LMFDB - Embedded newform 1815.4.a.bb.1.1

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			1.1
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	$-0.880342\pi$
$2$	−5.26184	−1.86034	−0.930171	+	0.367127i	$0.880342\pi$
$3$	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	$-0.453663\pi$
$7$	5.37309	0.290120	0.145060	+	0.989423i	$0.453663\pi$
$8$	−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	$-0.517498\pi$
$13$	−5.15059	−0.109886	−0.0549430	+	0.998489i	$0.517498\pi$
$14$	−28.2724	−0.539722
$15$	15.0000	0.258199
$16$	166.081	2.59502
$17$	107.538	1.53423	0.767113	−	0.641512i	$-0.221692\pi$
$17$	107.538	1.53423	0.767113	+	0.641512i	$0.221692\pi$
$18$	−47.3566	−0.620114
$19$	−70.1491	−0.847016	−0.423508	−	0.905892i	$-0.639201\pi$
$19$	−70.1491	−0.847016	−0.423508	+	0.905892i	$0.639201\pi$
$20$	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	$-0.628336\pi$
$29$	−122.545	−0.784691	−0.392345	+	0.919818i	$0.628336\pi$
$30$	−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	$-0.510495\pi$
$41$	−17.3082	−0.0659289	−0.0329645	+	0.999457i	$0.510495\pi$
$42$	−84.8171	−0.311608
$43$	−459.004	−1.62785	−0.813924	−	0.580972i	$-0.802673\pi$
$43$	−459.004	−1.62785	−0.813924	+	0.580972i	$0.802673\pi$
$44$	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	$-0.474845\pi$
$61$	75.2233	0.157891	0.0789455	+	0.996879i	$0.474845\pi$
$62$	−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	$-0.428624\pi$
$73$	277.379	0.444722	0.222361	+	0.974964i	$0.428624\pi$
$74$	−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	$-0.147028\pi$
$79$	1257.17	1.79041	0.895207	+	0.445651i	$0.147028\pi$
$80$	830.406	1.16053
$81$	81.0000	0.111111
$82$	91.0729	0.122650
$83$	−156.497	−0.206961	−0.103480	−	0.994632i	$-0.532998\pi$
$83$	−156.497	−0.206961	−0.103480	+	0.994632i	$0.532998\pi$
$84$	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	$-0.111446\pi$
$101$	1906.92	1.87867	0.939333	+	0.343008i	$0.111446\pi$
$102$	−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	$-0.169195\pi$
$107$	1908.21	1.72405	0.862027	+	0.506862i	$0.169195\pi$
$108$	531.548	0.473595
$109$	672.607	0.591046	0.295523	−	0.955336i	$-0.404506\pi$
$109$	672.607	0.591046	0.295523	+	0.955336i	$0.404506\pi$
$110$	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	$-0.606625\pi$
$127$	−941.002	−0.657484	−0.328742	+	0.944420i	$0.606625\pi$
$128$	−527.958	−0.364573
$129$	−1377.01	−0.939838
$130$	135.508	0.0914218
$131$	−1243.83	−0.829574	−0.414787	−	0.909919i	$-0.636144\pi$
$131$	−1243.83	−0.829574	−0.414787	+	0.909919i	$0.636144\pi$
$132$	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	$-0.218084\pi$
$139$	2537.94	1.54867	0.774335	+	0.632776i	$0.218084\pi$
$140$	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	$-0.242095\pi$
$149$	2635.22	1.44890	0.724448	+	0.689329i	$0.242095\pi$
$150$	−394.638	−0.214814
$151$	−2665.26	−1.43639	−0.718197	−	0.695839i	$-0.755032\pi$
$151$	−2665.26	−1.43639	−0.718197	+	0.695839i	$0.755032\pi$
$152$	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	$-0.442930\pi$
$167$	769.717	0.356662	0.178331	+	0.983971i	$0.442930\pi$
$168$	−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	$-0.653498\pi$
$173$	−2110.51	−0.927508	−0.463754	+	0.885964i	$0.653498\pi$
$174$	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	$-0.449152\pi$
$193$	852.978	0.318128	0.159064	+	0.987268i	$0.449152\pi$
$194$	4423.58	1.63708
$195$	−77.2589	−0.0283724
$196$	−6184.27	−2.25374
$197$	−766.748	−0.277302	−0.138651	−	0.990341i	$-0.544277\pi$
$197$	−766.748	−0.277302	−0.138651	+	0.990341i	$0.544277\pi$
$198$	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	$-0.750989\pi$
$211$	−4347.95	−1.41860	−0.709301	+	0.704906i	$0.750989\pi$
$212$	−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	$-0.520257\pi$
$227$	−435.002	−0.127190	−0.0635949	+	0.997976i	$0.520257\pi$
$228$	−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	$-0.489470\pi$
$233$	235.268	0.0661499	0.0330749	+	0.999453i	$0.489470\pi$
$234$	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	$-0.527947\pi$
$239$	−647.975	−0.175373	−0.0876863	+	0.996148i	$0.527947\pi$
$240$	2491.22	0.670031
$241$	2281.64	0.609848	0.304924	−	0.952377i	$-0.401369\pi$
$241$	2281.64	0.609848	0.304924	+	0.952377i	$0.401369\pi$
$242$	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	$-0.458046\pi$
$263$	1121.05	0.262841	0.131420	+	0.991327i	$0.458046\pi$
$264$	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	$-0.369026\pi$
$271$	3568.57	0.799908	0.399954	+	0.916535i	$0.369026\pi$
$272$	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	$-0.139698\pi$
$277$	8346.58	1.81046	0.905230	+	0.424922i	$0.139698\pi$
$278$	−13354.2	−2.88106
$279$	2270.19	0.487143
$280$	−1652.09	−0.352612
$281$	−1723.69	−0.365931	−0.182966	−	0.983119i	$-0.558570\pi$
$281$	−1723.69	−0.365931	−0.182966	+	0.983119i	$0.558570\pi$
$282$	−4902.75	−1.03530
$283$	−2086.76	−0.438321	−0.219160	−	0.975689i	$-0.570332\pi$
$283$	−2086.76	−0.438321	−0.219160	+	0.975689i	$0.570332\pi$
$284$	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	$-0.399070\pi$
$293$	3127.52	0.623588	0.311794	+	0.950150i	$0.399070\pi$
$294$	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	$-0.541455\pi$
$307$	−1397.12	−0.259732	−0.129866	+	0.991532i	$0.541455\pi$
$308$	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	$-0.324396\pi$
$337$	6484.89	1.04823	0.524117	+	0.851647i	$0.324396\pi$
$338$	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	$-0.562827\pi$
$347$	−2535.12	−0.392198	−0.196099	+	0.980584i	$0.562827\pi$
$348$	−7237.62	−1.11488
$349$	−9319.24	−1.42936	−0.714681	−	0.699451i	$-0.753428\pi$
$349$	−9319.24	−1.42936	−0.714681	+	0.699451i	$0.753428\pi$
$350$	−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	$-0.205109\pi$
$359$	10876.2	1.59896	0.799480	+	0.600693i	$0.205109\pi$
$360$	−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	$-0.219735\pi$
$373$	11108.9	1.54209	0.771044	+	0.636782i	$0.219735\pi$
$374$	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	$-0.502176\pi$
$409$	−113.092	−0.0136725	−0.00683626	+	0.999977i	$0.502176\pi$
$410$	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	$-0.312123\pi$
$431$	9959.87	1.11311	0.556555	+	0.830811i	$0.312123\pi$
$432$	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	$-0.616611\pi$
$439$	−6589.60	−0.716411	−0.358205	+	0.933643i	$0.616611\pi$
$440$	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	$-0.371486\pi$
$457$	7676.12	0.785720	0.392860	+	0.919598i	$0.371486\pi$
$458$	−22138.5	−2.25865
$459$	2903.53	0.295262
$460$	10290.8	1.04307
$461$	−7555.51	−0.763330	−0.381665	−	0.924301i	$-0.624649\pi$
$461$	−7555.51	−0.763330	−0.381665	+	0.924301i	$0.624649\pi$
$462$	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	$-0.566068\pi$
$479$	−4320.65	−0.412141	−0.206070	+	0.978537i	$0.566068\pi$
$480$	−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	$-0.508336\pi$
$491$	−569.771	−0.0523694	−0.0261847	+	0.999657i	$0.508336\pi$
$492$	−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	$-0.809047\pi$
$503$	−18622.7	−1.65079	−0.825395	+	0.564556i	$0.809047\pi$
$504$	−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	$-0.657003\pi$
$523$	−11326.3	−0.946966	−0.473483	+	0.880803i	$0.657003\pi$
$524$	−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	$-0.902478\pi$
$541$	−23994.8	−1.90687	−0.953433	+	0.301603i	$0.902478\pi$
$542$	−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	$-0.689487\pi$
$547$	−14347.6	−1.12150	−0.560748	+	0.827986i	$0.689487\pi$
$548$	−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	$-0.242179\pi$
$557$	19041.9	1.44853	0.724265	+	0.689521i	$0.242179\pi$
$558$	−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	$-0.309068\pi$
$563$	15082.0	1.12900	0.564502	+	0.825432i	$0.309068\pi$
$564$	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	$-0.336720\pi$
$569$	13321.8	0.981513	0.490756	+	0.871297i	$0.336720\pi$
$570$	5536.70	0.406854
$571$	24112.7	1.76722	0.883611	−	0.468223i	$-0.155105\pi$
$571$	24112.7	1.76722	0.883611	+	0.468223i	$0.155105\pi$
$572$	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	$-0.466981\pi$
$593$	2990.56	0.207095	0.103548	+	0.994625i	$0.466981\pi$
$594$	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	$-0.283756\pi$
$601$	18514.0	1.25658	0.628289	+	0.777980i	$0.283756\pi$
$602$	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	$-0.331190\pi$
$607$	15129.0	1.01164	0.505821	+	0.862638i	$0.331190\pi$
$608$	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	$-0.615518\pi$
$613$	−10775.7	−0.709992	−0.354996	+	0.934868i	$0.615518\pi$
$614$	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	$-0.485727\pi$
$659$	1516.65	0.0896515	0.0448258	+	0.998995i	$0.485727\pi$
$660$	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	$-0.648728\pi$
$673$	−15728.1	−0.900852	−0.450426	+	0.892814i	$0.648728\pi$
$674$	−34122.5	−1.95007
$675$	675.000	0.0384900
$676$	−42730.0	−2.43116
$677$	316.806	0.0179850	0.00899249	−	0.999960i	$-0.497138\pi$
$677$	316.806	0.0179850	0.00899249	+	0.999960i	$0.497138\pi$
$678$	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	$-0.824749\pi$
$701$	−31634.6	−1.70445	−0.852227	+	0.523172i	$0.824749\pi$
$702$	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	$-0.505349\pi$
$733$	−666.961	−0.0336082	−0.0168041	+	0.999859i	$0.505349\pi$
$734$	−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	$-0.280822\pi$
$739$	25530.9	1.27087	0.635433	+	0.772156i	$0.280822\pi$
$740$	14818.8	0.736151
$741$	1083.93	0.0537370
$742$	9108.97	0.450675
$743$	−11909.0	−0.588019	−0.294009	−	0.955803i	$-0.594990\pi$
$743$	−11909.0	−0.588019	−0.294009	+	0.955803i	$0.594990\pi$
$744$	−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	$-0.220300\pi$
$761$	32325.7	1.53982	0.769912	+	0.638150i	$0.220300\pi$
$762$	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	$-0.257270\pi$
$769$	29461.5	1.38155	0.690773	+	0.723072i	$0.257270\pi$
$770$	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	$-0.287731\pi$
$787$	27311.6	1.23705	0.618523	+	0.785767i	$0.287731\pi$
$788$	−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	$-0.697654\pi$
$809$	−26775.1	−1.16361	−0.581806	+	0.813327i	$0.697654\pi$
$810$	−2131.05	−0.0924411
$811$	22851.7	0.989434	0.494717	−	0.869054i	$-0.335272\pi$
$811$	22851.7	0.989434	0.494717	+	0.869054i	$0.335272\pi$
$812$	−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	$-0.422950\pi$
$821$	11277.7	0.479408	0.239704	+	0.970846i	$0.422950\pi$
$822$	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	$-0.437551\pi$
$827$	9271.96	0.389864	0.194932	+	0.980817i	$0.437551\pi$
$828$	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	$-0.365906\pi$
$853$	20374.6	0.817835	0.408917	+	0.912571i	$0.365906\pi$
$854$	−2126.74	−0.0852172
$855$	−3156.71	−0.126266
$856$	−117346.	−4.68550
$857$	−47483.4	−1.89265	−0.946326	−	0.323213i	$-0.895237\pi$
$857$	−47483.4	−1.89265	−0.946326	+	0.323213i	$0.895237\pi$
$858$	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	$-0.670463\pi$
$877$	−26506.3	−1.02059	−0.510294	+	0.860000i	$0.670463\pi$
$878$	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	$-0.628428\pi$
$887$	−20743.2	−0.785220	−0.392610	+	0.919705i	$0.628428\pi$
$888$	−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	$-0.408427\pi$
$919$	15809.4	0.567469	0.283734	+	0.958903i	$0.408427\pi$
$920$	−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	$-0.515009\pi$
$937$	−2703.76	−0.0942669	−0.0471334	+	0.998889i	$0.515009\pi$
$938$	25404.9	0.884326
$939$	18722.3	0.650670
$940$	30572.4	1.06081
$941$	42365.9	1.46768	0.733841	−	0.679322i	$-0.237726\pi$
$941$	42365.9	1.46768	0.733841	+	0.679322i	$0.237726\pi$
$942$	−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	$-0.351840\pi$
$953$	26409.0	0.897662	0.448831	+	0.893617i	$0.351840\pi$
$954$	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	$-0.325902\pi$
$967$	31278.1	1.04016	0.520081	+	0.854117i	$0.325902\pi$
$968$	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	$-0.893830\pi$
$997$	−59491.3	−1.88978	−0.944889	+	0.327392i	$0.893830\pi$
$998$	−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.1.1	✓	6
11.10	odd	2	inner	1815.4.a.bb.1.6	yes	6

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

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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