Properties

Label 684.3.y.a
Level $684$
Weight $3$
Character orbit 684.y
Analytic conductor $18.638$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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.

\(f(q)\) \(=\) \( q + ( -2 + 2 \zeta_{6} ) q^{5} - q^{7} +O(q^{10})\) \( q + ( -2 + 2 \zeta_{6} ) q^{5} - q^{7} -16 q^{11} + ( 18 - 9 \zeta_{6} ) q^{13} + ( 22 - 22 \zeta_{6} ) q^{17} + 19 q^{19} + 40 \zeta_{6} q^{23} + 21 \zeta_{6} q^{25} + ( -20 + 10 \zeta_{6} ) q^{29} + ( -29 + 58 \zeta_{6} ) q^{31} + ( 2 - 2 \zeta_{6} ) q^{35} + ( 9 - 18 \zeta_{6} ) q^{37} + ( 8 + 8 \zeta_{6} ) q^{41} + ( 49 - 49 \zeta_{6} ) q^{43} + 46 \zeta_{6} q^{47} -48 q^{49} + ( 56 - 28 \zeta_{6} ) q^{53} + ( 32 - 32 \zeta_{6} ) q^{55} + ( 38 + 38 \zeta_{6} ) q^{59} + 97 \zeta_{6} q^{61} + ( -18 + 36 \zeta_{6} ) q^{65} + ( -30 + 15 \zeta_{6} ) q^{67} + ( 28 + 28 \zeta_{6} ) q^{71} + ( -35 + 35 \zeta_{6} ) q^{73} + 16 q^{77} + ( -51 - 51 \zeta_{6} ) q^{79} + 146 q^{83} + 44 \zeta_{6} q^{85} + ( 44 - 22 \zeta_{6} ) q^{89} + ( -18 + 9 \zeta_{6} ) q^{91} + ( -38 + 38 \zeta_{6} ) q^{95} + ( 36 + 36 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7} + O(q^{10}) \) \( 2 q - 2 q^{5} - 2 q^{7} - 32 q^{11} + 27 q^{13} + 22 q^{17} + 38 q^{19} + 40 q^{23} + 21 q^{25} - 30 q^{29} + 2 q^{35} + 24 q^{41} + 49 q^{43} + 46 q^{47} - 96 q^{49} + 84 q^{53} + 32 q^{55} + 114 q^{59} + 97 q^{61} - 45 q^{67} + 84 q^{71} - 35 q^{73} + 32 q^{77} - 153 q^{79} + 292 q^{83} + 44 q^{85} + 66 q^{89} - 27 q^{91} - 38 q^{95} + 108 q^{97} + O(q^{100}) \)

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 −1.00000 1.73205i 0 −1.00000 0 0 0
217.1 0 0 0 −1.00000 + 1.73205i 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.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.a 2
3.b odd 2 1 228.3.l.b 2
12.b even 2 1 912.3.be.c 2
19.d odd 6 1 inner 684.3.y.a 2
57.f even 6 1 228.3.l.b 2
228.n odd 6 1 912.3.be.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.l.b 2 3.b odd 2 1
228.3.l.b 2 57.f even 6 1
684.3.y.a 2 1.a even 1 1 trivial
684.3.y.a 2 19.d odd 6 1 inner
912.3.be.c 2 12.b even 2 1
912.3.be.c 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 \(S_{3}^{\mathrm{new}}(684, [\chi])\):

\( T_{5}^{2} + 2 T_{5} + 4 \)
\( T_{7} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 + 2 T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 16 + T )^{2} \)
$13$ \( 243 - 27 T + T^{2} \)
$17$ \( 484 - 22 T + T^{2} \)
$19$ \( ( -19 + T )^{2} \)
$23$ \( 1600 - 40 T + T^{2} \)
$29$ \( 300 + 30 T + T^{2} \)
$31$ \( 2523 + T^{2} \)
$37$ \( 243 + T^{2} \)
$41$ \( 192 - 24 T + T^{2} \)
$43$ \( 2401 - 49 T + T^{2} \)
$47$ \( 2116 - 46 T + T^{2} \)
$53$ \( 2352 - 84 T + T^{2} \)
$59$ \( 4332 - 114 T + T^{2} \)
$61$ \( 9409 - 97 T + T^{2} \)
$67$ \( 675 + 45 T + T^{2} \)
$71$ \( 2352 - 84 T + T^{2} \)
$73$ \( 1225 + 35 T + T^{2} \)
$79$ \( 7803 + 153 T + T^{2} \)
$83$ \( ( -146 + T )^{2} \)
$89$ \( 1452 - 66 T + T^{2} \)
$97$ \( 3888 - 108 T + T^{2} \)
show more
show less