# Properties

 Label 720.6.f.f 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: $$\Q(\sqrt{-11})$$ Defining polynomial: $$x^{2} - x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 45 + 5 \beta ) q^{5} -9 \beta q^{7} +O(q^{10})$$ $$q + ( 45 + 5 \beta ) q^{5} -9 \beta q^{7} + 252 q^{11} + 18 \beta q^{13} -104 \beta q^{17} + 220 q^{19} -367 \beta q^{23} + ( 925 + 450 \beta ) q^{25} + 6930 q^{29} -6752 q^{31} + ( 1980 - 405 \beta ) q^{35} -2106 \beta q^{37} + 198 q^{41} -63 \beta q^{43} + 1589 \beta q^{47} + 13243 q^{49} -878 \beta q^{53} + ( 11340 + 1260 \beta ) q^{55} -24660 q^{59} -5698 q^{61} + ( -3960 + 810 \beta ) q^{65} -6579 \beta q^{67} + 53352 q^{71} -10692 \beta q^{73} -2268 \beta q^{77} -51920 q^{79} + 9323 \beta q^{83} + ( 22880 - 4680 \beta ) q^{85} + 9990 q^{89} + 7128 q^{91} + ( 9900 + 1100 \beta ) q^{95} + 15264 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 90q^{5} + O(q^{10})$$ $$2q + 90q^{5} + 504q^{11} + 440q^{19} + 1850q^{25} + 13860q^{29} - 13504q^{31} + 3960q^{35} + 396q^{41} + 26486q^{49} + 22680q^{55} - 49320q^{59} - 11396q^{61} - 7920q^{65} + 106704q^{71} - 103840q^{79} + 45760q^{85} + 19980q^{89} + 14256q^{91} + 19800q^{95} + O(q^{100})$$

## 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 − 1.65831i 0.5 + 1.65831i
0 0 0 45.0000 33.1662i 0 59.6992i 0 0 0
289.2 0 0 0 45.0000 + 33.1662i 0 59.6992i 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.f 2
3.b odd 2 1 80.6.c.a 2
4.b odd 2 1 45.6.b.b 2
5.b even 2 1 inner 720.6.f.f 2
12.b even 2 1 5.6.b.a 2
15.d odd 2 1 80.6.c.a 2
15.e even 4 2 400.6.a.t 2
20.d odd 2 1 45.6.b.b 2
20.e even 4 2 225.6.a.n 2
24.f even 2 1 320.6.c.f 2
24.h odd 2 1 320.6.c.g 2
60.h even 2 1 5.6.b.a 2
60.l odd 4 2 25.6.a.c 2
84.h odd 2 1 245.6.b.a 2
120.i odd 2 1 320.6.c.g 2
120.m even 2 1 320.6.c.f 2
420.o odd 2 1 245.6.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.b.a 2 12.b even 2 1
5.6.b.a 2 60.h even 2 1
25.6.a.c 2 60.l odd 4 2
45.6.b.b 2 4.b odd 2 1
45.6.b.b 2 20.d odd 2 1
80.6.c.a 2 3.b odd 2 1
80.6.c.a 2 15.d odd 2 1
225.6.a.n 2 20.e even 4 2
245.6.b.a 2 84.h odd 2 1
245.6.b.a 2 420.o odd 2 1
320.6.c.f 2 24.f even 2 1
320.6.c.f 2 120.m even 2 1
320.6.c.g 2 24.h odd 2 1
320.6.c.g 2 120.i odd 2 1
400.6.a.t 2 15.e even 4 2
720.6.f.f 2 1.a even 1 1 trivial
720.6.f.f 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 $$S_{6}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{2} + 3564$$ $$T_{11} - 252$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 90 T + 3125 T^{2}$$
$7$ $$1 - 30050 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 - 252 T + 161051 T^{2} )^{2}$$
$13$ $$1 - 728330 T^{2} + 137858491849 T^{4}$$
$17$ $$1 - 2363810 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 - 220 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 6946370 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 - 6930 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 + 6752 T + 28629151 T^{2} )^{2}$$
$37$ $$1 + 56462470 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 - 198 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 293842250 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 347593490 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 - 802472090 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 + 24660 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 + 5698 T + 844596301 T^{2} )^{2}$$
$67$ $$1 - 795787610 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 - 53352 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 + 883886830 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 + 51920 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 4053674810 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 - 9990 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 - 6923133890 T^{2} + 73742412689492826049 T^{4}$$