LMFDB - Embedded newform 1815.4.a.bb.1.6

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:	$6$
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} - 44x^{4} + 495x^{2} - 1200 $ x^6 - 44x^4 + 495x^2 - 1200
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$5.26184$ 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)

$f(q)$	$=$	$q+5.26184 q^{2} +3.00000 q^{3} +19.6870 q^{4} +5.00000 q^{5} +15.7855 q^{6} -5.37309 q^{7} +61.4950 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+5.26184 q^{2} +3.00000 q^{3} +19.6870 q^{4} +5.00000 q^{5} +15.7855 q^{6} -5.37309 q^{7} +61.4950 q^{8} +9.00000 q^{9} +26.3092 q^{10} +59.0609 q^{12} +5.15059 q^{13} -28.2724 q^{14} +15.0000 q^{15} +166.081 q^{16} -107.538 q^{17} +47.3566 q^{18} +70.1491 q^{19} +98.4349 q^{20} -16.1193 q^{21} +104.545 q^{23} +184.485 q^{24} +25.0000 q^{25} +27.1016 q^{26} +27.0000 q^{27} -105.780 q^{28} +122.545 q^{29} +78.9276 q^{30} +252.244 q^{31} +381.933 q^{32} -565.849 q^{34} -26.8655 q^{35} +177.183 q^{36} +150.545 q^{37} +369.113 q^{38} +15.4518 q^{39} +307.475 q^{40} +17.3082 q^{41} -84.8171 q^{42} +459.004 q^{43} +45.0000 q^{45} +550.098 q^{46} +310.585 q^{47} +498.244 q^{48} -314.130 q^{49} +131.546 q^{50} -322.615 q^{51} +101.400 q^{52} -322.186 q^{53} +142.070 q^{54} -330.418 q^{56} +210.447 q^{57} +644.812 q^{58} +425.601 q^{59} +295.305 q^{60} -75.2233 q^{61} +1327.27 q^{62} -48.3578 q^{63} +681.020 q^{64} +25.7530 q^{65} -898.576 q^{67} -2117.10 q^{68} +313.634 q^{69} -141.362 q^{70} +1192.10 q^{71} +553.455 q^{72} -277.379 q^{73} +792.142 q^{74} +75.0000 q^{75} +1381.02 q^{76} +81.3048 q^{78} -1257.17 q^{79} +830.406 q^{80} +81.0000 q^{81} +91.0729 q^{82} +156.497 q^{83} -317.340 q^{84} -537.691 q^{85} +2415.21 q^{86} +367.635 q^{87} -96.0978 q^{89} +236.783 q^{90} -27.6746 q^{91} +2058.17 q^{92} +756.731 q^{93} +1634.25 q^{94} +350.746 q^{95} +1145.80 q^{96} -840.690 q^{97} -1652.90 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9}+O(q^{10}) $ 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9	$ 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9} + 120 q^{12} + 116 q^{14} + 90 q^{15} + 164 q^{16} + 200 q^{20} + 56 q^{23} + 150 q^{25} + 292 q^{26} + 162 q^{27} + 576 q^{31} - 92 q^{34} + 360 q^{36} + 332 q^{37} + 496 q^{38} + 348 q^{42} + 270 q^{45} + 96 q^{47} + 492 q^{48} + 454 q^{49} + 308 q^{53} - 652 q^{56} + 1784 q^{58} + 2080 q^{59} + 600 q^{60} + 1928 q^{64} - 1168 q^{67} + 168 q^{69} + 580 q^{70} + 1064 q^{71} + 450 q^{75} + 876 q^{78} + 820 q^{80} + 486 q^{81} + 24 q^{82} + 5412 q^{86} - 684 q^{89} + 2744 q^{91} + 1368 q^{92} + 1728 q^{93} + 2812 q^{97}+O(q^{100}) $ 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9 + 120 * q^12 + 116 * q^14 + 90 * q^15 + 164 * q^16 + 200 * q^20 + 56 * q^23 + 150 * q^25 + 292 * q^26 + 162 * q^27 + 576 * q^31 - 92 * q^34 + 360 * q^36 + 332 * q^37 + 496 * q^38 + 348 * q^42 + 270 * q^45 + 96 * q^47 + 492 * q^48 + 454 * q^49 + 308 * q^53 - 652 * q^56 + 1784 * q^58 + 2080 * q^59 + 600 * q^60 + 1928 * q^64 - 1168 * q^67 + 168 * q^69 + 580 * q^70 + 1064 * q^71 + 450 * q^75 + 876 * q^78 + 820 * q^80 + 486 * q^81 + 24 * q^82 + 5412 * q^86 - 684 * q^89 + 2744 * q^91 + 1368 * q^92 + 1728 * q^93 + 2812 * 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 $

$2$	5.26184	1.86034	0.930171	−	0.367127i	$-0.119658\pi$
$2$	5.26184	1.86034	0.930171	+	0.367127i	$0.119658\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	19.6870	2.46087
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	15.7855	1.07407
$7$	−5.37309	−0.290120	−0.145060	−	0.989423i	$-0.546337\pi$
$7$	−5.37309	−0.290120	−0.145060	+	0.989423i	$0.546337\pi$
$8$	61.4950	2.71772
$9$	9.00000	0.333333
$10$	26.3092	0.831970
$11$	0	0
$11$	0	0
$12$	59.0609	1.42079
$13$	5.15059	0.109886	0.0549430	−	0.998489i	$-0.482502\pi$
$13$	5.15059	0.109886	0.0549430	+	0.998489i	$0.482502\pi$
$14$	−28.2724	−0.539722
$15$	15.0000	0.258199
$16$	166.081	2.59502
$17$	−107.538	−1.53423	−0.767113	−	0.641512i	$-0.778308\pi$
$17$	−107.538	−1.53423	−0.767113	+	0.641512i	$0.778308\pi$
$18$	47.3566	0.620114
$19$	70.1491	0.847016	0.423508	−	0.905892i	$-0.360799\pi$
$19$	70.1491	0.847016	0.423508	+	0.905892i	$0.360799\pi$
$20$	98.4349	1.10054
$21$	−16.1193	−0.167501
$22$	0	0
$23$	104.545	0.947786	0.473893	−	0.880582i	$-0.342848\pi$
$23$	104.545	0.947786	0.473893	+	0.880582i	$0.342848\pi$
$24$	184.485	1.56908
$25$	25.0000	0.200000
$26$	27.1016	0.204425
$27$	27.0000	0.192450
$28$	−105.780	−0.713947
$29$	122.545	0.784691	0.392345	−	0.919818i	$-0.371664\pi$
$29$	122.545	0.784691	0.392345	+	0.919818i	$0.371664\pi$
$30$	78.9276	0.480338
$31$	252.244	1.46143	0.730715	−	0.682683i	$-0.239187\pi$
$31$	252.244	1.46143	0.730715	+	0.682683i	$0.239187\pi$
$32$	381.933	2.10990
$33$	0	0
$34$	−565.849	−2.85419
$35$	−26.8655	−0.129745
$36$	177.183	0.820291
$37$	150.545	0.668903	0.334451	−	0.942413i	$-0.391449\pi$
$37$	150.545	0.668903	0.334451	+	0.942413i	$0.391449\pi$
$38$	369.113	1.57574
$39$	15.4518	0.0634427
$40$	307.475	1.21540
$41$	17.3082	0.0659289	0.0329645	−	0.999457i	$-0.489505\pi$
$41$	17.3082	0.0659289	0.0329645	+	0.999457i	$0.489505\pi$
$42$	−84.8171	−0.311608
$43$	459.004	1.62785	0.813924	−	0.580972i	$-0.197327\pi$
$43$	459.004	1.62785	0.813924	+	0.580972i	$0.197327\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	550.098	1.76321
$47$	310.585	0.963904	0.481952	−	0.876198i	$-0.339928\pi$
$47$	310.585	0.963904	0.481952	+	0.876198i	$0.339928\pi$
$48$	498.244	1.49823
$49$	−314.130	−0.915831
$50$	131.546	0.372068
$51$	−322.615	−0.885786
$52$	101.400	0.270415
$53$	−322.186	−0.835013	−0.417507	−	0.908674i	$-0.637096\pi$
$53$	−322.186	−0.835013	−0.417507	+	0.908674i	$0.637096\pi$
$54$	142.070	0.358023
$55$	0	0
$56$	−330.418	−0.788464
$57$	210.447	0.489025
$58$	644.812	1.45979
$59$	425.601	0.939128	0.469564	−	0.882898i	$-0.344411\pi$
$59$	425.601	0.939128	0.469564	+	0.882898i	$0.344411\pi$
$60$	295.305	0.635394
$61$	−75.2233	−0.157891	−0.0789455	−	0.996879i	$-0.525155\pi$
$61$	−75.2233	−0.157891	−0.0789455	+	0.996879i	$0.525155\pi$
$62$	1327.27	2.71876
$63$	−48.3578	−0.0967065
$64$	681.020	1.33012
$65$	25.7530	0.0491425
$66$	0	0
$67$	−898.576	−1.63849	−0.819243	−	0.573447i	$-0.805606\pi$
$67$	−898.576	−1.63849	−0.819243	+	0.573447i	$0.805606\pi$
$68$	−2117.10	−3.77554
$69$	313.634	0.547204
$70$	−141.362	−0.241371
$71$	1192.10	1.99263	0.996315	−	0.0857641i	$-0.0273331\pi$
$71$	1192.10	1.99263	0.996315	+	0.0857641i	$0.0273331\pi$
$72$	553.455	0.905907
$73$	−277.379	−0.444722	−0.222361	−	0.974964i	$-0.571376\pi$
$73$	−277.379	−0.444722	−0.222361	+	0.974964i	$0.571376\pi$
$74$	792.142	1.24439
$75$	75.0000	0.115470
$76$	1381.02	2.08440
$77$	0	0
$78$	81.3048	0.118025
$79$	−1257.17	−1.79041	−0.895207	−	0.445651i	$-0.852972\pi$
$79$	−1257.17	−1.79041	−0.895207	+	0.445651i	$0.852972\pi$
$80$	830.406	1.16053
$81$	81.0000	0.111111
$82$	91.0729	0.122650
$83$	156.497	0.206961	0.103480	−	0.994632i	$-0.467002\pi$
$83$	156.497	0.206961	0.103480	+	0.994632i	$0.467002\pi$
$84$	−317.340	−0.412198
$85$	−537.691	−0.686127
$86$	2415.21	3.02835
$87$	367.635	0.453042
$88$	0	0
$89$	−96.0978	−0.114453	−0.0572267	−	0.998361i	$-0.518226\pi$
$89$	−96.0978	−0.114453	−0.0572267	+	0.998361i	$0.518226\pi$
$90$	236.783	0.277323
$91$	−27.6746	−0.0318801
$92$	2058.17	2.33238
$93$	756.731	0.843757
$94$	1634.25	1.79319
$95$	350.746	0.378797
$96$	1145.80	1.21815
$97$	−840.690	−0.879991	−0.439995	−	0.898000i	$-0.645020\pi$
$97$	−840.690	−0.879991	−0.439995	+	0.898000i	$0.645020\pi$
$98$	−1652.90	−1.70376
$99$	0	0
$100$	492.174	0.492174
$101$	−1906.92	−1.87867	−0.939333	−	0.343008i	$-0.888554\pi$
$101$	−1906.92	−1.87867	−0.939333	+	0.343008i	$0.888554\pi$
$102$	−1697.55	−1.64787
$103$	142.967	0.136767	0.0683834	−	0.997659i	$-0.478216\pi$
$103$	142.967	0.136767	0.0683834	+	0.997659i	$0.478216\pi$
$104$	316.736	0.298639
$105$	−80.5964	−0.0749086
$106$	−1695.29	−1.55341
$107$	−1908.21	−1.72405	−0.862027	−	0.506862i	$-0.830805\pi$
$107$	−1908.21	−1.72405	−0.862027	+	0.506862i	$0.830805\pi$
$108$	531.548	0.473595
$109$	−672.607	−0.591046	−0.295523	−	0.955336i	$-0.595494\pi$
$109$	−672.607	−0.591046	−0.295523	+	0.955336i	$0.595494\pi$
$110$	0	0
$111$	451.634	0.386191
$112$	−892.369	−0.752866
$113$	−1635.11	−1.36123	−0.680613	−	0.732644i	$-0.738286\pi$
$113$	−1635.11	−1.36123	−0.680613	+	0.732644i	$0.738286\pi$
$114$	1107.34	0.909754
$115$	522.724	0.423863
$116$	2412.54	1.93102
$117$	46.3553	0.0366286
$118$	2239.45	1.74710
$119$	577.813	0.445109
$120$	922.425	0.701713
$121$	0	0
$122$	−395.813	−0.293731
$123$	51.9246	0.0380641
$124$	4965.92	3.59639
$125$	125.000	0.0894427
$126$	−254.451	−0.179907
$127$	941.002	0.657484	0.328742	−	0.944420i	$-0.393375\pi$
$127$	941.002	0.657484	0.328742	+	0.944420i	$0.393375\pi$
$128$	527.958	0.364573
$129$	1377.01	0.939838
$130$	135.508	0.0914218
$131$	1243.83	0.829574	0.414787	−	0.909919i	$-0.363856\pi$
$131$	1243.83	0.829574	0.414787	+	0.909919i	$0.363856\pi$
$132$	0	0
$133$	−376.917	−0.245736
$134$	−4728.16	−3.04814
$135$	135.000	0.0860663
$136$	−6613.06	−4.16960
$137$	−1127.14	−0.702904	−0.351452	−	0.936206i	$-0.614312\pi$
$137$	−1127.14	−0.702904	−0.351452	+	0.936206i	$0.614312\pi$
$138$	1650.29	1.01799
$139$	−2537.94	−1.54867	−0.774335	−	0.632776i	$-0.781916\pi$
$139$	−2537.94	−1.54867	−0.774335	+	0.632776i	$0.781916\pi$
$140$	−528.899	−0.319287
$141$	931.756	0.556510
$142$	6272.67	3.70697
$143$	0	0
$144$	1494.73	0.865006
$145$	612.725	0.350924
$146$	−1459.52	−0.827335
$147$	−942.390	−0.528755
$148$	2963.77	1.64608
$149$	−2635.22	−1.44890	−0.724448	−	0.689329i	$-0.757905\pi$
$149$	−2635.22	−1.44890	−0.724448	+	0.689329i	$0.757905\pi$
$150$	394.638	0.214814
$151$	2665.26	1.43639	0.718197	−	0.695839i	$-0.244968\pi$
$151$	2665.26	1.43639	0.718197	+	0.695839i	$0.244968\pi$
$152$	4313.82	2.30195
$153$	−967.844	−0.511409
$154$	0	0
$155$	1261.22	0.653571
$156$	304.199	0.156124
$157$	2341.32	1.19017	0.595087	−	0.803661i	$-0.297118\pi$
$157$	2341.32	1.19017	0.595087	+	0.803661i	$0.297118\pi$
$158$	−6615.03	−3.33078
$159$	−966.559	−0.482095
$160$	1909.66	0.943576
$161$	−561.728	−0.274971
$162$	426.209	0.206705
$163$	−3408.54	−1.63790	−0.818949	−	0.573866i	$-0.805443\pi$
$163$	−3408.54	−1.63790	−0.818949	+	0.573866i	$0.805443\pi$
$164$	340.746	0.162243
$165$	0	0
$166$	823.460	0.385017
$167$	−769.717	−0.356662	−0.178331	−	0.983971i	$-0.557070\pi$
$167$	−769.717	−0.356662	−0.178331	+	0.983971i	$0.557070\pi$
$168$	−991.255	−0.455220
$169$	−2170.47	−0.987925
$170$	−2829.25	−1.27643
$171$	631.342	0.282339
$172$	9036.40	4.00592
$173$	2110.51	0.927508	0.463754	−	0.885964i	$-0.346502\pi$
$173$	2110.51	0.927508	0.463754	+	0.885964i	$0.346502\pi$
$174$	1934.44	0.842812
$175$	−134.327	−0.0580239
$176$	0	0
$177$	1276.80	0.542206
$178$	−505.651	−0.212922
$179$	1473.25	0.615173	0.307587	−	0.951520i	$-0.400479\pi$
$179$	1473.25	0.615173	0.307587	+	0.951520i	$0.400479\pi$
$180$	885.914	0.366845
$181$	−2290.00	−0.940413	−0.470206	−	0.882557i	$-0.655821\pi$
$181$	−2290.00	−0.940413	−0.470206	+	0.882557i	$0.655821\pi$
$182$	−145.619	−0.0593078
$183$	−225.670	−0.0911584
$184$	6428.98	2.57582
$185$	752.724	0.299142
$186$	3981.80	1.56968
$187$	0	0
$188$	6114.48	2.37205
$189$	−145.073	−0.0558335
$190$	1845.57	0.704692
$191$	3880.34	1.47001	0.735003	−	0.678063i	$-0.237180\pi$
$191$	3880.34	1.47001	0.735003	+	0.678063i	$0.237180\pi$
$192$	2043.06	0.767944
$193$	−852.978	−0.318128	−0.159064	−	0.987268i	$-0.550848\pi$
$193$	−852.978	−0.318128	−0.159064	+	0.987268i	$0.550848\pi$
$194$	−4423.58	−1.63708
$195$	77.2589	0.0283724
$196$	−6184.27	−2.25374
$197$	766.748	0.277302	0.138651	−	0.990341i	$-0.455723\pi$
$197$	766.748	0.277302	0.138651	+	0.990341i	$0.455723\pi$
$198$	0	0
$199$	1242.15	0.442479	0.221240	−	0.975219i	$-0.428990\pi$
$199$	1242.15	0.442479	0.221240	+	0.975219i	$0.428990\pi$
$200$	1537.38	0.543544
$201$	−2695.73	−0.945980
$202$	−10033.9	−3.49496
$203$	−658.445	−0.227654
$204$	−6351.31	−2.17981
$205$	86.5409	0.0294843
$206$	752.271	0.254433
$207$	940.902	0.315929
$208$	855.416	0.285156
$209$	0	0
$210$	−424.085	−0.139356
$211$	4347.95	1.41860	0.709301	−	0.704906i	$-0.249011\pi$
$211$	4347.95	1.41860	0.709301	+	0.704906i	$0.249011\pi$
$212$	−6342.88	−2.05486
$213$	3576.31	1.15045
$214$	−10040.7	−3.20733
$215$	2295.02	0.727996
$216$	1660.37	0.523026
$217$	−1355.33	−0.423989
$218$	−3539.15	−1.09955
$219$	−832.136	−0.256760
$220$	0	0
$221$	−553.886	−0.168590
$222$	2376.43	0.718447
$223$	−1123.30	−0.337317	−0.168658	−	0.985675i	$-0.553943\pi$
$223$	−1123.30	−0.337317	−0.168658	+	0.985675i	$0.553943\pi$
$224$	−2052.16	−0.612123
$225$	225.000	0.0666667
$226$	−8603.70	−2.53234
$227$	435.002	0.127190	0.0635949	−	0.997976i	$-0.479743\pi$
$227$	435.002	0.127190	0.0635949	+	0.997976i	$0.479743\pi$
$228$	4143.07	1.20343
$229$	4207.36	1.21411	0.607053	−	0.794661i	$-0.292351\pi$
$229$	4207.36	1.21411	0.607053	+	0.794661i	$0.292351\pi$
$230$	2750.49	0.788530
$231$	0	0
$232$	7535.91	2.13257
$233$	−235.268	−0.0661499	−0.0330749	−	0.999453i	$-0.510530\pi$
$233$	−235.268	−0.0661499	−0.0330749	+	0.999453i	$0.510530\pi$
$234$	243.914	0.0681418
$235$	1552.93	0.431071
$236$	8378.80	2.31107
$237$	−3771.51	−1.03370
$238$	3040.36	0.828055
$239$	647.975	0.175373	0.0876863	−	0.996148i	$-0.472053\pi$
$239$	647.975	0.175373	0.0876863	+	0.996148i	$0.472053\pi$
$240$	2491.22	0.670031
$241$	−2281.64	−0.609848	−0.304924	−	0.952377i	$-0.598631\pi$
$241$	−2281.64	−0.609848	−0.304924	+	0.952377i	$0.598631\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	−1480.92	−0.388550
$245$	−1570.65	−0.409572
$246$	273.219	0.0708122
$247$	361.309	0.0930752
$248$	15511.7	3.97176
$249$	469.490	0.119489
$250$	657.730	0.166394
$251$	−4869.34	−1.22450	−0.612251	−	0.790664i	$-0.709736\pi$
$251$	−4869.34	−1.22450	−0.612251	+	0.790664i	$0.709736\pi$
$252$	−952.019	−0.237982
$253$	0	0
$254$	4951.41	1.22314
$255$	−1613.07	−0.396136
$256$	−2670.13	−0.651887
$257$	−1381.17	−0.335234	−0.167617	−	0.985852i	$-0.553607\pi$
$257$	−1381.17	−0.335234	−0.167617	+	0.985852i	$0.553607\pi$
$258$	7245.62	1.74842
$259$	−808.890	−0.194062
$260$	506.998	0.120933
$261$	1102.91	0.261564
$262$	6544.85	1.54329
$263$	−1121.05	−0.262841	−0.131420	−	0.991327i	$-0.541954\pi$
$263$	−1121.05	−0.262841	−0.131420	+	0.991327i	$0.541954\pi$
$264$	0	0
$265$	−1610.93	−0.373429
$266$	−1983.28	−0.457153
$267$	−288.293	−0.0660797
$268$	−17690.2	−4.03210
$269$	−1161.36	−0.263232	−0.131616	−	0.991301i	$-0.542017\pi$
$269$	−1161.36	−0.263232	−0.131616	+	0.991301i	$0.542017\pi$
$270$	710.349	0.160113
$271$	−3568.57	−0.799908	−0.399954	−	0.916535i	$-0.630974\pi$
$271$	−3568.57	−0.799908	−0.399954	+	0.916535i	$0.630974\pi$
$272$	−17860.1	−3.98135
$273$	−83.0238	−0.0184060
$274$	−5930.81	−1.30764
$275$	0	0
$276$	6174.51	1.34660
$277$	−8346.58	−1.81046	−0.905230	−	0.424922i	$-0.860302\pi$
$277$	−8346.58	−1.81046	−0.905230	+	0.424922i	$0.860302\pi$
$278$	−13354.2	−2.88106
$279$	2270.19	0.487143
$280$	−1652.09	−0.352612
$281$	1723.69	0.365931	0.182966	−	0.983119i	$-0.441430\pi$
$281$	1723.69	0.365931	0.182966	+	0.983119i	$0.441430\pi$
$282$	4902.75	1.03530
$283$	2086.76	0.438321	0.219160	−	0.975689i	$-0.429668\pi$
$283$	2086.76	0.438321	0.219160	+	0.975689i	$0.429668\pi$
$284$	23468.9	4.90361
$285$	1052.24	0.218699
$286$	0	0
$287$	−92.9985	−0.0191273
$288$	3437.39	0.703300
$289$	6651.47	1.35385
$290$	3224.06	0.652839
$291$	−2522.07	−0.508063
$292$	−5460.75	−1.09440
$293$	−3127.52	−0.623588	−0.311794	−	0.950150i	$-0.600930\pi$
$293$	−3127.52	−0.623588	−0.311794	+	0.950150i	$0.600930\pi$
$294$	−4958.71	−0.983665
$295$	2128.01	0.419991
$296$	9257.75	1.81789
$297$	0	0
$298$	−13866.1	−2.69544
$299$	538.467	0.104148
$300$	1476.52	0.284157
$301$	−2466.27	−0.472271
$302$	14024.2	2.67219
$303$	−5720.75	−1.08465
$304$	11650.4	2.19802
$305$	−376.116	−0.0706110
$306$	−5092.64	−0.951395
$307$	1397.12	0.259732	0.129866	−	0.991532i	$-0.458545\pi$
$307$	1397.12	0.259732	0.129866	+	0.991532i	$0.458545\pi$
$308$	0	0
$309$	428.902	0.0789623
$310$	6636.33	1.21587
$311$	5405.45	0.985579	0.492790	−	0.870148i	$-0.335977\pi$
$311$	5405.45	0.985579	0.492790	+	0.870148i	$0.335977\pi$
$312$	950.207	0.172419
$313$	6240.77	1.12699	0.563497	−	0.826118i	$-0.309456\pi$
$313$	6240.77	1.12699	0.563497	+	0.826118i	$0.309456\pi$
$314$	12319.6	2.21413
$315$	−241.789	−0.0432485
$316$	−24749.9	−4.40598
$317$	−6919.66	−1.22601	−0.613007	−	0.790077i	$-0.710040\pi$
$317$	−6919.66	−1.22601	−0.613007	+	0.790077i	$0.710040\pi$
$318$	−5085.88	−0.896862
$319$	0	0
$320$	3405.10	0.594847
$321$	−5724.64	−0.995383
$322$	−2955.72	−0.511541
$323$	−7543.71	−1.29951
$324$	1594.64	0.273430
$325$	128.765	0.0219772
$326$	−17935.2	−3.04705
$327$	−2017.82	−0.341241
$328$	1064.37	0.179176
$329$	−1668.80	−0.279648
$330$	0	0
$331$	−3134.58	−0.520519	−0.260260	−	0.965539i	$-0.583808\pi$
$331$	−3134.58	−0.520519	−0.260260	+	0.965539i	$0.583808\pi$
$332$	3080.94	0.509303
$333$	1354.90	0.222968
$334$	−4050.13	−0.663513
$335$	−4492.88	−0.732753
$336$	−2677.11	−0.434667
$337$	−6484.89	−1.04823	−0.524117	−	0.851647i	$-0.675604\pi$
$337$	−6484.89	−1.04823	−0.524117	+	0.851647i	$0.675604\pi$
$338$	−11420.7	−1.83788
$339$	−4905.34	−0.785904
$340$	−10585.5	−1.68847
$341$	0	0
$342$	3322.02	0.525247
$343$	3530.82	0.555820
$344$	28226.5	4.42404
$345$	1568.17	0.244717
$346$	11105.1	1.72548
$347$	2535.12	0.392198	0.196099	−	0.980584i	$-0.437173\pi$
$347$	2535.12	0.392198	0.196099	+	0.980584i	$0.437173\pi$
$348$	7237.62	1.11488
$349$	9319.24	1.42936	0.714681	−	0.699451i	$-0.246572\pi$
$349$	9319.24	1.42936	0.714681	+	0.699451i	$0.246572\pi$
$350$	−706.809	−0.107944
$351$	139.066	0.0211476
$352$	0	0
$353$	−3958.44	−0.596846	−0.298423	−	0.954434i	$-0.596461\pi$
$353$	−3958.44	−0.596846	−0.298423	+	0.954434i	$0.596461\pi$
$354$	6718.34	1.00869
$355$	5960.52	0.891132
$356$	−1891.88	−0.281655
$357$	1733.44	0.256984
$358$	7752.02	1.14443
$359$	−10876.2	−1.59896	−0.799480	−	0.600693i	$-0.794891\pi$
$359$	−10876.2	−1.59896	−0.799480	+	0.600693i	$0.794891\pi$
$360$	2767.28	0.405134
$361$	−1938.10	−0.282564
$362$	−12049.6	−1.74949
$363$	0	0
$364$	−544.829	−0.0784528
$365$	−1386.89	−0.198886
$366$	−1187.44	−0.169586
$367$	4192.86	0.596364	0.298182	−	0.954509i	$-0.403620\pi$
$367$	4192.86	0.596364	0.298182	+	0.954509i	$0.403620\pi$
$368$	17362.9	2.45952
$369$	155.774	0.0219763
$370$	3960.71	0.556507
$371$	1731.14	0.242254
$372$	14897.7	2.07638
$373$	−11108.9	−1.54209	−0.771044	−	0.636782i	$-0.780265\pi$
$373$	−11108.9	−1.54209	−0.771044	+	0.636782i	$0.780265\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	19099.4	2.61962
$377$	631.179	0.0862265
$378$	−763.353	−0.103869
$379$	−4716.38	−0.639219	−0.319610	−	0.947549i	$-0.603552\pi$
$379$	−4716.38	−0.639219	−0.319610	+	0.947549i	$0.603552\pi$
$380$	6905.12	0.932171
$381$	2823.01	0.379599
$382$	20417.7	2.73472
$383$	10117.1	1.34977	0.674883	−	0.737925i	$-0.264194\pi$
$383$	10117.1	1.34977	0.674883	+	0.737925i	$0.264194\pi$
$384$	1583.87	0.210486
$385$	0	0
$386$	−4488.23	−0.591827
$387$	4131.04	0.542616
$388$	−16550.6	−2.16554
$389$	11322.0	1.47571	0.737853	−	0.674961i	$-0.235840\pi$
$389$	11322.0	1.47571	0.737853	+	0.674961i	$0.235840\pi$
$390$	406.524	0.0527824
$391$	−11242.6	−1.45412
$392$	−19317.4	−2.48897
$393$	3731.50	0.478955
$394$	4034.51	0.515877
$395$	−6285.85	−0.800697
$396$	0	0
$397$	5400.42	0.682718	0.341359	−	0.939933i	$-0.389113\pi$
$397$	5400.42	0.682718	0.341359	+	0.939933i	$0.389113\pi$
$398$	6535.97	0.823163
$399$	−1130.75	−0.141876
$400$	4152.03	0.519004
$401$	13782.4	1.71637	0.858183	−	0.513344i	$-0.171594\pi$
$401$	13782.4	1.71637	0.858183	+	0.513344i	$0.171594\pi$
$402$	−14184.5	−1.75985
$403$	1299.20	0.160591
$404$	−37541.4	−4.62315
$405$	405.000	0.0496904
$406$	−3464.64	−0.423515
$407$	0	0
$408$	−19839.2	−2.40732
$409$	113.092	0.0136725	0.00683626	−	0.999977i	$-0.497824\pi$
$409$	113.092	0.0136725	0.00683626	+	0.999977i	$0.497824\pi$
$410$	455.365	0.0548509
$411$	−3381.41	−0.405822
$412$	2814.59	0.336565
$413$	−2286.79	−0.272459
$414$	4950.88	0.587735
$415$	782.483	0.0925556
$416$	1967.18	0.231848
$417$	−7613.82	−0.894125
$418$	0	0
$419$	7352.48	0.857261	0.428630	−	0.903480i	$-0.358996\pi$
$419$	7352.48	0.857261	0.428630	+	0.903480i	$0.358996\pi$
$420$	−1586.70	−0.184340
$421$	3306.97	0.382831	0.191416	−	0.981509i	$-0.438692\pi$
$421$	3306.97	0.382831	0.191416	+	0.981509i	$0.438692\pi$
$422$	22878.2	2.63908
$423$	2795.27	0.321301
$424$	−19812.9	−2.26933
$425$	−2688.46	−0.306845
$426$	18818.0	2.14022
$427$	404.181	0.0458073
$428$	−37566.9	−4.24268
$429$	0	0
$430$	12076.0	1.35432
$431$	−9959.87	−1.11311	−0.556555	−	0.830811i	$-0.687877\pi$
$431$	−9959.87	−1.11311	−0.556555	+	0.830811i	$0.687877\pi$
$432$	4484.19	0.499412
$433$	10913.1	1.21120	0.605598	−	0.795771i	$-0.292934\pi$
$433$	10913.1	1.21120	0.605598	+	0.795771i	$0.292934\pi$
$434$	−7131.52	−0.788765
$435$	1838.18	0.202606
$436$	−13241.6	−1.45449
$437$	7333.72	0.802790
$438$	−4378.57	−0.477662
$439$	6589.60	0.716411	0.358205	−	0.933643i	$-0.383389\pi$
$439$	6589.60	0.716411	0.358205	+	0.933643i	$0.383389\pi$
$440$	0	0
$441$	−2827.17	−0.305277
$442$	−2914.46	−0.313635
$443$	−2597.43	−0.278572	−0.139286	−	0.990252i	$-0.544481\pi$
$443$	−2597.43	−0.278572	−0.139286	+	0.990252i	$0.544481\pi$
$444$	8891.31	0.950367
$445$	−480.489	−0.0511851
$446$	−5910.62	−0.627524
$447$	−7905.66	−0.836521
$448$	−3659.18	−0.385893
$449$	6460.30	0.679021	0.339511	−	0.940602i	$-0.389739\pi$
$449$	6460.30	0.679021	0.339511	+	0.940602i	$0.389739\pi$
$450$	1183.91	0.124023
$451$	0	0
$452$	−32190.4	−3.34980
$453$	7995.77	0.829303
$454$	2288.91	0.236617
$455$	−138.373	−0.0142572
$456$	12941.5	1.32903
$457$	−7676.12	−0.785720	−0.392860	−	0.919598i	$-0.628514\pi$
$457$	−7676.12	−0.785720	−0.392860	+	0.919598i	$0.628514\pi$
$458$	22138.5	2.25865
$459$	−2903.53	−0.295262
$460$	10290.8	1.04307
$461$	7555.51	0.763330	0.381665	−	0.924301i	$-0.375351\pi$
$461$	7555.51	0.763330	0.381665	+	0.924301i	$0.375351\pi$
$462$	0	0
$463$	5480.50	0.550109	0.275055	−	0.961429i	$-0.411304\pi$
$463$	5480.50	0.550109	0.275055	+	0.961429i	$0.411304\pi$
$464$	20352.4	2.03629
$465$	3783.66	0.377339
$466$	−1237.94	−0.123061
$467$	−10339.7	−1.02455	−0.512275	−	0.858822i	$-0.671197\pi$
$467$	−10339.7	−1.02455	−0.512275	+	0.858822i	$0.671197\pi$
$468$	912.596	0.0901384
$469$	4828.13	0.475357
$470$	8171.25	0.801940
$471$	7023.95	0.687147
$472$	26172.3	2.55229
$473$	0	0
$474$	−19845.1	−1.92303
$475$	1753.73	0.169403
$476$	11375.4	1.09536
$477$	−2899.68	−0.278338
$478$	3409.54	0.326253
$479$	4320.65	0.412141	0.206070	−	0.978537i	$-0.433932\pi$
$479$	4320.65	0.412141	0.206070	+	0.978537i	$0.433932\pi$
$480$	5728.99	0.544774
$481$	775.394	0.0735030
$482$	−12005.6	−1.13453
$483$	−1685.18	−0.158755
$484$	0	0
$485$	−4203.45	−0.393544
$486$	1278.63	0.119341
$487$	−9426.09	−0.877078	−0.438539	−	0.898712i	$-0.644504\pi$
$487$	−9426.09	−0.877078	−0.438539	+	0.898712i	$0.644504\pi$
$488$	−4625.86	−0.429104
$489$	−10225.6	−0.945641
$490$	−8264.51	−0.761944
$491$	569.771	0.0523694	0.0261847	−	0.999657i	$-0.491664\pi$
$491$	569.771	0.0523694	0.0261847	+	0.999657i	$0.491664\pi$
$492$	1022.24	0.0936708
$493$	−13178.3	−1.20389
$494$	1901.15	0.173152
$495$	0	0
$496$	41892.9	3.79244
$497$	−6405.29	−0.578101
$498$	2470.38	0.222290
$499$	17072.5	1.53161	0.765804	−	0.643074i	$-0.222341\pi$
$499$	17072.5	1.53161	0.765804	+	0.643074i	$0.222341\pi$
$500$	2460.87	0.220107
$501$	−2309.15	−0.205919
$502$	−25621.7	−2.27799
$503$	18622.7	1.65079	0.825395	−	0.564556i	$-0.190953\pi$
$503$	18622.7	1.65079	0.825395	+	0.564556i	$0.190953\pi$
$504$	−2973.76	−0.262821
$505$	−9534.58	−0.840165
$506$	0	0
$507$	−6511.41	−0.570379
$508$	18525.5	1.61798
$509$	16294.6	1.41895	0.709473	−	0.704733i	$-0.248933\pi$
$509$	16294.6	1.41895	0.709473	+	0.704733i	$0.248933\pi$
$510$	−8487.74	−0.736948
$511$	1490.38	0.129023
$512$	−18273.5	−1.57731
$513$	1894.03	0.163008
$514$	−7267.50	−0.623649
$515$	714.836	0.0611640
$516$	27109.2	2.31282
$517$	0	0
$518$	−4256.25	−0.361021
$519$	6331.52	0.535497
$520$	1583.68	0.133556
$521$	−20991.0	−1.76513	−0.882565	−	0.470189i	$-0.844186\pi$
$521$	−20991.0	−1.76513	−0.882565	+	0.470189i	$0.844186\pi$
$522$	5803.31	0.486598
$523$	11326.3	0.946966	0.473483	−	0.880803i	$-0.342997\pi$
$523$	11326.3	0.946966	0.473483	+	0.880803i	$0.342997\pi$
$524$	24487.3	2.04147
$525$	−402.982	−0.0335001
$526$	−5898.80	−0.488973
$527$	−27125.8	−2.24216
$528$	0	0
$529$	−1237.41	−0.101702
$530$	−8476.47	−0.694706
$531$	3830.41	0.313043
$532$	−7420.36	−0.604725
$533$	89.1474	0.00724466
$534$	−1516.95	−0.122931
$535$	−9541.06	−0.771021
$536$	−55257.9	−4.45295
$537$	4419.76	0.355171
$538$	−6110.90	−0.489702
$539$	0	0
$540$	2657.74	0.211798
$541$	23994.8	1.90687	0.953433	−	0.301603i	$-0.0975218\pi$
$541$	23994.8	1.90687	0.953433	+	0.301603i	$0.0975218\pi$
$542$	−18777.2	−1.48810
$543$	−6870.01	−0.542948
$544$	−41072.4	−3.23707
$545$	−3363.03	−0.264324
$546$	−436.858	−0.0342414
$547$	14347.6	1.12150	0.560748	−	0.827986i	$-0.310513\pi$
$547$	14347.6	1.12150	0.560748	+	0.827986i	$0.310513\pi$
$548$	−22189.9	−1.72976
$549$	−677.009	−0.0526303
$550$	0	0
$551$	8596.42	0.664646
$552$	19286.9	1.48715
$553$	6754.89	0.519434
$554$	−43918.4	−3.36807
$555$	2258.17	0.172710
$556$	−49964.3	−3.81108
$557$	−19041.9	−1.44853	−0.724265	−	0.689521i	$-0.757821\pi$
$557$	−19041.9	−1.44853	−0.724265	+	0.689521i	$0.757821\pi$
$558$	11945.4	0.906253
$559$	2364.14	0.178878
$560$	−4461.85	−0.336692
$561$	0	0
$562$	9069.77	0.680757
$563$	−15082.0	−1.12900	−0.564502	−	0.825432i	$-0.690932\pi$
$563$	−15082.0	−1.12900	−0.564502	+	0.825432i	$0.690932\pi$
$564$	18343.4	1.36950
$565$	−8175.56	−0.608758
$566$	10980.2	0.815426
$567$	−435.220	−0.0322355
$568$	73308.5	5.41541
$569$	−13321.8	−0.981513	−0.490756	−	0.871297i	$-0.663280\pi$
$569$	−13321.8	−0.981513	−0.490756	+	0.871297i	$0.663280\pi$
$570$	5536.70	0.406854
$571$	−24112.7	−1.76722	−0.883611	−	0.468223i	$-0.844895\pi$
$571$	−24112.7	−1.76722	−0.883611	+	0.468223i	$0.844895\pi$
$572$	0	0
$573$	11641.0	0.848709
$574$	−489.343	−0.0355833
$575$	2613.62	0.189557
$576$	6129.18	0.443372
$577$	−4355.07	−0.314218	−0.157109	−	0.987581i	$-0.550217\pi$
$577$	−4355.07	−0.314218	−0.157109	+	0.987581i	$0.550217\pi$
$578$	34999.0	2.51863
$579$	−2558.93	−0.183671
$580$	12062.7	0.863580
$581$	−840.870	−0.0600433
$582$	−13270.7	−0.945171
$583$	0	0
$584$	−17057.4	−1.20863
$585$	231.777	0.0163808
$586$	−16456.5	−1.16009
$587$	14765.2	1.03820	0.519101	−	0.854713i	$-0.326267\pi$
$587$	14765.2	1.03820	0.519101	+	0.854713i	$0.326267\pi$
$588$	−18552.8	−1.30120
$589$	17694.7	1.23785
$590$	11197.2	0.781327
$591$	2300.24	0.160100
$592$	25002.6	1.73581
$593$	−2990.56	−0.207095	−0.103548	−	0.994625i	$-0.533019\pi$
$593$	−2990.56	−0.207095	−0.103548	+	0.994625i	$0.533019\pi$
$594$	0	0
$595$	2889.06	0.199059
$596$	−51879.5	−3.56555
$597$	3726.44	0.255466
$598$	2833.33	0.193752
$599$	−1019.09	−0.0695138	−0.0347569	−	0.999396i	$-0.511066\pi$
$599$	−1019.09	−0.0695138	−0.0347569	+	0.999396i	$0.511066\pi$
$600$	4612.13	0.313815
$601$	−18514.0	−1.25658	−0.628289	−	0.777980i	$-0.716244\pi$
$601$	−18514.0	−1.25658	−0.628289	+	0.777980i	$0.716244\pi$
$602$	−12977.1	−0.878585
$603$	−8087.18	−0.546162
$604$	52470.9	3.53478
$605$	0	0
$606$	−30101.7	−2.01782
$607$	−15129.0	−1.01164	−0.505821	−	0.862638i	$-0.668810\pi$
$607$	−15129.0	−1.01164	−0.505821	+	0.862638i	$0.668810\pi$
$608$	26792.2	1.78712
$609$	−1975.34	−0.131436
$610$	−1979.06	−0.131361
$611$	1599.70	0.105920
$612$	−19053.9	−1.25851
$613$	10775.7	0.709992	0.354996	−	0.934868i	$-0.384482\pi$
$613$	10775.7	0.709992	0.354996	+	0.934868i	$0.384482\pi$
$614$	7351.42	0.483191
$615$	259.623	0.0170228
$616$	0	0
$617$	13290.2	0.867170	0.433585	−	0.901113i	$-0.357248\pi$
$617$	13290.2	0.867170	0.433585	+	0.901113i	$0.357248\pi$
$618$	2256.81	0.146897
$619$	−21299.7	−1.38305	−0.691526	−	0.722352i	$-0.743061\pi$
$619$	−21299.7	−1.38305	−0.691526	+	0.722352i	$0.743061\pi$
$620$	24829.6	1.60835
$621$	2822.71	0.182401
$622$	28442.6	1.83351
$623$	516.342	0.0332052
$624$	2566.25	0.164635
$625$	625.000	0.0400000
$626$	32837.9	2.09659
$627$	0	0
$628$	46093.4	2.92887
$629$	−16189.3	−1.02625
$630$	−1272.26	−0.0804570
$631$	13486.2	0.850834	0.425417	−	0.904997i	$-0.360127\pi$
$631$	13486.2	0.850834	0.425417	+	0.904997i	$0.360127\pi$
$632$	−77309.7	−4.86584
$633$	13043.8	0.819030
$634$	−36410.1	−2.28081
$635$	4705.01	0.294036
$636$	−19028.6	−1.18637
$637$	−1617.95	−0.100637
$638$	0	0
$639$	10728.9	0.664210
$640$	2639.79	0.163042
$641$	3234.75	0.199321	0.0996605	−	0.995021i	$-0.468224\pi$
$641$	3234.75	0.199321	0.0996605	+	0.995021i	$0.468224\pi$
$642$	−30122.1	−1.85175
$643$	−19623.4	−1.20353	−0.601766	−	0.798672i	$-0.705536\pi$
$643$	−19623.4	−1.20353	−0.601766	+	0.798672i	$0.705536\pi$
$644$	−11058.7	−0.676669
$645$	6885.06	0.420308
$646$	−39693.8	−2.41754
$647$	−14293.8	−0.868541	−0.434271	−	0.900782i	$-0.642994\pi$
$647$	−14293.8	−0.868541	−0.434271	+	0.900782i	$0.642994\pi$
$648$	4981.10	0.301969
$649$	0	0
$650$	677.540	0.0408851
$651$	−4065.98	−0.244790
$652$	−67103.9	−4.03066
$653$	−3963.73	−0.237539	−0.118769	−	0.992922i	$-0.537895\pi$
$653$	−3963.73	−0.237539	−0.118769	+	0.992922i	$0.537895\pi$
$654$	−10617.5	−0.634825
$655$	6219.16	0.370997
$656$	2874.56	0.171087
$657$	−2496.41	−0.148241
$658$	−8780.97	−0.520240
$659$	−1516.65	−0.0896515	−0.0448258	−	0.998995i	$-0.514273\pi$
$659$	−1516.65	−0.0896515	−0.0448258	+	0.998995i	$0.514273\pi$
$660$	0	0
$661$	106.966	0.00629425	0.00314713	−	0.999995i	$-0.498998\pi$
$661$	106.966	0.00629425	0.00314713	+	0.999995i	$0.498998\pi$
$662$	−16493.6	−0.968344
$663$	−1661.66	−0.0973354
$664$	9623.76	0.562461
$665$	−1884.59	−0.109896
$666$	7129.28	0.414796
$667$	12811.4	0.743719
$668$	−15153.4	−0.877699
$669$	−3369.90	−0.194750
$670$	−23640.8	−1.36317
$671$	0	0
$672$	−6156.48	−0.353410
$673$	15728.1	0.900852	0.450426	−	0.892814i	$-0.351272\pi$
$673$	15728.1	0.900852	0.450426	+	0.892814i	$0.351272\pi$
$674$	−34122.5	−1.95007
$675$	675.000	0.0384900
$676$	−42730.0	−2.43116
$677$	−316.806	−0.0179850	−0.00899249	−	0.999960i	$-0.502862\pi$
$677$	−316.806	−0.0179850	−0.00899249	+	0.999960i	$0.502862\pi$
$678$	−25811.1	−1.46205
$679$	4517.10	0.255303
$680$	−33065.3	−1.86470
$681$	1305.01	0.0734331
$682$	0	0
$683$	−11621.2	−0.651057	−0.325528	−	0.945532i	$-0.605542\pi$
$683$	−11621.2	−0.651057	−0.325528	+	0.945532i	$0.605542\pi$
$684$	12429.2	0.694799
$685$	−5635.68	−0.314348
$686$	18578.6	1.03402
$687$	12622.1	0.700965
$688$	76231.9	4.22429
$689$	−1659.45	−0.0917562
$690$	8251.46	0.455258
$691$	22281.8	1.22669	0.613344	−	0.789816i	$-0.289824\pi$
$691$	22281.8	1.22669	0.613344	+	0.789816i	$0.289824\pi$
$692$	41549.5	2.28248
$693$	0	0
$694$	13339.4	0.729622
$695$	−12689.7	−0.692586
$696$	22607.7	1.23124
$697$	−1861.29	−0.101150
$698$	49036.3	2.65910
$699$	−705.804	−0.0381917
$700$	−2644.50	−0.142789
$701$	31634.6	1.70445	0.852227	−	0.523172i	$-0.175251\pi$
$701$	31634.6	1.70445	0.852227	+	0.523172i	$0.175251\pi$
$702$	731.743	0.0393417
$703$	10560.6	0.566571
$704$	0	0
$705$	4658.78	0.248879
$706$	−20828.7	−1.11034
$707$	10246.0	0.545038
$708$	25136.4	1.33430
$709$	−7245.11	−0.383774	−0.191887	−	0.981417i	$-0.561461\pi$
$709$	−7245.11	−0.383774	−0.191887	+	0.981417i	$0.561461\pi$
$710$	31363.3	1.65781
$711$	−11314.5	−0.596804
$712$	−5909.54	−0.311052
$713$	26370.7	1.38512
$714$	9121.08	0.478078
$715$	0	0
$716$	29003.9	1.51386
$717$	1943.93	0.101251
$718$	−57229.1	−2.97461
$719$	−33452.4	−1.73514	−0.867569	−	0.497317i	$-0.834319\pi$
$719$	−33452.4	−1.73514	−0.867569	+	0.497317i	$0.834319\pi$
$720$	7473.65	0.386842
$721$	−768.176	−0.0396787
$722$	−10198.0	−0.525665
$723$	−6844.92	−0.352096
$724$	−45083.3	−2.31424
$725$	3063.63	0.156938
$726$	0	0
$727$	7395.29	0.377271	0.188636	−	0.982047i	$-0.439594\pi$
$727$	7395.29	0.377271	0.188636	+	0.982047i	$0.439594\pi$
$728$	−1701.85	−0.0866411
$729$	729.000	0.0370370
$730$	−7297.61	−0.369996
$731$	−49360.5	−2.49749
$732$	−4442.76	−0.224329
$733$	666.961	0.0336082	0.0168041	−	0.999859i	$-0.494651\pi$
$733$	666.961	0.0336082	0.0168041	+	0.999859i	$0.494651\pi$
$734$	22062.2	1.10944
$735$	−4711.95	−0.236466
$736$	39929.0	1.99973
$737$	0	0
$738$	819.656	0.0408834
$739$	−25530.9	−1.27087	−0.635433	−	0.772156i	$-0.719178\pi$
$739$	−25530.9	−1.27087	−0.635433	+	0.772156i	$0.719178\pi$
$740$	14818.8	0.736151
$741$	1083.93	0.0537370
$742$	9108.97	0.450675
$743$	11909.0	0.588019	0.294009	−	0.955803i	$-0.405010\pi$
$743$	11909.0	0.588019	0.294009	+	0.955803i	$0.405010\pi$
$744$	46535.2	2.29309
$745$	−13176.1	−0.647966
$746$	−58453.4	−2.86881
$747$	1408.47	0.0689869
$748$	0	0
$749$	10253.0	0.500182
$750$	1973.19	0.0960676
$751$	−16437.5	−0.798685	−0.399343	−	0.916802i	$-0.630762\pi$
$751$	−16437.5	−0.798685	−0.399343	+	0.916802i	$0.630762\pi$
$752$	51582.4	2.50135
$753$	−14608.0	−0.706966
$754$	3321.17	0.160411
$755$	13326.3	0.642375
$756$	−2856.06	−0.137399
$757$	38642.1	1.85531	0.927657	−	0.373434i	$-0.121820\pi$
$757$	38642.1	1.85531	0.927657	+	0.373434i	$0.121820\pi$
$758$	−24816.8	−1.18917
$759$	0	0
$760$	21569.1	1.02946
$761$	−32325.7	−1.53982	−0.769912	−	0.638150i	$-0.779700\pi$
$761$	−32325.7	−1.53982	−0.769912	+	0.638150i	$0.779700\pi$
$762$	14854.2	0.706183
$763$	3613.98	0.171474
$764$	76392.1	3.61750
$765$	−4839.22	−0.228709
$766$	53234.6	2.51102
$767$	2192.10	0.103197
$768$	−8010.39	−0.376367
$769$	−29461.5	−1.38155	−0.690773	−	0.723072i	$-0.742730\pi$
$769$	−29461.5	−1.38155	−0.690773	+	0.723072i	$0.742730\pi$
$770$	0	0
$771$	−4143.51	−0.193547
$772$	−16792.6	−0.782872
$773$	28275.2	1.31564	0.657820	−	0.753175i	$-0.271479\pi$
$773$	28275.2	1.31564	0.657820	+	0.753175i	$0.271479\pi$
$774$	21736.9	1.00945
$775$	6306.09	0.292286
$776$	−51698.2	−2.39157
$777$	−2426.67	−0.112042
$778$	59574.8	2.74532
$779$	1214.15	0.0558428
$780$	1520.99	0.0698209
$781$	0	0
$782$	−59156.5	−2.70516
$783$	3308.72	0.151014
$784$	−52171.1	−2.37660
$785$	11706.6	0.532262
$786$	19634.6	0.891019
$787$	−27311.6	−1.23705	−0.618523	−	0.785767i	$-0.712269\pi$
$787$	−27311.6	−1.23705	−0.618523	+	0.785767i	$0.712269\pi$
$788$	15094.9	0.682405
$789$	−3363.16	−0.151751
$790$	−33075.1	−1.48957
$791$	8785.61	0.394918
$792$	0	0
$793$	−387.444	−0.0173500
$794$	28416.1	1.27009
$795$	−4832.80	−0.215600
$796$	24454.1	1.08888
$797$	−35217.6	−1.56521	−0.782605	−	0.622518i	$-0.786110\pi$
$797$	−35217.6	−1.56521	−0.782605	+	0.622518i	$0.786110\pi$
$798$	−5949.84	−0.263937
$799$	−33399.8	−1.47885
$800$	9548.32	0.421980
$801$	−864.880	−0.0381511
$802$	72521.1	3.19303
$803$	0	0
$804$	−53070.7	−2.32794
$805$	−2808.64	−0.122971
$806$	6836.21	0.298753
$807$	−3484.08	−0.151977
$808$	−117266.	−5.10569
$809$	26775.1	1.16361	0.581806	−	0.813327i	$-0.302346\pi$
$809$	26775.1	1.16361	0.581806	+	0.813327i	$0.302346\pi$
$810$	2131.05	0.0924411
$811$	−22851.7	−0.989434	−0.494717	−	0.869054i	$-0.664728\pi$
$811$	−22851.7	−0.989434	−0.494717	+	0.869054i	$0.664728\pi$
$812$	−12962.8	−0.560228
$813$	−10705.7	−0.461827
$814$	0	0
$815$	−17042.7	−0.732491
$816$	−53580.2	−2.29863
$817$	32198.7	1.37881
$818$	595.074	0.0254356
$819$	−249.071	−0.0106267
$820$	1703.73	0.0725571
$821$	−11277.7	−0.479408	−0.239704	−	0.970846i	$-0.577050\pi$
$821$	−11277.7	−0.479408	−0.239704	+	0.970846i	$0.577050\pi$
$822$	−17792.4	−0.754967
$823$	3325.42	0.140847	0.0704234	−	0.997517i	$-0.477565\pi$
$823$	3325.42	0.140847	0.0704234	+	0.997517i	$0.477565\pi$
$824$	8791.77	0.371694
$825$	0	0
$826$	−12032.7	−0.506868
$827$	−9271.96	−0.389864	−0.194932	−	0.980817i	$-0.562449\pi$
$827$	−9271.96	−0.389864	−0.194932	+	0.980817i	$0.562449\pi$
$828$	18523.5	0.777460
$829$	7773.46	0.325674	0.162837	−	0.986653i	$-0.447936\pi$
$829$	7773.46	0.325674	0.162837	+	0.986653i	$0.447936\pi$
$830$	4117.30	0.172185
$831$	−25039.7	−1.04527
$832$	3507.66	0.146161
$833$	33781.0	1.40509
$834$	−40062.7	−1.66338
$835$	−3848.59	−0.159504
$836$	0	0
$837$	6810.58	0.281252
$838$	38687.6	1.59480
$839$	8879.30	0.365372	0.182686	−	0.983171i	$-0.441521\pi$
$839$	8879.30	0.365372	0.182686	+	0.983171i	$0.441521\pi$
$840$	−4956.27	−0.203581
$841$	−9371.72	−0.384260
$842$	17400.8	0.712197
$843$	5171.07	0.211270
$844$	85597.9	3.49100
$845$	−10852.4	−0.441814
$846$	14708.3	0.597731
$847$	0	0
$848$	−53509.1	−2.16687
$849$	6260.27	0.253065
$850$	−14146.2	−0.570837
$851$	15738.7	0.633976
$852$	70406.8	2.83110
$853$	−20374.6	−0.817835	−0.408917	−	0.912571i	$-0.634094\pi$
$853$	−20374.6	−0.817835	−0.408917	+	0.912571i	$0.634094\pi$
$854$	2126.74	0.0852172
$855$	3156.71	0.126266
$856$	−117346.	−4.68550
$857$	47483.4	1.89265	0.946326	−	0.323213i	$-0.104763\pi$
$857$	47483.4	1.89265	0.946326	+	0.323213i	$0.104763\pi$
$858$	0	0
$859$	−10722.1	−0.425882	−0.212941	−	0.977065i	$-0.568304\pi$
$859$	−10722.1	−0.425882	−0.212941	+	0.977065i	$0.568304\pi$
$860$	45182.0	1.79150
$861$	−278.995	−0.0110431
$862$	−52407.3	−2.07076
$863$	18040.1	0.711578	0.355789	−	0.934566i	$-0.384212\pi$
$863$	18040.1	0.711578	0.355789	+	0.934566i	$0.384212\pi$
$864$	10312.2	0.406050
$865$	10552.5	0.414794
$866$	57422.8	2.25324
$867$	19954.4	0.781647
$868$	−26682.3	−1.04338
$869$	0	0
$870$	9672.19	0.376917
$871$	−4628.20	−0.180047
$872$	−41362.0	−1.60630
$873$	−7566.21	−0.293330
$874$	38588.9	1.49346
$875$	−671.636	−0.0259491
$876$	−16382.2	−0.631855
$877$	26506.3	1.02059	0.510294	−	0.860000i	$-0.329537\pi$
$877$	26506.3	1.02059	0.510294	+	0.860000i	$0.329537\pi$
$878$	34673.4	1.33277
$879$	−9382.55	−0.360029
$880$	0	0
$881$	14870.8	0.568685	0.284342	−	0.958723i	$-0.408225\pi$
$881$	14870.8	0.568685	0.284342	+	0.958723i	$0.408225\pi$
$882$	−14876.1	−0.567919
$883$	42210.9	1.60873	0.804366	−	0.594134i	$-0.202505\pi$
$883$	42210.9	1.60873	0.804366	+	0.594134i	$0.202505\pi$
$884$	−10904.3	−0.414878
$885$	6384.02	0.242482
$886$	−13667.3	−0.518240
$887$	20743.2	0.785220	0.392610	−	0.919705i	$-0.371572\pi$
$887$	20743.2	0.785220	0.392610	+	0.919705i	$0.371572\pi$
$888$	27773.2	1.04956
$889$	−5056.09	−0.190749
$890$	−2528.26	−0.0952218
$891$	0	0
$892$	−22114.3	−0.830093
$893$	21787.3	0.816443
$894$	−41598.3	−1.55622
$895$	7366.26	0.275114
$896$	−2836.76	−0.105770
$897$	1615.40	0.0601301
$898$	33993.1	1.26321
$899$	30911.2	1.14677
$900$	4429.57	0.164058
$901$	34647.4	1.28110
$902$	0	0
$903$	−7398.81	−0.272666
$904$	−100551.	−3.69943
$905$	−11450.0	−0.420565
$906$	42072.5	1.54279
$907$	34322.7	1.25652	0.628261	−	0.778002i	$-0.283767\pi$
$907$	34322.7	1.25652	0.628261	+	0.778002i	$0.283767\pi$
$908$	8563.87	0.312998
$909$	−17162.2	−0.626222
$910$	−728.097	−0.0265233
$911$	46372.6	1.68649	0.843245	−	0.537529i	$-0.180642\pi$
$911$	46372.6	1.68649	0.843245	+	0.537529i	$0.180642\pi$
$912$	34951.3	1.26903
$913$	0	0
$914$	−40390.5	−1.46171
$915$	−1128.35	−0.0407673
$916$	82830.3	2.98776
$917$	−6683.23	−0.240676
$918$	−15277.9	−0.549288
$919$	−15809.4	−0.567469	−0.283734	−	0.958903i	$-0.591573\pi$
$919$	−15809.4	−0.567469	−0.283734	+	0.958903i	$0.591573\pi$
$920$	32144.9	1.15194
$921$	4191.36	0.149957
$922$	39755.9	1.42005
$923$	6140.05	0.218962
$924$	0	0
$925$	3763.62	0.133781
$926$	28837.5	1.02339
$927$	1286.70	0.0455889
$928$	46804.0	1.65562
$929$	37806.1	1.33517	0.667587	−	0.744532i	$-0.267327\pi$
$929$	37806.1	1.33517	0.667587	+	0.744532i	$0.267327\pi$
$930$	19909.0	0.701980
$931$	−22035.9	−0.775723
$932$	−4631.72	−0.162786
$933$	16216.4	0.569025
$934$	−54405.9	−1.90601
$935$	0	0
$936$	2850.62	0.0995464
$937$	2703.76	0.0942669	0.0471334	−	0.998889i	$-0.484991\pi$
$937$	2703.76	0.0942669	0.0471334	+	0.998889i	$0.484991\pi$
$938$	25404.9	0.884326
$939$	18722.3	0.650670
$940$	30572.4	1.06081
$941$	−42365.9	−1.46768	−0.733841	−	0.679322i	$-0.762274\pi$
$941$	−42365.9	−1.46768	−0.733841	+	0.679322i	$0.762274\pi$
$942$	36958.9	1.27833
$943$	1809.48	0.0624865
$944$	70684.3	2.43705
$945$	−725.367	−0.0249695
$946$	0	0
$947$	−7085.97	−0.243150	−0.121575	−	0.992582i	$-0.538794\pi$
$947$	−7085.97	−0.243150	−0.121575	+	0.992582i	$0.538794\pi$
$948$	−74249.6	−2.54379
$949$	−1428.66	−0.0488687
$950$	9227.84	0.315148
$951$	−20759.0	−0.707840
$952$	35532.6	1.20968
$953$	−26409.0	−0.897662	−0.448831	−	0.893617i	$-0.648160\pi$
$953$	−26409.0	−0.897662	−0.448831	+	0.893617i	$0.648160\pi$
$954$	−15257.6	−0.517803
$955$	19401.7	0.657407
$956$	12756.7	0.431569
$957$	0	0
$958$	22734.6	0.766723
$959$	6056.21	0.203926
$960$	10215.3	0.343435
$961$	33835.9	1.13578
$962$	4080.00	0.136741
$963$	−17173.9	−0.574685
$964$	−44918.6	−1.50076
$965$	−4264.89	−0.142271
$966$	−8867.17	−0.295338
$967$	−31278.1	−1.04016	−0.520081	−	0.854117i	$-0.674098\pi$
$967$	−31278.1	−1.04016	−0.520081	+	0.854117i	$0.674098\pi$
$968$	0	0
$969$	−22631.1	−0.750275
$970$	−22117.9	−0.732126
$971$	−10461.4	−0.345750	−0.172875	−	0.984944i	$-0.555306\pi$
$971$	−10461.4	−0.345750	−0.172875	+	0.984944i	$0.555306\pi$
$972$	4783.93	0.157865
$973$	13636.6	0.449300
$974$	−49598.6	−1.63167
$975$	386.294	0.0126885
$976$	−12493.2	−0.409730
$977$	−38539.6	−1.26202	−0.631008	−	0.775776i	$-0.717359\pi$
$977$	−38539.6	−1.26202	−0.631008	+	0.775776i	$0.717359\pi$
$978$	−53805.6	−1.75922
$979$	0	0
$980$	−30921.3	−1.00790
$981$	−6053.46	−0.197015
$982$	2998.04	0.0974250
$983$	48220.9	1.56461	0.782304	−	0.622897i	$-0.214045\pi$
$983$	48220.9	1.56461	0.782304	+	0.622897i	$0.214045\pi$
$984$	3193.10	0.103448
$985$	3833.74	0.124013
$986$	−69342.0	−2.23965
$987$	−5006.41	−0.161455
$988$	7113.09	0.229046
$989$	47986.4	1.54285
$990$	0	0
$991$	19959.7	0.639798	0.319899	−	0.947452i	$-0.396351\pi$
$991$	19959.7	0.639798	0.319899	+	0.947452i	$0.396351\pi$
$992$	96340.1	3.08347
$993$	−9403.73	−0.300522
$994$	−33703.6	−1.07547
$995$	6210.73	0.197883
$996$	9242.83	0.294046
$997$	59491.3	1.88978	0.944889	−	0.327392i	$-0.106170\pi$
$997$	59491.3	1.88978	0.944889	+	0.327392i	$0.106170\pi$
$998$	89833.0	2.84931
$999$	4064.71	0.128730

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.bb.1.6	yes	6
11.10	odd	2	inner	1815.4.a.bb.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.4.a.bb.1.1	✓	6	11.10	odd	2	inner
1815.4.a.bb.1.6	yes	6	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)

\(f(q)\)	\(=\)	\(q+5.26184 q^{2} +3.00000 q^{3} +19.6870 q^{4} +5.00000 q^{5} +15.7855 q^{6} -5.37309 q^{7} +61.4950 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+5.26184 q^{2} +3.00000 q^{3} +19.6870 q^{4} +5.00000 q^{5} +15.7855 q^{6} -5.37309 q^{7} +61.4950 q^{8} +9.00000 q^{9} +26.3092 q^{10} +59.0609 q^{12} +5.15059 q^{13} -28.2724 q^{14} +15.0000 q^{15} +166.081 q^{16} -107.538 q^{17} +47.3566 q^{18} +70.1491 q^{19} +98.4349 q^{20} -16.1193 q^{21} +104.545 q^{23} +184.485 q^{24} +25.0000 q^{25} +27.1016 q^{26} +27.0000 q^{27} -105.780 q^{28} +122.545 q^{29} +78.9276 q^{30} +252.244 q^{31} +381.933 q^{32} -565.849 q^{34} -26.8655 q^{35} +177.183 q^{36} +150.545 q^{37} +369.113 q^{38} +15.4518 q^{39} +307.475 q^{40} +17.3082 q^{41} -84.8171 q^{42} +459.004 q^{43} +45.0000 q^{45} +550.098 q^{46} +310.585 q^{47} +498.244 q^{48} -314.130 q^{49} +131.546 q^{50} -322.615 q^{51} +101.400 q^{52} -322.186 q^{53} +142.070 q^{54} -330.418 q^{56} +210.447 q^{57} +644.812 q^{58} +425.601 q^{59} +295.305 q^{60} -75.2233 q^{61} +1327.27 q^{62} -48.3578 q^{63} +681.020 q^{64} +25.7530 q^{65} -898.576 q^{67} -2117.10 q^{68} +313.634 q^{69} -141.362 q^{70} +1192.10 q^{71} +553.455 q^{72} -277.379 q^{73} +792.142 q^{74} +75.0000 q^{75} +1381.02 q^{76} +81.3048 q^{78} -1257.17 q^{79} +830.406 q^{80} +81.0000 q^{81} +91.0729 q^{82} +156.497 q^{83} -317.340 q^{84} -537.691 q^{85} +2415.21 q^{86} +367.635 q^{87} -96.0978 q^{89} +236.783 q^{90} -27.6746 q^{91} +2058.17 q^{92} +756.731 q^{93} +1634.25 q^{94} +350.746 q^{95} +1145.80 q^{96} -840.690 q^{97} -1652.90 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9}+O(q^{10}) \) 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9	\( 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9} + 120 q^{12} + 116 q^{14} + 90 q^{15} + 164 q^{16} + 200 q^{20} + 56 q^{23} + 150 q^{25} + 292 q^{26} + 162 q^{27} + 576 q^{31} - 92 q^{34} + 360 q^{36} + 332 q^{37} + 496 q^{38} + 348 q^{42} + 270 q^{45} + 96 q^{47} + 492 q^{48} + 454 q^{49} + 308 q^{53} - 652 q^{56} + 1784 q^{58} + 2080 q^{59} + 600 q^{60} + 1928 q^{64} - 1168 q^{67} + 168 q^{69} + 580 q^{70} + 1064 q^{71} + 450 q^{75} + 876 q^{78} + 820 q^{80} + 486 q^{81} + 24 q^{82} + 5412 q^{86} - 684 q^{89} + 2744 q^{91} + 1368 q^{92} + 1728 q^{93} + 2812 q^{97}+O(q^{100}) \) 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9 + 120 * q^12 + 116 * q^14 + 90 * q^15 + 164 * q^16 + 200 * q^20 + 56 * q^23 + 150 * q^25 + 292 * q^26 + 162 * q^27 + 576 * q^31 - 92 * q^34 + 360 * q^36 + 332 * q^37 + 496 * q^38 + 348 * q^42 + 270 * q^45 + 96 * q^47 + 492 * q^48 + 454 * q^49 + 308 * q^53 - 652 * q^56 + 1784 * q^58 + 2080 * q^59 + 600 * q^60 + 1928 * q^64 - 1168 * q^67 + 168 * q^69 + 580 * q^70 + 1064 * q^71 + 450 * q^75 + 876 * q^78 + 820 * q^80 + 486 * q^81 + 24 * q^82 + 5412 * q^86 - 684 * q^89 + 2744 * q^91 + 1368 * q^92 + 1728 * q^93 + 2812 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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