Embedded newform 245.2.s.a.118.4

• Label: 245.2.s.a.118.4
• Level: $245$
• Weight: $2$
• Character: 245.118
• 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			118.4
Character	\(\chi\)	\(=\)	245.118
Dual form			245.2.s.a.27.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.27213 + 0.795054i) q^{2} +(-1.66209 - 1.04436i) q^{3} +(2.96682 - 2.36596i) q^{4} +(-1.11505 + 1.93821i) q^{5} +(4.60681 + 1.05147i) q^{6} +(1.15495 - 2.38036i) q^{7} +(-2.29851 + 3.65806i) q^{8} +(0.370199 + 0.768725i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 10 q^{2} - 14 q^{3} - 14 q^{5} - 56 q^{6} - 18 q^{7} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.27213	+	0.795054i	−1.60664	+	0.562188i	−0.976729	−	0.214478i	\(-0.931195\pi\)
							−0.629913	+	0.776666i	\(0.716909\pi\)
\(3\)	−1.66209	−	1.04436i	−0.959607	−	0.602961i	−0.0414340	−	0.999141i	\(-0.513193\pi\)
							−0.918173	+	0.396180i	\(0.870335\pi\)
\(5\)	−1.11505	+	1.93821i	−0.498666	+	0.866794i
							\(7\)	1.15495	−	2.38036i	0.436529	−	0.899690i
\(11\)	3.44400	+	1.65854i	1.03840	+	0.500069i								0.873798	−	0.486290i	\(-0.161650\pi\)
							0.164607	+	0.986359i	\(0.447364\pi\)
\(13\)	−0.443151	+	0.155065i	−0.122908	+	0.0430074i	−0.391032	−	0.920377i	\(-0.627882\pi\)
							0.268124	+	0.963384i	\(0.413596\pi\)
\(17\)	−4.96562	+	0.559490i	−1.20434	+	0.135696i	−0.691235	−	0.722630i	\(-0.742933\pi\)
							−0.513104	+	0.858327i	\(0.671504\pi\)
\(19\)	−6.30901			−1.44739			−0.723693	−	0.690122i	\(-0.757557\pi\)
							−0.723693	+	0.690122i	\(0.757557\pi\)
\(23\)	−0.290766	+	2.58062i	−0.0606290	+	0.538097i	0.926282	+	0.376831i	\(0.122986\pi\)
							−0.986911	+	0.161266i	\(0.948442\pi\)
\(29\)	−6.19979	−	4.94417i	−1.15127	−	0.918109i	−0.153726	−	0.988113i	\(-0.549127\pi\)
							−0.997547	+	0.0700041i	\(0.977699\pi\)
\(31\)			4.17280i			0.749458i	0.927134	+	0.374729i	\(0.122264\pi\)
							−0.927134	+	0.374729i	\(0.877736\pi\)
\(37\)	0.497357	+	4.41416i	0.0817649	+	0.725684i	0.965982	+	0.258608i	\(0.0832639\pi\)
							−0.884217	+	0.467076i	\(0.845308\pi\)
\(41\)	−4.43529	+	1.01233i	−0.692676	+	0.158099i	−0.554345	−	0.832287i	\(-0.687031\pi\)
							−0.138331	+	0.990386i	\(0.544174\pi\)
\(43\)	−4.01059	+	2.52002i	−0.611609	+	0.384299i	−0.801926	−	0.597423i	\(-0.796191\pi\)
							0.190317	+	0.981723i	\(0.439048\pi\)
\(47\)	1.01363	+	2.89678i	0.147853	+	0.422539i	0.993862	−	0.110630i	\(-0.0352868\pi\)
							−0.846009	+	0.533169i	\(0.821001\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.s.a.27.4	✓	312	245.27	even	28	inner
245.2.s.a.118.4	yes	312	1.1	even	1	trivial
245.2.s.a.167.4	yes	312	5.2	odd	4	inner
245.2.s.a.223.4	yes	312	49.27	odd	14	inner