Embedded newform 440.2.b.b.89.1

• Label: 440.2.b.b.89.1
• Level: $440$
• Weight: $2$
• Character: 440.89
• Analytic conductor: $3.513$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.51341768894\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.1
Root			\(-1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	440.89
Dual form			440.2.b.b.89.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +(1.00000 + 2.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} +6.00000i q^{13} +(4.00000 - 2.00000i) q^{15} +2.00000i q^{17} -4.00000 q^{19} +8.00000 q^{21} -6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000i q^{27} +2.00000 q^{29} +8.00000 q^{31} -2.00000i q^{33} +(-8.00000 + 4.00000i) q^{35} -8.00000i q^{37} +12.0000 q^{39} +6.00000 q^{41} -12.0000i q^{43} +(-1.00000 - 2.00000i) q^{45} +10.0000i q^{47} -9.00000 q^{49} +4.00000 q^{51} +(1.00000 + 2.00000i) q^{55} +8.00000i q^{57} +4.00000 q^{59} -10.0000 q^{61} -4.00000i q^{63} +(-12.0000 + 6.00000i) q^{65} +2.00000i q^{67} -12.0000 q^{69} -8.00000 q^{71} -2.00000i q^{73} +(8.00000 + 6.00000i) q^{75} +4.00000i q^{77} -4.00000 q^{79} -11.0000 q^{81} +4.00000i q^{83} +(-4.00000 + 2.00000i) q^{85} -4.00000i q^{87} +14.0000 q^{89} -24.0000 q^{91} -16.0000i q^{93} +(-4.00000 - 8.00000i) q^{95} -4.00000i q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^5 - 2 * q^9 + 2 * q^11 + 8 * q^15 - 8 * q^19 + 16 * q^21 - 6 * q^25 + 4 * q^29 + 16 * q^31 - 16 * q^35 + 24 * q^39 + 12 * q^41 - 2 * q^45 - 18 * q^49 + 8 * q^51 + 2 * q^55 + 8 * q^59 - 20 * q^61 - 24 * q^65 - 24 * q^69 - 16 * q^71 + 16 * q^75 - 8 * q^79 - 22 * q^81 - 8 * q^85 + 28 * q^89 - 48 * q^91 - 8 * q^95 - 2 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).

\(n\)	\(111\)	\(177\)	\(221\)	\(321\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		0			0
							\(3\)		−	2.00000i		−	1.15470i	−0.816497	−	0.577350i	\(-0.804087\pi\)
0.816497	−	0.577350i	\(-0.195913\pi\)
\(5\)	1.00000	+	2.00000i	0.447214	+	0.894427i
							\(7\)			4.00000i			1.51186i	0.654654	+	0.755929i	\(0.272814\pi\)
−0.654654	+	0.755929i	\(0.727186\pi\)
\(11\)	1.00000			0.301511
							\(13\)			6.00000i			1.66410i	0.554700	+	0.832050i	\(0.312833\pi\)
−0.554700	+	0.832050i	\(0.687167\pi\)
\(17\)			2.00000i			0.485071i	0.970143	+	0.242536i	\(0.0779791\pi\)
							−0.970143	+	0.242536i	\(0.922021\pi\)
\(19\)	−4.00000			−0.917663			−0.458831	−	0.888523i	\(-0.651732\pi\)
							−0.458831	+	0.888523i	\(0.651732\pi\)
\(23\)		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	\(-0.784877\pi\)
							0.780189	−	0.625543i	\(-0.215123\pi\)
\(29\)	2.00000			0.371391			0.185695	−	0.982607i	\(-0.440546\pi\)
							0.185695	+	0.982607i	\(0.440546\pi\)
\(31\)	8.00000			1.43684			0.718421	−	0.695608i	\(-0.244865\pi\)
							0.718421	+	0.695608i	\(0.244865\pi\)
\(37\)		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	\(-0.771573\pi\)
							0.753371	−	0.657596i	\(-0.228427\pi\)
\(41\)	6.00000			0.937043			0.468521	−	0.883452i	\(-0.344787\pi\)
							0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)		−	12.0000i		−	1.82998i	−0.403473	−	0.914991i	\(-0.632197\pi\)
							0.403473	−	0.914991i	\(-0.367803\pi\)
\(47\)			10.0000i			1.45865i	0.684167	+	0.729325i	\(0.260166\pi\)
							−0.684167	+	0.729325i	\(0.739834\pi\)
\(53\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(59\)	4.00000			0.520756			0.260378	−	0.965507i	\(-0.416153\pi\)
							0.260378	+	0.965507i	\(0.416153\pi\)
\(61\)	−10.0000			−1.28037			−0.640184	−	0.768221i	\(-0.721142\pi\)
							−0.640184	+	0.768221i	\(0.721142\pi\)
\(67\)			2.00000i			0.244339i	0.992509	+	0.122169i	\(0.0389851\pi\)
							−0.992509	+	0.122169i	\(0.961015\pi\)
\(71\)	−8.00000			−0.949425			−0.474713	−	0.880141i	\(-0.657448\pi\)
							−0.474713	+	0.880141i	\(0.657448\pi\)
\(73\)		−	2.00000i		−	0.234082i	−0.993127	−	0.117041i	\(-0.962659\pi\)
							0.993127	−	0.117041i	\(-0.0373409\pi\)
\(79\)	−4.00000			−0.450035			−0.225018	−	0.974355i	\(-0.572244\pi\)
							−0.225018	+	0.974355i	\(0.572244\pi\)
\(83\)			4.00000i			0.439057i	0.975606	+	0.219529i	\(0.0704519\pi\)
							−0.975606	+	0.219529i	\(0.929548\pi\)
\(89\)	14.0000			1.48400			0.741999	−	0.670402i	\(-0.233878\pi\)
							0.741999	+	0.670402i	\(0.233878\pi\)
\(97\)		−	4.00000i		−	0.406138i	−0.979164	−	0.203069i	\(-0.934908\pi\)
							0.979164	−	0.203069i	\(-0.0650917\pi\)

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	440.2.b.b.89.1	✓	2
3.2	odd	2		3960.2.d.b.3169.1		2
4.3	odd	2		880.2.b.d.529.2		2
5.2	odd	4		2200.2.a.b.1.1		1
5.3	odd	4		2200.2.a.j.1.1		1
5.4	even	2	inner	440.2.b.b.89.2	yes	2
15.14	odd	2		3960.2.d.b.3169.2		2
20.3	even	4		4400.2.a.c.1.1		1
20.7	even	4		4400.2.a.bb.1.1		1
20.19	odd	2		880.2.b.d.529.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.b.b.89.1	✓	2	1.1	even	1	trivial
440.2.b.b.89.2	yes	2	5.4	even	2	inner
880.2.b.d.529.1		2	20.19	odd	2
880.2.b.d.529.2		2	4.3	odd	2
2200.2.a.b.1.1		1	5.2	odd	4
2200.2.a.j.1.1		1	5.3	odd	4
3960.2.d.b.3169.1		2	3.2	odd	2
3960.2.d.b.3169.2		2	15.14	odd	2
4400.2.a.c.1.1		1	20.3	even	4
4400.2.a.bb.1.1		1	20.7	even	4