# Properties

 Label 720.2.f.f 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: $$1$$ Twist minimal: no (minimal twist has level 30) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 + i ) q^{5} -2 i q^{7} +O(q^{10})$$ $$q + ( 2 + i ) q^{5} -2 i q^{7} + 2 q^{11} -6 i q^{13} -2 i q^{17} + 4 i q^{23} + ( 3 + 4 i ) q^{25} + 8 q^{31} + ( 2 - 4 i ) q^{35} + 2 i q^{37} -2 q^{41} -4 i q^{43} -8 i q^{47} + 3 q^{49} + 6 i q^{53} + ( 4 + 2 i ) q^{55} -10 q^{59} + 2 q^{61} + ( 6 - 12 i ) q^{65} + 8 i q^{67} + 12 q^{71} + 4 i q^{73} -4 i q^{77} + 4 i q^{83} + ( 2 - 4 i ) q^{85} -10 q^{89} -12 q^{91} -8 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{5} + O(q^{10})$$ $$2q + 4q^{5} + 4q^{11} + 6q^{25} + 16q^{31} + 4q^{35} - 4q^{41} + 6q^{49} + 8q^{55} - 20q^{59} + 4q^{61} + 12q^{65} + 24q^{71} + 4q^{85} - 20q^{89} - 24q^{91} + O(q^{100})$$

## 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 2.00000 1.00000i 0 2.00000i 0 0 0
289.2 0 0 0 2.00000 + 1.00000i 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.f 2
3.b odd 2 1 240.2.f.a 2
4.b odd 2 1 90.2.c.a 2
5.b even 2 1 inner 720.2.f.f 2
5.c odd 4 1 3600.2.a.o 1
5.c odd 4 1 3600.2.a.bg 1
8.b even 2 1 2880.2.f.c 2
8.d odd 2 1 2880.2.f.e 2
12.b even 2 1 30.2.c.a 2
15.d odd 2 1 240.2.f.a 2
15.e even 4 1 1200.2.a.g 1
15.e even 4 1 1200.2.a.m 1
20.d odd 2 1 90.2.c.a 2
20.e even 4 1 450.2.a.b 1
20.e even 4 1 450.2.a.f 1
24.f even 2 1 960.2.f.h 2
24.h odd 2 1 960.2.f.i 2
36.f odd 6 2 810.2.i.b 4
36.h even 6 2 810.2.i.e 4
40.e odd 2 1 2880.2.f.e 2
40.f even 2 1 2880.2.f.c 2
48.i odd 4 1 3840.2.d.j 2
48.i odd 4 1 3840.2.d.x 2
48.k even 4 1 3840.2.d.g 2
48.k even 4 1 3840.2.d.y 2
60.h even 2 1 30.2.c.a 2
60.l odd 4 1 150.2.a.a 1
60.l odd 4 1 150.2.a.c 1
84.h odd 2 1 1470.2.g.g 2
84.j odd 6 2 1470.2.n.a 4
84.n even 6 2 1470.2.n.h 4
120.i odd 2 1 960.2.f.i 2
120.m even 2 1 960.2.f.h 2
120.q odd 4 1 4800.2.a.l 1
120.q odd 4 1 4800.2.a.cg 1
120.w even 4 1 4800.2.a.m 1
120.w even 4 1 4800.2.a.cj 1
180.n even 6 2 810.2.i.e 4
180.p odd 6 2 810.2.i.b 4
240.t even 4 1 3840.2.d.g 2
240.t even 4 1 3840.2.d.y 2
240.bm odd 4 1 3840.2.d.j 2
240.bm odd 4 1 3840.2.d.x 2
420.o odd 2 1 1470.2.g.g 2
420.w even 4 1 7350.2.a.bg 1
420.w even 4 1 7350.2.a.cc 1
420.ba even 6 2 1470.2.n.h 4
420.be odd 6 2 1470.2.n.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 12.b even 2 1
30.2.c.a 2 60.h even 2 1
90.2.c.a 2 4.b odd 2 1
90.2.c.a 2 20.d odd 2 1
150.2.a.a 1 60.l odd 4 1
150.2.a.c 1 60.l odd 4 1
240.2.f.a 2 3.b odd 2 1
240.2.f.a 2 15.d odd 2 1
450.2.a.b 1 20.e even 4 1
450.2.a.f 1 20.e even 4 1
720.2.f.f 2 1.a even 1 1 trivial
720.2.f.f 2 5.b even 2 1 inner
810.2.i.b 4 36.f odd 6 2
810.2.i.b 4 180.p odd 6 2
810.2.i.e 4 36.h even 6 2
810.2.i.e 4 180.n even 6 2
960.2.f.h 2 24.f even 2 1
960.2.f.h 2 120.m even 2 1
960.2.f.i 2 24.h odd 2 1
960.2.f.i 2 120.i odd 2 1
1200.2.a.g 1 15.e even 4 1
1200.2.a.m 1 15.e even 4 1
1470.2.g.g 2 84.h odd 2 1
1470.2.g.g 2 420.o odd 2 1
1470.2.n.a 4 84.j odd 6 2
1470.2.n.a 4 420.be odd 6 2
1470.2.n.h 4 84.n even 6 2
1470.2.n.h 4 420.ba even 6 2
2880.2.f.c 2 8.b even 2 1
2880.2.f.c 2 40.f even 2 1
2880.2.f.e 2 8.d odd 2 1
2880.2.f.e 2 40.e odd 2 1
3600.2.a.o 1 5.c odd 4 1
3600.2.a.bg 1 5.c odd 4 1
3840.2.d.g 2 48.k even 4 1
3840.2.d.g 2 240.t even 4 1
3840.2.d.j 2 48.i odd 4 1
3840.2.d.j 2 240.bm odd 4 1
3840.2.d.x 2 48.i odd 4 1
3840.2.d.x 2 240.bm odd 4 1
3840.2.d.y 2 48.k even 4 1
3840.2.d.y 2 240.t even 4 1
4800.2.a.l 1 120.q odd 4 1
4800.2.a.m 1 120.w even 4 1
4800.2.a.cg 1 120.q odd 4 1
4800.2.a.cj 1 120.w even 4 1
7350.2.a.bg 1 420.w even 4 1
7350.2.a.cc 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 $$S_{2}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{2} + 4$$ $$T_{11} - 2$$ $$T_{13}^{2} + 36$$

## Hecke characteristic polynomials

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