↑

Properties

Label 60.5.f.a
Level $60$
Weight $5$
Character orbit 60.f
Analytic conductor $6.202$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,5,Mod(19,60)]
 
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, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.19");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 60.f (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: \(6.20219778503\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 14 q^{4} - 24 q^{5} - 18 q^{6} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 14 q^{4} - 24 q^{5} - 18 q^{6} + 648 q^{9} + 274 q^{10} - 36 q^{14} + 594 q^{16} - 12 q^{20} - 594 q^{24} + 1208 q^{25} - 2868 q^{26} - 1680 q^{29} + 468 q^{30} + 3076 q^{34} + 378 q^{36} - 7222 q^{40} - 4848 q^{41} - 3828 q^{44} - 648 q^{45} - 15280 q^{46} + 5416 q^{49} + 14472 q^{50} - 486 q^{54} + 32172 q^{56} - 7506 q^{60} + 2896 q^{61} - 18298 q^{64} - 2688 q^{65} - 15588 q^{66} + 9792 q^{69} + 27608 q^{70} + 31836 q^{74} + 50136 q^{76} - 27348 q^{80} + 17496 q^{81} - 4284 q^{84} - 15680 q^{85} - 58152 q^{86} - 38544 q^{89} + 7398 q^{90} + 4808 q^{94} + 21978 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −3.90764 − 0.854601i −5.19615 14.5393 + 6.67895i −9.11612 − 23.2787i 20.3047 + 4.44064i −10.5267 −51.1066 − 38.5243i 27.0000 15.7285 + 98.7553i
19.2 −3.90764 + 0.854601i −5.19615 14.5393 − 6.67895i −9.11612 + 23.2787i 20.3047 − 4.44064i −10.5267 −51.1066 + 38.5243i 27.0000 15.7285 − 98.7553i
19.3 −3.76887 − 1.34002i 5.19615 12.4087 + 10.1007i 11.9809 + 21.9421i −19.5836 − 6.96293i −63.7886 −33.2317 − 54.6960i 27.0000 −15.7516 − 98.7516i
19.4 −3.76887 + 1.34002i 5.19615 12.4087 − 10.1007i 11.9809 − 21.9421i −19.5836 + 6.96293i −63.7886 −33.2317 + 54.6960i 27.0000 −15.7516 + 98.7516i
19.5 −3.44337 − 2.03548i 5.19615 7.71364 + 14.0178i −20.5121 − 14.2918i −17.8923 − 10.5767i 51.0440 1.97210 − 63.9696i 27.0000 41.5401 + 90.9638i
19.6 −3.44337 + 2.03548i 5.19615 7.71364 − 14.0178i −20.5121 + 14.2918i −17.8923 + 10.5767i 51.0440 1.97210 + 63.9696i 27.0000 41.5401 − 90.9638i
19.7 −2.59896 − 3.04063i −5.19615 −2.49086 + 15.8049i 24.9405 − 1.72355i 13.5046 + 15.7996i −2.65040 54.5305 − 33.5025i 27.0000 −70.0600 − 71.3554i
19.8 −2.59896 + 3.04063i −5.19615 −2.49086 − 15.8049i 24.9405 + 1.72355i 13.5046 − 15.7996i −2.65040 54.5305 + 33.5025i 27.0000 −70.0600 + 71.3554i
19.9 −0.988963 − 3.87582i 5.19615 −14.0439 + 7.66608i 11.6321 + 22.1290i −5.13880 − 20.1393i 66.7450 43.6012 + 46.8501i 27.0000 74.2643 − 66.9688i
19.10 −0.988963 + 3.87582i 5.19615 −14.0439 − 7.66608i 11.6321 − 22.1290i −5.13880 + 20.1393i 66.7450 43.6012 − 46.8501i 27.0000 74.2643 + 66.9688i
19.11 −0.828581 − 3.91324i −5.19615 −14.6269 + 6.48488i −24.9254 + 1.93048i 4.30543 + 20.3338i 67.1774 37.4965 + 51.8654i 27.0000 28.2071 + 95.9394i
19.12 −0.828581 + 3.91324i −5.19615 −14.6269 − 6.48488i −24.9254 − 1.93048i 4.30543 − 20.3338i 67.1774 37.4965 − 51.8654i 27.0000 28.2071 − 95.9394i
19.13 0.828581 − 3.91324i 5.19615 −14.6269 − 6.48488i −24.9254 + 1.93048i 4.30543 − 20.3338i −67.1774 −37.4965 + 51.8654i 27.0000 −13.0983 + 99.1385i
19.14 0.828581 + 3.91324i 5.19615 −14.6269 + 6.48488i −24.9254 − 1.93048i 4.30543 + 20.3338i −67.1774 −37.4965 − 51.8654i 27.0000 −13.0983 − 99.1385i
19.15 0.988963 − 3.87582i −5.19615 −14.0439 − 7.66608i 11.6321 + 22.1290i −5.13880 + 20.1393i −66.7450 −43.6012 + 46.8501i 27.0000 97.2718 − 23.1992i
19.16 0.988963 + 3.87582i −5.19615 −14.0439 + 7.66608i 11.6321 − 22.1290i −5.13880 − 20.1393i −66.7450 −43.6012 − 46.8501i 27.0000 97.2718 + 23.1992i
19.17 2.59896 − 3.04063i 5.19615 −2.49086 − 15.8049i 24.9405 − 1.72355i 13.5046 − 15.7996i 2.65040 −54.5305 − 33.5025i 27.0000 59.5786 − 80.3143i
19.18 2.59896 + 3.04063i 5.19615 −2.49086 + 15.8049i 24.9405 + 1.72355i 13.5046 + 15.7996i 2.65040 −54.5305 + 33.5025i 27.0000 59.5786 + 80.3143i
19.19 3.44337 − 2.03548i −5.19615 7.71364 − 14.0178i −20.5121 − 14.2918i −17.8923 + 10.5767i −51.0440 −1.97210 − 63.9696i 27.0000 −99.7214 − 7.45996i
19.20 3.44337 + 2.03548i −5.19615 7.71364 + 14.0178i −20.5121 + 14.2918i −17.8923 − 10.5767i −51.0440 −1.97210 + 63.9696i 27.0000 −99.7214 + 7.45996i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.24
Significant digits:
Format:

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 60.5.f.a ✓ 24
3.b odd 2 1 180.5.f.i 24
4.b odd 2 1 inner 60.5.f.a ✓ 24
5.b even 2 1 inner 60.5.f.a ✓ 24
5.c odd 4 2 300.5.c.e 24
8.b even 2 1 960.5.j.d 24
8.d odd 2 1 960.5.j.d 24
12.b even 2 1 180.5.f.i 24
15.d odd 2 1 180.5.f.i 24
20.d odd 2 1 inner 60.5.f.a ✓ 24
20.e even 4 2 300.5.c.e 24
40.e odd 2 1 960.5.j.d 24
40.f even 2 1 960.5.j.d 24
60.h even 2 1 180.5.f.i 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.f.a ✓ 24 1.a even 1 1 trivial
60.5.f.a ✓ 24 4.b odd 2 1 inner
60.5.f.a ✓ 24 5.b even 2 1 inner
60.5.f.a ✓ 24 20.d odd 2 1 inner
180.5.f.i 24 3.b odd 2 1
180.5.f.i 24 12.b even 2 1
180.5.f.i 24 15.d odd 2 1
180.5.f.i 24 60.h even 2 1
300.5.c.e 24 5.c odd 4 2
300.5.c.e 24 20.e even 4 2
960.5.j.d 24 8.b even 2 1
960.5.j.d 24 8.d odd 2 1
960.5.j.d 24 40.e odd 2 1
960.5.j.d 24 40.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(60, [\chi])\).