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Properties

Label 456.2.j.e
Level $456$
Weight $2$
Character orbit 456.j
Analytic conductor $3.641$
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] = [456,2,Mod(419,456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(456, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("456.419");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 456.j (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: \(3.64117833217\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{3} - 6 q^{4} + q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{3} - 6 q^{4} + q^{6} - 6 q^{9} + 4 q^{10} - 33 q^{12} - 30 q^{16} - 19 q^{18} + 24 q^{19} - 28 q^{22} + 21 q^{24} + 96 q^{25} + 34 q^{28} + 8 q^{30} + 24 q^{33} + 58 q^{34} - 9 q^{36} - 24 q^{40} - 31 q^{42} + 16 q^{43} - 6 q^{46} + 3 q^{48} - 36 q^{49} + 38 q^{51} - 6 q^{52} - 8 q^{54} + 6 q^{57} - 42 q^{58} - 50 q^{60} + 18 q^{64} + 10 q^{66} + 20 q^{67} - 96 q^{70} - 33 q^{72} - 12 q^{73} - 30 q^{75} - 6 q^{76} - 17 q^{78} + 34 q^{81} + 68 q^{82} - 29 q^{84} - 76 q^{88} - 70 q^{90} - 36 q^{91} - 12 q^{94} + 29 q^{96} - 104 q^{97} + 56 q^{99}+O(q^{100}) \) 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 −1.34037 − 0.450999i −1.59320 + 0.679496i 1.59320 + 1.20901i 2.84644 2.44193 − 0.192246i − 0.466225i −1.59022 − 2.33906i 2.07657 − 2.16515i −3.81529 − 1.28374i
419.2 −1.34037 + 0.450999i −1.59320 − 0.679496i 1.59320 − 1.20901i 2.84644 2.44193 + 0.192246i 0.466225i −1.59022 + 2.33906i 2.07657 + 2.16515i −3.81529 + 1.28374i
419.3 −1.16880 − 0.796176i −0.732208 + 1.56967i 0.732208 + 1.86115i −4.19368 2.10554 − 1.25167i − 1.07444i 0.625992 − 2.75828i −1.92774 − 2.29865i 4.90159 + 3.33891i
419.4 −1.16880 + 0.796176i −0.732208 − 1.56967i 0.732208 − 1.86115i −4.19368 2.10554 + 1.25167i 1.07444i 0.625992 + 2.75828i −1.92774 + 2.29865i 4.90159 − 3.33891i
419.5 −0.960810 − 1.03771i 0.153689 − 1.72522i −0.153689 + 1.99409i −1.97287 −1.93794 + 1.49812i − 2.35621i 2.21695 − 1.75645i −2.95276 − 0.530294i 1.89556 + 2.04727i
419.6 −0.960810 + 1.03771i 0.153689 + 1.72522i −0.153689 − 1.99409i −1.97287 −1.93794 − 1.49812i 2.35621i 2.21695 + 1.75645i −2.95276 + 0.530294i 1.89556 − 2.04727i
419.7 −0.836171 − 1.14053i 0.601637 + 1.62420i −0.601637 + 1.90736i 3.49684 1.34939 − 2.04430i − 4.78800i 2.67848 − 0.908693i −2.27607 + 1.95436i −2.92395 − 3.98826i
419.8 −0.836171 + 1.14053i 0.601637 − 1.62420i −0.601637 − 1.90736i 3.49684 1.34939 + 2.04430i 4.78800i 2.67848 + 0.908693i −2.27607 − 1.95436i −2.92395 + 3.98826i
419.9 −0.575131 − 1.29198i 1.33845 − 1.09934i −1.33845 + 1.48612i −2.88790 −2.19012 − 1.09699i 4.39097i 2.68983 + 0.874542i 0.582891 − 2.94283i 1.66092 + 3.73112i
419.10 −0.575131 + 1.29198i 1.33845 + 1.09934i −1.33845 − 1.48612i −2.88790 −2.19012 + 1.09699i − 4.39097i 2.68983 − 0.874542i 0.582891 + 2.94283i 1.66092 − 3.73112i
419.11 −0.366311 − 1.36595i 1.73163 + 0.0380449i −1.73163 + 1.00072i 1.96235 −0.582348 − 2.37926i 1.36779i 2.00125 + 1.99875i 2.99711 + 0.131760i −0.718829 − 2.68047i
419.12 −0.366311 + 1.36595i 1.73163 − 0.0380449i −1.73163 − 1.00072i 1.96235 −0.582348 + 2.37926i − 1.36779i 2.00125 − 1.99875i 2.99711 − 0.131760i −0.718829 + 2.68047i
419.13 0.366311 − 1.36595i 1.73163 + 0.0380449i −1.73163 − 1.00072i −1.96235 0.686283 − 2.35139i − 1.36779i −2.00125 + 1.99875i 2.99711 + 0.131760i −0.718829 + 2.68047i
419.14 0.366311 + 1.36595i 1.73163 − 0.0380449i −1.73163 + 1.00072i −1.96235 0.686283 + 2.35139i 1.36779i −2.00125 − 1.99875i 2.99711 − 0.131760i −0.718829 − 2.68047i
419.15 0.575131 − 1.29198i 1.33845 − 1.09934i −1.33845 − 1.48612i 2.88790 −0.650551 − 2.36152i − 4.39097i −2.68983 + 0.874542i 0.582891 − 2.94283i 1.66092 − 3.73112i
419.16 0.575131 + 1.29198i 1.33845 + 1.09934i −1.33845 + 1.48612i 2.88790 −0.650551 + 2.36152i 4.39097i −2.68983 − 0.874542i 0.582891 + 2.94283i 1.66092 + 3.73112i
419.17 0.836171 − 1.14053i 0.601637 + 1.62420i −0.601637 − 1.90736i −3.49684 2.35553 + 0.671922i 4.78800i −2.67848 − 0.908693i −2.27607 + 1.95436i −2.92395 + 3.98826i
419.18 0.836171 + 1.14053i 0.601637 − 1.62420i −0.601637 + 1.90736i −3.49684 2.35553 − 0.671922i − 4.78800i −2.67848 + 0.908693i −2.27607 − 1.95436i −2.92395 − 3.98826i
419.19 0.960810 − 1.03771i 0.153689 − 1.72522i −0.153689 − 1.99409i 1.97287 −1.64261 − 1.81709i 2.35621i −2.21695 − 1.75645i −2.95276 − 0.530294i 1.89556 − 2.04727i
419.20 0.960810 + 1.03771i 0.153689 + 1.72522i −0.153689 + 1.99409i 1.97287 −1.64261 + 1.81709i − 2.35621i −2.21695 + 1.75645i −2.95276 + 0.530294i 1.89556 + 2.04727i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 456.2.j.e ✓ 24
3.b odd 2 1 inner 456.2.j.e ✓ 24
4.b odd 2 1 1824.2.j.d 24
8.b even 2 1 1824.2.j.d 24
8.d odd 2 1 inner 456.2.j.e ✓ 24
12.b even 2 1 1824.2.j.d 24
24.f even 2 1 inner 456.2.j.e ✓ 24
24.h odd 2 1 1824.2.j.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.j.e ✓ 24 1.a even 1 1 trivial
456.2.j.e ✓ 24 3.b odd 2 1 inner
456.2.j.e ✓ 24 8.d odd 2 1 inner
456.2.j.e ✓ 24 24.f even 2 1 inner
1824.2.j.d 24 4.b odd 2 1
1824.2.j.d 24 8.b even 2 1
1824.2.j.d 24 12.b even 2 1
1824.2.j.d 24 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(456, [\chi])\):

\( T_{5}^{12} - 54T_{5}^{10} + 1146T_{5}^{8} - 12228T_{5}^{6} + 69093T_{5}^{4} - 195710T_{5}^{2} + 217800 \) Copy content Toggle raw display
\( T_{7}^{12} + 51T_{7}^{10} + 834T_{7}^{8} + 4782T_{7}^{6} + 9885T_{7}^{4} + 7231T_{7}^{2} + 1152 \) Copy content Toggle raw display