Embedded newform 121.3.h.a.2.16

• Label: 121.3.h.a.2.16
• 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			2.16
Character	\(\chi\)	\(=\)	121.2
Dual form			121.3.h.a.61.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.76028 + 0.0502871i) q^{2} +(-1.16832 + 0.848837i) q^{3} +(-0.897425 - 0.0513166i) q^{4} +(-4.08088 + 6.76738i) q^{5} +(-2.09926 + 1.43544i) q^{6} +(9.01268 + 1.82621i) q^{7} +(-8.59528 - 0.738250i) q^{8} +(-2.13670 + 6.57608i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 39 q^{2} - 34 q^{3} - 75 q^{4} - 43 q^{5} - 15 q^{6} - 54 q^{7} - 59 q^{8} - 594 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.76028	+	0.0502871i	0.880139	+	0.0251436i	0.465709	−	0.884938i	\(-0.345799\pi\)
							0.414430	+	0.910081i	\(0.363981\pi\)
\(3\)	−1.16832	+	0.848837i	−0.389441	+	0.282946i	−0.765226	−	0.643761i	\(-0.777373\pi\)
							0.375785	+	0.926707i	\(0.377373\pi\)
\(5\)	−4.08088	+	6.76738i	−0.816176	+	1.35348i	0.116660	+	0.993172i	\(0.462781\pi\)
							−0.932835	+	0.360303i	\(0.882673\pi\)
\(7\)	9.01268	+	1.82621i	1.28753	+	0.260886i	0.793505	−	0.608564i	\(-0.208254\pi\)
							0.494020	+	0.869450i	\(0.335527\pi\)
\(11\)	10.9985	−	0.183547i	0.999861	−	0.0166860i
							\(13\)	−14.5245	+	3.81783i	−1.11727	+	0.293680i	−0.766158	−	0.642653i	\(-0.777834\pi\)
−0.351115	+	0.936332i	\(0.614197\pi\)
\(17\)	0.981775	+	5.67314i	0.0577515	+	0.333714i	0.999999	−	0.00120256i	\(-0.000382787\pi\)
							−0.942248	+	0.334917i	\(0.891292\pi\)
\(19\)	26.5740	−	25.8257i	1.39863	−	1.35925i	0.544934	−	0.838479i	\(-0.316555\pi\)
							0.853698	−	0.520768i	\(-0.174354\pi\)
\(23\)	−0.191520	+	0.419371i	−0.00832696	+	0.0182335i	−0.913750	−	0.406277i	\(-0.866827\pi\)
							0.905423	+	0.424511i	\(0.139554\pi\)
\(29\)	38.0054	+	20.0534i	1.31053	+	0.691497i	0.968693	−	0.248262i	\(-0.0798593\pi\)
							0.341839	+	0.939759i	\(0.388950\pi\)
\(31\)	−21.5021	+	17.5822i	−0.693617	+	0.567168i	−0.912746	−	0.408528i	\(-0.866042\pi\)
							0.219129	+	0.975696i	\(0.429679\pi\)
\(37\)	−7.14306	−	18.3468i	−0.193056	−	0.495860i	0.801749	−	0.597661i	\(-0.203903\pi\)
							−0.994805	+	0.101801i	\(0.967539\pi\)
\(41\)	26.3406	+	20.3116i	0.642454	+	0.495406i	0.879379	−	0.476122i	\(-0.157958\pi\)
							−0.236926	+	0.971528i	\(0.576140\pi\)
\(43\)	50.1275	+	43.4358i	1.16576	+	1.01013i	0.999712	+	0.0240160i	\(0.00764527\pi\)
							0.166045	+	0.986118i	\(0.446900\pi\)
\(47\)	−28.7111	+	10.2441i	−0.610873	+	0.217959i	−0.623325	−	0.781963i	\(-0.714218\pi\)
							0.0124510	+	0.999922i	\(0.496037\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.3.h.a.2.16	✓	840
121.61	odd	110	inner	121.3.h.a.61.16	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.3.h.a.2.16	✓	840	1.1	even	1	trivial
121.3.h.a.61.16	yes	840	121.61	odd	110	inner