Embedded newform 121.3.h.a.2.10

• Label: 121.3.h.a.2.10
• 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			2.10
Character	\(\chi\)	\(=\)	121.2
Dual form			121.3.h.a.61.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06069 - 0.0303015i) q^{2} +(-3.99954 + 2.90584i) q^{3} +(-2.86933 - 0.164074i) q^{4} +(-2.07765 + 3.44539i) q^{5} +(4.33032 - 2.96100i) q^{6} +(0.996879 + 0.201994i) q^{7} +(7.26742 + 0.624199i) q^{8} +(4.77129 - 14.6845i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 39 q^{2} - 34 q^{3} - 75 q^{4} - 43 q^{5} - 15 q^{6} - 54 q^{7} - 59 q^{8} - 594 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06069	−	0.0303015i	−0.530345	−	0.0151507i	−0.237639	−	0.971353i	\(-0.576374\pi\)
							−0.292705	+	0.956203i	\(0.594555\pi\)
\(3\)	−3.99954	+	2.90584i	−1.33318	+	0.968612i	−0.333515	+	0.942745i	\(0.608235\pi\)
							−0.999665	+	0.0258672i	\(0.991765\pi\)
\(5\)	−2.07765	+	3.44539i	−0.415529	+	0.689078i	−0.991746	−	0.128219i	\(-0.959074\pi\)
							0.576216	+	0.817297i	\(0.304529\pi\)
\(7\)	0.996879	+	0.201994i	0.142411	+	0.0288563i	0.269271	−	0.963064i	\(-0.413217\pi\)
							−0.126860	+	0.991921i	\(0.540490\pi\)
\(11\)	−0.572581	−	10.9851i	−0.0520528	−	0.998644i
							\(13\)	−7.11756	+	1.87088i	−0.547505	+	0.143914i	−0.517574	−	0.855638i	\(-0.673165\pi\)
−0.0299304	+	0.999552i	\(0.509529\pi\)
\(17\)	1.03894	+	6.00343i	0.0611138	+	0.353143i	0.999961	+	0.00880959i	\(0.00280422\pi\)
							−0.938847	+	0.344334i	\(0.888105\pi\)
\(19\)	2.29264	−	2.22808i	0.120666	−	0.117268i	−0.634134	−	0.773223i	\(-0.718643\pi\)
							0.754799	+	0.655956i	\(0.227734\pi\)
\(23\)	6.53666	−	14.3133i	0.284203	−	0.622317i	−0.712656	−	0.701513i	\(-0.752508\pi\)
							0.996859	+	0.0791963i	\(0.0252354\pi\)
\(29\)	−44.5887	−	23.5270i	−1.53754	−	0.811277i	−0.538041	−	0.842919i	\(-0.680835\pi\)
							−0.999499	+	0.0316420i	\(0.989926\pi\)
\(31\)	16.0554	−	13.1284i	0.517915	−	0.423497i	−0.337589	−	0.941294i	\(-0.609611\pi\)
							0.855504	+	0.517797i	\(0.173248\pi\)
\(37\)	−9.64031	−	24.7609i	−0.260549	−	0.669214i	0.739443	−	0.673219i	\(-0.235089\pi\)
							−0.999992	+	0.00400464i	\(0.998725\pi\)
\(41\)	−11.1222	−	8.57650i	−0.271273	−	0.209183i	0.465987	−	0.884791i	\(-0.345699\pi\)
							−0.737261	+	0.675608i	\(0.763881\pi\)
\(43\)	25.6726	+	22.2455i	0.597038	+	0.517337i	0.900128	−	0.435625i	\(-0.143473\pi\)
							−0.303090	+	0.952962i	\(0.598018\pi\)
\(47\)	72.1325	−	25.7368i	1.53473	−	0.547592i	0.572521	−	0.819890i	\(-0.305965\pi\)
							0.962213	+	0.272298i	\(0.0877836\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.3.h.a.2.10	✓	840
121.61	odd	110	inner	121.3.h.a.61.10	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.3.h.a.2.10	✓	840	1.1	even	1	trivial
121.3.h.a.61.10	yes	840	121.61	odd	110	inner