Properties

Label 1089.3.c.b
Level $1089$
Weight $3$
Character orbit 1089.c
Analytic conductor $29.673$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} -7 q^{4} -2 \beta_{1} q^{5} + 7 \beta_{2} q^{7} + 3 \beta_{3} q^{8} +O(q^{10})\) \( q -\beta_{3} q^{2} -7 q^{4} -2 \beta_{1} q^{5} + 7 \beta_{2} q^{7} + 3 \beta_{3} q^{8} + 22 \beta_{2} q^{10} + 9 \beta_{2} q^{13} + 7 \beta_{1} q^{14} + 5 q^{16} -3 \beta_{2} q^{19} + 14 \beta_{1} q^{20} -6 \beta_{1} q^{23} + 63 q^{25} + 9 \beta_{1} q^{26} -49 \beta_{2} q^{28} -4 \beta_{3} q^{29} -44 q^{31} + 7 \beta_{3} q^{32} -28 \beta_{3} q^{35} + 44 q^{37} -3 \beta_{1} q^{38} -66 \beta_{2} q^{40} -12 \beta_{3} q^{41} -21 \beta_{2} q^{43} + 66 \beta_{2} q^{46} -2 \beta_{1} q^{47} -49 q^{49} -63 \beta_{3} q^{50} -63 \beta_{2} q^{52} + 14 \beta_{1} q^{53} -21 \beta_{1} q^{56} -44 q^{58} -4 \beta_{1} q^{59} + 49 \beta_{2} q^{61} + 44 \beta_{3} q^{62} + 97 q^{64} -36 \beta_{3} q^{65} -88 q^{67} -308 q^{70} + 10 \beta_{1} q^{71} -29 \beta_{2} q^{73} -44 \beta_{3} q^{74} + 21 \beta_{2} q^{76} -9 \beta_{2} q^{79} -10 \beta_{1} q^{80} -132 q^{82} + 28 \beta_{3} q^{83} -21 \beta_{1} q^{86} -24 \beta_{1} q^{89} -126 q^{91} + 42 \beta_{1} q^{92} + 22 \beta_{2} q^{94} + 12 \beta_{3} q^{95} + 70 q^{97} + 49 \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{4} + O(q^{10}) \) \( 4 q - 28 q^{4} + 20 q^{16} + 252 q^{25} - 176 q^{31} + 176 q^{37} - 196 q^{49} - 176 q^{58} + 388 q^{64} - 352 q^{67} - 1232 q^{70} - 528 q^{82} - 504 q^{91} + 280 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 5 \)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{3} - 3 \nu^{2} + 19 \nu - 9 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 4 \nu^{3} - 6 \nu^{2} + 44 \nu - 21 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - 2 \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 2 \beta_{2} + 2 \beta_{1} - 9\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-8 \beta_{3} + 19 \beta_{2} + 3 \beta_{1} - 14\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
604.1
0.500000 + 0.244099i
0.500000 + 3.07253i
0.500000 0.244099i
0.500000 3.07253i
3.31662i 0 −7.00000 −9.38083 0 9.89949i 9.94987i 0 31.1127i
604.2 3.31662i 0 −7.00000 9.38083 0 9.89949i 9.94987i 0 31.1127i
604.3 3.31662i 0 −7.00000 −9.38083 0 9.89949i 9.94987i 0 31.1127i
604.4 3.31662i 0 −7.00000 9.38083 0 9.89949i 9.94987i 0 31.1127i
\(n\): 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
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.3.c.b 4
3.b odd 2 1 inner 1089.3.c.b 4
11.b odd 2 1 inner 1089.3.c.b 4
33.d even 2 1 inner 1089.3.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.3.c.b 4 1.a even 1 1 trivial
1089.3.c.b 4 3.b odd 2 1 inner
1089.3.c.b 4 11.b odd 2 1 inner
1089.3.c.b 4 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 11 \) acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 11 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( -88 + T^{2} )^{2} \)
$7$ \( ( 98 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( 162 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( 18 + T^{2} )^{2} \)
$23$ \( ( -792 + T^{2} )^{2} \)
$29$ \( ( 176 + T^{2} )^{2} \)
$31$ \( ( 44 + T )^{4} \)
$37$ \( ( -44 + T )^{4} \)
$41$ \( ( 1584 + T^{2} )^{2} \)
$43$ \( ( 882 + T^{2} )^{2} \)
$47$ \( ( -88 + T^{2} )^{2} \)
$53$ \( ( -4312 + T^{2} )^{2} \)
$59$ \( ( -352 + T^{2} )^{2} \)
$61$ \( ( 4802 + T^{2} )^{2} \)
$67$ \( ( 88 + T )^{4} \)
$71$ \( ( -2200 + T^{2} )^{2} \)
$73$ \( ( 1682 + T^{2} )^{2} \)
$79$ \( ( 162 + T^{2} )^{2} \)
$83$ \( ( 8624 + T^{2} )^{2} \)
$89$ \( ( -12672 + T^{2} )^{2} \)
$97$ \( ( -70 + T )^{4} \)
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