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Properties

Label 144.5.m.b
Level $144$
Weight $5$
Character orbit 144.m
Analytic conductor $14.885$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,5,Mod(19,144)]
 
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([2, 3, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.19");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 144.m (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: \(14.8852746841\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{4} - 496 q^{10} + 352 q^{16} + 704 q^{19} + 1096 q^{22} - 960 q^{28} - 2232 q^{34} - 3648 q^{37} + 4872 q^{40} - 10048 q^{43} + 4640 q^{46} + 10976 q^{49} - 5064 q^{52} + 11776 q^{55} - 5352 q^{58} + 3776 q^{61} + 2568 q^{64} - 6656 q^{67} - 15672 q^{70} - 10680 q^{76} + 3400 q^{82} - 11200 q^{85} + 34416 q^{88} + 1344 q^{91} + 22512 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −3.96065 − 0.559657i 0 15.3736 + 4.43322i 24.0193 − 24.0193i 0 −67.0350 −58.4083 − 26.1624i 0 −108.575 + 81.6897i
19.2 −3.79566 + 1.26211i 0 12.8141 − 9.58112i −23.4449 + 23.4449i 0 −16.4067 −36.5457 + 52.5396i 0 59.3989 − 118.579i
19.3 −3.45802 + 2.01050i 0 7.91575 − 13.9047i 20.3894 − 20.3894i 0 91.3104 0.582666 + 63.9973i 0 −29.5139 + 111.500i
19.4 −3.38412 − 2.13254i 0 6.90451 + 14.4336i −7.10657 + 7.10657i 0 31.2871 7.41450 − 63.5691i 0 39.2046 − 8.89441i
19.5 −2.11669 + 3.39406i 0 −7.03926 − 14.3683i 6.25652 − 6.25652i 0 −66.1838 63.6669 + 6.52162i 0 7.99189 + 34.4781i
19.6 −1.36037 − 3.76157i 0 −12.2988 + 10.2343i −16.5947 + 16.5947i 0 −37.2763 55.2278 + 32.3402i 0 84.9972 + 39.8471i
19.7 −1.27321 − 3.79196i 0 −12.7579 + 9.65591i 32.8166 − 32.8166i 0 27.9506 52.8582 + 36.0833i 0 −166.221 − 82.6566i
19.8 −1.02175 + 3.86730i 0 −13.9121 − 7.90280i −3.96490 + 3.96490i 0 36.3537 44.7771 − 45.7275i 0 −11.2824 − 19.3846i
19.9 1.02175 − 3.86730i 0 −13.9121 − 7.90280i 3.96490 − 3.96490i 0 36.3537 −44.7771 + 45.7275i 0 −11.2824 − 19.3846i
19.10 1.27321 + 3.79196i 0 −12.7579 + 9.65591i −32.8166 + 32.8166i 0 27.9506 −52.8582 − 36.0833i 0 −166.221 − 82.6566i
19.11 1.36037 + 3.76157i 0 −12.2988 + 10.2343i 16.5947 − 16.5947i 0 −37.2763 −55.2278 − 32.3402i 0 84.9972 + 39.8471i
19.12 2.11669 − 3.39406i 0 −7.03926 − 14.3683i −6.25652 + 6.25652i 0 −66.1838 −63.6669 − 6.52162i 0 7.99189 + 34.4781i
19.13 3.38412 + 2.13254i 0 6.90451 + 14.4336i 7.10657 − 7.10657i 0 31.2871 −7.41450 + 63.5691i 0 39.2046 − 8.89441i
19.14 3.45802 − 2.01050i 0 7.91575 − 13.9047i −20.3894 + 20.3894i 0 91.3104 −0.582666 − 63.9973i 0 −29.5139 + 111.500i
19.15 3.79566 − 1.26211i 0 12.8141 − 9.58112i 23.4449 − 23.4449i 0 −16.4067 36.5457 − 52.5396i 0 59.3989 − 118.579i
19.16 3.96065 + 0.559657i 0 15.3736 + 4.43322i −24.0193 + 24.0193i 0 −67.0350 58.4083 + 26.1624i 0 −108.575 + 81.6897i
91.1 −3.96065 + 0.559657i 0 15.3736 − 4.43322i 24.0193 + 24.0193i 0 −67.0350 −58.4083 + 26.1624i 0 −108.575 − 81.6897i
91.2 −3.79566 − 1.26211i 0 12.8141 + 9.58112i −23.4449 − 23.4449i 0 −16.4067 −36.5457 − 52.5396i 0 59.3989 + 118.579i
91.3 −3.45802 − 2.01050i 0 7.91575 + 13.9047i 20.3894 + 20.3894i 0 91.3104 0.582666 − 63.9973i 0 −29.5139 − 111.500i
91.4 −3.38412 + 2.13254i 0 6.90451 − 14.4336i −7.10657 − 7.10657i 0 31.2871 7.41450 + 63.5691i 0 39.2046 + 8.89441i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.5.m.b ✓ 32
3.b odd 2 1 inner 144.5.m.b ✓ 32
4.b odd 2 1 576.5.m.c 32
12.b even 2 1 576.5.m.c 32
16.e even 4 1 576.5.m.c 32
16.f odd 4 1 inner 144.5.m.b ✓ 32
48.i odd 4 1 576.5.m.c 32
48.k even 4 1 inner 144.5.m.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.5.m.b ✓ 32 1.a even 1 1 trivial
144.5.m.b ✓ 32 3.b odd 2 1 inner
144.5.m.b ✓ 32 16.f odd 4 1 inner
144.5.m.b ✓ 32 48.k even 4 1 inner
576.5.m.c 32 4.b odd 2 1
576.5.m.c 32 12.b even 2 1
576.5.m.c 32 16.e even 4 1
576.5.m.c 32 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} + 8190976 T_{5}^{28} + 20883917771008 T_{5}^{24} + \cdots + 96\!\cdots\!96 \) acting on \(S_{5}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display