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Properties

Label 945.2.p.b
Level $945$
Weight $2$
Character orbit 945.p
Analytic conductor $7.546$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(433,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.433");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.p (of order \(4\), 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: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{7} + 40 q^{16} + 8 q^{22} - 48 q^{25} - 20 q^{28} - 24 q^{37} - 40 q^{43} + 40 q^{46} - 80 q^{58} - 64 q^{67} - 4 q^{70} - 8 q^{85} - 48 q^{88} + 48 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1 −1.62321 + 1.62321i 0 − 3.26965i −0.248987 − 2.22216i 0 0.952683 − 2.46828i 2.06091 + 2.06091i 0 4.01120 + 3.20289i
433.2 −1.62321 + 1.62321i 0 − 3.26965i 0.248987 + 2.22216i 0 2.46828 − 0.952683i 2.06091 + 2.06091i 0 −4.01120 − 3.20289i
433.3 −1.50406 + 1.50406i 0 − 2.52441i −0.469512 − 2.18622i 0 1.79340 + 1.94518i 0.788742 + 0.788742i 0 3.99439 + 2.58204i
433.4 −1.50406 + 1.50406i 0 − 2.52441i 0.469512 + 2.18622i 0 −1.94518 − 1.79340i 0.788742 + 0.788742i 0 −3.99439 − 2.58204i
433.5 −1.27393 + 1.27393i 0 − 1.24581i −2.23606 + 0.00600881i 0 −1.54853 + 2.14524i −0.960789 − 0.960789i 0 2.84093 − 2.85624i
433.6 −1.27393 + 1.27393i 0 − 1.24581i 2.23606 − 0.00600881i 0 −2.14524 + 1.54853i −0.960789 − 0.960789i 0 −2.84093 + 2.85624i
433.7 −0.834614 + 0.834614i 0 0.606840i −1.65237 − 1.50655i 0 −2.61358 + 0.411321i −2.17570 − 2.17570i 0 2.63647 − 0.121701i
433.8 −0.834614 + 0.834614i 0 0.606840i 1.65237 + 1.50655i 0 −0.411321 + 2.61358i −2.17570 − 2.17570i 0 −2.63647 + 0.121701i
433.9 −0.515806 + 0.515806i 0 1.46789i −1.00702 + 1.99648i 0 −2.04999 − 1.67258i −1.78876 − 1.78876i 0 −0.510370 − 1.54922i
433.10 −0.515806 + 0.515806i 0 1.46789i 1.00702 − 1.99648i 0 1.67258 + 2.04999i −1.78876 − 1.78876i 0 0.510370 + 1.54922i
433.11 −0.132028 + 0.132028i 0 1.96514i −1.72430 − 1.42366i 0 2.54612 + 0.719217i −0.523508 − 0.523508i 0 0.415617 − 0.0396933i
433.12 −0.132028 + 0.132028i 0 1.96514i 1.72430 + 1.42366i 0 −0.719217 − 2.54612i −0.523508 − 0.523508i 0 −0.415617 + 0.0396933i
433.13 0.132028 − 0.132028i 0 1.96514i −1.72430 − 1.42366i 0 −0.719217 − 2.54612i 0.523508 + 0.523508i 0 −0.415617 + 0.0396933i
433.14 0.132028 − 0.132028i 0 1.96514i 1.72430 + 1.42366i 0 2.54612 + 0.719217i 0.523508 + 0.523508i 0 0.415617 − 0.0396933i
433.15 0.515806 − 0.515806i 0 1.46789i −1.00702 + 1.99648i 0 1.67258 + 2.04999i 1.78876 + 1.78876i 0 0.510370 + 1.54922i
433.16 0.515806 − 0.515806i 0 1.46789i 1.00702 − 1.99648i 0 −2.04999 − 1.67258i 1.78876 + 1.78876i 0 −0.510370 − 1.54922i
433.17 0.834614 − 0.834614i 0 0.606840i −1.65237 − 1.50655i 0 −0.411321 + 2.61358i 2.17570 + 2.17570i 0 −2.63647 + 0.121701i
433.18 0.834614 − 0.834614i 0 0.606840i 1.65237 + 1.50655i 0 −2.61358 + 0.411321i 2.17570 + 2.17570i 0 2.63647 − 0.121701i
433.19 1.27393 − 1.27393i 0 − 1.24581i −2.23606 + 0.00600881i 0 −2.14524 + 1.54853i 0.960789 + 0.960789i 0 −2.84093 + 2.85624i
433.20 1.27393 − 1.27393i 0 − 1.24581i 2.23606 − 0.00600881i 0 −1.54853 + 2.14524i 0.960789 + 0.960789i 0 2.84093 − 2.85624i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.b odd 2 1 inner
15.e even 4 1 inner
21.c even 2 1 inner
35.f even 4 1 inner
105.k odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.p.b ✓ 48
3.b odd 2 1 inner 945.2.p.b ✓ 48
5.c odd 4 1 inner 945.2.p.b ✓ 48
7.b odd 2 1 inner 945.2.p.b ✓ 48
15.e even 4 1 inner 945.2.p.b ✓ 48
21.c even 2 1 inner 945.2.p.b ✓ 48
35.f even 4 1 inner 945.2.p.b ✓ 48
105.k odd 4 1 inner 945.2.p.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.p.b ✓ 48 1.a even 1 1 trivial
945.2.p.b ✓ 48 3.b odd 2 1 inner
945.2.p.b ✓ 48 5.c odd 4 1 inner
945.2.p.b ✓ 48 7.b odd 2 1 inner
945.2.p.b ✓ 48 15.e even 4 1 inner
945.2.p.b ✓ 48 21.c even 2 1 inner
945.2.p.b ✓ 48 35.f even 4 1 inner
945.2.p.b ✓ 48 105.k odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 61T_{2}^{20} + 1208T_{2}^{16} + 8417T_{2}^{12} + 13921T_{2}^{8} + 3308T_{2}^{4} + 4 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display