Properties

Label 9280.2.a.bw
Level $9280$
Weight $2$
Character orbit 9280.a
Self dual yes
Analytic conductor $74.101$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9280,2,Mod(1,9280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9280.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9280.a (trivial)

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: yes
Analytic conductor: \(74.1011730757\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 580)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} + q^{5} + ( - \beta_{2} + 1) q^{7} + (\beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{3} + q^{5} + ( - \beta_{2} + 1) q^{7} + (\beta_1 + 4) q^{9} + (\beta_{2} + 3) q^{11} + (\beta_1 + 1) q^{13} + (\beta_{2} + 1) q^{15} + ( - \beta_{2} - \beta_1) q^{17} + ( - \beta_{2} - \beta_1 + 2) q^{19} + (2 \beta_{2} - \beta_1 - 5) q^{21} + ( - \beta_{2} + \beta_1) q^{23} + q^{25} + (2 \beta_{2} + 2 \beta_1 + 4) q^{27} + q^{29} + ( - \beta_{2} - \beta_1 - 2) q^{31} + (2 \beta_{2} + \beta_1 + 9) q^{33} + ( - \beta_{2} + 1) q^{35} + (3 \beta_{2} - \beta_1 + 4) q^{37} + (2 \beta_{2} + 2 \beta_1 + 4) q^{39} + ( - \beta_1 + 3) q^{41} + ( - \beta_{2} + 2 \beta_1 + 5) q^{43} + (\beta_1 + 4) q^{45} + ( - \beta_{2} + 3) q^{47} + ( - 4 \beta_{2} + \beta_1) q^{49} + ( - 3 \beta_1 - 9) q^{51} + (2 \beta_{2} - \beta_1 - 3) q^{53} + (\beta_{2} + 3) q^{55} + (2 \beta_{2} - 3 \beta_1 - 7) q^{57} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{59} + ( - 2 \beta_{2} + \beta_1 + 7) q^{61} + ( - 5 \beta_{2} + 1) q^{63} + (\beta_1 + 1) q^{65} + (\beta_{2} - \beta_1 - 4) q^{67} + (2 \beta_{2} + \beta_1 - 3) q^{69} + (2 \beta_{2} - \beta_1 + 3) q^{71} + (\beta_{2} - 2 \beta_1 - 1) q^{73} + (\beta_{2} + 1) q^{75} + ( - \beta_1 - 3) q^{77} + (\beta_{2} + 2 \beta_1 + 1) q^{79} + (4 \beta_{2} + 3 \beta_1 + 10) q^{81} + ( - 3 \beta_{2} + 3) q^{83} + ( - \beta_{2} - \beta_1) q^{85} + (\beta_{2} + 1) q^{87} + (2 \beta_{2} - 2 \beta_1 + 6) q^{89} + ( - 2 \beta_{2} - 2) q^{91} + ( - 2 \beta_{2} - 3 \beta_1 - 11) q^{93} + ( - \beta_{2} - \beta_1 + 2) q^{95} + ( - \beta_{2} - 13) q^{97} + (5 \beta_{2} + 4 \beta_1 + 15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} + 3 q^{5} + 4 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} + 3 q^{5} + 4 q^{7} + 11 q^{9} + 8 q^{11} + 2 q^{13} + 2 q^{15} + 2 q^{17} + 8 q^{19} - 16 q^{21} + 3 q^{25} + 8 q^{27} + 3 q^{29} - 4 q^{31} + 24 q^{33} + 4 q^{35} + 10 q^{37} + 8 q^{39} + 10 q^{41} + 14 q^{43} + 11 q^{45} + 10 q^{47} + 3 q^{49} - 24 q^{51} - 10 q^{53} + 8 q^{55} - 20 q^{57} + 8 q^{59} + 22 q^{61} + 8 q^{63} + 2 q^{65} - 12 q^{67} - 12 q^{69} + 8 q^{71} - 2 q^{73} + 2 q^{75} - 8 q^{77} + 23 q^{81} + 12 q^{83} + 2 q^{85} + 2 q^{87} + 18 q^{89} - 4 q^{91} - 28 q^{93} + 8 q^{95} - 38 q^{97} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.571993
−2.08613
2.51414
0 −2.67282 0 1.00000 0 4.67282 0 4.14399 0
1.2 0 1.35194 0 1.00000 0 0.648061 0 −1.17226 0
1.3 0 3.32088 0 1.00000 0 −1.32088 0 8.02827 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(29\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9280.2.a.bw 3
4.b odd 2 1 9280.2.a.bk 3
8.b even 2 1 2320.2.a.m 3
8.d odd 2 1 580.2.a.c 3
24.f even 2 1 5220.2.a.x 3
40.e odd 2 1 2900.2.a.g 3
40.k even 4 2 2900.2.c.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.a.c 3 8.d odd 2 1
2320.2.a.m 3 8.b even 2 1
2900.2.a.g 3 40.e odd 2 1
2900.2.c.f 6 40.k even 4 2
5220.2.a.x 3 24.f even 2 1
9280.2.a.bk 3 4.b odd 2 1
9280.2.a.bw 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9280))\):

\( T_{3}^{3} - 2T_{3}^{2} - 8T_{3} + 12 \) Copy content Toggle raw display
\( T_{7}^{3} - 4T_{7}^{2} - 4T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{3} - 8T_{11}^{2} + 12T_{11} + 12 \) Copy content Toggle raw display
\( T_{13}^{3} - 2T_{13}^{2} - 20T_{13} + 24 \) Copy content Toggle raw display
\( T_{19}^{3} - 8T_{19}^{2} - 16T_{19} + 164 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 2 T^{2} + \cdots + 12 \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 4 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{3} - 8 T^{2} + \cdots + 12 \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} + \cdots + 24 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} + \cdots + 108 \) Copy content Toggle raw display
$19$ \( T^{3} - 8 T^{2} + \cdots + 164 \) Copy content Toggle raw display
$23$ \( T^{3} - 24T + 36 \) Copy content Toggle raw display
$29$ \( (T - 1)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} + 4 T^{2} + \cdots + 36 \) Copy content Toggle raw display
$37$ \( T^{3} - 10 T^{2} + \cdots + 508 \) Copy content Toggle raw display
$41$ \( T^{3} - 10 T^{2} + \cdots + 24 \) Copy content Toggle raw display
$43$ \( T^{3} - 14 T^{2} + \cdots + 548 \) Copy content Toggle raw display
$47$ \( T^{3} - 10 T^{2} + \cdots - 12 \) Copy content Toggle raw display
$53$ \( T^{3} + 10 T^{2} + \cdots - 72 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} + \cdots + 1488 \) Copy content Toggle raw display
$61$ \( T^{3} - 22 T^{2} + \cdots - 104 \) Copy content Toggle raw display
$67$ \( T^{3} + 12 T^{2} + \cdots - 68 \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$73$ \( T^{3} + 2 T^{2} + \cdots - 324 \) Copy content Toggle raw display
$79$ \( T^{3} - 108T - 244 \) Copy content Toggle raw display
$83$ \( T^{3} - 12 T^{2} + \cdots + 108 \) Copy content Toggle raw display
$89$ \( T^{3} - 18 T^{2} + \cdots + 72 \) Copy content Toggle raw display
$97$ \( T^{3} + 38 T^{2} + \cdots + 1908 \) Copy content Toggle raw display
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