Embedded newform 1134.2.f.p.757.1

• Label: 1134.2.f.p.757.1
• Level: $1134$
• Weight: $2$
• Character: 1134.757
• 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.f (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 378)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			757.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	1134.757
Dual form			1134.2.f.p.379.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(2.00000 - 3.46410i) q^{5} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} +4.00000 q^{10} +(-2.00000 - 3.46410i) q^{11} +(-1.50000 + 2.59808i) q^{13} +(-0.500000 + 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +7.00000 q^{17} +2.00000 q^{19} +(2.00000 + 3.46410i) q^{20} +(2.00000 - 3.46410i) q^{22} +(-0.500000 + 0.866025i) q^{23} +(-5.50000 - 9.52628i) q^{25} -3.00000 q^{26} -1.00000 q^{28} +(0.500000 + 0.866025i) q^{29} +(4.50000 - 7.79423i) q^{31} +(0.500000 - 0.866025i) q^{32} +(3.50000 + 6.06218i) q^{34} +4.00000 q^{35} +2.00000 q^{37} +(1.00000 + 1.73205i) q^{38} +(-2.00000 + 3.46410i) q^{40} +(3.00000 - 5.19615i) q^{41} +(-5.50000 - 9.52628i) q^{43} +4.00000 q^{44} -1.00000 q^{46} +(-3.00000 - 5.19615i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(5.50000 - 9.52628i) q^{50} +(-1.50000 - 2.59808i) q^{52} +9.00000 q^{53} -16.0000 q^{55} +(-0.500000 - 0.866025i) q^{56} +(-0.500000 + 0.866025i) q^{58} +(-2.50000 + 4.33013i) q^{59} +(3.00000 + 5.19615i) q^{61} +9.00000 q^{62} +1.00000 q^{64} +(6.00000 + 10.3923i) q^{65} +(-3.50000 + 6.06218i) q^{67} +(-3.50000 + 6.06218i) q^{68} +(2.00000 + 3.46410i) q^{70} +7.00000 q^{71} -14.0000 q^{73} +(1.00000 + 1.73205i) q^{74} +(-1.00000 + 1.73205i) q^{76} +(2.00000 - 3.46410i) q^{77} +(3.00000 + 5.19615i) q^{79} -4.00000 q^{80} +6.00000 q^{82} +(-2.00000 - 3.46410i) q^{83} +(14.0000 - 24.2487i) q^{85} +(5.50000 - 9.52628i) q^{86} +(2.00000 + 3.46410i) q^{88} +3.00000 q^{89} -3.00000 q^{91} +(-0.500000 - 0.866025i) q^{92} +(3.00000 - 5.19615i) q^{94} +(4.00000 - 6.92820i) q^{95} +(4.00000 + 6.92820i) q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^2 - q^4 + 4 * q^5 + q^7 - 2 * q^8 + 8 * q^10 - 4 * q^11 - 3 * q^13 - q^14 - q^16 + 14 * q^17 + 4 * q^19 + 4 * q^20 + 4 * q^22 - q^23 - 11 * q^25 - 6 * q^26 - 2 * q^28 + q^29 + 9 * q^31 + q^32 + 7 * q^34 + 8 * q^35 + 4 * q^37 + 2 * q^38 - 4 * q^40 + 6 * q^41 - 11 * q^43 + 8 * q^44 - 2 * q^46 - 6 * q^47 - q^49 + 11 * q^50 - 3 * q^52 + 18 * q^53 - 32 * q^55 - q^56 - q^58 - 5 * q^59 + 6 * q^61 + 18 * q^62 + 2 * q^64 + 12 * q^65 - 7 * q^67 - 7 * q^68 + 4 * q^70 + 14 * q^71 - 28 * q^73 + 2 * q^74 - 2 * q^76 + 4 * q^77 + 6 * q^79 - 8 * q^80 + 12 * q^82 - 4 * q^83 + 28 * q^85 + 11 * q^86 + 4 * q^88 + 6 * q^89 - 6 * q^91 - q^92 + 6 * q^94 + 8 * q^95 + 8 * q^97 - 2 * q^98

Character values

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

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

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
\(5\)	2.00000	−	3.46410i	0.894427	−	1.54919i								0.0599153	−	0.998203i	\(-0.480917\pi\)
							0.834512	−	0.550990i	\(-0.185750\pi\)
\(7\)	0.500000	+	0.866025i	0.188982	+	0.327327i
							\(11\)	−2.00000	−	3.46410i	−0.603023	−	1.04447i	−0.992361	−	0.123371i	\(-0.960630\pi\)
0.389338	−	0.921095i	\(-0.372704\pi\)
\(13\)	−1.50000	+	2.59808i	−0.416025	+	0.720577i	−0.995535	−	0.0943882i	\(-0.969911\pi\)
							0.579510	+	0.814965i	\(0.303244\pi\)
\(17\)	7.00000			1.69775			0.848875	−	0.528594i	\(-0.177281\pi\)
							0.848875	+	0.528594i	\(0.177281\pi\)
\(19\)	2.00000			0.458831			0.229416	−	0.973329i	\(-0.426318\pi\)
							0.229416	+	0.973329i	\(0.426318\pi\)
\(23\)	−0.500000	+	0.866025i	−0.104257	+	0.180579i	−0.913434	−	0.406986i	\(-0.866580\pi\)
							0.809177	+	0.587565i	\(0.199913\pi\)
\(29\)	0.500000	+	0.866025i	0.0928477	+	0.160817i	0.908708	−	0.417432i	\(-0.137070\pi\)
							−0.815861	+	0.578249i	\(0.803736\pi\)
\(31\)	4.50000	−	7.79423i	0.808224	−	1.39988i	−0.105869	−	0.994380i	\(-0.533762\pi\)
							0.914093	−	0.405505i	\(-0.132904\pi\)
\(37\)	2.00000			0.328798			0.164399	−	0.986394i	\(-0.447432\pi\)
							0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)	3.00000	−	5.19615i	0.468521	−	0.811503i	−0.530831	−	0.847477i	\(-0.678120\pi\)
							0.999353	+	0.0359748i	\(0.0114536\pi\)
\(43\)	−5.50000	−	9.52628i	−0.838742	−	1.45274i	−0.890947	−	0.454108i	\(-0.849958\pi\)
							0.0522047	−	0.998636i	\(-0.483375\pi\)
\(47\)	−3.00000	−	5.19615i	−0.437595	−	0.757937i	0.559908	−	0.828554i	\(-0.310836\pi\)
							−0.997503	+	0.0706177i	\(0.977503\pi\)
\(53\)	9.00000			1.23625			0.618123	−	0.786082i	\(-0.287894\pi\)
							0.618123	+	0.786082i	\(0.287894\pi\)
\(59\)	−2.50000	+	4.33013i	−0.325472	+	0.563735i	−0.981608	−	0.190909i	\(-0.938857\pi\)
							0.656136	+	0.754643i	\(0.272190\pi\)
\(61\)	3.00000	+	5.19615i	0.384111	+	0.665299i	0.991645	−	0.128994i	\(-0.0411748\pi\)
							−0.607535	+	0.794293i	\(0.707841\pi\)
\(67\)	−3.50000	+	6.06218i	−0.427593	+	0.740613i	−0.996659	−	0.0816792i	\(-0.973972\pi\)
							0.569066	+	0.822292i	\(0.307305\pi\)
\(71\)	7.00000			0.830747			0.415374	−	0.909651i	\(-0.363651\pi\)
							0.415374	+	0.909651i	\(0.363651\pi\)
\(73\)	−14.0000			−1.63858			−0.819288	−	0.573382i	\(-0.805631\pi\)
							−0.819288	+	0.573382i	\(0.805631\pi\)
\(79\)	3.00000	+	5.19615i	0.337526	+	0.584613i	0.983967	−	0.178352i	\(-0.0570765\pi\)
							−0.646440	+	0.762964i	\(0.723743\pi\)
\(83\)	−2.00000	−	3.46410i	−0.219529	−	0.380235i	0.735135	−	0.677920i	\(-0.237119\pi\)
							−0.954664	+	0.297686i	\(0.903785\pi\)
\(89\)	3.00000			0.317999			0.159000	−	0.987279i	\(-0.449173\pi\)
							0.159000	+	0.987279i	\(0.449173\pi\)
\(97\)	4.00000	+	6.92820i	0.406138	+	0.703452i	0.994453	−	0.105180i	\(-0.0335417\pi\)
							−0.588315	+	0.808632i	\(0.700208\pi\)

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.f.p.757.1		2
3.2	odd	2		1134.2.f.a.757.1		2
9.2	odd	6		1134.2.f.a.379.1		2
9.4	even	3		378.2.a.a.1.1	✓	1
9.5	odd	6		378.2.a.h.1.1	yes	1
9.7	even	3	inner	1134.2.f.p.379.1		2
36.23	even	6		3024.2.a.bd.1.1		1
36.31	odd	6		3024.2.a.a.1.1		1
45.4	even	6		9450.2.a.dv.1.1		1
45.14	odd	6		9450.2.a.bc.1.1		1
63.13	odd	6		2646.2.a.o.1.1		1
63.41	even	6		2646.2.a.p.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.a.a.1.1	✓	1	9.4	even	3
378.2.a.h.1.1	yes	1	9.5	odd	6
1134.2.f.a.379.1		2	9.2	odd	6
1134.2.f.a.757.1		2	3.2	odd	2
1134.2.f.p.379.1		2	9.7	even	3	inner
1134.2.f.p.757.1		2	1.1	even	1	trivial
2646.2.a.o.1.1		1	63.13	odd	6
2646.2.a.p.1.1		1	63.41	even	6
3024.2.a.a.1.1		1	36.31	odd	6
3024.2.a.bd.1.1		1	36.23	even	6
9450.2.a.bc.1.1		1	45.14	odd	6
9450.2.a.dv.1.1		1	45.4	even	6