Newform orbit 165.2.f.b

• Label: 165.2.f.b
• Level: $165$
• Weight: $2$
• Character orbit: 165.f
• Analytic conductor: $1.318$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	165.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.31753163335\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.619810816.2
Defining polynomial:	\( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \) :

\(\beta_{1}\)	\(=\)	\( ( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 378\nu + 63 ) / 319 \)
\(\beta_{2}\)	\(=\)	\( ( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319 \)
\(\beta_{3}\)	\(=\)	\( ( -67\nu^{7} + 63\nu^{6} - 64\nu^{5} + 118\nu^{4} - 1068\nu^{3} + 1545\nu^{2} - 1263\nu + 268 ) / 319 \)
\(\beta_{4}\)	\(=\)	\( ( 83\nu^{7} - 59\nu^{6} + 65\nu^{5} - 70\nu^{4} + 1304\nu^{3} - 1614\nu^{2} + 1198\nu + 306 ) / 319 \)
\(\beta_{5}\)	\(=\)	\( ( -172\nu^{7} - 43\nu^{6} + 69\nu^{5} + 441\nu^{4} - 2218\nu^{3} + 662\nu^{2} + 619\nu - 269 ) / 319 \)
\(\beta_{6}\)	\(=\)	\( ( -196\nu^{7} - 49\nu^{6} - 92\nu^{5} + 369\nu^{4} - 2572\nu^{3} + 1244\nu^{2} - 1038\nu - 173 ) / 319 \)
\(\beta_{7}\)	\(=\)	\( \nu^{7} - 2\nu^{4} + 14\nu^{3} - 8\nu^{2} + \nu + 2 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).

\(n\)	\(46\)	\(56\)	\(67\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

131.1

−1.49094	+	1.49094i
−1.49094	−	1.49094i
1.18254	+	1.18254i
1.18254	−	1.18254i
0.561103	−	0.561103i
0.561103	+	0.561103i
−0.252709	−	0.252709i
−0.252709	+	0.252709i

−2.44579

−1.29021

−

1.15558i

3.98187

1.00000i

3.15558

+

2.82630i

0.981874i

−4.84724

0.329281

+

2.98187i

−

2.44579i

131.2

−2.44579

−1.29021

+

1.15558i

3.98187

−

1.00000i

3.15558

−

2.82630i

−

0.981874i

−4.84724

0.329281

−

2.98187i

2.44579i

131.3

−0.796815

−1.55654

−

0.759725i

−1.36509

−

1.00000i

1.24027

+

0.605361i

4.36509i

2.68135

1.84564

+

2.36509i

0.796815i

131.4

−0.796815

−1.55654

+

0.759725i

−1.36509

1.00000i

1.24027

−

0.605361i

−

4.36509i

2.68135

1.84564

−

2.36509i

−

0.796815i

131.5

1.37033

1.70032

−

0.329998i

−0.122207

1.00000i

2.33000

−

0.452205i

−

3.12221i

−2.90812

2.78220

−

1.12221i

1.37033i

131.6

1.37033

1.70032

+

0.329998i

−0.122207

−

1.00000i

2.33000

+

0.452205i

3.12221i

−2.90812

2.78220

+

1.12221i

−

1.37033i

131.7

1.87228

0.146426

−

1.72585i

1.50542

−

1.00000i

0.274150

−

3.23127i

1.49458i

−0.925994

−2.95712

−

0.505418i

−

1.87228i

131.8

1.87228

0.146426

+

1.72585i

1.50542

1.00000i

0.274150

+

3.23127i

−

1.49458i

−0.925994

−2.95712

+

0.505418i

1.87228i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
33.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	165.2.f.b	yes	8
3.b	odd	2	1		165.2.f.a	✓	8
4.b	odd	2	1		2640.2.f.c		8
5.b	even	2	1		825.2.f.d		8
5.c	odd	4	1		825.2.d.c		8
5.c	odd	4	1		825.2.d.e		8
11.b	odd	2	1		165.2.f.a	✓	8
12.b	even	2	1		2640.2.f.d		8
15.d	odd	2	1		825.2.f.c		8
15.e	even	4	1		825.2.d.b		8
15.e	even	4	1		825.2.d.d		8
33.d	even	2	1	inner	165.2.f.b	yes	8
44.c	even	2	1		2640.2.f.d		8
55.d	odd	2	1		825.2.f.c		8
55.e	even	4	1		825.2.d.b		8
55.e	even	4	1		825.2.d.d		8
132.d	odd	2	1		2640.2.f.c		8
165.d	even	2	1		825.2.f.d		8
165.l	odd	4	1		825.2.d.c		8
165.l	odd	4	1		825.2.d.e		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
165.2.f.a	✓	8	3.b	odd	2	1
165.2.f.a	✓	8	11.b	odd	2	1
165.2.f.b	yes	8	1.a	even	1	1	trivial
165.2.f.b	yes	8	33.d	even	2	1	inner
825.2.d.b		8	15.e	even	4	1
825.2.d.b		8	55.e	even	4	1
825.2.d.c		8	5.c	odd	4	1
825.2.d.c		8	165.l	odd	4	1
825.2.d.d		8	15.e	even	4	1
825.2.d.d		8	55.e	even	4	1
825.2.d.e		8	5.c	odd	4	1
825.2.d.e		8	165.l	odd	4	1
825.2.f.c		8	15.d	odd	2	1
825.2.f.c		8	55.d	odd	2	1
825.2.f.d		8	5.b	even	2	1
825.2.f.d		8	165.d	even	2	1
2640.2.f.c		8	4.b	odd	2	1
2640.2.f.c		8	132.d	odd	2	1
2640.2.f.d		8	12.b	even	2	1
2640.2.f.d		8	44.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 6T_{2}^{2} + 2T_{2} + 5 \) acting on \(S_{2}^{\mathrm{new}}(165, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} - 6 T^{2} + 2 T + 5)^{2} \)
$3$	T^8 + 2*T^7 - 6*T^5 - 10*T^4 - 18*T^3 + 54*T + 81
$5$	\( (T^{2} + 1)^{4} \)
$7$	T^8 + 32*T^6 + 280*T^4 + 656*T^2 + 400
$11$	T^8 - 12*T^7 + 76*T^6 - 348*T^5 + 1270*T^4 - 3828*T^3 + 9196*T^2 - 15972*T + 14641