LMFDB - Embedded newform 2151.2.a.k.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + \cdots + 1 $ x^20 - 4x^19 - 21x^18 + 96x^17 + 164x^16 - 936x^15 - 540x^14 + 4804x^13 + 229x^12 - 14020x^11 + 3356x^10 + 23404x^9 - 9429x^8 - 21252x^7 + 10479x^6 + 9108x^5 - 4844x^4 - 1184x^3 + 640x^2 - 56*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$2.41299$ 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)

$f(q)$	$=$	$q+2.41299 q^{2} +3.82251 q^{4} -0.666388 q^{5} +0.549662 q^{7} +4.39769 q^{8} +O(q^{10})$	\(q+2.41299 q^{2} +3.82251 q^{4} -0.666388 q^{5} +0.549662 q^{7} +4.39769 q^{8} -1.60799 q^{10} +0.262501 q^{11} +3.05335 q^{13} +1.32633 q^{14} +2.96655 q^{16} +2.00829 q^{17} +1.22323 q^{19} -2.54727 q^{20} +0.633411 q^{22} +6.88399 q^{23} -4.55593 q^{25} +7.36769 q^{26} +2.10109 q^{28} +5.64739 q^{29} +2.28051 q^{31} -1.63713 q^{32} +4.84598 q^{34} -0.366288 q^{35} +8.46115 q^{37} +2.95163 q^{38} -2.93057 q^{40} +1.90913 q^{41} -7.39552 q^{43} +1.00341 q^{44} +16.6110 q^{46} +1.67407 q^{47} -6.69787 q^{49} -10.9934 q^{50} +11.6714 q^{52} +7.14980 q^{53} -0.174927 q^{55} +2.41724 q^{56} +13.6271 q^{58} +3.67831 q^{59} -13.7223 q^{61} +5.50285 q^{62} -9.88347 q^{64} -2.03472 q^{65} -2.80719 q^{67} +7.67670 q^{68} -0.883849 q^{70} -4.78655 q^{71} +2.16854 q^{73} +20.4166 q^{74} +4.67580 q^{76} +0.144287 q^{77} -5.81088 q^{79} -1.97687 q^{80} +4.60671 q^{82} -3.57961 q^{83} -1.33830 q^{85} -17.8453 q^{86} +1.15440 q^{88} -9.71189 q^{89} +1.67831 q^{91} +26.3141 q^{92} +4.03951 q^{94} -0.815145 q^{95} +7.62712 q^{97} -16.1619 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) $ 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8	$ 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) $ 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8 + 4 * q^10 + 12 * q^11 - 4 * q^13 + 20 * q^14 + 22 * q^16 + 24 * q^17 - 4 * q^19 + 40 * q^20 - 6 * q^22 + 12 * q^23 + 22 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 - 4 * q^31 + 28 * q^32 + 8 * q^34 + 20 * q^35 - 10 * q^37 + 26 * q^38 + 6 * q^40 + 66 * q^41 + 8 * q^43 + 36 * q^44 - 12 * q^46 + 28 * q^47 + 18 * q^49 + 28 * q^50 - 18 * q^52 + 28 * q^53 - 4 * q^55 + 60 * q^56 + 54 * q^59 - 4 * q^61 + 20 * q^62 + 22 * q^64 + 42 * q^65 + 12 * q^67 + 12 * q^68 + 20 * q^70 + 36 * q^71 + 14 * q^73 - 50 * q^76 + 8 * q^77 - 12 * q^79 + 88 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 + 18 * q^86 - 10 * q^88 + 130 * q^89 - 6 * q^91 - 46 * q^92 - 26 * q^94 - 2 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.41299	1.70624	0.853120	−	0.521715i	$-0.174708\pi$
$2$	2.41299	1.70624	0.853120	+	0.521715i	$0.174708\pi$
$3$	0	0
$3$	0	0
$4$	3.82251	1.91125
$5$	−0.666388	−0.298018	−0.149009	−	0.988836i	$-0.547608\pi$
$5$	−0.666388	−0.298018	−0.149009	+	0.988836i	$0.547608\pi$
$6$	0	0
$7$	0.549662	0.207753	0.103876	−	0.994590i	$-0.466875\pi$
$7$	0.549662	0.207753	0.103876	+	0.994590i	$0.466875\pi$
$8$	4.39769	1.55482
$9$	0	0
$10$	−1.60799	−0.508490
$11$	0.262501	0.0791470	0.0395735	−	0.999217i	$-0.487400\pi$
$11$	0.262501	0.0791470	0.0395735	+	0.999217i	$0.487400\pi$
$12$	0	0
$13$	3.05335	0.846846	0.423423	−	0.905932i	$-0.360828\pi$
$13$	3.05335	0.846846	0.423423	+	0.905932i	$0.360828\pi$
$14$	1.32633	0.354476
$15$	0	0
$16$	2.96655	0.741637
$17$	2.00829	0.487082	0.243541	−	0.969891i	$-0.421691\pi$
$17$	2.00829	0.487082	0.243541	+	0.969891i	$0.421691\pi$
$18$	0	0
$19$	1.22323	0.280628	0.140314	−	0.990107i	$-0.455189\pi$
$19$	1.22323	0.280628	0.140314	+	0.990107i	$0.455189\pi$
$20$	−2.54727	−0.569588
$21$	0	0
$22$	0.633411	0.135044
$23$	6.88399	1.43541	0.717706	−	0.696347i	$-0.245192\pi$
$23$	6.88399	1.43541	0.717706	+	0.696347i	$0.245192\pi$
$24$	0	0
$25$	−4.55593	−0.911185
$26$	7.36769	1.44492
$27$	0	0
$28$	2.10109	0.397068
$29$	5.64739	1.04869	0.524347	−	0.851505i	$-0.324310\pi$
$29$	5.64739	1.04869	0.524347	+	0.851505i	$0.324310\pi$
$30$	0	0
$31$	2.28051	0.409592	0.204796	−	0.978805i	$-0.434347\pi$
$31$	2.28051	0.409592	0.204796	+	0.978805i	$0.434347\pi$
$32$	−1.63713	−0.289407
$33$	0	0
$34$	4.84598	0.831079
$35$	−0.366288	−0.0619140
$36$	0	0
$37$	8.46115	1.39100	0.695502	−	0.718524i	$-0.255182\pi$
$37$	8.46115	1.39100	0.695502	+	0.718524i	$0.255182\pi$
$38$	2.95163	0.478818
$39$	0	0
$40$	−2.93057	−0.463363
$41$	1.90913	0.298156	0.149078	−	0.988825i	$-0.452369\pi$
$41$	1.90913	0.298156	0.149078	+	0.988825i	$0.452369\pi$
$42$	0	0
$43$	−7.39552	−1.12781	−0.563903	−	0.825841i	$-0.690701\pi$
$43$	−7.39552	−1.12781	−0.563903	+	0.825841i	$0.690701\pi$
$44$	1.00341	0.151270
$45$	0	0
$46$	16.6110	2.44916
$47$	1.67407	0.244188	0.122094	−	0.992519i	$-0.461039\pi$
$47$	1.67407	0.244188	0.122094	+	0.992519i	$0.461039\pi$
$48$	0	0
$49$	−6.69787	−0.956839
$50$	−10.9934	−1.55470
$51$	0	0
$52$	11.6714	1.61854
$53$	7.14980	0.982100	0.491050	−	0.871131i	$-0.336613\pi$
$53$	7.14980	0.982100	0.491050	+	0.871131i	$0.336613\pi$
$54$	0	0
$55$	−0.174927	−0.0235872
$56$	2.41724	0.323017
$57$	0	0
$58$	13.6271	1.78932
$59$	3.67831	0.478875	0.239437	−	0.970912i	$-0.423037\pi$
$59$	3.67831	0.478875	0.239437	+	0.970912i	$0.423037\pi$
$60$	0	0
$61$	−13.7223	−1.75697	−0.878483	−	0.477773i	$-0.841444\pi$
$61$	−13.7223	−1.75697	−0.878483	+	0.477773i	$0.841444\pi$
$62$	5.50285	0.698862
$63$	0	0
$64$	−9.88347	−1.23543
$65$	−2.03472	−0.252375
$66$	0	0
$67$	−2.80719	−0.342953	−0.171477	−	0.985188i	$-0.554854\pi$
$67$	−2.80719	−0.342953	−0.171477	+	0.985188i	$0.554854\pi$
$68$	7.67670	0.930937
$69$	0	0
$70$	−0.883849	−0.105640
$71$	−4.78655	−0.568059	−0.284030	−	0.958816i	$-0.591671\pi$
$71$	−4.78655	−0.568059	−0.284030	+	0.958816i	$0.591671\pi$
$72$	0	0
$73$	2.16854	0.253808	0.126904	−	0.991915i	$-0.459496\pi$
$73$	2.16854	0.253808	0.126904	+	0.991915i	$0.459496\pi$
$74$	20.4166	2.37339
$75$	0	0
$76$	4.67580	0.536351
$77$	0.144287	0.0164430
$78$	0	0
$79$	−5.81088	−0.653775	−0.326888	−	0.945063i	$-0.606000\pi$
$79$	−5.81088	−0.653775	−0.326888	+	0.945063i	$0.606000\pi$
$80$	−1.97687	−0.221021
$81$	0	0
$82$	4.60671	0.508726
$83$	−3.57961	−0.392914	−0.196457	−	0.980512i	$-0.562944\pi$
$83$	−3.57961	−0.392914	−0.196457	+	0.980512i	$0.562944\pi$
$84$	0	0
$85$	−1.33830	−0.145159
$86$	−17.8453	−1.92431
$87$	0	0
$88$	1.15440	0.123059
$89$	−9.71189	−1.02946	−0.514729	−	0.857353i	$-0.672108\pi$
$89$	−9.71189	−1.02946	−0.514729	+	0.857353i	$0.672108\pi$
$90$	0	0
$91$	1.67831	0.175935
$92$	26.3141	2.74344
$93$	0	0
$94$	4.03951	0.416644
$95$	−0.815145	−0.0836321
$96$	0	0
$97$	7.62712	0.774417	0.387209	−	0.921992i	$-0.373439\pi$
$97$	7.62712	0.774417	0.387209	+	0.921992i	$0.373439\pi$
$98$	−16.1619	−1.63260
$99$	0	0
$100$	−17.4151	−1.74151
$101$	−1.34774	−0.134106	−0.0670528	−	0.997749i	$-0.521360\pi$
$101$	−1.34774	−0.134106	−0.0670528	+	0.997749i	$0.521360\pi$
$102$	0	0
$103$	18.5433	1.82712	0.913561	−	0.406702i	$-0.133321\pi$
$103$	18.5433	1.82712	0.913561	+	0.406702i	$0.133321\pi$
$104$	13.4277	1.31669
$105$	0	0
$106$	17.2524	1.67570
$107$	−15.2283	−1.47217	−0.736086	−	0.676888i	$-0.763328\pi$
$107$	−15.2283	−1.47217	−0.736086	+	0.676888i	$0.763328\pi$
$108$	0	0
$109$	−2.17137	−0.207979	−0.103990	−	0.994578i	$-0.533161\pi$
$109$	−2.17137	−0.207979	−0.103990	+	0.994578i	$0.533161\pi$
$110$	−0.422098	−0.0402454
$111$	0	0
$112$	1.63060	0.154077
$113$	−0.665124	−0.0625696	−0.0312848	−	0.999511i	$-0.509960\pi$
$113$	−0.665124	−0.0625696	−0.0312848	+	0.999511i	$0.509960\pi$
$114$	0	0
$115$	−4.58741	−0.427778
$116$	21.5872	2.00432
$117$	0	0
$118$	8.87571	0.817075
$119$	1.10388	0.101193
$120$	0	0
$121$	−10.9311	−0.993736
$122$	−33.1118	−2.99781
$123$	0	0
$124$	8.71727	0.782834
$125$	6.36796	0.569567
$126$	0	0
$127$	−9.90489	−0.878917	−0.439459	−	0.898263i	$-0.644830\pi$
$127$	−9.90489	−0.878917	−0.439459	+	0.898263i	$0.644830\pi$
$128$	−20.5744	−1.81854
$129$	0	0
$130$	−4.90974	−0.430613
$131$	4.12041	0.360002	0.180001	−	0.983666i	$-0.442390\pi$
$131$	4.12041	0.360002	0.180001	+	0.983666i	$0.442390\pi$
$132$	0	0
$133$	0.672362	0.0583012
$134$	−6.77372	−0.585160
$135$	0	0
$136$	8.83183	0.757323
$137$	11.0196	0.941472	0.470736	−	0.882274i	$-0.343988\pi$
$137$	11.0196	0.941472	0.470736	+	0.882274i	$0.343988\pi$
$138$	0	0
$139$	−1.99415	−0.169142	−0.0845708	−	0.996417i	$-0.526952\pi$
$139$	−1.99415	−0.169142	−0.0845708	+	0.996417i	$0.526952\pi$
$140$	−1.40014	−0.118333
$141$	0	0
$142$	−11.5499	−0.969245
$143$	0.801506	0.0670253
$144$	0	0
$145$	−3.76335	−0.312529
$146$	5.23265	0.433057
$147$	0	0
$148$	32.3428	2.65856
$149$	−14.7199	−1.20590	−0.602952	−	0.797778i	$-0.706009\pi$
$149$	−14.7199	−1.20590	−0.602952	+	0.797778i	$0.706009\pi$
$150$	0	0
$151$	−1.49585	−0.121730	−0.0608652	−	0.998146i	$-0.519386\pi$
$151$	−1.49585	−0.121730	−0.0608652	+	0.998146i	$0.519386\pi$
$152$	5.37938	0.436325
$153$	0	0
$154$	0.348162	0.0280557
$155$	−1.51971	−0.122066
$156$	0	0
$157$	−1.34647	−0.107460	−0.0537300	−	0.998555i	$-0.517111\pi$
$157$	−1.34647	−0.107460	−0.0537300	+	0.998555i	$0.517111\pi$
$158$	−14.0216	−1.11550
$159$	0	0
$160$	1.09097	0.0862484
$161$	3.78387	0.298211
$162$	0	0
$163$	−10.0209	−0.784899	−0.392450	−	0.919773i	$-0.628372\pi$
$163$	−10.0209	−0.784899	−0.392450	+	0.919773i	$0.628372\pi$
$164$	7.29767	0.569853
$165$	0	0
$166$	−8.63756	−0.670405
$167$	−3.14438	−0.243319	−0.121660	−	0.992572i	$-0.538822\pi$
$167$	−3.14438	−0.243319	−0.121660	+	0.992572i	$0.538822\pi$
$168$	0	0
$169$	−3.67706	−0.282851
$170$	−3.22930	−0.247676
$171$	0	0
$172$	−28.2694	−2.15553
$173$	7.87801	0.598954	0.299477	−	0.954104i	$-0.403188\pi$
$173$	7.87801	0.598954	0.299477	+	0.954104i	$0.403188\pi$
$174$	0	0
$175$	−2.50422	−0.189301
$176$	0.778721	0.0586983
$177$	0	0
$178$	−23.4347	−1.75650
$179$	−1.92732	−0.144055	−0.0720275	−	0.997403i	$-0.522947\pi$
$179$	−1.92732	−0.144055	−0.0720275	+	0.997403i	$0.522947\pi$
$180$	0	0
$181$	−9.05976	−0.673406	−0.336703	−	0.941611i	$-0.609312\pi$
$181$	−9.05976	−0.673406	−0.336703	+	0.941611i	$0.609312\pi$
$182$	4.04974	0.300187
$183$	0	0
$184$	30.2736	2.23180
$185$	−5.63841	−0.414544
$186$	0	0
$187$	0.527178	0.0385511
$188$	6.39915	0.466706
$189$	0	0
$190$	−1.96693	−0.142696
$191$	19.2051	1.38963	0.694816	−	0.719188i	$-0.255486\pi$
$191$	19.2051	1.38963	0.694816	+	0.719188i	$0.255486\pi$
$192$	0	0
$193$	−15.4298	−1.11066	−0.555332	−	0.831629i	$-0.687409\pi$
$193$	−15.4298	−1.11066	−0.555332	+	0.831629i	$0.687409\pi$
$194$	18.4042	1.32134
$195$	0	0
$196$	−25.6027	−1.82876
$197$	17.3806	1.23831	0.619157	−	0.785267i	$-0.287474\pi$
$197$	17.3806	1.23831	0.619157	+	0.785267i	$0.287474\pi$
$198$	0	0
$199$	−10.3156	−0.731252	−0.365626	−	0.930762i	$-0.619145\pi$
$199$	−10.3156	−0.731252	−0.365626	+	0.930762i	$0.619145\pi$
$200$	−20.0355	−1.41673
$201$	0	0
$202$	−3.25209	−0.228816
$203$	3.10415	0.217869
$204$	0	0
$205$	−1.27222	−0.0888560
$206$	44.7447	3.11751
$207$	0	0
$208$	9.05790	0.628053
$209$	0.321098	0.0222108
$210$	0	0
$211$	5.32226	0.366400	0.183200	−	0.983076i	$-0.441354\pi$
$211$	5.32226	0.366400	0.183200	+	0.983076i	$0.441354\pi$
$212$	27.3301	1.87704
$213$	0	0
$214$	−36.7456	−2.51188
$215$	4.92829	0.336107
$216$	0	0
$217$	1.25351	0.0850938
$218$	−5.23948	−0.354862
$219$	0	0
$220$	−0.668661	−0.0450811
$221$	6.13201	0.412484
$222$	0	0
$223$	−7.17603	−0.480542	−0.240271	−	0.970706i	$-0.577236\pi$
$223$	−7.17603	−0.480542	−0.240271	+	0.970706i	$0.577236\pi$
$224$	−0.899869	−0.0601250
$225$	0	0
$226$	−1.60494	−0.106759
$227$	−26.6721	−1.77029	−0.885144	−	0.465318i	$-0.845940\pi$
$227$	−26.6721	−1.77029	−0.885144	+	0.465318i	$0.845940\pi$
$228$	0	0
$229$	4.07227	0.269103	0.134552	−	0.990907i	$-0.457041\pi$
$229$	4.07227	0.269103	0.134552	+	0.990907i	$0.457041\pi$
$230$	−11.0694	−0.729892
$231$	0	0
$232$	24.8354	1.63053
$233$	−14.5101	−0.950590	−0.475295	−	0.879827i	$-0.657659\pi$
$233$	−14.5101	−0.950590	−0.475295	+	0.879827i	$0.657659\pi$
$234$	0	0
$235$	−1.11558	−0.0727725
$236$	14.0604	0.915252
$237$	0	0
$238$	2.66365	0.172659
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−10.1087	−0.651160	−0.325580	−	0.945515i	$-0.605559\pi$
$241$	−10.1087	−0.651160	−0.325580	+	0.945515i	$0.605559\pi$
$242$	−26.3766	−1.69555
$243$	0	0
$244$	−52.4538	−3.35801
$245$	4.46338	0.285155
$246$	0	0
$247$	3.73494	0.237649
$248$	10.0290	0.636841
$249$	0	0
$250$	15.3658	0.971818
$251$	22.5850	1.42555	0.712775	−	0.701393i	$-0.247438\pi$
$251$	22.5850	1.42555	0.712775	+	0.701393i	$0.247438\pi$
$252$	0	0
$253$	1.80705	0.113608
$254$	−23.9004	−1.49964
$255$	0	0
$256$	−29.8789	−1.86743
$257$	6.41138	0.399931	0.199966	−	0.979803i	$-0.435917\pi$
$257$	6.41138	0.399931	0.199966	+	0.979803i	$0.435917\pi$
$258$	0	0
$259$	4.65077	0.288985
$260$	−7.77771	−0.482353
$261$	0	0
$262$	9.94250	0.614250
$263$	−3.66746	−0.226145	−0.113073	−	0.993587i	$-0.536069\pi$
$263$	−3.66746	−0.226145	−0.113073	+	0.993587i	$0.536069\pi$
$264$	0	0
$265$	−4.76454	−0.292683
$266$	1.62240	0.0994758
$267$	0	0
$268$	−10.7305	−0.655471
$269$	19.7784	1.20591	0.602954	−	0.797776i	$-0.293990\pi$
$269$	19.7784	1.20591	0.602954	+	0.797776i	$0.293990\pi$
$270$	0	0
$271$	−1.14335	−0.0694534	−0.0347267	−	0.999397i	$-0.511056\pi$
$271$	−1.14335	−0.0694534	−0.0347267	+	0.999397i	$0.511056\pi$
$272$	5.95769	0.361238
$273$	0	0
$274$	26.5903	1.60638
$275$	−1.19593	−0.0721176
$276$	0	0
$277$	−10.1579	−0.610331	−0.305165	−	0.952299i	$-0.598712\pi$
$277$	−10.1579	−0.610331	−0.305165	+	0.952299i	$0.598712\pi$
$278$	−4.81186	−0.288596
$279$	0	0
$280$	−1.61082	−0.0962650
$281$	−12.7932	−0.763176	−0.381588	−	0.924333i	$-0.624623\pi$
$281$	−12.7932	−0.763176	−0.381588	+	0.924333i	$0.624623\pi$
$282$	0	0
$283$	−5.32245	−0.316387	−0.158193	−	0.987408i	$-0.550567\pi$
$283$	−5.32245	−0.316387	−0.158193	+	0.987408i	$0.550567\pi$
$284$	−18.2966	−1.08571
$285$	0	0
$286$	1.93402	0.114361
$287$	1.04938	0.0619428
$288$	0	0
$289$	−12.9668	−0.762751
$290$	−9.08092	−0.533250
$291$	0	0
$292$	8.28924	0.485091
$293$	7.15519	0.418011	0.209005	−	0.977915i	$-0.432977\pi$
$293$	7.15519	0.418011	0.209005	+	0.977915i	$0.432977\pi$
$294$	0	0
$295$	−2.45118	−0.142713
$296$	37.2095	2.16276
$297$	0	0
$298$	−35.5190	−2.05756
$299$	21.0192	1.21557
$300$	0	0
$301$	−4.06504	−0.234305
$302$	−3.60946	−0.207701
$303$	0	0
$304$	3.62877	0.208124
$305$	9.14441	0.523607
$306$	0	0
$307$	−20.5867	−1.17494	−0.587472	−	0.809244i	$-0.699877\pi$
$307$	−20.5867	−1.17494	−0.587472	+	0.809244i	$0.699877\pi$
$308$	0.551537	0.0314267
$309$	0	0
$310$	−3.66703	−0.208273
$311$	−20.1654	−1.14347	−0.571736	−	0.820437i	$-0.693730\pi$
$311$	−20.1654	−1.14347	−0.571736	+	0.820437i	$0.693730\pi$
$312$	0	0
$313$	17.7852	1.00528	0.502639	−	0.864496i	$-0.332362\pi$
$313$	17.7852	1.00528	0.502639	+	0.864496i	$0.332362\pi$
$314$	−3.24902	−0.183353
$315$	0	0
$316$	−22.2121	−1.24953
$317$	32.3858	1.81897	0.909485	−	0.415737i	$-0.136476\pi$
$317$	32.3858	1.81897	0.909485	+	0.415737i	$0.136476\pi$
$318$	0	0
$319$	1.48244	0.0830009
$320$	6.58623	0.368181
$321$	0	0
$322$	9.13043	0.508819
$323$	2.45660	0.136689
$324$	0	0
$325$	−13.9108	−0.771634
$326$	−24.1804	−1.33923
$327$	0	0
$328$	8.39577	0.463579
$329$	0.920173	0.0507308
$330$	0	0
$331$	−4.21115	−0.231466	−0.115733	−	0.993280i	$-0.536922\pi$
$331$	−4.21115	−0.231466	−0.115733	+	0.993280i	$0.536922\pi$
$332$	−13.6831	−0.750958
$333$	0	0
$334$	−7.58735	−0.415161
$335$	1.87068	0.102206
$336$	0	0
$337$	7.22450	0.393543	0.196772	−	0.980449i	$-0.436954\pi$
$337$	7.22450	0.393543	0.196772	+	0.980449i	$0.436954\pi$
$338$	−8.87271	−0.482612
$339$	0	0
$340$	−5.11567	−0.277436
$341$	0.598636	0.0324180
$342$	0	0
$343$	−7.52920	−0.406539
$344$	−32.5232	−1.75353
$345$	0	0
$346$	19.0095	1.02196
$347$	12.1793	0.653820	0.326910	−	0.945055i	$-0.393993\pi$
$347$	12.1793	0.653820	0.326910	+	0.945055i	$0.393993\pi$
$348$	0	0
$349$	7.63405	0.408641	0.204321	−	0.978904i	$-0.434501\pi$
$349$	7.63405	0.408641	0.204321	+	0.978904i	$0.434501\pi$
$350$	−6.04265	−0.322993
$351$	0	0
$352$	−0.429748	−0.0229057
$353$	7.39637	0.393669	0.196835	−	0.980437i	$-0.436934\pi$
$353$	7.39637	0.393669	0.196835	+	0.980437i	$0.436934\pi$
$354$	0	0
$355$	3.18970	0.169292
$356$	−37.1238	−1.96756
$357$	0	0
$358$	−4.65061	−0.245792
$359$	−9.14233	−0.482514	−0.241257	−	0.970461i	$-0.577560\pi$
$359$	−9.14233	−0.482514	−0.241257	+	0.970461i	$0.577560\pi$
$360$	0	0
$361$	−17.5037	−0.921248
$362$	−21.8611	−1.14899
$363$	0	0
$364$	6.41535	0.336256
$365$	−1.44509	−0.0756393
$366$	0	0
$367$	19.7352	1.03017	0.515086	−	0.857139i	$-0.327760\pi$
$367$	19.7352	1.03017	0.515086	+	0.857139i	$0.327760\pi$
$368$	20.4217	1.06455
$369$	0	0
$370$	−13.6054	−0.707311
$371$	3.92997	0.204034
$372$	0	0
$373$	−0.398821	−0.0206502	−0.0103251	−	0.999947i	$-0.503287\pi$
$373$	−0.398821	−0.0206502	−0.0103251	+	0.999947i	$0.503287\pi$
$374$	1.27207	0.0657774
$375$	0	0
$376$	7.36204	0.379668
$377$	17.2434	0.888082
$378$	0	0
$379$	−26.9887	−1.38632	−0.693158	−	0.720785i	$-0.743781\pi$
$379$	−26.9887	−1.38632	−0.693158	+	0.720785i	$0.743781\pi$
$380$	−3.11590	−0.159842
$381$	0	0
$382$	46.3416	2.37104
$383$	−0.514040	−0.0262662	−0.0131331	−	0.999914i	$-0.504181\pi$
$383$	−0.514040	−0.0262662	−0.0131331	+	0.999914i	$0.504181\pi$
$384$	0	0
$385$	−0.0961510	−0.00490031
$386$	−37.2320	−1.89506
$387$	0	0
$388$	29.1547	1.48011
$389$	10.7557	0.545337	0.272669	−	0.962108i	$-0.412094\pi$
$389$	10.7557	0.545337	0.272669	+	0.962108i	$0.412094\pi$
$390$	0	0
$391$	13.8251	0.699163
$392$	−29.4551	−1.48771
$393$	0	0
$394$	41.9391	2.11286
$395$	3.87230	0.194837
$396$	0	0
$397$	16.2650	0.816316	0.408158	−	0.912911i	$-0.366171\pi$
$397$	16.2650	0.816316	0.408158	+	0.912911i	$0.366171\pi$
$398$	−24.8914	−1.24769
$399$	0	0
$400$	−13.5154	−0.675769
$401$	8.64206	0.431564	0.215782	−	0.976442i	$-0.430770\pi$
$401$	8.64206	0.431564	0.215782	+	0.976442i	$0.430770\pi$
$402$	0	0
$403$	6.96320	0.346862
$404$	−5.15177	−0.256310
$405$	0	0
$406$	7.49028	0.371736
$407$	2.22106	0.110094
$408$	0	0
$409$	9.97211	0.493089	0.246544	−	0.969131i	$-0.420705\pi$
$409$	9.97211	0.493089	0.246544	+	0.969131i	$0.420705\pi$
$410$	−3.06986	−0.151610
$411$	0	0
$412$	70.8818	3.49209
$413$	2.02183	0.0994876
$414$	0	0
$415$	2.38541	0.117095
$416$	−4.99873	−0.245083
$417$	0	0
$418$	0.774806	0.0378970
$419$	3.70284	0.180895	0.0904477	−	0.995901i	$-0.471170\pi$
$419$	3.70284	0.180895	0.0904477	+	0.995901i	$0.471170\pi$
$420$	0	0
$421$	−29.7692	−1.45086	−0.725432	−	0.688294i	$-0.758360\pi$
$421$	−29.7692	−1.45086	−0.725432	+	0.688294i	$0.758360\pi$
$422$	12.8426	0.625166
$423$	0	0
$424$	31.4426	1.52699
$425$	−9.14962	−0.443822
$426$	0	0
$427$	−7.54265	−0.365015
$428$	−58.2101	−2.81369
$429$	0	0
$430$	11.8919	0.573478
$431$	27.9651	1.34703	0.673516	−	0.739172i	$-0.264783\pi$
$431$	27.9651	1.34703	0.673516	+	0.739172i	$0.264783\pi$
$432$	0	0
$433$	36.2370	1.74144	0.870720	−	0.491779i	$-0.163653\pi$
$433$	36.2370	1.74144	0.870720	+	0.491779i	$0.163653\pi$
$434$	3.02471	0.145190
$435$	0	0
$436$	−8.30006	−0.397501
$437$	8.42069	0.402816
$438$	0	0
$439$	−2.01152	−0.0960048	−0.0480024	−	0.998847i	$-0.515286\pi$
$439$	−2.01152	−0.0960048	−0.0480024	+	0.998847i	$0.515286\pi$
$440$	−0.769276	−0.0366738
$441$	0	0
$442$	14.7965	0.703796
$443$	−25.4185	−1.20767	−0.603834	−	0.797110i	$-0.706361\pi$
$443$	−25.4185	−1.20767	−0.603834	+	0.797110i	$0.706361\pi$
$444$	0	0
$445$	6.47189	0.306797
$446$	−17.3157	−0.819920
$447$	0	0
$448$	−5.43257	−0.256665
$449$	−15.7520	−0.743381	−0.371690	−	0.928357i	$-0.621222\pi$
$449$	−15.7520	−0.743381	−0.371690	+	0.928357i	$0.621222\pi$
$450$	0	0
$451$	0.501149	0.0235982
$452$	−2.54244	−0.119586
$453$	0	0
$454$	−64.3593	−3.02053
$455$	−1.11841	−0.0524317
$456$	0	0
$457$	4.73035	0.221277	0.110638	−	0.993861i	$-0.464710\pi$
$457$	4.73035	0.221277	0.110638	+	0.993861i	$0.464710\pi$
$458$	9.82634	0.459155
$459$	0	0
$460$	−17.5354	−0.817593
$461$	26.6833	1.24277	0.621383	−	0.783507i	$-0.286571\pi$
$461$	26.6833	1.24277	0.621383	+	0.783507i	$0.286571\pi$
$462$	0	0
$463$	26.2745	1.22108	0.610539	−	0.791986i	$-0.290953\pi$
$463$	26.2745	1.22108	0.610539	+	0.791986i	$0.290953\pi$
$464$	16.7532	0.777750
$465$	0	0
$466$	−35.0127	−1.62193
$467$	−18.8007	−0.869994	−0.434997	−	0.900432i	$-0.643251\pi$
$467$	−18.8007	−0.869994	−0.434997	+	0.900432i	$0.643251\pi$
$468$	0	0
$469$	−1.54301	−0.0712494
$470$	−2.69188	−0.124167
$471$	0	0
$472$	16.1760	0.744563
$473$	−1.94133	−0.0892625
$474$	0	0
$475$	−5.57294	−0.255704
$476$	4.21959	0.193405
$477$	0	0
$478$	2.41299	0.110367
$479$	−32.0039	−1.46230	−0.731148	−	0.682218i	$-0.761015\pi$
$479$	−32.0039	−1.46230	−0.731148	+	0.682218i	$0.761015\pi$
$480$	0	0
$481$	25.8348	1.17797
$482$	−24.3922	−1.11103
$483$	0	0
$484$	−41.7842	−1.89928
$485$	−5.08263	−0.230790
$486$	0	0
$487$	4.76961	0.216132	0.108066	−	0.994144i	$-0.465534\pi$
$487$	4.76961	0.216132	0.108066	+	0.994144i	$0.465534\pi$
$488$	−60.3466	−2.73176
$489$	0	0
$490$	10.7701	0.486543
$491$	9.99992	0.451290	0.225645	−	0.974210i	$-0.427551\pi$
$491$	9.99992	0.451290	0.225645	+	0.974210i	$0.427551\pi$
$492$	0	0
$493$	11.3416	0.510800
$494$	9.01237	0.405486
$495$	0	0
$496$	6.76525	0.303769
$497$	−2.63099	−0.118016
$498$	0	0
$499$	29.8371	1.33569	0.667847	−	0.744299i	$-0.267216\pi$
$499$	29.8371	1.33569	0.667847	+	0.744299i	$0.267216\pi$
$500$	24.3416	1.08859
$501$	0	0
$502$	54.4972	2.43233
$503$	−10.1386	−0.452057	−0.226029	−	0.974121i	$-0.572574\pi$
$503$	−10.1386	−0.452057	−0.226029	+	0.974121i	$0.572574\pi$
$504$	0	0
$505$	0.898121	0.0399659
$506$	4.36040	0.193843
$507$	0	0
$508$	−37.8615	−1.67983
$509$	−16.9917	−0.753145	−0.376572	−	0.926387i	$-0.622897\pi$
$509$	−16.9917	−0.753145	−0.376572	+	0.926387i	$0.622897\pi$
$510$	0	0
$511$	1.19196	0.0527293
$512$	−30.9485	−1.36774
$513$	0	0
$514$	15.4706	0.682378
$515$	−12.3570	−0.544515
$516$	0	0
$517$	0.439445	0.0193268
$518$	11.2222	0.493077
$519$	0	0
$520$	−8.94804	−0.392398
$521$	−4.77975	−0.209405	−0.104702	−	0.994504i	$-0.533389\pi$
$521$	−4.77975	−0.209405	−0.104702	+	0.994504i	$0.533389\pi$
$522$	0	0
$523$	13.7212	0.599985	0.299992	−	0.953942i	$-0.403016\pi$
$523$	13.7212	0.599985	0.299992	+	0.953942i	$0.403016\pi$
$524$	15.7503	0.688055
$525$	0	0
$526$	−8.84954	−0.385858
$527$	4.57993	0.199505
$528$	0	0
$529$	24.3893	1.06041
$530$	−11.4968	−0.499388
$531$	0	0
$532$	2.57011	0.111428
$533$	5.82925	0.252493
$534$	0	0
$535$	10.1479	0.438733
$536$	−12.3452	−0.533229
$537$	0	0
$538$	47.7249	2.05757
$539$	−1.75820	−0.0757309
$540$	0	0
$541$	−14.2257	−0.611612	−0.305806	−	0.952094i	$-0.598926\pi$
$541$	−14.2257	−0.611612	−0.305806	+	0.952094i	$0.598926\pi$
$542$	−2.75888	−0.118504
$543$	0	0
$544$	−3.28784	−0.140965
$545$	1.44697	0.0619815
$546$	0	0
$547$	−26.3222	−1.12546	−0.562728	−	0.826642i	$-0.690248\pi$
$547$	−26.3222	−1.12546	−0.562728	+	0.826642i	$0.690248\pi$
$548$	42.1227	1.79939
$549$	0	0
$550$	−2.88577	−0.123050
$551$	6.90804	0.294293
$552$	0	0
$553$	−3.19402	−0.135824
$554$	−24.5110	−1.04137
$555$	0	0
$556$	−7.62265	−0.323273
$557$	−34.8025	−1.47463	−0.737315	−	0.675549i	$-0.763907\pi$
$557$	−34.8025	−1.47463	−0.737315	+	0.675549i	$0.763907\pi$
$558$	0	0
$559$	−22.5811	−0.955079
$560$	−1.08661	−0.0459177
$561$	0	0
$562$	−30.8697	−1.30216
$563$	−40.1796	−1.69337	−0.846685	−	0.532095i	$-0.821405\pi$
$563$	−40.1796	−1.69337	−0.846685	+	0.532095i	$0.821405\pi$
$564$	0	0
$565$	0.443231	0.0186469
$566$	−12.8430	−0.539832
$567$	0	0
$568$	−21.0498	−0.883228
$569$	29.3863	1.23194	0.615968	−	0.787771i	$-0.288765\pi$
$569$	29.3863	1.23194	0.615968	+	0.787771i	$0.288765\pi$
$570$	0	0
$571$	−12.7435	−0.533300	−0.266650	−	0.963793i	$-0.585917\pi$
$571$	−12.7435	−0.533300	−0.266650	+	0.963793i	$0.585917\pi$
$572$	3.06376	0.128102
$573$	0	0
$574$	2.53214	0.105689
$575$	−31.3630	−1.30793
$576$	0	0
$577$	−32.1954	−1.34031	−0.670157	−	0.742219i	$-0.733773\pi$
$577$	−32.1954	−1.34031	−0.670157	+	0.742219i	$0.733773\pi$
$578$	−31.2886	−1.30144
$579$	0	0
$580$	−14.3854	−0.597323
$581$	−1.96758	−0.0816289
$582$	0	0
$583$	1.87683	0.0777302
$584$	9.53654	0.394625
$585$	0	0
$586$	17.2654	0.713226
$587$	10.6677	0.440303	0.220152	−	0.975466i	$-0.429345\pi$
$587$	10.6677	0.440303	0.220152	+	0.975466i	$0.429345\pi$
$588$	0	0
$589$	2.78959	0.114943
$590$	−5.91467	−0.243503
$591$	0	0
$592$	25.1004	1.03162
$593$	−23.8504	−0.979418	−0.489709	−	0.871886i	$-0.662897\pi$
$593$	−23.8504	−0.979418	−0.489709	+	0.871886i	$0.662897\pi$
$594$	0	0
$595$	−0.735613	−0.0301572
$596$	−56.2670	−2.30479
$597$	0	0
$598$	50.7191	2.07406
$599$	17.6222	0.720023	0.360012	−	0.932948i	$-0.382773\pi$
$599$	17.6222	0.720023	0.360012	+	0.932948i	$0.382773\pi$
$600$	0	0
$601$	12.5309	0.511148	0.255574	−	0.966790i	$-0.417736\pi$
$601$	12.5309	0.511148	0.255574	+	0.966790i	$0.417736\pi$
$602$	−9.80889	−0.399780
$603$	0	0
$604$	−5.71789	−0.232658
$605$	7.28435	0.296151
$606$	0	0
$607$	17.3888	0.705790	0.352895	−	0.935663i	$-0.385197\pi$
$607$	17.3888	0.705790	0.352895	+	0.935663i	$0.385197\pi$
$608$	−2.00259	−0.0812156
$609$	0	0
$610$	22.0653	0.893400
$611$	5.11152	0.206790
$612$	0	0
$613$	−29.0189	−1.17206	−0.586031	−	0.810288i	$-0.699310\pi$
$613$	−29.0189	−1.17206	−0.586031	+	0.810288i	$0.699310\pi$
$614$	−49.6754	−2.00474
$615$	0	0
$616$	0.634528	0.0255659
$617$	34.7121	1.39745	0.698727	−	0.715388i	$-0.253750\pi$
$617$	34.7121	1.39745	0.698727	+	0.715388i	$0.253750\pi$
$618$	0	0
$619$	43.2472	1.73825	0.869125	−	0.494592i	$-0.164683\pi$
$619$	43.2472	1.73825	0.869125	+	0.494592i	$0.164683\pi$
$620$	−5.80909	−0.233299
$621$	0	0
$622$	−48.6588	−1.95104
$623$	−5.33826	−0.213873
$624$	0	0
$625$	18.5361	0.741444
$626$	42.9154	1.71525
$627$	0	0
$628$	−5.14689	−0.205383
$629$	16.9924	0.677533
$630$	0	0
$631$	−29.9746	−1.19327	−0.596635	−	0.802513i	$-0.703496\pi$
$631$	−29.9746	−1.19327	−0.596635	+	0.802513i	$0.703496\pi$
$632$	−25.5544	−1.01650
$633$	0	0
$634$	78.1466	3.10360
$635$	6.60050	0.261933
$636$	0	0
$637$	−20.4509	−0.810296
$638$	3.57712	0.141619
$639$	0	0
$640$	13.7106	0.541957
$641$	9.69551	0.382949	0.191475	−	0.981498i	$-0.438673\pi$
$641$	9.69551	0.382949	0.191475	+	0.981498i	$0.438673\pi$
$642$	0	0
$643$	4.85463	0.191448	0.0957240	−	0.995408i	$-0.469483\pi$
$643$	4.85463	0.191448	0.0957240	+	0.995408i	$0.469483\pi$
$644$	14.4639	0.569956
$645$	0	0
$646$	5.92774	0.233224
$647$	16.8226	0.661366	0.330683	−	0.943742i	$-0.392721\pi$
$647$	16.8226	0.661366	0.330683	+	0.943742i	$0.392721\pi$
$648$	0	0
$649$	0.965559	0.0379015
$650$	−33.5667	−1.31659
$651$	0	0
$652$	−38.3051	−1.50014
$653$	6.22373	0.243553	0.121777	−	0.992558i	$-0.461141\pi$
$653$	6.22373	0.243553	0.121777	+	0.992558i	$0.461141\pi$
$654$	0	0
$655$	−2.74579	−0.107287
$656$	5.66353	0.221124
$657$	0	0
$658$	2.22037	0.0865589
$659$	4.01398	0.156363	0.0781813	−	0.996939i	$-0.475089\pi$
$659$	4.01398	0.156363	0.0781813	+	0.996939i	$0.475089\pi$
$660$	0	0
$661$	8.75196	0.340412	0.170206	−	0.985409i	$-0.445557\pi$
$661$	8.75196	0.340412	0.170206	+	0.985409i	$0.445557\pi$
$662$	−10.1615	−0.394936
$663$	0	0
$664$	−15.7420	−0.610909
$665$	−0.448054	−0.0173748
$666$	0	0
$667$	38.8766	1.50531
$668$	−12.0194	−0.465045
$669$	0	0
$670$	4.51393	0.174388
$671$	−3.60213	−0.139059
$672$	0	0
$673$	20.6931	0.797662	0.398831	−	0.917024i	$-0.369416\pi$
$673$	20.6931	0.797662	0.398831	+	0.917024i	$0.369416\pi$
$674$	17.4326	0.671479
$675$	0	0
$676$	−14.0556	−0.540600
$677$	−3.24148	−0.124580	−0.0622900	−	0.998058i	$-0.519840\pi$
$677$	−3.24148	−0.124580	−0.0622900	+	0.998058i	$0.519840\pi$
$678$	0	0
$679$	4.19234	0.160887
$680$	−5.88543	−0.225696
$681$	0	0
$682$	1.44450	0.0553128
$683$	28.0620	1.07376	0.536881	−	0.843658i	$-0.319602\pi$
$683$	28.0620	1.07376	0.536881	+	0.843658i	$0.319602\pi$
$684$	0	0
$685$	−7.34336	−0.280575
$686$	−18.1679	−0.693652
$687$	0	0
$688$	−21.9392	−0.836423
$689$	21.8308	0.831688
$690$	0	0
$691$	8.47107	0.322255	0.161127	−	0.986934i	$-0.448487\pi$
$691$	8.47107	0.322255	0.161127	+	0.986934i	$0.448487\pi$
$692$	30.1137	1.14475
$693$	0	0
$694$	29.3885	1.11557
$695$	1.32888	0.0504072
$696$	0	0
$697$	3.83409	0.145227
$698$	18.4209	0.697240
$699$	0	0
$700$	−9.57240	−0.361803
$701$	3.89669	0.147176	0.0735879	−	0.997289i	$-0.476555\pi$
$701$	3.89669	0.147176	0.0735879	+	0.997289i	$0.476555\pi$
$702$	0	0
$703$	10.3499	0.390354
$704$	−2.59442	−0.0977809
$705$	0	0
$706$	17.8473	0.671694
$707$	−0.740804	−0.0278608
$708$	0	0
$709$	−11.0463	−0.414853	−0.207426	−	0.978251i	$-0.566509\pi$
$709$	−11.0463	−0.414853	−0.207426	+	0.978251i	$0.566509\pi$
$710$	7.69671	0.288852
$711$	0	0
$712$	−42.7099	−1.60062
$713$	15.6990	0.587933
$714$	0	0
$715$	−0.534114	−0.0199747
$716$	−7.36721	−0.275326
$717$	0	0
$718$	−22.0603	−0.823284
$719$	19.2426	0.717628	0.358814	−	0.933409i	$-0.383181\pi$
$719$	19.2426	0.717628	0.358814	+	0.933409i	$0.383181\pi$
$720$	0	0
$721$	10.1925	0.379589
$722$	−42.2362	−1.57187
$723$	0	0
$724$	−34.6310	−1.28705
$725$	−25.7291	−0.955554
$726$	0	0
$727$	32.7014	1.21283	0.606414	−	0.795149i	$-0.292607\pi$
$727$	32.7014	1.21283	0.606414	+	0.795149i	$0.292607\pi$
$728$	7.38068	0.273546
$729$	0	0
$730$	−3.48697	−0.129059
$731$	−14.8524	−0.549334
$732$	0	0
$733$	27.7761	1.02593	0.512966	−	0.858409i	$-0.328547\pi$
$733$	27.7761	1.02593	0.512966	+	0.858409i	$0.328547\pi$
$734$	47.6209	1.75772
$735$	0	0
$736$	−11.2700	−0.415418
$737$	−0.736890	−0.0271437
$738$	0	0
$739$	−15.1638	−0.557811	−0.278905	−	0.960319i	$-0.589972\pi$
$739$	−15.1638	−0.557811	−0.278905	+	0.960319i	$0.589972\pi$
$740$	−21.5529	−0.792299
$741$	0	0
$742$	9.48297	0.348131
$743$	−21.1140	−0.774599	−0.387299	−	0.921954i	$-0.626592\pi$
$743$	−21.1140	−0.774599	−0.387299	+	0.921954i	$0.626592\pi$
$744$	0	0
$745$	9.80918	0.359381
$746$	−0.962350	−0.0352341
$747$	0	0
$748$	2.01514	0.0736809
$749$	−8.37039	−0.305848
$750$	0	0
$751$	3.25042	0.118609	0.0593047	−	0.998240i	$-0.481112\pi$
$751$	3.25042	0.118609	0.0593047	+	0.998240i	$0.481112\pi$
$752$	4.96621	0.181099
$753$	0	0
$754$	41.6082	1.51528
$755$	0.996816	0.0362778
$756$	0	0
$757$	44.0840	1.60226	0.801130	−	0.598491i	$-0.204233\pi$
$757$	44.0840	1.60226	0.801130	+	0.598491i	$0.204233\pi$
$758$	−65.1234	−2.36539
$759$	0	0
$760$	−3.58475	−0.130033
$761$	38.8537	1.40845	0.704223	−	0.709979i	$-0.251296\pi$
$761$	38.8537	1.40845	0.704223	+	0.709979i	$0.251296\pi$
$762$	0	0
$763$	−1.19352	−0.0432082
$764$	73.4116	2.65594
$765$	0	0
$766$	−1.24037	−0.0448164
$767$	11.2312	0.405534
$768$	0	0
$769$	4.35090	0.156897	0.0784487	−	0.996918i	$-0.475003\pi$
$769$	4.35090	0.156897	0.0784487	+	0.996918i	$0.475003\pi$
$770$	−0.232011	−0.00836110
$771$	0	0
$772$	−58.9806	−2.12276
$773$	−20.0826	−0.722321	−0.361161	−	0.932504i	$-0.617619\pi$
$773$	−20.0826	−0.722321	−0.361161	+	0.932504i	$0.617619\pi$
$774$	0	0
$775$	−10.3898	−0.373214
$776$	33.5417	1.20408
$777$	0	0
$778$	25.9535	0.930476
$779$	2.33531	0.0836710
$780$	0	0
$781$	−1.25647	−0.0449602
$782$	33.3597	1.19294
$783$	0	0
$784$	−19.8696	−0.709627
$785$	0.897272	0.0320250
$786$	0	0
$787$	45.6772	1.62822	0.814109	−	0.580712i	$-0.197226\pi$
$787$	45.6772	1.62822	0.814109	+	0.580712i	$0.197226\pi$
$788$	66.4374	2.36673
$789$	0	0
$790$	9.34382	0.332438
$791$	−0.365594	−0.0129990
$792$	0	0
$793$	−41.8991	−1.48788
$794$	39.2472	1.39283
$795$	0	0
$796$	−39.4314	−1.39761
$797$	42.3136	1.49883	0.749413	−	0.662103i	$-0.230336\pi$
$797$	42.3136	1.49883	0.749413	+	0.662103i	$0.230336\pi$
$798$	0	0
$799$	3.36202	0.118940
$800$	7.45865	0.263703
$801$	0	0
$802$	20.8532	0.736352
$803$	0.569242	0.0200881
$804$	0	0
$805$	−2.52152	−0.0888721
$806$	16.8021	0.591829
$807$	0	0
$808$	−5.92696	−0.208510
$809$	−2.33468	−0.0820831	−0.0410416	−	0.999157i	$-0.513068\pi$
$809$	−2.33468	−0.0820831	−0.0410416	+	0.999157i	$0.513068\pi$
$810$	0	0
$811$	−21.1927	−0.744175	−0.372088	−	0.928198i	$-0.621358\pi$
$811$	−21.1927	−0.744175	−0.372088	+	0.928198i	$0.621358\pi$
$812$	11.8656	0.416403
$813$	0	0
$814$	5.35938	0.187846
$815$	6.67783	0.233914
$816$	0	0
$817$	−9.04642	−0.316494
$818$	24.0626	0.841328
$819$	0	0
$820$	−4.86308	−0.169826
$821$	−50.6599	−1.76804	−0.884022	−	0.467446i	$-0.845174\pi$
$821$	−50.6599	−1.76804	−0.884022	+	0.467446i	$0.845174\pi$
$822$	0	0
$823$	−12.9252	−0.450544	−0.225272	−	0.974296i	$-0.572327\pi$
$823$	−12.9252	−0.450544	−0.225272	+	0.974296i	$0.572327\pi$
$824$	81.5475	2.84084
$825$	0	0
$826$	4.87864	0.169750
$827$	−5.83428	−0.202878	−0.101439	−	0.994842i	$-0.532345\pi$
$827$	−5.83428	−0.202878	−0.101439	+	0.994842i	$0.532345\pi$
$828$	0	0
$829$	25.8365	0.897340	0.448670	−	0.893697i	$-0.351898\pi$
$829$	25.8365	0.897340	0.448670	+	0.893697i	$0.351898\pi$
$830$	5.75597	0.199793
$831$	0	0
$832$	−30.1777	−1.04622
$833$	−13.4513	−0.466059
$834$	0	0
$835$	2.09538	0.0725135
$836$	1.22740	0.0424506
$837$	0	0
$838$	8.93490	0.308651
$839$	20.5187	0.708386	0.354193	−	0.935172i	$-0.384756\pi$
$839$	20.5187	0.708386	0.354193	+	0.935172i	$0.384756\pi$
$840$	0	0
$841$	2.89298	0.0997578
$842$	−71.8328	−2.47552
$843$	0	0
$844$	20.3444	0.700283
$845$	2.45035	0.0842947
$846$	0	0
$847$	−6.00841	−0.206451
$848$	21.2102	0.728362
$849$	0	0
$850$	−22.0779	−0.757267
$851$	58.2465	1.99666
$852$	0	0
$853$	12.0881	0.413888	0.206944	−	0.978353i	$-0.433648\pi$
$853$	12.0881	0.413888	0.206944	+	0.978353i	$0.433648\pi$
$854$	−18.2003	−0.622802
$855$	0	0
$856$	−66.9691	−2.28896
$857$	47.5584	1.62457	0.812283	−	0.583264i	$-0.198225\pi$
$857$	47.5584	1.62457	0.812283	+	0.583264i	$0.198225\pi$
$858$	0	0
$859$	21.0304	0.717547	0.358774	−	0.933425i	$-0.383195\pi$
$859$	21.0304	0.717547	0.358774	+	0.933425i	$0.383195\pi$
$860$	18.8384	0.642385
$861$	0	0
$862$	67.4795	2.29836
$863$	−55.2047	−1.87919	−0.939595	−	0.342288i	$-0.888798\pi$
$863$	−55.2047	−1.87919	−0.939595	+	0.342288i	$0.888798\pi$
$864$	0	0
$865$	−5.24981	−0.178499
$866$	87.4395	2.97131
$867$	0	0
$868$	4.79155	0.162636
$869$	−1.52536	−0.0517443
$870$	0	0
$871$	−8.57134	−0.290429
$872$	−9.54899	−0.323369
$873$	0	0
$874$	20.3190	0.687301
$875$	3.50022	0.118329
$876$	0	0
$877$	−4.74754	−0.160313	−0.0801565	−	0.996782i	$-0.525542\pi$
$877$	−4.74754	−0.160313	−0.0801565	+	0.996782i	$0.525542\pi$
$878$	−4.85378	−0.163807
$879$	0	0
$880$	−0.518931	−0.0174931
$881$	17.3193	0.583501	0.291750	−	0.956494i	$-0.405762\pi$
$881$	17.3193	0.583501	0.291750	+	0.956494i	$0.405762\pi$
$882$	0	0
$883$	−19.1558	−0.644644	−0.322322	−	0.946630i	$-0.604463\pi$
$883$	−19.1558	−0.644644	−0.322322	+	0.946630i	$0.604463\pi$
$884$	23.4397	0.788361
$885$	0	0
$886$	−61.3345	−2.06057
$887$	11.6930	0.392614	0.196307	−	0.980542i	$-0.437105\pi$
$887$	11.6930	0.392614	0.196307	+	0.980542i	$0.437105\pi$
$888$	0	0
$889$	−5.44434	−0.182597
$890$	15.6166	0.523469
$891$	0	0
$892$	−27.4304	−0.918438
$893$	2.04777	0.0685261
$894$	0	0
$895$	1.28435	0.0429309
$896$	−11.3090	−0.377807
$897$	0	0
$898$	−38.0093	−1.26839
$899$	12.8789	0.429536
$900$	0	0
$901$	14.3589	0.478363
$902$	1.20927	0.0402642
$903$	0	0
$904$	−2.92501	−0.0972844
$905$	6.03731	0.200687
$906$	0	0
$907$	1.98748	0.0659932	0.0329966	−	0.999455i	$-0.489495\pi$
$907$	1.98748	0.0659932	0.0329966	+	0.999455i	$0.489495\pi$
$908$	−101.954	−3.38347
$909$	0	0
$910$	−2.69870	−0.0894610
$911$	17.2834	0.572625	0.286312	−	0.958136i	$-0.407570\pi$
$911$	17.2834	0.572625	0.286312	+	0.958136i	$0.407570\pi$
$912$	0	0
$913$	−0.939651	−0.0310979
$914$	11.4143	0.377551
$915$	0	0
$916$	15.5663	0.514324
$917$	2.26483	0.0747914
$918$	0	0
$919$	33.1032	1.09197	0.545987	−	0.837794i	$-0.316155\pi$
$919$	33.1032	1.09197	0.545987	+	0.837794i	$0.316155\pi$
$920$	−20.1740	−0.665117
$921$	0	0
$922$	64.3865	2.12046
$923$	−14.6150	−0.481059
$924$	0	0
$925$	−38.5484	−1.26746
$926$	63.3999	2.08345
$927$	0	0
$928$	−9.24552	−0.303499
$929$	−35.7910	−1.17427	−0.587133	−	0.809490i	$-0.699743\pi$
$929$	−35.7910	−1.17427	−0.587133	+	0.809490i	$0.699743\pi$
$930$	0	0
$931$	−8.19303	−0.268516
$932$	−55.4651	−1.81682
$933$	0	0
$934$	−45.3659	−1.48442
$935$	−0.351305	−0.0114889
$936$	0	0
$937$	39.1301	1.27832	0.639162	−	0.769072i	$-0.279281\pi$
$937$	39.1301	1.27832	0.639162	+	0.769072i	$0.279281\pi$
$938$	−3.72326	−0.121569
$939$	0	0
$940$	−4.26432	−0.139087
$941$	−40.8856	−1.33283	−0.666417	−	0.745579i	$-0.732173\pi$
$941$	−40.8856	−1.33283	−0.666417	+	0.745579i	$0.732173\pi$
$942$	0	0
$943$	13.1425	0.427977
$944$	10.9119	0.355151
$945$	0	0
$946$	−4.68441	−0.152303
$947$	−3.52599	−0.114579	−0.0572896	−	0.998358i	$-0.518246\pi$
$947$	−3.52599	−0.114579	−0.0572896	+	0.998358i	$0.518246\pi$
$948$	0	0
$949$	6.62129	0.214936
$950$	−13.4474	−0.436292
$951$	0	0
$952$	4.85452	0.157336
$953$	−9.66364	−0.313036	−0.156518	−	0.987675i	$-0.550027\pi$
$953$	−9.66364	−0.313036	−0.156518	+	0.987675i	$0.550027\pi$
$954$	0	0
$955$	−12.7980	−0.414135
$956$	3.82251	0.123629
$957$	0	0
$958$	−77.2251	−2.49503
$959$	6.05708	0.195593
$960$	0	0
$961$	−25.7993	−0.832234
$962$	62.3391	2.00989
$963$	0	0
$964$	−38.6406	−1.24453
$965$	10.2823	0.330997
$966$	0	0
$967$	39.4687	1.26923	0.634615	−	0.772829i	$-0.281159\pi$
$967$	39.4687	1.26923	0.634615	+	0.772829i	$0.281159\pi$
$968$	−48.0715	−1.54508
$969$	0	0
$970$	−12.2643	−0.393783
$971$	14.0810	0.451882	0.225941	−	0.974141i	$-0.427454\pi$
$971$	14.0810	0.451882	0.225941	+	0.974141i	$0.427454\pi$
$972$	0	0
$973$	−1.09611	−0.0351396
$974$	11.5090	0.368772
$975$	0	0
$976$	−40.7080	−1.30303
$977$	24.7863	0.792983	0.396491	−	0.918038i	$-0.370228\pi$
$977$	24.7863	0.792983	0.396491	+	0.918038i	$0.370228\pi$
$978$	0	0
$979$	−2.54938	−0.0814785
$980$	17.0613	0.545004
$981$	0	0
$982$	24.1297	0.770009
$983$	−17.2441	−0.550000	−0.275000	−	0.961444i	$-0.588678\pi$
$983$	−17.2441	−0.550000	−0.275000	+	0.961444i	$0.588678\pi$
$984$	0	0
$985$	−11.5822	−0.369040
$986$	27.3671	0.871547
$987$	0	0
$988$	14.2768	0.454207
$989$	−50.9107	−1.61887
$990$	0	0
$991$	−56.9132	−1.80791	−0.903953	−	0.427631i	$-0.859348\pi$
$991$	−56.9132	−1.80791	−0.903953	+	0.427631i	$0.859348\pi$
$992$	−3.73350	−0.118539
$993$	0	0
$994$	−6.34853	−0.201363
$995$	6.87418	0.217926
$996$	0	0
$997$	−15.5192	−0.491498	−0.245749	−	0.969333i	$-0.579034\pi$
$997$	−15.5192	−0.491498	−0.245749	+	0.969333i	$0.579034\pi$
$998$	71.9966	2.27901
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.k.1.18	yes	20
3.2	odd	2		2151.2.a.j.1.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.3	✓	20	3.2	odd	2
2151.2.a.k.1.18	yes	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.41299 q^{2} +3.82251 q^{4} -0.666388 q^{5} +0.549662 q^{7} +4.39769 q^{8} +O(q^{10})\)	\(q+2.41299 q^{2} +3.82251 q^{4} -0.666388 q^{5} +0.549662 q^{7} +4.39769 q^{8} -1.60799 q^{10} +0.262501 q^{11} +3.05335 q^{13} +1.32633 q^{14} +2.96655 q^{16} +2.00829 q^{17} +1.22323 q^{19} -2.54727 q^{20} +0.633411 q^{22} +6.88399 q^{23} -4.55593 q^{25} +7.36769 q^{26} +2.10109 q^{28} +5.64739 q^{29} +2.28051 q^{31} -1.63713 q^{32} +4.84598 q^{34} -0.366288 q^{35} +8.46115 q^{37} +2.95163 q^{38} -2.93057 q^{40} +1.90913 q^{41} -7.39552 q^{43} +1.00341 q^{44} +16.6110 q^{46} +1.67407 q^{47} -6.69787 q^{49} -10.9934 q^{50} +11.6714 q^{52} +7.14980 q^{53} -0.174927 q^{55} +2.41724 q^{56} +13.6271 q^{58} +3.67831 q^{59} -13.7223 q^{61} +5.50285 q^{62} -9.88347 q^{64} -2.03472 q^{65} -2.80719 q^{67} +7.67670 q^{68} -0.883849 q^{70} -4.78655 q^{71} +2.16854 q^{73} +20.4166 q^{74} +4.67580 q^{76} +0.144287 q^{77} -5.81088 q^{79} -1.97687 q^{80} +4.60671 q^{82} -3.57961 q^{83} -1.33830 q^{85} -17.8453 q^{86} +1.15440 q^{88} -9.71189 q^{89} +1.67831 q^{91} +26.3141 q^{92} +4.03951 q^{94} -0.815145 q^{95} +7.62712 q^{97} -16.1619 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) 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

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