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Properties

Label 760.2.bo.d
Level $760$
Weight $2$
Character orbit 760.bo
Analytic conductor $6.069$
Analytic rank $0$
Dimension $36$
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: \(36\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 3 q^{3} - 6 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 3 q^{3} - 6 q^{7} + 9 q^{9} - 12 q^{11} + 15 q^{13} - 3 q^{15} - 12 q^{17} + 12 q^{19} + 3 q^{21} - 18 q^{23} + 6 q^{27} - 3 q^{29} + 30 q^{31} + 21 q^{33} - 9 q^{35} + 48 q^{37} + 36 q^{39} + 9 q^{41} + 6 q^{43} - 27 q^{45} + 9 q^{47} - 42 q^{49} + 87 q^{51} + 12 q^{53} + 3 q^{55} - 66 q^{57} + 12 q^{59} - 6 q^{61} - 51 q^{63} - 12 q^{65} - 36 q^{67} - 3 q^{69} + 12 q^{71} - 54 q^{73} - 6 q^{75} + 138 q^{77} + 15 q^{79} - 63 q^{81} - 24 q^{83} - 12 q^{85} - 33 q^{87} + 24 q^{89} - 108 q^{93} + 15 q^{95} + 27 q^{97} - 114 q^{99}+O(q^{100}) \) 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 −0.528887 + 2.99947i 0 0.766044 + 0.642788i 0 −0.707937 − 1.22618i 0 −5.89799 − 2.14669i 0
81.2 0 −0.392555 + 2.22629i 0 0.766044 + 0.642788i 0 1.81590 + 3.14523i 0 −1.98320 − 0.721825i 0
81.3 0 −0.191183 + 1.08425i 0 0.766044 + 0.642788i 0 −0.458826 − 0.794709i 0 1.68003 + 0.611480i 0
81.4 0 −0.0896758 + 0.508577i 0 0.766044 + 0.642788i 0 −2.26115 − 3.91643i 0 2.56847 + 0.934846i 0
81.5 0 0.246316 − 1.39693i 0 0.766044 + 0.642788i 0 1.47970 + 2.56291i 0 0.928346 + 0.337890i 0
81.6 0 0.455985 − 2.58602i 0 0.766044 + 0.642788i 0 −1.46008 − 2.52892i 0 −3.66049 − 1.33231i 0
161.1 0 −2.51614 + 2.11130i 0 −0.939693 − 0.342020i 0 −1.25203 + 2.16857i 0 1.35247 − 7.67023i 0
161.2 0 −1.38947 + 1.16590i 0 −0.939693 − 0.342020i 0 1.02465 − 1.77475i 0 0.0503479 − 0.285537i 0
161.3 0 −0.243215 + 0.204082i 0 −0.939693 − 0.342020i 0 1.79028 − 3.10085i 0 −0.503440 + 2.85515i 0
161.4 0 0.0937682 − 0.0786809i 0 −0.939693 − 0.342020i 0 −0.425202 + 0.736472i 0 −0.518343 + 2.93967i 0
161.5 0 1.27742 − 1.07188i 0 −0.939693 − 0.342020i 0 −2.16626 + 3.75207i 0 −0.0380746 + 0.215932i 0
161.6 0 2.27764 − 1.91117i 0 −0.939693 − 0.342020i 0 1.73429 − 3.00387i 0 1.01414 − 5.75146i 0
321.1 0 −2.51614 − 2.11130i 0 −0.939693 + 0.342020i 0 −1.25203 − 2.16857i 0 1.35247 + 7.67023i 0
321.2 0 −1.38947 − 1.16590i 0 −0.939693 + 0.342020i 0 1.02465 + 1.77475i 0 0.0503479 + 0.285537i 0
321.3 0 −0.243215 − 0.204082i 0 −0.939693 + 0.342020i 0 1.79028 + 3.10085i 0 −0.503440 − 2.85515i 0
321.4 0 0.0937682 + 0.0786809i 0 −0.939693 + 0.342020i 0 −0.425202 − 0.736472i 0 −0.518343 − 2.93967i 0
321.5 0 1.27742 + 1.07188i 0 −0.939693 + 0.342020i 0 −2.16626 − 3.75207i 0 −0.0380746 − 0.215932i 0
321.6 0 2.27764 + 1.91117i 0 −0.939693 + 0.342020i 0 1.73429 + 3.00387i 0 1.01414 + 5.75146i 0
441.1 0 −0.528887 − 2.99947i 0 0.766044 − 0.642788i 0 −0.707937 + 1.22618i 0 −5.89799 + 2.14669i 0
441.2 0 −0.392555 − 2.22629i 0 0.766044 − 0.642788i 0 1.81590 − 3.14523i 0 −1.98320 + 0.721825i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.6
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.d ✓ 36
19.e even 9 1 inner 760.2.bo.d ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.bo.d ✓ 36 1.a even 1 1 trivial
760.2.bo.d ✓ 36 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} + 3 T_{3}^{35} - 17 T_{3}^{33} - 15 T_{3}^{32} + 51 T_{3}^{31} + 983 T_{3}^{30} + 1767 T_{3}^{29} + \cdots + 576 \) acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display