Embedded newform 80.6.q.a.29.13

• Label: 80.6.q.a.29.13
• Level: $80$
• Weight: $6$
• Character: 80.29
• Analytic conductor: $12.831$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	80.q (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8307055850\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			29.13
Character	\(\chi\)	\(=\)	80.29
Dual form			80.6.q.a.69.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.30125 - 3.67413i) q^{2} +(-2.38770 + 2.38770i) q^{3} +(5.00149 + 31.6067i) q^{4} +(4.21193 + 55.7428i) q^{5} +(19.0429 - 1.49737i) q^{6} -204.311 q^{7} +(94.6147 - 154.325i) q^{8} +231.598i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 4 q^{4} - 2 q^{5} - 4 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(21\)	\(31\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.30125	−	3.67413i	−0.760361	−	0.649501i
							\(3\)	−2.38770	+	2.38770i	−0.153171	+	0.153171i	−0.779533	−	0.626361i	\(-0.784543\pi\)
0.626361	+	0.779533i	\(0.284543\pi\)
\(5\)	4.21193	+	55.7428i	0.0753454	+	0.997157i
							\(7\)	−204.311			−1.57596			−0.787982	−	0.615698i	\(-0.788874\pi\)
−0.787982	+	0.615698i	\(0.788874\pi\)
\(11\)	7.30519	−	7.30519i	0.0182033	−	0.0182033i	−0.697947	−	0.716150i	\(-0.745903\pi\)
							0.716150	+	0.697947i	\(0.245903\pi\)
\(13\)	822.372	−	822.372i	1.34961	−	1.34961i	0.463537	−	0.886077i	\(-0.346580\pi\)
							0.886077	−	0.463537i	\(-0.153420\pi\)
\(17\)		−	790.697i		−	0.663572i	−0.943355	−	0.331786i	\(-0.892349\pi\)
							0.943355	−	0.331786i	\(-0.107651\pi\)
\(19\)	−939.325	−	939.325i	−0.596942	−	0.596942i	0.342556	−	0.939498i	\(-0.388707\pi\)
							−0.939498	+	0.342556i	\(0.888707\pi\)
\(23\)	−2289.29			−0.902364			−0.451182	−	0.892432i	\(-0.648997\pi\)
							−0.451182	+	0.892432i	\(0.648997\pi\)
\(29\)	−886.548	−	886.548i	−0.195752	−	0.195752i	0.602424	−	0.798176i	\(-0.294202\pi\)
							−0.798176	+	0.602424i	\(0.794202\pi\)
\(31\)	5111.23			0.955258			0.477629	−	0.878562i	\(-0.341496\pi\)
							0.477629	+	0.878562i	\(0.341496\pi\)
\(37\)	−629.784	−	629.784i	−0.0756288	−	0.0756288i	0.668281	−	0.743909i	\(-0.267031\pi\)
							−0.743909	+	0.668281i	\(0.767031\pi\)
\(41\)		−	12446.4i		−	1.15633i	−0.815919	−	0.578166i	\(-0.803768\pi\)
							0.815919	−	0.578166i	\(-0.196232\pi\)
\(43\)	−9649.01	−	9649.01i	−0.795814	−	0.795814i	0.186618	−	0.982432i	\(-0.440247\pi\)
							−0.982432	+	0.186618i	\(0.940247\pi\)
\(47\)		−	3430.22i		−	0.226505i	−0.993566	−	0.113252i	\(-0.963873\pi\)
							0.993566	−	0.113252i	\(-0.0361268\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.q.a.29.13	✓	116
5.4	even	2	inner	80.6.q.a.29.46	yes	116
16.5	even	4	inner	80.6.q.a.69.46	yes	116
80.69	even	4	inner	80.6.q.a.69.13	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.q.a.29.13	✓	116	1.1	even	1	trivial
80.6.q.a.29.46	yes	116	5.4	even	2	inner
80.6.q.a.69.13	yes	116	80.69	even	4	inner
80.6.q.a.69.46	yes	116	16.5	even	4	inner