Newform orbit 169.4.c.i

• Label: 169.4.c.i
• Level: $169$
• Weight: $4$
• Character orbit: 169.c
• Analytic conductor: $9.971$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.97132279097\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - 2 \beta_{2} q^{2} + 7 \beta_1 q^{3} + (4 \beta_1 - 4) q^{4} + 8 \beta_{3} q^{5} + (14 \beta_{3} - 14 \beta_{2}) q^{6} + (13 \beta_{3} - 13 \beta_{2}) q^{7} - 8 \beta_{3} q^{8} + (22 \beta_1 - 22) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 14 q^{3} - 8 q^{4} - 44 q^{9}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( \zeta_{12}^{2} \)
\(\beta_{2}\)	\(=\)	\( \zeta_{12}^{3} + \zeta_{12} \)
\(\beta_{3}\)	\(=\)	\( -\zeta_{12}^{3} + 2\zeta_{12} \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(-1 + \beta_{1}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

22.1

0.866025	−	0.500000i
−0.866025	+	0.500000i
0.866025	+	0.500000i
−0.866025	−	0.500000i

−1.73205

+

3.00000i

3.50000

−

6.06218i

−2.00000

−

3.46410i

13.8564

12.1244

+

21.0000i

11.2583

+

19.5000i

−13.8564

−11.0000

−

19.0526i

−24.0000

+

41.5692i

22.2

1.73205

−

3.00000i

3.50000

−

6.06218i

−2.00000

−

3.46410i

−13.8564

−12.1244

−

21.0000i

−11.2583

−

19.5000i

13.8564

−11.0000

−

19.0526i

−24.0000

+

41.5692i

146.1

−1.73205

−

3.00000i

3.50000

+

6.06218i

−2.00000

+

3.46410i

13.8564

12.1244

−

21.0000i

11.2583

−

19.5000i

−13.8564

−11.0000

+

19.0526i

−24.0000

−

41.5692i

146.2

1.73205

+

3.00000i

3.50000

+

6.06218i

−2.00000

+

3.46410i

−13.8564

−12.1244

+

21.0000i

−11.2583

+

19.5000i

13.8564

−11.0000

+

19.0526i

−24.0000

−

41.5692i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner
13.c	even	3	1	inner
13.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	169.4.c.i		4
13.b	even	2	1	inner	169.4.c.i		4
13.c	even	3	1		169.4.a.h		2
13.c	even	3	1	inner	169.4.c.i		4
13.d	odd	4	1		13.4.e.a	✓	2
13.d	odd	4	1		169.4.e.b		2
13.e	even	6	1		169.4.a.h		2
13.e	even	6	1	inner	169.4.c.i		4
13.f	odd	12	1		13.4.e.a	✓	2
13.f	odd	12	2		169.4.b.b		2
13.f	odd	12	1		169.4.e.b		2
39.f	even	4	1		117.4.q.c		2
39.h	odd	6	1		1521.4.a.q		2
39.i	odd	6	1		1521.4.a.q		2
39.k	even	12	1		117.4.q.c		2
52.f	even	4	1		208.4.w.a		2
52.l	even	12	1		208.4.w.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.4.e.a	✓	2	13.d	odd	4	1
13.4.e.a	✓	2	13.f	odd	12	1
117.4.q.c		2	39.f	even	4	1
117.4.q.c		2	39.k	even	12	1
169.4.a.h		2	13.c	even	3	1
169.4.a.h		2	13.e	even	6	1
169.4.b.b		2	13.f	odd	12	2
169.4.c.i		4	1.a	even	1	1	trivial
169.4.c.i		4	13.b	even	2	1	inner
169.4.c.i		4	13.c	even	3	1	inner
169.4.c.i		4	13.e	even	6	1	inner
169.4.e.b		2	13.d	odd	4	1
169.4.e.b		2	13.f	odd	12	1
208.4.w.a		2	52.f	even	4	1
208.4.w.a		2	52.l	even	12	1
1521.4.a.q		2	39.h	odd	6	1
1521.4.a.q		2	39.i	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 12T_{2}^{2} + 144 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} + 12T^{2} + 144 \)
$3$	\( (T^{2} - 7 T + 49)^{2} \)
$5$	\( (T^{2} - 192)^{2} \)
$7$	\( T^{4} + 507 T^{2} + 257049 \)
$11$	\( T^{4} + 507 T^{2} + 257049 \)