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Properties

Label 153.2.s.a
Level $153$
Weight $2$
Character orbit 153.s
Analytic conductor $1.222$
Analytic rank $0$
Dimension $256$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [153,2,Mod(5,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, base_ring=CyclotomicField(48))
 
chi = DirichletCharacter(H, H._module([40, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 153.s (of order \(48\), degree \(16\), 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: \(1.22171115093\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(16\) over \(\Q(\zeta_{48})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{48}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9} - 32 q^{10} - 24 q^{11} + 32 q^{12} - 8 q^{13} - 24 q^{14} - 40 q^{15} - 64 q^{18} - 32 q^{19} - 24 q^{20} + 32 q^{21} - 8 q^{22}+ \cdots + 88 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.18373 − 1.67563i 1.29886 − 1.14584i 1.44329 + 5.38642i −0.164844 + 0.187968i −4.75638 + 0.325791i 2.51761 − 2.20789i 3.76722 − 9.09488i 0.374089 − 2.97658i 0.674941 − 0.134254i
5.2 −1.93474 − 1.48458i −1.55420 − 0.764493i 1.02161 + 3.81269i 0.541710 − 0.617701i 1.87203 + 3.78643i −3.71094 + 3.25440i 1.81720 − 4.38711i 1.83110 + 2.37636i −1.96509 + 0.390881i
5.3 −1.55959 − 1.19672i 1.16178 + 1.28463i 0.482560 + 1.80094i −1.78450 + 2.03483i −0.274574 − 3.39382i −1.16370 + 1.02054i −0.101958 + 0.246149i −0.300523 + 2.98491i 5.21821 − 1.03797i
5.4 −1.50245 − 1.15287i −1.48352 + 0.893963i 0.410611 + 1.53242i −1.19430 + 1.36184i 3.25954 + 0.367171i 2.49823 − 2.19089i −0.299691 + 0.723517i 1.40166 − 2.65242i 3.36440 − 0.669221i
5.5 −1.12404 − 0.862508i 1.69438 − 0.359292i 0.00191254 + 0.00713769i 2.50849 − 2.86038i −2.21444 − 1.05755i −2.28298 + 2.00212i −1.08038 + 2.60828i 2.74182 − 1.21755i −5.28675 + 1.05160i
5.6 −1.06065 − 0.813864i −0.369214 − 1.69224i −0.0550376 − 0.205403i 0.984560 − 1.12268i −0.985649 + 2.09536i 2.03121 − 1.78132i −1.13203 + 2.73296i −2.72736 + 1.24960i −1.95798 + 0.389466i
5.7 −0.490187 − 0.376133i 0.343811 + 1.69758i −0.418832 − 1.56310i 1.49606 − 1.70593i 0.469987 − 0.961452i 1.62514 − 1.42521i −0.855523 + 2.06542i −2.76359 + 1.16730i −1.37500 + 0.273505i
5.8 −0.183891 − 0.141104i −1.43824 − 0.965129i −0.503733 − 1.87996i −2.35903 + 2.68996i 0.128295 + 0.380420i −0.826331 + 0.724673i −0.350042 + 0.845077i 1.13705 + 2.77617i 0.813369 − 0.161789i
5.9 −0.0707646 − 0.0542996i 1.70868 + 0.283550i −0.515579 − 1.92417i −1.47631 + 1.68341i −0.105518 − 0.112846i 3.03807 − 2.66432i −0.136265 + 0.328973i 2.83920 + 0.968993i 0.195879 − 0.0389627i
5.10 0.501640 + 0.384922i 0.437145 − 1.67598i −0.414160 − 1.54567i 0.147778 − 0.168508i 0.864411 − 0.672472i −1.44130 + 1.26399i 0.871146 − 2.10313i −2.61781 − 1.46529i 0.138994 − 0.0276476i
5.11 0.564127 + 0.432870i −1.71373 + 0.251259i −0.386775 − 1.44347i 1.71199 − 1.95215i −1.07552 − 0.600080i 0.0419494 − 0.0367886i 0.950868 − 2.29560i 2.87374 − 0.861179i 1.81081 − 0.360192i
5.12 1.00677 + 0.772524i 1.70648 + 0.296551i −0.100840 − 0.376340i 0.0953134 − 0.108684i 1.48894 + 1.61685i −2.38350 + 2.09027i 1.16047 − 2.80161i 2.82411 + 1.01211i 0.179920 − 0.0357883i
5.13 1.40306 + 1.07661i 0.184815 + 1.72216i 0.291859 + 1.08923i 0.349197 − 0.398183i −1.59478 + 2.61527i 0.487217 − 0.427278i 0.590386 − 1.42532i −2.93169 + 0.636563i 0.918630 − 0.182727i
5.14 1.78424 + 1.36910i 0.827829 − 1.52141i 0.791458 + 2.95376i −2.27494 + 2.59407i 3.56001 − 1.58119i 2.00390 − 1.75737i −0.910532 + 2.19822i −1.62940 − 2.51894i −7.61057 + 1.51384i
5.15 1.78977 + 1.37334i −1.65992 + 0.494620i 0.799582 + 2.98408i −1.53223 + 1.74718i −3.65017 − 1.39438i −0.511998 + 0.449011i −0.940452 + 2.27045i 2.51070 − 1.64206i −5.14182 + 1.02277i
5.16 2.00872 + 1.54135i −0.894687 − 1.48308i 1.14157 + 4.26041i 2.50546 − 2.85693i 0.488768 − 4.35812i −1.62924 + 1.42880i −2.33580 + 5.63913i −1.39907 + 2.65379i 9.43628 − 1.87699i
11.1 −1.63540 − 2.13130i 1.11938 + 1.32173i −1.35025 + 5.03920i −2.13674 − 0.140050i 0.986361 − 4.54731i 0.0884715 + 1.34981i 7.98434 − 3.30722i −0.493955 + 2.95906i 3.19595 + 4.78307i
11.2 −1.53625 − 2.00208i −1.72157 + 0.190287i −1.13062 + 4.21953i 3.62526 + 0.237612i 3.02572 + 3.15438i 0.115282 + 1.75887i 5.52180 − 2.28721i 2.92758 − 0.655185i −5.09359 − 7.62309i
11.3 −1.34967 − 1.75893i −0.353226 − 1.69565i −0.754572 + 2.81610i −1.95144 − 0.127904i −2.50579 + 2.90987i −0.0117682 − 0.179547i 1.87511 − 0.776696i −2.75046 + 1.19790i 2.40883 + 3.60506i
11.4 −1.03701 − 1.35146i −0.776090 + 1.54845i −0.233410 + 0.871097i −0.974191 − 0.0638519i 2.89747 − 0.556901i −0.213223 − 3.25316i −1.72831 + 0.715890i −1.79537 − 2.40347i 0.923953 + 1.38279i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
17.e odd 16 1 inner
153.s even 48 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.2.s.a ✓ 256
3.b odd 2 1 459.2.y.a 256
9.c even 3 1 459.2.y.a 256
9.d odd 6 1 inner 153.2.s.a ✓ 256
17.e odd 16 1 inner 153.2.s.a ✓ 256
51.i even 16 1 459.2.y.a 256
153.s even 48 1 inner 153.2.s.a ✓ 256
153.t odd 48 1 459.2.y.a 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.2.s.a ✓ 256 1.a even 1 1 trivial
153.2.s.a ✓ 256 9.d odd 6 1 inner
153.2.s.a ✓ 256 17.e odd 16 1 inner
153.2.s.a ✓ 256 153.s even 48 1 inner
459.2.y.a 256 3.b odd 2 1
459.2.y.a 256 9.c even 3 1
459.2.y.a 256 51.i even 16 1
459.2.y.a 256 153.t odd 48 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(153, [\chi])\).