# Properties

 Label 720.2.x.c Level $720$ Weight $2$ Character orbit 720.x Analytic conductor $5.749$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.x (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.74922894553$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + 2 i ) q^{5} +O(q^{10})$$ $$q + ( 1 + 2 i ) q^{5} + ( 1 - i ) q^{13} + ( 5 + 5 i ) q^{17} + ( -3 + 4 i ) q^{25} + 10 i q^{29} + ( -7 - 7 i ) q^{37} + 10 q^{41} + 7 i q^{49} + ( -5 + 5 i ) q^{53} + 12 q^{61} + ( 3 + i ) q^{65} + ( 11 - 11 i ) q^{73} + ( -5 + 15 i ) q^{85} -10 i q^{89} + ( -13 - 13 i ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + O(q^{10})$$ $$2q + 2q^{5} + 2q^{13} + 10q^{17} - 6q^{25} - 14q^{37} + 20q^{41} - 10q^{53} + 24q^{61} + 6q^{65} + 22q^{73} - 10q^{85} - 26q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$-1$$ $$i$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 1.00000i − 1.00000i
0 0 0 1.00000 + 2.00000i 0 0 0 0 0
703.1 0 0 0 1.00000 2.00000i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.x.c yes 2
3.b odd 2 1 720.2.x.b 2
4.b odd 2 1 CM 720.2.x.c yes 2
5.b even 2 1 3600.2.x.a 2
5.c odd 4 1 inner 720.2.x.c yes 2
5.c odd 4 1 3600.2.x.a 2
12.b even 2 1 720.2.x.b 2
15.d odd 2 1 3600.2.x.b 2
15.e even 4 1 720.2.x.b 2
15.e even 4 1 3600.2.x.b 2
20.d odd 2 1 3600.2.x.a 2
20.e even 4 1 inner 720.2.x.c yes 2
20.e even 4 1 3600.2.x.a 2
60.h even 2 1 3600.2.x.b 2
60.l odd 4 1 720.2.x.b 2
60.l odd 4 1 3600.2.x.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.x.b 2 3.b odd 2 1
720.2.x.b 2 12.b even 2 1
720.2.x.b 2 15.e even 4 1
720.2.x.b 2 60.l odd 4 1
720.2.x.c yes 2 1.a even 1 1 trivial
720.2.x.c yes 2 4.b odd 2 1 CM
720.2.x.c yes 2 5.c odd 4 1 inner
720.2.x.c yes 2 20.e even 4 1 inner
3600.2.x.a 2 5.b even 2 1
3600.2.x.a 2 5.c odd 4 1
3600.2.x.a 2 20.d odd 2 1
3600.2.x.a 2 20.e even 4 1
3600.2.x.b 2 15.d odd 2 1
3600.2.x.b 2 15.e even 4 1
3600.2.x.b 2 60.h even 2 1
3600.2.x.b 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}$$ $$T_{11}$$ $$T_{13}^{2} - 2 T_{13} + 2$$ $$T_{17}^{2} - 10 T_{17} + 50$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$2 - 2 T + T^{2}$$
$17$ $$50 - 10 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$100 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$98 + 14 T + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$50 + 10 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -12 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$242 - 22 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$100 + T^{2}$$
$97$ $$338 + 26 T + T^{2}$$