Properties

Label 882.4.g.m
Level $882$
Weight $4$
Character orbit 882.g
Analytic conductor $52.040$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(52.0396846251\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 22 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})\) \( q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 22 \zeta_{6} q^{5} + 8 q^{8} + ( 44 - 44 \zeta_{6} ) q^{10} + ( 26 - 26 \zeta_{6} ) q^{11} -54 q^{13} -16 \zeta_{6} q^{16} + ( 74 - 74 \zeta_{6} ) q^{17} -116 \zeta_{6} q^{19} -88 q^{20} -52 q^{22} -58 \zeta_{6} q^{23} + ( -359 + 359 \zeta_{6} ) q^{25} + 108 \zeta_{6} q^{26} -208 q^{29} + ( 252 - 252 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} -148 q^{34} -50 \zeta_{6} q^{37} + ( -232 + 232 \zeta_{6} ) q^{38} + 176 \zeta_{6} q^{40} -126 q^{41} + 164 q^{43} + 104 \zeta_{6} q^{44} + ( -116 + 116 \zeta_{6} ) q^{46} -444 \zeta_{6} q^{47} + 718 q^{50} + ( 216 - 216 \zeta_{6} ) q^{52} + ( 12 - 12 \zeta_{6} ) q^{53} + 572 q^{55} + 416 \zeta_{6} q^{58} + ( 124 - 124 \zeta_{6} ) q^{59} + 162 \zeta_{6} q^{61} -504 q^{62} + 64 q^{64} -1188 \zeta_{6} q^{65} + ( 860 - 860 \zeta_{6} ) q^{67} + 296 \zeta_{6} q^{68} + 238 q^{71} + ( 146 - 146 \zeta_{6} ) q^{73} + ( -100 + 100 \zeta_{6} ) q^{74} + 464 q^{76} + 984 \zeta_{6} q^{79} + ( 352 - 352 \zeta_{6} ) q^{80} + 252 \zeta_{6} q^{82} -656 q^{83} + 1628 q^{85} -328 \zeta_{6} q^{86} + ( 208 - 208 \zeta_{6} ) q^{88} -954 \zeta_{6} q^{89} + 232 q^{92} + ( -888 + 888 \zeta_{6} ) q^{94} + ( 2552 - 2552 \zeta_{6} ) q^{95} + 526 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 4q^{4} + 22q^{5} + 16q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{4} + 22q^{5} + 16q^{8} + 44q^{10} + 26q^{11} - 108q^{13} - 16q^{16} + 74q^{17} - 116q^{19} - 176q^{20} - 104q^{22} - 58q^{23} - 359q^{25} + 108q^{26} - 416q^{29} + 252q^{31} - 32q^{32} - 296q^{34} - 50q^{37} - 232q^{38} + 176q^{40} - 252q^{41} + 328q^{43} + 104q^{44} - 116q^{46} - 444q^{47} + 1436q^{50} + 216q^{52} + 12q^{53} + 1144q^{55} + 416q^{58} + 124q^{59} + 162q^{61} - 1008q^{62} + 128q^{64} - 1188q^{65} + 860q^{67} + 296q^{68} + 476q^{71} + 146q^{73} - 100q^{74} + 928q^{76} + 984q^{79} + 352q^{80} + 252q^{82} - 1312q^{83} + 3256q^{85} - 328q^{86} + 208q^{88} - 954q^{89} + 464q^{92} - 888q^{94} + 2552q^{95} + 1052q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 1.73205i 0 −2.00000 + 3.46410i 11.0000 + 19.0526i 0 0 8.00000 0 22.0000 38.1051i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i 11.0000 19.0526i 0 0 8.00000 0 22.0000 + 38.1051i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.g.m 2
3.b odd 2 1 882.4.g.n 2
7.b odd 2 1 882.4.g.a 2
7.c even 3 1 126.4.a.f yes 1
7.c even 3 1 inner 882.4.g.m 2
7.d odd 6 1 882.4.a.s 1
7.d odd 6 1 882.4.g.a 2
21.c even 2 1 882.4.g.x 2
21.g even 6 1 882.4.a.a 1
21.g even 6 1 882.4.g.x 2
21.h odd 6 1 126.4.a.e 1
21.h odd 6 1 882.4.g.n 2
28.g odd 6 1 1008.4.a.a 1
84.n even 6 1 1008.4.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.e 1 21.h odd 6 1
126.4.a.f yes 1 7.c even 3 1
882.4.a.a 1 21.g even 6 1
882.4.a.s 1 7.d odd 6 1
882.4.g.a 2 7.b odd 2 1
882.4.g.a 2 7.d odd 6 1
882.4.g.m 2 1.a even 1 1 trivial
882.4.g.m 2 7.c even 3 1 inner
882.4.g.n 2 3.b odd 2 1
882.4.g.n 2 21.h odd 6 1
882.4.g.x 2 21.c even 2 1
882.4.g.x 2 21.g even 6 1
1008.4.a.a 1 28.g odd 6 1
1008.4.a.w 1 84.n even 6 1

Hecke kernels

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

\( T_{5}^{2} - 22 T_{5} + 484 \)
\( T_{11}^{2} - 26 T_{11} + 676 \)
\( T_{13} + 54 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 484 - 22 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 676 - 26 T + T^{2} \)
$13$ \( ( 54 + T )^{2} \)
$17$ \( 5476 - 74 T + T^{2} \)
$19$ \( 13456 + 116 T + T^{2} \)
$23$ \( 3364 + 58 T + T^{2} \)
$29$ \( ( 208 + T )^{2} \)
$31$ \( 63504 - 252 T + T^{2} \)
$37$ \( 2500 + 50 T + T^{2} \)
$41$ \( ( 126 + T )^{2} \)
$43$ \( ( -164 + T )^{2} \)
$47$ \( 197136 + 444 T + T^{2} \)
$53$ \( 144 - 12 T + T^{2} \)
$59$ \( 15376 - 124 T + T^{2} \)
$61$ \( 26244 - 162 T + T^{2} \)
$67$ \( 739600 - 860 T + T^{2} \)
$71$ \( ( -238 + T )^{2} \)
$73$ \( 21316 - 146 T + T^{2} \)
$79$ \( 968256 - 984 T + T^{2} \)
$83$ \( ( 656 + T )^{2} \)
$89$ \( 910116 + 954 T + T^{2} \)
$97$ \( ( -526 + T )^{2} \)
show more
show less