LMFDB - Newform orbit 529.2.c.d

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

−0.0336545

−

0.234072i

0.198939

−

0.435615i

1.86533

−

0.547710i

−0.991025

1.14370i

−0.108660

−

0.0319056i

−2.14200

1.37658i

−0.387454

−

0.848406i

1.81440

2.09393i

0.301061

0.193480i

170.1

2.41153

−

0.708089i

−1.04408

−

1.20493i

3.63158

−

2.33387i

0.210468

1.46384i

−3.37102

−

2.16642i

0.510424

−

1.11767i

3.81329

−

4.40077i

0.0651865

−

0.453382i

1.54408

3.38106i

177.1

−1.44306

1.66538i

−2.11435

1.35881i

−0.406440

−

2.82685i

−0.513652

−

1.12474i

0.788201

−

5.48206i

2.27667

−

0.668491i

1.58671

1.01971i

1.37787

−

3.01713i

2.61435

0.767644i

255.1

0.402869

0.258908i

−0.313607

−

2.18119i

−0.735560

−

1.61065i

2.48926

0.730913i

0.438384

−

0.959928i

2.22514

−

2.56794i

0.256983

−

1.78736i

−1.78074

0.522874i

0.813607

0.938953i

266.1

−1.44306

−

1.66538i

−2.11435

−

1.35881i

−0.406440

2.82685i

−0.513652

1.12474i

0.788201

5.48206i

2.27667

0.668491i

1.58671

−

1.01971i

1.37787

3.01713i

2.61435

−

0.767644i

334.1

0.402869

−

0.258908i

−0.313607

2.18119i

−0.735560

1.61065i

2.48926

−

0.730913i

0.438384

0.959928i

2.22514

2.56794i

0.256983

1.78736i

−1.78074

−

0.522874i

0.813607

−

0.938953i

399.1

−0.0336545

0.234072i

0.198939

0.435615i

1.86533

0.547710i

−0.991025

−

1.14370i

−0.108660

0.0319056i

−2.14200

−

1.37658i

−0.387454

0.848406i

1.81440

−

2.09393i

0.301061

−

0.193480i

466.1

0.662317

−

1.45027i

−0.226900

−

0.0666238i

−0.354905

−

0.409583i

−2.69505

1.73201i

−0.246902

0.284941i

0.129769

−

0.902563i

2.23047

−

0.654925i

−2.47672

−

1.59169i

0.726900

5.05570i

487.1

0.662317

1.45027i

−0.226900

0.0666238i

−0.354905

0.409583i

−2.69505

−

1.73201i

−0.246902

−

0.284941i

0.129769

0.902563i

2.23047

0.654925i

−2.47672

1.59169i

0.726900

−

5.05570i

501.1

2.41153

0.708089i

−1.04408

1.20493i

3.63158

2.33387i

0.210468

−

1.46384i

−3.37102

2.16642i

0.510424

1.11767i

3.81329

4.40077i

0.0651865

0.453382i

1.54408

−

3.38106i

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.d		10
23.b	odd	2	1		529.2.c.e		10
23.c	even	11	2		23.2.c.a	✓	10
23.c	even	11	1		529.2.a.i		5
23.c	even	11	2		529.2.c.b		10
23.c	even	11	1	inner	529.2.c.d		10
23.c	even	11	2		529.2.c.g		10
23.c	even	11	2		529.2.c.i		10
23.d	odd	22	1		529.2.a.j		5
23.d	odd	22	2		529.2.c.a		10
23.d	odd	22	2		529.2.c.c		10
23.d	odd	22	1		529.2.c.e		10
23.d	odd	22	2		529.2.c.f		10
23.d	odd	22	2		529.2.c.h		10
69.g	even	22	1		4761.2.a.bn		5
69.h	odd	22	2		207.2.i.c		10
69.h	odd	22	1		4761.2.a.bo		5
92.g	odd	22	2		368.2.m.c		10
92.g	odd	22	1		8464.2.a.bs		5
92.h	even	22	1		8464.2.a.bt		5
115.j	even	22	2		575.2.k.b		10
115.k	odd	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.c	even	11	2
207.2.i.c		10	69.h	odd	22	2
368.2.m.c		10	92.g	odd	22	2
529.2.a.i		5	23.c	even	11	1
529.2.a.j		5	23.d	odd	22	1
529.2.c.a		10	23.d	odd	22	2
529.2.c.b		10	23.c	even	11	2
529.2.c.c		10	23.d	odd	22	2
529.2.c.d		10	1.a	even	1	1	trivial
529.2.c.d		10	23.c	even	11	1	inner
529.2.c.e		10	23.b	odd	2	1
529.2.c.e		10	23.d	odd	22	1
529.2.c.f		10	23.d	odd	22	2
529.2.c.g		10	23.c	even	11	2
529.2.c.h		10	23.d	odd	22	2
529.2.c.i		10	23.c	even	11	2
575.2.k.b		10	115.j	even	22	2
575.2.p.b		20	115.k	odd	44	4
4761.2.a.bn		5	69.g	even	22	1
4761.2.a.bo		5	69.h	odd	22	1
8464.2.a.bs		5	92.g	odd	22	1
8464.2.a.bt		5	92.h	even	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} + 5T_{2}^{8} - 9T_{2}^{7} + 36T_{2}^{6} - 78T_{2}^{5} + 125T_{2}^{4} - 71T_{2}^{3} + 20T_{2}^{2} - 3T_{2} + 1 $

$ T_{5}^{10} + 3T_{5}^{9} - 2T_{5}^{8} - 28T_{5}^{7} - 18T_{5}^{6} + T_{5}^{5} + 201T_{5}^{4} + 350T_{5}^{3} + 742T_{5}^{2} + 598T_{5} + 529 $

$ T_{7}^{10} - 6 T_{7}^{9} + 14 T_{7}^{8} + 4 T_{7}^{7} - 35 T_{7}^{6} - 142 T_{7}^{5} + 621 T_{7}^{4} + \cdots + 529 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ 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
$3$	$ T^{10} + 7 T^{9} + \cdots + 1 $ T^10 + 7T^9 + 27T^8 + 68T^7 + 124T^6 + 142T^5 + 103T^4 + 28T^3 + 20T^2 + 8*T + 1
$5$	$ T^{10} + 3 T^{9} + \cdots + 529 $ T^10 + 3T^9 - 2T^8 - 28T^7 - 18T^6 + T^5 + 201T^4 + 350T^3 + 742T^2 + 598T + 529
$7$	$ T^{10} - 6 T^{9} + \cdots + 529 $ T^10 - 6T^9 + 14T^8 + 4T^7 - 35T^6 - 142T^5 + 621T^4 - 1064T^3 + 1346T^2 - 805*T + 529
$11$	$ T^{10} - 7 T^{9} + \cdots + 529 $ T^10 - 7T^9 + 38T^8 - 211T^7 + 883T^6 - 3013T^5 + 8353T^4 - 14284T^3 + 14859T^2 - 4485*T + 529
$13$	$ T^{10} - 8 T^{9} + \cdots + 1 $ T^10 - 8T^9 + 42T^8 - 160T^7 + 499T^6 - 934T^5 + 861T^4 + 570T^3 + 126T^2 + 15*T + 1
$17$	$ T^{10} - T^{9} + \cdots + 529 $ T^10 - T^9 - 43T^8 - 89T^7 + 1167T^6 + 1022T^5 + 6986T^4 + 1000T^3 + 12497T^2 - 3312T + 529
$19$	$ T^{10} - 2 T^{9} + \cdots + 541696 $ T^10 - 2T^9 + 48T^8 - 8T^7 - 688T^6 - 3552T^5 + 8512T^4 + 35072T^3 + 62208T^2 - 11776*T + 541696
$23$	$ T^{10} $ T^10
$29$	$ T^{10} + 8 T^{9} + \cdots + 4932841 $ T^10 + 8T^9 + 42T^8 + 83T^7 - 414T^6 + 1682T^5 + 17647T^4 + 51295T^3 + 633066T^2 - 2092182*T + 4932841
$31$	$ T^{10} + 12 T^{9} + \cdots + 17161 $ T^10 + 12T^9 + 78T^8 + 298T^7 + 606T^6 + 430T^5 - 285T^4 + 683T^3 + 6139T^2 + 7074*T + 17161
$37$	$ T^{10} - 25 T^{9} + \cdots + 529 $ T^10 - 25T^9 + 284T^8 - 1985T^7 + 10003T^6 - 39205T^5 + 118231T^4 - 254142T^3 + 354117T^2 + 17871*T + 529
$41$	$ T^{10} + 26 T^{9} + \cdots + 1849 $ T^10 + 26T^9 + 280T^8 + 1626T^7 + 5679T^6 + 11958T^5 + 14744T^4 + 9850T^3 + 8611T^2 - 1075*T + 1849
$43$	$ T^{10} - 22 T^{9} + \cdots + 64009 $ T^10 - 22T^9 + 220T^8 - 1331T^7 + 5566T^6 - 17567T^5 + 44044T^4 - 86273T^3 + 123541T^2 - 108537*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 + 48T^9 + 1138T^8 + 17279T^7 + 183824T^6 + 1423643T^5 + 8197754T^4 + 35754581T^3 + 122227037T^2 + 320709953*T + 517426009
$59$	$ T^{10} - T^{9} + \cdots + 4489 $ T^10 - T^9 - 65T^8 - 133T^7 + 5600T^6 - 41581T^5 + 209254T^4 - 423787T^3 + 1142450T^2 + 93532T + 4489
$61$	$ T^{10} - 36 T^{9} + \cdots + 529 $ T^10 - 36T^9 + 658T^8 - 6671T^7 + 40990T^6 - 154661T^5 + 385850T^4 - 513797T^3 + 275181T^2 - 4393*T + 529
$67$	$ T^{10} + \cdots + 1113757129 $ T^10 + 10T^9 - 21T^8 - 199T^7 + 9230T^6 + 183391T^5 + 2100583T^4 + 16463787T^3 + 104349598T^2 + 388795450*T + 1113757129
$71$	$ T^{10} + 14 T^{9} + \cdots + 529 $ T^10 + 14T^9 + 86T^8 + 214T^7 + 906T^6 - 1132T^5 + 1213T^4 - 2345T^3 + 2975T^2 - 1932*T + 529
$73$	$ T^{10} + 3 T^{9} + \cdots + 982081 $ T^10 + 3T^9 + 31T^8 + 731T^7 + 6560T^6 + 45365T^5 + 321456T^4 + 1460721T^3 + 3040922T^2 + 1791728*T + 982081
$79$	$ T^{10} + \cdots + 517426009 $ T^10 - 18T^9 + 390T^8 - 5590T^7 + 75199T^6 - 696277T^5 + 6322540T^4 - 50760034T^3 + 240079244T^2 - 542538697*T + 517426009
$83$	$ T^{10} + 15 T^{9} + \cdots + 529 $ T^10 + 15T^9 + 225T^8 + 867T^7 + 2346T^6 + 3004T^5 + 5482T^4 + 3316T^3 - 750T^2 - 1702*T + 529
$89$	$ T^{10} + \cdots + 78310985281 $ T^10 + 19T^9 + 350T^8 + 7519T^7 + 95121T^6 + 741267T^5 + 12326229T^4 + 123422048T^3 + 216876775T^2 + 4293040781*T + 78310985281
$97$	$ T^{10} - 21 T^{9} + \cdots + 2374681 $ T^10 - 21T^9 + 551T^8 - 9371T^7 + 116183T^6 - 861090T^5 + 4165492T^4 - 12702282T^3 + 23055209T^2 - 12999876*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)

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

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