Embedded newform 605.2.m.f.112.2

• Label: 605.2.m.f.112.2
• Level: $605$
• Weight: $2$
• Character: 605.112
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.m (of order \(20\), degree \(8\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(10\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			112.2
Character	\(\chi\)	\(=\)	605.112
Dual form			605.2.m.f.578.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.868868 - 1.70525i) q^{2} +(-0.0409975 + 0.258848i) q^{3} +(-0.977373 + 1.34524i) q^{4} +(-2.23450 + 0.0835971i) q^{5} +(0.477022 - 0.154994i) q^{6} +(-4.53756 + 0.718679i) q^{7} +(-0.637391 - 0.100953i) q^{8} +(2.78785 + 0.905827i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} - 4 q^{5}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.868868	−	1.70525i	−0.614382	−	1.20579i	−0.963246	−	0.268620i	\(-0.913432\pi\)
							0.348864	−	0.937173i	\(-0.386568\pi\)
\(3\)	−0.0409975	+	0.258848i	−0.0236699	+	0.149446i	−0.996693	−	0.0812631i	\(-0.974105\pi\)
							0.973023	+	0.230709i	\(0.0741046\pi\)
\(5\)	−2.23450	+	0.0835971i	−0.999301	+	0.0373858i
							\(7\)	−4.53756	+	0.718679i	−1.71504	+	0.271635i	−0.935142	−	0.354274i	\(-0.884728\pi\)
−0.779896	+	0.625909i	\(0.784728\pi\)
\(11\)		0			0
							\(13\)	1.46048	−	0.744154i	0.405065	−	0.206391i	−0.239574	−	0.970878i	\(-0.577008\pi\)
0.644639	+	0.764487i	\(0.277008\pi\)
\(17\)	1.28378	+	0.654118i	0.311362	+	0.158647i	0.602688	−	0.797977i	\(-0.294096\pi\)
							−0.291326	+	0.956624i	\(0.594096\pi\)
\(19\)	2.92802	−	2.12733i	0.671734	−	0.488043i	−0.198872	−	0.980026i	\(-0.563728\pi\)
							0.870605	+	0.491983i	\(0.163728\pi\)
\(23\)	3.55745	−	3.55745i	0.741780	−	0.741780i	−0.231141	−	0.972920i	\(-0.574246\pi\)
							0.972920	+	0.231141i	\(0.0742458\pi\)
\(29\)	1.47841	+	1.07413i	0.274534	+	0.199460i	0.716530	−	0.697557i	\(-0.245729\pi\)
							−0.441996	+	0.897017i	\(0.645729\pi\)
\(31\)	−1.44391	+	4.44388i	−0.259333	+	0.798145i	0.733612	+	0.679569i	\(0.237833\pi\)
							−0.992945	+	0.118576i	\(0.962167\pi\)
\(37\)	1.61967	+	10.2262i	0.266272	+	1.68118i	0.651730	+	0.758451i	\(0.274044\pi\)
							−0.385458	+	0.922726i	\(0.625956\pi\)
\(41\)	3.77016	+	5.18919i	0.588801	+	0.810415i	0.994626	−	0.103536i	\(-0.0330157\pi\)
							−0.405825	+	0.913951i	\(0.633016\pi\)
\(43\)	4.54091	+	4.54091i	0.692482	+	0.692482i	0.962777	−	0.270295i	\(-0.0871213\pi\)
							−0.270295	+	0.962777i	\(0.587121\pi\)
\(47\)	−8.63553	−	1.36773i	−1.25962	−	0.199504i	−0.509307	−	0.860585i	\(-0.670098\pi\)
							−0.750315	+	0.661081i	\(0.770098\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.m.f.112.2		80
5.3	odd	4	inner	605.2.m.f.233.2		80
11.2	odd	10	inner	605.2.m.f.602.9		80
11.3	even	5	inner	605.2.m.f.282.2		80
11.4	even	5		605.2.e.a.362.9	yes	20
11.5	even	5	inner	605.2.m.f.457.9		80
11.6	odd	10	inner	605.2.m.f.457.2		80
11.7	odd	10		605.2.e.a.362.2	✓	20
11.8	odd	10	inner	605.2.m.f.282.9		80
11.9	even	5	inner	605.2.m.f.602.2		80
11.10	odd	2	inner	605.2.m.f.112.9		80
55.3	odd	20	inner	605.2.m.f.403.9		80
55.8	even	20	inner	605.2.m.f.403.2		80
55.13	even	20	inner	605.2.m.f.118.2		80
55.18	even	20		605.2.e.a.483.9	yes	20
55.28	even	20	inner	605.2.m.f.578.2		80
55.38	odd	20	inner	605.2.m.f.578.9		80
55.43	even	4	inner	605.2.m.f.233.9		80
55.48	odd	20		605.2.e.a.483.2	yes	20
55.53	odd	20	inner	605.2.m.f.118.9		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.e.a.362.2	✓	20	11.7	odd	10
605.2.e.a.362.9	yes	20	11.4	even	5
605.2.e.a.483.2	yes	20	55.48	odd	20
605.2.e.a.483.9	yes	20	55.18	even	20
605.2.m.f.112.2		80	1.1	even	1	trivial
605.2.m.f.112.9		80	11.10	odd	2	inner
605.2.m.f.118.2		80	55.13	even	20	inner
605.2.m.f.118.9		80	55.53	odd	20	inner
605.2.m.f.233.2		80	5.3	odd	4	inner
605.2.m.f.233.9		80	55.43	even	4	inner
605.2.m.f.282.2		80	11.3	even	5	inner
605.2.m.f.282.9		80	11.8	odd	10	inner
605.2.m.f.403.2		80	55.8	even	20	inner
605.2.m.f.403.9		80	55.3	odd	20	inner
605.2.m.f.457.2		80	11.6	odd	10	inner
605.2.m.f.457.9		80	11.5	even	5	inner
605.2.m.f.578.2		80	55.28	even	20	inner
605.2.m.f.578.9		80	55.38	odd	20	inner
605.2.m.f.602.2		80	11.9	even	5	inner
605.2.m.f.602.9		80	11.2	odd	10	inner