Embedded newform 3.67.b.a.2.1

• Label: 3.67.b.a.2.1
• Level: $3$
• Weight: $67$
• Character: 3.2
• Self dual: yes
• Analytic conductor: $82.760$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3 \)
Weight:	\( k \)	\(=\)	\( 67 \)
Character orbit:	\([\chi]\)	\(=\)	3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(82.7604085389\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2.1
Character	\(\chi\)	\(=\)	3.2

$q$-expansion

\(f(q)\)

\(=\)

\(q-5.55906e15 q^{3} +7.37870e19 q^{4} -1.54595e28 q^{7} +3.09032e31 q^{9} -4.10186e35 q^{12} -1.09938e37 q^{13} +5.44452e39 q^{16} -8.32798e41 q^{19} +8.59401e43 q^{21} +1.35525e46 q^{25} -1.71793e47 q^{27} -1.14071e48 q^{28} -2.08297e49 q^{31} +2.28025e51 q^{36} -7.41794e51 q^{37} +6.11153e52 q^{39} -1.24524e54 q^{43} -3.02664e55 q^{48} +1.79227e56 q^{49} -8.11200e56 q^{52} +4.62958e57 q^{57} +7.67187e58 q^{61} -4.77746e59 q^{63} +4.01735e59 q^{64} +2.36444e59 q^{67} +5.64366e61 q^{73} -7.53393e61 q^{75} -6.14497e61 q^{76} +6.24528e62 q^{79} +9.55005e62 q^{81} +6.34126e63 q^{84} +1.69958e65 q^{91} +1.15793e65 q^{93} +2.44251e65 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(-1\)

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	1.00000			\(0\)
			−1.00000			\(\pi\)
\(3\)	−5.55906e15	−1.00000
			\(5\)	0	0	1.00000			\(0\)
−1.00000						\(\pi\)
\(7\)	−1.54595e28	−1.99967	−0.999837	−	0.0180798i	\(-0.994245\pi\)
			−0.999837	+	0.0180798i	\(0.994245\pi\)
\(11\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(13\)	−1.09938e37	−1.90993	−0.954966	−	0.296716i	\(-0.904108\pi\)
			−0.954966	+	0.296716i	\(0.904108\pi\)
\(17\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(19\)	−8.32798e41	−0.526831	−0.263415	−	0.964683i	\(-0.584849\pi\)
			−0.263415	+	0.964683i	\(0.584849\pi\)
\(23\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(29\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(31\)	−2.08297e49	−1.26983	−0.634917	−	0.772581i	\(-0.718966\pi\)
			−0.634917	+	0.772581i	\(0.718966\pi\)
\(37\)	−7.41794e51	−1.31712	−0.658562	−	0.752527i	\(-0.728835\pi\)
			−0.658562	+	0.752527i	\(0.728835\pi\)
\(41\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(43\)	−1.24524e54	−1.55165	−0.775825	−	0.630948i	\(-0.782666\pi\)
			−0.775825	+	0.630948i	\(0.782666\pi\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(53\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(59\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(61\)	7.67187e58	0.931143	0.465572	−	0.885010i	\(-0.345849\pi\)
			0.465572	+	0.885010i	\(0.345849\pi\)
\(67\)	2.36444e59	0.129795	0.0648977	−	0.997892i	\(-0.479328\pi\)
			0.0648977	+	0.997892i	\(0.479328\pi\)
\(71\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(73\)	5.64366e61	1.82771	0.913853	−	0.406046i	\(-0.133093\pi\)
			0.913853	+	0.406046i	\(0.133093\pi\)
\(79\)	6.24528e62	1.49231	0.746153	−	0.665775i	\(-0.231899\pi\)
			0.746153	+	0.665775i	\(0.231899\pi\)
\(83\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(89\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(97\)	2.44251e65	0.667374	0.333687	−	0.942684i	\(-0.391707\pi\)
			0.333687	+	0.942684i	\(0.391707\pi\)

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	3.67.b.a.2.1	✓	1
3.2	odd	2	CM	3.67.b.a.2.1	✓	1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.67.b.a.2.1	✓	1	1.1	even	1	trivial
3.67.b.a.2.1	✓	1	3.2	odd	2	CM