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Properties

Label 462.2.w.a
Level $462$
Weight $2$
Character orbit 462.w
Analytic conductor $3.689$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [462,2,Mod(29,462)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(462, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.w (of order \(10\), degree \(4\), 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: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{2} - 4 q^{3} - 12 q^{4} + 6 q^{6} - 12 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{2} - 4 q^{3} - 12 q^{4} + 6 q^{6} - 12 q^{8} + 10 q^{9} - 8 q^{11} + 6 q^{12} + 36 q^{15} - 12 q^{16} - 24 q^{17} + 30 q^{19} - 8 q^{22} - 4 q^{24} + 18 q^{25} - 20 q^{26} - 22 q^{27} + 8 q^{29} - 24 q^{30} - 32 q^{31} + 48 q^{32} - 14 q^{33} - 4 q^{34} + 6 q^{35} - 10 q^{36} - 20 q^{37} + 20 q^{38} - 6 q^{39} + 22 q^{44} + 12 q^{45} + 20 q^{46} + 20 q^{47} - 4 q^{48} + 12 q^{49} + 28 q^{50} + 8 q^{51} + 20 q^{52} + 20 q^{53} + 18 q^{54} + 16 q^{55} - 2 q^{57} + 8 q^{58} + 30 q^{59} - 4 q^{60} - 20 q^{61} + 8 q^{62} + 4 q^{63} - 12 q^{64} - 14 q^{66} + 36 q^{67} - 24 q^{68} - 70 q^{69} - 4 q^{70} + 10 q^{72} - 20 q^{73} - 20 q^{74} - 30 q^{75} - 16 q^{77} - 16 q^{78} - 20 q^{79} + 26 q^{81} - 10 q^{82} - 46 q^{83} - 10 q^{84} + 10 q^{85} - 30 q^{86} + 8 q^{87} - 8 q^{88} - 38 q^{90} - 36 q^{91} + 10 q^{92} - 36 q^{93} + 50 q^{95} - 4 q^{96} - 2 q^{97} - 48 q^{98} + 70 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −0.809017 − 0.587785i −1.67853 − 0.427250i 0.309017 + 0.951057i 0.354145 + 0.487439i 1.10683 + 1.33227i −0.951057 + 0.309017i 0.309017 − 0.951057i 2.63492 + 1.43430i − 0.602508i
29.2 −0.809017 − 0.587785i −1.63583 − 0.569257i 0.309017 + 0.951057i −0.732542 − 1.00826i 0.988815 + 1.42206i 0.951057 − 0.309017i 0.309017 − 0.951057i 2.35189 + 1.86242i 1.24628i
29.3 −0.809017 − 0.587785i −1.48497 + 0.891547i 0.309017 + 0.951057i 0.456676 + 0.628560i 1.72541 + 0.151568i 0.951057 − 0.309017i 0.309017 − 0.951057i 1.41029 − 2.64785i − 0.776943i
29.4 −0.809017 − 0.587785i −1.31903 + 1.12257i 0.309017 + 0.951057i −1.88925 − 2.60033i 1.72695 − 0.132877i −0.951057 + 0.309017i 0.309017 − 0.951057i 0.479659 − 2.96141i 3.21418i
29.5 −0.809017 − 0.587785i −0.911344 − 1.47291i 0.309017 + 0.951057i 2.04373 + 2.81296i −0.128460 + 1.72728i 0.951057 − 0.309017i 0.309017 − 0.951057i −1.33891 + 2.68465i − 3.47700i
29.6 −0.809017 − 0.587785i −0.520106 − 1.65212i 0.309017 + 0.951057i −1.54008 − 2.11973i −0.550316 + 1.64230i −0.951057 + 0.309017i 0.309017 − 0.951057i −2.45898 + 1.71855i 2.62013i
29.7 −0.809017 − 0.587785i 0.261877 − 1.71214i 0.309017 + 0.951057i 1.43689 + 1.97771i −1.21823 + 1.23122i −0.951057 + 0.309017i 0.309017 − 0.951057i −2.86284 − 0.896741i − 2.44458i
29.8 −0.809017 − 0.587785i 0.455621 + 1.67105i 0.309017 + 0.951057i −1.96876 − 2.70976i 0.613613 − 1.61972i 0.951057 − 0.309017i 0.309017 − 0.951057i −2.58482 + 1.52273i 3.34945i
29.9 −0.809017 − 0.587785i 0.856700 + 1.50535i 0.309017 + 0.951057i 1.72358 + 2.37230i 0.191735 − 1.72141i 0.951057 − 0.309017i 0.309017 − 0.951057i −1.53213 + 2.57926i − 2.93233i
29.10 −0.809017 − 0.587785i 1.61613 + 0.622992i 0.309017 + 0.951057i −1.77599 − 2.44445i −0.941292 − 1.45395i −0.951057 + 0.309017i 0.309017 − 0.951057i 2.22376 + 2.01367i 3.02150i
29.11 −0.809017 − 0.587785i 1.63204 − 0.580034i 0.309017 + 0.951057i −0.571630 − 0.786781i −1.66128 − 0.490033i 0.951057 − 0.309017i 0.309017 − 0.951057i 2.32712 − 1.89328i 0.972514i
29.12 −0.809017 − 0.587785i 1.72744 − 0.126348i 0.309017 + 0.951057i 2.46323 + 3.39035i −1.47179 − 0.913144i −0.951057 + 0.309017i 0.309017 − 0.951057i 2.96807 − 0.436515i − 4.19070i
239.1 −0.809017 + 0.587785i −1.67853 + 0.427250i 0.309017 − 0.951057i 0.354145 − 0.487439i 1.10683 − 1.33227i −0.951057 − 0.309017i 0.309017 + 0.951057i 2.63492 − 1.43430i 0.602508i
239.2 −0.809017 + 0.587785i −1.63583 + 0.569257i 0.309017 − 0.951057i −0.732542 + 1.00826i 0.988815 − 1.42206i 0.951057 + 0.309017i 0.309017 + 0.951057i 2.35189 − 1.86242i − 1.24628i
239.3 −0.809017 + 0.587785i −1.48497 − 0.891547i 0.309017 − 0.951057i 0.456676 − 0.628560i 1.72541 − 0.151568i 0.951057 + 0.309017i 0.309017 + 0.951057i 1.41029 + 2.64785i 0.776943i
239.4 −0.809017 + 0.587785i −1.31903 − 1.12257i 0.309017 − 0.951057i −1.88925 + 2.60033i 1.72695 + 0.132877i −0.951057 − 0.309017i 0.309017 + 0.951057i 0.479659 + 2.96141i − 3.21418i
239.5 −0.809017 + 0.587785i −0.911344 + 1.47291i 0.309017 − 0.951057i 2.04373 − 2.81296i −0.128460 − 1.72728i 0.951057 + 0.309017i 0.309017 + 0.951057i −1.33891 − 2.68465i 3.47700i
239.6 −0.809017 + 0.587785i −0.520106 + 1.65212i 0.309017 − 0.951057i −1.54008 + 2.11973i −0.550316 − 1.64230i −0.951057 − 0.309017i 0.309017 + 0.951057i −2.45898 − 1.71855i − 2.62013i
239.7 −0.809017 + 0.587785i 0.261877 + 1.71214i 0.309017 − 0.951057i 1.43689 − 1.97771i −1.21823 − 1.23122i −0.951057 − 0.309017i 0.309017 + 0.951057i −2.86284 + 0.896741i 2.44458i
239.8 −0.809017 + 0.587785i 0.455621 − 1.67105i 0.309017 − 0.951057i −1.96876 + 2.70976i 0.613613 + 1.61972i 0.951057 + 0.309017i 0.309017 + 0.951057i −2.58482 − 1.52273i − 3.34945i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.w.a ✓ 48
3.b odd 2 1 462.2.w.b yes 48
11.d odd 10 1 462.2.w.b yes 48
33.f even 10 1 inner 462.2.w.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.w.a ✓ 48 1.a even 1 1 trivial
462.2.w.a ✓ 48 33.f even 10 1 inner
462.2.w.b yes 48 3.b odd 2 1
462.2.w.b yes 48 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} - 39 T_{5}^{46} + 20 T_{5}^{45} + 976 T_{5}^{44} - 780 T_{5}^{43} - 19614 T_{5}^{42} + \cdots + 2900399739136 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display