Embedded newform 64.2.i.a.37.3

• Label: 64.2.i.a.37.3
• Level: $64$
• Weight: $2$
• Character: 64.37
• Analytic conductor: $0.511$
• Analytic rank: $0$
• Dimension: $56$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	64.i (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.511042572936\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(7\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			37.3
Character	\(\chi\)	\(=\)	64.37
Dual form			64.2.i.a.45.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.809383 - 1.15970i) q^{2} +(-1.77714 + 2.65968i) q^{3} +(-0.689800 + 1.87728i) q^{4} +(-0.159018 + 0.799435i) q^{5} +(4.52282 - 0.0917514i) q^{6} +(-0.742008 + 1.79137i) q^{7} +(2.73539 - 0.719478i) q^{8} +(-2.76763 - 6.68165i) q^{9} +(1.05581 - 0.462637i) q^{10} +(1.29226 - 0.863458i) q^{11} +(-3.76710 - 5.17084i) q^{12} +(1.07211 + 5.38985i) q^{13} +(2.67801 - 0.589395i) q^{14} +(-1.84365 - 1.84365i) q^{15} +(-3.04835 - 2.58989i) q^{16} +(1.43298 - 1.43298i) q^{17} +(-5.50862 + 8.61762i) q^{18} +(0.0883441 - 0.0175727i) q^{19} +(-1.39107 - 0.849970i) q^{20} +(-3.44581 - 5.15702i) q^{21} +(-2.04728 - 0.799759i) q^{22} +(3.31519 - 1.37320i) q^{23} +(-2.94760 + 8.55389i) q^{24} +(4.00559 + 1.65917i) q^{25} +(5.38285 - 5.60577i) q^{26} +(13.2776 + 2.64108i) q^{27} +(-2.85106 - 2.62864i) q^{28} +(-1.04042 - 0.695186i) q^{29} +(-0.645858 + 3.63029i) q^{30} +2.58743i q^{31} +(-0.536209 + 5.63138i) q^{32} +4.97148i q^{33} +(-2.82165 - 0.501994i) q^{34} +(-1.31409 - 0.878046i) q^{35} +(14.4524 - 0.586615i) q^{36} +(1.60572 + 0.319397i) q^{37} +(-0.0918832 - 0.0882294i) q^{38} +(-16.2406 - 6.72707i) q^{39} +(0.140201 + 2.30118i) q^{40} +(0.605183 - 0.250675i) q^{41} +(-3.19161 + 8.17011i) q^{42} +(-5.04263 - 7.54683i) q^{43} +(0.729554 + 3.02154i) q^{44} +(5.78164 - 1.15004i) q^{45} +(-4.27576 - 2.73318i) q^{46} +(-3.86580 + 3.86580i) q^{47} +(12.3057 - 3.50504i) q^{48} +(2.29133 + 2.29133i) q^{49} +(-1.31792 - 5.98817i) q^{50} +(1.26466 + 6.35787i) q^{51} +(-10.8578 - 1.70527i) q^{52} +(-8.70867 + 5.81895i) q^{53} +(-7.68381 - 17.5357i) q^{54} +(0.484787 + 1.17038i) q^{55} +(-0.740832 + 5.43394i) q^{56} +(-0.110262 + 0.266197i) q^{57} +(0.0358915 + 1.76924i) q^{58} +(1.17041 - 5.88407i) q^{59} +(4.73279 - 2.18929i) q^{60} +(3.52821 - 5.28034i) q^{61} +(3.00064 - 2.09422i) q^{62} +14.0229 q^{63} +(6.96470 - 3.93610i) q^{64} -4.47932 q^{65} +(5.76542 - 4.02383i) q^{66} +(3.15404 - 4.72036i) q^{67} +(1.70163 + 3.67856i) q^{68} +(-2.23930 + 11.2577i) q^{69} +(0.0453323 + 2.23462i) q^{70} +(5.01419 - 12.1053i) q^{71} +(-12.3778 - 16.2857i) q^{72} +(-1.75399 - 4.23450i) q^{73} +(-0.929237 - 2.12066i) q^{74} +(-11.5314 + 7.70502i) q^{75} +(-0.0279508 + 0.177968i) q^{76} +(0.587905 + 2.95560i) q^{77} +(5.34347 + 24.2789i) q^{78} +(7.46021 + 7.46021i) q^{79} +(2.55519 - 2.02512i) q^{80} +(-15.2789 + 15.2789i) q^{81} +(-0.780532 - 0.498937i) q^{82} +(-2.65306 + 0.527725i) q^{83} +(12.0581 - 2.91144i) q^{84} +(0.917704 + 1.37344i) q^{85} +(-4.67063 + 11.9562i) q^{86} +(3.69795 - 1.53174i) q^{87} +(2.91358 - 3.29164i) q^{88} +(7.09322 + 2.93811i) q^{89} +(-6.01326 - 5.77414i) q^{90} +(-10.4507 - 2.07877i) q^{91} +(0.291057 + 7.17078i) q^{92} +(-6.88175 - 4.59824i) q^{93} +(7.61208 + 1.35425i) q^{94} +0.0734197i q^{95} +(-14.0248 - 11.4339i) q^{96} -11.4524i q^{97} +(0.802689 - 4.51182i) q^{98} +(-9.34580 - 6.24466i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	56 * q - 8 * q^2 - 8 * q^3 - 8 * q^4 - 8 * q^5 - 8 * q^6 - 8 * q^7 - 8 * q^8 - 8 * q^9 - 8 * q^10 - 8 * q^11 - 8 * q^12 - 8 * q^13 - 8 * q^14 - 8 * q^15 - 8 * q^16 - 8 * q^17 - 8 * q^18 - 8 * q^19 - 8 * q^20 - 8 * q^21 - 8 * q^23 + 32 * q^24 - 8 * q^25 + 32 * q^26 - 8 * q^27 + 32 * q^28 - 8 * q^29 + 72 * q^30 + 32 * q^32 + 32 * q^34 - 8 * q^35 + 72 * q^36 - 8 * q^37 + 32 * q^38 - 8 * q^39 + 32 * q^40 - 8 * q^41 + 32 * q^42 - 8 * q^43 - 8 * q^45 - 8 * q^46 - 8 * q^47 - 8 * q^48 - 8 * q^49 - 32 * q^50 + 24 * q^51 - 56 * q^52 - 8 * q^53 - 72 * q^54 + 56 * q^55 - 64 * q^56 - 8 * q^57 - 80 * q^58 + 56 * q^59 - 104 * q^60 - 8 * q^61 - 40 * q^62 + 64 * q^63 - 104 * q^64 - 16 * q^65 - 88 * q^66 + 72 * q^67 - 56 * q^68 - 8 * q^69 - 104 * q^70 + 56 * q^71 - 80 * q^72 - 8 * q^73 - 64 * q^74 + 56 * q^75 - 72 * q^76 - 8 * q^77 - 32 * q^78 + 24 * q^79 + 32 * q^80 - 8 * q^81 + 72 * q^82 - 8 * q^83 + 104 * q^84 - 8 * q^85 + 96 * q^86 - 8 * q^87 + 72 * q^88 - 8 * q^89 + 136 * q^90 - 8 * q^91 + 144 * q^92 + 16 * q^93 + 88 * q^94 + 128 * q^96 + 128 * q^98 + 16 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(63\)
\(\chi(n)\)	\(e\left(\frac{9}{16}\right)\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.809383	−	1.15970i	−0.572320	−	0.820030i
							\(3\)	−1.77714	+	2.65968i	−1.02603	+	1.53557i	−0.193880	+	0.981025i	\(0.562107\pi\)
−0.832154	+	0.554544i	\(0.812893\pi\)
\(5\)	−0.159018	+	0.799435i	−0.0711148	+	0.357518i	−0.999915	−	0.0130634i	\(-0.995842\pi\)
							0.928800	+	0.370582i	\(0.120842\pi\)
\(7\)	−0.742008	+	1.79137i	−0.280453	+	0.677073i	−0.999846	−	0.0175299i	\(-0.994420\pi\)
							0.719394	+	0.694603i	\(0.244420\pi\)
\(11\)	1.29226	−	0.863458i	0.389630	−	0.260342i	−0.345290	−	0.938496i	\(-0.612220\pi\)
							0.734920	+	0.678154i	\(0.237220\pi\)
\(13\)	1.07211	+	5.38985i	0.297349	+	1.49487i	0.783716	+	0.621119i	\(0.213322\pi\)
							−0.486367	+	0.873755i	\(0.661678\pi\)
\(17\)	1.43298	−	1.43298i	0.347548	−	0.347548i	−0.511648	−	0.859195i	\(-0.670965\pi\)
							0.859195	+	0.511648i	\(0.170965\pi\)
\(19\)	0.0883441	−	0.0175727i	0.0202675	−	0.00403146i	−0.184946	−	0.982749i	\(-0.559211\pi\)
							0.205214	+	0.978717i	\(0.434211\pi\)
\(23\)	3.31519	−	1.37320i	0.691266	−	0.286332i	−0.00926165	−	0.999957i	\(-0.502948\pi\)
							0.700527	+	0.713625i	\(0.252948\pi\)
\(29\)	−1.04042	−	0.695186i	−0.193201	−	0.129093i	0.455207	−	0.890385i	\(-0.349565\pi\)
							−0.648409	+	0.761293i	\(0.724565\pi\)
\(31\)			2.58743i			0.464717i	0.972630	+	0.232358i	\(0.0746442\pi\)
							−0.972630	+	0.232358i	\(0.925356\pi\)
\(37\)	1.60572	+	0.319397i	0.263979	+	0.0525086i	0.325305	−	0.945609i	\(-0.394533\pi\)
							−0.0613263	+	0.998118i	\(0.519533\pi\)
\(41\)	0.605183	−	0.250675i	0.0945137	−	0.0391488i	−0.334926	−	0.942245i	\(-0.608711\pi\)
							0.429439	+	0.903096i	\(0.358711\pi\)
\(43\)	−5.04263	−	7.54683i	−0.768994	−	1.15088i	−0.984672	−	0.174416i	\(-0.944196\pi\)
							0.215678	−	0.976465i	\(-0.430804\pi\)
\(47\)	−3.86580	+	3.86580i	−0.563885	+	0.563885i	−0.930409	−	0.366524i	\(-0.880548\pi\)
							0.366524	+	0.930409i	\(0.380548\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.2.i.a.37.3	✓	56
3.2	odd	2		576.2.bd.a.37.5		56
4.3	odd	2		256.2.i.a.241.7		56
8.3	odd	2		512.2.i.a.225.1		56
8.5	even	2		512.2.i.b.225.7		56
64.13	even	16		512.2.i.b.289.7		56
64.19	odd	16		256.2.i.a.17.7		56
64.45	even	16	inner	64.2.i.a.45.3	yes	56
64.51	odd	16		512.2.i.a.289.1		56
192.173	odd	16		576.2.bd.a.109.5		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.2.i.a.37.3	✓	56	1.1	even	1	trivial
64.2.i.a.45.3	yes	56	64.45	even	16	inner
256.2.i.a.17.7		56	64.19	odd	16
256.2.i.a.241.7		56	4.3	odd	2
512.2.i.a.225.1		56	8.3	odd	2
512.2.i.a.289.1		56	64.51	odd	16
512.2.i.b.225.7		56	8.5	even	2
512.2.i.b.289.7		56	64.13	even	16
576.2.bd.a.37.5		56	3.2	odd	2
576.2.bd.a.109.5		56	192.173	odd	16