Properties

Label 273.2.a.e
Level $273$
Weight $2$
Character orbit 273.a
Self dual yes
Analytic conductor $2.180$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} - 1) q^{5} + \beta_1 q^{6} + q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} - 1) q^{5} + \beta_1 q^{6} + q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + q^{9} + ( - \beta_{3} - 3 \beta_1) q^{10} + ( - \beta_{3} - \beta_1) q^{11} + (\beta_{2} + 2) q^{12} + q^{13} + \beta_1 q^{14} + ( - \beta_{2} - 1) q^{15} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{16} - 2 \beta_1 q^{17} + \beta_1 q^{18} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{19} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 8) q^{20} + q^{21} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{22} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{23} + (\beta_{3} + 2 \beta_1) q^{24} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{25} + \beta_1 q^{26} + q^{27} + (\beta_{2} + 2) q^{28} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{29} + ( - \beta_{3} - 3 \beta_1) q^{30} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{31} + (2 \beta_{2} + \beta_1 + 2) q^{32} + ( - \beta_{3} - \beta_1) q^{33} + ( - 2 \beta_{2} - 8) q^{34} + ( - \beta_{2} - 1) q^{35} + (\beta_{2} + 2) q^{36} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 4) q^{37}+ \cdots + ( - \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 4 q^{3} + 7 q^{4} - 3 q^{5} + q^{6} + 4 q^{7} + 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 4 q^{3} + 7 q^{4} - 3 q^{5} + q^{6} + 4 q^{7} + 3 q^{8} + 4 q^{9} - 4 q^{10} - 2 q^{11} + 7 q^{12} + 4 q^{13} + q^{14} - 3 q^{15} + 9 q^{16} - 2 q^{17} + q^{18} + 7 q^{19} - 32 q^{20} + 4 q^{21} - 8 q^{22} + 3 q^{23} + 3 q^{24} + 9 q^{25} + q^{26} + 4 q^{27} + 7 q^{28} + q^{29} - 4 q^{30} + 3 q^{31} + 7 q^{32} - 2 q^{33} - 30 q^{34} - 3 q^{35} + 7 q^{36} + 10 q^{37} + 6 q^{38} + 4 q^{39} - 14 q^{40} - 16 q^{41} + q^{42} + 3 q^{43} - 12 q^{44} - 3 q^{45} - 18 q^{46} + 5 q^{47} + 9 q^{48} + 4 q^{49} + 13 q^{50} - 2 q^{51} + 7 q^{52} + 5 q^{53} + q^{54} + 10 q^{55} + 3 q^{56} + 7 q^{57} - 4 q^{58} - 20 q^{59} - 32 q^{60} + 12 q^{61} - 54 q^{62} + 4 q^{63} + 5 q^{64} - 3 q^{65} - 8 q^{66} - 22 q^{67} - 10 q^{68} + 3 q^{69} - 4 q^{70} + 3 q^{72} - 13 q^{73} - 6 q^{74} + 9 q^{75} - 6 q^{76} - 2 q^{77} + q^{78} + 11 q^{79} - 42 q^{80} + 4 q^{81} + 10 q^{82} + q^{83} + 7 q^{84} + 8 q^{85} - 10 q^{86} + q^{87} - 60 q^{88} - 5 q^{89} - 4 q^{90} + 4 q^{91} + 34 q^{92} + 3 q^{93} + 34 q^{94} + 13 q^{95} + 7 q^{96} - 17 q^{97} + q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + 2\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + \beta_{2} + 2\beta _1 + 1 \) 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
−0.787711
1.52616
−2.10710
2.36865
−2.37951 1.00000 3.66208 −2.66208 −2.37951 1.00000 −3.95493 1.00000 6.33445
1.2 −0.670843 1.00000 −1.54997 2.54997 −0.670843 1.00000 2.38147 1.00000 −1.71063
1.3 1.43986 1.00000 0.0731828 0.926817 1.43986 1.00000 −2.77434 1.00000 1.33448
1.4 2.61050 1.00000 4.81471 −3.81471 2.61050 1.00000 7.34780 1.00000 −9.95830
\(n\): 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.e 4
3.b odd 2 1 819.2.a.k 4
4.b odd 2 1 4368.2.a.br 4
5.b even 2 1 6825.2.a.bg 4
7.b odd 2 1 1911.2.a.s 4
13.b even 2 1 3549.2.a.w 4
21.c even 2 1 5733.2.a.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.a.e 4 1.a even 1 1 trivial
819.2.a.k 4 3.b odd 2 1
1911.2.a.s 4 7.b odd 2 1
3549.2.a.w 4 13.b even 2 1
4368.2.a.br 4 4.b odd 2 1
5733.2.a.bf 4 21.c even 2 1
6825.2.a.bg 4 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} - 7T_{2}^{2} + 5T_{2} + 6 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(273))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 7 T^{2} + \cdots + 6 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 3 T^{3} + \cdots + 24 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 96 \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + \cdots + 96 \) Copy content Toggle raw display
$19$ \( T^{4} - 7 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{4} - 3 T^{3} + \cdots - 288 \) Copy content Toggle raw display
$29$ \( T^{4} - T^{3} + \cdots + 72 \) Copy content Toggle raw display
$31$ \( T^{4} - 3 T^{3} + \cdots + 3968 \) Copy content Toggle raw display
$37$ \( T^{4} - 10 T^{3} + \cdots - 128 \) Copy content Toggle raw display
$41$ \( T^{4} + 16 T^{3} + \cdots - 1392 \) Copy content Toggle raw display
$43$ \( T^{4} - 3 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$47$ \( T^{4} - 5 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$53$ \( T^{4} - 5 T^{3} + \cdots - 24 \) Copy content Toggle raw display
$59$ \( T^{4} + 20 T^{3} + \cdots - 1536 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + \cdots + 496 \) Copy content Toggle raw display
$67$ \( T^{4} + 22 T^{3} + \cdots - 15488 \) Copy content Toggle raw display
$71$ \( T^{4} - 232 T^{2} + \cdots + 10176 \) Copy content Toggle raw display
$73$ \( T^{4} + 13 T^{3} + \cdots - 11672 \) Copy content Toggle raw display
$79$ \( T^{4} - 11 T^{3} + \cdots - 3456 \) Copy content Toggle raw display
$83$ \( T^{4} - T^{3} + \cdots - 48 \) Copy content Toggle raw display
$89$ \( T^{4} + 5 T^{3} + \cdots - 1704 \) Copy content Toggle raw display
$97$ \( T^{4} + 17 T^{3} + \cdots - 1528 \) Copy content Toggle raw display
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