Properties

Label 1764.4.k.c
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.

\(f(q)\) \(=\) \( q -14 \zeta_{6} q^{5} +O(q^{10})\) \( q -14 \zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{11} -54 q^{13} + ( 14 - 14 \zeta_{6} ) q^{17} + 92 \zeta_{6} q^{19} -152 \zeta_{6} q^{23} + ( -71 + 71 \zeta_{6} ) q^{25} + 106 q^{29} + ( -144 + 144 \zeta_{6} ) q^{31} -158 \zeta_{6} q^{37} -390 q^{41} -508 q^{43} + 528 \zeta_{6} q^{47} + ( 606 - 606 \zeta_{6} ) q^{53} -56 q^{55} + ( 364 - 364 \zeta_{6} ) q^{59} + 678 \zeta_{6} q^{61} + 756 \zeta_{6} q^{65} + ( -844 + 844 \zeta_{6} ) q^{67} + 8 q^{71} + ( -422 + 422 \zeta_{6} ) q^{73} -384 \zeta_{6} q^{79} -548 q^{83} -196 q^{85} -1194 \zeta_{6} q^{89} + ( 1288 - 1288 \zeta_{6} ) q^{95} + 1502 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 14q^{5} + O(q^{10}) \) \( 2q - 14q^{5} + 4q^{11} - 108q^{13} + 14q^{17} + 92q^{19} - 152q^{23} - 71q^{25} + 212q^{29} - 144q^{31} - 158q^{37} - 780q^{41} - 1016q^{43} + 528q^{47} + 606q^{53} - 112q^{55} + 364q^{59} + 678q^{61} + 756q^{65} - 844q^{67} + 16q^{71} - 422q^{73} - 384q^{79} - 1096q^{83} - 392q^{85} - 1194q^{89} + 1288q^{95} + 3004q^{97} + O(q^{100}) \)

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 −7.00000 12.1244i 0 0 0 0 0
1549.1 0 0 0 −7.00000 + 12.1244i 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.c 2
3.b odd 2 1 588.4.i.h 2
7.b odd 2 1 1764.4.k.n 2
7.c even 3 1 1764.4.a.l 1
7.c even 3 1 inner 1764.4.k.c 2
7.d odd 6 1 252.4.a.a 1
7.d odd 6 1 1764.4.k.n 2
21.c even 2 1 588.4.i.a 2
21.g even 6 1 84.4.a.b 1
21.g even 6 1 588.4.i.a 2
21.h odd 6 1 588.4.a.a 1
21.h odd 6 1 588.4.i.h 2
28.f even 6 1 1008.4.a.d 1
84.j odd 6 1 336.4.a.e 1
84.n even 6 1 2352.4.a.v 1
105.p even 6 1 2100.4.a.g 1
105.w odd 12 2 2100.4.k.g 2
168.ba even 6 1 1344.4.a.b 1
168.be odd 6 1 1344.4.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.b 1 21.g even 6 1
252.4.a.a 1 7.d odd 6 1
336.4.a.e 1 84.j odd 6 1
588.4.a.a 1 21.h odd 6 1
588.4.i.a 2 21.c even 2 1
588.4.i.a 2 21.g even 6 1
588.4.i.h 2 3.b odd 2 1
588.4.i.h 2 21.h odd 6 1
1008.4.a.d 1 28.f even 6 1
1344.4.a.b 1 168.ba even 6 1
1344.4.a.p 1 168.be odd 6 1
1764.4.a.l 1 7.c even 3 1
1764.4.k.c 2 1.a even 1 1 trivial
1764.4.k.c 2 7.c even 3 1 inner
1764.4.k.n 2 7.b odd 2 1
1764.4.k.n 2 7.d odd 6 1
2100.4.a.g 1 105.p even 6 1
2100.4.k.g 2 105.w odd 12 2
2352.4.a.v 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 \(S_{4}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{2} + 14 T_{5} + 196 \)
\( T_{11}^{2} - 4 T_{11} + 16 \)
\( T_{13} + 54 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 196 + 14 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 16 - 4 T + T^{2} \)
$13$ \( ( 54 + T )^{2} \)
$17$ \( 196 - 14 T + T^{2} \)
$19$ \( 8464 - 92 T + T^{2} \)
$23$ \( 23104 + 152 T + T^{2} \)
$29$ \( ( -106 + T )^{2} \)
$31$ \( 20736 + 144 T + T^{2} \)
$37$ \( 24964 + 158 T + T^{2} \)
$41$ \( ( 390 + T )^{2} \)
$43$ \( ( 508 + T )^{2} \)
$47$ \( 278784 - 528 T + T^{2} \)
$53$ \( 367236 - 606 T + T^{2} \)
$59$ \( 132496 - 364 T + T^{2} \)
$61$ \( 459684 - 678 T + T^{2} \)
$67$ \( 712336 + 844 T + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 178084 + 422 T + T^{2} \)
$79$ \( 147456 + 384 T + T^{2} \)
$83$ \( ( 548 + T )^{2} \)
$89$ \( 1425636 + 1194 T + T^{2} \)
$97$ \( ( -1502 + T )^{2} \)
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