Properties

Label 273.1.x.a
Level $273$
Weight $1$
Character orbit 273.x
Analytic conductor $0.136$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 273.x (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.136244748449\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.24069811311.4

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - q^{3} -\zeta_{6}^{2} q^{4} -\zeta_{6} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} -\zeta_{6}^{2} q^{4} -\zeta_{6} q^{7} + q^{9} + \zeta_{6}^{2} q^{12} -\zeta_{6}^{2} q^{13} -\zeta_{6} q^{16} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{19} + \zeta_{6} q^{21} + \zeta_{6} q^{25} - q^{27} - q^{28} -\zeta_{6}^{2} q^{36} + ( -1 + \zeta_{6}^{2} ) q^{37} + \zeta_{6}^{2} q^{39} + \zeta_{6} q^{43} + \zeta_{6} q^{48} + \zeta_{6}^{2} q^{49} -\zeta_{6} q^{52} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{57} + q^{61} -\zeta_{6} q^{63} - q^{64} + ( 1 - \zeta_{6}^{2} ) q^{73} -\zeta_{6} q^{75} + ( 1 + \zeta_{6} ) q^{76} -2 \zeta_{6}^{2} q^{79} + q^{81} + q^{84} - q^{91} + ( -1 + \zeta_{6}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + q^{4} - q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + q^{4} - q^{7} + 2q^{9} - q^{12} + q^{13} - q^{16} + q^{21} + q^{25} - 2q^{27} - 2q^{28} + q^{36} - 3q^{37} - q^{39} + q^{43} + q^{48} - q^{49} - q^{52} + 2q^{61} - q^{63} - 2q^{64} + 3q^{73} - q^{75} + 3q^{76} + 2q^{79} + 2q^{81} + 2q^{84} - 2q^{91} - 3q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\) \(\zeta_{6}^{2}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.00000 0.500000 + 0.866025i 0 0 −0.500000 + 0.866025i 0 1.00000 0
212.1 0 −1.00000 0.500000 0.866025i 0 0 −0.500000 0.866025i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
91.u even 6 1 inner
273.x odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.1.x.a 2
3.b odd 2 1 CM 273.1.x.a 2
7.b odd 2 1 1911.1.x.a 2
7.c even 3 1 273.1.bp.a yes 2
7.c even 3 1 1911.1.bc.b 2
7.d odd 6 1 1911.1.bc.a 2
7.d odd 6 1 1911.1.bp.a 2
13.b even 2 1 3549.1.x.a 2
13.c even 3 1 3549.1.w.b 2
13.c even 3 1 3549.1.bp.a 2
13.d odd 4 2 3549.1.bm.b 4
13.e even 6 1 273.1.bp.a yes 2
13.e even 6 1 3549.1.w.a 2
13.f odd 12 2 3549.1.s.c 4
13.f odd 12 2 3549.1.bk.e 4
21.c even 2 1 1911.1.x.a 2
21.g even 6 1 1911.1.bc.a 2
21.g even 6 1 1911.1.bp.a 2
21.h odd 6 1 273.1.bp.a yes 2
21.h odd 6 1 1911.1.bc.b 2
39.d odd 2 1 3549.1.x.a 2
39.f even 4 2 3549.1.bm.b 4
39.h odd 6 1 273.1.bp.a yes 2
39.h odd 6 1 3549.1.w.a 2
39.i odd 6 1 3549.1.w.b 2
39.i odd 6 1 3549.1.bp.a 2
39.k even 12 2 3549.1.s.c 4
39.k even 12 2 3549.1.bk.e 4
91.g even 3 1 3549.1.x.a 2
91.h even 3 1 3549.1.w.a 2
91.k even 6 1 1911.1.bc.b 2
91.k even 6 1 3549.1.w.b 2
91.l odd 6 1 1911.1.bc.a 2
91.p odd 6 1 1911.1.x.a 2
91.r even 6 1 3549.1.bp.a 2
91.t odd 6 1 1911.1.bp.a 2
91.u even 6 1 inner 273.1.x.a 2
91.x odd 12 2 3549.1.bk.e 4
91.z odd 12 2 3549.1.s.c 4
91.bd odd 12 2 3549.1.bm.b 4
273.s odd 6 1 3549.1.w.a 2
273.u even 6 1 1911.1.bp.a 2
273.w odd 6 1 3549.1.bp.a 2
273.x odd 6 1 inner 273.1.x.a 2
273.y even 6 1 1911.1.x.a 2
273.bm odd 6 1 3549.1.x.a 2
273.bp odd 6 1 1911.1.bc.b 2
273.bp odd 6 1 3549.1.w.b 2
273.br even 6 1 1911.1.bc.a 2
273.bv even 12 2 3549.1.bk.e 4
273.bw even 12 2 3549.1.bm.b 4
273.cd even 12 2 3549.1.s.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.1.x.a 2 1.a even 1 1 trivial
273.1.x.a 2 3.b odd 2 1 CM
273.1.x.a 2 91.u even 6 1 inner
273.1.x.a 2 273.x odd 6 1 inner
273.1.bp.a yes 2 7.c even 3 1
273.1.bp.a yes 2 13.e even 6 1
273.1.bp.a yes 2 21.h odd 6 1
273.1.bp.a yes 2 39.h odd 6 1
1911.1.x.a 2 7.b odd 2 1
1911.1.x.a 2 21.c even 2 1
1911.1.x.a 2 91.p odd 6 1
1911.1.x.a 2 273.y even 6 1
1911.1.bc.a 2 7.d odd 6 1
1911.1.bc.a 2 21.g even 6 1
1911.1.bc.a 2 91.l odd 6 1
1911.1.bc.a 2 273.br even 6 1
1911.1.bc.b 2 7.c even 3 1
1911.1.bc.b 2 21.h odd 6 1
1911.1.bc.b 2 91.k even 6 1
1911.1.bc.b 2 273.bp odd 6 1
1911.1.bp.a 2 7.d odd 6 1
1911.1.bp.a 2 21.g even 6 1
1911.1.bp.a 2 91.t odd 6 1
1911.1.bp.a 2 273.u even 6 1
3549.1.s.c 4 13.f odd 12 2
3549.1.s.c 4 39.k even 12 2
3549.1.s.c 4 91.z odd 12 2
3549.1.s.c 4 273.cd even 12 2
3549.1.w.a 2 13.e even 6 1
3549.1.w.a 2 39.h odd 6 1
3549.1.w.a 2 91.h even 3 1
3549.1.w.a 2 273.s odd 6 1
3549.1.w.b 2 13.c even 3 1
3549.1.w.b 2 39.i odd 6 1
3549.1.w.b 2 91.k even 6 1
3549.1.w.b 2 273.bp odd 6 1
3549.1.x.a 2 13.b even 2 1
3549.1.x.a 2 39.d odd 2 1
3549.1.x.a 2 91.g even 3 1
3549.1.x.a 2 273.bm odd 6 1
3549.1.bk.e 4 13.f odd 12 2
3549.1.bk.e 4 39.k even 12 2
3549.1.bk.e 4 91.x odd 12 2
3549.1.bk.e 4 273.bv even 12 2
3549.1.bm.b 4 13.d odd 4 2
3549.1.bm.b 4 39.f even 4 2
3549.1.bm.b 4 91.bd odd 12 2
3549.1.bm.b 4 273.bw even 12 2
3549.1.bp.a 2 13.c even 3 1
3549.1.bp.a 2 39.i odd 6 1
3549.1.bp.a 2 91.r even 6 1
3549.1.bp.a 2 273.w odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(273, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 1 - T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 3 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( 3 + 3 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -1 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 3 - 3 T + T^{2} \)
$79$ \( 4 - 2 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 3 + 3 T + T^{2} \)
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