Properties

Label 688.2.a.b
Level 688
Weight 2
Character orbit 688.a
Self dual yes
Analytic conductor 5.494
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.49370765906\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} - 4q^{5} + q^{9} + O(q^{10}) \) \( q + 2q^{3} - 4q^{5} + q^{9} - 3q^{11} - 5q^{13} - 8q^{15} - 3q^{17} + 2q^{19} + q^{23} + 11q^{25} - 4q^{27} - 6q^{29} + q^{31} - 6q^{33} - 10q^{39} + 5q^{41} + q^{43} - 4q^{45} - 4q^{47} - 7q^{49} - 6q^{51} - 5q^{53} + 12q^{55} + 4q^{57} + 12q^{59} + 2q^{61} + 20q^{65} + 3q^{67} + 2q^{69} - 2q^{71} + 2q^{73} + 22q^{75} + 8q^{79} - 11q^{81} - 15q^{83} + 12q^{85} - 12q^{87} - 4q^{89} + 2q^{93} - 8q^{95} + 7q^{97} - 3q^{99} + O(q^{100}) \)

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 2.00000 0 −4.00000 0 0 0 1.00000 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 688.2.a.b 1
3.b odd 2 1 6192.2.a.ba 1
4.b odd 2 1 43.2.a.a 1
8.b even 2 1 2752.2.a.b 1
8.d odd 2 1 2752.2.a.f 1
12.b even 2 1 387.2.a.e 1
20.d odd 2 1 1075.2.a.h 1
20.e even 4 2 1075.2.b.b 2
28.d even 2 1 2107.2.a.a 1
44.c even 2 1 5203.2.a.a 1
52.b odd 2 1 7267.2.a.a 1
60.h even 2 1 9675.2.a.b 1
172.d even 2 1 1849.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.a.a 1 4.b odd 2 1
387.2.a.e 1 12.b even 2 1
688.2.a.b 1 1.a even 1 1 trivial
1075.2.a.h 1 20.d odd 2 1
1075.2.b.b 2 20.e even 4 2
1849.2.a.d 1 172.d even 2 1
2107.2.a.a 1 28.d even 2 1
2752.2.a.b 1 8.b even 2 1
2752.2.a.f 1 8.d odd 2 1
5203.2.a.a 1 44.c even 2 1
6192.2.a.ba 1 3.b odd 2 1
7267.2.a.a 1 52.b odd 2 1
9675.2.a.b 1 60.h even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(43\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(688))\):

\( T_{3} - 2 \)
\( T_{5} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T + 3 T^{2} \)
$5$ \( 1 + 4 T + 5 T^{2} \)
$7$ \( 1 + 7 T^{2} \)
$11$ \( 1 + 3 T + 11 T^{2} \)
$13$ \( 1 + 5 T + 13 T^{2} \)
$17$ \( 1 + 3 T + 17 T^{2} \)
$19$ \( 1 - 2 T + 19 T^{2} \)
$23$ \( 1 - T + 23 T^{2} \)
$29$ \( 1 + 6 T + 29 T^{2} \)
$31$ \( 1 - T + 31 T^{2} \)
$37$ \( 1 + 37 T^{2} \)
$41$ \( 1 - 5 T + 41 T^{2} \)
$43$ \( 1 - T \)
$47$ \( 1 + 4 T + 47 T^{2} \)
$53$ \( 1 + 5 T + 53 T^{2} \)
$59$ \( 1 - 12 T + 59 T^{2} \)
$61$ \( 1 - 2 T + 61 T^{2} \)
$67$ \( 1 - 3 T + 67 T^{2} \)
$71$ \( 1 + 2 T + 71 T^{2} \)
$73$ \( 1 - 2 T + 73 T^{2} \)
$79$ \( 1 - 8 T + 79 T^{2} \)
$83$ \( 1 + 15 T + 83 T^{2} \)
$89$ \( 1 + 4 T + 89 T^{2} \)
$97$ \( 1 - 7 T + 97 T^{2} \)
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