# Properties

 Label 60.7.f.a Level $60$ Weight $7$ Character orbit 60.f Analytic conductor $13.803$ Analytic rank $0$ Dimension $36$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 60.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.8032450172$$ Analytic rank: $$0$$ Dimension: $$36$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 66q^{4} + 44q^{5} - 162q^{6} + 8748q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 66q^{4} + 44q^{5} - 162q^{6} + 8748q^{9} - 1422q^{10} + 15484q^{14} + 1266q^{16} - 2188q^{20} - 9234q^{24} + 13812q^{25} - 32052q^{26} - 36920q^{29} - 24300q^{30} - 156204q^{34} - 16038q^{36} + 7674q^{40} + 341848q^{41} + 74892q^{44} + 10692q^{45} + 478080q^{46} + 482556q^{49} - 191448q^{50} - 39366q^{54} - 926132q^{56} + 126846q^{60} - 455976q^{61} - 618q^{64} - 624192q^{65} - 127332q^{66} - 541728q^{69} - 1871304q^{70} - 612324q^{74} - 1566456q^{76} + 2360972q^{80} + 2125764q^{81} + 858276q^{84} + 2280864q^{85} + 5157592q^{86} - 1806104q^{89} - 345546q^{90} - 3121992q^{94} - 4040118q^{96} + O(q^{100})$$

## 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 −7.93273 1.03526i 15.5885 61.8565 + 16.4248i −80.2400 + 95.8465i −123.659 16.1381i −143.606 −473.687 194.331i 243.000 735.748 677.255i
19.2 −7.93273 + 1.03526i 15.5885 61.8565 16.4248i −80.2400 95.8465i −123.659 + 16.1381i −143.606 −473.687 + 194.331i 243.000 735.748 + 677.255i
19.3 −7.65640 2.31937i −15.5885 53.2410 + 35.5161i 87.3911 + 89.3745i 119.352 + 36.1554i −36.3122 −325.260 395.411i 243.000 −461.809 886.980i
19.4 −7.65640 + 2.31937i −15.5885 53.2410 35.5161i 87.3911 89.3745i 119.352 36.1554i −36.3122 −325.260 + 395.411i 243.000 −461.809 + 886.980i
19.5 −7.34967 3.15947i 15.5885 44.0354 + 46.4422i 111.509 56.4858i −114.570 49.2513i 231.647 −176.913 480.464i 243.000 −998.024 + 62.8414i
19.6 −7.34967 + 3.15947i 15.5885 44.0354 46.4422i 111.509 + 56.4858i −114.570 + 49.2513i 231.647 −176.913 + 480.464i 243.000 −998.024 62.8414i
19.7 −6.43144 4.75779i −15.5885 18.7270 + 61.1989i −121.123 + 30.8887i 100.256 + 74.1665i −549.446 170.730 482.696i 243.000 925.960 + 377.621i
19.8 −6.43144 + 4.75779i −15.5885 18.7270 61.1989i −121.123 30.8887i 100.256 74.1665i −549.446 170.730 + 482.696i 243.000 925.960 377.621i
19.9 −5.26743 6.02115i −15.5885 −8.50840 + 63.4319i −13.2381 124.297i 82.1111 + 93.8604i 380.004 426.750 282.893i 243.000 −678.680 + 734.434i
19.10 −5.26743 + 6.02115i −15.5885 −8.50840 63.4319i −13.2381 + 124.297i 82.1111 93.8604i 380.004 426.750 + 282.893i 243.000 −678.680 734.434i
19.11 −4.30858 6.74063i 15.5885 −26.8722 + 58.0851i −108.897 61.3721i −67.1642 105.076i −118.127 507.312 69.1287i 243.000 55.5036 + 998.458i
19.12 −4.30858 + 6.74063i 15.5885 −26.8722 58.0851i −108.897 + 61.3721i −67.1642 + 105.076i −118.127 507.312 + 69.1287i 243.000 55.5036 998.458i
19.13 −3.70638 7.08962i 15.5885 −36.5255 + 52.5537i 108.680 + 61.7555i −57.7768 110.516i −629.059 507.963 + 64.1675i 243.000 35.0151 999.387i
19.14 −3.70638 + 7.08962i 15.5885 −36.5255 52.5537i 108.680 61.7555i −57.7768 + 110.516i −629.059 507.963 64.1675i 243.000 35.0151 + 999.387i
19.15 −1.63894 7.83032i −15.5885 −58.6278 + 25.6668i −44.2074 + 116.922i 25.5485 + 122.063i 42.6864 297.066 + 417.008i 243.000 987.988 + 154.531i
19.16 −1.63894 + 7.83032i −15.5885 −58.6278 25.6668i −44.2074 116.922i 25.5485 122.063i 42.6864 297.066 417.008i 243.000 987.988 154.531i
19.17 −0.294917 7.99456i 15.5885 −63.8260 + 4.71546i 71.1255 102.792i −4.59730 124.623i 496.078 56.5214 + 508.871i 243.000 −842.752 538.302i
19.18 −0.294917 + 7.99456i 15.5885 −63.8260 4.71546i 71.1255 + 102.792i −4.59730 + 124.623i 496.078 56.5214 508.871i 243.000 −842.752 + 538.302i
19.19 0.294917 7.99456i −15.5885 −63.8260 4.71546i 71.1255 102.792i −4.59730 + 124.623i −496.078 −56.5214 + 508.871i 243.000 −800.800 598.932i
19.20 0.294917 + 7.99456i −15.5885 −63.8260 + 4.71546i 71.1255 + 102.792i −4.59730 124.623i −496.078 −56.5214 508.871i 243.000 −800.800 + 598.932i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.36 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 60.7.f.a 36
3.b odd 2 1 180.7.f.g 36
4.b odd 2 1 inner 60.7.f.a 36
5.b even 2 1 inner 60.7.f.a 36
12.b even 2 1 180.7.f.g 36
15.d odd 2 1 180.7.f.g 36
20.d odd 2 1 inner 60.7.f.a 36
60.h even 2 1 180.7.f.g 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.f.a 36 1.a even 1 1 trivial
60.7.f.a 36 4.b odd 2 1 inner
60.7.f.a 36 5.b even 2 1 inner
60.7.f.a 36 20.d odd 2 1 inner
180.7.f.g 36 3.b odd 2 1
180.7.f.g 36 12.b even 2 1
180.7.f.g 36 15.d odd 2 1
180.7.f.g 36 60.h even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(60, [\chi])$$.