Embedded newform 105.4.i.d.16.5

• Label: 105.4.i.d.16.5
• 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			16.5
Root			\(2.90324 - 5.02855i\) of defining polynomial
Character	\(\chi\)	\(=\)	105.16
Dual form			105.4.i.d.46.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.40324 - 4.16253i) q^{2} +(-1.50000 - 2.59808i) q^{3} +(-7.55109 - 13.0789i) q^{4} +(-2.50000 + 4.33013i) q^{5} -14.4194 q^{6} +(-18.4976 - 0.916253i) q^{7} -34.1365 q^{8} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{2} - 15 q^{3} - 25 q^{4} - 25 q^{5} + 18 q^{6} - 32 q^{7} + 42 q^{8} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.40324	−	4.16253i	0.849673	−	1.47168i	−0.0318283	−	0.999493i	\(-0.510133\pi\)
							0.881501	−	0.472183i	\(-0.156534\pi\)
\(3\)	−1.50000	−	2.59808i	−0.288675	−	0.500000i
							\(5\)	−2.50000	+	4.33013i	−0.223607	+	0.387298i
\(7\)	−18.4976	−	0.916253i	−0.998775	−	0.0494730i
							\(11\)	−26.6281	−	46.1213i	−0.729881	−	1.26419i	−0.956933	−	0.290308i	\(-0.906242\pi\)
0.227053	−	0.973882i	\(-0.427091\pi\)
\(13\)	60.9548			1.30045			0.650224	−	0.759743i	\(-0.274675\pi\)
							0.650224	+	0.759743i	\(0.274675\pi\)
\(17\)	−15.5599	−	26.9505i	−0.221990	−	0.384498i	0.733422	−	0.679773i	\(-0.237922\pi\)
							−0.955412	+	0.295276i	\(0.904589\pi\)
\(19\)	8.95663	−	15.5133i	0.108147	−	0.187316i	−0.806873	−	0.590725i	\(-0.798842\pi\)
							0.915020	+	0.403409i	\(0.132175\pi\)
\(23\)	17.4112	−	30.1570i	0.157847	−	0.273399i	−0.776245	−	0.630431i	\(-0.782878\pi\)
							0.934092	+	0.357032i	\(0.116211\pi\)
\(29\)	141.423			0.905571			0.452786	−	0.891619i	\(-0.350430\pi\)
							0.452786	+	0.891619i	\(0.350430\pi\)
\(31\)	−58.8697	−	101.965i	−0.341075	−	0.590759i	0.643558	−	0.765397i	\(-0.277458\pi\)
							−0.984633	+	0.174639i	\(0.944124\pi\)
\(37\)	−87.6595	+	151.831i	−0.389490	+	0.674617i	−0.992381	−	0.123207i	\(-0.960682\pi\)
							0.602891	+	0.797824i	\(0.294015\pi\)
\(41\)	411.157			1.56615			0.783073	−	0.621930i	\(-0.213651\pi\)
							0.783073	+	0.621930i	\(0.213651\pi\)
\(43\)	−498.509			−1.76795			−0.883976	−	0.467532i	\(-0.845143\pi\)
							−0.883976	+	0.467532i	\(0.845143\pi\)
\(47\)	145.131	−	251.375i	0.450416	−	0.780144i	−0.547996	−	0.836481i	\(-0.684609\pi\)
							0.998412	+	0.0563376i	\(0.0179423\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.i.d.16.5	✓	10
3.2	odd	2		315.4.j.h.226.1		10
7.2	even	3		735.4.a.ba.1.1		5
7.4	even	3	inner	105.4.i.d.46.5	yes	10
7.5	odd	6		735.4.a.z.1.1		5
21.2	odd	6		2205.4.a.br.1.5		5
21.5	even	6		2205.4.a.bs.1.5		5
21.11	odd	6		315.4.j.h.46.1		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.i.d.16.5	✓	10	1.1	even	1	trivial
105.4.i.d.46.5	yes	10	7.4	even	3	inner
315.4.j.h.46.1		10	21.11	odd	6
315.4.j.h.226.1		10	3.2	odd	2
735.4.a.z.1.1		5	7.5	odd	6
735.4.a.ba.1.1		5	7.2	even	3
2205.4.a.br.1.5		5	21.2	odd	6
2205.4.a.bs.1.5		5	21.5	even	6