Newform orbit 1045.2.a.k

• Label: 1045.2.a.k
• 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:	\( x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} - \beta_{8} q^{6} + (\beta_{4} + 1) q^{7} + (\beta_{3} + \beta_1 + 1) q^{8} + (\beta_{7} - \beta_{2} + 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{2} + 3 q^{3} + 9 q^{4} + 9 q^{5} + 13 q^{7} + 9 q^{8} + 12 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} - 5\nu - 1 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{8} - 2\nu^{7} - 9\nu^{6} + 16\nu^{5} + 21\nu^{4} - 31\nu^{3} - 12\nu^{2} + 15\nu - 1 ) / 2 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{8} - 4\nu^{7} - 5\nu^{6} + 32\nu^{5} - 7\nu^{4} - 59\nu^{3} + 24\nu^{2} + 25\nu - 7 ) / 2 \)
\(\beta_{6}\)	\(=\)	\( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 24\nu^{4} + 8\nu^{3} - 45\nu^{2} + 19 \)
\(\beta_{7}\)	\(=\)	\( -\nu^{7} + 2\nu^{6} + 9\nu^{5} - 16\nu^{4} - 21\nu^{3} + 31\nu^{2} + 13\nu - 15 \)
\(\beta_{8}\)	\(=\)	\( ( \nu^{8} - 4\nu^{7} - 3\nu^{6} + 30\nu^{5} - 25\nu^{4} - 47\nu^{3} + 64\nu^{2} + 15\nu - 27 ) / 2 \)

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

−2.35228
−1.35249
−1.24377
−0.730132
0.682920
1.22961
1.92391
2.14345
2.69878

−2.35228

0.154676

3.53320

1.00000

−0.363841

1.56958

−3.60652

−2.97608

−2.35228

1.2

−1.35249

−2.73889

−0.170779

1.00000

3.70431

2.99075

2.93595

4.50151

−1.35249

1.3

−1.24377

3.09454

−0.453038

1.00000

−3.84889

3.83759

3.05101

6.57616

−1.24377

1.4

−0.730132

0.820975

−1.46691

1.00000

−0.599420

−1.34825

2.53130

−2.32600

−0.730132

1.5

0.682920

2.48419

−1.53362

1.00000

1.69650

0.856948

−2.41318

3.17118

0.682920

1.6

1.22961

−2.38474

−0.488068

1.00000

−2.93229

1.13367

−3.05934

2.68698

1.22961

1.7

1.92391

0.621031

1.70143

1.00000

1.19481

5.15278

−0.574421

−2.61432

1.92391

1.8

2.14345

2.53942

2.59437

1.00000

5.44311

−1.53582

1.27399

3.44866

2.14345

1.9

2.69878

−1.59120

5.28341

1.00000

−4.29429

0.342738

8.86121

−0.468091

2.69878

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.k	✓	9
3.b	odd	2	1		9405.2.a.bh		9
5.b	even	2	1		5225.2.a.p		9

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 3T_{2}^{8} - 9T_{2}^{7} + 29T_{2}^{6} + 23T_{2}^{5} - 84T_{2}^{4} - 23T_{2}^{3} + 89T_{2}^{2} + 8T_{2} - 27 \) 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 + 29*T^6 + 23*T^5 - 84*T^4 - 23*T^3 + 89*T^2 + 8*T - 27
$3$	T^9 - 3*T^8 - 15*T^7 + 46*T^6 + 63*T^5 - 213*T^4 - 32*T^3 + 272*T^2 - 144*T + 16
$5$	\( (T - 1)^{9} \)
$7$	T^9 - 13*T^8 + 55*T^7 - 56*T^6 - 169*T^5 + 389*T^4 - 56*T^3 - 408*T^2 + 320*T - 64
$11$	\( (T - 1)^{9} \)