Embedded newform 155.2.p.c.68.6

• Label: 155.2.p.c.68.6
• Level: $155$
• Weight: $2$
• Character: 155.68
• Analytic conductor: $1.238$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 155 = 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	155.p (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.23768123133\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			68.6
Character	\(\chi\)	\(=\)	155.68
Dual form			155.2.p.c.57.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0988180 + 0.0988180i) q^{2} +(2.41036 + 0.645854i) q^{3} +1.98047i q^{4} +(-2.03273 - 0.931676i) q^{5} +(-0.302009 + 0.174365i) q^{6} +(1.97917 + 0.530317i) q^{7} +(-0.393342 - 0.393342i) q^{8} +(2.79463 + 1.61348i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 6 q^{3} - 4 q^{5} - 48 q^{6} - 8 q^{7} + 36 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).

\(n\)	\(32\)	\(96\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0988180	+	0.0988180i	−0.0698749	+	0.0698749i	−0.741181	−	0.671306i	\(-0.765734\pi\)
							0.671306	+	0.741181i	\(0.265734\pi\)
\(3\)	2.41036	+	0.645854i	1.39162	+	0.372884i	0.875329	−	0.483528i	\(-0.160645\pi\)
							0.516293	+	0.856412i	\(0.327312\pi\)
\(5\)	−2.03273	−	0.931676i	−0.909063	−	0.416658i
							\(7\)	1.97917	+	0.530317i	0.748056	+	0.200441i	0.612656	−	0.790350i	\(-0.290101\pi\)
0.135400	+	0.990791i	\(0.456768\pi\)
\(11\)	−2.20455	−	1.27280i	−0.664698	−	0.383763i	0.129367	−	0.991597i	\(-0.458706\pi\)
							−0.794065	+	0.607833i	\(0.792039\pi\)
\(13\)	0.123621	+	0.461358i	0.0342862	+	0.127958i	0.980948	−	0.194272i	\(-0.0622344\pi\)
							−0.946662	+	0.322230i	\(0.895568\pi\)
\(17\)	1.29208	−	4.82210i	0.313375	−	1.16953i	−0.612118	−	0.790766i	\(-0.709682\pi\)
							0.925493	−	0.378765i	\(-0.123651\pi\)
\(19\)	4.05525	−	2.34130i	0.930338	−	0.537131i	0.0434197	−	0.999057i	\(-0.486175\pi\)
							0.886919	+	0.461926i	\(0.152841\pi\)
\(23\)	−3.82307	+	3.82307i	−0.797164	+	0.797164i	−0.982647	−	0.185483i	\(-0.940615\pi\)
							0.185483	+	0.982647i	\(0.440615\pi\)
\(29\)	2.03317			0.377551			0.188775	−	0.982020i	\(-0.439548\pi\)
							0.188775	+	0.982020i	\(0.439548\pi\)
\(31\)	0.314089	−	5.55890i	0.0564121	−	0.998408i
							\(37\)	−0.00153688	+	0.00573572i	−0.000252662	+	0.000942946i	−0.966052	−	0.258348i	\(-0.916822\pi\)
0.965799	+	0.259290i	\(0.0834887\pi\)
\(41\)	2.31704	−	4.01323i	0.361861	−	0.626761i	−0.626406	−	0.779497i	\(-0.715475\pi\)
							0.988267	+	0.152736i	\(0.0488083\pi\)
\(43\)	−11.1984	−	3.00060i	−1.70774	−	0.457587i	−0.732870	−	0.680368i	\(-0.761820\pi\)
							−0.974868	+	0.222781i	\(0.928486\pi\)
\(47\)	−7.40468	+	7.40468i	−1.08008	+	1.08008i	−0.0835834	+	0.996501i	\(0.526637\pi\)
							−0.996501	+	0.0835834i	\(0.973363\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	155.2.p.c.68.6	yes	48
5.2	odd	4	inner	155.2.p.c.37.6	✓	48
5.3	odd	4		775.2.bj.f.657.7		48
5.4	even	2		775.2.bj.f.68.7		48
31.26	odd	6	inner	155.2.p.c.88.6	yes	48
155.57	even	12	inner	155.2.p.c.57.6	yes	48
155.88	even	12		775.2.bj.f.57.7		48
155.119	odd	6		775.2.bj.f.243.7		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.p.c.37.6	✓	48	5.2	odd	4	inner
155.2.p.c.57.6	yes	48	155.57	even	12	inner
155.2.p.c.68.6	yes	48	1.1	even	1	trivial
155.2.p.c.88.6	yes	48	31.26	odd	6	inner
775.2.bj.f.57.7		48	155.88	even	12
775.2.bj.f.68.7		48	5.4	even	2
775.2.bj.f.243.7		48	155.119	odd	6
775.2.bj.f.657.7		48	5.3	odd	4