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Properties

Label 756.2.bo.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 54 q + 3 q^{3} - 3 q^{5} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q + 3 q^{3} - 3 q^{5} - 9 q^{9} + 9 q^{13} + 3 q^{15} + 3 q^{21} + 12 q^{23} + 27 q^{25} + 39 q^{27} + 30 q^{29} - 9 q^{31} - 6 q^{33} + 6 q^{35} - 18 q^{39} - 27 q^{41} + 9 q^{43} - 81 q^{45} + 9 q^{47} + 12 q^{53} - 21 q^{57} - 24 q^{59} - 3 q^{63} - 78 q^{65} - 9 q^{67} + 6 q^{69} - 30 q^{71} + 57 q^{75} + 9 q^{77} + 99 q^{81} + 78 q^{83} + 18 q^{85} + 21 q^{87} + 12 q^{89} - 51 q^{93} - 36 q^{95} + 36 q^{97} + 15 q^{99}+O(q^{100}) \) 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 −1.72800 + 0.118316i 0 0.626647 − 3.55389i 0 −0.766044 + 0.642788i 0 2.97200 − 0.408902i 0
85.2 0 −1.46947 + 0.916867i 0 −0.137580 + 0.780255i 0 −0.766044 + 0.642788i 0 1.31871 − 2.69462i 0
85.3 0 −1.24763 − 1.20143i 0 0.123908 − 0.702715i 0 −0.766044 + 0.642788i 0 0.113154 + 2.99787i 0
85.4 0 −0.377044 + 1.69051i 0 0.491885 − 2.78962i 0 −0.766044 + 0.642788i 0 −2.71568 − 1.27480i 0
85.5 0 −0.368051 − 1.69249i 0 −0.269336 + 1.52748i 0 −0.766044 + 0.642788i 0 −2.72908 + 1.24585i 0
85.6 0 0.425197 + 1.67905i 0 −0.440759 + 2.49967i 0 −0.766044 + 0.642788i 0 −2.63841 + 1.42785i 0
85.7 0 0.794215 − 1.53923i 0 0.457658 − 2.59551i 0 −0.766044 + 0.642788i 0 −1.73845 − 2.44496i 0
85.8 0 1.64827 + 0.532173i 0 0.150583 − 0.853998i 0 −0.766044 + 0.642788i 0 2.43358 + 1.75433i 0
85.9 0 1.70918 − 0.280534i 0 −0.389665 + 2.20990i 0 −0.766044 + 0.642788i 0 2.84260 − 0.958966i 0
169.1 0 −1.72800 − 0.118316i 0 0.626647 + 3.55389i 0 −0.766044 − 0.642788i 0 2.97200 + 0.408902i 0
169.2 0 −1.46947 − 0.916867i 0 −0.137580 − 0.780255i 0 −0.766044 − 0.642788i 0 1.31871 + 2.69462i 0
169.3 0 −1.24763 + 1.20143i 0 0.123908 + 0.702715i 0 −0.766044 − 0.642788i 0 0.113154 − 2.99787i 0
169.4 0 −0.377044 − 1.69051i 0 0.491885 + 2.78962i 0 −0.766044 − 0.642788i 0 −2.71568 + 1.27480i 0
169.5 0 −0.368051 + 1.69249i 0 −0.269336 − 1.52748i 0 −0.766044 − 0.642788i 0 −2.72908 − 1.24585i 0
169.6 0 0.425197 − 1.67905i 0 −0.440759 − 2.49967i 0 −0.766044 − 0.642788i 0 −2.63841 − 1.42785i 0
169.7 0 0.794215 + 1.53923i 0 0.457658 + 2.59551i 0 −0.766044 − 0.642788i 0 −1.73845 + 2.44496i 0
169.8 0 1.64827 − 0.532173i 0 0.150583 + 0.853998i 0 −0.766044 − 0.642788i 0 2.43358 − 1.75433i 0
169.9 0 1.70918 + 0.280534i 0 −0.389665 − 2.20990i 0 −0.766044 − 0.642788i 0 2.84260 + 0.958966i 0
337.1 0 −1.70391 − 0.310978i 0 −2.38170 − 1.99849i 0 0.939693 + 0.342020i 0 2.80659 + 1.05975i 0
337.2 0 −1.09514 + 1.34189i 0 0.684464 + 0.574334i 0 0.939693 + 0.342020i 0 −0.601340 − 2.93911i 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.b ✓ 54
27.e even 9 1 inner 756.2.bo.b ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.bo.b ✓ 54 1.a even 1 1 trivial
756.2.bo.b ✓ 54 27.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{54} + 3 T_{5}^{53} - 9 T_{5}^{52} - 2 T_{5}^{51} + 300 T_{5}^{50} + 816 T_{5}^{49} + \cdots + 2804931592849 \) acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\). Copy content Toggle raw display