Newform orbit 5265.2.a.bh

• Label: 5265.2.a.bh
• Level: $5265$
• Weight: $2$
• Character orbit: 5265.a
• Self dual: yes
• Analytic conductor: $42.041$
• Analytic rank: $0$
• Dimension: $13$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5265.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(42.0412366642\)
Analytic rank:	\(0\)
Dimension:	\(13\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial:	x^13 - x^12 - 20*x^11 + 17*x^10 + 150*x^9 - 109*x^8 - 522*x^7 + 334*x^6 + 846*x^5 - 490*x^4 - 512*x^3 + 275*x^2 - 3*x - 2
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{5} + 1) q^{7} + (\beta_{3} + \beta_{2} + \beta_1) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + q^{2} + 15 q^{4} - 13 q^{5} + 10 q^{7} + 6 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of x^13 - x^12 - 20*x^11 + 17*x^10 + 150*x^9 - 109*x^8 - 522*x^7 + 334*x^6 + 846*x^5 - 490*x^4 - 512*x^3 + 275*x^2 - 3*x - 2

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} - \nu^{2} - 5\nu + 3 \)
\(\beta_{4}\)	\(=\)	\( \nu^{5} - 9\nu^{3} - \nu^{2} + 15\nu \)
\(\beta_{5}\)	\(=\)	(v^10 - v^9 - 18*v^8 + 14*v^7 + 112*v^6 - 66*v^5 - 276*v^4 + 129*v^3 + 231*v^2 - 88*v - 5) / 3
\(\beta_{6}\)	\(=\)	(v^10 - v^9 - 18*v^8 + 14*v^7 + 115*v^6 - 66*v^5 - 303*v^4 + 123*v^3 + 279*v^2 - 73*v - 8) / 3
\(\beta_{7}\)	\(=\)	(v^12 - 2*v^11 - 18*v^10 + 33*v^9 + 119*v^8 - 192*v^7 - 352*v^6 + 462*v^5 + 456*v^4 - 400*v^3 - 220*v^2 + 33*v + 32) / 6
\(\beta_{8}\)	\(=\)	(-v^12 + 20*v^10 + 3*v^9 - 147*v^8 - 38*v^7 + 490*v^6 + 150*v^5 - 756*v^4 - 218*v^3 + 450*v^2 + 85*v - 26) / 6
\(\beta_{9}\)	\(=\)	(v^12 - 2*v^11 - 18*v^10 + 33*v^9 + 119*v^8 - 192*v^7 - 352*v^6 + 468*v^5 + 450*v^4 - 454*v^3 - 184*v^2 + 129*v + 2) / 6
\(\beta_{10}\)	\(=\)	(-v^12 + 22*v^10 + v^9 - 183*v^8 - 4*v^7 + 714*v^6 - 48*v^5 - 1314*v^4 + 238*v^3 + 918*v^2 - 265*v - 12) / 6
\(\beta_{11}\)	\(=\)	(2*v^12 - v^11 - 39*v^10 + 15*v^9 + 280*v^8 - 87*v^7 - 911*v^6 + 267*v^5 + 1353*v^4 - 431*v^3 - 737*v^2 + 285*v - 8) / 6
\(\beta_{12}\)	\(=\)	(-v^12 + 2*v^11 + 22*v^10 - 37*v^9 - 185*v^8 + 248*v^7 + 734*v^6 - 738*v^5 - 1356*v^4 + 976*v^3 + 934*v^2 - 463*v - 10) / 6

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

−2.47342
−2.41029
−1.75512
−1.56171
−1.28768
−0.0755671
0.102888
0.477012
1.33986
1.53724
1.70702
2.68668
2.71308

−2.47342

0

4.11780

−1.00000

0

4.34305

−5.23822

0

2.47342

1.2

−2.41029

0

3.80950

−1.00000

0

0.995005

−4.36143

0

2.41029

1.3

−1.75512

0

1.08045

−1.00000

0

−3.67006

1.61392

0

1.75512

1.4

−1.56171

0

0.438929

−1.00000

0

4.94615

2.43794

0

1.56171

1.5

−1.28768

0

−0.341875

−1.00000

0

−0.783370

3.01559

0

1.28768

1.6

−0.0755671

0

−1.99429

−1.00000

0

0.0318269

0.301837

0

0.0755671

1.7

0.102888

0

−1.98941

−1.00000

0

4.83328

−0.410464

0

−0.102888

1.8

0.477012

0

−1.77246

−1.00000

0

−0.671928

−1.79951

0

−0.477012

1.9

1.33986

0

−0.204778

−1.00000

0

0.373916

−2.95409

0

−1.33986

1.10

1.53724

0

0.363114

−1.00000

0

3.39823

−2.51629

0

−1.53724

1.11

1.70702

0

0.913910

−1.00000

0

−2.70418

−1.85397

0

−1.70702

1.12

2.68668

0

5.21827

−1.00000

0

−3.56183

8.64648

0

−2.68668

1.13

2.71308

0

5.36083

−1.00000

0

2.46990

9.11822

0

−2.71308

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(5\)	\(1\)
\(13\)	\(1\)

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	5265.2.a.bh		13
3.b	odd	2	1		5265.2.a.bg		13
9.c	even	3	2		1755.2.i.g		26
9.d	odd	6	2		585.2.i.g	✓	26

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
585.2.i.g	✓	26	9.d	odd	6	2
1755.2.i.g		26	9.c	even	3	2
5265.2.a.bg		13	3.b	odd	2	1
5265.2.a.bh		13	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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5265))\):

T2^13 - T2^12 - 20*T2^11 + 17*T2^10 + 150*T2^9 - 109*T2^8 - 522*T2^7 + 334*T2^6 + 846*T2^5 - 490*T2^4 - 512*T2^3 + 275*T2^2 - 3*T2 - 2

T7^13 - 10*T7^12 - 10*T7^11 + 348*T7^10 - 447*T7^9 - 4071*T7^8 + 8041*T7^7 + 17387*T7^6 - 38194*T7^5 - 17572*T7^4 + 33520*T7^3 + 6976*T7^2 - 6288*T7 + 192

T11^13 + 11*T11^12 - 26*T11^11 - 580*T11^10 - 480*T11^9 + 9314*T11^8 + 15036*T11^7 - 44927*T11^6 - 68388*T11^5 + 53399*T11^4 + 91942*T11^3 + 16727*T11^2 - 7911*T11 - 1832

T17^13 + 3*T17^12 - 136*T17^11 - 299*T17^10 + 6617*T17^9 + 9864*T17^8 - 137036*T17^7 - 122583*T17^6 + 1123661*T17^5 + 471252*T17^4 - 2680492*T17^3 + 467440*T17^2 + 1403840*T17 - 564848

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^13 - T^12 - 20*T^11 + 17*T^10 + 150*T^9 - 109*T^8 - 522*T^7 + 334*T^6 + 846*T^5 - 490*T^4 - 512*T^3 + 275*T^2 - 3*T - 2
$3$	\( T^{13} \)
$5$	\( (T + 1)^{13} \)
$7$	T^13 - 10*T^12 - 10*T^11 + 348*T^10 - 447*T^9 - 4071*T^8 + 8041*T^7 + 17387*T^6 - 38194*T^5 - 17572*T^4 + 33520*T^3 + 6976*T^2 - 6288*T + 192
$11$	T^13 + 11*T^12 - 26*T^11 - 580*T^10 - 480*T^9 + 9314*T^8 + 15036*T^7 - 44927*T^6 - 68388*T^5 + 53399*T^4 + 91942*T^3 + 16727*T^2 - 7911*T - 1832