Embedded newform 71.5.f.a.23.15

• Label: 71.5.f.a.23.15
• Level: $71$
• Weight: $5$
• Character: 71.23
• Analytic conductor: $7.339$
• Analytic rank: $0$
• Dimension: $138$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	71.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.33926737895\)
Analytic rank:	\(0\)
Dimension:	\(138\)
Relative dimension:	\(23\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			23.15
Character	\(\chi\)	\(=\)	71.23
Dual form			71.5.f.a.34.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.14448 + 1.43513i) q^{2} +(-2.33687 + 10.2385i) q^{3} +(2.81057 - 12.3139i) q^{4} -19.2615 q^{5} +(-17.3681 + 8.36402i) q^{6} +(-74.1615 - 59.1418i) q^{7} +(47.3497 - 22.8024i) q^{8} +(-26.3877 - 12.7076i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 11 q^{2} + 20 q^{3} - 187 q^{4} - 18 q^{5} + 93 q^{6} - 7 q^{7} - 123 q^{8} - 329 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{3}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.14448	+	1.43513i	0.286119	+	0.358782i	0.904032	−	0.427464i	\(-0.140593\pi\)
							−0.617913	+	0.786246i	\(0.712022\pi\)
\(3\)	−2.33687	+	10.2385i	−0.259653	+	1.13761i	0.661972	+	0.749529i	\(0.269720\pi\)
							−0.921624	+	0.388083i	\(0.873137\pi\)
\(5\)	−19.2615			−0.770459			−0.385230	−	0.922821i	\(-0.625878\pi\)
							−0.385230	+	0.922821i	\(0.625878\pi\)
\(7\)	−74.1615	−	59.1418i	−1.51350	−	1.20698i	−0.913408	−	0.407046i	\(-0.866559\pi\)
							−0.600093	−	0.799930i	\(-0.704870\pi\)
\(11\)	−2.80804	+	5.83096i	−0.0232069	+	0.0481897i	−0.912250	−	0.409634i	\(-0.865657\pi\)
							0.889043	+	0.457823i	\(0.151371\pi\)
\(13\)	−31.6303	−	65.6810i	−0.187162	−	0.388645i	0.786183	−	0.617994i	\(-0.212055\pi\)
							−0.973344	+	0.229349i	\(0.926340\pi\)
\(17\)			280.746i			0.971440i	0.874115	+	0.485720i	\(0.161442\pi\)
							−0.874115	+	0.485720i	\(0.838558\pi\)
\(19\)	−111.881	−	490.182i	−0.309919	−	1.35784i	−0.854638	−	0.519225i	\(-0.826221\pi\)
							0.544719	−	0.838619i	\(-0.316636\pi\)
\(23\)	−354.915	−	283.035i	−0.670917	−	0.535038i	0.227723	−	0.973726i	\(-0.426872\pi\)
							−0.898639	+	0.438688i	\(0.855443\pi\)
\(29\)	−102.187	+	447.711i	−0.121507	+	0.532355i	0.877135	+	0.480244i	\(0.159452\pi\)
							−0.998641	+	0.0521108i	\(0.983405\pi\)
\(31\)	283.101	+	587.866i	0.294590	+	0.611723i	0.994758	−	0.102255i	\(-0.0326058\pi\)
							−0.700168	+	0.713978i	\(0.746892\pi\)
\(37\)	−796.719	−	999.054i	−0.581972	−	0.729769i	0.400476	−	0.916307i	\(-0.368845\pi\)
							−0.982448	+	0.186538i	\(0.940273\pi\)
\(41\)	175.617	+	364.673i	0.104472	+	0.216938i	0.946652	−	0.322259i	\(-0.104442\pi\)
							−0.842180	+	0.539197i	\(0.818728\pi\)
\(43\)	345.424	+	433.148i	0.186817	+	0.234261i	0.866416	−	0.499323i	\(-0.166418\pi\)
							−0.679599	+	0.733583i	\(0.737846\pi\)
\(47\)	2938.01	−	670.582i	1.33002	−	0.303568i	0.502291	−	0.864699i	\(-0.332491\pi\)
							0.827727	+	0.561131i	\(0.189634\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	71.5.f.a.23.15	✓	138
71.34	odd	14	inner	71.5.f.a.34.15	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.5.f.a.23.15	✓	138	1.1	even	1	trivial
71.5.f.a.34.15	yes	138	71.34	odd	14	inner