Properties

Label 175.4.o.a
Level $175$
Weight $4$
Character orbit 175.o
Analytic conductor $10.325$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,4,Mod(68,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 10]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.68");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.o (of order \(12\), degree \(4\), 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: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{11} - 148 q^{16} + 756 q^{21} - 1524 q^{26} - 1584 q^{31} + 1752 q^{36} + 2324 q^{46} + 1752 q^{51} + 7560 q^{56} - 1644 q^{61} - 15504 q^{66} - 5600 q^{71} + 11424 q^{81} + 3440 q^{86} + 15456 q^{91} - 19560 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 −1.24155 4.63351i 1.84884 + 0.495394i −12.9998 + 7.50544i 0 9.18166i 4.86236 17.8706i 23.7807 + 23.7807i −20.2099 11.6682i 0
68.2 −0.951619 3.55149i −7.36493 1.97343i −4.77930 + 2.75933i 0 28.0344i 0.130416 + 18.5198i −6.45116 6.45116i 26.9651 + 15.5683i 0
68.3 −0.575247 2.14685i 9.51748 + 2.55020i 2.65014 1.53006i 0 21.8996i 9.19207 + 16.0781i −17.3821 17.3821i 60.6963 + 35.0430i 0
68.4 −0.320201 1.19501i 0.325765 + 0.0872885i 5.60269 3.23471i 0 0.417242i −17.8942 4.77454i −12.6579 12.6579i −23.2842 13.4431i 0
68.5 0.320201 + 1.19501i −0.325765 0.0872885i 5.60269 3.23471i 0 0.417242i 17.8942 + 4.77454i 12.6579 + 12.6579i −23.2842 13.4431i 0
68.6 0.575247 + 2.14685i −9.51748 2.55020i 2.65014 1.53006i 0 21.8996i −9.19207 16.0781i 17.3821 + 17.3821i 60.6963 + 35.0430i 0
68.7 0.951619 + 3.55149i 7.36493 + 1.97343i −4.77930 + 2.75933i 0 28.0344i −0.130416 18.5198i 6.45116 + 6.45116i 26.9651 + 15.5683i 0
68.8 1.24155 + 4.63351i −1.84884 0.495394i −12.9998 + 7.50544i 0 9.18166i −4.86236 + 17.8706i −23.7807 23.7807i −20.2099 11.6682i 0
82.1 −4.63351 + 1.24155i −0.495394 + 1.84884i 12.9998 7.50544i 0 9.18166i −17.8706 4.86236i −23.7807 + 23.7807i 20.2099 + 11.6682i 0
82.2 −3.55149 + 0.951619i 1.97343 7.36493i 4.77930 2.75933i 0 28.0344i 18.5198 0.130416i 6.45116 6.45116i −26.9651 15.5683i 0
82.3 −2.14685 + 0.575247i −2.55020 + 9.51748i −2.65014 + 1.53006i 0 21.8996i 16.0781 9.19207i 17.3821 17.3821i −60.6963 35.0430i 0
82.4 −1.19501 + 0.320201i −0.0872885 + 0.325765i −5.60269 + 3.23471i 0 0.417242i −4.77454 + 17.8942i 12.6579 12.6579i 23.2842 + 13.4431i 0
82.5 1.19501 0.320201i 0.0872885 0.325765i −5.60269 + 3.23471i 0 0.417242i 4.77454 17.8942i −12.6579 + 12.6579i 23.2842 + 13.4431i 0
82.6 2.14685 0.575247i 2.55020 9.51748i −2.65014 + 1.53006i 0 21.8996i −16.0781 + 9.19207i −17.3821 + 17.3821i −60.6963 35.0430i 0
82.7 3.55149 0.951619i −1.97343 + 7.36493i 4.77930 2.75933i 0 28.0344i −18.5198 + 0.130416i −6.45116 + 6.45116i −26.9651 15.5683i 0
82.8 4.63351 1.24155i 0.495394 1.84884i 12.9998 7.50544i 0 9.18166i 17.8706 + 4.86236i 23.7807 23.7807i 20.2099 + 11.6682i 0
143.1 −4.63351 1.24155i −0.495394 1.84884i 12.9998 + 7.50544i 0 9.18166i −17.8706 + 4.86236i −23.7807 23.7807i 20.2099 11.6682i 0
143.2 −3.55149 0.951619i 1.97343 + 7.36493i 4.77930 + 2.75933i 0 28.0344i 18.5198 + 0.130416i 6.45116 + 6.45116i −26.9651 + 15.5683i 0
143.3 −2.14685 0.575247i −2.55020 9.51748i −2.65014 1.53006i 0 21.8996i 16.0781 + 9.19207i 17.3821 + 17.3821i −60.6963 + 35.0430i 0
143.4 −1.19501 0.320201i −0.0872885 0.325765i −5.60269 3.23471i 0 0.417242i −4.77454 17.8942i 12.6579 + 12.6579i 23.2842 13.4431i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.d odd 6 1 inner
35.i odd 6 1 inner
35.k even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.4.o.a 32
5.b even 2 1 inner 175.4.o.a 32
5.c odd 4 2 inner 175.4.o.a 32
7.d odd 6 1 inner 175.4.o.a 32
35.i odd 6 1 inner 175.4.o.a 32
35.k even 12 2 inner 175.4.o.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.4.o.a 32 1.a even 1 1 trivial
175.4.o.a 32 5.b even 2 1 inner
175.4.o.a 32 5.c odd 4 2 inner
175.4.o.a 32 7.d odd 6 1 inner
175.4.o.a 32 35.i odd 6 1 inner
175.4.o.a 32 35.k even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 739 T_{2}^{28} + 430246 T_{2}^{24} - 80374023 T_{2}^{20} + 11478799782 T_{2}^{16} - 296436161763 T_{2}^{12} + 6269585321601 T_{2}^{8} - 14542274767104 T_{2}^{4} + \cdots + 30601961865216 \) acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display