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

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

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(882))\):

\( T_{5} - 9 \) Copy content Toggle raw display
\( T_{11} - 57 \) Copy content Toggle raw display
\( T_{13} + 70 \) Copy content Toggle raw display

Hecke characteristic polynomials

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