# Properties

 Label 720.6.f.a 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 10) 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.

 $$f(q)$$ $$=$$ $$q + ( -55 - 10 i ) q^{5} + 158 i q^{7} +O(q^{10})$$ $$q + ( -55 - 10 i ) q^{5} + 158 i q^{7} -148 q^{11} + 684 i q^{13} + 2048 i q^{17} + 2220 q^{19} -1246 i q^{23} + ( 2925 + 1100 i ) q^{25} -270 q^{29} + 2048 q^{31} + ( 1580 - 8690 i ) q^{35} + 4372 i q^{37} + 2398 q^{41} -2294 i q^{43} + 10682 i q^{47} -8157 q^{49} -2964 i q^{53} + ( 8140 + 1480 i ) q^{55} + 39740 q^{59} -42298 q^{61} + ( 6840 - 37620 i ) q^{65} + 32098 i q^{67} -4248 q^{71} + 30104 i q^{73} -23384 i q^{77} + 35280 q^{79} -27826 i q^{83} + ( 20480 - 112640 i ) q^{85} -85210 q^{89} -108072 q^{91} + ( -122100 - 22200 i ) q^{95} + 97232 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 110q^{5} + O(q^{10})$$ $$2q - 110q^{5} - 296q^{11} + 4440q^{19} + 5850q^{25} - 540q^{29} + 4096q^{31} + 3160q^{35} + 4796q^{41} - 16314q^{49} + 16280q^{55} + 79480q^{59} - 84596q^{61} + 13680q^{65} - 8496q^{71} + 70560q^{79} + 40960q^{85} - 170420q^{89} - 216144q^{91} - 244200q^{95} + O(q^{100})$$

## 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 0 158.000i 0 0 0
289.2 0 0 0 −55.0000 + 10.0000i 0 158.000i 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.a 2
3.b odd 2 1 80.6.c.c 2
4.b odd 2 1 90.6.c.a 2
5.b even 2 1 inner 720.6.f.a 2
12.b even 2 1 10.6.b.a 2
15.d odd 2 1 80.6.c.c 2
15.e even 4 1 400.6.a.c 1
15.e even 4 1 400.6.a.k 1
20.d odd 2 1 90.6.c.a 2
20.e even 4 1 450.6.a.c 1
20.e even 4 1 450.6.a.w 1
24.f even 2 1 320.6.c.b 2
24.h odd 2 1 320.6.c.a 2
60.h even 2 1 10.6.b.a 2
60.l odd 4 1 50.6.a.c 1
60.l odd 4 1 50.6.a.e 1
120.i odd 2 1 320.6.c.a 2
120.m even 2 1 320.6.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.b.a 2 12.b even 2 1
10.6.b.a 2 60.h even 2 1
50.6.a.c 1 60.l odd 4 1
50.6.a.e 1 60.l odd 4 1
80.6.c.c 2 3.b odd 2 1
80.6.c.c 2 15.d odd 2 1
90.6.c.a 2 4.b odd 2 1
90.6.c.a 2 20.d odd 2 1
320.6.c.a 2 24.h odd 2 1
320.6.c.a 2 120.i odd 2 1
320.6.c.b 2 24.f even 2 1
320.6.c.b 2 120.m even 2 1
400.6.a.c 1 15.e even 4 1
400.6.a.k 1 15.e even 4 1
450.6.a.c 1 20.e even 4 1
450.6.a.w 1 20.e even 4 1
720.6.f.a 2 1.a even 1 1 trivial
720.6.f.a 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 $$S_{6}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{2} + 24964$$ $$T_{11} + 148$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 110 T + 3125 T^{2}$$
$7$ $$1 - 8650 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 + 148 T + 161051 T^{2} )^{2}$$
$13$ $$1 - 274730 T^{2} + 137858491849 T^{4}$$
$17$ $$1 + 1354590 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 - 2220 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 11320170 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 + 270 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 - 2048 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 119573530 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 - 2398 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 288754450 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 344584890 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 - 827605690 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 - 39740 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 + 42298 T + 844596301 T^{2} )^{2}$$
$67$ $$1 - 1669968610 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 + 4248 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 3239892370 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 - 35280 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 7103795010 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 + 85210 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 - 7720618690 T^{2} + 73742412689492826049 T^{4}$$