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Properties

Label 1120.2.bi.a
Level $1120$
Weight $2$
Character orbit 1120.bi
Analytic conductor $8.943$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1120,2,Mod(463,1120)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1120, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1120.463"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.bi (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.94324502638\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 8 q^{17} - 8 q^{25} + 64 q^{43} + 32 q^{51} - 8 q^{65} - 40 q^{73} - 112 q^{75} - 72 q^{81} - 80 q^{83} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
463.1 0 −2.03802 − 2.03802i 0 1.56810 − 1.59407i 0 0.707107 + 0.707107i 0 5.30708i 0
463.2 0 −2.03802 − 2.03802i 0 −1.56810 + 1.59407i 0 −0.707107 − 0.707107i 0 5.30708i 0
463.3 0 −2.03800 − 2.03800i 0 1.22676 − 1.86951i 0 −0.707107 − 0.707107i 0 5.30689i 0
463.4 0 −2.03800 − 2.03800i 0 −1.22676 + 1.86951i 0 0.707107 + 0.707107i 0 5.30689i 0
463.5 0 −1.94776 − 1.94776i 0 2.03914 + 0.917560i 0 −0.707107 − 0.707107i 0 4.58750i 0
463.6 0 −1.94776 − 1.94776i 0 −2.03914 − 0.917560i 0 0.707107 + 0.707107i 0 4.58750i 0
463.7 0 −1.73985 − 1.73985i 0 −0.452388 − 2.18983i 0 −0.707107 − 0.707107i 0 3.05418i 0
463.8 0 −1.73985 − 1.73985i 0 0.452388 + 2.18983i 0 0.707107 + 0.707107i 0 3.05418i 0
463.9 0 −1.33262 − 1.33262i 0 −2.22159 + 0.254081i 0 −0.707107 − 0.707107i 0 0.551751i 0
463.10 0 −1.33262 − 1.33262i 0 2.22159 − 0.254081i 0 0.707107 + 0.707107i 0 0.551751i 0
463.11 0 −0.683238 − 0.683238i 0 −2.03938 − 0.917023i 0 0.707107 + 0.707107i 0 − 2.06637i 0
463.12 0 −0.683238 − 0.683238i 0 2.03938 + 0.917023i 0 −0.707107 − 0.707107i 0 − 2.06637i 0
463.13 0 −0.570077 − 0.570077i 0 −0.776798 + 2.09680i 0 −0.707107 − 0.707107i 0 − 2.35002i 0
463.14 0 −0.570077 − 0.570077i 0 0.776798 − 2.09680i 0 0.707107 + 0.707107i 0 − 2.35002i 0
463.15 0 −0.532572 − 0.532572i 0 1.25149 + 1.85304i 0 0.707107 + 0.707107i 0 − 2.43273i 0
463.16 0 −0.532572 − 0.532572i 0 −1.25149 − 1.85304i 0 −0.707107 − 0.707107i 0 − 2.43273i 0
463.17 0 −0.311152 − 0.311152i 0 −1.92300 + 1.14109i 0 0.707107 + 0.707107i 0 − 2.80637i 0
463.18 0 −0.311152 − 0.311152i 0 1.92300 − 1.14109i 0 −0.707107 − 0.707107i 0 − 2.80637i 0
463.19 0 0.161197 + 0.161197i 0 0.0283969 + 2.23589i 0 −0.707107 − 0.707107i 0 − 2.94803i 0
463.20 0 0.161197 + 0.161197i 0 −0.0283969 − 2.23589i 0 0.707107 + 0.707107i 0 − 2.94803i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 463.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.bi.a 72
4.b odd 2 1 280.2.w.a ✓ 72
5.c odd 4 1 inner 1120.2.bi.a 72
8.b even 2 1 280.2.w.a ✓ 72
8.d odd 2 1 inner 1120.2.bi.a 72
20.e even 4 1 280.2.w.a ✓ 72
40.i odd 4 1 280.2.w.a ✓ 72
40.k even 4 1 inner 1120.2.bi.a 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.w.a ✓ 72 4.b odd 2 1
280.2.w.a ✓ 72 8.b even 2 1
280.2.w.a ✓ 72 20.e even 4 1
280.2.w.a ✓ 72 40.i odd 4 1
1120.2.bi.a 72 1.a even 1 1 trivial
1120.2.bi.a 72 5.c odd 4 1 inner
1120.2.bi.a 72 8.d odd 2 1 inner
1120.2.bi.a 72 40.k even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1120, [\chi])\).