Properties

Label 882.5.c.c
Level $882$
Weight $5$
Character orbit 882.c
Analytic conductor $91.172$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 882.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(91.1723074400\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
Defining polynomial: \(x^{4} + 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} + 8 q^{4} + ( -5 \beta_{1} - 13 \beta_{3} ) q^{5} + 16 \beta_{2} q^{8} +O(q^{10})\) \( q + 2 \beta_{2} q^{2} + 8 q^{4} + ( -5 \beta_{1} - 13 \beta_{3} ) q^{5} + 16 \beta_{2} q^{8} + ( -16 \beta_{1} - 36 \beta_{3} ) q^{10} + ( -6 + 103 \beta_{2} ) q^{11} + ( -101 \beta_{1} + 97 \beta_{3} ) q^{13} + 64 q^{16} + ( -282 \beta_{1} - 177 \beta_{3} ) q^{17} + ( -207 \beta_{1} + 10 \beta_{3} ) q^{19} + ( -40 \beta_{1} - 104 \beta_{3} ) q^{20} + ( 412 - 12 \beta_{2} ) q^{22} + ( -26 - 144 \beta_{2} ) q^{23} + ( 237 - 274 \beta_{2} ) q^{25} + ( 396 \beta_{1} - 8 \beta_{3} ) q^{26} + ( 352 + 22 \beta_{2} ) q^{29} + ( 698 \beta_{1} - 74 \beta_{3} ) q^{31} + 128 \beta_{2} q^{32} + ( 210 \beta_{1} - 918 \beta_{3} ) q^{34} + ( -848 - 36 \beta_{2} ) q^{37} + ( 434 \beta_{1} - 394 \beta_{3} ) q^{38} + ( -128 \beta_{1} - 288 \beta_{3} ) q^{40} + ( 251 \beta_{1} - 460 \beta_{3} ) q^{41} + ( -506 + 1175 \beta_{2} ) q^{43} + ( -48 + 824 \beta_{2} ) q^{44} + ( -576 - 52 \beta_{2} ) q^{46} + ( -1732 \beta_{1} + 1418 \beta_{3} ) q^{47} + ( -1096 + 474 \beta_{2} ) q^{50} + ( -808 \beta_{1} + 776 \beta_{3} ) q^{52} + ( 4170 + 706 \beta_{2} ) q^{53} + ( -794 \beta_{1} - 1776 \beta_{3} ) q^{55} + ( 88 + 704 \beta_{2} ) q^{58} + ( 1379 \beta_{1} - 2820 \beta_{3} ) q^{59} + ( 1523 \beta_{1} + 365 \beta_{3} ) q^{61} + ( -1544 \beta_{1} + 1248 \beta_{3} ) q^{62} + 512 q^{64} + ( 1512 + 938 \beta_{2} ) q^{65} + ( -5204 - 786 \beta_{2} ) q^{67} + ( -2256 \beta_{1} - 1416 \beta_{3} ) q^{68} + ( 496 - 1244 \beta_{2} ) q^{71} + ( 1385 \beta_{1} - 2736 \beta_{3} ) q^{73} + ( -144 - 1696 \beta_{2} ) q^{74} + ( -1656 \beta_{1} + 80 \beta_{3} ) q^{76} + ( -7404 - 2270 \beta_{2} ) q^{79} + ( -320 \beta_{1} - 832 \beta_{3} ) q^{80} + ( -1422 \beta_{1} - 418 \beta_{3} ) q^{82} + ( -4703 \beta_{1} - 1164 \beta_{3} ) q^{83} + ( -7422 - 5442 \beta_{2} ) q^{85} + ( 4700 - 1012 \beta_{2} ) q^{86} + ( 3296 - 96 \beta_{2} ) q^{88} + ( 3573 \beta_{1} - 3746 \beta_{3} ) q^{89} + ( -208 - 1152 \beta_{2} ) q^{92} + ( 6300 \beta_{1} - 628 \beta_{3} ) q^{94} + ( -1810 - 1476 \beta_{2} ) q^{95} + ( -5324 \beta_{1} + 141 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{4} + O(q^{10}) \) \( 4q + 32q^{4} - 24q^{11} + 256q^{16} + 1648q^{22} - 104q^{23} + 948q^{25} + 1408q^{29} - 3392q^{37} - 2024q^{43} - 192q^{44} - 2304q^{46} - 4384q^{50} + 16680q^{53} + 352q^{58} + 2048q^{64} + 6048q^{65} - 20816q^{67} + 1984q^{71} - 576q^{74} - 29616q^{79} - 29688q^{85} + 18800q^{86} + 13184q^{88} - 832q^{92} - 7240q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 3 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 3 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
685.1
1.84776i
1.84776i
0.765367i
0.765367i
−2.82843 0 8.00000 0.710974i 0 0 −22.6274 0 2.01094i
685.2 −2.82843 0 8.00000 0.710974i 0 0 −22.6274 0 2.01094i
685.3 2.82843 0 8.00000 27.8477i 0 0 22.6274 0 78.7652i
685.4 2.82843 0 8.00000 27.8477i 0 0 22.6274 0 78.7652i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 882.5.c.c 4
3.b odd 2 1 98.5.b.a 4
7.b odd 2 1 inner 882.5.c.c 4
12.b even 2 1 784.5.c.a 4
21.c even 2 1 98.5.b.a 4
21.g even 6 2 98.5.d.c 8
21.h odd 6 2 98.5.d.c 8
84.h odd 2 1 784.5.c.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.5.b.a 4 3.b odd 2 1
98.5.b.a 4 21.c even 2 1
98.5.d.c 8 21.g even 6 2
98.5.d.c 8 21.h odd 6 2
784.5.c.a 4 12.b even 2 1
784.5.c.a 4 84.h odd 2 1
882.5.c.c 4 1.a even 1 1 trivial
882.5.c.c 4 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} + 776 T_{5}^{2} + 392 \)
\( T_{11}^{2} + 12 T_{11} - 21182 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -8 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 392 + 776 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -21182 + 12 T + T^{2} )^{2} \)
$13$ \( 707030408 + 78440 T^{2} + T^{4} \)
$17$ \( 43821617058 + 443412 T^{2} + T^{4} \)
$19$ \( 2981309762 + 171796 T^{2} + T^{4} \)
$23$ \( ( -40796 + 52 T + T^{2} )^{2} \)
$29$ \( ( 122936 - 704 T + T^{2} )^{2} \)
$31$ \( 286409447552 + 1970720 T^{2} + T^{4} \)
$37$ \( ( 716512 + 1696 T + T^{2} )^{2} \)
$41$ \( 288069342722 + 1098404 T^{2} + T^{4} \)
$43$ \( ( -2505214 + 1012 T + T^{2} )^{2} \)
$47$ \( 30777535627808 + 20042192 T^{2} + T^{4} \)
$53$ \( ( 16392028 - 8340 T + T^{2} )^{2} \)
$59$ \( 382444812731522 + 39416164 T^{2} + T^{4} \)
$61$ \( 21754848065672 + 9811016 T^{2} + T^{4} \)
$67$ \( ( 25846024 + 10408 T + T^{2} )^{2} \)
$71$ \( ( -2849056 - 992 T + T^{2} )^{2} \)
$73$ \( 345644675616962 + 37615684 T^{2} + T^{4} \)
$79$ \( ( 44513416 + 14808 T + T^{2} )^{2} \)
$83$ \( 2011288822677218 + 93892420 T^{2} + T^{4} \)
$89$ \( 1571934000441218 + 107195380 T^{2} + T^{4} \)
$97$ \( 1439024660341058 + 113459428 T^{2} + T^{4} \)
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