Properties

Label 32.6.a.a
Level $32$
Weight $6$
Character orbit 32.a
Self dual yes
Analytic conductor $5.132$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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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: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 8q^{3} + 14q^{5} - 208q^{7} - 179q^{9} + O(q^{10}) \) \( q - 8q^{3} + 14q^{5} - 208q^{7} - 179q^{9} - 536q^{11} + 694q^{13} - 112q^{15} - 1278q^{17} + 1112q^{19} + 1664q^{21} + 3216q^{23} - 2929q^{25} + 3376q^{27} + 2918q^{29} - 2624q^{31} + 4288q^{33} - 2912q^{35} - 9458q^{37} - 5552q^{39} + 170q^{41} - 19928q^{43} - 2506q^{45} + 32q^{47} + 26457q^{49} + 10224q^{51} - 22178q^{53} - 7504q^{55} - 8896q^{57} + 41480q^{59} + 15462q^{61} + 37232q^{63} + 9716q^{65} - 20744q^{67} - 25728q^{69} + 28592q^{71} - 53670q^{73} + 23432q^{75} + 111488q^{77} - 69152q^{79} + 16489q^{81} - 37800q^{83} - 17892q^{85} - 23344q^{87} - 126806q^{89} - 144352q^{91} + 20992q^{93} + 15568q^{95} + 62290q^{97} + 95944q^{99} + O(q^{100}) \)

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 −8.00000 0 14.0000 0 −208.000 0 −179.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 8 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(32))\).