# Properties

 Label 60.5.g.a Level $60$ Weight $5$ Character orbit 60.g Analytic conductor $6.202$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.20219778503$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ Defining polynomial: $$x^{2} + 5$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 6 + 3 \beta ) q^{3} -5 \beta q^{5} + 74 q^{7} + ( -9 + 36 \beta ) q^{9} +O(q^{10})$$ $$q + ( 6 + 3 \beta ) q^{3} -5 \beta q^{5} + 74 q^{7} + ( -9 + 36 \beta ) q^{9} + 54 \beta q^{11} -16 q^{13} + ( 75 - 30 \beta ) q^{15} + 78 \beta q^{17} + 374 q^{19} + ( 444 + 222 \beta ) q^{21} -294 \beta q^{23} -125 q^{25} + ( -594 + 189 \beta ) q^{27} -618 \beta q^{29} -1426 q^{31} + ( -810 + 324 \beta ) q^{33} -370 \beta q^{35} + 272 q^{37} + ( -96 - 48 \beta ) q^{39} -348 \beta q^{41} -1936 q^{43} + ( 900 + 45 \beta ) q^{45} -1434 \beta q^{47} + 3075 q^{49} + ( -1170 + 468 \beta ) q^{51} + 2454 \beta q^{53} + 1350 q^{55} + ( 2244 + 1122 \beta ) q^{57} -2154 \beta q^{59} + 2234 q^{61} + ( -666 + 2664 \beta ) q^{63} + 80 \beta q^{65} -5416 q^{67} + ( 4410 - 1764 \beta ) q^{69} + 696 \beta q^{71} -538 q^{73} + ( -750 - 375 \beta ) q^{75} + 3996 \beta q^{77} + 9962 q^{79} + ( -6399 - 648 \beta ) q^{81} + 2082 \beta q^{83} + 1950 q^{85} + ( 9270 - 3708 \beta ) q^{87} -2352 \beta q^{89} -1184 q^{91} + ( -8556 - 4278 \beta ) q^{93} -1870 \beta q^{95} -10726 q^{97} + ( -9720 - 486 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 12q^{3} + 148q^{7} - 18q^{9} + O(q^{10})$$ $$2q + 12q^{3} + 148q^{7} - 18q^{9} - 32q^{13} + 150q^{15} + 748q^{19} + 888q^{21} - 250q^{25} - 1188q^{27} - 2852q^{31} - 1620q^{33} + 544q^{37} - 192q^{39} - 3872q^{43} + 1800q^{45} + 6150q^{49} - 2340q^{51} + 2700q^{55} + 4488q^{57} + 4468q^{61} - 1332q^{63} - 10832q^{67} + 8820q^{69} - 1076q^{73} - 1500q^{75} + 19924q^{79} - 12798q^{81} + 3900q^{85} + 18540q^{87} - 2368q^{91} - 17112q^{93} - 21452q^{97} - 19440q^{99} + O(q^{100})$$

## 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 6.00000 6.70820i 0 11.1803i 0 74.0000 0 −9.00000 80.4984i 0
41.2 0 6.00000 + 6.70820i 0 11.1803i 0 74.0000 0 −9.00000 + 80.4984i 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 60.5.g.a 2
3.b odd 2 1 inner 60.5.g.a 2
4.b odd 2 1 240.5.l.a 2
5.b even 2 1 300.5.g.d 2
5.c odd 4 2 300.5.b.c 4
12.b even 2 1 240.5.l.a 2
15.d odd 2 1 300.5.g.d 2
15.e even 4 2 300.5.b.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.g.a 2 1.a even 1 1 trivial
60.5.g.a 2 3.b odd 2 1 inner
240.5.l.a 2 4.b odd 2 1
240.5.l.a 2 12.b even 2 1
300.5.b.c 4 5.c odd 4 2
300.5.b.c 4 15.e even 4 2
300.5.g.d 2 5.b even 2 1
300.5.g.d 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} - 74$$ acting on $$S_{5}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$81 - 12 T + T^{2}$$
$5$ $$125 + T^{2}$$
$7$ $$( -74 + T )^{2}$$
$11$ $$14580 + T^{2}$$
$13$ $$( 16 + T )^{2}$$
$17$ $$30420 + T^{2}$$
$19$ $$( -374 + T )^{2}$$
$23$ $$432180 + T^{2}$$
$29$ $$1909620 + T^{2}$$
$31$ $$( 1426 + T )^{2}$$
$37$ $$( -272 + T )^{2}$$
$41$ $$605520 + T^{2}$$
$43$ $$( 1936 + T )^{2}$$
$47$ $$10281780 + T^{2}$$
$53$ $$30110580 + T^{2}$$
$59$ $$23198580 + T^{2}$$
$61$ $$( -2234 + T )^{2}$$
$67$ $$( 5416 + T )^{2}$$
$71$ $$2422080 + T^{2}$$
$73$ $$( 538 + T )^{2}$$
$79$ $$( -9962 + T )^{2}$$
$83$ $$21673620 + T^{2}$$
$89$ $$27659520 + T^{2}$$
$97$ $$( 10726 + T )^{2}$$