Properties

Label 882.4.a.d
Level $882$
Weight $4$
Character orbit 882.a
Self dual yes
Analytic conductor $52.040$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q - 2q^{2} + 4q^{4} + 2q^{5} - 8q^{8} + O(q^{10}) \) \( q - 2q^{2} + 4q^{4} + 2q^{5} - 8q^{8} - 4q^{10} + 8q^{11} + 42q^{13} + 16q^{16} - 2q^{17} + 124q^{19} + 8q^{20} - 16q^{22} - 76q^{23} - 121q^{25} - 84q^{26} - 254q^{29} + 72q^{31} - 32q^{32} + 4q^{34} + 398q^{37} - 248q^{38} - 16q^{40} + 462q^{41} + 212q^{43} + 32q^{44} + 152q^{46} - 264q^{47} + 242q^{50} + 168q^{52} + 162q^{53} + 16q^{55} + 508q^{58} - 772q^{59} - 30q^{61} - 144q^{62} + 64q^{64} + 84q^{65} - 764q^{67} - 8q^{68} + 236q^{71} - 418q^{73} - 796q^{74} + 496q^{76} + 552q^{79} + 32q^{80} - 924q^{82} + 1036q^{83} - 4q^{85} - 424q^{86} - 64q^{88} + 30q^{89} - 304q^{92} + 528q^{94} + 248q^{95} + 1190q^{97} + O(q^{100}) \)

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 2.00000 0 0 −8.00000 0 −4.00000
\(n\): 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.d 1
3.b odd 2 1 294.4.a.h 1
7.b odd 2 1 126.4.a.c 1
7.c even 3 2 882.4.g.r 2
7.d odd 6 2 882.4.g.s 2
12.b even 2 1 2352.4.a.ba 1
21.c even 2 1 42.4.a.b 1
21.g even 6 2 294.4.e.a 2
21.h odd 6 2 294.4.e.d 2
28.d even 2 1 1008.4.a.j 1
84.h odd 2 1 336.4.a.d 1
105.g even 2 1 1050.4.a.d 1
105.k odd 4 2 1050.4.g.n 2
168.e odd 2 1 1344.4.a.t 1
168.i even 2 1 1344.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.b 1 21.c even 2 1
126.4.a.c 1 7.b odd 2 1
294.4.a.h 1 3.b odd 2 1
294.4.e.a 2 21.g even 6 2
294.4.e.d 2 21.h odd 6 2
336.4.a.d 1 84.h odd 2 1
882.4.a.d 1 1.a even 1 1 trivial
882.4.g.r 2 7.c even 3 2
882.4.g.s 2 7.d odd 6 2
1008.4.a.j 1 28.d even 2 1
1050.4.a.d 1 105.g even 2 1
1050.4.g.n 2 105.k odd 4 2
1344.4.a.f 1 168.i even 2 1
1344.4.a.t 1 168.e odd 2 1
2352.4.a.ba 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(882))\):

\( T_{5} - 2 \)
\( T_{11} - 8 \)
\( T_{13} - 42 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( -2 + T \)
$7$ \( T \)
$11$ \( -8 + T \)
$13$ \( -42 + T \)
$17$ \( 2 + T \)
$19$ \( -124 + T \)
$23$ \( 76 + T \)
$29$ \( 254 + T \)
$31$ \( -72 + T \)
$37$ \( -398 + T \)
$41$ \( -462 + T \)
$43$ \( -212 + T \)
$47$ \( 264 + T \)
$53$ \( -162 + T \)
$59$ \( 772 + T \)
$61$ \( 30 + T \)
$67$ \( 764 + T \)
$71$ \( -236 + T \)
$73$ \( 418 + T \)
$79$ \( -552 + T \)
$83$ \( -1036 + T \)
$89$ \( -30 + T \)
$97$ \( -1190 + T \)
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