Embedded newform 342.2.u.a.289.1

• Label: 342.2.u.a.289.1
• Level: $342$
• Weight: $2$
• Character: 342.289
• Analytic conductor: $2.731$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	342.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.73088374913\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{18})\)
Defining polynomial:	\( x^{6} - x^{3} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 114)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			289.1
Root			\(0.939693 + 0.342020i\) of defining polynomial
Character	\(\chi\)	\(=\)	342.289
Dual form			342.2.u.a.271.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.939693 - 0.342020i) q^{2} +(0.766044 + 0.642788i) q^{4} +(-0.907604 + 0.761570i) q^{5} +(-0.266044 + 0.460802i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(1.11334 - 0.405223i) q^{10} +(0.939693 + 1.62760i) q^{11} +(-0.673648 + 3.82045i) q^{13} +(0.407604 - 0.342020i) q^{14} +(0.173648 + 0.984808i) q^{16} +(1.09240 + 0.397600i) q^{17} +(-3.93969 + 1.86516i) q^{19} -1.18479 q^{20} +(-0.326352 - 1.85083i) q^{22} +(5.13429 + 4.30818i) q^{23} +(-0.624485 + 3.54163i) q^{25} +(1.93969 - 3.35965i) q^{26} +(-0.500000 + 0.181985i) q^{28} +(-3.77972 + 1.37570i) q^{29} +(0.979055 - 1.69577i) q^{31} +(0.173648 - 0.984808i) q^{32} +(-0.890530 - 0.747243i) q^{34} +(-0.109470 - 0.620838i) q^{35} +6.88713 q^{37} +(4.34002 - 0.405223i) q^{38} +(1.11334 + 0.405223i) q^{40} +(1.56031 + 8.84894i) q^{41} +(1.85844 - 1.55942i) q^{43} +(-0.326352 + 1.85083i) q^{44} +(-3.35117 - 5.80439i) q^{46} +(1.91875 - 0.698367i) q^{47} +(3.35844 + 5.81699i) q^{49} +(1.79813 - 3.11446i) q^{50} +(-2.97178 + 2.49362i) q^{52} +(-9.93629 - 8.33754i) q^{53} +(-2.09240 - 0.761570i) q^{55} +0.532089 q^{56} +4.02229 q^{58} +(-2.51842 - 0.916629i) q^{59} +(-8.69253 - 7.29390i) q^{61} +(-1.50000 + 1.25865i) q^{62} +(-0.500000 + 0.866025i) q^{64} +(-2.29813 - 3.98048i) q^{65} +(10.4966 - 3.82045i) q^{67} +(0.581252 + 1.00676i) q^{68} +(-0.109470 + 0.620838i) q^{70} +(-4.65136 + 3.90295i) q^{71} +(-0.0569038 - 0.322718i) q^{73} +(-6.47178 - 2.35554i) q^{74} +(-4.21688 - 1.10359i) q^{76} -1.00000 q^{77} +(-2.80154 - 15.8883i) q^{79} +(-0.907604 - 0.761570i) q^{80} +(1.56031 - 8.84894i) q^{82} +(5.78699 - 10.0234i) q^{83} +(-1.29426 + 0.471073i) q^{85} +(-2.27972 + 0.829748i) q^{86} +(0.939693 - 1.62760i) q^{88} +(0.618089 - 3.50535i) q^{89} +(-1.58125 - 1.32683i) q^{91} +(1.16385 + 6.60051i) q^{92} -2.04189 q^{94} +(2.15523 - 4.69318i) q^{95} +(-5.52481 - 2.01087i) q^{97} +(-1.16637 - 6.61484i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q - 9 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^13 + 6 * q^14 + 3 * q^17 - 18 * q^19 - 3 * q^22 + 21 * q^23 + 9 * q^25 + 6 * q^26 - 3 * q^28 + 3 * q^29 + 9 * q^31 + 12 * q^34 - 18 * q^35 - 18 * q^37 + 6 * q^38 + 15 * q^41 + 3 * q^43 - 3 * q^44 + 6 * q^46 + 9 * q^47 + 12 * q^49 - 3 * q^50 - 3 * q^52 - 12 * q^53 - 9 * q^55 - 6 * q^56 + 12 * q^58 - 27 * q^59 + 3 * q^61 - 9 * q^62 - 3 * q^64 + 21 * q^67 + 6 * q^68 - 18 * q^70 - 39 * q^71 + 36 * q^73 - 24 * q^74 - 9 * q^76 - 6 * q^77 - 45 * q^79 - 9 * q^80 + 15 * q^82 + 27 * q^83 - 18 * q^85 + 12 * q^86 + 30 * q^89 - 12 * q^91 + 3 * q^92 - 6 * q^94 - 6 * q^97 + 12 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(325\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{9}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.939693	−	0.342020i	−0.664463	−	0.241845i
							\(3\)		0			0
\(5\)	−0.907604	+	0.761570i	−0.405893	+	0.340584i								−0.822766	−	0.568380i	\(-0.807570\pi\)
							0.416873	+	0.908965i	\(0.363126\pi\)
\(7\)	−0.266044	+	0.460802i	−0.100555	+	0.174167i	−0.911914	−	0.410382i	\(-0.865395\pi\)
							0.811358	+	0.584549i	\(0.198729\pi\)
\(11\)	0.939693	+	1.62760i	0.283328	+	0.490738i	0.972202	−	0.234142i	\(-0.0752282\pi\)
							−0.688874	+	0.724881i	\(0.741895\pi\)
\(13\)	−0.673648	+	3.82045i	−0.186836	+	1.05960i	0.736737	+	0.676180i	\(0.236366\pi\)
							−0.923573	+	0.383422i	\(0.874745\pi\)
\(17\)	1.09240	+	0.397600i	0.264945	+	0.0964321i	0.471077	−	0.882092i	\(-0.343865\pi\)
							−0.206132	+	0.978524i	\(0.566088\pi\)
\(19\)	−3.93969	+	1.86516i	−0.903827	+	0.427897i
							\(23\)	5.13429	+	4.30818i	1.07057	+	0.898317i	0.995104	−	0.0988312i	\(-0.0315104\pi\)
0.0754683	+	0.997148i	\(0.475955\pi\)
\(29\)	−3.77972	+	1.37570i	−0.701875	+	0.255462i	−0.668211	−	0.743971i	\(-0.732940\pi\)
							−0.0336640	+	0.999433i	\(0.510718\pi\)
\(31\)	0.979055	−	1.69577i	0.175844	−	0.304570i	−0.764609	−	0.644494i	\(-0.777068\pi\)
							0.940453	+	0.339924i	\(0.110401\pi\)
\(37\)	6.88713			1.13224			0.566118	−	0.824324i	\(-0.308445\pi\)
							0.566118	+	0.824324i	\(0.308445\pi\)
\(41\)	1.56031	+	8.84894i	0.243679	+	1.38197i	0.823540	+	0.567258i	\(0.191996\pi\)
							−0.579861	+	0.814715i	\(0.696893\pi\)
\(43\)	1.85844	−	1.55942i	0.283410	−	0.237809i	−0.489989	−	0.871728i	\(-0.662999\pi\)
							0.773399	+	0.633919i	\(0.218555\pi\)
\(47\)	1.91875	−	0.698367i	0.279878	−	0.101867i	−0.198267	−	0.980148i	\(-0.563531\pi\)
							0.478145	+	0.878281i	\(0.341309\pi\)
\(53\)	−9.93629	−	8.33754i	−1.36485	−	1.14525i	−0.974450	−	0.224605i	\(-0.927891\pi\)
							−0.390404	−	0.920643i	\(-0.627665\pi\)
\(59\)	−2.51842	−	0.916629i	−0.327870	−	0.119335i	0.172840	−	0.984950i	\(-0.444706\pi\)
							−0.500710	+	0.865615i	\(0.666928\pi\)
\(61\)	−8.69253	−	7.29390i	−1.11296	−	0.933888i	−0.114737	−	0.993396i	\(-0.536603\pi\)
							−0.998228	+	0.0595075i	\(0.981047\pi\)
\(67\)	10.4966	−	3.82045i	1.28236	−	0.466742i	0.391150	−	0.920327i	\(-0.372077\pi\)
							0.891213	+	0.453585i	\(0.149855\pi\)
\(71\)	−4.65136	+	3.90295i	−0.552015	+	0.463195i	−0.875623	−	0.482996i	\(-0.839549\pi\)
							0.323608	+	0.946191i	\(0.395104\pi\)
\(73\)	−0.0569038	−	0.322718i	−0.00666009	−	0.0377712i	0.981297	−	0.192502i	\(-0.0616601\pi\)
							−0.987957	+	0.154731i	\(0.950549\pi\)
\(79\)	−2.80154	−	15.8883i	−0.315198	−	1.78757i	−0.571109	−	0.820874i	\(-0.693487\pi\)
							0.255912	−	0.966700i	\(-0.417624\pi\)
\(83\)	5.78699	−	10.0234i	0.635205	−	1.10021i	−0.351267	−	0.936275i	\(-0.614249\pi\)
							0.986472	−	0.163931i	\(-0.0524175\pi\)
\(89\)	0.618089	−	3.50535i	0.0655173	−	0.371567i	−0.934366	−	0.356314i	\(-0.884033\pi\)
							0.999884	−	0.0152532i	\(-0.00485542\pi\)
\(97\)	−5.52481	−	2.01087i	−0.560960	−	0.204173i	0.0459494	−	0.998944i	\(-0.485369\pi\)
							−0.606909	+	0.794771i	\(0.707591\pi\)

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	342.2.u.a.289.1		6
3.2	odd	2		114.2.i.d.61.1	yes	6
12.11	even	2		912.2.bo.f.289.1		6
19.5	even	9	inner	342.2.u.a.271.1		6
19.9	even	9		6498.2.a.bs.1.2		3
19.10	odd	18		6498.2.a.bn.1.2		3
57.5	odd	18		114.2.i.d.43.1	✓	6
57.29	even	18		2166.2.a.u.1.2		3
57.47	odd	18		2166.2.a.o.1.2		3
228.119	even	18		912.2.bo.f.385.1		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.d.43.1	✓	6	57.5	odd	18
114.2.i.d.61.1	yes	6	3.2	odd	2
342.2.u.a.271.1		6	19.5	even	9	inner
342.2.u.a.289.1		6	1.1	even	1	trivial
912.2.bo.f.289.1		6	12.11	even	2
912.2.bo.f.385.1		6	228.119	even	18
2166.2.a.o.1.2		3	57.47	odd	18
2166.2.a.u.1.2		3	57.29	even	18
6498.2.a.bn.1.2		3	19.10	odd	18
6498.2.a.bs.1.2		3	19.9	even	9