Properties

Label 1089.3.b.f
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: 8.0.1105425137664.10
Defining polynomial: \(x^{8} - 14 x^{6} + 71 x^{4} - 80 x^{2} + 121\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -4 - \beta_{2} ) q^{4} + ( -\beta_{6} - \beta_{7} ) q^{5} + ( \beta_{3} + \beta_{5} ) q^{7} + ( -5 \beta_{1} + \beta_{4} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -4 - \beta_{2} ) q^{4} + ( -\beta_{6} - \beta_{7} ) q^{5} + ( \beta_{3} + \beta_{5} ) q^{7} + ( -5 \beta_{1} + \beta_{4} ) q^{8} + ( 5 \beta_{3} + \beta_{5} ) q^{10} + \beta_{5} q^{13} + ( 3 \beta_{6} + 5 \beta_{7} ) q^{14} + ( 21 + 4 \beta_{2} ) q^{16} + ( -4 \beta_{1} - \beta_{4} ) q^{17} -5 \beta_{3} q^{19} + ( 11 \beta_{6} + 9 \beta_{7} ) q^{20} + 5 \beta_{7} q^{23} + ( -6 - 5 \beta_{2} ) q^{25} + 3 \beta_{7} q^{26} + ( -15 \beta_{3} - \beta_{5} ) q^{28} -10 \beta_{1} q^{29} + ( -25 - \beta_{2} ) q^{31} + 21 \beta_{1} q^{32} + ( 35 + \beta_{2} ) q^{34} + ( -16 \beta_{1} + \beta_{4} ) q^{35} + ( -15 - 5 \beta_{2} ) q^{37} + ( -15 \beta_{6} - 10 \beta_{7} ) q^{38} + ( -31 \beta_{3} - 5 \beta_{5} ) q^{40} + ( 10 \beta_{1} - 5 \beta_{4} ) q^{41} + ( 5 \beta_{3} - 6 \beta_{5} ) q^{43} + ( -10 \beta_{3} - 5 \beta_{5} ) q^{46} + ( -10 \beta_{6} + 5 \beta_{7} ) q^{47} + ( 24 - \beta_{2} ) q^{49} + ( -31 \beta_{1} + 5 \beta_{4} ) q^{50} + ( -6 \beta_{3} + \beta_{5} ) q^{52} + ( 5 \beta_{6} - 15 \beta_{7} ) q^{53} + ( -33 \beta_{6} - 13 \beta_{7} ) q^{56} + ( 80 + 10 \beta_{2} ) q^{58} -2 \beta_{6} q^{59} + ( -20 \beta_{3} + 5 \beta_{5} ) q^{61} + ( -30 \beta_{1} + \beta_{4} ) q^{62} + ( -84 - 5 \beta_{2} ) q^{64} -6 \beta_{1} q^{65} + ( 20 - 10 \beta_{2} ) q^{67} + ( 24 \beta_{1} - 5 \beta_{4} ) q^{68} + ( 125 + 19 \beta_{2} ) q^{70} + ( 8 \beta_{6} + 25 \beta_{7} ) q^{71} + ( 14 \beta_{3} + 4 \beta_{5} ) q^{73} + ( -40 \beta_{1} + 5 \beta_{4} ) q^{74} + ( 45 \beta_{3} + 10 \beta_{5} ) q^{76} + ( -15 \beta_{3} + 5 \beta_{5} ) q^{79} + ( -49 \beta_{6} - 41 \beta_{7} ) q^{80} + ( -65 - 25 \beta_{2} ) q^{82} + ( -16 \beta_{1} - 5 \beta_{4} ) q^{83} + ( -14 \beta_{3} - 4 \beta_{5} ) q^{85} + ( 15 \beta_{6} - 8 \beta_{7} ) q^{86} + ( -15 \beta_{6} + 10 \beta_{7} ) q^{89} + ( 60 - 6 \beta_{2} ) q^{91} + ( -30 \beta_{6} - 15 \beta_{7} ) q^{92} + ( 20 \beta_{3} - 5 \beta_{5} ) q^{94} + ( 50 \beta_{1} - 5 \beta_{4} ) q^{95} + ( -55 + 15 \beta_{2} ) q^{97} + ( 19 \beta_{1} + \beta_{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} + O(q^{10}) \) \( 8 q - 32 q^{4} + 168 q^{16} - 48 q^{25} - 200 q^{31} + 280 q^{34} - 120 q^{37} + 192 q^{49} + 640 q^{58} - 672 q^{64} + 160 q^{67} + 1000 q^{70} - 520 q^{82} + 480 q^{91} - 440 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 14 x^{6} + 71 x^{4} - 80 x^{2} + 121\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{6} + 19 \nu^{4} - 93 \nu^{2} + 70 \)\()/73\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{6} + 38 \nu^{4} - 40 \nu^{2} - 371 \)\()/73\)
\(\beta_{3}\)\(=\)\((\)\( 13 \nu^{7} - 160 \nu^{5} + 714 \nu^{3} - 1623 \nu \)\()/803\)
\(\beta_{4}\)\(=\)\((\)\( -14 \nu^{6} + 206 \nu^{4} - 1016 \nu^{2} + 782 \)\()/73\)
\(\beta_{5}\)\(=\)\((\)\( -25 \nu^{7} + 493 \nu^{5} - 3535 \nu^{3} + 8248 \nu \)\()/803\)
\(\beta_{6}\)\(=\)\((\)\( 39 \nu^{7} - 480 \nu^{5} + 2142 \nu^{3} - 51 \nu \)\()/803\)
\(\beta_{7}\)\(=\)\((\)\( -43 \nu^{7} + 591 \nu^{5} - 2547 \nu^{3} + 118 \nu \)\()/803\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - 3 \beta_{3}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1} + 7\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(9 \beta_{7} + 13 \beta_{6} - 3 \beta_{5} - 15 \beta_{3}\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{4} + 5 \beta_{2} - 24 \beta_{1} + 27\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(105 \beta_{7} + 124 \beta_{6} - 9 \beta_{5} - 42 \beta_{3}\)\()/6\)
\(\nu^{6}\)\(=\)\((\)\(19 \beta_{4} + \beta_{2} - 208 \beta_{1} + 1\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(798 \beta_{7} + 937 \beta_{6} + 54 \beta_{5} + 303 \beta_{3}\)\()/6\)

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
2.65356 + 0.707107i
−2.65356 0.707107i
−0.979091 0.707107i
0.979091 + 0.707107i
0.979091 0.707107i
−0.979091 + 0.707107i
−2.65356 + 0.707107i
2.65356 0.707107i
3.75270i 0 −10.0828 7.83670i 0 −8.18030 22.8268i 0 −29.4088
485.2 3.75270i 0 −10.0828 7.83670i 0 8.18030 22.8268i 0 29.4088
485.3 1.38464i 0 2.08276 0.765629i 0 −8.89285 8.42246i 0 −1.06012
485.4 1.38464i 0 2.08276 0.765629i 0 8.89285 8.42246i 0 1.06012
485.5 1.38464i 0 2.08276 0.765629i 0 8.89285 8.42246i 0 1.06012
485.6 1.38464i 0 2.08276 0.765629i 0 −8.89285 8.42246i 0 −1.06012
485.7 3.75270i 0 −10.0828 7.83670i 0 8.18030 22.8268i 0 29.4088
485.8 3.75270i 0 −10.0828 7.83670i 0 −8.18030 22.8268i 0 −29.4088
\(n\): 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.f 8
3.b odd 2 1 inner 1089.3.b.f 8
11.b odd 2 1 inner 1089.3.b.f 8
33.d even 2 1 inner 1089.3.b.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.3.b.f 8 1.a even 1 1 trivial
1089.3.b.f 8 3.b odd 2 1 inner
1089.3.b.f 8 11.b odd 2 1 inner
1089.3.b.f 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 \(S_{3}^{\mathrm{new}}(1089, [\chi])\):

\( T_{2}^{4} + 16 T_{2}^{2} + 27 \)
\( T_{7}^{4} - 146 T_{7}^{2} + 5292 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 27 + 16 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( ( 36 + 62 T^{2} + T^{4} )^{2} \)
$7$ \( ( 5292 - 146 T^{2} + T^{4} )^{2} \)
$11$ \( T^{8} \)
$13$ \( ( 972 - 126 T^{2} + T^{4} )^{2} \)
$17$ \( ( 52272 + 556 T^{2} + T^{4} )^{2} \)
$19$ \( ( 67500 - 800 T^{2} + T^{4} )^{2} \)
$23$ \( ( 202500 + 950 T^{2} + T^{4} )^{2} \)
$29$ \( ( 270000 + 1600 T^{2} + T^{4} )^{2} \)
$31$ \( ( 588 + 50 T + T^{2} )^{4} \)
$37$ \( ( -700 + 30 T + T^{2} )^{4} \)
$41$ \( ( 13230000 + 7300 T^{2} + T^{4} )^{2} \)
$43$ \( ( 484812 - 5696 T^{2} + T^{4} )^{2} \)
$47$ \( ( 2722500 + 5150 T^{2} + T^{4} )^{2} \)
$53$ \( ( 9922500 + 10350 T^{2} + T^{4} )^{2} \)
$59$ \( ( 72 + T^{2} )^{4} \)
$61$ \( ( 73507500 - 17150 T^{2} + T^{4} )^{2} \)
$67$ \( ( -3300 - 40 T + T^{2} )^{4} \)
$71$ \( ( 127644804 + 23654 T^{2} + T^{4} )^{2} \)
$73$ \( ( 1881792 - 7616 T^{2} + T^{4} )^{2} \)
$79$ \( ( 29767500 - 11250 T^{2} + T^{4} )^{2} \)
$83$ \( ( 15431472 + 11356 T^{2} + T^{4} )^{2} \)
$89$ \( ( 9922500 + 13700 T^{2} + T^{4} )^{2} \)
$97$ \( ( -5300 + 110 T + T^{2} )^{4} \)
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