Properties

Label 369.3.g.a
Level $369$
Weight $3$
Character orbit 369.g
Analytic conductor $10.055$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [369,3,Mod(278,369)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(369, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("369.278");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 369 = 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 369.g (of order \(4\), 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: \(10.0545217549\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 112 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 112 q^{4} + 32 q^{10} - 24 q^{13} + 224 q^{16} - 40 q^{19} + 120 q^{22} + 344 q^{25} - 120 q^{28} + 8 q^{31} + 176 q^{34} + 168 q^{37} + 128 q^{40} + 32 q^{52} - 408 q^{55} + 8 q^{58} + 448 q^{64} + 40 q^{67} + 560 q^{70} - 496 q^{76} + 72 q^{79} - 736 q^{82} - 368 q^{85} + 8 q^{88} - 1256 q^{94} - 384 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
278.1 −3.87511 0 11.0165 −2.44085 0 −6.91222 + 6.91222i −27.1895 0 9.45854
278.2 −3.75560 0 10.1045 −1.37623 0 1.88410 1.88410i −22.9260 0 5.16856
278.3 −3.17750 0 6.09651 9.24351 0 −8.22970 + 8.22970i −6.66167 0 −29.3713
278.4 −3.11478 0 5.70186 2.46109 0 5.80271 5.80271i −5.30091 0 −7.66577
278.5 −3.02801 0 5.16885 −7.81511 0 7.82646 7.82646i −3.53928 0 23.6642
278.6 −2.76859 0 3.66508 −6.51777 0 0.248864 0.248864i 0.927255 0 18.0450
278.7 −2.38729 0 1.69916 4.01428 0 −3.72450 + 3.72450i 5.49277 0 −9.58325
278.8 −2.36522 0 1.59427 6.17585 0 8.22083 8.22083i 5.69007 0 −14.6073
278.9 −1.65305 0 −1.26743 −2.76858 0 −7.08063 + 7.08063i 8.70732 0 4.57659
278.10 −1.29986 0 −2.31036 −8.88333 0 −5.37438 + 5.37438i 8.20259 0 11.5471
278.11 −1.14055 0 −2.69914 −1.53165 0 0.674505 0.674505i 7.64072 0 1.74693
278.12 −0.955263 0 −3.08747 0.755334 0 3.41913 3.41913i 6.77040 0 −0.721543
278.13 −0.474550 0 −3.77480 3.29396 0 −2.12416 + 2.12416i 3.68953 0 −1.56315
278.14 −0.304166 0 −3.90748 8.85950 0 5.36900 5.36900i 2.40519 0 −2.69476
278.15 0.304166 0 −3.90748 −8.85950 0 5.36900 5.36900i −2.40519 0 −2.69476
278.16 0.474550 0 −3.77480 −3.29396 0 −2.12416 + 2.12416i −3.68953 0 −1.56315
278.17 0.955263 0 −3.08747 −0.755334 0 3.41913 3.41913i −6.77040 0 −0.721543
278.18 1.14055 0 −2.69914 1.53165 0 0.674505 0.674505i −7.64072 0 1.74693
278.19 1.29986 0 −2.31036 8.88333 0 −5.37438 + 5.37438i −8.20259 0 11.5471
278.20 1.65305 0 −1.26743 2.76858 0 −7.08063 + 7.08063i −8.70732 0 4.57659
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 278.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
41.c even 4 1 inner
123.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 369.3.g.a 56
3.b odd 2 1 inner 369.3.g.a 56
41.c even 4 1 inner 369.3.g.a 56
123.f odd 4 1 inner 369.3.g.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
369.3.g.a 56 1.a even 1 1 trivial
369.3.g.a 56 3.b odd 2 1 inner
369.3.g.a 56 41.c even 4 1 inner
369.3.g.a 56 123.f odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(369, [\chi])\).