Embedded newform 61.2.i.a.12.2

• Label: 61.2.i.a.12.2
• Level: $61$
• Weight: $2$
• Character: 61.12
• Analytic conductor: $0.487$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	61.i (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.487087452330\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			12.2
Character	\(\chi\)	\(=\)	61.12
Dual form			61.2.i.a.56.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.94561 - 0.866241i) q^{2} +(1.86941 - 1.35820i) q^{3} +(1.69676 + 1.88444i) q^{4} +(-1.20434 + 0.255990i) q^{5} +(-4.81367 + 1.02318i) q^{6} +(0.404506 - 3.84862i) q^{7} +(-0.352600 - 1.08519i) q^{8} +(0.722916 - 2.22491i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 10 q^{2} - 4 q^{3} - 10 q^{4} + 2 q^{5} + q^{6} + q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/61\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{2}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.94561	−	0.866241i	−1.37575	−	0.612525i	−0.420224	−	0.907421i	\(-0.638048\pi\)
							−0.955529	+	0.294896i	\(0.904715\pi\)
\(3\)	1.86941	−	1.35820i	1.07930	−	0.784160i	0.101742	−	0.994811i	\(-0.467558\pi\)
							0.977561	+	0.210651i	\(0.0675584\pi\)
\(5\)	−1.20434	+	0.255990i	−0.538595	+	0.114482i	−0.469175	−	0.883105i	\(-0.655448\pi\)
							−0.0694209	+	0.997587i	\(0.522115\pi\)
\(7\)	0.404506	−	3.84862i	0.152889	−	1.45464i	−0.601846	−	0.798612i	\(-0.705568\pi\)
							0.754735	−	0.656029i	\(-0.227765\pi\)
\(11\)	−1.98434			−0.598301			−0.299151	−	0.954206i	\(-0.596703\pi\)
							−0.299151	+	0.954206i	\(0.596703\pi\)
\(13\)	2.72937	+	4.72741i	0.756991	+	1.31115i	0.944378	+	0.328862i	\(0.106665\pi\)
							−0.187387	+	0.982286i	\(0.560002\pi\)
\(17\)	3.60066	+	3.99894i	0.873288	+	0.969885i	0.999756	−	0.0220861i	\(-0.00703079\pi\)
							−0.126468	+	0.991971i	\(0.540364\pi\)
\(19\)	0.472909	+	4.49943i	0.108493	+	1.03224i	0.904361	+	0.426769i	\(0.140348\pi\)
							−0.795868	+	0.605470i	\(0.792985\pi\)
\(23\)	2.24547	−	6.91085i	0.468213	−	1.44101i	−0.386683	−	0.922213i	\(-0.626379\pi\)
							0.854896	−	0.518800i	\(-0.173621\pi\)
\(29\)	−0.207283	+	0.359024i	−0.0384914	+	0.0666691i	−0.884629	−	0.466295i	\(-0.845589\pi\)
							0.846138	+	0.532964i	\(0.178922\pi\)
\(31\)	−2.26627	+	1.00901i	−0.407035	+	0.181224i	−0.600033	−	0.799976i	\(-0.704846\pi\)
							0.192998	+	0.981199i	\(0.438179\pi\)
\(37\)	−5.96555	−	4.33423i	−0.980730	−	0.712542i	−0.0228586	−	0.999739i	\(-0.507277\pi\)
							−0.957872	+	0.287196i	\(0.907277\pi\)
\(41\)	−3.12409	−	2.26979i	−0.487902	−	0.354481i	0.316475	−	0.948601i	\(-0.397501\pi\)
							−0.804377	+	0.594120i	\(0.797501\pi\)
\(43\)	−1.69649	+	1.88414i	−0.258712	+	0.287329i	−0.858482	−	0.512844i	\(-0.828592\pi\)
							0.599770	+	0.800172i	\(0.295259\pi\)
\(47\)	1.97577	−	3.42213i	0.288195	−	0.499169i	−0.685184	−	0.728370i	\(-0.740278\pi\)
							0.973379	+	0.229201i	\(0.0736114\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.2.i.a.12.2	✓	32
3.2	odd	2		549.2.bl.b.73.3		32
4.3	odd	2		976.2.bw.c.561.1		32
61.19	even	30		3721.2.a.l.1.13		16
61.42	even	15		3721.2.a.j.1.4		16
61.56	even	15	inner	61.2.i.a.56.2	yes	32
183.56	odd	30		549.2.bl.b.361.3		32
244.239	odd	30		976.2.bw.c.849.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.2.i.a.12.2	✓	32	1.1	even	1	trivial
61.2.i.a.56.2	yes	32	61.56	even	15	inner
549.2.bl.b.73.3		32	3.2	odd	2
549.2.bl.b.361.3		32	183.56	odd	30
976.2.bw.c.561.1		32	4.3	odd	2
976.2.bw.c.849.1		32	244.239	odd	30
3721.2.a.j.1.4		16	61.42	even	15
3721.2.a.l.1.13		16	61.19	even	30