Properties

Label 8.12.a.a
Level 8
Weight 12
Character orbit 8.a
Self dual yes
Analytic conductor 6.147
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 8.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.14674544448\)
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

\(f(q)\) \(=\) \( q - 36q^{3} - 3490q^{5} - 55464q^{7} - 175851q^{9} + O(q^{10}) \) \( q - 36q^{3} - 3490q^{5} - 55464q^{7} - 175851q^{9} - 597004q^{11} + 1373878q^{13} + 125640q^{15} + 10140850q^{17} - 7297396q^{19} + 1996704q^{21} - 32057464q^{23} - 36648025q^{25} + 12707928q^{27} - 13605402q^{29} + 233160800q^{31} + 21492144q^{33} + 193569360q^{35} - 257786178q^{37} - 49459608q^{39} - 221438598q^{41} - 1697758892q^{43} + 613719990q^{45} + 527509392q^{47} + 1098928553q^{49} - 365070600q^{51} + 3277379822q^{53} + 2083543960q^{55} + 262706256q^{57} - 3001908988q^{59} - 11630023610q^{61} + 9753399864q^{63} - 4794834220q^{65} - 17189000548q^{67} + 1154068704q^{69} + 26169539608q^{71} - 7039021094q^{73} + 1319328900q^{75} + 33112229856q^{77} - 4199910416q^{79} + 30693991689q^{81} - 39739936436q^{83} - 35391566500q^{85} + 489794472q^{87} + 10565331594q^{89} - 76200769392q^{91} - 8393788800q^{93} + 25467912040q^{95} - 69851645662q^{97} + 104983750404q^{99} + O(q^{100}) \)

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 −36.0000 0 −3490.00 0 −55464.0 0 −175851. 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 8.12.a.a 1
3.b odd 2 1 72.12.a.c 1
4.b odd 2 1 16.12.a.b 1
5.b even 2 1 200.12.a.b 1
5.c odd 4 2 200.12.c.b 2
8.b even 2 1 64.12.a.e 1
8.d odd 2 1 64.12.a.c 1
12.b even 2 1 144.12.a.j 1
16.e even 4 2 256.12.b.g 2
16.f odd 4 2 256.12.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.12.a.a 1 1.a even 1 1 trivial
16.12.a.b 1 4.b odd 2 1
64.12.a.c 1 8.d odd 2 1
64.12.a.e 1 8.b even 2 1
72.12.a.c 1 3.b odd 2 1
144.12.a.j 1 12.b even 2 1
200.12.a.b 1 5.b even 2 1
200.12.c.b 2 5.c odd 4 2
256.12.b.a 2 16.f odd 4 2
256.12.b.g 2 16.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 \( T_{3} + 36 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(8))\).