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Properties

Label 324.5.d.g
Level $324$
Weight $5$
Character orbit 324.d
Analytic conductor $33.492$
Analytic rank $0$
Dimension $24$
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] = [324,5,Mod(163,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.163");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 324.d (of order \(2\), degree \(1\), 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: \(33.4918680392\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{4} + 198 q^{10} - 120 q^{13} - 366 q^{16} - 852 q^{22} + 2160 q^{25} + 996 q^{28} + 702 q^{34} + 2808 q^{37} - 6102 q^{40} - 60 q^{46} - 11208 q^{49} - 9270 q^{52} + 14178 q^{58} + 2808 q^{61} - 6474 q^{64} + 15060 q^{70} - 17160 q^{73} + 276 q^{76} + 32568 q^{82} + 6024 q^{85} - 54012 q^{88} + 52272 q^{94} + 39216 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1 −3.84702 − 1.09564i 0 13.5991 + 8.42992i 19.3338 0 − 81.9386i −43.0800 − 47.3299i 0 −74.3777 − 21.1830i
163.2 −3.84702 + 1.09564i 0 13.5991 − 8.42992i 19.3338 0 81.9386i −43.0800 + 47.3299i 0 −74.3777 + 21.1830i
163.3 −3.71114 − 1.49247i 0 11.5451 + 11.0775i 6.28175 0 40.2802i −26.3124 − 58.3409i 0 −23.3124 − 9.37533i
163.4 −3.71114 + 1.49247i 0 11.5451 − 11.0775i 6.28175 0 − 40.2802i −26.3124 + 58.3409i 0 −23.3124 + 9.37533i
163.5 −3.39472 − 2.11563i 0 7.04819 + 14.3639i −35.3640 0 64.8206i 6.46226 − 63.6729i 0 120.051 + 74.8174i
163.6 −3.39472 + 2.11563i 0 7.04819 − 14.3639i −35.3640 0 − 64.8206i 6.46226 + 63.6729i 0 120.051 − 74.8174i
163.7 −2.17965 − 3.35397i 0 −6.49828 + 14.6210i −38.2904 0 − 64.2502i 63.2023 − 10.0734i 0 83.4596 + 128.425i
163.8 −2.17965 + 3.35397i 0 −6.49828 − 14.6210i −38.2904 0 64.2502i 63.2023 + 10.0734i 0 83.4596 − 128.425i
163.9 −1.87362 − 3.53406i 0 −8.97910 + 13.2430i 21.1272 0 − 7.89411i 63.6247 + 6.92038i 0 −39.5844 − 74.6649i
163.10 −1.87362 + 3.53406i 0 −8.97910 − 13.2430i 21.1272 0 7.89411i 63.6247 − 6.92038i 0 −39.5844 + 74.6649i
163.11 −0.626498 − 3.95063i 0 −15.2150 + 4.95012i 26.7134 0 21.8965i 29.0883 + 57.0076i 0 −16.7359 − 105.535i
163.12 −0.626498 + 3.95063i 0 −15.2150 − 4.95012i 26.7134 0 − 21.8965i 29.0883 − 57.0076i 0 −16.7359 + 105.535i
163.13 0.626498 − 3.95063i 0 −15.2150 − 4.95012i −26.7134 0 − 21.8965i −29.0883 + 57.0076i 0 −16.7359 + 105.535i
163.14 0.626498 + 3.95063i 0 −15.2150 + 4.95012i −26.7134 0 21.8965i −29.0883 − 57.0076i 0 −16.7359 − 105.535i
163.15 1.87362 − 3.53406i 0 −8.97910 − 13.2430i −21.1272 0 7.89411i −63.6247 + 6.92038i 0 −39.5844 + 74.6649i
163.16 1.87362 + 3.53406i 0 −8.97910 + 13.2430i −21.1272 0 − 7.89411i −63.6247 − 6.92038i 0 −39.5844 − 74.6649i
163.17 2.17965 − 3.35397i 0 −6.49828 − 14.6210i 38.2904 0 64.2502i −63.2023 − 10.0734i 0 83.4596 − 128.425i
163.18 2.17965 + 3.35397i 0 −6.49828 + 14.6210i 38.2904 0 − 64.2502i −63.2023 + 10.0734i 0 83.4596 + 128.425i
163.19 3.39472 − 2.11563i 0 7.04819 − 14.3639i 35.3640 0 − 64.8206i −6.46226 − 63.6729i 0 120.051 − 74.8174i
163.20 3.39472 + 2.11563i 0 7.04819 + 14.3639i 35.3640 0 64.8206i −6.46226 + 63.6729i 0 120.051 + 74.8174i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.5.d.g ✓ 24
3.b odd 2 1 inner 324.5.d.g ✓ 24
4.b odd 2 1 inner 324.5.d.g ✓ 24
12.b even 2 1 inner 324.5.d.g ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.5.d.g ✓ 24 1.a even 1 1 trivial
324.5.d.g ✓ 24 3.b odd 2 1 inner
324.5.d.g ✓ 24 4.b odd 2 1 inner
324.5.d.g ✓ 24 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 4290 T_{5}^{10} + 6920343 T_{5}^{8} - 5241178364 T_{5}^{6} + 1898862580959 T_{5}^{4} + \cdots + 86\!\cdots\!61 \) acting on \(S_{5}^{\mathrm{new}}(324, [\chi])\). Copy content Toggle raw display