Properties

Label 882.4.a.h
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

\(f(q)\) \(=\) \( q + 2q^{2} + 4q^{4} - 15q^{5} + 8q^{8} + O(q^{10}) \) \( q + 2q^{2} + 4q^{4} - 15q^{5} + 8q^{8} - 30q^{10} + 9q^{11} + 88q^{13} + 16q^{16} - 84q^{17} - 104q^{19} - 60q^{20} + 18q^{22} + 84q^{23} + 100q^{25} + 176q^{26} - 51q^{29} - 185q^{31} + 32q^{32} - 168q^{34} + 44q^{37} - 208q^{38} - 120q^{40} - 168q^{41} + 326q^{43} + 36q^{44} + 168q^{46} - 138q^{47} + 200q^{50} + 352q^{52} - 639q^{53} - 135q^{55} - 102q^{58} + 159q^{59} - 722q^{61} - 370q^{62} + 64q^{64} - 1320q^{65} - 166q^{67} - 336q^{68} - 1086q^{71} - 218q^{73} + 88q^{74} - 416q^{76} - 583q^{79} - 240q^{80} - 336q^{82} - 597q^{83} + 1260q^{85} + 652q^{86} + 72q^{88} - 1038q^{89} + 336q^{92} - 276q^{94} + 1560q^{95} + 169q^{97} + O(q^{100}) \)

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 −15.0000 0 0 8.00000 0 −30.0000
\(n\): 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.h 1
3.b odd 2 1 294.4.a.g 1
7.b odd 2 1 882.4.a.r 1
7.c even 3 2 882.4.g.l 2
7.d odd 6 2 126.4.g.a 2
12.b even 2 1 2352.4.a.q 1
21.c even 2 1 294.4.a.a 1
21.g even 6 2 42.4.e.b 2
21.h odd 6 2 294.4.e.e 2
84.h odd 2 1 2352.4.a.u 1
84.j odd 6 2 336.4.q.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 21.g even 6 2
126.4.g.a 2 7.d odd 6 2
294.4.a.a 1 21.c even 2 1
294.4.a.g 1 3.b odd 2 1
294.4.e.e 2 21.h odd 6 2
336.4.q.d 2 84.j odd 6 2
882.4.a.h 1 1.a even 1 1 trivial
882.4.a.r 1 7.b odd 2 1
882.4.g.l 2 7.c even 3 2
2352.4.a.q 1 12.b even 2 1
2352.4.a.u 1 84.h odd 2 1

Hecke kernels

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

\( T_{5} + 15 \)
\( T_{11} - 9 \)
\( T_{13} - 88 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( T \)
$5$ \( 15 + T \)
$7$ \( T \)
$11$ \( -9 + T \)
$13$ \( -88 + T \)
$17$ \( 84 + T \)
$19$ \( 104 + T \)
$23$ \( -84 + T \)
$29$ \( 51 + T \)
$31$ \( 185 + T \)
$37$ \( -44 + T \)
$41$ \( 168 + T \)
$43$ \( -326 + T \)
$47$ \( 138 + T \)
$53$ \( 639 + T \)
$59$ \( -159 + T \)
$61$ \( 722 + T \)
$67$ \( 166 + T \)
$71$ \( 1086 + T \)
$73$ \( 218 + T \)
$79$ \( 583 + T \)
$83$ \( 597 + T \)
$89$ \( 1038 + T \)
$97$ \( -169 + T \)
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