Newform orbit 8619.2.a.bm

• Label: 8619.2.a.bm
• Level: $8619$
• Weight: $2$
• Character orbit: 8619.a
• Self dual: yes
• Analytic conductor: $68.823$
• Analytic rank: $1$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8619.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(68.8230615021\)
Analytic rank:	\(1\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - 19x^{10} + 110x^{8} - 272x^{6} + 301x^{4} - 129x^{2} + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 663)
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 + b7 * q^2 - q^3 + (-b2 + 1) * q^4 + (-b10 - b7 - b1) * q^5 - b7 * q^6 + b10 * q^7 + (b11 + b10 + 2*b7 + 2*b1) * q^8 + q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} + 16 q^{4} + 12 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 19x^{10} + 110x^{8} - 272x^{6} + 301x^{4} - 129x^{2} + 9 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( 8\nu^{10} - 131\nu^{8} + 547\nu^{6} - 925\nu^{4} + 839\nu^{2} - 363 ) / 87 \)
\(\beta_{3}\)	\(=\)	\( ( -10\nu^{10} + 142\nu^{8} - 314\nu^{6} - 475\nu^{4} + 1322\nu^{2} - 438 ) / 87 \)
\(\beta_{4}\)	\(=\)	\( ( -17\nu^{10} + 311\nu^{8} - 1630\nu^{6} + 3151\nu^{4} - 1772\nu^{2} - 66 ) / 87 \)
\(\beta_{5}\)	\(=\)	\( ( -25\nu^{10} + 442\nu^{8} - 2177\nu^{6} + 4076\nu^{4} - 2524\nu^{2} + 36 ) / 87 \)
\(\beta_{6}\)	\(=\)	\( ( -34\nu^{11} + 622\nu^{9} - 3347\nu^{7} + 7607\nu^{5} - 7459\nu^{3} + 2130\nu ) / 261 \)
\(\beta_{7}\)	\(=\)	\( ( 41\nu^{11} - 704\nu^{9} + 3184\nu^{7} - 4621\nu^{5} + 113\nu^{3} + 2283\nu ) / 261 \)
\(\beta_{8}\)	\(=\)	\( ( 47\nu^{10} - 824\nu^{8} + 3964\nu^{6} - 7033\nu^{4} + 4157\nu^{2} - 360 ) / 87 \)
\(\beta_{9}\)	\(=\)	\( ( 74\nu^{11} - 1277\nu^{9} + 5995\nu^{7} - 11014\nu^{5} + 8696\nu^{3} - 3162\nu ) / 261 \)
\(\beta_{10}\)	\(=\)	\( ( -91\nu^{11} + 1588\nu^{9} - 7538\nu^{7} + 12860\nu^{5} - 6292\nu^{3} - 732\nu ) / 261 \)
\(\beta_{11}\)	\(=\)	\( ( -95\nu^{11} + 1784\nu^{9} - 9943\nu^{7} + 21892\nu^{5} - 17717\nu^{3} + 2973\nu ) / 261 \)

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

−0.292541
1.78316
−0.952347
1.17972
1.53857
3.32699
−3.32699
−1.53857
−1.17972
0.952347
−1.78316
0.292541

−2.53399

−1.00000

4.42112

1.50290

2.53399

1.32363

−6.13509

1.00000

−3.80833

1.2

−2.43081

−1.00000

3.90886

2.70824

2.43081

−2.06058

−4.64008

1.00000

−6.58322

1.3

−1.85567

−1.00000

1.44352

1.11030

1.85567

1.69772

1.03264

1.00000

−2.06034

1.4

−1.60153

−1.00000

0.564914

−2.21119

1.60153

2.63300

2.29834

1.00000

3.54130

1.5

−0.983829

−1.00000

−1.03208

2.12454

0.983829

−2.67929

2.98305

1.00000

−2.09019

1.6

−0.832869

−1.00000

−1.30633

−2.40228

0.832869

−0.0918385

2.75374

1.00000

2.00078

1.7

0.832869

−1.00000

−1.30633

2.40228

−0.832869

0.0918385

−2.75374

1.00000

2.00078

1.8

0.983829

−1.00000

−1.03208

−2.12454

−0.983829

2.67929

−2.98305

1.00000

−2.09019

1.9

1.60153

−1.00000

0.564914

2.21119

−1.60153

−2.63300

−2.29834

1.00000

3.54130

1.10

1.85567

−1.00000

1.44352

−1.11030

−1.85567

−1.69772

−1.03264

1.00000

−2.06034

1.11

2.43081

−1.00000

3.90886

−2.70824

−2.43081

2.06058

4.64008

1.00000

−6.58322

1.12

2.53399

−1.00000

4.42112

−1.50290

−2.53399

−1.32363

6.13509

1.00000

−3.80833

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(1\)
\(13\)	\(-1\)
\(17\)	\(-1\)

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8619.2.a.bm		12
13.b	even	2	1	inner	8619.2.a.bm		12
13.f	odd	12	2		663.2.z.c	✓	12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
663.2.z.c	✓	12	13.f	odd	12	2
8619.2.a.bm		12	1.a	even	1	1	trivial
8619.2.a.bm		12	13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):

\( T_{2}^{12} - 20T_{2}^{10} + 152T_{2}^{8} - 550T_{2}^{6} + 976T_{2}^{4} - 783T_{2}^{2} + 225 \)

\( T_{5}^{12} - 26T_{5}^{10} + 269T_{5}^{8} - 1405T_{5}^{6} + 3856T_{5}^{4} - 5175T_{5}^{2} + 2601 \)

\( T_{7}^{12} - 23T_{7}^{10} + 200T_{7}^{8} - 814T_{7}^{6} + 1540T_{7}^{4} - 1080T_{7}^{2} + 9 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^12 - 20*T^10 + 152*T^8 - 550*T^6 + 976*T^4 - 783*T^2 + 225
$3$	\( (T + 1)^{12} \)
$5$	T^12 - 26*T^10 + 269*T^8 - 1405*T^6 + 3856*T^4 - 5175*T^2 + 2601
$7$	T^12 - 23*T^10 + 200*T^8 - 814*T^6 + 1540*T^4 - 1080*T^2 + 9
$11$	T^12 - 35*T^10 + 350*T^8 - 964*T^6 + 745*T^4 - 153*T^2 + 9