# Properties

 Label 720.4.f.f Level $720$ Weight $4$ Character orbit 720.f Analytic conductor $42.481$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 720.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$42.4813752041$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 5 - 10 i ) q^{5} + 26 i q^{7} +O(q^{10})$$ $$q + ( 5 - 10 i ) q^{5} + 26 i q^{7} -28 q^{11} + 12 i q^{13} -64 i q^{17} -60 q^{19} -58 i q^{23} + ( -75 - 100 i ) q^{25} + 90 q^{29} + 128 q^{31} + ( 260 + 130 i ) q^{35} -236 i q^{37} -242 q^{41} -362 i q^{43} -226 i q^{47} -333 q^{49} + 108 i q^{53} + ( -140 + 280 i ) q^{55} + 20 q^{59} + 542 q^{61} + ( 120 + 60 i ) q^{65} -434 i q^{67} -1128 q^{71} + 632 i q^{73} -728 i q^{77} -720 q^{79} -478 i q^{83} + ( -640 - 320 i ) q^{85} -490 q^{89} -312 q^{91} + ( -300 + 600 i ) q^{95} -1456 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{5} + O(q^{10})$$ $$2 q + 10 q^{5} - 56 q^{11} - 120 q^{19} - 150 q^{25} + 180 q^{29} + 256 q^{31} + 520 q^{35} - 484 q^{41} - 666 q^{49} - 280 q^{55} + 40 q^{59} + 1084 q^{61} + 240 q^{65} - 2256 q^{71} - 1440 q^{79} - 1280 q^{85} - 980 q^{89} - 624 q^{91} - 600 q^{95} + O(q^{100})$$

## 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 5.00000 10.0000i 0 26.0000i 0 0 0
289.2 0 0 0 5.00000 + 10.0000i 0 26.0000i 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 720.4.f.f 2
3.b odd 2 1 80.4.c.a 2
4.b odd 2 1 90.4.c.b 2
5.b even 2 1 inner 720.4.f.f 2
12.b even 2 1 10.4.b.a 2
15.d odd 2 1 80.4.c.a 2
15.e even 4 1 400.4.a.h 1
15.e even 4 1 400.4.a.n 1
20.d odd 2 1 90.4.c.b 2
20.e even 4 1 450.4.a.j 1
20.e even 4 1 450.4.a.k 1
24.f even 2 1 320.4.c.d 2
24.h odd 2 1 320.4.c.c 2
60.h even 2 1 10.4.b.a 2
60.l odd 4 1 50.4.a.b 1
60.l odd 4 1 50.4.a.d 1
84.h odd 2 1 490.4.c.b 2
120.i odd 2 1 320.4.c.c 2
120.m even 2 1 320.4.c.d 2
120.q odd 4 1 1600.4.a.u 1
120.q odd 4 1 1600.4.a.bh 1
120.w even 4 1 1600.4.a.t 1
120.w even 4 1 1600.4.a.bg 1
420.o odd 2 1 490.4.c.b 2
420.w even 4 1 2450.4.a.o 1
420.w even 4 1 2450.4.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 12.b even 2 1
10.4.b.a 2 60.h even 2 1
50.4.a.b 1 60.l odd 4 1
50.4.a.d 1 60.l odd 4 1
80.4.c.a 2 3.b odd 2 1
80.4.c.a 2 15.d odd 2 1
90.4.c.b 2 4.b odd 2 1
90.4.c.b 2 20.d odd 2 1
320.4.c.c 2 24.h odd 2 1
320.4.c.c 2 120.i odd 2 1
320.4.c.d 2 24.f even 2 1
320.4.c.d 2 120.m even 2 1
400.4.a.h 1 15.e even 4 1
400.4.a.n 1 15.e even 4 1
450.4.a.j 1 20.e even 4 1
450.4.a.k 1 20.e even 4 1
490.4.c.b 2 84.h odd 2 1
490.4.c.b 2 420.o odd 2 1
720.4.f.f 2 1.a even 1 1 trivial
720.4.f.f 2 5.b even 2 1 inner
1600.4.a.t 1 120.w even 4 1
1600.4.a.u 1 120.q odd 4 1
1600.4.a.bg 1 120.w even 4 1
1600.4.a.bh 1 120.q odd 4 1
2450.4.a.o 1 420.w even 4 1
2450.4.a.bb 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 $$S_{4}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{2} + 676$$ $$T_{11} + 28$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$125 - 10 T + T^{2}$$
$7$ $$676 + T^{2}$$
$11$ $$( 28 + T )^{2}$$
$13$ $$144 + T^{2}$$
$17$ $$4096 + T^{2}$$
$19$ $$( 60 + T )^{2}$$
$23$ $$3364 + T^{2}$$
$29$ $$( -90 + T )^{2}$$
$31$ $$( -128 + T )^{2}$$
$37$ $$55696 + T^{2}$$
$41$ $$( 242 + T )^{2}$$
$43$ $$131044 + T^{2}$$
$47$ $$51076 + T^{2}$$
$53$ $$11664 + T^{2}$$
$59$ $$( -20 + T )^{2}$$
$61$ $$( -542 + T )^{2}$$
$67$ $$188356 + T^{2}$$
$71$ $$( 1128 + T )^{2}$$
$73$ $$399424 + T^{2}$$
$79$ $$( 720 + T )^{2}$$
$83$ $$228484 + T^{2}$$
$89$ $$( 490 + T )^{2}$$
$97$ $$2119936 + T^{2}$$