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Properties

Label 760.2.bo.c
Level $760$
Weight $2$
Character orbit 760.bo
Analytic conductor $6.069$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [760,2,Mod(81,760)]
 
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([0, 0, 0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("760.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.bo (of order \(9\), 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: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 12 q^{9} - 6 q^{11} + 3 q^{13} - 24 q^{17} - 6 q^{19} + 27 q^{21} - 6 q^{23} + 21 q^{27} - 9 q^{29} + 48 q^{33} + 3 q^{35} - 60 q^{37} + 48 q^{39} + 42 q^{41} + 30 q^{43} + 18 q^{45} + 15 q^{47} - 15 q^{49} - 18 q^{51} - 12 q^{53} - 6 q^{55} + 48 q^{57} + 3 q^{59} - 60 q^{61} - 27 q^{63} - 6 q^{65} + 9 q^{67} + 39 q^{69} - 18 q^{71} - 54 q^{73} - 6 q^{75} - 18 q^{77} - 51 q^{79} - 6 q^{81} - 6 q^{83} + 24 q^{85} - 33 q^{87} + 36 q^{89} - 120 q^{91} + 30 q^{93} - 15 q^{95} + 114 q^{97} - 51 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 0 −0.512909 + 2.90885i 0 −0.766044 − 0.642788i 0 −0.446702 − 0.773710i 0 −5.37927 − 1.95789i 0
81.2 0 −0.222422 + 1.26142i 0 −0.766044 − 0.642788i 0 0.588103 + 1.01862i 0 1.27737 + 0.464925i 0
81.3 0 −0.0716428 + 0.406306i 0 −0.766044 − 0.642788i 0 −2.13453 − 3.69712i 0 2.65913 + 0.967843i 0
81.4 0 0.287892 − 1.63272i 0 −0.766044 − 0.642788i 0 1.60956 + 2.78783i 0 0.236201 + 0.0859700i 0
81.5 0 0.345434 − 1.95906i 0 −0.766044 − 0.642788i 0 1.32327 + 2.29196i 0 −0.899495 − 0.327389i 0
161.1 0 −1.98505 + 1.66566i 0 0.939693 + 0.342020i 0 −2.57170 + 4.45432i 0 0.645079 − 3.65842i 0
161.2 0 −1.64417 + 1.37962i 0 0.939693 + 0.342020i 0 1.08122 − 1.87273i 0 0.278990 − 1.58223i 0
161.3 0 −0.242746 + 0.203688i 0 0.939693 + 0.342020i 0 0.506149 − 0.876675i 0 −0.503508 + 2.85553i 0
161.4 0 0.776793 − 0.651807i 0 0.939693 + 0.342020i 0 −0.412065 + 0.713718i 0 −0.342389 + 1.94179i 0
161.5 0 2.32913 − 1.95437i 0 0.939693 + 0.342020i 0 1.22275 − 2.11787i 0 1.08433 − 6.14955i 0
321.1 0 −1.98505 − 1.66566i 0 0.939693 − 0.342020i 0 −2.57170 − 4.45432i 0 0.645079 + 3.65842i 0
321.2 0 −1.64417 − 1.37962i 0 0.939693 − 0.342020i 0 1.08122 + 1.87273i 0 0.278990 + 1.58223i 0
321.3 0 −0.242746 − 0.203688i 0 0.939693 − 0.342020i 0 0.506149 + 0.876675i 0 −0.503508 − 2.85553i 0
321.4 0 0.776793 + 0.651807i 0 0.939693 − 0.342020i 0 −0.412065 − 0.713718i 0 −0.342389 − 1.94179i 0
321.5 0 2.32913 + 1.95437i 0 0.939693 − 0.342020i 0 1.22275 + 2.11787i 0 1.08433 + 6.14955i 0
441.1 0 −0.512909 − 2.90885i 0 −0.766044 + 0.642788i 0 −0.446702 + 0.773710i 0 −5.37927 + 1.95789i 0
441.2 0 −0.222422 − 1.26142i 0 −0.766044 + 0.642788i 0 0.588103 − 1.01862i 0 1.27737 − 0.464925i 0
441.3 0 −0.0716428 − 0.406306i 0 −0.766044 + 0.642788i 0 −2.13453 + 3.69712i 0 2.65913 − 0.967843i 0
441.4 0 0.287892 + 1.63272i 0 −0.766044 + 0.642788i 0 1.60956 − 2.78783i 0 0.236201 − 0.0859700i 0
441.5 0 0.345434 + 1.95906i 0 −0.766044 + 0.642788i 0 1.32327 − 2.29196i 0 −0.899495 + 0.327389i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.bo.c ✓ 30
19.e even 9 1 inner 760.2.bo.c ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.bo.c ✓ 30 1.a even 1 1 trivial
760.2.bo.c ✓ 30 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{30} - 6 T_{3}^{28} - 13 T_{3}^{27} + 33 T_{3}^{26} + 195 T_{3}^{25} + 619 T_{3}^{24} + \cdots + 87616 \) acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display