Properties

Label 2320.2.d.d
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: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 290)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{2} + \beta_1) q^{5} + \beta_{2} q^{7} + (\beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{2} + \beta_1) q^{5} + \beta_{2} q^{7} + (\beta_{3} - 1) q^{9} - 2 q^{11} + \beta_1 q^{13} + ( - \beta_{3} + 3) q^{15} + ( - \beta_{2} - 2 \beta_1) q^{17} - 2 q^{19} + q^{21} + ( - 2 \beta_{2} + \beta_1) q^{23} - 5 q^{25} + ( - \beta_{2} + \beta_1) q^{27} + q^{29} + \beta_{3} q^{31} + 2 \beta_1 q^{33} + (\beta_{3} + 2) q^{35} + ( - 4 \beta_{2} + 4 \beta_1) q^{37} + ( - \beta_{3} + 4) q^{39} + 2 \beta_{3} q^{41} + ( - \beta_{2} - 2 \beta_1) q^{43} + ( - 2 \beta_{2} - 3 \beta_1) q^{45} + 4 \beta_1 q^{47} + ( - \beta_{3} + 4) q^{49} + (2 \beta_{3} - 9) q^{51} + (3 \beta_{2} + 4 \beta_1) q^{53} + (2 \beta_{2} - 2 \beta_1) q^{55} + 2 \beta_1 q^{57} + (\beta_{3} - 1) q^{59} + ( - \beta_{3} - 7) q^{61} + (3 \beta_{2} - \beta_1) q^{63} + (\beta_{3} - 3) q^{65} + ( - 4 \beta_{2} + 4 \beta_1) q^{67} + ( - \beta_{3} + 2) q^{69} + (2 \beta_{3} - 2) q^{71} + ( - 3 \beta_{2} + 6 \beta_1) q^{73} + 5 \beta_1 q^{75} - 2 \beta_{2} q^{77} + (3 \beta_{3} - 5) q^{79} + 2 \beta_{3} q^{81} + ( - 6 \beta_{2} + 2 \beta_1) q^{83} + ( - 3 \beta_{3} + 4) q^{85} - \beta_1 q^{87} - 2 q^{89} - q^{91} + ( - \beta_{2} + 3 \beta_1) q^{93} + (2 \beta_{2} - 2 \beta_1) q^{95} + ( - 2 \beta_{2} - 5 \beta_1) q^{97} + ( - 2 \beta_{3} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{9} - 8 q^{11} + 10 q^{15} - 8 q^{19} + 4 q^{21} - 20 q^{25} + 4 q^{29} + 2 q^{31} + 10 q^{35} + 14 q^{39} + 4 q^{41} + 14 q^{49} - 32 q^{51} - 2 q^{59} - 30 q^{61} - 10 q^{65} + 6 q^{69} - 4 q^{71} - 14 q^{79} + 4 q^{81} + 10 q^{85} - 8 q^{89} - 4 q^{91} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 5\beta_1 ) / 3 \) 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
1.61803i
0.618034i
0.618034i
1.61803i
0 2.61803i 0 2.23607i 0 0.381966i 0 −3.85410 0
929.2 0 0.381966i 0 2.23607i 0 2.61803i 0 2.85410 0
929.3 0 0.381966i 0 2.23607i 0 2.61803i 0 2.85410 0
929.4 0 2.61803i 0 2.23607i 0 0.381966i 0 −3.85410 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.d 4
4.b odd 2 1 290.2.b.a 4
5.b even 2 1 inner 2320.2.d.d 4
12.b even 2 1 2610.2.e.e 4
20.d odd 2 1 290.2.b.a 4
20.e even 4 1 1450.2.a.k 2
20.e even 4 1 1450.2.a.l 2
60.h even 2 1 2610.2.e.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.b.a 4 4.b odd 2 1
290.2.b.a 4 20.d odd 2 1
1450.2.a.k 2 20.e even 4 1
1450.2.a.l 2 20.e even 4 1
2320.2.d.d 4 1.a even 1 1 trivial
2320.2.d.d 4 5.b even 2 1 inner
2610.2.e.e 4 12.b even 2 1
2610.2.e.e 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 \(S_{2}^{\mathrm{new}}(2320, [\chi])\):

\( T_{3}^{4} + 7T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 7T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 7T^{2} + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 7T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T + 2)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 7T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{4} + 43T^{2} + 361 \) Copy content Toggle raw display
$19$ \( (T + 2)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 27T^{2} + 81 \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - T - 11)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T - 44)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 43T^{2} + 361 \) Copy content Toggle raw display
$47$ \( T^{4} + 112T^{2} + 256 \) Copy content Toggle raw display
$53$ \( T^{4} + 223 T^{2} + 11881 \) Copy content Toggle raw display
$59$ \( (T^{2} + T - 11)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 15 T + 45)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2 T - 44)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 243T^{2} + 6561 \) Copy content Toggle raw display
$79$ \( (T^{2} + 7 T - 89)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 232T^{2} + 1936 \) Copy content Toggle raw display
$89$ \( (T + 2)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 243T^{2} + 9801 \) Copy content Toggle raw display
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