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Properties

Label 405.2.f.b
Level $405$
Weight $2$
Character orbit 405.f
Analytic conductor $3.234$
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] = [405,2,Mod(242,405)] 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("405.242"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] 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: \(3.23394128186\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{13} - 24 q^{16} + 36 q^{37} - 36 q^{40} - 24 q^{43} - 48 q^{46} - 36 q^{52} - 60 q^{58} + 72 q^{70} + 12 q^{73} + 48 q^{76} - 96 q^{82} + 60 q^{85} + 96 q^{88} - 48 q^{91} + 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
242.1 −1.85830 + 1.85830i 0 − 4.90659i −1.97703 + 1.04468i 0 0.484523 + 0.484523i 5.40134 + 5.40134i 0 1.73258 − 5.61526i
242.2 −1.58458 + 1.58458i 0 − 3.02177i 2.21170 + 0.329208i 0 −2.52674 − 2.52674i 1.61908 + 1.61908i 0 −4.02627 + 2.98296i
242.3 −1.33432 + 1.33432i 0 − 1.56081i −0.800519 − 2.08786i 0 0.625021 + 0.625021i −0.586021 − 0.586021i 0 3.85402 + 1.71773i
242.4 −1.01382 + 1.01382i 0 − 0.0556605i 0.721513 + 2.11646i 0 3.46834 + 3.46834i −1.97121 − 1.97121i 0 −2.87720 − 1.41423i
242.5 −0.456847 + 0.456847i 0 1.58258i −2.23576 − 0.0369307i 0 0.169667 + 0.169667i −1.63669 − 1.63669i 0 1.03827 − 1.00453i
242.6 −0.137378 + 0.137378i 0 1.96225i 0.199238 − 2.22717i 0 −2.22081 − 2.22081i −0.544328 − 0.544328i 0 0.278595 + 0.333337i
242.7 0.137378 − 0.137378i 0 1.96225i −0.199238 + 2.22717i 0 −2.22081 − 2.22081i 0.544328 + 0.544328i 0 0.278595 + 0.333337i
242.8 0.456847 − 0.456847i 0 1.58258i 2.23576 + 0.0369307i 0 0.169667 + 0.169667i 1.63669 + 1.63669i 0 1.03827 − 1.00453i
242.9 1.01382 − 1.01382i 0 − 0.0556605i −0.721513 − 2.11646i 0 3.46834 + 3.46834i 1.97121 + 1.97121i 0 −2.87720 − 1.41423i
242.10 1.33432 − 1.33432i 0 − 1.56081i 0.800519 + 2.08786i 0 0.625021 + 0.625021i 0.586021 + 0.586021i 0 3.85402 + 1.71773i
242.11 1.58458 − 1.58458i 0 − 3.02177i −2.21170 − 0.329208i 0 −2.52674 − 2.52674i −1.61908 − 1.61908i 0 −4.02627 + 2.98296i
242.12 1.85830 − 1.85830i 0 − 4.90659i 1.97703 − 1.04468i 0 0.484523 + 0.484523i −5.40134 − 5.40134i 0 1.73258 − 5.61526i
323.1 −1.85830 − 1.85830i 0 4.90659i −1.97703 − 1.04468i 0 0.484523 − 0.484523i 5.40134 − 5.40134i 0 1.73258 + 5.61526i
323.2 −1.58458 − 1.58458i 0 3.02177i 2.21170 − 0.329208i 0 −2.52674 + 2.52674i 1.61908 − 1.61908i 0 −4.02627 − 2.98296i
323.3 −1.33432 − 1.33432i 0 1.56081i −0.800519 + 2.08786i 0 0.625021 − 0.625021i −0.586021 + 0.586021i 0 3.85402 − 1.71773i
323.4 −1.01382 − 1.01382i 0 0.0556605i 0.721513 − 2.11646i 0 3.46834 − 3.46834i −1.97121 + 1.97121i 0 −2.87720 + 1.41423i
323.5 −0.456847 − 0.456847i 0 − 1.58258i −2.23576 + 0.0369307i 0 0.169667 − 0.169667i −1.63669 + 1.63669i 0 1.03827 + 1.00453i
323.6 −0.137378 − 0.137378i 0 − 1.96225i 0.199238 + 2.22717i 0 −2.22081 + 2.22081i −0.544328 + 0.544328i 0 0.278595 − 0.333337i
323.7 0.137378 + 0.137378i 0 − 1.96225i −0.199238 − 2.22717i 0 −2.22081 + 2.22081i 0.544328 − 0.544328i 0 0.278595 − 0.333337i
323.8 0.456847 + 0.456847i 0 − 1.58258i 2.23576 − 0.0369307i 0 0.169667 − 0.169667i 1.63669 − 1.63669i 0 1.03827 + 1.00453i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 242.12
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 405.2.f.b ✓ 24
3.b odd 2 1 inner 405.2.f.b ✓ 24
5.c odd 4 1 inner 405.2.f.b ✓ 24
9.c even 3 1 405.2.m.d 24
9.c even 3 1 405.2.m.e 24
9.d odd 6 1 405.2.m.d 24
9.d odd 6 1 405.2.m.e 24
15.e even 4 1 inner 405.2.f.b ✓ 24
45.k odd 12 1 405.2.m.d 24
45.k odd 12 1 405.2.m.e 24
45.l even 12 1 405.2.m.d 24
45.l even 12 1 405.2.m.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.f.b ✓ 24 1.a even 1 1 trivial
405.2.f.b ✓ 24 3.b odd 2 1 inner
405.2.f.b ✓ 24 5.c odd 4 1 inner
405.2.f.b ✓ 24 15.e even 4 1 inner
405.2.m.d 24 9.c even 3 1
405.2.m.d 24 9.d odd 6 1
405.2.m.d 24 45.k odd 12 1
405.2.m.d 24 45.l even 12 1
405.2.m.e 24 9.c even 3 1
405.2.m.e 24 9.d odd 6 1
405.2.m.e 24 45.k odd 12 1
405.2.m.e 24 45.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 90T_{2}^{20} + 2505T_{2}^{16} + 24680T_{2}^{12} + 68712T_{2}^{8} + 11328T_{2}^{4} + 16 \) acting on \(S_{2}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display