Embedded newform 245.2.s.a.118.21

• Label: 245.2.s.a.118.21
• 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			118.21
Character	\(\chi\)	\(=\)	245.118
Dual form			245.2.s.a.27.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.61654 - 0.565653i) q^{2} +(-1.61562 - 1.01516i) q^{3} +(0.729584 - 0.581824i) q^{4} +(2.01094 - 0.977810i) q^{5} +(-3.18595 - 0.727172i) q^{6} +(1.92380 - 1.81631i) q^{7} +(-0.972077 + 1.54705i) q^{8} +(0.278023 + 0.577322i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 10 q^{2} - 14 q^{3} - 14 q^{5} - 56 q^{6} - 18 q^{7} - 6 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.61654	−	0.565653i	1.14307	−	0.399977i	0.308687	−	0.951164i	\(-0.400111\pi\)
							0.834382	+	0.551187i	\(0.185825\pi\)
\(3\)	−1.61562	−	1.01516i	−0.932778	−	0.586104i	−0.0223410	−	0.999750i	\(-0.507112\pi\)
							−0.910437	+	0.413647i	\(0.864255\pi\)
\(5\)	2.01094	−	0.977810i	0.899321	−	0.437290i
							\(7\)	1.92380	−	1.81631i	0.727127	−	0.686502i
\(11\)	−4.10833	−	1.97847i	−1.23871	−	0.596530i								−0.304245	−	0.952594i	\(-0.598404\pi\)
							−0.934462	+	0.356064i	\(0.884118\pi\)
\(13\)	5.29554	−	1.85299i	1.46872	−	0.513927i	0.526864	−	0.849950i	\(-0.323368\pi\)
							0.941854	+	0.336023i	\(0.109082\pi\)
\(17\)	−2.86306	+	0.322590i	−0.694395	+	0.0782395i	−0.452105	−	0.891964i	\(-0.649327\pi\)
							−0.242289	+	0.970204i	\(0.577898\pi\)
\(19\)	2.58217			0.592391			0.296196	−	0.955127i	\(-0.404282\pi\)
							0.296196	+	0.955127i	\(0.404282\pi\)
\(23\)	−0.798874	+	7.09020i	−0.166577	+	1.47841i	0.580821	+	0.814032i	\(0.302732\pi\)
							−0.747397	+	0.664377i	\(0.768697\pi\)
\(29\)	4.39018	+	3.50105i	0.815236	+	0.650129i	0.939664	−	0.342100i	\(-0.111138\pi\)
							−0.124428	+	0.992229i	\(0.539710\pi\)
\(31\)		−	0.00527715i		−	0.000947804i	−1.00000		0.000473902i	\(-0.999849\pi\)
							1.00000		0.000473902i	\(-0.000150848\pi\)
\(37\)	0.459367	+	4.07699i	0.0755194	+	0.670253i	0.973256	+	0.229722i	\(0.0737816\pi\)
							−0.897737	+	0.440532i	\(0.854790\pi\)
\(41\)	5.77966	−	1.31917i	0.902631	−	0.206020i	0.254073	−	0.967185i	\(-0.418230\pi\)
							0.648558	+	0.761165i	\(0.275372\pi\)
\(43\)	−5.93007	+	3.72611i	−0.904327	+	0.568226i	−0.901983	−	0.431771i	\(-0.857889\pi\)
							−0.00234341	+	0.999997i	\(0.500746\pi\)
\(47\)	0.0854191	+	0.244114i	0.0124597	+	0.0356077i	0.949945	−	0.312416i	\(-0.101138\pi\)
							−0.937486	+	0.348024i	\(0.886853\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.s.a.118.21	yes	312
5.2	odd	4	inner	245.2.s.a.167.21	yes	312
49.27	odd	14	inner	245.2.s.a.223.21	yes	312
245.27	even	28	inner	245.2.s.a.27.21	✓	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.s.a.27.21	✓	312	245.27	even	28	inner
245.2.s.a.118.21	yes	312	1.1	even	1	trivial
245.2.s.a.167.21	yes	312	5.2	odd	4	inner
245.2.s.a.223.21	yes	312	49.27	odd	14	inner