Newform orbit 3483.2.a.q

• Label: 3483.2.a.q
• Level: $3483$
• Weight: $2$
• Character orbit: 3483.a
• Self dual: yes
• Analytic conductor: $27.812$
• Analytic rank: $1$
• Dimension: $16$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3483 = 3^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3483.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(27.8118950240\)
Analytic rank:	\(1\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\( x^{16} - 20x^{14} + 158x^{12} - 627x^{10} + 1313x^{8} - 1392x^{6} + 656x^{4} - 104x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{15} q^{5} + ( - \beta_{4} - 1) q^{7} + \beta_{3} q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4} - 8 q^{7}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 20x^{14} + 158x^{12} - 627x^{10} + 1313x^{8} - 1392x^{6} + 656x^{4} - 104x^{2} + 4 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} - 4\nu \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{14} - 16\nu^{12} + 84\nu^{10} - 101\nu^{8} - 471\nu^{6} + 1484\nu^{4} - 1178\nu^{2} + 114 ) / 30 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{14} - 11\nu^{12} + 4\nu^{10} + 349\nu^{8} - 1486\nu^{6} + 2129\nu^{4} - 868\nu^{2} + 14 ) / 30 \)
\(\beta_{6}\)	\(=\)	\( ( 3\nu^{14} - 53\nu^{12} + 362\nu^{10} - 1233\nu^{8} + 2302\nu^{6} - 2433\nu^{4} + 1256\nu^{2} - 158 ) / 30 \)
\(\beta_{7}\)	\(=\)	\( ( 7\nu^{15} - 132\nu^{13} + 968\nu^{11} - 3497\nu^{9} + 6523\nu^{7} - 6082\nu^{5} + 2694\nu^{3} - 572\nu ) / 60 \)
\(\beta_{8}\)	\(=\)	\( ( -\nu^{14} + 21\nu^{12} - 174\nu^{10} + 721\nu^{8} - 1564\nu^{6} + 1701\nu^{4} - 802\nu^{2} + 86 ) / 10 \)
\(\beta_{9}\)	\(=\)	\( ( 7\nu^{14} - 132\nu^{12} + 968\nu^{10} - 3497\nu^{8} + 6493\nu^{6} - 5782\nu^{4} + 1854\nu^{2} - 32 ) / 30 \)
\(\beta_{10}\)	\(=\)	\( ( -7\nu^{14} + 132\nu^{12} - 968\nu^{10} + 3497\nu^{8} - 6493\nu^{6} + 5812\nu^{4} - 2034\nu^{2} + 152 ) / 30 \)
\(\beta_{11}\)	\(=\)	\( ( -7\nu^{15} + 137\nu^{13} - 1048\nu^{11} + 3947\nu^{9} - 7508\nu^{7} + 6427\nu^{5} - 1514\nu^{3} - 218\nu ) / 30 \)
\(\beta_{12}\)	\(=\)	\( ( \nu^{15} - 21\nu^{13} + 176\nu^{11} - 749\nu^{9} + 1696\nu^{7} - 1933\nu^{5} + 906\nu^{3} - 86\nu ) / 6 \)
\(\beta_{13}\)	\(=\)	\( ( 9\nu^{15} - 184\nu^{13} + 1486\nu^{11} - 6009\nu^{9} + 12701\nu^{7} - 13314\nu^{5} + 5938\nu^{3} - 724\nu ) / 30 \)
\(\beta_{14}\)	\(=\)	\( ( -27\nu^{15} + 532\nu^{13} - 4108\nu^{11} + 15717\nu^{9} - 30923\nu^{7} + 29142\nu^{5} - 10474\nu^{3} + 412\nu ) / 60 \)
\(\beta_{15}\)	\(=\)	\( ( -8\nu^{15} + 158\nu^{13} - 1227\nu^{11} + 4753\nu^{9} - 9597\nu^{7} + 9548\nu^{5} - 3896\nu^{3} + 363\nu ) / 15 \)

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

−2.41520
−2.23850
−1.87072
−1.75847
−1.25173
−0.865402
−0.438565
−0.236709
0.236709
0.438565
0.865402
1.25173
1.75847
1.87072
2.23850
2.41520

−2.41520

0

3.83319

−1.24897

0

−4.20797

−4.42753

0

3.01650

1.2

−2.23850

0

3.01088

3.92170

0

0.804832

−2.26285

0

−8.77871

1.3

−1.87072

0

1.49960

0.383483

0

0.346798

0.936116

0

−0.717390

1.4

−1.75847

0

1.09222

−2.55003

0

0.179977

1.59630

0

4.48416

1.5

−1.25173

0

−0.433183

0.431421

0

−3.35608

3.04568

0

−0.540021

1.6

−0.865402

0

−1.25108

−0.127304

0

3.94666

2.81349

0

0.110169

1.7

−0.438565

0

−1.80766

2.78425

0

1.03813

1.66991

0

−1.22107

1.8

−0.236709

0

−1.94397

−2.73062

0

−2.75235

0.933573

0

0.646362

1.9

0.236709

0

−1.94397

2.73062

0

−2.75235

−0.933573

0

0.646362

1.10

0.438565

0

−1.80766

−2.78425

0

1.03813

−1.66991

0

−1.22107

1.11

0.865402

0

−1.25108

0.127304

0

3.94666

−2.81349

0

0.110169

1.12

1.25173

0

−0.433183

−0.431421

0

−3.35608

−3.04568

0

−0.540021

1.13

1.75847

0

1.09222

2.55003

0

0.179977

−1.59630

0

4.48416

1.14

1.87072

0

1.49960

−0.383483

0

0.346798

−0.936116

0

−0.717390

1.15

2.23850

0

3.01088

−3.92170

0

0.804832

2.26285

0

−8.77871

1.16

2.41520

0

3.83319

1.24897

0

−4.20797

4.42753

0

3.01650

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( -1 \)
\(43\)	\( -1 \)

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3483.2.a.q	✓	16
3.b	odd	2	1	inner	3483.2.a.q	✓	16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3483.2.a.q	✓	16	1.a	even	1	1	trivial
3483.2.a.q	✓	16	3.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3483))\):

\( T_{2}^{16} - 20T_{2}^{14} + 158T_{2}^{12} - 627T_{2}^{10} + 1313T_{2}^{8} - 1392T_{2}^{6} + 656T_{2}^{4} - 104T_{2}^{2} + 4 \)

\( T_{5}^{16} - 39T_{5}^{14} + 562T_{5}^{12} - 3744T_{5}^{10} + 11385T_{5}^{8} - 12672T_{5}^{6} + 3484T_{5}^{4} - 300T_{5}^{2} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^16 - 20*T^14 + 158*T^12 - 627*T^10 + 1313*T^8 - 1392*T^6 + 656*T^4 - 104*T^2 + 4
$3$	\( T^{16} \)
$5$	T^16 - 39*T^14 + 562*T^12 - 3744*T^10 + 11385*T^8 - 12672*T^6 + 3484*T^4 - 300*T^2 + 4
$7$	(T^8 + 4*T^7 - 19*T^6 - 74*T^5 + 67*T^4 + 182*T^3 - 232*T^2 + 80*T - 8)^2
$11$	T^16 - 86*T^14 + 2753*T^12 - 41406*T^10 + 311564*T^8 - 1112022*T^6 + 1475873*T^4 - 318014*T^2 + 3481