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Properties

Label 756.2.bq.a
Level $756$
Weight $2$
Character orbit 756.bq
Analytic conductor $6.037$
Analytic rank $0$
Dimension $144$
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(25,756)]
 
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, 10, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.bq (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: \(144\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 6 q^{9} + 6 q^{11} - 12 q^{15} + 48 q^{17} + 33 q^{21} - 21 q^{23} + 6 q^{29} + 18 q^{33} - 9 q^{35} + 9 q^{39} - 12 q^{41} - 12 q^{45} + 18 q^{47} - 18 q^{49} - 9 q^{51} + 15 q^{53} + 3 q^{57} - 15 q^{59} - 36 q^{61} + 3 q^{63} + 36 q^{65} + 30 q^{69} - 12 q^{71} + 18 q^{73} - 51 q^{75} - 3 q^{77} + 18 q^{79} - 6 q^{81} - 36 q^{85} + 33 q^{87} + 144 q^{89} + 9 q^{91} + 48 q^{93} - 30 q^{95} - 72 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 −1.73181 − 0.0288799i 0 2.86672 + 1.04340i 0 2.28241 + 1.33814i 0 2.99833 + 0.100029i 0
25.2 0 −1.67230 − 0.451029i 0 −0.947523 − 0.344870i 0 −1.17643 + 2.36981i 0 2.59315 + 1.50851i 0
25.3 0 −1.63641 − 0.567600i 0 −2.66918 − 0.971502i 0 −0.492567 − 2.59950i 0 2.35566 + 1.85765i 0
25.4 0 −1.47274 + 0.911614i 0 −3.21812 − 1.17130i 0 −2.64556 + 0.0321766i 0 1.33792 − 2.68514i 0
25.5 0 −1.38976 + 1.03372i 0 3.13641 + 1.14156i 0 −2.54733 − 0.714936i 0 0.862856 − 2.87324i 0
25.6 0 −1.36056 − 1.07186i 0 2.22913 + 0.811338i 0 −2.26967 − 1.35962i 0 0.702252 + 2.91665i 0
25.7 0 −1.26745 + 1.18050i 0 −0.0700823 − 0.0255079i 0 1.65572 − 2.06363i 0 0.212841 − 2.99244i 0
25.8 0 −1.22028 + 1.22920i 0 −0.946810 − 0.344611i 0 0.0913265 + 2.64417i 0 −0.0218551 − 2.99992i 0
25.9 0 −1.05320 − 1.37506i 0 −1.13419 − 0.412813i 0 2.58962 − 0.542074i 0 −0.781555 + 2.89641i 0
25.10 0 −0.277537 − 1.70967i 0 −0.774544 − 0.281911i 0 2.22920 + 1.42502i 0 −2.84595 + 0.948993i 0
25.11 0 −0.247313 − 1.71430i 0 −0.839073 − 0.305398i 0 −1.83758 + 1.90350i 0 −2.87767 + 0.847940i 0
25.12 0 −0.0456125 + 1.73145i 0 2.25335 + 0.820154i 0 −0.197469 − 2.63837i 0 −2.99584 − 0.157952i 0
25.13 0 0.0685939 + 1.73069i 0 −3.85868 − 1.40444i 0 2.56653 − 0.642577i 0 −2.99059 + 0.237430i 0
25.14 0 0.280333 + 1.70921i 0 −0.659448 − 0.240019i 0 −2.50760 + 0.843777i 0 −2.84283 + 0.958297i 0
25.15 0 0.568135 − 1.63622i 0 0.924390 + 0.336451i 0 −1.51517 − 2.16893i 0 −2.35445 − 1.85919i 0
25.16 0 0.898399 − 1.48084i 0 −3.88836 − 1.41525i 0 0.934219 − 2.47533i 0 −1.38576 − 2.66077i 0
25.17 0 0.910101 + 1.47367i 0 2.21872 + 0.807546i 0 2.43789 + 1.02795i 0 −1.34343 + 2.68238i 0
25.18 0 0.939844 − 1.45489i 0 2.93786 + 1.06929i 0 0.0512030 + 2.64526i 0 −1.23339 − 2.73473i 0
25.19 0 1.52533 + 0.820597i 0 −1.98623 − 0.722928i 0 0.905880 + 2.48584i 0 1.65324 + 2.50336i 0
25.20 0 1.52539 − 0.820472i 0 −3.26720 − 1.18916i 0 −0.0267571 + 2.64562i 0 1.65365 − 2.50309i 0
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.w 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.bq.a yes 144
7.c even 3 1 756.2.bp.a ✓ 144
27.e even 9 1 756.2.bp.a ✓ 144
189.w even 9 1 inner 756.2.bq.a yes 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.bp.a ✓ 144 7.c even 3 1
756.2.bp.a ✓ 144 27.e even 9 1
756.2.bq.a yes 144 1.a even 1 1 trivial
756.2.bq.a yes 144 189.w even 9 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(756, [\chi])\).