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Properties

Label 360.2.bd.a
Level $360$
Weight $2$
Character orbit 360.bd
Analytic conductor $2.875$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,2,Mod(59,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.bd (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: \(2.87461447277\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_1) q^{2} + (\beta_{4} - \beta_{3}) q^{3} + 2 \beta_{2} q^{4} + (\beta_{6} + \beta_{5} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{6}+ \cdots - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_1) q^{2} + (\beta_{4} - \beta_{3}) q^{3} + 2 \beta_{2} q^{4} + (\beta_{6} + \beta_{5} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{6}+ \cdots + ( - 18 \beta_{2} + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 12 q^{9} - 8 q^{10} + 36 q^{11} + 24 q^{14} - 16 q^{16} - 40 q^{19} - 16 q^{25} - 36 q^{30} + 24 q^{36} - 8 q^{40} + 12 q^{41} + 32 q^{46} + 4 q^{49} + 36 q^{51} + 48 q^{56} - 12 q^{59} - 64 q^{64} + 12 q^{65} + 36 q^{75} - 40 q^{76} - 36 q^{81} + 12 q^{90} - 48 q^{91} + 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{5} + 2\beta_1 ) / 3 \) 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

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
0.258819 + 0.965926i
0.965926 − 0.258819i
−0.258819 − 0.965926i
−0.965926 + 0.258819i
0.258819 − 0.965926i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 − 0.258819i
−1.22474 + 0.707107i −0.866025 − 1.50000i 1.00000 − 1.73205i −0.448288 + 2.19067i 2.12132 + 1.22474i −1.22474 − 2.12132i 2.82843i −1.50000 + 2.59808i −1.00000 − 3.00000i
59.2 −1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 − 1.73205i 1.67303 − 1.48356i −2.12132 − 1.22474i −1.22474 − 2.12132i 2.82843i −1.50000 + 2.59808i −1.00000 + 3.00000i
59.3 1.22474 − 0.707107i −0.866025 − 1.50000i 1.00000 − 1.73205i 0.448288 − 2.19067i −2.12132 − 1.22474i 1.22474 + 2.12132i − 2.82843i −1.50000 + 2.59808i −1.00000 − 3.00000i
59.4 1.22474 − 0.707107i 0.866025 + 1.50000i 1.00000 − 1.73205i −1.67303 + 1.48356i 2.12132 + 1.22474i 1.22474 + 2.12132i − 2.82843i −1.50000 + 2.59808i −1.00000 + 3.00000i
299.1 −1.22474 − 0.707107i −0.866025 + 1.50000i 1.00000 + 1.73205i −0.448288 − 2.19067i 2.12132 − 1.22474i −1.22474 + 2.12132i − 2.82843i −1.50000 − 2.59808i −1.00000 + 3.00000i
299.2 −1.22474 − 0.707107i 0.866025 − 1.50000i 1.00000 + 1.73205i 1.67303 + 1.48356i −2.12132 + 1.22474i −1.22474 + 2.12132i − 2.82843i −1.50000 − 2.59808i −1.00000 − 3.00000i
299.3 1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 + 1.73205i 0.448288 + 2.19067i −2.12132 + 1.22474i 1.22474 − 2.12132i 2.82843i −1.50000 − 2.59808i −1.00000 + 3.00000i
299.4 1.22474 + 0.707107i 0.866025 − 1.50000i 1.00000 + 1.73205i −1.67303 − 1.48356i 2.12132 − 1.22474i 1.22474 − 2.12132i 2.82843i −1.50000 − 2.59808i −1.00000 − 3.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
9.d odd 6 1 inner
40.e odd 2 1 inner
45.h odd 6 1 inner
72.l even 6 1 inner
360.bd even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.bd.a ✓ 8
5.b even 2 1 inner 360.2.bd.a ✓ 8
8.d odd 2 1 inner 360.2.bd.a ✓ 8
9.d odd 6 1 inner 360.2.bd.a ✓ 8
40.e odd 2 1 inner 360.2.bd.a ✓ 8
45.h odd 6 1 inner 360.2.bd.a ✓ 8
72.l even 6 1 inner 360.2.bd.a ✓ 8
360.bd even 6 1 inner 360.2.bd.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bd.a ✓ 8 1.a even 1 1 trivial
360.2.bd.a ✓ 8 5.b even 2 1 inner
360.2.bd.a ✓ 8 8.d odd 2 1 inner
360.2.bd.a ✓ 8 9.d odd 6 1 inner
360.2.bd.a ✓ 8 40.e odd 2 1 inner
360.2.bd.a ✓ 8 45.h odd 6 1 inner
360.2.bd.a ✓ 8 72.l even 6 1 inner
360.2.bd.a ✓ 8 360.bd even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 6T_{7}^{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 8 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 9 T + 27)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 27)^{4} \) Copy content Toggle raw display
$19$ \( (T + 5)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 72 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 54 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{2} - 3 T + 3)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 81 T^{2} + 6561)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 50 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} - 54 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 54 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
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