Embedded newform 57.3.k.a.13.3

• Label: 57.3.k.a.13.3
• Level: $57$
• Weight: $3$
• Character: 57.13
• 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			13.3
Root			\(3.09812i\) of defining polynomial
Character	\(\chi\)	\(=\)	57.13
Dual form			57.3.k.a.22.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.99143 + 2.37330i) q^{2} +(0.592396 - 1.62760i) q^{3} +(-0.972139 + 5.51328i) q^{4} +(0.0525425 + 0.297983i) q^{5} +(5.04248 - 1.83531i) q^{6} +(-1.79984 + 3.11741i) q^{7} +(-4.28839 + 2.47590i) q^{8} +(-2.29813 - 1.92836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} - 27 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.99143	+	2.37330i	0.995716	+	1.18665i	0.982410	+	0.186736i	\(0.0597909\pi\)
							0.0133054	+	0.999911i	\(0.495765\pi\)
\(3\)	0.592396	−	1.62760i	0.197465	−	0.542532i
							\(5\)	0.0525425	+	0.297983i	0.0105085	+	0.0595967i	0.989611	−	0.143770i	\(-0.0459227\pi\)
−0.979103	+	0.203367i	\(0.934812\pi\)
\(7\)	−1.79984	+	3.11741i	−0.257120	+	0.445345i	−0.965469	−	0.260517i	\(-0.916107\pi\)
							0.708349	+	0.705862i	\(0.249440\pi\)
\(11\)	−9.85364	−	17.0670i	−0.895785	−	1.55155i	−0.832830	−	0.553529i	\(-0.813281\pi\)
							−0.0629550	−	0.998016i	\(-0.520052\pi\)
\(13\)	−0.921973	−	2.53310i	−0.0709210	−	0.194854i	0.899168	−	0.437604i	\(-0.144173\pi\)
							−0.970089	+	0.242750i	\(0.921951\pi\)
\(17\)	−2.16287	+	1.81486i	−0.127228	+	0.106757i	−0.704181	−	0.710020i	\(-0.748686\pi\)
							0.576954	+	0.816777i	\(0.304241\pi\)
\(19\)	0.848719	+	18.9810i	0.0446694	+	0.999002i
							\(23\)	0.575632	−	3.26457i	0.0250275	−	0.141938i	−0.969734	−	0.244166i	\(-0.921486\pi\)
0.994761	+	0.102228i	\(0.0325970\pi\)
\(29\)	−29.1735	+	34.7677i	−1.00598	+	1.19888i	−0.0260298	+	0.999661i	\(0.508286\pi\)
							−0.979954	+	0.199224i	\(0.936158\pi\)
\(31\)	−15.0386	−	8.68256i	−0.485117	−	0.280083i	0.237429	−	0.971405i	\(-0.423695\pi\)
							−0.722547	+	0.691322i	\(0.757029\pi\)
\(37\)			50.1934i			1.35658i	0.734795	+	0.678289i	\(0.237278\pi\)
							−0.734795	+	0.678289i	\(0.762722\pi\)
\(41\)	20.9363	−	57.5219i	0.510641	−	1.40297i	−0.369930	−	0.929060i	\(-0.620618\pi\)
							0.880571	−	0.473915i	\(-0.157159\pi\)
\(43\)	4.12466	+	23.3921i	0.0959223	+	0.544002i	0.994461	+	0.105108i	\(0.0335188\pi\)
							−0.898539	+	0.438895i	\(0.855370\pi\)
\(47\)	−24.4852	−	20.5455i	−0.520962	−	0.437139i	0.344005	−	0.938968i	\(-0.388216\pi\)
							−0.864967	+	0.501829i	\(0.832661\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.a.13.3	✓	18
3.2	odd	2		171.3.ba.c.127.1		18
19.3	odd	18	inner	57.3.k.a.22.3	yes	18
57.41	even	18		171.3.ba.c.136.1		18

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.a.13.3	✓	18	1.1	even	1	trivial
57.3.k.a.22.3	yes	18	19.3	odd	18	inner
171.3.ba.c.127.1		18	3.2	odd	2
171.3.ba.c.136.1		18	57.41	even	18