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Properties

Label 144.7.j.a
Level $144$
Weight $7$
Character orbit 144.j
Analytic conductor $33.128$
Analytic rank $0$
Dimension $96$
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] = [144,7,Mod(53,144)]
 
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, 2]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.53");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 144.j (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: \(33.1277880413\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 3000 q^{10} - 3792 q^{16} + 7872 q^{19} + 84168 q^{22} - 21240 q^{28} - 256344 q^{34} + 328776 q^{40} + 390528 q^{43} - 212232 q^{46} - 1613472 q^{49} + 244728 q^{52} - 1233048 q^{58} + 652992 q^{61} - 577584 q^{64} - 603840 q^{67} - 789912 q^{70} + 1023168 q^{76} - 1721856 q^{79} + 964080 q^{82} - 744000 q^{85} - 2665104 q^{88} - 3700800 q^{91} + 2825496 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −7.96863 − 0.707795i 0 62.9981 + 11.2803i −2.05325 + 2.05325i 0 176.771i −494.024 − 134.478i 0 17.8148 − 14.9083i
53.2 −7.96219 − 0.776837i 0 62.7930 + 12.3706i 71.4717 − 71.4717i 0 − 315.658i −490.360 − 147.277i 0 −624.593 + 513.549i
53.3 −7.86323 − 1.47298i 0 59.6606 + 23.1648i −141.033 + 141.033i 0 − 48.8223i −435.004 − 270.029i 0 1316.71 − 901.234i
53.4 −7.70043 + 2.16874i 0 54.5931 − 33.4005i 15.2729 − 15.2729i 0 498.002i −347.953 + 375.596i 0 −84.4848 + 150.731i
53.5 −7.69485 − 2.18844i 0 54.4215 + 33.6794i 8.97801 − 8.97801i 0 132.433i −345.060 − 378.256i 0 −88.7322 + 49.4366i
53.6 −7.68760 + 2.21376i 0 54.1985 − 34.0371i 158.404 − 158.404i 0 − 133.106i −341.306 + 381.646i 0 −867.078 + 1568.42i
53.7 −7.19036 + 3.50696i 0 39.4025 − 50.4326i −109.398 + 109.398i 0 − 428.243i −106.453 + 500.811i 0 402.956 − 1170.26i
53.8 −6.70691 − 4.36088i 0 25.9654 + 58.4961i −140.082 + 140.082i 0 − 123.966i 80.9474 − 505.561i 0 1550.39 − 328.635i
53.9 −6.68684 + 4.39161i 0 25.4276 − 58.7319i 68.0621 − 68.0621i 0 − 538.920i 87.8974 + 504.399i 0 −156.218 + 754.022i
53.10 −6.44786 + 4.73552i 0 19.1497 − 61.0679i −0.837641 + 0.837641i 0 466.887i 165.713 + 484.441i 0 1.43433 − 9.36765i
53.11 −6.33958 − 4.87953i 0 16.3804 + 61.8683i 95.8031 − 95.8031i 0 356.832i 198.043 − 472.147i 0 −1074.82 + 139.877i
53.12 −6.06497 − 5.21691i 0 9.56773 + 63.2808i 108.765 − 108.765i 0 − 659.213i 272.102 − 433.710i 0 −1227.07 + 92.2395i
53.13 −5.55737 + 5.75462i 0 −2.23126 − 63.9611i −100.508 + 100.508i 0 − 48.6019i 380.472 + 342.615i 0 −19.8249 − 1136.94i
53.14 −5.55195 − 5.75985i 0 −2.35175 + 63.9568i −41.7984 + 41.7984i 0 − 240.685i 381.438 − 341.539i 0 472.815 + 8.68999i
53.15 −4.52535 − 6.59706i 0 −23.0424 + 59.7080i 143.326 − 143.326i 0 548.969i 498.172 − 118.188i 0 −1594.13 − 296.930i
53.16 −3.74981 + 7.06675i 0 −35.8778 − 52.9980i 168.179 − 168.179i 0 260.196i 509.058 − 54.8067i 0 557.840 + 1819.12i
53.17 −3.65312 + 7.11721i 0 −37.3094 − 52.0001i 78.6085 − 78.6085i 0 − 255.650i 506.391 − 75.5764i 0 272.307 + 846.640i
53.18 −3.63712 − 7.12540i 0 −37.5427 + 51.8319i −67.0992 + 67.0992i 0 212.431i 505.870 + 78.9887i 0 722.157 + 234.061i
53.19 −2.59168 + 7.56857i 0 −50.5664 − 39.2306i −102.459 + 102.459i 0 388.352i 427.971 − 281.042i 0 −509.925 − 1041.01i
53.20 −2.38971 − 7.63474i 0 −52.5785 + 36.4897i −125.068 + 125.068i 0 645.212i 404.237 + 314.223i 0 1253.74 + 655.987i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.48
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 144.7.j.a ✓ 96
3.b odd 2 1 inner 144.7.j.a ✓ 96
4.b odd 2 1 576.7.j.a 96
12.b even 2 1 576.7.j.a 96
16.e even 4 1 inner 144.7.j.a ✓ 96
16.f odd 4 1 576.7.j.a 96
48.i odd 4 1 inner 144.7.j.a ✓ 96
48.k even 4 1 576.7.j.a 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.7.j.a ✓ 96 1.a even 1 1 trivial
144.7.j.a ✓ 96 3.b odd 2 1 inner
144.7.j.a ✓ 96 16.e even 4 1 inner
144.7.j.a ✓ 96 48.i odd 4 1 inner
576.7.j.a 96 4.b odd 2 1
576.7.j.a 96 12.b even 2 1
576.7.j.a 96 16.f odd 4 1
576.7.j.a 96 48.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(144, [\chi])\).