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Properties

Label 760.2.bx.a
Level $760$
Weight $2$
Character orbit 760.bx
Analytic conductor $6.069$
Analytic rank $0$
Dimension $24$
CM discriminant -40
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [760,2,Mod(59,760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(760, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 9, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("760.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.bx (of order \(18\), degree \(6\), 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: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 48 q^{14} + 48 q^{26} + 60 q^{35} - 12 q^{41} + 24 q^{44} - 84 q^{49} + 96 q^{64} - 180 q^{65} - 96 q^{74} + 72 q^{76} + 168 q^{89} + 96 q^{91} + 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −1.39273 − 0.245576i 0 1.87939 + 0.684040i −2.10122 + 0.764780i 0 −1.61753 − 2.80164i −2.44949 − 1.41421i −0.520945 − 2.95442i 3.11424 − 0.549124i
59.2 −1.39273 − 0.245576i 0 1.87939 + 0.684040i 2.10122 − 0.764780i 0 2.58491 + 4.47719i −2.44949 − 1.41421i −0.520945 − 2.95442i −3.11424 + 0.549124i
59.3 1.39273 + 0.245576i 0 1.87939 + 0.684040i −2.10122 + 0.764780i 0 −2.58491 − 4.47719i 2.44949 + 1.41421i −0.520945 − 2.95442i −3.11424 + 0.549124i
59.4 1.39273 + 0.245576i 0 1.87939 + 0.684040i 2.10122 − 0.764780i 0 1.61753 + 2.80164i 2.44949 + 1.41421i −0.520945 − 2.95442i 3.11424 − 0.549124i
219.1 −1.39273 + 0.245576i 0 1.87939 − 0.684040i −2.10122 − 0.764780i 0 −1.61753 + 2.80164i −2.44949 + 1.41421i −0.520945 + 2.95442i 3.11424 + 0.549124i
219.2 −1.39273 + 0.245576i 0 1.87939 − 0.684040i 2.10122 + 0.764780i 0 2.58491 − 4.47719i −2.44949 + 1.41421i −0.520945 + 2.95442i −3.11424 − 0.549124i
219.3 1.39273 − 0.245576i 0 1.87939 − 0.684040i −2.10122 − 0.764780i 0 −2.58491 + 4.47719i 2.44949 − 1.41421i −0.520945 + 2.95442i −3.11424 − 0.549124i
219.4 1.39273 − 0.245576i 0 1.87939 − 0.684040i 2.10122 + 0.764780i 0 1.61753 − 2.80164i 2.44949 − 1.41421i −0.520945 + 2.95442i 3.11424 + 0.549124i
299.1 −0.483690 − 1.32893i 0 −1.53209 + 1.28558i −1.71293 − 1.43732i 0 −2.62197 − 4.54138i 2.44949 + 1.41421i 2.81908 + 1.02606i −1.08156 + 2.97157i
299.2 −0.483690 − 1.32893i 0 −1.53209 + 1.28558i 1.71293 + 1.43732i 0 0.803888 + 1.39238i 2.44949 + 1.41421i 2.81908 + 1.02606i 1.08156 − 2.97157i
299.3 0.483690 + 1.32893i 0 −1.53209 + 1.28558i −1.71293 − 1.43732i 0 −0.803888 − 1.39238i −2.44949 − 1.41421i 2.81908 + 1.02606i 1.08156 − 2.97157i
299.4 0.483690 + 1.32893i 0 −1.53209 + 1.28558i 1.71293 + 1.43732i 0 2.62197 + 4.54138i −2.44949 − 1.41421i 2.81908 + 1.02606i −1.08156 + 2.97157i
459.1 −0.909039 + 1.08335i 0 −0.347296 − 1.96962i −0.388289 + 2.20210i 0 1.00444 + 1.73974i 2.44949 + 1.41421i −2.29813 + 1.92836i −2.03267 − 2.42245i
459.2 −0.909039 + 1.08335i 0 −0.347296 − 1.96962i 0.388289 − 2.20210i 0 1.78102 + 3.08481i 2.44949 + 1.41421i −2.29813 + 1.92836i 2.03267 + 2.42245i
459.3 0.909039 − 1.08335i 0 −0.347296 − 1.96962i −0.388289 + 2.20210i 0 −1.78102 − 3.08481i −2.44949 − 1.41421i −2.29813 + 1.92836i 2.03267 + 2.42245i
459.4 0.909039 − 1.08335i 0 −0.347296 − 1.96962i 0.388289 − 2.20210i 0 −1.00444 − 1.73974i −2.44949 − 1.41421i −2.29813 + 1.92836i −2.03267 − 2.42245i
659.1 −0.909039 − 1.08335i 0 −0.347296 + 1.96962i −0.388289 − 2.20210i 0 1.00444 − 1.73974i 2.44949 − 1.41421i −2.29813 − 1.92836i −2.03267 + 2.42245i
659.2 −0.909039 − 1.08335i 0 −0.347296 + 1.96962i 0.388289 + 2.20210i 0 1.78102 − 3.08481i 2.44949 − 1.41421i −2.29813 − 1.92836i 2.03267 − 2.42245i
659.3 0.909039 + 1.08335i 0 −0.347296 + 1.96962i −0.388289 − 2.20210i 0 −1.78102 + 3.08481i −2.44949 + 1.41421i −2.29813 − 1.92836i 2.03267 − 2.42245i
659.4 0.909039 + 1.08335i 0 −0.347296 + 1.96962i 0.388289 + 2.20210i 0 −1.00444 + 1.73974i −2.44949 + 1.41421i −2.29813 − 1.92836i −2.03267 + 2.42245i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
5.b even 2 1 inner
8.d odd 2 1 inner
19.f odd 18 1 inner
95.o odd 18 1 inner
152.v even 18 1 inner
760.bx even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.bx.a ✓ 24
5.b even 2 1 inner 760.2.bx.a ✓ 24
8.d odd 2 1 inner 760.2.bx.a ✓ 24
19.f odd 18 1 inner 760.2.bx.a ✓ 24
40.e odd 2 1 CM 760.2.bx.a ✓ 24
95.o odd 18 1 inner 760.2.bx.a ✓ 24
152.v even 18 1 inner 760.2.bx.a ✓ 24
760.bx even 18 1 inner 760.2.bx.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.bx.a ✓ 24 1.a even 1 1 trivial
760.2.bx.a ✓ 24 5.b even 2 1 inner
760.2.bx.a ✓ 24 8.d odd 2 1 inner
760.2.bx.a ✓ 24 19.f odd 18 1 inner
760.2.bx.a ✓ 24 40.e odd 2 1 CM
760.2.bx.a ✓ 24 95.o odd 18 1 inner
760.2.bx.a ✓ 24 152.v even 18 1 inner
760.2.bx.a ✓ 24 760.bx even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display