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Properties

Label 72.5.m.a
Level $72$
Weight $5$
Character orbit 72.m
Analytic conductor $7.443$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,5,Mod(41,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.41");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 72.m (of order \(6\), degree \(2\), 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: \(7.44263734204\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} - 100 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} - 100 q^{9} + 252 q^{11} - 80 q^{15} - 408 q^{19} + 24 q^{21} + 720 q^{23} + 1500 q^{25} - 1280 q^{27} + 2376 q^{29} - 1104 q^{31} - 1412 q^{33} - 4184 q^{39} + 1980 q^{41} + 1476 q^{43} - 4696 q^{45} + 4536 q^{47} - 6084 q^{49} - 7828 q^{51} + 2544 q^{55} - 1204 q^{57} + 10332 q^{59} + 2784 q^{61} + 9072 q^{63} + 17280 q^{65} - 2604 q^{67} + 5680 q^{69} + 5112 q^{73} - 15412 q^{75} - 28368 q^{77} + 3480 q^{79} - 26548 q^{81} - 23400 q^{83} + 7392 q^{85} - 3192 q^{87} - 14208 q^{91} + 39488 q^{93} + 57528 q^{95} - 4020 q^{97} + 50744 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 0 −8.92719 − 1.14252i 0 −23.0826 − 13.3268i 0 28.8348 + 49.9434i 0 78.3893 + 20.3989i 0
41.2 0 −7.33134 + 5.22030i 0 −7.42683 − 4.28788i 0 −2.37612 − 4.11556i 0 26.4970 − 76.5435i 0
41.3 0 −6.25234 − 6.47366i 0 −1.90728 − 1.10117i 0 −9.68825 − 16.7805i 0 −2.81653 + 80.9510i 0
41.4 0 −5.78971 − 6.89052i 0 38.2277 + 22.0708i 0 −25.3787 − 43.9571i 0 −13.9586 + 79.7882i 0
41.5 0 −5.34380 + 7.24181i 0 18.9074 + 10.9162i 0 7.93787 + 13.7488i 0 −23.8877 − 77.3975i 0
41.6 0 2.36760 − 8.68300i 0 −20.4039 − 11.7802i 0 0.0859679 + 0.148901i 0 −69.7889 − 41.1158i 0
41.7 0 2.41484 + 8.66998i 0 −25.9859 − 15.0030i 0 −37.7553 − 65.3941i 0 −69.3371 + 41.8733i 0
41.8 0 3.21271 + 8.40705i 0 36.5152 + 21.0820i 0 −13.5757 − 23.5138i 0 −60.3569 + 54.0189i 0
41.9 0 4.50339 − 7.79227i 0 20.8375 + 12.0305i 0 37.9185 + 65.6768i 0 −40.4389 − 70.1833i 0
41.10 0 5.77378 + 6.90387i 0 −33.3715 − 19.2670i 0 45.3759 + 78.5934i 0 −14.3269 + 79.7229i 0
41.11 0 8.44225 − 3.11903i 0 −14.6470 − 8.45647i 0 −42.9574 − 74.4045i 0 61.5433 − 52.6632i 0
41.12 0 8.92978 + 1.12208i 0 12.3374 + 7.12298i 0 11.5784 + 20.0543i 0 78.4819 + 20.0399i 0
65.1 0 −8.92719 + 1.14252i 0 −23.0826 + 13.3268i 0 28.8348 − 49.9434i 0 78.3893 − 20.3989i 0
65.2 0 −7.33134 − 5.22030i 0 −7.42683 + 4.28788i 0 −2.37612 + 4.11556i 0 26.4970 + 76.5435i 0
65.3 0 −6.25234 + 6.47366i 0 −1.90728 + 1.10117i 0 −9.68825 + 16.7805i 0 −2.81653 − 80.9510i 0
65.4 0 −5.78971 + 6.89052i 0 38.2277 − 22.0708i 0 −25.3787 + 43.9571i 0 −13.9586 − 79.7882i 0
65.5 0 −5.34380 − 7.24181i 0 18.9074 − 10.9162i 0 7.93787 − 13.7488i 0 −23.8877 + 77.3975i 0
65.6 0 2.36760 + 8.68300i 0 −20.4039 + 11.7802i 0 0.0859679 − 0.148901i 0 −69.7889 + 41.1158i 0
65.7 0 2.41484 − 8.66998i 0 −25.9859 + 15.0030i 0 −37.7553 + 65.3941i 0 −69.3371 − 41.8733i 0
65.8 0 3.21271 − 8.40705i 0 36.5152 − 21.0820i 0 −13.5757 + 23.5138i 0 −60.3569 − 54.0189i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.5.m.a ✓ 24
3.b odd 2 1 216.5.m.a 24
4.b odd 2 1 144.5.q.d 24
9.c even 3 1 216.5.m.a 24
9.c even 3 1 648.5.e.c 24
9.d odd 6 1 inner 72.5.m.a ✓ 24
9.d odd 6 1 648.5.e.c 24
12.b even 2 1 432.5.q.d 24
36.f odd 6 1 432.5.q.d 24
36.f odd 6 1 1296.5.e.j 24
36.h even 6 1 144.5.q.d 24
36.h even 6 1 1296.5.e.j 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.5.m.a ✓ 24 1.a even 1 1 trivial
72.5.m.a ✓ 24 9.d odd 6 1 inner
144.5.q.d 24 4.b odd 2 1
144.5.q.d 24 36.h even 6 1
216.5.m.a 24 3.b odd 2 1
216.5.m.a 24 9.c even 3 1
432.5.q.d 24 12.b even 2 1
432.5.q.d 24 36.f odd 6 1
648.5.e.c 24 9.c even 3 1
648.5.e.c 24 9.d odd 6 1
1296.5.e.j 24 36.f odd 6 1
1296.5.e.j 24 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(72, [\chi])\).