Properties

Label 1764.4.k.j
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 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.

\(f(q)\) \(=\) \( q + 4 \zeta_{6} q^{5} +O(q^{10})\) \( q + 4 \zeta_{6} q^{5} + ( -20 + 20 \zeta_{6} ) q^{11} -4 q^{13} + ( 24 - 24 \zeta_{6} ) q^{17} -44 \zeta_{6} q^{19} + 72 \zeta_{6} q^{23} + ( 109 - 109 \zeta_{6} ) q^{25} + 38 q^{29} + ( -184 + 184 \zeta_{6} ) q^{31} + 30 \zeta_{6} q^{37} + 216 q^{41} -164 q^{43} + 520 \zeta_{6} q^{47} + ( -146 + 146 \zeta_{6} ) q^{53} -80 q^{55} + ( 460 - 460 \zeta_{6} ) q^{59} -628 \zeta_{6} q^{61} -16 \zeta_{6} q^{65} + ( -556 + 556 \zeta_{6} ) q^{67} -592 q^{71} + ( -1024 + 1024 \zeta_{6} ) q^{73} + 104 \zeta_{6} q^{79} + 324 q^{83} + 96 q^{85} + 896 \zeta_{6} q^{89} + ( 176 - 176 \zeta_{6} ) q^{95} -920 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} + O(q^{10}) \) \( 2q + 4q^{5} - 20q^{11} - 8q^{13} + 24q^{17} - 44q^{19} + 72q^{23} + 109q^{25} + 76q^{29} - 184q^{31} + 30q^{37} + 432q^{41} - 328q^{43} + 520q^{47} - 146q^{53} - 160q^{55} + 460q^{59} - 628q^{61} - 16q^{65} - 556q^{67} - 1184q^{71} - 1024q^{73} + 104q^{79} + 648q^{83} + 192q^{85} + 896q^{89} + 176q^{95} - 1840q^{97} + O(q^{100}) \)

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 1764.4.k.j 2
3.b odd 2 1 588.4.i.g 2
7.b odd 2 1 1764.4.k.g 2
7.c even 3 1 1764.4.a.d 1
7.c even 3 1 inner 1764.4.k.j 2
7.d odd 6 1 1764.4.a.i 1
7.d odd 6 1 1764.4.k.g 2
21.c even 2 1 588.4.i.b 2
21.g even 6 1 588.4.a.e yes 1
21.g even 6 1 588.4.i.b 2
21.h odd 6 1 588.4.a.b 1
21.h odd 6 1 588.4.i.g 2
84.j odd 6 1 2352.4.a.g 1
84.n even 6 1 2352.4.a.be 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.4.a.b 1 21.h odd 6 1
588.4.a.e yes 1 21.g even 6 1
588.4.i.b 2 21.c even 2 1
588.4.i.b 2 21.g even 6 1
588.4.i.g 2 3.b odd 2 1
588.4.i.g 2 21.h odd 6 1
1764.4.a.d 1 7.c even 3 1
1764.4.a.i 1 7.d odd 6 1
1764.4.k.g 2 7.b odd 2 1
1764.4.k.g 2 7.d odd 6 1
1764.4.k.j 2 1.a even 1 1 trivial
1764.4.k.j 2 7.c even 3 1 inner
2352.4.a.g 1 84.j odd 6 1
2352.4.a.be 1 84.n even 6 1

Hecke kernels

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

\( T_{5}^{2} - 4 T_{5} + 16 \)
\( T_{11}^{2} + 20 T_{11} + 400 \)
\( T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 16 - 4 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 400 + 20 T + T^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( 576 - 24 T + T^{2} \)
$19$ \( 1936 + 44 T + T^{2} \)
$23$ \( 5184 - 72 T + T^{2} \)
$29$ \( ( -38 + T )^{2} \)
$31$ \( 33856 + 184 T + T^{2} \)
$37$ \( 900 - 30 T + T^{2} \)
$41$ \( ( -216 + T )^{2} \)
$43$ \( ( 164 + T )^{2} \)
$47$ \( 270400 - 520 T + T^{2} \)
$53$ \( 21316 + 146 T + T^{2} \)
$59$ \( 211600 - 460 T + T^{2} \)
$61$ \( 394384 + 628 T + T^{2} \)
$67$ \( 309136 + 556 T + T^{2} \)
$71$ \( ( 592 + T )^{2} \)
$73$ \( 1048576 + 1024 T + T^{2} \)
$79$ \( 10816 - 104 T + T^{2} \)
$83$ \( ( -324 + T )^{2} \)
$89$ \( 802816 - 896 T + T^{2} \)
$97$ \( ( 920 + T )^{2} \)
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