Embedded newform 570.4.i.h.391.1

• Label: 570.4.i.h.391.1
• Level: $570$
• Weight: $4$
• Character: 570.391
• Analytic conductor: $33.631$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	570.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(33.6310887033\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{385})\)
Defining polynomial:	\( x^{4} - x^{3} + 97x^{2} + 96x + 9216 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			391.1
Root			\(5.15535 + 8.92934i\) of defining polynomial
Character	\(\chi\)	\(=\)	570.391
Dual form			570.4.i.h.121.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(1.50000 - 2.59808i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(2.50000 - 4.33013i) q^{5} +(3.00000 + 5.19615i) q^{6} -10.6214 q^{7} +8.00000 q^{8} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 6 q^{3} - 8 q^{4} + 10 q^{5} + 12 q^{6} + 36 q^{7} + 32 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(211\)	\(457\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	+	1.73205i	−0.353553	+	0.612372i
							\(3\)	1.50000	−	2.59808i	0.288675	−	0.500000i
\(5\)	2.50000	−	4.33013i	0.223607	−	0.387298i
							\(7\)	−10.6214			−0.573503			−0.286751	−	0.958005i	\(-0.592575\pi\)
−0.286751	+	0.958005i	\(0.592575\pi\)
\(11\)	−15.3107			−0.419668			−0.209834	−	0.977737i	\(-0.567292\pi\)
							−0.209834	+	0.977737i	\(0.567292\pi\)
\(13\)	23.0000	+	39.8372i	0.490696	+	0.849911i	0.999943	−	0.0107098i	\(-0.00340911\pi\)
							−0.509246	+	0.860621i	\(0.670076\pi\)
\(17\)	−11.5339	+	19.9774i	−0.164552	+	0.285013i	−0.936496	−	0.350678i	\(-0.885951\pi\)
							0.771944	+	0.635691i	\(0.219285\pi\)
\(19\)	4.56787	−	82.6930i	0.0551549	−	0.998478i
							\(23\)	−92.4857	−	160.190i	−0.838461	−	1.45226i	−0.891181	−	0.453647i	\(-0.850123\pi\)
0.0527208	−	0.998609i	\(-0.483211\pi\)
\(29\)	111.830	+	193.696i	0.716082	+	1.24029i	0.962541	+	0.271137i	\(0.0873995\pi\)
							−0.246459	+	0.969153i	\(0.579267\pi\)
\(31\)	32.5143			0.188379			0.0941895	−	0.995554i	\(-0.469974\pi\)
							0.0941895	+	0.995554i	\(0.469974\pi\)
\(37\)	−310.621			−1.38016			−0.690079	−	0.723734i	\(-0.742424\pi\)
							−0.690079	+	0.723734i	\(0.742424\pi\)
\(41\)	−174.729	+	302.639i	−0.665561	+	1.15279i	0.313572	+	0.949565i	\(0.398474\pi\)
							−0.979133	+	0.203221i	\(0.934859\pi\)
\(43\)	−92.2428	+	159.769i	−0.327137	+	0.566618i	−0.981943	−	0.189179i	\(-0.939417\pi\)
							0.654805	+	0.755798i	\(0.272751\pi\)
\(47\)	−112.398	−	194.679i	−0.348829	−	0.604189i	0.637213	−	0.770688i	\(-0.280087\pi\)
							−0.986042	+	0.166498i	\(0.946754\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	570.4.i.h.391.1	yes	4
19.7	even	3	inner	570.4.i.h.121.1	✓	4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.4.i.h.121.1	✓	4	19.7	even	3	inner
570.4.i.h.391.1	yes	4	1.1	even	1	trivial