Properties

Label 882.4.a.f
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 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 4q^{4} + 9q^{5} - 8q^{8} + O(q^{10}) \) \( q - 2q^{2} + 4q^{4} + 9q^{5} - 8q^{8} - 18q^{10} + 57q^{11} - 70q^{13} + 16q^{16} - 51q^{17} + 5q^{19} + 36q^{20} - 114q^{22} - 69q^{23} - 44q^{25} + 140q^{26} - 114q^{29} + 23q^{31} - 32q^{32} + 102q^{34} - 253q^{37} - 10q^{38} - 72q^{40} + 42q^{41} - 124q^{43} + 228q^{44} + 138q^{46} - 201q^{47} + 88q^{50} - 280q^{52} + 393q^{53} + 513q^{55} + 228q^{58} - 219q^{59} - 709q^{61} - 46q^{62} + 64q^{64} - 630q^{65} + 419q^{67} - 204q^{68} + 96q^{71} - 313q^{73} + 506q^{74} + 20q^{76} + 461q^{79} + 144q^{80} - 84q^{82} + 588q^{83} - 459q^{85} + 248q^{86} - 456q^{88} + 1017q^{89} - 276q^{92} + 402q^{94} + 45q^{95} - 1834q^{97} + O(q^{100}) \)

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 9.00000 0 0 −8.00000 0 −18.0000
\(n\): 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.f 1
3.b odd 2 1 98.4.a.d 1
7.b odd 2 1 882.4.a.c 1
7.c even 3 2 126.4.g.d 2
7.d odd 6 2 882.4.g.u 2
12.b even 2 1 784.4.a.p 1
15.d odd 2 1 2450.4.a.q 1
21.c even 2 1 98.4.a.f 1
21.g even 6 2 98.4.c.a 2
21.h odd 6 2 14.4.c.a 2
84.h odd 2 1 784.4.a.c 1
84.n even 6 2 112.4.i.a 2
105.g even 2 1 2450.4.a.d 1
105.o odd 6 2 350.4.e.e 2
105.x even 12 4 350.4.j.b 4
168.s odd 6 2 448.4.i.b 2
168.v even 6 2 448.4.i.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 21.h odd 6 2
98.4.a.d 1 3.b odd 2 1
98.4.a.f 1 21.c even 2 1
98.4.c.a 2 21.g even 6 2
112.4.i.a 2 84.n even 6 2
126.4.g.d 2 7.c even 3 2
350.4.e.e 2 105.o odd 6 2
350.4.j.b 4 105.x even 12 4
448.4.i.b 2 168.s odd 6 2
448.4.i.e 2 168.v even 6 2
784.4.a.c 1 84.h odd 2 1
784.4.a.p 1 12.b even 2 1
882.4.a.c 1 7.b odd 2 1
882.4.a.f 1 1.a even 1 1 trivial
882.4.g.u 2 7.d odd 6 2
2450.4.a.d 1 105.g even 2 1
2450.4.a.q 1 15.d odd 2 1

Hecke kernels

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

\( T_{5} - 9 \)
\( T_{11} - 57 \)
\( T_{13} + 70 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( -9 + T \)
$7$ \( T \)
$11$ \( -57 + T \)
$13$ \( 70 + T \)
$17$ \( 51 + T \)
$19$ \( -5 + T \)
$23$ \( 69 + T \)
$29$ \( 114 + T \)
$31$ \( -23 + T \)
$37$ \( 253 + T \)
$41$ \( -42 + T \)
$43$ \( 124 + T \)
$47$ \( 201 + T \)
$53$ \( -393 + T \)
$59$ \( 219 + T \)
$61$ \( 709 + T \)
$67$ \( -419 + T \)
$71$ \( -96 + T \)
$73$ \( 313 + T \)
$79$ \( -461 + T \)
$83$ \( -588 + T \)
$89$ \( -1017 + T \)
$97$ \( 1834 + T \)
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