Embedded newform 3960.2.d.a.3169.2

• Label: 3960.2.d.a.3169.2
• Level: $3960$
• Weight: $2$
• Character: 3960.3169
• Analytic conductor: $31.621$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3960.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(31.6207592004\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3169.2
Root			\(-1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	3960.3169
Dual form			3960.2.d.a.3169.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +1.00000 q^{11} +1.00000i q^{17} -1.00000 q^{19} +(3.00000 - 4.00000i) q^{25} -1.00000 q^{29} -1.00000 q^{31} +(1.00000 + 2.00000i) q^{35} -1.00000i q^{37} +6.00000i q^{43} +8.00000i q^{47} +6.00000 q^{49} -9.00000i q^{53} +(-2.00000 + 1.00000i) q^{55} +4.00000 q^{59} -7.00000 q^{61} +4.00000i q^{67} -5.00000 q^{71} +14.0000i q^{73} -1.00000i q^{77} -4.00000 q^{79} -16.0000i q^{83} +(-1.00000 - 2.00000i) q^{85} -7.00000 q^{89} +(2.00000 - 1.00000i) q^{95} +16.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 2 q^{11} - 2 q^{19} + 6 q^{25} - 2 q^{29} - 2 q^{31} + 2 q^{35} + 12 q^{49} - 4 q^{55} + 8 q^{59} - 14 q^{61} - 10 q^{71} - 8 q^{79} - 2 q^{85} - 14 q^{89} + 4 q^{95}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times\).

\(n\)	\(991\)	\(1981\)	\(2377\)	\(2521\)	\(3521\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(1\)

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
							\(3\)		0			0
\(5\)	−2.00000	+	1.00000i	−0.894427	+	0.447214i
							\(7\)		−	1.00000i		−	0.377964i	−0.981981	−	0.188982i	\(-0.939481\pi\)
0.981981	−	0.188982i	\(-0.0605189\pi\)
\(11\)	1.00000			0.301511
							\(13\)		0			0		1.00000			\(0\)
−1.00000			\(\pi\)
\(17\)			1.00000i			0.242536i	0.992620	+	0.121268i	\(0.0386960\pi\)
							−0.992620	+	0.121268i	\(0.961304\pi\)
\(19\)	−1.00000			−0.229416			−0.114708	−	0.993399i	\(-0.536593\pi\)
							−0.114708	+	0.993399i	\(0.536593\pi\)
\(23\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(29\)	−1.00000			−0.185695			−0.0928477	−	0.995680i	\(-0.529597\pi\)
							−0.0928477	+	0.995680i	\(0.529597\pi\)
\(31\)	−1.00000			−0.179605			−0.0898027	−	0.995960i	\(-0.528624\pi\)
							−0.0898027	+	0.995960i	\(0.528624\pi\)
\(37\)		−	1.00000i		−	0.164399i	−0.996616	−	0.0821995i	\(-0.973806\pi\)
							0.996616	−	0.0821995i	\(-0.0261945\pi\)
\(41\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(43\)			6.00000i			0.914991i	0.889212	+	0.457496i	\(0.151253\pi\)
							−0.889212	+	0.457496i	\(0.848747\pi\)
\(47\)			8.00000i			1.16692i	0.812142	+	0.583460i	\(0.198301\pi\)
							−0.812142	+	0.583460i	\(0.801699\pi\)
\(53\)		−	9.00000i		−	1.23625i	−0.786082	−	0.618123i	\(-0.787894\pi\)
							0.786082	−	0.618123i	\(-0.212106\pi\)
\(59\)	4.00000			0.520756			0.260378	−	0.965507i	\(-0.416153\pi\)
							0.260378	+	0.965507i	\(0.416153\pi\)
\(61\)	−7.00000			−0.896258			−0.448129	−	0.893969i	\(-0.647910\pi\)
							−0.448129	+	0.893969i	\(0.647910\pi\)
\(67\)			4.00000i			0.488678i	0.969690	+	0.244339i	\(0.0785709\pi\)
							−0.969690	+	0.244339i	\(0.921429\pi\)
\(71\)	−5.00000			−0.593391			−0.296695	−	0.954972i	\(-0.595885\pi\)
							−0.296695	+	0.954972i	\(0.595885\pi\)
\(73\)			14.0000i			1.63858i	0.573382	+	0.819288i	\(0.305631\pi\)
							−0.573382	+	0.819288i	\(0.694369\pi\)
\(79\)	−4.00000			−0.450035			−0.225018	−	0.974355i	\(-0.572244\pi\)
							−0.225018	+	0.974355i	\(0.572244\pi\)
\(83\)		−	16.0000i		−	1.75623i	−0.478451	−	0.878114i	\(-0.658802\pi\)
							0.478451	−	0.878114i	\(-0.341198\pi\)
\(89\)	−7.00000			−0.741999			−0.370999	−	0.928633i	\(-0.620985\pi\)
							−0.370999	+	0.928633i	\(0.620985\pi\)
\(97\)			16.0000i			1.62455i	0.583272	+	0.812277i	\(0.301772\pi\)
							−0.583272	+	0.812277i	\(0.698228\pi\)

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	3960.2.d.a.3169.2		2
3.2	odd	2		440.2.b.c.89.1	✓	2
5.4	even	2	inner	3960.2.d.a.3169.1		2
12.11	even	2		880.2.b.g.529.2		2
15.2	even	4		2200.2.a.d.1.1		1
15.8	even	4		2200.2.a.h.1.1		1
15.14	odd	2		440.2.b.c.89.2	yes	2
60.23	odd	4		4400.2.a.j.1.1		1
60.47	odd	4		4400.2.a.u.1.1		1
60.59	even	2		880.2.b.g.529.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.b.c.89.1	✓	2	3.2	odd	2
440.2.b.c.89.2	yes	2	15.14	odd	2
880.2.b.g.529.1		2	60.59	even	2
880.2.b.g.529.2		2	12.11	even	2
2200.2.a.d.1.1		1	15.2	even	4
2200.2.a.h.1.1		1	15.8	even	4
3960.2.d.a.3169.1		2	5.4	even	2	inner
3960.2.d.a.3169.2		2	1.1	even	1	trivial
4400.2.a.j.1.1		1	60.23	odd	4
4400.2.a.u.1.1		1	60.47	odd	4