Properties

Label 882.4.a.w
Level $882$
Weight $4$
Character orbit 882.a
Self dual yes
Analytic conductor $52.040$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{22}) \)
Defining polynomial: \(x^{2} - 22\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 98)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{22}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{2} + 4 q^{4} + \beta q^{5} -8 q^{8} +O(q^{10})\) \( q -2 q^{2} + 4 q^{4} + \beta q^{5} -8 q^{8} -2 \beta q^{10} -20 q^{11} -7 \beta q^{13} + 16 q^{16} + 6 \beta q^{17} -\beta q^{19} + 4 \beta q^{20} + 40 q^{22} -48 q^{23} -37 q^{25} + 14 \beta q^{26} + 166 q^{29} + 22 \beta q^{31} -32 q^{32} -12 \beta q^{34} -78 q^{37} + 2 \beta q^{38} -8 \beta q^{40} + 42 \beta q^{41} + 436 q^{43} -80 q^{44} + 96 q^{46} + 22 \beta q^{47} + 74 q^{50} -28 \beta q^{52} -62 q^{53} -20 \beta q^{55} -332 q^{58} -71 \beta q^{59} -29 \beta q^{61} -44 \beta q^{62} + 64 q^{64} -616 q^{65} + 580 q^{67} + 24 \beta q^{68} + 544 q^{71} + 64 \beta q^{73} + 156 q^{74} -4 \beta q^{76} -680 q^{79} + 16 \beta q^{80} -84 \beta q^{82} + 21 \beta q^{83} + 528 q^{85} -872 q^{86} + 160 q^{88} -160 \beta q^{89} -192 q^{92} -44 \beta q^{94} -88 q^{95} + 70 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 8q^{4} - 16q^{8} + O(q^{10}) \) \( 2q - 4q^{2} + 8q^{4} - 16q^{8} - 40q^{11} + 32q^{16} + 80q^{22} - 96q^{23} - 74q^{25} + 332q^{29} - 64q^{32} - 156q^{37} + 872q^{43} - 160q^{44} + 192q^{46} + 148q^{50} - 124q^{53} - 664q^{58} + 128q^{64} - 1232q^{65} + 1160q^{67} + 1088q^{71} + 312q^{74} - 1360q^{79} + 1056q^{85} - 1744q^{86} + 320q^{88} - 384q^{92} - 176q^{95} + O(q^{100}) \)

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
−4.69042
4.69042
−2.00000 0 4.00000 −9.38083 0 0 −8.00000 0 18.7617
1.2 −2.00000 0 4.00000 9.38083 0 0 −8.00000 0 −18.7617
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.a.w 2
3.b odd 2 1 98.4.a.h 2
7.b odd 2 1 inner 882.4.a.w 2
7.c even 3 2 882.4.g.bi 4
7.d odd 6 2 882.4.g.bi 4
12.b even 2 1 784.4.a.z 2
15.d odd 2 1 2450.4.a.bs 2
21.c even 2 1 98.4.a.h 2
21.g even 6 2 98.4.c.g 4
21.h odd 6 2 98.4.c.g 4
84.h odd 2 1 784.4.a.z 2
105.g even 2 1 2450.4.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.h 2 3.b odd 2 1
98.4.a.h 2 21.c even 2 1
98.4.c.g 4 21.g even 6 2
98.4.c.g 4 21.h odd 6 2
784.4.a.z 2 12.b even 2 1
784.4.a.z 2 84.h odd 2 1
882.4.a.w 2 1.a even 1 1 trivial
882.4.a.w 2 7.b odd 2 1 inner
882.4.g.bi 4 7.c even 3 2
882.4.g.bi 4 7.d odd 6 2
2450.4.a.bs 2 15.d odd 2 1
2450.4.a.bs 2 105.g even 2 1

Hecke kernels

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

\( T_{5}^{2} - 88 \)
\( T_{11} + 20 \)
\( T_{13}^{2} - 4312 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -88 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 20 + T )^{2} \)
$13$ \( -4312 + T^{2} \)
$17$ \( -3168 + T^{2} \)
$19$ \( -88 + T^{2} \)
$23$ \( ( 48 + T )^{2} \)
$29$ \( ( -166 + T )^{2} \)
$31$ \( -42592 + T^{2} \)
$37$ \( ( 78 + T )^{2} \)
$41$ \( -155232 + T^{2} \)
$43$ \( ( -436 + T )^{2} \)
$47$ \( -42592 + T^{2} \)
$53$ \( ( 62 + T )^{2} \)
$59$ \( -443608 + T^{2} \)
$61$ \( -74008 + T^{2} \)
$67$ \( ( -580 + T )^{2} \)
$71$ \( ( -544 + T )^{2} \)
$73$ \( -360448 + T^{2} \)
$79$ \( ( 680 + T )^{2} \)
$83$ \( -38808 + T^{2} \)
$89$ \( -2252800 + T^{2} \)
$97$ \( -431200 + T^{2} \)
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