Embedded newform 490.2.w.a.103.25

• Label: 490.2.w.a.103.25
• Level: $490$
• Weight: $2$
• Character: 490.103
• Analytic conductor: $3.913$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	490.w (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.91266969904\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(28\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			103.25
Character	\(\chi\)	\(=\)	490.103
Dual form			490.2.w.a.157.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.593820 - 0.804598i) q^{2} +(1.94056 - 0.367174i) q^{3} +(-0.294755 - 0.955573i) q^{4} +(0.178478 - 2.22893i) q^{5} +(0.856917 - 1.77941i) q^{6} +(2.63441 + 0.244673i) q^{7} +(-0.943883 - 0.330279i) q^{8} +(0.838339 - 0.329024i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 12 q^{5} - 28 q^{6} - 8 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.593820	−	0.804598i	0.419894	−	0.568937i
							\(3\)	1.94056	−	0.367174i	1.12038	−	0.211988i	0.407455	−	0.913225i	\(-0.366416\pi\)
0.712928	+	0.701237i	\(0.247369\pi\)
\(5\)	0.178478	−	2.22893i	0.0798180	−	0.996809i
							\(7\)	2.63441	+	0.244673i	0.995715	+	0.0924779i
\(11\)	−0.564085	+	1.43727i	−0.170078	+	0.433352i								−0.990344	−	0.138630i	\(-0.955730\pi\)
							0.820266	+	0.571982i	\(0.193825\pi\)
\(13\)	−2.14816	−	0.242040i	−0.595793	−	0.0671297i	−0.191081	−	0.981574i	\(-0.561199\pi\)
							−0.404711	+	0.914444i	\(0.632628\pi\)
\(17\)	2.71473	+	0.101578i	0.658419	+	0.0246363i	0.364514	−	0.931198i	\(-0.381235\pi\)
							0.293906	+	0.955834i	\(0.405045\pi\)
\(19\)	0.197255	−	0.341656i	0.0452534	−	0.0783812i	−0.842511	−	0.538678i	\(-0.818924\pi\)
							0.887765	+	0.460297i	\(0.152257\pi\)
\(23\)	0.204161	+	5.45631i	0.0425704	+	1.13772i	0.842870	+	0.538118i	\(0.180864\pi\)
							−0.800299	+	0.599601i	\(0.795326\pi\)
\(29\)	−4.56911	+	1.04287i	−0.848463	+	0.193656i	−0.624581	−	0.780960i	\(-0.714730\pi\)
							−0.223882	+	0.974616i	\(0.571873\pi\)
\(31\)	1.90489	−	1.09979i	0.342128	−	0.197528i	−0.319085	−	0.947726i	\(-0.603375\pi\)
							0.661213	+	0.750198i	\(0.270042\pi\)
\(37\)	7.58302	−	4.00774i	1.24664	−	0.658869i	0.291940	−	0.956436i	\(-0.405699\pi\)
							0.954701	+	0.297567i	\(0.0961753\pi\)
\(41\)	−3.78644	−	7.86262i	−0.591342	−	1.22794i	−0.955056	−	0.296425i	\(-0.904206\pi\)
							0.363714	−	0.931511i	\(-0.381509\pi\)
\(43\)	3.37865	+	9.65563i	0.515239	+	1.47247i	0.848275	+	0.529555i	\(0.177641\pi\)
							−0.333036	+	0.942914i	\(0.608073\pi\)
\(47\)	2.60478	+	1.92241i	0.379946	+	0.280413i	0.767004	−	0.641643i	\(-0.221747\pi\)
							−0.387058	+	0.922055i	\(0.626509\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.2.w.a.103.25	✓	672
5.2	odd	4	inner	490.2.w.a.397.18	yes	672
49.10	odd	42	inner	490.2.w.a.353.18	yes	672
245.157	even	84	inner	490.2.w.a.157.25	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.w.a.103.25	✓	672	1.1	even	1	trivial
490.2.w.a.157.25	yes	672	245.157	even	84	inner
490.2.w.a.353.18	yes	672	49.10	odd	42	inner
490.2.w.a.397.18	yes	672	5.2	odd	4	inner