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Properties

Label 378.4.h.b
Level $378$
Weight $4$
Character orbit 378.h
Analytic conductor $22.303$
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] = [378,4,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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{2} - 48 q^{4} + 20 q^{5} - 16 q^{7} - 192 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{2} - 48 q^{4} + 20 q^{5} - 16 q^{7} - 192 q^{8} + 20 q^{10} - 8 q^{11} + 80 q^{13} + 2 q^{14} - 192 q^{16} - 92 q^{17} + 54 q^{19} - 40 q^{20} - 8 q^{22} + 262 q^{23} + 356 q^{25} - 160 q^{26} + 68 q^{28} + 278 q^{29} + 110 q^{31} + 384 q^{32} + 184 q^{34} + 493 q^{35} - 21 q^{37} + 216 q^{38} - 160 q^{40} - 465 q^{41} + 159 q^{43} + 16 q^{44} + 262 q^{46} - 339 q^{47} - 744 q^{49} + 356 q^{50} - 640 q^{52} + 78 q^{53} - 1532 q^{55} + 128 q^{56} + 1112 q^{58} - 811 q^{59} + 989 q^{61} + 440 q^{62} + 1536 q^{64} - 312 q^{65} + 40 q^{67} + 736 q^{68} + 82 q^{70} - 980 q^{71} + 1510 q^{73} - 84 q^{74} + 216 q^{76} - 2209 q^{77} - 406 q^{79} - 160 q^{80} + 930 q^{82} + 7 q^{83} - 581 q^{85} + 636 q^{86} + 64 q^{88} - 675 q^{89} - 232 q^{91} - 524 q^{92} + 678 q^{94} + 1219 q^{95} + 2836 q^{97} + 414 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −15.6004 0 −11.1234 + 14.8077i −8.00000 0 −15.6004 − 27.0206i
289.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −15.4679 0 −14.9229 − 10.9685i −8.00000 0 −15.4679 − 26.7912i
289.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −11.3529 0 5.95018 + 17.5384i −8.00000 0 −11.3529 − 19.6638i
289.4 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −10.1344 0 2.82404 − 18.3037i −8.00000 0 −10.1344 − 17.5532i
289.5 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −3.73108 0 7.36279 − 16.9938i −8.00000 0 −3.73108 − 6.46242i
289.6 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −2.57874 0 17.5633 − 5.87629i −8.00000 0 −2.57874 − 4.46651i
289.7 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 3.46326 0 −11.1560 + 14.7832i −8.00000 0 3.46326 + 5.99855i
289.8 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 8.48072 0 −16.0459 − 9.24821i −8.00000 0 8.48072 + 14.6890i
289.9 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 10.2352 0 17.3134 + 6.57624i −8.00000 0 10.2352 + 17.7278i
289.10 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 11.3995 0 −18.5126 + 0.532294i −8.00000 0 11.3995 + 19.7445i
289.11 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 16.6414 0 9.00740 − 16.1823i −8.00000 0 16.6414 + 28.8237i
289.12 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 18.6453 0 3.73979 + 18.1387i −8.00000 0 18.6453 + 32.2946i
361.1 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −15.6004 0 −11.1234 − 14.8077i −8.00000 0 −15.6004 + 27.0206i
361.2 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −15.4679 0 −14.9229 + 10.9685i −8.00000 0 −15.4679 + 26.7912i
361.3 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −11.3529 0 5.95018 − 17.5384i −8.00000 0 −11.3529 + 19.6638i
361.4 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −10.1344 0 2.82404 + 18.3037i −8.00000 0 −10.1344 + 17.5532i
361.5 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −3.73108 0 7.36279 + 16.9938i −8.00000 0 −3.73108 + 6.46242i
361.6 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −2.57874 0 17.5633 + 5.87629i −8.00000 0 −2.57874 + 4.46651i
361.7 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 3.46326 0 −11.1560 − 14.7832i −8.00000 0 3.46326 − 5.99855i
361.8 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 8.48072 0 −16.0459 + 9.24821i −8.00000 0 8.48072 − 14.6890i
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.b 24
3.b odd 2 1 126.4.h.a yes 24
7.c even 3 1 378.4.e.a 24
9.c even 3 1 378.4.e.a 24
9.d odd 6 1 126.4.e.b ✓ 24
21.h odd 6 1 126.4.e.b ✓ 24
63.g even 3 1 inner 378.4.h.b 24
63.n odd 6 1 126.4.h.a yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.e.b ✓ 24 9.d odd 6 1
126.4.e.b ✓ 24 21.h odd 6 1
126.4.h.a yes 24 3.b odd 2 1
126.4.h.a yes 24 63.n odd 6 1
378.4.e.a 24 7.c even 3 1
378.4.e.a 24 9.c even 3 1
378.4.h.b 24 1.a even 1 1 trivial
378.4.h.b 24 63.g even 3 1 inner