Embedded newform 2790.2.d.h.559.1

• Label: 2790.2.d.h.559.1
• Level: $2790$
• Weight: $2$
• Character: 2790.559
• Analytic conductor: $22.278$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.2782621639\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.1
Root			\(1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	2790.559
Dual form			2790.2.d.h.559.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} +2.00000i q^{7} +1.00000i q^{8} +(-1.00000 - 2.00000i) q^{10} +4.00000 q^{11} +2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} -4.00000 q^{19} +(-2.00000 + 1.00000i) q^{20} -4.00000i q^{22} +(3.00000 - 4.00000i) q^{25} +2.00000 q^{26} -2.00000i q^{28} +2.00000 q^{29} +1.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +(2.00000 + 4.00000i) q^{35} -2.00000i q^{37} +4.00000i q^{38} +(1.00000 + 2.00000i) q^{40} +4.00000i q^{43} -4.00000 q^{44} +3.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} -2.00000i q^{52} +2.00000i q^{53} +(8.00000 - 4.00000i) q^{55} -2.00000 q^{56} -2.00000i q^{58} +14.0000 q^{59} +10.0000 q^{61} -1.00000i q^{62} -1.00000 q^{64} +(2.00000 + 4.00000i) q^{65} -2.00000i q^{67} +6.00000i q^{68} +(4.00000 - 2.00000i) q^{70} +6.00000 q^{71} -6.00000i q^{73} -2.00000 q^{74} +4.00000 q^{76} +8.00000i q^{77} +4.00000 q^{79} +(2.00000 - 1.00000i) q^{80} +12.0000i q^{83} +(-6.00000 - 12.0000i) q^{85} +4.00000 q^{86} +4.00000i q^{88} +6.00000 q^{89} -4.00000 q^{91} +(-8.00000 + 4.00000i) q^{95} -12.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 2 * q^4 + 4 * q^5 - 2 * q^10 + 8 * q^11 + 4 * q^14 + 2 * q^16 - 8 * q^19 - 4 * q^20 + 6 * q^25 + 4 * q^26 + 4 * q^29 + 2 * q^31 - 12 * q^34 + 4 * q^35 + 2 * q^40 - 8 * q^44 + 6 * q^49 - 8 * q^50 + 16 * q^55 - 4 * q^56 + 28 * q^59 + 20 * q^61 - 2 * q^64 + 4 * q^65 + 8 * q^70 + 12 * q^71 - 4 * q^74 + 8 * q^76 + 8 * q^79 + 4 * q^80 - 12 * q^85 + 8 * q^86 + 12 * q^89 - 8 * q^91 - 16 * q^95

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2790\mathbb{Z}\right)^\times\).

\(n\)	\(1117\)	\(1801\)	\(2171\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.00000i		−	0.707107i
							\(3\)		0			0
\(5\)	2.00000	−	1.00000i	0.894427	−	0.447214i
							\(7\)			2.00000i			0.755929i	0.925820	+	0.377964i	\(0.123376\pi\)
−0.925820	+	0.377964i	\(0.876624\pi\)
\(11\)	4.00000			1.20605			0.603023	−	0.797724i	\(-0.293963\pi\)
							0.603023	+	0.797724i	\(0.293963\pi\)
\(13\)			2.00000i			0.554700i	0.960769	+	0.277350i	\(0.0894562\pi\)
							−0.960769	+	0.277350i	\(0.910544\pi\)
\(17\)		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	\(-0.740633\pi\)
							0.685994	−	0.727607i	\(-0.259367\pi\)
\(19\)	−4.00000			−0.917663			−0.458831	−	0.888523i	\(-0.651732\pi\)
							−0.458831	+	0.888523i	\(0.651732\pi\)
\(23\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(29\)	2.00000			0.371391			0.185695	−	0.982607i	\(-0.440546\pi\)
							0.185695	+	0.982607i	\(0.440546\pi\)
\(31\)	1.00000			0.179605
							\(37\)		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	\(-0.947432\pi\)
0.986394	−	0.164399i	\(-0.0525685\pi\)
\(41\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(43\)			4.00000i			0.609994i	0.952353	+	0.304997i	\(0.0986555\pi\)
							−0.952353	+	0.304997i	\(0.901344\pi\)
\(47\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(53\)			2.00000i			0.274721i	0.990521	+	0.137361i	\(0.0438619\pi\)
							−0.990521	+	0.137361i	\(0.956138\pi\)
\(59\)	14.0000			1.82264			0.911322	−	0.411693i	\(-0.135063\pi\)
							0.911322	+	0.411693i	\(0.135063\pi\)
\(61\)	10.0000			1.28037			0.640184	−	0.768221i	\(-0.278858\pi\)
							0.640184	+	0.768221i	\(0.278858\pi\)
\(67\)		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	\(-0.961015\pi\)
							0.992509	−	0.122169i	\(-0.0389851\pi\)
\(71\)	6.00000			0.712069			0.356034	−	0.934473i	\(-0.384129\pi\)
							0.356034	+	0.934473i	\(0.384129\pi\)
\(73\)		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	\(-0.885800\pi\)
							0.936329	−	0.351123i	\(-0.114200\pi\)
\(79\)	4.00000			0.450035			0.225018	−	0.974355i	\(-0.427756\pi\)
							0.225018	+	0.974355i	\(0.427756\pi\)
\(83\)			12.0000i			1.31717i	0.752506	+	0.658586i	\(0.228845\pi\)
							−0.752506	+	0.658586i	\(0.771155\pi\)
\(89\)	6.00000			0.635999			0.317999	−	0.948091i	\(-0.396989\pi\)
							0.317999	+	0.948091i	\(0.396989\pi\)
\(97\)		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	\(-0.791488\pi\)
							0.793011	−	0.609208i	\(-0.208512\pi\)

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	2790.2.d.h.559.1	yes	2
3.2	odd	2		2790.2.d.a.559.2	yes	2
5.4	even	2	inner	2790.2.d.h.559.2	yes	2
15.14	odd	2		2790.2.d.a.559.1	✓	2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2790.2.d.a.559.1	✓	2	15.14	odd	2
2790.2.d.a.559.2	yes	2	3.2	odd	2
2790.2.d.h.559.1	yes	2	1.1	even	1	trivial
2790.2.d.h.559.2	yes	2	5.4	even	2	inner