Embedded newform 114.2.i.c.25.1

• Label: 114.2.i.c.25.1
• Level: $114$
• Weight: $2$
• Character: 114.25
• 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			25.1
Root			\(-0.766044 + 0.642788i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.25
Dual form			114.2.i.c.73.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.766044 - 0.642788i) q^{2} +(0.939693 + 0.342020i) q^{3} +(0.173648 - 0.984808i) q^{4} +(-0.613341 - 3.47843i) q^{5} +(0.939693 - 0.342020i) q^{6} +(-1.85844 + 3.21891i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(0.766044 + 0.642788i) q^{9} +(-2.70574 - 2.27038i) q^{10} +(2.64543 + 4.58202i) q^{11} +(0.500000 - 0.866025i) q^{12} +(0.213011 - 0.0775297i) q^{13} +(0.645430 + 3.66041i) q^{14} +(0.613341 - 3.47843i) q^{15} +(-0.939693 - 0.342020i) q^{16} +(-1.26604 + 1.06234i) q^{17} +1.00000 q^{18} +(-4.17752 + 1.24432i) q^{19} -3.53209 q^{20} +(-2.84730 + 2.38917i) q^{21} +(4.97178 + 1.80958i) q^{22} +(1.50727 - 8.54818i) q^{23} +(-0.173648 - 0.984808i) q^{24} +(-7.02481 + 2.55682i) q^{25} +(0.113341 - 0.196312i) q^{26} +(0.500000 + 0.866025i) q^{27} +(2.84730 + 2.38917i) q^{28} +(0.0923963 + 0.0775297i) q^{29} +(-1.76604 - 3.05888i) q^{30} +(-1.56031 + 2.70253i) q^{31} +(-0.939693 + 0.342020i) q^{32} +(0.918748 + 5.21048i) q^{33} +(-0.286989 + 1.62760i) q^{34} +(12.3366 + 4.49016i) q^{35} +(0.766044 - 0.642788i) q^{36} +5.12836 q^{37} +(-2.40033 + 3.63846i) q^{38} +0.226682 q^{39} +(-2.70574 + 2.27038i) q^{40} +(-6.67752 - 2.43042i) q^{41} +(-0.645430 + 3.66041i) q^{42} +(-0.929892 - 5.27368i) q^{43} +(4.97178 - 1.80958i) q^{44} +(1.76604 - 3.05888i) q^{45} +(-4.34002 - 7.51714i) q^{46} +(-1.92262 - 1.61327i) q^{47} +(-0.766044 - 0.642788i) q^{48} +(-3.40760 - 5.90214i) q^{49} +(-3.73783 + 6.47410i) q^{50} +(-1.55303 + 0.565258i) q^{51} +(-0.0393628 - 0.223238i) q^{52} +(1.03074 - 5.84564i) q^{53} +(0.939693 + 0.342020i) q^{54} +(14.3157 - 12.0123i) q^{55} +3.71688 q^{56} +(-4.35117 - 0.259515i) q^{57} +0.120615 q^{58} +(0.167718 - 0.140732i) q^{59} +(-3.31908 - 1.20805i) q^{60} +(-0.273318 + 1.55007i) q^{61} +(0.541889 + 3.07321i) q^{62} +(-3.49273 + 1.27125i) q^{63} +(-0.500000 + 0.866025i) q^{64} +(-0.400330 - 0.693392i) q^{65} +(4.05303 + 3.40090i) q^{66} +(11.8589 + 9.95080i) q^{67} +(0.826352 + 1.43128i) q^{68} +(4.34002 - 7.51714i) q^{69} +(12.3366 - 4.49016i) q^{70} +(-0.235300 - 1.33445i) q^{71} +(0.173648 - 0.984808i) q^{72} +(-2.27972 - 0.829748i) q^{73} +(3.92855 - 3.29644i) q^{74} -7.47565 q^{75} +(0.500000 + 4.33013i) q^{76} -19.6655 q^{77} +(0.173648 - 0.145708i) q^{78} +(2.69207 + 0.979832i) q^{79} +(-0.613341 + 3.47843i) q^{80} +(0.173648 + 0.984808i) q^{81} +(-6.67752 + 2.43042i) q^{82} +(-0.960637 + 1.66387i) q^{83} +(1.85844 + 3.21891i) q^{84} +(4.47178 + 3.75227i) q^{85} +(-4.10220 - 3.44215i) q^{86} +(0.0603074 + 0.104455i) q^{87} +(2.64543 - 4.58202i) q^{88} +(11.4226 - 4.15749i) q^{89} +(-0.613341 - 3.47843i) q^{90} +(-0.146307 + 0.829748i) q^{91} +(-8.15657 - 2.96875i) q^{92} +(-2.39053 + 2.00589i) q^{93} -2.50980 q^{94} +(6.89053 + 13.7680i) q^{95} -1.00000 q^{96} +(13.4081 - 11.2507i) q^{97} +(-6.40420 - 2.33094i) q^{98} +(-0.918748 + 5.21048i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q + 3 * q^5 - 3 * q^7 - 3 * q^8 - 6 * q^10 + 3 * q^12 + 9 * q^13 - 12 * q^14 - 3 * q^15 - 3 * q^17 + 6 * q^18 - 12 * q^20 - 15 * q^21 + 15 * q^22 + 27 * q^23 - 15 * q^25 - 6 * q^26 + 3 * q^27 + 15 * q^28 - 3 * q^29 - 6 * q^30 - 15 * q^31 + 3 * q^33 + 6 * q^34 + 12 * q^35 - 6 * q^37 - 12 * q^39 - 6 * q^40 - 15 * q^41 + 12 * q^42 + 3 * q^43 + 15 * q^44 + 6 * q^45 - 6 * q^46 + 15 * q^47 - 24 * q^49 - 3 * q^50 + 3 * q^51 - 9 * q^52 + 6 * q^53 + 27 * q^55 + 6 * q^56 + 12 * q^58 - 27 * q^59 - 3 * q^60 - 15 * q^61 - 3 * q^62 - 3 * q^63 - 3 * q^64 + 12 * q^65 + 12 * q^66 - 3 * q^67 + 6 * q^68 + 6 * q^69 + 12 * q^70 + 3 * q^71 + 12 * q^73 + 24 * q^74 - 6 * q^75 + 3 * q^76 - 42 * q^77 + 27 * q^79 + 3 * q^80 - 15 * q^82 + 3 * q^83 + 3 * q^84 + 12 * q^85 - 24 * q^86 + 6 * q^87 + 42 * q^89 + 3 * q^90 - 42 * q^91 - 27 * q^92 + 3 * q^93 - 18 * q^94 + 24 * q^95 - 6 * q^96 + 18 * q^97 - 3 * q^99

Character values

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

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{7}{9}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.766044	−	0.642788i	0.541675	−	0.454519i
							\(3\)	0.939693	+	0.342020i	0.542532	+	0.197465i
\(5\)	−0.613341	−	3.47843i	−0.274294	−	1.55560i								−0.741194	−	0.671290i	\(-0.765740\pi\)
							0.466900	−	0.884310i	\(-0.345371\pi\)
\(7\)	−1.85844	+	3.21891i	−0.702425	+	1.21664i	0.265188	+	0.964197i	\(0.414566\pi\)
							−0.967613	+	0.252438i	\(0.918767\pi\)
\(11\)	2.64543	+	4.58202i	0.797627	+	1.38153i	0.921158	+	0.389190i	\(0.127245\pi\)
							−0.123531	+	0.992341i	\(0.539422\pi\)
\(13\)	0.213011	−	0.0775297i	0.0590786	−	0.0215029i	−0.312312	−	0.949980i	\(-0.601103\pi\)
							0.371390	+	0.928477i	\(0.378881\pi\)
\(17\)	−1.26604	+	1.06234i	−0.307061	+	0.257655i	−0.783276	−	0.621674i	\(-0.786453\pi\)
							0.476215	+	0.879329i	\(0.342008\pi\)
\(19\)	−4.17752	+	1.24432i	−0.958388	+	0.285467i
							\(23\)	1.50727	−	8.54818i	0.314288	−	1.78242i	−0.261894	−	0.965097i	\(-0.584347\pi\)
0.576182	−	0.817321i	\(-0.304542\pi\)
\(29\)	0.0923963	+	0.0775297i	0.0171576	+	0.0143969i	0.651326	−	0.758798i	\(-0.274213\pi\)
							−0.634169	+	0.773195i	\(0.718657\pi\)
\(31\)	−1.56031	+	2.70253i	−0.280239	+	0.485389i	−0.971444	−	0.237271i	\(-0.923747\pi\)
							0.691204	+	0.722660i	\(0.257081\pi\)
\(37\)	5.12836			0.843096			0.421548	−	0.906806i	\(-0.361487\pi\)
							0.421548	+	0.906806i	\(0.361487\pi\)
\(41\)	−6.67752	−	2.43042i	−1.04285	−	0.379568i	−0.236892	−	0.971536i	\(-0.576129\pi\)
							−0.805961	+	0.591968i	\(0.798351\pi\)
\(43\)	−0.929892	−	5.27368i	−0.141807	−	0.804229i	−0.969875	−	0.243602i	\(-0.921671\pi\)
							0.828068	−	0.560627i	\(-0.189440\pi\)
\(47\)	−1.92262	−	1.61327i	−0.280443	−	0.235319i	0.491706	−	0.870761i	\(-0.336374\pi\)
							−0.772149	+	0.635442i	\(0.780818\pi\)
\(53\)	1.03074	−	5.84564i	0.141584	−	0.802961i	−0.828463	−	0.560043i	\(-0.810784\pi\)
							0.970047	−	0.242918i	\(-0.0781044\pi\)
\(59\)	0.167718	−	0.140732i	0.0218351	−	0.0183218i	−0.631805	−	0.775128i	\(-0.717686\pi\)
							0.653640	+	0.756806i	\(0.273241\pi\)
\(61\)	−0.273318	+	1.55007i	−0.0349948	+	0.198466i	−0.997293	−	0.0735316i	\(-0.976573\pi\)
							0.962298	+	0.271997i	\(0.0876841\pi\)
\(67\)	11.8589	+	9.95080i	1.44880	+	1.21568i	0.933453	+	0.358701i	\(0.116780\pi\)
							0.515343	+	0.856984i	\(0.327664\pi\)
\(71\)	−0.235300	−	1.33445i	−0.0279249	−	0.158370i	0.967657	−	0.252271i	\(-0.0811775\pi\)
							−0.995582	+	0.0939008i	\(0.970066\pi\)
\(73\)	−2.27972	−	0.829748i	−0.266820	−	0.0971147i	0.205145	−	0.978731i	\(-0.434233\pi\)
							−0.471966	+	0.881617i	\(0.656455\pi\)
\(79\)	2.69207	+	0.979832i	0.302881	+	0.110240i	0.488990	−	0.872289i	\(-0.337365\pi\)
							−0.186109	+	0.982529i	\(0.559588\pi\)
\(83\)	−0.960637	+	1.66387i	−0.105444	+	0.182634i	−0.913919	−	0.405896i	\(-0.866960\pi\)
							0.808476	+	0.588530i	\(0.200293\pi\)
\(89\)	11.4226	−	4.15749i	1.21080	−	0.440693i	0.343816	−	0.939037i	\(-0.388280\pi\)
							0.866980	+	0.498344i	\(0.166058\pi\)
\(97\)	13.4081	−	11.2507i	1.36138	−	1.14234i	0.385831	−	0.922570i	\(-0.373915\pi\)
							0.975552	−	0.219767i	\(-0.0705296\pi\)

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.c.25.1	✓	6
3.2	odd	2		342.2.u.b.253.1		6
4.3	odd	2		912.2.bo.d.481.1		6
19.4	even	9		2166.2.a.r.1.1		3
19.15	odd	18		2166.2.a.p.1.1		3
19.16	even	9	inner	114.2.i.c.73.1	yes	6
57.23	odd	18		6498.2.a.bp.1.3		3
57.35	odd	18		342.2.u.b.73.1		6
57.53	even	18		6498.2.a.bu.1.3		3
76.35	odd	18		912.2.bo.d.529.1		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.c.25.1	✓	6	1.1	even	1	trivial
114.2.i.c.73.1	yes	6	19.16	even	9	inner
342.2.u.b.73.1		6	57.35	odd	18
342.2.u.b.253.1		6	3.2	odd	2
912.2.bo.d.481.1		6	4.3	odd	2
912.2.bo.d.529.1		6	76.35	odd	18
2166.2.a.p.1.1		3	19.15	odd	18
2166.2.a.r.1.1		3	19.4	even	9
6498.2.a.bp.1.3		3	57.23	odd	18
6498.2.a.bu.1.3		3	57.53	even	18