Embedded newform 507.2.t.a.4.3

• Label: 507.2.t.a.4.3
• Level: $507$
• Weight: $2$
• Character: 507.4
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.t (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(14\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			4.3
Character	\(\chi\)	\(=\)	507.4
Dual form			507.2.t.a.127.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.22635 - 0.0897189i) q^{2} +(0.278217 - 0.960518i) q^{3} +(2.95507 + 0.238558i) q^{4} +(1.89329 + 0.718032i) q^{5} +(-0.705586 + 2.11349i) q^{6} +(-0.669684 - 1.57181i) q^{7} +(-2.13381 - 0.259091i) q^{8} +(-0.845190 - 0.534466i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 14 q^{3} - 12 q^{4} + 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{78}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.22635	−	0.0897189i	−1.57427	−	0.0634408i	−0.762331	−	0.647187i	\(-0.775945\pi\)
							−0.811936	+	0.583746i	\(0.801586\pi\)
\(3\)	0.278217	−	0.960518i	0.160629	−	0.554555i
							\(5\)	1.89329	+	0.718032i	0.846707	+	0.321114i	0.739528	−	0.673126i	\(-0.235049\pi\)
0.107179	+	0.994240i	\(0.465818\pi\)
\(7\)	−0.669684	−	1.57181i	−0.253117	−	0.594087i	0.744147	−	0.668016i	\(-0.232856\pi\)
							−0.997263	+	0.0739295i	\(0.976446\pi\)
\(11\)	2.15305	+	3.40478i	0.649169	+	1.02658i	0.996061	+	0.0886732i	\(0.0282627\pi\)
							−0.346892	+	0.937905i	\(0.612763\pi\)
\(13\)	3.56212	−	0.557956i	0.987954	−	0.154749i
							\(17\)	3.74310	−	1.59478i	0.907834	−	0.386792i	0.112983	−	0.993597i	\(-0.463959\pi\)
0.794851	+	0.606805i	\(0.207549\pi\)
\(19\)	0.892394	−	0.515224i	0.204729	−	0.118200i	−0.394130	−	0.919055i	\(-0.628954\pi\)
							0.598859	+	0.800854i	\(0.295621\pi\)
\(23\)	−4.45782	+	7.72116i	−0.929519	+	1.60997i	−0.145392	+	0.989374i	\(0.546444\pi\)
							−0.784127	+	0.620600i	\(0.786889\pi\)
\(29\)	−0.183582	+	4.55555i	−0.0340904	+	0.845945i	0.888202	+	0.459453i	\(0.151954\pi\)
							−0.922293	+	0.386492i	\(0.873687\pi\)
\(31\)	−0.0212241	−	0.0239571i	−0.00381196	−	0.00430282i	0.746602	−	0.665271i	\(-0.231684\pi\)
							−0.750414	+	0.660969i	\(0.770146\pi\)
\(37\)	6.15543	−	1.25664i	1.01195	−	0.206590i	0.334683	−	0.942331i	\(-0.391371\pi\)
							0.677263	+	0.735741i	\(0.263166\pi\)
\(41\)	−0.0212979	−	0.00616901i	−0.00332617	−	0.000963437i	0.276554	−	0.960998i	\(-0.410808\pi\)
							−0.279880	+	0.960035i	\(0.590295\pi\)
\(43\)	0.529302	−	2.59269i	0.0807178	−	0.395382i	−0.919281	−	0.393603i	\(-0.871229\pi\)
							0.999998	−	0.00177910i	\(-0.000566304\pi\)
\(47\)	9.53495	−	6.58150i	1.39082	−	0.960011i	0.391612	−	0.920131i	\(-0.371918\pi\)
							0.999204	−	0.0398804i	\(-0.0126977\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.t.a.4.3	✓	336
169.127	even	78	inner	507.2.t.a.127.3	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.t.a.4.3	✓	336	1.1	even	1	trivial
507.2.t.a.127.3	yes	336	169.127	even	78	inner