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Properties

Label 336.3.bn.h
Level $336$
Weight $3$
Character orbit 336.bn
Analytic conductor $9.155$
Analytic rank $0$
Dimension $32$
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] = [336,3,Mod(65,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.bn (of order \(6\), degree \(2\), not 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: \(9.15533688251\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 168)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{7} - 8 q^{9} - 8 q^{13} - 16 q^{15} - 16 q^{19} + 60 q^{21} + 76 q^{25} + 24 q^{27} + 36 q^{31} + 32 q^{33} + 28 q^{37} - 4 q^{39} + 240 q^{43} + 28 q^{45} - 56 q^{49} - 36 q^{51} + 56 q^{55} - 176 q^{57} - 152 q^{61} + 28 q^{63} + 88 q^{67} - 224 q^{69} - 156 q^{73} + 312 q^{75} - 84 q^{79} - 164 q^{81} + 304 q^{85} + 176 q^{87} - 400 q^{91} + 8 q^{93} - 40 q^{97} - 736 q^{99}+O(q^{100}) \) 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 −2.88154 − 0.834716i 0 −5.08175 − 2.93395i 0 −6.96219 + 0.726610i 0 7.60650 + 4.81053i 0
65.2 0 −2.73739 − 1.22748i 0 2.11473 + 1.22094i 0 2.19666 − 6.64640i 0 5.98656 + 6.72020i 0
65.3 0 −2.61180 + 1.47597i 0 0.609275 + 0.351765i 0 0.672075 + 6.96766i 0 4.64302 − 7.70989i 0
65.4 0 −2.35296 + 1.86107i 0 5.36241 + 3.09599i 0 −4.86547 − 5.03261i 0 2.07285 − 8.75804i 0
65.5 0 −1.81766 − 2.38665i 0 8.14580 + 4.70298i 0 1.50029 + 6.83733i 0 −2.39220 + 8.67625i 0
65.6 0 −1.52370 + 2.58425i 0 −3.33237 − 1.92395i 0 6.64746 + 2.19347i 0 −4.35671 − 7.87522i 0
65.7 0 −1.15807 − 2.76747i 0 −8.14580 − 4.70298i 0 1.50029 + 6.83733i 0 −6.31776 + 6.40983i 0
65.8 0 0.305660 − 2.98439i 0 −2.11473 − 1.22094i 0 2.19666 − 6.64640i 0 −8.81314 − 1.82442i 0
65.9 0 0.414229 + 2.97126i 0 −4.60124 − 2.65653i 0 4.50663 − 5.35634i 0 −8.65683 + 2.46157i 0
65.10 0 0.717883 − 2.91284i 0 5.08175 + 2.93395i 0 −6.96219 + 0.726610i 0 −7.96929 − 4.18216i 0
65.11 0 0.811802 + 2.88808i 0 4.52182 + 2.61068i 0 −6.69544 + 2.04232i 0 −7.68195 + 4.68909i 0
65.12 0 2.09525 + 2.14708i 0 −4.52182 − 2.61068i 0 −6.69544 + 2.04232i 0 −0.219895 + 8.99731i 0
65.13 0 2.36608 + 1.84436i 0 4.60124 + 2.65653i 0 4.50663 − 5.35634i 0 2.19664 + 8.72782i 0
65.14 0 2.58413 − 1.52390i 0 −0.609275 − 0.351765i 0 0.672075 + 6.96766i 0 4.35545 − 7.87592i 0
65.15 0 2.78821 − 1.10719i 0 −5.36241 − 3.09599i 0 −4.86547 − 5.03261i 0 6.54826 − 6.17416i 0
65.16 0 2.99987 − 0.0274332i 0 3.33237 + 1.92395i 0 6.64746 + 2.19347i 0 8.99849 − 0.164592i 0
305.1 0 −2.88154 + 0.834716i 0 −5.08175 + 2.93395i 0 −6.96219 − 0.726610i 0 7.60650 − 4.81053i 0
305.2 0 −2.73739 + 1.22748i 0 2.11473 − 1.22094i 0 2.19666 + 6.64640i 0 5.98656 − 6.72020i 0
305.3 0 −2.61180 − 1.47597i 0 0.609275 − 0.351765i 0 0.672075 − 6.96766i 0 4.64302 + 7.70989i 0
305.4 0 −2.35296 − 1.86107i 0 5.36241 − 3.09599i 0 −4.86547 + 5.03261i 0 2.07285 + 8.75804i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.3.bn.h 32
3.b odd 2 1 inner 336.3.bn.h 32
4.b odd 2 1 168.3.bf.a ✓ 32
7.c even 3 1 inner 336.3.bn.h 32
12.b even 2 1 168.3.bf.a ✓ 32
21.h odd 6 1 inner 336.3.bn.h 32
28.f even 6 1 1176.3.d.g 16
28.g odd 6 1 168.3.bf.a ✓ 32
28.g odd 6 1 1176.3.d.f 16
84.j odd 6 1 1176.3.d.g 16
84.n even 6 1 168.3.bf.a ✓ 32
84.n even 6 1 1176.3.d.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.bf.a ✓ 32 4.b odd 2 1
168.3.bf.a ✓ 32 12.b even 2 1
168.3.bf.a ✓ 32 28.g odd 6 1
168.3.bf.a ✓ 32 84.n even 6 1
336.3.bn.h 32 1.a even 1 1 trivial
336.3.bn.h 32 3.b odd 2 1 inner
336.3.bn.h 32 7.c even 3 1 inner
336.3.bn.h 32 21.h odd 6 1 inner
1176.3.d.f 16 28.g odd 6 1
1176.3.d.f 16 84.n even 6 1
1176.3.d.g 16 28.f even 6 1
1176.3.d.g 16 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{32} - 238 T_{5}^{30} + 34461 T_{5}^{28} - 3150698 T_{5}^{26} + 210290618 T_{5}^{24} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
\( T_{13}^{8} + 2 T_{13}^{7} - 863 T_{13}^{6} - 1024 T_{13}^{5} + 254320 T_{13}^{4} + 127520 T_{13}^{3} + \cdots + 611008128 \) Copy content Toggle raw display