Properties

Label 882.5.c.b
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: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 14)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 8 q^{4} + ( -9 \beta_{2} - 2 \beta_{3} ) q^{5} + 8 \beta_{1} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 8 q^{4} + ( -9 \beta_{2} - 2 \beta_{3} ) q^{5} + 8 \beta_{1} q^{8} + ( 16 \beta_{2} + 9 \beta_{3} ) q^{10} + ( -27 + 6 \beta_{1} ) q^{11} + ( 8 \beta_{2} - 36 \beta_{3} ) q^{13} + 64 q^{16} + ( 153 \beta_{2} - 32 \beta_{3} ) q^{17} + ( 5 \beta_{2} + 90 \beta_{3} ) q^{19} + ( -72 \beta_{2} - 16 \beta_{3} ) q^{20} + ( 48 - 27 \beta_{1} ) q^{22} + ( -243 - 240 \beta_{1} ) q^{23} + ( 286 + 108 \beta_{1} ) q^{25} + ( 288 \beta_{2} - 8 \beta_{3} ) q^{26} + ( -810 - 24 \beta_{1} ) q^{29} + ( 91 \beta_{2} + 180 \beta_{3} ) q^{31} + 64 \beta_{1} q^{32} + ( 256 \beta_{2} - 153 \beta_{3} ) q^{34} + ( 223 - 270 \beta_{1} ) q^{37} + ( -720 \beta_{2} - 5 \beta_{3} ) q^{38} + ( 128 \beta_{2} + 72 \beta_{3} ) q^{40} + ( -72 \beta_{2} - 208 \beta_{3} ) q^{41} + ( 586 - 648 \beta_{1} ) q^{43} + ( -216 + 48 \beta_{1} ) q^{44} + ( -1920 - 243 \beta_{1} ) q^{46} + ( -117 \beta_{2} + 316 \beta_{3} ) q^{47} + ( 864 + 286 \beta_{1} ) q^{50} + ( 64 \beta_{2} - 288 \beta_{3} ) q^{52} + ( 1377 - 774 \beta_{1} ) q^{53} + ( 339 \beta_{2} + 108 \beta_{3} ) q^{55} + ( -192 - 810 \beta_{1} ) q^{58} + ( 2061 \beta_{2} + 146 \beta_{3} ) q^{59} + ( 1281 \beta_{2} - 738 \beta_{3} ) q^{61} + ( -1440 \beta_{2} - 91 \beta_{3} ) q^{62} + 512 q^{64} + ( -1512 + 924 \beta_{1} ) q^{65} + ( 2531 - 1674 \beta_{1} ) q^{67} + ( 1224 \beta_{2} - 256 \beta_{3} ) q^{68} + ( -4698 + 456 \beta_{1} ) q^{71} + ( 2879 \beta_{2} - 900 \beta_{3} ) q^{73} + ( -2160 + 223 \beta_{1} ) q^{74} + ( 40 \beta_{2} + 720 \beta_{3} ) q^{76} + ( -397 + 972 \beta_{1} ) q^{79} + ( -576 \beta_{2} - 128 \beta_{3} ) q^{80} + ( 1664 \beta_{2} + 72 \beta_{3} ) q^{82} + ( -2448 \beta_{2} - 100 \beta_{3} ) q^{83} + ( 2595 - 54 \beta_{1} ) q^{85} + ( -5184 + 586 \beta_{1} ) q^{86} + ( 384 - 216 \beta_{1} ) q^{88} + ( 2079 \beta_{2} - 564 \beta_{3} ) q^{89} + ( -1944 - 1920 \beta_{1} ) q^{92} + ( -2528 \beta_{2} + 117 \beta_{3} ) q^{94} + ( 4455 - 2460 \beta_{1} ) q^{95} + ( 216 \beta_{2} - 1224 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{4} + O(q^{10}) \) \( 4q + 32q^{4} - 108q^{11} + 256q^{16} + 192q^{22} - 972q^{23} + 1144q^{25} - 3240q^{29} + 892q^{37} + 2344q^{43} - 864q^{44} - 7680q^{46} + 3456q^{50} + 5508q^{53} - 768q^{58} + 2048q^{64} - 6048q^{65} + 10124q^{67} - 18792q^{71} - 8640q^{74} - 1588q^{79} + 10380q^{85} - 20736q^{86} + 1536q^{88} - 7776q^{92} + 17820q^{95} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{1}\)

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
0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
−0.707107 + 1.22474i
−2.82843 0 8.00000 25.3864i 0 0 −22.6274 0 71.8036i
685.2 −2.82843 0 8.00000 25.3864i 0 0 −22.6274 0 71.8036i
685.3 2.82843 0 8.00000 5.79050i 0 0 22.6274 0 16.3780i
685.4 2.82843 0 8.00000 5.79050i 0 0 22.6274 0 16.3780i
\(n\): 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.b 4
3.b odd 2 1 98.5.b.b 4
7.b odd 2 1 inner 882.5.c.b 4
7.c even 3 1 126.5.n.a 4
7.d odd 6 1 126.5.n.a 4
12.b even 2 1 784.5.c.b 4
21.c even 2 1 98.5.b.b 4
21.g even 6 1 14.5.d.a 4
21.g even 6 1 98.5.d.a 4
21.h odd 6 1 14.5.d.a 4
21.h odd 6 1 98.5.d.a 4
84.h odd 2 1 784.5.c.b 4
84.j odd 6 1 112.5.s.b 4
84.n even 6 1 112.5.s.b 4
105.o odd 6 1 350.5.k.a 4
105.p even 6 1 350.5.k.a 4
105.w odd 12 2 350.5.i.a 8
105.x even 12 2 350.5.i.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.5.d.a 4 21.g even 6 1
14.5.d.a 4 21.h odd 6 1
98.5.b.b 4 3.b odd 2 1
98.5.b.b 4 21.c even 2 1
98.5.d.a 4 21.g even 6 1
98.5.d.a 4 21.h odd 6 1
112.5.s.b 4 84.j odd 6 1
112.5.s.b 4 84.n even 6 1
126.5.n.a 4 7.c even 3 1
126.5.n.a 4 7.d odd 6 1
350.5.i.a 8 105.w odd 12 2
350.5.i.a 8 105.x even 12 2
350.5.k.a 4 105.o odd 6 1
350.5.k.a 4 105.p even 6 1
784.5.c.b 4 12.b even 2 1
784.5.c.b 4 84.h odd 2 1
882.5.c.b 4 1.a even 1 1 trivial
882.5.c.b 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 \(S_{5}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} + 678 T_{5}^{2} + 21609 \)
\( T_{11}^{2} + 54 T_{11} + 441 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -8 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 21609 + 678 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 441 + 54 T + T^{2} )^{2} \)
$13$ \( 955551744 + 62592 T^{2} + T^{4} \)
$17$ \( 2084013801 + 189606 T^{2} + T^{4} \)
$19$ \( 37762205625 + 388950 T^{2} + T^{4} \)
$23$ \( ( -401751 + 486 T + T^{2} )^{2} \)
$29$ \( ( 651492 + 1620 T + T^{2} )^{2} \)
$31$ \( 566643101049 + 1604886 T^{2} + T^{4} \)
$37$ \( ( -533471 - 446 T + T^{2} )^{2} \)
$41$ \( 1046087110656 + 2107776 T^{2} + T^{4} \)
$43$ \( ( -3015836 - 1172 T + T^{2} )^{2} \)
$47$ \( 5548271897529 + 4875222 T^{2} + T^{4} \)
$53$ \( ( -2896479 - 2754 T + T^{2} )^{2} \)
$59$ \( 149611524833241 + 26509494 T^{2} + T^{4} \)
$61$ \( 66399241936329 + 35988678 T^{2} + T^{4} \)
$67$ \( ( -16012247 - 5062 T + T^{2} )^{2} \)
$71$ \( ( 20407716 + 9396 T + T^{2} )^{2} \)
$73$ \( 29440640401929 + 88611846 T^{2} + T^{4} \)
$79$ \( ( -7400663 + 794 T + T^{2} )^{2} \)
$83$ \( 314640617324544 + 36436224 T^{2} + T^{4} \)
$89$ \( 28434692391561 + 41202054 T^{2} + T^{4} \)
$97$ \( 1282804193857536 + 72192384 T^{2} + T^{4} \)
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