Properties

Label 2320.2.a.u
Level $2320$
Weight $2$
Character orbit 2320.a
Self dual yes
Analytic conductor $18.525$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.580484.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 7x^{3} - 3x^{2} + 8x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1160)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{3} + q^{5} + ( - \beta_{3} + \beta_{2} - 1) q^{7} + ( - \beta_{3} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{3} + q^{5} + ( - \beta_{3} + \beta_{2} - 1) q^{7} + ( - \beta_{3} - \beta_1 + 2) q^{9} + ( - \beta_{2} + \beta_1 - 2) q^{11} + \beta_{4} q^{13} + (\beta_{3} - 1) q^{15} + ( - \beta_{4} + \beta_1) q^{17} + ( - \beta_{2} - \beta_1 - 2) q^{19} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + \cdots - 1) q^{21}+ \cdots + ( - 2 \beta_{4} - \beta_{2} + \beta_1 - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{3} + 5 q^{5} - 7 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 3 q^{3} + 5 q^{5} - 7 q^{7} + 8 q^{9} - 10 q^{11} + q^{13} - 3 q^{15} - q^{17} - 10 q^{19} - 8 q^{21} - q^{23} + 5 q^{25} - 12 q^{27} + 5 q^{29} - 7 q^{31} - 8 q^{33} - 7 q^{35} - 8 q^{37} - 3 q^{39} - 4 q^{41} - 9 q^{43} + 8 q^{45} - 8 q^{47} + 16 q^{49} - 2 q^{51} - 9 q^{53} - 10 q^{55} + 2 q^{57} - 29 q^{59} + q^{61} - 16 q^{63} + q^{65} - 24 q^{67} + 17 q^{69} + 12 q^{71} - 9 q^{73} - 3 q^{75} + 2 q^{77} - 13 q^{79} - 3 q^{81} - 10 q^{83} - q^{85} - 3 q^{87} + 4 q^{89} + 4 q^{91} - 26 q^{93} - 10 q^{95} - 15 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 7x^{3} - 3x^{2} + 8x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + \nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{4} + \nu^{3} + 7\nu^{2} - 3\nu - 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{4} + \beta_{3} - 4\beta_{2} + 6\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{4} + 4\beta_{3} - 4\beta_{2} + 5\beta _1 + 11 \) 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
1.12721
−1.44568
2.60873
−0.366905
−1.92335
0 −3.04362 0 1.00000 0 −3.43080 0 6.26362 0
1.2 0 −2.50610 0 1.00000 0 3.62303 0 3.28054 0
1.3 0 −0.857770 0 1.00000 0 −2.23769 0 −2.26423 0
1.4 0 1.02751 0 1.00000 0 −0.377000 0 −1.94422 0
1.5 0 2.37998 0 1.00000 0 −4.57754 0 2.66429 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.u 5
4.b odd 2 1 1160.2.a.j 5
8.b even 2 1 9280.2.a.cl 5
8.d odd 2 1 9280.2.a.cg 5
20.d odd 2 1 5800.2.a.t 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.2.a.j 5 4.b odd 2 1
2320.2.a.u 5 1.a even 1 1 trivial
5800.2.a.t 5 20.d odd 2 1
9280.2.a.cg 5 8.d odd 2 1
9280.2.a.cl 5 8.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2320))\):

\( T_{3}^{5} + 3T_{3}^{4} - 7T_{3}^{3} - 20T_{3}^{2} + 8T_{3} + 16 \) Copy content Toggle raw display
\( T_{7}^{5} + 7T_{7}^{4} - T_{7}^{3} - 88T_{7}^{2} - 160T_{7} - 48 \) Copy content Toggle raw display
\( T_{11}^{5} + 10T_{11}^{4} + 12T_{11}^{3} - 112T_{11}^{2} - 224T_{11} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + 3 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T - 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 7 T^{4} + \cdots - 48 \) Copy content Toggle raw display
$11$ \( T^{5} + 10 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{5} - T^{4} + \cdots - 536 \) Copy content Toggle raw display
$17$ \( T^{5} + T^{4} + \cdots - 72 \) Copy content Toggle raw display
$19$ \( T^{5} + 10 T^{4} + \cdots + 1856 \) Copy content Toggle raw display
$23$ \( T^{5} + T^{4} + \cdots - 16 \) Copy content Toggle raw display
$29$ \( (T - 1)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} + 7 T^{4} + \cdots - 16 \) Copy content Toggle raw display
$37$ \( T^{5} + 8 T^{4} + \cdots + 8032 \) Copy content Toggle raw display
$41$ \( T^{5} + 4 T^{4} + \cdots + 992 \) Copy content Toggle raw display
$43$ \( T^{5} + 9 T^{4} + \cdots + 464 \) Copy content Toggle raw display
$47$ \( T^{5} + 8 T^{4} + \cdots - 4096 \) Copy content Toggle raw display
$53$ \( T^{5} + 9 T^{4} + \cdots + 29992 \) Copy content Toggle raw display
$59$ \( T^{5} + 29 T^{4} + \cdots - 38336 \) Copy content Toggle raw display
$61$ \( T^{5} - T^{4} + \cdots - 2984 \) Copy content Toggle raw display
$67$ \( T^{5} + 24 T^{4} + \cdots - 15616 \) Copy content Toggle raw display
$71$ \( T^{5} - 12 T^{4} + \cdots - 16384 \) Copy content Toggle raw display
$73$ \( T^{5} + 9 T^{4} + \cdots - 5832 \) Copy content Toggle raw display
$79$ \( T^{5} + 13 T^{4} + \cdots - 15376 \) Copy content Toggle raw display
$83$ \( T^{5} + 10 T^{4} + \cdots - 64 \) Copy content Toggle raw display
$89$ \( T^{5} - 4 T^{4} + \cdots + 50656 \) Copy content Toggle raw display
$97$ \( T^{5} + 15 T^{4} + \cdots + 33944 \) Copy content Toggle raw display
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