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Properties

Label 387.8.a.h
Level $387$
Weight $8$
Character orbit 387.a
Self dual yes
Analytic conductor $120.893$
Analytic rank $0$
Dimension $26$
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] = [387,8,Mod(1,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 387.a (trivial)

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: yes
Analytic conductor: \(120.893004862\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 1766 q^{4} + 698 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q + 1766 q^{4} + 698 q^{7} + 15724 q^{10} + 22564 q^{13} + 82582 q^{16} - 58402 q^{19} + 226668 q^{22} + 541708 q^{25} + 81316 q^{28} + 222358 q^{31} + 715456 q^{34} + 1779428 q^{37} + 2177152 q^{40} - 2067182 q^{43} + 4380228 q^{46} + 7431816 q^{49} + 7385696 q^{52} + 2601026 q^{55} + 5804116 q^{58} + 7381340 q^{61} + 24358326 q^{64} - 2946542 q^{67} + 15154464 q^{70} + 9959584 q^{73} - 26462084 q^{76} - 1647804 q^{79} + 15947976 q^{82} + 23958072 q^{85} + 58319776 q^{88} - 9312030 q^{91} + 16336660 q^{94} + 37732732 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −22.4038 0 373.928 −94.8869 0 −918.674 −5509.71 0 2125.82
1.2 −20.1889 0 279.590 −273.206 0 1735.33 −3060.43 0 5515.73
1.3 −19.8297 0 265.215 −281.857 0 −174.893 −2720.93 0 5589.13
1.4 −16.2101 0 134.768 408.682 0 −1268.57 −109.714 0 −6624.79
1.5 −16.1346 0 132.324 407.685 0 1649.92 −69.7638 0 −6577.81
1.6 −13.3151 0 49.2926 62.5971 0 256.235 1048.00 0 −833.489
1.7 −12.8116 0 36.1382 −537.658 0 18.6121 1176.90 0 6888.28
1.8 −11.9013 0 13.6398 51.2055 0 −1740.00 1361.03 0 −609.409
1.9 −9.98699 0 −28.2601 −54.5062 0 −860.873 1560.57 0 544.353
1.10 −8.14806 0 −61.6092 −526.518 0 1056.67 1544.95 0 4290.10
1.11 −7.96103 0 −64.6220 392.364 0 430.566 1533.47 0 −3123.62
1.12 −2.46533 0 −121.922 −239.564 0 −643.728 616.141 0 590.605
1.13 −1.58661 0 −125.483 −54.9039 0 808.402 402.178 0 87.1110
1.14 1.58661 0 −125.483 54.9039 0 808.402 −402.178 0 87.1110
1.15 2.46533 0 −121.922 239.564 0 −643.728 −616.141 0 590.605
1.16 7.96103 0 −64.6220 −392.364 0 430.566 −1533.47 0 −3123.62
1.17 8.14806 0 −61.6092 526.518 0 1056.67 −1544.95 0 4290.10
1.18 9.98699 0 −28.2601 54.5062 0 −860.873 −1560.57 0 544.353
1.19 11.9013 0 13.6398 −51.2055 0 −1740.00 −1361.03 0 −609.409
1.20 12.8116 0 36.1382 537.658 0 18.6121 −1176.90 0 6888.28
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.26
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(43\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.8.a.h ✓ 26
3.b odd 2 1 inner 387.8.a.h ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.8.a.h ✓ 26 1.a even 1 1 trivial
387.8.a.h ✓ 26 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{26} - 2547 T_{2}^{24} + 2840431 T_{2}^{22} - 1828645293 T_{2}^{20} + 754437694788 T_{2}^{18} + \cdots - 14\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(387))\). Copy content Toggle raw display