Properties

Label 378.4.h.a
Level $378$
Weight $4$
Character orbit 378.h
Analytic conductor $22.303$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,4,Mod(289,378)]
 
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([4, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 378.h (of order \(3\), 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: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{2} - 48 q^{4} + 20 q^{5} + 28 q^{7} + 192 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{2} - 48 q^{4} + 20 q^{5} + 28 q^{7} + 192 q^{8} - 20 q^{10} - 8 q^{11} - 56 q^{13} - 46 q^{14} - 192 q^{16} - 92 q^{17} - 174 q^{19} - 40 q^{20} + 8 q^{22} + 10 q^{23} + 844 q^{25} - 112 q^{26} - 20 q^{28} + 152 q^{29} - 140 q^{31} - 384 q^{32} - 184 q^{34} + 331 q^{35} + 189 q^{37} + 696 q^{38} + 160 q^{40} - 465 q^{41} - 117 q^{43} + 16 q^{44} - 10 q^{46} + 273 q^{47} + 900 q^{49} - 844 q^{50} + 448 q^{52} + 78 q^{53} + 2144 q^{55} + 224 q^{56} - 608 q^{58} - 397 q^{59} - 1847 q^{61} + 560 q^{62} + 1536 q^{64} - 996 q^{65} - 628 q^{67} + 736 q^{68} + 134 q^{70} - 44 q^{71} - 838 q^{73} - 756 q^{74} - 696 q^{76} - 1147 q^{77} + 16 q^{79} - 160 q^{80} - 930 q^{82} - 947 q^{83} - 139 q^{85} + 468 q^{86} - 64 q^{88} - 207 q^{89} - 518 q^{91} - 20 q^{92} + 546 q^{94} + 2731 q^{95} - 2308 q^{97} - 462 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i −18.7213 0 −15.3695 10.3334i 8.00000 0 18.7213 + 32.4263i
289.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i −15.6350 0 17.1578 + 6.97217i 8.00000 0 15.6350 + 27.0807i
289.3 −1.00000 1.73205i 0 −2.00000 + 3.46410i −15.5553 0 6.99950 17.1466i 8.00000 0 15.5553 + 26.9425i
289.4 −1.00000 1.73205i 0 −2.00000 + 3.46410i −3.49343 0 −11.9430 + 14.1550i 8.00000 0 3.49343 + 6.05080i
289.5 −1.00000 1.73205i 0 −2.00000 + 3.46410i −1.46289 0 −3.71747 + 18.1433i 8.00000 0 1.46289 + 2.53380i
289.6 −1.00000 1.73205i 0 −2.00000 + 3.46410i 0.521453 0 −3.56519 18.1739i 8.00000 0 −0.521453 0.903183i
289.7 −1.00000 1.73205i 0 −2.00000 + 3.46410i 1.52253 0 18.1220 + 3.82009i 8.00000 0 −1.52253 2.63709i
289.8 −1.00000 1.73205i 0 −2.00000 + 3.46410i 3.11229 0 8.38218 + 16.5148i 8.00000 0 −3.11229 5.39064i
289.9 −1.00000 1.73205i 0 −2.00000 + 3.46410i 4.38272 0 −16.3832 8.63665i 8.00000 0 −4.38272 7.59110i
289.10 −1.00000 1.73205i 0 −2.00000 + 3.46410i 14.9231 0 14.0821 12.0289i 8.00000 0 −14.9231 25.8476i
289.11 −1.00000 1.73205i 0 −2.00000 + 3.46410i 19.8520 0 −18.1695 + 3.58722i 8.00000 0 −19.8520 34.3846i
289.12 −1.00000 1.73205i 0 −2.00000 + 3.46410i 20.5539 0 18.4043 2.06929i 8.00000 0 −20.5539 35.6003i
361.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −18.7213 0 −15.3695 + 10.3334i 8.00000 0 18.7213 32.4263i
361.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i −15.6350 0 17.1578 6.97217i 8.00000 0 15.6350 27.0807i
361.3 −1.00000 + 1.73205i 0 −2.00000 3.46410i −15.5553 0 6.99950 + 17.1466i 8.00000 0 15.5553 26.9425i
361.4 −1.00000 + 1.73205i 0 −2.00000 3.46410i −3.49343 0 −11.9430 14.1550i 8.00000 0 3.49343 6.05080i
361.5 −1.00000 + 1.73205i 0 −2.00000 3.46410i −1.46289 0 −3.71747 18.1433i 8.00000 0 1.46289 2.53380i
361.6 −1.00000 + 1.73205i 0 −2.00000 3.46410i 0.521453 0 −3.56519 + 18.1739i 8.00000 0 −0.521453 + 0.903183i
361.7 −1.00000 + 1.73205i 0 −2.00000 3.46410i 1.52253 0 18.1220 3.82009i 8.00000 0 −1.52253 + 2.63709i
361.8 −1.00000 + 1.73205i 0 −2.00000 3.46410i 3.11229 0 8.38218 16.5148i 8.00000 0 −3.11229 + 5.39064i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.4.h.a 24
3.b odd 2 1 126.4.h.b yes 24
7.c even 3 1 378.4.e.b 24
9.c even 3 1 378.4.e.b 24
9.d odd 6 1 126.4.e.a 24
21.h odd 6 1 126.4.e.a 24
63.g even 3 1 inner 378.4.h.a 24
63.n odd 6 1 126.4.h.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.e.a 24 9.d odd 6 1
126.4.e.a 24 21.h odd 6 1
126.4.h.b yes 24 3.b odd 2 1
126.4.h.b yes 24 63.n odd 6 1
378.4.e.b 24 7.c even 3 1
378.4.e.b 24 9.c even 3 1
378.4.h.a 24 1.a even 1 1 trivial
378.4.h.a 24 63.g even 3 1 inner