Newform orbit 2873.2.a.r

• Label: 2873.2.a.r
• Level: $2873$
• Weight: $2$
• Character orbit: 2873.a
• Self dual: yes
• Analytic conductor: $22.941$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2873 = 13^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2873.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(22.9410205007\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	\( x^{10} - 18x^{8} + 107x^{6} - 237x^{4} + 188x^{2} - 25 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{3} \)
Twist minimal:	no (minimal twist has level 221)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b1 * q^2 - b3 * q^3 + (b2 + 2) * q^4 - b4 * q^5 + (-b8 + b7 + b4 + b1) * q^6 + (b8 - b1) * q^7 + (-b8 + b7 - b6 - b4 + 2*b1) * q^8 + (b2 + 2) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} + 16 q^{4} + 16 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 18x^{8} + 107x^{6} - 237x^{4} + 188x^{2} - 25 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 4 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{8} - 17\nu^{6} + 86\nu^{4} - 115\nu^{2} + 21 ) / 4 \)
\(\beta_{4}\)	\(=\)	\( ( -3\nu^{9} + 49\nu^{7} - 236\nu^{5} + 281\nu^{3} + \nu ) / 10 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{8} - 17\nu^{6} + 90\nu^{4} - 147\nu^{2} + 49 ) / 4 \)
\(\beta_{6}\)	\(=\)	\( ( 7\nu^{9} - 111\nu^{7} + 514\nu^{5} - 569\nu^{3} - 9\nu ) / 20 \)
\(\beta_{7}\)	\(=\)	\( ( 3\nu^{9} - 49\nu^{7} + 246\nu^{5} - 371\nu^{3} + 119\nu ) / 10 \)
\(\beta_{8}\)	\(=\)	\( ( \nu^{9} - 17\nu^{7} + 90\nu^{5} - 151\nu^{3} + 73\nu ) / 4 \)
\(\beta_{9}\)	\(=\)	\( \nu^{8} - 16\nu^{6} + 76\nu^{4} - 94\nu^{2} + 12 \)

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

−2.76630
−2.59662
−1.36079
−1.26044
−0.405836
0.405836
1.26044
1.36079
2.59662
2.76630

−2.76630

2.94150

5.65243

0.444845

−8.13708

3.46814

−10.1037

5.65243

−1.23058

1.2

−2.59662

−2.78252

4.74243

−3.58446

7.22515

1.13991

−7.12104

4.74243

9.30747

1.3

−1.36079

−1.68871

−0.148260

−1.63814

2.29797

−0.615063

2.92332

−0.148260

2.22916

1.4

−1.26044

1.60894

−0.411297

3.67268

−2.02797

1.74356

3.03929

−0.411297

−4.62919

1.5

−0.405836

−1.07921

−1.83530

1.66784

0.437984

−4.71745

1.55650

−1.83530

−0.676869

1.6

0.405836

−1.07921

−1.83530

−1.66784

−0.437984

4.71745

−1.55650

−1.83530

−0.676869

1.7

1.26044

1.60894

−0.411297

−3.67268

2.02797

−1.74356

−3.03929

−0.411297

−4.62919

1.8

1.36079

−1.68871

−0.148260

1.63814

−2.29797

0.615063

−2.92332

−0.148260

2.22916

1.9

2.59662

−2.78252

4.74243

3.58446

−7.22515

−1.13991

7.12104

4.74243

9.30747

1.10

2.76630

2.94150

5.65243

−0.444845

8.13708

−3.46814

10.1037

5.65243

−1.23058

Atkin-Lehner signs

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

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	2873.2.a.r		10
13.b	even	2	1	inner	2873.2.a.r		10
13.d	odd	4	2		221.2.c.d	✓	10
39.f	even	4	2		1989.2.b.j		10

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
221.2.c.d	✓	10	13.d	odd	4	2
1989.2.b.j		10	39.f	even	4	2
2873.2.a.r		10	1.a	even	1	1	trivial
2873.2.a.r		10	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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2873))\):

\( T_{2}^{10} - 18T_{2}^{8} + 107T_{2}^{6} - 237T_{2}^{4} + 188T_{2}^{2} - 25 \)

\( T_{3}^{5} + T_{3}^{4} - 11T_{3}^{3} - 12T_{3}^{2} + 22T_{3} + 24 \)

\( T_{5}^{10} - 32T_{5}^{8} + 331T_{5}^{6} - 1208T_{5}^{4} + 1520T_{5}^{2} - 256 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^10 - 18*T^8 + 107*T^6 - 237*T^4 + 188*T^2 - 25
$3$	(T^5 + T^4 - 11*T^3 - 12*T^2 + 22*T + 24)^2
$5$	T^10 - 32*T^8 + 331*T^6 - 1208*T^4 + 1520*T^2 - 256
$7$	T^10 - 39*T^8 + 435*T^6 - 1456*T^4 + 1548*T^2 - 400
$11$	T^10 - 53*T^8 + 795*T^6 - 2680*T^4 + 1996*T^2 - 16