Properties

Label 882.4.a.a
Level $882$
Weight $4$
Character orbit 882.a
Self dual yes
Analytic conductor $52.040$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(52.0396846251\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 2q^{2} + 4q^{4} - 22q^{5} - 8q^{8} + O(q^{10}) \) \( q - 2q^{2} + 4q^{4} - 22q^{5} - 8q^{8} + 44q^{10} + 26q^{11} + 54q^{13} + 16q^{16} - 74q^{17} - 116q^{19} - 88q^{20} - 52q^{22} - 58q^{23} + 359q^{25} - 108q^{26} + 208q^{29} + 252q^{31} - 32q^{32} + 148q^{34} + 50q^{37} + 232q^{38} + 176q^{40} - 126q^{41} + 164q^{43} + 104q^{44} + 116q^{46} + 444q^{47} - 718q^{50} + 216q^{52} + 12q^{53} - 572q^{55} - 416q^{58} - 124q^{59} + 162q^{61} - 504q^{62} + 64q^{64} - 1188q^{65} - 860q^{67} - 296q^{68} - 238q^{71} + 146q^{73} - 100q^{74} - 464q^{76} - 984q^{79} - 352q^{80} + 252q^{82} - 656q^{83} + 1628q^{85} - 328q^{86} - 208q^{88} + 954q^{89} - 232q^{92} - 888q^{94} + 2552q^{95} - 526q^{97} + O(q^{100}) \)

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.

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
−2.00000 0 4.00000 −22.0000 0 0 −8.00000 0 44.0000
\(n\): 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 882.4.a.a 1
3.b odd 2 1 882.4.a.s 1
7.b odd 2 1 126.4.a.e 1
7.c even 3 2 882.4.g.x 2
7.d odd 6 2 882.4.g.n 2
21.c even 2 1 126.4.a.f yes 1
21.g even 6 2 882.4.g.m 2
21.h odd 6 2 882.4.g.a 2
28.d even 2 1 1008.4.a.w 1
84.h odd 2 1 1008.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.e 1 7.b odd 2 1
126.4.a.f yes 1 21.c even 2 1
882.4.a.a 1 1.a even 1 1 trivial
882.4.a.s 1 3.b odd 2 1
882.4.g.a 2 21.h odd 6 2
882.4.g.m 2 21.g even 6 2
882.4.g.n 2 7.d odd 6 2
882.4.g.x 2 7.c even 3 2
1008.4.a.a 1 84.h odd 2 1
1008.4.a.w 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(882))\):

\( T_{5} + 22 \)
\( T_{11} - 26 \)
\( T_{13} - 54 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( 22 + T \)
$7$ \( T \)
$11$ \( -26 + T \)
$13$ \( -54 + T \)
$17$ \( 74 + T \)
$19$ \( 116 + T \)
$23$ \( 58 + T \)
$29$ \( -208 + T \)
$31$ \( -252 + T \)
$37$ \( -50 + T \)
$41$ \( 126 + T \)
$43$ \( -164 + T \)
$47$ \( -444 + T \)
$53$ \( -12 + T \)
$59$ \( 124 + T \)
$61$ \( -162 + T \)
$67$ \( 860 + T \)
$71$ \( 238 + T \)
$73$ \( -146 + T \)
$79$ \( 984 + T \)
$83$ \( 656 + T \)
$89$ \( -954 + T \)
$97$ \( 526 + T \)
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