Embedded newform 576.2.y.a.239.14

• Label: 576.2.y.a.239.14
• 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			239.14
Character	\(\chi\)	\(=\)	576.239
Dual form			576.2.y.a.335.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.679468 - 1.59321i) q^{3} +(-2.39818 + 0.642590i) q^{5} +(1.93190 + 3.34616i) q^{7} +(-2.07665 - 2.16507i) q^{9} +(4.01810 + 1.07665i) q^{11} +(3.17330 - 0.850284i) q^{13} +(-0.605703 + 4.25743i) q^{15} +1.33161i q^{17} +(6.09021 - 6.09021i) q^{19} +(6.64380 - 0.804327i) q^{21} +(-0.521462 - 0.301066i) q^{23} +(1.00822 - 0.582095i) q^{25} +(-4.86043 + 1.83744i) q^{27} +(0.272752 + 0.0730837i) q^{29} +(5.84901 + 3.37693i) q^{31} +(4.44549 - 5.67013i) q^{33} +(-6.78326 - 6.78326i) q^{35} +(0.00346351 - 0.00346351i) q^{37} +(0.801475 - 5.63348i) q^{39} +(-0.614318 + 1.06403i) q^{41} +(0.151042 - 0.563698i) q^{43} +(6.37143 + 3.85780i) q^{45} +(-1.24972 - 2.16458i) q^{47} +(-3.96450 + 6.86672i) q^{49} +(2.12154 + 0.904787i) q^{51} +(-3.24883 - 3.24883i) q^{53} -10.3280 q^{55} +(-5.56489 - 13.8411i) q^{57} +(-1.44745 - 5.40194i) q^{59} +(0.528668 - 1.97301i) q^{61} +(3.23278 - 11.1315i) q^{63} +(-7.06377 + 4.07827i) q^{65} +(2.65385 + 9.90429i) q^{67} +(-0.833979 + 0.626235i) q^{69} +7.85052i q^{71} +7.41855i q^{73} +(-0.242349 - 2.00182i) q^{75} +(4.15995 + 15.5252i) q^{77} +(0.839597 - 0.484742i) q^{79} +(-0.375072 + 8.99218i) q^{81} +(0.171300 - 0.639302i) q^{83} +(-0.855680 - 3.19344i) q^{85} +(0.301764 - 0.384894i) q^{87} +11.2223 q^{89} +(8.97570 + 8.97570i) q^{91} +(9.35438 - 7.02420i) q^{93} +(-10.6919 + 18.5189i) q^{95} +(-3.24710 - 5.62414i) q^{97} +(-6.01316 - 10.9353i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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
							\(3\)	0.679468	−	1.59321i	0.392291	−	0.919841i
\(5\)	−2.39818	+	0.642590i	−1.07250	+	0.287375i								−0.751519	−	0.659711i	\(-0.770679\pi\)
							−0.320979	+	0.947086i	\(0.604012\pi\)
\(7\)	1.93190	+	3.34616i	0.730191	+	1.26473i	0.956801	+	0.290742i	\(0.0939023\pi\)
							−0.226610	+	0.973985i	\(0.572764\pi\)
\(11\)	4.01810	+	1.07665i	1.21150	+	0.324621i	0.807351	−	0.590071i	\(-0.200900\pi\)
							0.404151	+	0.914692i	\(0.367567\pi\)
\(13\)	3.17330	−	0.850284i	0.880116	−	0.235826i	0.209659	−	0.977775i	\(-0.432765\pi\)
							0.670457	+	0.741948i	\(0.266098\pi\)
\(17\)			1.33161i			0.322963i	0.986876	+	0.161482i	\(0.0516272\pi\)
							−0.986876	+	0.161482i	\(0.948373\pi\)
\(19\)	6.09021	−	6.09021i	1.39719	−	1.39719i	0.589208	−	0.807981i	\(-0.299440\pi\)
							0.807981	−	0.589208i	\(-0.200560\pi\)
\(23\)	−0.521462	−	0.301066i	−0.108732	−	0.0627767i	0.444648	−	0.895706i	\(-0.353329\pi\)
							−0.553380	+	0.832929i	\(0.686662\pi\)
\(29\)	0.272752	+	0.0730837i	0.0506488	+	0.0135713i	0.284054	−	0.958808i	\(-0.408320\pi\)
							−0.233406	+	0.972379i	\(0.574987\pi\)
\(31\)	5.84901	+	3.37693i	1.05051	+	0.606514i	0.922793	−	0.385296i	\(-0.125901\pi\)
							0.127721	+	0.991810i	\(0.459234\pi\)
\(37\)	0.00346351	−	0.00346351i	0.000569398	−	0.000569398i	−0.706822	−	0.707391i	\(-0.749872\pi\)
							0.707391	+	0.706822i	\(0.249872\pi\)
\(41\)	−0.614318	+	1.06403i	−0.0959403	+	0.166174i	−0.910001	−	0.414607i	\(-0.863919\pi\)
							0.814060	+	0.580780i	\(0.197252\pi\)
\(43\)	0.151042	−	0.563698i	0.0230337	−	0.0859631i	−0.953452	−	0.301544i	\(-0.902498\pi\)
							0.976486	+	0.215581i	\(0.0691646\pi\)
\(47\)	−1.24972	−	2.16458i	−0.182290	−	0.315736i	0.760370	−	0.649490i	\(-0.225018\pi\)
							−0.942660	+	0.333754i	\(0.891684\pi\)
\(53\)	−3.24883	−	3.24883i	−0.446262	−	0.446262i	0.447848	−	0.894110i	\(-0.352191\pi\)
							−0.894110	+	0.447848i	\(0.852191\pi\)
\(59\)	−1.44745	−	5.40194i	−0.188441	−	0.703273i	−0.993868	−	0.110577i	\(-0.964730\pi\)
							0.805426	−	0.592696i	\(-0.201936\pi\)
\(61\)	0.528668	−	1.97301i	0.0676890	−	0.252619i	−0.923787	−	0.382906i	\(-0.874923\pi\)
							0.991476	+	0.130287i	\(0.0415900\pi\)
\(67\)	2.65385	+	9.90429i	0.324219	+	1.21000i	0.915095	+	0.403239i	\(0.132116\pi\)
							−0.590876	+	0.806763i	\(0.701218\pi\)
\(71\)			7.85052i			0.931685i	0.884868	+	0.465843i	\(0.154249\pi\)
							−0.884868	+	0.465843i	\(0.845751\pi\)
\(73\)			7.41855i			0.868275i	0.900847	+	0.434138i	\(0.142947\pi\)
							−0.900847	+	0.434138i	\(0.857053\pi\)
\(79\)	0.839597	−	0.484742i	0.0944620	−	0.0545377i	−0.452025	−	0.892005i	\(-0.649298\pi\)
							0.546487	+	0.837468i	\(0.315965\pi\)
\(83\)	0.171300	−	0.639302i	0.0188027	−	0.0701725i	−0.955887	−	0.293735i	\(-0.905102\pi\)
							0.974690	+	0.223562i	\(0.0717685\pi\)
\(89\)	11.2223			1.18956			0.594781	−	0.803887i	\(-0.297239\pi\)
							0.594781	+	0.803887i	\(0.297239\pi\)
\(97\)	−3.24710	−	5.62414i	−0.329693	−	0.571045i	0.652758	−	0.757567i	\(-0.273612\pi\)
							−0.982451	+	0.186521i	\(0.940279\pi\)

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.239.14		88
3.2	odd	2		1728.2.z.a.1583.17		88
4.3	odd	2		144.2.u.a.131.3	yes	88
9.2	odd	6	inner	576.2.y.a.47.19		88
9.7	even	3		1728.2.z.a.1007.17		88
12.11	even	2		432.2.v.a.179.20		88
16.5	even	4		144.2.u.a.59.12	yes	88
16.11	odd	4	inner	576.2.y.a.527.19		88
36.7	odd	6		432.2.v.a.35.11		88
36.11	even	6		144.2.u.a.83.12	yes	88
48.5	odd	4		432.2.v.a.395.11		88
48.11	even	4		1728.2.z.a.719.17		88
144.11	even	12	inner	576.2.y.a.335.14		88
144.43	odd	12		1728.2.z.a.143.17		88
144.101	odd	12		144.2.u.a.11.3	✓	88
144.133	even	12		432.2.v.a.251.20		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.3	✓	88	144.101	odd	12
144.2.u.a.59.12	yes	88	16.5	even	4
144.2.u.a.83.12	yes	88	36.11	even	6
144.2.u.a.131.3	yes	88	4.3	odd	2
432.2.v.a.35.11		88	36.7	odd	6
432.2.v.a.179.20		88	12.11	even	2
432.2.v.a.251.20		88	144.133	even	12
432.2.v.a.395.11		88	48.5	odd	4
576.2.y.a.47.19		88	9.2	odd	6	inner
576.2.y.a.239.14		88	1.1	even	1	trivial
576.2.y.a.335.14		88	144.11	even	12	inner
576.2.y.a.527.19		88	16.11	odd	4	inner
1728.2.z.a.143.17		88	144.43	odd	12
1728.2.z.a.719.17		88	48.11	even	4
1728.2.z.a.1007.17		88	9.7	even	3
1728.2.z.a.1583.17		88	3.2	odd	2