Embedded newform 114.2.i.b.55.1

• Label: 114.2.i.b.55.1
• Level: $114$
• Weight: $2$
• Character: 114.55
• 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			55.1
Root			\(-0.173648 + 0.984808i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.55
Dual form			114.2.i.b.85.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.173648 - 0.984808i) q^{2} +(0.766044 + 0.642788i) q^{3} +(-0.939693 - 0.342020i) q^{4} +(1.55303 - 0.565258i) q^{5} +(0.766044 - 0.642788i) q^{6} +(-0.0923963 - 0.160035i) q^{7} +(-0.500000 + 0.866025i) q^{8} +(0.173648 + 0.984808i) q^{9} +(-0.286989 - 1.62760i) q^{10} +(2.17365 - 3.76487i) q^{11} +(-0.500000 - 0.866025i) q^{12} +(-4.96064 + 4.16247i) q^{13} +(-0.173648 + 0.0632028i) q^{14} +(1.55303 + 0.565258i) q^{15} +(0.766044 + 0.642788i) q^{16} +(-0.368241 + 2.08840i) q^{17} +1.00000 q^{18} +(-4.11721 - 1.43128i) q^{19} -1.65270 q^{20} +(0.0320889 - 0.181985i) q^{21} +(-3.33022 - 2.79439i) q^{22} +(-0.0996702 - 0.0362770i) q^{23} +(-0.939693 + 0.342020i) q^{24} +(-1.73783 + 1.45821i) q^{25} +(3.23783 + 5.60808i) q^{26} +(-0.500000 + 0.866025i) q^{27} +(0.0320889 + 0.181985i) q^{28} +(0.692066 + 3.92490i) q^{29} +(0.826352 - 1.43128i) q^{30} +(1.61334 + 2.79439i) q^{31} +(0.766044 - 0.642788i) q^{32} +(4.08512 - 1.48686i) q^{33} +(1.99273 + 0.725293i) q^{34} +(-0.233956 - 0.196312i) q^{35} +(0.173648 - 0.984808i) q^{36} +4.06418 q^{37} +(-2.12449 + 3.80612i) q^{38} -6.47565 q^{39} +(-0.286989 + 1.62760i) q^{40} +(-6.61721 - 5.55250i) q^{41} +(-0.173648 - 0.0632028i) q^{42} +(-0.0393628 + 0.0143269i) q^{43} +(-3.33022 + 2.79439i) q^{44} +(0.826352 + 1.43128i) q^{45} +(-0.0530334 + 0.0918566i) q^{46} +(-1.37551 - 7.80093i) q^{47} +(0.173648 + 0.984808i) q^{48} +(3.48293 - 6.03260i) q^{49} +(1.13429 + 1.96464i) q^{50} +(-1.62449 + 1.36310i) q^{51} +(6.08512 - 2.21480i) q^{52} +(8.65657 + 3.15074i) q^{53} +(0.766044 + 0.642788i) q^{54} +(1.24763 - 7.07564i) q^{55} +0.184793 q^{56} +(-2.23396 - 3.74292i) q^{57} +3.98545 q^{58} +(1.75237 - 9.93821i) q^{59} +(-1.26604 - 1.06234i) q^{60} +(-3.37939 - 1.23000i) q^{61} +(3.03209 - 1.10359i) q^{62} +(0.141559 - 0.118782i) q^{63} +(-0.500000 - 0.866025i) q^{64} +(-5.35117 + 9.26849i) q^{65} +(-0.754900 - 4.28125i) q^{66} +(-1.38666 - 7.86414i) q^{67} +(1.06031 - 1.83651i) q^{68} +(-0.0530334 - 0.0918566i) q^{69} +(-0.233956 + 0.196312i) q^{70} +(3.79813 - 1.38241i) q^{71} +(-0.939693 - 0.342020i) q^{72} +(11.6420 + 9.76882i) q^{73} +(0.705737 - 4.00243i) q^{74} -2.26857 q^{75} +(3.37939 + 2.75314i) q^{76} -0.803348 q^{77} +(-1.12449 + 6.37727i) q^{78} +(9.49660 + 7.96859i) q^{79} +(1.55303 + 0.565258i) q^{80} +(-0.939693 + 0.342020i) q^{81} +(-6.61721 + 5.55250i) q^{82} +(-4.22803 - 7.32316i) q^{83} +(-0.0923963 + 0.160035i) q^{84} +(0.608593 + 3.45150i) q^{85} +(0.00727396 + 0.0412527i) q^{86} +(-1.99273 + 3.45150i) q^{87} +(2.17365 + 3.76487i) q^{88} +(-13.7404 + 11.5295i) q^{89} +(1.55303 - 0.565258i) q^{90} +(1.12449 + 0.409279i) q^{91} +(0.0812519 + 0.0681784i) q^{92} +(-0.560307 + 3.17766i) q^{93} -7.92127 q^{94} +(-7.20321 + 0.104455i) q^{95} +1.00000 q^{96} +(1.85591 - 10.5254i) q^{97} +(-5.33615 - 4.47756i) q^{98} +(4.08512 + 1.48686i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q - 3 * q^5 + 3 * q^7 - 3 * q^8 + 6 * q^10 + 12 * q^11 - 3 * q^12 - 21 * q^13 - 3 * q^15 + 3 * q^17 + 6 * q^18 + 6 * q^19 - 12 * q^20 - 9 * q^21 + 3 * q^22 - 15 * q^23 + 9 * q^25 - 3 * q^27 - 9 * q^28 + 15 * q^29 + 6 * q^30 + 3 * q^31 + 3 * q^33 - 6 * q^34 - 6 * q^35 + 6 * q^37 + 6 * q^40 - 9 * q^41 - 9 * q^43 + 3 * q^44 + 6 * q^45 + 12 * q^46 - 21 * q^47 - 3 * q^50 + 3 * q^51 + 15 * q^52 + 30 * q^53 - 9 * q^55 - 6 * q^56 - 18 * q^57 - 12 * q^58 + 27 * q^59 - 3 * q^60 - 9 * q^61 + 9 * q^62 + 9 * q^63 - 3 * q^64 - 6 * q^65 - 6 * q^66 - 15 * q^67 + 12 * q^68 + 12 * q^69 - 6 * q^70 + 9 * q^71 + 12 * q^73 - 6 * q^74 + 6 * q^75 + 9 * q^76 + 42 * q^77 + 6 * q^78 + 15 * q^79 - 3 * q^80 - 9 * q^82 - 3 * q^83 + 3 * q^84 - 36 * q^85 + 18 * q^86 + 6 * q^87 + 12 * q^88 - 48 * q^89 - 3 * q^90 - 6 * q^91 + 3 * q^92 - 9 * q^93 - 30 * q^94 - 48 * q^95 + 6 * q^96 + 18 * q^97 - 36 * q^98 + 3 * q^99

Character values

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

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{9}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.173648	−	0.984808i	0.122788	−	0.696364i
							\(3\)	0.766044	+	0.642788i	0.442276	+	0.371114i
\(5\)	1.55303	−	0.565258i	0.694538	−	0.252791i								0.0294608	−	0.999566i	\(-0.490621\pi\)
							0.665077	+	0.746775i	\(0.268399\pi\)
\(7\)	−0.0923963	−	0.160035i	−0.0349225	−	0.0604876i	0.848036	−	0.529939i	\(-0.177785\pi\)
							−0.882958	+	0.469451i	\(0.844452\pi\)
\(11\)	2.17365	−	3.76487i	0.655380	−	1.13515i	−0.326419	−	0.945225i	\(-0.605842\pi\)
							0.981798	−	0.189926i	\(-0.0608247\pi\)
\(13\)	−4.96064	+	4.16247i	−1.37583	+	1.15446i	−0.405108	+	0.914269i	\(0.632766\pi\)
							−0.970725	+	0.240192i	\(0.922790\pi\)
\(17\)	−0.368241	+	2.08840i	−0.0893115	+	0.506511i	0.907031	+	0.421064i	\(0.138343\pi\)
							−0.996343	+	0.0854474i	\(0.972768\pi\)
\(19\)	−4.11721	−	1.43128i	−0.944553	−	0.328359i
							\(23\)	−0.0996702	−	0.0362770i	−0.0207827	−	0.00756428i	0.331608	−	0.943417i	\(-0.392409\pi\)
−0.352391	+	0.935853i	\(0.614631\pi\)
\(29\)	0.692066	+	3.92490i	0.128514	+	0.728836i	0.979159	+	0.203096i	\(0.0651004\pi\)
							−0.850645	+	0.525740i	\(0.823788\pi\)
\(31\)	1.61334	+	2.79439i	0.289765	+	0.501887i	0.973753	−	0.227606i	\(-0.0730897\pi\)
							−0.683989	+	0.729492i	\(0.739756\pi\)
\(37\)	4.06418			0.668147			0.334073	−	0.942547i	\(-0.391577\pi\)
							0.334073	+	0.942547i	\(0.391577\pi\)
\(41\)	−6.61721	−	5.55250i	−1.03343	−	0.867155i	−0.0421791	−	0.999110i	\(-0.513430\pi\)
							−0.991256	+	0.131955i	\(0.957874\pi\)
\(43\)	−0.0393628	+	0.0143269i	−0.00600278	+	0.00218483i	−0.345020	−	0.938595i	\(-0.612128\pi\)
							0.339017	+	0.940780i	\(0.389906\pi\)
\(47\)	−1.37551	−	7.80093i	−0.200639	−	1.13788i	−0.904155	−	0.427204i	\(-0.859499\pi\)
							0.703516	−	0.710679i	\(-0.251612\pi\)
\(53\)	8.65657	+	3.15074i	1.18907	+	0.432787i	0.859398	−	0.511308i	\(-0.170839\pi\)
							0.329674	+	0.944095i	\(0.393061\pi\)
\(59\)	1.75237	−	9.93821i	0.228140	−	1.29384i	−0.628452	−	0.777849i	\(-0.716311\pi\)
							0.856591	−	0.515996i	\(-0.172578\pi\)
\(61\)	−3.37939	−	1.23000i	−0.432686	−	0.157485i	0.116489	−	0.993192i	\(-0.462836\pi\)
							−0.549175	+	0.835707i	\(0.685058\pi\)
\(67\)	−1.38666	−	7.86414i	−0.169407	−	0.960757i	−0.944403	−	0.328790i	\(-0.893359\pi\)
							0.774996	−	0.631967i	\(-0.217752\pi\)
\(71\)	3.79813	−	1.38241i	0.450755	−	0.164062i	−0.106659	−	0.994296i	\(-0.534015\pi\)
							0.557415	+	0.830234i	\(0.311793\pi\)
\(73\)	11.6420	+	9.76882i	1.36260	+	1.14335i	0.975171	+	0.221453i	\(0.0710798\pi\)
							0.387425	+	0.921901i	\(0.373365\pi\)
\(79\)	9.49660	+	7.96859i	1.06845	+	0.896536i	0.994911	−	0.100756i	\(-0.0321261\pi\)
							0.0735394	+	0.997292i	\(0.476571\pi\)
\(83\)	−4.22803	−	7.32316i	−0.464086	−	0.803821i	0.535074	−	0.844805i	\(-0.320284\pi\)
							−0.999160	+	0.0409847i	\(0.986951\pi\)
\(89\)	−13.7404	+	11.5295i	−1.45647	+	1.22213i	−0.528795	+	0.848750i	\(0.677356\pi\)
							−0.927679	+	0.373378i	\(0.878200\pi\)
\(97\)	1.85591	−	10.5254i	0.188440	−	1.06869i	−0.733016	−	0.680211i	\(-0.761888\pi\)
							0.921456	−	0.388483i	\(-0.127001\pi\)

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.b.55.1	✓	6
3.2	odd	2		342.2.u.d.55.1		6
4.3	odd	2		912.2.bo.c.625.1		6
19.3	odd	18		2166.2.a.n.1.2		3
19.9	even	9	inner	114.2.i.b.85.1	yes	6
19.16	even	9		2166.2.a.t.1.2		3
57.35	odd	18		6498.2.a.bo.1.2		3
57.41	even	18		6498.2.a.bt.1.2		3
57.47	odd	18		342.2.u.d.199.1		6
76.47	odd	18		912.2.bo.c.769.1		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.b.55.1	✓	6	1.1	even	1	trivial
114.2.i.b.85.1	yes	6	19.9	even	9	inner
342.2.u.d.55.1		6	3.2	odd	2
342.2.u.d.199.1		6	57.47	odd	18
912.2.bo.c.625.1		6	4.3	odd	2
912.2.bo.c.769.1		6	76.47	odd	18
2166.2.a.n.1.2		3	19.3	odd	18
2166.2.a.t.1.2		3	19.16	even	9
6498.2.a.bo.1.2		3	57.35	odd	18
6498.2.a.bt.1.2		3	57.41	even	18