Properties

Label 882.4.a.bg
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 14 \beta q^{5} + 8 q^{8} +O(q^{10})\) \( q + 2 q^{2} + 4 q^{4} + 14 \beta q^{5} + 8 q^{8} + 28 \beta q^{10} + 14 q^{11} + 36 \beta q^{13} + 16 q^{16} -\beta q^{17} -\beta q^{19} + 56 \beta q^{20} + 28 q^{22} -140 q^{23} + 267 q^{25} + 72 \beta q^{26} + 286 q^{29} -66 \beta q^{31} + 32 q^{32} -2 \beta q^{34} -38 q^{37} -2 \beta q^{38} + 112 \beta q^{40} + 89 \beta q^{41} -34 q^{43} + 56 q^{44} -280 q^{46} -370 \beta q^{47} + 534 q^{50} + 144 \beta q^{52} + 74 q^{53} + 196 \beta q^{55} + 572 q^{58} -307 \beta q^{59} + 10 \beta q^{61} -132 \beta q^{62} + 64 q^{64} + 1008 q^{65} + 684 q^{67} -4 \beta q^{68} -588 q^{71} -191 \beta q^{73} -76 q^{74} -4 \beta q^{76} + 1220 q^{79} + 224 \beta q^{80} + 178 \beta q^{82} -299 \beta q^{83} -28 q^{85} -68 q^{86} + 112 q^{88} -437 \beta q^{89} -560 q^{92} -740 \beta q^{94} -28 q^{95} + 1049 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 8q^{4} + 16q^{8} + O(q^{10}) \) \( 2q + 4q^{2} + 8q^{4} + 16q^{8} + 28q^{11} + 32q^{16} + 56q^{22} - 280q^{23} + 534q^{25} + 572q^{29} + 64q^{32} - 76q^{37} - 68q^{43} + 112q^{44} - 560q^{46} + 1068q^{50} + 148q^{53} + 1144q^{58} + 128q^{64} + 2016q^{65} + 1368q^{67} - 1176q^{71} - 152q^{74} + 2440q^{79} - 56q^{85} - 136q^{86} + 224q^{88} - 1120q^{92} - 56q^{95} + O(q^{100}) \)

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 −19.7990 0 0 8.00000 0 −39.5980
1.2 2.00000 0 4.00000 19.7990 0 0 8.00000 0 39.5980
\(n\): 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.bg 2
3.b odd 2 1 98.4.a.g 2
7.b odd 2 1 inner 882.4.a.bg 2
7.c even 3 2 882.4.g.ba 4
7.d odd 6 2 882.4.g.ba 4
12.b even 2 1 784.4.a.y 2
15.d odd 2 1 2450.4.a.bx 2
21.c even 2 1 98.4.a.g 2
21.g even 6 2 98.4.c.h 4
21.h odd 6 2 98.4.c.h 4
84.h odd 2 1 784.4.a.y 2
105.g even 2 1 2450.4.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g 2 3.b odd 2 1
98.4.a.g 2 21.c even 2 1
98.4.c.h 4 21.g even 6 2
98.4.c.h 4 21.h odd 6 2
784.4.a.y 2 12.b even 2 1
784.4.a.y 2 84.h odd 2 1
882.4.a.bg 2 1.a even 1 1 trivial
882.4.a.bg 2 7.b odd 2 1 inner
882.4.g.ba 4 7.c even 3 2
882.4.g.ba 4 7.d odd 6 2
2450.4.a.bx 2 15.d odd 2 1
2450.4.a.bx 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(882))\):

\( T_{5}^{2} - 392 \)
\( T_{11} - 14 \)
\( T_{13}^{2} - 2592 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -392 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -14 + T )^{2} \)
$13$ \( -2592 + T^{2} \)
$17$ \( -2 + T^{2} \)
$19$ \( -2 + T^{2} \)
$23$ \( ( 140 + T )^{2} \)
$29$ \( ( -286 + T )^{2} \)
$31$ \( -8712 + T^{2} \)
$37$ \( ( 38 + T )^{2} \)
$41$ \( -15842 + T^{2} \)
$43$ \( ( 34 + T )^{2} \)
$47$ \( -273800 + T^{2} \)
$53$ \( ( -74 + T )^{2} \)
$59$ \( -188498 + T^{2} \)
$61$ \( -200 + T^{2} \)
$67$ \( ( -684 + T )^{2} \)
$71$ \( ( 588 + T )^{2} \)
$73$ \( -72962 + T^{2} \)
$79$ \( ( -1220 + T )^{2} \)
$83$ \( -178802 + T^{2} \)
$89$ \( -381938 + T^{2} \)
$97$ \( -2200802 + T^{2} \)
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