Embedded newform 242.4.c.n.9.2

• Label: 242.4.c.n.9.2
• Level: $242$
• Weight: $4$
• Character: 242.9
• Analytic conductor: $14.278$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	242.c (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.2784622214\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	\( x^{8} - x^{7} + 71x^{6} - 141x^{5} + 2911x^{4} + 2710x^{3} + 75340x^{2} + 169400x + 5856400 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 22)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			9.2
Root			\(-4.79501 - 3.48378i\) of defining polynomial
Character	\(\chi\)	\(=\)	242.9
Dual form			242.4.c.n.27.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.618034 - 1.90211i) q^{2} +(3.48599 - 2.53272i) q^{3} +(-3.23607 - 2.35114i) q^{4} +(-2.49134 - 7.66756i) q^{5} +(-2.66306 - 8.19605i) q^{6} +(-21.0985 - 15.3289i) q^{7} +(-6.47214 + 4.70228i) q^{8} +(-2.60601 + 8.02046i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} - 7 q^{3} - 8 q^{4} - 30 q^{5} + 6 q^{6} - 4 q^{7} - 16 q^{8} - 81 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/242\mathbb{Z}\right)^\times\).

\(n\)	\(123\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.618034	−	1.90211i	0.218508	−	0.672499i
							\(3\)	3.48599	−	2.53272i	0.670879	−	0.487422i	−0.199440	−	0.979910i	\(-0.563912\pi\)
0.870319	+	0.492488i	\(0.163912\pi\)
\(5\)	−2.49134	−	7.66756i	−0.222832	−	0.685807i	−0.998504	−	0.0546712i	\(-0.982589\pi\)
							0.775672	−	0.631136i	\(-0.217411\pi\)
\(7\)	−21.0985	−	15.3289i	−1.13921	−	0.827685i	−0.152202	−	0.988349i	\(-0.548636\pi\)
							−0.987009	+	0.160665i	\(0.948636\pi\)
\(11\)		0			0
							\(13\)	1.00858	−	3.10408i	0.0215176	−	0.0662243i	−0.939721	−	0.341942i	\(-0.888915\pi\)
0.961239	+	0.275717i	\(0.0889154\pi\)
\(17\)	6.44970	+	19.8501i	0.0920166	+	0.283198i	0.986465	−	0.163974i	\(-0.0524313\pi\)
							−0.894448	+	0.447172i	\(0.852431\pi\)
\(19\)	−101.912	+	74.0431i	−1.23053	+	0.894035i	−0.996930	−	0.0782975i	\(-0.975052\pi\)
							−0.233603	+	0.972332i	\(0.575052\pi\)
\(23\)	97.8394			0.886997			0.443498	−	0.896275i	\(-0.353737\pi\)
							0.443498	+	0.896275i	\(0.353737\pi\)
\(29\)	−213.446	−	155.078i	−1.36676	−	0.993008i	−0.997982	−	0.0634903i	\(-0.979777\pi\)
							−0.368776	−	0.929518i	\(-0.620223\pi\)
\(31\)	61.6067	−	189.606i	0.356932	−	1.09852i	−0.597949	−	0.801534i	\(-0.704017\pi\)
							0.954881	−	0.296990i	\(-0.0959827\pi\)
\(37\)	−295.812	−	214.920i	−1.31436	−	0.954935i	−0.999984	−	0.00563504i	\(-0.998206\pi\)
							−0.314371	−	0.949300i	\(-0.601794\pi\)
\(41\)	221.454	−	160.896i	0.843545	−	0.612871i	−0.0798140	−	0.996810i	\(-0.525433\pi\)
							0.923359	+	0.383939i	\(0.125433\pi\)
\(43\)	−388.059			−1.37624			−0.688121	−	0.725596i	\(-0.741564\pi\)
							−0.688121	+	0.725596i	\(0.741564\pi\)
\(47\)	41.9509	−	30.4791i	0.130195	−	0.0945922i	−0.520782	−	0.853690i	\(-0.674359\pi\)
							0.650977	+	0.759098i	\(0.274359\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.4.c.n.9.2		8
11.2	odd	10		22.4.c.b.3.1	✓	8
11.3	even	5		242.4.c.q.81.1		8
11.4	even	5		242.4.a.o.1.2		4
11.5	even	5	inner	242.4.c.n.27.2		8
11.6	odd	10		242.4.c.r.27.2		8
11.7	odd	10		242.4.a.n.1.2		4
11.8	odd	10		22.4.c.b.15.1	yes	8
11.9	even	5		242.4.c.q.3.1		8
11.10	odd	2		242.4.c.r.9.2		8
33.2	even	10		198.4.f.d.91.1		8
33.8	even	10		198.4.f.d.37.1		8
33.26	odd	10		2178.4.a.bt.1.4		4
33.29	even	10		2178.4.a.by.1.4		4
44.7	even	10		1936.4.a.bn.1.3		4
44.15	odd	10		1936.4.a.bm.1.3		4
44.19	even	10		176.4.m.b.81.2		8
44.35	even	10		176.4.m.b.113.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.b.3.1	✓	8	11.2	odd	10
22.4.c.b.15.1	yes	8	11.8	odd	10
176.4.m.b.81.2		8	44.19	even	10
176.4.m.b.113.2		8	44.35	even	10
198.4.f.d.37.1		8	33.8	even	10
198.4.f.d.91.1		8	33.2	even	10
242.4.a.n.1.2		4	11.7	odd	10
242.4.a.o.1.2		4	11.4	even	5
242.4.c.n.9.2		8	1.1	even	1	trivial
242.4.c.n.27.2		8	11.5	even	5	inner
242.4.c.q.3.1		8	11.9	even	5
242.4.c.q.81.1		8	11.3	even	5
242.4.c.r.9.2		8	11.10	odd	2
242.4.c.r.27.2		8	11.6	odd	10
1936.4.a.bm.1.3		4	44.15	odd	10
1936.4.a.bn.1.3		4	44.7	even	10
2178.4.a.bt.1.4		4	33.26	odd	10
2178.4.a.by.1.4		4	33.29	even	10