Newform orbit 5472.2.d.g

• Label: 5472.2.d.g
• Level: $5472$
• Weight: $2$
• Character orbit: 5472.d
• Analytic conductor: $43.694$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(43.6941399860\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\( x^{16} - 4x^{14} + 5x^{12} + 4x^{10} - 20x^{8} + 16x^{6} + 80x^{4} - 256x^{2} + 256 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{14} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - \beta_{11} q^{5} + ( - \beta_{5} - \beta_{4}) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 5x^{12} + 4x^{10} - 20x^{8} + 16x^{6} + 80x^{4} - 256x^{2} + 256 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{13} - \nu^{9} + 8\nu^{5} - 8\nu^{3} - 32\nu ) / 64 \)
\(\beta_{2}\)	\(=\)	\( ( -2\nu^{14} - 3\nu^{12} - 4\nu^{10} - 15\nu^{8} + 70\nu^{6} - 52\nu^{4} - 176\nu^{2} + 64 ) / 320 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{12} + 2\nu^{10} + 3\nu^{8} - 6\nu^{6} - 4\nu^{4} - 64 ) / 32 \)
\(\beta_{4}\)	\(=\)	\( ( 4\nu^{14} - 9\nu^{12} - 2\nu^{10} + 35\nu^{8} - 30\nu^{6} + 4\nu^{4} + 272\nu^{2} - 448 ) / 320 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{14} - 4\nu^{12} + 5\nu^{10} + 4\nu^{8} - 20\nu^{6} + 16\nu^{4} + 144\nu^{2} - 256 ) / 64 \)
\(\beta_{6}\)	\(=\)	\( ( 7\nu^{15} + 18\nu^{13} - 21\nu^{11} - 150\nu^{9} + 60\nu^{7} + 32\nu^{5} + 736\nu^{3} + 896\nu ) / 1280 \)
\(\beta_{7}\)	\(=\)	\( ( 2\nu^{15} - 7\nu^{13} + 14\nu^{11} - 15\nu^{9} - 20\nu^{7} + 32\nu^{5} + 216\nu^{3} - 864\nu ) / 320 \)
\(\beta_{8}\)	\(=\)	\( ( 2\nu^{14} + \nu^{12} + 5\nu^{8} + 6\nu^{6} - 36\nu^{4} + 80\nu^{2} + 64 ) / 64 \)
\(\beta_{9}\)	\(=\)	\( ( -2\nu^{14} + 5\nu^{12} - 15\nu^{8} + 26\nu^{6} + 36\nu^{4} - 192\nu^{2} + 256 ) / 64 \)
\(\beta_{10}\)	\(=\)	\( ( \nu^{15} - \nu^{13} - 3\nu^{11} + 7\nu^{9} + 4\nu^{7} - 8\nu^{5} + 72\nu^{3} - 96\nu ) / 64 \)
\(\beta_{11}\)	\(=\)	\( ( -5\nu^{15} + 6\nu^{13} - 9\nu^{11} - 18\nu^{9} + 20\nu^{7} - 16\nu^{5} - 224\nu^{3} + 128\nu ) / 256 \)
\(\beta_{12}\)	\(=\)	\( ( -13\nu^{15} + 28\nu^{13} - \nu^{11} - 60\nu^{9} + 100\nu^{7} + 192\nu^{5} - 944\nu^{3} + 1216\nu ) / 640 \)
\(\beta_{13}\)	\(=\)	\( ( -3\nu^{14} + 7\nu^{12} - \nu^{10} - 13\nu^{8} + 34\nu^{6} + 12\nu^{4} - 208\nu^{2} + 352 ) / 32 \)
\(\beta_{14}\)	\(=\)	\( ( -7\nu^{15} + 26\nu^{13} - 19\nu^{11} - 46\nu^{9} + 124\nu^{7} - 48\nu^{5} - 736\nu^{3} + 1152\nu ) / 256 \)
\(\beta_{15}\)	\(=\)	\( ( 49\nu^{15} - 114\nu^{13} + 13\nu^{11} + 310\nu^{9} - 700\nu^{7} - 96\nu^{5} + 3232\nu^{3} - 6528\nu ) / 1280 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times\).

\(n\)	\(1217\)	\(2053\)	\(3745\)	\(4447\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

2015.1

−1.39770	−	0.215490i
1.39770	−	0.215490i
−0.310269	−	1.37976i
0.310269	−	1.37976i
1.33768	+	0.458926i
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−1.07740	−	0.916088i
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1.07740	+	0.916088i
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−1.33768	−	0.458926i
1.33768	−	0.458926i
0.310269	+	1.37976i
−0.310269	+	1.37976i
1.39770	+	0.215490i
−1.39770	+	0.215490i

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0.712386i

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Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	5472.2.d.g	✓	16
3.b	odd	2	1	inner	5472.2.d.g	✓	16
4.b	odd	2	1	inner	5472.2.d.g	✓	16
12.b	even	2	1	inner	5472.2.d.g	✓	16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
5472.2.d.g	✓	16	1.a	even	1	1	trivial
5472.2.d.g	✓	16	3.b	odd	2	1	inner
5472.2.d.g	✓	16	4.b	odd	2	1	inner
5472.2.d.g	✓	16	12.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 \(S_{2}^{\mathrm{new}}(5472, [\chi])\):

\( T_{5}^{8} + 32T_{5}^{6} + 253T_{5}^{4} + 68T_{5}^{2} + 4 \)

\( T_{11}^{8} - 32T_{11}^{6} + 253T_{11}^{4} - 68T_{11}^{2} + 4 \)

\( T_{23}^{8} - 112T_{23}^{6} + 3544T_{23}^{4} - 36160T_{23}^{2} + 110224 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{16} \)
$3$	\( T^{16} \)
$5$	(T^8 + 32*T^6 + 253*T^4 + 68*T^2 + 4)^2
$7$	(T^8 + 26*T^6 + 217*T^4 + 608*T^2 + 256)^2
$11$	(T^8 - 32*T^6 + 253*T^4 - 68*T^2 + 4)^2