Embedded newform 135.2.q.a.113.3

• Label: 135.2.q.a.113.3
• Level: $135$
• Weight: $2$
• Character: 135.113
• Analytic conductor: $1.078$
• Analytic rank: $0$
• Dimension: $192$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	135.q (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.07798042729\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(16\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			113.3
Character	\(\chi\)	\(=\)	135.113
Dual form			135.2.q.a.92.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.19031 - 0.191628i) q^{2} +(-1.58451 + 0.699511i) q^{3} +(2.79113 + 0.492152i) q^{4} +(-2.15257 + 0.605349i) q^{5} +(3.60463 - 1.22851i) q^{6} +(0.000530458 + 0.000371431i) q^{7} +(-1.77162 - 0.474704i) q^{8} +(2.02137 - 2.21677i) q^{9} +(4.83080 - 0.913412i) q^{10} +(0.925209 - 2.54199i) q^{11} +(-4.76685 + 1.17261i) q^{12} +(-0.277011 - 3.16625i) q^{13} +(-0.00109069 - 0.000915200i) q^{14} +(2.98733 - 2.46493i) q^{15} +(-1.53710 - 0.559459i) q^{16} +(6.58840 - 1.76536i) q^{17} +(-4.85222 + 4.46807i) q^{18} +(1.52337 - 0.879517i) q^{19} +(-6.30603 + 0.630218i) q^{20} +(-0.00110034 - 0.000217476i) q^{21} +(-2.51361 + 5.39046i) q^{22} +(-2.93096 - 4.18584i) q^{23} +(3.13922 - 0.487092i) q^{24} +(4.26710 - 2.60611i) q^{25} +6.98815i q^{26} +(-1.65223 + 4.92647i) q^{27} +(0.00129778 + 0.00129778i) q^{28} +(-4.37718 + 3.67289i) q^{29} +(-7.01553 + 4.82651i) q^{30} +(-0.944975 + 5.35922i) q^{31} +(6.58407 + 3.07020i) q^{32} +(0.312144 + 4.67501i) q^{33} +(-14.7689 + 2.60416i) q^{34} +(-0.00136669 - 0.000478418i) q^{35} +(6.73289 - 5.19248i) q^{36} +(-0.0678960 - 0.253391i) q^{37} +(-3.50519 + 1.63450i) q^{38} +(2.65375 + 4.82319i) q^{39} +(4.10090 - 0.0506152i) q^{40} +(7.56097 - 9.01082i) q^{41} +(0.00236841 + 0.000687195i) q^{42} +(-3.67622 - 7.88368i) q^{43} +(3.83342 - 6.63969i) q^{44} +(-3.00922 + 5.99538i) q^{45} +(5.61759 + 9.72995i) q^{46} +(-0.403711 + 0.576559i) q^{47} +(2.82690 - 0.188748i) q^{48} +(-2.39414 - 6.57785i) q^{49} +(-9.84570 + 4.89050i) q^{50} +(-9.20452 + 7.40589i) q^{51} +(0.785101 - 8.97374i) q^{52} +(5.73786 - 5.73786i) q^{53} +(4.56295 - 10.4739i) q^{54} +(-0.452784 + 6.03188i) q^{55} +(-0.000763450 - 0.000909845i) q^{56} +(-1.79857 + 2.45922i) q^{57} +(10.2912 - 7.20599i) q^{58} +(-2.11198 + 0.768697i) q^{59} +(9.55114 - 5.40973i) q^{60} +(-1.14333 - 6.48417i) q^{61} +(3.09676 - 11.5573i) q^{62} +(0.00189563 - 0.000425105i) q^{63} +(-10.9996 - 6.35064i) q^{64} +(2.51297 + 6.64787i) q^{65} +(0.212168 - 10.2996i) q^{66} +(2.47165 - 0.216242i) q^{67} +(19.2579 - 1.68485i) q^{68} +(7.57218 + 4.58228i) q^{69} +(0.00290181 + 0.00130978i) q^{70} +(-10.4316 - 6.02267i) q^{71} +(-4.63341 + 2.96772i) q^{72} +(-0.375408 + 1.40104i) q^{73} +(0.100157 + 0.568017i) q^{74} +(-4.93828 + 7.11431i) q^{75} +(4.68478 - 1.70512i) q^{76} +(0.00143496 - 0.00100477i) q^{77} +(-4.88829 - 11.0728i) q^{78} +(3.36157 + 4.00616i) q^{79} +(3.64738 + 0.273791i) q^{80} +(-0.828138 - 8.96182i) q^{81} +(-18.2876 + 18.2876i) q^{82} +(-1.39592 + 15.9554i) q^{83} +(-0.00296416 - 0.00114854i) q^{84} +(-13.1133 + 7.78833i) q^{85} +(6.54134 + 17.9722i) q^{86} +(4.36647 - 8.88163i) q^{87} +(-2.84581 + 4.06424i) q^{88} +(2.99426 + 5.18621i) q^{89} +(7.74000 - 12.5551i) q^{90} +(0.00102910 - 0.00178245i) q^{91} +(-6.12062 - 13.1257i) q^{92} +(-2.25151 - 9.15278i) q^{93} +(0.994737 - 1.18548i) q^{94} +(-2.74674 + 2.81539i) q^{95} +(-12.5802 - 0.259149i) q^{96} +(7.62753 - 3.55678i) q^{97} +(3.98342 + 14.8663i) q^{98} +(-3.76482 - 7.18927i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	192 * q - 12 * q^2 - 12 * q^3 - 12 * q^5 - 36 * q^6 - 12 * q^7 - 18 * q^8 - 6 * q^10 - 36 * q^11 - 12 * q^12 - 12 * q^13 - 12 * q^15 - 24 * q^16 - 18 * q^17 - 54 * q^18 + 36 * q^20 - 24 * q^21 - 12 * q^22 - 36 * q^23 - 30 * q^25 - 36 * q^27 - 24 * q^28 + 60 * q^30 - 24 * q^31 - 48 * q^32 - 6 * q^33 + 36 * q^35 + 12 * q^36 - 6 * q^37 + 12 * q^38 - 36 * q^40 + 24 * q^41 - 24 * q^42 - 12 * q^43 + 18 * q^45 - 12 * q^46 - 6 * q^47 + 12 * q^48 + 36 * q^50 + 144 * q^51 + 12 * q^52 - 24 * q^55 + 180 * q^56 - 12 * q^57 - 12 * q^58 - 36 * q^60 - 60 * q^61 - 18 * q^62 - 54 * q^63 - 84 * q^65 + 72 * q^66 + 24 * q^67 - 60 * q^68 - 12 * q^70 - 36 * q^71 + 180 * q^72 - 6 * q^73 - 60 * q^75 - 72 * q^76 + 132 * q^77 + 78 * q^78 + 12 * q^81 - 24 * q^82 + 48 * q^83 - 12 * q^85 + 12 * q^86 + 144 * q^87 - 48 * q^88 + 48 * q^90 - 12 * q^91 + 258 * q^92 + 180 * q^93 + 18 * q^95 - 12 * q^96 + 24 * q^97 + 324 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\)	\(56\)	\(82\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{3}{4}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.19031	−	0.191628i	−1.54878	−	0.135501i	−0.719566	−	0.694425i	\(-0.755659\pi\)
							−0.829219	+	0.558923i	\(0.811215\pi\)
\(3\)	−1.58451	+	0.699511i	−0.914820	+	0.403863i
							\(5\)	−2.15257	+	0.605349i	−0.962658	+	0.270720i
\(7\)	0.000530458	0	0.000371431i	0.000200494	0	0.000140388i								0.573677	−	0.819082i	\(-0.305517\pi\)
							−0.573476	+	0.819222i	\(0.694405\pi\)
\(11\)	0.925209	−	2.54199i	0.278961	−	0.766439i	−0.718520	−	0.695506i	\(-0.755180\pi\)
							0.997481	−	0.0709326i	\(-0.0225975\pi\)
\(13\)	−0.277011	−	3.16625i	−0.0768289	−	0.878159i	−0.932625	−	0.360846i	\(-0.882488\pi\)
							0.855797	−	0.517313i	\(-0.173068\pi\)
\(17\)	6.58840	−	1.76536i	1.59792	−	0.428162i	0.653507	−	0.756921i	\(-0.273297\pi\)
							0.944414	+	0.328759i	\(0.106630\pi\)
\(19\)	1.52337	−	0.879517i	0.349485	−	0.201775i	−0.314974	−	0.949100i	\(-0.601996\pi\)
							0.664458	+	0.747325i	\(0.268662\pi\)
\(23\)	−2.93096	−	4.18584i	−0.611147	−	0.872808i	0.387652	−	0.921806i	\(-0.373286\pi\)
							−0.998799	+	0.0489979i	\(0.984397\pi\)
\(29\)	−4.37718	+	3.67289i	−0.812822	+	0.682038i	−0.951279	−	0.308330i	\(-0.900230\pi\)
							0.138458	+	0.990368i	\(0.455785\pi\)
\(31\)	−0.944975	+	5.35922i	−0.169723	+	0.962544i	0.774338	+	0.632772i	\(0.218083\pi\)
							−0.944061	+	0.329772i	\(0.893028\pi\)
\(37\)	−0.0678960	−	0.253391i	−0.0111620	−	0.0416573i	0.960120	−	0.279587i	\(-0.0901977\pi\)
							−0.971282	+	0.237930i	\(0.923531\pi\)
\(41\)	7.56097	−	9.01082i	1.18083	−	1.40725i	0.287537	−	0.957770i	\(-0.407164\pi\)
							0.893289	−	0.449483i	\(-0.148392\pi\)
\(43\)	−3.67622	−	7.88368i	−0.560618	−	1.20225i	−0.958574	−	0.284845i	\(-0.908058\pi\)
							0.397955	−	0.917405i	\(-0.369720\pi\)
\(47\)	−0.403711	+	0.576559i	−0.0588873	+	0.0840997i	−0.847516	−	0.530770i	\(-0.821903\pi\)
							0.788628	+	0.614870i	\(0.210792\pi\)
\(53\)	5.73786	−	5.73786i	0.788156	−	0.788156i	−0.193036	−	0.981192i	\(-0.561833\pi\)
							0.981192	+	0.193036i	\(0.0618333\pi\)
\(59\)	−2.11198	+	0.768697i	−0.274956	+	0.100076i	−0.475818	−	0.879544i	\(-0.657848\pi\)
							0.200862	+	0.979620i	\(0.435626\pi\)
\(61\)	−1.14333	−	6.48417i	−0.146389	−	0.830213i	−0.966241	−	0.257638i	\(-0.917056\pi\)
							0.819852	−	0.572575i	\(-0.194055\pi\)
\(67\)	2.47165	−	0.216242i	0.301960	−	0.0264181i	0.0648313	−	0.997896i	\(-0.479349\pi\)
							0.237129	+	0.971478i	\(0.423794\pi\)
\(71\)	−10.4316	−	6.02267i	−1.23800	−	0.714759i	−0.269314	−	0.963052i	\(-0.586797\pi\)
							−0.968685	+	0.248294i	\(0.920130\pi\)
\(73\)	−0.375408	+	1.40104i	−0.0439381	+	0.163979i	−0.984409	−	0.175895i	\(-0.943718\pi\)
							0.940471	+	0.339875i	\(0.110385\pi\)
\(79\)	3.36157	+	4.00616i	0.378206	+	0.450729i	0.921247	−	0.388978i	\(-0.127172\pi\)
							−0.543041	+	0.839706i	\(0.682727\pi\)
\(83\)	−1.39592	+	15.9554i	−0.153222	+	1.75134i	0.398372	+	0.917224i	\(0.369575\pi\)
							−0.551594	+	0.834113i	\(0.685980\pi\)
\(89\)	2.99426	+	5.18621i	0.317391	+	0.549737i	0.979943	−	0.199279i	\(-0.0638600\pi\)
							−0.662552	+	0.749016i	\(0.730527\pi\)
\(97\)	7.62753	−	3.55678i	0.774459	−	0.361136i	0.00512815	−	0.999987i	\(-0.498368\pi\)
							0.769331	+	0.638851i	\(0.220590\pi\)

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	135.2.q.a.113.3	yes	192
3.2	odd	2		405.2.r.a.368.14		192
5.2	odd	4	inner	135.2.q.a.32.14	✓	192
5.3	odd	4		675.2.ba.b.32.3		192
5.4	even	2		675.2.ba.b.518.14		192
15.2	even	4		405.2.r.a.287.3		192
27.11	odd	18	inner	135.2.q.a.38.14	yes	192
27.16	even	9		405.2.r.a.278.3		192
135.38	even	36		675.2.ba.b.632.14		192
135.92	even	36	inner	135.2.q.a.92.3	yes	192
135.97	odd	36		405.2.r.a.197.14		192
135.119	odd	18		675.2.ba.b.443.3		192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.q.a.32.14	✓	192	5.2	odd	4	inner
135.2.q.a.38.14	yes	192	27.11	odd	18	inner
135.2.q.a.92.3	yes	192	135.92	even	36	inner
135.2.q.a.113.3	yes	192	1.1	even	1	trivial
405.2.r.a.197.14		192	135.97	odd	36
405.2.r.a.278.3		192	27.16	even	9
405.2.r.a.287.3		192	15.2	even	4
405.2.r.a.368.14		192	3.2	odd	2
675.2.ba.b.32.3		192	5.3	odd	4
675.2.ba.b.443.3		192	135.119	odd	18
675.2.ba.b.518.14		192	5.4	even	2
675.2.ba.b.632.14		192	135.38	even	36