Properties

Label 882.2.g.l
Level $882$
Weight $2$
Character orbit 882.g
Analytic conductor $7.043$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{2} + \beta_{2} q^{4} + 2 \beta_{1} q^{5} - q^{8} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{2} + \beta_{2} q^{4} + 2 \beta_{1} q^{5} - q^{8} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{10} + 2 \beta_{2} q^{11} + ( -1 - \beta_{2} ) q^{16} + ( \beta_{1} + \beta_{3} ) q^{17} + 5 \beta_{1} q^{19} + 2 \beta_{3} q^{20} -2 q^{22} + ( -4 - 4 \beta_{2} ) q^{23} + 3 \beta_{2} q^{25} -2 q^{29} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{31} -\beta_{2} q^{32} + \beta_{3} q^{34} + ( -10 - 10 \beta_{2} ) q^{37} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{38} -2 \beta_{1} q^{40} -7 \beta_{3} q^{41} + 2 q^{43} + ( -2 - 2 \beta_{2} ) q^{44} -4 \beta_{2} q^{46} + 2 \beta_{1} q^{47} -3 q^{50} + 2 \beta_{2} q^{53} + 4 \beta_{3} q^{55} + ( -2 - 2 \beta_{2} ) q^{58} + ( \beta_{1} + \beta_{3} ) q^{59} -2 \beta_{1} q^{61} + 6 \beta_{3} q^{62} + q^{64} + 12 \beta_{2} q^{67} -\beta_{1} q^{68} + 12 q^{71} + ( -\beta_{1} - \beta_{3} ) q^{73} -10 \beta_{2} q^{74} + 5 \beta_{3} q^{76} + ( 4 + 4 \beta_{2} ) q^{79} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{80} + 7 \beta_{1} q^{82} + 7 \beta_{3} q^{83} -4 q^{85} + ( 2 + 2 \beta_{2} ) q^{86} -2 \beta_{2} q^{88} -5 \beta_{1} q^{89} + 4 q^{92} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{94} + 20 \beta_{2} q^{95} -7 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 2q^{4} - 4q^{8} + O(q^{10}) \) \( 4q + 2q^{2} - 2q^{4} - 4q^{8} - 4q^{11} - 2q^{16} - 8q^{22} - 8q^{23} - 6q^{25} - 8q^{29} + 2q^{32} - 20q^{37} + 8q^{43} - 4q^{44} + 8q^{46} - 12q^{50} - 4q^{53} - 4q^{58} + 4q^{64} - 24q^{67} + 48q^{71} + 20q^{74} + 8q^{79} - 16q^{85} + 4q^{86} + 4q^{88} + 16q^{92} - 40q^{95} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.41421 2.44949i 0 0 −1.00000 0 1.41421 2.44949i
361.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.41421 + 2.44949i 0 0 −1.00000 0 −1.41421 + 2.44949i
667.1 0.500000 0.866025i 0 −0.500000 0.866025i −1.41421 + 2.44949i 0 0 −1.00000 0 1.41421 + 2.44949i
667.2 0.500000 0.866025i 0 −0.500000 0.866025i 1.41421 2.44949i 0 0 −1.00000 0 −1.41421 2.44949i
\(n\): 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
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.g.l 4
3.b odd 2 1 98.2.c.c 4
7.b odd 2 1 inner 882.2.g.l 4
7.c even 3 1 882.2.a.n 2
7.c even 3 1 inner 882.2.g.l 4
7.d odd 6 1 882.2.a.n 2
7.d odd 6 1 inner 882.2.g.l 4
12.b even 2 1 784.2.i.m 4
21.c even 2 1 98.2.c.c 4
21.g even 6 1 98.2.a.b 2
21.g even 6 1 98.2.c.c 4
21.h odd 6 1 98.2.a.b 2
21.h odd 6 1 98.2.c.c 4
28.f even 6 1 7056.2.a.cl 2
28.g odd 6 1 7056.2.a.cl 2
84.h odd 2 1 784.2.i.m 4
84.j odd 6 1 784.2.a.l 2
84.j odd 6 1 784.2.i.m 4
84.n even 6 1 784.2.a.l 2
84.n even 6 1 784.2.i.m 4
105.o odd 6 1 2450.2.a.bj 2
105.p even 6 1 2450.2.a.bj 2
105.w odd 12 2 2450.2.c.v 4
105.x even 12 2 2450.2.c.v 4
168.s odd 6 1 3136.2.a.bn 2
168.v even 6 1 3136.2.a.bm 2
168.ba even 6 1 3136.2.a.bn 2
168.be odd 6 1 3136.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.a.b 2 21.g even 6 1
98.2.a.b 2 21.h odd 6 1
98.2.c.c 4 3.b odd 2 1
98.2.c.c 4 21.c even 2 1
98.2.c.c 4 21.g even 6 1
98.2.c.c 4 21.h odd 6 1
784.2.a.l 2 84.j odd 6 1
784.2.a.l 2 84.n even 6 1
784.2.i.m 4 12.b even 2 1
784.2.i.m 4 84.h odd 2 1
784.2.i.m 4 84.j odd 6 1
784.2.i.m 4 84.n even 6 1
882.2.a.n 2 7.c even 3 1
882.2.a.n 2 7.d odd 6 1
882.2.g.l 4 1.a even 1 1 trivial
882.2.g.l 4 7.b odd 2 1 inner
882.2.g.l 4 7.c even 3 1 inner
882.2.g.l 4 7.d odd 6 1 inner
2450.2.a.bj 2 105.o odd 6 1
2450.2.a.bj 2 105.p even 6 1
2450.2.c.v 4 105.w odd 12 2
2450.2.c.v 4 105.x even 12 2
3136.2.a.bm 2 168.v even 6 1
3136.2.a.bm 2 168.be odd 6 1
3136.2.a.bn 2 168.s odd 6 1
3136.2.a.bn 2 168.ba even 6 1
7056.2.a.cl 2 28.f even 6 1
7056.2.a.cl 2 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 64 + 8 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 4 + 2 T + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( 4 + 2 T^{2} + T^{4} \)
$19$ \( 2500 + 50 T^{2} + T^{4} \)
$23$ \( ( 16 + 4 T + T^{2} )^{2} \)
$29$ \( ( 2 + T )^{4} \)
$31$ \( 5184 + 72 T^{2} + T^{4} \)
$37$ \( ( 100 + 10 T + T^{2} )^{2} \)
$41$ \( ( -98 + T^{2} )^{2} \)
$43$ \( ( -2 + T )^{4} \)
$47$ \( 64 + 8 T^{2} + T^{4} \)
$53$ \( ( 4 + 2 T + T^{2} )^{2} \)
$59$ \( 4 + 2 T^{2} + T^{4} \)
$61$ \( 64 + 8 T^{2} + T^{4} \)
$67$ \( ( 144 + 12 T + T^{2} )^{2} \)
$71$ \( ( -12 + T )^{4} \)
$73$ \( 4 + 2 T^{2} + T^{4} \)
$79$ \( ( 16 - 4 T + T^{2} )^{2} \)
$83$ \( ( -98 + T^{2} )^{2} \)
$89$ \( 2500 + 50 T^{2} + T^{4} \)
$97$ \( ( -98 + T^{2} )^{2} \)
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