Newform orbit 1045.2.a.j

• Label: 1045.2.a.j
• Level: $1045$
• Weight: $2$
• Character orbit: 1045.a
• Self dual: yes
• Analytic conductor: $8.344$
• Analytic rank: $0$
• Dimension: $9$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(8.34436701122\)
Analytic rank:	\(0\)
Dimension:	\(9\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial:	\( x^{9} - 2x^{8} - 11x^{7} + 11x^{6} + 34x^{5} - 20x^{4} - 36x^{3} + 13x^{2} + 10x - 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} + \beta_{5} q^{3} + ( - \beta_{8} - \beta_1 + 1) q^{4} - q^{5} + (\beta_{7} - \beta_{6} + \beta_{3} + 1) q^{6} + ( - \beta_{3} - 1) q^{7} + ( - \beta_{8} - \beta_{6} - \beta_1 + 2) q^{8} + (\beta_{7} + \beta_{4} + 2) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{2} + 3 q^{3} + 9 q^{4} - 9 q^{5} + 6 q^{6} - 9 q^{7} + 15 q^{8} + 16 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 11x^{7} + 11x^{6} + 34x^{5} - 20x^{4} - 36x^{3} + 13x^{2} + 10x - 2 \) :

\(\beta_{1}\)	\(=\)	\( ( -4\nu^{8} + 6\nu^{7} + 47\nu^{6} - 35\nu^{5} - 110\nu^{4} + 112\nu^{3} + 26\nu^{2} - 126\nu + 13 ) / 29 \)
\(\beta_{2}\)	\(=\)	\( ( 8\nu^{8} - 12\nu^{7} - 94\nu^{6} + 41\nu^{5} + 307\nu^{4} - 50\nu^{3} - 400\nu^{2} + 78\nu + 177 ) / 29 \)
\(\beta_{3}\)	\(=\)	\( ( -5\nu^{8} + 22\nu^{7} + 37\nu^{6} - 196\nu^{5} - 94\nu^{4} + 488\nu^{3} + 76\nu^{2} - 288\nu + 9 ) / 29 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{8} - 16\nu^{7} + 39\nu^{6} + 103\nu^{5} - 277\nu^{4} - 202\nu^{3} + 472\nu^{2} + 162\nu - 170 ) / 29 \)
\(\beta_{5}\)	\(=\)	\( ( 14\nu^{8} - 21\nu^{7} - 150\nu^{6} + 50\nu^{5} + 356\nu^{4} + 43\nu^{3} - 178\nu^{2} - 110\nu - 2 ) / 29 \)
\(\beta_{6}\)	\(=\)	\( ( -20\nu^{8} + 30\nu^{7} + 235\nu^{6} - 117\nu^{5} - 695\nu^{4} + 154\nu^{3} + 565\nu^{2} - 108\nu + 7 ) / 29 \)
\(\beta_{7}\)	\(=\)	\( ( 21\nu^{8} - 46\nu^{7} - 196\nu^{6} + 220\nu^{5} + 418\nu^{4} - 327\nu^{3} - 180\nu^{2} + 96\nu - 3 ) / 29 \)
\(\beta_{8}\)	\(=\)	\( ( 16\nu^{8} - 53\nu^{7} - 130\nu^{6} + 372\nu^{5} + 324\nu^{4} - 738\nu^{3} - 249\nu^{2} + 388\nu + 35 ) / 29 \)

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

−0.600178
−1.98926
3.65050
1.56050
0.180277
−1.15934
0.724133
−1.58924
1.22261

−2.21339

0.921304

2.89909

−1.00000

−2.03920

−4.71795

−1.99002

−2.15120

2.21339

1.2

−1.53772

−2.83656

0.364584

−1.00000

4.36184

−2.01727

2.51481

5.04608

1.53772

1.3

−1.31671

2.24751

−0.266275

−1.00000

−2.95932

2.74324

2.98403

2.05132

1.31671

1.4

−0.0815632

−0.583675

−1.99335

−1.00000

0.0476064

−3.83288

0.325710

−2.65932

0.0815632

1.5

0.287410

−0.930455

−1.91740

−1.00000

−0.267422

0.300889

−1.12590

−2.13425

−0.287410

1.6

0.749568

3.38871

−1.43815

−1.00000

2.54007

1.17594

−2.57713

8.48335

−0.749568

1.7

1.79978

−2.53737

1.23919

−1.00000

−4.56670

0.104026

−1.36928

3.43826

−1.79978

1.8

2.64409

0.185485

4.99120

−1.00000

0.490440

1.87571

7.90900

−2.96560

−2.64409

1.9

2.66854

3.14505

5.12110

−1.00000

8.39270

−4.63172

8.32878

6.89136

−2.66854

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\(1\)
\(11\)	\(1\)
\(19\)	\(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1045.2.a.j	✓	9
3.b	odd	2	1		9405.2.a.bi		9
5.b	even	2	1		5225.2.a.q		9

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1045.2.a.j	✓	9	1.a	even	1	1	trivial
5225.2.a.q		9	5.b	even	2	1
9405.2.a.bi		9	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 3T_{2}^{8} - 9T_{2}^{7} + 27T_{2}^{6} + 25T_{2}^{5} - 70T_{2}^{4} - 21T_{2}^{3} + 51T_{2}^{2} - 8T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^9 - 3*T^8 - 9*T^7 + 27*T^6 + 25*T^5 - 70*T^4 - 21*T^3 + 51*T^2 - 8*T - 1
$3$	T^9 - 3*T^8 - 17*T^7 + 46*T^6 + 89*T^5 - 193*T^4 - 140*T^3 + 152*T^2 + 64*T - 16
$5$	\( (T + 1)^{9} \)
$7$	T^9 + 9*T^8 + 3*T^7 - 136*T^6 - 137*T^5 + 701*T^4 + 316*T^3 - 1236*T^2 + 432*T - 32
$11$	\( (T + 1)^{9} \)