Embedded newform 245.2.s.a.223.2

• Label: 245.2.s.a.223.2
• Level: $245$
• Weight: $2$
• Character: 245.223
• 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			223.2
Character	\(\chi\)	\(=\)	245.223
Dual form			245.2.s.a.167.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.879310 + 2.51292i) q^{2} +(0.183309 + 0.291735i) q^{3} +(-3.97793 - 3.17230i) q^{4} +(0.772116 - 2.09853i) q^{5} +(-0.894292 + 0.204116i) q^{6} +(0.182770 - 2.63943i) q^{7} +(6.96106 - 4.37392i) q^{8} +(1.25014 - 2.59595i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 10 q^{2} - 14 q^{3} - 14 q^{5} - 56 q^{6} - 18 q^{7} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.879310	+	2.51292i	−0.621766	+	1.77690i	0.00781454	+	0.999969i	\(0.497513\pi\)
							−0.629580	+	0.776935i	\(0.716773\pi\)
\(3\)	0.183309	+	0.291735i	0.105834	+	0.168433i	0.895442	−	0.445177i	\(-0.146859\pi\)
							−0.789609	+	0.613611i	\(0.789717\pi\)
\(5\)	0.772116	−	2.09853i	0.345301	−	0.938492i
							\(7\)	0.182770	−	2.63943i	0.0690804	−	0.997611i
\(11\)	−4.64275	+	2.23583i	−1.39984	+	0.674128i								−0.973129	−	0.230259i	\(-0.926043\pi\)
							−0.426712	+	0.904388i	\(0.640328\pi\)
\(13\)	1.82910	−	5.22726i	0.507300	−	1.44978i	−0.350874	−	0.936423i	\(-0.614115\pi\)
							0.858174	−	0.513359i	\(-0.171599\pi\)
\(17\)	−0.0741536	+	0.658131i	−0.0179849	+	0.159620i	−0.999494	−	0.0317926i	\(-0.989878\pi\)
							0.981510	+	0.191413i	\(0.0613070\pi\)
\(19\)	−0.915129			−0.209945			−0.104972	−	0.994475i	\(-0.533475\pi\)
							−0.104972	+	0.994475i	\(0.533475\pi\)
\(23\)	−3.14974	+	0.354890i	−0.656765	+	0.0739997i	−0.434057	−	0.900885i	\(-0.642918\pi\)
							−0.222708	+	0.974885i	\(0.571490\pi\)
\(29\)	−1.80580	+	1.44007i	−0.335328	+	0.267415i	−0.776648	−	0.629935i	\(-0.783082\pi\)
							0.441320	+	0.897350i	\(0.354510\pi\)
\(31\)			3.94256i			0.708105i	0.935226	+	0.354052i	\(0.115197\pi\)
							−0.935226	+	0.354052i	\(0.884803\pi\)
\(37\)	4.62590	+	0.521213i	0.760493	+	0.0856869i	0.483689	−	0.875240i	\(-0.339296\pi\)
							0.276803	+	0.960927i	\(0.410725\pi\)
\(41\)	8.84126	+	2.01796i	1.38077	+	0.315152i	0.847502	−	0.530792i	\(-0.178106\pi\)
							0.533271	+	0.845945i	\(0.320963\pi\)
\(43\)	2.07153	−	3.29683i	0.315906	−	0.502761i	−0.650595	−	0.759425i	\(-0.725480\pi\)
							0.966501	+	0.256664i	\(0.0826232\pi\)
\(47\)	5.03425	+	1.76156i	0.734320	+	0.256950i	0.671427	−	0.741070i	\(-0.265682\pi\)
							0.0628931	+	0.998020i	\(0.479967\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.s.a.223.2	yes	312
5.2	odd	4	inner	245.2.s.a.27.2	✓	312
49.20	odd	14	inner	245.2.s.a.118.2	yes	312
245.167	even	28	inner	245.2.s.a.167.2	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.s.a.27.2	✓	312	5.2	odd	4	inner
245.2.s.a.118.2	yes	312	49.20	odd	14	inner
245.2.s.a.167.2	yes	312	245.167	even	28	inner
245.2.s.a.223.2	yes	312	1.1	even	1	trivial