Newform orbit 130.2.p.a

• Label: 130.2.p.a
• Level: $130$
• Weight: $2$
• Character orbit: 130.p
• Analytic conductor: $1.038$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 130 = 2 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	130.p (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.03805522628\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(3\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	\( x^{12} + 24x^{10} + 192x^{8} + 680x^{6} + 1104x^{4} + 672x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{2} + (\beta_{11} + \beta_{8} + \beta_{7} + \beta_{5}) q^{3} + \beta_{3} q^{4} + ( - \beta_{11} + \beta_{10} + \beta_{7} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{5} - \beta_{8} q^{6} + ( - \beta_{7} + \beta_{2} - \beta_1) q^{7} + (\beta_{6} + \beta_{4}) q^{8} + (\beta_{11} - 2 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 4) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{4} - 24 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 24x^{10} + 192x^{8} + 680x^{6} + 1104x^{4} + 672x^{2} + 4 \) :

\(\beta_{1}\)	\(=\)	\( ( 7\nu^{10} + 136\nu^{8} + 668\nu^{6} + 702\nu^{4} + 176\nu^{2} + 380\nu + 1728 ) / 760 \)
\(\beta_{2}\)	\(=\)	\( ( 7\nu^{10} + 136\nu^{8} + 668\nu^{6} + 702\nu^{4} + 176\nu^{2} - 380\nu + 1728 ) / 760 \)
\(\beta_{3}\)	\(=\)	\( ( 4\nu^{11} + 87\nu^{9} + 576\nu^{7} + 1504\nu^{5} + 1542\nu^{3} + 456\nu + 20 ) / 40 \)
\(\beta_{4}\)	\(=\)	\( ( - 60 \nu^{11} + 36 \nu^{10} - 1315 \nu^{9} + 808 \nu^{8} - 8820 \nu^{7} + 5634 \nu^{6} - 23280 \nu^{5} + 15336 \nu^{4} - 23250 \nu^{3} + 14368 \nu^{2} - 5040 \nu + 744 ) / 760 \)
\(\beta_{5}\)	\(=\)	\( ( - 11 \nu^{11} + 89 \nu^{10} - 173 \nu^{9} + 1892 \nu^{8} - 154 \nu^{7} + 11886 \nu^{6} + 5194 \nu^{5} + 27654 \nu^{4} + 19402 \nu^{3} + 21292 \nu^{2} + 19216 \nu + 1016 ) / 760 \)
\(\beta_{6}\)	\(=\)	\( ( - 60 \nu^{11} - 36 \nu^{10} - 1315 \nu^{9} - 808 \nu^{8} - 8820 \nu^{7} - 5634 \nu^{6} - 23280 \nu^{5} - 15336 \nu^{4} - 23250 \nu^{3} - 14368 \nu^{2} - 5040 \nu - 744 ) / 760 \)
\(\beta_{7}\)	\(=\)	\( ( 11 \nu^{11} + 82 \nu^{10} + 173 \nu^{9} + 1756 \nu^{8} + 154 \nu^{7} + 11218 \nu^{6} - 5194 \nu^{5} + 26952 \nu^{4} - 19402 \nu^{3} + 21116 \nu^{2} - 19596 \nu - 712 ) / 760 \)
\(\beta_{8}\)	\(=\)	\( ( - 36 \nu^{11} + 125 \nu^{10} - 808 \nu^{9} + 2700 \nu^{8} - 5634 \nu^{7} + 17520 \nu^{6} - 15336 \nu^{5} + 42990 \nu^{4} - 14368 \nu^{3} + 35280 \nu^{2} - 1504 \nu + 240 ) / 760 \)
\(\beta_{9}\)	\(=\)	\( ( 3 \nu^{11} - 134 \nu^{10} + 99 \nu^{9} - 2902 \nu^{8} + 1182 \nu^{7} - 18976 \nu^{6} + 6218 \nu^{5} - 47584 \nu^{4} + 14054 \nu^{3} - 41152 \nu^{2} + 10892 \nu - 1756 ) / 760 \)
\(\beta_{10}\)	\(=\)	\( ( - 105 \nu^{11} - 178 \nu^{10} - 2135 \nu^{9} - 3784 \nu^{8} - 11920 \nu^{7} - 23772 \nu^{6} - 18890 \nu^{5} - 55308 \nu^{4} + 9330 \nu^{3} - 42204 \nu^{2} + 28040 \nu - 132 ) / 760 \)
\(\beta_{11}\)	\(=\)	\( ( -421\nu^{10} - 9048\nu^{8} - 58144\nu^{6} - 140586\nu^{4} - 112968\nu^{2} + 380\nu - 784 ) / 760 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(-\beta_{4} - \beta_{6}\)	\(\beta_{4}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

7.1

		1.55227i
		0.0775341i
	−	1.62980i
		3.51623i
	−	1.32083i
	−	2.19540i
	−	1.55227i
	−	0.0775341i
		1.62980i
	−	3.51623i
		1.32083i
		2.19540i

−0.866025

−

0.500000i

−0.776134

−

2.89657i

0.500000

+

0.866025i

2.12044

−

0.709753i

−0.776134

+

2.89657i

−0.638961

−

1.10671i

−

1.00000i

−5.18966

+

2.99625i

−2.19123

−

0.445554i

7.2

−0.866025

−

0.500000i

−0.0387670

−

0.144681i

0.500000

+

0.866025i

0.105914

+

2.23356i

−0.0387670

+

0.144681i

1.10259

+

1.90974i

−

1.00000i

2.57865

−

1.48878i

1.02506

−

1.98727i

7.3

−0.866025

−

0.500000i

0.814901

+

3.04125i

0.500000

+

0.866025i

−2.22635

+

0.208246i

0.814901

−

3.04125i

0.402397

+

0.696972i

−

1.00000i

−5.98707

+

3.45663i

2.03220

+

0.932829i

37.1

0.866025

−

0.500000i

−1.75811

−

0.471085i

0.500000

−

0.866025i

−1.28703

−

1.82854i

−1.75811

+

0.471085i

1.48736

−

2.57618i

−

1.00000i

0.270964

+

0.156441i

−2.02887

−

0.940048i

37.2

0.866025

−

0.500000i

0.660414

+

0.176957i

0.500000

−

0.866025i

0.483457

+

2.18318i

0.660414

−

0.176957i

0.189447

−

0.328132i

−

1.00000i

−2.19324

−

1.26627i

1.51028

+

1.64896i

37.3

0.866025

−

0.500000i

1.09770

+

0.294128i

0.500000

−

0.866025i

0.803571

−

2.08669i

1.09770

−

0.294128i

−2.54283

+

4.40431i

−

1.00000i

−1.47964

−

0.854273i

−0.347432

−

2.20891i

93.1

−0.866025

+

0.500000i

−0.776134

+

2.89657i

0.500000

−

0.866025i

2.12044

+

0.709753i

−0.776134

−

2.89657i

−0.638961

+

1.10671i

1.00000i

−5.18966

−

2.99625i

−2.19123

+

0.445554i

93.2

−0.866025

+

0.500000i

−0.0387670

+

0.144681i

0.500000

−

0.866025i

0.105914

−

2.23356i

−0.0387670

−

0.144681i

1.10259

−

1.90974i

1.00000i

2.57865

+

1.48878i

1.02506

+

1.98727i

93.3

−0.866025

+

0.500000i

0.814901

−

3.04125i

0.500000

−

0.866025i

−2.22635

−

0.208246i

0.814901

+

3.04125i

0.402397

−

0.696972i

1.00000i

−5.98707

−

3.45663i

2.03220

−

0.932829i

123.1

0.866025

+

0.500000i

−1.75811

+

0.471085i

0.500000

+

0.866025i

−1.28703

+

1.82854i

−1.75811

−

0.471085i

1.48736

+

2.57618i

1.00000i

0.270964

−

0.156441i

−2.02887

+

0.940048i

123.2

0.866025

+

0.500000i

0.660414

−

0.176957i

0.500000

+

0.866025i

0.483457

−

2.18318i

0.660414

+

0.176957i

0.189447

+

0.328132i

1.00000i

−2.19324

+

1.26627i

1.51028

−

1.64896i

123.3

0.866025

+

0.500000i

1.09770

−

0.294128i

0.500000

+

0.866025i

0.803571

+

2.08669i

1.09770

+

0.294128i

−2.54283

−

4.40431i

1.00000i

−1.47964

+

0.854273i

−0.347432

+

2.20891i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
65.t	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	130.2.p.a	✓	12
5.b	even	2	1		650.2.t.e		12
5.c	odd	4	1		130.2.s.a	yes	12
5.c	odd	4	1		650.2.w.e		12
13.f	odd	12	1		130.2.s.a	yes	12
65.o	even	12	1		650.2.t.e		12
65.s	odd	12	1		650.2.w.e		12
65.t	even	12	1	inner	130.2.p.a	✓	12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
130.2.p.a	✓	12	1.a	even	1	1	trivial
130.2.p.a	✓	12	65.t	even	12	1	inner
130.2.s.a	yes	12	5.c	odd	4	1
130.2.s.a	yes	12	13.f	odd	12	1
650.2.t.e		12	5.b	even	2	1
650.2.t.e		12	65.o	even	12	1
650.2.w.e		12	5.c	odd	4	1
650.2.w.e		12	65.s	odd	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 12 T_{3}^{10} + 4 T_{3}^{9} + 24 T_{3}^{8} + 24 T_{3}^{7} - 280 T_{3}^{6} + 48 T_{3}^{5} + 600 T_{3}^{4} - 568 T_{3}^{3} + 144 T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(130, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} - T^{2} + 1)^{3} \)
$3$	T^12 + 12*T^10 + 4*T^9 + 24*T^8 + 24*T^7 - 280*T^6 + 48*T^5 + 600*T^4 - 568*T^3 + 144*T^2 + 4
$5$	T^12 + 6*T^10 + 8*T^9 - 9*T^8 + 24*T^7 - 124*T^6 + 120*T^5 - 225*T^4 + 1000*T^3 + 3750*T^2 + 15625
$7$	T^12 + 21*T^10 - 64*T^9 + 414*T^8 - 720*T^7 + 1617*T^6 - 1152*T^5 + 2538*T^4 - 2128*T^3 + 1953*T^2 - 624*T + 169
$11$	T^12 - 6*T^11 + 9*T^10 - 28*T^9 + 30*T^8 + 426*T^7 + 293*T^6 + 3462*T^5 + 4848*T^4 - 10748*T^3 + 4719*T^2 - 7470*T + 6889