Embedded newform 105.4.s.b.101.11

• Label: 105.4.s.b.101.11
• Level: $105$
• Weight: $4$
• Character: 105.101
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			101.11
Character	\(\chi\)	\(=\)	105.101
Dual form			105.4.s.b.26.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.35336 - 0.781362i) q^{2} +(-2.92325 - 4.29588i) q^{3} +(-2.77895 + 4.81328i) q^{4} +(2.50000 + 4.33013i) q^{5} +(-7.31285 - 3.52975i) q^{6} +(-17.7664 - 5.23007i) q^{7} +21.1872i q^{8} +(-9.90921 + 25.1159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2 q^{3} + 64 q^{4} + 80 q^{5} + 28 q^{6} + 46 q^{7} - 98 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.35336	−	0.781362i	0.478485	−	0.276253i	−0.241300	−	0.970451i	\(-0.577574\pi\)
							0.719785	+	0.694197i	\(0.244241\pi\)
\(3\)	−2.92325	−	4.29588i	−0.562580	−	0.826743i
							\(5\)	2.50000	+	4.33013i	0.223607	+	0.387298i
\(7\)	−17.7664	−	5.23007i	−0.959298	−	0.282397i
							\(11\)	−26.1426	−	15.0935i	−0.716573	−	0.413713i	0.0969173	−	0.995292i	\(-0.469102\pi\)
−0.813490	+	0.581579i	\(0.802435\pi\)
\(13\)			59.7880i			1.27555i	0.770221	+	0.637777i	\(0.220146\pi\)
							−0.770221	+	0.637777i	\(0.779854\pi\)
\(17\)	−59.7827	+	103.547i	−0.852907	+	1.47728i	0.0256655	+	0.999671i	\(0.491830\pi\)
							−0.878573	+	0.477608i	\(0.841504\pi\)
\(19\)	102.212	−	59.0123i	1.23416	−	0.712545i	0.266269	−	0.963899i	\(-0.414209\pi\)
							0.967895	+	0.251354i	\(0.0808757\pi\)
\(23\)	−65.0867	+	37.5778i	−0.590066	+	0.340675i	−0.765124	−	0.643884i	\(-0.777322\pi\)
							0.175058	+	0.984558i	\(0.443989\pi\)
\(29\)		−	189.451i		−	1.21311i	−0.795042	−	0.606554i	\(-0.792551\pi\)
							0.795042	−	0.606554i	\(-0.207449\pi\)
\(31\)	−211.357	−	122.027i	−1.22454	−	0.706990i	−0.258660	−	0.965968i	\(-0.583281\pi\)
							−0.965883	+	0.258978i	\(0.916614\pi\)
\(37\)	−61.8489	−	107.125i	−0.274808	−	0.475981i	0.695279	−	0.718740i	\(-0.255281\pi\)
							−0.970087	+	0.242759i	\(0.921948\pi\)
\(41\)	−344.535			−1.31237			−0.656186	−	0.754599i	\(-0.727831\pi\)
							−0.656186	+	0.754599i	\(0.727831\pi\)
\(43\)	88.4283			0.313609			0.156805	−	0.987630i	\(-0.449881\pi\)
							0.156805	+	0.987630i	\(0.449881\pi\)
\(47\)	278.061	+	481.616i	0.862966	+	1.49470i	0.869053	+	0.494720i	\(0.164729\pi\)
							−0.00608661	+	0.999981i	\(0.501937\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.s.b.101.11	yes	32
3.2	odd	2		105.4.s.a.101.6	yes	32
7.5	odd	6		105.4.s.a.26.6	✓	32
21.5	even	6	inner	105.4.s.b.26.11	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.s.a.26.6	✓	32	7.5	odd	6
105.4.s.a.101.6	yes	32	3.2	odd	2
105.4.s.b.26.11	yes	32	21.5	even	6	inner
105.4.s.b.101.11	yes	32	1.1	even	1	trivial