Properties

Label 675.2.h.b
Level $675$
Weight $2$
Character orbit 675.h
Analytic conductor $5.390$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [675,2,Mod(136,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.136");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.h (of order \(5\), 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 16 q^{4} + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 16 q^{4} + 20 q^{7} + 20 q^{10} - 8 q^{13} - 12 q^{16} + 2 q^{19} - 8 q^{22} - 30 q^{28} - 2 q^{31} - 6 q^{34} + 2 q^{37} - 50 q^{40} + 40 q^{43} - 36 q^{46} - 28 q^{49} - 30 q^{52} - 30 q^{55} - 52 q^{58} - 28 q^{61} + 46 q^{64} - 10 q^{67} + 60 q^{70} - 46 q^{73} + 260 q^{76} - 42 q^{79} + 136 q^{82} + 100 q^{85} - 68 q^{88} - 56 q^{91} + 182 q^{94} - 58 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1 −1.95344 + 1.41926i 0 1.18360 3.64275i −1.22237 1.87238i 0 0.0108985 1.36561 + 4.20292i 0 5.04522 + 1.92272i
136.2 −1.92473 + 1.39840i 0 1.13104 3.48099i 0.154240 + 2.23074i 0 1.19432 1.22050 + 3.75632i 0 −3.41634 4.07790i
136.3 −1.17685 + 0.855035i 0 0.0358679 0.110390i 2.21985 0.268806i 0 −0.960718 −0.846861 2.60637i 0 −2.38260 + 2.21440i
136.4 −0.634202 + 0.460775i 0 −0.428135 + 1.31766i −2.22112 0.258104i 0 4.26911 −0.820110 2.52404i 0 1.52757 0.859747i
136.5 −0.324639 + 0.235864i 0 −0.568275 + 1.74897i −0.0534510 + 2.23543i 0 −3.13164 −0.476037 1.46509i 0 −0.509905 0.738315i
136.6 0.324639 0.235864i 0 −0.568275 + 1.74897i 0.0534510 2.23543i 0 −3.13164 0.476037 + 1.46509i 0 −0.509905 0.738315i
136.7 0.634202 0.460775i 0 −0.428135 + 1.31766i 2.22112 + 0.258104i 0 4.26911 0.820110 + 2.52404i 0 1.52757 0.859747i
136.8 1.17685 0.855035i 0 0.0358679 0.110390i −2.21985 + 0.268806i 0 −0.960718 0.846861 + 2.60637i 0 −2.38260 + 2.21440i
136.9 1.92473 1.39840i 0 1.13104 3.48099i −0.154240 2.23074i 0 1.19432 −1.22050 3.75632i 0 −3.41634 4.07790i
136.10 1.95344 1.41926i 0 1.18360 3.64275i 1.22237 + 1.87238i 0 0.0108985 −1.36561 4.20292i 0 5.04522 + 1.92272i
271.1 −0.813148 + 2.50261i 0 −3.98383 2.89442i −1.55748 1.60445i 0 3.64251 6.22536 4.52299i 0 5.28178 2.59311i
271.2 −0.626486 + 1.92813i 0 −1.70715 1.24032i 1.52132 1.63878i 0 −1.89708 0.180674 0.131267i 0 2.20668 + 3.95997i
271.3 −0.601697 + 1.85183i 0 −1.44921 1.05292i 1.17636 + 1.90162i 0 1.11838 −0.328714 + 0.238825i 0 −4.22930 + 1.03423i
271.4 −0.402108 + 1.23756i 0 0.248165 + 0.180302i −2.22558 0.216297i 0 −2.28898 −2.42839 + 1.76433i 0 1.16261 2.66732i
271.5 −0.0972368 + 0.299264i 0 1.53793 + 1.11737i 1.59400 1.56817i 0 3.04319 −0.993071 + 0.721508i 0 0.314302 + 0.629511i
271.6 0.0972368 0.299264i 0 1.53793 + 1.11737i −1.59400 + 1.56817i 0 3.04319 0.993071 0.721508i 0 0.314302 + 0.629511i
271.7 0.402108 1.23756i 0 0.248165 + 0.180302i 2.22558 + 0.216297i 0 −2.28898 2.42839 1.76433i 0 1.16261 2.66732i
271.8 0.601697 1.85183i 0 −1.44921 1.05292i −1.17636 1.90162i 0 1.11838 0.328714 0.238825i 0 −4.22930 + 1.03423i
271.9 0.626486 1.92813i 0 −1.70715 1.24032i −1.52132 + 1.63878i 0 −1.89708 −0.180674 + 0.131267i 0 2.20668 + 3.95997i
271.10 0.813148 2.50261i 0 −3.98383 2.89442i 1.55748 + 1.60445i 0 3.64251 −6.22536 + 4.52299i 0 5.28178 2.59311i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 136.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.d even 5 1 inner
75.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.h.b 40
3.b odd 2 1 inner 675.2.h.b 40
25.d even 5 1 inner 675.2.h.b 40
75.j odd 10 1 inner 675.2.h.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
675.2.h.b 40 1.a even 1 1 trivial
675.2.h.b 40 3.b odd 2 1 inner
675.2.h.b 40 25.d even 5 1 inner
675.2.h.b 40 75.j odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 18 T_{2}^{38} + 194 T_{2}^{36} + 1663 T_{2}^{34} + 12932 T_{2}^{32} + 80442 T_{2}^{30} + 418854 T_{2}^{28} + 1915409 T_{2}^{26} + 7601379 T_{2}^{24} + 23323386 T_{2}^{22} + 54398827 T_{2}^{20} + \cdots + 15625 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display