Embedded newform 22.14.c.a.3.4

• Label: 22.14.c.a.3.4
• Level: $22$
• Weight: $14$
• Character: 22.3
• Analytic conductor: $23.591$
• Analytic rank: $0$
• Dimension: $24$
• 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:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			3.4
Character	\(\chi\)	\(=\)	22.3
Dual form			22.14.c.a.15.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(51.7771 - 37.6183i) q^{2} +(346.581 + 1066.67i) q^{3} +(1265.73 - 3895.53i) q^{4} +(-24923.9 - 18108.3i) q^{5} +(58071.1 + 42191.1i) q^{6} +(-7979.10 + 24557.2i) q^{7} +(-81007.0 - 249314. i) q^{8} +(272175. - 197747. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 384 q^{2} + 2708 q^{3} - 24576 q^{4} + 32920 q^{5} + 47488 q^{6} - 509572 q^{7} + 1572864 q^{8} - 3517894 q^{9}+O(q^{10}) \)

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
							\(3\)	346.581	+	1066.67i	0.274484	+	0.844774i	0.989356	+	0.145519i	\(0.0464850\pi\)
−0.714872	+	0.699256i	\(0.753515\pi\)
\(5\)	−24923.9	−	18108.3i	−0.713363	−	0.518289i	0.170894	−	0.985289i	\(-0.445335\pi\)
							−0.884257	+	0.467001i	\(0.845335\pi\)
\(7\)	−7979.10	+	24557.2i	−0.0256340	+	0.0788934i	−0.963055	−	0.269304i	\(-0.913206\pi\)
							0.937421	+	0.348198i	\(0.113206\pi\)
\(11\)	−2.54998e6	+	5.29342e6i	−0.433995	+	0.900915i
							\(13\)	−2.21672e7	+	1.61054e7i	−1.27373	+	0.925422i	−0.999345	−	0.0361969i	\(-0.988476\pi\)
−0.274389	+	0.961619i	\(0.588476\pi\)
\(17\)	−4.73985e7	−	3.44370e7i	−0.476262	−	0.346025i	0.323615	−	0.946189i	\(-0.395102\pi\)
							−0.799877	+	0.600164i	\(0.795102\pi\)
\(19\)	−8.90824e7	−	2.74167e8i	−0.434403	−	1.33696i	−0.893697	−	0.448672i	\(-0.851897\pi\)
							0.459293	−	0.888285i	\(-0.348103\pi\)
\(23\)	−1.25809e9			−1.77207			−0.886036	−	0.463617i	\(-0.846551\pi\)
							−0.886036	+	0.463617i	\(0.846551\pi\)
\(29\)	6.92509e7	−	2.13132e8i	0.0216191	−	0.0665369i	−0.939665	−	0.342096i	\(-0.888863\pi\)
							0.961284	+	0.275559i	\(0.0888631\pi\)
\(31\)	−6.90221e9	+	5.01475e9i	−1.39681	+	1.01484i	−0.401731	+	0.915758i	\(0.631591\pi\)
							−0.995079	+	0.0990837i	\(0.968409\pi\)
\(37\)	1.06317e9	−	3.27210e9i	0.0681226	−	0.209660i	−0.911200	−	0.411964i	\(-0.864843\pi\)
							0.979323	+	0.202304i	\(0.0648429\pi\)
\(41\)	6.01470e9	+	1.85114e10i	0.197751	+	0.608616i	0.999933	+	0.0115365i	\(0.00367225\pi\)
							−0.802182	+	0.597079i	\(0.796328\pi\)
\(43\)	1.14408e10			0.276000			0.138000	−	0.990432i	\(-0.455933\pi\)
							0.138000	+	0.990432i	\(0.455933\pi\)
\(47\)	−8.18692e9	−	2.51967e10i	−0.110786	−	0.340964i	0.880259	−	0.474494i	\(-0.157369\pi\)
							−0.991045	+	0.133530i	\(0.957369\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.14.c.a.3.4	✓	24
11.4	even	5	inner	22.14.c.a.15.4	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.14.c.a.3.4	✓	24	1.1	even	1	trivial
22.14.c.a.15.4	yes	24	11.4	even	5	inner