Embedded newform 1134.2.h.j.109.1

• Label: 1134.2.h.j.109.1
• Level: $1134$
• Weight: $2$
• Character: 1134.109
• Analytic conductor: $9.055$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1134.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.05503558921\)
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 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	1134.109
Dual form			1134.2.h.j.541.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} -3.00000 q^{5} +(-2.00000 + 1.73205i) q^{7} -1.00000 q^{8} +(-1.50000 + 2.59808i) q^{10} +3.00000 q^{11} +(-1.00000 + 1.73205i) q^{13} +(0.500000 + 2.59808i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(3.00000 - 5.19615i) q^{17} +(-1.00000 - 1.73205i) q^{19} +(1.50000 + 2.59808i) q^{20} +(1.50000 - 2.59808i) q^{22} +6.00000 q^{23} +4.00000 q^{25} +(1.00000 + 1.73205i) q^{26} +(2.50000 + 0.866025i) q^{28} +(4.50000 + 7.79423i) q^{29} +(3.50000 + 6.06218i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-3.00000 - 5.19615i) q^{34} +(6.00000 - 5.19615i) q^{35} +(5.00000 + 8.66025i) q^{37} -2.00000 q^{38} +3.00000 q^{40} +(2.00000 + 3.46410i) q^{43} +(-1.50000 - 2.59808i) q^{44} +(3.00000 - 5.19615i) q^{46} +(6.00000 - 10.3923i) q^{47} +(1.00000 - 6.92820i) q^{49} +(2.00000 - 3.46410i) q^{50} +2.00000 q^{52} +(-1.50000 + 2.59808i) q^{53} -9.00000 q^{55} +(2.00000 - 1.73205i) q^{56} +9.00000 q^{58} +(-1.50000 - 2.59808i) q^{59} +(2.00000 - 3.46410i) q^{61} +7.00000 q^{62} +1.00000 q^{64} +(3.00000 - 5.19615i) q^{65} +(-1.00000 - 1.73205i) q^{67} -6.00000 q^{68} +(-1.50000 - 7.79423i) q^{70} +(-1.00000 + 1.73205i) q^{73} +10.0000 q^{74} +(-1.00000 + 1.73205i) q^{76} +(-6.00000 + 5.19615i) q^{77} +(-2.50000 + 4.33013i) q^{79} +(1.50000 - 2.59808i) q^{80} +(4.50000 + 7.79423i) q^{83} +(-9.00000 + 15.5885i) q^{85} +4.00000 q^{86} -3.00000 q^{88} +(-3.00000 - 5.19615i) q^{89} +(-1.00000 - 5.19615i) q^{91} +(-3.00000 - 5.19615i) q^{92} +(-6.00000 - 10.3923i) q^{94} +(3.00000 + 5.19615i) q^{95} +(6.50000 + 11.2583i) q^{97} +(-5.50000 - 4.33013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^2 - q^4 - 6 * q^5 - 4 * q^7 - 2 * q^8 - 3 * q^10 + 6 * q^11 - 2 * q^13 + q^14 - q^16 + 6 * q^17 - 2 * q^19 + 3 * q^20 + 3 * q^22 + 12 * q^23 + 8 * q^25 + 2 * q^26 + 5 * q^28 + 9 * q^29 + 7 * q^31 + q^32 - 6 * q^34 + 12 * q^35 + 10 * q^37 - 4 * q^38 + 6 * q^40 + 4 * q^43 - 3 * q^44 + 6 * q^46 + 12 * q^47 + 2 * q^49 + 4 * q^50 + 4 * q^52 - 3 * q^53 - 18 * q^55 + 4 * q^56 + 18 * q^58 - 3 * q^59 + 4 * q^61 + 14 * q^62 + 2 * q^64 + 6 * q^65 - 2 * q^67 - 12 * q^68 - 3 * q^70 - 2 * q^73 + 20 * q^74 - 2 * q^76 - 12 * q^77 - 5 * q^79 + 3 * q^80 + 9 * q^83 - 18 * q^85 + 8 * q^86 - 6 * q^88 - 6 * q^89 - 2 * q^91 - 6 * q^92 - 12 * q^94 + 6 * q^95 + 13 * q^97 - 11 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(407\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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
							\(3\)		0			0
\(5\)	−3.00000			−1.34164										−0.670820	−	0.741620i	\(-0.734058\pi\)
							−0.670820	+	0.741620i	\(0.734058\pi\)
\(7\)	−2.00000	+	1.73205i	−0.755929	+	0.654654i
							\(11\)	3.00000			0.904534			0.452267	−	0.891883i	\(-0.350615\pi\)
0.452267	+	0.891883i	\(0.350615\pi\)
\(13\)	−1.00000	+	1.73205i	−0.277350	+	0.480384i	−0.970725	−	0.240192i	\(-0.922790\pi\)
							0.693375	+	0.720577i	\(0.256123\pi\)
\(17\)	3.00000	−	5.19615i	0.727607	−	1.26025i	−0.230285	−	0.973123i	\(-0.573966\pi\)
							0.957892	−	0.287129i	\(-0.0927008\pi\)
\(19\)	−1.00000	−	1.73205i	−0.229416	−	0.397360i	0.728219	−	0.685344i	\(-0.240348\pi\)
							−0.957635	+	0.287984i	\(0.907015\pi\)
\(23\)	6.00000			1.25109			0.625543	−	0.780189i	\(-0.284877\pi\)
							0.625543	+	0.780189i	\(0.284877\pi\)
\(29\)	4.50000	+	7.79423i	0.835629	+	1.44735i	0.893517	+	0.449029i	\(0.148230\pi\)
							−0.0578882	+	0.998323i	\(0.518437\pi\)
\(31\)	3.50000	+	6.06218i	0.628619	+	1.08880i	0.987829	+	0.155543i	\(0.0497126\pi\)
							−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	5.00000	+	8.66025i	0.821995	+	1.42374i	0.904194	+	0.427121i	\(0.140472\pi\)
							−0.0821995	+	0.996616i	\(0.526194\pi\)
\(41\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(43\)	2.00000	+	3.46410i	0.304997	+	0.528271i	0.977261	−	0.212041i	\(-0.0680112\pi\)
							−0.672264	+	0.740312i	\(0.734678\pi\)
\(47\)	6.00000	−	10.3923i	0.875190	−	1.51587i	0.0186297	−	0.999826i	\(-0.494070\pi\)
							0.856560	−	0.516047i	\(-0.172597\pi\)
\(53\)	−1.50000	+	2.59808i	−0.206041	+	0.356873i	−0.950464	−	0.310835i	\(-0.899391\pi\)
							0.744423	+	0.667708i	\(0.232725\pi\)
\(59\)	−1.50000	−	2.59808i	−0.195283	−	0.338241i	0.751710	−	0.659494i	\(-0.229229\pi\)
							−0.946993	+	0.321253i	\(0.895896\pi\)
\(61\)	2.00000	−	3.46410i	0.256074	−	0.443533i	−0.709113	−	0.705095i	\(-0.750904\pi\)
							0.965187	+	0.261562i	\(0.0842377\pi\)
\(67\)	−1.00000	−	1.73205i	−0.122169	−	0.211604i	0.798454	−	0.602056i	\(-0.205652\pi\)
							−0.920623	+	0.390453i	\(0.872318\pi\)
\(71\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(73\)	−1.00000	+	1.73205i	−0.117041	+	0.202721i	−0.918594	−	0.395203i	\(-0.870674\pi\)
							0.801553	+	0.597924i	\(0.204008\pi\)
\(79\)	−2.50000	+	4.33013i	−0.281272	+	0.487177i	−0.971698	−	0.236225i	\(-0.924090\pi\)
							0.690426	+	0.723403i	\(0.257423\pi\)
\(83\)	4.50000	+	7.79423i	0.493939	+	0.855528i	0.999976	−	0.00698436i	\(-0.00222321\pi\)
							−0.506036	+	0.862512i	\(0.668890\pi\)
\(89\)	−3.00000	−	5.19615i	−0.317999	−	0.550791i	0.662071	−	0.749441i	\(-0.269678\pi\)
							−0.980071	+	0.198650i	\(0.936344\pi\)
\(97\)	6.50000	+	11.2583i	0.659975	+	1.14311i	0.980622	+	0.195911i	\(0.0627665\pi\)
							−0.320647	+	0.947199i	\(0.603900\pi\)

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	1134.2.h.j.109.1		2
3.2	odd	2		1134.2.h.f.109.1		2
7.2	even	3		1134.2.e.g.919.1		2
9.2	odd	6		1134.2.e.k.865.1		2
9.4	even	3		126.2.g.d.109.1	yes	2
9.5	odd	6		126.2.g.a.109.1	yes	2
9.7	even	3		1134.2.e.g.865.1		2
21.2	odd	6		1134.2.e.k.919.1		2
36.23	even	6		1008.2.s.b.865.1		2
36.31	odd	6		1008.2.s.o.865.1		2
63.2	odd	6		1134.2.h.f.541.1		2
63.4	even	3		882.2.a.a.1.1		1
63.5	even	6		882.2.g.e.667.1		2
63.13	odd	6		882.2.g.g.361.1		2
63.16	even	3	inner	1134.2.h.j.541.1		2
63.23	odd	6		126.2.g.a.37.1	✓	2
63.31	odd	6		882.2.a.e.1.1		1
63.32	odd	6		882.2.a.j.1.1		1
63.40	odd	6		882.2.g.g.667.1		2
63.41	even	6		882.2.g.e.361.1		2
63.58	even	3		126.2.g.d.37.1	yes	2
63.59	even	6		882.2.a.h.1.1		1
252.23	even	6		1008.2.s.b.289.1		2
252.31	even	6		7056.2.a.bx.1.1		1
252.59	odd	6		7056.2.a.h.1.1		1
252.67	odd	6		7056.2.a.e.1.1		1
252.95	even	6		7056.2.a.by.1.1		1
252.247	odd	6		1008.2.s.o.289.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.g.a.37.1	✓	2	63.23	odd	6
126.2.g.a.109.1	yes	2	9.5	odd	6
126.2.g.d.37.1	yes	2	63.58	even	3
126.2.g.d.109.1	yes	2	9.4	even	3
882.2.a.a.1.1		1	63.4	even	3
882.2.a.e.1.1		1	63.31	odd	6
882.2.a.h.1.1		1	63.59	even	6
882.2.a.j.1.1		1	63.32	odd	6
882.2.g.e.361.1		2	63.41	even	6
882.2.g.e.667.1		2	63.5	even	6
882.2.g.g.361.1		2	63.13	odd	6
882.2.g.g.667.1		2	63.40	odd	6
1008.2.s.b.289.1		2	252.23	even	6
1008.2.s.b.865.1		2	36.23	even	6
1008.2.s.o.289.1		2	252.247	odd	6
1008.2.s.o.865.1		2	36.31	odd	6
1134.2.e.g.865.1		2	9.7	even	3
1134.2.e.g.919.1		2	7.2	even	3
1134.2.e.k.865.1		2	9.2	odd	6
1134.2.e.k.919.1		2	21.2	odd	6
1134.2.h.f.109.1		2	3.2	odd	2
1134.2.h.f.541.1		2	63.2	odd	6
1134.2.h.j.109.1		2	1.1	even	1	trivial
1134.2.h.j.541.1		2	63.16	even	3	inner
7056.2.a.e.1.1		1	252.67	odd	6
7056.2.a.h.1.1		1	252.59	odd	6
7056.2.a.bx.1.1		1	252.31	even	6
7056.2.a.by.1.1		1	252.95	even	6