Embedded newform 624.2.q.e.289.1

• Label: 624.2.q.e.289.1
• Level: $624$
• Weight: $2$
• Character: 624.289
• Analytic conductor: $4.983$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.98266508613\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	624.289
Dual form			624.2.q.e.529.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} +3.00000 q^{5} +(-2.00000 + 3.46410i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(2.00000 + 3.46410i) q^{11} +(-2.50000 + 2.59808i) q^{13} +(-1.50000 - 2.59808i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(-2.00000 + 3.46410i) q^{19} +4.00000 q^{21} +(-4.00000 - 6.92820i) q^{23} +4.00000 q^{25} +1.00000 q^{27} +(2.50000 + 4.33013i) q^{29} +8.00000 q^{31} +(2.00000 - 3.46410i) q^{33} +(-6.00000 + 10.3923i) q^{35} +(-3.50000 - 6.06218i) q^{37} +(3.50000 + 0.866025i) q^{39} +(4.50000 + 7.79423i) q^{41} +(4.00000 - 6.92820i) q^{43} +(-1.50000 + 2.59808i) q^{45} +4.00000 q^{47} +(-4.50000 - 7.79423i) q^{49} +3.00000 q^{51} -5.00000 q^{53} +(6.00000 + 10.3923i) q^{55} +4.00000 q^{57} +(2.00000 - 3.46410i) q^{59} +(2.50000 - 4.33013i) q^{61} +(-2.00000 - 3.46410i) q^{63} +(-7.50000 + 7.79423i) q^{65} +(4.00000 + 6.92820i) q^{67} +(-4.00000 + 6.92820i) q^{69} +(-2.00000 + 3.46410i) q^{71} +11.0000 q^{73} +(-2.00000 - 3.46410i) q^{75} -16.0000 q^{77} +4.00000 q^{79} +(-0.500000 - 0.866025i) q^{81} +(-4.50000 + 7.79423i) q^{85} +(2.50000 - 4.33013i) q^{87} +(3.00000 + 5.19615i) q^{89} +(-4.00000 - 13.8564i) q^{91} +(-4.00000 - 6.92820i) q^{93} +(-6.00000 + 10.3923i) q^{95} +(7.00000 - 12.1244i) q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^3 + 6 * q^5 - 4 * q^7 - q^9 + 4 * q^11 - 5 * q^13 - 3 * q^15 - 3 * q^17 - 4 * q^19 + 8 * q^21 - 8 * q^23 + 8 * q^25 + 2 * q^27 + 5 * q^29 + 16 * q^31 + 4 * q^33 - 12 * q^35 - 7 * q^37 + 7 * q^39 + 9 * q^41 + 8 * q^43 - 3 * q^45 + 8 * q^47 - 9 * q^49 + 6 * q^51 - 10 * q^53 + 12 * q^55 + 8 * q^57 + 4 * q^59 + 5 * q^61 - 4 * q^63 - 15 * q^65 + 8 * q^67 - 8 * q^69 - 4 * q^71 + 22 * q^73 - 4 * q^75 - 32 * q^77 + 8 * q^79 - q^81 - 9 * q^85 + 5 * q^87 + 6 * q^89 - 8 * q^91 - 8 * q^93 - 12 * q^95 + 14 * q^97 - 8 * q^99

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		0			0
							\(3\)	−0.500000	−	0.866025i	−0.288675	−	0.500000i
\(5\)	3.00000			1.34164										0.670820	−	0.741620i	\(-0.265942\pi\)
							0.670820	+	0.741620i	\(0.265942\pi\)
\(7\)	−2.00000	+	3.46410i	−0.755929	+	1.30931i	0.188982	+	0.981981i	\(0.439481\pi\)
							−0.944911	+	0.327327i	\(0.893852\pi\)
\(11\)	2.00000	+	3.46410i	0.603023	+	1.04447i	0.992361	+	0.123371i	\(0.0393705\pi\)
							−0.389338	+	0.921095i	\(0.627296\pi\)
\(13\)	−2.50000	+	2.59808i	−0.693375	+	0.720577i
							\(17\)	−1.50000	+	2.59808i	−0.363803	+	0.630126i	−0.988583	−	0.150675i	\(-0.951855\pi\)
0.624780	+	0.780801i	\(0.285189\pi\)
\(19\)	−2.00000	+	3.46410i	−0.458831	+	0.794719i	−0.998899	−	0.0469020i	\(-0.985065\pi\)
							0.540068	+	0.841621i	\(0.318398\pi\)
\(23\)	−4.00000	−	6.92820i	−0.834058	−	1.44463i	−0.894795	−	0.446476i	\(-0.852679\pi\)
							0.0607377	−	0.998154i	\(-0.480655\pi\)
\(29\)	2.50000	+	4.33013i	0.464238	+	0.804084i	0.999167	−	0.0408130i	\(-0.0129948\pi\)
							−0.534928	+	0.844897i	\(0.679661\pi\)
\(31\)	8.00000			1.43684			0.718421	−	0.695608i	\(-0.244865\pi\)
							0.718421	+	0.695608i	\(0.244865\pi\)
\(37\)	−3.50000	−	6.06218i	−0.575396	−	0.996616i	−0.995998	−	0.0893706i	\(-0.971514\pi\)
							0.420602	−	0.907245i	\(-0.361819\pi\)
\(41\)	4.50000	+	7.79423i	0.702782	+	1.21725i	0.967486	+	0.252924i	\(0.0813924\pi\)
							−0.264704	+	0.964330i	\(0.585274\pi\)
\(43\)	4.00000	−	6.92820i	0.609994	−	1.05654i	−0.381246	−	0.924473i	\(-0.624505\pi\)
							0.991241	−	0.132068i	\(-0.0421616\pi\)
\(47\)	4.00000			0.583460			0.291730	−	0.956501i	\(-0.405769\pi\)
							0.291730	+	0.956501i	\(0.405769\pi\)
\(53\)	−5.00000			−0.686803			−0.343401	−	0.939189i	\(-0.611579\pi\)
							−0.343401	+	0.939189i	\(0.611579\pi\)
\(59\)	2.00000	−	3.46410i	0.260378	−	0.450988i	−0.705965	−	0.708247i	\(-0.749486\pi\)
							0.966342	+	0.257260i	\(0.0828195\pi\)
\(61\)	2.50000	−	4.33013i	0.320092	−	0.554416i	−0.660415	−	0.750901i	\(-0.729619\pi\)
							0.980507	+	0.196485i	\(0.0629528\pi\)
\(67\)	4.00000	+	6.92820i	0.488678	+	0.846415i	0.999915	−	0.0130248i	\(-0.00414604\pi\)
							−0.511237	+	0.859440i	\(0.670813\pi\)
\(71\)	−2.00000	+	3.46410i	−0.237356	+	0.411113i	−0.959955	−	0.280155i	\(-0.909614\pi\)
							0.722599	+	0.691268i	\(0.242948\pi\)
\(73\)	11.0000			1.28745			0.643726	−	0.765256i	\(-0.277388\pi\)
							0.643726	+	0.765256i	\(0.277388\pi\)
\(79\)	4.00000			0.450035			0.225018	−	0.974355i	\(-0.427756\pi\)
							0.225018	+	0.974355i	\(0.427756\pi\)
\(83\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(89\)	3.00000	+	5.19615i	0.317999	+	0.550791i	0.980071	−	0.198650i	\(-0.0636557\pi\)
							−0.662071	+	0.749441i	\(0.730322\pi\)
\(97\)	7.00000	−	12.1244i	0.710742	−	1.23104i	−0.253837	−	0.967247i	\(-0.581693\pi\)
							0.964579	−	0.263795i	\(-0.0849741\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.2.q.e.289.1		2
3.2	odd	2		1872.2.t.a.289.1		2
4.3	odd	2		312.2.q.c.289.1	yes	2
12.11	even	2		936.2.t.b.289.1		2
13.3	even	3		8112.2.a.bh.1.1		1
13.9	even	3	inner	624.2.q.e.529.1		2
13.10	even	6		8112.2.a.r.1.1		1
39.35	odd	6		1872.2.t.a.1153.1		2
52.3	odd	6		4056.2.a.j.1.1		1
52.11	even	12		4056.2.c.b.337.2		2
52.15	even	12		4056.2.c.b.337.1		2
52.23	odd	6		4056.2.a.b.1.1		1
52.35	odd	6		312.2.q.c.217.1	✓	2
156.35	even	6		936.2.t.b.217.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.q.c.217.1	✓	2	52.35	odd	6
312.2.q.c.289.1	yes	2	4.3	odd	2
624.2.q.e.289.1		2	1.1	even	1	trivial
624.2.q.e.529.1		2	13.9	even	3	inner
936.2.t.b.217.1		2	156.35	even	6
936.2.t.b.289.1		2	12.11	even	2
1872.2.t.a.289.1		2	3.2	odd	2
1872.2.t.a.1153.1		2	39.35	odd	6
4056.2.a.b.1.1		1	52.23	odd	6
4056.2.a.j.1.1		1	52.3	odd	6
4056.2.c.b.337.1		2	52.15	even	12
4056.2.c.b.337.2		2	52.11	even	12
8112.2.a.r.1.1		1	13.10	even	6
8112.2.a.bh.1.1		1	13.3	even	3