LMFDB - Embedded newform 2254.4.a.y.1.10

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			1.10
Root			$-7.59976$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} +9.59976 q^{3} +4.00000 q^{4} +12.3184 q^{5} +19.1995 q^{6} +8.00000 q^{8} +65.1554 q^{9} +24.6368 q^{10} +3.20522 q^{11} +38.3990 q^{12} +90.5065 q^{13} +118.254 q^{15} +16.0000 q^{16} -39.3727 q^{17} +130.311 q^{18} -31.2763 q^{19} +49.2736 q^{20} +6.41044 q^{22} +23.0000 q^{23} +76.7981 q^{24} +26.7429 q^{25} +181.013 q^{26} +366.283 q^{27} -100.977 q^{29} +236.507 q^{30} +101.739 q^{31} +32.0000 q^{32} +30.7694 q^{33} -78.7453 q^{34} +260.622 q^{36} -276.916 q^{37} -62.5526 q^{38} +868.841 q^{39} +98.5472 q^{40} -428.691 q^{41} +189.291 q^{43} +12.8209 q^{44} +802.610 q^{45} +46.0000 q^{46} -461.455 q^{47} +153.596 q^{48} +53.4858 q^{50} -377.968 q^{51} +362.026 q^{52} +532.272 q^{53} +732.566 q^{54} +39.4832 q^{55} -300.245 q^{57} -201.953 q^{58} -535.315 q^{59} +473.015 q^{60} -61.3049 q^{61} +203.477 q^{62} +64.0000 q^{64} +1114.90 q^{65} +61.5387 q^{66} +324.862 q^{67} -157.491 q^{68} +220.794 q^{69} +339.474 q^{71} +521.243 q^{72} -493.299 q^{73} -553.833 q^{74} +256.726 q^{75} -125.105 q^{76} +1737.68 q^{78} -1027.27 q^{79} +197.094 q^{80} +1757.03 q^{81} -857.383 q^{82} -967.752 q^{83} -485.008 q^{85} +378.582 q^{86} -969.351 q^{87} +25.6418 q^{88} +141.069 q^{89} +1605.22 q^{90} +92.0000 q^{92} +976.665 q^{93} -922.909 q^{94} -385.274 q^{95} +307.192 q^{96} +533.840 q^{97} +208.837 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	9.59976	1.84747	0.923737	−	0.383026i	$-0.125118\pi$
$3$	9.59976	1.84747	0.923737	+	0.383026i	$0.125118\pi$
$4$	4.00000	0.500000
$5$	12.3184	1.10179	0.550895	−	0.834574i	$-0.314286\pi$
$5$	12.3184	1.10179	0.550895	+	0.834574i	$0.314286\pi$
$6$	19.1995	1.30636
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	65.1554	2.41316
$10$	24.6368	0.779084
$11$	3.20522	0.0878555	0.0439277	−	0.999035i	$-0.486013\pi$
$11$	3.20522	0.0878555	0.0439277	+	0.999035i	$0.486013\pi$
$12$	38.3990	0.923737
$13$	90.5065	1.93092	0.965461	−	0.260548i	$-0.0839031\pi$
$13$	90.5065	1.93092	0.965461	+	0.260548i	$0.0839031\pi$
$14$	0	0
$15$	118.254	2.03553
$16$	16.0000	0.250000
$17$	−39.3727	−0.561722	−0.280861	−	0.959748i	$-0.590620\pi$
$17$	−39.3727	−0.561722	−0.280861	+	0.959748i	$0.590620\pi$
$18$	130.311	1.70636
$19$	−31.2763	−0.377646	−0.188823	−	0.982011i	$-0.560467\pi$
$19$	−31.2763	−0.377646	−0.188823	+	0.982011i	$0.560467\pi$
$20$	49.2736	0.550895
$21$	0	0
$22$	6.41044	0.0621232
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	76.7981	0.653181
$25$	26.7429	0.213943
$26$	181.013	1.36537
$27$	366.283	2.61078
$28$	0	0
$29$	−100.977	−0.646582	−0.323291	−	0.946300i	$-0.604789\pi$
$29$	−100.977	−0.646582	−0.323291	+	0.946300i	$0.604789\pi$
$30$	236.507	1.43934
$31$	101.739	0.589444	0.294722	−	0.955583i	$-0.404773\pi$
$31$	101.739	0.589444	0.294722	+	0.955583i	$0.404773\pi$
$32$	32.0000	0.176777
$33$	30.7694	0.162311
$34$	−78.7453	−0.397197
$35$	0	0
$36$	260.622	1.20658
$37$	−276.916	−1.23040	−0.615200	−	0.788371i	$-0.710925\pi$
$37$	−276.916	−1.23040	−0.615200	+	0.788371i	$0.710925\pi$
$38$	−62.5526	−0.267036
$39$	868.841	3.56733
$40$	98.5472	0.389542
$41$	−428.691	−1.63294	−0.816468	−	0.577391i	$-0.804071\pi$
$41$	−428.691	−1.63294	−0.816468	+	0.577391i	$0.804071\pi$
$42$	0	0
$43$	189.291	0.671316	0.335658	−	0.941984i	$-0.391041\pi$
$43$	189.291	0.671316	0.335658	+	0.941984i	$0.391041\pi$
$44$	12.8209	0.0439277
$45$	802.610	2.65880
$46$	46.0000	0.147442
$47$	−461.455	−1.43213	−0.716065	−	0.698034i	$-0.754058\pi$
$47$	−461.455	−1.43213	−0.716065	+	0.698034i	$0.754058\pi$
$48$	153.596	0.461869
$49$	0	0
$50$	53.4858	0.151281
$51$	−377.968	−1.03777
$52$	362.026	0.965461
$53$	532.272	1.37949	0.689746	−	0.724051i	$-0.257722\pi$
$53$	532.272	1.37949	0.689746	+	0.724051i	$0.257722\pi$
$54$	732.566	1.84610
$55$	39.4832	0.0967984
$56$	0	0
$57$	−300.245	−0.697692
$58$	−201.953	−0.457203
$59$	−535.315	−1.18122	−0.590611	−	0.806956i	$-0.701113\pi$
$59$	−535.315	−1.18122	−0.590611	+	0.806956i	$0.701113\pi$
$60$	473.015	1.01777
$61$	−61.3049	−0.128677	−0.0643384	−	0.997928i	$-0.520494\pi$
$61$	−61.3049	−0.128677	−0.0643384	+	0.997928i	$0.520494\pi$
$62$	203.477	0.416800
$63$	0	0
$64$	64.0000	0.125000
$65$	1114.90	2.12747
$66$	61.5387	0.114771
$67$	324.862	0.592361	0.296180	−	0.955132i	$-0.404287\pi$
$67$	324.862	0.592361	0.296180	+	0.955132i	$0.404287\pi$
$68$	−157.491	−0.280861
$69$	220.794	0.385225
$70$	0	0
$71$	339.474	0.567439	0.283719	−	0.958907i	$-0.408432\pi$
$71$	339.474	0.567439	0.283719	+	0.958907i	$0.408432\pi$
$72$	521.243	0.853182
$73$	−493.299	−0.790908	−0.395454	−	0.918486i	$-0.629413\pi$
$73$	−493.299	−0.790908	−0.395454	+	0.918486i	$0.629413\pi$
$74$	−553.833	−0.870023
$75$	256.726	0.395255
$76$	−125.105	−0.188823
$77$	0	0
$78$	1737.68	2.52248
$79$	−1027.27	−1.46300	−0.731502	−	0.681839i	$-0.761180\pi$
$79$	−1027.27	−1.46300	−0.731502	+	0.681839i	$0.761180\pi$
$80$	197.094	0.275448
$81$	1757.03	2.41019
$82$	−857.383	−1.15466
$83$	−967.752	−1.27981	−0.639907	−	0.768452i	$-0.721027\pi$
$83$	−967.752	−1.27981	−0.639907	+	0.768452i	$0.721027\pi$
$84$	0	0
$85$	−485.008	−0.618900
$86$	378.582	0.474692
$87$	−969.351	−1.19454
$88$	25.6418	0.0310616
$89$	141.069	0.168015	0.0840075	−	0.996465i	$-0.473228\pi$
$89$	141.069	0.168015	0.0840075	+	0.996465i	$0.473228\pi$
$90$	1605.22	1.88006
$91$	0	0
$92$	92.0000	0.104257
$93$	976.665	1.08898
$94$	−922.909	−1.01267
$95$	−385.274	−0.416087
$96$	307.192	0.326590
$97$	533.840	0.558796	0.279398	−	0.960175i	$-0.409865\pi$
$97$	533.840	0.558796	0.279398	+	0.960175i	$0.409865\pi$
$98$	0	0
$99$	208.837	0.212010
$100$	106.972	0.106972
$101$	778.628	0.767093	0.383546	−	0.923522i	$-0.374703\pi$
$101$	778.628	0.767093	0.383546	+	0.923522i	$0.374703\pi$
$102$	−755.936	−0.733812
$103$	−713.698	−0.682745	−0.341372	−	0.939928i	$-0.610892\pi$
$103$	−713.698	−0.682745	−0.341372	+	0.939928i	$0.610892\pi$
$104$	724.052	0.682684
$105$	0	0
$106$	1064.54	0.975449
$107$	−507.532	−0.458551	−0.229276	−	0.973362i	$-0.573636\pi$
$107$	−507.532	−0.458551	−0.229276	+	0.973362i	$0.573636\pi$
$108$	1465.13	1.30539
$109$	−1436.73	−1.26251	−0.631257	−	0.775574i	$-0.717461\pi$
$109$	−1436.73	−1.26251	−0.631257	+	0.775574i	$0.717461\pi$
$110$	78.9664	0.0684468
$111$	−2658.33	−2.27313
$112$	0	0
$113$	1213.13	1.00993	0.504963	−	0.863141i	$-0.331506\pi$
$113$	1213.13	1.00993	0.504963	+	0.863141i	$0.331506\pi$
$114$	−600.490	−0.493343
$115$	283.323	0.229739
$116$	−403.906	−0.323291
$117$	5896.99	4.65963
$118$	−1070.63	−0.835250
$119$	0	0
$120$	946.029	0.719669
$121$	−1320.73	−0.992281
$122$	−122.610	−0.0909882
$123$	−4115.33	−3.01681
$124$	406.954	0.294722
$125$	−1210.37	−0.866070
$126$	0	0
$127$	−2138.33	−1.49406	−0.747031	−	0.664789i	$-0.768522\pi$
$127$	−2138.33	−1.49406	−0.747031	+	0.664789i	$0.768522\pi$
$128$	128.000	0.0883883
$129$	1817.15	1.24024
$130$	2229.79	1.50435
$131$	2057.53	1.37227	0.686134	−	0.727475i	$-0.259306\pi$
$131$	2057.53	1.37227	0.686134	+	0.727475i	$0.259306\pi$
$132$	123.077	0.0811554
$133$	0	0
$134$	649.724	0.418862
$135$	4512.02	2.87654
$136$	−314.981	−0.198599
$137$	1729.65	1.07864	0.539321	−	0.842100i	$-0.318681\pi$
$137$	1729.65	1.07864	0.539321	+	0.842100i	$0.318681\pi$
$138$	441.589	0.272395
$139$	2294.70	1.40024	0.700121	−	0.714024i	$-0.253129\pi$
$139$	2294.70	1.40024	0.700121	+	0.714024i	$0.253129\pi$
$140$	0	0
$141$	−4429.85	−2.64582
$142$	678.948	0.401240
$143$	290.093	0.169642
$144$	1042.49	0.603291
$145$	−1243.87	−0.712399
$146$	−986.598	−0.559256
$147$	0	0
$148$	−1107.67	−0.615200
$149$	−2915.09	−1.60277	−0.801387	−	0.598146i	$-0.795904\pi$
$149$	−2915.09	−1.60277	−0.801387	+	0.598146i	$0.795904\pi$
$150$	513.451	0.279487
$151$	2462.75	1.32726	0.663628	−	0.748063i	$-0.269016\pi$
$151$	2462.75	1.32726	0.663628	+	0.748063i	$0.269016\pi$
$152$	−250.211	−0.133518
$153$	−2565.34	−1.35553
$154$	0	0
$155$	1253.26	0.649444
$156$	3475.36	1.78366
$157$	−611.315	−0.310753	−0.155377	−	0.987855i	$-0.549659\pi$
$157$	−611.315	−0.310753	−0.155377	+	0.987855i	$0.549659\pi$
$158$	−2054.55	−1.03450
$159$	5109.68	2.54858
$160$	394.189	0.194771
$161$	0	0
$162$	3514.06	1.70426
$163$	1426.67	0.685555	0.342778	−	0.939417i	$-0.388632\pi$
$163$	1426.67	0.685555	0.342778	+	0.939417i	$0.388632\pi$
$164$	−1714.77	−0.816468
$165$	379.029	0.178833
$166$	−1935.50	−0.904966
$167$	2650.58	1.22819	0.614095	−	0.789232i	$-0.289521\pi$
$167$	2650.58	1.22819	0.614095	+	0.789232i	$0.289521\pi$
$168$	0	0
$169$	5994.43	2.72846
$170$	−970.016	−0.437629
$171$	−2037.82	−0.911322
$172$	757.163	0.335658
$173$	978.955	0.430223	0.215112	−	0.976589i	$-0.430988\pi$
$173$	978.955	0.430223	0.215112	+	0.976589i	$0.430988\pi$
$174$	−1938.70	−0.844671
$175$	0	0
$176$	51.2835	0.0219639
$177$	−5138.90	−2.18228
$178$	282.139	0.118804
$179$	4511.37	1.88377	0.941887	−	0.335929i	$-0.109050\pi$
$179$	4511.37	1.88377	0.941887	+	0.335929i	$0.109050\pi$
$180$	3210.44	1.32940
$181$	−1920.55	−0.788693	−0.394346	−	0.918962i	$-0.629029\pi$
$181$	−1920.55	−0.788693	−0.394346	+	0.918962i	$0.629029\pi$
$182$	0	0
$183$	−588.512	−0.237727
$184$	184.000	0.0737210
$185$	−3411.17	−1.35564
$186$	1953.33	0.770028
$187$	−126.198	−0.0493504
$188$	−1845.82	−0.716065
$189$	0	0
$190$	−770.548	−0.294218
$191$	−3574.68	−1.35421	−0.677106	−	0.735885i	$-0.736766\pi$
$191$	−3574.68	−1.35421	−0.677106	+	0.735885i	$0.736766\pi$
$192$	614.385	0.230934
$193$	2888.94	1.07746	0.538731	−	0.842478i	$-0.318904\pi$
$193$	2888.94	1.07746	0.538731	+	0.842478i	$0.318904\pi$
$194$	1067.68	0.395129
$195$	10702.7	3.93045
$196$	0	0
$197$	3758.27	1.35921	0.679607	−	0.733576i	$-0.262150\pi$
$197$	3758.27	1.35921	0.679607	+	0.733576i	$0.262150\pi$
$198$	417.675	0.149913
$199$	−1703.40	−0.606788	−0.303394	−	0.952865i	$-0.598120\pi$
$199$	−1703.40	−0.606788	−0.303394	+	0.952865i	$0.598120\pi$
$200$	213.943	0.0756404
$201$	3118.60	1.09437
$202$	1557.26	0.542416
$203$	0	0
$204$	−1511.87	−0.518884
$205$	−5280.79	−1.79915
$206$	−1427.40	−0.482774
$207$	1498.57	0.503179
$208$	1448.10	0.482730
$209$	−100.248	−0.0331783
$210$	0	0
$211$	−4098.54	−1.33723	−0.668614	−	0.743609i	$-0.733112\pi$
$211$	−4098.54	−1.33723	−0.668614	+	0.743609i	$0.733112\pi$
$212$	2129.09	0.689746
$213$	3258.87	1.04833
$214$	−1015.06	−0.324245
$215$	2331.76	0.739650
$216$	2930.26	0.923051
$217$	0	0
$218$	−2873.46	−0.892732
$219$	−4735.55	−1.46118
$220$	157.933	0.0483992
$221$	−3563.48	−1.08464
$222$	−5316.66	−1.60735
$223$	159.396	0.0478652	0.0239326	−	0.999714i	$-0.492381\pi$
$223$	159.396	0.0478652	0.0239326	+	0.999714i	$0.492381\pi$
$224$	0	0
$225$	1742.45	0.516280
$226$	2426.26	0.714126
$227$	824.662	0.241122	0.120561	−	0.992706i	$-0.461531\pi$
$227$	824.662	0.241122	0.120561	+	0.992706i	$0.461531\pi$
$228$	−1200.98	−0.348846
$229$	4701.27	1.35663	0.678316	−	0.734770i	$-0.262710\pi$
$229$	4701.27	1.35663	0.678316	+	0.734770i	$0.262710\pi$
$230$	566.646	0.162450
$231$	0	0
$232$	−807.813	−0.228601
$233$	−5078.00	−1.42777	−0.713886	−	0.700262i	$-0.753067\pi$
$233$	−5078.00	−1.42777	−0.713886	+	0.700262i	$0.753067\pi$
$234$	11794.0	3.29486
$235$	−5684.38	−1.57791
$236$	−2141.26	−0.590611
$237$	−9861.58	−2.70286
$238$	0	0
$239$	−871.048	−0.235747	−0.117873	−	0.993029i	$-0.537608\pi$
$239$	−871.048	−0.235747	−0.117873	+	0.993029i	$0.537608\pi$
$240$	1892.06	0.508883
$241$	−472.461	−0.126282	−0.0631409	−	0.998005i	$-0.520112\pi$
$241$	−472.461	−0.126282	−0.0631409	+	0.998005i	$0.520112\pi$
$242$	−2641.45	−0.701649
$243$	6977.44	1.84199
$244$	−245.219	−0.0643384
$245$	0	0
$246$	−8230.67	−2.13320
$247$	−2830.71	−0.729206
$248$	813.908	0.208400
$249$	−9290.19	−2.36443
$250$	−2420.74	−0.612404
$251$	6104.29	1.53506	0.767528	−	0.641015i	$-0.221486\pi$
$251$	6104.29	1.53506	0.767528	+	0.641015i	$0.221486\pi$
$252$	0	0
$253$	73.7201	0.0183191
$254$	−4276.66	−1.05646
$255$	−4655.96	−1.14340
$256$	256.000	0.0625000
$257$	43.7778	0.0106256	0.00531281	−	0.999986i	$-0.498309\pi$
$257$	43.7778	0.0106256	0.00531281	+	0.999986i	$0.498309\pi$
$258$	3634.29	0.876981
$259$	0	0
$260$	4459.58	1.06374
$261$	−6579.17	−1.56031
$262$	4115.06	0.970340
$263$	−1834.78	−0.430179	−0.215090	−	0.976594i	$-0.569004\pi$
$263$	−1834.78	−0.430179	−0.215090	+	0.976594i	$0.569004\pi$
$264$	246.155	0.0573855
$265$	6556.73	1.51991
$266$	0	0
$267$	1354.23	0.310403
$268$	1299.45	0.296180
$269$	−2092.73	−0.474335	−0.237167	−	0.971469i	$-0.576219\pi$
$269$	−2092.73	−0.474335	−0.237167	+	0.971469i	$0.576219\pi$
$270$	9024.03	2.03402
$271$	−2432.91	−0.545347	−0.272673	−	0.962107i	$-0.587908\pi$
$271$	−2432.91	−0.545347	−0.272673	+	0.962107i	$0.587908\pi$
$272$	−629.963	−0.140430
$273$	0	0
$274$	3459.30	0.762715
$275$	85.7169	0.0187961
$276$	883.178	0.192613
$277$	−78.7657	−0.0170851	−0.00854255	−	0.999964i	$-0.502719\pi$
$277$	−78.7657	−0.0170851	−0.00854255	+	0.999964i	$0.502719\pi$
$278$	4589.39	0.990121
$279$	6628.81	1.42243
$280$	0	0
$281$	1626.25	0.345245	0.172622	−	0.984988i	$-0.444776\pi$
$281$	1626.25	0.345245	0.172622	+	0.984988i	$0.444776\pi$
$282$	−8859.71	−1.87088
$283$	−2019.49	−0.424191	−0.212096	−	0.977249i	$-0.568029\pi$
$283$	−2019.49	−0.424191	−0.212096	+	0.977249i	$0.568029\pi$
$284$	1357.90	0.283719
$285$	−3698.54	−0.768711
$286$	580.187	0.119955
$287$	0	0
$288$	2084.97	0.426591
$289$	−3362.79	−0.684468
$290$	−2487.74	−0.503742
$291$	5124.74	1.03236
$292$	−1973.20	−0.395454
$293$	213.222	0.0425138	0.0212569	−	0.999774i	$-0.493233\pi$
$293$	213.222	0.0425138	0.0212569	+	0.999774i	$0.493233\pi$
$294$	0	0
$295$	−6594.22	−1.30146
$296$	−2215.33	−0.435012
$297$	1174.02	0.229372
$298$	−5830.18	−1.13333
$299$	2081.65	0.402625
$300$	1026.90	0.197627
$301$	0	0
$302$	4925.49	0.938511
$303$	7474.64	1.41718
$304$	−500.421	−0.0944116
$305$	−755.178	−0.141775
$306$	−5130.68	−0.958502
$307$	554.869	0.103153	0.0515766	−	0.998669i	$-0.483575\pi$
$307$	554.869	0.103153	0.0515766	+	0.998669i	$0.483575\pi$
$308$	0	0
$309$	−6851.33	−1.26135
$310$	2506.51	0.459227
$311$	9101.92	1.65956	0.829779	−	0.558092i	$-0.188466\pi$
$311$	9101.92	1.65956	0.829779	+	0.558092i	$0.188466\pi$
$312$	6950.73	1.26124
$313$	3349.81	0.604928	0.302464	−	0.953161i	$-0.402191\pi$
$313$	3349.81	0.604928	0.302464	+	0.953161i	$0.402191\pi$
$314$	−1222.63	−0.219736
$315$	0	0
$316$	−4109.10	−0.731502
$317$	494.837	0.0876746	0.0438373	−	0.999039i	$-0.486042\pi$
$317$	494.837	0.0876746	0.0438373	+	0.999039i	$0.486042\pi$
$318$	10219.4	1.80212
$319$	−323.652	−0.0568058
$320$	788.377	0.137724
$321$	−4872.19	−0.847162
$322$	0	0
$323$	1231.43	0.212132
$324$	7028.12	1.20510
$325$	2420.41	0.413108
$326$	2853.34	0.484761
$327$	−13792.3	−2.33246
$328$	−3429.53	−0.577330
$329$	0	0
$330$	758.058	0.126454
$331$	−5534.52	−0.919048	−0.459524	−	0.888165i	$-0.651980\pi$
$331$	−5534.52	−0.919048	−0.459524	+	0.888165i	$0.651980\pi$
$332$	−3871.01	−0.639907
$333$	−18042.6	−2.96915
$334$	5301.15	0.868462
$335$	4001.78	0.652658
$336$	0	0
$337$	−6104.34	−0.986720	−0.493360	−	0.869825i	$-0.664231\pi$
$337$	−6104.34	−0.986720	−0.493360	+	0.869825i	$0.664231\pi$
$338$	11988.9	1.92931
$339$	11645.8	1.86581
$340$	−1940.03	−0.309450
$341$	326.094	0.0517859
$342$	−4075.64	−0.644402
$343$	0	0
$344$	1514.33	0.237346
$345$	2719.83	0.424438
$346$	1957.91	0.304214
$347$	4679.43	0.723934	0.361967	−	0.932191i	$-0.382105\pi$
$347$	4679.43	0.723934	0.361967	+	0.932191i	$0.382105\pi$
$348$	−3877.41	−0.597272
$349$	10570.3	1.62124	0.810622	−	0.585570i	$-0.199129\pi$
$349$	10570.3	1.62124	0.810622	+	0.585570i	$0.199129\pi$
$350$	0	0
$351$	33151.0	5.04122
$352$	102.567	0.0155308
$353$	1138.62	0.171678	0.0858391	−	0.996309i	$-0.472643\pi$
$353$	1138.62	0.171678	0.0858391	+	0.996309i	$0.472643\pi$
$354$	−10277.8	−1.54310
$355$	4181.78	0.625199
$356$	564.278	0.0840075
$357$	0	0
$358$	9022.74	1.33203
$359$	7926.33	1.16528	0.582640	−	0.812730i	$-0.302020\pi$
$359$	7926.33	1.16528	0.582640	+	0.812730i	$0.302020\pi$
$360$	6420.88	0.940028
$361$	−5880.79	−0.857383
$362$	−3841.10	−0.557690
$363$	−12678.7	−1.83321
$364$	0	0
$365$	−6076.65	−0.871415
$366$	−1177.02	−0.168098
$367$	7201.35	1.02427	0.512136	−	0.858905i	$-0.328854\pi$
$367$	7201.35	1.02427	0.512136	+	0.858905i	$0.328854\pi$
$368$	368.000	0.0521286
$369$	−27931.6	−3.94054
$370$	−6822.33	−0.958584
$371$	0	0
$372$	3906.66	0.544492
$373$	5391.22	0.748383	0.374192	−	0.927351i	$-0.377920\pi$
$373$	5391.22	0.748383	0.374192	+	0.927351i	$0.377920\pi$
$374$	−252.396	−0.0348960
$375$	−11619.3	−1.60004
$376$	−3691.64	−0.506334
$377$	−9139.04	−1.24850
$378$	0	0
$379$	−12296.3	−1.66653	−0.833267	−	0.552871i	$-0.813532\pi$
$379$	−12296.3	−1.66653	−0.833267	+	0.552871i	$0.813532\pi$
$380$	−1541.10	−0.208044
$381$	−20527.4	−2.76024
$382$	−7149.36	−0.957573
$383$	−9230.14	−1.23143	−0.615715	−	0.787969i	$-0.711133\pi$
$383$	−9230.14	−1.23143	−0.615715	+	0.787969i	$0.711133\pi$
$384$	1228.77	0.163295
$385$	0	0
$386$	5777.88	0.761881
$387$	12333.3	1.61999
$388$	2135.36	0.279398
$389$	−12094.1	−1.57634	−0.788169	−	0.615459i	$-0.788971\pi$
$389$	−12094.1	−1.57634	−0.788169	+	0.615459i	$0.788971\pi$
$390$	21405.5	2.77925
$391$	−905.571	−0.117127
$392$	0	0
$393$	19751.8	2.53523
$394$	7516.53	0.961110
$395$	−12654.4	−1.61192
$396$	835.350	0.106005
$397$	−4796.97	−0.606430	−0.303215	−	0.952922i	$-0.598060\pi$
$397$	−4796.97	−0.606430	−0.303215	+	0.952922i	$0.598060\pi$
$398$	−3406.80	−0.429064
$399$	0	0
$400$	427.887	0.0534858
$401$	5211.25	0.648971	0.324486	−	0.945891i	$-0.394809\pi$
$401$	5211.25	0.648971	0.324486	+	0.945891i	$0.394809\pi$
$402$	6237.19	0.773838
$403$	9208.00	1.13817
$404$	3114.51	0.383546
$405$	21643.8	2.65553
$406$	0	0
$407$	−887.578	−0.108097
$408$	−3023.75	−0.366906
$409$	7034.91	0.850499	0.425250	−	0.905076i	$-0.360186\pi$
$409$	7034.91	0.850499	0.425250	+	0.905076i	$0.360186\pi$
$410$	−10561.6	−1.27219
$411$	16604.2	1.99276
$412$	−2854.79	−0.341372
$413$	0	0
$414$	2997.15	0.355802
$415$	−11921.2	−1.41009
$416$	2896.21	0.341342
$417$	22028.5	2.58691
$418$	−200.495	−0.0234606
$419$	8463.00	0.986742	0.493371	−	0.869819i	$-0.335765\pi$
$419$	8463.00	0.986742	0.493371	+	0.869819i	$0.335765\pi$
$420$	0	0
$421$	−12704.8	−1.47077	−0.735386	−	0.677649i	$-0.762999\pi$
$421$	−12704.8	−1.47077	−0.735386	+	0.677649i	$0.762999\pi$
$422$	−8197.08	−0.945564
$423$	−30066.3	−3.45596
$424$	4258.17	0.487724
$425$	−1052.94	−0.120177
$426$	6517.74	0.741280
$427$	0	0
$428$	−2030.13	−0.229276
$429$	2784.83	0.313409
$430$	4663.52	0.523011
$431$	−14888.0	−1.66387	−0.831935	−	0.554873i	$-0.812767\pi$
$431$	−14888.0	−1.66387	−0.831935	+	0.554873i	$0.812767\pi$
$432$	5860.52	0.652696
$433$	−15105.4	−1.67649	−0.838244	−	0.545295i	$-0.816418\pi$
$433$	−15105.4	−1.67649	−0.838244	+	0.545295i	$0.816418\pi$
$434$	0	0
$435$	−11940.9	−1.31614
$436$	−5746.93	−0.631257
$437$	−719.355	−0.0787447
$438$	−9471.11	−1.03321
$439$	−10682.1	−1.16135	−0.580673	−	0.814137i	$-0.697211\pi$
$439$	−10682.1	−1.16135	−0.580673	+	0.814137i	$0.697211\pi$
$440$	315.865	0.0342234
$441$	0	0
$442$	−7126.96	−0.766957
$443$	4901.10	0.525640	0.262820	−	0.964845i	$-0.415347\pi$
$443$	4901.10	0.525640	0.262820	+	0.964845i	$0.415347\pi$
$444$	−10633.3	−1.13657
$445$	1737.75	0.185117
$446$	318.792	0.0338458
$447$	−27984.2	−2.96109
$448$	0	0
$449$	13352.4	1.40342	0.701712	−	0.712460i	$-0.252419\pi$
$449$	13352.4	1.40342	0.701712	+	0.712460i	$0.252419\pi$
$450$	3484.89	0.365065
$451$	−1374.05	−0.143462
$452$	4852.52	0.504963
$453$	23641.8	2.45207
$454$	1649.32	0.170499
$455$	0	0
$456$	−2401.96	−0.246671
$457$	6832.77	0.699395	0.349698	−	0.936863i	$-0.386284\pi$
$457$	6832.77	0.699395	0.349698	+	0.936863i	$0.386284\pi$
$458$	9402.54	0.959283
$459$	−14421.5	−1.46653
$460$	1133.29	0.114870
$461$	1494.84	0.151023	0.0755115	−	0.997145i	$-0.475941\pi$
$461$	1494.84	0.151023	0.0755115	+	0.997145i	$0.475941\pi$
$462$	0	0
$463$	3632.90	0.364655	0.182328	−	0.983238i	$-0.441637\pi$
$463$	3632.90	0.364655	0.182328	+	0.983238i	$0.441637\pi$
$464$	−1615.63	−0.161646
$465$	12031.0	1.19983
$466$	−10156.0	−1.00959
$467$	8702.16	0.862287	0.431143	−	0.902283i	$-0.358110\pi$
$467$	8702.16	0.862287	0.431143	+	0.902283i	$0.358110\pi$
$468$	23588.0	2.32982
$469$	0	0
$470$	−11368.8	−1.11575
$471$	−5868.48	−0.574109
$472$	−4282.52	−0.417625
$473$	606.719	0.0589788
$474$	−19723.2	−1.91121
$475$	−836.420	−0.0807949
$476$	0	0
$477$	34680.4	3.32894
$478$	−1742.10	−0.166698
$479$	10930.5	1.04265	0.521324	−	0.853359i	$-0.325438\pi$
$479$	10930.5	1.04265	0.521324	+	0.853359i	$0.325438\pi$
$480$	3784.12	0.359834
$481$	−25062.7	−2.37580
$482$	−944.922	−0.0892947
$483$	0	0
$484$	−5282.91	−0.496141
$485$	6576.05	0.615677
$486$	13954.9	1.30248
$487$	12266.0	1.14132	0.570661	−	0.821186i	$-0.306687\pi$
$487$	12266.0	1.14132	0.570661	+	0.821186i	$0.306687\pi$
$488$	−490.439	−0.0454941
$489$	13695.7	1.26655
$490$	0	0
$491$	−1818.20	−0.167117	−0.0835584	−	0.996503i	$-0.526629\pi$
$491$	−1818.20	−0.167117	−0.0835584	+	0.996503i	$0.526629\pi$
$492$	−16461.3	−1.50840
$493$	3975.72	0.363199
$494$	−5661.42	−0.515626
$495$	2572.54	0.233590
$496$	1627.82	0.147361
$497$	0	0
$498$	−18580.4	−1.67190
$499$	4982.41	0.446980	0.223490	−	0.974706i	$-0.428255\pi$
$499$	4982.41	0.446980	0.223490	+	0.974706i	$0.428255\pi$
$500$	−4841.48	−0.433035
$501$	25444.9	2.26905
$502$	12208.6	1.08545
$503$	21067.0	1.86746	0.933730	−	0.357979i	$-0.116534\pi$
$503$	21067.0	1.86746	0.933730	+	0.357979i	$0.116534\pi$
$504$	0	0
$505$	9591.45	0.845176
$506$	147.440	0.0129536
$507$	57545.1	5.04076
$508$	−8553.31	−0.747031
$509$	−6133.98	−0.534153	−0.267077	−	0.963675i	$-0.586058\pi$
$509$	−6133.98	−0.534153	−0.267077	+	0.963675i	$0.586058\pi$
$510$	−9311.92	−0.808508
$511$	0	0
$512$	512.000	0.0441942
$513$	−11456.0	−0.985953
$514$	87.5557	0.00751345
$515$	−8791.61	−0.752242
$516$	7268.59	0.620120
$517$	−1479.06	−0.125820
$518$	0	0
$519$	9397.74	0.794826
$520$	8919.16	0.752175
$521$	−18238.5	−1.53367	−0.766835	−	0.641844i	$-0.778170\pi$
$521$	−18238.5	−1.53367	−0.766835	+	0.641844i	$0.778170\pi$
$522$	−13158.3	−1.10330
$523$	4632.05	0.387276	0.193638	−	0.981073i	$-0.437971\pi$
$523$	4632.05	0.387276	0.193638	+	0.981073i	$0.437971\pi$
$524$	8230.11	0.686134
$525$	0	0
$526$	−3669.55	−0.304183
$527$	−4005.72	−0.331104
$528$	492.310	0.0405777
$529$	529.000	0.0434783
$530$	13113.5	1.07474
$531$	−34878.7	−2.85048
$532$	0	0
$533$	−38799.4	−3.15307
$534$	2708.46	0.219488
$535$	−6251.98	−0.505228
$536$	2598.89	0.209431
$537$	43308.1	3.48023
$538$	−4185.46	−0.335405
$539$	0	0
$540$	18048.1	1.43827
$541$	21368.5	1.69816	0.849078	−	0.528268i	$-0.177158\pi$
$541$	21368.5	1.69816	0.849078	+	0.528268i	$0.177158\pi$
$542$	−4865.83	−0.385618
$543$	−18436.8	−1.45709
$544$	−1259.93	−0.0992994
$545$	−17698.2	−1.39103
$546$	0	0
$547$	−1245.08	−0.0973234	−0.0486617	−	0.998815i	$-0.515496\pi$
$547$	−1245.08	−0.0973234	−0.0486617	+	0.998815i	$0.515496\pi$
$548$	6918.60	0.539321
$549$	−3994.34	−0.310518
$550$	171.434	0.0132908
$551$	3158.18	0.244179
$552$	1766.36	0.136198
$553$	0	0
$554$	−157.531	−0.0120810
$555$	−32746.4	−2.50452
$556$	9178.79	0.700121
$557$	17190.3	1.30768	0.653840	−	0.756633i	$-0.273157\pi$
$557$	17190.3	1.30768	0.653840	+	0.756633i	$0.273157\pi$
$558$	13257.6	1.00581
$559$	17132.0	1.29626
$560$	0	0
$561$	−1211.47	−0.0911735
$562$	3252.49	0.244125
$563$	−10898.7	−0.815854	−0.407927	−	0.913015i	$-0.633748\pi$
$563$	−10898.7	−0.815854	−0.407927	+	0.913015i	$0.633748\pi$
$564$	−17719.4	−1.32291
$565$	14943.8	1.11273
$566$	−4038.98	−0.299948
$567$	0	0
$568$	2715.79	0.200620
$569$	−10552.2	−0.777456	−0.388728	−	0.921353i	$-0.627085\pi$
$569$	−10552.2	−0.777456	−0.388728	+	0.921353i	$0.627085\pi$
$570$	−7397.08	−0.543561
$571$	−10692.3	−0.783643	−0.391822	−	0.920041i	$-0.628155\pi$
$571$	−10692.3	−0.783643	−0.391822	+	0.920041i	$0.628155\pi$
$572$	1160.37	0.0848210
$573$	−34316.1	−2.50187
$574$	0	0
$575$	615.087	0.0446103
$576$	4169.95	0.301645
$577$	−7386.53	−0.532938	−0.266469	−	0.963843i	$-0.585857\pi$
$577$	−7386.53	−0.532938	−0.266469	+	0.963843i	$0.585857\pi$
$578$	−6725.59	−0.483992
$579$	27733.1	1.99059
$580$	−4975.48	−0.356199
$581$	0	0
$582$	10249.5	0.729990
$583$	1706.05	0.121196
$584$	−3946.39	−0.279628
$585$	72641.4	5.13394
$586$	426.444	0.0300618
$587$	14194.6	0.998079	0.499040	−	0.866579i	$-0.333686\pi$
$587$	14194.6	0.998079	0.499040	+	0.866579i	$0.333686\pi$
$588$	0	0
$589$	−3182.01	−0.222601
$590$	−13188.4	−0.920271
$591$	36078.5	2.51112
$592$	−4430.66	−0.307600
$593$	−4223.73	−0.292492	−0.146246	−	0.989248i	$-0.546719\pi$
$593$	−4223.73	−0.292492	−0.146246	+	0.989248i	$0.546719\pi$
$594$	2348.03	0.162190
$595$	0	0
$596$	−11660.4	−0.801387
$597$	−16352.2	−1.12102
$598$	4163.30	0.284699
$599$	−5658.02	−0.385944	−0.192972	−	0.981204i	$-0.561813\pi$
$599$	−5658.02	−0.385944	−0.192972	+	0.981204i	$0.561813\pi$
$600$	2053.80	0.139744
$601$	−9080.45	−0.616305	−0.308152	−	0.951337i	$-0.599711\pi$
$601$	−9080.45	−0.616305	−0.308152	+	0.951337i	$0.599711\pi$
$602$	0	0
$603$	21166.5	1.42946
$604$	9850.99	0.663628
$605$	−16269.2	−1.09329
$606$	14949.3	1.00210
$607$	10474.0	0.700375	0.350187	−	0.936680i	$-0.386118\pi$
$607$	10474.0	0.700375	0.350187	+	0.936680i	$0.386118\pi$
$608$	−1000.84	−0.0667591
$609$	0	0
$610$	−1510.36	−0.100250
$611$	−41764.6	−2.76533
$612$	−10261.4	−0.677763
$613$	−14206.8	−0.936064	−0.468032	−	0.883711i	$-0.655037\pi$
$613$	−14206.8	−0.936064	−0.468032	+	0.883711i	$0.655037\pi$
$614$	1109.74	0.0729403
$615$	−50694.3	−3.32389
$616$	0	0
$617$	−1625.86	−0.106085	−0.0530427	−	0.998592i	$-0.516892\pi$
$617$	−1625.86	−0.106085	−0.0530427	+	0.998592i	$0.516892\pi$
$618$	−13702.7	−0.891912
$619$	−22233.0	−1.44365	−0.721826	−	0.692074i	$-0.756697\pi$
$619$	−22233.0	−1.44365	−0.721826	+	0.692074i	$0.756697\pi$
$620$	5013.02	0.324722
$621$	8424.50	0.544386
$622$	18203.8	1.17349
$623$	0	0
$624$	13901.5	0.891832
$625$	−18252.7	−1.16817
$626$	6699.62	0.427748
$627$	−962.352	−0.0612961
$628$	−2445.26	−0.155377
$629$	10902.9	0.691142
$630$	0	0
$631$	19654.9	1.24002	0.620008	−	0.784596i	$-0.287129\pi$
$631$	19654.9	1.24002	0.620008	+	0.784596i	$0.287129\pi$
$632$	−8218.19	−0.517250
$633$	−39345.0	−2.47050
$634$	989.675	0.0619953
$635$	−26340.8	−1.64615
$636$	20438.7	1.27429
$637$	0	0
$638$	−647.305	−0.0401678
$639$	22118.6	1.36932
$640$	1576.75	0.0973855
$641$	−5791.55	−0.356868	−0.178434	−	0.983952i	$-0.557103\pi$
$641$	−5791.55	−0.356868	−0.178434	+	0.983952i	$0.557103\pi$
$642$	−9744.38	−0.599034
$643$	28777.7	1.76498	0.882491	−	0.470330i	$-0.155865\pi$
$643$	28777.7	1.76498	0.882491	+	0.470330i	$0.155865\pi$
$644$	0	0
$645$	22384.3	1.36648
$646$	2462.86	0.150000
$647$	3658.70	0.222316	0.111158	−	0.993803i	$-0.464544\pi$
$647$	3658.70	0.222316	0.111158	+	0.993803i	$0.464544\pi$
$648$	14056.2	0.852132
$649$	−1715.80	−0.103777
$650$	4840.82	0.292111
$651$	0	0
$652$	5706.68	0.342778
$653$	22504.0	1.34862	0.674311	−	0.738447i	$-0.264441\pi$
$653$	22504.0	1.34862	0.674311	+	0.738447i	$0.264441\pi$
$654$	−27584.6	−1.64930
$655$	25345.4	1.51195
$656$	−6859.06	−0.408234
$657$	−32141.1	−1.90859
$658$	0	0
$659$	−24842.1	−1.46845	−0.734227	−	0.678904i	$-0.762455\pi$
$659$	−24842.1	−1.46845	−0.734227	+	0.678904i	$0.762455\pi$
$660$	1516.12	0.0894163
$661$	−19488.7	−1.14678	−0.573390	−	0.819282i	$-0.694372\pi$
$661$	−19488.7	−1.14678	−0.573390	+	0.819282i	$0.694372\pi$
$662$	−11069.0	−0.649865
$663$	−34208.6	−2.00385
$664$	−7742.02	−0.452483
$665$	0	0
$666$	−36085.2	−2.09951
$667$	−2322.46	−0.134822
$668$	10602.3	0.614095
$669$	1530.16	0.0884297
$670$	8003.55	0.461499
$671$	−196.496	−0.0113050
$672$	0	0
$673$	−3510.32	−0.201059	−0.100530	−	0.994934i	$-0.532054\pi$
$673$	−3510.32	−0.201059	−0.100530	+	0.994934i	$0.532054\pi$
$674$	−12208.7	−0.697716
$675$	9795.47	0.558560
$676$	23977.7	1.36423
$677$	−8630.86	−0.489972	−0.244986	−	0.969527i	$-0.578783\pi$
$677$	−8630.86	−0.489972	−0.244986	+	0.969527i	$0.578783\pi$
$678$	23291.5	1.31933
$679$	0	0
$680$	−3880.07	−0.218814
$681$	7916.55	0.445467
$682$	652.189	0.0366182
$683$	−5485.63	−0.307323	−0.153662	−	0.988124i	$-0.549107\pi$
$683$	−5485.63	−0.307323	−0.153662	+	0.988124i	$0.549107\pi$
$684$	−8151.29	−0.455661
$685$	21306.5	1.18844
$686$	0	0
$687$	45131.1	2.50634
$688$	3028.65	0.167829
$689$	48174.0	2.66369
$690$	5439.67	0.300123
$691$	24519.4	1.34987	0.674937	−	0.737875i	$-0.264171\pi$
$691$	24519.4	1.34987	0.674937	+	0.737875i	$0.264171\pi$
$692$	3915.82	0.215112
$693$	0	0
$694$	9358.86	0.511898
$695$	28267.0	1.54277
$696$	−7754.81	−0.422335
$697$	16878.7	0.917256
$698$	21140.6	1.14639
$699$	−48747.6	−2.63777
$700$	0	0
$701$	−3991.48	−0.215059	−0.107529	−	0.994202i	$-0.534294\pi$
$701$	−3991.48	−0.215059	−0.107529	+	0.994202i	$0.534294\pi$
$702$	66301.9	3.56468
$703$	8660.92	0.464656
$704$	205.134	0.0109819
$705$	−54568.7	−2.91514
$706$	2277.23	0.121395
$707$	0	0
$708$	−20555.6	−1.09114
$709$	26142.2	1.38476	0.692378	−	0.721535i	$-0.256563\pi$
$709$	26142.2	1.38476	0.692378	+	0.721535i	$0.256563\pi$
$710$	8363.55	0.442082
$711$	−66932.4	−3.53047
$712$	1128.56	0.0594022
$713$	2339.99	0.122908
$714$	0	0
$715$	3573.48	0.186910
$716$	18045.5	0.941887
$717$	−8361.85	−0.435536
$718$	15852.7	0.823977
$719$	6281.19	0.325798	0.162899	−	0.986643i	$-0.447916\pi$
$719$	6281.19	0.325798	0.162899	+	0.986643i	$0.447916\pi$
$720$	12841.8	0.664700
$721$	0	0
$722$	−11761.6	−0.606262
$723$	−4535.51	−0.233302
$724$	−7682.20	−0.394346
$725$	−2700.41	−0.138332
$726$	−25357.3	−1.29628
$727$	16949.5	0.864681	0.432340	−	0.901711i	$-0.357688\pi$
$727$	16949.5	0.864681	0.432340	+	0.901711i	$0.357688\pi$
$728$	0	0
$729$	19542.0	0.992834
$730$	−12153.3	−0.616184
$731$	−7452.88	−0.377093
$732$	−2354.05	−0.118864
$733$	25101.0	1.26484	0.632420	−	0.774626i	$-0.282062\pi$
$733$	25101.0	1.26484	0.632420	+	0.774626i	$0.282062\pi$
$734$	14402.7	0.724269
$735$	0	0
$736$	736.000	0.0368605
$737$	1041.25	0.0520422
$738$	−55863.1	−2.78638
$739$	15306.6	0.761924	0.380962	−	0.924591i	$-0.375593\pi$
$739$	15306.6	0.761924	0.380962	+	0.924591i	$0.375593\pi$
$740$	−13644.7	−0.677821
$741$	−27174.1	−1.34719
$742$	0	0
$743$	−9114.17	−0.450022	−0.225011	−	0.974356i	$-0.572242\pi$
$743$	−9114.17	−0.450022	−0.225011	+	0.974356i	$0.572242\pi$
$744$	7813.32	0.385014
$745$	−35909.2	−1.76592
$746$	10782.4	0.529187
$747$	−63054.3	−3.08840
$748$	−504.792	−0.0246752
$749$	0	0
$750$	−23238.5	−1.13140
$751$	18894.4	0.918062	0.459031	−	0.888420i	$-0.348197\pi$
$751$	18894.4	0.918062	0.459031	+	0.888420i	$0.348197\pi$
$752$	−7383.27	−0.358032
$753$	58599.7	2.83598
$754$	−18278.1	−0.882823
$755$	30337.1	1.46236
$756$	0	0
$757$	40455.6	1.94238	0.971192	−	0.238299i	$-0.0765897\pi$
$757$	40455.6	1.94238	0.971192	+	0.238299i	$0.0765897\pi$
$758$	−24592.5	−1.17842
$759$	707.695	0.0338441
$760$	−3082.19	−0.147109
$761$	−18058.1	−0.860189	−0.430095	−	0.902784i	$-0.641520\pi$
$761$	−18058.1	−0.860189	−0.430095	+	0.902784i	$0.641520\pi$
$762$	−41054.9	−1.95179
$763$	0	0
$764$	−14298.7	−0.677106
$765$	−31600.9	−1.49351
$766$	−18460.3	−0.870753
$767$	−48449.5	−2.28085
$768$	2457.54	0.115467
$769$	−8782.33	−0.411832	−0.205916	−	0.978570i	$-0.566017\pi$
$769$	−8782.33	−0.411832	−0.205916	+	0.978570i	$0.566017\pi$
$770$	0	0
$771$	420.257	0.0196306
$772$	11555.8	0.538731
$773$	14634.2	0.680926	0.340463	−	0.940258i	$-0.389416\pi$
$773$	14634.2	0.680926	0.340463	+	0.940258i	$0.389416\pi$
$774$	24666.6	1.14551
$775$	2720.78	0.126108
$776$	4270.72	0.197564
$777$	0	0
$778$	−24188.2	−1.11464
$779$	13407.9	0.616672
$780$	42810.9	1.96523
$781$	1088.09	0.0498526
$782$	−1811.14	−0.0828214
$783$	−36986.0	−1.68809
$784$	0	0
$785$	−7530.42	−0.342385
$786$	39503.6	1.79268
$787$	−36107.0	−1.63542	−0.817710	−	0.575630i	$-0.804757\pi$
$787$	−36107.0	−1.63542	−0.817710	+	0.575630i	$0.804757\pi$
$788$	15033.1	0.679607
$789$	−17613.4	−0.794745
$790$	−25308.7	−1.13980
$791$	0	0
$792$	1670.70	0.0749567
$793$	−5548.49	−0.248465
$794$	−9593.93	−0.428811
$795$	62943.1	2.80800
$796$	−6813.59	−0.303394
$797$	−8052.00	−0.357862	−0.178931	−	0.983862i	$-0.557264\pi$
$797$	−8052.00	−0.357862	−0.178931	+	0.983862i	$0.557264\pi$
$798$	0	0
$799$	18168.7	0.804458
$800$	855.773	0.0378202
$801$	9191.43	0.405447
$802$	10422.5	0.458892
$803$	−1581.13	−0.0694856
$804$	12474.4	0.547186
$805$	0	0
$806$	18416.0	0.804809
$807$	−20089.7	−0.876321
$808$	6229.02	0.271208
$809$	10909.1	0.474094	0.237047	−	0.971498i	$-0.423820\pi$
$809$	10909.1	0.474094	0.237047	+	0.971498i	$0.423820\pi$
$810$	43287.6	1.87774
$811$	−20492.9	−0.887302	−0.443651	−	0.896200i	$-0.646317\pi$
$811$	−20492.9	−0.887302	−0.443651	+	0.896200i	$0.646317\pi$
$812$	0	0
$813$	−23355.4	−1.00751
$814$	−1775.16	−0.0764363
$815$	17574.3	0.755339
$816$	−6047.49	−0.259442
$817$	−5920.32	−0.253520
$818$	14069.8	0.601394
$819$	0	0
$820$	−21123.2	−0.899577
$821$	−34410.3	−1.46276	−0.731381	−	0.681969i	$-0.761124\pi$
$821$	−34410.3	−1.46276	−0.731381	+	0.681969i	$0.761124\pi$
$822$	33208.4	1.40910
$823$	−24532.0	−1.03904	−0.519521	−	0.854458i	$-0.673890\pi$
$823$	−24532.0	−1.03904	−0.519521	+	0.854458i	$0.673890\pi$
$824$	−5709.58	−0.241387
$825$	822.862	0.0347253
$826$	0	0
$827$	15326.2	0.644430	0.322215	−	0.946666i	$-0.395573\pi$
$827$	15326.2	0.644430	0.322215	+	0.946666i	$0.395573\pi$
$828$	5994.30	0.251590
$829$	22832.6	0.956586	0.478293	−	0.878200i	$-0.341256\pi$
$829$	22832.6	0.956586	0.478293	+	0.878200i	$0.341256\pi$
$830$	−23842.3	−0.997083
$831$	−756.132	−0.0315643
$832$	5792.42	0.241365
$833$	0	0
$834$	44057.1	1.82922
$835$	32650.9	1.35321
$836$	−400.990	−0.0165892
$837$	37265.1	1.53891
$838$	16926.0	0.697732
$839$	−1197.07	−0.0492581	−0.0246291	−	0.999697i	$-0.507840\pi$
$839$	−1197.07	−0.0492581	−0.0246291	+	0.999697i	$0.507840\pi$
$840$	0	0
$841$	−14192.7	−0.581931
$842$	−25409.6	−1.03999
$843$	15611.6	0.637831
$844$	−16394.2	−0.668614
$845$	73841.7	3.00619
$846$	−60132.5	−2.44373
$847$	0	0
$848$	8516.35	0.344873
$849$	−19386.6	−0.783682
$850$	−2105.88	−0.0849777
$851$	−6369.08	−0.256556
$852$	13035.5	0.524164
$853$	37476.2	1.50429	0.752146	−	0.658997i	$-0.229019\pi$
$853$	37476.2	1.50429	0.752146	+	0.658997i	$0.229019\pi$
$854$	0	0
$855$	−25102.7	−1.00409
$856$	−4060.26	−0.162122
$857$	−27855.2	−1.11029	−0.555143	−	0.831755i	$-0.687337\pi$
$857$	−27855.2	−1.11029	−0.555143	+	0.831755i	$0.687337\pi$
$858$	5569.65	0.221614
$859$	3897.45	0.154807	0.0774035	−	0.997000i	$-0.475337\pi$
$859$	3897.45	0.154807	0.0774035	+	0.997000i	$0.475337\pi$
$860$	9327.04	0.369825
$861$	0	0
$862$	−29775.9	−1.17653
$863$	−2633.38	−0.103872	−0.0519359	−	0.998650i	$-0.516539\pi$
$863$	−2633.38	−0.103872	−0.0519359	+	0.998650i	$0.516539\pi$
$864$	11721.0	0.461526
$865$	12059.2	0.474016
$866$	−30210.8	−1.18546
$867$	−32282.0	−1.26454
$868$	0	0
$869$	−3292.64	−0.128533
$870$	−23881.7	−0.930650
$871$	29402.1	1.14380
$872$	−11493.9	−0.446366
$873$	34782.6	1.34847
$874$	−1438.71	−0.0556809
$875$	0	0
$876$	−18942.2	−0.730591
$877$	−31560.8	−1.21520	−0.607601	−	0.794243i	$-0.707868\pi$
$877$	−31560.8	−1.21520	−0.607601	+	0.794243i	$0.707868\pi$
$878$	−21364.3	−0.821196
$879$	2046.88	0.0785432
$880$	631.731	0.0241996
$881$	28596.5	1.09358	0.546788	−	0.837271i	$-0.315850\pi$
$881$	28596.5	1.09358	0.546788	+	0.837271i	$0.315850\pi$
$882$	0	0
$883$	10131.2	0.386119	0.193059	−	0.981187i	$-0.438159\pi$
$883$	10131.2	0.386119	0.193059	+	0.981187i	$0.438159\pi$
$884$	−14253.9	−0.542321
$885$	−63303.0	−2.40441
$886$	9802.21	0.371684
$887$	3743.52	0.141708	0.0708540	−	0.997487i	$-0.477428\pi$
$887$	3743.52	0.141708	0.0708540	+	0.997487i	$0.477428\pi$
$888$	−21266.6	−0.803673
$889$	0	0
$890$	3475.50	0.130898
$891$	5631.67	0.211749
$892$	637.583	0.0239326
$893$	14432.6	0.540838
$894$	−55968.3	−2.09380
$895$	55572.9	2.07553
$896$	0	0
$897$	19983.3	0.743840
$898$	26704.8	0.992371
$899$	−10273.2	−0.381124
$900$	6969.78	0.258140
$901$	−20957.0	−0.774892
$902$	−2748.10	−0.101443
$903$	0	0
$904$	9705.04	0.357063
$905$	−23658.1	−0.868975
$906$	47283.6	1.73388
$907$	−42283.7	−1.54797	−0.773984	−	0.633205i	$-0.781739\pi$
$907$	−42283.7	−1.54797	−0.773984	+	0.633205i	$0.781739\pi$
$908$	3298.65	0.120561
$909$	50731.8	1.85112
$910$	0	0
$911$	−20097.2	−0.730899	−0.365449	−	0.930831i	$-0.619085\pi$
$911$	−20097.2	−0.730899	−0.365449	+	0.930831i	$0.619085\pi$
$912$	−4803.92	−0.174423
$913$	−3101.86	−0.112439
$914$	13665.5	0.494547
$915$	−7249.53	−0.261926
$916$	18805.1	0.678316
$917$	0	0
$918$	−28843.1	−1.03700
$919$	−50782.5	−1.82281	−0.911405	−	0.411511i	$-0.865001\pi$
$919$	−50782.5	−1.82281	−0.911405	+	0.411511i	$0.865001\pi$
$920$	2266.59	0.0812251
$921$	5326.61	0.190573
$922$	2989.68	0.106789
$923$	30724.6	1.09568
$924$	0	0
$925$	−7405.55	−0.263236
$926$	7265.81	0.257850
$927$	−46501.3	−1.64758
$928$	−3231.25	−0.114301
$929$	13395.8	0.473093	0.236546	−	0.971620i	$-0.423984\pi$
$929$	13395.8	0.473093	0.236546	+	0.971620i	$0.423984\pi$
$930$	24061.9	0.848410
$931$	0	0
$932$	−20312.0	−0.713886
$933$	87376.3	3.06599
$934$	17404.3	0.609729
$935$	−1554.56	−0.0543738
$936$	47175.9	1.64743
$937$	−1753.27	−0.0611280	−0.0305640	−	0.999533i	$-0.509730\pi$
$937$	−1753.27	−0.0611280	−0.0305640	+	0.999533i	$0.509730\pi$
$938$	0	0
$939$	32157.4	1.11759
$940$	−22737.5	−0.788953
$941$	−3805.28	−0.131826	−0.0659132	−	0.997825i	$-0.520996\pi$
$941$	−3805.28	−0.131826	−0.0659132	+	0.997825i	$0.520996\pi$
$942$	−11737.0	−0.405956
$943$	−9859.90	−0.340491
$944$	−8565.04	−0.295305
$945$	0	0
$946$	1213.44	0.0417043
$947$	8663.09	0.297268	0.148634	−	0.988892i	$-0.452512\pi$
$947$	8663.09	0.297268	0.148634	+	0.988892i	$0.452512\pi$
$948$	−39446.3	−1.35143
$949$	−44646.8	−1.52718
$950$	−1672.84	−0.0571306
$951$	4750.32	0.161977
$952$	0	0
$953$	−40285.4	−1.36933	−0.684666	−	0.728857i	$-0.740052\pi$
$953$	−40285.4	−1.36933	−0.684666	+	0.728857i	$0.740052\pi$
$954$	69360.7	2.35392
$955$	−44034.3	−1.49206
$956$	−3484.19	−0.117873
$957$	−3106.98	−0.104947
$958$	21861.0	0.737263
$959$	0	0
$960$	7568.23	0.254441
$961$	−19440.3	−0.652555
$962$	−50125.5	−1.67995
$963$	−33068.5	−1.10656
$964$	−1889.84	−0.0631409
$965$	35587.1	1.18714
$966$	0	0
$967$	11328.2	0.376723	0.188362	−	0.982100i	$-0.439682\pi$
$967$	11328.2	0.376723	0.188362	+	0.982100i	$0.439682\pi$
$968$	−10565.8	−0.350824
$969$	11821.5	0.391909
$970$	13152.1	0.435349
$971$	−16804.9	−0.555402	−0.277701	−	0.960668i	$-0.589572\pi$
$971$	−16804.9	−0.555402	−0.277701	+	0.960668i	$0.589572\pi$
$972$	27909.8	0.920994
$973$	0	0
$974$	24531.9	0.807037
$975$	23235.3	0.763206
$976$	−980.878	−0.0321692
$977$	−10358.8	−0.339208	−0.169604	−	0.985512i	$-0.554249\pi$
$977$	−10358.8	−0.339208	−0.169604	+	0.985512i	$0.554249\pi$
$978$	27391.4	0.895583
$979$	452.159	0.0147610
$980$	0	0
$981$	−93610.8	−3.04665
$982$	−3636.41	−0.118169
$983$	17315.0	0.561813	0.280906	−	0.959735i	$-0.409365\pi$
$983$	17315.0	0.561813	0.280906	+	0.959735i	$0.409365\pi$
$984$	−32922.7	−1.06660
$985$	46295.8	1.49757
$986$	7951.44	0.256821
$987$	0	0
$988$	−11322.8	−0.364603
$989$	4353.69	0.139979
$990$	5145.09	0.165173
$991$	−5746.76	−0.184210	−0.0921048	−	0.995749i	$-0.529359\pi$
$991$	−5746.76	−0.184210	−0.0921048	+	0.995749i	$0.529359\pi$
$992$	3255.63	0.104200
$993$	−53130.1	−1.69792
$994$	0	0
$995$	−20983.1	−0.668553
$996$	−37160.8	−1.18221
$997$	−36523.2	−1.16018	−0.580091	−	0.814552i	$-0.696983\pi$
$997$	−36523.2	−1.16018	−0.580091	+	0.814552i	$0.696983\pi$
$998$	9964.81	0.316063
$999$	−101430.	−3.21231

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.y.1.10		11
7.2	even	3		322.4.e.a.277.2	yes	22
7.4	even	3		322.4.e.a.93.2	✓	22
7.6	odd	2		2254.4.a.v.1.2		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.a.93.2	✓	22	7.4	even	3
322.4.e.a.277.2	yes	22	7.2	even	3
2254.4.a.v.1.2		11	7.6	odd	2
2254.4.a.y.1.10		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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +9.59976 q^{3} +4.00000 q^{4} +12.3184 q^{5} +19.1995 q^{6} +8.00000 q^{8} +65.1554 q^{9} +24.6368 q^{10} +3.20522 q^{11} +38.3990 q^{12} +90.5065 q^{13} +118.254 q^{15} +16.0000 q^{16} -39.3727 q^{17} +130.311 q^{18} -31.2763 q^{19} +49.2736 q^{20} +6.41044 q^{22} +23.0000 q^{23} +76.7981 q^{24} +26.7429 q^{25} +181.013 q^{26} +366.283 q^{27} -100.977 q^{29} +236.507 q^{30} +101.739 q^{31} +32.0000 q^{32} +30.7694 q^{33} -78.7453 q^{34} +260.622 q^{36} -276.916 q^{37} -62.5526 q^{38} +868.841 q^{39} +98.5472 q^{40} -428.691 q^{41} +189.291 q^{43} +12.8209 q^{44} +802.610 q^{45} +46.0000 q^{46} -461.455 q^{47} +153.596 q^{48} +53.4858 q^{50} -377.968 q^{51} +362.026 q^{52} +532.272 q^{53} +732.566 q^{54} +39.4832 q^{55} -300.245 q^{57} -201.953 q^{58} -535.315 q^{59} +473.015 q^{60} -61.3049 q^{61} +203.477 q^{62} +64.0000 q^{64} +1114.90 q^{65} +61.5387 q^{66} +324.862 q^{67} -157.491 q^{68} +220.794 q^{69} +339.474 q^{71} +521.243 q^{72} -493.299 q^{73} -553.833 q^{74} +256.726 q^{75} -125.105 q^{76} +1737.68 q^{78} -1027.27 q^{79} +197.094 q^{80} +1757.03 q^{81} -857.383 q^{82} -967.752 q^{83} -485.008 q^{85} +378.582 q^{86} -969.351 q^{87} +25.6418 q^{88} +141.069 q^{89} +1605.22 q^{90} +92.0000 q^{92} +976.665 q^{93} -922.909 q^{94} -385.274 q^{95} +307.192 q^{96} +533.840 q^{97} +208.837 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 22 q^{2} + 18 q^{3} + 44 q^{4} + 33 q^{5} + 36 q^{6} + 88 q^{8} + 171 q^{9} + 66 q^{10} + 8 q^{11} + 72 q^{12} + 185 q^{13} - 186 q^{15} + 176 q^{16} + 107 q^{17} + 342 q^{18} + 114 q^{19} + 132 q^{20}+ \cdots - 1729 q^{99}+O(q^{100}) \) 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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