Embedded newform 324.2.e.d.217.1

• Label: 324.2.e.d.217.1
• Level: $324$
• Weight: $2$
• Character: 324.217
• Analytic conductor: $2.587$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.58715302549\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	324.217
Dual form			324.2.e.d.109.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.50000 + 2.59808i) q^{5} +(-1.00000 + 1.73205i) q^{7} +(-3.00000 + 5.19615i) q^{11} +(-2.50000 - 4.33013i) q^{13} +3.00000 q^{17} +2.00000 q^{19} +(3.00000 + 5.19615i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(1.50000 - 2.59808i) q^{29} +(2.00000 + 3.46410i) q^{31} -6.00000 q^{35} +5.00000 q^{37} +(-3.00000 - 5.19615i) q^{41} +(5.00000 - 8.66025i) q^{43} +(1.50000 + 2.59808i) q^{49} +6.00000 q^{53} -18.0000 q^{55} +(-6.00000 - 10.3923i) q^{59} +(-2.50000 + 4.33013i) q^{61} +(7.50000 - 12.9904i) q^{65} +(-1.00000 - 1.73205i) q^{67} -6.00000 q^{71} -1.00000 q^{73} +(-6.00000 - 10.3923i) q^{77} +(5.00000 - 8.66025i) q^{79} +(4.50000 + 7.79423i) q^{85} +3.00000 q^{89} +10.0000 q^{91} +(3.00000 + 5.19615i) q^{95} +(5.00000 - 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 3 * q^5 - 2 * q^7 - 6 * q^11 - 5 * q^13 + 6 * q^17 + 4 * q^19 + 6 * q^23 - 4 * q^25 + 3 * q^29 + 4 * q^31 - 12 * q^35 + 10 * q^37 - 6 * q^41 + 10 * q^43 + 3 * q^49 + 12 * q^53 - 36 * q^55 - 12 * q^59 - 5 * q^61 + 15 * q^65 - 2 * q^67 - 12 * q^71 - 2 * q^73 - 12 * q^77 + 10 * q^79 + 9 * q^85 + 6 * q^89 + 20 * q^91 + 6 * q^95 + 10 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\)	\(163\)	\(245\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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	+	2.59808i	0.670820	+	1.16190i								0.977672	+	0.210138i	\(0.0673912\pi\)
							−0.306851	+	0.951757i	\(0.599275\pi\)
\(7\)	−1.00000	+	1.73205i	−0.377964	+	0.654654i	−0.990766	−	0.135583i	\(-0.956709\pi\)
							0.612801	+	0.790237i	\(0.290043\pi\)
\(11\)	−3.00000	+	5.19615i	−0.904534	+	1.56670i	−0.0829925	+	0.996550i	\(0.526448\pi\)
							−0.821541	+	0.570149i	\(0.806886\pi\)
\(13\)	−2.50000	−	4.33013i	−0.693375	−	1.20096i	−0.970725	−	0.240192i	\(-0.922790\pi\)
							0.277350	−	0.960769i	\(-0.410544\pi\)
\(17\)	3.00000			0.727607			0.363803	−	0.931476i	\(-0.381478\pi\)
							0.363803	+	0.931476i	\(0.381478\pi\)
\(19\)	2.00000			0.458831			0.229416	−	0.973329i	\(-0.426318\pi\)
							0.229416	+	0.973329i	\(0.426318\pi\)
\(23\)	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	\(0.0484560\pi\)
							−0.362892	+	0.931831i	\(0.618211\pi\)
\(29\)	1.50000	−	2.59808i	0.278543	−	0.482451i	−0.692480	−	0.721437i	\(-0.743482\pi\)
							0.971023	+	0.238987i	\(0.0768152\pi\)
\(31\)	2.00000	+	3.46410i	0.359211	+	0.622171i	0.987829	−	0.155543i	\(-0.0497126\pi\)
							−0.628619	+	0.777714i	\(0.716379\pi\)
\(37\)	5.00000			0.821995			0.410997	−	0.911636i	\(-0.365181\pi\)
							0.410997	+	0.911636i	\(0.365181\pi\)
\(41\)	−3.00000	−	5.19615i	−0.468521	−	0.811503i	0.530831	−	0.847477i	\(-0.321880\pi\)
							−0.999353	+	0.0359748i	\(0.988546\pi\)
\(43\)	5.00000	−	8.66025i	0.762493	−	1.32068i	−0.179069	−	0.983836i	\(-0.557309\pi\)
							0.941562	−	0.336840i	\(-0.109358\pi\)
\(47\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(53\)	6.00000			0.824163			0.412082	−	0.911147i	\(-0.364802\pi\)
							0.412082	+	0.911147i	\(0.364802\pi\)
\(59\)	−6.00000	−	10.3923i	−0.781133	−	1.35296i	−0.931282	−	0.364299i	\(-0.881308\pi\)
							0.150148	−	0.988663i	\(-0.452025\pi\)
\(61\)	−2.50000	+	4.33013i	−0.320092	+	0.554416i	−0.980507	−	0.196485i	\(-0.937047\pi\)
							0.660415	+	0.750901i	\(0.270381\pi\)
\(67\)	−1.00000	−	1.73205i	−0.122169	−	0.211604i	0.798454	−	0.602056i	\(-0.205652\pi\)
							−0.920623	+	0.390453i	\(0.872318\pi\)
\(71\)	−6.00000			−0.712069			−0.356034	−	0.934473i	\(-0.615871\pi\)
							−0.356034	+	0.934473i	\(0.615871\pi\)
\(73\)	−1.00000			−0.117041			−0.0585206	−	0.998286i	\(-0.518638\pi\)
							−0.0585206	+	0.998286i	\(0.518638\pi\)
\(79\)	5.00000	−	8.66025i	0.562544	−	0.974355i	−0.434730	−	0.900561i	\(-0.643156\pi\)
							0.997274	−	0.0737937i	\(-0.0235106\pi\)
\(83\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(89\)	3.00000			0.317999			0.159000	−	0.987279i	\(-0.449173\pi\)
							0.159000	+	0.987279i	\(0.449173\pi\)
\(97\)	5.00000	−	8.66025i	0.507673	−	0.879316i	−0.492287	−	0.870433i	\(-0.663839\pi\)
							0.999961	−	0.00888289i	\(-0.00282755\pi\)

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	324.2.e.d.217.1		2
3.2	odd	2		324.2.e.a.217.1		2
4.3	odd	2		1296.2.i.p.865.1		2
9.2	odd	6		324.2.a.d.1.1	yes	1
9.4	even	3	inner	324.2.e.d.109.1		2
9.5	odd	6		324.2.e.a.109.1		2
9.7	even	3		324.2.a.b.1.1	✓	1
12.11	even	2		1296.2.i.d.865.1		2
36.7	odd	6		1296.2.a.a.1.1		1
36.11	even	6		1296.2.a.j.1.1		1
36.23	even	6		1296.2.i.d.433.1		2
36.31	odd	6		1296.2.i.p.433.1		2
45.2	even	12		8100.2.d.a.649.2		2
45.7	odd	12		8100.2.d.j.649.2		2
45.29	odd	6		8100.2.a.a.1.1		1
45.34	even	6		8100.2.a.f.1.1		1
45.38	even	12		8100.2.d.a.649.1		2
45.43	odd	12		8100.2.d.j.649.1		2
72.11	even	6		5184.2.a.d.1.1		1
72.29	odd	6		5184.2.a.g.1.1		1
72.43	odd	6		5184.2.a.z.1.1		1
72.61	even	6		5184.2.a.bc.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.2.a.b.1.1	✓	1	9.7	even	3
324.2.a.d.1.1	yes	1	9.2	odd	6
324.2.e.a.109.1		2	9.5	odd	6
324.2.e.a.217.1		2	3.2	odd	2
324.2.e.d.109.1		2	9.4	even	3	inner
324.2.e.d.217.1		2	1.1	even	1	trivial
1296.2.a.a.1.1		1	36.7	odd	6
1296.2.a.j.1.1		1	36.11	even	6
1296.2.i.d.433.1		2	36.23	even	6
1296.2.i.d.865.1		2	12.11	even	2
1296.2.i.p.433.1		2	36.31	odd	6
1296.2.i.p.865.1		2	4.3	odd	2
5184.2.a.d.1.1		1	72.11	even	6
5184.2.a.g.1.1		1	72.29	odd	6
5184.2.a.z.1.1		1	72.43	odd	6
5184.2.a.bc.1.1		1	72.61	even	6
8100.2.a.a.1.1		1	45.29	odd	6
8100.2.a.f.1.1		1	45.34	even	6
8100.2.d.a.649.1		2	45.38	even	12
8100.2.d.a.649.2		2	45.2	even	12
8100.2.d.j.649.1		2	45.43	odd	12
8100.2.d.j.649.2		2	45.7	odd	12