Embedded newform 121.2.g.a.4.3

• Label: 121.2.g.a.4.3
• Level: $121$
• Weight: $2$
• Character: 121.4
• Analytic conductor: $0.966$
• Analytic rank: $0$
• Dimension: $400$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	121.g (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.966189864457\)
Analytic rank:	\(0\)
Dimension:	\(400\)
Relative dimension:	\(10\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label			4.3
Character	\(\chi\)	\(=\)	121.4
Dual form			121.2.g.a.91.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.94119 - 0.111001i) q^{2} +(-0.618438 + 1.90336i) q^{3} +(1.76894 + 0.202967i) q^{4} +(0.816229 + 1.54693i) q^{5} +(1.41178 - 3.62613i) q^{6} +(-4.27807 - 1.80793i) q^{7} +(0.420440 + 0.0727601i) q^{8} +(-0.813251 - 0.590861i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 44 q^{2} - 32 q^{3} - 34 q^{4} - 43 q^{5} - 22 q^{6} - 44 q^{7} - 44 q^{8} - 110 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.94119	−	0.111001i	−1.37263	−	0.0784899i	−0.644855	−	0.764305i	\(-0.723082\pi\)
							−0.727776	+	0.685815i	\(0.759446\pi\)
\(3\)	−0.618438	+	1.90336i	−0.357055	+	1.09890i	0.597753	+	0.801681i	\(0.296060\pi\)
							−0.954808	+	0.297223i	\(0.903940\pi\)
\(5\)	0.816229	+	1.54693i	0.365029	+	0.691807i	0.996500	−	0.0835927i	\(-0.0266395\pi\)
							−0.631471	+	0.775399i	\(0.717549\pi\)
\(7\)	−4.27807	−	1.80793i	−1.61696	−	0.683332i	−0.621943	−	0.783063i	\(-0.713656\pi\)
							−0.995014	+	0.0997311i	\(0.968202\pi\)
\(11\)	−3.19120	−	0.903469i	−0.962182	−	0.272406i
							\(13\)	−4.45448	+	2.51556i	−1.23545	+	0.697690i	−0.964291	−	0.264845i	\(-0.914679\pi\)
−0.271158	+	0.962535i	\(0.587407\pi\)
\(17\)	0.127466	−	0.0454799i	0.0309151	−	0.0110305i	−0.320547	−	0.947233i	\(-0.603867\pi\)
							0.351462	+	0.936202i	\(0.385685\pi\)
\(19\)	0.0115425	−	0.404039i	0.00264802	−	0.0926928i	−0.997193	−	0.0748717i	\(-0.976145\pi\)
							0.999841	−	0.0178211i	\(-0.00567292\pi\)
\(23\)	2.79795	+	3.22901i	0.583413	+	0.673294i	0.968335	−	0.249655i	\(-0.0803171\pi\)
							−0.384922	+	0.922949i	\(0.625772\pi\)
\(29\)	3.70208	+	5.41411i	0.687458	+	1.00538i	0.998488	+	0.0549752i	\(0.0175080\pi\)
							−0.311030	+	0.950400i	\(0.600674\pi\)
\(31\)	0.883940	−	4.36242i	0.158760	−	0.783513i	−0.819001	−	0.573792i	\(-0.805472\pi\)
							0.977761	−	0.209721i	\(-0.0672556\pi\)
\(37\)	−6.34491	+	5.82330i	−1.04310	+	0.957345i	−0.999131	−	0.0416876i	\(-0.986727\pi\)
							−0.0439659	+	0.999033i	\(0.513999\pi\)
\(41\)	1.82401	+	6.93925i	0.284862	+	1.08373i	0.943528	+	0.331293i	\(0.107485\pi\)
							−0.658665	+	0.752436i	\(0.728879\pi\)
\(43\)	1.01625	+	7.06820i	0.154977	+	1.07789i	0.907720	+	0.419576i	\(0.137821\pi\)
							−0.752743	+	0.658314i	\(0.771270\pi\)
\(47\)	1.48465	−	1.21399i	0.216558	−	0.177079i	−0.518511	−	0.855071i	\(-0.673513\pi\)
							0.735069	+	0.677992i	\(0.237150\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.2.g.a.4.3	✓	400
121.91	even	55	inner	121.2.g.a.91.3	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.2.g.a.4.3	✓	400	1.1	even	1	trivial
121.2.g.a.91.3	yes	400	121.91	even	55	inner