Embedded newform 245.2.x.a.17.21

• Label: 245.2.x.a.17.21
• Level: $245$
• Weight: $2$
• Character: 245.17
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.x (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(624\)
Relative dimension:	\(26\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			17.21
Character	\(\chi\)	\(=\)	245.17
Dual form			245.2.x.a.173.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.10637 - 1.28562i) q^{2} +(1.04169 + 1.97097i) q^{3} +(-0.130691 - 0.867081i) q^{4} +(-1.87221 - 1.22264i) q^{5} +(3.68642 + 0.841402i) q^{6} +(2.58701 + 0.554395i) q^{7} +(1.61299 + 1.01351i) q^{8} +(-1.10966 + 1.62757i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{25}{42}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.10637	−	1.28562i	0.782320	−	0.909072i	−0.215402	−	0.976525i	\(-0.569106\pi\)
							0.997722	+	0.0674529i	\(0.0214872\pi\)
\(3\)	1.04169	+	1.97097i	0.601420	+	1.13794i	0.977417	+	0.211321i	\(0.0677767\pi\)
							−0.375997	+	0.926621i	\(0.622700\pi\)
\(5\)	−1.87221	−	1.22264i	−0.837276	−	0.546780i
							\(7\)	2.58701	+	0.554395i	0.977800	+	0.209542i
\(11\)	−0.325668	+	0.222037i	−0.0981927	+	0.0669466i								−0.611421	−	0.791305i	\(-0.709402\pi\)
							0.513228	+	0.858252i	\(0.328449\pi\)
\(13\)	0.523756	+	1.49681i	0.145264	+	0.415140i	0.993409	−	0.114624i	\(-0.0365663\pi\)
							−0.848145	+	0.529764i	\(0.822281\pi\)
\(17\)	−6.65737	−	2.90458i	−1.61465	−	0.704464i	−0.617993	−	0.786183i	\(-0.712054\pi\)
							−0.996655	+	0.0817194i	\(0.973959\pi\)
\(19\)	−2.53768	−	4.39539i	−0.582184	−	1.00837i	−0.995220	−	0.0976573i	\(-0.968865\pi\)
							0.413036	−	0.910715i	\(-0.364468\pi\)
\(23\)	0.947297	+	2.17123i	0.197525	+	0.452732i	0.986932	−	0.161138i	\(-0.0515165\pi\)
							−0.789407	+	0.613870i	\(0.789612\pi\)
\(29\)	−4.74109	−	3.78089i	−0.880399	−	0.702094i	0.0750728	−	0.997178i	\(-0.476081\pi\)
							−0.955471	+	0.295084i	\(0.904653\pi\)
\(31\)	−3.05869	−	1.76594i	−0.549357	−	0.317172i	0.199505	−	0.979897i	\(-0.436066\pi\)
							−0.748863	+	0.662725i	\(0.769400\pi\)
\(37\)	0.881230	+	1.19402i	0.144873	+	0.196296i	0.871026	−	0.491237i	\(-0.163455\pi\)
							−0.726152	+	0.687534i	\(0.758693\pi\)
\(41\)	−6.67214	+	1.52287i	−1.04201	+	0.237833i	−0.709088	−	0.705120i	\(-0.750893\pi\)
							−0.332925	+	0.942953i	\(0.608036\pi\)
\(43\)	−3.52905	−	5.61645i	−0.538175	−	0.856501i	0.461353	−	0.887217i	\(-0.347364\pi\)
							−0.999528	+	0.0307155i	\(0.990221\pi\)
\(47\)	−5.20546	−	4.47966i	−0.759295	−	0.653426i	0.184921	−	0.982753i	\(-0.440797\pi\)
							−0.944216	+	0.329327i	\(0.893178\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.17.21	✓	624
5.3	odd	4	inner	245.2.x.a.213.6	yes	624
49.26	odd	42	inner	245.2.x.a.222.6	yes	624
245.173	even	84	inner	245.2.x.a.173.21	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.17.21	✓	624	1.1	even	1	trivial
245.2.x.a.173.21	yes	624	245.173	even	84	inner
245.2.x.a.213.6	yes	624	5.3	odd	4	inner
245.2.x.a.222.6	yes	624	49.26	odd	42	inner