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Properties

Label 238.2.p.b
Level $238$
Weight $2$
Character orbit 238.p
Analytic conductor $1.900$
Analytic rank $0$
Dimension $48$
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] = [238,2,Mod(27,238)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(238, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("238.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 238 = 2 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 238.p (of order \(16\), degree \(8\), 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: \(1.90043956811\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 16 q^{11} - 8 q^{14} + 64 q^{15} - 80 q^{18} - 48 q^{21} + 16 q^{22} - 16 q^{25} + 8 q^{28} + 32 q^{29} + 16 q^{39} - 24 q^{42} - 16 q^{43} + 16 q^{44} - 16 q^{46} + 16 q^{49} + 32 q^{51} - 48 q^{53} + 64 q^{57} + 16 q^{58} + 32 q^{60} - 104 q^{63} - 48 q^{65} + 16 q^{70} - 112 q^{71} - 16 q^{72} - 40 q^{77} - 64 q^{78} + 32 q^{79} + 112 q^{81} - 176 q^{85} + 32 q^{86} + 16 q^{88} + 80 q^{91} - 32 q^{93} - 32 q^{95} - 96 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 0.382683 − 0.923880i −1.67987 + 2.51410i −0.707107 − 0.707107i −0.693461 − 3.48626i 1.67987 + 2.51410i −1.54068 + 2.15088i −0.923880 + 0.382683i −2.35069 − 5.67506i −3.48626 − 0.693461i
27.2 0.382683 − 0.923880i −0.750734 + 1.12355i −0.707107 − 0.707107i 0.521884 + 2.62369i 0.750734 + 1.12355i −2.64166 − 0.147081i −0.923880 + 0.382683i 0.449281 + 1.08466i 2.62369 + 0.521884i
27.3 0.382683 − 0.923880i −0.706508 + 1.05736i −0.707107 − 0.707107i 0.195254 + 0.981609i 0.706508 + 1.05736i 2.03267 + 1.69359i −0.923880 + 0.382683i 0.529186 + 1.27757i 0.981609 + 0.195254i
27.4 0.382683 − 0.923880i 0.706508 − 1.05736i −0.707107 − 0.707107i −0.195254 − 0.981609i −0.706508 − 1.05736i 2.52605 − 0.786807i −0.923880 + 0.382683i 0.529186 + 1.27757i −0.981609 − 0.195254i
27.5 0.382683 − 0.923880i 0.750734 − 1.12355i −0.707107 − 0.707107i −0.521884 − 2.62369i −0.750734 − 1.12355i −2.49686 − 0.875034i −0.923880 + 0.382683i 0.449281 + 1.08466i −2.62369 − 0.521884i
27.6 0.382683 − 0.923880i 1.67987 − 2.51410i −0.707107 − 0.707107i 0.693461 + 3.48626i −1.67987 − 2.51410i −0.600296 − 2.57675i −0.923880 + 0.382683i −2.35069 − 5.67506i 3.48626 + 0.693461i
41.1 −0.382683 + 0.923880i −2.36903 − 1.58294i −0.707107 − 0.707107i −2.73760 + 0.544542i 2.36903 − 1.58294i 2.43407 + 1.03696i 0.923880 − 0.382683i 1.95857 + 4.72841i 0.544542 − 2.73760i
41.2 −0.382683 + 0.923880i −1.74276 − 1.16448i −0.707107 − 0.707107i 1.25354 − 0.249344i 1.74276 − 1.16448i −2.18163 + 1.49683i 0.923880 − 0.382683i 0.533163 + 1.28717i −0.249344 + 1.25354i
41.3 −0.382683 + 0.923880i −0.227054 − 0.151713i −0.707107 − 0.707107i −2.10612 + 0.418933i 0.227054 − 0.151713i −1.07719 − 2.41654i 0.923880 − 0.382683i −1.11951 − 2.70274i 0.418933 − 2.10612i
41.4 −0.382683 + 0.923880i 0.227054 + 0.151713i −0.707107 − 0.707107i 2.10612 − 0.418933i −0.227054 + 0.151713i 1.91997 − 1.82037i 0.923880 − 0.382683i −1.11951 − 2.70274i −0.418933 + 2.10612i
41.5 −0.382683 + 0.923880i 1.74276 + 1.16448i −0.707107 − 0.707107i −1.25354 + 0.249344i −1.74276 + 1.16448i 1.44275 + 2.21776i 0.923880 − 0.382683i 0.533163 + 1.28717i 0.249344 − 1.25354i
41.6 −0.382683 + 0.923880i 2.36903 + 1.58294i −0.707107 − 0.707107i 2.73760 − 0.544542i −2.36903 + 1.58294i −2.64562 + 0.0265465i 0.923880 − 0.382683i 1.95857 + 4.72841i −0.544542 + 2.73760i
97.1 0.382683 + 0.923880i −1.67987 − 2.51410i −0.707107 + 0.707107i −0.693461 + 3.48626i 1.67987 − 2.51410i −1.54068 − 2.15088i −0.923880 − 0.382683i −2.35069 + 5.67506i −3.48626 + 0.693461i
97.2 0.382683 + 0.923880i −0.750734 − 1.12355i −0.707107 + 0.707107i 0.521884 − 2.62369i 0.750734 − 1.12355i −2.64166 + 0.147081i −0.923880 − 0.382683i 0.449281 − 1.08466i 2.62369 − 0.521884i
97.3 0.382683 + 0.923880i −0.706508 − 1.05736i −0.707107 + 0.707107i 0.195254 − 0.981609i 0.706508 − 1.05736i 2.03267 − 1.69359i −0.923880 − 0.382683i 0.529186 − 1.27757i 0.981609 − 0.195254i
97.4 0.382683 + 0.923880i 0.706508 + 1.05736i −0.707107 + 0.707107i −0.195254 + 0.981609i −0.706508 + 1.05736i 2.52605 + 0.786807i −0.923880 − 0.382683i 0.529186 − 1.27757i −0.981609 + 0.195254i
97.5 0.382683 + 0.923880i 0.750734 + 1.12355i −0.707107 + 0.707107i −0.521884 + 2.62369i −0.750734 + 1.12355i −2.49686 + 0.875034i −0.923880 − 0.382683i 0.449281 − 1.08466i −2.62369 + 0.521884i
97.6 0.382683 + 0.923880i 1.67987 + 2.51410i −0.707107 + 0.707107i 0.693461 − 3.48626i −1.67987 + 2.51410i −0.600296 + 2.57675i −0.923880 − 0.382683i −2.35069 + 5.67506i 3.48626 − 0.693461i
125.1 0.923880 + 0.382683i −0.611623 − 3.07484i 0.707107 + 0.707107i 2.98323 + 1.99333i 0.611623 − 3.07484i 1.97624 + 1.75911i 0.382683 + 0.923880i −6.30889 + 2.61323i 1.99333 + 2.98323i
125.2 0.923880 + 0.382683i −0.459580 − 2.31047i 0.707107 + 0.707107i −1.50717 − 1.00706i 0.459580 − 2.31047i −1.50331 − 2.17717i 0.382683 + 0.923880i −2.35540 + 0.975640i −1.00706 − 1.50717i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
17.e odd 16 1 inner
119.p even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 238.2.p.b ✓ 48
7.b odd 2 1 inner 238.2.p.b ✓ 48
17.e odd 16 1 inner 238.2.p.b ✓ 48
119.p even 16 1 inner 238.2.p.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
238.2.p.b ✓ 48 1.a even 1 1 trivial
238.2.p.b ✓ 48 7.b odd 2 1 inner
238.2.p.b ✓ 48 17.e odd 16 1 inner
238.2.p.b ✓ 48 119.p even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 64 T_{3}^{44} - 168 T_{3}^{42} + 2048 T_{3}^{40} + 30992 T_{3}^{38} + 407400 T_{3}^{36} + \cdots + 5345344 \) acting on \(S_{2}^{\mathrm{new}}(238, [\chi])\). Copy content Toggle raw display