# Properties

 Label 60.4.e.a Level $60$ Weight $4$ Character orbit 60.e Analytic conductor $3.540$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 60.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.54011460034$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 6q^{4} + 30q^{6} + 20q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 6q^{4} + 30q^{6} + 20q^{9} - 30q^{10} - 188q^{12} + 72q^{13} + 306q^{16} + 256q^{18} - 68q^{21} - 300q^{22} - 434q^{24} - 600q^{25} + 300q^{28} - 40q^{30} + 848q^{33} - 468q^{34} - 294q^{36} + 504q^{37} - 210q^{40} - 228q^{42} - 220q^{45} + 684q^{46} + 1212q^{48} - 2256q^{49} + 576q^{52} - 1054q^{54} + 1416q^{57} + 3108q^{58} + 490q^{60} + 1992q^{61} - 1842q^{64} - 472q^{66} - 1548q^{69} + 540q^{70} + 312q^{72} - 2304q^{73} - 420q^{76} - 2792q^{78} + 3840q^{81} + 600q^{82} - 176q^{84} + 240q^{85} - 372q^{88} - 1170q^{90} - 4384q^{93} + 1044q^{94} - 3846q^{96} - 2448q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −2.81775 0.245521i −1.56674 4.95432i 7.87944 + 1.38363i 5.00000i 3.19829 + 14.3447i 5.80602i −21.8626 5.83329i −22.0907 + 15.5243i −1.22760 + 14.0888i
11.2 −2.81775 + 0.245521i −1.56674 + 4.95432i 7.87944 1.38363i 5.00000i 3.19829 14.3447i 5.80602i −21.8626 + 5.83329i −22.0907 15.5243i −1.22760 14.0888i
11.3 −2.72356 0.763049i 4.56698 + 2.47844i 6.83551 + 4.15641i 5.00000i −10.5473 10.2350i 20.9745i −15.4454 16.5360i 14.7146 + 22.6380i −3.81524 + 13.6178i
11.4 −2.72356 + 0.763049i 4.56698 2.47844i 6.83551 4.15641i 5.00000i −10.5473 + 10.2350i 20.9745i −15.4454 + 16.5360i 14.7146 22.6380i −3.81524 13.6178i
11.5 −2.29793 1.64910i −5.00938 1.38062i 2.56093 + 7.57903i 5.00000i 9.23440 + 11.4335i 0.228949i 6.61376 21.6393i 23.1878 + 13.8321i 8.24551 11.4896i
11.6 −2.29793 + 1.64910i −5.00938 + 1.38062i 2.56093 7.57903i 5.00000i 9.23440 11.4335i 0.228949i 6.61376 + 21.6393i 23.1878 13.8321i 8.24551 + 11.4896i
11.7 −1.91554 2.08103i −2.09257 + 4.75617i −0.661378 + 7.97261i 5.00000i 13.9062 4.75594i 32.2690i 17.8582 13.8955i −18.2423 19.9053i −10.4052 + 9.57772i
11.8 −1.91554 + 2.08103i −2.09257 4.75617i −0.661378 7.97261i 5.00000i 13.9062 + 4.75594i 32.2690i 17.8582 + 13.8955i −18.2423 + 19.9053i −10.4052 9.57772i
11.9 −0.622346 2.75911i 1.95519 + 4.81427i −7.22537 + 3.43424i 5.00000i 12.0663 8.39074i 20.3642i 13.9721 + 17.7983i −19.3544 + 18.8257i 13.7955 3.11173i
11.10 −0.622346 + 2.75911i 1.95519 4.81427i −7.22537 3.43424i 5.00000i 12.0663 + 8.39074i 20.3642i 13.9721 17.7983i −19.3544 18.8257i 13.7955 + 3.11173i
11.11 −0.235442 2.81861i −5.18580 0.327893i −7.88913 + 1.32724i 5.00000i 0.296752 + 14.6939i 26.3120i 5.59840 + 21.9239i 26.7850 + 3.40077i −14.0931 + 1.17721i
11.12 −0.235442 + 2.81861i −5.18580 + 0.327893i −7.88913 1.32724i 5.00000i 0.296752 14.6939i 26.3120i 5.59840 21.9239i 26.7850 3.40077i −14.0931 1.17721i
11.13 0.235442 2.81861i 5.18580 + 0.327893i −7.88913 1.32724i 5.00000i 2.14516 14.5395i 26.3120i −5.59840 + 21.9239i 26.7850 + 3.40077i −14.0931 1.17721i
11.14 0.235442 + 2.81861i 5.18580 0.327893i −7.88913 + 1.32724i 5.00000i 2.14516 + 14.5395i 26.3120i −5.59840 21.9239i 26.7850 3.40077i −14.0931 + 1.17721i
11.15 0.622346 2.75911i −1.95519 4.81427i −7.22537 3.43424i 5.00000i −14.4999 + 2.39845i 20.3642i −13.9721 + 17.7983i −19.3544 + 18.8257i 13.7955 + 3.11173i
11.16 0.622346 + 2.75911i −1.95519 + 4.81427i −7.22537 + 3.43424i 5.00000i −14.4999 2.39845i 20.3642i −13.9721 17.7983i −19.3544 18.8257i 13.7955 3.11173i
11.17 1.91554 2.08103i 2.09257 4.75617i −0.661378 7.97261i 5.00000i −5.88931 13.4654i 32.2690i −17.8582 13.8955i −18.2423 19.9053i −10.4052 9.57772i
11.18 1.91554 + 2.08103i 2.09257 + 4.75617i −0.661378 + 7.97261i 5.00000i −5.88931 + 13.4654i 32.2690i −17.8582 + 13.8955i −18.2423 + 19.9053i −10.4052 + 9.57772i
11.19 2.29793 1.64910i 5.00938 + 1.38062i 2.56093 7.57903i 5.00000i 13.7880 5.08841i 0.228949i −6.61376 21.6393i 23.1878 + 13.8321i 8.24551 + 11.4896i
11.20 2.29793 + 1.64910i 5.00938 1.38062i 2.56093 + 7.57903i 5.00000i 13.7880 + 5.08841i 0.228949i −6.61376 + 21.6393i 23.1878 13.8321i 8.24551 11.4896i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.4.e.a 24
3.b odd 2 1 inner 60.4.e.a 24
4.b odd 2 1 inner 60.4.e.a 24
8.b even 2 1 960.4.h.d 24
8.d odd 2 1 960.4.h.d 24
12.b even 2 1 inner 60.4.e.a 24
24.f even 2 1 960.4.h.d 24
24.h odd 2 1 960.4.h.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.e.a 24 1.a even 1 1 trivial
60.4.e.a 24 3.b odd 2 1 inner
60.4.e.a 24 4.b odd 2 1 inner
60.4.e.a 24 12.b even 2 1 inner
960.4.h.d 24 8.b even 2 1
960.4.h.d 24 8.d odd 2 1
960.4.h.d 24 24.f even 2 1
960.4.h.d 24 24.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(60, [\chi])$$.