Newform orbit 4719.2.a.bl

• Label: 4719.2.a.bl
• Level: $4719$
• Weight: $2$
• Character orbit: 4719.a
• Self dual: yes
• Analytic conductor: $37.681$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4719 = 3 \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4719.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(37.6814047138\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	\( x^{10} - 2x^{9} - 14x^{8} + 24x^{7} + 67x^{6} - 88x^{5} - 123x^{4} + 96x^{3} + 64x^{2} - 24x - 12 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index:	\( 2 \)
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 - b3 * q^5 - b1 * q^6 - b7 * q^7 + (-b9 + b8 + b2 + b1 + 1) * q^8 + q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} - 10 q^{3} + 12 q^{4} + 4 q^{5} - 2 q^{6} - 4 q^{7} + 12 q^{8} + 10 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 14x^{8} + 24x^{7} + 67x^{6} - 88x^{5} - 123x^{4} + 96x^{3} + 64x^{2} - 24x - 12 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{9} - 2\nu^{8} - 12\nu^{7} + 24\nu^{6} + 41\nu^{5} - 94\nu^{4} - 25\nu^{3} + 130\nu^{2} - 24\nu - 32 ) / 2 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{8} - 2\nu^{7} - 14\nu^{6} + 22\nu^{5} + 69\nu^{4} - 68\nu^{3} - 131\nu^{2} + 42\nu + 46 ) / 2 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{9} - 3\nu^{8} - 12\nu^{7} + 38\nu^{6} + 43\nu^{5} - 153\nu^{4} - 37\nu^{3} + 201\nu^{2} - 20\nu - 48 ) / 2 \)
\(\beta_{6}\)	\(=\)	\( ( 2\nu^{9} - 3\nu^{8} - 28\nu^{7} + 34\nu^{6} + 130\nu^{5} - 113\nu^{4} - 214\nu^{3} + 95\nu^{2} + 68\nu - 8 ) / 2 \)
\(\beta_{7}\)	\(=\)	\( -\nu^{9} + 3\nu^{8} + 13\nu^{7} - 37\nu^{6} - 56\nu^{5} + 139\nu^{4} + 83\nu^{3} - 152\nu^{2} - 8\nu + 31 \)
\(\beta_{8}\)	\(=\)	\( ( 4\nu^{9} - 9\nu^{8} - 52\nu^{7} + 108\nu^{6} + 218\nu^{5} - 399\nu^{4} - 298\nu^{3} + 443\nu^{2} + 26\nu - 84 ) / 2 \)
\(\beta_{9}\)	\(=\)	\( ( 4\nu^{9} - 9\nu^{8} - 52\nu^{7} + 108\nu^{6} + 218\nu^{5} - 399\nu^{4} - 300\nu^{3} + 445\nu^{2} + 36\nu - 88 ) / 2 \)

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

−2.37832
−2.14679
−1.40292
−0.480927
−0.438799
0.685651
0.786976
2.29348
2.33850
2.74314

−2.37832

−1.00000

3.65639

2.08204

2.37832

3.68137

−3.93941

1.00000

−4.95175

1.2

−2.14679

−1.00000

2.60869

−0.116817

2.14679

−0.435337

−1.30672

1.00000

0.250781

1.3

−1.40292

−1.00000

−0.0318104

0.0564647

1.40292

−1.58803

2.85047

1.00000

−0.0792156

1.4

−0.480927

−1.00000

−1.76871

−3.33435

0.480927

1.18974

1.81247

1.00000

1.60358

1.5

−0.438799

−1.00000

−1.80746

−0.863934

0.438799

−3.99516

1.67071

1.00000

0.379094

1.6

0.685651

−1.00000

−1.52988

4.19312

−0.685651

−0.240133

−2.42027

1.00000

2.87501

1.7

0.786976

−1.00000

−1.38067

1.47962

−0.786976

−1.39844

−2.66050

1.00000

1.16442

1.8

2.29348

−1.00000

3.26006

−4.09776

−2.29348

−1.95782

2.88993

1.00000

−9.39814

1.9

2.33850

−1.00000

3.46859

1.15335

−2.33850

4.62845

3.43430

1.00000

2.69710

1.10

2.74314

−1.00000

5.52480

3.44828

−2.74314

−3.88464

9.66902

1.00000

9.45911

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(1\)
\(11\)	\(1\)
\(13\)	\(-1\)

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	4719.2.a.bl	yes	10
11.b	odd	2	1		4719.2.a.bj	✓	10

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4719.2.a.bj	✓	10	11.b	odd	2	1
4719.2.a.bl	yes	10	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4719))\):

\( T_{2}^{10} - 2T_{2}^{9} - 14T_{2}^{8} + 24T_{2}^{7} + 67T_{2}^{6} - 88T_{2}^{5} - 123T_{2}^{4} + 96T_{2}^{3} + 64T_{2}^{2} - 24T_{2} - 12 \)

T5^10 - 4*T5^9 - 25*T5^8 + 114*T5^7 + 100*T5^6 - 822*T5^5 + 661*T5^4 + 564*T5^3 - 580*T5^2 - 40*T5 + 4

T7^10 + 4*T7^9 - 30*T7^8 - 146*T7^7 + 122*T7^6 + 1308*T7^5 + 1539*T7^4 - 806*T7^3 - 2249*T7^2 - 1072*T7 - 143

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^10 - 2*T^9 - 14*T^8 + 24*T^7 + 67*T^6 - 88*T^5 - 123*T^4 + 96*T^3 + 64*T^2 - 24*T - 12
$3$	\( (T + 1)^{10} \)
$5$	T^10 - 4*T^9 - 25*T^8 + 114*T^7 + 100*T^6 - 822*T^5 + 661*T^4 + 564*T^3 - 580*T^2 - 40*T + 4
$7$	T^10 + 4*T^9 - 30*T^8 - 146*T^7 + 122*T^6 + 1308*T^5 + 1539*T^4 - 806*T^3 - 2249*T^2 - 1072*T - 143
$11$	\( T^{10} \)