LMFDB - Embedded newform 2169.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2169,2,Mod(1,2169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2169, 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("2169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2169 = 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2169.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:	$17.3195521984$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} + \cdots - 1 $ x^12 - 3x^11 - 14x^10 + 44x^9 + 65x^8 - 219x^7 - 123x^6 + 444x^5 + 105x^4 - 328x^3 - 45x^2 + 18*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.49073$ of defining polynomial
Character	$\chi$	$=$	2169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49073 q^{2} +4.20371 q^{4} +3.14843 q^{5} +0.136122 q^{7} -5.48885 q^{8} +O(q^{10})$	\(q-2.49073 q^{2} +4.20371 q^{4} +3.14843 q^{5} +0.136122 q^{7} -5.48885 q^{8} -7.84189 q^{10} +0.905365 q^{11} -0.123706 q^{13} -0.339044 q^{14} +5.26378 q^{16} -1.26034 q^{17} -2.13460 q^{19} +13.2351 q^{20} -2.25502 q^{22} -6.64978 q^{23} +4.91264 q^{25} +0.308118 q^{26} +0.572220 q^{28} -5.36862 q^{29} -9.78467 q^{31} -2.13295 q^{32} +3.13917 q^{34} +0.428573 q^{35} +5.76688 q^{37} +5.31669 q^{38} -17.2813 q^{40} -6.43642 q^{41} -3.18712 q^{43} +3.80590 q^{44} +16.5628 q^{46} -12.9849 q^{47} -6.98147 q^{49} -12.2360 q^{50} -0.520025 q^{52} -3.90862 q^{53} +2.85048 q^{55} -0.747155 q^{56} +13.3718 q^{58} -8.15085 q^{59} -14.3712 q^{61} +24.3709 q^{62} -5.21498 q^{64} -0.389481 q^{65} -4.89534 q^{67} -5.29812 q^{68} -1.06746 q^{70} -4.32869 q^{71} +5.64935 q^{73} -14.3637 q^{74} -8.97323 q^{76} +0.123241 q^{77} +1.43490 q^{79} +16.5727 q^{80} +16.0314 q^{82} +11.7625 q^{83} -3.96811 q^{85} +7.93825 q^{86} -4.96941 q^{88} +13.7381 q^{89} -0.0168392 q^{91} -27.9538 q^{92} +32.3419 q^{94} -6.72063 q^{95} +13.6204 q^{97} +17.3889 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8} - 7 q^{10} - 22 q^{11} - 5 q^{13} - 6 q^{14} + 15 q^{16} + 4 q^{17} - 6 q^{19} - 10 q^{20} - 12 q^{22} - 32 q^{23} + 4 q^{25} - 8 q^{26} - 11 q^{28} - 6 q^{29} + 8 q^{31} - q^{32} - 19 q^{34} - 15 q^{35} - 8 q^{37} + 10 q^{38} - 52 q^{40} + q^{41} - 2 q^{43} - 42 q^{44} - 25 q^{46} - 34 q^{47} - 9 q^{49} + 27 q^{50} - 41 q^{52} - 5 q^{53} - 3 q^{55} - q^{56} - 33 q^{58} - 26 q^{59} - 26 q^{61} + 17 q^{62} + 13 q^{64} + 25 q^{65} + 6 q^{67} + 35 q^{68} - 4 q^{70} - 94 q^{71} - 22 q^{73} - 26 q^{74} - 20 q^{76} + 7 q^{77} + 9 q^{79} - 19 q^{80} + 15 q^{82} + 8 q^{83} + 4 q^{85} - 9 q^{86} + 6 q^{88} + 3 q^{89} - 20 q^{91} - 36 q^{92} + 48 q^{94} - 33 q^{95} - 29 q^{97} - 28 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8 - 7 * q^10 - 22 * q^11 - 5 * q^13 - 6 * q^14 + 15 * q^16 + 4 * q^17 - 6 * q^19 - 10 * q^20 - 12 * q^22 - 32 * q^23 + 4 * q^25 - 8 * q^26 - 11 * q^28 - 6 * q^29 + 8 * q^31 - q^32 - 19 * q^34 - 15 * q^35 - 8 * q^37 + 10 * q^38 - 52 * q^40 + q^41 - 2 * q^43 - 42 * q^44 - 25 * q^46 - 34 * q^47 - 9 * q^49 + 27 * q^50 - 41 * q^52 - 5 * q^53 - 3 * q^55 - q^56 - 33 * q^58 - 26 * q^59 - 26 * q^61 + 17 * q^62 + 13 * q^64 + 25 * q^65 + 6 * q^67 + 35 * q^68 - 4 * q^70 - 94 * q^71 - 22 * q^73 - 26 * q^74 - 20 * q^76 + 7 * q^77 + 9 * q^79 - 19 * q^80 + 15 * q^82 + 8 * q^83 + 4 * q^85 - 9 * q^86 + 6 * q^88 + 3 * q^89 - 20 * q^91 - 36 * q^92 + 48 * q^94 - 33 * q^95 - 29 * q^97 - 28 * 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 $

$2$	−2.49073	−1.76121	−0.880604	−	0.473852i	$-0.842863\pi$
$2$	−2.49073	−1.76121	−0.880604	+	0.473852i	$0.842863\pi$
$3$	0	0
$3$	0	0
$4$	4.20371	2.10186
$5$	3.14843	1.40802	0.704011	−	0.710189i	$-0.251390\pi$
$5$	3.14843	1.40802	0.704011	+	0.710189i	$0.251390\pi$
$6$	0	0
$7$	0.136122	0.0514495	0.0257247	−	0.999669i	$-0.491811\pi$
$7$	0.136122	0.0514495	0.0257247	+	0.999669i	$0.491811\pi$
$8$	−5.48885	−1.94060
$9$	0	0
$10$	−7.84189	−2.47982
$11$	0.905365	0.272978	0.136489	−	0.990642i	$-0.456418\pi$
$11$	0.905365	0.272978	0.136489	+	0.990642i	$0.456418\pi$
$12$	0	0
$13$	−0.123706	−0.0343099	−0.0171549	−	0.999853i	$-0.505461\pi$
$13$	−0.123706	−0.0343099	−0.0171549	+	0.999853i	$0.505461\pi$
$14$	−0.339044	−0.0906133
$15$	0	0
$16$	5.26378	1.31595
$17$	−1.26034	−0.305678	−0.152839	−	0.988251i	$-0.548842\pi$
$17$	−1.26034	−0.305678	−0.152839	+	0.988251i	$0.548842\pi$
$18$	0	0
$19$	−2.13460	−0.489710	−0.244855	−	0.969560i	$-0.578740\pi$
$19$	−2.13460	−0.489710	−0.244855	+	0.969560i	$0.578740\pi$
$20$	13.2351	2.95946
$21$	0	0
$22$	−2.25502	−0.480771
$23$	−6.64978	−1.38657	−0.693287	−	0.720661i	$-0.743838\pi$
$23$	−6.64978	−1.38657	−0.693287	+	0.720661i	$0.743838\pi$
$24$	0	0
$25$	4.91264	0.982528
$26$	0.308118	0.0604269
$27$	0	0
$28$	0.572220	0.108139
$29$	−5.36862	−0.996928	−0.498464	−	0.866911i	$-0.666102\pi$
$29$	−5.36862	−0.996928	−0.498464	+	0.866911i	$0.666102\pi$
$30$	0	0
$31$	−9.78467	−1.75738	−0.878689	−	0.477395i	$-0.841581\pi$
$31$	−9.78467	−1.75738	−0.878689	+	0.477395i	$0.841581\pi$
$32$	−2.13295	−0.377055
$33$	0	0
$34$	3.13917	0.538363
$35$	0.428573	0.0724420
$36$	0	0
$37$	5.76688	0.948070	0.474035	−	0.880506i	$-0.342797\pi$
$37$	5.76688	0.948070	0.474035	+	0.880506i	$0.342797\pi$
$38$	5.31669	0.862481
$39$	0	0
$40$	−17.2813	−2.73241
$41$	−6.43642	−1.00520	−0.502600	−	0.864519i	$-0.667623\pi$
$41$	−6.43642	−1.00520	−0.502600	+	0.864519i	$0.667623\pi$
$42$	0	0
$43$	−3.18712	−0.486032	−0.243016	−	0.970022i	$-0.578137\pi$
$43$	−3.18712	−0.486032	−0.243016	+	0.970022i	$0.578137\pi$
$44$	3.80590	0.573761
$45$	0	0
$46$	16.5628	2.44205
$47$	−12.9849	−1.89404	−0.947022	−	0.321168i	$-0.895925\pi$
$47$	−12.9849	−1.89404	−0.947022	+	0.321168i	$0.895925\pi$
$48$	0	0
$49$	−6.98147	−0.997353
$50$	−12.2360	−1.73044
$51$	0	0
$52$	−0.520025	−0.0721145
$53$	−3.90862	−0.536890	−0.268445	−	0.963295i	$-0.586510\pi$
$53$	−3.90862	−0.536890	−0.268445	+	0.963295i	$0.586510\pi$
$54$	0	0
$55$	2.85048	0.384359
$56$	−0.747155	−0.0998429
$57$	0	0
$58$	13.3718	1.75580
$59$	−8.15085	−1.06115	−0.530575	−	0.847638i	$-0.678024\pi$
$59$	−8.15085	−1.06115	−0.530575	+	0.847638i	$0.678024\pi$
$60$	0	0
$61$	−14.3712	−1.84004	−0.920021	−	0.391870i	$-0.871828\pi$
$61$	−14.3712	−1.84004	−0.920021	+	0.391870i	$0.871828\pi$
$62$	24.3709	3.09511
$63$	0	0
$64$	−5.21498	−0.651873
$65$	−0.389481	−0.0483091
$66$	0	0
$67$	−4.89534	−0.598062	−0.299031	−	0.954243i	$-0.596663\pi$
$67$	−4.89534	−0.598062	−0.299031	+	0.954243i	$0.596663\pi$
$68$	−5.29812	−0.642491
$69$	0	0
$70$	−1.06746	−0.127586
$71$	−4.32869	−0.513720	−0.256860	−	0.966449i	$-0.582688\pi$
$71$	−4.32869	−0.513720	−0.256860	+	0.966449i	$0.582688\pi$
$72$	0	0
$73$	5.64935	0.661206	0.330603	−	0.943770i	$-0.392748\pi$
$73$	5.64935	0.661206	0.330603	+	0.943770i	$0.392748\pi$
$74$	−14.3637	−1.66975
$75$	0	0
$76$	−8.97323	−1.02930
$77$	0.123241	0.0140446
$78$	0	0
$79$	1.43490	0.161439	0.0807195	−	0.996737i	$-0.474278\pi$
$79$	1.43490	0.161439	0.0807195	+	0.996737i	$0.474278\pi$
$80$	16.5727	1.85288
$81$	0	0
$82$	16.0314	1.77037
$83$	11.7625	1.29110	0.645549	−	0.763718i	$-0.276628\pi$
$83$	11.7625	1.29110	0.645549	+	0.763718i	$0.276628\pi$
$84$	0	0
$85$	−3.96811	−0.430402
$86$	7.93825	0.856004
$87$	0	0
$88$	−4.96941	−0.529741
$89$	13.7381	1.45624	0.728118	−	0.685452i	$-0.240395\pi$
$89$	13.7381	1.45624	0.728118	+	0.685452i	$0.240395\pi$
$90$	0	0
$91$	−0.0168392	−0.00176523
$92$	−27.9538	−2.91438
$93$	0	0
$94$	32.3419	3.33581
$95$	−6.72063	−0.689522
$96$	0	0
$97$	13.6204	1.38294	0.691472	−	0.722404i	$-0.256963\pi$
$97$	13.6204	1.38294	0.691472	+	0.722404i	$0.256963\pi$
$98$	17.3889	1.75655
$99$	0	0
$100$	20.6513	2.06513
$101$	0.0787660	0.00783751	0.00391876	−	0.999992i	$-0.498753\pi$
$101$	0.0787660	0.00783751	0.00391876	+	0.999992i	$0.498753\pi$
$102$	0	0
$103$	−6.36701	−0.627360	−0.313680	−	0.949529i	$-0.601562\pi$
$103$	−6.36701	−0.627360	−0.313680	+	0.949529i	$0.601562\pi$
$104$	0.679004	0.0665818
$105$	0	0
$106$	9.73529	0.945575
$107$	1.99570	0.192932	0.0964660	−	0.995336i	$-0.469246\pi$
$107$	1.99570	0.192932	0.0964660	+	0.995336i	$0.469246\pi$
$108$	0	0
$109$	14.8223	1.41972	0.709861	−	0.704341i	$-0.248758\pi$
$109$	14.8223	1.41972	0.709861	+	0.704341i	$0.248758\pi$
$110$	−7.09977	−0.676937
$111$	0	0
$112$	0.716519	0.0677047
$113$	12.3669	1.16338	0.581690	−	0.813410i	$-0.302392\pi$
$113$	12.3669	1.16338	0.581690	+	0.813410i	$0.302392\pi$
$114$	0	0
$115$	−20.9364	−1.95233
$116$	−22.5681	−2.09540
$117$	0	0
$118$	20.3015	1.86891
$119$	−0.171561	−0.0157270
$120$	0	0
$121$	−10.1803	−0.925483
$122$	35.7947	3.24070
$123$	0	0
$124$	−41.1319	−3.69376
$125$	−0.275046	−0.0246009
$126$	0	0
$127$	15.9678	1.41691	0.708456	−	0.705754i	$-0.249392\pi$
$127$	15.9678	1.41691	0.708456	+	0.705754i	$0.249392\pi$
$128$	17.2550	1.52514
$129$	0	0
$130$	0.970089	0.0850824
$131$	12.1390	1.06059	0.530293	−	0.847814i	$-0.322082\pi$
$131$	12.1390	1.06059	0.530293	+	0.847814i	$0.322082\pi$
$132$	0	0
$133$	−0.290566	−0.0251953
$134$	12.1930	1.05331
$135$	0	0
$136$	6.91783	0.593199
$137$	−0.846052	−0.0722831	−0.0361416	−	0.999347i	$-0.511507\pi$
$137$	−0.846052	−0.0722831	−0.0361416	+	0.999347i	$0.511507\pi$
$138$	0	0
$139$	15.6472	1.32717	0.663587	−	0.748099i	$-0.269033\pi$
$139$	15.6472	1.32717	0.663587	+	0.748099i	$0.269033\pi$
$140$	1.80160	0.152263
$141$	0	0
$142$	10.7816	0.904769
$143$	−0.111999	−0.00936584
$144$	0	0
$145$	−16.9027	−1.40370
$146$	−14.0710	−1.16452
$147$	0	0
$148$	24.2423	1.99271
$149$	0.542212	0.0444198	0.0222099	−	0.999753i	$-0.492930\pi$
$149$	0.542212	0.0444198	0.0222099	+	0.999753i	$0.492930\pi$
$150$	0	0
$151$	8.53046	0.694199	0.347100	−	0.937828i	$-0.387167\pi$
$151$	8.53046	0.694199	0.347100	+	0.937828i	$0.387167\pi$
$152$	11.7165	0.950331
$153$	0	0
$154$	−0.306958	−0.0247354
$155$	−30.8064	−2.47443
$156$	0	0
$157$	−16.2204	−1.29453	−0.647263	−	0.762267i	$-0.724086\pi$
$157$	−16.2204	−1.29453	−0.647263	+	0.762267i	$0.724086\pi$
$158$	−3.57394	−0.284328
$159$	0	0
$160$	−6.71544	−0.530902
$161$	−0.905184	−0.0713385
$162$	0	0
$163$	1.36363	0.106808	0.0534039	−	0.998573i	$-0.482993\pi$
$163$	1.36363	0.106808	0.0534039	+	0.998573i	$0.482993\pi$
$164$	−27.0569	−2.11279
$165$	0	0
$166$	−29.2971	−2.27389
$167$	−13.4045	−1.03727	−0.518636	−	0.854995i	$-0.673560\pi$
$167$	−13.4045	−1.03727	−0.518636	+	0.854995i	$0.673560\pi$
$168$	0	0
$169$	−12.9847	−0.998823
$170$	9.88346	0.758027
$171$	0	0
$172$	−13.3978	−1.02157
$173$	13.6007	1.03404	0.517022	−	0.855972i	$-0.327041\pi$
$173$	13.6007	1.03404	0.517022	+	0.855972i	$0.327041\pi$
$174$	0	0
$175$	0.668721	0.0505505
$176$	4.76565	0.359224
$177$	0	0
$178$	−34.2179	−2.56474
$179$	−23.9070	−1.78689	−0.893445	−	0.449172i	$-0.851719\pi$
$179$	−23.9070	−1.78689	−0.893445	+	0.449172i	$0.851719\pi$
$180$	0	0
$181$	−11.6044	−0.862545	−0.431273	−	0.902222i	$-0.641935\pi$
$181$	−11.6044	−0.862545	−0.431273	+	0.902222i	$0.641935\pi$
$182$	0.0419418	0.00310893
$183$	0	0
$184$	36.4996	2.69079
$185$	18.1567	1.33490
$186$	0	0
$187$	−1.14107	−0.0834433
$188$	−54.5849	−3.98101
$189$	0	0
$190$	16.7393	1.21439
$191$	4.85929	0.351606	0.175803	−	0.984425i	$-0.443748\pi$
$191$	4.85929	0.351606	0.175803	+	0.984425i	$0.443748\pi$
$192$	0	0
$193$	25.6451	1.84597	0.922987	−	0.384831i	$-0.125740\pi$
$193$	25.6451	1.84597	0.922987	+	0.384831i	$0.125740\pi$
$194$	−33.9247	−2.43565
$195$	0	0
$196$	−29.3481	−2.09629
$197$	−1.46990	−0.104726	−0.0523629	−	0.998628i	$-0.516675\pi$
$197$	−1.46990	−0.104726	−0.0523629	+	0.998628i	$0.516675\pi$
$198$	0	0
$199$	7.46714	0.529332	0.264666	−	0.964340i	$-0.414738\pi$
$199$	7.46714	0.529332	0.264666	+	0.964340i	$0.414738\pi$
$200$	−26.9647	−1.90669
$201$	0	0
$202$	−0.196185	−0.0138035
$203$	−0.730790	−0.0512914
$204$	0	0
$205$	−20.2646	−1.41534
$206$	15.8585	1.10491
$207$	0	0
$208$	−0.651162	−0.0451500
$209$	−1.93259	−0.133680
$210$	0	0
$211$	−20.1549	−1.38752	−0.693759	−	0.720208i	$-0.744047\pi$
$211$	−20.1549	−1.38752	−0.693759	+	0.720208i	$0.744047\pi$
$212$	−16.4307	−1.12847
$213$	0	0
$214$	−4.97075	−0.339793
$215$	−10.0345	−0.684344
$216$	0	0
$217$	−1.33191	−0.0904162
$218$	−36.9184	−2.50043
$219$	0	0
$220$	11.9826	0.807868
$221$	0.155912	0.0104878
$222$	0	0
$223$	−8.91317	−0.596870	−0.298435	−	0.954430i	$-0.596465\pi$
$223$	−8.91317	−0.596870	−0.298435	+	0.954430i	$0.596465\pi$
$224$	−0.290342	−0.0193993
$225$	0	0
$226$	−30.8026	−2.04896
$227$	17.2725	1.14642	0.573209	−	0.819409i	$-0.305698\pi$
$227$	17.2725	1.14642	0.573209	+	0.819409i	$0.305698\pi$
$228$	0	0
$229$	−12.9392	−0.855044	−0.427522	−	0.904005i	$-0.640613\pi$
$229$	−12.9392	−0.855044	−0.427522	+	0.904005i	$0.640613\pi$
$230$	52.1468	3.43846
$231$	0	0
$232$	29.4675	1.93464
$233$	24.8301	1.62668	0.813338	−	0.581791i	$-0.197648\pi$
$233$	24.8301	1.62668	0.813338	+	0.581791i	$0.197648\pi$
$234$	0	0
$235$	−40.8822	−2.66686
$236$	−34.2639	−2.23039
$237$	0	0
$238$	0.427311	0.0276985
$239$	−7.52122	−0.486507	−0.243254	−	0.969963i	$-0.578215\pi$
$239$	−7.52122	−0.486507	−0.243254	+	0.969963i	$0.578215\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	25.3564	1.62997
$243$	0	0
$244$	−60.4123	−3.86750
$245$	−21.9807	−1.40430
$246$	0	0
$247$	0.264062	0.0168019
$248$	53.7065	3.41037
$249$	0	0
$250$	0.685064	0.0433272
$251$	30.0279	1.89534	0.947671	−	0.319250i	$-0.103431\pi$
$251$	30.0279	1.89534	0.947671	+	0.319250i	$0.103431\pi$
$252$	0	0
$253$	−6.02048	−0.378504
$254$	−39.7714	−2.49548
$255$	0	0
$256$	−32.5475	−2.03422
$257$	9.93458	0.619702	0.309851	−	0.950785i	$-0.399721\pi$
$257$	9.93458	0.619702	0.309851	+	0.950785i	$0.399721\pi$
$258$	0	0
$259$	0.785002	0.0487777
$260$	−1.63726	−0.101539
$261$	0	0
$262$	−30.2348	−1.86791
$263$	−18.4575	−1.13814	−0.569069	−	0.822290i	$-0.692696\pi$
$263$	−18.4575	−1.13814	−0.569069	+	0.822290i	$0.692696\pi$
$264$	0	0
$265$	−12.3060	−0.755953
$266$	0.723721	0.0443742
$267$	0	0
$268$	−20.5786	−1.25704
$269$	1.14902	0.0700569	0.0350285	−	0.999386i	$-0.488848\pi$
$269$	1.14902	0.0700569	0.0350285	+	0.999386i	$0.488848\pi$
$270$	0	0
$271$	12.2034	0.741303	0.370651	−	0.928772i	$-0.379134\pi$
$271$	12.2034	0.741303	0.370651	+	0.928772i	$0.379134\pi$
$272$	−6.63417	−0.402256
$273$	0	0
$274$	2.10728	0.127306
$275$	4.44773	0.268208
$276$	0	0
$277$	−4.31327	−0.259159	−0.129580	−	0.991569i	$-0.541363\pi$
$277$	−4.31327	−0.259159	−0.129580	+	0.991569i	$0.541363\pi$
$278$	−38.9728	−2.33743
$279$	0	0
$280$	−2.35237	−0.140581
$281$	17.1399	1.02248	0.511240	−	0.859438i	$-0.329186\pi$
$281$	17.1399	1.02248	0.511240	+	0.859438i	$0.329186\pi$
$282$	0	0
$283$	13.8905	0.825704	0.412852	−	0.910798i	$-0.364533\pi$
$283$	13.8905	0.825704	0.412852	+	0.910798i	$0.364533\pi$
$284$	−18.1966	−1.07977
$285$	0	0
$286$	0.278959	0.0164952
$287$	−0.876142	−0.0517170
$288$	0	0
$289$	−15.4115	−0.906561
$290$	42.1001	2.47220
$291$	0	0
$292$	23.7482	1.38976
$293$	0.325090	0.0189920	0.00949598	−	0.999955i	$-0.496977\pi$
$293$	0.325090	0.0189920	0.00949598	+	0.999955i	$0.496977\pi$
$294$	0	0
$295$	−25.6624	−1.49412
$296$	−31.6535	−1.83982
$297$	0	0
$298$	−1.35050	−0.0782325
$299$	0.822618	0.0475732
$300$	0	0
$301$	−0.433839	−0.0250061
$302$	−21.2470	−1.22263
$303$	0	0
$304$	−11.2360	−0.644432
$305$	−45.2467	−2.59082
$306$	0	0
$307$	−14.3600	−0.819566	−0.409783	−	0.912183i	$-0.634396\pi$
$307$	−14.3600	−0.819566	−0.409783	+	0.912183i	$0.634396\pi$
$308$	0.518068	0.0295197
$309$	0	0
$310$	76.7302	4.35798
$311$	11.2072	0.635502	0.317751	−	0.948174i	$-0.397072\pi$
$311$	11.2072	0.635502	0.317751	+	0.948174i	$0.397072\pi$
$312$	0	0
$313$	19.3860	1.09576	0.547879	−	0.836557i	$-0.315435\pi$
$313$	19.3860	1.09576	0.547879	+	0.836557i	$0.315435\pi$
$314$	40.4005	2.27993
$315$	0	0
$316$	6.03191	0.339322
$317$	−5.19578	−0.291824	−0.145912	−	0.989298i	$-0.546612\pi$
$317$	−5.19578	−0.291824	−0.145912	+	0.989298i	$0.546612\pi$
$318$	0	0
$319$	−4.86056	−0.272139
$320$	−16.4190	−0.917852
$321$	0	0
$322$	2.25457	0.125642
$323$	2.69032	0.149693
$324$	0	0
$325$	−0.607724	−0.0337104
$326$	−3.39643	−0.188111
$327$	0	0
$328$	35.3285	1.95069
$329$	−1.76754	−0.0974476
$330$	0	0
$331$	0.0173250	0.000952268 0	0.000476134	−	1.00000i	$-0.499848\pi$
$331$	0.0173250	0.000952268 0	0.000476134		1.00000i	$0.499848\pi$
$332$	49.4461	2.71371
$333$	0	0
$334$	33.3869	1.82685
$335$	−15.4127	−0.842084
$336$	0	0
$337$	0.347899	0.0189513	0.00947563	−	0.999955i	$-0.496984\pi$
$337$	0.347899	0.0189513	0.00947563	+	0.999955i	$0.496984\pi$
$338$	32.3413	1.75914
$339$	0	0
$340$	−16.6808	−0.904642
$341$	−8.85870	−0.479725
$342$	0	0
$343$	−1.90319	−0.102763
$344$	17.4936	0.943194
$345$	0	0
$346$	−33.8756	−1.82117
$347$	−6.37601	−0.342282	−0.171141	−	0.985247i	$-0.554745\pi$
$347$	−6.37601	−0.342282	−0.171141	+	0.985247i	$0.554745\pi$
$348$	0	0
$349$	−0.970720	−0.0519614	−0.0259807	−	0.999662i	$-0.508271\pi$
$349$	−0.970720	−0.0519614	−0.0259807	+	0.999662i	$0.508271\pi$
$350$	−1.66560	−0.0890301
$351$	0	0
$352$	−1.93110	−0.102928
$353$	−30.5488	−1.62595	−0.812975	−	0.582299i	$-0.802153\pi$
$353$	−30.5488	−1.62595	−0.812975	+	0.582299i	$0.802153\pi$
$354$	0	0
$355$	−13.6286	−0.723330
$356$	57.7511	3.06080
$357$	0	0
$358$	59.5457	3.14709
$359$	−8.22124	−0.433901	−0.216950	−	0.976183i	$-0.569611\pi$
$359$	−8.22124	−0.433901	−0.216950	+	0.976183i	$0.569611\pi$
$360$	0	0
$361$	−14.4435	−0.760184
$362$	28.9033	1.51912
$363$	0	0
$364$	−0.0707871	−0.00371025
$365$	17.7866	0.930994
$366$	0	0
$367$	25.1108	1.31077	0.655387	−	0.755294i	$-0.272506\pi$
$367$	25.1108	1.31077	0.655387	+	0.755294i	$0.272506\pi$
$368$	−35.0030	−1.82466
$369$	0	0
$370$	−45.2232	−2.35104
$371$	−0.532051	−0.0276227
$372$	0	0
$373$	−27.0231	−1.39920	−0.699601	−	0.714534i	$-0.746639\pi$
$373$	−27.0231	−1.39920	−0.699601	+	0.714534i	$0.746639\pi$
$374$	2.84209	0.146961
$375$	0	0
$376$	71.2722	3.67558
$377$	0.664131	0.0342045
$378$	0	0
$379$	22.9961	1.18123	0.590616	−	0.806953i	$-0.298885\pi$
$379$	22.9961	1.18123	0.590616	+	0.806953i	$0.298885\pi$
$380$	−28.2516	−1.44928
$381$	0	0
$382$	−12.1032	−0.619252
$383$	−3.72670	−0.190425	−0.0952126	−	0.995457i	$-0.530353\pi$
$383$	−3.72670	−0.190425	−0.0952126	+	0.995457i	$0.530353\pi$
$384$	0	0
$385$	0.388015	0.0197751
$386$	−63.8749	−3.25115
$387$	0	0
$388$	57.2563	2.90675
$389$	−4.72525	−0.239580	−0.119790	−	0.992799i	$-0.538222\pi$
$389$	−4.72525	−0.239580	−0.119790	+	0.992799i	$0.538222\pi$
$390$	0	0
$391$	8.38100	0.423845
$392$	38.3202	1.93546
$393$	0	0
$394$	3.66111	0.184444
$395$	4.51769	0.227310
$396$	0	0
$397$	32.2227	1.61721	0.808606	−	0.588350i	$-0.200222\pi$
$397$	32.2227	1.61721	0.808606	+	0.588350i	$0.200222\pi$
$398$	−18.5986	−0.932264
$399$	0	0
$400$	25.8591	1.29295
$401$	6.02370	0.300809	0.150405	−	0.988625i	$-0.451942\pi$
$401$	6.02370	0.300809	0.150405	+	0.988625i	$0.451942\pi$
$402$	0	0
$403$	1.21042	0.0602955
$404$	0.331110	0.0164733
$405$	0	0
$406$	1.82020	0.0903349
$407$	5.22113	0.258802
$408$	0	0
$409$	18.5405	0.916767	0.458384	−	0.888754i	$-0.348429\pi$
$409$	18.5405	0.916767	0.458384	+	0.888754i	$0.348429\pi$
$410$	50.4737	2.49272
$411$	0	0
$412$	−26.7651	−1.31862
$413$	−1.10951	−0.0545956
$414$	0	0
$415$	37.0334	1.81790
$416$	0.263858	0.0129367
$417$	0	0
$418$	4.81355	0.235438
$419$	−29.7394	−1.45286	−0.726432	−	0.687238i	$-0.758823\pi$
$419$	−29.7394	−1.45286	−0.726432	+	0.687238i	$0.758823\pi$
$420$	0	0
$421$	2.65516	0.129405	0.0647023	−	0.997905i	$-0.479390\pi$
$421$	2.65516	0.129405	0.0647023	+	0.997905i	$0.479390\pi$
$422$	50.2002	2.44371
$423$	0	0
$424$	21.4538	1.04189
$425$	−6.19161	−0.300337
$426$	0	0
$427$	−1.95624	−0.0946691
$428$	8.38937	0.405515
$429$	0	0
$430$	24.9931	1.20527
$431$	−27.9512	−1.34636	−0.673181	−	0.739478i	$-0.735073\pi$
$431$	−27.9512	−1.34636	−0.673181	+	0.739478i	$0.735073\pi$
$432$	0	0
$433$	13.5815	0.652687	0.326344	−	0.945251i	$-0.394183\pi$
$433$	13.5815	0.652687	0.326344	+	0.945251i	$0.394183\pi$
$434$	3.31743	0.159242
$435$	0	0
$436$	62.3089	2.98405
$437$	14.1946	0.679019
$438$	0	0
$439$	27.3113	1.30350	0.651748	−	0.758435i	$-0.274036\pi$
$439$	27.3113	1.30350	0.651748	+	0.758435i	$0.274036\pi$
$440$	−15.6459	−0.745887
$441$	0	0
$442$	−0.388334	−0.0184712
$443$	−4.06001	−0.192897	−0.0964484	−	0.995338i	$-0.530748\pi$
$443$	−4.06001	−0.192897	−0.0964484	+	0.995338i	$0.530748\pi$
$444$	0	0
$445$	43.2535	2.05041
$446$	22.2003	1.05121
$447$	0	0
$448$	−0.709876	−0.0335385
$449$	1.53571	0.0724746	0.0362373	−	0.999343i	$-0.488463\pi$
$449$	1.53571	0.0724746	0.0362373	+	0.999343i	$0.488463\pi$
$450$	0	0
$451$	−5.82731	−0.274397
$452$	51.9869	2.44526
$453$	0	0
$454$	−43.0211	−2.01908
$455$	−0.0530171	−0.00248548
$456$	0	0
$457$	−4.50705	−0.210831	−0.105415	−	0.994428i	$-0.533617\pi$
$457$	−4.50705	−0.210831	−0.105415	+	0.994428i	$0.533617\pi$
$458$	32.2279	1.50591
$459$	0	0
$460$	−88.0106	−4.10352
$461$	−17.5820	−0.818875	−0.409438	−	0.912338i	$-0.634275\pi$
$461$	−17.5820	−0.818875	−0.409438	+	0.912338i	$0.634275\pi$
$462$	0	0
$463$	−42.1082	−1.95693	−0.978466	−	0.206407i	$-0.933823\pi$
$463$	−42.1082	−1.95693	−0.978466	+	0.206407i	$0.933823\pi$
$464$	−28.2592	−1.31190
$465$	0	0
$466$	−61.8451	−2.86492
$467$	−10.6368	−0.492210	−0.246105	−	0.969243i	$-0.579151\pi$
$467$	−10.6368	−0.492210	−0.246105	+	0.969243i	$0.579151\pi$
$468$	0	0
$469$	−0.666367	−0.0307699
$470$	101.826	4.69689
$471$	0	0
$472$	44.7388	2.05927
$473$	−2.88551	−0.132676
$474$	0	0
$475$	−10.4865	−0.481154
$476$	−0.721193	−0.0330558
$477$	0	0
$478$	18.7333	0.856841
$479$	−24.3303	−1.11168	−0.555839	−	0.831290i	$-0.687603\pi$
$479$	−24.3303	−1.11168	−0.555839	+	0.831290i	$0.687603\pi$
$480$	0	0
$481$	−0.713398	−0.0325282
$482$	−2.49073	−0.113449
$483$	0	0
$484$	−42.7951	−1.94523
$485$	42.8830	1.94722
$486$	0	0
$487$	15.1777	0.687768	0.343884	−	0.939012i	$-0.388257\pi$
$487$	15.1777	0.687768	0.343884	+	0.939012i	$0.388257\pi$
$488$	78.8812	3.57079
$489$	0	0
$490$	54.7479	2.47326
$491$	−10.0194	−0.452169	−0.226084	−	0.974108i	$-0.572592\pi$
$491$	−10.0194	−0.452169	−0.226084	+	0.974108i	$0.572592\pi$
$492$	0	0
$493$	6.76630	0.304739
$494$	−0.657707	−0.0295916
$495$	0	0
$496$	−51.5044	−2.31261
$497$	−0.589231	−0.0264306
$498$	0	0
$499$	−7.82989	−0.350514	−0.175257	−	0.984523i	$-0.556076\pi$
$499$	−7.82989	−0.350514	−0.175257	+	0.984523i	$0.556076\pi$
$500$	−1.15621	−0.0517075
$501$	0	0
$502$	−74.7911	−3.33809
$503$	7.39455	0.329707	0.164853	−	0.986318i	$-0.447285\pi$
$503$	7.39455	0.329707	0.164853	+	0.986318i	$0.447285\pi$
$504$	0	0
$505$	0.247990	0.0110354
$506$	14.9954	0.666625
$507$	0	0
$508$	67.1241	2.97815
$509$	−13.7526	−0.609572	−0.304786	−	0.952421i	$-0.598585\pi$
$509$	−13.7526	−0.609572	−0.304786	+	0.952421i	$0.598585\pi$
$510$	0	0
$511$	0.769003	0.0340187
$512$	46.5568	2.05754
$513$	0	0
$514$	−24.7443	−1.09143
$515$	−20.0461	−0.883337
$516$	0	0
$517$	−11.7561	−0.517032
$518$	−1.95523	−0.0859077
$519$	0	0
$520$	2.13780	0.0937487
$521$	14.2825	0.625730	0.312865	−	0.949798i	$-0.398711\pi$
$521$	14.2825	0.625730	0.312865	+	0.949798i	$0.398711\pi$
$522$	0	0
$523$	−29.4674	−1.28852	−0.644260	−	0.764806i	$-0.722835\pi$
$523$	−29.4674	−1.28852	−0.644260	+	0.764806i	$0.722835\pi$
$524$	51.0287	2.22920
$525$	0	0
$526$	45.9726	2.00450
$527$	12.3320	0.537192
$528$	0	0
$529$	21.2195	0.922589
$530$	30.6509	1.33139
$531$	0	0
$532$	−1.22146	−0.0529569
$533$	0.796224	0.0344883
$534$	0	0
$535$	6.28334	0.271653
$536$	26.8698	1.16060
$537$	0	0
$538$	−2.86189	−0.123385
$539$	−6.32078	−0.272255
$540$	0	0
$541$	31.6583	1.36110	0.680549	−	0.732703i	$-0.261741\pi$
$541$	31.6583	1.36110	0.680549	+	0.732703i	$0.261741\pi$
$542$	−30.3953	−1.30559
$543$	0	0
$544$	2.68824	0.115257
$545$	46.6672	1.99900
$546$	0	0
$547$	−31.5186	−1.34764	−0.673820	−	0.738896i	$-0.735348\pi$
$547$	−31.5186	−1.34764	−0.673820	+	0.738896i	$0.735348\pi$
$548$	−3.55656	−0.151929
$549$	0	0
$550$	−11.0781	−0.472371
$551$	11.4598	0.488205
$552$	0	0
$553$	0.195322	0.00830595
$554$	10.7432	0.456434
$555$	0	0
$556$	65.7762	2.78953
$557$	32.3589	1.37109	0.685545	−	0.728030i	$-0.259564\pi$
$557$	32.3589	1.37109	0.685545	+	0.728030i	$0.259564\pi$
$558$	0	0
$559$	0.394267	0.0166757
$560$	2.25591	0.0953298
$561$	0	0
$562$	−42.6907	−1.80080
$563$	−20.8428	−0.878419	−0.439209	−	0.898385i	$-0.644741\pi$
$563$	−20.8428	−0.878419	−0.439209	+	0.898385i	$0.644741\pi$
$564$	0	0
$565$	38.9364	1.63807
$566$	−34.5974	−1.45424
$567$	0	0
$568$	23.7595	0.996926
$569$	−26.6787	−1.11843	−0.559215	−	0.829022i	$-0.688897\pi$
$569$	−26.6787	−1.11843	−0.559215	+	0.829022i	$0.688897\pi$
$570$	0	0
$571$	0.500659	0.0209519	0.0104760	−	0.999945i	$-0.496665\pi$
$571$	0.500659	0.0209519	0.0104760	+	0.999945i	$0.496665\pi$
$572$	−0.470813	−0.0196857
$573$	0	0
$574$	2.18223	0.0910844
$575$	−32.6680	−1.36235
$576$	0	0
$577$	−9.59281	−0.399354	−0.199677	−	0.979862i	$-0.563989\pi$
$577$	−9.59281	−0.399354	−0.199677	+	0.979862i	$0.563989\pi$
$578$	38.3859	1.59664
$579$	0	0
$580$	−71.0543	−2.95037
$581$	1.60114	0.0664263
$582$	0	0
$583$	−3.53873	−0.146559
$584$	−31.0084	−1.28314
$585$	0	0
$586$	−0.809710	−0.0334488
$587$	−19.6676	−0.811770	−0.405885	−	0.913924i	$-0.633037\pi$
$587$	−19.6676	−0.811770	−0.405885	+	0.913924i	$0.633037\pi$
$588$	0	0
$589$	20.8863	0.860605
$590$	63.9181	2.63146
$591$	0	0
$592$	30.3556	1.24761
$593$	−35.9622	−1.47679	−0.738394	−	0.674369i	$-0.764416\pi$
$593$	−35.9622	−1.47679	−0.738394	+	0.674369i	$0.764416\pi$
$594$	0	0
$595$	−0.540148	−0.0221439
$596$	2.27931	0.0933640
$597$	0	0
$598$	−2.04892	−0.0837864
$599$	8.60123	0.351437	0.175718	−	0.984440i	$-0.443775\pi$
$599$	8.60123	0.351437	0.175718	+	0.984440i	$0.443775\pi$
$600$	0	0
$601$	36.8567	1.50342	0.751708	−	0.659496i	$-0.229230\pi$
$601$	36.8567	1.50342	0.751708	+	0.659496i	$0.229230\pi$
$602$	1.08057	0.0440409
$603$	0	0
$604$	35.8596	1.45911
$605$	−32.0521	−1.30310
$606$	0	0
$607$	−38.6665	−1.56942	−0.784712	−	0.619861i	$-0.787189\pi$
$607$	−38.6665	−1.56942	−0.784712	+	0.619861i	$0.787189\pi$
$608$	4.55298	0.184648
$609$	0	0
$610$	112.697	4.56298
$611$	1.60631	0.0649845
$612$	0	0
$613$	−38.3341	−1.54830	−0.774150	−	0.633002i	$-0.781822\pi$
$613$	−38.3341	−1.54830	−0.774150	+	0.633002i	$0.781822\pi$
$614$	35.7667	1.44343
$615$	0	0
$616$	−0.676449	−0.0272549
$617$	44.3829	1.78679	0.893395	−	0.449272i	$-0.148317\pi$
$617$	44.3829	1.78679	0.893395	+	0.449272i	$0.148317\pi$
$618$	0	0
$619$	−15.6150	−0.627619	−0.313810	−	0.949486i	$-0.601605\pi$
$619$	−15.6150	−0.627619	−0.313810	+	0.949486i	$0.601605\pi$
$620$	−129.501	−5.20089
$621$	0	0
$622$	−27.9141	−1.11925
$623$	1.87007	0.0749226
$624$	0	0
$625$	−25.4292	−1.01717
$626$	−48.2851	−1.92986
$627$	0	0
$628$	−68.1858	−2.72091
$629$	−7.26825	−0.289804
$630$	0	0
$631$	−7.52472	−0.299555	−0.149777	−	0.988720i	$-0.547856\pi$
$631$	−7.52472	−0.299555	−0.149777	+	0.988720i	$0.547856\pi$
$632$	−7.87595	−0.313288
$633$	0	0
$634$	12.9413	0.513963
$635$	50.2736	1.99505
$636$	0	0
$637$	0.863650	0.0342191
$638$	12.1063	0.479294
$639$	0	0
$640$	54.3262	2.14743
$641$	−2.42509	−0.0957851	−0.0478926	−	0.998852i	$-0.515251\pi$
$641$	−2.42509	−0.0957851	−0.0478926	+	0.998852i	$0.515251\pi$
$642$	0	0
$643$	12.1224	0.478061	0.239030	−	0.971012i	$-0.423170\pi$
$643$	12.1224	0.478061	0.239030	+	0.971012i	$0.423170\pi$
$644$	−3.80514	−0.149943
$645$	0	0
$646$	−6.70085	−0.263642
$647$	−12.4628	−0.489964	−0.244982	−	0.969528i	$-0.578782\pi$
$647$	−12.4628	−0.489964	−0.244982	+	0.969528i	$0.578782\pi$
$648$	0	0
$649$	−7.37950	−0.289671
$650$	1.51367	0.0593711
$651$	0	0
$652$	5.73231	0.224495
$653$	49.4695	1.93589	0.967945	−	0.251163i	$-0.0808131\pi$
$653$	49.4695	1.93589	0.967945	+	0.251163i	$0.0808131\pi$
$654$	0	0
$655$	38.2187	1.49333
$656$	−33.8799	−1.32279
$657$	0	0
$658$	4.40245	0.171626
$659$	−19.5067	−0.759874	−0.379937	−	0.925012i	$-0.624054\pi$
$659$	−19.5067	−0.759874	−0.379937	+	0.925012i	$0.624054\pi$
$660$	0	0
$661$	33.2398	1.29288	0.646439	−	0.762966i	$-0.276258\pi$
$661$	33.2398	1.29288	0.646439	+	0.762966i	$0.276258\pi$
$662$	−0.0431518	−0.00167714
$663$	0	0
$664$	−64.5624	−2.50551
$665$	−0.914829	−0.0354756
$666$	0	0
$667$	35.7001	1.38231
$668$	−56.3487	−2.18020
$669$	0	0
$670$	38.3887	1.48309
$671$	−13.0112	−0.502291
$672$	0	0
$673$	28.8764	1.11310	0.556552	−	0.830813i	$-0.312124\pi$
$673$	28.8764	1.11310	0.556552	+	0.830813i	$0.312124\pi$
$674$	−0.866521	−0.0333771
$675$	0	0
$676$	−54.5840	−2.09938
$677$	−29.4480	−1.13178	−0.565889	−	0.824482i	$-0.691467\pi$
$677$	−29.4480	−1.13178	−0.565889	+	0.824482i	$0.691467\pi$
$678$	0	0
$679$	1.85404	0.0711517
$680$	21.7803	0.835237
$681$	0	0
$682$	22.0646	0.844897
$683$	−19.6959	−0.753643	−0.376821	−	0.926286i	$-0.622983\pi$
$683$	−19.6959	−0.753643	−0.376821	+	0.926286i	$0.622983\pi$
$684$	0	0
$685$	−2.66374	−0.101776
$686$	4.74033	0.180987
$687$	0	0
$688$	−16.7763	−0.639592
$689$	0.483520	0.0184206
$690$	0	0
$691$	15.6587	0.595683	0.297842	−	0.954615i	$-0.403733\pi$
$691$	15.6587	0.595683	0.297842	+	0.954615i	$0.403733\pi$
$692$	57.1735	2.17341
$693$	0	0
$694$	15.8809	0.602830
$695$	49.2641	1.86869
$696$	0	0
$697$	8.11209	0.307267
$698$	2.41780	0.0915149
$699$	0	0
$700$	2.81111	0.106250
$701$	−32.6726	−1.23403	−0.617014	−	0.786952i	$-0.711658\pi$
$701$	−32.6726	−1.23403	−0.617014	+	0.786952i	$0.711658\pi$
$702$	0	0
$703$	−12.3100	−0.464279
$704$	−4.72146	−0.177947
$705$	0	0
$706$	76.0887	2.86364
$707$	0.0107218	0.000403236 0
$708$	0	0
$709$	−5.63690	−0.211698	−0.105849	−	0.994382i	$-0.533756\pi$
$709$	−5.63690	−0.211698	−0.105849	+	0.994382i	$0.533756\pi$
$710$	33.9451	1.27394
$711$	0	0
$712$	−75.4064	−2.82597
$713$	65.0659	2.43674
$714$	0	0
$715$	−0.352622	−0.0131873
$716$	−100.498	−3.75579
$717$	0	0
$718$	20.4769	0.764190
$719$	−13.2196	−0.493006	−0.246503	−	0.969142i	$-0.579282\pi$
$719$	−13.2196	−0.493006	−0.246503	+	0.969142i	$0.579282\pi$
$720$	0	0
$721$	−0.866693	−0.0322773
$722$	35.9748	1.33884
$723$	0	0
$724$	−48.7814	−1.81295
$725$	−26.3741	−0.979509
$726$	0	0
$727$	21.6517	0.803018	0.401509	−	0.915855i	$-0.368486\pi$
$727$	21.6517	0.803018	0.401509	+	0.915855i	$0.368486\pi$
$728$	0.0924277	0.00342560
$729$	0	0
$730$	−44.3016	−1.63967
$731$	4.01687	0.148569
$732$	0	0
$733$	−37.5385	−1.38651	−0.693257	−	0.720690i	$-0.743825\pi$
$733$	−37.5385	−1.38651	−0.693257	+	0.720690i	$0.743825\pi$
$734$	−62.5441	−2.30855
$735$	0	0
$736$	14.1836	0.522815
$737$	−4.43208	−0.163258
$738$	0	0
$739$	−28.8751	−1.06219	−0.531093	−	0.847313i	$-0.678219\pi$
$739$	−28.8751	−1.06219	−0.531093	+	0.847313i	$0.678219\pi$
$740$	76.3254	2.80578
$741$	0	0
$742$	1.32519	0.0486493
$743$	24.5272	0.899817	0.449909	−	0.893075i	$-0.351457\pi$
$743$	24.5272	0.899817	0.449909	+	0.893075i	$0.351457\pi$
$744$	0	0
$745$	1.70712	0.0625440
$746$	67.3070	2.46429
$747$	0	0
$748$	−4.79673	−0.175386
$749$	0.271660	0.00992624
$750$	0	0
$751$	38.3355	1.39888	0.699442	−	0.714689i	$-0.253432\pi$
$751$	38.3355	1.39888	0.699442	+	0.714689i	$0.253432\pi$
$752$	−68.3498	−2.49246
$753$	0	0
$754$	−1.65417	−0.0602412
$755$	26.8576	0.977448
$756$	0	0
$757$	31.6239	1.14939	0.574694	−	0.818368i	$-0.305121\pi$
$757$	31.6239	1.14939	0.574694	+	0.818368i	$0.305121\pi$
$758$	−57.2771	−2.08040
$759$	0	0
$760$	36.8885	1.33809
$761$	28.4924	1.03285	0.516424	−	0.856333i	$-0.327263\pi$
$761$	28.4924	1.03285	0.516424	+	0.856333i	$0.327263\pi$
$762$	0	0
$763$	2.01765	0.0730440
$764$	20.4271	0.739026
$765$	0	0
$766$	9.28218	0.335379
$767$	1.00831	0.0364080
$768$	0	0
$769$	−8.72245	−0.314539	−0.157270	−	0.987556i	$-0.550269\pi$
$769$	−8.72245	−0.314539	−0.157270	+	0.987556i	$0.550269\pi$
$770$	−0.966439	−0.0348280
$771$	0	0
$772$	107.805	3.87997
$773$	42.2655	1.52019	0.760093	−	0.649814i	$-0.225153\pi$
$773$	42.2655	1.52019	0.760093	+	0.649814i	$0.225153\pi$
$774$	0	0
$775$	−48.0685	−1.72667
$776$	−74.7604	−2.68374
$777$	0	0
$778$	11.7693	0.421950
$779$	13.7392	0.492256
$780$	0	0
$781$	−3.91904	−0.140234
$782$	−20.8748	−0.746480
$783$	0	0
$784$	−36.7490	−1.31246
$785$	−51.0687	−1.82272
$786$	0	0
$787$	17.5127	0.624261	0.312130	−	0.950039i	$-0.398957\pi$
$787$	17.5127	0.624261	0.312130	+	0.950039i	$0.398957\pi$
$788$	−6.17903	−0.220119
$789$	0	0
$790$	−11.2523	−0.400340
$791$	1.68341	0.0598553
$792$	0	0
$793$	1.77780	0.0631316
$794$	−80.2580	−2.84825
$795$	0	0
$796$	31.3897	1.11258
$797$	44.6851	1.58283	0.791413	−	0.611281i	$-0.209346\pi$
$797$	44.6851	1.58283	0.791413	+	0.611281i	$0.209346\pi$
$798$	0	0
$799$	16.3654	0.578968
$800$	−10.4784	−0.370467
$801$	0	0
$802$	−15.0034	−0.529788
$803$	5.11472	0.180495
$804$	0	0
$805$	−2.84991	−0.100446
$806$	−3.01483	−0.106193
$807$	0	0
$808$	−0.432335	−0.0152095
$809$	−37.7898	−1.32862	−0.664309	−	0.747458i	$-0.731274\pi$
$809$	−37.7898	−1.32862	−0.664309	+	0.747458i	$0.731274\pi$
$810$	0	0
$811$	17.5453	0.616097	0.308049	−	0.951371i	$-0.400324\pi$
$811$	17.5453	0.616097	0.308049	+	0.951371i	$0.400324\pi$
$812$	−3.07203	−0.107807
$813$	0	0
$814$	−13.0044	−0.455804
$815$	4.29330	0.150388
$816$	0	0
$817$	6.80322	0.238015
$818$	−46.1792	−1.61462
$819$	0	0
$820$	−85.1868	−2.97485
$821$	21.1741	0.738980	0.369490	−	0.929235i	$-0.379532\pi$
$821$	21.1741	0.738980	0.369490	+	0.929235i	$0.379532\pi$
$822$	0	0
$823$	−47.1268	−1.64274	−0.821369	−	0.570397i	$-0.806789\pi$
$823$	−47.1268	−1.64274	−0.821369	+	0.570397i	$0.806789\pi$
$824$	34.9475	1.21746
$825$	0	0
$826$	2.76350	0.0961543
$827$	13.3177	0.463103	0.231552	−	0.972823i	$-0.425620\pi$
$827$	13.3177	0.463103	0.231552	+	0.972823i	$0.425620\pi$
$828$	0	0
$829$	−41.6063	−1.44505	−0.722523	−	0.691347i	$-0.757017\pi$
$829$	−41.6063	−1.44505	−0.722523	+	0.691347i	$0.757017\pi$
$830$	−92.2400	−3.20170
$831$	0	0
$832$	0.645125	0.0223657
$833$	8.79904	0.304869
$834$	0	0
$835$	−42.2032	−1.46050
$836$	−8.12405	−0.280976
$837$	0	0
$838$	74.0726	2.55880
$839$	−20.2863	−0.700359	−0.350180	−	0.936683i	$-0.613879\pi$
$839$	−20.2863	−0.700359	−0.350180	+	0.936683i	$0.613879\pi$
$840$	0	0
$841$	−0.177928	−0.00613546
$842$	−6.61327	−0.227908
$843$	0	0
$844$	−84.7252	−2.91636
$845$	−40.8815	−1.40637
$846$	0	0
$847$	−1.38577	−0.0476156
$848$	−20.5741	−0.706518
$849$	0	0
$850$	15.4216	0.528957
$851$	−38.3485	−1.31457
$852$	0	0
$853$	−26.7153	−0.914715	−0.457358	−	0.889283i	$-0.651204\pi$
$853$	−26.7153	−0.914715	−0.457358	+	0.889283i	$0.651204\pi$
$854$	4.87246	0.166732
$855$	0	0
$856$	−10.9541	−0.374404
$857$	1.26642	0.0432602	0.0216301	−	0.999766i	$-0.493114\pi$
$857$	1.26642	0.0432602	0.0216301	+	0.999766i	$0.493114\pi$
$858$	0	0
$859$	−45.0089	−1.53569	−0.767843	−	0.640638i	$-0.778670\pi$
$859$	−45.0089	−1.53569	−0.767843	+	0.640638i	$0.778670\pi$
$860$	−42.1820	−1.43839
$861$	0	0
$862$	69.6188	2.37122
$863$	−26.1118	−0.888855	−0.444428	−	0.895815i	$-0.646593\pi$
$863$	−26.1118	−0.888855	−0.444428	+	0.895815i	$0.646593\pi$
$864$	0	0
$865$	42.8210	1.45596
$866$	−33.8279	−1.14952
$867$	0	0
$868$	−5.59898	−0.190042
$869$	1.29911	0.0440693
$870$	0	0
$871$	0.605584	0.0205194
$872$	−81.3576	−2.75511
$873$	0	0
$874$	−35.3548	−1.19589
$875$	−0.0374399	−0.00126570
$876$	0	0
$877$	18.2107	0.614931	0.307466	−	0.951559i	$-0.400519\pi$
$877$	18.2107	0.614931	0.307466	+	0.951559i	$0.400519\pi$
$878$	−68.0249	−2.29573
$879$	0	0
$880$	15.0043	0.505796
$881$	−7.40050	−0.249329	−0.124665	−	0.992199i	$-0.539786\pi$
$881$	−7.40050	−0.249329	−0.124665	+	0.992199i	$0.539786\pi$
$882$	0	0
$883$	−1.57266	−0.0529243	−0.0264621	−	0.999650i	$-0.508424\pi$
$883$	−1.57266	−0.0529243	−0.0264621	+	0.999650i	$0.508424\pi$
$884$	0.655410	0.0220438
$885$	0	0
$886$	10.1124	0.339732
$887$	1.08367	0.0363861	0.0181930	−	0.999834i	$-0.494209\pi$
$887$	1.08367	0.0363861	0.0181930	+	0.999834i	$0.494209\pi$
$888$	0	0
$889$	2.17358	0.0728994
$890$	−107.733	−3.61121
$891$	0	0
$892$	−37.4684	−1.25453
$893$	27.7175	0.927532
$894$	0	0
$895$	−75.2695	−2.51598
$896$	2.34879	0.0784676
$897$	0	0
$898$	−3.82503	−0.127643
$899$	52.5301	1.75198
$900$	0	0
$901$	4.92620	0.164115
$902$	14.5142	0.483271
$903$	0	0
$904$	−67.8801	−2.25766
$905$	−36.5356	−1.21448
$906$	0	0
$907$	39.1868	1.30118	0.650588	−	0.759431i	$-0.274522\pi$
$907$	39.1868	1.30118	0.650588	+	0.759431i	$0.274522\pi$
$908$	72.6088	2.40961
$909$	0	0
$910$	0.132051	0.00437745
$911$	−11.8818	−0.393662	−0.196831	−	0.980437i	$-0.563065\pi$
$911$	−11.8818	−0.393662	−0.196831	+	0.980437i	$0.563065\pi$
$912$	0	0
$913$	10.6493	0.352441
$914$	11.2258	0.371317
$915$	0	0
$916$	−54.3925	−1.79718
$917$	1.65239	0.0545666
$918$	0	0
$919$	−28.1347	−0.928077	−0.464039	−	0.885815i	$-0.653600\pi$
$919$	−28.1347	−0.928077	−0.464039	+	0.885815i	$0.653600\pi$
$920$	114.917	3.78869
$921$	0	0
$922$	43.7919	1.44221
$923$	0.535485	0.0176257
$924$	0	0
$925$	28.3306	0.931505
$926$	104.880	3.44657
$927$	0	0
$928$	11.4510	0.375897
$929$	−59.5689	−1.95439	−0.977197	−	0.212335i	$-0.931893\pi$
$929$	−59.5689	−1.95439	−0.977197	+	0.212335i	$0.931893\pi$
$930$	0	0
$931$	14.9026	0.488413
$932$	104.379	3.41904
$933$	0	0
$934$	26.4932	0.866885
$935$	−3.59259	−0.117490
$936$	0	0
$937$	25.8722	0.845209	0.422605	−	0.906314i	$-0.361116\pi$
$937$	25.8722	0.845209	0.422605	+	0.906314i	$0.361116\pi$
$938$	1.65974	0.0541923
$939$	0	0
$940$	−171.857	−5.60535
$941$	43.2143	1.40875	0.704373	−	0.709830i	$-0.251228\pi$
$941$	43.2143	1.40875	0.704373	+	0.709830i	$0.251228\pi$
$942$	0	0
$943$	42.8008	1.39378
$944$	−42.9043	−1.39642
$945$	0	0
$946$	7.18702	0.233670
$947$	53.0980	1.72545	0.862726	−	0.505672i	$-0.168755\pi$
$947$	53.0980	1.72545	0.862726	+	0.505672i	$0.168755\pi$
$948$	0	0
$949$	−0.698859	−0.0226859
$950$	26.1190	0.847412
$951$	0	0
$952$	0.941672	0.0305198
$953$	28.8397	0.934210	0.467105	−	0.884202i	$-0.345297\pi$
$953$	28.8397	0.934210	0.467105	+	0.884202i	$0.345297\pi$
$954$	0	0
$955$	15.2992	0.495069
$956$	−31.6171	−1.02257
$957$	0	0
$958$	60.6000	1.95790
$959$	−0.115167	−0.00371893
$960$	0	0
$961$	64.7397	2.08838
$962$	1.77688	0.0572889
$963$	0	0
$964$	4.20371	0.135393
$965$	80.7419	2.59917
$966$	0	0
$967$	6.08196	0.195583	0.0977914	−	0.995207i	$-0.468822\pi$
$967$	6.08196	0.195583	0.0977914	+	0.995207i	$0.468822\pi$
$968$	55.8782	1.79599
$969$	0	0
$970$	−106.810	−3.42945
$971$	−2.57155	−0.0825251	−0.0412626	−	0.999148i	$-0.513138\pi$
$971$	−2.57155	−0.0825251	−0.0412626	+	0.999148i	$0.513138\pi$
$972$	0	0
$973$	2.12993	0.0682824
$974$	−37.8035	−1.21130
$975$	0	0
$976$	−75.6468	−2.42139
$977$	−52.2605	−1.67196	−0.835980	−	0.548760i	$-0.815100\pi$
$977$	−52.2605	−1.67196	−0.835980	+	0.548760i	$0.815100\pi$
$978$	0	0
$979$	12.4380	0.397520
$980$	−92.4006	−2.95163
$981$	0	0
$982$	24.9555	0.796363
$983$	−5.96528	−0.190263	−0.0951314	−	0.995465i	$-0.530327\pi$
$983$	−5.96528	−0.190263	−0.0951314	+	0.995465i	$0.530327\pi$
$984$	0	0
$985$	−4.62787	−0.147456
$986$	−16.8530	−0.536709
$987$	0	0
$988$	1.11004	0.0353152
$989$	21.1937	0.673920
$990$	0	0
$991$	1.36771	0.0434466	0.0217233	−	0.999764i	$-0.493085\pi$
$991$	1.36771	0.0434466	0.0217233	+	0.999764i	$0.493085\pi$
$992$	20.8702	0.662629
$993$	0	0
$994$	1.46761	0.0465499
$995$	23.5098	0.745311
$996$	0	0
$997$	−17.2016	−0.544780	−0.272390	−	0.962187i	$-0.587814\pi$
$997$	−17.2016	−0.544780	−0.272390	+	0.962187i	$0.587814\pi$
$998$	19.5021	0.617329
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.2.a.h.1.2		12
3.2	odd	2		241.2.a.b.1.11	✓	12
12.11	even	2		3856.2.a.n.1.5		12
15.14	odd	2		6025.2.a.h.1.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.11	✓	12	3.2	odd	2
2169.2.a.h.1.2		12	1.1	even	1	trivial
3856.2.a.n.1.5		12	12.11	even	2
6025.2.a.h.1.2		12	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2169 = 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2169.a (trivial)

\(f(q)\)	\(=\)	\(q-2.49073 q^{2} +4.20371 q^{4} +3.14843 q^{5} +0.136122 q^{7} -5.48885 q^{8} +O(q^{10})\)	\(q-2.49073 q^{2} +4.20371 q^{4} +3.14843 q^{5} +0.136122 q^{7} -5.48885 q^{8} -7.84189 q^{10} +0.905365 q^{11} -0.123706 q^{13} -0.339044 q^{14} +5.26378 q^{16} -1.26034 q^{17} -2.13460 q^{19} +13.2351 q^{20} -2.25502 q^{22} -6.64978 q^{23} +4.91264 q^{25} +0.308118 q^{26} +0.572220 q^{28} -5.36862 q^{29} -9.78467 q^{31} -2.13295 q^{32} +3.13917 q^{34} +0.428573 q^{35} +5.76688 q^{37} +5.31669 q^{38} -17.2813 q^{40} -6.43642 q^{41} -3.18712 q^{43} +3.80590 q^{44} +16.5628 q^{46} -12.9849 q^{47} -6.98147 q^{49} -12.2360 q^{50} -0.520025 q^{52} -3.90862 q^{53} +2.85048 q^{55} -0.747155 q^{56} +13.3718 q^{58} -8.15085 q^{59} -14.3712 q^{61} +24.3709 q^{62} -5.21498 q^{64} -0.389481 q^{65} -4.89534 q^{67} -5.29812 q^{68} -1.06746 q^{70} -4.32869 q^{71} +5.64935 q^{73} -14.3637 q^{74} -8.97323 q^{76} +0.123241 q^{77} +1.43490 q^{79} +16.5727 q^{80} +16.0314 q^{82} +11.7625 q^{83} -3.96811 q^{85} +7.93825 q^{86} -4.96941 q^{88} +13.7381 q^{89} -0.0168392 q^{91} -27.9538 q^{92} +32.3419 q^{94} -6.72063 q^{95} +13.6204 q^{97} +17.3889 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8} - 7 q^{10} - 22 q^{11} - 5 q^{13} - 6 q^{14} + 15 q^{16} + 4 q^{17} - 6 q^{19} - 10 q^{20} - 12 q^{22} - 32 q^{23} + 4 q^{25} - 8 q^{26} - 11 q^{28} - 6 q^{29} + 8 q^{31} - q^{32} - 19 q^{34} - 15 q^{35} - 8 q^{37} + 10 q^{38} - 52 q^{40} + q^{41} - 2 q^{43} - 42 q^{44} - 25 q^{46} - 34 q^{47} - 9 q^{49} + 27 q^{50} - 41 q^{52} - 5 q^{53} - 3 q^{55} - q^{56} - 33 q^{58} - 26 q^{59} - 26 q^{61} + 17 q^{62} + 13 q^{64} + 25 q^{65} + 6 q^{67} + 35 q^{68} - 4 q^{70} - 94 q^{71} - 22 q^{73} - 26 q^{74} - 20 q^{76} + 7 q^{77} + 9 q^{79} - 19 q^{80} + 15 q^{82} + 8 q^{83} + 4 q^{85} - 9 q^{86} + 6 q^{88} + 3 q^{89} - 20 q^{91} - 36 q^{92} + 48 q^{94} - 33 q^{95} - 29 q^{97} - 28 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8 - 7 * q^10 - 22 * q^11 - 5 * q^13 - 6 * q^14 + 15 * q^16 + 4 * q^17 - 6 * q^19 - 10 * q^20 - 12 * q^22 - 32 * q^23 + 4 * q^25 - 8 * q^26 - 11 * q^28 - 6 * q^29 + 8 * q^31 - q^32 - 19 * q^34 - 15 * q^35 - 8 * q^37 + 10 * q^38 - 52 * q^40 + q^41 - 2 * q^43 - 42 * q^44 - 25 * q^46 - 34 * q^47 - 9 * q^49 + 27 * q^50 - 41 * q^52 - 5 * q^53 - 3 * q^55 - q^56 - 33 * q^58 - 26 * q^59 - 26 * q^61 + 17 * q^62 + 13 * q^64 + 25 * q^65 + 6 * q^67 + 35 * q^68 - 4 * q^70 - 94 * q^71 - 22 * q^73 - 26 * q^74 - 20 * q^76 + 7 * q^77 + 9 * q^79 - 19 * q^80 + 15 * q^82 + 8 * q^83 + 4 * q^85 - 9 * q^86 + 6 * q^88 + 3 * q^89 - 20 * q^91 - 36 * q^92 + 48 * q^94 - 33 * q^95 - 29 * q^97 - 28 * q^98

Introduction

L-functions

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Database

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