Properties

Label 464.2.y.e
Level $464$
Weight $2$
Character orbit 464.y
Analytic conductor $3.705$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [464,2,Mod(33,464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(464, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("464.33");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 464.y (of order \(14\), degree \(6\), 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: \(3.70505865379\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{14})\)
Twist minimal: no (minimal twist has level 232)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{5} + 4 q^{7} + 6 q^{9} + 10 q^{13} - 14 q^{15} + 14 q^{21} - 4 q^{23} - 48 q^{25} - 4 q^{29} + 10 q^{33} - 8 q^{35} - 38 q^{45} + 14 q^{47} - 18 q^{49} + 56 q^{51} - 48 q^{53} + 28 q^{55} - 12 q^{57} + 128 q^{59} - 28 q^{61} - 42 q^{63} - 28 q^{65} + 4 q^{67} + 28 q^{69} + 14 q^{71} - 28 q^{73} + 14 q^{77} - 32 q^{81} - 80 q^{83} + 112 q^{87} + 42 q^{89} + 28 q^{91} + 6 q^{93} + 70 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1 0 −1.42415 2.95728i 0 −0.136467 + 0.597901i 0 −0.294470 + 0.141809i 0 −4.84685 + 6.07775i 0
33.2 0 −0.857903 1.78145i 0 −0.419644 + 1.83858i 0 2.09267 1.00778i 0 −0.567114 + 0.711138i 0
33.3 0 −0.846741 1.75828i 0 0.850744 3.72735i 0 −2.85493 + 1.37486i 0 −0.504093 + 0.632112i 0
33.4 0 −0.201238 0.417875i 0 −0.296646 + 1.29969i 0 −2.78319 + 1.34032i 0 1.73635 2.17731i 0
33.5 0 0.186194 + 0.386637i 0 0.478397 2.09600i 0 1.33207 0.641493i 0 1.75565 2.20152i 0
33.6 0 0.661075 + 1.37274i 0 −0.510076 + 2.23479i 0 −3.30380 + 1.59102i 0 0.423086 0.530533i 0
33.7 0 1.19841 + 2.48852i 0 −0.882074 + 3.86462i 0 3.53526 1.70249i 0 −2.88609 + 3.61904i 0
33.8 0 1.28436 + 2.66699i 0 0.805849 3.53066i 0 1.47444 0.710053i 0 −3.59282 + 4.50525i 0
129.1 0 −2.26073 1.80288i 0 −0.608430 0.293004i 0 2.74476 3.44182i 0 1.19300 + 5.22686i 0
129.2 0 −1.96464 1.56675i 0 1.24505 + 0.599586i 0 −1.40365 + 1.76012i 0 0.737555 + 3.23144i 0
129.3 0 −0.884679 0.705508i 0 0.449269 + 0.216356i 0 −2.01029 + 2.52083i 0 −0.382647 1.67649i 0
129.4 0 −0.0131000 0.0104469i 0 3.41718 + 1.64563i 0 3.15424 3.95529i 0 −0.667500 2.92451i 0
129.5 0 0.212201 + 0.169225i 0 −1.28207 0.617411i 0 −1.27595 + 1.59999i 0 −0.651170 2.85296i 0
129.6 0 0.728009 + 0.580567i 0 −2.19459 1.05686i 0 0.988136 1.23908i 0 −0.474625 2.07947i 0
129.7 0 1.79043 + 1.42782i 0 2.68076 + 1.29099i 0 −0.839902 + 1.05320i 0 0.499400 + 2.18801i 0
129.8 0 2.39252 + 1.90797i 0 −1.10330 0.531322i 0 0.889636 1.11557i 0 1.41624 + 6.20496i 0
209.1 0 −2.64359 0.603381i 0 −2.76955 + 3.47291i 0 −0.416896 + 1.82654i 0 3.92157 + 1.88853i 0
209.2 0 −2.35308 0.537075i 0 1.76863 2.21779i 0 0.833292 3.65089i 0 2.54562 + 1.22591i 0
209.3 0 −2.16441 0.494012i 0 0.486884 0.610533i 0 −0.478099 + 2.09469i 0 1.73771 + 0.836835i 0
209.4 0 −0.0690613 0.0157628i 0 −2.25017 + 2.82162i 0 0.686126 3.00612i 0 −2.69839 1.29947i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 464.2.y.e 48
4.b odd 2 1 232.2.q.a 48
29.e even 14 1 inner 464.2.y.e 48
116.h odd 14 1 232.2.q.a 48
116.l even 28 1 6728.2.a.be 24
116.l even 28 1 6728.2.a.bf 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.2.q.a 48 4.b odd 2 1
232.2.q.a 48 116.h odd 14 1
464.2.y.e 48 1.a even 1 1 trivial
464.2.y.e 48 29.e even 14 1 inner
6728.2.a.be 24 116.l even 28 1
6728.2.a.bf 24 116.l even 28 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 15 T_{3}^{46} + 170 T_{3}^{44} - 1954 T_{3}^{42} + 22159 T_{3}^{40} - 1120 T_{3}^{39} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(464, [\chi])\). Copy content Toggle raw display