LMFDB - Embedded newform 2254.4.a.l.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2254,4,Mod(1,2254)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2254, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2254.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.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:	$132.990305153$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 92x^{3} - 28x^{2} + 1593x - 1782 $ x^5 - 92x^3 - 28x^2 + 1593*x - 1782
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.26012$ of defining polynomial
Character	$\chi$	$=$	2254.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} +0.260120 q^{3} +4.00000 q^{4} -15.2484 q^{5} +0.520240 q^{6} +8.00000 q^{8} -26.9323 q^{9} -30.4969 q^{10} +63.1379 q^{11} +1.04048 q^{12} -40.8864 q^{13} -3.96643 q^{15} +16.0000 q^{16} +57.0717 q^{17} -53.8647 q^{18} -34.4579 q^{19} -60.9938 q^{20} +126.276 q^{22} +23.0000 q^{23} +2.08096 q^{24} +107.515 q^{25} -81.7727 q^{26} -14.0289 q^{27} +180.951 q^{29} -7.93285 q^{30} +5.92805 q^{31} +32.0000 q^{32} +16.4234 q^{33} +114.143 q^{34} -107.729 q^{36} -161.864 q^{37} -68.9158 q^{38} -10.6354 q^{39} -121.988 q^{40} +104.432 q^{41} +16.5725 q^{43} +252.552 q^{44} +410.676 q^{45} +46.0000 q^{46} -333.754 q^{47} +4.16192 q^{48} +215.030 q^{50} +14.8455 q^{51} -163.545 q^{52} +296.103 q^{53} -28.0578 q^{54} -962.756 q^{55} -8.96319 q^{57} +361.902 q^{58} +805.644 q^{59} -15.8657 q^{60} -715.603 q^{61} +11.8561 q^{62} +64.0000 q^{64} +623.454 q^{65} +32.8469 q^{66} +301.232 q^{67} +228.287 q^{68} +5.98276 q^{69} -871.046 q^{71} -215.459 q^{72} -894.771 q^{73} -323.728 q^{74} +27.9668 q^{75} -137.832 q^{76} -21.2707 q^{78} -1246.19 q^{79} -243.975 q^{80} +723.524 q^{81} +208.863 q^{82} +1223.93 q^{83} -870.255 q^{85} +33.1451 q^{86} +47.0690 q^{87} +505.104 q^{88} +86.3579 q^{89} +821.353 q^{90} +92.0000 q^{92} +1.54200 q^{93} -667.507 q^{94} +525.429 q^{95} +8.32384 q^{96} -343.939 q^{97} -1700.45 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 10 q^{2} - 5 q^{3} + 20 q^{4} - 22 q^{5} - 10 q^{6} + 40 q^{8} + 54 q^{9} - 44 q^{10} + 42 q^{11} - 20 q^{12} - 107 q^{13} + 122 q^{15} + 80 q^{16} - 218 q^{17} + 108 q^{18} - 194 q^{19} - 88 q^{20}+ \cdots - 2616 q^{99}+O(q^{100}) $ 5 * q + 10 * q^2 - 5 * q^3 + 20 * q^4 - 22 * q^5 - 10 * q^6 + 40 * q^8 + 54 * q^9 - 44 * q^10 + 42 * q^11 - 20 * q^12 - 107 * q^13 + 122 * q^15 + 80 * q^16 - 218 * q^17 + 108 * q^18 - 194 * q^19 - 88 * q^20 + 84 * q^22 + 115 * q^23 - 40 * q^24 + 219 * q^25 - 214 * q^26 - 203 * q^27 + 33 * q^29 + 244 * q^30 - 19 * q^31 + 160 * q^32 - 240 * q^33 - 436 * q^34 + 216 * q^36 + 214 * q^37 - 388 * q^38 - 243 * q^39 - 176 * q^40 - 395 * q^41 - 284 * q^43 + 168 * q^44 - 470 * q^45 + 230 * q^46 + 493 * q^47 - 80 * q^48 + 438 * q^50 - 244 * q^51 - 428 * q^52 - 70 * q^53 - 406 * q^54 + 186 * q^55 + 22 * q^57 + 66 * q^58 - 226 * q^59 + 488 * q^60 - 1606 * q^61 - 38 * q^62 + 320 * q^64 + 362 * q^65 - 480 * q^66 + 298 * q^67 - 872 * q^68 - 115 * q^69 - 969 * q^71 + 432 * q^72 - 697 * q^73 + 428 * q^74 - 1065 * q^75 - 776 * q^76 - 486 * q^78 + 186 * q^79 - 352 * q^80 - 335 * q^81 - 790 * q^82 + 1104 * q^83 - 1344 * q^85 - 568 * q^86 + 1603 * q^87 + 336 * q^88 + 814 * q^89 - 940 * q^90 + 460 * q^92 - 3423 * q^93 + 986 * q^94 - 2812 * q^95 - 160 * q^96 + 256 * q^97 - 2616 * q^99

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0.260120	0.0500601	0.0250301	−	0.999687i	$-0.492032\pi$
$3$	0.260120	0.0500601	0.0250301	+	0.999687i	$0.492032\pi$
$4$	4.00000	0.500000
$5$	−15.2484	−1.36386	−0.681931	−	0.731416i	$-0.738860\pi$
$5$	−15.2484	−1.36386	−0.681931	+	0.731416i	$0.738860\pi$
$6$	0.520240	0.0353978
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−26.9323	−0.997494
$10$	−30.4969	−0.964397
$11$	63.1379	1.73062	0.865309	−	0.501238i	$-0.167122\pi$
$11$	63.1379	1.73062	0.865309	+	0.501238i	$0.167122\pi$
$12$	1.04048	0.0250301
$13$	−40.8864	−0.872295	−0.436148	−	0.899875i	$-0.643657\pi$
$13$	−40.8864	−0.872295	−0.436148	+	0.899875i	$0.643657\pi$
$14$	0	0
$15$	−3.96643	−0.0682751
$16$	16.0000	0.250000
$17$	57.0717	0.814231	0.407115	−	0.913377i	$-0.366535\pi$
$17$	57.0717	0.814231	0.407115	+	0.913377i	$0.366535\pi$
$18$	−53.8647	−0.705335
$19$	−34.4579	−0.416062	−0.208031	−	0.978122i	$-0.566706\pi$
$19$	−34.4579	−0.416062	−0.208031	+	0.978122i	$0.566706\pi$
$20$	−60.9938	−0.681931
$21$	0	0
$22$	126.276	1.22373
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	2.08096	0.0176989
$25$	107.515	0.860121
$26$	−81.7727	−0.616806
$27$	−14.0289	−0.0999948
$28$	0	0
$29$	180.951	1.15868	0.579340	−	0.815086i	$-0.303310\pi$
$29$	180.951	1.15868	0.579340	+	0.815086i	$0.303310\pi$
$30$	−7.93285	−0.0482778
$31$	5.92805	0.0343455	0.0171727	−	0.999853i	$-0.494533\pi$
$31$	5.92805	0.0343455	0.0171727	+	0.999853i	$0.494533\pi$
$32$	32.0000	0.176777
$33$	16.4234	0.0866350
$34$	114.143	0.575748
$35$	0	0
$36$	−107.729	−0.498747
$37$	−161.864	−0.719196	−0.359598	−	0.933107i	$-0.617086\pi$
$37$	−161.864	−0.719196	−0.359598	+	0.933107i	$0.617086\pi$
$38$	−68.9158	−0.294200
$39$	−10.6354	−0.0436672
$40$	−121.988	−0.482198
$41$	104.432	0.397792	0.198896	−	0.980021i	$-0.436264\pi$
$41$	104.432	0.397792	0.198896	+	0.980021i	$0.436264\pi$
$42$	0	0
$43$	16.5725	0.0587742	0.0293871	−	0.999568i	$-0.490644\pi$
$43$	16.5725	0.0587742	0.0293871	+	0.999568i	$0.490644\pi$
$44$	252.552	0.865309
$45$	410.676	1.36044
$46$	46.0000	0.147442
$47$	−333.754	−1.03581	−0.517904	−	0.855439i	$-0.673287\pi$
$47$	−333.754	−1.03581	−0.517904	+	0.855439i	$0.673287\pi$
$48$	4.16192	0.0125150
$49$	0	0
$50$	215.030	0.608198
$51$	14.8455	0.0407605
$52$	−163.545	−0.436148
$53$	296.103	0.767412	0.383706	−	0.923455i	$-0.374648\pi$
$53$	296.103	0.767412	0.383706	+	0.923455i	$0.374648\pi$
$54$	−28.0578	−0.0707070
$55$	−962.756	−2.36033
$56$	0	0
$57$	−8.96319	−0.0208281
$58$	361.902	0.819311
$59$	805.644	1.77773	0.888864	−	0.458171i	$-0.151495\pi$
$59$	805.644	1.77773	0.888864	+	0.458171i	$0.151495\pi$
$60$	−15.8657	−0.0341376
$61$	−715.603	−1.50202	−0.751012	−	0.660288i	$-0.770434\pi$
$61$	−715.603	−1.50202	−0.751012	+	0.660288i	$0.770434\pi$
$62$	11.8561	0.0242859
$63$	0	0
$64$	64.0000	0.125000
$65$	623.454	1.18969
$66$	32.8469	0.0612602
$67$	301.232	0.549275	0.274637	−	0.961548i	$-0.411442\pi$
$67$	301.232	0.549275	0.274637	+	0.961548i	$0.411442\pi$
$68$	228.287	0.407115
$69$	5.98276	0.0104383
$70$	0	0
$71$	−871.046	−1.45597	−0.727987	−	0.685591i	$-0.759544\pi$
$71$	−871.046	−1.45597	−0.727987	+	0.685591i	$0.759544\pi$
$72$	−215.459	−0.352667
$73$	−894.771	−1.43459	−0.717295	−	0.696770i	$-0.754620\pi$
$73$	−894.771	−1.43459	−0.717295	+	0.696770i	$0.754620\pi$
$74$	−323.728	−0.508548
$75$	27.9668	0.0430578
$76$	−137.832	−0.208031
$77$	0	0
$78$	−21.2707	−0.0308774
$79$	−1246.19	−1.77477	−0.887386	−	0.461028i	$-0.847481\pi$
$79$	−1246.19	−1.77477	−0.887386	+	0.461028i	$0.847481\pi$
$80$	−243.975	−0.340966
$81$	723.524	0.992488
$82$	208.863	0.281281
$83$	1223.93	1.61860	0.809300	−	0.587395i	$-0.199846\pi$
$83$	1223.93	1.61860	0.809300	+	0.587395i	$0.199846\pi$
$84$	0	0
$85$	−870.255	−1.11050
$86$	33.1451	0.0415596
$87$	47.0690	0.0580037
$88$	505.104	0.611866
$89$	86.3579	0.102853	0.0514265	−	0.998677i	$-0.483623\pi$
$89$	86.3579	0.102853	0.0514265	+	0.998677i	$0.483623\pi$
$90$	821.353	0.961980
$91$	0	0
$92$	92.0000	0.104257
$93$	1.54200	0.00171934
$94$	−667.507	−0.732426
$95$	525.429	0.567452
$96$	8.32384	0.00884946
$97$	−343.939	−0.360018	−0.180009	−	0.983665i	$-0.557613\pi$
$97$	−343.939	−0.360018	−0.180009	+	0.983665i	$0.557613\pi$
$98$	0	0
$99$	−1700.45	−1.72628
$100$	430.061	0.430061
$101$	−458.505	−0.451713	−0.225856	−	0.974161i	$-0.572518\pi$
$101$	−458.505	−0.451713	−0.225856	+	0.974161i	$0.572518\pi$
$102$	29.6910	0.0288220
$103$	574.886	0.549954	0.274977	−	0.961451i	$-0.411330\pi$
$103$	574.886	0.549954	0.274977	+	0.961451i	$0.411330\pi$
$104$	−327.091	−0.308403
$105$	0	0
$106$	592.205	0.542642
$107$	−1451.76	−1.31165	−0.655825	−	0.754913i	$-0.727679\pi$
$107$	−1451.76	−1.31165	−0.655825	+	0.754913i	$0.727679\pi$
$108$	−56.1155	−0.0499974
$109$	−1812.89	−1.59306	−0.796530	−	0.604598i	$-0.793334\pi$
$109$	−1812.89	−1.59306	−0.796530	+	0.604598i	$0.793334\pi$
$110$	−1925.51	−1.66900
$111$	−42.1040	−0.0360030
$112$	0	0
$113$	−1693.79	−1.41007	−0.705035	−	0.709173i	$-0.749069\pi$
$113$	−1693.79	−1.41007	−0.705035	+	0.709173i	$0.749069\pi$
$114$	−17.9264	−0.0147277
$115$	−350.714	−0.284385
$116$	723.804	0.579340
$117$	1101.17	0.870109
$118$	1611.29	1.25704
$119$	0	0
$120$	−31.7314	−0.0241389
$121$	2655.40	1.99504
$122$	−1431.21	−1.06209
$123$	27.1647	0.0199135
$124$	23.7122	0.0171727
$125$	266.616	0.190775
$126$	0	0
$127$	251.950	0.176039	0.0880194	−	0.996119i	$-0.471946\pi$
$127$	251.950	0.176039	0.0880194	+	0.996119i	$0.471946\pi$
$128$	128.000	0.0883883
$129$	4.31085	0.00294224
$130$	1246.91	0.841238
$131$	1844.44	1.23015	0.615074	−	0.788469i	$-0.289126\pi$
$131$	1844.44	1.23015	0.615074	+	0.788469i	$0.289126\pi$
$132$	65.6938	0.0433175
$133$	0	0
$134$	602.465	0.388396
$135$	213.919	0.136379
$136$	456.574	0.287874
$137$	−1169.93	−0.729592	−0.364796	−	0.931087i	$-0.618861\pi$
$137$	−1169.93	−0.729592	−0.364796	+	0.931087i	$0.618861\pi$
$138$	11.9655	0.00738096
$139$	1618.44	0.987585	0.493792	−	0.869580i	$-0.335610\pi$
$139$	1618.44	0.987585	0.493792	+	0.869580i	$0.335610\pi$
$140$	0	0
$141$	−86.8160	−0.0518526
$142$	−1742.09	−1.02953
$143$	−2581.48	−1.50961
$144$	−430.917	−0.249373
$145$	−2759.22	−1.58028
$146$	−1789.54	−1.01441
$147$	0	0
$148$	−647.455	−0.359598
$149$	−1745.63	−0.959785	−0.479892	−	0.877327i	$-0.659324\pi$
$149$	−1745.63	−0.959785	−0.479892	+	0.877327i	$0.659324\pi$
$150$	55.9337	0.0304464
$151$	−2520.41	−1.35833	−0.679167	−	0.733984i	$-0.737659\pi$
$151$	−2520.41	−1.35833	−0.679167	+	0.733984i	$0.737659\pi$
$152$	−275.663	−0.147100
$153$	−1537.07	−0.812190
$154$	0	0
$155$	−90.3936	−0.0468425
$156$	−42.5414	−0.0218336
$157$	−1122.56	−0.570635	−0.285318	−	0.958433i	$-0.592099\pi$
$157$	−1122.56	−0.570635	−0.285318	+	0.958433i	$0.592099\pi$
$158$	−2492.37	−1.25495
$159$	77.0222	0.0384167
$160$	−487.950	−0.241099
$161$	0	0
$162$	1447.05	0.701795
$163$	149.438	0.0718091	0.0359045	−	0.999355i	$-0.488569\pi$
$163$	149.438	0.0718091	0.0359045	+	0.999355i	$0.488569\pi$
$164$	417.726	0.198896
$165$	−250.432	−0.118158
$166$	2447.86	1.14452
$167$	−1950.78	−0.903927	−0.451963	−	0.892037i	$-0.649276\pi$
$167$	−1950.78	−0.903927	−0.451963	+	0.892037i	$0.649276\pi$
$168$	0	0
$169$	−525.306	−0.239101
$170$	−1740.51	−0.785241
$171$	928.032	0.415020
$172$	66.2902	0.0293871
$173$	−2850.79	−1.25284	−0.626420	−	0.779486i	$-0.715480\pi$
$173$	−2850.79	−1.25284	−0.626420	+	0.779486i	$0.715480\pi$
$174$	94.1379	0.0410148
$175$	0	0
$176$	1010.21	0.432655
$177$	209.564	0.0889933
$178$	172.716	0.0727280
$179$	1847.62	0.771493	0.385747	−	0.922605i	$-0.373944\pi$
$179$	1847.62	0.771493	0.385747	+	0.922605i	$0.373944\pi$
$180$	1642.71	0.680222
$181$	−3260.42	−1.33892	−0.669461	−	0.742847i	$-0.733475\pi$
$181$	−3260.42	−1.33892	−0.669461	+	0.742847i	$0.733475\pi$
$182$	0	0
$183$	−186.143	−0.0751915
$184$	184.000	0.0737210
$185$	2468.17	0.980884
$186$	3.08401	0.00121576
$187$	3603.39	1.40912
$188$	−1335.01	−0.517904
$189$	0	0
$190$	1050.86	0.401249
$191$	1616.13	0.612247	0.306123	−	0.951992i	$-0.400968\pi$
$191$	1616.13	0.612247	0.306123	+	0.951992i	$0.400968\pi$
$192$	16.6477	0.00625751
$193$	−4746.87	−1.77040	−0.885200	−	0.465211i	$-0.845979\pi$
$193$	−4746.87	−1.77040	−0.885200	+	0.465211i	$0.845979\pi$
$194$	−687.878	−0.254571
$195$	162.173	0.0595560
$196$	0	0
$197$	−1471.77	−0.532280	−0.266140	−	0.963934i	$-0.585748\pi$
$197$	−1471.77	−0.532280	−0.266140	+	0.963934i	$0.585748\pi$
$198$	−3400.90	−1.22067
$199$	−3296.74	−1.17437	−0.587184	−	0.809453i	$-0.699764\pi$
$199$	−3296.74	−1.17437	−0.587184	+	0.809453i	$0.699764\pi$
$200$	860.121	0.304099
$201$	78.3566	0.0274967
$202$	−917.011	−0.319409
$203$	0	0
$204$	59.3820	0.0203802
$205$	−1592.42	−0.542533
$206$	1149.77	0.388876
$207$	−619.444	−0.207992
$208$	−654.182	−0.218074
$209$	−2175.60	−0.720045
$210$	0	0
$211$	−2436.09	−0.794821	−0.397410	−	0.917641i	$-0.630091\pi$
$211$	−2436.09	−0.794821	−0.397410	+	0.917641i	$0.630091\pi$
$212$	1184.41	0.383706
$213$	−226.577	−0.0728862
$214$	−2903.51	−0.927477
$215$	−252.706	−0.0801599
$216$	−112.231	−0.0353535
$217$	0	0
$218$	−3625.79	−1.12646
$219$	−232.748	−0.0718157
$220$	−3851.02	−1.18016
$221$	−2333.45	−0.710250
$222$	−84.2080	−0.0254580
$223$	−1464.12	−0.439662	−0.219831	−	0.975538i	$-0.570551\pi$
$223$	−1464.12	−0.439662	−0.219831	+	0.975538i	$0.570551\pi$
$224$	0	0
$225$	−2895.64	−0.857966
$226$	−3387.57	−0.997070
$227$	3419.39	0.999794	0.499897	−	0.866085i	$-0.333371\pi$
$227$	3419.39	0.999794	0.499897	+	0.866085i	$0.333371\pi$
$228$	−35.8527	−0.0104141
$229$	3437.40	0.991922	0.495961	−	0.868345i	$-0.334816\pi$
$229$	3437.40	0.991922	0.495961	+	0.868345i	$0.334816\pi$
$230$	−701.429	−0.201091
$231$	0	0
$232$	1447.61	0.409656
$233$	3049.48	0.857415	0.428708	−	0.903443i	$-0.358969\pi$
$233$	3049.48	0.857415	0.428708	+	0.903443i	$0.358969\pi$
$234$	2202.33	0.615260
$235$	5089.22	1.41270
$236$	3222.58	0.888864
$237$	−324.158	−0.0888453
$238$	0	0
$239$	1961.10	0.530767	0.265383	−	0.964143i	$-0.414501\pi$
$239$	1961.10	0.530767	0.265383	+	0.964143i	$0.414501\pi$
$240$	−63.4628	−0.0170688
$241$	−6121.70	−1.63624	−0.818119	−	0.575049i	$-0.804983\pi$
$241$	−6121.70	−1.63624	−0.818119	+	0.575049i	$0.804983\pi$
$242$	5310.80	1.41071
$243$	566.983	0.149679
$244$	−2862.41	−0.751012
$245$	0	0
$246$	54.3295	0.0140810
$247$	1408.86	0.362929
$248$	47.4244	0.0121430
$249$	318.369	0.0810273
$250$	533.233	0.134898
$251$	−4112.95	−1.03429	−0.517145	−	0.855898i	$-0.673005\pi$
$251$	−4112.95	−1.03429	−0.517145	+	0.855898i	$0.673005\pi$
$252$	0	0
$253$	1452.17	0.360859
$254$	503.899	0.124478
$255$	−226.371	−0.0555917
$256$	256.000	0.0625000
$257$	2135.52	0.518328	0.259164	−	0.965833i	$-0.416553\pi$
$257$	2135.52	0.518328	0.259164	+	0.965833i	$0.416553\pi$
$258$	8.62170	0.00208048
$259$	0	0
$260$	2493.81	0.594845
$261$	−4873.43	−1.15578
$262$	3688.88	0.869846
$263$	7005.80	1.64257	0.821286	−	0.570517i	$-0.193257\pi$
$263$	7005.80	1.64257	0.821286	+	0.570517i	$0.193257\pi$
$264$	131.388	0.0306301
$265$	−4515.10	−1.04664
$266$	0	0
$267$	22.4634	0.00514883
$268$	1204.93	0.274637
$269$	−6566.34	−1.48831	−0.744157	−	0.668004i	$-0.767149\pi$
$269$	−6566.34	−1.48831	−0.744157	+	0.668004i	$0.767149\pi$
$270$	427.837	0.0964346
$271$	5759.43	1.29100	0.645499	−	0.763761i	$-0.276649\pi$
$271$	5759.43	1.29100	0.645499	+	0.763761i	$0.276649\pi$
$272$	913.147	0.203558
$273$	0	0
$274$	−2339.87	−0.515900
$275$	6788.29	1.48854
$276$	23.9310	0.00521913
$277$	356.886	0.0774123	0.0387061	−	0.999251i	$-0.487676\pi$
$277$	356.886	0.0774123	0.0387061	+	0.999251i	$0.487676\pi$
$278$	3236.88	0.698328
$279$	−159.656	−0.0342594
$280$	0	0
$281$	−925.745	−0.196531	−0.0982657	−	0.995160i	$-0.531329\pi$
$281$	−925.745	−0.196531	−0.0982657	+	0.995160i	$0.531329\pi$
$282$	−173.632	−0.0366654
$283$	−6928.18	−1.45526	−0.727629	−	0.685971i	$-0.759378\pi$
$283$	−6928.18	−1.45526	−0.727629	+	0.685971i	$0.759378\pi$
$284$	−3484.19	−0.727987
$285$	136.675	0.0284067
$286$	−5162.96	−1.06746
$287$	0	0
$288$	−861.835	−0.176334
$289$	−1655.82	−0.337028
$290$	−5518.44	−1.11743
$291$	−89.4655	−0.0180225
$292$	−3579.08	−0.717295
$293$	−148.190	−0.0295472	−0.0147736	−	0.999891i	$-0.504703\pi$
$293$	−148.190	−0.0295472	−0.0147736	+	0.999891i	$0.504703\pi$
$294$	0	0
$295$	−12284.8	−2.42458
$296$	−1294.91	−0.254274
$297$	−885.755	−0.173053
$298$	−3491.27	−0.678670
$299$	−940.386	−0.181886
$300$	111.867	0.0215289
$301$	0	0
$302$	−5040.83	−0.960487
$303$	−119.266	−0.0226128
$304$	−551.326	−0.104016
$305$	10911.8	2.04856
$306$	−3074.15	−0.574305
$307$	10096.8	1.87705	0.938527	−	0.345207i	$-0.112191\pi$
$307$	10096.8	1.87705	0.938527	+	0.345207i	$0.112191\pi$
$308$	0	0
$309$	149.539	0.0275308
$310$	−180.787	−0.0331227
$311$	1063.81	0.193965	0.0969826	−	0.995286i	$-0.469081\pi$
$311$	1063.81	0.193965	0.0969826	+	0.995286i	$0.469081\pi$
$312$	−85.0829	−0.0154387
$313$	−1953.88	−0.352844	−0.176422	−	0.984315i	$-0.556452\pi$
$313$	−1953.88	−0.352844	−0.176422	+	0.984315i	$0.556452\pi$
$314$	−2245.11	−0.403500
$315$	0	0
$316$	−4984.75	−0.887386
$317$	−2939.51	−0.520819	−0.260409	−	0.965498i	$-0.583858\pi$
$317$	−2939.51	−0.520819	−0.260409	+	0.965498i	$0.583858\pi$
$318$	154.044	0.0271647
$319$	11424.9	2.00523
$320$	−975.901	−0.170483
$321$	−377.631	−0.0656614
$322$	0	0
$323$	−1966.57	−0.338771
$324$	2894.10	0.496244
$325$	−4395.90	−0.750280
$326$	298.876	0.0507767
$327$	−471.570	−0.0797488
$328$	835.452	0.140641
$329$	0	0
$330$	−500.864	−0.0835505
$331$	−8808.46	−1.46271	−0.731354	−	0.681998i	$-0.761111\pi$
$331$	−8808.46	−1.46271	−0.731354	+	0.681998i	$0.761111\pi$
$332$	4895.72	0.809300
$333$	4359.37	0.717393
$334$	−3901.56	−0.639173
$335$	−4593.33	−0.749135
$336$	0	0
$337$	3966.73	0.641191	0.320596	−	0.947216i	$-0.396117\pi$
$337$	3966.73	0.641191	0.320596	+	0.947216i	$0.396117\pi$
$338$	−1050.61	−0.169070
$339$	−440.587	−0.0705883
$340$	−3481.02	−0.555250
$341$	374.285	0.0594389
$342$	1856.06	0.293463
$343$	0	0
$344$	132.580	0.0207798
$345$	−91.2278	−0.0142363
$346$	−5701.57	−0.885891
$347$	5747.24	0.889129	0.444565	−	0.895747i	$-0.353358\pi$
$347$	5747.24	0.889129	0.444565	+	0.895747i	$0.353358\pi$
$348$	188.276	0.0290018
$349$	−8367.21	−1.28334	−0.641671	−	0.766980i	$-0.721759\pi$
$349$	−8367.21	−1.28334	−0.641671	+	0.766980i	$0.721759\pi$
$350$	0	0
$351$	573.590	0.0872249
$352$	2020.41	0.305933
$353$	−4056.10	−0.611571	−0.305785	−	0.952100i	$-0.598919\pi$
$353$	−4056.10	−0.611571	−0.305785	+	0.952100i	$0.598919\pi$
$354$	419.128	0.0629278
$355$	13282.1	1.98575
$356$	345.431	0.0514265
$357$	0	0
$358$	3695.23	0.545528
$359$	−2580.13	−0.379315	−0.189658	−	0.981850i	$-0.560738\pi$
$359$	−2580.13	−0.379315	−0.189658	+	0.981850i	$0.560738\pi$
$360$	3285.41	0.480990
$361$	−5671.65	−0.826892
$362$	−6520.83	−0.946761
$363$	690.722	0.0998720
$364$	0	0
$365$	13643.9	1.95658
$366$	−372.285	−0.0531684
$367$	−938.253	−0.133451	−0.0667254	−	0.997771i	$-0.521255\pi$
$367$	−938.253	−0.133451	−0.0667254	+	0.997771i	$0.521255\pi$
$368$	368.000	0.0521286
$369$	−2812.59	−0.396795
$370$	4936.34	0.693590
$371$	0	0
$372$	6.16802	0.000859669 0
$373$	10951.6	1.52025	0.760125	−	0.649777i	$-0.225138\pi$
$373$	10951.6	1.52025	0.760125	+	0.649777i	$0.225138\pi$
$374$	7206.78	0.996401
$375$	69.3522	0.00955023
$376$	−2670.03	−0.366213
$377$	−7398.42	−1.01071
$378$	0	0
$379$	9520.76	1.29037	0.645183	−	0.764028i	$-0.276781\pi$
$379$	9520.76	1.29037	0.645183	+	0.764028i	$0.276781\pi$
$380$	2101.72	0.283726
$381$	65.5371	0.00881252
$382$	3232.26	0.432924
$383$	3099.25	0.413484	0.206742	−	0.978396i	$-0.433714\pi$
$383$	3099.25	0.413484	0.206742	+	0.978396i	$0.433714\pi$
$384$	33.2954	0.00442473
$385$	0	0
$386$	−9493.74	−1.25186
$387$	−446.337	−0.0586269
$388$	−1375.76	−0.180009
$389$	4368.35	0.569368	0.284684	−	0.958621i	$-0.408111\pi$
$389$	4368.35	0.569368	0.284684	+	0.958621i	$0.408111\pi$
$390$	324.345	0.0421125
$391$	1312.65	0.169779
$392$	0	0
$393$	479.775	0.0615813
$394$	−2943.54	−0.376379
$395$	19002.4	2.42054
$396$	−6801.81	−0.863141
$397$	−6306.06	−0.797210	−0.398605	−	0.917123i	$-0.630505\pi$
$397$	−6306.06	−0.797210	−0.398605	+	0.917123i	$0.630505\pi$
$398$	−6593.47	−0.830404
$399$	0	0
$400$	1720.24	0.215030
$401$	−83.0874	−0.0103471	−0.00517355	−	0.999987i	$-0.501647\pi$
$401$	−83.0874	−0.0103471	−0.00517355	+	0.999987i	$0.501647\pi$
$402$	156.713	0.0194431
$403$	−242.376	−0.0299594
$404$	−1834.02	−0.225856
$405$	−11032.6	−1.35362
$406$	0	0
$407$	−10219.7	−1.24465
$408$	118.764	0.0144110
$409$	−1594.33	−0.192750	−0.0963749	−	0.995345i	$-0.530725\pi$
$409$	−1594.33	−0.192750	−0.0963749	+	0.995345i	$0.530725\pi$
$410$	−3184.84	−0.383629
$411$	−304.323	−0.0365235
$412$	2299.55	0.274977
$413$	0	0
$414$	−1238.89	−0.147072
$415$	−18663.0	−2.20755
$416$	−1308.36	−0.154201
$417$	420.988	0.0494386
$418$	−4351.20	−0.509149
$419$	3260.67	0.380177	0.190089	−	0.981767i	$-0.439122\pi$
$419$	3260.67	0.380177	0.190089	+	0.981767i	$0.439122\pi$
$420$	0	0
$421$	−7104.66	−0.822471	−0.411235	−	0.911529i	$-0.634903\pi$
$421$	−7104.66	−0.822471	−0.411235	+	0.911529i	$0.634903\pi$
$422$	−4872.18	−0.562023
$423$	8988.76	1.03321
$424$	2368.82	0.271321
$425$	6136.08	0.700337
$426$	−453.153	−0.0515384
$427$	0	0
$428$	−5807.02	−0.655825
$429$	−671.495	−0.0755712
$430$	−505.411	−0.0566816
$431$	10760.4	1.20258	0.601290	−	0.799031i	$-0.294654\pi$
$431$	10760.4	1.20258	0.601290	+	0.799031i	$0.294654\pi$
$432$	−224.462	−0.0249987
$433$	11537.4	1.28049	0.640245	−	0.768171i	$-0.278833\pi$
$433$	11537.4	1.28049	0.640245	+	0.768171i	$0.278833\pi$
$434$	0	0
$435$	−717.729	−0.0791091
$436$	−7251.57	−0.796530
$437$	−792.532	−0.0867550
$438$	−465.496	−0.0507814
$439$	10785.2	1.17255	0.586277	−	0.810111i	$-0.300593\pi$
$439$	10785.2	1.17255	0.586277	+	0.810111i	$0.300593\pi$
$440$	−7702.05	−0.834501
$441$	0	0
$442$	−4666.91	−0.502222
$443$	2483.46	0.266349	0.133175	−	0.991093i	$-0.457483\pi$
$443$	2483.46	0.266349	0.133175	+	0.991093i	$0.457483\pi$
$444$	−168.416	−0.0180015
$445$	−1316.82	−0.140277
$446$	−2928.24	−0.310888
$447$	−454.074	−0.0480469
$448$	0	0
$449$	−3010.79	−0.316454	−0.158227	−	0.987403i	$-0.550578\pi$
$449$	−3010.79	−0.316454	−0.158227	+	0.987403i	$0.550578\pi$
$450$	−5791.27	−0.606674
$451$	6593.59	0.688426
$452$	−6775.14	−0.705035
$453$	−655.610	−0.0679983
$454$	6838.79	0.706961
$455$	0	0
$456$	−71.7055	−0.00736385
$457$	−12906.5	−1.32110	−0.660549	−	0.750783i	$-0.729677\pi$
$457$	−12906.5	−1.32110	−0.660549	+	0.750783i	$0.729677\pi$
$458$	6874.81	0.701395
$459$	−800.652	−0.0814188
$460$	−1402.86	−0.142193
$461$	−1928.01	−0.194786	−0.0973931	−	0.995246i	$-0.531050\pi$
$461$	−1928.01	−0.194786	−0.0973931	+	0.995246i	$0.531050\pi$
$462$	0	0
$463$	2880.38	0.289121	0.144560	−	0.989496i	$-0.453823\pi$
$463$	2880.38	0.289121	0.144560	+	0.989496i	$0.453823\pi$
$464$	2895.22	0.289670
$465$	−23.5132	−0.00234494
$466$	6098.95	0.606284
$467$	−6861.47	−0.679895	−0.339947	−	0.940444i	$-0.610409\pi$
$467$	−6861.47	−0.679895	−0.339947	+	0.940444i	$0.610409\pi$
$468$	4404.66	0.435055
$469$	0	0
$470$	10178.4	0.998929
$471$	−291.999	−0.0285661
$472$	6445.16	0.628522
$473$	1046.36	0.101716
$474$	−648.316	−0.0628231
$475$	−3704.75	−0.357864
$476$	0	0
$477$	−7974.73	−0.765488
$478$	3922.21	0.375309
$479$	4580.05	0.436885	0.218443	−	0.975850i	$-0.429902\pi$
$479$	4580.05	0.436885	0.218443	+	0.975850i	$0.429902\pi$
$480$	−126.926	−0.0120695
$481$	6618.02	0.627351
$482$	−12243.4	−1.15700
$483$	0	0
$484$	10621.6	0.997521
$485$	5244.54	0.491015
$486$	1133.97	0.105839
$487$	8844.29	0.822943	0.411471	−	0.911423i	$-0.365015\pi$
$487$	8844.29	0.822943	0.411471	+	0.911423i	$0.365015\pi$
$488$	−5724.82	−0.531046
$489$	38.8718	0.00359477
$490$	0	0
$491$	−4687.22	−0.430817	−0.215408	−	0.976524i	$-0.569108\pi$
$491$	−4687.22	−0.430817	−0.215408	+	0.976524i	$0.569108\pi$
$492$	108.659	0.00995675
$493$	10327.2	0.943434
$494$	2817.72	0.256630
$495$	25929.3	2.35441
$496$	94.8488	0.00858637
$497$	0	0
$498$	636.738	0.0572950
$499$	−10245.1	−0.919107	−0.459553	−	0.888150i	$-0.651991\pi$
$499$	−10245.1	−0.919107	−0.459553	+	0.888150i	$0.651991\pi$
$500$	1066.47	0.0953876
$501$	−507.436	−0.0452507
$502$	−8225.89	−0.731354
$503$	10020.9	0.888286	0.444143	−	0.895956i	$-0.353508\pi$
$503$	10020.9	0.888286	0.444143	+	0.895956i	$0.353508\pi$
$504$	0	0
$505$	6991.50	0.616074
$506$	2904.35	0.255166
$507$	−136.643	−0.0119694
$508$	1007.80	0.0880194
$509$	15666.6	1.36427	0.682133	−	0.731228i	$-0.261053\pi$
$509$	15666.6	1.36427	0.682133	+	0.731228i	$0.261053\pi$
$510$	−452.741	−0.0393093
$511$	0	0
$512$	512.000	0.0441942
$513$	483.406	0.0416041
$514$	4271.05	0.366513
$515$	−8766.13	−0.750062
$516$	17.2434	0.00147112
$517$	−21072.5	−1.79259
$518$	0	0
$519$	−741.546	−0.0627173
$520$	4987.63	0.420619
$521$	−12571.8	−1.05716	−0.528581	−	0.848883i	$-0.677276\pi$
$521$	−12571.8	−1.05716	−0.528581	+	0.848883i	$0.677276\pi$
$522$	−9746.86	−0.817258
$523$	−12266.4	−1.02557	−0.512785	−	0.858517i	$-0.671386\pi$
$523$	−12266.4	−1.02557	−0.512785	+	0.858517i	$0.671386\pi$
$524$	7377.76	0.615074
$525$	0	0
$526$	14011.6	1.16147
$527$	338.324	0.0279651
$528$	262.775	0.0216587
$529$	529.000	0.0434783
$530$	−9030.21	−0.740089
$531$	−21697.9	−1.77327
$532$	0	0
$533$	−4269.83	−0.346992
$534$	44.9268	0.00364077
$535$	22137.0	1.78891
$536$	2409.86	0.194198
$537$	480.602	0.0386210
$538$	−13132.7	−1.05240
$539$	0	0
$540$	855.675	0.0681896
$541$	−17141.8	−1.36226	−0.681131	−	0.732161i	$-0.738512\pi$
$541$	−17141.8	−1.36226	−0.681131	+	0.732161i	$0.738512\pi$
$542$	11518.9	0.912874
$543$	−848.100	−0.0670266
$544$	1826.29	0.143937
$545$	27643.8	2.17272
$546$	0	0
$547$	−13619.5	−1.06459	−0.532294	−	0.846560i	$-0.678670\pi$
$547$	−13619.5	−1.06459	−0.532294	+	0.846560i	$0.678670\pi$
$548$	−4679.73	−0.364796
$549$	19272.8	1.49826
$550$	13576.6	1.05256
$551$	−6235.19	−0.482083
$552$	47.8621	0.00369048
$553$	0	0
$554$	713.772	0.0547387
$555$	642.021	0.0491032
$556$	6473.76	0.493792
$557$	−14755.1	−1.12243	−0.561214	−	0.827671i	$-0.689666\pi$
$557$	−14755.1	−1.12243	−0.561214	+	0.827671i	$0.689666\pi$
$558$	−319.313	−0.0242251
$559$	−677.591	−0.0512684
$560$	0	0
$561$	937.314	0.0705409
$562$	−1851.49	−0.138969
$563$	−21018.7	−1.57341	−0.786706	−	0.617327i	$-0.788215\pi$
$563$	−21018.7	−1.57341	−0.786706	+	0.617327i	$0.788215\pi$
$564$	−347.264	−0.0259263
$565$	25827.6	1.92314
$566$	−13856.4	−1.02902
$567$	0	0
$568$	−6968.37	−0.514765
$569$	13750.9	1.01313	0.506563	−	0.862203i	$-0.330916\pi$
$569$	13750.9	1.01313	0.506563	+	0.862203i	$0.330916\pi$
$570$	273.349	0.0200866
$571$	10636.3	0.779535	0.389768	−	0.920913i	$-0.372555\pi$
$571$	10636.3	0.779535	0.389768	+	0.920913i	$0.372555\pi$
$572$	−10325.9	−0.754805
$573$	420.388	0.0306491
$574$	0	0
$575$	2472.85	0.179348
$576$	−1723.67	−0.124687
$577$	−7290.64	−0.526019	−0.263010	−	0.964793i	$-0.584715\pi$
$577$	−7290.64	−0.526019	−0.263010	+	0.964793i	$0.584715\pi$
$578$	−3311.64	−0.238315
$579$	−1234.76	−0.0886264
$580$	−11036.9	−0.790141
$581$	0	0
$582$	−178.931	−0.0127439
$583$	18695.3	1.32810
$584$	−7158.17	−0.507204
$585$	−16791.1	−1.18671
$586$	−296.379	−0.0208930
$587$	6315.96	0.444102	0.222051	−	0.975035i	$-0.428725\pi$
$587$	6315.96	0.444102	0.222051	+	0.975035i	$0.428725\pi$
$588$	0	0
$589$	−204.268	−0.0142899
$590$	−24569.7	−1.71444
$591$	−382.836	−0.0266460
$592$	−2589.82	−0.179799
$593$	−19795.1	−1.37081	−0.685403	−	0.728164i	$-0.740374\pi$
$593$	−19795.1	−1.37081	−0.685403	+	0.728164i	$0.740374\pi$
$594$	−1771.51	−0.122367
$595$	0	0
$596$	−6982.54	−0.479892
$597$	−857.547	−0.0587890
$598$	−1880.77	−0.128613
$599$	27201.1	1.85544	0.927720	−	0.373277i	$-0.121766\pi$
$599$	27201.1	1.85544	0.927720	+	0.373277i	$0.121766\pi$
$600$	223.735	0.0152232
$601$	−11695.1	−0.793763	−0.396882	−	0.917870i	$-0.629908\pi$
$601$	−11695.1	−0.793763	−0.396882	+	0.917870i	$0.629908\pi$
$602$	0	0
$603$	−8112.89	−0.547898
$604$	−10081.7	−0.679167
$605$	−40490.7	−2.72096
$606$	−238.533	−0.0159897
$607$	4523.78	0.302495	0.151247	−	0.988496i	$-0.451671\pi$
$607$	4523.78	0.302495	0.151247	+	0.988496i	$0.451671\pi$
$608$	−1102.65	−0.0735501
$609$	0	0
$610$	21823.7	1.44855
$611$	13646.0	0.903530
$612$	−6148.30	−0.406095
$613$	27979.4	1.84352	0.921758	−	0.387765i	$-0.126753\pi$
$613$	27979.4	1.84352	0.921758	+	0.387765i	$0.126753\pi$
$614$	20193.6	1.32728
$615$	−414.220	−0.0271593
$616$	0	0
$617$	27419.8	1.78911	0.894553	−	0.446961i	$-0.147494\pi$
$617$	27419.8	1.78911	0.894553	+	0.446961i	$0.147494\pi$
$618$	299.079	0.0194672
$619$	−4941.83	−0.320887	−0.160443	−	0.987045i	$-0.551292\pi$
$619$	−4941.83	−0.320887	−0.160443	+	0.987045i	$0.551292\pi$
$620$	−361.574	−0.0234213
$621$	−322.664	−0.0208504
$622$	2127.62	0.137154
$623$	0	0
$624$	−170.166	−0.0109168
$625$	−17504.9	−1.12031
$626$	−3907.77	−0.249498
$627$	−565.917	−0.0360455
$628$	−4490.22	−0.285318
$629$	−9237.84	−0.585591
$630$	0	0
$631$	−13077.3	−0.825037	−0.412519	−	0.910949i	$-0.635351\pi$
$631$	−13077.3	−0.825037	−0.412519	+	0.910949i	$0.635351\pi$
$632$	−9969.49	−0.627477
$633$	−633.675	−0.0397888
$634$	−5879.03	−0.368274
$635$	−3841.84	−0.240093
$636$	308.089	0.0192084
$637$	0	0
$638$	22849.7	1.41792
$639$	23459.3	1.45233
$640$	−1951.80	−0.120550
$641$	−1272.72	−0.0784235	−0.0392117	−	0.999231i	$-0.512485\pi$
$641$	−1272.72	−0.0784235	−0.0392117	+	0.999231i	$0.512485\pi$
$642$	−755.262	−0.0464296
$643$	−113.036	−0.00693269	−0.00346634	−	0.999994i	$-0.501103\pi$
$643$	−113.036	−0.00693269	−0.00346634	+	0.999994i	$0.501103\pi$
$644$	0	0
$645$	−65.7338	−0.00401281
$646$	−3933.14	−0.239547
$647$	1829.78	0.111184	0.0555919	−	0.998454i	$-0.482295\pi$
$647$	1829.78	0.111184	0.0555919	+	0.998454i	$0.482295\pi$
$648$	5788.19	0.350898
$649$	50866.7	3.07657
$650$	−8791.81	−0.530528
$651$	0	0
$652$	597.751	0.0359045
$653$	1936.86	0.116072	0.0580361	−	0.998314i	$-0.481516\pi$
$653$	1936.86	0.116072	0.0580361	+	0.998314i	$0.481516\pi$
$654$	−943.139	−0.0563909
$655$	−28124.8	−1.67775
$656$	1670.90	0.0994480
$657$	24098.3	1.43099
$658$	0	0
$659$	−22889.3	−1.35302	−0.676509	−	0.736434i	$-0.736508\pi$
$659$	−22889.3	−1.35302	−0.676509	+	0.736434i	$0.736508\pi$
$660$	−1001.73	−0.0590791
$661$	−17818.0	−1.04847	−0.524237	−	0.851573i	$-0.675649\pi$
$661$	−17818.0	−1.04847	−0.524237	+	0.851573i	$0.675649\pi$
$662$	−17616.9	−1.03429
$663$	−606.978	−0.0355552
$664$	9791.45	0.572262
$665$	0	0
$666$	8718.74	0.507274
$667$	4161.87	0.241602
$668$	−7803.11	−0.451963
$669$	−380.847	−0.0220095
$670$	−9186.65	−0.529718
$671$	−45181.7	−2.59943
$672$	0	0
$673$	20993.6	1.20244	0.601222	−	0.799082i	$-0.294681\pi$
$673$	20993.6	1.20244	0.601222	+	0.799082i	$0.294681\pi$
$674$	7933.46	0.453391
$675$	−1508.32	−0.0860076
$676$	−2101.22	−0.119551
$677$	−5364.75	−0.304555	−0.152278	−	0.988338i	$-0.548661\pi$
$677$	−5364.75	−0.304555	−0.152278	+	0.988338i	$0.548661\pi$
$678$	−881.175	−0.0499134
$679$	0	0
$680$	−6962.04	−0.392621
$681$	889.453	0.0500498
$682$	748.570	0.0420297
$683$	−31720.5	−1.77709	−0.888543	−	0.458793i	$-0.848282\pi$
$683$	−31720.5	−1.77709	−0.888543	+	0.458793i	$0.848282\pi$
$684$	3712.13	0.207510
$685$	17839.7	0.995063
$686$	0	0
$687$	894.138	0.0496557
$688$	265.161	0.0146935
$689$	−12106.6	−0.669409
$690$	−182.456	−0.0100666
$691$	12501.1	0.688228	0.344114	−	0.938928i	$-0.388179\pi$
$691$	12501.1	0.688228	0.344114	+	0.938928i	$0.388179\pi$
$692$	−11403.1	−0.626420
$693$	0	0
$694$	11494.5	0.628709
$695$	−24678.7	−1.34693
$696$	376.552	0.0205074
$697$	5960.09	0.323894
$698$	−16734.4	−0.907460
$699$	793.230	0.0429223
$700$	0	0
$701$	−4917.13	−0.264932	−0.132466	−	0.991188i	$-0.542290\pi$
$701$	−4917.13	−0.264932	−0.132466	+	0.991188i	$0.542290\pi$
$702$	1147.18	0.0616773
$703$	5577.48	0.299230
$704$	4040.83	0.216327
$705$	1323.81	0.0707199
$706$	−8112.20	−0.432446
$707$	0	0
$708$	838.257	0.0444966
$709$	2043.21	0.108229	0.0541145	−	0.998535i	$-0.482766\pi$
$709$	2043.21	0.108229	0.0541145	+	0.998535i	$0.482766\pi$
$710$	26564.2	1.40414
$711$	33562.7	1.77032
$712$	690.863	0.0363640
$713$	136.345	0.00716153
$714$	0	0
$715$	39363.6	2.05890
$716$	7390.46	0.385747
$717$	510.122	0.0265703
$718$	−5160.27	−0.268216
$719$	−24690.8	−1.28069	−0.640343	−	0.768089i	$-0.721208\pi$
$719$	−24690.8	−1.28069	−0.640343	+	0.768089i	$0.721208\pi$
$720$	6570.82	0.340111
$721$	0	0
$722$	−11343.3	−0.584701
$723$	−1592.38	−0.0819103
$724$	−13041.7	−0.669461
$725$	19455.0	0.996606
$726$	1381.44	0.0706202
$727$	23243.0	1.18574	0.592872	−	0.805297i	$-0.297994\pi$
$727$	23243.0	1.18574	0.592872	+	0.805297i	$0.297994\pi$
$728$	0	0
$729$	−19387.7	−0.984995
$730$	27287.7	1.38351
$731$	945.823	0.0478557
$732$	−744.570	−0.0375958
$733$	35315.4	1.77954	0.889772	−	0.456406i	$-0.150863\pi$
$733$	35315.4	1.77954	0.889772	+	0.456406i	$0.150863\pi$
$734$	−1876.51	−0.0943639
$735$	0	0
$736$	736.000	0.0368605
$737$	19019.2	0.950585
$738$	−5625.17	−0.280576
$739$	10783.2	0.536763	0.268382	−	0.963313i	$-0.413511\pi$
$739$	10783.2	0.536763	0.268382	+	0.963313i	$0.413511\pi$
$740$	9872.69	0.490442
$741$	366.472	0.0181683
$742$	0	0
$743$	−4972.73	−0.245534	−0.122767	−	0.992436i	$-0.539177\pi$
$743$	−4972.73	−0.245534	−0.122767	+	0.992436i	$0.539177\pi$
$744$	12.3360	0.000607878 0
$745$	26618.2	1.30901
$746$	21903.2	1.07498
$747$	−32963.3	−1.61454
$748$	14413.6	0.704562
$749$	0	0
$750$	138.704	0.00675303
$751$	−29604.7	−1.43847	−0.719234	−	0.694768i	$-0.755507\pi$
$751$	−29604.7	−1.43847	−0.719234	+	0.694768i	$0.755507\pi$
$752$	−5340.06	−0.258952
$753$	−1069.86	−0.0517767
$754$	−14796.8	−0.714681
$755$	38432.4	1.85258
$756$	0	0
$757$	−9958.05	−0.478113	−0.239056	−	0.971006i	$-0.576838\pi$
$757$	−9958.05	−0.478113	−0.239056	+	0.971006i	$0.576838\pi$
$758$	19041.5	0.912426
$759$	377.739	0.0180646
$760$	4203.44	0.200625
$761$	−28654.4	−1.36494	−0.682472	−	0.730911i	$-0.739095\pi$
$761$	−28654.4	−1.36494	−0.682472	+	0.730911i	$0.739095\pi$
$762$	131.074	0.00623139
$763$	0	0
$764$	6464.52	0.306123
$765$	23438.0	1.10772
$766$	6198.50	0.292377
$767$	−32939.9	−1.55070
$768$	66.5907	0.00312876
$769$	−1580.21	−0.0741011	−0.0370506	−	0.999313i	$-0.511796\pi$
$769$	−1580.21	−0.0741011	−0.0370506	+	0.999313i	$0.511796\pi$
$770$	0	0
$771$	555.493	0.0259476
$772$	−18987.5	−0.885200
$773$	−36297.7	−1.68892	−0.844461	−	0.535617i	$-0.820079\pi$
$773$	−36297.7	−1.68892	−0.844461	+	0.535617i	$0.820079\pi$
$774$	−892.675	−0.0414555
$775$	637.356	0.0295413
$776$	−2751.51	−0.127286
$777$	0	0
$778$	8736.69	0.402604
$779$	−3598.49	−0.165506
$780$	648.691	0.0297780
$781$	−54996.1	−2.51974
$782$	2625.30	0.120052
$783$	−2538.54	−0.115862
$784$	0	0
$785$	17117.2	0.778268
$786$	959.551	0.0435446
$787$	−3840.61	−0.173956	−0.0869778	−	0.996210i	$-0.527721\pi$
$787$	−3840.61	−0.173956	−0.0869778	+	0.996210i	$0.527721\pi$
$788$	−5887.07	−0.266140
$789$	1822.35	0.0822273
$790$	38004.8	1.71158
$791$	0	0
$792$	−13603.6	−0.610333
$793$	29258.4	1.31021
$794$	−12612.1	−0.563712
$795$	−1174.47	−0.0523951
$796$	−13186.9	−0.587184
$797$	24193.7	1.07526	0.537632	−	0.843179i	$-0.319319\pi$
$797$	24193.7	1.07526	0.537632	+	0.843179i	$0.319319\pi$
$798$	0	0
$799$	−19047.9	−0.843386
$800$	3440.49	0.152049
$801$	−2325.82	−0.102595
$802$	−166.175	−0.00731650
$803$	−56494.0	−2.48273
$804$	313.426	0.0137484
$805$	0	0
$806$	−484.753	−0.0211845
$807$	−1708.04	−0.0745052
$808$	−3668.04	−0.159705
$809$	−16476.8	−0.716063	−0.358031	−	0.933710i	$-0.616552\pi$
$809$	−16476.8	−0.716063	−0.358031	+	0.933710i	$0.616552\pi$
$810$	−22065.2	−0.957152
$811$	−2891.58	−0.125200	−0.0625999	−	0.998039i	$-0.519939\pi$
$811$	−2891.58	−0.125200	−0.0625999	+	0.998039i	$0.519939\pi$
$812$	0	0
$813$	1498.14	0.0646275
$814$	−20439.5	−0.880103
$815$	−2278.70	−0.0979377
$816$	237.528	0.0101901
$817$	−571.055	−0.0244537
$818$	−3188.66	−0.136295
$819$	0	0
$820$	−6369.68	−0.271267
$821$	36315.1	1.54373	0.771867	−	0.635785i	$-0.219323\pi$
$821$	36315.1	1.54373	0.771867	+	0.635785i	$0.219323\pi$
$822$	−608.646	−0.0258260
$823$	−18744.4	−0.793913	−0.396956	−	0.917838i	$-0.629934\pi$
$823$	−18744.4	−0.793913	−0.396956	+	0.917838i	$0.629934\pi$
$824$	4599.09	0.194438
$825$	1765.77	0.0745166
$826$	0	0
$827$	25294.4	1.06357	0.531784	−	0.846880i	$-0.321522\pi$
$827$	25294.4	1.06357	0.531784	+	0.846880i	$0.321522\pi$
$828$	−2477.78	−0.103996
$829$	39438.4	1.65230	0.826148	−	0.563453i	$-0.190527\pi$
$829$	39438.4	1.65230	0.826148	+	0.563453i	$0.190527\pi$
$830$	−37326.1	−1.56097
$831$	92.8332	0.00387527
$832$	−2616.73	−0.109037
$833$	0	0
$834$	841.977	0.0349584
$835$	29746.3	1.23283
$836$	−8702.40	−0.360023
$837$	−83.1639	−0.00343437
$838$	6521.34	0.268826
$839$	30165.0	1.24125	0.620627	−	0.784106i	$-0.286878\pi$
$839$	30165.0	1.24125	0.620627	+	0.784106i	$0.286878\pi$
$840$	0	0
$841$	8354.24	0.342541
$842$	−14209.3	−0.581574
$843$	−240.805	−0.00983838
$844$	−9744.35	−0.397410
$845$	8010.10	0.326101
$846$	17977.5	0.730591
$847$	0	0
$848$	4737.64	0.191853
$849$	−1802.16	−0.0728503
$850$	12272.2	0.495213
$851$	−3722.87	−0.149963
$852$	−906.306	−0.0364431
$853$	−8977.27	−0.360347	−0.180173	−	0.983635i	$-0.557666\pi$
$853$	−8977.27	−0.360347	−0.180173	+	0.983635i	$0.557666\pi$
$854$	0	0
$855$	−14151.0	−0.566030
$856$	−11614.0	−0.463738
$857$	40593.6	1.61803	0.809015	−	0.587789i	$-0.200001\pi$
$857$	40593.6	1.61803	0.809015	+	0.587789i	$0.200001\pi$
$858$	−1342.99	−0.0534369
$859$	14117.7	0.560755	0.280377	−	0.959890i	$-0.409540\pi$
$859$	14117.7	0.560755	0.280377	+	0.959890i	$0.409540\pi$
$860$	−1010.82	−0.0400799
$861$	0	0
$862$	21520.9	0.850352
$863$	−7080.02	−0.279266	−0.139633	−	0.990203i	$-0.544592\pi$
$863$	−7080.02	−0.279266	−0.139633	+	0.990203i	$0.544592\pi$
$864$	−448.924	−0.0176767
$865$	43470.1	1.70870
$866$	23074.8	0.905443
$867$	−430.712	−0.0168717
$868$	0	0
$869$	−78681.7	−3.07145
$870$	−1435.46	−0.0559386
$871$	−12316.3	−0.479129
$872$	−14503.1	−0.563232
$873$	9263.09	0.359116
$874$	−1585.06	−0.0613450
$875$	0	0
$876$	−930.991	−0.0359079
$877$	−29775.4	−1.14646	−0.573229	−	0.819395i	$-0.694309\pi$
$877$	−29775.4	−1.14646	−0.573229	+	0.819395i	$0.694309\pi$
$878$	21570.5	0.829120
$879$	−38.5471	−0.00147914
$880$	−15404.1	−0.590082
$881$	−32974.9	−1.26101	−0.630506	−	0.776184i	$-0.717153\pi$
$881$	−32974.9	−1.26101	−0.630506	+	0.776184i	$0.717153\pi$
$882$	0	0
$883$	−8286.80	−0.315825	−0.157912	−	0.987453i	$-0.550476\pi$
$883$	−8286.80	−0.315825	−0.157912	+	0.987453i	$0.550476\pi$
$884$	−9333.82	−0.355125
$885$	−3195.53	−0.121375
$886$	4966.91	0.188337
$887$	13354.7	0.505533	0.252767	−	0.967527i	$-0.418659\pi$
$887$	13354.7	0.505533	0.252767	+	0.967527i	$0.418659\pi$
$888$	−336.832	−0.0127290
$889$	0	0
$890$	−2633.65	−0.0991911
$891$	45681.8	1.71762
$892$	−5856.48	−0.219831
$893$	11500.4	0.430960
$894$	−908.149	−0.0339743
$895$	−28173.3	−1.05221
$896$	0	0
$897$	−244.613	−0.00910524
$898$	−6021.57	−0.223767
$899$	1072.69	0.0397954
$900$	−11582.5	−0.428983
$901$	16899.1	0.624850
$902$	13187.2	0.486791
$903$	0	0
$904$	−13550.3	−0.498535
$905$	49716.3	1.82611
$906$	−1311.22	−0.0480821
$907$	53901.6	1.97329	0.986645	−	0.162885i	$-0.0520801\pi$
$907$	53901.6	1.97329	0.986645	+	0.162885i	$0.0520801\pi$
$908$	13677.6	0.499897
$909$	12348.6	0.450581
$910$	0	0
$911$	−14861.5	−0.540488	−0.270244	−	0.962792i	$-0.587104\pi$
$911$	−14861.5	−0.540488	−0.270244	+	0.962792i	$0.587104\pi$
$912$	−143.411	−0.00520703
$913$	77276.5	2.80118
$914$	−25813.1	−0.934158
$915$	2838.38	0.102551
$916$	13749.6	0.495961
$917$	0	0
$918$	−1601.30	−0.0575718
$919$	39976.2	1.43492	0.717460	−	0.696599i	$-0.245304\pi$
$919$	39976.2	1.43492	0.717460	+	0.696599i	$0.245304\pi$
$920$	−2805.71	−0.100545
$921$	2626.38	0.0939655
$922$	−3856.02	−0.137735
$923$	35613.9	1.27004
$924$	0	0
$925$	−17402.8	−0.618596
$926$	5760.77	0.204439
$927$	−15483.0	−0.548576
$928$	5790.43	0.204828
$929$	7480.52	0.264185	0.132093	−	0.991237i	$-0.457830\pi$
$929$	7480.52	0.264185	0.132093	+	0.991237i	$0.457830\pi$
$930$	−47.0264	−0.00165812
$931$	0	0
$932$	12197.9	0.428708
$933$	276.718	0.00970992
$934$	−13722.9	−0.480758
$935$	−54946.1	−1.92185
$936$	8809.32	0.307630
$937$	18057.0	0.629558	0.314779	−	0.949165i	$-0.398070\pi$
$937$	18057.0	0.629558	0.314779	+	0.949165i	$0.398070\pi$
$938$	0	0
$939$	−508.244	−0.0176634
$940$	20356.9	0.706350
$941$	18921.5	0.655496	0.327748	−	0.944765i	$-0.393710\pi$
$941$	18921.5	0.655496	0.327748	+	0.944765i	$0.393710\pi$
$942$	−583.999	−0.0201993
$943$	2401.93	0.0829453
$944$	12890.3	0.444432
$945$	0	0
$946$	2092.71	0.0719238
$947$	−10147.2	−0.348193	−0.174097	−	0.984729i	$-0.555701\pi$
$947$	−10147.2	−0.348193	−0.174097	+	0.984729i	$0.555701\pi$
$948$	−1296.63	−0.0444226
$949$	36583.9	1.25139
$950$	−7409.49	−0.253048
$951$	−764.626	−0.0260722
$952$	0	0
$953$	6284.28	0.213607	0.106804	−	0.994280i	$-0.465938\pi$
$953$	6284.28	0.213607	0.106804	+	0.994280i	$0.465938\pi$
$954$	−15949.5	−0.541282
$955$	−24643.5	−0.835020
$956$	7844.42	0.265383
$957$	2971.84	0.100382
$958$	9160.11	0.308925
$959$	0	0
$960$	−253.851	−0.00853439
$961$	−29755.9	−0.998820
$962$	13236.0	0.443604
$963$	39099.2	1.30836
$964$	−24486.8	−0.818119
$965$	72382.4	2.41458
$966$	0	0
$967$	−10750.9	−0.357524	−0.178762	−	0.983892i	$-0.557209\pi$
$967$	−10750.9	−0.357524	−0.178762	+	0.983892i	$0.557209\pi$
$968$	21243.2	0.705354
$969$	−511.544	−0.0169589
$970$	10489.1	0.347200
$971$	−21337.3	−0.705198	−0.352599	−	0.935774i	$-0.614702\pi$
$971$	−21337.3	−0.705198	−0.352599	+	0.935774i	$0.614702\pi$
$972$	2267.93	0.0748394
$973$	0	0
$974$	17688.6	0.581908
$975$	−1143.46	−0.0375591
$976$	−11449.6	−0.375506
$977$	−713.051	−0.0233496	−0.0116748	−	0.999932i	$-0.503716\pi$
$977$	−713.051	−0.0233496	−0.0116748	+	0.999932i	$0.503716\pi$
$978$	77.7435	0.00254189
$979$	5452.46	0.177999
$980$	0	0
$981$	48825.4	1.58907
$982$	−9374.43	−0.304634
$983$	−31306.7	−1.01580	−0.507898	−	0.861417i	$-0.669577\pi$
$983$	−31306.7	−1.01580	−0.507898	+	0.861417i	$0.669577\pi$
$984$	217.318	0.00704049
$985$	22442.2	0.725956
$986$	20654.4	0.667108
$987$	0	0
$988$	5635.43	0.181465
$989$	381.168	0.0122553
$990$	51858.5	1.66482
$991$	13538.1	0.433956	0.216978	−	0.976176i	$-0.430380\pi$
$991$	13538.1	0.433956	0.216978	+	0.976176i	$0.430380\pi$
$992$	189.698	0.00607148
$993$	−2291.26	−0.0732234
$994$	0	0
$995$	50270.1	1.60168
$996$	1273.48	0.0405137
$997$	−44983.5	−1.42893	−0.714464	−	0.699672i	$-0.753329\pi$
$997$	−44983.5	−1.42893	−0.714464	+	0.699672i	$0.753329\pi$
$998$	−20490.2	−0.649907
$999$	2270.77	0.0719158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.l.1.3		5
7.6	odd	2		322.4.a.h.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.a.h.1.3	✓	5	7.6	odd	2
2254.4.a.l.1.3		5	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:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +0.260120 q^{3} +4.00000 q^{4} -15.2484 q^{5} +0.520240 q^{6} +8.00000 q^{8} -26.9323 q^{9} -30.4969 q^{10} +63.1379 q^{11} +1.04048 q^{12} -40.8864 q^{13} -3.96643 q^{15} +16.0000 q^{16} +57.0717 q^{17} -53.8647 q^{18} -34.4579 q^{19} -60.9938 q^{20} +126.276 q^{22} +23.0000 q^{23} +2.08096 q^{24} +107.515 q^{25} -81.7727 q^{26} -14.0289 q^{27} +180.951 q^{29} -7.93285 q^{30} +5.92805 q^{31} +32.0000 q^{32} +16.4234 q^{33} +114.143 q^{34} -107.729 q^{36} -161.864 q^{37} -68.9158 q^{38} -10.6354 q^{39} -121.988 q^{40} +104.432 q^{41} +16.5725 q^{43} +252.552 q^{44} +410.676 q^{45} +46.0000 q^{46} -333.754 q^{47} +4.16192 q^{48} +215.030 q^{50} +14.8455 q^{51} -163.545 q^{52} +296.103 q^{53} -28.0578 q^{54} -962.756 q^{55} -8.96319 q^{57} +361.902 q^{58} +805.644 q^{59} -15.8657 q^{60} -715.603 q^{61} +11.8561 q^{62} +64.0000 q^{64} +623.454 q^{65} +32.8469 q^{66} +301.232 q^{67} +228.287 q^{68} +5.98276 q^{69} -871.046 q^{71} -215.459 q^{72} -894.771 q^{73} -323.728 q^{74} +27.9668 q^{75} -137.832 q^{76} -21.2707 q^{78} -1246.19 q^{79} -243.975 q^{80} +723.524 q^{81} +208.863 q^{82} +1223.93 q^{83} -870.255 q^{85} +33.1451 q^{86} +47.0690 q^{87} +505.104 q^{88} +86.3579 q^{89} +821.353 q^{90} +92.0000 q^{92} +1.54200 q^{93} -667.507 q^{94} +525.429 q^{95} +8.32384 q^{96} -343.939 q^{97} -1700.45 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 10 q^{2} - 5 q^{3} + 20 q^{4} - 22 q^{5} - 10 q^{6} + 40 q^{8} + 54 q^{9} - 44 q^{10} + 42 q^{11} - 20 q^{12} - 107 q^{13} + 122 q^{15} + 80 q^{16} - 218 q^{17} + 108 q^{18} - 194 q^{19} - 88 q^{20}+ \cdots - 2616 q^{99}+O(q^{100}) \) 5 * q + 10 * q^2 - 5 * q^3 + 20 * q^4 - 22 * q^5 - 10 * q^6 + 40 * q^8 + 54 * q^9 - 44 * q^10 + 42 * q^11 - 20 * q^12 - 107 * q^13 + 122 * q^15 + 80 * q^16 - 218 * q^17 + 108 * q^18 - 194 * q^19 - 88 * q^20 + 84 * q^22 + 115 * q^23 - 40 * q^24 + 219 * q^25 - 214 * q^26 - 203 * q^27 + 33 * q^29 + 244 * q^30 - 19 * q^31 + 160 * q^32 - 240 * q^33 - 436 * q^34 + 216 * q^36 + 214 * q^37 - 388 * q^38 - 243 * q^39 - 176 * q^40 - 395 * q^41 - 284 * q^43 + 168 * q^44 - 470 * q^45 + 230 * q^46 + 493 * q^47 - 80 * q^48 + 438 * q^50 - 244 * q^51 - 428 * q^52 - 70 * q^53 - 406 * q^54 + 186 * q^55 + 22 * q^57 + 66 * q^58 - 226 * q^59 + 488 * q^60 - 1606 * q^61 - 38 * q^62 + 320 * q^64 + 362 * q^65 - 480 * q^66 + 298 * q^67 - 872 * q^68 - 115 * q^69 - 969 * q^71 + 432 * q^72 - 697 * q^73 + 428 * q^74 - 1065 * q^75 - 776 * q^76 - 486 * q^78 + 186 * q^79 - 352 * q^80 - 335 * q^81 - 790 * q^82 + 1104 * q^83 - 1344 * q^85 - 568 * q^86 + 1603 * q^87 + 336 * q^88 + 814 * q^89 - 940 * q^90 + 460 * q^92 - 3423 * q^93 + 986 * q^94 - 2812 * q^95 - 160 * q^96 + 256 * q^97 - 2616 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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