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Properties

Label 756.2.bo.a
Level $756$
Weight $2$
Character orbit 756.bo
Analytic conductor $6.037$
Analytic rank $0$
Dimension $54$
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] = [756,2,Mod(85,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.85");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 54 q - 3 q^{3} - 3 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q - 3 q^{3} - 3 q^{5} + 3 q^{9} - 9 q^{13} + 3 q^{15} - 3 q^{21} - 45 q^{25} + 15 q^{27} + 6 q^{29} - 9 q^{31} + 6 q^{33} + 6 q^{35} + 30 q^{39} + 9 q^{41} + 9 q^{43} - 21 q^{45} - 27 q^{47} - 108 q^{51} - 84 q^{53} - 57 q^{57} - 66 q^{59} + 3 q^{63} + 30 q^{65} + 63 q^{67} + 42 q^{69} + 42 q^{71} + 3 q^{75} - 9 q^{77} + 36 q^{79} + 3 q^{81} + 12 q^{83} + 18 q^{85} + 39 q^{87} + 12 q^{89} + 21 q^{93} + 120 q^{95} + 18 q^{97} + 39 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1 0 −1.72198 + 0.186471i 0 −0.509284 + 2.88830i 0 0.766044 − 0.642788i 0 2.93046 − 0.642199i 0
85.2 0 −1.57816 − 0.713728i 0 0.103874 − 0.589101i 0 0.766044 − 0.642788i 0 1.98118 + 2.25276i 0
85.3 0 −1.42882 + 0.979012i 0 −0.0286316 + 0.162378i 0 0.766044 − 0.642788i 0 1.08307 − 2.79767i 0
85.4 0 −0.576212 − 1.63340i 0 0.746163 − 4.23170i 0 0.766044 − 0.642788i 0 −2.33596 + 1.88236i 0
85.5 0 −0.550546 − 1.64222i 0 −0.678318 + 3.84693i 0 0.766044 − 0.642788i 0 −2.39380 + 1.80824i 0
85.6 0 −0.195914 + 1.72094i 0 0.348691 − 1.97752i 0 0.766044 − 0.642788i 0 −2.92324 − 0.674312i 0
85.7 0 1.33395 + 1.10480i 0 −0.457942 + 2.59712i 0 0.766044 − 0.642788i 0 0.558836 + 2.94749i 0
85.8 0 1.48997 − 0.883169i 0 −0.192063 + 1.08924i 0 0.766044 − 0.642788i 0 1.44003 − 2.63179i 0
85.9 0 1.61438 − 0.627515i 0 0.586259 − 3.32484i 0 0.766044 − 0.642788i 0 2.21245 − 2.02610i 0
169.1 0 −1.72198 − 0.186471i 0 −0.509284 − 2.88830i 0 0.766044 + 0.642788i 0 2.93046 + 0.642199i 0
169.2 0 −1.57816 + 0.713728i 0 0.103874 + 0.589101i 0 0.766044 + 0.642788i 0 1.98118 − 2.25276i 0
169.3 0 −1.42882 − 0.979012i 0 −0.0286316 − 0.162378i 0 0.766044 + 0.642788i 0 1.08307 + 2.79767i 0
169.4 0 −0.576212 + 1.63340i 0 0.746163 + 4.23170i 0 0.766044 + 0.642788i 0 −2.33596 − 1.88236i 0
169.5 0 −0.550546 + 1.64222i 0 −0.678318 − 3.84693i 0 0.766044 + 0.642788i 0 −2.39380 − 1.80824i 0
169.6 0 −0.195914 − 1.72094i 0 0.348691 + 1.97752i 0 0.766044 + 0.642788i 0 −2.92324 + 0.674312i 0
169.7 0 1.33395 − 1.10480i 0 −0.457942 − 2.59712i 0 0.766044 + 0.642788i 0 0.558836 − 2.94749i 0
169.8 0 1.48997 + 0.883169i 0 −0.192063 − 1.08924i 0 0.766044 + 0.642788i 0 1.44003 + 2.63179i 0
169.9 0 1.61438 + 0.627515i 0 0.586259 + 3.32484i 0 0.766044 + 0.642788i 0 2.21245 + 2.02610i 0
337.1 0 −1.69816 + 0.340939i 0 −2.75833 − 2.31451i 0 −0.939693 − 0.342020i 0 2.76752 − 1.15794i 0
337.2 0 −1.55705 − 0.758672i 0 0.267700 + 0.224627i 0 −0.939693 − 0.342020i 0 1.84883 + 2.36259i 0
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.9
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.bo.a ✓ 54
27.e even 9 1 inner 756.2.bo.a ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.bo.a ✓ 54 1.a even 1 1 trivial
756.2.bo.a ✓ 54 27.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{54} + 3 T_{5}^{53} + 27 T_{5}^{52} + 42 T_{5}^{51} + 216 T_{5}^{50} - 846 T_{5}^{49} + \cdots + 347818037121 \) acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\). Copy content Toggle raw display