Properties

Label 684.3.y.e
Level $684$
Weight $3$
Character orbit 684.y
Analytic conductor $18.638$
Analytic rank $0$
Dimension $2$
CM no
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.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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 6 - 6 \zeta_{6} ) q^{5} -5 q^{7} +O(q^{10})\) \( q + ( 6 - 6 \zeta_{6} ) q^{5} -5 q^{7} + ( -22 + 11 \zeta_{6} ) q^{13} + ( 6 - 6 \zeta_{6} ) q^{17} + ( -5 - 16 \zeta_{6} ) q^{19} -24 \zeta_{6} q^{23} -11 \zeta_{6} q^{25} + ( -36 + 18 \zeta_{6} ) q^{29} + ( -17 + 34 \zeta_{6} ) q^{31} + ( -30 + 30 \zeta_{6} ) q^{35} + ( -35 + 70 \zeta_{6} ) q^{37} + ( 24 + 24 \zeta_{6} ) q^{41} + ( 25 - 25 \zeta_{6} ) q^{43} -42 \zeta_{6} q^{47} -24 q^{49} + ( -72 + 36 \zeta_{6} ) q^{53} + ( -42 - 42 \zeta_{6} ) q^{59} -43 \zeta_{6} q^{61} + ( -66 + 132 \zeta_{6} ) q^{65} + ( 66 - 33 \zeta_{6} ) q^{67} + ( -36 - 36 \zeta_{6} ) q^{71} + ( -11 + 11 \zeta_{6} ) q^{73} + ( 1 + \zeta_{6} ) q^{79} -126 q^{83} -36 \zeta_{6} q^{85} + ( 12 - 6 \zeta_{6} ) q^{89} + ( 110 - 55 \zeta_{6} ) q^{91} + ( -126 + 30 \zeta_{6} ) q^{95} + ( -76 - 76 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} - 10 q^{7} + O(q^{10}) \) \( 2 q + 6 q^{5} - 10 q^{7} - 33 q^{13} + 6 q^{17} - 26 q^{19} - 24 q^{23} - 11 q^{25} - 54 q^{29} - 30 q^{35} + 72 q^{41} + 25 q^{43} - 42 q^{47} - 48 q^{49} - 108 q^{53} - 126 q^{59} - 43 q^{61} + 99 q^{67} - 108 q^{71} - 11 q^{73} + 3 q^{79} - 252 q^{83} - 36 q^{85} + 18 q^{89} + 165 q^{91} - 222 q^{95} - 228 q^{97} + O(q^{100}) \)

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 3.00000 + 5.19615i 0 −5.00000 0 0 0
217.1 0 0 0 3.00000 5.19615i 0 −5.00000 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.d odd 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.e 2
3.b odd 2 1 228.3.l.a 2
12.b even 2 1 912.3.be.a 2
19.d odd 6 1 inner 684.3.y.e 2
57.f even 6 1 228.3.l.a 2
228.n odd 6 1 912.3.be.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.l.a 2 3.b odd 2 1
228.3.l.a 2 57.f even 6 1
684.3.y.e 2 1.a even 1 1 trivial
684.3.y.e 2 19.d odd 6 1 inner
912.3.be.a 2 12.b even 2 1
912.3.be.a 2 228.n odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 6 T_{5} + 36 \)
\( T_{7} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 36 - 6 T + T^{2} \)
$7$ \( ( 5 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 363 + 33 T + T^{2} \)
$17$ \( 36 - 6 T + T^{2} \)
$19$ \( 361 + 26 T + T^{2} \)
$23$ \( 576 + 24 T + T^{2} \)
$29$ \( 972 + 54 T + T^{2} \)
$31$ \( 867 + T^{2} \)
$37$ \( 3675 + T^{2} \)
$41$ \( 1728 - 72 T + T^{2} \)
$43$ \( 625 - 25 T + T^{2} \)
$47$ \( 1764 + 42 T + T^{2} \)
$53$ \( 3888 + 108 T + T^{2} \)
$59$ \( 5292 + 126 T + T^{2} \)
$61$ \( 1849 + 43 T + T^{2} \)
$67$ \( 3267 - 99 T + T^{2} \)
$71$ \( 3888 + 108 T + T^{2} \)
$73$ \( 121 + 11 T + T^{2} \)
$79$ \( 3 - 3 T + T^{2} \)
$83$ \( ( 126 + T )^{2} \)
$89$ \( 108 - 18 T + T^{2} \)
$97$ \( 17328 + 228 T + T^{2} \)
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