Embedded newform 507.2.p.a.25.11

• Label: 507.2.p.a.25.11
• Level: $507$
• Weight: $2$
• Character: 507.25
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.p (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label			25.11
Character	\(\chi\)	\(=\)	507.25
Dual form			507.2.p.a.142.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.35653 + 0.514462i) q^{2} +(0.568065 + 0.822984i) q^{3} +(0.0784691 + 0.0695176i) q^{4} +(-3.71157 - 0.450666i) q^{5} +(0.347200 + 1.40865i) q^{6} +(-1.65187 - 3.14737i) q^{7} +(-1.27776 - 2.43458i) q^{8} +(-0.354605 + 0.935016i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{3} + 12 q^{4} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{26}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.35653	+	0.514462i	0.959209	+	0.363780i	0.784006	−	0.620753i	\(-0.213173\pi\)
							0.175202	+	0.984532i	\(0.443942\pi\)
\(3\)	0.568065	+	0.822984i	0.327972	+	0.475150i
							\(5\)	−3.71157	−	0.450666i	−1.65987	−	0.201544i	−0.763796	−	0.645458i	\(-0.776667\pi\)
−0.896070	+	0.443914i	\(0.853590\pi\)
\(7\)	−1.65187	−	3.14737i	−0.624347	−	1.18959i	−0.968107	−	0.250537i	\(-0.919393\pi\)
							0.343760	−	0.939058i	\(-0.388299\pi\)
\(11\)	−5.08753	+	1.92944i	−1.53395	+	0.581750i	−0.969795	−	0.243921i	\(-0.921566\pi\)
							−0.564152	+	0.825671i	\(0.690797\pi\)
\(13\)	3.35751	−	1.31420i	0.931206	−	0.364493i
							\(17\)	0.269896	−	0.141652i	0.0654595	−	0.0343558i	−0.431676	−	0.902029i	\(-0.642078\pi\)
0.497135	+	0.867673i	\(0.334385\pi\)
\(19\)			2.95763i			0.678526i	0.940691	+	0.339263i	\(0.110178\pi\)
							−0.940691	+	0.339263i	\(0.889822\pi\)
\(23\)	−3.75769			−0.783532			−0.391766	−	0.920065i	\(-0.628136\pi\)
							−0.391766	+	0.920065i	\(0.628136\pi\)
\(29\)	0.158852	−	0.418857i	0.0294980	−	0.0777798i	−0.919458	−	0.393188i	\(-0.871372\pi\)
							0.948956	+	0.315408i	\(0.102141\pi\)
\(31\)	0.151068	+	0.612907i	0.0271326	+	0.110081i	0.982947	−	0.183890i	\(-0.0588691\pi\)
							−0.955814	+	0.293972i	\(0.905023\pi\)
\(37\)	−0.749751	−	3.04186i	−0.123258	−	0.500079i	−0.999823	−	0.0187952i	\(-0.994017\pi\)
							0.876565	−	0.481283i	\(-0.159829\pi\)
\(41\)	5.06711	−	3.49757i	0.791349	−	0.546229i	−0.102428	−	0.994740i	\(-0.532661\pi\)
							0.893777	+	0.448512i	\(0.148046\pi\)
\(43\)	−1.41292	−	0.348253i	−0.215468	−	0.0531080i	0.130104	−	0.991500i	\(-0.458469\pi\)
							−0.345572	+	0.938392i	\(0.612315\pi\)
\(47\)	−8.16423	−	9.21551i	−1.19088	−	1.34422i	−0.924886	−	0.380244i	\(-0.875840\pi\)
							−0.265989	−	0.963976i	\(-0.585698\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.p.a.25.11	✓	168
169.142	even	26	inner	507.2.p.a.142.11	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.p.a.25.11	✓	168	1.1	even	1	trivial
507.2.p.a.142.11	yes	168	169.142	even	26	inner