LMFDB - Embedded newform 50.6.b.b.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,6,Mod(49,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("50.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	50.b (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:	$8.01919099065$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	50.49
Dual form			50.6.b.b.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -6.00000i q^{3} -16.0000 q^{4} +24.0000 q^{6} -118.000i q^{7} -64.0000i q^{8} +207.000 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} -6.00000i q^{3} -16.0000 q^{4} +24.0000 q^{6} -118.000i q^{7} -64.0000i q^{8} +207.000 q^{9} +192.000 q^{11} +96.0000i q^{12} -1106.00i q^{13} +472.000 q^{14} +256.000 q^{16} +762.000i q^{17} +828.000i q^{18} +2740.00 q^{19} -708.000 q^{21} +768.000i q^{22} -1566.00i q^{23} -384.000 q^{24} +4424.00 q^{26} -2700.00i q^{27} +1888.00i q^{28} -5910.00 q^{29} -6868.00 q^{31} +1024.00i q^{32} -1152.00i q^{33} -3048.00 q^{34} -3312.00 q^{36} -5518.00i q^{37} +10960.0i q^{38} -6636.00 q^{39} -378.000 q^{41} -2832.00i q^{42} +2434.00i q^{43} -3072.00 q^{44} +6264.00 q^{46} +13122.0i q^{47} -1536.00i q^{48} +2883.00 q^{49} +4572.00 q^{51} +17696.0i q^{52} +9174.00i q^{53} +10800.0 q^{54} -7552.00 q^{56} -16440.0i q^{57} -23640.0i q^{58} +34980.0 q^{59} -9838.00 q^{61} -27472.0i q^{62} -24426.0i q^{63} -4096.00 q^{64} +4608.00 q^{66} +33722.0i q^{67} -12192.0i q^{68} -9396.00 q^{69} +70212.0 q^{71} -13248.0i q^{72} -21986.0i q^{73} +22072.0 q^{74} -43840.0 q^{76} -22656.0i q^{77} -26544.0i q^{78} -4520.00 q^{79} +34101.0 q^{81} -1512.00i q^{82} +109074. i q^{83} +11328.0 q^{84} -9736.00 q^{86} +35460.0i q^{87} -12288.0i q^{88} -38490.0 q^{89} -130508. q^{91} +25056.0i q^{92} +41208.0i q^{93} -52488.0 q^{94} +6144.00 q^{96} -1918.00i q^{97} +11532.0i q^{98} +39744.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 48 q^{6} + 414 q^{9}+O(q^{10}) $ 2 * q - 32 * q^4 + 48 * q^6 + 414 * q^9	$ 2 q - 32 q^{4} + 48 q^{6} + 414 q^{9} + 384 q^{11} + 944 q^{14} + 512 q^{16} + 5480 q^{19} - 1416 q^{21} - 768 q^{24} + 8848 q^{26} - 11820 q^{29} - 13736 q^{31} - 6096 q^{34} - 6624 q^{36} - 13272 q^{39} - 756 q^{41} - 6144 q^{44} + 12528 q^{46} + 5766 q^{49} + 9144 q^{51} + 21600 q^{54} - 15104 q^{56} + 69960 q^{59} - 19676 q^{61} - 8192 q^{64} + 9216 q^{66} - 18792 q^{69} + 140424 q^{71} + 44144 q^{74} - 87680 q^{76} - 9040 q^{79} + 68202 q^{81} + 22656 q^{84} - 19472 q^{86} - 76980 q^{89} - 261016 q^{91} - 104976 q^{94} + 12288 q^{96} + 79488 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 48 * q^6 + 414 * q^9 + 384 * q^11 + 944 * q^14 + 512 * q^16 + 5480 * q^19 - 1416 * q^21 - 768 * q^24 + 8848 * q^26 - 11820 * q^29 - 13736 * q^31 - 6096 * q^34 - 6624 * q^36 - 13272 * q^39 - 756 * q^41 - 6144 * q^44 + 12528 * q^46 + 5766 * q^49 + 9144 * q^51 + 21600 * q^54 - 15104 * q^56 + 69960 * q^59 - 19676 * q^61 - 8192 * q^64 + 9216 * q^66 - 18792 * q^69 + 140424 * q^71 + 44144 * q^74 - 87680 * q^76 - 9040 * q^79 + 68202 * q^81 + 22656 * q^84 - 19472 * q^86 - 76980 * q^89 - 261016 * q^91 - 104976 * q^94 + 12288 * q^96 + 79488 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$.

$n$	$27$
$\chi(n)$	$-1$

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
$3$	− 6.00000i	− 0.384900i	−0.981307	−	0.192450i	$-0.938357\pi$
$3$	− 6.00000i	− 0.384900i	0.981307	−	0.192450i	$-0.0616434\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	24.0000	0.272166
$7$	− 118.000i	− 0.910200i	−0.890440	−	0.455100i	$-0.849603\pi$
$7$	− 118.000i	− 0.910200i	0.890440	−	0.455100i	$-0.150397\pi$
$8$	− 64.0000i	− 0.353553i
$9$	207.000	0.851852
$10$	0	0
$11$	192.000	0.478431	0.239216	−	0.970966i	$-0.423110\pi$
$11$	192.000	0.478431	0.239216	+	0.970966i	$0.423110\pi$
$12$	96.0000i	0.192450i
$13$	− 1106.00i	− 1.81508i	−0.419961	−	0.907542i	$-0.637956\pi$
$13$	− 1106.00i	− 1.81508i	0.419961	−	0.907542i	$-0.362044\pi$
$14$	472.000	0.643609
$15$	0	0
$16$	256.000	0.250000
$17$	762.000i	0.639488i	0.947504	+	0.319744i	$0.103597\pi$
$17$	762.000i	0.639488i	−0.947504	+	0.319744i	$0.896403\pi$
$18$	828.000i	0.602350i
$19$	2740.00	1.74127	0.870636	−	0.491928i	$-0.163708\pi$
$19$	2740.00	1.74127	0.870636	+	0.491928i	$0.163708\pi$
$20$	0	0
$21$	−708.000	−0.350336
$22$	768.000i	0.338302i
$23$	− 1566.00i	− 0.617266i	−0.951181	−	0.308633i	$-0.900129\pi$
$23$	− 1566.00i	− 0.617266i	0.951181	−	0.308633i	$-0.0998714\pi$
$24$	−384.000	−0.136083
$25$	0	0
$26$	4424.00	1.28346
$27$	− 2700.00i	− 0.712778i
$28$	1888.00i	0.455100i
$29$	−5910.00	−1.30495	−0.652473	−	0.757812i	$-0.726268\pi$
$29$	−5910.00	−1.30495	−0.652473	+	0.757812i	$0.726268\pi$
$30$	0	0
$31$	−6868.00	−1.28359	−0.641795	−	0.766877i	$-0.721810\pi$
$31$	−6868.00	−1.28359	−0.641795	+	0.766877i	$0.721810\pi$
$32$	1024.00i	0.176777i
$33$	− 1152.00i	− 0.184148i
$34$	−3048.00	−0.452187
$35$	0	0
$36$	−3312.00	−0.425926
$37$	− 5518.00i	− 0.662640i	−0.943519	−	0.331320i	$-0.892506\pi$
$37$	− 5518.00i	− 0.662640i	0.943519	−	0.331320i	$-0.107494\pi$
$38$	10960.0i	1.23127i
$39$	−6636.00	−0.698626
$40$	0	0
$41$	−378.000	−0.0351182	−0.0175591	−	0.999846i	$-0.505590\pi$
$41$	−378.000	−0.0351182	−0.0175591	+	0.999846i	$0.505590\pi$
$42$	− 2832.00i	− 0.247725i
$43$	2434.00i	0.200747i	0.994950	+	0.100374i	$0.0320038\pi$
$43$	2434.00i	0.200747i	−0.994950	+	0.100374i	$0.967996\pi$
$44$	−3072.00	−0.239216
$45$	0	0
$46$	6264.00	0.436473
$47$	13122.0i	0.866474i	0.901280	+	0.433237i	$0.142629\pi$
$47$	13122.0i	0.866474i	−0.901280	+	0.433237i	$0.857371\pi$
$48$	− 1536.00i	− 0.0962250i
$49$	2883.00	0.171536
$50$	0	0
$51$	4572.00	0.246139
$52$	17696.0i	0.907542i
$53$	9174.00i	0.448610i	0.974519	+	0.224305i	$0.0720112\pi$
$53$	9174.00i	0.448610i	−0.974519	+	0.224305i	$0.927989\pi$
$54$	10800.0	0.504010
$55$	0	0
$56$	−7552.00	−0.321804
$57$	− 16440.0i	− 0.670216i
$58$	− 23640.0i	− 0.922736i
$59$	34980.0	1.30825	0.654124	−	0.756388i	$-0.273038\pi$
$59$	34980.0	1.30825	0.654124	+	0.756388i	$0.273038\pi$
$60$	0	0
$61$	−9838.00	−0.338518	−0.169259	−	0.985572i	$-0.554137\pi$
$61$	−9838.00	−0.338518	−0.169259	+	0.985572i	$0.554137\pi$
$62$	− 27472.0i	− 0.907635i
$63$	− 24426.0i	− 0.775356i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	4608.00	0.130212
$67$	33722.0i	0.917754i	0.888500	+	0.458877i	$0.151748\pi$
$67$	33722.0i	0.917754i	−0.888500	+	0.458877i	$0.848252\pi$
$68$	− 12192.0i	− 0.319744i
$69$	−9396.00	−0.237586
$70$	0	0
$71$	70212.0	1.65297	0.826486	−	0.562957i	$-0.190336\pi$
$71$	70212.0	1.65297	0.826486	+	0.562957i	$0.190336\pi$
$72$	− 13248.0i	− 0.301175i
$73$	− 21986.0i	− 0.482880i	−0.970416	−	0.241440i	$-0.922380\pi$
$73$	− 21986.0i	− 0.482880i	0.970416	−	0.241440i	$-0.0776197\pi$
$74$	22072.0	0.468557
$75$	0	0
$76$	−43840.0	−0.870636
$77$	− 22656.0i	− 0.435468i
$78$	− 26544.0i	− 0.494003i
$79$	−4520.00	−0.0814837	−0.0407418	−	0.999170i	$-0.512972\pi$
$79$	−4520.00	−0.0814837	−0.0407418	+	0.999170i	$0.512972\pi$
$80$	0	0
$81$	34101.0	0.577503
$82$	− 1512.00i	− 0.0248323i
$83$	109074.i	1.73790i	0.494896	+	0.868952i	$0.335206\pi$
$83$	109074.i	1.73790i	−0.494896	+	0.868952i	$0.664794\pi$
$84$	11328.0	0.175168
$85$	0	0
$86$	−9736.00	−0.141950
$87$	35460.0i	0.502274i
$88$	− 12288.0i	− 0.169151i
$89$	−38490.0	−0.515078	−0.257539	−	0.966268i	$-0.582912\pi$
$89$	−38490.0	−0.515078	−0.257539	+	0.966268i	$0.582912\pi$
$90$	0	0
$91$	−130508.	−1.65209
$92$	25056.0i	0.308633i
$93$	41208.0i	0.494054i
$94$	−52488.0	−0.612689
$95$	0	0
$96$	6144.00	0.0680414
$97$	− 1918.00i	− 0.0206976i	−0.999946	−	0.0103488i	$-0.996706\pi$
$97$	− 1918.00i	− 0.0206976i	0.999946	−	0.0103488i	$-0.00329418\pi$
$98$	11532.0i	0.121294i
$99$	39744.0	0.407553
$100$	0	0
$101$	77622.0	0.757149	0.378575	−	0.925571i	$-0.376414\pi$
$101$	77622.0	0.757149	0.378575	+	0.925571i	$0.376414\pi$
$102$	18288.0i	0.174047i
$103$	46714.0i	0.433864i	0.976187	+	0.216932i	$0.0696051\pi$
$103$	46714.0i	0.433864i	−0.976187	+	0.216932i	$0.930395\pi$
$104$	−70784.0	−0.641729
$105$	0	0
$106$	−36696.0	−0.317215
$107$	− 1038.00i	− 0.00876472i	−0.999990	−	0.00438236i	$-0.998605\pi$
$107$	− 1038.00i	− 0.00876472i	0.999990	−	0.00438236i	$-0.00139495\pi$
$108$	43200.0i	0.356389i
$109$	−206930.	−1.66823	−0.834117	−	0.551587i	$-0.814023\pi$
$109$	−206930.	−1.66823	−0.834117	+	0.551587i	$0.814023\pi$
$110$	0	0
$111$	−33108.0	−0.255050
$112$	− 30208.0i	− 0.227550i
$113$	− 139386.i	− 1.02689i	−0.858123	−	0.513444i	$-0.828369\pi$
$113$	− 139386.i	− 1.02689i	0.858123	−	0.513444i	$-0.171631\pi$
$114$	65760.0	0.473914
$115$	0	0
$116$	94560.0	0.652473
$117$	− 228942.i	− 1.54618i
$118$	139920.i	0.925070i
$119$	89916.0	0.582062
$120$	0	0
$121$	−124187.	−0.771104
$122$	− 39352.0i	− 0.239369i
$123$	2268.00i	0.0135170i
$124$	109888.	0.641795
$125$	0	0
$126$	97704.0	0.548259
$127$	299882.i	1.64984i	0.565252	+	0.824919i	$0.308779\pi$
$127$	299882.i	1.64984i	−0.565252	+	0.824919i	$0.691221\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	14604.0	0.0772676
$130$	0	0
$131$	7872.00	0.0400781	0.0200390	−	0.999799i	$-0.493621\pi$
$131$	7872.00	0.0400781	0.0200390	+	0.999799i	$0.493621\pi$
$132$	18432.0i	0.0920741i
$133$	− 323320.i	− 1.58491i
$134$	−134888.	−0.648950
$135$	0	0
$136$	48768.0	0.226093
$137$	− 164238.i	− 0.747605i	−0.927508	−	0.373803i	$-0.878054\pi$
$137$	− 164238.i	− 0.747605i	0.927508	−	0.373803i	$-0.121946\pi$
$138$	− 37584.0i	− 0.167998i
$139$	282100.	1.23841	0.619207	−	0.785228i	$-0.287454\pi$
$139$	282100.	1.23841	0.619207	+	0.785228i	$0.287454\pi$
$140$	0	0
$141$	78732.0	0.333506
$142$	280848.i	1.16883i
$143$	− 212352.i	− 0.868393i
$144$	52992.0	0.212963
$145$	0	0
$146$	87944.0	0.341448
$147$	− 17298.0i	− 0.0660241i
$148$	88288.0i	0.331320i
$149$	388950.	1.43525	0.717626	−	0.696429i	$-0.245229\pi$
$149$	388950.	1.43525	0.717626	+	0.696429i	$0.245229\pi$
$150$	0	0
$151$	−97948.0	−0.349585	−0.174793	−	0.984605i	$-0.555926\pi$
$151$	−97948.0	−0.349585	−0.174793	+	0.984605i	$0.555926\pi$
$152$	− 175360.i	− 0.615633i
$153$	157734.i	0.544749i
$154$	90624.0	0.307923
$155$	0	0
$156$	106176.	0.349313
$157$	− 3718.00i	− 0.0120382i	−0.999982	−	0.00601908i	$-0.998084\pi$
$157$	− 3718.00i	− 0.0120382i	0.999982	−	0.00601908i	$-0.00191594\pi$
$158$	− 18080.0i	− 0.0576177i
$159$	55044.0	0.172670
$160$	0	0
$161$	−184788.	−0.561835
$162$	136404.i	0.408357i
$163$	43234.0i	0.127455i	0.997967	+	0.0637274i	$0.0202988\pi$
$163$	43234.0i	0.127455i	−0.997967	+	0.0637274i	$0.979701\pi$
$164$	6048.00	0.0175591
$165$	0	0
$166$	−436296.	−1.22888
$167$	186522.i	0.517534i	0.965940	+	0.258767i	$0.0833162\pi$
$167$	186522.i	0.517534i	−0.965940	+	0.258767i	$0.916684\pi$
$168$	45312.0i	0.123863i
$169$	−851943.	−2.29453
$170$	0	0
$171$	567180.	1.48331
$172$	− 38944.0i	− 0.100374i
$173$	374454.i	0.951225i	0.879655	+	0.475612i	$0.157774\pi$
$173$	374454.i	0.951225i	−0.879655	+	0.475612i	$0.842226\pi$
$174$	−141840.	−0.355161
$175$	0	0
$176$	49152.0	0.119608
$177$	− 209880.i	− 0.503545i
$178$	− 153960.i	− 0.364215i
$179$	−272100.	−0.634740	−0.317370	−	0.948302i	$-0.602800\pi$
$179$	−272100.	−0.634740	−0.317370	+	0.948302i	$0.602800\pi$
$180$	0	0
$181$	−75418.0	−0.171111	−0.0855556	−	0.996333i	$-0.527267\pi$
$181$	−75418.0	−0.171111	−0.0855556	+	0.996333i	$0.527267\pi$
$182$	− 522032.i	− 1.16820i
$183$	59028.0i	0.130296i
$184$	−100224.	−0.218236
$185$	0	0
$186$	−164832.	−0.349349
$187$	146304.i	0.305951i
$188$	− 209952.i	− 0.433237i
$189$	−318600.	−0.648771
$190$	0	0
$191$	−356988.	−0.708060	−0.354030	−	0.935234i	$-0.615189\pi$
$191$	−356988.	−0.708060	−0.354030	+	0.935234i	$0.615189\pi$
$192$	24576.0i	0.0481125i
$193$	438694.i	0.847751i	0.905720	+	0.423876i	$0.139331\pi$
$193$	438694.i	0.847751i	−0.905720	+	0.423876i	$0.860669\pi$
$194$	7672.00	0.0146354
$195$	0	0
$196$	−46128.0	−0.0857678
$197$	− 156798.i	− 0.287856i	−0.989588	−	0.143928i	$-0.954027\pi$
$197$	− 156798.i	− 0.287856i	0.989588	−	0.143928i	$-0.0459733\pi$
$198$	158976.i	0.288183i
$199$	162520.	0.290920	0.145460	−	0.989364i	$-0.453534\pi$
$199$	162520.	0.290920	0.145460	+	0.989364i	$0.453534\pi$
$200$	0	0
$201$	202332.	0.353244
$202$	310488.i	0.535385i
$203$	697380.i	1.18776i
$204$	−73152.0	−0.123070
$205$	0	0
$206$	−186856.	−0.306788
$207$	− 324162.i	− 0.525819i
$208$	− 283136.i	− 0.453771i
$209$	526080.	0.833079
$210$	0	0
$211$	−181648.	−0.280882	−0.140441	−	0.990089i	$-0.544852\pi$
$211$	−181648.	−0.280882	−0.140441	+	0.990089i	$0.544852\pi$
$212$	− 146784.i	− 0.224305i
$213$	− 421272.i	− 0.636229i
$214$	4152.00	0.00619759
$215$	0	0
$216$	−172800.	−0.252005
$217$	810424.i	1.16832i
$218$	− 827720.i	− 1.17962i
$219$	−131916.	−0.185861
$220$	0	0
$221$	842772.	1.16073
$222$	− 132432.i	− 0.180348i
$223$	288274.i	0.388189i	0.980983	+	0.194095i	$0.0621769\pi$
$223$	288274.i	0.388189i	−0.980983	+	0.194095i	$0.937823\pi$
$224$	120832.	0.160902
$225$	0	0
$226$	557544.	0.726119
$227$	1.12552e6i	1.44974i	0.688887	+	0.724869i	$0.258100\pi$
$227$	1.12552e6i	1.44974i	−0.688887	+	0.724869i	$0.741900\pi$
$228$	263040.i	0.335108i
$229$	415810.	0.523970	0.261985	−	0.965072i	$-0.415623\pi$
$229$	415810.	0.523970	0.261985	+	0.965072i	$0.415623\pi$
$230$	0	0
$231$	−135936.	−0.167612
$232$	378240.i	0.461368i
$233$	− 770586.i	− 0.929889i	−0.885340	−	0.464945i	$-0.846074\pi$
$233$	− 770586.i	− 0.929889i	0.885340	−	0.464945i	$-0.153926\pi$
$234$	915768.	1.09332
$235$	0	0
$236$	−559680.	−0.654124
$237$	27120.0i	0.0313631i
$238$	359664.i	0.411580i
$239$	595320.	0.674149	0.337074	−	0.941478i	$-0.390563\pi$
$239$	595320.	0.674149	0.337074	+	0.941478i	$0.390563\pi$
$240$	0	0
$241$	273902.	0.303775	0.151888	−	0.988398i	$-0.451465\pi$
$241$	273902.	0.303775	0.151888	+	0.988398i	$0.451465\pi$
$242$	− 496748.i	− 0.545253i
$243$	− 860706.i	− 0.935059i
$244$	157408.	0.169259
$245$	0	0
$246$	−9072.00	−0.00955796
$247$	− 3.03044e6i	− 3.16055i
$248$	439552.i	0.453817i
$249$	654444.	0.668920
$250$	0	0
$251$	850752.	0.852351	0.426176	−	0.904640i	$-0.359861\pi$
$251$	850752.	0.852351	0.426176	+	0.904640i	$0.359861\pi$
$252$	390816.i	0.387678i
$253$	− 300672.i	− 0.295319i
$254$	−1.19953e6	−1.16661
$255$	0	0
$256$	65536.0	0.0625000
$257$	825402.i	0.779530i	0.920914	+	0.389765i	$0.127444\pi$
$257$	825402.i	0.779530i	−0.920914	+	0.389765i	$0.872556\pi$
$258$	58416.0i	0.0546365i
$259$	−651124.	−0.603135
$260$	0	0
$261$	−1.22337e6	−1.11162
$262$	31488.0i	0.0283395i
$263$	− 1.36465e6i	− 1.21655i	−0.793726	−	0.608276i	$-0.791861\pi$
$263$	− 1.36465e6i	− 1.21655i	0.793726	−	0.608276i	$-0.208139\pi$
$264$	−73728.0	−0.0651062
$265$	0	0
$266$	1.29328e6	1.12070
$267$	230940.i	0.198254i
$268$	− 539552.i	− 0.458877i
$269$	113310.	0.0954745	0.0477373	−	0.998860i	$-0.484799\pi$
$269$	113310.	0.0954745	0.0477373	+	0.998860i	$0.484799\pi$
$270$	0	0
$271$	−849628.	−0.702758	−0.351379	−	0.936233i	$-0.614287\pi$
$271$	−849628.	−0.702758	−0.351379	+	0.936233i	$0.614287\pi$
$272$	195072.i	0.159872i
$273$	783048.i	0.635890i
$274$	656952.	0.528637
$275$	0	0
$276$	150336.	0.118793
$277$	438602.i	0.343456i	0.985144	+	0.171728i	$0.0549350\pi$
$277$	438602.i	0.343456i	−0.985144	+	0.171728i	$0.945065\pi$
$278$	1.12840e6i	0.875691i
$279$	−1.42168e6	−1.09343
$280$	0	0
$281$	−1.45670e6	−1.10053	−0.550267	−	0.834989i	$-0.685474\pi$
$281$	−1.45670e6	−1.10053	−0.550267	+	0.834989i	$0.685474\pi$
$282$	314928.i	0.235824i
$283$	120394.i	0.0893591i	0.999001	+	0.0446795i	$0.0142267\pi$
$283$	120394.i	0.0893591i	−0.999001	+	0.0446795i	$0.985773\pi$
$284$	−1.12339e6	−0.826486
$285$	0	0
$286$	849408.	0.614047
$287$	44604.0i	0.0319646i
$288$	211968.i	0.150588i
$289$	839213.	0.591055
$290$	0	0
$291$	−11508.0	−0.00796650
$292$	351776.i	0.241440i
$293$	2.64209e6i	1.79796i	0.437993	+	0.898978i	$0.355689\pi$
$293$	2.64209e6i	1.79796i	−0.437993	+	0.898978i	$0.644311\pi$
$294$	69192.0	0.0466861
$295$	0	0
$296$	−353152.	−0.234278
$297$	− 518400.i	− 0.341015i
$298$	1.55580e6i	1.01488i
$299$	−1.73200e6	−1.12039
$300$	0	0
$301$	287212.	0.182720
$302$	− 391792.i	− 0.247194i
$303$	− 465732.i	− 0.291427i
$304$	701440.	0.435318
$305$	0	0
$306$	−630936.	−0.385196
$307$	− 1.44756e6i	− 0.876577i	−0.898834	−	0.438288i	$-0.855585\pi$
$307$	− 1.44756e6i	− 0.876577i	0.898834	−	0.438288i	$-0.144415\pi$
$308$	362496.i	0.217734i
$309$	280284.	0.166994
$310$	0	0
$311$	−928068.	−0.544100	−0.272050	−	0.962283i	$-0.587702\pi$
$311$	−928068.	−0.544100	−0.272050	+	0.962283i	$0.587702\pi$
$312$	424704.i	0.247002i
$313$	− 2.29563e6i	− 1.32446i	−0.749299	−	0.662232i	$-0.769609\pi$
$313$	− 2.29563e6i	− 1.32446i	0.749299	−	0.662232i	$-0.230391\pi$
$314$	14872.0	0.00851227
$315$	0	0
$316$	72320.0	0.0407418
$317$	2.73652e6i	1.52950i	0.644324	+	0.764752i	$0.277139\pi$
$317$	2.73652e6i	1.52950i	−0.644324	+	0.764752i	$0.722861\pi$
$318$	220176.i	0.122096i
$319$	−1.13472e6	−0.624327
$320$	0	0
$321$	−6228.00	−0.00337354
$322$	− 739152.i	− 0.397278i
$323$	2.08788e6i	1.11352i
$324$	−545616.	−0.288752
$325$	0	0
$326$	−172936.	−0.0901242
$327$	1.24158e6i	0.642104i
$328$	24192.0i	0.0124162i
$329$	1.54840e6	0.788665
$330$	0	0
$331$	3.81879e6	1.91583	0.957913	−	0.287059i	$-0.0926776\pi$
$331$	3.81879e6	1.91583	0.957913	+	0.287059i	$0.0926776\pi$
$332$	− 1.74518e6i	− 0.868952i
$333$	− 1.14223e6i	− 0.564471i
$334$	−746088.	−0.365952
$335$	0	0
$336$	−181248.	−0.0875841
$337$	− 2.21088e6i	− 1.06045i	−0.847857	−	0.530225i	$-0.822108\pi$
$337$	− 2.21088e6i	− 1.06045i	0.847857	−	0.530225i	$-0.177892\pi$
$338$	− 3.40777e6i	− 1.62248i
$339$	−836316.	−0.395249
$340$	0	0
$341$	−1.31866e6	−0.614109
$342$	2.26872e6i	1.04886i
$343$	− 2.32342e6i	− 1.06633i
$344$	155776.	0.0709748
$345$	0	0
$346$	−1.49782e6	−0.672618
$347$	− 2.32724e6i	− 1.03757i	−0.854905	−	0.518785i	$-0.826385\pi$
$347$	− 2.32724e6i	− 1.03757i	0.854905	−	0.518785i	$-0.173615\pi$
$348$	− 567360.i	− 0.251137i
$349$	311290.	0.136805	0.0684024	−	0.997658i	$-0.478210\pi$
$349$	311290.	0.136805	0.0684024	+	0.997658i	$0.478210\pi$
$350$	0	0
$351$	−2.98620e6	−1.29375
$352$	196608.i	0.0845755i
$353$	3.08657e6i	1.31838i	0.751977	+	0.659189i	$0.229100\pi$
$353$	3.08657e6i	1.31838i	−0.751977	+	0.659189i	$0.770900\pi$
$354$	839520.	0.356060
$355$	0	0
$356$	615840.	0.257539
$357$	− 539496.i	− 0.224036i
$358$	− 1.08840e6i	− 0.448829i
$359$	3.53076e6	1.44588	0.722940	−	0.690911i	$-0.242790\pi$
$359$	3.53076e6	1.44588	0.722940	+	0.690911i	$0.242790\pi$
$360$	0	0
$361$	5.03150e6	2.03203
$362$	− 301672.i	− 0.120994i
$363$	745122.i	0.296798i
$364$	2.08813e6	0.826045
$365$	0	0
$366$	−236112.	−0.0921330
$367$	35762.0i	0.0138598i	0.999976	+	0.00692989i	$0.00220587\pi$
$367$	35762.0i	0.0138598i	−0.999976	+	0.00692989i	$0.997794\pi$
$368$	− 400896.i	− 0.154316i
$369$	−78246.0	−0.0299155
$370$	0	0
$371$	1.08253e6	0.408325
$372$	− 659328.i	− 0.247027i
$373$	1.71525e6i	0.638346i	0.947696	+	0.319173i	$0.103405\pi$
$373$	1.71525e6i	0.638346i	−0.947696	+	0.319173i	$0.896595\pi$
$374$	−585216.	−0.216340
$375$	0	0
$376$	839808.	0.306345
$377$	6.53646e6i	2.36859i
$378$	− 1.27440e6i	− 0.458750i
$379$	3.10174e6	1.10919	0.554597	−	0.832119i	$-0.312873\pi$
$379$	3.10174e6	1.10919	0.554597	+	0.832119i	$0.312873\pi$
$380$	0	0
$381$	1.79929e6	0.635023
$382$	− 1.42795e6i	− 0.500674i
$383$	− 5.31949e6i	− 1.85299i	−0.376309	−	0.926494i	$-0.622807\pi$
$383$	− 5.31949e6i	− 1.85299i	0.376309	−	0.926494i	$-0.377193\pi$
$384$	−98304.0	−0.0340207
$385$	0	0
$386$	−1.75478e6	−0.599451
$387$	503838.i	0.171007i
$388$	30688.0i	0.0103488i
$389$	−1.16145e6	−0.389158	−0.194579	−	0.980887i	$-0.562334\pi$
$389$	−1.16145e6	−0.389158	−0.194579	+	0.980887i	$0.562334\pi$
$390$	0	0
$391$	1.19329e6	0.394734
$392$	− 184512.i	− 0.0606470i
$393$	− 47232.0i	− 0.0154261i
$394$	627192.	0.203545
$395$	0	0
$396$	−635904.	−0.203776
$397$	628562.i	0.200157i	0.994980	+	0.100079i	$0.0319095\pi$
$397$	628562.i	0.200157i	−0.994980	+	0.100079i	$0.968091\pi$
$398$	650080.i	0.205712i
$399$	−1.93992e6	−0.610031
$400$	0	0
$401$	−2.72432e6	−0.846052	−0.423026	−	0.906118i	$-0.639032\pi$
$401$	−2.72432e6	−0.846052	−0.423026	+	0.906118i	$0.639032\pi$
$402$	809328.i	0.249781i
$403$	7.59601e6i	2.32982i
$404$	−1.24195e6	−0.378575
$405$	0	0
$406$	−2.78952e6	−0.839875
$407$	− 1.05946e6i	− 0.317027i
$408$	− 292608.i	− 0.0870233i
$409$	−1.78019e6	−0.526209	−0.263104	−	0.964767i	$-0.584746\pi$
$409$	−1.78019e6	−0.526209	−0.263104	+	0.964767i	$0.584746\pi$
$410$	0	0
$411$	−985428.	−0.287753
$412$	− 747424.i	− 0.216932i
$413$	− 4.12764e6i	− 1.19077i
$414$	1.29665e6	0.371810
$415$	0	0
$416$	1.13254e6	0.320865
$417$	− 1.69260e6i	− 0.476666i
$418$	2.10432e6i	0.589076i
$419$	−650580.	−0.181036	−0.0905181	−	0.995895i	$-0.528852\pi$
$419$	−650580.	−0.181036	−0.0905181	+	0.995895i	$0.528852\pi$
$420$	0	0
$421$	−3.54060e6	−0.973579	−0.486790	−	0.873519i	$-0.661832\pi$
$421$	−3.54060e6	−0.973579	−0.486790	+	0.873519i	$0.661832\pi$
$422$	− 726592.i	− 0.198614i
$423$	2.71625e6i	0.738107i
$424$	587136.	0.158608
$425$	0	0
$426$	1.68509e6	0.449882
$427$	1.16088e6i	0.308119i
$428$	16608.0i	0.00438236i
$429$	−1.27411e6	−0.334245
$430$	0	0
$431$	−548748.	−0.142292	−0.0711459	−	0.997466i	$-0.522666\pi$
$431$	−548748.	−0.142292	−0.0711459	+	0.997466i	$0.522666\pi$
$432$	− 691200.i	− 0.178195i
$433$	1.49241e6i	0.382534i	0.981538	+	0.191267i	$0.0612596\pi$
$433$	1.49241e6i	0.382534i	−0.981538	+	0.191267i	$0.938740\pi$
$434$	−3.24170e6	−0.826129
$435$	0	0
$436$	3.31088e6	0.834117
$437$	− 4.29084e6i	− 1.07483i
$438$	− 527664.i	− 0.131423i
$439$	−4.86212e6	−1.20411	−0.602053	−	0.798456i	$-0.705650\pi$
$439$	−4.86212e6	−1.20411	−0.602053	+	0.798456i	$0.705650\pi$
$440$	0	0
$441$	596781.	0.146123
$442$	3.37109e6i	0.820757i
$443$	1.86155e6i	0.450678i	0.974280	+	0.225339i	$0.0723490\pi$
$443$	1.86155e6i	0.450678i	−0.974280	+	0.225339i	$0.927651\pi$
$444$	529728.	0.127525
$445$	0	0
$446$	−1.15310e6	−0.274491
$447$	− 2.33370e6i	− 0.552429i
$448$	483328.i	0.113775i
$449$	−3.73719e6	−0.874841	−0.437421	−	0.899257i	$-0.644108\pi$
$449$	−3.73719e6	−0.874841	−0.437421	+	0.899257i	$0.644108\pi$
$450$	0	0
$451$	−72576.0	−0.0168016
$452$	2.23018e6i	0.513444i
$453$	587688.i	0.134555i
$454$	−4.50209e6	−1.02512
$455$	0	0
$456$	−1.05216e6	−0.236957
$457$	− 6.48276e6i	− 1.45201i	−0.687690	−	0.726005i	$-0.741375\pi$
$457$	− 6.48276e6i	− 1.45201i	0.687690	−	0.726005i	$-0.258625\pi$
$458$	1.66324e6i	0.370503i
$459$	2.05740e6	0.455813
$460$	0	0
$461$	1.50910e6	0.330724	0.165362	−	0.986233i	$-0.447121\pi$
$461$	1.50910e6	0.330724	0.165362	+	0.986233i	$0.447121\pi$
$462$	− 543744.i	− 0.118519i
$463$	− 8.68401e6i	− 1.88264i	−0.337513	−	0.941321i	$-0.609586\pi$
$463$	− 8.68401e6i	− 1.88264i	0.337513	−	0.941321i	$-0.390414\pi$
$464$	−1.51296e6	−0.326236
$465$	0	0
$466$	3.08234e6	0.657531
$467$	6.96412e6i	1.47766i	0.673893	+	0.738829i	$0.264621\pi$
$467$	6.96412e6i	1.47766i	−0.673893	+	0.738829i	$0.735379\pi$
$468$	3.66307e6i	0.773091i
$469$	3.97920e6	0.835340
$470$	0	0
$471$	−22308.0	−0.00463349
$472$	− 2.23872e6i	− 0.462535i
$473$	467328.i	0.0960437i
$474$	−108480.	−0.0221771
$475$	0	0
$476$	−1.43866e6	−0.291031
$477$	1.89902e6i	0.382149i
$478$	2.38128e6i	0.476695i
$479$	5.51052e6	1.09737	0.548686	−	0.836029i	$-0.315128\pi$
$479$	5.51052e6	1.09737	0.548686	+	0.836029i	$0.315128\pi$
$480$	0	0
$481$	−6.10291e6	−1.20275
$482$	1.09561e6i	0.214802i
$483$	1.10873e6i	0.216251i
$484$	1.98699e6	0.385552
$485$	0	0
$486$	3.44282e6	0.661187
$487$	5.51808e6i	1.05430i	0.849771	+	0.527152i	$0.176740\pi$
$487$	5.51808e6i	1.05430i	−0.849771	+	0.527152i	$0.823260\pi$
$488$	629632.i	0.119684i
$489$	259404.	0.0490574
$490$	0	0
$491$	−1.51277e6	−0.283184	−0.141592	−	0.989925i	$-0.545222\pi$
$491$	−1.51277e6	−0.283184	−0.141592	+	0.989925i	$0.545222\pi$
$492$	− 36288.0i	− 0.00675850i
$493$	− 4.50342e6i	− 0.834498i
$494$	1.21218e7	2.23485
$495$	0	0
$496$	−1.75821e6	−0.320897
$497$	− 8.28502e6i	− 1.50454i
$498$	2.61778e6i	0.472998i
$499$	1.93042e6	0.347057	0.173528	−	0.984829i	$-0.444483\pi$
$499$	1.93042e6	0.347057	0.173528	+	0.984829i	$0.444483\pi$
$500$	0	0
$501$	1.11913e6	0.199199
$502$	3.40301e6i	0.602703i
$503$	− 6.73105e6i	− 1.18621i	−0.805124	−	0.593106i	$-0.797901\pi$
$503$	− 6.73105e6i	− 1.18621i	0.805124	−	0.593106i	$-0.202099\pi$
$504$	−1.56326e6	−0.274130
$505$	0	0
$506$	1.20269e6	0.208822
$507$	5.11166e6i	0.883165i
$508$	− 4.79811e6i	− 0.824919i
$509$	556650.	0.0952331	0.0476165	−	0.998866i	$-0.484837\pi$
$509$	556650.	0.0952331	0.0476165	+	0.998866i	$0.484837\pi$
$510$	0	0
$511$	−2.59435e6	−0.439517
$512$	262144.i	0.0441942i
$513$	− 7.39800e6i	− 1.24114i
$514$	−3.30161e6	−0.551211
$515$	0	0
$516$	−233664.	−0.0386338
$517$	2.51942e6i	0.414548i
$518$	− 2.60450e6i	− 0.426481i
$519$	2.24672e6	0.366127
$520$	0	0
$521$	1.01110e7	1.63192	0.815962	−	0.578106i	$-0.196208\pi$
$521$	1.01110e7	1.63192	0.815962	+	0.578106i	$0.196208\pi$
$522$	− 4.89348e6i	− 0.786034i
$523$	7.03719e6i	1.12498i	0.826804	+	0.562491i	$0.190157\pi$
$523$	7.03719e6i	1.12498i	−0.826804	+	0.562491i	$0.809843\pi$
$524$	−125952.	−0.0200390
$525$	0	0
$526$	5.45858e6	0.860232
$527$	− 5.23342e6i	− 0.820840i
$528$	− 294912.i	− 0.0460371i
$529$	3.98399e6	0.618983
$530$	0	0
$531$	7.24086e6	1.11443
$532$	5.17312e6i	0.792453i
$533$	418068.i	0.0637425i
$534$	−923760.	−0.140186
$535$	0	0
$536$	2.15821e6	0.324475
$537$	1.63260e6i	0.244312i
$538$	453240.i	0.0675107i
$539$	553536.	0.0820680
$540$	0	0
$541$	−4.23114e6	−0.621533	−0.310766	−	0.950486i	$-0.600586\pi$
$541$	−4.23114e6	−0.621533	−0.310766	+	0.950486i	$0.600586\pi$
$542$	− 3.39851e6i	− 0.496925i
$543$	452508.i	0.0658608i
$544$	−780288.	−0.113047
$545$	0	0
$546$	−3.13219e6	−0.449642
$547$	4.44024e6i	0.634510i	0.948340	+	0.317255i	$0.102761\pi$
$547$	4.44024e6i	0.634510i	−0.948340	+	0.317255i	$0.897239\pi$
$548$	2.62781e6i	0.373803i
$549$	−2.03647e6	−0.288367
$550$	0	0
$551$	−1.61934e7	−2.27227
$552$	601344.i	0.0839992i
$553$	533360.i	0.0741665i
$554$	−1.75441e6	−0.242860
$555$	0	0
$556$	−4.51360e6	−0.619207
$557$	− 9.01448e6i	− 1.23113i	−0.788088	−	0.615563i	$-0.788929\pi$
$557$	− 9.01448e6i	− 1.23113i	0.788088	−	0.615563i	$-0.211071\pi$
$558$	− 5.68670e6i	− 0.773170i
$559$	2.69200e6	0.364373
$560$	0	0
$561$	877824.	0.117761
$562$	− 5.82679e6i	− 0.778196i
$563$	9.81287e6i	1.30474i	0.757899	+	0.652372i	$0.226226\pi$
$563$	9.81287e6i	1.30474i	−0.757899	+	0.652372i	$0.773774\pi$
$564$	−1.25971e6	−0.166753
$565$	0	0
$566$	−481576.	−0.0631864
$567$	− 4.02392e6i	− 0.525644i
$568$	− 4.49357e6i	− 0.584414i
$569$	−1.33152e7	−1.72412	−0.862061	−	0.506804i	$-0.830827\pi$
$569$	−1.33152e7	−1.72412	−0.862061	+	0.506804i	$0.830827\pi$
$570$	0	0
$571$	9.95895e6	1.27827	0.639136	−	0.769094i	$-0.279292\pi$
$571$	9.95895e6	1.27827	0.639136	+	0.769094i	$0.279292\pi$
$572$	3.39763e6i	0.434196i
$573$	2.14193e6i	0.272533i
$574$	−178416.	−0.0226024
$575$	0	0
$576$	−847872.	−0.106481
$577$	4.50372e6i	0.563160i	0.959538	+	0.281580i	$0.0908585\pi$
$577$	4.50372e6i	0.563160i	−0.959538	+	0.281580i	$0.909141\pi$
$578$	3.35685e6i	0.417939i
$579$	2.63216e6	0.326300
$580$	0	0
$581$	1.28707e7	1.58184
$582$	− 46032.0i	− 0.00563316i
$583$	1.76141e6i	0.214629i
$584$	−1.40710e6	−0.170724
$585$	0	0
$586$	−1.05684e7	−1.27135
$587$	625842.i	0.0749669i	0.999297	+	0.0374834i	$0.0119341\pi$
$587$	625842.i	0.0749669i	−0.999297	+	0.0374834i	$0.988066\pi$
$588$	276768.i	0.0330121i
$589$	−1.88183e7	−2.23508
$590$	0	0
$591$	−940788.	−0.110796
$592$	− 1.41261e6i	− 0.165660i
$593$	2.50385e6i	0.292397i	0.989255	+	0.146198i	$0.0467038\pi$
$593$	2.50385e6i	0.292397i	−0.989255	+	0.146198i	$0.953296\pi$
$594$	2.07360e6	0.241134
$595$	0	0
$596$	−6.22320e6	−0.717626
$597$	− 975120.i	− 0.111975i
$598$	− 6.92798e6i	− 0.792235i
$599$	756480.	0.0861451	0.0430725	−	0.999072i	$-0.486285\pi$
$599$	756480.	0.0861451	0.0430725	+	0.999072i	$0.486285\pi$
$600$	0	0
$601$	−1.38565e7	−1.56483	−0.782413	−	0.622760i	$-0.786011\pi$
$601$	−1.38565e7	−1.56483	−0.782413	+	0.622760i	$0.786011\pi$
$602$	1.14885e6i	0.129203i
$603$	6.98045e6i	0.781791i
$604$	1.56717e6	0.174793
$605$	0	0
$606$	1.86293e6	0.206070
$607$	1.13772e7i	1.25333i	0.779291	+	0.626663i	$0.215580\pi$
$607$	1.13772e7i	1.25333i	−0.779291	+	0.626663i	$0.784420\pi$
$608$	2.80576e6i	0.307816i
$609$	4.18428e6	0.457170
$610$	0	0
$611$	1.45129e7	1.57272
$612$	− 2.52374e6i	− 0.272375i
$613$	7.00161e6i	0.752570i	0.926504	+	0.376285i	$0.122799\pi$
$613$	7.00161e6i	0.752570i	−0.926504	+	0.376285i	$0.877201\pi$
$614$	5.79023e6	0.619833
$615$	0	0
$616$	−1.44998e6	−0.153961
$617$	7.90300e6i	0.835755i	0.908503	+	0.417878i	$0.137226\pi$
$617$	7.90300e6i	0.835755i	−0.908503	+	0.417878i	$0.862774\pi$
$618$	1.12114e6i	0.118083i
$619$	−4.02362e6	−0.422076	−0.211038	−	0.977478i	$-0.567684\pi$
$619$	−4.02362e6	−0.422076	−0.211038	+	0.977478i	$0.567684\pi$
$620$	0	0
$621$	−4.22820e6	−0.439974
$622$	− 3.71227e6i	− 0.384737i
$623$	4.54182e6i	0.468824i
$624$	−1.69882e6	−0.174657
$625$	0	0
$626$	9.18250e6	0.936538
$627$	− 3.15648e6i	− 0.320652i
$628$	59488.0i	0.00601908i
$629$	4.20472e6	0.423750
$630$	0	0
$631$	−1.00227e7	−1.00210	−0.501049	−	0.865419i	$-0.667052\pi$
$631$	−1.00227e7	−1.00210	−0.501049	+	0.865419i	$0.667052\pi$
$632$	289280.i	0.0288088i
$633$	1.08989e6i	0.108112i
$634$	−1.09461e7	−1.08152
$635$	0	0
$636$	−880704.	−0.0863351
$637$	− 3.18860e6i	− 0.311352i
$638$	− 4.53888e6i	− 0.441466i
$639$	1.45339e7	1.40809
$640$	0	0
$641$	6.37390e6	0.612718	0.306359	−	0.951916i	$-0.400889\pi$
$641$	6.37390e6	0.612718	0.306359	+	0.951916i	$0.400889\pi$
$642$	− 24912.0i	− 0.00238545i
$643$	− 5.00457e6i	− 0.477352i	−0.971099	−	0.238676i	$-0.923287\pi$
$643$	− 5.00457e6i	− 0.477352i	0.971099	−	0.238676i	$-0.0767134\pi$
$644$	2.95661e6	0.280918
$645$	0	0
$646$	−8.35152e6	−0.787380
$647$	− 8.71928e6i	− 0.818879i	−0.912337	−	0.409440i	$-0.865724\pi$
$647$	− 8.71928e6i	− 0.818879i	0.912337	−	0.409440i	$-0.134276\pi$
$648$	− 2.18246e6i	− 0.204178i
$649$	6.71616e6	0.625906
$650$	0	0
$651$	4.86254e6	0.449688
$652$	− 691744.i	− 0.0637274i
$653$	1.58477e6i	0.145440i	0.997352	+	0.0727201i	$0.0231680\pi$
$653$	1.58477e6i	0.145440i	−0.997352	+	0.0727201i	$0.976832\pi$
$654$	−4.96632e6	−0.454036
$655$	0	0
$656$	−96768.0	−0.00877955
$657$	− 4.55110e6i	− 0.411342i
$658$	6.19358e6i	0.557670i
$659$	−1.26410e7	−1.13388	−0.566940	−	0.823759i	$-0.691873\pi$
$659$	−1.26410e7	−1.13388	−0.566940	+	0.823759i	$0.691873\pi$
$660$	0	0
$661$	−3.61572e6	−0.321878	−0.160939	−	0.986964i	$-0.551452\pi$
$661$	−3.61572e6	−0.321878	−0.160939	+	0.986964i	$0.551452\pi$
$662$	1.52752e7i	1.35469i
$663$	− 5.05663e6i	− 0.446763i
$664$	6.98074e6	0.614442
$665$	0	0
$666$	4.56890e6	0.399141
$667$	9.25506e6i	0.805498i
$668$	− 2.98435e6i	− 0.258767i
$669$	1.72964e6	0.149414
$670$	0	0
$671$	−1.88890e6	−0.161958
$672$	− 724992.i	− 0.0619313i
$673$	− 1.11313e7i	− 0.947349i	−0.880700	−	0.473675i	$-0.842927\pi$
$673$	− 1.11313e7i	− 0.947349i	0.880700	−	0.473675i	$-0.157073\pi$
$674$	8.84351e6	0.749851
$675$	0	0
$676$	1.36311e7	1.14727
$677$	− 235518.i	− 0.0197493i	−0.999951	−	0.00987467i	$-0.996857\pi$
$677$	− 235518.i	− 0.0197493i	0.999951	−	0.00987467i	$-0.00314326\pi$
$678$	− 3.34526e6i	− 0.279483i
$679$	−226324.	−0.0188389
$680$	0	0
$681$	6.75313e6	0.558004
$682$	− 5.27462e6i	− 0.434241i
$683$	− 2.05830e7i	− 1.68833i	−0.536084	−	0.844164i	$-0.680097\pi$
$683$	− 2.05830e7i	− 1.68833i	0.536084	−	0.844164i	$-0.319903\pi$
$684$	−9.07488e6	−0.741653
$685$	0	0
$686$	9.29368e6	0.754011
$687$	− 2.49486e6i	− 0.201676i
$688$	623104.i	0.0501868i
$689$	1.01464e7	0.814265
$690$	0	0
$691$	−9.54825e6	−0.760727	−0.380363	−	0.924837i	$-0.624201\pi$
$691$	−9.54825e6	−0.760727	−0.380363	+	0.924837i	$0.624201\pi$
$692$	− 5.99126e6i	− 0.475612i
$693$	− 4.68979e6i	− 0.370954i
$694$	9.30895e6	0.733672
$695$	0	0
$696$	2.26944e6	0.177581
$697$	− 288036.i	− 0.0224577i
$698$	1.24516e6i	0.0967357i
$699$	−4.62352e6	−0.357915
$700$	0	0
$701$	1.29304e6	0.0993843	0.0496921	−	0.998765i	$-0.484176\pi$
$701$	1.29304e6	0.0993843	0.0496921	+	0.998765i	$0.484176\pi$
$702$	− 1.19448e7i	− 0.914821i
$703$	− 1.51193e7i	− 1.15384i
$704$	−786432.	−0.0598039
$705$	0	0
$706$	−1.23463e7	−0.932234
$707$	− 9.15940e6i	− 0.689157i
$708$	3.35808e6i	0.251772i
$709$	2.12720e7	1.58926	0.794628	−	0.607097i	$-0.207666\pi$
$709$	2.12720e7	1.58926	0.794628	+	0.607097i	$0.207666\pi$
$710$	0	0
$711$	−935640.	−0.0694120
$712$	2.46336e6i	0.182108i
$713$	1.07553e7i	0.792316i
$714$	2.15798e6	0.158417
$715$	0	0
$716$	4.35360e6	0.317370
$717$	− 3.57192e6i	− 0.259480i
$718$	1.41230e7i	1.02239i
$719$	−8.31732e6	−0.600014	−0.300007	−	0.953937i	$-0.596989\pi$
$719$	−8.31732e6	−0.600014	−0.300007	+	0.953937i	$0.596989\pi$
$720$	0	0
$721$	5.51225e6	0.394903
$722$	2.01260e7i	1.43686i
$723$	− 1.64341e6i	− 0.116923i
$724$	1.20669e6	0.0855556
$725$	0	0
$726$	−2.98049e6	−0.209868
$727$	− 4.36740e6i	− 0.306469i	−0.988190	−	0.153235i	$-0.951031\pi$
$727$	− 4.36740e6i	− 0.306469i	0.988190	−	0.153235i	$-0.0489690\pi$
$728$	8.35251e6i	0.584102i
$729$	3.12231e6	0.217599
$730$	0	0
$731$	−1.85471e6	−0.128375
$732$	− 944448.i	− 0.0651479i
$733$	4.05645e6i	0.278860i	0.990232	+	0.139430i	$0.0445271\pi$
$733$	4.05645e6i	0.278860i	−0.990232	+	0.139430i	$0.955473\pi$
$734$	−143048.	−0.00980035
$735$	0	0
$736$	1.60358e6	0.109118
$737$	6.47462e6i	0.439082i
$738$	− 312984.i	− 0.0211535i
$739$	−768260.	−0.0517484	−0.0258742	−	0.999665i	$-0.508237\pi$
$739$	−768260.	−0.0517484	−0.0258742	+	0.999665i	$0.508237\pi$
$740$	0	0
$741$	−1.81826e7	−1.21650
$742$	4.33013e6i	0.288729i
$743$	− 6.18781e6i	− 0.411211i	−0.978635	−	0.205605i	$-0.934084\pi$
$743$	− 6.18781e6i	− 0.411211i	0.978635	−	0.205605i	$-0.0659164\pi$
$744$	2.63731e6	0.174674
$745$	0	0
$746$	−6.86102e6	−0.451379
$747$	2.25783e7i	1.48044i
$748$	− 2.34086e6i	− 0.152976i
$749$	−122484.	−0.00797765
$750$	0	0
$751$	1.81698e7	1.17557	0.587787	−	0.809016i	$-0.299999\pi$
$751$	1.81698e7	1.17557	0.587787	+	0.809016i	$0.299999\pi$
$752$	3.35923e6i	0.216618i
$753$	− 5.10451e6i	− 0.328070i
$754$	−2.61458e7	−1.67484
$755$	0	0
$756$	5.09760e6	0.324385
$757$	1.93494e7i	1.22724i	0.789603	+	0.613618i	$0.210286\pi$
$757$	1.93494e7i	1.22724i	−0.789603	+	0.613618i	$0.789714\pi$
$758$	1.24070e7i	0.784318i
$759$	−1.80403e6	−0.113668
$760$	0	0
$761$	−3.01992e7	−1.89031	−0.945155	−	0.326621i	$-0.894090\pi$
$761$	−3.01992e7	−1.89031	−0.945155	+	0.326621i	$0.894090\pi$
$762$	7.19717e6i	0.449029i
$763$	2.44177e7i	1.51843i
$764$	5.71181e6	0.354030
$765$	0	0
$766$	2.12779e7	1.31026
$767$	− 3.86879e7i	− 2.37458i
$768$	− 393216.i	− 0.0240563i
$769$	−2.15854e7	−1.31627	−0.658134	−	0.752901i	$-0.728654\pi$
$769$	−2.15854e7	−1.31627	−0.658134	+	0.752901i	$0.728654\pi$
$770$	0	0
$771$	4.95241e6	0.300041
$772$	− 7.01910e6i	− 0.423876i
$773$	− 3.90895e6i	− 0.235294i	−0.993055	−	0.117647i	$-0.962465\pi$
$773$	− 3.90895e6i	− 0.235294i	0.993055	−	0.117647i	$-0.0375351\pi$
$774$	−2.01535e6	−0.120920
$775$	0	0
$776$	−122752.	−0.00731769
$777$	3.90674e6i	0.232147i
$778$	− 4.64580e6i	− 0.275177i
$779$	−1.03572e6	−0.0611503
$780$	0	0
$781$	1.34807e7	0.790833
$782$	4.77317e6i	0.279119i
$783$	1.59570e7i	0.930137i
$784$	738048.	0.0428839
$785$	0	0
$786$	188928.	0.0109079
$787$	− 2.65082e7i	− 1.52561i	−0.646628	−	0.762806i	$-0.723821\pi$
$787$	− 2.65082e7i	− 1.52561i	0.646628	−	0.762806i	$-0.276179\pi$
$788$	2.50877e6i	0.143928i
$789$	−8.18788e6	−0.468251
$790$	0	0
$791$	−1.64475e7	−0.934674
$792$	− 2.54362e6i	− 0.144092i
$793$	1.08808e7i	0.614439i
$794$	−2.51425e6	−0.141533
$795$	0	0
$796$	−2.60032e6	−0.145460
$797$	1.07940e7i	0.601919i	0.953637	+	0.300960i	$0.0973070\pi$
$797$	1.07940e7i	0.601919i	−0.953637	+	0.300960i	$0.902693\pi$
$798$	− 7.75968e6i	− 0.431357i
$799$	−9.99896e6	−0.554100
$800$	0	0
$801$	−7.96743e6	−0.438770
$802$	− 1.08973e7i	− 0.598249i
$803$	− 4.22131e6i	− 0.231025i
$804$	−3.23731e6	−0.176622
$805$	0	0
$806$	−3.03840e7	−1.64743
$807$	− 679860.i	− 0.0367482i
$808$	− 4.96781e6i	− 0.267693i
$809$	1.11446e7	0.598675	0.299338	−	0.954147i	$-0.403234\pi$
$809$	1.11446e7	0.598675	0.299338	+	0.954147i	$0.403234\pi$
$810$	0	0
$811$	−1.14866e7	−0.613253	−0.306626	−	0.951830i	$-0.599200\pi$
$811$	−1.14866e7	−0.613253	−0.306626	+	0.951830i	$0.599200\pi$
$812$	− 1.11581e7i	− 0.593881i
$813$	5.09777e6i	0.270492i
$814$	4.23782e6	0.224172
$815$	0	0
$816$	1.17043e6	0.0615348
$817$	6.66916e6i	0.349555i
$818$	− 7.12076e6i	− 0.372086i
$819$	−2.70152e7	−1.40734
$820$	0	0
$821$	3.04347e7	1.57584	0.787918	−	0.615781i	$-0.211159\pi$
$821$	3.04347e7	1.57584	0.787918	+	0.615781i	$0.211159\pi$
$822$	− 3.94171e6i	− 0.203472i
$823$	− 4.09773e6i	− 0.210884i	−0.994425	−	0.105442i	$-0.966374\pi$
$823$	− 4.09773e6i	− 0.210884i	0.994425	−	0.105442i	$-0.0336257\pi$
$824$	2.98970e6	0.153394
$825$	0	0
$826$	1.65106e7	0.841999
$827$	− 1.70652e7i	− 0.867654i	−0.900996	−	0.433827i	$-0.857163\pi$
$827$	− 1.70652e7i	− 0.867654i	0.900996	−	0.433827i	$-0.142837\pi$
$828$	5.18659e6i	0.262909i
$829$	2.47617e7	1.25139	0.625697	−	0.780066i	$-0.284815\pi$
$829$	2.47617e7	1.25139	0.625697	+	0.780066i	$0.284815\pi$
$830$	0	0
$831$	2.63161e6	0.132196
$832$	4.53018e6i	0.226886i
$833$	2.19685e6i	0.109695i
$834$	6.77040e6	0.337054
$835$	0	0
$836$	−8.41728e6	−0.416539
$837$	1.85436e7i	0.914914i
$838$	− 2.60232e6i	− 0.128012i
$839$	−3.16529e7	−1.55242	−0.776208	−	0.630476i	$-0.782860\pi$
$839$	−3.16529e7	−1.55242	−0.776208	+	0.630476i	$0.782860\pi$
$840$	0	0
$841$	1.44170e7	0.702884
$842$	− 1.41624e7i	− 0.688425i
$843$	8.74019e6i	0.423596i
$844$	2.90637e6	0.140441
$845$	0	0
$846$	−1.08650e7	−0.521921
$847$	1.46541e7i	0.701859i
$848$	2.34854e6i	0.112153i
$849$	722364.	0.0343943
$850$	0	0
$851$	−8.64119e6	−0.409025
$852$	6.74035e6i	0.318115i
$853$	− 2.82671e7i	− 1.33017i	−0.746765	−	0.665087i	$-0.768394\pi$
$853$	− 2.82671e7i	− 1.33017i	0.746765	−	0.665087i	$-0.231606\pi$
$854$	−4.64354e6	−0.217873
$855$	0	0
$856$	−66432.0	−0.00309880
$857$	2.60870e7i	1.21331i	0.794966	+	0.606655i	$0.207489\pi$
$857$	2.60870e7i	1.21331i	−0.794966	+	0.606655i	$0.792511\pi$
$858$	− 5.09645e6i	− 0.236347i
$859$	3.38111e7	1.56342	0.781710	−	0.623642i	$-0.214348\pi$
$859$	3.38111e7	1.56342	0.781710	+	0.623642i	$0.214348\pi$
$860$	0	0
$861$	267624.	0.0123032
$862$	− 2.19499e6i	− 0.100615i
$863$	− 2.22817e7i	− 1.01841i	−0.860646	−	0.509204i	$-0.829940\pi$
$863$	− 2.22817e7i	− 1.01841i	0.860646	−	0.509204i	$-0.170060\pi$
$864$	2.76480e6	0.126003
$865$	0	0
$866$	−5.96966e6	−0.270492
$867$	− 5.03528e6i	− 0.227497i
$868$	− 1.29668e7i	− 0.584162i
$869$	−867840.	−0.0389843
$870$	0	0
$871$	3.72965e7	1.66580
$872$	1.32435e7i	0.589810i
$873$	− 397026.i	− 0.0176313i
$874$	1.71634e7	0.760018
$875$	0	0
$876$	2.11066e6	0.0929303
$877$	− 3.46748e7i	− 1.52235i	−0.648545	−	0.761177i	$-0.724622\pi$
$877$	− 3.46748e7i	− 1.52235i	0.648545	−	0.761177i	$-0.275378\pi$
$878$	− 1.94485e7i	− 0.851431i
$879$	1.58526e7	0.692034
$880$	0	0
$881$	1.42603e7	0.618998	0.309499	−	0.950900i	$-0.399839\pi$
$881$	1.42603e7	0.618998	0.309499	+	0.950900i	$0.399839\pi$
$882$	2.38712e6i	0.103325i
$883$	3.75177e7i	1.61933i	0.586895	+	0.809663i	$0.300350\pi$
$883$	3.75177e7i	1.61933i	−0.586895	+	0.809663i	$0.699650\pi$
$884$	−1.34844e7	−0.580363
$885$	0	0
$886$	−7.44622e6	−0.318677
$887$	4.07657e7i	1.73975i	0.493275	+	0.869873i	$0.335800\pi$
$887$	4.07657e7i	1.73975i	−0.493275	+	0.869873i	$0.664200\pi$
$888$	2.11891e6i	0.0901738i
$889$	3.53861e7	1.50168
$890$	0	0
$891$	6.54739e6	0.276296
$892$	− 4.61238e6i	− 0.194095i
$893$	3.59543e7i	1.50877i
$894$	9.33480e6	0.390626
$895$	0	0
$896$	−1.93331e6	−0.0804511
$897$	1.03920e7i	0.431238i
$898$	− 1.49488e7i	− 0.618606i
$899$	4.05899e7	1.67501
$900$	0	0
$901$	−6.99059e6	−0.286881
$902$	− 290304.i	− 0.0118806i
$903$	− 1.72327e6i	− 0.0703290i
$904$	−8.92070e6	−0.363060
$905$	0	0
$906$	−2.35075e6	−0.0951451
$907$	− 3.57116e7i	− 1.44142i	−0.693235	−	0.720712i	$-0.743815\pi$
$907$	− 3.57116e7i	− 1.44142i	0.693235	−	0.720712i	$-0.256185\pi$
$908$	− 1.80084e7i	− 0.724869i
$909$	1.60678e7	0.644979
$910$	0	0
$911$	−2.11389e7	−0.843893	−0.421947	−	0.906621i	$-0.638653\pi$
$911$	−2.11389e7	−0.843893	−0.421947	+	0.906621i	$0.638653\pi$
$912$	− 4.20864e6i	− 0.167554i
$913$	2.09422e7i	0.831468i
$914$	2.59310e7	1.02673
$915$	0	0
$916$	−6.65296e6	−0.261985
$917$	− 928896.i	− 0.0364791i
$918$	8.22960e6i	0.322309i
$919$	−1.85996e7	−0.726465	−0.363233	−	0.931698i	$-0.618327\pi$
$919$	−1.85996e7	−0.726465	−0.363233	+	0.931698i	$0.618327\pi$
$920$	0	0
$921$	−8.68535e6	−0.337395
$922$	6.03641e6i	0.233857i
$923$	− 7.76545e7i	− 3.00028i
$924$	2.17498e6	0.0838059
$925$	0	0
$926$	3.47360e7	1.33123
$927$	9.66980e6i	0.369588i
$928$	− 6.05184e6i	− 0.230684i
$929$	−4.45110e7	−1.69211	−0.846055	−	0.533096i	$-0.821028\pi$
$929$	−4.45110e7	−1.69211	−0.846055	+	0.533096i	$0.821028\pi$
$930$	0	0
$931$	7.89942e6	0.298690
$932$	1.23294e7i	0.464945i
$933$	5.56841e6i	0.209424i
$934$	−2.78565e7	−1.04486
$935$	0	0
$936$	−1.46523e7	−0.546658
$937$	− 2.19419e7i	− 0.816441i	−0.912883	−	0.408221i	$-0.866149\pi$
$937$	− 2.19419e7i	− 0.816441i	0.912883	−	0.408221i	$-0.133851\pi$
$938$	1.59168e7i	0.590675i
$939$	−1.37738e7	−0.509787
$940$	0	0
$941$	−7.77722e6	−0.286319	−0.143160	−	0.989700i	$-0.545726\pi$
$941$	−7.77722e6	−0.286319	−0.143160	+	0.989700i	$0.545726\pi$
$942$	− 89232.0i	− 0.00327637i
$943$	591948.i	0.0216773i
$944$	8.95488e6	0.327062
$945$	0	0
$946$	−1.86931e6	−0.0679132
$947$	3.17199e7i	1.14936i	0.818378	+	0.574681i	$0.194874\pi$
$947$	3.17199e7i	1.14936i	−0.818378	+	0.574681i	$0.805126\pi$
$948$	− 433920.i	− 0.0156815i
$949$	−2.43165e7	−0.876468
$950$	0	0
$951$	1.64191e7	0.588707
$952$	− 5.75462e6i	− 0.205790i
$953$	5.60285e6i	0.199838i	0.994996	+	0.0999188i	$0.0318583\pi$
$953$	5.60285e6i	0.199838i	−0.994996	+	0.0999188i	$0.968142\pi$
$954$	−7.59607e6	−0.270220
$955$	0	0
$956$	−9.52512e6	−0.337074
$957$	6.80832e6i	0.240304i
$958$	2.20421e7i	0.775959i
$959$	−1.93801e7	−0.680470
$960$	0	0
$961$	1.85403e7	0.647601
$962$	− 2.44116e7i	− 0.850470i
$963$	− 214866.i	− 0.00746624i
$964$	−4.38243e6	−0.151888
$965$	0	0
$966$	−4.43491e6	−0.152912
$967$	− 2.03532e7i	− 0.699949i	−0.936759	−	0.349975i	$-0.886190\pi$
$967$	− 2.03532e7i	− 0.699949i	0.936759	−	0.349975i	$-0.113810\pi$
$968$	7.94797e6i	0.272626i
$969$	1.25273e7	0.428595
$970$	0	0
$971$	−2.34306e7	−0.797510	−0.398755	−	0.917057i	$-0.630558\pi$
$971$	−2.34306e7	−0.797510	−0.398755	+	0.917057i	$0.630558\pi$
$972$	1.37713e7i	0.467530i
$973$	− 3.32878e7i	− 1.12721i
$974$	−2.20723e7	−0.745505
$975$	0	0
$976$	−2.51853e6	−0.0846296
$977$	− 4.30412e7i	− 1.44261i	−0.692619	−	0.721303i	$-0.743543\pi$
$977$	− 4.30412e7i	− 1.44261i	0.692619	−	0.721303i	$-0.256457\pi$
$978$	1.03762e6i	0.0346888i
$979$	−7.39008e6	−0.246429
$980$	0	0
$981$	−4.28345e7	−1.42109
$982$	− 6.05107e6i	− 0.200241i
$983$	4.75003e7i	1.56788i	0.620837	+	0.783940i	$0.286793\pi$
$983$	4.75003e7i	1.56788i	−0.620837	+	0.783940i	$0.713207\pi$
$984$	145152.	0.00477898
$985$	0	0
$986$	1.80137e7	0.590079
$987$	− 9.29038e6i	− 0.303557i
$988$	4.84870e7i	1.58028i
$989$	3.81164e6	0.123914
$990$	0	0
$991$	2.09231e7	0.676770	0.338385	−	0.941008i	$-0.390119\pi$
$991$	2.09231e7	0.676770	0.338385	+	0.941008i	$0.390119\pi$
$992$	− 7.03283e6i	− 0.226909i
$993$	− 2.29128e7i	− 0.737402i
$994$	3.31401e7	1.06387
$995$	0	0
$996$	−1.04711e7	−0.334460
$997$	2.96332e7i	0.944148i	0.881559	+	0.472074i	$0.156495\pi$
$997$	2.96332e7i	0.944148i	−0.881559	+	0.472074i	$0.843505\pi$
$998$	7.72168e6i	0.245406i
$999$	−1.48986e7	−0.472315

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.b.b.49.2		2
3.2	odd	2		450.6.c.f.199.1		2
4.3	odd	2		400.6.c.i.49.2		2
5.2	odd	4		50.6.a.b.1.1		1
5.3	odd	4		10.6.a.c.1.1	✓	1
5.4	even	2	inner	50.6.b.b.49.1		2
15.2	even	4		450.6.a.u.1.1		1
15.8	even	4		90.6.a.b.1.1		1
15.14	odd	2		450.6.c.f.199.2		2
20.3	even	4		80.6.a.c.1.1		1
20.7	even	4		400.6.a.i.1.1		1
20.19	odd	2		400.6.c.i.49.1		2
35.13	even	4		490.6.a.k.1.1		1
40.3	even	4		320.6.a.k.1.1		1
40.13	odd	4		320.6.a.f.1.1		1
60.23	odd	4		720.6.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.a.c.1.1	✓	1	5.3	odd	4
50.6.a.b.1.1		1	5.2	odd	4
50.6.b.b.49.1		2	5.4	even	2	inner
50.6.b.b.49.2		2	1.1	even	1	trivial
80.6.a.c.1.1		1	20.3	even	4
90.6.a.b.1.1		1	15.8	even	4
320.6.a.f.1.1		1	40.13	odd	4
320.6.a.k.1.1		1	40.3	even	4
400.6.a.i.1.1		1	20.7	even	4
400.6.c.i.49.1		2	20.19	odd	2
400.6.c.i.49.2		2	4.3	odd	2
450.6.a.u.1.1		1	15.2	even	4
450.6.c.f.199.1		2	3.2	odd	2
450.6.c.f.199.2		2	15.14	odd	2
490.6.a.k.1.1		1	35.13	even	4
720.6.a.v.1.1		1	60.23	odd	4

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:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -6.00000i q^{3} -16.0000 q^{4} +24.0000 q^{6} -118.000i q^{7} -64.0000i q^{8} +207.000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} -6.00000i q^{3} -16.0000 q^{4} +24.0000 q^{6} -118.000i q^{7} -64.0000i q^{8} +207.000 q^{9} +192.000 q^{11} +96.0000i q^{12} -1106.00i q^{13} +472.000 q^{14} +256.000 q^{16} +762.000i q^{17} +828.000i q^{18} +2740.00 q^{19} -708.000 q^{21} +768.000i q^{22} -1566.00i q^{23} -384.000 q^{24} +4424.00 q^{26} -2700.00i q^{27} +1888.00i q^{28} -5910.00 q^{29} -6868.00 q^{31} +1024.00i q^{32} -1152.00i q^{33} -3048.00 q^{34} -3312.00 q^{36} -5518.00i q^{37} +10960.0i q^{38} -6636.00 q^{39} -378.000 q^{41} -2832.00i q^{42} +2434.00i q^{43} -3072.00 q^{44} +6264.00 q^{46} +13122.0i q^{47} -1536.00i q^{48} +2883.00 q^{49} +4572.00 q^{51} +17696.0i q^{52} +9174.00i q^{53} +10800.0 q^{54} -7552.00 q^{56} -16440.0i q^{57} -23640.0i q^{58} +34980.0 q^{59} -9838.00 q^{61} -27472.0i q^{62} -24426.0i q^{63} -4096.00 q^{64} +4608.00 q^{66} +33722.0i q^{67} -12192.0i q^{68} -9396.00 q^{69} +70212.0 q^{71} -13248.0i q^{72} -21986.0i q^{73} +22072.0 q^{74} -43840.0 q^{76} -22656.0i q^{77} -26544.0i q^{78} -4520.00 q^{79} +34101.0 q^{81} -1512.00i q^{82} +109074. i q^{83} +11328.0 q^{84} -9736.00 q^{86} +35460.0i q^{87} -12288.0i q^{88} -38490.0 q^{89} -130508. q^{91} +25056.0i q^{92} +41208.0i q^{93} -52488.0 q^{94} +6144.00 q^{96} -1918.00i q^{97} +11532.0i q^{98} +39744.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 48 q^{6} + 414 q^{9}+O(q^{10}) \) 2 * q - 32 * q^4 + 48 * q^6 + 414 * q^9	\( 2 q - 32 q^{4} + 48 q^{6} + 414 q^{9} + 384 q^{11} + 944 q^{14} + 512 q^{16} + 5480 q^{19} - 1416 q^{21} - 768 q^{24} + 8848 q^{26} - 11820 q^{29} - 13736 q^{31} - 6096 q^{34} - 6624 q^{36} - 13272 q^{39} - 756 q^{41} - 6144 q^{44} + 12528 q^{46} + 5766 q^{49} + 9144 q^{51} + 21600 q^{54} - 15104 q^{56} + 69960 q^{59} - 19676 q^{61} - 8192 q^{64} + 9216 q^{66} - 18792 q^{69} + 140424 q^{71} + 44144 q^{74} - 87680 q^{76} - 9040 q^{79} + 68202 q^{81} + 22656 q^{84} - 19472 q^{86} - 76980 q^{89} - 261016 q^{91} - 104976 q^{94} + 12288 q^{96} + 79488 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 48 * q^6 + 414 * q^9 + 384 * q^11 + 944 * q^14 + 512 * q^16 + 5480 * q^19 - 1416 * q^21 - 768 * q^24 + 8848 * q^26 - 11820 * q^29 - 13736 * q^31 - 6096 * q^34 - 6624 * q^36 - 13272 * q^39 - 756 * q^41 - 6144 * q^44 + 12528 * q^46 + 5766 * q^49 + 9144 * q^51 + 21600 * q^54 - 15104 * q^56 + 69960 * q^59 - 19676 * q^61 - 8192 * q^64 + 9216 * q^66 - 18792 * q^69 + 140424 * q^71 + 44144 * q^74 - 87680 * q^76 - 9040 * q^79 + 68202 * q^81 + 22656 * q^84 - 19472 * q^86 - 76980 * q^89 - 261016 * q^91 - 104976 * q^94 + 12288 * q^96 + 79488 * q^99

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