Embedded newform 891.2.e.f.298.1

• Label: 891.2.e.f.298.1
• Level: $891$
• Weight: $2$
• Character: 891.298
• Analytic conductor: $7.115$
• Analytic rank: $0$
• 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:	\(0\)
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 297)
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.f.595.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.00000 + 1.73205i) q^{5} +(2.50000 - 4.33013i) q^{7} -3.00000 q^{8} -2.00000 q^{10} +(-0.500000 + 0.866025i) q^{11} +(1.00000 + 1.73205i) q^{13} +(2.50000 + 4.33013i) q^{14} +(0.500000 - 0.866025i) q^{16} +7.00000 q^{17} +(-1.00000 + 1.73205i) q^{20} +(-0.500000 - 0.866025i) q^{22} +(0.500000 + 0.866025i) q^{23} +(0.500000 - 0.866025i) q^{25} -2.00000 q^{26} +5.00000 q^{28} +(-1.50000 + 2.59808i) q^{29} +(4.00000 + 6.92820i) q^{31} +(-2.50000 - 4.33013i) q^{32} +(-3.50000 + 6.06218i) q^{34} +10.0000 q^{35} -3.00000 q^{37} +(-3.00000 - 5.19615i) q^{40} +(5.50000 + 9.52628i) q^{41} +(4.50000 - 7.79423i) q^{43} -1.00000 q^{44} -1.00000 q^{46} +(0.500000 - 0.866025i) q^{47} +(-9.00000 - 15.5885i) q^{49} +(0.500000 + 0.866025i) q^{50} +(-1.00000 + 1.73205i) q^{52} -12.0000 q^{53} -2.00000 q^{55} +(-7.50000 + 12.9904i) q^{56} +(-1.50000 - 2.59808i) q^{58} +(-2.50000 - 4.33013i) q^{59} +(-3.00000 + 5.19615i) q^{61} -8.00000 q^{62} +7.00000 q^{64} +(-2.00000 + 3.46410i) q^{65} +(2.00000 + 3.46410i) q^{67} +(3.50000 + 6.06218i) q^{68} +(-5.00000 + 8.66025i) q^{70} +4.00000 q^{73} +(1.50000 - 2.59808i) q^{74} +(2.50000 + 4.33013i) q^{77} +(-2.50000 + 4.33013i) q^{79} +2.00000 q^{80} -11.0000 q^{82} +(3.00000 - 5.19615i) q^{83} +(7.00000 + 12.1244i) q^{85} +(4.50000 + 7.79423i) q^{86} +(1.50000 - 2.59808i) q^{88} -6.00000 q^{89} +10.0000 q^{91} +(-0.500000 + 0.866025i) q^{92} +(0.500000 + 0.866025i) q^{94} +(-5.50000 + 9.52628i) q^{97} +18.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^2 + q^4 + 2 * q^5 + 5 * q^7 - 6 * q^8 - 4 * q^10 - q^11 + 2 * q^13 + 5 * q^14 + q^16 + 14 * q^17 - 2 * q^20 - q^22 + q^23 + q^25 - 4 * q^26 + 10 * q^28 - 3 * q^29 + 8 * q^31 - 5 * q^32 - 7 * q^34 + 20 * q^35 - 6 * q^37 - 6 * q^40 + 11 * q^41 + 9 * q^43 - 2 * q^44 - 2 * q^46 + q^47 - 18 * q^49 + q^50 - 2 * q^52 - 24 * q^53 - 4 * q^55 - 15 * q^56 - 3 * q^58 - 5 * q^59 - 6 * q^61 - 16 * q^62 + 14 * q^64 - 4 * q^65 + 4 * q^67 + 7 * q^68 - 10 * q^70 + 8 * q^73 + 3 * q^74 + 5 * q^77 - 5 * q^79 + 4 * q^80 - 22 * q^82 + 6 * q^83 + 14 * q^85 + 9 * q^86 + 3 * q^88 - 12 * q^89 + 20 * q^91 - q^92 + q^94 - 11 * q^97 + 36 * 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
							\(5\)	1.00000	+	1.73205i	0.447214	+	0.774597i	0.998203	−	0.0599153i	\(-0.0190830\pi\)
−0.550990	+	0.834512i	\(0.685750\pi\)
\(7\)	2.50000	−	4.33013i	0.944911	−	1.63663i	0.188982	−	0.981981i	\(-0.439481\pi\)
							0.755929	−	0.654654i	\(-0.227186\pi\)
\(11\)	−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	\(0.743877\pi\)
\(17\)	7.00000			1.69775			0.848875	−	0.528594i	\(-0.177281\pi\)
							0.848875	+	0.528594i	\(0.177281\pi\)
\(19\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(23\)	0.500000	+	0.866025i	0.104257	+	0.180579i	0.913434	−	0.406986i	\(-0.133420\pi\)
							−0.809177	+	0.587565i	\(0.800087\pi\)
\(29\)	−1.50000	+	2.59808i	−0.278543	+	0.482451i	−0.971023	−	0.238987i	\(-0.923185\pi\)
							0.692480	+	0.721437i	\(0.256518\pi\)
\(31\)	4.00000	+	6.92820i	0.718421	+	1.24434i	0.961625	+	0.274367i	\(0.0884683\pi\)
							−0.243204	+	0.969975i	\(0.578198\pi\)
\(37\)	−3.00000			−0.493197			−0.246598	−	0.969118i	\(-0.579313\pi\)
							−0.246598	+	0.969118i	\(0.579313\pi\)
\(41\)	5.50000	+	9.52628i	0.858956	+	1.48775i	0.872926	+	0.487852i	\(0.162220\pi\)
							−0.0139704	+	0.999902i	\(0.504447\pi\)
\(43\)	4.50000	−	7.79423i	0.686244	−	1.18861i	−0.286801	−	0.957990i	\(-0.592592\pi\)
							0.973044	−	0.230618i	\(-0.0740749\pi\)
\(47\)	0.500000	−	0.866025i	0.0729325	−	0.126323i	−0.827253	−	0.561830i	\(-0.810098\pi\)
							0.900185	+	0.435507i	\(0.143431\pi\)
\(53\)	−12.0000			−1.64833			−0.824163	−	0.566352i	\(-0.808354\pi\)
							−0.824163	+	0.566352i	\(0.808354\pi\)
\(59\)	−2.50000	−	4.33013i	−0.325472	−	0.563735i	0.656136	−	0.754643i	\(-0.272190\pi\)
							−0.981608	+	0.190909i	\(0.938857\pi\)
\(61\)	−3.00000	+	5.19615i	−0.384111	+	0.665299i	−0.991645	−	0.128994i	\(-0.958825\pi\)
							0.607535	+	0.794293i	\(0.292159\pi\)
\(67\)	2.00000	+	3.46410i	0.244339	+	0.423207i	0.961946	−	0.273241i	\(-0.0880957\pi\)
							−0.717607	+	0.696449i	\(0.754762\pi\)
\(71\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(73\)	4.00000			0.468165			0.234082	−	0.972217i	\(-0.424791\pi\)
							0.234082	+	0.972217i	\(0.424791\pi\)
\(79\)	−2.50000	+	4.33013i	−0.281272	+	0.487177i	−0.971698	−	0.236225i	\(-0.924090\pi\)
							0.690426	+	0.723403i	\(0.257423\pi\)
\(83\)	3.00000	−	5.19615i	0.329293	−	0.570352i	−0.653079	−	0.757290i	\(-0.726523\pi\)
							0.982372	+	0.186938i	\(0.0598564\pi\)
\(89\)	−6.00000			−0.635999			−0.317999	−	0.948091i	\(-0.603011\pi\)
							−0.317999	+	0.948091i	\(0.603011\pi\)
\(97\)	−5.50000	+	9.52628i	−0.558440	+	0.967247i	0.439187	+	0.898396i	\(0.355267\pi\)
							−0.997627	+	0.0688512i	\(0.978067\pi\)

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.f.298.1		2
3.2	odd	2		891.2.e.h.298.1		2
9.2	odd	6		297.2.a.b.1.1	✓	1
9.4	even	3	inner	891.2.e.f.595.1		2
9.5	odd	6		891.2.e.h.595.1		2
9.7	even	3		297.2.a.c.1.1	yes	1
36.7	odd	6		4752.2.a.g.1.1		1
36.11	even	6		4752.2.a.r.1.1		1
45.29	odd	6		7425.2.a.s.1.1		1
45.34	even	6		7425.2.a.k.1.1		1
99.43	odd	6		3267.2.a.c.1.1		1
99.65	even	6		3267.2.a.j.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.a.b.1.1	✓	1	9.2	odd	6
297.2.a.c.1.1	yes	1	9.7	even	3
891.2.e.f.298.1		2	1.1	even	1	trivial
891.2.e.f.595.1		2	9.4	even	3	inner
891.2.e.h.298.1		2	3.2	odd	2
891.2.e.h.595.1		2	9.5	odd	6
3267.2.a.c.1.1		1	99.43	odd	6
3267.2.a.j.1.1		1	99.65	even	6
4752.2.a.g.1.1		1	36.7	odd	6
4752.2.a.r.1.1		1	36.11	even	6
7425.2.a.k.1.1		1	45.34	even	6
7425.2.a.s.1.1		1	45.29	odd	6