Newform orbit 5929.2.a.cd

• Label: 5929.2.a.cd
• Level: $5929$
• Weight: $2$
• Character orbit: 5929.a
• Self dual: yes
• Analytic conductor: $47.343$
• Analytic rank: $1$
• Dimension: $14$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5929 = 7^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5929.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(47.3433033584\)
Analytic rank:	\(1\)
Dimension:	\(14\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial:	\( x^{14} - 22x^{12} + 187x^{10} - 779x^{8} + 1669x^{6} - 1743x^{4} + 715x^{2} - 27 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 847)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b1 * q^2 - b7 * q^3 + (b2 + 1) * q^4 + (-b3 - 1) * q^5 + b11 * q^6 + (b12 + b11 - b5 + b1) * q^8 + (-b9 + b7 - b2 + 1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 6 q^{3} + 16 q^{4} - 8 q^{5} + 12 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 22x^{12} + 187x^{10} - 779x^{8} + 1669x^{6} - 1743x^{4} + 715x^{2} - 27 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \)
\(\beta_{3}\)	\(=\)	\( ( -5\nu^{12} + 92\nu^{10} - 600\nu^{8} + 1697\nu^{6} - 2232\nu^{4} + 1550\nu^{2} - 427 ) / 133 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{10} - 17\nu^{8} + 99\nu^{6} - 226\nu^{4} + 165\nu^{2} - 9 ) / 7 \)
\(\beta_{5}\)	\(=\)	\( ( 2\nu^{13} - 52\nu^{11} + 525\nu^{9} - 2556\nu^{7} + 6057\nu^{5} - 6320\nu^{3} + 2276\nu ) / 133 \)
\(\beta_{6}\)	\(=\)	\( ( -10\nu^{13} + 203\nu^{11} - 1523\nu^{9} + 5142\nu^{7} - 7295\nu^{5} + 1846\nu^{3} + 2566\nu ) / 133 \)
\(\beta_{7}\)	\(=\)	\( ( 15\nu^{12} - 295\nu^{10} + 2123\nu^{8} - 6839\nu^{6} + 9660\nu^{4} - 4593\nu^{2} - 11 ) / 133 \)
\(\beta_{8}\)	\(=\)	\( ( -\nu^{13} + 22\nu^{11} - 184\nu^{9} + 728\nu^{7} - 1372\nu^{5} + 1065\nu^{3} - 220\nu ) / 21 \)
\(\beta_{9}\)	\(=\)	\( ( -12\nu^{12} + 255\nu^{10} - 2048\nu^{8} + 7698\nu^{6} - 13352\nu^{4} + 8166\nu^{2} + 157 ) / 133 \)
\(\beta_{10}\)	\(=\)	\( ( -43\nu^{13} + 871\nu^{11} - 6490\nu^{9} + 21590\nu^{7} - 30181\nu^{5} + 10404\nu^{3} + 4361\nu ) / 399 \)
\(\beta_{11}\)	\(=\)	\( ( -15\nu^{13} + 295\nu^{11} - 2123\nu^{9} + 6839\nu^{7} - 9660\nu^{5} + 4593\nu^{3} + 11\nu ) / 133 \)
\(\beta_{12}\)	\(=\)	\( ( 17\nu^{13} - 347\nu^{11} + 2648\nu^{9} - 9395\nu^{7} + 15717\nu^{5} - 10780\nu^{3} + 1600\nu ) / 133 \)
\(\beta_{13}\)	\(=\)	\( ( 27\nu^{12} - 550\nu^{10} + 4171\nu^{8} - 14537\nu^{6} + 23145\nu^{4} - 13823\nu^{2} + 1029 ) / 133 \)

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

−2.65570
−2.45891
−1.89070
−1.66077
−1.22946
−1.00735
−0.204615
0.204615
1.00735
1.22946
1.66077
1.89070
2.45891
2.65570

−2.65570

0.499027

5.05272

−3.52145

−1.32526

0

−8.10710

−2.75097

9.35189

1.2

−2.45891

−0.553859

4.04622

1.24104

1.36189

0

−5.03146

−2.69324

−3.05159

1.3

−1.89070

−2.84299

1.57473

3.47694

5.37522

0

0.804055

5.08258

−6.57383

1.4

−1.66077

2.33876

0.758144

−2.77312

−3.88414

0

2.06243

2.46981

4.60550

1.5

−1.22946

−3.25006

−0.488438

−2.58745

3.99581

0

3.05942

7.56289

3.18116

1.6

−1.00735

−0.595841

−0.985244

−1.58705

0.600222

0

3.00719

−2.64497

1.59872

1.7

−0.204615

1.40496

−1.95813

1.75109

−0.287475

0

0.809894

−1.02609

−0.358300

1.8

0.204615

1.40496

−1.95813

1.75109

0.287475

0

−0.809894

−1.02609

0.358300

1.9

1.00735

−0.595841

−0.985244

−1.58705

−0.600222

0

−3.00719

−2.64497

−1.59872

1.10

1.22946

−3.25006

−0.488438

−2.58745

−3.99581

0

−3.05942

7.56289

−3.18116

1.11

1.66077

2.33876

0.758144

−2.77312

3.88414

0

−2.06243

2.46981

−4.60550

1.12

1.89070

−2.84299

1.57473

3.47694

−5.37522

0

−0.804055

5.08258

6.57383

1.13

2.45891

−0.553859

4.04622

1.24104

−1.36189

0

5.03146

−2.69324

3.05159

1.14

2.65570

0.499027

5.05272

−3.52145

1.32526

0

8.10710

−2.75097

−9.35189

Atkin-Lehner signs

\( p \)	Sign
\(7\)	\(1\)
\(11\)	\(1\)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	5929.2.a.cd		14
7.b	odd	2	1		5929.2.a.ce		14
7.c	even	3	2		847.2.e.j	✓	28
11.b	odd	2	1	inner	5929.2.a.cd		14
77.b	even	2	1		5929.2.a.ce		14
77.h	odd	6	2		847.2.e.j	✓	28
77.m	even	15	8		847.2.n.n		112
77.o	odd	30	8		847.2.n.n		112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
847.2.e.j	✓	28	7.c	even	3	2
847.2.e.j	✓	28	77.h	odd	6	2
847.2.n.n		112	77.m	even	15	8
847.2.n.n		112	77.o	odd	30	8
5929.2.a.cd		14	1.a	even	1	1	trivial
5929.2.a.cd		14	11.b	odd	2	1	inner
5929.2.a.ce		14	7.b	odd	2	1
5929.2.a.ce		14	77.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5929))\):

\( T_{2}^{14} - 22T_{2}^{12} + 187T_{2}^{10} - 779T_{2}^{8} + 1669T_{2}^{6} - 1743T_{2}^{4} + 715T_{2}^{2} - 27 \)

\( T_{3}^{7} + 3T_{3}^{6} - 9T_{3}^{5} - 22T_{3}^{4} + 23T_{3}^{3} + 25T_{3}^{2} - 5T_{3} - 5 \)

\( T_{5}^{7} + 4T_{5}^{6} - 15T_{5}^{5} - 69T_{5}^{4} + 35T_{5}^{3} + 275T_{5}^{2} + T_{5} - 303 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^14 - 22*T^12 + 187*T^10 - 779*T^8 + 1669*T^6 - 1743*T^4 + 715*T^2 - 27
$3$	(T^7 + 3*T^6 - 9*T^5 - 22*T^4 + 23*T^3 + 25*T^2 - 5*T - 5)^2
$5$	(T^7 + 4*T^6 - 15*T^5 - 69*T^4 + 35*T^3 + 275*T^2 + T - 303)^2
$7$	\( T^{14} \)
$11$	\( T^{14} \)