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Properties

Label 315.3.f.a
Level $315$
Weight $3$
Character orbit 315.f
Analytic conductor $8.583$
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] = [315,3,Mod(134,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.134");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 315.f (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: \(8.58312832735\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 40 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 40 q^{4} + 24 q^{10} + 280 q^{16} - 144 q^{19} + 8 q^{25} - 160 q^{31} + 96 q^{34} - 64 q^{40} - 128 q^{46} - 168 q^{49} + 296 q^{55} - 144 q^{61} + 776 q^{64} - 280 q^{70} - 928 q^{76} - 192 q^{79} - 128 q^{85} - 656 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1 −3.87138 0 10.9876 2.56984 − 4.28905i 0 2.64575i −27.0516 0 −9.94881 + 16.6045i
134.2 −3.87138 0 10.9876 2.56984 + 4.28905i 0 − 2.64575i −27.0516 0 −9.94881 − 16.6045i
134.3 −3.37722 0 7.40561 −4.63101 − 1.88515i 0 2.64575i −11.5015 0 15.6399 + 6.36657i
134.4 −3.37722 0 7.40561 −4.63101 + 1.88515i 0 − 2.64575i −11.5015 0 15.6399 − 6.36657i
134.5 −2.04616 0 0.186775 −2.46039 − 4.35276i 0 2.64575i 7.80247 0 5.03435 + 8.90644i
134.6 −2.04616 0 0.186775 −2.46039 + 4.35276i 0 − 2.64575i 7.80247 0 5.03435 − 8.90644i
134.7 −1.76508 0 −0.884492 3.39097 − 3.67441i 0 − 2.64575i 8.62152 0 −5.98534 + 6.48563i
134.8 −1.76508 0 −0.884492 3.39097 + 3.67441i 0 2.64575i 8.62152 0 −5.98534 − 6.48563i
134.9 −0.432442 0 −3.81299 −4.90663 − 0.961744i 0 − 2.64575i 3.37867 0 2.12184 + 0.415899i
134.10 −0.432442 0 −3.81299 −4.90663 + 0.961744i 0 2.64575i 3.37867 0 2.12184 − 0.415899i
134.11 −0.342800 0 −3.88249 2.51446 − 4.32175i 0 2.64575i 2.70211 0 −0.861956 + 1.48149i
134.12 −0.342800 0 −3.88249 2.51446 + 4.32175i 0 − 2.64575i 2.70211 0 −0.861956 − 1.48149i
134.13 0.342800 0 −3.88249 −2.51446 − 4.32175i 0 − 2.64575i −2.70211 0 −0.861956 − 1.48149i
134.14 0.342800 0 −3.88249 −2.51446 + 4.32175i 0 2.64575i −2.70211 0 −0.861956 + 1.48149i
134.15 0.432442 0 −3.81299 4.90663 − 0.961744i 0 2.64575i −3.37867 0 2.12184 − 0.415899i
134.16 0.432442 0 −3.81299 4.90663 + 0.961744i 0 − 2.64575i −3.37867 0 2.12184 + 0.415899i
134.17 1.76508 0 −0.884492 −3.39097 − 3.67441i 0 2.64575i −8.62152 0 −5.98534 − 6.48563i
134.18 1.76508 0 −0.884492 −3.39097 + 3.67441i 0 − 2.64575i −8.62152 0 −5.98534 + 6.48563i
134.19 2.04616 0 0.186775 2.46039 − 4.35276i 0 − 2.64575i −7.80247 0 5.03435 − 8.90644i
134.20 2.04616 0 0.186775 2.46039 + 4.35276i 0 2.64575i −7.80247 0 5.03435 + 8.90644i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 134.24
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 315.3.f.a ✓ 24
3.b odd 2 1 inner 315.3.f.a ✓ 24
5.b even 2 1 inner 315.3.f.a ✓ 24
5.c odd 4 2 1575.3.c.e 24
15.d odd 2 1 inner 315.3.f.a ✓ 24
15.e even 4 2 1575.3.c.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.3.f.a ✓ 24 1.a even 1 1 trivial
315.3.f.a ✓ 24 3.b odd 2 1 inner
315.3.f.a ✓ 24 5.b even 2 1 inner
315.3.f.a ✓ 24 15.d odd 2 1 inner
1575.3.c.e 24 5.c odd 4 2
1575.3.c.e 24 15.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(315, [\chi])\).