Properties

Label 363.3.g.h
Level $363$
Weight $3$
Character orbit 363.g
Analytic conductor $9.891$
Analytic rank $0$
Dimension $32$
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] = [363,3,Mod(40,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.40");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.g (of order \(10\), 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: \(9.89103359628\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 24 q^{4} - 16 q^{5} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 24 q^{4} - 16 q^{5} - 24 q^{9} - 136 q^{14} - 72 q^{16} + 16 q^{20} + 88 q^{25} + 120 q^{26} + 128 q^{31} + 384 q^{34} + 72 q^{36} - 80 q^{37} + 216 q^{38} - 72 q^{42} + 192 q^{45} + 32 q^{47} - 288 q^{48} + 152 q^{49} - 80 q^{53} - 2784 q^{56} + 176 q^{58} + 64 q^{59} - 192 q^{60} - 8 q^{64} + 1856 q^{67} + 96 q^{69} - 304 q^{70} + 128 q^{71} + 48 q^{75} + 1632 q^{78} - 80 q^{80} - 72 q^{81} - 528 q^{82} - 24 q^{86} - 2880 q^{89} - 80 q^{91} + 1248 q^{92} + 96 q^{93} - 416 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1 −2.19808 + 3.02540i 0.535233 1.64728i −3.08541 9.49592i 0.450251 0.327126i 3.80719 + 5.24014i −11.4499 + 3.72029i 21.2846 + 6.91580i −2.42705 1.76336i 2.08124i
40.2 −1.82934 + 2.51787i −0.535233 + 1.64728i −1.75712 5.40785i −4.58660 + 3.33236i −3.16851 4.36108i −9.13303 + 2.96750i 4.99088 + 1.62164i −2.42705 1.76336i 17.6445i
40.3 −1.22082 + 1.68031i −0.535233 + 1.64728i −0.0969856 0.298491i 1.35053 0.981217i −2.11452 2.91039i 2.87537 0.934266i −7.28135 2.36585i −2.42705 1.76336i 3.46720i
40.4 −0.0729469 + 0.100403i 0.535233 1.64728i 1.23131 + 3.78958i −3.68632 + 2.67827i 0.126348 + 0.173903i 4.82146 1.56659i −0.942426 0.306213i −2.42705 1.76336i 0.565488i
40.5 0.0729469 0.100403i 0.535233 1.64728i 1.23131 + 3.78958i −3.68632 + 2.67827i −0.126348 0.173903i −4.82146 + 1.56659i 0.942426 + 0.306213i −2.42705 1.76336i 0.565488i
40.6 1.22082 1.68031i −0.535233 + 1.64728i −0.0969856 0.298491i 1.35053 0.981217i 2.11452 + 2.91039i −2.87537 + 0.934266i 7.28135 + 2.36585i −2.42705 1.76336i 3.46720i
40.7 1.82934 2.51787i −0.535233 + 1.64728i −1.75712 5.40785i −4.58660 + 3.33236i 3.16851 + 4.36108i 9.13303 2.96750i −4.99088 1.62164i −2.42705 1.76336i 17.6445i
40.8 2.19808 3.02540i 0.535233 1.64728i −3.08541 9.49592i 0.450251 0.327126i −3.80719 5.24014i 11.4499 3.72029i −21.2846 6.91580i −2.42705 1.76336i 2.08124i
94.1 −3.55657 + 1.15560i −1.40126 1.01807i 8.07771 5.86880i −0.171980 + 0.529301i 6.16016 + 2.00156i 7.07642 + 9.73985i −13.1546 + 18.1058i 0.927051 + 2.85317i 2.08124i
94.2 −2.95993 + 0.961741i 1.40126 + 1.01807i 4.60019 3.34223i 1.75192 5.39187i −5.12675 1.66578i 5.64452 + 7.76902i −3.08454 + 4.24550i 0.927051 + 2.85317i 17.6445i
94.3 −1.97533 + 0.641823i 1.40126 + 1.01807i 0.253912 0.184478i −0.515856 + 1.58764i −3.42137 1.11167i −1.77708 2.44594i 4.50012 6.19388i 0.927051 + 2.85317i 3.46720i
94.4 −0.118031 + 0.0383504i −1.40126 1.01807i −3.22361 + 2.34209i 1.40805 4.33353i 0.204435 + 0.0664249i −2.97983 4.10138i 0.582452 0.801676i 0.927051 + 2.85317i 0.565488i
94.5 0.118031 0.0383504i −1.40126 1.01807i −3.22361 + 2.34209i 1.40805 4.33353i −0.204435 0.0664249i 2.97983 + 4.10138i −0.582452 + 0.801676i 0.927051 + 2.85317i 0.565488i
94.6 1.97533 0.641823i 1.40126 + 1.01807i 0.253912 0.184478i −0.515856 + 1.58764i 3.42137 + 1.11167i 1.77708 + 2.44594i −4.50012 + 6.19388i 0.927051 + 2.85317i 3.46720i
94.7 2.95993 0.961741i 1.40126 + 1.01807i 4.60019 3.34223i 1.75192 5.39187i 5.12675 + 1.66578i −5.64452 7.76902i 3.08454 4.24550i 0.927051 + 2.85317i 17.6445i
94.8 3.55657 1.15560i −1.40126 1.01807i 8.07771 5.86880i −0.171980 + 0.529301i −6.16016 2.00156i −7.07642 9.73985i 13.1546 18.1058i 0.927051 + 2.85317i 2.08124i
112.1 −3.55657 1.15560i −1.40126 + 1.01807i 8.07771 + 5.86880i −0.171980 0.529301i 6.16016 2.00156i 7.07642 9.73985i −13.1546 18.1058i 0.927051 2.85317i 2.08124i
112.2 −2.95993 0.961741i 1.40126 1.01807i 4.60019 + 3.34223i 1.75192 + 5.39187i −5.12675 + 1.66578i 5.64452 7.76902i −3.08454 4.24550i 0.927051 2.85317i 17.6445i
112.3 −1.97533 0.641823i 1.40126 1.01807i 0.253912 + 0.184478i −0.515856 1.58764i −3.42137 + 1.11167i −1.77708 + 2.44594i 4.50012 + 6.19388i 0.927051 2.85317i 3.46720i
112.4 −0.118031 0.0383504i −1.40126 + 1.01807i −3.22361 2.34209i 1.40805 + 4.33353i 0.204435 0.0664249i −2.97983 + 4.10138i 0.582452 + 0.801676i 0.927051 2.85317i 0.565488i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.g.h 32
11.b odd 2 1 inner 363.3.g.h 32
11.c even 5 1 363.3.c.d 8
11.c even 5 3 inner 363.3.g.h 32
11.d odd 10 1 363.3.c.d 8
11.d odd 10 3 inner 363.3.g.h 32
33.f even 10 1 1089.3.c.j 8
33.h odd 10 1 1089.3.c.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.c.d 8 11.c even 5 1
363.3.c.d 8 11.d odd 10 1
363.3.g.h 32 1.a even 1 1 trivial
363.3.g.h 32 11.b odd 2 1 inner
363.3.g.h 32 11.c even 5 3 inner
363.3.g.h 32 11.d odd 10 3 inner
1089.3.c.j 8 33.f even 10 1
1089.3.c.j 8 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 28 T_{2}^{30} + 546 T_{2}^{28} - 9212 T_{2}^{26} + 144443 T_{2}^{24} - 1551116 T_{2}^{22} + 14465556 T_{2}^{20} - 120719536 T_{2}^{18} + 848100901 T_{2}^{16} - 3507362544 T_{2}^{14} + \cdots + 6561 \) acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display