# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.f (of order $$2$$, degree $$1$$, not 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + 2 i ) q^{5} + 2 i q^{7} +O(q^{10})$$ $$q + ( 1 + 2 i ) q^{5} + 2 i q^{7} -4 q^{11} + 4 i q^{13} -4 q^{19} -2 i q^{23} + ( -3 + 4 i ) q^{25} + 2 q^{29} + ( -4 + 2 i ) q^{35} -4 i q^{37} -2 q^{41} + 6 i q^{43} + 6 i q^{47} + 3 q^{49} + 4 i q^{53} + ( -4 - 8 i ) q^{55} + 12 q^{59} -10 q^{61} + ( -8 + 4 i ) q^{65} + 14 i q^{67} + 8 q^{71} + 8 i q^{73} -8 i q^{77} + 16 q^{79} + 2 i q^{83} + 6 q^{89} -8 q^{91} + ( -4 - 8 i ) q^{95} -16 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + O(q^{10})$$ $$2q + 2q^{5} - 8q^{11} - 8q^{19} - 6q^{25} + 4q^{29} - 8q^{35} - 4q^{41} + 6q^{49} - 8q^{55} + 24q^{59} - 20q^{61} - 16q^{65} + 16q^{71} + 32q^{79} + 12q^{89} - 16q^{91} - 8q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 − 1.00000i 1.00000i
0 0 0 1.00000 2.00000i 0 2.00000i 0 0 0
289.2 0 0 0 1.00000 + 2.00000i 0 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
5.b 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.f.e 2
3.b odd 2 1 80.2.c.a 2
4.b odd 2 1 360.2.f.c 2
5.b even 2 1 inner 720.2.f.e 2
5.c odd 4 1 3600.2.a.k 1
5.c odd 4 1 3600.2.a.bb 1
8.b even 2 1 2880.2.f.i 2
8.d odd 2 1 2880.2.f.h 2
12.b even 2 1 40.2.c.a 2
15.d odd 2 1 80.2.c.a 2
15.e even 4 1 400.2.a.b 1
15.e even 4 1 400.2.a.g 1
20.d odd 2 1 360.2.f.c 2
20.e even 4 1 1800.2.a.j 1
20.e even 4 1 1800.2.a.s 1
24.f even 2 1 320.2.c.c 2
24.h odd 2 1 320.2.c.b 2
40.e odd 2 1 2880.2.f.h 2
40.f even 2 1 2880.2.f.i 2
48.i odd 4 1 1280.2.f.b 2
48.i odd 4 1 1280.2.f.e 2
48.k even 4 1 1280.2.f.a 2
48.k even 4 1 1280.2.f.f 2
60.h even 2 1 40.2.c.a 2
60.l odd 4 1 200.2.a.b 1
60.l odd 4 1 200.2.a.d 1
84.h odd 2 1 1960.2.g.b 2
120.i odd 2 1 320.2.c.b 2
120.m even 2 1 320.2.c.c 2
120.q odd 4 1 1600.2.a.f 1
120.q odd 4 1 1600.2.a.v 1
120.w even 4 1 1600.2.a.d 1
120.w even 4 1 1600.2.a.u 1
240.t even 4 1 1280.2.f.a 2
240.t even 4 1 1280.2.f.f 2
240.bm odd 4 1 1280.2.f.b 2
240.bm odd 4 1 1280.2.f.e 2
420.o odd 2 1 1960.2.g.b 2
420.w even 4 1 9800.2.a.d 1
420.w even 4 1 9800.2.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.c.a 2 12.b even 2 1
40.2.c.a 2 60.h even 2 1
80.2.c.a 2 3.b odd 2 1
80.2.c.a 2 15.d odd 2 1
200.2.a.b 1 60.l odd 4 1
200.2.a.d 1 60.l odd 4 1
320.2.c.b 2 24.h odd 2 1
320.2.c.b 2 120.i odd 2 1
320.2.c.c 2 24.f even 2 1
320.2.c.c 2 120.m even 2 1
360.2.f.c 2 4.b odd 2 1
360.2.f.c 2 20.d odd 2 1
400.2.a.b 1 15.e even 4 1
400.2.a.g 1 15.e even 4 1
720.2.f.e 2 1.a even 1 1 trivial
720.2.f.e 2 5.b even 2 1 inner
1280.2.f.a 2 48.k even 4 1
1280.2.f.a 2 240.t even 4 1
1280.2.f.b 2 48.i odd 4 1
1280.2.f.b 2 240.bm odd 4 1
1280.2.f.e 2 48.i odd 4 1
1280.2.f.e 2 240.bm odd 4 1
1280.2.f.f 2 48.k even 4 1
1280.2.f.f 2 240.t even 4 1
1600.2.a.d 1 120.w even 4 1
1600.2.a.f 1 120.q odd 4 1
1600.2.a.u 1 120.w even 4 1
1600.2.a.v 1 120.q odd 4 1
1800.2.a.j 1 20.e even 4 1
1800.2.a.s 1 20.e even 4 1
1960.2.g.b 2 84.h odd 2 1
1960.2.g.b 2 420.o odd 2 1
2880.2.f.h 2 8.d odd 2 1
2880.2.f.h 2 40.e odd 2 1
2880.2.f.i 2 8.b even 2 1
2880.2.f.i 2 40.f even 2 1
3600.2.a.k 1 5.c odd 4 1
3600.2.a.bb 1 5.c odd 4 1
9800.2.a.d 1 420.w even 4 1
9800.2.a.bf 1 420.w even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ $$T_{11} + 4$$ $$T_{13}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 - 2 T + T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$4 + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$16 + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$36 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$16 + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$196 + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$64 + T^{2}$$
$79$ $$( -16 + T )^{2}$$
$83$ $$4 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$256 + T^{2}$$