Properties

Label 882.6.a.e
Level $882$
Weight $6$
Character orbit 882.a
Self dual yes
Analytic conductor $141.459$
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] = [882,6,Mod(1,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 882.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: \(141.458529075\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 16 q^{4} - 6 q^{5} - 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 16 q^{4} - 6 q^{5} - 64 q^{8} + 24 q^{10} + 666 q^{11} + 559 q^{13} + 256 q^{16} - 1740 q^{17} - 1157 q^{19} - 96 q^{20} - 2664 q^{22} + 3468 q^{23} - 3089 q^{25} - 2236 q^{26} - 3372 q^{29} - 6293 q^{31} - 1024 q^{32} + 6960 q^{34} + 3131 q^{37} + 4628 q^{38} + 384 q^{40} - 4866 q^{41} - 11407 q^{43} + 10656 q^{44} - 13872 q^{46} + 2310 q^{47} + 12356 q^{50} + 8944 q^{52} + 28296 q^{53} - 3996 q^{55} + 13488 q^{58} + 20544 q^{59} + 4630 q^{61} + 25172 q^{62} + 4096 q^{64} - 3354 q^{65} - 18745 q^{67} - 27840 q^{68} + 38226 q^{71} - 70589 q^{73} - 12524 q^{74} - 18512 q^{76} - 62293 q^{79} - 1536 q^{80} + 19464 q^{82} + 79818 q^{83} + 10440 q^{85} + 45628 q^{86} - 42624 q^{88} - 18120 q^{89} + 55488 q^{92} - 9240 q^{94} + 6942 q^{95} - 124754 q^{97}+O(q^{100}) \) 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
−4.00000 0 16.0000 −6.00000 0 0 −64.0000 0 24.0000
\(n\): 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 882.6.a.e 1
3.b odd 2 1 294.6.a.j 1
7.b odd 2 1 882.6.a.f 1
7.d odd 6 2 126.6.g.c 2
21.c even 2 1 294.6.a.l 1
21.g even 6 2 42.6.e.a 2
21.h odd 6 2 294.6.e.e 2
84.j odd 6 2 336.6.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.a 2 21.g even 6 2
126.6.g.c 2 7.d odd 6 2
294.6.a.j 1 3.b odd 2 1
294.6.a.l 1 21.c even 2 1
294.6.e.e 2 21.h odd 6 2
336.6.q.c 2 84.j odd 6 2
882.6.a.e 1 1.a even 1 1 trivial
882.6.a.f 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(882))\):

\( T_{5} + 6 \) Copy content Toggle raw display
\( T_{11} - 666 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 6 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 666 \) Copy content Toggle raw display
$13$ \( T - 559 \) Copy content Toggle raw display
$17$ \( T + 1740 \) Copy content Toggle raw display
$19$ \( T + 1157 \) Copy content Toggle raw display
$23$ \( T - 3468 \) Copy content Toggle raw display
$29$ \( T + 3372 \) Copy content Toggle raw display
$31$ \( T + 6293 \) Copy content Toggle raw display
$37$ \( T - 3131 \) Copy content Toggle raw display
$41$ \( T + 4866 \) Copy content Toggle raw display
$43$ \( T + 11407 \) Copy content Toggle raw display
$47$ \( T - 2310 \) Copy content Toggle raw display
$53$ \( T - 28296 \) Copy content Toggle raw display
$59$ \( T - 20544 \) Copy content Toggle raw display
$61$ \( T - 4630 \) Copy content Toggle raw display
$67$ \( T + 18745 \) Copy content Toggle raw display
$71$ \( T - 38226 \) Copy content Toggle raw display
$73$ \( T + 70589 \) Copy content Toggle raw display
$79$ \( T + 62293 \) Copy content Toggle raw display
$83$ \( T - 79818 \) Copy content Toggle raw display
$89$ \( T + 18120 \) Copy content Toggle raw display
$97$ \( T + 124754 \) Copy content Toggle raw display
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