Embedded newform 507.2.t.b.127.4

• Label: 507.2.t.b.127.4
• Level: $507$
• Weight: $2$
• Character: 507.127
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $360$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.t (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(15\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			127.4
Character	\(\chi\)	\(=\)	507.127
Dual form			507.2.t.b.4.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.79660 + 0.0724007i) q^{2} +(-0.278217 - 0.960518i) q^{3} +(1.22903 - 0.0992178i) q^{4} +(1.73729 - 0.658868i) q^{5} +(0.569389 + 1.70553i) q^{6} +(1.36244 - 3.19777i) q^{7} +(1.36900 - 0.166227i) q^{8} +(-0.845190 + 0.534466i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 15 q^{3} - 14 q^{4} + 3 q^{7} + 15 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{77}{78}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.79660	+	0.0724007i	−1.27039	+	0.0511950i	−0.666278	−	0.745703i	\(-0.732114\pi\)
							−0.604113	+	0.796898i	\(0.706473\pi\)
\(3\)	−0.278217	−	0.960518i	−0.160629	−	0.554555i
							\(5\)	1.73729	−	0.658868i	0.776941	−	0.294655i	0.0659193	−	0.997825i	\(-0.479002\pi\)
0.711022	+	0.703170i	\(0.248233\pi\)
\(7\)	1.36244	−	3.19777i	0.514954	−	1.20864i	−0.436179	−	0.899860i	\(-0.643668\pi\)
							0.951133	−	0.308782i	\(-0.0999213\pi\)
\(11\)	2.46967	−	3.90546i	0.744632	−	1.17754i	−0.234063	−	0.972221i	\(-0.575202\pi\)
							0.978695	−	0.205320i	\(-0.0658235\pi\)
\(13\)	−3.37805	+	1.26048i	−0.936901	+	0.349594i
							\(17\)	4.77680	+	2.03520i	1.15854	+	0.493609i	0.883780	−	0.467902i	\(-0.154990\pi\)
0.274764	+	0.961512i	\(0.411400\pi\)
\(19\)	4.15553	+	2.39920i	0.953344	+	0.550414i	0.894118	−	0.447831i	\(-0.147803\pi\)
							0.0592260	+	0.998245i	\(0.481137\pi\)
\(23\)	0.442066	+	0.765680i	0.0921771	+	0.159655i	0.908427	−	0.418044i	\(-0.137284\pi\)
							−0.816250	+	0.577699i	\(0.803951\pi\)
\(29\)	−0.225646	−	5.59934i	−0.0419014	−	1.03977i	−0.874237	−	0.485500i	\(-0.838638\pi\)
							0.832336	−	0.554272i	\(-0.187003\pi\)
\(31\)	2.03491	−	2.29694i	0.365481	−	0.412543i	−0.536787	−	0.843718i	\(-0.680362\pi\)
							0.902269	+	0.431175i	\(0.141901\pi\)
\(37\)	−10.1464	−	2.07141i	−1.66806	−	0.340538i	−0.729062	−	0.684448i	\(-0.760043\pi\)
							−0.939002	+	0.343910i	\(0.888248\pi\)
\(41\)	4.35141	−	1.26040i	0.679576	−	0.196842i	0.0795605	−	0.996830i	\(-0.474648\pi\)
							0.600015	+	0.799989i	\(0.295161\pi\)
\(43\)	−0.221516	−	1.08506i	−0.0337809	−	0.165470i	0.959401	−	0.282046i	\(-0.0910131\pi\)
							−0.993182	+	0.116576i	\(0.962808\pi\)
\(47\)	−2.51231	−	1.73412i	−0.366457	−	0.252947i	0.370578	−	0.928801i	\(-0.379160\pi\)
							−0.737036	+	0.675854i	\(0.763775\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.t.b.127.4	yes	360
169.4	even	78	inner	507.2.t.b.4.4	✓	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.t.b.4.4	✓	360	169.4	even	78	inner
507.2.t.b.127.4	yes	360	1.1	even	1	trivial