Embedded newform 490.2.w.a.467.17

• Label: 490.2.w.a.467.17
• Level: $490$
• Weight: $2$
• Character: 490.467
• 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			467.17
Character	\(\chi\)	\(=\)	490.467
Dual form			490.2.w.a.213.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.757972 - 0.652287i) q^{2} +(-2.15321 - 1.13800i) q^{3} +(0.149042 - 0.988831i) q^{4} +(-1.64676 - 1.51268i) q^{5} +(-2.37438 + 0.541936i) q^{6} +(-2.63596 - 0.227416i) q^{7} +(-0.532032 - 0.846724i) q^{8} +(1.65130 + 2.42201i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 12 q^{5} - 28 q^{6} - 8 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{17}{42}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.757972	−	0.652287i	0.535967	−	0.461237i
							\(3\)	−2.15321	−	1.13800i	−1.24316	−	0.657027i	−0.289285	−	0.957243i	\(-0.593418\pi\)
−0.953871	+	0.300216i	\(0.902941\pi\)
\(5\)	−1.64676	−	1.51268i	−0.736452	−	0.676490i
							\(7\)	−2.63596	−	0.227416i	−0.996299	−	0.0859551i
\(11\)	3.22385	+	2.19798i	0.972027	+	0.662717i								0.941569	−	0.336819i	\(-0.109351\pi\)
							0.0304580	+	0.999536i	\(0.490303\pi\)
\(13\)	−1.32759	−	0.464544i	−0.368207	−	0.128841i	0.139832	−	0.990175i	\(-0.455344\pi\)
							−0.508039	+	0.861334i	\(0.669629\pi\)
\(17\)	0.920827	+	2.11056i	0.223333	+	0.511885i	0.991769	−	0.128042i	\(-0.0408692\pi\)
							−0.768435	+	0.639927i	\(0.778964\pi\)
\(19\)	−1.07174	+	1.85630i	−0.245873	+	0.425865i	−0.962377	−	0.271718i	\(-0.912408\pi\)
							0.716503	+	0.697584i	\(0.245741\pi\)
\(23\)	−4.35218	−	1.89884i	−0.907493	−	0.395935i	−0.106295	−	0.994335i	\(-0.533899\pi\)
							−0.801198	+	0.598399i	\(0.795804\pi\)
\(29\)	3.62725	−	2.89264i	0.673564	−	0.537149i	−0.225897	−	0.974151i	\(-0.572531\pi\)
							0.899460	+	0.437002i	\(0.143960\pi\)
\(31\)	−4.55708	+	2.63103i	−0.818476	+	0.472547i	−0.849891	−	0.526959i	\(-0.823332\pi\)
							0.0314146	+	0.999506i	\(0.489999\pi\)
\(37\)	0.790883	+	0.583698i	0.130020	+	0.0959594i	0.656889	−	0.753987i	\(-0.271872\pi\)
							−0.526869	+	0.849947i	\(0.676634\pi\)
\(41\)	−7.90570	−	1.80442i	−1.23466	−	0.281804i	−0.445104	−	0.895479i	\(-0.646833\pi\)
							−0.789559	+	0.613675i	\(0.789690\pi\)
\(43\)	−5.71404	−	3.59037i	−0.871383	−	0.547526i	0.0205011	−	0.999790i	\(-0.493474\pi\)
							−0.891884	+	0.452263i	\(0.850617\pi\)
\(47\)	−3.70329	−	4.30330i	−0.540180	−	0.627701i	0.419851	−	0.907593i	\(-0.362082\pi\)
							−0.960031	+	0.279892i	\(0.909701\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.2.w.a.467.17	yes	672
5.3	odd	4	inner	490.2.w.a.173.12	yes	672
49.17	odd	42	inner	490.2.w.a.17.12	✓	672
245.213	even	84	inner	490.2.w.a.213.17	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.w.a.17.12	✓	672	49.17	odd	42	inner
490.2.w.a.173.12	yes	672	5.3	odd	4	inner
490.2.w.a.213.17	yes	672	245.213	even	84	inner
490.2.w.a.467.17	yes	672	1.1	even	1	trivial