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Properties

Label 378.3.r.a
Level $378$
Weight $3$
Character orbit 378.r
Analytic conductor $10.300$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,3,Mod(233,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.233");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 378.r (of order \(6\), degree \(2\), not 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: \(10.2997539928\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 64 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 64 q^{4} + 2 q^{7} + 36 q^{11} + 10 q^{13} - 36 q^{14} + 128 q^{16} + 54 q^{17} + 28 q^{19} + 126 q^{23} + 80 q^{25} + 72 q^{26} - 4 q^{28} - 36 q^{29} + 16 q^{31} + 90 q^{35} + 22 q^{37} - 72 q^{41} + 16 q^{43} - 72 q^{44} - 12 q^{46} + 2 q^{49} + 288 q^{50} - 20 q^{52} + 72 q^{53} - 24 q^{55} + 72 q^{56} - 24 q^{58} + 124 q^{61} - 256 q^{64} - 140 q^{67} - 108 q^{68} + 72 q^{70} + 196 q^{73} - 216 q^{74} - 56 q^{76} - 486 q^{77} + 76 q^{79} + 60 q^{85} + 144 q^{86} + 486 q^{89} - 122 q^{91} - 252 q^{92} - 336 q^{94} - 38 q^{97} - 288 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
233.1 − 1.41421i 0 −2.00000 −6.63902 + 3.83304i 0 2.93662 − 6.35423i 2.82843i 0 5.42073 + 9.38899i
233.2 − 1.41421i 0 −2.00000 −2.90307 + 1.67609i 0 5.64984 − 4.13271i 2.82843i 0 2.37035 + 4.10556i
233.3 − 1.41421i 0 −2.00000 −2.36201 + 1.36371i 0 −2.54299 − 6.52175i 2.82843i 0 1.92857 + 3.34039i
233.4 − 1.41421i 0 −2.00000 −1.22482 + 0.707152i 0 3.10205 + 6.27513i 2.82843i 0 1.00006 + 1.73216i
233.5 − 1.41421i 0 −2.00000 −0.857116 + 0.494856i 0 −6.76024 + 1.81637i 2.82843i 0 0.699833 + 1.21215i
233.6 − 1.41421i 0 −2.00000 2.07383 − 1.19732i 0 −5.45012 + 4.39274i 2.82843i 0 −1.69327 − 2.93283i
233.7 − 1.41421i 0 −2.00000 5.54397 − 3.20081i 0 4.41544 + 5.43175i 2.82843i 0 −4.52663 − 7.84035i
233.8 − 1.41421i 0 −2.00000 6.36825 − 3.67671i 0 2.82364 − 6.40524i 2.82843i 0 −5.19965 − 9.00606i
233.9 1.41421i 0 −2.00000 −8.39861 + 4.84894i 0 −3.70991 + 5.93604i − 2.82843i 0 −6.85744 − 11.8774i
233.10 1.41421i 0 −2.00000 −5.46142 + 3.15315i 0 −3.23416 − 6.20807i − 2.82843i 0 −4.45923 − 7.72361i
233.11 1.41421i 0 −2.00000 −2.56482 + 1.48080i 0 6.84216 + 1.47812i − 2.82843i 0 −2.09417 − 3.62721i
233.12 1.41421i 0 −2.00000 −1.84316 + 1.06415i 0 6.16730 − 3.31125i − 2.82843i 0 −1.50493 − 2.60662i
233.13 1.41421i 0 −2.00000 1.51694 − 0.875808i 0 −1.24611 + 6.88819i − 2.82843i 0 1.23858 + 2.14528i
233.14 1.41421i 0 −2.00000 2.12968 − 1.22957i 0 −6.56056 − 2.44111i − 2.82843i 0 1.73888 + 3.01183i
233.15 1.41421i 0 −2.00000 7.20455 − 4.15955i 0 5.54044 + 4.27827i − 2.82843i 0 5.88249 + 10.1888i
233.16 1.41421i 0 −2.00000 7.41683 − 4.28211i 0 −6.97339 + 0.609798i − 2.82843i 0 6.05582 + 10.4890i
305.1 − 1.41421i 0 −2.00000 −8.39861 − 4.84894i 0 −3.70991 − 5.93604i 2.82843i 0 −6.85744 + 11.8774i
305.2 − 1.41421i 0 −2.00000 −5.46142 − 3.15315i 0 −3.23416 + 6.20807i 2.82843i 0 −4.45923 + 7.72361i
305.3 − 1.41421i 0 −2.00000 −2.56482 − 1.48080i 0 6.84216 − 1.47812i 2.82843i 0 −2.09417 + 3.62721i
305.4 − 1.41421i 0 −2.00000 −1.84316 − 1.06415i 0 6.16730 + 3.31125i 2.82843i 0 −1.50493 + 2.60662i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 233.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.3.r.a 32
3.b odd 2 1 126.3.r.a yes 32
7.c even 3 1 378.3.i.a 32
9.c even 3 1 126.3.i.a ✓ 32
9.d odd 6 1 378.3.i.a 32
21.h odd 6 1 126.3.i.a ✓ 32
63.h even 3 1 126.3.r.a yes 32
63.j odd 6 1 inner 378.3.r.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.i.a ✓ 32 9.c even 3 1
126.3.i.a ✓ 32 21.h odd 6 1
126.3.r.a yes 32 3.b odd 2 1
126.3.r.a yes 32 63.h even 3 1
378.3.i.a 32 7.c even 3 1
378.3.i.a 32 9.d odd 6 1
378.3.r.a 32 1.a even 1 1 trivial
378.3.r.a 32 63.j odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(378, [\chi])\).