Embedded newform 105.4.i.d.16.3

• Label: 105.4.i.d.16.3
• Level: $105$
• Weight: $4$
• Character: 105.16
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Relative dimension:	\(5\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	x^10 - 2*x^9 + 34*x^8 + 16*x^7 + 791*x^6 - 132*x^5 + 4906*x^4 - 1674*x^3 + 25257*x^2 - 12852*x + 7056
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			16.3
Root			\(0.269375 - 0.466571i\) of defining polynomial
Character	\(\chi\)	\(=\)	105.16
Dual form			105.4.i.d.46.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.230625 + 0.399454i) q^{2} +(-1.50000 - 2.59808i) q^{3} +(3.89362 + 6.74396i) q^{4} +(-2.50000 + 4.33013i) q^{5} +1.38375 q^{6} +(-18.0800 - 4.01437i) q^{7} -7.28187 q^{8} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{2} - 15 q^{3} - 25 q^{4} - 25 q^{5} + 18 q^{6} - 32 q^{7} + 42 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.230625	+	0.399454i	−0.0815382	+	0.141228i	−0.903911	−	0.427721i	\(-0.859317\pi\)
							0.822373	+	0.568949i	\(0.192650\pi\)
\(3\)	−1.50000	−	2.59808i	−0.288675	−	0.500000i
							\(5\)	−2.50000	+	4.33013i	−0.223607	+	0.387298i
\(7\)	−18.0800	−	4.01437i	−0.976226	−	0.216756i
							\(11\)	12.1628	+	21.0667i	0.333385	+	0.577440i	0.983173	−	0.182676i	\(-0.0584758\pi\)
−0.649788	+	0.760115i	\(0.725142\pi\)
\(13\)	−66.1258			−1.41077			−0.705384	−	0.708825i	\(-0.749226\pi\)
							−0.705384	+	0.708825i	\(0.749226\pi\)
\(17\)	9.80377	+	16.9806i	0.139868	+	0.242259i	0.927447	−	0.373955i	\(-0.121999\pi\)
							−0.787578	+	0.616215i	\(0.788665\pi\)
\(19\)	−32.3499	+	56.0317i	−0.390609	+	0.676555i	−0.992530	−	0.122001i	\(-0.961069\pi\)
							0.601921	+	0.798556i	\(0.294402\pi\)
\(23\)	−73.7290	+	127.702i	−0.668415	+	1.15773i	0.309932	+	0.950759i	\(0.399694\pi\)
							−0.978347	+	0.206970i	\(0.933640\pi\)
\(29\)	166.165			1.06400			0.532002	−	0.846743i	\(-0.321440\pi\)
							0.532002	+	0.846743i	\(0.321440\pi\)
\(31\)	−94.9410	−	164.443i	−0.550062	−	0.952735i	−0.998269	−	0.0588054i	\(-0.981271\pi\)
							0.448208	−	0.893929i	\(-0.352062\pi\)
\(37\)	37.9478	−	65.7275i	0.168610	−	0.292041i	−0.769321	−	0.638862i	\(-0.779405\pi\)
							0.937931	+	0.346821i	\(0.112739\pi\)
\(41\)	186.313			0.709688			0.354844	−	0.934926i	\(-0.384534\pi\)
							0.354844	+	0.934926i	\(0.384534\pi\)
\(43\)	110.890			0.393269			0.196634	−	0.980477i	\(-0.436999\pi\)
							0.196634	+	0.980477i	\(0.436999\pi\)
\(47\)	190.247	−	329.517i	0.590433	−	1.02266i	−0.403741	−	0.914873i	\(-0.632290\pi\)
							0.994174	−	0.107787i	\(-0.0343764\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.i.d.16.3	✓	10
3.2	odd	2		315.4.j.h.226.3		10
7.2	even	3		735.4.a.ba.1.3		5
7.4	even	3	inner	105.4.i.d.46.3	yes	10
7.5	odd	6		735.4.a.z.1.3		5
21.2	odd	6		2205.4.a.br.1.3		5
21.5	even	6		2205.4.a.bs.1.3		5
21.11	odd	6		315.4.j.h.46.3		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.i.d.16.3	✓	10	1.1	even	1	trivial
105.4.i.d.46.3	yes	10	7.4	even	3	inner
315.4.j.h.46.3		10	21.11	odd	6
315.4.j.h.226.3		10	3.2	odd	2
735.4.a.z.1.3		5	7.5	odd	6
735.4.a.ba.1.3		5	7.2	even	3
2205.4.a.br.1.3		5	21.2	odd	6
2205.4.a.bs.1.3		5	21.5	even	6