Embedded newform 576.2.bb.e.49.9

• Label: 576.2.bb.e.49.9
• Level: $576$
• Weight: $2$
• Character: 576.49
• 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			49.9
Character	\(\chi\)	\(=\)	576.49
Dual form			576.2.bb.e.529.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.222657 + 1.71768i) q^{3} +(2.97932 - 0.798307i) q^{5} +(-1.78208 + 1.02889i) q^{7} +(-2.90085 + 0.764906i) q^{9} +(-0.119598 + 0.446347i) q^{11} +(1.52075 + 5.67550i) q^{13} +(2.03460 + 4.93977i) q^{15} +0.0443921 q^{17} +(1.10726 - 1.10726i) q^{19} +(-2.16409 - 2.83196i) q^{21} +(7.89263 + 4.55681i) q^{23} +(3.90893 - 2.25682i) q^{25} +(-1.95976 - 4.81242i) q^{27} +(-6.95662 - 1.86402i) q^{29} +(-0.542236 + 0.939180i) q^{31} +(-0.793310 - 0.106049i) q^{33} +(-4.48803 + 4.48803i) q^{35} +(-0.769054 - 0.769054i) q^{37} +(-9.41009 + 3.87585i) q^{39} +(5.77193 + 3.33242i) q^{41} +(2.96351 - 11.0600i) q^{43} +(-8.03193 + 4.59467i) q^{45} +(1.22453 + 2.12095i) q^{47} +(-1.38279 + 2.39506i) q^{49} +(0.00988420 + 0.0762514i) q^{51} +(2.44801 + 2.44801i) q^{53} +1.42529i q^{55} +(2.14846 + 1.65538i) q^{57} +(-3.59122 + 0.962265i) q^{59} +(-1.18758 - 0.318210i) q^{61} +(4.38255 - 4.34777i) q^{63} +(9.06158 + 15.6951i) q^{65} +(-1.48102 - 5.52723i) q^{67} +(-6.06980 + 14.5716i) q^{69} +6.88571i q^{71} -13.1963i q^{73} +(4.74685 + 6.21180i) q^{75} +(-0.246106 - 0.918480i) q^{77} +(-3.46441 - 6.00054i) q^{79} +(7.82984 - 4.43775i) q^{81} +(0.588112 + 0.157584i) q^{83} +(0.132258 - 0.0354385i) q^{85} +(1.65285 - 12.3643i) q^{87} -5.30004i q^{89} +(-8.54954 - 8.54954i) q^{91} +(-1.73394 - 0.722273i) q^{93} +(2.41495 - 4.18282i) q^{95} +(-5.88304 - 10.1897i) q^{97} +(0.00552307 - 1.38627i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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 \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

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

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
							\(3\)	0.222657	+	1.71768i	0.128551	+	0.991703i
\(5\)	2.97932	−	0.798307i	1.33239	−	0.357014i								0.478786	−	0.877932i	\(-0.341077\pi\)
							0.853607	+	0.520918i	\(0.174410\pi\)
\(7\)	−1.78208	+	1.02889i	−0.673564	+	0.388882i	−0.797426	−	0.603417i	\(-0.793805\pi\)
							0.123862	+	0.992299i	\(0.460472\pi\)
\(11\)	−0.119598	+	0.446347i	−0.0360602	+	0.134579i	−0.981609	−	0.190901i	\(-0.938859\pi\)
							0.945549	+	0.325480i	\(0.105526\pi\)
\(13\)	1.52075	+	5.67550i	0.421779	+	1.57410i	0.770857	+	0.637008i	\(0.219828\pi\)
							−0.349078	+	0.937094i	\(0.613505\pi\)
\(17\)	0.0443921			0.0107667			0.00538333	−	0.999986i	\(-0.498286\pi\)
							0.00538333	+	0.999986i	\(0.498286\pi\)
\(19\)	1.10726	−	1.10726i	0.254023	−	0.254023i	−0.568595	−	0.822618i	\(-0.692513\pi\)
							0.822618	+	0.568595i	\(0.192513\pi\)
\(23\)	7.89263	+	4.55681i	1.64573	+	0.950161i	0.978743	+	0.205089i	\(0.0657485\pi\)
							0.666984	+	0.745072i	\(0.267585\pi\)
\(29\)	−6.95662	−	1.86402i	−1.29181	−	0.346140i	−0.453464	−	0.891275i	\(-0.649812\pi\)
							−0.838349	+	0.545134i	\(0.816479\pi\)
\(31\)	−0.542236	+	0.939180i	−0.0973884	+	0.168682i	−0.910603	−	0.413282i	\(-0.864382\pi\)
							0.813214	+	0.581964i	\(0.197716\pi\)
\(37\)	−0.769054	−	0.769054i	−0.126432	−	0.126432i	0.641060	−	0.767491i	\(-0.278495\pi\)
							−0.767491	+	0.641060i	\(0.778495\pi\)
\(41\)	5.77193	+	3.33242i	0.901423	+	0.520437i	0.877662	−	0.479281i	\(-0.159102\pi\)
							0.0237617	+	0.999718i	\(0.492436\pi\)
\(43\)	2.96351	−	11.0600i	0.451932	−	1.68663i	−0.245023	−	0.969517i	\(-0.578796\pi\)
							0.696955	−	0.717115i	\(-0.254538\pi\)
\(47\)	1.22453	+	2.12095i	0.178617	+	0.309373i	0.941407	−	0.337273i	\(-0.109504\pi\)
							−0.762790	+	0.646646i	\(0.776171\pi\)
\(53\)	2.44801	+	2.44801i	0.336260	+	0.336260i	0.854958	−	0.518698i	\(-0.173583\pi\)
							−0.518698	+	0.854958i	\(0.673583\pi\)
\(59\)	−3.59122	+	0.962265i	−0.467537	+	0.125276i	−0.484894	−	0.874573i	\(-0.661142\pi\)
							0.0173563	+	0.999849i	\(0.494475\pi\)
\(61\)	−1.18758	−	0.318210i	−0.152054	−	0.0407426i	0.181989	−	0.983301i	\(-0.441746\pi\)
							−0.334043	+	0.942558i	\(0.608413\pi\)
\(67\)	−1.48102	−	5.52723i	−0.180935	−	0.675258i	−0.995464	−	0.0951361i	\(-0.969671\pi\)
							0.814529	−	0.580122i	\(-0.196995\pi\)
\(71\)			6.88571i			0.817184i	0.912717	+	0.408592i	\(0.133980\pi\)
							−0.912717	+	0.408592i	\(0.866020\pi\)
\(73\)		−	13.1963i		−	1.54451i	−0.635312	−	0.772255i	\(-0.719129\pi\)
							0.635312	−	0.772255i	\(-0.280871\pi\)
\(79\)	−3.46441	−	6.00054i	−0.389777	−	0.675113i	0.602643	−	0.798011i	\(-0.294115\pi\)
							−0.992419	+	0.122898i	\(0.960781\pi\)
\(83\)	0.588112	+	0.157584i	0.0645537	+	0.0172971i	0.290951	−	0.956738i	\(-0.406028\pi\)
							−0.226398	+	0.974035i	\(0.572695\pi\)
\(89\)		−	5.30004i		−	0.561803i	−0.959737	−	0.280902i	\(-0.909367\pi\)
							0.959737	−	0.280902i	\(-0.0906335\pi\)
\(97\)	−5.88304	−	10.1897i	−0.597333	−	1.03461i	−0.993213	−	0.116309i	\(-0.962894\pi\)
							0.395880	−	0.918302i	\(-0.370439\pi\)

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.49.9		72
3.2	odd	2		1728.2.bc.e.1009.3		72
4.3	odd	2		144.2.x.e.85.11	yes	72
9.2	odd	6		1728.2.bc.e.1585.16		72
9.7	even	3	inner	576.2.bb.e.241.2		72
12.11	even	2		432.2.y.e.37.8		72
16.3	odd	4		144.2.x.e.13.14	✓	72
16.13	even	4	inner	576.2.bb.e.337.2		72
36.7	odd	6		144.2.x.e.133.14	yes	72
36.11	even	6		432.2.y.e.181.5		72
48.29	odd	4		1728.2.bc.e.145.16		72
48.35	even	4		432.2.y.e.253.5		72
144.29	odd	12		1728.2.bc.e.721.3		72
144.61	even	12	inner	576.2.bb.e.529.9		72
144.83	even	12		432.2.y.e.397.8		72
144.115	odd	12		144.2.x.e.61.11	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.e.13.14	✓	72	16.3	odd	4
144.2.x.e.61.11	yes	72	144.115	odd	12
144.2.x.e.85.11	yes	72	4.3	odd	2
144.2.x.e.133.14	yes	72	36.7	odd	6
432.2.y.e.37.8		72	12.11	even	2
432.2.y.e.181.5		72	36.11	even	6
432.2.y.e.253.5		72	48.35	even	4
432.2.y.e.397.8		72	144.83	even	12
576.2.bb.e.49.9		72	1.1	even	1	trivial
576.2.bb.e.241.2		72	9.7	even	3	inner
576.2.bb.e.337.2		72	16.13	even	4	inner
576.2.bb.e.529.9		72	144.61	even	12	inner
1728.2.bc.e.145.16		72	48.29	odd	4
1728.2.bc.e.721.3		72	144.29	odd	12
1728.2.bc.e.1009.3		72	3.2	odd	2
1728.2.bc.e.1585.16		72	9.2	odd	6