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Properties

Label 225.9.d.c
Level $225$
Weight $9$
Character orbit 225.d
Analytic conductor $91.660$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [225,9,Mod(224,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 9, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.224"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 225.d (of order \(2\), degree \(1\), not 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: \(91.6601872638\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 5048 q^{4} + 595144 q^{16} - 249920 q^{19} - 1241936 q^{31} - 14972992 q^{34} - 14323536 q^{46} - 42020312 q^{49} + 56906192 q^{61} + 338779896 q^{64} - 511347840 q^{76} + 127619024 q^{79} - 593698576 q^{91}+ \cdots - 51203776 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
224.1 −31.8503 0 758.442 0 0 1917.57i −16002.9 0 0
224.2 −31.8503 0 758.442 0 0 − 1917.57i −16002.9 0 0
224.3 −25.3359 0 385.909 0 0 2907.18i −3291.35 0 0
224.4 −25.3359 0 385.909 0 0 − 2907.18i −3291.35 0 0
224.5 −23.2704 0 285.513 0 0 − 4399.16i −686.784 0 0
224.6 −23.2704 0 285.513 0 0 4399.16i −686.784 0 0
224.7 −19.1733 0 111.617 0 0 − 367.611i 2768.31 0 0
224.8 −19.1733 0 111.617 0 0 367.611i 2768.31 0 0
224.9 −13.3807 0 −76.9563 0 0 − 256.182i 4455.20 0 0
224.10 −13.3807 0 −76.9563 0 0 256.182i 4455.20 0 0
224.11 −7.31273 0 −202.524 0 0 3662.21i 3353.06 0 0
224.12 −7.31273 0 −202.524 0 0 − 3662.21i 3353.06 0 0
224.13 7.31273 0 −202.524 0 0 3662.21i −3353.06 0 0
224.14 7.31273 0 −202.524 0 0 − 3662.21i −3353.06 0 0
224.15 13.3807 0 −76.9563 0 0 − 256.182i −4455.20 0 0
224.16 13.3807 0 −76.9563 0 0 256.182i −4455.20 0 0
224.17 19.1733 0 111.617 0 0 − 367.611i −2768.31 0 0
224.18 19.1733 0 111.617 0 0 367.611i −2768.31 0 0
224.19 23.2704 0 285.513 0 0 − 4399.16i 686.784 0 0
224.20 23.2704 0 285.513 0 0 4399.16i 686.784 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 224.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.9.d.c 24
3.b odd 2 1 inner 225.9.d.c 24
5.b even 2 1 inner 225.9.d.c 24
5.c odd 4 1 45.9.c.a ✓ 12
5.c odd 4 1 225.9.c.d 12
15.d odd 2 1 inner 225.9.d.c 24
15.e even 4 1 45.9.c.a ✓ 12
15.e even 4 1 225.9.c.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.9.c.a ✓ 12 5.c odd 4 1
45.9.c.a ✓ 12 15.e even 4 1
225.9.c.d 12 5.c odd 4 1
225.9.c.d 12 15.e even 4 1
225.9.d.c 24 1.a even 1 1 trivial
225.9.d.c 24 3.b odd 2 1 inner
225.9.d.c 24 5.b even 2 1 inner
225.9.d.c 24 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 2798 T_{2}^{10} + 2962185 T_{2}^{8} - 1494134240 T_{2}^{6} + 366509587840 T_{2}^{4} + \cdots + 12\!\cdots\!44 \) acting on \(S_{9}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display