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Properties

Label 144.6.k.b
Level $144$
Weight $6$
Character orbit 144.k
Analytic conductor $23.095$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,6,Mod(37,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.37");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 144.k (of order \(4\), 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: \(23.0952700531\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 44 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 44 q^{4} - 1168 q^{10} - 384 q^{16} + 2360 q^{19} - 14616 q^{22} + 6112 q^{28} - 23064 q^{31} - 7896 q^{34} - 21296 q^{37} - 28504 q^{40} - 32728 q^{43} - 9376 q^{46} - 96040 q^{49} - 17000 q^{52} - 6248 q^{58} + 48080 q^{61} - 181592 q^{64} - 77024 q^{67} + 204136 q^{70} - 32216 q^{76} + 427320 q^{79} - 333400 q^{82} + 132400 q^{85} + 115312 q^{88} - 229384 q^{91} + 271024 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −5.64667 − 0.339366i 0 31.7697 + 3.83257i −40.1411 − 40.1411i 0 30.7493i −178.092 − 32.4228i 0 213.041 + 240.286i
37.2 −5.33138 + 1.89115i 0 24.8471 − 20.1648i 67.9227 + 67.9227i 0 195.797i −94.3348 + 154.496i 0 −490.573 − 233.670i
37.3 −5.20588 − 2.21333i 0 22.2024 + 23.0446i 42.9922 + 42.9922i 0 − 49.0467i −64.5776 − 169.109i 0 −128.656 − 318.968i
37.4 −4.72946 + 3.10358i 0 12.7356 − 29.3565i 11.5050 + 11.5050i 0 − 185.403i 30.8780 + 178.366i 0 −90.1189 − 18.7057i
37.5 −4.03450 + 3.96520i 0 0.554358 − 31.9952i −42.6260 − 42.6260i 0 10.7824i 124.631 + 131.283i 0 340.995 + 2.95387i
37.6 −3.91062 − 4.08743i 0 −1.41417 + 31.9687i −1.95618 − 1.95618i 0 − 146.383i 136.200 − 119.237i 0 −0.345879 + 15.6456i
37.7 −2.98465 − 4.80540i 0 −14.1837 + 28.6849i −21.0344 − 21.0344i 0 199.282i 180.176 − 17.4562i 0 −38.2982 + 163.859i
37.8 −1.94518 + 5.31190i 0 −24.4325 − 20.6652i 9.33200 + 9.33200i 0 93.5449i 157.297 − 89.5854i 0 −67.7231 + 31.4182i
37.9 −0.644434 − 5.62003i 0 −31.1694 + 7.24347i 63.9508 + 63.9508i 0 − 35.8385i 60.7951 + 170.505i 0 318.193 − 400.617i
37.10 −0.212943 − 5.65284i 0 −31.9093 + 2.40747i −64.0661 − 64.0661i 0 − 211.484i 20.4039 + 179.866i 0 −348.513 + 375.798i
37.11 0.212943 + 5.65284i 0 −31.9093 + 2.40747i 64.0661 + 64.0661i 0 − 211.484i −20.4039 − 179.866i 0 −348.513 + 375.798i
37.12 0.644434 + 5.62003i 0 −31.1694 + 7.24347i −63.9508 − 63.9508i 0 − 35.8385i −60.7951 − 170.505i 0 318.193 − 400.617i
37.13 1.94518 − 5.31190i 0 −24.4325 − 20.6652i −9.33200 − 9.33200i 0 93.5449i −157.297 + 89.5854i 0 −67.7231 + 31.4182i
37.14 2.98465 + 4.80540i 0 −14.1837 + 28.6849i 21.0344 + 21.0344i 0 199.282i −180.176 + 17.4562i 0 −38.2982 + 163.859i
37.15 3.91062 + 4.08743i 0 −1.41417 + 31.9687i 1.95618 + 1.95618i 0 − 146.383i −136.200 + 119.237i 0 −0.345879 + 15.6456i
37.16 4.03450 − 3.96520i 0 0.554358 − 31.9952i 42.6260 + 42.6260i 0 10.7824i −124.631 − 131.283i 0 340.995 + 2.95387i
37.17 4.72946 − 3.10358i 0 12.7356 − 29.3565i −11.5050 − 11.5050i 0 − 185.403i −30.8780 − 178.366i 0 −90.1189 − 18.7057i
37.18 5.20588 + 2.21333i 0 22.2024 + 23.0446i −42.9922 − 42.9922i 0 − 49.0467i 64.5776 + 169.109i 0 −128.656 − 318.968i
37.19 5.33138 − 1.89115i 0 24.8471 − 20.1648i −67.9227 − 67.9227i 0 195.797i 94.3348 − 154.496i 0 −490.573 − 233.670i
37.20 5.64667 + 0.339366i 0 31.7697 + 3.83257i 40.1411 + 40.1411i 0 30.7493i 178.092 + 32.4228i 0 213.041 + 240.286i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.6.k.b ✓ 40
3.b odd 2 1 inner 144.6.k.b ✓ 40
4.b odd 2 1 576.6.k.b 40
12.b even 2 1 576.6.k.b 40
16.e even 4 1 inner 144.6.k.b ✓ 40
16.f odd 4 1 576.6.k.b 40
48.i odd 4 1 inner 144.6.k.b ✓ 40
48.k even 4 1 576.6.k.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.6.k.b ✓ 40 1.a even 1 1 trivial
144.6.k.b ✓ 40 3.b odd 2 1 inner
144.6.k.b ✓ 40 16.e even 4 1 inner
144.6.k.b ✓ 40 48.i odd 4 1 inner
576.6.k.b 40 4.b odd 2 1
576.6.k.b 40 12.b even 2 1
576.6.k.b 40 16.f odd 4 1
576.6.k.b 40 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} + 257566336 T_{5}^{36} + \cdots + 70\!\cdots\!64 \) acting on \(S_{6}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display