Properties

Label 704.2.a.f
Level $704$
Weight $2$
Character orbit 704.a
Self dual yes
Analytic conductor $5.621$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(5.62146830230\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + 3 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + 3 q^{5} + 2 q^{7} - 2 q^{9} + q^{11} + 4 q^{13} - 3 q^{15} + 6 q^{17} - 8 q^{19} - 2 q^{21} - 3 q^{23} + 4 q^{25} + 5 q^{27} + 5 q^{31} - q^{33} + 6 q^{35} + q^{37} - 4 q^{39} + 10 q^{43} - 6 q^{45} - 3 q^{49} - 6 q^{51} + 6 q^{53} + 3 q^{55} + 8 q^{57} - 3 q^{59} + 4 q^{61} - 4 q^{63} + 12 q^{65} + q^{67} + 3 q^{69} + 15 q^{71} - 4 q^{73} - 4 q^{75} + 2 q^{77} + 2 q^{79} + q^{81} - 6 q^{83} + 18 q^{85} - 9 q^{89} + 8 q^{91} - 5 q^{93} - 24 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100}) \) 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 −1.00000 0 3.00000 0 2.00000 0 −2.00000 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 704.2.a.f 1
3.b odd 2 1 6336.2.a.j 1
4.b odd 2 1 704.2.a.i 1
8.b even 2 1 44.2.a.a 1
8.d odd 2 1 176.2.a.a 1
11.b odd 2 1 7744.2.a.m 1
12.b even 2 1 6336.2.a.i 1
16.e even 4 2 2816.2.c.e 2
16.f odd 4 2 2816.2.c.k 2
24.f even 2 1 1584.2.a.p 1
24.h odd 2 1 396.2.a.c 1
40.e odd 2 1 4400.2.a.v 1
40.f even 2 1 1100.2.a.b 1
40.i odd 4 2 1100.2.b.c 2
40.k even 4 2 4400.2.b.k 2
44.c even 2 1 7744.2.a.bc 1
56.e even 2 1 8624.2.a.w 1
56.h odd 2 1 2156.2.a.a 1
56.j odd 6 2 2156.2.i.c 2
56.p even 6 2 2156.2.i.b 2
72.j odd 6 2 3564.2.i.a 2
72.n even 6 2 3564.2.i.j 2
88.b odd 2 1 484.2.a.a 1
88.g even 2 1 1936.2.a.c 1
88.o even 10 4 484.2.e.a 4
88.p odd 10 4 484.2.e.b 4
104.e even 2 1 7436.2.a.d 1
120.i odd 2 1 9900.2.a.h 1
120.w even 4 2 9900.2.c.g 2
264.m even 2 1 4356.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 8.b even 2 1
176.2.a.a 1 8.d odd 2 1
396.2.a.c 1 24.h odd 2 1
484.2.a.a 1 88.b odd 2 1
484.2.e.a 4 88.o even 10 4
484.2.e.b 4 88.p odd 10 4
704.2.a.f 1 1.a even 1 1 trivial
704.2.a.i 1 4.b odd 2 1
1100.2.a.b 1 40.f even 2 1
1100.2.b.c 2 40.i odd 4 2
1584.2.a.p 1 24.f even 2 1
1936.2.a.c 1 88.g even 2 1
2156.2.a.a 1 56.h odd 2 1
2156.2.i.b 2 56.p even 6 2
2156.2.i.c 2 56.j odd 6 2
2816.2.c.e 2 16.e even 4 2
2816.2.c.k 2 16.f odd 4 2
3564.2.i.a 2 72.j odd 6 2
3564.2.i.j 2 72.n even 6 2
4356.2.a.j 1 264.m even 2 1
4400.2.a.v 1 40.e odd 2 1
4400.2.b.k 2 40.k even 4 2
6336.2.a.i 1 12.b even 2 1
6336.2.a.j 1 3.b odd 2 1
7436.2.a.d 1 104.e even 2 1
7744.2.a.m 1 11.b odd 2 1
7744.2.a.bc 1 44.c even 2 1
8624.2.a.w 1 56.e even 2 1
9900.2.a.h 1 120.i odd 2 1
9900.2.c.g 2 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(704))\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T - 3 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T - 4 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T + 8 \) Copy content Toggle raw display
$23$ \( T + 3 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 5 \) Copy content Toggle raw display
$37$ \( T - 1 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 10 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 3 \) Copy content Toggle raw display
$61$ \( T - 4 \) Copy content Toggle raw display
$67$ \( T - 1 \) Copy content Toggle raw display
$71$ \( T - 15 \) Copy content Toggle raw display
$73$ \( T + 4 \) Copy content Toggle raw display
$79$ \( T - 2 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T + 9 \) Copy content Toggle raw display
$97$ \( T + 7 \) Copy content Toggle raw display
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