Embedded newform 245.2.s.a.167.4

• Label: 245.2.s.a.167.4
• Level: $245$
• Weight: $2$
• Character: 245.167
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.s (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(26\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			167.4
Character	\(\chi\)	\(=\)	245.167
Dual form			245.2.s.a.223.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.795054 - 2.27213i) q^{2} +(-1.04436 + 1.66209i) q^{3} +(-2.96682 + 2.36596i) q^{4} +(-1.84558 - 1.26246i) q^{5} +(4.60681 + 1.05147i) q^{6} +(2.38036 + 1.15495i) q^{7} +(3.65806 + 2.29851i) q^{8} +(-0.370199 - 0.768725i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 10 q^{2} - 14 q^{3} - 14 q^{5} - 56 q^{6} - 18 q^{7} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.795054	−	2.27213i	−0.562188	−	1.60664i	−0.776666	−	0.629913i	\(-0.783091\pi\)
							0.214478	−	0.976729i	\(-0.431195\pi\)
\(3\)	−1.04436	+	1.66209i	−0.602961	+	0.959607i	0.396180	+	0.918173i	\(0.370335\pi\)
							−0.999141	+	0.0414340i	\(0.986807\pi\)
\(5\)	−1.84558	−	1.26246i	−0.825371	−	0.564591i
							\(7\)	2.38036	+	1.15495i	0.899690	+	0.436529i
\(11\)	3.44400	+	1.65854i	1.03840	+	0.500069i								0.873798	−	0.486290i	\(-0.161650\pi\)
							0.164607	+	0.986359i	\(0.447364\pi\)
\(13\)	0.155065	+	0.443151i	0.0430074	+	0.122908i	0.963384	−	0.268124i	\(-0.0864038\pi\)
							−0.920377	+	0.391032i	\(0.872118\pi\)
\(17\)	−0.559490	−	4.96562i	−0.135696	−	1.20434i	−0.858327	−	0.513104i	\(-0.828496\pi\)
							0.722630	−	0.691235i	\(-0.242933\pi\)
\(19\)	6.30901			1.44739			0.723693	−	0.690122i	\(-0.242443\pi\)
							0.723693	+	0.690122i	\(0.242443\pi\)
\(23\)	2.58062	+	0.290766i	0.538097	+	0.0606290i	0.376831	−	0.926282i	\(-0.377014\pi\)
							0.161266	+	0.986911i	\(0.448442\pi\)
\(29\)	6.19979	+	4.94417i	1.15127	+	0.918109i	0.997547	−	0.0700041i	\(-0.0223012\pi\)
							0.153726	+	0.988113i	\(0.450873\pi\)
\(31\)			4.17280i			0.749458i	0.927134	+	0.374729i	\(0.122264\pi\)
							−0.927134	+	0.374729i	\(0.877736\pi\)
\(37\)	−4.41416	+	0.497357i	−0.725684	+	0.0817649i	−0.467076	−	0.884217i	\(-0.654692\pi\)
							−0.258608	+	0.965982i	\(0.583264\pi\)
\(41\)	−4.43529	+	1.01233i	−0.692676	+	0.158099i	−0.554345	−	0.832287i	\(-0.687031\pi\)
							−0.138331	+	0.990386i	\(0.544174\pi\)
\(43\)	2.52002	+	4.01059i	0.384299	+	0.611609i	0.981723	−	0.190317i	\(-0.0609517\pi\)
							−0.597423	+	0.801926i	\(0.703809\pi\)
\(47\)	−2.89678	+	1.01363i	−0.422539	+	0.147853i	−0.533169	−	0.846009i	\(-0.678999\pi\)
							0.110630	+	0.993862i	\(0.464713\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.s.a.167.4	yes	312
5.3	odd	4	inner	245.2.s.a.118.4	yes	312
49.27	odd	14	inner	245.2.s.a.27.4	✓	312
245.223	even	28	inner	245.2.s.a.223.4	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.s.a.27.4	✓	312	49.27	odd	14	inner
245.2.s.a.118.4	yes	312	5.3	odd	4	inner
245.2.s.a.167.4	yes	312	1.1	even	1	trivial
245.2.s.a.223.4	yes	312	245.223	even	28	inner