Properties

Label 882.4.a.z
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: \(\Q(\sqrt{1345}) \)
Defining polynomial: \(x^{2} - x - 336\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{1345})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{2} + 4 q^{4} + ( 3 - \beta ) q^{5} -8 q^{8} +O(q^{10})\) \( q -2 q^{2} + 4 q^{4} + ( 3 - \beta ) q^{5} -8 q^{8} + ( -6 + 2 \beta ) q^{10} + ( -33 - \beta ) q^{11} + ( -20 - \beta ) q^{13} + 16 q^{16} + ( -48 + 4 \beta ) q^{17} + ( -20 - 3 \beta ) q^{19} + ( 12 - 4 \beta ) q^{20} + ( 66 + 2 \beta ) q^{22} + ( -72 - 4 \beta ) q^{23} + ( 220 - 5 \beta ) q^{25} + ( 40 + 2 \beta ) q^{26} + ( -33 - 11 \beta ) q^{29} + ( 259 + 2 \beta ) q^{31} -32 q^{32} + ( 96 - 8 \beta ) q^{34} + ( 8 - 9 \beta ) q^{37} + ( 40 + 6 \beta ) q^{38} + ( -24 + 8 \beta ) q^{40} + ( -216 + 6 \beta ) q^{41} + ( -46 - 15 \beta ) q^{43} + ( -132 - 4 \beta ) q^{44} + ( 144 + 8 \beta ) q^{46} + ( 294 - 12 \beta ) q^{47} + ( -440 + 10 \beta ) q^{50} + ( -80 - 4 \beta ) q^{52} + ( 123 - 3 \beta ) q^{53} + ( 237 + 31 \beta ) q^{55} + ( 66 + 22 \beta ) q^{58} + ( 9 - 25 \beta ) q^{59} + ( -110 - 4 \beta ) q^{61} + ( -518 - 4 \beta ) q^{62} + 64 q^{64} + ( 276 + 18 \beta ) q^{65} + ( 338 + 11 \beta ) q^{67} + ( -192 + 16 \beta ) q^{68} + ( -246 + 20 \beta ) q^{71} + ( 478 - 35 \beta ) q^{73} + ( -16 + 18 \beta ) q^{74} + ( -80 - 12 \beta ) q^{76} + ( -259 - 8 \beta ) q^{79} + ( 48 - 16 \beta ) q^{80} + ( 432 - 12 \beta ) q^{82} + ( -123 + 25 \beta ) q^{83} + ( -1488 + 56 \beta ) q^{85} + ( 92 + 30 \beta ) q^{86} + ( 264 + 8 \beta ) q^{88} + ( 366 + 42 \beta ) q^{89} + ( -288 - 16 \beta ) q^{92} + ( -588 + 24 \beta ) q^{94} + ( 948 + 14 \beta ) q^{95} + ( -959 - 35 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 8q^{4} + 5q^{5} - 16q^{8} + O(q^{10}) \) \( 2q - 4q^{2} + 8q^{4} + 5q^{5} - 16q^{8} - 10q^{10} - 67q^{11} - 41q^{13} + 32q^{16} - 92q^{17} - 43q^{19} + 20q^{20} + 134q^{22} - 148q^{23} + 435q^{25} + 82q^{26} - 77q^{29} + 520q^{31} - 64q^{32} + 184q^{34} + 7q^{37} + 86q^{38} - 40q^{40} - 426q^{41} - 107q^{43} - 268q^{44} + 296q^{46} + 576q^{47} - 870q^{50} - 164q^{52} + 243q^{53} + 505q^{55} + 154q^{58} - 7q^{59} - 224q^{61} - 1040q^{62} + 128q^{64} + 570q^{65} + 687q^{67} - 368q^{68} - 472q^{71} + 921q^{73} - 14q^{74} - 172q^{76} - 526q^{79} + 80q^{80} + 852q^{82} - 221q^{83} - 2920q^{85} + 214q^{86} + 536q^{88} + 774q^{89} - 592q^{92} - 1152q^{94} + 1910q^{95} - 1953q^{97} + O(q^{100}) \)

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
18.8371
−17.8371
−2.00000 0 4.00000 −15.8371 0 0 −8.00000 0 31.6742
1.2 −2.00000 0 4.00000 20.8371 0 0 −8.00000 0 −41.6742
\(n\): 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.z 2
3.b odd 2 1 294.4.a.m 2
7.b odd 2 1 882.4.a.v 2
7.c even 3 2 882.4.g.bf 4
7.d odd 6 2 126.4.g.g 4
12.b even 2 1 2352.4.a.ca 2
21.c even 2 1 294.4.a.n 2
21.g even 6 2 42.4.e.c 4
21.h odd 6 2 294.4.e.l 4
84.h odd 2 1 2352.4.a.bq 2
84.j odd 6 2 336.4.q.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.c 4 21.g even 6 2
126.4.g.g 4 7.d odd 6 2
294.4.a.m 2 3.b odd 2 1
294.4.a.n 2 21.c even 2 1
294.4.e.l 4 21.h odd 6 2
336.4.q.j 4 84.j odd 6 2
882.4.a.v 2 7.b odd 2 1
882.4.a.z 2 1.a even 1 1 trivial
882.4.g.bf 4 7.c even 3 2
2352.4.a.bq 2 84.h odd 2 1
2352.4.a.ca 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(882))\):

\( T_{5}^{2} - 5 T_{5} - 330 \)
\( T_{11}^{2} + 67 T_{11} + 786 \)
\( T_{13}^{2} + 41 T_{13} + 84 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -330 - 5 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 786 + 67 T + T^{2} \)
$13$ \( 84 + 41 T + T^{2} \)
$17$ \( -3264 + 92 T + T^{2} \)
$19$ \( -2564 + 43 T + T^{2} \)
$23$ \( 96 + 148 T + T^{2} \)
$29$ \( -39204 + 77 T + T^{2} \)
$31$ \( 66255 - 520 T + T^{2} \)
$37$ \( -27224 - 7 T + T^{2} \)
$41$ \( 33264 + 426 T + T^{2} \)
$43$ \( -72794 + 107 T + T^{2} \)
$47$ \( 34524 - 576 T + T^{2} \)
$53$ \( 11736 - 243 T + T^{2} \)
$59$ \( -210144 + 7 T + T^{2} \)
$61$ \( 7164 + 224 T + T^{2} \)
$67$ \( 77306 - 687 T + T^{2} \)
$71$ \( -78804 + 472 T + T^{2} \)
$73$ \( -199846 - 921 T + T^{2} \)
$79$ \( 47649 + 526 T + T^{2} \)
$83$ \( -197946 + 221 T + T^{2} \)
$89$ \( -443376 - 774 T + T^{2} \)
$97$ \( 541646 + 1953 T + T^{2} \)
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