1.94765	−0.0694262	−0.0347131	−	0.999397i	$-0.511052\pi$
$787$	−1.94765	−0.0694262	−0.0347131	+	0.999397i	$0.511052\pi$
$788$	−0.333070	−0.0118651
$789$	0	0
$790$	0.162736	0.00578988
$791$	10.7231	0.381269
$792$	0	0
$793$	16.8046	0.596747
$794$	1.72288	0.0611428
$795$	0	0
$796$	−1.78268	−0.0631853
$797$	40.1953	1.42379	0.711895	−	0.702286i	$-0.247837\pi$
$797$	40.1953	1.42379	0.711895	+	0.702286i	$0.247837\pi$
$798$	0	0
$799$	−79.7835	−2.82254
$800$	1.47216	0.0520488
$801$	0	0
$802$	1.15097	0.0406422
$803$	−82.8018	−2.92201
$804$	0	0
$805$	3.81272	0.134381
$806$	0.122323	0.00430864
$807$	0	0
$808$	−1.55261	−0.0546207
$809$	20.0648	0.705442	0.352721	−	0.935729i	$-0.385257\pi$
$809$	20.0648	0.705442	0.352721	+	0.935729i	$0.385257\pi$
$810$	0	0
$811$	−32.2480	−1.13238	−0.566190	−	0.824275i	$-0.691583\pi$
$811$	−32.2480	−1.13238	−0.566190	+	0.824275i	$0.691583\pi$
$812$	−20.1926	−0.708622
$813$	0	0
$814$	2.68035	0.0939463
$815$	53.7939	1.88432
$816$	0	0
$817$	−19.5923	−0.685449
$818$	2.13583	0.0746774
$819$	0	0
$820$	−39.4916	−1.37911
$821$	25.4724	0.888994	0.444497	−	0.895780i	$-0.353382\pi$
$821$	25.4724	0.888994	0.444497	+	0.895780i	$0.353382\pi$
$822$	0	0
$823$	27.8403	0.970453	0.485226	−	0.874389i	$-0.338737\pi$
$823$	27.8403	0.970453	0.485226	+	0.874389i	$0.338737\pi$
$824$	2.67928	0.0933373
$825$	0	0
$826$	1.30545	0.0454225
$827$	52.1370	1.81298	0.906491	−	0.422225i	$-0.138751\pi$
$827$	52.1370	1.81298	0.906491	+	0.422225i	$0.138751\pi$
$828$	0	0
$829$	−50.4899	−1.75359	−0.876794	−	0.480866i	$-0.840322\pi$
$829$	−50.4899	−1.75359	−0.876794	+	0.480866i	$0.840322\pi$
$830$	−2.19332	−0.0761312
$831$	0	0
$832$	−15.5005	−0.537382
$833$	26.3130	0.911690
$834$	0	0
$835$	32.3137	1.11826
$836$	−38.5649	−1.33380
$837$	0	0
$838$	−1.85667	−0.0641378
$839$	24.1351	0.833236	0.416618	−	0.909082i	$-0.363215\pi$
$839$	24.1351	0.833236	0.416618	+	0.909082i	$0.363215\pi$
$840$	0	0
$841$	9.87565	0.340540
$842$	−2.70747	−0.0933055
$843$	0	0
$844$	35.2451	1.21319
$845$	23.0869	0.794213
$846$	0	0
$847$	−32.7702	−1.12600
$848$	36.3704	1.24897
$849$	0	0
$850$	−0.744038	−0.0255203
$851$	−5.25941	−0.180290
$852$	0	0
$853$	1.52495	0.0522131	0.0261066	−	0.999659i	$-0.491689\pi$
$853$	1.52495	0.0522131	0.0261066	+	0.999659i	$0.491689\pi$
$854$	−1.16221	−0.0397700
$855$	0	0
$856$	0.605467	0.0206944
$857$	9.11787	0.311460	0.155730	−	0.987800i	$-0.450227\pi$
$857$	9.11787	0.311460	0.155730	+	0.987800i	$0.450227\pi$
$858$	0	0
$859$	48.8861	1.66797	0.833986	−	0.551786i	$-0.186053\pi$
$859$	48.8861	1.66797	0.833986	+	0.551786i	$0.186053\pi$
$860$	28.6367	0.976504
$861$	0	0
$862$	0.592769	0.0201898
$863$	23.8511	0.811900	0.405950	−	0.913895i	$-0.366941\pi$
$863$	23.8511	0.811900	0.405950	+	0.913895i	$0.366941\pi$
$864$	0	0
$865$	−29.6376	−1.00771
$866$	−1.44918	−0.0492452
$867$	0	0
$868$	2.37533	0.0806239
$869$	4.24168	0.143889
$870$	0	0
$871$	16.7368	0.567105
$872$	−2.04197	−0.0691498
$873$	0	0
$874$	−0.269513	−0.00911640
$875$	−14.6121	−0.493979
$876$	0	0
$877$	28.3993	0.958976	0.479488	−	0.877548i	$-0.340822\pi$
$877$	28.3993	0.958976	0.479488	+	0.877548i	$0.340822\pi$
$878$	−0.751162	−0.0253505
$879$	0	0
$880$	56.1661	1.89336
$881$	−27.8057	−0.936798	−0.468399	−	0.883517i	$-0.655169\pi$
$881$	−27.8057	−0.936798	−0.468399	+	0.883517i	$0.655169\pi$
$882$	0	0
$883$	27.9154	0.939429	0.469714	−	0.882818i	$-0.344357\pi$
$883$	27.9154	0.939429	0.469714	+	0.882818i	$0.344357\pi$
$884$	23.8135	0.800934
$885$	0	0
$886$	1.15658	0.0388562
$887$	−27.5730	−0.925809	−0.462905	−	0.886408i	$-0.653193\pi$
$887$	−27.5730	−0.925809	−0.462905	+	0.886408i	$0.653193\pi$
$888$	0	0
$889$	−13.6171	−0.456702
$890$	1.80235	0.0604148
$891$	0	0
$892$	−30.0854	−1.00734
$893$	−45.8162	−1.53318
$894$	0	0
$895$	16.9238	0.565699
$896$	4.34230	0.145066
$897$	0	0
$898$	−1.59691	−0.0532896
$899$	−4.57308	−0.152521
$900$	0	0
$901$	−55.4747	−1.84813
$902$	3.66607	0.122067
$903$	0	0
$904$	2.21975	0.0738279
$905$	6.23993	0.207422
$906$	0	0
$907$	−8.86552	−0.294375	−0.147187	−	0.989109i	$-0.547022\pi$
$907$	−8.86552	−0.294375	−0.147187	+	0.989109i	$0.547022\pi$
$908$	12.2230	0.405634
$909$	0	0
$910$	0.689010	0.0228405
$911$	19.0682	0.631757	0.315879	−	0.948800i	$-0.397701\pi$
$911$	19.0682	0.631757	0.315879	+	0.948800i	$0.397701\pi$
$912$	0	0
$913$	−57.1684	−1.89200
$914$	2.63327	0.0871008
$915$	0	0
$916$	−9.37619	−0.309798
$917$	−2.39527	−0.0790989
$918$	0	0
$919$	46.0109	1.51776	0.758879	−	0.651231i	$-0.225747\pi$
$919$	46.0109	1.51776	0.758879	+	0.651231i	$0.225747\pi$
$920$	0.789258	0.0260211
$921$	0	0
$922$	1.75365	0.0577533
$923$	−11.4512	−0.376919
$924$	0	0
$925$	−8.33794	−0.274150
$926$	−0.854846	−0.0280920
$927$	0	0
$928$	−6.27373	−0.205945
$929$	21.4459	0.703619	0.351809	−	0.936072i	$-0.385567\pi$
$929$	21.4459	0.703619	0.351809	+	0.936072i	$0.385567\pi$
$930$	0	0
$931$	15.1104	0.495223
$932$	−42.6435	−1.39684
$933$	0	0
$934$	3.23181	0.105748
$935$	−85.6684	−2.80166
$936$	0	0
$937$	−23.3147	−0.761657	−0.380828	−	0.924646i	$-0.624361\pi$
$937$	−23.3147	−0.761657	−0.380828	+	0.924646i	$0.624361\pi$
$938$	−1.15752	−0.0377945
$939$	0	0
$940$	66.9662	2.18420
$941$	−11.4999	−0.374887	−0.187443	−	0.982275i	$-0.560020\pi$
$941$	−11.4999	−0.374887	−0.187443	+	0.982275i	$0.560020\pi$
$942$	0	0
$943$	−7.19360	−0.234256
$944$	−37.7352	−1.22818
$945$	0	0
$946$	−2.65839	−0.0864318
$947$	34.0166	1.10539	0.552696	−	0.833383i	$-0.313599\pi$
$947$	34.0166	1.10539	0.552696	+	0.833383i	$0.313599\pi$
$948$	0	0
$949$	29.3614	0.953110
$950$	−0.427268	−0.0138624
$951$	0	0
$952$	−3.29977	−0.106946
$953$	14.5269	0.470572	0.235286	−	0.971926i	$-0.424397\pi$
$953$	14.5269	0.470572	0.235286	+	0.971926i	$0.424397\pi$
$954$	0	0
$955$	−13.2256	−0.427969
$956$	1.99290	0.0644551
$957$	0	0
$958$	3.63107	0.117315
$959$	−10.9254	−0.352798
$960$	0	0
$961$	−30.4621	−0.982647
$962$	−0.950447	−0.0306436
$963$	0	0
$964$	−50.0286	−1.61131
$965$	46.7267	1.50419
$966$	0	0
$967$	−16.5188	−0.531209	−0.265604	−	0.964082i	$-0.585571\pi$
$967$	−16.5188	−0.531209	−0.265604	+	0.964082i	$0.585571\pi$
$968$	−6.78366	−0.218035
$969$	0	0
$970$	1.49488	0.0479976
$971$	−55.8116	−1.79108	−0.895539	−	0.444982i	$-0.853210\pi$
$971$	−55.8116	−1.79108	−0.895539	+	0.444982i	$0.853210\pi$
$972$	0	0
$973$	18.6808	0.598879
$974$	2.02212	0.0647929
$975$	0	0
$976$	33.5946	1.07534
$977$	−40.0390	−1.28096	−0.640481	−	0.767974i	$-0.721265\pi$
$977$	−40.0390	−1.28096	−0.640481	+	0.767974i	$0.721265\pi$
$978$	0	0
$979$	46.9778	1.50142
$980$	−22.0858	−0.705504
$981$	0	0
$982$	−1.29797	−0.0414199
$983$	14.1711	0.451989	0.225995	−	0.974129i	$-0.427437\pi$
$983$	14.1711	0.451989	0.225995	+	0.974129i	$0.427437\pi$
$984$	0	0
$985$	−0.424883	−0.0135379
$986$	3.17078	0.100978
$987$	0	0
$988$	13.6750	0.435061
$989$	5.21632	0.165869
$990$	0	0
$991$	22.0470	0.700347	0.350173	−	0.936685i	$-0.386123\pi$
$991$	22.0470	0.700347	0.350173	+	0.936685i	$0.386123\pi$
$992$	0.738000	0.0234315
$993$	0	0
$994$	0.791966	0.0251196
$995$	−2.27408	−0.0720933
$996$	0	0
$997$	−8.42514	−0.266827	−0.133413	−	0.991060i	$-0.542594\pi$
$997$	−8.42514	−0.266827	−0.133413	+	0.991060i	$0.542594\pi$
$998$	2.85446	0.0903563
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.i.1.9		17
3.2	odd	2		239.2.a.b.1.9	✓	17
12.11	even	2		3824.2.a.p.1.15		17
15.14	odd	2		5975.2.a.g.1.9		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.2.a.b.1.9	✓	17	3.2	odd	2
2151.2.a.i.1.9		17	1.1	even	1	trivial
3824.2.a.p.1.15		17	12.11	even	2
5975.2.a.g.1.9		17	15.14	odd	2

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)

\(f(q)\)	\(=\)	\(q+0.0842488 q^{2} -1.99290 q^{4} -2.54226 q^{5} -1.62505 q^{7} -0.336397 q^{8} +O(q^{10})\)	\(q+0.0842488 q^{2} -1.99290 q^{4} -2.54226 q^{5} -1.62505 q^{7} -0.336397 q^{8} -0.214182 q^{10} -5.58262 q^{11} +1.97958 q^{13} -0.136909 q^{14} +3.95746 q^{16} -6.03619 q^{17} -3.46632 q^{19} +5.06647 q^{20} -0.470329 q^{22} +0.922884 q^{23} +1.46308 q^{25} +0.166778 q^{26} +3.23857 q^{28} -6.23503 q^{29} +0.733449 q^{31} +1.00621 q^{32} -0.508542 q^{34} +4.13131 q^{35} -5.69889 q^{37} -0.292033 q^{38} +0.855209 q^{40} -7.79470 q^{41} +5.65220 q^{43} +11.1256 q^{44} +0.0777518 q^{46} +13.2175 q^{47} -4.35920 q^{49} +0.123263 q^{50} -3.94512 q^{52} +9.19034 q^{53} +14.1925 q^{55} +0.546664 q^{56} -0.525294 q^{58} -9.53520 q^{59} +8.48893 q^{61} +0.0617922 q^{62} -7.83015 q^{64} -5.03262 q^{65} +8.45471 q^{67} +12.0295 q^{68} +0.348058 q^{70} -5.78462 q^{71} +14.8321 q^{73} -0.480124 q^{74} +6.90803 q^{76} +9.07205 q^{77} -0.759801 q^{79} -10.0609 q^{80} -0.656694 q^{82} +10.2404 q^{83} +15.3456 q^{85} +0.476191 q^{86} +1.87798 q^{88} -8.41501 q^{89} -3.21693 q^{91} -1.83922 q^{92} +1.11356 q^{94} +8.81228 q^{95} -6.97946 q^{97} -0.367257 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8	\( 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8} + 5 q^{10} + q^{11} + 15 q^{13} + 3 q^{14} + 24 q^{16} - 4 q^{17} + 24 q^{19} - 4 q^{20} - 10 q^{22} + 9 q^{23} + 39 q^{25} + 12 q^{26} - 7 q^{28} + 2 q^{29} + 28 q^{31} + 31 q^{32} + 29 q^{34} + 24 q^{35} + 11 q^{37} + 19 q^{38} - 18 q^{40} - 20 q^{41} - 9 q^{43} + 43 q^{44} - 18 q^{46} + 18 q^{47} + 60 q^{49} + 61 q^{50} - q^{52} + 12 q^{53} - 10 q^{55} + 60 q^{56} - 38 q^{58} - q^{59} + 24 q^{61} + 33 q^{62} + 21 q^{64} - 2 q^{65} + 16 q^{67} + 10 q^{68} + 7 q^{70} - 12 q^{71} + 30 q^{73} + 21 q^{74} + 75 q^{76} + 15 q^{77} - 10 q^{79} - 32 q^{80} + 50 q^{82} + 16 q^{83} - 18 q^{85} + 3 q^{86} - 28 q^{88} - 65 q^{89} + 47 q^{91} - 24 q^{92} + 32 q^{94} + 37 q^{95} + 87 q^{97} + 9 q^{98}+O(q^{100}) \) 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8 + 5 * q^10 + q^11 + 15 * q^13 + 3 * q^14 + 24 * q^16 - 4 * q^17 + 24 * q^19 - 4 * q^20 - 10 * q^22 + 9 * q^23 + 39 * q^25 + 12 * q^26 - 7 * q^28 + 2 * q^29 + 28 * q^31 + 31 * q^32 + 29 * q^34 + 24 * q^35 + 11 * q^37 + 19 * q^38 - 18 * q^40 - 20 * q^41 - 9 * q^43 + 43 * q^44 - 18 * q^46 + 18 * q^47 + 60 * q^49 + 61 * q^50 - q^52 + 12 * q^53 - 10 * q^55 + 60 * q^56 - 38 * q^58 - q^59 + 24 * q^61 + 33 * q^62 + 21 * q^64 - 2 * q^65 + 16 * q^67 + 10 * q^68 + 7 * q^70 - 12 * q^71 + 30 * q^73 + 21 * q^74 + 75 * q^76 + 15 * q^77 - 10 * q^79 - 32 * q^80 + 50 * q^82 + 16 * q^83 - 18 * q^85 + 3 * q^86 - 28 * q^88 - 65 * q^89 + 47 * q^91 - 24 * q^92 + 32 * q^94 + 37 * q^95 + 87 * q^97 + 9 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more