Embedded newform 57.3.k.b.34.4

• Label: 57.3.k.b.34.4
• Level: $57$
• Weight: $3$
• Character: 57.34
• 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			34.4
Character	\(\chi\)	\(=\)	57.34
Dual form			57.3.k.b.52.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.04608 - 2.87407i) q^{2} +(1.70574 - 0.300767i) q^{3} +(-4.10184 - 3.44186i) q^{4} +(0.191054 - 0.160314i) q^{5} +(0.919905 - 5.21704i) q^{6} +(-3.85502 + 6.67710i) q^{7} +(-3.58795 + 2.07151i) q^{8} +(2.81908 - 1.02606i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} + 27 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{11}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.04608	−	2.87407i	0.523039	−	1.43704i	−0.344082	−	0.938939i	\(-0.611810\pi\)
							0.867121	−	0.498097i	\(-0.165968\pi\)
\(3\)	1.70574	−	0.300767i	0.568579	−	0.100256i
							\(5\)	0.191054	−	0.160314i	0.0382109	−	0.0320627i	−0.623482	−	0.781838i	\(-0.714283\pi\)
0.661693	+	0.749775i	\(0.269838\pi\)
\(7\)	−3.85502	+	6.67710i	−0.550718	+	0.953871i	0.447505	+	0.894281i	\(0.352313\pi\)
							−0.998223	+	0.0595898i	\(0.981021\pi\)
\(11\)	4.20135	+	7.27695i	0.381941	+	0.661541i	0.991340	−	0.131323i	\(-0.0419225\pi\)
							−0.609399	+	0.792864i	\(0.708589\pi\)
\(13\)	−16.3284	−	2.87913i	−1.25603	−	0.221472i	−0.494255	−	0.869317i	\(-0.664559\pi\)
							−0.761772	+	0.647845i	\(0.775670\pi\)
\(17\)	10.7142	+	3.89963i	0.630244	+	0.229390i	0.637338	−	0.770585i	\(-0.280036\pi\)
							−0.00709330	+	0.999975i	\(0.502258\pi\)
\(19\)	18.9991	+	0.183151i	0.999954	+	0.00963951i
							\(23\)	0.904232	+	0.758740i	0.0393144	+	0.0329887i	0.662233	−	0.749298i	\(-0.269609\pi\)
−0.622919	+	0.782287i	\(0.714053\pi\)
\(29\)	−8.37573	−	23.0121i	−0.288818	−	0.793521i	−0.996233	−	0.0867224i	\(-0.972361\pi\)
							0.707414	−	0.706799i	\(-0.249862\pi\)
\(31\)	−49.2588	−	28.4396i	−1.58899	−	0.917406i	−0.993473	−	0.114067i	\(-0.963612\pi\)
							−0.595521	−	0.803340i	\(-0.703054\pi\)
\(37\)			0.192130i			0.00519269i	0.999997	+	0.00259635i	\(0.000826443\pi\)
							−0.999997	+	0.00259635i	\(0.999174\pi\)
\(41\)	−47.4095	+	8.35957i	−1.15633	+	0.203892i	−0.718737	−	0.695282i	\(-0.755280\pi\)
							−0.437592	+	0.899174i	\(0.644169\pi\)
\(43\)	−34.4710	+	28.9246i	−0.801650	+	0.672664i	−0.948599	−	0.316480i	\(-0.897499\pi\)
							0.146949	+	0.989144i	\(0.453055\pi\)
\(47\)	78.2386	−	28.4765i	1.66465	−	0.605884i	0.673568	−	0.739125i	\(-0.264761\pi\)
							0.991084	+	0.133241i	\(0.0425386\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.b.34.4	✓	24
3.2	odd	2		171.3.ba.d.91.1		24
19.14	odd	18	inner	57.3.k.b.52.4	yes	24
57.14	even	18		171.3.ba.d.109.1		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.b.34.4	✓	24	1.1	even	1	trivial
57.3.k.b.52.4	yes	24	19.14	odd	18	inner
171.3.ba.d.91.1		24	3.2	odd	2
171.3.ba.d.109.1		24	57.14	even	18