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Properties

Label 760.2.bo.a
Level $760$
Weight $2$
Character orbit 760.bo
Analytic conductor $6.069$
Analytic rank $0$
Dimension $24$
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: \(24\)
Relative dimension: \(4\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{7} + 12 q^{11} - 9 q^{13} + 6 q^{17} + 6 q^{19} + 3 q^{21} + 3 q^{23} + 21 q^{27} + 15 q^{29} + 12 q^{31} - 12 q^{33} - 6 q^{35} - 12 q^{37} - 36 q^{39} + 9 q^{41} + 18 q^{43} - 3 q^{45} + 3 q^{47} + 18 q^{49} - 48 q^{51} - 24 q^{53} - 6 q^{57} - 3 q^{59} + 30 q^{61} - 12 q^{63} - 15 q^{65} + 15 q^{67} - 9 q^{69} - 54 q^{71} + 24 q^{73} - 6 q^{75} - 45 q^{79} + 6 q^{81} + 6 q^{85} + 15 q^{87} - 54 q^{89} + 12 q^{91} - 24 q^{93} + 3 q^{95} - 30 q^{97} + 66 q^{99}+O(q^{100}) \) 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 −0.427728 + 2.42577i 0 0.766044 + 0.642788i 0 −0.704590 − 1.22039i 0 −2.88232 − 1.04908i 0
81.2 0 −0.147298 + 0.835369i 0 0.766044 + 0.642788i 0 2.24662 + 3.89126i 0 2.14293 + 0.779964i 0
81.3 0 0.0677501 − 0.384230i 0 0.766044 + 0.642788i 0 −0.807489 − 1.39861i 0 2.67604 + 0.973997i 0
81.4 0 0.333628 − 1.89210i 0 0.766044 + 0.642788i 0 0.439110 + 0.760561i 0 −0.649656 − 0.236456i 0
161.1 0 −2.36047 + 1.98067i 0 −0.939693 − 0.342020i 0 1.24303 − 2.15298i 0 1.12782 − 6.39620i 0
161.2 0 −1.29467 + 1.08636i 0 −0.939693 − 0.342020i 0 0.409579 − 0.709412i 0 −0.0249433 + 0.141460i 0
161.3 0 1.11964 − 0.939486i 0 −0.939693 − 0.342020i 0 −0.724975 + 1.25569i 0 −0.149994 + 0.850657i 0
161.4 0 1.76946 − 1.48476i 0 −0.939693 − 0.342020i 0 0.838414 − 1.45218i 0 0.405555 − 2.30002i 0
321.1 0 −2.36047 − 1.98067i 0 −0.939693 + 0.342020i 0 1.24303 + 2.15298i 0 1.12782 + 6.39620i 0
321.2 0 −1.29467 − 1.08636i 0 −0.939693 + 0.342020i 0 0.409579 + 0.709412i 0 −0.0249433 − 0.141460i 0
321.3 0 1.11964 + 0.939486i 0 −0.939693 + 0.342020i 0 −0.724975 − 1.25569i 0 −0.149994 − 0.850657i 0
321.4 0 1.76946 + 1.48476i 0 −0.939693 + 0.342020i 0 0.838414 + 1.45218i 0 0.405555 + 2.30002i 0
441.1 0 −0.427728 − 2.42577i 0 0.766044 − 0.642788i 0 −0.704590 + 1.22039i 0 −2.88232 + 1.04908i 0
441.2 0 −0.147298 − 0.835369i 0 0.766044 − 0.642788i 0 2.24662 − 3.89126i 0 2.14293 − 0.779964i 0
441.3 0 0.0677501 + 0.384230i 0 0.766044 − 0.642788i 0 −0.807489 + 1.39861i 0 2.67604 − 0.973997i 0
441.4 0 0.333628 + 1.89210i 0 0.766044 − 0.642788i 0 0.439110 − 0.760561i 0 −0.649656 + 0.236456i 0
481.1 0 −1.08573 − 0.395173i 0 0.173648 + 0.984808i 0 −0.484924 + 0.839913i 0 −1.27549 − 1.07026i 0
481.2 0 −0.895948 − 0.326098i 0 0.173648 + 0.984808i 0 1.61342 − 2.79453i 0 −1.60175 − 1.34403i 0
481.3 0 0.655038 + 0.238414i 0 0.173648 + 0.984808i 0 −1.90545 + 3.30033i 0 −1.92590 − 1.61602i 0
481.4 0 2.26633 + 0.824878i 0 0.173648 + 0.984808i 0 0.837257 − 1.45017i 0 2.15771 + 1.81053i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.4
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.a ✓ 24
19.e even 9 1 inner 760.2.bo.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.bo.a ✓ 24 1.a even 1 1 trivial
760.2.bo.a ✓ 24 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 13 T_{3}^{21} + 39 T_{3}^{20} - 57 T_{3}^{19} + 337 T_{3}^{18} - 411 T_{3}^{17} + \cdots + 2601 \) acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display