Newform orbit 1011.2.a.d

• Label: 1011.2.a.d
• Level: $1011$
• Weight: $2$
• Character orbit: 1011.a
• Self dual: yes
• Analytic conductor: $8.073$
• Analytic rank: $1$
• Dimension: $10$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1011 = 3 \cdot 337 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1011.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(8.07287564435\)
Analytic rank:	\(1\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	\( x^{10} - 2x^{9} - 10x^{8} + 18x^{7} + 33x^{6} - 50x^{5} - 38x^{4} + 43x^{3} + 9x^{2} - 9x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
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 * q^4 + (-b9 + b7 - b4 + b3 + b2 - 1) * q^5 + b1 * q^6 + (b9 - b7 - b6 + b4 - b3 - b2 + 1) * q^7 + (-b3 - b2) * q^8 + q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{2} - 10 q^{3} + 4 q^{4} - 8 q^{5} + 2 q^{6} + 9 q^{7} - 6 q^{8} + 10 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 10x^{8} + 18x^{7} + 33x^{6} - 50x^{5} - 38x^{4} + 43x^{3} + 9x^{2} - 9x + 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 2 \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{9} - \nu^{8} - 11\nu^{7} + 7\nu^{6} + 42\nu^{5} - 12\nu^{4} - 60\nu^{3} + \nu^{2} + 18\nu - 1 ) / 2 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{9} + \nu^{8} + 13\nu^{7} - 11\nu^{6} - 56\nu^{5} + 36\nu^{4} + 90\nu^{3} - 35\nu^{2} - 42\nu + 7 ) / 2 \)
\(\beta_{6}\)	\(=\)	\( -\nu^{9} + 2\nu^{8} + 10\nu^{7} - 17\nu^{6} - 34\nu^{5} + 43\nu^{4} + 42\nu^{3} - 30\nu^{2} - 11\nu + 4 \)
\(\beta_{7}\)	\(=\)	\( -\nu^{9} + 2\nu^{8} + 10\nu^{7} - 18\nu^{6} - 33\nu^{5} + 49\nu^{4} + 39\nu^{3} - 38\nu^{2} - 12\nu + 5 \)
\(\beta_{8}\)	\(=\)	\( -\nu^{9} + 2\nu^{8} + 10\nu^{7} - 18\nu^{6} - 33\nu^{5} + 50\nu^{4} + 38\nu^{3} - 42\nu^{2} - 10\nu + 6 \)
\(\beta_{9}\)	\(=\)	\( ( -3\nu^{9} + 5\nu^{8} + 31\nu^{7} - 43\nu^{6} - 106\nu^{5} + 108\nu^{4} + 128\nu^{3} - 69\nu^{2} - 36\nu + 7 ) / 2 \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

2.42743
2.20978
1.73841
0.815847
0.341574
0.143977
−0.610688
−1.13095
−1.92428
−2.01110

−2.42743

−1.00000

3.89244

−3.30246

2.42743

1.66763

−4.59377

1.00000

8.01650

1.2

−2.20978

−1.00000

2.88312

1.89576

2.20978

3.59682

−1.95149

1.00000

−4.18921

1.3

−1.73841

−1.00000

1.02207

1.52059

1.73841

−1.71352

1.70005

1.00000

−2.64342

1.4

−0.815847

−1.00000

−1.33439

−4.03357

0.815847

1.78232

2.72035

1.00000

3.29077

1.5

−0.341574

−1.00000

−1.88333

−1.60963

0.341574

2.78735

1.32644

1.00000

0.549810

1.6

−0.143977

−1.00000

−1.97927

−0.0724815

0.143977

−1.86348

0.572924

1.00000

0.0104357

1.7

0.610688

−1.00000

−1.62706

2.64058

−0.610688

−0.324890

−2.21500

1.00000

1.61257

1.8

1.13095

−1.00000

−0.720949

−1.39264

−1.13095

1.68257

−3.07726

1.00000

−1.57500

1.9

1.92428

−1.00000

1.70284

−2.99833

−1.92428

3.85734

−0.571821

1.00000

−5.76962

1.10

2.01110

−1.00000

2.04454

−0.647820

−2.01110

−2.47215

0.0895749

1.00000

−1.30283

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( +1 \)
\(337\)	\( +1 \)

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	1011.2.a.d	✓	10
3.b	odd	2	1		3033.2.a.e		10

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1011.2.a.d	✓	10	1.a	even	1	1	trivial
3033.2.a.e		10	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 2T_{2}^{9} - 10T_{2}^{8} - 18T_{2}^{7} + 33T_{2}^{6} + 50T_{2}^{5} - 38T_{2}^{4} - 43T_{2}^{3} + 9T_{2}^{2} + 9T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1011))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^10 + 2*T^9 - 10*T^8 - 18*T^7 + 33*T^6 + 50*T^5 - 38*T^4 - 43*T^3 + 9*T^2 + 9*T + 1
$3$	\( (T + 1)^{10} \)
$5$	T^10 + 8*T^9 + 5*T^8 - 95*T^7 - 179*T^6 + 291*T^5 + 805*T^4 - 43*T^3 - 980*T^2 - 512*T - 32
$7$	T^10 - 9*T^9 + 12*T^8 + 97*T^7 - 266*T^6 - 249*T^5 + 1250*T^4 - 207*T^3 - 1924*T^2 + 968*T + 496
$11$	T^10 + 12*T^9 + 16*T^8 - 294*T^7 - 1070*T^6 + 1122*T^5 + 9385*T^4 + 6749*T^3 - 20262*T^2 - 31674*T - 10726