Embedded newform 507.2.s.a.5.3

• Label: 507.2.s.a.5.3
• Level: $507$
• Weight: $2$
• Character: 507.5
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $1392$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.s (of order \(52\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(1392\)
Relative dimension:	\(58\) over \(\Q(\zeta_{52})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{52}]$

Embedding invariants

Embedding label			5.3
Character	\(\chi\)	\(=\)	507.5
Dual form			507.2.s.a.203.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.465956 + 2.54264i) q^{2} +(-0.218600 + 1.71820i) q^{3} +(-4.37786 - 1.66030i) q^{4} +(1.00192 + 0.0606047i) q^{5} +(-4.26690 - 1.35643i) q^{6} +(-1.49633 - 0.904565i) q^{7} +(3.58682 - 5.93332i) q^{8} +(-2.90443 - 0.751197i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1392 q - 22 q^{3} - 52 q^{4} - 22 q^{6} - 56 q^{7} - 22 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{52}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.465956	+	2.54264i	−0.329480	+	1.79792i	0.234487	+	0.972119i	\(0.424659\pi\)
							−0.563968	+	0.825797i	\(0.690726\pi\)
\(3\)	−0.218600	+	1.71820i	−0.126209	+	0.992004i
							\(5\)	1.00192	+	0.0606047i	0.448070	+	0.0271033i	0.282873	−	0.959157i	\(-0.408712\pi\)
0.165197	+	0.986261i	\(0.447174\pi\)
\(7\)	−1.49633	−	0.904565i	−0.565561	−	0.341893i	0.205498	−	0.978658i	\(-0.434119\pi\)
							−0.771059	+	0.636764i	\(0.780272\pi\)
\(11\)	−0.519207	−	2.83322i	−0.156547	−	0.854249i	−0.964317	−	0.264751i	\(-0.914710\pi\)
							0.807770	−	0.589498i	\(-0.200674\pi\)
\(13\)	−1.89643	−	3.06652i	−0.525975	−	0.850500i
							\(17\)	−1.68280	+	0.414772i	−0.408138	+	0.100597i	−0.438039	−	0.898956i	\(-0.644327\pi\)
0.0299010	+	0.999553i	\(0.490481\pi\)
\(19\)	0.428458	−	0.428458i	0.0982950	−	0.0982950i	−0.656249	−	0.754544i	\(-0.727858\pi\)
							0.754544	+	0.656249i	\(0.227858\pi\)
\(23\)	−3.29987			−0.688070			−0.344035	−	0.938957i	\(-0.611794\pi\)
							−0.344035	+	0.938957i	\(0.611794\pi\)
\(29\)	−5.98741	+	4.13281i	−1.11183	+	0.767444i	−0.974791	−	0.223122i	\(-0.928375\pi\)
							−0.137043	+	0.990565i	\(0.543760\pi\)
\(31\)	1.70538	+	1.33608i	0.306296	+	0.239967i	0.758083	−	0.652159i	\(-0.226136\pi\)
							−0.451787	+	0.892126i	\(0.649213\pi\)
\(37\)	−1.68884	+	2.15564i	−0.277643	+	0.354385i	−0.905803	−	0.423699i	\(-0.860731\pi\)
							0.628160	+	0.778084i	\(0.283808\pi\)
\(41\)	1.99469	+	6.40120i	0.311519	+	0.999699i	0.969173	+	0.246380i	\(0.0792410\pi\)
							−0.657655	+	0.753320i	\(0.728451\pi\)
\(43\)	−5.57958	−	0.677484i	−0.850878	−	0.103315i	−0.316533	−	0.948582i	\(-0.602519\pi\)
							−0.534346	+	0.845266i	\(0.679442\pi\)
\(47\)	−8.71152	−	3.92074i	−1.27071	−	0.571898i	−0.340952	−	0.940081i	\(-0.610749\pi\)
							−0.929753	+	0.368183i	\(0.879980\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.s.a.5.3	✓	1392
3.2	odd	2	inner	507.2.s.a.5.56	yes	1392
169.34	odd	52	inner	507.2.s.a.203.56	yes	1392
507.203	even	52	inner	507.2.s.a.203.3	yes	1392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.s.a.5.3	✓	1392	1.1	even	1	trivial
507.2.s.a.5.56	yes	1392	3.2	odd	2	inner
507.2.s.a.203.3	yes	1392	507.203	even	52	inner
507.2.s.a.203.56	yes	1392	169.34	odd	52	inner