LMFDB - Embedded newform 2151.2.a.k.1.8

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			1.8
Root			$-0.510429$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.510429 q^{2} -1.73946 q^{4} +1.97336 q^{5} -1.30841 q^{7} +1.90873 q^{8} +O(q^{10})$	\(q-0.510429 q^{2} -1.73946 q^{4} +1.97336 q^{5} -1.30841 q^{7} +1.90873 q^{8} -1.00726 q^{10} -4.91929 q^{11} +1.24517 q^{13} +0.667852 q^{14} +2.50465 q^{16} -0.353562 q^{17} +3.33866 q^{19} -3.43259 q^{20} +2.51095 q^{22} -2.97413 q^{23} -1.10584 q^{25} -0.635571 q^{26} +2.27594 q^{28} +7.75908 q^{29} -3.52991 q^{31} -5.09591 q^{32} +0.180468 q^{34} -2.58198 q^{35} +7.67166 q^{37} -1.70415 q^{38} +3.76662 q^{40} +10.3589 q^{41} -11.1008 q^{43} +8.55693 q^{44} +1.51808 q^{46} +3.54382 q^{47} -5.28805 q^{49} +0.564451 q^{50} -2.16593 q^{52} +6.99423 q^{53} -9.70755 q^{55} -2.49741 q^{56} -3.96046 q^{58} -1.32123 q^{59} +13.0637 q^{61} +1.80177 q^{62} -2.40821 q^{64} +2.45717 q^{65} -12.2786 q^{67} +0.615007 q^{68} +1.31792 q^{70} +12.6828 q^{71} +11.4497 q^{73} -3.91584 q^{74} -5.80746 q^{76} +6.43647 q^{77} +4.16632 q^{79} +4.94259 q^{80} -5.28750 q^{82} +11.7987 q^{83} -0.697706 q^{85} +5.66617 q^{86} -9.38960 q^{88} -3.43216 q^{89} -1.62920 q^{91} +5.17339 q^{92} -1.80887 q^{94} +6.58838 q^{95} +0.321470 q^{97} +2.69918 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.510429	−0.360928	−0.180464	−	0.983582i	$-0.557760\pi$
$2$	−0.510429	−0.360928	−0.180464	+	0.983582i	$0.557760\pi$
$3$	0	0
$3$	0	0
$4$	−1.73946	−0.869731
$5$	1.97336	0.882515	0.441257	−	0.897381i	$-0.354533\pi$
$5$	1.97336	0.882515	0.441257	+	0.897381i	$0.354533\pi$
$6$	0	0
$7$	−1.30841	−0.494534	−0.247267	−	0.968947i	$-0.579532\pi$
$7$	−1.30841	−0.494534	−0.247267	+	0.968947i	$0.579532\pi$
$8$	1.90873	0.674838
$9$	0	0
$10$	−1.00726	−0.318524
$11$	−4.91929	−1.48322	−0.741611	−	0.670830i	$-0.765938\pi$
$11$	−4.91929	−1.48322	−0.741611	+	0.670830i	$0.765938\pi$
$12$	0	0
$13$	1.24517	0.345348	0.172674	−	0.984979i	$-0.444759\pi$
$13$	1.24517	0.345348	0.172674	+	0.984979i	$0.444759\pi$
$14$	0.667852	0.178491
$15$	0	0
$16$	2.50465	0.626163
$17$	−0.353562	−0.0857513	−0.0428757	−	0.999080i	$-0.513652\pi$
$17$	−0.353562	−0.0857513	−0.0428757	+	0.999080i	$0.513652\pi$
$18$	0	0
$19$	3.33866	0.765940	0.382970	−	0.923761i	$-0.374901\pi$
$19$	3.33866	0.765940	0.382970	+	0.923761i	$0.374901\pi$
$20$	−3.43259	−0.767551
$21$	0	0
$22$	2.51095	0.535336
$23$	−2.97413	−0.620149	−0.310075	−	0.950712i	$-0.600354\pi$
$23$	−2.97413	−0.620149	−0.310075	+	0.950712i	$0.600354\pi$
$24$	0	0
$25$	−1.10584	−0.221167
$26$	−0.635571	−0.124646
$27$	0	0
$28$	2.27594	0.430112
$29$	7.75908	1.44082	0.720412	−	0.693546i	$-0.243953\pi$
$29$	7.75908	1.44082	0.720412	+	0.693546i	$0.243953\pi$
$30$	0	0
$31$	−3.52991	−0.633990	−0.316995	−	0.948427i	$-0.602674\pi$
$31$	−3.52991	−0.633990	−0.316995	+	0.948427i	$0.602674\pi$
$32$	−5.09591	−0.900838
$33$	0	0
$34$	0.180468	0.0309500
$35$	−2.58198	−0.436434
$36$	0	0
$37$	7.67166	1.26121	0.630607	−	0.776102i	$-0.282806\pi$
$37$	7.67166	1.26121	0.630607	+	0.776102i	$0.282806\pi$
$38$	−1.70415	−0.276449
$39$	0	0
$40$	3.76662	0.595555
$41$	10.3589	1.61779	0.808897	−	0.587950i	$-0.200065\pi$
$41$	10.3589	1.61779	0.808897	+	0.587950i	$0.200065\pi$
$42$	0	0
$43$	−11.1008	−1.69286	−0.846428	−	0.532503i	$-0.821252\pi$
$43$	−11.1008	−1.69286	−0.846428	+	0.532503i	$0.821252\pi$
$44$	8.55693	1.29001
$45$	0	0
$46$	1.51808	0.223829
$47$	3.54382	0.516919	0.258460	−	0.966022i	$-0.416785\pi$
$47$	3.54382	0.516919	0.258460	+	0.966022i	$0.416785\pi$
$48$	0	0
$49$	−5.28805	−0.755436
$50$	0.564451	0.0798255
$51$	0	0
$52$	−2.16593	−0.300360
$53$	6.99423	0.960731	0.480366	−	0.877068i	$-0.340504\pi$
$53$	6.99423	0.960731	0.480366	+	0.877068i	$0.340504\pi$
$54$	0	0
$55$	−9.70755	−1.30897
$56$	−2.49741	−0.333730
$57$	0	0
$58$	−3.96046	−0.520034
$59$	−1.32123	−0.172009	−0.0860047	−	0.996295i	$-0.527410\pi$
$59$	−1.32123	−0.172009	−0.0860047	+	0.996295i	$0.527410\pi$
$60$	0	0
$61$	13.0637	1.67263	0.836317	−	0.548246i	$-0.184704\pi$
$61$	13.0637	1.67263	0.836317	+	0.548246i	$0.184704\pi$
$62$	1.80177	0.228825
$63$	0	0
$64$	−2.40821	−0.301026
$65$	2.45717	0.304775
$66$	0	0
$67$	−12.2786	−1.50007	−0.750034	−	0.661400i	$-0.769963\pi$
$67$	−12.2786	−1.50007	−0.750034	+	0.661400i	$0.769963\pi$
$68$	0.615007	0.0745806
$69$	0	0
$70$	1.31792	0.157521
$71$	12.6828	1.50518	0.752588	−	0.658492i	$-0.228805\pi$
$71$	12.6828	1.50518	0.752588	+	0.658492i	$0.228805\pi$
$72$	0	0
$73$	11.4497	1.34008	0.670040	−	0.742325i	$-0.266277\pi$
$73$	11.4497	1.34008	0.670040	+	0.742325i	$0.266277\pi$
$74$	−3.91584	−0.455207
$75$	0	0
$76$	−5.80746	−0.666162
$77$	6.43647	0.733504
$78$	0	0
$79$	4.16632	0.468748	0.234374	−	0.972147i	$-0.424696\pi$
$79$	4.16632	0.468748	0.234374	+	0.972147i	$0.424696\pi$
$80$	4.94259	0.552598
$81$	0	0
$82$	−5.28750	−0.583907
$83$	11.7987	1.29508	0.647541	−	0.762031i	$-0.275798\pi$
$83$	11.7987	1.29508	0.647541	+	0.762031i	$0.275798\pi$
$84$	0	0
$85$	−0.697706	−0.0756768
$86$	5.66617	0.610999
$87$	0	0
$88$	−9.38960	−1.00094
$89$	−3.43216	−0.363809	−0.181904	−	0.983316i	$-0.558226\pi$
$89$	−3.43216	−0.363809	−0.181904	+	0.983316i	$0.558226\pi$
$90$	0	0
$91$	−1.62920	−0.170786
$92$	5.17339	0.539363
$93$	0	0
$94$	−1.80887	−0.186570
$95$	6.58838	0.675954
$96$	0	0
$97$	0.321470	0.0326403	0.0163202	−	0.999867i	$-0.494805\pi$
$97$	0.321470	0.0326403	0.0163202	+	0.999867i	$0.494805\pi$
$98$	2.69918	0.272658
$99$	0	0
$100$	1.92356	0.192356
$101$	−5.75176	−0.572321	−0.286161	−	0.958182i	$-0.592379\pi$
$101$	−5.75176	−0.572321	−0.286161	+	0.958182i	$0.592379\pi$
$102$	0	0
$103$	−13.1895	−1.29960	−0.649799	−	0.760106i	$-0.725147\pi$
$103$	−13.1895	−1.29960	−0.649799	+	0.760106i	$0.725147\pi$
$104$	2.37669	0.233054
$105$	0	0
$106$	−3.57006	−0.346755
$107$	4.87136	0.470932	0.235466	−	0.971883i	$-0.424338\pi$
$107$	4.87136	0.470932	0.235466	+	0.971883i	$0.424338\pi$
$108$	0	0
$109$	−4.66428	−0.446757	−0.223379	−	0.974732i	$-0.571709\pi$
$109$	−4.66428	−0.446757	−0.223379	+	0.974732i	$0.571709\pi$
$110$	4.95502	0.472442
$111$	0	0
$112$	−3.27712	−0.309659
$113$	−2.06815	−0.194555	−0.0972775	−	0.995257i	$-0.531013\pi$
$113$	−2.06815	−0.194555	−0.0972775	+	0.995257i	$0.531013\pi$
$114$	0	0
$115$	−5.86904	−0.547291
$116$	−13.4966	−1.25313
$117$	0	0
$118$	0.674394	0.0620830
$119$	0.462605	0.0424069
$120$	0	0
$121$	13.1995	1.19995
$122$	−6.66809	−0.603700
$123$	0	0
$124$	6.14014	0.551401
$125$	−12.0490	−1.07770
$126$	0	0
$127$	19.8504	1.76144	0.880721	−	0.473635i	$-0.157058\pi$
$127$	19.8504	1.76144	0.880721	+	0.473635i	$0.157058\pi$
$128$	11.4210	1.00949
$129$	0	0
$130$	−1.25421	−0.110002
$131$	16.1821	1.41384	0.706918	−	0.707295i	$-0.250085\pi$
$131$	16.1821	1.41384	0.706918	+	0.707295i	$0.250085\pi$
$132$	0	0
$133$	−4.36834	−0.378783
$134$	6.26734	0.541416
$135$	0	0
$136$	−0.674854	−0.0578682
$137$	−5.15387	−0.440324	−0.220162	−	0.975463i	$-0.570659\pi$
$137$	−5.15387	−0.440324	−0.220162	+	0.975463i	$0.570659\pi$
$138$	0	0
$139$	−9.66367	−0.819662	−0.409831	−	0.912162i	$-0.634412\pi$
$139$	−9.66367	−0.819662	−0.409831	+	0.912162i	$0.634412\pi$
$140$	4.49125	0.379580
$141$	0	0
$142$	−6.47369	−0.543260
$143$	−6.12535	−0.512228
$144$	0	0
$145$	15.3115	1.27155
$146$	−5.84424	−0.483672
$147$	0	0
$148$	−13.3446	−1.09692
$149$	18.4553	1.51192	0.755958	−	0.654620i	$-0.227171\pi$
$149$	18.4553	1.51192	0.755958	+	0.654620i	$0.227171\pi$
$150$	0	0
$151$	21.9774	1.78849	0.894246	−	0.447576i	$-0.147713\pi$
$151$	21.9774	1.78849	0.894246	+	0.447576i	$0.147713\pi$
$152$	6.37259	0.516885
$153$	0	0
$154$	−3.28536	−0.264742
$155$	−6.96579	−0.559506
$156$	0	0
$157$	17.6338	1.40733	0.703664	−	0.710532i	$-0.251546\pi$
$157$	17.6338	1.40733	0.703664	+	0.710532i	$0.251546\pi$
$158$	−2.12661	−0.169184
$159$	0	0
$160$	−10.0561	−0.795003
$161$	3.89139	0.306685
$162$	0	0
$163$	14.0892	1.10355	0.551775	−	0.833993i	$-0.313951\pi$
$163$	14.0892	1.10355	0.551775	+	0.833993i	$0.313951\pi$
$164$	−18.0190	−1.40705
$165$	0	0
$166$	−6.02242	−0.467431
$167$	15.5247	1.20133	0.600667	−	0.799499i	$-0.294902\pi$
$167$	15.5247	1.20133	0.600667	+	0.799499i	$0.294902\pi$
$168$	0	0
$169$	−11.4496	−0.880735
$170$	0.356129	0.0273139
$171$	0	0
$172$	19.3094	1.47233
$173$	24.7262	1.87990	0.939951	−	0.341310i	$-0.110871\pi$
$173$	24.7262	1.87990	0.939951	+	0.341310i	$0.110871\pi$
$174$	0	0
$175$	1.44689	0.109375
$176$	−12.3211	−0.928740
$177$	0	0
$178$	1.75188	0.131309
$179$	20.0246	1.49671	0.748354	−	0.663299i	$-0.230844\pi$
$179$	20.0246	1.49671	0.748354	+	0.663299i	$0.230844\pi$
$180$	0	0
$181$	−9.47348	−0.704158	−0.352079	−	0.935970i	$-0.614525\pi$
$181$	−9.47348	−0.704158	−0.352079	+	0.935970i	$0.614525\pi$
$182$	0.831589	0.0616415
$183$	0	0
$184$	−5.67681	−0.418500
$185$	15.1390	1.11304
$186$	0	0
$187$	1.73927	0.127188
$188$	−6.16434	−0.449581
$189$	0	0
$190$	−3.36290	−0.243970
$191$	−11.6015	−0.839453	−0.419726	−	0.907651i	$-0.637874\pi$
$191$	−11.6015	−0.839453	−0.419726	+	0.907651i	$0.637874\pi$
$192$	0	0
$193$	−19.7244	−1.41979	−0.709896	−	0.704307i	$-0.751258\pi$
$193$	−19.7244	−1.41979	−0.709896	+	0.704307i	$0.751258\pi$
$194$	−0.164087	−0.0117808
$195$	0	0
$196$	9.19837	0.657026
$197$	3.37371	0.240367	0.120184	−	0.992752i	$-0.461652\pi$
$197$	3.37371	0.240367	0.120184	+	0.992752i	$0.461652\pi$
$198$	0	0
$199$	15.5040	1.09905	0.549524	−	0.835478i	$-0.314809\pi$
$199$	15.5040	1.09905	0.549524	+	0.835478i	$0.314809\pi$
$200$	−2.11074	−0.149252
$201$	0	0
$202$	2.93586	0.206567
$203$	−10.1521	−0.712537
$204$	0	0
$205$	20.4419	1.42773
$206$	6.73229	0.469061
$207$	0	0
$208$	3.11872	0.216244
$209$	−16.4238	−1.13606
$210$	0	0
$211$	8.23128	0.566665	0.283332	−	0.959022i	$-0.408560\pi$
$211$	8.23128	0.566665	0.283332	+	0.959022i	$0.408560\pi$
$212$	−12.1662	−0.835578
$213$	0	0
$214$	−2.48648	−0.169973
$215$	−21.9059	−1.49397
$216$	0	0
$217$	4.61858	0.313530
$218$	2.38078	0.161247
$219$	0	0
$220$	16.8859	1.13845
$221$	−0.440244	−0.0296140
$222$	0	0
$223$	8.24532	0.552148	0.276074	−	0.961136i	$-0.410967\pi$
$223$	8.24532	0.552148	0.276074	+	0.961136i	$0.410967\pi$
$224$	6.66756	0.445495
$225$	0	0
$226$	1.05564	0.0702203
$227$	−4.91650	−0.326319	−0.163160	−	0.986600i	$-0.552169\pi$
$227$	−4.91650	−0.326319	−0.163160	+	0.986600i	$0.552169\pi$
$228$	0	0
$229$	0.733030	0.0484400	0.0242200	−	0.999707i	$-0.492290\pi$
$229$	0.733030	0.0484400	0.0242200	+	0.999707i	$0.492290\pi$
$230$	2.99573	0.197532
$231$	0	0
$232$	14.8100	0.972323
$233$	16.6162	1.08856	0.544281	−	0.838903i	$-0.316803\pi$
$233$	16.6162	1.08856	0.544281	+	0.838903i	$0.316803\pi$
$234$	0	0
$235$	6.99324	0.456189
$236$	2.29823	0.149602
$237$	0	0
$238$	−0.236127	−0.0153058
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−9.85009	−0.634500	−0.317250	−	0.948342i	$-0.602759\pi$
$241$	−9.85009	−0.634500	−0.317250	+	0.948342i	$0.602759\pi$
$242$	−6.73738	−0.433095
$243$	0	0
$244$	−22.7238	−1.45474
$245$	−10.4352	−0.666684
$246$	0	0
$247$	4.15719	0.264516
$248$	−6.73764	−0.427841
$249$	0	0
$250$	6.15018	0.388971
$251$	13.2011	0.833249	0.416624	−	0.909079i	$-0.363213\pi$
$251$	13.2011	0.833249	0.416624	+	0.909079i	$0.363213\pi$
$252$	0	0
$253$	14.6306	0.919819
$254$	−10.1322	−0.635753
$255$	0	0
$256$	−1.01321	−0.0633257
$257$	−17.8240	−1.11183	−0.555914	−	0.831240i	$-0.687632\pi$
$257$	−17.8240	−1.11183	−0.555914	+	0.831240i	$0.687632\pi$
$258$	0	0
$259$	−10.0377	−0.623713
$260$	−4.27416	−0.265072
$261$	0	0
$262$	−8.25981	−0.510293
$263$	−7.70672	−0.475217	−0.237608	−	0.971361i	$-0.576363\pi$
$263$	−7.70672	−0.475217	−0.237608	+	0.971361i	$0.576363\pi$
$264$	0	0
$265$	13.8022	0.847860
$266$	2.22973	0.136713
$267$	0	0
$268$	21.3581	1.30466
$269$	16.4022	1.00006	0.500029	−	0.866009i	$-0.333323\pi$
$269$	16.4022	1.00006	0.500029	+	0.866009i	$0.333323\pi$
$270$	0	0
$271$	−29.1174	−1.76876	−0.884378	−	0.466772i	$-0.845417\pi$
$271$	−29.1174	−1.76876	−0.884378	+	0.466772i	$0.845417\pi$
$272$	−0.885549	−0.0536943
$273$	0	0
$274$	2.63068	0.158925
$275$	5.43994	0.328041
$276$	0	0
$277$	−14.5818	−0.876134	−0.438067	−	0.898942i	$-0.644337\pi$
$277$	−14.5818	−0.876134	−0.438067	+	0.898942i	$0.644337\pi$
$278$	4.93262	0.295839
$279$	0	0
$280$	−4.92829	−0.294522
$281$	−12.4685	−0.743807	−0.371903	−	0.928271i	$-0.621295\pi$
$281$	−12.4685	−0.743807	−0.371903	+	0.928271i	$0.621295\pi$
$282$	0	0
$283$	−12.8582	−0.764343	−0.382172	−	0.924091i	$-0.624824\pi$
$283$	−12.8582	−0.764343	−0.382172	+	0.924091i	$0.624824\pi$
$284$	−22.0613	−1.30910
$285$	0	0
$286$	3.12656	0.184877
$287$	−13.5538	−0.800054
$288$	0	0
$289$	−16.8750	−0.992647
$290$	−7.81542	−0.458937
$291$	0	0
$292$	−19.9162	−1.16551
$293$	16.1140	0.941389	0.470694	−	0.882296i	$-0.344003\pi$
$293$	16.1140	0.941389	0.470694	+	0.882296i	$0.344003\pi$
$294$	0	0
$295$	−2.60726	−0.151801
$296$	14.6431	0.851115
$297$	0	0
$298$	−9.42012	−0.545693
$299$	−3.70330	−0.214167
$300$	0	0
$301$	14.5244	0.837175
$302$	−11.2179	−0.645516
$303$	0	0
$304$	8.36217	0.479604
$305$	25.7794	1.47612
$306$	0	0
$307$	−16.7468	−0.955791	−0.477896	−	0.878417i	$-0.658600\pi$
$307$	−16.7468	−0.955791	−0.477896	+	0.878417i	$0.658600\pi$
$308$	−11.1960	−0.637951
$309$	0	0
$310$	3.55554	0.201941
$311$	13.2617	0.752004	0.376002	−	0.926619i	$-0.377299\pi$
$311$	13.2617	0.752004	0.376002	+	0.926619i	$0.377299\pi$
$312$	0	0
$313$	−5.47215	−0.309304	−0.154652	−	0.987969i	$-0.549426\pi$
$313$	−5.47215	−0.309304	−0.154652	+	0.987969i	$0.549426\pi$
$314$	−9.00079	−0.507944
$315$	0	0
$316$	−7.24716	−0.407684
$317$	−2.83295	−0.159114	−0.0795572	−	0.996830i	$-0.525351\pi$
$317$	−2.83295	−0.159114	−0.0795572	+	0.996830i	$0.525351\pi$
$318$	0	0
$319$	−38.1692	−2.13706
$320$	−4.75227	−0.265660
$321$	0	0
$322$	−1.98628	−0.110691
$323$	−1.18042	−0.0656804
$324$	0	0
$325$	−1.37695	−0.0763797
$326$	−7.19153	−0.398302
$327$	0	0
$328$	19.7724	1.09175
$329$	−4.63678	−0.255634
$330$	0	0
$331$	−18.6580	−1.02553	−0.512767	−	0.858528i	$-0.671380\pi$
$331$	−18.6580	−1.02553	−0.512767	+	0.858528i	$0.671380\pi$
$332$	−20.5235	−1.12637
$333$	0	0
$334$	−7.92424	−0.433595
$335$	−24.2301	−1.32383
$336$	0	0
$337$	12.6388	0.688478	0.344239	−	0.938882i	$-0.388137\pi$
$337$	12.6388	0.688478	0.344239	+	0.938882i	$0.388137\pi$
$338$	5.84418	0.317882
$339$	0	0
$340$	1.21363	0.0658185
$341$	17.3647	0.940349
$342$	0	0
$343$	16.0779	0.868123
$344$	−21.1884	−1.14240
$345$	0	0
$346$	−12.6210	−0.678509
$347$	−27.2616	−1.46348	−0.731739	−	0.681585i	$-0.761291\pi$
$347$	−27.2616	−1.46348	−0.731739	+	0.681585i	$0.761291\pi$
$348$	0	0
$349$	−0.710696	−0.0380427	−0.0190213	−	0.999819i	$-0.506055\pi$
$349$	−0.710696	−0.0380427	−0.0190213	+	0.999819i	$0.506055\pi$
$350$	−0.738536	−0.0394764
$351$	0	0
$352$	25.0683	1.33614
$353$	−10.8851	−0.579353	−0.289677	−	0.957125i	$-0.593548\pi$
$353$	−10.8851	−0.579353	−0.289677	+	0.957125i	$0.593548\pi$
$354$	0	0
$355$	25.0278	1.32834
$356$	5.97012	0.316416
$357$	0	0
$358$	−10.2211	−0.540204
$359$	−34.6231	−1.82734	−0.913669	−	0.406460i	$-0.866763\pi$
$359$	−34.6231	−1.82734	−0.913669	+	0.406460i	$0.866763\pi$
$360$	0	0
$361$	−7.85338	−0.413336
$362$	4.83554	0.254150
$363$	0	0
$364$	2.83393	0.148538
$365$	22.5943	1.18264
$366$	0	0
$367$	21.6577	1.13052	0.565261	−	0.824912i	$-0.308775\pi$
$367$	21.6577	1.13052	0.565261	+	0.824912i	$0.308775\pi$
$368$	−7.44917	−0.388315
$369$	0	0
$370$	−7.72737	−0.401727
$371$	−9.15135	−0.475114
$372$	0	0
$373$	27.2659	1.41177	0.705887	−	0.708324i	$-0.250548\pi$
$373$	27.2659	1.41177	0.705887	+	0.708324i	$0.250548\pi$
$374$	−0.887776	−0.0459058
$375$	0	0
$376$	6.76419	0.348837
$377$	9.66137	0.497586
$378$	0	0
$379$	3.94609	0.202697	0.101348	−	0.994851i	$-0.467684\pi$
$379$	3.94609	0.202697	0.101348	+	0.994851i	$0.467684\pi$
$380$	−11.4602	−0.587898
$381$	0	0
$382$	5.92172	0.302982
$383$	19.0059	0.971157	0.485578	−	0.874193i	$-0.338609\pi$
$383$	19.0059	0.971157	0.485578	+	0.874193i	$0.338609\pi$
$384$	0	0
$385$	12.7015	0.647328
$386$	10.0679	0.512442
$387$	0	0
$388$	−0.559185	−0.0283883
$389$	19.7747	1.00262	0.501309	−	0.865268i	$-0.332852\pi$
$389$	19.7747	1.00262	0.501309	+	0.865268i	$0.332852\pi$
$390$	0	0
$391$	1.05154	0.0531786
$392$	−10.0935	−0.509797
$393$	0	0
$394$	−1.72204	−0.0867552
$395$	8.22166	0.413677
$396$	0	0
$397$	−33.9206	−1.70243	−0.851213	−	0.524821i	$-0.824132\pi$
$397$	−33.9206	−1.70243	−0.851213	+	0.524821i	$0.824132\pi$
$398$	−7.91368	−0.396677
$399$	0	0
$400$	−2.76974	−0.138487
$401$	4.19358	0.209417	0.104709	−	0.994503i	$-0.466609\pi$
$401$	4.19358	0.209417	0.104709	+	0.994503i	$0.466609\pi$
$402$	0	0
$403$	−4.39533	−0.218947
$404$	10.0050	0.497766
$405$	0	0
$406$	5.18192	0.257174
$407$	−37.7392	−1.87066
$408$	0	0
$409$	−16.3446	−0.808187	−0.404094	−	0.914718i	$-0.632413\pi$
$409$	−16.3446	−0.808187	−0.404094	+	0.914718i	$0.632413\pi$
$410$	−10.4342	−0.515306
$411$	0	0
$412$	22.9426	1.13030
$413$	1.72871	0.0850645
$414$	0	0
$415$	23.2832	1.14293
$416$	−6.34527	−0.311102
$417$	0	0
$418$	8.38320	0.410036
$419$	21.6479	1.05757	0.528786	−	0.848755i	$-0.322648\pi$
$419$	21.6479	1.05757	0.528786	+	0.848755i	$0.322648\pi$
$420$	0	0
$421$	−0.857053	−0.0417702	−0.0208851	−	0.999782i	$-0.506648\pi$
$421$	−0.857053	−0.0417702	−0.0208851	+	0.999782i	$0.506648\pi$
$422$	−4.20148	−0.204525
$423$	0	0
$424$	13.3501	0.648338
$425$	0.390982	0.0189654
$426$	0	0
$427$	−17.0927	−0.827174
$428$	−8.47355	−0.409585
$429$	0	0
$430$	11.1814	0.539216
$431$	−3.54552	−0.170782	−0.0853908	−	0.996348i	$-0.527214\pi$
$431$	−3.54552	−0.170782	−0.0853908	+	0.996348i	$0.527214\pi$
$432$	0	0
$433$	−6.61178	−0.317742	−0.158871	−	0.987299i	$-0.550785\pi$
$433$	−6.61178	−0.317742	−0.158871	+	0.987299i	$0.550785\pi$
$434$	−2.35746	−0.113162
$435$	0	0
$436$	8.11334	0.388559
$437$	−9.92960	−0.474997
$438$	0	0
$439$	−31.0403	−1.48147	−0.740736	−	0.671796i	$-0.765523\pi$
$439$	−31.0403	−1.48147	−0.740736	+	0.671796i	$0.765523\pi$
$440$	−18.5291	−0.883340
$441$	0	0
$442$	0.224713	0.0106885
$443$	10.6671	0.506811	0.253406	−	0.967360i	$-0.418449\pi$
$443$	10.6671	0.506811	0.253406	+	0.967360i	$0.418449\pi$
$444$	0	0
$445$	−6.77291	−0.321066
$446$	−4.20865	−0.199285
$447$	0	0
$448$	3.15093	0.148868
$449$	−8.35664	−0.394374	−0.197187	−	0.980366i	$-0.563181\pi$
$449$	−8.35664	−0.394374	−0.197187	+	0.980366i	$0.563181\pi$
$450$	0	0
$451$	−50.9586	−2.39955
$452$	3.59746	0.169210
$453$	0	0
$454$	2.50952	0.117778
$455$	−3.21500	−0.150721
$456$	0	0
$457$	1.46849	0.0686930	0.0343465	−	0.999410i	$-0.489065\pi$
$457$	1.46849	0.0686930	0.0343465	+	0.999410i	$0.489065\pi$
$458$	−0.374160	−0.0174833
$459$	0	0
$460$	10.2090	0.475996
$461$	−42.2951	−1.96988	−0.984940	−	0.172896i	$-0.944687\pi$
$461$	−42.2951	−1.96988	−0.984940	+	0.172896i	$0.944687\pi$
$462$	0	0
$463$	26.1426	1.21495	0.607475	−	0.794339i	$-0.292182\pi$
$463$	26.1426	1.21495	0.607475	+	0.794339i	$0.292182\pi$
$464$	19.4338	0.902192
$465$	0	0
$466$	−8.48138	−0.392892
$467$	−19.1402	−0.885705	−0.442852	−	0.896595i	$-0.646033\pi$
$467$	−19.1402	−0.885705	−0.442852	+	0.896595i	$0.646033\pi$
$468$	0	0
$469$	16.0655	0.741834
$470$	−3.56955	−0.164651
$471$	0	0
$472$	−2.52187	−0.116078
$473$	54.6081	2.51088
$474$	0	0
$475$	−3.69201	−0.169401
$476$	−0.804684	−0.0368826
$477$	0	0
$478$	−0.510429	−0.0233465
$479$	26.2033	1.19726	0.598629	−	0.801026i	$-0.295712\pi$
$479$	26.2033	1.19726	0.598629	+	0.801026i	$0.295712\pi$
$480$	0	0
$481$	9.55252	0.435557
$482$	5.02777	0.229009
$483$	0	0
$484$	−22.9599	−1.04363
$485$	0.634377	0.0288056
$486$	0	0
$487$	−40.7502	−1.84657	−0.923284	−	0.384119i	$-0.874505\pi$
$487$	−40.7502	−1.84657	−0.923284	+	0.384119i	$0.874505\pi$
$488$	24.9351	1.12876
$489$	0	0
$490$	5.32645	0.240625
$491$	−6.89804	−0.311304	−0.155652	−	0.987812i	$-0.549748\pi$
$491$	−6.89804	−0.311304	−0.155652	+	0.987812i	$0.549748\pi$
$492$	0	0
$493$	−2.74331	−0.123553
$494$	−2.12195	−0.0954711
$495$	0	0
$496$	−8.84120	−0.396982
$497$	−16.5944	−0.744360
$498$	0	0
$499$	21.4957	0.962281	0.481140	−	0.876644i	$-0.340223\pi$
$499$	21.4957	0.962281	0.481140	+	0.876644i	$0.340223\pi$
$500$	20.9588	0.937308
$501$	0	0
$502$	−6.73825	−0.300743
$503$	−24.9171	−1.11100	−0.555499	−	0.831517i	$-0.687473\pi$
$503$	−24.9171	−1.11100	−0.555499	+	0.831517i	$0.687473\pi$
$504$	0	0
$505$	−11.3503	−0.505082
$506$	−7.46789	−0.331988
$507$	0	0
$508$	−34.5291	−1.53198
$509$	−36.1616	−1.60284	−0.801418	−	0.598105i	$-0.795921\pi$
$509$	−36.1616	−1.60284	−0.801418	+	0.598105i	$0.795921\pi$
$510$	0	0
$511$	−14.9809	−0.662715
$512$	−22.3249	−0.986630
$513$	0	0
$514$	9.09787	0.401290
$515$	−26.0276	−1.14691
$516$	0	0
$517$	−17.4331	−0.766706
$518$	5.12354	0.225115
$519$	0	0
$520$	4.69008	0.205673
$521$	27.6929	1.21325	0.606625	−	0.794988i	$-0.292523\pi$
$521$	27.6929	1.21325	0.606625	+	0.794988i	$0.292523\pi$
$522$	0	0
$523$	24.9136	1.08939	0.544697	−	0.838633i	$-0.316645\pi$
$523$	24.9136	1.08939	0.544697	+	0.838633i	$0.316645\pi$
$524$	−28.1482	−1.22966
$525$	0	0
$526$	3.93373	0.171519
$527$	1.24804	0.0543655
$528$	0	0
$529$	−14.1545	−0.615415
$530$	−7.04502	−0.306016
$531$	0	0
$532$	7.59857	0.329440
$533$	12.8986	0.558702
$534$	0	0
$535$	9.61297	0.415605
$536$	−23.4365	−1.01230
$537$	0	0
$538$	−8.37214	−0.360949
$539$	26.0135	1.12048
$540$	0	0
$541$	38.8319	1.66951	0.834757	−	0.550619i	$-0.185608\pi$
$541$	38.8319	1.66951	0.834757	+	0.550619i	$0.185608\pi$
$542$	14.8624	0.638393
$543$	0	0
$544$	1.80172	0.0772480
$545$	−9.20432	−0.394270
$546$	0	0
$547$	22.6886	0.970093	0.485046	−	0.874488i	$-0.338803\pi$
$547$	22.6886	0.970093	0.485046	+	0.874488i	$0.338803\pi$
$548$	8.96496	0.382964
$549$	0	0
$550$	−2.77670	−0.118399
$551$	25.9049	1.10359
$552$	0	0
$553$	−5.45127	−0.231812
$554$	7.44296	0.316221
$555$	0	0
$556$	16.8096	0.712885
$557$	−42.1075	−1.78415	−0.892077	−	0.451883i	$-0.850752\pi$
$557$	−42.1075	−1.78415	−0.892077	+	0.451883i	$0.850752\pi$
$558$	0	0
$559$	−13.8224	−0.584624
$560$	−6.46696	−0.273279
$561$	0	0
$562$	6.36427	0.268461
$563$	−39.8210	−1.67825	−0.839127	−	0.543936i	$-0.816933\pi$
$563$	−39.8210	−1.67825	−0.839127	+	0.543936i	$0.816933\pi$
$564$	0	0
$565$	−4.08121	−0.171698
$566$	6.56322	0.275873
$567$	0	0
$568$	24.2081	1.01575
$569$	32.6387	1.36828	0.684142	−	0.729349i	$-0.260177\pi$
$569$	32.6387	1.36828	0.684142	+	0.729349i	$0.260177\pi$
$570$	0	0
$571$	−9.33796	−0.390781	−0.195391	−	0.980725i	$-0.562597\pi$
$571$	−9.33796	−0.390781	−0.195391	+	0.980725i	$0.562597\pi$
$572$	10.6548	0.445501
$573$	0	0
$574$	6.91824	0.288762
$575$	3.28890	0.137157
$576$	0	0
$577$	−21.3634	−0.889371	−0.444686	−	0.895687i	$-0.646685\pi$
$577$	−21.3634	−0.889371	−0.444686	+	0.895687i	$0.646685\pi$
$578$	8.61349	0.358274
$579$	0	0
$580$	−26.6337	−1.10591
$581$	−15.4376	−0.640462
$582$	0	0
$583$	−34.4067	−1.42498
$584$	21.8543	0.904337
$585$	0	0
$586$	−8.22504	−0.339773
$587$	−34.4889	−1.42351	−0.711755	−	0.702427i	$-0.752100\pi$
$587$	−34.4889	−1.42351	−0.711755	+	0.702427i	$0.752100\pi$
$588$	0	0
$589$	−11.7851	−0.485599
$590$	1.33082	0.0547891
$591$	0	0
$592$	19.2149	0.789726
$593$	38.2385	1.57027	0.785134	−	0.619326i	$-0.212594\pi$
$593$	38.2385	1.57027	0.785134	+	0.619326i	$0.212594\pi$
$594$	0	0
$595$	0.912888	0.0374247
$596$	−32.1023	−1.31496
$597$	0	0
$598$	1.89027	0.0772989
$599$	1.50195	0.0613680	0.0306840	−	0.999529i	$-0.490231\pi$
$599$	1.50195	0.0613680	0.0306840	+	0.999529i	$0.490231\pi$
$600$	0	0
$601$	1.49985	0.0611800	0.0305900	−	0.999532i	$-0.490261\pi$
$601$	1.49985	0.0611800	0.0305900	+	0.999532i	$0.490261\pi$
$602$	−7.41370	−0.302160
$603$	0	0
$604$	−38.2288	−1.55551
$605$	26.0473	1.05897
$606$	0	0
$607$	−12.2316	−0.496464	−0.248232	−	0.968701i	$-0.579849\pi$
$607$	−12.2316	−0.496464	−0.248232	+	0.968701i	$0.579849\pi$
$608$	−17.0135	−0.689988
$609$	0	0
$610$	−13.1586	−0.532774
$611$	4.41266	0.178517
$612$	0	0
$613$	23.9996	0.969335	0.484667	−	0.874698i	$-0.338941\pi$
$613$	23.9996	0.969335	0.484667	+	0.874698i	$0.338941\pi$
$614$	8.54806	0.344972
$615$	0	0
$616$	12.2855	0.494996
$617$	−4.04588	−0.162881	−0.0814405	−	0.996678i	$-0.525952\pi$
$617$	−4.04588	−0.162881	−0.0814405	+	0.996678i	$0.525952\pi$
$618$	0	0
$619$	11.8427	0.476000	0.238000	−	0.971265i	$-0.423508\pi$
$619$	11.8427	0.476000	0.238000	+	0.971265i	$0.423508\pi$
$620$	12.1167	0.486620
$621$	0	0
$622$	−6.76917	−0.271419
$623$	4.49069	0.179916
$624$	0	0
$625$	−18.2479	−0.729918
$626$	2.79315	0.111637
$627$	0	0
$628$	−30.6733	−1.22400
$629$	−2.71241	−0.108151
$630$	0	0
$631$	41.6870	1.65953	0.829767	−	0.558110i	$-0.188473\pi$
$631$	41.6870	1.65953	0.829767	+	0.558110i	$0.188473\pi$
$632$	7.95238	0.316329
$633$	0	0
$634$	1.44602	0.0574288
$635$	39.1721	1.55450
$636$	0	0
$637$	−6.58452	−0.260888
$638$	19.4827	0.771326
$639$	0	0
$640$	22.5378	0.890887
$641$	−41.8667	−1.65363	−0.826817	−	0.562471i	$-0.809851\pi$
$641$	−41.8667	−1.65363	−0.826817	+	0.562471i	$0.809851\pi$
$642$	0	0
$643$	−31.1939	−1.23017	−0.615085	−	0.788461i	$-0.710878\pi$
$643$	−31.1939	−1.23017	−0.615085	+	0.788461i	$0.710878\pi$
$644$	−6.76893	−0.266733
$645$	0	0
$646$	0.602521	0.0237059
$647$	11.5784	0.455195	0.227597	−	0.973755i	$-0.426913\pi$
$647$	11.5784	0.455195	0.227597	+	0.973755i	$0.426913\pi$
$648$	0	0
$649$	6.49951	0.255128
$650$	0.702838	0.0275676
$651$	0	0
$652$	−24.5076	−0.959792
$653$	3.04123	0.119012	0.0595062	−	0.998228i	$-0.481047\pi$
$653$	3.04123	0.119012	0.0595062	+	0.998228i	$0.481047\pi$
$654$	0	0
$655$	31.9332	1.24773
$656$	25.9455	1.01300
$657$	0	0
$658$	2.36675	0.0922655
$659$	15.2081	0.592423	0.296212	−	0.955122i	$-0.404277\pi$
$659$	15.2081	0.592423	0.296212	+	0.955122i	$0.404277\pi$
$660$	0	0
$661$	2.84857	0.110797	0.0553983	−	0.998464i	$-0.482357\pi$
$661$	2.84857	0.110797	0.0553983	+	0.998464i	$0.482357\pi$
$662$	9.52356	0.370144
$663$	0	0
$664$	22.5206	0.873970
$665$	−8.62033	−0.334282
$666$	0	0
$667$	−23.0765	−0.893526
$668$	−27.0046	−1.04484
$669$	0	0
$670$	12.3677	0.477808
$671$	−64.2641	−2.48089
$672$	0	0
$673$	−34.2012	−1.31836	−0.659181	−	0.751985i	$-0.729097\pi$
$673$	−34.2012	−1.31836	−0.659181	+	0.751985i	$0.729097\pi$
$674$	−6.45120	−0.248491
$675$	0	0
$676$	19.9161	0.766003
$677$	−24.7943	−0.952922	−0.476461	−	0.879196i	$-0.658081\pi$
$677$	−24.7943	−0.952922	−0.476461	+	0.879196i	$0.658081\pi$
$678$	0	0
$679$	−0.420616	−0.0161417
$680$	−1.33173	−0.0510696
$681$	0	0
$682$	−8.86342	−0.339398
$683$	2.51062	0.0960662	0.0480331	−	0.998846i	$-0.484705\pi$
$683$	2.51062	0.0960662	0.0480331	+	0.998846i	$0.484705\pi$
$684$	0	0
$685$	−10.1704	−0.388593
$686$	−8.20661	−0.313330
$687$	0	0
$688$	−27.8037	−1.06000
$689$	8.70900	0.331786
$690$	0	0
$691$	13.0442	0.496226	0.248113	−	0.968731i	$-0.420190\pi$
$691$	13.0442	0.496226	0.248113	+	0.968731i	$0.420190\pi$
$692$	−43.0104	−1.63501
$693$	0	0
$694$	13.9151	0.528210
$695$	−19.0699	−0.723364
$696$	0	0
$697$	−3.66252	−0.138728
$698$	0.362760	0.0137307
$699$	0	0
$700$	−2.51682	−0.0951267
$701$	−8.84820	−0.334192	−0.167096	−	0.985941i	$-0.553439\pi$
$701$	−8.84820	−0.334192	−0.167096	+	0.985941i	$0.553439\pi$
$702$	0	0
$703$	25.6130	0.966014
$704$	11.8467	0.446489
$705$	0	0
$706$	5.55605	0.209105
$707$	7.52568	0.283032
$708$	0	0
$709$	29.5125	1.10837	0.554183	−	0.832395i	$-0.313031\pi$
$709$	29.5125	1.10837	0.554183	+	0.832395i	$0.313031\pi$
$710$	−12.7749	−0.479435
$711$	0	0
$712$	−6.55107	−0.245512
$713$	10.4984	0.393169
$714$	0	0
$715$	−12.0875	−0.452049
$716$	−34.8320	−1.30173
$717$	0	0
$718$	17.6726	0.659537
$719$	−10.5343	−0.392861	−0.196431	−	0.980518i	$-0.562935\pi$
$719$	−10.5343	−0.392861	−0.196431	+	0.980518i	$0.562935\pi$
$720$	0	0
$721$	17.2573	0.642695
$722$	4.00859	0.149184
$723$	0	0
$724$	16.4788	0.612429
$725$	−8.58028	−0.318664
$726$	0	0
$727$	13.6078	0.504684	0.252342	−	0.967638i	$-0.418799\pi$
$727$	13.6078	0.504684	0.252342	+	0.967638i	$0.418799\pi$
$728$	−3.10970	−0.115253
$729$	0	0
$730$	−11.5328	−0.426848
$731$	3.92482	0.145165
$732$	0	0
$733$	26.1557	0.966083	0.483041	−	0.875597i	$-0.339532\pi$
$733$	26.1557	0.966083	0.483041	+	0.875597i	$0.339532\pi$
$734$	−11.0547	−0.408037
$735$	0	0
$736$	15.1559	0.558654
$737$	60.4019	2.22493
$738$	0	0
$739$	18.5120	0.680976	0.340488	−	0.940249i	$-0.389408\pi$
$739$	18.5120	0.680976	0.340488	+	0.940249i	$0.389408\pi$
$740$	−26.3337	−0.968045
$741$	0	0
$742$	4.67111	0.171482
$743$	3.57989	0.131333	0.0656667	−	0.997842i	$-0.479083\pi$
$743$	3.57989	0.131333	0.0656667	+	0.997842i	$0.479083\pi$
$744$	0	0
$745$	36.4190	1.33429
$746$	−13.9173	−0.509549
$747$	0	0
$748$	−3.02540	−0.110620
$749$	−6.37376	−0.232892
$750$	0	0
$751$	24.0523	0.877681	0.438841	−	0.898565i	$-0.355389\pi$
$751$	24.0523	0.877681	0.438841	+	0.898565i	$0.355389\pi$
$752$	8.87604	0.323676
$753$	0	0
$754$	−4.93144	−0.179593
$755$	43.3693	1.57837
$756$	0	0
$757$	1.29824	0.0471855	0.0235927	−	0.999722i	$-0.492490\pi$
$757$	1.29824	0.0471855	0.0235927	+	0.999722i	$0.492490\pi$
$758$	−2.01420	−0.0731589
$759$	0	0
$760$	12.5754	0.456159
$761$	−1.07425	−0.0389414	−0.0194707	−	0.999810i	$-0.506198\pi$
$761$	−1.07425	−0.0389414	−0.0194707	+	0.999810i	$0.506198\pi$
$762$	0	0
$763$	6.10281	0.220937
$764$	20.1803	0.730098
$765$	0	0
$766$	−9.70117	−0.350517
$767$	−1.64515	−0.0594031
$768$	0	0
$769$	12.5505	0.452584	0.226292	−	0.974060i	$-0.427340\pi$
$769$	12.5505	0.452584	0.226292	+	0.974060i	$0.427340\pi$
$770$	−6.48321	−0.233639
$771$	0	0
$772$	34.3098	1.23484
$773$	−8.72300	−0.313744	−0.156872	−	0.987619i	$-0.550141\pi$
$773$	−8.72300	−0.313744	−0.156872	+	0.987619i	$0.550141\pi$
$774$	0	0
$775$	3.90350	0.140218
$776$	0.613599	0.0220269
$777$	0	0
$778$	−10.0936	−0.361873
$779$	34.5849	1.23913
$780$	0	0
$781$	−62.3906	−2.23251
$782$	−0.536736	−0.0191936
$783$	0	0
$784$	−13.2447	−0.473026
$785$	34.7979	1.24199
$786$	0	0
$787$	−13.4078	−0.477937	−0.238969	−	0.971027i	$-0.576809\pi$
$787$	−13.4078	−0.477937	−0.238969	+	0.971027i	$0.576809\pi$
$788$	−5.86845	−0.209055
$789$	0	0
$790$	−4.19658	−0.149307
$791$	2.70599	0.0962140
$792$	0	0
$793$	16.2665	0.577641
$794$	17.3140	0.614453
$795$	0	0
$796$	−26.9686	−0.955876
$797$	28.0679	0.994215	0.497107	−	0.867689i	$-0.334395\pi$
$797$	28.0679	0.994215	0.497107	+	0.867689i	$0.334395\pi$
$798$	0	0
$799$	−1.25296	−0.0443265
$800$	5.63524	0.199236
$801$	0	0
$802$	−2.14053	−0.0755846
$803$	−56.3242	−1.98764
$804$	0	0
$805$	7.67913	0.270654
$806$	2.24351	0.0790241
$807$	0	0
$808$	−10.9786	−0.386224
$809$	56.3769	1.98211	0.991053	−	0.133468i	$-0.0426112\pi$
$809$	56.3769	1.98211	0.991053	+	0.133468i	$0.0426112\pi$
$810$	0	0
$811$	39.6998	1.39405	0.697025	−	0.717047i	$-0.254507\pi$
$811$	39.6998	1.39405	0.697025	+	0.717047i	$0.254507\pi$
$812$	17.6592	0.619715
$813$	0	0
$814$	19.2632	0.675174
$815$	27.8031	0.973900
$816$	0	0
$817$	−37.0617	−1.29663
$818$	8.34275	0.291697
$819$	0	0
$820$	−35.5580	−1.24174
$821$	0.289440	0.0101015	0.00505077	−	0.999987i	$-0.498392\pi$
$821$	0.289440	0.0101015	0.00505077	+	0.999987i	$0.498392\pi$
$822$	0	0
$823$	41.7663	1.45588	0.727942	−	0.685639i	$-0.240477\pi$
$823$	41.7663	1.45588	0.727942	+	0.685639i	$0.240477\pi$
$824$	−25.1752	−0.877018
$825$	0	0
$826$	−0.882386	−0.0307021
$827$	7.69068	0.267431	0.133716	−	0.991020i	$-0.457309\pi$
$827$	7.69068	0.267431	0.133716	+	0.991020i	$0.457309\pi$
$828$	0	0
$829$	−5.46418	−0.189779	−0.0948894	−	0.995488i	$-0.530250\pi$
$829$	−5.46418	−0.189779	−0.0948894	+	0.995488i	$0.530250\pi$
$830$	−11.8844	−0.412515
$831$	0	0
$832$	−2.99863	−0.103959
$833$	1.86965	0.0647796
$834$	0	0
$835$	30.6358	1.06020
$836$	28.5686	0.988067
$837$	0	0
$838$	−11.0497	−0.381707
$839$	−45.3313	−1.56501	−0.782506	−	0.622643i	$-0.786059\pi$
$839$	−45.3313	−1.56501	−0.782506	+	0.622643i	$0.786059\pi$
$840$	0	0
$841$	31.2033	1.07598
$842$	0.437465	0.0150760
$843$	0	0
$844$	−14.3180	−0.492846
$845$	−22.5941	−0.777262
$846$	0	0
$847$	−17.2703	−0.593416
$848$	17.5181	0.601575
$849$	0	0
$850$	−0.199568	−0.00684514
$851$	−22.8165	−0.782141
$852$	0	0
$853$	−0.228143	−0.00781146	−0.00390573	−	0.999992i	$-0.501243\pi$
$853$	−0.228143	−0.00781146	−0.00390573	+	0.999992i	$0.501243\pi$
$854$	8.72462	0.298550
$855$	0	0
$856$	9.29812	0.317803
$857$	−44.4962	−1.51996	−0.759980	−	0.649946i	$-0.774791\pi$
$857$	−44.4962	−1.51996	−0.759980	+	0.649946i	$0.774791\pi$
$858$	0	0
$859$	−32.6841	−1.11517	−0.557583	−	0.830121i	$-0.688271\pi$
$859$	−32.6841	−1.11517	−0.557583	+	0.830121i	$0.688271\pi$
$860$	38.1045	1.29935
$861$	0	0
$862$	1.80973	0.0616398
$863$	−39.9025	−1.35830	−0.679148	−	0.734001i	$-0.737651\pi$
$863$	−39.9025	−1.35830	−0.679148	+	0.734001i	$0.737651\pi$
$864$	0	0
$865$	48.7939	1.65904
$866$	3.37484	0.114682
$867$	0	0
$868$	−8.03385	−0.272687
$869$	−20.4954	−0.695257
$870$	0	0
$871$	−15.2889	−0.518045
$872$	−8.90285	−0.301489
$873$	0	0
$874$	5.06835	0.171440
$875$	15.7651	0.532959
$876$	0	0
$877$	−4.27574	−0.144382	−0.0721908	−	0.997391i	$-0.522999\pi$
$877$	−4.27574	−0.144382	−0.0721908	+	0.997391i	$0.522999\pi$
$878$	15.8439	0.534705
$879$	0	0
$880$	−24.3141	−0.819627
$881$	−38.9219	−1.31131	−0.655655	−	0.755060i	$-0.727608\pi$
$881$	−38.9219	−1.31131	−0.655655	+	0.755060i	$0.727608\pi$
$882$	0	0
$883$	8.70306	0.292881	0.146441	−	0.989219i	$-0.453218\pi$
$883$	8.70306	0.292881	0.146441	+	0.989219i	$0.453218\pi$
$884$	0.765788	0.0257562
$885$	0	0
$886$	−5.44482	−0.182922
$887$	−9.32121	−0.312976	−0.156488	−	0.987680i	$-0.550017\pi$
$887$	−9.32121	−0.312976	−0.156488	+	0.987680i	$0.550017\pi$
$888$	0	0
$889$	−25.9726	−0.871093
$890$	3.45709	0.115882
$891$	0	0
$892$	−14.3424	−0.480220
$893$	11.8316	0.395929
$894$	0	0
$895$	39.5158	1.32087
$896$	−14.9434	−0.499225
$897$	0	0
$898$	4.26547	0.142341
$899$	−27.3888	−0.913469
$900$	0	0
$901$	−2.47289	−0.0823839
$902$	26.0108	0.866064
$903$	0	0
$904$	−3.94753	−0.131293
$905$	−18.6946	−0.621430
$906$	0	0
$907$	10.7219	0.356016	0.178008	−	0.984029i	$-0.443035\pi$
$907$	10.7219	0.356016	0.178008	+	0.984029i	$0.443035\pi$
$908$	8.55206	0.283810
$909$	0	0
$910$	1.64103	0.0543996
$911$	44.4272	1.47194	0.735970	−	0.677014i	$-0.236726\pi$
$911$	44.4272	1.47194	0.735970	+	0.677014i	$0.236726\pi$
$912$	0	0
$913$	−58.0415	−1.92089
$914$	−0.749560	−0.0247932
$915$	0	0
$916$	−1.27508	−0.0421297
$917$	−21.1729	−0.699190
$918$	0	0
$919$	11.3809	0.375421	0.187710	−	0.982224i	$-0.439893\pi$
$919$	11.3809	0.375421	0.187710	+	0.982224i	$0.439893\pi$
$920$	−11.2024	−0.369333
$921$	0	0
$922$	21.5887	0.710984
$923$	15.7923	0.519809
$924$	0	0
$925$	−8.48361	−0.278939
$926$	−13.3440	−0.438509
$927$	0	0
$928$	−39.5395	−1.29795
$929$	−12.7353	−0.417831	−0.208916	−	0.977934i	$-0.566993\pi$
$929$	−12.7353	−0.417831	−0.208916	+	0.977934i	$0.566993\pi$
$930$	0	0
$931$	−17.6550	−0.578619
$932$	−28.9032	−0.946756
$933$	0	0
$934$	9.76974	0.319676
$935$	3.43222	0.112246
$936$	0	0
$937$	−22.8143	−0.745312	−0.372656	−	0.927970i	$-0.621553\pi$
$937$	−22.8143	−0.745312	−0.372656	+	0.927970i	$0.621553\pi$
$938$	−8.20028	−0.267749
$939$	0	0
$940$	−12.1645	−0.396762
$941$	41.4690	1.35185	0.675925	−	0.736971i	$-0.263744\pi$
$941$	41.4690	1.35185	0.675925	+	0.736971i	$0.263744\pi$
$942$	0	0
$943$	−30.8088	−1.00327
$944$	−3.30922	−0.107706
$945$	0	0
$946$	−27.8736	−0.906247
$947$	27.4984	0.893578	0.446789	−	0.894639i	$-0.352568\pi$
$947$	27.4984	0.893578	0.446789	+	0.894639i	$0.352568\pi$
$948$	0	0
$949$	14.2568	0.462794
$950$	1.88451	0.0611415
$951$	0	0
$952$	0.882988	0.0286178
$953$	22.5938	0.731887	0.365943	−	0.930637i	$-0.380746\pi$
$953$	22.5938	0.731887	0.365943	+	0.930637i	$0.380746\pi$
$954$	0	0
$955$	−22.8939	−0.740829
$956$	−1.73946	−0.0562582
$957$	0	0
$958$	−13.3749	−0.432124
$959$	6.74339	0.217755
$960$	0	0
$961$	−18.5397	−0.598056
$962$	−4.87588	−0.157205
$963$	0	0
$964$	17.1339	0.551845
$965$	−38.9233	−1.25299
$966$	0	0
$967$	1.80872	0.0581645	0.0290823	−	0.999577i	$-0.490742\pi$
$967$	1.80872	0.0581645	0.0290823	+	0.999577i	$0.490742\pi$
$968$	25.1942	0.809772
$969$	0	0
$970$	−0.323804	−0.0103967
$971$	−16.7063	−0.536132	−0.268066	−	0.963401i	$-0.586384\pi$
$971$	−16.7063	−0.536132	−0.268066	+	0.963401i	$0.586384\pi$
$972$	0	0
$973$	12.6441	0.405351
$974$	20.8001	0.666477
$975$	0	0
$976$	32.7200	1.04734
$977$	40.8168	1.30585	0.652923	−	0.757424i	$-0.273543\pi$
$977$	40.8168	1.30585	0.652923	+	0.757424i	$0.273543\pi$
$978$	0	0
$979$	16.8838	0.539609
$980$	18.1517	0.579836
$981$	0	0
$982$	3.52096	0.112358
$983$	30.0618	0.958822	0.479411	−	0.877591i	$-0.340850\pi$
$983$	30.0618	0.958822	0.479411	+	0.877591i	$0.340850\pi$
$984$	0	0
$985$	6.65756	0.212128
$986$	1.40027	0.0445936
$987$	0	0
$988$	−7.23128	−0.230058
$989$	33.0152	1.04982
$990$	0	0
$991$	−1.64335	−0.0522028	−0.0261014	−	0.999659i	$-0.508309\pi$
$991$	−1.64335	−0.0522028	−0.0261014	+	0.999659i	$0.508309\pi$
$992$	17.9881	0.571122
$993$	0	0
$994$	8.47026	0.268660
$995$	30.5950	0.969926
$996$	0	0
$997$	−44.6421	−1.41383	−0.706915	−	0.707299i	$-0.749914\pi$
$997$	−44.6421	−1.41383	−0.706915	+	0.707299i	$0.749914\pi$
$998$	−10.9720	−0.347314
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.k.1.8	yes	20
3.2	odd	2		2151.2.a.j.1.13	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.13	✓	20	3.2	odd	2
2151.2.a.k.1.8	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)

\(f(q)\)	\(=\)	\(q-0.510429 q^{2} -1.73946 q^{4} +1.97336 q^{5} -1.30841 q^{7} +1.90873 q^{8} +O(q^{10})\)	\(q-0.510429 q^{2} -1.73946 q^{4} +1.97336 q^{5} -1.30841 q^{7} +1.90873 q^{8} -1.00726 q^{10} -4.91929 q^{11} +1.24517 q^{13} +0.667852 q^{14} +2.50465 q^{16} -0.353562 q^{17} +3.33866 q^{19} -3.43259 q^{20} +2.51095 q^{22} -2.97413 q^{23} -1.10584 q^{25} -0.635571 q^{26} +2.27594 q^{28} +7.75908 q^{29} -3.52991 q^{31} -5.09591 q^{32} +0.180468 q^{34} -2.58198 q^{35} +7.67166 q^{37} -1.70415 q^{38} +3.76662 q^{40} +10.3589 q^{41} -11.1008 q^{43} +8.55693 q^{44} +1.51808 q^{46} +3.54382 q^{47} -5.28805 q^{49} +0.564451 q^{50} -2.16593 q^{52} +6.99423 q^{53} -9.70755 q^{55} -2.49741 q^{56} -3.96046 q^{58} -1.32123 q^{59} +13.0637 q^{61} +1.80177 q^{62} -2.40821 q^{64} +2.45717 q^{65} -12.2786 q^{67} +0.615007 q^{68} +1.31792 q^{70} +12.6828 q^{71} +11.4497 q^{73} -3.91584 q^{74} -5.80746 q^{76} +6.43647 q^{77} +4.16632 q^{79} +4.94259 q^{80} -5.28750 q^{82} +11.7987 q^{83} -0.697706 q^{85} +5.66617 q^{86} -9.38960 q^{88} -3.43216 q^{89} -1.62920 q^{91} +5.17339 q^{92} -1.80887 q^{94} +6.58838 q^{95} +0.321470 q^{97} +2.69918 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) \) 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8 + 4 * q^10 + 12 * q^11 - 4 * q^13 + 20 * q^14 + 22 * q^16 + 24 * q^17 - 4 * q^19 + 40 * q^20 - 6 * q^22 + 12 * q^23 + 22 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 - 4 * q^31 + 28 * q^32 + 8 * q^34 + 20 * q^35 - 10 * q^37 + 26 * q^38 + 6 * q^40 + 66 * q^41 + 8 * q^43 + 36 * q^44 - 12 * q^46 + 28 * q^47 + 18 * q^49 + 28 * q^50 - 18 * q^52 + 28 * q^53 - 4 * q^55 + 60 * q^56 + 54 * q^59 - 4 * q^61 + 20 * q^62 + 22 * q^64 + 42 * q^65 + 12 * q^67 + 12 * q^68 + 20 * q^70 + 36 * q^71 + 14 * q^73 - 50 * q^76 + 8 * q^77 - 12 * q^79 + 88 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 + 18 * q^86 - 10 * q^88 + 130 * q^89 - 6 * q^91 - 46 * q^92 - 26 * q^94 - 2 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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