Embedded newform 432.2.u.c.385.2

• Label: 432.2.u.c.385.2
• Level: $432$
• Weight: $2$
• Character: 432.385
• Analytic conductor: $3.450$
• Analytic rank: $0$
• Dimension: $12$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.u (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.44953736732\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{9})\)
Coefficient field:	12.0.1952986685049.1
Defining polynomial:	x^12 - 6*x^11 + 27*x^10 - 80*x^9 + 186*x^8 - 330*x^7 + 463*x^6 - 504*x^5 + 420*x^4 - 258*x^3 + 108*x^2 - 27*x + 3
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			385.2
Root			\(0.500000 + 1.27297i\) of defining polynomial
Character	\(\chi\)	\(=\)	432.385
Dual form			432.2.u.c.193.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.68842 - 0.386327i) q^{3} +(-0.477505 - 2.70806i) q^{5} +(-1.82076 - 1.52780i) q^{7} +(2.70150 - 1.30456i) q^{9} +(0.0434396 - 0.246358i) q^{11} +(-2.45446 - 0.893351i) q^{13} +(-1.85243 - 4.38787i) q^{15} +(0.146688 + 0.254072i) q^{17} +(-1.39237 + 2.41166i) q^{19} +(-3.66443 - 1.87615i) q^{21} +(5.12472 - 4.30015i) q^{23} +(-2.40714 + 0.876128i) q^{25} +(4.05728 - 3.24631i) q^{27} +(0.333645 - 0.121437i) q^{29} +(-2.11847 + 1.77761i) q^{31} +(-0.0218307 - 0.432738i) q^{33} +(-3.26796 + 5.66027i) q^{35} +(3.49619 + 6.05558i) q^{37} +(-4.48928 - 0.560124i) q^{39} +(9.13156 + 3.32362i) q^{41} +(-0.0452712 + 0.256746i) q^{43} +(-4.82282 - 6.69291i) q^{45} +(8.75249 + 7.34421i) q^{47} +(-0.234540 - 1.33014i) q^{49} +(0.345826 + 0.372309i) q^{51} +5.43137 q^{53} -0.687897 q^{55} +(-1.41922 + 4.60980i) q^{57} +(-1.03788 - 5.88612i) q^{59} +(-9.07515 - 7.61495i) q^{61} +(-6.91190 - 1.75206i) q^{63} +(-1.24723 + 7.07342i) q^{65} +(1.70113 + 0.619160i) q^{67} +(6.99139 - 9.24026i) q^{69} +(-0.185255 - 0.320871i) q^{71} +(-2.51339 + 4.35333i) q^{73} +(-3.72579 + 2.40921i) q^{75} +(-0.455479 + 0.382193i) q^{77} +(0.754406 - 0.274581i) q^{79} +(5.59624 - 7.04855i) q^{81} +(-2.58947 + 0.942488i) q^{83} +(0.617998 - 0.518562i) q^{85} +(0.516417 - 0.333932i) q^{87} +(-5.22533 + 9.05054i) q^{89} +(3.10412 + 5.37650i) q^{91} +(-2.89012 + 3.81976i) q^{93} +(7.19580 + 2.61906i) q^{95} +(-2.57600 + 14.6092i) q^{97} +(-0.204037 - 0.722208i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	12 * q + 6 * q^3 - 3 * q^5 + 6 * q^7 - 3 * q^11 - 6 * q^13 - 9 * q^15 + 9 * q^17 + 3 * q^19 - 12 * q^21 + 12 * q^23 + 3 * q^25 + 9 * q^27 - 6 * q^29 - 3 * q^31 - 12 * q^35 - 3 * q^37 - 33 * q^39 + 15 * q^41 - 3 * q^43 - 9 * q^45 + 15 * q^47 + 12 * q^49 + 18 * q^51 - 18 * q^53 + 12 * q^55 - 3 * q^57 + 12 * q^59 + 12 * q^61 - 9 * q^63 + 3 * q^65 + 15 * q^67 + 9 * q^69 - 27 * q^71 + 6 * q^73 - 39 * q^75 + 15 * q^77 + 42 * q^79 + 36 * q^81 - 39 * q^83 - 27 * q^85 - 9 * q^87 + 9 * q^89 - 6 * q^91 - 39 * q^93 + 33 * q^95 + 3 * q^97 + 27 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(1\)	\(1\)	\(e\left(\frac{8}{9}\right)\)

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
							\(3\)	1.68842	−	0.386327i	0.974808	−	0.223046i
\(5\)	−0.477505	−	2.70806i	−0.213547	−	1.21108i								−0.883411	−	0.468600i	\(-0.844759\pi\)
							0.669864	−	0.742484i	\(-0.266352\pi\)
\(7\)	−1.82076	−	1.52780i	−0.688183	−	0.577454i	0.230202	−	0.973143i	\(-0.426061\pi\)
							−0.918385	+	0.395689i	\(0.870506\pi\)
\(11\)	0.0434396	−	0.246358i	0.0130975	−	0.0742798i	−0.977558	−	0.210665i	\(-0.932437\pi\)
							0.990656	+	0.136385i	\(0.0435483\pi\)
\(13\)	−2.45446	−	0.893351i	−0.680745	−	0.247771i	−0.0215777	−	0.999767i	\(-0.506869\pi\)
							−0.659167	+	0.751996i	\(0.729091\pi\)
\(17\)	0.146688	+	0.254072i	0.0355772	+	0.0616215i	0.883266	−	0.468873i	\(-0.155340\pi\)
							−0.847689	+	0.530494i	\(0.822006\pi\)
\(19\)	−1.39237	+	2.41166i	−0.319432	+	0.553273i	−0.980370	−	0.197168i	\(-0.936825\pi\)
							0.660937	+	0.750441i	\(0.270159\pi\)
\(23\)	5.12472	−	4.30015i	1.06858	−	0.896643i	0.0736543	−	0.997284i	\(-0.476534\pi\)
							0.994923	+	0.100641i	\(0.0320894\pi\)
\(29\)	0.333645	−	0.121437i	0.0619562	−	0.0225502i	−0.310856	−	0.950457i	\(-0.600616\pi\)
							0.372812	+	0.927907i	\(0.378394\pi\)
\(31\)	−2.11847	+	1.77761i	−0.380488	+	0.319268i	−0.812894	−	0.582411i	\(-0.802109\pi\)
							0.432406	+	0.901679i	\(0.357665\pi\)
\(37\)	3.49619	+	6.05558i	0.574770	+	0.995531i	0.996067	+	0.0886080i	\(0.0282418\pi\)
							−0.421297	+	0.906923i	\(0.638425\pi\)
\(41\)	9.13156	+	3.32362i	1.42611	+	0.519062i	0.935814	−	0.352494i	\(-0.114666\pi\)
							0.490296	+	0.871556i	\(0.336888\pi\)
\(43\)	−0.0452712	+	0.256746i	−0.00690379	+	0.0391534i	−0.988065	−	0.154037i	\(-0.950772\pi\)
							0.981161	+	0.193191i	\(0.0618836\pi\)
\(47\)	8.75249	+	7.34421i	1.27668	+	1.07126i	0.993692	+	0.112140i	\(0.0357704\pi\)
							0.282989	+	0.959123i	\(0.408674\pi\)
\(53\)	5.43137			0.746056			0.373028	−	0.927820i	\(-0.378320\pi\)
							0.373028	+	0.927820i	\(0.378320\pi\)
\(59\)	−1.03788	−	5.88612i	−0.135121	−	0.766308i	−0.974776	−	0.223188i	\(-0.928354\pi\)
							0.839655	−	0.543121i	\(-0.182757\pi\)
\(61\)	−9.07515	−	7.61495i	−1.16195	−	0.974995i	−0.162023	−	0.986787i	\(-0.551802\pi\)
							−0.999930	+	0.0117924i	\(0.996246\pi\)
\(67\)	1.70113	+	0.619160i	0.207826	+	0.0756424i	0.443835	−	0.896108i	\(-0.353617\pi\)
							−0.236010	+	0.971751i	\(0.575840\pi\)
\(71\)	−0.185255	−	0.320871i	−0.0219857	−	0.0380804i	0.854823	−	0.518919i	\(-0.173666\pi\)
							−0.876809	+	0.480839i	\(0.840332\pi\)
\(73\)	−2.51339	+	4.35333i	−0.294171	+	0.509518i	−0.974792	−	0.223117i	\(-0.928377\pi\)
							0.680621	+	0.732636i	\(0.261710\pi\)
\(79\)	0.754406	−	0.274581i	0.0848773	−	0.0308928i	−0.299233	−	0.954180i	\(-0.596731\pi\)
							0.384110	+	0.923287i	\(0.374508\pi\)
\(83\)	−2.58947	+	0.942488i	−0.284231	+	0.103452i	−0.480201	−	0.877158i	\(-0.659436\pi\)
							0.195971	+	0.980610i	\(0.437214\pi\)
\(89\)	−5.22533	+	9.05054i	−0.553884	+	0.959356i	0.444105	+	0.895975i	\(0.353522\pi\)
							−0.997989	+	0.0633809i	\(0.979812\pi\)
\(97\)	−2.57600	+	14.6092i	−0.261553	+	1.48334i	0.517120	+	0.855913i	\(0.327004\pi\)
							−0.778673	+	0.627430i	\(0.784107\pi\)

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	432.2.u.c.385.2		12
4.3	odd	2		27.2.e.a.7.2	yes	12
12.11	even	2		81.2.e.a.19.1		12
20.3	even	4		675.2.u.b.574.2		24
20.7	even	4		675.2.u.b.574.3		24
20.19	odd	2		675.2.l.c.601.1		12
27.4	even	9	inner	432.2.u.c.193.2		12
36.7	odd	6		243.2.e.d.136.1		12
36.11	even	6		243.2.e.a.136.2		12
36.23	even	6		243.2.e.b.217.2		12
36.31	odd	6		243.2.e.c.217.1		12
108.7	odd	18		729.2.c.e.487.4		12
108.11	even	18		729.2.c.b.244.3		12
108.23	even	18		81.2.e.a.64.1		12
108.31	odd	18		27.2.e.a.4.2	✓	12
108.43	odd	18		729.2.c.e.244.4		12
108.47	even	18		729.2.c.b.487.3		12
108.59	even	18		243.2.e.b.28.2		12
108.67	odd	18		243.2.e.d.109.1		12
108.79	odd	18		729.2.a.a.1.3		6
108.83	even	18		729.2.a.d.1.4		6
108.95	even	18		243.2.e.a.109.2		12
108.103	odd	18		243.2.e.c.28.1		12
540.139	odd	18		675.2.l.c.301.1		12
540.247	even	36		675.2.u.b.274.2		24
540.463	even	36		675.2.u.b.274.3		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.4.2	✓	12	108.31	odd	18
27.2.e.a.7.2	yes	12	4.3	odd	2
81.2.e.a.19.1		12	12.11	even	2
81.2.e.a.64.1		12	108.23	even	18
243.2.e.a.109.2		12	108.95	even	18
243.2.e.a.136.2		12	36.11	even	6
243.2.e.b.28.2		12	108.59	even	18
243.2.e.b.217.2		12	36.23	even	6
243.2.e.c.28.1		12	108.103	odd	18
243.2.e.c.217.1		12	36.31	odd	6
243.2.e.d.109.1		12	108.67	odd	18
243.2.e.d.136.1		12	36.7	odd	6
432.2.u.c.193.2		12	27.4	even	9	inner
432.2.u.c.385.2		12	1.1	even	1	trivial
675.2.l.c.301.1		12	540.139	odd	18
675.2.l.c.601.1		12	20.19	odd	2
675.2.u.b.274.2		24	540.247	even	36
675.2.u.b.274.3		24	540.463	even	36
675.2.u.b.574.2		24	20.3	even	4
675.2.u.b.574.3		24	20.7	even	4
729.2.a.a.1.3		6	108.79	odd	18
729.2.a.d.1.4		6	108.83	even	18
729.2.c.b.244.3		12	108.11	even	18
729.2.c.b.487.3		12	108.47	even	18
729.2.c.e.244.4		12	108.43	odd	18
729.2.c.e.487.4		12	108.7	odd	18