Newform orbit 2695.2.a.u

• Label: 2695.2.a.u
• Level: $2695$
• Weight: $2$
• Character orbit: 2695.a
• Self dual: yes
• Analytic conductor: $21.520$
• Analytic rank: $1$
• Dimension: $10$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2695 = 5 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2695.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(21.5196833447\)
Analytic rank:	\(1\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	\( x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 56x^{6} - 92x^{5} - 86x^{4} + 116x^{3} + 31x^{2} - 30x - 7 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - b1 * q^2 + (-b9 - 1) * q^3 + (b4 + b3 + 1) * q^4 + q^5 + (b9 + b7 + b6 + 2*b1 - 1) * q^6 + (-b7 - b6 - b4 - b3 - b2 - b1) * q^8 + (b9 + b8 - b6 - b4 - b2 + 1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{2} - 8 q^{3} + 10 q^{4} + 10 q^{5} - 4 q^{6} - 6 q^{8} + 10 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 56x^{6} - 92x^{5} - 86x^{4} + 116x^{3} + 31x^{2} - 30x - 7 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{9} - \nu^{8} + 10\nu^{7} + 17\nu^{6} - 27\nu^{5} - 66\nu^{4} + 20\nu^{3} + 32\nu^{2} - 34\nu + 16 ) / 11 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{9} - \nu^{8} + 10\nu^{7} + 6\nu^{6} - 5\nu^{5} + 22\nu^{4} - 123\nu^{3} - 133\nu^{2} + 131\nu + 38 ) / 11 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{9} + \nu^{8} - 10\nu^{7} - 6\nu^{6} + 5\nu^{5} - 22\nu^{4} + 123\nu^{3} + 144\nu^{2} - 131\nu - 71 ) / 11 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{9} - \nu^{8} + 21\nu^{7} + 6\nu^{6} - 126\nu^{5} - 11\nu^{4} + 251\nu^{3} + 32\nu^{2} - 111\nu - 28 ) / 11 \)
\(\beta_{6}\)	\(=\)	\( ( 5\nu^{9} - 6\nu^{8} - 61\nu^{7} + 58\nu^{6} + 245\nu^{5} - 176\nu^{4} - 342\nu^{3} + 170\nu^{2} + 71\nu + 8 ) / 11 \)
\(\beta_{7}\)	\(=\)	\( ( -4\nu^{9} + 7\nu^{8} + 51\nu^{7} - 75\nu^{6} - 218\nu^{5} + 242\nu^{4} + 333\nu^{3} - 213\nu^{2} - 92\nu + 9 ) / 11 \)
\(\beta_{8}\)	\(=\)	\( ( 3\nu^{9} - 8\nu^{8} - 41\nu^{7} + 103\nu^{6} + 180\nu^{5} - 407\nu^{4} - 247\nu^{3} + 476\nu^{2} - 8\nu - 70 ) / 11 \)
\(\beta_{9}\)	\(=\)	\( ( 6\nu^{9} - 5\nu^{8} - 82\nu^{7} + 52\nu^{6} + 371\nu^{5} - 154\nu^{4} - 604\nu^{3} + 83\nu^{2} + 226\nu + 36 ) / 11 \)

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

2.62366
2.18262
2.13489
0.911764
0.777582
−0.231979
−0.480122
−1.45836
−2.20075
−2.25931

−2.62366

−3.16125

4.88361

1.00000

8.29406

0

−7.56562

6.99351

−2.62366

1.2

−2.18262

1.88048

2.76381

1.00000

−4.10437

0

−1.66711

0.536219

−2.18262

1.3

−2.13489

−0.317256

2.55774

1.00000

0.677306

0

−1.19071

−2.89935

−2.13489

1.4

−0.911764

1.93300

−1.16869

1.00000

−1.76244

0

2.88909

0.736495

−0.911764

1.5

−0.777582

−3.22419

−1.39537

1.00000

2.50707

0

2.64018

7.39539

−0.777582

1.6

0.231979

−0.535939

−1.94619

1.00000

−0.124327

0

−0.915432

−2.71277

0.231979

1.7

0.480122

−0.720208

−1.76948

1.00000

−0.345788

0

−1.80981

−2.48130

0.480122

1.8

1.45836

0.658575

0.126803

1.00000

0.960437

0

−2.73179

−2.56628

1.45836

1.9

2.20075

−1.61883

2.84330

1.00000

−3.56264

0

1.85588

−0.379385

2.20075

1.10

2.25931

−2.89439

3.10446

1.00000

−6.53930

0

2.49532

5.37747

2.25931

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\(-1\)
\(7\)	\(1\)
\(11\)	\(-1\)

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	2695.2.a.u	✓	10
7.b	odd	2	1		2695.2.a.v	yes	10

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2695.2.a.u	✓	10	1.a	even	1	1	trivial
2695.2.a.v	yes	10	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2695))\):

\( T_{2}^{10} + 2T_{2}^{9} - 13T_{2}^{8} - 24T_{2}^{7} + 56T_{2}^{6} + 92T_{2}^{5} - 86T_{2}^{4} - 116T_{2}^{3} + 31T_{2}^{2} + 30T_{2} - 7 \)

\( T_{3}^{10} + 8T_{3}^{9} + 12T_{3}^{8} - 48T_{3}^{7} - 134T_{3}^{6} + 28T_{3}^{5} + 300T_{3}^{4} + 172T_{3}^{3} - 74T_{3}^{2} - 76T_{3} - 14 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^10 + 2*T^9 - 13*T^8 - 24*T^7 + 56*T^6 + 92*T^5 - 86*T^4 - 116*T^3 + 31*T^2 + 30*T - 7
$3$	T^10 + 8*T^9 + 12*T^8 - 48*T^7 - 134*T^6 + 28*T^5 + 300*T^4 + 172*T^3 - 74*T^2 - 76*T - 14
$5$	\( (T - 1)^{10} \)
$7$	\( T^{10} \)
$11$	\( (T - 1)^{10} \)