Embedded newform 121.3.h.a.2.9

• Label: 121.3.h.a.2.9
• 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			2.9
Character	\(\chi\)	\(=\)	121.2
Dual form			121.3.h.a.61.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.21524 - 0.0347167i) q^{2} +(0.594871 - 0.432199i) q^{3} +(-2.51787 - 0.143977i) q^{4} +(-0.691538 + 1.14679i) q^{5} +(-0.737917 + 0.504575i) q^{6} +(4.58308 + 0.928653i) q^{7} +(7.89994 + 0.678526i) q^{8} +(-2.61408 + 8.04530i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 39 q^{2} - 34 q^{3} - 75 q^{4} - 43 q^{5} - 15 q^{6} - 54 q^{7} - 59 q^{8} - 594 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.21524	−	0.0347167i	−0.607622	−	0.0173584i	−0.276606	−	0.960983i	\(-0.589210\pi\)
							−0.331016	+	0.943625i	\(0.607391\pi\)
\(3\)	0.594871	−	0.432199i	0.198290	−	0.144066i	−0.484210	−	0.874952i	\(-0.660893\pi\)
							0.682500	+	0.730886i	\(0.260893\pi\)
\(5\)	−0.691538	+	1.14679i	−0.138308	+	0.229357i	−0.917789	−	0.397067i	\(-0.870028\pi\)
							0.779482	+	0.626425i	\(0.215482\pi\)
\(7\)	4.58308	+	0.928653i	0.654726	+	0.132665i	0.514548	−	0.857462i	\(-0.327960\pi\)
							0.140178	+	0.990126i	\(0.455233\pi\)
\(11\)	9.71367	+	5.16184i	0.883061	+	0.469258i
							\(13\)	10.3311	−	2.71557i	0.794700	−	0.208890i	0.165578	−	0.986197i	\(-0.447051\pi\)
0.629122	+	0.777307i	\(0.283415\pi\)
\(17\)	2.99335	+	17.2969i	0.176080	+	1.01747i	0.931907	+	0.362697i	\(0.118144\pi\)
							−0.755828	+	0.654771i	\(0.772765\pi\)
\(19\)	−12.6744	+	12.3175i	−0.667073	+	0.648288i	−0.950507	−	0.310703i	\(-0.899435\pi\)
							0.283434	+	0.958992i	\(0.408526\pi\)
\(23\)	13.8927	−	30.4207i	0.604029	−	1.32264i	−0.322555	−	0.946551i	\(-0.604542\pi\)
							0.926584	−	0.376088i	\(-0.122731\pi\)
\(29\)	10.1812	+	5.37209i	0.351077	+	0.185244i	0.632922	−	0.774216i	\(-0.281855\pi\)
							−0.281845	+	0.959460i	\(0.590946\pi\)
\(31\)	−15.7361	+	12.8673i	−0.507615	+	0.415075i	−0.851832	−	0.523816i	\(-0.824508\pi\)
							0.344217	+	0.938890i	\(0.388144\pi\)
\(37\)	−0.744992	−	1.91350i	−0.0201349	−	0.0517161i	0.921438	−	0.388526i	\(-0.127016\pi\)
							−0.941573	+	0.336810i	\(0.890652\pi\)
\(41\)	−43.3443	−	33.4235i	−1.05718	−	0.815206i	−0.0738910	−	0.997266i	\(-0.523542\pi\)
							−0.983287	+	0.182060i	\(0.941724\pi\)
\(43\)	−50.1729	−	43.4751i	−1.16681	−	1.01105i	−0.999685	−	0.0251136i	\(-0.992005\pi\)
							−0.167128	−	0.985935i	\(-0.553449\pi\)
\(47\)	65.3864	−	23.3298i	1.39120	−	0.496379i	0.469037	−	0.883179i	\(-0.344601\pi\)
							0.922164	+	0.386799i	\(0.126419\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.3.h.a.2.9	✓	840
121.61	odd	110	inner	121.3.h.a.61.9	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.3.h.a.2.9	✓	840	1.1	even	1	trivial
121.3.h.a.61.9	yes	840	121.61	odd	110	inner