Properties

Label 273.2.a.a
Level $273$
Weight $2$
Character orbit 273.a
Self dual yes
Analytic conductor $2.180$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [273,2,Mod(1,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.a (trivial)

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: yes
Analytic conductor: \(2.17991597518\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 2 q^{2} - q^{3} + 2 q^{4} - q^{5} + 2 q^{6} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - q^{3} + 2 q^{4} - q^{5} + 2 q^{6} + q^{7} + q^{9} + 2 q^{10} - 2 q^{11} - 2 q^{12} + q^{13} - 2 q^{14} + q^{15} - 4 q^{16} - 4 q^{17} - 2 q^{18} + 3 q^{19} - 2 q^{20} - q^{21} + 4 q^{22} - 9 q^{23} - 4 q^{25} - 2 q^{26} - q^{27} + 2 q^{28} - q^{29} - 2 q^{30} - 5 q^{31} + 8 q^{32} + 2 q^{33} + 8 q^{34} - q^{35} + 2 q^{36} - 8 q^{37} - 6 q^{38} - q^{39} + 6 q^{41} + 2 q^{42} - 9 q^{43} - 4 q^{44} - q^{45} + 18 q^{46} - 3 q^{47} + 4 q^{48} + q^{49} + 8 q^{50} + 4 q^{51} + 2 q^{52} + 3 q^{53} + 2 q^{54} + 2 q^{55} - 3 q^{57} + 2 q^{58} + 2 q^{60} + 10 q^{61} + 10 q^{62} + q^{63} - 8 q^{64} - q^{65} - 4 q^{66} - 2 q^{67} - 8 q^{68} + 9 q^{69} + 2 q^{70} + 12 q^{71} + 5 q^{73} + 16 q^{74} + 4 q^{75} + 6 q^{76} - 2 q^{77} + 2 q^{78} - 13 q^{79} + 4 q^{80} + q^{81} - 12 q^{82} - 11 q^{83} - 2 q^{84} + 4 q^{85} + 18 q^{86} + q^{87} + q^{89} + 2 q^{90} + q^{91} - 18 q^{92} + 5 q^{93} + 6 q^{94} - 3 q^{95} - 8 q^{96} + q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−2.00000 −1.00000 2.00000 −1.00000 2.00000 1.00000 0 1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.a.a 1
3.b odd 2 1 819.2.a.e 1
4.b odd 2 1 4368.2.a.q 1
5.b even 2 1 6825.2.a.l 1
7.b odd 2 1 1911.2.a.a 1
13.b even 2 1 3549.2.a.d 1
21.c even 2 1 5733.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.a.a 1 1.a even 1 1 trivial
819.2.a.e 1 3.b odd 2 1
1911.2.a.a 1 7.b odd 2 1
3549.2.a.d 1 13.b even 2 1
4368.2.a.q 1 4.b odd 2 1
5733.2.a.m 1 21.c even 2 1
6825.2.a.l 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(273))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T + 2 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T + 4 \) Copy content Toggle raw display
$19$ \( T - 3 \) Copy content Toggle raw display
$23$ \( T + 9 \) Copy content Toggle raw display
$29$ \( T + 1 \) Copy content Toggle raw display
$31$ \( T + 5 \) Copy content Toggle raw display
$37$ \( T + 8 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 9 \) Copy content Toggle raw display
$47$ \( T + 3 \) Copy content Toggle raw display
$53$ \( T - 3 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 10 \) Copy content Toggle raw display
$67$ \( T + 2 \) Copy content Toggle raw display
$71$ \( T - 12 \) Copy content Toggle raw display
$73$ \( T - 5 \) Copy content Toggle raw display
$79$ \( T + 13 \) Copy content Toggle raw display
$83$ \( T + 11 \) Copy content Toggle raw display
$89$ \( T - 1 \) Copy content Toggle raw display
$97$ \( T - 1 \) Copy content Toggle raw display
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