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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,4,Mod(59,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.59");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \( T_{7}^{6} - 866T_{7}^{4} + 120448T_{7}^{2} - 4313088 \) acting on \(S_{4}^{\mathrm{new}}(60, [\chi])\). Copy content Toggle raw display