Embedded newform 245.2.s.a.188.4

• Label: 245.2.s.a.188.4
• Level: $245$
• Weight: $2$
• Character: 245.188
• 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			188.4
Character	\(\chi\)	\(=\)	245.188
Dual form			245.2.s.a.202.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.19532 - 0.247354i) q^{2} +(0.431359 - 1.23275i) q^{3} +(2.80841 + 0.641001i) q^{4} +(2.21325 - 0.318608i) q^{5} +(-1.25190 + 2.59960i) q^{6} +(-0.614798 + 2.57333i) q^{7} +(-1.83633 - 0.642561i) q^{8} +(1.01188 + 0.806951i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 10 q^{2} - 14 q^{3} - 14 q^{5} - 56 q^{6} - 18 q^{7} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.19532	−	0.247354i	−1.55233	−	0.174905i	−0.706250	−	0.707963i	\(-0.749614\pi\)
							−0.846079	+	0.533057i	\(0.821043\pi\)
\(3\)	0.431359	−	1.23275i	0.249045	−	0.711730i	−0.749706	−	0.661771i	\(-0.769805\pi\)
							0.998751	−	0.0499594i	\(-0.0159092\pi\)
\(5\)	2.21325	−	0.318608i	0.989797	−	0.142486i
							\(7\)	−0.614798	+	2.57333i	−0.232372	+	0.972627i
\(11\)	2.84609	+	3.56889i	0.858129	+	1.07606i								0.996325	+	0.0856553i	\(0.0272984\pi\)
							−0.138195	+	0.990405i	\(0.544130\pi\)
\(13\)	−2.63787	−	0.297216i	−0.731613	−	0.0824329i	−0.261704	−	0.965148i	\(-0.584284\pi\)
							−0.469908	+	0.882715i	\(0.655713\pi\)
\(17\)	−2.81511	+	4.48022i	−0.682765	+	1.08661i	0.308048	+	0.951371i	\(0.400324\pi\)
							−0.990812	+	0.135243i	\(0.956819\pi\)
\(19\)	1.22166			0.280268			0.140134	−	0.990133i	\(-0.455247\pi\)
							0.140134	+	0.990133i	\(0.455247\pi\)
\(23\)	6.24916	−	3.92661i	1.30304	−	0.818755i	0.311714	−	0.950176i	\(-0.399097\pi\)
							0.991327	+	0.131421i	\(0.0419540\pi\)
\(29\)	2.84020	−	0.648256i	0.527411	−	0.120378i	0.0494793	−	0.998775i	\(-0.484244\pi\)
							0.477932	+	0.878397i	\(0.341387\pi\)
\(31\)			3.79879i			0.682282i	0.940012	+	0.341141i	\(0.110813\pi\)
							−0.940012	+	0.341141i	\(0.889187\pi\)
\(37\)	−9.54677	−	5.99863i	−1.56948	−	0.986169i	−0.984554	−	0.175082i	\(-0.943981\pi\)
							−0.584925	−	0.811087i	\(-0.698876\pi\)
\(41\)	−3.24418	−	6.73661i	−0.506656	−	1.05208i	−0.984783	−	0.173791i	\(-0.944398\pi\)
							0.478127	−	0.878291i	\(-0.341316\pi\)
\(43\)	−2.31140	−	6.60560i	−0.352485	−	1.00734i	−0.974509	−	0.224347i	\(-0.927975\pi\)
							0.622024	−	0.782998i	\(-0.286310\pi\)
\(47\)	−0.542023	+	4.81059i	−0.0790622	+	0.701697i	0.890184	+	0.455602i	\(0.150576\pi\)
							−0.969246	+	0.246095i	\(0.920853\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.s.a.188.4	yes	312
5.2	odd	4	inner	245.2.s.a.237.23	yes	312
49.6	odd	14	inner	245.2.s.a.153.23	✓	312
245.202	even	28	inner	245.2.s.a.202.4	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.s.a.153.23	✓	312	49.6	odd	14	inner
245.2.s.a.188.4	yes	312	1.1	even	1	trivial
245.2.s.a.202.4	yes	312	245.202	even	28	inner
245.2.s.a.237.23	yes	312	5.2	odd	4	inner