Properties

Label 1764.4.k.m
Level $1764$
Weight $4$
Character orbit 1764.k
Analytic conductor $104.079$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.k (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(104.079369250\)
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 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.

\(f(q)\) \(=\) \( q + 8 \zeta_{6} q^{5} +O(q^{10})\) \( q + 8 \zeta_{6} q^{5} + ( -40 + 40 \zeta_{6} ) q^{11} + 12 q^{13} + ( 58 - 58 \zeta_{6} ) q^{17} + 26 \zeta_{6} q^{19} -64 \zeta_{6} q^{23} + ( 61 - 61 \zeta_{6} ) q^{25} + 62 q^{29} + ( 252 - 252 \zeta_{6} ) q^{31} -26 \zeta_{6} q^{37} + 6 q^{41} + 416 q^{43} + 396 \zeta_{6} q^{47} + ( -450 + 450 \zeta_{6} ) q^{53} -320 q^{55} + ( -274 + 274 \zeta_{6} ) q^{59} -576 \zeta_{6} q^{61} + 96 \zeta_{6} q^{65} + ( 476 - 476 \zeta_{6} ) q^{67} + 448 q^{71} + ( -158 + 158 \zeta_{6} ) q^{73} + 936 \zeta_{6} q^{79} + 530 q^{83} + 464 q^{85} + 390 \zeta_{6} q^{89} + ( -208 + 208 \zeta_{6} ) q^{95} -214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{5} + O(q^{10}) \) \( 2q + 8q^{5} - 40q^{11} + 24q^{13} + 58q^{17} + 26q^{19} - 64q^{23} + 61q^{25} + 124q^{29} + 252q^{31} - 26q^{37} + 12q^{41} + 832q^{43} + 396q^{47} - 450q^{53} - 640q^{55} - 274q^{59} - 576q^{61} + 96q^{65} + 476q^{67} + 896q^{71} - 158q^{73} + 936q^{79} + 1060q^{83} + 928q^{85} + 390q^{89} - 208q^{95} - 428q^{97} + O(q^{100}) \)

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 4.00000 + 6.92820i 0 0 0 0 0
1549.1 0 0 0 4.00000 6.92820i 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 1764.4.k.m 2
3.b odd 2 1 196.4.e.a 2
7.b odd 2 1 1764.4.k.d 2
7.c even 3 1 1764.4.a.c 1
7.c even 3 1 inner 1764.4.k.m 2
7.d odd 6 1 252.4.a.d 1
7.d odd 6 1 1764.4.k.d 2
21.c even 2 1 196.4.e.f 2
21.g even 6 1 28.4.a.a 1
21.g even 6 1 196.4.e.f 2
21.h odd 6 1 196.4.a.d 1
21.h odd 6 1 196.4.e.a 2
28.f even 6 1 1008.4.a.o 1
84.j odd 6 1 112.4.a.g 1
84.n even 6 1 784.4.a.a 1
105.p even 6 1 700.4.a.n 1
105.w odd 12 2 700.4.e.a 2
168.ba even 6 1 448.4.a.p 1
168.be odd 6 1 448.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 21.g even 6 1
112.4.a.g 1 84.j odd 6 1
196.4.a.d 1 21.h odd 6 1
196.4.e.a 2 3.b odd 2 1
196.4.e.a 2 21.h odd 6 1
196.4.e.f 2 21.c even 2 1
196.4.e.f 2 21.g even 6 1
252.4.a.d 1 7.d odd 6 1
448.4.a.a 1 168.be odd 6 1
448.4.a.p 1 168.ba even 6 1
700.4.a.n 1 105.p even 6 1
700.4.e.a 2 105.w odd 12 2
784.4.a.a 1 84.n even 6 1
1008.4.a.o 1 28.f even 6 1
1764.4.a.c 1 7.c even 3 1
1764.4.k.d 2 7.b odd 2 1
1764.4.k.d 2 7.d odd 6 1
1764.4.k.m 2 1.a even 1 1 trivial
1764.4.k.m 2 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{2} - 8 T_{5} + 64 \)
\( T_{11}^{2} + 40 T_{11} + 1600 \)
\( T_{13} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 64 - 8 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 1600 + 40 T + T^{2} \)
$13$ \( ( -12 + T )^{2} \)
$17$ \( 3364 - 58 T + T^{2} \)
$19$ \( 676 - 26 T + T^{2} \)
$23$ \( 4096 + 64 T + T^{2} \)
$29$ \( ( -62 + T )^{2} \)
$31$ \( 63504 - 252 T + T^{2} \)
$37$ \( 676 + 26 T + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( ( -416 + T )^{2} \)
$47$ \( 156816 - 396 T + T^{2} \)
$53$ \( 202500 + 450 T + T^{2} \)
$59$ \( 75076 + 274 T + T^{2} \)
$61$ \( 331776 + 576 T + T^{2} \)
$67$ \( 226576 - 476 T + T^{2} \)
$71$ \( ( -448 + T )^{2} \)
$73$ \( 24964 + 158 T + T^{2} \)
$79$ \( 876096 - 936 T + T^{2} \)
$83$ \( ( -530 + T )^{2} \)
$89$ \( 152100 - 390 T + T^{2} \)
$97$ \( ( 214 + T )^{2} \)
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