Embedded newform 61.2.i.a.15.1

• Label: 61.2.i.a.15.1
• Level: $61$
• Weight: $2$
• Character: 61.15
• Analytic conductor: $0.487$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	61.i (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.487087452330\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			15.1
Character	\(\chi\)	\(=\)	61.15
Dual form			61.2.i.a.57.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.261670 - 2.48962i) q^{2} +(1.09846 - 0.798080i) q^{3} +(-4.17345 + 0.887094i) q^{4} +(2.01896 + 2.24228i) q^{5} +(-2.27435 - 2.52592i) q^{6} +(-2.42543 + 1.07987i) q^{7} +(1.75345 + 5.39656i) q^{8} +(-0.357362 + 1.09985i) q^{9} +(5.05413 - 5.61318i) q^{10} -2.81637 q^{11} +(-3.87641 + 4.30519i) q^{12} +(3.25426 - 5.63654i) q^{13} +(3.32313 + 5.75582i) q^{14} +(4.00727 + 0.851773i) q^{15} +(5.18094 - 2.30670i) q^{16} +(0.988198 - 0.210048i) q^{17} +(2.83171 + 0.601899i) q^{18} +(-2.84524 - 1.26678i) q^{19} +(-10.4151 - 7.56704i) q^{20} +(-1.80242 + 3.12188i) q^{21} +(0.736958 + 7.01169i) q^{22} +(-0.957874 + 2.94803i) q^{23} +(6.23298 + 4.52853i) q^{24} +(-0.428988 + 4.08155i) q^{25} +(-14.8844 - 6.62696i) q^{26} +(1.74394 + 5.36731i) q^{27} +(9.16444 - 6.65836i) q^{28} +(-1.25301 - 2.17028i) q^{29} +(1.07201 - 10.1995i) q^{30} +(0.444151 - 4.22582i) q^{31} +(-1.42423 - 2.46684i) q^{32} +(-3.09368 + 2.24769i) q^{33} +(-0.781521 - 2.40527i) q^{34} +(-7.31821 - 3.25828i) q^{35} +(0.515764 - 4.90717i) q^{36} +(-5.69250 - 4.13585i) q^{37} +(-2.40930 + 7.41505i) q^{38} +(-0.923728 - 8.78869i) q^{39} +(-8.56046 + 14.8272i) q^{40} +(8.51930 + 6.18964i) q^{41} +(8.24394 + 3.67044i) q^{42} +(6.04008 + 1.28386i) q^{43} +(11.7540 - 2.49838i) q^{44} +(-3.18766 + 1.41924i) q^{45} +(7.59013 + 1.61333i) q^{46} +(0.0929380 + 0.160973i) q^{47} +(3.85013 - 6.66863i) q^{48} +(0.0326575 - 0.0362698i) q^{49} +10.2738 q^{50} +(0.917864 - 1.01939i) q^{51} +(-8.58134 + 26.4106i) q^{52} +(1.96750 + 6.05535i) q^{53} +(12.9062 - 5.74622i) q^{54} +(-5.68613 - 6.31509i) q^{55} +(-10.0804 - 11.1955i) q^{56} +(-4.13639 + 0.879216i) q^{57} +(-5.07529 + 3.68741i) q^{58} +(-0.425310 - 4.04655i) q^{59} -17.4798 q^{60} +(-1.85551 - 7.58664i) q^{61} -10.6369 q^{62} +(-0.320936 - 3.05350i) q^{63} +(3.40743 - 2.47565i) q^{64} +(19.2089 - 4.08298i) q^{65} +(6.40541 + 7.11393i) q^{66} +(-0.557122 - 0.618747i) q^{67} +(-3.93786 + 1.75325i) q^{68} +(1.30058 + 4.00276i) q^{69} +(-6.19692 + 19.0722i) q^{70} +(4.21558 - 4.68187i) q^{71} -6.56200 q^{72} +(-0.628481 + 0.697999i) q^{73} +(-8.80714 + 15.2544i) q^{74} +(2.78618 + 4.82580i) q^{75} +(12.9982 + 2.76286i) q^{76} +(6.83089 - 3.04131i) q^{77} +(-21.6388 + 4.59947i) q^{78} +(-11.6299 - 2.47200i) q^{79} +(15.6324 + 6.95998i) q^{80} +(3.39244 + 2.46475i) q^{81} +(13.1806 - 22.8295i) q^{82} +(-0.517961 - 4.92807i) q^{83} +(4.75290 - 14.6279i) q^{84} +(2.46612 + 1.79174i) q^{85} +(1.61581 - 15.3735i) q^{86} +(-3.10844 - 1.38397i) q^{87} +(-4.93835 - 15.1987i) q^{88} +(-7.02573 + 5.10449i) q^{89} +(4.36749 + 7.56471i) q^{90} +(-1.80624 + 17.1852i) q^{91} +(1.38246 - 13.1532i) q^{92} +(-2.88466 - 4.99637i) q^{93} +(0.376444 - 0.273502i) q^{94} +(-2.90394 - 8.93742i) q^{95} +(-3.53320 - 1.57308i) q^{96} +(-0.337747 + 3.21345i) q^{97} +(-0.0988435 - 0.0718140i) q^{98} +(1.00646 - 3.09757i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	32 * q - 10 * q^2 - 4 * q^3 - 10 * q^4 + 2 * q^5 + q^6 + q^7 - 4 * q^8 - 2 * q^9 + 15 * q^10 - 18 * q^11 - 53 * q^12 + 11 * q^14 + 2 * q^15 + 6 * q^16 - 24 * q^17 - 15 * q^18 + 9 * q^19 - 4 * q^20 - 3 * q^21 + q^22 - 2 * q^23 + 15 * q^24 + 28 * q^25 + 16 * q^26 + 35 * q^27 + 4 * q^28 - 4 * q^29 + 54 * q^30 - 11 * q^31 + 34 * q^32 - 35 * q^33 + 18 * q^34 - 58 * q^35 + 65 * q^36 - 14 * q^37 - 24 * q^38 + 17 * q^39 - 60 * q^40 + 11 * q^41 + 73 * q^42 + 40 * q^43 + 29 * q^44 + 12 * q^45 - 89 * q^46 + 40 * q^47 + 43 * q^48 + q^49 - 56 * q^50 - 9 * q^51 - 67 * q^52 + 17 * q^53 - q^54 - 60 * q^55 - 102 * q^56 - 38 * q^57 + 73 * q^58 - 11 * q^59 - 20 * q^60 - 55 * q^61 - 74 * q^62 - 58 * q^63 + 6 * q^64 + 59 * q^65 + 34 * q^66 - 13 * q^67 - 3 * q^68 - 32 * q^69 + 44 * q^70 + 63 * q^71 + 18 * q^72 - 46 * q^73 - 10 * q^74 + q^75 + 55 * q^76 - 31 * q^77 - 103 * q^78 - 49 * q^79 + 74 * q^80 + 48 * q^81 + 39 * q^82 + 39 * q^83 + 21 * q^85 + 74 * q^86 + 17 * q^87 + 70 * q^88 + 32 * q^89 - 60 * q^90 + 70 * q^91 + 77 * q^92 + 67 * q^93 - 64 * q^94 + 47 * q^95 - 16 * q^96 + 37 * q^97 + 127 * q^98 - 31 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/61\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{15}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.261670	−	2.48962i	−0.185028	−	1.76043i	−0.555436	−	0.831559i	\(-0.687449\pi\)
							0.370408	−	0.928869i	\(-0.379218\pi\)
\(3\)	1.09846	−	0.798080i	0.634198	−	0.460772i	−0.223654	−	0.974669i	\(-0.571799\pi\)
							0.857852	+	0.513897i	\(0.171799\pi\)
\(5\)	2.01896	+	2.24228i	0.902907	+	1.00278i	0.999972	+	0.00750126i	\(0.00238775\pi\)
							−0.0970654	+	0.995278i	\(0.530946\pi\)
\(7\)	−2.42543	+	1.07987i	−0.916725	+	0.408152i	−0.810197	−	0.586158i	\(-0.800640\pi\)
							−0.106528	+	0.994310i	\(0.533973\pi\)
\(11\)	−2.81637			−0.849167			−0.424583	−	0.905389i	\(-0.639579\pi\)
							−0.424583	+	0.905389i	\(0.639579\pi\)
\(13\)	3.25426	−	5.63654i	0.902569	−	1.56329i	0.0784206	−	0.996920i	\(-0.475012\pi\)
							0.824148	−	0.566374i	\(-0.191654\pi\)
\(17\)	0.988198	−	0.210048i	0.239673	−	0.0509441i	−0.0865089	−	0.996251i	\(-0.527571\pi\)
							0.326182	+	0.945307i	\(0.394238\pi\)
\(19\)	−2.84524	−	1.26678i	−0.652743	−	0.290620i	0.0535227	−	0.998567i	\(-0.482955\pi\)
							−0.706266	+	0.707947i	\(0.749622\pi\)
\(23\)	−0.957874	+	2.94803i	−0.199730	+	0.614707i	0.800158	+	0.599789i	\(0.204749\pi\)
							−0.999889	+	0.0149182i	\(0.995251\pi\)
\(29\)	−1.25301	−	2.17028i	−0.232678	−	0.403010i	0.725917	−	0.687782i	\(-0.241415\pi\)
							−0.958595	+	0.284772i	\(0.908082\pi\)
\(31\)	0.444151	−	4.22582i	0.0797720	−	0.758979i	−0.879386	−	0.476109i	\(-0.842047\pi\)
							0.959158	−	0.282870i	\(-0.0912866\pi\)
\(37\)	−5.69250	−	4.13585i	−0.935842	−	0.679929i	0.0115743	−	0.999933i	\(-0.496316\pi\)
							−0.947416	+	0.320004i	\(0.896316\pi\)
\(41\)	8.51930	+	6.18964i	1.33049	+	0.966659i	0.999737	+	0.0229423i	\(0.00730339\pi\)
							0.330755	+	0.943717i	\(0.392697\pi\)
\(43\)	6.04008	+	1.28386i	0.921103	+	0.195787i	0.643975	−	0.765046i	\(-0.277284\pi\)
							0.277128	+	0.960833i	\(0.410617\pi\)
\(47\)	0.0929380	+	0.160973i	0.0135564	+	0.0234804i	0.872724	−	0.488214i	\(-0.162351\pi\)
							−0.859168	+	0.511694i	\(0.829018\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.2.i.a.15.1	✓	32
3.2	odd	2		549.2.bl.b.442.4		32
4.3	odd	2		976.2.bw.c.625.1		32
61.22	even	15		3721.2.a.j.1.16		16
61.39	even	30		3721.2.a.l.1.1		16
61.57	even	15	inner	61.2.i.a.57.1	yes	32
183.179	odd	30		549.2.bl.b.118.4		32
244.179	odd	30		976.2.bw.c.545.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.2.i.a.15.1	✓	32	1.1	even	1	trivial
61.2.i.a.57.1	yes	32	61.57	even	15	inner
549.2.bl.b.118.4		32	183.179	odd	30
549.2.bl.b.442.4		32	3.2	odd	2
976.2.bw.c.545.1		32	244.179	odd	30
976.2.bw.c.625.1		32	4.3	odd	2
3721.2.a.j.1.16		16	61.22	even	15
3721.2.a.l.1.1		16	61.39	even	30