Properties

Label 882.4.a.be
Level $882$
Weight $4$
Character orbit 882.a
Self dual yes
Analytic conductor $52.040$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 5 \beta q^{5} + 8 q^{8} +O(q^{10})\) \( q + 2 q^{2} + 4 q^{4} + 5 \beta q^{5} + 8 q^{8} + 10 \beta q^{10} -40 q^{11} -45 \beta q^{13} + 16 q^{16} -\beta q^{17} + 8 \beta q^{19} + 20 \beta q^{20} -80 q^{22} -68 q^{23} -75 q^{25} -90 \beta q^{26} -110 q^{29} -84 \beta q^{31} + 32 q^{32} -2 \beta q^{34} -20 q^{37} + 16 \beta q^{38} + 40 \beta q^{40} + 35 \beta q^{41} -340 q^{43} -160 q^{44} -136 q^{46} -64 \beta q^{47} -150 q^{50} -180 \beta q^{52} -628 q^{53} -200 \beta q^{55} -220 q^{58} + 620 \beta q^{59} + 649 \beta q^{61} -168 \beta q^{62} + 64 q^{64} -450 q^{65} + 540 q^{67} -4 \beta q^{68} + 420 q^{71} + 205 \beta q^{73} -40 q^{74} + 32 \beta q^{76} -760 q^{79} + 80 \beta q^{80} + 70 \beta q^{82} -668 \beta q^{83} -10 q^{85} -680 q^{86} -320 q^{88} -815 \beta q^{89} -272 q^{92} -128 \beta q^{94} + 80 q^{95} -355 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 8q^{4} + 16q^{8} + O(q^{10}) \) \( 2q + 4q^{2} + 8q^{4} + 16q^{8} - 80q^{11} + 32q^{16} - 160q^{22} - 136q^{23} - 150q^{25} - 220q^{29} + 64q^{32} - 40q^{37} - 680q^{43} - 320q^{44} - 272q^{46} - 300q^{50} - 1256q^{53} - 440q^{58} + 128q^{64} - 900q^{65} + 1080q^{67} + 840q^{71} - 80q^{74} - 1520q^{79} - 20q^{85} - 1360q^{86} - 640q^{88} - 544q^{92} + 160q^{95} + O(q^{100}) \)

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
−1.41421
1.41421
2.00000 0 4.00000 −7.07107 0 0 8.00000 0 −14.1421
1.2 2.00000 0 4.00000 7.07107 0 0 8.00000 0 14.1421
\(n\): 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.be yes 2
3.b odd 2 1 882.4.a.y 2
7.b odd 2 1 inner 882.4.a.be yes 2
7.c even 3 2 882.4.g.bc 4
7.d odd 6 2 882.4.g.bc 4
21.c even 2 1 882.4.a.y 2
21.g even 6 2 882.4.g.bg 4
21.h odd 6 2 882.4.g.bg 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.4.a.y 2 3.b odd 2 1
882.4.a.y 2 21.c even 2 1
882.4.a.be yes 2 1.a even 1 1 trivial
882.4.a.be yes 2 7.b odd 2 1 inner
882.4.g.bc 4 7.c even 3 2
882.4.g.bc 4 7.d odd 6 2
882.4.g.bg 4 21.g even 6 2
882.4.g.bg 4 21.h odd 6 2

Hecke kernels

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

\( T_{5}^{2} - 50 \)
\( T_{11} + 40 \)
\( T_{13}^{2} - 4050 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -50 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 40 + T )^{2} \)
$13$ \( -4050 + T^{2} \)
$17$ \( -2 + T^{2} \)
$19$ \( -128 + T^{2} \)
$23$ \( ( 68 + T )^{2} \)
$29$ \( ( 110 + T )^{2} \)
$31$ \( -14112 + T^{2} \)
$37$ \( ( 20 + T )^{2} \)
$41$ \( -2450 + T^{2} \)
$43$ \( ( 340 + T )^{2} \)
$47$ \( -8192 + T^{2} \)
$53$ \( ( 628 + T )^{2} \)
$59$ \( -768800 + T^{2} \)
$61$ \( -842402 + T^{2} \)
$67$ \( ( -540 + T )^{2} \)
$71$ \( ( -420 + T )^{2} \)
$73$ \( -84050 + T^{2} \)
$79$ \( ( 760 + T )^{2} \)
$83$ \( -892448 + T^{2} \)
$89$ \( -1328450 + T^{2} \)
$97$ \( -252050 + T^{2} \)
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