Embedded newform 576.2.y.a.239.7

• Label: 576.2.y.a.239.7
• Level: $576$
• Weight: $2$
• Character: 576.239
• Analytic conductor: $4.599$
• Analytic rank: $0$
• Dimension: $88$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	576.y (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.59938315643\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			239.7
Character	\(\chi\)	\(=\)	576.239
Dual form			576.2.y.a.335.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.10465 + 1.33407i) q^{3} +(0.664471 - 0.178044i) q^{5} +(-0.645693 - 1.11837i) q^{7} +(-0.559489 - 2.94737i) q^{9} +(-3.21069 - 0.860301i) q^{11} +(-4.74727 + 1.27203i) q^{13} +(-0.496485 + 1.08313i) q^{15} -5.58523i q^{17} +(2.49649 - 2.49649i) q^{19} +(2.20525 + 0.374013i) q^{21} +(2.36529 + 1.36560i) q^{23} +(-3.92031 + 2.26339i) q^{25} +(4.55004 + 2.50942i) q^{27} +(-2.95682 - 0.792277i) q^{29} +(-5.28160 - 3.04933i) q^{31} +(4.69439 - 3.33295i) q^{33} +(-0.628165 - 0.628165i) q^{35} +(0.507420 - 0.507420i) q^{37} +(3.54711 - 7.73835i) q^{39} +(4.89892 - 8.48518i) q^{41} +(-0.254540 + 0.949956i) q^{43} +(-0.896527 - 1.85883i) q^{45} +(-6.13774 - 10.6309i) q^{47} +(2.66616 - 4.61793i) q^{49} +(7.45109 + 6.16973i) q^{51} +(-0.601793 - 0.601793i) q^{53} -2.28658 q^{55} +(0.572741 + 6.08824i) q^{57} +(-1.28003 - 4.77715i) q^{59} +(-2.90167 + 10.8292i) q^{61} +(-2.93500 + 2.52881i) q^{63} +(-2.92795 + 1.69045i) q^{65} +(0.0295686 + 0.110351i) q^{67} +(-4.43463 + 1.64695i) q^{69} -0.0447904i q^{71} +13.2931i q^{73} +(1.31105 - 7.73022i) q^{75} +(1.11098 + 4.14624i) q^{77} +(-2.50052 + 1.44368i) q^{79} +(-8.37394 + 3.29804i) q^{81} +(-1.01705 + 3.79568i) q^{83} +(-0.994419 - 3.71122i) q^{85} +(4.32321 - 3.06941i) q^{87} +12.7362 q^{89} +(4.48788 + 4.48788i) q^{91} +(9.90236 - 3.67758i) q^{93} +(1.21436 - 2.10333i) q^{95} +(4.41066 + 7.63949i) q^{97} +(-0.739279 + 9.94440i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	88 * q + 4 * q^3 - 6 * q^5 + 4 * q^7 + 6 * q^11 - 2 * q^13 + 8 * q^19 + 2 * q^21 + 12 * q^23 + 16 * q^27 - 6 * q^29 - 8 * q^33 - 8 * q^37 + 32 * q^39 + 2 * q^43 + 6 * q^45 - 24 * q^49 + 12 * q^51 + 16 * q^55 + 42 * q^59 - 2 * q^61 - 12 * q^65 + 2 * q^67 - 10 * q^69 + 56 * q^75 - 6 * q^77 - 8 * q^81 - 54 * q^83 + 8 * q^85 - 48 * q^87 - 20 * q^91 - 34 * q^93 - 4 * q^97 - 46 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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
							\(3\)	−1.10465	+	1.33407i	−0.637771	+	0.770226i
\(5\)	0.664471	−	0.178044i	0.297161	−	0.0796239i								−0.107158	−	0.994242i	\(-0.534175\pi\)
							0.404319	+	0.914618i	\(0.367509\pi\)
\(7\)	−0.645693	−	1.11837i	−0.244049	−	0.422705i	0.717815	−	0.696234i	\(-0.245142\pi\)
							−0.961864	+	0.273529i	\(0.911809\pi\)
\(11\)	−3.21069	−	0.860301i	−0.968058	−	0.259390i	−0.260051	−	0.965595i	\(-0.583739\pi\)
							−0.708008	+	0.706205i	\(0.750406\pi\)
\(13\)	−4.74727	+	1.27203i	−1.31666	+	0.352797i	−0.847725	−	0.530437i	\(-0.822028\pi\)
							−0.468933	+	0.883234i	\(0.655361\pi\)
\(17\)		−	5.58523i		−	1.35462i	−0.735699	−	0.677308i	\(-0.763146\pi\)
							0.735699	−	0.677308i	\(-0.236854\pi\)
\(19\)	2.49649	−	2.49649i	0.572733	−	0.572733i	−0.360158	−	0.932891i	\(-0.617277\pi\)
							0.932891	+	0.360158i	\(0.117277\pi\)
\(23\)	2.36529	+	1.36560i	0.493197	+	0.284747i	0.725900	−	0.687801i	\(-0.241424\pi\)
							−0.232703	+	0.972548i	\(0.574757\pi\)
\(29\)	−2.95682	−	0.792277i	−0.549067	−	0.147122i	−0.0263884	−	0.999652i	\(-0.508401\pi\)
							−0.522679	+	0.852530i	\(0.675067\pi\)
\(31\)	−5.28160	−	3.04933i	−0.948604	−	0.547677i	−0.0559568	−	0.998433i	\(-0.517821\pi\)
							−0.892647	+	0.450757i	\(0.851154\pi\)
\(37\)	0.507420	−	0.507420i	0.0834193	−	0.0834193i	−0.664166	−	0.747585i	\(-0.731213\pi\)
							0.747585	+	0.664166i	\(0.231213\pi\)
\(41\)	4.89892	−	8.48518i	0.765083	−	1.32516i	−0.175120	−	0.984547i	\(-0.556031\pi\)
							0.940203	−	0.340616i	\(-0.110635\pi\)
\(43\)	−0.254540	+	0.949956i	−0.0388170	+	0.144867i	−0.982615	−	0.185657i	\(-0.940559\pi\)
							0.943798	+	0.330524i	\(0.107225\pi\)
\(47\)	−6.13774	−	10.6309i	−0.895281	−	1.55067i	−0.833456	−	0.552586i	\(-0.813641\pi\)
							−0.0618250	−	0.998087i	\(-0.519692\pi\)
\(53\)	−0.601793	−	0.601793i	−0.0826626	−	0.0826626i	0.664567	−	0.747229i	\(-0.268616\pi\)
							−0.747229	+	0.664567i	\(0.768616\pi\)
\(59\)	−1.28003	−	4.77715i	−0.166646	−	0.621932i	−0.997824	−	0.0659263i	\(-0.979000\pi\)
							0.831178	−	0.556006i	\(-0.187667\pi\)
\(61\)	−2.90167	+	10.8292i	−0.371520	+	1.38653i	0.486842	+	0.873490i	\(0.338149\pi\)
							−0.858363	+	0.513043i	\(0.828518\pi\)
\(67\)	0.0295686	+	0.110351i	0.00361238	+	0.0134816i	0.967708	−	0.252072i	\(-0.0811120\pi\)
							−0.964096	+	0.265554i	\(0.914445\pi\)
\(71\)		−	0.0447904i		−	0.00531565i	−0.999996	−	0.00265782i	\(-0.999154\pi\)
							0.999996	−	0.00265782i	\(-0.000846013\pi\)
\(73\)			13.2931i			1.55585i	0.628360	+	0.777923i	\(0.283726\pi\)
							−0.628360	+	0.777923i	\(0.716274\pi\)
\(79\)	−2.50052	+	1.44368i	−0.281331	+	0.162426i	−0.634026	−	0.773312i	\(-0.718599\pi\)
							0.352695	+	0.935738i	\(0.385265\pi\)
\(83\)	−1.01705	+	3.79568i	−0.111636	+	0.416630i	−0.999013	−	0.0444135i	\(-0.985858\pi\)
							0.887378	+	0.461043i	\(0.152525\pi\)
\(89\)	12.7362			1.35003			0.675017	−	0.737802i	\(-0.264136\pi\)
							0.675017	+	0.737802i	\(0.264136\pi\)
\(97\)	4.41066	+	7.63949i	0.447835	+	0.775673i	0.998245	−	0.0592215i	\(-0.0188618\pi\)
							−0.550410	+	0.834895i	\(0.685529\pi\)

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	576.2.y.a.239.7		88
3.2	odd	2		1728.2.z.a.1583.10		88
4.3	odd	2		144.2.u.a.131.15	yes	88
9.2	odd	6	inner	576.2.y.a.47.5		88
9.7	even	3		1728.2.z.a.1007.10		88
12.11	even	2		432.2.v.a.179.8		88
16.5	even	4		144.2.u.a.59.8	yes	88
16.11	odd	4	inner	576.2.y.a.527.5		88
36.7	odd	6		432.2.v.a.35.15		88
36.11	even	6		144.2.u.a.83.8	yes	88
48.5	odd	4		432.2.v.a.395.15		88
48.11	even	4		1728.2.z.a.719.10		88
144.11	even	12	inner	576.2.y.a.335.7		88
144.43	odd	12		1728.2.z.a.143.10		88
144.101	odd	12		144.2.u.a.11.15	✓	88
144.133	even	12		432.2.v.a.251.8		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.15	✓	88	144.101	odd	12
144.2.u.a.59.8	yes	88	16.5	even	4
144.2.u.a.83.8	yes	88	36.11	even	6
144.2.u.a.131.15	yes	88	4.3	odd	2
432.2.v.a.35.15		88	36.7	odd	6
432.2.v.a.179.8		88	12.11	even	2
432.2.v.a.251.8		88	144.133	even	12
432.2.v.a.395.15		88	48.5	odd	4
576.2.y.a.47.5		88	9.2	odd	6	inner
576.2.y.a.239.7		88	1.1	even	1	trivial
576.2.y.a.335.7		88	144.11	even	12	inner
576.2.y.a.527.5		88	16.11	odd	4	inner
1728.2.z.a.143.10		88	144.43	odd	12
1728.2.z.a.719.10		88	48.11	even	4
1728.2.z.a.1007.10		88	9.7	even	3
1728.2.z.a.1583.10		88	3.2	odd	2