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Properties

Label 756.2.bf.a
Level $756$
Weight $2$
Character orbit 756.bf
Analytic conductor $6.037$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [756,2,Mod(271,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.271");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.bf (of order \(6\), degree \(2\), 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: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{7} - 4 q^{10} - 28 q^{16} + 6 q^{19} + 20 q^{22} + 20 q^{25} + 28 q^{28} - 8 q^{34} - 2 q^{37} + 8 q^{40} - 12 q^{46} - 10 q^{49} + 20 q^{52} - 16 q^{55} + 40 q^{58} + 48 q^{64} - 42 q^{67} - 56 q^{70} - 18 q^{73} + 40 q^{76} + 6 q^{79} + 88 q^{82} - 8 q^{85} + 76 q^{88} - 8 q^{91} + 8 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1 −1.41410 + 0.0181665i 0 1.99934 − 0.0513784i −0.766614 − 0.442605i 0 2.19153 + 1.48229i −2.82633 + 0.108975i 0 1.09211 + 0.611960i
271.2 −1.33776 + 0.458698i 0 1.57919 − 1.22725i 0.945181 + 0.545701i 0 −2.64237 − 0.133712i −1.54964 + 2.36614i 0 −1.51474 − 0.296462i
271.3 −1.14872 − 0.824895i 0 0.639096 + 1.89514i −1.99770 − 1.15337i 0 −0.0928632 − 2.64412i 0.829152 − 2.70416i 0 1.34338 + 2.97278i
271.4 −0.940173 + 1.05644i 0 −0.232150 − 1.98648i −1.00309 − 0.579135i 0 0.250854 − 2.63383i 2.31687 + 1.62238i 0 1.55490 − 0.515224i
271.5 −0.922848 − 1.07161i 0 −0.296703 + 1.97787i 3.59250 + 2.07413i 0 2.54686 − 0.716577i 2.39332 − 1.50732i 0 −1.09267 − 5.76388i
271.6 −0.829453 + 1.14543i 0 −0.624014 − 1.90016i 3.11886 + 1.80067i 0 0.838804 + 2.50926i 2.69409 + 0.861330i 0 −4.64949 + 2.07885i
271.7 −0.670549 − 1.24514i 0 −1.10073 + 1.66985i −1.51137 − 0.872592i 0 −1.00609 + 2.44699i 2.81728 + 0.250838i 0 −0.0730451 + 2.46698i
271.8 −0.134101 + 1.40784i 0 −1.96403 − 0.377586i −2.47070 − 1.42646i 0 −2.58673 + 0.555723i 0.794961 − 2.71441i 0 2.33955 − 3.28706i
271.9 0.134101 − 1.40784i 0 −1.96403 − 0.377586i 2.47070 + 1.42646i 0 −2.58673 + 0.555723i −0.794961 + 2.71441i 0 2.33955 − 3.28706i
271.10 0.670549 + 1.24514i 0 −1.10073 + 1.66985i 1.51137 + 0.872592i 0 −1.00609 + 2.44699i −2.81728 − 0.250838i 0 −0.0730451 + 2.46698i
271.11 0.829453 − 1.14543i 0 −0.624014 − 1.90016i −3.11886 − 1.80067i 0 0.838804 + 2.50926i −2.69409 − 0.861330i 0 −4.64949 + 2.07885i
271.12 0.922848 + 1.07161i 0 −0.296703 + 1.97787i −3.59250 − 2.07413i 0 2.54686 − 0.716577i −2.39332 + 1.50732i 0 −1.09267 − 5.76388i
271.13 0.940173 − 1.05644i 0 −0.232150 − 1.98648i 1.00309 + 0.579135i 0 0.250854 − 2.63383i −2.31687 − 1.62238i 0 1.55490 − 0.515224i
271.14 1.14872 + 0.824895i 0 0.639096 + 1.89514i 1.99770 + 1.15337i 0 −0.0928632 − 2.64412i −0.829152 + 2.70416i 0 1.34338 + 2.97278i
271.15 1.33776 − 0.458698i 0 1.57919 − 1.22725i −0.945181 − 0.545701i 0 −2.64237 − 0.133712i 1.54964 − 2.36614i 0 −1.51474 − 0.296462i
271.16 1.41410 − 0.0181665i 0 1.99934 − 0.0513784i 0.766614 + 0.442605i 0 2.19153 + 1.48229i 2.82633 − 0.108975i 0 1.09211 + 0.611960i
703.1 −1.41410 − 0.0181665i 0 1.99934 + 0.0513784i −0.766614 + 0.442605i 0 2.19153 − 1.48229i −2.82633 − 0.108975i 0 1.09211 − 0.611960i
703.2 −1.33776 − 0.458698i 0 1.57919 + 1.22725i 0.945181 − 0.545701i 0 −2.64237 + 0.133712i −1.54964 − 2.36614i 0 −1.51474 + 0.296462i
703.3 −1.14872 + 0.824895i 0 0.639096 − 1.89514i −1.99770 + 1.15337i 0 −0.0928632 + 2.64412i 0.829152 + 2.70416i 0 1.34338 − 2.97278i
703.4 −0.940173 − 1.05644i 0 −0.232150 + 1.98648i −1.00309 + 0.579135i 0 0.250854 + 2.63383i 2.31687 − 1.62238i 0 1.55490 + 0.515224i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.f even 6 1 inner
84.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.bf.a ✓ 32
3.b odd 2 1 inner 756.2.bf.a ✓ 32
4.b odd 2 1 756.2.bf.d yes 32
7.d odd 6 1 756.2.bf.d yes 32
12.b even 2 1 756.2.bf.d yes 32
21.g even 6 1 756.2.bf.d yes 32
28.f even 6 1 inner 756.2.bf.a ✓ 32
84.j odd 6 1 inner 756.2.bf.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.bf.a ✓ 32 1.a even 1 1 trivial
756.2.bf.a ✓ 32 3.b odd 2 1 inner
756.2.bf.a ✓ 32 28.f even 6 1 inner
756.2.bf.a ✓ 32 84.j odd 6 1 inner
756.2.bf.d yes 32 4.b odd 2 1
756.2.bf.d yes 32 7.d odd 6 1
756.2.bf.d yes 32 12.b even 2 1
756.2.bf.d yes 32 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\):

\( T_{5}^{32} - 50 T_{5}^{30} + 1536 T_{5}^{28} - 29800 T_{5}^{26} + 422464 T_{5}^{24} - 4314432 T_{5}^{22} + \cdots + 1358954496 \) Copy content Toggle raw display
\( T_{11}^{32} - 86 T_{11}^{30} + 4600 T_{11}^{28} - 150840 T_{11}^{26} + 3570704 T_{11}^{24} + \cdots + 22419997720576 \) Copy content Toggle raw display
\( T_{19}^{16} - 3 T_{19}^{15} + 80 T_{19}^{14} - 177 T_{19}^{13} + 4280 T_{19}^{12} - 9285 T_{19}^{11} + \cdots + 29942784 \) Copy content Toggle raw display