Properties

Label 2320.2.a.s
Level $2320$
Weight $2$
Character orbit 2320.a
Self dual yes
Analytic conductor $18.525$
Analytic rank $1$
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] = [2320,2,Mod(1,2320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2320, 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("2320.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.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: \(18.5252932689\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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 145)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 2 \) 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
2.17009
−1.48119
0.311108
0 −1.70928 0 −1.00000 0 −0.630898 0 −0.0783777 0
1.2 0 0.806063 0 −1.00000 0 4.15633 0 −2.35026 0
1.3 0 2.90321 0 −1.00000 0 −1.52543 0 5.42864 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 2320.2.a.s 3
4.b odd 2 1 145.2.a.d 3
8.b even 2 1 9280.2.a.bm 3
8.d odd 2 1 9280.2.a.bu 3
12.b even 2 1 1305.2.a.o 3
20.d odd 2 1 725.2.a.d 3
20.e even 4 2 725.2.b.d 6
28.d even 2 1 7105.2.a.p 3
60.h even 2 1 6525.2.a.bh 3
116.d odd 2 1 4205.2.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.d 3 4.b odd 2 1
725.2.a.d 3 20.d odd 2 1
725.2.b.d 6 20.e even 4 2
1305.2.a.o 3 12.b even 2 1
2320.2.a.s 3 1.a even 1 1 trivial
4205.2.a.e 3 116.d odd 2 1
6525.2.a.bh 3 60.h even 2 1
7105.2.a.p 3 28.d even 2 1
9280.2.a.bm 3 8.b even 2 1
9280.2.a.bu 3 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2320))\):

\( T_{3}^{3} - 2T_{3}^{2} - 4T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{3} - 2T_{7}^{2} - 8T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{3} + 8T_{11}^{2} + 16T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 2 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$11$ \( T^{3} + 8 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{3} + 6 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$17$ \( T^{3} - 40T + 76 \) Copy content Toggle raw display
$19$ \( T^{3} - 28T - 52 \) Copy content Toggle raw display
$23$ \( T^{3} + 14 T^{2} + \cdots + 76 \) Copy content Toggle raw display
$29$ \( (T - 1)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} + 12 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$37$ \( T^{3} - 4 T^{2} + \cdots + 68 \) Copy content Toggle raw display
$41$ \( T^{3} + 10 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$43$ \( T^{3} - 10 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$47$ \( T^{3} + 18 T^{2} + \cdots - 92 \) Copy content Toggle raw display
$53$ \( T^{3} - 10 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$59$ \( T^{3} + 4 T^{2} + \cdots + 80 \) Copy content Toggle raw display
$61$ \( T^{3} - 6 T^{2} + \cdots + 40 \) Copy content Toggle raw display
$67$ \( T^{3} - 10 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$71$ \( T^{3} + 24 T^{2} + \cdots + 368 \) Copy content Toggle raw display
$73$ \( T^{3} + 4 T^{2} + \cdots - 1108 \) Copy content Toggle raw display
$79$ \( T^{3} + 8 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$83$ \( T^{3} - 2 T^{2} + \cdots - 52 \) Copy content Toggle raw display
$89$ \( T^{3} - 22 T^{2} + \cdots - 200 \) Copy content Toggle raw display
$97$ \( T^{3} + 36 T^{2} + \cdots + 452 \) Copy content Toggle raw display
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