Properties

Label 882.4.a.l
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 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} + 4q^{4} - 6q^{5} + 8q^{8} + O(q^{10}) \) \( q + 2q^{2} + 4q^{4} - 6q^{5} + 8q^{8} - 12q^{10} + 30q^{11} - 53q^{13} + 16q^{16} - 84q^{17} + 97q^{19} - 24q^{20} + 60q^{22} - 84q^{23} - 89q^{25} - 106q^{26} + 180q^{29} - 179q^{31} + 32q^{32} - 168q^{34} - 145q^{37} + 194q^{38} - 48q^{40} + 126q^{41} - 325q^{43} + 120q^{44} - 168q^{46} - 366q^{47} - 178q^{50} - 212q^{52} + 768q^{53} - 180q^{55} + 360q^{58} - 264q^{59} - 818q^{61} - 358q^{62} + 64q^{64} + 318q^{65} - 523q^{67} - 336q^{68} + 342q^{71} + 43q^{73} - 290q^{74} + 388q^{76} - 1171q^{79} - 96q^{80} + 252q^{82} - 810q^{83} + 504q^{85} - 650q^{86} + 240q^{88} - 600q^{89} - 336q^{92} - 732q^{94} - 582q^{95} - 386q^{97} + O(q^{100}) \)

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 −6.00000 0 0 8.00000 0 −12.0000
\(n\): 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.l 1
3.b odd 2 1 294.4.a.c 1
7.b odd 2 1 882.4.a.o 1
7.c even 3 2 882.4.g.g 2
7.d odd 6 2 126.4.g.b 2
12.b even 2 1 2352.4.a.bf 1
21.c even 2 1 294.4.a.d 1
21.g even 6 2 42.4.e.a 2
21.h odd 6 2 294.4.e.i 2
84.h odd 2 1 2352.4.a.f 1
84.j odd 6 2 336.4.q.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.a 2 21.g even 6 2
126.4.g.b 2 7.d odd 6 2
294.4.a.c 1 3.b odd 2 1
294.4.a.d 1 21.c even 2 1
294.4.e.i 2 21.h odd 6 2
336.4.q.f 2 84.j odd 6 2
882.4.a.l 1 1.a even 1 1 trivial
882.4.a.o 1 7.b odd 2 1
882.4.g.g 2 7.c even 3 2
2352.4.a.f 1 84.h odd 2 1
2352.4.a.bf 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(882))\):

\( T_{5} + 6 \)
\( T_{11} - 30 \)
\( T_{13} + 53 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( T \)
$5$ \( 6 + T \)
$7$ \( T \)
$11$ \( -30 + T \)
$13$ \( 53 + T \)
$17$ \( 84 + T \)
$19$ \( -97 + T \)
$23$ \( 84 + T \)
$29$ \( -180 + T \)
$31$ \( 179 + T \)
$37$ \( 145 + T \)
$41$ \( -126 + T \)
$43$ \( 325 + T \)
$47$ \( 366 + T \)
$53$ \( -768 + T \)
$59$ \( 264 + T \)
$61$ \( 818 + T \)
$67$ \( 523 + T \)
$71$ \( -342 + T \)
$73$ \( -43 + T \)
$79$ \( 1171 + T \)
$83$ \( 810 + T \)
$89$ \( 600 + T \)
$97$ \( 386 + T \)
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