Embedded newform 121.3.h.a.2.17

• Label: 121.3.h.a.2.17
• Level: $121$
• Weight: $3$
• Character: 121.2
• Analytic conductor: $3.297$
• Analytic rank: $0$
• Dimension: $840$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	121.h (of order \(110\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.29701119876\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(21\) over \(\Q(\zeta_{110})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{110}]$

Embedding invariants

Embedding label			2.17
Character	\(\chi\)	\(=\)	121.2
Dual form			121.3.h.a.61.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.10635 + 0.0601735i) q^{2} +(3.61347 - 2.62534i) q^{3} +(0.439597 + 0.0251371i) q^{4} +(-0.303526 + 0.503342i) q^{5} +(7.76920 - 5.31244i) q^{6} +(2.43403 + 0.493197i) q^{7} +(-7.47347 - 0.641897i) q^{8} +(3.38362 - 10.4137i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 39 q^{2} - 34 q^{3} - 75 q^{4} - 43 q^{5} - 15 q^{6} - 54 q^{7} - 59 q^{8} - 594 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{110}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.10635	+	0.0601735i	1.05317	+	0.0300867i	0.550859	−	0.834598i	\(-0.314300\pi\)
							0.502314	+	0.864685i	\(0.332482\pi\)
\(3\)	3.61347	−	2.62534i	1.20449	−	0.875114i	0.209772	−	0.977750i	\(-0.432728\pi\)
							0.994719	+	0.102636i	\(0.0327278\pi\)
\(5\)	−0.303526	+	0.503342i	−0.0607053	+	0.100668i	−0.885221	−	0.465170i	\(-0.845993\pi\)
							0.824516	+	0.565839i	\(0.191448\pi\)
\(7\)	2.43403	+	0.493197i	0.347718	+	0.0704568i	0.369300	−	0.929310i	\(-0.379598\pi\)
							−0.0215819	+	0.999767i	\(0.506870\pi\)
\(11\)	4.48676	+	10.0434i	0.407887	+	0.913032i
							\(13\)	−2.14188	+	0.563000i	−0.164760	+	0.0433077i	−0.335674	−	0.941978i	\(-0.608964\pi\)
0.170915	+	0.985286i	\(0.445328\pi\)
\(17\)	−2.33607	−	13.4989i	−0.137416	−	0.794051i	−0.970536	−	0.240958i	\(-0.922538\pi\)
							0.833120	−	0.553093i	\(-0.186553\pi\)
\(19\)	−14.5926	+	14.1817i	−0.768034	+	0.746406i	−0.972610	−	0.232444i	\(-0.925328\pi\)
							0.204576	+	0.978851i	\(0.434419\pi\)
\(23\)	−1.88647	+	4.13079i	−0.0820203	+	0.179599i	−0.946199	−	0.323586i	\(-0.895111\pi\)
							0.864178	+	0.503185i	\(0.167839\pi\)
\(29\)	−9.82180	−	5.18243i	−0.338683	−	0.178705i	0.288692	−	0.957422i	\(-0.406780\pi\)
							−0.627375	+	0.778717i	\(0.715871\pi\)
\(31\)	26.7248	−	21.8528i	0.862092	−	0.704929i	−0.0947616	−	0.995500i	\(-0.530209\pi\)
							0.956853	+	0.290571i	\(0.0938452\pi\)
\(37\)	−2.75762	−	7.08290i	−0.0745304	−	0.191430i	0.889670	−	0.456604i	\(-0.150934\pi\)
							−0.964201	+	0.265174i	\(0.914571\pi\)
\(41\)	6.21916	+	4.79569i	0.151687	+	0.116968i	0.683684	−	0.729778i	\(-0.260377\pi\)
							−0.531997	+	0.846746i	\(0.678558\pi\)
\(43\)	−36.3740	−	31.5183i	−0.845907	−	0.732983i	0.119749	−	0.992804i	\(-0.461791\pi\)
							−0.965657	+	0.259821i	\(0.916336\pi\)
\(47\)	−31.2565	+	11.1523i	−0.665031	+	0.237283i	−0.646925	−	0.762554i	\(-0.723945\pi\)
							−0.0181058	+	0.999836i	\(0.505764\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.3.h.a.2.17	✓	840
121.61	odd	110	inner	121.3.h.a.61.17	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.3.h.a.2.17	✓	840	1.1	even	1	trivial
121.3.h.a.61.17	yes	840	121.61	odd	110	inner