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Properties

Label 351.3.n.a
Level $351$
Weight $3$
Character orbit 351.n
Analytic conductor $9.564$
Analytic rank $0$
Dimension $52$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [351,3,Mod(116,351)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(351, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("351.116");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 351.n (of order \(6\), degree \(2\), not 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: \(9.56405727905\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 117)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 50 q^{4} + 8 q^{10} - 6 q^{13} + 6 q^{14} - 90 q^{16} + 14 q^{22} - 138 q^{23} - 92 q^{25} - 48 q^{29} - 324 q^{38} - 68 q^{40} + 62 q^{43} + 70 q^{49} - 4 q^{52} + 92 q^{55} + 276 q^{56} + 12 q^{61}+ \cdots - 504 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1 −1.92341 + 3.33144i 0 −5.39901 − 9.35136i 0.506509 + 0.877299i 0 3.89680 + 2.24982i 26.1508 0 −3.89690
116.2 −1.77860 + 3.08062i 0 −4.32683 − 7.49428i −4.26663 − 7.39002i 0 −7.99720 − 4.61719i 16.5540 0 30.3545
116.3 −1.68517 + 2.91881i 0 −3.67962 − 6.37330i 3.61500 + 6.26136i 0 2.20723 + 1.27434i 11.3218 0 −24.3676
116.4 −1.55290 + 2.68970i 0 −2.82298 − 4.88955i 0.611466 + 1.05909i 0 8.52943 + 4.92447i 5.11202 0 −3.79817
116.5 −1.38552 + 2.39980i 0 −1.83936 − 3.18586i −2.12220 − 3.67577i 0 4.71279 + 2.72093i −0.890291 0 11.7615
116.6 −1.33789 + 2.31729i 0 −1.57990 − 2.73646i 0.194524 + 0.336926i 0 −8.30078 − 4.79246i −2.24820 0 −1.04101
116.7 −1.11386 + 1.92926i 0 −0.481371 − 0.833760i 4.12144 + 7.13855i 0 −8.40220 − 4.85102i −6.76616 0 −18.3629
116.8 −1.04133 + 1.80365i 0 −0.168757 − 0.292296i −2.59043 − 4.48676i 0 1.68126 + 0.970675i −7.62775 0 10.7900
116.9 −0.647986 + 1.12234i 0 1.16023 + 2.00957i 2.25417 + 3.90433i 0 −0.162124 − 0.0936021i −8.19114 0 −5.84268
116.10 −0.641309 + 1.11078i 0 1.17745 + 2.03940i −0.763734 − 1.32283i 0 −5.50560 − 3.17866i −8.15090 0 1.95916
116.11 −0.433498 + 0.750840i 0 1.62416 + 2.81313i −4.23652 − 7.33786i 0 3.54375 + 2.04598i −6.28426 0 7.34608
116.12 −0.245962 + 0.426018i 0 1.87901 + 3.25453i 1.35391 + 2.34503i 0 8.51494 + 4.91611i −3.81635 0 −1.33204
116.13 −0.146647 + 0.254001i 0 1.95699 + 3.38960i 2.67644 + 4.63573i 0 8.20362 + 4.73636i −2.32113 0 −1.56997
116.14 0.146647 − 0.254001i 0 1.95699 + 3.38960i −2.67644 − 4.63573i 0 −8.20362 − 4.73636i 2.32113 0 −1.56997
116.15 0.245962 − 0.426018i 0 1.87901 + 3.25453i −1.35391 − 2.34503i 0 −8.51494 − 4.91611i 3.81635 0 −1.33204
116.16 0.433498 − 0.750840i 0 1.62416 + 2.81313i 4.23652 + 7.33786i 0 −3.54375 − 2.04598i 6.28426 0 7.34608
116.17 0.641309 − 1.11078i 0 1.17745 + 2.03940i 0.763734 + 1.32283i 0 5.50560 + 3.17866i 8.15090 0 1.95916
116.18 0.647986 − 1.12234i 0 1.16023 + 2.00957i −2.25417 − 3.90433i 0 0.162124 + 0.0936021i 8.19114 0 −5.84268
116.19 1.04133 − 1.80365i 0 −0.168757 − 0.292296i 2.59043 + 4.48676i 0 −1.68126 − 0.970675i 7.62775 0 10.7900
116.20 1.11386 − 1.92926i 0 −0.481371 − 0.833760i −4.12144 − 7.13855i 0 8.40220 + 4.85102i 6.76616 0 −18.3629
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
13.b even 2 1 inner
117.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 351.3.n.a 52
3.b odd 2 1 117.3.n.a ✓ 52
9.c even 3 1 117.3.n.a ✓ 52
9.d odd 6 1 inner 351.3.n.a 52
13.b even 2 1 inner 351.3.n.a 52
39.d odd 2 1 117.3.n.a ✓ 52
117.n odd 6 1 inner 351.3.n.a 52
117.t even 6 1 117.3.n.a ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.3.n.a ✓ 52 3.b odd 2 1
117.3.n.a ✓ 52 9.c even 3 1
117.3.n.a ✓ 52 39.d odd 2 1
117.3.n.a ✓ 52 117.t even 6 1
351.3.n.a 52 1.a even 1 1 trivial
351.3.n.a 52 9.d odd 6 1 inner
351.3.n.a 52 13.b even 2 1 inner
351.3.n.a 52 117.n odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(351, [\chi])\).