Newform orbit 1077.2.a.i

• Label: 1077.2.a.i
• Level: $1077$
• Weight: $2$
• Character orbit: 1077.a
• Self dual: yes
• Analytic conductor: $8.600$
• Analytic rank: $1$
• Dimension: $10$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1077 = 3 \cdot 359 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1077.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(8.59988829769\)
Analytic rank:	\(1\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	\( x^{10} - 13x^{8} - x^{7} + 57x^{6} + 5x^{5} - 96x^{4} - x^{3} + 46x^{2} - 15x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - b1 * q^2 - q^3 + (b2 + 1) * q^4 + (b7 + 1) * q^5 + b1 * q^6 + (b5 + b4 + b1 - 1) * q^7 + (-b9 - b8 - b7 - b4 - b2 - 1) * q^8 + q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 6 q^{4} + 6 q^{5} - 11 q^{7} - 3 q^{8} + 10 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 13x^{8} - x^{7} + 57x^{6} + 5x^{5} - 96x^{4} - x^{3} + 46x^{2} - 15x + 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{9} + \nu^{8} - 12\nu^{7} - 13\nu^{6} + 44\nu^{5} + 53\nu^{4} - 47\nu^{3} - 76\nu^{2} - 10\nu + 15 ) / 4 \)
\(\beta_{4}\)	\(=\)	\( ( -3\nu^{9} - \nu^{8} + 40\nu^{7} + 15\nu^{6} - 178\nu^{5} - 65\nu^{4} + 297\nu^{3} + 82\nu^{2} - 132\nu + 17 ) / 4 \)
\(\beta_{5}\)	\(=\)	\( ( 3\nu^{9} + 3\nu^{8} - 40\nu^{7} - 39\nu^{6} + 176\nu^{5} + 155\nu^{4} - 285\nu^{3} - 192\nu^{2} + 110\nu + 1 ) / 4 \)
\(\beta_{6}\)	\(=\)	\( ( 2\nu^{9} + \nu^{8} - 26\nu^{7} - 14\nu^{6} + 113\nu^{5} + 57\nu^{4} - 188\nu^{3} - 69\nu^{2} + 89\nu - 7 ) / 2 \)
\(\beta_{7}\)	\(=\)	\( ( 5\nu^{9} + \nu^{8} - 64\nu^{7} - 17\nu^{6} + 272\nu^{5} + 73\nu^{4} - 431\nu^{3} - 84\nu^{2} + 178\nu - 29 ) / 4 \)
\(\beta_{8}\)	\(=\)	\( ( 5\nu^{9} + \nu^{8} - 64\nu^{7} - 21\nu^{6} + 272\nu^{5} + 109\nu^{4} - 423\nu^{3} - 164\nu^{2} + 150\nu - 5 ) / 4 \)
\(\beta_{9}\)	\(=\)	\( ( -7\nu^{9} - \nu^{8} + 88\nu^{7} + 23\nu^{6} - 366\nu^{5} - 117\nu^{4} + 561\nu^{3} + 162\nu^{2} - 212\nu + 25 ) / 4 \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

2.51242
1.91892
1.66541
0.474183
0.338744
0.0922564
−1.18707
−1.59901
−1.98320
−2.23265

−2.51242

−1.00000

4.31226

2.08086

2.51242

−2.62582

−5.80936

1.00000

−5.22800

1.2

−1.91892

−1.00000

1.68224

−0.986126

1.91892

−0.773935

0.609749

1.00000

1.89229

1.3

−1.66541

−1.00000

0.773599

2.24950

1.66541

1.84117

2.04246

1.00000

−3.74635

1.4

−0.474183

−1.00000

−1.77515

1.06149

0.474183

−3.43869

1.79011

1.00000

−0.503338

1.5

−0.338744

−1.00000

−1.88525

2.75527

0.338744

−0.778249

1.31611

1.00000

−0.933330

1.6

−0.0922564

−1.00000

−1.99149

−2.40616

0.0922564

2.85477

0.368240

1.00000

0.221984

1.7

1.18707

−1.00000

−0.590861

3.91311

−1.18707

−3.88863

−3.07554

1.00000

4.64515

1.8

1.59901

−1.00000

0.556833

−0.231919

−1.59901

1.51410

−2.30764

1.00000

−0.370841

1.9

1.98320

−1.00000

1.93307

−1.84893

−1.98320

−1.14204

−0.132731

1.00000

−3.66679

1.10

2.23265

−1.00000

2.98475

−0.587093

−2.23265

−4.56268

2.19860

1.00000

−1.31078

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( +1 \)
\(359\)	\( +1 \)

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	1077.2.a.i	✓	10
3.b	odd	2	1		3231.2.a.n		10

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1077.2.a.i	✓	10	1.a	even	1	1	trivial
3231.2.a.n		10	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1077))\):

\( T_{2}^{10} - 13T_{2}^{8} + T_{2}^{7} + 57T_{2}^{6} - 5T_{2}^{5} - 96T_{2}^{4} + T_{2}^{3} + 46T_{2}^{2} + 15T_{2} + 1 \)

\( T_{5}^{10} - 6T_{5}^{9} - 4T_{5}^{8} + 69T_{5}^{7} - 34T_{5}^{6} - 255T_{5}^{5} + 154T_{5}^{4} + 359T_{5}^{3} - 92T_{5}^{2} - 176T_{5} - 32 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^10 - 13*T^8 + T^7 + 57*T^6 - 5*T^5 - 96*T^4 + T^3 + 46*T^2 + 15*T + 1
$3$	\( (T + 1)^{10} \)
$5$	T^10 - 6*T^9 - 4*T^8 + 69*T^7 - 34*T^6 - 255*T^5 + 154*T^4 + 359*T^3 - 92*T^2 - 176*T - 32
$7$	T^10 + 11*T^9 + 25*T^8 - 108*T^7 - 474*T^6 - 5*T^5 + 1888*T^4 + 1592*T^3 - 1753*T^2 - 2672*T - 877
$11$	T^10 + 12*T^9 + 12*T^8 - 282*T^7 - 626*T^6 + 2300*T^5 + 5677*T^4 - 6855*T^3 - 17126*T^2 + 3621*T + 10082