Embedded newform 210.3.o.b.31.2

• Label: 210.3.o.b.31.2
• Level: $210$
• Weight: $3$
• Character: 210.31
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial:	x^16 + 92*x^14 - 112*x^13 + 5846*x^12 - 7728*x^11 + 197216*x^10 - 298200*x^9 + 4836403*x^8 - 6808704*x^7 + 64376800*x^6 - 91953512*x^5 + 595763862*x^4 - 630430976*x^3 + 1087013404*x^2 + 294123256*x + 101626561
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{8}\cdot 7 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.2
Root			\(2.96377 + 5.13339i\) of defining polynomial
Character	\(\chi\)	\(=\)	210.31
Dual form			210.3.o.b.61.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{2} +(-1.50000 - 0.866025i) q^{3} +(-1.00000 + 1.73205i) q^{4} +(-1.93649 + 1.11803i) q^{5} +2.44949i q^{6} +(5.73733 + 4.01037i) q^{7} +2.82843 q^{8} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 24 q^{3} - 16 q^{4} + 4 q^{7} + 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.707107	−	1.22474i	−0.353553	−	0.612372i
							\(3\)	−1.50000	−	0.866025i	−0.500000	−	0.288675i
\(5\)	−1.93649	+	1.11803i	−0.387298	+	0.223607i
							\(7\)	5.73733	+	4.01037i	0.819618	+	0.572910i
\(11\)	8.69259	−	15.0560i	0.790236	−	1.36873i								−0.135585	−	0.990766i	\(-0.543291\pi\)
							0.925821	−	0.377963i	\(-0.123375\pi\)
\(13\)		−	7.22559i		−	0.555815i	−0.960608	−	0.277907i	\(-0.910359\pi\)
							0.960608	−	0.277907i	\(-0.0896408\pi\)
\(17\)	−2.29669	−	1.32600i	−0.135100	−	0.0779998i	0.430927	−	0.902387i	\(-0.358187\pi\)
							−0.566027	+	0.824387i	\(0.691520\pi\)
\(19\)	2.20128	−	1.27091i	0.115857	−	0.0668900i	−0.440952	−	0.897531i	\(-0.645359\pi\)
							0.556809	+	0.830641i	\(0.312026\pi\)
\(23\)	−20.0507	−	34.7289i	−0.871771	−	1.50995i	−0.860163	−	0.510019i	\(-0.829639\pi\)
							−0.0116074	−	0.999933i	\(-0.503695\pi\)
\(29\)	47.0080			1.62096			0.810482	−	0.585763i	\(-0.199205\pi\)
							0.810482	+	0.585763i	\(0.199205\pi\)
\(31\)	34.9025	+	20.1510i	1.12589	+	0.650031i	0.942897	−	0.333084i	\(-0.108089\pi\)
							0.182990	+	0.983115i	\(0.441422\pi\)
\(37\)	−16.2514	−	28.1482i	−0.439226	−	0.760762i	0.558404	−	0.829569i	\(-0.311414\pi\)
							−0.997630	+	0.0688071i	\(0.978081\pi\)
\(41\)		−	70.6679i		−	1.72361i	−0.507241	−	0.861804i	\(-0.669335\pi\)
							0.507241	−	0.861804i	\(-0.330665\pi\)
\(43\)	37.3732			0.869144			0.434572	−	0.900637i	\(-0.356900\pi\)
							0.434572	+	0.900637i	\(0.356900\pi\)
\(47\)	−28.9672	+	16.7242i	−0.616324	+	0.355835i	−0.775436	−	0.631426i	\(-0.782470\pi\)
							0.159113	+	0.987260i	\(0.449137\pi\)