Embedded newform 57.3.k.b.52.1

• Label: 57.3.k.b.52.1
• Level: $57$
• Weight: $3$
• Character: 57.52
• Analytic conductor: $1.553$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	57.k (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.55313750685\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			52.1
Character	\(\chi\)	\(=\)	57.52
Dual form			57.3.k.b.34.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28897 - 3.54141i) q^{2} +(1.70574 + 0.300767i) q^{3} +(-7.81596 + 6.55837i) q^{4} +(-6.09501 - 5.11432i) q^{5} +(-1.13350 - 6.42839i) q^{6} +(-2.17630 - 3.76946i) q^{7} +(20.2453 + 11.6886i) q^{8} +(2.81908 + 1.02606i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} + 27 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{7}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28897	−	3.54141i	−0.644484	−	1.77070i	−0.637161	−	0.770731i	\(-0.719891\pi\)
							−0.00732300	−	0.999973i	\(-0.502331\pi\)
\(3\)	1.70574	+	0.300767i	0.568579	+	0.100256i
							\(5\)	−6.09501	−	5.11432i	−1.21900	−	1.02286i	−0.998876	−	0.0473932i	\(-0.984909\pi\)
−0.220126	−	0.975471i	\(-0.570647\pi\)
\(7\)	−2.17630	−	3.76946i	−0.310900	−	0.538495i	0.667657	−	0.744469i	\(-0.267297\pi\)
							−0.978558	+	0.205974i	\(0.933964\pi\)
\(11\)	4.81596	−	8.34149i	0.437815	−	0.758317i	−0.559706	−	0.828691i	\(-0.689086\pi\)
							0.997521	+	0.0703739i	\(0.0224192\pi\)
\(13\)	11.2788	−	1.98875i	0.867597	−	0.152981i	0.277904	−	0.960609i	\(-0.410360\pi\)
							0.589692	+	0.807628i	\(0.299249\pi\)
\(17\)	3.02124	−	1.09964i	0.177720	−	0.0646847i	−0.251628	−	0.967824i	\(-0.580966\pi\)
							0.429347	+	0.903139i	\(0.358744\pi\)
\(19\)	4.60796	−	18.4328i	0.242524	−	0.970145i
							\(23\)	−19.2204	+	16.1278i	−0.835668	+	0.701209i	−0.956585	−	0.291454i	\(-0.905861\pi\)
0.120917	+	0.992663i	\(0.461417\pi\)
\(29\)	7.54374	−	20.7263i	0.260129	−	0.714698i	−0.739029	−	0.673673i	\(-0.764715\pi\)
							0.999158	−	0.0410250i	\(-0.0130623\pi\)
\(31\)	3.93885	−	2.27410i	0.127060	−	0.0733580i	−0.435123	−	0.900371i	\(-0.643295\pi\)
							0.562183	+	0.827013i	\(0.309962\pi\)
\(37\)			38.4931i			1.04035i	0.854059	+	0.520177i	\(0.174134\pi\)
							−0.854059	+	0.520177i	\(0.825866\pi\)
\(41\)	20.0544	+	3.53613i	0.489132	+	0.0862472i	0.412774	−	0.910833i	\(-0.364560\pi\)
							0.0763576	+	0.997080i	\(0.475671\pi\)
\(43\)	49.0120	+	41.1259i	1.13981	+	0.956417i	0.999433	−	0.0336807i	\(-0.0107229\pi\)
							0.140381	+	0.990098i	\(0.455167\pi\)
\(47\)	−5.05126	−	1.83851i	−0.107474	−	0.0391172i	0.287724	−	0.957713i	\(-0.407102\pi\)
							−0.395197	+	0.918596i	\(0.629324\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.b.52.1	yes	24
3.2	odd	2		171.3.ba.d.109.4		24
19.15	odd	18	inner	57.3.k.b.34.1	✓	24
57.53	even	18		171.3.ba.d.91.4		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.b.34.1	✓	24	19.15	odd	18	inner
57.3.k.b.52.1	yes	24	1.1	even	1	trivial
171.3.ba.d.91.4		24	57.53	even	18
171.3.ba.d.109.4		24	3.2	odd	2