Properties

Label 882.4.a.bf
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: \(\Q(\sqrt{58}) \)
Defining polynomial: \(x^{2} - 58\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{58}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + \beta q^{5} + 8 q^{8} +O(q^{10})\) \( q + 2 q^{2} + 4 q^{4} + \beta q^{5} + 8 q^{8} + 2 \beta q^{10} + 2 q^{11} -2 \beta q^{13} + 16 q^{16} -3 \beta q^{17} + 10 \beta q^{19} + 4 \beta q^{20} + 4 q^{22} + 30 q^{23} + 107 q^{25} -4 \beta q^{26} + 212 q^{29} + 14 \beta q^{31} + 32 q^{32} -6 \beta q^{34} + 246 q^{37} + 20 \beta q^{38} + 8 \beta q^{40} -21 \beta q^{41} -284 q^{43} + 8 q^{44} + 60 q^{46} + 4 \beta q^{47} + 214 q^{50} -8 \beta q^{52} + 548 q^{53} + 2 \beta q^{55} + 424 q^{58} -44 \beta q^{59} -34 \beta q^{61} + 28 \beta q^{62} + 64 q^{64} -464 q^{65} + 652 q^{67} -12 \beta q^{68} + 770 q^{71} -64 \beta q^{73} + 492 q^{74} + 40 \beta q^{76} + 472 q^{79} + 16 \beta q^{80} -42 \beta q^{82} + 12 \beta q^{83} -696 q^{85} -568 q^{86} + 16 q^{88} + 47 \beta q^{89} + 120 q^{92} + 8 \beta q^{94} + 2320 q^{95} + 20 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 8q^{4} + 16q^{8} + O(q^{10}) \) \( 2q + 4q^{2} + 8q^{4} + 16q^{8} + 4q^{11} + 32q^{16} + 8q^{22} + 60q^{23} + 214q^{25} + 424q^{29} + 64q^{32} + 492q^{37} - 568q^{43} + 16q^{44} + 120q^{46} + 428q^{50} + 1096q^{53} + 848q^{58} + 128q^{64} - 928q^{65} + 1304q^{67} + 1540q^{71} + 984q^{74} + 944q^{79} - 1392q^{85} - 1136q^{86} + 32q^{88} + 240q^{92} + 4640q^{95} + O(q^{100}) \)

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
−7.61577
7.61577
2.00000 0 4.00000 −15.2315 0 0 8.00000 0 −30.4631
1.2 2.00000 0 4.00000 15.2315 0 0 8.00000 0 30.4631
\(n\): 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.bf yes 2
3.b odd 2 1 882.4.a.x 2
7.b odd 2 1 inner 882.4.a.bf yes 2
7.c even 3 2 882.4.g.bb 4
7.d odd 6 2 882.4.g.bb 4
21.c even 2 1 882.4.a.x 2
21.g even 6 2 882.4.g.bh 4
21.h odd 6 2 882.4.g.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.4.a.x 2 3.b odd 2 1
882.4.a.x 2 21.c even 2 1
882.4.a.bf yes 2 1.a even 1 1 trivial
882.4.a.bf yes 2 7.b odd 2 1 inner
882.4.g.bb 4 7.c even 3 2
882.4.g.bb 4 7.d odd 6 2
882.4.g.bh 4 21.g even 6 2
882.4.g.bh 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(882))\):

\( T_{5}^{2} - 232 \)
\( T_{11} - 2 \)
\( T_{13}^{2} - 928 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -232 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( -928 + T^{2} \)
$17$ \( -2088 + T^{2} \)
$19$ \( -23200 + T^{2} \)
$23$ \( ( -30 + T )^{2} \)
$29$ \( ( -212 + T )^{2} \)
$31$ \( -45472 + T^{2} \)
$37$ \( ( -246 + T )^{2} \)
$41$ \( -102312 + T^{2} \)
$43$ \( ( 284 + T )^{2} \)
$47$ \( -3712 + T^{2} \)
$53$ \( ( -548 + T )^{2} \)
$59$ \( -449152 + T^{2} \)
$61$ \( -268192 + T^{2} \)
$67$ \( ( -652 + T )^{2} \)
$71$ \( ( -770 + T )^{2} \)
$73$ \( -950272 + T^{2} \)
$79$ \( ( -472 + T )^{2} \)
$83$ \( -33408 + T^{2} \)
$89$ \( -512488 + T^{2} \)
$97$ \( -92800 + T^{2} \)
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