# 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 $4$

# 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: $$\Z[a_1, \ldots, a_{25}]$$ 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
 0.5 + 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.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

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