Newform orbit 273.2.c.b

• Label: 273.2.c.b
• Level: $273$
• Weight: $2$
• Character orbit: 273.c
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17991597518\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	6.0.350464.1
Defining polynomial:	\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{2} + q^{3} + (\beta_{2} - \beta_1) q^{4} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{5} - \beta_{4} q^{6} - \beta_{3} q^{7} + ( - \beta_{5} + \beta_{3}) q^{8} + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 2 q^{4} + 6 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \)
\(\beta_{2}\)	\(=\)	\( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \)
\(\beta_{3}\)	\(=\)	\( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \)
\(\beta_{4}\)	\(=\)	\( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \)
\(\beta_{5}\)	\(=\)	\( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

64.1

−0.854638	−	0.854638i
0.403032	−	0.403032i
1.45161	+	1.45161i
1.45161	−	1.45161i
0.403032	+	0.403032i
−0.854638	+	0.854638i

−

2.17009i

1.00000

−2.70928

0.630898i

−

2.17009i

−

1.00000i

1.53919i

1.00000

1.36910

64.2

−

1.48119i

1.00000

−0.193937

4.15633i

−

1.48119i

1.00000i

−

2.67513i

1.00000

6.15633

64.3

−

0.311108i

1.00000

1.90321

1.52543i

−

0.311108i

−

1.00000i

−

1.21432i

1.00000

0.474572

64.4

0.311108i

1.00000

1.90321

−

1.52543i

0.311108i

1.00000i

1.21432i

1.00000

0.474572

64.5

1.48119i

1.00000

−0.193937

−

4.15633i

1.48119i

−

1.00000i

2.67513i

1.00000

6.15633

64.6

2.17009i

1.00000

−2.70928

−

0.630898i

2.17009i

1.00000i

−

1.53919i

1.00000

1.36910

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	273.2.c.b	✓	6
3.b	odd	2	1		819.2.c.c		6
4.b	odd	2	1		4368.2.h.o		6
7.b	odd	2	1		1911.2.c.h		6
13.b	even	2	1	inner	273.2.c.b	✓	6
13.d	odd	4	1		3549.2.a.k		3
13.d	odd	4	1		3549.2.a.q		3
39.d	odd	2	1		819.2.c.c		6
52.b	odd	2	1		4368.2.h.o		6
91.b	odd	2	1		1911.2.c.h		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
273.2.c.b	✓	6	1.a	even	1	1	trivial
273.2.c.b	✓	6	13.b	even	2	1	inner
819.2.c.c		6	3.b	odd	2	1
819.2.c.c		6	39.d	odd	2	1
1911.2.c.h		6	7.b	odd	2	1
1911.2.c.h		6	91.b	odd	2	1
3549.2.a.k		3	13.d	odd	4	1
3549.2.a.q		3	13.d	odd	4	1
4368.2.h.o		6	4.b	odd	2	1
4368.2.h.o		6	52.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 7T_{2}^{4} + 11T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{6} + 7 T^{4} + 11 T^{2} + 1 \)
$3$	\( (T - 1)^{6} \)
$5$	\( T^{6} + 20 T^{4} + 48 T^{2} + 16 \)
$7$	\( (T^{2} + 1)^{3} \)
$11$	T^6 + 44*T^4 + 384*T^2 + 400