Properties

Label 1764.2.k.k
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 84)
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.

\(f(q)\) \(=\) \( q + 4 \zeta_{6} q^{5} +O(q^{10})\) \( q + 4 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{11} -6 q^{13} + ( -4 + 4 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} + 2 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + 2 q^{29} -2 \zeta_{6} q^{37} -4 q^{43} + 12 \zeta_{6} q^{47} + ( -6 + 6 \zeta_{6} ) q^{53} + 8 q^{55} + ( -8 + 8 \zeta_{6} ) q^{59} -6 \zeta_{6} q^{61} -24 \zeta_{6} q^{65} + ( 8 - 8 \zeta_{6} ) q^{67} -14 q^{71} + ( 2 - 2 \zeta_{6} ) q^{73} -12 \zeta_{6} q^{79} + 4 q^{83} -16 q^{85} + ( -16 + 16 \zeta_{6} ) q^{95} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} + O(q^{10}) \) \( 2q + 4q^{5} + 2q^{11} - 12q^{13} - 4q^{17} + 4q^{19} + 2q^{23} - 11q^{25} + 4q^{29} - 2q^{37} - 8q^{43} + 12q^{47} - 6q^{53} + 16q^{55} - 8q^{59} - 6q^{61} - 24q^{65} + 8q^{67} - 28q^{71} + 2q^{73} - 12q^{79} + 8q^{83} - 32q^{85} - 16q^{95} - 4q^{97} + O(q^{100}) \)

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 2.00000 + 3.46410i 0 0 0 0 0
1549.1 0 0 0 2.00000 3.46410i 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.k 2
3.b odd 2 1 588.2.i.e 2
7.b odd 2 1 1764.2.k.a 2
7.c even 3 1 252.2.a.a 1
7.c even 3 1 inner 1764.2.k.k 2
7.d odd 6 1 1764.2.a.k 1
7.d odd 6 1 1764.2.k.a 2
12.b even 2 1 2352.2.q.b 2
21.c even 2 1 588.2.i.d 2
21.g even 6 1 588.2.a.d 1
21.g even 6 1 588.2.i.d 2
21.h odd 6 1 84.2.a.a 1
21.h odd 6 1 588.2.i.e 2
28.f even 6 1 7056.2.a.cd 1
28.g odd 6 1 1008.2.a.a 1
35.j even 6 1 6300.2.a.w 1
35.l odd 12 2 6300.2.k.g 2
56.k odd 6 1 4032.2.a.bn 1
56.p even 6 1 4032.2.a.bm 1
63.g even 3 1 2268.2.j.n 2
63.h even 3 1 2268.2.j.n 2
63.j odd 6 1 2268.2.j.a 2
63.n odd 6 1 2268.2.j.a 2
84.h odd 2 1 2352.2.q.z 2
84.j odd 6 1 2352.2.a.a 1
84.j odd 6 1 2352.2.q.z 2
84.n even 6 1 336.2.a.f 1
84.n even 6 1 2352.2.q.b 2
105.o odd 6 1 2100.2.a.r 1
105.x even 12 2 2100.2.k.i 2
168.s odd 6 1 1344.2.a.k 1
168.v even 6 1 1344.2.a.a 1
168.ba even 6 1 9408.2.a.bn 1
168.be odd 6 1 9408.2.a.df 1
336.bt odd 12 2 5376.2.c.q 2
336.bu even 12 2 5376.2.c.p 2
420.ba even 6 1 8400.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.a.a 1 21.h odd 6 1
252.2.a.a 1 7.c even 3 1
336.2.a.f 1 84.n even 6 1
588.2.a.d 1 21.g even 6 1
588.2.i.d 2 21.c even 2 1
588.2.i.d 2 21.g even 6 1
588.2.i.e 2 3.b odd 2 1
588.2.i.e 2 21.h odd 6 1
1008.2.a.a 1 28.g odd 6 1
1344.2.a.a 1 168.v even 6 1
1344.2.a.k 1 168.s odd 6 1
1764.2.a.k 1 7.d odd 6 1
1764.2.k.a 2 7.b odd 2 1
1764.2.k.a 2 7.d odd 6 1
1764.2.k.k 2 1.a even 1 1 trivial
1764.2.k.k 2 7.c even 3 1 inner
2100.2.a.r 1 105.o odd 6 1
2100.2.k.i 2 105.x even 12 2
2268.2.j.a 2 63.j odd 6 1
2268.2.j.a 2 63.n odd 6 1
2268.2.j.n 2 63.g even 3 1
2268.2.j.n 2 63.h even 3 1
2352.2.a.a 1 84.j odd 6 1
2352.2.q.b 2 12.b even 2 1
2352.2.q.b 2 84.n even 6 1
2352.2.q.z 2 84.h odd 2 1
2352.2.q.z 2 84.j odd 6 1
4032.2.a.bm 1 56.p even 6 1
4032.2.a.bn 1 56.k odd 6 1
5376.2.c.p 2 336.bu even 12 2
5376.2.c.q 2 336.bt odd 12 2
6300.2.a.w 1 35.j even 6 1
6300.2.k.g 2 35.l odd 12 2
7056.2.a.cd 1 28.f even 6 1
8400.2.a.e 1 420.ba even 6 1
9408.2.a.bn 1 168.ba even 6 1
9408.2.a.df 1 168.be odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 4 T_{5} + 16 \)
\( T_{11}^{2} - 2 T_{11} + 4 \)
\( T_{13} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 6 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \)
$19$ \( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 2 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 31 T^{2} + 961 T^{4} \)
$37$ \( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 8 T - 3 T^{2} - 536 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 14 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 4 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 89 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 + 2 T + 97 T^{2} )^{2} \)
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