Embedded newform 3312.2.a.n.1.1

• Label: 3312.2.a.n.1.1
• Level: $3312$
• Weight: $2$
• Character: 3312.1
• Self dual: yes
• Analytic conductor: $26.446$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3312.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(26.4464531494\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	3312.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+2.00000 q^{5} +2.00000 q^{7} -6.00000 q^{11} -2.00000 q^{13} -1.00000 q^{23} -1.00000 q^{25} -6.00000 q^{29} -8.00000 q^{31} +4.00000 q^{35} -10.0000 q^{41} +12.0000 q^{43} -8.00000 q^{47} -3.00000 q^{49} -2.00000 q^{53} -12.0000 q^{55} -12.0000 q^{59} +4.00000 q^{61} -4.00000 q^{65} +12.0000 q^{67} -10.0000 q^{73} -12.0000 q^{77} +6.00000 q^{79} +14.0000 q^{83} -4.00000 q^{91} -6.00000 q^{97} +O(q^{100})\)

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
			\(3\)	0	0
\(5\)	2.00000	0.894427				0.447214	−	0.894427i	\(-0.352416\pi\)
			0.447214	+	0.894427i	\(0.352416\pi\)
\(7\)	2.00000	0.755929	0.377964	−	0.925820i	\(-0.376624\pi\)
			0.377964	+	0.925820i	\(0.376624\pi\)
\(11\)	−6.00000	−1.80907	−0.904534	−	0.426401i	\(-0.859781\pi\)
			−0.904534	+	0.426401i	\(0.859781\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	−1.00000	−0.208514
			\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
−0.557086	+	0.830455i				\(0.688081\pi\)
\(31\)	−8.00000	−1.43684	−0.718421	−	0.695608i	\(-0.755135\pi\)
			−0.718421	+	0.695608i	\(0.755135\pi\)
\(37\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(41\)	−10.0000	−1.56174	−0.780869	−	0.624695i	\(-0.785223\pi\)
			−0.780869	+	0.624695i	\(0.785223\pi\)
\(43\)	12.0000	1.82998	0.914991	−	0.403473i	\(-0.132197\pi\)
			0.914991	+	0.403473i	\(0.132197\pi\)
\(47\)	−8.00000	−1.16692	−0.583460	−	0.812142i	\(-0.698301\pi\)
			−0.583460	+	0.812142i	\(0.698301\pi\)
\(53\)	−2.00000	−0.274721	−0.137361	−	0.990521i	\(-0.543862\pi\)
			−0.137361	+	0.990521i	\(0.543862\pi\)
\(59\)	−12.0000	−1.56227	−0.781133	−	0.624364i	\(-0.785358\pi\)
			−0.781133	+	0.624364i	\(0.785358\pi\)
\(61\)	4.00000	0.512148	0.256074	−	0.966657i	\(-0.417571\pi\)
			0.256074	+	0.966657i	\(0.417571\pi\)
\(67\)	12.0000	1.46603	0.733017	−	0.680211i	\(-0.238112\pi\)
			0.733017	+	0.680211i	\(0.238112\pi\)
\(71\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(73\)	−10.0000	−1.17041	−0.585206	−	0.810885i	\(-0.698986\pi\)
			−0.585206	+	0.810885i	\(0.698986\pi\)
\(79\)	6.00000	0.675053	0.337526	−	0.941316i	\(-0.390410\pi\)
			0.337526	+	0.941316i	\(0.390410\pi\)
\(83\)	14.0000	1.53670	0.768350	−	0.640030i	\(-0.221078\pi\)
			0.768350	+	0.640030i	\(0.221078\pi\)
\(89\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(97\)	−6.00000	−0.609208	−0.304604	−	0.952479i	\(-0.598524\pi\)
			−0.304604	+	0.952479i	\(0.598524\pi\)

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	3312.2.a.n.1.1		1
3.2	odd	2		1104.2.a.e.1.1		1
4.3	odd	2		414.2.a.d.1.1		1
12.11	even	2		138.2.a.a.1.1	✓	1
24.5	odd	2		4416.2.a.m.1.1		1
24.11	even	2		4416.2.a.z.1.1		1
60.23	odd	4		3450.2.d.j.2899.2		2
60.47	odd	4		3450.2.d.j.2899.1		2
60.59	even	2		3450.2.a.y.1.1		1
84.83	odd	2		6762.2.a.q.1.1		1
92.91	even	2		9522.2.a.i.1.1		1
276.275	odd	2		3174.2.a.b.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.a.a.1.1	✓	1	12.11	even	2
414.2.a.d.1.1		1	4.3	odd	2
1104.2.a.e.1.1		1	3.2	odd	2
3174.2.a.b.1.1		1	276.275	odd	2
3312.2.a.n.1.1		1	1.1	even	1	trivial
3450.2.a.y.1.1		1	60.59	even	2
3450.2.d.j.2899.1		2	60.47	odd	4
3450.2.d.j.2899.2		2	60.23	odd	4
4416.2.a.m.1.1		1	24.5	odd	2
4416.2.a.z.1.1		1	24.11	even	2
6762.2.a.q.1.1		1	84.83	odd	2
9522.2.a.i.1.1		1	92.91	even	2