Embedded newform 121.2.g.a.5.9

• Label: 121.2.g.a.5.9
• Level: $121$
• Weight: $2$
• Character: 121.5
• 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			5.9
Character	\(\chi\)	\(=\)	121.5
Dual form			121.2.g.a.97.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.11220 - 1.84437i) q^{2} +(-1.96002 - 1.42404i) q^{3} +(-1.23139 - 2.33374i) q^{4} +(-0.598559 + 0.489439i) q^{5} +(-4.80638 + 2.03119i) q^{6} +(0.115990 - 0.150419i) q^{7} +(-1.37332 - 0.0785293i) q^{8} +(0.886746 + 2.72912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 44 q^{2} - 32 q^{3} - 34 q^{4} - 43 q^{5} - 22 q^{6} - 44 q^{7} - 44 q^{8} - 110 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{37}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.11220	−	1.84437i	0.786441	−	1.30417i	−0.161866	−	0.986813i	\(-0.551751\pi\)
							0.948307	−	0.317354i	\(-0.102794\pi\)
\(3\)	−1.96002	−	1.42404i	−1.13162	−	0.822169i	−0.145689	−	0.989330i	\(-0.546540\pi\)
							−0.985930	+	0.167161i	\(0.946540\pi\)
\(5\)	−0.598559	+	0.489439i	−0.267684	+	0.218884i	−0.757322	−	0.653042i	\(-0.773493\pi\)
							0.489638	+	0.871926i	\(0.337129\pi\)
\(7\)	0.115990	−	0.150419i	0.0438401	−	0.0568529i	−0.769472	−	0.638681i	\(-0.779480\pi\)
							0.813312	+	0.581828i	\(0.197662\pi\)
\(11\)	2.29277	−	2.39650i	0.691297	−	0.722570i
							\(13\)	2.06686	−	0.357684i	0.573244	−	0.0992038i	0.123473	−	0.992348i	\(-0.460597\pi\)
0.449771	+	0.893144i	\(0.351506\pi\)
\(17\)	−6.15427	+	0.706137i	−1.49263	+	0.171263i	−0.821557	−	0.570127i	\(-0.806894\pi\)
							−0.671074	+	0.741390i	\(0.734167\pi\)
\(19\)	2.72445	−	1.53857i	0.625031	−	0.352971i	−0.146455	−	0.989217i	\(-0.546786\pi\)
							0.771486	+	0.636246i	\(0.219514\pi\)
\(23\)	4.76869	+	1.40021i	0.994340	+	0.291965i	0.738132	−	0.674657i	\(-0.235708\pi\)
							0.256209	+	0.966621i	\(0.417527\pi\)
\(29\)	1.41987	+	7.00731i	0.263662	+	1.30123i	0.864815	+	0.502091i	\(0.167436\pi\)
							−0.601152	+	0.799134i	\(0.705291\pi\)
\(31\)	3.28705	−	1.61613i	0.590371	−	0.290266i	−0.121483	−	0.992594i	\(-0.538765\pi\)
							0.711854	+	0.702328i	\(0.247856\pi\)
\(37\)	3.32744	+	3.42386i	0.547028	+	0.562878i	0.933112	−	0.359585i	\(-0.117082\pi\)
							−0.386084	+	0.922463i	\(0.626173\pi\)
\(41\)	−0.239441	−	2.78776i	−0.0373944	−	0.435375i	−0.990957	−	0.134180i	\(-0.957160\pi\)
							0.953563	−	0.301195i	\(-0.0973855\pi\)
\(43\)	1.53815	−	0.988506i	0.234565	−	0.150746i	−0.418079	−	0.908411i	\(-0.637296\pi\)
							0.652644	+	0.757665i	\(0.273660\pi\)
\(47\)	−8.88617	+	2.06639i	−1.29618	+	0.301414i	−0.817171	−	0.576395i	\(-0.804459\pi\)
							−0.479011	+	0.877809i	\(0.659004\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.2.g.a.5.9	✓	400
121.97	even	55	inner	121.2.g.a.97.9	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.2.g.a.5.9	✓	400	1.1	even	1	trivial
121.2.g.a.97.9	yes	400	121.97	even	55	inner