Embedded newform 151.2.d.b.59.3

• Label: 151.2.d.b.59.3
• Level: $151$
• Weight: $2$
• Character: 151.59
• 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			59.3
Character	\(\chi\)	\(=\)	151.59
Dual form			151.2.d.b.64.3

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.19627 q^{2} +(2.19258 + 1.59300i) q^{3} -0.568943 q^{4} +(0.154342 + 0.475015i) q^{5} +(-2.62292 - 1.90566i) q^{6} +(0.379296 + 1.16735i) q^{7} +3.07314 q^{8} +(1.34270 + 4.13242i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2 q^{2} + q^{3} + 22 q^{4} - 19 q^{6} + 2 q^{7} + 18 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19627			−0.845889			−0.422945	−	0.906156i	\(-0.639003\pi\)
							−0.422945	+	0.906156i	\(0.639003\pi\)
\(3\)	2.19258	+	1.59300i	1.26589	+	0.919722i	0.999031	−	0.0440145i	\(-0.0140148\pi\)
							0.266857	+	0.963736i	\(0.414015\pi\)
\(5\)	0.154342	+	0.475015i	0.0690237	+	0.212433i	0.979618	−	0.200867i	\(-0.0643760\pi\)
							−0.910595	+	0.413300i	\(0.864376\pi\)
\(7\)	0.379296	+	1.16735i	0.143360	+	0.441218i	0.996797	−	0.0799795i	\(-0.0254855\pi\)
							−0.853436	+	0.521198i	\(0.825485\pi\)
\(11\)	−1.79540	+	1.30443i	−0.541334	+	0.393302i	−0.824580	−	0.565745i	\(-0.808589\pi\)
							0.283246	+	0.959047i	\(0.408589\pi\)
\(13\)	−1.56545	−	1.13736i	−0.434177	−	0.315448i	0.349140	−	0.937071i	\(-0.386474\pi\)
							−0.783317	+	0.621623i	\(0.786474\pi\)
\(17\)	−0.683587	+	2.10386i	−0.165794	+	0.510262i	−0.999094	−	0.0425593i	\(-0.986449\pi\)
							0.833300	+	0.552821i	\(0.186449\pi\)
\(19\)	7.16922			1.64473			0.822366	−	0.568959i	\(-0.192654\pi\)
							0.822366	+	0.568959i	\(0.192654\pi\)
\(23\)	−4.22376			−0.880716			−0.440358	−	0.897822i	\(-0.645148\pi\)
							−0.440358	+	0.897822i	\(0.645148\pi\)
\(29\)	1.67343	−	1.21582i	0.310748	−	0.225772i	−0.421469	−	0.906843i	\(-0.638485\pi\)
							0.732217	+	0.681071i	\(0.238485\pi\)
\(31\)	2.28582	−	7.03503i	0.410545	−	1.26353i	−0.505630	−	0.862751i	\(-0.668740\pi\)
							0.916175	−	0.400778i	\(-0.131260\pi\)
\(37\)	−4.26091	−	3.09573i	−0.700489	−	0.508935i	0.179602	−	0.983739i	\(-0.442519\pi\)
							−0.880092	+	0.474804i	\(0.842519\pi\)
\(41\)	−7.56611	−	5.49710i	−1.18163	−	0.858503i	−0.189274	−	0.981924i	\(-0.560613\pi\)
							−0.992354	+	0.123421i	\(0.960613\pi\)
\(43\)	1.86651	−	5.74452i	0.284640	−	0.876031i	−0.701867	−	0.712308i	\(-0.747650\pi\)
							0.986506	−	0.163723i	\(-0.0523502\pi\)
\(47\)	5.38326	+	3.91117i	0.785229	+	0.570502i	0.906544	−	0.422112i	\(-0.138711\pi\)
							−0.121315	+	0.992614i	\(0.538711\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	151.2.d.b.59.3	✓	32
151.64	even	5	inner	151.2.d.b.64.3	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
151.2.d.b.59.3	✓	32	1.1	even	1	trivial
151.2.d.b.64.3	yes	32	151.64	even	5	inner