Embedded newform 22.14.c.a.5.2

• Label: 22.14.c.a.5.2
• Level: $22$
• Weight: $14$
• Character: 22.5
• 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			5.2
Character	\(\chi\)	\(=\)	22.5
Dual form			22.14.c.a.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-19.7771 - 60.8676i) q^{2} +(-351.775 - 255.580i) q^{3} +(-3313.73 + 2407.57i) q^{4} +(-3087.88 + 9503.52i) q^{5} +(-8599.44 + 26466.3i) q^{6} +(327454. - 237910. i) q^{7} +(212079. + 154084. i) q^{8} +(-434248. - 1.33648e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 384 q^{2} + 2708 q^{3} - 24576 q^{4} + 32920 q^{5} + 47488 q^{6} - 509572 q^{7} + 1572864 q^{8} - 3517894 q^{9}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	−351.775	−	255.580i	−0.278597	−	0.202413i	0.439708	−	0.898141i	\(-0.355082\pi\)
−0.718305	+	0.695728i	\(0.755082\pi\)
\(5\)	−3087.88	+	9503.52i	−0.0883803	+	0.272007i	−0.985472	−	0.169838i	\(-0.945676\pi\)
							0.897092	+	0.441844i	\(0.145676\pi\)
\(7\)	327454.	−	237910.i	1.05199	−	0.764319i	0.0794039	−	0.996843i	\(-0.474698\pi\)
							0.972591	+	0.232524i	\(0.0746983\pi\)
\(11\)	−5.87323e6	−	166817.i	−0.999597	−	0.0283915i
							\(13\)	−1.05335e6	−	3.24189e6i	−0.0605260	−	0.186280i	0.916222	−	0.400672i	\(-0.131223\pi\)
−0.976748	+	0.214392i	\(0.931223\pi\)
\(17\)	−5.11907e6	+	1.57549e7i	−0.0514367	+	0.158306i	−0.973475	−	0.228792i	\(-0.926522\pi\)
							0.922039	+	0.387098i	\(0.126522\pi\)
\(19\)	−3.12533e7	−	2.27068e7i	−0.152404	−	0.110728i	0.508970	−	0.860784i	\(-0.330026\pi\)
							−0.661374	+	0.750056i	\(0.730026\pi\)
\(23\)	−2.88671e8			−0.406605			−0.203302	−	0.979116i	\(-0.565167\pi\)
							−0.203302	+	0.979116i	\(0.565167\pi\)
\(29\)	−3.78401e9	+	2.74925e9i	−1.18131	+	0.858275i	−0.992319	−	0.123703i	\(-0.960523\pi\)
							−0.188995	+	0.981978i	\(0.560523\pi\)
\(31\)	1.04768e9	+	3.22444e9i	0.212021	+	0.652534i	0.999352	+	0.0360031i	\(0.0114626\pi\)
							−0.787330	+	0.616531i	\(0.788537\pi\)
\(37\)	−2.33170e10	+	1.69408e10i	−1.49404	+	1.08548i	−0.521354	+	0.853341i	\(0.674573\pi\)
							−0.972682	+	0.232140i	\(0.925427\pi\)
\(41\)	−4.51438e10	−	3.27989e10i	−1.48424	−	1.07836i	−0.976161	−	0.217048i	\(-0.930357\pi\)
							−0.508075	−	0.861313i	\(-0.669643\pi\)
\(43\)	−3.66013e9			−0.0882980			−0.0441490	−	0.999025i	\(-0.514058\pi\)
							−0.0441490	+	0.999025i	\(0.514058\pi\)
\(47\)	2.33864e10	+	1.69912e10i	0.316466	+	0.229926i	0.734666	−	0.678429i	\(-0.237339\pi\)
							−0.418200	+	0.908355i	\(0.637339\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.14.c.a.5.2	✓	24
11.9	even	5	inner	22.14.c.a.9.2	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.14.c.a.5.2	✓	24	1.1	even	1	trivial
22.14.c.a.9.2	yes	24	11.9	even	5	inner