Embedded newform 507.2.s.a.5.19

• Label: 507.2.s.a.5.19
• 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			5.19
Character	\(\chi\)	\(=\)	507.5
Dual form			507.2.s.a.203.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.230989 + 1.26047i) q^{2} +(-0.545148 - 1.64402i) q^{3} +(0.334608 + 0.126900i) q^{4} +(2.48769 + 0.150478i) q^{5} +(2.19816 - 0.307389i) q^{6} +(0.0931907 + 0.0563358i) q^{7} +(-1.56314 + 2.58575i) q^{8} +(-2.40563 + 1.79247i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1392 q - 22 q^{3} - 52 q^{4} - 22 q^{6} - 56 q^{7} - 22 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.230989	+	1.26047i	−0.163334	+	0.891286i	0.795208	+	0.606336i	\(0.207361\pi\)
							−0.958543	+	0.284950i	\(0.908023\pi\)
\(3\)	−0.545148	−	1.64402i	−0.314741	−	0.949178i
							\(5\)	2.48769	+	0.150478i	1.11253	+	0.0672957i	0.606399	−	0.795161i	\(-0.292614\pi\)
0.506131	+	0.862456i	\(0.331075\pi\)
\(7\)	0.0931907	+	0.0563358i	0.0352228	+	0.0212929i	0.534840	−	0.844954i	\(-0.320372\pi\)
							−0.499617	+	0.866246i	\(0.666526\pi\)
\(11\)	0.554986	+	3.02846i	0.167335	+	0.913116i	0.954910	+	0.296897i	\(0.0959518\pi\)
							−0.787575	+	0.616219i	\(0.788664\pi\)
\(13\)	−1.68648	+	3.18681i	−0.467746	+	0.883863i
							\(17\)	5.19473	−	1.28039i	1.25991	−	0.310539i	0.447845	−	0.894111i	\(-0.352192\pi\)
0.812061	+	0.583572i	\(0.198345\pi\)
\(19\)	−0.0284434	+	0.0284434i	−0.00652535	+	0.00652535i	−0.710362	−	0.703837i	\(-0.751469\pi\)
							0.703837	+	0.710362i	\(0.251469\pi\)
\(23\)	7.14268			1.48935			0.744676	−	0.667426i	\(-0.232604\pi\)
							0.744676	+	0.667426i	\(0.232604\pi\)
\(29\)	4.76575	−	3.28956i	0.884978	−	0.610856i	−0.0364797	−	0.999334i	\(-0.511614\pi\)
							0.921458	+	0.388478i	\(0.126999\pi\)
\(31\)	−6.19203	−	4.85115i	−1.11212	−	0.871291i	−0.119535	−	0.992830i	\(-0.538140\pi\)
							−0.992587	+	0.121539i	\(0.961217\pi\)
\(37\)	−0.102179	+	0.130421i	−0.0167981	+	0.0214412i	−0.795510	−	0.605941i	\(-0.792797\pi\)
							0.778711	+	0.627382i	\(0.215874\pi\)
\(41\)	2.75904	+	8.85407i	0.430889	+	1.38277i	0.873834	+	0.486224i	\(0.161626\pi\)
							−0.442945	+	0.896549i	\(0.646066\pi\)
\(43\)	2.40885	+	0.292487i	0.367346	+	0.0446038i	0.302128	−	0.953267i	\(-0.402303\pi\)
							0.0652172	+	0.997871i	\(0.479226\pi\)
\(47\)	−9.73273	−	4.38034i	−1.41966	−	0.638939i	−0.452193	−	0.891920i	\(-0.649358\pi\)
							−0.967471	+	0.252981i	\(0.918589\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.s.a.5.19	✓	1392
3.2	odd	2	inner	507.2.s.a.5.40	yes	1392
169.34	odd	52	inner	507.2.s.a.203.40	yes	1392
507.203	even	52	inner	507.2.s.a.203.19	yes	1392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.s.a.5.19	✓	1392	1.1	even	1	trivial
507.2.s.a.5.40	yes	1392	3.2	odd	2	inner
507.2.s.a.203.19	yes	1392	507.203	even	52	inner
507.2.s.a.203.40	yes	1392	169.34	odd	52	inner