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Properties

Label 363.4.d.e
Level $363$
Weight $4$
Character orbit 363.d
Analytic conductor $21.418$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,4,Mod(362,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.362");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.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: \(21.4176933321\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{3} + 160 q^{4} + 64 q^{9} - 92 q^{12} - 56 q^{15} + 904 q^{16} - 1576 q^{25} - 1120 q^{27} - 384 q^{31} + 2424 q^{34} + 2864 q^{36} + 960 q^{37} - 152 q^{42} - 64 q^{45} - 1892 q^{48} - 136 q^{49}+ \cdots + 14824 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
362.1 −5.30886 −5.19217 − 0.203515i 20.1840 − 14.8127i 27.5645 + 1.08043i − 9.97480i −64.6829 26.9172 + 2.11337i 78.6385i
362.2 −5.30886 −5.19217 + 0.203515i 20.1840 14.8127i 27.5645 − 1.08043i 9.97480i −64.6829 26.9172 − 2.11337i − 78.6385i
362.3 −5.22678 3.03140 + 4.22026i 19.3192 − 9.40366i −15.8445 − 22.0584i 24.5139i −59.1628 −8.62120 + 25.5866i 49.1508i
362.4 −5.22678 3.03140 − 4.22026i 19.3192 9.40366i −15.8445 + 22.0584i − 24.5139i −59.1628 −8.62120 − 25.5866i − 49.1508i
362.5 −4.71918 3.92886 + 3.40060i 14.2706 19.6157i −18.5410 − 16.0480i 7.51156i −29.5922 3.87181 + 26.7209i − 92.5699i
362.6 −4.71918 3.92886 − 3.40060i 14.2706 − 19.6157i −18.5410 + 16.0480i − 7.51156i −29.5922 3.87181 − 26.7209i 92.5699i
362.7 −4.00331 −4.84768 − 1.87084i 8.02649 7.39054i 19.4067 + 7.48955i 7.67817i −0.106061 19.9999 + 18.1385i − 29.5866i
362.8 −4.00331 −4.84768 + 1.87084i 8.02649 − 7.39054i 19.4067 − 7.48955i − 7.67817i −0.106061 19.9999 − 18.1385i 29.5866i
362.9 −3.13301 1.63897 + 4.93090i 1.81573 1.61815i −5.13491 − 15.4485i − 14.9583i 19.3754 −21.6275 + 16.1632i − 5.06967i
362.10 −3.13301 1.63897 − 4.93090i 1.81573 − 1.61815i −5.13491 + 15.4485i 14.9583i 19.3754 −21.6275 − 16.1632i 5.06967i
362.11 −2.65152 4.96436 − 1.53464i −0.969417 11.9200i −13.1631 + 4.06913i − 10.9714i 23.7826 22.2898 − 15.2370i − 31.6061i
362.12 −2.65152 4.96436 + 1.53464i −0.969417 − 11.9200i −13.1631 − 4.06913i 10.9714i 23.7826 22.2898 + 15.2370i 31.6061i
362.13 −1.91863 −2.77646 + 4.39218i −4.31885 − 10.4360i 5.32701 − 8.42699i − 27.3772i 23.6353 −11.5825 − 24.3894i 20.0229i
362.14 −1.91863 −2.77646 − 4.39218i −4.31885 10.4360i 5.32701 + 8.42699i 27.3772i 23.6353 −11.5825 + 24.3894i − 20.0229i
362.15 −1.71840 4.33004 − 2.87241i −5.04710 18.0808i −7.44074 + 4.93596i 28.2971i 22.4201 10.4985 − 24.8753i − 31.0701i
362.16 −1.71840 4.33004 + 2.87241i −5.04710 − 18.0808i −7.44074 − 4.93596i − 28.2971i 22.4201 10.4985 + 24.8753i 31.0701i
362.17 −1.40461 −3.70570 + 3.64250i −6.02706 4.83460i 5.20508 − 5.11630i 4.83009i 19.7026 0.464443 − 26.9960i − 6.79075i
362.18 −1.40461 −3.70570 − 3.64250i −6.02706 − 4.83460i 5.20508 + 5.11630i − 4.83009i 19.7026 0.464443 + 26.9960i 6.79075i
362.19 −0.863967 0.628371 + 5.15802i −7.25356 17.1130i −0.542892 − 4.45636i − 27.0279i 13.1786 −26.2103 + 6.48230i − 14.7851i
362.20 −0.863967 0.628371 − 5.15802i −7.25356 − 17.1130i −0.542892 + 4.45636i 27.0279i 13.1786 −26.2103 − 6.48230i 14.7851i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 362.40
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.4.d.e ✓ 40
3.b odd 2 1 inner 363.4.d.e ✓ 40
11.b odd 2 1 inner 363.4.d.e ✓ 40
33.d even 2 1 inner 363.4.d.e ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.d.e ✓ 40 1.a even 1 1 trivial
363.4.d.e ✓ 40 3.b odd 2 1 inner
363.4.d.e ✓ 40 11.b odd 2 1 inner
363.4.d.e ✓ 40 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 120 T_{2}^{18} + 5967 T_{2}^{16} - 159812 T_{2}^{14} + 2517195 T_{2}^{12} + \cdots + 303595776 \) acting on \(S_{4}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display