Embedded newform 3648.2.a.i.1.1

• Label: 3648.2.a.i.1.1
• Level: $3648$
• Weight: $2$
• Character: 3648.1
• Self dual: yes
• Analytic conductor: $29.129$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3648.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(29.1294266574\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	3648.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} -4.00000 q^{7} +1.00000 q^{9} +4.00000 q^{13} +6.00000 q^{17} -1.00000 q^{19} +4.00000 q^{21} -6.00000 q^{23} -5.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +2.00000 q^{31} +4.00000 q^{37} -4.00000 q^{39} +6.00000 q^{41} +4.00000 q^{43} +6.00000 q^{47} +9.00000 q^{49} -6.00000 q^{51} -6.00000 q^{53} +1.00000 q^{57} +12.0000 q^{59} -14.0000 q^{61} -4.00000 q^{63} -8.00000 q^{67} +6.00000 q^{69} +14.0000 q^{73} +5.00000 q^{75} -10.0000 q^{79} +1.00000 q^{81} +12.0000 q^{83} +6.00000 q^{87} -6.00000 q^{89} -16.0000 q^{91} -2.00000 q^{93} -10.0000 q^{97} +O(q^{100})\)

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
			\(3\)	−1.00000	−0.577350
\(5\)	0	0					−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(7\)	−4.00000	−1.51186	−0.755929	−	0.654654i	\(-0.772814\pi\)
			−0.755929	+	0.654654i	\(0.772814\pi\)
\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	4.00000	1.10940	0.554700	−	0.832050i	\(-0.312833\pi\)
			0.554700	+	0.832050i	\(0.312833\pi\)
\(17\)	6.00000	1.45521	0.727607	−	0.685994i	\(-0.240633\pi\)
			0.727607	+	0.685994i	\(0.240633\pi\)
\(19\)	−1.00000	−0.229416
			\(23\)	−6.00000	−1.25109	−0.625543	−	0.780189i	\(-0.715123\pi\)
−0.625543	+	0.780189i				\(0.715123\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	2.00000	0.359211	0.179605	−	0.983739i	\(-0.442518\pi\)
			0.179605	+	0.983739i	\(0.442518\pi\)
\(37\)	4.00000	0.657596	0.328798	−	0.944400i	\(-0.393356\pi\)
			0.328798	+	0.944400i	\(0.393356\pi\)
\(41\)	6.00000	0.937043	0.468521	−	0.883452i	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	6.00000	0.875190	0.437595	−	0.899172i	\(-0.355830\pi\)
			0.437595	+	0.899172i	\(0.355830\pi\)
\(53\)	−6.00000	−0.824163	−0.412082	−	0.911147i	\(-0.635198\pi\)
			−0.412082	+	0.911147i	\(0.635198\pi\)
\(59\)	12.0000	1.56227	0.781133	−	0.624364i	\(-0.214642\pi\)
			0.781133	+	0.624364i	\(0.214642\pi\)
\(61\)	−14.0000	−1.79252	−0.896258	−	0.443533i	\(-0.853725\pi\)
			−0.896258	+	0.443533i	\(0.853725\pi\)
\(67\)	−8.00000	−0.977356	−0.488678	−	0.872464i	\(-0.662521\pi\)
			−0.488678	+	0.872464i	\(0.662521\pi\)
\(71\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(73\)	14.0000	1.63858	0.819288	−	0.573382i	\(-0.194369\pi\)
			0.819288	+	0.573382i	\(0.194369\pi\)
\(79\)	−10.0000	−1.12509	−0.562544	−	0.826767i	\(-0.690177\pi\)
			−0.562544	+	0.826767i	\(0.690177\pi\)
\(83\)	12.0000	1.31717	0.658586	−	0.752506i	\(-0.271155\pi\)
			0.658586	+	0.752506i	\(0.271155\pi\)
\(89\)	−6.00000	−0.635999	−0.317999	−	0.948091i	\(-0.603011\pi\)
			−0.317999	+	0.948091i	\(0.603011\pi\)
\(97\)	−10.0000	−1.01535	−0.507673	−	0.861550i	\(-0.669494\pi\)
			−0.507673	+	0.861550i	\(0.669494\pi\)

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	3648.2.a.i.1.1		1
4.3	odd	2		3648.2.a.bc.1.1		1
8.3	odd	2		912.2.a.c.1.1		1
8.5	even	2		114.2.a.c.1.1	✓	1
24.5	odd	2		342.2.a.c.1.1		1
24.11	even	2		2736.2.a.o.1.1		1
40.13	odd	4		2850.2.d.p.799.1		2
40.29	even	2		2850.2.a.g.1.1		1
40.37	odd	4		2850.2.d.p.799.2		2
56.13	odd	2		5586.2.a.u.1.1		1
120.29	odd	2		8550.2.a.bj.1.1		1
152.37	odd	2		2166.2.a.a.1.1		1
456.341	even	2		6498.2.a.t.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.a.c.1.1	✓	1	8.5	even	2
342.2.a.c.1.1		1	24.5	odd	2
912.2.a.c.1.1		1	8.3	odd	2
2166.2.a.a.1.1		1	152.37	odd	2
2736.2.a.o.1.1		1	24.11	even	2
2850.2.a.g.1.1		1	40.29	even	2
2850.2.d.p.799.1		2	40.13	odd	4
2850.2.d.p.799.2		2	40.37	odd	4
3648.2.a.i.1.1		1	1.1	even	1	trivial
3648.2.a.bc.1.1		1	4.3	odd	2
5586.2.a.u.1.1		1	56.13	odd	2
6498.2.a.t.1.1		1	456.341	even	2
8550.2.a.bj.1.1		1	120.29	odd	2