# Properties

 Label 60.6.h.c Level $60$ Weight $6$ Character orbit 60.h Analytic conductor $9.623$ Analytic rank $0$ Dimension $48$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 60.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.62302918878$$ Analytic rank: $$0$$ Dimension: $$48$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 8q^{4} + 420q^{6} + 1440q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 8q^{4} + 420q^{6} + 1440q^{9} - 1600q^{10} - 10608q^{16} - 1296q^{21} - 8568q^{24} + 1360q^{25} + 13260q^{30} + 23744q^{34} + 15312q^{36} - 35840q^{40} + 71760q^{45} - 126152q^{46} + 191888q^{49} - 19908q^{54} + 64800q^{60} + 12224q^{61} + 69856q^{64} + 21696q^{66} - 640176q^{69} - 81560q^{70} + 56640q^{76} - 123120q^{81} - 312q^{84} + 117440q^{85} + 401280q^{90} + 558088q^{94} + 646128q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1 −5.44678 1.52727i −11.2078 + 10.8345i 27.3349 + 16.6374i 42.6056 36.1906i 77.5935 41.8958i −128.907 −123.478 132.368i 8.22823 242.861i −287.336 + 132.052i
59.2 −5.44678 1.52727i 11.2078 + 10.8345i 27.3349 + 16.6374i 42.6056 + 36.1906i −44.4991 76.1303i 128.907 −123.478 132.368i 8.22823 + 242.861i −176.791 262.193i
59.3 −5.44678 + 1.52727i −11.2078 10.8345i 27.3349 16.6374i 42.6056 + 36.1906i 77.5935 + 41.8958i −128.907 −123.478 + 132.368i 8.22823 + 242.861i −287.336 132.052i
59.4 −5.44678 + 1.52727i 11.2078 10.8345i 27.3349 16.6374i 42.6056 36.1906i −44.4991 + 76.1303i 128.907 −123.478 + 132.368i 8.22823 242.861i −176.791 + 262.193i
59.5 −5.20228 2.22177i −15.5789 + 0.545517i 22.1275 + 23.1165i −17.3193 + 53.1511i 82.2579 + 31.7747i 106.220 −63.7541 169.421i 242.405 16.9971i 208.189 238.028i
59.6 −5.20228 2.22177i 15.5789 + 0.545517i 22.1275 + 23.1165i −17.3193 53.1511i −79.8339 37.4506i −106.220 −63.7541 169.421i 242.405 + 16.9971i −27.9895 + 314.987i
59.7 −5.20228 + 2.22177i −15.5789 0.545517i 22.1275 23.1165i −17.3193 53.1511i 82.2579 31.7747i 106.220 −63.7541 + 169.421i 242.405 + 16.9971i 208.189 + 238.028i
59.8 −5.20228 + 2.22177i 15.5789 0.545517i 22.1275 23.1165i −17.3193 + 53.1511i −79.8339 + 37.4506i −106.220 −63.7541 + 169.421i 242.405 16.9971i −27.9895 314.987i
59.9 −4.16295 3.83012i −7.28769 13.7800i 2.66035 + 31.8892i 45.7166 32.1713i −22.4409 + 85.2784i 213.200 111.065 142.943i −136.779 + 200.849i −313.536 41.1725i
59.10 −4.16295 3.83012i 7.28769 13.7800i 2.66035 + 31.8892i 45.7166 + 32.1713i −83.1175 + 29.4530i −213.200 111.065 142.943i −136.779 200.849i −67.0961 309.028i
59.11 −4.16295 + 3.83012i −7.28769 + 13.7800i 2.66035 31.8892i 45.7166 + 32.1713i −22.4409 85.2784i 213.200 111.065 + 142.943i −136.779 200.849i −313.536 + 41.1725i
59.12 −4.16295 + 3.83012i 7.28769 + 13.7800i 2.66035 31.8892i 45.7166 32.1713i −83.1175 29.4530i −213.200 111.065 + 142.943i −136.779 + 200.849i −67.0961 + 309.028i
59.13 −3.28930 4.60223i −3.33289 + 15.2280i −10.3610 + 30.2762i −17.3746 + 53.1331i 81.0456 34.7508i −121.108 173.418 51.9042i −220.784 101.506i 301.681 94.8087i
59.14 −3.28930 4.60223i 3.33289 + 15.2280i −10.3610 + 30.2762i −17.3746 53.1331i 59.1198 65.4282i 121.108 173.418 51.9042i −220.784 + 101.506i −187.380 + 254.733i
59.15 −3.28930 + 4.60223i −3.33289 15.2280i −10.3610 30.2762i −17.3746 53.1331i 81.0456 + 34.7508i −121.108 173.418 + 51.9042i −220.784 + 101.506i 301.681 + 94.8087i
59.16 −3.28930 + 4.60223i 3.33289 15.2280i −10.3610 30.2762i −17.3746 + 53.1331i 59.1198 + 65.4282i 121.108 173.418 + 51.9042i −220.784 101.506i −187.380 254.733i
59.17 −3.08938 4.73875i −14.4591 5.82522i −12.9114 + 29.2796i −43.0117 35.7070i 17.0656 + 86.5145i −168.076 178.637 29.2717i 175.134 + 168.455i −36.3270 + 314.134i
59.18 −3.08938 4.73875i 14.4591 5.82522i −12.9114 + 29.2796i −43.0117 + 35.7070i −72.2740 50.5219i 168.076 178.637 29.2717i 175.134 168.455i 302.086 + 93.5090i
59.19 −3.08938 + 4.73875i −14.4591 + 5.82522i −12.9114 29.2796i −43.0117 + 35.7070i 17.0656 86.5145i −168.076 178.637 + 29.2717i 175.134 168.455i −36.3270 314.134i
59.20 −3.08938 + 4.73875i 14.4591 + 5.82522i −12.9114 29.2796i −43.0117 35.7070i −72.2740 + 50.5219i 168.076 178.637 + 29.2717i 175.134 + 168.455i 302.086 93.5090i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 59.48 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
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.6.h.c 48
3.b odd 2 1 inner 60.6.h.c 48
4.b odd 2 1 inner 60.6.h.c 48
5.b even 2 1 inner 60.6.h.c 48
12.b even 2 1 inner 60.6.h.c 48
15.d odd 2 1 inner 60.6.h.c 48
20.d odd 2 1 inner 60.6.h.c 48
60.h even 2 1 inner 60.6.h.c 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.h.c 48 1.a even 1 1 trivial
60.6.h.c 48 3.b odd 2 1 inner
60.6.h.c 48 4.b odd 2 1 inner
60.6.h.c 48 5.b even 2 1 inner
60.6.h.c 48 12.b even 2 1 inner
60.6.h.c 48 15.d odd 2 1 inner
60.6.h.c 48 20.d odd 2 1 inner
60.6.h.c 48 60.h even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{12} - 124828 T_{7}^{10} + 6013060404 T_{7}^{8} -$$$$14\!\cdots\!44$$$$T_{7}^{6} +$$$$18\!\cdots\!28$$$$T_{7}^{4} -$$$$11\!\cdots\!20$$$$T_{7}^{2} +$$$$30\!\cdots\!00$$ acting on $$S_{6}^{\mathrm{new}}(60, [\chi])$$.