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Properties

Label 450.2.s.d
Level $450$
Weight $2$
Character orbit 450.s
Analytic conductor $3.593$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,2,Mod(17,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.s (of order \(20\), degree \(8\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{7} - 12 q^{10} - 36 q^{13} + 8 q^{16} + 40 q^{19} + 44 q^{22} + 56 q^{25} + 24 q^{28} - 32 q^{31} - 20 q^{34} - 16 q^{37} + 24 q^{40} - 40 q^{43} - 32 q^{46} + 36 q^{52} - 24 q^{55} - 88 q^{58} - 20 q^{61} - 48 q^{67} - 72 q^{70} + 32 q^{73} + 64 q^{76} - 40 q^{79} - 4 q^{82} - 72 q^{85} - 16 q^{88} + 76 q^{91} + 84 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −0.891007 − 0.453990i 0 0.587785 + 0.809017i −2.20443 − 0.374841i 0 0.647181 + 0.647181i −0.156434 − 0.987688i 0 1.79398 + 1.33477i
17.2 −0.891007 − 0.453990i 0 0.587785 + 0.809017i 0.886894 + 2.05266i 0 −3.38856 − 3.38856i −0.156434 − 0.987688i 0 0.141660 − 2.23158i
17.3 0.891007 + 0.453990i 0 0.587785 + 0.809017i −0.886894 − 2.05266i 0 −3.38856 − 3.38856i 0.156434 + 0.987688i 0 0.141660 − 2.23158i
17.4 0.891007 + 0.453990i 0 0.587785 + 0.809017i 2.20443 + 0.374841i 0 0.647181 + 0.647181i 0.156434 + 0.987688i 0 1.79398 + 1.33477i
53.1 −0.891007 + 0.453990i 0 0.587785 − 0.809017i −2.20443 + 0.374841i 0 0.647181 − 0.647181i −0.156434 + 0.987688i 0 1.79398 − 1.33477i
53.2 −0.891007 + 0.453990i 0 0.587785 − 0.809017i 0.886894 − 2.05266i 0 −3.38856 + 3.38856i −0.156434 + 0.987688i 0 0.141660 + 2.23158i
53.3 0.891007 − 0.453990i 0 0.587785 − 0.809017i −0.886894 + 2.05266i 0 −3.38856 + 3.38856i 0.156434 − 0.987688i 0 0.141660 + 2.23158i
53.4 0.891007 − 0.453990i 0 0.587785 − 0.809017i 2.20443 − 0.374841i 0 0.647181 − 0.647181i 0.156434 − 0.987688i 0 1.79398 − 1.33477i
197.1 −0.453990 − 0.891007i 0 −0.587785 + 0.809017i −1.38876 − 1.75253i 0 1.03549 + 1.03549i 0.987688 + 0.156434i 0 −0.931033 + 2.03302i
197.2 −0.453990 − 0.891007i 0 −0.587785 + 0.809017i 2.19254 − 0.439035i 0 −1.14821 − 1.14821i 0.987688 + 0.156434i 0 −1.38658 − 1.75425i
197.3 0.453990 + 0.891007i 0 −0.587785 + 0.809017i −2.19254 + 0.439035i 0 −1.14821 − 1.14821i −0.987688 − 0.156434i 0 −1.38658 − 1.75425i
197.4 0.453990 + 0.891007i 0 −0.587785 + 0.809017i 1.38876 + 1.75253i 0 1.03549 + 1.03549i −0.987688 − 0.156434i 0 −0.931033 + 2.03302i
233.1 −0.453990 + 0.891007i 0 −0.587785 − 0.809017i −1.38876 + 1.75253i 0 1.03549 − 1.03549i 0.987688 − 0.156434i 0 −0.931033 − 2.03302i
233.2 −0.453990 + 0.891007i 0 −0.587785 − 0.809017i 2.19254 + 0.439035i 0 −1.14821 + 1.14821i 0.987688 − 0.156434i 0 −1.38658 + 1.75425i
233.3 0.453990 − 0.891007i 0 −0.587785 − 0.809017i −2.19254 − 0.439035i 0 −1.14821 + 1.14821i −0.987688 + 0.156434i 0 −1.38658 + 1.75425i
233.4 0.453990 − 0.891007i 0 −0.587785 − 0.809017i 1.38876 − 1.75253i 0 1.03549 − 1.03549i −0.987688 + 0.156434i 0 −0.931033 − 2.03302i
287.1 −0.987688 + 0.156434i 0 0.951057 − 0.309017i −2.20655 − 0.362124i 0 0.865154 + 0.865154i −0.891007 + 0.453990i 0 2.23603 + 0.0124851i
287.2 −0.987688 + 0.156434i 0 0.951057 − 0.309017i 2.17909 + 0.501584i 0 3.18852 + 3.18852i −0.891007 + 0.453990i 0 −2.23072 − 0.154525i
287.3 0.987688 − 0.156434i 0 0.951057 − 0.309017i −2.17909 − 0.501584i 0 3.18852 + 3.18852i 0.891007 − 0.453990i 0 −2.23072 − 0.154525i
287.4 0.987688 − 0.156434i 0 0.951057 − 0.309017i 2.20655 + 0.362124i 0 0.865154 + 0.865154i 0.891007 − 0.453990i 0 2.23603 + 0.0124851i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.s.d ✓ 32
3.b odd 2 1 inner 450.2.s.d ✓ 32
25.f odd 20 1 inner 450.2.s.d ✓ 32
75.l even 20 1 inner 450.2.s.d ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.s.d ✓ 32 1.a even 1 1 trivial
450.2.s.d ✓ 32 3.b odd 2 1 inner
450.2.s.d ✓ 32 25.f odd 20 1 inner
450.2.s.d ✓ 32 75.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} - 2 T_{7}^{15} + 2 T_{7}^{14} - 16 T_{7}^{13} + 791 T_{7}^{12} - 1852 T_{7}^{11} + \cdots + 929296 \) acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display