Newform orbit 503.2.a.e

• Label: 503.2.a.e
• Level: $503$
• Weight: $2$
• Character orbit: 503.a
• Self dual: yes
• Analytic conductor: $4.016$
• Analytic rank: $1$
• Dimension: $10$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	503.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(4.01647522167\)
Analytic rank:	\(1\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	\( x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
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 - b9 * q^2 + (b1 - 1) * q^3 + b5 * q^4 - b2 * q^5 + (b9 + b6 - b1) * q^6 + (b9 - b5 - b4 + b2 - b1) * q^7 + (b9 - b8 - b5 + b3 + b2) * q^8 + (-b6 + b4 - b3 - b1 - 1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{2} - 8 q^{3} + 4 q^{4} - q^{5} - 2 q^{6} - 5 q^{7} - 3 q^{8} - 2 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( 3\nu^{9} - 8\nu^{8} - 22\nu^{7} + 57\nu^{6} + 46\nu^{5} - 113\nu^{4} - 34\nu^{3} + 65\nu^{2} + 12\nu - 5 \)
\(\beta_{3}\)	\(=\)	\( 5\nu^{9} - 13\nu^{8} - 37\nu^{7} + 92\nu^{6} + 78\nu^{5} - 181\nu^{4} - 57\nu^{3} + 104\nu^{2} + 20\nu - 8 \)
\(\beta_{4}\)	\(=\)	\( -6\nu^{9} + 15\nu^{8} + 47\nu^{7} - 108\nu^{6} - 113\nu^{5} + 221\nu^{4} + 108\nu^{3} - 139\nu^{2} - 44\nu + 13 \)
\(\beta_{5}\)	\(=\)	\( 10\nu^{9} - 25\nu^{8} - 78\nu^{7} + 180\nu^{6} + 184\nu^{5} - 368\nu^{4} - 165\nu^{3} + 230\nu^{2} + 60\nu - 21 \)
\(\beta_{6}\)	\(=\)	\( -11\nu^{9} + 28\nu^{8} + 84\nu^{7} - 200\nu^{6} - 191\nu^{5} + 402\nu^{4} + 165\nu^{3} - 244\nu^{2} - 63\nu + 22 \)
\(\beta_{7}\)	\(=\)	\( -18\nu^{9} + 45\nu^{8} + 139\nu^{7} - 321\nu^{6} - 321\nu^{5} + 645\nu^{4} + 278\nu^{3} - 394\nu^{2} - 102\nu + 38 \)
\(\beta_{8}\)	\(=\)	\( -21\nu^{9} + 53\nu^{8} + 162\nu^{7} - 380\nu^{6} - 375\nu^{5} + 770\nu^{4} + 331\nu^{3} - 475\nu^{2} - 127\nu + 45 \)
\(\beta_{9}\)	\(=\)	\( -22\nu^{9} + 55\nu^{8} + 170\nu^{7} - 392\nu^{6} - 394\nu^{5} + 785\nu^{4} + 346\nu^{3} - 473\nu^{2} - 130\nu + 42 \)

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

−0.858231
2.78533
−0.489003
1.07636
−2.07227
1.95007
1.31567
0.208270
−1.40552
−0.510671

−2.58686

−1.85823

4.69185

1.44291

4.80698

−1.96509

−6.96343

0.453023

−3.73261

1.2

−2.03947

1.78533

2.15945

−0.701114

−3.64112

−2.02991

−0.325186

0.187388

1.42990

1.3

−1.62786

−1.48900

0.649933

−1.79865

2.42389

0.552233

2.19772

−0.782869

2.92795

1.4

−1.37178

0.0763625

−0.118218

1.17276

−0.104753

0.469303

2.90573

−2.99417

−1.60876

1.5

−0.756417

−3.07227

−1.42783

0.386144

2.32392

0.194914

2.59287

6.43884

−0.292086

1.6

−0.392284

0.950069

−1.84611

−2.28693

−0.372697

2.71022

1.50877

−2.09737

0.897127

1.7

0.0830530

0.315672

−1.99310

2.25024

0.0262175

−3.20647

−0.331639

−2.90035

0.186890

1.8

1.17266

−0.791730

−0.624870

0.178789

−0.928430

−0.0809018

−3.07808

−2.37316

0.209658

1.9

1.36113

−2.40552

−0.147314

0.590303

−3.27424

1.95900

−2.92278

2.78655

0.803481

1.10

2.15783

−1.51067

2.65622

−2.23445

−3.25977

−3.60329

1.41602

−0.717874

−4.82156

Atkin-Lehner signs

\( p \)	Sign
\(503\)	\(1\)

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	503.2.a.e	✓	10
3.b	odd	2	1		4527.2.a.k		10
4.b	odd	2	1		8048.2.a.p		10

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
503.2.a.e	✓	10	1.a	even	1	1	trivial
4527.2.a.k		10	3.b	odd	2	1
8048.2.a.p		10	4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(503))\):

\( T_{2}^{10} + 4T_{2}^{9} - 4T_{2}^{8} - 31T_{2}^{7} - 13T_{2}^{6} + 66T_{2}^{5} + 56T_{2}^{4} - 37T_{2}^{3} - 46T_{2}^{2} - 8T_{2} + 1 \)

\( T_{3}^{10} + 8T_{3}^{9} + 18T_{3}^{8} - 10T_{3}^{7} - 85T_{3}^{6} - 75T_{3}^{5} + 54T_{3}^{4} + 86T_{3}^{3} + 5T_{3}^{2} - 14T_{3} + 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^10 + 4*T^9 - 4*T^8 - 31*T^7 - 13*T^6 + 66*T^5 + 56*T^4 - 37*T^3 - 46*T^2 - 8*T + 1
$3$	T^10 + 8*T^9 + 18*T^8 - 10*T^7 - 85*T^6 - 75*T^5 + 54*T^4 + 86*T^3 + 5*T^2 - 14*T + 1
$5$	T^10 + T^9 - 11*T^8 - 7*T^7 + 41*T^6 + 7*T^5 - 59*T^4 + 16*T^3 + 18*T^2 - 9*T + 1
$7$	T^10 + 5*T^9 - 9*T^8 - 64*T^7 + 8*T^6 + 228*T^5 + 23*T^4 - 214*T^3 + 86*T^2 - 4*T - 1
$11$	T^10 + 3*T^9 - 41*T^8 - 80*T^7 + 515*T^6 + 537*T^5 - 1961*T^4 - 1781*T^3 + 1973*T^2 + 1920*T + 311