LMFDB - Embedded newform 150.6.c.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,6,Mod(49,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, 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("150.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	150.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:	$24.0575729719$
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:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	150.49
Dual form			150.6.c.a.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +1.00000i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +1.00000i q^{7} -64.0000i q^{8} -81.0000 q^{9} -210.000 q^{11} -144.000i q^{12} -667.000i q^{13} -4.00000 q^{14} +256.000 q^{16} -114.000i q^{17} -324.000i q^{18} -581.000 q^{19} -9.00000 q^{21} -840.000i q^{22} -4350.00i q^{23} +576.000 q^{24} +2668.00 q^{26} -729.000i q^{27} -16.0000i q^{28} +126.000 q^{29} +7583.00 q^{31} +1024.00i q^{32} -1890.00i q^{33} +456.000 q^{34} +1296.00 q^{36} +3742.00i q^{37} -2324.00i q^{38} +6003.00 q^{39} -2856.00 q^{41} -36.0000i q^{42} -18241.0i q^{43} +3360.00 q^{44} +17400.0 q^{46} +23370.0i q^{47} +2304.00i q^{48} +16806.0 q^{49} +1026.00 q^{51} +10672.0i q^{52} -21684.0i q^{53} +2916.00 q^{54} +64.0000 q^{56} -5229.00i q^{57} +504.000i q^{58} +32310.0 q^{59} -7165.00 q^{61} +30332.0i q^{62} -81.0000i q^{63} -4096.00 q^{64} +7560.00 q^{66} -59579.0i q^{67} +1824.00i q^{68} +39150.0 q^{69} -43080.0 q^{71} +5184.00i q^{72} -28942.0i q^{73} -14968.0 q^{74} +9296.00 q^{76} -210.000i q^{77} +24012.0i q^{78} -27608.0 q^{79} +6561.00 q^{81} -11424.0i q^{82} -1782.00i q^{83} +144.000 q^{84} +72964.0 q^{86} +1134.00i q^{87} +13440.0i q^{88} -50208.0 q^{89} +667.000 q^{91} +69600.0i q^{92} +68247.0i q^{93} -93480.0 q^{94} -9216.00 q^{96} -142793. i q^{97} +67224.0i q^{98} +17010.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} - 72 q^{6} - 162 q^{9}+O(q^{10}) $ 2 * q - 32 * q^4 - 72 * q^6 - 162 * q^9	$ 2 q - 32 q^{4} - 72 q^{6} - 162 q^{9} - 420 q^{11} - 8 q^{14} + 512 q^{16} - 1162 q^{19} - 18 q^{21} + 1152 q^{24} + 5336 q^{26} + 252 q^{29} + 15166 q^{31} + 912 q^{34} + 2592 q^{36} + 12006 q^{39} - 5712 q^{41} + 6720 q^{44} + 34800 q^{46} + 33612 q^{49} + 2052 q^{51} + 5832 q^{54} + 128 q^{56} + 64620 q^{59} - 14330 q^{61} - 8192 q^{64} + 15120 q^{66} + 78300 q^{69} - 86160 q^{71} - 29936 q^{74} + 18592 q^{76} - 55216 q^{79} + 13122 q^{81} + 288 q^{84} + 145928 q^{86} - 100416 q^{89} + 1334 q^{91} - 186960 q^{94} - 18432 q^{96} + 34020 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 - 72 * q^6 - 162 * q^9 - 420 * q^11 - 8 * q^14 + 512 * q^16 - 1162 * q^19 - 18 * q^21 + 1152 * q^24 + 5336 * q^26 + 252 * q^29 + 15166 * q^31 + 912 * q^34 + 2592 * q^36 + 12006 * q^39 - 5712 * q^41 + 6720 * q^44 + 34800 * q^46 + 33612 * q^49 + 2052 * q^51 + 5832 * q^54 + 128 * q^56 + 64620 * q^59 - 14330 * q^61 - 8192 * q^64 + 15120 * q^66 + 78300 * q^69 - 86160 * q^71 - 29936 * q^74 + 18592 * q^76 - 55216 * q^79 + 13122 * q^81 + 288 * q^84 + 145928 * q^86 - 100416 * q^89 + 1334 * q^91 - 186960 * q^94 - 18432 * q^96 + 34020 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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
$7$	1.00000i	0.00771356i	0.999993	+	0.00385678i	$0.00122765\pi$
$7$	1.00000i	0.00771356i	−0.999993	+	0.00385678i	$0.998772\pi$
$8$	− 64.0000i	− 0.353553i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	−210.000	−0.523284	−0.261642	−	0.965165i	$-0.584264\pi$
$11$	−210.000	−0.523284	−0.261642	+	0.965165i	$0.584264\pi$
$12$	− 144.000i	− 0.288675i
$13$	− 667.000i	− 1.09463i	−0.836927	−	0.547315i	$-0.815650\pi$
$13$	− 667.000i	− 1.09463i	0.836927	−	0.547315i	$-0.184350\pi$
$14$	−4.00000	−0.00545431
$15$	0	0
$16$	256.000	0.250000
$17$	− 114.000i	− 0.0956715i	−0.998855	−	0.0478357i	$-0.984768\pi$
$17$	− 114.000i	− 0.0956715i	0.998855	−	0.0478357i	$-0.0152324\pi$
$18$	− 324.000i	− 0.235702i
$19$	−581.000	−0.369226	−0.184613	−	0.982811i	$-0.559103\pi$
$19$	−581.000	−0.369226	−0.184613	+	0.982811i	$0.559103\pi$
$20$	0	0
$21$	−9.00000	−0.00445343
$22$	− 840.000i	− 0.370018i
$23$	− 4350.00i	− 1.71463i	−0.514795	−	0.857314i	$-0.672132\pi$
$23$	− 4350.00i	− 1.71463i	0.514795	−	0.857314i	$-0.327868\pi$
$24$	576.000	0.204124
$25$	0	0
$26$	2668.00	0.774020
$27$	− 729.000i	− 0.192450i
$28$	− 16.0000i	− 0.00385678i
$29$	126.000	0.0278212	0.0139106	−	0.999903i	$-0.495572\pi$
$29$	126.000	0.0278212	0.0139106	+	0.999903i	$0.495572\pi$
$30$	0	0
$31$	7583.00	1.41722	0.708609	−	0.705601i	$-0.249323\pi$
$31$	7583.00	1.41722	0.708609	+	0.705601i	$0.249323\pi$
$32$	1024.00i	0.176777i
$33$	− 1890.00i	− 0.302118i
$34$	456.000	0.0676500
$35$	0	0
$36$	1296.00	0.166667
$37$	3742.00i	0.449365i	0.974432	+	0.224683i	$0.0721345\pi$
$37$	3742.00i	0.449365i	−0.974432	+	0.224683i	$0.927865\pi$
$38$	− 2324.00i	− 0.261082i
$39$	6003.00	0.631985
$40$	0	0
$41$	−2856.00	−0.265337	−0.132669	−	0.991160i	$-0.542355\pi$
$41$	−2856.00	−0.265337	−0.132669	+	0.991160i	$0.542355\pi$
$42$	− 36.0000i	− 0.00314905i
$43$	− 18241.0i	− 1.50445i	−0.658907	−	0.752225i	$-0.728981\pi$
$43$	− 18241.0i	− 1.50445i	0.658907	−	0.752225i	$-0.271019\pi$
$44$	3360.00	0.261642
$45$	0	0
$46$	17400.0	1.21242
$47$	23370.0i	1.54317i	0.636126	+	0.771586i	$0.280536\pi$
$47$	23370.0i	1.54317i	−0.636126	+	0.771586i	$0.719464\pi$
$48$	2304.00i	0.144338i
$49$	16806.0	0.999941
$50$	0	0
$51$	1026.00	0.0552360
$52$	10672.0i	0.547315i
$53$	− 21684.0i	− 1.06035i	−0.847888	−	0.530176i	$-0.822126\pi$
$53$	− 21684.0i	− 1.06035i	0.847888	−	0.530176i	$-0.177874\pi$
$54$	2916.00	0.136083
$55$	0	0
$56$	64.0000	0.00272716
$57$	− 5229.00i	− 0.213173i
$58$	504.000i	0.0196725i
$59$	32310.0	1.20839	0.604195	−	0.796837i	$-0.293495\pi$
$59$	32310.0	1.20839	0.604195	+	0.796837i	$0.293495\pi$
$60$	0	0
$61$	−7165.00	−0.246542	−0.123271	−	0.992373i	$-0.539339\pi$
$61$	−7165.00	−0.246542	−0.123271	+	0.992373i	$0.539339\pi$
$62$	30332.0i	1.00212i
$63$	− 81.0000i	− 0.00257119i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	7560.00	0.213630
$67$	− 59579.0i	− 1.62146i	−0.585420	−	0.810730i	$-0.699071\pi$
$67$	− 59579.0i	− 1.62146i	0.585420	−	0.810730i	$-0.300929\pi$
$68$	1824.00i	0.0478357i
$69$	39150.0	0.989940
$70$	0	0
$71$	−43080.0	−1.01421	−0.507107	−	0.861883i	$-0.669285\pi$
$71$	−43080.0	−1.01421	−0.507107	+	0.861883i	$0.669285\pi$
$72$	5184.00i	0.117851i
$73$	− 28942.0i	− 0.635655i	−0.948149	−	0.317827i	$-0.897047\pi$
$73$	− 28942.0i	− 0.635655i	0.948149	−	0.317827i	$-0.102953\pi$
$74$	−14968.0	−0.317749
$75$	0	0
$76$	9296.00	0.184613
$77$	− 210.000i	− 0.00403638i
$78$	24012.0i	0.446881i
$79$	−27608.0	−0.497700	−0.248850	−	0.968542i	$-0.580053\pi$
$79$	−27608.0	−0.497700	−0.248850	+	0.968542i	$0.580053\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	− 11424.0i	− 0.187622i
$83$	− 1782.00i	− 0.0283931i	−0.999899	−	0.0141965i	$-0.995481\pi$
$83$	− 1782.00i	− 0.0283931i	0.999899	−	0.0141965i	$-0.00451905\pi$
$84$	144.000	0.00222671
$85$	0	0
$86$	72964.0	1.06381
$87$	1134.00i	0.0160626i
$88$	13440.0i	0.185009i
$89$	−50208.0	−0.671890	−0.335945	−	0.941882i	$-0.609056\pi$
$89$	−50208.0	−0.671890	−0.335945	+	0.941882i	$0.609056\pi$
$90$	0	0
$91$	667.000	0.00844350
$92$	69600.0i	0.857314i
$93$	68247.0i	0.818231i
$94$	−93480.0	−1.09119
$95$	0	0
$96$	−9216.00	−0.102062
$97$	− 142793.i	− 1.54091i	−0.637494	−	0.770456i	$-0.720029\pi$
$97$	− 142793.i	− 1.54091i	0.637494	−	0.770456i	$-0.279971\pi$
$98$	67224.0i	0.707065i
$99$	17010.0	0.174428
$100$	0	0
$101$	−164052.	−1.60021	−0.800107	−	0.599857i	$-0.795224\pi$
$101$	−164052.	−1.60021	−0.800107	+	0.599857i	$0.795224\pi$
$102$	4104.00i	0.0390577i
$103$	− 107236.i	− 0.995973i	−0.867185	−	0.497986i	$-0.834073\pi$
$103$	− 107236.i	− 0.995973i	0.867185	−	0.497986i	$-0.165927\pi$
$104$	−42688.0	−0.387010
$105$	0	0
$106$	86736.0	0.749782
$107$	− 21012.0i	− 0.177422i	−0.996057	−	0.0887111i	$-0.971725\pi$
$107$	− 21012.0i	− 0.177422i	0.996057	−	0.0887111i	$-0.0282748\pi$
$108$	11664.0i	0.0962250i
$109$	−237545.	−1.91505	−0.957524	−	0.288354i	$-0.906892\pi$
$109$	−237545.	−1.91505	−0.957524	+	0.288354i	$0.906892\pi$
$110$	0	0
$111$	−33678.0	−0.259441
$112$	256.000i	0.00192839i
$113$	233388.i	1.71942i	0.510781	+	0.859711i	$0.329356\pi$
$113$	233388.i	1.71942i	−0.510781	+	0.859711i	$0.670644\pi$
$114$	20916.0	0.150736
$115$	0	0
$116$	−2016.00	−0.0139106
$117$	54027.0i	0.364877i
$118$	129240.i	0.854460i
$119$	114.000	0.000737968 0
$120$	0	0
$121$	−116951.	−0.726174
$122$	− 28660.0i	− 0.174332i
$123$	− 25704.0i	− 0.153193i
$124$	−121328.	−0.708609
$125$	0	0
$126$	324.000	0.00181810
$127$	− 138800.i	− 0.763625i	−0.924240	−	0.381813i	$-0.875300\pi$
$127$	− 138800.i	− 0.763625i	0.924240	−	0.381813i	$-0.124700\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	164169.	0.868594
$130$	0	0
$131$	−161340.	−0.821417	−0.410709	−	0.911767i	$-0.634719\pi$
$131$	−161340.	−0.821417	−0.410709	+	0.911767i	$0.634719\pi$
$132$	30240.0i	0.151059i
$133$	− 581.000i	− 0.00284805i
$134$	238316.	1.14655
$135$	0	0
$136$	−7296.00	−0.0338250
$137$	− 69054.0i	− 0.314331i	−0.987572	−	0.157166i	$-0.949764\pi$
$137$	− 69054.0i	− 0.314331i	0.987572	−	0.157166i	$-0.0502356\pi$
$138$	156600.i	0.699994i
$139$	−224396.	−0.985095	−0.492547	−	0.870286i	$-0.663934\pi$
$139$	−224396.	−0.985095	−0.492547	+	0.870286i	$0.663934\pi$
$140$	0	0
$141$	−210330.	−0.890950
$142$	− 172320.i	− 0.717158i
$143$	140070.i	0.572803i
$144$	−20736.0	−0.0833333
$145$	0	0
$146$	115768.	0.449476
$147$	151254.i	0.577316i
$148$	− 59872.0i	− 0.224683i
$149$	407646.	1.50424	0.752121	−	0.659025i	$-0.229031\pi$
$149$	407646.	1.50424	0.752121	+	0.659025i	$0.229031\pi$
$150$	0	0
$151$	216053.	0.771113	0.385556	−	0.922684i	$-0.374010\pi$
$151$	216053.	0.771113	0.385556	+	0.922684i	$0.374010\pi$
$152$	37184.0i	0.130541i
$153$	9234.00i	0.0318905i
$154$	840.000	0.00285415
$155$	0	0
$156$	−96048.0	−0.315993
$157$	188617.i	0.610705i	0.952239	+	0.305353i	$0.0987743\pi$
$157$	188617.i	0.610705i	−0.952239	+	0.305353i	$0.901226\pi$
$158$	− 110432.i	− 0.351927i
$159$	195156.	0.612194
$160$	0	0
$161$	4350.00	0.0132259
$162$	26244.0i	0.0785674i
$163$	− 422251.i	− 1.24481i	−0.782697	−	0.622403i	$-0.786157\pi$
$163$	− 422251.i	− 1.24481i	0.782697	−	0.622403i	$-0.213843\pi$
$164$	45696.0	0.132669
$165$	0	0
$166$	7128.00	0.0200769
$167$	− 41700.0i	− 0.115703i	−0.998325	−	0.0578515i	$-0.981575\pi$
$167$	− 41700.0i	− 0.115703i	0.998325	−	0.0578515i	$-0.0184250\pi$
$168$	576.000i	0.00157452i
$169$	−73596.0	−0.198215
$170$	0	0
$171$	47061.0	0.123075
$172$	291856.i	0.752225i
$173$	266142.i	0.676080i	0.941132	+	0.338040i	$0.109764\pi$
$173$	266142.i	0.676080i	−0.941132	+	0.338040i	$0.890236\pi$
$174$	−4536.00	−0.0113579
$175$	0	0
$176$	−53760.0	−0.130821
$177$	290790.i	0.697664i
$178$	− 200832.i	− 0.475098i
$179$	51846.0	0.120944	0.0604718	−	0.998170i	$-0.480739\pi$
$179$	51846.0	0.120944	0.0604718	+	0.998170i	$0.480739\pi$
$180$	0	0
$181$	499349.	1.13294	0.566471	−	0.824082i	$-0.308308\pi$
$181$	499349.	1.13294	0.566471	+	0.824082i	$0.308308\pi$
$182$	2668.00i	0.00597045i
$183$	− 64485.0i	− 0.142341i
$184$	−278400.	−0.606212
$185$	0	0
$186$	−272988.	−0.578577
$187$	23940.0i	0.0500634i
$188$	− 373920.i	− 0.771586i
$189$	729.000	0.00148448
$190$	0	0
$191$	705906.	1.40011	0.700057	−	0.714087i	$-0.253158\pi$
$191$	705906.	1.40011	0.700057	+	0.714087i	$0.253158\pi$
$192$	− 36864.0i	− 0.0721688i
$193$	− 519973.i	− 1.00482i	−0.864630	−	0.502409i	$-0.832447\pi$
$193$	− 519973.i	− 1.00482i	0.864630	−	0.502409i	$-0.167553\pi$
$194$	571172.	1.08959
$195$	0	0
$196$	−268896.	−0.499970
$197$	200394.i	0.367891i	0.982936	+	0.183945i	$0.0588870\pi$
$197$	200394.i	0.367891i	−0.982936	+	0.183945i	$0.941113\pi$
$198$	68040.0i	0.123339i
$199$	−553673.	−0.991107	−0.495554	−	0.868577i	$-0.665035\pi$
$199$	−553673.	−0.991107	−0.495554	+	0.868577i	$0.665035\pi$
$200$	0	0
$201$	536211.	0.936150
$202$	− 656208.i	− 1.13152i
$203$	126.000i	0 0.000214600i
$204$	−16416.0	−0.0276180
$205$	0	0
$206$	428944.	0.704259
$207$	352350.i	0.571542i
$208$	− 170752.i	− 0.273658i
$209$	122010.	0.193210
$210$	0	0
$211$	777365.	1.20204	0.601020	−	0.799234i	$-0.294761\pi$
$211$	777365.	1.20204	0.601020	+	0.799234i	$0.294761\pi$
$212$	346944.i	0.530176i
$213$	− 387720.i	− 0.585557i
$214$	84048.0	0.125456
$215$	0	0
$216$	−46656.0	−0.0680414
$217$	7583.00i	0.0109318i
$218$	− 950180.i	− 1.35414i
$219$	260478.	0.366996
$220$	0	0
$221$	−76038.0	−0.104725
$222$	− 134712.i	− 0.183453i
$223$	− 776185.i	− 1.04521i	−0.852575	−	0.522604i	$-0.824960\pi$
$223$	− 776185.i	− 1.04521i	0.852575	−	0.522604i	$-0.175040\pi$
$224$	−1024.00	−0.00136358
$225$	0	0
$226$	−933552.	−1.21581
$227$	− 919992.i	− 1.18500i	−0.805569	−	0.592501i	$-0.798140\pi$
$227$	− 919992.i	− 1.18500i	0.805569	−	0.592501i	$-0.201860\pi$
$228$	83664.0i	0.106586i
$229$	1.13220e6	1.42670	0.713350	−	0.700808i	$-0.247177\pi$
$229$	1.13220e6	1.42670	0.713350	+	0.700808i	$0.247177\pi$
$230$	0	0
$231$	1890.00	0.00233041
$232$	− 8064.00i	− 0.00983627i
$233$	823128.i	0.993293i	0.867953	+	0.496647i	$0.165436\pi$
$233$	823128.i	0.993293i	−0.867953	+	0.496647i	$0.834564\pi$
$234$	−216108.	−0.258007
$235$	0	0
$236$	−516960.	−0.604195
$237$	− 248472.i	− 0.287347i
$238$	456.000i	0 0.000521822i
$239$	−1.32836e6	−1.50426	−0.752129	−	0.659016i	$-0.770973\pi$
$239$	−1.32836e6	−1.50426	−0.752129	+	0.659016i	$0.770973\pi$
$240$	0	0
$241$	94937.0	0.105291	0.0526457	−	0.998613i	$-0.483235\pi$
$241$	94937.0	0.105291	0.0526457	+	0.998613i	$0.483235\pi$
$242$	− 467804.i	− 0.513482i
$243$	59049.0i	0.0641500i
$244$	114640.	0.123271
$245$	0	0
$246$	102816.	0.108324
$247$	387527.i	0.404166i
$248$	− 485312.i	− 0.501062i
$249$	16038.0	0.0163928
$250$	0	0
$251$	−128124.	−0.128365	−0.0641824	−	0.997938i	$-0.520444\pi$
$251$	−128124.	−0.128365	−0.0641824	+	0.997938i	$0.520444\pi$
$252$	1296.00i	0.00128559i
$253$	913500.i	0.897237i
$254$	555200.	0.539964
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.72285e6i	1.62710i	0.581493	+	0.813552i	$0.302469\pi$
$257$	1.72285e6i	1.62710i	−0.581493	+	0.813552i	$0.697531\pi$
$258$	656676.i	0.614189i
$259$	−3742.00	−0.00346621
$260$	0	0
$261$	−10206.0	−0.00927373
$262$	− 645360.i	− 0.580830i
$263$	692160.i	0.617045i	0.951217	+	0.308523i	$0.0998346\pi$
$263$	692160.i	0.617045i	−0.951217	+	0.308523i	$0.900165\pi$
$264$	−120960.	−0.106815
$265$	0	0
$266$	2324.00	0.00201387
$267$	− 451872.i	− 0.387916i
$268$	953264.i	0.810730i
$269$	−690894.	−0.582144	−0.291072	−	0.956701i	$-0.594012\pi$
$269$	−690894.	−0.582144	−0.291072	+	0.956701i	$0.594012\pi$
$270$	0	0
$271$	−1.04690e6	−0.865925	−0.432963	−	0.901412i	$-0.642532\pi$
$271$	−1.04690e6	−0.865925	−0.432963	+	0.901412i	$0.642532\pi$
$272$	− 29184.0i	− 0.0239179i
$273$	6003.00i	0.00487486i
$274$	276216.	0.222266
$275$	0	0
$276$	−626400.	−0.494970
$277$	1.07137e6i	0.838955i	0.907766	+	0.419478i	$0.137787\pi$
$277$	1.07137e6i	0.838955i	−0.907766	+	0.419478i	$0.862213\pi$
$278$	− 897584.i	− 0.696567i
$279$	−614223.	−0.472406
$280$	0	0
$281$	−1.40416e6	−1.06084	−0.530422	−	0.847734i	$-0.677966\pi$
$281$	−1.40416e6	−1.06084	−0.530422	+	0.847734i	$0.677966\pi$
$282$	− 841320.i	− 0.629997i
$283$	1.68203e6i	1.24844i	0.781248	+	0.624221i	$0.214584\pi$
$283$	1.68203e6i	1.24844i	−0.781248	+	0.624221i	$0.785416\pi$
$284$	689280.	0.507107
$285$	0	0
$286$	−560280.	−0.405033
$287$	− 2856.00i	− 0.00204670i
$288$	− 82944.0i	− 0.0589256i
$289$	1.40686e6	0.990847
$290$	0	0
$291$	1.28514e6	0.889645
$292$	463072.i	0.317827i
$293$	1.76293e6i	1.19968i	0.800120	+	0.599839i	$0.204769\pi$
$293$	1.76293e6i	1.19968i	−0.800120	+	0.599839i	$0.795231\pi$
$294$	−605016.	−0.408224
$295$	0	0
$296$	239488.	0.158875
$297$	153090.i	0.100706i
$298$	1.63058e6i	1.06366i
$299$	−2.90145e6	−1.87688
$300$	0	0
$301$	18241.0	0.0116047
$302$	864212.i	0.545259i
$303$	− 1.47647e6i	− 0.923884i
$304$	−148736.	−0.0923065
$305$	0	0
$306$	−36936.0	−0.0225500
$307$	− 471905.i	− 0.285765i	−0.989740	−	0.142882i	$-0.954363\pi$
$307$	− 471905.i	− 0.285765i	0.989740	−	0.142882i	$-0.0456371\pi$
$308$	3360.00i	0.00201819i
$309$	965124.	0.575025
$310$	0	0
$311$	−1.64439e6	−0.964060	−0.482030	−	0.876155i	$-0.660100\pi$
$311$	−1.64439e6	−0.964060	−0.482030	+	0.876155i	$0.660100\pi$
$312$	− 384192.i	− 0.223440i
$313$	2.37908e6i	1.37262i	0.727311	+	0.686308i	$0.240770\pi$
$313$	2.37908e6i	1.37262i	−0.727311	+	0.686308i	$0.759230\pi$
$314$	−754468.	−0.431834
$315$	0	0
$316$	441728.	0.248850
$317$	− 1.32679e6i	− 0.741574i	−0.928718	−	0.370787i	$-0.879088\pi$
$317$	− 1.32679e6i	− 0.741574i	0.928718	−	0.370787i	$-0.120912\pi$
$318$	780624.i	0.432887i
$319$	−26460.0	−0.0145584
$320$	0	0
$321$	189108.	0.102435
$322$	17400.0i	0.00935211i
$323$	66234.0i	0.0353244i
$324$	−104976.	−0.0555556
$325$	0	0
$326$	1.68900e6	0.880211
$327$	− 2.13790e6i	− 1.10565i
$328$	182784.i	0.0938110i
$329$	−23370.0	−0.0119033
$330$	0	0
$331$	661172.	0.331699	0.165850	−	0.986151i	$-0.446963\pi$
$331$	661172.	0.331699	0.165850	+	0.986151i	$0.446963\pi$
$332$	28512.0i	0.0141965i
$333$	− 303102.i	− 0.149788i
$334$	166800.	0.0818144
$335$	0	0
$336$	−2304.00	−0.00111336
$337$	1.53675e6i	0.737104i	0.929607	+	0.368552i	$0.120146\pi$
$337$	1.53675e6i	0.737104i	−0.929607	+	0.368552i	$0.879854\pi$
$338$	− 294384.i	− 0.140159i
$339$	−2.10049e6	−0.992709
$340$	0	0
$341$	−1.59243e6	−0.741608
$342$	188244.i	0.0870274i
$343$	33613.0i	0.0154267i
$344$	−1.16742e6	−0.531903
$345$	0	0
$346$	−1.06457e6	−0.478061
$347$	− 3.86576e6i	− 1.72350i	−0.507334	−	0.861749i	$-0.669369\pi$
$347$	− 3.86576e6i	− 1.72350i	0.507334	−	0.861749i	$-0.330631\pi$
$348$	− 18144.0i	− 0.00803128i
$349$	4.23391e6	1.86071	0.930354	−	0.366663i	$-0.119500\pi$
$349$	4.23391e6	1.86071	0.930354	+	0.366663i	$0.119500\pi$
$350$	0	0
$351$	−486243.	−0.210662
$352$	− 215040.i	− 0.0925044i
$353$	− 2.41459e6i	− 1.03135i	−0.856784	−	0.515675i	$-0.827541\pi$
$353$	− 2.41459e6i	− 1.03135i	0.856784	−	0.515675i	$-0.172459\pi$
$354$	−1.16316e6	−0.493323
$355$	0	0
$356$	803328.	0.335945
$357$	1026.00i	0 0.000426066i
$358$	207384.i	0.0855200i
$359$	−2.94802e6	−1.20724	−0.603620	−	0.797272i	$-0.706276\pi$
$359$	−2.94802e6	−1.20724	−0.603620	+	0.797272i	$0.706276\pi$
$360$	0	0
$361$	−2.13854e6	−0.863672
$362$	1.99740e6i	0.801111i
$363$	− 1.05256e6i	− 0.419257i
$364$	−10672.0	−0.00422175
$365$	0	0
$366$	257940.	0.100651
$367$	− 679793.i	− 0.263458i	−0.991286	−	0.131729i	$-0.957947\pi$
$367$	− 679793.i	− 0.263458i	0.991286	−	0.131729i	$-0.0420529\pi$
$368$	− 1.11360e6i	− 0.428657i
$369$	231336.	0.0884458
$370$	0	0
$371$	21684.0	0.00817908
$372$	− 1.09195e6i	− 0.409116i
$373$	− 3.60888e6i	− 1.34308i	−0.740970	−	0.671538i	$-0.765634\pi$
$373$	− 3.60888e6i	− 1.34308i	0.740970	−	0.671538i	$-0.234366\pi$
$374$	−95760.0	−0.0354002
$375$	0	0
$376$	1.49568e6	0.545593
$377$	− 84042.0i	− 0.0304539i
$378$	2916.00i	0.00104968i
$379$	386809.	0.138324	0.0691622	−	0.997605i	$-0.477967\pi$
$379$	386809.	0.138324	0.0691622	+	0.997605i	$0.477967\pi$
$380$	0	0
$381$	1.24920e6	0.440879
$382$	2.82362e6i	0.990030i
$383$	− 645060.i	− 0.224700i	−0.993669	−	0.112350i	$-0.964162\pi$
$383$	− 645060.i	− 0.224700i	0.993669	−	0.112350i	$-0.0358378\pi$
$384$	147456.	0.0510310
$385$	0	0
$386$	2.07989e6	0.710514
$387$	1.47752e6i	0.501483i
$388$	2.28469e6i	0.770456i
$389$	812976.	0.272398	0.136199	−	0.990682i	$-0.456511\pi$
$389$	812976.	0.272398	0.136199	+	0.990682i	$0.456511\pi$
$390$	0	0
$391$	−495900.	−0.164041
$392$	− 1.07558e6i	− 0.353532i
$393$	− 1.45206e6i	− 0.474245i
$394$	−801576.	−0.260138
$395$	0	0
$396$	−272160.	−0.0872140
$397$	988813.i	0.314875i	0.987529	+	0.157437i	$0.0503232\pi$
$397$	988813.i	0.314875i	−0.987529	+	0.157437i	$0.949677\pi$
$398$	− 2.21469e6i	− 0.700819i
$399$	5229.00	0.00164432
$400$	0	0
$401$	2.14580e6	0.666391	0.333195	−	0.942858i	$-0.391873\pi$
$401$	2.14580e6	0.666391	0.333195	+	0.942858i	$0.391873\pi$
$402$	2.14484e6i	0.661958i
$403$	− 5.05786e6i	− 1.55133i
$404$	2.62483e6	0.800107
$405$	0	0
$406$	−504.000	−0.000151745 0
$407$	− 785820.i	− 0.235146i
$408$	− 65664.0i	− 0.0195289i
$409$	5.77658e6	1.70751	0.853753	−	0.520678i	$-0.174321\pi$
$409$	5.77658e6	1.70751	0.853753	+	0.520678i	$0.174321\pi$
$410$	0	0
$411$	621486.	0.181479
$412$	1.71578e6i	0.497986i
$413$	32310.0i	0.00932099i
$414$	−1.40940e6	−0.404141
$415$	0	0
$416$	683008.	0.193505
$417$	− 2.01956e6i	− 0.568745i
$418$	488040.i	0.136620i
$419$	5.07318e6	1.41171	0.705855	−	0.708357i	$-0.250563\pi$
$419$	5.07318e6	1.41171	0.705855	+	0.708357i	$0.250563\pi$
$420$	0	0
$421$	−3.01297e6	−0.828494	−0.414247	−	0.910164i	$-0.635955\pi$
$421$	−3.01297e6	−0.828494	−0.414247	+	0.910164i	$0.635955\pi$
$422$	3.10946e6i	0.849970i
$423$	− 1.89297e6i	− 0.514390i
$424$	−1.38778e6	−0.374891
$425$	0	0
$426$	1.55088e6	0.414051
$427$	− 7165.00i	− 0.00190172i
$428$	336192.i	0.0887111i
$429$	−1.26063e6	−0.330708
$430$	0	0
$431$	5.59449e6	1.45067	0.725333	−	0.688398i	$-0.241686\pi$
$431$	5.59449e6	1.45067	0.725333	+	0.688398i	$0.241686\pi$
$432$	− 186624.i	− 0.0481125i
$433$	5.63454e6i	1.44424i	0.691769	+	0.722119i	$0.256832\pi$
$433$	5.63454e6i	1.44424i	−0.691769	+	0.722119i	$0.743168\pi$
$434$	−30332.0	−0.00772995
$435$	0	0
$436$	3.80072e6	0.957524
$437$	2.52735e6i	0.633085i
$438$	1.04191e6i	0.259505i
$439$	−860663.	−0.213143	−0.106572	−	0.994305i	$-0.533987\pi$
$439$	−860663.	−0.213143	−0.106572	+	0.994305i	$0.533987\pi$
$440$	0	0
$441$	−1.36129e6	−0.333314
$442$	− 304152.i	− 0.0740517i
$443$	− 2.75335e6i	− 0.666580i	−0.942824	−	0.333290i	$-0.891841\pi$
$443$	− 2.75335e6i	− 0.666580i	0.942824	−	0.333290i	$-0.108159\pi$
$444$	538848.	0.129721
$445$	0	0
$446$	3.10474e6	0.739074
$447$	3.66881e6i	0.868474i
$448$	− 4096.00i	0 0.000964195i
$449$	2.83168e6	0.662869	0.331434	−	0.943478i	$-0.392467\pi$
$449$	2.83168e6	0.662869	0.331434	+	0.943478i	$0.392467\pi$
$450$	0	0
$451$	599760.	0.138847
$452$	− 3.73421e6i	− 0.859711i
$453$	1.94448e6i	0.445202i
$454$	3.67997e6	0.837924
$455$	0	0
$456$	−334656.	−0.0753679
$457$	4.66235e6i	1.04427i	0.852862	+	0.522136i	$0.174865\pi$
$457$	4.66235e6i	1.04427i	−0.852862	+	0.522136i	$0.825135\pi$
$458$	4.52878e6i	1.00883i
$459$	−83106.0	−0.0184120
$460$	0	0
$461$	−5.27154e6	−1.15527	−0.577637	−	0.816294i	$-0.696025\pi$
$461$	−5.27154e6	−1.15527	−0.577637	+	0.816294i	$0.696025\pi$
$462$	7560.00i	0.00164785i
$463$	− 7.16954e6i	− 1.55431i	−0.629307	−	0.777157i	$-0.716661\pi$
$463$	− 7.16954e6i	− 1.55431i	0.629307	−	0.777157i	$-0.283339\pi$
$464$	32256.0	0.00695530
$465$	0	0
$466$	−3.29251e6	−0.702365
$467$	− 4.30441e6i	− 0.913316i	−0.889642	−	0.456658i	$-0.849046\pi$
$467$	− 4.30441e6i	− 0.913316i	0.889642	−	0.456658i	$-0.150954\pi$
$468$	− 864432.i	− 0.182438i
$469$	59579.0	0.0125072
$470$	0	0
$471$	−1.69755e6	−0.352591
$472$	− 2.06784e6i	− 0.427230i
$473$	3.83061e6i	0.787254i
$474$	993888.	0.203185
$475$	0	0
$476$	−1824.00	−0.000368984 0
$477$	1.75640e6i	0.353450i
$478$	− 5.31346e6i	− 1.06367i
$479$	−190506.	−0.0379376	−0.0189688	−	0.999820i	$-0.506038\pi$
$479$	−190506.	−0.0379376	−0.0189688	+	0.999820i	$0.506038\pi$
$480$	0	0
$481$	2.49591e6	0.491889
$482$	379748.i	0.0744523i
$483$	39150.0i	0.00763597i
$484$	1.87122e6	0.363087
$485$	0	0
$486$	−236196.	−0.0453609
$487$	128887.i	0.0246256i	0.999924	+	0.0123128i	$0.00391938\pi$
$487$	128887.i	0.0246256i	−0.999924	+	0.0123128i	$0.996081\pi$
$488$	458560.i	0.0871659i
$489$	3.80026e6	0.718689
$490$	0	0
$491$	−304296.	−0.0569630	−0.0284815	−	0.999594i	$-0.509067\pi$
$491$	−304296.	−0.0569630	−0.0284815	+	0.999594i	$0.509067\pi$
$492$	411264.i	0.0765963i
$493$	− 14364.0i	− 0.00266169i
$494$	−1.55011e6	−0.285788
$495$	0	0
$496$	1.94125e6	0.354305
$497$	− 43080.0i	− 0.00782321i
$498$	64152.0i	0.0115914i
$499$	−6.41974e6	−1.15416	−0.577080	−	0.816688i	$-0.695808\pi$
$499$	−6.41974e6	−1.15416	−0.577080	+	0.816688i	$0.695808\pi$
$500$	0	0
$501$	375300.	0.0668012
$502$	− 512496.i	− 0.0907677i
$503$	− 1.55534e6i	− 0.274098i	−0.990564	−	0.137049i	$-0.956238\pi$
$503$	− 1.55534e6i	− 0.274098i	0.990564	−	0.137049i	$-0.0437618\pi$
$504$	−5184.00	−0.000909052 0
$505$	0	0
$506$	−3.65400e6	−0.634442
$507$	− 662364.i	− 0.114440i
$508$	2.22080e6i	0.381813i
$509$	541506.	0.0926422	0.0463211	−	0.998927i	$-0.485250\pi$
$509$	541506.	0.0926422	0.0463211	+	0.998927i	$0.485250\pi$
$510$	0	0
$511$	28942.0	0.00490316
$512$	262144.i	0.0441942i
$513$	423549.i	0.0710576i
$514$	−6.89141e6	−1.15054
$515$	0	0
$516$	−2.62670e6	−0.434297
$517$	− 4.90770e6i	− 0.807517i
$518$	− 14968.0i	− 0.00245098i
$519$	−2.39528e6	−0.390335
$520$	0	0
$521$	−888762.	−0.143447	−0.0717235	−	0.997425i	$-0.522850\pi$
$521$	−888762.	−0.143447	−0.0717235	+	0.997425i	$0.522850\pi$
$522$	− 40824.0i	− 0.00655752i
$523$	7.69950e6i	1.23086i	0.788192	+	0.615430i	$0.211018\pi$
$523$	7.69950e6i	1.23086i	−0.788192	+	0.615430i	$0.788982\pi$
$524$	2.58144e6	0.410709
$525$	0	0
$526$	−2.76864e6	−0.436317
$527$	− 864462.i	− 0.135587i
$528$	− 483840.i	− 0.0755296i
$529$	−1.24862e7	−1.93995
$530$	0	0
$531$	−2.61711e6	−0.402796
$532$	9296.00i	0.00142402i
$533$	1.90495e6i	0.290446i
$534$	1.80749e6	0.274298
$535$	0	0
$536$	−3.81306e6	−0.573273
$537$	466614.i	0.0698268i
$538$	− 2.76358e6i	− 0.411638i
$539$	−3.52926e6	−0.523253
$540$	0	0
$541$	−3.01878e6	−0.443443	−0.221721	−	0.975110i	$-0.571168\pi$
$541$	−3.01878e6	−0.443443	−0.221721	+	0.975110i	$0.571168\pi$
$542$	− 4.18758e6i	− 0.612301i
$543$	4.49414e6i	0.654104i
$544$	116736.	0.0169125
$545$	0	0
$546$	−24012.0	−0.00344704
$547$	− 7.23481e6i	− 1.03385i	−0.856030	−	0.516926i	$-0.827076\pi$
$547$	− 7.23481e6i	− 1.03385i	0.856030	−	0.516926i	$-0.172924\pi$
$548$	1.10486e6i	0.157166i
$549$	580365.	0.0821808
$550$	0	0
$551$	−73206.0	−0.0102723
$552$	− 2.50560e6i	− 0.349997i
$553$	− 27608.0i	− 0.00383904i
$554$	−4.28547e6	−0.593231
$555$	0	0
$556$	3.59034e6	0.492547
$557$	9.88712e6i	1.35030i	0.737678	+	0.675152i	$0.235922\pi$
$557$	9.88712e6i	1.35030i	−0.737678	+	0.675152i	$0.764078\pi$
$558$	− 2.45689e6i	− 0.334042i
$559$	−1.21667e7	−1.64682
$560$	0	0
$561$	−215460.	−0.0289041
$562$	− 5.61665e6i	− 0.750130i
$563$	946362.i	0.125831i	0.998019	+	0.0629153i	$0.0200398\pi$
$563$	946362.i	0.125831i	−0.998019	+	0.0629153i	$0.979960\pi$
$564$	3.36528e6	0.445475
$565$	0	0
$566$	−6.72813e6	−0.882782
$567$	6561.00i	0 0.000857062i
$568$	2.75712e6i	0.358579i
$569$	4.51552e6	0.584692	0.292346	−	0.956313i	$-0.405564\pi$
$569$	4.51552e6	0.584692	0.292346	+	0.956313i	$0.405564\pi$
$570$	0	0
$571$	7.57824e6	0.972699	0.486349	−	0.873764i	$-0.338328\pi$
$571$	7.57824e6	0.972699	0.486349	+	0.873764i	$0.338328\pi$
$572$	− 2.24112e6i	− 0.286401i
$573$	6.35315e6i	0.808356i
$574$	11424.0	0.00144723
$575$	0	0
$576$	331776.	0.0416667
$577$	− 1.11688e7i	− 1.39659i	−0.715811	−	0.698295i	$-0.753943\pi$
$577$	− 1.11688e7i	− 1.39659i	0.715811	−	0.698295i	$-0.246057\pi$
$578$	5.62744e6i	0.700635i
$579$	4.67976e6	0.580132
$580$	0	0
$581$	1782.00	0.000219012 0
$582$	5.14055e6i	0.629074i
$583$	4.55364e6i	0.554865i
$584$	−1.85229e6	−0.224738
$585$	0	0
$586$	−7.05170e6	−0.848301
$587$	− 1.01053e6i	− 0.121047i	−0.998167	−	0.0605236i	$-0.980723\pi$
$587$	− 1.01053e6i	− 0.121047i	0.998167	−	0.0605236i	$-0.0192770\pi$
$588$	− 2.42006e6i	− 0.288658i
$589$	−4.40572e6	−0.523274
$590$	0	0
$591$	−1.80355e6	−0.212402
$592$	957952.i	0.112341i
$593$	1.13674e7i	1.32746i	0.747970	+	0.663732i	$0.231029\pi$
$593$	1.13674e7i	1.32746i	−0.747970	+	0.663732i	$0.768971\pi$
$594$	−612360.	−0.0712100
$595$	0	0
$596$	−6.52234e6	−0.752121
$597$	− 4.98306e6i	− 0.572216i
$598$	− 1.16058e7i	− 1.32716i
$599$	−2.56732e6	−0.292356	−0.146178	−	0.989258i	$-0.546697\pi$
$599$	−2.56732e6	−0.292356	−0.146178	+	0.989258i	$0.546697\pi$
$600$	0	0
$601$	1.59837e7	1.80506	0.902530	−	0.430626i	$-0.141707\pi$
$601$	1.59837e7	1.80506	0.902530	+	0.430626i	$0.141707\pi$
$602$	72964.0i	0.00820573i
$603$	4.82590e6i	0.540487i
$604$	−3.45685e6	−0.385556
$605$	0	0
$606$	5.90587e6	0.653285
$607$	1.01626e7i	1.11952i	0.828655	+	0.559760i	$0.189107\pi$
$607$	1.01626e7i	1.11952i	−0.828655	+	0.559760i	$0.810893\pi$
$608$	− 594944.i	− 0.0652705i
$609$	−1134.00	−0.000123900 0
$610$	0	0
$611$	1.55878e7	1.68920
$612$	− 147744.i	− 0.0159452i
$613$	5.83042e6i	0.626684i	0.949640	+	0.313342i	$0.101449\pi$
$613$	5.83042e6i	0.626684i	−0.949640	+	0.313342i	$0.898551\pi$
$614$	1.88762e6	0.202066
$615$	0	0
$616$	−13440.0	−0.00142708
$617$	1.51029e7i	1.59716i	0.601891	+	0.798578i	$0.294414\pi$
$617$	1.51029e7i	1.59716i	−0.601891	+	0.798578i	$0.705586\pi$
$618$	3.86050e6i	0.406604i
$619$	5.38515e6	0.564900	0.282450	−	0.959282i	$-0.408853\pi$
$619$	5.38515e6	0.564900	0.282450	+	0.959282i	$0.408853\pi$
$620$	0	0
$621$	−3.17115e6	−0.329980
$622$	− 6.57756e6i	− 0.681693i
$623$	− 50208.0i	− 0.00518266i
$624$	1.53677e6	0.157996
$625$	0	0
$626$	−9.51633e6	−0.970585
$627$	1.09809e6i	0.111550i
$628$	− 3.01787e6i	− 0.305353i
$629$	426588.	0.0429914
$630$	0	0
$631$	−5.59345e6	−0.559251	−0.279625	−	0.960109i	$-0.590210\pi$
$631$	−5.59345e6	−0.559251	−0.279625	+	0.960109i	$0.590210\pi$
$632$	1.76691e6i	0.175963i
$633$	6.99628e6i	0.693998i
$634$	5.30717e6	0.524372
$635$	0	0
$636$	−3.12250e6	−0.306097
$637$	− 1.12096e7i	− 1.09457i
$638$	− 105840.i	− 0.0102943i
$639$	3.48948e6	0.338072
$640$	0	0
$641$	−1.00072e7	−0.961985	−0.480993	−	0.876725i	$-0.659724\pi$
$641$	−1.00072e7	−0.961985	−0.480993	+	0.876725i	$0.659724\pi$
$642$	756432.i	0.0724323i
$643$	− 9.95036e6i	− 0.949098i	−0.880229	−	0.474549i	$-0.842611\pi$
$643$	− 9.95036e6i	− 0.949098i	0.880229	−	0.474549i	$-0.157389\pi$
$644$	−69600.0	−0.00661294
$645$	0	0
$646$	−264936.	−0.0249781
$647$	9.16904e6i	0.861119i	0.902562	+	0.430560i	$0.141684\pi$
$647$	9.16904e6i	0.861119i	−0.902562	+	0.430560i	$0.858316\pi$
$648$	− 419904.i	− 0.0392837i
$649$	−6.78510e6	−0.632331
$650$	0	0
$651$	−68247.0	−0.00631148
$652$	6.75602e6i	0.622403i
$653$	− 1.77127e7i	− 1.62555i	−0.582575	−	0.812777i	$-0.697955\pi$
$653$	− 1.77127e7i	− 1.62555i	0.582575	−	0.812777i	$-0.302045\pi$
$654$	8.55162e6	0.781815
$655$	0	0
$656$	−731136.	−0.0663344
$657$	2.34430e6i	0.211885i
$658$	− 93480.0i	− 0.00841693i
$659$	2.07949e7	1.86527	0.932637	−	0.360816i	$-0.117502\pi$
$659$	2.07949e7	1.86527	0.932637	+	0.360816i	$0.117502\pi$
$660$	0	0
$661$	−1.51745e6	−0.135086	−0.0675429	−	0.997716i	$-0.521516\pi$
$661$	−1.51745e6	−0.135086	−0.0675429	+	0.997716i	$0.521516\pi$
$662$	2.64469e6i	0.234547i
$663$	− 684342.i	− 0.0604630i
$664$	−114048.	−0.0100385
$665$	0	0
$666$	1.21241e6	0.105916
$667$	− 548100.i	− 0.0477029i
$668$	667200.i	0.0578515i
$669$	6.98566e6	0.603452
$670$	0	0
$671$	1.50465e6	0.129012
$672$	− 9216.00i	0 0.000787262i
$673$	− 2.90483e6i	− 0.247219i	−0.992331	−	0.123610i	$-0.960553\pi$
$673$	− 2.90483e6i	− 0.247219i	0.992331	−	0.123610i	$-0.0394471\pi$
$674$	−6.14700e6	−0.521211
$675$	0	0
$676$	1.17754e6	0.0991077
$677$	7.06049e6i	0.592057i	0.955179	+	0.296028i	$0.0956623\pi$
$677$	7.06049e6i	0.592057i	−0.955179	+	0.296028i	$0.904338\pi$
$678$	− 8.40197e6i	− 0.701951i
$679$	142793.	0.0118859
$680$	0	0
$681$	8.27993e6	0.684162
$682$	− 6.36972e6i	− 0.524396i
$683$	3.81415e6i	0.312857i	0.987689	+	0.156429i	$0.0499981\pi$
$683$	3.81415e6i	0.312857i	−0.987689	+	0.156429i	$0.950002\pi$
$684$	−752976.	−0.0615376
$685$	0	0
$686$	−134452.	−0.0109083
$687$	1.01898e7i	0.823705i
$688$	− 4.66970e6i	− 0.376112i
$689$	−1.44632e7	−1.16069
$690$	0	0
$691$	−8.95166e6	−0.713195	−0.356597	−	0.934258i	$-0.616063\pi$
$691$	−8.95166e6	−0.713195	−0.356597	+	0.934258i	$0.616063\pi$
$692$	− 4.25827e6i	− 0.338040i
$693$	17010.0i	0.00134546i
$694$	1.54630e7	1.21870
$695$	0	0
$696$	72576.0	0.00567897
$697$	325584.i	0.0253852i
$698$	1.69356e7i	1.31572i
$699$	−7.40815e6	−0.573478
$700$	0	0
$701$	8.45297e6	0.649702	0.324851	−	0.945765i	$-0.394686\pi$
$701$	8.45297e6	0.649702	0.324851	+	0.945765i	$0.394686\pi$
$702$	− 1.94497e6i	− 0.148960i
$703$	− 2.17410e6i	− 0.165917i
$704$	860160.	0.0654105
$705$	0	0
$706$	9.65834e6	0.729274
$707$	− 164052.i	− 0.0123433i
$708$	− 4.65264e6i	− 0.348832i
$709$	1.18033e7	0.881834	0.440917	−	0.897548i	$-0.354653\pi$
$709$	1.18033e7	0.881834	0.440917	+	0.897548i	$0.354653\pi$
$710$	0	0
$711$	2.23625e6	0.165900
$712$	3.21331e6i	0.237549i
$713$	− 3.29861e7i	− 2.43000i
$714$	−4104.00	−0.000301274 0
$715$	0	0
$716$	−829536.	−0.0604718
$717$	− 1.19553e7i	− 0.868484i
$718$	− 1.17921e7i	− 0.853648i
$719$	2.68089e7	1.93400	0.967000	−	0.254778i	$-0.0820024\pi$
$719$	2.68089e7	1.93400	0.967000	+	0.254778i	$0.0820024\pi$
$720$	0	0
$721$	107236.	0.00768250
$722$	− 8.55415e6i	− 0.610709i
$723$	854433.i	0.0607900i
$724$	−7.98958e6	−0.566471
$725$	0	0
$726$	4.21024e6	0.296459
$727$	2.00638e6i	0.140792i	0.997519	+	0.0703958i	$0.0224262\pi$
$727$	2.00638e6i	0.140792i	−0.997519	+	0.0703958i	$0.977574\pi$
$728$	− 42688.0i	− 0.00298523i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	−2.07947e6	−0.143933
$732$	1.03176e6i	0.0711707i
$733$	− 9.16554e6i	− 0.630083i	−0.949078	−	0.315042i	$-0.897981\pi$
$733$	− 9.16554e6i	− 0.630083i	0.949078	−	0.315042i	$-0.102019\pi$
$734$	2.71917e6	0.186293
$735$	0	0
$736$	4.45440e6	0.303106
$737$	1.25116e7i	0.848484i
$738$	925344.i	0.0625406i
$739$	−1.55897e7	−1.05009	−0.525047	−	0.851073i	$-0.675952\pi$
$739$	−1.55897e7	−1.05009	−0.525047	+	0.851073i	$0.675952\pi$
$740$	0	0
$741$	−3.48774e6	−0.233345
$742$	86736.0i	0.00578349i
$743$	− 3.78304e6i	− 0.251402i	−0.992068	−	0.125701i	$-0.959882\pi$
$743$	− 3.78304e6i	− 0.251402i	0.992068	−	0.125701i	$-0.0401180\pi$
$744$	4.36781e6	0.289289
$745$	0	0
$746$	1.44355e7	0.949698
$747$	144342.i	0.00946436i
$748$	− 383040.i	− 0.0250317i
$749$	21012.0	0.00136856
$750$	0	0
$751$	1.24726e7	0.806972	0.403486	−	0.914986i	$-0.367798\pi$
$751$	1.24726e7	0.806972	0.403486	+	0.914986i	$0.367798\pi$
$752$	5.98272e6i	0.385793i
$753$	− 1.15312e6i	− 0.0741115i
$754$	336168.	0.0215342
$755$	0	0
$756$	−11664.0	−0.000742238 0
$757$	− 1.87121e6i	− 0.118682i	−0.998238	−	0.0593408i	$-0.981100\pi$
$757$	− 1.87121e6i	− 0.118682i	0.998238	−	0.0593408i	$-0.0188999\pi$
$758$	1.54724e6i	0.0978101i
$759$	−8.22150e6	−0.518020
$760$	0	0
$761$	2.53638e7	1.58764	0.793821	−	0.608152i	$-0.208089\pi$
$761$	2.53638e7	1.58764	0.793821	+	0.608152i	$0.208089\pi$
$762$	4.99680e6i	0.311749i
$763$	− 237545.i	− 0.0147718i
$764$	−1.12945e7	−0.700057
$765$	0	0
$766$	2.58024e6	0.158887
$767$	− 2.15508e7i	− 1.32274i
$768$	589824.i	0.0360844i
$769$	1.43729e7	0.876452	0.438226	−	0.898865i	$-0.355607\pi$
$769$	1.43729e7	0.876452	0.438226	+	0.898865i	$0.355607\pi$
$770$	0	0
$771$	−1.55057e7	−0.939409
$772$	8.31957e6i	0.502409i
$773$	− 1.64185e7i	− 0.988289i	−0.869380	−	0.494145i	$-0.835481\pi$
$773$	− 1.64185e7i	− 0.988289i	0.869380	−	0.494145i	$-0.164519\pi$
$774$	−5.91008e6	−0.354602
$775$	0	0
$776$	−9.13875e6	−0.544794
$777$	− 33678.0i	− 0.00200121i
$778$	3.25190e6i	0.192614i
$779$	1.65934e6	0.0979695
$780$	0	0
$781$	9.04680e6	0.530722
$782$	− 1.98360e6i	− 0.115994i
$783$	− 91854.0i	− 0.00535419i
$784$	4.30234e6	0.249985
$785$	0	0
$786$	5.80824e6	0.335342
$787$	1.46320e7i	0.842106i	0.907036	+	0.421053i	$0.138339\pi$
$787$	1.46320e7i	0.842106i	−0.907036	+	0.421053i	$0.861661\pi$
$788$	− 3.20630e6i	− 0.183945i
$789$	−6.22944e6	−0.356251
$790$	0	0
$791$	−233388.	−0.0132629
$792$	− 1.08864e6i	− 0.0616696i
$793$	4.77906e6i	0.269873i
$794$	−3.95525e6	−0.222650
$795$	0	0
$796$	8.85877e6	0.495554
$797$	− 2.61796e7i	− 1.45988i	−0.683511	−	0.729941i	$-0.739548\pi$
$797$	− 2.61796e7i	− 1.45988i	0.683511	−	0.729941i	$-0.260452\pi$
$798$	20916.0i	0.00116271i
$799$	2.66418e6	0.147637
$800$	0	0
$801$	4.06685e6	0.223963
$802$	8.58322e6i	0.471210i
$803$	6.07782e6i	0.332628i
$804$	−8.57938e6	−0.468075
$805$	0	0
$806$	2.02314e7	1.09696
$807$	− 6.21805e6i	− 0.336101i
$808$	1.04993e7i	0.565761i
$809$	−2.99051e7	−1.60648	−0.803238	−	0.595658i	$-0.796891\pi$
$809$	−2.99051e7	−1.60648	−0.803238	+	0.595658i	$0.796891\pi$
$810$	0	0
$811$	−3.12873e7	−1.67038	−0.835190	−	0.549961i	$-0.814642\pi$
$811$	−3.12873e7	−1.67038	−0.835190	+	0.549961i	$0.814642\pi$
$812$	− 2016.00i	0 0.000107300i
$813$	− 9.42206e6i	− 0.499942i
$814$	3.14328e6	0.166273
$815$	0	0
$816$	262656.	0.0138090
$817$	1.05980e7i	0.555482i
$818$	2.31063e7i	1.20739i
$819$	−54027.0	−0.00281450
$820$	0	0
$821$	−3.24291e7	−1.67910	−0.839550	−	0.543282i	$-0.817182\pi$
$821$	−3.24291e7	−1.67910	−0.839550	+	0.543282i	$0.817182\pi$
$822$	2.48594e6i	0.128325i
$823$	1.91700e7i	0.986560i	0.869871	+	0.493280i	$0.164202\pi$
$823$	1.91700e7i	0.986560i	−0.869871	+	0.493280i	$0.835798\pi$
$824$	−6.86310e6	−0.352130
$825$	0	0
$826$	−129240.	−0.00659093
$827$	98052.0i	0.00498532i	0.999997	+	0.00249266i	$0.000793438\pi$
$827$	98052.0i	0.00498532i	−0.999997	+	0.00249266i	$0.999207\pi$
$828$	− 5.63760e6i	− 0.285771i
$829$	−5.77426e6	−0.291817	−0.145908	−	0.989298i	$-0.546610\pi$
$829$	−5.77426e6	−0.291817	−0.145908	+	0.989298i	$0.546610\pi$
$830$	0	0
$831$	−9.64230e6	−0.484371
$832$	2.73203e6i	0.136829i
$833$	− 1.91588e6i	− 0.0956658i
$834$	8.07826e6	0.402163
$835$	0	0
$836$	−1.95216e6	−0.0966050
$837$	− 5.52801e6i	− 0.272744i
$838$	2.02927e7i	0.998229i
$839$	1.69354e7	0.830599	0.415299	−	0.909685i	$-0.363677\pi$
$839$	1.69354e7	0.830599	0.415299	+	0.909685i	$0.363677\pi$
$840$	0	0
$841$	−2.04953e7	−0.999226
$842$	− 1.20519e7i	− 0.585834i
$843$	− 1.26375e7i	− 0.612478i
$844$	−1.24378e7	−0.601020
$845$	0	0
$846$	7.57188e6	0.363729
$847$	− 116951.i	− 0.00560138i
$848$	− 5.55110e6i	− 0.265088i
$849$	−1.51383e7	−0.720788
$850$	0	0
$851$	1.62777e7	0.770494
$852$	6.20352e6i	0.292779i
$853$	5.04391e6i	0.237353i	0.992933	+	0.118677i	$0.0378652\pi$
$853$	5.04391e6i	0.237353i	−0.992933	+	0.118677i	$0.962135\pi$
$854$	28660.0	0.00134472
$855$	0	0
$856$	−1.34477e6	−0.0627282
$857$	− 1.28829e7i	− 0.599188i	−0.954067	−	0.299594i	$-0.903149\pi$
$857$	− 1.28829e7i	− 0.599188i	0.954067	−	0.299594i	$-0.0968512\pi$
$858$	− 5.04252e6i	− 0.233846i
$859$	3.59522e7	1.66243	0.831213	−	0.555954i	$-0.187647\pi$
$859$	3.59522e7	1.66243	0.831213	+	0.555954i	$0.187647\pi$
$860$	0	0
$861$	25704.0	0.00118166
$862$	2.23780e7i	1.02578i
$863$	2.20345e7i	1.00711i	0.863964	+	0.503554i	$0.167975\pi$
$863$	2.20345e7i	1.00711i	−0.863964	+	0.503554i	$0.832025\pi$
$864$	746496.	0.0340207
$865$	0	0
$866$	−2.25382e7	−1.02123
$867$	1.26617e7i	0.572066i
$868$	− 121328.i	− 0.00546590i
$869$	5.79768e6	0.260438
$870$	0	0
$871$	−3.97392e7	−1.77490
$872$	1.52029e7i	0.677072i
$873$	1.15662e7i	0.513637i
$874$	−1.01094e7	−0.447658
$875$	0	0
$876$	−4.16765e6	−0.183498
$877$	− 1.34531e7i	− 0.590641i	−0.955398	−	0.295320i	$-0.904574\pi$
$877$	− 1.34531e7i	− 0.590641i	0.955398	−	0.295320i	$-0.0954264\pi$
$878$	− 3.44265e6i	− 0.150715i
$879$	−1.58663e7	−0.692635
$880$	0	0
$881$	2.36331e7	1.02584	0.512922	−	0.858435i	$-0.328563\pi$
$881$	2.36331e7	1.02584	0.512922	+	0.858435i	$0.328563\pi$
$882$	− 5.44514e6i	− 0.235688i
$883$	3.49588e7i	1.50888i	0.656370	+	0.754439i	$0.272091\pi$
$883$	3.49588e7i	1.50888i	−0.656370	+	0.754439i	$0.727909\pi$
$884$	1.21661e6	0.0523625
$885$	0	0
$886$	1.10134e7	0.471343
$887$	− 3.11168e7i	− 1.32796i	−0.747749	−	0.663982i	$-0.768865\pi$
$887$	− 3.11168e7i	− 1.32796i	0.747749	−	0.663982i	$-0.231135\pi$
$888$	2.15539e6i	0.0917263i
$889$	138800.	0.00589027
$890$	0	0
$891$	−1.37781e6	−0.0581427
$892$	1.24190e7i	0.522604i
$893$	− 1.35780e7i	− 0.569779i
$894$	−1.46753e7	−0.614104
$895$	0	0
$896$	16384.0	0.000681789 0
$897$	− 2.61130e7i	− 1.08362i
$898$	1.13267e7i	0.468719i
$899$	955458.	0.0394287
$900$	0	0
$901$	−2.47198e6	−0.101445
$902$	2.39904e6i	0.0981796i
$903$	164169.i	0.00669995i
$904$	1.49368e7	0.607907
$905$	0	0
$906$	−7.77791e6	−0.314805
$907$	1.97320e7i	0.796438i	0.917290	+	0.398219i	$0.130372\pi$
$907$	1.97320e7i	0.796438i	−0.917290	+	0.398219i	$0.869628\pi$
$908$	1.47199e7i	0.592501i
$909$	1.32882e7	0.533405
$910$	0	0
$911$	−4.40720e7	−1.75941	−0.879704	−	0.475521i	$-0.842260\pi$
$911$	−4.40720e7	−1.75941	−0.879704	+	0.475521i	$0.842260\pi$
$912$	− 1.33862e6i	− 0.0532932i
$913$	374220.i	0.0148576i
$914$	−1.86494e7	−0.738412
$915$	0	0
$916$	−1.81151e7	−0.713350
$917$	− 161340.i	− 0.00633605i
$918$	− 332424.i	− 0.0130192i
$919$	4.33621e7	1.69364	0.846821	−	0.531879i	$-0.178514\pi$
$919$	4.33621e7	1.69364	0.846821	+	0.531879i	$0.178514\pi$
$920$	0	0
$921$	4.24714e6	0.164986
$922$	− 2.10862e7i	− 0.816902i
$923$	2.87344e7i	1.11019i
$924$	−30240.0	−0.00116520
$925$	0	0
$926$	2.86781e7	1.09907
$927$	8.68612e6i	0.331991i
$928$	129024.i	0.00491814i
$929$	−1.65235e7	−0.628147	−0.314074	−	0.949399i	$-0.601694\pi$
$929$	−1.65235e7	−0.628147	−0.314074	+	0.949399i	$0.601694\pi$
$930$	0	0
$931$	−9.76429e6	−0.369204
$932$	− 1.31700e7i	− 0.496647i
$933$	− 1.47995e7i	− 0.556600i
$934$	1.72176e7	0.645812
$935$	0	0
$936$	3.45773e6	0.129003
$937$	− 2.03546e7i	− 0.757381i	−0.925523	−	0.378691i	$-0.876374\pi$
$937$	− 2.03546e7i	− 0.757381i	0.925523	−	0.378691i	$-0.123626\pi$
$938$	238316.i	0.00884395i
$939$	−2.14117e7	−0.792480
$940$	0	0
$941$	−5.12406e7	−1.88643	−0.943215	−	0.332184i	$-0.892214\pi$
$941$	−5.12406e7	−1.88643	−0.943215	+	0.332184i	$0.892214\pi$
$942$	− 6.79021e6i	− 0.249319i
$943$	1.24236e7i	0.454955i
$944$	8.27136e6	0.302097
$945$	0	0
$946$	−1.53224e7	−0.556673
$947$	− 2.79736e7i	− 1.01361i	−0.862059	−	0.506807i	$-0.830826\pi$
$947$	− 2.79736e7i	− 1.01361i	0.862059	−	0.506807i	$-0.169174\pi$
$948$	3.97555e6i	0.143673i
$949$	−1.93043e7	−0.695807
$950$	0	0
$951$	1.19411e7	0.428148
$952$	− 7296.00i	0 0.000260911i
$953$	2.68337e7i	0.957080i	0.878066	+	0.478540i	$0.158834\pi$
$953$	2.68337e7i	0.957080i	−0.878066	+	0.478540i	$0.841166\pi$
$954$	−7.02562e6	−0.249927
$955$	0	0
$956$	2.12538e7	0.752129
$957$	− 238140.i	− 0.00840529i
$958$	− 762024.i	− 0.0268259i
$959$	69054.0	0.00242461
$960$	0	0
$961$	2.88727e7	1.00851
$962$	9.98366e6i	0.347818i
$963$	1.70197e6i	0.0591407i
$964$	−1.51899e6	−0.0526457
$965$	0	0
$966$	−156600.	−0.00539944
$967$	5.24831e7i	1.80490i	0.430794	+	0.902450i	$0.358234\pi$
$967$	5.24831e7i	1.80490i	−0.430794	+	0.902450i	$0.641766\pi$
$968$	7.48486e6i	0.256741i
$969$	−596106.	−0.0203945
$970$	0	0
$971$	5.01029e7	1.70536	0.852678	−	0.522437i	$-0.174977\pi$
$971$	5.01029e7	1.70536	0.852678	+	0.522437i	$0.174977\pi$
$972$	− 944784.i	− 0.0320750i
$973$	− 224396.i	− 0.00759859i
$974$	−515548.	−0.0174129
$975$	0	0
$976$	−1.83424e6	−0.0616356
$977$	4.50181e7i	1.50887i	0.656377	+	0.754433i	$0.272088\pi$
$977$	4.50181e7i	1.50887i	−0.656377	+	0.754433i	$0.727912\pi$
$978$	1.52010e7i	0.508190i
$979$	1.05437e7	0.351589
$980$	0	0
$981$	1.92411e7	0.638349
$982$	− 1.21718e6i	− 0.0402789i
$983$	− 3.26907e7i	− 1.07905i	−0.841971	−	0.539523i	$-0.818605\pi$
$983$	− 3.26907e7i	− 1.07905i	0.841971	−	0.539523i	$-0.181395\pi$
$984$	−1.64506e6	−0.0541618
$985$	0	0
$986$	57456.0	0.00188210
$987$	− 210330.i	− 0.00687240i
$988$	− 6.20043e6i	− 0.202083i
$989$	−7.93484e7	−2.57957
$990$	0	0
$991$	6.24606e6	0.202033	0.101016	−	0.994885i	$-0.467791\pi$
$991$	6.24606e6	0.202033	0.101016	+	0.994885i	$0.467791\pi$
$992$	7.76499e6i	0.250531i
$993$	5.95055e6i	0.191507i
$994$	172320.	0.00553184
$995$	0	0
$996$	−256608.	−0.00819638
$997$	− 2.85010e7i	− 0.908077i	−0.890982	−	0.454039i	$-0.849983\pi$
$997$	− 2.85010e7i	− 0.908077i	0.890982	−	0.454039i	$-0.150017\pi$
$998$	− 2.56790e7i	− 0.816115i
$999$	2.72792e6	0.0864804

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.c.a.49.2		2
3.2	odd	2		450.6.c.k.199.1		2
5.2	odd	4		150.6.a.e.1.1	✓	1
5.3	odd	4		150.6.a.k.1.1	yes	1
5.4	even	2	inner	150.6.c.a.49.1		2
15.2	even	4		450.6.a.r.1.1		1
15.8	even	4		450.6.a.g.1.1		1
15.14	odd	2		450.6.c.k.199.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.a.e.1.1	✓	1	5.2	odd	4
150.6.a.k.1.1	yes	1	5.3	odd	4
150.6.c.a.49.1		2	5.4	even	2	inner
150.6.c.a.49.2		2	1.1	even	1	trivial
450.6.a.g.1.1		1	15.8	even	4
450.6.a.r.1.1		1	15.2	even	4
450.6.c.k.199.1		2	3.2	odd	2
450.6.c.k.199.2		2	15.14	odd	2

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:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +1.00000i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +1.00000i q^{7} -64.0000i q^{8} -81.0000 q^{9} -210.000 q^{11} -144.000i q^{12} -667.000i q^{13} -4.00000 q^{14} +256.000 q^{16} -114.000i q^{17} -324.000i q^{18} -581.000 q^{19} -9.00000 q^{21} -840.000i q^{22} -4350.00i q^{23} +576.000 q^{24} +2668.00 q^{26} -729.000i q^{27} -16.0000i q^{28} +126.000 q^{29} +7583.00 q^{31} +1024.00i q^{32} -1890.00i q^{33} +456.000 q^{34} +1296.00 q^{36} +3742.00i q^{37} -2324.00i q^{38} +6003.00 q^{39} -2856.00 q^{41} -36.0000i q^{42} -18241.0i q^{43} +3360.00 q^{44} +17400.0 q^{46} +23370.0i q^{47} +2304.00i q^{48} +16806.0 q^{49} +1026.00 q^{51} +10672.0i q^{52} -21684.0i q^{53} +2916.00 q^{54} +64.0000 q^{56} -5229.00i q^{57} +504.000i q^{58} +32310.0 q^{59} -7165.00 q^{61} +30332.0i q^{62} -81.0000i q^{63} -4096.00 q^{64} +7560.00 q^{66} -59579.0i q^{67} +1824.00i q^{68} +39150.0 q^{69} -43080.0 q^{71} +5184.00i q^{72} -28942.0i q^{73} -14968.0 q^{74} +9296.00 q^{76} -210.000i q^{77} +24012.0i q^{78} -27608.0 q^{79} +6561.00 q^{81} -11424.0i q^{82} -1782.00i q^{83} +144.000 q^{84} +72964.0 q^{86} +1134.00i q^{87} +13440.0i q^{88} -50208.0 q^{89} +667.000 q^{91} +69600.0i q^{92} +68247.0i q^{93} -93480.0 q^{94} -9216.00 q^{96} -142793. i q^{97} +67224.0i q^{98} +17010.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} - 72 q^{6} - 162 q^{9}+O(q^{10}) \) 2 * q - 32 * q^4 - 72 * q^6 - 162 * q^9	\( 2 q - 32 q^{4} - 72 q^{6} - 162 q^{9} - 420 q^{11} - 8 q^{14} + 512 q^{16} - 1162 q^{19} - 18 q^{21} + 1152 q^{24} + 5336 q^{26} + 252 q^{29} + 15166 q^{31} + 912 q^{34} + 2592 q^{36} + 12006 q^{39} - 5712 q^{41} + 6720 q^{44} + 34800 q^{46} + 33612 q^{49} + 2052 q^{51} + 5832 q^{54} + 128 q^{56} + 64620 q^{59} - 14330 q^{61} - 8192 q^{64} + 15120 q^{66} + 78300 q^{69} - 86160 q^{71} - 29936 q^{74} + 18592 q^{76} - 55216 q^{79} + 13122 q^{81} + 288 q^{84} + 145928 q^{86} - 100416 q^{89} + 1334 q^{91} - 186960 q^{94} - 18432 q^{96} + 34020 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 - 72 * q^6 - 162 * q^9 - 420 * q^11 - 8 * q^14 + 512 * q^16 - 1162 * q^19 - 18 * q^21 + 1152 * q^24 + 5336 * q^26 + 252 * q^29 + 15166 * q^31 + 912 * q^34 + 2592 * q^36 + 12006 * q^39 - 5712 * q^41 + 6720 * q^44 + 34800 * q^46 + 33612 * q^49 + 2052 * q^51 + 5832 * q^54 + 128 * q^56 + 64620 * q^59 - 14330 * q^61 - 8192 * q^64 + 15120 * q^66 + 78300 * q^69 - 86160 * q^71 - 29936 * q^74 + 18592 * q^76 - 55216 * q^79 + 13122 * q^81 + 288 * q^84 + 145928 * q^86 - 100416 * q^89 + 1334 * q^91 - 186960 * q^94 - 18432 * q^96 + 34020 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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