Embedded newform 22.14.c.b.3.2

• Label: 22.14.c.b.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	22.3
Dual form			22.14.c.b.15.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-51.7771 + 37.6183i) q^{2} +(-630.449 - 1940.32i) q^{3} +(1265.73 - 3895.53i) q^{4} +(52985.6 + 38496.3i) q^{5} +(105634. + 76747.8i) q^{6} +(86177.6 - 265227. i) q^{7} +(81007.0 + 249314. i) q^{8} +(-2.07755e6 + 1.50943e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 448 q^{2} - 1248 q^{3} - 28672 q^{4} + 71554 q^{5} + 99008 q^{6} + 361734 q^{7} - 1835008 q^{8} - 8615367 q^{9}+O(q^{10}) \)

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
							\(3\)	−630.449	−	1940.32i	−0.499300	−	1.53669i	−0.810147	−	0.586227i	\(-0.800613\pi\)
0.310847	−	0.950460i	\(-0.399387\pi\)
\(5\)	52985.6	+	38496.3i	1.51654	+	1.10183i	0.963171	+	0.268889i	\(0.0866565\pi\)
							0.553365	+	0.832939i	\(0.313344\pi\)
\(7\)	86177.6	−	265227.i	0.276858	−	0.852081i	−0.711864	−	0.702317i	\(-0.752149\pi\)
							0.988722	−	0.149764i	\(-0.0478513\pi\)
\(11\)	590643.	+	5.84584e6i	0.100525	+	0.994935i
							\(13\)	−1.33355e7	+	9.68879e6i	−0.766261	+	0.556721i	−0.900824	−	0.434184i	\(-0.857037\pi\)
0.134563	+	0.990905i	\(0.457037\pi\)
\(17\)	3.24758e7	+	2.35950e7i	0.326318	+	0.237084i	0.738867	−	0.673851i	\(-0.235361\pi\)
							−0.412548	+	0.910936i	\(0.635361\pi\)
\(19\)	8.74928e7	+	2.69275e8i	0.426652	+	1.31310i	0.901403	+	0.432980i	\(0.142538\pi\)
							−0.474751	+	0.880120i	\(0.657462\pi\)
\(23\)	6.47831e8			0.912496			0.456248	−	0.889853i	\(-0.349193\pi\)
							0.456248	+	0.889853i	\(0.349193\pi\)
\(29\)	−2.55043e8	+	7.84943e8i	−0.0796209	+	0.245048i	−0.982942	−	0.183918i	\(-0.941122\pi\)
							0.903321	+	0.428966i	\(0.141122\pi\)
\(31\)	−6.31221e8	+	4.58609e8i	−0.127741	+	0.0928093i	−0.649821	−	0.760087i	\(-0.725156\pi\)
							0.522080	+	0.852896i	\(0.325156\pi\)
\(37\)	5.58691e8	−	1.71947e9i	0.0357981	−	0.110175i	−0.931561	−	0.363586i	\(-0.881552\pi\)
							0.967359	+	0.253411i	\(0.0815523\pi\)
\(41\)	−1.43226e10	−	4.40804e10i	−0.470898	−	1.44927i	−0.851411	−	0.524499i	\(-0.824253\pi\)
							0.380513	−	0.924776i	\(-0.375747\pi\)
\(43\)	4.84943e10			1.16989			0.584946	−	0.811072i	\(-0.301116\pi\)
							0.584946	+	0.811072i	\(0.301116\pi\)
\(47\)	3.83939e10	+	1.18164e11i	0.519548	+	1.59901i	0.774851	+	0.632144i	\(0.217825\pi\)
							−0.255302	+	0.966861i	\(0.582175\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.14.c.b.3.2	✓	28
11.4	even	5	inner	22.14.c.b.15.2	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.14.c.b.3.2	✓	28	1.1	even	1	trivial
22.14.c.b.15.2	yes	28	11.4	even	5	inner