Embedded newform 363.3.g.f.118.3

• Label: 363.3.g.f.118.3
• Level: $363$
• Weight: $3$
• Character: 363.118
• Analytic conductor: $9.891$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	363.g (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89103359628\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 2*x^15 + 3*x^14 - 4*x^13 + 77*x^12 + 88*x^11 - 577*x^10 + 578*x^9 + 1520*x^8 + 1868*x^7 - 1619*x^6 - 16804*x^5 + 32427*x^4 + 43316*x^3 - 71672*x^2 + 83521
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			118.3
Root			\(1.60675 + 1.36085i\) of defining polynomial
Character	\(\chi\)	\(=\)	363.118
Dual form			363.3.g.f.40.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.184008 + 0.253266i) q^{2} +(0.535233 + 1.64728i) q^{3} +(1.20578 - 3.71102i) q^{4} +(5.99919 + 4.35866i) q^{5} +(-0.318712 + 0.438669i) q^{6} +(9.53633 + 3.09854i) q^{7} +(2.35267 - 0.764430i) q^{8} +(-2.42705 + 1.76336i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} - 4 q^{5} + 30 q^{7} + 40 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.184008	+	0.253266i	0.0920041	+	0.126633i	0.852538	−	0.522666i	\(-0.175062\pi\)
							−0.760533	+	0.649299i	\(0.775062\pi\)
\(3\)	0.535233	+	1.64728i	0.178411	+	0.549093i
							\(5\)	5.99919	+	4.35866i	1.19984	+	0.871733i	0.994269	−	0.106910i	\(-0.0340956\pi\)
0.205569	+	0.978643i	\(0.434096\pi\)
\(7\)	9.53633	+	3.09854i	1.36233	+	0.442649i	0.896821	−	0.442393i	\(-0.145870\pi\)
							0.465511	+	0.885042i	\(0.345870\pi\)
\(11\)		0			0
							\(13\)	−2.00157	−	2.75492i	−0.153967	−	0.211917i	0.725065	−	0.688681i	\(-0.241810\pi\)
−0.879031	+	0.476764i	\(0.841810\pi\)
\(17\)	9.14407	−	12.5857i	0.537886	−	0.740337i	−0.450420	−	0.892817i	\(-0.648726\pi\)
							0.988307	+	0.152480i	\(0.0487258\pi\)
\(19\)	−29.1422	+	9.46888i	−1.53380	+	0.498362i	−0.949658	−	0.313289i	\(-0.898569\pi\)
							−0.584142	+	0.811651i	\(0.698569\pi\)
\(23\)	7.67868			0.333856			0.166928	−	0.985969i	\(-0.446615\pi\)
							0.166928	+	0.985969i	\(0.446615\pi\)
\(29\)	−3.21080	−	1.04325i	−0.110717	−	0.0359743i	0.253134	−	0.967431i	\(-0.418539\pi\)
							−0.363852	+	0.931457i	\(0.618539\pi\)
\(31\)	3.27045	−	2.37612i	0.105498	−	0.0766490i	−0.533785	−	0.845620i	\(-0.679231\pi\)
							0.639284	+	0.768971i	\(0.279231\pi\)
\(37\)	−0.734229	+	2.25972i	−0.0198440	+	0.0610736i	−0.960488	−	0.278321i	\(-0.910222\pi\)
							0.940644	+	0.339394i	\(0.110222\pi\)
\(41\)	−6.69014	+	2.17376i	−0.163174	+	0.0530185i	−0.389465	−	0.921041i	\(-0.627340\pi\)
							0.226291	+	0.974060i	\(0.427340\pi\)
\(43\)		−	3.99630i		−	0.0929371i	−0.998920	−	0.0464686i	\(-0.985203\pi\)
							0.998920	−	0.0464686i	\(-0.0147967\pi\)
\(47\)	15.2775	+	47.0193i	0.325053	+	1.00041i	0.971417	+	0.237381i	\(0.0762890\pi\)
							−0.646363	+	0.763030i	\(0.723711\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.3.g.f.118.3		16
11.2	odd	10		363.3.c.e.241.9		16
11.3	even	5		363.3.g.a.94.3		16
11.4	even	5		33.3.g.a.7.2	✓	16
11.5	even	5		363.3.g.g.112.2		16
11.6	odd	10		363.3.g.a.112.3		16
11.7	odd	10	inner	363.3.g.f.40.3		16
11.8	odd	10		363.3.g.g.94.2		16
11.9	even	5		363.3.c.e.241.8		16
11.10	odd	2		33.3.g.a.19.2	yes	16
33.2	even	10		1089.3.c.m.604.8		16
33.20	odd	10		1089.3.c.m.604.9		16
33.26	odd	10		99.3.k.c.73.3		16
33.32	even	2		99.3.k.c.19.3		16
44.15	odd	10		528.3.bf.b.337.2		16
44.43	even	2		528.3.bf.b.481.2		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.7.2	✓	16	11.4	even	5
33.3.g.a.19.2	yes	16	11.10	odd	2
99.3.k.c.19.3		16	33.32	even	2
99.3.k.c.73.3		16	33.26	odd	10
363.3.c.e.241.8		16	11.9	even	5
363.3.c.e.241.9		16	11.2	odd	10
363.3.g.a.94.3		16	11.3	even	5
363.3.g.a.112.3		16	11.6	odd	10
363.3.g.f.40.3		16	11.7	odd	10	inner
363.3.g.f.118.3		16	1.1	even	1	trivial
363.3.g.g.94.2		16	11.8	odd	10
363.3.g.g.112.2		16	11.5	even	5
528.3.bf.b.337.2		16	44.15	odd	10
528.3.bf.b.481.2		16	44.43	even	2
1089.3.c.m.604.8		16	33.2	even	10
1089.3.c.m.604.9		16	33.20	odd	10