# Properties

 Label 60.7.c.a Level $60$ Weight $7$ Character orbit 60.c Analytic conductor $13.803$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$13.8032450172$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 20q^{2} - 246q^{4} + 162q^{6} - 340q^{8} - 5832q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 20q^{2} - 246q^{4} + 162q^{6} - 340q^{8} - 5832q^{9} - 750q^{10} + 5040q^{13} - 2596q^{14} + 4194q^{16} - 4860q^{18} + 7000q^{20} + 19440q^{21} + 45780q^{22} + 24786q^{24} + 75000q^{25} + 75852q^{26} + 54300q^{28} + 132800q^{29} - 10700q^{32} - 173484q^{34} + 59778q^{36} - 69840q^{37} + 215800q^{38} - 14250q^{40} - 70448q^{41} - 189540q^{42} - 395668q^{44} - 158760q^{46} + 252720q^{48} - 642984q^{49} + 62500q^{50} - 210240q^{52} - 644320q^{53} - 39366q^{54} - 917708q^{56} + 408240q^{57} - 1345020q^{58} + 222750q^{60} - 222864q^{61} + 1948520q^{62} + 935922q^{64} + 266000q^{65} - 640548q^{66} + 572680q^{68} - 541728q^{69} + 220500q^{70} + 82620q^{72} + 771120q^{73} - 589164q^{74} - 191544q^{76} + 1383840q^{77} - 693360q^{78} - 946000q^{80} + 1417176q^{81} + 2672520q^{82} + 1256796q^{84} - 372000q^{85} + 1781528q^{86} + 956940q^{88} - 1566224q^{89} + 182250q^{90} - 3040560q^{92} + 1496880q^{93} - 3788352q^{94} - 413262q^{96} - 1666800q^{97} - 2709660q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 −7.80071 1.77450i 15.5885i 57.7023 + 27.6847i 55.9017 −27.6617 + 121.601i 264.480i −400.993 318.353i −243.000 −436.073 99.1975i
31.2 −7.80071 + 1.77450i 15.5885i 57.7023 27.6847i 55.9017 −27.6617 121.601i 264.480i −400.993 + 318.353i −243.000 −436.073 + 99.1975i
31.3 −6.61106 4.50487i 15.5885i 23.4123 + 59.5639i −55.9017 70.2239 103.056i 489.339i 113.547 499.250i −243.000 369.570 + 251.830i
31.4 −6.61106 + 4.50487i 15.5885i 23.4123 59.5639i −55.9017 70.2239 + 103.056i 489.339i 113.547 + 499.250i −243.000 369.570 251.830i
31.5 −3.53904 7.17462i 15.5885i −38.9505 + 50.7825i 55.9017 −111.841 + 55.1681i 99.9522i 502.192 + 99.7338i −243.000 −197.838 401.074i
31.6 −3.53904 + 7.17462i 15.5885i −38.9505 50.7825i 55.9017 −111.841 55.1681i 99.9522i 502.192 99.7338i −243.000 −197.838 + 401.074i
31.7 −3.08457 7.38143i 15.5885i −44.9709 + 45.5370i 55.9017 115.065 48.0836i 200.003i 474.844 + 191.487i −243.000 −172.433 412.634i
31.8 −3.08457 + 7.38143i 15.5885i −44.9709 45.5370i 55.9017 115.065 + 48.0836i 200.003i 474.844 191.487i −243.000 −172.433 + 412.634i
31.9 −1.36400 7.88286i 15.5885i −60.2790 + 21.5045i −55.9017 122.882 21.2627i 86.8984i 251.737 + 445.839i −243.000 76.2500 + 440.665i
31.10 −1.36400 + 7.88286i 15.5885i −60.2790 21.5045i −55.9017 122.882 + 21.2627i 86.8984i 251.737 445.839i −243.000 76.2500 440.665i
31.11 0.331992 7.99311i 15.5885i −63.7796 5.30729i −55.9017 −124.600 5.17524i 552.003i −63.5960 + 508.035i −243.000 −18.5589 + 446.828i
31.12 0.331992 + 7.99311i 15.5885i −63.7796 + 5.30729i −55.9017 −124.600 + 5.17524i 552.003i −63.5960 508.035i −243.000 −18.5589 446.828i
31.13 2.59351 7.56794i 15.5885i −50.5474 39.2551i −55.9017 −117.972 40.4289i 671.100i −428.176 + 280.731i −243.000 −144.982 + 423.061i
31.14 2.59351 + 7.56794i 15.5885i −50.5474 + 39.2551i −55.9017 −117.972 + 40.4289i 671.100i −428.176 280.731i −243.000 −144.982 423.061i
31.15 3.43879 7.22321i 15.5885i −40.3494 49.6782i 55.9017 112.599 + 53.6055i 472.024i −497.589 + 120.619i −243.000 192.234 403.790i
31.16 3.43879 + 7.22321i 15.5885i −40.3494 + 49.6782i 55.9017 112.599 53.6055i 472.024i −497.589 120.619i −243.000 192.234 + 403.790i
31.17 5.21792 6.06410i 15.5885i −9.54653 63.2840i 55.9017 −94.5299 81.3394i 234.050i −433.573 272.320i −243.000 291.691 338.993i
31.18 5.21792 + 6.06410i 15.5885i −9.54653 + 63.2840i 55.9017 −94.5299 + 81.3394i 234.050i −433.573 + 272.320i −243.000 291.691 + 338.993i
31.19 5.40467 5.89827i 15.5885i −5.57916 63.7564i −55.9017 91.9449 + 84.2504i 125.427i −406.206 311.674i −243.000 −302.130 + 329.723i
31.20 5.40467 + 5.89827i 15.5885i −5.57916 + 63.7564i −55.9017 91.9449 84.2504i 125.427i −406.206 + 311.674i −243.000 −302.130 329.723i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.24 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.7.c.a 24
3.b odd 2 1 180.7.c.b 24
4.b odd 2 1 inner 60.7.c.a 24
12.b even 2 1 180.7.c.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.c.a 24 1.a even 1 1 trivial
60.7.c.a 24 4.b odd 2 1 inner
180.7.c.b 24 3.b odd 2 1
180.7.c.b 24 12.b even 2 1

## Hecke kernels

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