Embedded newform 22.14.c.b.5.5

• Label: 22.14.c.b.5.5
• Level: $22$
• Weight: $14$
• Character: 22.5
• 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			5.5
Character	\(\chi\)	\(=\)	22.5
Dual form			22.14.c.b.9.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(19.7771 + 60.8676i) q^{2} +(697.531 + 506.786i) q^{3} +(-3313.73 + 2407.57i) q^{4} +(7802.80 - 24014.5i) q^{5} +(-17051.7 + 52479.8i) q^{6} +(9628.39 - 6995.43i) q^{7} +(-212079. - 154084. i) q^{8} +(-262955. - 809293. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 448 q^{2} - 1248 q^{3} - 28672 q^{4} + 71554 q^{5} + 99008 q^{6} + 361734 q^{7} - 1835008 q^{8} - 8615367 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	19.7771	+	60.8676i	0.218508	+	0.672499i
							\(3\)	697.531	+	506.786i	0.552428	+	0.401362i	0.828680	−	0.559723i	\(-0.189092\pi\)
−0.276252	+	0.961085i	\(0.589092\pi\)
\(5\)	7802.80	−	24014.5i	0.223329	−	0.687336i	−0.775128	−	0.631804i	\(-0.782315\pi\)
							0.998457	−	0.0555318i	\(-0.0176854\pi\)
\(7\)	9628.39	−	6995.43i	0.0309326	−	0.0224738i	−0.572212	−	0.820106i	\(-0.693914\pi\)
							0.603144	+	0.797632i	\(0.293914\pi\)
\(11\)	−5.75635e6	−	1.17778e6i	−0.979703	−	0.200453i
							\(13\)	−7.65534e6	−	2.35607e7i	−0.439879	−	1.35381i	−0.888004	−	0.459837i	\(-0.847908\pi\)
0.448125	−	0.893971i	\(-0.352092\pi\)
\(17\)	−2.05669e7	+	6.32984e7i	−0.206657	+	0.636026i	0.792984	+	0.609243i	\(0.208526\pi\)
							−0.999641	+	0.0267834i	\(0.991474\pi\)
\(19\)	−1.68420e8	−	1.22364e8i	−0.821288	−	0.596701i	0.0957928	−	0.995401i	\(-0.469461\pi\)
							−0.917081	+	0.398700i	\(0.869461\pi\)
\(23\)	−1.61296e8			−0.227192			−0.113596	−	0.993527i	\(-0.536237\pi\)
							−0.113596	+	0.993527i	\(0.536237\pi\)
\(29\)	3.74579e9	−	2.72147e9i	1.16938	−	0.849605i	0.178446	−	0.983950i	\(-0.442893\pi\)
							0.990935	+	0.134345i	\(0.0428930\pi\)
\(31\)	−5.65131e8	−	1.73930e9i	−0.114366	−	0.351984i	0.877448	−	0.479672i	\(-0.159244\pi\)
							−0.991814	+	0.127688i	\(0.959244\pi\)
\(37\)	7.48615e9	−	5.43901e9i	0.479675	−	0.348505i	−0.321525	−	0.946901i	\(-0.604195\pi\)
							0.801200	+	0.598397i	\(0.204195\pi\)
\(41\)	−1.29280e10	−	9.39272e9i	−0.425045	−	0.308814i	0.354619	−	0.935011i	\(-0.384610\pi\)
							−0.779665	+	0.626197i	\(0.784610\pi\)
\(43\)	2.18383e10			0.526834			0.263417	−	0.964682i	\(-0.415151\pi\)
							0.263417	+	0.964682i	\(0.415151\pi\)
\(47\)	−8.88102e10	−	6.45244e10i	−1.20178	−	0.873148i	−0.207326	−	0.978272i	\(-0.566476\pi\)
							−0.994459	+	0.105124i	\(0.966476\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.14.c.b.5.5	✓	28
11.9	even	5	inner	22.14.c.b.9.5	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.14.c.b.5.5	✓	28	1.1	even	1	trivial
22.14.c.b.9.5	yes	28	11.9	even	5	inner