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Properties

Label 420.2.s.a
Level $420$
Weight $2$
Character orbit 420.s
Analytic conductor $3.354$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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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] = [420,2,Mod(113,420)] 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("420.113"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.s (of order \(4\), degree \(2\), 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: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{3} + 24 q^{13} + 8 q^{15} + 4 q^{21} - 32 q^{25} + 8 q^{27} - 16 q^{31} - 20 q^{33} + 32 q^{37} - 8 q^{43} + 20 q^{45} + 52 q^{51} - 8 q^{55} - 28 q^{57} + 8 q^{63} - 24 q^{67} - 28 q^{75} - 12 q^{81}+ \cdots - 104 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 0 −1.65123 − 0.522921i 0 1.89765 − 1.18276i 0 −0.707107 + 0.707107i 0 2.45311 + 1.72692i 0
113.2 0 −1.62919 + 0.587996i 0 −0.677300 + 2.13102i 0 −0.707107 + 0.707107i 0 2.30852 − 1.91591i 0
113.3 0 −1.38370 − 1.04182i 0 1.37156 + 1.76602i 0 0.707107 − 0.707107i 0 0.829228 + 2.88312i 0
113.4 0 −1.27746 + 1.16966i 0 −2.20774 − 0.354811i 0 0.707107 − 0.707107i 0 0.263792 − 2.98838i 0
113.5 0 −1.04182 − 1.38370i 0 −1.37156 − 1.76602i 0 0.707107 − 0.707107i 0 −0.829228 + 2.88312i 0
113.6 0 −0.522921 − 1.65123i 0 −1.89765 + 1.18276i 0 −0.707107 + 0.707107i 0 −2.45311 + 1.72692i 0
113.7 0 −0.210923 + 1.71916i 0 −0.161137 − 2.23025i 0 −0.707107 + 0.707107i 0 −2.91102 − 0.725222i 0
113.8 0 0.587996 − 1.62919i 0 0.677300 − 2.13102i 0 −0.707107 + 0.707107i 0 −2.30852 − 1.91591i 0
113.9 0 0.625101 + 1.61532i 0 −0.398641 + 2.20025i 0 0.707107 − 0.707107i 0 −2.21850 + 2.01947i 0
113.10 0 1.16966 − 1.27746i 0 2.20774 + 0.354811i 0 0.707107 − 0.707107i 0 −0.263792 − 2.98838i 0
113.11 0 1.61532 + 0.625101i 0 0.398641 − 2.20025i 0 0.707107 − 0.707107i 0 2.21850 + 2.01947i 0
113.12 0 1.71916 − 0.210923i 0 0.161137 + 2.23025i 0 −0.707107 + 0.707107i 0 2.91102 − 0.725222i 0
197.1 0 −1.65123 + 0.522921i 0 1.89765 + 1.18276i 0 −0.707107 − 0.707107i 0 2.45311 − 1.72692i 0
197.2 0 −1.62919 − 0.587996i 0 −0.677300 − 2.13102i 0 −0.707107 − 0.707107i 0 2.30852 + 1.91591i 0
197.3 0 −1.38370 + 1.04182i 0 1.37156 − 1.76602i 0 0.707107 + 0.707107i 0 0.829228 − 2.88312i 0
197.4 0 −1.27746 − 1.16966i 0 −2.20774 + 0.354811i 0 0.707107 + 0.707107i 0 0.263792 + 2.98838i 0
197.5 0 −1.04182 + 1.38370i 0 −1.37156 + 1.76602i 0 0.707107 + 0.707107i 0 −0.829228 − 2.88312i 0
197.6 0 −0.522921 + 1.65123i 0 −1.89765 − 1.18276i 0 −0.707107 − 0.707107i 0 −2.45311 − 1.72692i 0
197.7 0 −0.210923 − 1.71916i 0 −0.161137 + 2.23025i 0 −0.707107 − 0.707107i 0 −2.91102 + 0.725222i 0
197.8 0 0.587996 + 1.62919i 0 0.677300 + 2.13102i 0 −0.707107 − 0.707107i 0 −2.30852 + 1.91591i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 113.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.s.a ✓ 24
3.b odd 2 1 inner 420.2.s.a ✓ 24
5.b even 2 1 2100.2.s.b 24
5.c odd 4 1 inner 420.2.s.a ✓ 24
5.c odd 4 1 2100.2.s.b 24
15.d odd 2 1 2100.2.s.b 24
15.e even 4 1 inner 420.2.s.a ✓ 24
15.e even 4 1 2100.2.s.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.s.a ✓ 24 1.a even 1 1 trivial
420.2.s.a ✓ 24 3.b odd 2 1 inner
420.2.s.a ✓ 24 5.c odd 4 1 inner
420.2.s.a ✓ 24 15.e even 4 1 inner
2100.2.s.b 24 5.b even 2 1
2100.2.s.b 24 5.c odd 4 1
2100.2.s.b 24 15.d odd 2 1
2100.2.s.b 24 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(420, [\chi])\).