Embedded newform 23.4.c.a.6.2

• Label: 23.4.c.a.6.2
• Level: $23$
• Weight: $4$
• Character: 23.6
• Analytic conductor: $1.357$
• Analytic rank: $0$
• Dimension: $50$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	23.c (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.35704393013\)
Analytic rank:	\(0\)
Dimension:	\(50\)
Relative dimension:	\(5\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			6.2
Character	\(\chi\)	\(=\)	23.6
Dual form			23.4.c.a.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.58245 + 3.46509i) q^{2} +(-7.04544 - 2.06873i) q^{3} +(-4.26380 - 4.92069i) q^{4} +(-4.88780 + 3.14120i) q^{5} +(18.3174 - 21.1394i) q^{6} +(-2.25200 + 15.6630i) q^{7} +(-5.44232 + 1.59801i) q^{8} +(22.6448 + 14.5529i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 11 q^{2} - 13 q^{3} - 27 q^{4} - 19 q^{5} - 4 q^{6} - 19 q^{7} + 28 q^{8} + 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{9}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.58245	+	3.46509i	−0.559482	+	1.22509i	0.392730	+	0.919654i	\(0.371531\pi\)
							−0.952211	+	0.305440i	\(0.901196\pi\)
\(3\)	−7.04544	−	2.06873i	−1.35590	−	0.398127i	−0.478582	−	0.878043i	\(-0.658849\pi\)
							−0.877314	+	0.479916i	\(0.840667\pi\)
\(5\)	−4.88780	+	3.14120i	−0.437178	+	0.280957i	−0.740660	−	0.671880i	\(-0.765487\pi\)
							0.303482	+	0.952837i	\(0.401851\pi\)
\(7\)	−2.25200	+	15.6630i	−0.121597	+	0.845724i	0.834151	+	0.551536i	\(0.185958\pi\)
							−0.955748	+	0.294188i	\(0.904951\pi\)
\(11\)	27.3406	+	59.8675i	0.749408	+	1.64098i	0.767426	+	0.641138i	\(0.221537\pi\)
							−0.0180172	+	0.999838i	\(0.505735\pi\)
\(13\)	−10.3660	−	72.0973i	−0.221155	−	1.53817i	−0.733679	−	0.679496i	\(-0.762199\pi\)
							0.512524	−	0.858673i	\(-0.328710\pi\)
\(17\)	−25.6631	+	29.6168i	−0.366131	+	0.422537i	−0.908684	−	0.417484i	\(-0.862912\pi\)
							0.542554	+	0.840021i	\(0.317457\pi\)
\(19\)	25.7943	+	29.7682i	0.311454	+	0.359437i	0.889797	−	0.456357i	\(-0.150846\pi\)
							−0.578343	+	0.815794i	\(0.696300\pi\)
\(23\)	−90.0070	+	63.7631i	−0.815990	+	0.578067i
							\(29\)	76.4670	−	88.2477i	0.489640	−	0.565075i	−0.456129	−	0.889914i	\(-0.650764\pi\)
0.945770	+	0.324838i	\(0.105310\pi\)
\(31\)	−44.2678	+	12.9982i	−0.256475	+	0.0753079i	−0.407443	−	0.913231i	\(-0.633579\pi\)
							0.150967	+	0.988539i	\(0.451761\pi\)
\(37\)	232.264	+	149.267i	1.03200	+	0.663224i	0.942995	−	0.332807i	\(-0.107996\pi\)
							0.0890023	+	0.996031i	\(0.471632\pi\)
\(41\)	−221.026	+	142.045i	−0.841913	+	0.541065i	−0.889043	−	0.457824i	\(-0.848629\pi\)
							0.0471298	+	0.998889i	\(0.484993\pi\)
\(43\)	−158.552	−	46.5550i	−0.562301	−	0.165106i	−0.0117842	−	0.999931i	\(-0.503751\pi\)
							−0.550517	+	0.834824i	\(0.685569\pi\)
\(47\)	168.338			0.522438			0.261219	−	0.965280i	\(-0.415876\pi\)
							0.261219	+	0.965280i	\(0.415876\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.4.c.a.6.2	yes	50
3.2	odd	2		207.4.i.a.190.4		50
23.2	even	11		529.4.a.n.1.5		25
23.4	even	11	inner	23.4.c.a.4.2	✓	50
23.21	odd	22		529.4.a.m.1.5		25
69.50	odd	22		207.4.i.a.73.4		50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.c.a.4.2	✓	50	23.4	even	11	inner
23.4.c.a.6.2	yes	50	1.1	even	1	trivial
207.4.i.a.73.4		50	69.50	odd	22
207.4.i.a.190.4		50	3.2	odd	2
529.4.a.m.1.5		25	23.21	odd	22
529.4.a.n.1.5		25	23.2	even	11