Embedded newform 121.4.c.c.81.2

• Label: 121.4.c.c.81.2
• Level: $121$
• Weight: $4$
• Character: 121.81
• Analytic conductor: $7.139$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	121.c (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.13923111069\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	8.0.324000000.3
Defining polynomial:	\( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 11)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			81.2
Root			\(-0.535233 + 1.64728i\) of defining polynomial
Character	\(\chi\)	\(=\)	121.81
Dual form			121.4.c.c.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.592242 + 0.430289i) q^{2} +(1.83192 - 5.63806i) q^{3} +(-2.30653 - 7.09878i) q^{4} +(10.4011 - 7.55681i) q^{5} +(3.51093 - 2.55084i) q^{6} +(5.23110 + 16.0997i) q^{7} +(3.49823 - 10.7664i) q^{8} +(-6.58831 - 4.78668i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} - 44 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.592242	+	0.430289i	0.209389	+	0.152130i	0.687537	−	0.726149i	\(-0.258692\pi\)
							−0.478148	+	0.878279i	\(0.658692\pi\)
\(3\)	1.83192	−	5.63806i	0.352552	−	1.08504i	−0.604863	−	0.796330i	\(-0.706772\pi\)
							0.957415	−	0.288715i	\(-0.0932279\pi\)
\(5\)	10.4011	−	7.55681i	0.930298	−	0.675901i	−0.0157675	−	0.999876i	\(-0.505019\pi\)
							0.946066	+	0.323974i	\(0.105019\pi\)
\(7\)	5.23110	+	16.0997i	0.282453	+	0.869301i	0.987151	+	0.159793i	\(0.0510827\pi\)
							−0.704698	+	0.709508i	\(0.748917\pi\)
\(11\)		0			0
							\(13\)	−60.3859	−	43.8729i	−1.28831	−	0.936012i	−0.288540	−	0.957468i	\(-0.593170\pi\)
−0.999770	+	0.0214564i	\(0.993170\pi\)
\(17\)	66.9742	−	48.6596i	0.955507	−	0.694216i	0.00340407	−	0.999994i	\(-0.498916\pi\)
							0.952103	+	0.305778i	\(0.0989164\pi\)
\(19\)	−20.9894	+	64.5987i	−0.253436	+	0.779997i	0.740697	+	0.671839i	\(0.234495\pi\)
							−0.994134	+	0.108158i	\(0.965505\pi\)
\(23\)	13.3538			0.121064			0.0605319	−	0.998166i	\(-0.480720\pi\)
							0.0605319	+	0.998166i	\(0.480720\pi\)
\(29\)	52.2223	+	160.724i	0.334394	+	1.02916i	0.967020	+	0.254702i	\(0.0819773\pi\)
							−0.632625	+	0.774458i	\(0.718023\pi\)
\(31\)	52.9885	+	38.4984i	0.307001	+	0.223049i	0.730608	−	0.682797i	\(-0.239237\pi\)
							−0.423608	+	0.905846i	\(0.639237\pi\)
\(37\)	12.6253	+	38.8568i	0.0560970	+	0.172649i	0.975179	−	0.221417i	\(-0.0710682\pi\)
							−0.919082	+	0.394066i	\(0.871068\pi\)
\(41\)	84.9575	−	261.472i	0.323613	−	0.995978i	−0.648450	−	0.761257i	\(-0.724582\pi\)
							0.972063	−	0.234721i	\(-0.0754176\pi\)
\(43\)	−2.28719			−0.00811146			−0.00405573	−	0.999992i	\(-0.501291\pi\)
							−0.00405573	+	0.999992i	\(0.501291\pi\)
\(47\)	22.2017	−	68.3297i	0.0689031	−	0.212062i	−0.910676	−	0.413121i	\(-0.864438\pi\)
							0.979579	+	0.201060i	\(0.0644385\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.4.c.c.81.2		8
11.2	odd	10		121.4.c.f.27.2		8
11.3	even	5	inner	121.4.c.c.3.2		8
11.4	even	5	inner	121.4.c.c.9.1		8
11.5	even	5		11.4.a.a.1.1	✓	2
11.6	odd	10		121.4.a.c.1.2		2
11.7	odd	10		121.4.c.f.9.2		8
11.8	odd	10		121.4.c.f.3.1		8
11.9	even	5	inner	121.4.c.c.27.1		8
11.10	odd	2		121.4.c.f.81.1		8
33.5	odd	10		99.4.a.c.1.2		2
33.17	even	10		1089.4.a.v.1.1		2
44.27	odd	10		176.4.a.i.1.1		2
44.39	even	10		1936.4.a.w.1.1		2
55.27	odd	20		275.4.b.c.199.2		4
55.38	odd	20		275.4.b.c.199.3		4
55.49	even	10		275.4.a.b.1.2		2
77.27	odd	10		539.4.a.e.1.1		2
88.5	even	10		704.4.a.p.1.1		2
88.27	odd	10		704.4.a.n.1.2		2
132.71	even	10		1584.4.a.bc.1.2		2
143.38	even	10		1859.4.a.a.1.2		2
165.104	odd	10		2475.4.a.q.1.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.1	✓	2	11.5	even	5
99.4.a.c.1.2		2	33.5	odd	10
121.4.a.c.1.2		2	11.6	odd	10
121.4.c.c.3.2		8	11.3	even	5	inner
121.4.c.c.9.1		8	11.4	even	5	inner
121.4.c.c.27.1		8	11.9	even	5	inner
121.4.c.c.81.2		8	1.1	even	1	trivial
121.4.c.f.3.1		8	11.8	odd	10
121.4.c.f.9.2		8	11.7	odd	10
121.4.c.f.27.2		8	11.2	odd	10
121.4.c.f.81.1		8	11.10	odd	2
176.4.a.i.1.1		2	44.27	odd	10
275.4.a.b.1.2		2	55.49	even	10
275.4.b.c.199.2		4	55.27	odd	20
275.4.b.c.199.3		4	55.38	odd	20
539.4.a.e.1.1		2	77.27	odd	10
704.4.a.n.1.2		2	88.27	odd	10
704.4.a.p.1.1		2	88.5	even	10
1089.4.a.v.1.1		2	33.17	even	10
1584.4.a.bc.1.2		2	132.71	even	10
1859.4.a.a.1.2		2	143.38	even	10
1936.4.a.w.1.1		2	44.39	even	10
2475.4.a.q.1.1		2	165.104	odd	10