Properties

Label 45.2.j.a
Level $45$
Weight $2$
Character orbit 45.j
Analytic conductor $0.359$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,2,Mod(4,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 45.j (of order \(6\), degree \(2\), 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: \(0.359326809096\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( - \beta_{7} + \beta_{5}) q^{3} - \beta_{3} q^{4} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 1) q^{6} + (\beta_{6} - 2 \beta_1) q^{7} + (\beta_{7} - \beta_{6} - \beta_{5}) q^{8} + ( - \beta_{4} - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + ( - \beta_{7} + \beta_{5}) q^{3} - \beta_{3} q^{4} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 1) q^{6} + (\beta_{6} - 2 \beta_1) q^{7} + (\beta_{7} - \beta_{6} - \beta_{5}) q^{8} + ( - \beta_{4} - \beta_{3}) q^{9} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 1) q^{10}+ \cdots + (6 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{6} - 8 q^{10} - 12 q^{11} + 12 q^{15} + 4 q^{16} - 8 q^{19} + 12 q^{20} + 12 q^{21} + 12 q^{24} - 4 q^{25} + 24 q^{26} + 12 q^{29} - 12 q^{30} + 4 q^{31} - 4 q^{34} - 24 q^{35} - 12 q^{39} - 4 q^{40} - 24 q^{41} - 24 q^{44} - 16 q^{46} - 4 q^{49} + 24 q^{50} - 36 q^{51} + 24 q^{55} - 12 q^{56} + 24 q^{59} + 24 q^{60} - 8 q^{61} + 32 q^{64} + 60 q^{66} + 12 q^{69} + 12 q^{70} + 72 q^{71} - 12 q^{74} - 48 q^{75} - 12 q^{76} + 16 q^{79} - 48 q^{80} + 36 q^{81} - 36 q^{84} + 16 q^{85} - 36 q^{86} - 24 q^{89} - 36 q^{90} - 24 q^{91} + 20 q^{94} + 12 q^{95} - 60 q^{96} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} + \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{24}^{5} - \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{5} + 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} - \beta_{5} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + 2\beta_{5} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -2\beta_{7} + 3\beta_{6} + \beta_{5} - \beta_1 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−1.67303 + 0.965926i 1.67303 0.448288i 0.866025 1.50000i 0.358719 + 2.20711i −2.36603 + 2.36603i −0.776457 + 0.448288i 0.517638i 2.59808 1.50000i −2.73205 3.34607i
4.2 −0.448288 + 0.258819i 0.448288 + 1.67303i −0.866025 + 1.50000i −0.358719 2.20711i −0.633975 0.633975i 2.89778 1.67303i 1.93185i −2.59808 + 1.50000i 0.732051 + 0.896575i
4.3 0.448288 0.258819i −0.448288 1.67303i −0.866025 + 1.50000i 2.09077 0.792893i −0.633975 0.633975i −2.89778 + 1.67303i 1.93185i −2.59808 + 1.50000i 0.732051 0.896575i
4.4 1.67303 0.965926i −1.67303 + 0.448288i 0.866025 1.50000i −2.09077 + 0.792893i −2.36603 + 2.36603i 0.776457 0.448288i 0.517638i 2.59808 1.50000i −2.73205 + 3.34607i
34.1 −1.67303 0.965926i 1.67303 + 0.448288i 0.866025 + 1.50000i 0.358719 2.20711i −2.36603 2.36603i −0.776457 0.448288i 0.517638i 2.59808 + 1.50000i −2.73205 + 3.34607i
34.2 −0.448288 0.258819i 0.448288 1.67303i −0.866025 1.50000i −0.358719 + 2.20711i −0.633975 + 0.633975i 2.89778 + 1.67303i 1.93185i −2.59808 1.50000i 0.732051 0.896575i
34.3 0.448288 + 0.258819i −0.448288 + 1.67303i −0.866025 1.50000i 2.09077 + 0.792893i −0.633975 + 0.633975i −2.89778 1.67303i 1.93185i −2.59808 1.50000i 0.732051 + 0.896575i
34.4 1.67303 + 0.965926i −1.67303 0.448288i 0.866025 + 1.50000i −2.09077 0.792893i −2.36603 2.36603i 0.776457 + 0.448288i 0.517638i 2.59808 + 1.50000i −2.73205 3.34607i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.2.j.a 8
3.b odd 2 1 135.2.j.a 8
4.b odd 2 1 720.2.by.d 8
5.b even 2 1 inner 45.2.j.a 8
5.c odd 4 2 225.2.e.d 8
9.c even 3 1 inner 45.2.j.a 8
9.c even 3 1 405.2.b.d 4
9.d odd 6 1 135.2.j.a 8
9.d odd 6 1 405.2.b.c 4
12.b even 2 1 2160.2.by.c 8
15.d odd 2 1 135.2.j.a 8
15.e even 4 2 675.2.e.d 8
20.d odd 2 1 720.2.by.d 8
36.f odd 6 1 720.2.by.d 8
36.h even 6 1 2160.2.by.c 8
45.h odd 6 1 135.2.j.a 8
45.h odd 6 1 405.2.b.c 4
45.j even 6 1 inner 45.2.j.a 8
45.j even 6 1 405.2.b.d 4
45.k odd 12 2 225.2.e.d 8
45.k odd 12 2 2025.2.a.t 4
45.l even 12 2 675.2.e.d 8
45.l even 12 2 2025.2.a.r 4
60.h even 2 1 2160.2.by.c 8
180.n even 6 1 2160.2.by.c 8
180.p odd 6 1 720.2.by.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.j.a 8 1.a even 1 1 trivial
45.2.j.a 8 5.b even 2 1 inner
45.2.j.a 8 9.c even 3 1 inner
45.2.j.a 8 45.j even 6 1 inner
135.2.j.a 8 3.b odd 2 1
135.2.j.a 8 9.d odd 6 1
135.2.j.a 8 15.d odd 2 1
135.2.j.a 8 45.h odd 6 1
225.2.e.d 8 5.c odd 4 2
225.2.e.d 8 45.k odd 12 2
405.2.b.c 4 9.d odd 6 1
405.2.b.c 4 45.h odd 6 1
405.2.b.d 4 9.c even 3 1
405.2.b.d 4 45.j even 6 1
675.2.e.d 8 15.e even 4 2
675.2.e.d 8 45.l even 12 2
720.2.by.d 8 4.b odd 2 1
720.2.by.d 8 20.d odd 2 1
720.2.by.d 8 36.f odd 6 1
720.2.by.d 8 180.p odd 6 1
2025.2.a.r 4 45.l even 12 2
2025.2.a.t 4 45.k odd 12 2
2160.2.by.c 8 12.b even 2 1
2160.2.by.c 8 36.h even 6 1
2160.2.by.c 8 60.h even 2 1
2160.2.by.c 8 180.n even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} - 12 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( (T^{4} + 6 T^{3} + 30 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 2)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( (T^{4} - 6 T^{3} + 39 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{3} + 6 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 84 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$47$ \( T^{8} - 28 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$53$ \( (T^{4} + 16 T^{2} + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 12 T^{3} + \cdots + 576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4 T^{3} + \cdots + 5041)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 84 T^{6} + \cdots + 2313441 \) Copy content Toggle raw display
$71$ \( (T^{2} - 18 T + 54)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + 60 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 172 T^{6} + \cdots + 47458321 \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T - 99)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} - 336 T^{6} + \cdots + 592240896 \) Copy content Toggle raw display
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