Properties

Label 740.2.x.a
Level $740$
Weight $2$
Character orbit 740.x
Analytic conductor $5.909$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(101,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.x (of order \(6\), 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: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{7} - 18 q^{9} + 4 q^{11} - 6 q^{13} - 12 q^{19} + 6 q^{21} + 14 q^{25} + 2 q^{33} + 6 q^{35} + 22 q^{37} - 6 q^{39} - 14 q^{41} + 44 q^{47} - 22 q^{49} - 8 q^{53} + 12 q^{55} - 54 q^{57} + 16 q^{63} + 6 q^{65} - 12 q^{67} + 18 q^{69} - 26 q^{71} - 4 q^{73} + 14 q^{77} - 54 q^{79} - 22 q^{81} - 28 q^{83} - 4 q^{85} - 48 q^{87} + 36 q^{89} + 36 q^{91} - 18 q^{93} + 12 q^{95} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 0 −1.45802 2.52536i 0 −0.866025 + 0.500000i 0 −0.977514 1.69310i 0 −2.75162 + 4.76595i 0
101.2 0 −1.43328 2.48251i 0 0.866025 0.500000i 0 1.51361 + 2.62165i 0 −2.60857 + 4.51817i 0
101.3 0 −1.39306 2.41285i 0 0.866025 0.500000i 0 −1.45145 2.51399i 0 −2.38122 + 4.12440i 0
101.4 0 −0.924212 1.60078i 0 −0.866025 + 0.500000i 0 1.23057 + 2.13141i 0 −0.208337 + 0.360850i 0
101.5 0 −0.579869 1.00436i 0 −0.866025 + 0.500000i 0 −0.997854 1.72833i 0 0.827505 1.43328i 0
101.6 0 −0.376553 0.652209i 0 0.866025 0.500000i 0 0.571101 + 0.989176i 0 1.21642 2.10689i 0
101.7 0 −0.242376 0.419807i 0 −0.866025 + 0.500000i 0 2.17692 + 3.77053i 0 1.38251 2.39457i 0
101.8 0 0.0959198 + 0.166138i 0 0.866025 0.500000i 0 −1.15672 2.00350i 0 1.48160 2.56620i 0
101.9 0 0.559168 + 0.968507i 0 −0.866025 + 0.500000i 0 −1.54298 2.67252i 0 0.874663 1.51496i 0
101.10 0 0.764283 + 1.32378i 0 0.866025 0.500000i 0 2.10769 + 3.65062i 0 0.331743 0.574597i 0
101.11 0 0.926709 + 1.60511i 0 −0.866025 + 0.500000i 0 −1.34484 2.32933i 0 −0.217581 + 0.376861i 0
101.12 0 1.05756 + 1.83175i 0 0.866025 0.500000i 0 −2.39121 4.14170i 0 −0.736869 + 1.27629i 0
101.13 0 1.28513 + 2.22590i 0 0.866025 0.500000i 0 1.17302 + 2.03172i 0 −1.80310 + 3.12305i 0
101.14 0 1.71860 + 2.97669i 0 −0.866025 + 0.500000i 0 0.0896750 + 0.155322i 0 −4.40714 + 7.63339i 0
381.1 0 −1.45802 + 2.52536i 0 −0.866025 0.500000i 0 −0.977514 + 1.69310i 0 −2.75162 4.76595i 0
381.2 0 −1.43328 + 2.48251i 0 0.866025 + 0.500000i 0 1.51361 2.62165i 0 −2.60857 4.51817i 0
381.3 0 −1.39306 + 2.41285i 0 0.866025 + 0.500000i 0 −1.45145 + 2.51399i 0 −2.38122 4.12440i 0
381.4 0 −0.924212 + 1.60078i 0 −0.866025 0.500000i 0 1.23057 2.13141i 0 −0.208337 0.360850i 0
381.5 0 −0.579869 + 1.00436i 0 −0.866025 0.500000i 0 −0.997854 + 1.72833i 0 0.827505 + 1.43328i 0
381.6 0 −0.376553 + 0.652209i 0 0.866025 + 0.500000i 0 0.571101 0.989176i 0 1.21642 + 2.10689i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.x.a 28
37.e even 6 1 inner 740.2.x.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.x.a 28 1.a even 1 1 trivial
740.2.x.a 28 37.e even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(740, [\chi])\).