Embedded newform 61.2.i.a.15.4

• Label: 61.2.i.a.15.4
• 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			15.4
Character	\(\chi\)	\(=\)	61.15
Dual form			61.2.i.a.57.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.283167 + 2.69415i) q^{2} +(0.768086 - 0.558047i) q^{3} +(-5.22198 + 1.10997i) q^{4} +(-0.799461 - 0.887892i) q^{5} +(1.72096 + 1.91132i) q^{6} +(3.76417 - 1.67592i) q^{7} +(-2.79486 - 8.60169i) q^{8} +(-0.648512 + 1.99591i) q^{9} +(2.16573 - 2.40529i) q^{10} -2.89804 q^{11} +(-3.39151 + 3.76666i) q^{12} +(0.647782 - 1.12199i) q^{13} +(5.58107 + 9.66669i) q^{14} +(-1.10954 - 0.235840i) q^{15} +(12.6287 - 5.62265i) q^{16} +(-1.17068 + 0.248836i) q^{17} +(-5.56093 - 1.18201i) q^{18} +(-1.65370 - 0.736273i) q^{19} +(5.16030 + 3.74918i) q^{20} +(1.95597 - 3.38783i) q^{21} +(-0.820628 - 7.80776i) q^{22} +(-0.656179 + 2.01951i) q^{23} +(-6.94684 - 5.04717i) q^{24} +(0.373429 - 3.55294i) q^{25} +(3.20624 + 1.42751i) q^{26} +(1.49585 + 4.60375i) q^{27} +(-17.7962 + 12.9297i) q^{28} +(-1.94793 - 3.37391i) q^{29} +(0.321204 - 3.05605i) q^{30} +(-0.841767 + 8.00888i) q^{31} +(9.67994 + 16.7661i) q^{32} +(-2.22594 + 1.61724i) q^{33} +(-1.00190 - 3.08353i) q^{34} +(-4.49734 - 2.00235i) q^{35} +(1.17112 - 11.1424i) q^{36} +(-6.55459 - 4.76219i) q^{37} +(1.51536 - 4.66380i) q^{38} +(-0.128572 - 1.22328i) q^{39} +(-5.40299 + 9.35825i) q^{40} +(-0.547742 - 0.397958i) q^{41} +(9.68120 + 4.31035i) q^{42} +(4.01799 + 0.854051i) q^{43} +(15.1335 - 3.21672i) q^{44} +(2.29062 - 1.01985i) q^{45} +(-5.62668 - 1.19599i) q^{46} +(5.64965 + 9.78548i) q^{47} +(6.56220 - 11.3661i) q^{48} +(6.67638 - 7.41487i) q^{49} +9.67790 q^{50} +(-0.760321 + 0.844422i) q^{51} +(-2.13733 + 6.57803i) q^{52} +(1.92613 + 5.92803i) q^{53} +(-11.9796 + 5.33367i) q^{54} +(2.31687 + 2.57314i) q^{55} +(-24.9361 - 27.6943i) q^{56} +(-1.68105 + 0.357319i) q^{57} +(8.53823 - 6.20339i) q^{58} +(-0.516423 - 4.91343i) q^{59} +6.05577 q^{60} +(6.31565 + 4.59484i) q^{61} -21.8155 q^{62} +(0.903877 + 8.59982i) q^{63} +(-20.0621 + 14.5760i) q^{64} +(-1.51408 + 0.321828i) q^{65} +(-4.98741 - 5.53908i) q^{66} +(1.00359 + 1.11460i) q^{67} +(5.83707 - 2.59883i) q^{68} +(0.622980 + 1.91734i) q^{69} +(4.12113 - 12.6835i) q^{70} +(0.220953 - 0.245393i) q^{71} +18.9807 q^{72} +(-0.931658 + 1.03471i) q^{73} +(10.9740 - 19.0076i) q^{74} +(-1.69588 - 2.93735i) q^{75} +(9.45280 + 2.00925i) q^{76} +(-10.9087 + 4.85687i) q^{77} +(3.25929 - 0.692783i) q^{78} +(5.69982 + 1.21153i) q^{79} +(-15.0884 - 6.71781i) q^{80} +(-1.37543 - 0.999308i) q^{81} +(0.917056 - 1.58839i) q^{82} +(-0.0378638 - 0.360250i) q^{83} +(-6.45364 + 19.8622i) q^{84} +(1.15685 + 0.840504i) q^{85} +(-1.16318 + 11.0669i) q^{86} +(-3.37897 - 1.50441i) q^{87} +(8.09961 + 24.9280i) q^{88} +(6.33347 - 4.60153i) q^{89} +(3.39625 + 5.88248i) q^{90} +(0.557998 - 5.30900i) q^{91} +(1.18496 - 11.2742i) q^{92} +(3.82278 + 6.62125i) q^{93} +(-24.7638 + 17.9919i) q^{94} +(0.668335 + 2.05692i) q^{95} +(16.7913 + 7.47598i) q^{96} +(1.27519 - 12.1326i) q^{97} +(21.8673 + 15.8875i) q^{98} +(1.87941 - 5.78423i) 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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.283167	+	2.69415i	0.200229	+	1.90505i	0.386063	+	0.922472i	\(0.373835\pi\)
							−0.185834	+	0.982581i	\(0.559499\pi\)
\(3\)	0.768086	−	0.558047i	0.443454	−	0.322189i	−0.343552	−	0.939134i	\(-0.611630\pi\)
							0.787006	+	0.616945i	\(0.211630\pi\)
\(5\)	−0.799461	−	0.887892i	−0.357530	−	0.397077i	0.537368	−	0.843348i	\(-0.319419\pi\)
							−0.894898	+	0.446271i	\(0.852752\pi\)
\(7\)	3.76417	−	1.67592i	1.42272	−	0.633437i	0.456168	−	0.889894i	\(-0.349222\pi\)
							0.966556	+	0.256456i	\(0.0825550\pi\)
\(11\)	−2.89804			−0.873791			−0.436896	−	0.899512i	\(-0.643922\pi\)
							−0.436896	+	0.899512i	\(0.643922\pi\)
\(13\)	0.647782	−	1.12199i	0.179662	−	0.311184i	−0.762103	−	0.647456i	\(-0.775833\pi\)
							0.941765	+	0.336272i	\(0.109166\pi\)
\(17\)	−1.17068	+	0.248836i	−0.283932	+	0.0603516i	−0.347676	−	0.937615i	\(-0.613029\pi\)
							0.0637442	+	0.997966i	\(0.479696\pi\)
\(19\)	−1.65370	−	0.736273i	−0.379384	−	0.168913i	0.208180	−	0.978091i	\(-0.433246\pi\)
							−0.587564	+	0.809178i	\(0.699913\pi\)
\(23\)	−0.656179	+	2.01951i	−0.136823	+	0.421097i	−0.995869	−	0.0908007i	\(-0.971057\pi\)
							0.859046	+	0.511898i	\(0.171057\pi\)
\(29\)	−1.94793	−	3.37391i	−0.361721	−	0.626519i	0.626524	−	0.779403i	\(-0.284477\pi\)
							−0.988244	+	0.152884i	\(0.951144\pi\)
\(31\)	−0.841767	+	8.00888i	−0.151186	+	1.43844i	0.611283	+	0.791412i	\(0.290654\pi\)
							−0.762469	+	0.647025i	\(0.776013\pi\)
\(37\)	−6.55459	−	4.76219i	−1.07757	−	0.782899i	−0.100311	−	0.994956i	\(-0.531984\pi\)
							−0.977257	+	0.212057i	\(0.931984\pi\)
\(41\)	−0.547742	−	0.397958i	−0.0855429	−	0.0621505i	0.544192	−	0.838961i	\(-0.316836\pi\)
							−0.629735	+	0.776810i	\(0.716836\pi\)
\(43\)	4.01799	+	0.854051i	0.612738	+	0.130242i	0.503819	−	0.863809i	\(-0.331928\pi\)
							0.108919	+	0.994051i	\(0.465261\pi\)
\(47\)	5.64965	+	9.78548i	0.824086	+	1.42736i	0.902616	+	0.430448i	\(0.141644\pi\)
							−0.0785293	+	0.996912i	\(0.525022\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.2.i.a.15.4	✓	32
3.2	odd	2		549.2.bl.b.442.1		32
4.3	odd	2		976.2.bw.c.625.2		32
61.22	even	15		3721.2.a.j.1.1		16
61.39	even	30		3721.2.a.l.1.16		16
61.57	even	15	inner	61.2.i.a.57.4	yes	32
183.179	odd	30		549.2.bl.b.118.1		32
244.179	odd	30		976.2.bw.c.545.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.2.i.a.15.4	✓	32	1.1	even	1	trivial
61.2.i.a.57.4	yes	32	61.57	even	15	inner
549.2.bl.b.118.1		32	183.179	odd	30
549.2.bl.b.442.1		32	3.2	odd	2
976.2.bw.c.545.2		32	244.179	odd	30
976.2.bw.c.625.2		32	4.3	odd	2
3721.2.a.j.1.1		16	61.22	even	15
3721.2.a.l.1.16		16	61.39	even	30