Properties

Label 3040.2.j.d
Level $3040$
Weight $2$
Character orbit 3040.j
Analytic conductor $24.275$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3040,2,Mod(2431,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3040.2431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.j (of order \(2\), degree \(1\), 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: \(24.2745222145\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} - \beta_{7} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} - \beta_{7} q^{7} + ( - 2 \beta_{7} + \beta_{2}) q^{11} + (2 \beta_{4} + \beta_{3}) q^{13} - \beta_1 q^{15} - 3 q^{17} + (2 \beta_{2} - \beta_1) q^{19} + ( - \beta_{4} + 2 \beta_{3}) q^{21} + (\beta_{7} + 3 \beta_{2}) q^{23} + q^{25} + 3 \beta_1 q^{27} + ( - 2 \beta_{4} + 3 \beta_{3}) q^{29} + (\beta_{5} - 4 \beta_1) q^{31} + ( - 3 \beta_{4} + 3 \beta_{3}) q^{33} - \beta_{7} q^{35} + ( - \beta_{4} - \beta_{3}) q^{37} + (\beta_{7} - 5 \beta_{2}) q^{39} + ( - 2 \beta_{4} + 2 \beta_{3}) q^{41} + (2 \beta_{7} - 4 \beta_{2}) q^{43} + ( - 4 \beta_{7} + 3 \beta_{2}) q^{47} - \beta_{6} q^{49} + 3 \beta_1 q^{51} + ( - 2 \beta_{4} - \beta_{3}) q^{53} + ( - 2 \beta_{7} + \beta_{2}) q^{55} + ( - 2 \beta_{4} - 2 \beta_{3} + 3) q^{57} + (2 \beta_{5} + 3 \beta_1) q^{59} + ( - \beta_{6} + 10) q^{61} + (2 \beta_{4} + \beta_{3}) q^{65} + (3 \beta_{5} + 3 \beta_1) q^{67} + ( - 2 \beta_{4} - 5 \beta_{3}) q^{69} + 2 \beta_{5} q^{71} + (\beta_{6} - 5) q^{73} - \beta_1 q^{75} + ( - \beta_{6} - 12) q^{77} + (3 \beta_{5} + 2 \beta_1) q^{79} - 9 q^{81} + ( - 2 \beta_{7} - \beta_{2}) q^{83} - 3 q^{85} + ( - 5 \beta_{7} + \beta_{2}) q^{87} + 2 \beta_{4} q^{89} + (4 \beta_{5} + \beta_1) q^{91} + ( - \beta_{6} + 12) q^{93} + (2 \beta_{2} - \beta_1) q^{95} + ( - \beta_{4} + 3 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} - 24 q^{17} + 8 q^{25} + 24 q^{57} + 80 q^{61} - 40 q^{73} - 96 q^{77} - 72 q^{81} - 24 q^{85} + 96 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{5} + 2\zeta_{24}^{4} + \zeta_{24}^{3} - \zeta_{24} - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{5} + 2\zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24} - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -4\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{6} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( 2\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{4} + \beta_{3} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( 2\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(1217\) \(1921\) \(2661\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
0.965926 + 0.258819i
−0.965926 + 0.258819i
−0.965926 0.258819i
0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 + 0.965926i
−0.258819 0.965926i
0.258819 0.965926i
0 −1.73205 0 1.00000 0 3.44949i 0 0 0
2431.2 0 −1.73205 0 1.00000 0 1.44949i 0 0 0
2431.3 0 −1.73205 0 1.00000 0 1.44949i 0 0 0
2431.4 0 −1.73205 0 1.00000 0 3.44949i 0 0 0
2431.5 0 1.73205 0 1.00000 0 3.44949i 0 0 0
2431.6 0 1.73205 0 1.00000 0 1.44949i 0 0 0
2431.7 0 1.73205 0 1.00000 0 1.44949i 0 0 0
2431.8 0 1.73205 0 1.00000 0 3.44949i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2431.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3040.2.j.d 8
4.b odd 2 1 inner 3040.2.j.d 8
19.b odd 2 1 inner 3040.2.j.d 8
76.d even 2 1 inner 3040.2.j.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.j.d 8 1.a even 1 1 trivial
3040.2.j.d 8 4.b odd 2 1 inner
3040.2.j.d 8 19.b odd 2 1 inner
3040.2.j.d 8 76.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(3040, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 14 T^{2} + 25)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 58 T^{2} + 625)^{2} \) Copy content Toggle raw display
$17$ \( (T + 3)^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 26 T^{2} + 361)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 110 T^{2} + 1849)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 106 T^{2} + 2209)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 112 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 120 T^{2} + 144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 200 T^{2} + 8464)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 58 T^{2} + 625)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 118 T^{2} + 25)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 20 T + 76)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 198 T^{2} + 2025)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 32)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 10 T + 1)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 168 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 80 T^{2} + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 40 T^{2} + 16)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 88 T^{2} + 400)^{2} \) Copy content Toggle raw display
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