LMFDB - Embedded newform 750.6.c.a.499.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [750,6,Mod(499,750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(750, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("750.499");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 750 = 2 \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	750.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$120.287864860$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1810x^{4} + 801025 $ x^8 + 1810*x^4 + 801025
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			499.5
Root			$3.72585 + 3.72585i$ of defining polynomial
Character	$\chi$	$=$	750.499
Dual form			750.6.c.a.499.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} -121.516i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} -121.516i q^{7} -64.0000i q^{8} -81.0000 q^{9} -181.652 q^{11} -144.000i q^{12} -775.680i q^{13} +486.063 q^{14} +256.000 q^{16} -973.827i q^{17} -324.000i q^{18} -573.377 q^{19} +1093.64 q^{21} -726.607i q^{22} -1098.54i q^{23} +576.000 q^{24} +3102.72 q^{26} -729.000i q^{27} +1944.25i q^{28} -447.258 q^{29} -12.1857 q^{31} +1024.00i q^{32} -1634.87i q^{33} +3895.31 q^{34} +1296.00 q^{36} +8772.25i q^{37} -2293.51i q^{38} +6981.12 q^{39} -11607.8 q^{41} +4374.57i q^{42} -6856.51i q^{43} +2906.43 q^{44} +4394.17 q^{46} -1423.86i q^{47} +2304.00i q^{48} +2040.91 q^{49} +8764.45 q^{51} +12410.9i q^{52} +17405.8i q^{53} +2916.00 q^{54} -7777.01 q^{56} -5160.40i q^{57} -1789.03i q^{58} -1303.16 q^{59} -1336.54 q^{61} -48.7429i q^{62} +9842.78i q^{63} -4096.00 q^{64} +6539.46 q^{66} +36470.3i q^{67} +15581.2i q^{68} +9886.87 q^{69} -54059.4 q^{71} +5184.00i q^{72} +10191.7i q^{73} -35089.0 q^{74} +9174.04 q^{76} +22073.6i q^{77} +27924.5i q^{78} +48131.3 q^{79} +6561.00 q^{81} -46431.1i q^{82} +90905.1i q^{83} -17498.3 q^{84} +27426.1 q^{86} -4025.32i q^{87} +11625.7i q^{88} -8917.29 q^{89} -94257.4 q^{91} +17576.7i q^{92} -109.672i q^{93} +5695.45 q^{94} -9216.00 q^{96} +55700.1i q^{97} +8163.63i q^{98} +14713.8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9}+O(q^{10}) $ 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9	$ 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9} - 520 q^{11} + 688 q^{14} + 2048 q^{16} + 2672 q^{19} + 1548 q^{21} + 4608 q^{24} - 1216 q^{26} - 7072 q^{29} - 26636 q^{31} - 8592 q^{34} + 10368 q^{36} - 2736 q^{39} + 21172 q^{41} + 8320 q^{44} - 7680 q^{46} + 75868 q^{49} - 19332 q^{51} + 23328 q^{54} - 11008 q^{56} + 31180 q^{59} - 76620 q^{61} - 32768 q^{64} + 18720 q^{66} - 17280 q^{69} - 164820 q^{71} - 114944 q^{74} - 42752 q^{76} + 342456 q^{79} + 52488 q^{81} - 24768 q^{84} + 52432 q^{86} + 356856 q^{89} - 340324 q^{91} + 118720 q^{94} - 73728 q^{96} + 42120 q^{99}+O(q^{100}) $ 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9 - 520 * q^11 + 688 * q^14 + 2048 * q^16 + 2672 * q^19 + 1548 * q^21 + 4608 * q^24 - 1216 * q^26 - 7072 * q^29 - 26636 * q^31 - 8592 * q^34 + 10368 * q^36 - 2736 * q^39 + 21172 * q^41 + 8320 * q^44 - 7680 * q^46 + 75868 * q^49 - 19332 * q^51 + 23328 * q^54 - 11008 * q^56 + 31180 * q^59 - 76620 * q^61 - 32768 * q^64 + 18720 * q^66 - 17280 * q^69 - 164820 * q^71 - 114944 * q^74 - 42752 * q^76 + 342456 * q^79 + 52488 * q^81 - 24768 * q^84 + 52432 * q^86 + 356856 * q^89 - 340324 * q^91 + 118720 * q^94 - 73728 * q^96 + 42120 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/750\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$-1$	$1$

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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	9.00000i	0.577350i
$3$	9.00000i	0.577350i
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−36.0000	−0.408248
$7$	− 121.516i	− 0.937320i	−0.883379	−	0.468660i	$-0.844737\pi$
$7$	− 121.516i	− 0.937320i	0.883379	−	0.468660i	$-0.155263\pi$
$8$	− 64.0000i	− 0.353553i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	−181.652	−0.452645	−0.226323	−	0.974052i	$-0.572670\pi$
$11$	−181.652	−0.452645	−0.226323	+	0.974052i	$0.572670\pi$
$12$	− 144.000i	− 0.288675i
$13$	− 775.680i	− 1.27299i	−0.771282	−	0.636494i	$-0.780384\pi$
$13$	− 775.680i	− 1.27299i	0.771282	−	0.636494i	$-0.219616\pi$
$14$	486.063	0.662785
$15$	0	0
$16$	256.000	0.250000
$17$	− 973.827i	− 0.817259i	−0.912700	−	0.408629i	$-0.866007\pi$
$17$	− 973.827i	− 0.817259i	0.912700	−	0.408629i	$-0.133993\pi$
$18$	− 324.000i	− 0.235702i
$19$	−573.377	−0.364382	−0.182191	−	0.983263i	$-0.558319\pi$
$19$	−573.377	−0.364382	−0.182191	+	0.983263i	$0.558319\pi$
$20$	0	0
$21$	1093.64	0.541162
$22$	− 726.607i	− 0.320069i
$23$	− 1098.54i	− 0.433009i	−0.976282	−	0.216504i	$-0.930534\pi$
$23$	− 1098.54i	− 0.433009i	0.976282	−	0.216504i	$-0.0694656\pi$
$24$	576.000	0.204124
$25$	0	0
$26$	3102.72	0.900138
$27$	− 729.000i	− 0.192450i
$28$	1944.25i	0.468660i
$29$	−447.258	−0.0987559	−0.0493779	−	0.998780i	$-0.515724\pi$
$29$	−447.258	−0.0987559	−0.0493779	+	0.998780i	$0.515724\pi$
$30$	0	0
$31$	−12.1857	−0.00227744	−0.00113872	−	0.999999i	$-0.500362\pi$
$31$	−12.1857	−0.00227744	−0.00113872	+	0.999999i	$0.500362\pi$
$32$	1024.00i	0.176777i
$33$	− 1634.87i	− 0.261335i
$34$	3895.31	0.577889
$35$	0	0
$36$	1296.00	0.166667
$37$	8772.25i	1.05343i	0.850041	+	0.526716i	$0.176577\pi$
$37$	8772.25i	1.05343i	−0.850041	+	0.526716i	$0.823423\pi$
$38$	− 2293.51i	− 0.257657i
$39$	6981.12	0.734960
$40$	0	0
$41$	−11607.8	−1.07842	−0.539212	−	0.842170i	$-0.681278\pi$
$41$	−11607.8	−1.07842	−0.539212	+	0.842170i	$0.681278\pi$
$42$	4374.57i	0.382659i
$43$	− 6856.51i	− 0.565500i	−0.959194	−	0.282750i	$-0.908753\pi$
$43$	− 6856.51i	− 0.565500i	0.959194	−	0.282750i	$-0.0912466\pi$
$44$	2906.43	0.226323
$45$	0	0
$46$	4394.17	0.306184
$47$	− 1423.86i	− 0.0940207i	−0.998894	−	0.0470103i	$-0.985031\pi$
$47$	− 1423.86i	− 0.0940207i	0.998894	−	0.0470103i	$-0.0149694\pi$
$48$	2304.00i	0.144338i
$49$	2040.91	0.121432
$50$	0	0
$51$	8764.45	0.471845
$52$	12410.9i	0.636494i
$53$	17405.8i	0.851148i	0.904924	+	0.425574i	$0.139928\pi$
$53$	17405.8i	0.851148i	−0.904924	+	0.425574i	$0.860072\pi$
$54$	2916.00	0.136083
$55$	0	0
$56$	−7777.01	−0.331393
$57$	− 5160.40i	− 0.210376i
$58$	− 1789.03i	− 0.0698309i
$59$	−1303.16	−0.0487378	−0.0243689	−	0.999703i	$-0.507758\pi$
$59$	−1303.16	−0.0487378	−0.0243689	+	0.999703i	$0.507758\pi$
$60$	0	0
$61$	−1336.54	−0.0459893	−0.0229947	−	0.999736i	$-0.507320\pi$
$61$	−1336.54	−0.0459893	−0.0229947	+	0.999736i	$0.507320\pi$
$62$	− 48.7429i	− 0.00161039i
$63$	9842.78i	0.312440i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	6539.46	0.184792
$67$	36470.3i	0.992550i	0.868165	+	0.496275i	$0.165299\pi$
$67$	36470.3i	0.992550i	−0.868165	+	0.496275i	$0.834701\pi$
$68$	15581.2i	0.408629i
$69$	9886.87	0.249998
$70$	0	0
$71$	−54059.4	−1.27270	−0.636349	−	0.771402i	$-0.719556\pi$
$71$	−54059.4	−1.27270	−0.636349	+	0.771402i	$0.719556\pi$
$72$	5184.00i	0.117851i
$73$	10191.7i	0.223840i	0.993717	+	0.111920i	$0.0357001\pi$
$73$	10191.7i	0.223840i	−0.993717	+	0.111920i	$0.964300\pi$
$74$	−35089.0	−0.744889
$75$	0	0
$76$	9174.04	0.182191
$77$	22073.6i	0.424273i
$78$	27924.5i	0.519695i
$79$	48131.3	0.867681	0.433840	−	0.900990i	$-0.357158\pi$
$79$	48131.3	0.867681	0.433840	+	0.900990i	$0.357158\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	− 46431.1i	− 0.762561i
$83$	90905.1i	1.44841i	0.689582	+	0.724207i	$0.257794\pi$
$83$	90905.1i	1.44841i	−0.689582	+	0.724207i	$0.742206\pi$
$84$	−17498.3	−0.270581
$85$	0	0
$86$	27426.1	0.399869
$87$	− 4025.32i	− 0.0570167i
$88$	11625.7i	0.160034i
$89$	−8917.29	−0.119332	−0.0596661	−	0.998218i	$-0.519004\pi$
$89$	−8917.29	−0.119332	−0.0596661	+	0.998218i	$0.519004\pi$
$90$	0	0
$91$	−94257.4	−1.19320
$92$	17576.7i	0.216504i
$93$	− 109.672i	− 0.00131488i
$94$	5695.45	0.0664827
$95$	0	0
$96$	−9216.00	−0.102062
$97$	55700.1i	0.601072i	0.953770	+	0.300536i	$0.0971656\pi$
$97$	55700.1i	0.601072i	−0.953770	+	0.300536i	$0.902834\pi$
$98$	8163.63i	0.0858654i
$99$	14713.8	0.150882
$100$	0	0
$101$	24249.0	0.236532	0.118266	−	0.992982i	$-0.462266\pi$
$101$	24249.0	0.236532	0.118266	+	0.992982i	$0.462266\pi$
$102$	35057.8i	0.333645i
$103$	195610.i	1.81676i	0.418143	+	0.908381i	$0.362681\pi$
$103$	195610.i	1.81676i	−0.418143	+	0.908381i	$0.637319\pi$
$104$	−49643.5	−0.450069
$105$	0	0
$106$	−69623.3	−0.601852
$107$	24155.7i	0.203967i	0.994786	+	0.101983i	$0.0325189\pi$
$107$	24155.7i	0.203967i	−0.994786	+	0.101983i	$0.967481\pi$
$108$	11664.0i	0.0962250i
$109$	183775.	1.48157	0.740783	−	0.671744i	$-0.234455\pi$
$109$	183775.	1.48157	0.740783	+	0.671744i	$0.234455\pi$
$110$	0	0
$111$	−78950.3	−0.608200
$112$	− 31108.0i	− 0.234330i
$113$	− 52987.5i	− 0.390371i	−0.980766	−	0.195185i	$-0.937469\pi$
$113$	− 52987.5i	− 0.390371i	0.980766	−	0.195185i	$-0.0625308\pi$
$114$	20641.6	0.148758
$115$	0	0
$116$	7156.12	0.0493779
$117$	62830.1i	0.424329i
$118$	− 5212.62i	− 0.0344628i
$119$	−118335.	−0.766033
$120$	0	0
$121$	−128054.	−0.795112
$122$	− 5346.16i	− 0.0325194i
$123$	− 104470.i	− 0.622629i
$124$	194.972	0.00113872
$125$	0	0
$126$	−39371.1	−0.220928
$127$	147089.i	0.809227i	0.914488	+	0.404614i	$0.132594\pi$
$127$	147089.i	0.809227i	−0.914488	+	0.404614i	$0.867406\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	61708.6	0.326491
$130$	0	0
$131$	−94832.5	−0.482813	−0.241407	−	0.970424i	$-0.577609\pi$
$131$	−94832.5	−0.482813	−0.241407	+	0.970424i	$0.577609\pi$
$132$	26157.9i	0.130667i
$133$	69674.4i	0.341542i
$134$	−145881.	−0.701839
$135$	0	0
$136$	−62325.0	−0.288945
$137$	− 209273.i	− 0.952603i	−0.879282	−	0.476302i	$-0.841977\pi$
$137$	− 209273.i	− 0.952603i	0.879282	−	0.476302i	$-0.158023\pi$
$138$	39547.5i	0.176775i
$139$	−79747.4	−0.350090	−0.175045	−	0.984560i	$-0.556007\pi$
$139$	−79747.4	−0.350090	−0.175045	+	0.984560i	$0.556007\pi$
$140$	0	0
$141$	12814.8	0.0542829
$142$	− 216238.i	− 0.899933i
$143$	140904.i	0.576212i
$144$	−20736.0	−0.0833333
$145$	0	0
$146$	−40766.7	−0.158279
$147$	18368.2i	0.0701088i
$148$	− 140356.i	− 0.526716i
$149$	−293213.	−1.08198	−0.540988	−	0.841030i	$-0.681950\pi$
$149$	−293213.	−1.08198	−0.540988	+	0.841030i	$0.681950\pi$
$150$	0	0
$151$	−278690.	−0.994672	−0.497336	−	0.867558i	$-0.665688\pi$
$151$	−278690.	−0.994672	−0.497336	+	0.867558i	$0.665688\pi$
$152$	36696.1i	0.128828i
$153$	78880.0i	0.272420i
$154$	−88294.3	−0.300006
$155$	0	0
$156$	−111698.	−0.367480
$157$	− 116196.i	− 0.376220i	−0.982148	−	0.188110i	$-0.939764\pi$
$157$	− 116196.i	− 0.376220i	0.982148	−	0.188110i	$-0.0602362\pi$
$158$	192525.i	0.613543i
$159$	−156652.	−0.491410
$160$	0	0
$161$	−133490.	−0.405868
$162$	26244.0i	0.0785674i
$163$	− 60540.0i	− 0.178473i	−0.996010	−	0.0892367i	$-0.971557\pi$
$163$	− 60540.0i	− 0.178473i	0.996010	−	0.0892367i	$-0.0284428\pi$
$164$	185725.	0.539212
$165$	0	0
$166$	−363620.	−1.02418
$167$	− 613483.i	− 1.70220i	−0.525001	−	0.851102i	$-0.675935\pi$
$167$	− 613483.i	− 1.70220i	0.525001	−	0.851102i	$-0.324065\pi$
$168$	− 69993.1i	− 0.191330i
$169$	−230387.	−0.620498
$170$	0	0
$171$	46443.6	0.121461
$172$	109704.i	0.282750i
$173$	− 58116.5i	− 0.147633i	−0.997272	−	0.0738167i	$-0.976482\pi$
$173$	− 58116.5i	− 0.147633i	0.997272	−	0.0738167i	$-0.0235180\pi$
$174$	16101.3	0.0403169
$175$	0	0
$176$	−46502.9	−0.113161
$177$	− 11728.4i	− 0.0281388i
$178$	− 35669.1i	− 0.0843806i
$179$	496418.	1.15802	0.579008	−	0.815322i	$-0.303440\pi$
$179$	496418.	1.15802	0.579008	+	0.815322i	$0.303440\pi$
$180$	0	0
$181$	563042.	1.27745	0.638725	−	0.769435i	$-0.279462\pi$
$181$	563042.	1.27745	0.638725	+	0.769435i	$0.279462\pi$
$182$	− 377030.i	− 0.843717i
$183$	− 12028.8i	− 0.0265519i
$184$	−70306.6	−0.153092
$185$	0	0
$186$	438.686	0.000929762 0
$187$	176897.i	0.369928i
$188$	22781.8i	0.0470103i
$189$	−88585.0	−0.180387
$190$	0	0
$191$	−562696.	−1.11607	−0.558033	−	0.829819i	$-0.688444\pi$
$191$	−562696.	−1.11607	−0.558033	+	0.829819i	$0.688444\pi$
$192$	− 36864.0i	− 0.0721688i
$193$	528085.i	1.02049i	0.860028	+	0.510247i	$0.170446\pi$
$193$	528085.i	1.02049i	−0.860028	+	0.510247i	$0.829554\pi$
$194$	−222800.	−0.425022
$195$	0	0
$196$	−32654.5	−0.0607160
$197$	895318.i	1.64366i	0.569733	+	0.821830i	$0.307047\pi$
$197$	895318.i	1.64366i	−0.569733	+	0.821830i	$0.692953\pi$
$198$	58855.2i	0.106690i
$199$	907778.	1.62498	0.812488	−	0.582978i	$-0.198113\pi$
$199$	907778.	1.62498	0.812488	+	0.582978i	$0.198113\pi$
$200$	0	0
$201$	−328233.	−0.573049
$202$	96995.9i	0.167253i
$203$	54348.9i	0.0925658i
$204$	−140231.	−0.235922
$205$	0	0
$206$	−782440.	−1.28465
$207$	88981.8i	0.144336i
$208$	− 198574.i	− 0.318247i
$209$	104155.	0.164936
$210$	0	0
$211$	−826317.	−1.27773	−0.638867	−	0.769317i	$-0.720597\pi$
$211$	−826317.	−1.27773	−0.638867	+	0.769317i	$0.720597\pi$
$212$	− 278493.i	− 0.425574i
$213$	− 486534.i	− 0.734792i
$214$	−96622.7	−0.144226
$215$	0	0
$216$	−46656.0	−0.0680414
$217$	1480.76i	0.00213469i
$218$	735101.i	1.04763i
$219$	−91725.1	−0.129234
$220$	0	0
$221$	−755379.	−1.04036
$222$	− 315801.i	− 0.430062i
$223$	− 198690.i	− 0.267555i	−0.991011	−	0.133778i	$-0.957289\pi$
$223$	− 198690.i	− 0.267555i	0.991011	−	0.133778i	$-0.0427108\pi$
$224$	124432.	0.165696
$225$	0	0
$226$	211950.	0.276034
$227$	− 474777.i	− 0.611541i	−0.952105	−	0.305770i	$-0.901086\pi$
$227$	− 474777.i	− 0.611541i	0.952105	−	0.305770i	$-0.0989140\pi$
$228$	82566.3i	0.105188i
$229$	235056.	0.296199	0.148099	−	0.988972i	$-0.452684\pi$
$229$	235056.	0.296199	0.148099	+	0.988972i	$0.452684\pi$
$230$	0	0
$231$	−198662.	−0.244954
$232$	28624.5i	0.0349155i
$233$	− 451110.i	− 0.544368i	−0.962245	−	0.272184i	$-0.912254\pi$
$233$	− 451110.i	− 0.544368i	0.962245	−	0.272184i	$-0.0877460\pi$
$234$	−251320.	−0.300046
$235$	0	0
$236$	20850.5	0.0243689
$237$	433182.i	0.500956i
$238$	− 473342.i	− 0.541667i
$239$	922502.	1.04465	0.522327	−	0.852745i	$-0.325064\pi$
$239$	922502.	1.04465	0.522327	+	0.852745i	$0.325064\pi$
$240$	0	0
$241$	−1.23628e6	−1.37111	−0.685555	−	0.728020i	$-0.740441\pi$
$241$	−1.23628e6	−1.37111	−0.685555	+	0.728020i	$0.740441\pi$
$242$	− 512215.i	− 0.562229i
$243$	59049.0i	0.0641500i
$244$	21384.6	0.0229947
$245$	0	0
$246$	417880.	0.440265
$247$	444757.i	0.463853i
$248$	779.887i	0 0.000805197i
$249$	−818146.	−0.836243
$250$	0	0
$251$	−1.89718e6	−1.90075	−0.950373	−	0.311114i	$-0.899298\pi$
$251$	−1.89718e6	−1.90075	−0.950373	+	0.311114i	$0.899298\pi$
$252$	− 157484.i	− 0.156220i
$253$	199552.i	0.195999i
$254$	−588356.	−0.572210
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 211391.i	− 0.199643i	−0.995005	−	0.0998215i	$-0.968173\pi$
$257$	− 211391.i	− 0.199643i	0.995005	−	0.0998215i	$-0.0318272\pi$
$258$	246835.i	0.230864i
$259$	1.06597e6	0.987403
$260$	0	0
$261$	36227.9	0.0329186
$262$	− 379330.i	− 0.341400i
$263$	− 11971.3i	− 0.0106721i	−0.999986	−	0.00533607i	$-0.998301\pi$
$263$	− 11971.3i	− 0.0106721i	0.999986	−	0.00533607i	$-0.00169853\pi$
$264$	−104631.	−0.0923958
$265$	0	0
$266$	−278698.	−0.241507
$267$	− 80255.6i	− 0.0688965i
$268$	− 583525.i	− 0.496275i
$269$	−600883.	−0.506302	−0.253151	−	0.967427i	$-0.581467\pi$
$269$	−600883.	−0.506302	−0.253151	+	0.967427i	$0.581467\pi$
$270$	0	0
$271$	−1.06392e6	−0.880010	−0.440005	−	0.897995i	$-0.645023\pi$
$271$	−1.06392e6	−0.880010	−0.440005	+	0.897995i	$0.645023\pi$
$272$	− 249300.i	− 0.204315i
$273$	− 848317.i	− 0.688892i
$274$	837093.	0.673592
$275$	0	0
$276$	−158190.	−0.124999
$277$	− 355101.i	− 0.278069i	−0.990288	−	0.139035i	$-0.955600\pi$
$277$	− 355101.i	− 0.278069i	0.990288	−	0.139035i	$-0.0443999\pi$
$278$	− 318990.i	− 0.247551i
$279$	987.044	0.000759147 0
$280$	0	0
$281$	−1.30120e6	−0.983053	−0.491527	−	0.870862i	$-0.663561\pi$
$281$	−1.30120e6	−0.983053	−0.491527	+	0.870862i	$0.663561\pi$
$282$	51259.0i	0.0383838i
$283$	− 1.56975e6i	− 1.16510i	−0.812795	−	0.582550i	$-0.802055\pi$
$283$	− 1.56975e6i	− 1.16510i	0.812795	−	0.582550i	$-0.197945\pi$
$284$	864950.	0.636349
$285$	0	0
$286$	−563615.	−0.407443
$287$	1.41053e6i	1.01083i
$288$	− 82944.0i	− 0.0589256i
$289$	471517.	0.332088
$290$	0	0
$291$	−501301.	−0.347029
$292$	− 163067.i	− 0.111920i
$293$	− 476235.i	− 0.324080i	−0.986784	−	0.162040i	$-0.948193\pi$
$293$	− 476235.i	− 0.324080i	0.986784	−	0.162040i	$-0.0518073\pi$
$294$	−73472.7	−0.0495744
$295$	0	0
$296$	561424.	0.372445
$297$	132424.i	0.0871116i
$298$	− 1.17285e6i	− 0.765073i
$299$	−852117.	−0.551215
$300$	0	0
$301$	−833175.	−0.530054
$302$	− 1.11476e6i	− 0.703339i
$303$	218241.i	0.136562i
$304$	−146785.	−0.0910954
$305$	0	0
$306$	−315520.	−0.192630
$307$	− 279929.i	− 0.169513i	−0.996402	−	0.0847563i	$-0.972989\pi$
$307$	− 279929.i	− 0.169513i	0.996402	−	0.0847563i	$-0.0270112\pi$
$308$	− 353177.i	− 0.212137i
$309$	−1.76049e6	−1.04891
$310$	0	0
$311$	1.68143e6	0.985774	0.492887	−	0.870093i	$-0.335942\pi$
$311$	1.68143e6	0.985774	0.492887	+	0.870093i	$0.335942\pi$
$312$	− 446792.i	− 0.259848i
$313$	− 2.34993e6i	− 1.35579i	−0.735157	−	0.677897i	$-0.762892\pi$
$313$	− 2.34993e6i	− 1.35579i	0.735157	−	0.677897i	$-0.237108\pi$
$314$	464784.	0.266028
$315$	0	0
$316$	−770101.	−0.433840
$317$	3.36066e6i	1.87835i	0.343443	+	0.939174i	$0.388407\pi$
$317$	3.36066e6i	1.87835i	−0.343443	+	0.939174i	$0.611593\pi$
$318$	− 626610.i	− 0.347480i
$319$	81245.2	0.0447014
$320$	0	0
$321$	−217401.	−0.117760
$322$	− 533961.i	− 0.286992i
$323$	558371.i	0.297794i
$324$	−104976.	−0.0555556
$325$	0	0
$326$	242160.	0.126200
$327$	1.65398e6i	0.855383i
$328$	742898.i	0.381281i
$329$	−173022.	−0.0881274
$330$	0	0
$331$	1.08666e6	0.545157	0.272579	−	0.962134i	$-0.412124\pi$
$331$	1.08666e6	0.545157	0.272579	+	0.962134i	$0.412124\pi$
$332$	− 1.45448e6i	− 0.724207i
$333$	− 710552.i	− 0.351144i
$334$	2.45393e6	1.20364
$335$	0	0
$336$	279972.	0.135290
$337$	2.38152e6i	1.14230i	0.820847	+	0.571148i	$0.193502\pi$
$337$	2.38152e6i	1.14230i	−0.820847	+	0.571148i	$0.806498\pi$
$338$	− 921546.i	− 0.438758i
$339$	476887.	0.225381
$340$	0	0
$341$	2213.56	0.00103087
$342$	185774.i	0.0858856i
$343$	− 2.29032e6i	− 1.05114i
$344$	−438817.	−0.199934
$345$	0	0
$346$	232466.	0.104393
$347$	1.86294e6i	0.830570i	0.909691	+	0.415285i	$0.136318\pi$
$347$	1.86294e6i	0.830570i	−0.909691	+	0.415285i	$0.863682\pi$
$348$	64405.1i	0.0285084i
$349$	−3.30379e6	−1.45194	−0.725969	−	0.687727i	$-0.758609\pi$
$349$	−3.30379e6	−1.45194	−0.725969	+	0.687727i	$0.758609\pi$
$350$	0	0
$351$	−565471.	−0.244987
$352$	− 186011.i	− 0.0800171i
$353$	2.53715e6i	1.08370i	0.840475	+	0.541851i	$0.182276\pi$
$353$	2.53715e6i	1.08370i	−0.840475	+	0.541851i	$0.817724\pi$
$354$	46913.6	0.0198971
$355$	0	0
$356$	142677.	0.0596661
$357$	− 1.06502e6i	− 0.442269i
$358$	1.98567e6i	0.818842i
$359$	−4.02351e6	−1.64766	−0.823832	−	0.566835i	$-0.808168\pi$
$359$	−4.02351e6	−1.64766	−0.823832	+	0.566835i	$0.808168\pi$
$360$	0	0
$361$	−2.14734e6	−0.867226
$362$	2.25217e6i	0.903294i
$363$	− 1.15248e6i	− 0.459058i
$364$	1.50812e6	0.596598
$365$	0	0
$366$	48115.4	0.0187751
$367$	2.78489e6i	1.07930i	0.841888	+	0.539652i	$0.181444\pi$
$367$	2.78489e6i	1.07930i	−0.841888	+	0.539652i	$0.818556\pi$
$368$	− 281227.i	− 0.108252i
$369$	940231.	0.359475
$370$	0	0
$371$	2.11508e6	0.797797
$372$	1754.75i	0 0.000657441i
$373$	2.86936e6i	1.06786i	0.845530	+	0.533928i	$0.179285\pi$
$373$	2.86936e6i	1.06786i	−0.845530	+	0.533928i	$0.820715\pi$
$374$	−707590.	−0.261579
$375$	0	0
$376$	−91127.2	−0.0332413
$377$	346929.i	0.125715i
$378$	− 354340.i	− 0.127553i
$379$	4.08007e6	1.45905	0.729524	−	0.683955i	$-0.239742\pi$
$379$	4.08007e6	1.45905	0.729524	+	0.683955i	$0.239742\pi$
$380$	0	0
$381$	−1.32380e6	−0.467208
$382$	− 2.25078e6i	− 0.789178i
$383$	5.20039e6i	1.81150i	0.423810	+	0.905751i	$0.360693\pi$
$383$	5.20039e6i	1.81150i	−0.423810	+	0.905751i	$0.639307\pi$
$384$	147456.	0.0510310
$385$	0	0
$386$	−2.11234e6	−0.721599
$387$	555378.i	0.188500i
$388$	− 891202.i	− 0.300536i
$389$	−1.51547e6	−0.507777	−0.253888	−	0.967234i	$-0.581710\pi$
$389$	−1.51547e6	−0.507777	−0.253888	+	0.967234i	$0.581710\pi$
$390$	0	0
$391$	−1.06979e6	−0.353880
$392$	− 130618.i	− 0.0429327i
$393$	− 853493.i	− 0.278752i
$394$	−3.58127e6	−1.16224
$395$	0	0
$396$	−235421.	−0.0754409
$397$	20619.2i	0.00656592i	0.999995	+	0.00328296i	$0.00104500\pi$
$397$	20619.2i	0.00656592i	−0.999995	+	0.00328296i	$0.998955\pi$
$398$	3.63111e6i	1.14903i
$399$	−627070.	−0.197189
$400$	0	0
$401$	2.73600e6	0.849678	0.424839	−	0.905269i	$-0.360331\pi$
$401$	2.73600e6	0.849678	0.424839	+	0.905269i	$0.360331\pi$
$402$	− 1.31293e6i	− 0.405207i
$403$	9452.23i	0.00289916i
$404$	−387984.	−0.118266
$405$	0	0
$406$	−217396.	−0.0654539
$407$	− 1.59350e6i	− 0.476831i
$408$	− 560925.i	− 0.166822i
$409$	3.06698e6	0.906573	0.453286	−	0.891365i	$-0.350252\pi$
$409$	3.06698e6	0.906573	0.453286	+	0.891365i	$0.350252\pi$
$410$	0	0
$411$	1.88346e6	0.549986
$412$	− 3.12976e6i	− 0.908381i
$413$	158354.i	0.0456829i
$414$	−355927.	−0.102061
$415$	0	0
$416$	794296.	0.225035
$417$	− 717727.i	− 0.202125i
$418$	416620.i	0.116627i
$419$	−1.72617e6	−0.480339	−0.240170	−	0.970731i	$-0.577203\pi$
$419$	−1.72617e6	−0.480339	−0.240170	+	0.970731i	$0.577203\pi$
$420$	0	0
$421$	4.87229e6	1.33976	0.669882	−	0.742468i	$-0.266345\pi$
$421$	4.87229e6	1.33976	0.669882	+	0.742468i	$0.266345\pi$
$422$	− 3.30527e6i	− 0.903494i
$423$	115333.i	0.0313402i
$424$	1.11397e6	0.300926
$425$	0	0
$426$	1.94614e6	0.519577
$427$	162411.i	0.0431067i
$428$	− 386491.i	− 0.101983i
$429$	−1.26813e6	−0.332676
$430$	0	0
$431$	−2.20696e6	−0.572271	−0.286136	−	0.958189i	$-0.592371\pi$
$431$	−2.20696e6	−0.572271	−0.286136	+	0.958189i	$0.592371\pi$
$432$	− 186624.i	− 0.0481125i
$433$	2.00729e6i	0.514507i	0.966344	+	0.257253i	$0.0828175\pi$
$433$	2.00729e6i	0.514507i	−0.966344	+	0.257253i	$0.917182\pi$
$434$	−5923.04	−0.00150945
$435$	0	0
$436$	−2.94041e6	−0.740783
$437$	629879.i	0.157781i
$438$	− 366900.i	− 0.0913824i
$439$	−161997.	−0.0401186	−0.0200593	−	0.999799i	$-0.506386\pi$
$439$	−161997.	−0.0401186	−0.0200593	+	0.999799i	$0.506386\pi$
$440$	0	0
$441$	−165314.	−0.0404773
$442$	− 3.02151e6i	− 0.735646i
$443$	4.88655e6i	1.18302i	0.806296	+	0.591512i	$0.201469\pi$
$443$	4.88655e6i	1.18302i	−0.806296	+	0.591512i	$0.798531\pi$
$444$	1.26320e6	0.304100
$445$	0	0
$446$	794759.	0.189190
$447$	− 2.63892e6i	− 0.624679i
$448$	497729.i	0.117165i
$449$	1.73015e6	0.405012	0.202506	−	0.979281i	$-0.435091\pi$
$449$	1.73015e6	0.405012	0.202506	+	0.979281i	$0.435091\pi$
$450$	0	0
$451$	2.10858e6	0.488144
$452$	847800.i	0.195185i
$453$	− 2.50821e6i	− 0.574274i
$454$	1.89911e6	0.432425
$455$	0	0
$456$	−330265.	−0.0743791
$457$	− 438403.i	− 0.0981936i	−0.998794	−	0.0490968i	$-0.984366\pi$
$457$	− 438403.i	− 0.0981936i	0.998794	−	0.0490968i	$-0.0156343\pi$
$458$	940225.i	0.209444i
$459$	−709920.	−0.157282
$460$	0	0
$461$	−242647.	−0.0531768	−0.0265884	−	0.999646i	$-0.508464\pi$
$461$	−242647.	−0.0531768	−0.0265884	+	0.999646i	$0.508464\pi$
$462$	− 794648.i	− 0.173209i
$463$	− 5.98909e6i	− 1.29840i	−0.760618	−	0.649199i	$-0.775104\pi$
$463$	− 5.98909e6i	− 1.29840i	0.760618	−	0.649199i	$-0.224896\pi$
$464$	−114498.	−0.0246890
$465$	0	0
$466$	1.80444e6	0.384927
$467$	3.29307e6i	0.698729i	0.936987	+	0.349364i	$0.113602\pi$
$467$	3.29307e6i	0.698729i	−0.936987	+	0.349364i	$0.886398\pi$
$468$	− 1.00528e6i	− 0.212165i
$469$	4.43172e6	0.930337
$470$	0	0
$471$	1.04576e6	0.217211
$472$	83401.9i	0.0172314i
$473$	1.24550e6i	0.255971i
$474$	−1.73273e6	−0.354229
$475$	0	0
$476$	1.89337e6	0.383016
$477$	− 1.40987e6i	− 0.283716i
$478$	3.69001e6i	0.738682i
$479$	−2.48892e6	−0.495647	−0.247824	−	0.968805i	$-0.579715\pi$
$479$	−2.48892e6	−0.495647	−0.247824	+	0.968805i	$0.579715\pi$
$480$	0	0
$481$	6.80446e6	1.34101
$482$	− 4.94510e6i	− 0.969522i
$483$	− 1.20141e6i	− 0.234328i
$484$	2.04886e6	0.397556
$485$	0	0
$486$	−236196.	−0.0453609
$487$	8.13337e6i	1.55399i	0.629507	+	0.776994i	$0.283257\pi$
$487$	8.13337e6i	1.55399i	−0.629507	+	0.776994i	$0.716743\pi$
$488$	85538.5i	0.0162597i
$489$	544860.	0.103042
$490$	0	0
$491$	−2.23929e6	−0.419186	−0.209593	−	0.977789i	$-0.567214\pi$
$491$	−2.23929e6	−0.419186	−0.209593	+	0.977789i	$0.567214\pi$
$492$	1.67152e6i	0.311314i
$493$	435552.i	0.0807091i
$494$	−1.77903e6	−0.327994
$495$	0	0
$496$	−3119.55	−0.000569361 0
$497$	6.56907e6i	1.19292i
$498$	− 3.27258e6i	− 0.591313i
$499$	2.06377e6	0.371031	0.185516	−	0.982641i	$-0.440604\pi$
$499$	2.06377e6	0.371031	0.185516	+	0.982641i	$0.440604\pi$
$500$	0	0
$501$	5.52135e6	0.982768
$502$	− 7.58871e6i	− 1.34403i
$503$	5.71915e6i	1.00789i	0.863737	+	0.503943i	$0.168118\pi$
$503$	5.71915e6i	1.00789i	−0.863737	+	0.503943i	$0.831882\pi$
$504$	629938.	0.110464
$505$	0	0
$506$	−798208.	−0.138593
$507$	− 2.07348e6i	− 0.358245i
$508$	− 2.35342e6i	− 0.404614i
$509$	−6.52514e6	−1.11634	−0.558168	−	0.829728i	$-0.688496\pi$
$509$	−6.52514e6	−1.11634	−0.558168	+	0.829728i	$0.688496\pi$
$510$	0	0
$511$	1.23845e6	0.209810
$512$	262144.i	0.0441942i
$513$	417992.i	0.0701253i
$514$	845565.	0.141169
$515$	0	0
$516$	−987338.	−0.163246
$517$	258647.i	0.0425580i
$518$	4.26387e6i	0.698199i
$519$	523049.	0.0852361
$520$	0	0
$521$	3.48021e6	0.561709	0.280855	−	0.959750i	$-0.409382\pi$
$521$	3.48021e6	0.561709	0.280855	+	0.959750i	$0.409382\pi$
$522$	144912.i	0.0232770i
$523$	− 4.85070e6i	− 0.775443i	−0.921777	−	0.387722i	$-0.873262\pi$
$523$	− 4.85070e6i	− 0.775443i	0.921777	−	0.387722i	$-0.126738\pi$
$524$	1.51732e6	0.241407
$525$	0	0
$526$	47885.1	0.00754634
$527$	11866.8i	0.00186126i
$528$	− 418526.i	− 0.0653337i
$529$	5.22955e6	0.812503
$530$	0	0
$531$	105556.	0.0162459
$532$	− 1.11479e6i	− 0.170771i
$533$	9.00393e6i	1.37282i
$534$	321022.	0.0487172
$535$	0	0
$536$	2.33410e6	0.350919
$537$	4.46776e6i	0.668581i
$538$	− 2.40353e6i	− 0.358009i
$539$	−370735.	−0.0549656
$540$	0	0
$541$	1.21646e7	1.78691	0.893456	−	0.449152i	$-0.148274\pi$
$541$	1.21646e7	1.78691	0.893456	+	0.449152i	$0.148274\pi$
$542$	− 4.25570e6i	− 0.622261i
$543$	5.06737e6i	0.737536i
$544$	997199.	0.144472
$545$	0	0
$546$	3.39327e6	0.487120
$547$	6.48941e6i	0.927335i	0.886009	+	0.463668i	$0.153467\pi$
$547$	6.48941e6i	0.927335i	−0.886009	+	0.463668i	$0.846533\pi$
$548$	3.34837e6i	0.476302i
$549$	108260.	0.0153298
$550$	0	0
$551$	256447.	0.0359848
$552$	− 632760.i	− 0.0883876i
$553$	− 5.84871e6i	− 0.813294i
$554$	1.42041e6	0.196625
$555$	0	0
$556$	1.27596e6	0.175045
$557$	− 1.06703e6i	− 0.145726i	−0.997342	−	0.0728629i	$-0.976786\pi$
$557$	− 1.06703e6i	− 0.145726i	0.997342	−	0.0728629i	$-0.0232136\pi$
$558$	3948.18i	0 0.000536798i
$559$	−5.31846e6	−0.719874
$560$	0	0
$561$	−1.59208e6	−0.213578
$562$	− 5.20479e6i	− 0.695124i
$563$	7.78244e6i	1.03477i	0.855752	+	0.517386i	$0.173095\pi$
$563$	7.78244e6i	1.03477i	−0.855752	+	0.517386i	$0.826905\pi$
$564$	−205036.	−0.0271414
$565$	0	0
$566$	6.27898e6	0.823850
$567$	− 797265.i	− 0.104147i
$568$	3.45980e6i	0.449966i
$569$	8.69392e6	1.12573	0.562866	−	0.826548i	$-0.309699\pi$
$569$	8.69392e6	1.12573	0.562866	+	0.826548i	$0.309699\pi$
$570$	0	0
$571$	5.15530e6	0.661704	0.330852	−	0.943683i	$-0.392664\pi$
$571$	5.15530e6	0.661704	0.330852	+	0.943683i	$0.392664\pi$
$572$	− 2.25446e6i	− 0.288106i
$573$	− 5.06426e6i	− 0.644361i
$574$	−5.64212e6	−0.714764
$575$	0	0
$576$	331776.	0.0416667
$577$	− 1.10792e7i	− 1.38538i	−0.721233	−	0.692692i	$-0.756424\pi$
$577$	− 1.10792e7i	− 1.38538i	0.721233	−	0.692692i	$-0.243576\pi$
$578$	1.88607e6i	0.234822i
$579$	−4.75277e6	−0.589183
$580$	0	0
$581$	1.10464e7	1.35763
$582$	− 2.00520e6i	− 0.245387i
$583$	− 3.16180e6i	− 0.385268i
$584$	652267.	0.0791395
$585$	0	0
$586$	1.90494e6	0.229159
$587$	− 2.70928e6i	− 0.324533i	−0.986747	−	0.162267i	$-0.948120\pi$
$587$	− 2.70928e6i	− 0.324533i	0.986747	−	0.162267i	$-0.0518805\pi$
$588$	− 293891.i	− 0.0350544i
$589$	6987.02	0.000829858 0
$590$	0	0
$591$	−8.05786e6	−0.948967
$592$	2.24570e6i	0.263358i
$593$	4.21114e6i	0.491771i	0.969299	+	0.245886i	$0.0790787\pi$
$593$	4.21114e6i	0.491771i	−0.969299	+	0.245886i	$0.920921\pi$
$594$	−529697.	−0.0615972
$595$	0	0
$596$	4.69141e6	0.540988
$597$	8.17000e6i	0.938180i
$598$	− 3.40847e6i	− 0.389768i
$599$	1.27576e7	1.45279	0.726396	−	0.687276i	$-0.241194\pi$
$599$	1.27576e7	1.45279	0.726396	+	0.687276i	$0.241194\pi$
$600$	0	0
$601$	1.01360e7	1.14467	0.572337	−	0.820018i	$-0.306037\pi$
$601$	1.01360e7	1.14467	0.572337	+	0.820018i	$0.306037\pi$
$602$	− 3.33270e6i	− 0.374805i
$603$	− 2.95409e6i	− 0.330850i
$604$	4.45905e6	0.497336
$605$	0	0
$606$	−872963.	−0.0965638
$607$	1.64475e6i	0.181188i	0.995888	+	0.0905938i	$0.0288765\pi$
$607$	1.64475e6i	0.181188i	−0.995888	+	0.0905938i	$0.971123\pi$
$608$	− 587138.i	− 0.0644142i
$609$	−489140.	−0.0534429
$610$	0	0
$611$	−1.10446e6	−0.119687
$612$	− 1.26208e6i	− 0.136210i
$613$	− 7.44887e6i	− 0.800644i	−0.916375	−	0.400322i	$-0.868898\pi$
$613$	− 7.44887e6i	− 0.800644i	0.916375	−	0.400322i	$-0.131102\pi$
$614$	1.11972e6	0.119864
$615$	0	0
$616$	1.41271e6	0.150003
$617$	4.07579e6i	0.431021i	0.976502	+	0.215511i	$0.0691416\pi$
$617$	4.07579e6i	0.431021i	−0.976502	+	0.215511i	$0.930858\pi$
$618$	− 7.04196e6i	− 0.741690i
$619$	−1.52740e7	−1.60224	−0.801119	−	0.598505i	$-0.795761\pi$
$619$	−1.52740e7	−1.60224	−0.801119	+	0.598505i	$0.795761\pi$
$620$	0	0
$621$	−800837.	−0.0833326
$622$	6.72571e6i	0.697047i
$623$	1.08359e6i	0.111852i
$624$	1.78717e6	0.183740
$625$	0	0
$626$	9.39971e6	0.958691
$627$	937395.i	0.0952256i
$628$	1.85914e6i	0.188110i
$629$	8.54266e6	0.860927
$630$	0	0
$631$	−4.21288e6	−0.421216	−0.210608	−	0.977571i	$-0.567544\pi$
$631$	−4.21288e6	−0.421216	−0.210608	+	0.977571i	$0.567544\pi$
$632$	− 3.08040e6i	− 0.306771i
$633$	− 7.43685e6i	− 0.737700i
$634$	−1.34426e7	−1.32819
$635$	0	0
$636$	2.50644e6	0.245705
$637$	− 1.58309e6i	− 0.154582i
$638$	324981.i	0.0316086i
$639$	4.37881e6	0.424232
$640$	0	0
$641$	4.38889e6	0.421900	0.210950	−	0.977497i	$-0.432344\pi$
$641$	4.38889e6	0.421900	0.210950	+	0.977497i	$0.432344\pi$
$642$	− 869605.i	− 0.0832692i
$643$	1.40652e7i	1.34159i	0.741644	+	0.670793i	$0.234046\pi$
$643$	1.40652e7i	1.34159i	−0.741644	+	0.670793i	$0.765954\pi$
$644$	2.13584e6	0.202934
$645$	0	0
$646$	−2.23348e6	−0.210572
$647$	6.58712e6i	0.618635i	0.950959	+	0.309318i	$0.100101\pi$
$647$	6.58712e6i	0.618635i	−0.950959	+	0.309318i	$0.899899\pi$
$648$	− 419904.i	− 0.0392837i
$649$	236720.	0.0220609
$650$	0	0
$651$	−13326.8	−0.00123246
$652$	968640.i	0.0892367i
$653$	1.94602e7i	1.78593i	0.450128	+	0.892964i	$0.351378\pi$
$653$	1.94602e7i	1.78593i	−0.450128	+	0.892964i	$0.648622\pi$
$654$	−6.61591e6	−0.604847
$655$	0	0
$656$	−2.97159e6	−0.269606
$657$	− 825526.i	− 0.0746135i
$658$	− 692087.i	− 0.0623155i
$659$	296399.	0.0265867	0.0132933	−	0.999912i	$-0.495768\pi$
$659$	296399.	0.0265867	0.0132933	+	0.999912i	$0.495768\pi$
$660$	0	0
$661$	−1.82870e7	−1.62794	−0.813970	−	0.580906i	$-0.802698\pi$
$661$	−1.82870e7	−1.62794	−0.813970	+	0.580906i	$0.802698\pi$
$662$	4.34662e6i	0.385484i
$663$	− 6.79841e6i	− 0.600653i
$664$	5.81793e6	0.512092
$665$	0	0
$666$	2.84221e6	0.248296
$667$	491331.i	0.0427622i
$668$	9.81573e6i	0.851102i
$669$	1.78821e6	0.154473
$670$	0	0
$671$	242785.	0.0208168
$672$	1.11989e6i	0.0956648i
$673$	− 1.09926e7i	− 0.935541i	−0.883850	−	0.467770i	$-0.845057\pi$
$673$	− 1.09926e7i	− 0.935541i	0.883850	−	0.467770i	$-0.154943\pi$
$674$	−9.52606e6	−0.807726
$675$	0	0
$676$	3.68619e6	0.310249
$677$	9.07079e6i	0.760630i	0.924857	+	0.380315i	$0.124184\pi$
$677$	9.07079e6i	0.760630i	−0.924857	+	0.380315i	$0.875816\pi$
$678$	1.90755e6i	0.159368i
$679$	6.76844e6	0.563397
$680$	0	0
$681$	4.27300e6	0.353073
$682$	8854.24i	0 0.000728938i
$683$	8.50582e6i	0.697693i	0.937180	+	0.348847i	$0.113427\pi$
$683$	8.50582e6i	0.697693i	−0.937180	+	0.348847i	$0.886573\pi$
$684$	−743097.	−0.0607303
$685$	0	0
$686$	9.16128e6	0.743268
$687$	2.11551e6i	0.171010i
$688$	− 1.75527e6i	− 0.141375i
$689$	1.35014e7	1.08350
$690$	0	0
$691$	1.89850e7	1.51257	0.756285	−	0.654243i	$-0.227013\pi$
$691$	1.89850e7	1.51257	0.756285	+	0.654243i	$0.227013\pi$
$692$	929864.i	0.0738167i
$693$	− 1.78796e6i	− 0.141424i
$694$	−7.45177e6	−0.587301
$695$	0	0
$696$	−257620.	−0.0201585
$697$	1.13040e7i	0.881352i
$698$	− 1.32151e7i	− 1.02668i
$699$	4.05999e6	0.314291
$700$	0	0
$701$	2.33898e7	1.79776	0.898878	−	0.438199i	$-0.144384\pi$
$701$	2.33898e7	1.79776	0.898878	+	0.438199i	$0.144384\pi$
$702$	− 2.26188e6i	− 0.173232i
$703$	− 5.02981e6i	− 0.383852i
$704$	744046.	0.0565807
$705$	0	0
$706$	−1.01486e7	−0.766293
$707$	− 2.94663e6i	− 0.221706i
$708$	187654.i	0.0140694i
$709$	−2.29816e7	−1.71698	−0.858490	−	0.512830i	$-0.828597\pi$
$709$	−2.29816e7	−1.71698	−0.858490	+	0.512830i	$0.828597\pi$
$710$	0	0
$711$	−3.89864e6	−0.289227
$712$	570706.i	0.0421903i
$713$	13386.5i	0 0.000986153i
$714$	4.26008e6	0.312732
$715$	0	0
$716$	−7.94269e6	−0.579008
$717$	8.30252e6i	0.603132i
$718$	− 1.60940e7i	− 1.16507i
$719$	−2.01205e7	−1.45150	−0.725751	−	0.687958i	$-0.758508\pi$
$719$	−2.01205e7	−1.45150	−0.725751	+	0.687958i	$0.758508\pi$
$720$	0	0
$721$	2.37697e7	1.70289
$722$	− 8.58935e6i	− 0.613221i
$723$	− 1.11265e7i	− 0.791611i
$724$	−9.00867e6	−0.638725
$725$	0	0
$726$	4.60993e6	0.324603
$727$	1.34321e7i	0.942554i	0.881985	+	0.471277i	$0.156207\pi$
$727$	1.34321e7i	0.942554i	−0.881985	+	0.471277i	$0.843793\pi$
$728$	6.03247e6i	0.421859i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	−6.67706e6	−0.462160
$732$	192462.i	0.0132760i
$733$	− 1.65480e7i	− 1.13759i	−0.822480	−	0.568794i	$-0.807410\pi$
$733$	− 1.65480e7i	− 1.13759i	0.822480	−	0.568794i	$-0.192590\pi$
$734$	−1.11396e7	−0.763183
$735$	0	0
$736$	1.12491e6	0.0765459
$737$	− 6.62490e6i	− 0.449273i
$738$	3.76092e6i	0.254187i
$739$	7.62922e6	0.513888	0.256944	−	0.966426i	$-0.417284\pi$
$739$	7.62922e6	0.513888	0.256944	+	0.966426i	$0.417284\pi$
$740$	0	0
$741$	−4.00282e6	−0.267806
$742$	8.46033e6i	0.564128i
$743$	1.34847e7i	0.896127i	0.894002	+	0.448064i	$0.147886\pi$
$743$	1.34847e7i	0.896127i	−0.894002	+	0.448064i	$0.852114\pi$
$744$	−7018.98	−0.000464881 0
$745$	0	0
$746$	−1.14774e7	−0.755089
$747$	− 7.36331e6i	− 0.482805i
$748$	− 2.83036e6i	− 0.184964i
$749$	2.93530e6	0.191182
$750$	0	0
$751$	−1.37365e7	−0.888743	−0.444372	−	0.895843i	$-0.646573\pi$
$751$	−1.37365e7	−0.888743	−0.444372	+	0.895843i	$0.646573\pi$
$752$	− 364509.i	− 0.0235052i
$753$	− 1.70746e7i	− 1.09740i
$754$	−1.38772e6	−0.0888939
$755$	0	0
$756$	1.41736e6	0.0901936
$757$	− 832827.i	− 0.0528220i	−0.999651	−	0.0264110i	$-0.991592\pi$
$757$	− 832827.i	− 0.0528220i	0.999651	−	0.0264110i	$-0.00840786\pi$
$758$	1.63203e7i	1.03170i
$759$	−1.79597e6	−0.113160
$760$	0	0
$761$	−2.49849e7	−1.56392	−0.781961	−	0.623327i	$-0.785780\pi$
$761$	−2.49849e7	−1.56392	−0.781961	+	0.623327i	$0.785780\pi$
$762$	− 5.29520e6i	− 0.330366i
$763$	− 2.23316e7i	− 1.38870i
$764$	9.00313e6	0.558033
$765$	0	0
$766$	−2.08016e7	−1.28093
$767$	1.01083e6i	0.0620427i
$768$	589824.i	0.0360844i
$769$	−261268.	−0.0159320	−0.00796600	−	0.999968i	$-0.502536\pi$
$769$	−261268.	−0.0159320	−0.00796600	+	0.999968i	$0.502536\pi$
$770$	0	0
$771$	1.90252e6	0.115264
$772$	− 8.44936e6i	− 0.510247i
$773$	− 2.07535e7i	− 1.24923i	−0.780934	−	0.624614i	$-0.785256\pi$
$773$	− 2.07535e7i	− 1.24923i	0.780934	−	0.624614i	$-0.214744\pi$
$774$	−2.22151e6	−0.133290
$775$	0	0
$776$	3.56481e6	0.212511
$777$	9.59371e6i	0.570077i
$778$	− 6.06187e6i	− 0.359052i
$779$	6.65564e6	0.392958
$780$	0	0
$781$	9.81998e6	0.576080
$782$	− 4.27916e6i	− 0.250231i
$783$	326051.i	0.0190056i
$784$	522473.	0.0303580
$785$	0	0
$786$	3.41397e6	0.197108
$787$	− 1.35079e7i	− 0.777411i	−0.921362	−	0.388706i	$-0.872922\pi$
$787$	− 1.35079e7i	− 0.777411i	0.921362	−	0.388706i	$-0.127078\pi$
$788$	− 1.43251e7i	− 0.821830i
$789$	107742.	0.00616156
$790$	0	0
$791$	−6.43882e6	−0.365902
$792$	− 941683.i	− 0.0533448i
$793$	1.03673e6i	0.0585438i
$794$	−82476.8	−0.00464280
$795$	0	0
$796$	−1.45244e7	−0.812488
$797$	− 1.48523e7i	− 0.828226i	−0.910225	−	0.414113i	$-0.864092\pi$
$797$	− 1.48523e7i	− 0.828226i	0.910225	−	0.414113i	$-0.135908\pi$
$798$	− 2.50828e6i	− 0.139434i
$799$	−1.38660e6	−0.0768392
$800$	0	0
$801$	722300.	0.0397774
$802$	1.09440e7i	0.600813i
$803$	− 1.85134e6i	− 0.101320i
$804$	5.25172e6	0.286525
$805$	0	0
$806$	−37808.9	−0.00205001
$807$	− 5.40795e6i	− 0.292314i
$808$	− 1.55193e6i	− 0.0836267i
$809$	−1.02647e7	−0.551412	−0.275706	−	0.961242i	$-0.588912\pi$
$809$	−1.02647e7	−0.551412	−0.275706	+	0.961242i	$0.588912\pi$
$810$	0	0
$811$	−9.19932e6	−0.491138	−0.245569	−	0.969379i	$-0.578975\pi$
$811$	−9.19932e6	−0.491138	−0.245569	+	0.969379i	$0.578975\pi$
$812$	− 869582.i	− 0.0462829i
$813$	− 9.57532e6i	− 0.508074i
$814$	6.37398e6	0.337171
$815$	0	0
$816$	2.24370e6	0.117961
$817$	3.93137e6i	0.206058i
$818$	1.22679e7i	0.641044i
$819$	7.63485e6	0.397732
$820$	0	0
$821$	−1.14810e7	−0.594460	−0.297230	−	0.954806i	$-0.596063\pi$
$821$	−1.14810e7	−0.594460	−0.297230	+	0.954806i	$0.596063\pi$
$822$	7.53383e6i	0.388899i
$823$	6.34501e6i	0.326537i	0.986582	+	0.163269i	$0.0522037\pi$
$823$	6.34501e6i	0.326537i	−0.986582	+	0.163269i	$0.947796\pi$
$824$	1.25190e7	0.642323
$825$	0	0
$826$	−633416.	−0.0323027
$827$	− 8.47717e6i	− 0.431010i	−0.976503	−	0.215505i	$-0.930860\pi$
$827$	− 8.47717e6i	− 0.431010i	0.976503	−	0.215505i	$-0.0691398\pi$
$828$	− 1.42371e6i	− 0.0721681i
$829$	−3.07145e6	−0.155224	−0.0776118	−	0.996984i	$-0.524729\pi$
$829$	−3.07145e6	−0.155224	−0.0776118	+	0.996984i	$0.524729\pi$
$830$	0	0
$831$	3.19591e6	0.160543
$832$	3.17719e6i	0.159123i
$833$	− 1.98749e6i	− 0.0992414i
$834$	2.87091e6	0.142924
$835$	0	0
$836$	−1.66648e6	−0.0824678
$837$	8883.40i	0 0.000438294i
$838$	− 6.90467e6i	− 0.339651i
$839$	1.09247e7	0.535802	0.267901	−	0.963447i	$-0.413670\pi$
$839$	1.09247e7	0.535802	0.267901	+	0.963447i	$0.413670\pi$
$840$	0	0
$841$	−2.03111e7	−0.990247
$842$	1.94892e7i	0.947356i
$843$	− 1.17108e7i	− 0.567566i
$844$	1.32211e7	0.638867
$845$	0	0
$846$	−461331.	−0.0221609
$847$	1.55605e7i	0.745274i
$848$	4.45589e6i	0.212787i
$849$	1.41277e7	0.672671
$850$	0	0
$851$	9.63668e6	0.456146
$852$	7.78455e6i	0.367396i
$853$	− 1.07711e7i	− 0.506860i	−0.967354	−	0.253430i	$-0.918441\pi$
$853$	− 1.07711e7i	− 0.506860i	0.967354	−	0.253430i	$-0.0815587\pi$
$854$	−649642.	−0.0304810
$855$	0	0
$856$	1.54596e6	0.0721132
$857$	− 2.21421e6i	− 0.102983i	−0.998673	−	0.0514917i	$-0.983602\pi$
$857$	− 2.21421e6i	− 0.102983i	0.998673	−	0.0514917i	$-0.0163976\pi$
$858$	− 5.07253e6i	− 0.235238i
$859$	−1.83525e7	−0.848618	−0.424309	−	0.905517i	$-0.639483\pi$
$859$	−1.83525e7	−0.848618	−0.424309	+	0.905517i	$0.639483\pi$
$860$	0	0
$861$	−1.26948e7	−0.583602
$862$	− 8.82786e6i	− 0.404657i
$863$	3.69703e7i	1.68976i	0.534954	+	0.844881i	$0.320329\pi$
$863$	3.69703e7i	1.68976i	−0.534954	+	0.844881i	$0.679671\pi$
$864$	746496.	0.0340207
$865$	0	0
$866$	−8.02917e6	−0.363811
$867$	4.24365e6i	0.191731i
$868$	− 23692.1i	− 0.00106735i
$869$	−8.74314e6	−0.392752
$870$	0	0
$871$	2.82893e7	1.26350
$872$	− 1.17616e7i	− 0.523813i
$873$	− 4.51171e6i	− 0.200357i
$874$	−2.51951e6	−0.111568
$875$	0	0
$876$	1.46760e6	0.0646171
$877$	− 2.21326e7i	− 0.971705i	−0.874041	−	0.485852i	$-0.838509\pi$
$877$	− 2.21326e7i	− 0.971705i	0.874041	−	0.485852i	$-0.161491\pi$
$878$	− 647988.i	− 0.0283681i
$879$	4.28611e6	0.187108
$880$	0	0
$881$	−1.98284e7	−0.860693	−0.430346	−	0.902664i	$-0.641609\pi$
$881$	−1.98284e7	−0.860693	−0.430346	+	0.902664i	$0.641609\pi$
$882$	− 661254.i	− 0.0286218i
$883$	2.92615e7i	1.26298i	0.775385	+	0.631489i	$0.217556\pi$
$883$	2.92615e7i	1.26298i	−0.775385	+	0.631489i	$0.782444\pi$
$884$	1.20861e7	0.520180
$885$	0	0
$886$	−1.95462e7	−0.836524
$887$	− 1.80630e7i	− 0.770870i	−0.922735	−	0.385435i	$-0.874051\pi$
$887$	− 1.80630e7i	− 0.770870i	0.922735	−	0.385435i	$-0.125949\pi$
$888$	5.05282e6i	0.215031i
$889$	1.78736e7	0.758505
$890$	0	0
$891$	−1.19182e6	−0.0502939
$892$	3.17904e6i	0.133778i
$893$	816410.i	0.0342594i
$894$	1.05557e7	0.441715
$895$	0	0
$896$	−1.99091e6	−0.0828481
$897$	− 7.66905e6i	− 0.318244i
$898$	6.92059e6i	0.286386i
$899$	5450.16	0.000224911 0
$900$	0	0
$901$	1.69503e7	0.695608
$902$	8.43430e6i	0.345170i
$903$	− 7.49857e6i	− 0.306027i
$904$	−3.39120e6	−0.138017
$905$	0	0
$906$	1.00329e7	0.406073
$907$	7.19621e6i	0.290460i	0.989398	+	0.145230i	$0.0463922\pi$
$907$	7.19621e6i	0.290460i	−0.989398	+	0.145230i	$0.953608\pi$
$908$	7.59644e6i	0.305770i
$909$	−1.96417e6	−0.0788440
$910$	0	0
$911$	−4.38536e7	−1.75069	−0.875346	−	0.483497i	$-0.839366\pi$
$911$	−4.38536e7	−1.75069	−0.875346	+	0.483497i	$0.839366\pi$
$912$	− 1.32106e6i	− 0.0525940i
$913$	− 1.65131e7i	− 0.655618i
$914$	1.75361e6	0.0694334
$915$	0	0
$916$	−3.76090e6	−0.148099
$917$	1.15236e7i	0.452550i
$918$	− 2.83968e6i	− 0.111215i
$919$	−2.83993e7	−1.10922	−0.554612	−	0.832109i	$-0.687133\pi$
$919$	−2.83993e7	−1.10922	−0.554612	+	0.832109i	$0.687133\pi$
$920$	0	0
$921$	2.51936e6	0.0978682
$922$	− 970588.i	− 0.0376017i
$923$	4.19328e7i	1.62013i
$924$	3.17859e6	0.122477
$925$	0	0
$926$	2.39563e7	0.918106
$927$	− 1.58444e7i	− 0.605587i
$928$	− 457992.i	− 0.0174577i
$929$	1.58918e7	0.604134	0.302067	−	0.953287i	$-0.402323\pi$
$929$	1.58918e7	0.604134	0.302067	+	0.953287i	$0.402323\pi$
$930$	0	0
$931$	−1.17021e6	−0.0442476
$932$	7.21777e6i	0.272184i
$933$	1.51329e7i	0.569137i
$934$	−1.31723e7	−0.494076
$935$	0	0
$936$	4.02113e6	0.150023
$937$	− 2.35736e7i	− 0.877158i	−0.898693	−	0.438579i	$-0.855482\pi$
$937$	− 2.35736e7i	− 0.877158i	0.898693	−	0.438579i	$-0.144518\pi$
$938$	1.77269e7i	0.657847i
$939$	2.11494e7	0.782768
$940$	0	0
$941$	−2.35257e7	−0.866102	−0.433051	−	0.901369i	$-0.642563\pi$
$941$	−2.35257e7	−0.866102	−0.433051	+	0.901369i	$0.642563\pi$
$942$	4.18306e6i	0.153591i
$943$	1.27516e7i	0.466967i
$944$	−333608.	−0.0121845
$945$	0	0
$946$	−4.98199e6	−0.180999
$947$	58494.7i	0.00211954i	0.999999	+	0.00105977i	$0.000337335\pi$
$947$	58494.7i	0.00211954i	−0.999999	+	0.00105977i	$0.999663\pi$
$948$	− 6.93091e6i	− 0.250478i
$949$	7.90548e6	0.284946
$950$	0	0
$951$	−3.02459e7	−1.08446
$952$	7.57347e6i	0.270833i
$953$	8.44025e6i	0.301039i	0.988607	+	0.150520i	$0.0480947\pi$
$953$	8.44025e6i	0.301039i	−0.988607	+	0.150520i	$0.951905\pi$
$954$	5.63949e6	0.200617
$955$	0	0
$956$	−1.47600e7	−0.522327
$957$	731207.i	0.0258083i
$958$	− 9.95569e6i	− 0.350476i
$959$	−2.54300e7	−0.892894
$960$	0	0
$961$	−2.86290e7	−0.999995
$962$	2.72178e7i	0.948235i
$963$	− 1.95661e6i	− 0.0679890i
$964$	1.97804e7	0.685555
$965$	0	0
$966$	4.80564e6	0.165695
$967$	− 2.11888e7i	− 0.728685i	−0.931265	−	0.364342i	$-0.881294\pi$
$967$	− 2.11888e7i	− 0.728685i	0.931265	−	0.364342i	$-0.118706\pi$
$968$	8.19543e6i	0.281115i
$969$	−5.02534e6	−0.171932
$970$	0	0
$971$	1.53719e7	0.523215	0.261608	−	0.965174i	$-0.415747\pi$
$971$	1.53719e7	0.523215	0.261608	+	0.965174i	$0.415747\pi$
$972$	− 944784.i	− 0.0320750i
$973$	9.69057e6i	0.328146i
$974$	−3.25335e7	−1.09884
$975$	0	0
$976$	−342154.	−0.0114973
$977$	− 3.99264e7i	− 1.33821i	−0.743168	−	0.669104i	$-0.766678\pi$
$977$	− 3.99264e7i	− 1.33821i	0.743168	−	0.669104i	$-0.233322\pi$
$978$	2.17944e6i	0.0728615i
$979$	1.61984e6	0.0540152
$980$	0	0
$981$	−1.48858e7	−0.493855
$982$	− 8.95717e6i	− 0.296410i
$983$	− 2.80060e7i	− 0.924417i	−0.886771	−	0.462209i	$-0.847057\pi$
$983$	− 2.80060e7i	− 0.924417i	0.886771	−	0.462209i	$-0.152943\pi$
$984$	−6.68608e6	−0.220132
$985$	0	0
$986$	−1.74221e6	−0.0570700
$987$	− 1.55720e6i	− 0.0508804i
$988$	− 7.11612e6i	− 0.231927i
$989$	−7.53216e6	−0.244866
$990$	0	0
$991$	−4.81595e7	−1.55775	−0.778874	−	0.627180i	$-0.784209\pi$
$991$	−4.81595e7	−1.55775	−0.778874	+	0.627180i	$0.784209\pi$
$992$	− 12478.2i	0 0.000402599i
$993$	9.77990e6i	0.314747i
$994$	−2.62763e7	−0.843525
$995$	0	0
$996$	1.30903e7	0.418121
$997$	− 2.88893e7i	− 0.920449i	−0.887803	−	0.460225i	$-0.847769\pi$
$997$	− 2.88893e7i	− 0.920449i	0.887803	−	0.460225i	$-0.152231\pi$
$998$	8.25510e6i	0.262359i
$999$	6.39497e6	0.202733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	750.6.c.a.499.5		8
5.2	odd	4		750.6.a.d.1.4	✓	4
5.3	odd	4		750.6.a.e.1.1	yes	4
5.4	even	2	inner	750.6.c.a.499.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.6.a.d.1.4	✓	4	5.2	odd	4
750.6.a.e.1.1	yes	4	5.3	odd	4
750.6.c.a.499.4		8	5.4	even	2	inner
750.6.c.a.499.5		8	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:	\( N \)	\(=\)	\( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	750.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} -121.516i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} -121.516i q^{7} -64.0000i q^{8} -81.0000 q^{9} -181.652 q^{11} -144.000i q^{12} -775.680i q^{13} +486.063 q^{14} +256.000 q^{16} -973.827i q^{17} -324.000i q^{18} -573.377 q^{19} +1093.64 q^{21} -726.607i q^{22} -1098.54i q^{23} +576.000 q^{24} +3102.72 q^{26} -729.000i q^{27} +1944.25i q^{28} -447.258 q^{29} -12.1857 q^{31} +1024.00i q^{32} -1634.87i q^{33} +3895.31 q^{34} +1296.00 q^{36} +8772.25i q^{37} -2293.51i q^{38} +6981.12 q^{39} -11607.8 q^{41} +4374.57i q^{42} -6856.51i q^{43} +2906.43 q^{44} +4394.17 q^{46} -1423.86i q^{47} +2304.00i q^{48} +2040.91 q^{49} +8764.45 q^{51} +12410.9i q^{52} +17405.8i q^{53} +2916.00 q^{54} -7777.01 q^{56} -5160.40i q^{57} -1789.03i q^{58} -1303.16 q^{59} -1336.54 q^{61} -48.7429i q^{62} +9842.78i q^{63} -4096.00 q^{64} +6539.46 q^{66} +36470.3i q^{67} +15581.2i q^{68} +9886.87 q^{69} -54059.4 q^{71} +5184.00i q^{72} +10191.7i q^{73} -35089.0 q^{74} +9174.04 q^{76} +22073.6i q^{77} +27924.5i q^{78} +48131.3 q^{79} +6561.00 q^{81} -46431.1i q^{82} +90905.1i q^{83} -17498.3 q^{84} +27426.1 q^{86} -4025.32i q^{87} +11625.7i q^{88} -8917.29 q^{89} -94257.4 q^{91} +17576.7i q^{92} -109.672i q^{93} +5695.45 q^{94} -9216.00 q^{96} +55700.1i q^{97} +8163.63i q^{98} +14713.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9}+O(q^{10}) \) 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9	\( 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9} - 520 q^{11} + 688 q^{14} + 2048 q^{16} + 2672 q^{19} + 1548 q^{21} + 4608 q^{24} - 1216 q^{26} - 7072 q^{29} - 26636 q^{31} - 8592 q^{34} + 10368 q^{36} - 2736 q^{39} + 21172 q^{41} + 8320 q^{44} - 7680 q^{46} + 75868 q^{49} - 19332 q^{51} + 23328 q^{54} - 11008 q^{56} + 31180 q^{59} - 76620 q^{61} - 32768 q^{64} + 18720 q^{66} - 17280 q^{69} - 164820 q^{71} - 114944 q^{74} - 42752 q^{76} + 342456 q^{79} + 52488 q^{81} - 24768 q^{84} + 52432 q^{86} + 356856 q^{89} - 340324 q^{91} + 118720 q^{94} - 73728 q^{96} + 42120 q^{99}+O(q^{100}) \) 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9 - 520 * q^11 + 688 * q^14 + 2048 * q^16 + 2672 * q^19 + 1548 * q^21 + 4608 * q^24 - 1216 * q^26 - 7072 * q^29 - 26636 * q^31 - 8592 * q^34 + 10368 * q^36 - 2736 * q^39 + 21172 * q^41 + 8320 * q^44 - 7680 * q^46 + 75868 * q^49 - 19332 * q^51 + 23328 * q^54 - 11008 * q^56 + 31180 * q^59 - 76620 * q^61 - 32768 * q^64 + 18720 * q^66 - 17280 * q^69 - 164820 * q^71 - 114944 * q^74 - 42752 * q^76 + 342456 * q^79 + 52488 * q^81 - 24768 * q^84 + 52432 * q^86 + 356856 * q^89 - 340324 * q^91 + 118720 * q^94 - 73728 * q^96 + 42120 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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