Embedded newform 138.2.e.a.121.1

• Label: 138.2.e.a.121.1
• Level: $138$
• Weight: $2$
• Character: 138.121
• Analytic conductor: $1.102$
• Analytic rank: $0$
• Dimension: $10$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	138.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.10193554789\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\Q(\zeta_{22})\)
Defining polynomial:	\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			121.1
Root			\(-0.415415 + 0.909632i\) of defining polynomial
Character	\(\chi\)	\(=\)	138.121
Dual form			138.2.e.a.73.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.415415 - 0.909632i) q^{2} +(-0.959493 - 0.281733i) q^{3} +(-0.654861 - 0.755750i) q^{4} +(1.27310 - 0.818172i) q^{5} +(-0.654861 + 0.755750i) q^{6} +(0.369215 - 2.56794i) q^{7} +(-0.959493 + 0.281733i) q^{8} +(0.841254 + 0.540641i) q^{9} +(-0.215370 - 1.49793i) q^{10} +(-2.08004 - 4.55466i) q^{11} +(0.415415 + 0.909632i) q^{12} +(0.686393 + 4.77397i) q^{13} +(-2.18251 - 1.40261i) q^{14} +(-1.45204 + 0.426356i) q^{15} +(-0.142315 + 0.989821i) q^{16} +(0.565599 - 0.652736i) q^{17} +(0.841254 - 0.540641i) q^{18} +(4.58506 + 5.29144i) q^{19} +(-1.45204 - 0.426356i) q^{20} +(-1.07773 + 2.35990i) q^{21} -5.00714 q^{22} +(4.66533 + 1.11118i) q^{23} +1.00000 q^{24} +(-1.12570 + 2.46493i) q^{25} +(4.62769 + 1.35881i) q^{26} +(-0.654861 - 0.755750i) q^{27} +(-2.18251 + 1.40261i) q^{28} +(-2.03899 + 2.35312i) q^{29} +(-0.215370 + 1.49793i) q^{30} +(1.12671 - 0.330833i) q^{31} +(0.841254 + 0.540641i) q^{32} +(0.712591 + 4.95618i) q^{33} +(-0.358791 - 0.785643i) q^{34} +(-1.63097 - 3.57133i) q^{35} +(-0.142315 - 0.989821i) q^{36} +(-0.506538 - 0.325532i) q^{37} +(6.71797 - 1.97257i) q^{38} +(0.686393 - 4.77397i) q^{39} +(-0.991025 + 1.14370i) q^{40} +(4.45305 - 2.86180i) q^{41} +(1.69894 + 1.96068i) q^{42} +(-1.40014 - 0.411119i) q^{43} +(-2.08004 + 4.55466i) q^{44} +1.51334 q^{45} +(2.94881 - 3.78213i) q^{46} -12.6797 q^{47} +(0.415415 - 0.909632i) q^{48} +(0.258432 + 0.0758824i) q^{49} +(1.77455 + 2.04794i) q^{50} +(-0.726585 + 0.466948i) q^{51} +(3.15843 - 3.64502i) q^{52} +(-0.602392 + 4.18973i) q^{53} +(-0.959493 + 0.281733i) q^{54} +(-6.37459 - 4.09670i) q^{55} +(0.369215 + 2.56794i) q^{56} +(-2.90856 - 6.36886i) q^{57} +(1.29345 + 2.83225i) q^{58} +(-0.566615 - 3.94090i) q^{59} +(1.27310 + 0.818172i) q^{60} +(6.96812 - 2.04602i) q^{61} +(0.167117 - 1.16233i) q^{62} +(1.69894 - 1.96068i) q^{63} +(0.841254 - 0.540641i) q^{64} +(4.77977 + 5.51615i) q^{65} +(4.80432 + 1.41067i) q^{66} +(-6.70359 + 14.6788i) q^{67} -0.863693 q^{68} +(-4.16330 - 2.38054i) q^{69} -3.92613 q^{70} +(2.44456 - 5.35285i) q^{71} +(-0.959493 - 0.281733i) q^{72} +(-9.41782 - 10.8687i) q^{73} +(-0.506538 + 0.325532i) q^{74} +(1.77455 - 2.04794i) q^{75} +(0.996429 - 6.93032i) q^{76} +(-12.4641 + 3.65979i) q^{77} +(-4.05742 - 2.60754i) q^{78} +(-1.47156 - 10.2349i) q^{79} +(0.628663 + 1.37658i) q^{80} +(0.415415 + 0.909632i) q^{81} +(-0.753323 - 5.23948i) q^{82} +(2.22176 + 1.42784i) q^{83} +(2.48926 - 0.730913i) q^{84} +(0.186014 - 1.29376i) q^{85} +(-0.955607 + 1.10283i) q^{86} +(2.61935 - 1.68335i) q^{87} +(3.27898 + 3.78415i) q^{88} +(5.02538 + 1.47559i) q^{89} +(0.628663 - 1.37658i) q^{90} +12.5127 q^{91} +(-2.21537 - 4.25348i) q^{92} -1.17428 q^{93} +(-5.26732 + 11.5338i) q^{94} +(10.1666 + 2.98517i) q^{95} +(-0.654861 - 0.755750i) q^{96} +(12.5782 - 8.08355i) q^{97} +(0.176382 - 0.203555i) q^{98} +(0.712591 - 4.95618i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	10 * q - q^2 - q^3 - q^4 + 8 * q^5 - q^6 + 8 * q^7 - q^8 - q^9 - 3 * q^10 + 7 * q^11 - q^12 + 3 * q^13 - 3 * q^14 - 3 * q^15 - q^16 + 4 * q^17 - q^18 - 3 * q^20 - 3 * q^21 - 26 * q^22 - 12 * q^23 + 10 * q^24 - 15 * q^25 + 3 * q^26 - q^27 - 3 * q^28 - 25 * q^29 - 3 * q^30 + 6 * q^31 - q^32 - 4 * q^33 - 7 * q^34 + 2 * q^35 - q^36 + 9 * q^37 + 11 * q^38 + 3 * q^39 - 3 * q^40 + 24 * q^41 + 8 * q^42 - 30 * q^43 + 7 * q^44 - 14 * q^45 + 21 * q^46 - 48 * q^47 - q^48 + 9 * q^49 + 7 * q^50 + 15 * q^51 + 14 * q^52 + 15 * q^53 - q^54 - 23 * q^55 + 8 * q^56 - 11 * q^57 - 3 * q^58 + 5 * q^59 + 8 * q^60 + 12 * q^61 + 28 * q^62 + 8 * q^63 - q^64 - 13 * q^65 + 18 * q^66 + 18 * q^67 - 18 * q^68 - q^69 + 2 * q^70 + 28 * q^71 - q^72 + 19 * q^73 + 9 * q^74 + 7 * q^75 + 22 * q^76 - 12 * q^77 - 8 * q^78 - 52 * q^79 + 8 * q^80 - q^81 - 20 * q^82 + 7 * q^83 - 3 * q^84 + 23 * q^85 + 14 * q^86 + 30 * q^87 - 4 * q^88 + 3 * q^89 + 8 * q^90 + 42 * q^91 - 23 * q^92 - 16 * q^93 + 29 * q^94 + 22 * q^95 - q^96 + 51 * q^97 - 2 * q^98 - 4 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{11}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.415415	−	0.909632i	0.293743	−	0.643207i
							\(3\)	−0.959493	−	0.281733i	−0.553964	−	0.162658i
\(5\)	1.27310	−	0.818172i	0.569348	−	0.365898i								−0.224072	−	0.974573i	\(-0.571935\pi\)
							0.793420	+	0.608675i	\(0.208299\pi\)
\(7\)	0.369215	−	2.56794i	0.139550	−	0.970592i	−0.792915	−	0.609332i	\(-0.791438\pi\)
							0.932465	−	0.361260i	\(-0.117653\pi\)
\(11\)	−2.08004	−	4.55466i	−0.627156	−	1.37328i	−0.910198	−	0.414173i	\(-0.864071\pi\)
							0.283042	−	0.959108i	\(-0.408656\pi\)
\(13\)	0.686393	+	4.77397i	0.190371	+	1.32406i	0.831027	+	0.556232i	\(0.187753\pi\)
							−0.640656	+	0.767828i	\(0.721337\pi\)
\(17\)	0.565599	−	0.652736i	0.137178	−	0.158312i	−0.683004	−	0.730415i	\(-0.739327\pi\)
							0.820182	+	0.572103i	\(0.193872\pi\)
\(19\)	4.58506	+	5.29144i	1.05189	+	1.21394i	0.976214	+	0.216807i	\(0.0695644\pi\)
							0.0756708	+	0.997133i	\(0.475890\pi\)
\(23\)	4.66533	+	1.11118i	0.972788	+	0.231696i
							\(29\)	−2.03899	+	2.35312i	−0.378631	+	0.436963i	−0.912795	−	0.408417i	\(-0.866081\pi\)
0.534165	+	0.845381i	\(0.320626\pi\)
\(31\)	1.12671	−	0.330833i	0.202364	−	0.0594193i	−0.178980	−	0.983853i	\(-0.557280\pi\)
							0.381343	+	0.924433i	\(0.375462\pi\)
\(37\)	−0.506538	−	0.325532i	−0.0832743	−	0.0535171i	0.498341	−	0.866981i	\(-0.333943\pi\)
							−0.581615	+	0.813464i	\(0.697579\pi\)
\(41\)	4.45305	−	2.86180i	0.695450	−	0.446939i	−0.144570	−	0.989495i	\(-0.546180\pi\)
							0.840020	+	0.542556i	\(0.182543\pi\)
\(43\)	−1.40014	−	0.411119i	−0.213520	−	0.0626951i	0.173223	−	0.984883i	\(-0.444582\pi\)
							−0.386743	+	0.922188i	\(0.626400\pi\)
\(47\)	−12.6797			−1.84952			−0.924760	−	0.380552i	\(-0.875734\pi\)
							−0.924760	+	0.380552i	\(0.875734\pi\)
\(53\)	−0.602392	+	4.18973i	−0.0827449	+	0.575504i	0.905700	+	0.423920i	\(0.139346\pi\)
							−0.988444	+	0.151583i	\(0.951563\pi\)
\(59\)	−0.566615	−	3.94090i	−0.0737670	−	0.513061i	−0.992885	−	0.119078i	\(-0.962006\pi\)
							0.919118	−	0.393983i	\(-0.128903\pi\)
\(61\)	6.96812	−	2.04602i	0.892176	−	0.261966i	0.196655	−	0.980473i	\(-0.436992\pi\)
							0.695520	+	0.718506i	\(0.255174\pi\)
\(67\)	−6.70359	+	14.6788i	−0.818973	+	1.79330i	−0.256360	+	0.966581i	\(0.582523\pi\)
							−0.562614	+	0.826720i	\(0.690204\pi\)
\(71\)	2.44456	−	5.35285i	0.290116	−	0.635266i	−0.707315	−	0.706899i	\(-0.750094\pi\)
							0.997431	+	0.0716327i	\(0.0228209\pi\)
\(73\)	−9.41782	−	10.8687i	−1.10227	−	1.27209i	−0.959308	−	0.282362i	\(-0.908882\pi\)
							−0.142964	−	0.989728i	\(-0.545663\pi\)
\(79\)	−1.47156	−	10.2349i	−0.165564	−	1.15152i	−0.887919	−	0.459999i	\(-0.847850\pi\)
							0.722356	−	0.691522i	\(-0.243059\pi\)
\(83\)	2.22176	+	1.42784i	0.243870	+	0.156725i	0.656866	−	0.754007i	\(-0.271882\pi\)
							−0.412997	+	0.910732i	\(0.635518\pi\)
\(89\)	5.02538	+	1.47559i	0.532689	+	0.156412i	0.537005	−	0.843579i	\(-0.319556\pi\)
							−0.00431578	+	0.999991i	\(0.501374\pi\)
\(97\)	12.5782	−	8.08355i	1.27713	−	0.820760i	0.286597	−	0.958051i	\(-0.407476\pi\)
							0.990531	+	0.137291i	\(0.0438397\pi\)

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	138.2.e.a.121.1	yes	10
3.2	odd	2		414.2.i.d.397.1		10
23.2	even	11		3174.2.a.bc.1.5		5
23.4	even	11	inner	138.2.e.a.73.1	✓	10
23.21	odd	22		3174.2.a.bd.1.1		5
69.2	odd	22		9522.2.a.bt.1.1		5
69.44	even	22		9522.2.a.bq.1.5		5
69.50	odd	22		414.2.i.d.73.1		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.a.73.1	✓	10	23.4	even	11	inner
138.2.e.a.121.1	yes	10	1.1	even	1	trivial
414.2.i.d.73.1		10	69.50	odd	22
414.2.i.d.397.1		10	3.2	odd	2
3174.2.a.bc.1.5		5	23.2	even	11
3174.2.a.bd.1.1		5	23.21	odd	22
9522.2.a.bq.1.5		5	69.44	even	22
9522.2.a.bt.1.1		5	69.2	odd	22