# Properties

 Label 720.6.f.g Level $720$ Weight $6$ Character orbit 720.f Analytic conductor $115.476$ 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: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 720.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$115.476350265$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 55 - 10 i ) q^{5} + 4 i q^{7} +O(q^{10})$$ $$q + ( 55 - 10 i ) q^{5} + 4 i q^{7} -500 q^{11} + 288 i q^{13} -1516 i q^{17} -1344 q^{19} + 4100 i q^{23} + ( 2925 - 1100 i ) q^{25} -2646 q^{29} + 5612 q^{31} + ( 40 + 220 i ) q^{35} -7288 i q^{37} + 18986 q^{41} -2404 i q^{43} + 8900 i q^{47} + 16791 q^{49} + 39804 i q^{53} + ( -27500 + 5000 i ) q^{55} + 28300 q^{59} + 18290 q^{61} + ( 2880 + 15840 i ) q^{65} -65956 i q^{67} -28800 q^{71} + 30808 i q^{73} -2000 i q^{77} + 60228 q^{79} + 2468 i q^{83} + ( -15160 - 83380 i ) q^{85} + 22678 q^{89} -1152 q^{91} + ( -73920 + 13440 i ) q^{95} -36968 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 110q^{5} + O(q^{10})$$ $$2q + 110q^{5} - 1000q^{11} - 2688q^{19} + 5850q^{25} - 5292q^{29} + 11224q^{31} + 80q^{35} + 37972q^{41} + 33582q^{49} - 55000q^{55} + 56600q^{59} + 36580q^{61} + 5760q^{65} - 57600q^{71} + 120456q^{79} - 30320q^{85} + 45356q^{89} - 2304q^{91} - 147840q^{95} + O(q^{100})$$

## 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 55.0000 10.0000i 0 4.00000i 0 0 0
289.2 0 0 0 55.0000 + 10.0000i 0 4.00000i 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 720.6.f.g 2
3.b odd 2 1 240.6.f.a 2
4.b odd 2 1 90.6.c.b 2
5.b even 2 1 inner 720.6.f.g 2
12.b even 2 1 30.6.c.a 2
15.d odd 2 1 240.6.f.a 2
20.d odd 2 1 90.6.c.b 2
20.e even 4 1 450.6.a.f 1
20.e even 4 1 450.6.a.s 1
60.h even 2 1 30.6.c.a 2
60.l odd 4 1 150.6.a.f 1
60.l odd 4 1 150.6.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.a 2 12.b even 2 1
30.6.c.a 2 60.h even 2 1
90.6.c.b 2 4.b odd 2 1
90.6.c.b 2 20.d odd 2 1
150.6.a.f 1 60.l odd 4 1
150.6.a.j 1 60.l odd 4 1
240.6.f.a 2 3.b odd 2 1
240.6.f.a 2 15.d odd 2 1
450.6.a.f 1 20.e even 4 1
450.6.a.s 1 20.e even 4 1
720.6.f.g 2 1.a even 1 1 trivial
720.6.f.g 2 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{2} + 16$$ $$T_{11} + 500$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$3125 - 110 T + T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$( 500 + T )^{2}$$
$13$ $$82944 + T^{2}$$
$17$ $$2298256 + T^{2}$$
$19$ $$( 1344 + T )^{2}$$
$23$ $$16810000 + T^{2}$$
$29$ $$( 2646 + T )^{2}$$
$31$ $$( -5612 + T )^{2}$$
$37$ $$53114944 + T^{2}$$
$41$ $$( -18986 + T )^{2}$$
$43$ $$5779216 + T^{2}$$
$47$ $$79210000 + T^{2}$$
$53$ $$1584358416 + T^{2}$$
$59$ $$( -28300 + T )^{2}$$
$61$ $$( -18290 + T )^{2}$$
$67$ $$4350193936 + T^{2}$$
$71$ $$( 28800 + T )^{2}$$
$73$ $$949132864 + T^{2}$$
$79$ $$( -60228 + T )^{2}$$
$83$ $$6091024 + T^{2}$$
$89$ $$( -22678 + T )^{2}$$
$97$ $$1366633024 + T^{2}$$