Embedded newform 70.6.e.a.11.1

• Label: 70.6.e.a.11.1
• Level: $70$
• Weight: $6$
• Character: 70.11
• Analytic conductor: $11.227$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 70 = 2 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	70.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.2268673869\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			11.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	70.11
Dual form			70.6.e.a.51.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.00000 + 3.46410i) q^{2} +(2.50000 - 4.33013i) q^{3} +(-8.00000 + 13.8564i) q^{4} +(12.5000 + 21.6506i) q^{5} +20.0000 q^{6} +(-129.500 + 6.06218i) q^{7} -64.0000 q^{8} +(109.000 + 188.794i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 5 q^{3} - 16 q^{4} + 25 q^{5} + 40 q^{6} - 259 q^{7} - 128 q^{8} + 218 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(57\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00000	+	3.46410i	0.353553	+	0.612372i
							\(3\)	2.50000	−	4.33013i	0.160375	−	0.277778i	−0.774628	−	0.632417i	\(-0.782063\pi\)
0.935003	+	0.354639i	\(0.115396\pi\)
\(5\)	12.5000	+	21.6506i	0.223607	+	0.387298i
							\(7\)	−129.500	+	6.06218i	−0.998906	+	0.0467610i
\(11\)	−99.0000	+	171.473i	−0.246691	+	0.427282i								−0.962606	−	0.270906i	\(-0.912677\pi\)
							0.715915	+	0.698188i	\(0.246010\pi\)
\(13\)	−340.000			−0.557982			−0.278991	−	0.960294i	\(-0.590000\pi\)
							−0.278991	+	0.960294i	\(0.590000\pi\)
\(17\)	−924.000	+	1600.41i	−0.775443	+	1.34311i	0.159103	+	0.987262i	\(0.449140\pi\)
							−0.934545	+	0.355844i	\(0.884193\pi\)
\(19\)	605.000	+	1047.89i	0.384478	+	0.665935i	0.991697	−	0.128599i	\(-0.0410482\pi\)
							−0.607219	+	0.794535i	\(0.707715\pi\)
\(23\)	−1411.50	−	2444.79i	−0.556367	−	0.963656i	−0.997796	−	0.0663595i	\(-0.978862\pi\)
							0.441429	−	0.897296i	\(-0.354472\pi\)
\(29\)	−4539.00			−1.00222			−0.501112	−	0.865382i	\(-0.667076\pi\)
							−0.501112	+	0.865382i	\(0.667076\pi\)
\(31\)	356.000	−	616.610i	0.0665343	−	0.115241i	−0.830839	−	0.556512i	\(-0.812139\pi\)
							0.897374	+	0.441272i	\(0.145472\pi\)
\(37\)	3662.00	+	6342.77i	0.439758	+	0.761684i	0.997671	−	0.0682162i	\(-0.0217308\pi\)
							−0.557912	+	0.829900i	\(0.688397\pi\)
\(41\)	15633.0			1.45239			0.726194	−	0.687490i	\(-0.241287\pi\)
							0.726194	+	0.687490i	\(0.241287\pi\)
\(43\)	15827.0			1.30535			0.652676	−	0.757637i	\(-0.273646\pi\)
							0.652676	+	0.757637i	\(0.273646\pi\)
\(47\)	1596.00	+	2764.35i	0.105387	+	0.182536i	0.913896	−	0.405948i	\(-0.133058\pi\)
							−0.808509	+	0.588484i	\(0.799725\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	70.6.e.a.11.1	✓	2
7.2	even	3	inner	70.6.e.a.51.1	yes	2
7.3	odd	6		490.6.a.f.1.1		1
7.4	even	3		490.6.a.d.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.e.a.11.1	✓	2	1.1	even	1	trivial
70.6.e.a.51.1	yes	2	7.2	even	3	inner
490.6.a.d.1.1		1	7.4	even	3
490.6.a.f.1.1		1	7.3	odd	6