Embedded newform 114.2.i.d.73.1

• Label: 114.2.i.d.73.1
• Level: $114$
• Weight: $2$
• Character: 114.73
• 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			73.1
Root			\(-0.766044 - 0.642788i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.73
Dual form			114.2.i.d.25.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.766044 - 0.642788i) q^{2} +(0.939693 - 0.342020i) q^{3} +(0.173648 + 0.984808i) q^{4} +(0.386659 - 2.19285i) q^{5} +(-0.939693 - 0.342020i) q^{6} +(0.326352 + 0.565258i) q^{7} +(0.500000 - 0.866025i) q^{8} +(0.766044 - 0.642788i) q^{9} +(-1.70574 + 1.43128i) q^{10} +(0.766044 - 1.32683i) q^{11} +(0.500000 + 0.866025i) q^{12} +(0.439693 + 0.160035i) q^{13} +(0.113341 - 0.642788i) q^{14} +(-0.386659 - 2.19285i) q^{15} +(-0.939693 + 0.342020i) q^{16} +(-1.61334 - 1.35375i) q^{17} -1.00000 q^{18} +(-2.23396 + 3.74292i) q^{19} +2.22668 q^{20} +(0.500000 + 0.419550i) q^{21} +(-1.43969 + 0.524005i) q^{22} +(1.02481 + 5.81201i) q^{23} +(0.173648 - 0.984808i) q^{24} +(0.0393628 + 0.0143269i) q^{25} +(-0.233956 - 0.405223i) q^{26} +(0.500000 - 0.866025i) q^{27} +(-0.500000 + 0.419550i) q^{28} +(-6.38326 + 5.35619i) q^{29} +(-1.11334 + 1.92836i) q^{30} +(4.31908 + 7.48086i) q^{31} +(0.939693 + 0.342020i) q^{32} +(0.266044 - 1.50881i) q^{33} +(0.365715 + 2.07407i) q^{34} +(1.36571 - 0.497079i) q^{35} +(0.766044 + 0.642788i) q^{36} -4.67499 q^{37} +(4.11721 - 1.43128i) q^{38} +0.467911 q^{39} +(-1.70574 - 1.43128i) q^{40} +(-3.26604 + 1.18874i) q^{41} +(-0.113341 - 0.642788i) q^{42} +(1.78699 - 10.1345i) q^{43} +(1.43969 + 0.524005i) q^{44} +(-1.11334 - 1.92836i) q^{45} +(2.95084 - 5.11100i) q^{46} +(-3.55303 + 2.98135i) q^{47} +(-0.766044 + 0.642788i) q^{48} +(3.28699 - 5.69323i) q^{49} +(-0.0209445 - 0.0362770i) q^{50} +(-1.97906 - 0.720317i) q^{51} +(-0.0812519 + 0.460802i) q^{52} +(-2.07532 - 11.7697i) q^{53} +(-0.939693 + 0.342020i) q^{54} +(-2.61334 - 2.19285i) q^{55} +0.652704 q^{56} +(-0.819078 + 4.28125i) q^{57} +8.33275 q^{58} +(10.9042 + 9.14971i) q^{59} +(2.09240 - 0.761570i) q^{60} +(-1.58378 - 8.98205i) q^{61} +(1.50000 - 8.50692i) q^{62} +(0.613341 + 0.223238i) q^{63} +(-0.500000 - 0.866025i) q^{64} +(0.520945 - 0.902302i) q^{65} +(-1.17365 + 0.984808i) q^{66} +(0.190722 - 0.160035i) q^{67} +(1.05303 - 1.82391i) q^{68} +(2.95084 + 5.11100i) q^{69} +(-1.36571 - 0.497079i) q^{70} +(-0.772441 + 4.38073i) q^{71} +(-0.173648 - 0.984808i) q^{72} +(8.54323 - 3.10948i) q^{73} +(3.58125 + 3.00503i) q^{74} +0.0418891 q^{75} +(-4.07398 - 1.55007i) q^{76} +1.00000 q^{77} +(-0.358441 - 0.300767i) q^{78} +(-11.3302 + 4.12386i) q^{79} +(0.386659 + 2.19285i) q^{80} +(0.173648 - 0.984808i) q^{81} +(3.26604 + 1.18874i) q^{82} +(-1.85457 - 3.21221i) q^{83} +(-0.326352 + 0.565258i) q^{84} +(-3.59240 + 3.01438i) q^{85} +(-7.88326 + 6.61484i) q^{86} +(-4.16637 + 7.21637i) q^{87} +(-0.766044 - 1.32683i) q^{88} +(-15.7554 - 5.73448i) q^{89} +(-0.386659 + 2.19285i) q^{90} +(0.0530334 + 0.300767i) q^{91} +(-5.54576 + 2.01849i) q^{92} +(6.61721 + 5.55250i) q^{93} +4.63816 q^{94} +(7.34389 + 6.34597i) q^{95} +1.00000 q^{96} +(1.89053 + 1.58634i) q^{97} +(-6.17752 + 2.24843i) q^{98} +(-0.266044 - 1.50881i) 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 \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{9}\right)\)

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
\(5\)	0.386659	−	2.19285i	0.172919	−	0.980674i								−0.767599	−	0.640930i	\(-0.778549\pi\)
							0.940518	−	0.339743i	\(-0.110340\pi\)
\(7\)	0.326352	+	0.565258i	0.123349	+	0.213647i	0.921087	−	0.389358i	\(-0.127303\pi\)
							−0.797737	+	0.603005i	\(0.793970\pi\)
\(11\)	0.766044	−	1.32683i	0.230971	−	0.400054i	−0.727123	−	0.686507i	\(-0.759143\pi\)
							0.958094	+	0.286453i	\(0.0924764\pi\)
\(13\)	0.439693	+	0.160035i	0.121949	+	0.0443857i	0.402274	−	0.915519i	\(-0.368220\pi\)
							−0.280325	+	0.959905i	\(0.590442\pi\)
\(17\)	−1.61334	−	1.35375i	−0.391293	−	0.328333i	0.425824	−	0.904806i	\(-0.359984\pi\)
							−0.817116	+	0.576473i	\(0.804429\pi\)
\(19\)	−2.23396	+	3.74292i	−0.512505	+	0.858685i
							\(23\)	1.02481	+	5.81201i	0.213689	+	1.21189i	0.883168	+	0.469058i	\(0.155406\pi\)
−0.669479	+	0.742831i	\(0.733483\pi\)
\(29\)	−6.38326	+	5.35619i	−1.18534	+	0.994619i	−0.185412	+	0.982661i	\(0.559362\pi\)
							−0.999928	+	0.0119582i	\(0.996193\pi\)
\(31\)	4.31908	+	7.48086i	0.775729	+	1.34360i	0.934384	+	0.356268i	\(0.115951\pi\)
							−0.158654	+	0.987334i	\(0.550716\pi\)
\(37\)	−4.67499			−0.768564			−0.384282	−	0.923216i	\(-0.625551\pi\)
							−0.384282	+	0.923216i	\(0.625551\pi\)
\(41\)	−3.26604	+	1.18874i	−0.510070	+	0.185650i	−0.584218	−	0.811597i	\(-0.698599\pi\)
							0.0741475	+	0.997247i	\(0.476376\pi\)
\(43\)	1.78699	−	10.1345i	0.272513	−	1.54550i	−0.474238	−	0.880396i	\(-0.657277\pi\)
							0.746752	−	0.665103i	\(-0.231612\pi\)
\(47\)	−3.55303	+	2.98135i	−0.518263	+	0.434874i	−0.864026	−	0.503448i	\(-0.832065\pi\)
							0.345763	+	0.938322i	\(0.387620\pi\)
\(53\)	−2.07532	−	11.7697i	−0.285067	−	1.61670i	−0.705045	−	0.709163i	\(-0.749073\pi\)
							0.419977	−	0.907535i	\(-0.362038\pi\)
\(59\)	10.9042	+	9.14971i	1.41961	+	1.19119i	0.951551	+	0.307492i	\(0.0994896\pi\)
							0.468055	+	0.883699i	\(0.344955\pi\)
\(61\)	−1.58378	−	8.98205i	−0.202782	−	1.15003i	−0.900892	−	0.434043i	\(-0.857086\pi\)
							0.698110	−	0.715991i	\(-0.254025\pi\)
\(67\)	0.190722	−	0.160035i	0.0233004	−	0.0195514i	−0.631063	−	0.775732i	\(-0.717381\pi\)
							0.654363	+	0.756180i	\(0.272937\pi\)
\(71\)	−0.772441	+	4.38073i	−0.0916719	+	0.519897i	0.904045	+	0.427438i	\(0.140584\pi\)
							−0.995716	+	0.0924590i	\(0.970527\pi\)
\(73\)	8.54323	−	3.10948i	0.999910	−	0.363937i	0.210360	−	0.977624i	\(-0.432536\pi\)
							0.789550	+	0.613687i	\(0.210314\pi\)
\(79\)	−11.3302	+	4.12386i	−1.27475	+	0.463971i	−0.888692	−	0.458504i	\(-0.848385\pi\)
							−0.386057	+	0.922475i	\(0.626163\pi\)
\(83\)	−1.85457	−	3.21221i	−0.203566	−	0.352586i	0.746109	−	0.665824i	\(-0.231920\pi\)
							−0.949675	+	0.313238i	\(0.898586\pi\)
\(89\)	−15.7554	−	5.73448i	−1.67007	−	0.607854i	−0.678169	−	0.734906i	\(-0.737226\pi\)
							−0.991896	+	0.127051i	\(0.959449\pi\)
\(97\)	1.89053	+	1.58634i	0.191954	+	0.161069i	0.733700	−	0.679474i	\(-0.237792\pi\)
							−0.541746	+	0.840543i	\(0.682236\pi\)

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.73.1	yes	6
3.2	odd	2		342.2.u.a.73.1		6
4.3	odd	2		912.2.bo.f.529.1		6
19.5	even	9		2166.2.a.o.1.3		3
19.6	even	9	inner	114.2.i.d.25.1	✓	6
19.14	odd	18		2166.2.a.u.1.3		3
57.5	odd	18		6498.2.a.bs.1.1		3
57.14	even	18		6498.2.a.bn.1.1		3
57.44	odd	18		342.2.u.a.253.1		6
76.63	odd	18		912.2.bo.f.481.1		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.d.25.1	✓	6	19.6	even	9	inner
114.2.i.d.73.1	yes	6	1.1	even	1	trivial
342.2.u.a.73.1		6	3.2	odd	2
342.2.u.a.253.1		6	57.44	odd	18
912.2.bo.f.481.1		6	76.63	odd	18
912.2.bo.f.529.1		6	4.3	odd	2
2166.2.a.o.1.3		3	19.5	even	9
2166.2.a.u.1.3		3	19.14	odd	18
6498.2.a.bn.1.1		3	57.14	even	18
6498.2.a.bs.1.1		3	57.5	odd	18