# Properties

 Label 60.3.g.a Level $60$ Weight $3$ Character orbit 60.g Analytic conductor $1.635$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 60.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.63488158616$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ Defining polynomial: $$x^{2} + 5$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 + \beta ) q^{3} + \beta q^{5} + 2 q^{7} + ( -1 + 4 \beta ) q^{9} +O(q^{10})$$ $$q + ( 2 + \beta ) q^{3} + \beta q^{5} + 2 q^{7} + ( -1 + 4 \beta ) q^{9} -6 \beta q^{11} + 8 q^{13} + ( -5 + 2 \beta ) q^{15} -6 \beta q^{17} -34 q^{19} + ( 4 + 2 \beta ) q^{21} -18 \beta q^{23} -5 q^{25} + ( -22 + 7 \beta ) q^{27} + 18 \beta q^{29} + 14 q^{31} + ( 30 - 12 \beta ) q^{33} + 2 \beta q^{35} + 56 q^{37} + ( 16 + 8 \beta ) q^{39} + 12 \beta q^{41} + 8 q^{43} + ( -20 - \beta ) q^{45} + 18 \beta q^{47} -45 q^{49} + ( 30 - 12 \beta ) q^{51} + 18 \beta q^{53} + 30 q^{55} + ( -68 - 34 \beta ) q^{57} -6 \beta q^{59} -46 q^{61} + ( -2 + 8 \beta ) q^{63} + 8 \beta q^{65} + 32 q^{67} + ( 90 - 36 \beta ) q^{69} -24 \beta q^{71} -106 q^{73} + ( -10 - 5 \beta ) q^{75} -12 \beta q^{77} -22 q^{79} + ( -79 - 8 \beta ) q^{81} + 54 \beta q^{83} + 30 q^{85} + ( -90 + 36 \beta ) q^{87} -48 \beta q^{89} + 16 q^{91} + ( 28 + 14 \beta ) q^{93} -34 \beta q^{95} + 122 q^{97} + ( 120 + 6 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{3} + 4q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 4q^{3} + 4q^{7} - 2q^{9} + 16q^{13} - 10q^{15} - 68q^{19} + 8q^{21} - 10q^{25} - 44q^{27} + 28q^{31} + 60q^{33} + 112q^{37} + 32q^{39} + 16q^{43} - 40q^{45} - 90q^{49} + 60q^{51} + 60q^{55} - 136q^{57} - 92q^{61} - 4q^{63} + 64q^{67} + 180q^{69} - 212q^{73} - 20q^{75} - 44q^{79} - 158q^{81} + 60q^{85} - 180q^{87} + 32q^{91} + 56q^{93} + 244q^{97} + 240q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$37$$ $$41$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1
 − 2.23607i 2.23607i
0 2.00000 2.23607i 0 2.23607i 0 2.00000 0 −1.00000 8.94427i 0
41.2 0 2.00000 + 2.23607i 0 2.23607i 0 2.00000 0 −1.00000 + 8.94427i 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.3.g.a 2
3.b odd 2 1 inner 60.3.g.a 2
4.b odd 2 1 240.3.l.a 2
5.b even 2 1 300.3.g.d 2
5.c odd 4 2 300.3.b.c 4
8.b even 2 1 960.3.l.a 2
8.d odd 2 1 960.3.l.d 2
9.c even 3 2 1620.3.o.b 4
9.d odd 6 2 1620.3.o.b 4
12.b even 2 1 240.3.l.a 2
15.d odd 2 1 300.3.g.d 2
15.e even 4 2 300.3.b.c 4
20.d odd 2 1 1200.3.l.r 2
20.e even 4 2 1200.3.c.e 4
24.f even 2 1 960.3.l.d 2
24.h odd 2 1 960.3.l.a 2
60.h even 2 1 1200.3.l.r 2
60.l odd 4 2 1200.3.c.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.g.a 2 1.a even 1 1 trivial
60.3.g.a 2 3.b odd 2 1 inner
240.3.l.a 2 4.b odd 2 1
240.3.l.a 2 12.b even 2 1
300.3.b.c 4 5.c odd 4 2
300.3.b.c 4 15.e even 4 2
300.3.g.d 2 5.b even 2 1
300.3.g.d 2 15.d odd 2 1
960.3.l.a 2 8.b even 2 1
960.3.l.a 2 24.h odd 2 1
960.3.l.d 2 8.d odd 2 1
960.3.l.d 2 24.f even 2 1
1200.3.c.e 4 20.e even 4 2
1200.3.c.e 4 60.l odd 4 2
1200.3.l.r 2 20.d odd 2 1
1200.3.l.r 2 60.h even 2 1
1620.3.o.b 4 9.c even 3 2
1620.3.o.b 4 9.d odd 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 - 4 T + T^{2}$$
$5$ $$5 + T^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$180 + T^{2}$$
$13$ $$( -8 + T )^{2}$$
$17$ $$180 + T^{2}$$
$19$ $$( 34 + T )^{2}$$
$23$ $$1620 + T^{2}$$
$29$ $$1620 + T^{2}$$
$31$ $$( -14 + T )^{2}$$
$37$ $$( -56 + T )^{2}$$
$41$ $$720 + T^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$1620 + T^{2}$$
$53$ $$1620 + T^{2}$$
$59$ $$180 + T^{2}$$
$61$ $$( 46 + T )^{2}$$
$67$ $$( -32 + T )^{2}$$
$71$ $$2880 + T^{2}$$
$73$ $$( 106 + T )^{2}$$
$79$ $$( 22 + T )^{2}$$
$83$ $$14580 + T^{2}$$
$89$ $$11520 + T^{2}$$
$97$ $$( -122 + T )^{2}$$
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