# Properties

 Label 720.6.f.d 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: $$\Q(\sqrt{-31})$$ Defining polynomial: $$x^{2} - x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-31}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 5 - 5 \beta ) q^{5} -11 \beta q^{7} +O(q^{10})$$ $$q + ( 5 - 5 \beta ) q^{5} -11 \beta q^{7} -100 q^{11} -66 \beta q^{13} + 88 \beta q^{17} -2244 q^{19} + 307 \beta q^{23} + ( -3075 - 50 \beta ) q^{25} -7854 q^{29} + 2144 q^{31} + ( -6820 - 55 \beta ) q^{35} -934 \beta q^{37} + 7414 q^{41} + 1595 \beta q^{43} + 847 \beta q^{47} + 1803 q^{49} -2178 \beta q^{53} + ( -500 + 500 \beta ) q^{55} -25972 q^{59} -3058 q^{61} + ( -40920 - 330 \beta ) q^{65} + 5279 \beta q^{67} + 37608 q^{71} -2156 \beta q^{73} + 1100 \beta q^{77} + 79728 q^{79} -1463 \beta q^{83} + ( 54560 + 440 \beta ) q^{85} -826 q^{89} -90024 q^{91} + ( -11220 + 11220 \beta ) q^{95} + 3376 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 10q^{5} + O(q^{10})$$ $$2q + 10q^{5} - 200q^{11} - 4488q^{19} - 6150q^{25} - 15708q^{29} + 4288q^{31} - 13640q^{35} + 14828q^{41} + 3606q^{49} - 1000q^{55} - 51944q^{59} - 6116q^{61} - 81840q^{65} + 75216q^{71} + 159456q^{79} + 109120q^{85} - 1652q^{89} - 180048q^{91} - 22440q^{95} + O(q^{100})$$

## 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
 0.5 + 2.78388i 0.5 − 2.78388i
0 0 0 5.00000 55.6776i 0 122.491i 0 0 0
289.2 0 0 0 5.00000 + 55.6776i 0 122.491i 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.d 2
3.b odd 2 1 80.6.c.b 2
4.b odd 2 1 180.6.d.b 2
5.b even 2 1 inner 720.6.f.d 2
12.b even 2 1 20.6.c.a 2
15.d odd 2 1 80.6.c.b 2
15.e even 4 2 400.6.a.s 2
20.d odd 2 1 180.6.d.b 2
20.e even 4 2 900.6.a.q 2
24.f even 2 1 320.6.c.e 2
24.h odd 2 1 320.6.c.d 2
60.h even 2 1 20.6.c.a 2
60.l odd 4 2 100.6.a.d 2
120.i odd 2 1 320.6.c.d 2
120.m even 2 1 320.6.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.c.a 2 12.b even 2 1
20.6.c.a 2 60.h even 2 1
80.6.c.b 2 3.b odd 2 1
80.6.c.b 2 15.d odd 2 1
100.6.a.d 2 60.l odd 4 2
180.6.d.b 2 4.b odd 2 1
180.6.d.b 2 20.d odd 2 1
320.6.c.d 2 24.h odd 2 1
320.6.c.d 2 120.i odd 2 1
320.6.c.e 2 24.f even 2 1
320.6.c.e 2 120.m even 2 1
400.6.a.s 2 15.e even 4 2
720.6.f.d 2 1.a even 1 1 trivial
720.6.f.d 2 5.b even 2 1 inner
900.6.a.q 2 20.e even 4 2

## Hecke kernels

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

 $$T_{7}^{2} + 15004$$ $$T_{11} + 100$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 10 T + 3125 T^{2}$$
$7$ $$1 - 18610 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 + 100 T + 161051 T^{2} )^{2}$$
$13$ $$1 - 202442 T^{2} + 137858491849 T^{4}$$
$17$ $$1 - 1879458 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 + 2244 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 1185810 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 + 7854 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 - 2144 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 30515770 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 - 7414 T + 115856201 T^{2} )^{2}$$
$43$ $$1 + 21442214 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 369731298 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 - 248174170 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 + 25972 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 + 3058 T + 844596301 T^{2} )^{2}$$
$67$ $$1 + 755362070 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 - 37608 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 3569749522 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 - 79728 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 7612675530 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 + 826 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 - 15761405890 T^{2} + 73742412689492826049 T^{4}$$