Properties

Label 684.2.k.b
Level $684$
Weight $2$
Character orbit 684.k
Analytic conductor $5.462$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{5} + 4 q^{11} + \zeta_{6} q^{13} + ( - 3 \zeta_{6} + 3) q^{17} + (2 \zeta_{6} - 5) q^{19} + 5 \zeta_{6} q^{23} + 4 \zeta_{6} q^{25} + 7 \zeta_{6} q^{29} + 4 q^{31} + 10 q^{37} + (5 \zeta_{6} - 5) q^{41} + ( - 5 \zeta_{6} + 5) q^{43} - 7 \zeta_{6} q^{47} - 7 q^{49} + 11 \zeta_{6} q^{53} + (4 \zeta_{6} - 4) q^{55} + ( - 3 \zeta_{6} + 3) q^{59} - 11 \zeta_{6} q^{61} - q^{65} + 3 \zeta_{6} q^{67} + ( - 11 \zeta_{6} + 11) q^{71} + (15 \zeta_{6} - 15) q^{73} + ( - 13 \zeta_{6} + 13) q^{79} + 3 \zeta_{6} q^{85} + 3 \zeta_{6} q^{89} + ( - 5 \zeta_{6} + 3) q^{95} + ( - 5 \zeta_{6} + 5) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + 8 q^{11} + q^{13} + 3 q^{17} - 8 q^{19} + 5 q^{23} + 4 q^{25} + 7 q^{29} + 8 q^{31} + 20 q^{37} - 5 q^{41} + 5 q^{43} - 7 q^{47} - 14 q^{49} + 11 q^{53} - 4 q^{55} + 3 q^{59} - 11 q^{61} - 2 q^{65} + 3 q^{67} + 11 q^{71} - 15 q^{73} + 13 q^{79} + 3 q^{85} + 3 q^{89} + q^{95} + 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
505.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −0.500000 + 0.866025i 0 0 0 0 0
577.1 0 0 0 −0.500000 0.866025i 0 0 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.k.b 2
3.b odd 2 1 76.2.e.a 2
4.b odd 2 1 2736.2.s.g 2
12.b even 2 1 304.2.i.a 2
15.d odd 2 1 1900.2.i.a 2
15.e even 4 2 1900.2.s.a 4
19.c even 3 1 inner 684.2.k.b 2
24.f even 2 1 1216.2.i.g 2
24.h odd 2 1 1216.2.i.c 2
57.d even 2 1 1444.2.e.b 2
57.f even 6 1 1444.2.a.c 1
57.f even 6 1 1444.2.e.b 2
57.h odd 6 1 76.2.e.a 2
57.h odd 6 1 1444.2.a.b 1
76.g odd 6 1 2736.2.s.g 2
228.m even 6 1 304.2.i.a 2
228.m even 6 1 5776.2.a.k 1
228.n odd 6 1 5776.2.a.f 1
285.n odd 6 1 1900.2.i.a 2
285.v even 12 2 1900.2.s.a 4
456.u even 6 1 1216.2.i.g 2
456.x odd 6 1 1216.2.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.e.a 2 3.b odd 2 1
76.2.e.a 2 57.h odd 6 1
304.2.i.a 2 12.b even 2 1
304.2.i.a 2 228.m even 6 1
684.2.k.b 2 1.a even 1 1 trivial
684.2.k.b 2 19.c even 3 1 inner
1216.2.i.c 2 24.h odd 2 1
1216.2.i.c 2 456.x odd 6 1
1216.2.i.g 2 24.f even 2 1
1216.2.i.g 2 456.u even 6 1
1444.2.a.b 1 57.h odd 6 1
1444.2.a.c 1 57.f even 6 1
1444.2.e.b 2 57.d even 2 1
1444.2.e.b 2 57.f even 6 1
1900.2.i.a 2 15.d odd 2 1
1900.2.i.a 2 285.n odd 6 1
1900.2.s.a 4 15.e even 4 2
1900.2.s.a 4 285.v even 12 2
2736.2.s.g 2 4.b odd 2 1
2736.2.s.g 2 76.g odd 6 1
5776.2.a.f 1 228.n odd 6 1
5776.2.a.k 1 228.m even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(684, [\chi])\):

\( T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$29$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( (T - 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$47$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$53$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$59$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$67$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$71$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$73$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$79$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
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