Newform 6027.2.a.bd

• Label: 6027.2.a.bd
• Level: 6027
• Weight: 2
• Character orbit: 6027.a
• Self dual: Yes
• Analytic conductor: 48.126
• Analytic rank: 0
• Dimension: 10
• CM: No
• Inner twists: 1

Newspace parameters

Level:	\( N \)	=	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	6027.a (trivial)

Newform invariants

Self dual:	Yes
Analytic conductor:	\(48.1258372982\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
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.

\(f(q)\)	\(=\)	\( q\) \( + \beta_{1} q^{2} \) \(- q^{3}\) \( + ( 2 + \beta_{2} ) q^{4} \) \( + ( -1 - \beta_{4} ) q^{5} \) \( -\beta_{1} q^{6} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} + \beta_{8} ) q^{8} \) \(+ q^{9}\) \(+O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\(10q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 10q^{3} \) \(\mathstrut +\mathstrut 18q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(4\) \(x^{9}\mathstrut -\mathstrut \) \(11\) \(x^{8}\mathstrut +\mathstrut \) \(56\) \(x^{7}\mathstrut +\mathstrut \) \(26\) \(x^{6}\mathstrut -\mathstrut \) \(266\) \(x^{5}\mathstrut +\mathstrut \) \(52\) \(x^{4}\mathstrut +\mathstrut \) \(526\) \(x^{3}\mathstrut -\mathstrut \) \(255\) \(x^{2}\mathstrut -\mathstrut \) \(372\) \(x\mathstrut +\mathstrut \) \(239\):

\(\beta_{0}\)	\(=\)	\( 1 \)
\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 4 \)
\(\beta_{3}\)	\(=\)	\((\)\( \nu^{8} - 4 \nu^{7} - 8 \nu^{6} + 46 \nu^{5} - 2 \nu^{4} - 152 \nu^{3} + 90 \nu^{2} + 154 \nu - 121 \)\()/2\)
\(\beta_{4}\)	\(=\)	\((\)\( \nu^{9} - 4 \nu^{8} - 8 \nu^{7} + 46 \nu^{6} - 4 \nu^{5} - 148 \nu^{4} + 110 \nu^{3} + 120 \nu^{2} - 163 \nu + 52 \)\()/2\)
\(\beta_{5}\)	\(=\)	\((\)\( -\nu^{9} + 5 \nu^{8} + 4 \nu^{7} - 54 \nu^{6} + 50 \nu^{5} + 148 \nu^{4} - 262 \nu^{3} - 46 \nu^{2} + 315 \nu - 153 \)\()/2\)
\(\beta_{6}\)	\(=\)	\( -\nu^{8} + 4 \nu^{7} + 9 \nu^{6} - 47 \nu^{5} - 9 \nu^{4} + 160 \nu^{3} - 59 \nu^{2} - 165 \nu + 102 \)
\(\beta_{7}\)	\(=\)	\( \nu^{8} - 4 \nu^{7} - 9 \nu^{6} + 48 \nu^{5} + 8 \nu^{4} - 169 \nu^{3} + 67 \nu^{2} + 180 \nu - 114 \)
\(\beta_{8}\)	\(=\)	\( -\nu^{8} + 4 \nu^{7} + 9 \nu^{6} - 48 \nu^{5} - 8 \nu^{4} + 170 \nu^{3} - 68 \nu^{2} - 185 \nu + 117 \)
\(\beta_{9}\)	\(=\)	\((\)\( -2 \nu^{9} + 9 \nu^{8} + 14 \nu^{7} - 104 \nu^{6} + 28 \nu^{5} + 342 \nu^{4} - 268 \nu^{3} - 318 \nu^{2} + 348 \nu - 57 \)\()/2\)

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

−2.58399
−1.69562
−1.49626
−1.44855
0.775610
1.32624
1.55584
2.13183
2.62834
2.80657

−2.58399

−1.00000

4.67700

−0.798812

2.58399

0

−6.91733

1.00000

2.06412

1.2

−1.69562

−1.00000

0.875138

0.871357

1.69562

0

1.90734

1.00000

−1.47749

1.3

−1.49626

−1.00000

0.238800

0.660548

1.49626

0

2.63522

1.00000

−0.988352

1.4

−1.44855

−1.00000

0.0982847

−3.93238

1.44855

0

2.75472

1.00000

5.69623

1.5

0.775610

−1.00000

−1.39843

−2.32372

−0.775610

0

−2.63586

1.00000

−1.80230

1.6

1.32624

−1.00000

−0.241096

0.903630

−1.32624

0

−2.97222

1.00000

1.19843

1.7

1.55584

−1.00000

0.420641

3.26436

−1.55584

0

−2.45723

1.00000

5.07882

1.8

2.13183

−1.00000

2.54468

−2.36072

−2.13183

0

1.16116

1.00000

−5.03264

1.9

2.62834

−1.00000

4.90817

1.82844

−2.62834

0

7.64366

1.00000

4.80575

1.10

2.80657

−1.00000

5.87681

−4.11270

−2.80657

0

10.8805

1.00000

−11.5426

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(1\)
\(7\)	\(-1\)
\(41\)	\(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2}^{10} - \cdots\)

\(T_{5}^{10} + \cdots\)

\(T_{13}^{10} - \cdots\)