Properties

Label 18.8.a.a
Level $18$
Weight $8$
Character orbit 18.a
Self dual yes
Analytic conductor $5.623$
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] = [18,8,Mod(1,18)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(18, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("18.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 18.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: \(5.62293045871\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
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)
 
\(f(q)\) \(=\) \( q - 8 q^{2} + 64 q^{4} + 114 q^{5} - 1576 q^{7} - 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} + 64 q^{4} + 114 q^{5} - 1576 q^{7} - 512 q^{8} - 912 q^{10} - 7332 q^{11} - 3802 q^{13} + 12608 q^{14} + 4096 q^{16} + 6606 q^{17} + 24860 q^{19} + 7296 q^{20} + 58656 q^{22} - 41448 q^{23} - 65129 q^{25} + 30416 q^{26} - 100864 q^{28} + 41610 q^{29} + 33152 q^{31} - 32768 q^{32} - 52848 q^{34} - 179664 q^{35} - 36466 q^{37} - 198880 q^{38} - 58368 q^{40} + 639078 q^{41} - 156412 q^{43} - 469248 q^{44} + 331584 q^{46} + 433776 q^{47} + 1660233 q^{49} + 521032 q^{50} - 243328 q^{52} - 786078 q^{53} - 835848 q^{55} + 806912 q^{56} - 332880 q^{58} - 745140 q^{59} - 1660618 q^{61} - 265216 q^{62} + 262144 q^{64} - 433428 q^{65} - 3290836 q^{67} + 422784 q^{68} + 1437312 q^{70} - 5716152 q^{71} + 2659898 q^{73} + 291728 q^{74} + 1591040 q^{76} + 11555232 q^{77} + 3807440 q^{79} + 466944 q^{80} - 5112624 q^{82} - 2229468 q^{83} + 753084 q^{85} + 1251296 q^{86} + 3753984 q^{88} - 5991210 q^{89} + 5991952 q^{91} - 2652672 q^{92} - 3470208 q^{94} + 2834040 q^{95} - 4060126 q^{97} - 13281864 q^{98}+O(q^{100}) \) 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 114.000 0 −1576.00 −512.000 0 −912.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 18.8.a.a 1
3.b odd 2 1 6.8.a.a 1
4.b odd 2 1 144.8.a.h 1
5.b even 2 1 450.8.a.ba 1
5.c odd 4 2 450.8.c.a 2
8.b even 2 1 576.8.a.h 1
8.d odd 2 1 576.8.a.i 1
9.c even 3 2 162.8.c.i 2
9.d odd 6 2 162.8.c.d 2
12.b even 2 1 48.8.a.b 1
15.d odd 2 1 150.8.a.e 1
15.e even 4 2 150.8.c.k 2
21.c even 2 1 294.8.a.l 1
21.g even 6 2 294.8.e.d 2
21.h odd 6 2 294.8.e.c 2
24.f even 2 1 192.8.a.n 1
24.h odd 2 1 192.8.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.8.a.a 1 3.b odd 2 1
18.8.a.a 1 1.a even 1 1 trivial
48.8.a.b 1 12.b even 2 1
144.8.a.h 1 4.b odd 2 1
150.8.a.e 1 15.d odd 2 1
150.8.c.k 2 15.e even 4 2
162.8.c.d 2 9.d odd 6 2
162.8.c.i 2 9.c even 3 2
192.8.a.f 1 24.h odd 2 1
192.8.a.n 1 24.f even 2 1
294.8.a.l 1 21.c even 2 1
294.8.e.c 2 21.h odd 6 2
294.8.e.d 2 21.g even 6 2
450.8.a.ba 1 5.b even 2 1
450.8.c.a 2 5.c odd 4 2
576.8.a.h 1 8.b even 2 1
576.8.a.i 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 114 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(18))\). 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
$5$ \( T - 114 \) Copy content Toggle raw display
$7$ \( T + 1576 \) Copy content Toggle raw display
$11$ \( T + 7332 \) Copy content Toggle raw display
$13$ \( T + 3802 \) Copy content Toggle raw display
$17$ \( T - 6606 \) Copy content Toggle raw display
$19$ \( T - 24860 \) Copy content Toggle raw display
$23$ \( T + 41448 \) Copy content Toggle raw display
$29$ \( T - 41610 \) Copy content Toggle raw display
$31$ \( T - 33152 \) Copy content Toggle raw display
$37$ \( T + 36466 \) Copy content Toggle raw display
$41$ \( T - 639078 \) Copy content Toggle raw display
$43$ \( T + 156412 \) Copy content Toggle raw display
$47$ \( T - 433776 \) Copy content Toggle raw display
$53$ \( T + 786078 \) Copy content Toggle raw display
$59$ \( T + 745140 \) Copy content Toggle raw display
$61$ \( T + 1660618 \) Copy content Toggle raw display
$67$ \( T + 3290836 \) Copy content Toggle raw display
$71$ \( T + 5716152 \) Copy content Toggle raw display
$73$ \( T - 2659898 \) Copy content Toggle raw display
$79$ \( T - 3807440 \) Copy content Toggle raw display
$83$ \( T + 2229468 \) Copy content Toggle raw display
$89$ \( T + 5991210 \) Copy content Toggle raw display
$97$ \( T + 4060126 \) Copy content Toggle raw display
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