LMFDB - Newform orbit 529.2.c.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [529,2,Mod(118,529)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(529, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([16]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("529.118");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 529 = 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	529.c (of order $11$, degree $10$, 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:	$4.22408626693$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 23)
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{22}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\zeta_{22}^{8} - \zeta_{22}^{7} + \cdots + 1) q^{2}+ \cdots + ( - \zeta_{22}^{9} - 2 \zeta_{22}^{7} + \cdots + 2) q^{9}+O(q^{10}) $ q + (z^8 - z^7 + z^6 + z^4 + z^2 - z + 1) * q^2 + (-z^8 - z^6 + z^3 + z) * q^3 + (z^8 + z^5 + z^4 - z^3 + z^2 - z - 1) * q^4 + (-z^4 + z^3 + z^2 + z - 1) * q^5 + (2z^6 + z^4 + z^3 + z^2 + 2) q^6 + (z^8 - z^7 - 2z^5 - z^3 + z^2) q^7 + (z^9 - z^8 + 2z^7 - 2z^6 + z^5 - z^4 - 2z^2 + 3z - 2) * q^8 + (-z^9 - 2z^7 + z^6 - z^5 + 2z^4 - 2z^3 + z^2 - z + 2) q^9	$ q + (\zeta_{22}^{8} - \zeta_{22}^{7} + \cdots + 1) q^{2}+ \cdots + (3 \zeta_{22}^{9} + \cdots + 3 \zeta_{22}) q^{99}+O(q^{100}) $ q + (z^8 - z^7 + z^6 + z^4 + z^2 - z + 1) * q^2 + (-z^8 - z^6 + z^3 + z) * q^3 + (z^8 + z^5 + z^4 - z^3 + z^2 - z - 1) * q^4 + (-z^4 + z^3 + z^2 + z - 1) * q^5 + (2z^6 + z^4 + z^3 + z^2 + 2) q^6 + (z^8 - z^7 - 2z^5 - z^3 + z^2) q^7 + (z^9 - z^8 + 2z^7 - 2z^6 + z^5 - z^4 - 2z^2 + 3z - 2) * q^8 + (-z^9 - 2z^7 + z^6 - z^5 + 2z^4 - 2z^3 + z^2 - z + 2) q^9 + (-z^8 + 2z^7 + 2z^5 - z^4 + 2z - 2) q^10 + (z^8 - 3z^7 + z^6 - z^3 + 1) q^11 + (z^9 + 3z^8 + 3z^6 + z^5 - z^3 + 2z^2 - 2z + 1) * q^12 + (z^9 - z^8 - z^4 + z^3 + 2z - 2) q^13 + (-3z^9 - z^7 - z^5 + z^4 - z^3 + z^2 + 1) q^14 + (-2z^9 - 2z^7 + 2z^6 + 2z^4 + z^2 - 2z + 1) q^15 + (3z^9 - 2z^8 + z^7 + z^3 - 2z^2 + 3z) * q^16 + (-z^9 + z^8 - z^5 + 2z^4 - 4z^3 + 2z^2 - z) q^17 + (-z^9 + 2z^8 - 2z^7 + z^6 + 3z^4 + 2z^2 + 3) * q^18 + (2z^9 + 2z^7 + 4z^5 - 2z^4 + 2z^3 - 2z^2 + 2z - 4) q^19 + (z^9 - 2z^8 + 4z^7 - z^6 + 2z^5 - 2z^4 + z^3 - 4z^2 + 2z - 1) * q^20 + (-2z^8 - 2z^7 - z^6 - z^2 - 2z - 2) q^21 + (-2z^9 + 3z^8 - 3z^7 - z^6 + z^5 + 3z^4 - 3z^3 + 2z^2 + 2) * q^22 + (z^8 - 2z^6 + 2z^5 - z^3 - 4) * q^24 + (z^8 - 2z^7 - z^6 - z^2 - 2z + 1) * q^25 + (z^9 - 2z^8 + z^7 + 2z^5 - 2z^4 - z^2 + 2z - 1) * q^26 + (z^9 - 3z^8 + z^7 + z^3 - z^2) q^27 + (-3z^9 + 3z^6 + 2z^4 + z^3 - z^2 + z + 2) q^28 + (z^9 - z^8 - 3z^6 - z^5 + z^4 - z^3 + z^2 - z - 3) q^29 + (3z^8 + z^6 + z^5 + z^4 + 3z^2) * q^30 + (-z^9 + 2z^8 - 2z^7 + 2z^6 - 2z^5 + z^4 - 2z) q^31 + (2z^9 + z^7 + z^6 + 3z^5 - z^4 + z^3 - 3z^2 - z - 1) q^32 + (-z^8 - 2z^7 + z^6 - 2z^5 - z^4 - z + 1) * q^33 + (-4z^8 + 4z^7 - 4z^6 + 2z^3 - 5z^2 + 5z - 2) * q^34 + (2z^9 - 4z^8 - 4z^6 + 2z^5 + 2z^3 - z^2 + z - 2) q^35 + (2z^9 + 2z^8 + z^7 + z^6 + z^5 + 2z^4 + 2z^3 + 2z - 2) q^36 + (z^9 - z^7 + 2z^6 + 2z^5 + z^4 - z^3 - 2z^2 - 2z + 1) * q^37 + (6z^9 - 4z^8 + 6z^7 - 6z^6 + 4z^5 - 6z^4 - 8z^2 + 4z - 8) * q^38 + (-2z^9 - z^7 + 2z^6 + 2z^4 - z^3 - 2z) * q^39 + (z^9 - z^8 - 3z^5 + 3z^3 - 3z) q^40 + (3z^9 - 3z^6 + z^3 + z^2 + z) * q^41 + (-5z^9 + 2z^8 - 5z^7 - 4z^5 + z^4 - 4z^3 + 4z^2 - z + 4) * q^42 + (-z^9 - 2z^8 + z^6 - z^3 + 2z + 1) * q^43 + (-2z^8 + 6z^7 - 5z^6 + 4z^5 + 4z^3 - 5z^2 + 6z - 2) q^44 + (-3z^9 + 2z^7 + z^6 - z^5 - 2z^4 + 3z^2 + 1) * q^45 + (-2z^8 + 3z^7 - z^6 + z^5 - 3z^4 + 2z^3 - 3) * q^47 + (-3z^8 + 3z^7 - 2z^6 + 2z^5 + 2z^4 + 2z^3 - 2z^2 + 3z - 3) * q^48 + (-z^9 + 3z^8 - 3z^7 - 2z^5 + 2z^4 + 3z^2 - 3z + 1) * q^49 + (-5z^9 + 5z^8 - 5z^7 - 4z^5 + 4z^4 - 4z^3 + 4z^2 - 4z + 4) * q^50 + (z^9 + 2z^8 - 2z^7 - z^6 - z^4 - 2z^3 + 3z^2 - 2z - 1) q^51 + (-z^9 + z^8 + z^6 + z^5 - 2z^4 - 2z^2 + z + 1) * q^52 + (-z^9 - z^8 + 2z^7 + 5z^6 - z^5 + 5z^4 + 2z^3 - z^2 - z) * q^53 + (z^9 + z^8 - 4z^7 + 4z^6 - z^5 - z^4 + 3z^2 - 3z + 3) * q^54 + (-4z^9 + 2z^7 + z^6 - 4z^5 + z^4 - z^3 + 4z^2 - z - 2) * q^55 + (4z^9 - z^8 + z^7 - z^6 + z^5 - z^4 + 4z^3) * q^56 + (4z^9 + 4z^8 + 4z^7 + 4z^6 + 4z^5 + 2z^3 - 2) * q^57 + (-4z^9 - 2z^8 + z^7 - 2z^6 - 4z^5 - z^3 - 2z^2 + 2z + 1) * q^58 + (-2z^8 - 3z^6 - 2z^4 - 3z + 3) * q^59 + (5z^9 + 6z^7 - 6z^6 + 3z^5 - 3z^4 + 3z^3 - 3z^2 + 6z - 6) * q^60 + (-2z^9 + 3z^8 - 6z^7 + 6z^6 - 3z^5 + 2z^4 + 5z^2 - 4z + 5) * q^61 + (-3z^9 + 3z^8 - 2z^7 - 2z^5 - 2z^3 + 3z^2 - 3z) q^62 + (-2z^9 + 2z^8 - z^5 - 3z^4 - z^3 - 3z^2 - z) * q^63 + (z^9 - 2z^8 + 2z^7 - z^6 - 5z^4 + z^3 - 6z^2 + z - 5) * q^64 + (-2z^9 + 2z^8 - 2z^7 - 2z^5 + 6z^4 - z^3 + z^2 - 6z + 2) * q^65 + (-5z^9 + 4z^8 - 8z^7 + 4z^6 - 2z^5 + 2z^4 - 4z^3 + 8z^2 - 4z + 5) q^66 + (6z^8 - 5z^7 + 4z^6 - 7z^5 + z^4 - 7z^3 + 4z^2 - 5z + 6) q^67 + (3z^9 - 5z^8 - 2z^7 + 6z^6 - 6z^5 + 2z^4 + 5z^3 - 3z^2 + 3) * q^68 + (-z^9 + z^8 - 2z^7 + 2z^6 - 2z^5 + 2z^4 - z^3 + z^2 + 5) * q^70 + (2z^8 + 2z^6 - 3z^5 - 3z^3 + 2z^2 + 2) q^71 + (3z^9 - 3z^8 + 4z^7 - z^6 + 4z^5 - 4z^4 + z^3 - 4z^2 + 3z - 3) q^72 + (-5z^8 + z^5 - 4z^4 + 2z^3 - 2z^2 + 4z - 1) q^73 + (2z^9 + 4z^8 - 4z^7 - 2z^6 - 3z^3 - 3z) * q^74 + (z^9 - z^8 - z^5 - 3z^4 - z^3 - 3z^2 - z) * q^75 + (2z^9 - 8z^8 + 2z^7 - 8z^6 - 2z^5 - 8z^4 + 2z^3 - 8z^2 + 2z) q^76 + (z^9 - z^8 + 4z^7 - 4z^6 + z^5 - z^4 - z^2 - z - 1) * q^77 + (z^9 + z^7 - z^6 - z^4 + z^3 + z - 1) * q^78 + (-2z^9 + 5z^8 - 2z^7 + 6z^6 - 2z^5 + 5z^4 - 2z^3 - 8z + 8) * q^79 + (-2z^9 + z^8 + z^6 - 2z^5 + 2z^3 + 6z^2 - 6z - 2) q^80 + (2z^9 + 2z^8 + 3z^7 + 2z^6 + 2z^5 - z^3 - 5z^2 + 5z + 1) q^81 + (-2z^9 + 3z^8 - z^7 + z^6 - z^5 + 3z^4 - 2z^3 - 2z + 2) q^82 + (-5z^9 - 4z^7 + z^6 - 4z^5 + 3z^4 - 3z^3 + 4z^2 - z + 4) * q^83 + (-3z^9 + 3z^8 - 3z^5 + 3z^4 + 5z^2 + 6z + 5) * q^84 + (3z^9 - 4z^8 + 5z^7 - 5z^6 + 2z^5 - 5z^4 + 5z^3 - 4z^2 + 3z) q^85 + (3z^9 - 3z^8 + 2z^6 + 3z^5 + 3z^3 + 3z + 2) * q^86 + (-3z^9 + 2z^8 - 2z^7 + 3z^6 - 7z^3 - z^2 - 7z) * q^87 + (7z^9 - 10z^8 + 7z^7 + 2z^5 - 5z^4 + 6z^3 - 6z^2 + 5z - 2) * q^88 + (8z^9 - z^8 + 4z^7 - 5z^6 - 3z^5 + 3z^4 + 5z^3 - 4z^2 + z - 8) q^89 + (-4z^8 + 4z^7 + 2z^6 + 3z^5 - 2z^4 + 3z^3 + 2z^2 + 4z - 4) * q^90 + (2z^9 - 4z^8 + z^7 - 5z^6 + 5z^5 - z^4 + 4z^3 - 2z^2 - 2) * q^91 + (2z^9 + z^7 + z^6 - z^5 - z^4 - 2z^2) * q^93 + (-4z^8 + 2z^7 - 5z^4 + 2z - 4) * q^94 + (4z^8 + 2z^7 + 2z^6 - 4z^5 + 4z^4 - 2z^3 - 2z^2 - 4z) * q^95 + (8z^9 + z^8 + 8z^7 + 3z^5 - 4z^4 + 4z^3 - 4z^2 + 4z - 3) q^96 + (5z^9 - 7z^8 + 7z^7 - 5z^6 + 2z^3 - 6z^2 + 2z) q^97 + (-6z^9 + 6z^8 - 2z^5 + 4z^4 - 3z^3 + 4z^2 - 2z) q^98 + (3z^9 - 4z^8 - 5z^6 + 6z^5 - 5z^4 - 4z^2 + 3z) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{2} + 4 q^{3} - 14 q^{4} - 8 q^{5} + 17 q^{6} - 6 q^{7} - 7 q^{8} + 9 q^{9}+O(q^{10}) $ 10 * q + 4 * q^2 + 4 * q^3 - 14 * q^4 - 8 * q^5 + 17 * q^6 - 6 * q^7 - 7 * q^8 + 9 * q^9	\( 10 q + 4 q^{2} + 4 q^{3} - 14 q^{4} - 8 q^{5} + 17 q^{6} - 6 q^{7} - 7 q^{8} + 9 q^{9} - 12 q^{10} + 4 q^{11} + q^{12} - 14 q^{13} + 2 q^{14} - q^{15} + 12 q^{16} - 12 q^{17} + 19 q^{18} - 24 q^{19} + 9 q^{20} - 20 q^{21} + 6 q^{22} - 38 q^{24} + 7 q^{25} + q^{26} + 7 q^{27} + 15 q^{28} - 30 q^{29} - 7 q^{30} - 12 q^{31} - q^{32} + 6 q^{33} + 4 q^{34} - 4 q^{35} - 17 q^{36} + 8 q^{37} - 36 q^{38} - 10 q^{39} - q^{40} + 7 q^{41} + 14 q^{42} + 11 q^{43} + 12 q^{44} + 6 q^{45} - 18 q^{47} - 15 q^{48} - 7 q^{49} + 5 q^{50} - 18 q^{51} + 13 q^{52} - 7 q^{53} + 16 q^{54} - 34 q^{55} + 13 q^{56} - 14 q^{57} + 10 q^{58} + 34 q^{59} - 25 q^{60} + 19 q^{61} - 18 q^{62} - q^{63} - 31 q^{64} - 2 q^{65} + 9 q^{66} + 21 q^{67} + 30 q^{68} + 38 q^{70} + 8 q^{71} - 3 q^{72} + 8 q^{73} - 10 q^{74} + 5 q^{75} + 38 q^{76} + 2 q^{77} - 4 q^{78} + 48 q^{79} - 36 q^{80} + 22 q^{81} + 5 q^{82} + 15 q^{83} + 39 q^{84} + 36 q^{85} + 33 q^{86} - 23 q^{87} + 28 q^{88} - 58 q^{89} - 24 q^{90} + 4 q^{91} + 4 q^{93} - 27 q^{94} - 16 q^{95} + 4 q^{96} + 34 q^{97} - 27 q^{98} + 30 q^{99}+O(q^{100}) \) 10 * q + 4 * q^2 + 4 * q^3 - 14 * q^4 - 8 * q^5 + 17 * q^6 - 6 * q^7 - 7 * q^8 + 9 * q^9 - 12 * q^10 + 4 * q^11 + q^12 - 14 * q^13 + 2 * q^14 - q^15 + 12 * q^16 - 12 * q^17 + 19 * q^18 - 24 * q^19 + 9 * q^20 - 20 * q^21 + 6 * q^22 - 38 * q^24 + 7 * q^25 + q^26 + 7 * q^27 + 15 * q^28 - 30 * q^29 - 7 * q^30 - 12 * q^31 - q^32 + 6 * q^33 + 4 * q^34 - 4 * q^35 - 17 * q^36 + 8 * q^37 - 36 * q^38 - 10 * q^39 - q^40 + 7 * q^41 + 14 * q^42 + 11 * q^43 + 12 * q^44 + 6 * q^45 - 18 * q^47 - 15 * q^48 - 7 * q^49 + 5 * q^50 - 18 * q^51 + 13 * q^52 - 7 * q^53 + 16 * q^54 - 34 * q^55 + 13 * q^56 - 14 * q^57 + 10 * q^58 + 34 * q^59 - 25 * q^60 + 19 * q^61 - 18 * q^62 - q^63 - 31 * q^64 - 2 * q^65 + 9 * q^66 + 21 * q^67 + 30 * q^68 + 38 * q^70 + 8 * q^71 - 3 * q^72 + 8 * q^73 - 10 * q^74 + 5 * q^75 + 38 * q^76 + 2 * q^77 - 4 * q^78 + 48 * q^79 - 36 * q^80 + 22 * q^81 + 5 * q^82 + 15 * q^83 + 39 * q^84 + 36 * q^85 + 33 * q^86 - 23 * q^87 + 28 * q^88 - 58 * q^89 - 24 * q^90 + 4 * q^91 + 4 * q^93 - 27 * q^94 - 16 * q^95 + 4 * q^96 + 34 * q^97 - 27 * q^98 + 30 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/529\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$\zeta_{22}^{4}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

118.1

−0.415415	−	0.909632i
0.654861	−	0.755750i
−0.841254	−	0.540641i
0.142315	−	0.989821i
−0.841254	+	0.540641i
0.142315	+	0.989821i
−0.415415	+	0.909632i
0.959493	−	0.281733i
0.959493	+	0.281733i
0.654861	+	0.755750i

0.357685

2.48775i

0.662317

−

1.45027i

−4.14200

1.21620i

−0.968468

1.11767i

3.84482

1.12894i

−1.03365

0.664289i

−2.41899

−

5.29684i

0.299955

0.346167i

−3.12690

−

2.00954i

170.1

−0.459493

0.134919i

−1.44306

−

1.66538i

−1.48958

0.957293i

−0.369215

−

2.56794i

0.887769

0.570534i

1.41153

−

3.09082i

1.18251

−

1.36469i

−0.264125

1.83703i

0.516117

1.13014i

177.1

−0.154861

0.178719i

0.402869

−

0.258908i

0.276671

1.92429i

−0.628663

−

1.37658i

−0.0161168

0.112095i

−2.44306

0.717348i

−0.784630

−

0.504251i

−1.15098

2.52028i

0.343376

0.100824i

255.1

1.34125

0.861971i

−0.0336545

−

0.234072i

0.225136

0.492980i

−3.07385

−

0.902563i

0.156624

−

0.342959i

−0.597131

0.689126i

0.330830

−

2.30097i

2.82482

−

0.829443i

−3.34482

−

3.86013i

266.1

−0.154861

−

0.178719i

0.402869

0.258908i

0.276671

−

1.92429i

−0.628663

1.37658i

−0.0161168

−

0.112095i

−2.44306

−

0.717348i

−0.784630

0.504251i

−1.15098

−

2.52028i

0.343376

−

0.100824i

334.1

1.34125

−

0.861971i

−0.0336545

0.234072i

0.225136

−

0.492980i

−3.07385

0.902563i

0.156624

0.342959i

−0.597131

−

0.689126i

0.330830

2.30097i

2.82482

0.829443i

−3.34482

3.86013i

399.1

0.357685

−

2.48775i

0.662317

1.45027i

−4.14200

−

1.21620i

−0.968468

−

1.11767i

3.84482

−

1.12894i

−1.03365

−

0.664289i

−2.41899

5.29684i

0.299955

−

0.346167i

−3.12690

2.00954i

466.1

0.915415

−

2.00448i

2.41153

0.708089i

−1.87023

−

2.15836i

1.04019

−

0.668491i

3.62690

−

4.18567i

−0.337683

2.34863i

−1.80972

0.531382i

2.79032

1.79323i

−0.387769

−

2.69699i

487.1

0.915415

2.00448i

2.41153

−

0.708089i

−1.87023

2.15836i

1.04019

0.668491i

3.62690

4.18567i

−0.337683

−

2.34863i

−1.80972

−

0.531382i

2.79032

−

1.79323i

−0.387769

2.69699i

501.1

−0.459493

−

0.134919i

−1.44306

1.66538i

−1.48958

−

0.957293i

−0.369215

2.56794i

0.887769

−

0.570534i

1.41153

3.09082i

1.18251

1.36469i

−0.264125

−

1.83703i

0.516117

−

1.13014i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.c	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	529.2.c.f		10
23.b	odd	2	1		529.2.c.g		10
23.c	even	11	1		529.2.a.j		5
23.c	even	11	2		529.2.c.a		10
23.c	even	11	2		529.2.c.c		10
23.c	even	11	2		529.2.c.e		10
23.c	even	11	1	inner	529.2.c.f		10
23.c	even	11	2		529.2.c.h		10
23.d	odd	22	2		23.2.c.a	✓	10
23.d	odd	22	1		529.2.a.i		5
23.d	odd	22	2		529.2.c.b		10
23.d	odd	22	2		529.2.c.d		10
23.d	odd	22	1		529.2.c.g		10
23.d	odd	22	2		529.2.c.i		10
69.g	even	22	2		207.2.i.c		10
69.g	even	22	1		4761.2.a.bo		5
69.h	odd	22	1		4761.2.a.bn		5
92.g	odd	22	1		8464.2.a.bt		5
92.h	even	22	2		368.2.m.c		10
92.h	even	22	1		8464.2.a.bs		5
115.i	odd	22	2		575.2.k.b		10
115.l	even	44	4		575.2.p.b		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
23.2.c.a	✓	10	23.d	odd	22	2
207.2.i.c		10	69.g	even	22	2
368.2.m.c		10	92.h	even	22	2
529.2.a.i		5	23.d	odd	22	1
529.2.a.j		5	23.c	even	11	1
529.2.c.a		10	23.c	even	11	2
529.2.c.b		10	23.d	odd	22	2
529.2.c.c		10	23.c	even	11	2
529.2.c.d		10	23.d	odd	22	2
529.2.c.e		10	23.c	even	11	2
529.2.c.f		10	1.a	even	1	1	trivial
529.2.c.f		10	23.c	even	11	1	inner
529.2.c.g		10	23.b	odd	2	1
529.2.c.g		10	23.d	odd	22	1
529.2.c.h		10	23.c	even	11	2
529.2.c.i		10	23.d	odd	22	2
575.2.k.b		10	115.i	odd	22	2
575.2.p.b		20	115.l	even	44	4
4761.2.a.bn		5	69.h	odd	22	1
4761.2.a.bo		5	69.g	even	22	1
8464.2.a.bs		5	92.h	even	22	1
8464.2.a.bt		5	92.g	odd	22	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(529, [\chi])$:

$ T_{2}^{10} - 4T_{2}^{9} + 16T_{2}^{8} - 31T_{2}^{7} + 47T_{2}^{6} - 23T_{2}^{5} - 18T_{2}^{4} + 39T_{2}^{3} + 31T_{2}^{2} + 8T_{2} + 1 $

$ T_{5}^{10} + 8 T_{5}^{9} + 31 T_{5}^{8} + 83 T_{5}^{7} + 147 T_{5}^{6} + 120 T_{5}^{5} + 36 T_{5}^{4} + \cdots + 529 $

$ T_{7}^{10} + 6 T_{7}^{9} + 25 T_{7}^{8} + 106 T_{7}^{7} + 361 T_{7}^{6} + 967 T_{7}^{5} + 2084 T_{7}^{4} + \cdots + 529 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} - 4 T^{9} + \cdots + 1 $ T^10 - 4T^9 + 16T^8 - 31T^7 + 47T^6 - 23T^5 - 18T^4 + 39T^3 + 31T^2 + 8*T + 1
$3$	$ T^{10} - 4 T^{9} + \cdots + 1 $ T^10 - 4T^9 + 5T^8 - 9T^7 + 36T^6 - 78T^5 + 125T^4 - 71T^3 + 20T^2 - 3*T + 1
$5$	$ T^{10} + 8 T^{9} + \cdots + 529 $ T^10 + 8T^9 + 31T^8 + 83T^7 + 147T^6 + 120T^5 + 36T^4 - 64T^3 + 236T^2 + 414*T + 529
$7$	$ T^{10} + 6 T^{9} + \cdots + 529 $ T^10 + 6T^9 + 25T^8 + 106T^7 + 361T^6 + 967T^5 + 2084T^4 + 3154T^3 + 3117T^2 + 1817*T + 529
$11$	$ T^{10} - 4 T^{9} + \cdots + 529 $ T^10 - 4T^9 + 16T^8 - 64T^7 + 311T^6 - 342T^5 + 3216T^4 - 4900T^3 + 11823T^2 - 4623*T + 529
$13$	$ T^{10} + 14 T^{9} + \cdots + 1 $ T^10 + 14T^9 + 97T^8 + 412T^7 + 1137T^6 + 2003T^5 + 1950T^4 + 526T^3 + 115T^2 + 15*T + 1
$17$	$ T^{10} + 12 T^{9} + \cdots + 529 $ T^10 + 12T^9 + 78T^8 + 386T^7 + 705T^6 - 1990T^5 + 375T^4 + 4192T^3 + 12244T^2 - 2001*T + 529
$19$	$ T^{10} + 24 T^{9} + \cdots + 541696 $ T^10 + 24T^9 + 312T^8 + 2560T^7 + 15152T^6 + 67264T^5 + 224640T^4 + 556288T^3 + 968960T^2 + 1048064*T + 541696
$23$	$ T^{10} $ T^10
$29$	$ T^{10} + 30 T^{9} + \cdots + 4932841 $ T^10 + 30T^9 + 460T^8 + 4670T^7 + 34522T^6 + 190706T^5 + 783896T^4 + 2339361T^3 + 4859629T^2 + 6580823*T + 4932841
$31$	$ T^{10} + 12 T^{9} + \cdots + 17161 $ T^10 + 12T^9 + 78T^8 + 353T^7 + 1332T^6 + 3862T^5 + 10055T^4 + 22639T^3 + 36400T^2 + 35894*T + 17161
$37$	$ T^{10} - 8 T^{9} + \cdots + 529 $ T^10 - 8T^9 + 42T^8 - 314T^7 + 785T^6 + 2850T^5 - 1394T^4 - 67630T^3 + 364743T^2 + 11477*T + 529
$41$	$ T^{10} - 7 T^{9} + \cdots + 1849 $ T^10 - 7T^9 + 82T^8 - 607T^7 + 2379T^6 - 5598T^5 + 6384T^4 + 2623T^3 + 6719T^2 - 6278*T + 1849
$43$	$ T^{10} - 11 T^{9} + \cdots + 64009 $ T^10 - 11T^9 + 77T^8 - 451T^7 + 1903T^6 - 5302T^5 + 8954T^4 - 1936T^3 - 10043T^2 + 11132*T + 64009
$47$	$ (T^{5} + 9 T^{4} - 5 T^{3} + \cdots + 1)^{2} $ (T^5 + 9T^4 - 5T^3 - 97T^2 - 106T + 1)^2
$53$	$ T^{10} + \cdots + 517426009 $ T^10 + 7T^9 + 93T^8 + 563T^7 + 3083T^6 + 34924T^5 + 471640T^4 + 7146882T^3 + 61674523T^2 + 249034156*T + 517426009
$59$	$ T^{10} - 34 T^{9} + \cdots + 4489 $ T^10 - 34T^9 + 529T^8 - 5248T^7 + 37522T^6 - 186616T^5 + 615143T^4 - 1218273T^3 + 1240471T^2 - 64923*T + 4489
$61$	$ T^{10} - 19 T^{9} + \cdots + 529 $ T^10 - 19T^9 + 251T^8 - 1073T^7 - 4319T^6 + 15082T^5 + 131464T^4 + 100670T^3 + 273663T^2 + 15778*T + 529
$67$	$ T^{10} + \cdots + 1113757129 $ T^10 - 21T^9 + 210T^8 - 2309T^7 + 19416T^6 - 109339T^5 + 1345840T^4 - 5559069T^3 + 25422453T^2 - 199370302*T + 1113757129
$71$	$ T^{10} - 8 T^{9} + \cdots + 529 $ T^10 - 8T^9 - 24T^8 + 71T^7 + 1412T^6 + 3422T^5 + 4447T^4 + 3617T^3 + 2722T^2 + 1104*T + 529
$73$	$ T^{10} - 8 T^{9} + \cdots + 982081 $ T^10 - 8T^9 + 97T^8 - 1172T^7 + 10102T^6 - 52634T^5 + 189709T^4 - 654227T^3 + 1285861T^2 + 2064253*T + 982081
$79$	$ T^{10} + \cdots + 517426009 $ T^10 - 48T^9 + 1248T^8 - 20656T^7 + 240936T^6 - 2015190T^5 + 12620975T^4 - 60507595T^3 + 240329461T^2 - 556664584*T + 517426009
$83$	$ T^{10} - 15 T^{9} + \cdots + 529 $ T^10 - 15T^9 + 49T^8 + 574T^7 + 3270T^6 + 9074T^5 + 15228T^4 + 14383T^3 + 7093T^2 + 1955*T + 529
$89$	$ T^{10} + \cdots + 78310985281 $ T^10 + 58T^9 + 1670T^8 + 31322T^7 + 447803T^6 + 5390694T^5 + 54170658T^4 + 428521740T^3 + 2907268149T^2 + 18504486125*T + 78310985281
$97$	$ T^{10} - 34 T^{9} + \cdots + 2374681 $ T^10 - 34T^9 + 672T^8 - 5754T^7 + 30141T^6 + 58090T^5 - 581085T^4 - 2633852T^3 + 14562758T^2 + 1286735*T + 2374681
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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:	\( N \)	\(=\)	\( 529 = 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	529.c (of order \(11\), degree \(10\), not minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{22}^{8} - \zeta_{22}^{7} + \cdots + 1) q^{2}+ \cdots + ( - \zeta_{22}^{9} - 2 \zeta_{22}^{7} + \cdots + 2) q^{9}+O(q^{10}) \) q + (z^8 - z^7 + z^6 + z^4 + z^2 - z + 1) * q^2 + (-z^8 - z^6 + z^3 + z) * q^3 + (z^8 + z^5 + z^4 - z^3 + z^2 - z - 1) * q^4 + (-z^4 + z^3 + z^2 + z - 1) * q^5 + (2z^6 + z^4 + z^3 + z^2 + 2) q^6 + (z^8 - z^7 - 2z^5 - z^3 + z^2) q^7 + (z^9 - z^8 + 2z^7 - 2z^6 + z^5 - z^4 - 2z^2 + 3z - 2) * q^8 + (-z^9 - 2z^7 + z^6 - z^5 + 2z^4 - 2z^3 + z^2 - z + 2) q^9	\( q + (\zeta_{22}^{8} - \zeta_{22}^{7} + \cdots + 1) q^{2}+ \cdots + (3 \zeta_{22}^{9} + \cdots + 3 \zeta_{22}) q^{99}+O(q^{100}) \) q + (z^8 - z^7 + z^6 + z^4 + z^2 - z + 1) * q^2 + (-z^8 - z^6 + z^3 + z) * q^3 + (z^8 + z^5 + z^4 - z^3 + z^2 - z - 1) * q^4 + (-z^4 + z^3 + z^2 + z - 1) * q^5 + (2z^6 + z^4 + z^3 + z^2 + 2) q^6 + (z^8 - z^7 - 2z^5 - z^3 + z^2) q^7 + (z^9 - z^8 + 2z^7 - 2z^6 + z^5 - z^4 - 2z^2 + 3z - 2) * q^8 + (-z^9 - 2z^7 + z^6 - z^5 + 2z^4 - 2z^3 + z^2 - z + 2) q^9 + (-z^8 + 2z^7 + 2z^5 - z^4 + 2z - 2) q^10 + (z^8 - 3z^7 + z^6 - z^3 + 1) q^11 + (z^9 + 3z^8 + 3z^6 + z^5 - z^3 + 2z^2 - 2z + 1) * q^12 + (z^9 - z^8 - z^4 + z^3 + 2z - 2) q^13 + (-3z^9 - z^7 - z^5 + z^4 - z^3 + z^2 + 1) q^14 + (-2z^9 - 2z^7 + 2z^6 + 2z^4 + z^2 - 2z + 1) q^15 + (3z^9 - 2z^8 + z^7 + z^3 - 2z^2 + 3z) * q^16 + (-z^9 + z^8 - z^5 + 2z^4 - 4z^3 + 2z^2 - z) q^17 + (-z^9 + 2z^8 - 2z^7 + z^6 + 3z^4 + 2z^2 + 3) * q^18 + (2z^9 + 2z^7 + 4z^5 - 2z^4 + 2z^3 - 2z^2 + 2z - 4) q^19 + (z^9 - 2z^8 + 4z^7 - z^6 + 2z^5 - 2z^4 + z^3 - 4z^2 + 2z - 1) * q^20 + (-2z^8 - 2z^7 - z^6 - z^2 - 2z - 2) q^21 + (-2z^9 + 3z^8 - 3z^7 - z^6 + z^5 + 3z^4 - 3z^3 + 2z^2 + 2) * q^22 + (z^8 - 2z^6 + 2z^5 - z^3 - 4) * q^24 + (z^8 - 2z^7 - z^6 - z^2 - 2z + 1) * q^25 + (z^9 - 2z^8 + z^7 + 2z^5 - 2z^4 - z^2 + 2z - 1) * q^26 + (z^9 - 3z^8 + z^7 + z^3 - z^2) q^27 + (-3z^9 + 3z^6 + 2z^4 + z^3 - z^2 + z + 2) q^28 + (z^9 - z^8 - 3z^6 - z^5 + z^4 - z^3 + z^2 - z - 3) q^29 + (3z^8 + z^6 + z^5 + z^4 + 3z^2) * q^30 + (-z^9 + 2z^8 - 2z^7 + 2z^6 - 2z^5 + z^4 - 2z) q^31 + (2z^9 + z^7 + z^6 + 3z^5 - z^4 + z^3 - 3z^2 - z - 1) q^32 + (-z^8 - 2z^7 + z^6 - 2z^5 - z^4 - z + 1) * q^33 + (-4z^8 + 4z^7 - 4z^6 + 2z^3 - 5z^2 + 5z - 2) * q^34 + (2z^9 - 4z^8 - 4z^6 + 2z^5 + 2z^3 - z^2 + z - 2) q^35 + (2z^9 + 2z^8 + z^7 + z^6 + z^5 + 2z^4 + 2z^3 + 2z - 2) q^36 + (z^9 - z^7 + 2z^6 + 2z^5 + z^4 - z^3 - 2z^2 - 2z + 1) * q^37 + (6z^9 - 4z^8 + 6z^7 - 6z^6 + 4z^5 - 6z^4 - 8z^2 + 4z - 8) * q^38 + (-2z^9 - z^7 + 2z^6 + 2z^4 - z^3 - 2z) * q^39 + (z^9 - z^8 - 3z^5 + 3z^3 - 3z) q^40 + (3z^9 - 3z^6 + z^3 + z^2 + z) * q^41 + (-5z^9 + 2z^8 - 5z^7 - 4z^5 + z^4 - 4z^3 + 4z^2 - z + 4) * q^42 + (-z^9 - 2z^8 + z^6 - z^3 + 2z + 1) * q^43 + (-2z^8 + 6z^7 - 5z^6 + 4z^5 + 4z^3 - 5z^2 + 6z - 2) q^44 + (-3z^9 + 2z^7 + z^6 - z^5 - 2z^4 + 3z^2 + 1) * q^45 + (-2z^8 + 3z^7 - z^6 + z^5 - 3z^4 + 2z^3 - 3) * q^47 + (-3z^8 + 3z^7 - 2z^6 + 2z^5 + 2z^4 + 2z^3 - 2z^2 + 3z - 3) * q^48 + (-z^9 + 3z^8 - 3z^7 - 2z^5 + 2z^4 + 3z^2 - 3z + 1) * q^49 + (-5z^9 + 5z^8 - 5z^7 - 4z^5 + 4z^4 - 4z^3 + 4z^2 - 4z + 4) * q^50 + (z^9 + 2z^8 - 2z^7 - z^6 - z^4 - 2z^3 + 3z^2 - 2z - 1) q^51 + (-z^9 + z^8 + z^6 + z^5 - 2z^4 - 2z^2 + z + 1) * q^52 + (-z^9 - z^8 + 2z^7 + 5z^6 - z^5 + 5z^4 + 2z^3 - z^2 - z) * q^53 + (z^9 + z^8 - 4z^7 + 4z^6 - z^5 - z^4 + 3z^2 - 3z + 3) * q^54 + (-4z^9 + 2z^7 + z^6 - 4z^5 + z^4 - z^3 + 4z^2 - z - 2) * q^55 + (4z^9 - z^8 + z^7 - z^6 + z^5 - z^4 + 4z^3) * q^56 + (4z^9 + 4z^8 + 4z^7 + 4z^6 + 4z^5 + 2z^3 - 2) * q^57 + (-4z^9 - 2z^8 + z^7 - 2z^6 - 4z^5 - z^3 - 2z^2 + 2z + 1) * q^58 + (-2z^8 - 3z^6 - 2z^4 - 3z + 3) * q^59 + (5z^9 + 6z^7 - 6z^6 + 3z^5 - 3z^4 + 3z^3 - 3z^2 + 6z - 6) * q^60 + (-2z^9 + 3z^8 - 6z^7 + 6z^6 - 3z^5 + 2z^4 + 5z^2 - 4z + 5) * q^61 + (-3z^9 + 3z^8 - 2z^7 - 2z^5 - 2z^3 + 3z^2 - 3z) q^62 + (-2z^9 + 2z^8 - z^5 - 3z^4 - z^3 - 3z^2 - z) * q^63 + (z^9 - 2z^8 + 2z^7 - z^6 - 5z^4 + z^3 - 6z^2 + z - 5) * q^64 + (-2z^9 + 2z^8 - 2z^7 - 2z^5 + 6z^4 - z^3 + z^2 - 6z + 2) * q^65 + (-5z^9 + 4z^8 - 8z^7 + 4z^6 - 2z^5 + 2z^4 - 4z^3 + 8z^2 - 4z + 5) q^66 + (6z^8 - 5z^7 + 4z^6 - 7z^5 + z^4 - 7z^3 + 4z^2 - 5z + 6) q^67 + (3z^9 - 5z^8 - 2z^7 + 6z^6 - 6z^5 + 2z^4 + 5z^3 - 3z^2 + 3) * q^68 + (-z^9 + z^8 - 2z^7 + 2z^6 - 2z^5 + 2z^4 - z^3 + z^2 + 5) * q^70 + (2z^8 + 2z^6 - 3z^5 - 3z^3 + 2z^2 + 2) q^71 + (3z^9 - 3z^8 + 4z^7 - z^6 + 4z^5 - 4z^4 + z^3 - 4z^2 + 3z - 3) q^72 + (-5z^8 + z^5 - 4z^4 + 2z^3 - 2z^2 + 4z - 1) q^73 + (2z^9 + 4z^8 - 4z^7 - 2z^6 - 3z^3 - 3z) * q^74 + (z^9 - z^8 - z^5 - 3z^4 - z^3 - 3z^2 - z) * q^75 + (2z^9 - 8z^8 + 2z^7 - 8z^6 - 2z^5 - 8z^4 + 2z^3 - 8z^2 + 2z) q^76 + (z^9 - z^8 + 4z^7 - 4z^6 + z^5 - z^4 - z^2 - z - 1) * q^77 + (z^9 + z^7 - z^6 - z^4 + z^3 + z - 1) * q^78 + (-2z^9 + 5z^8 - 2z^7 + 6z^6 - 2z^5 + 5z^4 - 2z^3 - 8z + 8) * q^79 + (-2z^9 + z^8 + z^6 - 2z^5 + 2z^3 + 6z^2 - 6z - 2) q^80 + (2z^9 + 2z^8 + 3z^7 + 2z^6 + 2z^5 - z^3 - 5z^2 + 5z + 1) q^81 + (-2z^9 + 3z^8 - z^7 + z^6 - z^5 + 3z^4 - 2z^3 - 2z + 2) q^82 + (-5z^9 - 4z^7 + z^6 - 4z^5 + 3z^4 - 3z^3 + 4z^2 - z + 4) * q^83 + (-3z^9 + 3z^8 - 3z^5 + 3z^4 + 5z^2 + 6z + 5) * q^84 + (3z^9 - 4z^8 + 5z^7 - 5z^6 + 2z^5 - 5z^4 + 5z^3 - 4z^2 + 3z) q^85 + (3z^9 - 3z^8 + 2z^6 + 3z^5 + 3z^3 + 3z + 2) * q^86 + (-3z^9 + 2z^8 - 2z^7 + 3z^6 - 7z^3 - z^2 - 7z) * q^87 + (7z^9 - 10z^8 + 7z^7 + 2z^5 - 5z^4 + 6z^3 - 6z^2 + 5z - 2) * q^88 + (8z^9 - z^8 + 4z^7 - 5z^6 - 3z^5 + 3z^4 + 5z^3 - 4z^2 + z - 8) q^89 + (-4z^8 + 4z^7 + 2z^6 + 3z^5 - 2z^4 + 3z^3 + 2z^2 + 4z - 4) * q^90 + (2z^9 - 4z^8 + z^7 - 5z^6 + 5z^5 - z^4 + 4z^3 - 2z^2 - 2) * q^91 + (2z^9 + z^7 + z^6 - z^5 - z^4 - 2z^2) * q^93 + (-4z^8 + 2z^7 - 5z^4 + 2z - 4) * q^94 + (4z^8 + 2z^7 + 2z^6 - 4z^5 + 4z^4 - 2z^3 - 2z^2 - 4z) * q^95 + (8z^9 + z^8 + 8z^7 + 3z^5 - 4z^4 + 4z^3 - 4z^2 + 4z - 3) q^96 + (5z^9 - 7z^8 + 7z^7 - 5z^6 + 2z^3 - 6z^2 + 2z) q^97 + (-6z^9 + 6z^8 - 2z^5 + 4z^4 - 3z^3 + 4z^2 - 2z) q^98 + (3z^9 - 4z^8 - 5z^6 + 6z^5 - 5z^4 - 4z^2 + 3z) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 4 q^{3} - 14 q^{4} - 8 q^{5} + 17 q^{6} - 6 q^{7} - 7 q^{8} + 9 q^{9}+O(q^{10}) \) 10 * q + 4 * q^2 + 4 * q^3 - 14 * q^4 - 8 * q^5 + 17 * q^6 - 6 * q^7 - 7 * q^8 + 9 * q^9	\( 10 q + 4 q^{2} + 4 q^{3} - 14 q^{4} - 8 q^{5} + 17 q^{6} - 6 q^{7} - 7 q^{8} + 9 q^{9} - 12 q^{10} + 4 q^{11} + q^{12} - 14 q^{13} + 2 q^{14} - q^{15} + 12 q^{16} - 12 q^{17} + 19 q^{18} - 24 q^{19} + 9 q^{20} - 20 q^{21} + 6 q^{22} - 38 q^{24} + 7 q^{25} + q^{26} + 7 q^{27} + 15 q^{28} - 30 q^{29} - 7 q^{30} - 12 q^{31} - q^{32} + 6 q^{33} + 4 q^{34} - 4 q^{35} - 17 q^{36} + 8 q^{37} - 36 q^{38} - 10 q^{39} - q^{40} + 7 q^{41} + 14 q^{42} + 11 q^{43} + 12 q^{44} + 6 q^{45} - 18 q^{47} - 15 q^{48} - 7 q^{49} + 5 q^{50} - 18 q^{51} + 13 q^{52} - 7 q^{53} + 16 q^{54} - 34 q^{55} + 13 q^{56} - 14 q^{57} + 10 q^{58} + 34 q^{59} - 25 q^{60} + 19 q^{61} - 18 q^{62} - q^{63} - 31 q^{64} - 2 q^{65} + 9 q^{66} + 21 q^{67} + 30 q^{68} + 38 q^{70} + 8 q^{71} - 3 q^{72} + 8 q^{73} - 10 q^{74} + 5 q^{75} + 38 q^{76} + 2 q^{77} - 4 q^{78} + 48 q^{79} - 36 q^{80} + 22 q^{81} + 5 q^{82} + 15 q^{83} + 39 q^{84} + 36 q^{85} + 33 q^{86} - 23 q^{87} + 28 q^{88} - 58 q^{89} - 24 q^{90} + 4 q^{91} + 4 q^{93} - 27 q^{94} - 16 q^{95} + 4 q^{96} + 34 q^{97} - 27 q^{98} + 30 q^{99}+O(q^{100}) \) 10 * q + 4 * q^2 + 4 * q^3 - 14 * q^4 - 8 * q^5 + 17 * q^6 - 6 * q^7 - 7 * q^8 + 9 * q^9 - 12 * q^10 + 4 * q^11 + q^12 - 14 * q^13 + 2 * q^14 - q^15 + 12 * q^16 - 12 * q^17 + 19 * q^18 - 24 * q^19 + 9 * q^20 - 20 * q^21 + 6 * q^22 - 38 * q^24 + 7 * q^25 + q^26 + 7 * q^27 + 15 * q^28 - 30 * q^29 - 7 * q^30 - 12 * q^31 - q^32 + 6 * q^33 + 4 * q^34 - 4 * q^35 - 17 * q^36 + 8 * q^37 - 36 * q^38 - 10 * q^39 - q^40 + 7 * q^41 + 14 * q^42 + 11 * q^43 + 12 * q^44 + 6 * q^45 - 18 * q^47 - 15 * q^48 - 7 * q^49 + 5 * q^50 - 18 * q^51 + 13 * q^52 - 7 * q^53 + 16 * q^54 - 34 * q^55 + 13 * q^56 - 14 * q^57 + 10 * q^58 + 34 * q^59 - 25 * q^60 + 19 * q^61 - 18 * q^62 - q^63 - 31 * q^64 - 2 * q^65 + 9 * q^66 + 21 * q^67 + 30 * q^68 + 38 * q^70 + 8 * q^71 - 3 * q^72 + 8 * q^73 - 10 * q^74 + 5 * q^75 + 38 * q^76 + 2 * q^77 - 4 * q^78 + 48 * q^79 - 36 * q^80 + 22 * q^81 + 5 * q^82 + 15 * q^83 + 39 * q^84 + 36 * q^85 + 33 * q^86 - 23 * q^87 + 28 * q^88 - 58 * q^89 - 24 * q^90 + 4 * q^91 + 4 * q^93 - 27 * q^94 - 16 * q^95 + 4 * q^96 + 34 * q^97 - 27 * q^98 + 30 * q^99

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