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Properties

Label 462.2.n.f
Level $462$
Weight $2$
Character orbit 462.n
Analytic conductor $3.689$
Analytic rank $0$
Dimension $24$
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] = [462,2,Mod(65,462)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(462, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.n (of order \(6\), degree \(2\), 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: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{2} - 12 q^{4} - 24 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{2} - 12 q^{4} - 24 q^{8} - 8 q^{9} + 4 q^{11} + 4 q^{15} - 12 q^{16} - 4 q^{17} + 8 q^{18} + 8 q^{22} + 20 q^{25} - 12 q^{27} - 40 q^{29} + 2 q^{30} - 28 q^{31} + 12 q^{32} + 24 q^{33} - 8 q^{34} + 32 q^{35} + 16 q^{36} - 16 q^{37} + 8 q^{41} + 6 q^{42} + 4 q^{44} - 28 q^{45} - 8 q^{49} + 40 q^{50} + 34 q^{51} - 6 q^{54} + 8 q^{55} - 40 q^{57} - 20 q^{58} - 2 q^{60} - 56 q^{62} + 14 q^{63} + 24 q^{64} + 44 q^{65} - 24 q^{66} - 4 q^{68} - 20 q^{69} + 64 q^{70} + 8 q^{72} + 16 q^{74} - 26 q^{75} + 24 q^{81} + 4 q^{82} - 144 q^{83} + 6 q^{84} - 8 q^{87} - 4 q^{88} - 56 q^{90} - 32 q^{91} - 22 q^{93} + 44 q^{95} + 64 q^{97} + 8 q^{98} + 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0.500000 − 0.866025i −1.72951 − 0.0938734i −0.500000 − 0.866025i −3.23623 − 1.86844i −0.946049 + 1.45086i 0.830341 − 2.51208i −1.00000 2.98238 + 0.324709i −3.23623 + 1.86844i
65.2 0.500000 − 0.866025i −1.64448 − 0.543759i −0.500000 − 0.866025i 1.63087 + 0.941585i −1.29315 + 1.15228i 1.76745 + 1.96879i −1.00000 2.40865 + 1.78841i 1.63087 − 0.941585i
65.3 0.500000 − 0.866025i −1.20836 − 1.24091i −0.500000 − 0.866025i −1.00577 − 0.580683i −1.67884 + 0.426020i −2.19957 + 1.47034i −1.00000 −0.0797091 + 2.99894i −1.00577 + 0.580683i
65.4 0.500000 − 0.866025i −0.993776 + 1.41859i −0.500000 − 0.866025i 0.246102 + 0.142087i 0.731651 + 1.56993i −2.09441 − 1.61662i −1.00000 −1.02482 − 2.81953i 0.246102 − 0.142087i
65.5 0.500000 − 0.866025i −0.470476 − 1.66693i −0.500000 − 0.866025i 1.00577 + 0.580683i −1.67884 − 0.426020i 2.19957 − 1.47034i −1.00000 −2.55730 + 1.56850i 1.00577 − 0.580683i
65.6 0.500000 − 0.866025i 0.125098 + 1.72753i −0.500000 − 0.866025i 3.72427 + 2.15021i 1.55863 + 0.755426i 2.62857 + 0.301041i −1.00000 −2.96870 + 0.432220i 3.72427 − 2.15021i
65.7 0.500000 − 0.866025i 0.301239 + 1.70565i −0.500000 − 0.866025i −1.38736 − 0.800992i 1.62776 + 0.591946i −0.229478 + 2.63578i −1.00000 −2.81851 + 1.02762i −1.38736 + 0.800992i
65.8 0.500000 − 0.866025i 0.351333 − 1.69604i −0.500000 − 0.866025i −1.63087 − 0.941585i −1.29315 − 1.15228i −1.76745 − 1.96879i −1.00000 −2.75313 − 1.19175i −1.63087 + 0.941585i
65.9 0.500000 − 0.866025i 0.783456 − 1.54473i −0.500000 − 0.866025i 3.23623 + 1.86844i −0.946049 − 1.45086i −0.830341 + 2.51208i −1.00000 −1.77239 − 2.42046i 3.23623 − 1.86844i
65.10 0.500000 − 0.866025i 1.32652 + 1.11371i −0.500000 − 0.866025i 1.38736 + 0.800992i 1.62776 − 0.591946i 0.229478 − 2.63578i −1.00000 0.519310 + 2.95471i 1.38736 − 0.800992i
65.11 0.500000 − 0.866025i 1.43353 + 0.972102i −0.500000 − 0.866025i −3.72427 − 2.15021i 1.55863 − 0.755426i −2.62857 − 0.301041i −1.00000 1.11004 + 2.78708i −3.72427 + 2.15021i
65.12 0.500000 − 0.866025i 1.72543 − 0.151338i −0.500000 − 0.866025i −0.246102 − 0.142087i 0.731651 − 1.56993i 2.09441 + 1.61662i −1.00000 2.95419 − 0.522246i −0.246102 + 0.142087i
263.1 0.500000 + 0.866025i −1.72951 + 0.0938734i −0.500000 + 0.866025i −3.23623 + 1.86844i −0.946049 − 1.45086i 0.830341 + 2.51208i −1.00000 2.98238 − 0.324709i −3.23623 − 1.86844i
263.2 0.500000 + 0.866025i −1.64448 + 0.543759i −0.500000 + 0.866025i 1.63087 − 0.941585i −1.29315 − 1.15228i 1.76745 − 1.96879i −1.00000 2.40865 − 1.78841i 1.63087 + 0.941585i
263.3 0.500000 + 0.866025i −1.20836 + 1.24091i −0.500000 + 0.866025i −1.00577 + 0.580683i −1.67884 − 0.426020i −2.19957 − 1.47034i −1.00000 −0.0797091 − 2.99894i −1.00577 − 0.580683i
263.4 0.500000 + 0.866025i −0.993776 − 1.41859i −0.500000 + 0.866025i 0.246102 − 0.142087i 0.731651 − 1.56993i −2.09441 + 1.61662i −1.00000 −1.02482 + 2.81953i 0.246102 + 0.142087i
263.5 0.500000 + 0.866025i −0.470476 + 1.66693i −0.500000 + 0.866025i 1.00577 − 0.580683i −1.67884 + 0.426020i 2.19957 + 1.47034i −1.00000 −2.55730 − 1.56850i 1.00577 + 0.580683i
263.6 0.500000 + 0.866025i 0.125098 − 1.72753i −0.500000 + 0.866025i 3.72427 − 2.15021i 1.55863 − 0.755426i 2.62857 − 0.301041i −1.00000 −2.96870 − 0.432220i 3.72427 + 2.15021i
263.7 0.500000 + 0.866025i 0.301239 − 1.70565i −0.500000 + 0.866025i −1.38736 + 0.800992i 1.62776 − 0.591946i −0.229478 − 2.63578i −1.00000 −2.81851 − 1.02762i −1.38736 − 0.800992i
263.8 0.500000 + 0.866025i 0.351333 + 1.69604i −0.500000 + 0.866025i −1.63087 + 0.941585i −1.29315 + 1.15228i −1.76745 + 1.96879i −1.00000 −2.75313 + 1.19175i −1.63087 − 0.941585i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
33.d even 2 1 inner
231.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.n.f yes 24
3.b odd 2 1 462.2.n.e ✓ 24
7.c even 3 1 inner 462.2.n.f yes 24
11.b odd 2 1 462.2.n.e ✓ 24
21.h odd 6 1 462.2.n.e ✓ 24
33.d even 2 1 inner 462.2.n.f yes 24
77.h odd 6 1 462.2.n.e ✓ 24
231.l even 6 1 inner 462.2.n.f yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.n.e ✓ 24 3.b odd 2 1
462.2.n.e ✓ 24 11.b odd 2 1
462.2.n.e ✓ 24 21.h odd 6 1
462.2.n.e ✓ 24 77.h odd 6 1
462.2.n.f yes 24 1.a even 1 1 trivial
462.2.n.f yes 24 7.c even 3 1 inner
462.2.n.f yes 24 33.d even 2 1 inner
462.2.n.f yes 24 231.l even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\):

\( T_{5}^{24} - 40 T_{5}^{22} + 1079 T_{5}^{20} - 15752 T_{5}^{18} + 164601 T_{5}^{16} - 922588 T_{5}^{14} + \cdots + 65536 \) Copy content Toggle raw display
\( T_{17}^{12} + 2 T_{17}^{11} + 47 T_{17}^{10} - 58 T_{17}^{9} + 1365 T_{17}^{8} - 1830 T_{17}^{7} + \cdots + 589824 \) Copy content Toggle raw display