Embedded newform 245.2.x.a.208.1

• Label: 245.2.x.a.208.1
• 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			208.1
Character	\(\chi\)	\(=\)	245.208
Dual form			245.2.x.a.192.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.514212 + 2.71768i) q^{2} +(0.0531741 - 1.42111i) q^{3} +(-5.25961 - 2.06425i) q^{4} +(-2.17848 - 0.504189i) q^{5} +(3.83477 + 0.875262i) q^{6} +(2.44292 - 1.01595i) q^{7} +(5.37142 - 8.54857i) q^{8} +(0.974891 + 0.0730580i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.514212	+	2.71768i	−0.363603	+	1.92169i	0.0236610	+	0.999720i	\(0.492468\pi\)
							−0.387264	+	0.921969i	\(0.626580\pi\)
\(3\)	0.0531741	−	1.42111i	0.0307001	−	0.820477i	−0.895861	−	0.444334i	\(-0.853440\pi\)
							0.926561	−	0.376143i	\(-0.122750\pi\)
\(5\)	−2.17848	−	0.504189i	−0.974248	−	0.225480i
							\(7\)	2.44292	−	1.01595i	0.923337	−	0.383991i
\(11\)	−0.0463726	−	0.618799i	−0.0139819	−	0.186575i								−0.999822	−	0.0188852i	\(-0.993988\pi\)
							0.985840	−	0.167690i	\(-0.0536307\pi\)
\(13\)	0.334260	−	0.116963i	0.0927070	−	0.0324396i	−0.283527	−	0.958964i	\(-0.591505\pi\)
							0.376234	+	0.926525i	\(0.377219\pi\)
\(17\)	−1.29584	−	1.75580i	−0.314287	−	0.425843i	0.618770	−	0.785572i	\(-0.287631\pi\)
							−0.933057	+	0.359729i	\(0.882869\pi\)
\(19\)	3.50827	−	6.07651i	0.804853	−	1.39405i	−0.111537	−	0.993760i	\(-0.535577\pi\)
							0.916390	−	0.400287i	\(-0.131089\pi\)
\(23\)	−5.29739	−	3.90965i	−1.10458	−	0.815219i	−0.120461	−	0.992718i	\(-0.538437\pi\)
							−0.984121	+	0.177499i	\(0.943199\pi\)
\(29\)	0.477635	+	0.380901i	0.0886945	+	0.0707315i	0.666833	−	0.745207i	\(-0.267649\pi\)
							−0.578139	+	0.815938i	\(0.696221\pi\)
\(31\)	3.02044	−	1.74385i	0.542488	−	0.313205i	−0.203599	−	0.979054i	\(-0.565264\pi\)
							0.746087	+	0.665849i	\(0.231930\pi\)
\(37\)	0.631460	+	0.275503i	0.103811	+	0.0452924i	0.451156	−	0.892445i	\(-0.351012\pi\)
							−0.347345	+	0.937738i	\(0.612917\pi\)
\(41\)	−2.54010	+	0.579762i	−0.396697	+	0.0905436i	−0.416216	−	0.909266i	\(-0.636644\pi\)
							0.0195183	+	0.999810i	\(0.493787\pi\)
\(43\)	−6.37745	+	4.00722i	−0.972552	+	0.611095i	−0.921820	−	0.387617i	\(-0.873298\pi\)
							−0.0507316	+	0.998712i	\(0.516155\pi\)
\(47\)	4.39813	+	0.832171i	0.641533	+	0.121385i	0.496492	−	0.868041i	\(-0.334621\pi\)
							0.145041	+	0.989426i	\(0.453669\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.208.1	yes	624
5.2	odd	4	inner	245.2.x.a.12.1	✓	624
49.45	odd	42	inner	245.2.x.a.143.1	yes	624
245.192	even	84	inner	245.2.x.a.192.1	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.12.1	✓	624	5.2	odd	4	inner
245.2.x.a.143.1	yes	624	49.45	odd	42	inner
245.2.x.a.192.1	yes	624	245.192	even	84	inner
245.2.x.a.208.1	yes	624	1.1	even	1	trivial