Properties

Label 43.5.b.a
Level $43$
Weight $5$
Character orbit 43.b
Self dual yes
Analytic conductor $4.445$
Analytic rank $0$
Dimension $1$
CM discriminant -43
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 43.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(4.44490841261\)
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

\(f(q)\) \(=\) \( q + 16q^{4} + 81q^{9} + O(q^{10}) \) \( q + 16q^{4} + 81q^{9} + 199q^{11} - 49q^{13} + 256q^{16} - 497q^{17} - 1049q^{23} + 625q^{25} - 1561q^{31} + 1296q^{36} - 1841q^{41} + 1849q^{43} + 3184q^{44} + 1666q^{47} + 2401q^{49} - 784q^{52} - 1649q^{53} - 4046q^{59} + 4096q^{64} - 697q^{67} - 7952q^{68} - 12286q^{79} + 6561q^{81} + 1351q^{83} - 16784q^{92} + 18431q^{97} + 16119q^{99} + O(q^{100}) \)

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 16.0000 0 0 0 0 81.0000 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 43.5.b.a 1
3.b odd 2 1 387.5.b.a 1
4.b odd 2 1 688.5.b.a 1
43.b odd 2 1 CM 43.5.b.a 1
129.d even 2 1 387.5.b.a 1
172.d even 2 1 688.5.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.5.b.a 1 1.a even 1 1 trivial
43.5.b.a 1 43.b odd 2 1 CM
387.5.b.a 1 3.b odd 2 1
387.5.b.a 1 129.d even 2 1
688.5.b.a 1 4.b odd 2 1
688.5.b.a 1 172.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{5}^{\mathrm{new}}(43, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -199 + T \)
$13$ \( 49 + T \)
$17$ \( 497 + T \)
$19$ \( T \)
$23$ \( 1049 + T \)
$29$ \( T \)
$31$ \( 1561 + T \)
$37$ \( T \)
$41$ \( 1841 + T \)
$43$ \( -1849 + T \)
$47$ \( -1666 + T \)
$53$ \( 1649 + T \)
$59$ \( 4046 + T \)
$61$ \( T \)
$67$ \( 697 + T \)
$71$ \( T \)
$73$ \( T \)
$79$ \( 12286 + T \)
$83$ \( -1351 + T \)
$89$ \( T \)
$97$ \( -18431 + T \)
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