# Properties

 Label 1764.2.e.b Level $1764$ Weight $2$ Character orbit 1764.e Analytic conductor $14.086$ Analytic rank $0$ Dimension $2$ CM discriminant -4 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.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.0856109166$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -2 q^{4} + \beta q^{5} -2 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} -2 q^{4} + \beta q^{5} -2 \beta q^{8} -2 q^{10} + 4 q^{13} + 4 q^{16} -5 \beta q^{17} -2 \beta q^{20} + 3 q^{25} + 4 \beta q^{26} -7 \beta q^{29} + 4 \beta q^{32} + 10 q^{34} + 2 q^{37} + 4 q^{40} + \beta q^{41} + 3 \beta q^{50} -8 q^{52} + 5 \beta q^{53} + 14 q^{58} + 10 q^{61} -8 q^{64} + 4 \beta q^{65} + 10 \beta q^{68} + 16 q^{73} + 2 \beta q^{74} + 4 \beta q^{80} -2 q^{82} + 10 q^{85} + 13 \beta q^{89} -8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} + O(q^{10})$$ $$2q - 4q^{4} - 4q^{10} + 8q^{13} + 8q^{16} + 6q^{25} + 20q^{34} + 4q^{37} + 8q^{40} - 16q^{52} + 28q^{58} + 20q^{61} - 16q^{64} + 32q^{73} - 4q^{82} + 20q^{85} - 16q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1079.1
 − 1.41421i 1.41421i
1.41421i 0 −2.00000 1.41421i 0 0 2.82843i 0 −2.00000
1079.2 1.41421i 0 −2.00000 1.41421i 0 0 2.82843i 0 −2.00000
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.e.b 2
3.b odd 2 1 inner 1764.2.e.b 2
4.b odd 2 1 CM 1764.2.e.b 2
7.b odd 2 1 36.2.b.a 2
12.b even 2 1 inner 1764.2.e.b 2
21.c even 2 1 36.2.b.a 2
28.d even 2 1 36.2.b.a 2
35.c odd 2 1 900.2.e.b 2
35.f even 4 2 900.2.h.a 4
56.e even 2 1 576.2.c.b 2
56.h odd 2 1 576.2.c.b 2
63.l odd 6 2 324.2.h.c 4
63.o even 6 2 324.2.h.c 4
84.h odd 2 1 36.2.b.a 2
105.g even 2 1 900.2.e.b 2
105.k odd 4 2 900.2.h.a 4
112.j even 4 2 2304.2.f.d 4
112.l odd 4 2 2304.2.f.d 4
140.c even 2 1 900.2.e.b 2
140.j odd 4 2 900.2.h.a 4
168.e odd 2 1 576.2.c.b 2
168.i even 2 1 576.2.c.b 2
252.s odd 6 2 324.2.h.c 4
252.bi even 6 2 324.2.h.c 4
336.v odd 4 2 2304.2.f.d 4
336.y even 4 2 2304.2.f.d 4
420.o odd 2 1 900.2.e.b 2
420.w even 4 2 900.2.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.b.a 2 7.b odd 2 1
36.2.b.a 2 21.c even 2 1
36.2.b.a 2 28.d even 2 1
36.2.b.a 2 84.h odd 2 1
324.2.h.c 4 63.l odd 6 2
324.2.h.c 4 63.o even 6 2
324.2.h.c 4 252.s odd 6 2
324.2.h.c 4 252.bi even 6 2
576.2.c.b 2 56.e even 2 1
576.2.c.b 2 56.h odd 2 1
576.2.c.b 2 168.e odd 2 1
576.2.c.b 2 168.i even 2 1
900.2.e.b 2 35.c odd 2 1
900.2.e.b 2 105.g even 2 1
900.2.e.b 2 140.c even 2 1
900.2.e.b 2 420.o odd 2 1
900.2.h.a 4 35.f even 4 2
900.2.h.a 4 105.k odd 4 2
900.2.h.a 4 140.j odd 4 2
900.2.h.a 4 420.w even 4 2
1764.2.e.b 2 1.a even 1 1 trivial
1764.2.e.b 2 3.b odd 2 1 inner
1764.2.e.b 2 4.b odd 2 1 CM
1764.2.e.b 2 12.b even 2 1 inner
2304.2.f.d 4 112.j even 4 2
2304.2.f.d 4 112.l odd 4 2
2304.2.f.d 4 336.v odd 4 2
2304.2.f.d 4 336.y even 4 2

## Hecke kernels

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

 $$T_{5}^{2} + 2$$ $$T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$2 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$50 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$98 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$2 + T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$50 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -10 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -16 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$338 + T^{2}$$
$97$ $$( 8 + T )^{2}$$