Properties

Label 1764.2.k.h
Level $1764$
Weight $2$
Character orbit 1764.k
Analytic conductor $14.086$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q + 2 q^{13} -8 \zeta_{6} q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + ( 4 - 4 \zeta_{6} ) q^{31} + 10 \zeta_{6} q^{37} + 8 q^{43} -14 \zeta_{6} q^{61} + ( 16 - 16 \zeta_{6} ) q^{67} + ( 10 - 10 \zeta_{6} ) q^{73} + 4 \zeta_{6} q^{79} + 14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 4q^{13} - 8q^{19} + 5q^{25} + 4q^{31} + 10q^{37} + 16q^{43} - 14q^{61} + 16q^{67} + 10q^{73} + 4q^{79} + 28q^{97} + O(q^{100}) \)

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 0 0 0 0 0 0
1549.1 0 0 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.k.h 2
3.b odd 2 1 CM 1764.2.k.h 2
7.b odd 2 1 1764.2.k.g 2
7.c even 3 1 36.2.a.a 1
7.c even 3 1 inner 1764.2.k.h 2
7.d odd 6 1 1764.2.a.e 1
7.d odd 6 1 1764.2.k.g 2
21.c even 2 1 1764.2.k.g 2
21.g even 6 1 1764.2.a.e 1
21.g even 6 1 1764.2.k.g 2
21.h odd 6 1 36.2.a.a 1
21.h odd 6 1 inner 1764.2.k.h 2
28.f even 6 1 7056.2.a.bb 1
28.g odd 6 1 144.2.a.a 1
35.j even 6 1 900.2.a.g 1
35.l odd 12 2 900.2.d.b 2
56.k odd 6 1 576.2.a.f 1
56.p even 6 1 576.2.a.e 1
63.g even 3 1 324.2.e.c 2
63.h even 3 1 324.2.e.c 2
63.j odd 6 1 324.2.e.c 2
63.n odd 6 1 324.2.e.c 2
77.h odd 6 1 4356.2.a.g 1
84.j odd 6 1 7056.2.a.bb 1
84.n even 6 1 144.2.a.a 1
91.r even 6 1 6084.2.a.i 1
91.z odd 12 2 6084.2.b.f 2
105.o odd 6 1 900.2.a.g 1
105.x even 12 2 900.2.d.b 2
112.u odd 12 2 2304.2.d.a 2
112.w even 12 2 2304.2.d.q 2
140.p odd 6 1 3600.2.a.e 1
140.w even 12 2 3600.2.f.m 2
168.s odd 6 1 576.2.a.e 1
168.v even 6 1 576.2.a.f 1
231.l even 6 1 4356.2.a.g 1
252.o even 6 1 1296.2.i.h 2
252.u odd 6 1 1296.2.i.h 2
252.bb even 6 1 1296.2.i.h 2
252.bl odd 6 1 1296.2.i.h 2
273.w odd 6 1 6084.2.a.i 1
273.cd even 12 2 6084.2.b.f 2
336.bt odd 12 2 2304.2.d.q 2
336.bu even 12 2 2304.2.d.a 2
420.ba even 6 1 3600.2.a.e 1
420.bp odd 12 2 3600.2.f.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.a.a 1 7.c even 3 1
36.2.a.a 1 21.h odd 6 1
144.2.a.a 1 28.g odd 6 1
144.2.a.a 1 84.n even 6 1
324.2.e.c 2 63.g even 3 1
324.2.e.c 2 63.h even 3 1
324.2.e.c 2 63.j odd 6 1
324.2.e.c 2 63.n odd 6 1
576.2.a.e 1 56.p even 6 1
576.2.a.e 1 168.s odd 6 1
576.2.a.f 1 56.k odd 6 1
576.2.a.f 1 168.v even 6 1
900.2.a.g 1 35.j even 6 1
900.2.a.g 1 105.o odd 6 1
900.2.d.b 2 35.l odd 12 2
900.2.d.b 2 105.x even 12 2
1296.2.i.h 2 252.o even 6 1
1296.2.i.h 2 252.u odd 6 1
1296.2.i.h 2 252.bb even 6 1
1296.2.i.h 2 252.bl odd 6 1
1764.2.a.e 1 7.d odd 6 1
1764.2.a.e 1 21.g even 6 1
1764.2.k.g 2 7.b odd 2 1
1764.2.k.g 2 7.d odd 6 1
1764.2.k.g 2 21.c even 2 1
1764.2.k.g 2 21.g even 6 1
1764.2.k.h 2 1.a even 1 1 trivial
1764.2.k.h 2 3.b odd 2 1 CM
1764.2.k.h 2 7.c even 3 1 inner
1764.2.k.h 2 21.h odd 6 1 inner
2304.2.d.a 2 112.u odd 12 2
2304.2.d.a 2 336.bu even 12 2
2304.2.d.q 2 112.w even 12 2
2304.2.d.q 2 336.bt odd 12 2
3600.2.a.e 1 140.p odd 6 1
3600.2.a.e 1 420.ba even 6 1
3600.2.f.m 2 140.w even 12 2
3600.2.f.m 2 420.bp odd 12 2
4356.2.a.g 1 77.h odd 6 1
4356.2.a.g 1 231.l even 6 1
6084.2.a.i 1 91.r even 6 1
6084.2.a.i 1 273.w odd 6 1
6084.2.b.f 2 91.z odd 12 2
6084.2.b.f 2 273.cd even 12 2
7056.2.a.bb 1 28.f even 6 1
7056.2.a.bb 1 84.j odd 6 1

Hecke kernels

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

\( T_{5} \)
\( T_{11} \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 64 + 8 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 16 - 4 T + T^{2} \)
$37$ \( 100 - 10 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 196 + 14 T + T^{2} \)
$67$ \( 256 - 16 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 100 - 10 T + T^{2} \)
$79$ \( 16 - 4 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -14 + T )^{2} \)
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