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Properties

Label 468.3.e.m
Level $468$
Weight $3$
Character orbit 468.e
Analytic conductor $12.752$
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] = [468,3,Mod(415,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("468.415");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 468.e (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: \(12.7520763721\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 156)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4} + 28 q^{10} + 48 q^{13} + 40 q^{14} + 100 q^{16} - 32 q^{17} + 84 q^{22} - 312 q^{25} + 16 q^{26} + 80 q^{29} - 120 q^{38} - 204 q^{40} + 392 q^{49} + 28 q^{52} + 224 q^{53} - 800 q^{56} - 96 q^{61} + 352 q^{62} - 184 q^{64} + 112 q^{65} + 344 q^{68} - 232 q^{74} + 16 q^{77} + 20 q^{82} - 92 q^{88} + 616 q^{92} - 684 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1 −1.99484 − 0.143522i 0 3.95880 + 0.572609i 8.96077i 0 7.15347 −7.81501 − 1.71044i 0 1.28607 − 17.8753i
415.2 −1.99484 + 0.143522i 0 3.95880 − 0.572609i − 8.96077i 0 7.15347 −7.81501 + 1.71044i 0 1.28607 + 17.8753i
415.3 −1.86766 − 0.715427i 0 2.97633 + 2.67235i 4.65917i 0 2.30148 −3.64691 − 7.12040i 0 3.33330 − 8.70176i
415.4 −1.86766 + 0.715427i 0 2.97633 − 2.67235i − 4.65917i 0 2.30148 −3.64691 + 7.12040i 0 3.33330 + 8.70176i
415.5 −1.86140 − 0.731561i 0 2.92964 + 2.72346i − 3.15488i 0 −3.05810 −3.46086 − 7.21266i 0 −2.30799 + 5.87250i
415.6 −1.86140 + 0.731561i 0 2.92964 − 2.72346i 3.15488i 0 −3.05810 −3.46086 + 7.21266i 0 −2.30799 − 5.87250i
415.7 −1.31729 − 1.50490i 0 −0.529474 + 3.96480i − 7.41736i 0 −13.3843 6.66412 − 4.42600i 0 −11.1624 + 9.77084i
415.8 −1.31729 + 1.50490i 0 −0.529474 − 3.96480i 7.41736i 0 −13.3843 6.66412 + 4.42600i 0 −11.1624 − 9.77084i
415.9 −0.570481 − 1.91691i 0 −3.34910 + 2.18712i 7.79890i 0 −7.84779 6.10312 + 5.17222i 0 14.9498 − 4.44912i
415.10 −0.570481 + 1.91691i 0 −3.34910 − 2.18712i − 7.79890i 0 −7.84779 6.10312 − 5.17222i 0 14.9498 + 4.44912i
415.11 −0.0830862 − 1.99827i 0 −3.98619 + 0.332058i 0.451006i 0 −9.24405 0.994741 + 7.93791i 0 0.901234 − 0.0374724i
415.12 −0.0830862 + 1.99827i 0 −3.98619 − 0.332058i − 0.451006i 0 −9.24405 0.994741 − 7.93791i 0 0.901234 + 0.0374724i
415.13 0.0830862 − 1.99827i 0 −3.98619 − 0.332058i 0.451006i 0 9.24405 −0.994741 + 7.93791i 0 0.901234 + 0.0374724i
415.14 0.0830862 + 1.99827i 0 −3.98619 + 0.332058i − 0.451006i 0 9.24405 −0.994741 − 7.93791i 0 0.901234 − 0.0374724i
415.15 0.570481 − 1.91691i 0 −3.34910 − 2.18712i 7.79890i 0 7.84779 −6.10312 + 5.17222i 0 14.9498 + 4.44912i
415.16 0.570481 + 1.91691i 0 −3.34910 + 2.18712i − 7.79890i 0 7.84779 −6.10312 − 5.17222i 0 14.9498 − 4.44912i
415.17 1.31729 − 1.50490i 0 −0.529474 − 3.96480i − 7.41736i 0 13.3843 −6.66412 − 4.42600i 0 −11.1624 − 9.77084i
415.18 1.31729 + 1.50490i 0 −0.529474 + 3.96480i 7.41736i 0 13.3843 −6.66412 + 4.42600i 0 −11.1624 + 9.77084i
415.19 1.86140 − 0.731561i 0 2.92964 − 2.72346i − 3.15488i 0 3.05810 3.46086 − 7.21266i 0 −2.30799 − 5.87250i
415.20 1.86140 + 0.731561i 0 2.92964 + 2.72346i 3.15488i 0 3.05810 3.46086 + 7.21266i 0 −2.30799 + 5.87250i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
13.b even 2 1 inner
52.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.3.e.m 24
3.b odd 2 1 156.3.e.c ✓ 24
4.b odd 2 1 inner 468.3.e.m 24
12.b even 2 1 156.3.e.c ✓ 24
13.b even 2 1 inner 468.3.e.m 24
39.d odd 2 1 156.3.e.c ✓ 24
52.b odd 2 1 inner 468.3.e.m 24
156.h even 2 1 156.3.e.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.e.c ✓ 24 3.b odd 2 1
156.3.e.c ✓ 24 12.b even 2 1
156.3.e.c ✓ 24 39.d odd 2 1
156.3.e.c ✓ 24 156.h even 2 1
468.3.e.m 24 1.a even 1 1 trivial
468.3.e.m 24 4.b odd 2 1 inner
468.3.e.m 24 13.b even 2 1 inner
468.3.e.m 24 52.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(468, [\chi])\):

\( T_{5}^{12} + 228T_{5}^{10} + 19120T_{5}^{8} + 715392T_{5}^{6} + 11384576T_{5}^{4} + 60341248T_{5}^{2} + 11808768 \) Copy content Toggle raw display
\( T_{7}^{12} - 392T_{7}^{10} + 53872T_{7}^{8} - 3286144T_{7}^{6} + 88136960T_{7}^{4} - 833523712T_{7}^{2} + 2389782528 \) Copy content Toggle raw display
\( T_{11}^{12} - 972 T_{11}^{10} + 358928 T_{11}^{8} - 63684224 T_{11}^{6} + 5535424512 T_{11}^{4} + \cdots + 1494427041792 \) Copy content Toggle raw display