Embedded newform 121.3.h.a.2.15

• Label: 121.3.h.a.2.15
• Level: $121$
• Weight: $3$
• Character: 121.2
• Analytic conductor: $3.297$
• Analytic rank: $0$
• Dimension: $840$
• CM: no
• 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			2.15
Character	\(\chi\)	\(=\)	121.2
Dual form			121.3.h.a.61.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.72705 + 0.0493378i) q^{2} +(-3.03901 + 2.20797i) q^{3} +(-1.01321 - 0.0579376i) q^{4} +(-0.589093 + 0.976900i) q^{5} +(-5.35745 + 3.66333i) q^{6} +(-4.45396 - 0.902490i) q^{7} +(-8.63267 - 0.741461i) q^{8} +(1.57929 - 4.86057i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 39 q^{2} - 34 q^{3} - 75 q^{4} - 43 q^{5} - 15 q^{6} - 54 q^{7} - 59 q^{8} - 594 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.72705	+	0.0493378i	0.863524	+	0.0246689i	0.457517	−	0.889201i	\(-0.348739\pi\)
							0.406007	+	0.913870i	\(0.366921\pi\)
\(3\)	−3.03901	+	2.20797i	−1.01300	+	0.735990i	−0.964837	−	0.262850i	\(-0.915338\pi\)
							−0.0481661	+	0.998839i	\(0.515338\pi\)
\(5\)	−0.589093	+	0.976900i	−0.117819	+	0.195380i	−0.909668	−	0.415336i	\(-0.863664\pi\)
							0.791849	+	0.610716i	\(0.209118\pi\)
\(7\)	−4.45396	−	0.902490i	−0.636280	−	0.128927i	−0.130304	−	0.991474i	\(-0.541595\pi\)
							−0.505977	+	0.862547i	\(0.668868\pi\)
\(11\)	−9.31573	+	5.84955i	−0.846884	+	0.531777i
							\(13\)	7.65350	−	2.01175i	0.588731	−	0.154750i	0.0522084	−	0.998636i	\(-0.483374\pi\)
0.536523	+	0.843886i	\(0.319738\pi\)
\(17\)	3.23112	+	18.6708i	0.190066	+	1.09828i	0.913180	+	0.407556i	\(0.133619\pi\)
							−0.723114	+	0.690728i	\(0.757290\pi\)
\(19\)	−3.87608	+	3.76693i	−0.204004	+	0.198259i	−0.791854	−	0.610711i	\(-0.790884\pi\)
							0.587850	+	0.808970i	\(0.299975\pi\)
\(23\)	3.71742	−	8.14001i	0.161627	−	0.353913i	−0.811440	−	0.584435i	\(-0.801316\pi\)
							0.973067	+	0.230522i	\(0.0740433\pi\)
\(29\)	7.82815	+	4.13049i	0.269936	+	0.142431i	0.596169	−	0.802859i	\(-0.296689\pi\)
							−0.326233	+	0.945289i	\(0.605780\pi\)
\(31\)	3.27464	−	2.67766i	0.105634	−	0.0863761i	−0.578721	−	0.815526i	\(-0.696448\pi\)
							0.684354	+	0.729150i	\(0.260084\pi\)
\(37\)	20.2408	+	51.9880i	0.547048	+	1.40508i	0.885739	+	0.464184i	\(0.153652\pi\)
							−0.338690	+	0.940898i	\(0.609984\pi\)
\(41\)	−40.8931	−	31.5333i	−0.997393	−	0.769105i	−0.0243440	−	0.999704i	\(-0.507750\pi\)
							−0.973049	+	0.230599i	\(0.925932\pi\)
\(43\)	−30.3233	−	26.2753i	−0.705194	−	0.611054i	0.226623	−	0.973983i	\(-0.427231\pi\)
							−0.931817	+	0.362929i	\(0.881777\pi\)
\(47\)	−51.4609	+	18.3612i	−1.09491	+	0.390664i	−0.820906	−	0.571064i	\(-0.806531\pi\)
							−0.274006	+	0.961728i	\(0.588349\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.3.h.a.2.15	✓	840
121.61	odd	110	inner	121.3.h.a.61.15	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.3.h.a.2.15	✓	840	1.1	even	1	trivial
121.3.h.a.61.15	yes	840	121.61	odd	110	inner