Embedded newform 245.2.x.a.173.4

• Label: 245.2.x.a.173.4
• Level: $245$
• Weight: $2$
• Character: 245.173
• 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			173.4
Character	\(\chi\)	\(=\)	245.173
Dual form			245.2.x.a.17.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41047 - 1.63900i) q^{2} +(-1.54126 + 2.91620i) q^{3} +(-0.398801 + 2.64587i) q^{4} +(0.229203 + 2.22429i) q^{5} +(6.95356 - 1.58710i) q^{6} +(-1.10254 + 2.40508i) q^{7} +(1.23725 - 0.777415i) q^{8} +(-4.43880 - 6.51053i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{17}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41047	−	1.63900i	−0.997355	−	1.15895i	−0.987279	−	0.158999i	\(-0.949173\pi\)
							−0.0100764	−	0.999949i	\(-0.503207\pi\)
\(3\)	−1.54126	+	2.91620i	−0.889846	+	1.68367i	−0.174787	+	0.984606i	\(0.555924\pi\)
							−0.715059	+	0.699064i	\(0.753600\pi\)
\(5\)	0.229203	+	2.22429i	0.102503	+	0.994733i
							\(7\)	−1.10254	+	2.40508i	−0.416721	+	0.909035i
\(11\)	−1.30512	−	0.889817i	−0.393509	−	0.268290i								0.350360	−	0.936615i	\(-0.386059\pi\)
							−0.743869	+	0.668325i	\(0.767011\pi\)
\(13\)	1.74957	−	4.99999i	0.485244	−	1.38675i	−0.397807	−	0.917469i	\(-0.630228\pi\)
							0.883050	−	0.469278i	\(-0.155486\pi\)
\(17\)	−2.60638	+	1.13715i	−0.632140	+	0.275800i	−0.691433	−	0.722441i	\(-0.743020\pi\)
							0.0592929	+	0.998241i	\(0.481115\pi\)
\(19\)	0.453967	−	0.786294i	0.104147	−	0.180388i	−0.809242	−	0.587475i	\(-0.800122\pi\)
							0.913389	+	0.407087i	\(0.133455\pi\)
\(23\)	−1.57408	+	3.60783i	−0.328219	+	0.752285i	0.671716	+	0.740809i	\(0.265558\pi\)
							−0.999934	+	0.0114757i	\(0.996347\pi\)
\(29\)	−1.41901	+	1.13162i	−0.263503	+	0.210137i	−0.746325	−	0.665582i	\(-0.768183\pi\)
							0.482822	+	0.875719i	\(0.339612\pi\)
\(31\)	2.29047	−	1.32240i	0.411381	−	0.237511i	−0.280002	−	0.959999i	\(-0.590335\pi\)
							0.691383	+	0.722489i	\(0.257002\pi\)
\(37\)	0.0340207	−	0.0460964i	0.00559297	−	0.00757820i	−0.801792	−	0.597603i	\(-0.796120\pi\)
							0.807385	+	0.590025i	\(0.200882\pi\)
\(41\)	−3.78615	−	0.864165i	−0.591298	−	0.134960i	−0.0836063	−	0.996499i	\(-0.526644\pi\)
							−0.507691	+	0.861539i	\(0.669501\pi\)
\(43\)	−1.89628	+	3.01791i	−0.289180	+	0.460227i	−0.959445	−	0.281894i	\(-0.909037\pi\)
							0.670266	+	0.742121i	\(0.266180\pi\)
\(47\)	−1.71251	+	1.47373i	−0.249795	+	0.214966i	−0.768269	−	0.640128i	\(-0.778881\pi\)
							0.518473	+	0.855094i	\(0.326501\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.173.4	yes	624
5.2	odd	4	inner	245.2.x.a.222.23	yes	624
49.17	odd	42	inner	245.2.x.a.213.23	yes	624
245.17	even	84	inner	245.2.x.a.17.4	✓	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.17.4	✓	624	245.17	even	84	inner
245.2.x.a.173.4	yes	624	1.1	even	1	trivial
245.2.x.a.213.23	yes	624	49.17	odd	42	inner
245.2.x.a.222.23	yes	624	5.2	odd	4	inner