Embedded newform 242.4.c.q.3.2

• Label: 242.4.c.q.3.2
• Level: $242$
• Weight: $4$
• Character: 242.3
• 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			3.2
Root			\(5.60402 - 4.07156i\) of defining polynomial
Character	\(\chi\)	\(=\)	242.3
Dual form			242.4.c.q.81.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.61803 + 1.17557i) q^{2} +(2.64055 + 8.12677i) q^{3} +(1.23607 - 3.80423i) q^{4} +(-10.3036 - 7.48598i) q^{5} +(-13.8261 - 10.0452i) q^{6} +(-7.24988 + 22.3128i) q^{7} +(2.47214 + 7.60845i) q^{8} +(-37.2284 + 27.0480i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 3 q^{3} - 8 q^{4} + 5 q^{5} - 14 q^{6} + q^{7} - 16 q^{8} - 21 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(123\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.61803	+	1.17557i	−0.572061	+	0.415627i
							\(3\)	2.64055	+	8.12677i	0.508173	+	1.56400i	0.795369	+	0.606125i	\(0.207277\pi\)
−0.287196	+	0.957872i	\(0.592723\pi\)
\(5\)	−10.3036	−	7.48598i	−0.921579	−	0.669566i	0.0223375	−	0.999750i	\(-0.492889\pi\)
							−0.943917	+	0.330184i	\(0.892889\pi\)
\(7\)	−7.24988	+	22.3128i	−0.391457	+	1.20478i	0.540230	+	0.841518i	\(0.318337\pi\)
							−0.931687	+	0.363263i	\(0.881663\pi\)
\(11\)		0			0
							\(13\)	9.27574	−	6.73922i	0.197894	−	0.143779i	−0.484425	−	0.874833i	\(-0.660971\pi\)
0.682320	+	0.731054i	\(0.260971\pi\)
\(17\)	−52.9924	−	38.5012i	−0.756032	−	0.549289i	0.141659	−	0.989916i	\(-0.454756\pi\)
							−0.897691	+	0.440626i	\(0.854756\pi\)
\(19\)	2.24041	+	6.89528i	0.0270519	+	0.0832571i	0.963671	−	0.267092i	\(-0.0860628\pi\)
							−0.936619	+	0.350349i	\(0.886063\pi\)
\(23\)	−104.072			−0.943504			−0.471752	−	0.881731i	\(-0.656378\pi\)
							−0.471752	+	0.881731i	\(0.656378\pi\)
\(29\)	39.3536	−	121.118i	0.251993	−	0.775554i	−0.742415	−	0.669941i	\(-0.766319\pi\)
							0.994407	−	0.105613i	\(-0.0336805\pi\)
\(31\)	233.653	−	169.759i	1.35372	−	0.983536i	0.354905	−	0.934903i	\(-0.384513\pi\)
							0.998817	−	0.0486336i	\(-0.0154867\pi\)
\(37\)	−26.3908	+	81.2224i	−0.117260	+	0.360889i	−0.992412	−	0.122960i	\(-0.960761\pi\)
							0.875152	+	0.483849i	\(0.160761\pi\)
\(41\)	41.8544	+	128.815i	0.159428	+	0.490670i	0.998583	−	0.0532236i	\(-0.0169496\pi\)
							−0.839154	+	0.543893i	\(0.816950\pi\)
\(43\)	−353.691			−1.25436			−0.627178	−	0.778876i	\(-0.715790\pi\)
							−0.627178	+	0.778876i	\(0.715790\pi\)
\(47\)	−41.5948	−	128.016i	−0.129090	−	0.397298i	0.865534	−	0.500850i	\(-0.166979\pi\)
							−0.994624	+	0.103552i	\(0.966979\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.4.c.q.3.2		8
11.2	odd	10		242.4.a.n.1.4		4
11.3	even	5		242.4.c.n.27.1		8
11.4	even	5	inner	242.4.c.q.81.2		8
11.5	even	5		242.4.c.n.9.1		8
11.6	odd	10		242.4.c.r.9.1		8
11.7	odd	10		22.4.c.b.15.2	yes	8
11.8	odd	10		242.4.c.r.27.1		8
11.9	even	5		242.4.a.o.1.4		4
11.10	odd	2		22.4.c.b.3.2	✓	8
33.2	even	10		2178.4.a.by.1.2		4
33.20	odd	10		2178.4.a.bt.1.2		4
33.29	even	10		198.4.f.d.37.2		8
33.32	even	2		198.4.f.d.91.2		8
44.7	even	10		176.4.m.b.81.1		8
44.31	odd	10		1936.4.a.bm.1.1		4
44.35	even	10		1936.4.a.bn.1.1		4
44.43	even	2		176.4.m.b.113.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.b.3.2	✓	8	11.10	odd	2
22.4.c.b.15.2	yes	8	11.7	odd	10
176.4.m.b.81.1		8	44.7	even	10
176.4.m.b.113.1		8	44.43	even	2
198.4.f.d.37.2		8	33.29	even	10
198.4.f.d.91.2		8	33.32	even	2
242.4.a.n.1.4		4	11.2	odd	10
242.4.a.o.1.4		4	11.9	even	5
242.4.c.n.9.1		8	11.5	even	5
242.4.c.n.27.1		8	11.3	even	5
242.4.c.q.3.2		8	1.1	even	1	trivial
242.4.c.q.81.2		8	11.4	even	5	inner
242.4.c.r.9.1		8	11.6	odd	10
242.4.c.r.27.1		8	11.8	odd	10
1936.4.a.bm.1.1		4	44.31	odd	10
1936.4.a.bn.1.1		4	44.35	even	10
2178.4.a.bt.1.2		4	33.20	odd	10
2178.4.a.by.1.2		4	33.2	even	10