Embedded newform 13.7.f.a.6.2

• Label: 13.7.f.a.6.2
• Level: $13$
• Weight: $7$
• Character: 13.6
• Analytic conductor: $2.991$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	13.f (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.99070308706\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			6.2
Character	\(\chi\)	\(=\)	13.6
Dual form			13.7.f.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.74273 - 10.2360i) q^{2} +(-22.3986 + 38.7956i) q^{3} +(-41.8275 + 24.1491i) q^{4} +(-35.0169 + 35.0169i) q^{5} +(458.545 + 122.867i) q^{6} +(-112.597 + 420.218i) q^{7} +(-117.658 - 117.658i) q^{8} +(-638.899 - 1106.61i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{2} - 2 q^{3} - 6 q^{4} - 114 q^{5} + 634 q^{6} + 316 q^{7} - 1398 q^{8} - 1946 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.74273	−	10.2360i	−0.342841	−	1.27950i	−0.895114	−	0.445837i	\(-0.852906\pi\)
							0.552273	−	0.833663i	\(-0.313760\pi\)
\(3\)	−22.3986	+	38.7956i	−0.829580	+	1.43687i	0.0687888	+	0.997631i	\(0.478087\pi\)
							−0.898368	+	0.439243i	\(0.855247\pi\)
\(5\)	−35.0169	+	35.0169i	−0.280136	+	0.280136i	−0.833163	−	0.553027i	\(-0.813472\pi\)
							0.553027	+	0.833163i	\(0.313472\pi\)
\(7\)	−112.597	+	420.218i	−0.328271	+	1.22513i	0.582711	+	0.812679i	\(0.301992\pi\)
							−0.910982	+	0.412446i	\(0.864675\pi\)
\(11\)	−1837.55	+	492.370i	−1.38058	+	0.369925i	−0.871331	−	0.490695i	\(-0.836743\pi\)
							−0.509248	+	0.860620i	\(0.670076\pi\)
\(13\)	2158.38	−	410.139i	0.982421	−	0.186681i
							\(17\)	2973.01	−	1716.47i	0.605132	−	0.349373i	−0.165926	−	0.986138i	\(-0.553061\pi\)
0.771058	+	0.636765i	\(0.219728\pi\)
\(19\)	1319.54	+	353.570i	0.192381	+	0.0515483i	0.353723	−	0.935350i	\(-0.384916\pi\)
							−0.161342	+	0.986899i	\(0.551582\pi\)
\(23\)	4730.68	+	2731.26i	0.388812	+	0.224481i	0.681645	−	0.731683i	\(-0.261265\pi\)
							−0.292833	+	0.956164i	\(0.594598\pi\)
\(29\)	6245.69	−	10817.9i	0.256086	−	0.443555i	−0.709104	−	0.705104i	\(-0.750900\pi\)
							0.965190	+	0.261550i	\(0.0842335\pi\)
\(31\)	−39371.6	+	39371.6i	−1.32160	+	1.32160i	−0.409111	+	0.912485i	\(0.634161\pi\)
							−0.912485	+	0.409111i	\(0.865839\pi\)
\(37\)	−41656.4	+	11161.8i	−0.822388	+	0.220358i	−0.645390	−	0.763853i	\(-0.723305\pi\)
							−0.176998	+	0.984211i	\(0.556638\pi\)
\(41\)	9281.22	+	34638.0i	0.134665	+	0.502575i	0.999999	+	0.00138043i	\(0.000439406\pi\)
							−0.865334	+	0.501195i	\(0.832894\pi\)
\(43\)	80790.5	−	46644.4i	1.01614	−	0.586671i	0.103159	−	0.994665i	\(-0.467105\pi\)
							0.912985	+	0.407994i	\(0.133772\pi\)
\(47\)	31648.7	+	31648.7i	0.304833	+	0.304833i	0.842901	−	0.538068i	\(-0.180846\pi\)
							−0.538068	+	0.842901i	\(0.680846\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.7.f.a.6.2	✓	24
13.11	odd	12	inner	13.7.f.a.11.2	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.7.f.a.6.2	✓	24	1.1	even	1	trivial
13.7.f.a.11.2	yes	24	13.11	odd	12	inner