Properties

Label 60.4.h.c
Level $60$
Weight $4$
Character orbit 60.h
Analytic conductor $3.540$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 60.h (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.

\(\operatorname{Tr}(f)(q) = \) \( 24q + 56q^{4} + 12q^{6} + 192q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 56q^{4} + 12q^{6} + 192q^{9} - 32q^{10} - 240q^{16} - 264q^{21} + 168q^{24} - 88q^{25} - 252q^{30} - 1088q^{34} - 1104q^{36} + 704q^{40} + 456q^{45} + 3368q^{46} - 1304q^{49} + 468q^{54} + 2496q^{60} + 2080q^{61} + 1376q^{64} - 672q^{66} + 2568q^{69} - 2632q^{70} - 1536q^{76} - 5112q^{81} - 2328q^{84} - 6944q^{85} - 1152q^{90} - 4840q^{94} - 2832q^{96} + O(q^{100}) \)

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 −2.76761 0.583363i −3.16559 4.12057i 7.31938 + 3.22905i 4.82699 10.0847i 6.35734 + 13.2508i −10.1458 −18.3735 13.2066i −6.95813 + 26.0880i −19.2422 + 25.0945i
59.2 −2.76761 0.583363i 3.16559 4.12057i 7.31938 + 3.22905i 4.82699 + 10.0847i −11.1649 + 9.55745i 10.1458 −18.3735 13.2066i −6.95813 26.0880i −7.47622 30.7263i
59.3 −2.76761 + 0.583363i −3.16559 + 4.12057i 7.31938 3.22905i 4.82699 + 10.0847i 6.35734 13.2508i −10.1458 −18.3735 + 13.2066i −6.95813 26.0880i −19.2422 25.0945i
59.4 −2.76761 + 0.583363i 3.16559 + 4.12057i 7.31938 3.22905i 4.82699 10.0847i −11.1649 9.55745i 10.1458 −18.3735 + 13.2066i −6.95813 + 26.0880i −7.47622 + 30.7263i
59.5 −2.40208 1.49331i −3.96106 + 3.36304i 3.54002 + 7.17414i −9.10892 6.48287i 14.5369 2.16320i 26.5236 2.20980 22.5193i 4.37993 26.6424i 12.1994 + 29.1749i
59.6 −2.40208 1.49331i 3.96106 + 3.36304i 3.54002 + 7.17414i −9.10892 + 6.48287i −4.49272 13.9934i −26.5236 2.20980 22.5193i 4.37993 + 26.6424i 31.5614 1.96993i
59.7 −2.40208 + 1.49331i −3.96106 3.36304i 3.54002 7.17414i −9.10892 + 6.48287i 14.5369 + 2.16320i 26.5236 2.20980 + 22.5193i 4.37993 + 26.6424i 12.1994 29.1749i
59.8 −2.40208 + 1.49331i 3.96106 3.36304i 3.54002 7.17414i −9.10892 6.48287i −4.49272 + 13.9934i −26.5236 2.20980 + 22.5193i 4.37993 26.6424i 31.5614 + 1.96993i
59.9 −1.43885 2.43510i −5.17582 0.459239i −3.85940 + 7.00750i 8.70218 + 7.01941i 6.32895 + 13.2644i −7.71743 22.6171 0.684751i 26.5782 + 4.75387i 4.57180 31.2906i
59.10 −1.43885 2.43510i 5.17582 0.459239i −3.85940 + 7.00750i 8.70218 7.01941i −8.56554 11.9428i 7.71743 22.6171 0.684751i 26.5782 4.75387i −29.6141 11.0907i
59.11 −1.43885 + 2.43510i −5.17582 + 0.459239i −3.85940 7.00750i 8.70218 7.01941i 6.32895 13.2644i −7.71743 22.6171 + 0.684751i 26.5782 4.75387i 4.57180 + 31.2906i
59.12 −1.43885 + 2.43510i 5.17582 + 0.459239i −3.85940 7.00750i 8.70218 + 7.01941i −8.56554 + 11.9428i 7.71743 22.6171 + 0.684751i 26.5782 + 4.75387i −29.6141 + 11.0907i
59.13 1.43885 2.43510i −5.17582 0.459239i −3.85940 7.00750i −8.70218 + 7.01941i −8.56554 + 11.9428i −7.71743 −22.6171 0.684751i 26.5782 + 4.75387i 4.57180 + 31.2906i
59.14 1.43885 2.43510i 5.17582 0.459239i −3.85940 7.00750i −8.70218 7.01941i 6.32895 13.2644i 7.71743 −22.6171 0.684751i 26.5782 4.75387i −29.6141 + 11.0907i
59.15 1.43885 + 2.43510i −5.17582 + 0.459239i −3.85940 + 7.00750i −8.70218 7.01941i −8.56554 11.9428i −7.71743 −22.6171 + 0.684751i 26.5782 4.75387i 4.57180 31.2906i
59.16 1.43885 + 2.43510i 5.17582 + 0.459239i −3.85940 + 7.00750i −8.70218 + 7.01941i 6.32895 + 13.2644i 7.71743 −22.6171 + 0.684751i 26.5782 + 4.75387i −29.6141 11.0907i
59.17 2.40208 1.49331i −3.96106 + 3.36304i 3.54002 7.17414i 9.10892 6.48287i −4.49272 + 13.9934i 26.5236 −2.20980 22.5193i 4.37993 26.6424i 12.1994 29.1749i
59.18 2.40208 1.49331i 3.96106 + 3.36304i 3.54002 7.17414i 9.10892 + 6.48287i 14.5369 + 2.16320i −26.5236 −2.20980 22.5193i 4.37993 + 26.6424i 31.5614 + 1.96993i
59.19 2.40208 + 1.49331i −3.96106 3.36304i 3.54002 + 7.17414i 9.10892 + 6.48287i −4.49272 13.9934i 26.5236 −2.20980 + 22.5193i 4.37993 + 26.6424i 12.1994 + 29.1749i
59.20 2.40208 + 1.49331i 3.96106 3.36304i 3.54002 + 7.17414i 9.10892 6.48287i 14.5369 2.16320i −26.5236 −2.20980 + 22.5193i 4.37993 26.6424i 31.5614 1.96993i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.24
Significant digits:
Format:

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 60.4.h.c 24
3.b odd 2 1 inner 60.4.h.c 24
4.b odd 2 1 inner 60.4.h.c 24
5.b even 2 1 inner 60.4.h.c 24
12.b even 2 1 inner 60.4.h.c 24
15.d odd 2 1 inner 60.4.h.c 24
20.d odd 2 1 inner 60.4.h.c 24
60.h even 2 1 inner 60.4.h.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.h.c 24 1.a even 1 1 trivial
60.4.h.c 24 3.b odd 2 1 inner
60.4.h.c 24 4.b odd 2 1 inner
60.4.h.c 24 5.b even 2 1 inner
60.4.h.c 24 12.b even 2 1 inner
60.4.h.c 24 15.d odd 2 1 inner
60.4.h.c 24 20.d odd 2 1 inner
60.4.h.c 24 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 866 T_{7}^{4} + 120448 T_{7}^{2} - 4313088 \) acting on \(S_{4}^{\mathrm{new}}(60, [\chi])\).