Embedded newform 13.7.f.a.6.4

• Label: 13.7.f.a.6.4
• 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			6.4
Character	\(\chi\)	\(=\)	13.6
Dual form			13.7.f.a.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.76668 + 6.59335i) q^{2} +(-10.7542 + 18.6268i) q^{3} +(15.0745 - 8.70327i) q^{4} +(-144.857 + 144.857i) q^{5} +(-141.812 - 37.9984i) q^{6} +(49.5658 - 184.982i) q^{7} +(392.922 + 392.922i) q^{8} +(133.195 + 230.701i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{2} - 2 q^{3} - 6 q^{4} - 114 q^{5} + 634 q^{6} + 316 q^{7} - 1398 q^{8} - 1946 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.76668	+	6.59335i	0.220835	+	0.824169i	0.984030	+	0.178001i	\(0.0569629\pi\)
							−0.763195	+	0.646168i	\(0.776370\pi\)
\(3\)	−10.7542	+	18.6268i	−0.398303	+	0.689881i	−0.993517	−	0.113687i	\(-0.963734\pi\)
							0.595214	+	0.803567i	\(0.297067\pi\)
\(5\)	−144.857	+	144.857i	−1.15886	+	1.15886i	−0.174134	+	0.984722i	\(0.555713\pi\)
							−0.984722	+	0.174134i	\(0.944287\pi\)
\(7\)	49.5658	−	184.982i	0.144507	−	0.539306i	−0.855270	−	0.518182i	\(-0.826609\pi\)
							0.999777	−	0.0211237i	\(-0.00672440\pi\)
\(11\)	1439.16	−	385.622i	1.08126	−	0.289724i	0.326150	−	0.945318i	\(-0.394248\pi\)
							0.755114	+	0.655594i	\(0.227582\pi\)
\(13\)	1708.06	−	1381.79i	0.777449	−	0.628946i
							\(17\)	−5325.85	+	3074.88i	−1.08403	+	0.625866i	−0.931981	−	0.362507i	\(-0.881921\pi\)
−0.152051	+	0.988373i	\(0.548588\pi\)
\(19\)	2668.98	+	715.152i	0.389121	+	0.104265i	0.448075	−	0.893996i	\(-0.352110\pi\)
							−0.0589537	+	0.998261i	\(0.518776\pi\)
\(23\)	−1100.31	−	635.262i	−0.0904337	−	0.0522119i	0.454101	−	0.890950i	\(-0.349960\pi\)
							−0.544535	+	0.838738i	\(0.683294\pi\)
\(29\)	14334.4	−	24827.9i	0.587741	−	1.01800i	−0.406786	−	0.913523i	\(-0.633351\pi\)
							0.994528	−	0.104474i	\(-0.0333160\pi\)
\(31\)	−2357.22	+	2357.22i	−0.0791253	+	0.0791253i	−0.745562	−	0.666436i	\(-0.767819\pi\)
							0.666436	+	0.745562i	\(0.267819\pi\)
\(37\)	37580.6	−	10069.7i	0.741923	−	0.198798i	0.131991	−	0.991251i	\(-0.457863\pi\)
							0.609933	+	0.792453i	\(0.291196\pi\)
\(41\)	−18395.3	−	68652.4i	−0.266905	−	0.996102i	−0.961074	−	0.276290i	\(-0.910895\pi\)
							0.694169	−	0.719812i	\(-0.255772\pi\)
\(43\)	−51822.0	+	29919.5i	−0.651792	+	0.376312i	−0.789142	−	0.614210i	\(-0.789475\pi\)
							0.137351	+	0.990522i	\(0.456141\pi\)
\(47\)	137657.	+	137657.i	1.32589	+	1.32589i	0.908924	+	0.416962i	\(0.136905\pi\)
							0.416962	+	0.908924i	\(0.363095\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.7.f.a.6.4	✓	24
13.11	odd	12	inner	13.7.f.a.11.4	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.7.f.a.6.4	✓	24	1.1	even	1	trivial
13.7.f.a.11.4	yes	24	13.11	odd	12	inner