Properties

Label 882.3.b.g
Level $882$
Weight $3$
Character orbit 882.b
Analytic conductor $24.033$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{37})\)
Defining polynomial: \(x^{4} - 2 x^{3} - 13 x^{2} + 14 x + 123\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -2 q^{4} -\beta_{2} q^{5} + 2 \beta_{1} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} -2 q^{4} -\beta_{2} q^{5} + 2 \beta_{1} q^{8} -\beta_{3} q^{10} + 2 \beta_{1} q^{11} + \beta_{3} q^{13} + 4 q^{16} -3 \beta_{2} q^{17} -2 \beta_{3} q^{19} + 2 \beta_{2} q^{20} + 4 q^{22} -30 \beta_{1} q^{23} -49 q^{25} -2 \beta_{2} q^{26} + 11 \beta_{1} q^{29} + 2 \beta_{3} q^{31} -4 \beta_{1} q^{32} -3 \beta_{3} q^{34} -6 q^{37} + 4 \beta_{2} q^{38} + 2 \beta_{3} q^{40} -3 \beta_{2} q^{41} -68 q^{43} -4 \beta_{1} q^{44} -60 q^{46} + 8 \beta_{2} q^{47} + 49 \beta_{1} q^{50} -2 \beta_{3} q^{52} + 29 \beta_{1} q^{53} + 2 \beta_{3} q^{55} + 22 q^{58} + 8 \beta_{2} q^{59} + 8 \beta_{3} q^{61} -4 \beta_{2} q^{62} -8 q^{64} -74 \beta_{1} q^{65} -104 q^{67} + 6 \beta_{2} q^{68} + 50 \beta_{1} q^{71} + 5 \beta_{3} q^{73} + 6 \beta_{1} q^{74} + 4 \beta_{3} q^{76} -20 q^{79} -4 \beta_{2} q^{80} -3 \beta_{3} q^{82} -222 q^{85} + 68 \beta_{1} q^{86} -8 q^{88} -11 \beta_{2} q^{89} + 60 \beta_{1} q^{92} + 8 \beta_{3} q^{94} + 148 \beta_{1} q^{95} -13 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} + 16q^{16} + 16q^{22} - 196q^{25} - 24q^{37} - 272q^{43} - 240q^{46} + 88q^{58} - 32q^{64} - 416q^{67} - 80q^{79} - 888q^{85} - 32q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 13 x^{2} + 14 x + 123\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{3} - 3 \nu^{2} - 5 \nu + 3 \)\()/45\)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 7 \)
\(\beta_{3}\)\(=\)\((\)\( -8 \nu^{3} + 12 \nu^{2} + 200 \nu - 102 \)\()/45\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 4 \beta_{1} + 2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 4 \beta_{2} + 4 \beta_{1} + 30\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{3} + 3 \beta_{2} + 53 \beta_{1} + 22\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
3.54138 + 1.41421i
−2.54138 + 1.41421i
−2.54138 1.41421i
3.54138 1.41421i
1.41421i 0 −2.00000 8.60233i 0 0 2.82843i 0 −12.1655
197.2 1.41421i 0 −2.00000 8.60233i 0 0 2.82843i 0 12.1655
197.3 1.41421i 0 −2.00000 8.60233i 0 0 2.82843i 0 12.1655
197.4 1.41421i 0 −2.00000 8.60233i 0 0 2.82843i 0 −12.1655
\(n\): e.g. 2-40 or 990-1000
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
21.c even 2 1 inner

Twists

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

Hecke kernels

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

\( T_{5}^{2} + 74 \)
\( T_{11}^{2} + 8 \)
\( T_{13}^{2} - 148 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 74 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( 8 + T^{2} )^{2} \)
$13$ \( ( -148 + T^{2} )^{2} \)
$17$ \( ( 666 + T^{2} )^{2} \)
$19$ \( ( -592 + T^{2} )^{2} \)
$23$ \( ( 1800 + T^{2} )^{2} \)
$29$ \( ( 242 + T^{2} )^{2} \)
$31$ \( ( -592 + T^{2} )^{2} \)
$37$ \( ( 6 + T )^{4} \)
$41$ \( ( 666 + T^{2} )^{2} \)
$43$ \( ( 68 + T )^{4} \)
$47$ \( ( 4736 + T^{2} )^{2} \)
$53$ \( ( 1682 + T^{2} )^{2} \)
$59$ \( ( 4736 + T^{2} )^{2} \)
$61$ \( ( -9472 + T^{2} )^{2} \)
$67$ \( ( 104 + T )^{4} \)
$71$ \( ( 5000 + T^{2} )^{2} \)
$73$ \( ( -3700 + T^{2} )^{2} \)
$79$ \( ( 20 + T )^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 8954 + T^{2} )^{2} \)
$97$ \( ( -25012 + T^{2} )^{2} \)
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