Properties

Label 43.7.b.a
Level 43
Weight 7
Character orbit 43.b
Self dual yes
Analytic conductor 9.892
Analytic rank 0
Dimension 1
CM discriminant -43
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 43.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(9.89232559565\)
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

\(f(q)\) \(=\) \( q + 64q^{4} + 729q^{9} + O(q^{10}) \) \( q + 64q^{4} + 729q^{9} - 1638q^{11} + 3706q^{13} + 4096q^{16} + 7074q^{17} - 4734q^{23} + 15625q^{25} - 47918q^{31} + 46656q^{36} - 137358q^{41} - 79507q^{43} - 104832q^{44} + 42354q^{47} + 117649q^{49} + 237184q^{52} - 280854q^{53} + 406458q^{59} + 262144q^{64} - 471926q^{67} + 452736q^{68} + 259378q^{79} + 531441q^{81} - 681174q^{83} - 302976q^{92} - 1741246q^{97} - 1194102q^{99} + O(q^{100}) \)

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 64.0000 0 0 0 0 729.000 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 43.7.b.a 1
43.b odd 2 1 CM 43.7.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.7.b.a 1 1.a even 1 1 trivial
43.7.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 \(S_{7}^{\mathrm{new}}(43, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 8 T )( 1 + 8 T ) \)
$3$ \( ( 1 - 27 T )( 1 + 27 T ) \)
$5$ \( ( 1 - 125 T )( 1 + 125 T ) \)
$7$ \( ( 1 - 343 T )( 1 + 343 T ) \)
$11$ \( 1 + 1638 T + 1771561 T^{2} \)
$13$ \( 1 - 3706 T + 4826809 T^{2} \)
$17$ \( 1 - 7074 T + 24137569 T^{2} \)
$19$ \( ( 1 - 6859 T )( 1 + 6859 T ) \)
$23$ \( 1 + 4734 T + 148035889 T^{2} \)
$29$ \( ( 1 - 24389 T )( 1 + 24389 T ) \)
$31$ \( 1 + 47918 T + 887503681 T^{2} \)
$37$ \( ( 1 - 50653 T )( 1 + 50653 T ) \)
$41$ \( 1 + 137358 T + 4750104241 T^{2} \)
$43$ \( 1 + 79507 T \)
$47$ \( 1 - 42354 T + 10779215329 T^{2} \)
$53$ \( 1 + 280854 T + 22164361129 T^{2} \)
$59$ \( 1 - 406458 T + 42180533641 T^{2} \)
$61$ \( ( 1 - 226981 T )( 1 + 226981 T ) \)
$67$ \( 1 + 471926 T + 90458382169 T^{2} \)
$71$ \( ( 1 - 357911 T )( 1 + 357911 T ) \)
$73$ \( ( 1 - 389017 T )( 1 + 389017 T ) \)
$79$ \( 1 - 259378 T + 243087455521 T^{2} \)
$83$ \( 1 + 681174 T + 326940373369 T^{2} \)
$89$ \( ( 1 - 704969 T )( 1 + 704969 T ) \)
$97$ \( 1 + 1741246 T + 832972004929 T^{2} \)
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