Embedded newform 71.2.d.a.20.1

• Label: 71.2.d.a.20.1
• Level: $71$
• Weight: $2$
• Character: 71.20
• Analytic conductor: $0.567$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	71.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.566937854351\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			20.1
Character	\(\chi\)	\(=\)	71.20
Dual form			71.2.d.a.32.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.512695 - 2.24626i) q^{2} +(-2.31785 + 1.11622i) q^{3} +(-2.98090 + 1.43552i) q^{4} -3.13999 q^{5} +(3.69567 + 4.63422i) q^{6} +(0.916758 - 4.01658i) q^{7} +(1.87978 + 2.35717i) q^{8} +(2.25602 - 2.82895i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 3 q^{2} - 10 q^{3} - 7 q^{4} - 10 q^{5} + 5 q^{6} - 3 q^{7} + 17 q^{8} - 9 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.512695	−	2.24626i	−0.362530	−	1.58835i	−0.746750	−	0.665105i	\(-0.768387\pi\)
							0.384220	−	0.923242i	\(-0.374470\pi\)
\(3\)	−2.31785	+	1.11622i	−1.33821	+	0.644449i	−0.959668	−	0.281135i	\(-0.909289\pi\)
							−0.378543	+	0.925584i	\(0.623575\pi\)
\(5\)	−3.13999			−1.40424			−0.702122	−	0.712057i	\(-0.747764\pi\)
							−0.702122	+	0.712057i	\(0.747764\pi\)
\(7\)	0.916758	−	4.01658i	0.346502	−	1.51812i	−0.438558	−	0.898703i	\(-0.644510\pi\)
							0.785059	−	0.619421i	\(-0.212632\pi\)
\(11\)	0.612193	−	0.767666i	0.184583	−	0.231460i	−0.680927	−	0.732351i	\(-0.738423\pi\)
							0.865510	+	0.500891i	\(0.166994\pi\)
\(13\)	0.587823	+	0.737107i	0.163033	+	0.204437i	0.856637	−	0.515920i	\(-0.172550\pi\)
							−0.693604	+	0.720356i	\(0.743978\pi\)
\(17\)	−5.47189			−1.32713			−0.663565	−	0.748119i	\(-0.730957\pi\)
							−0.663565	+	0.748119i	\(0.730957\pi\)
\(19\)	0.919657	+	0.442884i	0.210984	+	0.101604i	0.536391	−	0.843969i	\(-0.319787\pi\)
							−0.325407	+	0.945574i	\(0.605501\pi\)
\(23\)	1.39455	−	6.10994i	0.290785	−	1.27401i	−0.592651	−	0.805459i	\(-0.701919\pi\)
							0.883436	−	0.468552i	\(-0.155224\pi\)
\(29\)	−1.18607	+	0.571181i	−0.220247	+	0.106066i	−0.540754	−	0.841181i	\(-0.681861\pi\)
							0.320507	+	0.947246i	\(0.396147\pi\)
\(31\)	−0.194559	−	0.243969i	−0.0349438	−	0.0438182i	0.764053	−	0.645154i	\(-0.223207\pi\)
							−0.798996	+	0.601336i	\(0.794635\pi\)
\(37\)	0.410227	+	1.79732i	0.0674409	+	0.295478i	0.997389	−	0.0722123i	\(-0.0230059\pi\)
							−0.929948	+	0.367690i	\(0.880149\pi\)
\(41\)	−2.48629	−	3.11771i	−0.388293	−	0.486904i	0.548815	−	0.835944i	\(-0.315079\pi\)
							−0.937108	+	0.349040i	\(0.886508\pi\)
\(43\)	−2.79402	−	12.2414i	−0.426083	−	1.86679i	−0.494565	−	0.869141i	\(-0.664672\pi\)
							0.0684813	−	0.997652i	\(-0.478185\pi\)
\(47\)	3.78687	+	1.82366i	0.552372	+	0.266009i	0.689181	−	0.724589i	\(-0.257970\pi\)
							−0.136809	+	0.990597i	\(0.543685\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	71.2.d.a.20.1	✓	30
3.2	odd	2		639.2.j.c.91.5		30
71.23	odd	14		5041.2.a.m.1.13		15
71.32	even	7	inner	71.2.d.a.32.1	yes	30
71.48	even	7		5041.2.a.l.1.13		15
213.32	odd	14		639.2.j.c.316.5		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.2.d.a.20.1	✓	30	1.1	even	1	trivial
71.2.d.a.32.1	yes	30	71.32	even	7	inner
639.2.j.c.91.5		30	3.2	odd	2
639.2.j.c.316.5		30	213.32	odd	14
5041.2.a.l.1.13		15	71.48	even	7
5041.2.a.m.1.13		15	71.23	odd	14