Properties

Label 2320.2.d.e
Level $2320$
Weight $2$
Character orbit 2320.d
Analytic conductor $18.525$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2320,2,Mod(929,2320)]
 
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, 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("2320.929");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.d (of order \(2\), degree \(1\), not 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: \(18.5252932689\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 580)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 5\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(1857\) \(2031\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
929.1
1.93185i
0.517638i
1.93185i
0.517638i
0 1.41421i 0 −1.73205 1.41421i 0 1.41421i 0 1.00000 0
929.2 0 1.41421i 0 1.73205 1.41421i 0 1.41421i 0 1.00000 0
929.3 0 1.41421i 0 −1.73205 + 1.41421i 0 1.41421i 0 1.00000 0
929.4 0 1.41421i 0 1.73205 + 1.41421i 0 1.41421i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2320.2.d.e 4
4.b odd 2 1 580.2.c.a 4
5.b even 2 1 inner 2320.2.d.e 4
12.b even 2 1 5220.2.g.c 4
20.d odd 2 1 580.2.c.a 4
20.e even 4 2 2900.2.a.h 4
60.h even 2 1 5220.2.g.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.c.a 4 4.b odd 2 1
580.2.c.a 4 20.d odd 2 1
2320.2.d.e 4 1.a even 1 1 trivial
2320.2.d.e 4 5.b even 2 1 inner
2900.2.a.h 4 20.e even 4 2
5220.2.g.c 4 12.b even 2 1
5220.2.g.c 4 60.h even 2 1

Hecke kernels

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

\( T_{3}^{2} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 2T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 16T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 28T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 28T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 196T^{2} + 8836 \) Copy content Toggle raw display
$53$ \( T^{4} + 16T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T + 2)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 172T^{2} + 484 \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 84T^{2} + 36 \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T - 50)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 148T^{2} + 676 \) Copy content Toggle raw display
$89$ \( (T^{2} - 8 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 228T^{2} + 6084 \) Copy content Toggle raw display
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