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Properties

Label 2240.2.g.j
Level $2240$
Weight $2$
Character orbit 2240.g
Analytic conductor $17.886$
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] = [2240,2,Mod(449,2240)] 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("2240.449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-4,0,0,0,-12,0,0,0,0,0,12,0,0,0,16] 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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{3} + ( - \beta_{3} - \beta_{2} - 1) q^{5} - \beta_{2} q^{7} - 3 q^{9} + (2 \beta_{3} - 2 \beta_1) q^{11} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{13} + ( - 2 \beta_{3} + 3 \beta_{2} + 3) q^{15}+ \cdots + ( - 6 \beta_{3} + 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - 12 q^{9} + 12 q^{15} + 16 q^{19} + 8 q^{29} + 16 q^{31} - 4 q^{35} + 24 q^{39} - 24 q^{41} + 12 q^{45} - 4 q^{49} - 24 q^{55} - 16 q^{59} - 24 q^{61} - 4 q^{65} - 48 q^{69} - 24 q^{71} - 8 q^{79}+ \cdots - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

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

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−1.22474 − 1.22474i
1.22474 − 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
0 − 2.44949i 0 −2.22474 + 0.224745i 0 − 1.00000i 0 −3.00000 0
449.2 0 − 2.44949i 0 0.224745 + 2.22474i 0 1.00000i 0 −3.00000 0
449.3 0 2.44949i 0 −2.22474 − 0.224745i 0 1.00000i 0 −3.00000 0
449.4 0 2.44949i 0 0.224745 − 2.22474i 0 − 1.00000i 0 −3.00000 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 2240.2.g.j 4
4.b odd 2 1 2240.2.g.i 4
5.b even 2 1 inner 2240.2.g.j 4
8.b even 2 1 70.2.c.a ✓ 4
8.d odd 2 1 560.2.g.e 4
20.d odd 2 1 2240.2.g.i 4
24.f even 2 1 5040.2.t.t 4
24.h odd 2 1 630.2.g.g 4
40.e odd 2 1 560.2.g.e 4
40.f even 2 1 70.2.c.a ✓ 4
40.i odd 4 1 350.2.a.g 2
40.i odd 4 1 350.2.a.h 2
40.k even 4 1 2800.2.a.bl 2
40.k even 4 1 2800.2.a.bm 2
56.h odd 2 1 490.2.c.e 4
56.j odd 6 2 490.2.i.f 8
56.p even 6 2 490.2.i.c 8
120.i odd 2 1 630.2.g.g 4
120.m even 2 1 5040.2.t.t 4
120.w even 4 1 3150.2.a.bs 2
120.w even 4 1 3150.2.a.bt 2
280.c odd 2 1 490.2.c.e 4
280.s even 4 1 2450.2.a.bl 2
280.s even 4 1 2450.2.a.bq 2
280.bf even 6 2 490.2.i.c 8
280.bk odd 6 2 490.2.i.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a ✓ 4 8.b even 2 1
70.2.c.a ✓ 4 40.f even 2 1
350.2.a.g 2 40.i odd 4 1
350.2.a.h 2 40.i odd 4 1
490.2.c.e 4 56.h odd 2 1
490.2.c.e 4 280.c odd 2 1
490.2.i.c 8 56.p even 6 2
490.2.i.c 8 280.bf even 6 2
490.2.i.f 8 56.j odd 6 2
490.2.i.f 8 280.bk odd 6 2
560.2.g.e 4 8.d odd 2 1
560.2.g.e 4 40.e odd 2 1
630.2.g.g 4 24.h odd 2 1
630.2.g.g 4 120.i odd 2 1
2240.2.g.i 4 4.b odd 2 1
2240.2.g.i 4 20.d odd 2 1
2240.2.g.j 4 1.a even 1 1 trivial
2240.2.g.j 4 5.b even 2 1 inner
2450.2.a.bl 2 280.s even 4 1
2450.2.a.bq 2 280.s even 4 1
2800.2.a.bl 2 40.k even 4 1
2800.2.a.bm 2 40.k even 4 1
3150.2.a.bs 2 120.w even 4 1
3150.2.a.bt 2 120.w even 4 1
5040.2.t.t 4 24.f even 2 1
5040.2.t.t 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 \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{2} + 6 \) Copy content Toggle raw display
\( T_{11}^{2} - 24 \) Copy content Toggle raw display
\( T_{19}^{2} - 8T_{19} + 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 20T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 8 T + 10)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 80T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{4} + 80T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{4} + 120T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T^{2} + 8 T + 10)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12 T + 30)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 20)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$89$ \( (T - 10)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 264T^{2} + 3600 \) Copy content Toggle raw display
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