Properties

Label 704.6.a.b
Level $704$
Weight $6$
Character orbit 704.a
Self dual yes
Analytic conductor $112.910$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 704.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(112.910209148\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 21 q^{3} - 81 q^{5} - 98 q^{7} + 198 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 21 q^{3} - 81 q^{5} - 98 q^{7} + 198 q^{9} + 121 q^{11} - 824 q^{13} + 1701 q^{15} + 978 q^{17} - 2140 q^{19} + 2058 q^{21} - 3699 q^{23} + 3436 q^{25} + 945 q^{27} - 3480 q^{29} + 7813 q^{31} - 2541 q^{33} + 7938 q^{35} + 13597 q^{37} + 17304 q^{39} + 6492 q^{41} + 14234 q^{43} - 16038 q^{45} + 20352 q^{47} - 7203 q^{49} - 20538 q^{51} + 366 q^{53} - 9801 q^{55} + 44940 q^{57} + 9825 q^{59} - 26132 q^{61} - 19404 q^{63} + 66744 q^{65} + 17093 q^{67} + 77679 q^{69} + 23583 q^{71} - 35176 q^{73} - 72156 q^{75} - 11858 q^{77} + 42490 q^{79} - 67959 q^{81} + 22674 q^{83} - 79218 q^{85} + 73080 q^{87} - 17145 q^{89} + 80752 q^{91} - 164073 q^{93} + 173340 q^{95} - 30727 q^{97} + 23958 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −21.0000 0 −81.0000 0 −98.0000 0 198.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)

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 704.6.a.b 1
4.b odd 2 1 704.6.a.i 1
8.b even 2 1 176.6.a.d 1
8.d odd 2 1 22.6.a.a 1
24.f even 2 1 198.6.a.d 1
40.e odd 2 1 550.6.a.g 1
40.k even 4 2 550.6.b.g 2
56.e even 2 1 1078.6.a.b 1
88.g even 2 1 242.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.a 1 8.d odd 2 1
176.6.a.d 1 8.b even 2 1
198.6.a.d 1 24.f even 2 1
242.6.a.c 1 88.g even 2 1
550.6.a.g 1 40.e odd 2 1
550.6.b.g 2 40.k even 4 2
704.6.a.b 1 1.a even 1 1 trivial
704.6.a.i 1 4.b odd 2 1
1078.6.a.b 1 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 21 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(704))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 21 \) Copy content Toggle raw display
$5$ \( T + 81 \) Copy content Toggle raw display
$7$ \( T + 98 \) Copy content Toggle raw display
$11$ \( T - 121 \) Copy content Toggle raw display
$13$ \( T + 824 \) Copy content Toggle raw display
$17$ \( T - 978 \) Copy content Toggle raw display
$19$ \( T + 2140 \) Copy content Toggle raw display
$23$ \( T + 3699 \) Copy content Toggle raw display
$29$ \( T + 3480 \) Copy content Toggle raw display
$31$ \( T - 7813 \) Copy content Toggle raw display
$37$ \( T - 13597 \) Copy content Toggle raw display
$41$ \( T - 6492 \) Copy content Toggle raw display
$43$ \( T - 14234 \) Copy content Toggle raw display
$47$ \( T - 20352 \) Copy content Toggle raw display
$53$ \( T - 366 \) Copy content Toggle raw display
$59$ \( T - 9825 \) Copy content Toggle raw display
$61$ \( T + 26132 \) Copy content Toggle raw display
$67$ \( T - 17093 \) Copy content Toggle raw display
$71$ \( T - 23583 \) Copy content Toggle raw display
$73$ \( T + 35176 \) Copy content Toggle raw display
$79$ \( T - 42490 \) Copy content Toggle raw display
$83$ \( T - 22674 \) Copy content Toggle raw display
$89$ \( T + 17145 \) Copy content Toggle raw display
$97$ \( T + 30727 \) Copy content Toggle raw display
show more
show less