Properties

Label 684.1.h.a
Level $684$
Weight $1$
Character orbit 684.h
Self dual yes
Analytic conductor $0.341$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -19
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 684.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.341360468641\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.76.1
Artin image $D_6$
Artin field Galois closure of 6.0.155952.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} - q^{7} + O(q^{10}) \) \( q + q^{5} - q^{7} + q^{11} + q^{17} + q^{19} - 2q^{23} - q^{35} - q^{43} + q^{47} + q^{55} - q^{61} - q^{73} - q^{77} - 2q^{83} + q^{85} + q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0
0 0 0 1.00000 0 −1.00000 0 0 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.1.h.a 1
3.b odd 2 1 76.1.c.a 1
4.b odd 2 1 2736.1.o.b 1
12.b even 2 1 304.1.e.a 1
15.d odd 2 1 1900.1.e.a 1
15.e even 4 2 1900.1.g.a 2
19.b odd 2 1 CM 684.1.h.a 1
21.c even 2 1 3724.1.e.c 1
21.g even 6 2 3724.1.bc.b 2
21.h odd 6 2 3724.1.bc.c 2
24.f even 2 1 1216.1.e.b 1
24.h odd 2 1 1216.1.e.a 1
57.d even 2 1 76.1.c.a 1
57.f even 6 2 1444.1.h.a 2
57.h odd 6 2 1444.1.h.a 2
57.j even 18 6 1444.1.j.a 6
57.l odd 18 6 1444.1.j.a 6
76.d even 2 1 2736.1.o.b 1
228.b odd 2 1 304.1.e.a 1
285.b even 2 1 1900.1.e.a 1
285.j odd 4 2 1900.1.g.a 2
399.h odd 2 1 3724.1.e.c 1
399.s odd 6 2 3724.1.bc.b 2
399.w even 6 2 3724.1.bc.c 2
456.l odd 2 1 1216.1.e.b 1
456.p even 2 1 1216.1.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.1.c.a 1 3.b odd 2 1
76.1.c.a 1 57.d even 2 1
304.1.e.a 1 12.b even 2 1
304.1.e.a 1 228.b odd 2 1
684.1.h.a 1 1.a even 1 1 trivial
684.1.h.a 1 19.b odd 2 1 CM
1216.1.e.a 1 24.h odd 2 1
1216.1.e.a 1 456.p even 2 1
1216.1.e.b 1 24.f even 2 1
1216.1.e.b 1 456.l odd 2 1
1444.1.h.a 2 57.f even 6 2
1444.1.h.a 2 57.h odd 6 2
1444.1.j.a 6 57.j even 18 6
1444.1.j.a 6 57.l odd 18 6
1900.1.e.a 1 15.d odd 2 1
1900.1.e.a 1 285.b even 2 1
1900.1.g.a 2 15.e even 4 2
1900.1.g.a 2 285.j odd 4 2
2736.1.o.b 1 4.b odd 2 1
2736.1.o.b 1 76.d even 2 1
3724.1.e.c 1 21.c even 2 1
3724.1.e.c 1 399.h odd 2 1
3724.1.bc.b 2 21.g even 6 2
3724.1.bc.b 2 399.s odd 6 2
3724.1.bc.c 2 21.h odd 6 2
3724.1.bc.c 2 399.w even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 1 \) acting on \(S_{1}^{\mathrm{new}}(684, [\chi])\).

Hecke characteristic polynomials

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