Embedded newform 57.3.k.a.10.3

• Label: 57.3.k.a.10.3
• Level: $57$
• Weight: $3$
• Character: 57.10
• Analytic conductor: $1.553$
• Analytic rank: $0$
• Dimension: $18$
• CM: no
• 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:	\(18\)
Relative dimension:	\(3\) over \(\Q(\zeta_{18})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial:	x^18 + 48*x^16 + 936*x^14 + 9539*x^12 + 54576*x^10 + 176517*x^8 + 313396*x^6 + 277917*x^4 + 96435*x^2 + 8427
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			10.3
Root			\(-3.14274i\) of defining polynomial
Character	\(\chi\)	\(=\)	57.10
Dual form			57.3.k.a.40.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.09500 - 0.545732i) q^{2} +(1.11334 + 1.32683i) q^{3} +(5.52242 - 2.00999i) q^{4} +(-8.12052 - 2.95563i) q^{5} +(4.16988 + 3.49894i) q^{6} +(-1.67445 + 2.90024i) q^{7} +(5.10816 - 2.94920i) q^{8} +(-0.520945 + 2.95442i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} - 27 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{17}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.09500	−	0.545732i	1.54750	−	0.272866i	0.666326	−	0.745660i	\(-0.267866\pi\)
							0.881173	+	0.472795i	\(0.156755\pi\)
\(3\)	1.11334	+	1.32683i	0.371114	+	0.442276i
							\(5\)	−8.12052	−	2.95563i	−1.62410	−	0.591125i	−0.639945	−	0.768420i	\(-0.721043\pi\)
−0.984158	+	0.177295i	\(0.943265\pi\)
\(7\)	−1.67445	+	2.90024i	−0.239208	+	0.414320i	−0.960487	−	0.278324i	\(-0.910221\pi\)
							0.721279	+	0.692644i	\(0.243554\pi\)
\(11\)	4.16140	+	7.20775i	0.378309	+	0.655250i	0.990816	−	0.135215i	\(-0.0431725\pi\)
							−0.612508	+	0.790465i	\(0.709839\pi\)
\(13\)	14.5259	−	17.3113i	1.11738	−	1.33164i	0.179861	−	0.983692i	\(-0.442435\pi\)
							0.937515	−	0.347944i	\(-0.113120\pi\)
\(17\)	−2.74042	−	15.5417i	−0.161201	−	0.914217i	−0.952895	−	0.303300i	\(-0.901912\pi\)
							0.791694	−	0.610918i	\(-0.209199\pi\)
\(19\)	6.76727	+	17.7540i	0.356172	+	0.934420i
							\(23\)	−0.844215	+	0.307269i	−0.0367050	+	0.0133595i	−0.360307	−	0.932834i	\(-0.617328\pi\)
0.323602	+	0.946193i	\(0.395106\pi\)
\(29\)	−22.9627	−	4.04894i	−0.791816	−	0.139618i	−0.236910	−	0.971532i	\(-0.576135\pi\)
							−0.554906	+	0.831913i	\(0.687246\pi\)
\(31\)	−3.33002	−	1.92259i	−0.107420	−	0.0620190i	0.445327	−	0.895368i	\(-0.353087\pi\)
							−0.552748	+	0.833349i	\(0.686421\pi\)
\(37\)		−	17.1090i		−	0.462405i	−0.972906	−	0.231202i	\(-0.925734\pi\)
							0.972906	−	0.231202i	\(-0.0742660\pi\)
\(41\)	−29.9401	−	35.6812i	−0.730246	−	0.870273i	0.265338	−	0.964156i	\(-0.414517\pi\)
							−0.995583	+	0.0938828i	\(0.970072\pi\)
\(43\)	32.8903	+	11.9711i	0.764891	+	0.278398i	0.694858	−	0.719147i	\(-0.255467\pi\)
							0.0700332	+	0.997545i	\(0.477689\pi\)
\(47\)	5.96974	−	33.8561i	0.127016	−	0.720342i	−0.853074	−	0.521789i	\(-0.825265\pi\)
							0.980090	−	0.198553i	\(-0.0636242\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.a.10.3	✓	18
3.2	odd	2		171.3.ba.c.10.1		18
19.2	odd	18	inner	57.3.k.a.40.3	yes	18
57.2	even	18		171.3.ba.c.154.1		18

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.a.10.3	✓	18	1.1	even	1	trivial
57.3.k.a.40.3	yes	18	19.2	odd	18	inner
171.3.ba.c.10.1		18	3.2	odd	2
171.3.ba.c.154.1		18	57.2	even	18