Embedded newform 22.14.c.b.3.7

• Label: 22.14.c.b.3.7
• Level: $22$
• Weight: $14$
• Character: 22.3
• Analytic conductor: $23.591$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	22.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.5908043694\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			3.7
Character	\(\chi\)	\(=\)	22.3
Dual form			22.14.c.b.15.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-51.7771 + 37.6183i) q^{2} +(751.070 + 2311.56i) q^{3} +(1265.73 - 3895.53i) q^{4} +(25138.8 + 18264.4i) q^{5} +(-125845. - 91431.7i) q^{6} +(81934.7 - 252169. i) q^{7} +(81007.0 + 249314. i) q^{8} +(-3.48935e6 + 2.53516e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 448 q^{2} - 1248 q^{3} - 28672 q^{4} + 71554 q^{5} + 99008 q^{6} + 361734 q^{7} - 1835008 q^{8} - 8615367 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−51.7771	+	37.6183i	−0.572061	+	0.415627i
							\(3\)	751.070	+	2311.56i	0.594829	+	1.83070i	0.555574	+	0.831467i	\(0.312499\pi\)
0.0392556	+	0.999229i	\(0.487501\pi\)
\(5\)	25138.8	+	18264.4i	0.719514	+	0.522758i	0.886229	−	0.463247i	\(-0.153316\pi\)
							−0.166715	+	0.986005i	\(0.553316\pi\)
\(7\)	81934.7	−	252169.i	0.263227	−	0.810130i	−0.728869	−	0.684653i	\(-0.759954\pi\)
							0.992097	−	0.125477i	\(-0.0400461\pi\)
\(11\)	−3.82444e6	−	4.46054e6i	−0.650901	−	0.759162i
							\(13\)	−1.61606e7	+	1.17414e7i	−0.928594	+	0.674663i	−0.945648	−	0.325192i	\(-0.894571\pi\)
0.0170543	+	0.999855i	\(0.494571\pi\)
\(17\)	−1.83807e7	−	1.33543e7i	−0.184690	−	0.134185i	0.491599	−	0.870822i	\(-0.336413\pi\)
							−0.676289	+	0.736637i	\(0.736413\pi\)
\(19\)	3.55579e7	+	1.09436e8i	0.173396	+	0.533657i	0.999557	−	0.0297775i	\(-0.00947988\pi\)
							−0.826161	+	0.563434i	\(0.809480\pi\)
\(23\)	−1.17317e9			−1.65246			−0.826230	−	0.563334i	\(-0.809519\pi\)
							−0.826230	+	0.563334i	\(0.809519\pi\)
\(29\)	−9.71136e8	+	2.98885e9i	−0.303175	+	0.933076i	0.677177	+	0.735820i	\(0.263203\pi\)
							−0.980352	+	0.197256i	\(0.936797\pi\)
\(31\)	−1.53534e9	+	1.11549e9i	−0.310710	+	0.225744i	−0.732201	−	0.681089i	\(-0.761507\pi\)
							0.421491	+	0.906832i	\(0.361507\pi\)
\(37\)	7.23826e9	−	2.22771e10i	0.463792	−	1.42740i	−0.396704	−	0.917946i	\(-0.629846\pi\)
							0.860496	−	0.509457i	\(-0.170154\pi\)
\(41\)	1.07991e10	+	3.32364e10i	0.355054	+	1.09274i	0.955978	+	0.293438i	\(0.0947993\pi\)
							−0.600924	+	0.799306i	\(0.705201\pi\)
\(43\)	4.35443e10			1.05048			0.525238	−	0.850955i	\(-0.323976\pi\)
							0.525238	+	0.850955i	\(0.323976\pi\)
\(47\)	2.94433e10	+	9.06171e10i	0.398428	+	1.22624i	0.926259	+	0.376887i	\(0.123006\pi\)
							−0.527831	+	0.849350i	\(0.676994\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.14.c.b.3.7	✓	28
11.4	even	5	inner	22.14.c.b.15.7	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.14.c.b.3.7	✓	28	1.1	even	1	trivial
22.14.c.b.15.7	yes	28	11.4	even	5	inner