Embedded newform 71.2.d.a.20.4

• Label: 71.2.d.a.20.4
• 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			20.4
Character	\(\chi\)	\(=\)	71.20
Dual form			71.2.d.a.32.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0919395 + 0.402813i) q^{2} +(0.860332 - 0.414314i) q^{3} +(1.64813 - 0.793699i) q^{4} -2.73724 q^{5} +(0.245990 + 0.308461i) q^{6} +(0.0252603 - 0.110672i) q^{7} +(0.986458 + 1.23698i) q^{8} +(-1.30195 + 1.63260i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 3 q^{2} - 10 q^{3} - 7 q^{4} - 10 q^{5} + 5 q^{6} - 3 q^{7} + 17 q^{8} - 9 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0919395	+	0.402813i	0.0650110	+	0.284832i	0.996975	−	0.0777180i	\(-0.0247634\pi\)
							−0.931964	+	0.362550i	\(0.881906\pi\)
\(3\)	0.860332	−	0.414314i	0.496713	−	0.239204i	−0.168717	−	0.985665i	\(-0.553962\pi\)
							0.665430	+	0.746460i	\(0.268248\pi\)
\(5\)	−2.73724			−1.22413			−0.612066	−	0.790806i	\(-0.709661\pi\)
							−0.612066	+	0.790806i	\(0.709661\pi\)
\(7\)	0.0252603	−	0.110672i	0.00954748	−	0.0418302i	−0.969930	−	0.243385i	\(-0.921742\pi\)
							0.979477	+	0.201555i	\(0.0645993\pi\)
\(11\)	−0.721709	+	0.904995i	−0.217604	+	0.272866i	−0.878637	−	0.477490i	\(-0.841547\pi\)
							0.661034	+	0.750356i	\(0.270118\pi\)
\(13\)	−0.294972	−	0.369884i	−0.0818106	−	0.102587i	0.739242	−	0.673440i	\(-0.235184\pi\)
							−0.821052	+	0.570853i	\(0.806613\pi\)
\(17\)	−3.18709			−0.772983			−0.386492	−	0.922293i	\(-0.626313\pi\)
							−0.386492	+	0.922293i	\(0.626313\pi\)
\(19\)	−2.70157	−	1.30101i	−0.619782	−	0.298471i	0.0975202	−	0.995234i	\(-0.468909\pi\)
							−0.717302	+	0.696762i	\(0.754623\pi\)
\(23\)	0.765590	−	3.35427i	0.159637	−	0.699414i	−0.830231	−	0.557420i	\(-0.811791\pi\)
							0.989868	−	0.141994i	\(-0.0453514\pi\)
\(29\)	8.81520	−	4.24518i	1.63694	−	0.788310i	0.637096	−	0.770785i	\(-0.280136\pi\)
							0.999846	−	0.0175249i	\(-0.00557863\pi\)
\(31\)	5.34724	+	6.70523i	0.960393	+	1.20429i	0.978874	+	0.204466i	\(0.0655457\pi\)
							−0.0184806	+	0.999829i	\(0.505883\pi\)
\(37\)	−1.76001	−	7.71113i	−0.289345	−	1.26770i	−0.885427	−	0.464778i	\(-0.846134\pi\)
							0.596083	−	0.802923i	\(-0.296723\pi\)
\(41\)	3.04436	+	3.81751i	0.475449	+	0.596195i	0.960496	−	0.278294i	\(-0.0897689\pi\)
							−0.485047	+	0.874488i	\(0.661197\pi\)
\(43\)	1.13032	+	4.95224i	0.172372	+	0.755209i	0.985018	+	0.172452i	\(0.0551689\pi\)
							−0.812646	+	0.582757i	\(0.801974\pi\)
\(47\)	−9.84988	−	4.74345i	−1.43675	−	0.691903i	−0.456513	−	0.889717i	\(-0.650902\pi\)
							−0.980240	+	0.197813i	\(0.936616\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	71.2.d.a.20.4	✓	30
3.2	odd	2		639.2.j.c.91.2		30
71.23	odd	14		5041.2.a.m.1.5		15
71.32	even	7	inner	71.2.d.a.32.4	yes	30
71.48	even	7		5041.2.a.l.1.5		15
213.32	odd	14		639.2.j.c.316.2		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.2.d.a.20.4	✓	30	1.1	even	1	trivial
71.2.d.a.32.4	yes	30	71.32	even	7	inner
639.2.j.c.91.2		30	3.2	odd	2
639.2.j.c.316.2		30	213.32	odd	14
5041.2.a.l.1.5		15	71.48	even	7
5041.2.a.m.1.5		15	71.23	odd	14