LMFDB - Embedded newform 460.4.a.c.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [460,4,Mod(1,460)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("460.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(460, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 460 = 2^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	460.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$27.1408786026$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 118x^{4} + 155x^{3} + 3095x^{2} - 6472x + 2800 $ x^6 - x^5 - 118x^4 + 155x^3 + 3095x^2 - 6472x + 2800
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-6.89901$ of defining polynomial
Character	$\chi$	$=$	460.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.89901 q^{3} +5.00000 q^{5} +27.4822 q^{7} +20.5963 q^{9} +6.96617 q^{11} -30.3927 q^{13} +34.4950 q^{15} +82.0584 q^{17} +59.3015 q^{19} +189.600 q^{21} -23.0000 q^{23} +25.0000 q^{25} -44.1793 q^{27} -121.972 q^{29} -134.077 q^{31} +48.0596 q^{33} +137.411 q^{35} +205.934 q^{37} -209.679 q^{39} -7.47209 q^{41} -6.53173 q^{43} +102.981 q^{45} +153.360 q^{47} +412.269 q^{49} +566.121 q^{51} +532.263 q^{53} +34.8308 q^{55} +409.122 q^{57} -662.392 q^{59} -728.732 q^{61} +566.030 q^{63} -151.963 q^{65} -442.249 q^{67} -158.677 q^{69} -271.491 q^{71} +141.301 q^{73} +172.475 q^{75} +191.445 q^{77} +898.207 q^{79} -860.893 q^{81} +743.376 q^{83} +410.292 q^{85} -841.488 q^{87} +676.060 q^{89} -835.256 q^{91} -925.000 q^{93} +296.508 q^{95} +538.136 q^{97} +143.477 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 30 q^{5} + 24 q^{7} + 75 q^{9} + 117 q^{11} - 59 q^{13} - 5 q^{15} + 88 q^{17} + 105 q^{19} + 44 q^{21} - 138 q^{23} + 150 q^{25} + 164 q^{27} - 71 q^{29} + 396 q^{31} - 85 q^{33} + 120 q^{35}+ \cdots + 4077 q^{99}+O(q^{100}) $ 6 * q - q^3 + 30 * q^5 + 24 * q^7 + 75 * q^9 + 117 * q^11 - 59 * q^13 - 5 * q^15 + 88 * q^17 + 105 * q^19 + 44 * q^21 - 138 * q^23 + 150 * q^25 + 164 * q^27 - 71 * q^29 + 396 * q^31 - 85 * q^33 + 120 * q^35 + 57 * q^37 + 718 * q^39 + 692 * q^41 + 778 * q^43 + 375 * q^45 + 248 * q^47 + 958 * q^49 + 545 * q^51 - 49 * q^53 + 585 * q^55 + 396 * q^57 + 1539 * q^59 + 461 * q^61 + 921 * q^63 - 295 * q^65 + 2815 * q^67 + 23 * q^69 + 378 * q^71 + 518 * q^73 - 25 * q^75 - 286 * q^77 + 1694 * q^79 + 498 * q^81 + 1757 * q^83 + 440 * q^85 + 2918 * q^87 + 688 * q^89 + 2363 * q^91 + 36 * q^93 + 525 * q^95 + 1029 * q^97 + 4077 * 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 $

$2$	0	0
$2$	0	0
$3$	6.89901	1.32771	0.663857	−	0.747859i	$-0.268918\pi$
$3$	6.89901	1.32771	0.663857	+	0.747859i	$0.268918\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	27.4822	1.48390	0.741948	−	0.670457i	$-0.233902\pi$
$7$	27.4822	1.48390	0.741948	+	0.670457i	$0.233902\pi$
$8$	0	0
$9$	20.5963	0.762825
$10$	0	0
$11$	6.96617	0.190943	0.0954717	−	0.995432i	$-0.469564\pi$
$11$	6.96617	0.190943	0.0954717	+	0.995432i	$0.469564\pi$
$12$	0	0
$13$	−30.3927	−0.648416	−0.324208	−	0.945986i	$-0.605098\pi$
$13$	−30.3927	−0.648416	−0.324208	+	0.945986i	$0.605098\pi$
$14$	0	0
$15$	34.4950	0.593772
$16$	0	0
$17$	82.0584	1.17071	0.585355	−	0.810777i	$-0.300955\pi$
$17$	82.0584	1.17071	0.585355	+	0.810777i	$0.300955\pi$
$18$	0	0
$19$	59.3015	0.716037	0.358019	−	0.933714i	$-0.383452\pi$
$19$	59.3015	0.716037	0.358019	+	0.933714i	$0.383452\pi$
$20$	0	0
$21$	189.600	1.97019
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−44.1793	−0.314900
$28$	0	0
$29$	−121.972	−0.781024	−0.390512	−	0.920598i	$-0.627702\pi$
$29$	−121.972	−0.781024	−0.390512	+	0.920598i	$0.627702\pi$
$30$	0	0
$31$	−134.077	−0.776806	−0.388403	−	0.921490i	$-0.626973\pi$
$31$	−134.077	−0.776806	−0.388403	+	0.921490i	$0.626973\pi$
$32$	0	0
$33$	48.0596	0.253518
$34$	0	0
$35$	137.411	0.663619
$36$	0	0
$37$	205.934	0.915008	0.457504	−	0.889208i	$-0.348744\pi$
$37$	205.934	0.915008	0.457504	+	0.889208i	$0.348744\pi$
$38$	0	0
$39$	−209.679	−0.860911
$40$	0	0
$41$	−7.47209	−0.0284621	−0.0142310	−	0.999899i	$-0.504530\pi$
$41$	−7.47209	−0.0284621	−0.0142310	+	0.999899i	$0.504530\pi$
$42$	0	0
$43$	−6.53173	−0.0231647	−0.0115823	−	0.999933i	$-0.503687\pi$
$43$	−6.53173	−0.0231647	−0.0115823	+	0.999933i	$0.503687\pi$
$44$	0	0
$45$	102.981	0.341146
$46$	0	0
$47$	153.360	0.475956	0.237978	−	0.971271i	$-0.423515\pi$
$47$	153.360	0.475956	0.237978	+	0.971271i	$0.423515\pi$
$48$	0	0
$49$	412.269	1.20195
$50$	0	0
$51$	566.121	1.55437
$52$	0	0
$53$	532.263	1.37947	0.689736	−	0.724061i	$-0.257727\pi$
$53$	532.263	1.37947	0.689736	+	0.724061i	$0.257727\pi$
$54$	0	0
$55$	34.8308	0.0853925
$56$	0	0
$57$	409.122	0.950693
$58$	0	0
$59$	−662.392	−1.46163	−0.730815	−	0.682576i	$-0.760860\pi$
$59$	−662.392	−1.46163	−0.730815	+	0.682576i	$0.760860\pi$
$60$	0	0
$61$	−728.732	−1.52958	−0.764792	−	0.644278i	$-0.777158\pi$
$61$	−728.732	−1.52958	−0.764792	+	0.644278i	$0.777158\pi$
$62$	0	0
$63$	566.030	1.13195
$64$	0	0
$65$	−151.963	−0.289980
$66$	0	0
$67$	−442.249	−0.806407	−0.403203	−	0.915110i	$-0.632103\pi$
$67$	−442.249	−0.806407	−0.403203	+	0.915110i	$0.632103\pi$
$68$	0	0
$69$	−158.677	−0.276848
$70$	0	0
$71$	−271.491	−0.453803	−0.226902	−	0.973918i	$-0.572860\pi$
$71$	−271.491	−0.453803	−0.226902	+	0.973918i	$0.572860\pi$
$72$	0	0
$73$	141.301	0.226549	0.113274	−	0.993564i	$-0.463866\pi$
$73$	141.301	0.226549	0.113274	+	0.993564i	$0.463866\pi$
$74$	0	0
$75$	172.475	0.265543
$76$	0	0
$77$	191.445	0.283340
$78$	0	0
$79$	898.207	1.27919	0.639596	−	0.768711i	$-0.279102\pi$
$79$	898.207	1.27919	0.639596	+	0.768711i	$0.279102\pi$
$80$	0	0
$81$	−860.893	−1.18092
$82$	0	0
$83$	743.376	0.983086	0.491543	−	0.870853i	$-0.336433\pi$
$83$	743.376	0.983086	0.491543	+	0.870853i	$0.336433\pi$
$84$	0	0
$85$	410.292	0.523558
$86$	0	0
$87$	−841.488	−1.03698
$88$	0	0
$89$	676.060	0.805193	0.402597	−	0.915378i	$-0.368108\pi$
$89$	676.060	0.805193	0.402597	+	0.915378i	$0.368108\pi$
$90$	0	0
$91$	−835.256	−0.962182
$92$	0	0
$93$	−925.000	−1.03138
$94$	0	0
$95$	296.508	0.320221
$96$	0	0
$97$	538.136	0.563293	0.281646	−	0.959518i	$-0.409120\pi$
$97$	538.136	0.563293	0.281646	+	0.959518i	$0.409120\pi$
$98$	0	0
$99$	143.477	0.145657
$100$	0	0
$101$	163.980	0.161550	0.0807752	−	0.996732i	$-0.474260\pi$
$101$	163.980	0.161550	0.0807752	+	0.996732i	$0.474260\pi$
$102$	0	0
$103$	−163.679	−0.156580	−0.0782900	−	0.996931i	$-0.524946\pi$
$103$	−163.679	−0.156580	−0.0782900	+	0.996931i	$0.524946\pi$
$104$	0	0
$105$	947.998	0.881096
$106$	0	0
$107$	−318.343	−0.287620	−0.143810	−	0.989605i	$-0.545935\pi$
$107$	−318.343	−0.287620	−0.143810	+	0.989605i	$0.545935\pi$
$108$	0	0
$109$	131.706	0.115735	0.0578674	−	0.998324i	$-0.481570\pi$
$109$	131.706	0.115735	0.0578674	+	0.998324i	$0.481570\pi$
$110$	0	0
$111$	1420.74	1.21487
$112$	0	0
$113$	1587.72	1.32177	0.660887	−	0.750486i	$-0.270180\pi$
$113$	1587.72	1.32177	0.660887	+	0.750486i	$0.270180\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	−625.976	−0.494628
$118$	0	0
$119$	2255.14	1.73721
$120$	0	0
$121$	−1282.47	−0.963541
$122$	0	0
$123$	−51.5500	−0.0377895
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−222.079	−0.155168	−0.0775841	−	0.996986i	$-0.524721\pi$
$127$	−222.079	−0.155168	−0.0775841	+	0.996986i	$0.524721\pi$
$128$	0	0
$129$	−45.0625	−0.0307560
$130$	0	0
$131$	−2120.63	−1.41435	−0.707175	−	0.707038i	$-0.750031\pi$
$131$	−2120.63	−1.41435	−0.707175	+	0.707038i	$0.750031\pi$
$132$	0	0
$133$	1629.73	1.06253
$134$	0	0
$135$	−220.896	−0.140828
$136$	0	0
$137$	1407.06	0.877471	0.438735	−	0.898616i	$-0.355427\pi$
$137$	1407.06	0.877471	0.438735	+	0.898616i	$0.355427\pi$
$138$	0	0
$139$	−53.5769	−0.0326931	−0.0163465	−	0.999866i	$-0.505203\pi$
$139$	−53.5769	−0.0326931	−0.0163465	+	0.999866i	$0.505203\pi$
$140$	0	0
$141$	1058.03	0.631933
$142$	0	0
$143$	−211.720	−0.123811
$144$	0	0
$145$	−609.862	−0.349285
$146$	0	0
$147$	2844.24	1.59585
$148$	0	0
$149$	−1687.74	−0.927952	−0.463976	−	0.885848i	$-0.653578\pi$
$149$	−1687.74	−0.927952	−0.463976	+	0.885848i	$0.653578\pi$
$150$	0	0
$151$	−1882.53	−1.01456	−0.507278	−	0.861782i	$-0.669348\pi$
$151$	−1882.53	−1.01456	−0.507278	+	0.861782i	$0.669348\pi$
$152$	0	0
$153$	1690.10	0.893048
$154$	0	0
$155$	−670.386	−0.347398
$156$	0	0
$157$	−3743.42	−1.90291	−0.951457	−	0.307782i	$-0.900413\pi$
$157$	−3743.42	−1.90291	−0.951457	+	0.307782i	$0.900413\pi$
$158$	0	0
$159$	3672.09	1.83154
$160$	0	0
$161$	−632.090	−0.309414
$162$	0	0
$163$	1115.93	0.536236	0.268118	−	0.963386i	$-0.413598\pi$
$163$	1115.93	0.536236	0.268118	+	0.963386i	$0.413598\pi$
$164$	0	0
$165$	240.298	0.113377
$166$	0	0
$167$	−3248.82	−1.50540	−0.752699	−	0.658364i	$-0.771249\pi$
$167$	−3248.82	−1.50540	−0.752699	+	0.658364i	$0.771249\pi$
$168$	0	0
$169$	−1273.29	−0.579557
$170$	0	0
$171$	1221.39	0.546211
$172$	0	0
$173$	1125.61	0.494672	0.247336	−	0.968930i	$-0.420445\pi$
$173$	1125.61	0.494672	0.247336	+	0.968930i	$0.420445\pi$
$174$	0	0
$175$	687.054	0.296779
$176$	0	0
$177$	−4569.85	−1.94063
$178$	0	0
$179$	3060.31	1.27787	0.638934	−	0.769262i	$-0.279376\pi$
$179$	3060.31	1.27787	0.638934	+	0.769262i	$0.279376\pi$
$180$	0	0
$181$	−1527.33	−0.627212	−0.313606	−	0.949553i	$-0.601537\pi$
$181$	−1527.33	−0.627212	−0.313606	+	0.949553i	$0.601537\pi$
$182$	0	0
$183$	−5027.53	−2.03085
$184$	0	0
$185$	1029.67	0.409204
$186$	0	0
$187$	571.632	0.223540
$188$	0	0
$189$	−1214.14	−0.467279
$190$	0	0
$191$	−673.024	−0.254965	−0.127483	−	0.991841i	$-0.540690\pi$
$191$	−673.024	−0.254965	−0.127483	+	0.991841i	$0.540690\pi$
$192$	0	0
$193$	−3217.73	−1.20009	−0.600045	−	0.799966i	$-0.704851\pi$
$193$	−3217.73	−1.20009	−0.600045	+	0.799966i	$0.704851\pi$
$194$	0	0
$195$	−1048.40	−0.385011
$196$	0	0
$197$	−1204.46	−0.435606	−0.217803	−	0.975993i	$-0.569889\pi$
$197$	−1204.46	−0.435606	−0.217803	+	0.975993i	$0.569889\pi$
$198$	0	0
$199$	3839.18	1.36760	0.683799	−	0.729670i	$-0.260327\pi$
$199$	3839.18	1.36760	0.683799	+	0.729670i	$0.260327\pi$
$200$	0	0
$201$	−3051.08	−1.07068
$202$	0	0
$203$	−3352.06	−1.15896
$204$	0	0
$205$	−37.3605	−0.0127286
$206$	0	0
$207$	−473.715	−0.159060
$208$	0	0
$209$	413.104	0.136723
$210$	0	0
$211$	753.803	0.245943	0.122971	−	0.992410i	$-0.460758\pi$
$211$	753.803	0.245943	0.122971	+	0.992410i	$0.460758\pi$
$212$	0	0
$213$	−1873.02	−0.602521
$214$	0	0
$215$	−32.6587	−0.0103595
$216$	0	0
$217$	−3684.73	−1.15270
$218$	0	0
$219$	974.838	0.300792
$220$	0	0
$221$	−2493.97	−0.759107
$222$	0	0
$223$	240.057	0.0720869	0.0360434	−	0.999350i	$-0.488525\pi$
$223$	240.057	0.0720869	0.0360434	+	0.999350i	$0.488525\pi$
$224$	0	0
$225$	514.907	0.152565
$226$	0	0
$227$	−4341.80	−1.26950	−0.634748	−	0.772719i	$-0.718896\pi$
$227$	−4341.80	−1.26950	−0.634748	+	0.772719i	$0.718896\pi$
$228$	0	0
$229$	−4424.87	−1.27687	−0.638436	−	0.769675i	$-0.720418\pi$
$229$	−4424.87	−1.27687	−0.638436	+	0.769675i	$0.720418\pi$
$230$	0	0
$231$	1320.78	0.376195
$232$	0	0
$233$	−4962.74	−1.39536	−0.697682	−	0.716407i	$-0.745785\pi$
$233$	−4962.74	−1.39536	−0.697682	+	0.716407i	$0.745785\pi$
$234$	0	0
$235$	766.802	0.212854
$236$	0	0
$237$	6196.74	1.69840
$238$	0	0
$239$	−3778.60	−1.02267	−0.511334	−	0.859382i	$-0.670848\pi$
$239$	−3778.60	−1.02267	−0.511334	+	0.859382i	$0.670848\pi$
$240$	0	0
$241$	−4607.64	−1.23155	−0.615777	−	0.787921i	$-0.711158\pi$
$241$	−4607.64	−1.23155	−0.615777	+	0.787921i	$0.711158\pi$
$242$	0	0
$243$	−4746.46	−1.25303
$244$	0	0
$245$	2061.34	0.537528
$246$	0	0
$247$	−1802.33	−0.464290
$248$	0	0
$249$	5128.56	1.30526
$250$	0	0
$251$	270.435	0.0680068	0.0340034	−	0.999422i	$-0.489174\pi$
$251$	270.435	0.0680068	0.0340034	+	0.999422i	$0.489174\pi$
$252$	0	0
$253$	−160.222	−0.0398145
$254$	0	0
$255$	2830.61	0.695135
$256$	0	0
$257$	4429.73	1.07517	0.537586	−	0.843209i	$-0.319336\pi$
$257$	4429.73	1.07517	0.537586	+	0.843209i	$0.319336\pi$
$258$	0	0
$259$	5659.50	1.35778
$260$	0	0
$261$	−2512.18	−0.595785
$262$	0	0
$263$	−1283.91	−0.301025	−0.150513	−	0.988608i	$-0.548092\pi$
$263$	−1283.91	−0.301025	−0.150513	+	0.988608i	$0.548092\pi$
$264$	0	0
$265$	2661.32	0.616919
$266$	0	0
$267$	4664.14	1.06907
$268$	0	0
$269$	−4859.68	−1.10149	−0.550743	−	0.834675i	$-0.685656\pi$
$269$	−4859.68	−1.10149	−0.550743	+	0.834675i	$0.685656\pi$
$270$	0	0
$271$	6717.67	1.50579	0.752895	−	0.658140i	$-0.228657\pi$
$271$	6717.67	1.50579	0.752895	+	0.658140i	$0.228657\pi$
$272$	0	0
$273$	−5762.43	−1.27750
$274$	0	0
$275$	174.154	0.0381887
$276$	0	0
$277$	−2079.18	−0.450996	−0.225498	−	0.974244i	$-0.572401\pi$
$277$	−2079.18	−0.450996	−0.225498	+	0.974244i	$0.572401\pi$
$278$	0	0
$279$	−2761.49	−0.592567
$280$	0	0
$281$	729.615	0.154894	0.0774470	−	0.996996i	$-0.475323\pi$
$281$	729.615	0.154894	0.0774470	+	0.996996i	$0.475323\pi$
$282$	0	0
$283$	−3340.79	−0.701730	−0.350865	−	0.936426i	$-0.614112\pi$
$283$	−3340.79	−0.701730	−0.350865	+	0.936426i	$0.614112\pi$
$284$	0	0
$285$	2045.61	0.425163
$286$	0	0
$287$	−205.349	−0.0422348
$288$	0	0
$289$	1820.58	0.370563
$290$	0	0
$291$	3712.60	0.747892
$292$	0	0
$293$	−250.069	−0.0498606	−0.0249303	−	0.999689i	$-0.507936\pi$
$293$	−250.069	−0.0498606	−0.0249303	+	0.999689i	$0.507936\pi$
$294$	0	0
$295$	−3311.96	−0.653660
$296$	0	0
$297$	−307.760	−0.0601281
$298$	0	0
$299$	699.031	0.135204
$300$	0	0
$301$	−179.506	−0.0343740
$302$	0	0
$303$	1131.30	0.214493
$304$	0	0
$305$	−3643.66	−0.684050
$306$	0	0
$307$	9936.59	1.84727	0.923634	−	0.383276i	$-0.125204\pi$
$307$	9936.59	1.84727	0.923634	+	0.383276i	$0.125204\pi$
$308$	0	0
$309$	−1129.22	−0.207893
$310$	0	0
$311$	2850.50	0.519733	0.259866	−	0.965645i	$-0.416321\pi$
$311$	2850.50	0.519733	0.259866	+	0.965645i	$0.416321\pi$
$312$	0	0
$313$	−10118.9	−1.82733	−0.913666	−	0.406466i	$-0.866761\pi$
$313$	−10118.9	−1.82733	−0.913666	+	0.406466i	$0.866761\pi$
$314$	0	0
$315$	2830.15	0.506225
$316$	0	0
$317$	1098.47	0.194625	0.0973124	−	0.995254i	$-0.468975\pi$
$317$	1098.47	0.194625	0.0973124	+	0.995254i	$0.468975\pi$
$318$	0	0
$319$	−849.680	−0.149131
$320$	0	0
$321$	−2196.25	−0.381877
$322$	0	0
$323$	4866.19	0.838272
$324$	0	0
$325$	−759.816	−0.129683
$326$	0	0
$327$	908.637	0.153663
$328$	0	0
$329$	4214.67	0.706269
$330$	0	0
$331$	959.705	0.159366	0.0796830	−	0.996820i	$-0.474609\pi$
$331$	959.705	0.159366	0.0796830	+	0.996820i	$0.474609\pi$
$332$	0	0
$333$	4241.47	0.697991
$334$	0	0
$335$	−2211.24	−0.360636
$336$	0	0
$337$	3593.01	0.580783	0.290391	−	0.956908i	$-0.406215\pi$
$337$	3593.01	0.580783	0.290391	+	0.956908i	$0.406215\pi$
$338$	0	0
$339$	10953.7	1.75494
$340$	0	0
$341$	−934.004	−0.148326
$342$	0	0
$343$	1903.65	0.299673
$344$	0	0
$345$	−793.386	−0.123810
$346$	0	0
$347$	12835.4	1.98570	0.992850	−	0.119369i	$-0.0380873\pi$
$347$	12835.4	1.98570	0.992850	+	0.119369i	$0.0380873\pi$
$348$	0	0
$349$	−2354.83	−0.361178	−0.180589	−	0.983559i	$-0.557800\pi$
$349$	−2354.83	−0.361178	−0.180589	+	0.983559i	$0.557800\pi$
$350$	0	0
$351$	1342.72	0.204186
$352$	0	0
$353$	−8242.78	−1.24283	−0.621415	−	0.783482i	$-0.713442\pi$
$353$	−8242.78	−1.24283	−0.621415	+	0.783482i	$0.713442\pi$
$354$	0	0
$355$	−1357.45	−0.202947
$356$	0	0
$357$	15558.2	2.30652
$358$	0	0
$359$	10359.5	1.52298	0.761492	−	0.648174i	$-0.224467\pi$
$359$	10359.5	1.52298	0.761492	+	0.648174i	$0.224467\pi$
$360$	0	0
$361$	−3342.33	−0.487291
$362$	0	0
$363$	−8847.79	−1.27931
$364$	0	0
$365$	706.506	0.101316
$366$	0	0
$367$	−10578.5	−1.50461	−0.752307	−	0.658813i	$-0.771059\pi$
$367$	−10578.5	−1.50461	−0.752307	+	0.658813i	$0.771059\pi$
$368$	0	0
$369$	−153.897	−0.0217116
$370$	0	0
$371$	14627.7	2.04699
$372$	0	0
$373$	1976.38	0.274351	0.137175	−	0.990547i	$-0.456198\pi$
$373$	1976.38	0.274351	0.137175	+	0.990547i	$0.456198\pi$
$374$	0	0
$375$	862.376	0.118754
$376$	0	0
$377$	3707.06	0.506428
$378$	0	0
$379$	7643.33	1.03591	0.517957	−	0.855407i	$-0.326693\pi$
$379$	7643.33	1.03591	0.517957	+	0.855407i	$0.326693\pi$
$380$	0	0
$381$	−1532.13	−0.206019
$382$	0	0
$383$	−5786.70	−0.772027	−0.386013	−	0.922493i	$-0.626148\pi$
$383$	−5786.70	−0.772027	−0.386013	+	0.922493i	$0.626148\pi$
$384$	0	0
$385$	957.226	0.126714
$386$	0	0
$387$	−134.529	−0.0176706
$388$	0	0
$389$	13071.0	1.70366	0.851832	−	0.523815i	$-0.175492\pi$
$389$	13071.0	1.70366	0.851832	+	0.523815i	$0.175492\pi$
$390$	0	0
$391$	−1887.34	−0.244110
$392$	0	0
$393$	−14630.2	−1.87785
$394$	0	0
$395$	4491.04	0.572072
$396$	0	0
$397$	−12417.1	−1.56976	−0.784881	−	0.619647i	$-0.787276\pi$
$397$	−12417.1	−1.56976	−0.784881	+	0.619647i	$0.787276\pi$
$398$	0	0
$399$	11243.5	1.41073
$400$	0	0
$401$	246.358	0.0306796	0.0153398	−	0.999882i	$-0.495117\pi$
$401$	246.358	0.0306796	0.0153398	+	0.999882i	$0.495117\pi$
$402$	0	0
$403$	4074.96	0.503693
$404$	0	0
$405$	−4304.46	−0.528125
$406$	0	0
$407$	1434.57	0.174715
$408$	0	0
$409$	11251.1	1.36022	0.680109	−	0.733111i	$-0.261932\pi$
$409$	11251.1	1.36022	0.680109	+	0.733111i	$0.261932\pi$
$410$	0	0
$411$	9707.34	1.16503
$412$	0	0
$413$	−18204.0	−2.16891
$414$	0	0
$415$	3716.88	0.439649
$416$	0	0
$417$	−369.628	−0.0434071
$418$	0	0
$419$	8009.61	0.933879	0.466939	−	0.884289i	$-0.345357\pi$
$419$	8009.61	0.933879	0.466939	+	0.884289i	$0.345357\pi$
$420$	0	0
$421$	−13518.8	−1.56501	−0.782503	−	0.622646i	$-0.786058\pi$
$421$	−13518.8	−1.56501	−0.782503	+	0.622646i	$0.786058\pi$
$422$	0	0
$423$	3158.65	0.363071
$424$	0	0
$425$	2051.46	0.234142
$426$	0	0
$427$	−20027.1	−2.26974
$428$	0	0
$429$	−1460.66	−0.164385
$430$	0	0
$431$	3042.78	0.340059	0.170030	−	0.985439i	$-0.445614\pi$
$431$	3042.78	0.340059	0.170030	+	0.985439i	$0.445614\pi$
$432$	0	0
$433$	16035.2	1.77969	0.889844	−	0.456265i	$-0.150813\pi$
$433$	16035.2	1.77969	0.889844	+	0.456265i	$0.150813\pi$
$434$	0	0
$435$	−4207.44	−0.463750
$436$	0	0
$437$	−1363.94	−0.149304
$438$	0	0
$439$	−2517.90	−0.273742	−0.136871	−	0.990589i	$-0.543705\pi$
$439$	−2517.90	−0.273742	−0.136871	+	0.990589i	$0.543705\pi$
$440$	0	0
$441$	8491.21	0.916878
$442$	0	0
$443$	6703.37	0.718931	0.359466	−	0.933158i	$-0.382959\pi$
$443$	6703.37	0.718931	0.359466	+	0.933158i	$0.382959\pi$
$444$	0	0
$445$	3380.30	0.360093
$446$	0	0
$447$	−11643.7	−1.23206
$448$	0	0
$449$	10022.1	1.05339	0.526697	−	0.850053i	$-0.323430\pi$
$449$	10022.1	1.05339	0.526697	+	0.850053i	$0.323430\pi$
$450$	0	0
$451$	−52.0519	−0.00543465
$452$	0	0
$453$	−12987.6	−1.34704
$454$	0	0
$455$	−4176.28	−0.430301
$456$	0	0
$457$	8575.29	0.877758	0.438879	−	0.898546i	$-0.355376\pi$
$457$	8575.29	0.877758	0.438879	+	0.898546i	$0.355376\pi$
$458$	0	0
$459$	−3625.28	−0.368657
$460$	0	0
$461$	16130.9	1.62970	0.814848	−	0.579674i	$-0.196820\pi$
$461$	16130.9	1.62970	0.814848	+	0.579674i	$0.196820\pi$
$462$	0	0
$463$	−3806.34	−0.382064	−0.191032	−	0.981584i	$-0.561183\pi$
$463$	−3806.34	−0.382064	−0.191032	+	0.981584i	$0.561183\pi$
$464$	0	0
$465$	−4625.00	−0.461246
$466$	0	0
$467$	7533.36	0.746472	0.373236	−	0.927737i	$-0.378248\pi$
$467$	7533.36	0.746472	0.373236	+	0.927737i	$0.378248\pi$
$468$	0	0
$469$	−12153.9	−1.19662
$470$	0	0
$471$	−25825.9	−2.52653
$472$	0	0
$473$	−45.5012	−0.00442314
$474$	0	0
$475$	1482.54	0.143207
$476$	0	0
$477$	10962.7	1.05230
$478$	0	0
$479$	−5389.36	−0.514084	−0.257042	−	0.966400i	$-0.582748\pi$
$479$	−5389.36	−0.514084	−0.257042	+	0.966400i	$0.582748\pi$
$480$	0	0
$481$	−6258.87	−0.593305
$482$	0	0
$483$	−4360.79	−0.410813
$484$	0	0
$485$	2690.68	0.251912
$486$	0	0
$487$	−15848.1	−1.47463	−0.737317	−	0.675547i	$-0.763908\pi$
$487$	−15848.1	−1.47463	−0.737317	+	0.675547i	$0.763908\pi$
$488$	0	0
$489$	7698.81	0.711968
$490$	0	0
$491$	7934.19	0.729257	0.364629	−	0.931153i	$-0.381196\pi$
$491$	7934.19	0.729257	0.364629	+	0.931153i	$0.381196\pi$
$492$	0	0
$493$	−10008.9	−0.914353
$494$	0	0
$495$	717.386	0.0651396
$496$	0	0
$497$	−7461.15	−0.673397
$498$	0	0
$499$	−7349.90	−0.659372	−0.329686	−	0.944091i	$-0.606943\pi$
$499$	−7349.90	−0.659372	−0.329686	+	0.944091i	$0.606943\pi$
$500$	0	0
$501$	−22413.7	−1.99874
$502$	0	0
$503$	12987.8	1.15129	0.575646	−	0.817699i	$-0.304751\pi$
$503$	12987.8	1.15129	0.575646	+	0.817699i	$0.304751\pi$
$504$	0	0
$505$	819.899	0.0722475
$506$	0	0
$507$	−8784.41	−0.769486
$508$	0	0
$509$	13937.8	1.21372	0.606858	−	0.794810i	$-0.292430\pi$
$509$	13937.8	1.21372	0.606858	+	0.794810i	$0.292430\pi$
$510$	0	0
$511$	3883.26	0.336175
$512$	0	0
$513$	−2619.90	−0.225480
$514$	0	0
$515$	−818.393	−0.0700247
$516$	0	0
$517$	1068.33	0.0908806
$518$	0	0
$519$	7765.56	0.656783
$520$	0	0
$521$	17345.9	1.45861	0.729307	−	0.684186i	$-0.239842\pi$
$521$	17345.9	1.45861	0.729307	+	0.684186i	$0.239842\pi$
$522$	0	0
$523$	19808.3	1.65613	0.828067	−	0.560629i	$-0.189441\pi$
$523$	19808.3	1.65613	0.828067	+	0.560629i	$0.189441\pi$
$524$	0	0
$525$	4739.99	0.394038
$526$	0	0
$527$	−11002.2	−0.909415
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−13642.8	−1.11497
$532$	0	0
$533$	227.097	0.0184553
$534$	0	0
$535$	−1591.71	−0.128628
$536$	0	0
$537$	21113.1	1.69664
$538$	0	0
$539$	2871.93	0.229504
$540$	0	0
$541$	4536.27	0.360498	0.180249	−	0.983621i	$-0.442310\pi$
$541$	4536.27	0.360498	0.180249	+	0.983621i	$0.442310\pi$
$542$	0	0
$543$	−10537.1	−0.832759
$544$	0	0
$545$	658.528	0.0517582
$546$	0	0
$547$	−11026.2	−0.861879	−0.430940	−	0.902381i	$-0.641818\pi$
$547$	−11026.2	−0.861879	−0.430940	+	0.902381i	$0.641818\pi$
$548$	0	0
$549$	−15009.2	−1.16681
$550$	0	0
$551$	−7233.15	−0.559242
$552$	0	0
$553$	24684.7	1.89819
$554$	0	0
$555$	7103.69	0.543306
$556$	0	0
$557$	5223.45	0.397351	0.198676	−	0.980065i	$-0.436336\pi$
$557$	5223.45	0.397351	0.198676	+	0.980065i	$0.436336\pi$
$558$	0	0
$559$	198.517	0.0150203
$560$	0	0
$561$	3943.70	0.296797
$562$	0	0
$563$	2920.23	0.218602	0.109301	−	0.994009i	$-0.465139\pi$
$563$	2920.23	0.218602	0.109301	+	0.994009i	$0.465139\pi$
$564$	0	0
$565$	7938.61	0.591115
$566$	0	0
$567$	−23659.2	−1.75237
$568$	0	0
$569$	16443.8	1.21153	0.605766	−	0.795643i	$-0.292867\pi$
$569$	16443.8	1.21153	0.605766	+	0.795643i	$0.292867\pi$
$570$	0	0
$571$	2873.85	0.210625	0.105312	−	0.994439i	$-0.466416\pi$
$571$	2873.85	0.210625	0.105312	+	0.994439i	$0.466416\pi$
$572$	0	0
$573$	−4643.20	−0.338521
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	15321.5	1.10545	0.552725	−	0.833364i	$-0.313588\pi$
$577$	15321.5	1.10545	0.552725	+	0.833364i	$0.313588\pi$
$578$	0	0
$579$	−22199.1	−1.59338
$580$	0	0
$581$	20429.6	1.45880
$582$	0	0
$583$	3707.84	0.263401
$584$	0	0
$585$	−3129.88	−0.221204
$586$	0	0
$587$	−2506.36	−0.176232	−0.0881162	−	0.996110i	$-0.528085\pi$
$587$	−2506.36	−0.176232	−0.0881162	+	0.996110i	$0.528085\pi$
$588$	0	0
$589$	−7950.98	−0.556222
$590$	0	0
$591$	−8309.59	−0.578360
$592$	0	0
$593$	17131.2	1.18633	0.593164	−	0.805081i	$-0.297878\pi$
$593$	17131.2	1.18633	0.593164	+	0.805081i	$0.297878\pi$
$594$	0	0
$595$	11275.7	0.776906
$596$	0	0
$597$	26486.5	1.81578
$598$	0	0
$599$	7851.10	0.535538	0.267769	−	0.963483i	$-0.413714\pi$
$599$	7851.10	0.535538	0.267769	+	0.963483i	$0.413714\pi$
$600$	0	0
$601$	503.477	0.0341718	0.0170859	−	0.999854i	$-0.494561\pi$
$601$	503.477	0.0341718	0.0170859	+	0.999854i	$0.494561\pi$
$602$	0	0
$603$	−9108.68	−0.615148
$604$	0	0
$605$	−6412.36	−0.430908
$606$	0	0
$607$	−15682.9	−1.04868	−0.524340	−	0.851509i	$-0.675688\pi$
$607$	−15682.9	−1.04868	−0.524340	+	0.851509i	$0.675688\pi$
$608$	0	0
$609$	−23125.9	−1.53877
$610$	0	0
$611$	−4661.03	−0.308617
$612$	0	0
$613$	−21907.3	−1.44344	−0.721719	−	0.692187i	$-0.756648\pi$
$613$	−21907.3	−1.44344	−0.721719	+	0.692187i	$0.756648\pi$
$614$	0	0
$615$	−257.750	−0.0169000
$616$	0	0
$617$	6995.09	0.456421	0.228211	−	0.973612i	$-0.426713\pi$
$617$	6995.09	0.456421	0.228211	+	0.973612i	$0.426713\pi$
$618$	0	0
$619$	3225.25	0.209425	0.104712	−	0.994503i	$-0.466608\pi$
$619$	3225.25	0.209425	0.104712	+	0.994503i	$0.466608\pi$
$620$	0	0
$621$	1016.12	0.0656612
$622$	0	0
$623$	18579.6	1.19482
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	2850.01	0.181529
$628$	0	0
$629$	16898.6	1.07121
$630$	0	0
$631$	−20496.7	−1.29312	−0.646560	−	0.762863i	$-0.723793\pi$
$631$	−20496.7	−1.29312	−0.646560	+	0.762863i	$0.723793\pi$
$632$	0	0
$633$	5200.49	0.326542
$634$	0	0
$635$	−1110.40	−0.0693933
$636$	0	0
$637$	−12529.9	−0.779363
$638$	0	0
$639$	−5591.70	−0.346173
$640$	0	0
$641$	5340.21	0.329057	0.164529	−	0.986372i	$-0.447390\pi$
$641$	5340.21	0.329057	0.164529	+	0.986372i	$0.447390\pi$
$642$	0	0
$643$	23089.8	1.41613	0.708066	−	0.706146i	$-0.249568\pi$
$643$	23089.8	1.41613	0.708066	+	0.706146i	$0.249568\pi$
$644$	0	0
$645$	−225.312	−0.0137545
$646$	0	0
$647$	12559.3	0.763150	0.381575	−	0.924338i	$-0.375382\pi$
$647$	12559.3	0.763150	0.381575	+	0.924338i	$0.375382\pi$
$648$	0	0
$649$	−4614.33	−0.279089
$650$	0	0
$651$	−25421.0	−1.53046
$652$	0	0
$653$	−4975.31	−0.298161	−0.149080	−	0.988825i	$-0.547631\pi$
$653$	−4975.31	−0.298161	−0.149080	+	0.988825i	$0.547631\pi$
$654$	0	0
$655$	−10603.1	−0.632517
$656$	0	0
$657$	2910.28	0.172817
$658$	0	0
$659$	4502.16	0.266129	0.133065	−	0.991107i	$-0.457518\pi$
$659$	4502.16	0.266129	0.133065	+	0.991107i	$0.457518\pi$
$660$	0	0
$661$	−7540.37	−0.443701	−0.221851	−	0.975081i	$-0.571210\pi$
$661$	−7540.37	−0.443701	−0.221851	+	0.975081i	$0.571210\pi$
$662$	0	0
$663$	−17205.9	−1.00788
$664$	0	0
$665$	8148.67	0.475176
$666$	0	0
$667$	2805.36	0.162855
$668$	0	0
$669$	1656.15	0.0957108
$670$	0	0
$671$	−5076.47	−0.292064
$672$	0	0
$673$	−12774.4	−0.731676	−0.365838	−	0.930679i	$-0.619218\pi$
$673$	−12774.4	−0.731676	−0.365838	+	0.930679i	$0.619218\pi$
$674$	0	0
$675$	−1104.48	−0.0629800
$676$	0	0
$677$	−20628.0	−1.17104	−0.585522	−	0.810657i	$-0.699110\pi$
$677$	−20628.0	−1.17104	−0.585522	+	0.810657i	$0.699110\pi$
$678$	0	0
$679$	14789.1	0.835868
$680$	0	0
$681$	−29954.1	−1.68553
$682$	0	0
$683$	−22883.8	−1.28203	−0.641013	−	0.767530i	$-0.721485\pi$
$683$	−22883.8	−1.28203	−0.641013	+	0.767530i	$0.721485\pi$
$684$	0	0
$685$	7035.32	0.392417
$686$	0	0
$687$	−30527.2	−1.69532
$688$	0	0
$689$	−16176.9	−0.894471
$690$	0	0
$691$	−10354.2	−0.570030	−0.285015	−	0.958523i	$-0.591999\pi$
$691$	−10354.2	−0.570030	−0.285015	+	0.958523i	$0.591999\pi$
$692$	0	0
$693$	3943.06	0.216139
$694$	0	0
$695$	−267.885	−0.0146208
$696$	0	0
$697$	−613.148	−0.0333208
$698$	0	0
$699$	−34238.0	−1.85265
$700$	0	0
$701$	−31487.4	−1.69652	−0.848262	−	0.529576i	$-0.822351\pi$
$701$	−31487.4	−1.69652	−0.848262	+	0.529576i	$0.822351\pi$
$702$	0	0
$703$	12212.2	0.655179
$704$	0	0
$705$	5290.17	0.282609
$706$	0	0
$707$	4506.52	0.239724
$708$	0	0
$709$	−7078.79	−0.374964	−0.187482	−	0.982268i	$-0.560033\pi$
$709$	−7078.79	−0.374964	−0.187482	+	0.982268i	$0.560033\pi$
$710$	0	0
$711$	18499.7	0.975800
$712$	0	0
$713$	3083.78	0.161975
$714$	0	0
$715$	−1058.60	−0.0553699
$716$	0	0
$717$	−26068.6	−1.35781
$718$	0	0
$719$	−15970.7	−0.828384	−0.414192	−	0.910190i	$-0.635936\pi$
$719$	−15970.7	−0.828384	−0.414192	+	0.910190i	$0.635936\pi$
$720$	0	0
$721$	−4498.24	−0.232349
$722$	0	0
$723$	−31788.2	−1.63515
$724$	0	0
$725$	−3049.31	−0.156205
$726$	0	0
$727$	2312.68	0.117981	0.0589907	−	0.998259i	$-0.481212\pi$
$727$	2312.68	0.117981	0.0589907	+	0.998259i	$0.481212\pi$
$728$	0	0
$729$	−9501.78	−0.482741
$730$	0	0
$731$	−535.984	−0.0271191
$732$	0	0
$733$	3638.22	0.183330	0.0916648	−	0.995790i	$-0.470781\pi$
$733$	3638.22	0.183330	0.0916648	+	0.995790i	$0.470781\pi$
$734$	0	0
$735$	14221.2	0.713684
$736$	0	0
$737$	−3080.78	−0.153978
$738$	0	0
$739$	−11875.5	−0.591133	−0.295566	−	0.955322i	$-0.595508\pi$
$739$	−11875.5	−0.591133	−0.295566	+	0.955322i	$0.595508\pi$
$740$	0	0
$741$	−12434.3	−0.616444
$742$	0	0
$743$	30642.9	1.51302	0.756512	−	0.653979i	$-0.226902\pi$
$743$	30642.9	1.51302	0.756512	+	0.653979i	$0.226902\pi$
$744$	0	0
$745$	−8438.69	−0.414993
$746$	0	0
$747$	15310.8	0.749923
$748$	0	0
$749$	−8748.74	−0.426799
$750$	0	0
$751$	−27743.7	−1.34805	−0.674023	−	0.738711i	$-0.735435\pi$
$751$	−27743.7	−1.34805	−0.674023	+	0.738711i	$0.735435\pi$
$752$	0	0
$753$	1865.73	0.0902936
$754$	0	0
$755$	−9412.64	−0.453723
$756$	0	0
$757$	2576.75	0.123717	0.0618583	−	0.998085i	$-0.480297\pi$
$757$	2576.75	0.123717	0.0618583	+	0.998085i	$0.480297\pi$
$758$	0	0
$759$	−1105.37	−0.0528622
$760$	0	0
$761$	−6965.38	−0.331793	−0.165897	−	0.986143i	$-0.553052\pi$
$761$	−6965.38	−0.331793	−0.165897	+	0.986143i	$0.553052\pi$
$762$	0	0
$763$	3619.55	0.171739
$764$	0	0
$765$	8450.49	0.399383
$766$	0	0
$767$	20131.9	0.947743
$768$	0	0
$769$	−27746.0	−1.30110	−0.650550	−	0.759464i	$-0.725461\pi$
$769$	−27746.0	−1.30110	−0.650550	+	0.759464i	$0.725461\pi$
$770$	0	0
$771$	30560.7	1.42752
$772$	0	0
$773$	−5839.57	−0.271714	−0.135857	−	0.990728i	$-0.543379\pi$
$773$	−5839.57	−0.271714	−0.135857	+	0.990728i	$0.543379\pi$
$774$	0	0
$775$	−3351.93	−0.155361
$776$	0	0
$777$	39044.9	1.80274
$778$	0	0
$779$	−443.107	−0.0203799
$780$	0	0
$781$	−1891.25	−0.0866508
$782$	0	0
$783$	5388.65	0.245944
$784$	0	0
$785$	−18717.1	−0.851009
$786$	0	0
$787$	37705.5	1.70782	0.853912	−	0.520418i	$-0.174224\pi$
$787$	37705.5	1.70782	0.853912	+	0.520418i	$0.174224\pi$
$788$	0	0
$789$	−8857.74	−0.399675
$790$	0	0
$791$	43634.0	1.96138
$792$	0	0
$793$	22148.1	0.991806
$794$	0	0
$795$	18360.4	0.819092
$796$	0	0
$797$	14236.4	0.632724	0.316362	−	0.948639i	$-0.397539\pi$
$797$	14236.4	0.632724	0.316362	+	0.948639i	$0.397539\pi$
$798$	0	0
$799$	12584.5	0.557206
$800$	0	0
$801$	13924.3	0.614222
$802$	0	0
$803$	984.328	0.0432580
$804$	0	0
$805$	−3160.45	−0.138374
$806$	0	0
$807$	−33526.9	−1.46246
$808$	0	0
$809$	1302.69	0.0566131	0.0283065	−	0.999599i	$-0.490989\pi$
$809$	1302.69	0.0566131	0.0283065	+	0.999599i	$0.490989\pi$
$810$	0	0
$811$	32808.1	1.42053	0.710263	−	0.703936i	$-0.248576\pi$
$811$	32808.1	1.42053	0.710263	+	0.703936i	$0.248576\pi$
$812$	0	0
$813$	46345.2	1.99926
$814$	0	0
$815$	5579.65	0.239812
$816$	0	0
$817$	−387.342	−0.0165868
$818$	0	0
$819$	−17203.2	−0.733977
$820$	0	0
$821$	−26353.5	−1.12027	−0.560136	−	0.828401i	$-0.689251\pi$
$821$	−26353.5	−1.12027	−0.560136	+	0.828401i	$0.689251\pi$
$822$	0	0
$823$	9755.27	0.413180	0.206590	−	0.978428i	$-0.433763\pi$
$823$	9755.27	0.413180	0.206590	+	0.978428i	$0.433763\pi$
$824$	0	0
$825$	1201.49	0.0507037
$826$	0	0
$827$	−40323.2	−1.69549	−0.847747	−	0.530401i	$-0.822041\pi$
$827$	−40323.2	−1.69549	−0.847747	+	0.530401i	$0.822041\pi$
$828$	0	0
$829$	7309.11	0.306220	0.153110	−	0.988209i	$-0.451071\pi$
$829$	7309.11	0.306220	0.153110	+	0.988209i	$0.451071\pi$
$830$	0	0
$831$	−14344.3	−0.598794
$832$	0	0
$833$	33830.1	1.40714
$834$	0	0
$835$	−16244.1	−0.673235
$836$	0	0
$837$	5923.43	0.244616
$838$	0	0
$839$	−32282.8	−1.32840	−0.664199	−	0.747556i	$-0.731227\pi$
$839$	−32282.8	−1.32840	−0.664199	+	0.747556i	$0.731227\pi$
$840$	0	0
$841$	−9511.75	−0.390001
$842$	0	0
$843$	5033.62	0.205655
$844$	0	0
$845$	−6366.43	−0.259186
$846$	0	0
$847$	−35245.1	−1.42979
$848$	0	0
$849$	−23048.2	−0.931697
$850$	0	0
$851$	−4736.47	−0.190792
$852$	0	0
$853$	−16639.7	−0.667918	−0.333959	−	0.942588i	$-0.608385\pi$
$853$	−16639.7	−0.667918	−0.333959	+	0.942588i	$0.608385\pi$
$854$	0	0
$855$	6106.96	0.244273
$856$	0	0
$857$	44900.7	1.78971	0.894854	−	0.446359i	$-0.147279\pi$
$857$	44900.7	1.78971	0.894854	+	0.446359i	$0.147279\pi$
$858$	0	0
$859$	30010.0	1.19200	0.596001	−	0.802984i	$-0.296756\pi$
$859$	30010.0	1.19200	0.596001	+	0.802984i	$0.296756\pi$
$860$	0	0
$861$	−1416.71	−0.0560757
$862$	0	0
$863$	35854.6	1.41426	0.707129	−	0.707085i	$-0.249990\pi$
$863$	35854.6	1.41426	0.707129	+	0.707085i	$0.249990\pi$
$864$	0	0
$865$	5628.03	0.221224
$866$	0	0
$867$	12560.2	0.492002
$868$	0	0
$869$	6257.06	0.244253
$870$	0	0
$871$	13441.1	0.522887
$872$	0	0
$873$	11083.6	0.429694
$874$	0	0
$875$	3435.27	0.132724
$876$	0	0
$877$	6060.53	0.233352	0.116676	−	0.993170i	$-0.462776\pi$
$877$	6060.53	0.233352	0.116676	+	0.993170i	$0.462776\pi$
$878$	0	0
$879$	−1725.23	−0.0662007
$880$	0	0
$881$	−23430.7	−0.896027	−0.448014	−	0.894027i	$-0.647868\pi$
$881$	−23430.7	−0.896027	−0.448014	+	0.894027i	$0.647868\pi$
$882$	0	0
$883$	−31448.9	−1.19857	−0.599286	−	0.800535i	$-0.704549\pi$
$883$	−31448.9	−1.19857	−0.599286	+	0.800535i	$0.704549\pi$
$884$	0	0
$885$	−22849.2	−0.867874
$886$	0	0
$887$	11413.0	0.432031	0.216015	−	0.976390i	$-0.430694\pi$
$887$	11413.0	0.432031	0.216015	+	0.976390i	$0.430694\pi$
$888$	0	0
$889$	−6103.22	−0.230253
$890$	0	0
$891$	−5997.12	−0.225490
$892$	0	0
$893$	9094.51	0.340802
$894$	0	0
$895$	15301.6	0.571480
$896$	0	0
$897$	4822.62	0.179512
$898$	0	0
$899$	16353.7	0.606704
$900$	0	0
$901$	43676.7	1.61496
$902$	0	0
$903$	−1238.41	−0.0456388
$904$	0	0
$905$	−7636.64	−0.280498
$906$	0	0
$907$	−40591.2	−1.48601	−0.743004	−	0.669287i	$-0.766600\pi$
$907$	−40591.2	−1.48601	−0.743004	+	0.669287i	$0.766600\pi$
$908$	0	0
$909$	3377.37	0.123235
$910$	0	0
$911$	28262.2	1.02785	0.513924	−	0.857836i	$-0.328191\pi$
$911$	28262.2	1.02785	0.513924	+	0.857836i	$0.328191\pi$
$912$	0	0
$913$	5178.48	0.187714
$914$	0	0
$915$	−25137.6	−0.908224
$916$	0	0
$917$	−58279.4	−2.09875
$918$	0	0
$919$	36673.3	1.31637	0.658183	−	0.752858i	$-0.271325\pi$
$919$	36673.3	1.31637	0.658183	+	0.752858i	$0.271325\pi$
$920$	0	0
$921$	68552.6	2.45264
$922$	0	0
$923$	8251.33	0.294253
$924$	0	0
$925$	5148.34	0.183002
$926$	0	0
$927$	−3371.17	−0.119443
$928$	0	0
$929$	41751.4	1.47451	0.737255	−	0.675614i	$-0.236121\pi$
$929$	41751.4	1.47451	0.737255	+	0.675614i	$0.236121\pi$
$930$	0	0
$931$	24448.2	0.860641
$932$	0	0
$933$	19665.6	0.690057
$934$	0	0
$935$	2858.16	0.0999699
$936$	0	0
$937$	40051.3	1.39639	0.698195	−	0.715908i	$-0.253987\pi$
$937$	40051.3	1.39639	0.698195	+	0.715908i	$0.253987\pi$
$938$	0	0
$939$	−69810.5	−2.42617
$940$	0	0
$941$	−49386.7	−1.71090	−0.855452	−	0.517882i	$-0.826721\pi$
$941$	−49386.7	−1.71090	−0.855452	+	0.517882i	$0.826721\pi$
$942$	0	0
$943$	171.858	0.00593475
$944$	0	0
$945$	−6070.70	−0.208974
$946$	0	0
$947$	−2219.40	−0.0761573	−0.0380786	−	0.999275i	$-0.512124\pi$
$947$	−2219.40	−0.0761573	−0.0380786	+	0.999275i	$0.512124\pi$
$948$	0	0
$949$	−4294.52	−0.146898
$950$	0	0
$951$	7578.33	0.258406
$952$	0	0
$953$	23576.1	0.801368	0.400684	−	0.916216i	$-0.368773\pi$
$953$	23576.1	0.801368	0.400684	+	0.916216i	$0.368773\pi$
$954$	0	0
$955$	−3365.12	−0.114024
$956$	0	0
$957$	−5861.95	−0.198004
$958$	0	0
$959$	38669.1	1.30208
$960$	0	0
$961$	−11814.3	−0.396573
$962$	0	0
$963$	−6556.68	−0.219404
$964$	0	0
$965$	−16088.7	−0.536696
$966$	0	0
$967$	20542.6	0.683151	0.341575	−	0.939854i	$-0.389040\pi$
$967$	20542.6	0.683151	0.341575	+	0.939854i	$0.389040\pi$
$968$	0	0
$969$	33571.9	1.11299
$970$	0	0
$971$	6788.31	0.224353	0.112177	−	0.993688i	$-0.464218\pi$
$971$	6788.31	0.224353	0.112177	+	0.993688i	$0.464218\pi$
$972$	0	0
$973$	−1472.41	−0.0485131
$974$	0	0
$975$	−5241.98	−0.172182
$976$	0	0
$977$	52034.9	1.70394	0.851968	−	0.523594i	$-0.175409\pi$
$977$	52034.9	1.70394	0.851968	+	0.523594i	$0.175409\pi$
$978$	0	0
$979$	4709.55	0.153746
$980$	0	0
$981$	2712.65	0.0882855
$982$	0	0
$983$	23833.1	0.773303	0.386652	−	0.922226i	$-0.373632\pi$
$983$	23833.1	0.773303	0.386652	+	0.922226i	$0.373632\pi$
$984$	0	0
$985$	−6022.31	−0.194809
$986$	0	0
$987$	29077.1	0.937724
$988$	0	0
$989$	150.230	0.00483016
$990$	0	0
$991$	53905.7	1.72792	0.863961	−	0.503559i	$-0.167976\pi$
$991$	53905.7	1.72792	0.863961	+	0.503559i	$0.167976\pi$
$992$	0	0
$993$	6621.01	0.211593
$994$	0	0
$995$	19195.9	0.611608
$996$	0	0
$997$	−45593.4	−1.44830	−0.724150	−	0.689642i	$-0.757768\pi$
$997$	−45593.4	−1.44830	−0.724150	+	0.689642i	$0.757768\pi$
$998$	0	0
$999$	−9097.99	−0.288136

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.4.a.c.1.5	✓	6
4.3	odd	2		1840.4.a.t.1.2		6
5.2	odd	4		2300.4.c.f.1749.3		12
5.3	odd	4		2300.4.c.f.1749.10		12
5.4	even	2		2300.4.a.f.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.4.a.c.1.5	✓	6	1.1	even	1	trivial
1840.4.a.t.1.2		6	4.3	odd	2
2300.4.a.f.1.2		6	5.4	even	2
2300.4.c.f.1749.3		12	5.2	odd	4
2300.4.c.f.1749.10		12	5.3	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	460.a (trivial)

\(f(q)\)	\(=\)	\(q+6.89901 q^{3} +5.00000 q^{5} +27.4822 q^{7} +20.5963 q^{9} +6.96617 q^{11} -30.3927 q^{13} +34.4950 q^{15} +82.0584 q^{17} +59.3015 q^{19} +189.600 q^{21} -23.0000 q^{23} +25.0000 q^{25} -44.1793 q^{27} -121.972 q^{29} -134.077 q^{31} +48.0596 q^{33} +137.411 q^{35} +205.934 q^{37} -209.679 q^{39} -7.47209 q^{41} -6.53173 q^{43} +102.981 q^{45} +153.360 q^{47} +412.269 q^{49} +566.121 q^{51} +532.263 q^{53} +34.8308 q^{55} +409.122 q^{57} -662.392 q^{59} -728.732 q^{61} +566.030 q^{63} -151.963 q^{65} -442.249 q^{67} -158.677 q^{69} -271.491 q^{71} +141.301 q^{73} +172.475 q^{75} +191.445 q^{77} +898.207 q^{79} -860.893 q^{81} +743.376 q^{83} +410.292 q^{85} -841.488 q^{87} +676.060 q^{89} -835.256 q^{91} -925.000 q^{93} +296.508 q^{95} +538.136 q^{97} +143.477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 30 q^{5} + 24 q^{7} + 75 q^{9} + 117 q^{11} - 59 q^{13} - 5 q^{15} + 88 q^{17} + 105 q^{19} + 44 q^{21} - 138 q^{23} + 150 q^{25} + 164 q^{27} - 71 q^{29} + 396 q^{31} - 85 q^{33} + 120 q^{35}+ \cdots + 4077 q^{99}+O(q^{100}) \) 6 * q - q^3 + 30 * q^5 + 24 * q^7 + 75 * q^9 + 117 * q^11 - 59 * q^13 - 5 * q^15 + 88 * q^17 + 105 * q^19 + 44 * q^21 - 138 * q^23 + 150 * q^25 + 164 * q^27 - 71 * q^29 + 396 * q^31 - 85 * q^33 + 120 * q^35 + 57 * q^37 + 718 * q^39 + 692 * q^41 + 778 * q^43 + 375 * q^45 + 248 * q^47 + 958 * q^49 + 545 * q^51 - 49 * q^53 + 585 * q^55 + 396 * q^57 + 1539 * q^59 + 461 * q^61 + 921 * q^63 - 295 * q^65 + 2815 * q^67 + 23 * q^69 + 378 * q^71 + 518 * q^73 - 25 * q^75 - 286 * q^77 + 1694 * q^79 + 498 * q^81 + 1757 * q^83 + 440 * q^85 + 2918 * q^87 + 688 * q^89 + 2363 * q^91 + 36 * q^93 + 525 * q^95 + 1029 * q^97 + 4077 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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