Properties

Label 688.3.b.a
Level 688
Weight 3
Character orbit 688.b
Self dual yes
Analytic conductor 18.747
Analytic rank 0
Dimension 1
CM discriminant -43
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 688.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(18.7466421880\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 9q^{9} + O(q^{10}) \) \( q + 9q^{9} + 21q^{11} - 17q^{13} - 9q^{17} - 3q^{23} + 25q^{25} - 19q^{31} + 39q^{41} + 43q^{43} + 78q^{47} + 49q^{49} + 63q^{53} + 54q^{59} - 91q^{67} + 14q^{79} + 81q^{81} - 123q^{83} - 193q^{97} + 189q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).

\(n\) \(431\) \(433\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0
0 0 0 0 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.b odd 2 1 CM by \(\Q(\sqrt{-43}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.3.b.a 1
4.b odd 2 1 43.3.b.a 1
12.b even 2 1 387.3.b.a 1
43.b odd 2 1 CM 688.3.b.a 1
172.d even 2 1 43.3.b.a 1
516.h odd 2 1 387.3.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.3.b.a 1 4.b odd 2 1
43.3.b.a 1 172.d even 2 1
387.3.b.a 1 12.b even 2 1
387.3.b.a 1 516.h odd 2 1
688.3.b.a 1 1.a even 1 1 trivial
688.3.b.a 1 43.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(688, [\chi])\):

\( T_{3} \)
\( T_{11} - 21 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T )( 1 + 3 T ) \)
$5$ \( ( 1 - 5 T )( 1 + 5 T ) \)
$7$ \( ( 1 - 7 T )( 1 + 7 T ) \)
$11$ \( 1 - 21 T + 121 T^{2} \)
$13$ \( 1 + 17 T + 169 T^{2} \)
$17$ \( 1 + 9 T + 289 T^{2} \)
$19$ \( ( 1 - 19 T )( 1 + 19 T ) \)
$23$ \( 1 + 3 T + 529 T^{2} \)
$29$ \( ( 1 - 29 T )( 1 + 29 T ) \)
$31$ \( 1 + 19 T + 961 T^{2} \)
$37$ \( ( 1 - 37 T )( 1 + 37 T ) \)
$41$ \( 1 - 39 T + 1681 T^{2} \)
$43$ \( 1 - 43 T \)
$47$ \( 1 - 78 T + 2209 T^{2} \)
$53$ \( 1 - 63 T + 2809 T^{2} \)
$59$ \( 1 - 54 T + 3481 T^{2} \)
$61$ \( ( 1 - 61 T )( 1 + 61 T ) \)
$67$ \( 1 + 91 T + 4489 T^{2} \)
$71$ \( ( 1 - 71 T )( 1 + 71 T ) \)
$73$ \( ( 1 - 73 T )( 1 + 73 T ) \)
$79$ \( 1 - 14 T + 6241 T^{2} \)
$83$ \( 1 + 123 T + 6889 T^{2} \)
$89$ \( ( 1 - 89 T )( 1 + 89 T ) \)
$97$ \( 1 + 193 T + 9409 T^{2} \)
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