Newform orbit 2023.2.a.n

• Label: 2023.2.a.n
• Level: $2023$
• Weight: $2$
• Character orbit: 2023.a
• Self dual: yes
• Analytic conductor: $16.154$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2023 = 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2023.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(16.1537363289\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	\( x^{10} - 16x^{8} + 88x^{6} - 2x^{5} - 192x^{4} + 16x^{3} + 136x^{2} - 40x - 7 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 119)
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.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + \beta_{8} q^{3} + ( - \beta_{8} + \beta_{7} + 2) q^{4} + (\beta_{7} + \beta_{4} + \beta_{3} + 1) q^{5} + ( - \beta_{8} + \beta_{7} - \beta_{5} + 2) q^{6} + q^{7} + (\beta_{8} + \beta_{5} - \beta_{4}) q^{8} + (\beta_{7} - \beta_{5} + \beta_{4} + 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{3} + 12 q^{4} + 8 q^{5} + 14 q^{6} + 10 q^{7} + 10 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 16x^{8} + 88x^{6} - 2x^{5} - 192x^{4} + 16x^{3} + 136x^{2} - 40x - 7 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{9} + \nu^{8} + 15\nu^{7} - 15\nu^{6} - 67\nu^{5} + 69\nu^{4} + 75\nu^{3} - 91\nu^{2} + 27\nu + 7 ) / 12 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{9} - \nu^{8} - 15\nu^{7} + 15\nu^{6} + 73\nu^{5} - 69\nu^{4} - 123\nu^{3} + 91\nu^{2} + 45\nu - 13 ) / 12 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{9} + \nu^{8} + 15\nu^{7} - 12\nu^{6} - 73\nu^{5} + 39\nu^{4} + 117\nu^{3} - 19\nu^{2} - 18\nu - 5 ) / 6 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{9} + 2\nu^{8} - 15\nu^{7} - 27\nu^{6} + 76\nu^{5} + 111\nu^{4} - 150\nu^{3} - 137\nu^{2} + 87\nu + 2 ) / 6 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{8} - \nu^{7} - 14\nu^{6} + 13\nu^{5} + 62\nu^{4} - 51\nu^{3} - 87\nu^{2} + 52\nu + 7 ) / 2 \)
\(\beta_{7}\)	\(=\)	\( ( -2\nu^{9} - \nu^{8} + 30\nu^{7} + 15\nu^{6} - 149\nu^{5} - 72\nu^{4} + 273\nu^{3} + 124\nu^{2} - 129\nu - 31 ) / 6 \)
\(\beta_{8}\)	\(=\)	\( ( -2\nu^{9} - \nu^{8} + 30\nu^{7} + 15\nu^{6} - 149\nu^{5} - 72\nu^{4} + 273\nu^{3} + 118\nu^{2} - 129\nu - 7 ) / 6 \)
\(\beta_{9}\)	\(=\)	\( ( -5\nu^{9} - \nu^{8} + 81\nu^{7} + 21\nu^{6} - 443\nu^{5} - 135\nu^{4} + 909\nu^{3} + 283\nu^{2} - 489\nu - 25 ) / 12 \)

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

−2.51056
−2.32367
−1.57229
−1.40508
−0.124341
0.582556
0.710316
1.88449
2.13581
2.62277

−2.51056

−2.61726

4.30293

2.99641

6.57081

1.00000

−5.78166

3.85007

−7.52267

1.2

−2.32367

−2.10665

3.39944

−1.47195

4.89517

1.00000

−3.25185

1.43798

3.42033

1.3

−1.57229

2.03431

0.472100

3.39483

−3.19853

1.00000

2.40230

1.13841

−5.33766

1.4

−1.40508

0.642386

−0.0257627

−0.296288

−0.902601

1.00000

2.84635

−2.58734

0.416307

1.5

−0.124341

1.72112

−1.98454

−3.22248

−0.214006

1.00000

0.495440

−0.0377310

0.400685

1.6

0.582556

−0.863302

−1.66063

−1.99820

−0.502922

1.00000

−2.13252

−2.25471

−1.16407

1.7

0.710316

2.99685

−1.49545

3.77118

2.12871

1.00000

−2.48287

5.98113

2.67873

1.8

1.88449

−1.32311

1.55131

1.38113

−2.49340

1.00000

−0.845554

−1.24937

2.60272

1.9

2.13581

3.08855

2.56169

1.09005

6.59655

1.00000

1.19966

6.53913

2.32813

1.10

2.62277

0.427112

4.87891

2.35533

1.12022

1.00000

7.55071

−2.81758

6.17749

Atkin-Lehner signs

\( p \)	Sign
\(7\)	\(-1\)
\(17\)	\(1\)

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	2023.2.a.n		10
17.b	even	2	1		2023.2.a.m		10
17.d	even	8	2		119.2.g.a	✓	20
51.g	odd	8	2		1071.2.n.c		20
119.l	odd	8	2		833.2.g.h		20
119.q	even	24	4		833.2.o.g		40
119.r	odd	24	4		833.2.o.f		40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
119.2.g.a	✓	20	17.d	even	8	2
833.2.g.h		20	119.l	odd	8	2
833.2.o.f		40	119.r	odd	24	4
833.2.o.g		40	119.q	even	24	4
1071.2.n.c		20	51.g	odd	8	2
2023.2.a.m		10	17.b	even	2	1
2023.2.a.n		10	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2023))\):

\( T_{2}^{10} - 16T_{2}^{8} + 88T_{2}^{6} - 2T_{2}^{5} - 192T_{2}^{4} + 16T_{2}^{3} + 136T_{2}^{2} - 40T_{2} - 7 \)

\( T_{3}^{10} - 4T_{3}^{9} - 12T_{3}^{8} + 56T_{3}^{7} + 37T_{3}^{6} - 248T_{3}^{5} + 384T_{3}^{3} - 79T_{3}^{2} - 160T_{3} + 56 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^10 - 16*T^8 + 88*T^6 - 2*T^5 - 192*T^4 + 16*T^3 + 136*T^2 - 40*T - 7
$3$	T^10 - 4*T^9 - 12*T^8 + 56*T^7 + 37*T^6 - 248*T^5 + 384*T^3 - 79*T^2 - 160*T + 56
$5$	T^10 - 8*T^9 + 2*T^8 + 124*T^7 - 239*T^6 - 444*T^5 + 1334*T^4 + 128*T^3 - 1983*T^2 + 728*T + 382
$7$	\( (T - 1)^{10} \)
$11$	T^10 - 8*T^9 - 14*T^8 + 184*T^7 + 100*T^6 - 1376*T^5 - 1128*T^4 + 2432*T^3 + 3376*T^2 + 1280*T + 128