LMFDB - Embedded newform 847.1.bb.a.377.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,1,Mod(20,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(110))

chi = DirichletCharacter(H, H._module([55, 76]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("847.20");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	847.bb (of order $110$, degree $40$, 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:	$0.422708065700$
Analytic rank:	$0$
Dimension:	$40$
Coefficient field:	$\Q(\zeta_{55})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{40} - x^{39} + x^{35} - x^{34} + x^{30} - x^{28} + x^{25} - x^{23} + x^{20} - x^{17} + x^{15} + \cdots + 1 $ x^40 - x^39 + x^35 - x^34 + x^30 - x^28 + x^25 - x^23 + x^20 - x^17 + x^15 - x^12 + x^10 - x^6 + x^5 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{55}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{55} - \cdots)$

Embedding invariants

Embedding label			377.1
Root			$0.974012 + 0.226497i$ of defining polynomial
Character	$\chi$	$=$	847.377
Dual form			847.1.bb.a.510.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0556279 - 0.0129357i) q^{2} +(-0.894471 - 0.439782i) q^{4} +(-0.0285561 + 0.999592i) q^{7} +(0.0882815 + 0.0721874i) q^{8} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})$	\(q+(-0.0556279 - 0.0129357i) q^{2} +(-0.894471 - 0.439782i) q^{4} +(-0.0285561 + 0.999592i) q^{7} +(0.0882815 + 0.0721874i) q^{8} +(-0.809017 + 0.587785i) q^{9} +(0.0855750 + 0.996332i) q^{11} +(0.0145189 - 0.0552358i) q^{14} +(0.604679 + 0.784162i) q^{16} +(0.0526073 - 0.0222320i) q^{18} +(0.00812790 - 0.0565308i) q^{22} +(1.08316 + 0.318044i) q^{23} +(-0.736741 + 0.676175i) q^{25} +(0.465145 - 0.881548i) q^{28} +(0.534591 + 0.490643i) q^{29} +(-0.0708664 - 0.155176i) q^{32} +(0.982140 - 0.169966i) q^{36} +(-0.608982 - 0.105389i) q^{37} +(1.02742 - 0.660282i) q^{43} +(0.361624 - 0.928824i) q^{44} +(-0.0561397 - 0.0317035i) q^{46} +(-0.998369 - 0.0570888i) q^{49} +(0.0497301 - 0.0280839i) q^{50} +(-1.06344 + 1.37909i) q^{53} +(-0.0746790 + 0.0861841i) q^{56} +(-0.0233913 - 0.0342087i) q^{58} +(-0.564443 - 0.825472i) q^{63} +(-0.194714 - 0.960954i) q^{64} +(-0.260098 - 0.300169i) q^{67} +(0.812697 - 0.458951i) q^{71} +(-0.113852 - 0.00651029i) q^{72} +(0.0325131 + 0.0137402i) q^{74} +(-0.998369 + 0.0570888i) q^{77} +(-1.39160 + 0.0795747i) q^{79} +(0.309017 - 0.951057i) q^{81} +(-0.0656944 + 0.0234397i) q^{86} +(-0.0643679 + 0.0941351i) q^{88} +(-0.828984 - 0.760835i) q^{92} +(0.0547987 + 0.0160903i) q^{98} +(-0.654861 - 0.755750i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 2 q^{2} + 3 q^{4} + q^{7} - q^{8} - 10 q^{9}+O(q^{10}) $ 40 * q + 2 * q^2 + 3 * q^4 + q^7 - q^8 - 10 * q^9	$ 40 q + 2 q^{2} + 3 q^{4} + q^{7} - q^{8} - 10 q^{9} + q^{11} - 3 q^{14} - 3 q^{18} + 2 q^{22} - 9 q^{23} + q^{25} - 2 q^{28} + 2 q^{29} - 4 q^{32} + 3 q^{36} - 3 q^{37} + 2 q^{43} - 13 q^{44} - q^{46} + q^{49} + 2 q^{50} - 3 q^{53} - 7 q^{56} - 12 q^{58} + q^{63} - 9 q^{64} + 2 q^{67} - 3 q^{71} - q^{72} + 4 q^{74} + q^{77} - 3 q^{79} - 10 q^{81} - q^{86} - 12 q^{88} - 4 q^{92} - 9 q^{98} - 4 q^{99}+O(q^{100}) $ 40 * q + 2 * q^2 + 3 * q^4 + q^7 - q^8 - 10 * q^9 + q^11 - 3 * q^14 - 3 * q^18 + 2 * q^22 - 9 * q^23 + q^25 - 2 * q^28 + 2 * q^29 - 4 * q^32 + 3 * q^36 - 3 * q^37 + 2 * q^43 - 13 * q^44 - q^46 + q^49 + 2 * q^50 - 3 * q^53 - 7 * q^56 - 12 * q^58 + q^63 - 9 * q^64 + 2 * q^67 - 3 * q^71 - q^72 + 4 * q^74 + q^77 - 3 * q^79 - 10 * q^81 - q^86 - 12 * q^88 - 4 * q^92 - 9 * q^98 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$.

$n$	$122$	$365$
$\chi(n)$	$-1$	$e\left(\frac{4}{55}\right)$

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 $

$2$	−0.0556279	−	0.0129357i	−0.0556279	−	0.0129357i	0.198590	−	0.980083i	$-0.436364\pi$
$2$	−0.0556279	−	0.0129357i	−0.0556279	−	0.0129357i	−0.254218	+	0.967147i	$0.581818\pi$
$3$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$3$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$4$	−0.894471	−	0.439782i	−0.894471	−	0.439782i
$5$		0			0		−0.362808	−	0.931864i	$-0.618182\pi$
$5$		0			0		0.362808	+	0.931864i	$0.381818\pi$
$6$		0			0
$7$	−0.0285561	+	0.999592i	−0.0285561	+	0.999592i
$7$	−0.0285561	+	0.999592i	−0.0285561	+	0.999592i
$8$	0.0882815	+	0.0721874i	0.0882815	+	0.0721874i
$9$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$10$		0			0
$11$	0.0855750	+	0.996332i	0.0855750	+	0.996332i
$11$	0.0855750	+	0.996332i	0.0855750	+	0.996332i
$12$		0			0
$13$		0			0		−0.466667	−	0.884433i	$-0.654545\pi$
$13$		0			0		0.466667	+	0.884433i	$0.345455\pi$
$14$	0.0145189	−	0.0552358i	0.0145189	−	0.0552358i
$15$		0			0
$16$	0.604679	+	0.784162i	0.604679	+	0.784162i
$17$		0			0		0.198590	−	0.980083i	$-0.436364\pi$
$17$		0			0		−0.198590	+	0.980083i	$0.563636\pi$
$18$	0.0526073	−	0.0222320i	0.0526073	−	0.0222320i
$19$		0			0		−0.993482	−	0.113991i	$-0.963636\pi$
$19$		0			0		0.993482	+	0.113991i	$0.0363636\pi$
$20$		0			0
$21$		0			0
$22$	0.00812790	−	0.0565308i	0.00812790	−	0.0565308i
$23$	1.08316	+	0.318044i	1.08316	+	0.318044i	0.774142	−	0.633012i	$-0.218182\pi$
$23$	1.08316	+	0.318044i	1.08316	+	0.318044i	0.309017	+	0.951057i	$0.400000\pi$
$24$		0			0
$25$	−0.736741	+	0.676175i	−0.736741	+	0.676175i
$26$		0			0
$27$		0			0
$28$	0.465145	−	0.881548i	0.465145	−	0.881548i
$29$	0.534591	+	0.490643i	0.534591	+	0.490643i	0.897398	−	0.441221i	$-0.145455\pi$
$29$	0.534591	+	0.490643i	0.534591	+	0.490643i	−0.362808	+	0.931864i	$0.618182\pi$
$30$		0			0
$31$		0			0		−0.696938	−	0.717132i	$-0.745455\pi$
$31$		0			0		0.696938	+	0.717132i	$0.254545\pi$
$32$	−0.0708664	−	0.155176i	−0.0708664	−	0.155176i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.982140	−	0.169966i	0.982140	−	0.169966i
$37$	−0.608982	−	0.105389i	−0.608982	−	0.105389i	−0.142315	−	0.989821i	$-0.545455\pi$
$37$	−0.608982	−	0.105389i	−0.608982	−	0.105389i	−0.466667	+	0.884433i	$0.654545\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.516397	−	0.856349i	$-0.327273\pi$
$41$		0			0		−0.516397	+	0.856349i	$0.672727\pi$
$42$		0			0
$43$	1.02742	−	0.660282i	1.02742	−	0.660282i	0.0855750	−	0.996332i	$-0.472727\pi$
$43$	1.02742	−	0.660282i	1.02742	−	0.660282i	0.941844	+	0.336049i	$0.109091\pi$
$44$	0.361624	−	0.928824i	0.361624	−	0.928824i
$45$		0			0
$46$	−0.0561397	−	0.0317035i	−0.0561397	−	0.0317035i
$47$		0			0		−0.921124	−	0.389270i	$-0.872727\pi$
$47$		0			0		0.921124	+	0.389270i	$0.127273\pi$
$48$		0			0
$49$	−0.998369	−	0.0570888i	−0.998369	−	0.0570888i
$50$	0.0497301	−	0.0280839i	0.0497301	−	0.0280839i
$51$		0			0
$52$		0			0
$53$	−1.06344	+	1.37909i	−1.06344	+	1.37909i	−0.142315	+	0.989821i	$0.545455\pi$
$53$	−1.06344	+	1.37909i	−1.06344	+	1.37909i	−0.921124	+	0.389270i	$0.872727\pi$
$54$		0			0
$55$		0			0
$56$	−0.0746790	+	0.0861841i	−0.0746790	+	0.0861841i
$57$		0			0
$58$	−0.0233913	−	0.0342087i	−0.0233913	−	0.0342087i
$59$		0			0		−0.516397	−	0.856349i	$-0.672727\pi$
$59$		0			0		0.516397	+	0.856349i	$0.327273\pi$
$60$		0			0
$61$		0			0		0.974012	−	0.226497i	$-0.0727273\pi$
$61$		0			0		−0.974012	+	0.226497i	$0.927273\pi$
$62$		0			0
$63$	−0.564443	−	0.825472i	−0.564443	−	0.825472i
$64$	−0.194714	−	0.960954i	−0.194714	−	0.960954i
$65$		0			0
$66$		0			0
$67$	−0.260098	−	0.300169i	−0.260098	−	0.300169i	0.610648	−	0.791902i	$-0.290909\pi$
$67$	−0.260098	−	0.300169i	−0.260098	−	0.300169i	−0.870746	+	0.491733i	$0.836364\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.812697	−	0.458951i	0.812697	−	0.458951i	−0.0285561	−	0.999592i	$-0.509091\pi$
$71$	0.812697	−	0.458951i	0.812697	−	0.458951i	0.841254	+	0.540641i	$0.181818\pi$
$72$	−0.113852	−	0.00651029i	−0.113852	−	0.00651029i
$73$		0			0		0.564443	−	0.825472i	$-0.309091\pi$
$73$		0			0		−0.564443	+	0.825472i	$0.690909\pi$
$74$	0.0325131	+	0.0137402i	0.0325131	+	0.0137402i
$75$		0			0
$76$		0			0
$77$	−0.998369	+	0.0570888i	−0.998369	+	0.0570888i
$78$		0			0
$79$	−1.39160	+	0.0795747i	−1.39160	+	0.0795747i	−0.736741	−	0.676175i	$-0.763636\pi$
$79$	−1.39160	+	0.0795747i	−1.39160	+	0.0795747i	−0.654861	+	0.755750i	$0.727273\pi$
$80$		0			0
$81$	0.309017	−	0.951057i	0.309017	−	0.951057i
$82$		0			0
$83$		0			0		−0.941844	−	0.336049i	$-0.890909\pi$
$83$		0			0		0.941844	+	0.336049i	$0.109091\pi$
$84$		0			0
$85$		0			0
$86$	−0.0656944	+	0.0234397i	−0.0656944	+	0.0234397i
$87$		0			0
$88$	−0.0643679	+	0.0941351i	−0.0643679	+	0.0941351i
$89$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$89$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$90$		0			0
$91$		0			0
$92$	−0.828984	−	0.760835i	−0.828984	−	0.760835i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.362808	−	0.931864i	$-0.381818\pi$
$97$		0			0		−0.362808	+	0.931864i	$0.618182\pi$
$98$	0.0547987	+	0.0160903i	0.0547987	+	0.0160903i
$99$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$100$	0.956363	−	0.280814i	0.956363	−	0.280814i
$101$		0			0		0.0855750	−	0.996332i	$-0.472727\pi$
$101$		0			0		−0.0855750	+	0.996332i	$0.527273\pi$
$102$		0			0
$103$		0			0		0.921124	−	0.389270i	$-0.127273\pi$
$103$		0			0		−0.921124	+	0.389270i	$0.872727\pi$
$104$		0			0
$105$		0			0
$106$	0.0769963	−	0.0629596i	0.0769963	−	0.0629596i
$107$	0.374586	−	1.42507i	0.374586	−	1.42507i	−0.466667	−	0.884433i	$-0.654545\pi$
$107$	0.374586	−	1.42507i	0.374586	−	1.42507i	0.841254	−	0.540641i	$-0.181818\pi$
$108$		0			0
$109$	0.262179	−	1.82350i	0.262179	−	1.82350i	−0.254218	−	0.967147i	$-0.581818\pi$
$109$	0.262179	−	1.82350i	0.262179	−	1.82350i	0.516397	−	0.856349i	$-0.327273\pi$
$110$		0			0
$111$		0			0
$112$	−0.801109	+	0.582040i	−0.801109	+	0.582040i
$113$	1.50805	+	1.23312i	1.50805	+	1.23312i	0.897398	+	0.441221i	$0.145455\pi$
$113$	1.50805	+	1.23312i	1.50805	+	1.23312i	0.610648	+	0.791902i	$0.290909\pi$
$114$		0			0
$115$		0			0
$116$	−0.262400	−	0.673969i	−0.262400	−	0.673969i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.985354	+	0.170522i	−0.985354	+	0.170522i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	0.0207207	+	0.0532207i	0.0207207	+	0.0532207i
$127$	−0.0435095	−	0.506572i	−0.0435095	−	0.506572i	−0.985354	−	0.170522i	$-0.945455\pi$
$127$	−0.0435095	−	0.506572i	−0.0435095	−	0.506572i	0.941844	−	0.336049i	$-0.109091\pi$
$128$	−0.00647049	+	0.226497i	−0.00647049	+	0.226497i
$129$		0			0
$130$		0			0
$131$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$131$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$132$		0			0
$133$		0			0
$134$	0.0105858	+	0.0200623i	0.0105858	+	0.0200623i
$135$		0			0
$136$		0			0
$137$	1.09599	+	1.42130i	1.09599	+	1.42130i	0.897398	+	0.441221i	$0.145455\pi$
$137$	1.09599	+	1.42130i	1.09599	+	1.42130i	0.198590	+	0.980083i	$0.436364\pi$
$138$		0			0
$139$		0			0		0.921124	−	0.389270i	$-0.127273\pi$
$139$		0			0		−0.921124	+	0.389270i	$0.872727\pi$
$140$		0			0
$141$		0			0
$142$	−0.0511455	+	0.0150177i	−0.0511455	+	0.0150177i
$143$		0			0
$144$	−0.950114	−	0.278979i	−0.950114	−	0.278979i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.498369	+	0.362086i	0.498369	+	0.362086i
$149$	0.611204	−	1.15836i	0.611204	−	1.15836i	−0.362808	−	0.931864i	$-0.618182\pi$
$149$	0.611204	−	1.15836i	0.611204	−	1.15836i	0.974012	−	0.226497i	$-0.0727273\pi$
$150$		0			0
$151$	1.38943	−	0.683135i	1.38943	−	0.683135i	0.415415	−	0.909632i	$-0.363636\pi$
$151$	1.38943	−	0.683135i	1.38943	−	0.683135i	0.974012	+	0.226497i	$0.0727273\pi$
$152$		0			0
$153$		0			0
$154$	0.0562756	+	0.00973888i	0.0562756	+	0.00973888i
$155$		0			0
$156$		0			0
$157$		0			0		0.985354	−	0.170522i	$-0.0545455\pi$
$157$		0			0		−0.985354	+	0.170522i	$0.945455\pi$
$158$	0.0784412	+	0.0135748i	0.0784412	+	0.0135748i
$159$		0			0
$160$		0			0
$161$	−0.348845	+	1.07363i	−0.348845	+	1.07363i
$162$	−0.0294925	+	0.0489079i	−0.0294925	+	0.0489079i
$163$	1.47108	−	0.0841194i	1.47108	−	0.0841194i	0.696938	−	0.717132i	$-0.254545\pi$
$163$	1.47108	−	0.0841194i	1.47108	−	0.0841194i	0.774142	+	0.633012i	$0.218182\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.870746	−	0.491733i	$-0.836364\pi$
$167$		0			0		0.870746	+	0.491733i	$0.163636\pi$
$168$		0			0
$169$	−0.564443	+	0.825472i	−0.564443	+	0.825472i
$170$		0			0
$171$		0			0
$172$	−1.20938	+	0.138763i	−1.20938	+	0.138763i
$173$		0			0		−0.0285561	−	0.999592i	$-0.509091\pi$
$173$		0			0		0.0285561	+	0.999592i	$0.490909\pi$
$174$		0			0
$175$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$176$	−0.729540	+	0.669565i	−0.729540	+	0.669565i
$177$		0			0
$178$		0			0
$179$	0.286984	+	0.419700i	0.286984	+	0.419700i	0.941844	−	0.336049i	$-0.109091\pi$
$179$	0.286984	+	0.419700i	0.286984	+	0.419700i	−0.654861	+	0.755750i	$0.727273\pi$
$180$		0			0
$181$		0			0		0.974012	−	0.226497i	$-0.0727273\pi$
$181$		0			0		−0.974012	+	0.226497i	$0.927273\pi$
$182$		0			0
$183$		0			0
$184$	0.0726641	+	0.106268i	0.0726641	+	0.106268i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.38479	−	0.158890i	1.38479	−	0.158890i	0.610648	−	0.791902i	$-0.290909\pi$
$191$	1.38479	−	0.158890i	1.38479	−	0.158890i	0.774142	+	0.633012i	$0.218182\pi$
$192$		0			0
$193$	−1.94485	−	0.111210i	−1.94485	−	0.111210i	−0.959493	−	0.281733i	$-0.909091\pi$
$193$	−1.94485	−	0.111210i	−1.94485	−	0.111210i	−0.985354	+	0.170522i	$0.945455\pi$
$194$		0			0
$195$		0			0
$196$	0.867906	+	0.490129i	0.867906	+	0.490129i
$197$	1.50988	+	0.970340i	1.50988	+	0.970340i	0.993482	+	0.113991i	$0.0363636\pi$
$197$	1.50988	+	0.970340i	1.50988	+	0.970340i	0.516397	+	0.856349i	$0.327273\pi$
$198$	0.0266524	+	0.0505118i	0.0266524	+	0.0505118i
$199$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$199$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$200$	−0.113852	+	0.00651029i	−0.113852	+	0.00651029i
$201$		0			0
$202$		0			0
$203$	−0.505709	+	0.520362i	−0.505709	+	0.520362i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.06324	+	0.379362i	−1.06324	+	0.379362i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.37346	−	1.41326i	−1.37346	−	1.41326i	−0.809017	−	0.587785i	$-0.800000\pi$
$211$	−1.37346	−	1.41326i	−1.37346	−	1.41326i	−0.564443	−	0.825472i	$-0.690909\pi$
$212$	1.55771	−	0.765877i	1.55771	−	0.765877i
$213$		0			0
$214$	−0.0392718	+	0.0744283i	−0.0392718	+	0.0744283i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.0381727	+	0.0980457i	−0.0381727	+	0.0980457i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.993482	−	0.113991i	$-0.963636\pi$
$223$		0			0		0.993482	+	0.113991i	$0.0363636\pi$
$224$	0.157136	−	0.0664063i	0.157136	−	0.0664063i
$225$	0.198590	−	0.980083i	0.198590	−	0.980083i
$226$	−0.0679381	−	0.0881037i	−0.0679381	−	0.0881037i
$227$		0			0		0.774142	−	0.633012i	$-0.218182\pi$
$227$		0			0		−0.774142	+	0.633012i	$0.781818\pi$
$228$		0			0
$229$		0			0		−0.466667	−	0.884433i	$-0.654545\pi$
$229$		0			0		0.466667	+	0.884433i	$0.345455\pi$
$230$		0			0
$231$		0			0
$232$	0.0117762	+	0.0819054i	0.0117762	+	0.0819054i
$233$	−0.672156	+	0.488350i	−0.672156	+	0.488350i	−0.870746	−	0.491733i	$-0.836364\pi$
$233$	−0.672156	+	0.488350i	−0.672156	+	0.488350i	0.198590	+	0.980083i	$0.436364\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.122736	+	0.377742i	0.122736	+	0.377742i	0.993482	−	0.113991i	$-0.0363636\pi$
$239$	0.122736	+	0.377742i	0.122736	+	0.377742i	−0.870746	+	0.491733i	$0.836364\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	0.0570190	+	0.00326046i	0.0570190	+	0.00326046i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$251$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$252$	0.141851	+	0.986593i	0.141851	+	0.986593i
$253$	−0.224186	+	1.10640i	−0.224186	+	1.10640i
$254$	−0.00413252	+	0.0287423i	−0.00413252	+	0.0287423i
$255$		0			0
$256$	−0.245967	+	0.935755i	−0.245967	+	0.935755i
$257$		0			0		0.774142	−	0.633012i	$-0.218182\pi$
$257$		0			0		−0.774142	+	0.633012i	$0.781818\pi$
$258$		0			0
$259$	0.122736	−	0.605724i	0.122736	−	0.605724i
$260$		0			0
$261$	−0.720886	−	0.0827139i	−0.720886	−	0.0827139i
$262$		0			0
$263$	0.895528	−	0.262951i	0.895528	−	0.262951i	0.198590	−	0.980083i	$-0.436364\pi$
$263$	0.895528	−	0.262951i	0.895528	−	0.262951i	0.696938	+	0.717132i	$0.254545\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.100641	+	0.382879i	0.100641	+	0.382879i
$269$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$269$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$270$		0			0
$271$		0			0		−0.736741	−	0.676175i	$-0.763636\pi$
$271$		0			0		0.736741	+	0.676175i	$0.236364\pi$
$272$		0			0
$273$		0			0
$274$	−0.0425820	−	0.0932415i	−0.0425820	−	0.0932415i
$275$	−0.736741	−	0.676175i	−0.736741	−	0.676175i
$276$		0			0
$277$	1.87141	−	0.667718i	1.87141	−	0.667718i	0.897398	−	0.441221i	$-0.145455\pi$
$277$	1.87141	−	0.667718i	1.87141	−	0.667718i	0.974012	−	0.226497i	$-0.0727273\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.21371	+	1.24888i	−1.21371	+	1.24888i	−0.254218	+	0.967147i	$0.581818\pi$
$281$	−1.21371	+	1.24888i	−1.21371	+	1.24888i	−0.959493	+	0.281733i	$0.909091\pi$
$282$		0			0
$283$		0			0		0.516397	−	0.856349i	$-0.327273\pi$
$283$		0			0		−0.516397	+	0.856349i	$0.672727\pi$
$284$	−0.928773	+	0.0531092i	−0.928773	+	0.0531092i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.148542	+	0.0838856i	0.148542	+	0.0838856i
$289$	−0.921124	−	0.389270i	−0.921124	−	0.389270i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.993482	−	0.113991i	$-0.0363636\pi$
$293$		0			0		−0.993482	+	0.113991i	$0.963636\pi$
$294$		0			0
$295$		0			0
$296$	−0.0461541	−	0.0532647i	−0.0461541	−	0.0532647i
$297$		0			0
$298$	−0.0489842	+	0.0565308i	−0.0489842	+	0.0565308i
$299$		0			0
$300$		0			0
$301$	0.630674	+	1.04586i	0.630674	+	1.04586i
$302$	−0.0861277	+	0.0200281i	−0.0861277	+	0.0200281i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$307$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$308$	0.918119	+	0.388000i	0.918119	+	0.388000i
$309$		0			0
$310$		0			0
$311$		0			0		−0.0285561	−	0.999592i	$-0.509091\pi$
$311$		0			0		0.0285561	+	0.999592i	$0.490909\pi$
$312$		0			0
$313$		0			0		0.870746	−	0.491733i	$-0.163636\pi$
$313$		0			0		−0.870746	+	0.491733i	$0.836364\pi$
$314$		0			0
$315$		0			0
$316$	1.27974	+	0.540824i	1.27974	+	0.540824i
$317$	1.40890	+	0.795641i	1.40890	+	0.795641i	0.993482	−	0.113991i	$-0.0363636\pi$
$317$	1.40890	+	0.795641i	1.40890	+	0.795641i	0.415415	+	0.909632i	$0.363636\pi$
$318$		0			0
$319$	−0.443096	+	0.574616i	−0.443096	+	0.574616i
$320$		0			0
$321$		0			0
$322$	0.0332937	−	0.0552115i	0.0332937	−	0.0552115i
$323$		0			0
$324$	−0.694664	+	0.714793i	−0.694664	+	0.714793i
$325$		0			0
$326$	−0.0829211	−	0.0143501i	−0.0829211	−	0.0143501i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.429039	+	0.939463i	0.429039	+	0.939463i	0.993482	+	0.113991i	$0.0363636\pi$
$331$	0.429039	+	0.939463i	0.429039	+	0.939463i	−0.564443	+	0.825472i	$0.690909\pi$
$332$		0			0
$333$	0.554623	−	0.272690i	0.554623	−	0.272690i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.262555	−	0.998864i	−0.262555	−	0.998864i	−0.959493	−	0.281733i	$-0.909091\pi$
$337$	−0.262555	−	0.998864i	−0.262555	−	0.998864i	0.696938	−	0.717132i	$-0.254545\pi$
$338$	0.0420768	−	0.0386178i	0.0420768	−	0.0386178i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.0855750	−	0.996332i	0.0855750	−	0.996332i
$344$	0.138366	+	0.0158760i	0.138366	+	0.0158760i
$345$		0			0
$346$		0			0
$347$	1.15027	+	1.49170i	1.15027	+	1.49170i	0.841254	+	0.540641i	$0.181818\pi$
$347$	1.15027	+	1.49170i	1.15027	+	1.49170i	0.309017	+	0.951057i	$0.400000\pi$
$348$		0			0
$349$		0			0		0.254218	−	0.967147i	$-0.418182\pi$
$349$		0			0		−0.254218	+	0.967147i	$0.581818\pi$
$350$	0.0266524	+	0.0505118i	0.0266524	+	0.0505118i
$351$		0			0
$352$	0.148542	−	0.0838856i	0.148542	−	0.0838856i
$353$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$353$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.0105352	−	0.0270594i	−0.0105352	−	0.0270594i
$359$	0.926829	+	0.455691i	0.926829	+	0.455691i	0.841254	−	0.540641i	$-0.181818\pi$
$359$	0.926829	+	0.455691i	0.926829	+	0.455691i	0.0855750	+	0.996332i	$0.472727\pi$
$360$		0			0
$361$	0.974012	+	0.226497i	0.974012	+	0.226497i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.897398	−	0.441221i	$-0.854545\pi$
$367$		0			0		0.897398	+	0.441221i	$0.145455\pi$
$368$	0.405565	+	1.04169i	0.405565	+	1.04169i
$369$		0			0
$370$		0			0
$371$	−1.34816	−	1.10239i	−1.34816	−	1.10239i
$372$		0			0
$373$	0.273100	+	1.89945i	0.273100	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$373$	0.273100	+	1.89945i	0.273100	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.945456	+	1.22609i	0.945456	+	1.22609i	0.974012	+	0.226497i	$0.0727273\pi$
$379$	0.945456	+	1.22609i	0.945456	+	1.22609i	−0.0285561	+	0.999592i	$0.509091\pi$
$380$		0			0
$381$		0			0
$382$	−0.0790882	−	0.00907453i	−0.0790882	−	0.00907453i
$383$		0			0		0.0855750	−	0.996332i	$-0.472727\pi$
$383$		0			0		−0.0855750	+	0.996332i	$0.527273\pi$
$384$		0			0
$385$		0			0
$386$	0.106749	+	0.0313444i	0.106749	+	0.0313444i
$387$	−0.443096	+	1.13808i	−0.443096	+	1.13808i
$388$		0			0
$389$	−0.393602	−	1.49742i	−0.393602	−	1.49742i	−0.809017	−	0.587785i	$-0.800000\pi$
$389$	−0.393602	−	1.49742i	−0.393602	−	1.49742i	0.415415	−	0.909632i	$-0.363636\pi$
$390$		0			0
$391$		0			0
$392$	−0.0840164	−	0.0771096i	−0.0840164	−	0.0771096i
$393$		0			0
$394$	−0.0714393	−	0.0735093i	−0.0714393	−	0.0735093i
$395$		0			0
$396$	0.253389	+	0.963992i	0.253389	+	0.963992i
$397$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$397$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$398$		0			0
$399$		0			0
$400$	−0.975722	−	0.168855i	−0.975722	−	0.168855i
$401$	−1.85610	−	0.662255i	−1.85610	−	0.662255i	−0.985354	−	0.170522i	$-0.945455\pi$
$401$	−1.85610	−	0.662255i	−1.85610	−	0.662255i	−0.870746	−	0.491733i	$-0.836364\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	0.0348627	−	0.0224049i	0.0348627	−	0.0224049i
$407$	0.0528883	−	0.615767i	0.0528883	−	0.615767i
$408$		0			0
$409$		0			0		−0.870746	−	0.491733i	$-0.836364\pi$
$409$		0			0		0.870746	+	0.491733i	$0.163636\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.0640528	−	0.00734938i	0.0640528	−	0.00734938i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$419$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$420$		0			0
$421$	−0.582954	−	0.852543i	−0.582954	−	0.852543i	0.415415	−	0.909632i	$-0.363636\pi$
$421$	−0.582954	−	0.852543i	−0.582954	−	0.852543i	−0.998369	+	0.0570888i	$0.981818\pi$
$422$	0.0581212	+	0.0963832i	0.0581212	+	0.0963832i
$423$		0			0
$424$	−0.193435	+	0.0449814i	−0.193435	+	0.0449814i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.961778	+	1.10995i	−0.961778	+	1.10995i
$429$		0			0
$430$		0			0
$431$	1.21334	−	1.57348i	1.21334	−	1.57348i	0.516397	−	0.856349i	$-0.327273\pi$
$431$	1.21334	−	1.57348i	1.21334	−	1.57348i	0.696938	−	0.717132i	$-0.254545\pi$
$432$		0			0
$433$		0			0		0.993482	−	0.113991i	$-0.0363636\pi$
$433$		0			0		−0.993482	+	0.113991i	$0.963636\pi$
$434$		0			0
$435$		0			0
$436$	−1.03645	+	1.51576i	−1.03645	+	1.51576i
$437$		0			0
$438$		0			0
$439$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$439$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$440$		0			0
$441$	0.841254	−	0.540641i	0.841254	−	0.540641i
$442$		0			0
$443$	0.972732	−	1.61310i	0.972732	−	1.61310i	0.198590	−	0.980083i	$-0.436364\pi$
$443$	0.972732	−	1.61310i	0.972732	−	1.61310i	0.774142	−	0.633012i	$-0.218182\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.966122	−	0.167194i	0.966122	−	0.167194i
$449$	−1.73511	+	0.619086i	−1.73511	+	0.619086i	−0.998369	−	0.0570888i	$-0.981818\pi$
$449$	−1.73511	+	0.619086i	−1.73511	+	0.619086i	−0.736741	+	0.676175i	$0.763636\pi$
$450$	−0.0237252	+	0.0519510i	−0.0237252	+	0.0519510i
$451$		0			0
$452$	−0.806599	−	1.76620i	−0.806599	−	1.76620i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$457$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$461$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$462$		0			0
$463$	−0.164217	+	0.0482185i	−0.164217	+	0.0482185i	−0.362808	−	0.931864i	$-0.618182\pi$
$463$	−0.164217	+	0.0482185i	−0.164217	+	0.0482185i	0.198590	+	0.980083i	$0.436364\pi$
$464$	−0.0614876	+	0.715887i	−0.0614876	+	0.715887i
$465$		0			0
$466$	0.0437077	−	0.0184710i	0.0437077	−	0.0184710i
$467$		0			0		0.198590	−	0.980083i	$-0.436364\pi$
$467$		0			0		−0.198590	+	0.980083i	$0.563636\pi$
$468$		0			0
$469$	0.307474	−	0.251420i	0.307474	−	0.251420i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.745782	+	0.967147i	0.745782	+	0.967147i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.0497301	−	1.74078i	0.0497301	−	1.74078i
$478$	−0.00194117	−	0.0226006i	−0.00194117	−	0.0226006i
$479$		0			0		−0.362808	−	0.931864i	$-0.618182\pi$
$479$		0			0		0.362808	+	0.931864i	$0.381818\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.956363	+	0.280814i	0.956363	+	0.280814i
$485$		0			0
$486$		0			0
$487$	0.377401	+	1.16152i	0.377401	+	1.16152i	0.941844	+	0.336049i	$0.109091\pi$
$487$	0.377401	+	1.16152i	0.377401	+	1.16152i	−0.564443	+	0.825472i	$0.690909\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.0547987	−	1.91820i	0.0547987	−	1.91820i	−0.254218	−	0.967147i	$-0.581818\pi$
$491$	0.0547987	−	1.91820i	0.0547987	−	1.91820i	0.309017	−	0.951057i	$-0.400000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.435557	+	0.825472i	0.435557	+	0.825472i
$498$		0			0
$499$	−1.25259	+	1.02424i	−1.25259	+	1.02424i	−0.254218	+	0.967147i	$0.581818\pi$
$499$	−1.25259	+	1.02424i	−1.25259	+	1.02424i	−0.998369	+	0.0570888i	$0.981818\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.993482	−	0.113991i	$-0.963636\pi$
$503$		0			0		0.993482	+	0.113991i	$0.0363636\pi$
$504$	0.00975880	−	0.113620i	0.00975880	−	0.113620i
$505$		0			0
$506$	0.0267831	−	0.0586468i	0.0267831	−	0.0586468i
$507$		0			0
$508$	−0.183863	+	0.472248i	−0.183863	+	0.472248i
$509$		0			0		0.736741	−	0.676175i	$-0.236364\pi$
$509$		0			0		−0.736741	+	0.676175i	$0.763636\pi$
$510$		0			0
$511$		0			0
$512$	0.131529	−	0.249275i	0.131529	−	0.249275i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−0.0146630	+	0.0321075i	−0.0146630	+	0.0321075i
$519$		0			0
$520$		0			0
$521$		0			0		−0.985354	−	0.170522i	$-0.945455\pi$
$521$		0			0		0.985354	+	0.170522i	$0.0545455\pi$
$522$	0.0390314	+	0.0139264i	0.0390314	+	0.0139264i
$523$		0			0		0.696938	−	0.717132i	$-0.254545\pi$
$523$		0			0		−0.696938	+	0.717132i	$0.745455\pi$
$524$		0			0
$525$		0			0
$526$	−0.0532178	+	0.00304310i	−0.0532178	+	0.00304310i
$527$		0			0
$528$		0			0
$529$	0.230827	+	0.148344i	0.230827	+	0.148344i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.00129341	−	0.0452752i	−0.00129341	−	0.0452752i
$537$		0			0
$538$		0			0
$539$	−0.0285561	−	0.999592i	−0.0285561	−	0.999592i
$540$		0			0
$541$	−0.391364	−	1.93146i	−0.391364	−	1.93146i	−0.362808	−	0.931864i	$-0.618182\pi$
$541$	−0.391364	−	1.93146i	−0.391364	−	1.93146i	−0.0285561	−	0.999592i	$-0.509091\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.468956	−	0.685827i	−0.468956	−	0.685827i	0.516397	−	0.856349i	$-0.327273\pi$
$547$	−0.468956	−	0.685827i	−0.468956	−	0.685827i	−0.985354	+	0.170522i	$0.945455\pi$
$548$	−0.355267	−	1.75331i	−0.355267	−	1.75331i
$549$		0			0
$550$	0.0322365	+	0.0471444i	0.0322365	+	0.0471444i
$551$		0			0
$552$		0			0
$553$	−0.0398036	−	1.39331i	−0.0398036	−	1.39331i
$554$	−0.112740	+	0.0129357i	−0.112740	+	0.0129357i
$555$		0			0
$556$		0			0
$557$	0.913288	−	1.33564i	0.913288	−	1.33564i	−0.0285561	−	0.999592i	$-0.509091\pi$
$557$	0.913288	−	1.33564i	0.913288	−	1.33564i	0.941844	−	0.336049i	$-0.109091\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.0836713	−	0.0537723i	0.0836713	−	0.0537723i
$563$		0			0		0.998369	−	0.0570888i	$-0.0181818\pi$
$563$		0			0		−0.998369	+	0.0570888i	$0.981818\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.941844	+	0.336049i	0.941844	+	0.336049i
$568$	0.104877	+	0.0181496i	0.104877	+	0.0181496i
$569$	−1.65786	+	0.286905i	−1.65786	+	0.286905i	−0.921124	−	0.389270i	$-0.872727\pi$
$569$	−1.65786	+	0.286905i	−1.65786	+	0.286905i	−0.736741	+	0.676175i	$0.763636\pi$
$570$		0			0
$571$	0.745586	−	1.63260i	0.745586	−	1.63260i	−0.0285561	−	0.999592i	$-0.509091\pi$
$571$	0.745586	−	1.63260i	0.745586	−	1.63260i	0.774142	−	0.633012i	$-0.218182\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.01306	+	0.498089i	−1.01306	+	0.498089i
$576$	0.722362	+	0.662978i	0.722362	+	0.662978i
$577$		0			0		0.466667	−	0.884433i	$-0.345455\pi$
$577$		0			0		−0.466667	+	0.884433i	$0.654545\pi$
$578$	0.0462047	+	0.0335697i	0.0462047	+	0.0335697i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.46504	−	0.941522i	−1.46504	−	0.941522i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.921124	−	0.389270i	$-0.127273\pi$
$587$		0			0		−0.921124	+	0.389270i	$0.872727\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.285597	−	0.541267i	−0.285597	−	0.541267i
$593$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$593$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$594$		0			0
$595$		0			0
$596$	−1.05613	+	0.767324i	−1.05613	+	0.767324i
$597$		0			0
$598$		0			0
$599$	−0.157650	−	1.83549i	−0.157650	−	1.83549i	−0.466667	−	0.884433i	$-0.654545\pi$
$599$	−0.157650	−	1.83549i	−0.157650	−	1.83549i	0.309017	−	0.951057i	$-0.400000\pi$
$600$		0			0
$601$		0			0		−0.897398	−	0.441221i	$-0.854545\pi$
$601$		0			0		0.897398	+	0.441221i	$0.145455\pi$
$602$	−0.0215542	−	0.0663369i	−0.0215542	−	0.0663369i
$603$	0.386859	+	0.0899602i	0.386859	+	0.0899602i
$604$	−1.54323			−1.54323
$605$		0			0
$606$		0			0
$607$		0			0		−0.974012	−	0.226497i	$-0.927273\pi$
$607$		0			0		0.974012	+	0.226497i	$0.0727273\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.45824	+	1.19240i	1.45824	+	1.19240i	0.941844	+	0.336049i	$0.109091\pi$
$613$	1.45824	+	1.19240i	1.45824	+	1.19240i	0.516397	+	0.856349i	$0.327273\pi$
$614$		0			0
$615$		0			0
$616$	−0.0922586	−	0.0670298i	−0.0922586	−	0.0670298i
$617$	−0.0243572	+	0.169408i	−0.0243572	+	0.169408i	−0.998369	−	0.0570888i	$-0.981818\pi$
$617$	−0.0243572	+	0.169408i	−0.0243572	+	0.169408i	0.974012	+	0.226497i	$0.0727273\pi$
$618$		0			0
$619$		0			0		0.254218	−	0.967147i	$-0.418182\pi$
$619$		0			0		−0.254218	+	0.967147i	$0.581818\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.0855750	−	0.996332i	0.0855750	−	0.996332i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.500990	+	1.90596i	0.500990	+	1.90596i	0.415415	+	0.909632i	$0.363636\pi$
$631$	0.500990	+	1.90596i	0.500990	+	1.90596i	0.0855750	+	0.996332i	$0.472727\pi$
$632$	−0.128597	−	0.0934312i	−0.128597	−	0.0934312i
$633$		0			0
$634$	−0.0680817	−	0.0624849i	−0.0680817	−	0.0624849i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	0.0320815	−	0.0262329i	0.0320815	−	0.0262329i
$639$	−0.387721	+	0.848991i	−0.387721	+	0.848991i
$640$		0			0
$641$	−1.85610	+	0.321211i	−1.85610	+	0.321211i	−0.985354	−	0.170522i	$-0.945455\pi$
$641$	−1.85610	+	0.321211i	−1.85610	+	0.321211i	−0.870746	+	0.491733i	$0.836364\pi$
$642$		0			0
$643$		0			0		−0.941844	−	0.336049i	$-0.890909\pi$
$643$		0			0		0.941844	+	0.336049i	$0.109091\pi$
$644$	0.784197	−	0.806920i	0.784197	−	0.806920i
$645$		0			0
$646$		0			0
$647$		0			0		0.998369	−	0.0570888i	$-0.0181818\pi$
$647$		0			0		−0.998369	+	0.0570888i	$0.981818\pi$
$648$	0.0959348	−	0.0616536i	0.0959348	−	0.0616536i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.35283	−	0.571712i	−1.35283	−	0.571712i
$653$	0.286984	−	0.419700i	0.286984	−	0.419700i	−0.654861	−	0.755750i	$-0.727273\pi$
$653$	0.286984	−	0.419700i	0.286984	−	0.419700i	0.941844	+	0.336049i	$0.109091\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.959493	−	0.281733i	$-0.909091\pi$
$659$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.142315	−	0.989821i	$-0.545455\pi$
$660$		0			0
$661$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$661$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$662$	−0.0117139	−	0.0578103i	−0.0117139	−	0.0578103i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.0343799	+	0.00799471i	−0.0343799	+	0.00799471i
$667$	0.423000	+	0.701468i	0.423000	+	0.701468i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.15027	−	1.49170i	1.15027	−	1.49170i	0.309017	−	0.951057i	$-0.400000\pi$
$673$	1.15027	−	1.49170i	1.15027	−	1.49170i	0.841254	−	0.540641i	$-0.181818\pi$
$674$	0.00168438	+	0.0589610i	0.00168438	+	0.0589610i
$675$		0			0
$676$	0.867906	−	0.490129i	0.867906	−	0.490129i
$677$		0			0		−0.998369	−	0.0570888i	$-0.981818\pi$
$677$		0			0		0.998369	+	0.0570888i	$0.0181818\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.65786	+	1.06545i	−1.65786	+	1.06545i	−0.736741	+	0.676175i	$0.763636\pi$
$683$	−1.65786	+	1.06545i	−1.65786	+	1.06545i	−0.921124	+	0.389270i	$0.872727\pi$
$684$		0			0
$685$		0			0
$686$	−0.0176486	+	0.0543168i	−0.0176486	+	0.0543168i
$687$		0			0
$688$	1.13903	+	0.406404i	1.13903	+	0.406404i
$689$		0			0
$690$		0			0
$691$		0			0		0.941844	−	0.336049i	$-0.109091\pi$
$691$		0			0		−0.941844	+	0.336049i	$0.890909\pi$
$692$		0			0
$693$	0.774142	−	0.633012i	0.774142	−	0.633012i
$694$	−0.0446909	−	0.0978595i	−0.0446909	−	0.0978595i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.253389	+	0.963992i	0.253389	+	0.963992i
$701$	−1.14068	+	1.04691i	−1.14068	+	1.04691i	−0.142315	+	0.989821i	$0.545455\pi$
$701$	−1.14068	+	1.04691i	−1.14068	+	1.04691i	−0.998369	+	0.0570888i	$0.981818\pi$
$702$		0			0
$703$		0			0
$704$	0.940766	−	0.276234i	0.940766	−	0.276234i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.374083	−	1.84617i	0.374083	−	1.84617i	−0.142315	−	0.989821i	$-0.545455\pi$
$709$	0.374083	−	1.84617i	0.374083	−	1.84617i	0.516397	−	0.856349i	$-0.327273\pi$
$710$		0			0
$711$	1.07906	−	0.882340i	1.07906	−	0.882340i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.0721221	−	0.501620i	−0.0721221	−	0.501620i
$717$		0			0
$718$	−0.0456628	−	0.0373383i	−0.0456628	−	0.0373383i
$719$		0			0		0.0285561	−	0.999592i	$-0.490909\pi$
$719$		0			0		−0.0285561	+	0.999592i	$0.509091\pi$
$720$		0			0
$721$		0			0
$722$	−0.0512523	−	0.0251991i	−0.0512523	−	0.0251991i
$723$		0			0
$724$		0			0
$725$	−0.725615			−0.725615
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	0.309017	+	0.951057i	0.309017	+	0.951057i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.0285561	−	0.999592i	$-0.490909\pi$
$733$		0			0		−0.0285561	+	0.999592i	$0.509091\pi$
$734$		0			0
$735$		0			0
$736$	−0.0274068	−	0.190619i	−0.0274068	−	0.190619i
$737$	0.276810	−	0.284831i	0.276810	−	0.284831i
$738$		0			0
$739$	0.755084	+	1.43104i	0.755084	+	1.43104i	0.897398	+	0.441221i	$0.145455\pi$
$739$	0.755084	+	1.43104i	0.755084	+	1.43104i	−0.142315	+	0.989821i	$0.545455\pi$
$740$		0			0
$741$		0			0
$742$	0.0607352	+	0.0787628i	0.0607352	+	0.0787628i
$743$	−0.321326	+	1.58581i	−0.321326	+	1.58581i	0.415415	+	0.909632i	$0.363636\pi$
$743$	−0.321326	+	1.58581i	−0.321326	+	1.58581i	−0.736741	+	0.676175i	$0.763636\pi$
$744$		0			0
$745$		0			0
$746$	0.00937879	−	0.109195i	0.00937879	−	0.109195i
$747$		0			0
$748$		0			0
$749$	1.41380	+	0.415128i	1.41380	+	0.415128i
$750$		0			0
$751$	−1.32230	+	1.21360i	−1.32230	+	1.21360i	−0.362808	+	0.931864i	$0.618182\pi$
$751$	−1.32230	+	1.21360i	−1.32230	+	1.21360i	−0.959493	+	0.281733i	$0.909091\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.119281	+	0.122737i	0.119281	+	0.122737i	0.774142	−	0.633012i	$-0.218182\pi$
$757$	0.119281	+	0.122737i	0.119281	+	0.122737i	−0.654861	+	0.755750i	$0.727273\pi$
$758$	−0.0367334	−	0.0804349i	−0.0367334	−	0.0804349i
$759$		0			0
$760$		0			0
$761$		0			0		0.941844	−	0.336049i	$-0.109091\pi$
$761$		0			0		−0.941844	+	0.336049i	$0.890909\pi$
$762$		0			0
$763$	1.81527	+	0.314144i	1.81527	+	0.314144i
$764$	−1.30853	−	0.466883i	−1.30853	−	0.466883i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$769$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$770$		0			0
$771$		0			0
$772$	1.69070	+	0.954783i	1.69070	+	0.954783i
$773$		0			0		−0.921124	−	0.389270i	$-0.872727\pi$
$773$		0			0		0.921124	+	0.389270i	$0.127273\pi$
$774$	0.0393703	−	0.0575773i	0.0393703	−	0.0575773i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	0.00252509	+	0.0883896i	0.00252509	+	0.0883896i
$779$		0			0
$780$		0			0
$781$	0.526814	+	0.770442i	0.526814	+	0.770442i
$782$		0			0
$783$		0			0
$784$	−0.558926	−	0.817403i	−0.558926	−	0.817403i
$785$		0			0
$786$		0			0
$787$		0			0		0.974012	−	0.226497i	$-0.0727273\pi$
$787$		0			0		−0.974012	+	0.226497i	$0.927273\pi$
$788$	−0.923805	−	1.53196i	−0.923805	−	1.53196i
$789$		0			0
$790$		0			0
$791$	−1.27568	+	1.47222i	−1.27568	+	1.47222i
$792$	−0.00325647	−	0.113991i	−0.00325647	−	0.113991i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.870746	−	0.491733i	$-0.163636\pi$
$797$		0			0		−0.870746	+	0.491733i	$0.836364\pi$
$798$		0			0
$799$		0			0
$800$	0.157136	+	0.0664063i	0.157136	+	0.0664063i
$801$		0			0
$802$	0.0946841	+	0.0608498i	0.0946841	+	0.0608498i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.478868	−	0.170860i	−0.478868	−	0.170860i	0.0855750	−	0.996332i	$-0.472727\pi$
$809$	−0.478868	−	0.170860i	−0.478868	−	0.170860i	−0.564443	+	0.825472i	$0.690909\pi$
$810$		0			0
$811$		0			0		0.985354	−	0.170522i	$-0.0545455\pi$
$811$		0			0		−0.985354	+	0.170522i	$0.945455\pi$
$812$	0.681188	−	0.243047i	0.681188	−	0.243047i
$813$		0			0
$814$	−0.0109074	+	0.0335697i	−0.0109074	+	0.0335697i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.0723581	+	0.275279i	0.0723581	+	0.275279i	0.993482	−	0.113991i	$-0.0363636\pi$
$821$	0.0723581	+	0.275279i	0.0723581	+	0.275279i	−0.921124	+	0.389270i	$0.872727\pi$
$822$		0			0
$823$	0.587035	−	1.50779i	0.587035	−	1.50779i	−0.254218	−	0.967147i	$-0.581818\pi$
$823$	0.587035	−	1.50779i	0.587035	−	1.50779i	0.841254	−	0.540641i	$-0.181818\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.112079	+	1.30492i	−0.112079	+	1.30492i	0.696938	+	0.717132i	$0.254545\pi$
$827$	−0.112079	+	1.30492i	−0.112079	+	1.30492i	−0.809017	+	0.587785i	$0.800000\pi$
$828$	1.11787	+	0.128264i	1.11787	+	0.128264i
$829$		0			0		0.921124	−	0.389270i	$-0.127273\pi$
$829$		0			0		−0.921124	+	0.389270i	$0.872727\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.774142	−	0.633012i	$-0.781818\pi$
$839$		0			0		0.774142	+	0.633012i	$0.218182\pi$
$840$		0			0
$841$	−0.0405182	−	0.471745i	−0.0405182	−	0.471745i
$842$	0.0214002	+	0.0549661i	0.0214002	+	0.0549661i
$843$		0			0
$844$	0.606996	+	1.86814i	0.606996	+	1.86814i
$845$		0			0
$846$		0			0
$847$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$848$	−1.72447			−1.72447
$849$		0			0
$850$		0			0
$851$	−0.626106	−	0.307836i	−0.626106	−	0.307836i
$852$		0			0
$853$		0			0		−0.0855750	−	0.996332i	$-0.527273\pi$
$853$		0			0		0.0855750	+	0.996332i	$0.472727\pi$
$854$		0			0
$855$		0			0
$856$	0.135941	−	0.0987672i	0.135941	−	0.0987672i
$857$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$857$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$858$		0			0
$859$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$859$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$860$		0			0
$861$		0			0
$862$	−0.0878493	+	0.0718340i	−0.0878493	+	0.0718340i
$863$	−0.173809	−	0.225399i	−0.173809	−	0.225399i	0.696938	−	0.717132i	$-0.254545\pi$
$863$	−0.173809	−	0.225399i	−0.173809	−	0.225399i	−0.870746	+	0.491733i	$0.836364\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.198369	−	1.37969i	−0.198369	−	1.37969i
$870$		0			0
$871$		0			0
$872$	0.154779	−	0.142055i	0.154779	−	0.142055i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.837573	+	0.411807i	−0.837573	+	0.411807i	−0.809017	−	0.587785i	$-0.800000\pi$
$877$	−0.837573	+	0.411807i	−0.837573	+	0.411807i	−0.0285561	+	0.999592i	$0.509091\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$881$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$882$	−0.0537907	+	0.0191925i	−0.0537907	+	0.0191925i
$883$	−0.818662	+	0.141675i	−0.818662	+	0.141675i	−0.564443	−	0.825472i	$-0.690909\pi$
$883$	−0.818662	+	0.141675i	−0.818662	+	0.141675i	−0.254218	+	0.967147i	$0.581818\pi$
$884$		0			0
$885$		0			0
$886$	−0.0749775	+	0.0771501i	−0.0749775	+	0.0771501i
$887$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$887$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$888$		0			0
$889$	0.507607	−	0.0290260i	0.507607	−	0.0290260i
$890$		0			0
$891$	0.974012	+	0.226497i	0.974012	+	0.226497i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−0.226220	−	0.0129357i	−0.226220	−	0.0129357i
$897$		0			0
$898$	0.104529	−	0.0119936i	0.104529	−	0.0119936i
$899$		0			0
$900$	−0.608656	+	0.789319i	−0.608656	+	0.789319i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.0441166	+	0.217724i	0.0441166	+	0.217724i
$905$		0			0
$906$		0			0
$907$	1.89740	−	0.441221i	1.89740	−	0.441221i	0.897398	−	0.441221i	$-0.145455\pi$
$907$	1.89740	−	0.441221i	1.89740	−	0.441221i	1.00000			$0$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.260098	−	1.28364i	−0.260098	−	1.28364i	−0.870746	−	0.491733i	$-0.836364\pi$
$911$	−0.260098	−	1.28364i	−0.260098	−	1.28364i	0.610648	−	0.791902i	$-0.290909\pi$
$912$		0			0
$913$		0			0
$914$	0.0231148	+	0.0266759i	0.0231148	+	0.0266759i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.99348	+	0.113991i	1.99348	+	0.113991i	1.00000			$0$
$919$	1.99348	+	0.113991i	1.99348	+	0.113991i	0.993482	+	0.113991i	$0.0363636\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.519923	−	0.334134i	0.519923	−	0.334134i
$926$	0.00975880		0.000558028i	0.00975880		0.000558028i
$927$		0			0
$928$	0.0382514	−	0.117726i	0.0382514	−	0.117726i
$929$		0			0		0.696938	−	0.717132i	$-0.254545\pi$
$929$		0			0		−0.696938	+	0.717132i	$0.745455\pi$
$930$		0			0
$931$		0			0
$932$	0.815991	−	0.141213i	0.815991	−	0.141213i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.696938	−	0.717132i	$-0.745455\pi$
$937$		0			0		0.696938	+	0.717132i	$0.254545\pi$
$938$	−0.0203564	+	0.0100086i	−0.0203564	+	0.0100086i
$939$		0			0
$940$		0			0
$941$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$941$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−0.0289755	−	0.0634475i	−0.0289755	−	0.0634475i
$947$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.959493	−	0.281733i	$-0.909091\pi$
$947$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.654861	+	0.755750i	$0.727273\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.52561	+	1.24748i	−1.52561	+	1.24748i	−0.654861	+	0.755750i	$0.727273\pi$
$953$	−1.52561	+	1.24748i	−1.52561	+	1.24748i	−0.870746	+	0.491733i	$0.836364\pi$
$954$	−0.0252846	+	0.0961927i	−0.0252846	+	0.0961927i
$955$		0			0
$956$	0.0563404	−	0.391856i	0.0563404	−	0.391856i
$957$		0			0
$958$		0			0
$959$	−1.45202	+	1.05496i	−1.45202	+	1.05496i
$960$		0			0
$961$	−0.0285561	+	0.999592i	−0.0285561	+	0.999592i
$962$		0			0
$963$	0.534591	+	1.37309i	0.534591	+	1.37309i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.84225			−1.84225			−0.921124	−	0.389270i	$-0.872727\pi$
$967$	−1.84225			−1.84225			−0.921124	+	0.389270i	$0.872727\pi$
$968$	−0.0992981	−	0.0560762i	−0.0992981	−	0.0560762i
$969$		0			0
$970$		0			0
$971$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$971$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$972$		0			0
$973$		0			0
$974$	−0.00596892	−	0.0694949i	−0.00596892	−	0.0694949i
$975$		0			0
$976$		0			0
$977$	−0.835549	+	0.607062i	−0.835549	+	0.607062i	−0.921124	−	0.389270i	$-0.872727\pi$
$977$	−0.835549	+	0.607062i	−0.835549	+	0.607062i	0.0855750	+	0.996332i	$0.472727\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.859717	+	1.62934i	0.859717	+	1.62934i
$982$	−0.0278616	+	0.105997i	−0.0278616	+	0.105997i
$983$		0			0		0.774142	−	0.633012i	$-0.218182\pi$
$983$		0			0		−0.774142	+	0.633012i	$0.781818\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.32286	−	0.388426i	1.32286	−	0.388426i
$990$		0			0
$991$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.654861	−	0.755750i	$-0.727273\pi$
$991$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.959493	+	0.281733i	$0.909091\pi$
$992$		0			0
$993$		0			0
$994$	−0.0135510	−	0.0515535i	−0.0135510	−	0.0515535i
$995$		0			0
$996$		0			0
$997$		0			0		−0.736741	−	0.676175i	$-0.763636\pi$
$997$		0			0		0.736741	+	0.676175i	$0.236364\pi$
$998$	0.0829280	−	0.0407729i	0.0829280	−	0.0407729i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.1.bb.a.377.1	✓	40
7.6	odd	2	CM	847.1.bb.a.377.1	✓	40
121.26	even	55	inner	847.1.bb.a.510.1	yes	40
847.510	odd	110	inner	847.1.bb.a.510.1	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.1.bb.a.377.1	✓	40	1.1	even	1	trivial
847.1.bb.a.377.1	✓	40	7.6	odd	2	CM
847.1.bb.a.510.1	yes	40	121.26	even	55	inner
847.1.bb.a.510.1	yes	40	847.510	odd	110	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	847.bb (of order \(110\), degree \(40\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.0556279 - 0.0129357i) q^{2} +(-0.894471 - 0.439782i) q^{4} +(-0.0285561 + 0.999592i) q^{7} +(0.0882815 + 0.0721874i) q^{8} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})\)	\(q+(-0.0556279 - 0.0129357i) q^{2} +(-0.894471 - 0.439782i) q^{4} +(-0.0285561 + 0.999592i) q^{7} +(0.0882815 + 0.0721874i) q^{8} +(-0.809017 + 0.587785i) q^{9} +(0.0855750 + 0.996332i) q^{11} +(0.0145189 - 0.0552358i) q^{14} +(0.604679 + 0.784162i) q^{16} +(0.0526073 - 0.0222320i) q^{18} +(0.00812790 - 0.0565308i) q^{22} +(1.08316 + 0.318044i) q^{23} +(-0.736741 + 0.676175i) q^{25} +(0.465145 - 0.881548i) q^{28} +(0.534591 + 0.490643i) q^{29} +(-0.0708664 - 0.155176i) q^{32} +(0.982140 - 0.169966i) q^{36} +(-0.608982 - 0.105389i) q^{37} +(1.02742 - 0.660282i) q^{43} +(0.361624 - 0.928824i) q^{44} +(-0.0561397 - 0.0317035i) q^{46} +(-0.998369 - 0.0570888i) q^{49} +(0.0497301 - 0.0280839i) q^{50} +(-1.06344 + 1.37909i) q^{53} +(-0.0746790 + 0.0861841i) q^{56} +(-0.0233913 - 0.0342087i) q^{58} +(-0.564443 - 0.825472i) q^{63} +(-0.194714 - 0.960954i) q^{64} +(-0.260098 - 0.300169i) q^{67} +(0.812697 - 0.458951i) q^{71} +(-0.113852 - 0.00651029i) q^{72} +(0.0325131 + 0.0137402i) q^{74} +(-0.998369 + 0.0570888i) q^{77} +(-1.39160 + 0.0795747i) q^{79} +(0.309017 - 0.951057i) q^{81} +(-0.0656944 + 0.0234397i) q^{86} +(-0.0643679 + 0.0941351i) q^{88} +(-0.828984 - 0.760835i) q^{92} +(0.0547987 + 0.0160903i) q^{98} +(-0.654861 - 0.755750i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 2 q^{2} + 3 q^{4} + q^{7} - q^{8} - 10 q^{9}+O(q^{10}) \) 40 * q + 2 * q^2 + 3 * q^4 + q^7 - q^8 - 10 * q^9	\( 40 q + 2 q^{2} + 3 q^{4} + q^{7} - q^{8} - 10 q^{9} + q^{11} - 3 q^{14} - 3 q^{18} + 2 q^{22} - 9 q^{23} + q^{25} - 2 q^{28} + 2 q^{29} - 4 q^{32} + 3 q^{36} - 3 q^{37} + 2 q^{43} - 13 q^{44} - q^{46} + q^{49} + 2 q^{50} - 3 q^{53} - 7 q^{56} - 12 q^{58} + q^{63} - 9 q^{64} + 2 q^{67} - 3 q^{71} - q^{72} + 4 q^{74} + q^{77} - 3 q^{79} - 10 q^{81} - q^{86} - 12 q^{88} - 4 q^{92} - 9 q^{98} - 4 q^{99}+O(q^{100}) \) 40 * q + 2 * q^2 + 3 * q^4 + q^7 - q^8 - 10 * q^9 + q^11 - 3 * q^14 - 3 * q^18 + 2 * q^22 - 9 * q^23 + q^25 - 2 * q^28 + 2 * q^29 - 4 * q^32 + 3 * q^36 - 3 * q^37 + 2 * q^43 - 13 * q^44 - q^46 + q^49 + 2 * q^50 - 3 * q^53 - 7 * q^56 - 12 * q^58 + q^63 - 9 * q^64 + 2 * q^67 - 3 * q^71 - q^72 + 4 * q^74 + q^77 - 3 * q^79 - 10 * q^81 - q^86 - 12 * q^88 - 4 * q^92 - 9 * q^98 - 4 * q^99

Introduction

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