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Properties

Label 828.2.e.f
Level $828$
Weight $2$
Character orbit 828.e
Analytic conductor $6.612$
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] = [828,2,Mod(91,828)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(828, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("828.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 828.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: \(6.61161328736\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 276)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} - 4 q^{8} + 8 q^{16} - 24 q^{25} - 40 q^{26} + 32 q^{29} + 36 q^{32} - 16 q^{41} + 40 q^{49} + 12 q^{50} - 40 q^{52} + 24 q^{58} + 40 q^{62} + 48 q^{64} + 72 q^{70} - 16 q^{77} - 40 q^{82} - 64 q^{85} - 44 q^{92} + 72 q^{94} + 52 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −1.40342 − 0.174423i 0 1.93915 + 0.489577i − 2.63558i 0 −0.526424 −2.63604 − 1.02531i 0 −0.459706 + 3.69881i
91.2 −1.40342 − 0.174423i 0 1.93915 + 0.489577i 2.63558i 0 0.526424 −2.63604 − 1.02531i 0 0.459706 − 3.69881i
91.3 −1.40342 + 0.174423i 0 1.93915 − 0.489577i − 2.63558i 0 0.526424 −2.63604 + 1.02531i 0 0.459706 + 3.69881i
91.4 −1.40342 + 0.174423i 0 1.93915 − 0.489577i 2.63558i 0 −0.526424 −2.63604 + 1.02531i 0 −0.459706 − 3.69881i
91.5 −1.11292 − 0.872582i 0 0.477201 + 1.94224i − 0.970352i 0 −4.31859 1.16367 − 2.57796i 0 −0.846712 + 1.07993i
91.6 −1.11292 − 0.872582i 0 0.477201 + 1.94224i 0.970352i 0 4.31859 1.16367 − 2.57796i 0 0.846712 − 1.07993i
91.7 −1.11292 + 0.872582i 0 0.477201 − 1.94224i − 0.970352i 0 4.31859 1.16367 + 2.57796i 0 0.846712 + 1.07993i
91.8 −1.11292 + 0.872582i 0 0.477201 − 1.94224i 0.970352i 0 −4.31859 1.16367 + 2.57796i 0 −0.846712 − 1.07993i
91.9 −0.714279 − 1.22058i 0 −0.979610 + 1.74366i − 3.78153i 0 −1.02234 2.82799 − 0.0497743i 0 −4.61564 + 2.70107i
91.10 −0.714279 − 1.22058i 0 −0.979610 + 1.74366i 3.78153i 0 1.02234 2.82799 − 0.0497743i 0 4.61564 − 2.70107i
91.11 −0.714279 + 1.22058i 0 −0.979610 − 1.74366i − 3.78153i 0 1.02234 2.82799 + 0.0497743i 0 4.61564 + 2.70107i
91.12 −0.714279 + 1.22058i 0 −0.979610 − 1.74366i 3.78153i 0 −1.02234 2.82799 + 0.0497743i 0 −4.61564 − 2.70107i
91.13 0.279557 − 1.38631i 0 −1.84370 − 0.775104i − 1.27568i 0 2.49131 −1.58995 + 2.33924i 0 −1.76848 − 0.356624i
91.14 0.279557 − 1.38631i 0 −1.84370 − 0.775104i 1.27568i 0 −2.49131 −1.58995 + 2.33924i 0 1.76848 + 0.356624i
91.15 0.279557 + 1.38631i 0 −1.84370 + 0.775104i − 1.27568i 0 −2.49131 −1.58995 − 2.33924i 0 1.76848 − 0.356624i
91.16 0.279557 + 1.38631i 0 −1.84370 + 0.775104i 1.27568i 0 2.49131 −1.58995 − 2.33924i 0 −1.76848 + 0.356624i
91.17 0.588134 − 1.28612i 0 −1.30820 − 1.51282i − 2.28672i 0 −3.41490 −2.71506 + 0.792755i 0 −2.94099 − 1.34490i
91.18 0.588134 − 1.28612i 0 −1.30820 − 1.51282i 2.28672i 0 3.41490 −2.71506 + 0.792755i 0 2.94099 + 1.34490i
91.19 0.588134 + 1.28612i 0 −1.30820 + 1.51282i − 2.28672i 0 3.41490 −2.71506 − 0.792755i 0 2.94099 − 1.34490i
91.20 0.588134 + 1.28612i 0 −1.30820 + 1.51282i 2.28672i 0 −3.41490 −2.71506 − 0.792755i 0 −2.94099 + 1.34490i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 828.2.e.f 24
3.b odd 2 1 276.2.e.a ✓ 24
4.b odd 2 1 inner 828.2.e.f 24
12.b even 2 1 276.2.e.a ✓ 24
23.b odd 2 1 inner 828.2.e.f 24
24.f even 2 1 4416.2.i.d 24
24.h odd 2 1 4416.2.i.d 24
69.c even 2 1 276.2.e.a ✓ 24
92.b even 2 1 inner 828.2.e.f 24
276.h odd 2 1 276.2.e.a ✓ 24
552.b even 2 1 4416.2.i.d 24
552.h odd 2 1 4416.2.i.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.e.a ✓ 24 3.b odd 2 1
276.2.e.a ✓ 24 12.b even 2 1
276.2.e.a ✓ 24 69.c even 2 1
276.2.e.a ✓ 24 276.h odd 2 1
828.2.e.f 24 1.a even 1 1 trivial
828.2.e.f 24 4.b odd 2 1 inner
828.2.e.f 24 23.b odd 2 1 inner
828.2.e.f 24 92.b even 2 1 inner
4416.2.i.d 24 24.f even 2 1
4416.2.i.d 24 24.h odd 2 1
4416.2.i.d 24 552.b even 2 1
4416.2.i.d 24 552.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 \(S_{2}^{\mathrm{new}}(828, [\chi])\):

\( T_{5}^{12} + 36T_{5}^{10} + 482T_{5}^{8} + 3048T_{5}^{6} + 9312T_{5}^{4} + 12320T_{5}^{2} + 5536 \) Copy content Toggle raw display
\( T_{7}^{12} - 52T_{7}^{10} + 990T_{7}^{8} - 8328T_{7}^{6} + 28760T_{7}^{4} - 27328T_{7}^{2} + 5536 \) Copy content Toggle raw display