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Properties

Label 273.2.n.c
Level $273$
Weight $2$
Character orbit 273.n
Analytic conductor $2.180$
Analytic rank $0$
Dimension $48$
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] = [273,2,Mod(8,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("273.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.n (of order \(4\), 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: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{3} + 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{3} + 4 q^{6} + 8 q^{13} - 16 q^{15} - 72 q^{16} - 12 q^{18} + 40 q^{19} + 16 q^{22} + 8 q^{24} - 16 q^{27} + 44 q^{33} - 32 q^{34} - 8 q^{37} - 4 q^{39} - 48 q^{40} - 8 q^{42} + 44 q^{45} - 32 q^{46} + 80 q^{48} - 72 q^{52} + 44 q^{54} - 80 q^{55} - 52 q^{57} + 16 q^{58} + 44 q^{60} - 64 q^{61} + 24 q^{63} - 152 q^{66} + 56 q^{67} + 16 q^{70} + 16 q^{72} + 32 q^{73} + 104 q^{76} - 44 q^{78} + 8 q^{79} + 12 q^{84} - 96 q^{85} - 72 q^{87} - 8 q^{91} - 8 q^{93} + 160 q^{94} + 8 q^{96} - 32 q^{97} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −1.89860 + 1.89860i −1.57000 − 0.731512i − 5.20935i −0.367301 + 0.367301i 4.36964 − 1.59195i 0.707107 − 0.707107i 6.09327 + 6.09327i 1.92978 + 2.29694i − 1.39472i
8.2 −1.76503 + 1.76503i −0.743367 + 1.56442i − 4.23065i −0.269592 + 0.269592i −1.44918 − 4.07331i −0.707107 + 0.707107i 3.93715 + 3.93715i −1.89481 − 2.32587i − 0.951676i
8.3 −1.72156 + 1.72156i 0.578040 − 1.63275i − 3.92752i −1.26580 + 1.26580i 1.81574 + 3.80600i −0.707107 + 0.707107i 3.31834 + 3.31834i −2.33174 − 1.88759i − 4.35829i
8.4 −1.65298 + 1.65298i 0.818165 + 1.52663i − 3.46469i 1.96311 − 1.96311i −3.87590 − 1.17108i 0.707107 − 0.707107i 2.42111 + 2.42111i −1.66121 + 2.49807i 6.48995i
8.5 −1.48770 + 1.48770i 1.53135 − 0.809292i − 2.42651i 0.854208 − 0.854208i −1.07421 + 3.48218i 0.707107 − 0.707107i 0.634513 + 0.634513i 1.69009 − 2.47863i 2.54161i
8.6 −1.34745 + 1.34745i −0.469658 − 1.66716i − 1.63125i 3.06725 − 3.06725i 2.87926 + 1.61358i −0.707107 + 0.707107i −0.496878 − 0.496878i −2.55884 + 1.56599i 8.26592i
8.7 −1.08486 + 1.08486i 1.14443 + 1.30011i − 0.353851i −0.879295 + 0.879295i −2.65199 − 0.168890i −0.707107 + 0.707107i −1.78584 − 1.78584i −0.380562 + 2.97576i − 1.90783i
8.8 −0.946941 + 0.946941i −1.73159 − 0.0399728i 0.206606i 1.18824 − 1.18824i 1.67756 − 1.60186i 0.707107 − 0.707107i −2.08953 − 2.08953i 2.99680 + 0.138433i 2.25040i
8.9 −0.636874 + 0.636874i 1.71860 − 0.215472i 1.18878i 1.55282 − 1.55282i −0.957301 + 1.23176i −0.707107 + 0.707107i −2.03085 − 2.03085i 2.90714 − 0.740618i 1.97791i
8.10 −0.217303 + 0.217303i 0.368455 + 1.69241i 1.90556i −1.58813 + 1.58813i −0.447831 − 0.287699i 0.707107 − 0.707107i −0.848689 − 0.848689i −2.72848 + 1.24715i − 0.690209i
8.11 −0.150864 + 0.150864i −1.22804 − 1.22144i 1.95448i 0.482343 − 0.482343i 0.369539 0.000996021i −0.707107 + 0.707107i −0.596590 − 0.596590i 0.0161717 + 2.99996i 0.145537i
8.12 −0.0762157 + 0.0762157i 1.58361 − 0.701549i 1.98838i −2.41817 + 2.41817i −0.0672271 + 0.174165i 0.707107 − 0.707107i −0.303978 − 0.303978i 2.01566 − 2.22196i − 0.368606i
8.13 0.0762157 − 0.0762157i 1.58361 + 0.701549i 1.98838i 2.41817 − 2.41817i 0.174165 − 0.0672271i 0.707107 − 0.707107i 0.303978 + 0.303978i 2.01566 + 2.22196i − 0.368606i
8.14 0.150864 − 0.150864i −1.22804 + 1.22144i 1.95448i −0.482343 + 0.482343i −0.000996021 0.369539i −0.707107 + 0.707107i 0.596590 + 0.596590i 0.0161717 − 2.99996i 0.145537i
8.15 0.217303 − 0.217303i 0.368455 − 1.69241i 1.90556i 1.58813 − 1.58813i −0.287699 − 0.447831i 0.707107 − 0.707107i 0.848689 + 0.848689i −2.72848 − 1.24715i − 0.690209i
8.16 0.636874 − 0.636874i 1.71860 + 0.215472i 1.18878i −1.55282 + 1.55282i 1.23176 − 0.957301i −0.707107 + 0.707107i 2.03085 + 2.03085i 2.90714 + 0.740618i 1.97791i
8.17 0.946941 − 0.946941i −1.73159 + 0.0399728i 0.206606i −1.18824 + 1.18824i −1.60186 + 1.67756i 0.707107 − 0.707107i 2.08953 + 2.08953i 2.99680 − 0.138433i 2.25040i
8.18 1.08486 − 1.08486i 1.14443 − 1.30011i − 0.353851i 0.879295 − 0.879295i −0.168890 − 2.65199i −0.707107 + 0.707107i 1.78584 + 1.78584i −0.380562 − 2.97576i − 1.90783i
8.19 1.34745 − 1.34745i −0.469658 + 1.66716i − 1.63125i −3.06725 + 3.06725i 1.61358 + 2.87926i −0.707107 + 0.707107i 0.496878 + 0.496878i −2.55884 − 1.56599i 8.26592i
8.20 1.48770 − 1.48770i 1.53135 + 0.809292i − 2.42651i −0.854208 + 0.854208i 3.48218 − 1.07421i 0.707107 − 0.707107i −0.634513 − 0.634513i 1.69009 + 2.47863i 2.54161i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.n.c ✓ 48
3.b odd 2 1 inner 273.2.n.c ✓ 48
13.d odd 4 1 inner 273.2.n.c ✓ 48
39.f even 4 1 inner 273.2.n.c ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.n.c ✓ 48 1.a even 1 1 trivial
273.2.n.c ✓ 48 3.b odd 2 1 inner
273.2.n.c ✓ 48 13.d odd 4 1 inner
273.2.n.c ✓ 48 39.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 198 T_{2}^{44} + 16135 T_{2}^{40} + 700244 T_{2}^{36} + 17482631 T_{2}^{32} + 253103622 T_{2}^{28} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display