Properties

Label 378.8.a.b
Level $378$
Weight $8$
Character orbit 378.a
Self dual yes
Analytic conductor $118.082$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,8,Mod(1,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, 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("378.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 378.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: \(118.081539633\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 8 q^{2} + 64 q^{4} + 40 q^{5} - 343 q^{7} - 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} + 64 q^{4} + 40 q^{5} - 343 q^{7} - 512 q^{8} - 320 q^{10} - 7096 q^{11} - 699 q^{13} + 2744 q^{14} + 4096 q^{16} + 38657 q^{17} - 3454 q^{19} + 2560 q^{20} + 56768 q^{22} + 92789 q^{23} - 76525 q^{25} + 5592 q^{26} - 21952 q^{28} - 100961 q^{29} + 232677 q^{31} - 32768 q^{32} - 309256 q^{34} - 13720 q^{35} - 63286 q^{37} + 27632 q^{38} - 20480 q^{40} - 798762 q^{41} + 189911 q^{43} - 454144 q^{44} - 742312 q^{46} + 548526 q^{47} + 117649 q^{49} + 612200 q^{50} - 44736 q^{52} + 1123569 q^{53} - 283840 q^{55} + 175616 q^{56} + 807688 q^{58} + 948703 q^{59} - 2268546 q^{61} - 1861416 q^{62} + 262144 q^{64} - 27960 q^{65} - 1751285 q^{67} + 2474048 q^{68} + 109760 q^{70} + 5424563 q^{71} + 1325794 q^{73} + 506288 q^{74} - 221056 q^{76} + 2433928 q^{77} - 7744362 q^{79} + 163840 q^{80} + 6390096 q^{82} - 7856212 q^{83} + 1546280 q^{85} - 1519288 q^{86} + 3633152 q^{88} - 2469531 q^{89} + 239757 q^{91} + 5938496 q^{92} - 4388208 q^{94} - 138160 q^{95} + 11162296 q^{97} - 941192 q^{98}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−8.00000 0 64.0000 40.0000 0 −343.000 −512.000 0 −320.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.8.a.b 1
3.b odd 2 1 378.8.a.c yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.8.a.b 1 1.a even 1 1 trivial
378.8.a.c yes 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 40 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(378))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 8 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 40 \) Copy content Toggle raw display
$7$ \( T + 343 \) Copy content Toggle raw display
$11$ \( T + 7096 \) Copy content Toggle raw display
$13$ \( T + 699 \) Copy content Toggle raw display
$17$ \( T - 38657 \) Copy content Toggle raw display
$19$ \( T + 3454 \) Copy content Toggle raw display
$23$ \( T - 92789 \) Copy content Toggle raw display
$29$ \( T + 100961 \) Copy content Toggle raw display
$31$ \( T - 232677 \) Copy content Toggle raw display
$37$ \( T + 63286 \) Copy content Toggle raw display
$41$ \( T + 798762 \) Copy content Toggle raw display
$43$ \( T - 189911 \) Copy content Toggle raw display
$47$ \( T - 548526 \) Copy content Toggle raw display
$53$ \( T - 1123569 \) Copy content Toggle raw display
$59$ \( T - 948703 \) Copy content Toggle raw display
$61$ \( T + 2268546 \) Copy content Toggle raw display
$67$ \( T + 1751285 \) Copy content Toggle raw display
$71$ \( T - 5424563 \) Copy content Toggle raw display
$73$ \( T - 1325794 \) Copy content Toggle raw display
$79$ \( T + 7744362 \) Copy content Toggle raw display
$83$ \( T + 7856212 \) Copy content Toggle raw display
$89$ \( T + 2469531 \) Copy content Toggle raw display
$97$ \( T - 11162296 \) Copy content Toggle raw display
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