LMFDB - Embedded newform 2254.4.a.u.1.4

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} - 234 x^{9} - 105 x^{8} + 18997 x^{7} + 16513 x^{6} - 621598 x^{5} - 743169 x^{4} + \cdots - 12103441 $ x^11 - 234x^9 - 105x^8 + 18997x^7 + 16513x^6 - 621598x^5 - 743169x^4 + 7229916x^3 + 7324254x^2 - 26794327*x - 12103441
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.73826$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} -3.73826 q^{3} +4.00000 q^{4} -15.3436 q^{5} +7.47652 q^{6} -8.00000 q^{8} -13.0254 q^{9} +30.6872 q^{10} +50.1520 q^{11} -14.9530 q^{12} +48.5248 q^{13} +57.3583 q^{15} +16.0000 q^{16} +123.081 q^{17} +26.0508 q^{18} -76.1082 q^{19} -61.3743 q^{20} -100.304 q^{22} +23.0000 q^{23} +29.9061 q^{24} +110.425 q^{25} -97.0497 q^{26} +149.625 q^{27} -154.546 q^{29} -114.717 q^{30} +262.327 q^{31} -32.0000 q^{32} -187.481 q^{33} -246.162 q^{34} -52.1016 q^{36} +313.312 q^{37} +152.216 q^{38} -181.399 q^{39} +122.749 q^{40} +249.375 q^{41} +192.473 q^{43} +200.608 q^{44} +199.856 q^{45} -46.0000 q^{46} +410.610 q^{47} -59.8122 q^{48} -220.851 q^{50} -460.109 q^{51} +194.099 q^{52} -11.5547 q^{53} -299.251 q^{54} -769.512 q^{55} +284.512 q^{57} +309.092 q^{58} -430.746 q^{59} +229.433 q^{60} -561.392 q^{61} -524.653 q^{62} +64.0000 q^{64} -744.545 q^{65} +374.963 q^{66} -659.887 q^{67} +492.324 q^{68} -85.9800 q^{69} -920.555 q^{71} +104.203 q^{72} +351.457 q^{73} -626.624 q^{74} -412.799 q^{75} -304.433 q^{76} +362.797 q^{78} -586.480 q^{79} -245.497 q^{80} -207.653 q^{81} -498.749 q^{82} +538.459 q^{83} -1888.50 q^{85} -384.946 q^{86} +577.734 q^{87} -401.216 q^{88} +431.970 q^{89} -399.713 q^{90} +92.0000 q^{92} -980.646 q^{93} -821.220 q^{94} +1167.77 q^{95} +119.624 q^{96} +953.090 q^{97} -653.251 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 22 q^{2} + 44 q^{4} + 23 q^{5} - 88 q^{8} + 171 q^{9} - 46 q^{10} - 48 q^{11} + 77 q^{13} + 104 q^{15} + 176 q^{16} + 97 q^{17} - 342 q^{18} + 138 q^{19} + 92 q^{20} + 96 q^{22} + 253 q^{23} + 30 q^{25}+ \cdots - 2545 q^{99}+O(q^{100}) $ 11 * q - 22 * q^2 + 44 * q^4 + 23 * q^5 - 88 * q^8 + 171 * q^9 - 46 * q^10 - 48 * q^11 + 77 * q^13 + 104 * q^15 + 176 * q^16 + 97 * q^17 - 342 * q^18 + 138 * q^19 + 92 * q^20 + 96 * q^22 + 253 * q^23 + 30 * q^25 - 154 * q^26 + 315 * q^27 + 137 * q^29 - 208 * q^30 + 234 * q^31 - 352 * q^32 + 28 * q^33 - 194 * q^34 + 684 * q^36 + 87 * q^37 - 276 * q^38 + 129 * q^39 - 184 * q^40 + 612 * q^41 + 187 * q^43 - 192 * q^44 + 597 * q^45 - 506 * q^46 + 793 * q^47 - 60 * q^50 - 1257 * q^51 + 308 * q^52 + 575 * q^53 - 630 * q^54 - 549 * q^55 + 39 * q^57 - 274 * q^58 + 1241 * q^59 + 416 * q^60 + 187 * q^61 - 468 * q^62 + 704 * q^64 - 937 * q^65 - 56 * q^66 + 2275 * q^67 + 388 * q^68 + 115 * q^71 - 1368 * q^72 + 2385 * q^73 - 174 * q^74 + 196 * q^75 + 552 * q^76 - 258 * q^78 - 1225 * q^79 + 368 * q^80 + 3635 * q^81 - 1224 * q^82 - 597 * q^83 - 2493 * q^85 - 374 * q^86 + 1428 * q^87 + 384 * q^88 + 602 * q^89 - 1194 * q^90 + 1012 * q^92 - 1360 * q^93 - 1586 * q^94 + 4835 * q^95 - 1039 * q^97 - 2545 * 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
$3$	−3.73826	−0.719429	−0.359714	−	0.933062i	$-0.617126\pi$
$3$	−3.73826	−0.719429	−0.359714	+	0.933062i	$0.617126\pi$
$4$	4.00000	0.500000
$5$	−15.3436	−1.37237	−0.686186	−	0.727426i	$-0.740716\pi$
$5$	−15.3436	−1.37237	−0.686186	+	0.727426i	$0.740716\pi$
$6$	7.47652	0.508713
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	−13.0254	−0.482422
$10$	30.6872	0.970413
$11$	50.1520	1.37467	0.687337	−	0.726339i	$-0.258780\pi$
$11$	50.1520	1.37467	0.687337	+	0.726339i	$0.258780\pi$
$12$	−14.9530	−0.359714
$13$	48.5248	1.03526	0.517630	−	0.855605i	$-0.326815\pi$
$13$	48.5248	1.03526	0.517630	+	0.855605i	$0.326815\pi$
$14$	0	0
$15$	57.3583	0.987323
$16$	16.0000	0.250000
$17$	123.081	1.75597	0.877986	−	0.478686i	$-0.158887\pi$
$17$	123.081	1.75597	0.877986	+	0.478686i	$0.158887\pi$
$18$	26.0508	0.341124
$19$	−76.1082	−0.918970	−0.459485	−	0.888186i	$-0.651966\pi$
$19$	−76.1082	−0.918970	−0.459485	+	0.888186i	$0.651966\pi$
$20$	−61.3743	−0.686186
$21$	0	0
$22$	−100.304	−0.972041
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	29.9061	0.254356
$25$	110.425	0.883403
$26$	−97.0497	−0.732039
$27$	149.625	1.06650
$28$	0	0
$29$	−154.546	−0.989603	−0.494802	−	0.869006i	$-0.664759\pi$
$29$	−154.546	−0.989603	−0.494802	+	0.869006i	$0.664759\pi$
$30$	−114.717	−0.698143
$31$	262.327	1.51985	0.759924	−	0.650012i	$-0.225236\pi$
$31$	262.327	1.51985	0.759924	+	0.650012i	$0.225236\pi$
$32$	−32.0000	−0.176777
$33$	−187.481	−0.988980
$34$	−246.162	−1.24166
$35$	0	0
$36$	−52.1016	−0.241211
$37$	313.312	1.39211	0.696057	−	0.717987i	$-0.254936\pi$
$37$	313.312	1.39211	0.696057	+	0.717987i	$0.254936\pi$
$38$	152.216	0.649810
$39$	−181.399	−0.744795
$40$	122.749	0.485207
$41$	249.375	0.949896	0.474948	−	0.880014i	$-0.342467\pi$
$41$	249.375	0.949896	0.474948	+	0.880014i	$0.342467\pi$
$42$	0	0
$43$	192.473	0.682602	0.341301	−	0.939954i	$-0.389133\pi$
$43$	192.473	0.682602	0.341301	+	0.939954i	$0.389133\pi$
$44$	200.608	0.687337
$45$	199.856	0.662063
$46$	−46.0000	−0.147442
$47$	410.610	1.27433	0.637166	−	0.770727i	$-0.280107\pi$
$47$	410.610	1.27433	0.637166	+	0.770727i	$0.280107\pi$
$48$	−59.8122	−0.179857
$49$	0	0
$50$	−220.851	−0.624660
$51$	−460.109	−1.26330
$52$	194.099	0.517630
$53$	−11.5547	−0.0299465	−0.0149733	−	0.999888i	$-0.504766\pi$
$53$	−11.5547	−0.0299465	−0.0149733	+	0.999888i	$0.504766\pi$
$54$	−299.251	−0.754127
$55$	−769.512	−1.88656
$56$	0	0
$57$	284.512	0.661133
$58$	309.092	0.699755
$59$	−430.746	−0.950482	−0.475241	−	0.879856i	$-0.657639\pi$
$59$	−430.746	−0.950482	−0.475241	+	0.879856i	$0.657639\pi$
$60$	229.433	0.493662
$61$	−561.392	−1.17834	−0.589171	−	0.808008i	$-0.700546\pi$
$61$	−561.392	−1.17834	−0.589171	+	0.808008i	$0.700546\pi$
$62$	−524.653	−1.07469
$63$	0	0
$64$	64.0000	0.125000
$65$	−744.545	−1.42076
$66$	374.963	0.699314
$67$	−659.887	−1.20325	−0.601627	−	0.798777i	$-0.705481\pi$
$67$	−659.887	−1.20325	−0.601627	+	0.798777i	$0.705481\pi$
$68$	492.324	0.877986
$69$	−85.9800	−0.150011
$70$	0	0
$71$	−920.555	−1.53873	−0.769364	−	0.638810i	$-0.779427\pi$
$71$	−920.555	−1.53873	−0.769364	+	0.638810i	$0.779427\pi$
$72$	104.203	0.170562
$73$	351.457	0.563492	0.281746	−	0.959489i	$-0.409087\pi$
$73$	351.457	0.563492	0.281746	+	0.959489i	$0.409087\pi$
$74$	−626.624	−0.984373
$75$	−412.799	−0.635546
$76$	−304.433	−0.459485
$77$	0	0
$78$	362.797	0.526650
$79$	−586.480	−0.835242	−0.417621	−	0.908621i	$-0.637136\pi$
$79$	−586.480	−0.835242	−0.417621	+	0.908621i	$0.637136\pi$
$80$	−245.497	−0.343093
$81$	−207.653	−0.284846
$82$	−498.749	−0.671678
$83$	538.459	0.712091	0.356046	−	0.934469i	$-0.384125\pi$
$83$	538.459	0.712091	0.356046	+	0.934469i	$0.384125\pi$
$84$	0	0
$85$	−1888.50	−2.40985
$86$	−384.946	−0.482672
$87$	577.734	0.711949
$88$	−401.216	−0.486021
$89$	431.970	0.514481	0.257240	−	0.966347i	$-0.417187\pi$
$89$	431.970	0.514481	0.257240	+	0.966347i	$0.417187\pi$
$90$	−399.713	−0.468149
$91$	0	0
$92$	92.0000	0.104257
$93$	−980.646	−1.09342
$94$	−821.220	−0.901089
$95$	1167.77	1.26117
$96$	119.624	0.127178
$97$	953.090	0.997646	0.498823	−	0.866704i	$-0.333766\pi$
$97$	953.090	0.997646	0.498823	+	0.866704i	$0.333766\pi$
$98$	0	0
$99$	−653.251	−0.663173
$100$	441.702	0.441702
$101$	−879.033	−0.866010	−0.433005	−	0.901392i	$-0.642547\pi$
$101$	−879.033	−0.866010	−0.433005	+	0.901392i	$0.642547\pi$
$102$	920.218	0.893286
$103$	706.202	0.675574	0.337787	−	0.941223i	$-0.390322\pi$
$103$	706.202	0.675574	0.337787	+	0.941223i	$0.390322\pi$
$104$	−388.199	−0.366019
$105$	0	0
$106$	23.1095	0.0211754
$107$	1528.18	1.38070	0.690351	−	0.723475i	$-0.257456\pi$
$107$	1528.18	1.38070	0.690351	+	0.723475i	$0.257456\pi$
$108$	598.502	0.533249
$109$	−593.239	−0.521303	−0.260651	−	0.965433i	$-0.583937\pi$
$109$	−593.239	−0.521303	−0.260651	+	0.965433i	$0.583937\pi$
$110$	1539.02	1.33400
$111$	−1171.24	−1.00153
$112$	0	0
$113$	−56.3707	−0.0469284	−0.0234642	−	0.999725i	$-0.507470\pi$
$113$	−56.3707	−0.0469284	−0.0234642	+	0.999725i	$0.507470\pi$
$114$	−569.025	−0.467492
$115$	−352.902	−0.286159
$116$	−618.185	−0.494802
$117$	−632.056	−0.499432
$118$	861.493	0.672092
$119$	0	0
$120$	−458.866	−0.349072
$121$	1184.23	0.889728
$122$	1122.78	0.833214
$123$	−932.227	−0.683383
$124$	1049.31	0.759924
$125$	223.626	0.160014
$126$	0	0
$127$	−971.337	−0.678679	−0.339339	−	0.940664i	$-0.610204\pi$
$127$	−971.337	−0.678679	−0.339339	+	0.940664i	$0.610204\pi$
$128$	−128.000	−0.0883883
$129$	−719.515	−0.491083
$130$	1489.09	1.00463
$131$	1420.67	0.947515	0.473757	−	0.880655i	$-0.342897\pi$
$131$	1420.67	0.947515	0.473757	+	0.880655i	$0.342897\pi$
$132$	−749.926	−0.494490
$133$	0	0
$134$	1319.77	0.850829
$135$	−2295.79	−1.46363
$136$	−984.648	−0.620830
$137$	1660.08	1.03526	0.517630	−	0.855605i	$-0.326814\pi$
$137$	1660.08	1.03526	0.517630	+	0.855605i	$0.326814\pi$
$138$	171.960	0.106074
$139$	1238.35	0.755653	0.377826	−	0.925876i	$-0.376672\pi$
$139$	1238.35	0.755653	0.377826	+	0.925876i	$0.376672\pi$
$140$	0	0
$141$	−1534.97	−0.916791
$142$	1841.11	1.08805
$143$	2433.62	1.42314
$144$	−208.406	−0.120606
$145$	2371.29	1.35810
$146$	−702.913	−0.398449
$147$	0	0
$148$	1253.25	0.696057
$149$	−2631.75	−1.44699	−0.723494	−	0.690331i	$-0.757465\pi$
$149$	−2631.75	−1.44699	−0.723494	+	0.690331i	$0.757465\pi$
$150$	825.598	0.449399
$151$	−240.094	−0.129395	−0.0646974	−	0.997905i	$-0.520608\pi$
$151$	−240.094	−0.129395	−0.0646974	+	0.997905i	$0.520608\pi$
$152$	608.866	0.324905
$153$	−1603.18	−0.847120
$154$	0	0
$155$	−4025.03	−2.08579
$156$	−725.594	−0.372398
$157$	−1883.73	−0.957567	−0.478784	−	0.877933i	$-0.658922\pi$
$157$	−1883.73	−0.957567	−0.478784	+	0.877933i	$0.658922\pi$
$158$	1172.96	0.590605
$159$	43.1946	0.0215444
$160$	490.995	0.242603
$161$	0	0
$162$	415.306	0.201417
$163$	−3371.63	−1.62016	−0.810080	−	0.586319i	$-0.800576\pi$
$163$	−3371.63	−1.62016	−0.810080	+	0.586319i	$0.800576\pi$
$164$	997.498	0.474948
$165$	2876.64	1.35725
$166$	−1076.92	−0.503525
$167$	286.494	0.132752	0.0663760	−	0.997795i	$-0.478856\pi$
$167$	286.494	0.132752	0.0663760	+	0.997795i	$0.478856\pi$
$168$	0	0
$169$	157.660	0.0717614
$170$	3777.01	1.70402
$171$	991.340	0.443331
$172$	769.892	0.341301
$173$	−4065.65	−1.78674	−0.893370	−	0.449322i	$-0.851665\pi$
$173$	−4065.65	−1.78674	−0.893370	+	0.449322i	$0.851665\pi$
$174$	−1155.47	−0.503424
$175$	0	0
$176$	802.433	0.343668
$177$	1610.24	0.683804
$178$	−863.941	−0.363793
$179$	611.659	0.255405	0.127703	−	0.991813i	$-0.459240\pi$
$179$	611.659	0.255405	0.127703	+	0.991813i	$0.459240\pi$
$180$	799.425	0.331031
$181$	−1876.36	−0.770545	−0.385272	−	0.922803i	$-0.625892\pi$
$181$	−1876.36	−0.770545	−0.385272	+	0.922803i	$0.625892\pi$
$182$	0	0
$183$	2098.63	0.847733
$184$	−184.000	−0.0737210
$185$	−4807.33	−1.91050
$186$	1961.29	0.773166
$187$	6172.76	2.41389
$188$	1642.44	0.637166
$189$	0	0
$190$	−2335.54	−0.891780
$191$	4580.79	1.73536	0.867682	−	0.497120i	$-0.165609\pi$
$191$	4580.79	1.73536	0.867682	+	0.497120i	$0.165609\pi$
$192$	−239.249	−0.0899286
$193$	1265.94	0.472147	0.236074	−	0.971735i	$-0.424139\pi$
$193$	1265.94	0.472147	0.236074	+	0.971735i	$0.424139\pi$
$194$	−1906.18	−0.705442
$195$	2783.30	1.02214
$196$	0	0
$197$	3145.60	1.13764	0.568819	−	0.822462i	$-0.307400\pi$
$197$	3145.60	1.13764	0.568819	+	0.822462i	$0.307400\pi$
$198$	1306.50	0.468934
$199$	3249.56	1.15756	0.578782	−	0.815482i	$-0.303528\pi$
$199$	3249.56	1.15756	0.578782	+	0.815482i	$0.303528\pi$
$200$	−883.403	−0.312330
$201$	2466.83	0.865655
$202$	1758.07	0.612362
$203$	0	0
$204$	−1840.44	−0.631648
$205$	−3826.30	−1.30361
$206$	−1412.40	−0.477703
$207$	−299.584	−0.100592
$208$	776.397	0.258815
$209$	−3816.98	−1.26328
$210$	0	0
$211$	2447.79	0.798638	0.399319	−	0.916812i	$-0.369247\pi$
$211$	2447.79	0.798638	0.399319	+	0.916812i	$0.369247\pi$
$212$	−46.2189	−0.0149733
$213$	3441.27	1.10701
$214$	−3056.37	−0.976304
$215$	−2953.23	−0.936783
$216$	−1197.00	−0.377064
$217$	0	0
$218$	1186.48	0.368617
$219$	−1313.84	−0.405392
$220$	−3078.05	−0.943281
$221$	5972.49	1.81789
$222$	2342.49	0.708186
$223$	−1501.46	−0.450876	−0.225438	−	0.974258i	$-0.572381\pi$
$223$	−1501.46	−0.450876	−0.225438	+	0.974258i	$0.572381\pi$
$224$	0	0
$225$	−1438.34	−0.426173
$226$	112.741	0.0331834
$227$	−3673.27	−1.07403	−0.537013	−	0.843574i	$-0.680447\pi$
$227$	−3673.27	−1.07403	−0.537013	+	0.843574i	$0.680447\pi$
$228$	1138.05	0.330567
$229$	4350.70	1.25547	0.627735	−	0.778427i	$-0.283982\pi$
$229$	4350.70	1.25547	0.627735	+	0.778427i	$0.283982\pi$
$230$	705.805	0.202345
$231$	0	0
$232$	1236.37	0.349878
$233$	−3366.22	−0.946474	−0.473237	−	0.880935i	$-0.656915\pi$
$233$	−3366.22	−0.946474	−0.473237	+	0.880935i	$0.656915\pi$
$234$	1264.11	0.353152
$235$	−6300.22	−1.74886
$236$	−1722.99	−0.475241
$237$	2192.41	0.600897
$238$	0	0
$239$	−4025.61	−1.08952	−0.544760	−	0.838592i	$-0.683379\pi$
$239$	−4025.61	−1.08952	−0.544760	+	0.838592i	$0.683379\pi$
$240$	917.733	0.246831
$241$	−2041.55	−0.545676	−0.272838	−	0.962060i	$-0.587962\pi$
$241$	−2041.55	−0.545676	−0.272838	+	0.962060i	$0.587962\pi$
$242$	−2368.46	−0.629133
$243$	−3263.63	−0.861571
$244$	−2245.57	−0.589171
$245$	0	0
$246$	1864.45	0.483225
$247$	−3693.14	−0.951372
$248$	−2098.61	−0.537347
$249$	−2012.90	−0.512299
$250$	−447.253	−0.113147
$251$	4772.70	1.20020	0.600099	−	0.799925i	$-0.295128\pi$
$251$	4772.70	1.20020	0.600099	+	0.799925i	$0.295128\pi$
$252$	0	0
$253$	1153.50	0.286639
$254$	1942.67	0.479898
$255$	7059.72	1.73371
$256$	256.000	0.0625000
$257$	−2558.07	−0.620887	−0.310443	−	0.950592i	$-0.600478\pi$
$257$	−2558.07	−0.620887	−0.310443	+	0.950592i	$0.600478\pi$
$258$	1439.03	0.347248
$259$	0	0
$260$	−2978.18	−0.710380
$261$	2013.03	0.477407
$262$	−2841.34	−0.669994
$263$	1259.39	0.295276	0.147638	−	0.989041i	$-0.452833\pi$
$263$	1259.39	0.295276	0.147638	+	0.989041i	$0.452833\pi$
$264$	1499.85	0.349657
$265$	177.291	0.0410977
$266$	0	0
$267$	−1614.82	−0.370132
$268$	−2639.55	−0.601627
$269$	7288.59	1.65202	0.826009	−	0.563656i	$-0.190606\pi$
$269$	7288.59	1.65202	0.826009	+	0.563656i	$0.190606\pi$
$270$	4591.58	1.03494
$271$	−4735.04	−1.06138	−0.530688	−	0.847567i	$-0.678067\pi$
$271$	−4735.04	−1.06138	−0.530688	+	0.847567i	$0.678067\pi$
$272$	1969.30	0.438993
$273$	0	0
$274$	−3320.17	−0.732039
$275$	5538.06	1.21439
$276$	−343.920	−0.0750056
$277$	6353.58	1.37816	0.689079	−	0.724686i	$-0.258015\pi$
$277$	6353.58	1.37816	0.689079	+	0.724686i	$0.258015\pi$
$278$	−2476.71	−0.534327
$279$	−3416.91	−0.733208
$280$	0	0
$281$	−470.688	−0.0999250	−0.0499625	−	0.998751i	$-0.515910\pi$
$281$	−470.688	−0.0999250	−0.0499625	+	0.998751i	$0.515910\pi$
$282$	3069.93	0.648269
$283$	−5705.87	−1.19851	−0.599256	−	0.800558i	$-0.704537\pi$
$283$	−5705.87	−1.19851	−0.599256	+	0.800558i	$0.704537\pi$
$284$	−3682.22	−0.769364
$285$	−4365.44	−0.907320
$286$	−4867.24	−1.00631
$287$	0	0
$288$	416.813	0.0852810
$289$	10235.9	2.08344
$290$	−4742.58	−0.960324
$291$	−3562.90	−0.717735
$292$	1405.83	0.281746
$293$	−3850.89	−0.767820	−0.383910	−	0.923371i	$-0.625423\pi$
$293$	−3850.89	−0.767820	−0.383910	+	0.923371i	$0.625423\pi$
$294$	0	0
$295$	6609.19	1.30441
$296$	−2506.50	−0.492186
$297$	7504.02	1.46609
$298$	5263.50	1.02317
$299$	1116.07	0.215866
$300$	−1651.20	−0.317773
$301$	0	0
$302$	480.189	0.0914959
$303$	3286.05	0.623032
$304$	−1217.73	−0.229742
$305$	8613.76	1.61712
$306$	3206.36	0.599004
$307$	3517.53	0.653929	0.326964	−	0.945037i	$-0.393974\pi$
$307$	3517.53	0.653929	0.326964	+	0.945037i	$0.393974\pi$
$308$	0	0
$309$	−2639.97	−0.486027
$310$	8050.06	1.47488
$311$	9947.46	1.81373	0.906863	−	0.421426i	$-0.138470\pi$
$311$	9947.46	1.81373	0.906863	+	0.421426i	$0.138470\pi$
$312$	1451.19	0.263325
$313$	3354.69	0.605809	0.302904	−	0.953021i	$-0.402044\pi$
$313$	3354.69	0.605809	0.302904	+	0.953021i	$0.402044\pi$
$314$	3767.46	0.677102
$315$	0	0
$316$	−2345.92	−0.417621
$317$	9148.17	1.62086	0.810430	−	0.585836i	$-0.199234\pi$
$317$	9148.17	1.62086	0.810430	+	0.585836i	$0.199234\pi$
$318$	−86.3892	−0.0152342
$319$	−7750.81	−1.36038
$320$	−981.989	−0.171546
$321$	−5712.75	−0.993317
$322$	0	0
$323$	−9367.48	−1.61369
$324$	−830.612	−0.142423
$325$	5358.37	0.914551
$326$	6743.25	1.14563
$327$	2217.68	0.375040
$328$	−1995.00	−0.335839
$329$	0	0
$330$	−5753.27	−0.959719
$331$	7072.35	1.17442	0.587208	−	0.809436i	$-0.300227\pi$
$331$	7072.35	1.17442	0.587208	+	0.809436i	$0.300227\pi$
$332$	2153.84	0.356046
$333$	−4081.02	−0.671587
$334$	−572.988	−0.0938698
$335$	10125.0	1.65131
$336$	0	0
$337$	−9017.84	−1.45767	−0.728833	−	0.684692i	$-0.759937\pi$
$337$	−9017.84	−1.45767	−0.728833	+	0.684692i	$0.759937\pi$
$338$	−315.319	−0.0507429
$339$	210.728	0.0337616
$340$	−7554.01	−1.20492
$341$	13156.2	2.08929
$342$	−1982.68	−0.313483
$343$	0	0
$344$	−1539.78	−0.241336
$345$	1319.24	0.205871
$346$	8131.31	1.26342
$347$	9367.01	1.44913	0.724564	−	0.689207i	$-0.242041\pi$
$347$	9367.01	1.44913	0.724564	+	0.689207i	$0.242041\pi$
$348$	2310.94	0.355975
$349$	820.873	0.125904	0.0629518	−	0.998017i	$-0.479949\pi$
$349$	820.873	0.125904	0.0629518	+	0.998017i	$0.479949\pi$
$350$	0	0
$351$	7260.55	1.10410
$352$	−1604.87	−0.243010
$353$	5781.36	0.871702	0.435851	−	0.900019i	$-0.356447\pi$
$353$	5781.36	0.871702	0.435851	+	0.900019i	$0.356447\pi$
$354$	−3220.49	−0.483522
$355$	14124.6	2.11171
$356$	1727.88	0.257240
$357$	0	0
$358$	−1223.32	−0.180599
$359$	−1847.18	−0.271561	−0.135781	−	0.990739i	$-0.543354\pi$
$359$	−1847.18	−0.271561	−0.135781	+	0.990739i	$0.543354\pi$
$360$	−1598.85	−0.234074
$361$	−1066.54	−0.155495
$362$	3752.72	0.544857
$363$	−4426.95	−0.640096
$364$	0	0
$365$	−5392.60	−0.773320
$366$	−4197.26	−0.599438
$367$	3727.30	0.530146	0.265073	−	0.964228i	$-0.414604\pi$
$367$	3727.30	0.530146	0.265073	+	0.964228i	$0.414604\pi$
$368$	368.000	0.0521286
$369$	−3248.20	−0.458251
$370$	9614.66	1.35093
$371$	0	0
$372$	−3922.58	−0.546711
$373$	−3540.61	−0.491490	−0.245745	−	0.969334i	$-0.579033\pi$
$373$	−3540.61	−0.491490	−0.245745	+	0.969334i	$0.579033\pi$
$374$	−12345.5	−1.70688
$375$	−835.974	−0.115119
$376$	−3284.88	−0.450544
$377$	−7499.33	−1.02450
$378$	0	0
$379$	3016.44	0.408823	0.204412	−	0.978885i	$-0.434472\pi$
$379$	3016.44	0.408823	0.204412	+	0.978885i	$0.434472\pi$
$380$	4671.09	0.630584
$381$	3631.11	0.488261
$382$	−9161.59	−1.22709
$383$	8440.45	1.12608	0.563038	−	0.826431i	$-0.309633\pi$
$383$	8440.45	1.12608	0.563038	+	0.826431i	$0.309633\pi$
$384$	478.497	0.0635891
$385$	0	0
$386$	−2531.88	−0.333859
$387$	−2507.04	−0.329302
$388$	3812.36	0.498823
$389$	−1589.08	−0.207120	−0.103560	−	0.994623i	$-0.533023\pi$
$389$	−1589.08	−0.207120	−0.103560	+	0.994623i	$0.533023\pi$
$390$	−5566.60	−0.722759
$391$	2830.86	0.366146
$392$	0	0
$393$	−5310.83	−0.681669
$394$	−6291.20	−0.804432
$395$	8998.69	1.14626
$396$	−2613.00	−0.331587
$397$	−10782.7	−1.36314	−0.681569	−	0.731753i	$-0.738702\pi$
$397$	−10782.7	−1.36314	−0.681569	+	0.731753i	$0.738702\pi$
$398$	−6499.12	−0.818522
$399$	0	0
$400$	1766.81	0.220851
$401$	−4015.39	−0.500048	−0.250024	−	0.968240i	$-0.580438\pi$
$401$	−4015.39	−0.500048	−0.250024	+	0.968240i	$0.580438\pi$
$402$	−4933.66	−0.612111
$403$	12729.4	1.57344
$404$	−3516.13	−0.433005
$405$	3186.14	0.390915
$406$	0	0
$407$	15713.2	1.91370
$408$	3680.87	0.446643
$409$	8257.36	0.998290	0.499145	−	0.866519i	$-0.333648\pi$
$409$	8257.36	0.998290	0.499145	+	0.866519i	$0.333648\pi$
$410$	7652.60	0.921792
$411$	−6205.83	−0.744795
$412$	2824.81	0.337787
$413$	0	0
$414$	599.169	0.0711293
$415$	−8261.89	−0.977254
$416$	−1552.79	−0.183010
$417$	−4629.29	−0.543638
$418$	7633.97	0.893276
$419$	7604.71	0.886669	0.443335	−	0.896356i	$-0.353795\pi$
$419$	7604.71	0.886669	0.443335	+	0.896356i	$0.353795\pi$
$420$	0	0
$421$	−2514.29	−0.291067	−0.145533	−	0.989353i	$-0.546490\pi$
$421$	−2514.29	−0.291067	−0.145533	+	0.989353i	$0.546490\pi$
$422$	−4895.57	−0.564722
$423$	−5348.36	−0.614766
$424$	92.4379	0.0105877
$425$	13591.3	1.55123
$426$	−6882.55	−0.782771
$427$	0	0
$428$	6112.74	0.690351
$429$	−9097.51	−1.02385
$430$	5906.45	0.662406
$431$	4342.25	0.485287	0.242643	−	0.970116i	$-0.421985\pi$
$431$	4342.25	0.485287	0.242643	+	0.970116i	$0.421985\pi$
$432$	2394.01	0.266624
$433$	3224.64	0.357890	0.178945	−	0.983859i	$-0.442732\pi$
$433$	3224.64	0.357890	0.178945	+	0.983859i	$0.442732\pi$
$434$	0	0
$435$	−8864.50	−0.977059
$436$	−2372.96	−0.260651
$437$	−1750.49	−0.191618
$438$	2627.67	0.286655
$439$	−9373.54	−1.01908	−0.509538	−	0.860448i	$-0.670184\pi$
$439$	−9373.54	−1.01908	−0.509538	+	0.860448i	$0.670184\pi$
$440$	6156.10	0.667001
$441$	0	0
$442$	−11945.0	−1.28544
$443$	−5579.73	−0.598422	−0.299211	−	0.954187i	$-0.596723\pi$
$443$	−5579.73	−0.598422	−0.299211	+	0.954187i	$0.596723\pi$
$444$	−4684.97	−0.500763
$445$	−6627.97	−0.706059
$446$	3002.92	0.318817
$447$	9838.16	1.04100
$448$	0	0
$449$	−14442.9	−1.51805	−0.759025	−	0.651062i	$-0.774324\pi$
$449$	−14442.9	−1.51805	−0.759025	+	0.651062i	$0.774324\pi$
$450$	2876.67	0.301350
$451$	12506.6	1.30580
$452$	−225.483	−0.0234642
$453$	897.536	0.0930903
$454$	7346.55	0.759451
$455$	0	0
$456$	−2276.10	−0.233746
$457$	−8445.79	−0.864502	−0.432251	−	0.901753i	$-0.642281\pi$
$457$	−8445.79	−0.864502	−0.432251	+	0.901753i	$0.642281\pi$
$458$	−8701.40	−0.887751
$459$	18416.0	1.87274
$460$	−1411.61	−0.143080
$461$	5265.12	0.531933	0.265966	−	0.963982i	$-0.414309\pi$
$461$	5265.12	0.531933	0.265966	+	0.963982i	$0.414309\pi$
$462$	0	0
$463$	−1969.69	−0.197709	−0.0988543	−	0.995102i	$-0.531518\pi$
$463$	−1969.69	−0.197709	−0.0988543	+	0.995102i	$0.531518\pi$
$464$	−2472.74	−0.247401
$465$	15046.6	1.50058
$466$	6732.44	0.669258
$467$	10344.8	1.02505	0.512526	−	0.858672i	$-0.328710\pi$
$467$	10344.8	1.02505	0.512526	+	0.858672i	$0.328710\pi$
$468$	−2528.22	−0.249716
$469$	0	0
$470$	12600.4	1.23663
$471$	7041.88	0.688902
$472$	3445.97	0.336046
$473$	9652.92	0.938354
$474$	−4384.83	−0.424898
$475$	−8404.28	−0.811821
$476$	0	0
$477$	150.505	0.0144469
$478$	8051.22	0.770407
$479$	11773.7	1.12308	0.561540	−	0.827450i	$-0.310209\pi$
$479$	11773.7	1.12308	0.561540	+	0.827450i	$0.310209\pi$
$480$	−1835.47	−0.174536
$481$	15203.4	1.44120
$482$	4083.10	0.385851
$483$	0	0
$484$	4736.91	0.444864
$485$	−14623.8	−1.36914
$486$	6527.25	0.609222
$487$	525.788	0.0489234	0.0244617	−	0.999701i	$-0.492213\pi$
$487$	525.788	0.0489234	0.0244617	+	0.999701i	$0.492213\pi$
$488$	4491.13	0.416607
$489$	12604.0	1.16559
$490$	0	0
$491$	−5414.47	−0.497661	−0.248831	−	0.968547i	$-0.580046\pi$
$491$	−5414.47	−0.497661	−0.248831	+	0.968547i	$0.580046\pi$
$492$	−3728.91	−0.341691
$493$	−19021.7	−1.73772
$494$	7386.28	0.672721
$495$	10023.2	0.910120
$496$	4197.23	0.379962
$497$	0	0
$498$	4025.80	0.362250
$499$	−2641.01	−0.236929	−0.118465	−	0.992958i	$-0.537797\pi$
$499$	−2641.01	−0.236929	−0.118465	+	0.992958i	$0.537797\pi$
$500$	894.505	0.0800070
$501$	−1070.99	−0.0955056
$502$	−9545.39	−0.848669
$503$	−19739.9	−1.74982	−0.874911	−	0.484284i	$-0.839080\pi$
$503$	−19739.9	−1.74982	−0.874911	+	0.484284i	$0.839080\pi$
$504$	0	0
$505$	13487.5	1.18849
$506$	−2306.99	−0.202685
$507$	−589.373	−0.0516272
$508$	−3885.35	−0.339339
$509$	7582.71	0.660310	0.330155	−	0.943927i	$-0.392899\pi$
$509$	7582.71	0.660310	0.330155	+	0.943927i	$0.392899\pi$
$510$	−14119.4	−1.22592
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−11387.7	−0.980078
$514$	5116.14	0.439033
$515$	−10835.7	−0.927138
$516$	−2878.06	−0.245542
$517$	20592.9	1.75179
$518$	0	0
$519$	15198.5	1.28543
$520$	5956.36	0.502314
$521$	−3199.47	−0.269042	−0.134521	−	0.990911i	$-0.542950\pi$
$521$	−3199.47	−0.269042	−0.134521	+	0.990911i	$0.542950\pi$
$522$	−4026.05	−0.337578
$523$	−14099.5	−1.17883	−0.589416	−	0.807830i	$-0.700642\pi$
$523$	−14099.5	−1.17883	−0.589416	+	0.807830i	$0.700642\pi$
$524$	5682.68	0.473757
$525$	0	0
$526$	−2518.79	−0.208792
$527$	32287.4	2.66881
$528$	−2999.70	−0.247245
$529$	529.000	0.0434783
$530$	−354.582	−0.0290605
$531$	5610.65	0.458534
$532$	0	0
$533$	12100.9	0.983389
$534$	3229.64	0.261723
$535$	−23447.8	−1.89484
$536$	5279.09	0.425414
$537$	−2286.54	−0.183746
$538$	−14577.2	−1.16815
$539$	0	0
$540$	−9183.16	−0.731815
$541$	−23848.0	−1.89521	−0.947604	−	0.319448i	$-0.896502\pi$
$541$	−23848.0	−1.89521	−0.947604	+	0.319448i	$0.896502\pi$
$542$	9470.08	0.750507
$543$	7014.32	0.554352
$544$	−3938.59	−0.310415
$545$	9102.41	0.715421
$546$	0	0
$547$	22391.6	1.75027	0.875135	−	0.483879i	$-0.160773\pi$
$547$	22391.6	1.75027	0.875135	+	0.483879i	$0.160773\pi$
$548$	6640.34	0.517630
$549$	7312.36	0.568458
$550$	−11076.1	−0.858704
$551$	11762.2	0.909415
$552$	687.840	0.0530370
$553$	0	0
$554$	−12707.2	−0.974505
$555$	17971.1	1.37447
$556$	4953.41	0.377826
$557$	11947.3	0.908842	0.454421	−	0.890787i	$-0.349846\pi$
$557$	11947.3	0.908842	0.454421	+	0.890787i	$0.349846\pi$
$558$	6833.82	0.518457
$559$	9339.72	0.706669
$560$	0	0
$561$	−23075.4	−1.73662
$562$	941.377	0.0706576
$563$	−1037.06	−0.0776322	−0.0388161	−	0.999246i	$-0.512359\pi$
$563$	−1037.06	−0.0776322	−0.0388161	+	0.999246i	$0.512359\pi$
$564$	−6139.87	−0.458396
$565$	864.928	0.0644032
$566$	11411.7	0.847475
$567$	0	0
$568$	7364.44	0.544023
$569$	14072.6	1.03683	0.518415	−	0.855129i	$-0.326522\pi$
$569$	14072.6	1.03683	0.518415	+	0.855129i	$0.326522\pi$
$570$	8730.88	0.641572
$571$	−23568.7	−1.72735	−0.863676	−	0.504047i	$-0.831844\pi$
$571$	−23568.7	−1.72735	−0.863676	+	0.504047i	$0.831844\pi$
$572$	9734.48	0.711572
$573$	−17124.2	−1.24847
$574$	0	0
$575$	2539.78	0.184202
$576$	−833.626	−0.0603028
$577$	11196.9	0.807856	0.403928	−	0.914791i	$-0.367645\pi$
$577$	11196.9	0.807856	0.403928	+	0.914791i	$0.367645\pi$
$578$	−20471.9	−1.47321
$579$	−4732.42	−0.339676
$580$	9485.16	0.679052
$581$	0	0
$582$	7125.80	0.507515
$583$	−579.493	−0.0411667
$584$	−2811.65	−0.199224
$585$	9697.99	0.685406
$586$	7701.77	0.542931
$587$	17414.2	1.22447	0.612234	−	0.790677i	$-0.290271\pi$
$587$	17414.2	1.22447	0.612234	+	0.790677i	$0.290271\pi$
$588$	0	0
$589$	−19965.2	−1.39669
$590$	−13218.4	−0.922360
$591$	−11759.1	−0.818450
$592$	5012.99	0.348028
$593$	28003.8	1.93925	0.969626	−	0.244592i	$-0.0786539\pi$
$593$	28003.8	1.93925	0.969626	+	0.244592i	$0.0786539\pi$
$594$	−15008.0	−1.03668
$595$	0	0
$596$	−10527.0	−0.723494
$597$	−12147.7	−0.832785
$598$	−2232.14	−0.152641
$599$	−4079.06	−0.278240	−0.139120	−	0.990275i	$-0.544427\pi$
$599$	−4079.06	−0.278240	−0.139120	+	0.990275i	$0.544427\pi$
$600$	3302.39	0.224699
$601$	−8199.39	−0.556506	−0.278253	−	0.960508i	$-0.589755\pi$
$601$	−8199.39	−0.556506	−0.278253	+	0.960508i	$0.589755\pi$
$602$	0	0
$603$	8595.29	0.580476
$604$	−960.378	−0.0646974
$605$	−18170.3	−1.22104
$606$	−6572.11	−0.440550
$607$	−4077.30	−0.272640	−0.136320	−	0.990665i	$-0.543528\pi$
$607$	−4077.30	−0.272640	−0.136320	+	0.990665i	$0.543528\pi$
$608$	2435.46	0.162452
$609$	0	0
$610$	−17227.5	−1.14348
$611$	19924.8	1.31926
$612$	−6412.72	−0.423560
$613$	−17662.5	−1.16376	−0.581879	−	0.813276i	$-0.697682\pi$
$613$	−17662.5	−1.16376	−0.581879	+	0.813276i	$0.697682\pi$
$614$	−7035.06	−0.462397
$615$	14303.7	0.937855
$616$	0	0
$617$	167.228	0.0109114	0.00545570	−	0.999985i	$-0.498263\pi$
$617$	167.228	0.0109114	0.00545570	+	0.999985i	$0.498263\pi$
$618$	5279.93	0.343673
$619$	−26088.7	−1.69401	−0.847006	−	0.531584i	$-0.821597\pi$
$619$	−26088.7	−1.69401	−0.847006	+	0.531584i	$0.821597\pi$
$620$	−16100.1	−1.04290
$621$	3441.38	0.222380
$622$	−19894.9	−1.28250
$623$	0	0
$624$	−2902.38	−0.186199
$625$	−17234.4	−1.10300
$626$	−6709.37	−0.428371
$627$	14268.9	0.908842
$628$	−7534.92	−0.478784
$629$	38562.8	2.44451
$630$	0	0
$631$	17963.2	1.13329	0.566643	−	0.823963i	$-0.308242\pi$
$631$	17963.2	1.13329	0.566643	+	0.823963i	$0.308242\pi$
$632$	4691.84	0.295303
$633$	−9150.47	−0.574563
$634$	−18296.3	−1.14612
$635$	14903.8	0.931400
$636$	172.778	0.0107722
$637$	0	0
$638$	15501.6	0.961935
$639$	11990.6	0.742317
$640$	1963.98	0.121302
$641$	5509.28	0.339475	0.169738	−	0.985489i	$-0.445708\pi$
$641$	5509.28	0.339475	0.169738	+	0.985489i	$0.445708\pi$
$642$	11425.5	0.702381
$643$	−4311.62	−0.264438	−0.132219	−	0.991221i	$-0.542210\pi$
$643$	−4311.62	−0.264438	−0.132219	+	0.991221i	$0.542210\pi$
$644$	0	0
$645$	11039.9	0.673949
$646$	18735.0	1.14105
$647$	−14142.2	−0.859331	−0.429665	−	0.902988i	$-0.641368\pi$
$647$	−14142.2	−0.859331	−0.429665	+	0.902988i	$0.641368\pi$
$648$	1661.22	0.100708
$649$	−21602.8	−1.30660
$650$	−10716.7	−0.646685
$651$	0	0
$652$	−13486.5	−0.810080
$653$	27908.6	1.67251	0.836253	−	0.548344i	$-0.184741\pi$
$653$	27908.6	1.67251	0.836253	+	0.548344i	$0.184741\pi$
$654$	−4435.37	−0.265193
$655$	−21798.1	−1.30034
$656$	3989.99	0.237474
$657$	−4577.86	−0.271841
$658$	0	0
$659$	7297.00	0.431337	0.215668	−	0.976467i	$-0.430807\pi$
$659$	7297.00	0.431337	0.215668	+	0.976467i	$0.430807\pi$
$660$	11506.5	0.678624
$661$	30889.9	1.81767	0.908835	−	0.417157i	$-0.136973\pi$
$661$	30889.9	1.81767	0.908835	+	0.417157i	$0.136973\pi$
$662$	−14144.7	−0.830437
$663$	−22326.7	−1.30784
$664$	−4307.67	−0.251762
$665$	0	0
$666$	8162.03	0.474883
$667$	−3554.56	−0.206347
$668$	1145.98	0.0663760
$669$	5612.86	0.324373
$670$	−20250.0	−1.16765
$671$	−28155.0	−1.61984
$672$	0	0
$673$	−7671.51	−0.439398	−0.219699	−	0.975568i	$-0.570508\pi$
$673$	−7671.51	−0.439398	−0.219699	+	0.975568i	$0.570508\pi$
$674$	18035.7	1.03073
$675$	16522.4	0.942147
$676$	630.639	0.0358807
$677$	−10663.3	−0.605352	−0.302676	−	0.953093i	$-0.597880\pi$
$677$	−10663.3	−0.605352	−0.302676	+	0.953093i	$0.597880\pi$
$678$	−421.457	−0.0238731
$679$	0	0
$680$	15108.0	0.852009
$681$	13731.7	0.772685
$682$	−26312.4	−1.47735
$683$	−16289.0	−0.912562	−0.456281	−	0.889836i	$-0.650819\pi$
$683$	−16289.0	−0.912562	−0.456281	+	0.889836i	$0.650819\pi$
$684$	3965.36	0.221666
$685$	−25471.6	−1.42076
$686$	0	0
$687$	−16264.1	−0.903221
$688$	3079.57	0.170650
$689$	−560.691	−0.0310024
$690$	−2638.48	−0.145573
$691$	−16771.9	−0.923348	−0.461674	−	0.887050i	$-0.652751\pi$
$691$	−16771.9	−0.923348	−0.461674	+	0.887050i	$0.652751\pi$
$692$	−16262.6	−0.893370
$693$	0	0
$694$	−18734.0	−1.02469
$695$	−19000.8	−1.03704
$696$	−4621.87	−0.251712
$697$	30693.3	1.66799
$698$	−1641.75	−0.0890273
$699$	12583.8	0.680921
$700$	0	0
$701$	−12347.4	−0.665272	−0.332636	−	0.943055i	$-0.607938\pi$
$701$	−12347.4	−0.665272	−0.332636	+	0.943055i	$0.607938\pi$
$702$	−14521.1	−0.780717
$703$	−23845.6	−1.27931
$704$	3209.73	0.171834
$705$	23551.9	1.25818
$706$	−11562.7	−0.616386
$707$	0	0
$708$	6440.97	0.341902
$709$	−516.206	−0.0273435	−0.0136717	−	0.999907i	$-0.504352\pi$
$709$	−516.206	−0.0273435	−0.0136717	+	0.999907i	$0.504352\pi$
$710$	−28249.2	−1.49320
$711$	7639.13	0.402939
$712$	−3455.76	−0.181896
$713$	6033.51	0.316910
$714$	0	0
$715$	−37340.4	−1.95308
$716$	2446.63	0.127703
$717$	15048.8	0.783832
$718$	3694.36	0.192023
$719$	35577.3	1.84535	0.922676	−	0.385576i	$-0.125997\pi$
$719$	35577.3	1.84535	0.922676	+	0.385576i	$0.125997\pi$
$720$	3197.70	0.165516
$721$	0	0
$722$	2133.08	0.109952
$723$	7631.85	0.392575
$724$	−7505.43	−0.385272
$725$	−17065.8	−0.874219
$726$	8853.91	0.452616
$727$	19868.9	1.01361	0.506806	−	0.862060i	$-0.330826\pi$
$727$	19868.9	1.01361	0.506806	+	0.862060i	$0.330826\pi$
$728$	0	0
$729$	17806.9	0.904685
$730$	10785.2	0.546820
$731$	23689.8	1.19863
$732$	8394.52	0.423867
$733$	−20954.6	−1.05590	−0.527950	−	0.849275i	$-0.677039\pi$
$733$	−20954.6	−1.05590	−0.527950	+	0.849275i	$0.677039\pi$
$734$	−7454.61	−0.374870
$735$	0	0
$736$	−736.000	−0.0368605
$737$	−33094.7	−1.65408
$738$	6496.41	0.324033
$739$	−14005.9	−0.697181	−0.348590	−	0.937275i	$-0.613340\pi$
$739$	−14005.9	−0.697181	−0.348590	+	0.937275i	$0.613340\pi$
$740$	−19229.3	−0.955248
$741$	13805.9	0.684444
$742$	0	0
$743$	−1643.38	−0.0811439	−0.0405719	−	0.999177i	$-0.512918\pi$
$743$	−1643.38	−0.0811439	−0.0405719	+	0.999177i	$0.512918\pi$
$744$	7845.17	0.386583
$745$	40380.4	1.98580
$746$	7081.22	0.347536
$747$	−7013.65	−0.343529
$748$	24691.1	1.20694
$749$	0	0
$750$	1671.95	0.0814012
$751$	−6205.21	−0.301506	−0.150753	−	0.988571i	$-0.548170\pi$
$751$	−6205.21	−0.301506	−0.150753	+	0.988571i	$0.548170\pi$
$752$	6569.76	0.318583
$753$	−17841.6	−0.863457
$754$	14998.7	0.724428
$755$	3683.91	0.177578
$756$	0	0
$757$	40899.2	1.96368	0.981841	−	0.189705i	$-0.0607532\pi$
$757$	40899.2	1.96368	0.981841	+	0.189705i	$0.0607532\pi$
$758$	−6032.88	−0.289082
$759$	−4312.07	−0.206217
$760$	−9342.18	−0.445890
$761$	−15510.4	−0.738832	−0.369416	−	0.929264i	$-0.620442\pi$
$761$	−15510.4	−0.738832	−0.369416	+	0.929264i	$0.620442\pi$
$762$	−7262.22	−0.345253
$763$	0	0
$764$	18323.2	0.867682
$765$	24598.5	1.16256
$766$	−16880.9	−0.796256
$767$	−20901.9	−0.983995
$768$	−956.995	−0.0449643
$769$	10646.4	0.499246	0.249623	−	0.968343i	$-0.419693\pi$
$769$	10646.4	0.499246	0.249623	+	0.968343i	$0.419693\pi$
$770$	0	0
$771$	9562.73	0.446684
$772$	5063.76	0.236074
$773$	10230.6	0.476026	0.238013	−	0.971262i	$-0.423504\pi$
$773$	10230.6	0.476026	0.238013	+	0.971262i	$0.423504\pi$
$774$	5014.08	0.232852
$775$	28967.5	1.34264
$776$	−7624.72	−0.352721
$777$	0	0
$778$	3178.16	0.146456
$779$	−18979.4	−0.872926
$780$	11133.2	0.511068
$781$	−46167.7	−2.11525
$782$	−5661.73	−0.258904
$783$	−23124.0	−1.05541
$784$	0	0
$785$	28903.2	1.31414
$786$	10621.7	0.482013
$787$	−3275.04	−0.148339	−0.0741694	−	0.997246i	$-0.523631\pi$
$787$	−3275.04	−0.148339	−0.0741694	+	0.997246i	$0.523631\pi$
$788$	12582.4	0.568819
$789$	−4707.94	−0.212430
$790$	−17997.4	−0.810530
$791$	0	0
$792$	5226.01	0.234467
$793$	−27241.4	−1.21989
$794$	21565.3	0.963885
$795$	−662.760	−0.0295669
$796$	12998.2	0.578782
$797$	20657.2	0.918087	0.459044	−	0.888414i	$-0.348192\pi$
$797$	20657.2	0.918087	0.459044	+	0.888414i	$0.348192\pi$
$798$	0	0
$799$	50538.3	2.23769
$800$	−3533.61	−0.156165
$801$	−5626.59	−0.248197
$802$	8030.79	0.353587
$803$	17626.3	0.774617
$804$	9867.31	0.432828
$805$	0	0
$806$	−25458.7	−1.11259
$807$	−27246.6	−1.18851
$808$	7032.26	0.306181
$809$	22546.1	0.979827	0.489914	−	0.871771i	$-0.337028\pi$
$809$	22546.1	0.979827	0.489914	+	0.871771i	$0.337028\pi$
$810$	−6372.28	−0.276419
$811$	−42856.7	−1.85561	−0.927806	−	0.373064i	$-0.878307\pi$
$811$	−42856.7	−1.85561	−0.927806	+	0.373064i	$0.878307\pi$
$812$	0	0
$813$	17700.8	0.763585
$814$	−31426.5	−1.35319
$815$	51732.8	2.22346
$816$	−7361.74	−0.315824
$817$	−14648.8	−0.627290
$818$	−16514.7	−0.705897
$819$	0	0
$820$	−15305.2	−0.651805
$821$	−22567.9	−0.959350	−0.479675	−	0.877446i	$-0.659245\pi$
$821$	−22567.9	−0.959350	−0.479675	+	0.877446i	$0.659245\pi$
$822$	12411.7	0.526650
$823$	−3575.51	−0.151439	−0.0757196	−	0.997129i	$-0.524125\pi$
$823$	−3575.51	−0.151439	−0.0757196	+	0.997129i	$0.524125\pi$
$824$	−5649.61	−0.238851
$825$	−20702.7	−0.873668
$826$	0	0
$827$	−25118.0	−1.05615	−0.528076	−	0.849197i	$-0.677086\pi$
$827$	−25118.0	−1.05615	−0.528076	+	0.849197i	$0.677086\pi$
$828$	−1198.34	−0.0502960
$829$	−10559.6	−0.442399	−0.221200	−	0.975229i	$-0.570997\pi$
$829$	−10559.6	−0.442399	−0.221200	+	0.975229i	$0.570997\pi$
$830$	16523.8	0.691023
$831$	−23751.4	−0.991487
$832$	3105.59	0.129407
$833$	0	0
$834$	9258.57	0.384410
$835$	−4395.84	−0.182185
$836$	−15267.9	−0.631642
$837$	39250.7	1.62091
$838$	−15209.4	−0.626970
$839$	28017.9	1.15290	0.576452	−	0.817131i	$-0.304437\pi$
$839$	28017.9	1.15290	0.576452	+	0.817131i	$0.304437\pi$
$840$	0	0
$841$	−504.491	−0.0206852
$842$	5028.58	0.205815
$843$	1759.56	0.0718889
$844$	9791.15	0.399319
$845$	−2419.06	−0.0984832
$846$	10696.7	0.434705
$847$	0	0
$848$	−184.876	−0.00748663
$849$	21330.0	0.862243
$850$	−27182.5	−1.09689
$851$	7206.18	0.290276
$852$	13765.1	0.553503
$853$	14286.2	0.573448	0.286724	−	0.958013i	$-0.407434\pi$
$853$	14286.2	0.573448	0.286724	+	0.958013i	$0.407434\pi$
$854$	0	0
$855$	−15210.7	−0.608415
$856$	−12225.5	−0.488152
$857$	−27467.7	−1.09484	−0.547421	−	0.836857i	$-0.684390\pi$
$857$	−27467.7	−1.09484	−0.547421	+	0.836857i	$0.684390\pi$
$858$	18195.0	0.723971
$859$	−3545.68	−0.140835	−0.0704173	−	0.997518i	$-0.522433\pi$
$859$	−3545.68	−0.140835	−0.0704173	+	0.997518i	$0.522433\pi$
$860$	−11812.9	−0.468391
$861$	0	0
$862$	−8684.49	−0.343150
$863$	−13581.5	−0.535714	−0.267857	−	0.963459i	$-0.586315\pi$
$863$	−13581.5	−0.535714	−0.267857	+	0.963459i	$0.586315\pi$
$864$	−4788.01	−0.188532
$865$	62381.7	2.45207
$866$	−6449.29	−0.253067
$867$	−38264.6	−1.49889
$868$	0	0
$869$	−29413.1	−1.14818
$870$	17729.0	0.690885
$871$	−32020.9	−1.24568
$872$	4745.91	0.184308
$873$	−12414.4	−0.481287
$874$	3500.98	0.135495
$875$	0	0
$876$	−5255.35	−0.202696
$877$	−29529.1	−1.13698	−0.568488	−	0.822692i	$-0.692471\pi$
$877$	−29529.1	−1.13698	−0.568488	+	0.822692i	$0.692471\pi$
$878$	18747.1	0.720596
$879$	14395.6	0.552392
$880$	−12312.2	−0.471641
$881$	43648.0	1.66917	0.834585	−	0.550879i	$-0.185708\pi$
$881$	43648.0	1.66917	0.834585	+	0.550879i	$0.185708\pi$
$882$	0	0
$883$	15451.0	0.588865	0.294433	−	0.955672i	$-0.404869\pi$
$883$	15451.0	0.588865	0.294433	+	0.955672i	$0.404869\pi$
$884$	23889.9	0.908943
$885$	−24706.9	−0.938433
$886$	11159.5	0.423148
$887$	−45587.9	−1.72570	−0.862849	−	0.505462i	$-0.831322\pi$
$887$	−45587.9	−1.72570	−0.862849	+	0.505462i	$0.831322\pi$
$888$	9369.94	0.354093
$889$	0	0
$890$	13255.9	0.499259
$891$	−10414.2	−0.391571
$892$	−6005.85	−0.225438
$893$	−31250.8	−1.17107
$894$	−19676.3	−0.736101
$895$	−9385.03	−0.350511
$896$	0	0
$897$	−4172.17	−0.155301
$898$	28885.9	1.07342
$899$	−40541.6	−1.50405
$900$	−5753.34	−0.213087
$901$	−1422.17	−0.0525852
$902$	−25013.3	−0.923338
$903$	0	0
$904$	450.965	0.0165917
$905$	28790.0	1.05747
$906$	−1795.07	−0.0658248
$907$	29939.6	1.09606	0.548031	−	0.836458i	$-0.315378\pi$
$907$	29939.6	1.09606	0.548031	+	0.836458i	$0.315378\pi$
$908$	−14693.1	−0.537013
$909$	11449.8	0.417783
$910$	0	0
$911$	9534.22	0.346743	0.173371	−	0.984857i	$-0.444534\pi$
$911$	9534.22	0.346743	0.173371	+	0.984857i	$0.444534\pi$
$912$	4552.20	0.165283
$913$	27004.8	0.978893
$914$	16891.6	0.611295
$915$	−32200.5	−1.16340
$916$	17402.8	0.627735
$917$	0	0
$918$	−36832.1	−1.32423
$919$	1921.78	0.0689812	0.0344906	−	0.999405i	$-0.489019\pi$
$919$	1921.78	0.0689812	0.0344906	+	0.999405i	$0.489019\pi$
$920$	2823.22	0.101173
$921$	−13149.4	−0.470455
$922$	−10530.2	−0.376133
$923$	−44669.8	−1.59298
$924$	0	0
$925$	34597.6	1.22980
$926$	3939.37	0.139801
$927$	−9198.56	−0.325912
$928$	4945.48	0.174939
$929$	−39700.7	−1.40209	−0.701043	−	0.713119i	$-0.747282\pi$
$929$	−39700.7	−1.40209	−0.701043	+	0.713119i	$0.747282\pi$
$930$	−30093.2	−1.06107
$931$	0	0
$932$	−13464.9	−0.473237
$933$	−37186.2	−1.30485
$934$	−20689.6	−0.724821
$935$	−94712.3	−3.31275
$936$	5056.44	0.176576
$937$	52946.6	1.84599	0.922993	−	0.384816i	$-0.125735\pi$
$937$	52946.6	1.84599	0.922993	+	0.384816i	$0.125735\pi$
$938$	0	0
$939$	−12540.7	−0.435836
$940$	−25200.9	−0.874428
$941$	−7982.65	−0.276543	−0.138271	−	0.990394i	$-0.544155\pi$
$941$	−7982.65	−0.276543	−0.138271	+	0.990394i	$0.544155\pi$
$942$	−14083.8	−0.487127
$943$	5735.61	0.198067
$944$	−6891.94	−0.237620
$945$	0	0
$946$	−19305.8	−0.663517
$947$	−4772.39	−0.163761	−0.0818806	−	0.996642i	$-0.526093\pi$
$947$	−4772.39	−0.163761	−0.0818806	+	0.996642i	$0.526093\pi$
$948$	8769.65	0.300448
$949$	17054.4	0.583360
$950$	16808.6	0.574044
$951$	−34198.2	−1.16609
$952$	0	0
$953$	37000.3	1.25767	0.628834	−	0.777540i	$-0.283533\pi$
$953$	37000.3	1.25767	0.628834	+	0.777540i	$0.283533\pi$
$954$	−301.010	−0.0102155
$955$	−70285.8	−2.38156
$956$	−16102.4	−0.544760
$957$	28974.5	0.978698
$958$	−23547.4	−0.794137
$959$	0	0
$960$	3670.93	0.123415
$961$	39024.3	1.30994
$962$	−30406.8	−1.01908
$963$	−19905.2	−0.666082
$964$	−8166.20	−0.272838
$965$	−19424.1	−0.647961
$966$	0	0
$967$	23119.4	0.768841	0.384420	−	0.923158i	$-0.374401\pi$
$967$	23119.4	0.768841	0.384420	+	0.923158i	$0.374401\pi$
$968$	−9473.82	−0.314566
$969$	35018.1	1.16093
$970$	29247.6	0.968128
$971$	36166.0	1.19529	0.597643	−	0.801762i	$-0.296104\pi$
$971$	36166.0	1.19529	0.597643	+	0.801762i	$0.296104\pi$
$972$	−13054.5	−0.430785
$973$	0	0
$974$	−1051.58	−0.0345941
$975$	−20031.0	−0.657954
$976$	−8982.27	−0.294585
$977$	37011.8	1.21199	0.605994	−	0.795469i	$-0.292775\pi$
$977$	37011.8	1.21199	0.605994	+	0.795469i	$0.292775\pi$
$978$	−25208.0	−0.824196
$979$	21664.2	0.707243
$980$	0	0
$981$	7727.18	0.251488
$982$	10828.9	0.351899
$983$	−48524.7	−1.57446	−0.787232	−	0.616657i	$-0.788487\pi$
$983$	−48524.7	−1.57446	−0.787232	+	0.616657i	$0.788487\pi$
$984$	7457.82	0.241612
$985$	−48264.8	−1.56126
$986$	38043.4	1.22875
$987$	0	0
$988$	−14772.6	−0.475686
$989$	4426.88	0.142332
$990$	−20046.4	−0.643552
$991$	7123.34	0.228335	0.114168	−	0.993461i	$-0.463580\pi$
$991$	7123.34	0.228335	0.114168	+	0.993461i	$0.463580\pi$
$992$	−8394.45	−0.268674
$993$	−26438.3	−0.844909
$994$	0	0
$995$	−49859.9	−1.58861
$996$	−8051.60	−0.256149
$997$	−101.084	−0.00321101	−0.00160550	−	0.999999i	$-0.500511\pi$
$997$	−101.084	−0.00321101	−0.00160550	+	0.999999i	$0.500511\pi$
$998$	5282.01	0.167534
$999$	46879.5	1.48469

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.u.1.4		11
7.3	odd	6		322.4.e.d.93.4	✓	22
7.5	odd	6		322.4.e.d.277.4	yes	22
7.6	odd	2		2254.4.a.r.1.8		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.d.93.4	✓	22	7.3	odd	6
322.4.e.d.277.4	yes	22	7.5	odd	6
2254.4.a.r.1.8		11	7.6	odd	2
2254.4.a.u.1.4		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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.73826 q^{3} +4.00000 q^{4} -15.3436 q^{5} +7.47652 q^{6} -8.00000 q^{8} -13.0254 q^{9} +30.6872 q^{10} +50.1520 q^{11} -14.9530 q^{12} +48.5248 q^{13} +57.3583 q^{15} +16.0000 q^{16} +123.081 q^{17} +26.0508 q^{18} -76.1082 q^{19} -61.3743 q^{20} -100.304 q^{22} +23.0000 q^{23} +29.9061 q^{24} +110.425 q^{25} -97.0497 q^{26} +149.625 q^{27} -154.546 q^{29} -114.717 q^{30} +262.327 q^{31} -32.0000 q^{32} -187.481 q^{33} -246.162 q^{34} -52.1016 q^{36} +313.312 q^{37} +152.216 q^{38} -181.399 q^{39} +122.749 q^{40} +249.375 q^{41} +192.473 q^{43} +200.608 q^{44} +199.856 q^{45} -46.0000 q^{46} +410.610 q^{47} -59.8122 q^{48} -220.851 q^{50} -460.109 q^{51} +194.099 q^{52} -11.5547 q^{53} -299.251 q^{54} -769.512 q^{55} +284.512 q^{57} +309.092 q^{58} -430.746 q^{59} +229.433 q^{60} -561.392 q^{61} -524.653 q^{62} +64.0000 q^{64} -744.545 q^{65} +374.963 q^{66} -659.887 q^{67} +492.324 q^{68} -85.9800 q^{69} -920.555 q^{71} +104.203 q^{72} +351.457 q^{73} -626.624 q^{74} -412.799 q^{75} -304.433 q^{76} +362.797 q^{78} -586.480 q^{79} -245.497 q^{80} -207.653 q^{81} -498.749 q^{82} +538.459 q^{83} -1888.50 q^{85} -384.946 q^{86} +577.734 q^{87} -401.216 q^{88} +431.970 q^{89} -399.713 q^{90} +92.0000 q^{92} -980.646 q^{93} -821.220 q^{94} +1167.77 q^{95} +119.624 q^{96} +953.090 q^{97} -653.251 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 22 q^{2} + 44 q^{4} + 23 q^{5} - 88 q^{8} + 171 q^{9} - 46 q^{10} - 48 q^{11} + 77 q^{13} + 104 q^{15} + 176 q^{16} + 97 q^{17} - 342 q^{18} + 138 q^{19} + 92 q^{20} + 96 q^{22} + 253 q^{23} + 30 q^{25}+ \cdots - 2545 q^{99}+O(q^{100}) \) 11 * q - 22 * q^2 + 44 * q^4 + 23 * q^5 - 88 * q^8 + 171 * q^9 - 46 * q^10 - 48 * q^11 + 77 * q^13 + 104 * q^15 + 176 * q^16 + 97 * q^17 - 342 * q^18 + 138 * q^19 + 92 * q^20 + 96 * q^22 + 253 * q^23 + 30 * q^25 - 154 * q^26 + 315 * q^27 + 137 * q^29 - 208 * q^30 + 234 * q^31 - 352 * q^32 + 28 * q^33 - 194 * q^34 + 684 * q^36 + 87 * q^37 - 276 * q^38 + 129 * q^39 - 184 * q^40 + 612 * q^41 + 187 * q^43 - 192 * q^44 + 597 * q^45 - 506 * q^46 + 793 * q^47 - 60 * q^50 - 1257 * q^51 + 308 * q^52 + 575 * q^53 - 630 * q^54 - 549 * q^55 + 39 * q^57 - 274 * q^58 + 1241 * q^59 + 416 * q^60 + 187 * q^61 - 468 * q^62 + 704 * q^64 - 937 * q^65 - 56 * q^66 + 2275 * q^67 + 388 * q^68 + 115 * q^71 - 1368 * q^72 + 2385 * q^73 - 174 * q^74 + 196 * q^75 + 552 * q^76 - 258 * q^78 - 1225 * q^79 + 368 * q^80 + 3635 * q^81 - 1224 * q^82 - 597 * q^83 - 2493 * q^85 - 374 * q^86 + 1428 * q^87 + 384 * q^88 + 602 * q^89 - 1194 * q^90 + 1012 * q^92 - 1360 * q^93 - 1586 * q^94 + 4835 * q^95 - 1039 * q^97 - 2545 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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