Embedded newform 105.4.q.b.4.2

• Label: 105.4.q.b.4.2
• Level: $105$
• Weight: $4$
• Character: 105.4
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4.2
Character	\(\chi\)	\(=\)	105.4
Dual form			105.4.q.b.79.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.10597 + 2.37058i) q^{2} +(-2.59808 - 1.50000i) q^{3} +(7.23930 - 12.5388i) q^{4} +(-2.66182 + 10.8589i) q^{5} +14.2235 q^{6} +(15.6813 + 9.85381i) q^{7} +30.7161i q^{8} +(4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 62 q^{4} - 4 q^{5} + 108 q^{6} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.10597	+	2.37058i	−1.45168	+	0.838127i	−0.998577	−	0.0533323i	\(-0.983016\pi\)
							−0.453101	+	0.891459i	\(0.649682\pi\)
\(3\)	−2.59808	−	1.50000i	−0.500000	−	0.288675i
							\(5\)	−2.66182	+	10.8589i	−0.238081	+	0.971245i
\(7\)	15.6813	+	9.85381i	0.846710	+	0.532055i
							\(11\)	−18.3468	+	31.7776i	−0.502888	+	0.871027i	0.497107	+	0.867689i	\(0.334396\pi\)
−0.999994	+	0.00333759i	\(0.998938\pi\)
\(13\)			11.1505i			0.237891i	0.992901	+	0.118946i	\(0.0379514\pi\)
							−0.992901	+	0.118946i	\(0.962049\pi\)
\(17\)	−31.6218	−	18.2569i	−0.451143	−	0.260467i	0.257170	−	0.966366i	\(-0.417210\pi\)
							−0.708313	+	0.705899i	\(0.750543\pi\)
\(19\)	−75.6660	−	131.057i	−0.913630	−	1.58245i	−0.808894	−	0.587954i	\(-0.799934\pi\)
							−0.104736	−	0.994500i	\(-0.533400\pi\)
\(23\)	59.1041	−	34.1237i	0.535828	−	0.309361i	−0.207558	−	0.978223i	\(-0.566552\pi\)
							0.743387	+	0.668862i	\(0.233218\pi\)
\(29\)	−167.883			−1.07500			−0.537500	−	0.843263i	\(-0.680631\pi\)
							−0.537500	+	0.843263i	\(0.680631\pi\)
\(31\)	−130.648	+	226.289i	−0.756939	+	1.31106i	0.187466	+	0.982271i	\(0.439973\pi\)
							−0.944405	+	0.328785i	\(0.893361\pi\)
\(37\)	−166.019	+	95.8511i	−0.737658	+	0.425887i	−0.821217	−	0.570616i	\(-0.806705\pi\)
							0.0835593	+	0.996503i	\(0.473371\pi\)
\(41\)	142.625			0.543274			0.271637	−	0.962400i	\(-0.412435\pi\)
							0.271637	+	0.962400i	\(0.412435\pi\)
\(43\)		−	122.050i		−	0.432849i	−0.976299	−	0.216424i	\(-0.930561\pi\)
							0.976299	−	0.216424i	\(-0.0694395\pi\)
\(47\)	−259.383	+	149.755i	−0.804996	+	0.464765i	−0.845215	−	0.534426i	\(-0.820528\pi\)
							0.0402189	+	0.999191i	\(0.487194\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.q.b.4.2	✓	44
5.4	even	2	inner	105.4.q.b.4.21	yes	44
7.2	even	3	inner	105.4.q.b.79.21	yes	44
35.9	even	6	inner	105.4.q.b.79.2	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.q.b.4.2	✓	44	1.1	even	1	trivial
105.4.q.b.4.21	yes	44	5.4	even	2	inner
105.4.q.b.79.2	yes	44	35.9	even	6	inner
105.4.q.b.79.21	yes	44	7.2	even	3	inner