Properties

Label 684.3.h.a
Level $684$
Weight $3$
Character orbit 684.h
Self dual yes
Analytic conductor $18.638$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -4 - \beta ) q^{5} + ( 4 - 3 \beta ) q^{7} +O(q^{10})\) \( q + ( -4 - \beta ) q^{5} + ( 4 - 3 \beta ) q^{7} + ( 4 - 5 \beta ) q^{11} + ( 4 + 7 \beta ) q^{17} -19 q^{19} + 30 q^{23} + ( 5 + 9 \beta ) q^{25} + ( 26 + 11 \beta ) q^{35} + ( 44 - 3 \beta ) q^{43} + ( 44 - 13 \beta ) q^{47} + ( 93 - 15 \beta ) q^{49} + ( 54 + 21 \beta ) q^{55} + ( -44 - 15 \beta ) q^{61} + ( -4 + 33 \beta ) q^{73} + ( 226 - 17 \beta ) q^{77} -90 q^{83} + ( -114 - 39 \beta ) q^{85} + ( 76 + 19 \beta ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{5} + 5 q^{7} + O(q^{10}) \) \( 2 q - 9 q^{5} + 5 q^{7} + 3 q^{11} + 15 q^{17} - 38 q^{19} + 60 q^{23} + 19 q^{25} + 63 q^{35} + 85 q^{43} + 75 q^{47} + 171 q^{49} + 129 q^{55} - 103 q^{61} + 25 q^{73} + 435 q^{77} - 180 q^{83} - 267 q^{85} + 171 q^{95} + O(q^{100}) \)

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
4.27492
−3.27492
0 0 0 −8.27492 0 −8.82475 0 0 0
37.2 0 0 0 −0.725083 0 13.8248 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 684.3.h.a 2
3.b odd 2 1 76.3.c.b 2
4.b odd 2 1 2736.3.o.c 2
12.b even 2 1 304.3.e.e 2
15.d odd 2 1 1900.3.e.a 2
15.e even 4 2 1900.3.g.a 4
19.b odd 2 1 CM 684.3.h.a 2
24.f even 2 1 1216.3.e.e 2
24.h odd 2 1 1216.3.e.f 2
57.d even 2 1 76.3.c.b 2
76.d even 2 1 2736.3.o.c 2
228.b odd 2 1 304.3.e.e 2
285.b even 2 1 1900.3.e.a 2
285.j odd 4 2 1900.3.g.a 4
456.l odd 2 1 1216.3.e.e 2
456.p even 2 1 1216.3.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.c.b 2 3.b odd 2 1
76.3.c.b 2 57.d even 2 1
304.3.e.e 2 12.b even 2 1
304.3.e.e 2 228.b odd 2 1
684.3.h.a 2 1.a even 1 1 trivial
684.3.h.a 2 19.b odd 2 1 CM
1216.3.e.e 2 24.f even 2 1
1216.3.e.e 2 456.l odd 2 1
1216.3.e.f 2 24.h odd 2 1
1216.3.e.f 2 456.p even 2 1
1900.3.e.a 2 15.d odd 2 1
1900.3.e.a 2 285.b even 2 1
1900.3.g.a 4 15.e even 4 2
1900.3.g.a 4 285.j odd 4 2
2736.3.o.c 2 4.b odd 2 1
2736.3.o.c 2 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 9 T_{5} + 6 \) acting on \(S_{3}^{\mathrm{new}}(684, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 6 + 9 T + T^{2} \)
$7$ \( -122 - 5 T + T^{2} \)
$11$ \( -354 - 3 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( -642 - 15 T + T^{2} \)
$19$ \( ( 19 + T )^{2} \)
$23$ \( ( -30 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( 1678 - 85 T + T^{2} \)
$47$ \( -1002 - 75 T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( -554 + 103 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( -15362 - 25 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( ( 90 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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