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Properties

Label 462.4.g.b
Level $462$
Weight $4$
Character orbit 462.g
Analytic conductor $27.259$
Analytic rank $0$
Dimension $36$
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] = [462,4,Mod(419,462)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(462, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.419");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 462.g (of order \(2\), degree \(1\), 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: \(27.2588824227\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 144 q^{4} - 36 q^{7} - 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 144 q^{4} - 36 q^{7} - 132 q^{9} - 468 q^{15} + 576 q^{16} + 8 q^{18} + 14 q^{21} + 792 q^{22} + 552 q^{25} + 144 q^{28} - 184 q^{30} + 528 q^{36} - 108 q^{37} - 708 q^{39} + 260 q^{42} + 2688 q^{43} - 912 q^{46} - 2796 q^{49} + 1544 q^{51} - 1784 q^{57} - 1128 q^{58} + 1872 q^{60} + 1030 q^{63} - 2304 q^{64} + 2196 q^{67} - 2544 q^{70} - 32 q^{72} - 384 q^{78} - 4488 q^{79} - 2176 q^{81} - 56 q^{84} + 8328 q^{85} - 3168 q^{88} - 5580 q^{91} + 2956 q^{93} - 44 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1 − 2.00000i −4.49783 + 2.60183i −4.00000 9.65815 5.20365 + 8.99567i −2.29801 + 18.3771i 8.00000i 13.4610 − 23.4052i − 19.3163i
419.2 − 2.00000i 4.49783 − 2.60183i −4.00000 −9.65815 −5.20365 − 8.99567i −2.29801 − 18.3771i 8.00000i 13.4610 − 23.4052i 19.3163i
419.3 2.00000i −4.49783 − 2.60183i −4.00000 9.65815 5.20365 − 8.99567i −2.29801 − 18.3771i − 8.00000i 13.4610 + 23.4052i 19.3163i
419.4 2.00000i 4.49783 + 2.60183i −4.00000 −9.65815 −5.20365 + 8.99567i −2.29801 + 18.3771i − 8.00000i 13.4610 + 23.4052i − 19.3163i
419.5 − 2.00000i −2.85163 − 4.34375i −4.00000 20.5252 −8.68750 + 5.70327i 4.68998 − 17.9166i 8.00000i −10.7364 + 24.7736i − 41.0504i
419.6 − 2.00000i 2.85163 + 4.34375i −4.00000 −20.5252 8.68750 − 5.70327i 4.68998 + 17.9166i 8.00000i −10.7364 + 24.7736i 41.0504i
419.7 2.00000i −2.85163 + 4.34375i −4.00000 20.5252 −8.68750 − 5.70327i 4.68998 + 17.9166i − 8.00000i −10.7364 − 24.7736i 41.0504i
419.8 2.00000i 2.85163 − 4.34375i −4.00000 −20.5252 8.68750 + 5.70327i 4.68998 − 17.9166i − 8.00000i −10.7364 − 24.7736i − 41.0504i
419.9 − 2.00000i −0.124204 − 5.19467i −4.00000 16.3702 −10.3893 + 0.248407i −2.94673 + 18.2843i 8.00000i −26.9691 + 1.29039i − 32.7404i
419.10 − 2.00000i 0.124204 + 5.19467i −4.00000 −16.3702 10.3893 − 0.248407i −2.94673 − 18.2843i 8.00000i −26.9691 + 1.29039i 32.7404i
419.11 2.00000i −0.124204 + 5.19467i −4.00000 16.3702 −10.3893 − 0.248407i −2.94673 − 18.2843i − 8.00000i −26.9691 − 1.29039i 32.7404i
419.12 2.00000i 0.124204 − 5.19467i −4.00000 −16.3702 10.3893 + 0.248407i −2.94673 + 18.2843i − 8.00000i −26.9691 − 1.29039i − 32.7404i
419.13 − 2.00000i −1.54198 + 4.96209i −4.00000 −4.78654 9.92417 + 3.08396i 7.29318 + 17.0238i 8.00000i −22.2446 − 15.3029i 9.57308i
419.14 − 2.00000i 1.54198 − 4.96209i −4.00000 4.78654 −9.92417 − 3.08396i 7.29318 − 17.0238i 8.00000i −22.2446 − 15.3029i − 9.57308i
419.15 2.00000i −1.54198 − 4.96209i −4.00000 −4.78654 9.92417 − 3.08396i 7.29318 − 17.0238i − 8.00000i −22.2446 + 15.3029i − 9.57308i
419.16 2.00000i 1.54198 + 4.96209i −4.00000 4.78654 −9.92417 + 3.08396i 7.29318 + 17.0238i − 8.00000i −22.2446 + 15.3029i 9.57308i
419.17 − 2.00000i −3.74770 + 3.59927i −4.00000 12.8313 7.19853 + 7.49540i 12.5066 − 13.6596i 8.00000i 1.09055 − 26.9780i − 25.6627i
419.18 − 2.00000i 3.74770 − 3.59927i −4.00000 −12.8313 −7.19853 − 7.49540i 12.5066 + 13.6596i 8.00000i 1.09055 − 26.9780i 25.6627i
419.19 2.00000i −3.74770 − 3.59927i −4.00000 12.8313 7.19853 − 7.49540i 12.5066 + 13.6596i − 8.00000i 1.09055 + 26.9780i 25.6627i
419.20 2.00000i 3.74770 + 3.59927i −4.00000 −12.8313 −7.19853 + 7.49540i 12.5066 − 13.6596i − 8.00000i 1.09055 + 26.9780i − 25.6627i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.4.g.b ✓ 36
3.b odd 2 1 inner 462.4.g.b ✓ 36
7.b odd 2 1 inner 462.4.g.b ✓ 36
21.c even 2 1 inner 462.4.g.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.g.b ✓ 36 1.a even 1 1 trivial
462.4.g.b ✓ 36 3.b odd 2 1 inner
462.4.g.b ✓ 36 7.b odd 2 1 inner
462.4.g.b ✓ 36 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} - 1263 T_{5}^{16} + 617385 T_{5}^{14} - 148057125 T_{5}^{12} + 18135575298 T_{5}^{10} + \cdots - 19311649357824 \) acting on \(S_{4}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display