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Properties

Label 2880.3.l.g
Level $2880$
Weight $3$
Character orbit 2880.l
Analytic conductor $78.474$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [2880,3,Mod(1601,2880)] 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("2880.1601"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2880.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,16,0,0,0,0,0,24,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(78.4743161358\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + (\beta_{3} + 4) q^{7} + (2 \beta_{2} - 7 \beta_1) q^{11} + ( - 5 \beta_{3} + 6) q^{13} + ( - 8 \beta_{2} + 4 \beta_1) q^{17} - 2 \beta_{3} q^{19} + (6 \beta_{2} - 18 \beta_1) q^{23}+ \cdots + (18 \beta_{3} - 58) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{7} + 24 q^{13} - 20 q^{25} - 56 q^{31} - 152 q^{37} + 48 q^{43} - 92 q^{49} + 40 q^{55} + 168 q^{61} - 208 q^{67} + 216 q^{73} - 56 q^{79} - 160 q^{85} - 104 q^{91} - 232 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 7\nu ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 7\beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1601.1
−1.58114 − 0.707107i
1.58114 + 0.707107i
−1.58114 + 0.707107i
1.58114 − 0.707107i
0 0 0 − 2.23607i 0 0.837722 0 0 0
1601.2 0 0 0 − 2.23607i 0 7.16228 0 0 0
1601.3 0 0 0 2.23607i 0 0.837722 0 0 0
1601.4 0 0 0 2.23607i 0 7.16228 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.3.l.g 4
3.b odd 2 1 inner 2880.3.l.g 4
4.b odd 2 1 2880.3.l.c 4
8.b even 2 1 45.3.c.a ✓ 4
8.d odd 2 1 720.3.l.a 4
12.b even 2 1 2880.3.l.c 4
24.f even 2 1 720.3.l.a 4
24.h odd 2 1 45.3.c.a ✓ 4
40.e odd 2 1 3600.3.l.v 4
40.f even 2 1 225.3.c.c 4
40.i odd 4 2 225.3.d.b 8
40.k even 4 2 3600.3.c.i 8
72.j odd 6 2 405.3.i.d 8
72.n even 6 2 405.3.i.d 8
120.i odd 2 1 225.3.c.c 4
120.m even 2 1 3600.3.l.v 4
120.q odd 4 2 3600.3.c.i 8
120.w even 4 2 225.3.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.c.a ✓ 4 8.b even 2 1
45.3.c.a ✓ 4 24.h odd 2 1
225.3.c.c 4 40.f even 2 1
225.3.c.c 4 120.i odd 2 1
225.3.d.b 8 40.i odd 4 2
225.3.d.b 8 120.w even 4 2
405.3.i.d 8 72.j odd 6 2
405.3.i.d 8 72.n even 6 2
720.3.l.a 4 8.d odd 2 1
720.3.l.a 4 24.f even 2 1
2880.3.l.c 4 4.b odd 2 1
2880.3.l.c 4 12.b even 2 1
2880.3.l.g 4 1.a even 1 1 trivial
2880.3.l.g 4 3.b odd 2 1 inner
3600.3.c.i 8 40.k even 4 2
3600.3.c.i 8 120.q odd 4 2
3600.3.l.v 4 40.e odd 2 1
3600.3.l.v 4 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2880, [\chi])\):

\( T_{7}^{2} - 8T_{7} + 6 \) Copy content Toggle raw display
\( T_{19}^{2} - 40 \) 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
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 8 T + 6)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 236T^{2} + 6084 \) Copy content Toggle raw display
$13$ \( (T^{2} - 12 T - 214)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 704 T^{2} + 82944 \) Copy content Toggle raw display
$19$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1656 T^{2} + 219024 \) Copy content Toggle raw display
$29$ \( T^{4} + 2024T^{2} + 144 \) Copy content Toggle raw display
$31$ \( (T^{2} + 28 T - 1764)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 76 T + 634)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 644 T^{2} + 101124 \) Copy content Toggle raw display
$43$ \( (T^{2} - 24 T - 856)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2096 T^{2} + 788544 \) Copy content Toggle raw display
$53$ \( T^{4} + 2576 T^{2} + 419904 \) Copy content Toggle raw display
$59$ \( T^{4} + 5996 T^{2} + 3392964 \) Copy content Toggle raw display
$61$ \( (T^{2} - 84 T + 1724)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 104 T + 2064)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 19424 T^{2} + 93083904 \) Copy content Toggle raw display
$73$ \( (T^{2} - 108 T + 2876)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 28 T - 6564)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 7056 T^{2} + 4981824 \) Copy content Toggle raw display
$89$ \( T^{4} + 14436 T^{2} + 33385284 \) Copy content Toggle raw display
$97$ \( (T^{2} + 116 T + 124)^{2} \) Copy content Toggle raw display
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