Newform orbit 4719.2.a.bm

• Label: 4719.2.a.bm
• 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:	\( x^{10} - 2x^{9} - 14x^{8} + 28x^{7} + 63x^{6} - 128x^{5} - 103x^{4} + 236x^{3} + 20x^{2} - 152x + 52 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index:	\( 2^{2} \)
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.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + \beta_{4} q^{5} + \beta_1 q^{6} + (\beta_{8} + \beta_{2} + 1) q^{7} + (\beta_{6} + \beta_{4} - \beta_{3}) q^{8} + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 10 q^{3} + 12 q^{4} - 4 q^{5} + 2 q^{6} + 12 q^{7} + 10 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 14x^{8} + 28x^{7} + 63x^{6} - 128x^{5} - 103x^{4} + 236x^{3} + 20x^{2} - 152x + 52 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{9} - 3\nu^{8} - 10\nu^{7} + 34\nu^{6} + 19\nu^{5} - 101\nu^{4} + 17\nu^{3} + 77\nu^{2} - 42\nu + 8 ) / 2 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{9} + 2\nu^{8} + 12\nu^{7} - 24\nu^{6} - 41\nu^{5} + 82\nu^{4} + 45\nu^{3} - 94\nu^{2} - 8\nu + 24 ) / 2 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{9} - \nu^{8} - 14\nu^{7} + 12\nu^{6} + 65\nu^{5} - 43\nu^{4} - 125\nu^{3} + 67\nu^{2} + 86\nu - 48 ) / 2 \)
\(\beta_{6}\)	\(=\)	\( ( 2\nu^{9} - 5\nu^{8} - 22\nu^{7} + 58\nu^{6} + 60\nu^{5} - 183\nu^{4} - 26\nu^{3} + 171\nu^{2} - 42\nu - 16 ) / 2 \)
\(\beta_{7}\)	\(=\)	\( \nu^{8} - 3\nu^{7} - 10\nu^{6} + 34\nu^{5} + 19\nu^{4} - 102\nu^{3} + 17\nu^{2} + 84\nu - 41 \)
\(\beta_{8}\)	\(=\)	\( ( -\nu^{9} + 2\nu^{8} + 14\nu^{7} - 26\nu^{6} - 65\nu^{5} + 104\nu^{4} + 125\nu^{3} - 156\nu^{2} - 82\nu + 78 ) / 2 \)
\(\beta_{9}\)	\(=\)	\( ( -\nu^{9} + 6\nu^{8} + 2\nu^{7} - 66\nu^{6} + 73\nu^{5} + 180\nu^{4} - 301\nu^{3} - 86\nu^{2} + 286\nu - 92 ) / 2 \)

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

−2.66620
−1.68586
−1.66271
−1.24010
0.544904
0.821931
1.11406
1.91576
2.25317
2.60505

−2.66620

1.00000

5.10861

−3.50021

−2.66620

4.42664

−8.28816

1.00000

9.33226

1.2

−1.68586

1.00000

0.842122

0.241170

−1.68586

1.30424

1.95202

1.00000

−0.406579

1.3

−1.66271

1.00000

0.764621

1.79195

−1.66271

1.81315

2.05408

1.00000

−2.97950

1.4

−1.24010

1.00000

−0.462151

−2.78511

−1.24010

−1.25904

3.05331

1.00000

3.45382

1.5

0.544904

1.00000

−1.70308

1.91240

0.544904

4.69645

−2.01782

1.00000

1.04208

1.6

0.821931

1.00000

−1.32443

−1.58046

0.821931

−4.58419

−2.73245

1.00000

−1.29903

1.7

1.11406

1.00000

−0.758867

−0.813844

1.11406

−2.39093

−3.07355

1.00000

−0.906673

1.8

1.91576

1.00000

1.67013

−4.04318

1.91576

4.99326

−0.631953

1.00000

−7.74576

1.9

2.25317

1.00000

3.07678

2.65916

2.25317

0.156861

2.42616

1.00000

5.99153

1.10

2.60505

1.00000

4.78627

2.11814

2.60505

2.84356

7.25836

1.00000

5.51785

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.bm	yes	10
11.b	odd	2	1		4719.2.a.bk	✓	10

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4719.2.a.bk	✓	10	11.b	odd	2	1
4719.2.a.bm	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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4719))\):

T2^10 - 2*T2^9 - 14*T2^8 + 28*T2^7 + 63*T2^6 - 128*T2^5 - 103*T2^4 + 236*T2^3 + 20*T2^2 - 152*T2 + 52

T5^10 + 4*T5^9 - 21*T5^8 - 74*T5^7 + 180*T5^6 + 458*T5^5 - 731*T5^4 - 1052*T5^3 + 1096*T5^2 + 784*T5 - 236

T7^10 - 12*T7^9 + 18*T7^8 + 282*T7^7 - 1142*T7^6 - 316*T7^5 + 7131*T7^4 - 7050*T7^3 - 7529*T7^2 + 10960*T7 - 1511

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^10 - 2*T^9 - 14*T^8 + 28*T^7 + 63*T^6 - 128*T^5 - 103*T^4 + 236*T^3 + 20*T^2 - 152*T + 52
$3$	\( (T - 1)^{10} \)
$5$	T^10 + 4*T^9 - 21*T^8 - 74*T^7 + 180*T^6 + 458*T^5 - 731*T^4 - 1052*T^3 + 1096*T^2 + 784*T - 236
$7$	T^10 - 12*T^9 + 18*T^8 + 282*T^7 - 1142*T^6 - 316*T^5 + 7131*T^4 - 7050*T^3 - 7529*T^2 + 10960*T - 1511
$11$	\( T^{10} \)