Embedded newform 22.14.c.b.3.4

• Label: 22.14.c.b.3.4
• 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			3.4
Character	\(\chi\)	\(=\)	22.3
Dual form			22.14.c.b.15.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-51.7771 + 37.6183i) q^{2} +(41.6348 + 128.139i) q^{3} +(1265.73 - 3895.53i) q^{4} +(2573.60 + 1869.83i) q^{5} +(-6976.08 - 5068.42i) q^{6} +(-27076.8 + 83333.8i) q^{7} +(81007.0 + 249314. i) q^{8} +(1.27515e6 - 926449. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 448 q^{2} - 1248 q^{3} - 28672 q^{4} + 71554 q^{5} + 99008 q^{6} + 361734 q^{7} - 1835008 q^{8} - 8615367 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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
							\(3\)	41.6348	+	128.139i	0.0329737	+	0.101483i	0.966189	−	0.257835i	\(-0.0830093\pi\)
−0.933215	+	0.359318i	\(0.883009\pi\)
\(5\)	2573.60	+	1869.83i	0.0736607	+	0.0535176i	0.624006	−	0.781419i	\(-0.285504\pi\)
							−0.550346	+	0.834937i	\(0.685504\pi\)
\(7\)	−27076.8	+	83333.8i	−0.0869881	+	0.267722i	−0.985083	−	0.172080i	\(-0.944951\pi\)
							0.898095	+	0.439802i	\(0.144951\pi\)
\(11\)	−5.72968e6	+	1.30135e6i	−0.975164	+	0.221484i
							\(13\)	−1.02703e7	+	7.46184e6i	−0.590137	+	0.428760i	−0.842364	−	0.538908i	\(-0.818837\pi\)
0.252227	+	0.967668i	\(0.418837\pi\)
\(17\)	−3.54532e7	−	2.57582e7i	−0.356235	−	0.258820i	0.395245	−	0.918576i	\(-0.370660\pi\)
							−0.751480	+	0.659756i	\(0.770660\pi\)
\(19\)	−1.24283e7	−	3.82505e7i	−0.0606058	−	0.186526i	0.916170	−	0.400790i	\(-0.131264\pi\)
							−0.976776	+	0.214265i	\(0.931264\pi\)
\(23\)	6.03786e8			0.850457			0.425229	−	0.905086i	\(-0.360194\pi\)
							0.425229	+	0.905086i	\(0.360194\pi\)
\(29\)	1.42239e9	−	4.37766e9i	0.444049	−	1.36664i	−0.439474	−	0.898255i	\(-0.644835\pi\)
							0.883523	−	0.468387i	\(-0.155165\pi\)
\(31\)	6.94431e9	−	5.04534e9i	1.40533	−	1.02103i	0.411348	−	0.911478i	\(-0.365058\pi\)
							0.993981	−	0.109553i	\(-0.0349419\pi\)
\(37\)	3.43384e9	−	1.05683e10i	0.220023	−	0.677162i	−0.778736	−	0.627352i	\(-0.784139\pi\)
							0.998759	−	0.0498095i	\(-0.0158614\pi\)
\(41\)	−4.97323e9	−	1.53060e10i	−0.163510	−	0.503231i	0.835414	−	0.549622i	\(-0.185228\pi\)
							−0.998923	+	0.0463905i	\(0.985228\pi\)
\(43\)	−2.23742e10			−0.539763			−0.269881	−	0.962894i	\(-0.586984\pi\)
							−0.269881	+	0.962894i	\(0.586984\pi\)
\(47\)	−1.99260e10	−	6.13260e10i	−0.269640	−	0.829867i	−0.990588	−	0.136878i	\(-0.956293\pi\)
							0.720948	−	0.692990i	\(-0.243707\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.14.c.b.3.4	✓	28
11.4	even	5	inner	22.14.c.b.15.4	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.14.c.b.3.4	✓	28	1.1	even	1	trivial
22.14.c.b.15.4	yes	28	11.4	even	5	inner