Properties

Label 1764.2.k.i
Level $1764$
Weight $2$
Character orbit 1764.k
Analytic conductor $14.086$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.k (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 588)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{5} +O(q^{10})\) \( q + 2 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{11} -4 q^{13} + ( 6 - 6 \zeta_{6} ) q^{17} -8 \zeta_{6} q^{19} -6 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + 10 q^{29} + ( -4 + 4 \zeta_{6} ) q^{31} -6 \zeta_{6} q^{37} + 6 q^{41} + 4 q^{43} + 8 \zeta_{6} q^{47} + ( 2 - 2 \zeta_{6} ) q^{53} + 4 q^{55} + ( -4 + 4 \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} -8 \zeta_{6} q^{65} + ( 8 - 8 \zeta_{6} ) q^{67} + 10 q^{71} + ( -4 + 4 \zeta_{6} ) q^{73} -4 \zeta_{6} q^{79} -12 q^{83} + 12 q^{85} -14 \zeta_{6} q^{89} + ( 16 - 16 \zeta_{6} ) q^{95} + 4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + O(q^{10}) \) \( 2q + 2q^{5} + 2q^{11} - 8q^{13} + 6q^{17} - 8q^{19} - 6q^{23} + q^{25} + 20q^{29} - 4q^{31} - 6q^{37} + 12q^{41} + 8q^{43} + 8q^{47} + 2q^{53} + 8q^{55} - 4q^{59} + 8q^{61} - 8q^{65} + 8q^{67} + 20q^{71} - 4q^{73} - 4q^{79} - 24q^{83} + 24q^{85} - 14q^{89} + 16q^{95} + 8q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 1.00000 + 1.73205i 0 0 0 0 0
1549.1 0 0 0 1.00000 1.73205i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.k.i 2
3.b odd 2 1 588.2.i.a 2
7.b odd 2 1 1764.2.k.c 2
7.c even 3 1 1764.2.a.b 1
7.c even 3 1 inner 1764.2.k.i 2
7.d odd 6 1 1764.2.a.i 1
7.d odd 6 1 1764.2.k.c 2
12.b even 2 1 2352.2.q.p 2
21.c even 2 1 588.2.i.g 2
21.g even 6 1 588.2.a.b 1
21.g even 6 1 588.2.i.g 2
21.h odd 6 1 588.2.a.e yes 1
21.h odd 6 1 588.2.i.a 2
28.f even 6 1 7056.2.a.bu 1
28.g odd 6 1 7056.2.a.n 1
84.h odd 2 1 2352.2.q.k 2
84.j odd 6 1 2352.2.a.p 1
84.j odd 6 1 2352.2.q.k 2
84.n even 6 1 2352.2.a.j 1
84.n even 6 1 2352.2.q.p 2
168.s odd 6 1 9408.2.a.l 1
168.v even 6 1 9408.2.a.ca 1
168.ba even 6 1 9408.2.a.cu 1
168.be odd 6 1 9408.2.a.bf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.a.b 1 21.g even 6 1
588.2.a.e yes 1 21.h odd 6 1
588.2.i.a 2 3.b odd 2 1
588.2.i.a 2 21.h odd 6 1
588.2.i.g 2 21.c even 2 1
588.2.i.g 2 21.g even 6 1
1764.2.a.b 1 7.c even 3 1
1764.2.a.i 1 7.d odd 6 1
1764.2.k.c 2 7.b odd 2 1
1764.2.k.c 2 7.d odd 6 1
1764.2.k.i 2 1.a even 1 1 trivial
1764.2.k.i 2 7.c even 3 1 inner
2352.2.a.j 1 84.n even 6 1
2352.2.a.p 1 84.j odd 6 1
2352.2.q.k 2 84.h odd 2 1
2352.2.q.k 2 84.j odd 6 1
2352.2.q.p 2 12.b even 2 1
2352.2.q.p 2 84.n even 6 1
7056.2.a.n 1 28.g odd 6 1
7056.2.a.bu 1 28.f even 6 1
9408.2.a.l 1 168.s odd 6 1
9408.2.a.bf 1 168.be odd 6 1
9408.2.a.ca 1 168.v even 6 1
9408.2.a.cu 1 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{11}^{2} - 2 T_{11} + 4 \)
\( T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 4 - 2 T + T^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( 36 - 6 T + T^{2} \)
$19$ \( 64 + 8 T + T^{2} \)
$23$ \( 36 + 6 T + T^{2} \)
$29$ \( ( -10 + T )^{2} \)
$31$ \( 16 + 4 T + T^{2} \)
$37$ \( 36 + 6 T + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( 64 - 8 T + T^{2} \)
$53$ \( 4 - 2 T + T^{2} \)
$59$ \( 16 + 4 T + T^{2} \)
$61$ \( 64 - 8 T + T^{2} \)
$67$ \( 64 - 8 T + T^{2} \)
$71$ \( ( -10 + T )^{2} \)
$73$ \( 16 + 4 T + T^{2} \)
$79$ \( 16 + 4 T + T^{2} \)
$83$ \( ( 12 + T )^{2} \)
$89$ \( 196 + 14 T + T^{2} \)
$97$ \( ( -4 + T )^{2} \)
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