LMFDB - Embedded newform 592.3.k.c.561.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,3,Mod(401,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("592.401");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	592.k (of order $4$, degree $2$, 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:	$16.1308316501$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 74)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			561.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	592.561
Dual form			592.3.k.c.401.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{3} +(3.00000 + 3.00000i) q^{5} +4.00000 q^{7} -7.00000 q^{9} +O(q^{10})$	\(q+4.00000i q^{3} +(3.00000 + 3.00000i) q^{5} +4.00000 q^{7} -7.00000 q^{9} +4.00000i q^{11} +(3.00000 + 3.00000i) q^{13} +(-12.0000 + 12.0000i) q^{15} +(23.0000 + 23.0000i) q^{17} +(-10.0000 - 10.0000i) q^{19} +16.0000i q^{21} +(10.0000 + 10.0000i) q^{23} -7.00000i q^{25} +8.00000i q^{27} +(-19.0000 + 19.0000i) q^{29} +(18.0000 - 18.0000i) q^{31} -16.0000 q^{33} +(12.0000 + 12.0000i) q^{35} +37.0000i q^{37} +(-12.0000 + 12.0000i) q^{39} -74.0000i q^{41} +(-42.0000 - 42.0000i) q^{43} +(-21.0000 - 21.0000i) q^{45} +44.0000 q^{47} -33.0000 q^{49} +(-92.0000 + 92.0000i) q^{51} -80.0000 q^{53} +(-12.0000 + 12.0000i) q^{55} +(40.0000 - 40.0000i) q^{57} +(54.0000 + 54.0000i) q^{59} +(-3.00000 + 3.00000i) q^{61} -28.0000 q^{63} +18.0000i q^{65} -12.0000i q^{67} +(-40.0000 + 40.0000i) q^{69} -124.000 q^{71} +10.0000i q^{73} +28.0000 q^{75} +16.0000i q^{77} +(-14.0000 - 14.0000i) q^{79} -95.0000 q^{81} +64.0000 q^{83} +138.000i q^{85} +(-76.0000 - 76.0000i) q^{87} +(17.0000 - 17.0000i) q^{89} +(12.0000 + 12.0000i) q^{91} +(72.0000 + 72.0000i) q^{93} -60.0000i q^{95} +(129.000 + 129.000i) q^{97} -28.0000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5} + 8 q^{7} - 14 q^{9}+O(q^{10}) $ 2 * q + 6 * q^5 + 8 * q^7 - 14 * q^9	$ 2 q + 6 q^{5} + 8 q^{7} - 14 q^{9} + 6 q^{13} - 24 q^{15} + 46 q^{17} - 20 q^{19} + 20 q^{23} - 38 q^{29} + 36 q^{31} - 32 q^{33} + 24 q^{35} - 24 q^{39} - 84 q^{43} - 42 q^{45} + 88 q^{47} - 66 q^{49} - 184 q^{51} - 160 q^{53} - 24 q^{55} + 80 q^{57} + 108 q^{59} - 6 q^{61} - 56 q^{63} - 80 q^{69} - 248 q^{71} + 56 q^{75} - 28 q^{79} - 190 q^{81} + 128 q^{83} - 152 q^{87} + 34 q^{89} + 24 q^{91} + 144 q^{93} + 258 q^{97}+O(q^{100}) $ 2 * q + 6 * q^5 + 8 * q^7 - 14 * q^9 + 6 * q^13 - 24 * q^15 + 46 * q^17 - 20 * q^19 + 20 * q^23 - 38 * q^29 + 36 * q^31 - 32 * q^33 + 24 * q^35 - 24 * q^39 - 84 * q^43 - 42 * q^45 + 88 * q^47 - 66 * q^49 - 184 * q^51 - 160 * q^53 - 24 * q^55 + 80 * q^57 + 108 * q^59 - 6 * q^61 - 56 * q^63 - 80 * q^69 - 248 * q^71 + 56 * q^75 - 28 * q^79 - 190 * q^81 + 128 * q^83 - 152 * q^87 + 34 * q^89 + 24 * q^91 + 144 * q^93 + 258 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/592\mathbb{Z}\right)^\times$.

$n$	$113$	$149$	$223$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$			4.00000i			1.33333i	0.745356	+	0.666667i	$0.232280\pi$
$3$			4.00000i			1.33333i	−0.745356	+	0.666667i	$0.767720\pi$
$4$		0			0
$5$	3.00000	+	3.00000i	0.600000	+	0.600000i	0.940312	−	0.340312i	$-0.110533\pi$
$5$	3.00000	+	3.00000i	0.600000	+	0.600000i	−0.340312	+	0.940312i	$0.610533\pi$
$6$		0			0
$7$	4.00000			0.571429			0.285714	−	0.958315i	$-0.407769\pi$
$7$	4.00000			0.571429			0.285714	+	0.958315i	$0.407769\pi$
$8$		0			0
$9$	−7.00000			−0.777778
$10$		0			0
$11$			4.00000i			0.363636i	0.983332	+	0.181818i	$0.0581982\pi$
$11$			4.00000i			0.363636i	−0.983332	+	0.181818i	$0.941802\pi$
$12$		0			0
$13$	3.00000	+	3.00000i	0.230769	+	0.230769i	0.813014	−	0.582245i	$-0.197825\pi$
$13$	3.00000	+	3.00000i	0.230769	+	0.230769i	−0.582245	+	0.813014i	$0.697825\pi$
$14$		0			0
$15$	−12.0000	+	12.0000i	−0.800000	+	0.800000i
$16$		0			0
$17$	23.0000	+	23.0000i	1.35294	+	1.35294i	0.882353	+	0.470588i	$0.155958\pi$
$17$	23.0000	+	23.0000i	1.35294	+	1.35294i	0.470588	+	0.882353i	$0.344042\pi$
$18$		0			0
$19$	−10.0000	−	10.0000i	−0.526316	−	0.526316i	0.393156	−	0.919472i	$-0.371383\pi$
$19$	−10.0000	−	10.0000i	−0.526316	−	0.526316i	−0.919472	+	0.393156i	$0.871383\pi$
$20$		0			0
$21$			16.0000i			0.761905i
$22$		0			0
$23$	10.0000	+	10.0000i	0.434783	+	0.434783i	0.890252	−	0.455469i	$-0.150528\pi$
$23$	10.0000	+	10.0000i	0.434783	+	0.434783i	−0.455469	+	0.890252i	$0.650528\pi$
$24$		0			0
$25$		−	7.00000i		−	0.280000i
$26$		0			0
$27$			8.00000i			0.296296i
$28$		0			0
$29$	−19.0000	+	19.0000i	−0.655172	+	0.655172i	−0.954234	−	0.299061i	$-0.903326\pi$
$29$	−19.0000	+	19.0000i	−0.655172	+	0.655172i	0.299061	+	0.954234i	$0.403326\pi$
$30$		0			0
$31$	18.0000	−	18.0000i	0.580645	−	0.580645i	−0.354435	−	0.935081i	$-0.615327\pi$
$31$	18.0000	−	18.0000i	0.580645	−	0.580645i	0.935081	+	0.354435i	$0.115327\pi$
$32$		0			0
$33$	−16.0000			−0.484848
$34$		0			0
$35$	12.0000	+	12.0000i	0.342857	+	0.342857i
$36$		0			0
$37$			37.0000i			1.00000i
$37$			37.0000i			1.00000i
$38$		0			0
$39$	−12.0000	+	12.0000i	−0.307692	+	0.307692i
$40$		0			0
$41$		−	74.0000i		−	1.80488i	−0.430818	−	0.902439i	$-0.641775\pi$
$41$		−	74.0000i		−	1.80488i	0.430818	−	0.902439i	$-0.358225\pi$
$42$		0			0
$43$	−42.0000	−	42.0000i	−0.976744	−	0.976744i	0.0229915	−	0.999736i	$-0.492681\pi$
$43$	−42.0000	−	42.0000i	−0.976744	−	0.976744i	−0.999736	+	0.0229915i	$0.992681\pi$
$44$		0			0
$45$	−21.0000	−	21.0000i	−0.466667	−	0.466667i
$46$		0			0
$47$	44.0000			0.936170			0.468085	−	0.883683i	$-0.344944\pi$
$47$	44.0000			0.936170			0.468085	+	0.883683i	$0.344944\pi$
$48$		0			0
$49$	−33.0000			−0.673469
$50$		0			0
$51$	−92.0000	+	92.0000i	−1.80392	+	1.80392i
$52$		0			0
$53$	−80.0000			−1.50943			−0.754717	−	0.656051i	$-0.772226\pi$
$53$	−80.0000			−1.50943			−0.754717	+	0.656051i	$0.772226\pi$
$54$		0			0
$55$	−12.0000	+	12.0000i	−0.218182	+	0.218182i
$56$		0			0
$57$	40.0000	−	40.0000i	0.701754	−	0.701754i
$58$		0			0
$59$	54.0000	+	54.0000i	0.915254	+	0.915254i	0.996679	−	0.0814252i	$-0.0259472\pi$
$59$	54.0000	+	54.0000i	0.915254	+	0.915254i	−0.0814252	+	0.996679i	$0.525947\pi$
$60$		0			0
$61$	−3.00000	+	3.00000i	−0.0491803	+	0.0491803i	−0.731269	−	0.682089i	$-0.761072\pi$
$61$	−3.00000	+	3.00000i	−0.0491803	+	0.0491803i	0.682089	+	0.731269i	$0.261072\pi$
$62$		0			0
$63$	−28.0000			−0.444444
$64$		0			0
$65$			18.0000i			0.276923i
$66$		0			0
$67$		−	12.0000i		−	0.179104i	−0.995982	−	0.0895522i	$-0.971456\pi$
$67$		−	12.0000i		−	0.179104i	0.995982	−	0.0895522i	$-0.0285436\pi$
$68$		0			0
$69$	−40.0000	+	40.0000i	−0.579710	+	0.579710i
$70$		0			0
$71$	−124.000			−1.74648			−0.873239	−	0.487291i	$-0.837985\pi$
$71$	−124.000			−1.74648			−0.873239	+	0.487291i	$0.837985\pi$
$72$		0			0
$73$			10.0000i			0.136986i	0.997652	+	0.0684932i	$0.0218191\pi$
$73$			10.0000i			0.136986i	−0.997652	+	0.0684932i	$0.978181\pi$
$74$		0			0
$75$	28.0000			0.373333
$76$		0			0
$77$			16.0000i			0.207792i
$78$		0			0
$79$	−14.0000	−	14.0000i	−0.177215	−	0.177215i	0.612926	−	0.790141i	$-0.289993\pi$
$79$	−14.0000	−	14.0000i	−0.177215	−	0.177215i	−0.790141	+	0.612926i	$0.789993\pi$
$80$		0			0
$81$	−95.0000			−1.17284
$82$		0			0
$83$	64.0000			0.771084			0.385542	−	0.922690i	$-0.374014\pi$
$83$	64.0000			0.771084			0.385542	+	0.922690i	$0.374014\pi$
$84$		0			0
$85$			138.000i			1.62353i
$86$		0			0
$87$	−76.0000	−	76.0000i	−0.873563	−	0.873563i
$88$		0			0
$89$	17.0000	−	17.0000i	0.191011	−	0.191011i	−0.605122	−	0.796133i	$-0.706876\pi$
$89$	17.0000	−	17.0000i	0.191011	−	0.191011i	0.796133	+	0.605122i	$0.206876\pi$
$90$		0			0
$91$	12.0000	+	12.0000i	0.131868	+	0.131868i
$92$		0			0
$93$	72.0000	+	72.0000i	0.774194	+	0.774194i
$94$		0			0
$95$		−	60.0000i		−	0.631579i
$96$		0			0
$97$	129.000	+	129.000i	1.32990	+	1.32990i	0.905455	+	0.424442i	$0.139530\pi$
$97$	129.000	+	129.000i	1.32990	+	1.32990i	0.424442	+	0.905455i	$0.360470\pi$
$98$		0			0
$99$		−	28.0000i		−	0.282828i
$100$		0			0
$101$		−	118.000i		−	1.16832i	−0.811640	−	0.584158i	$-0.801425\pi$
$101$		−	118.000i		−	1.16832i	0.811640	−	0.584158i	$-0.198575\pi$
$102$		0			0
$103$	42.0000	−	42.0000i	0.407767	−	0.407767i	−0.473192	−	0.880959i	$-0.656898\pi$
$103$	42.0000	−	42.0000i	0.407767	−	0.407767i	0.880959	+	0.473192i	$0.156898\pi$
$104$		0			0
$105$	−48.0000	+	48.0000i	−0.457143	+	0.457143i
$106$		0			0
$107$	24.0000			0.224299			0.112150	−	0.993691i	$-0.464226\pi$
$107$	24.0000			0.224299			0.112150	+	0.993691i	$0.464226\pi$
$108$		0			0
$109$	−91.0000	−	91.0000i	−0.834862	−	0.834862i	0.153315	−	0.988177i	$-0.451005\pi$
$109$	−91.0000	−	91.0000i	−0.834862	−	0.834862i	−0.988177	+	0.153315i	$0.951005\pi$
$110$		0			0
$111$	−148.000			−1.33333
$112$		0			0
$113$	7.00000	−	7.00000i	0.0619469	−	0.0619469i	−0.675455	−	0.737402i	$-0.736053\pi$
$113$	7.00000	−	7.00000i	0.0619469	−	0.0619469i	0.737402	+	0.675455i	$0.236053\pi$
$114$		0			0
$115$			60.0000i			0.521739i
$116$		0			0
$117$	−21.0000	−	21.0000i	−0.179487	−	0.179487i
$118$		0			0
$119$	92.0000	+	92.0000i	0.773109	+	0.773109i
$120$		0			0
$121$	105.000			0.867769
$122$		0			0
$123$	296.000			2.40650
$124$		0			0
$125$	96.0000	−	96.0000i	0.768000	−	0.768000i
$126$		0			0
$127$	−76.0000			−0.598425			−0.299213	−	0.954186i	$-0.596724\pi$
$127$	−76.0000			−0.598425			−0.299213	+	0.954186i	$0.596724\pi$
$128$		0			0
$129$	168.000	−	168.000i	1.30233	−	1.30233i
$130$		0			0
$131$	−70.0000	+	70.0000i	−0.534351	+	0.534351i	−0.921864	−	0.387513i	$-0.873334\pi$
$131$	−70.0000	+	70.0000i	−0.534351	+	0.534351i	0.387513	+	0.921864i	$0.373334\pi$
$132$		0			0
$133$	−40.0000	−	40.0000i	−0.300752	−	0.300752i
$134$		0			0
$135$	−24.0000	+	24.0000i	−0.177778	+	0.177778i
$136$		0			0
$137$	216.000			1.57664			0.788321	−	0.615264i	$-0.210951\pi$
$137$	216.000			1.57664			0.788321	+	0.615264i	$0.210951\pi$
$138$		0			0
$139$			180.000i			1.29496i	0.762081	+	0.647482i	$0.224178\pi$
$139$			180.000i			1.29496i	−0.762081	+	0.647482i	$0.775822\pi$
$140$		0			0
$141$			176.000i			1.24823i
$142$		0			0
$143$	−12.0000	+	12.0000i	−0.0839161	+	0.0839161i
$144$		0			0
$145$	−114.000			−0.786207
$146$		0			0
$147$		−	132.000i		−	0.897959i
$148$		0			0
$149$	−34.0000			−0.228188			−0.114094	−	0.993470i	$-0.536396\pi$
$149$	−34.0000			−0.228188			−0.114094	+	0.993470i	$0.536396\pi$
$150$		0			0
$151$		−	72.0000i		−	0.476821i	−0.971164	−	0.238411i	$-0.923374\pi$
$151$		−	72.0000i		−	0.476821i	0.971164	−	0.238411i	$-0.0766264\pi$
$152$		0			0
$153$	−161.000	−	161.000i	−1.05229	−	1.05229i
$154$		0			0
$155$	108.000			0.696774
$156$		0			0
$157$	−14.0000			−0.0891720			−0.0445860	−	0.999006i	$-0.514197\pi$
$157$	−14.0000			−0.0891720			−0.0445860	+	0.999006i	$0.514197\pi$
$158$		0			0
$159$		−	320.000i		−	2.01258i
$160$		0			0
$161$	40.0000	+	40.0000i	0.248447	+	0.248447i
$162$		0			0
$163$	66.0000	−	66.0000i	0.404908	−	0.404908i	−0.475051	−	0.879959i	$-0.657570\pi$
$163$	66.0000	−	66.0000i	0.404908	−	0.404908i	0.879959	+	0.475051i	$0.157570\pi$
$164$		0			0
$165$	−48.0000	−	48.0000i	−0.290909	−	0.290909i
$166$		0			0
$167$	230.000	+	230.000i	1.37725	+	1.37725i	0.849242	+	0.528003i	$0.177059\pi$
$167$	230.000	+	230.000i	1.37725	+	1.37725i	0.528003	+	0.849242i	$0.322941\pi$
$168$		0			0
$169$		−	151.000i		−	0.893491i
$170$		0			0
$171$	70.0000	+	70.0000i	0.409357	+	0.409357i
$172$		0			0
$173$			88.0000i			0.508671i	0.967116	+	0.254335i	$0.0818567\pi$
$173$			88.0000i			0.508671i	−0.967116	+	0.254335i	$0.918143\pi$
$174$		0			0
$175$		−	28.0000i		−	0.160000i
$176$		0			0
$177$	−216.000	+	216.000i	−1.22034	+	1.22034i
$178$		0			0
$179$	166.000	−	166.000i	0.927374	−	0.927374i	−0.0701614	−	0.997536i	$-0.522351\pi$
$179$	166.000	−	166.000i	0.927374	−	0.927374i	0.997536	+	0.0701614i	$0.0223514\pi$
$180$		0			0
$181$	304.000			1.67956			0.839779	−	0.542928i	$-0.182684\pi$
$181$	304.000			1.67956			0.839779	+	0.542928i	$0.182684\pi$
$182$		0			0
$183$	−12.0000	−	12.0000i	−0.0655738	−	0.0655738i
$184$		0			0
$185$	−111.000	+	111.000i	−0.600000	+	0.600000i
$186$		0			0
$187$	−92.0000	+	92.0000i	−0.491979	+	0.491979i
$188$		0			0
$189$			32.0000i			0.169312i
$190$		0			0
$191$	−186.000	−	186.000i	−0.973822	−	0.973822i	0.0258440	−	0.999666i	$-0.491773\pi$
$191$	−186.000	−	186.000i	−0.973822	−	0.973822i	−0.999666	+	0.0258440i	$0.991773\pi$
$192$		0			0
$193$	−9.00000	−	9.00000i	−0.0466321	−	0.0466321i	0.683406	−	0.730038i	$-0.260498\pi$
$193$	−9.00000	−	9.00000i	−0.0466321	−	0.0466321i	−0.730038	+	0.683406i	$0.760498\pi$
$194$		0			0
$195$	−72.0000			−0.369231
$196$		0			0
$197$	34.0000			0.172589			0.0862944	−	0.996270i	$-0.472497\pi$
$197$	34.0000			0.172589			0.0862944	+	0.996270i	$0.472497\pi$
$198$		0			0
$199$	−46.0000	+	46.0000i	−0.231156	+	0.231156i	−0.813175	−	0.582019i	$-0.802263\pi$
$199$	−46.0000	+	46.0000i	−0.231156	+	0.231156i	0.582019	+	0.813175i	$0.302263\pi$
$200$		0			0
$201$	48.0000			0.238806
$202$		0			0
$203$	−76.0000	+	76.0000i	−0.374384	+	0.374384i
$204$		0			0
$205$	222.000	−	222.000i	1.08293	−	1.08293i
$206$		0			0
$207$	−70.0000	−	70.0000i	−0.338164	−	0.338164i
$208$		0			0
$209$	40.0000	−	40.0000i	0.191388	−	0.191388i
$210$		0			0
$211$	208.000			0.985782			0.492891	−	0.870091i	$-0.335940\pi$
$211$	208.000			0.985782			0.492891	+	0.870091i	$0.335940\pi$
$212$		0			0
$213$		−	496.000i		−	2.32864i
$214$		0			0
$215$		−	252.000i		−	1.17209i
$216$		0			0
$217$	72.0000	−	72.0000i	0.331797	−	0.331797i
$218$		0			0
$219$	−40.0000			−0.182648
$220$		0			0
$221$			138.000i			0.624434i
$222$		0			0
$223$	124.000			0.556054			0.278027	−	0.960573i	$-0.410320\pi$
$223$	124.000			0.556054			0.278027	+	0.960573i	$0.410320\pi$
$224$		0			0
$225$			49.0000i			0.217778i
$226$		0			0
$227$	90.0000	+	90.0000i	0.396476	+	0.396476i	0.876988	−	0.480512i	$-0.159549\pi$
$227$	90.0000	+	90.0000i	0.396476	+	0.396476i	−0.480512	+	0.876988i	$0.659549\pi$
$228$		0			0
$229$	−2.00000			−0.00873362			−0.00436681	−	0.999990i	$-0.501390\pi$
$229$	−2.00000			−0.00873362			−0.00436681	+	0.999990i	$0.501390\pi$
$230$		0			0
$231$	−64.0000			−0.277056
$232$		0			0
$233$		−	42.0000i		−	0.180258i	−0.995930	−	0.0901288i	$-0.971272\pi$
$233$		−	42.0000i		−	0.180258i	0.995930	−	0.0901288i	$-0.0287279\pi$
$234$		0			0
$235$	132.000	+	132.000i	0.561702	+	0.561702i
$236$		0			0
$237$	56.0000	−	56.0000i	0.236287	−	0.236287i
$238$		0			0
$239$	−22.0000	−	22.0000i	−0.0920502	−	0.0920502i	0.659582	−	0.751632i	$-0.270733\pi$
$239$	−22.0000	−	22.0000i	−0.0920502	−	0.0920502i	−0.751632	+	0.659582i	$0.770733\pi$
$240$		0			0
$241$	−175.000	−	175.000i	−0.726141	−	0.726141i	0.243708	−	0.969849i	$-0.421636\pi$
$241$	−175.000	−	175.000i	−0.726141	−	0.726141i	−0.969849	+	0.243708i	$0.921636\pi$
$242$		0			0
$243$		−	308.000i		−	1.26749i
$244$		0			0
$245$	−99.0000	−	99.0000i	−0.404082	−	0.404082i
$246$		0			0
$247$		−	60.0000i		−	0.242915i
$248$		0			0
$249$			256.000i			1.02811i
$250$		0			0
$251$	198.000	−	198.000i	0.788845	−	0.788845i	−0.192460	−	0.981305i	$-0.561647\pi$
$251$	198.000	−	198.000i	0.788845	−	0.788845i	0.981305	+	0.192460i	$0.0616466\pi$
$252$		0			0
$253$	−40.0000	+	40.0000i	−0.158103	+	0.158103i
$254$		0			0
$255$	−552.000			−2.16471
$256$		0			0
$257$	−159.000	−	159.000i	−0.618677	−	0.618677i	0.326515	−	0.945192i	$-0.394126\pi$
$257$	−159.000	−	159.000i	−0.618677	−	0.618677i	−0.945192	+	0.326515i	$0.894126\pi$
$258$		0			0
$259$			148.000i			0.571429i
$260$		0			0
$261$	133.000	−	133.000i	0.509579	−	0.509579i
$262$		0			0
$263$		−	88.0000i		−	0.334601i	−0.985906	−	0.167300i	$-0.946495\pi$
$263$		−	88.0000i		−	0.334601i	0.985906	−	0.167300i	$-0.0535050\pi$
$264$		0			0
$265$	−240.000	−	240.000i	−0.905660	−	0.905660i
$266$		0			0
$267$	68.0000	+	68.0000i	0.254682	+	0.254682i
$268$		0			0
$269$	178.000			0.661710			0.330855	−	0.943682i	$-0.392663\pi$
$269$	178.000			0.661710			0.330855	+	0.943682i	$0.392663\pi$
$270$		0			0
$271$	−404.000			−1.49077			−0.745387	−	0.666631i	$-0.767735\pi$
$271$	−404.000			−1.49077			−0.745387	+	0.666631i	$0.767735\pi$
$272$		0			0
$273$	−48.0000	+	48.0000i	−0.175824	+	0.175824i
$274$		0			0
$275$	28.0000			0.101818
$276$		0			0
$277$	−147.000	+	147.000i	−0.530686	+	0.530686i	−0.920776	−	0.390091i	$-0.872444\pi$
$277$	−147.000	+	147.000i	−0.530686	+	0.530686i	0.390091	+	0.920776i	$0.372444\pi$
$278$		0			0
$279$	−126.000	+	126.000i	−0.451613	+	0.451613i
$280$		0			0
$281$	135.000	+	135.000i	0.480427	+	0.480427i	0.905268	−	0.424841i	$-0.139670\pi$
$281$	135.000	+	135.000i	0.480427	+	0.480427i	−0.424841	+	0.905268i	$0.639670\pi$
$282$		0			0
$283$	258.000	−	258.000i	0.911661	−	0.911661i	−0.0847421	−	0.996403i	$-0.527007\pi$
$283$	258.000	−	258.000i	0.911661	−	0.911661i	0.996403	+	0.0847421i	$0.0270066\pi$
$284$		0			0
$285$	240.000			0.842105
$286$		0			0
$287$		−	296.000i		−	1.03136i
$288$		0			0
$289$			769.000i			2.66090i
$290$		0			0
$291$	−516.000	+	516.000i	−1.77320	+	1.77320i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$			324.000i			1.09831i
$296$		0			0
$297$	−32.0000			−0.107744
$298$		0			0
$299$			60.0000i			0.200669i
$300$		0			0
$301$	−168.000	−	168.000i	−0.558140	−	0.558140i
$302$		0			0
$303$	472.000			1.55776
$304$		0			0
$305$	−18.0000			−0.0590164
$306$		0			0
$307$			364.000i			1.18567i	0.805325	+	0.592834i	$0.201991\pi$
$307$			364.000i			1.18567i	−0.805325	+	0.592834i	$0.798009\pi$
$308$		0			0
$309$	168.000	+	168.000i	0.543689	+	0.543689i
$310$		0			0
$311$	238.000	−	238.000i	0.765273	−	0.765273i	−0.211997	−	0.977270i	$-0.567997\pi$
$311$	238.000	−	238.000i	0.765273	−	0.765273i	0.977270	+	0.211997i	$0.0679967\pi$
$312$		0			0
$313$	87.0000	+	87.0000i	0.277955	+	0.277955i	0.832292	−	0.554337i	$-0.187028\pi$
$313$	87.0000	+	87.0000i	0.277955	+	0.277955i	−0.554337	+	0.832292i	$0.687028\pi$
$314$		0			0
$315$	−84.0000	−	84.0000i	−0.266667	−	0.266667i
$316$		0			0
$317$		−	264.000i		−	0.832808i	−0.909180	−	0.416404i	$-0.863290\pi$
$317$		−	264.000i		−	0.832808i	0.909180	−	0.416404i	$-0.136710\pi$
$318$		0			0
$319$	−76.0000	−	76.0000i	−0.238245	−	0.238245i
$320$		0			0
$321$			96.0000i			0.299065i
$322$		0			0
$323$		−	460.000i		−	1.42415i
$324$		0			0
$325$	21.0000	−	21.0000i	0.0646154	−	0.0646154i
$326$		0			0
$327$	364.000	−	364.000i	1.11315	−	1.11315i
$328$		0			0
$329$	176.000			0.534954
$330$		0			0
$331$	326.000	+	326.000i	0.984894	+	0.984894i	0.999888	−	0.0149933i	$-0.00477271\pi$
$331$	326.000	+	326.000i	0.984894	+	0.984894i	−0.0149933	+	0.999888i	$0.504773\pi$
$332$		0			0
$333$		−	259.000i		−	0.777778i
$334$		0			0
$335$	36.0000	−	36.0000i	0.107463	−	0.107463i
$336$		0			0
$337$			230.000i			0.682493i	0.939974	+	0.341246i	$0.110849\pi$
$337$			230.000i			0.682493i	−0.939974	+	0.341246i	$0.889151\pi$
$338$		0			0
$339$	28.0000	+	28.0000i	0.0825959	+	0.0825959i
$340$		0			0
$341$	72.0000	+	72.0000i	0.211144	+	0.211144i
$342$		0			0
$343$	−328.000			−0.956268
$344$		0			0
$345$	−240.000			−0.695652
$346$		0			0
$347$	−362.000	+	362.000i	−1.04323	+	1.04323i	−0.0442052	+	0.999022i	$0.514076\pi$
$347$	−362.000	+	362.000i	−1.04323	+	1.04323i	−0.999022	+	0.0442052i	$0.985924\pi$
$348$		0			0
$349$	−48.0000			−0.137536			−0.0687679	−	0.997633i	$-0.521907\pi$
$349$	−48.0000			−0.137536			−0.0687679	+	0.997633i	$0.521907\pi$
$350$		0			0
$351$	−24.0000	+	24.0000i	−0.0683761	+	0.0683761i
$352$		0			0
$353$	343.000	−	343.000i	0.971671	−	0.971671i	−0.0279383	−	0.999610i	$-0.508894\pi$
$353$	343.000	−	343.000i	0.971671	−	0.971671i	0.999610	+	0.0279383i	$0.00889418\pi$
$354$		0			0
$355$	−372.000	−	372.000i	−1.04789	−	1.04789i
$356$		0			0
$357$	−368.000	+	368.000i	−1.03081	+	1.03081i
$358$		0			0
$359$	404.000			1.12535			0.562674	−	0.826679i	$-0.309773\pi$
$359$	404.000			1.12535			0.562674	+	0.826679i	$0.309773\pi$
$360$		0			0
$361$		−	161.000i		−	0.445983i
$362$		0			0
$363$			420.000i			1.15702i
$364$		0			0
$365$	−30.0000	+	30.0000i	−0.0821918	+	0.0821918i
$366$		0			0
$367$	−492.000			−1.34060			−0.670300	−	0.742090i	$-0.733834\pi$
$367$	−492.000			−1.34060			−0.670300	+	0.742090i	$0.733834\pi$
$368$		0			0
$369$			518.000i			1.40379i
$370$		0			0
$371$	−320.000			−0.862534
$372$		0			0
$373$			488.000i			1.30831i	0.756360	+	0.654155i	$0.226976\pi$
$373$			488.000i			1.30831i	−0.756360	+	0.654155i	$0.773024\pi$
$374$		0			0
$375$	384.000	+	384.000i	1.02400	+	1.02400i
$376$		0			0
$377$	−114.000			−0.302387
$378$		0			0
$379$	440.000			1.16095			0.580475	−	0.814278i	$-0.302867\pi$
$379$	440.000			1.16095			0.580475	+	0.814278i	$0.302867\pi$
$380$		0			0
$381$		−	304.000i		−	0.797900i
$382$		0			0
$383$	450.000	+	450.000i	1.17493	+	1.17493i	0.981018	+	0.193917i	$0.0621192\pi$
$383$	450.000	+	450.000i	1.17493	+	1.17493i	0.193917	+	0.981018i	$0.437881\pi$
$384$		0			0
$385$	−48.0000	+	48.0000i	−0.124675	+	0.124675i
$386$		0			0
$387$	294.000	+	294.000i	0.759690	+	0.759690i
$388$		0			0
$389$	21.0000	+	21.0000i	0.0539846	+	0.0539846i	0.733584	−	0.679599i	$-0.237846\pi$
$389$	21.0000	+	21.0000i	0.0539846	+	0.0539846i	−0.679599	+	0.733584i	$0.737846\pi$
$390$		0			0
$391$			460.000i			1.17647i
$392$		0			0
$393$	−280.000	−	280.000i	−0.712468	−	0.712468i
$394$		0			0
$395$		−	84.0000i		−	0.212658i
$396$		0			0
$397$			646.000i			1.62720i	0.581422	+	0.813602i	$0.302496\pi$
$397$			646.000i			1.62720i	−0.581422	+	0.813602i	$0.697504\pi$
$398$		0			0
$399$	160.000	−	160.000i	0.401003	−	0.401003i
$400$		0			0
$401$	273.000	−	273.000i	0.680798	−	0.680798i	−0.279382	−	0.960180i	$-0.590130\pi$
$401$	273.000	−	273.000i	0.680798	−	0.680798i	0.960180	+	0.279382i	$0.0901296\pi$
$402$		0			0
$403$	108.000			0.267990
$404$		0			0
$405$	−285.000	−	285.000i	−0.703704	−	0.703704i
$406$		0			0
$407$	−148.000			−0.363636
$408$		0			0
$409$	39.0000	−	39.0000i	0.0953545	−	0.0953545i	−0.657820	−	0.753175i	$-0.728521\pi$
$409$	39.0000	−	39.0000i	0.0953545	−	0.0953545i	0.753175	+	0.657820i	$0.228521\pi$
$410$		0			0
$411$			864.000i			2.10219i
$412$		0			0
$413$	216.000	+	216.000i	0.523002	+	0.523002i
$414$		0			0
$415$	192.000	+	192.000i	0.462651	+	0.462651i
$416$		0			0
$417$	−720.000			−1.72662
$418$		0			0
$419$	−464.000			−1.10740			−0.553699	−	0.832717i	$-0.686784\pi$
$419$	−464.000			−1.10740			−0.553699	+	0.832717i	$0.686784\pi$
$420$		0			0
$421$	−469.000	+	469.000i	−1.11401	+	1.11401i	−0.121412	+	0.992602i	$0.538742\pi$
$421$	−469.000	+	469.000i	−1.11401	+	1.11401i	−0.992602	+	0.121412i	$0.961258\pi$
$422$		0			0
$423$	−308.000			−0.728132
$424$		0			0
$425$	161.000	−	161.000i	0.378824	−	0.378824i
$426$		0			0
$427$	−12.0000	+	12.0000i	−0.0281030	+	0.0281030i
$428$		0			0
$429$	−48.0000	−	48.0000i	−0.111888	−	0.111888i
$430$		0			0
$431$	−450.000	+	450.000i	−1.04408	+	1.04408i	−0.0451011	+	0.998982i	$0.514361\pi$
$431$	−450.000	+	450.000i	−1.04408	+	1.04408i	−0.998982	+	0.0451011i	$0.985639\pi$
$432$		0			0
$433$	200.000			0.461894			0.230947	−	0.972966i	$-0.425818\pi$
$433$	200.000			0.461894			0.230947	+	0.972966i	$0.425818\pi$
$434$		0			0
$435$		−	456.000i		−	1.04828i
$436$		0			0
$437$		−	200.000i		−	0.457666i
$438$		0			0
$439$	−246.000	+	246.000i	−0.560364	+	0.560364i	−0.929411	−	0.369046i	$-0.879684\pi$
$439$	−246.000	+	246.000i	−0.560364	+	0.560364i	0.369046	+	0.929411i	$0.379684\pi$
$440$		0			0
$441$	231.000			0.523810
$442$		0			0
$443$		−	668.000i		−	1.50790i	−0.656931	−	0.753950i	$-0.728146\pi$
$443$		−	668.000i		−	1.50790i	0.656931	−	0.753950i	$-0.271854\pi$
$444$		0			0
$445$	102.000			0.229213
$446$		0			0
$447$		−	136.000i		−	0.304251i
$448$		0			0
$449$	199.000	+	199.000i	0.443207	+	0.443207i	0.893088	−	0.449881i	$-0.148534\pi$
$449$	199.000	+	199.000i	0.443207	+	0.443207i	−0.449881	+	0.893088i	$0.648534\pi$
$450$		0			0
$451$	296.000			0.656319
$452$		0			0
$453$	288.000			0.635762
$454$		0			0
$455$			72.0000i			0.158242i
$456$		0			0
$457$	−255.000	−	255.000i	−0.557987	−	0.557987i	0.370747	−	0.928734i	$-0.379102\pi$
$457$	−255.000	−	255.000i	−0.557987	−	0.557987i	−0.928734	+	0.370747i	$0.879102\pi$
$458$		0			0
$459$	−184.000	+	184.000i	−0.400871	+	0.400871i
$460$		0			0
$461$	−555.000	−	555.000i	−1.20390	−	1.20390i	−0.972969	−	0.230935i	$-0.925821\pi$
$461$	−555.000	−	555.000i	−1.20390	−	1.20390i	−0.230935	−	0.972969i	$-0.574179\pi$
$462$		0			0
$463$	−554.000	−	554.000i	−1.19654	−	1.19654i	−0.975194	−	0.221350i	$-0.928954\pi$
$463$	−554.000	−	554.000i	−1.19654	−	1.19654i	−0.221350	−	0.975194i	$-0.571046\pi$
$464$		0			0
$465$			432.000i			0.929032i
$466$		0			0
$467$	−158.000	−	158.000i	−0.338330	−	0.338330i	0.517409	−	0.855738i	$-0.326897\pi$
$467$	−158.000	−	158.000i	−0.338330	−	0.338330i	−0.855738	+	0.517409i	$0.826897\pi$
$468$		0			0
$469$		−	48.0000i		−	0.102345i
$470$		0			0
$471$		−	56.0000i		−	0.118896i
$472$		0			0
$473$	168.000	−	168.000i	0.355180	−	0.355180i
$474$		0			0
$475$	−70.0000	+	70.0000i	−0.147368	+	0.147368i
$476$		0			0
$477$	560.000			1.17400
$478$		0			0
$479$	−290.000	−	290.000i	−0.605428	−	0.605428i	0.336320	−	0.941748i	$-0.390818\pi$
$479$	−290.000	−	290.000i	−0.605428	−	0.605428i	−0.941748	+	0.336320i	$0.890818\pi$
$480$		0			0
$481$	−111.000	+	111.000i	−0.230769	+	0.230769i
$482$		0			0
$483$	−160.000	+	160.000i	−0.331263	+	0.331263i
$484$		0			0
$485$			774.000i			1.59588i
$486$		0			0
$487$	−266.000	−	266.000i	−0.546201	−	0.546201i	0.379139	−	0.925340i	$-0.376220\pi$
$487$	−266.000	−	266.000i	−0.546201	−	0.546201i	−0.925340	+	0.379139i	$0.876220\pi$
$488$		0			0
$489$	264.000	+	264.000i	0.539877	+	0.539877i
$490$		0			0
$491$	−312.000			−0.635438			−0.317719	−	0.948185i	$-0.602917\pi$
$491$	−312.000			−0.635438			−0.317719	+	0.948185i	$0.602917\pi$
$492$		0			0
$493$	−874.000			−1.77282
$494$		0			0
$495$	84.0000	−	84.0000i	0.169697	−	0.169697i
$496$		0			0
$497$	−496.000			−0.997988
$498$		0			0
$499$	210.000	−	210.000i	0.420842	−	0.420842i	−0.464652	−	0.885493i	$-0.653820\pi$
$499$	210.000	−	210.000i	0.420842	−	0.420842i	0.885493	+	0.464652i	$0.153820\pi$
$500$		0			0
$501$	−920.000	+	920.000i	−1.83633	+	1.83633i
$502$		0			0
$503$	−298.000	−	298.000i	−0.592445	−	0.592445i	0.345846	−	0.938291i	$-0.387592\pi$
$503$	−298.000	−	298.000i	−0.592445	−	0.592445i	−0.938291	+	0.345846i	$0.887592\pi$
$504$		0			0
$505$	354.000	−	354.000i	0.700990	−	0.700990i
$506$		0			0
$507$	604.000			1.19132
$508$		0			0
$509$		−	808.000i		−	1.58743i	−0.608292	−	0.793713i	$-0.708145\pi$
$509$		−	808.000i		−	1.58743i	0.608292	−	0.793713i	$-0.291855\pi$
$510$		0			0
$511$			40.0000i			0.0782779i
$512$		0			0
$513$	80.0000	−	80.0000i	0.155945	−	0.155945i
$514$		0			0
$515$	252.000			0.489320
$516$		0			0
$517$			176.000i			0.340426i
$518$		0			0
$519$	−352.000			−0.678227
$520$		0			0
$521$			170.000i			0.326296i	0.986602	+	0.163148i	$0.0521647\pi$
$521$			170.000i			0.326296i	−0.986602	+	0.163148i	$0.947835\pi$
$522$		0			0
$523$	−654.000	−	654.000i	−1.25048	−	1.25048i	−0.955506	−	0.294972i	$-0.904690\pi$
$523$	−654.000	−	654.000i	−1.25048	−	1.25048i	−0.294972	−	0.955506i	$-0.595310\pi$
$524$		0			0
$525$	112.000			0.213333
$526$		0			0
$527$	828.000			1.57116
$528$		0			0
$529$		−	329.000i		−	0.621928i
$530$		0			0
$531$	−378.000	−	378.000i	−0.711864	−	0.711864i
$532$		0			0
$533$	222.000	−	222.000i	0.416510	−	0.416510i
$534$		0			0
$535$	72.0000	+	72.0000i	0.134579	+	0.134579i
$536$		0			0
$537$	664.000	+	664.000i	1.23650	+	1.23650i
$538$		0			0
$539$		−	132.000i		−	0.244898i
$540$		0			0
$541$	147.000	+	147.000i	0.271719	+	0.271719i	0.829792	−	0.558073i	$-0.188459\pi$
$541$	147.000	+	147.000i	0.271719	+	0.271719i	−0.558073	+	0.829792i	$0.688459\pi$
$542$		0			0
$543$			1216.00i			2.23941i
$544$		0			0
$545$		−	546.000i		−	1.00183i
$546$		0			0
$547$	294.000	−	294.000i	0.537477	−	0.537477i	−0.385310	−	0.922787i	$-0.625905\pi$
$547$	294.000	−	294.000i	0.537477	−	0.537477i	0.922787	+	0.385310i	$0.125905\pi$
$548$		0			0
$549$	21.0000	−	21.0000i	0.0382514	−	0.0382514i
$550$		0			0
$551$	380.000			0.689655
$552$		0			0
$553$	−56.0000	−	56.0000i	−0.101266	−	0.101266i
$554$		0			0
$555$	−444.000	−	444.000i	−0.800000	−	0.800000i
$556$		0			0
$557$	−179.000	+	179.000i	−0.321364	+	0.321364i	−0.849290	−	0.527926i	$-0.822970\pi$
$557$	−179.000	+	179.000i	−0.321364	+	0.321364i	0.527926	+	0.849290i	$0.322970\pi$
$558$		0			0
$559$		−	252.000i		−	0.450805i
$560$		0			0
$561$	−368.000	−	368.000i	−0.655971	−	0.655971i
$562$		0			0
$563$	−282.000	−	282.000i	−0.500888	−	0.500888i	0.410826	−	0.911714i	$-0.365240\pi$
$563$	−282.000	−	282.000i	−0.500888	−	0.500888i	−0.911714	+	0.410826i	$0.865240\pi$
$564$		0			0
$565$	42.0000			0.0743363
$566$		0			0
$567$	−380.000			−0.670194
$568$		0			0
$569$	375.000	−	375.000i	0.659051	−	0.659051i	−0.296105	−	0.955156i	$-0.595688\pi$
$569$	375.000	−	375.000i	0.659051	−	0.659051i	0.955156	+	0.296105i	$0.0956877\pi$
$570$		0			0
$571$	184.000			0.322242			0.161121	−	0.986935i	$-0.448489\pi$
$571$	184.000			0.322242			0.161121	+	0.986935i	$0.448489\pi$
$572$		0			0
$573$	744.000	−	744.000i	1.29843	−	1.29843i
$574$		0			0
$575$	70.0000	−	70.0000i	0.121739	−	0.121739i
$576$		0			0
$577$	−623.000	−	623.000i	−1.07972	−	1.07972i	−0.996534	−	0.0831889i	$-0.973490\pi$
$577$	−623.000	−	623.000i	−1.07972	−	1.07972i	−0.0831889	−	0.996534i	$-0.526510\pi$
$578$		0			0
$579$	36.0000	−	36.0000i	0.0621762	−	0.0621762i
$580$		0			0
$581$	256.000			0.440620
$582$		0			0
$583$		−	320.000i		−	0.548885i
$584$		0			0
$585$		−	126.000i		−	0.215385i
$586$		0			0
$587$	382.000	−	382.000i	0.650767	−	0.650767i	−0.302411	−	0.953178i	$-0.597792\pi$
$587$	382.000	−	382.000i	0.650767	−	0.650767i	0.953178	+	0.302411i	$0.0977915\pi$
$588$		0			0
$589$	−360.000			−0.611205
$590$		0			0
$591$			136.000i			0.230118i
$592$		0			0
$593$	734.000			1.23777			0.618887	−	0.785480i	$-0.287584\pi$
$593$	734.000			1.23777			0.618887	+	0.785480i	$0.287584\pi$
$594$		0			0
$595$			552.000i			0.927731i
$596$		0			0
$597$	−184.000	−	184.000i	−0.308208	−	0.308208i
$598$		0			0
$599$	1004.00			1.67613			0.838063	−	0.545573i	$-0.183688\pi$
$599$	1004.00			1.67613			0.838063	+	0.545573i	$0.183688\pi$
$600$		0			0
$601$	306.000			0.509151			0.254576	−	0.967053i	$-0.418064\pi$
$601$	306.000			0.509151			0.254576	+	0.967053i	$0.418064\pi$
$602$		0			0
$603$			84.0000i			0.139303i
$604$		0			0
$605$	315.000	+	315.000i	0.520661	+	0.520661i
$606$		0			0
$607$	−298.000	+	298.000i	−0.490939	+	0.490939i	−0.908602	−	0.417663i	$-0.862849\pi$
$607$	−298.000	+	298.000i	−0.490939	+	0.490939i	0.417663	+	0.908602i	$0.362849\pi$
$608$		0			0
$609$	−304.000	−	304.000i	−0.499179	−	0.499179i
$610$		0			0
$611$	132.000	+	132.000i	0.216039	+	0.216039i
$612$		0			0
$613$		−	470.000i		−	0.766721i	−0.923599	−	0.383361i	$-0.874767\pi$
$613$		−	470.000i		−	0.766721i	0.923599	−	0.383361i	$-0.125233\pi$
$614$		0			0
$615$	888.000	+	888.000i	1.44390	+	1.44390i
$616$		0			0
$617$		−	304.000i		−	0.492707i	−0.969180	−	0.246353i	$-0.920768\pi$
$617$		−	304.000i		−	0.492707i	0.969180	−	0.246353i	$-0.0792324\pi$
$618$		0			0
$619$			956.000i			1.54443i	0.635363	+	0.772213i	$0.280850\pi$
$619$			956.000i			1.54443i	−0.635363	+	0.772213i	$0.719150\pi$
$620$		0			0
$621$	−80.0000	+	80.0000i	−0.128824	+	0.128824i
$622$		0			0
$623$	68.0000	−	68.0000i	0.109149	−	0.109149i
$624$		0			0
$625$	401.000			0.641600
$626$		0			0
$627$	160.000	+	160.000i	0.255183	+	0.255183i
$628$		0			0
$629$	−851.000	+	851.000i	−1.35294	+	1.35294i
$630$		0			0
$631$	362.000	−	362.000i	0.573693	−	0.573693i	−0.359466	−	0.933158i	$-0.617041\pi$
$631$	362.000	−	362.000i	0.573693	−	0.573693i	0.933158	+	0.359466i	$0.117041\pi$
$632$		0			0
$633$			832.000i			1.31438i
$634$		0			0
$635$	−228.000	−	228.000i	−0.359055	−	0.359055i
$636$		0			0
$637$	−99.0000	−	99.0000i	−0.155416	−	0.155416i
$638$		0			0
$639$	868.000			1.35837
$640$		0			0
$641$	−398.000			−0.620905			−0.310452	−	0.950589i	$-0.600481\pi$
$641$	−398.000			−0.620905			−0.310452	+	0.950589i	$0.600481\pi$
$642$		0			0
$643$	−218.000	+	218.000i	−0.339036	+	0.339036i	−0.856004	−	0.516969i	$-0.827060\pi$
$643$	−218.000	+	218.000i	−0.339036	+	0.339036i	0.516969	+	0.856004i	$0.327060\pi$
$644$		0			0
$645$	1008.00			1.56279
$646$		0			0
$647$	358.000	−	358.000i	0.553323	−	0.553323i	−0.374075	−	0.927398i	$-0.622040\pi$
$647$	358.000	−	358.000i	0.553323	−	0.553323i	0.927398	+	0.374075i	$0.122040\pi$
$648$		0			0
$649$	−216.000	+	216.000i	−0.332820	+	0.332820i
$650$		0			0
$651$	288.000	+	288.000i	0.442396	+	0.442396i
$652$		0			0
$653$	555.000	−	555.000i	0.849923	−	0.849923i	−0.140200	−	0.990123i	$-0.544774\pi$
$653$	555.000	−	555.000i	0.849923	−	0.849923i	0.990123	+	0.140200i	$0.0447745\pi$
$654$		0			0
$655$	−420.000			−0.641221
$656$		0			0
$657$		−	70.0000i		−	0.106545i
$658$		0			0
$659$			788.000i			1.19575i	0.801589	+	0.597876i	$0.203988\pi$
$659$			788.000i			1.19575i	−0.801589	+	0.597876i	$0.796012\pi$
$660$		0			0
$661$	−661.000	+	661.000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$661$	−661.000	+	661.000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$662$		0			0
$663$	−552.000			−0.832579
$664$		0			0
$665$		−	240.000i		−	0.360902i
$666$		0			0
$667$	−380.000			−0.569715
$668$		0			0
$669$			496.000i			0.741405i
$670$		0			0
$671$	−12.0000	−	12.0000i	−0.0178838	−	0.0178838i
$672$		0			0
$673$	50.0000			0.0742942			0.0371471	−	0.999310i	$-0.488173\pi$
$673$	50.0000			0.0742942			0.0371471	+	0.999310i	$0.488173\pi$
$674$		0			0
$675$	56.0000			0.0829630
$676$		0			0
$677$			296.000i			0.437223i	0.975812	+	0.218612i	$0.0701527\pi$
$677$			296.000i			0.437223i	−0.975812	+	0.218612i	$0.929847\pi$
$678$		0			0
$679$	516.000	+	516.000i	0.759941	+	0.759941i
$680$		0			0
$681$	−360.000	+	360.000i	−0.528634	+	0.528634i
$682$		0			0
$683$	−6.00000	−	6.00000i	−0.00878477	−	0.00878477i	0.702701	−	0.711486i	$-0.251977\pi$
$683$	−6.00000	−	6.00000i	−0.00878477	−	0.00878477i	−0.711486	+	0.702701i	$0.751977\pi$
$684$		0			0
$685$	648.000	+	648.000i	0.945985	+	0.945985i
$686$		0			0
$687$		−	8.00000i		−	0.0116448i
$688$		0			0
$689$	−240.000	−	240.000i	−0.348331	−	0.348331i
$690$		0			0
$691$		−	708.000i		−	1.02460i	−0.858806	−	0.512301i	$-0.828793\pi$
$691$		−	708.000i		−	1.02460i	0.858806	−	0.512301i	$-0.171207\pi$
$692$		0			0
$693$		−	112.000i		−	0.161616i
$694$		0			0
$695$	−540.000	+	540.000i	−0.776978	+	0.776978i
$696$		0			0
$697$	1702.00	−	1702.00i	2.44189	−	2.44189i
$698$		0			0
$699$	168.000			0.240343
$700$		0			0
$701$	291.000	+	291.000i	0.415121	+	0.415121i	0.883518	−	0.468397i	$-0.155168\pi$
$701$	291.000	+	291.000i	0.415121	+	0.415121i	−0.468397	+	0.883518i	$0.655168\pi$
$702$		0			0
$703$	370.000	−	370.000i	0.526316	−	0.526316i
$704$		0			0
$705$	−528.000	+	528.000i	−0.748936	+	0.748936i
$706$		0			0
$707$		−	472.000i		−	0.667610i
$708$		0			0
$709$	−77.0000	−	77.0000i	−0.108604	−	0.108604i	0.650717	−	0.759320i	$-0.274469\pi$
$709$	−77.0000	−	77.0000i	−0.108604	−	0.108604i	−0.759320	+	0.650717i	$0.774469\pi$
$710$		0			0
$711$	98.0000	+	98.0000i	0.137834	+	0.137834i
$712$		0			0
$713$	360.000			0.504909
$714$		0			0
$715$	−72.0000			−0.100699
$716$		0			0
$717$	88.0000	−	88.0000i	0.122734	−	0.122734i
$718$		0			0
$719$	412.000			0.573018			0.286509	−	0.958078i	$-0.407505\pi$
$719$	412.000			0.573018			0.286509	+	0.958078i	$0.407505\pi$
$720$		0			0
$721$	168.000	−	168.000i	0.233010	−	0.233010i
$722$		0			0
$723$	700.000	−	700.000i	0.968188	−	0.968188i
$724$		0			0
$725$	133.000	+	133.000i	0.183448	+	0.183448i
$726$		0			0
$727$	262.000	−	262.000i	0.360385	−	0.360385i	−0.503570	−	0.863955i	$-0.667980\pi$
$727$	262.000	−	262.000i	0.360385	−	0.360385i	0.863955	+	0.503570i	$0.167980\pi$
$728$		0			0
$729$	377.000			0.517147
$730$		0			0
$731$		−	1932.00i		−	2.64295i
$732$		0			0
$733$		−	682.000i		−	0.930423i	−0.885200	−	0.465211i	$-0.845978\pi$
$733$		−	682.000i		−	0.930423i	0.885200	−	0.465211i	$-0.154022\pi$
$734$		0			0
$735$	396.000	−	396.000i	0.538776	−	0.538776i
$736$		0			0
$737$	48.0000			0.0651289
$738$		0			0
$739$			100.000i			0.135318i	0.997709	+	0.0676590i	$0.0215530\pi$
$739$			100.000i			0.135318i	−0.997709	+	0.0676590i	$0.978447\pi$
$740$		0			0
$741$	240.000			0.323887
$742$		0			0
$743$		−	800.000i		−	1.07672i	−0.842716	−	0.538358i	$-0.819045\pi$
$743$		−	800.000i		−	1.07672i	0.842716	−	0.538358i	$-0.180955\pi$
$744$		0			0
$745$	−102.000	−	102.000i	−0.136913	−	0.136913i
$746$		0			0
$747$	−448.000			−0.599732
$748$		0			0
$749$	96.0000			0.128171
$750$		0			0
$751$			536.000i			0.713715i	0.934159	+	0.356858i	$0.116152\pi$
$751$			536.000i			0.713715i	−0.934159	+	0.356858i	$0.883848\pi$
$752$		0			0
$753$	792.000	+	792.000i	1.05179	+	1.05179i
$754$		0			0
$755$	216.000	−	216.000i	0.286093	−	0.286093i
$756$		0			0
$757$	−157.000	−	157.000i	−0.207398	−	0.207398i	0.595763	−	0.803160i	$-0.296850\pi$
$757$	−157.000	−	157.000i	−0.207398	−	0.207398i	−0.803160	+	0.595763i	$0.796850\pi$
$758$		0			0
$759$	−160.000	−	160.000i	−0.210804	−	0.210804i
$760$		0			0
$761$		−	272.000i		−	0.357424i	−0.983901	−	0.178712i	$-0.942807\pi$
$761$		−	272.000i		−	0.357424i	0.983901	−	0.178712i	$-0.0571931\pi$
$762$		0			0
$763$	−364.000	−	364.000i	−0.477064	−	0.477064i
$764$		0			0
$765$		−	966.000i		−	1.26275i
$766$		0			0
$767$			324.000i			0.422425i
$768$		0			0
$769$	−761.000	+	761.000i	−0.989597	+	0.989597i	−0.999946	−	0.0103496i	$-0.996706\pi$
$769$	−761.000	+	761.000i	−0.989597	+	0.989597i	0.0103496	+	0.999946i	$0.496706\pi$
$770$		0			0
$771$	636.000	−	636.000i	0.824903	−	0.824903i
$772$		0			0
$773$	−1282.00			−1.65847			−0.829237	−	0.558898i	$-0.811225\pi$
$773$	−1282.00			−1.65847			−0.829237	+	0.558898i	$0.811225\pi$
$774$		0			0
$775$	−126.000	−	126.000i	−0.162581	−	0.162581i
$776$		0			0
$777$	−592.000			−0.761905
$778$		0			0
$779$	−740.000	+	740.000i	−0.949936	+	0.949936i
$780$		0			0
$781$		−	496.000i		−	0.635083i
$782$		0			0
$783$	−152.000	−	152.000i	−0.194125	−	0.194125i
$784$		0			0
$785$	−42.0000	−	42.0000i	−0.0535032	−	0.0535032i
$786$		0			0
$787$	376.000			0.477764			0.238882	−	0.971049i	$-0.423219\pi$
$787$	376.000			0.477764			0.238882	+	0.971049i	$0.423219\pi$
$788$		0			0
$789$	352.000			0.446134
$790$		0			0
$791$	28.0000	−	28.0000i	0.0353982	−	0.0353982i
$792$		0			0
$793$	−18.0000			−0.0226986
$794$		0			0
$795$	960.000	−	960.000i	1.20755	−	1.20755i
$796$		0			0
$797$	−69.0000	+	69.0000i	−0.0865747	+	0.0865747i	−0.749068	−	0.662493i	$-0.769498\pi$
$797$	−69.0000	+	69.0000i	−0.0865747	+	0.0865747i	0.662493	+	0.749068i	$0.269498\pi$
$798$		0			0
$799$	1012.00	+	1012.00i	1.26658	+	1.26658i
$800$		0			0
$801$	−119.000	+	119.000i	−0.148564	+	0.148564i
$802$		0			0
$803$	−40.0000			−0.0498132
$804$		0			0
$805$			240.000i			0.298137i
$806$		0			0
$807$			712.000i			0.882280i
$808$		0			0
$809$	631.000	−	631.000i	0.779975	−	0.779975i	−0.199851	−	0.979826i	$-0.564046\pi$
$809$	631.000	−	631.000i	0.779975	−	0.779975i	0.979826	+	0.199851i	$0.0640458\pi$
$810$		0			0
$811$	−1184.00			−1.45993			−0.729963	−	0.683487i	$-0.760463\pi$
$811$	−1184.00			−1.45993			−0.729963	+	0.683487i	$0.760463\pi$
$812$		0			0
$813$		−	1616.00i		−	1.98770i
$814$		0			0
$815$	396.000			0.485890
$816$		0			0
$817$			840.000i			1.02815i
$818$		0			0
$819$	−84.0000	−	84.0000i	−0.102564	−	0.102564i
$820$		0			0
$821$	784.000			0.954933			0.477467	−	0.878650i	$-0.341555\pi$
$821$	784.000			0.954933			0.477467	+	0.878650i	$0.341555\pi$
$822$		0			0
$823$	−1116.00			−1.35601			−0.678007	−	0.735055i	$-0.737156\pi$
$823$	−1116.00			−1.35601			−0.678007	+	0.735055i	$0.737156\pi$
$824$		0			0
$825$			112.000i			0.135758i
$826$		0			0
$827$	−382.000	−	382.000i	−0.461911	−	0.461911i	0.437371	−	0.899281i	$-0.355910\pi$
$827$	−382.000	−	382.000i	−0.461911	−	0.461911i	−0.899281	+	0.437371i	$0.855910\pi$
$828$		0			0
$829$	603.000	−	603.000i	0.727382	−	0.727382i	−0.242715	−	0.970098i	$-0.578038\pi$
$829$	603.000	−	603.000i	0.727382	−	0.727382i	0.970098	+	0.242715i	$0.0780381\pi$
$830$		0			0
$831$	−588.000	−	588.000i	−0.707581	−	0.707581i
$832$		0			0
$833$	−759.000	−	759.000i	−0.911164	−	0.911164i
$834$		0			0
$835$			1380.00i			1.65269i
$836$		0			0
$837$	144.000	+	144.000i	0.172043	+	0.172043i
$838$		0			0
$839$		−	48.0000i		−	0.0572110i	−0.999591	−	0.0286055i	$-0.990893\pi$
$839$		−	48.0000i		−	0.0572110i	0.999591	−	0.0286055i	$-0.00910665\pi$
$840$		0			0
$841$			119.000i			0.141498i
$842$		0			0
$843$	−540.000	+	540.000i	−0.640569	+	0.640569i
$844$		0			0
$845$	453.000	−	453.000i	0.536095	−	0.536095i
$846$		0			0
$847$	420.000			0.495868
$848$		0			0
$849$	1032.00	+	1032.00i	1.21555	+	1.21555i
$850$		0			0
$851$	−370.000	+	370.000i	−0.434783	+	0.434783i
$852$		0			0
$853$	−163.000	+	163.000i	−0.191090	+	0.191090i	−0.796167	−	0.605077i	$-0.793142\pi$
$853$	−163.000	+	163.000i	−0.191090	+	0.191090i	0.605077	+	0.796167i	$0.293142\pi$
$854$		0			0
$855$			420.000i			0.491228i
$856$		0			0
$857$	−815.000	−	815.000i	−0.950992	−	0.950992i	0.0478621	−	0.998854i	$-0.484759\pi$
$857$	−815.000	−	815.000i	−0.950992	−	0.950992i	−0.998854	+	0.0478621i	$0.984759\pi$
$858$		0			0
$859$	−962.000	−	962.000i	−1.11991	−	1.11991i	−0.991754	−	0.128152i	$-0.959095\pi$
$859$	−962.000	−	962.000i	−1.11991	−	1.11991i	−0.128152	−	0.991754i	$-0.540905\pi$
$860$		0			0
$861$	1184.00			1.37515
$862$		0			0
$863$	−156.000			−0.180765			−0.0903824	−	0.995907i	$-0.528809\pi$
$863$	−156.000			−0.180765			−0.0903824	+	0.995907i	$0.528809\pi$
$864$		0			0
$865$	−264.000	+	264.000i	−0.305202	+	0.305202i
$866$		0			0
$867$	−3076.00			−3.54787
$868$		0			0
$869$	56.0000	−	56.0000i	0.0644419	−	0.0644419i
$870$		0			0
$871$	36.0000	−	36.0000i	0.0413318	−	0.0413318i
$872$		0			0
$873$	−903.000	−	903.000i	−1.03436	−	1.03436i
$874$		0			0
$875$	384.000	−	384.000i	0.438857	−	0.438857i
$876$		0			0
$877$	210.000			0.239453			0.119726	−	0.992807i	$-0.461798\pi$
$877$	210.000			0.239453			0.119726	+	0.992807i	$0.461798\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			720.000i			0.817253i	0.912702	+	0.408627i	$0.133992\pi$
$881$			720.000i			0.817253i	−0.912702	+	0.408627i	$0.866008\pi$
$882$		0			0
$883$	294.000	−	294.000i	0.332956	−	0.332956i	−0.520752	−	0.853708i	$-0.674348\pi$
$883$	294.000	−	294.000i	0.332956	−	0.332956i	0.853708	+	0.520752i	$0.174348\pi$
$884$		0			0
$885$	−1296.00			−1.46441
$886$		0			0
$887$			904.000i			1.01917i	0.860422	+	0.509583i	$0.170200\pi$
$887$			904.000i			1.01917i	−0.860422	+	0.509583i	$0.829800\pi$
$888$		0			0
$889$	−304.000			−0.341957
$890$		0			0
$891$		−	380.000i		−	0.426487i
$892$		0			0
$893$	−440.000	−	440.000i	−0.492721	−	0.492721i
$894$		0			0
$895$	996.000			1.11285
$896$		0			0
$897$	−240.000			−0.267559
$898$		0			0
$899$			684.000i			0.760845i
$900$		0			0
$901$	−1840.00	−	1840.00i	−2.04218	−	2.04218i
$902$		0			0
$903$	672.000	−	672.000i	0.744186	−	0.744186i
$904$		0			0
$905$	912.000	+	912.000i	1.00773	+	1.00773i
$906$		0			0
$907$	−618.000	−	618.000i	−0.681367	−	0.681367i	0.278941	−	0.960308i	$-0.410017\pi$
$907$	−618.000	−	618.000i	−0.681367	−	0.681367i	−0.960308	+	0.278941i	$0.910017\pi$
$908$		0			0
$909$			826.000i			0.908691i
$910$		0			0
$911$	882.000	+	882.000i	0.968167	+	0.968167i	0.999509	−	0.0313419i	$-0.00997806\pi$
$911$	882.000	+	882.000i	0.968167	+	0.968167i	−0.0313419	+	0.999509i	$0.509978\pi$
$912$		0			0
$913$			256.000i			0.280394i
$914$		0			0
$915$		−	72.0000i		−	0.0786885i
$916$		0			0
$917$	−280.000	+	280.000i	−0.305344	+	0.305344i
$918$		0			0
$919$	1218.00	−	1218.00i	1.32535	−	1.32535i	0.415980	−	0.909374i	$-0.363439\pi$
$919$	1218.00	−	1218.00i	1.32535	−	1.32535i	0.909374	−	0.415980i	$-0.136561\pi$
$920$		0			0
$921$	−1456.00			−1.58089
$922$		0			0
$923$	−372.000	−	372.000i	−0.403034	−	0.403034i
$924$		0			0
$925$	259.000			0.280000
$926$		0			0
$927$	−294.000	+	294.000i	−0.317152	+	0.317152i
$928$		0			0
$929$			666.000i			0.716900i	0.933549	+	0.358450i	$0.116695\pi$
$929$			666.000i			0.716900i	−0.933549	+	0.358450i	$0.883305\pi$
$930$		0			0
$931$	330.000	+	330.000i	0.354458	+	0.354458i
$932$		0			0
$933$	952.000	+	952.000i	1.02036	+	1.02036i
$934$		0			0
$935$	−552.000			−0.590374
$936$		0			0
$937$	−424.000			−0.452508			−0.226254	−	0.974068i	$-0.572648\pi$
$937$	−424.000			−0.452508			−0.226254	+	0.974068i	$0.572648\pi$
$938$		0			0
$939$	−348.000	+	348.000i	−0.370607	+	0.370607i
$940$		0			0
$941$	−912.000			−0.969182			−0.484591	−	0.874741i	$-0.661032\pi$
$941$	−912.000			−0.969182			−0.484591	+	0.874741i	$0.661032\pi$
$942$		0			0
$943$	740.000	−	740.000i	0.784730	−	0.784730i
$944$		0			0
$945$	−96.0000	+	96.0000i	−0.101587	+	0.101587i
$946$		0			0
$947$	134.000	+	134.000i	0.141499	+	0.141499i	0.774308	−	0.632809i	$-0.218098\pi$
$947$	134.000	+	134.000i	0.141499	+	0.141499i	−0.632809	+	0.774308i	$0.718098\pi$
$948$		0			0
$949$	−30.0000	+	30.0000i	−0.0316122	+	0.0316122i
$950$		0			0
$951$	1056.00			1.11041
$952$		0			0
$953$			1264.00i			1.32634i	0.748470	+	0.663169i	$0.230789\pi$
$953$			1264.00i			1.32634i	−0.748470	+	0.663169i	$0.769211\pi$
$954$		0			0
$955$		−	1116.00i		−	1.16859i
$956$		0			0
$957$	304.000	−	304.000i	0.317659	−	0.317659i
$958$		0			0
$959$	864.000			0.900938
$960$		0			0
$961$			313.000i			0.325702i
$962$		0			0
$963$	−168.000			−0.174455
$964$		0			0
$965$		−	54.0000i		−	0.0559585i
$966$		0			0
$967$	−1006.00	−	1006.00i	−1.04033	−	1.04033i	−0.999152	−	0.0411791i	$-0.986889\pi$
$967$	−1006.00	−	1006.00i	−1.04033	−	1.04033i	−0.0411791	−	0.999152i	$-0.513111\pi$
$968$		0			0
$969$	1840.00			1.89886
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$			720.000i			0.739979i
$974$		0			0
$975$	84.0000	+	84.0000i	0.0861538	+	0.0861538i
$976$		0			0
$977$	−921.000	+	921.000i	−0.942682	+	0.942682i	−0.998444	−	0.0557624i	$-0.982241\pi$
$977$	−921.000	+	921.000i	−0.942682	+	0.942682i	0.0557624	+	0.998444i	$0.482241\pi$
$978$		0			0
$979$	68.0000	+	68.0000i	0.0694586	+	0.0694586i
$980$		0			0
$981$	637.000	+	637.000i	0.649337	+	0.649337i
$982$		0			0
$983$			1480.00i			1.50560i	0.658252	+	0.752798i	$0.271296\pi$
$983$			1480.00i			1.50560i	−0.658252	+	0.752798i	$0.728704\pi$
$984$		0			0
$985$	102.000	+	102.000i	0.103553	+	0.103553i
$986$		0			0
$987$			704.000i			0.713273i
$988$		0			0
$989$		−	840.000i		−	0.849343i
$990$		0			0
$991$	−1038.00	+	1038.00i	−1.04743	+	1.04743i	−0.0486090	+	0.998818i	$0.515479\pi$
$991$	−1038.00	+	1038.00i	−1.04743	+	1.04743i	−0.998818	+	0.0486090i	$0.984521\pi$
$992$		0			0
$993$	−1304.00	+	1304.00i	−1.31319	+	1.31319i
$994$		0			0
$995$	−276.000			−0.277387
$996$		0			0
$997$	355.000	+	355.000i	0.356068	+	0.356068i	0.862361	−	0.506293i	$-0.168985\pi$
$997$	355.000	+	355.000i	0.356068	+	0.356068i	−0.506293	+	0.862361i	$0.668985\pi$
$998$		0			0
$999$	−296.000			−0.296296

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.3.k.c.561.1		2
4.3	odd	2		74.3.d.b.43.1	yes	2
12.11	even	2		666.3.i.a.487.1		2
37.31	odd	4	inner	592.3.k.c.401.1		2
148.31	even	4		74.3.d.b.31.1	✓	2
444.179	odd	4		666.3.i.a.253.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.3.d.b.31.1	✓	2	148.31	even	4
74.3.d.b.43.1	yes	2	4.3	odd	2
592.3.k.c.401.1		2	37.31	odd	4	inner
592.3.k.c.561.1		2	1.1	even	1	trivial
666.3.i.a.253.1		2	444.179	odd	4
666.3.i.a.487.1		2	12.11	even	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:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	592.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{3} +(3.00000 + 3.00000i) q^{5} +4.00000 q^{7} -7.00000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{3} +(3.00000 + 3.00000i) q^{5} +4.00000 q^{7} -7.00000 q^{9} +4.00000i q^{11} +(3.00000 + 3.00000i) q^{13} +(-12.0000 + 12.0000i) q^{15} +(23.0000 + 23.0000i) q^{17} +(-10.0000 - 10.0000i) q^{19} +16.0000i q^{21} +(10.0000 + 10.0000i) q^{23} -7.00000i q^{25} +8.00000i q^{27} +(-19.0000 + 19.0000i) q^{29} +(18.0000 - 18.0000i) q^{31} -16.0000 q^{33} +(12.0000 + 12.0000i) q^{35} +37.0000i q^{37} +(-12.0000 + 12.0000i) q^{39} -74.0000i q^{41} +(-42.0000 - 42.0000i) q^{43} +(-21.0000 - 21.0000i) q^{45} +44.0000 q^{47} -33.0000 q^{49} +(-92.0000 + 92.0000i) q^{51} -80.0000 q^{53} +(-12.0000 + 12.0000i) q^{55} +(40.0000 - 40.0000i) q^{57} +(54.0000 + 54.0000i) q^{59} +(-3.00000 + 3.00000i) q^{61} -28.0000 q^{63} +18.0000i q^{65} -12.0000i q^{67} +(-40.0000 + 40.0000i) q^{69} -124.000 q^{71} +10.0000i q^{73} +28.0000 q^{75} +16.0000i q^{77} +(-14.0000 - 14.0000i) q^{79} -95.0000 q^{81} +64.0000 q^{83} +138.000i q^{85} +(-76.0000 - 76.0000i) q^{87} +(17.0000 - 17.0000i) q^{89} +(12.0000 + 12.0000i) q^{91} +(72.0000 + 72.0000i) q^{93} -60.0000i q^{95} +(129.000 + 129.000i) q^{97} -28.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} + 8 q^{7} - 14 q^{9}+O(q^{10}) \) 2 * q + 6 * q^5 + 8 * q^7 - 14 * q^9	\( 2 q + 6 q^{5} + 8 q^{7} - 14 q^{9} + 6 q^{13} - 24 q^{15} + 46 q^{17} - 20 q^{19} + 20 q^{23} - 38 q^{29} + 36 q^{31} - 32 q^{33} + 24 q^{35} - 24 q^{39} - 84 q^{43} - 42 q^{45} + 88 q^{47} - 66 q^{49} - 184 q^{51} - 160 q^{53} - 24 q^{55} + 80 q^{57} + 108 q^{59} - 6 q^{61} - 56 q^{63} - 80 q^{69} - 248 q^{71} + 56 q^{75} - 28 q^{79} - 190 q^{81} + 128 q^{83} - 152 q^{87} + 34 q^{89} + 24 q^{91} + 144 q^{93} + 258 q^{97}+O(q^{100}) \) 2 * q + 6 * q^5 + 8 * q^7 - 14 * q^9 + 6 * q^13 - 24 * q^15 + 46 * q^17 - 20 * q^19 + 20 * q^23 - 38 * q^29 + 36 * q^31 - 32 * q^33 + 24 * q^35 - 24 * q^39 - 84 * q^43 - 42 * q^45 + 88 * q^47 - 66 * q^49 - 184 * q^51 - 160 * q^53 - 24 * q^55 + 80 * q^57 + 108 * q^59 - 6 * q^61 - 56 * q^63 - 80 * q^69 - 248 * q^71 + 56 * q^75 - 28 * q^79 - 190 * q^81 + 128 * q^83 - 152 * q^87 + 34 * q^89 + 24 * q^91 + 144 * q^93 + 258 * q^97

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