Newform orbit 169.4.c.e

• Label: 169.4.c.e
• Level: $169$
• Weight: $4$
• Character orbit: 169.c
• Analytic conductor: $9.971$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

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:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - 5 \zeta_{6} + 5) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} - 17 \zeta_{6} q^{4} - 7 q^{5} - 35 \zeta_{6} q^{6} + 13 \zeta_{6} q^{7} - 45 q^{8} - 22 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{2} + 7 q^{3} - 17 q^{4} - 14 q^{5} - 35 q^{6} + 13 q^{7} - 90 q^{8} - 22 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

22.1

0.500000	+	0.866025i
0.500000	−	0.866025i

2.50000

−

4.33013i

3.50000

−

6.06218i

−8.50000

−

14.7224i

−7.00000

−17.5000

−

30.3109i

6.50000

+

11.2583i

−45.0000

−11.0000

−

19.0526i

−17.5000

+

30.3109i

146.1

2.50000

+

4.33013i

3.50000

+

6.06218i

−8.50000

+

14.7224i

−7.00000

−17.5000

+

30.3109i

6.50000

−

11.2583i

−45.0000

−11.0000

+

19.0526i

−17.5000

−

30.3109i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	169.4.c.e		2
13.b	even	2	1		169.4.c.a		2
13.c	even	3	1		13.4.a.a	✓	1
13.c	even	3	1	inner	169.4.c.e		2
13.d	odd	4	2		169.4.e.e		4
13.e	even	6	1		169.4.a.e		1
13.e	even	6	1		169.4.c.a		2
13.f	odd	12	2		169.4.b.a		2
13.f	odd	12	2		169.4.e.e		4
39.h	odd	6	1		1521.4.a.a		1
39.i	odd	6	1		117.4.a.b		1
52.j	odd	6	1		208.4.a.g		1
65.n	even	6	1		325.4.a.d		1
65.q	odd	12	2		325.4.b.b		2
91.n	odd	6	1		637.4.a.a		1
104.n	odd	6	1		832.4.a.a		1
104.r	even	6	1		832.4.a.r		1
143.k	odd	6	1		1573.4.a.a		1
156.p	even	6	1		1872.4.a.k		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.4.a.a	✓	1	13.c	even	3	1
117.4.a.b		1	39.i	odd	6	1
169.4.a.e		1	13.e	even	6	1
169.4.b.a		2	13.f	odd	12	2
169.4.c.a		2	13.b	even	2	1
169.4.c.a		2	13.e	even	6	1
169.4.c.e		2	1.a	even	1	1	trivial
169.4.c.e		2	13.c	even	3	1	inner
169.4.e.e		4	13.d	odd	4	2
169.4.e.e		4	13.f	odd	12	2
208.4.a.g		1	52.j	odd	6	1
325.4.a.d		1	65.n	even	6	1
325.4.b.b		2	65.q	odd	12	2
637.4.a.a		1	91.n	odd	6	1
832.4.a.a		1	104.n	odd	6	1
832.4.a.r		1	104.r	even	6	1
1521.4.a.a		1	39.h	odd	6	1
1573.4.a.a		1	143.k	odd	6	1
1872.4.a.k		1	156.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 5T_{2} + 25 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 5T + 25 \)
$3$	\( T^{2} - 7T + 49 \)
$5$	\( (T + 7)^{2} \)
$7$	\( T^{2} - 13T + 169 \)
$11$	\( T^{2} - 26T + 676 \)