Embedded newform 71.4.g.a.2.16

• Label: 71.4.g.a.2.16
• Level: $71$
• Weight: $4$
• Character: 71.2
• Analytic conductor: $4.189$
• Analytic rank: $0$
• Dimension: $408$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	71.g (of order \(35\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.18913561041\)
Analytic rank:	\(0\)
Dimension:	\(408\)
Relative dimension:	\(17\) over \(\Q(\zeta_{35})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{35}]$

Embedding invariants

Embedding label			2.16
Character	\(\chi\)	\(=\)	71.2
Dual form			71.4.g.a.36.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.219654 - 4.89097i) q^{2} +(-3.24713 - 2.83693i) q^{3} +(-15.9055 - 1.43152i) q^{4} +(6.12849 + 18.8616i) q^{5} +(-14.5886 + 15.2584i) q^{6} +(-30.0980 - 8.30653i) q^{7} +(-5.23770 + 38.6662i) q^{8} +(-1.12864 - 8.33193i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 408 q - 28 q^{2} - 35 q^{3} + 54 q^{4} - 26 q^{5} - 86 q^{6} - 11 q^{7} - 43 q^{8} + 142 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/71\mathbb{Z}\right)^\times\).

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{3}{35}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.219654	−	4.89097i	0.0776593	−	1.72922i	−0.461608	−	0.887084i	\(-0.652727\pi\)
							0.539268	−	0.842134i	\(-0.318701\pi\)
\(3\)	−3.24713	−	2.83693i	−0.624910	−	0.545967i	0.286180	−	0.958176i	\(-0.407615\pi\)
							−0.911089	+	0.412209i	\(0.864757\pi\)
\(5\)	6.12849	+	18.8616i	0.548149	+	1.68703i	0.713383	+	0.700774i	\(0.247162\pi\)
							−0.165234	+	0.986254i	\(0.552838\pi\)
\(7\)	−30.0980	−	8.30653i	−1.62514	−	0.448510i	−0.669443	−	0.742863i	\(-0.733467\pi\)
							−0.955697	+	0.294353i	\(0.904896\pi\)
\(11\)	−7.22542	+	13.4271i	−0.198050	+	0.368038i	−0.960178	−	0.279388i	\(-0.909869\pi\)
							0.762129	+	0.647425i	\(0.224154\pi\)
\(13\)	−19.3524	−	35.9628i	−0.412876	−	0.767253i	0.586203	−	0.810165i	\(-0.300622\pi\)
							−0.999079	+	0.0429117i	\(0.986337\pi\)
\(17\)	27.7358	+	20.1512i	0.395701	+	0.287494i	0.767788	−	0.640704i	\(-0.221358\pi\)
							−0.372086	+	0.928198i	\(0.621358\pi\)
\(19\)	−18.7812	−	43.9407i	−0.226773	−	0.530563i	0.767043	−	0.641595i	\(-0.221727\pi\)
							−0.993817	+	0.111033i	\(0.964584\pi\)
\(23\)	−77.8145	−	97.5763i	−0.705454	−	0.884612i	0.291964	−	0.956429i	\(-0.405691\pi\)
							−0.997418	+	0.0718179i	\(0.977120\pi\)
\(29\)	−30.3506	+	18.1337i	−0.194344	+	0.116115i	−0.606775	−	0.794874i	\(-0.707537\pi\)
							0.412431	+	0.910989i	\(0.364680\pi\)
\(31\)	55.2350	−	10.0237i	0.320016	−	0.0580743i	−0.0161721	−	0.999869i	\(-0.505148\pi\)
							0.336188	+	0.941795i	\(0.390862\pi\)
\(37\)	174.423	−	218.719i	0.774998	−	0.971817i	−0.224999	−	0.974359i	\(-0.572238\pi\)
							0.999997	+	0.00254241i	\(0.000809275\pi\)
\(41\)	239.666	+	115.417i	0.912915	+	0.439637i	0.830536	−	0.556964i	\(-0.188034\pi\)
							0.0823788	+	0.996601i	\(0.473748\pi\)
\(43\)	−495.553	−	185.984i	−1.75747	−	0.659589i	−0.999977	−	0.00674291i	\(-0.997854\pi\)
							−0.757491	−	0.652846i	\(-0.773575\pi\)
\(47\)	403.108	−	352.185i	1.25105	−	1.09301i	0.259170	−	0.965832i	\(-0.416551\pi\)
							0.991880	−	0.127178i	\(-0.0405918\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	71.4.g.a.2.16	✓	408
71.36	even	35	inner	71.4.g.a.36.16	yes	408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.4.g.a.2.16	✓	408	1.1	even	1	trivial
71.4.g.a.36.16	yes	408	71.36	even	35	inner