LMFDB - Newform orbit 529.2.c.h

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

−0.0681534

−

0.474017i

0.915415

−

2.00448i

1.69894

−

0.498853i

−1.69894

1.96068i

−1.01255

−

0.297310i

2.85848

−

1.83703i

−0.750131

−

1.64256i

−1.21537

−

1.40261i

1.04518

0.671699i

170.1

−1.52977

0.449181i

−0.154861

−

0.178719i

0.455922

−

0.293003i

−0.455922

−

3.17101i

0.317178

0.203838i

0.378794

−

0.829443i

1.52231

−

1.75684i

0.418986

−

2.91411i

2.12181

4.64612i

177.1

1.64589

−

1.89945i

1.34125

−

0.861971i

−0.614354

−

4.27292i

0.614354

1.34525i

0.570276

−

3.96635i

1.17894

−

0.346167i

−4.89867

−

3.14818i

−0.190279

0.416652i

3.56639

1.04719i

255.1

1.85380

1.19136i

0.357685

2.48775i

1.18639

2.59784i

−1.18639

−

0.348356i

−2.30075

5.03793i

−1.55384

1.79323i

−0.268423

1.86692i

−3.18251

0.934468i

−1.78431

−

2.05921i

266.1

1.64589

1.89945i

1.34125

0.861971i

−0.614354

4.27292i

0.614354

−

1.34525i

0.570276

3.96635i

1.17894

0.346167i

−4.89867

3.14818i

−0.190279

−

0.416652i

3.56639

−

1.04719i

334.1

1.85380

−

1.19136i

0.357685

−

2.48775i

1.18639

−

2.59784i

−1.18639

0.348356i

−2.30075

−

5.03793i

−1.55384

−

1.79323i

−0.268423

−

1.86692i

−3.18251

−

0.934468i

−1.78431

2.05921i

399.1

−0.0681534

0.474017i

0.915415

2.00448i

1.69894

0.498853i

−1.69894

−

1.96068i

−1.01255

0.297310i

2.85848

1.83703i

−0.750131

1.64256i

−1.21537

1.40261i

1.04518

−

0.671699i

466.1

0.0982369

−

0.215109i

−0.459493

−

0.134919i

1.27310

1.46924i

−1.27310

0.818172i

−0.0741615

0.0855869i

−0.362362

2.52028i

0.894911

−

0.262769i

−2.33083

−

1.49793i

0.0509305

0.354230i

487.1

0.0982369

0.215109i

−0.459493

0.134919i

1.27310

−

1.46924i

−1.27310

−

0.818172i

−0.0741615

−

0.0855869i

−0.362362

−

2.52028i

0.894911

0.262769i

−2.33083

1.49793i

0.0509305

−

0.354230i

501.1

−1.52977

−

0.449181i

−0.154861

0.178719i

0.455922

0.293003i

−0.455922

3.17101i

0.317178

−

0.203838i

0.378794

0.829443i

1.52231

1.75684i

0.418986

2.91411i

2.12181

−

4.64612i

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.h		10
23.b	odd	2	1		529.2.c.i		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	2		529.2.c.f		10
23.c	even	11	1	inner	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	2		529.2.c.g		10
23.d	odd	22	1		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	23.c	even	11	2
529.2.c.g		10	23.d	odd	22	2
529.2.c.h		10	1.a	even	1	1	trivial
529.2.c.h		10	23.c	even	11	1	inner
529.2.c.i		10	23.b	odd	2	1
529.2.c.i		10	23.d	odd	22	1
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} + 5T_{2}^{8} + 13T_{2}^{7} - 30T_{2}^{6} - T_{2}^{5} + 70T_{2}^{4} - 5T_{2}^{3} + 20T_{2}^{2} - 3T_{2} + 1 $

$ T_{5}^{10} + 8 T_{5}^{9} + 42 T_{5}^{8} + 149 T_{5}^{7} + 389 T_{5}^{6} + 736 T_{5}^{5} + 1092 T_{5}^{4} + \cdots + 529 $

$ T_{7}^{10} - 5 T_{7}^{9} + 14 T_{7}^{8} - 37 T_{7}^{7} + 130 T_{7}^{6} - 199 T_{7}^{5} + 511 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 + 13T^7 - 30T^6 - T^5 + 70T^4 - 5T^3 + 20T^2 - 3T + 1
$3$	$ 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
$5$	$ T^{10} + 8 T^{9} + \cdots + 529 $ T^10 + 8T^9 + 42T^8 + 149T^7 + 389T^6 + 736T^5 + 1092T^4 + 1465T^3 + 1754T^2 + 1426*T + 529
$7$	$ T^{10} - 5 T^{9} + \cdots + 529 $ T^10 - 5T^9 + 14T^8 - 37T^7 + 130T^6 - 199T^5 + 511T^4 - 1235T^3 + 1599T^2 - 1219*T + 529
$11$	$ T^{10} + 7 T^{9} + \cdots + 529 $ T^10 + 7T^9 + 16T^8 + 35T^7 + 201T^6 + 450T^5 - 392T^4 - 3195T^3 + 10305T^2 - 828*T + 529
$13$	$ T^{10} + 3 T^{9} + \cdots + 1 $ T^10 + 3T^9 + 20T^8 + 71T^7 + 246T^6 + 573T^5 + 1389T^4 - 13T^3 - 17T^2 - 7*T + 1
$17$	$ T^{10} + 12 T^{9} + \cdots + 529 $ T^10 + 12T^9 + 111T^8 + 562T^7 + 2289T^6 + 3587T^5 + 2894T^4 + 8042T^3 + 9967T^2 - 4025*T + 529
$19$	$ T^{10} + 2 T^{9} + \cdots + 541696 $ T^10 + 2T^9 + 48T^8 + 360T^7 + 2128T^6 + 4608T^5 + 4992T^4 - 25216T^3 - 67328T^2 - 117760*T + 541696
$23$	$ T^{10} $ T^10
$29$	$ T^{10} - 3 T^{9} + \cdots + 4932841 $ T^10 - 3T^9 + 9T^8 - 225T^7 + 906T^6 - 5468T^5 + 28680T^4 + 23960T^3 + 559773T^2 + 2525277*T + 4932841
$31$	$ T^{10} - 10 T^{9} + \cdots + 17161 $ T^10 - 10T^9 + 34T^8 - 76T^7 + 430T^6 - 1880T^5 + 5028T^4 - 9789T^3 + 17667T^2 - 17423*T + 17161
$37$	$ T^{10} + 3 T^{9} + \cdots + 529 $ T^10 + 3T^9 + 42T^8 - 61T^7 + 785T^6 - 7864T^5 + 9342T^4 - 48853T^3 + 363731T^2 + 23368*T + 529
$41$	$ T^{10} + 4 T^{9} + \cdots + 1849 $ T^10 + 4T^9 + 27T^8 - 189T^7 + 1224T^6 - 4311T^5 + 10355T^4 - 1733T^3 + 2462T^2 + 6020*T + 1849
$43$	$ T^{10} + 22 T^{8} + \cdots + 64009 $ T^10 + 22T^8 - 66T^7 + 473T^6 - 2398T^5 + 8107T^4 - 13552T^3 + 48400T^2 - 97405T + 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 + 18T^9 + 126T^8 + 1124T^7 + 17361T^6 + 165736T^5 + 1179787T^4 + 6992046T^3 + 25893492T^2 + 59119453*T + 517426009
$59$	$ T^{10} - T^{9} + \cdots + 4489 $ T^10 - T^9 + 100T^8 + 450T^7 + 705T^6 + 25068T^5 + 176925T^4 + 536194T^3 + 1120340T^2 - 120198T + 4489
$61$	$ T^{10} - 30 T^{9} + \cdots + 529 $ T^10 - 30T^9 + 416T^8 - 3152T^7 + 18935T^6 - 52986T^5 + 13929T^4 + 58914T^3 + 265820T^2 - 9775*T + 529
$67$	$ T^{10} + \cdots + 1113757129 $ T^10 + T^9 + 144T^8 + 1431T^7 + 6843T^6 + 115875T^5 + 1442937T^4 + 6116320T^3 + 34232925T^2 + 329625121T + 1113757129
$71$	$ T^{10} - 8 T^{9} + \cdots + 529 $ T^10 - 8T^9 + 108T^8 - 490T^7 + 1852T^6 - 2738T^5 + 718T^4 + 1087T^3 - 61T^2 - 667*T + 529
$73$	$ T^{10} + 25 T^{9} + \cdots + 982081 $ T^10 + 25T^9 + 328T^8 + 2722T^7 + 17659T^6 + 80422T^5 + 316099T^4 + 677554T^3 + 860722T^2 - 2285246*T + 982081
$79$	$ T^{10} + \cdots + 517426009 $ T^10 + 18T^9 + 126T^8 - 471T^7 - 8654T^6 + 234024T^5 + 4584177T^4 + 27899207T^3 + 96955120T^2 + 110914372*T + 517426009
$83$	$ T^{10} + 29 T^{9} + \cdots + 529 $ T^10 + 29T^9 + 280T^8 + 926T^7 + 1686T^6 + 2661T^5 + 6615T^4 + 5066T^3 + 5828T^2 + 2208*T + 529
$89$	$ T^{10} + \cdots + 78310985281 $ T^10 - 63T^9 + 2198T^8 - 53103T^7 + 969995T^6 - 14007828T^5 + 161216982T^4 - 1435693833T^3 + 9279247719T^2 - 38489331140*T + 78310985281
$97$	$ T^{10} - 12 T^{9} + \cdots + 2374681 $ T^10 - 12T^9 + 243T^8 - 5974T^7 + 82787T^6 - 556645T^5 + 2504844T^4 - 7855882T^3 + 10579273T^2 + 6338133*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)

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

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