Embedded newform 245.2.x.a.103.16

• Label: 245.2.x.a.103.16
• Level: $245$
• Weight: $2$
• Character: 245.103
• 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			103.16
Character	\(\chi\)	\(=\)	245.103
Dual form			245.2.x.a.157.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.358222 - 0.485374i) q^{2} +(-0.0479283 + 0.00906852i) q^{3} +(0.482246 + 1.56340i) q^{4} +(-2.11468 + 0.726738i) q^{5} +(-0.0127674 + 0.0265117i) q^{6} +(-0.483606 + 2.60118i) q^{7} +(2.07038 + 0.724458i) q^{8} +(-2.79041 + 1.09515i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.358222	−	0.485374i	0.253302	−	0.343211i	−0.659434	−	0.751763i	\(-0.729204\pi\)
							0.912735	+	0.408551i	\(0.133966\pi\)
\(3\)	−0.0479283	+	0.00906852i	−0.0276714	+	0.00523571i	−0.199729	−	0.979851i	\(-0.564006\pi\)
							0.172057	+	0.985087i	\(0.444959\pi\)
\(5\)	−2.11468	+	0.726738i	−0.945712	+	0.325007i
							\(7\)	−0.483606	+	2.60118i	−0.182786	+	0.983153i
\(11\)	−0.343394	+	0.874953i	−0.103537	+	0.263808i								−0.973250	−	0.229751i	\(-0.926209\pi\)
							0.869712	+	0.493559i	\(0.164304\pi\)
\(13\)	5.89181	+	0.663848i	1.63410	+	0.184118i	0.880775	−	0.473535i	\(-0.157022\pi\)
							0.753320	+	0.657654i	\(0.228451\pi\)
\(17\)	−4.55630	−	0.170485i	−1.10506	−	0.0413486i	−0.521376	−	0.853327i	\(-0.674581\pi\)
							−0.583688	+	0.811978i	\(0.698391\pi\)
\(19\)	4.20317	−	7.28010i	0.964273	−	1.67017i	0.252717	−	0.967540i	\(-0.418676\pi\)
							0.711556	−	0.702629i	\(-0.247991\pi\)
\(23\)	0.248707	+	6.64685i	0.0518591	+	1.38596i	0.745914	+	0.666042i	\(0.232013\pi\)
							−0.694055	+	0.719922i	\(0.744178\pi\)
\(29\)	1.80121	−	0.411115i	0.334477	−	0.0763422i	−0.0519841	−	0.998648i	\(-0.516555\pi\)
							0.386461	+	0.922306i	\(0.373697\pi\)
\(31\)	3.48192	−	2.01029i	0.625371	−	0.361058i	−0.153586	−	0.988135i	\(-0.549082\pi\)
							0.778957	+	0.627077i	\(0.215749\pi\)
\(37\)	3.67030	−	1.93981i	0.603394	−	0.318903i	−0.137524	−	0.990498i	\(-0.543914\pi\)
							0.740918	+	0.671595i	\(0.234391\pi\)
\(41\)	−0.726736	−	1.50908i	−0.113497	−	0.235679i	0.836481	−	0.547996i	\(-0.184609\pi\)
							−0.949978	+	0.312317i	\(0.898895\pi\)
\(43\)	1.02076	+	2.91716i	0.155664	+	0.444863i	0.995133	−	0.0985460i	\(-0.0314191\pi\)
							−0.839468	+	0.543409i	\(0.817133\pi\)
\(47\)	−3.22383	−	2.37929i	−0.470243	−	0.347055i	0.332779	−	0.943005i	\(-0.392014\pi\)
							−0.803022	+	0.595950i	\(0.796776\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.103.16	✓	624
5.2	odd	4	inner	245.2.x.a.152.16	yes	624
49.10	odd	42	inner	245.2.x.a.108.16	yes	624
245.157	even	84	inner	245.2.x.a.157.16	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.103.16	✓	624	1.1	even	1	trivial
245.2.x.a.108.16	yes	624	49.10	odd	42	inner
245.2.x.a.152.16	yes	624	5.2	odd	4	inner
245.2.x.a.157.16	yes	624	245.157	even	84	inner