Embedded newform 114.4.h.b.65.10

• Label: 114.4.h.b.65.10
• Level: $114$
• Weight: $4$
• Character: 114.65
• Analytic conductor: $6.726$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	114.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.72621774065\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 5*x^19 + 10*x^18 - 183*x^17 + 864*x^16 - 495*x^15 - 1530*x^14 + 118935*x^13 - 524961*x^12 + 4026996*x^11 - 36285408*x^10 + 108728892*x^9 - 382696569*x^8 + 2340997605*x^7 - 813104730*x^6 - 7102708965*x^5 + 334731302496*x^4 - 1914244636149*x^3 + 2824295364810*x^2 - 38127987424935*x + 205891132094649
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{4}\cdot 3^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			65.10
Root			\(1.24390 - 5.04507i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.65
Dual form			114.4.h.b.107.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(4.99111 + 1.44529i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(-1.54776 - 0.893602i) q^{5} +(7.49442 - 7.19956i) q^{6} +19.7997 q^{7} -8.00000 q^{8} +(22.8223 + 14.4272i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{2} + q^{3} - 40 q^{4} - 8 q^{6} - 10 q^{7} - 160 q^{8} - 37 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	−	1.73205i	0.353553	−	0.612372i
							\(3\)	4.99111	+	1.44529i	0.960539	+	0.278146i
\(5\)	−1.54776	−	0.893602i	−0.138436	−	0.0799262i								0.429182	−	0.903218i	\(-0.358802\pi\)
							−0.567619	+	0.823292i	\(0.692135\pi\)
\(7\)	19.7997			1.06908			0.534541	−	0.845143i	\(-0.320485\pi\)
							0.534541	+	0.845143i	\(0.320485\pi\)
\(11\)		−	43.6809i		−	1.19730i	−0.801011	−	0.598649i	\(-0.795704\pi\)
							0.801011	−	0.598649i	\(-0.204296\pi\)
\(13\)	21.5591	−	12.4472i	0.459956	−	0.265556i	−0.252070	−	0.967709i	\(-0.581111\pi\)
							0.712026	+	0.702153i	\(0.247778\pi\)
\(17\)	47.2760	+	27.2948i	0.674478	+	0.389410i	0.797771	−	0.602960i	\(-0.206012\pi\)
							−0.123294	+	0.992370i	\(0.539346\pi\)
\(19\)	−49.7813	+	66.1878i	−0.601084	+	0.799186i
							\(23\)	−48.5966	+	28.0573i	−0.440570	+	0.254363i	−0.703839	−	0.710359i	\(-0.748532\pi\)
0.263270	+	0.964722i	\(0.415199\pi\)
\(29\)	−81.7025	−	141.513i	−0.523164	−	0.906147i	−0.999637	−	0.0269579i	\(-0.991418\pi\)
							0.476472	−	0.879190i	\(-0.341915\pi\)
\(31\)			148.462i			0.860148i	0.902794	+	0.430074i	\(0.141512\pi\)
							−0.902794	+	0.430074i	\(0.858488\pi\)
\(37\)			54.7219i			0.243141i	0.992583	+	0.121571i	\(0.0387931\pi\)
							−0.992583	+	0.121571i	\(0.961207\pi\)
\(41\)	−242.840	+	420.611i	−0.925006	+	1.60216i	−0.133455	+	0.991055i	\(0.542607\pi\)
							−0.791551	+	0.611103i	\(0.790726\pi\)
\(43\)	103.055	−	178.496i	0.365481	−	0.633032i	−0.623372	−	0.781926i	\(-0.714238\pi\)
							0.988853	+	0.148893i	\(0.0475710\pi\)
\(47\)	−434.840	+	251.055i	−1.34953	+	0.779151i	−0.988183	−	0.153280i	\(-0.951016\pi\)
							−0.361347	+	0.932432i	\(0.617683\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.4.h.b.65.10	yes	20
3.2	odd	2		114.4.h.a.65.8	✓	20
19.12	odd	6		114.4.h.a.107.8	yes	20
57.50	even	6	inner	114.4.h.b.107.10	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.h.a.65.8	✓	20	3.2	odd	2
114.4.h.a.107.8	yes	20	19.12	odd	6
114.4.h.b.65.10	yes	20	1.1	even	1	trivial
114.4.h.b.107.10	yes	20	57.50	even	6	inner