Embedded newform 1728.2.s.d.575.1

• Label: 1728.2.s.d.575.1
• Level: $1728$
• Weight: $2$
• Character: 1728.575
• Analytic conductor: $13.798$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.7981494693\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			575.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	1728.575
Dual form			1728.2.s.d.1151.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.50000 - 0.866025i) q^{5} +(1.50000 + 0.866025i) q^{7} +(-1.50000 + 2.59808i) q^{11} +(-2.50000 - 4.33013i) q^{13} +6.92820i q^{17} +3.46410i q^{19} +(4.50000 + 7.79423i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(1.50000 + 0.866025i) q^{29} +(4.50000 - 2.59808i) q^{31} +3.00000 q^{35} -2.00000 q^{37} +(4.50000 - 2.59808i) q^{41} +(4.50000 + 2.59808i) q^{43} +(1.50000 - 2.59808i) q^{47} +(-2.00000 - 3.46410i) q^{49} +5.19615i q^{55} +(1.50000 + 2.59808i) q^{59} +(-0.500000 + 0.866025i) q^{61} +(-7.50000 - 4.33013i) q^{65} +(7.50000 - 4.33013i) q^{67} -12.0000 q^{71} -2.00000 q^{73} +(-4.50000 + 2.59808i) q^{77} +(7.50000 + 4.33013i) q^{79} +(-7.50000 + 12.9904i) q^{83} +(6.00000 + 10.3923i) q^{85} -6.92820i q^{89} -8.66025i q^{91} +(3.00000 + 5.19615i) q^{95} +(2.50000 - 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 3 * q^5 + 3 * q^7 - 3 * q^11 - 5 * q^13 + 9 * q^23 - 2 * q^25 + 3 * q^29 + 9 * q^31 + 6 * q^35 - 4 * q^37 + 9 * q^41 + 9 * q^43 + 3 * q^47 - 4 * q^49 + 3 * q^59 - q^61 - 15 * q^65 + 15 * q^67 - 24 * q^71 - 4 * q^73 - 9 * q^77 + 15 * q^79 - 15 * q^83 + 12 * q^85 + 6 * q^95 + 5 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(703\)	\(1217\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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			0
\(5\)	1.50000	−	0.866025i	0.670820	−	0.387298i								−0.125567	−	0.992085i	\(-0.540075\pi\)
							0.796387	+	0.604787i	\(0.206742\pi\)
\(7\)	1.50000	+	0.866025i	0.566947	+	0.327327i	0.755929	−	0.654654i	\(-0.227186\pi\)
							−0.188982	+	0.981981i	\(0.560519\pi\)
\(11\)	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	\(-0.982718\pi\)
							0.546259	+	0.837616i	\(0.316051\pi\)
\(13\)	−2.50000	−	4.33013i	−0.693375	−	1.20096i	−0.970725	−	0.240192i	\(-0.922790\pi\)
							0.277350	−	0.960769i	\(-0.410544\pi\)
\(17\)			6.92820i			1.68034i	0.542326	+	0.840168i	\(0.317544\pi\)
							−0.542326	+	0.840168i	\(0.682456\pi\)
\(19\)			3.46410i			0.794719i	0.917663	+	0.397360i	\(0.130073\pi\)
							−0.917663	+	0.397360i	\(0.869927\pi\)
\(23\)	4.50000	+	7.79423i	0.938315	+	1.62521i	0.768613	+	0.639713i	\(0.220947\pi\)
							0.169701	+	0.985496i	\(0.445720\pi\)
\(29\)	1.50000	+	0.866025i	0.278543	+	0.160817i	0.632764	−	0.774345i	\(-0.281920\pi\)
							−0.354221	+	0.935162i	\(0.615254\pi\)
\(31\)	4.50000	−	2.59808i	0.808224	−	0.466628i	−0.0381148	−	0.999273i	\(-0.512135\pi\)
							0.846339	+	0.532645i	\(0.178802\pi\)
\(37\)	−2.00000			−0.328798			−0.164399	−	0.986394i	\(-0.552568\pi\)
							−0.164399	+	0.986394i	\(0.552568\pi\)
\(41\)	4.50000	−	2.59808i	0.702782	−	0.405751i	−0.105601	−	0.994409i	\(-0.533677\pi\)
							0.808383	+	0.588657i	\(0.200343\pi\)
\(43\)	4.50000	+	2.59808i	0.686244	+	0.396203i	0.802203	−	0.597051i	\(-0.203661\pi\)
							−0.115960	+	0.993254i	\(0.536994\pi\)
\(47\)	1.50000	−	2.59808i	0.218797	−	0.378968i	−0.735643	−	0.677369i	\(-0.763120\pi\)
							0.954441	+	0.298401i	\(0.0964533\pi\)
\(53\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(59\)	1.50000	+	2.59808i	0.195283	+	0.338241i	0.946993	−	0.321253i	\(-0.104104\pi\)
							−0.751710	+	0.659494i	\(0.770771\pi\)
\(61\)	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	−0.896258	−	0.443533i	\(-0.853725\pi\)
							0.832240	+	0.554416i	\(0.187058\pi\)
\(67\)	7.50000	−	4.33013i	0.916271	−	0.529009i	0.0338274	−	0.999428i	\(-0.489230\pi\)
							0.882443	+	0.470418i	\(0.155897\pi\)
\(71\)	−12.0000			−1.42414			−0.712069	−	0.702109i	\(-0.752242\pi\)
							−0.712069	+	0.702109i	\(0.752242\pi\)
\(73\)	−2.00000			−0.234082			−0.117041	−	0.993127i	\(-0.537341\pi\)
							−0.117041	+	0.993127i	\(0.537341\pi\)
\(79\)	7.50000	+	4.33013i	0.843816	+	0.487177i	0.858559	−	0.512714i	\(-0.171360\pi\)
							−0.0147436	+	0.999891i	\(0.504693\pi\)
\(83\)	−7.50000	+	12.9904i	−0.823232	+	1.42588i	0.0800311	+	0.996792i	\(0.474498\pi\)
							−0.903263	+	0.429087i	\(0.858835\pi\)
\(89\)		−	6.92820i		−	0.734388i	−0.930144	−	0.367194i	\(-0.880318\pi\)
							0.930144	−	0.367194i	\(-0.119682\pi\)
\(97\)	2.50000	−	4.33013i	0.253837	−	0.439658i	−0.710742	−	0.703452i	\(-0.751641\pi\)
							0.964579	+	0.263795i	\(0.0849741\pi\)

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	1728.2.s.d.575.1		2
3.2	odd	2		576.2.s.c.191.1		2
4.3	odd	2		1728.2.s.c.575.1		2
8.3	odd	2		432.2.s.a.143.1		2
8.5	even	2		432.2.s.b.143.1		2
9.2	odd	6		5184.2.c.b.5183.1		2
9.4	even	3		576.2.s.b.383.1		2
9.5	odd	6		1728.2.s.c.1151.1		2
9.7	even	3		5184.2.c.d.5183.2		2
12.11	even	2		576.2.s.b.191.1		2
24.5	odd	2		144.2.s.c.47.1	yes	2
24.11	even	2		144.2.s.b.47.1	✓	2
36.7	odd	6		5184.2.c.b.5183.2		2
36.11	even	6		5184.2.c.d.5183.1		2
36.23	even	6	inner	1728.2.s.d.1151.1		2
36.31	odd	6		576.2.s.c.383.1		2
72.5	odd	6		432.2.s.a.287.1		2
72.11	even	6		1296.2.c.a.1295.2		2
72.13	even	6		144.2.s.b.95.1	yes	2
72.29	odd	6		1296.2.c.c.1295.2		2
72.43	odd	6		1296.2.c.c.1295.1		2
72.59	even	6		432.2.s.b.287.1		2
72.61	even	6		1296.2.c.a.1295.1		2
72.67	odd	6		144.2.s.c.95.1	yes	2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.s.b.47.1	✓	2	24.11	even	2
144.2.s.b.95.1	yes	2	72.13	even	6
144.2.s.c.47.1	yes	2	24.5	odd	2
144.2.s.c.95.1	yes	2	72.67	odd	6
432.2.s.a.143.1		2	8.3	odd	2
432.2.s.a.287.1		2	72.5	odd	6
432.2.s.b.143.1		2	8.5	even	2
432.2.s.b.287.1		2	72.59	even	6
576.2.s.b.191.1		2	12.11	even	2
576.2.s.b.383.1		2	9.4	even	3
576.2.s.c.191.1		2	3.2	odd	2
576.2.s.c.383.1		2	36.31	odd	6
1296.2.c.a.1295.1		2	72.61	even	6
1296.2.c.a.1295.2		2	72.11	even	6
1296.2.c.c.1295.1		2	72.43	odd	6
1296.2.c.c.1295.2		2	72.29	odd	6
1728.2.s.c.575.1		2	4.3	odd	2
1728.2.s.c.1151.1		2	9.5	odd	6
1728.2.s.d.575.1		2	1.1	even	1	trivial
1728.2.s.d.1151.1		2	36.23	even	6	inner
5184.2.c.b.5183.1		2	9.2	odd	6
5184.2.c.b.5183.2		2	36.7	odd	6
5184.2.c.d.5183.1		2	36.11	even	6
5184.2.c.d.5183.2		2	9.7	even	3