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Properties

Label 2880.1.j.b
Level $2880$
Weight $1$
Character orbit 2880.j
Analytic conductor $1.437$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -4, -15, 60
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2880,1,Mod(1279,2880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2880.1279"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2880.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{15})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.1194393600.9

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - i q^{5} - 2 i q^{17} - q^{25} - q^{49} - 2 i q^{53} + 2 q^{61} - 2 q^{85} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{25} - 2 q^{49} + 4 q^{61} - 4 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1279.1
1.00000i
− 1.00000i
0 0 0 − 1.00000i 0 0 0 0 0
1279.2 0 0 0 1.00000i 0 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
60.h even 2 1 RM by \(\Q(\sqrt{15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.1.j.b 2
3.b odd 2 1 inner 2880.1.j.b 2
4.b odd 2 1 CM 2880.1.j.b 2
5.b even 2 1 inner 2880.1.j.b 2
8.b even 2 1 180.1.f.a ✓ 2
8.d odd 2 1 180.1.f.a ✓ 2
12.b even 2 1 inner 2880.1.j.b 2
15.d odd 2 1 CM 2880.1.j.b 2
20.d odd 2 1 inner 2880.1.j.b 2
24.f even 2 1 180.1.f.a ✓ 2
24.h odd 2 1 180.1.f.a ✓ 2
40.e odd 2 1 180.1.f.a ✓ 2
40.f even 2 1 180.1.f.a ✓ 2
40.i odd 4 1 900.1.c.a 1
40.i odd 4 1 900.1.c.b 1
40.k even 4 1 900.1.c.a 1
40.k even 4 1 900.1.c.b 1
60.h even 2 1 RM 2880.1.j.b 2
72.j odd 6 2 1620.1.p.b 4
72.l even 6 2 1620.1.p.b 4
72.n even 6 2 1620.1.p.b 4
72.p odd 6 2 1620.1.p.b 4
120.i odd 2 1 180.1.f.a ✓ 2
120.m even 2 1 180.1.f.a ✓ 2
120.q odd 4 1 900.1.c.a 1
120.q odd 4 1 900.1.c.b 1
120.w even 4 1 900.1.c.a 1
120.w even 4 1 900.1.c.b 1
360.z odd 6 2 1620.1.p.b 4
360.bd even 6 2 1620.1.p.b 4
360.bh odd 6 2 1620.1.p.b 4
360.bk even 6 2 1620.1.p.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.1.f.a ✓ 2 8.b even 2 1
180.1.f.a ✓ 2 8.d odd 2 1
180.1.f.a ✓ 2 24.f even 2 1
180.1.f.a ✓ 2 24.h odd 2 1
180.1.f.a ✓ 2 40.e odd 2 1
180.1.f.a ✓ 2 40.f even 2 1
180.1.f.a ✓ 2 120.i odd 2 1
180.1.f.a ✓ 2 120.m even 2 1
900.1.c.a 1 40.i odd 4 1
900.1.c.a 1 40.k even 4 1
900.1.c.a 1 120.q odd 4 1
900.1.c.a 1 120.w even 4 1
900.1.c.b 1 40.i odd 4 1
900.1.c.b 1 40.k even 4 1
900.1.c.b 1 120.q odd 4 1
900.1.c.b 1 120.w even 4 1
1620.1.p.b 4 72.j odd 6 2
1620.1.p.b 4 72.l even 6 2
1620.1.p.b 4 72.n even 6 2
1620.1.p.b 4 72.p odd 6 2
1620.1.p.b 4 360.z odd 6 2
1620.1.p.b 4 360.bd even 6 2
1620.1.p.b 4 360.bh odd 6 2
1620.1.p.b 4 360.bk even 6 2
2880.1.j.b 2 1.a even 1 1 trivial
2880.1.j.b 2 3.b odd 2 1 inner
2880.1.j.b 2 4.b odd 2 1 CM
2880.1.j.b 2 5.b even 2 1 inner
2880.1.j.b 2 12.b even 2 1 inner
2880.1.j.b 2 15.d odd 2 1 CM
2880.1.j.b 2 20.d odd 2 1 inner
2880.1.j.b 2 60.h even 2 1 RM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(2880, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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