LMFDB - Embedded newform 495.4.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,4,Mod(1,495)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("495.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	495.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:	$29.2059454528$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1539480.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 25x^{2} + 9x + 96 $ x^4 - x^3 - 25x^2 + 9x + 96
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30365$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30365 q^{2} -2.69320 q^{4} -5.00000 q^{5} +13.1506 q^{7} +24.6334 q^{8} +O(q^{10})$	\(q-2.30365 q^{2} -2.69320 q^{4} -5.00000 q^{5} +13.1506 q^{7} +24.6334 q^{8} +11.5182 q^{10} -11.0000 q^{11} -59.5317 q^{13} -30.2943 q^{14} -35.2010 q^{16} +0.315106 q^{17} +32.8179 q^{19} +13.4660 q^{20} +25.3401 q^{22} +72.2810 q^{23} +25.0000 q^{25} +137.140 q^{26} -35.4172 q^{28} +261.137 q^{29} +155.691 q^{31} -115.976 q^{32} -0.725893 q^{34} -65.7529 q^{35} +151.261 q^{37} -75.6010 q^{38} -123.167 q^{40} -209.059 q^{41} +111.040 q^{43} +29.6252 q^{44} -166.510 q^{46} -633.073 q^{47} -170.062 q^{49} -57.5912 q^{50} +160.331 q^{52} -331.397 q^{53} +55.0000 q^{55} +323.943 q^{56} -601.569 q^{58} -814.568 q^{59} -263.071 q^{61} -358.657 q^{62} +548.777 q^{64} +297.659 q^{65} -661.609 q^{67} -0.848644 q^{68} +151.471 q^{70} -15.2662 q^{71} +785.036 q^{73} -348.453 q^{74} -88.3853 q^{76} -144.656 q^{77} -1075.89 q^{79} +176.005 q^{80} +481.599 q^{82} +1187.89 q^{83} -1.57553 q^{85} -255.797 q^{86} -270.967 q^{88} -403.471 q^{89} -782.876 q^{91} -194.667 q^{92} +1458.38 q^{94} -164.090 q^{95} -304.398 q^{97} +391.764 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 19 q^{4} - 20 q^{5} + 9 q^{7} - 33 q^{8}+O(q^{10}) $ 4 * q - q^2 + 19 * q^4 - 20 * q^5 + 9 * q^7 - 33 * q^8	$ 4 q - q^{2} + 19 q^{4} - 20 q^{5} + 9 q^{7} - 33 q^{8} + 5 q^{10} - 44 q^{11} + 70 q^{13} + 49 q^{14} - 37 q^{16} - 103 q^{17} - 205 q^{19} - 95 q^{20} + 11 q^{22} + 56 q^{23} + 100 q^{25} + 86 q^{26} - 551 q^{28} + 79 q^{29} + 49 q^{31} - 225 q^{32} - 939 q^{34} - 45 q^{35} + 289 q^{37} - 145 q^{38} + 165 q^{40} - 736 q^{41} - 152 q^{43} - 209 q^{44} - 334 q^{46} - 412 q^{47} + 37 q^{49} - 25 q^{50} + 1598 q^{52} - 1685 q^{53} + 220 q^{55} + 257 q^{56} + 609 q^{58} + 842 q^{59} - 1097 q^{61} - 1359 q^{62} - 165 q^{64} - 350 q^{65} - 122 q^{67} + 757 q^{68} - 245 q^{70} + 521 q^{71} - 590 q^{73} - 3257 q^{74} - 2825 q^{76} - 99 q^{77} - 1118 q^{79} + 185 q^{80} - 402 q^{82} + 122 q^{83} + 515 q^{85} - 3452 q^{86} + 363 q^{88} + 181 q^{89} - 2190 q^{91} - 430 q^{92} + 1034 q^{94} + 1025 q^{95} + 1474 q^{97} + 872 q^{98}+O(q^{100}) $ 4 * q - q^2 + 19 * q^4 - 20 * q^5 + 9 * q^7 - 33 * q^8 + 5 * q^10 - 44 * q^11 + 70 * q^13 + 49 * q^14 - 37 * q^16 - 103 * q^17 - 205 * q^19 - 95 * q^20 + 11 * q^22 + 56 * q^23 + 100 * q^25 + 86 * q^26 - 551 * q^28 + 79 * q^29 + 49 * q^31 - 225 * q^32 - 939 * q^34 - 45 * q^35 + 289 * q^37 - 145 * q^38 + 165 * q^40 - 736 * q^41 - 152 * q^43 - 209 * q^44 - 334 * q^46 - 412 * q^47 + 37 * q^49 - 25 * q^50 + 1598 * q^52 - 1685 * q^53 + 220 * q^55 + 257 * q^56 + 609 * q^58 + 842 * q^59 - 1097 * q^61 - 1359 * q^62 - 165 * q^64 - 350 * q^65 - 122 * q^67 + 757 * q^68 - 245 * q^70 + 521 * q^71 - 590 * q^73 - 3257 * q^74 - 2825 * q^76 - 99 * q^77 - 1118 * q^79 + 185 * q^80 - 402 * q^82 + 122 * q^83 + 515 * q^85 - 3452 * q^86 + 363 * q^88 + 181 * q^89 - 2190 * q^91 - 430 * q^92 + 1034 * q^94 + 1025 * q^95 + 1474 * q^97 + 872 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.30365	−0.814463	−0.407231	−	0.913325i	$-0.633506\pi$
$2$	−2.30365	−0.814463	−0.407231	+	0.913325i	$0.633506\pi$
$3$	0	0
$3$	0	0
$4$	−2.69320	−0.336650
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	13.1506	0.710064	0.355032	−	0.934854i	$-0.384470\pi$
$7$	13.1506	0.710064	0.355032	+	0.934854i	$0.384470\pi$
$8$	24.6334	1.08865
$9$	0	0
$10$	11.5182	0.364239
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−59.5317	−1.27009	−0.635044	−	0.772476i	$-0.719018\pi$
$13$	−59.5317	−1.27009	−0.635044	+	0.772476i	$0.719018\pi$
$14$	−30.2943	−0.578321
$15$	0	0
$16$	−35.2010	−0.550016
$17$	0.315106	0.00449555	0.00224778	−	0.999997i	$-0.499285\pi$
$17$	0.315106	0.00449555	0.00224778	+	0.999997i	$0.499285\pi$
$18$	0	0
$19$	32.8179	0.396261	0.198130	−	0.980176i	$-0.436513\pi$
$19$	32.8179	0.396261	0.198130	+	0.980176i	$0.436513\pi$
$20$	13.4660	0.150555
$21$	0	0
$22$	25.3401	0.245570
$23$	72.2810	0.655288	0.327644	−	0.944801i	$-0.393745\pi$
$23$	72.2810	0.655288	0.327644	+	0.944801i	$0.393745\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	137.140	1.03444
$27$	0	0
$28$	−35.4172	−0.239043
$29$	261.137	1.67214	0.836069	−	0.548625i	$-0.184848\pi$
$29$	261.137	1.67214	0.836069	+	0.548625i	$0.184848\pi$
$30$	0	0
$31$	155.691	0.902030	0.451015	−	0.892516i	$-0.351062\pi$
$31$	155.691	0.902030	0.451015	+	0.892516i	$0.351062\pi$
$32$	−115.976	−0.640684
$33$	0	0
$34$	−0.725893	−0.00366146
$35$	−65.7529	−0.317550
$36$	0	0
$37$	151.261	0.672087	0.336044	−	0.941846i	$-0.390911\pi$
$37$	151.261	0.672087	0.336044	+	0.941846i	$0.390911\pi$
$38$	−75.6010	−0.322740
$39$	0	0
$40$	−123.167	−0.486860
$41$	−209.059	−0.796331	−0.398165	−	0.917314i	$-0.630353\pi$
$41$	−209.059	−0.796331	−0.398165	+	0.917314i	$0.630353\pi$
$42$	0	0
$43$	111.040	0.393801	0.196901	−	0.980423i	$-0.436912\pi$
$43$	111.040	0.393801	0.196901	+	0.980423i	$0.436912\pi$
$44$	29.6252	0.101504
$45$	0	0
$46$	−166.510	−0.533708
$47$	−633.073	−1.96475	−0.982375	−	0.186923i	$-0.940148\pi$
$47$	−633.073	−1.96475	−0.982375	+	0.186923i	$0.940148\pi$
$48$	0	0
$49$	−170.062	−0.495809
$50$	−57.5912	−0.162893
$51$	0	0
$52$	160.331	0.427575
$53$	−331.397	−0.858884	−0.429442	−	0.903094i	$-0.641290\pi$
$53$	−331.397	−0.858884	−0.429442	+	0.903094i	$0.641290\pi$
$54$	0	0
$55$	55.0000	0.134840
$56$	323.943	0.773013
$57$	0	0
$58$	−601.569	−1.36189
$59$	−814.568	−1.79742	−0.898709	−	0.438545i	$-0.855494\pi$
$59$	−814.568	−1.79742	−0.898709	+	0.438545i	$0.855494\pi$
$60$	0	0
$61$	−263.071	−0.552178	−0.276089	−	0.961132i	$-0.589038\pi$
$61$	−263.071	−0.552178	−0.276089	+	0.961132i	$0.589038\pi$
$62$	−358.657	−0.734670
$63$	0	0
$64$	548.777	1.07183
$65$	297.659	0.568000
$66$	0	0
$67$	−661.609	−1.20639	−0.603197	−	0.797592i	$-0.706107\pi$
$67$	−661.609	−1.20639	−0.603197	+	0.797592i	$0.706107\pi$
$68$	−0.848644	−0.00151343
$69$	0	0
$70$	151.471	0.258633
$71$	−15.2662	−0.0255178	−0.0127589	−	0.999919i	$-0.504061\pi$
$71$	−15.2662	−0.0255178	−0.0127589	+	0.999919i	$0.504061\pi$
$72$	0	0
$73$	785.036	1.25865	0.629325	−	0.777142i	$-0.283331\pi$
$73$	785.036	1.25865	0.629325	+	0.777142i	$0.283331\pi$
$74$	−348.453	−0.547390
$75$	0	0
$76$	−88.3853	−0.133401
$77$	−144.656	−0.214092
$78$	0	0
$79$	−1075.89	−1.53224	−0.766122	−	0.642696i	$-0.777816\pi$
$79$	−1075.89	−1.53224	−0.766122	+	0.642696i	$0.777816\pi$
$80$	176.005	0.245975
$81$	0	0
$82$	481.599	0.648582
$83$	1187.89	1.57094	0.785471	−	0.618899i	$-0.212421\pi$
$83$	1187.89	1.57094	0.785471	+	0.618899i	$0.212421\pi$
$84$	0	0
$85$	−1.57553	−0.00201047
$86$	−255.797	−0.320736
$87$	0	0
$88$	−270.967	−0.328241
$89$	−403.471	−0.480538	−0.240269	−	0.970706i	$-0.577236\pi$
$89$	−403.471	−0.480538	−0.240269	+	0.970706i	$0.577236\pi$
$90$	0	0
$91$	−782.876	−0.901843
$92$	−194.667	−0.220603
$93$	0	0
$94$	1458.38	1.60022
$95$	−164.090	−0.177213
$96$	0	0
$97$	−304.398	−0.318628	−0.159314	−	0.987228i	$-0.550928\pi$
$97$	−304.398	−0.318628	−0.159314	+	0.987228i	$0.550928\pi$
$98$	391.764	0.403818
$99$	0	0
$100$	−67.3301	−0.0673301
$101$	−186.726	−0.183959	−0.0919797	−	0.995761i	$-0.529319\pi$
$101$	−186.726	−0.183959	−0.0919797	+	0.995761i	$0.529319\pi$
$102$	0	0
$103$	−453.907	−0.434221	−0.217110	−	0.976147i	$-0.569663\pi$
$103$	−453.907	−0.434221	−0.217110	+	0.976147i	$0.569663\pi$
$104$	−1466.47	−1.38268
$105$	0	0
$106$	763.422	0.699529
$107$	−759.577	−0.686271	−0.343136	−	0.939286i	$-0.611489\pi$
$107$	−759.577	−0.686271	−0.343136	+	0.939286i	$0.611489\pi$
$108$	0	0
$109$	−854.771	−0.751121	−0.375560	−	0.926798i	$-0.622550\pi$
$109$	−854.771	−0.751121	−0.375560	+	0.926798i	$0.622550\pi$
$110$	−126.701	−0.109822
$111$	0	0
$112$	−462.914	−0.390547
$113$	−1614.53	−1.34409	−0.672044	−	0.740511i	$-0.734583\pi$
$113$	−1614.53	−1.34409	−0.672044	+	0.740511i	$0.734583\pi$
$114$	0	0
$115$	−361.405	−0.293054
$116$	−703.296	−0.562926
$117$	0	0
$118$	1876.48	1.46393
$119$	4.14382	0.00319213
$120$	0	0
$121$	121.000	0.0909091
$122$	606.024	0.449728
$123$	0	0
$124$	−419.307	−0.303669
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2379.56	−1.66261	−0.831306	−	0.555816i	$-0.812406\pi$
$127$	−2379.56	−1.66261	−0.831306	+	0.555816i	$0.812406\pi$
$128$	−336.379	−0.232281
$129$	0	0
$130$	−685.701	−0.462615
$131$	1229.50	0.820012	0.410006	−	0.912083i	$-0.365527\pi$
$131$	1229.50	0.820012	0.410006	+	0.912083i	$0.365527\pi$
$132$	0	0
$133$	431.575	0.281370
$134$	1524.12	0.982564
$135$	0	0
$136$	7.76212	0.00489409
$137$	1452.18	0.905604	0.452802	−	0.891611i	$-0.350424\pi$
$137$	1452.18	0.905604	0.452802	+	0.891611i	$0.350424\pi$
$138$	0	0
$139$	−1659.52	−1.01265	−0.506325	−	0.862343i	$-0.668996\pi$
$139$	−1659.52	−1.01265	−0.506325	+	0.862343i	$0.668996\pi$
$140$	177.086	0.106903
$141$	0	0
$142$	35.1680	0.0207833
$143$	654.849	0.382946
$144$	0	0
$145$	−1305.69	−0.747803
$146$	−1808.45	−1.02512
$147$	0	0
$148$	−407.378	−0.226258
$149$	−1881.39	−1.03442	−0.517212	−	0.855857i	$-0.673030\pi$
$149$	−1881.39	−1.03442	−0.517212	+	0.855857i	$0.673030\pi$
$150$	0	0
$151$	−3322.65	−1.79068	−0.895342	−	0.445379i	$-0.853069\pi$
$151$	−3322.65	−1.79068	−0.895342	+	0.445379i	$0.853069\pi$
$152$	808.417	0.431390
$153$	0	0
$154$	333.237	0.174370
$155$	−778.455	−0.403400
$156$	0	0
$157$	3062.06	1.55655	0.778276	−	0.627922i	$-0.216095\pi$
$157$	3062.06	1.55655	0.778276	+	0.627922i	$0.216095\pi$
$158$	2478.48	1.24796
$159$	0	0
$160$	579.881	0.286523
$161$	950.536	0.465296
$162$	0	0
$163$	3278.90	1.57560	0.787801	−	0.615930i	$-0.211219\pi$
$163$	3278.90	1.57560	0.787801	+	0.615930i	$0.211219\pi$
$164$	563.039	0.268085
$165$	0	0
$166$	−2736.49	−1.27947
$167$	419.454	0.194361	0.0971805	−	0.995267i	$-0.469018\pi$
$167$	419.454	0.194361	0.0971805	+	0.995267i	$0.469018\pi$
$168$	0	0
$169$	1347.03	0.613121
$170$	3.62947	0.00163745
$171$	0	0
$172$	−299.053	−0.132573
$173$	1374.15	0.603901	0.301950	−	0.953324i	$-0.402362\pi$
$173$	1374.15	0.603901	0.301950	+	0.953324i	$0.402362\pi$
$174$	0	0
$175$	328.764	0.142013
$176$	387.211	0.165836
$177$	0	0
$178$	929.456	0.391380
$179$	−3296.73	−1.37659	−0.688295	−	0.725431i	$-0.741640\pi$
$179$	−3296.73	−1.37659	−0.688295	+	0.725431i	$0.741640\pi$
$180$	0	0
$181$	−2288.22	−0.939680	−0.469840	−	0.882752i	$-0.655688\pi$
$181$	−2288.22	−0.939680	−0.469840	+	0.882752i	$0.655688\pi$
$182$	1803.47	0.734518
$183$	0	0
$184$	1780.52	0.713381
$185$	−756.307	−0.300567
$186$	0	0
$187$	−3.46616	−0.00135546
$188$	1704.99	0.661433
$189$	0	0
$190$	378.005	0.144334
$191$	4115.73	1.55918	0.779590	−	0.626290i	$-0.215427\pi$
$191$	4115.73	1.55918	0.779590	+	0.626290i	$0.215427\pi$
$192$	0	0
$193$	3659.72	1.36493	0.682467	−	0.730916i	$-0.260907\pi$
$193$	3659.72	1.36493	0.682467	+	0.730916i	$0.260907\pi$
$194$	701.225	0.259510
$195$	0	0
$196$	458.013	0.166914
$197$	−2404.37	−0.869566	−0.434783	−	0.900535i	$-0.643175\pi$
$197$	−2404.37	−0.869566	−0.434783	+	0.900535i	$0.643175\pi$
$198$	0	0
$199$	−1370.55	−0.488221	−0.244111	−	0.969747i	$-0.578496\pi$
$199$	−1370.55	−0.488221	−0.244111	+	0.969747i	$0.578496\pi$
$200$	615.835	0.217730
$201$	0	0
$202$	430.150	0.149828
$203$	3434.11	1.18733
$204$	0	0
$205$	1045.30	0.356130
$206$	1045.64	0.353657
$207$	0	0
$208$	2095.58	0.698569
$209$	−360.997	−0.119477
$210$	0	0
$211$	934.820	0.305003	0.152502	−	0.988303i	$-0.451267\pi$
$211$	934.820	0.305003	0.152502	+	0.988303i	$0.451267\pi$
$212$	892.519	0.289144
$213$	0	0
$214$	1749.80	0.558943
$215$	−555.200	−0.176113
$216$	0	0
$217$	2047.43	0.640499
$218$	1969.09	0.611760
$219$	0	0
$220$	−148.126	−0.0453939
$221$	−18.7588	−0.00570974
$222$	0	0
$223$	2151.48	0.646071	0.323035	−	0.946387i	$-0.395297\pi$
$223$	2151.48	0.646071	0.323035	+	0.946387i	$0.395297\pi$
$224$	−1525.15	−0.454927
$225$	0	0
$226$	3719.30	1.09471
$227$	−2697.19	−0.788628	−0.394314	−	0.918976i	$-0.629018\pi$
$227$	−2697.19	−0.788628	−0.394314	+	0.918976i	$0.629018\pi$
$228$	0	0
$229$	−6218.07	−1.79433	−0.897166	−	0.441694i	$-0.854378\pi$
$229$	−6218.07	−1.79433	−0.897166	+	0.441694i	$0.854378\pi$
$230$	832.550	0.238681
$231$	0	0
$232$	6432.70	1.82038
$233$	−3082.76	−0.866774	−0.433387	−	0.901208i	$-0.642682\pi$
$233$	−3082.76	−0.866774	−0.433387	+	0.901208i	$0.642682\pi$
$234$	0	0
$235$	3165.37	0.878663
$236$	2193.80	0.605101
$237$	0	0
$238$	−9.54591	−0.00259987
$239$	−168.639	−0.0456418	−0.0228209	−	0.999740i	$-0.507265\pi$
$239$	−168.639	−0.0456418	−0.0228209	+	0.999740i	$0.507265\pi$
$240$	0	0
$241$	2588.87	0.691965	0.345982	−	0.938241i	$-0.387546\pi$
$241$	2588.87	0.691965	0.345982	+	0.938241i	$0.387546\pi$
$242$	−278.742	−0.0740421
$243$	0	0
$244$	708.504	0.185891
$245$	850.312	0.221732
$246$	0	0
$247$	−1953.71	−0.503285
$248$	3835.20	0.981997
$249$	0	0
$250$	287.956	0.0728478
$251$	1117.67	0.281061	0.140531	−	0.990076i	$-0.455119\pi$
$251$	1117.67	0.281061	0.140531	+	0.990076i	$0.455119\pi$
$252$	0	0
$253$	−795.091	−0.197577
$254$	5481.67	1.35413
$255$	0	0
$256$	−3615.31	−0.882645
$257$	−3833.21	−0.930384	−0.465192	−	0.885210i	$-0.654015\pi$
$257$	−3833.21	−0.930384	−0.465192	+	0.885210i	$0.654015\pi$
$258$	0	0
$259$	1989.17	0.477225
$260$	−801.655	−0.191217
$261$	0	0
$262$	−2832.33	−0.667870
$263$	−3099.11	−0.726614	−0.363307	−	0.931670i	$-0.618352\pi$
$263$	−3099.11	−0.726614	−0.363307	+	0.931670i	$0.618352\pi$
$264$	0	0
$265$	1656.98	0.384105
$266$	−994.196	−0.229166
$267$	0	0
$268$	1781.85	0.406133
$269$	6221.46	1.41015	0.705073	−	0.709135i	$-0.250914\pi$
$269$	6221.46	1.41015	0.705073	+	0.709135i	$0.250914\pi$
$270$	0	0
$271$	1180.29	0.264567	0.132283	−	0.991212i	$-0.457769\pi$
$271$	1180.29	0.264567	0.132283	+	0.991212i	$0.457769\pi$
$272$	−11.0921	−0.00247263
$273$	0	0
$274$	−3345.30	−0.737581
$275$	−275.000	−0.0603023
$276$	0	0
$277$	−4624.95	−1.00320	−0.501600	−	0.865100i	$-0.667255\pi$
$277$	−4624.95	−1.00320	−0.501600	+	0.865100i	$0.667255\pi$
$278$	3822.94	0.824766
$279$	0	0
$280$	−1619.72	−0.345702
$281$	−7501.11	−1.59245	−0.796225	−	0.605000i	$-0.793173\pi$
$281$	−7501.11	−1.59245	−0.796225	+	0.605000i	$0.793173\pi$
$282$	0	0
$283$	−540.526	−0.113537	−0.0567685	−	0.998387i	$-0.518080\pi$
$283$	−540.526	−0.113537	−0.0567685	+	0.998387i	$0.518080\pi$
$284$	41.1150	0.00859059
$285$	0	0
$286$	−1508.54	−0.311895
$287$	−2749.25	−0.565446
$288$	0	0
$289$	−4912.90	−0.999980
$290$	3007.84	0.609058
$291$	0	0
$292$	−2114.26	−0.423725
$293$	3385.58	0.675044	0.337522	−	0.941318i	$-0.390411\pi$
$293$	3385.58	0.675044	0.337522	+	0.941318i	$0.390411\pi$
$294$	0	0
$295$	4072.84	0.803830
$296$	3726.08	0.731669
$297$	0	0
$298$	4334.05	0.842500
$299$	−4303.01	−0.832273
$300$	0	0
$301$	1460.24	0.279624
$302$	7654.21	1.45845
$303$	0	0
$304$	−1155.23	−0.217950
$305$	1315.36	0.246941
$306$	0	0
$307$	−32.5963	−0.00605984	−0.00302992	−	0.999995i	$-0.500964\pi$
$307$	−32.5963	−0.00605984	−0.00302992	+	0.999995i	$0.500964\pi$
$308$	389.589	0.0720743
$309$	0	0
$310$	1793.29	0.328554
$311$	1071.50	0.195367	0.0976836	−	0.995218i	$-0.468857\pi$
$311$	1071.50	0.195367	0.0976836	+	0.995218i	$0.468857\pi$
$312$	0	0
$313$	4808.89	0.868417	0.434208	−	0.900812i	$-0.357028\pi$
$313$	4808.89	0.868417	0.434208	+	0.900812i	$0.357028\pi$
$314$	−7053.90	−1.26775
$315$	0	0
$316$	2897.59	0.515830
$317$	5615.88	0.995014	0.497507	−	0.867460i	$-0.334249\pi$
$317$	5615.88	0.995014	0.497507	+	0.867460i	$0.334249\pi$
$318$	0	0
$319$	−2872.51	−0.504168
$320$	−2743.88	−0.479337
$321$	0	0
$322$	−2189.70	−0.378967
$323$	10.3411	0.00178141
$324$	0	0
$325$	−1488.29	−0.254017
$326$	−7553.43	−1.28327
$327$	0	0
$328$	−5149.84	−0.866927
$329$	−8325.27	−1.39510
$330$	0	0
$331$	−1355.05	−0.225015	−0.112508	−	0.993651i	$-0.535888\pi$
$331$	−1355.05	−0.225015	−0.112508	+	0.993651i	$0.535888\pi$
$332$	−3199.24	−0.528858
$333$	0	0
$334$	−966.274	−0.158300
$335$	3308.05	0.539516
$336$	0	0
$337$	−4072.44	−0.658278	−0.329139	−	0.944281i	$-0.606759\pi$
$337$	−4072.44	−0.658278	−0.329139	+	0.944281i	$0.606759\pi$
$338$	−3103.08	−0.499364
$339$	0	0
$340$	4.24322	0.000676826 0
$341$	−1712.60	−0.271972
$342$	0	0
$343$	−6747.06	−1.06212
$344$	2735.29	0.428712
$345$	0	0
$346$	−3165.56	−0.491855
$347$	5212.03	0.806329	0.403165	−	0.915128i	$-0.367910\pi$
$347$	5212.03	0.806329	0.403165	+	0.915128i	$0.367910\pi$
$348$	0	0
$349$	8528.85	1.30813	0.654067	−	0.756436i	$-0.273061\pi$
$349$	8528.85	1.30813	0.654067	+	0.756436i	$0.273061\pi$
$350$	−757.357	−0.115664
$351$	0	0
$352$	1275.74	0.193174
$353$	9548.25	1.43967	0.719833	−	0.694147i	$-0.244218\pi$
$353$	9548.25	1.43967	0.719833	+	0.694147i	$0.244218\pi$
$354$	0	0
$355$	76.3311	0.0114119
$356$	1086.63	0.161773
$357$	0	0
$358$	7594.52	1.12118
$359$	−1150.37	−0.169121	−0.0845604	−	0.996418i	$-0.526949\pi$
$359$	−1150.37	−0.169121	−0.0845604	+	0.996418i	$0.526949\pi$
$360$	0	0
$361$	−5781.98	−0.842978
$362$	5271.26	0.765335
$363$	0	0
$364$	2108.44	0.303606
$365$	−3925.18	−0.562885
$366$	0	0
$367$	−10086.2	−1.43459	−0.717293	−	0.696772i	$-0.754619\pi$
$367$	−10086.2	−1.43459	−0.717293	+	0.696772i	$0.754619\pi$
$368$	−2544.37	−0.360419
$369$	0	0
$370$	1742.27	0.244800
$371$	−4358.06	−0.609863
$372$	0	0
$373$	192.336	0.0266991	0.0133495	−	0.999911i	$-0.495751\pi$
$373$	192.336	0.0266991	0.0133495	+	0.999911i	$0.495751\pi$
$374$	7.98482	0.00110397
$375$	0	0
$376$	−15594.7	−2.13893
$377$	−15546.0	−2.12376
$378$	0	0
$379$	5563.29	0.754003	0.377002	−	0.926213i	$-0.376955\pi$
$379$	5563.29	0.754003	0.377002	+	0.926213i	$0.376955\pi$
$380$	441.927	0.0596589
$381$	0	0
$382$	−9481.19	−1.26989
$383$	−441.480	−0.0588997	−0.0294498	−	0.999566i	$-0.509376\pi$
$383$	−441.480	−0.0588997	−0.0294498	+	0.999566i	$0.509376\pi$
$384$	0	0
$385$	723.281	0.0957450
$386$	−8430.71	−1.11169
$387$	0	0
$388$	819.804	0.107266
$389$	11903.2	1.55145	0.775726	−	0.631070i	$-0.217384\pi$
$389$	11903.2	1.55145	0.775726	+	0.631070i	$0.217384\pi$
$390$	0	0
$391$	22.7762	0.00294588
$392$	−4189.21	−0.539763
$393$	0	0
$394$	5538.83	0.708229
$395$	5379.46	0.685240
$396$	0	0
$397$	9182.52	1.16085	0.580425	−	0.814314i	$-0.302886\pi$
$397$	9182.52	1.16085	0.580425	+	0.814314i	$0.302886\pi$
$398$	3157.27	0.397638
$399$	0	0
$400$	−880.026	−0.110003
$401$	−14020.6	−1.74602	−0.873010	−	0.487703i	$-0.837835\pi$
$401$	−14020.6	−1.74602	−0.873010	+	0.487703i	$0.837835\pi$
$402$	0	0
$403$	−9268.56	−1.14566
$404$	502.890	0.0619300
$405$	0	0
$406$	−7910.97	−0.967032
$407$	−1663.88	−0.202642
$408$	0	0
$409$	−14457.1	−1.74782	−0.873909	−	0.486089i	$-0.838423\pi$
$409$	−14457.1	−1.74782	−0.873909	+	0.486089i	$0.838423\pi$
$410$	−2408.00	−0.290055
$411$	0	0
$412$	1222.46	0.146181
$413$	−10712.0	−1.27628
$414$	0	0
$415$	−5939.46	−0.702546
$416$	6904.27	0.813725
$417$	0	0
$418$	831.611	0.0973096
$419$	9808.29	1.14360	0.571798	−	0.820395i	$-0.306246\pi$
$419$	9808.29	1.14360	0.571798	+	0.820395i	$0.306246\pi$
$420$	0	0
$421$	2119.58	0.245373	0.122687	−	0.992445i	$-0.460849\pi$
$421$	2119.58	0.245373	0.122687	+	0.992445i	$0.460849\pi$
$422$	−2153.50	−0.248414
$423$	0	0
$424$	−8163.42	−0.935026
$425$	7.87765	0.000899110 0
$426$	0	0
$427$	−3459.54	−0.392081
$428$	2045.69	0.231033
$429$	0	0
$430$	1278.99	0.143438
$431$	−15401.4	−1.72125	−0.860627	−	0.509236i	$-0.829928\pi$
$431$	−15401.4	−1.72125	−0.860627	+	0.509236i	$0.829928\pi$
$432$	0	0
$433$	12399.3	1.37614	0.688072	−	0.725642i	$-0.258457\pi$
$433$	12399.3	1.37614	0.688072	+	0.725642i	$0.258457\pi$
$434$	−4716.55	−0.521663
$435$	0	0
$436$	2302.07	0.252865
$437$	2372.11	0.259665
$438$	0	0
$439$	16988.5	1.84696	0.923480	−	0.383647i	$-0.125332\pi$
$439$	16988.5	1.84696	0.923480	+	0.383647i	$0.125332\pi$
$440$	1354.84	0.146794
$441$	0	0
$442$	43.2137	0.00465037
$443$	−9276.05	−0.994850	−0.497425	−	0.867507i	$-0.665721\pi$
$443$	−9276.05	−0.994850	−0.497425	+	0.867507i	$0.665721\pi$
$444$	0	0
$445$	2017.36	0.214903
$446$	−4956.25	−0.526201
$447$	0	0
$448$	7216.73	0.761068
$449$	−14861.9	−1.56209	−0.781043	−	0.624478i	$-0.785312\pi$
$449$	−14861.9	−1.56209	−0.781043	+	0.624478i	$0.785312\pi$
$450$	0	0
$451$	2299.65	0.240103
$452$	4348.25	0.452488
$453$	0	0
$454$	6213.37	0.642308
$455$	3914.38	0.403317
$456$	0	0
$457$	8738.15	0.894428	0.447214	−	0.894427i	$-0.352416\pi$
$457$	8738.15	0.894428	0.447214	+	0.894427i	$0.352416\pi$
$458$	14324.3	1.46142
$459$	0	0
$460$	973.336	0.0986566
$461$	13.0828	0.00132175	0.000660874	−	1.00000i	$-0.499790\pi$
$461$	13.0828	0.00132175	0.000660874		1.00000i	$0.499790\pi$
$462$	0	0
$463$	12934.7	1.29833	0.649166	−	0.760647i	$-0.275118\pi$
$463$	12934.7	1.29833	0.649166	+	0.760647i	$0.275118\pi$
$464$	−9192.31	−0.919703
$465$	0	0
$466$	7101.60	0.705955
$467$	11982.9	1.18737	0.593684	−	0.804698i	$-0.297673\pi$
$467$	11982.9	1.18737	0.593684	+	0.804698i	$0.297673\pi$
$468$	0	0
$469$	−8700.54	−0.856618
$470$	−7291.89	−0.715638
$471$	0	0
$472$	−20065.6	−1.95676
$473$	−1221.44	−0.118735
$474$	0	0
$475$	820.448	0.0792521
$476$	−11.1602	−0.00107463
$477$	0	0
$478$	388.486	0.0371735
$479$	4569.04	0.435835	0.217917	−	0.975967i	$-0.430074\pi$
$479$	4569.04	0.435835	0.217917	+	0.975967i	$0.430074\pi$
$480$	0	0
$481$	−9004.86	−0.853609
$482$	−5963.84	−0.563580
$483$	0	0
$484$	−325.877	−0.0306046
$485$	1521.99	0.142495
$486$	0	0
$487$	194.834	0.0181289	0.00906443	−	0.999959i	$-0.497115\pi$
$487$	194.834	0.0181289	0.00906443	+	0.999959i	$0.497115\pi$
$488$	−6480.34	−0.601129
$489$	0	0
$490$	−1958.82	−0.180593
$491$	2362.26	0.217123	0.108561	−	0.994090i	$-0.465376\pi$
$491$	2362.26	0.217123	0.108561	+	0.994090i	$0.465376\pi$
$492$	0	0
$493$	82.2859	0.00751718
$494$	4500.66	0.409907
$495$	0	0
$496$	−5480.49	−0.496131
$497$	−200.759	−0.0181193
$498$	0	0
$499$	−10071.4	−0.903524	−0.451762	−	0.892139i	$-0.649204\pi$
$499$	−10071.4	−0.903524	−0.451762	+	0.892139i	$0.649204\pi$
$500$	336.650	0.0301109
$501$	0	0
$502$	−2574.71	−0.228914
$503$	−12093.0	−1.07197	−0.535983	−	0.844229i	$-0.680059\pi$
$503$	−12093.0	−1.07197	−0.535983	+	0.844229i	$0.680059\pi$
$504$	0	0
$505$	933.628	0.0822691
$506$	1831.61	0.160919
$507$	0	0
$508$	6408.63	0.559719
$509$	11545.7	1.00541	0.502707	−	0.864457i	$-0.332337\pi$
$509$	11545.7	1.00541	0.502707	+	0.864457i	$0.332337\pi$
$510$	0	0
$511$	10323.7	0.893722
$512$	11019.4	0.951163
$513$	0	0
$514$	8830.36	0.757764
$515$	2269.53	0.194190
$516$	0	0
$517$	6963.80	0.592394
$518$	−4582.36	−0.388682
$519$	0	0
$520$	7332.34	0.618355
$521$	−7278.47	−0.612045	−0.306023	−	0.952024i	$-0.598998\pi$
$521$	−7278.47	−0.612045	−0.306023	+	0.952024i	$0.598998\pi$
$522$	0	0
$523$	11961.3	1.00006	0.500032	−	0.866007i	$-0.333322\pi$
$523$	11961.3	1.00006	0.500032	+	0.866007i	$0.333322\pi$
$524$	−3311.28	−0.276057
$525$	0	0
$526$	7139.26	0.591800
$527$	49.0591	0.00405512
$528$	0	0
$529$	−6942.46	−0.570598
$530$	−3817.11	−0.312839
$531$	0	0
$532$	−1162.32	−0.0947234
$533$	12445.7	1.01141
$534$	0	0
$535$	3797.88	0.306910
$536$	−16297.7	−1.31334
$537$	0	0
$538$	−14332.1	−1.14851
$539$	1870.69	0.149492
$540$	0	0
$541$	−13421.1	−1.06658	−0.533288	−	0.845934i	$-0.679044\pi$
$541$	−13421.1	−1.06658	−0.533288	+	0.845934i	$0.679044\pi$
$542$	−2718.98	−0.215480
$543$	0	0
$544$	−36.5448	−0.00288023
$545$	4273.85	0.335911
$546$	0	0
$547$	4928.89	0.385273	0.192636	−	0.981270i	$-0.438296\pi$
$547$	4928.89	0.385273	0.192636	+	0.981270i	$0.438296\pi$
$548$	−3911.00	−0.304872
$549$	0	0
$550$	633.503	0.0491140
$551$	8569.99	0.662602
$552$	0	0
$553$	−14148.6	−1.08799
$554$	10654.3	0.817069
$555$	0	0
$556$	4469.41	0.340909
$557$	−21545.6	−1.63899	−0.819493	−	0.573089i	$-0.805745\pi$
$557$	−21545.6	−1.63899	−0.819493	+	0.573089i	$0.805745\pi$
$558$	0	0
$559$	−6610.41	−0.500162
$560$	2314.57	0.174658
$561$	0	0
$562$	17279.9	1.29699
$563$	−4151.42	−0.310766	−0.155383	−	0.987854i	$-0.549661\pi$
$563$	−4151.42	−0.310766	−0.155383	+	0.987854i	$0.549661\pi$
$564$	0	0
$565$	8072.64	0.601095
$566$	1245.18	0.0924716
$567$	0	0
$568$	−376.059	−0.0277800
$569$	−3669.56	−0.270362	−0.135181	−	0.990821i	$-0.543162\pi$
$569$	−3669.56	−0.270362	−0.135181	+	0.990821i	$0.543162\pi$
$570$	0	0
$571$	−14356.6	−1.05220	−0.526101	−	0.850422i	$-0.676346\pi$
$571$	−14356.6	−1.05220	−0.526101	+	0.850422i	$0.676346\pi$
$572$	−1763.64	−0.128919
$573$	0	0
$574$	6333.30	0.460535
$575$	1807.02	0.131058
$576$	0	0
$577$	−10064.1	−0.726125	−0.363063	−	0.931765i	$-0.618269\pi$
$577$	−10064.1	−0.726125	−0.363063	+	0.931765i	$0.618269\pi$
$578$	11317.6	0.814446
$579$	0	0
$580$	3516.48	0.251748
$581$	15621.5	1.11547
$582$	0	0
$583$	3645.36	0.258963
$584$	19338.1	1.37023
$585$	0	0
$586$	−7799.20	−0.549799
$587$	6069.76	0.426790	0.213395	−	0.976966i	$-0.431548\pi$
$587$	6069.76	0.426790	0.213395	+	0.976966i	$0.431548\pi$
$588$	0	0
$589$	5109.46	0.357439
$590$	−9382.39	−0.654690
$591$	0	0
$592$	−5324.56	−0.369659
$593$	10565.2	0.731637	0.365819	−	0.930686i	$-0.380789\pi$
$593$	10565.2	0.731637	0.365819	+	0.930686i	$0.380789\pi$
$594$	0	0
$595$	−20.7191	−0.00142756
$596$	5066.95	0.348239
$597$	0	0
$598$	9912.63	0.677855
$599$	4810.98	0.328166	0.164083	−	0.986447i	$-0.447534\pi$
$599$	4810.98	0.328166	0.164083	+	0.986447i	$0.447534\pi$
$600$	0	0
$601$	6860.33	0.465622	0.232811	−	0.972522i	$-0.425208\pi$
$601$	6860.33	0.465622	0.232811	+	0.972522i	$0.425208\pi$
$602$	−3363.88	−0.227743
$603$	0	0
$604$	8948.56	0.602834
$605$	−605.000	−0.0406558
$606$	0	0
$607$	−13571.4	−0.907490	−0.453745	−	0.891132i	$-0.649912\pi$
$607$	−13571.4	−0.907490	−0.453745	+	0.891132i	$0.649912\pi$
$608$	−3806.10	−0.253878
$609$	0	0
$610$	−3030.12	−0.201125
$611$	37687.9	2.49540
$612$	0	0
$613$	−1406.62	−0.0926803	−0.0463402	−	0.998926i	$-0.514756\pi$
$613$	−1406.62	−0.0926803	−0.0463402	+	0.998926i	$0.514756\pi$
$614$	75.0904	0.00493551
$615$	0	0
$616$	−3563.37	−0.233072
$617$	−7788.23	−0.508172	−0.254086	−	0.967182i	$-0.581775\pi$
$617$	−7788.23	−0.508172	−0.254086	+	0.967182i	$0.581775\pi$
$618$	0	0
$619$	−7131.66	−0.463079	−0.231539	−	0.972826i	$-0.574376\pi$
$619$	−7131.66	−0.463079	−0.231539	+	0.972826i	$0.574376\pi$
$620$	2096.54	0.135805
$621$	0	0
$622$	−2468.36	−0.159119
$623$	−5305.88	−0.341213
$624$	0	0
$625$	625.000	0.0400000
$626$	−11078.0	−0.707293
$627$	0	0
$628$	−8246.73	−0.524014
$629$	47.6634	0.00302140
$630$	0	0
$631$	28826.0	1.81862	0.909308	−	0.416124i	$-0.136612\pi$
$631$	28826.0	1.81862	0.909308	+	0.416124i	$0.136612\pi$
$632$	−26502.8	−1.66808
$633$	0	0
$634$	−12937.0	−0.810402
$635$	11897.8	0.743542
$636$	0	0
$637$	10124.1	0.629720
$638$	6617.26	0.410627
$639$	0	0
$640$	1681.90	0.103879
$641$	−11535.8	−0.710821	−0.355410	−	0.934710i	$-0.615659\pi$
$641$	−11535.8	−0.710821	−0.355410	+	0.934710i	$0.615659\pi$
$642$	0	0
$643$	1123.87	0.0689283	0.0344642	−	0.999406i	$-0.489028\pi$
$643$	1123.87	0.0689283	0.0344642	+	0.999406i	$0.489028\pi$
$644$	−2559.99	−0.156642
$645$	0	0
$646$	−23.8223	−0.00145089
$647$	−4567.53	−0.277539	−0.138770	−	0.990325i	$-0.544315\pi$
$647$	−4567.53	−0.277539	−0.138770	+	0.990325i	$0.544315\pi$
$648$	0	0
$649$	8960.24	0.541942
$650$	3428.51	0.206888
$651$	0	0
$652$	−8830.74	−0.530427
$653$	−19663.0	−1.17836	−0.589181	−	0.808001i	$-0.700550\pi$
$653$	−19663.0	−1.17836	−0.589181	+	0.808001i	$0.700550\pi$
$654$	0	0
$655$	−6147.48	−0.366721
$656$	7359.10	0.437995
$657$	0	0
$658$	19178.5	1.13626
$659$	5440.78	0.321612	0.160806	−	0.986986i	$-0.448591\pi$
$659$	5440.78	0.321612	0.160806	+	0.986986i	$0.448591\pi$
$660$	0	0
$661$	−14661.0	−0.862703	−0.431351	−	0.902184i	$-0.641963\pi$
$661$	−14661.0	−0.862703	−0.431351	+	0.902184i	$0.641963\pi$
$662$	3121.55	0.183267
$663$	0	0
$664$	29261.8	1.71021
$665$	−2157.87	−0.125833
$666$	0	0
$667$	18875.3	1.09573
$668$	−1129.67	−0.0654317
$669$	0	0
$670$	−7620.58	−0.439416
$671$	2893.78	0.166488
$672$	0	0
$673$	7054.34	0.404049	0.202024	−	0.979380i	$-0.435248\pi$
$673$	7054.34	0.404049	0.202024	+	0.979380i	$0.435248\pi$
$674$	9381.46	0.536143
$675$	0	0
$676$	−3627.82	−0.206407
$677$	−16034.4	−0.910271	−0.455136	−	0.890422i	$-0.650409\pi$
$677$	−16034.4	−0.910271	−0.455136	+	0.890422i	$0.650409\pi$
$678$	0	0
$679$	−4003.00	−0.226246
$680$	−38.8106	−0.00218870
$681$	0	0
$682$	3945.23	0.221511
$683$	11575.3	0.648486	0.324243	−	0.945974i	$-0.394890\pi$
$683$	11575.3	0.648486	0.324243	+	0.945974i	$0.394890\pi$
$684$	0	0
$685$	−7260.88	−0.404999
$686$	15542.9	0.865057
$687$	0	0
$688$	−3908.73	−0.216597
$689$	19728.6	1.09086
$690$	0	0
$691$	−14758.9	−0.812525	−0.406262	−	0.913756i	$-0.633168\pi$
$691$	−14758.9	−0.812525	−0.406262	+	0.913756i	$0.633168\pi$
$692$	−3700.87	−0.203303
$693$	0	0
$694$	−12006.7	−0.656725
$695$	8297.58	0.452871
$696$	0	0
$697$	−65.8758	−0.00357995
$698$	−19647.5	−1.06543
$699$	0	0
$700$	−885.429	−0.0478087
$701$	14332.4	0.772224	0.386112	−	0.922452i	$-0.373818\pi$
$701$	14332.4	0.772224	0.386112	+	0.922452i	$0.373818\pi$
$702$	0	0
$703$	4964.09	0.266322
$704$	−6036.55	−0.323169
$705$	0	0
$706$	−21995.8	−1.17255
$707$	−2455.55	−0.130623
$708$	0	0
$709$	−14159.3	−0.750020	−0.375010	−	0.927021i	$-0.622361\pi$
$709$	−14159.3	−0.750020	−0.375010	+	0.927021i	$0.622361\pi$
$710$	−175.840	−0.00929459
$711$	0	0
$712$	−9938.86	−0.523138
$713$	11253.5	0.591089
$714$	0	0
$715$	−3274.25	−0.171259
$716$	8878.77	0.463429
$717$	0	0
$718$	2650.05	0.137743
$719$	−2750.64	−0.142672	−0.0713362	−	0.997452i	$-0.522726\pi$
$719$	−2750.64	−0.142672	−0.0713362	+	0.997452i	$0.522726\pi$
$720$	0	0
$721$	−5969.13	−0.308325
$722$	13319.7	0.686574
$723$	0	0
$724$	6162.64	0.316344
$725$	6528.43	0.334428
$726$	0	0
$727$	17477.0	0.891591	0.445795	−	0.895135i	$-0.352921\pi$
$727$	17477.0	0.891591	0.445795	+	0.895135i	$0.352921\pi$
$728$	−19284.9	−0.981794
$729$	0	0
$730$	9042.23	0.458449
$731$	34.9894	0.00177035
$732$	0	0
$733$	−34334.1	−1.73010	−0.865048	−	0.501690i	$-0.832712\pi$
$733$	−34334.1	−1.73010	−0.865048	+	0.501690i	$0.832712\pi$
$734$	23235.0	1.16842
$735$	0	0
$736$	−8382.87	−0.419833
$737$	7277.70	0.363742
$738$	0	0
$739$	2372.52	0.118098	0.0590491	−	0.998255i	$-0.481193\pi$
$739$	2372.52	0.118098	0.0590491	+	0.998255i	$0.481193\pi$
$740$	2036.89	0.101186
$741$	0	0
$742$	10039.4	0.496710
$743$	24756.2	1.22236	0.611182	−	0.791490i	$-0.290694\pi$
$743$	24756.2	1.22236	0.611182	+	0.791490i	$0.290694\pi$
$744$	0	0
$745$	9406.93	0.462608
$746$	−443.073	−0.0217454
$747$	0	0
$748$	9.33508	0.000456316 0
$749$	−9988.87	−0.487297
$750$	0	0
$751$	3961.78	0.192500	0.0962500	−	0.995357i	$-0.469315\pi$
$751$	3961.78	0.192500	0.0962500	+	0.995357i	$0.469315\pi$
$752$	22284.8	1.08064
$753$	0	0
$754$	35812.4	1.72972
$755$	16613.2	0.800818
$756$	0	0
$757$	−6139.00	−0.294750	−0.147375	−	0.989081i	$-0.547082\pi$
$757$	−6139.00	−0.294750	−0.147375	+	0.989081i	$0.547082\pi$
$758$	−12815.9	−0.614108
$759$	0	0
$760$	−4042.08	−0.192923
$761$	−9191.35	−0.437827	−0.218913	−	0.975744i	$-0.570251\pi$
$761$	−9191.35	−0.437827	−0.218913	+	0.975744i	$0.570251\pi$
$762$	0	0
$763$	−11240.7	−0.533344
$764$	−11084.5	−0.524899
$765$	0	0
$766$	1017.02	0.0479716
$767$	48492.6	2.28288
$768$	0	0
$769$	−2314.79	−0.108548	−0.0542741	−	0.998526i	$-0.517284\pi$
$769$	−2314.79	−0.108548	−0.0542741	+	0.998526i	$0.517284\pi$
$770$	−1666.19	−0.0779808
$771$	0	0
$772$	−9856.36	−0.459505
$773$	−7953.06	−0.370054	−0.185027	−	0.982733i	$-0.559237\pi$
$773$	−7953.06	−0.370054	−0.185027	+	0.982733i	$0.559237\pi$
$774$	0	0
$775$	3892.28	0.180406
$776$	−7498.34	−0.346875
$777$	0	0
$778$	−27420.7	−1.26360
$779$	−6860.89	−0.315555
$780$	0	0
$781$	167.928	0.00769392
$782$	−52.4683	−0.00239931
$783$	0	0
$784$	5986.38	0.272703
$785$	−15310.3	−0.696111
$786$	0	0
$787$	−11035.5	−0.499841	−0.249920	−	0.968266i	$-0.580404\pi$
$787$	−11035.5	−0.499841	−0.249920	+	0.968266i	$0.580404\pi$
$788$	6475.46	0.292740
$789$	0	0
$790$	−12392.4	−0.558103
$791$	−21232.0	−0.954389
$792$	0	0
$793$	15661.1	0.701314
$794$	−21153.3	−0.945470
$795$	0	0
$796$	3691.18	0.164360
$797$	18269.9	0.811984	0.405992	−	0.913877i	$-0.366926\pi$
$797$	18269.9	0.811984	0.405992	+	0.913877i	$0.366926\pi$
$798$	0	0
$799$	−199.485	−0.00883263
$800$	−2899.41	−0.128137
$801$	0	0
$802$	32298.5	1.42207
$803$	−8635.39	−0.379497
$804$	0	0
$805$	−4752.68	−0.208087
$806$	21351.5	0.933095
$807$	0	0
$808$	−4599.68	−0.200268
$809$	−26864.0	−1.16747	−0.583737	−	0.811943i	$-0.698410\pi$
$809$	−26864.0	−1.16747	−0.583737	+	0.811943i	$0.698410\pi$
$810$	0	0
$811$	−18456.1	−0.799114	−0.399557	−	0.916708i	$-0.630836\pi$
$811$	−18456.1	−0.799114	−0.399557	+	0.916708i	$0.630836\pi$
$812$	−9248.74	−0.399713
$813$	0	0
$814$	3832.99	0.165044
$815$	−16394.5	−0.704631
$816$	0	0
$817$	3644.11	0.156048
$818$	33304.1	1.42353
$819$	0	0
$820$	−2815.19	−0.119891
$821$	−32876.4	−1.39756	−0.698779	−	0.715337i	$-0.746273\pi$
$821$	−32876.4	−1.39756	−0.698779	+	0.715337i	$0.746273\pi$
$822$	0	0
$823$	−15535.6	−0.658004	−0.329002	−	0.944329i	$-0.606712\pi$
$823$	−15535.6	−0.658004	−0.329002	+	0.944329i	$0.606712\pi$
$824$	−11181.3	−0.472716
$825$	0	0
$826$	24676.8	1.03948
$827$	7730.74	0.325059	0.162530	−	0.986704i	$-0.448035\pi$
$827$	7730.74	0.325059	0.162530	+	0.986704i	$0.448035\pi$
$828$	0	0
$829$	44577.2	1.86759	0.933794	−	0.357810i	$-0.116476\pi$
$829$	44577.2	1.86759	0.933794	+	0.357810i	$0.116476\pi$
$830$	13682.4	0.572198
$831$	0	0
$832$	−32669.6	−1.36132
$833$	−53.5877	−0.00222893
$834$	0	0
$835$	−2097.27	−0.0869209
$836$	972.239	0.0402220
$837$	0	0
$838$	−22594.9	−0.931416
$839$	−20585.6	−0.847073	−0.423537	−	0.905879i	$-0.639211\pi$
$839$	−20585.6	−0.847073	−0.423537	+	0.905879i	$0.639211\pi$
$840$	0	0
$841$	43803.7	1.79604
$842$	−4882.77	−0.199847
$843$	0	0
$844$	−2517.66	−0.102679
$845$	−6735.14	−0.274196
$846$	0	0
$847$	1591.22	0.0645513
$848$	11665.5	0.472400
$849$	0	0
$850$	−18.1473	−0.000732292 0
$851$	10933.3	0.440411
$852$	0	0
$853$	35960.9	1.44347	0.721735	−	0.692170i	$-0.243345\pi$
$853$	35960.9	1.44347	0.721735	+	0.692170i	$0.243345\pi$
$854$	7969.56	0.319336
$855$	0	0
$856$	−18710.9	−0.747111
$857$	−10177.0	−0.405646	−0.202823	−	0.979215i	$-0.565012\pi$
$857$	−10177.0	−0.405646	−0.202823	+	0.979215i	$0.565012\pi$
$858$	0	0
$859$	37146.9	1.47548	0.737740	−	0.675085i	$-0.235893\pi$
$859$	37146.9	1.47548	0.737740	+	0.675085i	$0.235893\pi$
$860$	1495.27	0.0592886
$861$	0	0
$862$	35479.5	1.40190
$863$	−176.455	−0.00696015	−0.00348007	−	0.999994i	$-0.501108\pi$
$863$	−176.455	−0.00696015	−0.00348007	+	0.999994i	$0.501108\pi$
$864$	0	0
$865$	−6870.76	−0.270073
$866$	−28563.5	−1.12082
$867$	0	0
$868$	−5514.13	−0.215624
$869$	11834.8	0.461989
$870$	0	0
$871$	39386.7	1.53223
$872$	−21055.9	−0.817709
$873$	0	0
$874$	−5464.51	−0.211487
$875$	−1643.82	−0.0635101
$876$	0	0
$877$	3639.38	0.140129	0.0700644	−	0.997542i	$-0.477680\pi$
$877$	3639.38	0.140129	0.0700644	+	0.997542i	$0.477680\pi$
$878$	−39135.5	−1.50428
$879$	0	0
$880$	−1936.06	−0.0741642
$881$	31931.9	1.22113	0.610565	−	0.791967i	$-0.290943\pi$
$881$	31931.9	1.22113	0.610565	+	0.791967i	$0.290943\pi$
$882$	0	0
$883$	5974.79	0.227710	0.113855	−	0.993497i	$-0.463680\pi$
$883$	5974.79	0.227710	0.113855	+	0.993497i	$0.463680\pi$
$884$	50.5212	0.00192219
$885$	0	0
$886$	21368.8	0.810268
$887$	−7750.43	−0.293387	−0.146693	−	0.989182i	$-0.546863\pi$
$887$	−7750.43	−0.293387	−0.146693	+	0.989182i	$0.546863\pi$
$888$	0	0
$889$	−31292.5	−1.18056
$890$	−4647.28	−0.175030
$891$	0	0
$892$	−5794.37	−0.217500
$893$	−20776.2	−0.778553
$894$	0	0
$895$	16483.7	0.615629
$896$	−4423.58	−0.164935
$897$	0	0
$898$	34236.6	1.27226
$899$	40656.7	1.50832
$900$	0	0
$901$	−104.425	−0.00386116
$902$	−5297.59	−0.195555
$903$	0	0
$904$	−39771.3	−1.46324
$905$	11441.1	0.420238
$906$	0	0
$907$	25477.1	0.932693	0.466346	−	0.884602i	$-0.345570\pi$
$907$	25477.1	0.932693	0.466346	+	0.884602i	$0.345570\pi$
$908$	7264.07	0.265492
$909$	0	0
$910$	−9017.36	−0.328486
$911$	19229.5	0.699342	0.349671	−	0.936873i	$-0.386293\pi$
$911$	19229.5	0.699342	0.349671	+	0.936873i	$0.386293\pi$
$912$	0	0
$913$	−13066.8	−0.473657
$914$	−20129.6	−0.728478
$915$	0	0
$916$	16746.5	0.604062
$917$	16168.6	0.582261
$918$	0	0
$919$	−3232.39	−0.116025	−0.0580124	−	0.998316i	$-0.518476\pi$
$919$	−3232.39	−0.116025	−0.0580124	+	0.998316i	$0.518476\pi$
$920$	−8902.62	−0.319033
$921$	0	0
$922$	−30.1381	−0.00107651
$923$	908.824	0.0324099
$924$	0	0
$925$	3781.54	0.134417
$926$	−29797.0	−1.05744
$927$	0	0
$928$	−30285.7	−1.07131
$929$	−41921.6	−1.48052	−0.740260	−	0.672321i	$-0.765298\pi$
$929$	−41921.6	−1.48052	−0.740260	+	0.672321i	$0.765298\pi$
$930$	0	0
$931$	−5581.10	−0.196470
$932$	8302.50	0.291800
$933$	0	0
$934$	−27604.3	−0.967067
$935$	17.3308	0.000606180 0
$936$	0	0
$937$	34983.9	1.21972	0.609858	−	0.792511i	$-0.291227\pi$
$937$	34983.9	1.21972	0.609858	+	0.792511i	$0.291227\pi$
$938$	20043.0	0.697683
$939$	0	0
$940$	−8524.97	−0.295802
$941$	20711.8	0.717519	0.358760	−	0.933430i	$-0.383200\pi$
$941$	20711.8	0.717519	0.358760	+	0.933430i	$0.383200\pi$
$942$	0	0
$943$	−15111.0	−0.521826
$944$	28673.6	0.988609
$945$	0	0
$946$	2813.77	0.0967056
$947$	23659.4	0.811854	0.405927	−	0.913905i	$-0.366949\pi$
$947$	23659.4	0.811854	0.405927	+	0.913905i	$0.366949\pi$
$948$	0	0
$949$	−46734.5	−1.59860
$950$	−1890.03	−0.0645479
$951$	0	0
$952$	102.076	0.00347512
$953$	−10969.7	−0.372869	−0.186434	−	0.982467i	$-0.559693\pi$
$953$	−10969.7	−0.372869	−0.186434	+	0.982467i	$0.559693\pi$
$954$	0	0
$955$	−20578.6	−0.697287
$956$	454.180	0.0153653
$957$	0	0
$958$	−10525.5	−0.354971
$959$	19096.9	0.643037
$960$	0	0
$961$	−5551.31	−0.186342
$962$	20744.0	0.695233
$963$	0	0
$964$	−6972.34	−0.232950
$965$	−18298.6	−0.610417
$966$	0	0
$967$	−38508.5	−1.28061	−0.640305	−	0.768121i	$-0.721192\pi$
$967$	−38508.5	−1.28061	−0.640305	+	0.768121i	$0.721192\pi$
$968$	2980.64	0.0989684
$969$	0	0
$970$	−3506.13	−0.116057
$971$	57757.1	1.90887	0.954436	−	0.298417i	$-0.0964587\pi$
$971$	57757.1	1.90887	0.954436	+	0.298417i	$0.0964587\pi$
$972$	0	0
$973$	−21823.6	−0.719046
$974$	−448.828	−0.0147653
$975$	0	0
$976$	9260.38	0.303707
$977$	−22681.9	−0.742742	−0.371371	−	0.928485i	$-0.621112\pi$
$977$	−22681.9	−0.742742	−0.371371	+	0.928485i	$0.621112\pi$
$978$	0	0
$979$	4438.18	0.144888
$980$	−2290.06	−0.0746463
$981$	0	0
$982$	−5441.82	−0.176839
$983$	−20920.1	−0.678787	−0.339394	−	0.940644i	$-0.610222\pi$
$983$	−20920.1	−0.678787	−0.339394	+	0.940644i	$0.610222\pi$
$984$	0	0
$985$	12021.9	0.388882
$986$	−189.558	−0.00612247
$987$	0	0
$988$	5261.73	0.169431
$989$	8026.08	0.258053
$990$	0	0
$991$	10809.1	0.346480	0.173240	−	0.984880i	$-0.444576\pi$
$991$	10809.1	0.346480	0.173240	+	0.984880i	$0.444576\pi$
$992$	−18056.5	−0.577916
$993$	0	0
$994$	462.479	0.0147575
$995$	6852.77	0.218339
$996$	0	0
$997$	−26066.0	−0.828001	−0.414001	−	0.910277i	$-0.635869\pi$
$997$	−26066.0	−0.828001	−0.414001	+	0.910277i	$0.635869\pi$
$998$	23201.0	0.735886
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.n.1.2		4
3.2	odd	2		55.4.a.d.1.3	✓	4
5.4	even	2		2475.4.a.bc.1.3		4
12.11	even	2		880.4.a.z.1.2		4
15.2	even	4		275.4.b.e.199.6		8
15.8	even	4		275.4.b.e.199.3		8
15.14	odd	2		275.4.a.e.1.2		4
33.32	even	2		605.4.a.j.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.a.d.1.3	✓	4	3.2	odd	2
275.4.a.e.1.2		4	15.14	odd	2
275.4.b.e.199.3		8	15.8	even	4
275.4.b.e.199.6		8	15.2	even	4
495.4.a.n.1.2		4	1.1	even	1	trivial
605.4.a.j.1.2		4	33.32	even	2
880.4.a.z.1.2		4	12.11	even	2
2475.4.a.bc.1.3		4	5.4	even	2

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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30365 q^{2} -2.69320 q^{4} -5.00000 q^{5} +13.1506 q^{7} +24.6334 q^{8} +O(q^{10})\)	\(q-2.30365 q^{2} -2.69320 q^{4} -5.00000 q^{5} +13.1506 q^{7} +24.6334 q^{8} +11.5182 q^{10} -11.0000 q^{11} -59.5317 q^{13} -30.2943 q^{14} -35.2010 q^{16} +0.315106 q^{17} +32.8179 q^{19} +13.4660 q^{20} +25.3401 q^{22} +72.2810 q^{23} +25.0000 q^{25} +137.140 q^{26} -35.4172 q^{28} +261.137 q^{29} +155.691 q^{31} -115.976 q^{32} -0.725893 q^{34} -65.7529 q^{35} +151.261 q^{37} -75.6010 q^{38} -123.167 q^{40} -209.059 q^{41} +111.040 q^{43} +29.6252 q^{44} -166.510 q^{46} -633.073 q^{47} -170.062 q^{49} -57.5912 q^{50} +160.331 q^{52} -331.397 q^{53} +55.0000 q^{55} +323.943 q^{56} -601.569 q^{58} -814.568 q^{59} -263.071 q^{61} -358.657 q^{62} +548.777 q^{64} +297.659 q^{65} -661.609 q^{67} -0.848644 q^{68} +151.471 q^{70} -15.2662 q^{71} +785.036 q^{73} -348.453 q^{74} -88.3853 q^{76} -144.656 q^{77} -1075.89 q^{79} +176.005 q^{80} +481.599 q^{82} +1187.89 q^{83} -1.57553 q^{85} -255.797 q^{86} -270.967 q^{88} -403.471 q^{89} -782.876 q^{91} -194.667 q^{92} +1458.38 q^{94} -164.090 q^{95} -304.398 q^{97} +391.764 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 19 q^{4} - 20 q^{5} + 9 q^{7} - 33 q^{8}+O(q^{10}) \) 4 * q - q^2 + 19 * q^4 - 20 * q^5 + 9 * q^7 - 33 * q^8	\( 4 q - q^{2} + 19 q^{4} - 20 q^{5} + 9 q^{7} - 33 q^{8} + 5 q^{10} - 44 q^{11} + 70 q^{13} + 49 q^{14} - 37 q^{16} - 103 q^{17} - 205 q^{19} - 95 q^{20} + 11 q^{22} + 56 q^{23} + 100 q^{25} + 86 q^{26} - 551 q^{28} + 79 q^{29} + 49 q^{31} - 225 q^{32} - 939 q^{34} - 45 q^{35} + 289 q^{37} - 145 q^{38} + 165 q^{40} - 736 q^{41} - 152 q^{43} - 209 q^{44} - 334 q^{46} - 412 q^{47} + 37 q^{49} - 25 q^{50} + 1598 q^{52} - 1685 q^{53} + 220 q^{55} + 257 q^{56} + 609 q^{58} + 842 q^{59} - 1097 q^{61} - 1359 q^{62} - 165 q^{64} - 350 q^{65} - 122 q^{67} + 757 q^{68} - 245 q^{70} + 521 q^{71} - 590 q^{73} - 3257 q^{74} - 2825 q^{76} - 99 q^{77} - 1118 q^{79} + 185 q^{80} - 402 q^{82} + 122 q^{83} + 515 q^{85} - 3452 q^{86} + 363 q^{88} + 181 q^{89} - 2190 q^{91} - 430 q^{92} + 1034 q^{94} + 1025 q^{95} + 1474 q^{97} + 872 q^{98}+O(q^{100}) \) 4 * q - q^2 + 19 * q^4 - 20 * q^5 + 9 * q^7 - 33 * q^8 + 5 * q^10 - 44 * q^11 + 70 * q^13 + 49 * q^14 - 37 * q^16 - 103 * q^17 - 205 * q^19 - 95 * q^20 + 11 * q^22 + 56 * q^23 + 100 * q^25 + 86 * q^26 - 551 * q^28 + 79 * q^29 + 49 * q^31 - 225 * q^32 - 939 * q^34 - 45 * q^35 + 289 * q^37 - 145 * q^38 + 165 * q^40 - 736 * q^41 - 152 * q^43 - 209 * q^44 - 334 * q^46 - 412 * q^47 + 37 * q^49 - 25 * q^50 + 1598 * q^52 - 1685 * q^53 + 220 * q^55 + 257 * q^56 + 609 * q^58 + 842 * q^59 - 1097 * q^61 - 1359 * q^62 - 165 * q^64 - 350 * q^65 - 122 * q^67 + 757 * q^68 - 245 * q^70 + 521 * q^71 - 590 * q^73 - 3257 * q^74 - 2825 * q^76 - 99 * q^77 - 1118 * q^79 + 185 * q^80 - 402 * q^82 + 122 * q^83 + 515 * q^85 - 3452 * q^86 + 363 * q^88 + 181 * q^89 - 2190 * q^91 - 430 * q^92 + 1034 * q^94 + 1025 * q^95 + 1474 * q^97 + 872 * q^98

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