# Properties

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

# Related objects

## 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 252) 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.

 $$f(q)$$ $$=$$ $$q +O(q^{10})$$ $$q -5 q^{13} -\zeta_{6} q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + ( 11 - 11 \zeta_{6} ) q^{31} -11 \zeta_{6} q^{37} -13 q^{43} + 14 \zeta_{6} q^{61} + ( -5 + 5 \zeta_{6} ) q^{67} + ( 17 - 17 \zeta_{6} ) q^{73} -17 \zeta_{6} q^{79} -14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 10q^{13} - q^{19} + 5q^{25} + 11q^{31} - 11q^{37} - 26q^{43} + 14q^{61} - 5q^{67} + 17q^{73} - 17q^{79} - 28q^{97} + O(q^{100})$$

## 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
 0.5 + 0.866025i 0.5 − 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.f 2
3.b odd 2 1 CM 1764.2.k.f 2
7.b odd 2 1 252.2.k.b 2
7.c even 3 1 1764.2.a.d 1
7.c even 3 1 inner 1764.2.k.f 2
7.d odd 6 1 252.2.k.b 2
7.d odd 6 1 1764.2.a.f 1
21.c even 2 1 252.2.k.b 2
21.g even 6 1 252.2.k.b 2
21.g even 6 1 1764.2.a.f 1
21.h odd 6 1 1764.2.a.d 1
21.h odd 6 1 inner 1764.2.k.f 2
28.d even 2 1 1008.2.s.i 2
28.f even 6 1 1008.2.s.i 2
28.f even 6 1 7056.2.a.be 1
28.g odd 6 1 7056.2.a.z 1
63.i even 6 1 2268.2.l.e 2
63.k odd 6 1 2268.2.i.c 2
63.l odd 6 1 2268.2.i.c 2
63.l odd 6 1 2268.2.l.e 2
63.o even 6 1 2268.2.i.c 2
63.o even 6 1 2268.2.l.e 2
63.s even 6 1 2268.2.i.c 2
63.t odd 6 1 2268.2.l.e 2
84.h odd 2 1 1008.2.s.i 2
84.j odd 6 1 1008.2.s.i 2
84.j odd 6 1 7056.2.a.be 1
84.n even 6 1 7056.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.k.b 2 7.b odd 2 1
252.2.k.b 2 7.d odd 6 1
252.2.k.b 2 21.c even 2 1
252.2.k.b 2 21.g even 6 1
1008.2.s.i 2 28.d even 2 1
1008.2.s.i 2 28.f even 6 1
1008.2.s.i 2 84.h odd 2 1
1008.2.s.i 2 84.j odd 6 1
1764.2.a.d 1 7.c even 3 1
1764.2.a.d 1 21.h odd 6 1
1764.2.a.f 1 7.d odd 6 1
1764.2.a.f 1 21.g even 6 1
1764.2.k.f 2 1.a even 1 1 trivial
1764.2.k.f 2 3.b odd 2 1 CM
1764.2.k.f 2 7.c even 3 1 inner
1764.2.k.f 2 21.h odd 6 1 inner
2268.2.i.c 2 63.k odd 6 1
2268.2.i.c 2 63.l odd 6 1
2268.2.i.c 2 63.o even 6 1
2268.2.i.c 2 63.s even 6 1
2268.2.l.e 2 63.i even 6 1
2268.2.l.e 2 63.l odd 6 1
2268.2.l.e 2 63.o even 6 1
2268.2.l.e 2 63.t odd 6 1
7056.2.a.z 1 28.g odd 6 1
7056.2.a.z 1 84.n even 6 1
7056.2.a.be 1 28.f even 6 1
7056.2.a.be 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 $$S_{2}^{\mathrm{new}}(1764, [\chi])$$:

 $$T_{5}$$ $$T_{11}$$ $$T_{13} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 5 T^{2} + 25 T^{4}$$
$7$ 1
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 + 5 T + 13 T^{2} )^{2}$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$( 1 - 7 T + 31 T^{2} )( 1 - 4 T + 31 T^{2} )$$
$37$ $$( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 + 13 T + 43 T^{2} )^{2}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 - 53 T^{2} + 2809 T^{4}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} )$$
$67$ $$( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 - 10 T + 73 T^{2} )( 1 - 7 T + 73 T^{2} )$$
$79$ $$( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$1 - 89 T^{2} + 7921 T^{4}$$
$97$ $$( 1 + 14 T + 97 T^{2} )^{2}$$