Properties

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

Related objects

Downloads

Learn more

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.

\(f(q)\) \(=\) \( q + ( \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{2} + 2 \zeta_{24}^{4} q^{4} -2 \zeta_{24}^{2} q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{2} + 2 \zeta_{24}^{4} q^{4} -2 \zeta_{24}^{2} q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} + ( -2 \zeta_{24} - 2 \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} + ( -22 \zeta_{24}^{2} + 22 \zeta_{24}^{6} ) q^{17} + ( -4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{19} -4 \zeta_{24}^{6} q^{20} + 4 q^{22} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{23} -21 \zeta_{24}^{4} q^{25} -18 \zeta_{24}^{2} q^{26} + ( -25 \zeta_{24} + 25 \zeta_{24}^{3} + 25 \zeta_{24}^{5} ) q^{29} + ( -24 \zeta_{24} - 24 \zeta_{24}^{7} ) q^{31} + ( -4 \zeta_{24} + 4 \zeta_{24}^{7} ) q^{32} + ( -22 \zeta_{24} - 22 \zeta_{24}^{3} + 22 \zeta_{24}^{5} ) q^{34} + ( -64 + 64 \zeta_{24}^{4} ) q^{37} + ( -8 \zeta_{24}^{2} + 8 \zeta_{24}^{6} ) q^{38} + ( 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{40} + 20 \zeta_{24}^{6} q^{41} + 44 q^{43} + ( 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{44} + 4 \zeta_{24}^{4} q^{46} -68 \zeta_{24}^{2} q^{47} + ( 21 \zeta_{24} - 21 \zeta_{24}^{3} - 21 \zeta_{24}^{5} ) q^{50} + ( -18 \zeta_{24} - 18 \zeta_{24}^{7} ) q^{52} + ( -13 \zeta_{24} + 13 \zeta_{24}^{7} ) q^{53} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{55} + ( -50 + 50 \zeta_{24}^{4} ) q^{58} + ( -100 \zeta_{24}^{2} + 100 \zeta_{24}^{6} ) q^{59} + ( 37 \zeta_{24}^{3} - 37 \zeta_{24}^{5} - 37 \zeta_{24}^{7} ) q^{61} -48 \zeta_{24}^{6} q^{62} -8 q^{64} + ( 18 \zeta_{24}^{3} + 18 \zeta_{24}^{5} - 18 \zeta_{24}^{7} ) q^{65} -120 \zeta_{24}^{4} q^{67} -44 \zeta_{24}^{2} q^{68} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{71} + ( -53 \zeta_{24} - 53 \zeta_{24}^{7} ) q^{73} + ( -64 \zeta_{24} + 64 \zeta_{24}^{7} ) q^{74} + ( -8 \zeta_{24} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} ) q^{76} + ( -92 + 92 \zeta_{24}^{4} ) q^{79} + ( 8 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{80} + ( -20 \zeta_{24}^{3} + 20 \zeta_{24}^{5} + 20 \zeta_{24}^{7} ) q^{82} + 112 \zeta_{24}^{6} q^{83} + 44 q^{85} + ( 44 \zeta_{24}^{3} + 44 \zeta_{24}^{5} - 44 \zeta_{24}^{7} ) q^{86} + 8 \zeta_{24}^{4} q^{88} + 20 \zeta_{24}^{2} q^{89} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{92} + ( -68 \zeta_{24} - 68 \zeta_{24}^{7} ) q^{94} + ( 8 \zeta_{24} - 8 \zeta_{24}^{7} ) q^{95} + ( 19 \zeta_{24} + 19 \zeta_{24}^{3} - 19 \zeta_{24}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + O(q^{10}) \) \( 8 q + 8 q^{4} - 16 q^{16} + 32 q^{22} - 84 q^{25} - 256 q^{37} + 352 q^{43} + 16 q^{46} - 200 q^{58} - 64 q^{64} - 480 q^{67} - 368 q^{79} + 352 q^{85} + 32 q^{88} + O(q^{100}) \)

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 −1.73205 + 1.00000i 0 0 2.82843i 0 1.41421 2.44949i
557.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 1.73205 1.00000i 0 0 2.82843i 0 −1.41421 + 2.44949i
557.3 1.22474 0.707107i 0 1.00000 1.73205i −1.73205 + 1.00000i 0 0 2.82843i 0 −1.41421 + 2.44949i
557.4 1.22474 0.707107i 0 1.00000 1.73205i 1.73205 1.00000i 0 0 2.82843i 0 1.41421 2.44949i
863.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −1.73205 1.00000i 0 0 2.82843i 0 1.41421 + 2.44949i
863.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 1.73205 + 1.00000i 0 0 2.82843i 0 −1.41421 2.44949i
863.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −1.73205 1.00000i 0 0 2.82843i 0 −1.41421 2.44949i
863.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 1.73205 + 1.00000i 0 0 2.82843i 0 1.41421 + 2.44949i
\(n\): 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.f 8
3.b odd 2 1 inner 882.3.s.f 8
7.b odd 2 1 inner 882.3.s.f 8
7.c even 3 1 882.3.b.i 4
7.c even 3 1 inner 882.3.s.f 8
7.d odd 6 1 882.3.b.i 4
7.d odd 6 1 inner 882.3.s.f 8
21.c even 2 1 inner 882.3.s.f 8
21.g even 6 1 882.3.b.i 4
21.g even 6 1 inner 882.3.s.f 8
21.h odd 6 1 882.3.b.i 4
21.h odd 6 1 inner 882.3.s.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.3.b.i 4 7.c even 3 1
882.3.b.i 4 7.d odd 6 1
882.3.b.i 4 21.g even 6 1
882.3.b.i 4 21.h odd 6 1
882.3.s.f 8 1.a even 1 1 trivial
882.3.s.f 8 3.b odd 2 1 inner
882.3.s.f 8 7.b odd 2 1 inner
882.3.s.f 8 7.c even 3 1 inner
882.3.s.f 8 7.d odd 6 1 inner
882.3.s.f 8 21.c even 2 1 inner
882.3.s.f 8 21.g even 6 1 inner
882.3.s.f 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 \(S_{3}^{\mathrm{new}}(882, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 64 - 8 T^{2} + T^{4} )^{2} \)
$13$ \( ( -162 + T^{2} )^{4} \)
$17$ \( ( 234256 - 484 T^{2} + T^{4} )^{2} \)
$19$ \( ( 1024 + 32 T^{2} + T^{4} )^{2} \)
$23$ \( ( 64 - 8 T^{2} + T^{4} )^{2} \)
$29$ \( ( 1250 + T^{2} )^{4} \)
$31$ \( ( 1327104 + 1152 T^{2} + T^{4} )^{2} \)
$37$ \( ( 4096 + 64 T + T^{2} )^{4} \)
$41$ \( ( 400 + T^{2} )^{4} \)
$43$ \( ( -44 + T )^{8} \)
$47$ \( ( 21381376 - 4624 T^{2} + T^{4} )^{2} \)
$53$ \( ( 114244 - 338 T^{2} + T^{4} )^{2} \)
$59$ \( ( 100000000 - 10000 T^{2} + T^{4} )^{2} \)
$61$ \( ( 7496644 + 2738 T^{2} + T^{4} )^{2} \)
$67$ \( ( 14400 + 120 T + T^{2} )^{4} \)
$71$ \( ( 72 + T^{2} )^{4} \)
$73$ \( ( 31561924 + 5618 T^{2} + T^{4} )^{2} \)
$79$ \( ( 8464 + 92 T + T^{2} )^{4} \)
$83$ \( ( 12544 + T^{2} )^{4} \)
$89$ \( ( 160000 - 400 T^{2} + T^{4} )^{2} \)
$97$ \( ( -722 + T^{2} )^{4} \)
show more
show less