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Properties

Label 357.2.p.a
Level $357$
Weight $2$
Character orbit 357.p
Analytic conductor $2.851$
Analytic rank $0$
Dimension $48$
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] = [357,2,Mod(16,357)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(357, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("357.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 357 = 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 357.p (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: \(2.85065935216\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 24 q^{4} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 24 q^{4} + 24 q^{9} + 16 q^{13} - 40 q^{16} + 8 q^{17} + 16 q^{19} + 36 q^{25} - 52 q^{26} + 24 q^{30} + 20 q^{32} - 8 q^{33} - 32 q^{34} - 36 q^{35} - 48 q^{36} + 40 q^{38} + 4 q^{42} + 40 q^{43} - 80 q^{50} - 12 q^{52} - 60 q^{53} - 24 q^{59} + 112 q^{64} + 4 q^{66} - 4 q^{67} + 88 q^{68} + 64 q^{69} - 24 q^{70} - 184 q^{76} - 72 q^{77} - 24 q^{81} - 88 q^{83} - 20 q^{84} + 16 q^{85} + 60 q^{86} + 36 q^{87} + 12 q^{89} - 24 q^{93} - 32 q^{94} + 48 q^{98}+O(q^{100}) \) 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 −1.34373 + 2.32741i −0.866025 + 0.500000i −2.61122 − 4.52277i 2.98396 + 1.72279i − 2.68746i −0.462813 − 2.60496i 8.66019 0.500000 − 0.866025i −8.01926 + 4.62992i
16.2 −1.34373 + 2.32741i 0.866025 − 0.500000i −2.61122 − 4.52277i −2.98396 − 1.72279i 2.68746i 0.462813 + 2.60496i 8.66019 0.500000 − 0.866025i 8.01926 − 4.62992i
16.3 −1.11091 + 1.92416i −0.866025 + 0.500000i −1.46826 − 2.54309i 0.551608 + 0.318471i − 2.22183i 1.27391 + 2.31887i 2.08076 0.500000 − 0.866025i −1.22558 + 0.707587i
16.4 −1.11091 + 1.92416i 0.866025 − 0.500000i −1.46826 − 2.54309i −0.551608 − 0.318471i 2.22183i −1.27391 − 2.31887i 2.08076 0.500000 − 0.866025i 1.22558 − 0.707587i
16.5 −0.955129 + 1.65433i −0.866025 + 0.500000i −0.824544 − 1.42815i −1.82126 − 1.05151i − 1.91026i −2.40522 + 1.10223i −0.670332 0.500000 − 0.866025i 3.47908 − 2.00865i
16.6 −0.955129 + 1.65433i 0.866025 − 0.500000i −0.824544 − 1.42815i 1.82126 + 1.05151i 1.91026i 2.40522 − 1.10223i −0.670332 0.500000 − 0.866025i −3.47908 + 2.00865i
16.7 −0.636934 + 1.10320i −0.866025 + 0.500000i 0.188631 + 0.326718i −2.74862 − 1.58692i − 1.27387i 1.56475 − 2.13344i −3.02832 0.500000 − 0.866025i 3.50138 − 2.02152i
16.8 −0.636934 + 1.10320i 0.866025 − 0.500000i 0.188631 + 0.326718i 2.74862 + 1.58692i 1.27387i −1.56475 + 2.13344i −3.02832 0.500000 − 0.866025i −3.50138 + 2.02152i
16.9 −0.427664 + 0.740736i −0.866025 + 0.500000i 0.634207 + 1.09848i 3.01361 + 1.73991i − 0.855328i 2.30724 + 1.29485i −2.79557 0.500000 − 0.866025i −2.57762 + 1.48819i
16.10 −0.427664 + 0.740736i 0.866025 − 0.500000i 0.634207 + 1.09848i −3.01361 − 1.73991i 0.855328i −2.30724 − 1.29485i −2.79557 0.500000 − 0.866025i 2.57762 − 1.48819i
16.11 0.0460015 − 0.0796769i −0.866025 + 0.500000i 0.995768 + 1.72472i −2.24725 − 1.29745i 0.0920029i −0.592654 + 2.57852i 0.367233 0.500000 − 0.866025i −0.206754 + 0.119369i
16.12 0.0460015 − 0.0796769i 0.866025 − 0.500000i 0.995768 + 1.72472i 2.24725 + 1.29745i − 0.0920029i 0.592654 − 2.57852i 0.367233 0.500000 − 0.866025i 0.206754 − 0.119369i
16.13 0.113439 − 0.196482i −0.866025 + 0.500000i 0.974263 + 1.68747i 0.103638 + 0.0598352i 0.226878i 2.28748 − 1.32945i 0.895833 0.500000 − 0.866025i 0.0235130 − 0.0135753i
16.14 0.113439 − 0.196482i 0.866025 − 0.500000i 0.974263 + 1.68747i −0.103638 − 0.0598352i − 0.226878i −2.28748 + 1.32945i 0.895833 0.500000 − 0.866025i −0.0235130 + 0.0135753i
16.15 0.262370 − 0.454438i −0.866025 + 0.500000i 0.862324 + 1.49359i 3.41917 + 1.97406i 0.524739i −2.63978 + 0.177717i 1.95447 0.500000 − 0.866025i 1.79417 − 1.03587i
16.16 0.262370 − 0.454438i 0.866025 − 0.500000i 0.862324 + 1.49359i −3.41917 − 1.97406i − 0.524739i 2.63978 − 0.177717i 1.95447 0.500000 − 0.866025i −1.79417 + 1.03587i
16.17 0.667155 − 1.15555i −0.866025 + 0.500000i 0.109807 + 0.190192i −0.786313 − 0.453978i 1.33431i −1.35936 − 2.26983i 2.96166 0.500000 − 0.866025i −1.04919 + 0.605748i
16.18 0.667155 − 1.15555i 0.866025 − 0.500000i 0.109807 + 0.190192i 0.786313 + 0.453978i − 1.33431i 1.35936 + 2.26983i 2.96166 0.500000 − 0.866025i 1.04919 − 0.605748i
16.19 0.909060 − 1.57454i −0.866025 + 0.500000i −0.652781 − 1.13065i 1.17508 + 0.678431i 1.81812i 0.327440 + 2.62541i 1.26257 0.500000 − 0.866025i 2.13643 − 1.23347i
16.20 0.909060 − 1.57454i 0.866025 − 0.500000i −0.652781 − 1.13065i −1.17508 − 0.678431i − 1.81812i −0.327440 − 2.62541i 1.26257 0.500000 − 0.866025i −2.13643 + 1.23347i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
17.b even 2 1 inner
119.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 357.2.p.a ✓ 48
7.c even 3 1 inner 357.2.p.a ✓ 48
17.b even 2 1 inner 357.2.p.a ✓ 48
119.j even 6 1 inner 357.2.p.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
357.2.p.a ✓ 48 1.a even 1 1 trivial
357.2.p.a ✓ 48 7.c even 3 1 inner
357.2.p.a ✓ 48 17.b even 2 1 inner
357.2.p.a ✓ 48 119.j even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(357, [\chi])\).