Embedded newform 576.2.bb.e.337.10

• Label: 576.2.bb.e.337.10
• Level: $576$
• Weight: $2$
• Character: 576.337
• 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			337.10
Character	\(\chi\)	\(=\)	576.337
Dual form			576.2.bb.e.241.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.241506 - 1.71513i) q^{3} +(0.531653 + 1.98415i) q^{5} +(-1.54969 + 0.894715i) q^{7} +(-2.88335 + 0.828427i) q^{9} +(-2.58637 - 0.693015i) q^{11} +(-4.63944 + 1.24314i) q^{13} +(3.27469 - 1.39104i) q^{15} -3.58889 q^{17} +(-4.85244 - 4.85244i) q^{19} +(1.90881 + 2.44185i) q^{21} +(0.446082 + 0.257545i) q^{23} +(0.675912 - 0.390238i) q^{25} +(2.11721 + 4.74525i) q^{27} +(1.72815 - 6.44956i) q^{29} +(-4.05128 + 7.01703i) q^{31} +(-0.563989 + 4.60332i) q^{33} +(-2.59915 - 2.59915i) q^{35} +(1.25948 - 1.25948i) q^{37} +(3.25259 + 7.65703i) q^{39} +(4.07959 + 2.35535i) q^{41} +(-6.57670 - 1.76222i) q^{43} +(-3.17667 - 5.28058i) q^{45} +(3.48945 + 6.04391i) q^{47} +(-1.89897 + 3.28911i) q^{49} +(0.866737 + 6.15542i) q^{51} +(5.26302 - 5.26302i) q^{53} -5.50019i q^{55} +(-7.15068 + 9.49446i) q^{57} +(1.81387 + 6.76946i) q^{59} +(-1.55082 + 5.78773i) q^{61} +(3.72710 - 3.86359i) q^{63} +(-4.93314 - 8.54446i) q^{65} +(-1.69184 + 0.453328i) q^{67} +(0.333993 - 0.827288i) q^{69} -7.58339i q^{71} +12.5473i q^{73} +(-0.832546 - 1.06503i) q^{75} +(4.62812 - 1.24010i) q^{77} +(4.01735 + 6.95825i) q^{79} +(7.62742 - 4.77729i) q^{81} +(2.14313 - 7.99826i) q^{83} +(-1.90804 - 7.12091i) q^{85} +(-11.4792 - 1.40641i) q^{87} -16.5414i q^{89} +(6.07746 - 6.07746i) q^{91} +(13.0135 + 5.25383i) q^{93} +(7.04818 - 12.2078i) q^{95} +(-4.15739 - 7.20082i) q^{97} +(8.03151 - 0.144412i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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 \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

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

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
							\(3\)	−0.241506	−	1.71513i	−0.139433	−	0.990231i
\(5\)	0.531653	+	1.98415i	0.237762	+	0.887341i								0.976884	+	0.213769i	\(0.0685741\pi\)
							−0.739122	+	0.673572i	\(0.764759\pi\)
\(7\)	−1.54969	+	0.894715i	−0.585729	+	0.338171i	−0.763407	−	0.645918i	\(-0.776475\pi\)
							0.177678	+	0.984089i	\(0.443141\pi\)
\(11\)	−2.58637	−	0.693015i	−0.779819	−	0.208952i	−0.153114	−	0.988209i	\(-0.548930\pi\)
							−0.626705	+	0.779257i	\(0.715597\pi\)
\(13\)	−4.63944	+	1.24314i	−1.28675	+	0.344784i	−0.836425	−	0.548081i	\(-0.815359\pi\)
							−0.450325	+	0.892865i	\(0.648692\pi\)
\(17\)	−3.58889			−0.870434			−0.435217	−	0.900326i	\(-0.643328\pi\)
							−0.435217	+	0.900326i	\(0.643328\pi\)
\(19\)	−4.85244	−	4.85244i	−1.11323	−	1.11323i	−0.992712	−	0.120514i	\(-0.961546\pi\)
							−0.120514	−	0.992712i	\(-0.538454\pi\)
\(23\)	0.446082	+	0.257545i	0.0930145	+	0.0537019i	0.545786	−	0.837925i	\(-0.316231\pi\)
							−0.452771	+	0.891627i	\(0.649565\pi\)
\(29\)	1.72815	−	6.44956i	0.320910	−	1.19765i	−0.597450	−	0.801906i	\(-0.703819\pi\)
							0.918360	−	0.395746i	\(-0.129514\pi\)
\(31\)	−4.05128	+	7.01703i	−0.727632	+	1.26030i	0.230249	+	0.973132i	\(0.426046\pi\)
							−0.957881	+	0.287164i	\(0.907287\pi\)
\(37\)	1.25948	−	1.25948i	0.207057	−	0.207057i	−0.595959	−	0.803015i	\(-0.703228\pi\)
							0.803015	+	0.595959i	\(0.203228\pi\)
\(41\)	4.07959	+	2.35535i	0.637126	+	0.367845i	0.783506	−	0.621384i	\(-0.213429\pi\)
							−0.146381	+	0.989228i	\(0.546762\pi\)
\(43\)	−6.57670	−	1.76222i	−1.00294	−	0.268736i	−0.280262	−	0.959924i	\(-0.590421\pi\)
							−0.722676	+	0.691187i	\(0.757088\pi\)
\(47\)	3.48945	+	6.04391i	0.508989	+	0.881595i	0.999946	+	0.0104109i	\(0.00331395\pi\)
							−0.490957	+	0.871184i	\(0.663353\pi\)
\(53\)	5.26302	−	5.26302i	0.722932	−	0.722932i	−0.246269	−	0.969201i	\(-0.579205\pi\)
							0.969201	+	0.246269i	\(0.0792048\pi\)
\(59\)	1.81387	+	6.76946i	0.236146	+	0.881308i	0.977629	+	0.210337i	\(0.0674561\pi\)
							−0.741483	+	0.670972i	\(0.765877\pi\)
\(61\)	−1.55082	+	5.78773i	−0.198562	+	0.741042i	0.792754	+	0.609541i	\(0.208646\pi\)
							−0.991316	+	0.131501i	\(0.958020\pi\)
\(67\)	−1.69184	+	0.453328i	−0.206692	+	0.0553829i	−0.360679	−	0.932690i	\(-0.617455\pi\)
							0.153988	+	0.988073i	\(0.450788\pi\)
\(71\)		−	7.58339i		−	0.899983i	−0.893033	−	0.449992i	\(-0.851427\pi\)
							0.893033	−	0.449992i	\(-0.148573\pi\)
\(73\)			12.5473i			1.46855i	0.678854	+	0.734273i	\(0.262477\pi\)
							−0.678854	+	0.734273i	\(0.737523\pi\)
\(79\)	4.01735	+	6.95825i	0.451987	+	0.782864i	0.998509	−	0.0545802i	\(-0.0173821\pi\)
							−0.546523	+	0.837444i	\(0.684049\pi\)
\(83\)	2.14313	−	7.99826i	0.235239	−	0.877923i	−0.742802	−	0.669511i	\(-0.766504\pi\)
							0.978041	−	0.208412i	\(-0.0668295\pi\)
\(89\)		−	16.5414i		−	1.75339i	−0.481047	−	0.876695i	\(-0.659743\pi\)
							0.481047	−	0.876695i	\(-0.340257\pi\)
\(97\)	−4.15739	−	7.20082i	−0.422119	−	0.731132i	0.574027	−	0.818836i	\(-0.305380\pi\)
							−0.996147	+	0.0877040i	\(0.972047\pi\)

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.337.10		72
3.2	odd	2		1728.2.bc.e.145.5		72
4.3	odd	2		144.2.x.e.13.2	✓	72
9.2	odd	6		1728.2.bc.e.721.14		72
9.7	even	3	inner	576.2.bb.e.529.1		72
12.11	even	2		432.2.y.e.253.17		72
16.5	even	4	inner	576.2.bb.e.49.1		72
16.11	odd	4		144.2.x.e.85.15	yes	72
36.7	odd	6		144.2.x.e.61.15	yes	72
36.11	even	6		432.2.y.e.397.4		72
48.5	odd	4		1728.2.bc.e.1009.14		72
48.11	even	4		432.2.y.e.37.4		72
144.11	even	12		432.2.y.e.181.17		72
144.43	odd	12		144.2.x.e.133.2	yes	72
144.101	odd	12		1728.2.bc.e.1585.5		72
144.133	even	12	inner	576.2.bb.e.241.10		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.e.13.2	✓	72	4.3	odd	2
144.2.x.e.61.15	yes	72	36.7	odd	6
144.2.x.e.85.15	yes	72	16.11	odd	4
144.2.x.e.133.2	yes	72	144.43	odd	12
432.2.y.e.37.4		72	48.11	even	4
432.2.y.e.181.17		72	144.11	even	12
432.2.y.e.253.17		72	12.11	even	2
432.2.y.e.397.4		72	36.11	even	6
576.2.bb.e.49.1		72	16.5	even	4	inner
576.2.bb.e.241.10		72	144.133	even	12	inner
576.2.bb.e.337.10		72	1.1	even	1	trivial
576.2.bb.e.529.1		72	9.7	even	3	inner
1728.2.bc.e.145.5		72	3.2	odd	2
1728.2.bc.e.721.14		72	9.2	odd	6
1728.2.bc.e.1009.14		72	48.5	odd	4
1728.2.bc.e.1585.5		72	144.101	odd	12