LMFDB - Embedded newform 2254.4.a.y.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:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4 x^{10} - 212 x^{9} + 487 x^{8} + 16315 x^{7} - 9025 x^{6} - 516068 x^{5} - 504693 x^{4} + \cdots - 11394027 $ x^11 - 4x^10 - 212x^9 + 487x^8 + 16315x^7 - 9025x^6 - 516068x^5 - 504693x^4 + 5020566x^3 + 8901846x^2 - 4796415x - 11394027
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$8.10595$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} -6.10595 q^{3} +4.00000 q^{4} -5.20425 q^{5} -12.2119 q^{6} +8.00000 q^{8} +10.2826 q^{9} -10.4085 q^{10} -17.8837 q^{11} -24.4238 q^{12} +64.1389 q^{13} +31.7769 q^{15} +16.0000 q^{16} +30.2364 q^{17} +20.5652 q^{18} +89.2372 q^{19} -20.8170 q^{20} -35.7673 q^{22} +23.0000 q^{23} -48.8476 q^{24} -97.9157 q^{25} +128.278 q^{26} +102.076 q^{27} -61.4085 q^{29} +63.5538 q^{30} -158.069 q^{31} +32.0000 q^{32} +109.197 q^{33} +60.4728 q^{34} +41.1305 q^{36} -110.955 q^{37} +178.474 q^{38} -391.629 q^{39} -41.6340 q^{40} -93.3461 q^{41} -388.470 q^{43} -71.5347 q^{44} -53.5133 q^{45} +46.0000 q^{46} +134.149 q^{47} -97.6952 q^{48} -195.831 q^{50} -184.622 q^{51} +256.556 q^{52} +145.407 q^{53} +204.151 q^{54} +93.0712 q^{55} -544.877 q^{57} -122.817 q^{58} -46.8331 q^{59} +127.108 q^{60} -336.268 q^{61} -316.137 q^{62} +64.0000 q^{64} -333.795 q^{65} +218.394 q^{66} -385.500 q^{67} +120.946 q^{68} -140.437 q^{69} -94.5483 q^{71} +82.2609 q^{72} +1085.04 q^{73} -221.910 q^{74} +597.869 q^{75} +356.949 q^{76} -783.258 q^{78} +725.264 q^{79} -83.2681 q^{80} -900.898 q^{81} -186.692 q^{82} +169.422 q^{83} -157.358 q^{85} -776.941 q^{86} +374.957 q^{87} -143.069 q^{88} -151.991 q^{89} -107.027 q^{90} +92.0000 q^{92} +965.159 q^{93} +268.297 q^{94} -464.413 q^{95} -195.390 q^{96} +344.898 q^{97} -183.891 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 22 q^{2} + 18 q^{3} + 44 q^{4} + 33 q^{5} + 36 q^{6} + 88 q^{8} + 171 q^{9} + 66 q^{10} + 8 q^{11} + 72 q^{12} + 185 q^{13} - 186 q^{15} + 176 q^{16} + 107 q^{17} + 342 q^{18} + 114 q^{19} + 132 q^{20}+ \cdots - 1729 q^{99}+O(q^{100}) $ 11 * q + 22 * q^2 + 18 * q^3 + 44 * q^4 + 33 * q^5 + 36 * q^6 + 88 * q^8 + 171 * q^9 + 66 * q^10 + 8 * q^11 + 72 * q^12 + 185 * q^13 - 186 * q^15 + 176 * q^16 + 107 * q^17 + 342 * q^18 + 114 * q^19 + 132 * q^20 + 16 * q^22 + 253 * q^23 + 144 * q^24 + 494 * q^25 + 370 * q^26 + 561 * q^27 - 79 * q^29 - 372 * q^30 + 488 * q^31 + 352 * q^32 + 304 * q^33 + 214 * q^34 + 684 * q^36 - 41 * q^37 + 228 * q^38 + 425 * q^39 + 264 * q^40 + 812 * q^41 + 219 * q^43 + 32 * q^44 + 619 * q^45 + 506 * q^46 + 1141 * q^47 + 288 * q^48 + 988 * q^50 + 83 * q^51 + 740 * q^52 + 579 * q^53 + 1122 * q^54 + 951 * q^55 + 383 * q^57 - 158 * q^58 + 1695 * q^59 - 744 * q^60 + 1105 * q^61 + 976 * q^62 + 704 * q^64 - 347 * q^65 + 608 * q^66 + 695 * q^67 + 428 * q^68 + 414 * q^69 + 65 * q^71 + 1368 * q^72 + 1935 * q^73 - 82 * q^74 + 1712 * q^75 + 456 * q^76 + 850 * q^78 + 447 * q^79 + 528 * q^80 + 2203 * q^81 + 1624 * q^82 + 1389 * q^83 + 153 * q^85 + 438 * q^86 + 3354 * q^87 + 64 * q^88 + 2698 * q^89 + 1238 * q^90 + 1012 * q^92 - 686 * q^93 + 2282 * q^94 + 827 * q^95 + 576 * q^96 + 1589 * q^97 - 1729 * 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
$3$	−6.10595	−1.17509	−0.587545	−	0.809191i	$-0.699906\pi$
$3$	−6.10595	−1.17509	−0.587545	+	0.809191i	$0.699906\pi$
$4$	4.00000	0.500000
$5$	−5.20425	−0.465483	−0.232741	−	0.972539i	$-0.574770\pi$
$5$	−5.20425	−0.465483	−0.232741	+	0.972539i	$0.574770\pi$
$6$	−12.2119	−0.830914
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	10.2826	0.380838
$10$	−10.4085	−0.329146
$11$	−17.8837	−0.490194	−0.245097	−	0.969499i	$-0.578820\pi$
$11$	−17.8837	−0.490194	−0.245097	+	0.969499i	$0.578820\pi$
$12$	−24.4238	−0.587545
$13$	64.1389	1.36838	0.684190	−	0.729304i	$-0.260156\pi$
$13$	64.1389	1.36838	0.684190	+	0.729304i	$0.260156\pi$
$14$	0	0
$15$	31.7769	0.546984
$16$	16.0000	0.250000
$17$	30.2364	0.431377	0.215688	−	0.976462i	$-0.430800\pi$
$17$	30.2364	0.431377	0.215688	+	0.976462i	$0.430800\pi$
$18$	20.5652	0.269293
$19$	89.2372	1.07750	0.538748	−	0.842467i	$-0.318898\pi$
$19$	89.2372	1.07750	0.538748	+	0.842467i	$0.318898\pi$
$20$	−20.8170	−0.232741
$21$	0	0
$22$	−35.7673	−0.346619
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−48.8476	−0.415457
$25$	−97.9157	−0.783326
$26$	128.278	0.967590
$27$	102.076	0.727572
$28$	0	0
$29$	−61.4085	−0.393216	−0.196608	−	0.980482i	$-0.562993\pi$
$29$	−61.4085	−0.393216	−0.196608	+	0.980482i	$0.562993\pi$
$30$	63.5538	0.386776
$31$	−158.069	−0.915806	−0.457903	−	0.889002i	$-0.651399\pi$
$31$	−158.069	−0.915806	−0.457903	+	0.889002i	$0.651399\pi$
$32$	32.0000	0.176777
$33$	109.197	0.576022
$34$	60.4728	0.305029
$35$	0	0
$36$	41.1305	0.190419
$37$	−110.955	−0.492998	−0.246499	−	0.969143i	$-0.579280\pi$
$37$	−110.955	−0.492998	−0.246499	+	0.969143i	$0.579280\pi$
$38$	178.474	0.761904
$39$	−391.629	−1.60797
$40$	−41.6340	−0.164573
$41$	−93.3461	−0.355566	−0.177783	−	0.984070i	$-0.556893\pi$
$41$	−93.3461	−0.355566	−0.177783	+	0.984070i	$0.556893\pi$
$42$	0	0
$43$	−388.470	−1.37770	−0.688851	−	0.724903i	$-0.741885\pi$
$43$	−388.470	−1.37770	−0.688851	+	0.724903i	$0.741885\pi$
$44$	−71.5347	−0.245097
$45$	−53.5133	−0.177273
$46$	46.0000	0.147442
$47$	134.149	0.416332	0.208166	−	0.978094i	$-0.433251\pi$
$47$	134.149	0.416332	0.208166	+	0.978094i	$0.433251\pi$
$48$	−97.6952	−0.293773
$49$	0	0
$50$	−195.831	−0.553895
$51$	−184.622	−0.506907
$52$	256.556	0.684190
$53$	145.407	0.376852	0.188426	−	0.982087i	$-0.439661\pi$
$53$	145.407	0.376852	0.188426	+	0.982087i	$0.439661\pi$
$54$	204.151	0.514471
$55$	93.0712	0.228177
$56$	0	0
$57$	−544.877	−1.26615
$58$	−122.817	−0.278046
$59$	−46.8331	−0.103341	−0.0516707	−	0.998664i	$-0.516455\pi$
$59$	−46.8331	−0.103341	−0.0516707	+	0.998664i	$0.516455\pi$
$60$	127.108	0.273492
$61$	−336.268	−0.705815	−0.352907	−	0.935658i	$-0.614807\pi$
$61$	−336.268	−0.705815	−0.352907	+	0.935658i	$0.614807\pi$
$62$	−316.137	−0.647572
$63$	0	0
$64$	64.0000	0.125000
$65$	−333.795	−0.636957
$66$	218.394	0.407309
$67$	−385.500	−0.702931	−0.351465	−	0.936201i	$-0.614317\pi$
$67$	−385.500	−0.702931	−0.351465	+	0.936201i	$0.614317\pi$
$68$	120.946	0.215688
$69$	−140.437	−0.245023
$70$	0	0
$71$	−94.5483	−0.158040	−0.0790199	−	0.996873i	$-0.525179\pi$
$71$	−94.5483	−0.158040	−0.0790199	+	0.996873i	$0.525179\pi$
$72$	82.2609	0.134646
$73$	1085.04	1.73965	0.869825	−	0.493360i	$-0.164231\pi$
$73$	1085.04	1.73965	0.869825	+	0.493360i	$0.164231\pi$
$74$	−221.910	−0.348602
$75$	597.869	0.920479
$76$	356.949	0.538748
$77$	0	0
$78$	−783.258	−1.13701
$79$	725.264	1.03289	0.516447	−	0.856319i	$-0.327254\pi$
$79$	725.264	1.03289	0.516447	+	0.856319i	$0.327254\pi$
$80$	−83.2681	−0.116371
$81$	−900.898	−1.23580
$82$	−186.692	−0.251423
$83$	169.422	0.224054	0.112027	−	0.993705i	$-0.464266\pi$
$83$	169.422	0.224054	0.112027	+	0.993705i	$0.464266\pi$
$84$	0	0
$85$	−157.358	−0.200798
$86$	−776.941	−0.974182
$87$	374.957	0.462064
$88$	−143.069	−0.173310
$89$	−151.991	−0.181023	−0.0905113	−	0.995895i	$-0.528850\pi$
$89$	−151.991	−0.181023	−0.0905113	+	0.995895i	$0.528850\pi$
$90$	−107.027	−0.125351
$91$	0	0
$92$	92.0000	0.104257
$93$	965.159	1.07615
$94$	268.297	0.294391
$95$	−464.413	−0.501555
$96$	−195.390	−0.207729
$97$	344.898	0.361022	0.180511	−	0.983573i	$-0.442225\pi$
$97$	344.898	0.361022	0.180511	+	0.983573i	$0.442225\pi$
$98$	0	0
$99$	−183.891	−0.186684
$100$	−391.663	−0.391663
$101$	−514.137	−0.506521	−0.253260	−	0.967398i	$-0.581503\pi$
$101$	−514.137	−0.506521	−0.253260	+	0.967398i	$0.581503\pi$
$102$	−369.244	−0.358437
$103$	360.320	0.344693	0.172347	−	0.985036i	$-0.444865\pi$
$103$	360.320	0.344693	0.172347	+	0.985036i	$0.444865\pi$
$104$	513.111	0.483795
$105$	0	0
$106$	290.814	0.266475
$107$	426.885	0.385687	0.192844	−	0.981229i	$-0.438229\pi$
$107$	426.885	0.385687	0.192844	+	0.981229i	$0.438229\pi$
$108$	408.302	0.363786
$109$	238.729	0.209781	0.104890	−	0.994484i	$-0.466551\pi$
$109$	238.729	0.209781	0.104890	+	0.994484i	$0.466551\pi$
$110$	186.142	0.161345
$111$	677.486	0.579317
$112$	0	0
$113$	793.212	0.660346	0.330173	−	0.943920i	$-0.392893\pi$
$113$	793.212	0.660346	0.330173	+	0.943920i	$0.392893\pi$
$114$	−1089.75	−0.895306
$115$	−119.698	−0.0970598
$116$	−245.634	−0.196608
$117$	659.516	0.521130
$118$	−93.6662	−0.0730735
$119$	0	0
$120$	254.215	0.193388
$121$	−1011.17	−0.759710
$122$	−672.536	−0.499086
$123$	569.966	0.417822
$124$	−632.275	−0.457903
$125$	1160.11	0.830107
$126$	0	0
$127$	2122.13	1.48274	0.741372	−	0.671095i	$-0.234176\pi$
$127$	2122.13	1.48274	0.741372	+	0.671095i	$0.234176\pi$
$128$	128.000	0.0883883
$129$	2371.98	1.61892
$130$	−667.590	−0.450396
$131$	−1194.11	−0.796412	−0.398206	−	0.917296i	$-0.630367\pi$
$131$	−1194.11	−0.796412	−0.398206	+	0.917296i	$0.630367\pi$
$132$	436.787	0.288011
$133$	0	0
$134$	−771.001	−0.497047
$135$	−531.227	−0.338672
$136$	241.891	0.152515
$137$	569.745	0.355303	0.177652	−	0.984093i	$-0.443150\pi$
$137$	569.745	0.355303	0.177652	+	0.984093i	$0.443150\pi$
$138$	−280.874	−0.173258
$139$	2162.93	1.31984	0.659919	−	0.751337i	$-0.270591\pi$
$139$	2162.93	1.31984	0.659919	+	0.751337i	$0.270591\pi$
$140$	0	0
$141$	−819.105	−0.489227
$142$	−189.097	−0.111751
$143$	−1147.04	−0.670771
$144$	164.522	0.0952094
$145$	319.585	0.183035
$146$	2170.08	1.23012
$147$	0	0
$148$	−443.820	−0.246499
$149$	−241.738	−0.132912	−0.0664562	−	0.997789i	$-0.521169\pi$
$149$	−241.738	−0.132912	−0.0664562	+	0.997789i	$0.521169\pi$
$150$	1195.74	0.650877
$151$	−135.991	−0.0732900	−0.0366450	−	0.999328i	$-0.511667\pi$
$151$	−135.991	−0.0732900	−0.0366450	+	0.999328i	$0.511667\pi$
$152$	713.897	0.380952
$153$	310.909	0.164284
$154$	0	0
$155$	822.630	0.426292
$156$	−1566.52	−0.803985
$157$	745.044	0.378733	0.189366	−	0.981907i	$-0.439357\pi$
$157$	745.044	0.378733	0.189366	+	0.981907i	$0.439357\pi$
$158$	1450.53	0.730366
$159$	−887.847	−0.442836
$160$	−166.536	−0.0822865
$161$	0	0
$162$	−1801.80	−0.873843
$163$	2319.40	1.11453	0.557267	−	0.830333i	$-0.311850\pi$
$163$	2319.40	1.11453	0.557267	+	0.830333i	$0.311850\pi$
$164$	−373.384	−0.177783
$165$	−568.288	−0.268128
$166$	338.844	0.158430
$167$	3260.16	1.51065	0.755326	−	0.655350i	$-0.227479\pi$
$167$	3260.16	1.51065	0.755326	+	0.655350i	$0.227479\pi$
$168$	0	0
$169$	1916.80	0.872462
$170$	−314.716	−0.141986
$171$	917.591	0.410351
$172$	−1553.88	−0.688851
$173$	−2889.71	−1.26995	−0.634973	−	0.772534i	$-0.718989\pi$
$173$	−2889.71	−1.26995	−0.634973	+	0.772534i	$0.718989\pi$
$174$	749.914	0.326729
$175$	0	0
$176$	−286.139	−0.122548
$177$	285.960	0.121436
$178$	−303.982	−0.128002
$179$	3767.12	1.57300	0.786501	−	0.617589i	$-0.211890\pi$
$179$	3767.12	1.57300	0.786501	+	0.617589i	$0.211890\pi$
$180$	−214.053	−0.0886366
$181$	778.254	0.319598	0.159799	−	0.987150i	$-0.448915\pi$
$181$	778.254	0.319598	0.159799	+	0.987150i	$0.448915\pi$
$182$	0	0
$183$	2053.23	0.829396
$184$	184.000	0.0737210
$185$	577.439	0.229482
$186$	1930.32	0.760956
$187$	−540.738	−0.211458
$188$	536.595	0.208166
$189$	0	0
$190$	−928.826	−0.354653
$191$	−3723.02	−1.41041	−0.705206	−	0.709003i	$-0.749145\pi$
$191$	−3723.02	−1.41041	−0.705206	+	0.709003i	$0.749145\pi$
$192$	−390.781	−0.146886
$193$	−3478.30	−1.29727	−0.648636	−	0.761099i	$-0.724660\pi$
$193$	−3478.30	−1.29727	−0.648636	+	0.761099i	$0.724660\pi$
$194$	689.797	0.255281
$195$	2038.14	0.748482
$196$	0	0
$197$	4939.36	1.78637	0.893185	−	0.449690i	$-0.148466\pi$
$197$	4939.36	1.78637	0.893185	+	0.449690i	$0.148466\pi$
$198$	−367.782	−0.132006
$199$	−4132.56	−1.47211	−0.736054	−	0.676923i	$-0.763313\pi$
$199$	−4132.56	−1.47211	−0.736054	+	0.676923i	$0.763313\pi$
$200$	−783.326	−0.276948
$201$	2353.85	0.826007
$202$	−1028.27	−0.358164
$203$	0	0
$204$	−738.488	−0.253453
$205$	485.797	0.165510
$206$	720.640	0.243735
$207$	236.500	0.0794101
$208$	1026.22	0.342095
$209$	−1595.89	−0.528181
$210$	0	0
$211$	−2124.78	−0.693250	−0.346625	−	0.938004i	$-0.612672\pi$
$211$	−2124.78	−0.693250	−0.346625	+	0.938004i	$0.612672\pi$
$212$	581.628	0.188426
$213$	577.307	0.185711
$214$	853.771	0.272722
$215$	2021.70	0.641296
$216$	816.604	0.257236
$217$	0	0
$218$	477.458	0.148337
$219$	−6625.21	−2.04425
$220$	372.285	0.114088
$221$	1939.33	0.590287
$222$	1354.97	0.409639
$223$	−5085.59	−1.52716	−0.763579	−	0.645715i	$-0.776559\pi$
$223$	−5085.59	−1.52716	−0.763579	+	0.645715i	$0.776559\pi$
$224$	0	0
$225$	−1006.83	−0.298320
$226$	1586.42	0.466935
$227$	2114.86	0.618363	0.309182	−	0.951003i	$-0.399945\pi$
$227$	2114.86	0.618363	0.309182	+	0.951003i	$0.399945\pi$
$228$	−2179.51	−0.633077
$229$	4315.54	1.24532	0.622662	−	0.782491i	$-0.286051\pi$
$229$	4315.54	1.24532	0.622662	+	0.782491i	$0.286051\pi$
$230$	−239.396	−0.0686317
$231$	0	0
$232$	−491.268	−0.139023
$233$	−952.586	−0.267837	−0.133918	−	0.990992i	$-0.542756\pi$
$233$	−952.586	−0.267837	−0.133918	+	0.990992i	$0.542756\pi$
$234$	1319.03	0.368495
$235$	−698.144	−0.193795
$236$	−187.332	−0.0516707
$237$	−4428.43	−1.21374
$238$	0	0
$239$	4070.42	1.10165	0.550824	−	0.834622i	$-0.314314\pi$
$239$	4070.42	1.10165	0.550824	+	0.834622i	$0.314314\pi$
$240$	508.430	0.136746
$241$	2869.06	0.766856	0.383428	−	0.923571i	$-0.374743\pi$
$241$	2869.06	0.766856	0.383428	+	0.923571i	$0.374743\pi$
$242$	−2022.35	−0.537196
$243$	2744.80	0.724605
$244$	−1345.07	−0.352907
$245$	0	0
$246$	1139.93	0.295445
$247$	5723.57	1.47442
$248$	−1264.55	−0.323786
$249$	−1034.48	−0.263284
$250$	2320.22	0.586974
$251$	−490.005	−0.123223	−0.0616113	−	0.998100i	$-0.519624\pi$
$251$	−490.005	−0.123223	−0.0616113	+	0.998100i	$0.519624\pi$
$252$	0	0
$253$	−411.324	−0.102212
$254$	4244.26	1.04846
$255$	960.819	0.235956
$256$	256.000	0.0625000
$257$	6181.91	1.50046	0.750228	−	0.661179i	$-0.229944\pi$
$257$	6181.91	1.50046	0.750228	+	0.661179i	$0.229944\pi$
$258$	4743.96	1.14475
$259$	0	0
$260$	−1335.18	−0.318478
$261$	−631.440	−0.149751
$262$	−2388.22	−0.563148
$263$	7467.05	1.75072	0.875358	−	0.483476i	$-0.160626\pi$
$263$	7467.05	1.75072	0.875358	+	0.483476i	$0.160626\pi$
$264$	873.574	0.203654
$265$	−756.734	−0.175418
$266$	0	0
$267$	928.049	0.212718
$268$	−1542.00	−0.351465
$269$	7812.67	1.77081	0.885403	−	0.464824i	$-0.153882\pi$
$269$	7812.67	1.77081	0.885403	+	0.464824i	$0.153882\pi$
$270$	−1062.45	−0.239477
$271$	3472.68	0.778415	0.389207	−	0.921150i	$-0.372749\pi$
$271$	3472.68	0.778415	0.389207	+	0.921150i	$0.372749\pi$
$272$	483.782	0.107844
$273$	0	0
$274$	1139.49	0.251237
$275$	1751.09	0.383981
$276$	−561.747	−0.122512
$277$	−1968.94	−0.427084	−0.213542	−	0.976934i	$-0.568500\pi$
$277$	−1968.94	−0.427084	−0.213542	+	0.976934i	$0.568500\pi$
$278$	4325.86	0.933266
$279$	−1625.36	−0.348773
$280$	0	0
$281$	8301.12	1.76229	0.881144	−	0.472847i	$-0.156774\pi$
$281$	8301.12	1.76229	0.881144	+	0.472847i	$0.156774\pi$
$282$	−1638.21	−0.345936
$283$	6468.15	1.35863	0.679314	−	0.733848i	$-0.262278\pi$
$283$	6468.15	1.35863	0.679314	+	0.733848i	$0.262278\pi$
$284$	−378.193	−0.0790199
$285$	2835.68	0.589373
$286$	−2294.08	−0.474307
$287$	0	0
$288$	329.044	0.0673232
$289$	−3998.76	−0.813914
$290$	639.171	0.129425
$291$	−2105.93	−0.424234
$292$	4340.17	0.869825
$293$	−7281.48	−1.45184	−0.725919	−	0.687780i	$-0.758585\pi$
$293$	−7281.48	−1.45184	−0.725919	+	0.687780i	$0.758585\pi$
$294$	0	0
$295$	243.731	0.0481037
$296$	−887.641	−0.174301
$297$	−1825.49	−0.356651
$298$	−483.476	−0.0939833
$299$	1475.19	0.285327
$300$	2391.47	0.460239
$301$	0	0
$302$	−271.982	−0.0518239
$303$	3139.30	0.595208
$304$	1427.79	0.269374
$305$	1750.02	0.328544
$306$	621.818	0.116167
$307$	4371.49	0.812684	0.406342	−	0.913721i	$-0.366804\pi$
$307$	4371.49	0.812684	0.406342	+	0.913721i	$0.366804\pi$
$308$	0	0
$309$	−2200.10	−0.405046
$310$	1645.26	0.301434
$311$	4794.62	0.874206	0.437103	−	0.899412i	$-0.356005\pi$
$311$	4794.62	0.874206	0.437103	+	0.899412i	$0.356005\pi$
$312$	−3133.03	−0.568503
$313$	−4250.11	−0.767509	−0.383755	−	0.923435i	$-0.625369\pi$
$313$	−4250.11	−0.767509	−0.383755	+	0.923435i	$0.625369\pi$
$314$	1490.09	0.267804
$315$	0	0
$316$	2901.06	0.516447
$317$	−4167.97	−0.738475	−0.369237	−	0.929335i	$-0.620381\pi$
$317$	−4167.97	−0.738475	−0.369237	+	0.929335i	$0.620381\pi$
$318$	−1775.69	−0.313132
$319$	1098.21	0.192752
$320$	−333.072	−0.0581853
$321$	−2606.54	−0.453218
$322$	0	0
$323$	2698.21	0.464806
$324$	−3603.59	−0.617900
$325$	−6280.21	−1.07189
$326$	4638.79	0.788095
$327$	−1457.67	−0.246511
$328$	−746.769	−0.125712
$329$	0	0
$330$	−1136.58	−0.189595
$331$	−9969.28	−1.65547	−0.827736	−	0.561118i	$-0.810371\pi$
$331$	−9969.28	−1.65547	−0.827736	+	0.561118i	$0.810371\pi$
$332$	677.689	0.112027
$333$	−1140.91	−0.187752
$334$	6520.32	1.06819
$335$	2006.24	0.327202
$336$	0	0
$337$	2931.11	0.473792	0.236896	−	0.971535i	$-0.423870\pi$
$337$	2931.11	0.473792	0.236896	+	0.971535i	$0.423870\pi$
$338$	3833.60	0.616924
$339$	−4843.31	−0.775966
$340$	−629.431	−0.100399
$341$	2826.85	0.448922
$342$	1835.18	0.290162
$343$	0	0
$344$	−3107.76	−0.487091
$345$	730.869	0.114054
$346$	−5779.43	−0.897988
$347$	9969.80	1.54238	0.771191	−	0.636603i	$-0.219661\pi$
$347$	9969.80	1.54238	0.771191	+	0.636603i	$0.219661\pi$
$348$	1499.83	0.231032
$349$	−10062.2	−1.54332	−0.771659	−	0.636037i	$-0.780573\pi$
$349$	−10062.2	−1.54332	−0.771659	+	0.636037i	$0.780573\pi$
$350$	0	0
$351$	6547.01	0.995594
$352$	−572.278	−0.0866548
$353$	−2932.91	−0.442218	−0.221109	−	0.975249i	$-0.570968\pi$
$353$	−2932.91	−0.442218	−0.221109	+	0.975249i	$0.570968\pi$
$354$	571.921	0.0858679
$355$	492.053	0.0735648
$356$	−607.964	−0.0905113
$357$	0	0
$358$	7534.23	1.11228
$359$	−4482.82	−0.659036	−0.329518	−	0.944149i	$-0.606886\pi$
$359$	−4482.82	−0.659036	−0.329518	+	0.944149i	$0.606886\pi$
$360$	−428.107	−0.0626756
$361$	1104.27	0.160996
$362$	1556.51	0.225990
$363$	6174.18	0.892728
$364$	0	0
$365$	−5646.83	−0.809777
$366$	4106.47	0.586472
$367$	9116.95	1.29673	0.648367	−	0.761328i	$-0.275452\pi$
$367$	9116.95	1.29673	0.648367	+	0.761328i	$0.275452\pi$
$368$	368.000	0.0521286
$369$	−959.842	−0.135413
$370$	1154.88	0.162268
$371$	0	0
$372$	3860.64	0.538077
$373$	7015.28	0.973827	0.486914	−	0.873450i	$-0.338123\pi$
$373$	7015.28	0.973827	0.486914	+	0.873450i	$0.338123\pi$
$374$	−1081.48	−0.149523
$375$	−7083.57	−0.975451
$376$	1073.19	0.147196
$377$	−3938.67	−0.538069
$378$	0	0
$379$	10704.4	1.45079	0.725394	−	0.688334i	$-0.241658\pi$
$379$	10704.4	1.45079	0.725394	+	0.688334i	$0.241658\pi$
$380$	−1857.65	−0.250778
$381$	−12957.6	−1.74236
$382$	−7446.05	−0.997312
$383$	8442.79	1.12639	0.563194	−	0.826325i	$-0.309572\pi$
$383$	8442.79	1.12639	0.563194	+	0.826325i	$0.309572\pi$
$384$	−781.561	−0.103864
$385$	0	0
$386$	−6956.60	−0.917310
$387$	−3994.49	−0.524681
$388$	1379.59	0.180511
$389$	4443.70	0.579189	0.289595	−	0.957149i	$-0.406480\pi$
$389$	4443.70	0.579189	0.289595	+	0.957149i	$0.406480\pi$
$390$	4076.27	0.529257
$391$	695.437	0.0899483
$392$	0	0
$393$	7291.18	0.935856
$394$	9878.72	1.26315
$395$	−3774.46	−0.480794
$396$	−735.564	−0.0933421
$397$	11776.5	1.48878	0.744390	−	0.667745i	$-0.232740\pi$
$397$	11776.5	1.48878	0.744390	+	0.667745i	$0.232740\pi$
$398$	−8265.12	−1.04094
$399$	0	0
$400$	−1566.65	−0.195831
$401$	−190.858	−0.0237681	−0.0118840	−	0.999929i	$-0.503783\pi$
$401$	−190.858	−0.0237681	−0.0118840	+	0.999929i	$0.503783\pi$
$402$	4707.69	0.584075
$403$	−10138.4	−1.25317
$404$	−2056.55	−0.253260
$405$	4688.50	0.575244
$406$	0	0
$407$	1984.28	0.241664
$408$	−1476.98	−0.179219
$409$	−2722.13	−0.329097	−0.164549	−	0.986369i	$-0.552617\pi$
$409$	−2722.13	−0.329097	−0.164549	+	0.986369i	$0.552617\pi$
$410$	971.593	0.117033
$411$	−3478.83	−0.417514
$412$	1441.28	0.172347
$413$	0	0
$414$	473.000	0.0561514
$415$	−881.716	−0.104293
$416$	2052.44	0.241898
$417$	−13206.8	−1.55093
$418$	−3191.78	−0.373481
$419$	873.607	0.101858	0.0509290	−	0.998702i	$-0.483782\pi$
$419$	873.607	0.101858	0.0509290	+	0.998702i	$0.483782\pi$
$420$	0	0
$421$	−7984.18	−0.924287	−0.462144	−	0.886805i	$-0.652920\pi$
$421$	−7984.18	−0.924287	−0.462144	+	0.886805i	$0.652920\pi$
$422$	−4249.55	−0.490202
$423$	1379.40	0.158555
$424$	1163.26	0.133237
$425$	−2960.62	−0.337909
$426$	1154.61	0.131318
$427$	0	0
$428$	1707.54	0.192844
$429$	7003.76	0.788216
$430$	4043.40	0.453465
$431$	6074.49	0.678882	0.339441	−	0.940627i	$-0.389762\pi$
$431$	6074.49	0.678882	0.339441	+	0.940627i	$0.389762\pi$
$432$	1633.21	0.181893
$433$	−332.495	−0.0369023	−0.0184511	−	0.999830i	$-0.505874\pi$
$433$	−332.495	−0.0369023	−0.0184511	+	0.999830i	$0.505874\pi$
$434$	0	0
$435$	−1951.37	−0.215083
$436$	954.915	0.104890
$437$	2052.45	0.224673
$438$	−13250.4	−1.44550
$439$	−1005.04	−0.109267	−0.0546333	−	0.998506i	$-0.517399\pi$
$439$	−1005.04	−0.109267	−0.0546333	+	0.998506i	$0.517399\pi$
$440$	744.569	0.0806726
$441$	0	0
$442$	3878.66	0.417396
$443$	−10495.2	−1.12560	−0.562800	−	0.826593i	$-0.690276\pi$
$443$	−10495.2	−1.12560	−0.562800	+	0.826593i	$0.690276\pi$
$444$	2709.95	0.289658
$445$	790.999	0.0842628
$446$	−10171.2	−1.07986
$447$	1476.04	0.156184
$448$	0	0
$449$	−3788.61	−0.398208	−0.199104	−	0.979978i	$-0.563803\pi$
$449$	−3788.61	−0.398208	−0.199104	+	0.979978i	$0.563803\pi$
$450$	−2013.66	−0.210944
$451$	1669.37	0.174296
$452$	3172.85	0.330173
$453$	830.354	0.0861224
$454$	4229.73	0.437249
$455$	0	0
$456$	−4359.02	−0.447653
$457$	−5627.87	−0.576062	−0.288031	−	0.957621i	$-0.593001\pi$
$457$	−5627.87	−0.576062	−0.288031	+	0.957621i	$0.593001\pi$
$458$	8631.09	0.880577
$459$	3086.40	0.313858
$460$	−478.791	−0.0485299
$461$	12436.8	1.25649	0.628243	−	0.778017i	$-0.283774\pi$
$461$	12436.8	1.25649	0.628243	+	0.778017i	$0.283774\pi$
$462$	0	0
$463$	7017.24	0.704360	0.352180	−	0.935932i	$-0.385440\pi$
$463$	7017.24	0.704360	0.352180	+	0.935932i	$0.385440\pi$
$464$	−982.536	−0.0983040
$465$	−5022.93	−0.500931
$466$	−1905.17	−0.189389
$467$	−8428.61	−0.835181	−0.417591	−	0.908635i	$-0.637125\pi$
$467$	−8428.61	−0.835181	−0.417591	+	0.908635i	$0.637125\pi$
$468$	2638.06	0.260565
$469$	0	0
$470$	−1396.29	−0.137034
$471$	−4549.20	−0.445045
$472$	−374.665	−0.0365367
$473$	6947.28	0.675341
$474$	−8856.85	−0.858246
$475$	−8737.72	−0.844030
$476$	0	0
$477$	1495.16	0.143519
$478$	8140.84	0.778982
$479$	13500.8	1.28782	0.643911	−	0.765101i	$-0.277311\pi$
$479$	13500.8	1.28782	0.643911	+	0.765101i	$0.277311\pi$
$480$	1016.86	0.0966940
$481$	−7116.54	−0.674608
$482$	5738.11	0.542249
$483$	0	0
$484$	−4044.70	−0.379855
$485$	−1794.94	−0.168049
$486$	5489.60	0.512373
$487$	13315.2	1.23896	0.619478	−	0.785014i	$-0.287344\pi$
$487$	13315.2	1.23896	0.619478	+	0.785014i	$0.287344\pi$
$488$	−2690.14	−0.249543
$489$	−14162.1	−1.30968
$490$	0	0
$491$	10760.2	0.989006	0.494503	−	0.869176i	$-0.335350\pi$
$491$	10760.2	0.989006	0.494503	+	0.869176i	$0.335350\pi$
$492$	2279.87	0.208911
$493$	−1856.77	−0.169624
$494$	11447.1	1.04257
$495$	957.015	0.0868982
$496$	−2529.10	−0.228951
$497$	0	0
$498$	−2068.97	−0.186170
$499$	−3699.17	−0.331859	−0.165929	−	0.986138i	$-0.553062\pi$
$499$	−3699.17	−0.331859	−0.165929	+	0.986138i	$0.553062\pi$
$500$	4640.44	0.415054
$501$	−19906.4	−1.77515
$502$	−980.011	−0.0871315
$503$	−9014.79	−0.799104	−0.399552	−	0.916710i	$-0.630834\pi$
$503$	−9014.79	−0.799104	−0.399552	+	0.916710i	$0.630834\pi$
$504$	0	0
$505$	2675.70	0.235777
$506$	−822.649	−0.0722751
$507$	−11703.9	−1.02522
$508$	8488.51	0.741372
$509$	16396.9	1.42786	0.713928	−	0.700219i	$-0.246915\pi$
$509$	16396.9	1.42786	0.713928	+	0.700219i	$0.246915\pi$
$510$	1921.64	0.166846
$511$	0	0
$512$	512.000	0.0441942
$513$	9108.93	0.783955
$514$	12363.8	1.06098
$515$	−1875.20	−0.160449
$516$	9487.92	0.809462
$517$	−2399.07	−0.204083
$518$	0	0
$519$	17644.4	1.49230
$520$	−2670.36	−0.225198
$521$	4259.59	0.358188	0.179094	−	0.983832i	$-0.442683\pi$
$521$	4259.59	0.358188	0.179094	+	0.983832i	$0.442683\pi$
$522$	−1262.88	−0.105890
$523$	15541.2	1.29937	0.649686	−	0.760203i	$-0.274901\pi$
$523$	15541.2	1.29937	0.649686	+	0.760203i	$0.274901\pi$
$524$	−4776.44	−0.398206
$525$	0	0
$526$	14934.1	1.23794
$527$	−4779.43	−0.395057
$528$	1747.15	0.144005
$529$	529.000	0.0434783
$530$	−1513.47	−0.124039
$531$	−481.566	−0.0393563
$532$	0	0
$533$	−5987.12	−0.486549
$534$	1856.10	0.150414
$535$	−2221.62	−0.179531
$536$	−3084.00	−0.248524
$537$	−23001.8	−1.84842
$538$	15625.3	1.25215
$539$	0	0
$540$	−2124.91	−0.169336
$541$	−15295.4	−1.21553	−0.607765	−	0.794117i	$-0.707934\pi$
$541$	−15295.4	−1.21553	−0.607765	+	0.794117i	$0.707934\pi$
$542$	6945.36	0.550422
$543$	−4751.98	−0.375556
$544$	967.565	0.0762573
$545$	−1242.41	−0.0976492
$546$	0	0
$547$	6475.62	0.506175	0.253087	−	0.967443i	$-0.418554\pi$
$547$	6475.62	0.506175	0.253087	+	0.967443i	$0.418554\pi$
$548$	2278.98	0.177652
$549$	−3457.71	−0.268801
$550$	3502.19	0.271516
$551$	−5479.92	−0.423688
$552$	−1123.49	−0.0866288
$553$	0	0
$554$	−3937.89	−0.301994
$555$	−3525.81	−0.269662
$556$	8651.73	0.659919
$557$	−9228.26	−0.702000	−0.351000	−	0.936375i	$-0.614158\pi$
$557$	−9228.26	−0.702000	−0.351000	+	0.936375i	$0.614158\pi$
$558$	−3250.72	−0.246620
$559$	−24916.1	−1.88522
$560$	0	0
$561$	3301.72	0.248482
$562$	16602.2	1.24613
$563$	−3158.50	−0.236438	−0.118219	−	0.992988i	$-0.537719\pi$
$563$	−3158.50	−0.236438	−0.118219	+	0.992988i	$0.537719\pi$
$564$	−3276.42	−0.244614
$565$	−4128.08	−0.307380
$566$	12936.3	0.960694
$567$	0	0
$568$	−756.387	−0.0558755
$569$	2719.58	0.200371	0.100185	−	0.994969i	$-0.468056\pi$
$569$	2719.58	0.200371	0.100185	+	0.994969i	$0.468056\pi$
$570$	5671.36	0.416749
$571$	−14509.7	−1.06342	−0.531710	−	0.846926i	$-0.678450\pi$
$571$	−14509.7	−1.06342	−0.531710	+	0.846926i	$0.678450\pi$
$572$	−4588.16	−0.335385
$573$	22732.6	1.65736
$574$	0	0
$575$	−2252.06	−0.163335
$576$	658.087	0.0476047
$577$	4303.60	0.310505	0.155252	−	0.987875i	$-0.450381\pi$
$577$	4303.60	0.310505	0.155252	+	0.987875i	$0.450381\pi$
$578$	−7997.52	−0.575524
$579$	21238.3	1.52441
$580$	1278.34	0.0915176
$581$	0	0
$582$	−4211.87	−0.299978
$583$	−2600.41	−0.184731
$584$	8680.33	0.615059
$585$	−3432.29	−0.242577
$586$	−14563.0	−1.02660
$587$	4929.62	0.346622	0.173311	−	0.984867i	$-0.444553\pi$
$587$	4929.62	0.346622	0.173311	+	0.984867i	$0.444553\pi$
$588$	0	0
$589$	−14105.6	−0.986776
$590$	487.462	0.0340144
$591$	−30159.5	−2.09915
$592$	−1775.28	−0.123249
$593$	1348.82	0.0934056	0.0467028	−	0.998909i	$-0.485129\pi$
$593$	1348.82	0.0934056	0.0467028	+	0.998909i	$0.485129\pi$
$594$	−3650.97	−0.252190
$595$	0	0
$596$	−966.952	−0.0664562
$597$	25233.2	1.72986
$598$	2950.39	0.201757
$599$	−9007.72	−0.614433	−0.307217	−	0.951640i	$-0.599398\pi$
$599$	−9007.72	−0.614433	−0.307217	+	0.951640i	$0.599398\pi$
$600$	4782.95	0.325438
$601$	18277.2	1.24050	0.620251	−	0.784403i	$-0.287031\pi$
$601$	18277.2	1.24050	0.620251	+	0.784403i	$0.287031\pi$
$602$	0	0
$603$	−3963.95	−0.267702
$604$	−543.964	−0.0366450
$605$	5262.41	0.353632
$606$	6278.59	0.420875
$607$	−2473.95	−0.165428	−0.0827139	−	0.996573i	$-0.526359\pi$
$607$	−2473.95	−0.165428	−0.0827139	+	0.996573i	$0.526359\pi$
$608$	2855.59	0.190476
$609$	0	0
$610$	3500.05	0.232316
$611$	8604.15	0.569700
$612$	1243.64	0.0821422
$613$	−18525.0	−1.22058	−0.610292	−	0.792177i	$-0.708948\pi$
$613$	−18525.0	−1.22058	−0.610292	+	0.792177i	$0.708948\pi$
$614$	8742.97	0.574654
$615$	−2966.25	−0.194489
$616$	0	0
$617$	2856.55	0.186386	0.0931930	−	0.995648i	$-0.470293\pi$
$617$	2856.55	0.186386	0.0931930	+	0.995648i	$0.470293\pi$
$618$	−4400.19	−0.286411
$619$	10015.9	0.650362	0.325181	−	0.945652i	$-0.394575\pi$
$619$	10015.9	0.650362	0.325181	+	0.945652i	$0.394575\pi$
$620$	3290.52	0.213146
$621$	2347.74	0.151709
$622$	9589.24	0.618157
$623$	0	0
$624$	−6266.06	−0.401992
$625$	6201.96	0.396926
$626$	−8500.22	−0.542711
$627$	9744.41	0.620661
$628$	2980.18	0.189366
$629$	−3354.88	−0.212668
$630$	0	0
$631$	−9494.45	−0.598998	−0.299499	−	0.954097i	$-0.596820\pi$
$631$	−9494.45	−0.598998	−0.299499	+	0.954097i	$0.596820\pi$
$632$	5802.11	0.365183
$633$	12973.8	0.814631
$634$	−8335.94	−0.522180
$635$	−11044.1	−0.690191
$636$	−3551.39	−0.221418
$637$	0	0
$638$	2196.42	0.136296
$639$	−972.204	−0.0601875
$640$	−666.144	−0.0411432
$641$	−17412.6	−1.07294	−0.536471	−	0.843919i	$-0.680243\pi$
$641$	−17412.6	−1.07294	−0.536471	+	0.843919i	$0.680243\pi$
$642$	−5213.08	−0.320473
$643$	−7267.11	−0.445703	−0.222851	−	0.974852i	$-0.571536\pi$
$643$	−7267.11	−0.445703	−0.222851	+	0.974852i	$0.571536\pi$
$644$	0	0
$645$	−12344.4	−0.753581
$646$	5396.42	0.328668
$647$	−16217.0	−0.985405	−0.492702	−	0.870198i	$-0.663991\pi$
$647$	−16217.0	−0.985405	−0.492702	+	0.870198i	$0.663991\pi$
$648$	−7207.19	−0.436921
$649$	837.547	0.0506573
$650$	−12560.4	−0.757939
$651$	0	0
$652$	9277.58	0.557267
$653$	20809.4	1.24707	0.623533	−	0.781797i	$-0.285697\pi$
$653$	20809.4	1.24707	0.623533	+	0.781797i	$0.285697\pi$
$654$	−2915.33	−0.174310
$655$	6214.46	0.370716
$656$	−1493.54	−0.0888915
$657$	11157.1	0.662524
$658$	0	0
$659$	−14682.0	−0.867876	−0.433938	−	0.900943i	$-0.642876\pi$
$659$	−14682.0	−0.867876	−0.433938	+	0.900943i	$0.642876\pi$
$660$	−2273.15	−0.134064
$661$	−30806.3	−1.81275	−0.906375	−	0.422474i	$-0.861162\pi$
$661$	−30806.3	−1.81275	−0.906375	+	0.422474i	$0.861162\pi$
$662$	−19938.6	−1.17060
$663$	−11841.4	−0.693641
$664$	1355.38	0.0792151
$665$	0	0
$666$	−2281.82	−0.132761
$667$	−1412.39	−0.0819912
$668$	13040.6	0.755326
$669$	31052.3	1.79455
$670$	4012.48	0.231367
$671$	6013.71	0.345986
$672$	0	0
$673$	12868.9	0.737085	0.368543	−	0.929611i	$-0.379857\pi$
$673$	12868.9	0.737085	0.368543	+	0.929611i	$0.379857\pi$
$674$	5862.23	0.335022
$675$	−9994.80	−0.569926
$676$	7667.20	0.436231
$677$	−34645.7	−1.96683	−0.983415	−	0.181372i	$-0.941946\pi$
$677$	−34645.7	−1.96683	−0.983415	+	0.181372i	$0.941946\pi$
$678$	−9686.62	−0.548691
$679$	0	0
$680$	−1258.86	−0.0709929
$681$	−12913.2	−0.726633
$682$	5653.70	0.317436
$683$	−33846.1	−1.89617	−0.948085	−	0.318017i	$-0.896983\pi$
$683$	−33846.1	−1.89617	−0.948085	+	0.318017i	$0.896983\pi$
$684$	3670.36	0.205175
$685$	−2965.10	−0.165388
$686$	0	0
$687$	−26350.5	−1.46337
$688$	−6215.53	−0.344425
$689$	9326.24	0.515677
$690$	1461.74	0.0806484
$691$	3306.85	0.182053	0.0910265	−	0.995848i	$-0.470985\pi$
$691$	3306.85	0.182053	0.0910265	+	0.995848i	$0.470985\pi$
$692$	−11558.9	−0.634973
$693$	0	0
$694$	19939.6	1.09063
$695$	−11256.4	−0.614362
$696$	2999.66	0.163364
$697$	−2822.45	−0.153383
$698$	−20124.4	−1.09129
$699$	5816.44	0.314733
$700$	0	0
$701$	−9580.87	−0.516212	−0.258106	−	0.966117i	$-0.583098\pi$
$701$	−9580.87	−0.516212	−0.258106	+	0.966117i	$0.583098\pi$
$702$	13094.0	0.703992
$703$	−9901.32	−0.531202
$704$	−1144.56	−0.0612742
$705$	4262.83	0.227727
$706$	−5865.81	−0.312695
$707$	0	0
$708$	1143.84	0.0607178
$709$	−932.829	−0.0494120	−0.0247060	−	0.999695i	$-0.507865\pi$
$709$	−932.829	−0.0494120	−0.0247060	+	0.999695i	$0.507865\pi$
$710$	984.107	0.0520181
$711$	7457.61	0.393365
$712$	−1215.93	−0.0640011
$713$	−3635.58	−0.190959
$714$	0	0
$715$	5969.48	0.312232
$716$	15068.5	0.786501
$717$	−24853.8	−1.29454
$718$	−8965.63	−0.466009
$719$	25295.4	1.31205	0.656023	−	0.754741i	$-0.272237\pi$
$719$	25295.4	1.31205	0.656023	+	0.754741i	$0.272237\pi$
$720$	−856.213	−0.0443183
$721$	0	0
$722$	2208.54	0.113841
$723$	−17518.3	−0.901125
$724$	3113.02	0.159799
$725$	6012.86	0.308016
$726$	12348.4	0.631254
$727$	−12222.1	−0.623512	−0.311756	−	0.950162i	$-0.600917\pi$
$727$	−12222.1	−0.623512	−0.311756	+	0.950162i	$0.600917\pi$
$728$	0	0
$729$	7564.64	0.384324
$730$	−11293.7	−0.572599
$731$	−11745.9	−0.594308
$732$	8212.94	0.414698
$733$	21001.3	1.05825	0.529127	−	0.848543i	$-0.322520\pi$
$733$	21001.3	1.05825	0.529127	+	0.848543i	$0.322520\pi$
$734$	18233.9	0.916929
$735$	0	0
$736$	736.000	0.0368605
$737$	6894.16	0.344572
$738$	−1919.68	−0.0957514
$739$	9387.70	0.467297	0.233648	−	0.972321i	$-0.424934\pi$
$739$	9387.70	0.467297	0.233648	+	0.972321i	$0.424934\pi$
$740$	2309.75	0.114741
$741$	−34947.8	−1.73258
$742$	0	0
$743$	14386.9	0.710370	0.355185	−	0.934796i	$-0.384418\pi$
$743$	14386.9	0.710370	0.355185	+	0.934796i	$0.384418\pi$
$744$	7721.27	0.380478
$745$	1258.07	0.0618684
$746$	14030.6	0.688600
$747$	1742.10	0.0853282
$748$	−2162.95	−0.105729
$749$	0	0
$750$	−14167.1	−0.689748
$751$	23158.3	1.12525	0.562623	−	0.826714i	$-0.309792\pi$
$751$	23158.3	1.12525	0.562623	+	0.826714i	$0.309792\pi$
$752$	2146.38	0.104083
$753$	2991.95	0.144798
$754$	−7877.34	−0.380472
$755$	707.732	0.0341152
$756$	0	0
$757$	−16683.3	−0.801011	−0.400505	−	0.916294i	$-0.631165\pi$
$757$	−16683.3	−0.801011	−0.400505	+	0.916294i	$0.631165\pi$
$758$	21408.8	1.02586
$759$	2511.53	0.120109
$760$	−3715.30	−0.177327
$761$	−33544.5	−1.59788	−0.798939	−	0.601411i	$-0.794605\pi$
$761$	−33544.5	−1.59788	−0.798939	+	0.601411i	$0.794605\pi$
$762$	−25915.2	−1.23203
$763$	0	0
$764$	−14892.1	−0.705206
$765$	−1618.05	−0.0764715
$766$	16885.6	0.796476
$767$	−3003.82	−0.141410
$768$	−1563.12	−0.0734432
$769$	−8595.19	−0.403056	−0.201528	−	0.979483i	$-0.564591\pi$
$769$	−8595.19	−0.403056	−0.201528	+	0.979483i	$0.564591\pi$
$770$	0	0
$771$	−37746.4	−1.76317
$772$	−13913.2	−0.648636
$773$	21270.6	0.989718	0.494859	−	0.868973i	$-0.335220\pi$
$773$	21270.6	0.989718	0.494859	+	0.868973i	$0.335220\pi$
$774$	−7988.98	−0.371005
$775$	15477.4	0.717374
$776$	2759.19	0.127641
$777$	0	0
$778$	8887.40	0.409549
$779$	−8329.94	−0.383121
$780$	8152.54	0.374241
$781$	1690.87	0.0774701
$782$	1390.87	0.0636030
$783$	−6268.30	−0.286093
$784$	0	0
$785$	−3877.40	−0.176293
$786$	14582.4	0.661750
$787$	−5126.37	−0.232192	−0.116096	−	0.993238i	$-0.537038\pi$
$787$	−5126.37	−0.232192	−0.116096	+	0.993238i	$0.537038\pi$
$788$	19757.4	0.893185
$789$	−45593.4	−2.05725
$790$	−7548.92	−0.339973
$791$	0	0
$792$	−1471.13	−0.0660028
$793$	−21567.9	−0.965822
$794$	23553.0	1.05273
$795$	4620.58	0.206132
$796$	−16530.2	−0.736054
$797$	−3947.55	−0.175445	−0.0877223	−	0.996145i	$-0.527959\pi$
$797$	−3947.55	−0.175445	−0.0877223	+	0.996145i	$0.527959\pi$
$798$	0	0
$799$	4056.17	0.179596
$800$	−3133.30	−0.138474
$801$	−1562.86	−0.0689402
$802$	−381.716	−0.0168066
$803$	−19404.5	−0.852766
$804$	9415.38	0.413004
$805$	0	0
$806$	−20276.7	−0.886125
$807$	−47703.8	−2.08086
$808$	−4113.10	−0.179082
$809$	14952.8	0.649830	0.324915	−	0.945743i	$-0.394664\pi$
$809$	14952.8	0.649830	0.324915	+	0.945743i	$0.394664\pi$
$810$	9377.01	0.406759
$811$	6095.32	0.263916	0.131958	−	0.991255i	$-0.457874\pi$
$811$	6095.32	0.263916	0.131958	+	0.991255i	$0.457874\pi$
$812$	0	0
$813$	−21204.0	−0.914707
$814$	3968.57	0.170882
$815$	−12070.7	−0.518796
$816$	−2953.95	−0.126727
$817$	−34666.0	−1.48447
$818$	−5444.27	−0.232707
$819$	0	0
$820$	1943.19	0.0827549
$821$	−116.814	−0.00496571	−0.00248285	−	0.999997i	$-0.500790\pi$
$821$	−116.814	−0.00496571	−0.00248285	+	0.999997i	$0.500790\pi$
$822$	−6957.66	−0.295227
$823$	10320.1	0.437104	0.218552	−	0.975825i	$-0.429867\pi$
$823$	10320.1	0.437104	0.218552	+	0.975825i	$0.429867\pi$
$824$	2882.56	0.121867
$825$	−10692.1	−0.451213
$826$	0	0
$827$	45399.2	1.90893	0.954464	−	0.298325i	$-0.0964281\pi$
$827$	45399.2	1.90893	0.954464	+	0.298325i	$0.0964281\pi$
$828$	946.000	0.0397051
$829$	28762.4	1.20502	0.602509	−	0.798112i	$-0.294168\pi$
$829$	28762.4	1.20502	0.602509	+	0.798112i	$0.294168\pi$
$830$	−1763.43	−0.0737465
$831$	12022.3	0.501863
$832$	4104.89	0.171047
$833$	0	0
$834$	−26413.5	−1.09667
$835$	−16966.7	−0.703182
$836$	−6383.55	−0.264091
$837$	−16134.9	−0.666314
$838$	1747.21	0.0720245
$839$	−31285.0	−1.28734	−0.643671	−	0.765303i	$-0.722589\pi$
$839$	−31285.0	−1.28734	−0.643671	+	0.765303i	$0.722589\pi$
$840$	0	0
$841$	−20618.0	−0.845381
$842$	−15968.4	−0.653570
$843$	−50686.2	−2.07085
$844$	−8499.11	−0.346625
$845$	−9975.51	−0.406116
$846$	2758.80	0.112115
$847$	0	0
$848$	2326.51	0.0942131
$849$	−39494.2	−1.59651
$850$	−5921.24	−0.238937
$851$	−2551.97	−0.102797
$852$	2309.23	0.0928555
$853$	29226.1	1.17313	0.586566	−	0.809901i	$-0.300479\pi$
$853$	29226.1	1.17313	0.586566	+	0.809901i	$0.300479\pi$
$854$	0	0
$855$	−4775.38	−0.191011
$856$	3415.08	0.136361
$857$	9051.41	0.360782	0.180391	−	0.983595i	$-0.442264\pi$
$857$	9051.41	0.360782	0.180391	+	0.983595i	$0.442264\pi$
$858$	14007.5	0.557353
$859$	10204.6	0.405326	0.202663	−	0.979249i	$-0.435040\pi$
$859$	10204.6	0.405326	0.202663	+	0.979249i	$0.435040\pi$
$860$	8086.79	0.320648
$861$	0	0
$862$	12149.0	0.480042
$863$	−16529.1	−0.651978	−0.325989	−	0.945373i	$-0.605697\pi$
$863$	−16529.1	−0.651978	−0.325989	+	0.945373i	$0.605697\pi$
$864$	3266.42	0.128618
$865$	15038.8	0.591138
$866$	−664.990	−0.0260938
$867$	24416.2	0.956423
$868$	0	0
$869$	−12970.4	−0.506318
$870$	−3902.74	−0.152087
$871$	−24725.6	−0.961876
$872$	1909.83	0.0741686
$873$	3546.46	0.137491
$874$	4104.91	0.158868
$875$	0	0
$876$	−26500.8	−1.02212
$877$	−14080.8	−0.542159	−0.271079	−	0.962557i	$-0.587381\pi$
$877$	−14080.8	−0.542159	−0.271079	+	0.962557i	$0.587381\pi$
$878$	−2010.08	−0.0772631
$879$	44460.3	1.70604
$880$	1489.14	0.0570442
$881$	49428.8	1.89024	0.945119	−	0.326727i	$-0.105946\pi$
$881$	49428.8	1.89024	0.945119	+	0.326727i	$0.105946\pi$
$882$	0	0
$883$	20305.2	0.773869	0.386934	−	0.922107i	$-0.373534\pi$
$883$	20305.2	0.773869	0.386934	+	0.922107i	$0.373534\pi$
$884$	7757.32	0.295143
$885$	−1488.21	−0.0565261
$886$	−20990.4	−0.795919
$887$	12794.6	0.484328	0.242164	−	0.970235i	$-0.422143\pi$
$887$	12794.6	0.484328	0.242164	+	0.970235i	$0.422143\pi$
$888$	5419.89	0.204819
$889$	0	0
$890$	1582.00	0.0595828
$891$	16111.4	0.605781
$892$	−20342.3	−0.763579
$893$	11971.0	0.448595
$894$	2952.08	0.110439
$895$	−19605.0	−0.732205
$896$	0	0
$897$	−9007.46	−0.335285
$898$	−7577.21	−0.281576
$899$	9706.76	0.360110
$900$	−4027.32	−0.149160
$901$	4396.58	0.162565
$902$	3338.74	0.123246
$903$	0	0
$904$	6345.70	0.233468
$905$	−4050.23	−0.148767
$906$	1660.71	0.0608977
$907$	−40019.0	−1.46506	−0.732530	−	0.680734i	$-0.761661\pi$
$907$	−40019.0	−1.46506	−0.732530	+	0.680734i	$0.761661\pi$
$908$	8459.45	0.309182
$909$	−5286.68	−0.192902
$910$	0	0
$911$	−42922.5	−1.56102	−0.780508	−	0.625146i	$-0.785039\pi$
$911$	−42922.5	−1.56102	−0.780508	+	0.625146i	$0.785039\pi$
$912$	−8718.04	−0.316539
$913$	−3029.89	−0.109830
$914$	−11255.7	−0.407338
$915$	−10685.6	−0.386069
$916$	17262.2	0.622662
$917$	0	0
$918$	6172.79	0.221931
$919$	−14773.7	−0.530294	−0.265147	−	0.964208i	$-0.585421\pi$
$919$	−14773.7	−0.530294	−0.265147	+	0.964208i	$0.585421\pi$
$920$	−957.583	−0.0343158
$921$	−26692.1	−0.954977
$922$	24873.6	0.888470
$923$	−6064.23	−0.216258
$924$	0	0
$925$	10864.3	0.386178
$926$	14034.5	0.498058
$927$	3705.03	0.131272
$928$	−1965.07	−0.0695114
$929$	−28082.1	−0.991757	−0.495879	−	0.868392i	$-0.665154\pi$
$929$	−28082.1	−0.991757	−0.495879	+	0.868392i	$0.665154\pi$
$930$	−10045.9	−0.354212
$931$	0	0
$932$	−3810.34	−0.133918
$933$	−29275.7	−1.02727
$934$	−16857.2	−0.590562
$935$	2814.14	0.0984301
$936$	5276.12	0.184247
$937$	−43990.5	−1.53373	−0.766866	−	0.641808i	$-0.778185\pi$
$937$	−43990.5	−1.53373	−0.766866	+	0.641808i	$0.778185\pi$
$938$	0	0
$939$	25951.0	0.901893
$940$	−2792.57	−0.0968976
$941$	−51810.9	−1.79489	−0.897443	−	0.441130i	$-0.854578\pi$
$941$	−51810.9	−1.79489	−0.897443	+	0.441130i	$0.854578\pi$
$942$	−9098.41	−0.314694
$943$	−2146.96	−0.0741407
$944$	−749.329	−0.0258354
$945$	0	0
$946$	13894.6	0.477538
$947$	29776.6	1.02176	0.510881	−	0.859651i	$-0.329319\pi$
$947$	29776.6	1.02176	0.510881	+	0.859651i	$0.329319\pi$
$948$	−17713.7	−0.606872
$949$	69593.4	2.38050
$950$	−17475.4	−0.596819
$951$	25449.4	0.867774
$952$	0	0
$953$	23763.0	0.807722	0.403861	−	0.914820i	$-0.367668\pi$
$953$	23763.0	0.807722	0.403861	+	0.914820i	$0.367668\pi$
$954$	2990.33	0.101484
$955$	19375.6	0.656522
$956$	16281.7	0.550824
$957$	−6705.61	−0.226501
$958$	27001.6	0.910627
$959$	0	0
$960$	2033.72	0.0683730
$961$	−4805.29	−0.161300
$962$	−14233.1	−0.477020
$963$	4389.50	0.146884
$964$	11476.2	0.383428
$965$	18102.0	0.603858
$966$	0	0
$967$	−37060.0	−1.23244	−0.616219	−	0.787575i	$-0.711336\pi$
$967$	−37060.0	−1.23244	−0.616219	+	0.787575i	$0.711336\pi$
$968$	−8089.39	−0.268598
$969$	−16475.1	−0.546189
$970$	−3589.88	−0.118829
$971$	38147.8	1.26078	0.630391	−	0.776278i	$-0.282894\pi$
$971$	38147.8	1.26078	0.630391	+	0.776278i	$0.282894\pi$
$972$	10979.2	0.362303
$973$	0	0
$974$	26630.5	0.876074
$975$	38346.6	1.25956
$976$	−5380.29	−0.176454
$977$	52503.0	1.71926	0.859632	−	0.510914i	$-0.170693\pi$
$977$	52503.0	1.71926	0.859632	+	0.510914i	$0.170693\pi$
$978$	−28324.2	−0.926083
$979$	2718.16	0.0887361
$980$	0	0
$981$	2454.76	0.0798923
$982$	21520.4	0.699333
$983$	15812.2	0.513054	0.256527	−	0.966537i	$-0.417422\pi$
$983$	15812.2	0.513054	0.256527	+	0.966537i	$0.417422\pi$
$984$	4559.73	0.147722
$985$	−25705.7	−0.831524
$986$	−3713.54	−0.119942
$987$	0	0
$988$	22894.3	0.737211
$989$	−8934.82	−0.287271
$990$	1914.03	0.0614463
$991$	34918.9	1.11931	0.559655	−	0.828726i	$-0.310934\pi$
$991$	34918.9	1.11931	0.559655	+	0.828726i	$0.310934\pi$
$992$	−5058.20	−0.161893
$993$	60871.9	1.94533
$994$	0	0
$995$	21506.9	0.685241
$996$	−4137.93	−0.131642
$997$	16065.5	0.510331	0.255165	−	0.966897i	$-0.417870\pi$
$997$	16065.5	0.510331	0.255165	+	0.966897i	$0.417870\pi$
$998$	−7398.34	−0.234660
$999$	−11325.8	−0.358691

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.y.1.3		11
7.2	even	3		322.4.e.a.277.9	yes	22
7.4	even	3		322.4.e.a.93.9	✓	22
7.6	odd	2		2254.4.a.v.1.9		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.a.93.9	✓	22	7.4	even	3
322.4.e.a.277.9	yes	22	7.2	even	3
2254.4.a.v.1.9		11	7.6	odd	2
2254.4.a.y.1.3		11	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -6.10595 q^{3} +4.00000 q^{4} -5.20425 q^{5} -12.2119 q^{6} +8.00000 q^{8} +10.2826 q^{9} -10.4085 q^{10} -17.8837 q^{11} -24.4238 q^{12} +64.1389 q^{13} +31.7769 q^{15} +16.0000 q^{16} +30.2364 q^{17} +20.5652 q^{18} +89.2372 q^{19} -20.8170 q^{20} -35.7673 q^{22} +23.0000 q^{23} -48.8476 q^{24} -97.9157 q^{25} +128.278 q^{26} +102.076 q^{27} -61.4085 q^{29} +63.5538 q^{30} -158.069 q^{31} +32.0000 q^{32} +109.197 q^{33} +60.4728 q^{34} +41.1305 q^{36} -110.955 q^{37} +178.474 q^{38} -391.629 q^{39} -41.6340 q^{40} -93.3461 q^{41} -388.470 q^{43} -71.5347 q^{44} -53.5133 q^{45} +46.0000 q^{46} +134.149 q^{47} -97.6952 q^{48} -195.831 q^{50} -184.622 q^{51} +256.556 q^{52} +145.407 q^{53} +204.151 q^{54} +93.0712 q^{55} -544.877 q^{57} -122.817 q^{58} -46.8331 q^{59} +127.108 q^{60} -336.268 q^{61} -316.137 q^{62} +64.0000 q^{64} -333.795 q^{65} +218.394 q^{66} -385.500 q^{67} +120.946 q^{68} -140.437 q^{69} -94.5483 q^{71} +82.2609 q^{72} +1085.04 q^{73} -221.910 q^{74} +597.869 q^{75} +356.949 q^{76} -783.258 q^{78} +725.264 q^{79} -83.2681 q^{80} -900.898 q^{81} -186.692 q^{82} +169.422 q^{83} -157.358 q^{85} -776.941 q^{86} +374.957 q^{87} -143.069 q^{88} -151.991 q^{89} -107.027 q^{90} +92.0000 q^{92} +965.159 q^{93} +268.297 q^{94} -464.413 q^{95} -195.390 q^{96} +344.898 q^{97} -183.891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 22 q^{2} + 18 q^{3} + 44 q^{4} + 33 q^{5} + 36 q^{6} + 88 q^{8} + 171 q^{9} + 66 q^{10} + 8 q^{11} + 72 q^{12} + 185 q^{13} - 186 q^{15} + 176 q^{16} + 107 q^{17} + 342 q^{18} + 114 q^{19} + 132 q^{20}+ \cdots - 1729 q^{99}+O(q^{100}) \) 11 * q + 22 * q^2 + 18 * q^3 + 44 * q^4 + 33 * q^5 + 36 * q^6 + 88 * q^8 + 171 * q^9 + 66 * q^10 + 8 * q^11 + 72 * q^12 + 185 * q^13 - 186 * q^15 + 176 * q^16 + 107 * q^17 + 342 * q^18 + 114 * q^19 + 132 * q^20 + 16 * q^22 + 253 * q^23 + 144 * q^24 + 494 * q^25 + 370 * q^26 + 561 * q^27 - 79 * q^29 - 372 * q^30 + 488 * q^31 + 352 * q^32 + 304 * q^33 + 214 * q^34 + 684 * q^36 - 41 * q^37 + 228 * q^38 + 425 * q^39 + 264 * q^40 + 812 * q^41 + 219 * q^43 + 32 * q^44 + 619 * q^45 + 506 * q^46 + 1141 * q^47 + 288 * q^48 + 988 * q^50 + 83 * q^51 + 740 * q^52 + 579 * q^53 + 1122 * q^54 + 951 * q^55 + 383 * q^57 - 158 * q^58 + 1695 * q^59 - 744 * q^60 + 1105 * q^61 + 976 * q^62 + 704 * q^64 - 347 * q^65 + 608 * q^66 + 695 * q^67 + 428 * q^68 + 414 * q^69 + 65 * q^71 + 1368 * q^72 + 1935 * q^73 - 82 * q^74 + 1712 * q^75 + 456 * q^76 + 850 * q^78 + 447 * q^79 + 528 * q^80 + 2203 * q^81 + 1624 * q^82 + 1389 * q^83 + 153 * q^85 + 438 * q^86 + 3354 * q^87 + 64 * q^88 + 2698 * q^89 + 1238 * q^90 + 1012 * q^92 - 686 * q^93 + 2282 * q^94 + 827 * q^95 + 576 * q^96 + 1589 * q^97 - 1729 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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