Properties

Label 1764.2.k.b
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
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.

\(f(q)\) \(=\) \( q -3 \zeta_{6} q^{5} +O(q^{10})\) \( q -3 \zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{11} -2 q^{13} + ( -3 + 3 \zeta_{6} ) q^{17} -\zeta_{6} q^{19} + 3 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + 6 q^{29} + ( -7 + 7 \zeta_{6} ) q^{31} + \zeta_{6} q^{37} + 6 q^{41} -4 q^{43} + 9 \zeta_{6} q^{47} + ( 3 - 3 \zeta_{6} ) q^{53} + 9 q^{55} + ( -9 + 9 \zeta_{6} ) q^{59} -\zeta_{6} q^{61} + 6 \zeta_{6} q^{65} + ( 7 - 7 \zeta_{6} ) q^{67} + ( -1 + \zeta_{6} ) q^{73} + 13 \zeta_{6} q^{79} + 12 q^{83} + 9 q^{85} -15 \zeta_{6} q^{89} + ( -3 + 3 \zeta_{6} ) q^{95} + 10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{5} + O(q^{10}) \) \( 2q - 3q^{5} - 3q^{11} - 4q^{13} - 3q^{17} - q^{19} + 3q^{23} - 4q^{25} + 12q^{29} - 7q^{31} + q^{37} + 12q^{41} - 8q^{43} + 9q^{47} + 3q^{53} + 18q^{55} - 9q^{59} - q^{61} + 6q^{65} + 7q^{67} - q^{73} + 13q^{79} + 24q^{83} + 18q^{85} - 15q^{89} - 3q^{95} + 20q^{97} + O(q^{100}) \)

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 −1.50000 2.59808i 0 0 0 0 0
1549.1 0 0 0 −1.50000 + 2.59808i 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.b 2
3.b odd 2 1 196.2.e.a 2
7.b odd 2 1 252.2.k.c 2
7.c even 3 1 1764.2.a.j 1
7.c even 3 1 inner 1764.2.k.b 2
7.d odd 6 1 252.2.k.c 2
7.d odd 6 1 1764.2.a.a 1
12.b even 2 1 784.2.i.d 2
21.c even 2 1 28.2.e.a 2
21.g even 6 1 28.2.e.a 2
21.g even 6 1 196.2.a.b 1
21.h odd 6 1 196.2.a.a 1
21.h odd 6 1 196.2.e.a 2
28.d even 2 1 1008.2.s.p 2
28.f even 6 1 1008.2.s.p 2
28.f even 6 1 7056.2.a.f 1
28.g odd 6 1 7056.2.a.bw 1
63.i even 6 1 2268.2.l.h 2
63.k odd 6 1 2268.2.i.h 2
63.l odd 6 1 2268.2.i.h 2
63.l odd 6 1 2268.2.l.a 2
63.o even 6 1 2268.2.i.a 2
63.o even 6 1 2268.2.l.h 2
63.s even 6 1 2268.2.i.a 2
63.t odd 6 1 2268.2.l.a 2
84.h odd 2 1 112.2.i.b 2
84.j odd 6 1 112.2.i.b 2
84.j odd 6 1 784.2.a.d 1
84.n even 6 1 784.2.a.g 1
84.n even 6 1 784.2.i.d 2
105.g even 2 1 700.2.i.c 2
105.k odd 4 2 700.2.r.b 4
105.o odd 6 1 4900.2.a.n 1
105.p even 6 1 700.2.i.c 2
105.p even 6 1 4900.2.a.g 1
105.w odd 12 2 700.2.r.b 4
105.w odd 12 2 4900.2.e.i 2
105.x even 12 2 4900.2.e.h 2
168.e odd 2 1 448.2.i.c 2
168.i even 2 1 448.2.i.e 2
168.s odd 6 1 3136.2.a.v 1
168.v even 6 1 3136.2.a.k 1
168.ba even 6 1 448.2.i.e 2
168.ba even 6 1 3136.2.a.h 1
168.be odd 6 1 448.2.i.c 2
168.be odd 6 1 3136.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.e.a 2 21.c even 2 1
28.2.e.a 2 21.g even 6 1
112.2.i.b 2 84.h odd 2 1
112.2.i.b 2 84.j odd 6 1
196.2.a.a 1 21.h odd 6 1
196.2.a.b 1 21.g even 6 1
196.2.e.a 2 3.b odd 2 1
196.2.e.a 2 21.h odd 6 1
252.2.k.c 2 7.b odd 2 1
252.2.k.c 2 7.d odd 6 1
448.2.i.c 2 168.e odd 2 1
448.2.i.c 2 168.be odd 6 1
448.2.i.e 2 168.i even 2 1
448.2.i.e 2 168.ba even 6 1
700.2.i.c 2 105.g even 2 1
700.2.i.c 2 105.p even 6 1
700.2.r.b 4 105.k odd 4 2
700.2.r.b 4 105.w odd 12 2
784.2.a.d 1 84.j odd 6 1
784.2.a.g 1 84.n even 6 1
784.2.i.d 2 12.b even 2 1
784.2.i.d 2 84.n even 6 1
1008.2.s.p 2 28.d even 2 1
1008.2.s.p 2 28.f even 6 1
1764.2.a.a 1 7.d odd 6 1
1764.2.a.j 1 7.c even 3 1
1764.2.k.b 2 1.a even 1 1 trivial
1764.2.k.b 2 7.c even 3 1 inner
2268.2.i.a 2 63.o even 6 1
2268.2.i.a 2 63.s even 6 1
2268.2.i.h 2 63.k odd 6 1
2268.2.i.h 2 63.l odd 6 1
2268.2.l.a 2 63.l odd 6 1
2268.2.l.a 2 63.t odd 6 1
2268.2.l.h 2 63.i even 6 1
2268.2.l.h 2 63.o even 6 1
3136.2.a.h 1 168.ba even 6 1
3136.2.a.k 1 168.v even 6 1
3136.2.a.s 1 168.be odd 6 1
3136.2.a.v 1 168.s odd 6 1
4900.2.a.g 1 105.p even 6 1
4900.2.a.n 1 105.o odd 6 1
4900.2.e.h 2 105.x even 12 2
4900.2.e.i 2 105.w odd 12 2
7056.2.a.f 1 28.f even 6 1
7056.2.a.bw 1 28.g odd 6 1

Hecke kernels

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

\( T_{5}^{2} + 3 T_{5} + 9 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \)
$37$ \( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \)
$67$ \( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \)
$83$ \( ( 1 - 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 10 T + 97 T^{2} )^{2} \)
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