Embedded newform 576.2.bb.e.241.17

• Label: 576.2.bb.e.241.17
• Level: $576$
• Weight: $2$
• Character: 576.241
• Analytic conductor: $4.599$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.59938315643\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			241.17
Character	\(\chi\)	\(=\)	576.241
Dual form			576.2.bb.e.337.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.53873 - 0.795173i) q^{3} +(0.646846 - 2.41406i) q^{5} +(2.82197 + 1.62927i) q^{7} +(1.73540 - 2.44712i) q^{9} +(1.32906 - 0.356120i) q^{11} +(-5.32945 - 1.42802i) q^{13} +(-0.924273 - 4.22895i) q^{15} +5.37452 q^{17} +(-4.71269 + 4.71269i) q^{19} +(5.63781 + 0.263051i) q^{21} +(-2.88877 + 1.66783i) q^{23} +(-1.07916 - 0.623053i) q^{25} +(0.724438 - 5.14540i) q^{27} +(0.814251 + 3.03883i) q^{29} +(-0.621800 - 1.07699i) q^{31} +(1.76189 - 1.60481i) q^{33} +(5.75853 - 5.75853i) q^{35} +(-5.86087 - 5.86087i) q^{37} +(-9.33612 + 2.04049i) q^{39} +(2.81108 - 1.62298i) q^{41} +(6.03232 - 1.61636i) q^{43} +(-4.78496 - 5.77227i) q^{45} +(-2.17485 + 3.76695i) q^{47} +(1.80902 + 3.13331i) q^{49} +(8.26995 - 4.27367i) q^{51} +(-0.134334 - 0.134334i) q^{53} -3.43879i q^{55} +(-3.50417 + 10.9990i) q^{57} +(-0.592405 + 2.21088i) q^{59} +(0.615960 + 2.29879i) q^{61} +(8.88426 - 4.07827i) q^{63} +(-6.89466 + 11.9419i) q^{65} +(-0.112274 - 0.0300838i) q^{67} +(-3.11884 + 4.86343i) q^{69} +3.21118i q^{71} +9.75441i q^{73} +(-2.15597 - 0.100594i) q^{75} +(4.33078 + 1.16043i) q^{77} +(-1.11184 + 1.92576i) q^{79} +(-2.97677 - 8.49346i) q^{81} +(1.54010 + 5.74772i) q^{83} +(3.47649 - 12.9744i) q^{85} +(3.66931 + 4.02848i) q^{87} -6.12376i q^{89} +(-12.7129 - 12.7129i) q^{91} +(-1.81318 - 1.16276i) q^{93} +(8.32833 + 14.4251i) q^{95} +(-2.21495 + 3.83640i) q^{97} +(1.43498 - 3.87037i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	72 * q - 2 * q^3 + 4 * q^5 + 2 * q^11 - 16 * q^13 + 20 * q^15 - 16 * q^17 - 28 * q^19 - 16 * q^21 - 8 * q^27 + 4 * q^29 - 28 * q^31 - 32 * q^33 + 16 * q^35 + 16 * q^37 + 10 * q^43 + 40 * q^45 + 56 * q^47 + 4 * q^49 + 54 * q^51 - 8 * q^53 + 14 * q^59 - 32 * q^61 + 108 * q^63 - 64 * q^65 + 18 * q^67 + 32 * q^69 - 86 * q^75 - 36 * q^77 - 44 * q^79 - 44 * q^81 - 20 * q^83 - 8 * q^85 + 80 * q^91 - 4 * q^93 - 48 * q^95 + 40 * q^97 - 28 * q^99

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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
							\(3\)	1.53873	−	0.795173i	0.888388	−	0.459093i
\(5\)	0.646846	−	2.41406i	0.289278	−	1.07960i								−0.656378	−	0.754433i	\(-0.727912\pi\)
							0.945656	−	0.325169i	\(-0.105421\pi\)
\(7\)	2.82197	+	1.62927i	1.06661	+	0.615805i	0.927252	−	0.374438i	\(-0.122164\pi\)
							0.139353	+	0.990243i	\(0.455498\pi\)
\(11\)	1.32906	−	0.356120i	0.400726	−	0.107374i	−0.0528265	−	0.998604i	\(-0.516823\pi\)
							0.453553	+	0.891229i	\(0.350156\pi\)
\(13\)	−5.32945	−	1.42802i	−1.47812	−	0.396062i	−0.572415	−	0.819964i	\(-0.693993\pi\)
							−0.905708	+	0.423903i	\(0.860660\pi\)
\(17\)	5.37452			1.30351			0.651756	−	0.758428i	\(-0.274032\pi\)
							0.651756	+	0.758428i	\(0.274032\pi\)
\(19\)	−4.71269	+	4.71269i	−1.08116	+	1.08116i	−0.0847630	+	0.996401i	\(0.527013\pi\)
							−0.996401	+	0.0847630i	\(0.972987\pi\)
\(23\)	−2.88877	+	1.66783i	−0.602351	+	0.347767i	−0.769966	−	0.638085i	\(-0.779727\pi\)
							0.167615	+	0.985853i	\(0.446393\pi\)
\(29\)	0.814251	+	3.03883i	0.151203	+	0.564296i	0.999401	+	0.0346168i	\(0.0110211\pi\)
							−0.848198	+	0.529679i	\(0.822312\pi\)
\(31\)	−0.621800	−	1.07699i	−0.111679	−	0.193433i	0.804769	−	0.593589i	\(-0.202289\pi\)
							−0.916447	+	0.400156i	\(0.868956\pi\)
\(37\)	−5.86087	−	5.86087i	−0.963521	−	0.963521i	0.0358368	−	0.999358i	\(-0.488590\pi\)
							−0.999358	+	0.0358368i	\(0.988590\pi\)
\(41\)	2.81108	−	1.62298i	0.439017	−	0.253467i	−0.264163	−	0.964478i	\(-0.585096\pi\)
							0.703181	+	0.711011i	\(0.251763\pi\)
\(43\)	6.03232	−	1.61636i	0.919920	−	0.246492i	0.232369	−	0.972628i	\(-0.425352\pi\)
							0.687551	+	0.726136i	\(0.258686\pi\)
\(47\)	−2.17485	+	3.76695i	−0.317234	+	0.549466i	−0.979910	−	0.199441i	\(-0.936088\pi\)
							0.662676	+	0.748907i	\(0.269421\pi\)
\(53\)	−0.134334	−	0.134334i	−0.0184522	−	0.0184522i	0.697820	−	0.716273i	\(-0.254153\pi\)
							−0.716273	+	0.697820i	\(0.754153\pi\)
\(59\)	−0.592405	+	2.21088i	−0.0771245	+	0.287833i	−0.993707	−	0.112014i	\(-0.964270\pi\)
							0.916582	+	0.399846i	\(0.130937\pi\)
\(61\)	0.615960	+	2.29879i	0.0788656	+	0.294330i	0.994082	−	0.108632i	\(-0.0346471\pi\)
							−0.915216	+	0.402963i	\(0.867980\pi\)
\(67\)	−0.112274	−	0.0300838i	−0.0137165	−	0.00367532i	0.251954	−	0.967739i	\(-0.418927\pi\)
							−0.265671	+	0.964064i	\(0.585593\pi\)
\(71\)			3.21118i			0.381097i	0.981678	+	0.190548i	\(0.0610266\pi\)
							−0.981678	+	0.190548i	\(0.938973\pi\)
\(73\)			9.75441i			1.14167i	0.821066	+	0.570833i	\(0.193380\pi\)
							−0.821066	+	0.570833i	\(0.806620\pi\)
\(79\)	−1.11184	+	1.92576i	−0.125092	+	0.216665i	−0.921769	−	0.387740i	\(-0.873256\pi\)
							0.796677	+	0.604405i	\(0.206589\pi\)
\(83\)	1.54010	+	5.74772i	0.169048	+	0.630895i	0.997489	+	0.0708190i	\(0.0225613\pi\)
							−0.828441	+	0.560076i	\(0.810772\pi\)
\(89\)		−	6.12376i		−	0.649118i	−0.945866	−	0.324559i	\(-0.894784\pi\)
							0.945866	−	0.324559i	\(-0.105216\pi\)
\(97\)	−2.21495	+	3.83640i	−0.224894	+	0.389528i	−0.956288	−	0.292428i	\(-0.905537\pi\)
							0.731394	+	0.681956i	\(0.238870\pi\)

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.bb.e.241.17		72
3.2	odd	2		1728.2.bc.e.1585.3		72
4.3	odd	2		144.2.x.e.133.8	yes	72
9.4	even	3	inner	576.2.bb.e.49.12		72
9.5	odd	6		1728.2.bc.e.1009.16		72
12.11	even	2		432.2.y.e.181.11		72
16.3	odd	4		144.2.x.e.61.16	yes	72
16.13	even	4	inner	576.2.bb.e.529.12		72
36.23	even	6		432.2.y.e.37.3		72
36.31	odd	6		144.2.x.e.85.16	yes	72
48.29	odd	4		1728.2.bc.e.721.16		72
48.35	even	4		432.2.y.e.397.3		72
144.13	even	12	inner	576.2.bb.e.337.17		72
144.67	odd	12		144.2.x.e.13.8	✓	72
144.77	odd	12		1728.2.bc.e.145.3		72
144.131	even	12		432.2.y.e.253.11		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.e.13.8	✓	72	144.67	odd	12
144.2.x.e.61.16	yes	72	16.3	odd	4
144.2.x.e.85.16	yes	72	36.31	odd	6
144.2.x.e.133.8	yes	72	4.3	odd	2
432.2.y.e.37.3		72	36.23	even	6
432.2.y.e.181.11		72	12.11	even	2
432.2.y.e.253.11		72	144.131	even	12
432.2.y.e.397.3		72	48.35	even	4
576.2.bb.e.49.12		72	9.4	even	3	inner
576.2.bb.e.241.17		72	1.1	even	1	trivial
576.2.bb.e.337.17		72	144.13	even	12	inner
576.2.bb.e.529.12		72	16.13	even	4	inner
1728.2.bc.e.145.3		72	144.77	odd	12
1728.2.bc.e.721.16		72	48.29	odd	4
1728.2.bc.e.1009.16		72	9.5	odd	6
1728.2.bc.e.1585.3		72	3.2	odd	2