Embedded newform 13.12.e.a.4.5

• Label: 13.12.e.a.4.5
• Level: $13$
• Weight: $12$
• Character: 13.4
• Analytic conductor: $9.988$
• Analytic rank: $0$
• Dimension: $22$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	13.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.98846134727\)
Analytic rank:	\(0\)
Dimension:	\(22\)
Relative dimension:	\(11\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4.5
Character	\(\chi\)	\(=\)	13.4
Dual form			13.12.e.a.10.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-18.1006 - 10.4504i) q^{2} +(-200.570 + 347.397i) q^{3} +(-805.579 - 1395.30i) q^{4} -5339.78i q^{5} +(7260.86 - 4192.06i) q^{6} +(-15062.6 + 8696.38i) q^{7} +76479.2i q^{8} +(8117.03 + 14059.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 242 q^{3} + 8191 q^{4} + 45528 q^{6} + 128496 q^{7} - 322217 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−18.1006	−	10.4504i	−0.399970	−	0.230923i	0.286501	−	0.958080i	\(-0.407508\pi\)
							−0.686471	+	0.727157i	\(0.740841\pi\)
\(3\)	−200.570	+	347.397i	−0.476539	+	0.825390i	−0.999639	−	0.0268816i	\(-0.991442\pi\)
							0.523099	+	0.852272i	\(0.324776\pi\)
\(5\)		−	5339.78i		−	0.764167i	−0.924128	−	0.382083i	\(-0.875207\pi\)
							0.924128	−	0.382083i	\(-0.124793\pi\)
\(7\)	−15062.6	+	8696.38i	−0.338735	+	0.195569i	−0.659712	−	0.751518i	\(-0.729322\pi\)
							0.320978	+	0.947087i	\(0.395989\pi\)
\(11\)	343296.	+	198202.i	0.642701	+	0.371064i	0.785654	−	0.618666i	\(-0.212327\pi\)
							−0.142953	+	0.989729i	\(0.545660\pi\)
\(13\)	1.33554e6	−	92086.9i	0.997631	−	0.0687875i
							\(17\)	3.31334e6	+	5.73888e6i	0.565975	+	0.980297i	0.996958	+	0.0779375i	\(0.0248335\pi\)
−0.430983	+	0.902360i	\(0.641833\pi\)
\(19\)	769443.	−	444238.i	0.0712905	−	0.0411596i	−0.463931	−	0.885871i	\(-0.653561\pi\)
							0.535222	+	0.844712i	\(0.320228\pi\)
\(23\)	9.87015e6	−	1.70956e7i	0.319757	−	0.553836i	−0.660680	−	0.750668i	\(-0.729732\pi\)
							0.980437	+	0.196832i	\(0.0630652\pi\)
\(29\)	−3.14945e7	+	5.45501e7i	−0.285132	+	0.493863i	−0.972641	−	0.232313i	\(-0.925371\pi\)
							0.687509	+	0.726175i	\(0.258704\pi\)
\(31\)		−	1.91221e7i		−	0.119962i	−0.998200	−	0.0599812i	\(-0.980896\pi\)
							0.998200	−	0.0599812i	\(-0.0191041\pi\)
\(37\)	1.13603e8	+	6.55887e7i	0.269327	+	0.155496i	0.628582	−	0.777743i	\(-0.283636\pi\)
							−0.359255	+	0.933240i	\(0.616969\pi\)
\(41\)	1.22889e9	+	7.09503e8i	1.65655	+	0.956407i	0.974291	+	0.225292i	\(0.0723338\pi\)
							0.682255	+	0.731115i	\(0.261000\pi\)
\(43\)	8.04107e7	+	1.39275e8i	0.0834137	+	0.144477i	0.904714	−	0.426019i	\(-0.140084\pi\)
							−0.821300	+	0.570496i	\(0.806751\pi\)
\(47\)		−	1.75296e9i		−	1.11490i	−0.830212	−	0.557448i	\(-0.811781\pi\)
							0.830212	−	0.557448i	\(-0.188219\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.e.a.4.5	✓	22
13.6	odd	12		169.12.a.g.1.14		22
13.7	odd	12		169.12.a.g.1.9		22
13.10	even	6	inner	13.12.e.a.10.5	yes	22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.e.a.4.5	✓	22	1.1	even	1	trivial
13.12.e.a.10.5	yes	22	13.10	even	6	inner
169.12.a.g.1.9		22	13.7	odd	12
169.12.a.g.1.14		22	13.6	odd	12