Properties

Label 2320.2.a.w
Level $2320$
Weight $2$
Character orbit 2320.a
Self dual yes
Analytic conductor $18.525$
Analytic rank $0$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.6083172.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 13x^{3} + 10x^{2} + 40x - 28 \) 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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + q^{5} + ( - \beta_{3} + \beta_{2}) q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + q^{5} + ( - \beta_{3} + \beta_{2}) q^{7} + (\beta_{2} + 2) q^{9} + ( - \beta_{3} + 1) q^{11} + ( - \beta_{3} - \beta_1 + 3) q^{13} + \beta_1 q^{15} + (\beta_{4} + 1) q^{17} + (\beta_{4} + \beta_{2}) q^{19} + ( - \beta_{4} + \beta_1 + 1) q^{21} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots + 1) q^{23}+ \cdots + ( - 3 \beta_{3} - 2 \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{3} + 5 q^{5} + q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{3} + 5 q^{5} + q^{7} + 12 q^{9} + 4 q^{11} + 13 q^{13} + q^{15} + 3 q^{17} + 8 q^{21} + 7 q^{23} + 5 q^{25} + 4 q^{27} - 5 q^{29} - 11 q^{31} + 4 q^{33} + q^{35} + 6 q^{37} - 21 q^{39} + 16 q^{41} - 9 q^{43} + 12 q^{45} - 2 q^{47} + 16 q^{49} - 2 q^{51} + 7 q^{53} + 4 q^{55} + 2 q^{57} - 5 q^{59} + 7 q^{61} + 28 q^{63} + 13 q^{65} + 4 q^{67} + 23 q^{69} - 12 q^{71} + 7 q^{73} + q^{75} + 26 q^{77} - 45 q^{79} - 7 q^{81} + 22 q^{83} + 3 q^{85} - q^{87} + 24 q^{89} + 20 q^{91} - 10 q^{93} + 17 q^{97} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 13x^{3} + 10x^{2} + 40x - 28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 7\nu + 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 11\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 2\beta_{3} + 9\beta_{2} + 3\beta _1 + 34 \) 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
−2.57991
−2.36347
0.689426
2.13670
3.11726
0 −2.57991 0 1.00000 0 2.42414 0 3.65592 0
1.2 0 −2.36347 0 1.00000 0 −2.16993 0 2.58601 0
1.3 0 0.689426 0 1.00000 0 −4.55109 0 −2.52469 0
1.4 0 2.13670 0 1.00000 0 4.33279 0 1.56547 0
1.5 0 3.11726 0 1.00000 0 0.964080 0 6.71730 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.w 5
4.b odd 2 1 1160.2.a.i 5
8.b even 2 1 9280.2.a.ch 5
8.d odd 2 1 9280.2.a.cj 5
20.d odd 2 1 5800.2.a.v 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.2.a.i 5 4.b odd 2 1
2320.2.a.w 5 1.a even 1 1 trivial
5800.2.a.v 5 20.d odd 2 1
9280.2.a.ch 5 8.b even 2 1
9280.2.a.cj 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2320))\):

\( T_{3}^{5} - T_{3}^{4} - 13T_{3}^{3} + 10T_{3}^{2} + 40T_{3} - 28 \) Copy content Toggle raw display
\( T_{7}^{5} - T_{7}^{4} - 25T_{7}^{3} + 28T_{7}^{2} + 100T_{7} - 100 \) Copy content Toggle raw display
\( T_{11}^{5} - 4T_{11}^{4} - 16T_{11}^{3} + 28T_{11}^{2} + 40T_{11} - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - T^{4} + \cdots - 28 \) Copy content Toggle raw display
$5$ \( (T - 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - T^{4} + \cdots - 100 \) Copy content Toggle raw display
$11$ \( T^{5} - 4 T^{4} + \cdots - 48 \) Copy content Toggle raw display
$13$ \( T^{5} - 13 T^{4} + \cdots + 824 \) Copy content Toggle raw display
$17$ \( T^{5} - 3 T^{4} + \cdots - 1964 \) Copy content Toggle raw display
$19$ \( T^{5} - 76 T^{3} + \cdots - 432 \) Copy content Toggle raw display
$23$ \( T^{5} - 7 T^{4} + \cdots + 900 \) Copy content Toggle raw display
$29$ \( (T + 1)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} + 11 T^{4} + \cdots + 36 \) Copy content Toggle raw display
$37$ \( T^{5} - 6 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$41$ \( T^{5} - 16 T^{4} + \cdots + 14816 \) Copy content Toggle raw display
$43$ \( T^{5} + 9 T^{4} + \cdots + 1156 \) Copy content Toggle raw display
$47$ \( T^{5} + 2 T^{4} + \cdots - 64 \) Copy content Toggle raw display
$53$ \( T^{5} - 7 T^{4} + \cdots - 33336 \) Copy content Toggle raw display
$59$ \( T^{5} + 5 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$61$ \( T^{5} - 7 T^{4} + \cdots - 11112 \) Copy content Toggle raw display
$67$ \( T^{5} - 4 T^{4} + \cdots + 192 \) Copy content Toggle raw display
$71$ \( T^{5} + 12 T^{4} + \cdots + 30464 \) Copy content Toggle raw display
$73$ \( T^{5} - 7 T^{4} + \cdots + 1588 \) Copy content Toggle raw display
$79$ \( T^{5} + 45 T^{4} + \cdots - 36204 \) Copy content Toggle raw display
$83$ \( T^{5} - 22 T^{4} + \cdots - 25776 \) Copy content Toggle raw display
$89$ \( T^{5} - 24 T^{4} + \cdots + 88672 \) Copy content Toggle raw display
$97$ \( T^{5} - 17 T^{4} + \cdots - 196 \) Copy content Toggle raw display
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