Embedded newform 151.2.k.a.40.2

• Label: 151.2.k.a.40.2
• Level: $151$
• Weight: $2$
• Character: 151.40
• Analytic conductor: $1.206$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	151.k (of order \(75\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.20574107052\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(12\) over \(\Q(\zeta_{75})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{75}]$

Embedding invariants

Embedding label			40.2
Character	\(\chi\)	\(=\)	151.40
Dual form			151.2.k.a.34.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.62886 - 1.80903i) q^{2} +(2.70249 - 1.06999i) q^{3} +(-0.410353 + 3.90424i) q^{4} +(0.462519 + 1.97196i) q^{5} +(-6.33760 - 3.14601i) q^{6} +(3.32919 - 2.01623i) q^{7} +(3.79253 - 2.75544i) q^{8} +(3.97165 - 3.72963i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 45 q^{2} - 40 q^{3} + 15 q^{4} - 40 q^{5} - 60 q^{6} - 40 q^{7} - 50 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{11}{75}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.62886	−	1.80903i	−1.15177	−	1.27918i	−0.954294	−	0.298869i	\(-0.903391\pi\)
							−0.197481	−	0.980307i	\(-0.563276\pi\)
\(3\)	2.70249	−	1.06999i	1.56028	−	0.617759i	0.579868	−	0.814710i	\(-0.303104\pi\)
							0.980413	+	0.196951i	\(0.0631040\pi\)
\(5\)	0.462519	+	1.97196i	0.206845	+	0.881887i	0.971423	+	0.237356i	\(0.0762808\pi\)
							−0.764578	+	0.644531i	\(0.777053\pi\)
\(7\)	3.32919	−	2.01623i	1.25831	−	0.762062i	0.278205	−	0.960522i	\(-0.410260\pi\)
							0.980109	+	0.198459i	\(0.0635938\pi\)
\(11\)	−2.73025	+	2.16397i	−0.823201	+	0.652461i	−0.940182	−	0.340673i	\(-0.889345\pi\)
							0.116981	+	0.993134i	\(0.462678\pi\)
\(13\)	−2.95766	−	0.123963i	−0.820307	−	0.0343810i	−0.371969	−	0.928245i	\(-0.621317\pi\)
							−0.448338	+	0.893864i	\(0.647984\pi\)
\(17\)	−0.0499799	+	2.38601i	−0.0121219	+	0.578694i	0.950934	+	0.309393i	\(0.100126\pi\)
							−0.963056	+	0.269301i	\(0.913207\pi\)
\(19\)	−4.37111	−	3.17580i	−1.00280	−	0.728577i	−0.0401139	−	0.999195i	\(-0.512772\pi\)
							−0.962687	+	0.270618i	\(0.912772\pi\)
\(23\)	−8.30866	+	3.69925i	−1.73248	+	0.771348i	−0.737047	+	0.675842i	\(0.763780\pi\)
							−0.995429	+	0.0955062i	\(0.969553\pi\)
\(29\)	2.13081	−	2.57571i	0.395681	−	0.478296i	−0.534620	−	0.845092i	\(-0.679545\pi\)
							0.930301	+	0.366796i	\(0.119545\pi\)
\(31\)	−2.49677	−	0.869467i	−0.448433	−	0.156161i	0.0952467	−	0.995454i	\(-0.469636\pi\)
							−0.543680	+	0.839293i	\(0.682969\pi\)
\(37\)	4.02617	+	7.70136i	0.661899	+	1.26610i	0.951116	+	0.308833i	\(0.0999383\pi\)
							−0.289217	+	0.957263i	\(0.593395\pi\)
\(41\)	−0.316694	−	5.03371i	−0.0494593	−	0.786133i	−0.942093	−	0.335351i	\(-0.891145\pi\)
							0.892634	−	0.450782i	\(-0.148855\pi\)
\(43\)	0.129529	+	0.0784457i	0.0197530	+	0.0119629i	0.527869	−	0.849326i	\(-0.322991\pi\)
							−0.508116	+	0.861289i	\(0.669658\pi\)
\(47\)	−0.202106	+	0.134279i	−0.0294802	+	0.0195866i	−0.568045	−	0.822998i	\(-0.692300\pi\)
							0.538565	+	0.842584i	\(0.318967\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	151.2.k.a.40.2	yes	480
151.34	even	75	inner	151.2.k.a.34.2	✓	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
151.2.k.a.34.2	✓	480	151.34	even	75	inner
151.2.k.a.40.2	yes	480	1.1	even	1	trivial