Embedded newform 507.2.t.a.4.6

• Label: 507.2.t.a.4.6
• 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			4.6
Character	\(\chi\)	\(=\)	507.4
Dual form			507.2.t.a.127.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.04226 - 0.0420016i) q^{2} +(0.278217 - 0.960518i) q^{3} +(-0.908975 - 0.0733801i) q^{4} +(0.954453 + 0.361976i) q^{5} +(-0.330318 + 0.989423i) q^{6} +(-1.88928 - 4.43431i) q^{7} +(3.01530 + 0.366124i) q^{8} +(-0.845190 - 0.534466i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 14 q^{3} - 12 q^{4} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.04226	−	0.0420016i	−0.736988	−	0.0296996i	−0.331066	−	0.943607i	\(-0.607408\pi\)
							−0.405922	+	0.913908i	\(0.633049\pi\)
\(3\)	0.278217	−	0.960518i	0.160629	−	0.554555i
							\(5\)	0.954453	+	0.361976i	0.426844	+	0.161881i	0.558666	−	0.829393i	\(-0.311313\pi\)
−0.131822	+	0.991273i	\(0.542083\pi\)
\(7\)	−1.88928	−	4.43431i	−0.714082	−	1.67601i	−0.736658	−	0.676266i	\(-0.763597\pi\)
							0.0225760	−	0.999745i	\(-0.492813\pi\)
\(11\)	−1.18977	−	1.88146i	−0.358728	−	0.567283i	0.616810	−	0.787112i	\(-0.288424\pi\)
							−0.975538	+	0.219829i	\(0.929450\pi\)
\(13\)	1.50001	+	3.27871i	0.416028	+	0.909352i
							\(17\)	−2.20569	+	0.939756i	−0.534958	+	0.227924i	−0.642513	−	0.766275i	\(-0.722108\pi\)
0.107555	+	0.994199i	\(0.465698\pi\)
\(19\)	−3.91340	+	2.25940i	−0.897796	+	0.518343i	−0.876484	−	0.481430i	\(-0.840117\pi\)
							−0.0213114	+	0.999773i	\(0.506784\pi\)
\(23\)	0.991139	−	1.71670i	0.206667	−	0.357957i	−0.743996	−	0.668184i	\(-0.767072\pi\)
							0.950663	+	0.310227i	\(0.100405\pi\)
\(29\)	0.312776	−	7.76145i	0.0580810	−	1.44127i	−0.663106	−	0.748525i	\(-0.730762\pi\)
							0.721187	−	0.692740i	\(-0.243597\pi\)
\(31\)	3.04514	+	3.43725i	0.546923	+	0.617349i	0.955362	−	0.295437i	\(-0.0954652\pi\)
							−0.408439	+	0.912786i	\(0.633927\pi\)
\(37\)	−10.1862	+	2.07953i	−1.67460	+	0.341872i	−0.941180	−	0.337905i	\(-0.890282\pi\)
							−0.733419	+	0.679777i	\(0.762077\pi\)
\(41\)	−7.29017	−	2.11162i	−1.13853	−	0.329780i	−0.345177	−	0.938538i	\(-0.612181\pi\)
							−0.793357	+	0.608757i	\(0.791668\pi\)
\(43\)	0.0692476	−	0.339197i	0.0105602	−	0.0517271i	−0.974169	−	0.225820i	\(-0.927494\pi\)
							0.984729	+	0.174092i	\(0.0556992\pi\)
\(47\)	0.134927	−	0.0931335i	0.0196812	−	0.0135849i	−0.558184	−	0.829717i	\(-0.688502\pi\)
							0.577865	+	0.816133i	\(0.303886\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.t.a.4.6	✓	336
169.127	even	78	inner	507.2.t.a.127.6	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.t.a.4.6	✓	336	1.1	even	1	trivial
507.2.t.a.127.6	yes	336	169.127	even	78	inner