Newform orbit 1045.2.a.h

• Label: 1045.2.a.h
• Level: $1045$
• Weight: $2$
• Character orbit: 1045.a
• Self dual: yes
• Analytic conductor: $8.344$
• Analytic rank: $0$
• Dimension: $7$
• 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:	\(7\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial:	\( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \)
\(\beta_{3}\)	\(=\)	\( -\nu^{6} + 11\nu^{4} + 2\nu^{3} - 33\nu^{2} - 10\nu + 21 \)
\(\beta_{4}\)	\(=\)	\( -2\nu^{6} + \nu^{5} + 20\nu^{4} - 5\nu^{3} - 52\nu^{2} - \nu + 27 \)
\(\beta_{5}\)	\(=\)	\( -3\nu^{6} + \nu^{5} + 30\nu^{4} - 3\nu^{3} - 78\nu^{2} - 10\nu + 41 \)
\(\beta_{6}\)	\(=\)	\( 4\nu^{6} - \nu^{5} - 41\nu^{4} + 2\nu^{3} + 111\nu^{2} + 15\nu - 62 \)

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

−2.40300
−1.54354
−1.08185
0.719047
0.745312
1.97792
2.58611

−2.40300

−1.17935

3.77440

−1.00000

2.83397

−2.15598

−4.26389

−1.60914

2.40300

1.2

−1.54354

2.58921

0.382516

−1.00000

−3.99655

−3.41636

2.49665

3.70400

1.54354

1.3

−1.08185

−0.870874

−0.829591

−1.00000

0.942160

4.02863

3.06121

−2.24158

1.08185

1.4

0.719047

0.163114

−1.48297

−1.00000

0.117287

1.05678

−2.50442

−2.97339

−0.719047

1.5

0.745312

−2.05058

−1.44451

−1.00000

−1.52832

−3.86782

−2.56724

1.20489

−0.745312

1.6

1.97792

2.65412

1.91218

−1.00000

5.24965

1.06653

−0.173707

4.04436

−1.97792

1.7

2.58611

1.69436

4.68797

−1.00000

4.38180

2.28823

6.95140

−0.129144

−2.58611

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.h	✓	7
3.b	odd	2	1		9405.2.a.bd		7
5.b	even	2	1		5225.2.a.m		7

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1045.2.a.h	✓	7	1.a	even	1	1	trivial
5225.2.a.m		7	5.b	even	2	1
9405.2.a.bd		7	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - T_{2}^{6} - 10T_{2}^{5} + 8T_{2}^{4} + 27T_{2}^{3} - 16T_{2}^{2} - 18T_{2} + 11 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^7 - T^6 - 10*T^5 + 8*T^4 + 27*T^3 - 16*T^2 - 18*T + 11
$3$	T^7 - 3*T^6 - 7*T^5 + 20*T^4 + 17*T^3 - 31*T^2 - 20*T + 4
$5$	\( (T + 1)^{7} \)
$7$	T^7 + T^6 - 27*T^5 - 18*T^4 + 205*T^3 + 3*T^2 - 460*T + 296
$11$	\( (T - 1)^{7} \)