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Properties

Label 153.2.h.b
Level $153$
Weight $2$
Character orbit 153.h
Analytic conductor $1.222$
Analytic rank $0$
Dimension $24$
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] = [153,2,Mod(16,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 153.h (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: \(1.22171115093\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} - 12 q^{4} + 12 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} - 12 q^{4} + 12 q^{8} + 10 q^{9} - 18 q^{13} + 24 q^{15} - 36 q^{16} - 40 q^{17} + 34 q^{18} + 12 q^{19} - 16 q^{21} - 6 q^{25} + 40 q^{26} + 12 q^{30} - 26 q^{32} - 20 q^{33} - 60 q^{35} - 4 q^{36} + 28 q^{38} - 58 q^{42} - 4 q^{47} + 6 q^{49} + 28 q^{50} - 8 q^{51} - 6 q^{52} + 96 q^{53} + 36 q^{55} - 24 q^{59} - 102 q^{60} + 36 q^{64} - 8 q^{66} + 12 q^{67} + 64 q^{68} - 52 q^{69} + 12 q^{70} + 120 q^{72} - 12 q^{76} - 4 q^{77} - 70 q^{81} - 42 q^{83} + 64 q^{84} + 6 q^{85} - 14 q^{86} + 42 q^{87} - 8 q^{89} + 74 q^{93} - 18 q^{94} - 60 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.34900 + 2.33654i −1.61642 + 0.622240i −2.63960 − 4.57192i −2.67207 + 1.54272i 0.726665 − 4.61623i 1.17677 + 0.679411i 8.84727 2.22563 − 2.01160i − 8.32450i
16.2 −1.34900 + 2.33654i 1.61642 − 0.622240i −2.63960 − 4.57192i 2.67207 − 1.54272i −0.726665 + 4.61623i −1.17677 − 0.679411i 8.84727 2.22563 − 2.01160i 8.32450i
16.3 −0.934128 + 1.61796i −0.760163 − 1.55633i −0.745192 − 1.29071i −0.732243 + 0.422761i 3.22816 + 0.223897i −3.70583 − 2.13956i −0.952094 −1.84430 + 2.36612i − 1.57965i
16.4 −0.934128 + 1.61796i 0.760163 + 1.55633i −0.745192 − 1.29071i 0.732243 − 0.422761i −3.22816 − 0.223897i 3.70583 + 2.13956i −0.952094 −1.84430 + 2.36612i 1.57965i
16.5 −0.318708 + 0.552019i −0.926618 + 1.46335i 0.796850 + 1.38019i −1.10930 + 0.640456i −0.512473 − 0.977890i −0.289268 − 0.167009i −2.29068 −1.28276 − 2.71192i − 0.816474i
16.6 −0.318708 + 0.552019i 0.926618 − 1.46335i 0.796850 + 1.38019i 1.10930 − 0.640456i 0.512473 + 0.977890i 0.289268 + 0.167009i −2.29068 −1.28276 − 2.71192i 0.816474i
16.7 −0.0709226 + 0.122841i −1.24014 − 1.20915i 0.989940 + 1.71463i −2.64720 + 1.52836i 0.236488 − 0.0665842i 3.75444 + 2.16763i −0.564526 0.0758917 + 2.99904i − 0.433582i
16.8 −0.0709226 + 0.122841i 1.24014 + 1.20915i 0.989940 + 1.71463i 2.64720 − 1.52836i −0.236488 + 0.0665842i −3.75444 − 2.16763i −0.564526 0.0758917 + 2.99904i 0.433582i
16.9 0.447831 − 0.775665i −1.70700 + 0.293488i 0.598895 + 1.03732i 2.05730 − 1.18778i −0.536801 + 1.45550i −0.883126 − 0.509873i 2.86414 2.82773 − 1.00197i − 2.12770i
16.10 0.447831 − 0.775665i 1.70700 − 0.293488i 0.598895 + 1.03732i −2.05730 + 1.18778i 0.536801 − 1.45550i 0.883126 + 0.509873i 2.86414 2.82773 − 1.00197i 2.12770i
16.11 1.22493 − 2.12164i −1.32246 − 1.11852i −2.00090 − 3.46565i 0.321210 − 0.185451i −3.99302 + 1.43567i 1.91642 + 1.10644i −4.90410 0.497805 + 2.95841i − 0.908655i
16.12 1.22493 − 2.12164i 1.32246 + 1.11852i −2.00090 − 3.46565i −0.321210 + 0.185451i 3.99302 − 1.43567i −1.91642 − 1.10644i −4.90410 0.497805 + 2.95841i 0.908655i
67.1 −1.34900 − 2.33654i −1.61642 − 0.622240i −2.63960 + 4.57192i −2.67207 − 1.54272i 0.726665 + 4.61623i 1.17677 − 0.679411i 8.84727 2.22563 + 2.01160i 8.32450i
67.2 −1.34900 − 2.33654i 1.61642 + 0.622240i −2.63960 + 4.57192i 2.67207 + 1.54272i −0.726665 − 4.61623i −1.17677 + 0.679411i 8.84727 2.22563 + 2.01160i − 8.32450i
67.3 −0.934128 − 1.61796i −0.760163 + 1.55633i −0.745192 + 1.29071i −0.732243 − 0.422761i 3.22816 − 0.223897i −3.70583 + 2.13956i −0.952094 −1.84430 − 2.36612i 1.57965i
67.4 −0.934128 − 1.61796i 0.760163 − 1.55633i −0.745192 + 1.29071i 0.732243 + 0.422761i −3.22816 + 0.223897i 3.70583 − 2.13956i −0.952094 −1.84430 − 2.36612i − 1.57965i
67.5 −0.318708 − 0.552019i −0.926618 − 1.46335i 0.796850 − 1.38019i −1.10930 − 0.640456i −0.512473 + 0.977890i −0.289268 + 0.167009i −2.29068 −1.28276 + 2.71192i 0.816474i
67.6 −0.318708 − 0.552019i 0.926618 + 1.46335i 0.796850 − 1.38019i 1.10930 + 0.640456i 0.512473 − 0.977890i 0.289268 − 0.167009i −2.29068 −1.28276 + 2.71192i − 0.816474i
67.7 −0.0709226 − 0.122841i −1.24014 + 1.20915i 0.989940 − 1.71463i −2.64720 − 1.52836i 0.236488 + 0.0665842i 3.75444 − 2.16763i −0.564526 0.0758917 − 2.99904i 0.433582i
67.8 −0.0709226 − 0.122841i 1.24014 − 1.20915i 0.989940 − 1.71463i 2.64720 + 1.52836i −0.236488 − 0.0665842i −3.75444 + 2.16763i −0.564526 0.0758917 − 2.99904i − 0.433582i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
17.b even 2 1 inner
153.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.2.h.b ✓ 24
3.b odd 2 1 459.2.h.b 24
9.c even 3 1 inner 153.2.h.b ✓ 24
9.c even 3 1 1377.2.d.f 12
9.d odd 6 1 459.2.h.b 24
9.d odd 6 1 1377.2.d.e 12
17.b even 2 1 inner 153.2.h.b ✓ 24
51.c odd 2 1 459.2.h.b 24
153.h even 6 1 inner 153.2.h.b ✓ 24
153.h even 6 1 1377.2.d.f 12
153.i odd 6 1 459.2.h.b 24
153.i odd 6 1 1377.2.d.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.2.h.b ✓ 24 1.a even 1 1 trivial
153.2.h.b ✓ 24 9.c even 3 1 inner
153.2.h.b ✓ 24 17.b even 2 1 inner
153.2.h.b ✓ 24 153.h even 6 1 inner
459.2.h.b 24 3.b odd 2 1
459.2.h.b 24 9.d odd 6 1
459.2.h.b 24 51.c odd 2 1
459.2.h.b 24 153.i odd 6 1
1377.2.d.e 12 9.d odd 6 1
1377.2.d.e 12 153.i odd 6 1
1377.2.d.f 12 9.c even 3 1
1377.2.d.f 12 153.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 2 T_{2}^{11} + 11 T_{2}^{10} + 12 T_{2}^{9} + 70 T_{2}^{8} + 79 T_{2}^{7} + 190 T_{2}^{6} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(153, [\chi])\). Copy content Toggle raw display