Embedded newform 90.2.l.a.47.2

• Label: 90.2.l.a.47.2
• Level: $90$
• Weight: $2$
• Character: 90.47
• Analytic conductor: $0.719$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	90.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.718653618192\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			47.2
Root			\(-0.965926 + 0.258819i\) of defining polynomial
Character	\(\chi\)	\(=\)	90.47
Dual form			90.2.l.a.23.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.258819 + 0.965926i) q^{2} +(1.10721 + 1.33195i) q^{3} +(-0.866025 + 0.500000i) q^{4} +(0.792893 - 2.09077i) q^{5} +(-1.00000 + 1.41421i) q^{6} +(-1.05902 + 0.283763i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(-0.548188 + 2.94949i) q^{9} +(2.22474 + 0.224745i) q^{10} +(-5.44949 - 3.14626i) q^{11} +(-1.62484 - 0.599900i) q^{12} +(3.34607 + 0.896575i) q^{13} +(-0.548188 - 0.949490i) q^{14} +(3.66270 - 1.25882i) q^{15} +(0.500000 - 0.866025i) q^{16} +(3.14626 - 3.14626i) q^{17} +(-2.99087 + 0.233875i) q^{18} +1.55051i q^{19} +(0.358719 + 2.20711i) q^{20} +(-1.55051 - 1.09638i) q^{21} +(1.62863 - 6.07812i) q^{22} +(0.258819 - 0.965926i) q^{23} +(0.158919 - 1.72474i) q^{24} +(-3.74264 - 3.31552i) q^{25} +3.46410i q^{26} +(-4.53553 + 2.53553i) q^{27} +(0.775255 - 0.775255i) q^{28} +(-1.57313 + 2.72474i) q^{29} +(2.16390 + 3.21209i) q^{30} +(2.22474 + 3.85337i) q^{31} +(0.965926 + 0.258819i) q^{32} +(-1.84304 - 10.7420i) q^{33} +(3.85337 + 2.22474i) q^{34} +(-0.246405 + 2.43916i) q^{35} +(-1.00000 - 2.82843i) q^{36} +(-3.00000 - 3.00000i) q^{37} +(-1.49768 + 0.401302i) q^{38} +(2.51059 + 5.44949i) q^{39} +(-2.03906 + 0.917738i) q^{40} +(-3.39898 + 1.96240i) q^{41} +(0.657717 - 1.78144i) q^{42} +(0.896575 + 3.34607i) q^{43} +6.29253 q^{44} +(5.73205 + 3.48477i) q^{45} +1.00000 q^{46} +(2.32937 + 8.69333i) q^{47} +(1.70711 - 0.292893i) q^{48} +(-5.02118 + 2.89898i) q^{49} +(2.23388 - 4.47323i) q^{50} +(7.67423 + 0.707107i) q^{51} +(-3.34607 + 0.896575i) q^{52} +(6.61037 + 6.61037i) q^{53} +(-3.62302 - 3.72474i) q^{54} +(-10.8990 + 8.89898i) q^{55} +(0.949490 + 0.548188i) q^{56} +(-2.06520 + 1.71673i) q^{57} +(-3.03906 - 0.814313i) q^{58} +(-5.90326 - 10.2247i) q^{59} +(-2.54258 + 2.92152i) q^{60} +(2.72474 - 4.71940i) q^{61} +(-3.14626 + 3.14626i) q^{62} +(-0.256415 - 3.27912i) q^{63} +1.00000i q^{64} +(4.52761 - 6.28497i) q^{65} +(9.89898 - 4.56048i) q^{66} +(0.978838 - 3.65307i) q^{67} +(-1.15161 + 4.29788i) q^{68} +(1.57313 - 0.724745i) q^{69} +(-2.41982 + 0.393292i) q^{70} +0.635674i q^{71} +(2.47323 - 1.69798i) q^{72} +(2.89898 - 2.89898i) q^{73} +(2.12132 - 3.67423i) q^{74} +(0.272229 - 8.65597i) q^{75} +(-0.775255 - 1.34278i) q^{76} +(6.66390 + 1.78559i) q^{77} +(-4.61401 + 3.83548i) q^{78} +(2.12132 + 1.22474i) q^{79} +(-1.41421 - 1.73205i) q^{80} +(-8.39898 - 3.23375i) q^{81} +(-2.77526 - 2.77526i) q^{82} +(-0.531752 + 0.142483i) q^{83} +(1.89097 + 0.174235i) q^{84} +(-4.08346 - 9.07277i) q^{85} +(-3.00000 + 1.73205i) q^{86} +(-5.37101 + 0.921519i) q^{87} +(1.62863 + 6.07812i) q^{88} -2.36773 q^{89} +(-1.88246 + 6.43866i) q^{90} -3.79796 q^{91} +(0.258819 + 0.965926i) q^{92} +(-2.66925 + 7.22973i) q^{93} +(-7.79423 + 4.50000i) q^{94} +(3.24176 + 1.22939i) q^{95} +(0.724745 + 1.57313i) q^{96} +(10.7902 - 2.89123i) q^{97} +(-4.09978 - 4.09978i) q^{98} +(12.2672 - 14.3485i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	8 * q + 4 * q^3 + 12 * q^5 - 8 * q^6 - 8 * q^7 + 8 * q^10 - 24 * q^11 - 4 * q^12 + 16 * q^15 + 4 * q^16 - 8 * q^18 - 32 * q^21 - 8 * q^22 + 4 * q^25 - 8 * q^27 + 16 * q^28 - 12 * q^30 + 8 * q^31 + 16 * q^33 - 8 * q^36 - 24 * q^37 + 12 * q^38 + 4 * q^40 + 12 * q^41 + 20 * q^42 + 32 * q^45 + 8 * q^46 + 8 * q^48 + 24 * q^50 + 32 * q^51 - 48 * q^55 - 12 * q^56 + 28 * q^57 - 4 * q^58 - 8 * q^60 + 12 * q^61 - 32 * q^63 - 24 * q^65 + 40 * q^66 + 4 * q^67 - 12 * q^68 - 16 * q^70 + 8 * q^72 - 16 * q^73 + 8 * q^75 - 16 * q^76 + 24 * q^77 - 24 * q^78 - 28 * q^81 - 32 * q^82 + 12 * q^83 - 20 * q^85 - 24 * q^86 - 8 * q^87 - 8 * q^88 - 20 * q^90 + 48 * q^91 - 20 * q^93 - 24 * q^95 - 4 * q^96 + 12 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{4}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.258819	+	0.965926i	0.183013	+	0.683013i
							\(3\)	1.10721	+	1.33195i	0.639246	+	0.769002i
\(5\)	0.792893	−	2.09077i	0.354593	−	0.935021i
							\(7\)	−1.05902	+	0.283763i	−0.400271	+	0.107252i	−0.453338	−	0.891339i	\(-0.649767\pi\)
0.0530669	+	0.998591i	\(0.483100\pi\)
\(11\)	−5.44949	−	3.14626i	−1.64308	−	0.948634i	−0.979729	−	0.200329i	\(-0.935799\pi\)
							−0.663354	−	0.748305i	\(-0.730868\pi\)
\(13\)	3.34607	+	0.896575i	0.928032	+	0.248665i	0.691015	−	0.722840i	\(-0.257164\pi\)
							0.237016	+	0.971506i	\(0.423830\pi\)
\(17\)	3.14626	−	3.14626i	0.763081	−	0.763081i	−0.213797	−	0.976878i	\(-0.568583\pi\)
							0.976878	+	0.213797i	\(0.0685831\pi\)
\(19\)			1.55051i			0.355711i	0.984057	+	0.177856i	\(0.0569160\pi\)
							−0.984057	+	0.177856i	\(0.943084\pi\)
\(23\)	0.258819	−	0.965926i	0.0539675	−	0.201409i	−0.933678	−	0.358113i	\(-0.883420\pi\)
							0.987646	+	0.156704i	\(0.0500868\pi\)
\(29\)	−1.57313	+	2.72474i	−0.292123	+	0.505972i	−0.974312	−	0.225204i	\(-0.927695\pi\)
							0.682188	+	0.731177i	\(0.261028\pi\)
\(31\)	2.22474	+	3.85337i	0.399576	+	0.692086i	0.993674	−	0.112307i	\(-0.0358240\pi\)
							−0.594098	+	0.804393i	\(0.702491\pi\)
\(37\)	−3.00000	−	3.00000i	−0.493197	−	0.493197i	0.416115	−	0.909312i	\(-0.363391\pi\)
							−0.909312	+	0.416115i	\(0.863391\pi\)
\(41\)	−3.39898	+	1.96240i	−0.530831	+	0.306476i	−0.741355	−	0.671113i	\(-0.765816\pi\)
							0.210524	+	0.977589i	\(0.432483\pi\)
\(43\)	0.896575	+	3.34607i	0.136726	+	0.510270i	0.999985	+	0.00550783i	\(0.00175320\pi\)
							−0.863258	+	0.504762i	\(0.831580\pi\)
\(47\)	2.32937	+	8.69333i	0.339774	+	1.26805i	0.898600	+	0.438768i	\(0.144585\pi\)
							−0.558827	+	0.829285i	\(0.688748\pi\)
\(53\)	6.61037	+	6.61037i	0.908004	+	0.908004i	0.996111	−	0.0881074i	\(-0.0280819\pi\)
							−0.0881074	+	0.996111i	\(0.528082\pi\)
\(59\)	−5.90326	−	10.2247i	−0.768539	−	1.33115i	−0.938355	−	0.345673i	\(-0.887651\pi\)
							0.169816	−	0.985476i	\(-0.445683\pi\)
\(61\)	2.72474	−	4.71940i	0.348868	−	0.604257i	−0.637181	−	0.770714i	\(-0.719900\pi\)
							0.986049	+	0.166458i	\(0.0532329\pi\)
\(67\)	0.978838	−	3.65307i	0.119584	−	0.446294i	−0.880005	−	0.474965i	\(-0.842461\pi\)
							0.999589	+	0.0286709i	\(0.00912748\pi\)
\(71\)			0.635674i			0.0754407i	0.999288	+	0.0377203i	\(0.0120096\pi\)
							−0.999288	+	0.0377203i	\(0.987990\pi\)
\(73\)	2.89898	−	2.89898i	0.339300	−	0.339300i	−0.516804	−	0.856104i	\(-0.672878\pi\)
							0.856104	+	0.516804i	\(0.172878\pi\)
\(79\)	2.12132	+	1.22474i	0.238667	+	0.137795i	0.614564	−	0.788867i	\(-0.289332\pi\)
							−0.375897	+	0.926662i	\(0.622665\pi\)
\(83\)	−0.531752	+	0.142483i	−0.0583674	+	0.0156395i	−0.287885	−	0.957665i	\(-0.592952\pi\)
							0.229517	+	0.973305i	\(0.426285\pi\)
\(89\)	−2.36773			−0.250978			−0.125489	−	0.992095i	\(-0.540050\pi\)
							−0.125489	+	0.992095i	\(0.540050\pi\)
\(97\)	10.7902	−	2.89123i	1.09558	−	0.293560i	0.334616	−	0.942355i	\(-0.391393\pi\)
							0.760963	+	0.648795i	\(0.224727\pi\)

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	90.2.l.a.47.2	yes	8
3.2	odd	2		270.2.m.a.197.1		8
4.3	odd	2		720.2.cu.a.497.2		8
5.2	odd	4		450.2.p.a.443.1		8
5.3	odd	4	inner	90.2.l.a.83.2	yes	8
5.4	even	2		450.2.p.a.407.1		8
9.2	odd	6		810.2.f.b.647.1		8
9.4	even	3		270.2.m.a.17.1		8
9.5	odd	6	inner	90.2.l.a.77.2	yes	8
9.7	even	3		810.2.f.b.647.4		8
15.2	even	4		1350.2.q.g.143.2		8
15.8	even	4		270.2.m.a.143.1		8
15.14	odd	2		1350.2.q.g.1007.2		8
20.3	even	4		720.2.cu.a.353.2		8
36.23	even	6		720.2.cu.a.257.2		8
45.4	even	6		1350.2.q.g.557.2		8
45.13	odd	12		270.2.m.a.233.1		8
45.14	odd	6		450.2.p.a.257.1		8
45.22	odd	12		1350.2.q.g.1043.2		8
45.23	even	12	inner	90.2.l.a.23.2	✓	8
45.32	even	12		450.2.p.a.293.1		8
45.38	even	12		810.2.f.b.323.3		8
45.43	odd	12		810.2.f.b.323.2		8
180.23	odd	12		720.2.cu.a.113.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.l.a.23.2	✓	8	45.23	even	12	inner
90.2.l.a.47.2	yes	8	1.1	even	1	trivial
90.2.l.a.77.2	yes	8	9.5	odd	6	inner
90.2.l.a.83.2	yes	8	5.3	odd	4	inner
270.2.m.a.17.1		8	9.4	even	3
270.2.m.a.143.1		8	15.8	even	4
270.2.m.a.197.1		8	3.2	odd	2
270.2.m.a.233.1		8	45.13	odd	12
450.2.p.a.257.1		8	45.14	odd	6
450.2.p.a.293.1		8	45.32	even	12
450.2.p.a.407.1		8	5.4	even	2
450.2.p.a.443.1		8	5.2	odd	4
720.2.cu.a.113.2		8	180.23	odd	12
720.2.cu.a.257.2		8	36.23	even	6
720.2.cu.a.353.2		8	20.3	even	4
720.2.cu.a.497.2		8	4.3	odd	2
810.2.f.b.323.2		8	45.43	odd	12
810.2.f.b.323.3		8	45.38	even	12
810.2.f.b.647.1		8	9.2	odd	6
810.2.f.b.647.4		8	9.7	even	3
1350.2.q.g.143.2		8	15.2	even	4
1350.2.q.g.557.2		8	45.4	even	6
1350.2.q.g.1007.2		8	15.14	odd	2
1350.2.q.g.1043.2		8	45.22	odd	12