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Properties

Label 387.8.a.g
Level $387$
Weight $8$
Character orbit 387.a
Self dual yes
Analytic conductor $120.893$
Analytic rank $1$
Dimension $22$
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: \(1\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 1126 q^{4} - 2046 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 1126 q^{4} - 2046 q^{7} + 3724 q^{10} - 49194 q^{13} + 95846 q^{16} - 22282 q^{19} + 56300 q^{22} + 152152 q^{25} - 488080 q^{28} - 395580 q^{31} - 1071216 q^{34} - 1089160 q^{37} - 1999072 q^{40} + 1749154 q^{43} - 4935696 q^{46} + 1592044 q^{49} - 10709736 q^{52} - 3122942 q^{55} + 1874340 q^{58} - 6583408 q^{61} - 1583466 q^{64} - 4045412 q^{67} - 23137184 q^{70} - 699724 q^{73} - 11947652 q^{76} - 13438716 q^{79} - 44492624 q^{82} + 18576868 q^{85} - 7006792 q^{88} + 8886970 q^{91} - 29917192 q^{94} - 21206182 q^{97}+O(q^{100}) \) 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 −21.7869 0 346.669 334.269 0 365.642 −4764.12 0 −7282.69
1.2 −18.6042 0 218.117 −192.260 0 13.5428 −1676.56 0 3576.85
1.3 −18.2140 0 203.750 232.766 0 −891.415 −1379.71 0 −4239.61
1.4 −17.8189 0 189.514 −476.391 0 −1549.64 −1096.12 0 8488.78
1.5 −13.8544 0 63.9455 −160.702 0 335.694 887.440 0 2226.43
1.6 −12.5078 0 28.4449 −71.4183 0 1020.55 1245.21 0 893.285
1.7 −8.23204 0 −60.2336 252.943 0 855.172 1549.55 0 −2082.24
1.8 −6.23720 0 −89.0973 −224.109 0 −945.870 1354.08 0 1397.81
1.9 −5.70759 0 −95.4234 446.683 0 −559.762 1275.21 0 −2549.48
1.10 −2.94880 0 −119.305 −268.950 0 1465.79 729.251 0 793.079
1.11 −2.14900 0 −123.382 −297.711 0 −1132.71 540.219 0 639.780
1.12 2.14900 0 −123.382 297.711 0 −1132.71 −540.219 0 639.780
1.13 2.94880 0 −119.305 268.950 0 1465.79 −729.251 0 793.079
1.14 5.70759 0 −95.4234 −446.683 0 −559.762 −1275.21 0 −2549.48
1.15 6.23720 0 −89.0973 224.109 0 −945.870 −1354.08 0 1397.81
1.16 8.23204 0 −60.2336 −252.943 0 855.172 −1549.55 0 −2082.24
1.17 12.5078 0 28.4449 71.4183 0 1020.55 −1245.21 0 893.285
1.18 13.8544 0 63.9455 160.702 0 335.694 −887.440 0 2226.43
1.19 17.8189 0 189.514 476.391 0 −1549.64 1096.12 0 8488.78
1.20 18.2140 0 203.750 −232.766 0 −891.415 1379.71 0 −4239.61
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
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.g ✓ 22
3.b odd 2 1 inner 387.8.a.g ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.8.a.g ✓ 22 1.a even 1 1 trivial
387.8.a.g ✓ 22 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} - 1971 T_{2}^{20} + 1630139 T_{2}^{18} - 736743457 T_{2}^{16} + 198621096520 T_{2}^{14} + \cdots - 17\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(387))\). Copy content Toggle raw display