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Properties

Label 273.2.u.c
Level $273$
Weight $2$
Character orbit 273.u
Analytic conductor $2.180$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [273,2,Mod(62,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.62");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.u (of order \(6\), degree \(2\), 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: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 40 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 40 q^{4} - 6 q^{9} + 18 q^{15} - 32 q^{16} + 24 q^{22} - 80 q^{25} + 12 q^{28} + 20 q^{30} + 50 q^{36} - 36 q^{37} - 60 q^{39} + 48 q^{43} - 84 q^{46} - 16 q^{49} + 52 q^{51} - 48 q^{58} - 36 q^{63} + 176 q^{64} + 60 q^{67} - 66 q^{72} + 166 q^{78} - 104 q^{79} - 30 q^{81} + 60 q^{84} + 132 q^{85} - 8 q^{91} - 162 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
62.1 −1.35301 − 2.34347i −1.31833 + 1.12339i −2.66125 + 4.60942i 0.967216i 4.41634 + 1.56953i −2.52716 + 0.783246i 8.99072 0.476002 − 2.96200i 2.26665 − 1.30865i
62.2 −1.35301 − 2.34347i 1.31833 − 1.12339i −2.66125 + 4.60942i − 0.967216i −4.41634 − 1.56953i −0.585267 − 2.58021i 8.99072 0.476002 − 2.96200i −2.26665 + 1.30865i
62.3 −1.25961 − 2.18170i −0.786041 − 1.54342i −2.17322 + 3.76413i 2.08918i −2.37718 + 3.65901i 2.30363 + 1.30126i 5.91120 −1.76428 + 2.42638i 4.55798 − 2.63155i
62.4 −1.25961 − 2.18170i 0.786041 + 1.54342i −2.17322 + 3.76413i − 2.08918i 2.37718 − 3.65901i 2.27874 + 1.34438i 5.91120 −1.76428 + 2.42638i −4.55798 + 2.63155i
62.5 −1.03726 − 1.79658i −1.69497 + 0.356466i −1.15180 + 1.99498i − 4.16126i 2.39854 + 2.67541i 1.87191 − 1.86974i 0.629840 2.74586 − 1.20840i −7.47605 + 4.31630i
62.6 −1.03726 − 1.79658i 1.69497 − 0.356466i −1.15180 + 1.99498i 4.16126i −2.39854 − 2.67541i −0.683288 + 2.55600i 0.629840 2.74586 − 1.20840i 7.47605 − 4.31630i
62.7 −0.879183 − 1.52279i −1.61457 − 0.627032i −0.545926 + 0.945572i − 0.0518379i 0.464663 + 3.00992i −1.90128 + 1.83988i −1.59686 2.21366 + 2.02477i −0.0789383 + 0.0455750i
62.8 −0.879183 − 1.52279i 1.61457 + 0.627032i −0.545926 + 0.945572i 0.0518379i −0.464663 − 3.00992i 0.642741 − 2.56649i −1.59686 2.21366 + 2.02477i 0.0789383 − 0.0455750i
62.9 −0.835434 − 1.44701i −0.710596 + 1.57957i −0.395901 + 0.685720i 2.15210i 2.87932 − 0.291388i −0.338674 − 2.62399i −2.01874 −1.99011 − 2.24488i 3.11411 − 1.79793i
62.10 −0.835434 − 1.44701i 0.710596 − 1.57957i −0.395901 + 0.685720i − 2.15210i −2.87932 + 0.291388i −2.44178 + 1.01869i −2.01874 −1.99011 − 2.24488i −3.11411 + 1.79793i
62.11 −0.584057 − 1.01162i −0.340045 − 1.69834i 0.317754 − 0.550366i 3.55045i −1.51947 + 1.33593i −1.36289 − 2.26771i −3.07858 −2.76874 + 1.15503i 3.59170 − 2.07367i
62.12 −0.584057 − 1.01162i 0.340045 + 1.69834i 0.317754 − 0.550366i − 3.55045i 1.51947 − 1.33593i −2.64534 − 0.0464413i −3.07858 −2.76874 + 1.15503i −3.59170 + 2.07367i
62.13 −0.376715 − 0.652489i −1.73035 − 0.0766697i 0.716172 − 1.24045i 2.14750i 0.601824 + 1.15792i 2.55544 + 0.685379i −2.58603 2.98824 + 0.265331i 1.40122 − 0.808996i
62.14 −0.376715 − 0.652489i 1.73035 + 0.0766697i 0.716172 − 1.24045i − 2.14750i −0.601824 − 1.15792i 1.87127 + 1.87038i −2.58603 2.98824 + 0.265331i −1.40122 + 0.808996i
62.15 −0.230023 − 0.398412i −0.688459 − 1.58935i 0.894179 − 1.54876i − 2.35202i −0.474854 + 0.639877i 1.77437 − 1.96255i −1.74282 −2.05205 + 2.18840i −0.937075 + 0.541020i
62.16 −0.230023 − 0.398412i 0.688459 + 1.58935i 0.894179 − 1.54876i 2.35202i 0.474854 − 0.639877i −0.812435 + 2.51793i −1.74282 −2.05205 + 2.18840i 0.937075 − 0.541020i
62.17 0.230023 + 0.398412i −1.03219 − 1.39090i 0.894179 − 1.54876i − 2.35202i 0.316723 − 0.731174i −0.812435 + 2.51793i 1.74282 −0.869185 + 2.87133i 0.937075 − 0.541020i
62.18 0.230023 + 0.398412i 1.03219 + 1.39090i 0.894179 − 1.54876i 2.35202i −0.316723 + 0.731174i 1.77437 − 1.96255i 1.74282 −0.869185 + 2.87133i −0.937075 + 0.541020i
62.19 0.376715 + 0.652489i −0.798779 + 1.53686i 0.716172 − 1.24045i − 2.14750i −1.30370 + 0.0577652i 2.55544 + 0.685379i 2.58603 −1.72391 − 2.45523i 1.40122 − 0.808996i
62.20 0.376715 + 0.652489i 0.798779 − 1.53686i 0.716172 − 1.24045i 2.14750i 1.30370 − 0.0577652i 1.87127 + 1.87038i 2.58603 −1.72391 − 2.45523i −1.40122 + 0.808996i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 62.32
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
13.e even 6 1 inner
21.c even 2 1 inner
39.h odd 6 1 inner
91.t odd 6 1 inner
273.u even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.u.c ✓ 64
3.b odd 2 1 inner 273.2.u.c ✓ 64
7.b odd 2 1 inner 273.2.u.c ✓ 64
13.e even 6 1 inner 273.2.u.c ✓ 64
21.c even 2 1 inner 273.2.u.c ✓ 64
39.h odd 6 1 inner 273.2.u.c ✓ 64
91.t odd 6 1 inner 273.2.u.c ✓ 64
273.u even 6 1 inner 273.2.u.c ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.u.c ✓ 64 1.a even 1 1 trivial
273.2.u.c ✓ 64 3.b odd 2 1 inner
273.2.u.c ✓ 64 7.b odd 2 1 inner
273.2.u.c ✓ 64 13.e even 6 1 inner
273.2.u.c ✓ 64 21.c even 2 1 inner
273.2.u.c ✓ 64 39.h odd 6 1 inner
273.2.u.c ✓ 64 91.t odd 6 1 inner
273.2.u.c ✓ 64 273.u even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\):

\( T_{2}^{32} + 26 T_{2}^{30} + 404 T_{2}^{28} + 4124 T_{2}^{26} + 31221 T_{2}^{24} + 177487 T_{2}^{22} + \cdots + 80089 \) Copy content Toggle raw display
\( T_{19}^{32} + 88 T_{19}^{30} + 4767 T_{19}^{28} + 159186 T_{19}^{26} + 3849175 T_{19}^{24} + \cdots + 937519681536 \) Copy content Toggle raw display