Properties

Label 882.4.a.bc
Level $882$
Weight $4$
Character orbit 882.a
Self dual yes
Analytic conductor $52.040$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 294)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 7\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + ( -6 + \beta ) q^{5} + 8 q^{8} +O(q^{10})\) \( q + 2 q^{2} + 4 q^{4} + ( -6 + \beta ) q^{5} + 8 q^{8} + ( -12 + 2 \beta ) q^{10} + ( 2 + 6 \beta ) q^{11} + ( 24 + 3 \beta ) q^{13} + 16 q^{16} + ( -42 + \beta ) q^{17} + ( -36 - 2 \beta ) q^{19} + ( -24 + 4 \beta ) q^{20} + ( 4 + 12 \beta ) q^{22} + ( 154 - 6 \beta ) q^{23} + ( 9 - 12 \beta ) q^{25} + ( 48 + 6 \beta ) q^{26} + ( 40 - 18 \beta ) q^{29} + ( 192 - 6 \beta ) q^{31} + 32 q^{32} + ( -84 + 2 \beta ) q^{34} + ( 268 - 12 \beta ) q^{37} + ( -72 - 4 \beta ) q^{38} + ( -48 + 8 \beta ) q^{40} + ( -378 - 5 \beta ) q^{41} + ( 200 + 24 \beta ) q^{43} + ( 8 + 24 \beta ) q^{44} + ( 308 - 12 \beta ) q^{46} + ( -156 + 10 \beta ) q^{47} + ( 18 - 24 \beta ) q^{50} + ( 96 + 12 \beta ) q^{52} + ( 26 + 24 \beta ) q^{53} + ( 576 - 34 \beta ) q^{55} + ( 80 - 36 \beta ) q^{58} + ( -432 - 2 \beta ) q^{59} + ( 708 - 13 \beta ) q^{61} + ( 384 - 12 \beta ) q^{62} + 64 q^{64} + ( 150 + 6 \beta ) q^{65} + ( 72 + 24 \beta ) q^{67} + ( -168 + 4 \beta ) q^{68} + ( 762 + 30 \beta ) q^{71} + ( 372 + 83 \beta ) q^{73} + ( 536 - 24 \beta ) q^{74} + ( -144 - 8 \beta ) q^{76} + ( 488 + 84 \beta ) q^{79} + ( -96 + 16 \beta ) q^{80} + ( -756 - 10 \beta ) q^{82} + ( 156 - 136 \beta ) q^{83} + ( 350 - 48 \beta ) q^{85} + ( 400 + 48 \beta ) q^{86} + ( 16 + 48 \beta ) q^{88} + ( -54 + 29 \beta ) q^{89} + ( 616 - 24 \beta ) q^{92} + ( -312 + 20 \beta ) q^{94} + ( 20 - 24 \beta ) q^{95} + ( -372 - 125 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 8q^{4} - 12q^{5} + 16q^{8} + O(q^{10}) \) \( 2q + 4q^{2} + 8q^{4} - 12q^{5} + 16q^{8} - 24q^{10} + 4q^{11} + 48q^{13} + 32q^{16} - 84q^{17} - 72q^{19} - 48q^{20} + 8q^{22} + 308q^{23} + 18q^{25} + 96q^{26} + 80q^{29} + 384q^{31} + 64q^{32} - 168q^{34} + 536q^{37} - 144q^{38} - 96q^{40} - 756q^{41} + 400q^{43} + 16q^{44} + 616q^{46} - 312q^{47} + 36q^{50} + 192q^{52} + 52q^{53} + 1152q^{55} + 160q^{58} - 864q^{59} + 1416q^{61} + 768q^{62} + 128q^{64} + 300q^{65} + 144q^{67} - 336q^{68} + 1524q^{71} + 744q^{73} + 1072q^{74} - 288q^{76} + 976q^{79} - 192q^{80} - 1512q^{82} + 312q^{83} + 700q^{85} + 800q^{86} + 32q^{88} - 108q^{89} + 1232q^{92} - 624q^{94} + 40q^{95} - 744q^{97} + O(q^{100}) \)

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 −15.8995 0 0 8.00000 0 −31.7990
1.2 2.00000 0 4.00000 3.89949 0 0 8.00000 0 7.79899
\(n\): 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.bc 2
3.b odd 2 1 294.4.a.j 2
7.b odd 2 1 882.4.a.bi 2
7.c even 3 2 882.4.g.bd 4
7.d odd 6 2 882.4.g.y 4
12.b even 2 1 2352.4.a.cd 2
21.c even 2 1 294.4.a.k yes 2
21.g even 6 2 294.4.e.n 4
21.h odd 6 2 294.4.e.o 4
84.h odd 2 1 2352.4.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.j 2 3.b odd 2 1
294.4.a.k yes 2 21.c even 2 1
294.4.e.n 4 21.g even 6 2
294.4.e.o 4 21.h odd 6 2
882.4.a.bc 2 1.a even 1 1 trivial
882.4.a.bi 2 7.b odd 2 1
882.4.g.y 4 7.d odd 6 2
882.4.g.bd 4 7.c even 3 2
2352.4.a.bn 2 84.h odd 2 1
2352.4.a.cd 2 12.b even 2 1

Hecke kernels

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

\( T_{5}^{2} + 12 T_{5} - 62 \)
\( T_{11}^{2} - 4 T_{11} - 3524 \)
\( T_{13}^{2} - 48 T_{13} - 306 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -62 + 12 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -3524 - 4 T + T^{2} \)
$13$ \( -306 - 48 T + T^{2} \)
$17$ \( 1666 + 84 T + T^{2} \)
$19$ \( 904 + 72 T + T^{2} \)
$23$ \( 20188 - 308 T + T^{2} \)
$29$ \( -30152 - 80 T + T^{2} \)
$31$ \( 33336 - 384 T + T^{2} \)
$37$ \( 57712 - 536 T + T^{2} \)
$41$ \( 140434 + 756 T + T^{2} \)
$43$ \( -16448 - 400 T + T^{2} \)
$47$ \( 14536 + 312 T + T^{2} \)
$53$ \( -55772 - 52 T + T^{2} \)
$59$ \( 186232 + 864 T + T^{2} \)
$61$ \( 484702 - 1416 T + T^{2} \)
$67$ \( -51264 - 144 T + T^{2} \)
$71$ \( 492444 - 1524 T + T^{2} \)
$73$ \( -536738 - 744 T + T^{2} \)
$79$ \( -453344 - 976 T + T^{2} \)
$83$ \( -1788272 - 312 T + T^{2} \)
$89$ \( -79502 + 108 T + T^{2} \)
$97$ \( -1392866 + 744 T + T^{2} \)
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