Properties

Label 684.3.y.d
Level $684$
Weight $3$
Character orbit 684.y
Analytic conductor $18.638$
Analytic rank $0$
Dimension $2$
CM discriminant -3
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.y (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 13 q^{7} +O(q^{10})\) \( q + 13 q^{7} + ( 14 - 7 \zeta_{6} ) q^{13} + ( -5 - 16 \zeta_{6} ) q^{19} + 25 \zeta_{6} q^{25} + ( -11 + 22 \zeta_{6} ) q^{31} + ( 7 - 14 \zeta_{6} ) q^{37} + ( 61 - 61 \zeta_{6} ) q^{43} + 120 q^{49} + 47 \zeta_{6} q^{61} + ( 90 - 45 \zeta_{6} ) q^{67} + ( 97 - 97 \zeta_{6} ) q^{73} + ( 91 + 91 \zeta_{6} ) q^{79} + ( 182 - 91 \zeta_{6} ) q^{91} + ( -112 - 112 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 26q^{7} + O(q^{10}) \) \( 2q + 26q^{7} + 21q^{13} - 26q^{19} + 25q^{25} + 61q^{43} + 240q^{49} + 47q^{61} + 135q^{67} + 97q^{73} + 273q^{79} + 273q^{91} - 336q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(\zeta_{6}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 13.0000 0 0 0
217.1 0 0 0 0 0 13.0000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
19.d odd 6 1 inner
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.y.d 2
3.b odd 2 1 CM 684.3.y.d 2
19.d odd 6 1 inner 684.3.y.d 2
57.f even 6 1 inner 684.3.y.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.y.d 2 1.a even 1 1 trivial
684.3.y.d 2 3.b odd 2 1 CM
684.3.y.d 2 19.d odd 6 1 inner
684.3.y.d 2 57.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(684, [\chi])\):

\( T_{5} \)
\( T_{7} - 13 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -13 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 147 - 21 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 361 + 26 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 363 + T^{2} \)
$37$ \( 147 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 3721 - 61 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 2209 - 47 T + T^{2} \)
$67$ \( 6075 - 135 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 9409 - 97 T + T^{2} \)
$79$ \( 24843 - 273 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 37632 + 336 T + T^{2} \)
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