Properties

Label 2880.2.b.a
Level $2880$
Weight $2$
Character orbit 2880.b
Analytic conductor $22.997$
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] = [2880,2,Mod(2591,2880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2880.2591");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.b (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: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q - q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + \beta_{3} q^{7} + (\beta_{3} - \beta_1) q^{11} + \beta_{2} q^{13} + ( - \beta_{4} + \beta_{2}) q^{17} + \beta_{5} q^{19} + q^{25} + (\beta_{6} - 2) q^{29} + ( - 2 \beta_{3} + \beta_1) q^{31} - \beta_{3} q^{35} + (2 \beta_{4} - \beta_{2}) q^{37} + (\beta_{4} - 2 \beta_{2}) q^{41} + ( - \beta_{7} + \beta_{5}) q^{43} + \beta_{7} q^{47} - 3 q^{49} + ( - \beta_{6} - 4) q^{53} + ( - \beta_{3} + \beta_1) q^{55} + ( - \beta_{3} - 5 \beta_1) q^{59} + 2 \beta_{4} q^{61} - \beta_{2} q^{65} + ( - \beta_{7} - \beta_{5}) q^{67} + ( - \beta_{7} - \beta_{5}) q^{71} - 2 q^{73} + ( - \beta_{6} - 10) q^{77} + (2 \beta_{3} - 3 \beta_1) q^{79} + (2 \beta_{3} - 2 \beta_1) q^{83} + (\beta_{4} - \beta_{2}) q^{85} + (\beta_{4} + 2 \beta_{2}) q^{89} + ( - 2 \beta_{7} + \beta_{5}) q^{91} - \beta_{5} q^{95} + (\beta_{6} + 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} + 8 q^{25} - 16 q^{29} - 24 q^{49} - 32 q^{53} - 16 q^{73} - 80 q^{77} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} + 16\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} + 6\nu^{6} - \nu^{5} - 13\nu^{3} + 36\nu^{2} - 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{7} - \nu^{5} - 29\nu^{3} - 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{7} - \nu^{5} - 13\nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{7} + 2\nu^{5} + 4\nu^{4} - 26\nu^{3} + 10\nu + 14 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 2\nu^{5} - 58\nu^{3} + 22\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{7} - 4\nu^{5} + 4\nu^{4} + 52\nu^{3} - 20\nu + 14 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 3\beta_{6} - \beta_{5} + 2\beta_{4} - 6\beta_{3} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - 3\beta_{2} + 9\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} - 3\beta_{6} + 2\beta_{5} + 4\beta_{4} - 6\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + 2\beta_{5} - 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{7} - 15\beta_{6} + 11\beta_{5} - 22\beta_{4} + 30\beta_{3} ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4\beta_{4} + 12\beta_{2} - 27\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 29\beta_{7} + 39\beta_{6} - 29\beta_{5} - 58\beta_{4} + 78\beta_{3} ) / 24 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1
1.14412 + 1.14412i
0.437016 + 0.437016i
−0.437016 + 0.437016i
−1.14412 + 1.14412i
−1.14412 1.14412i
−0.437016 0.437016i
0.437016 0.437016i
1.14412 1.14412i
0 0 0 −1.00000 0 3.16228i 0 0 0
2591.2 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.3 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.4 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.5 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.6 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.7 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.8 0 0 0 −1.00000 0 3.16228i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.b.a 8
3.b odd 2 1 2880.2.b.d yes 8
4.b odd 2 1 inner 2880.2.b.a 8
8.b even 2 1 2880.2.b.d yes 8
8.d odd 2 1 2880.2.b.d yes 8
12.b even 2 1 2880.2.b.d yes 8
24.f even 2 1 inner 2880.2.b.a 8
24.h odd 2 1 inner 2880.2.b.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2880.2.b.a 8 1.a even 1 1 trivial
2880.2.b.a 8 4.b odd 2 1 inner
2880.2.b.a 8 24.f even 2 1 inner
2880.2.b.a 8 24.h odd 2 1 inner
2880.2.b.d yes 8 3.b odd 2 1
2880.2.b.d yes 8 8.b even 2 1
2880.2.b.d yes 8 8.d odd 2 1
2880.2.b.d yes 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2880, [\chi])\):

\( T_{7}^{2} + 10 \) Copy content Toggle raw display
\( T_{29}^{2} + 4T_{29} - 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 10)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 28 T^{2} + 36)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 44 T^{2} + 324)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 56 T^{2} + 144)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 56 T^{2} + 144)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 36)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 88 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 140 T^{2} + 900)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 164 T^{2} + 6084)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 72)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 104 T^{2} + 144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 8 T - 24)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 220 T^{2} + 8100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 176 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 176 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 152 T^{2} + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 112 T^{2} + 576)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 260 T^{2} + 900)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 12 T - 4)^{4} \) Copy content Toggle raw display
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