Properties

Label 945.2.bb.a
Level $945$
Weight $2$
Character orbit 945.bb
Analytic conductor $7.546$
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] = [945,2,Mod(269,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.269");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.bb (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: \(7.54586299101\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{4} - 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{4} - 3 q^{5} - 3 q^{10} - 16 q^{16} - 6 q^{17} + 12 q^{19} - q^{25} - 6 q^{31} + 30 q^{32} - 12 q^{35} - 48 q^{38} + 24 q^{40} - 18 q^{46} - 6 q^{47} + 2 q^{49} - 72 q^{50} - 48 q^{53} + 12 q^{61} - 16 q^{64} + 30 q^{65} + 90 q^{68} - 24 q^{70} - 42 q^{77} + 10 q^{79} + 96 q^{80} + 10 q^{85} - 62 q^{91} + 84 q^{92} + 84 q^{94} + 36 q^{95} + 72 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1 −1.23319 2.13595i 0 −2.04153 + 3.53604i −2.20106 0.394102i 0 0.465281 + 2.60452i 5.13765 0 1.87255 + 5.18738i
269.2 −1.23319 2.13595i 0 −2.04153 + 3.53604i −0.759230 + 2.10323i 0 −0.465281 2.60452i 5.13765 0 5.42868 0.972008i
269.3 −1.04316 1.80680i 0 −1.17634 + 2.03749i 1.42338 1.72453i 0 2.64510 0.0586776i 0.735821 0 −4.60068 0.772800i
269.4 −1.04316 1.80680i 0 −1.17634 + 2.03749i 2.20517 0.370415i 0 −2.64510 + 0.0586776i 0.735821 0 −2.96960 3.59790i
269.5 −0.640286 1.10901i 0 0.180069 0.311888i −1.97382 1.05073i 0 −2.58281 0.573647i −3.02232 0 0.0985445 + 2.86175i
269.6 −0.640286 1.10901i 0 0.180069 0.311888i −0.0769535 + 2.23474i 0 2.58281 + 0.573647i −3.02232 0 2.52762 1.34553i
269.7 −0.104913 0.181715i 0 0.977986 1.69392i 0.346458 2.20906i 0 0.694926 2.55286i −0.830069 0 −0.437769 + 0.168804i
269.8 −0.104913 0.181715i 0 0.977986 1.69392i 2.08634 + 0.804491i 0 −0.694926 + 2.55286i −0.830069 0 −0.0726961 0.463521i
269.9 0.187042 + 0.323967i 0 0.930030 1.61086i −2.23199 + 0.135023i 0 2.46237 0.967860i 1.44399 0 −0.461219 0.697834i
269.10 0.187042 + 0.323967i 0 0.930030 1.61086i −1.23293 + 1.86545i 0 −2.46237 + 0.967860i 1.44399 0 −0.834952 0.0505098i
269.11 0.598512 + 1.03665i 0 0.283567 0.491153i 1.06736 1.96488i 0 −1.89415 1.84722i 3.07292 0 2.67572 0.0695206i
269.12 0.598512 + 1.03665i 0 0.283567 0.491153i 2.23531 + 0.0580779i 0 1.89415 + 1.84722i 3.07292 0 1.27766 + 2.35200i
269.13 0.921715 + 1.59646i 0 −0.699118 + 1.21091i −1.28751 1.82820i 0 −0.918398 + 2.48124i 1.10931 0 1.73193 3.74054i
269.14 0.921715 + 1.59646i 0 −0.699118 + 1.21091i 0.939513 + 2.02912i 0 0.918398 2.48124i 1.10931 0 −2.37344 + 3.37016i
269.15 1.31428 + 2.27640i 0 −2.45466 + 4.25159i −1.97038 1.05717i 0 1.84082 1.90036i −7.64730 0 −0.183092 5.87478i
269.16 1.31428 + 2.27640i 0 −2.45466 + 4.25159i −0.0696551 + 2.23498i 0 −1.84082 + 1.90036i −7.64730 0 −5.17926 + 2.77883i
404.1 −1.23319 + 2.13595i 0 −2.04153 3.53604i −2.20106 + 0.394102i 0 0.465281 2.60452i 5.13765 0 1.87255 5.18738i
404.2 −1.23319 + 2.13595i 0 −2.04153 3.53604i −0.759230 2.10323i 0 −0.465281 + 2.60452i 5.13765 0 5.42868 + 0.972008i
404.3 −1.04316 + 1.80680i 0 −1.17634 2.03749i 1.42338 + 1.72453i 0 2.64510 + 0.0586776i 0.735821 0 −4.60068 + 0.772800i
404.4 −1.04316 + 1.80680i 0 −1.17634 2.03749i 2.20517 + 0.370415i 0 −2.64510 0.0586776i 0.735821 0 −2.96960 + 3.59790i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
15.d odd 2 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.bb.a 32
3.b odd 2 1 945.2.bb.b yes 32
5.b even 2 1 945.2.bb.b yes 32
7.d odd 6 1 inner 945.2.bb.a 32
15.d odd 2 1 inner 945.2.bb.a 32
21.g even 6 1 945.2.bb.b yes 32
35.i odd 6 1 945.2.bb.b yes 32
105.p even 6 1 inner 945.2.bb.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.bb.a 32 1.a even 1 1 trivial
945.2.bb.a 32 7.d odd 6 1 inner
945.2.bb.a 32 15.d odd 2 1 inner
945.2.bb.a 32 105.p even 6 1 inner
945.2.bb.b yes 32 3.b odd 2 1
945.2.bb.b yes 32 5.b even 2 1
945.2.bb.b yes 32 21.g even 6 1
945.2.bb.b yes 32 35.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 12 T_{2}^{14} + 102 T_{2}^{12} - 3 T_{2}^{11} + 420 T_{2}^{10} - 66 T_{2}^{9} + 1257 T_{2}^{8} - 198 T_{2}^{7} + 1701 T_{2}^{6} - 378 T_{2}^{5} + 1656 T_{2}^{4} - 234 T_{2}^{3} + 162 T_{2}^{2} + 18 T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display