Embedded newform 245.2.t.a.184.18

• Label: 245.2.t.a.184.18
• Level: $245$
• Weight: $2$
• Character: 245.184
• 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.t (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(26\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			184.18
Character	\(\chi\)	\(=\)	245.184
Dual form			245.2.t.a.4.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.546727 + 0.801902i) q^{2} +(0.174206 + 0.187749i) q^{3} +(0.386547 - 0.984905i) q^{4} +(-0.549230 + 2.16757i) q^{5} +(-0.0553132 + 0.242343i) q^{6} +(2.49858 - 0.870109i) q^{7} +(2.89355 - 0.660435i) q^{8} +(0.219288 - 2.92620i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 50 q^{4} - 12 q^{5} - 10 q^{6} - 64 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.546727	+	0.801902i	0.386595	+	0.567030i	0.969371	−	0.245603i	\(-0.0789859\pi\)
							−0.582776	+	0.812633i	\(0.698034\pi\)
\(3\)	0.174206	+	0.187749i	0.100578	+	0.108397i	0.781334	−	0.624113i	\(-0.214539\pi\)
							−0.680757	+	0.732510i	\(0.738349\pi\)
\(5\)	−0.549230	+	2.16757i	−0.245623	+	0.969365i
							\(7\)	2.49858	−	0.870109i	0.944375	−	0.328870i
\(11\)	0.216858	+	2.89376i	0.0653850	+	0.872503i								0.929409	+	0.369052i	\(0.120317\pi\)
							−0.864024	+	0.503451i	\(0.832063\pi\)
\(13\)	−1.88622	+	3.91677i	−0.523143	+	1.08632i	0.457263	+	0.889331i	\(0.348830\pi\)
							−0.980406	+	0.196986i	\(0.936885\pi\)
\(17\)	0.403161	−	2.67480i	0.0977809	−	0.648734i	−0.885225	−	0.465162i	\(-0.845996\pi\)
							0.983006	−	0.183572i	\(-0.0587659\pi\)
\(19\)	−1.45779	+	2.52497i	−0.334440	+	0.579268i	−0.983377	−	0.181575i	\(-0.941881\pi\)
							0.648937	+	0.760842i	\(0.275214\pi\)
\(23\)	0.506624	+	3.36123i	0.105638	+	0.700865i	0.977472	+	0.211064i	\(0.0676927\pi\)
							−0.871834	+	0.489802i	\(0.837069\pi\)
\(29\)	5.00672	−	6.27822i	0.929724	−	1.16584i	−0.0561625	−	0.998422i	\(-0.517886\pi\)
							0.985886	−	0.167415i	\(-0.0535421\pi\)
\(31\)	−5.04241	−	8.73370i	−0.905643	−	1.56862i	−0.820052	−	0.572289i	\(-0.806056\pi\)
							−0.0855909	−	0.996330i	\(-0.527278\pi\)
\(37\)	−7.59724	+	2.98170i	−1.24898	+	0.490188i	−0.895419	−	0.445224i	\(-0.853124\pi\)
							−0.353559	+	0.935412i	\(0.615029\pi\)
\(41\)	2.24706	+	9.84501i	0.350932	+	1.53753i	0.775035	+	0.631919i	\(0.217732\pi\)
							−0.424103	+	0.905614i	\(0.639411\pi\)
\(43\)	−9.27606	−	2.11720i	−1.41459	−	0.322870i	−0.554145	−	0.832420i	\(-0.686955\pi\)
							−0.860441	+	0.509550i	\(0.829812\pi\)
\(47\)	−4.51322	−	6.61967i	−0.658320	−	0.965578i	−0.999671	−	0.0256424i	\(-0.991837\pi\)
							0.341351	−	0.939936i	\(-0.389116\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.t.a.184.18	yes	312
5.4	even	2	inner	245.2.t.a.184.9	yes	312
49.4	even	21	inner	245.2.t.a.4.9	✓	312
245.4	even	42	inner	245.2.t.a.4.18	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.t.a.4.9	✓	312	49.4	even	21	inner
245.2.t.a.4.18	yes	312	245.4	even	42	inner
245.2.t.a.184.9	yes	312	5.4	even	2	inner
245.2.t.a.184.18	yes	312	1.1	even	1	trivial