LMFDB - Embedded newform 2254.4.a.u.1.7

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			1.7
Root			$2.27413$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +2.27413 q^{3} +4.00000 q^{4} +14.8127 q^{5} -4.54825 q^{6} -8.00000 q^{8} -21.8284 q^{9} -29.6254 q^{10} -59.6112 q^{11} +9.09650 q^{12} -46.9140 q^{13} +33.6859 q^{15} +16.0000 q^{16} -54.1041 q^{17} +43.6567 q^{18} +120.503 q^{19} +59.2507 q^{20} +119.222 q^{22} +23.0000 q^{23} -18.1930 q^{24} +94.4156 q^{25} +93.8279 q^{26} -111.042 q^{27} -126.575 q^{29} -67.3718 q^{30} -107.903 q^{31} -32.0000 q^{32} -135.563 q^{33} +108.208 q^{34} -87.3134 q^{36} -70.4483 q^{37} -241.006 q^{38} -106.688 q^{39} -118.501 q^{40} -168.095 q^{41} +307.768 q^{43} -238.445 q^{44} -323.337 q^{45} -46.0000 q^{46} +436.953 q^{47} +36.3860 q^{48} -188.831 q^{50} -123.039 q^{51} -187.656 q^{52} +407.469 q^{53} +222.084 q^{54} -883.001 q^{55} +274.039 q^{57} +253.151 q^{58} -61.6093 q^{59} +134.744 q^{60} -877.506 q^{61} +215.806 q^{62} +64.0000 q^{64} -694.922 q^{65} +271.126 q^{66} +280.980 q^{67} -216.416 q^{68} +52.3049 q^{69} +1040.12 q^{71} +174.627 q^{72} +950.870 q^{73} +140.897 q^{74} +214.713 q^{75} +482.012 q^{76} +213.376 q^{78} +736.448 q^{79} +237.003 q^{80} +336.843 q^{81} +336.190 q^{82} +1194.91 q^{83} -801.426 q^{85} -615.536 q^{86} -287.848 q^{87} +476.889 q^{88} +633.067 q^{89} +646.673 q^{90} +92.0000 q^{92} -245.385 q^{93} -873.907 q^{94} +1784.97 q^{95} -72.7720 q^{96} +721.448 q^{97} +1301.21 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.27413	0.437656	0.218828	−	0.975763i	$-0.429777\pi$
$3$	2.27413	0.437656	0.218828	+	0.975763i	$0.429777\pi$
$4$	4.00000	0.500000
$5$	14.8127	1.32489	0.662443	−	0.749112i	$-0.269520\pi$
$5$	14.8127	1.32489	0.662443	+	0.749112i	$0.269520\pi$
$6$	−4.54825	−0.309469
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	−21.8284	−0.808458
$10$	−29.6254	−0.936836
$11$	−59.6112	−1.63395	−0.816974	−	0.576674i	$-0.804350\pi$
$11$	−59.6112	−1.63395	−0.816974	+	0.576674i	$0.804350\pi$
$12$	9.09650	0.218828
$13$	−46.9140	−1.00089	−0.500446	−	0.865768i	$-0.666831\pi$
$13$	−46.9140	−1.00089	−0.500446	+	0.865768i	$0.666831\pi$
$14$	0	0
$15$	33.6859	0.579844
$16$	16.0000	0.250000
$17$	−54.1041	−0.771892	−0.385946	−	0.922521i	$-0.626125\pi$
$17$	−54.1041	−0.771892	−0.385946	+	0.922521i	$0.626125\pi$
$18$	43.6567	0.571666
$19$	120.503	1.45502	0.727508	−	0.686099i	$-0.240678\pi$
$19$	120.503	1.45502	0.727508	+	0.686099i	$0.240678\pi$
$20$	59.2507	0.662443
$21$	0	0
$22$	119.222	1.15538
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−18.1930	−0.154735
$25$	94.4156	0.755325
$26$	93.8279	0.707737
$27$	−111.042	−0.791482
$28$	0	0
$29$	−126.575	−0.810499	−0.405249	−	0.914206i	$-0.632815\pi$
$29$	−126.575	−0.810499	−0.405249	+	0.914206i	$0.632815\pi$
$30$	−67.3718	−0.410012
$31$	−107.903	−0.625160	−0.312580	−	0.949891i	$-0.601193\pi$
$31$	−107.903	−0.625160	−0.312580	+	0.949891i	$0.601193\pi$
$32$	−32.0000	−0.176777
$33$	−135.563	−0.715107
$34$	108.208	0.545810
$35$	0	0
$36$	−87.3134	−0.404229
$37$	−70.4483	−0.313017	−0.156508	−	0.987677i	$-0.550024\pi$
$37$	−70.4483	−0.313017	−0.156508	+	0.987677i	$0.550024\pi$
$38$	−241.006	−1.02885
$39$	−106.688	−0.438046
$40$	−118.501	−0.468418
$41$	−168.095	−0.640294	−0.320147	−	0.947368i	$-0.603732\pi$
$41$	−168.095	−0.640294	−0.320147	+	0.947368i	$0.603732\pi$
$42$	0	0
$43$	307.768	1.09149	0.545746	−	0.837951i	$-0.316246\pi$
$43$	307.768	1.09149	0.545746	+	0.837951i	$0.316246\pi$
$44$	−238.445	−0.816974
$45$	−323.337	−1.07111
$46$	−46.0000	−0.147442
$47$	436.953	1.35609	0.678045	−	0.735021i	$-0.262828\pi$
$47$	436.953	1.35609	0.678045	+	0.735021i	$0.262828\pi$
$48$	36.3860	0.109414
$49$	0	0
$50$	−188.831	−0.534095
$51$	−123.039	−0.337823
$52$	−187.656	−0.500446
$53$	407.469	1.05604	0.528020	−	0.849232i	$-0.322934\pi$
$53$	407.469	1.05604	0.528020	+	0.849232i	$0.322934\pi$
$54$	222.084	0.559662
$55$	−883.001	−2.16480
$56$	0	0
$57$	274.039	0.636796
$58$	253.151	0.573109
$59$	−61.6093	−0.135947	−0.0679733	−	0.997687i	$-0.521653\pi$
$59$	−61.6093	−0.135947	−0.0679733	+	0.997687i	$0.521653\pi$
$60$	134.744	0.289922
$61$	−877.506	−1.84185	−0.920927	−	0.389735i	$-0.872566\pi$
$61$	−877.506	−1.84185	−0.920927	+	0.389735i	$0.872566\pi$
$62$	215.806	0.442055
$63$	0	0
$64$	64.0000	0.125000
$65$	−694.922	−1.32607
$66$	271.126	0.505657
$67$	280.980	0.512346	0.256173	−	0.966631i	$-0.417538\pi$
$67$	280.980	0.512346	0.256173	+	0.966631i	$0.417538\pi$
$68$	−216.416	−0.385946
$69$	52.3049	0.0912575
$70$	0	0
$71$	1040.12	1.73858	0.869288	−	0.494305i	$-0.164578\pi$
$71$	1040.12	1.73858	0.869288	+	0.494305i	$0.164578\pi$
$72$	174.627	0.285833
$73$	950.870	1.52453	0.762267	−	0.647263i	$-0.224086\pi$
$73$	950.870	1.52453	0.762267	+	0.647263i	$0.224086\pi$
$74$	140.897	0.221336
$75$	214.713	0.330572
$76$	482.012	0.727508
$77$	0	0
$78$	213.376	0.309745
$79$	736.448	1.04882	0.524410	−	0.851466i	$-0.324286\pi$
$79$	736.448	1.04882	0.524410	+	0.851466i	$0.324286\pi$
$80$	237.003	0.331222
$81$	336.843	0.462061
$82$	336.190	0.452756
$83$	1194.91	1.58022	0.790108	−	0.612968i	$-0.210024\pi$
$83$	1194.91	1.58022	0.790108	+	0.612968i	$0.210024\pi$
$84$	0	0
$85$	−801.426	−1.02267
$86$	−615.536	−0.771802
$87$	−287.848	−0.354719
$88$	476.889	0.577688
$89$	633.067	0.753988	0.376994	−	0.926216i	$-0.376958\pi$
$89$	633.067	0.753988	0.376994	+	0.926216i	$0.376958\pi$
$90$	646.673	0.757393
$91$	0	0
$92$	92.0000	0.104257
$93$	−245.385	−0.273605
$94$	−873.907	−0.958900
$95$	1784.97	1.92773
$96$	−72.7720	−0.0773673
$97$	721.448	0.755175	0.377587	−	0.925974i	$-0.376754\pi$
$97$	721.448	0.755175	0.377587	+	0.925974i	$0.376754\pi$
$98$	0	0
$99$	1301.21	1.32098
$100$	377.662	0.377662
$101$	1230.70	1.21247	0.606234	−	0.795287i	$-0.292680\pi$
$101$	1230.70	1.21247	0.606234	+	0.795287i	$0.292680\pi$
$102$	246.079	0.238877
$103$	−614.543	−0.587890	−0.293945	−	0.955822i	$-0.594968\pi$
$103$	−614.543	−0.587890	−0.293945	+	0.955822i	$0.594968\pi$
$104$	375.312	0.353869
$105$	0	0
$106$	−814.938	−0.746734
$107$	1588.79	1.43546	0.717732	−	0.696320i	$-0.245180\pi$
$107$	1588.79	1.43546	0.717732	+	0.696320i	$0.245180\pi$
$108$	−444.167	−0.395741
$109$	850.708	0.747551	0.373775	−	0.927519i	$-0.378063\pi$
$109$	850.708	0.747551	0.373775	+	0.927519i	$0.378063\pi$
$110$	1766.00	1.53074
$111$	−160.208	−0.136994
$112$	0	0
$113$	994.396	0.827831	0.413915	−	0.910315i	$-0.364161\pi$
$113$	994.396	0.827831	0.413915	+	0.910315i	$0.364161\pi$
$114$	−548.078	−0.450283
$115$	340.692	0.276258
$116$	−506.302	−0.405249
$117$	1024.05	0.809179
$118$	123.219	0.0961288
$119$	0	0
$120$	−269.487	−0.205006
$121$	2222.49	1.66979
$122$	1755.01	1.30239
$123$	−382.269	−0.280228
$124$	−431.612	−0.312580
$125$	−453.037	−0.324167
$126$	0	0
$127$	−2081.05	−1.45404	−0.727020	−	0.686617i	$-0.759095\pi$
$127$	−2081.05	−1.45404	−0.727020	+	0.686617i	$0.759095\pi$
$128$	−128.000	−0.0883883
$129$	699.903	0.477698
$130$	1389.84	0.937672
$131$	−1051.07	−0.701008	−0.350504	−	0.936561i	$-0.613990\pi$
$131$	−1051.07	−0.701008	−0.350504	+	0.936561i	$0.613990\pi$
$132$	−542.253	−0.357553
$133$	0	0
$134$	−561.960	−0.362283
$135$	−1644.83	−1.04862
$136$	432.832	0.272905
$137$	−1686.67	−1.05184	−0.525920	−	0.850534i	$-0.676279\pi$
$137$	−1686.67	−1.05184	−0.525920	+	0.850534i	$0.676279\pi$
$138$	−104.610	−0.0645288
$139$	−2497.61	−1.52406	−0.762032	−	0.647539i	$-0.775798\pi$
$139$	−2497.61	−1.52406	−0.762032	+	0.647539i	$0.775798\pi$
$140$	0	0
$141$	993.686	0.593500
$142$	−2080.23	−1.22936
$143$	2796.60	1.63541
$144$	−349.254	−0.202114
$145$	−1874.92	−1.07382
$146$	−1901.74	−1.07801
$147$	0	0
$148$	−281.793	−0.156508
$149$	−973.894	−0.535467	−0.267733	−	0.963493i	$-0.586275\pi$
$149$	−973.894	−0.535467	−0.267733	+	0.963493i	$0.586275\pi$
$150$	−429.426	−0.233750
$151$	−1152.88	−0.621327	−0.310664	−	0.950520i	$-0.600551\pi$
$151$	−1152.88	−0.621327	−0.310664	+	0.950520i	$0.600551\pi$
$152$	−964.025	−0.514426
$153$	1181.00	0.624042
$154$	0	0
$155$	−1598.33	−0.828266
$156$	−426.753	−0.219023
$157$	328.038	0.166753	0.0833767	−	0.996518i	$-0.473430\pi$
$157$	328.038	0.166753	0.0833767	+	0.996518i	$0.473430\pi$
$158$	−1472.90	−0.741628
$159$	926.635	0.462182
$160$	−474.006	−0.234209
$161$	0	0
$162$	−673.685	−0.326727
$163$	3364.13	1.61656	0.808279	−	0.588800i	$-0.200400\pi$
$163$	3364.13	1.61656	0.808279	+	0.588800i	$0.200400\pi$
$164$	−672.380	−0.320147
$165$	−2008.05	−0.947436
$166$	−2389.81	−1.11738
$167$	1197.69	0.554972	0.277486	−	0.960730i	$-0.410499\pi$
$167$	1197.69	0.554972	0.277486	+	0.960730i	$0.410499\pi$
$168$	0	0
$169$	3.91989	0.00178420
$170$	1602.85	0.723136
$171$	−2630.38	−1.17632
$172$	1231.07	0.545746
$173$	−1292.18	−0.567875	−0.283938	−	0.958843i	$-0.591641\pi$
$173$	−1292.18	−0.567875	−0.283938	+	0.958843i	$0.591641\pi$
$174$	575.697	0.250824
$175$	0	0
$176$	−953.778	−0.408487
$177$	−140.107	−0.0594978
$178$	−1266.13	−0.533150
$179$	2625.32	1.09623	0.548116	−	0.836402i	$-0.315345\pi$
$179$	2625.32	1.09623	0.548116	+	0.836402i	$0.315345\pi$
$180$	−1293.35	−0.535557
$181$	490.418	0.201395	0.100697	−	0.994917i	$-0.467893\pi$
$181$	490.418	0.201395	0.100697	+	0.994917i	$0.467893\pi$
$182$	0	0
$183$	−1995.56	−0.806098
$184$	−184.000	−0.0737210
$185$	−1043.53	−0.414712
$186$	490.770	0.193468
$187$	3225.20	1.26123
$188$	1747.81	0.678045
$189$	0	0
$190$	−3569.95	−1.36311
$191$	3128.91	1.18534	0.592671	−	0.805445i	$-0.298073\pi$
$191$	3128.91	1.18534	0.592671	+	0.805445i	$0.298073\pi$
$192$	145.544	0.0547069
$193$	5084.73	1.89641	0.948205	−	0.317659i	$-0.102897\pi$
$193$	5084.73	1.89641	0.948205	+	0.317659i	$0.102897\pi$
$194$	−1442.90	−0.533989
$195$	−1580.34	−0.580361
$196$	0	0
$197$	−4191.04	−1.51573	−0.757866	−	0.652410i	$-0.773758\pi$
$197$	−4191.04	−1.51573	−0.757866	+	0.652410i	$0.773758\pi$
$198$	−2602.43	−0.934073
$199$	4661.20	1.66042	0.830210	−	0.557450i	$-0.188220\pi$
$199$	4661.20	1.66042	0.830210	+	0.557450i	$0.188220\pi$
$200$	−755.325	−0.267048
$201$	638.983	0.224231
$202$	−2461.40	−0.857344
$203$	0	0
$204$	−492.158	−0.168911
$205$	−2489.94	−0.848317
$206$	1229.09	0.415701
$207$	−502.052	−0.168575
$208$	−750.623	−0.250223
$209$	−7183.33	−2.37742
$210$	0	0
$211$	−1063.04	−0.346836	−0.173418	−	0.984848i	$-0.555481\pi$
$211$	−1063.04	−0.346836	−0.173418	+	0.984848i	$0.555481\pi$
$212$	1629.88	0.528020
$213$	2365.35	0.760898
$214$	−3177.59	−1.01503
$215$	4558.87	1.44610
$216$	888.334	0.279831
$217$	0	0
$218$	−1701.42	−0.528598
$219$	2162.40	0.667221
$220$	−3532.00	−1.08240
$221$	2538.24	0.772580
$222$	320.416	0.0968691
$223$	−2777.01	−0.833912	−0.416956	−	0.908927i	$-0.636903\pi$
$223$	−2777.01	−0.833912	−0.416956	+	0.908927i	$0.636903\pi$
$224$	0	0
$225$	−2060.94	−0.610648
$226$	−1988.79	−0.585365
$227$	588.055	0.171941	0.0859704	−	0.996298i	$-0.472601\pi$
$227$	588.055	0.171941	0.0859704	+	0.996298i	$0.472601\pi$
$228$	1096.16	0.318398
$229$	−782.498	−0.225803	−0.112902	−	0.993606i	$-0.536014\pi$
$229$	−782.498	−0.225803	−0.112902	+	0.993606i	$0.536014\pi$
$230$	−681.383	−0.195344
$231$	0	0
$232$	1012.60	0.286555
$233$	285.260	0.0802059	0.0401030	−	0.999196i	$-0.487231\pi$
$233$	285.260	0.0802059	0.0401030	+	0.999196i	$0.487231\pi$
$234$	−2048.11	−0.572176
$235$	6472.45	1.79666
$236$	−246.437	−0.0679733
$237$	1674.77	0.459022
$238$	0	0
$239$	−1867.46	−0.505422	−0.252711	−	0.967542i	$-0.581322\pi$
$239$	−1867.46	−0.505422	−0.252711	+	0.967542i	$0.581322\pi$
$240$	538.974	0.144961
$241$	1257.32	0.336062	0.168031	−	0.985782i	$-0.446259\pi$
$241$	1257.32	0.336062	0.168031	+	0.985782i	$0.446259\pi$
$242$	−4444.98	−1.18072
$243$	3764.15	0.993705
$244$	−3510.02	−0.920927
$245$	0	0
$246$	764.538	0.198151
$247$	−5653.28	−1.45631
$248$	863.224	0.221027
$249$	2717.36	0.691590
$250$	906.074	0.229221
$251$	1582.01	0.397832	0.198916	−	0.980017i	$-0.436258\pi$
$251$	1582.01	0.397832	0.198916	+	0.980017i	$0.436258\pi$
$252$	0	0
$253$	−1371.06	−0.340702
$254$	4162.09	1.02816
$255$	−1822.54	−0.447577
$256$	256.000	0.0625000
$257$	5590.33	1.35687	0.678435	−	0.734661i	$-0.262659\pi$
$257$	5590.33	1.35687	0.678435	+	0.734661i	$0.262659\pi$
$258$	−1399.81	−0.337783
$259$	0	0
$260$	−2779.69	−0.663034
$261$	2762.93	0.655254
$262$	2102.13	0.495687
$263$	−2291.99	−0.537376	−0.268688	−	0.963227i	$-0.586590\pi$
$263$	−2291.99	−0.537376	−0.268688	+	0.963227i	$0.586590\pi$
$264$	1084.51	0.252828
$265$	6035.71	1.39913
$266$	0	0
$267$	1439.67	0.329987
$268$	1123.92	0.256173
$269$	2171.50	0.492188	0.246094	−	0.969246i	$-0.420853\pi$
$269$	2171.50	0.492188	0.246094	+	0.969246i	$0.420853\pi$
$270$	3289.65	0.741489
$271$	−7493.52	−1.67970	−0.839850	−	0.542818i	$-0.817357\pi$
$271$	−7493.52	−1.67970	−0.839850	+	0.542818i	$0.817357\pi$
$272$	−865.665	−0.192973
$273$	0	0
$274$	3373.35	0.743764
$275$	−5628.22	−1.23416
$276$	209.220	0.0456287
$277$	2702.11	0.586116	0.293058	−	0.956095i	$-0.405327\pi$
$277$	2702.11	0.586116	0.293058	+	0.956095i	$0.405327\pi$
$278$	4995.23	1.07768
$279$	2355.35	0.505415
$280$	0	0
$281$	−208.991	−0.0443679	−0.0221840	−	0.999754i	$-0.507062\pi$
$281$	−208.991	−0.0443679	−0.0221840	+	0.999754i	$0.507062\pi$
$282$	−1987.37	−0.419668
$283$	−6585.03	−1.38318	−0.691589	−	0.722292i	$-0.743089\pi$
$283$	−6585.03	−1.38318	−0.691589	+	0.722292i	$0.743089\pi$
$284$	4160.46	0.869288
$285$	4059.25	0.843682
$286$	−5593.19	−1.15641
$287$	0	0
$288$	698.507	0.142916
$289$	−1985.75	−0.404183
$290$	3749.84	0.759305
$291$	1640.66	0.330506
$292$	3803.48	0.762267
$293$	−848.877	−0.169256	−0.0846278	−	0.996413i	$-0.526970\pi$
$293$	−848.877	−0.169256	−0.0846278	+	0.996413i	$0.526970\pi$
$294$	0	0
$295$	−912.600	−0.180114
$296$	563.586	0.110668
$297$	6619.33	1.29324
$298$	1947.79	0.378632
$299$	−1079.02	−0.208700
$300$	858.852	0.165286
$301$	0	0
$302$	2305.77	0.439345
$303$	2798.76	0.530643
$304$	1928.05	0.363754
$305$	−12998.2	−2.44025
$306$	−2362.00	−0.441264
$307$	−3261.27	−0.606288	−0.303144	−	0.952945i	$-0.598036\pi$
$307$	−3261.27	−0.606288	−0.303144	+	0.952945i	$0.598036\pi$
$308$	0	0
$309$	−1397.55	−0.257293
$310$	3196.67	0.585673
$311$	−2018.43	−0.368022	−0.184011	−	0.982924i	$-0.558908\pi$
$311$	−2018.43	−0.368022	−0.184011	+	0.982924i	$0.558908\pi$
$312$	853.506	0.154873
$313$	5593.70	1.01014	0.505072	−	0.863078i	$-0.331466\pi$
$313$	5593.70	1.01014	0.505072	+	0.863078i	$0.331466\pi$
$314$	−656.076	−0.117913
$315$	0	0
$316$	2945.79	0.524410
$317$	−9353.21	−1.65719	−0.828594	−	0.559850i	$-0.810859\pi$
$317$	−9353.21	−1.65719	−0.828594	+	0.559850i	$0.810859\pi$
$318$	−1853.27	−0.326812
$319$	7545.31	1.32431
$320$	948.012	0.165611
$321$	3613.12	0.628239
$322$	0	0
$323$	−6519.70	−1.12311
$324$	1347.37	0.231031
$325$	−4429.41	−0.755998
$326$	−6728.25	−1.14308
$327$	1934.62	0.327170
$328$	1344.76	0.226378
$329$	0	0
$330$	4016.11	0.669938
$331$	−4495.66	−0.746538	−0.373269	−	0.927723i	$-0.621763\pi$
$331$	−4495.66	−0.746538	−0.373269	+	0.927723i	$0.621763\pi$
$332$	4779.62	0.790108
$333$	1537.77	0.253061
$334$	−2395.39	−0.392425
$335$	4162.07	0.678800
$336$	0	0
$337$	−5715.31	−0.923836	−0.461918	−	0.886923i	$-0.652838\pi$
$337$	−5715.31	−0.923836	−0.461918	+	0.886923i	$0.652838\pi$
$338$	−7.83979	−0.00126162
$339$	2261.38	0.362305
$340$	−3205.70	−0.511335
$341$	6432.22	1.02148
$342$	5260.77	0.831783
$343$	0	0
$344$	−2462.14	−0.385901
$345$	774.776	0.120906
$346$	2584.35	0.401548
$347$	−5558.95	−0.860000	−0.430000	−	0.902829i	$-0.641486\pi$
$347$	−5558.95	−0.860000	−0.430000	+	0.902829i	$0.641486\pi$
$348$	−1151.39	−0.177360
$349$	7062.30	1.08320	0.541599	−	0.840637i	$-0.317819\pi$
$349$	7062.30	1.08320	0.541599	+	0.840637i	$0.317819\pi$
$350$	0	0
$351$	5209.41	0.792187
$352$	1907.56	0.288844
$353$	1279.72	0.192954	0.0964771	−	0.995335i	$-0.469243\pi$
$353$	1279.72	0.192954	0.0964771	+	0.995335i	$0.469243\pi$
$354$	280.215	0.0420713
$355$	15406.9	2.30342
$356$	2532.27	0.376994
$357$	0	0
$358$	−5250.64	−0.775153
$359$	−9650.22	−1.41872	−0.709358	−	0.704848i	$-0.751015\pi$
$359$	−9650.22	−1.41872	−0.709358	+	0.704848i	$0.751015\pi$
$360$	2586.69	0.378696
$361$	7661.99	1.11707
$362$	−980.835	−0.142408
$363$	5054.22	0.730793
$364$	0	0
$365$	14084.9	2.01983
$366$	3991.11	0.569997
$367$	2122.80	0.301932	0.150966	−	0.988539i	$-0.451762\pi$
$367$	2122.80	0.301932	0.150966	+	0.988539i	$0.451762\pi$
$368$	368.000	0.0521286
$369$	3669.24	0.517650
$370$	2087.06	0.293246
$371$	0	0
$372$	−981.540	−0.136802
$373$	11578.5	1.60727	0.803636	−	0.595120i	$-0.202896\pi$
$373$	11578.5	1.60727	0.803636	+	0.595120i	$0.202896\pi$
$374$	−6450.41	−0.891826
$375$	−1030.26	−0.141873
$376$	−3495.63	−0.479450
$377$	5938.15	0.811221
$378$	0	0
$379$	−4208.09	−0.570331	−0.285165	−	0.958478i	$-0.592048\pi$
$379$	−4208.09	−0.570331	−0.285165	+	0.958478i	$0.592048\pi$
$380$	7139.90	0.963866
$381$	−4732.56	−0.636368
$382$	−6257.83	−0.838163
$383$	9135.91	1.21886	0.609430	−	0.792840i	$-0.291398\pi$
$383$	9135.91	1.21886	0.609430	+	0.792840i	$0.291398\pi$
$384$	−291.088	−0.0386837
$385$	0	0
$386$	−10169.5	−1.34096
$387$	−6718.07	−0.882425
$388$	2885.79	0.377587
$389$	−1722.19	−0.224469	−0.112235	−	0.993682i	$-0.535801\pi$
$389$	−1722.19	−0.224469	−0.112235	+	0.993682i	$0.535801\pi$
$390$	3160.68	0.410377
$391$	−1244.39	−0.160951
$392$	0	0
$393$	−2390.25	−0.306800
$394$	8382.08	1.07178
$395$	10908.8	1.38957
$396$	5204.85	0.660489
$397$	−9337.52	−1.18045	−0.590223	−	0.807241i	$-0.700960\pi$
$397$	−9337.52	−1.18045	−0.590223	+	0.807241i	$0.700960\pi$
$398$	−9322.40	−1.17409
$399$	0	0
$400$	1510.65	0.188831
$401$	1661.24	0.206879	0.103439	−	0.994636i	$-0.467015\pi$
$401$	1661.24	0.206879	0.103439	+	0.994636i	$0.467015\pi$
$402$	−1277.97	−0.158555
$403$	5062.16	0.625717
$404$	4922.80	0.606234
$405$	4989.54	0.612179
$406$	0	0
$407$	4199.50	0.511454
$408$	984.315	0.119438
$409$	−8058.84	−0.974288	−0.487144	−	0.873322i	$-0.661961\pi$
$409$	−8058.84	−0.974288	−0.487144	+	0.873322i	$0.661961\pi$
$410$	4979.88	0.599850
$411$	−3835.70	−0.460344
$412$	−2458.17	−0.293945
$413$	0	0
$414$	1004.10	0.119201
$415$	17699.8	2.09361
$416$	1501.25	0.176934
$417$	−5679.89	−0.667015
$418$	14366.7	1.68109
$419$	−8223.99	−0.958874	−0.479437	−	0.877576i	$-0.659159\pi$
$419$	−8223.99	−0.958874	−0.479437	+	0.877576i	$0.659159\pi$
$420$	0	0
$421$	3038.01	0.351695	0.175847	−	0.984417i	$-0.443733\pi$
$421$	3038.01	0.351695	0.175847	+	0.984417i	$0.443733\pi$
$422$	2126.07	0.245250
$423$	−9537.97	−1.09634
$424$	−3259.75	−0.373367
$425$	−5108.27	−0.583029
$426$	−4730.70	−0.538036
$427$	0	0
$428$	6355.18	0.717732
$429$	6359.81	0.715745
$430$	−9117.74	−1.02255
$431$	−3703.90	−0.413945	−0.206973	−	0.978347i	$-0.566361\pi$
$431$	−3703.90	−0.413945	−0.206973	+	0.978347i	$0.566361\pi$
$432$	−1776.67	−0.197870
$433$	308.018	0.0341856	0.0170928	−	0.999854i	$-0.494559\pi$
$433$	308.018	0.0341856	0.0170928	+	0.999854i	$0.494559\pi$
$434$	0	0
$435$	−4263.81	−0.469963
$436$	3402.83	0.373775
$437$	2771.57	0.303392
$438$	−4324.80	−0.471796
$439$	9772.63	1.06247	0.531233	−	0.847226i	$-0.321729\pi$
$439$	9772.63	1.06247	0.531233	+	0.847226i	$0.321729\pi$
$440$	7064.01	0.765371
$441$	0	0
$442$	−5076.47	−0.546297
$443$	−5835.74	−0.625879	−0.312939	−	0.949773i	$-0.601314\pi$
$443$	−5835.74	−0.625879	−0.312939	+	0.949773i	$0.601314\pi$
$444$	−640.833	−0.0684968
$445$	9377.41	0.998948
$446$	5554.02	0.589665
$447$	−2214.76	−0.234350
$448$	0	0
$449$	−826.304	−0.0868501	−0.0434250	−	0.999057i	$-0.513827\pi$
$449$	−826.304	−0.0868501	−0.0434250	+	0.999057i	$0.513827\pi$
$450$	4121.87	0.431793
$451$	10020.3	1.04621
$452$	3977.58	0.413915
$453$	−2621.80	−0.271927
$454$	−1176.11	−0.121581
$455$	0	0
$456$	−2192.31	−0.225141
$457$	18929.0	1.93755	0.968774	−	0.247946i	$-0.0797557\pi$
$457$	18929.0	1.93755	0.968774	+	0.247946i	$0.0797557\pi$
$458$	1565.00	0.159667
$459$	6007.81	0.610938
$460$	1362.77	0.138129
$461$	15366.0	1.55242	0.776208	−	0.630477i	$-0.217141\pi$
$461$	15366.0	1.55242	0.776208	+	0.630477i	$0.217141\pi$
$462$	0	0
$463$	−16891.5	−1.69550	−0.847750	−	0.530397i	$-0.822043\pi$
$463$	−16891.5	−1.69550	−0.847750	+	0.530397i	$0.822043\pi$
$464$	−2025.21	−0.202625
$465$	−3634.81	−0.362495
$466$	−570.519	−0.0567142
$467$	−15025.6	−1.48887	−0.744435	−	0.667695i	$-0.767281\pi$
$467$	−15025.6	−1.48887	−0.744435	+	0.667695i	$0.767281\pi$
$468$	4096.22	0.404589
$469$	0	0
$470$	−12944.9	−1.27043
$471$	746.000	0.0729806
$472$	492.875	0.0480644
$473$	−18346.4	−1.78344
$474$	−3349.55	−0.324578
$475$	11377.4	1.09901
$476$	0	0
$477$	−8894.38	−0.853764
$478$	3734.92	0.357387
$479$	8951.28	0.853850	0.426925	−	0.904287i	$-0.359597\pi$
$479$	8951.28	0.853850	0.426925	+	0.904287i	$0.359597\pi$
$480$	−1077.95	−0.102503
$481$	3305.01	0.313296
$482$	−2514.63	−0.237631
$483$	0	0
$484$	8889.96	0.834895
$485$	10686.6	1.00052
$486$	−7528.30	−0.702656
$487$	−13182.2	−1.22658	−0.613290	−	0.789858i	$-0.710154\pi$
$487$	−13182.2	−1.22658	−0.613290	+	0.789858i	$0.710154\pi$
$488$	7020.04	0.651194
$489$	7650.45	0.707495
$490$	0	0
$491$	13672.8	1.25671	0.628355	−	0.777927i	$-0.283729\pi$
$491$	13672.8	1.25671	0.628355	+	0.777927i	$0.283729\pi$
$492$	−1529.08	−0.140114
$493$	6848.24	0.625617
$494$	11306.6	1.02977
$495$	19274.5	1.75015
$496$	−1726.45	−0.156290
$497$	0	0
$498$	−5434.73	−0.489028
$499$	−15613.9	−1.40075	−0.700375	−	0.713775i	$-0.746984\pi$
$499$	−15613.9	−1.40075	−0.700375	+	0.713775i	$0.746984\pi$
$500$	−1812.15	−0.162083
$501$	2723.71	0.242887
$502$	−3164.02	−0.281309
$503$	12622.6	1.11891	0.559457	−	0.828859i	$-0.311010\pi$
$503$	12622.6	1.11891	0.559457	+	0.828859i	$0.311010\pi$
$504$	0	0
$505$	18230.0	1.60638
$506$	2742.11	0.240913
$507$	8.91433	0.000780866 0
$508$	−8324.19	−0.727020
$509$	−13626.9	−1.18664	−0.593322	−	0.804965i	$-0.702184\pi$
$509$	−13626.9	−1.18664	−0.593322	+	0.804965i	$0.702184\pi$
$510$	3645.09	0.316485
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−13380.9	−1.15162
$514$	−11180.7	−0.959452
$515$	−9103.03	−0.778888
$516$	2799.61	0.238849
$517$	−26047.3	−2.21578
$518$	0	0
$519$	−2938.57	−0.248534
$520$	5559.37	0.468836
$521$	5163.25	0.434177	0.217088	−	0.976152i	$-0.430344\pi$
$521$	5163.25	0.434177	0.217088	+	0.976152i	$0.430344\pi$
$522$	−5525.87	−0.463334
$523$	−7324.10	−0.612353	−0.306176	−	0.951975i	$-0.599050\pi$
$523$	−7324.10	−0.612353	−0.306176	+	0.951975i	$0.599050\pi$
$524$	−4204.26	−0.350504
$525$	0	0
$526$	4583.97	0.379982
$527$	5837.99	0.482556
$528$	−2169.01	−0.178777
$529$	529.000	0.0434783
$530$	−12071.4	−0.989338
$531$	1344.83	0.109907
$532$	0	0
$533$	7886.01	0.640865
$534$	−2879.34	−0.233336
$535$	23534.3	1.90183
$536$	−2247.84	−0.181142
$537$	5970.30	0.479772
$538$	−4343.00	−0.348030
$539$	0	0
$540$	−6579.31	−0.524312
$541$	−7362.33	−0.585086	−0.292543	−	0.956252i	$-0.594501\pi$
$541$	−7362.33	−0.585086	−0.292543	+	0.956252i	$0.594501\pi$
$542$	14987.0	1.18773
$543$	1115.27	0.0881415
$544$	1731.33	0.136452
$545$	12601.3	0.990420
$546$	0	0
$547$	1506.22	0.117735	0.0588677	−	0.998266i	$-0.481251\pi$
$547$	1506.22	0.117735	0.0588677	+	0.998266i	$0.481251\pi$
$548$	−6746.69	−0.525920
$549$	19154.5	1.48906
$550$	11256.4	0.872685
$551$	−15252.7	−1.17929
$552$	−418.439	−0.0322644
$553$	0	0
$554$	−5404.22	−0.414447
$555$	−2373.11	−0.181501
$556$	−9990.46	−0.762032
$557$	5643.78	0.429326	0.214663	−	0.976688i	$-0.431135\pi$
$557$	5643.78	0.429326	0.214663	+	0.976688i	$0.431135\pi$
$558$	−4710.69	−0.357383
$559$	−14438.6	−1.09247
$560$	0	0
$561$	7334.52	0.551985
$562$	417.983	0.0313729
$563$	18788.6	1.40648	0.703238	−	0.710954i	$-0.251737\pi$
$563$	18788.6	1.40648	0.703238	+	0.710954i	$0.251737\pi$
$564$	3974.75	0.296750
$565$	14729.7	1.09678
$566$	13170.1	0.978054
$567$	0	0
$568$	−8320.92	−0.614680
$569$	13699.1	1.00931	0.504654	−	0.863322i	$-0.331620\pi$
$569$	13699.1	1.00931	0.504654	+	0.863322i	$0.331620\pi$
$570$	−8118.51	−0.596573
$571$	739.836	0.0542227	0.0271113	−	0.999632i	$-0.491369\pi$
$571$	739.836	0.0542227	0.0271113	+	0.999632i	$0.491369\pi$
$572$	11186.4	0.817703
$573$	7115.54	0.518771
$574$	0	0
$575$	2171.56	0.157496
$576$	−1397.01	−0.101057
$577$	6479.15	0.467470	0.233735	−	0.972300i	$-0.424905\pi$
$577$	6479.15	0.467470	0.233735	+	0.972300i	$0.424905\pi$
$578$	3971.50	0.285801
$579$	11563.3	0.829974
$580$	−7499.69	−0.536909
$581$	0	0
$582$	−3281.33	−0.233703
$583$	−24289.7	−1.72552
$584$	−7606.96	−0.539004
$585$	15169.0	1.07207
$586$	1697.75	0.119682
$587$	−2529.43	−0.177855	−0.0889274	−	0.996038i	$-0.528344\pi$
$587$	−2529.43	−0.177855	−0.0889274	+	0.996038i	$0.528344\pi$
$588$	0	0
$589$	−13002.6	−0.909618
$590$	1825.20	0.127360
$591$	−9530.95	−0.663369
$592$	−1127.17	−0.0782542
$593$	20434.4	1.41507	0.707537	−	0.706677i	$-0.249806\pi$
$593$	20434.4	1.41507	0.707537	+	0.706677i	$0.249806\pi$
$594$	−13238.7	−0.914459
$595$	0	0
$596$	−3895.58	−0.267733
$597$	10600.2	0.726692
$598$	2158.04	0.147573
$599$	3042.51	0.207535	0.103768	−	0.994602i	$-0.466910\pi$
$599$	3042.51	0.207535	0.103768	+	0.994602i	$0.466910\pi$
$600$	−1717.70	−0.116875
$601$	13729.5	0.931845	0.465923	−	0.884825i	$-0.345723\pi$
$601$	13729.5	0.931845	0.465923	+	0.884825i	$0.345723\pi$
$602$	0	0
$603$	−6133.33	−0.414210
$604$	−4611.54	−0.310664
$605$	32921.0	2.21228
$606$	−5597.53	−0.375221
$607$	4927.30	0.329478	0.164739	−	0.986337i	$-0.447322\pi$
$607$	4927.30	0.329478	0.164739	+	0.986337i	$0.447322\pi$
$608$	−3856.10	−0.257213
$609$	0	0
$610$	25996.4	1.72552
$611$	−20499.2	−1.35730
$612$	4724.01	0.312021
$613$	22664.9	1.49336	0.746678	−	0.665186i	$-0.231648\pi$
$613$	22664.9	1.49336	0.746678	+	0.665186i	$0.231648\pi$
$614$	6522.53	0.428710
$615$	−5662.43	−0.371270
$616$	0	0
$617$	−1895.18	−0.123658	−0.0618292	−	0.998087i	$-0.519693\pi$
$617$	−1895.18	−0.123658	−0.0618292	+	0.998087i	$0.519693\pi$
$618$	2795.09	0.181934
$619$	18999.5	1.23369	0.616845	−	0.787085i	$-0.288411\pi$
$619$	18999.5	1.23369	0.616845	+	0.787085i	$0.288411\pi$
$620$	−6393.33	−0.414133
$621$	−2553.96	−0.165035
$622$	4036.87	0.260231
$623$	0	0
$624$	−1707.01	−0.109511
$625$	−18512.6	−1.18481
$626$	−11187.4	−0.714279
$627$	−16335.8	−1.04049
$628$	1312.15	0.0833767
$629$	3811.54	0.241615
$630$	0	0
$631$	−16757.5	−1.05722	−0.528611	−	0.848864i	$-0.677287\pi$
$631$	−16757.5	−1.05722	−0.528611	+	0.848864i	$0.677287\pi$
$632$	−5891.58	−0.370814
$633$	−2417.48	−0.151795
$634$	18706.4	1.17181
$635$	−30825.9	−1.92644
$636$	3706.54	0.231091
$637$	0	0
$638$	−15090.6	−0.936431
$639$	−22704.0	−1.40557
$640$	−1896.02	−0.117105
$641$	−7529.21	−0.463941	−0.231970	−	0.972723i	$-0.574517\pi$
$641$	−7529.21	−0.463941	−0.231970	+	0.972723i	$0.574517\pi$
$642$	−7226.24	−0.444232
$643$	13087.0	0.802645	0.401322	−	0.915937i	$-0.368551\pi$
$643$	13087.0	0.802645	0.401322	+	0.915937i	$0.368551\pi$
$644$	0	0
$645$	10367.4	0.632895
$646$	13039.4	0.794162
$647$	7695.41	0.467601	0.233800	−	0.972285i	$-0.424884\pi$
$647$	7695.41	0.467601	0.233800	+	0.972285i	$0.424884\pi$
$648$	−2694.74	−0.163363
$649$	3672.60	0.222130
$650$	8858.82	0.534572
$651$	0	0
$652$	13456.5	0.808279
$653$	14205.1	0.851282	0.425641	−	0.904892i	$-0.360049\pi$
$653$	14205.1	0.851282	0.425641	+	0.904892i	$0.360049\pi$
$654$	−3869.23	−0.231344
$655$	−15569.1	−0.928756
$656$	−2689.52	−0.160073
$657$	−20755.9	−1.23252
$658$	0	0
$659$	21377.7	1.26367	0.631834	−	0.775103i	$-0.282302\pi$
$659$	21377.7	1.26367	0.631834	+	0.775103i	$0.282302\pi$
$660$	−8032.22	−0.473718
$661$	6655.83	0.391652	0.195826	−	0.980639i	$-0.437261\pi$
$661$	6655.83	0.391652	0.195826	+	0.980639i	$0.437261\pi$
$662$	8991.33	0.527882
$663$	5772.26	0.338124
$664$	−9559.24	−0.558691
$665$	0	0
$666$	−3075.54	−0.178941
$667$	−2911.23	−0.169001
$668$	4790.78	0.277486
$669$	−6315.27	−0.364966
$670$	−8324.13	−0.479984
$671$	52309.1	3.00949
$672$	0	0
$673$	−25917.1	−1.48444	−0.742222	−	0.670154i	$-0.766228\pi$
$673$	−25917.1	−1.48444	−0.742222	+	0.670154i	$0.766228\pi$
$674$	11430.6	0.653250
$675$	−10484.1	−0.597826
$676$	15.6796	0.000892101 0
$677$	−12548.3	−0.712361	−0.356181	−	0.934417i	$-0.615921\pi$
$677$	−12548.3	−0.712361	−0.356181	+	0.934417i	$0.615921\pi$
$678$	−4522.76	−0.256188
$679$	0	0
$680$	6411.41	0.361568
$681$	1337.31	0.0752509
$682$	−12864.4	−0.722295
$683$	28292.0	1.58501	0.792505	−	0.609866i	$-0.208777\pi$
$683$	28292.0	1.58501	0.792505	+	0.609866i	$0.208777\pi$
$684$	−10521.5	−0.588159
$685$	−24984.2	−1.39357
$686$	0	0
$687$	−1779.50	−0.0988240
$688$	4924.29	0.272873
$689$	−19116.0	−1.05698
$690$	−1549.55	−0.0854933
$691$	−9848.08	−0.542169	−0.271084	−	0.962556i	$-0.587382\pi$
$691$	−9848.08	−0.542169	−0.271084	+	0.962556i	$0.587382\pi$
$692$	−5168.71	−0.283938
$693$	0	0
$694$	11117.9	0.608112
$695$	−36996.4	−2.01921
$696$	2302.79	0.125412
$697$	9094.62	0.494237
$698$	−14124.6	−0.765937
$699$	648.716	0.0351026
$700$	0	0
$701$	11704.8	0.630649	0.315324	−	0.948984i	$-0.397887\pi$
$701$	11704.8	0.630649	0.315324	+	0.948984i	$0.397887\pi$
$702$	−10418.8	−0.560161
$703$	−8489.23	−0.455445
$704$	−3815.11	−0.204244
$705$	14719.2	0.786320
$706$	−2559.45	−0.136439
$707$	0	0
$708$	−560.429	−0.0297489
$709$	14512.5	0.768728	0.384364	−	0.923182i	$-0.374421\pi$
$709$	14512.5	0.768728	0.384364	+	0.923182i	$0.374421\pi$
$710$	−30813.8	−1.62876
$711$	−16075.4	−0.847927
$712$	−5064.53	−0.266575
$713$	−2481.77	−0.130355
$714$	0	0
$715$	41425.1	2.16673
$716$	10501.3	0.548116
$717$	−4246.83	−0.221201
$718$	19300.4	1.00318
$719$	26255.7	1.36185	0.680927	−	0.732351i	$-0.261577\pi$
$719$	26255.7	1.36185	0.680927	+	0.732351i	$0.261577\pi$
$720$	−5173.38	−0.267779
$721$	0	0
$722$	−15324.0	−0.789889
$723$	2859.29	0.147079
$724$	1961.67	0.100697
$725$	−11950.7	−0.612190
$726$	−10108.4	−0.516748
$727$	9048.41	0.461605	0.230802	−	0.973001i	$-0.425865\pi$
$727$	9048.41	0.461605	0.230802	+	0.973001i	$0.425865\pi$
$728$	0	0
$729$	−534.604	−0.0271607
$730$	−28169.9	−1.42824
$731$	−16651.5	−0.842514
$732$	−7982.23	−0.403049
$733$	3121.90	0.157313	0.0786563	−	0.996902i	$-0.474937\pi$
$733$	3121.90	0.157313	0.0786563	+	0.996902i	$0.474937\pi$
$734$	−4245.59	−0.213498
$735$	0	0
$736$	−736.000	−0.0368605
$737$	−16749.5	−0.837146
$738$	−7338.48	−0.366034
$739$	19858.4	0.988500	0.494250	−	0.869320i	$-0.335443\pi$
$739$	19858.4	0.988500	0.494250	+	0.869320i	$0.335443\pi$
$740$	−4174.11	−0.207356
$741$	−12856.3	−0.637364
$742$	0	0
$743$	−5437.39	−0.268477	−0.134239	−	0.990949i	$-0.542859\pi$
$743$	−5437.39	−0.268477	−0.134239	+	0.990949i	$0.542859\pi$
$744$	1963.08	0.0967339
$745$	−14426.0	−0.709433
$746$	−23157.0	−1.13651
$747$	−26082.8	−1.27754
$748$	12900.8	0.630616
$749$	0	0
$750$	2060.53	0.100320
$751$	34447.1	1.67376	0.836880	−	0.547386i	$-0.184377\pi$
$751$	34447.1	1.67376	0.836880	+	0.547386i	$0.184377\pi$
$752$	6991.25	0.339022
$753$	3597.69	0.174113
$754$	−11876.3	−0.573620
$755$	−17077.3	−0.823188
$756$	0	0
$757$	27562.1	1.32333	0.661665	−	0.749800i	$-0.269850\pi$
$757$	27562.1	1.32333	0.661665	+	0.749800i	$0.269850\pi$
$758$	8416.19	0.403285
$759$	−3117.95	−0.149110
$760$	−14279.8	−0.681556
$761$	25522.4	1.21575	0.607876	−	0.794032i	$-0.292022\pi$
$761$	25522.4	1.21575	0.607876	+	0.794032i	$0.292022\pi$
$762$	9465.12	0.449980
$763$	0	0
$764$	12515.7	0.592671
$765$	17493.8	0.826785
$766$	−18271.8	−0.861864
$767$	2890.34	0.136068
$768$	582.176	0.0273535
$769$	21433.7	1.00509	0.502547	−	0.864550i	$-0.332396\pi$
$769$	21433.7	1.00509	0.502547	+	0.864550i	$0.332396\pi$
$770$	0	0
$771$	12713.1	0.593841
$772$	20338.9	0.948205
$773$	16285.3	0.757751	0.378876	−	0.925448i	$-0.376311\pi$
$773$	16285.3	0.757751	0.378876	+	0.925448i	$0.376311\pi$
$774$	13436.1	0.623969
$775$	−10187.7	−0.472199
$776$	−5771.58	−0.266995
$777$	0	0
$778$	3444.38	0.158724
$779$	−20256.0	−0.931637
$780$	−6321.36	−0.290181
$781$	−62002.5	−2.84075
$782$	2488.79	0.113809
$783$	14055.2	0.641495
$784$	0	0
$785$	4859.13	0.220929
$786$	4780.51	0.216940
$787$	−18939.3	−0.857832	−0.428916	−	0.903344i	$-0.641104\pi$
$787$	−18939.3	−0.857832	−0.428916	+	0.903344i	$0.641104\pi$
$788$	−16764.2	−0.757866
$789$	−5212.26	−0.235186
$790$	−21817.5	−0.982574
$791$	0	0
$792$	−10409.7	−0.467036
$793$	41167.3	1.84350
$794$	18675.0	0.834701
$795$	13726.0	0.612339
$796$	18644.8	0.830210
$797$	−6446.96	−0.286528	−0.143264	−	0.989684i	$-0.545760\pi$
$797$	−6446.96	−0.286528	−0.143264	+	0.989684i	$0.545760\pi$
$798$	0	0
$799$	−23640.9	−1.04675
$800$	−3021.30	−0.133524
$801$	−13818.8	−0.609567
$802$	−3322.48	−0.146286
$803$	−56682.5	−2.49101
$804$	2555.93	0.112115
$805$	0	0
$806$	−10124.3	−0.442449
$807$	4938.26	0.215409
$808$	−9845.60	−0.428672
$809$	9782.98	0.425156	0.212578	−	0.977144i	$-0.431814\pi$
$809$	9782.98	0.425156	0.212578	+	0.977144i	$0.431814\pi$
$810$	−9979.09	−0.432876
$811$	−40372.4	−1.74805	−0.874025	−	0.485881i	$-0.838499\pi$
$811$	−40372.4	−1.74805	−0.874025	+	0.485881i	$0.838499\pi$
$812$	0	0
$813$	−17041.2	−0.735130
$814$	−8399.01	−0.361652
$815$	49831.8	2.14175
$816$	−1968.63	−0.0844557
$817$	37087.0	1.58814
$818$	16117.7	0.688926
$819$	0	0
$820$	−9959.76	−0.424158
$821$	−42623.7	−1.81191	−0.905955	−	0.423374i	$-0.860845\pi$
$821$	−42623.7	−1.81191	−0.905955	+	0.423374i	$0.860845\pi$
$822$	7671.41	0.325512
$823$	41262.0	1.74763	0.873817	−	0.486255i	$-0.161637\pi$
$823$	41262.0	1.74763	0.873817	+	0.486255i	$0.161637\pi$
$824$	4916.34	0.207851
$825$	−12799.3	−0.540138
$826$	0	0
$827$	10967.3	0.461151	0.230575	−	0.973054i	$-0.425939\pi$
$827$	10967.3	0.461151	0.230575	+	0.973054i	$0.425939\pi$
$828$	−2008.21	−0.0842875
$829$	−15418.6	−0.645970	−0.322985	−	0.946404i	$-0.604686\pi$
$829$	−15418.6	−0.645970	−0.322985	+	0.946404i	$0.604686\pi$
$830$	−35399.5	−1.48040
$831$	6144.94	0.256517
$832$	−3002.49	−0.125111
$833$	0	0
$834$	11359.8	0.471651
$835$	17741.1	0.735276
$836$	−28733.3	−1.18871
$837$	11981.7	0.494803
$838$	16448.0	0.678026
$839$	−35087.3	−1.44380	−0.721899	−	0.691998i	$-0.756731\pi$
$839$	−35087.3	−1.44380	−0.721899	+	0.691998i	$0.756731\pi$
$840$	0	0
$841$	−8367.67	−0.343092
$842$	−6076.02	−0.248686
$843$	−475.273	−0.0194179
$844$	−4252.14	−0.173418
$845$	58.0641	0.00236387
$846$	19075.9	0.775230
$847$	0	0
$848$	6519.50	0.264010
$849$	−14975.2	−0.605355
$850$	10216.5	0.412264
$851$	−1620.31	−0.0652685
$852$	9461.41	0.380449
$853$	20895.1	0.838726	0.419363	−	0.907819i	$-0.362253\pi$
$853$	20895.1	0.838726	0.419363	+	0.907819i	$0.362253\pi$
$854$	0	0
$855$	−38963.0	−1.55849
$856$	−12710.4	−0.507513
$857$	−9860.33	−0.393025	−0.196513	−	0.980501i	$-0.562962\pi$
$857$	−9860.33	−0.393025	−0.196513	+	0.980501i	$0.562962\pi$
$858$	−12719.6	−0.506108
$859$	−27508.7	−1.09265	−0.546324	−	0.837574i	$-0.683973\pi$
$859$	−27508.7	−1.09265	−0.546324	+	0.837574i	$0.683973\pi$
$860$	18235.5	0.723052
$861$	0	0
$862$	7407.79	0.292703
$863$	44397.4	1.75122	0.875611	−	0.483017i	$-0.160459\pi$
$863$	44397.4	1.75122	0.875611	+	0.483017i	$0.160459\pi$
$864$	3553.34	0.139915
$865$	−19140.6	−0.752370
$866$	−616.035	−0.0241729
$867$	−4515.85	−0.176893
$868$	0	0
$869$	−43900.5	−1.71372
$870$	8527.61	0.332314
$871$	−13181.9	−0.512802
$872$	−6805.66	−0.264299
$873$	−15748.0	−0.610527
$874$	−5543.14	−0.214530
$875$	0	0
$876$	8649.59	0.333610
$877$	−21240.3	−0.817827	−0.408913	−	0.912573i	$-0.634092\pi$
$877$	−21240.3	−0.817827	−0.408913	+	0.912573i	$0.634092\pi$
$878$	−19545.3	−0.751277
$879$	−1930.45	−0.0740757
$880$	−14128.0	−0.541199
$881$	−24746.5	−0.946345	−0.473173	−	0.880970i	$-0.656891\pi$
$881$	−24746.5	−0.946345	−0.473173	+	0.880970i	$0.656891\pi$
$882$	0	0
$883$	−6682.34	−0.254676	−0.127338	−	0.991859i	$-0.540643\pi$
$883$	−6682.34	−0.254676	−0.127338	+	0.991859i	$0.540643\pi$
$884$	10152.9	0.386290
$885$	−2075.37	−0.0788278
$886$	11671.5	0.442563
$887$	−9897.96	−0.374680	−0.187340	−	0.982295i	$-0.559987\pi$
$887$	−9897.96	−0.374680	−0.187340	+	0.982295i	$0.559987\pi$
$888$	1281.67	0.0484346
$889$	0	0
$890$	−18754.8	−0.706363
$891$	−20079.6	−0.754985
$892$	−11108.0	−0.416956
$893$	52654.2	1.97313
$894$	4429.51	0.165710
$895$	38888.0	1.45238
$896$	0	0
$897$	−2453.83	−0.0913389
$898$	1652.61	0.0614123
$899$	13657.9	0.506691
$900$	−8243.75	−0.305324
$901$	−22045.7	−0.815149
$902$	−20040.7	−0.739780
$903$	0	0
$904$	−7955.17	−0.292682
$905$	7264.40	0.266825
$906$	5243.61	0.192282
$907$	13540.9	0.495722	0.247861	−	0.968796i	$-0.420272\pi$
$907$	13540.9	0.495722	0.247861	+	0.968796i	$0.420272\pi$
$908$	2352.22	0.0859704
$909$	−26864.1	−0.980228
$910$	0	0
$911$	43832.5	1.59411	0.797056	−	0.603906i	$-0.206390\pi$
$911$	43832.5	1.59411	0.797056	+	0.603906i	$0.206390\pi$
$912$	4384.62	0.159199
$913$	−71229.7	−2.58199
$914$	−37857.9	−1.37005
$915$	−29559.6	−1.06799
$916$	−3129.99	−0.112902
$917$	0	0
$918$	−12015.6	−0.431998
$919$	−445.356	−0.0159858	−0.00799290	−	0.999968i	$-0.502544\pi$
$919$	−445.356	−0.0159858	−0.00799290	+	0.999968i	$0.502544\pi$
$920$	−2725.53	−0.0976719
$921$	−7416.53	−0.265345
$922$	−30731.9	−1.09772
$923$	−48795.9	−1.74013
$924$	0	0
$925$	−6651.42	−0.236429
$926$	33783.1	1.19890
$927$	13414.5	0.475284
$928$	4050.41	0.143277
$929$	17694.9	0.624922	0.312461	−	0.949931i	$-0.398847\pi$
$929$	17694.9	0.624922	0.312461	+	0.949931i	$0.398847\pi$
$930$	7269.62	0.256323
$931$	0	0
$932$	1141.04	0.0401030
$933$	−4590.17	−0.161067
$934$	30051.2	1.05279
$935$	47773.9	1.67099
$936$	−8192.44	−0.286088
$937$	−17172.3	−0.598712	−0.299356	−	0.954141i	$-0.596772\pi$
$937$	−17172.3	−0.598712	−0.299356	+	0.954141i	$0.596772\pi$
$938$	0	0
$939$	12720.8	0.442095
$940$	25889.8	0.898332
$941$	−12155.3	−0.421098	−0.210549	−	0.977583i	$-0.567525\pi$
$941$	−12155.3	−0.421098	−0.210549	+	0.977583i	$0.567525\pi$
$942$	−1492.00	−0.0516051
$943$	−3866.19	−0.133510
$944$	−985.749	−0.0339867
$945$	0	0
$946$	36692.8	1.26108
$947$	25702.4	0.881960	0.440980	−	0.897517i	$-0.354631\pi$
$947$	25702.4	0.881960	0.440980	+	0.897517i	$0.354631\pi$
$948$	6699.10	0.229511
$949$	−44609.1	−1.52589
$950$	−22754.7	−0.777117
$951$	−21270.4	−0.725278
$952$	0	0
$953$	13029.3	0.442874	0.221437	−	0.975175i	$-0.428925\pi$
$953$	13029.3	0.442874	0.221437	+	0.975175i	$0.428925\pi$
$954$	17788.8	0.603703
$955$	46347.6	1.57044
$956$	−7469.83	−0.252711
$957$	17159.0	0.579593
$958$	−17902.6	−0.603763
$959$	0	0
$960$	2155.90	0.0724805
$961$	−18147.9	−0.609175
$962$	−6610.02	−0.221534
$963$	−34680.8	−1.16051
$964$	5029.26	0.168031
$965$	75318.5	2.51253
$966$	0	0
$967$	16413.3	0.545829	0.272915	−	0.962038i	$-0.412012\pi$
$967$	16413.3	0.545829	0.272915	+	0.962038i	$0.412012\pi$
$968$	−17779.9	−0.590360
$969$	−14826.6	−0.491537
$970$	−21373.2	−0.707475
$971$	30138.7	0.996084	0.498042	−	0.867153i	$-0.334053\pi$
$971$	30138.7	0.996084	0.498042	+	0.867153i	$0.334053\pi$
$972$	15056.6	0.496853
$973$	0	0
$974$	26364.5	0.867323
$975$	−10073.0	−0.330867
$976$	−14040.1	−0.460463
$977$	49194.1	1.61091	0.805454	−	0.592658i	$-0.201921\pi$
$977$	49194.1	1.61091	0.805454	+	0.592658i	$0.201921\pi$
$978$	−15300.9	−0.500275
$979$	−37737.8	−1.23198
$980$	0	0
$981$	−18569.6	−0.604363
$982$	−27345.6	−0.888628
$983$	−19382.8	−0.628905	−0.314453	−	0.949273i	$-0.601821\pi$
$983$	−19382.8	−0.628905	−0.314453	+	0.949273i	$0.601821\pi$
$984$	3058.15	0.0990756
$985$	−62080.6	−2.00817
$986$	−13696.5	−0.442378
$987$	0	0
$988$	−22613.1	−0.728157
$989$	7078.66	0.227592
$990$	−38548.9	−1.23754
$991$	−3361.80	−0.107761	−0.0538805	−	0.998547i	$-0.517159\pi$
$991$	−3361.80	−0.107761	−0.0538805	+	0.998547i	$0.517159\pi$
$992$	3452.90	0.110514
$993$	−10223.7	−0.326726
$994$	0	0
$995$	69044.9	2.19987
$996$	10869.5	0.345795
$997$	41154.1	1.30729	0.653643	−	0.756803i	$-0.273240\pi$
$997$	41154.1	1.30729	0.653643	+	0.756803i	$0.273240\pi$
$998$	31227.8	0.990479
$999$	7822.70	0.247747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.u.1.7		11
7.3	odd	6		322.4.e.d.93.7	✓	22
7.5	odd	6		322.4.e.d.277.7	yes	22
7.6	odd	2		2254.4.a.r.1.5		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.d.93.7	✓	22	7.3	odd	6
322.4.e.d.277.7	yes	22	7.5	odd	6
2254.4.a.r.1.5		11	7.6	odd	2
2254.4.a.u.1.7		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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +2.27413 q^{3} +4.00000 q^{4} +14.8127 q^{5} -4.54825 q^{6} -8.00000 q^{8} -21.8284 q^{9} -29.6254 q^{10} -59.6112 q^{11} +9.09650 q^{12} -46.9140 q^{13} +33.6859 q^{15} +16.0000 q^{16} -54.1041 q^{17} +43.6567 q^{18} +120.503 q^{19} +59.2507 q^{20} +119.222 q^{22} +23.0000 q^{23} -18.1930 q^{24} +94.4156 q^{25} +93.8279 q^{26} -111.042 q^{27} -126.575 q^{29} -67.3718 q^{30} -107.903 q^{31} -32.0000 q^{32} -135.563 q^{33} +108.208 q^{34} -87.3134 q^{36} -70.4483 q^{37} -241.006 q^{38} -106.688 q^{39} -118.501 q^{40} -168.095 q^{41} +307.768 q^{43} -238.445 q^{44} -323.337 q^{45} -46.0000 q^{46} +436.953 q^{47} +36.3860 q^{48} -188.831 q^{50} -123.039 q^{51} -187.656 q^{52} +407.469 q^{53} +222.084 q^{54} -883.001 q^{55} +274.039 q^{57} +253.151 q^{58} -61.6093 q^{59} +134.744 q^{60} -877.506 q^{61} +215.806 q^{62} +64.0000 q^{64} -694.922 q^{65} +271.126 q^{66} +280.980 q^{67} -216.416 q^{68} +52.3049 q^{69} +1040.12 q^{71} +174.627 q^{72} +950.870 q^{73} +140.897 q^{74} +214.713 q^{75} +482.012 q^{76} +213.376 q^{78} +736.448 q^{79} +237.003 q^{80} +336.843 q^{81} +336.190 q^{82} +1194.91 q^{83} -801.426 q^{85} -615.536 q^{86} -287.848 q^{87} +476.889 q^{88} +633.067 q^{89} +646.673 q^{90} +92.0000 q^{92} -245.385 q^{93} -873.907 q^{94} +1784.97 q^{95} -72.7720 q^{96} +721.448 q^{97} +1301.21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 22 q^{2} + 44 q^{4} + 23 q^{5} - 88 q^{8} + 171 q^{9} - 46 q^{10} - 48 q^{11} + 77 q^{13} + 104 q^{15} + 176 q^{16} + 97 q^{17} - 342 q^{18} + 138 q^{19} + 92 q^{20} + 96 q^{22} + 253 q^{23} + 30 q^{25}+ \cdots - 2545 q^{99}+O(q^{100}) \) 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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