Embedded newform 507.2.s.a.5.12

• Label: 507.2.s.a.5.12
• 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			5.12
Character	\(\chi\)	\(=\)	507.5
Dual form			507.2.s.a.203.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.320681 + 1.74990i) q^{2} +(-1.48788 + 0.886686i) q^{3} +(-1.08928 - 0.413111i) q^{4} +(-3.06762 - 0.185557i) q^{5} +(-1.07448 - 2.88799i) q^{6} +(1.60183 + 0.968339i) q^{7} +(-0.768515 + 1.27128i) q^{8} +(1.42758 - 2.63857i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1392 q - 22 q^{3} - 52 q^{4} - 22 q^{6} - 56 q^{7} - 22 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.320681	+	1.74990i	−0.226756	+	1.23737i	0.651304	+	0.758817i	\(0.274222\pi\)
							−0.878060	+	0.478550i	\(0.841163\pi\)
\(3\)	−1.48788	+	0.886686i	−0.859028	+	0.511928i
							\(5\)	−3.06762	−	0.185557i	−1.37188	−	0.0829836i	−0.642078	−	0.766639i	\(-0.721928\pi\)
−0.729805	+	0.683656i	\(0.760389\pi\)
\(7\)	1.60183	+	0.968339i	0.605434	+	0.365998i	0.786607	−	0.617453i	\(-0.211836\pi\)
							−0.181173	+	0.983451i	\(0.557989\pi\)
\(11\)	−0.489805	−	2.67278i	−0.147682	−	0.805873i	−0.971143	−	0.238500i	\(-0.923344\pi\)
							0.823461	−	0.567373i	\(-0.192040\pi\)
\(13\)	−3.00548	−	1.99176i	−0.833569	−	0.552415i
							\(17\)	−5.24683	+	1.29323i	−1.27254	+	0.313654i	−0.817041	−	0.576580i	\(-0.804387\pi\)
−0.455503	+	0.890234i	\(0.650541\pi\)
\(19\)	2.04029	−	2.04029i	0.468074	−	0.468074i	−0.433216	−	0.901290i	\(-0.642621\pi\)
							0.901290	+	0.433216i	\(0.142621\pi\)
\(23\)	5.44964			1.13633			0.568165	−	0.822915i	\(-0.307654\pi\)
							0.568165	+	0.822915i	\(0.307654\pi\)
\(29\)	6.93027	−	4.78362i	1.28692	−	0.888295i	0.289298	−	0.957239i	\(-0.406578\pi\)
							0.997621	+	0.0689438i	\(0.0219629\pi\)
\(31\)	4.70087	+	3.68290i	0.844302	+	0.661468i	0.942686	−	0.333680i	\(-0.108291\pi\)
							−0.0983847	+	0.995148i	\(0.531368\pi\)
\(37\)	−2.62988	+	3.35680i	−0.432350	+	0.551854i	−0.953415	−	0.301663i	\(-0.902458\pi\)
							0.521065	+	0.853517i	\(0.325535\pi\)
\(41\)	0.489819	+	1.57188i	0.0764969	+	0.245487i	0.985046	−	0.172291i	\(-0.0551168\pi\)
							−0.908549	+	0.417778i	\(0.862809\pi\)
\(43\)	−8.99765	−	1.09251i	−1.37213	−	0.166607i	−0.598946	−	0.800790i	\(-0.704413\pi\)
							−0.773183	+	0.634183i	\(0.781337\pi\)
\(47\)	−7.98296	−	3.59284i	−1.16443	−	0.524069i	−0.266343	−	0.963878i	\(-0.585815\pi\)
							−0.898091	+	0.439809i	\(0.855046\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.s.a.5.12	✓	1392
3.2	odd	2	inner	507.2.s.a.5.47	yes	1392
169.34	odd	52	inner	507.2.s.a.203.47	yes	1392
507.203	even	52	inner	507.2.s.a.203.12	yes	1392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.s.a.5.12	✓	1392	1.1	even	1	trivial
507.2.s.a.5.47	yes	1392	3.2	odd	2	inner
507.2.s.a.203.12	yes	1392	507.203	even	52	inner
507.2.s.a.203.47	yes	1392	169.34	odd	52	inner