Embedded newform 64.2.i.a.13.7

• Label: 64.2.i.a.13.7
• Level: $64$
• Weight: $2$
• Character: 64.13
• 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			13.7
Character	\(\chi\)	\(=\)	64.13
Dual form			64.2.i.a.5.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36881 + 0.355465i) q^{2} +(-2.00147 + 1.33734i) q^{3} +(1.74729 + 0.973128i) q^{4} +(0.756852 - 0.150547i) q^{5} +(-3.21501 + 1.11911i) q^{6} +(-1.69148 - 4.08359i) q^{7} +(2.04580 + 1.95313i) q^{8} +(1.06935 - 2.58163i) q^{9} +(1.08950 + 0.0629634i) q^{10} +(0.290161 - 0.434257i) q^{11} +(-4.79854 + 0.389032i) q^{12} +(-1.79553 - 0.357153i) q^{13} +(-0.863742 - 6.19092i) q^{14} +(-1.31348 + 1.31348i) q^{15} +(2.10604 + 3.40067i) q^{16} +(-3.04259 - 3.04259i) q^{17} +(2.38142 - 3.15366i) q^{18} +(-1.26776 + 6.37347i) q^{19} +(1.46894 + 0.473464i) q^{20} +(8.84657 + 5.91109i) q^{21} +(0.551539 - 0.491274i) q^{22} +(7.32197 + 3.03286i) q^{23} +(-6.70659 - 1.17320i) q^{24} +(-4.06924 + 1.68553i) q^{25} +(-2.33079 - 1.12712i) q^{26} +(-0.0965796 - 0.485538i) q^{27} +(1.01835 - 8.78123i) q^{28} +(-0.690042 - 1.03272i) q^{29} +(-2.26481 + 1.33101i) q^{30} -1.55847i q^{31} +(1.67396 + 5.40351i) q^{32} +1.25719i q^{33} +(-3.08320 - 5.24627i) q^{34} +(-1.89497 - 2.83602i) q^{35} +(4.38072 - 3.47025i) q^{36} +(0.371584 + 1.86808i) q^{37} +(-4.00087 + 8.27344i) q^{38} +(4.07133 - 1.68640i) q^{39} +(1.84240 + 1.17024i) q^{40} +(6.15380 + 2.54899i) q^{41} +(10.0081 + 11.2358i) q^{42} +(-7.03859 - 4.70304i) q^{43} +(0.929583 - 0.476409i) q^{44} +(0.420680 - 2.11490i) q^{45} +(8.94433 + 6.75412i) q^{46} +(1.12515 + 1.12515i) q^{47} +(-8.76302 - 3.98985i) q^{48} +(-8.86483 + 8.86483i) q^{49} +(-6.16917 + 0.860707i) q^{50} +(10.1586 + 2.02068i) q^{51} +(-2.78976 - 2.37133i) q^{52} +(3.92962 - 5.88109i) q^{53} +(0.0403925 - 0.698941i) q^{54} +(0.154233 - 0.372351i) q^{55} +(4.51535 - 11.6579i) q^{56} +(-5.98610 - 14.4517i) q^{57} +(-0.577441 - 1.65888i) q^{58} +(0.738882 - 0.146973i) q^{59} +(-3.57322 + 1.01685i) q^{60} +(-3.34952 + 2.23808i) q^{61} +(0.553981 - 2.13325i) q^{62} -12.3511 q^{63} +(0.370575 + 7.99141i) q^{64} -1.41272 q^{65} +(-0.446888 + 1.72086i) q^{66} +(3.05271 - 2.03976i) q^{67} +(-2.35546 - 8.27713i) q^{68} +(-18.7106 + 3.72178i) q^{69} +(-1.58575 - 4.55557i) q^{70} +(-0.317495 - 0.766500i) q^{71} +(7.22993 - 3.19293i) q^{72} +(0.292843 - 0.706986i) q^{73} +(-0.155407 + 2.68913i) q^{74} +(5.89032 - 8.81548i) q^{75} +(-8.41735 + 9.90261i) q^{76} +(-2.26413 - 0.450363i) q^{77} +(6.17234 - 0.861150i) q^{78} +(6.17863 - 6.17863i) q^{79} +(2.10592 + 2.25675i) q^{80} +(6.77032 + 6.77032i) q^{81} +(7.51732 + 5.67654i) q^{82} +(-0.663054 + 3.33340i) q^{83} +(9.70527 + 18.9372i) q^{84} +(-2.76085 - 1.84474i) q^{85} +(-7.96274 - 8.93954i) q^{86} +(2.76219 + 1.14414i) q^{87} +(1.44177 - 0.321679i) q^{88} +(12.3740 - 5.12547i) q^{89} +(1.32761 - 2.74537i) q^{90} +(1.57863 + 7.93632i) q^{91} +(9.84225 + 12.4245i) q^{92} +(2.08420 + 3.11923i) q^{93} +(1.14017 + 1.94007i) q^{94} +5.01463i q^{95} +(-10.5767 - 8.57629i) q^{96} +3.44120i q^{97} +(-15.2854 + 8.98315i) q^{98} +(-0.810809 - 1.21346i) 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 \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(63\)
\(\chi(n)\)	\(e\left(\frac{15}{16}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36881	+	0.355465i	0.967896	+	0.251351i
							\(3\)	−2.00147	+	1.33734i	−1.15555	+	0.772112i	−0.977297	−	0.211873i	\(-0.932044\pi\)
−0.178250	+	0.983985i	\(0.557044\pi\)
\(5\)	0.756852	−	0.150547i	0.338474	−	0.0673268i	−0.0229268	−	0.999737i	\(-0.507298\pi\)
							0.361401	+	0.932410i	\(0.382298\pi\)
\(7\)	−1.69148	−	4.08359i	−0.639318	−	1.54345i	−0.827590	−	0.561333i	\(-0.810289\pi\)
							0.188272	−	0.982117i	\(-0.439711\pi\)
\(11\)	0.290161	−	0.434257i	0.0874869	−	0.130933i	−0.785145	−	0.619312i	\(-0.787412\pi\)
							0.872632	+	0.488379i	\(0.162412\pi\)
\(13\)	−1.79553	−	0.357153i	−0.497991	−	0.0990565i	−0.0602967	−	0.998180i	\(-0.519205\pi\)
							−0.437694	+	0.899124i	\(0.644205\pi\)
\(17\)	−3.04259	−	3.04259i	−0.737937	−	0.737937i	0.234241	−	0.972179i	\(-0.424739\pi\)
							−0.972179	+	0.234241i	\(0.924739\pi\)
\(19\)	−1.26776	+	6.37347i	−0.290845	+	1.46217i	0.508375	+	0.861136i	\(0.330246\pi\)
							−0.799220	+	0.601039i	\(0.794754\pi\)
\(23\)	7.32197	+	3.03286i	1.52674	+	0.632395i	0.978928	−	0.204208i	\(-0.0654618\pi\)
							0.547810	+	0.836603i	\(0.315462\pi\)
\(29\)	−0.690042	−	1.03272i	−0.128138	−	0.191771i	0.761854	−	0.647749i	\(-0.224290\pi\)
							−0.889991	+	0.455978i	\(0.849290\pi\)
\(31\)		−	1.55847i		−	0.279910i	−0.990158	−	0.139955i	\(-0.955304\pi\)
							0.990158	−	0.139955i	\(-0.0446957\pi\)
\(37\)	0.371584	+	1.86808i	0.0610880	+	0.307110i	0.999234	−	0.0391295i	\(-0.0124585\pi\)
							−0.938146	+	0.346239i	\(0.887458\pi\)
\(41\)	6.15380	+	2.54899i	0.961062	+	0.398085i	0.807378	−	0.590035i	\(-0.200886\pi\)
							0.153685	+	0.988120i	\(0.450886\pi\)
\(43\)	−7.03859	−	4.70304i	−1.07338	−	0.717206i	−0.112350	−	0.993669i	\(-0.535838\pi\)
							−0.961025	+	0.276462i	\(0.910838\pi\)
\(47\)	1.12515	+	1.12515i	0.164120	+	0.164120i	0.784389	−	0.620269i	\(-0.212977\pi\)
							−0.620269	+	0.784389i	\(0.712977\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.2.i.a.13.7	yes	56
3.2	odd	2		576.2.bd.a.397.1		56
4.3	odd	2		256.2.i.a.145.6		56
8.3	odd	2		512.2.i.a.33.2		56
8.5	even	2		512.2.i.b.33.6		56
64.5	even	16	inner	64.2.i.a.5.7	✓	56
64.27	odd	16		512.2.i.a.481.2		56
64.37	even	16		512.2.i.b.481.6		56
64.59	odd	16		256.2.i.a.113.6		56
192.5	odd	16		576.2.bd.a.325.1		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.2.i.a.5.7	✓	56	64.5	even	16	inner
64.2.i.a.13.7	yes	56	1.1	even	1	trivial
256.2.i.a.113.6		56	64.59	odd	16
256.2.i.a.145.6		56	4.3	odd	2
512.2.i.a.33.2		56	8.3	odd	2
512.2.i.a.481.2		56	64.27	odd	16
512.2.i.b.33.6		56	8.5	even	2
512.2.i.b.481.6		56	64.37	even	16
576.2.bd.a.325.1		56	192.5	odd	16
576.2.bd.a.397.1		56	3.2	odd	2