Properties

Label 1089.3.b.h
Level $1089$
Weight $3$
Character orbit 1089.b
Analytic conductor $29.673$
Analytic rank $0$
Dimension $8$
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.7
Defining polynomial: \(x^{8} + 12 x^{6} + 23 x^{4} + 12 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{7} q^{2} + \beta_{3} q^{4} -\beta_{1} q^{5} + ( \beta_{2} + \beta_{4} ) q^{7} + \beta_{6} q^{8} +O(q^{10})\) \( q -\beta_{7} q^{2} + \beta_{3} q^{4} -\beta_{1} q^{5} + ( \beta_{2} + \beta_{4} ) q^{7} + \beta_{6} q^{8} + ( 2 \beta_{2} - \beta_{4} ) q^{10} + ( 5 \beta_{2} - 6 \beta_{4} ) q^{13} + \beta_{5} q^{14} + ( -1 + 4 \beta_{3} ) q^{16} + ( -10 \beta_{6} + 3 \beta_{7} ) q^{17} + ( -6 \beta_{2} - 2 \beta_{4} ) q^{19} + ( -\beta_{1} - 7 \beta_{5} ) q^{20} + ( -12 \beta_{1} + 13 \beta_{5} ) q^{23} + ( 17 - \beta_{3} ) q^{25} + ( 11 \beta_{1} - 28 \beta_{5} ) q^{26} + ( 5 \beta_{2} + 3 \beta_{4} ) q^{28} + 13 \beta_{7} q^{29} + ( 12 + 4 \beta_{3} ) q^{31} + ( 8 \beta_{6} + 17 \beta_{7} ) q^{32} + ( 22 - 3 \beta_{3} ) q^{34} + ( -5 \beta_{6} - \beta_{7} ) q^{35} + ( 2 + 9 \beta_{3} ) q^{37} + ( -4 \beta_{1} + 6 \beta_{5} ) q^{38} + ( 3 \beta_{2} + 2 \beta_{4} ) q^{40} + ( 26 \beta_{6} + 13 \beta_{7} ) q^{41} + ( 9 \beta_{2} - 19 \beta_{4} ) q^{43} + ( 37 \beta_{2} - 25 \beta_{4} ) q^{46} + ( -10 \beta_{1} - 6 \beta_{5} ) q^{47} + ( -41 + 2 \beta_{3} ) q^{49} + ( -\beta_{6} - 21 \beta_{7} ) q^{50} + ( -30 \beta_{2} + 15 \beta_{4} ) q^{52} + ( -3 \beta_{1} - 48 \beta_{5} ) q^{53} + ( 2 \beta_{1} + 3 \beta_{5} ) q^{56} + ( 52 - 13 \beta_{3} ) q^{58} + ( -6 \beta_{1} - 51 \beta_{5} ) q^{59} + ( 26 \beta_{2} - 2 \beta_{4} ) q^{61} + ( 4 \beta_{6} + 4 \beta_{7} ) q^{62} + ( 56 - \beta_{3} ) q^{64} + ( 8 \beta_{6} + 17 \beta_{7} ) q^{65} + ( 27 - 21 \beta_{3} ) q^{67} + ( -43 \beta_{6} - 22 \beta_{7} ) q^{68} + ( 1 + \beta_{3} ) q^{70} + ( 26 \beta_{1} + \beta_{5} ) q^{71} + ( 4 \beta_{2} - 52 \beta_{4} ) q^{73} + ( 9 \beta_{6} + 34 \beta_{7} ) q^{74} + ( -10 \beta_{2} - 18 \beta_{4} ) q^{76} + ( 44 \beta_{2} + 30 \beta_{4} ) q^{79} + ( -3 \beta_{1} - 28 \beta_{5} ) q^{80} + ( 26 - 13 \beta_{3} ) q^{82} + ( -20 \beta_{6} - 34 \beta_{7} ) q^{83} + ( -36 \beta_{2} - 17 \beta_{4} ) q^{85} + ( 28 \beta_{1} - 75 \beta_{5} ) q^{86} + ( 11 \beta_{1} + 19 \beta_{5} ) q^{89} + ( -15 - \beta_{3} ) q^{91} + ( 14 \beta_{1} - 97 \beta_{5} ) q^{92} + ( 14 \beta_{2} - 4 \beta_{4} ) q^{94} + ( 18 \beta_{6} - 2 \beta_{7} ) q^{95} + ( 44 - 7 \beta_{3} ) q^{97} + ( 2 \beta_{6} + 49 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q - 8 q^{16} + 136 q^{25} + 96 q^{31} + 176 q^{34} + 16 q^{37} - 328 q^{49} + 416 q^{58} + 448 q^{64} + 216 q^{67} + 8 q^{70} + 208 q^{82} - 120 q^{91} + 352 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 12 x^{6} + 23 x^{4} + 12 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 13 \nu^{5} + 36 \nu^{3} + 48 \nu \)\()/7\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{6} + 32 \nu^{4} + 24 \nu^{2} - 10 \)\()/7\)
\(\beta_{3}\)\(=\)\( -\nu^{6} - 12 \nu^{4} - 22 \nu^{2} - 6 \)
\(\beta_{4}\)\(=\)\((\)\( 8 \nu^{6} + 90 \nu^{4} + 120 \nu^{2} + 27 \)\()/7\)
\(\beta_{5}\)\(=\)\((\)\( -6 \nu^{7} - 71 \nu^{5} - 125 \nu^{3} - 43 \nu \)\()/7\)
\(\beta_{6}\)\(=\)\( \nu^{7} + 11 \nu^{5} + 12 \nu^{3} \)
\(\beta_{7}\)\(=\)\( \nu^{7} + 12 \nu^{5} + 23 \nu^{3} + 11 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} - \beta_{5} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{4} + \beta_{3} - 3 \beta_{2} - 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(11 \beta_{7} + \beta_{6} + 13 \beta_{5} - 6 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-21 \beta_{4} - 12 \beta_{3} + 28 \beta_{2} + 49\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-108 \beta_{7} - 13 \beta_{6} - 132 \beta_{5} + 55 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(104 \beta_{4} + 60 \beta_{3} - 135 \beta_{2} - 234\)
\(\nu^{7}\)\(=\)\((\)\(1056 \beta_{7} + 133 \beta_{6} + 1296 \beta_{5} - 533 \beta_{1}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
0.319918i
3.12580i
0.837556i
1.19395i
1.19395i
0.837556i
3.12580i
0.319918i
2.80588i 0 −3.87298 2.03151i 0 0.504017 0.356394i 0 −5.70017
485.2 2.80588i 0 −3.87298 2.03151i 0 −0.504017 0.356394i 0 5.70017
485.3 0.356394i 0 3.87298 3.44572i 0 −3.96812 2.80588i 0 −1.22803
485.4 0.356394i 0 3.87298 3.44572i 0 3.96812 2.80588i 0 1.22803
485.5 0.356394i 0 3.87298 3.44572i 0 3.96812 2.80588i 0 1.22803
485.6 0.356394i 0 3.87298 3.44572i 0 −3.96812 2.80588i 0 −1.22803
485.7 2.80588i 0 −3.87298 2.03151i 0 −0.504017 0.356394i 0 5.70017
485.8 2.80588i 0 −3.87298 2.03151i 0 0.504017 0.356394i 0 −5.70017
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 485.8
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.b.h 8
3.b odd 2 1 inner 1089.3.b.h 8
11.b odd 2 1 inner 1089.3.b.h 8
33.d even 2 1 inner 1089.3.b.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.3.b.h 8 1.a even 1 1 trivial
1089.3.b.h 8 3.b odd 2 1 inner
1089.3.b.h 8 11.b odd 2 1 inner
1089.3.b.h 8 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\):

\( T_{2}^{4} + 8 T_{2}^{2} + 1 \)
\( T_{7}^{4} - 16 T_{7}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 8 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( ( 49 + 16 T^{2} + T^{4} )^{2} \)
$7$ \( ( 4 - 16 T^{2} + T^{4} )^{2} \)
$11$ \( T^{8} \)
$13$ \( ( 11025 - 510 T^{2} + T^{4} )^{2} \)
$17$ \( ( 121801 + 992 T^{2} + T^{4} )^{2} \)
$19$ \( ( 7744 - 256 T^{2} + T^{4} )^{2} \)
$23$ \( ( 964324 + 2356 T^{2} + T^{4} )^{2} \)
$29$ \( ( 28561 + 1352 T^{2} + T^{4} )^{2} \)
$31$ \( ( -96 - 24 T + T^{2} )^{4} \)
$37$ \( ( -1211 - 4 T + T^{2} )^{4} \)
$41$ \( ( 3455881 + 5408 T^{2} + T^{4} )^{2} \)
$43$ \( ( 2439844 - 4096 T^{2} + T^{4} )^{2} \)
$47$ \( ( 258064 + 1984 T^{2} + T^{4} )^{2} \)
$53$ \( ( 23357889 + 9936 T^{2} + T^{4} )^{2} \)
$59$ \( ( 30935844 + 12204 T^{2} + T^{4} )^{2} \)
$61$ \( ( 4032064 - 4096 T^{2} + T^{4} )^{2} \)
$67$ \( ( -5886 - 54 T + T^{2} )^{4} \)
$71$ \( ( 21883684 + 10924 T^{2} + T^{4} )^{2} \)
$73$ \( ( 181494784 - 27136 T^{2} + T^{4} )^{2} \)
$79$ \( ( 1710864 - 20616 T^{2} + T^{4} )^{2} \)
$83$ \( ( 15085456 + 9728 T^{2} + T^{4} )^{2} \)
$89$ \( ( 85849 + 4216 T^{2} + T^{4} )^{2} \)
$97$ \( ( 1201 - 88 T + T^{2} )^{4} \)
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