Newform orbit 5225.2.a.z

• Label: 5225.2.a.z
• Level: $5225$
• Weight: $2$
• Character orbit: 5225.a
• Self dual: yes
• Analytic conductor: $41.722$
• Analytic rank: $1$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(41.7218350561\)
Analytic rank:	\(1\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\( x^{16} - 19x^{14} + 144x^{12} - 552x^{10} + 1119x^{8} - 1146x^{6} + 524x^{4} - 83x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{3} \)
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} - \beta_{11} q^{3} + \beta_{2} q^{4} + (\beta_{9} - \beta_{8} + \beta_{6} - 1) q^{6} - \beta_{5} q^{7} + \beta_{3} q^{8} + (\beta_{8} + 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{4} - 8 q^{6} + 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 19x^{14} + 144x^{12} - 552x^{10} + 1119x^{8} - 1146x^{6} + 524x^{4} - 83x^{2} + 4 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 2 \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} - 4\nu \)
\(\beta_{4}\)	\(=\)	\( \nu^{4} - 5\nu^{2} + 3 \)
\(\beta_{5}\)	\(=\)	\( \nu^{5} - 6\nu^{3} + 7\nu \)
\(\beta_{6}\)	\(=\)	\( \nu^{6} - 7\nu^{4} + 12\nu^{2} - 3 \)
\(\beta_{7}\)	\(=\)	\( \nu^{12} - 14\nu^{10} + 72\nu^{8} - 165\nu^{6} + 163\nu^{4} - 60\nu^{2} + 6 \)
\(\beta_{8}\)	\(=\)	\( \nu^{12} - 15\nu^{10} + 84\nu^{8} - 215\nu^{6} + 246\nu^{4} - 104\nu^{2} + 7 \)
\(\beta_{9}\)	\(=\)	\( \nu^{12} - 15\nu^{10} + 85\nu^{8} - 225\nu^{6} + 277\nu^{4} - 134\nu^{2} + 13 \)
\(\beta_{10}\)	\(=\)	\( ( \nu^{15} - 17\nu^{13} + 114\nu^{11} - 382\nu^{9} + 667\nu^{7} - 572\nu^{5} + 198\nu^{3} - 15\nu ) / 2 \)
\(\beta_{11}\)	\(=\)	\( ( \nu^{15} - 19\nu^{13} + 144\nu^{11} - 552\nu^{9} + 1117\nu^{7} - 1128\nu^{5} + 476\nu^{3} - 47\nu ) / 2 \)
\(\beta_{12}\)	\(=\)	\( ( \nu^{15} - 19\nu^{13} + 144\nu^{11} - 552\nu^{9} + 1119\nu^{7} - 1144\nu^{5} + 510\nu^{3} - 59\nu ) / 2 \)
\(\beta_{13}\)	\(=\)	\( \nu^{14} - 17\nu^{12} + 114\nu^{10} - 382\nu^{8} + 667\nu^{6} - 573\nu^{4} + 203\nu^{2} - 17 \)
\(\beta_{14}\)	\(=\)	\( -\nu^{15} + 19\nu^{13} - 143\nu^{11} + 538\nu^{9} - 1047\nu^{7} + 981\nu^{5} - 362\nu^{3} + 28\nu \)
\(\beta_{15}\)	\(=\)	\( \nu^{15} - 19\nu^{13} + 144\nu^{11} - 551\nu^{9} + 1107\nu^{7} - 1096\nu^{5} + 440\nu^{3} - 34\nu \)

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

−2.18657
−2.10564
−2.05535
−1.78099
−1.19140
−0.853553
−0.387488
−0.301154
0.301154
0.387488
0.853553
1.19140
1.78099
2.05535
2.10564
2.18657

−2.18657

−1.58567

2.78109

0

3.46718

2.56330

−1.70792

−0.485645

0

1.2

−2.10564

1.90142

2.43371

0

−4.00370

0.116917

−0.913237

0.615392

0

1.3

−2.05535

2.80376

2.22445

0

−5.76269

−1.02917

−0.461316

4.86106

0

1.4

−1.78099

−0.213501

1.17191

0

0.380243

−3.50934

1.47481

−2.95442

0

1.5

−1.19140

−2.62369

−0.580557

0

3.12588

0.593512

3.07449

3.88377

0

1.6

−0.853553

1.84364

−1.27145

0

−1.57364

2.69678

2.79235

0.399003

0

1.7

−0.387488

0.494325

−1.84985

0

−0.191545

2.37207

1.49177

−2.75564

0

1.8

−0.301154

−1.85377

−1.90931

0

0.558272

1.94668

1.17730

0.436480

0

1.9

0.301154

1.85377

−1.90931

0

0.558272

−1.94668

−1.17730

0.436480

0

1.10

0.387488

−0.494325

−1.84985

0

−0.191545

−2.37207

−1.49177

−2.75564

0

1.11

0.853553

−1.84364

−1.27145

0

−1.57364

−2.69678

−2.79235

0.399003

0

1.12

1.19140

2.62369

−0.580557

0

3.12588

−0.593512

−3.07449

3.88377

0

1.13

1.78099

0.213501

1.17191

0

0.380243

3.50934

−1.47481

−2.95442

0

1.14

2.05535

−2.80376

2.22445

0

−5.76269

1.02917

0.461316

4.86106

0

1.15

2.10564

−1.90142

2.43371

0

−4.00370

−0.116917

0.913237

0.615392

0

1.16

2.18657

1.58567

2.78109

0

3.46718

−2.56330

1.70792

−0.485645

0

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\(-1\)
\(11\)	\(1\)
\(19\)	\(-1\)

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	5225.2.a.z		16
5.b	even	2	1	inner	5225.2.a.z		16
5.c	odd	4	2		1045.2.b.b	✓	16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1045.2.b.b	✓	16	5.c	odd	4	2
5225.2.a.z		16	1.a	even	1	1	trivial
5225.2.a.z		16	5.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\):

\( T_{2}^{16} - 19T_{2}^{14} + 144T_{2}^{12} - 552T_{2}^{10} + 1119T_{2}^{8} - 1146T_{2}^{6} + 524T_{2}^{4} - 83T_{2}^{2} + 4 \)

\( T_{7}^{16} - 37T_{7}^{14} + 537T_{7}^{12} - 3908T_{7}^{10} + 14949T_{7}^{8} - 28337T_{7}^{6} + 21900T_{7}^{4} - 4976T_{7}^{2} + 64 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^16 - 19*T^14 + 144*T^12 - 552*T^10 + 1119*T^8 - 1146*T^6 + 524*T^4 - 83*T^2 + 4
$3$	T^16 - 28*T^14 + 316*T^12 - 1849*T^10 + 5983*T^8 - 10408*T^6 + 8359*T^4 - 1764*T^2 + 64
$5$	\( T^{16} \)
$7$	T^16 - 37*T^14 + 537*T^12 - 3908*T^10 + 14949*T^8 - 28337*T^6 + 21900*T^4 - 4976*T^2 + 64
$11$	\( (T + 1)^{16} \)