Properties

Label 261.3.p.a
Level $261$
Weight $3$
Character orbit 261.p
Analytic conductor $7.112$
Analytic rank $0$
Dimension $120$
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] = [261,3,Mod(53,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 261.p (of order \(14\), degree \(6\), 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: \(7.11173489980\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 40 q^{4} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 40 q^{4} - 16 q^{7} + 24 q^{10} - 4 q^{13} - 72 q^{16} - 40 q^{19} - 92 q^{22} + 244 q^{25} + 544 q^{28} + 56 q^{31} - 144 q^{34} - 88 q^{40} + 244 q^{43} + 496 q^{46} + 420 q^{49} + 108 q^{52} - 224 q^{55} - 168 q^{58} + 72 q^{61} + 204 q^{64} - 472 q^{67} - 3232 q^{70} - 908 q^{73} - 916 q^{76} - 76 q^{79} + 352 q^{82} - 804 q^{85} + 864 q^{88} + 172 q^{91} - 188 q^{94} + 884 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −3.01719 + 2.40613i 0 2.42390 10.6198i −3.61293 + 2.88122i 0 0.446262 + 1.95520i 11.5416 + 23.9664i 0 3.96833 17.3864i
53.2 −2.87082 + 2.28940i 0 2.11016 9.24521i −0.550556 + 0.439054i 0 0.216354 + 0.947909i 8.73539 + 18.1392i 0 0.575376 2.52089i
53.3 −2.31293 + 1.84450i 0 1.05738 4.63269i 2.04793 1.63317i 0 −2.50930 10.9939i 0.965025 + 2.00390i 0 −1.72433 + 7.55480i
53.4 −2.04594 + 1.63158i 0 0.633728 2.77654i 5.84586 4.66192i 0 1.69806 + 7.43967i −1.30805 2.71620i 0 −4.35398 + 19.0760i
53.5 −1.77322 + 1.41409i 0 0.254554 1.11527i −7.00899 + 5.58948i 0 −1.51727 6.64761i −2.81052 5.83611i 0 4.52440 19.8227i
53.6 −1.75160 + 1.39686i 0 0.226820 0.993762i 2.77979 2.21681i 0 1.32518 + 5.80597i −2.89742 6.01655i 0 −1.77253 + 7.76594i
53.7 −1.58966 + 1.26771i 0 0.0298453 0.130761i −5.19041 + 4.13921i 0 1.56263 + 6.84632i −3.41046 7.08190i 0 3.00366 13.1599i
53.8 −0.865482 + 0.690199i 0 −0.617399 + 2.70500i 1.36029 1.08479i 0 −1.33999 5.87088i −3.25387 6.75673i 0 −0.428582 + 1.87774i
53.9 −0.504871 + 0.402621i 0 −0.797293 + 3.49317i 4.97416 3.96676i 0 −2.49371 10.9257i −2.12462 4.41182i 0 −0.914207 + 4.00540i
53.10 −0.227275 + 0.181246i 0 −0.871280 + 3.81733i −1.73603 + 1.38444i 0 1.50188 + 6.58016i −0.998367 2.07313i 0 0.143633 0.629297i
53.11 0.227275 0.181246i 0 −0.871280 + 3.81733i 1.73603 1.38444i 0 1.50188 + 6.58016i 0.998367 + 2.07313i 0 0.143633 0.629297i
53.12 0.504871 0.402621i 0 −0.797293 + 3.49317i −4.97416 + 3.96676i 0 −2.49371 10.9257i 2.12462 + 4.41182i 0 −0.914207 + 4.00540i
53.13 0.865482 0.690199i 0 −0.617399 + 2.70500i −1.36029 + 1.08479i 0 −1.33999 5.87088i 3.25387 + 6.75673i 0 −0.428582 + 1.87774i
53.14 1.58966 1.26771i 0 0.0298453 0.130761i 5.19041 4.13921i 0 1.56263 + 6.84632i 3.41046 + 7.08190i 0 3.00366 13.1599i
53.15 1.75160 1.39686i 0 0.226820 0.993762i −2.77979 + 2.21681i 0 1.32518 + 5.80597i 2.89742 + 6.01655i 0 −1.77253 + 7.76594i
53.16 1.77322 1.41409i 0 0.254554 1.11527i 7.00899 5.58948i 0 −1.51727 6.64761i 2.81052 + 5.83611i 0 4.52440 19.8227i
53.17 2.04594 1.63158i 0 0.633728 2.77654i −5.84586 + 4.66192i 0 1.69806 + 7.43967i 1.30805 + 2.71620i 0 −4.35398 + 19.0760i
53.18 2.31293 1.84450i 0 1.05738 4.63269i −2.04793 + 1.63317i 0 −2.50930 10.9939i −0.965025 2.00390i 0 −1.72433 + 7.55480i
53.19 2.87082 2.28940i 0 2.11016 9.24521i 0.550556 0.439054i 0 0.216354 + 0.947909i −8.73539 18.1392i 0 0.575376 2.52089i
53.20 3.01719 2.40613i 0 2.42390 10.6198i 3.61293 2.88122i 0 0.446262 + 1.95520i −11.5416 23.9664i 0 3.96833 17.3864i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.d even 7 1 inner
87.j odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.3.p.a 120
3.b odd 2 1 inner 261.3.p.a 120
29.d even 7 1 inner 261.3.p.a 120
87.j odd 14 1 inner 261.3.p.a 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.3.p.a 120 1.a even 1 1 trivial
261.3.p.a 120 3.b odd 2 1 inner
261.3.p.a 120 29.d even 7 1 inner
261.3.p.a 120 87.j odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(261, [\chi])\).