# Properties

 Label 60.6.a.a Level $60$ Weight $6$ Character orbit 60.a Self dual yes Analytic conductor $9.623$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 60.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.62302918878$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 9q^{3} - 25q^{5} + 44q^{7} + 81q^{9} + O(q^{10})$$ $$q - 9q^{3} - 25q^{5} + 44q^{7} + 81q^{9} + 216q^{11} + 770q^{13} + 225q^{15} + 534q^{17} + 1580q^{19} - 396q^{21} + 2904q^{23} + 625q^{25} - 729q^{27} - 4566q^{29} + 2744q^{31} - 1944q^{33} - 1100q^{35} + 1442q^{37} - 6930q^{39} - 13350q^{41} + 17204q^{43} - 2025q^{45} - 10824q^{47} - 14871q^{49} - 4806q^{51} - 9942q^{53} - 5400q^{55} - 14220q^{57} - 15576q^{59} + 39302q^{61} + 3564q^{63} - 19250q^{65} + 55796q^{67} - 26136q^{69} + 57120q^{71} + 50402q^{73} - 5625q^{75} + 9504q^{77} - 10552q^{79} + 6561q^{81} + 108564q^{83} - 13350q^{85} + 41094q^{87} - 116430q^{89} + 33880q^{91} - 24696q^{93} - 39500q^{95} - 2782q^{97} + 17496q^{99} + O(q^{100})$$

## 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 −9.00000 0 −25.0000 0 44.0000 0 81.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 60.6.a.a 1
3.b odd 2 1 180.6.a.d 1
4.b odd 2 1 240.6.a.i 1
5.b even 2 1 300.6.a.d 1
5.c odd 4 2 300.6.d.e 2
8.b even 2 1 960.6.a.z 1
8.d odd 2 1 960.6.a.i 1
12.b even 2 1 720.6.a.p 1
15.d odd 2 1 900.6.a.f 1
15.e even 4 2 900.6.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.a.a 1 1.a even 1 1 trivial
180.6.a.d 1 3.b odd 2 1
240.6.a.i 1 4.b odd 2 1
300.6.a.d 1 5.b even 2 1
300.6.d.e 2 5.c odd 4 2
720.6.a.p 1 12.b even 2 1
900.6.a.f 1 15.d odd 2 1
900.6.d.b 2 15.e even 4 2
960.6.a.i 1 8.d odd 2 1
960.6.a.z 1 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} - 44$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(60))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$9 + T$$
$5$ $$25 + T$$
$7$ $$-44 + T$$
$11$ $$-216 + T$$
$13$ $$-770 + T$$
$17$ $$-534 + T$$
$19$ $$-1580 + T$$
$23$ $$-2904 + T$$
$29$ $$4566 + T$$
$31$ $$-2744 + T$$
$37$ $$-1442 + T$$
$41$ $$13350 + T$$
$43$ $$-17204 + T$$
$47$ $$10824 + T$$
$53$ $$9942 + T$$
$59$ $$15576 + T$$
$61$ $$-39302 + T$$
$67$ $$-55796 + T$$
$71$ $$-57120 + T$$
$73$ $$-50402 + T$$
$79$ $$10552 + T$$
$83$ $$-108564 + T$$
$89$ $$116430 + T$$
$97$ $$2782 + T$$