# Properties

 Label 60.4.j.b Level $60$ Weight $4$ Character orbit 60.j Analytic conductor $3.540$ Analytic rank $0$ Dimension $28$ 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.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.54011460034$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28q + 24q^{5} - 36q^{6} + 84q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q + 24q^{5} - 36q^{6} + 84q^{8} + 128q^{10} + 24q^{12} - 412q^{13} - 180q^{16} + 20q^{17} + 52q^{20} + 144q^{21} - 436q^{22} + 132q^{25} + 704q^{26} + 508q^{28} + 480q^{30} + 340q^{32} - 96q^{33} + 324q^{36} + 508q^{37} - 1792q^{38} - 2696q^{40} - 1696q^{41} - 1500q^{42} + 612q^{45} + 2584q^{46} + 528q^{48} + 832q^{50} + 504q^{52} + 1772q^{53} - 512q^{56} + 720q^{57} - 1060q^{58} - 84q^{60} + 2096q^{61} - 472q^{62} + 28q^{65} - 648q^{66} + 5872q^{68} + 2956q^{70} + 756q^{72} - 3348q^{73} - 3480q^{76} - 384q^{77} - 1032q^{78} - 4828q^{80} - 2268q^{81} - 928q^{82} - 476q^{85} - 3616q^{86} + 380q^{88} - 1116q^{90} + 472q^{92} - 2688q^{93} + 396q^{96} + 8300q^{97} + 72q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −2.82546 0.129446i 2.12132 + 2.12132i 7.96649 + 0.731491i −8.68710 + 7.03806i −5.71912 6.26831i 4.43008 4.43008i −22.4143 3.09803i 9.00000i 25.4561 18.7613i
7.2 −2.23795 + 1.72961i 2.12132 + 2.12132i 2.01687 7.74159i 7.18431 8.56655i −8.41648 1.07835i 13.0358 13.0358i 8.87628 + 20.8137i 9.00000i −1.26133 + 31.5976i
7.3 −2.18597 1.79487i −2.12132 2.12132i 1.55691 + 7.84704i −3.87875 10.4860i 0.829651 + 8.44462i −17.0336 + 17.0336i 10.6810 19.9478i 9.00000i −10.3421 + 29.8838i
7.4 −2.15080 + 1.83686i −2.12132 2.12132i 1.25186 7.90145i 8.47773 + 7.28890i 8.45911 + 0.665953i −13.7652 + 13.7652i 11.8214 + 19.2939i 9.00000i −31.6226 0.104521i
7.5 −1.79487 2.18597i 2.12132 + 2.12132i −1.55691 + 7.84704i −3.87875 10.4860i 0.829651 8.44462i 17.0336 17.0336i 19.9478 10.6810i 9.00000i −15.9601 + 27.2997i
7.6 −0.949450 + 2.66431i 2.12132 + 2.12132i −6.19709 5.05926i −10.6252 3.47923i −7.66594 + 3.63776i −24.7270 + 24.7270i 19.3633 11.7074i 9.00000i 19.3578 25.0055i
7.7 −0.431034 + 2.79539i −2.12132 2.12132i −7.62842 2.40982i 2.54602 10.8866i 6.84428 5.01556i 9.77420 9.77420i 10.0245 20.2857i 9.00000i 29.3348 + 11.8096i
7.8 −0.129446 2.82546i −2.12132 2.12132i −7.96649 + 0.731491i −8.68710 + 7.03806i −5.71912 + 6.26831i −4.43008 + 4.43008i 3.09803 + 22.4143i 9.00000i 21.0103 + 23.6340i
7.9 1.05416 + 2.62464i 2.12132 + 2.12132i −5.77750 + 5.53358i 10.9830 + 2.09135i −3.33150 + 7.80392i −5.27814 + 5.27814i −20.6141 9.33060i 9.00000i 6.08878 + 31.0311i
7.10 1.72961 2.23795i −2.12132 2.12132i −2.01687 7.74159i 7.18431 8.56655i −8.41648 + 1.07835i −13.0358 + 13.0358i −20.8137 8.87628i 9.00000i −6.74547 30.8950i
7.11 1.83686 2.15080i 2.12132 + 2.12132i −1.25186 7.90145i 8.47773 + 7.28890i 8.45911 0.665953i 13.7652 13.7652i −19.2939 11.8214i 9.00000i 31.2494 4.84515i
7.12 2.62464 + 1.05416i −2.12132 2.12132i 5.77750 + 5.53358i 10.9830 + 2.09135i −3.33150 7.80392i 5.27814 5.27814i 9.33060 + 20.6141i 9.00000i 26.6218 + 17.0669i
7.13 2.66431 0.949450i −2.12132 2.12132i 6.19709 5.05926i −10.6252 3.47923i −7.66594 3.63776i 24.7270 24.7270i 11.7074 19.3633i 9.00000i −31.6122 + 0.818373i
7.14 2.79539 0.431034i 2.12132 + 2.12132i 7.62842 2.40982i 2.54602 10.8866i 6.84428 + 5.01556i −9.77420 + 9.77420i 20.2857 10.0245i 9.00000i 2.42464 31.5297i
43.1 −2.82546 + 0.129446i 2.12132 2.12132i 7.96649 0.731491i −8.68710 7.03806i −5.71912 + 6.26831i 4.43008 + 4.43008i −22.4143 + 3.09803i 9.00000i 25.4561 + 18.7613i
43.2 −2.23795 1.72961i 2.12132 2.12132i 2.01687 + 7.74159i 7.18431 + 8.56655i −8.41648 + 1.07835i 13.0358 + 13.0358i 8.87628 20.8137i 9.00000i −1.26133 31.5976i
43.3 −2.18597 + 1.79487i −2.12132 + 2.12132i 1.55691 7.84704i −3.87875 + 10.4860i 0.829651 8.44462i −17.0336 17.0336i 10.6810 + 19.9478i 9.00000i −10.3421 29.8838i
43.4 −2.15080 1.83686i −2.12132 + 2.12132i 1.25186 + 7.90145i 8.47773 7.28890i 8.45911 0.665953i −13.7652 13.7652i 11.8214 19.2939i 9.00000i −31.6226 + 0.104521i
43.5 −1.79487 + 2.18597i 2.12132 2.12132i −1.55691 7.84704i −3.87875 + 10.4860i 0.829651 + 8.44462i 17.0336 + 17.0336i 19.9478 + 10.6810i 9.00000i −15.9601 27.2997i
43.6 −0.949450 2.66431i 2.12132 2.12132i −6.19709 + 5.05926i −10.6252 + 3.47923i −7.66594 3.63776i −24.7270 24.7270i 19.3633 + 11.7074i 9.00000i 19.3578 + 25.0055i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.14 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.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.4.j.b 28
3.b odd 2 1 180.4.k.f 28
4.b odd 2 1 inner 60.4.j.b 28
5.c odd 4 1 inner 60.4.j.b 28
12.b even 2 1 180.4.k.f 28
15.e even 4 1 180.4.k.f 28
20.e even 4 1 inner 60.4.j.b 28
60.l odd 4 1 180.4.k.f 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.j.b 28 1.a even 1 1 trivial
60.4.j.b 28 4.b odd 2 1 inner
60.4.j.b 28 5.c odd 4 1 inner
60.4.j.b 28 20.e even 4 1 inner
180.4.k.f 28 3.b odd 2 1
180.4.k.f 28 12.b even 2 1
180.4.k.f 28 15.e even 4 1
180.4.k.f 28 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{28} + 2132368 T_{7}^{24} +$$$$10\!\cdots\!08$$$$T_{7}^{20} +$$$$20\!\cdots\!84$$$$T_{7}^{16} +$$$$15\!\cdots\!24$$$$T_{7}^{12} +$$$$37\!\cdots\!28$$$$T_{7}^{8} +$$$$14\!\cdots\!88$$$$T_{7}^{4} +$$$$14\!\cdots\!76$$ acting on $$S_{4}^{\mathrm{new}}(60, [\chi])$$.