LMFDB - Embedded newform 1815.4.a.bk.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, 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("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1815.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:	$107.088466660$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 79x^{10} + 2212x^{8} - 26055x^{6} + 118317x^{4} - 109056x^{2} + 12996 $ x^12 - 79x^10 + 2212x^8 - 26055x^6 + 118317x^4 - 109056*x^2 + 12996
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.01393$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.01393 q^{2} +3.00000 q^{3} -6.97196 q^{4} +5.00000 q^{5} +3.04178 q^{6} +26.7663 q^{7} -15.1804 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+1.01393 q^{2} +3.00000 q^{3} -6.97196 q^{4} +5.00000 q^{5} +3.04178 q^{6} +26.7663 q^{7} -15.1804 q^{8} +9.00000 q^{9} +5.06963 q^{10} -20.9159 q^{12} +69.8371 q^{13} +27.1390 q^{14} +15.0000 q^{15} +40.3838 q^{16} -52.5972 q^{17} +9.12533 q^{18} +34.9915 q^{19} -34.8598 q^{20} +80.2989 q^{21} +4.15717 q^{23} -45.5413 q^{24} +25.0000 q^{25} +70.8096 q^{26} +27.0000 q^{27} -186.613 q^{28} +88.6779 q^{29} +15.2089 q^{30} -217.233 q^{31} +162.390 q^{32} -53.3296 q^{34} +133.832 q^{35} -62.7476 q^{36} +280.641 q^{37} +35.4787 q^{38} +209.511 q^{39} -75.9022 q^{40} +354.090 q^{41} +81.4171 q^{42} +121.619 q^{43} +45.0000 q^{45} +4.21506 q^{46} -138.480 q^{47} +121.151 q^{48} +373.435 q^{49} +25.3481 q^{50} -157.791 q^{51} -486.901 q^{52} -106.111 q^{53} +27.3760 q^{54} -406.324 q^{56} +104.974 q^{57} +89.9128 q^{58} -355.070 q^{59} -104.579 q^{60} -274.034 q^{61} -220.258 q^{62} +240.897 q^{63} -158.419 q^{64} +349.186 q^{65} -578.790 q^{67} +366.705 q^{68} +12.4715 q^{69} +135.695 q^{70} +680.655 q^{71} -136.624 q^{72} -891.010 q^{73} +284.549 q^{74} +75.0000 q^{75} -243.959 q^{76} +212.429 q^{78} -614.496 q^{79} +201.919 q^{80} +81.0000 q^{81} +359.021 q^{82} +985.150 q^{83} -559.840 q^{84} -262.986 q^{85} +123.313 q^{86} +266.034 q^{87} +1470.02 q^{89} +45.6266 q^{90} +1869.28 q^{91} -28.9836 q^{92} -651.700 q^{93} -140.408 q^{94} +174.957 q^{95} +487.169 q^{96} -60.0685 q^{97} +378.635 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 36 q^{3} + 62 q^{4} + 60 q^{5} + 108 q^{9}+O(q^{10}) $ 12 * q + 36 * q^3 + 62 * q^4 + 60 * q^5 + 108 * q^9	$ 12 q + 36 q^{3} + 62 q^{4} + 60 q^{5} + 108 q^{9} + 186 q^{12} + 16 q^{14} + 180 q^{15} + 610 q^{16} + 310 q^{20} - 44 q^{23} + 300 q^{25} + 1016 q^{26} + 324 q^{27} + 288 q^{31} + 710 q^{34} + 558 q^{36} - 488 q^{37} + 872 q^{38} + 48 q^{42} + 540 q^{45} + 264 q^{47} + 1830 q^{48} + 140 q^{49} + 1144 q^{53} + 3172 q^{56} - 488 q^{58} + 1400 q^{59} + 930 q^{60} + 3088 q^{64} + 2032 q^{67} - 132 q^{69} + 80 q^{70} + 4852 q^{71} + 900 q^{75} + 3048 q^{78} + 3050 q^{80} + 972 q^{81} + 816 q^{82} - 2232 q^{86} + 4932 q^{89} + 7336 q^{91} - 2526 q^{92} + 864 q^{93} + 1436 q^{97}+O(q^{100}) $ 12 * q + 36 * q^3 + 62 * q^4 + 60 * q^5 + 108 * q^9 + 186 * q^12 + 16 * q^14 + 180 * q^15 + 610 * q^16 + 310 * q^20 - 44 * q^23 + 300 * q^25 + 1016 * q^26 + 324 * q^27 + 288 * q^31 + 710 * q^34 + 558 * q^36 - 488 * q^37 + 872 * q^38 + 48 * q^42 + 540 * q^45 + 264 * q^47 + 1830 * q^48 + 140 * q^49 + 1144 * q^53 + 3172 * q^56 - 488 * q^58 + 1400 * q^59 + 930 * q^60 + 3088 * q^64 + 2032 * q^67 - 132 * q^69 + 80 * q^70 + 4852 * q^71 + 900 * q^75 + 3048 * q^78 + 3050 * q^80 + 972 * q^81 + 816 * q^82 - 2232 * q^86 + 4932 * q^89 + 7336 * q^91 - 2526 * q^92 + 864 * q^93 + 1436 * q^97

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 $

$2$	1.01393	0.358477	0.179238	−	0.983806i	$-0.442637\pi$
$2$	1.01393	0.358477	0.179238	+	0.983806i	$0.442637\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−6.97196	−0.871494
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	3.04178	0.206967
$7$	26.7663	1.44524	0.722622	−	0.691243i	$-0.242937\pi$
$7$	26.7663	1.44524	0.722622	+	0.691243i	$0.242937\pi$
$8$	−15.1804	−0.670887
$9$	9.00000	0.333333
$10$	5.06963	0.160316
$11$	0	0
$11$	0	0
$12$	−20.9159	−0.503158
$13$	69.8371	1.48995	0.744975	−	0.667093i	$-0.232462\pi$
$13$	69.8371	1.48995	0.744975	+	0.667093i	$0.232462\pi$
$14$	27.1390	0.518087
$15$	15.0000	0.258199
$16$	40.3838	0.630997
$17$	−52.5972	−0.750393	−0.375197	−	0.926945i	$-0.622425\pi$
$17$	−52.5972	−0.750393	−0.375197	+	0.926945i	$0.622425\pi$
$18$	9.12533	0.119492
$19$	34.9915	0.422505	0.211253	−	0.977432i	$-0.432246\pi$
$19$	34.9915	0.422505	0.211253	+	0.977432i	$0.432246\pi$
$20$	−34.8598	−0.389744
$21$	80.2989	0.834412
$22$	0	0
$23$	4.15717	0.0376883	0.0188441	−	0.999822i	$-0.494001\pi$
$23$	4.15717	0.0376883	0.0188441	+	0.999822i	$0.494001\pi$
$24$	−45.5413	−0.387337
$25$	25.0000	0.200000
$26$	70.8096	0.534112
$27$	27.0000	0.192450
$28$	−186.613	−1.25952
$29$	88.6779	0.567830	0.283915	−	0.958849i	$-0.408367\pi$
$29$	88.6779	0.567830	0.283915	+	0.958849i	$0.408367\pi$
$30$	15.2089	0.0925583
$31$	−217.233	−1.25859	−0.629295	−	0.777167i	$-0.716656\pi$
$31$	−217.233	−1.25859	−0.629295	+	0.777167i	$0.716656\pi$
$32$	162.390	0.897085
$33$	0	0
$34$	−53.3296	−0.268998
$35$	133.832	0.646333
$36$	−62.7476	−0.290498
$37$	280.641	1.24695	0.623474	−	0.781844i	$-0.285721\pi$
$37$	280.641	1.24695	0.623474	+	0.781844i	$0.285721\pi$
$38$	35.4787	0.151458
$39$	209.511	0.860223
$40$	−75.9022	−0.300030
$41$	354.090	1.34877	0.674385	−	0.738380i	$-0.264409\pi$
$41$	354.090	1.34877	0.674385	+	0.738380i	$0.264409\pi$
$42$	81.4171	0.299117
$43$	121.619	0.431320	0.215660	−	0.976468i	$-0.430810\pi$
$43$	121.619	0.431320	0.215660	+	0.976468i	$0.430810\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	4.21506	0.0135104
$47$	−138.480	−0.429773	−0.214887	−	0.976639i	$-0.568938\pi$
$47$	−138.480	−0.429773	−0.214887	+	0.976639i	$0.568938\pi$
$48$	121.151	0.364306
$49$	373.435	1.08873
$50$	25.3481	0.0716953
$51$	−157.791	−0.433240
$52$	−486.901	−1.29848
$53$	−106.111	−0.275008	−0.137504	−	0.990501i	$-0.543908\pi$
$53$	−106.111	−0.275008	−0.137504	+	0.990501i	$0.543908\pi$
$54$	27.3760	0.0689889
$55$	0	0
$56$	−406.324	−0.969596
$57$	104.974	0.243933
$58$	89.9128	0.203554
$59$	−355.070	−0.783495	−0.391747	−	0.920073i	$-0.628129\pi$
$59$	−355.070	−0.783495	−0.391747	+	0.920073i	$0.628129\pi$
$60$	−104.579	−0.225019
$61$	−274.034	−0.575189	−0.287594	−	0.957752i	$-0.592855\pi$
$61$	−274.034	−0.575189	−0.287594	+	0.957752i	$0.592855\pi$
$62$	−220.258	−0.451175
$63$	240.897	0.481748
$64$	−158.419	−0.309413
$65$	349.186	0.666325
$66$	0	0
$67$	−578.790	−1.05538	−0.527690	−	0.849437i	$-0.676942\pi$
$67$	−578.790	−1.05538	−0.527690	+	0.849437i	$0.676942\pi$
$68$	366.705	0.653963
$69$	12.4715	0.0217593
$70$	135.695	0.231695
$71$	680.655	1.13773	0.568866	−	0.822430i	$-0.307382\pi$
$71$	680.655	1.13773	0.568866	+	0.822430i	$0.307382\pi$
$72$	−136.624	−0.223629
$73$	−891.010	−1.42856	−0.714280	−	0.699861i	$-0.753245\pi$
$73$	−891.010	−1.42856	−0.714280	+	0.699861i	$0.753245\pi$
$74$	284.549	0.447002
$75$	75.0000	0.115470
$76$	−243.959	−0.368211
$77$	0	0
$78$	212.429	0.308370
$79$	−614.496	−0.875142	−0.437571	−	0.899184i	$-0.644161\pi$
$79$	−614.496	−0.875142	−0.437571	+	0.899184i	$0.644161\pi$
$80$	201.919	0.282190
$81$	81.0000	0.111111
$82$	359.021	0.483503
$83$	985.150	1.30282	0.651411	−	0.758725i	$-0.274177\pi$
$83$	985.150	1.30282	0.651411	+	0.758725i	$0.274177\pi$
$84$	−559.840	−0.727186
$85$	−262.986	−0.335586
$86$	123.313	0.154618
$87$	266.034	0.327837
$88$	0	0
$89$	1470.02	1.75081	0.875406	−	0.483388i	$-0.160594\pi$
$89$	1470.02	1.75081	0.875406	+	0.483388i	$0.160594\pi$
$90$	45.6266	0.0534386
$91$	1869.28	2.15334
$92$	−28.9836	−0.0328451
$93$	−651.700	−0.726647
$94$	−140.408	−0.154064
$95$	174.957	0.188950
$96$	487.169	0.517932
$97$	−60.0685	−0.0628766	−0.0314383	−	0.999506i	$-0.510009\pi$
$97$	−60.0685	−0.0628766	−0.0314383	+	0.999506i	$0.510009\pi$
$98$	378.635	0.390285
$99$	0	0
$100$	−174.299	−0.174299
$101$	883.955	0.870859	0.435430	−	0.900223i	$-0.356596\pi$
$101$	883.955	0.870859	0.435430	+	0.900223i	$0.356596\pi$
$102$	−159.989	−0.155306
$103$	−1431.63	−1.36954	−0.684771	−	0.728759i	$-0.740098\pi$
$103$	−1431.63	−1.36954	−0.684771	+	0.728759i	$0.740098\pi$
$104$	−1060.16	−0.999588
$105$	401.495	0.373161
$106$	−107.588	−0.0985839
$107$	898.800	0.812059	0.406029	−	0.913860i	$-0.366913\pi$
$107$	898.800	0.812059	0.406029	+	0.913860i	$0.366913\pi$
$108$	−188.243	−0.167719
$109$	−595.217	−0.523041	−0.261521	−	0.965198i	$-0.584224\pi$
$109$	−595.217	−0.523041	−0.261521	+	0.965198i	$0.584224\pi$
$110$	0	0
$111$	841.923	0.719926
$112$	1080.93	0.911945
$113$	1101.37	0.916884	0.458442	−	0.888724i	$-0.348408\pi$
$113$	1101.37	0.916884	0.458442	+	0.888724i	$0.348408\pi$
$114$	106.436	0.0874445
$115$	20.7858	0.0168547
$116$	−618.258	−0.494861
$117$	628.534	0.496650
$118$	−360.014	−0.280865
$119$	−1407.83	−1.08450
$120$	−227.707	−0.173222
$121$	0	0
$122$	−277.850	−0.206192
$123$	1062.27	0.778713
$124$	1514.54	1.09685
$125$	125.000	0.0894427
$126$	244.251	0.172696
$127$	−1552.44	−1.08470	−0.542349	−	0.840153i	$-0.682465\pi$
$127$	−1552.44	−1.08470	−0.542349	+	0.840153i	$0.682465\pi$
$128$	−1459.74	−1.00800
$129$	364.858	0.249023
$130$	354.048	0.238862
$131$	−2793.65	−1.86323	−0.931613	−	0.363451i	$-0.881598\pi$
$131$	−2793.65	−1.86323	−0.931613	+	0.363451i	$0.881598\pi$
$132$	0	0
$133$	936.593	0.610623
$134$	−586.850	−0.378329
$135$	135.000	0.0860663
$136$	798.448	0.503429
$137$	1661.03	1.03585	0.517926	−	0.855426i	$-0.326704\pi$
$137$	1661.03	1.03585	0.517926	+	0.855426i	$0.326704\pi$
$138$	12.6452	0.00780021
$139$	38.5605	0.0235299	0.0117649	−	0.999931i	$-0.496255\pi$
$139$	38.5605	0.0235299	0.0117649	+	0.999931i	$0.496255\pi$
$140$	−933.067	−0.563276
$141$	−415.439	−0.248130
$142$	690.134	0.407850
$143$	0	0
$144$	363.454	0.210332
$145$	443.389	0.253941
$146$	−903.417	−0.512105
$147$	1120.31	0.628580
$148$	−1956.62	−1.08671
$149$	1865.68	1.02579	0.512896	−	0.858451i	$-0.328573\pi$
$149$	1865.68	1.02579	0.512896	+	0.858451i	$0.328573\pi$
$150$	76.0444	0.0413933
$151$	1655.20	0.892041	0.446021	−	0.895023i	$-0.352841\pi$
$151$	1655.20	0.892041	0.446021	+	0.895023i	$0.352841\pi$
$152$	−531.186	−0.283453
$153$	−473.374	−0.250131
$154$	0	0
$155$	−1086.17	−0.562858
$156$	−1460.70	−0.749679
$157$	3439.73	1.74854	0.874269	−	0.485441i	$-0.161341\pi$
$157$	3439.73	1.74854	0.874269	+	0.485441i	$0.161341\pi$
$158$	−623.053	−0.313718
$159$	−318.332	−0.158776
$160$	811.949	0.401189
$161$	111.272	0.0544687
$162$	82.1279	0.0398307
$163$	3495.57	1.67972	0.839859	−	0.542805i	$-0.182638\pi$
$163$	3495.57	1.67972	0.839859	+	0.542805i	$0.182638\pi$
$164$	−2468.70	−1.17545
$165$	0	0
$166$	998.869	0.467032
$167$	843.585	0.390889	0.195445	−	0.980715i	$-0.437385\pi$
$167$	843.585	0.390889	0.195445	+	0.980715i	$0.437385\pi$
$168$	−1218.97	−0.559797
$169$	2680.23	1.21995
$170$	−266.648	−0.120300
$171$	314.923	0.140835
$172$	−847.925	−0.375893
$173$	2547.24	1.11944	0.559720	−	0.828682i	$-0.310909\pi$
$173$	2547.24	1.11944	0.559720	+	0.828682i	$0.310909\pi$
$174$	269.738	0.117522
$175$	669.158	0.289049
$176$	0	0
$177$	−1065.21	−0.452351
$178$	1490.49	0.627625
$179$	−1719.44	−0.717971	−0.358985	−	0.933343i	$-0.616877\pi$
$179$	−1719.44	−0.717971	−0.358985	+	0.933343i	$0.616877\pi$
$180$	−313.738	−0.129915
$181$	2019.74	0.829428	0.414714	−	0.909952i	$-0.363882\pi$
$181$	2019.74	0.829428	0.414714	+	0.909952i	$0.363882\pi$
$182$	1895.31	0.771923
$183$	−822.103	−0.332085
$184$	−63.1077	−0.0252846
$185$	1403.20	0.557652
$186$	−660.775	−0.260486
$187$	0	0
$188$	965.474	0.374545
$189$	722.690	0.278137
$190$	177.394	0.0677342
$191$	3751.50	1.42120	0.710599	−	0.703597i	$-0.248424\pi$
$191$	3751.50	1.42120	0.710599	+	0.703597i	$0.248424\pi$
$192$	−475.258	−0.178640
$193$	3104.73	1.15795	0.578973	−	0.815347i	$-0.303454\pi$
$193$	3104.73	1.15795	0.578973	+	0.815347i	$0.303454\pi$
$194$	−60.9050	−0.0225398
$195$	1047.56	0.384703
$196$	−2603.57	−0.948824
$197$	−4955.82	−1.79232	−0.896161	−	0.443729i	$-0.853655\pi$
$197$	−4955.82	−1.79232	−0.896161	+	0.443729i	$0.853655\pi$
$198$	0	0
$199$	−3660.41	−1.30392	−0.651960	−	0.758254i	$-0.726053\pi$
$199$	−3660.41	−1.30392	−0.651960	+	0.758254i	$0.726053\pi$
$200$	−379.511	−0.134177
$201$	−1736.37	−0.609324
$202$	896.264	0.312183
$203$	2373.58	0.820653
$204$	1100.12	0.377566
$205$	1770.45	0.603188
$206$	−1451.57	−0.490949
$207$	37.4145	0.0125628
$208$	2820.29	0.940153
$209$	0	0
$210$	407.085	0.133769
$211$	−2702.17	−0.881635	−0.440818	−	0.897597i	$-0.645311\pi$
$211$	−2702.17	−0.881635	−0.440818	+	0.897597i	$0.645311\pi$
$212$	739.798	0.239668
$213$	2041.97	0.656870
$214$	911.316	0.291104
$215$	608.097	0.192892
$216$	−409.872	−0.129112
$217$	−5814.54	−1.81897
$218$	−603.506	−0.187498
$219$	−2673.03	−0.824779
$220$	0	0
$221$	−3673.24	−1.11805
$222$	853.647	0.258077
$223$	3917.28	1.17633	0.588163	−	0.808743i	$-0.299852\pi$
$223$	3917.28	1.17633	0.588163	+	0.808743i	$0.299852\pi$
$224$	4346.57	1.29651
$225$	225.000	0.0666667
$226$	1116.70	0.328681
$227$	−305.094	−0.0892062	−0.0446031	−	0.999005i	$-0.514202\pi$
$227$	−305.094	−0.0892062	−0.0446031	+	0.999005i	$0.514202\pi$
$228$	−731.877	−0.212587
$229$	1523.35	0.439587	0.219794	−	0.975546i	$-0.429462\pi$
$229$	1523.35	0.439587	0.219794	+	0.975546i	$0.429462\pi$
$230$	21.0753	0.00604202
$231$	0	0
$232$	−1346.17	−0.380950
$233$	6384.95	1.79524	0.897622	−	0.440767i	$-0.145293\pi$
$233$	6384.95	1.79524	0.897622	+	0.440767i	$0.145293\pi$
$234$	637.287	0.178037
$235$	−692.398	−0.192200
$236$	2475.53	0.682811
$237$	−1843.49	−0.505263
$238$	−1427.44	−0.388769
$239$	−1404.06	−0.380005	−0.190002	−	0.981784i	$-0.560850\pi$
$239$	−1404.06	−0.380005	−0.190002	+	0.981784i	$0.560850\pi$
$240$	605.757	0.162923
$241$	221.355	0.0591648	0.0295824	−	0.999562i	$-0.490582\pi$
$241$	221.355	0.0591648	0.0295824	+	0.999562i	$0.490582\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	1910.56	0.501274
$245$	1867.18	0.486896
$246$	1077.06	0.279150
$247$	2443.71	0.629511
$248$	3297.70	0.844371
$249$	2955.45	0.752185
$250$	126.741	0.0320631
$251$	2674.70	0.672611	0.336306	−	0.941753i	$-0.390822\pi$
$251$	2674.70	0.672611	0.336306	+	0.941753i	$0.390822\pi$
$252$	−1679.52	−0.419841
$253$	0	0
$254$	−1574.06	−0.388839
$255$	−788.957	−0.193751
$256$	−212.715	−0.0519324
$257$	6508.15	1.57964	0.789820	−	0.613339i	$-0.210174\pi$
$257$	6508.15	1.57964	0.789820	+	0.613339i	$0.210174\pi$
$258$	369.939	0.0892689
$259$	7511.72	1.80214
$260$	−2434.51	−0.580699
$261$	798.101	0.189277
$262$	−2832.56	−0.667923
$263$	2103.53	0.493190	0.246595	−	0.969119i	$-0.420688\pi$
$263$	2103.53	0.493190	0.246595	+	0.969119i	$0.420688\pi$
$264$	0	0
$265$	−530.553	−0.122987
$266$	949.635	0.218894
$267$	4410.07	1.01083
$268$	4035.30	0.919758
$269$	−5923.42	−1.34259	−0.671296	−	0.741189i	$-0.734262\pi$
$269$	−5923.42	−1.34259	−0.671296	+	0.741189i	$0.734262\pi$
$270$	136.880	0.0308528
$271$	−5395.14	−1.20934	−0.604670	−	0.796476i	$-0.706695\pi$
$271$	−5395.14	−1.20934	−0.604670	+	0.796476i	$0.706695\pi$
$272$	−2124.07	−0.473496
$273$	5607.85	1.24323
$274$	1684.16	0.371329
$275$	0	0
$276$	−86.9508	−0.0189631
$277$	−3978.11	−0.862894	−0.431447	−	0.902138i	$-0.641997\pi$
$277$	−3978.11	−0.862894	−0.431447	+	0.902138i	$0.641997\pi$
$278$	39.0974	0.00843492
$279$	−1955.10	−0.419530
$280$	−2031.62	−0.433617
$281$	−6270.50	−1.33120	−0.665600	−	0.746309i	$-0.731824\pi$
$281$	−6270.50	−1.33120	−0.665600	+	0.746309i	$0.731824\pi$
$282$	−421.224	−0.0889487
$283$	4236.05	0.889778	0.444889	−	0.895586i	$-0.353243\pi$
$283$	4236.05	0.889778	0.444889	+	0.895586i	$0.353243\pi$
$284$	−4745.50	−0.991527
$285$	524.872	0.109090
$286$	0	0
$287$	9477.68	1.94930
$288$	1461.51	0.299028
$289$	−2146.54	−0.436910
$290$	449.564	0.0910320
$291$	−180.206	−0.0363018
$292$	6212.08	1.24498
$293$	2703.69	0.539082	0.269541	−	0.962989i	$-0.413128\pi$
$293$	2703.69	0.539082	0.269541	+	0.962989i	$0.413128\pi$
$294$	1135.91	0.225331
$295$	−1775.35	−0.350389
$296$	−4260.25	−0.836561
$297$	0	0
$298$	1891.67	0.367722
$299$	290.325	0.0561536
$300$	−522.897	−0.100632
$301$	3255.30	0.623364
$302$	1678.25	0.319776
$303$	2651.86	0.502791
$304$	1413.09	0.266599
$305$	−1370.17	−0.257232
$306$	−479.966	−0.0896662
$307$	−1431.38	−0.266101	−0.133051	−	0.991109i	$-0.542477\pi$
$307$	−1431.38	−0.266101	−0.133051	+	0.991109i	$0.542477\pi$
$308$	0	0
$309$	−4294.89	−0.790705
$310$	−1101.29	−0.201772
$311$	8548.96	1.55874	0.779368	−	0.626566i	$-0.215540\pi$
$311$	8548.96	1.55874	0.779368	+	0.626566i	$0.215540\pi$
$312$	−3180.48	−0.577112
$313$	6164.97	1.11330	0.556652	−	0.830745i	$-0.312085\pi$
$313$	6164.97	1.11330	0.556652	+	0.830745i	$0.312085\pi$
$314$	3487.63	0.626810
$315$	1204.48	0.215444
$316$	4284.24	0.762681
$317$	−5043.58	−0.893614	−0.446807	−	0.894630i	$-0.647439\pi$
$317$	−5043.58	−0.893614	−0.446807	+	0.894630i	$0.647439\pi$
$318$	−322.765	−0.0569174
$319$	0	0
$320$	−792.097	−0.138374
$321$	2696.40	0.468842
$322$	112.822	0.0195258
$323$	−1840.45	−0.317045
$324$	−564.728	−0.0968327
$325$	1745.93	0.297990
$326$	3544.24	0.602139
$327$	−1785.65	−0.301978
$328$	−5375.24	−0.904872
$329$	−3706.59	−0.621127
$330$	0	0
$331$	4707.26	0.781675	0.390837	−	0.920460i	$-0.372186\pi$
$331$	4707.26	0.781675	0.390837	+	0.920460i	$0.372186\pi$
$332$	−6868.42	−1.13540
$333$	2525.77	0.415649
$334$	855.332	0.140125
$335$	−2893.95	−0.471980
$336$	3242.78	0.526512
$337$	−11297.4	−1.82615	−0.913073	−	0.407796i	$-0.866297\pi$
$337$	−11297.4	−1.82615	−0.913073	+	0.407796i	$0.866297\pi$
$338$	2717.55	0.437323
$339$	3304.10	0.529363
$340$	1833.53	0.292461
$341$	0	0
$342$	319.309	0.0504861
$343$	814.634	0.128239
$344$	−1846.24	−0.289367
$345$	62.3575	0.00973106
$346$	2582.71	0.401293
$347$	7426.05	1.14885	0.574426	−	0.818557i	$-0.305225\pi$
$347$	7426.05	1.14885	0.574426	+	0.818557i	$0.305225\pi$
$348$	−1854.77	−0.285708
$349$	5123.99	0.785904	0.392952	−	0.919559i	$-0.371454\pi$
$349$	5123.99	0.785904	0.392952	+	0.919559i	$0.371454\pi$
$350$	678.476	0.103617
$351$	1885.60	0.286741
$352$	0	0
$353$	−11727.8	−1.76830	−0.884151	−	0.467202i	$-0.845262\pi$
$353$	−11727.8	−1.76830	−0.884151	+	0.467202i	$0.845262\pi$
$354$	−1080.04	−0.162157
$355$	3403.28	0.508809
$356$	−10248.9	−1.52582
$357$	−4223.49	−0.626137
$358$	−1743.38	−0.257376
$359$	−10749.0	−1.58025	−0.790124	−	0.612947i	$-0.789984\pi$
$359$	−10749.0	−1.58025	−0.790124	+	0.612947i	$0.789984\pi$
$360$	−683.120	−0.100010
$361$	−5634.60	−0.821489
$362$	2047.87	0.297331
$363$	0	0
$364$	−13032.6	−1.87662
$365$	−4455.05	−0.638871
$366$	−833.551	−0.119045
$367$	−13304.8	−1.89239	−0.946193	−	0.323603i	$-0.895106\pi$
$367$	−13304.8	−1.89239	−0.946193	+	0.323603i	$0.895106\pi$
$368$	167.882	0.0237812
$369$	3186.81	0.449590
$370$	1422.74	0.199905
$371$	−2840.19	−0.397453
$372$	4543.62	0.633269
$373$	1552.40	0.215496	0.107748	−	0.994178i	$-0.465636\pi$
$373$	1552.40	0.215496	0.107748	+	0.994178i	$0.465636\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	2102.18	0.288329
$377$	6193.01	0.846038
$378$	732.754	0.0997058
$379$	−2711.39	−0.367480	−0.183740	−	0.982975i	$-0.558820\pi$
$379$	−2711.39	−0.367480	−0.183740	+	0.982975i	$0.558820\pi$
$380$	−1219.80	−0.164669
$381$	−4657.32	−0.626251
$382$	3803.74	0.509466
$383$	13119.1	1.75027	0.875134	−	0.483880i	$-0.160773\pi$
$383$	13119.1	1.75027	0.875134	+	0.483880i	$0.160773\pi$
$384$	−4379.23	−0.581970
$385$	0	0
$386$	3147.97	0.415097
$387$	1094.57	0.143773
$388$	418.795	0.0547966
$389$	11082.0	1.44442	0.722210	−	0.691674i	$-0.243126\pi$
$389$	11082.0	1.44442	0.722210	+	0.691674i	$0.243126\pi$
$390$	1062.14	0.137907
$391$	−218.655	−0.0282810
$392$	−5668.91	−0.730416
$393$	−8380.96	−1.07573
$394$	−5024.83	−0.642506
$395$	−3072.48	−0.391375
$396$	0	0
$397$	2061.46	0.260608	0.130304	−	0.991474i	$-0.458405\pi$
$397$	2061.46	0.260608	0.130304	+	0.991474i	$0.458405\pi$
$398$	−3711.39	−0.467425
$399$	2809.78	0.352543
$400$	1009.60	0.126199
$401$	3876.70	0.482776	0.241388	−	0.970429i	$-0.422397\pi$
$401$	3876.70	0.482776	0.241388	+	0.970429i	$0.422397\pi$
$402$	−1760.55	−0.218428
$403$	−15171.0	−1.87523
$404$	−6162.89	−0.758949
$405$	405.000	0.0496904
$406$	2406.63	0.294185
$407$	0	0
$408$	2395.34	0.290655
$409$	−8810.21	−1.06513	−0.532563	−	0.846390i	$-0.678771\pi$
$409$	−8810.21	−1.06513	−0.532563	+	0.846390i	$0.678771\pi$
$410$	1795.10	0.216229
$411$	4983.10	0.598049
$412$	9981.27	1.19355
$413$	−9503.91	−1.13234
$414$	37.9355	0.00450345
$415$	4925.75	0.582640
$416$	11340.8	1.33661
$417$	115.681	0.0135850
$418$	0	0
$419$	2044.80	0.238413	0.119207	−	0.992869i	$-0.461965\pi$
$419$	2044.80	0.238413	0.119207	+	0.992869i	$0.461965\pi$
$420$	−2799.20	−0.325207
$421$	5012.51	0.580273	0.290136	−	0.956985i	$-0.406299\pi$
$421$	5012.51	0.580273	0.290136	+	0.956985i	$0.406299\pi$
$422$	−2739.80	−0.316046
$423$	−1246.32	−0.143258
$424$	1610.81	0.184499
$425$	−1314.93	−0.150079
$426$	2070.40	0.235472
$427$	−7334.89	−0.831288
$428$	−6266.40	−0.707705
$429$	0	0
$430$	616.565	0.0691474
$431$	−7690.80	−0.859519	−0.429760	−	0.902943i	$-0.641402\pi$
$431$	−7690.80	−0.859519	−0.429760	+	0.902943i	$0.641402\pi$
$432$	1090.36	0.121435
$433$	8498.85	0.943254	0.471627	−	0.881798i	$-0.343667\pi$
$433$	8498.85	0.943254	0.471627	+	0.881798i	$0.343667\pi$
$434$	−5895.50	−0.652058
$435$	1330.17	0.146613
$436$	4149.83	0.455827
$437$	145.466	0.0159235
$438$	−2710.25	−0.295664
$439$	−699.844	−0.0760860	−0.0380430	−	0.999276i	$-0.512112\pi$
$439$	−699.844	−0.0760860	−0.0380430	+	0.999276i	$0.512112\pi$
$440$	0	0
$441$	3360.92	0.362911
$442$	−3724.39	−0.400794
$443$	−12879.3	−1.38130	−0.690649	−	0.723190i	$-0.742675\pi$
$443$	−12879.3	−1.38130	−0.690649	+	0.723190i	$0.742675\pi$
$444$	−5869.85	−0.627411
$445$	7350.12	0.782987
$446$	3971.83	0.421685
$447$	5597.05	0.592241
$448$	−4240.30	−0.447177
$449$	−3417.46	−0.359198	−0.179599	−	0.983740i	$-0.557480\pi$
$449$	−3417.46	−0.359198	−0.179599	+	0.983740i	$0.557480\pi$
$450$	228.133	0.0238984
$451$	0	0
$452$	−7678.68	−0.799059
$453$	4965.60	0.515020
$454$	−309.342	−0.0319783
$455$	9346.41	0.963003
$456$	−1593.56	−0.163652
$457$	−3273.82	−0.335105	−0.167552	−	0.985863i	$-0.553586\pi$
$457$	−3273.82	−0.335105	−0.167552	+	0.985863i	$0.553586\pi$
$458$	1544.56	0.157582
$459$	−1420.12	−0.144413
$460$	−144.918	−0.0146888
$461$	−7490.69	−0.756782	−0.378391	−	0.925646i	$-0.623522\pi$
$461$	−7490.69	−0.756782	−0.378391	+	0.925646i	$0.623522\pi$
$462$	0	0
$463$	1879.17	0.188623	0.0943113	−	0.995543i	$-0.469935\pi$
$463$	1879.17	0.188623	0.0943113	+	0.995543i	$0.469935\pi$
$464$	3581.15	0.358299
$465$	−3258.50	−0.324966
$466$	6473.86	0.643553
$467$	15138.3	1.50004	0.750019	−	0.661416i	$-0.230044\pi$
$467$	15138.3	1.50004	0.750019	+	0.661416i	$0.230044\pi$
$468$	−4382.11	−0.432827
$469$	−15492.1	−1.52528
$470$	−702.040	−0.0688994
$471$	10319.2	1.00952
$472$	5390.12	0.525636
$473$	0	0
$474$	−1869.16	−0.181125
$475$	874.787	0.0845010
$476$	9815.34	0.945137
$477$	−954.995	−0.0916692
$478$	−1423.61	−0.136223
$479$	−12405.5	−1.18335	−0.591673	−	0.806178i	$-0.701532\pi$
$479$	−12405.5	−1.18335	−0.591673	+	0.806178i	$0.701532\pi$
$480$	2435.85	0.231626
$481$	19599.2	1.85789
$482$	224.437	0.0212092
$483$	333.816	0.0314475
$484$	0	0
$485$	−300.343	−0.0281193
$486$	246.384	0.0229963
$487$	−3903.66	−0.363227	−0.181614	−	0.983370i	$-0.558132\pi$
$487$	−3903.66	−0.363227	−0.181614	+	0.983370i	$0.558132\pi$
$488$	4159.96	0.385887
$489$	10486.7	0.969785
$490$	1893.18	0.174541
$491$	−11824.8	−1.08685	−0.543427	−	0.839456i	$-0.682874\pi$
$491$	−11824.8	−1.08685	−0.543427	+	0.839456i	$0.682874\pi$
$492$	−7406.10	−0.678644
$493$	−4664.20	−0.426096
$494$	2477.73	0.225665
$495$	0	0
$496$	−8772.71	−0.794166
$497$	18218.6	1.64430
$498$	2996.61	0.269641
$499$	−18169.0	−1.62997	−0.814987	−	0.579479i	$-0.803256\pi$
$499$	−18169.0	−1.62997	−0.814987	+	0.579479i	$0.803256\pi$
$500$	−871.494	−0.0779488
$501$	2530.75	0.225680
$502$	2711.94	0.241115
$503$	8675.27	0.769008	0.384504	−	0.923123i	$-0.374372\pi$
$503$	8675.27	0.769008	0.384504	+	0.923123i	$0.374372\pi$
$504$	−3656.92	−0.323199
$505$	4419.77	0.389460
$506$	0	0
$507$	8040.68	0.704337
$508$	10823.5	0.945309
$509$	−11648.2	−1.01434	−0.507169	−	0.861846i	$-0.669308\pi$
$509$	−11648.2	−1.01434	−0.507169	+	0.861846i	$0.669308\pi$
$510$	−799.944	−0.0694551
$511$	−23849.0	−2.06462
$512$	11462.3	0.989386
$513$	944.770	0.0813111
$514$	6598.78	0.566264
$515$	−7158.15	−0.612478
$516$	−2543.77	−0.217022
$517$	0	0
$518$	7616.32	0.646027
$519$	7641.72	0.646309
$520$	−5300.79	−0.447029
$521$	5529.52	0.464977	0.232488	−	0.972599i	$-0.425313\pi$
$521$	5529.52	0.464977	0.232488	+	0.972599i	$0.425313\pi$
$522$	809.215	0.0678513
$523$	−19225.5	−1.60740	−0.803702	−	0.595032i	$-0.797139\pi$
$523$	−19225.5	−1.60740	−0.803702	+	0.595032i	$0.797139\pi$
$524$	19477.2	1.62379
$525$	2007.47	0.166882
$526$	2132.82	0.176797
$527$	11425.9	0.944437
$528$	0	0
$529$	−12149.7	−0.998580
$530$	−537.941	−0.0440880
$531$	−3195.63	−0.261165
$532$	−6529.88	−0.532155
$533$	24728.6	2.00960
$534$	4471.48	0.362360
$535$	4494.00	0.363164
$536$	8786.29	0.708041
$537$	−5158.31	−0.414521
$538$	−6005.91	−0.481288
$539$	0	0
$540$	−941.214	−0.0750063
$541$	5802.66	0.461138	0.230569	−	0.973056i	$-0.425941\pi$
$541$	5802.66	0.461138	0.230569	+	0.973056i	$0.425941\pi$
$542$	−5470.27	−0.433520
$543$	6059.23	0.478870
$544$	−8541.24	−0.673166
$545$	−2976.09	−0.233911
$546$	5685.94	0.445670
$547$	−6121.62	−0.478504	−0.239252	−	0.970958i	$-0.576902\pi$
$547$	−6121.62	−0.478504	−0.239252	+	0.970958i	$0.576902\pi$
$548$	−11580.7	−0.902739
$549$	−2466.31	−0.191730
$550$	0	0
$551$	3102.97	0.239911
$552$	−189.323	−0.0145980
$553$	−16447.8	−1.26479
$554$	−4033.51	−0.309328
$555$	4209.61	0.321961
$556$	−268.842	−0.0205062
$557$	16550.4	1.25900	0.629500	−	0.777001i	$-0.283260\pi$
$557$	16550.4	1.25900	0.629500	+	0.777001i	$0.283260\pi$
$558$	−1982.33	−0.150392
$559$	8493.55	0.642646
$560$	5404.63	0.407834
$561$	0	0
$562$	−6357.82	−0.477204
$563$	12986.0	0.972106	0.486053	−	0.873929i	$-0.338436\pi$
$563$	12986.0	0.972106	0.486053	+	0.873929i	$0.338436\pi$
$564$	2896.42	0.216244
$565$	5506.83	0.410043
$566$	4295.04	0.318965
$567$	2168.07	0.160583
$568$	−10332.7	−0.763290
$569$	−16456.3	−1.21245	−0.606225	−	0.795293i	$-0.707317\pi$
$569$	−16456.3	−1.21245	−0.606225	+	0.795293i	$0.707317\pi$
$570$	532.181	0.0391063
$571$	20716.4	1.51831	0.759154	−	0.650911i	$-0.225613\pi$
$571$	20716.4	1.51831	0.759154	+	0.650911i	$0.225613\pi$
$572$	0	0
$573$	11254.5	0.820529
$574$	9609.66	0.698780
$575$	103.929	0.00753765
$576$	−1425.77	−0.103138
$577$	−16502.3	−1.19064	−0.595322	−	0.803487i	$-0.702975\pi$
$577$	−16502.3	−1.19064	−0.595322	+	0.803487i	$0.702975\pi$
$578$	−2176.43	−0.156622
$579$	9314.20	0.668541
$580$	−3091.29	−0.221308
$581$	26368.8	1.88290
$582$	−182.715	−0.0130134
$583$	0	0
$584$	13525.9	0.958402
$585$	3142.67	0.222108
$586$	2741.34	0.193248
$587$	8621.89	0.606241	0.303120	−	0.952952i	$-0.401972\pi$
$587$	8621.89	0.606241	0.303120	+	0.952952i	$0.401972\pi$
$588$	−7810.72	−0.547804
$589$	−7601.32	−0.531760
$590$	−1800.07	−0.125606
$591$	−14867.5	−1.03480
$592$	11333.3	0.786820
$593$	−21499.1	−1.48881	−0.744404	−	0.667730i	$-0.767266\pi$
$593$	−21499.1	−1.48881	−0.744404	+	0.667730i	$0.767266\pi$
$594$	0	0
$595$	−7039.16	−0.485004
$596$	−13007.5	−0.893971
$597$	−10981.2	−0.752818
$598$	294.368	0.0201297
$599$	−15079.5	−1.02860	−0.514302	−	0.857609i	$-0.671949\pi$
$599$	−15079.5	−1.02860	−0.514302	+	0.857609i	$0.671949\pi$
$600$	−1138.53	−0.0774674
$601$	−17855.8	−1.21190	−0.605950	−	0.795503i	$-0.707207\pi$
$601$	−17855.8	−1.21190	−0.605950	+	0.795503i	$0.707207\pi$
$602$	3300.63	0.223461
$603$	−5209.11	−0.351793
$604$	−11540.0	−0.777409
$605$	0	0
$606$	2688.79	0.180239
$607$	16657.8	1.11387	0.556937	−	0.830555i	$-0.311977\pi$
$607$	16657.8	1.11387	0.556937	+	0.830555i	$0.311977\pi$
$608$	5682.26	0.379023
$609$	7120.74	0.473804
$610$	−1389.25	−0.0922117
$611$	−9671.02	−0.640340
$612$	3300.35	0.217988
$613$	−592.062	−0.0390100	−0.0195050	−	0.999810i	$-0.506209\pi$
$613$	−592.062	−0.0390100	−0.0195050	+	0.999810i	$0.506209\pi$
$614$	−1451.31	−0.0953912
$615$	5311.35	0.348251
$616$	0	0
$617$	19382.1	1.26466	0.632329	−	0.774700i	$-0.282099\pi$
$617$	19382.1	1.26466	0.632329	+	0.774700i	$0.282099\pi$
$618$	−4354.70	−0.283449
$619$	14735.3	0.956803	0.478402	−	0.878141i	$-0.341216\pi$
$619$	14735.3	0.956803	0.478402	+	0.878141i	$0.341216\pi$
$620$	7572.71	0.490528
$621$	112.244	0.00725311
$622$	8668.00	0.558771
$623$	39347.1	2.53035
$624$	8460.87	0.542798
$625$	625.000	0.0400000
$626$	6250.81	0.399094
$627$	0	0
$628$	−23981.7	−1.52384
$629$	−14760.9	−0.935701
$630$	1221.26	0.0772318
$631$	−22038.6	−1.39040	−0.695199	−	0.718817i	$-0.744684\pi$
$631$	−22038.6	−1.39040	−0.695199	+	0.718817i	$0.744684\pi$
$632$	9328.32	0.587121
$633$	−8106.51	−0.509012
$634$	−5113.81	−0.320340
$635$	−7762.20	−0.485092
$636$	2219.39	0.138372
$637$	26079.6	1.62216
$638$	0	0
$639$	6125.90	0.379244
$640$	−7298.72	−0.450792
$641$	−10475.7	−0.645498	−0.322749	−	0.946485i	$-0.604607\pi$
$641$	−10475.7	−0.645498	−0.322749	+	0.946485i	$0.604607\pi$
$642$	2733.95	0.168069
$643$	−2465.20	−0.151194	−0.0755972	−	0.997138i	$-0.524086\pi$
$643$	−2465.20	−0.151194	−0.0755972	+	0.997138i	$0.524086\pi$
$644$	−775.784	−0.0474692
$645$	1824.29	0.111366
$646$	−1866.08	−0.113653
$647$	−3043.27	−0.184920	−0.0924600	−	0.995716i	$-0.529473\pi$
$647$	−3043.27	−0.184920	−0.0924600	+	0.995716i	$0.529473\pi$
$648$	−1229.62	−0.0745430
$649$	0	0
$650$	1770.24	0.106822
$651$	−17443.6	−1.05018
$652$	−24370.9	−1.46386
$653$	−15603.4	−0.935082	−0.467541	−	0.883971i	$-0.654860\pi$
$653$	−15603.4	−0.935082	−0.467541	+	0.883971i	$0.654860\pi$
$654$	−1810.52	−0.108252
$655$	−13968.3	−0.833260
$656$	14299.5	0.851070
$657$	−8019.09	−0.476186
$658$	−3758.20	−0.222660
$659$	3073.38	0.181672	0.0908361	−	0.995866i	$-0.471046\pi$
$659$	3073.38	0.181672	0.0908361	+	0.995866i	$0.471046\pi$
$660$	0	0
$661$	13066.4	0.768872	0.384436	−	0.923152i	$-0.374396\pi$
$661$	13066.4	0.768872	0.384436	+	0.923152i	$0.374396\pi$
$662$	4772.81	0.280212
$663$	−11019.7	−0.645505
$664$	−14955.0	−0.874047
$665$	4682.96	0.273079
$666$	2560.94	0.149001
$667$	368.649	0.0214005
$668$	−5881.43	−0.340658
$669$	11751.8	0.679152
$670$	−2934.25	−0.169194
$671$	0	0
$672$	13039.7	0.748539
$673$	−1697.77	−0.0972423	−0.0486211	−	0.998817i	$-0.515483\pi$
$673$	−1697.77	−0.0972423	−0.0486211	+	0.998817i	$0.515483\pi$
$674$	−11454.8	−0.654631
$675$	675.000	0.0384900
$676$	−18686.4	−1.06318
$677$	9583.59	0.544058	0.272029	−	0.962289i	$-0.412305\pi$
$677$	9583.59	0.544058	0.272029	+	0.962289i	$0.412305\pi$
$678$	3350.11	0.189764
$679$	−1607.81	−0.0908721
$680$	3992.24	0.225140
$681$	−915.282	−0.0515032
$682$	0	0
$683$	−6127.71	−0.343295	−0.171647	−	0.985158i	$-0.554909\pi$
$683$	−6127.71	−0.343295	−0.171647	+	0.985158i	$0.554909\pi$
$684$	−2195.63	−0.122737
$685$	8305.17	0.463247
$686$	825.978	0.0459708
$687$	4570.04	0.253796
$688$	4911.45	0.272162
$689$	−7410.46	−0.409747
$690$	63.2259	0.00348836
$691$	17689.2	0.973846	0.486923	−	0.873445i	$-0.338119\pi$
$691$	17689.2	0.973846	0.486923	+	0.873445i	$0.338119\pi$
$692$	−17759.2	−0.975585
$693$	0	0
$694$	7529.46	0.411836
$695$	192.802	0.0105229
$696$	−4038.51	−0.219942
$697$	−18624.1	−1.01211
$698$	5195.34	0.281728
$699$	19154.8	1.03648
$700$	−4665.34	−0.251905
$701$	33371.3	1.79803	0.899013	−	0.437923i	$-0.144286\pi$
$701$	33371.3	1.79803	0.899013	+	0.437923i	$0.144286\pi$
$702$	1911.86	0.102790
$703$	9820.04	0.526842
$704$	0	0
$705$	−2077.20	−0.110967
$706$	−11891.2	−0.633895
$707$	23660.2	1.25860
$708$	7426.60	0.394221
$709$	18842.9	0.998109	0.499055	−	0.866570i	$-0.333681\pi$
$709$	18842.9	0.998109	0.499055	+	0.866570i	$0.333681\pi$
$710$	3450.67	0.182396
$711$	−5530.46	−0.291714
$712$	−22315.6	−1.17460
$713$	−903.076	−0.0474340
$714$	−4282.31	−0.224456
$715$	0	0
$716$	11987.8	0.625707
$717$	−4212.18	−0.219396
$718$	−10898.7	−0.566482
$719$	−29484.0	−1.52930	−0.764650	−	0.644446i	$-0.777088\pi$
$719$	−29484.0	−1.52930	−0.764650	+	0.644446i	$0.777088\pi$
$720$	1817.27	0.0940635
$721$	−38319.5	−1.97932
$722$	−5713.06	−0.294485
$723$	664.064	0.0341588
$724$	−14081.6	−0.722842
$725$	2216.95	0.113566
$726$	0	0
$727$	−5365.41	−0.273717	−0.136858	−	0.990591i	$-0.543701\pi$
$727$	−5365.41	−0.273717	−0.136858	+	0.990591i	$0.543701\pi$
$728$	−28376.5	−1.44465
$729$	729.000	0.0370370
$730$	−4517.09	−0.229020
$731$	−6396.83	−0.323660
$732$	5731.67	0.289410
$733$	4282.37	0.215788	0.107894	−	0.994162i	$-0.465589\pi$
$733$	4282.37	0.215788	0.107894	+	0.994162i	$0.465589\pi$
$734$	−13490.1	−0.678376
$735$	5601.53	0.281109
$736$	675.082	0.0338096
$737$	0	0
$738$	3231.19	0.161168
$739$	14013.2	0.697541	0.348770	−	0.937208i	$-0.386599\pi$
$739$	14013.2	0.697541	0.348770	+	0.937208i	$0.386599\pi$
$740$	−9783.08	−0.485991
$741$	7331.12	0.363448
$742$	−2879.74	−0.142478
$743$	15738.3	0.777094	0.388547	−	0.921429i	$-0.372977\pi$
$743$	15738.3	0.777094	0.388547	+	0.921429i	$0.372977\pi$
$744$	9893.10	0.487498
$745$	9328.42	0.458748
$746$	1574.01	0.0772503
$747$	8866.35	0.434274
$748$	0	0
$749$	24057.6	1.17362
$750$	380.222	0.0185117
$751$	33634.3	1.63427	0.817134	−	0.576448i	$-0.195562\pi$
$751$	33634.3	1.63427	0.817134	+	0.576448i	$0.195562\pi$
$752$	−5592.34	−0.271186
$753$	8024.09	0.388332
$754$	6279.25	0.303285
$755$	8276.00	0.398933
$756$	−5038.56	−0.242395
$757$	−35870.7	−1.72225	−0.861125	−	0.508394i	$-0.830239\pi$
$757$	−35870.7	−1.72225	−0.861125	+	0.508394i	$0.830239\pi$
$758$	−2749.15	−0.131733
$759$	0	0
$760$	−2655.93	−0.126764
$761$	−14380.3	−0.685003	−0.342501	−	0.939517i	$-0.611274\pi$
$761$	−14380.3	−0.685003	−0.342501	+	0.939517i	$0.611274\pi$
$762$	−4722.17	−0.224496
$763$	−15931.8	−0.755922
$764$	−26155.3	−1.23857
$765$	−2366.87	−0.111862
$766$	13301.7	0.627430
$767$	−24797.1	−1.16737
$768$	−638.145	−0.0299832
$769$	−20128.4	−0.943887	−0.471944	−	0.881629i	$-0.656447\pi$
$769$	−20128.4	−0.943887	−0.471944	+	0.881629i	$0.656447\pi$
$770$	0	0
$771$	19524.5	0.912006
$772$	−21646.1	−1.00914
$773$	−2642.25	−0.122943	−0.0614716	−	0.998109i	$-0.519579\pi$
$773$	−2642.25	−0.122943	−0.0614716	+	0.998109i	$0.519579\pi$
$774$	1109.82	0.0515395
$775$	−5430.84	−0.251718
$776$	911.867	0.0421831
$777$	22535.2	1.04047
$778$	11236.3	0.517791
$779$	12390.1	0.569862
$780$	−7303.52	−0.335267
$781$	0	0
$782$	−221.700	−0.0101381
$783$	2394.30	0.109279
$784$	15080.7	0.686987
$785$	17198.7	0.781970
$786$	−8497.67	−0.385626
$787$	−34691.5	−1.57131	−0.785653	−	0.618667i	$-0.787673\pi$
$787$	−34691.5	−1.57131	−0.785653	+	0.618667i	$0.787673\pi$
$788$	34551.7	1.56200
$789$	6310.58	0.284743
$790$	−3115.26	−0.140299
$791$	29479.5	1.32512
$792$	0	0
$793$	−19137.8	−0.857002
$794$	2090.16	0.0934220
$795$	−1591.66	−0.0710067
$796$	25520.2	1.13636
$797$	−23851.9	−1.06007	−0.530037	−	0.847974i	$-0.677822\pi$
$797$	−23851.9	−1.06007	−0.530037	+	0.847974i	$0.677822\pi$
$798$	2848.91	0.126379
$799$	7283.64	0.322499
$800$	4059.74	0.179417
$801$	13230.2	0.583604
$802$	3930.69	0.173064
$803$	0	0
$804$	12105.9	0.531023
$805$	556.360	0.0243592
$806$	−15382.2	−0.672228
$807$	−17770.3	−0.775146
$808$	−13418.8	−0.584248
$809$	−43905.6	−1.90808	−0.954040	−	0.299679i	$-0.903121\pi$
$809$	−43905.6	−1.90808	−0.954040	+	0.299679i	$0.903121\pi$
$810$	410.640	0.0178129
$811$	17661.6	0.764714	0.382357	−	0.924015i	$-0.375112\pi$
$811$	17661.6	0.764714	0.382357	+	0.924015i	$0.375112\pi$
$812$	−16548.5	−0.715195
$813$	−16185.4	−0.698213
$814$	0	0
$815$	17477.8	0.751192
$816$	−6372.22	−0.273373
$817$	4255.64	0.182235
$818$	−8932.89	−0.381823
$819$	16823.5	0.717780
$820$	−12343.5	−0.525675
$821$	32488.2	1.38105	0.690527	−	0.723307i	$-0.257379\pi$
$821$	32488.2	1.38105	0.690527	+	0.723307i	$0.257379\pi$
$822$	5052.49	0.214387
$823$	−36508.1	−1.54628	−0.773142	−	0.634233i	$-0.781316\pi$
$823$	−36508.1	−1.54628	−0.773142	+	0.634233i	$0.781316\pi$
$824$	21732.8	0.918808
$825$	0	0
$826$	−9636.26	−0.405918
$827$	33488.5	1.40811	0.704056	−	0.710144i	$-0.251370\pi$
$827$	33488.5	1.40811	0.704056	+	0.710144i	$0.251370\pi$
$828$	−260.852	−0.0109484
$829$	−33617.7	−1.40843	−0.704216	−	0.709986i	$-0.748701\pi$
$829$	−33617.7	−1.40843	−0.704216	+	0.709986i	$0.748701\pi$
$830$	4994.34	0.208863
$831$	−11934.3	−0.498192
$832$	−11063.6	−0.461010
$833$	−19641.6	−0.816977
$834$	117.292	0.00486990
$835$	4217.92	0.174811
$836$	0	0
$837$	−5865.30	−0.242216
$838$	2073.28	0.0854656
$839$	−31608.2	−1.30064	−0.650320	−	0.759660i	$-0.725365\pi$
$839$	−31608.2	−1.30064	−0.650320	+	0.759660i	$0.725365\pi$
$840$	−6094.87	−0.250349
$841$	−16525.2	−0.677569
$842$	5082.31	0.208014
$843$	−18811.5	−0.768568
$844$	18839.4	0.768340
$845$	13401.1	0.545577
$846$	−1263.67	−0.0513546
$847$	0	0
$848$	−4285.15	−0.173529
$849$	12708.2	0.513714
$850$	−1333.24	−0.0537997
$851$	1166.67	0.0469953
$852$	−14236.5	−0.572458
$853$	−3970.30	−0.159367	−0.0796837	−	0.996820i	$-0.525391\pi$
$853$	−3970.30	−0.159367	−0.0796837	+	0.996820i	$0.525391\pi$
$854$	−7437.03	−0.297997
$855$	1574.62	0.0629833
$856$	−13644.2	−0.544800
$857$	−46820.9	−1.86624	−0.933121	−	0.359562i	$-0.882926\pi$
$857$	−46820.9	−1.86624	−0.933121	+	0.359562i	$0.882926\pi$
$858$	0	0
$859$	−31041.3	−1.23296	−0.616481	−	0.787369i	$-0.711442\pi$
$859$	−31041.3	−1.23296	−0.616481	+	0.787369i	$0.711442\pi$
$860$	−4239.62	−0.168105
$861$	28433.0	1.12543
$862$	−7797.89	−0.308118
$863$	−12259.2	−0.483554	−0.241777	−	0.970332i	$-0.577730\pi$
$863$	−12259.2	−0.483554	−0.241777	+	0.970332i	$0.577730\pi$
$864$	4384.52	0.172644
$865$	12736.2	0.500628
$866$	8617.20	0.338135
$867$	−6439.62	−0.252250
$868$	40538.7	1.58522
$869$	0	0
$870$	1348.69	0.0525574
$871$	−40421.1	−1.57246
$872$	9035.66	0.350902
$873$	−540.617	−0.0209589
$874$	147.491	0.00570820
$875$	3345.79	0.129267
$876$	18636.2	0.718790
$877$	22212.1	0.855244	0.427622	−	0.903958i	$-0.359351\pi$
$877$	22212.1	0.855244	0.427622	+	0.903958i	$0.359351\pi$
$878$	−709.590	−0.0272751
$879$	8111.06	0.311239
$880$	0	0
$881$	23811.3	0.910583	0.455292	−	0.890342i	$-0.349535\pi$
$881$	23811.3	0.910583	0.455292	+	0.890342i	$0.349535\pi$
$882$	3407.72	0.130095
$883$	−16203.1	−0.617529	−0.308764	−	0.951139i	$-0.599915\pi$
$883$	−16203.1	−0.617529	−0.308764	+	0.951139i	$0.599915\pi$
$884$	25609.6	0.974372
$885$	−5326.05	−0.202297
$886$	−13058.7	−0.495163
$887$	−12349.5	−0.467481	−0.233741	−	0.972299i	$-0.575097\pi$
$887$	−12349.5	−0.467481	−0.233741	+	0.972299i	$0.575097\pi$
$888$	−12780.8	−0.482989
$889$	−41553.1	−1.56765
$890$	7452.47	0.280683
$891$	0	0
$892$	−27311.1	−1.02516
$893$	−4845.61	−0.181581
$894$	5675.00	0.212305
$895$	−8597.18	−0.321086
$896$	−39071.9	−1.45681
$897$	870.975	0.0324203
$898$	−3465.05	−0.128764
$899$	−19263.8	−0.714665
$900$	−1568.69	−0.0580996
$901$	5581.11	0.206364
$902$	0	0
$903$	9765.90	0.359899
$904$	−16719.2	−0.615125
$905$	10098.7	0.370931
$906$	5034.75	0.184623
$907$	17340.0	0.634802	0.317401	−	0.948291i	$-0.397190\pi$
$907$	17340.0	0.634802	0.317401	+	0.948291i	$0.397190\pi$
$908$	2127.10	0.0777427
$909$	7955.59	0.290286
$910$	9476.56	0.345214
$911$	−27609.2	−1.00410	−0.502049	−	0.864839i	$-0.667420\pi$
$911$	−27609.2	−1.00410	−0.502049	+	0.864839i	$0.667420\pi$
$912$	4239.27	0.153921
$913$	0	0
$914$	−3319.41	−0.120127
$915$	−4110.52	−0.148513
$916$	−10620.7	−0.383098
$917$	−74775.8	−2.69282
$918$	−1439.90	−0.0517688
$919$	−24306.5	−0.872468	−0.436234	−	0.899833i	$-0.643688\pi$
$919$	−24306.5	−0.872468	−0.436234	+	0.899833i	$0.643688\pi$
$920$	−315.538	−0.0113076
$921$	−4294.14	−0.153634
$922$	−7595.00	−0.271289
$923$	47535.0	1.69516
$924$	0	0
$925$	7016.02	0.249390
$926$	1905.33	0.0676168
$927$	−12884.7	−0.456514
$928$	14400.4	0.509392
$929$	18634.1	0.658091	0.329045	−	0.944314i	$-0.393273\pi$
$929$	18634.1	0.658091	0.329045	+	0.944314i	$0.393273\pi$
$930$	−3303.88	−0.116493
$931$	13067.0	0.459995
$932$	−44515.6	−1.56454
$933$	25646.9	0.899937
$934$	15349.1	0.537729
$935$	0	0
$936$	−9541.43	−0.333196
$937$	−9050.19	−0.315536	−0.157768	−	0.987476i	$-0.550430\pi$
$937$	−9050.19	−0.315536	−0.157768	+	0.987476i	$0.550430\pi$
$938$	−15707.8	−0.546778
$939$	18494.9	0.642767
$940$	4827.37	0.167502
$941$	−13817.1	−0.478665	−0.239333	−	0.970938i	$-0.576929\pi$
$941$	−13817.1	−0.478665	−0.239333	+	0.970938i	$0.576929\pi$
$942$	10462.9	0.361889
$943$	1472.01	0.0508328
$944$	−14339.1	−0.494383
$945$	3613.45	0.124387
$946$	0	0
$947$	−36100.7	−1.23877	−0.619386	−	0.785087i	$-0.712618\pi$
$947$	−36100.7	−1.23877	−0.619386	+	0.785087i	$0.712618\pi$
$948$	12852.7	0.440334
$949$	−62225.6	−2.12848
$950$	886.969	0.0302916
$951$	−15130.7	−0.515928
$952$	21371.5	0.727578
$953$	8171.47	0.277754	0.138877	−	0.990310i	$-0.455651\pi$
$953$	8171.47	0.277754	0.138877	+	0.990310i	$0.455651\pi$
$954$	−968.294	−0.0328613
$955$	18757.5	0.635579
$956$	9789.05	0.331172
$957$	0	0
$958$	−12578.3	−0.424202
$959$	44459.7	1.49706
$960$	−2376.29	−0.0798901
$961$	17399.4	0.584047
$962$	19872.1	0.666010
$963$	8089.20	0.270686
$964$	−1543.28	−0.0515618
$965$	15523.7	0.517849
$966$	338.465	0.0112732
$967$	4535.04	0.150814	0.0754070	−	0.997153i	$-0.475974\pi$
$967$	4535.04	0.150814	0.0754070	+	0.997153i	$0.475974\pi$
$968$	0	0
$969$	−5521.36	−0.183046
$970$	−304.525	−0.0100801
$971$	−23286.0	−0.769602	−0.384801	−	0.922999i	$-0.625730\pi$
$971$	−23286.0	−0.769602	−0.384801	+	0.922999i	$0.625730\pi$
$972$	−1694.19	−0.0559064
$973$	1032.12	0.0340065
$974$	−3958.02	−0.130208
$975$	5237.79	0.172045
$976$	−11066.6	−0.362942
$977$	4004.13	0.131119	0.0655597	−	0.997849i	$-0.479117\pi$
$977$	4004.13	0.131119	0.0655597	+	0.997849i	$0.479117\pi$
$978$	10632.7	0.347645
$979$	0	0
$980$	−13017.9	−0.424327
$981$	−5356.96	−0.174347
$982$	−11989.5	−0.389612
$983$	13914.5	0.451479	0.225739	−	0.974188i	$-0.427520\pi$
$983$	13914.5	0.451479	0.225739	+	0.974188i	$0.427520\pi$
$984$	−16125.7	−0.522428
$985$	−24779.1	−0.801551
$986$	−4729.16	−0.152745
$987$	−11119.8	−0.358608
$988$	−17037.4	−0.548615
$989$	505.592	0.0162557
$990$	0	0
$991$	−60060.2	−1.92520	−0.962601	−	0.270924i	$-0.912671\pi$
$991$	−60060.2	−1.92520	−0.962601	+	0.270924i	$0.912671\pi$
$992$	−35276.5	−1.12906
$993$	14121.8	0.451300
$994$	18472.3	0.589443
$995$	−18302.1	−0.583130
$996$	−20605.3	−0.655525
$997$	−15023.2	−0.477220	−0.238610	−	0.971115i	$-0.576692\pi$
$997$	−15023.2	−0.477220	−0.238610	+	0.971115i	$0.576692\pi$
$998$	−18422.0	−0.584308
$999$	7577.30	0.239975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.bk.1.8	yes	12
11.10	odd	2	inner	1815.4.a.bk.1.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.4.a.bk.1.5	✓	12	11.10	odd	2	inner
1815.4.a.bk.1.8	yes	12	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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q+1.01393 q^{2} +3.00000 q^{3} -6.97196 q^{4} +5.00000 q^{5} +3.04178 q^{6} +26.7663 q^{7} -15.1804 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+1.01393 q^{2} +3.00000 q^{3} -6.97196 q^{4} +5.00000 q^{5} +3.04178 q^{6} +26.7663 q^{7} -15.1804 q^{8} +9.00000 q^{9} +5.06963 q^{10} -20.9159 q^{12} +69.8371 q^{13} +27.1390 q^{14} +15.0000 q^{15} +40.3838 q^{16} -52.5972 q^{17} +9.12533 q^{18} +34.9915 q^{19} -34.8598 q^{20} +80.2989 q^{21} +4.15717 q^{23} -45.5413 q^{24} +25.0000 q^{25} +70.8096 q^{26} +27.0000 q^{27} -186.613 q^{28} +88.6779 q^{29} +15.2089 q^{30} -217.233 q^{31} +162.390 q^{32} -53.3296 q^{34} +133.832 q^{35} -62.7476 q^{36} +280.641 q^{37} +35.4787 q^{38} +209.511 q^{39} -75.9022 q^{40} +354.090 q^{41} +81.4171 q^{42} +121.619 q^{43} +45.0000 q^{45} +4.21506 q^{46} -138.480 q^{47} +121.151 q^{48} +373.435 q^{49} +25.3481 q^{50} -157.791 q^{51} -486.901 q^{52} -106.111 q^{53} +27.3760 q^{54} -406.324 q^{56} +104.974 q^{57} +89.9128 q^{58} -355.070 q^{59} -104.579 q^{60} -274.034 q^{61} -220.258 q^{62} +240.897 q^{63} -158.419 q^{64} +349.186 q^{65} -578.790 q^{67} +366.705 q^{68} +12.4715 q^{69} +135.695 q^{70} +680.655 q^{71} -136.624 q^{72} -891.010 q^{73} +284.549 q^{74} +75.0000 q^{75} -243.959 q^{76} +212.429 q^{78} -614.496 q^{79} +201.919 q^{80} +81.0000 q^{81} +359.021 q^{82} +985.150 q^{83} -559.840 q^{84} -262.986 q^{85} +123.313 q^{86} +266.034 q^{87} +1470.02 q^{89} +45.6266 q^{90} +1869.28 q^{91} -28.9836 q^{92} -651.700 q^{93} -140.408 q^{94} +174.957 q^{95} +487.169 q^{96} -60.0685 q^{97} +378.635 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 36 q^{3} + 62 q^{4} + 60 q^{5} + 108 q^{9}+O(q^{10}) \) 12 * q + 36 * q^3 + 62 * q^4 + 60 * q^5 + 108 * q^9	\( 12 q + 36 q^{3} + 62 q^{4} + 60 q^{5} + 108 q^{9} + 186 q^{12} + 16 q^{14} + 180 q^{15} + 610 q^{16} + 310 q^{20} - 44 q^{23} + 300 q^{25} + 1016 q^{26} + 324 q^{27} + 288 q^{31} + 710 q^{34} + 558 q^{36} - 488 q^{37} + 872 q^{38} + 48 q^{42} + 540 q^{45} + 264 q^{47} + 1830 q^{48} + 140 q^{49} + 1144 q^{53} + 3172 q^{56} - 488 q^{58} + 1400 q^{59} + 930 q^{60} + 3088 q^{64} + 2032 q^{67} - 132 q^{69} + 80 q^{70} + 4852 q^{71} + 900 q^{75} + 3048 q^{78} + 3050 q^{80} + 972 q^{81} + 816 q^{82} - 2232 q^{86} + 4932 q^{89} + 7336 q^{91} - 2526 q^{92} + 864 q^{93} + 1436 q^{97}+O(q^{100}) \) 12 * q + 36 * q^3 + 62 * q^4 + 60 * q^5 + 108 * q^9 + 186 * q^12 + 16 * q^14 + 180 * q^15 + 610 * q^16 + 310 * q^20 - 44 * q^23 + 300 * q^25 + 1016 * q^26 + 324 * q^27 + 288 * q^31 + 710 * q^34 + 558 * q^36 - 488 * q^37 + 872 * q^38 + 48 * q^42 + 540 * q^45 + 264 * q^47 + 1830 * q^48 + 140 * q^49 + 1144 * q^53 + 3172 * q^56 - 488 * q^58 + 1400 * q^59 + 930 * q^60 + 3088 * q^64 + 2032 * q^67 - 132 * q^69 + 80 * q^70 + 4852 * q^71 + 900 * q^75 + 3048 * q^78 + 3050 * q^80 + 972 * q^81 + 816 * q^82 - 2232 * q^86 + 4932 * q^89 + 7336 * q^91 - 2526 * q^92 + 864 * q^93 + 1436 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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