# Properties

 Label 360.6.a.b Level 360 Weight 6 Character orbit 360.a Self dual yes Analytic conductor 57.738 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 360.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$57.7381751327$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 25q^{5} - 108q^{7} + O(q^{10})$$ $$q - 25q^{5} - 108q^{7} + 604q^{11} - 306q^{13} - 930q^{17} - 1324q^{19} + 852q^{23} + 625q^{25} - 5902q^{29} - 3320q^{31} + 2700q^{35} + 10774q^{37} + 17958q^{41} + 9264q^{43} + 9796q^{47} - 5143q^{49} + 31434q^{53} - 15100q^{55} - 33228q^{59} - 40210q^{61} + 7650q^{65} + 58864q^{67} + 55312q^{71} + 27258q^{73} - 65232q^{77} + 31456q^{79} - 24552q^{83} + 23250q^{85} + 90854q^{89} + 33048q^{91} + 33100q^{95} + 154706q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 −25.0000 0 −108.000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.6.a.b 1
3.b odd 2 1 40.6.a.b 1
4.b odd 2 1 720.6.a.h 1
12.b even 2 1 80.6.a.f 1
15.d odd 2 1 200.6.a.c 1
15.e even 4 2 200.6.c.c 2
24.f even 2 1 320.6.a.e 1
24.h odd 2 1 320.6.a.l 1
60.h even 2 1 400.6.a.f 1
60.l odd 4 2 400.6.c.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.b 1 3.b odd 2 1
80.6.a.f 1 12.b even 2 1
200.6.a.c 1 15.d odd 2 1
200.6.c.c 2 15.e even 4 2
320.6.a.e 1 24.f even 2 1
320.6.a.l 1 24.h odd 2 1
360.6.a.b 1 1.a even 1 1 trivial
400.6.a.f 1 60.h even 2 1
400.6.c.h 2 60.l odd 4 2
720.6.a.h 1 4.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(360))$$:

 $$T_{7} + 108$$ $$T_{11} - 604$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 25 T$$
$7$ $$1 + 108 T + 16807 T^{2}$$
$11$ $$1 - 604 T + 161051 T^{2}$$
$13$ $$1 + 306 T + 371293 T^{2}$$
$17$ $$1 + 930 T + 1419857 T^{2}$$
$19$ $$1 + 1324 T + 2476099 T^{2}$$
$23$ $$1 - 852 T + 6436343 T^{2}$$
$29$ $$1 + 5902 T + 20511149 T^{2}$$
$31$ $$1 + 3320 T + 28629151 T^{2}$$
$37$ $$1 - 10774 T + 69343957 T^{2}$$
$41$ $$1 - 17958 T + 115856201 T^{2}$$
$43$ $$1 - 9264 T + 147008443 T^{2}$$
$47$ $$1 - 9796 T + 229345007 T^{2}$$
$53$ $$1 - 31434 T + 418195493 T^{2}$$
$59$ $$1 + 33228 T + 714924299 T^{2}$$
$61$ $$1 + 40210 T + 844596301 T^{2}$$
$67$ $$1 - 58864 T + 1350125107 T^{2}$$
$71$ $$1 - 55312 T + 1804229351 T^{2}$$
$73$ $$1 - 27258 T + 2073071593 T^{2}$$
$79$ $$1 - 31456 T + 3077056399 T^{2}$$
$83$ $$1 + 24552 T + 3939040643 T^{2}$$
$89$ $$1 - 90854 T + 5584059449 T^{2}$$
$97$ $$1 - 154706 T + 8587340257 T^{2}$$