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Properties

Label 381.2.e.c
Level $381$
Weight $2$
Character orbit 381.e
Analytic conductor $3.042$
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] = [381,2,Mod(19,381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(381, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("381.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 381 = 3 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 381.e (of order \(3\), degree \(2\), 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: \(3.04230031701\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{3} + 36 q^{4} + 2 q^{5} - 9 q^{7} + 6 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{3} + 36 q^{4} + 2 q^{5} - 9 q^{7} + 6 q^{8} - 12 q^{9} - 14 q^{10} - 10 q^{11} - 18 q^{12} - 3 q^{13} + 4 q^{14} - q^{15} + 36 q^{16} - q^{17} - 22 q^{19} + 2 q^{20} - 9 q^{21} + 3 q^{22} - 3 q^{24} + 14 q^{25} - 2 q^{26} + 24 q^{27} - 16 q^{28} - 3 q^{29} + 7 q^{30} + q^{31} + 22 q^{32} + 20 q^{33} + 7 q^{34} + 7 q^{35} - 18 q^{36} - 8 q^{37} - 14 q^{38} - 3 q^{39} + 18 q^{40} + q^{41} + 4 q^{42} - 14 q^{43} - 28 q^{44} - q^{45} + 14 q^{46} + 6 q^{47} - 18 q^{48} - 47 q^{49} - 42 q^{50} + 2 q^{51} - 9 q^{52} + 2 q^{53} - 6 q^{55} - 3 q^{56} + 11 q^{57} - 2 q^{58} + 24 q^{59} - q^{60} + 6 q^{61} - 26 q^{62} + 18 q^{63} + 90 q^{64} - 10 q^{65} - 6 q^{66} + 2 q^{67} + 13 q^{68} - 21 q^{70} + 24 q^{71} - 3 q^{72} - 2 q^{73} + 18 q^{74} - 7 q^{75} - 62 q^{76} - 20 q^{77} - 2 q^{78} + 5 q^{79} - 24 q^{80} - 12 q^{81} - 4 q^{82} - 15 q^{83} - 16 q^{84} - 9 q^{85} + 9 q^{86} + 6 q^{87} + 10 q^{88} - 28 q^{89} + 7 q^{90} + 47 q^{91} + 41 q^{92} + q^{93} - 10 q^{94} + 16 q^{95} - 11 q^{96} - 32 q^{97} - 43 q^{98} - 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.72296 −0.500000 + 0.866025i 5.41450 0.385600 1.36148 − 2.35815i 0.325891 − 0.564460i −9.29755 −0.500000 − 0.866025i −1.04997
19.2 −2.20619 −0.500000 + 0.866025i 2.86727 −3.41983 1.10309 − 1.91062i −2.28584 + 3.95920i −1.91336 −0.500000 − 0.866025i 7.54479
19.3 −2.05821 −0.500000 + 0.866025i 2.23622 2.99246 1.02910 − 1.78246i −1.70861 + 2.95940i −0.486182 −0.500000 − 0.866025i −6.15909
19.4 −1.66125 −0.500000 + 0.866025i 0.759765 3.36192 0.830627 − 1.43869i 1.49307 − 2.58608i 2.06035 −0.500000 − 0.866025i −5.58500
19.5 −1.08356 −0.500000 + 0.866025i −0.825906 −0.465669 0.541778 − 0.938387i 0.945797 − 1.63817i 3.06203 −0.500000 − 0.866025i 0.504578
19.6 −0.469437 −0.500000 + 0.866025i −1.77963 −1.30855 0.234719 − 0.406545i −0.671297 + 1.16272i 1.77430 −0.500000 − 0.866025i 0.614281
19.7 0.335116 −0.500000 + 0.866025i −1.88770 2.84374 −0.167558 + 0.290219i −2.47401 + 4.28510i −1.30283 −0.500000 − 0.866025i 0.952981
19.8 1.05609 −0.500000 + 0.866025i −0.884683 −2.48736 −0.528043 + 0.914597i 1.45168 − 2.51438i −3.04647 −0.500000 − 0.866025i −2.62687
19.9 1.67813 −0.500000 + 0.866025i 0.816130 −3.26586 −0.839066 + 1.45331i −1.72172 + 2.98211i −1.98669 −0.500000 − 0.866025i −5.48055
19.10 1.99039 −0.500000 + 0.866025i 1.96164 2.75611 −0.995194 + 1.72373i −0.525228 + 0.909722i −0.0763490 −0.500000 − 0.866025i 5.48573
19.11 2.34409 −0.500000 + 0.866025i 3.49476 0.226115 −1.17204 + 2.03004i 2.48363 − 4.30177i 3.50384 −0.500000 − 0.866025i 0.530033
19.12 2.79779 −0.500000 + 0.866025i 5.82764 −0.618671 −1.39890 + 2.42296i −1.81336 + 3.14083i 10.7089 −0.500000 − 0.866025i −1.73091
361.1 −2.72296 −0.500000 − 0.866025i 5.41450 0.385600 1.36148 + 2.35815i 0.325891 + 0.564460i −9.29755 −0.500000 + 0.866025i −1.04997
361.2 −2.20619 −0.500000 − 0.866025i 2.86727 −3.41983 1.10309 + 1.91062i −2.28584 − 3.95920i −1.91336 −0.500000 + 0.866025i 7.54479
361.3 −2.05821 −0.500000 − 0.866025i 2.23622 2.99246 1.02910 + 1.78246i −1.70861 − 2.95940i −0.486182 −0.500000 + 0.866025i −6.15909
361.4 −1.66125 −0.500000 − 0.866025i 0.759765 3.36192 0.830627 + 1.43869i 1.49307 + 2.58608i 2.06035 −0.500000 + 0.866025i −5.58500
361.5 −1.08356 −0.500000 − 0.866025i −0.825906 −0.465669 0.541778 + 0.938387i 0.945797 + 1.63817i 3.06203 −0.500000 + 0.866025i 0.504578
361.6 −0.469437 −0.500000 − 0.866025i −1.77963 −1.30855 0.234719 + 0.406545i −0.671297 − 1.16272i 1.77430 −0.500000 + 0.866025i 0.614281
361.7 0.335116 −0.500000 − 0.866025i −1.88770 2.84374 −0.167558 − 0.290219i −2.47401 − 4.28510i −1.30283 −0.500000 + 0.866025i 0.952981
361.8 1.05609 −0.500000 − 0.866025i −0.884683 −2.48736 −0.528043 − 0.914597i 1.45168 + 2.51438i −3.04647 −0.500000 + 0.866025i −2.62687
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
127.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 381.2.e.c ✓ 24
127.c even 3 1 inner 381.2.e.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.2.e.c ✓ 24 1.a even 1 1 trivial
381.2.e.c ✓ 24 127.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 21 T_{2}^{10} - T_{2}^{9} + 165 T_{2}^{8} + 14 T_{2}^{7} - 603 T_{2}^{6} - 66 T_{2}^{5} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(381, [\chi])\). Copy content Toggle raw display