Embedded newform 624.2.cj.a.589.10

• Label: 624.2.cj.a.589.10
• Level: $624$
• Weight: $2$
• Character: 624.589
• Analytic conductor: $4.983$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.cj (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.98266508613\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(56\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			589.10
Character	\(\chi\)	\(=\)	624.589
Dual form			624.2.cj.a.517.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23025 + 0.697482i) q^{2} +(-0.965926 + 0.258819i) q^{3} +(1.02704 - 1.71616i) q^{4} +(0.334101 + 0.334101i) q^{5} +(1.00781 - 0.992129i) q^{6} +(0.820051 + 1.42037i) q^{7} +(-0.0665256 + 2.82764i) q^{8} +(0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\)	\(79\)	\(145\)	\(209\)	\(469\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23025	+	0.697482i	−0.869919	+	0.493194i
							\(3\)	−0.965926	+	0.258819i	−0.557678	+	0.149429i
\(5\)	0.334101	+	0.334101i	0.149415	+	0.149415i								0.777856	−	0.628442i	\(-0.216307\pi\)
							−0.628442	+	0.777856i	\(0.716307\pi\)
\(7\)	0.820051	+	1.42037i	0.309950	+	0.536849i	0.978351	−	0.206952i	\(-0.0663543\pi\)
							−0.668401	+	0.743801i	\(0.733021\pi\)
\(11\)	2.84810	−	0.763145i	0.858733	−	0.230097i	0.197524	−	0.980298i	\(-0.436710\pi\)
							0.661210	+	0.750201i	\(0.270043\pi\)
\(13\)	−0.734900	−	3.52986i	−0.203825	−	0.979007i
							\(17\)	1.61312	+	2.79401i	0.391239	+	0.677646i	0.992613	−	0.121321i	\(-0.0387131\pi\)
−0.601374	+	0.798968i	\(0.705380\pi\)
\(19\)	−4.05710	−	1.08710i	−0.930762	−	0.249397i	−0.238582	−	0.971122i	\(-0.576683\pi\)
							−0.692179	+	0.721725i	\(0.743349\pi\)
\(23\)	4.63820	+	2.67787i	0.967132	+	0.558374i	0.898361	−	0.439258i	\(-0.144759\pi\)
							0.0687714	+	0.997632i	\(0.478092\pi\)
\(29\)	1.54316	−	0.413489i	0.286558	−	0.0767829i	−0.112677	−	0.993632i	\(-0.535943\pi\)
							0.399235	+	0.916849i	\(0.369276\pi\)
\(31\)			2.93656i			0.527423i	0.964602	+	0.263711i	\(0.0849466\pi\)
							−0.964602	+	0.263711i	\(0.915053\pi\)
\(37\)	8.35221	−	2.23797i	1.37310	−	0.367920i	0.504487	−	0.863419i	\(-0.331682\pi\)
							0.868608	+	0.495499i	\(0.165015\pi\)
\(41\)	−2.57910	+	4.46712i	−0.402787	+	0.697648i	−0.994061	−	0.108823i	\(-0.965292\pi\)
							0.591274	+	0.806471i	\(0.298625\pi\)
\(43\)	12.2800	+	3.29041i	1.87268	+	0.501782i	0.999907	+	0.0136521i	\(0.00434573\pi\)
							0.872771	+	0.488130i	\(0.162321\pi\)
\(47\)			5.43782i			0.793188i	0.917994	+	0.396594i	\(0.129808\pi\)
							−0.917994	+	0.396594i	\(0.870192\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.2.cj.a.589.10	yes	224
13.10	even	6	inner	624.2.cj.a.205.30	✓	224
16.5	even	4	inner	624.2.cj.a.277.30	yes	224
208.101	even	12	inner	624.2.cj.a.517.10	yes	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.2.cj.a.205.30	✓	224	13.10	even	6	inner
624.2.cj.a.277.30	yes	224	16.5	even	4	inner
624.2.cj.a.517.10	yes	224	208.101	even	12	inner
624.2.cj.a.589.10	yes	224	1.1	even	1	trivial