LMFDB - Embedded newform 2151.2.a.j.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$17.1758214748$
Analytic rank:	$1$
Dimension:	$20$
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			1.16
Root			$-1.38158$ 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)

$f(q)$	$=$	$q+1.38158 q^{2} -0.0912347 q^{4} +1.35155 q^{5} -1.50510 q^{7} -2.88921 q^{8} +O(q^{10})$	\(q+1.38158 q^{2} -0.0912347 q^{4} +1.35155 q^{5} -1.50510 q^{7} -2.88921 q^{8} +1.86728 q^{10} +5.20916 q^{11} -6.03908 q^{13} -2.07942 q^{14} -3.80921 q^{16} -2.23551 q^{17} -5.35678 q^{19} -0.123308 q^{20} +7.19688 q^{22} -2.95682 q^{23} -3.17331 q^{25} -8.34348 q^{26} +0.137318 q^{28} +0.529336 q^{29} -3.72398 q^{31} +0.515693 q^{32} -3.08854 q^{34} -2.03422 q^{35} +5.79583 q^{37} -7.40083 q^{38} -3.90492 q^{40} +4.00191 q^{41} +4.50457 q^{43} -0.475257 q^{44} -4.08509 q^{46} -11.3720 q^{47} -4.73466 q^{49} -4.38418 q^{50} +0.550974 q^{52} +8.22971 q^{53} +7.04045 q^{55} +4.34856 q^{56} +0.731321 q^{58} -5.35349 q^{59} -14.0242 q^{61} -5.14497 q^{62} +8.33088 q^{64} -8.16213 q^{65} -9.35409 q^{67} +0.203956 q^{68} -2.81045 q^{70} +12.5000 q^{71} +10.4873 q^{73} +8.00741 q^{74} +0.488725 q^{76} -7.84033 q^{77} -5.72701 q^{79} -5.14834 q^{80} +5.52897 q^{82} +4.70998 q^{83} -3.02141 q^{85} +6.22343 q^{86} -15.0504 q^{88} -14.4107 q^{89} +9.08944 q^{91} +0.269765 q^{92} -15.7113 q^{94} -7.23997 q^{95} -12.2286 q^{97} -6.54132 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.38158	0.976925	0.488463	−	0.872585i	$-0.337558\pi$
$2$	1.38158	0.976925	0.488463	+	0.872585i	$0.337558\pi$
$3$	0	0
$3$	0	0
$4$	−0.0912347	−0.0456174
$5$	1.35155	0.604432	0.302216	−	0.953239i	$-0.402274\pi$
$5$	1.35155	0.604432	0.302216	+	0.953239i	$0.402274\pi$
$6$	0	0
$7$	−1.50510	−0.568876	−0.284438	−	0.958695i	$-0.591807\pi$
$7$	−1.50510	−0.568876	−0.284438	+	0.958695i	$0.591807\pi$
$8$	−2.88921	−1.02149
$9$	0	0
$10$	1.86728	0.590485
$11$	5.20916	1.57062	0.785311	−	0.619102i	$-0.212503\pi$
$11$	5.20916	1.57062	0.785311	+	0.619102i	$0.212503\pi$
$12$	0	0
$13$	−6.03908	−1.67494	−0.837470	−	0.546483i	$-0.815966\pi$
$13$	−6.03908	−1.67494	−0.837470	+	0.546483i	$0.815966\pi$
$14$	−2.07942	−0.555749
$15$	0	0
$16$	−3.80921	−0.952302
$17$	−2.23551	−0.542191	−0.271096	−	0.962552i	$-0.587386\pi$
$17$	−2.23551	−0.542191	−0.271096	+	0.962552i	$0.587386\pi$
$18$	0	0
$19$	−5.35678	−1.22893	−0.614465	−	0.788944i	$-0.710628\pi$
$19$	−5.35678	−1.22893	−0.614465	+	0.788944i	$0.710628\pi$
$20$	−0.123308	−0.0275726
$21$	0	0
$22$	7.19688	1.53438
$23$	−2.95682	−0.616541	−0.308270	−	0.951299i	$-0.599750\pi$
$23$	−2.95682	−0.616541	−0.308270	+	0.951299i	$0.599750\pi$
$24$	0	0
$25$	−3.17331	−0.634662
$26$	−8.34348	−1.63629
$27$	0	0
$28$	0.137318	0.0259506
$29$	0.529336	0.0982953	0.0491477	−	0.998792i	$-0.484350\pi$
$29$	0.529336	0.0982953	0.0491477	+	0.998792i	$0.484350\pi$
$30$	0	0
$31$	−3.72398	−0.668846	−0.334423	−	0.942423i	$-0.608541\pi$
$31$	−3.72398	−0.668846	−0.334423	+	0.942423i	$0.608541\pi$
$32$	0.515693	0.0911624
$33$	0	0
$34$	−3.08854	−0.529680
$35$	−2.03422	−0.343847
$36$	0	0
$37$	5.79583	0.952829	0.476415	−	0.879221i	$-0.341936\pi$
$37$	5.79583	0.952829	0.476415	+	0.879221i	$0.341936\pi$
$38$	−7.40083	−1.20057
$39$	0	0
$40$	−3.90492	−0.617422
$41$	4.00191	0.624994	0.312497	−	0.949919i	$-0.398835\pi$
$41$	4.00191	0.624994	0.312497	+	0.949919i	$0.398835\pi$
$42$	0	0
$43$	4.50457	0.686941	0.343470	−	0.939163i	$-0.388397\pi$
$43$	4.50457	0.686941	0.343470	+	0.939163i	$0.388397\pi$
$44$	−0.475257	−0.0716476
$45$	0	0
$46$	−4.08509	−0.602314
$47$	−11.3720	−1.65878	−0.829388	−	0.558673i	$-0.811311\pi$
$47$	−11.3720	−1.65878	−0.829388	+	0.558673i	$0.811311\pi$
$48$	0	0
$49$	−4.73466	−0.676381
$50$	−4.38418	−0.620017
$51$	0	0
$52$	0.550974	0.0764064
$53$	8.22971	1.13044	0.565219	−	0.824941i	$-0.308792\pi$
$53$	8.22971	1.13044	0.565219	+	0.824941i	$0.308792\pi$
$54$	0	0
$55$	7.04045	0.949335
$56$	4.34856	0.581101
$57$	0	0
$58$	0.731321	0.0960272
$59$	−5.35349	−0.696965	−0.348482	−	0.937315i	$-0.613303\pi$
$59$	−5.35349	−0.696965	−0.348482	+	0.937315i	$0.613303\pi$
$60$	0	0
$61$	−14.0242	−1.79562	−0.897808	−	0.440386i	$-0.854841\pi$
$61$	−14.0242	−1.79562	−0.897808	+	0.440386i	$0.854841\pi$
$62$	−5.14497	−0.653412
$63$	0	0
$64$	8.33088	1.04136
$65$	−8.16213	−1.01239
$66$	0	0
$67$	−9.35409	−1.14278	−0.571392	−	0.820677i	$-0.693596\pi$
$67$	−9.35409	−1.14278	−0.571392	+	0.820677i	$0.693596\pi$
$68$	0.203956	0.0247333
$69$	0	0
$70$	−2.81045	−0.335913
$71$	12.5000	1.48347	0.741736	−	0.670692i	$-0.234003\pi$
$71$	12.5000	1.48347	0.741736	+	0.670692i	$0.234003\pi$
$72$	0	0
$73$	10.4873	1.22745	0.613725	−	0.789520i	$-0.289670\pi$
$73$	10.4873	1.22745	0.613725	+	0.789520i	$0.289670\pi$
$74$	8.00741	0.930843
$75$	0	0
$76$	0.488725	0.0560606
$77$	−7.84033	−0.893488
$78$	0	0
$79$	−5.72701	−0.644339	−0.322169	−	0.946682i	$-0.604412\pi$
$79$	−5.72701	−0.644339	−0.322169	+	0.946682i	$0.604412\pi$
$80$	−5.14834	−0.575602
$81$	0	0
$82$	5.52897	0.610572
$83$	4.70998	0.516987	0.258494	−	0.966013i	$-0.416774\pi$
$83$	4.70998	0.516987	0.258494	+	0.966013i	$0.416774\pi$
$84$	0	0
$85$	−3.02141	−0.327718
$86$	6.22343	0.671090
$87$	0	0
$88$	−15.0504	−1.60437
$89$	−14.4107	−1.52753	−0.763766	−	0.645493i	$-0.776652\pi$
$89$	−14.4107	−1.52753	−0.763766	+	0.645493i	$0.776652\pi$
$90$	0	0
$91$	9.08944	0.952832
$92$	0.269765	0.0281250
$93$	0	0
$94$	−15.7113	−1.62050
$95$	−7.23997	−0.742805
$96$	0	0
$97$	−12.2286	−1.24162	−0.620812	−	0.783959i	$-0.713197\pi$
$97$	−12.2286	−1.24162	−0.620812	+	0.783959i	$0.713197\pi$
$98$	−6.54132	−0.660773
$99$	0	0
$100$	0.289516	0.0289516
$101$	−13.1550	−1.30897	−0.654486	−	0.756074i	$-0.727115\pi$
$101$	−13.1550	−1.30897	−0.654486	+	0.756074i	$0.727115\pi$
$102$	0	0
$103$	5.79068	0.570572	0.285286	−	0.958442i	$-0.407911\pi$
$103$	5.79068	0.570572	0.285286	+	0.958442i	$0.407911\pi$
$104$	17.4482	1.71093
$105$	0	0
$106$	11.3700	1.10435
$107$	−8.90040	−0.860434	−0.430217	−	0.902725i	$-0.641563\pi$
$107$	−8.90040	−0.860434	−0.430217	+	0.902725i	$0.641563\pi$
$108$	0	0
$109$	9.44148	0.904330	0.452165	−	0.891934i	$-0.350652\pi$
$109$	9.44148	0.904330	0.452165	+	0.891934i	$0.350652\pi$
$110$	9.72696	0.927429
$111$	0	0
$112$	5.73325	0.541741
$113$	−0.0439495	−0.00413442	−0.00206721	−	0.999998i	$-0.500658\pi$
$113$	−0.0439495	−0.00413442	−0.00206721	+	0.999998i	$0.500658\pi$
$114$	0	0
$115$	−3.99630	−0.372657
$116$	−0.0482939	−0.00448397
$117$	0	0
$118$	−7.39628	−0.680883
$119$	3.36468	0.308439
$120$	0	0
$121$	16.1354	1.46685
$122$	−19.3756	−1.75418
$123$	0	0
$124$	0.339756	0.0305110
$125$	−11.0466	−0.988042
$126$	0	0
$127$	1.52724	0.135520	0.0677602	−	0.997702i	$-0.478415\pi$
$127$	1.52724	0.135520	0.0677602	+	0.997702i	$0.478415\pi$
$128$	10.4784	0.926169
$129$	0	0
$130$	−11.2766	−0.989027
$131$	−2.05574	−0.179611	−0.0898055	−	0.995959i	$-0.528625\pi$
$131$	−2.05574	−0.179611	−0.0898055	+	0.995959i	$0.528625\pi$
$132$	0	0
$133$	8.06251	0.699108
$134$	−12.9234	−1.11641
$135$	0	0
$136$	6.45886	0.553843
$137$	−11.4680	−0.979774	−0.489887	−	0.871786i	$-0.662962\pi$
$137$	−11.4680	−0.979774	−0.489887	+	0.871786i	$0.662962\pi$
$138$	0	0
$139$	−16.4276	−1.39337	−0.696686	−	0.717376i	$-0.745343\pi$
$139$	−16.4276	−1.39337	−0.696686	+	0.717376i	$0.745343\pi$
$140$	0.185592	0.0156854
$141$	0	0
$142$	17.2697	1.44924
$143$	−31.4586	−2.63070
$144$	0	0
$145$	0.715426	0.0594129
$146$	14.4891	1.19913
$147$	0	0
$148$	−0.528781	−0.0434656
$149$	−14.8822	−1.21919	−0.609597	−	0.792711i	$-0.708669\pi$
$149$	−14.8822	−1.21919	−0.609597	+	0.792711i	$0.708669\pi$
$150$	0	0
$151$	14.5748	1.18608	0.593040	−	0.805173i	$-0.297928\pi$
$151$	14.5748	1.18608	0.593040	+	0.805173i	$0.297928\pi$
$152$	15.4769	1.25534
$153$	0	0
$154$	−10.8320	−0.872871
$155$	−5.03315	−0.404272
$156$	0	0
$157$	3.59395	0.286829	0.143414	−	0.989663i	$-0.454192\pi$
$157$	3.59395	0.286829	0.143414	+	0.989663i	$0.454192\pi$
$158$	−7.91232	−0.629471
$159$	0	0
$160$	0.696985	0.0551015
$161$	4.45033	0.350735
$162$	0	0
$163$	10.5991	0.830188	0.415094	−	0.909779i	$-0.363749\pi$
$163$	10.5991	0.830188	0.415094	+	0.909779i	$0.363749\pi$
$164$	−0.365113	−0.0285106
$165$	0	0
$166$	6.50721	0.505058
$167$	20.8147	1.61069	0.805344	−	0.592808i	$-0.201981\pi$
$167$	20.8147	1.61069	0.805344	+	0.592808i	$0.201981\pi$
$168$	0	0
$169$	23.4705	1.80542
$170$	−4.17432	−0.320156
$171$	0	0
$172$	−0.410973	−0.0313364
$173$	15.4900	1.17768	0.588841	−	0.808249i	$-0.299584\pi$
$173$	15.4900	1.17768	0.588841	+	0.808249i	$0.299584\pi$
$174$	0	0
$175$	4.77616	0.361043
$176$	−19.8428	−1.49571
$177$	0	0
$178$	−19.9096	−1.49228
$179$	−5.79634	−0.433239	−0.216619	−	0.976256i	$-0.569503\pi$
$179$	−5.79634	−0.433239	−0.216619	+	0.976256i	$0.569503\pi$
$180$	0	0
$181$	17.6304	1.31046	0.655230	−	0.755429i	$-0.272572\pi$
$181$	17.6304	1.31046	0.655230	+	0.755429i	$0.272572\pi$
$182$	12.5578	0.930846
$183$	0	0
$184$	8.54289	0.629790
$185$	7.83337	0.575921
$186$	0	0
$187$	−11.6451	−0.851577
$188$	1.03752	0.0756690
$189$	0	0
$190$	−10.0026	−0.725665
$191$	14.4507	1.04562	0.522810	−	0.852449i	$-0.324884\pi$
$191$	14.4507	1.04562	0.522810	+	0.852449i	$0.324884\pi$
$192$	0	0
$193$	16.9861	1.22268	0.611342	−	0.791367i	$-0.290630\pi$
$193$	16.9861	1.22268	0.611342	+	0.791367i	$0.290630\pi$
$194$	−16.8948	−1.21297
$195$	0	0
$196$	0.431966	0.0308547
$197$	16.5500	1.17914	0.589571	−	0.807717i	$-0.299297\pi$
$197$	16.5500	1.17914	0.589571	+	0.807717i	$0.299297\pi$
$198$	0	0
$199$	−21.6857	−1.53726	−0.768631	−	0.639693i	$-0.779062\pi$
$199$	−21.6857	−1.53726	−0.768631	+	0.639693i	$0.779062\pi$
$200$	9.16835	0.648300
$201$	0	0
$202$	−18.1747	−1.27877
$203$	−0.796706	−0.0559178
$204$	0	0
$205$	5.40879	0.377767
$206$	8.00029	0.557406
$207$	0	0
$208$	23.0041	1.59505
$209$	−27.9044	−1.93018
$210$	0	0
$211$	−2.61046	−0.179711	−0.0898556	−	0.995955i	$-0.528641\pi$
$211$	−2.61046	−0.179711	−0.0898556	+	0.995955i	$0.528641\pi$
$212$	−0.750835	−0.0515676
$213$	0	0
$214$	−12.2966	−0.840580
$215$	6.08816	0.415209
$216$	0	0
$217$	5.60497	0.380490
$218$	13.0442	0.883462
$219$	0	0
$220$	−0.642334	−0.0433061
$221$	13.5004	0.908138
$222$	0	0
$223$	17.8225	1.19348	0.596741	−	0.802434i	$-0.296462\pi$
$223$	17.8225	1.19348	0.596741	+	0.802434i	$0.296462\pi$
$224$	−0.776170	−0.0518601
$225$	0	0
$226$	−0.0607198	−0.00403902
$227$	4.02849	0.267380	0.133690	−	0.991023i	$-0.457317\pi$
$227$	4.02849	0.267380	0.133690	+	0.991023i	$0.457317\pi$
$228$	0	0
$229$	3.30206	0.218206	0.109103	−	0.994030i	$-0.465202\pi$
$229$	3.30206	0.218206	0.109103	+	0.994030i	$0.465202\pi$
$230$	−5.52121	−0.364058
$231$	0	0
$232$	−1.52936	−0.100408
$233$	23.4736	1.53781	0.768904	−	0.639365i	$-0.220803\pi$
$233$	23.4736	1.53781	0.768904	+	0.639365i	$0.220803\pi$
$234$	0	0
$235$	−15.3698	−1.00262
$236$	0.488424	0.0317937
$237$	0	0
$238$	4.64857	0.301322
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	16.9690	1.09307	0.546534	−	0.837437i	$-0.315947\pi$
$241$	16.9690	1.09307	0.546534	+	0.837437i	$0.315947\pi$
$242$	22.2923	1.43301
$243$	0	0
$244$	1.27950	0.0819113
$245$	−6.39914	−0.408826
$246$	0	0
$247$	32.3501	2.05839
$248$	10.7593	0.683219
$249$	0	0
$250$	−15.2618	−0.965243
$251$	−3.02008	−0.190626	−0.0953129	−	0.995447i	$-0.530385\pi$
$251$	−3.02008	−0.190626	−0.0953129	+	0.995447i	$0.530385\pi$
$252$	0	0
$253$	−15.4026	−0.968352
$254$	2.11000	0.132393
$255$	0	0
$256$	−2.18501	−0.136563
$257$	−13.4671	−0.840058	−0.420029	−	0.907511i	$-0.637980\pi$
$257$	−13.4671	−0.840058	−0.420029	+	0.907511i	$0.637980\pi$
$258$	0	0
$259$	−8.72333	−0.542041
$260$	0.744670	0.0461825
$261$	0	0
$262$	−2.84017	−0.175467
$263$	−6.69271	−0.412690	−0.206345	−	0.978479i	$-0.566157\pi$
$263$	−6.69271	−0.412690	−0.206345	+	0.978479i	$0.566157\pi$
$264$	0	0
$265$	11.1229	0.683273
$266$	11.1390	0.682977
$267$	0	0
$268$	0.853418	0.0521308
$269$	−23.4572	−1.43021	−0.715106	−	0.699016i	$-0.753621\pi$
$269$	−23.4572	−1.43021	−0.715106	+	0.699016i	$0.753621\pi$
$270$	0	0
$271$	2.79891	0.170022	0.0850108	−	0.996380i	$-0.472908\pi$
$271$	2.79891	0.170022	0.0850108	+	0.996380i	$0.472908\pi$
$272$	8.51553	0.516330
$273$	0	0
$274$	−15.8439	−0.957166
$275$	−16.5303	−0.996813
$276$	0	0
$277$	1.33556	0.0802459	0.0401230	−	0.999195i	$-0.487225\pi$
$277$	1.33556	0.0802459	0.0401230	+	0.999195i	$0.487225\pi$
$278$	−22.6961	−1.36122
$279$	0	0
$280$	5.87730	0.351236
$281$	−11.5507	−0.689057	−0.344529	−	0.938776i	$-0.611961\pi$
$281$	−11.5507	−0.689057	−0.344529	+	0.938776i	$0.611961\pi$
$282$	0	0
$283$	23.6047	1.40315	0.701576	−	0.712595i	$-0.252480\pi$
$283$	23.6047	1.40315	0.701576	+	0.712595i	$0.252480\pi$
$284$	−1.14043	−0.0676721
$285$	0	0
$286$	−43.4625	−2.56999
$287$	−6.02329	−0.355544
$288$	0	0
$289$	−12.0025	−0.706029
$290$	0.988418	0.0580419
$291$	0	0
$292$	−0.956809	−0.0559930
$293$	−3.75303	−0.219254	−0.109627	−	0.993973i	$-0.534966\pi$
$293$	−3.75303	−0.219254	−0.109627	+	0.993973i	$0.534966\pi$
$294$	0	0
$295$	−7.23552	−0.421268
$296$	−16.7454	−0.973305
$297$	0	0
$298$	−20.5609	−1.19106
$299$	17.8565	1.03267
$300$	0	0
$301$	−6.77985	−0.390784
$302$	20.1362	1.15871
$303$	0	0
$304$	20.4051	1.17031
$305$	−18.9545	−1.08533
$306$	0	0
$307$	−1.83405	−0.104675	−0.0523373	−	0.998629i	$-0.516667\pi$
$307$	−1.83405	−0.104675	−0.0523373	+	0.998629i	$0.516667\pi$
$308$	0.715310	0.0407586
$309$	0	0
$310$	−6.95370	−0.394943
$311$	−23.0474	−1.30690	−0.653450	−	0.756970i	$-0.726679\pi$
$311$	−23.0474	−1.30690	−0.653450	+	0.756970i	$0.726679\pi$
$312$	0	0
$313$	−12.5384	−0.708712	−0.354356	−	0.935111i	$-0.615300\pi$
$313$	−12.5384	−0.708712	−0.354356	+	0.935111i	$0.615300\pi$
$314$	4.96533	0.280210
$315$	0	0
$316$	0.522502	0.0293930
$317$	5.42002	0.304419	0.152209	−	0.988348i	$-0.451361\pi$
$317$	5.42002	0.304419	0.152209	+	0.988348i	$0.451361\pi$
$318$	0	0
$319$	2.75740	0.154385
$320$	11.2596	0.629432
$321$	0	0
$322$	6.14848	0.342642
$323$	11.9752	0.666315
$324$	0	0
$325$	19.1639	1.06302
$326$	14.6436	0.811031
$327$	0	0
$328$	−11.5624	−0.638425
$329$	17.1160	0.943637
$330$	0	0
$331$	−13.7273	−0.754523	−0.377261	−	0.926107i	$-0.623134\pi$
$331$	−13.7273	−0.754523	−0.377261	+	0.926107i	$0.623134\pi$
$332$	−0.429714	−0.0235836
$333$	0	0
$334$	28.7572	1.57352
$335$	−12.6425	−0.690736
$336$	0	0
$337$	−14.7765	−0.804926	−0.402463	−	0.915436i	$-0.631846\pi$
$337$	−14.7765	−0.804926	−0.402463	+	0.915436i	$0.631846\pi$
$338$	32.4264	1.76376
$339$	0	0
$340$	0.275658	0.0149496
$341$	−19.3988	−1.05050
$342$	0	0
$343$	17.6619	0.953652
$344$	−13.0147	−0.701703
$345$	0	0
$346$	21.4007	1.15051
$347$	9.24009	0.496034	0.248017	−	0.968756i	$-0.420221\pi$
$347$	9.24009	0.496034	0.248017	+	0.968756i	$0.420221\pi$
$348$	0	0
$349$	−33.5592	−1.79639	−0.898193	−	0.439602i	$-0.855119\pi$
$349$	−33.5592	−1.79639	−0.898193	+	0.439602i	$0.855119\pi$
$350$	6.59864	0.352712
$351$	0	0
$352$	2.68633	0.143182
$353$	−11.0743	−0.589423	−0.294712	−	0.955586i	$-0.595224\pi$
$353$	−11.0743	−0.589423	−0.294712	+	0.955586i	$0.595224\pi$
$354$	0	0
$355$	16.8943	0.896658
$356$	1.31476	0.0696820
$357$	0	0
$358$	−8.00811	−0.423242
$359$	10.6761	0.563464	0.281732	−	0.959493i	$-0.409091\pi$
$359$	10.6761	0.563464	0.281732	+	0.959493i	$0.409091\pi$
$360$	0	0
$361$	9.69513	0.510270
$362$	24.3579	1.28022
$363$	0	0
$364$	−0.829273	−0.0434657
$365$	14.1742	0.741910
$366$	0	0
$367$	20.7721	1.08430	0.542148	−	0.840283i	$-0.317611\pi$
$367$	20.7721	1.08430	0.542148	+	0.840283i	$0.317611\pi$
$368$	11.2632	0.587133
$369$	0	0
$370$	10.8224	0.562631
$371$	−12.3866	−0.643078
$372$	0	0
$373$	−37.5360	−1.94354	−0.971770	−	0.235931i	$-0.924186\pi$
$373$	−37.5360	−1.94354	−0.971770	+	0.235931i	$0.924186\pi$
$374$	−16.0887	−0.831927
$375$	0	0
$376$	32.8561	1.69442
$377$	−3.19671	−0.164639
$378$	0	0
$379$	−14.7017	−0.755174	−0.377587	−	0.925974i	$-0.623246\pi$
$379$	−14.7017	−0.755174	−0.377587	+	0.925974i	$0.623246\pi$
$380$	0.660537	0.0338848
$381$	0	0
$382$	19.9649	1.02149
$383$	1.17130	0.0598507	0.0299253	−	0.999552i	$-0.490473\pi$
$383$	1.17130	0.0598507	0.0299253	+	0.999552i	$0.490473\pi$
$384$	0	0
$385$	−10.5966	−0.540053
$386$	23.4676	1.19447
$387$	0	0
$388$	1.11567	0.0566396
$389$	34.5061	1.74953	0.874766	−	0.484546i	$-0.161015\pi$
$389$	34.5061	1.74953	0.874766	+	0.484546i	$0.161015\pi$
$390$	0	0
$391$	6.61002	0.334283
$392$	13.6794	0.690916
$393$	0	0
$394$	22.8652	1.15193
$395$	−7.74035	−0.389459
$396$	0	0
$397$	3.05365	0.153258	0.0766292	−	0.997060i	$-0.475584\pi$
$397$	3.05365	0.153258	0.0766292	+	0.997060i	$0.475584\pi$
$398$	−29.9606	−1.50179
$399$	0	0
$400$	12.0878	0.604389
$401$	−21.5513	−1.07622	−0.538109	−	0.842875i	$-0.680861\pi$
$401$	−21.5513	−1.07622	−0.538109	+	0.842875i	$0.680861\pi$
$402$	0	0
$403$	22.4894	1.12028
$404$	1.20019	0.0597118
$405$	0	0
$406$	−1.10071	−0.0546275
$407$	30.1914	1.49653
$408$	0	0
$409$	−2.09326	−0.103505	−0.0517526	−	0.998660i	$-0.516481\pi$
$409$	−2.09326	−0.103505	−0.0517526	+	0.998660i	$0.516481\pi$
$410$	7.47268	0.369050
$411$	0	0
$412$	−0.528311	−0.0260280
$413$	8.05755	0.396486
$414$	0	0
$415$	6.36578	0.312484
$416$	−3.11431	−0.152692
$417$	0	0
$418$	−38.5521	−1.88565
$419$	4.55800	0.222673	0.111336	−	0.993783i	$-0.464487\pi$
$419$	4.55800	0.222673	0.111336	+	0.993783i	$0.464487\pi$
$420$	0	0
$421$	31.1765	1.51945	0.759724	−	0.650245i	$-0.225334\pi$
$421$	31.1765	1.51945	0.759724	+	0.650245i	$0.225334\pi$
$422$	−3.60656	−0.175564
$423$	0	0
$424$	−23.7774	−1.15473
$425$	7.09397	0.344108
$426$	0	0
$427$	21.1079	1.02148
$428$	0.812026	0.0392507
$429$	0	0
$430$	8.41129	0.405628
$431$	−6.08401	−0.293056	−0.146528	−	0.989206i	$-0.546810\pi$
$431$	−6.08401	−0.293056	−0.146528	+	0.989206i	$0.546810\pi$
$432$	0	0
$433$	7.69823	0.369953	0.184977	−	0.982743i	$-0.440779\pi$
$433$	7.69823	0.369953	0.184977	+	0.982743i	$0.440779\pi$
$434$	7.74371	0.371710
$435$	0	0
$436$	−0.861391	−0.0412531
$437$	15.8391	0.757686
$438$	0	0
$439$	−11.9627	−0.570950	−0.285475	−	0.958386i	$-0.592151\pi$
$439$	−11.9627	−0.570950	−0.285475	+	0.958386i	$0.592151\pi$
$440$	−20.3413	−0.969736
$441$	0	0
$442$	18.6520	0.887183
$443$	33.6798	1.60017	0.800087	−	0.599884i	$-0.204787\pi$
$443$	33.6798	1.60017	0.800087	+	0.599884i	$0.204787\pi$
$444$	0	0
$445$	−19.4768	−0.923290
$446$	24.6232	1.16594
$447$	0	0
$448$	−12.5388	−0.592405
$449$	−8.14181	−0.384236	−0.192118	−	0.981372i	$-0.561536\pi$
$449$	−8.14181	−0.384236	−0.192118	+	0.981372i	$0.561536\pi$
$450$	0	0
$451$	20.8466	0.981629
$452$	0.00400972	0.000188601 0
$453$	0	0
$454$	5.56569	0.261211
$455$	12.2849	0.575923
$456$	0	0
$457$	−18.2011	−0.851411	−0.425705	−	0.904862i	$-0.639974\pi$
$457$	−18.2011	−0.851411	−0.425705	+	0.904862i	$0.639974\pi$
$458$	4.56206	0.213171
$459$	0	0
$460$	0.364602	0.0169996
$461$	−35.8143	−1.66804	−0.834018	−	0.551737i	$-0.813965\pi$
$461$	−35.8143	−1.66804	−0.834018	+	0.551737i	$0.813965\pi$
$462$	0	0
$463$	17.6283	0.819255	0.409628	−	0.912253i	$-0.365659\pi$
$463$	17.6283	0.819255	0.409628	+	0.912253i	$0.365659\pi$
$464$	−2.01635	−0.0936068
$465$	0	0
$466$	32.4307	1.50232
$467$	35.8370	1.65834	0.829169	−	0.558999i	$-0.188814\pi$
$467$	35.8370	1.65834	0.829169	+	0.558999i	$0.188814\pi$
$468$	0	0
$469$	14.0789	0.650102
$470$	−21.2347	−0.979482
$471$	0	0
$472$	15.4674	0.711943
$473$	23.4651	1.07892
$474$	0	0
$475$	16.9987	0.779955
$476$	−0.306975	−0.0140702
$477$	0	0
$478$	−1.38158	−0.0631920
$479$	18.6108	0.850351	0.425176	−	0.905111i	$-0.360212\pi$
$479$	18.6108	0.850351	0.425176	+	0.905111i	$0.360212\pi$
$480$	0	0
$481$	−35.0015	−1.59593
$482$	23.4440	1.06785
$483$	0	0
$484$	−1.47211	−0.0669139
$485$	−16.5276	−0.750478
$486$	0	0
$487$	13.7084	0.621187	0.310594	−	0.950543i	$-0.399472\pi$
$487$	13.7084	0.621187	0.310594	+	0.950543i	$0.399472\pi$
$488$	40.5189	1.83420
$489$	0	0
$490$	−8.84093	−0.399393
$491$	−26.8768	−1.21293	−0.606466	−	0.795109i	$-0.707413\pi$
$491$	−26.8768	−1.21293	−0.606466	+	0.795109i	$0.707413\pi$
$492$	0	0
$493$	−1.18334	−0.0532949
$494$	44.6942	2.01089
$495$	0	0
$496$	14.1854	0.636943
$497$	−18.8137	−0.843911
$498$	0	0
$499$	−35.7292	−1.59946	−0.799729	−	0.600361i	$-0.795024\pi$
$499$	−35.7292	−1.59946	−0.799729	+	0.600361i	$0.795024\pi$
$500$	1.00784	0.0450719
$501$	0	0
$502$	−4.17249	−0.186227
$503$	26.3328	1.17412	0.587062	−	0.809542i	$-0.300285\pi$
$503$	26.3328	1.17412	0.587062	+	0.809542i	$0.300285\pi$
$504$	0	0
$505$	−17.7797	−0.791185
$506$	−21.2799	−0.946007
$507$	0	0
$508$	−0.139337	−0.00618209
$509$	42.2347	1.87202	0.936011	−	0.351972i	$-0.114489\pi$
$509$	42.2347	1.87202	0.936011	+	0.351972i	$0.114489\pi$
$510$	0	0
$511$	−15.7845	−0.698266
$512$	−23.9756	−1.05958
$513$	0	0
$514$	−18.6060	−0.820673
$515$	7.82640	0.344872
$516$	0	0
$517$	−59.2386	−2.60531
$518$	−12.0520	−0.529534
$519$	0	0
$520$	23.5821	1.03414
$521$	−13.8354	−0.606142	−0.303071	−	0.952968i	$-0.598012\pi$
$521$	−13.8354	−0.606142	−0.303071	+	0.952968i	$0.598012\pi$
$522$	0	0
$523$	−22.9910	−1.00533	−0.502663	−	0.864483i	$-0.667646\pi$
$523$	−22.9910	−1.00533	−0.502663	+	0.864483i	$0.667646\pi$
$524$	0.187555	0.00819338
$525$	0	0
$526$	−9.24652	−0.403167
$527$	8.32499	0.362642
$528$	0	0
$529$	−14.2572	−0.619878
$530$	15.3672	0.667507
$531$	0	0
$532$	−0.735581	−0.0318915
$533$	−24.1679	−1.04683
$534$	0	0
$535$	−12.0294	−0.520074
$536$	27.0259	1.16734
$537$	0	0
$538$	−32.4080	−1.39721
$539$	−24.6636	−1.06234
$540$	0	0
$541$	−41.7557	−1.79522	−0.897610	−	0.440791i	$-0.854698\pi$
$541$	−41.7557	−1.79522	−0.897610	+	0.440791i	$0.854698\pi$
$542$	3.86692	0.166098
$543$	0	0
$544$	−1.15284	−0.0494275
$545$	12.7606	0.546606
$546$	0	0
$547$	−33.2023	−1.41963	−0.709814	−	0.704389i	$-0.751221\pi$
$547$	−33.2023	−1.41963	−0.709814	+	0.704389i	$0.751221\pi$
$548$	1.04628	0.0446947
$549$	0	0
$550$	−22.8379	−0.973812
$551$	−2.83554	−0.120798
$552$	0	0
$553$	8.61974	0.366549
$554$	1.84518	0.0783943
$555$	0	0
$556$	1.49877	0.0635620
$557$	−25.6256	−1.08579	−0.542895	−	0.839800i	$-0.682672\pi$
$557$	−25.6256	−1.08579	−0.542895	+	0.839800i	$0.682672\pi$
$558$	0	0
$559$	−27.2035	−1.15058
$560$	7.74878	0.327446
$561$	0	0
$562$	−15.9582	−0.673157
$563$	−8.59177	−0.362100	−0.181050	−	0.983474i	$-0.557950\pi$
$563$	−8.59177	−0.362100	−0.181050	+	0.983474i	$0.557950\pi$
$564$	0	0
$565$	−0.0594000	−0.00249898
$566$	32.6118	1.37077
$567$	0	0
$568$	−36.1150	−1.51535
$569$	2.27120	0.0952138	0.0476069	−	0.998866i	$-0.484841\pi$
$569$	2.27120	0.0952138	0.0476069	+	0.998866i	$0.484841\pi$
$570$	0	0
$571$	−38.3368	−1.60434	−0.802172	−	0.597093i	$-0.796322\pi$
$571$	−38.3368	−1.60434	−0.802172	+	0.597093i	$0.796322\pi$
$572$	2.87011	0.120005
$573$	0	0
$574$	−8.32166	−0.347340
$575$	9.38291	0.391295
$576$	0	0
$577$	22.0162	0.916547	0.458274	−	0.888811i	$-0.348468\pi$
$577$	22.0162	0.916547	0.458274	+	0.888811i	$0.348468\pi$
$578$	−16.5824	−0.689737
$579$	0	0
$580$	−0.0652717	−0.00271026
$581$	−7.08900	−0.294101
$582$	0	0
$583$	42.8699	1.77549
$584$	−30.3001	−1.25383
$585$	0	0
$586$	−5.18511	−0.214195
$587$	28.8423	1.19045	0.595224	−	0.803560i	$-0.297063\pi$
$587$	28.8423	1.19045	0.595224	+	0.803560i	$0.297063\pi$
$588$	0	0
$589$	19.9485	0.821965
$590$	−9.99645	−0.411547
$591$	0	0
$592$	−22.0775	−0.907381
$593$	−10.5494	−0.433213	−0.216607	−	0.976259i	$-0.569499\pi$
$593$	−10.5494	−0.433213	−0.216607	+	0.976259i	$0.569499\pi$
$594$	0	0
$595$	4.54753	0.186431
$596$	1.35777	0.0556164
$597$	0	0
$598$	24.6702	1.00884
$599$	7.21520	0.294805	0.147403	−	0.989077i	$-0.452909\pi$
$599$	7.21520	0.294805	0.147403	+	0.989077i	$0.452909\pi$
$600$	0	0
$601$	24.0334	0.980344	0.490172	−	0.871626i	$-0.336934\pi$
$601$	24.0334	0.980344	0.490172	+	0.871626i	$0.336934\pi$
$602$	−9.36690	−0.381767
$603$	0	0
$604$	−1.32973	−0.0541058
$605$	21.8078	0.886613
$606$	0	0
$607$	17.5172	0.711001	0.355500	−	0.934676i	$-0.384310\pi$
$607$	17.5172	0.711001	0.355500	+	0.934676i	$0.384310\pi$
$608$	−2.76245	−0.112032
$609$	0	0
$610$	−26.1871	−1.06029
$611$	68.6764	2.77835
$612$	0	0
$613$	23.5816	0.952453	0.476226	−	0.879323i	$-0.342004\pi$
$613$	23.5816	0.952453	0.476226	+	0.879323i	$0.342004\pi$
$614$	−2.53388	−0.102259
$615$	0	0
$616$	22.6523	0.912689
$617$	−26.2668	−1.05746	−0.528731	−	0.848789i	$-0.677332\pi$
$617$	−26.2668	−1.05746	−0.528731	+	0.848789i	$0.677332\pi$
$618$	0	0
$619$	−11.7447	−0.472060	−0.236030	−	0.971746i	$-0.575846\pi$
$619$	−11.7447	−0.472060	−0.236030	+	0.971746i	$0.575846\pi$
$620$	0.459198	0.0184418
$621$	0	0
$622$	−31.8419	−1.27674
$623$	21.6896	0.868975
$624$	0	0
$625$	0.936418	0.0374567
$626$	−17.3228	−0.692359
$627$	0	0
$628$	−0.327893	−0.0130844
$629$	−12.9567	−0.516616
$630$	0	0
$631$	3.30347	0.131509	0.0657545	−	0.997836i	$-0.479055\pi$
$631$	3.30347	0.131509	0.0657545	+	0.997836i	$0.479055\pi$
$632$	16.5465	0.658186
$633$	0	0
$634$	7.48820	0.297394
$635$	2.06414	0.0819129
$636$	0	0
$637$	28.5930	1.13290
$638$	3.80957	0.150822
$639$	0	0
$640$	14.1621	0.559806
$641$	−25.9601	−1.02536	−0.512680	−	0.858580i	$-0.671347\pi$
$641$	−25.9601	−1.02536	−0.512680	+	0.858580i	$0.671347\pi$
$642$	0	0
$643$	−24.7989	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$643$	−24.7989	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$644$	−0.406024	−0.0159996
$645$	0	0
$646$	16.5446	0.650940
$647$	35.1636	1.38242	0.691211	−	0.722653i	$-0.257077\pi$
$647$	35.1636	1.38242	0.691211	+	0.722653i	$0.257077\pi$
$648$	0	0
$649$	−27.8872	−1.09467
$650$	26.4764	1.03849
$651$	0	0
$652$	−0.967009	−0.0378710
$653$	−43.2533	−1.69263	−0.846316	−	0.532681i	$-0.821184\pi$
$653$	−43.2533	−1.69263	−0.846316	+	0.532681i	$0.821184\pi$
$654$	0	0
$655$	−2.77844	−0.108563
$656$	−15.2441	−0.595183
$657$	0	0
$658$	23.6472	0.921863
$659$	−20.3893	−0.794256	−0.397128	−	0.917763i	$-0.629993\pi$
$659$	−20.3893	−0.794256	−0.397128	+	0.917763i	$0.629993\pi$
$660$	0	0
$661$	−41.5166	−1.61481	−0.807404	−	0.589999i	$-0.799128\pi$
$661$	−41.5166	−1.61481	−0.807404	+	0.589999i	$0.799128\pi$
$662$	−18.9654	−0.737112
$663$	0	0
$664$	−13.6081	−0.528097
$665$	10.8969	0.422564
$666$	0	0
$667$	−1.56516	−0.0606031
$668$	−1.89902	−0.0734754
$669$	0	0
$670$	−17.4667	−0.674797
$671$	−73.0544	−2.82023
$672$	0	0
$673$	−29.0303	−1.11903	−0.559517	−	0.828819i	$-0.689013\pi$
$673$	−29.0303	−1.11903	−0.559517	+	0.828819i	$0.689013\pi$
$674$	−20.4149	−0.786352
$675$	0	0
$676$	−2.14133	−0.0823587
$677$	13.5700	0.521539	0.260769	−	0.965401i	$-0.416024\pi$
$677$	13.5700	0.521539	0.260769	+	0.965401i	$0.416024\pi$
$678$	0	0
$679$	18.4053	0.706330
$680$	8.72949	0.334761
$681$	0	0
$682$	−26.8010	−1.02626
$683$	−42.1926	−1.61446	−0.807228	−	0.590240i	$-0.799033\pi$
$683$	−42.1926	−1.61446	−0.807228	+	0.590240i	$0.799033\pi$
$684$	0	0
$685$	−15.4995	−0.592207
$686$	24.4013	0.931646
$687$	0	0
$688$	−17.1588	−0.654175
$689$	−49.6999	−1.89342
$690$	0	0
$691$	43.0952	1.63942	0.819709	−	0.572781i	$-0.194135\pi$
$691$	43.0952	1.63942	0.819709	+	0.572781i	$0.194135\pi$
$692$	−1.41322	−0.0537227
$693$	0	0
$694$	12.7659	0.484588
$695$	−22.2028	−0.842199
$696$	0	0
$697$	−8.94633	−0.338866
$698$	−46.3648	−1.75493
$699$	0	0
$700$	−0.435751	−0.0164698
$701$	47.5830	1.79718	0.898592	−	0.438785i	$-0.144591\pi$
$701$	47.5830	1.79718	0.898592	+	0.438785i	$0.144591\pi$
$702$	0	0
$703$	−31.0470	−1.17096
$704$	43.3969	1.63558
$705$	0	0
$706$	−15.3000	−0.575822
$707$	19.7996	0.744642
$708$	0	0
$709$	8.00508	0.300637	0.150319	−	0.988638i	$-0.451970\pi$
$709$	8.00508	0.300637	0.150319	+	0.988638i	$0.451970\pi$
$710$	23.3409	0.875968
$711$	0	0
$712$	41.6355	1.56036
$713$	11.0111	0.412371
$714$	0	0
$715$	−42.5179	−1.59008
$716$	0.528827	0.0197632
$717$	0	0
$718$	14.7499	0.550462
$719$	−3.65160	−0.136182	−0.0680908	−	0.997679i	$-0.521691\pi$
$719$	−3.65160	−0.136182	−0.0680908	+	0.997679i	$0.521691\pi$
$720$	0	0
$721$	−8.71556	−0.324585
$722$	13.3946	0.498496
$723$	0	0
$724$	−1.60851	−0.0597798
$725$	−1.67975	−0.0623843
$726$	0	0
$727$	18.3105	0.679100	0.339550	−	0.940588i	$-0.389725\pi$
$727$	18.3105	0.679100	0.339550	+	0.940588i	$0.389725\pi$
$728$	−26.2613	−0.973309
$729$	0	0
$730$	19.5828	0.724791
$731$	−10.0700	−0.372453
$732$	0	0
$733$	45.3607	1.67543	0.837717	−	0.546104i	$-0.183890\pi$
$733$	45.3607	1.67543	0.837717	+	0.546104i	$0.183890\pi$
$734$	28.6983	1.05928
$735$	0	0
$736$	−1.52481	−0.0562053
$737$	−48.7270	−1.79488
$738$	0	0
$739$	33.4799	1.23158	0.615790	−	0.787911i	$-0.288837\pi$
$739$	33.4799	1.23158	0.615790	+	0.787911i	$0.288837\pi$
$740$	−0.714675	−0.0262720
$741$	0	0
$742$	−17.1130	−0.628239
$743$	13.6930	0.502348	0.251174	−	0.967942i	$-0.419183\pi$
$743$	13.6930	0.502348	0.251174	+	0.967942i	$0.419183\pi$
$744$	0	0
$745$	−20.1140	−0.736921
$746$	−51.8590	−1.89869
$747$	0	0
$748$	1.06244	0.0388467
$749$	13.3960	0.489480
$750$	0	0
$751$	−27.4062	−1.00007	−0.500034	−	0.866006i	$-0.666679\pi$
$751$	−27.4062	−1.00007	−0.500034	+	0.866006i	$0.666679\pi$
$752$	43.3183	1.57965
$753$	0	0
$754$	−4.41651	−0.160840
$755$	19.6986	0.716905
$756$	0	0
$757$	−9.98983	−0.363087	−0.181543	−	0.983383i	$-0.558109\pi$
$757$	−9.98983	−0.363087	−0.181543	+	0.983383i	$0.558109\pi$
$758$	−20.3116	−0.737749
$759$	0	0
$760$	20.9178	0.758768
$761$	−38.8553	−1.40851	−0.704253	−	0.709949i	$-0.748718\pi$
$761$	−38.8553	−1.40851	−0.704253	+	0.709949i	$0.748718\pi$
$762$	0	0
$763$	−14.2104	−0.514451
$764$	−1.31841	−0.0476984
$765$	0	0
$766$	1.61825	0.0584696
$767$	32.3302	1.16737
$768$	0	0
$769$	19.8198	0.714720	0.357360	−	0.933967i	$-0.383677\pi$
$769$	19.8198	0.714720	0.357360	+	0.933967i	$0.383677\pi$
$770$	−14.6401	−0.527592
$771$	0	0
$772$	−1.54972	−0.0557756
$773$	−29.0662	−1.04544	−0.522720	−	0.852505i	$-0.675083\pi$
$773$	−29.0662	−1.04544	−0.522720	+	0.852505i	$0.675083\pi$
$774$	0	0
$775$	11.8173	0.424491
$776$	35.3309	1.26831
$777$	0	0
$778$	47.6730	1.70916
$779$	−21.4374	−0.768074
$780$	0	0
$781$	65.1143	2.32997
$782$	9.13227	0.326569
$783$	0	0
$784$	18.0353	0.644118
$785$	4.85741	0.173368
$786$	0	0
$787$	−40.4799	−1.44295	−0.721476	−	0.692440i	$-0.756536\pi$
$787$	−40.4799	−1.44295	−0.721476	+	0.692440i	$0.756536\pi$
$788$	−1.50994	−0.0537893
$789$	0	0
$790$	−10.6939	−0.380472
$791$	0.0661485	0.00235197
$792$	0	0
$793$	84.6934	3.00755
$794$	4.21886	0.149722
$795$	0	0
$796$	1.97849	0.0701258
$797$	−11.4938	−0.407132	−0.203566	−	0.979061i	$-0.565253\pi$
$797$	−11.4938	−0.407132	−0.203566	+	0.979061i	$0.565253\pi$
$798$	0	0
$799$	25.4222	0.899374
$800$	−1.63645	−0.0578573
$801$	0	0
$802$	−29.7748	−1.05139
$803$	54.6302	1.92786
$804$	0	0
$805$	6.01485	0.211995
$806$	31.0709	1.09443
$807$	0	0
$808$	38.0076	1.33710
$809$	−51.3120	−1.80403	−0.902016	−	0.431702i	$-0.857913\pi$
$809$	−51.3120	−1.80403	−0.902016	+	0.431702i	$0.857913\pi$
$810$	0	0
$811$	32.5242	1.14208	0.571040	−	0.820922i	$-0.306540\pi$
$811$	32.5242	1.14208	0.571040	+	0.820922i	$0.306540\pi$
$812$	0.0726873	0.00255082
$813$	0	0
$814$	41.7119	1.46200
$815$	14.3253	0.501792
$816$	0	0
$817$	−24.1300	−0.844203
$818$	−2.89201	−0.101117
$819$	0	0
$820$	−0.493470	−0.0172327
$821$	−11.2602	−0.392983	−0.196492	−	0.980506i	$-0.562955\pi$
$821$	−11.2602	−0.392983	−0.196492	+	0.980506i	$0.562955\pi$
$822$	0	0
$823$	−3.83791	−0.133781	−0.0668906	−	0.997760i	$-0.521308\pi$
$823$	−3.83791	−0.133781	−0.0668906	+	0.997760i	$0.521308\pi$
$824$	−16.7305	−0.582834
$825$	0	0
$826$	11.1322	0.387337
$827$	22.1887	0.771577	0.385788	−	0.922587i	$-0.373930\pi$
$827$	22.1887	0.771577	0.385788	+	0.922587i	$0.373930\pi$
$828$	0	0
$829$	−12.6872	−0.440645	−0.220323	−	0.975427i	$-0.570711\pi$
$829$	−12.6872	−0.440645	−0.220323	+	0.975427i	$0.570711\pi$
$830$	8.79484	0.305273
$831$	0	0
$832$	−50.3109	−1.74422
$833$	10.5844	0.366728
$834$	0	0
$835$	28.1321	0.973552
$836$	2.54585	0.0880499
$837$	0	0
$838$	6.29725	0.217535
$839$	31.9752	1.10391	0.551953	−	0.833875i	$-0.313883\pi$
$839$	31.9752	1.10391	0.551953	+	0.833875i	$0.313883\pi$
$840$	0	0
$841$	−28.7198	−0.990338
$842$	43.0728	1.48439
$843$	0	0
$844$	0.238164	0.00819795
$845$	31.7216	1.09126
$846$	0	0
$847$	−24.2854	−0.834457
$848$	−31.3487	−1.07652
$849$	0	0
$850$	9.80089	0.336168
$851$	−17.1373	−0.587458
$852$	0	0
$853$	−31.1065	−1.06507	−0.532533	−	0.846409i	$-0.678760\pi$
$853$	−31.1065	−1.06507	−0.532533	+	0.846409i	$0.678760\pi$
$854$	29.1623	0.997912
$855$	0	0
$856$	25.7151	0.878925
$857$	11.3097	0.386333	0.193166	−	0.981166i	$-0.438124\pi$
$857$	11.3097	0.386333	0.193166	+	0.981166i	$0.438124\pi$
$858$	0	0
$859$	32.1279	1.09619	0.548095	−	0.836416i	$-0.315353\pi$
$859$	32.1279	1.09619	0.548095	+	0.836416i	$0.315353\pi$
$860$	−0.555452	−0.0189408
$861$	0	0
$862$	−8.40555	−0.286294
$863$	15.1014	0.514056	0.257028	−	0.966404i	$-0.417257\pi$
$863$	15.1014	0.514056	0.257028	+	0.966404i	$0.417257\pi$
$864$	0	0
$865$	20.9355	0.711829
$866$	10.6357	0.361416
$867$	0	0
$868$	−0.511368	−0.0173569
$869$	−29.8329	−1.01201
$870$	0	0
$871$	56.4901	1.91410
$872$	−27.2784	−0.923764
$873$	0	0
$874$	21.8830	0.740202
$875$	16.6263	0.562073
$876$	0	0
$877$	−37.9980	−1.28310	−0.641551	−	0.767080i	$-0.721709\pi$
$877$	−37.9980	−1.28310	−0.641551	+	0.767080i	$0.721709\pi$
$878$	−16.5275	−0.557775
$879$	0	0
$880$	−26.8185	−0.904053
$881$	−31.7883	−1.07097	−0.535487	−	0.844543i	$-0.679872\pi$
$881$	−31.7883	−1.07097	−0.535487	+	0.844543i	$0.679872\pi$
$882$	0	0
$883$	−31.7099	−1.06712	−0.533562	−	0.845761i	$-0.679147\pi$
$883$	−31.7099	−1.06712	−0.533562	+	0.845761i	$0.679147\pi$
$884$	−1.23171	−0.0414269
$885$	0	0
$886$	46.5313	1.56325
$887$	−3.92551	−0.131806	−0.0659029	−	0.997826i	$-0.520993\pi$
$887$	−3.92551	−0.131806	−0.0659029	+	0.997826i	$0.520993\pi$
$888$	0	0
$889$	−2.29865	−0.0770943
$890$	−26.9088	−0.901985
$891$	0	0
$892$	−1.62603	−0.0544435
$893$	60.9173	2.03852
$894$	0	0
$895$	−7.83405	−0.261863
$896$	−15.7711	−0.526875
$897$	0	0
$898$	−11.2486	−0.375370
$899$	−1.97124	−0.0657444
$900$	0	0
$901$	−18.3976	−0.612914
$902$	28.8013	0.958978
$903$	0	0
$904$	0.126979	0.00422327
$905$	23.8285	0.792085
$906$	0	0
$907$	37.1647	1.23403	0.617016	−	0.786950i	$-0.288341\pi$
$907$	37.1647	1.23403	0.617016	+	0.786950i	$0.288341\pi$
$908$	−0.367538	−0.0121972
$909$	0	0
$910$	16.9725	0.562633
$911$	−15.0959	−0.500151	−0.250075	−	0.968226i	$-0.580455\pi$
$911$	−15.0959	−0.500151	−0.250075	+	0.968226i	$0.580455\pi$
$912$	0	0
$913$	24.5350	0.811991
$914$	−25.1463	−0.831765
$915$	0	0
$916$	−0.301262	−0.00995399
$917$	3.09410	0.102176
$918$	0	0
$919$	−17.8896	−0.590123	−0.295062	−	0.955478i	$-0.595340\pi$
$919$	−17.8896	−0.590123	−0.295062	+	0.955478i	$0.595340\pi$
$920$	11.5462	0.380665
$921$	0	0
$922$	−49.4803	−1.62955
$923$	−75.4883	−2.48473
$924$	0	0
$925$	−18.3920	−0.604724
$926$	24.3549	0.800351
$927$	0	0
$928$	0.272975	0.00896084
$929$	−32.0444	−1.05134	−0.525671	−	0.850688i	$-0.676186\pi$
$929$	−32.0444	−1.05134	−0.525671	+	0.850688i	$0.676186\pi$
$930$	0	0
$931$	25.3626	0.831225
$932$	−2.14161	−0.0701507
$933$	0	0
$934$	49.5117	1.62007
$935$	−15.7390	−0.514721
$936$	0	0
$937$	11.6221	0.379677	0.189839	−	0.981815i	$-0.439204\pi$
$937$	11.6221	0.379677	0.189839	+	0.981815i	$0.439204\pi$
$938$	19.4511	0.635101
$939$	0	0
$940$	1.40226	0.0457368
$941$	4.15109	0.135322	0.0676608	−	0.997708i	$-0.478446\pi$
$941$	4.15109	0.135322	0.0676608	+	0.997708i	$0.478446\pi$
$942$	0	0
$943$	−11.8330	−0.385334
$944$	20.3925	0.663721
$945$	0	0
$946$	32.4189	1.05403
$947$	7.73026	0.251200	0.125600	−	0.992081i	$-0.459914\pi$
$947$	7.73026	0.251200	0.125600	+	0.992081i	$0.459914\pi$
$948$	0	0
$949$	−63.3339	−2.05590
$950$	23.4851	0.761957
$951$	0	0
$952$	−9.72126	−0.315068
$953$	−40.5116	−1.31230	−0.656149	−	0.754631i	$-0.727816\pi$
$953$	−40.5116	−1.31230	−0.656149	+	0.754631i	$0.727816\pi$
$954$	0	0
$955$	19.5309	0.632006
$956$	0.0912347	0.00295074
$957$	0	0
$958$	25.7124	0.830729
$959$	17.2605	0.557370
$960$	0	0
$961$	−17.1320	−0.552645
$962$	−48.3574	−1.55911
$963$	0	0
$964$	−1.54816	−0.0498629
$965$	22.9575	0.739030
$966$	0	0
$967$	−32.0195	−1.02968	−0.514839	−	0.857287i	$-0.672148\pi$
$967$	−32.0195	−1.02968	−0.514839	+	0.857287i	$0.672148\pi$
$968$	−46.6185	−1.49837
$969$	0	0
$970$	−22.8342	−0.733161
$971$	−13.2739	−0.425980	−0.212990	−	0.977054i	$-0.568320\pi$
$971$	−13.2739	−0.425980	−0.212990	+	0.977054i	$0.568320\pi$
$972$	0	0
$973$	24.7253	0.792655
$974$	18.9393	0.606853
$975$	0	0
$976$	53.4211	1.70997
$977$	16.6713	0.533361	0.266680	−	0.963785i	$-0.414073\pi$
$977$	16.6713	0.533361	0.266680	+	0.963785i	$0.414073\pi$
$978$	0	0
$979$	−75.0677	−2.39917
$980$	0.583824	0.0186496
$981$	0	0
$982$	−37.1325	−1.18494
$983$	−3.93033	−0.125358	−0.0626790	−	0.998034i	$-0.519964\pi$
$983$	−3.93033	−0.125358	−0.0626790	+	0.998034i	$0.519964\pi$
$984$	0	0
$985$	22.3682	0.712711
$986$	−1.63488	−0.0520651
$987$	0	0
$988$	−2.95145	−0.0938981
$989$	−13.3192	−0.423527
$990$	0	0
$991$	−48.9367	−1.55452	−0.777262	−	0.629177i	$-0.783392\pi$
$991$	−48.9367	−1.55452	−0.777262	+	0.629177i	$0.783392\pi$
$992$	−1.92043	−0.0609736
$993$	0	0
$994$	−25.9927	−0.824438
$995$	−29.3094	−0.929171
$996$	0	0
$997$	−37.6435	−1.19218	−0.596091	−	0.802917i	$-0.703280\pi$
$997$	−37.6435	−1.19218	−0.596091	+	0.802917i	$0.703280\pi$
$998$	−49.3628	−1.56255
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.j.1.16	✓	20
3.2	odd	2		2151.2.a.k.1.5	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.16	✓	20	1.1	even	1	trivial
2151.2.a.k.1.5	yes	20	3.2	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)

\(f(q)\)	\(=\)	\(q+1.38158 q^{2} -0.0912347 q^{4} +1.35155 q^{5} -1.50510 q^{7} -2.88921 q^{8} +O(q^{10})\)	\(q+1.38158 q^{2} -0.0912347 q^{4} +1.35155 q^{5} -1.50510 q^{7} -2.88921 q^{8} +1.86728 q^{10} +5.20916 q^{11} -6.03908 q^{13} -2.07942 q^{14} -3.80921 q^{16} -2.23551 q^{17} -5.35678 q^{19} -0.123308 q^{20} +7.19688 q^{22} -2.95682 q^{23} -3.17331 q^{25} -8.34348 q^{26} +0.137318 q^{28} +0.529336 q^{29} -3.72398 q^{31} +0.515693 q^{32} -3.08854 q^{34} -2.03422 q^{35} +5.79583 q^{37} -7.40083 q^{38} -3.90492 q^{40} +4.00191 q^{41} +4.50457 q^{43} -0.475257 q^{44} -4.08509 q^{46} -11.3720 q^{47} -4.73466 q^{49} -4.38418 q^{50} +0.550974 q^{52} +8.22971 q^{53} +7.04045 q^{55} +4.34856 q^{56} +0.731321 q^{58} -5.35349 q^{59} -14.0242 q^{61} -5.14497 q^{62} +8.33088 q^{64} -8.16213 q^{65} -9.35409 q^{67} +0.203956 q^{68} -2.81045 q^{70} +12.5000 q^{71} +10.4873 q^{73} +8.00741 q^{74} +0.488725 q^{76} -7.84033 q^{77} -5.72701 q^{79} -5.14834 q^{80} +5.52897 q^{82} +4.70998 q^{83} -3.02141 q^{85} +6.22343 q^{86} -15.0504 q^{88} -14.4107 q^{89} +9.08944 q^{91} +0.269765 q^{92} -15.7113 q^{94} -7.23997 q^{95} -12.2286 q^{97} -6.54132 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{2} + 18 q^{4} - 16 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10}) \) 20 * q - 4 * q^2 + 18 * q^4 - 16 * q^5 - 4 * q^7 - 12 * q^8	\( 20 q - 4 q^{2} + 18 q^{4} - 16 q^{5} - 4 q^{7} - 12 q^{8} + 4 q^{10} - 12 q^{11} - 4 q^{13} - 20 q^{14} + 22 q^{16} - 24 q^{17} - 4 q^{19} - 40 q^{20} - 6 q^{22} - 12 q^{23} + 22 q^{25} - 30 q^{26} - 12 q^{28} - 24 q^{29} - 4 q^{31} - 28 q^{32} + 8 q^{34} - 20 q^{35} - 10 q^{37} - 26 q^{38} + 6 q^{40} - 66 q^{41} + 8 q^{43} - 36 q^{44} - 12 q^{46} - 28 q^{47} + 18 q^{49} - 28 q^{50} - 18 q^{52} - 28 q^{53} - 4 q^{55} - 60 q^{56} - 54 q^{59} - 4 q^{61} - 20 q^{62} + 22 q^{64} - 42 q^{65} + 12 q^{67} - 12 q^{68} + 20 q^{70} - 36 q^{71} + 14 q^{73} - 50 q^{76} - 8 q^{77} - 12 q^{79} - 88 q^{80} - 8 q^{82} - 20 q^{83} + 4 q^{85} - 18 q^{86} - 10 q^{88} - 130 q^{89} - 6 q^{91} + 46 q^{92} - 26 q^{94} - 2 q^{97} - 12 q^{98}+O(q^{100}) \) 20 * q - 4 * q^2 + 18 * q^4 - 16 * q^5 - 4 * q^7 - 12 * q^8 + 4 * q^10 - 12 * q^11 - 4 * q^13 - 20 * q^14 + 22 * q^16 - 24 * q^17 - 4 * q^19 - 40 * q^20 - 6 * q^22 - 12 * q^23 + 22 * q^25 - 30 * q^26 - 12 * q^28 - 24 * q^29 - 4 * q^31 - 28 * q^32 + 8 * q^34 - 20 * q^35 - 10 * q^37 - 26 * q^38 + 6 * q^40 - 66 * q^41 + 8 * q^43 - 36 * q^44 - 12 * q^46 - 28 * q^47 + 18 * q^49 - 28 * q^50 - 18 * q^52 - 28 * q^53 - 4 * q^55 - 60 * q^56 - 54 * q^59 - 4 * q^61 - 20 * q^62 + 22 * q^64 - 42 * q^65 + 12 * q^67 - 12 * q^68 + 20 * q^70 - 36 * q^71 + 14 * q^73 - 50 * q^76 - 8 * q^77 - 12 * q^79 - 88 * q^80 - 8 * q^82 - 20 * q^83 + 4 * q^85 - 18 * q^86 - 10 * q^88 - 130 * q^89 - 6 * q^91 + 46 * q^92 - 26 * q^94 - 2 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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