Embedded newform 245.2.x.a.208.11

• Label: 245.2.x.a.208.11
• Level: $245$
• Weight: $2$
• Character: 245.208
• 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			208.11
Character	\(\chi\)	\(=\)	245.208
Dual form			245.2.x.a.192.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.165138 + 0.872773i) q^{2} +(0.0256237 - 0.684807i) q^{3} +(1.12729 + 0.442427i) q^{4} +(-2.22776 + 0.192611i) q^{5} +(0.593449 + 0.135451i) q^{6} +(-1.59574 + 2.11036i) q^{7} +(-1.51746 + 2.41502i) q^{8} +(2.52331 + 0.189096i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{11}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.165138	+	0.872773i	−0.116770	+	0.617144i	0.874471	+	0.485078i	\(0.161209\pi\)
							−0.991241	+	0.132066i	\(0.957839\pi\)
\(3\)	0.0256237	−	0.684807i	0.0147938	−	0.395373i	−0.972155	−	0.234339i	\(-0.924707\pi\)
							0.986949	−	0.161034i	\(-0.0514830\pi\)
\(5\)	−2.22776	+	0.192611i	−0.996283	+	0.0861381i
							\(7\)	−1.59574	+	2.11036i	−0.603134	+	0.797640i
\(11\)	0.385192	+	5.14003i	0.116140	+	1.54978i								0.685735	+	0.727851i	\(0.259481\pi\)
							−0.569595	+	0.821925i	\(0.692900\pi\)
\(13\)	1.06627	−	0.373105i	0.295731	−	0.103481i	−0.178336	−	0.983970i	\(-0.557071\pi\)
							0.474066	+	0.880489i	\(0.342786\pi\)
\(17\)	2.08690	+	2.82765i	0.506148	+	0.685806i	0.980944	−	0.194290i	\(-0.0622403\pi\)
							−0.474797	+	0.880096i	\(0.657478\pi\)
\(19\)	3.18720	−	5.52038i	0.731193	−	1.26646i	−0.225181	−	0.974317i	\(-0.572297\pi\)
							0.956374	−	0.292146i	\(-0.0943694\pi\)
\(23\)	−2.40746	−	1.77679i	−0.501990	−	0.370486i	0.313192	−	0.949690i	\(-0.398602\pi\)
							−0.815183	+	0.579204i	\(0.803363\pi\)
\(29\)	−1.81794	−	1.44976i	−0.337583	−	0.269213i	0.439995	−	0.898000i	\(-0.354980\pi\)
							−0.777578	+	0.628787i	\(0.783552\pi\)
\(31\)	−5.59628	+	3.23101i	−1.00512	+	0.580307i	−0.909760	−	0.415135i	\(-0.863734\pi\)
							−0.0953621	+	0.995443i	\(0.530401\pi\)
\(37\)	−6.87036	−	2.99751i	−1.12948	−	0.492787i	−0.249774	−	0.968304i	\(-0.580356\pi\)
							−0.879707	+	0.475517i	\(0.842261\pi\)
\(41\)	2.71331	−	0.619296i	0.423748	−	0.0967178i	−0.00532841	−	0.999986i	\(-0.501696\pi\)
							0.429077	+	0.903268i	\(0.358839\pi\)
\(43\)	3.91470	−	2.45977i	0.596986	−	0.375111i	−0.199384	−	0.979921i	\(-0.563894\pi\)
							0.796369	+	0.604811i	\(0.206751\pi\)
\(47\)	−0.681094	−	0.128870i	−0.0993477	−	0.0187976i	0.136000	−	0.990709i	\(-0.456575\pi\)
							−0.235348	+	0.971911i	\(0.575623\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.208.11	yes	624
5.2	odd	4	inner	245.2.x.a.12.11	✓	624
49.45	odd	42	inner	245.2.x.a.143.11	yes	624
245.192	even	84	inner	245.2.x.a.192.11	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.12.11	✓	624	5.2	odd	4	inner
245.2.x.a.143.11	yes	624	49.45	odd	42	inner
245.2.x.a.192.11	yes	624	245.192	even	84	inner
245.2.x.a.208.11	yes	624	1.1	even	1	trivial