Properties

Label 2880.2.t.a
Level $2880$
Weight $2$
Character orbit 2880.t
Analytic conductor $22.997$
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] = [2880,2,Mod(721,2880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 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("2880.721");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.t (of order \(4\), degree \(2\), 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: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{8} q^{5} + (2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 2 \zeta_{8}) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{5} + (2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 2 \zeta_{8}) q^{7} + 2 \zeta_{8} q^{11} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{13} + (\zeta_{8}^{3} - \zeta_{8} - 4) q^{17} + (4 \zeta_{8}^{3} + \zeta_{8}^{2} - 1) q^{19} + ( - \zeta_{8}^{3} - 4 \zeta_{8}^{2} - \zeta_{8}) q^{23} + \zeta_{8}^{2} q^{25} + (2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{29} + (4 \zeta_{8}^{3} - 4 \zeta_{8} + 2) q^{31} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{35} + (6 \zeta_{8}^{2} - 2 \zeta_{8} + 6) q^{37} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2 \zeta_{8}) q^{41} + (2 \zeta_{8}^{2} - 8 \zeta_{8} + 2) q^{43} + (5 \zeta_{8}^{3} - 5 \zeta_{8}) q^{47} + (8 \zeta_{8}^{3} - 8 \zeta_{8} - 5) q^{49} + (4 \zeta_{8}^{2} + 4) q^{53} - 2 \zeta_{8}^{2} q^{55} + ( - 6 \zeta_{8}^{2} - 2 \zeta_{8} - 6) q^{59} + ( - 12 \zeta_{8}^{3} - \zeta_{8}^{2} + 1) q^{61} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} - 2) q^{65} + ( - 8 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{67} - 8 \zeta_{8}^{2} q^{71} + ( - 2 \zeta_{8}^{3} + 6 \zeta_{8}^{2} - 2 \zeta_{8}) q^{73} + (4 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 4) q^{77} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8} + 8) q^{79} + (10 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 4) q^{83} + (\zeta_{8}^{2} + 4 \zeta_{8} + 1) q^{85} + ( - 4 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 4 \zeta_{8}) q^{89} - 4 \zeta_{8} q^{91} + ( - \zeta_{8}^{3} + \zeta_{8} + 4) q^{95} + (8 \zeta_{8}^{3} - 8 \zeta_{8} - 2) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{13} - 16 q^{17} - 4 q^{19} + 8 q^{29} + 8 q^{31} + 8 q^{35} + 24 q^{37} + 8 q^{43} - 20 q^{49} + 16 q^{53} - 24 q^{59} + 4 q^{61} - 8 q^{65} + 8 q^{67} - 16 q^{77} + 32 q^{79} + 16 q^{83} + 4 q^{85} + 16 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{8}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
0 0 0 −0.707107 + 0.707107i 0 4.82843i 0 0 0
721.2 0 0 0 0.707107 0.707107i 0 0.828427i 0 0 0
2161.1 0 0 0 −0.707107 0.707107i 0 4.82843i 0 0 0
2161.2 0 0 0 0.707107 + 0.707107i 0 0.828427i 0 0 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
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.t.a 4
3.b odd 2 1 960.2.s.a 4
4.b odd 2 1 720.2.t.a 4
12.b even 2 1 240.2.s.a 4
16.e even 4 1 inner 2880.2.t.a 4
16.f odd 4 1 720.2.t.a 4
24.f even 2 1 1920.2.s.a 4
24.h odd 2 1 1920.2.s.b 4
48.i odd 4 1 960.2.s.a 4
48.i odd 4 1 1920.2.s.b 4
48.k even 4 1 240.2.s.a 4
48.k even 4 1 1920.2.s.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.s.a 4 12.b even 2 1
240.2.s.a 4 48.k even 4 1
720.2.t.a 4 4.b odd 2 1
720.2.t.a 4 16.f odd 4 1
960.2.s.a 4 3.b odd 2 1
960.2.s.a 4 48.i odd 4 1
1920.2.s.a 4 24.f even 2 1
1920.2.s.a 4 48.k even 4 1
1920.2.s.b 4 24.h odd 2 1
1920.2.s.b 4 48.i odd 4 1
2880.2.t.a 4 1.a even 1 1 trivial
2880.2.t.a 4 16.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 24T_{7}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(2880, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 16 \) Copy content Toggle raw display
$13$ \( T^{4} + 8 T^{3} + 32 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$17$ \( (T^{2} + 8 T + 14)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + 8 T^{2} - 56 T + 196 \) Copy content Toggle raw display
$23$ \( T^{4} + 36T^{2} + 196 \) Copy content Toggle raw display
$29$ \( T^{4} - 8 T^{3} + 32 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 28)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 24 T^{3} + 288 T^{2} + \cdots + 4624 \) Copy content Toggle raw display
$41$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$47$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 8 T + 32)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 24 T^{3} + 288 T^{2} + \cdots + 4624 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 20164 \) Copy content Toggle raw display
$67$ \( T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$71$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 88T^{2} + 784 \) Copy content Toggle raw display
$79$ \( (T^{2} - 16 T + 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 4624 \) Copy content Toggle raw display
$89$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$97$ \( (T^{2} + 4 T - 124)^{2} \) Copy content Toggle raw display
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