# Properties

 Label 720.2.o.a Level $720$ Weight $2$ Character orbit 720.o Analytic conductor $5.749$ Analytic rank $0$ Dimension $4$ CM discriminant -4 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.o (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.74922894553$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} +O(q^{10})$$ $$q + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} + 6 \zeta_{8}^{2} q^{13} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{17} + ( 4 + 3 \zeta_{8}^{2} ) q^{25} + ( 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{29} + 12 \zeta_{8}^{2} q^{37} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{41} -7 q^{49} + ( -9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{53} + 10 q^{61} + ( -6 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{65} -6 \zeta_{8}^{2} q^{73} + ( 9 + 3 \zeta_{8}^{2} ) q^{85} + ( 13 \zeta_{8} + 13 \zeta_{8}^{3} ) q^{89} -18 \zeta_{8}^{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 16q^{25} - 28q^{49} + 40q^{61} + 36q^{85} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
719.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 −2.12132 0.707107i 0 0 0 0 0
719.2 0 0 0 −2.12132 + 0.707107i 0 0 0 0 0
719.3 0 0 0 2.12132 0.707107i 0 0 0 0 0
719.4 0 0 0 2.12132 + 0.707107i 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 $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.o.a 4
3.b odd 2 1 inner 720.2.o.a 4
4.b odd 2 1 CM 720.2.o.a 4
5.b even 2 1 inner 720.2.o.a 4
5.c odd 4 1 3600.2.h.a 2
5.c odd 4 1 3600.2.h.c 2
8.b even 2 1 2880.2.o.b 4
8.d odd 2 1 2880.2.o.b 4
12.b even 2 1 inner 720.2.o.a 4
15.d odd 2 1 inner 720.2.o.a 4
15.e even 4 1 3600.2.h.a 2
15.e even 4 1 3600.2.h.c 2
20.d odd 2 1 inner 720.2.o.a 4
20.e even 4 1 3600.2.h.a 2
20.e even 4 1 3600.2.h.c 2
24.f even 2 1 2880.2.o.b 4
24.h odd 2 1 2880.2.o.b 4
40.e odd 2 1 2880.2.o.b 4
40.f even 2 1 2880.2.o.b 4
60.h even 2 1 inner 720.2.o.a 4
60.l odd 4 1 3600.2.h.a 2
60.l odd 4 1 3600.2.h.c 2
120.i odd 2 1 2880.2.o.b 4
120.m even 2 1 2880.2.o.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.o.a 4 1.a even 1 1 trivial
720.2.o.a 4 3.b odd 2 1 inner
720.2.o.a 4 4.b odd 2 1 CM
720.2.o.a 4 5.b even 2 1 inner
720.2.o.a 4 12.b even 2 1 inner
720.2.o.a 4 15.d odd 2 1 inner
720.2.o.a 4 20.d odd 2 1 inner
720.2.o.a 4 60.h even 2 1 inner
2880.2.o.b 4 8.b even 2 1
2880.2.o.b 4 8.d odd 2 1
2880.2.o.b 4 24.f even 2 1
2880.2.o.b 4 24.h odd 2 1
2880.2.o.b 4 40.e odd 2 1
2880.2.o.b 4 40.f even 2 1
2880.2.o.b 4 120.i odd 2 1
2880.2.o.b 4 120.m even 2 1
3600.2.h.a 2 5.c odd 4 1
3600.2.h.a 2 15.e even 4 1
3600.2.h.a 2 20.e even 4 1
3600.2.h.a 2 60.l odd 4 1
3600.2.h.c 2 5.c odd 4 1
3600.2.h.c 2 15.e even 4 1
3600.2.h.c 2 20.e even 4 1
3600.2.h.c 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}$$ acting on $$S_{2}^{\mathrm{new}}(720, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 - 8 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$( 36 + T^{2} )^{2}$$
$17$ $$( -18 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( 98 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 144 + T^{2} )^{2}$$
$41$ $$( 2 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( -162 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( -10 + T )^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( 36 + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 338 + T^{2} )^{2}$$
$97$ $$( 324 + T^{2} )^{2}$$