Properties

Label 43.9.b.a
Level 43
Weight 9
Character orbit 43.b
Self dual yes
Analytic conductor 17.517
Analytic rank 0
Dimension 1
CM discriminant -43
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 43.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(17.5172802326\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 256q^{4} + 6561q^{9} + O(q^{10}) \) \( q + 256q^{4} + 6561q^{9} + 10319q^{11} - 54721q^{13} + 65536q^{16} + 79967q^{17} + 540719q^{23} + 390625q^{25} + 589679q^{31} + 1679616q^{36} - 2262241q^{41} + 3418801q^{43} + 2641664q^{44} - 6983806q^{47} + 5764801q^{49} - 14008576q^{52} - 13061761q^{53} - 7864606q^{59} + 16777216q^{64} - 39816433q^{67} + 20471552q^{68} + 73045634q^{79} + 43046721q^{81} - 93091441q^{83} + 138424064q^{92} + 162643199q^{97} + 67702959q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
42.1
0
0 0 256.000 0 0 0 0 6561.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.b odd 2 1 CM by \(\Q(\sqrt{-43}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.9.b.a 1
43.b odd 2 1 CM 43.9.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.9.b.a 1 1.a even 1 1 trivial
43.9.b.a 1 43.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{9}^{\mathrm{new}}(43, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 16 T )( 1 + 16 T ) \)
$3$ \( ( 1 - 81 T )( 1 + 81 T ) \)
$5$ \( ( 1 - 625 T )( 1 + 625 T ) \)
$7$ \( ( 1 - 2401 T )( 1 + 2401 T ) \)
$11$ \( 1 - 10319 T + 214358881 T^{2} \)
$13$ \( 1 + 54721 T + 815730721 T^{2} \)
$17$ \( 1 - 79967 T + 6975757441 T^{2} \)
$19$ \( ( 1 - 130321 T )( 1 + 130321 T ) \)
$23$ \( 1 - 540719 T + 78310985281 T^{2} \)
$29$ \( ( 1 - 707281 T )( 1 + 707281 T ) \)
$31$ \( 1 - 589679 T + 852891037441 T^{2} \)
$37$ \( ( 1 - 1874161 T )( 1 + 1874161 T ) \)
$41$ \( 1 + 2262241 T + 7984925229121 T^{2} \)
$43$ \( 1 - 3418801 T \)
$47$ \( 1 + 6983806 T + 23811286661761 T^{2} \)
$53$ \( 1 + 13061761 T + 62259690411361 T^{2} \)
$59$ \( 1 + 7864606 T + 146830437604321 T^{2} \)
$61$ \( ( 1 - 13845841 T )( 1 + 13845841 T ) \)
$67$ \( 1 + 39816433 T + 406067677556641 T^{2} \)
$71$ \( ( 1 - 25411681 T )( 1 + 25411681 T ) \)
$73$ \( ( 1 - 28398241 T )( 1 + 28398241 T ) \)
$79$ \( 1 - 73045634 T + 1517108809906561 T^{2} \)
$83$ \( 1 + 93091441 T + 2252292232139041 T^{2} \)
$89$ \( ( 1 - 62742241 T )( 1 + 62742241 T ) \)
$97$ \( 1 - 162643199 T + 7837433594376961 T^{2} \)
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