Newform orbit 8281.2.a.cm

• Label: 8281.2.a.cm
• Level: $8281$
• Weight: $2$
• Character orbit: 8281.a
• Self dual: yes
• Analytic conductor: $66.124$
• Analytic rank: $1$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(66.1241179138\)
Analytic rank:	\(1\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - 32x^{10} + 393x^{8} - 2334x^{6} + 6955x^{4} - 9591x^{2} + 4537 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b2 * q^2 + b1 * q^3 + (-b11 + b6 - b3 - b2) * q^4 - b10 * q^5 + (-b5 + b4 - b1) * q^6 + (b11 + b8 - 3*b6 + 2*b3 - 1) * q^8 + (b11 + 2*b8 - b3 - 2*b2 + 2) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} - 18 q^{8} + 28 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 32x^{10} + 393x^{8} - 2334x^{6} + 6955x^{4} - 9591x^{2} + 4537 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( 112\nu^{10} - 214\nu^{8} - 19475\nu^{6} + 109519\nu^{4} - 25648\nu^{2} - 248469 ) / 145223 \)
\(\beta_{3}\)	\(=\)	\( ( -345\nu^{10} + 8439\nu^{8} - 60597\nu^{6} + 99608\nu^{4} + 79005\nu^{2} + 128726 ) / 145223 \)
\(\beta_{4}\)	\(=\)	\( ( -460\nu^{11} + 11252\nu^{9} - 80796\nu^{7} + 84403\nu^{5} + 686232\nu^{3} - 1038557\nu ) / 145223 \)
\(\beta_{5}\)	\(=\)	\( ( -44\nu^{11} + 882\nu^{9} - 4717\nu^{7} - 1932\nu^{5} + 54760\nu^{3} - 71947\nu ) / 11171 \)
\(\beta_{6}\)	\(=\)	\( ( -44\nu^{10} + 882\nu^{8} - 4717\nu^{6} - 1932\nu^{4} + 54760\nu^{2} - 60776 ) / 11171 \)
\(\beta_{7}\)	\(=\)	\( ( -1215\nu^{11} + 36034\nu^{9} - 383886\nu^{7} + 1746197\nu^{5} - 3061894\nu^{3} + 1122628\nu ) / 145223 \)
\(\beta_{8}\)	\(=\)	\( ( -120\nu^{10} + 3421\nu^{8} - 34191\nu^{6} + 141985\nu^{4} - 229453\nu^{2} + 96258 ) / 11171 \)
\(\beta_{9}\)	\(=\)	\( ( -120\nu^{11} + 3421\nu^{9} - 34191\nu^{7} + 141985\nu^{5} - 229453\nu^{3} + 96258\nu ) / 11171 \)
\(\beta_{10}\)	\(=\)	\( ( 2427\nu^{11} - 69469\nu^{9} + 728098\nu^{7} - 3398080\nu^{5} + 6850590\nu^{3} - 4532346\nu ) / 145223 \)
\(\beta_{11}\)	\(=\)	\( ( 2999\nu^{10} - 80935\nu^{8} + 789419\nu^{6} - 3372964\nu^{4} + 6138710\nu^{2} - 3597035 ) / 145223 \)

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

−3.08235
3.08235
−0.973776
0.973776
−3.14163
3.14163
−1.55274
1.55274
−2.28055
2.28055
−2.01721
2.01721

−2.62756

−3.08235

4.90408

1.94978

8.09905

0

−7.63063

6.50087

−5.12316

1.2

−2.62756

3.08235

4.90408

−1.94978

−8.09905

0

−7.63063

6.50087

5.12316

1.3

−1.31526

−0.973776

−0.270084

−3.52215

1.28077

0

2.98576

−2.05176

4.63255

1.4

−1.31526

0.973776

−0.270084

3.52215

−1.28077

0

2.98576

−2.05176

−4.63255

1.5

−0.676424

−3.14163

−1.54245

4.04809

2.12508

0

2.39620

6.86987

−2.73822

1.6

−0.676424

3.14163

−1.54245

−4.04809

−2.12508

0

2.39620

6.86987

2.73822

1.7

0.380581

−1.55274

−1.85516

3.06093

−0.590942

0

−1.46720

−0.589012

1.16493

1.8

0.380581

1.55274

−1.85516

−3.06093

0.590942

0

−1.46720

−0.589012

−1.16493

1.9

0.760304

−2.28055

−1.42194

0.280603

−1.73391

0

−2.60171

2.20091

0.213343

1.10

0.760304

2.28055

−1.42194

−0.280603

1.73391

0

−2.60171

2.20091

−0.213343

1.11

1.47836

−2.01721

0.185554

−2.82096

−2.98217

0

−2.68241

1.06914

−4.17040

1.12

1.47836

2.01721

0.185554

2.82096

2.98217

0

−2.68241

1.06914

4.17040

Atkin-Lehner signs

\( p \)	Sign
\(7\)	\(-1\)
\(13\)	\(-1\)

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8281.2.a.cm	✓	12
7.b	odd	2	1	inner	8281.2.a.cm	✓	12
13.b	even	2	1		8281.2.a.cr	yes	12
91.b	odd	2	1		8281.2.a.cr	yes	12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8281.2.a.cm	✓	12	1.a	even	1	1	trivial
8281.2.a.cm	✓	12	7.b	odd	2	1	inner
8281.2.a.cr	yes	12	13.b	even	2	1
8281.2.a.cr	yes	12	91.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{6} + 2T_{2}^{5} - 4T_{2}^{4} - 5T_{2}^{3} + 4T_{2}^{2} + 2T_{2} - 1 \)

\( T_{3}^{12} - 32T_{3}^{10} + 393T_{3}^{8} - 2334T_{3}^{6} + 6955T_{3}^{4} - 9591T_{3}^{2} + 4537 \)

\( T_{5}^{12} - 50T_{5}^{10} + 956T_{5}^{8} - 8697T_{5}^{6} + 37388T_{5}^{4} - 60512T_{5}^{2} + 4537 \)

\( T_{11}^{6} + 10T_{11}^{5} + 22T_{11}^{4} - 45T_{11}^{3} - 164T_{11}^{2} - 50T_{11} + 29 \)

\( T_{17}^{12} - 179T_{17}^{10} + 11978T_{17}^{8} - 369981T_{17}^{6} + 5249745T_{17}^{4} - 28700260T_{17}^{2} + 42688633 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	(T^6 + 2*T^5 - 4*T^4 - 5*T^3 + 4*T^2 + 2*T - 1)^2
$3$	T^12 - 32*T^10 + 393*T^8 - 2334*T^6 + 6955*T^4 - 9591*T^2 + 4537
$5$	T^12 - 50*T^10 + 956*T^8 - 8697*T^6 + 37388*T^4 - 60512*T^2 + 4537
$7$	\( T^{12} \)
$11$	(T^6 + 10*T^5 + 22*T^4 - 45*T^3 - 164*T^2 - 50*T + 29)^2