Properties

Label 1764.4.k.f
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 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.

\(f(q)\) \(=\) \( q -6 \zeta_{6} q^{5} +O(q^{10})\) \( q -6 \zeta_{6} q^{5} + ( 36 - 36 \zeta_{6} ) q^{11} -62 q^{13} + ( -114 + 114 \zeta_{6} ) q^{17} -76 \zeta_{6} q^{19} -24 \zeta_{6} q^{23} + ( 89 - 89 \zeta_{6} ) q^{25} -54 q^{29} + ( -112 + 112 \zeta_{6} ) q^{31} + 178 \zeta_{6} q^{37} + 378 q^{41} -172 q^{43} + 192 \zeta_{6} q^{47} + ( -402 + 402 \zeta_{6} ) q^{53} -216 q^{55} + ( -396 + 396 \zeta_{6} ) q^{59} + 254 \zeta_{6} q^{61} + 372 \zeta_{6} q^{65} + ( 1012 - 1012 \zeta_{6} ) q^{67} -840 q^{71} + ( 890 - 890 \zeta_{6} ) q^{73} -80 \zeta_{6} q^{79} -108 q^{83} + 684 q^{85} + 1638 \zeta_{6} q^{89} + ( -456 + 456 \zeta_{6} ) q^{95} -1010 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{5} + O(q^{10}) \) \( 2q - 6q^{5} + 36q^{11} - 124q^{13} - 114q^{17} - 76q^{19} - 24q^{23} + 89q^{25} - 108q^{29} - 112q^{31} + 178q^{37} + 756q^{41} - 344q^{43} + 192q^{47} - 402q^{53} - 432q^{55} - 396q^{59} + 254q^{61} + 372q^{65} + 1012q^{67} - 1680q^{71} + 890q^{73} - 80q^{79} - 216q^{83} + 1368q^{85} + 1638q^{89} - 456q^{95} - 2020q^{97} + O(q^{100}) \)

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 −3.00000 5.19615i 0 0 0 0 0
1549.1 0 0 0 −3.00000 + 5.19615i 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.f 2
3.b odd 2 1 588.4.i.c 2
7.b odd 2 1 1764.4.k.l 2
7.c even 3 1 1764.4.a.j 1
7.c even 3 1 inner 1764.4.k.f 2
7.d odd 6 1 252.4.a.b 1
7.d odd 6 1 1764.4.k.l 2
21.c even 2 1 588.4.i.f 2
21.g even 6 1 84.4.a.a 1
21.g even 6 1 588.4.i.f 2
21.h odd 6 1 588.4.a.d 1
21.h odd 6 1 588.4.i.c 2
28.f even 6 1 1008.4.a.h 1
84.j odd 6 1 336.4.a.k 1
84.n even 6 1 2352.4.a.d 1
105.p even 6 1 2100.4.a.l 1
105.w odd 12 2 2100.4.k.j 2
168.ba even 6 1 1344.4.a.q 1
168.be odd 6 1 1344.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.a 1 21.g even 6 1
252.4.a.b 1 7.d odd 6 1
336.4.a.k 1 84.j odd 6 1
588.4.a.d 1 21.h odd 6 1
588.4.i.c 2 3.b odd 2 1
588.4.i.c 2 21.h odd 6 1
588.4.i.f 2 21.c even 2 1
588.4.i.f 2 21.g even 6 1
1008.4.a.h 1 28.f even 6 1
1344.4.a.d 1 168.be odd 6 1
1344.4.a.q 1 168.ba even 6 1
1764.4.a.j 1 7.c even 3 1
1764.4.k.f 2 1.a even 1 1 trivial
1764.4.k.f 2 7.c even 3 1 inner
1764.4.k.l 2 7.b odd 2 1
1764.4.k.l 2 7.d odd 6 1
2100.4.a.l 1 105.p even 6 1
2100.4.k.j 2 105.w odd 12 2
2352.4.a.d 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 \(S_{4}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{2} + 6 T_{5} + 36 \)
\( T_{11}^{2} - 36 T_{11} + 1296 \)
\( T_{13} + 62 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 36 + 6 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 1296 - 36 T + T^{2} \)
$13$ \( ( 62 + T )^{2} \)
$17$ \( 12996 + 114 T + T^{2} \)
$19$ \( 5776 + 76 T + T^{2} \)
$23$ \( 576 + 24 T + T^{2} \)
$29$ \( ( 54 + T )^{2} \)
$31$ \( 12544 + 112 T + T^{2} \)
$37$ \( 31684 - 178 T + T^{2} \)
$41$ \( ( -378 + T )^{2} \)
$43$ \( ( 172 + T )^{2} \)
$47$ \( 36864 - 192 T + T^{2} \)
$53$ \( 161604 + 402 T + T^{2} \)
$59$ \( 156816 + 396 T + T^{2} \)
$61$ \( 64516 - 254 T + T^{2} \)
$67$ \( 1024144 - 1012 T + T^{2} \)
$71$ \( ( 840 + T )^{2} \)
$73$ \( 792100 - 890 T + T^{2} \)
$79$ \( 6400 + 80 T + T^{2} \)
$83$ \( ( 108 + T )^{2} \)
$89$ \( 2683044 - 1638 T + T^{2} \)
$97$ \( ( 1010 + T )^{2} \)
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