Properties

Label 345.3.d.a
Level $345$
Weight $3$
Character orbit 345.d
Analytic conductor $9.401$
Analytic rank $0$
Dimension $48$
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] = [345,3,Mod(229,345)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(345, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("345.229");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 345.d (of order \(2\), degree \(1\), 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: \(9.40056912043\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 92 q^{4} + 12 q^{6} - 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 92 q^{4} + 12 q^{6} - 144 q^{9} + 148 q^{16} - 12 q^{24} - 64 q^{25} + 136 q^{26} + 76 q^{29} - 68 q^{31} - 108 q^{35} + 276 q^{36} + 48 q^{39} + 20 q^{41} + 344 q^{46} + 412 q^{49} - 352 q^{50} - 36 q^{54} - 184 q^{55} - 396 q^{59} - 684 q^{64} - 144 q^{69} + 600 q^{70} + 156 q^{71} - 120 q^{75} + 432 q^{81} - 76 q^{85} + 112 q^{95} + 516 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1 3.87170i 1.73205i −10.9901 −3.20378 + 3.83872i 6.70598 6.81780 27.0634i −3.00000 14.8624 + 12.4041i
229.2 3.87170i 1.73205i −10.9901 3.20378 3.83872i 6.70598 −6.81780 27.0634i −3.00000 −14.8624 12.4041i
229.3 3.64193i 1.73205i −9.26363 −0.940387 4.91077i −6.30800 −2.95540 19.1697i −3.00000 −17.8847 + 3.42482i
229.4 3.64193i 1.73205i −9.26363 0.940387 + 4.91077i −6.30800 2.95540 19.1697i −3.00000 17.8847 3.42482i
229.5 3.09614i 1.73205i −5.58609 −4.94474 + 0.741284i −5.36267 −4.84190 4.91075i −3.00000 2.29512 + 15.3096i
229.6 3.09614i 1.73205i −5.58609 4.94474 0.741284i −5.36267 4.84190 4.91075i −3.00000 −2.29512 15.3096i
229.7 2.96442i 1.73205i −4.78781 −3.79693 3.25320i 5.13453 2.22266 2.33540i −3.00000 −9.64386 + 11.2557i
229.8 2.96442i 1.73205i −4.78781 3.79693 + 3.25320i 5.13453 −2.22266 2.33540i −3.00000 9.64386 11.2557i
229.9 2.73585i 1.73205i −3.48487 −0.698940 + 4.95091i 4.73863 −5.09771 1.40933i −3.00000 13.5449 + 1.91219i
229.10 2.73585i 1.73205i −3.48487 0.698940 4.95091i 4.73863 5.09771 1.40933i −3.00000 −13.5449 1.91219i
229.11 2.58060i 1.73205i −2.65950 −2.43175 + 4.36882i −4.46973 10.6156 3.45929i −3.00000 11.2742 + 6.27538i
229.12 2.58060i 1.73205i −2.65950 2.43175 4.36882i −4.46973 −10.6156 3.45929i −3.00000 −11.2742 6.27538i
229.13 2.02853i 1.73205i −0.114950 −4.74723 + 1.56966i 3.51352 11.5567 7.88096i −3.00000 3.18411 + 9.62991i
229.14 2.02853i 1.73205i −0.114950 4.74723 1.56966i 3.51352 −11.5567 7.88096i −3.00000 −3.18411 9.62991i
229.15 1.48820i 1.73205i 1.78525 −0.794027 + 4.93655i −2.57765 −12.7168 8.60963i −3.00000 7.34659 + 1.18167i
229.16 1.48820i 1.73205i 1.78525 0.794027 4.93655i −2.57765 12.7168 8.60963i −3.00000 −7.34659 1.18167i
229.17 1.17949i 1.73205i 2.60880 −3.95152 3.06357i −2.04294 0.208857 7.79502i −3.00000 −3.61345 + 4.66079i
229.18 1.17949i 1.73205i 2.60880 3.95152 + 3.06357i −2.04294 −0.208857 7.79502i −3.00000 3.61345 4.66079i
229.19 1.12284i 1.73205i 2.73924 −3.66308 3.40321i 1.94481 −10.5997 7.56706i −3.00000 −3.82125 + 4.11303i
229.20 1.12284i 1.73205i 2.73924 3.66308 + 3.40321i 1.94481 10.5997 7.56706i −3.00000 3.82125 4.11303i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.b odd 2 1 inner
115.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 345.3.d.a 48
5.b even 2 1 inner 345.3.d.a 48
23.b odd 2 1 inner 345.3.d.a 48
115.c odd 2 1 inner 345.3.d.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.3.d.a 48 1.a even 1 1 trivial
345.3.d.a 48 5.b even 2 1 inner
345.3.d.a 48 23.b odd 2 1 inner
345.3.d.a 48 115.c odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(345, [\chi])\).