Properties

Label 882.3.s.g
Level $882$
Weight $3$
Character orbit 882.s
Analytic conductor $24.033$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{2} + 2 \zeta_{24}^{4} q^{4} -4 \zeta_{24}^{2} q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{2} + 2 \zeta_{24}^{4} q^{4} -4 \zeta_{24}^{2} q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} + ( -4 \zeta_{24} - 4 \zeta_{24}^{7} ) q^{10} + ( 2 \zeta_{24} - 2 \zeta_{24}^{7} ) q^{11} + ( -9 \zeta_{24} - 9 \zeta_{24}^{3} + 9 \zeta_{24}^{5} ) q^{13} + ( -4 + 4 \zeta_{24}^{4} ) q^{16} + ( 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{17} + ( 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} - 16 \zeta_{24}^{7} ) q^{19} -8 \zeta_{24}^{6} q^{20} + 4 q^{22} + ( 26 \zeta_{24}^{3} + 26 \zeta_{24}^{5} - 26 \zeta_{24}^{7} ) q^{23} -9 \zeta_{24}^{4} q^{25} -18 \zeta_{24}^{2} q^{26} + ( 23 \zeta_{24} - 23 \zeta_{24}^{3} - 23 \zeta_{24}^{5} ) q^{29} + ( -36 \zeta_{24} - 36 \zeta_{24}^{7} ) q^{31} + ( -4 \zeta_{24} + 4 \zeta_{24}^{7} ) q^{32} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{34} + ( 32 - 32 \zeta_{24}^{4} ) q^{37} + ( 32 \zeta_{24}^{2} - 32 \zeta_{24}^{6} ) q^{38} + ( 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{40} -38 \zeta_{24}^{6} q^{41} + 20 q^{43} + ( 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{44} + 52 \zeta_{24}^{4} q^{46} + 20 \zeta_{24}^{2} q^{47} + ( 9 \zeta_{24} - 9 \zeta_{24}^{3} - 9 \zeta_{24}^{5} ) q^{50} + ( -18 \zeta_{24} - 18 \zeta_{24}^{7} ) q^{52} + ( -67 \zeta_{24} + 67 \zeta_{24}^{7} ) q^{53} + ( -8 \zeta_{24} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} ) q^{55} + ( 46 - 46 \zeta_{24}^{4} ) q^{58} + ( 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{59} + ( 59 \zeta_{24}^{3} - 59 \zeta_{24}^{5} - 59 \zeta_{24}^{7} ) q^{61} -72 \zeta_{24}^{6} q^{62} -8 q^{64} + ( 36 \zeta_{24}^{3} + 36 \zeta_{24}^{5} - 36 \zeta_{24}^{7} ) q^{65} + 48 \zeta_{24}^{4} q^{67} + 8 \zeta_{24}^{2} q^{68} + ( 54 \zeta_{24} - 54 \zeta_{24}^{3} - 54 \zeta_{24}^{5} ) q^{71} + ( -85 \zeta_{24} - 85 \zeta_{24}^{7} ) q^{73} + ( 32 \zeta_{24} - 32 \zeta_{24}^{7} ) q^{74} + ( 32 \zeta_{24} + 32 \zeta_{24}^{3} - 32 \zeta_{24}^{5} ) q^{76} + ( 148 - 148 \zeta_{24}^{4} ) q^{79} + ( 16 \zeta_{24}^{2} - 16 \zeta_{24}^{6} ) q^{80} + ( 38 \zeta_{24}^{3} - 38 \zeta_{24}^{5} - 38 \zeta_{24}^{7} ) q^{82} + 80 \zeta_{24}^{6} q^{83} -16 q^{85} + ( 20 \zeta_{24}^{3} + 20 \zeta_{24}^{5} - 20 \zeta_{24}^{7} ) q^{86} + 8 \zeta_{24}^{4} q^{88} + 106 \zeta_{24}^{2} q^{89} + ( -52 \zeta_{24} + 52 \zeta_{24}^{3} + 52 \zeta_{24}^{5} ) q^{92} + ( 20 \zeta_{24} + 20 \zeta_{24}^{7} ) q^{94} + ( -64 \zeta_{24} + 64 \zeta_{24}^{7} ) q^{95} + ( -109 \zeta_{24} - 109 \zeta_{24}^{3} + 109 \zeta_{24}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + O(q^{10}) \) \( 8 q + 8 q^{4} - 16 q^{16} + 32 q^{22} - 36 q^{25} + 128 q^{37} + 160 q^{43} + 208 q^{46} + 184 q^{58} - 64 q^{64} + 192 q^{67} + 592 q^{79} - 128 q^{85} + 32 q^{88} + O(q^{100}) \)

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1 + \zeta_{24}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
−0.965926 + 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 0.965926i
−1.22474 + 0.707107i 0 1.00000 1.73205i −3.46410 + 2.00000i 0 0 2.82843i 0 2.82843 4.89898i
557.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 3.46410 2.00000i 0 0 2.82843i 0 −2.82843 + 4.89898i
557.3 1.22474 0.707107i 0 1.00000 1.73205i −3.46410 + 2.00000i 0 0 2.82843i 0 −2.82843 + 4.89898i
557.4 1.22474 0.707107i 0 1.00000 1.73205i 3.46410 2.00000i 0 0 2.82843i 0 2.82843 4.89898i
863.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −3.46410 2.00000i 0 0 2.82843i 0 2.82843 + 4.89898i
863.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 3.46410 + 2.00000i 0 0 2.82843i 0 −2.82843 4.89898i
863.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −3.46410 2.00000i 0 0 2.82843i 0 −2.82843 4.89898i
863.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 3.46410 + 2.00000i 0 0 2.82843i 0 2.82843 + 4.89898i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 863.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.s.g 8
3.b odd 2 1 inner 882.3.s.g 8
7.b odd 2 1 inner 882.3.s.g 8
7.c even 3 1 882.3.b.h 4
7.c even 3 1 inner 882.3.s.g 8
7.d odd 6 1 882.3.b.h 4
7.d odd 6 1 inner 882.3.s.g 8
21.c even 2 1 inner 882.3.s.g 8
21.g even 6 1 882.3.b.h 4
21.g even 6 1 inner 882.3.s.g 8
21.h odd 6 1 882.3.b.h 4
21.h odd 6 1 inner 882.3.s.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.3.b.h 4 7.c even 3 1
882.3.b.h 4 7.d odd 6 1
882.3.b.h 4 21.g even 6 1
882.3.b.h 4 21.h odd 6 1
882.3.s.g 8 1.a even 1 1 trivial
882.3.s.g 8 3.b odd 2 1 inner
882.3.s.g 8 7.b odd 2 1 inner
882.3.s.g 8 7.c even 3 1 inner
882.3.s.g 8 7.d odd 6 1 inner
882.3.s.g 8 21.c even 2 1 inner
882.3.s.g 8 21.g even 6 1 inner
882.3.s.g 8 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} - 16 T_{5}^{2} + 256 \)
\( T_{11}^{4} - 8 T_{11}^{2} + 64 \)
\( T_{13}^{2} - 162 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( ( 256 - 16 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 64 - 8 T^{2} + T^{4} )^{2} \)
$13$ \( ( -162 + T^{2} )^{4} \)
$17$ \( ( 256 - 16 T^{2} + T^{4} )^{2} \)
$19$ \( ( 262144 + 512 T^{2} + T^{4} )^{2} \)
$23$ \( ( 1827904 - 1352 T^{2} + T^{4} )^{2} \)
$29$ \( ( 1058 + T^{2} )^{4} \)
$31$ \( ( 6718464 + 2592 T^{2} + T^{4} )^{2} \)
$37$ \( ( 1024 - 32 T + T^{2} )^{4} \)
$41$ \( ( 1444 + T^{2} )^{4} \)
$43$ \( ( -20 + T )^{8} \)
$47$ \( ( 160000 - 400 T^{2} + T^{4} )^{2} \)
$53$ \( ( 80604484 - 8978 T^{2} + T^{4} )^{2} \)
$59$ \( ( 256 - 16 T^{2} + T^{4} )^{2} \)
$61$ \( ( 48469444 + 6962 T^{2} + T^{4} )^{2} \)
$67$ \( ( 2304 - 48 T + T^{2} )^{4} \)
$71$ \( ( 5832 + T^{2} )^{4} \)
$73$ \( ( 208802500 + 14450 T^{2} + T^{4} )^{2} \)
$79$ \( ( 21904 - 148 T + T^{2} )^{4} \)
$83$ \( ( 6400 + T^{2} )^{4} \)
$89$ \( ( 126247696 - 11236 T^{2} + T^{4} )^{2} \)
$97$ \( ( -23762 + T^{2} )^{4} \)
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