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Properties

Label 462.2.u.a
Level $462$
Weight $2$
Character orbit 462.u
Analytic conductor $3.689$
Analytic rank $0$
Dimension $32$
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(13,462)]
 
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([0, 5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.u (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: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{4} - 10 q^{5} + 8 q^{6} - 10 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{4} - 10 q^{5} + 8 q^{6} - 10 q^{7} + 8 q^{9} - 4 q^{10} + 8 q^{11} - 2 q^{14} + 6 q^{15} - 8 q^{16} + 12 q^{17} + 16 q^{19} - 10 q^{20} + 8 q^{21} - 4 q^{22} + 8 q^{23} - 8 q^{24} + 6 q^{25} + 20 q^{29} + 50 q^{31} + 16 q^{33} + 32 q^{35} - 8 q^{36} - 16 q^{37} - 6 q^{40} - 40 q^{41} - 10 q^{42} + 12 q^{44} - 28 q^{49} + 40 q^{51} + 32 q^{54} + 40 q^{55} - 8 q^{56} + 10 q^{58} - 60 q^{59} + 4 q^{60} + 4 q^{61} - 20 q^{62} - 10 q^{63} + 8 q^{64} - 8 q^{66} - 16 q^{67} - 12 q^{68} - 30 q^{69} - 18 q^{70} - 48 q^{71} + 74 q^{73} - 40 q^{74} + 24 q^{76} - 70 q^{77} - 60 q^{79} - 8 q^{81} - 20 q^{82} - 4 q^{83} + 2 q^{84} - 10 q^{85} - 36 q^{86} - 20 q^{87} - 16 q^{88} + 4 q^{90} - 60 q^{91} - 8 q^{92} - 10 q^{93} - 20 q^{95} + 8 q^{96} - 60 q^{97} + 40 q^{98} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −0.951057 − 0.309017i 0.587785 + 0.809017i 0.809017 + 0.587785i −3.76443 + 1.22314i −0.309017 − 0.951057i −1.44620 − 2.21552i −0.587785 − 0.809017i −0.309017 + 0.951057i 3.95815
13.2 −0.951057 − 0.309017i 0.587785 + 0.809017i 0.809017 + 0.587785i −0.0789999 + 0.0256686i −0.309017 − 0.951057i −1.18860 + 2.36373i −0.587785 − 0.809017i −0.309017 + 0.951057i 0.0830654
13.3 −0.951057 − 0.309017i 0.587785 + 0.809017i 0.809017 + 0.587785i 1.43844 − 0.467378i −0.309017 − 0.951057i 2.01686 + 1.71239i −0.587785 − 0.809017i −0.309017 + 0.951057i −1.51247
13.4 −0.951057 − 0.309017i 0.587785 + 0.809017i 0.809017 + 0.587785i 2.13481 − 0.693642i −0.309017 − 0.951057i 1.63277 − 2.08184i −0.587785 − 0.809017i −0.309017 + 0.951057i −2.24467
13.5 0.951057 + 0.309017i −0.587785 − 0.809017i 0.809017 + 0.587785i −3.18685 + 1.03547i −0.309017 − 0.951057i 2.59478 + 0.516820i 0.587785 + 0.809017i −0.309017 + 0.951057i −3.35085
13.6 0.951057 + 0.309017i −0.587785 − 0.809017i 0.809017 + 0.587785i −2.46351 + 0.800443i −0.309017 − 0.951057i −2.50308 + 0.857092i 0.587785 + 0.809017i −0.309017 + 0.951057i −2.59029
13.7 0.951057 + 0.309017i −0.587785 − 0.809017i 0.809017 + 0.587785i 0.169394 − 0.0550394i −0.309017 − 0.951057i −0.364877 − 2.62047i 0.587785 + 0.809017i −0.309017 + 0.951057i 0.178111
13.8 0.951057 + 0.309017i −0.587785 − 0.809017i 0.809017 + 0.587785i 2.13310 − 0.693087i −0.309017 − 0.951057i 0.112437 + 2.64336i 0.587785 + 0.809017i −0.309017 + 0.951057i 2.24288
139.1 −0.587785 + 0.809017i −0.951057 − 0.309017i −0.309017 − 0.951057i −2.23816 − 3.08056i 0.809017 − 0.587785i −1.26203 − 2.32536i 0.951057 + 0.309017i 0.809017 + 0.587785i 3.80779
139.2 −0.587785 + 0.809017i −0.951057 − 0.309017i −0.309017 − 0.951057i −0.884493 − 1.21740i 0.809017 − 0.587785i −1.60058 + 2.10669i 0.951057 + 0.309017i 0.809017 + 0.587785i 1.50479
139.3 −0.587785 + 0.809017i −0.951057 − 0.309017i −0.309017 − 0.951057i 0.946241 + 1.30239i 0.809017 − 0.587785i −1.50361 + 2.17696i 0.951057 + 0.309017i 0.809017 + 0.587785i −1.60984
139.4 −0.587785 + 0.809017i −0.951057 − 0.309017i −0.309017 − 0.951057i 1.12216 + 1.54452i 0.809017 − 0.587785i 0.488107 − 2.60034i 0.951057 + 0.309017i 0.809017 + 0.587785i −1.90913
139.5 0.587785 − 0.809017i 0.951057 + 0.309017i −0.309017 − 0.951057i −2.18694 − 3.01006i 0.809017 − 0.587785i −2.63889 + 0.190406i −0.951057 − 0.309017i 0.809017 + 0.587785i −3.72064
139.6 0.587785 − 0.809017i 0.951057 + 0.309017i −0.309017 − 0.951057i −0.531600 − 0.731684i 0.809017 − 0.587785i 0.261116 − 2.63283i −0.951057 − 0.309017i 0.809017 + 0.587785i −0.904411
139.7 0.587785 − 0.809017i 0.951057 + 0.309017i −0.309017 − 0.951057i 1.08816 + 1.49773i 0.809017 − 0.587785i −2.23342 + 1.41839i −0.951057 − 0.309017i 0.809017 + 0.587785i 1.85129
139.8 0.587785 − 0.809017i 0.951057 + 0.309017i −0.309017 − 0.951057i 1.30266 + 1.79296i 0.809017 − 0.587785i 2.63520 − 0.236035i −0.951057 − 0.309017i 0.809017 + 0.587785i 2.21622
349.1 −0.587785 − 0.809017i −0.951057 + 0.309017i −0.309017 + 0.951057i −2.23816 + 3.08056i 0.809017 + 0.587785i −1.26203 + 2.32536i 0.951057 − 0.309017i 0.809017 − 0.587785i 3.80779
349.2 −0.587785 − 0.809017i −0.951057 + 0.309017i −0.309017 + 0.951057i −0.884493 + 1.21740i 0.809017 + 0.587785i −1.60058 − 2.10669i 0.951057 − 0.309017i 0.809017 − 0.587785i 1.50479
349.3 −0.587785 − 0.809017i −0.951057 + 0.309017i −0.309017 + 0.951057i 0.946241 − 1.30239i 0.809017 + 0.587785i −1.50361 − 2.17696i 0.951057 − 0.309017i 0.809017 − 0.587785i −1.60984
349.4 −0.587785 − 0.809017i −0.951057 + 0.309017i −0.309017 + 0.951057i 1.12216 − 1.54452i 0.809017 + 0.587785i 0.488107 + 2.60034i 0.951057 − 0.309017i 0.809017 − 0.587785i −1.90913
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.l 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.u.a ✓ 32
7.b odd 2 1 462.2.u.b yes 32
11.d odd 10 1 462.2.u.b yes 32
77.l even 10 1 inner 462.2.u.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.u.a ✓ 32 1.a even 1 1 trivial
462.2.u.a ✓ 32 77.l even 10 1 inner
462.2.u.b yes 32 7.b odd 2 1
462.2.u.b yes 32 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} + 10 T_{5}^{31} + 27 T_{5}^{30} - 80 T_{5}^{29} - 521 T_{5}^{28} + 340 T_{5}^{27} + \cdots + 885481 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display