Embedded newform 1134.2.e.n.865.1

• Label: 1134.2.e.n.865.1
• Level: $1134$
• Weight: $2$
• Character: 1134.865
• 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.e (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:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			865.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	1134.865
Dual form			1134.2.e.n.919.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +(2.50000 + 0.866025i) q^{7} +1.00000 q^{8} +(2.00000 + 3.46410i) q^{13} +(2.50000 + 0.866025i) q^{14} +1.00000 q^{16} +(-3.00000 + 5.19615i) q^{17} +(-1.00000 - 1.73205i) q^{19} +(1.50000 - 2.59808i) q^{23} +(2.50000 + 4.33013i) q^{25} +(2.00000 + 3.46410i) q^{26} +(2.50000 + 0.866025i) q^{28} +(3.00000 - 5.19615i) q^{29} +5.00000 q^{31} +1.00000 q^{32} +(-3.00000 + 5.19615i) q^{34} +(-4.00000 - 6.92820i) q^{37} +(-1.00000 - 1.73205i) q^{38} +(-1.50000 - 2.59808i) q^{41} +(-1.00000 + 1.73205i) q^{43} +(1.50000 - 2.59808i) q^{46} -3.00000 q^{47} +(5.50000 + 4.33013i) q^{49} +(2.50000 + 4.33013i) q^{50} +(2.00000 + 3.46410i) q^{52} +(-3.00000 + 5.19615i) q^{53} +(2.50000 + 0.866025i) q^{56} +(3.00000 - 5.19615i) q^{58} +12.0000 q^{59} +8.00000 q^{61} +5.00000 q^{62} +1.00000 q^{64} +8.00000 q^{67} +(-3.00000 + 5.19615i) q^{68} -15.0000 q^{71} +(-5.50000 + 9.52628i) q^{73} +(-4.00000 - 6.92820i) q^{74} +(-1.00000 - 1.73205i) q^{76} -1.00000 q^{79} +(-1.50000 - 2.59808i) q^{82} +(-1.00000 + 1.73205i) q^{86} +(-4.50000 - 7.79423i) q^{89} +(2.00000 + 10.3923i) q^{91} +(1.50000 - 2.59808i) q^{92} -3.00000 q^{94} +(-1.00000 + 1.73205i) q^{97} +(5.50000 + 4.33013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^2 + 2 * q^4 + 5 * q^7 + 2 * q^8 + 4 * q^13 + 5 * q^14 + 2 * q^16 - 6 * q^17 - 2 * q^19 + 3 * q^23 + 5 * q^25 + 4 * q^26 + 5 * q^28 + 6 * q^29 + 10 * q^31 + 2 * q^32 - 6 * q^34 - 8 * q^37 - 2 * q^38 - 3 * q^41 - 2 * q^43 + 3 * q^46 - 6 * q^47 + 11 * q^49 + 5 * q^50 + 4 * q^52 - 6 * q^53 + 5 * q^56 + 6 * q^58 + 24 * q^59 + 16 * q^61 + 10 * q^62 + 2 * q^64 + 16 * q^67 - 6 * q^68 - 30 * q^71 - 11 * q^73 - 8 * q^74 - 2 * q^76 - 2 * q^79 - 3 * q^82 - 2 * q^86 - 9 * q^89 + 4 * q^91 + 3 * q^92 - 6 * q^94 - 2 * q^97 + 11 * q^98

Character values

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

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

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000			0.707107
							\(3\)		0			0
\(5\)		0			0									−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(7\)	2.50000	+	0.866025i	0.944911	+	0.327327i
							\(11\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
−0.866025	+	0.500000i	\(0.833333\pi\)
\(13\)	2.00000	+	3.46410i	0.554700	+	0.960769i	0.997927	+	0.0643593i	\(0.0205004\pi\)
							−0.443227	+	0.896410i	\(0.646166\pi\)
\(17\)	−3.00000	+	5.19615i	−0.727607	+	1.26025i	0.230285	+	0.973123i	\(0.426034\pi\)
							−0.957892	+	0.287129i	\(0.907299\pi\)
\(19\)	−1.00000	−	1.73205i	−0.229416	−	0.397360i	0.728219	−	0.685344i	\(-0.240348\pi\)
							−0.957635	+	0.287984i	\(0.907015\pi\)
\(23\)	1.50000	−	2.59808i	0.312772	−	0.541736i	−0.666190	−	0.745782i	\(-0.732076\pi\)
							0.978961	+	0.204046i	\(0.0654092\pi\)
\(29\)	3.00000	−	5.19615i	0.557086	−	0.964901i	−0.440652	−	0.897678i	\(-0.645253\pi\)
							0.997738	−	0.0672232i	\(-0.0214140\pi\)
\(31\)	5.00000			0.898027			0.449013	−	0.893525i	\(-0.351776\pi\)
							0.449013	+	0.893525i	\(0.351776\pi\)
\(37\)	−4.00000	−	6.92820i	−0.657596	−	1.13899i	−0.981236	−	0.192809i	\(-0.938240\pi\)
							0.323640	−	0.946180i	\(-0.395093\pi\)
\(41\)	−1.50000	−	2.59808i	−0.234261	−	0.405751i	0.724797	−	0.688963i	\(-0.241934\pi\)
							−0.959058	+	0.283211i	\(0.908600\pi\)
\(43\)	−1.00000	+	1.73205i	−0.152499	+	0.264135i	−0.932145	−	0.362084i	\(-0.882065\pi\)
							0.779647	+	0.626219i	\(0.215399\pi\)
\(47\)	−3.00000			−0.437595			−0.218797	−	0.975770i	\(-0.570213\pi\)
							−0.218797	+	0.975770i	\(0.570213\pi\)
\(53\)	−3.00000	+	5.19615i	−0.412082	+	0.713746i	−0.995117	−	0.0987002i	\(-0.968532\pi\)
							0.583036	+	0.812447i	\(0.301865\pi\)
\(59\)	12.0000			1.56227			0.781133	−	0.624364i	\(-0.214642\pi\)
							0.781133	+	0.624364i	\(0.214642\pi\)
\(61\)	8.00000			1.02430			0.512148	−	0.858898i	\(-0.328850\pi\)
							0.512148	+	0.858898i	\(0.328850\pi\)
\(67\)	8.00000			0.977356			0.488678	−	0.872464i	\(-0.337479\pi\)
							0.488678	+	0.872464i	\(0.337479\pi\)
\(71\)	−15.0000			−1.78017			−0.890086	−	0.455792i	\(-0.849356\pi\)
							−0.890086	+	0.455792i	\(0.849356\pi\)
\(73\)	−5.50000	+	9.52628i	−0.643726	+	1.11497i	0.340868	+	0.940111i	\(0.389279\pi\)
							−0.984594	+	0.174855i	\(0.944054\pi\)
\(79\)	−1.00000			−0.112509			−0.0562544	−	0.998416i	\(-0.517916\pi\)
							−0.0562544	+	0.998416i	\(0.517916\pi\)
\(83\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(89\)	−4.50000	−	7.79423i	−0.476999	−	0.826187i	0.522654	−	0.852545i	\(-0.324942\pi\)
							−0.999653	+	0.0263586i	\(0.991609\pi\)
\(97\)	−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	1134.2.e.n.865.1		2
3.2	odd	2		1134.2.e.d.865.1		2
7.2	even	3		1134.2.h.c.541.1		2
9.2	odd	6		1134.2.g.g.487.1	yes	2
9.4	even	3		1134.2.h.c.109.1		2
9.5	odd	6		1134.2.h.m.109.1		2
9.7	even	3		1134.2.g.b.487.1	yes	2
21.2	odd	6		1134.2.h.m.541.1		2
63.2	odd	6		1134.2.g.g.163.1	yes	2
63.11	odd	6		7938.2.a.g.1.1		1
63.16	even	3		1134.2.g.b.163.1	✓	2
63.23	odd	6		1134.2.e.d.919.1		2
63.25	even	3		7938.2.a.y.1.1		1
63.38	even	6		7938.2.a.h.1.1		1
63.52	odd	6		7938.2.a.z.1.1		1
63.58	even	3	inner	1134.2.e.n.919.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1134.2.e.d.865.1		2	3.2	odd	2
1134.2.e.d.919.1		2	63.23	odd	6
1134.2.e.n.865.1		2	1.1	even	1	trivial
1134.2.e.n.919.1		2	63.58	even	3	inner
1134.2.g.b.163.1	✓	2	63.16	even	3
1134.2.g.b.487.1	yes	2	9.7	even	3
1134.2.g.g.163.1	yes	2	63.2	odd	6
1134.2.g.g.487.1	yes	2	9.2	odd	6
1134.2.h.c.109.1		2	9.4	even	3
1134.2.h.c.541.1		2	7.2	even	3
1134.2.h.m.109.1		2	9.5	odd	6
1134.2.h.m.541.1		2	21.2	odd	6
7938.2.a.g.1.1		1	63.11	odd	6
7938.2.a.h.1.1		1	63.38	even	6
7938.2.a.y.1.1		1	63.25	even	3
7938.2.a.z.1.1		1	63.52	odd	6