Embedded newform 980.2.x.j.667.5

• Label: 980.2.x.j.667.5
• Level: $980$
• Weight: $2$
• Character: 980.667
• Analytic conductor: $7.825$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.x (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82533939809\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			667.5
Character	\(\chi\)	\(=\)	980.667
Dual form			980.2.x.j.263.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.636013 - 1.26313i) q^{2} +(-0.378919 + 1.41415i) q^{3} +(-1.19098 + 1.60673i) q^{4} +(-2.12761 + 0.687942i) q^{5} +(2.02724 - 0.420792i) q^{6} +(2.78698 + 0.482452i) q^{8} +(0.741848 + 0.428306i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{2} + 16 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)	\(491\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.636013	−	1.26313i	−0.449729	−	0.893165i
							\(3\)	−0.378919	+	1.41415i	−0.218769	+	0.816457i	0.766036	+	0.642797i	\(0.222226\pi\)
−0.984806	+	0.173660i	\(0.944441\pi\)
\(5\)	−2.12761	+	0.687942i	−0.951497	+	0.307657i
							\(7\)		0			0
\(11\)	2.08340	−	1.20285i	0.628170	−	0.362674i								−0.151873	−	0.988400i	\(-0.548531\pi\)
							0.780043	+	0.625726i	\(0.215197\pi\)
\(13\)	2.78231	−	2.78231i	0.771674	−	0.771674i	−0.206725	−	0.978399i	\(-0.566281\pi\)
							0.978399	+	0.206725i	\(0.0662805\pi\)
\(17\)	−0.137577	+	0.513443i	−0.0333673	+	0.124528i	−0.980601	−	0.196017i	\(-0.937199\pi\)
							0.947233	+	0.320545i	\(0.103866\pi\)
\(19\)	−1.35048	+	2.33910i	−0.309821	+	0.536626i	−0.978323	−	0.207085i	\(-0.933602\pi\)
							0.668502	+	0.743710i	\(0.266936\pi\)
\(23\)	2.17466	−	0.582698i	0.453448	−	0.121501i	−0.0248645	−	0.999691i	\(-0.507915\pi\)
							0.478312	+	0.878190i	\(0.341249\pi\)
\(29\)			6.46566i			1.20064i	0.799759	+	0.600322i	\(0.204961\pi\)
							−0.799759	+	0.600322i	\(0.795039\pi\)
\(31\)	−8.37282	+	4.83405i	−1.50380	+	0.868221i	−0.503813	+	0.863813i	\(0.668070\pi\)
							−0.999990	+	0.00440848i	\(0.998597\pi\)
\(37\)	3.84256	−	1.02961i	0.631712	−	0.169267i	0.0712658	−	0.997457i	\(-0.477296\pi\)
							0.560447	+	0.828191i	\(0.310629\pi\)
\(41\)	8.86070			1.38381			0.691905	−	0.721989i	\(-0.256772\pi\)
							0.691905	+	0.721989i	\(0.256772\pi\)
\(43\)	−1.51081	−	1.51081i	−0.230397	−	0.230397i	0.582462	−	0.812858i	\(-0.302090\pi\)
							−0.812858	+	0.582462i	\(0.802090\pi\)
\(47\)	3.06605	+	11.4426i	0.447229	+	1.66908i	0.709984	+	0.704218i	\(0.248702\pi\)
							−0.262755	+	0.964863i	\(0.584631\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.x.j.667.5		64
4.3	odd	2	inner	980.2.x.j.667.8		64
5.3	odd	4	inner	980.2.x.j.863.3		64
7.2	even	3		980.2.k.i.687.5	✓	32
7.3	odd	6	inner	980.2.x.j.67.15		64
7.4	even	3	inner	980.2.x.j.67.16		64
7.5	odd	6		980.2.k.i.687.6	yes	32
7.6	odd	2	inner	980.2.x.j.667.6		64
20.3	even	4	inner	980.2.x.j.863.16		64
28.3	even	6	inner	980.2.x.j.67.4		64
28.11	odd	6	inner	980.2.x.j.67.3		64
28.19	even	6		980.2.k.i.687.13	yes	32
28.23	odd	6		980.2.k.i.687.14	yes	32
28.27	even	2	inner	980.2.x.j.667.7		64
35.3	even	12	inner	980.2.x.j.263.7		64
35.13	even	4	inner	980.2.x.j.863.4		64
35.18	odd	12	inner	980.2.x.j.263.8		64
35.23	odd	12		980.2.k.i.883.14	yes	32
35.33	even	12		980.2.k.i.883.13	yes	32
140.3	odd	12	inner	980.2.x.j.263.6		64
140.23	even	12		980.2.k.i.883.5	yes	32
140.83	odd	4	inner	980.2.x.j.863.15		64
140.103	odd	12		980.2.k.i.883.6	yes	32
140.123	even	12	inner	980.2.x.j.263.5		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.k.i.687.5	✓	32	7.2	even	3
980.2.k.i.687.6	yes	32	7.5	odd	6
980.2.k.i.687.13	yes	32	28.19	even	6
980.2.k.i.687.14	yes	32	28.23	odd	6
980.2.k.i.883.5	yes	32	140.23	even	12
980.2.k.i.883.6	yes	32	140.103	odd	12
980.2.k.i.883.13	yes	32	35.33	even	12
980.2.k.i.883.14	yes	32	35.23	odd	12
980.2.x.j.67.3		64	28.11	odd	6	inner
980.2.x.j.67.4		64	28.3	even	6	inner
980.2.x.j.67.15		64	7.3	odd	6	inner
980.2.x.j.67.16		64	7.4	even	3	inner
980.2.x.j.263.5		64	140.123	even	12	inner
980.2.x.j.263.6		64	140.3	odd	12	inner
980.2.x.j.263.7		64	35.3	even	12	inner
980.2.x.j.263.8		64	35.18	odd	12	inner
980.2.x.j.667.5		64	1.1	even	1	trivial
980.2.x.j.667.6		64	7.6	odd	2	inner
980.2.x.j.667.7		64	28.27	even	2	inner
980.2.x.j.667.8		64	4.3	odd	2	inner
980.2.x.j.863.3		64	5.3	odd	4	inner
980.2.x.j.863.4		64	35.13	even	4	inner
980.2.x.j.863.15		64	140.83	odd	4	inner
980.2.x.j.863.16		64	20.3	even	4	inner