Embedded newform 3450.2.a.c.1.1

• Label: 3450.2.a.c.1.1
• Level: $3450$
• Weight: $2$
• Character: 3450.1
• Self dual: yes
• Analytic conductor: $27.548$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(27.5483886973\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 690)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	3450.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{12} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{27} -8.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{41} +2.00000 q^{43} +2.00000 q^{44} +1.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +6.00000 q^{51} +2.00000 q^{53} +1.00000 q^{54} -4.00000 q^{57} -2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +2.00000 q^{66} +2.00000 q^{67} -6.00000 q^{68} +1.00000 q^{69} -10.0000 q^{71} -1.00000 q^{72} +10.0000 q^{73} -6.00000 q^{74} +4.00000 q^{76} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} -2.00000 q^{86} -2.00000 q^{88} +4.00000 q^{89} -1.00000 q^{92} +8.00000 q^{93} +4.00000 q^{94} +1.00000 q^{96} -16.0000 q^{97} +7.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−0.707107
			\(3\)	−1.00000	−0.577350
\(5\)	0	0
			\(7\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(11\)	2.00000	0.603023	0.301511	−	0.953463i	\(-0.402509\pi\)
			0.301511	+	0.953463i	\(0.402509\pi\)
\(13\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(17\)	−6.00000	−1.45521	−0.727607	−	0.685994i	\(-0.759367\pi\)
			−0.727607	+	0.685994i	\(0.759367\pi\)
\(19\)	4.00000	0.917663	0.458831	−	0.888523i	\(-0.348268\pi\)
			0.458831	+	0.888523i	\(0.348268\pi\)
\(23\)	−1.00000	−0.208514
			\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(31\)	−8.00000	−1.43684	−0.718421	−	0.695608i	\(-0.755135\pi\)
			−0.718421	+	0.695608i	\(0.755135\pi\)
\(37\)	6.00000	0.986394	0.493197	−	0.869918i	\(-0.335828\pi\)
			0.493197	+	0.869918i	\(0.335828\pi\)
\(41\)	−2.00000	−0.312348	−0.156174	−	0.987730i	\(-0.549916\pi\)
			−0.156174	+	0.987730i	\(0.549916\pi\)
\(43\)	2.00000	0.304997	0.152499	−	0.988304i	\(-0.451268\pi\)
			0.152499	+	0.988304i	\(0.451268\pi\)
\(47\)	−4.00000	−0.583460	−0.291730	−	0.956501i	\(-0.594231\pi\)
			−0.291730	+	0.956501i	\(0.594231\pi\)
\(53\)	2.00000	0.274721	0.137361	−	0.990521i	\(-0.456138\pi\)
			0.137361	+	0.990521i	\(0.456138\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	−2.00000	−0.256074	−0.128037	−	0.991769i	\(-0.540868\pi\)
			−0.128037	+	0.991769i	\(0.540868\pi\)
\(67\)	2.00000	0.244339	0.122169	−	0.992509i	\(-0.461015\pi\)
			0.122169	+	0.992509i	\(0.461015\pi\)
\(71\)	−10.0000	−1.18678	−0.593391	−	0.804914i	\(-0.702211\pi\)
			−0.593391	+	0.804914i	\(0.702211\pi\)
\(73\)	10.0000	1.17041	0.585206	−	0.810885i	\(-0.301014\pi\)
			0.585206	+	0.810885i	\(0.301014\pi\)
\(79\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(83\)	4.00000	0.439057	0.219529	−	0.975606i	\(-0.429548\pi\)
			0.219529	+	0.975606i	\(0.429548\pi\)
\(89\)	4.00000	0.423999	0.212000	−	0.977270i	\(-0.432002\pi\)
			0.212000	+	0.977270i	\(0.432002\pi\)
\(97\)	−16.0000	−1.62455	−0.812277	−	0.583272i	\(-0.801772\pi\)
			−0.812277	+	0.583272i	\(0.801772\pi\)

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	3450.2.a.c.1.1		1
5.2	odd	4		3450.2.d.r.2899.1		2
5.3	odd	4		3450.2.d.r.2899.2		2
5.4	even	2		690.2.a.i.1.1	✓	1
15.14	odd	2		2070.2.a.g.1.1		1
20.19	odd	2		5520.2.a.d.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.i.1.1	✓	1	5.4	even	2
2070.2.a.g.1.1		1	15.14	odd	2
3450.2.a.c.1.1		1	1.1	even	1	trivial
3450.2.d.r.2899.1		2	5.2	odd	4
3450.2.d.r.2899.2		2	5.3	odd	4
5520.2.a.d.1.1		1	20.19	odd	2