Properties

Label 882.4.a.u
Level $882$
Weight $4$
Character orbit 882.a
Self dual yes
Analytic conductor $52.040$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
Defining polynomial: \(x^{2} - x - 48\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{193})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{2} + 4 q^{4} + ( -3 - \beta ) q^{5} -8 q^{8} +O(q^{10})\) \( q -2 q^{2} + 4 q^{4} + ( -3 - \beta ) q^{5} -8 q^{8} + ( 6 + 2 \beta ) q^{10} + ( -9 - 7 \beta ) q^{11} + ( 22 + 5 \beta ) q^{13} + 16 q^{16} + ( -54 + 10 \beta ) q^{17} + ( 58 + 3 \beta ) q^{19} + ( -12 - 4 \beta ) q^{20} + ( 18 + 14 \beta ) q^{22} + ( 54 + 14 \beta ) q^{23} + ( -68 + 7 \beta ) q^{25} + ( -44 - 10 \beta ) q^{26} + ( 33 + 7 \beta ) q^{29} + ( -35 - 28 \beta ) q^{31} -32 q^{32} + ( 108 - 20 \beta ) q^{34} + ( 134 + 21 \beta ) q^{37} + ( -116 - 6 \beta ) q^{38} + ( 24 + 8 \beta ) q^{40} -168 q^{41} + ( 164 - 21 \beta ) q^{43} + ( -36 - 28 \beta ) q^{44} + ( -108 - 28 \beta ) q^{46} + ( -348 + 24 \beta ) q^{47} + ( 136 - 14 \beta ) q^{50} + ( 88 + 20 \beta ) q^{52} + ( -219 + 63 \beta ) q^{53} + ( 363 + 37 \beta ) q^{55} + ( -66 - 14 \beta ) q^{58} + ( -351 - 61 \beta ) q^{59} + ( 220 - 34 \beta ) q^{61} + ( 70 + 56 \beta ) q^{62} + 64 q^{64} + ( -306 - 42 \beta ) q^{65} + ( -538 + 35 \beta ) q^{67} + ( -216 + 40 \beta ) q^{68} + ( 840 - 28 \beta ) q^{71} + ( 46 + 97 \beta ) q^{73} + ( -268 - 42 \beta ) q^{74} + ( 232 + 12 \beta ) q^{76} + ( 311 - 98 \beta ) q^{79} + ( -48 - 16 \beta ) q^{80} + 336 q^{82} + ( -99 - 89 \beta ) q^{83} + ( -318 + 14 \beta ) q^{85} + ( -328 + 42 \beta ) q^{86} + ( 72 + 56 \beta ) q^{88} + ( -1170 - 54 \beta ) q^{89} + ( 216 + 56 \beta ) q^{92} + ( 696 - 48 \beta ) q^{94} + ( -318 - 70 \beta ) q^{95} + ( 109 - 155 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 8q^{4} - 7q^{5} - 16q^{8} + O(q^{10}) \) \( 2q - 4q^{2} + 8q^{4} - 7q^{5} - 16q^{8} + 14q^{10} - 25q^{11} + 49q^{13} + 32q^{16} - 98q^{17} + 119q^{19} - 28q^{20} + 50q^{22} + 122q^{23} - 129q^{25} - 98q^{26} + 73q^{29} - 98q^{31} - 64q^{32} + 196q^{34} + 289q^{37} - 238q^{38} + 56q^{40} - 336q^{41} + 307q^{43} - 100q^{44} - 244q^{46} - 672q^{47} + 258q^{50} + 196q^{52} - 375q^{53} + 763q^{55} - 146q^{58} - 763q^{59} + 406q^{61} + 196q^{62} + 128q^{64} - 654q^{65} - 1041q^{67} - 392q^{68} + 1652q^{71} + 189q^{73} - 578q^{74} + 476q^{76} + 524q^{79} - 112q^{80} + 672q^{82} - 287q^{83} - 622q^{85} - 614q^{86} + 200q^{88} - 2394q^{89} + 488q^{92} + 1344q^{94} - 706q^{95} + 63q^{97} + O(q^{100}) \)

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
7.44622
−6.44622
−2.00000 0 4.00000 −10.4462 0 0 −8.00000 0 20.8924
1.2 −2.00000 0 4.00000 3.44622 0 0 −8.00000 0 −6.89244
\(n\): 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.u 2
3.b odd 2 1 882.4.a.bh 2
7.b odd 2 1 882.4.a.ba 2
7.c even 3 2 882.4.g.bj 4
7.d odd 6 2 126.4.g.f yes 4
21.c even 2 1 882.4.a.bd 2
21.g even 6 2 126.4.g.e 4
21.h odd 6 2 882.4.g.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.g.e 4 21.g even 6 2
126.4.g.f yes 4 7.d odd 6 2
882.4.a.u 2 1.a even 1 1 trivial
882.4.a.ba 2 7.b odd 2 1
882.4.a.bd 2 21.c even 2 1
882.4.a.bh 2 3.b odd 2 1
882.4.g.z 4 21.h odd 6 2
882.4.g.bj 4 7.c even 3 2

Hecke kernels

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

\( T_{5}^{2} + 7 T_{5} - 36 \)
\( T_{11}^{2} + 25 T_{11} - 2208 \)
\( T_{13}^{2} - 49 T_{13} - 606 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -36 + 7 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -2208 + 25 T + T^{2} \)
$13$ \( -606 - 49 T + T^{2} \)
$17$ \( -2424 + 98 T + T^{2} \)
$19$ \( 3106 - 119 T + T^{2} \)
$23$ \( -5736 - 122 T + T^{2} \)
$29$ \( -1032 - 73 T + T^{2} \)
$31$ \( -35427 + 98 T + T^{2} \)
$37$ \( -398 - 289 T + T^{2} \)
$41$ \( ( 168 + T )^{2} \)
$43$ \( 2284 - 307 T + T^{2} \)
$47$ \( 85104 + 672 T + T^{2} \)
$53$ \( -156348 + 375 T + T^{2} \)
$59$ \( -33996 + 763 T + T^{2} \)
$61$ \( -14568 - 406 T + T^{2} \)
$67$ \( 211814 + 1041 T + T^{2} \)
$71$ \( 644448 - 1652 T + T^{2} \)
$73$ \( -445054 - 189 T + T^{2} \)
$79$ \( -394749 - 524 T + T^{2} \)
$83$ \( -361596 + 287 T + T^{2} \)
$89$ \( 1292112 + 2394 T + T^{2} \)
$97$ \( -1158214 - 63 T + T^{2} \)
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