Embedded newform 242.4.c.q.3.1

• Label: 242.4.c.q.3.1
• 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			3.1
Root			\(-4.79501 + 3.48378i\) of defining polynomial
Character	\(\chi\)	\(=\)	242.3
Dual form			242.4.c.q.81.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.61803 + 1.17557i) q^{2} +(-1.33153 - 4.09803i) q^{3} +(1.23607 - 3.80423i) q^{4} +(6.52241 + 4.73881i) q^{5} +(6.97198 + 5.06544i) q^{6} +(8.05890 - 24.8027i) q^{7} +(2.47214 + 7.60845i) q^{8} +(6.82261 - 4.95692i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 3 q^{3} - 8 q^{4} + 5 q^{5} - 14 q^{6} + q^{7} - 16 q^{8} - 21 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.33153	−	4.09803i	−0.256253	−	0.788665i	−0.993580	−	0.113130i	\(-0.963912\pi\)
0.737327	−	0.675536i	\(-0.236088\pi\)
\(5\)	6.52241	+	4.73881i	0.583382	+	0.423852i	0.839942	−	0.542676i	\(-0.182589\pi\)
							−0.256559	+	0.966528i	\(0.582589\pi\)
\(7\)	8.05890	−	24.8027i	0.435140	−	1.33922i	−0.457804	−	0.889053i	\(-0.651364\pi\)
							0.892943	−	0.450169i	\(-0.148636\pi\)
\(11\)		0			0
							\(13\)	−2.64049	+	1.91843i	−0.0563338	+	0.0409289i	−0.615596	−	0.788062i	\(-0.711085\pi\)
0.559262	+	0.828991i	\(0.311085\pi\)
\(17\)	−16.8855	−	12.2681i	−0.240902	−	0.175026i	0.460783	−	0.887513i	\(-0.347569\pi\)
							−0.701685	+	0.712487i	\(0.747569\pi\)
\(19\)	38.9268	+	119.804i	0.470022	+	1.44658i	0.852555	+	0.522637i	\(0.175052\pi\)
							−0.382534	+	0.923942i	\(0.624948\pi\)
\(23\)	97.8394			0.886997			0.443498	−	0.896275i	\(-0.353737\pi\)
							0.443498	+	0.896275i	\(0.353737\pi\)
\(29\)	81.5293	−	250.921i	0.522055	−	1.60672i	−0.248011	−	0.968757i	\(-0.579777\pi\)
							0.770066	−	0.637964i	\(-0.220223\pi\)
\(31\)	−161.288	+	117.183i	−0.934460	+	0.678925i	−0.947081	−	0.320995i	\(-0.895983\pi\)
							0.0126205	+	0.999920i	\(0.495983\pi\)
\(37\)	112.990	−	347.748i	0.502039	−	1.54512i	−0.303653	−	0.952783i	\(-0.598206\pi\)
							0.805692	−	0.592335i	\(-0.201794\pi\)
\(41\)	−84.5880	−	260.335i	−0.322205	−	0.991646i	−0.972686	−	0.232124i	\(-0.925433\pi\)
							0.650481	−	0.759523i	\(-0.274567\pi\)
\(43\)	−388.059			−1.37624			−0.688121	−	0.725596i	\(-0.741564\pi\)
							−0.688121	+	0.725596i	\(0.741564\pi\)
\(47\)	−16.0238	−	49.3162i	−0.0497301	−	0.153053i	0.923108	−	0.384542i	\(-0.125641\pi\)
							−0.972838	+	0.231488i	\(0.925641\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.b.3.1	✓	8	11.10	odd	2
22.4.c.b.15.1	yes	8	11.7	odd	10
176.4.m.b.81.2		8	44.7	even	10
176.4.m.b.113.2		8	44.43	even	2
198.4.f.d.37.1		8	33.29	even	10
198.4.f.d.91.1		8	33.32	even	2
242.4.a.n.1.2		4	11.2	odd	10
242.4.a.o.1.2		4	11.9	even	5
242.4.c.n.9.2		8	11.5	even	5
242.4.c.n.27.2		8	11.3	even	5
242.4.c.q.3.1		8	1.1	even	1	trivial
242.4.c.q.81.1		8	11.4	even	5	inner
242.4.c.r.9.2		8	11.6	odd	10
242.4.c.r.27.2		8	11.8	odd	10
1936.4.a.bm.1.3		4	44.31	odd	10
1936.4.a.bn.1.3		4	44.35	even	10
2178.4.a.bt.1.4		4	33.20	odd	10
2178.4.a.by.1.4		4	33.2	even	10