# Properties

 Label 60.5.f.a Level $60$ Weight $5$ Character orbit 60.f Analytic conductor $6.202$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.20219778503$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 14q^{4} - 24q^{5} - 18q^{6} + 648q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 14q^{4} - 24q^{5} - 18q^{6} + 648q^{9} + 274q^{10} - 36q^{14} + 594q^{16} - 12q^{20} - 594q^{24} + 1208q^{25} - 2868q^{26} - 1680q^{29} + 468q^{30} + 3076q^{34} + 378q^{36} - 7222q^{40} - 4848q^{41} - 3828q^{44} - 648q^{45} - 15280q^{46} + 5416q^{49} + 14472q^{50} - 486q^{54} + 32172q^{56} - 7506q^{60} + 2896q^{61} - 18298q^{64} - 2688q^{65} - 15588q^{66} + 9792q^{69} + 27608q^{70} + 31836q^{74} + 50136q^{76} - 27348q^{80} + 17496q^{81} - 4284q^{84} - 15680q^{85} - 58152q^{86} - 38544q^{89} + 7398q^{90} + 4808q^{94} + 21978q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −3.90764 0.854601i −5.19615 14.5393 + 6.67895i −9.11612 23.2787i 20.3047 + 4.44064i −10.5267 −51.1066 38.5243i 27.0000 15.7285 + 98.7553i
19.2 −3.90764 + 0.854601i −5.19615 14.5393 6.67895i −9.11612 + 23.2787i 20.3047 4.44064i −10.5267 −51.1066 + 38.5243i 27.0000 15.7285 98.7553i
19.3 −3.76887 1.34002i 5.19615 12.4087 + 10.1007i 11.9809 + 21.9421i −19.5836 6.96293i −63.7886 −33.2317 54.6960i 27.0000 −15.7516 98.7516i
19.4 −3.76887 + 1.34002i 5.19615 12.4087 10.1007i 11.9809 21.9421i −19.5836 + 6.96293i −63.7886 −33.2317 + 54.6960i 27.0000 −15.7516 + 98.7516i
19.5 −3.44337 2.03548i 5.19615 7.71364 + 14.0178i −20.5121 14.2918i −17.8923 10.5767i 51.0440 1.97210 63.9696i 27.0000 41.5401 + 90.9638i
19.6 −3.44337 + 2.03548i 5.19615 7.71364 14.0178i −20.5121 + 14.2918i −17.8923 + 10.5767i 51.0440 1.97210 + 63.9696i 27.0000 41.5401 90.9638i
19.7 −2.59896 3.04063i −5.19615 −2.49086 + 15.8049i 24.9405 1.72355i 13.5046 + 15.7996i −2.65040 54.5305 33.5025i 27.0000 −70.0600 71.3554i
19.8 −2.59896 + 3.04063i −5.19615 −2.49086 15.8049i 24.9405 + 1.72355i 13.5046 15.7996i −2.65040 54.5305 + 33.5025i 27.0000 −70.0600 + 71.3554i
19.9 −0.988963 3.87582i 5.19615 −14.0439 + 7.66608i 11.6321 + 22.1290i −5.13880 20.1393i 66.7450 43.6012 + 46.8501i 27.0000 74.2643 66.9688i
19.10 −0.988963 + 3.87582i 5.19615 −14.0439 7.66608i 11.6321 22.1290i −5.13880 + 20.1393i 66.7450 43.6012 46.8501i 27.0000 74.2643 + 66.9688i
19.11 −0.828581 3.91324i −5.19615 −14.6269 + 6.48488i −24.9254 + 1.93048i 4.30543 + 20.3338i 67.1774 37.4965 + 51.8654i 27.0000 28.2071 + 95.9394i
19.12 −0.828581 + 3.91324i −5.19615 −14.6269 6.48488i −24.9254 1.93048i 4.30543 20.3338i 67.1774 37.4965 51.8654i 27.0000 28.2071 95.9394i
19.13 0.828581 3.91324i 5.19615 −14.6269 6.48488i −24.9254 + 1.93048i 4.30543 20.3338i −67.1774 −37.4965 + 51.8654i 27.0000 −13.0983 + 99.1385i
19.14 0.828581 + 3.91324i 5.19615 −14.6269 + 6.48488i −24.9254 1.93048i 4.30543 + 20.3338i −67.1774 −37.4965 51.8654i 27.0000 −13.0983 99.1385i
19.15 0.988963 3.87582i −5.19615 −14.0439 7.66608i 11.6321 + 22.1290i −5.13880 + 20.1393i −66.7450 −43.6012 + 46.8501i 27.0000 97.2718 23.1992i
19.16 0.988963 + 3.87582i −5.19615 −14.0439 + 7.66608i 11.6321 22.1290i −5.13880 20.1393i −66.7450 −43.6012 46.8501i 27.0000 97.2718 + 23.1992i
19.17 2.59896 3.04063i 5.19615 −2.49086 15.8049i 24.9405 1.72355i 13.5046 15.7996i 2.65040 −54.5305 33.5025i 27.0000 59.5786 80.3143i
19.18 2.59896 + 3.04063i 5.19615 −2.49086 + 15.8049i 24.9405 + 1.72355i 13.5046 + 15.7996i 2.65040 −54.5305 + 33.5025i 27.0000 59.5786 + 80.3143i
19.19 3.44337 2.03548i −5.19615 7.71364 14.0178i −20.5121 14.2918i −17.8923 + 10.5767i −51.0440 −1.97210 63.9696i 27.0000 −99.7214 7.45996i
19.20 3.44337 + 2.03548i −5.19615 7.71364 + 14.0178i −20.5121 + 14.2918i −17.8923 10.5767i −51.0440 −1.97210 + 63.9696i 27.0000 −99.7214 + 7.45996i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d 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.f.a 24
3.b odd 2 1 180.5.f.i 24
4.b odd 2 1 inner 60.5.f.a 24
5.b even 2 1 inner 60.5.f.a 24
5.c odd 4 2 300.5.c.e 24
8.b even 2 1 960.5.j.d 24
8.d odd 2 1 960.5.j.d 24
12.b even 2 1 180.5.f.i 24
15.d odd 2 1 180.5.f.i 24
20.d odd 2 1 inner 60.5.f.a 24
20.e even 4 2 300.5.c.e 24
40.e odd 2 1 960.5.j.d 24
40.f even 2 1 960.5.j.d 24
60.h even 2 1 180.5.f.i 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.f.a 24 1.a even 1 1 trivial
60.5.f.a 24 4.b odd 2 1 inner
60.5.f.a 24 5.b even 2 1 inner
60.5.f.a 24 20.d odd 2 1 inner
180.5.f.i 24 3.b odd 2 1
180.5.f.i 24 12.b even 2 1
180.5.f.i 24 15.d odd 2 1
180.5.f.i 24 60.h even 2 1
300.5.c.e 24 5.c odd 4 2
300.5.c.e 24 20.e even 4 2
960.5.j.d 24 8.b even 2 1
960.5.j.d 24 8.d odd 2 1
960.5.j.d 24 40.e odd 2 1
960.5.j.d 24 40.f even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(60, [\chi])$$.