Embedded newform 57.3.h.b.11.6

• Label: 57.3.h.b.11.6
• Level: $57$
• Weight: $3$
• Character: 57.11
• Analytic conductor: $1.553$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	57.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.55313750685\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial:	\( x^{16} + 44x^{14} + 686x^{12} + 4668x^{10} + 13913x^{8} + 18672x^{6} + 10976x^{4} + 2816x^{2} + 256 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.6
Root			\(-0.669760i\) of defining polynomial
Character	\(\chi\)	\(=\)	57.11
Dual form			57.3.h.b.26.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.00605 + 1.15819i) q^{2} +(2.99269 + 0.209258i) q^{3} +(0.682817 + 1.18267i) q^{4} +(-2.24545 - 1.29641i) q^{5} +(5.76112 + 3.88589i) q^{6} -12.9625 q^{7} -6.10220i q^{8} +(8.91242 + 1.25249i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{3} + 24 q^{4} - 17 q^{6} - 68 q^{7} + 25 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00605	+	1.15819i	1.00302	+	0.579096i	0.909141	−	0.416487i	\(-0.136739\pi\)
							0.0938821	+	0.995583i	\(0.470072\pi\)
\(3\)	2.99269	+	0.209258i	0.997564	+	0.0697526i
							\(5\)	−2.24545	−	1.29641i	−0.449089	−	0.259282i	0.258356	−	0.966050i	\(-0.416819\pi\)
−0.707446	+	0.706768i	\(0.750153\pi\)
\(7\)	−12.9625			−1.85178			−0.925890	−	0.377793i	\(-0.876683\pi\)
							−0.925890	+	0.377793i	\(0.876683\pi\)
\(11\)			14.5113i			1.31920i	0.751615	+	0.659602i	\(0.229275\pi\)
							−0.751615	+	0.659602i	\(0.770725\pi\)
\(13\)	2.52007	+	4.36489i	0.193851	+	0.335761i	0.946523	−	0.322635i	\(-0.104569\pi\)
							−0.752672	+	0.658396i	\(0.771235\pi\)
\(17\)	15.9210	+	9.19202i	0.936532	+	0.540707i	0.888871	−	0.458157i	\(-0.151490\pi\)
							0.0476602	+	0.998864i	\(0.484824\pi\)
\(19\)	2.86357	−	18.7830i	0.150714	−	0.988577i
							\(23\)	8.57128	−	4.94863i	0.372664	−	0.215158i	−0.301957	−	0.953321i	\(-0.597640\pi\)
0.674622	+	0.738164i	\(0.264307\pi\)
\(29\)	−7.48634	+	4.32224i	−0.258150	+	0.149043i	−0.623490	−	0.781831i	\(-0.714286\pi\)
							0.365340	+	0.930874i	\(0.380953\pi\)
\(31\)	−13.3199			−0.429673			−0.214836	−	0.976650i	\(-0.568922\pi\)
							−0.214836	+	0.976650i	\(0.568922\pi\)
\(37\)	7.28478			0.196886			0.0984429	−	0.995143i	\(-0.468614\pi\)
							0.0984429	+	0.995143i	\(0.468614\pi\)
\(41\)	11.5014	+	6.64034i	0.280522	+	0.161960i	0.633660	−	0.773612i	\(-0.281552\pi\)
							−0.353138	+	0.935571i	\(0.614885\pi\)
\(43\)	−19.6988	+	34.1193i	−0.458111	+	0.793471i	−0.998861	−	0.0477119i	\(-0.984807\pi\)
							0.540750	+	0.841183i	\(0.318140\pi\)
\(47\)	−13.5960	+	7.84964i	−0.289276	+	0.167014i	−0.637615	−	0.770355i	\(-0.720079\pi\)
							0.348339	+	0.937369i	\(0.386746\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.h.b.11.6	yes	16
3.2	odd	2	inner	57.3.h.b.11.3	✓	16
19.7	even	3	inner	57.3.h.b.26.3	yes	16
57.26	odd	6	inner	57.3.h.b.26.6	yes	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.h.b.11.3	✓	16	3.2	odd	2	inner
57.3.h.b.11.6	yes	16	1.1	even	1	trivial
57.3.h.b.26.3	yes	16	19.7	even	3	inner
57.3.h.b.26.6	yes	16	57.26	odd	6	inner