Properties

Label 1764.2.l.c
Level $1764$
Weight $2$
Character orbit 1764.l
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.l (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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
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.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{6} ) q^{3} + 3 q^{5} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{6} ) q^{3} + 3 q^{5} + 3 \zeta_{6} q^{9} + 3 q^{11} + ( 1 - \zeta_{6} ) q^{13} + ( 3 + 3 \zeta_{6} ) q^{15} + ( -6 + 6 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} -3 q^{23} + 4 q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} -3 \zeta_{6} q^{29} -5 \zeta_{6} q^{31} + ( 3 + 3 \zeta_{6} ) q^{33} -2 \zeta_{6} q^{37} + ( 2 - \zeta_{6} ) q^{39} + ( -3 + 3 \zeta_{6} ) q^{41} + \zeta_{6} q^{43} + 9 \zeta_{6} q^{45} + ( 9 - 9 \zeta_{6} ) q^{47} + ( -12 + 6 \zeta_{6} ) q^{51} + ( 6 - 6 \zeta_{6} ) q^{53} + 9 q^{55} + ( -4 + 8 \zeta_{6} ) q^{57} + 3 \zeta_{6} q^{59} + ( 13 - 13 \zeta_{6} ) q^{61} + ( 3 - 3 \zeta_{6} ) q^{65} + 7 \zeta_{6} q^{67} + ( -3 - 3 \zeta_{6} ) q^{69} -12 q^{71} + ( 10 - 10 \zeta_{6} ) q^{73} + ( 4 + 4 \zeta_{6} ) q^{75} + ( -11 + 11 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + 9 \zeta_{6} q^{83} + ( -18 + 18 \zeta_{6} ) q^{85} + ( 3 - 6 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} + ( 5 - 10 \zeta_{6} ) q^{93} + 12 \zeta_{6} q^{95} -11 \zeta_{6} q^{97} + 9 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} + 6q^{5} + 3q^{9} + O(q^{10}) \) \( 2q + 3q^{3} + 6q^{5} + 3q^{9} + 6q^{11} + q^{13} + 9q^{15} - 6q^{17} + 4q^{19} - 6q^{23} + 8q^{25} - 3q^{29} - 5q^{31} + 9q^{33} - 2q^{37} + 3q^{39} - 3q^{41} + q^{43} + 9q^{45} + 9q^{47} - 18q^{51} + 6q^{53} + 18q^{55} + 3q^{59} + 13q^{61} + 3q^{65} + 7q^{67} - 9q^{69} - 24q^{71} + 10q^{73} + 12q^{75} - 11q^{79} - 9q^{81} + 9q^{83} - 18q^{85} - 6q^{89} + 12q^{95} - 11q^{97} + 9q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.50000 + 0.866025i 0 3.00000 0 0 0 1.50000 + 2.59808i 0
961.1 0 1.50000 0.866025i 0 3.00000 0 0 0 1.50000 2.59808i 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
63.g 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.l.c 2
3.b odd 2 1 5292.2.l.a 2
7.b odd 2 1 1764.2.l.a 2
7.c even 3 1 36.2.e.a 2
7.c even 3 1 1764.2.i.a 2
7.d odd 6 1 1764.2.i.c 2
7.d odd 6 1 1764.2.j.b 2
9.c even 3 1 1764.2.i.a 2
9.d odd 6 1 5292.2.i.c 2
21.c even 2 1 5292.2.l.c 2
21.g even 6 1 5292.2.i.a 2
21.g even 6 1 5292.2.j.a 2
21.h odd 6 1 108.2.e.a 2
21.h odd 6 1 5292.2.i.c 2
28.g odd 6 1 144.2.i.a 2
35.j even 6 1 900.2.i.b 2
35.l odd 12 2 900.2.s.b 4
56.k odd 6 1 576.2.i.e 2
56.p even 6 1 576.2.i.f 2
63.g even 3 1 324.2.a.c 1
63.g even 3 1 inner 1764.2.l.c 2
63.h even 3 1 36.2.e.a 2
63.i even 6 1 5292.2.j.a 2
63.j odd 6 1 108.2.e.a 2
63.k odd 6 1 1764.2.l.a 2
63.l odd 6 1 1764.2.i.c 2
63.n odd 6 1 324.2.a.a 1
63.n odd 6 1 5292.2.l.a 2
63.o even 6 1 5292.2.i.a 2
63.s even 6 1 5292.2.l.c 2
63.t odd 6 1 1764.2.j.b 2
84.n even 6 1 432.2.i.c 2
105.o odd 6 1 2700.2.i.b 2
105.x even 12 2 2700.2.s.b 4
168.s odd 6 1 1728.2.i.d 2
168.v even 6 1 1728.2.i.c 2
252.o even 6 1 1296.2.a.b 1
252.u odd 6 1 144.2.i.a 2
252.bb even 6 1 432.2.i.c 2
252.bl odd 6 1 1296.2.a.k 1
315.r even 6 1 900.2.i.b 2
315.v odd 6 1 8100.2.a.g 1
315.bo even 6 1 8100.2.a.j 1
315.br odd 6 1 2700.2.i.b 2
315.bt odd 12 2 900.2.s.b 4
315.bv even 12 2 2700.2.s.b 4
315.bx even 12 2 8100.2.d.c 2
315.ch odd 12 2 8100.2.d.h 2
504.w even 6 1 5184.2.a.e 1
504.ba odd 6 1 5184.2.a.f 1
504.bi odd 6 1 1728.2.i.d 2
504.bt even 6 1 1728.2.i.c 2
504.ce odd 6 1 576.2.i.e 2
504.cq even 6 1 576.2.i.f 2
504.cy even 6 1 5184.2.a.bb 1
504.db odd 6 1 5184.2.a.ba 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 7.c even 3 1
36.2.e.a 2 63.h even 3 1
108.2.e.a 2 21.h odd 6 1
108.2.e.a 2 63.j odd 6 1
144.2.i.a 2 28.g odd 6 1
144.2.i.a 2 252.u odd 6 1
324.2.a.a 1 63.n odd 6 1
324.2.a.c 1 63.g even 3 1
432.2.i.c 2 84.n even 6 1
432.2.i.c 2 252.bb even 6 1
576.2.i.e 2 56.k odd 6 1
576.2.i.e 2 504.ce odd 6 1
576.2.i.f 2 56.p even 6 1
576.2.i.f 2 504.cq even 6 1
900.2.i.b 2 35.j even 6 1
900.2.i.b 2 315.r even 6 1
900.2.s.b 4 35.l odd 12 2
900.2.s.b 4 315.bt odd 12 2
1296.2.a.b 1 252.o even 6 1
1296.2.a.k 1 252.bl odd 6 1
1728.2.i.c 2 168.v even 6 1
1728.2.i.c 2 504.bt even 6 1
1728.2.i.d 2 168.s odd 6 1
1728.2.i.d 2 504.bi odd 6 1
1764.2.i.a 2 7.c even 3 1
1764.2.i.a 2 9.c even 3 1
1764.2.i.c 2 7.d odd 6 1
1764.2.i.c 2 63.l odd 6 1
1764.2.j.b 2 7.d odd 6 1
1764.2.j.b 2 63.t odd 6 1
1764.2.l.a 2 7.b odd 2 1
1764.2.l.a 2 63.k odd 6 1
1764.2.l.c 2 1.a even 1 1 trivial
1764.2.l.c 2 63.g even 3 1 inner
2700.2.i.b 2 105.o odd 6 1
2700.2.i.b 2 315.br odd 6 1
2700.2.s.b 4 105.x even 12 2
2700.2.s.b 4 315.bv even 12 2
5184.2.a.e 1 504.w even 6 1
5184.2.a.f 1 504.ba odd 6 1
5184.2.a.ba 1 504.db odd 6 1
5184.2.a.bb 1 504.cy even 6 1
5292.2.i.a 2 21.g even 6 1
5292.2.i.a 2 63.o even 6 1
5292.2.i.c 2 9.d odd 6 1
5292.2.i.c 2 21.h odd 6 1
5292.2.j.a 2 21.g even 6 1
5292.2.j.a 2 63.i even 6 1
5292.2.l.a 2 3.b odd 2 1
5292.2.l.a 2 63.n odd 6 1
5292.2.l.c 2 21.c even 2 1
5292.2.l.c 2 63.s even 6 1
8100.2.a.g 1 315.v odd 6 1
8100.2.a.j 1 315.bo even 6 1
8100.2.d.c 2 315.bx even 12 2
8100.2.d.h 2 315.ch odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 3 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 - 3 T + T^{2} \)
$5$ \( ( -3 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( 1 - T + T^{2} \)
$17$ \( 36 + 6 T + T^{2} \)
$19$ \( 16 - 4 T + T^{2} \)
$23$ \( ( 3 + T )^{2} \)
$29$ \( 9 + 3 T + T^{2} \)
$31$ \( 25 + 5 T + T^{2} \)
$37$ \( 4 + 2 T + T^{2} \)
$41$ \( 9 + 3 T + T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( 81 - 9 T + T^{2} \)
$53$ \( 36 - 6 T + T^{2} \)
$59$ \( 9 - 3 T + T^{2} \)
$61$ \( 169 - 13 T + T^{2} \)
$67$ \( 49 - 7 T + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 100 - 10 T + T^{2} \)
$79$ \( 121 + 11 T + T^{2} \)
$83$ \( 81 - 9 T + T^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( 121 + 11 T + T^{2} \)
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