Embedded newform 151.2.d.b.8.3

• Label: 151.2.d.b.8.3
• Level: $151$
• Weight: $2$
• Character: 151.8
• Analytic conductor: $1.206$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	151.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.20574107052\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			8.3
Character	\(\chi\)	\(=\)	151.8
Dual form			151.2.d.b.19.3

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.35309 q^{2} +(-0.106673 - 0.328306i) q^{3} -0.169135 q^{4} +(-2.12080 + 1.54085i) q^{5} +(0.144339 + 0.444229i) q^{6} +(3.63219 - 2.63894i) q^{7} +2.93504 q^{8} +(2.33065 - 1.69331i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2 q^{2} + q^{3} + 22 q^{4} - 19 q^{6} + 2 q^{7} + 18 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/151\mathbb{Z}\right)^\times\).

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35309			−0.956782			−0.478391	−	0.878147i	\(-0.658780\pi\)
							−0.478391	+	0.878147i	\(0.658780\pi\)
\(3\)	−0.106673	−	0.328306i	−0.0615877	−	0.189547i	0.915529	−	0.402253i	\(-0.131773\pi\)
							−0.977116	+	0.212705i	\(0.931773\pi\)
\(5\)	−2.12080	+	1.54085i	−0.948453	+	0.689091i	−0.950440	−	0.310907i	\(-0.899367\pi\)
							0.00198778	+	0.999998i	\(0.499367\pi\)
\(7\)	3.63219	−	2.63894i	1.37284	−	0.997427i	0.375331	−	0.926891i	\(-0.377529\pi\)
							0.997509	−	0.0705355i	\(-0.0224708\pi\)
\(11\)	0.100510	−	0.309339i	0.0303050	−	0.0932693i	−0.934760	−	0.355280i	\(-0.884386\pi\)
							0.965065	+	0.262011i	\(0.0843856\pi\)
\(13\)	−1.52223	−	4.68493i	−0.422189	−	1.29937i	−0.905660	−	0.424005i	\(-0.860624\pi\)
							0.483470	−	0.875361i	\(-0.339376\pi\)
\(17\)	1.52148	+	1.10542i	0.369013	+	0.268104i	0.756802	−	0.653645i	\(-0.226761\pi\)
							−0.387789	+	0.921748i	\(0.626761\pi\)
\(19\)	1.61703			0.370972			0.185486	−	0.982647i	\(-0.440614\pi\)
							0.185486	+	0.982647i	\(0.440614\pi\)
\(23\)	4.76543			0.993662			0.496831	−	0.867847i	\(-0.334497\pi\)
							0.496831	+	0.867847i	\(0.334497\pi\)
\(29\)	−0.175307	+	0.539540i	−0.0325537	+	0.100190i	−0.966013	−	0.258493i	\(-0.916774\pi\)
							0.933459	+	0.358683i	\(0.116774\pi\)
\(31\)	0.471203	+	0.342349i	0.0846305	+	0.0614876i	0.629296	−	0.777166i	\(-0.283343\pi\)
							−0.544665	+	0.838653i	\(0.683343\pi\)
\(37\)	1.95209	+	6.00793i	0.320922	+	0.987697i	0.973248	+	0.229758i	\(0.0737935\pi\)
							−0.652326	+	0.757939i	\(0.726206\pi\)
\(41\)	−3.72180	−	11.4545i	−0.581248	−	1.78890i	−0.613842	−	0.789429i	\(-0.710377\pi\)
							0.0325944	−	0.999469i	\(-0.489623\pi\)
\(43\)	−4.65962	−	3.38541i	−0.710585	−	0.516270i	0.172777	−	0.984961i	\(-0.444726\pi\)
							−0.883362	+	0.468691i	\(0.844726\pi\)
\(47\)	1.08241	+	3.33131i	0.157885	+	0.485921i	0.998442	−	0.0558027i	\(-0.0177718\pi\)
							−0.840556	+	0.541724i	\(0.817772\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	151.2.d.b.8.3	✓	32
151.19	even	5	inner	151.2.d.b.19.3	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
151.2.d.b.8.3	✓	32	1.1	even	1	trivial
151.2.d.b.19.3	yes	32	151.19	even	5	inner