Embedded newform 114.2.i.c.55.1

• Label: 114.2.i.c.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.c.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} +(2.20574 - 0.802823i) q^{5} +(-0.766044 + 0.642788i) q^{6} +(-1.78699 - 3.09516i) q^{7} +(-0.500000 + 0.866025i) q^{8} +(0.173648 + 0.984808i) q^{9} +(-0.407604 - 2.31164i) q^{10} +(-1.35844 + 2.35289i) q^{11} +(0.500000 + 0.866025i) q^{12} +(4.14543 - 3.47843i) q^{13} +(-3.35844 + 1.22237i) q^{14} +(-2.20574 - 0.802823i) q^{15} +(0.766044 + 0.642788i) q^{16} +(-0.673648 + 3.82045i) q^{17} +1.00000 q^{18} +(1.01114 + 4.24000i) q^{19} -2.34730 q^{20} +(-0.620615 + 3.51968i) q^{21} +(2.08125 + 1.74638i) q^{22} +(7.73783 + 2.81634i) q^{23} +(0.939693 - 0.342020i) q^{24} +(0.390530 - 0.327693i) q^{25} +(-2.70574 - 4.68647i) q^{26} +(0.500000 - 0.866025i) q^{27} +(0.620615 + 3.51968i) q^{28} +(0.613341 + 3.47843i) q^{29} +(-1.17365 + 2.03282i) q^{30} +(-3.26604 - 5.65695i) q^{31} +(0.766044 - 0.642788i) q^{32} +(2.55303 - 0.929228i) q^{33} +(3.64543 + 1.32683i) q^{34} +(-6.42649 - 5.39246i) q^{35} +(0.173648 - 0.984808i) q^{36} +0.389185 q^{37} +(4.35117 - 0.259515i) q^{38} -5.41147 q^{39} +(-0.407604 + 2.31164i) q^{40} +(-1.48886 - 1.24930i) q^{41} +(3.35844 + 1.22237i) q^{42} +(-4.71941 + 1.71772i) q^{43} +(2.08125 - 1.74638i) q^{44} +(1.17365 + 2.03282i) q^{45} +(4.11721 - 7.13122i) q^{46} +(0.518418 + 2.94010i) q^{47} +(-0.173648 - 0.984808i) q^{48} +(-2.88666 + 4.99984i) q^{49} +(-0.254900 - 0.441500i) q^{50} +(2.97178 - 2.49362i) q^{51} +(-5.08512 + 1.85083i) q^{52} +(-7.80453 - 2.84062i) q^{53} +(-0.766044 - 0.642788i) q^{54} +(-1.10741 + 6.28044i) q^{55} +3.57398 q^{56} +(1.95084 - 3.89798i) q^{57} +3.53209 q^{58} +(0.474308 - 2.68993i) q^{59} +(1.79813 + 1.50881i) q^{60} +(-5.91147 - 2.15160i) q^{61} +(-6.13816 + 2.23411i) q^{62} +(2.73783 - 2.29731i) q^{63} +(-0.500000 - 0.866025i) q^{64} +(6.35117 - 11.0005i) q^{65} +(-0.471782 - 2.67561i) q^{66} +(-2.59374 - 14.7098i) q^{67} +(1.93969 - 3.35965i) q^{68} +(-4.11721 - 7.13122i) q^{69} +(-6.42649 + 5.39246i) q^{70} +(-8.47818 + 3.08580i) q^{71} +(-0.939693 - 0.342020i) q^{72} +(7.88326 + 6.61484i) q^{73} +(0.0675813 - 0.383273i) q^{74} -0.509800 q^{75} +(0.500000 - 4.33013i) q^{76} +9.71007 q^{77} +(-0.939693 + 5.32926i) q^{78} +(9.96451 + 8.36121i) q^{79} +(2.20574 + 0.802823i) q^{80} +(-0.939693 + 0.342020i) q^{81} +(-1.48886 + 1.24930i) q^{82} +(4.08512 + 7.07564i) q^{83} +(1.78699 - 3.09516i) q^{84} +(1.58125 + 8.96773i) q^{85} +(0.872111 + 4.94599i) q^{86} +(1.76604 - 3.05888i) q^{87} +(-1.35844 - 2.35289i) q^{88} +(8.98158 - 7.53644i) q^{89} +(2.20574 - 0.802823i) q^{90} +(-18.1741 - 6.61484i) q^{91} +(-6.30793 - 5.29298i) q^{92} +(-1.13429 + 6.43285i) q^{93} +2.98545 q^{94} +(5.63429 + 8.54055i) q^{95} -1.00000 q^{96} +(-1.49407 + 8.47329i) q^{97} +(4.42262 + 3.71102i) q^{98} +(-2.55303 - 0.929228i) 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 \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

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

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
\(5\)	2.20574	−	0.802823i	0.986436	−	0.359033i								0.202097	−	0.979366i	\(-0.435225\pi\)
							0.784339	+	0.620332i	\(0.213002\pi\)
\(7\)	−1.78699	−	3.09516i	−0.675418	−	1.16986i	−0.976346	−	0.216212i	\(-0.930630\pi\)
							0.300928	−	0.953647i	\(-0.402704\pi\)
\(11\)	−1.35844	+	2.35289i	−0.409585	+	0.709423i	−0.994843	−	0.101425i	\(-0.967660\pi\)
							0.585258	+	0.810847i	\(0.300993\pi\)
\(13\)	4.14543	−	3.47843i	1.14974	−	0.964742i	0.150022	−	0.988683i	\(-0.452066\pi\)
							0.999713	+	0.0239402i	\(0.00762112\pi\)
\(17\)	−0.673648	+	3.82045i	−0.163384	+	0.926595i	0.787332	+	0.616530i	\(0.211462\pi\)
							−0.950715	+	0.310065i	\(0.899649\pi\)
\(19\)	1.01114	+	4.24000i	0.231972	+	0.972722i
							\(23\)	7.73783	+	2.81634i	1.61345	+	0.587247i	0.982118	−	0.188267i	\(-0.0602870\pi\)
0.631330	+	0.775514i	\(0.282509\pi\)
\(29\)	0.613341	+	3.47843i	0.113895	+	0.645928i	0.987292	+	0.158919i	\(0.0508009\pi\)
							−0.873397	+	0.487009i	\(0.838088\pi\)
\(31\)	−3.26604	−	5.65695i	−0.586599	−	1.01602i	−0.994674	−	0.103071i	\(-0.967133\pi\)
							0.408075	−	0.912948i	\(-0.366200\pi\)
\(37\)	0.389185			0.0639817			0.0319908	−	0.999488i	\(-0.489815\pi\)
							0.0319908	+	0.999488i	\(0.489815\pi\)
\(41\)	−1.48886	−	1.24930i	−0.232520	−	0.195108i	0.519082	−	0.854725i	\(-0.326274\pi\)
							−0.751602	+	0.659617i	\(0.770718\pi\)
\(43\)	−4.71941	+	1.71772i	−0.719703	+	0.261950i	−0.675800	−	0.737085i	\(-0.736202\pi\)
							−0.0439033	+	0.999036i	\(0.513979\pi\)
\(47\)	0.518418	+	2.94010i	0.0756191	+	0.428857i	0.998989	+	0.0449466i	\(0.0143118\pi\)
							−0.923370	+	0.383911i	\(0.874577\pi\)
\(53\)	−7.80453	−	2.84062i	−1.07203	−	0.390189i	−0.255097	−	0.966915i	\(-0.582108\pi\)
							−0.816937	+	0.576727i	\(0.804330\pi\)
\(59\)	0.474308	−	2.68993i	0.0617496	−	0.350199i	−0.938241	−	0.345981i	\(-0.887546\pi\)
							0.999991	−	0.00421836i	\(-0.00134275\pi\)
\(61\)	−5.91147	−	2.15160i	−0.756887	−	0.275484i	−0.0653860	−	0.997860i	\(-0.520828\pi\)
							−0.691501	+	0.722376i	\(0.743050\pi\)
\(67\)	−2.59374	−	14.7098i	−0.316876	−	1.79709i	−0.561503	−	0.827475i	\(-0.689777\pi\)
							0.244627	−	0.969617i	\(-0.421335\pi\)
\(71\)	−8.47818	+	3.08580i	−1.00617	+	0.366218i	−0.791963	−	0.610570i	\(-0.790941\pi\)
							−0.214212	+	0.976787i	\(0.568718\pi\)
\(73\)	7.88326	+	6.61484i	0.922665	+	0.774208i	0.974486	−	0.224448i	\(-0.0720580\pi\)
							−0.0518207	+	0.998656i	\(0.516502\pi\)
\(79\)	9.96451	+	8.36121i	1.12109	+	0.940710i	0.998659	−	0.0517663i	\(-0.0164851\pi\)
							0.122435	+	0.992476i	\(0.460930\pi\)
\(83\)	4.08512	+	7.07564i	0.448400	+	0.776652i	0.998282	−	0.0585902i	\(-0.0186605\pi\)
							−0.549882	+	0.835243i	\(0.685327\pi\)
\(89\)	8.98158	−	7.53644i	0.952046	−	0.798861i	−0.0275951	−	0.999619i	\(-0.508785\pi\)
							0.979641	+	0.200758i	\(0.0643405\pi\)
\(97\)	−1.49407	+	8.47329i	−0.151700	+	0.860333i	0.810041	+	0.586373i	\(0.199445\pi\)
							−0.961741	+	0.273960i	\(0.911666\pi\)

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

    

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