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Properties

Label 169.3.f.g
Level $169$
Weight $3$
Character orbit 169.f
Analytic conductor $4.605$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,3,Mod(19,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 169.f (of order \(12\), degree \(4\), not 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: \(4.60491646769\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{3} - 84 q^{9} + 376 q^{14} - 188 q^{16} + 136 q^{22} + 120 q^{27} - 84 q^{29} - 176 q^{35} - 1048 q^{40} + 368 q^{42} + 368 q^{48} - 88 q^{53} + 704 q^{55} + 8 q^{61} - 1480 q^{66} + 168 q^{68}+ \cdots - 1132 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.834643 − 3.11493i 1.02304 − 1.77195i −5.54206 + 3.19971i −5.50472 + 5.50472i −6.37336 − 1.70774i −0.846854 + 3.16050i 5.47136 + 5.47136i 2.40680 + 4.16870i 21.7413 + 12.5523i
19.2 −0.739915 − 2.76140i −2.32422 + 4.02567i −3.61376 + 2.08640i 2.43155 − 2.43155i 12.8362 + 3.43945i 0.277954 − 1.03734i 0.349324 + 0.349324i −6.30402 − 10.9189i −8.51361 − 4.91533i
19.3 −0.609234 − 2.27369i 2.51175 − 4.35048i −1.33442 + 0.770425i 1.10964 − 1.10964i −11.4219 − 3.06049i −1.70128 + 6.34925i −4.09315 − 4.09315i −8.11776 − 14.0604i −3.19902 − 1.84696i
19.4 −0.515320 − 1.92320i 0.299764 − 0.519206i 0.0309536 − 0.0178711i 5.65111 − 5.65111i −1.15301 − 0.308949i −0.526568 + 1.96518i −5.68184 − 5.68184i 4.32028 + 7.48295i −13.7804 − 7.95610i
19.5 −0.429937 − 1.60455i −0.677024 + 1.17264i 1.07438 − 0.620291i −1.57408 + 1.57408i 2.17264 + 0.582156i −3.25921 + 12.1635i −6.15564 − 6.15564i 3.58328 + 6.20642i 3.20245 + 1.84893i
19.6 −0.104575 − 0.390278i −2.33330 + 4.04139i 3.32272 − 1.91837i 4.14024 − 4.14024i 1.82127 + 0.488008i 0.591332 − 2.20688i −2.23898 − 2.23898i −6.38858 − 11.0653i −2.04881 − 1.18288i
19.7 0.104575 + 0.390278i −2.33330 + 4.04139i 3.32272 − 1.91837i −4.14024 + 4.14024i −1.82127 − 0.488008i −0.591332 + 2.20688i 2.23898 + 2.23898i −6.38858 − 11.0653i −2.04881 − 1.18288i
19.8 0.429937 + 1.60455i −0.677024 + 1.17264i 1.07438 − 0.620291i 1.57408 − 1.57408i −2.17264 − 0.582156i 3.25921 − 12.1635i 6.15564 + 6.15564i 3.58328 + 6.20642i 3.20245 + 1.84893i
19.9 0.515320 + 1.92320i 0.299764 − 0.519206i 0.0309536 − 0.0178711i −5.65111 + 5.65111i 1.15301 + 0.308949i 0.526568 − 1.96518i 5.68184 + 5.68184i 4.32028 + 7.48295i −13.7804 − 7.95610i
19.10 0.609234 + 2.27369i 2.51175 − 4.35048i −1.33442 + 0.770425i −1.10964 + 1.10964i 11.4219 + 3.06049i 1.70128 − 6.34925i 4.09315 + 4.09315i −8.11776 − 14.0604i −3.19902 − 1.84696i
19.11 0.739915 + 2.76140i −2.32422 + 4.02567i −3.61376 + 2.08640i −2.43155 + 2.43155i −12.8362 − 3.43945i −0.277954 + 1.03734i −0.349324 − 0.349324i −6.30402 − 10.9189i −8.51361 − 4.91533i
19.12 0.834643 + 3.11493i 1.02304 − 1.77195i −5.54206 + 3.19971i 5.50472 − 5.50472i 6.37336 + 1.70774i 0.846854 − 3.16050i −5.47136 − 5.47136i 2.40680 + 4.16870i 21.7413 + 12.5523i
80.1 −3.11493 − 0.834643i 1.02304 + 1.77195i 5.54206 + 3.19971i 5.50472 − 5.50472i −1.70774 − 6.37336i −3.16050 + 0.846854i −5.47136 − 5.47136i 2.40680 − 4.16870i −21.7413 + 12.5523i
80.2 −2.76140 − 0.739915i −2.32422 − 4.02567i 3.61376 + 2.08640i −2.43155 + 2.43155i 3.43945 + 12.8362i 1.03734 − 0.277954i −0.349324 − 0.349324i −6.30402 + 10.9189i 8.51361 − 4.91533i
80.3 −2.27369 − 0.609234i 2.51175 + 4.35048i 1.33442 + 0.770425i −1.10964 + 1.10964i −3.06049 − 11.4219i −6.34925 + 1.70128i 4.09315 + 4.09315i −8.11776 + 14.0604i 3.19902 − 1.84696i
80.4 −1.92320 − 0.515320i 0.299764 + 0.519206i −0.0309536 − 0.0178711i −5.65111 + 5.65111i −0.308949 − 1.15301i −1.96518 + 0.526568i 5.68184 + 5.68184i 4.32028 − 7.48295i 13.7804 − 7.95610i
80.5 −1.60455 − 0.429937i −0.677024 − 1.17264i −1.07438 − 0.620291i 1.57408 − 1.57408i 0.582156 + 2.17264i −12.1635 + 3.25921i 6.15564 + 6.15564i 3.58328 − 6.20642i −3.20245 + 1.84893i
80.6 −0.390278 − 0.104575i −2.33330 − 4.04139i −3.32272 − 1.91837i −4.14024 + 4.14024i 0.488008 + 1.82127i 2.20688 − 0.591332i 2.23898 + 2.23898i −6.38858 + 11.0653i 2.04881 − 1.18288i
80.7 0.390278 + 0.104575i −2.33330 − 4.04139i −3.32272 − 1.91837i 4.14024 − 4.14024i −0.488008 − 1.82127i −2.20688 + 0.591332i −2.23898 − 2.23898i −6.38858 + 11.0653i 2.04881 − 1.18288i
80.8 1.60455 + 0.429937i −0.677024 − 1.17264i −1.07438 − 0.620291i −1.57408 + 1.57408i −0.582156 − 2.17264i 12.1635 − 3.25921i −6.15564 − 6.15564i 3.58328 − 6.20642i −3.20245 + 1.84893i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.d odd 4 2 inner
13.e even 6 1 inner
13.f odd 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.3.f.g 48
13.b even 2 1 inner 169.3.f.g 48
13.c even 3 1 169.3.d.e ✓ 24
13.c even 3 1 inner 169.3.f.g 48
13.d odd 4 2 inner 169.3.f.g 48
13.e even 6 1 169.3.d.e ✓ 24
13.e even 6 1 inner 169.3.f.g 48
13.f odd 12 2 169.3.d.e ✓ 24
13.f odd 12 2 inner 169.3.f.g 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.3.d.e ✓ 24 13.c even 3 1
169.3.d.e ✓ 24 13.e even 6 1
169.3.d.e ✓ 24 13.f odd 12 2
169.3.f.g 48 1.a even 1 1 trivial
169.3.f.g 48 13.b even 2 1 inner
169.3.f.g 48 13.c even 3 1 inner
169.3.f.g 48 13.d odd 4 2 inner
169.3.f.g 48 13.e even 6 1 inner
169.3.f.g 48 13.f odd 12 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 229 T_{2}^{44} + 34923 T_{2}^{40} - 2930264 T_{2}^{36} + 176369373 T_{2}^{32} + \cdots + 500246412961 \) acting on \(S_{3}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display