LMFDB - Embedded newform 2151.2.a.i.1.4

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			1.4
Root			$-2.08085$ 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)

$f(q)$	$=$	$q-2.08085 q^{2} +2.32992 q^{4} -2.29912 q^{5} +3.55134 q^{7} -0.686506 q^{8} +O(q^{10})$	\(q-2.08085 q^{2} +2.32992 q^{4} -2.29912 q^{5} +3.55134 q^{7} -0.686506 q^{8} +4.78411 q^{10} +2.81100 q^{11} +3.36181 q^{13} -7.38978 q^{14} -3.23132 q^{16} -1.06613 q^{17} +3.41036 q^{19} -5.35676 q^{20} -5.84925 q^{22} +4.88027 q^{23} +0.285951 q^{25} -6.99540 q^{26} +8.27432 q^{28} +10.1328 q^{29} +3.57478 q^{31} +8.09689 q^{32} +2.21844 q^{34} -8.16495 q^{35} +1.70546 q^{37} -7.09644 q^{38} +1.57836 q^{40} -5.33434 q^{41} -5.12092 q^{43} +6.54939 q^{44} -10.1551 q^{46} +7.02348 q^{47} +5.61200 q^{49} -0.595020 q^{50} +7.83273 q^{52} -4.51898 q^{53} -6.46282 q^{55} -2.43801 q^{56} -21.0848 q^{58} -8.30124 q^{59} -9.87602 q^{61} -7.43857 q^{62} -10.3857 q^{64} -7.72920 q^{65} +1.17442 q^{67} -2.48399 q^{68} +16.9900 q^{70} -3.94473 q^{71} -3.16042 q^{73} -3.54880 q^{74} +7.94586 q^{76} +9.98281 q^{77} +7.95744 q^{79} +7.42919 q^{80} +11.0999 q^{82} +9.16760 q^{83} +2.45115 q^{85} +10.6558 q^{86} -1.92977 q^{88} -13.1736 q^{89} +11.9389 q^{91} +11.3706 q^{92} -14.6148 q^{94} -7.84083 q^{95} -6.36314 q^{97} -11.6777 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.08085	−1.47138	−0.735690	−	0.677319i	$-0.763142\pi$
$2$	−2.08085	−1.47138	−0.735690	+	0.677319i	$0.763142\pi$
$3$	0	0
$3$	0	0
$4$	2.32992	1.16496
$5$	−2.29912	−1.02820	−0.514099	−	0.857731i	$-0.671874\pi$
$5$	−2.29912	−1.02820	−0.514099	+	0.857731i	$0.671874\pi$
$6$	0	0
$7$	3.55134	1.34228	0.671140	−	0.741331i	$-0.265805\pi$
$7$	3.55134	1.34228	0.671140	+	0.741331i	$0.265805\pi$
$8$	−0.686506	−0.242716
$9$	0	0
$10$	4.78411	1.51287
$11$	2.81100	0.847548	0.423774	−	0.905768i	$-0.360705\pi$
$11$	2.81100	0.847548	0.423774	+	0.905768i	$0.360705\pi$
$12$	0	0
$13$	3.36181	0.932398	0.466199	−	0.884680i	$-0.345623\pi$
$13$	3.36181	0.932398	0.466199	+	0.884680i	$0.345623\pi$
$14$	−7.38978	−1.97500
$15$	0	0
$16$	−3.23132	−0.807830
$17$	−1.06613	−0.258574	−0.129287	−	0.991607i	$-0.541269\pi$
$17$	−1.06613	−0.258574	−0.129287	+	0.991607i	$0.541269\pi$
$18$	0	0
$19$	3.41036	0.782391	0.391196	−	0.920308i	$-0.372062\pi$
$19$	3.41036	0.782391	0.391196	+	0.920308i	$0.372062\pi$
$20$	−5.35676	−1.19781
$21$	0	0
$22$	−5.84925	−1.24706
$23$	4.88027	1.01761	0.508803	−	0.860883i	$-0.330088\pi$
$23$	4.88027	1.01761	0.508803	+	0.860883i	$0.330088\pi$
$24$	0	0
$25$	0.285951	0.0571902
$26$	−6.99540	−1.37191
$27$	0	0
$28$	8.27432	1.56370
$29$	10.1328	1.88161	0.940807	−	0.338944i	$-0.110070\pi$
$29$	10.1328	1.88161	0.940807	+	0.338944i	$0.110070\pi$
$30$	0	0
$31$	3.57478	0.642050	0.321025	−	0.947071i	$-0.395973\pi$
$31$	3.57478	0.642050	0.321025	+	0.947071i	$0.395973\pi$
$32$	8.09689	1.43134
$33$	0	0
$34$	2.21844	0.380460
$35$	−8.16495	−1.38013
$36$	0	0
$37$	1.70546	0.280376	0.140188	−	0.990125i	$-0.455229\pi$
$37$	1.70546	0.280376	0.140188	+	0.990125i	$0.455229\pi$
$38$	−7.09644	−1.15119
$39$	0	0
$40$	1.57836	0.249560
$41$	−5.33434	−0.833084	−0.416542	−	0.909116i	$-0.636758\pi$
$41$	−5.33434	−0.833084	−0.416542	+	0.909116i	$0.636758\pi$
$42$	0	0
$43$	−5.12092	−0.780934	−0.390467	−	0.920617i	$-0.627686\pi$
$43$	−5.12092	−0.780934	−0.390467	+	0.920617i	$0.627686\pi$
$44$	6.54939	0.987358
$45$	0	0
$46$	−10.1551	−1.49728
$47$	7.02348	1.02448	0.512240	−	0.858842i	$-0.328816\pi$
$47$	7.02348	1.02448	0.512240	+	0.858842i	$0.328816\pi$
$48$	0	0
$49$	5.61200	0.801714
$50$	−0.595020	−0.0841486
$51$	0	0
$52$	7.83273	1.08620
$53$	−4.51898	−0.620729	−0.310365	−	0.950618i	$-0.600451\pi$
$53$	−4.51898	−0.620729	−0.310365	+	0.950618i	$0.600451\pi$
$54$	0	0
$55$	−6.46282	−0.871447
$56$	−2.43801	−0.325793
$57$	0	0
$58$	−21.0848	−2.76857
$59$	−8.30124	−1.08073	−0.540365	−	0.841431i	$-0.681714\pi$
$59$	−8.30124	−1.08073	−0.540365	+	0.841431i	$0.681714\pi$
$60$	0	0
$61$	−9.87602	−1.26449	−0.632247	−	0.774767i	$-0.717867\pi$
$61$	−9.87602	−1.26449	−0.632247	+	0.774767i	$0.717867\pi$
$62$	−7.43857	−0.944699
$63$	0	0
$64$	−10.3857	−1.29822
$65$	−7.72920	−0.958689
$66$	0	0
$67$	1.17442	0.143478	0.0717389	−	0.997423i	$-0.477145\pi$
$67$	1.17442	0.143478	0.0717389	+	0.997423i	$0.477145\pi$
$68$	−2.48399	−0.301228
$69$	0	0
$70$	16.9900	2.03069
$71$	−3.94473	−0.468154	−0.234077	−	0.972218i	$-0.575207\pi$
$71$	−3.94473	−0.468154	−0.234077	+	0.972218i	$0.575207\pi$
$72$	0	0
$73$	−3.16042	−0.369899	−0.184950	−	0.982748i	$-0.559212\pi$
$73$	−3.16042	−0.369899	−0.184950	+	0.982748i	$0.559212\pi$
$74$	−3.54880	−0.412539
$75$	0	0
$76$	7.94586	0.911453
$77$	9.98281	1.13765
$78$	0	0
$79$	7.95744	0.895281	0.447641	−	0.894214i	$-0.352264\pi$
$79$	7.95744	0.895281	0.447641	+	0.894214i	$0.352264\pi$
$80$	7.42919	0.830609
$81$	0	0
$82$	11.0999	1.22578
$83$	9.16760	1.00628	0.503138	−	0.864206i	$-0.332179\pi$
$83$	9.16760	1.00628	0.503138	+	0.864206i	$0.332179\pi$
$84$	0	0
$85$	2.45115	0.265865
$86$	10.6558	1.14905
$87$	0	0
$88$	−1.92977	−0.205714
$89$	−13.1736	−1.39640	−0.698200	−	0.715902i	$-0.746016\pi$
$89$	−13.1736	−1.39640	−0.698200	+	0.715902i	$0.746016\pi$
$90$	0	0
$91$	11.9389	1.25154
$92$	11.3706	1.18547
$93$	0	0
$94$	−14.6148	−1.50740
$95$	−7.84083	−0.804453
$96$	0	0
$97$	−6.36314	−0.646079	−0.323040	−	0.946385i	$-0.604705\pi$
$97$	−6.36314	−0.646079	−0.323040	+	0.946385i	$0.604705\pi$
$98$	−11.6777	−1.17963
$99$	0	0
$100$	0.666243	0.0666243
$101$	10.5905	1.05380	0.526898	−	0.849929i	$-0.323355\pi$
$101$	10.5905	1.05380	0.526898	+	0.849929i	$0.323355\pi$
$102$	0	0
$103$	−2.50099	−0.246430	−0.123215	−	0.992380i	$-0.539321\pi$
$103$	−2.50099	−0.246430	−0.123215	+	0.992380i	$0.539321\pi$
$104$	−2.30790	−0.226308
$105$	0	0
$106$	9.40330	0.913329
$107$	4.24940	0.410805	0.205402	−	0.978678i	$-0.434150\pi$
$107$	4.24940	0.410805	0.205402	+	0.978678i	$0.434150\pi$
$108$	0	0
$109$	−6.78293	−0.649687	−0.324843	−	0.945768i	$-0.605312\pi$
$109$	−6.78293	−0.649687	−0.324843	+	0.945768i	$0.605312\pi$
$110$	13.4481	1.28223
$111$	0	0
$112$	−11.4755	−1.08433
$113$	4.17583	0.392829	0.196414	−	0.980521i	$-0.437070\pi$
$113$	4.17583	0.392829	0.196414	+	0.980521i	$0.437070\pi$
$114$	0	0
$115$	−11.2203	−1.04630
$116$	23.6086	2.19200
$117$	0	0
$118$	17.2736	1.59016
$119$	−3.78618	−0.347078
$120$	0	0
$121$	−3.09829	−0.281662
$122$	20.5505	1.86055
$123$	0	0
$124$	8.32894	0.747961
$125$	10.8382	0.969395
$126$	0	0
$127$	2.91690	0.258833	0.129416	−	0.991590i	$-0.458690\pi$
$127$	2.91690	0.258833	0.129416	+	0.991590i	$0.458690\pi$
$128$	5.41732	0.478828
$129$	0	0
$130$	16.0833	1.41060
$131$	16.9226	1.47854	0.739268	−	0.673412i	$-0.235172\pi$
$131$	16.9226	1.47854	0.739268	+	0.673412i	$0.235172\pi$
$132$	0	0
$133$	12.1114	1.05019
$134$	−2.44378	−0.211110
$135$	0	0
$136$	0.731902	0.0627601
$137$	−2.89827	−0.247616	−0.123808	−	0.992306i	$-0.539511\pi$
$137$	−2.89827	−0.247616	−0.123808	+	0.992306i	$0.539511\pi$
$138$	0	0
$139$	15.6979	1.33148	0.665739	−	0.746185i	$-0.268116\pi$
$139$	15.6979	1.33148	0.665739	+	0.746185i	$0.268116\pi$
$140$	−19.0237	−1.60779
$141$	0	0
$142$	8.20838	0.688832
$143$	9.45004	0.790252
$144$	0	0
$145$	−23.2965	−1.93467
$146$	6.57635	0.544262
$147$	0	0
$148$	3.97358	0.326626
$149$	−8.45272	−0.692474	−0.346237	−	0.938147i	$-0.612541\pi$
$149$	−8.45272	−0.692474	−0.346237	+	0.938147i	$0.612541\pi$
$150$	0	0
$151$	21.0391	1.71214	0.856070	−	0.516860i	$-0.172899\pi$
$151$	21.0391	1.71214	0.856070	+	0.516860i	$0.172899\pi$
$152$	−2.34123	−0.189899
$153$	0	0
$154$	−20.7727	−1.67391
$155$	−8.21885	−0.660154
$156$	0	0
$157$	3.40689	0.271899	0.135950	−	0.990716i	$-0.456591\pi$
$157$	3.40689	0.271899	0.135950	+	0.990716i	$0.456591\pi$
$158$	−16.5582	−1.31730
$159$	0	0
$160$	−18.6157	−1.47170
$161$	17.3315	1.36591
$162$	0	0
$163$	−18.7701	−1.47019	−0.735094	−	0.677966i	$-0.762862\pi$
$163$	−18.7701	−1.47019	−0.735094	+	0.677966i	$0.762862\pi$
$164$	−12.4286	−0.970509
$165$	0	0
$166$	−19.0764	−1.48061
$167$	15.6894	1.21408	0.607042	−	0.794670i	$-0.292356\pi$
$167$	15.6894	1.21408	0.607042	+	0.794670i	$0.292356\pi$
$168$	0	0
$169$	−1.69825	−0.130634
$170$	−5.10047	−0.391188
$171$	0	0
$172$	−11.9313	−0.909755
$173$	−14.5730	−1.10797	−0.553984	−	0.832527i	$-0.686893\pi$
$173$	−14.5730	−1.10797	−0.553984	+	0.832527i	$0.686893\pi$
$174$	0	0
$175$	1.01551	0.0767653
$176$	−9.08324	−0.684675
$177$	0	0
$178$	27.4123	2.05464
$179$	−0.310125	−0.0231798	−0.0115899	−	0.999933i	$-0.503689\pi$
$179$	−0.310125	−0.0231798	−0.0115899	+	0.999933i	$0.503689\pi$
$180$	0	0
$181$	−1.05759	−0.0786102	−0.0393051	−	0.999227i	$-0.512514\pi$
$181$	−1.05759	−0.0786102	−0.0393051	+	0.999227i	$0.512514\pi$
$182$	−24.8430	−1.84149
$183$	0	0
$184$	−3.35033	−0.246990
$185$	−3.92106	−0.288282
$186$	0	0
$187$	−2.99688	−0.219154
$188$	16.3641	1.19348
$189$	0	0
$190$	16.3156	1.18366
$191$	−9.58937	−0.693862	−0.346931	−	0.937891i	$-0.612776\pi$
$191$	−9.58937	−0.693862	−0.346931	+	0.937891i	$0.612776\pi$
$192$	0	0
$193$	5.50045	0.395931	0.197965	−	0.980209i	$-0.436567\pi$
$193$	5.50045	0.395931	0.197965	+	0.980209i	$0.436567\pi$
$194$	13.2407	0.950628
$195$	0	0
$196$	13.0755	0.933964
$197$	−12.1490	−0.865580	−0.432790	−	0.901495i	$-0.642471\pi$
$197$	−12.1490	−0.865580	−0.432790	+	0.901495i	$0.642471\pi$
$198$	0	0
$199$	25.4760	1.80595	0.902974	−	0.429695i	$-0.141379\pi$
$199$	25.4760	1.80595	0.902974	+	0.429695i	$0.141379\pi$
$200$	−0.196307	−0.0138810
$201$	0	0
$202$	−22.0372	−1.55053
$203$	35.9850	2.52565
$204$	0	0
$205$	12.2643	0.856575
$206$	5.20418	0.362592
$207$	0	0
$208$	−10.8631	−0.753219
$209$	9.58653	0.663114
$210$	0	0
$211$	15.8300	1.08978	0.544891	−	0.838507i	$-0.316571\pi$
$211$	15.8300	1.08978	0.544891	+	0.838507i	$0.316571\pi$
$212$	−10.5288	−0.723124
$213$	0	0
$214$	−8.84234	−0.604450
$215$	11.7736	0.802954
$216$	0	0
$217$	12.6953	0.861810
$218$	14.1142	0.955936
$219$	0	0
$220$	−15.0578	−1.01520
$221$	−3.58411	−0.241094
$222$	0	0
$223$	6.82304	0.456905	0.228452	−	0.973555i	$-0.426634\pi$
$223$	6.82304	0.456905	0.228452	+	0.973555i	$0.426634\pi$
$224$	28.7548	1.92126
$225$	0	0
$226$	−8.68925	−0.578000
$227$	−13.0991	−0.869417	−0.434708	−	0.900571i	$-0.643149\pi$
$227$	−13.0991	−0.869417	−0.434708	+	0.900571i	$0.643149\pi$
$228$	0	0
$229$	19.6724	1.29999	0.649995	−	0.759938i	$-0.274771\pi$
$229$	19.6724	1.29999	0.649995	+	0.759938i	$0.274771\pi$
$230$	23.3477	1.53950
$231$	0	0
$232$	−6.95622	−0.456698
$233$	−3.39374	−0.222331	−0.111166	−	0.993802i	$-0.535458\pi$
$233$	−3.39374	−0.222331	−0.111166	+	0.993802i	$0.535458\pi$
$234$	0	0
$235$	−16.1478	−1.05337
$236$	−19.3412	−1.25900
$237$	0	0
$238$	7.87844	0.510684
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	20.0828	1.29365	0.646825	−	0.762639i	$-0.276097\pi$
$241$	20.0828	1.29365	0.646825	+	0.762639i	$0.276097\pi$
$242$	6.44706	0.414432
$243$	0	0
$244$	−23.0103	−1.47308
$245$	−12.9027	−0.824321
$246$	0	0
$247$	11.4650	0.729500
$248$	−2.45411	−0.155836
$249$	0	0
$250$	−22.5525	−1.42635
$251$	−22.5079	−1.42069	−0.710343	−	0.703855i	$-0.751460\pi$
$251$	−22.5079	−1.42069	−0.710343	+	0.703855i	$0.751460\pi$
$252$	0	0
$253$	13.7184	0.862470
$254$	−6.06961	−0.380841
$255$	0	0
$256$	9.49886	0.593679
$257$	5.87549	0.366503	0.183251	−	0.983066i	$-0.441338\pi$
$257$	5.87549	0.366503	0.183251	+	0.983066i	$0.441338\pi$
$258$	0	0
$259$	6.05666	0.376343
$260$	−18.0084	−1.11683
$261$	0	0
$262$	−35.2133	−2.17549
$263$	26.1843	1.61460	0.807298	−	0.590145i	$-0.200929\pi$
$263$	26.1843	1.61460	0.807298	+	0.590145i	$0.200929\pi$
$264$	0	0
$265$	10.3897	0.638233
$266$	−25.2018	−1.54522
$267$	0	0
$268$	2.73629	0.167146
$269$	−4.39722	−0.268103	−0.134052	−	0.990974i	$-0.542799\pi$
$269$	−4.39722	−0.268103	−0.134052	+	0.990974i	$0.542799\pi$
$270$	0	0
$271$	23.5374	1.42979	0.714897	−	0.699229i	$-0.246473\pi$
$271$	23.5374	1.42979	0.714897	+	0.699229i	$0.246473\pi$
$272$	3.44500	0.208884
$273$	0	0
$274$	6.03085	0.364337
$275$	0.803808	0.0484715
$276$	0	0
$277$	−8.51270	−0.511478	−0.255739	−	0.966746i	$-0.582319\pi$
$277$	−8.51270	−0.511478	−0.255739	+	0.966746i	$0.582319\pi$
$278$	−32.6649	−1.95911
$279$	0	0
$280$	5.60529	0.334980
$281$	−1.90942	−0.113907	−0.0569533	−	0.998377i	$-0.518139\pi$
$281$	−1.90942	−0.113907	−0.0569533	+	0.998377i	$0.518139\pi$
$282$	0	0
$283$	15.6197	0.928493	0.464246	−	0.885706i	$-0.346325\pi$
$283$	15.6197	0.928493	0.464246	+	0.885706i	$0.346325\pi$
$284$	−9.19090	−0.545380
$285$	0	0
$286$	−19.6641	−1.16276
$287$	−18.9441	−1.11823
$288$	0	0
$289$	−15.8634	−0.933140
$290$	48.4764	2.84663
$291$	0	0
$292$	−7.36352	−0.430917
$293$	27.9005	1.62996	0.814981	−	0.579488i	$-0.196747\pi$
$293$	27.9005	1.62996	0.814981	+	0.579488i	$0.196747\pi$
$294$	0	0
$295$	19.0855	1.11120
$296$	−1.17081	−0.0680518
$297$	0	0
$298$	17.5888	1.01889
$299$	16.4065	0.948814
$300$	0	0
$301$	−18.1861	−1.04823
$302$	−43.7792	−2.51921
$303$	0	0
$304$	−11.0200	−0.632039
$305$	22.7062	1.30015
$306$	0	0
$307$	13.9794	0.797846	0.398923	−	0.916984i	$-0.369384\pi$
$307$	13.9794	0.797846	0.398923	+	0.916984i	$0.369384\pi$
$308$	23.2591	1.32531
$309$	0	0
$310$	17.1022	0.971337
$311$	10.8223	0.613674	0.306837	−	0.951762i	$-0.400729\pi$
$311$	10.8223	0.613674	0.306837	+	0.951762i	$0.400729\pi$
$312$	0	0
$313$	−1.79652	−0.101545	−0.0507726	−	0.998710i	$-0.516168\pi$
$313$	−1.79652	−0.101545	−0.0507726	+	0.998710i	$0.516168\pi$
$314$	−7.08920	−0.400067
$315$	0	0
$316$	18.5402	1.04297
$317$	−19.6956	−1.10621	−0.553107	−	0.833110i	$-0.686558\pi$
$317$	−19.6956	−1.10621	−0.553107	+	0.833110i	$0.686558\pi$
$318$	0	0
$319$	28.4833	1.59476
$320$	23.8780	1.33482
$321$	0	0
$322$	−36.0641	−2.00977
$323$	−3.63588	−0.202306
$324$	0	0
$325$	0.961313	0.0533241
$326$	39.0577	2.16320
$327$	0	0
$328$	3.66206	0.202203
$329$	24.9427	1.37514
$330$	0	0
$331$	−13.4591	−0.739782	−0.369891	−	0.929075i	$-0.620605\pi$
$331$	−13.4591	−0.739782	−0.369891	+	0.929075i	$0.620605\pi$
$332$	21.3598	1.17227
$333$	0	0
$334$	−32.6473	−1.78638
$335$	−2.70013	−0.147524
$336$	0	0
$337$	−24.3551	−1.32671	−0.663353	−	0.748307i	$-0.730867\pi$
$337$	−24.3551	−1.32671	−0.663353	+	0.748307i	$0.730867\pi$
$338$	3.53379	0.192213
$339$	0	0
$340$	5.71098	0.309721
$341$	10.0487	0.544168
$342$	0	0
$343$	−4.92926	−0.266155
$344$	3.51554	0.189545
$345$	0	0
$346$	30.3242	1.63024
$347$	14.6470	0.786292	0.393146	−	0.919476i	$-0.371387\pi$
$347$	14.6470	0.786292	0.393146	+	0.919476i	$0.371387\pi$
$348$	0	0
$349$	−14.0511	−0.752137	−0.376068	−	0.926592i	$-0.622724\pi$
$349$	−14.0511	−0.752137	−0.376068	+	0.926592i	$0.622724\pi$
$350$	−2.11312	−0.112951
$351$	0	0
$352$	22.7603	1.21313
$353$	4.38331	0.233300	0.116650	−	0.993173i	$-0.462784\pi$
$353$	4.38331	0.233300	0.116650	+	0.993173i	$0.462784\pi$
$354$	0	0
$355$	9.06942	0.481355
$356$	−30.6934	−1.62675
$357$	0	0
$358$	0.645321	0.0341063
$359$	−0.402143	−0.0212243	−0.0106121	−	0.999944i	$-0.503378\pi$
$359$	−0.402143	−0.0212243	−0.0106121	+	0.999944i	$0.503378\pi$
$360$	0	0
$361$	−7.36942	−0.387864
$362$	2.20069	0.115666
$363$	0	0
$364$	27.8167	1.45799
$365$	7.26619	0.380330
$366$	0	0
$367$	−7.63378	−0.398480	−0.199240	−	0.979951i	$-0.563847\pi$
$367$	−7.63378	−0.398480	−0.199240	+	0.979951i	$0.563847\pi$
$368$	−15.7697	−0.822053
$369$	0	0
$370$	8.15911	0.424172
$371$	−16.0484	−0.833192
$372$	0	0
$373$	9.96039	0.515730	0.257865	−	0.966181i	$-0.416981\pi$
$373$	9.96039	0.515730	0.257865	+	0.966181i	$0.416981\pi$
$374$	6.23604	0.322458
$375$	0	0
$376$	−4.82166	−0.248658
$377$	34.0645	1.75441
$378$	0	0
$379$	−24.7492	−1.27128	−0.635640	−	0.771986i	$-0.719264\pi$
$379$	−24.7492	−1.27128	−0.635640	+	0.771986i	$0.719264\pi$
$380$	−18.2685	−0.937154
$381$	0	0
$382$	19.9540	1.02093
$383$	−35.1620	−1.79669	−0.898347	−	0.439286i	$-0.855232\pi$
$383$	−35.1620	−1.79669	−0.898347	+	0.439286i	$0.855232\pi$
$384$	0	0
$385$	−22.9517	−1.16973
$386$	−11.4456	−0.582564
$387$	0	0
$388$	−14.8256	−0.752655
$389$	−13.5109	−0.685029	−0.342514	−	0.939513i	$-0.611279\pi$
$389$	−13.5109	−0.685029	−0.342514	+	0.939513i	$0.611279\pi$
$390$	0	0
$391$	−5.20298	−0.263126
$392$	−3.85267	−0.194589
$393$	0	0
$394$	25.2802	1.27360
$395$	−18.2951	−0.920526
$396$	0	0
$397$	7.26244	0.364491	0.182246	−	0.983253i	$-0.441663\pi$
$397$	7.26244	0.364491	0.182246	+	0.983253i	$0.441663\pi$
$398$	−53.0117	−2.65724
$399$	0	0
$400$	−0.924000	−0.0462000
$401$	14.4128	0.719741	0.359871	−	0.933002i	$-0.382821\pi$
$401$	14.4128	0.719741	0.359871	+	0.933002i	$0.382821\pi$
$402$	0	0
$403$	12.0177	0.598646
$404$	24.6750	1.22763
$405$	0	0
$406$	−74.8792	−3.71619
$407$	4.79404	0.237632
$408$	0	0
$409$	20.9811	1.03745	0.518724	−	0.854942i	$-0.326407\pi$
$409$	20.9811	1.03745	0.518724	+	0.854942i	$0.326407\pi$
$410$	−25.5201	−1.26035
$411$	0	0
$412$	−5.82711	−0.287081
$413$	−29.4805	−1.45064
$414$	0	0
$415$	−21.0774	−1.03465
$416$	27.2202	1.33458
$417$	0	0
$418$	−19.9481	−0.975692
$419$	30.9958	1.51424	0.757122	−	0.653273i	$-0.226605\pi$
$419$	30.9958	1.51424	0.757122	+	0.653273i	$0.226605\pi$
$420$	0	0
$421$	−7.39624	−0.360471	−0.180235	−	0.983624i	$-0.557686\pi$
$421$	−7.39624	−0.360471	−0.180235	+	0.983624i	$0.557686\pi$
$422$	−32.9398	−1.60348
$423$	0	0
$424$	3.10230	0.150661
$425$	−0.304860	−0.0147879
$426$	0	0
$427$	−35.0731	−1.69731
$428$	9.90074	0.478570
$429$	0	0
$430$	−24.4991	−1.18145
$431$	10.8125	0.520819	0.260410	−	0.965498i	$-0.416142\pi$
$431$	10.8125	0.520819	0.260410	+	0.965498i	$0.416142\pi$
$432$	0	0
$433$	−9.84646	−0.473191	−0.236595	−	0.971608i	$-0.576032\pi$
$433$	−9.84646	−0.473191	−0.236595	+	0.971608i	$0.576032\pi$
$434$	−26.4169	−1.26805
$435$	0	0
$436$	−15.8037	−0.756858
$437$	16.6435	0.796166
$438$	0	0
$439$	16.8575	0.804563	0.402281	−	0.915516i	$-0.368217\pi$
$439$	16.8575	0.804563	0.402281	+	0.915516i	$0.368217\pi$
$440$	4.43676	0.211514
$441$	0	0
$442$	7.45798	0.354740
$443$	−9.03511	−0.429271	−0.214635	−	0.976694i	$-0.568856\pi$
$443$	−9.03511	−0.429271	−0.214635	+	0.976694i	$0.568856\pi$
$444$	0	0
$445$	30.2877	1.43578
$446$	−14.1977	−0.672280
$447$	0	0
$448$	−36.8832	−1.74257
$449$	3.28388	0.154976	0.0774880	−	0.996993i	$-0.475310\pi$
$449$	3.28388	0.154976	0.0774880	+	0.996993i	$0.475310\pi$
$450$	0	0
$451$	−14.9948	−0.706079
$452$	9.72932	0.457629
$453$	0	0
$454$	27.2572	1.27924
$455$	−27.4490	−1.28683
$456$	0	0
$457$	−8.26934	−0.386823	−0.193412	−	0.981118i	$-0.561955\pi$
$457$	−8.26934	−0.386823	−0.193412	+	0.981118i	$0.561955\pi$
$458$	−40.9353	−1.91278
$459$	0	0
$460$	−26.1424	−1.21890
$461$	37.9883	1.76929	0.884647	−	0.466262i	$-0.154400\pi$
$461$	37.9883	1.76929	0.884647	+	0.466262i	$0.154400\pi$
$462$	0	0
$463$	22.6423	1.05228	0.526139	−	0.850399i	$-0.323639\pi$
$463$	22.6423	1.05228	0.526139	+	0.850399i	$0.323639\pi$
$464$	−32.7423	−1.52002
$465$	0	0
$466$	7.06185	0.327134
$467$	−40.3525	−1.86729	−0.933645	−	0.358200i	$-0.883390\pi$
$467$	−40.3525	−1.86729	−0.933645	+	0.358200i	$0.883390\pi$
$468$	0	0
$469$	4.17075	0.192587
$470$	33.6011	1.54990
$471$	0	0
$472$	5.69885	0.262311
$473$	−14.3949	−0.661879
$474$	0	0
$475$	0.975198	0.0447451
$476$	−8.82147	−0.404332
$477$	0	0
$478$	2.08085	0.0951756
$479$	9.68556	0.442544	0.221272	−	0.975212i	$-0.428979\pi$
$479$	9.68556	0.442544	0.221272	+	0.975212i	$0.428979\pi$
$480$	0	0
$481$	5.73343	0.261422
$482$	−41.7893	−1.90345
$483$	0	0
$484$	−7.21875	−0.328125
$485$	14.6296	0.664297
$486$	0	0
$487$	−17.4084	−0.788851	−0.394425	−	0.918928i	$-0.629056\pi$
$487$	−17.4084	−0.788851	−0.394425	+	0.918928i	$0.629056\pi$
$488$	6.77994	0.306914
$489$	0	0
$490$	26.8484	1.21289
$491$	−26.4589	−1.19407	−0.597037	−	0.802214i	$-0.703655\pi$
$491$	−26.4589	−1.19407	−0.597037	+	0.802214i	$0.703655\pi$
$492$	0	0
$493$	−10.8028	−0.486536
$494$	−23.8569	−1.07337
$495$	0	0
$496$	−11.5513	−0.518667
$497$	−14.0091	−0.628393
$498$	0	0
$499$	−25.7968	−1.15482	−0.577412	−	0.816453i	$-0.695937\pi$
$499$	−25.7968	−1.15482	−0.577412	+	0.816453i	$0.695937\pi$
$500$	25.2520	1.12930
$501$	0	0
$502$	46.8355	2.09037
$503$	−4.34593	−0.193775	−0.0968876	−	0.995295i	$-0.530889\pi$
$503$	−4.34593	−0.193775	−0.0968876	+	0.995295i	$0.530889\pi$
$504$	0	0
$505$	−24.3489	−1.08351
$506$	−28.5459	−1.26902
$507$	0	0
$508$	6.79612	0.301529
$509$	−36.7424	−1.62858	−0.814290	−	0.580458i	$-0.802874\pi$
$509$	−36.7424	−1.62858	−0.814290	+	0.580458i	$0.802874\pi$
$510$	0	0
$511$	−11.2237	−0.496508
$512$	−30.6003	−1.35235
$513$	0	0
$514$	−12.2260	−0.539265
$515$	5.75008	0.253379
$516$	0	0
$517$	19.7430	0.868296
$518$	−12.6030	−0.553743
$519$	0	0
$520$	5.30614	0.232690
$521$	18.3808	0.805279	0.402640	−	0.915359i	$-0.368093\pi$
$521$	18.3808	0.805279	0.402640	+	0.915359i	$0.368093\pi$
$522$	0	0
$523$	9.68637	0.423556	0.211778	−	0.977318i	$-0.432075\pi$
$523$	9.68637	0.423556	0.211778	+	0.977318i	$0.432075\pi$
$524$	39.4283	1.72243
$525$	0	0
$526$	−54.4855	−2.37568
$527$	−3.81117	−0.166017
$528$	0	0
$529$	0.817006	0.0355220
$530$	−21.6193	−0.939082
$531$	0	0
$532$	28.2184	1.22342
$533$	−17.9330	−0.776766
$534$	0	0
$535$	−9.76987	−0.422388
$536$	−0.806244	−0.0348244
$537$	0	0
$538$	9.14994	0.394482
$539$	15.7753	0.679491
$540$	0	0
$541$	−2.20270	−0.0947013	−0.0473507	−	0.998878i	$-0.515078\pi$
$541$	−2.20270	−0.0947013	−0.0473507	+	0.998878i	$0.515078\pi$
$542$	−48.9777	−2.10377
$543$	0	0
$544$	−8.63231	−0.370107
$545$	15.5948	0.668006
$546$	0	0
$547$	9.28050	0.396806	0.198403	−	0.980121i	$-0.436425\pi$
$547$	9.28050	0.396806	0.198403	+	0.980121i	$0.436425\pi$
$548$	−6.75272	−0.288462
$549$	0	0
$550$	−1.67260	−0.0713199
$551$	34.5565	1.47216
$552$	0	0
$553$	28.2595	1.20172
$554$	17.7136	0.752579
$555$	0	0
$556$	36.5748	1.55112
$557$	−7.06913	−0.299529	−0.149764	−	0.988722i	$-0.547852\pi$
$557$	−7.06913	−0.299529	−0.149764	+	0.988722i	$0.547852\pi$
$558$	0	0
$559$	−17.2156	−0.728141
$560$	26.3836	1.11491
$561$	0	0
$562$	3.97321	0.167600
$563$	4.36983	0.184166	0.0920832	−	0.995751i	$-0.470647\pi$
$563$	4.36983	0.184166	0.0920832	+	0.995751i	$0.470647\pi$
$564$	0	0
$565$	−9.60072	−0.403905
$566$	−32.5021	−1.36617
$567$	0	0
$568$	2.70808	0.113629
$569$	−15.9279	−0.667734	−0.333867	−	0.942620i	$-0.608354\pi$
$569$	−15.9279	−0.667734	−0.333867	+	0.942620i	$0.608354\pi$
$570$	0	0
$571$	46.0244	1.92606	0.963031	−	0.269392i	$-0.0868227\pi$
$571$	46.0244	1.92606	0.963031	+	0.269392i	$0.0868227\pi$
$572$	22.0178	0.920610
$573$	0	0
$574$	39.4196	1.64534
$575$	1.39552	0.0581971
$576$	0	0
$577$	6.27285	0.261142	0.130571	−	0.991439i	$-0.458319\pi$
$577$	6.27285	0.261142	0.130571	+	0.991439i	$0.458319\pi$
$578$	33.0092	1.37300
$579$	0	0
$580$	−54.2789	−2.25381
$581$	32.5573	1.35070
$582$	0	0
$583$	−12.7028	−0.526098
$584$	2.16965	0.0897806
$585$	0	0
$586$	−58.0565	−2.39829
$587$	−16.1816	−0.667885	−0.333943	−	0.942593i	$-0.608379\pi$
$587$	−16.1816	−0.667885	−0.333943	+	0.942593i	$0.608379\pi$
$588$	0	0
$589$	12.1913	0.502334
$590$	−39.7141	−1.63500
$591$	0	0
$592$	−5.51089	−0.226496
$593$	−45.7193	−1.87747	−0.938733	−	0.344646i	$-0.887999\pi$
$593$	−45.7193	−1.87747	−0.938733	+	0.344646i	$0.887999\pi$
$594$	0	0
$595$	8.70487	0.356865
$596$	−19.6941	−0.806703
$597$	0	0
$598$	−34.1394	−1.39607
$599$	−16.8652	−0.689095	−0.344548	−	0.938769i	$-0.611968\pi$
$599$	−16.8652	−0.689095	−0.344548	+	0.938769i	$0.611968\pi$
$600$	0	0
$601$	19.7806	0.806868	0.403434	−	0.915009i	$-0.367816\pi$
$601$	19.7806	0.806868	0.403434	+	0.915009i	$0.367816\pi$
$602$	37.8425	1.54235
$603$	0	0
$604$	49.0194	1.99457
$605$	7.12333	0.289605
$606$	0	0
$607$	−20.2793	−0.823111	−0.411556	−	0.911385i	$-0.635014\pi$
$607$	−20.2793	−0.823111	−0.411556	+	0.911385i	$0.635014\pi$
$608$	27.6133	1.11987
$609$	0	0
$610$	−47.2480	−1.91301
$611$	23.6116	0.955222
$612$	0	0
$613$	−39.5977	−1.59934	−0.799669	−	0.600442i	$-0.794991\pi$
$613$	−39.5977	−1.59934	−0.799669	+	0.600442i	$0.794991\pi$
$614$	−29.0889	−1.17393
$615$	0	0
$616$	−6.85325	−0.276125
$617$	36.7192	1.47826	0.739129	−	0.673564i	$-0.235237\pi$
$617$	36.7192	1.47826	0.739129	+	0.673564i	$0.235237\pi$
$618$	0	0
$619$	4.17159	0.167670	0.0838351	−	0.996480i	$-0.473283\pi$
$619$	4.17159	0.167670	0.0838351	+	0.996480i	$0.473283\pi$
$620$	−19.1492	−0.769052
$621$	0	0
$622$	−22.5195	−0.902948
$623$	−46.7840	−1.87436
$624$	0	0
$625$	−26.3480	−1.05392
$626$	3.73827	0.149411
$627$	0	0
$628$	7.93776	0.316751
$629$	−1.81824	−0.0724978
$630$	0	0
$631$	−24.6747	−0.982283	−0.491142	−	0.871080i	$-0.663420\pi$
$631$	−24.6747	−0.982283	−0.491142	+	0.871080i	$0.663420\pi$
$632$	−5.46282	−0.217300
$633$	0	0
$634$	40.9835	1.62766
$635$	−6.70629	−0.266131
$636$	0	0
$637$	18.8665	0.747516
$638$	−59.2693	−2.34649
$639$	0	0
$640$	−12.4551	−0.492330
$641$	34.9681	1.38116	0.690578	−	0.723257i	$-0.257356\pi$
$641$	34.9681	1.38116	0.690578	+	0.723257i	$0.257356\pi$
$642$	0	0
$643$	−45.0952	−1.77838	−0.889190	−	0.457539i	$-0.848731\pi$
$643$	−45.0952	−1.77838	−0.889190	+	0.457539i	$0.848731\pi$
$644$	40.3809	1.59123
$645$	0	0
$646$	7.56570	0.297669
$647$	12.9959	0.510920	0.255460	−	0.966820i	$-0.417773\pi$
$647$	12.9959	0.510920	0.255460	+	0.966820i	$0.417773\pi$
$648$	0	0
$649$	−23.3348	−0.915970
$650$	−2.00034	−0.0784599
$651$	0	0
$652$	−43.7328	−1.71271
$653$	−39.0921	−1.52979	−0.764897	−	0.644153i	$-0.777210\pi$
$653$	−39.0921	−1.52979	−0.764897	+	0.644153i	$0.777210\pi$
$654$	0	0
$655$	−38.9071	−1.52023
$656$	17.2370	0.672991
$657$	0	0
$658$	−51.9020	−2.02335
$659$	43.6369	1.69985	0.849926	−	0.526902i	$-0.176647\pi$
$659$	43.6369	1.69985	0.849926	+	0.526902i	$0.176647\pi$
$660$	0	0
$661$	−6.89505	−0.268186	−0.134093	−	0.990969i	$-0.542812\pi$
$661$	−6.89505	−0.268186	−0.134093	+	0.990969i	$0.542812\pi$
$662$	28.0064	1.08850
$663$	0	0
$664$	−6.29361	−0.244240
$665$	−27.8454	−1.07980
$666$	0	0
$667$	49.4508	1.91474
$668$	36.5550	1.41436
$669$	0	0
$670$	5.61854	0.217063
$671$	−27.7615	−1.07172
$672$	0	0
$673$	−19.9581	−0.769327	−0.384663	−	0.923057i	$-0.625682\pi$
$673$	−19.9581	−0.769327	−0.384663	+	0.923057i	$0.625682\pi$
$674$	50.6792	1.95209
$675$	0	0
$676$	−3.95678	−0.152184
$677$	−9.14338	−0.351409	−0.175704	−	0.984443i	$-0.556220\pi$
$677$	−9.14338	−0.351409	−0.175704	+	0.984443i	$0.556220\pi$
$678$	0	0
$679$	−22.5977	−0.867219
$680$	−1.68273	−0.0645298
$681$	0	0
$682$	−20.9098	−0.800678
$683$	−31.3045	−1.19783	−0.598917	−	0.800811i	$-0.704402\pi$
$683$	−31.3045	−1.19783	−0.598917	+	0.800811i	$0.704402\pi$
$684$	0	0
$685$	6.66347	0.254598
$686$	10.2570	0.391615
$687$	0	0
$688$	16.5474	0.630862
$689$	−15.1919	−0.578767
$690$	0	0
$691$	30.3447	1.15437	0.577184	−	0.816614i	$-0.304151\pi$
$691$	30.3447	1.15437	0.577184	+	0.816614i	$0.304151\pi$
$692$	−33.9540	−1.29074
$693$	0	0
$694$	−30.4781	−1.15693
$695$	−36.0913	−1.36902
$696$	0	0
$697$	5.68708	0.215414
$698$	29.2381	1.10668
$699$	0	0
$700$	2.36605	0.0894284
$701$	13.7804	0.520480	0.260240	−	0.965544i	$-0.416198\pi$
$701$	13.7804	0.520480	0.260240	+	0.965544i	$0.416198\pi$
$702$	0	0
$703$	5.81624	0.219364
$704$	−29.1943	−1.10030
$705$	0	0
$706$	−9.12099	−0.343273
$707$	37.6105	1.41449
$708$	0	0
$709$	40.8315	1.53346	0.766730	−	0.641970i	$-0.221883\pi$
$709$	40.8315	1.53346	0.766730	+	0.641970i	$0.221883\pi$
$710$	−18.8720	−0.708255
$711$	0	0
$712$	9.04376	0.338929
$713$	17.4459	0.653354
$714$	0	0
$715$	−21.7268	−0.812535
$716$	−0.722564	−0.0270035
$717$	0	0
$718$	0.836797	0.0312290
$719$	−6.40614	−0.238909	−0.119454	−	0.992840i	$-0.538115\pi$
$719$	−6.40614	−0.238909	−0.119454	+	0.992840i	$0.538115\pi$
$720$	0	0
$721$	−8.88187	−0.330778
$722$	15.3346	0.570696
$723$	0	0
$724$	−2.46410	−0.0915777
$725$	2.89749	0.107610
$726$	0	0
$727$	−21.6716	−0.803756	−0.401878	−	0.915693i	$-0.631642\pi$
$727$	−21.6716	−0.803756	−0.401878	+	0.915693i	$0.631642\pi$
$728$	−8.19613	−0.303769
$729$	0	0
$730$	−15.1198	−0.559609
$731$	5.45955	0.201929
$732$	0	0
$733$	11.4385	0.422491	0.211245	−	0.977433i	$-0.432248\pi$
$733$	11.4385	0.422491	0.211245	+	0.977433i	$0.432248\pi$
$734$	15.8847	0.586315
$735$	0	0
$736$	39.5150	1.45654
$737$	3.30128	0.121604
$738$	0	0
$739$	30.6307	1.12677	0.563384	−	0.826195i	$-0.309499\pi$
$739$	30.6307	1.12677	0.563384	+	0.826195i	$0.309499\pi$
$740$	−9.13573	−0.335836
$741$	0	0
$742$	33.3943	1.22594
$743$	35.0301	1.28513	0.642565	−	0.766231i	$-0.277870\pi$
$743$	35.0301	1.28513	0.642565	+	0.766231i	$0.277870\pi$
$744$	0	0
$745$	19.4338	0.712000
$746$	−20.7260	−0.758834
$747$	0	0
$748$	−6.98248	−0.255305
$749$	15.0910	0.551415
$750$	0	0
$751$	−5.92782	−0.216309	−0.108155	−	0.994134i	$-0.534494\pi$
$751$	−5.92782	−0.216309	−0.108155	+	0.994134i	$0.534494\pi$
$752$	−22.6951	−0.827606
$753$	0	0
$754$	−70.8830	−2.58141
$755$	−48.3715	−1.76042
$756$	0	0
$757$	−31.4496	−1.14305	−0.571527	−	0.820583i	$-0.693649\pi$
$757$	−31.4496	−1.14305	−0.571527	+	0.820583i	$0.693649\pi$
$758$	51.4992	1.87053
$759$	0	0
$760$	5.38278	0.195254
$761$	2.18586	0.0792372	0.0396186	−	0.999215i	$-0.487386\pi$
$761$	2.18586	0.0792372	0.0396186	+	0.999215i	$0.487386\pi$
$762$	0	0
$763$	−24.0885	−0.872061
$764$	−22.3424	−0.808321
$765$	0	0
$766$	73.1667	2.64362
$767$	−27.9072	−1.00767
$768$	0	0
$769$	45.2552	1.63194	0.815972	−	0.578091i	$-0.196202\pi$
$769$	45.2552	1.63194	0.815972	+	0.578091i	$0.196202\pi$
$770$	47.7589	1.72111
$771$	0	0
$772$	12.8156	0.461243
$773$	17.9182	0.644471	0.322236	−	0.946660i	$-0.395566\pi$
$773$	17.9182	0.644471	0.322236	+	0.946660i	$0.395566\pi$
$774$	0	0
$775$	1.02221	0.0367190
$776$	4.36833	0.156814
$777$	0	0
$778$	28.1141	1.00794
$779$	−18.1920	−0.651798
$780$	0	0
$781$	−11.0886	−0.396783
$782$	10.8266	0.387158
$783$	0	0
$784$	−18.1342	−0.647649
$785$	−7.83284	−0.279566
$786$	0	0
$787$	−13.1564	−0.468974	−0.234487	−	0.972119i	$-0.575341\pi$
$787$	−13.1564	−0.468974	−0.234487	+	0.972119i	$0.575341\pi$
$788$	−28.3061	−1.00836
$789$	0	0
$790$	38.0693	1.35444
$791$	14.8298	0.527286
$792$	0	0
$793$	−33.2013	−1.17901
$794$	−15.1120	−0.536305
$795$	0	0
$796$	59.3571	2.10385
$797$	−17.7626	−0.629183	−0.314591	−	0.949227i	$-0.601868\pi$
$797$	−17.7626	−0.629183	−0.314591	+	0.949227i	$0.601868\pi$
$798$	0	0
$799$	−7.48792	−0.264903
$800$	2.31532	0.0818588
$801$	0	0
$802$	−29.9908	−1.05901
$803$	−8.88394	−0.313507
$804$	0	0
$805$	−39.8471	−1.40443
$806$	−25.0070	−0.880835
$807$	0	0
$808$	−7.27045	−0.255773
$809$	−2.36023	−0.0829812	−0.0414906	−	0.999139i	$-0.513211\pi$
$809$	−2.36023	−0.0829812	−0.0414906	+	0.999139i	$0.513211\pi$
$810$	0	0
$811$	−5.95783	−0.209208	−0.104604	−	0.994514i	$-0.533357\pi$
$811$	−5.95783	−0.209208	−0.104604	+	0.994514i	$0.533357\pi$
$812$	83.8420	2.94228
$813$	0	0
$814$	−9.97566	−0.349647
$815$	43.1547	1.51164
$816$	0	0
$817$	−17.4642	−0.610995
$818$	−43.6584	−1.52648
$819$	0	0
$820$	28.5748	0.997875
$821$	−1.24138	−0.0433243	−0.0216622	−	0.999765i	$-0.506896\pi$
$821$	−1.24138	−0.0433243	−0.0216622	+	0.999765i	$0.506896\pi$
$822$	0	0
$823$	48.5679	1.69297	0.846486	−	0.532411i	$-0.178714\pi$
$823$	48.5679	1.69297	0.846486	+	0.532411i	$0.178714\pi$
$824$	1.71695	0.0598127
$825$	0	0
$826$	61.3444	2.13444
$827$	−23.2368	−0.808021	−0.404011	−	0.914754i	$-0.632384\pi$
$827$	−23.2368	−0.808021	−0.404011	+	0.914754i	$0.632384\pi$
$828$	0	0
$829$	−11.8209	−0.410556	−0.205278	−	0.978704i	$-0.565810\pi$
$829$	−11.8209	−0.410556	−0.205278	+	0.978704i	$0.565810\pi$
$830$	43.8588	1.52236
$831$	0	0
$832$	−34.9148	−1.21045
$833$	−5.98310	−0.207302
$834$	0	0
$835$	−36.0719	−1.24832
$836$	22.3358	0.772500
$837$	0	0
$838$	−64.4975	−2.22803
$839$	−10.9478	−0.377959	−0.188979	−	0.981981i	$-0.560518\pi$
$839$	−10.9478	−0.377959	−0.188979	+	0.981981i	$0.560518\pi$
$840$	0	0
$841$	73.6736	2.54047
$842$	15.3904	0.530389
$843$	0	0
$844$	36.8826	1.26955
$845$	3.90448	0.134318
$846$	0	0
$847$	−11.0031	−0.378070
$848$	14.6023	0.501444
$849$	0	0
$850$	0.634367	0.0217586
$851$	8.32310	0.285312
$852$	0	0
$853$	−10.7425	−0.367817	−0.183908	−	0.982943i	$-0.558875\pi$
$853$	−10.7425	−0.367817	−0.183908	+	0.982943i	$0.558875\pi$
$854$	72.9816	2.49738
$855$	0	0
$856$	−2.91723	−0.0997090
$857$	−33.8828	−1.15742	−0.578708	−	0.815535i	$-0.696443\pi$
$857$	−33.8828	−1.15742	−0.578708	+	0.815535i	$0.696443\pi$
$858$	0	0
$859$	−31.8162	−1.08556	−0.542778	−	0.839876i	$-0.682628\pi$
$859$	−31.8162	−1.08556	−0.542778	+	0.839876i	$0.682628\pi$
$860$	27.4315	0.935408
$861$	0	0
$862$	−22.4991	−0.766323
$863$	29.4079	1.00106	0.500528	−	0.865721i	$-0.333139\pi$
$863$	29.4079	1.00106	0.500528	+	0.865721i	$0.333139\pi$
$864$	0	0
$865$	33.5052	1.13921
$866$	20.4890	0.696243
$867$	0	0
$868$	29.5789	1.00397
$869$	22.3683	0.758794
$870$	0	0
$871$	3.94816	0.133778
$872$	4.65652	0.157690
$873$	0	0
$874$	−34.6325	−1.17146
$875$	38.4900	1.30120
$876$	0	0
$877$	−31.5619	−1.06577	−0.532886	−	0.846187i	$-0.678892\pi$
$877$	−31.5619	−1.06577	−0.532886	+	0.846187i	$0.678892\pi$
$878$	−35.0778	−1.18382
$879$	0	0
$880$	20.8835	0.703981
$881$	34.9017	1.17587	0.587935	−	0.808908i	$-0.299941\pi$
$881$	34.9017	1.17587	0.587935	+	0.808908i	$0.299941\pi$
$882$	0	0
$883$	43.4568	1.46244	0.731219	−	0.682143i	$-0.238952\pi$
$883$	43.4568	1.46244	0.731219	+	0.682143i	$0.238952\pi$
$884$	−8.35068	−0.280864
$885$	0	0
$886$	18.8007	0.631621
$887$	−36.1264	−1.21301	−0.606503	−	0.795081i	$-0.707428\pi$
$887$	−36.1264	−1.21301	−0.606503	+	0.795081i	$0.707428\pi$
$888$	0	0
$889$	10.3589	0.347426
$890$	−63.0241	−2.11257
$891$	0	0
$892$	15.8971	0.532275
$893$	23.9526	0.801544
$894$	0	0
$895$	0.713014	0.0238334
$896$	19.2387	0.642721
$897$	0	0
$898$	−6.83325	−0.228028
$899$	36.2225	1.20809
$900$	0	0
$901$	4.81780	0.160504
$902$	31.2019	1.03891
$903$	0	0
$904$	−2.86673	−0.0953459
$905$	2.43153	0.0808269
$906$	0	0
$907$	−2.39627	−0.0795670	−0.0397835	−	0.999208i	$-0.512667\pi$
$907$	−2.39627	−0.0795670	−0.0397835	+	0.999208i	$0.512667\pi$
$908$	−30.5198	−1.01283
$909$	0	0
$910$	57.1171	1.89341
$911$	25.0496	0.829930	0.414965	−	0.909837i	$-0.363794\pi$
$911$	25.0496	0.829930	0.414965	+	0.909837i	$0.363794\pi$
$912$	0	0
$913$	25.7701	0.852867
$914$	17.2072	0.569164
$915$	0	0
$916$	45.8351	1.51443
$917$	60.0979	1.98461
$918$	0	0
$919$	16.3189	0.538310	0.269155	−	0.963097i	$-0.413256\pi$
$919$	16.3189	0.538310	0.269155	+	0.963097i	$0.413256\pi$
$920$	7.70281	0.253954
$921$	0	0
$922$	−79.0479	−2.60330
$923$	−13.2614	−0.436506
$924$	0	0
$925$	0.487678	0.0160348
$926$	−47.1151	−1.54830
$927$	0	0
$928$	82.0442	2.69323
$929$	25.5435	0.838054	0.419027	−	0.907974i	$-0.362371\pi$
$929$	25.5435	0.838054	0.419027	+	0.907974i	$0.362371\pi$
$930$	0	0
$931$	19.1390	0.627254
$932$	−7.90714	−0.259007
$933$	0	0
$934$	83.9672	2.74749
$935$	6.89019	0.225333
$936$	0	0
$937$	39.2599	1.28256	0.641282	−	0.767305i	$-0.278403\pi$
$937$	39.2599	1.28256	0.641282	+	0.767305i	$0.278403\pi$
$938$	−8.67869	−0.283369
$939$	0	0
$940$	−37.6231	−1.22713
$941$	35.4115	1.15438	0.577191	−	0.816609i	$-0.304149\pi$
$941$	35.4115	1.15438	0.577191	+	0.816609i	$0.304149\pi$
$942$	0	0
$943$	−26.0330	−0.847752
$944$	26.8240	0.873046
$945$	0	0
$946$	29.9536	0.973875
$947$	44.5990	1.44927	0.724637	−	0.689131i	$-0.242008\pi$
$947$	44.5990	1.44927	0.724637	+	0.689131i	$0.242008\pi$
$948$	0	0
$949$	−10.6247	−0.344893
$950$	−2.02924	−0.0658371
$951$	0	0
$952$	2.59923	0.0842416
$953$	15.8485	0.513384	0.256692	−	0.966493i	$-0.417367\pi$
$953$	15.8485	0.513384	0.256692	+	0.966493i	$0.417367\pi$
$954$	0	0
$955$	22.0471	0.713428
$956$	−2.32992	−0.0753549
$957$	0	0
$958$	−20.1541	−0.651151
$959$	−10.2927	−0.332370
$960$	0	0
$961$	−18.2209	−0.587772
$962$	−11.9304	−0.384651
$963$	0	0
$964$	46.7913	1.50705
$965$	−12.6462	−0.407095
$966$	0	0
$967$	−5.80008	−0.186518	−0.0932590	−	0.995642i	$-0.529728\pi$
$967$	−5.80008	−0.186518	−0.0932590	+	0.995642i	$0.529728\pi$
$968$	2.12699	0.0683641
$969$	0	0
$970$	−30.4420	−0.977433
$971$	−19.2997	−0.619356	−0.309678	−	0.950842i	$-0.600221\pi$
$971$	−19.2997	−0.619356	−0.309678	+	0.950842i	$0.600221\pi$
$972$	0	0
$973$	55.7485	1.78722
$974$	36.2242	1.16070
$975$	0	0
$976$	31.9126	1.02150
$977$	5.47406	0.175131	0.0875654	−	0.996159i	$-0.472091\pi$
$977$	5.47406	0.175131	0.0875654	+	0.996159i	$0.472091\pi$
$978$	0	0
$979$	−37.0310	−1.18352
$980$	−30.0621	−0.960299
$981$	0	0
$982$	55.0569	1.75694
$983$	9.53870	0.304237	0.152119	−	0.988362i	$-0.451390\pi$
$983$	9.53870	0.304237	0.152119	+	0.988362i	$0.451390\pi$
$984$	0	0
$985$	27.9320	0.889987
$986$	22.4790	0.715879
$987$	0	0
$988$	26.7125	0.849837
$989$	−24.9915	−0.794683
$990$	0	0
$991$	28.3930	0.901933	0.450967	−	0.892541i	$-0.351079\pi$
$991$	28.3930	0.901933	0.450967	+	0.892541i	$0.351079\pi$
$992$	28.9446	0.918993
$993$	0	0
$994$	29.1507	0.924605
$995$	−58.5725	−1.85687
$996$	0	0
$997$	−7.04870	−0.223235	−0.111617	−	0.993751i	$-0.535603\pi$
$997$	−7.04870	−0.223235	−0.111617	+	0.993751i	$0.535603\pi$
$998$	53.6792	1.69918
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.i.1.4		17
3.2	odd	2		239.2.a.b.1.14	✓	17
12.11	even	2		3824.2.a.p.1.17		17
15.14	odd	2		5975.2.a.g.1.4		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.2.a.b.1.14	✓	17	3.2	odd	2
2151.2.a.i.1.4		17	1.1	even	1	trivial
3824.2.a.p.1.17		17	12.11	even	2
5975.2.a.g.1.4		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)

\(f(q)\)	\(=\)	\(q-2.08085 q^{2} +2.32992 q^{4} -2.29912 q^{5} +3.55134 q^{7} -0.686506 q^{8} +O(q^{10})\)	\(q-2.08085 q^{2} +2.32992 q^{4} -2.29912 q^{5} +3.55134 q^{7} -0.686506 q^{8} +4.78411 q^{10} +2.81100 q^{11} +3.36181 q^{13} -7.38978 q^{14} -3.23132 q^{16} -1.06613 q^{17} +3.41036 q^{19} -5.35676 q^{20} -5.84925 q^{22} +4.88027 q^{23} +0.285951 q^{25} -6.99540 q^{26} +8.27432 q^{28} +10.1328 q^{29} +3.57478 q^{31} +8.09689 q^{32} +2.21844 q^{34} -8.16495 q^{35} +1.70546 q^{37} -7.09644 q^{38} +1.57836 q^{40} -5.33434 q^{41} -5.12092 q^{43} +6.54939 q^{44} -10.1551 q^{46} +7.02348 q^{47} +5.61200 q^{49} -0.595020 q^{50} +7.83273 q^{52} -4.51898 q^{53} -6.46282 q^{55} -2.43801 q^{56} -21.0848 q^{58} -8.30124 q^{59} -9.87602 q^{61} -7.43857 q^{62} -10.3857 q^{64} -7.72920 q^{65} +1.17442 q^{67} -2.48399 q^{68} +16.9900 q^{70} -3.94473 q^{71} -3.16042 q^{73} -3.54880 q^{74} +7.94586 q^{76} +9.98281 q^{77} +7.95744 q^{79} +7.42919 q^{80} +11.0999 q^{82} +9.16760 q^{83} +2.45115 q^{85} +10.6558 q^{86} -1.92977 q^{88} -13.1736 q^{89} +11.9389 q^{91} +11.3706 q^{92} -14.6148 q^{94} -7.84083 q^{95} -6.36314 q^{97} -11.6777 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 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

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