Properties

Label 32.6.a.b
Level 32
Weight 6
Character orbit 32.a
Self dual yes
Analytic conductor 5.132
Analytic rank 1
Dimension 1
CM discriminant -4
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 32.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.13228223402\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 82q^{5} - 243q^{9} + O(q^{10}) \) \( q - 82q^{5} - 243q^{9} - 1194q^{13} + 2242q^{17} + 3599q^{25} + 2950q^{29} - 12242q^{37} - 20950q^{41} + 19926q^{45} - 16807q^{49} + 7294q^{53} + 18950q^{61} + 97908q^{65} - 88806q^{73} + 59049q^{81} - 183844q^{85} + 51050q^{89} - 92142q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 −82.0000 0 0 0 −243.000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.6.a.b 1
3.b odd 2 1 288.6.a.i 1
4.b odd 2 1 CM 32.6.a.b 1
5.b even 2 1 800.6.a.d 1
5.c odd 4 2 800.6.c.c 2
8.b even 2 1 64.6.a.d 1
8.d odd 2 1 64.6.a.d 1
12.b even 2 1 288.6.a.i 1
16.e even 4 2 256.6.b.e 2
16.f odd 4 2 256.6.b.e 2
20.d odd 2 1 800.6.a.d 1
20.e even 4 2 800.6.c.c 2
24.f even 2 1 576.6.a.e 1
24.h odd 2 1 576.6.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.6.a.b 1 1.a even 1 1 trivial
32.6.a.b 1 4.b odd 2 1 CM
64.6.a.d 1 8.b even 2 1
64.6.a.d 1 8.d odd 2 1
256.6.b.e 2 16.e even 4 2
256.6.b.e 2 16.f odd 4 2
288.6.a.i 1 3.b odd 2 1
288.6.a.i 1 12.b even 2 1
576.6.a.e 1 24.f even 2 1
576.6.a.e 1 24.h odd 2 1
800.6.a.d 1 5.b even 2 1
800.6.a.d 1 20.d odd 2 1
800.6.c.c 2 5.c odd 4 2
800.6.c.c 2 20.e even 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(32))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 243 T^{2} \)
$5$ \( 1 + 82 T + 3125 T^{2} \)
$7$ \( 1 + 16807 T^{2} \)
$11$ \( 1 + 161051 T^{2} \)
$13$ \( 1 + 1194 T + 371293 T^{2} \)
$17$ \( 1 - 2242 T + 1419857 T^{2} \)
$19$ \( 1 + 2476099 T^{2} \)
$23$ \( 1 + 6436343 T^{2} \)
$29$ \( 1 - 2950 T + 20511149 T^{2} \)
$31$ \( 1 + 28629151 T^{2} \)
$37$ \( 1 + 12242 T + 69343957 T^{2} \)
$41$ \( 1 + 20950 T + 115856201 T^{2} \)
$43$ \( 1 + 147008443 T^{2} \)
$47$ \( 1 + 229345007 T^{2} \)
$53$ \( 1 - 7294 T + 418195493 T^{2} \)
$59$ \( 1 + 714924299 T^{2} \)
$61$ \( 1 - 18950 T + 844596301 T^{2} \)
$67$ \( 1 + 1350125107 T^{2} \)
$71$ \( 1 + 1804229351 T^{2} \)
$73$ \( 1 + 88806 T + 2073071593 T^{2} \)
$79$ \( 1 + 3077056399 T^{2} \)
$83$ \( 1 + 3939040643 T^{2} \)
$89$ \( 1 - 51050 T + 5584059449 T^{2} \)
$97$ \( 1 + 92142 T + 8587340257 T^{2} \)
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