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Properties

Label 378.4.l.a
Level $378$
Weight $4$
Character orbit 378.l
Analytic conductor $22.303$
Analytic rank $0$
Dimension $48$
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] = [378,4,Mod(143,378)]
 
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([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.143");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 378.l (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: \(22.3027219822\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 192 q^{4} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 192 q^{4} - 12 q^{7} + 24 q^{11} + 72 q^{13} + 132 q^{14} + 768 q^{16} - 144 q^{17} - 408 q^{23} - 600 q^{25} - 120 q^{26} + 48 q^{28} - 42 q^{29} + 780 q^{35} - 168 q^{37} + 618 q^{41} - 42 q^{43} - 96 q^{44} - 252 q^{46} - 396 q^{47} - 42 q^{49} - 1464 q^{50} - 288 q^{52} + 36 q^{53} - 528 q^{56} - 252 q^{58} + 3000 q^{59} - 2952 q^{62} - 3072 q^{64} + 1176 q^{67} + 576 q^{68} - 324 q^{70} - 1260 q^{74} - 6420 q^{77} - 2460 q^{79} + 720 q^{85} + 1200 q^{86} + 4398 q^{89} - 90 q^{91} + 1632 q^{92} + 1584 q^{97} + 1104 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1 − 2.00000i 0 −4.00000 −10.1776 + 17.6281i 0 4.40132 + 17.9897i 8.00000i 0 35.2562 + 20.3552i
143.2 − 2.00000i 0 −4.00000 9.84577 − 17.0534i 0 4.63716 + 17.9303i 8.00000i 0 −34.1068 − 19.6915i
143.3 2.00000i 0 −4.00000 −7.32164 + 12.6815i 0 15.8480 − 9.58338i − 8.00000i 0 −25.3629 − 14.6433i
143.4 − 2.00000i 0 −4.00000 −6.43016 + 11.1374i 0 11.1140 − 14.8148i 8.00000i 0 22.2747 + 12.8603i
143.5 2.00000i 0 −4.00000 −5.48053 + 9.49256i 0 −1.82034 − 18.4306i − 8.00000i 0 −18.9851 − 10.9611i
143.6 − 2.00000i 0 −4.00000 −1.88736 + 3.26901i 0 17.9293 + 4.64125i 8.00000i 0 6.53802 + 3.77473i
143.7 2.00000i 0 −4.00000 3.58162 − 6.20355i 0 −7.92464 − 16.7392i − 8.00000i 0 12.4071 + 7.16324i
143.8 2.00000i 0 −4.00000 −1.36122 + 2.35771i 0 1.29655 + 18.4748i − 8.00000i 0 −4.71542 − 2.72245i
143.9 2.00000i 0 −4.00000 3.13500 − 5.42997i 0 13.5626 + 12.6118i − 8.00000i 0 10.8599 + 6.26999i
143.10 − 2.00000i 0 −4.00000 0.213260 − 0.369378i 0 −18.0845 − 3.99406i 8.00000i 0 −0.738755 − 0.426520i
143.11 − 2.00000i 0 −4.00000 1.85594 − 3.21459i 0 16.4255 + 8.55580i 8.00000i 0 −6.42918 − 3.71189i
143.12 − 2.00000i 0 −4.00000 4.72900 − 8.19087i 0 18.4230 + 1.89542i 8.00000i 0 −16.3817 − 9.45800i
143.13 − 2.00000i 0 −4.00000 −1.39685 + 2.41942i 0 −17.1508 + 6.98919i 8.00000i 0 4.83883 + 2.79370i
143.14 2.00000i 0 −4.00000 −1.48002 + 2.56347i 0 −18.4948 + 0.970961i − 8.00000i 0 −5.12693 − 2.96004i
143.15 − 2.00000i 0 −4.00000 −3.16053 + 5.47419i 0 −1.17581 − 18.4829i 8.00000i 0 10.9484 + 6.32106i
143.16 2.00000i 0 −4.00000 −3.47705 + 6.02243i 0 14.1742 − 11.9202i − 8.00000i 0 −12.0449 − 6.95410i
143.17 2.00000i 0 −4.00000 −1.86960 + 3.23823i 0 −8.88689 + 16.2488i − 8.00000i 0 −6.47647 − 3.73919i
143.18 2.00000i 0 −4.00000 6.21279 − 10.7609i 0 −2.69645 + 18.3229i − 8.00000i 0 21.5217 + 12.4256i
143.19 − 2.00000i 0 −4.00000 7.48096 − 12.9574i 0 −11.8885 + 14.2008i 8.00000i 0 −25.9148 − 14.9619i
143.20 2.00000i 0 −4.00000 8.15472 − 14.1244i 0 −14.6316 − 11.3541i − 8.00000i 0 28.2488 + 16.3094i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.4.l.a 48
3.b odd 2 1 126.4.l.a ✓ 48
7.d odd 6 1 378.4.t.a 48
9.c even 3 1 126.4.t.a yes 48
9.d odd 6 1 378.4.t.a 48
21.g even 6 1 126.4.t.a yes 48
63.i even 6 1 inner 378.4.l.a 48
63.t odd 6 1 126.4.l.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.l.a ✓ 48 3.b odd 2 1
126.4.l.a ✓ 48 63.t odd 6 1
126.4.t.a yes 48 9.c even 3 1
126.4.t.a yes 48 21.g even 6 1
378.4.l.a 48 1.a even 1 1 trivial
378.4.l.a 48 63.i even 6 1 inner
378.4.t.a 48 7.d odd 6 1
378.4.t.a 48 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(378, [\chi])\).