Embedded newform 891.2.e.c.298.1

• Label: 891.2.e.c.298.1
• Level: $891$
• Weight: $2$
• Character: 891.298
• Analytic conductor: $7.115$
• Analytic rank: $1$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 891 = 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	891.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.11467082010\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			298.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	891.298
Dual form			891.2.e.c.595.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-2.00000 - 3.46410i) q^{5} +(1.00000 - 1.73205i) q^{7} -3.00000 q^{8} +4.00000 q^{10} +(-0.500000 + 0.866025i) q^{11} +(1.00000 + 1.73205i) q^{13} +(1.00000 + 1.73205i) q^{14} +(0.500000 - 0.866025i) q^{16} -2.00000 q^{17} -6.00000 q^{19} +(2.00000 - 3.46410i) q^{20} +(-0.500000 - 0.866025i) q^{22} +(2.00000 + 3.46410i) q^{23} +(-5.50000 + 9.52628i) q^{25} -2.00000 q^{26} +2.00000 q^{28} +(-3.00000 + 5.19615i) q^{29} +(-2.00000 - 3.46410i) q^{31} +(-2.50000 - 4.33013i) q^{32} +(1.00000 - 1.73205i) q^{34} -8.00000 q^{35} -6.00000 q^{37} +(3.00000 - 5.19615i) q^{38} +(6.00000 + 10.3923i) q^{40} +(-5.00000 - 8.66025i) q^{41} +(-3.00000 + 5.19615i) q^{43} -1.00000 q^{44} -4.00000 q^{46} +(-4.00000 + 6.92820i) q^{47} +(1.50000 + 2.59808i) q^{49} +(-5.50000 - 9.52628i) q^{50} +(-1.00000 + 1.73205i) q^{52} +4.00000 q^{55} +(-3.00000 + 5.19615i) q^{56} +(-3.00000 - 5.19615i) q^{58} +(2.00000 + 3.46410i) q^{59} +(3.00000 - 5.19615i) q^{61} +4.00000 q^{62} +7.00000 q^{64} +(4.00000 - 6.92820i) q^{65} +(-4.00000 - 6.92820i) q^{67} +(-1.00000 - 1.73205i) q^{68} +(4.00000 - 6.92820i) q^{70} -2.00000 q^{73} +(3.00000 - 5.19615i) q^{74} +(-3.00000 - 5.19615i) q^{76} +(1.00000 + 1.73205i) q^{77} +(5.00000 - 8.66025i) q^{79} -4.00000 q^{80} +10.0000 q^{82} +(6.00000 - 10.3923i) q^{83} +(4.00000 + 6.92820i) q^{85} +(-3.00000 - 5.19615i) q^{86} +(1.50000 - 2.59808i) q^{88} +4.00000 q^{91} +(-2.00000 + 3.46410i) q^{92} +(-4.00000 - 6.92820i) q^{94} +(12.0000 + 20.7846i) q^{95} +(-1.00000 + 1.73205i) q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^2 + q^4 - 4 * q^5 + 2 * q^7 - 6 * q^8 + 8 * q^10 - q^11 + 2 * q^13 + 2 * q^14 + q^16 - 4 * q^17 - 12 * q^19 + 4 * q^20 - q^22 + 4 * q^23 - 11 * q^25 - 4 * q^26 + 4 * q^28 - 6 * q^29 - 4 * q^31 - 5 * q^32 + 2 * q^34 - 16 * q^35 - 12 * q^37 + 6 * q^38 + 12 * q^40 - 10 * q^41 - 6 * q^43 - 2 * q^44 - 8 * q^46 - 8 * q^47 + 3 * q^49 - 11 * q^50 - 2 * q^52 + 8 * q^55 - 6 * q^56 - 6 * q^58 + 4 * q^59 + 6 * q^61 + 8 * q^62 + 14 * q^64 + 8 * q^65 - 8 * q^67 - 2 * q^68 + 8 * q^70 - 4 * q^73 + 6 * q^74 - 6 * q^76 + 2 * q^77 + 10 * q^79 - 8 * q^80 + 20 * q^82 + 12 * q^83 + 8 * q^85 - 6 * q^86 + 3 * q^88 + 8 * q^91 - 4 * q^92 - 8 * q^94 + 24 * q^95 - 2 * q^97 - 6 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).

\(n\)	\(244\)	\(650\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

Coefficient data

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

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.500000	+	0.866025i	−0.353553	+	0.612372i	−0.986869	−	0.161521i	\(-0.948360\pi\)
							0.633316	+	0.773893i	\(0.281693\pi\)
\(3\)		0			0
							\(5\)	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.834512	−	0.550990i	\(-0.814250\pi\)
−0.0599153	−	0.998203i	\(-0.519083\pi\)
\(7\)	1.00000	−	1.73205i	0.377964	−	0.654654i	−0.612801	−	0.790237i	\(-0.709957\pi\)
							0.990766	+	0.135583i	\(0.0432908\pi\)
\(11\)	−0.500000	+	0.866025i	−0.150756	+	0.261116i
							\(13\)	1.00000	+	1.73205i	0.277350	+	0.480384i	0.970725	−	0.240192i	\(-0.0772105\pi\)
−0.693375	+	0.720577i	\(0.743877\pi\)
\(17\)	−2.00000			−0.485071			−0.242536	−	0.970143i	\(-0.577979\pi\)
							−0.242536	+	0.970143i	\(0.577979\pi\)
\(19\)	−6.00000			−1.37649			−0.688247	−	0.725476i	\(-0.741620\pi\)
							−0.688247	+	0.725476i	\(0.741620\pi\)
\(23\)	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	\(-0.0297381\pi\)
							−0.578610	+	0.815604i	\(0.696405\pi\)
\(29\)	−3.00000	+	5.19615i	−0.557086	+	0.964901i	0.440652	+	0.897678i	\(0.354747\pi\)
							−0.997738	+	0.0672232i	\(0.978586\pi\)
\(31\)	−2.00000	−	3.46410i	−0.359211	−	0.622171i	0.628619	−	0.777714i	\(-0.283621\pi\)
							−0.987829	+	0.155543i	\(0.950287\pi\)
\(37\)	−6.00000			−0.986394			−0.493197	−	0.869918i	\(-0.664172\pi\)
							−0.493197	+	0.869918i	\(0.664172\pi\)
\(41\)	−5.00000	−	8.66025i	−0.780869	−	1.35250i	−0.931436	−	0.363905i	\(-0.881443\pi\)
							0.150567	−	0.988600i	\(-0.451890\pi\)
\(43\)	−3.00000	+	5.19615i	−0.457496	+	0.792406i	−0.998828	−	0.0484030i	\(-0.984587\pi\)
							0.541332	+	0.840809i	\(0.317920\pi\)
\(47\)	−4.00000	+	6.92820i	−0.583460	+	1.01058i	0.411606	+	0.911362i	\(0.364968\pi\)
							−0.995066	+	0.0992202i	\(0.968365\pi\)
\(53\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(59\)	2.00000	+	3.46410i	0.260378	+	0.450988i	0.966342	−	0.257260i	\(-0.0828195\pi\)
							−0.705965	+	0.708247i	\(0.749486\pi\)
\(61\)	3.00000	−	5.19615i	0.384111	−	0.665299i	−0.607535	−	0.794293i	\(-0.707841\pi\)
							0.991645	+	0.128994i	\(0.0411748\pi\)
\(67\)	−4.00000	−	6.92820i	−0.488678	−	0.846415i	0.511237	−	0.859440i	\(-0.329187\pi\)
							−0.999915	+	0.0130248i	\(0.995854\pi\)
\(71\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(73\)	−2.00000			−0.234082			−0.117041	−	0.993127i	\(-0.537341\pi\)
							−0.117041	+	0.993127i	\(0.537341\pi\)
\(79\)	5.00000	−	8.66025i	0.562544	−	0.974355i	−0.434730	−	0.900561i	\(-0.643156\pi\)
							0.997274	−	0.0737937i	\(-0.0235106\pi\)
\(83\)	6.00000	−	10.3923i	0.658586	−	1.14070i	−0.322396	−	0.946605i	\(-0.604488\pi\)
							0.980982	−	0.194099i	\(-0.0621783\pi\)
\(89\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(97\)	−1.00000	+	1.73205i	−0.101535	+	0.175863i	−0.912317	−	0.409484i	\(-0.865709\pi\)
							0.810782	+	0.585348i	\(0.199042\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.2.e.c.298.1		2
3.2	odd	2		891.2.e.j.298.1		2
9.2	odd	6		99.2.a.a.1.1	✓	1
9.4	even	3	inner	891.2.e.c.595.1		2
9.5	odd	6		891.2.e.j.595.1		2
9.7	even	3		99.2.a.c.1.1	yes	1
36.7	odd	6		1584.2.a.r.1.1		1
36.11	even	6		1584.2.a.b.1.1		1
45.2	even	12		2475.2.c.b.199.1		2
45.7	odd	12		2475.2.c.g.199.2		2
45.29	odd	6		2475.2.a.j.1.1		1
45.34	even	6		2475.2.a.c.1.1		1
45.38	even	12		2475.2.c.b.199.2		2
45.43	odd	12		2475.2.c.g.199.1		2
63.20	even	6		4851.2.a.g.1.1		1
63.34	odd	6		4851.2.a.o.1.1		1
72.11	even	6		6336.2.a.cm.1.1		1
72.29	odd	6		6336.2.a.cl.1.1		1
72.43	odd	6		6336.2.a.f.1.1		1
72.61	even	6		6336.2.a.b.1.1		1
99.43	odd	6		1089.2.a.d.1.1		1
99.65	even	6		1089.2.a.h.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.a.a.1.1	✓	1	9.2	odd	6
99.2.a.c.1.1	yes	1	9.7	even	3
891.2.e.c.298.1		2	1.1	even	1	trivial
891.2.e.c.595.1		2	9.4	even	3	inner
891.2.e.j.298.1		2	3.2	odd	2
891.2.e.j.595.1		2	9.5	odd	6
1089.2.a.d.1.1		1	99.43	odd	6
1089.2.a.h.1.1		1	99.65	even	6
1584.2.a.b.1.1		1	36.11	even	6
1584.2.a.r.1.1		1	36.7	odd	6
2475.2.a.c.1.1		1	45.34	even	6
2475.2.a.j.1.1		1	45.29	odd	6
2475.2.c.b.199.1		2	45.2	even	12
2475.2.c.b.199.2		2	45.38	even	12
2475.2.c.g.199.1		2	45.43	odd	12
2475.2.c.g.199.2		2	45.7	odd	12
4851.2.a.g.1.1		1	63.20	even	6
4851.2.a.o.1.1		1	63.34	odd	6
6336.2.a.b.1.1		1	72.61	even	6
6336.2.a.f.1.1		1	72.43	odd	6
6336.2.a.cl.1.1		1	72.29	odd	6
6336.2.a.cm.1.1		1	72.11	even	6