Properties

Label 882.6.a.c
Level $882$
Weight $6$
Character orbit 882.a
Self dual yes
Analytic conductor $141.459$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 16 q^{4} - 26 q^{5} - 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 16 q^{4} - 26 q^{5} - 64 q^{8} + 104 q^{10} - 470 q^{11} - 642 q^{13} + 256 q^{16} + 1010 q^{17} - 1532 q^{19} - 416 q^{20} + 1880 q^{22} + 430 q^{23} - 2449 q^{25} + 2568 q^{26} - 6736 q^{29} - 2268 q^{31} - 1024 q^{32} - 4040 q^{34} - 9574 q^{37} + 6128 q^{38} + 1664 q^{40} + 14406 q^{41} - 9748 q^{43} - 7520 q^{44} - 1720 q^{46} + 17004 q^{47} + 9796 q^{50} - 10272 q^{52} - 7596 q^{53} + 12220 q^{55} + 26944 q^{58} - 18908 q^{59} + 36762 q^{61} + 9072 q^{62} + 4096 q^{64} + 16692 q^{65} - 36788 q^{67} + 16160 q^{68} - 18326 q^{71} - 36382 q^{73} + 38296 q^{74} - 24512 q^{76} + 29784 q^{79} - 6656 q^{80} - 57624 q^{82} - 28240 q^{83} - 26260 q^{85} + 38992 q^{86} + 30080 q^{88} - 75954 q^{89} + 6880 q^{92} - 68016 q^{94} + 39832 q^{95} + 80690 q^{97}+O(q^{100}) \) 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 −26.0000 0 0 −64.0000 0 104.000
\(n\): 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 882.6.a.c 1
3.b odd 2 1 882.6.a.u 1
7.b odd 2 1 126.6.a.d 1
21.c even 2 1 126.6.a.j yes 1
28.d even 2 1 1008.6.a.r 1
84.h odd 2 1 1008.6.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.6.a.d 1 7.b odd 2 1
126.6.a.j yes 1 21.c even 2 1
882.6.a.c 1 1.a even 1 1 trivial
882.6.a.u 1 3.b odd 2 1
1008.6.a.i 1 84.h odd 2 1
1008.6.a.r 1 28.d even 2 1

Hecke kernels

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

\( T_{5} + 26 \) Copy content Toggle raw display
\( T_{11} + 470 \) 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
$5$ \( T + 26 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 470 \) Copy content Toggle raw display
$13$ \( T + 642 \) Copy content Toggle raw display
$17$ \( T - 1010 \) Copy content Toggle raw display
$19$ \( T + 1532 \) Copy content Toggle raw display
$23$ \( T - 430 \) Copy content Toggle raw display
$29$ \( T + 6736 \) Copy content Toggle raw display
$31$ \( T + 2268 \) Copy content Toggle raw display
$37$ \( T + 9574 \) Copy content Toggle raw display
$41$ \( T - 14406 \) Copy content Toggle raw display
$43$ \( T + 9748 \) Copy content Toggle raw display
$47$ \( T - 17004 \) Copy content Toggle raw display
$53$ \( T + 7596 \) Copy content Toggle raw display
$59$ \( T + 18908 \) Copy content Toggle raw display
$61$ \( T - 36762 \) Copy content Toggle raw display
$67$ \( T + 36788 \) Copy content Toggle raw display
$71$ \( T + 18326 \) Copy content Toggle raw display
$73$ \( T + 36382 \) Copy content Toggle raw display
$79$ \( T - 29784 \) Copy content Toggle raw display
$83$ \( T + 28240 \) Copy content Toggle raw display
$89$ \( T + 75954 \) Copy content Toggle raw display
$97$ \( T - 80690 \) Copy content Toggle raw display
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