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Properties

Label 369.2.u.c
Level $369$
Weight $2$
Character orbit 369.u
Analytic conductor $2.946$
Analytic rank $0$
Dimension $64$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [369,2,Mod(46,369)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(369, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("369.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 369 = 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 369.u (of order \(20\), degree \(8\), 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: \(2.94647983459\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 20 q^{4} - 32 q^{10} - 8 q^{13} - 28 q^{16} + 12 q^{19} - 12 q^{22} - 8 q^{25} + 80 q^{28} - 8 q^{34} - 72 q^{37} - 128 q^{40} + 20 q^{43} - 140 q^{46} - 40 q^{49} - 84 q^{52} + 40 q^{55} + 56 q^{58} - 40 q^{61} + 20 q^{64} + 8 q^{67} + 120 q^{70} - 56 q^{76} - 24 q^{79} + 216 q^{82} + 44 q^{85} + 72 q^{88} - 36 q^{94} + 128 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −1.61429 − 2.22187i 0 −1.71277 + 5.27136i −2.94681 − 0.957476i 0 −0.122786 − 0.775239i 9.25326 − 3.00657i 0 2.62960 + 8.09307i
46.2 −1.09881 − 1.51238i 0 −0.461878 + 1.42151i 0.800141 + 0.259982i 0 0.495810 + 3.13042i −0.898434 + 0.291919i 0 −0.486010 − 1.49579i
46.3 −0.815131 − 1.12193i 0 0.0237421 − 0.0730707i 0.0564606 + 0.0183452i 0 −0.662063 − 4.18010i −2.73915 + 0.890005i 0 −0.0254408 − 0.0782987i
46.4 −0.116858 − 0.160841i 0 0.605820 − 1.86452i −3.78572 − 1.23006i 0 0.289039 + 1.82492i −0.748846 + 0.243315i 0 0.244548 + 0.752641i
46.5 0.116858 + 0.160841i 0 0.605820 − 1.86452i 3.78572 + 1.23006i 0 0.289039 + 1.82492i 0.748846 − 0.243315i 0 0.244548 + 0.752641i
46.6 0.815131 + 1.12193i 0 0.0237421 − 0.0730707i −0.0564606 − 0.0183452i 0 −0.662063 − 4.18010i 2.73915 − 0.890005i 0 −0.0254408 − 0.0782987i
46.7 1.09881 + 1.51238i 0 −0.461878 + 1.42151i −0.800141 − 0.259982i 0 0.495810 + 3.13042i 0.898434 − 0.291919i 0 −0.486010 − 1.49579i
46.8 1.61429 + 2.22187i 0 −1.71277 + 5.27136i 2.94681 + 0.957476i 0 −0.122786 − 0.775239i −9.25326 + 3.00657i 0 2.62960 + 8.09307i
118.1 −1.47344 − 2.02802i 0 −1.32380 + 4.07424i 3.31238 + 1.07626i 0 −3.95176 + 0.625897i 5.44503 − 1.76920i 0 −2.69794 − 8.30339i
118.2 −1.27633 − 1.75671i 0 −0.838996 + 2.58216i −0.978587 − 0.317962i 0 4.45898 − 0.706233i 1.47667 − 0.479799i 0 0.690428 + 2.12492i
118.3 −0.801450 − 1.10310i 0 0.0435243 − 0.133954i 0.454690 + 0.147738i 0 0.725152 − 0.114853i −2.77619 + 0.902040i 0 −0.201441 − 0.619973i
118.4 −0.221409 − 0.304743i 0 0.574187 − 1.76717i −2.60139 − 0.845244i 0 −1.23237 + 0.195189i −1.38216 + 0.449090i 0 0.318390 + 0.979903i
118.5 0.221409 + 0.304743i 0 0.574187 − 1.76717i 2.60139 + 0.845244i 0 −1.23237 + 0.195189i 1.38216 − 0.449090i 0 0.318390 + 0.979903i
118.6 0.801450 + 1.10310i 0 0.0435243 − 0.133954i −0.454690 − 0.147738i 0 0.725152 − 0.114853i 2.77619 − 0.902040i 0 −0.201441 − 0.619973i
118.7 1.27633 + 1.75671i 0 −0.838996 + 2.58216i 0.978587 + 0.317962i 0 4.45898 − 0.706233i −1.47667 + 0.479799i 0 0.690428 + 2.12492i
118.8 1.47344 + 2.02802i 0 −1.32380 + 4.07424i −3.31238 − 1.07626i 0 −3.95176 + 0.625897i −5.44503 + 1.76920i 0 −2.69794 − 8.30339i
172.1 −1.47344 + 2.02802i 0 −1.32380 − 4.07424i 3.31238 − 1.07626i 0 −3.95176 − 0.625897i 5.44503 + 1.76920i 0 −2.69794 + 8.30339i
172.2 −1.27633 + 1.75671i 0 −0.838996 − 2.58216i −0.978587 + 0.317962i 0 4.45898 + 0.706233i 1.47667 + 0.479799i 0 0.690428 − 2.12492i
172.3 −0.801450 + 1.10310i 0 0.0435243 + 0.133954i 0.454690 − 0.147738i 0 0.725152 + 0.114853i −2.77619 − 0.902040i 0 −0.201441 + 0.619973i
172.4 −0.221409 + 0.304743i 0 0.574187 + 1.76717i −2.60139 + 0.845244i 0 −1.23237 − 0.195189i −1.38216 − 0.449090i 0 0.318390 − 0.979903i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
41.g even 20 1 inner
123.m odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 369.2.u.c ✓ 64
3.b odd 2 1 inner 369.2.u.c ✓ 64
41.g even 20 1 inner 369.2.u.c ✓ 64
123.m odd 20 1 inner 369.2.u.c ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
369.2.u.c ✓ 64 1.a even 1 1 trivial
369.2.u.c ✓ 64 3.b odd 2 1 inner
369.2.u.c ✓ 64 41.g even 20 1 inner
369.2.u.c ✓ 64 123.m odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 26 T_{2}^{62} + 407 T_{2}^{60} - 5000 T_{2}^{58} + 53357 T_{2}^{56} - 482536 T_{2}^{54} + \cdots + 2825761 \) acting on \(S_{2}^{\mathrm{new}}(369, [\chi])\). Copy content Toggle raw display