Properties

Label 684.3.h.d
Level $684$
Weight $3$
Character orbit 684.h
Self dual yes
Analytic conductor $18.638$
Analytic rank $0$
Dimension $4$
CM discriminant -19
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{19})\)
Defining polynomial: \(x^{4} - 11 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + ( -3 - \beta_{3} ) q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + ( -3 - \beta_{3} ) q^{7} -\beta_{2} q^{11} + ( 2 \beta_{1} + \beta_{2} ) q^{17} + 19 q^{19} + ( \beta_{1} + 3 \beta_{2} ) q^{23} + ( 42 + 3 \beta_{3} ) q^{25} + ( -8 \beta_{1} - 7 \beta_{2} ) q^{35} + ( 43 + \beta_{3} ) q^{43} + ( -7 \beta_{1} + 4 \beta_{2} ) q^{47} + ( 88 + 5 \beta_{3} ) q^{49} + ( -7 - 7 \beta_{3} ) q^{55} + ( 49 - 5 \beta_{3} ) q^{61} + ( 7 - 11 \beta_{3} ) q^{73} + ( 14 \beta_{1} - 3 \beta_{2} ) q^{77} + ( 4 \beta_{1} + 12 \beta_{2} ) q^{83} + ( 141 + 13 \beta_{3} ) q^{85} + 19 \beta_{1} q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{7} + O(q^{10}) \) \( 4 q - 10 q^{7} + 76 q^{19} + 162 q^{25} + 170 q^{43} + 342 q^{49} - 14 q^{55} + 206 q^{61} + 50 q^{73} + 538 q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 11 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - 6 \nu \)
\(\beta_{2}\)\(=\)\( -\nu^{3} + 12 \nu \)
\(\beta_{3}\)\(=\)\( 3 \nu^{2} - 17 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 17\)\()/3\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 2 \beta_{1}\)

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
−3.04547
1.31342
−1.31342
3.04547
0 0 0 −9.97368 0 −13.8248 0 0 0
37.2 0 0 0 −5.61478 0 8.82475 0 0 0
37.3 0 0 0 5.61478 0 8.82475 0 0 0
37.4 0 0 0 9.97368 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 \(\Q(\sqrt{-19}) \)
3.b odd 2 1 inner
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.h.d 4
3.b odd 2 1 inner 684.3.h.d 4
4.b odd 2 1 2736.3.o.k 4
12.b even 2 1 2736.3.o.k 4
19.b odd 2 1 CM 684.3.h.d 4
57.d even 2 1 inner 684.3.h.d 4
76.d even 2 1 2736.3.o.k 4
228.b odd 2 1 2736.3.o.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.h.d 4 1.a even 1 1 trivial
684.3.h.d 4 3.b odd 2 1 inner
684.3.h.d 4 19.b odd 2 1 CM
684.3.h.d 4 57.d even 2 1 inner
2736.3.o.k 4 4.b odd 2 1
2736.3.o.k 4 12.b even 2 1
2736.3.o.k 4 76.d even 2 1
2736.3.o.k 4 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 131 T_{5}^{2} + 3136 \) acting on \(S_{3}^{\mathrm{new}}(684, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 3136 - 131 T^{2} + T^{4} \)
$7$ \( ( -122 + 5 T + T^{2} )^{2} \)
$11$ \( 12544 - 251 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 4096 - 803 T^{2} + T^{4} \)
$19$ \( ( -19 + T )^{4} \)
$23$ \( ( -1216 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 1678 - 85 T + T^{2} )^{2} \)
$47$ \( 11669056 - 10043 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -554 - 103 T + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -15362 - 25 T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( ( -19456 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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