# Properties

 Label 720.2.w.d Level $720$ Weight $2$ Character orbit 720.w Analytic conductor $5.749$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.w (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.74922894553$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( 2 + 2 \zeta_{8}^{2} ) q^{7} +O(q^{10})$$ $$q + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( 2 + 2 \zeta_{8}^{2} ) q^{7} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{11} + ( 1 - \zeta_{8}^{2} ) q^{13} + 4 \zeta_{8}^{3} q^{17} -4 \zeta_{8} q^{23} + ( 4 + 3 \zeta_{8}^{2} ) q^{25} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{29} + 4 q^{31} + ( 6 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{35} + ( 1 + \zeta_{8}^{2} ) q^{37} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{41} + ( 8 - 8 \zeta_{8}^{2} ) q^{43} + 8 \zeta_{8}^{3} q^{47} + \zeta_{8}^{2} q^{49} + 4 \zeta_{8} q^{53} + ( -2 + 6 \zeta_{8}^{2} ) q^{55} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{59} + 8 q^{61} + ( \zeta_{8} - 3 \zeta_{8}^{3} ) q^{65} + ( -4 - 4 \zeta_{8}^{2} ) q^{67} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{71} + ( 1 - \zeta_{8}^{2} ) q^{73} + 8 \zeta_{8}^{3} q^{77} + 12 \zeta_{8}^{2} q^{79} -4 \zeta_{8} q^{83} + ( -8 + 4 \zeta_{8}^{2} ) q^{85} + ( 9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{89} + 4 q^{91} + ( -11 - 11 \zeta_{8}^{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{7} + O(q^{10})$$ $$4q + 8q^{7} + 4q^{13} + 16q^{25} + 16q^{31} + 4q^{37} + 32q^{43} - 8q^{55} + 32q^{61} - 16q^{67} + 4q^{73} - 32q^{85} + 16q^{91} - 44q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$1$$ $$\zeta_{8}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 0 0 −2.12132 0.707107i 0 2.00000 + 2.00000i 0 0 0
17.2 0 0 0 2.12132 + 0.707107i 0 2.00000 + 2.00000i 0 0 0
593.1 0 0 0 −2.12132 + 0.707107i 0 2.00000 2.00000i 0 0 0
593.2 0 0 0 2.12132 0.707107i 0 2.00000 2.00000i 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 inner
5.c odd 4 1 inner
15.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.w.d 4
3.b odd 2 1 inner 720.2.w.d 4
4.b odd 2 1 45.2.f.a 4
5.b even 2 1 3600.2.w.b 4
5.c odd 4 1 inner 720.2.w.d 4
5.c odd 4 1 3600.2.w.b 4
8.b even 2 1 2880.2.w.k 4
8.d odd 2 1 2880.2.w.b 4
12.b even 2 1 45.2.f.a 4
15.d odd 2 1 3600.2.w.b 4
15.e even 4 1 inner 720.2.w.d 4
15.e even 4 1 3600.2.w.b 4
20.d odd 2 1 225.2.f.a 4
20.e even 4 1 45.2.f.a 4
20.e even 4 1 225.2.f.a 4
24.f even 2 1 2880.2.w.b 4
24.h odd 2 1 2880.2.w.k 4
36.f odd 6 2 405.2.m.a 8
36.h even 6 2 405.2.m.a 8
40.i odd 4 1 2880.2.w.k 4
40.k even 4 1 2880.2.w.b 4
60.h even 2 1 225.2.f.a 4
60.l odd 4 1 45.2.f.a 4
60.l odd 4 1 225.2.f.a 4
120.q odd 4 1 2880.2.w.b 4
120.w even 4 1 2880.2.w.k 4
180.v odd 12 2 405.2.m.a 8
180.x even 12 2 405.2.m.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.f.a 4 4.b odd 2 1
45.2.f.a 4 12.b even 2 1
45.2.f.a 4 20.e even 4 1
45.2.f.a 4 60.l odd 4 1
225.2.f.a 4 20.d odd 2 1
225.2.f.a 4 20.e even 4 1
225.2.f.a 4 60.h even 2 1
225.2.f.a 4 60.l odd 4 1
405.2.m.a 8 36.f odd 6 2
405.2.m.a 8 36.h even 6 2
405.2.m.a 8 180.v odd 12 2
405.2.m.a 8 180.x even 12 2
720.2.w.d 4 1.a even 1 1 trivial
720.2.w.d 4 3.b odd 2 1 inner
720.2.w.d 4 5.c odd 4 1 inner
720.2.w.d 4 15.e even 4 1 inner
2880.2.w.b 4 8.d odd 2 1
2880.2.w.b 4 24.f even 2 1
2880.2.w.b 4 40.k even 4 1
2880.2.w.b 4 120.q odd 4 1
2880.2.w.k 4 8.b even 2 1
2880.2.w.k 4 24.h odd 2 1
2880.2.w.k 4 40.i odd 4 1
2880.2.w.k 4 120.w even 4 1
3600.2.w.b 4 5.b even 2 1
3600.2.w.b 4 5.c odd 4 1
3600.2.w.b 4 15.d odd 2 1
3600.2.w.b 4 15.e even 4 1

## Hecke kernels

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

 $$T_{7}^{2} - 4 T_{7} + 8$$ $$T_{13}^{2} - 2 T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 - 8 T^{2} + T^{4}$$
$7$ $$( 8 - 4 T + T^{2} )^{2}$$
$11$ $$( 8 + T^{2} )^{2}$$
$13$ $$( 2 - 2 T + T^{2} )^{2}$$
$17$ $$256 + T^{4}$$
$19$ $$T^{4}$$
$23$ $$256 + T^{4}$$
$29$ $$( -18 + T^{2} )^{2}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$( 2 - 2 T + T^{2} )^{2}$$
$41$ $$( 2 + T^{2} )^{2}$$
$43$ $$( 128 - 16 T + T^{2} )^{2}$$
$47$ $$4096 + T^{4}$$
$53$ $$256 + T^{4}$$
$59$ $$( -72 + T^{2} )^{2}$$
$61$ $$( -8 + T )^{4}$$
$67$ $$( 32 + 8 T + T^{2} )^{2}$$
$71$ $$( 32 + T^{2} )^{2}$$
$73$ $$( 2 - 2 T + T^{2} )^{2}$$
$79$ $$( 144 + T^{2} )^{2}$$
$83$ $$256 + T^{4}$$
$89$ $$( -162 + T^{2} )^{2}$$
$97$ $$( 242 + 22 T + T^{2} )^{2}$$