Embedded newform 22.14.c.b.9.3

• Label: 22.14.c.b.9.3
• Level: $22$
• Weight: $14$
• Character: 22.9
• 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			9.3
Character	\(\chi\)	\(=\)	22.9
Dual form			22.14.c.b.5.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(19.7771 - 60.8676i) q^{2} +(-501.424 + 364.306i) q^{3} +(-3313.73 - 2407.57i) q^{4} +(17458.1 + 53730.4i) q^{5} +(12257.7 + 37725.4i) q^{6} +(-385628. - 280175. i) q^{7} +(-212079. + 154084. i) q^{8} +(-373966. + 1.15095e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 448 q^{2} - 1248 q^{3} - 28672 q^{4} + 71554 q^{5} + 99008 q^{6} + 361734 q^{7} - 1835008 q^{8} - 8615367 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)

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
							\(3\)	−501.424	+	364.306i	−0.397115	+	0.288521i	−0.768365	−	0.640012i	\(-0.778929\pi\)
0.371250	+	0.928533i	\(0.378929\pi\)
\(5\)	17458.1	+	53730.4i	0.499679	+	1.53785i	0.809535	+	0.587071i	\(0.199719\pi\)
							−0.309856	+	0.950784i	\(0.600281\pi\)
\(7\)	−385628.	−	280175.i	−1.23889	−	0.900103i	−0.241362	−	0.970435i	\(-0.577594\pi\)
							−0.997524	+	0.0703325i	\(0.977594\pi\)
\(11\)	5.87360e6	+	153242.i	0.999660	+	0.0260811i
							\(13\)	6.48230e6	−	1.99505e7i	0.372475	−	1.14636i	−0.572692	−	0.819771i	\(-0.694101\pi\)
0.945166	−	0.326589i	\(-0.105899\pi\)
\(17\)	−3.88058e7	−	1.19432e8i	−0.389922	−	1.20006i	−0.932846	−	0.360276i	\(-0.882683\pi\)
							0.542923	−	0.839782i	\(-0.317317\pi\)
\(19\)	9.84864e7	−	7.15545e7i	0.480261	−	0.348930i	−0.321165	−	0.947023i	\(-0.604075\pi\)
							0.801427	+	0.598093i	\(0.204075\pi\)
\(23\)	−5.82343e8			−0.820253			−0.410126	−	0.912029i	\(-0.634515\pi\)
							−0.410126	+	0.912029i	\(0.634515\pi\)
\(29\)	−2.19265e9	−	1.59305e9i	−0.684514	−	0.497328i	0.190338	−	0.981719i	\(-0.439041\pi\)
							−0.874852	+	0.484390i	\(0.839041\pi\)
\(31\)	1.11280e9	−	3.42483e9i	0.225198	−	0.693088i	−0.773074	−	0.634316i	\(-0.781282\pi\)
							0.998271	−	0.0587714i	\(-0.0187183\pi\)
\(37\)	−8.13105e9	−	5.90756e9i	−0.520997	−	0.378527i	0.295982	−	0.955193i	\(-0.404353\pi\)
							−0.816979	+	0.576667i	\(0.804353\pi\)
\(41\)	−1.26905e10	+	9.22015e9i	−0.417236	+	0.303140i	−0.776525	−	0.630087i	\(-0.783019\pi\)
							0.359289	+	0.933226i	\(0.383019\pi\)
\(43\)	2.91315e10			0.702778			0.351389	−	0.936230i	\(-0.385709\pi\)
							0.351389	+	0.936230i	\(0.385709\pi\)
\(47\)	1.17541e11	−	8.53986e10i	1.59057	−	1.15562i	0.687456	−	0.726226i	\(-0.258728\pi\)
							0.903118	−	0.429393i	\(-0.141272\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.14.c.b.9.3	yes	28
11.5	even	5	inner	22.14.c.b.5.3	✓	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.14.c.b.5.3	✓	28	11.5	even	5	inner
22.14.c.b.9.3	yes	28	1.1	even	1	trivial