Embedded newform 507.2.s.a.5.1

• Label: 507.2.s.a.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	507.5
Dual form			507.2.s.a.203.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.504808 + 2.75465i) q^{2} +(1.69376 - 0.362196i) q^{3} +(-5.46324 - 2.07193i) q^{4} +(2.58865 + 0.156585i) q^{5} +(0.142700 + 4.84855i) q^{6} +(0.0295242 + 0.0178480i) q^{7} +(5.56770 - 9.21011i) q^{8} +(2.73763 - 1.22694i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1392 q - 22 q^{3} - 52 q^{4} - 22 q^{6} - 56 q^{7} - 22 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.504808	+	2.75465i	−0.356953	+	1.94783i	−0.0406523	+	0.999173i	\(0.512944\pi\)
							−0.316301	+	0.948659i	\(0.602441\pi\)
\(3\)	1.69376	−	0.362196i	0.977891	−	0.209114i
							\(5\)	2.58865	+	0.156585i	1.15768	+	0.0700268i	0.628030	−	0.778189i	\(-0.283861\pi\)
0.529651	+	0.848216i	\(0.322323\pi\)
\(7\)	0.0295242	+	0.0178480i	0.0111591	+	0.00674592i	0.522906	−	0.852390i	\(-0.324848\pi\)
							−0.511747	+	0.859136i	\(0.671002\pi\)
\(11\)	−0.190985	−	1.04217i	−0.0575842	−	0.314227i	0.942199	−	0.335054i	\(-0.108755\pi\)
							−0.999783	+	0.0208273i	\(0.993370\pi\)
\(13\)	2.58997	+	2.50840i	0.718329	+	0.695704i
							\(17\)	−0.0724170	+	0.0178492i	−0.0175637	+	0.00432907i	−0.248088	−	0.968738i	\(-0.579802\pi\)
0.230524	+	0.973067i	\(0.425956\pi\)
\(19\)	−2.75538	+	2.75538i	−0.632128	+	0.632128i	−0.948601	−	0.316473i	\(-0.897501\pi\)
							0.316473	+	0.948601i	\(0.397501\pi\)
\(23\)	6.71857			1.40092			0.700459	−	0.713693i	\(-0.252979\pi\)
							0.700459	+	0.713693i	\(0.252979\pi\)
\(29\)	−1.39633	+	0.963819i	−0.259292	+	0.178977i	−0.690618	−	0.723220i	\(-0.742661\pi\)
							0.431326	+	0.902196i	\(0.358046\pi\)
\(31\)	−3.81026	−	2.98515i	−0.684344	−	0.536149i	0.213236	−	0.977001i	\(-0.431600\pi\)
							−0.897580	+	0.440852i	\(0.854676\pi\)
\(37\)	−6.77525	+	8.64796i	−1.11384	+	1.42172i	−0.218782	+	0.975774i	\(0.570208\pi\)
							−0.895062	+	0.445943i	\(0.852869\pi\)
\(41\)	−1.39197	−	4.46698i	−0.217389	−	0.697624i	−0.997414	−	0.0718760i	\(-0.977101\pi\)
							0.780025	−	0.625748i	\(-0.215206\pi\)
\(43\)	8.12787	+	0.986902i	1.23949	+	0.150501i	0.713906	−	0.700242i	\(-0.246924\pi\)
							0.525582	+	0.850743i	\(0.323848\pi\)
\(47\)	−0.587558	−	0.264438i	−0.0857042	−	0.0385723i	0.367107	−	0.930179i	\(-0.380348\pi\)
							−0.452811	+	0.891606i	\(0.649579\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.s.a.5.1	✓	1392
3.2	odd	2	inner	507.2.s.a.5.58	yes	1392
169.34	odd	52	inner	507.2.s.a.203.58	yes	1392
507.203	even	52	inner	507.2.s.a.203.1	yes	1392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.s.a.5.1	✓	1392	1.1	even	1	trivial
507.2.s.a.5.58	yes	1392	3.2	odd	2	inner
507.2.s.a.203.1	yes	1392	507.203	even	52	inner
507.2.s.a.203.58	yes	1392	169.34	odd	52	inner