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Properties

Label 1530.2.u.c
Level $1530$
Weight $2$
Character orbit 1530.u
Analytic conductor $12.217$
Analytic rank $0$
Dimension $40$
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] = [1530,2,Mod(557,1530)] 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("1530.557"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1530, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1530.u (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40] 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: \(12.2171115093\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{10} + 8 q^{13} - 40 q^{16} + 16 q^{19} - 8 q^{31} + 32 q^{43} + 56 q^{49} + 8 q^{52} + 32 q^{55} - 64 q^{61} + 32 q^{67} + 32 q^{70} + 88 q^{79} + 72 q^{85} + 16 q^{88} + 56 q^{91} - 16 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1 −0.707107 − 0.707107i 0 1.00000i −0.844240 + 2.07057i 0 5.17497 0.707107 − 0.707107i 0 2.06108 − 0.867146i
557.2 −0.707107 − 0.707107i 0 1.00000i −1.92688 + 1.13451i 0 −2.08205 0.707107 − 0.707107i 0 2.16473 + 0.560290i
557.3 −0.707107 − 0.707107i 0 1.00000i −1.98486 − 1.02972i 0 1.10043 0.707107 − 0.707107i 0 0.675385 + 2.13163i
557.4 −0.707107 − 0.707107i 0 1.00000i −1.72967 − 1.41713i 0 3.01070 0.707107 − 0.707107i 0 0.220996 + 2.22512i
557.5 −0.707107 − 0.707107i 0 1.00000i −1.46080 + 1.69295i 0 −4.17229 0.707107 − 0.707107i 0 2.23003 − 0.164152i
557.6 −0.707107 − 0.707107i 0 1.00000i 1.66962 + 1.48741i 0 2.66970 0.707107 − 0.707107i 0 −0.128842 − 2.23235i
557.7 −0.707107 − 0.707107i 0 1.00000i 2.23606 − 0.00470086i 0 0.105515 0.707107 − 0.707107i 0 −1.58446 − 1.57781i
557.8 −0.707107 − 0.707107i 0 1.00000i 1.53663 + 1.62443i 0 −0.784820 0.707107 − 0.707107i 0 0.0620885 − 2.23521i
557.9 −0.707107 − 0.707107i 0 1.00000i −0.0740013 − 2.23484i 0 −0.956061 0.707107 − 0.707107i 0 −1.52795 + 1.63260i
557.10 −0.707107 − 0.707107i 0 1.00000i 1.16393 − 1.90926i 0 −4.06609 0.707107 − 0.707107i 0 −2.17307 + 0.527026i
557.11 0.707107 + 0.707107i 0 1.00000i 1.92688 − 1.13451i 0 −2.08205 −0.707107 + 0.707107i 0 2.16473 + 0.560290i
557.12 0.707107 + 0.707107i 0 1.00000i 0.844240 − 2.07057i 0 5.17497 −0.707107 + 0.707107i 0 2.06108 − 0.867146i
557.13 0.707107 + 0.707107i 0 1.00000i −1.66962 − 1.48741i 0 2.66970 −0.707107 + 0.707107i 0 −0.128842 − 2.23235i
557.14 0.707107 + 0.707107i 0 1.00000i 0.0740013 + 2.23484i 0 −0.956061 −0.707107 + 0.707107i 0 −1.52795 + 1.63260i
557.15 0.707107 + 0.707107i 0 1.00000i 1.98486 + 1.02972i 0 1.10043 −0.707107 + 0.707107i 0 0.675385 + 2.13163i
557.16 0.707107 + 0.707107i 0 1.00000i −1.16393 + 1.90926i 0 −4.06609 −0.707107 + 0.707107i 0 −2.17307 + 0.527026i
557.17 0.707107 + 0.707107i 0 1.00000i −1.53663 − 1.62443i 0 −0.784820 −0.707107 + 0.707107i 0 0.0620885 − 2.23521i
557.18 0.707107 + 0.707107i 0 1.00000i −2.23606 + 0.00470086i 0 0.105515 −0.707107 + 0.707107i 0 −1.58446 − 1.57781i
557.19 0.707107 + 0.707107i 0 1.00000i 1.46080 − 1.69295i 0 −4.17229 −0.707107 + 0.707107i 0 2.23003 − 0.164152i
557.20 0.707107 + 0.707107i 0 1.00000i 1.72967 + 1.41713i 0 3.01070 −0.707107 + 0.707107i 0 0.220996 + 2.22512i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 557.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
85.i odd 4 1 inner
255.r even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1530.2.u.c yes 40
3.b odd 2 1 inner 1530.2.u.c yes 40
5.c odd 4 1 1530.2.j.c ✓ 40
15.e even 4 1 1530.2.j.c ✓ 40
17.c even 4 1 1530.2.j.c ✓ 40
51.f odd 4 1 1530.2.j.c ✓ 40
85.i odd 4 1 inner 1530.2.u.c yes 40
255.r even 4 1 inner 1530.2.u.c yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1530.2.j.c ✓ 40 5.c odd 4 1
1530.2.j.c ✓ 40 15.e even 4 1
1530.2.j.c ✓ 40 17.c even 4 1
1530.2.j.c ✓ 40 51.f odd 4 1
1530.2.u.c yes 40 1.a even 1 1 trivial
1530.2.u.c yes 40 3.b odd 2 1 inner
1530.2.u.c yes 40 85.i odd 4 1 inner
1530.2.u.c yes 40 255.r even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} - 42T_{7}^{8} - 12T_{7}^{7} + 520T_{7}^{6} + 168T_{7}^{5} - 2104T_{7}^{4} - 992T_{7}^{3} + 1920T_{7}^{2} + 1024T_{7} - 128 \) acting on \(S_{2}^{\mathrm{new}}(1530, [\chi])\). Copy content Toggle raw display