Embedded newform 121.3.h.a.2.6

• Label: 121.3.h.a.2.6
• 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			2.6
Character	\(\chi\)	\(=\)	121.2
Dual form			121.3.h.a.61.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.18890 - 0.0625320i) q^{2} +(-1.30459 + 0.947839i) q^{3} +(0.793918 + 0.0453979i) q^{4} +(2.33368 - 3.86998i) q^{5} +(2.91489 - 1.99315i) q^{6} +(1.42055 + 0.287841i) q^{7} +(6.99209 + 0.600551i) q^{8} +(-1.97760 + 6.08643i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 39 q^{2} - 34 q^{3} - 75 q^{4} - 43 q^{5} - 15 q^{6} - 54 q^{7} - 59 q^{8} - 594 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.18890	−	0.0625320i	−1.09445	−	0.0312660i	−0.523329	−	0.852130i	\(-0.675310\pi\)
							−0.571123	+	0.820864i	\(0.693492\pi\)
\(3\)	−1.30459	+	0.947839i	−0.434863	+	0.315946i	−0.783590	−	0.621278i	\(-0.786614\pi\)
							0.348728	+	0.937224i	\(0.386614\pi\)
\(5\)	2.33368	−	3.86998i	0.466736	−	0.773995i	−0.530547	−	0.847655i	\(-0.678014\pi\)
							0.997283	+	0.0736603i	\(0.0234681\pi\)
\(7\)	1.42055	+	0.287841i	0.202936	+	0.0411202i	0.298991	−	0.954256i	\(-0.403350\pi\)
							−0.0960553	+	0.995376i	\(0.530623\pi\)
\(11\)	−10.6403	+	2.78998i	−0.967300	+	0.253635i
							\(13\)	−15.8213	+	4.15870i	−1.21703	+	0.319900i	−0.806106	−	0.591771i	\(-0.798429\pi\)
−0.410921	+	0.911671i	\(0.634793\pi\)
\(17\)	2.48157	+	14.3396i	0.145975	+	0.843508i	0.963518	+	0.267642i	\(0.0862444\pi\)
							−0.817544	+	0.575867i	\(0.804665\pi\)
\(19\)	−3.88744	+	3.77798i	−0.204602	+	0.198841i	−0.792111	−	0.610377i	\(-0.791018\pi\)
							0.587509	+	0.809218i	\(0.300109\pi\)
\(23\)	−13.7623	+	30.1353i	−0.598362	+	1.31023i	0.331894	+	0.943317i	\(0.392312\pi\)
							−0.930255	+	0.366913i	\(0.880415\pi\)
\(29\)	22.6257	+	11.9383i	0.780195	+	0.411667i	0.808919	−	0.587921i	\(-0.200053\pi\)
							−0.0287233	+	0.999587i	\(0.509144\pi\)
\(31\)	−9.34314	+	7.63985i	−0.301392	+	0.246447i	−0.771599	−	0.636109i	\(-0.780543\pi\)
							0.470207	+	0.882556i	\(0.344179\pi\)
\(37\)	−22.3625	−	57.4376i	−0.604392	−	1.55237i	−0.817859	−	0.575418i	\(-0.804839\pi\)
							0.213467	−	0.976950i	\(-0.431524\pi\)
\(41\)	43.1284	+	33.2570i	1.05191	+	0.811146i	0.982466	−	0.186439i	\(-0.0596948\pi\)
							0.0694466	+	0.997586i	\(0.477877\pi\)
\(43\)	4.24570	+	3.67892i	0.0987372	+	0.0855562i	0.702831	−	0.711357i	\(-0.251919\pi\)
							−0.604093	+	0.796913i	\(0.706465\pi\)
\(47\)	−76.8532	+	27.4212i	−1.63517	+	0.583429i	−0.984416	−	0.175856i	\(-0.943731\pi\)
							−0.650758	+	0.759285i	\(0.725549\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.3.h.a.2.6	✓	840
121.61	odd	110	inner	121.3.h.a.61.6	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.3.h.a.2.6	✓	840	1.1	even	1	trivial
121.3.h.a.61.6	yes	840	121.61	odd	110	inner