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Properties

Label 374.2.l.b
Level $374$
Weight $2$
Character orbit 374.l
Analytic conductor $2.986$
Analytic rank $0$
Dimension $40$
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] = [374,2,Mod(135,374)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(374, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("374.135");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 374 = 2 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 374.l (of order \(10\), 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: \(2.98640503560\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 10 q^{2} - 10 q^{4} + 10 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 10 q^{2} - 10 q^{4} + 10 q^{8} - 10 q^{9} - 4 q^{13} + 12 q^{15} - 10 q^{16} + 2 q^{17} - 30 q^{18} + 12 q^{21} - 6 q^{25} - 6 q^{26} - 12 q^{30} - 40 q^{32} + 40 q^{33} - 2 q^{34} - 18 q^{35} + 30 q^{36} + 8 q^{42} - 60 q^{43} + 2 q^{47} + 22 q^{49} - 34 q^{50} - 31 q^{51} + 6 q^{52} + 2 q^{53} - 68 q^{55} + 64 q^{59} + 2 q^{60} - 10 q^{64} + 24 q^{67} + 2 q^{68} + 38 q^{69} - 12 q^{70} + 10 q^{72} + 14 q^{77} - 114 q^{81} + 46 q^{83} + 2 q^{84} + 72 q^{85} - 10 q^{86} + 28 q^{87} + 56 q^{89} - 42 q^{93} + 18 q^{94} + 148 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
135.1 0.809017 − 0.587785i −2.40511 + 0.781467i 0.309017 − 0.951057i 0.0649435 − 0.0893870i −1.48644 + 2.04591i 0.258513 + 0.0839961i −0.309017 − 0.951057i 2.74680 − 1.99567i − 0.110488i
135.2 0.809017 − 0.587785i −1.90420 + 0.618711i 0.309017 − 0.951057i 2.44888 − 3.37060i −1.17686 + 1.61981i −3.63211 − 1.18014i −0.309017 − 0.951057i 0.816109 − 0.592938i − 4.16629i
135.3 0.809017 − 0.587785i −1.80695 + 0.587114i 0.309017 − 0.951057i −1.08511 + 1.49352i −1.11676 + 1.53708i 0.687837 + 0.223492i −0.309017 − 0.951057i 0.493314 − 0.358414i 1.84609i
135.4 0.809017 − 0.587785i −1.15193 + 0.374284i 0.309017 − 0.951057i −0.944411 + 1.29987i −0.711931 + 0.979889i 3.16255 + 1.02758i −0.309017 − 0.951057i −1.24020 + 0.901058i 1.60673i
135.5 0.809017 − 0.587785i −0.206332 + 0.0670415i 0.309017 − 0.951057i −2.12843 + 2.92953i −0.127520 + 0.175517i −4.65011 − 1.51091i −0.309017 − 0.951057i −2.38897 + 1.73569i 3.62109i
135.6 0.809017 − 0.587785i 0.206332 − 0.0670415i 0.309017 − 0.951057i 2.12843 − 2.92953i 0.127520 − 0.175517i 4.65011 + 1.51091i −0.309017 − 0.951057i −2.38897 + 1.73569i − 3.62109i
135.7 0.809017 − 0.587785i 1.15193 − 0.374284i 0.309017 − 0.951057i 0.944411 − 1.29987i 0.711931 − 0.979889i −3.16255 − 1.02758i −0.309017 − 0.951057i −1.24020 + 0.901058i − 1.60673i
135.8 0.809017 − 0.587785i 1.80695 − 0.587114i 0.309017 − 0.951057i 1.08511 − 1.49352i 1.11676 − 1.53708i −0.687837 − 0.223492i −0.309017 − 0.951057i 0.493314 − 0.358414i − 1.84609i
135.9 0.809017 − 0.587785i 1.90420 − 0.618711i 0.309017 − 0.951057i −2.44888 + 3.37060i 1.17686 − 1.61981i 3.63211 + 1.18014i −0.309017 − 0.951057i 0.816109 − 0.592938i 4.16629i
135.10 0.809017 − 0.587785i 2.40511 − 0.781467i 0.309017 − 0.951057i −0.0649435 + 0.0893870i 1.48644 − 2.04591i −0.258513 − 0.0839961i −0.309017 − 0.951057i 2.74680 − 1.99567i 0.110488i
169.1 0.809017 + 0.587785i −2.40511 − 0.781467i 0.309017 + 0.951057i 0.0649435 + 0.0893870i −1.48644 − 2.04591i 0.258513 − 0.0839961i −0.309017 + 0.951057i 2.74680 + 1.99567i 0.110488i
169.2 0.809017 + 0.587785i −1.90420 − 0.618711i 0.309017 + 0.951057i 2.44888 + 3.37060i −1.17686 − 1.61981i −3.63211 + 1.18014i −0.309017 + 0.951057i 0.816109 + 0.592938i 4.16629i
169.3 0.809017 + 0.587785i −1.80695 − 0.587114i 0.309017 + 0.951057i −1.08511 − 1.49352i −1.11676 − 1.53708i 0.687837 − 0.223492i −0.309017 + 0.951057i 0.493314 + 0.358414i − 1.84609i
169.4 0.809017 + 0.587785i −1.15193 − 0.374284i 0.309017 + 0.951057i −0.944411 − 1.29987i −0.711931 − 0.979889i 3.16255 − 1.02758i −0.309017 + 0.951057i −1.24020 − 0.901058i − 1.60673i
169.5 0.809017 + 0.587785i −0.206332 − 0.0670415i 0.309017 + 0.951057i −2.12843 − 2.92953i −0.127520 − 0.175517i −4.65011 + 1.51091i −0.309017 + 0.951057i −2.38897 − 1.73569i − 3.62109i
169.6 0.809017 + 0.587785i 0.206332 + 0.0670415i 0.309017 + 0.951057i 2.12843 + 2.92953i 0.127520 + 0.175517i 4.65011 − 1.51091i −0.309017 + 0.951057i −2.38897 − 1.73569i 3.62109i
169.7 0.809017 + 0.587785i 1.15193 + 0.374284i 0.309017 + 0.951057i 0.944411 + 1.29987i 0.711931 + 0.979889i −3.16255 + 1.02758i −0.309017 + 0.951057i −1.24020 − 0.901058i 1.60673i
169.8 0.809017 + 0.587785i 1.80695 + 0.587114i 0.309017 + 0.951057i 1.08511 + 1.49352i 1.11676 + 1.53708i −0.687837 + 0.223492i −0.309017 + 0.951057i 0.493314 + 0.358414i 1.84609i
169.9 0.809017 + 0.587785i 1.90420 + 0.618711i 0.309017 + 0.951057i −2.44888 − 3.37060i 1.17686 + 1.61981i 3.63211 − 1.18014i −0.309017 + 0.951057i 0.816109 + 0.592938i − 4.16629i
169.10 0.809017 + 0.587785i 2.40511 + 0.781467i 0.309017 + 0.951057i −0.0649435 − 0.0893870i 1.48644 + 2.04591i −0.258513 + 0.0839961i −0.309017 + 0.951057i 2.74680 + 1.99567i − 0.110488i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 135.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
17.b even 2 1 inner
187.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 374.2.l.b ✓ 40
11.c even 5 1 inner 374.2.l.b ✓ 40
17.b even 2 1 inner 374.2.l.b ✓ 40
187.j even 10 1 inner 374.2.l.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
374.2.l.b ✓ 40 1.a even 1 1 trivial
374.2.l.b ✓ 40 11.c even 5 1 inner
374.2.l.b ✓ 40 17.b even 2 1 inner
374.2.l.b ✓ 40 187.j even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 10 T_{3}^{38} + 191 T_{3}^{36} - 2533 T_{3}^{34} + 24811 T_{3}^{32} - 197325 T_{3}^{30} + \cdots + 707281 \) acting on \(S_{2}^{\mathrm{new}}(374, [\chi])\). Copy content Toggle raw display