Embedded newform 114.2.i.d.61.1

• Label: 114.2.i.d.61.1
• Level: $114$
• Weight: $2$
• Character: 114.61
• Analytic conductor: $0.910$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	114.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.910294583043\)
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			61.1
Root			\(0.939693 + 0.342020i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.61
Dual form			114.2.i.d.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.939693 + 0.342020i) q^{2} +(-0.173648 - 0.984808i) q^{3} +(0.766044 + 0.642788i) q^{4} +(0.907604 - 0.761570i) q^{5} +(0.173648 - 0.984808i) q^{6} +(-0.266044 + 0.460802i) q^{7} +(0.500000 + 0.866025i) q^{8} +(-0.939693 + 0.342020i) q^{9} +(1.11334 - 0.405223i) q^{10} +(-0.939693 - 1.62760i) q^{11} +(0.500000 - 0.866025i) q^{12} +(-0.673648 + 3.82045i) q^{13} +(-0.407604 + 0.342020i) q^{14} +(-0.907604 - 0.761570i) q^{15} +(0.173648 + 0.984808i) q^{16} +(-1.09240 - 0.397600i) q^{17} -1.00000 q^{18} +(-3.93969 + 1.86516i) q^{19} +1.18479 q^{20} +(0.500000 + 0.181985i) q^{21} +(-0.326352 - 1.85083i) q^{22} +(-5.13429 - 4.30818i) q^{23} +(0.766044 - 0.642788i) q^{24} +(-0.624485 + 3.54163i) q^{25} +(-1.93969 + 3.35965i) q^{26} +(0.500000 + 0.866025i) q^{27} +(-0.500000 + 0.181985i) q^{28} +(3.77972 - 1.37570i) q^{29} +(-0.592396 - 1.02606i) q^{30} +(0.979055 - 1.69577i) q^{31} +(-0.173648 + 0.984808i) q^{32} +(-1.43969 + 1.20805i) q^{33} +(-0.890530 - 0.747243i) q^{34} +(0.109470 + 0.620838i) q^{35} +(-0.939693 - 0.342020i) q^{36} +6.88713 q^{37} +(-4.34002 + 0.405223i) q^{38} +3.87939 q^{39} +(1.11334 + 0.405223i) q^{40} +(-1.56031 - 8.84894i) q^{41} +(0.407604 + 0.342020i) q^{42} +(1.85844 - 1.55942i) q^{43} +(0.326352 - 1.85083i) q^{44} +(-0.592396 + 1.02606i) q^{45} +(-3.35117 - 5.80439i) q^{46} +(-1.91875 + 0.698367i) q^{47} +(0.939693 - 0.342020i) q^{48} +(3.35844 + 5.81699i) q^{49} +(-1.79813 + 3.11446i) q^{50} +(-0.201867 + 1.14484i) q^{51} +(-2.97178 + 2.49362i) q^{52} +(9.93629 + 8.33754i) q^{53} +(0.173648 + 0.984808i) q^{54} +(-2.09240 - 0.761570i) q^{55} -0.532089 q^{56} +(2.52094 + 3.55596i) q^{57} +4.02229 q^{58} +(2.51842 + 0.916629i) q^{59} +(-0.205737 - 1.16679i) q^{60} +(-8.69253 - 7.29390i) q^{61} +(1.50000 - 1.25865i) q^{62} +(0.0923963 - 0.524005i) q^{63} +(-0.500000 + 0.866025i) q^{64} +(2.29813 + 3.98048i) q^{65} +(-1.76604 + 0.642788i) q^{66} +(10.4966 - 3.82045i) q^{67} +(-0.581252 - 1.00676i) q^{68} +(-3.35117 + 5.80439i) q^{69} +(-0.109470 + 0.620838i) q^{70} +(4.65136 - 3.90295i) q^{71} +(-0.766044 - 0.642788i) q^{72} +(-0.0569038 - 0.322718i) q^{73} +(6.47178 + 2.35554i) q^{74} +3.59627 q^{75} +(-4.21688 - 1.10359i) q^{76} +1.00000 q^{77} +(3.64543 + 1.32683i) q^{78} +(-2.80154 - 15.8883i) q^{79} +(0.907604 + 0.761570i) q^{80} +(0.766044 - 0.642788i) q^{81} +(1.56031 - 8.84894i) q^{82} +(-5.78699 + 10.0234i) q^{83} +(0.266044 + 0.460802i) q^{84} +(-1.29426 + 0.471073i) q^{85} +(2.27972 - 0.829748i) q^{86} +(-2.01114 - 3.48340i) q^{87} +(0.939693 - 1.62760i) q^{88} +(-0.618089 + 3.50535i) q^{89} +(-0.907604 + 0.761570i) q^{90} +(-1.58125 - 1.32683i) q^{91} +(-1.16385 - 6.60051i) q^{92} +(-1.84002 - 0.669713i) q^{93} -2.04189 q^{94} +(-2.15523 + 4.69318i) q^{95} +1.00000 q^{96} +(-5.52481 - 2.01087i) q^{97} +(1.16637 + 6.61484i) q^{98} +(1.43969 + 1.20805i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q + 9 * q^5 + 3 * q^7 + 3 * q^8 + 3 * q^12 - 3 * q^13 - 6 * q^14 - 9 * q^15 - 3 * q^17 - 6 * q^18 - 18 * q^19 + 3 * q^21 - 3 * q^22 - 21 * q^23 + 9 * q^25 - 6 * q^26 + 3 * q^27 - 3 * q^28 - 3 * q^29 + 9 * q^31 - 3 * q^33 + 12 * q^34 + 18 * q^35 - 18 * q^37 - 6 * q^38 + 12 * q^39 - 15 * q^41 + 6 * q^42 + 3 * q^43 + 3 * q^44 + 6 * q^46 - 9 * q^47 + 12 * q^49 + 3 * q^50 - 15 * q^51 - 3 * q^52 + 12 * q^53 - 9 * q^55 + 6 * q^56 + 12 * q^57 + 12 * q^58 + 27 * q^59 + 9 * q^60 + 3 * q^61 + 9 * q^62 - 3 * q^63 - 3 * q^64 - 6 * q^66 + 21 * q^67 - 6 * q^68 + 6 * q^69 - 18 * q^70 + 39 * q^71 + 36 * q^73 + 24 * q^74 - 6 * q^75 - 9 * q^76 + 6 * q^77 + 6 * q^78 - 45 * q^79 + 9 * q^80 + 15 * q^82 - 27 * q^83 - 3 * q^84 - 18 * q^85 - 12 * q^86 - 6 * q^87 - 30 * q^89 - 9 * q^90 - 12 * q^91 - 3 * q^92 + 9 * q^93 - 6 * q^94 + 6 * q^96 - 6 * q^97 - 12 * q^98 + 3 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{9}\right)\)

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
							\(3\)	−0.173648	−	0.984808i	−0.100256	−	0.568579i
\(5\)	0.907604	−	0.761570i	0.405893	−	0.340584i								−0.416873	−	0.908965i	\(-0.636874\pi\)
							0.822766	+	0.568380i	\(0.192430\pi\)
\(7\)	−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	0.688874	−	0.724881i	\(-0.258105\pi\)
							−0.972202	+	0.234142i	\(0.924772\pi\)
\(13\)	−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	0.206132	−	0.978524i	\(-0.433912\pi\)
							−0.471077	+	0.882092i	\(0.656135\pi\)
\(19\)	−3.93969	+	1.86516i	−0.903827	+	0.427897i
							\(23\)	−5.13429	−	4.30818i	−1.07057	−	0.898317i	−0.0754683	−	0.997148i	\(-0.524045\pi\)
−0.995104	+	0.0988312i	\(0.968490\pi\)
\(29\)	3.77972	−	1.37570i	0.701875	−	0.255462i	0.0336640	−	0.999433i	\(-0.489282\pi\)
							0.668211	+	0.743971i	\(0.267060\pi\)
\(31\)	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	\(-0.808004\pi\)
							0.579861	−	0.814715i	\(-0.303107\pi\)
\(43\)	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	−0.478145	−	0.878281i	\(-0.658691\pi\)
							0.198267	+	0.980148i	\(0.436469\pi\)
\(53\)	9.93629	+	8.33754i	1.36485	+	1.14525i	0.974450	+	0.224605i	\(0.0721093\pi\)
							0.390404	+	0.920643i	\(0.372335\pi\)
\(59\)	2.51842	+	0.916629i	0.327870	+	0.119335i	0.500710	−	0.865615i	\(-0.333072\pi\)
							−0.172840	+	0.984950i	\(0.555294\pi\)
\(61\)	−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	−0.323608	−	0.946191i	\(-0.604896\pi\)
							0.875623	+	0.482996i	\(0.160451\pi\)
\(73\)	−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	\(0.385751\pi\)
							−0.986472	+	0.163931i	\(0.947582\pi\)
\(89\)	−0.618089	+	3.50535i	−0.0655173	+	0.371567i	0.934366	+	0.356314i	\(0.115967\pi\)
							−0.999884	+	0.0152532i	\(0.995145\pi\)
\(97\)	−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	114.2.i.d.61.1	yes	6
3.2	odd	2		342.2.u.a.289.1		6
4.3	odd	2		912.2.bo.f.289.1		6
19.5	even	9	inner	114.2.i.d.43.1	✓	6
19.9	even	9		2166.2.a.o.1.2		3
19.10	odd	18		2166.2.a.u.1.2		3
57.5	odd	18		342.2.u.a.271.1		6
57.29	even	18		6498.2.a.bn.1.2		3
57.47	odd	18		6498.2.a.bs.1.2		3
76.43	odd	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	19.5	even	9	inner
114.2.i.d.61.1	yes	6	1.1	even	1	trivial
342.2.u.a.271.1		6	57.5	odd	18
342.2.u.a.289.1		6	3.2	odd	2
912.2.bo.f.289.1		6	4.3	odd	2
912.2.bo.f.385.1		6	76.43	odd	18
2166.2.a.o.1.2		3	19.9	even	9
2166.2.a.u.1.2		3	19.10	odd	18
6498.2.a.bn.1.2		3	57.29	even	18
6498.2.a.bs.1.2		3	57.47	odd	18