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Properties

Label 378.3.q.a
Level $378$
Weight $3$
Character orbit 378.q
Analytic conductor $10.300$
Analytic rank $0$
Dimension $24$
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(71,378)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.71");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 378.q (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: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{4} - 36 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{4} - 36 q^{5} - 48 q^{16} + 24 q^{19} - 72 q^{20} + 24 q^{22} + 72 q^{23} + 72 q^{25} + 108 q^{29} - 60 q^{31} - 48 q^{34} - 168 q^{37} - 144 q^{38} - 108 q^{41} + 60 q^{43} + 324 q^{47} - 84 q^{49} - 144 q^{50} + 264 q^{55} + 432 q^{59} - 192 q^{64} + 180 q^{65} + 72 q^{67} - 72 q^{68} + 24 q^{73} - 288 q^{74} + 24 q^{76} + 12 q^{79} - 288 q^{82} - 756 q^{83} - 156 q^{85} - 360 q^{86} - 48 q^{88} - 168 q^{91} + 144 q^{92} + 936 q^{95} + 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −1.22474 + 0.707107i 0 1.00000 − 1.73205i −7.45939 − 4.30668i 0 1.32288 + 2.29129i 2.82843i 0 12.1811
71.2 −1.22474 + 0.707107i 0 1.00000 − 1.73205i −7.44329 − 4.29739i 0 −1.32288 − 2.29129i 2.82843i 0 12.1548
71.3 −1.22474 + 0.707107i 0 1.00000 − 1.73205i −2.56117 − 1.47869i 0 1.32288 + 2.29129i 2.82843i 0 4.18237
71.4 −1.22474 + 0.707107i 0 1.00000 − 1.73205i −0.665254 − 0.384085i 0 −1.32288 − 2.29129i 2.82843i 0 1.08636
71.5 −1.22474 + 0.707107i 0 1.00000 − 1.73205i 3.60855 + 2.08340i 0 −1.32288 − 2.29129i 2.82843i 0 −5.89274
71.6 −1.22474 + 0.707107i 0 1.00000 − 1.73205i 5.52056 + 3.18729i 0 1.32288 + 2.29129i 2.82843i 0 −9.01503
71.7 1.22474 − 0.707107i 0 1.00000 − 1.73205i −6.70848 − 3.87314i 0 1.32288 + 2.29129i − 2.82843i 0 −10.9549
71.8 1.22474 − 0.707107i 0 1.00000 − 1.73205i −5.20486 − 3.00503i 0 −1.32288 − 2.29129i − 2.82843i 0 −8.49950
71.9 1.22474 − 0.707107i 0 1.00000 − 1.73205i −2.76866 − 1.59849i 0 −1.32288 − 2.29129i − 2.82843i 0 −4.52120
71.10 1.22474 − 0.707107i 0 1.00000 − 1.73205i −2.30097 − 1.32846i 0 1.32288 + 2.29129i − 2.82843i 0 −3.75747
71.11 1.22474 − 0.707107i 0 1.00000 − 1.73205i 3.47352 + 2.00544i 0 −1.32288 − 2.29129i − 2.82843i 0 5.67224
71.12 1.22474 − 0.707107i 0 1.00000 − 1.73205i 4.50944 + 2.60353i 0 1.32288 + 2.29129i − 2.82843i 0 7.36389
197.1 −1.22474 − 0.707107i 0 1.00000 + 1.73205i −7.45939 + 4.30668i 0 1.32288 − 2.29129i − 2.82843i 0 12.1811
197.2 −1.22474 − 0.707107i 0 1.00000 + 1.73205i −7.44329 + 4.29739i 0 −1.32288 + 2.29129i − 2.82843i 0 12.1548
197.3 −1.22474 − 0.707107i 0 1.00000 + 1.73205i −2.56117 + 1.47869i 0 1.32288 − 2.29129i − 2.82843i 0 4.18237
197.4 −1.22474 − 0.707107i 0 1.00000 + 1.73205i −0.665254 + 0.384085i 0 −1.32288 + 2.29129i − 2.82843i 0 1.08636
197.5 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 3.60855 − 2.08340i 0 −1.32288 + 2.29129i − 2.82843i 0 −5.89274
197.6 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 5.52056 − 3.18729i 0 1.32288 − 2.29129i − 2.82843i 0 −9.01503
197.7 1.22474 + 0.707107i 0 1.00000 + 1.73205i −6.70848 + 3.87314i 0 1.32288 − 2.29129i 2.82843i 0 −10.9549
197.8 1.22474 + 0.707107i 0 1.00000 + 1.73205i −5.20486 + 3.00503i 0 −1.32288 + 2.29129i 2.82843i 0 −8.49950
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d 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.q.a 24
3.b odd 2 1 126.3.q.a ✓ 24
9.c even 3 1 126.3.q.a ✓ 24
9.c even 3 1 1134.3.b.c 24
9.d odd 6 1 inner 378.3.q.a 24
9.d odd 6 1 1134.3.b.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.q.a ✓ 24 3.b odd 2 1
126.3.q.a ✓ 24 9.c even 3 1
378.3.q.a 24 1.a even 1 1 trivial
378.3.q.a 24 9.d odd 6 1 inner
1134.3.b.c 24 9.c even 3 1
1134.3.b.c 24 9.d odd 6 1

Hecke kernels

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