Embedded newform 768.2.c.g.767.3

• Label: 768.2.c.g.767.3
• Level: $768$
• Weight: $2$
• Character: 768.767
• Analytic conductor: $6.133$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -3
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.13251087523\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	no (minimal twist has level 192)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			767.3
Root			\(0.866025 + 0.500000i\) of defining polynomial
Character	\(\chi\)	\(=\)	768.767
Dual form			768.2.c.g.767.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -4.00000i q^{7} -3.00000 q^{9} +6.92820 q^{13} -3.46410i q^{19} +6.92820 q^{21} +5.00000 q^{25} -5.19615i q^{27} -4.00000i q^{31} +6.92820 q^{37} +12.0000i q^{39} +10.3923i q^{43} -9.00000 q^{49} +6.00000 q^{57} +6.92820 q^{61} +12.0000i q^{63} +3.46410i q^{67} -10.0000 q^{73} +8.66025i q^{75} -4.00000i q^{79} +9.00000 q^{81} -27.7128i q^{91} +6.92820 q^{93} +14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9} + 20 q^{25} - 36 q^{49} + 24 q^{57} - 40 q^{73} + 36 q^{81} + 56 q^{97}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\)	\(257\)	\(511\)	\(517\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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
			\(3\)	1.73205i	1.00000i
\(5\)	0	0				1.00000			\(0\)
			−1.00000			\(\pi\)
\(7\)	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	\(-0.727186\pi\)
			0.654654	−	0.755929i	\(-0.272814\pi\)
\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	6.92820	1.92154	0.960769	−	0.277350i	\(-0.0894562\pi\)
			0.960769	+	0.277350i	\(0.0894562\pi\)
\(17\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(19\)	− 3.46410i	− 0.794719i	−0.917663	−	0.397360i	\(-0.869927\pi\)
			0.917663	−	0.397360i	\(-0.130073\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(31\)	− 4.00000i	− 0.718421i	−0.933257	−	0.359211i	\(-0.883046\pi\)
			0.933257	−	0.359211i	\(-0.116954\pi\)
\(37\)	6.92820	1.13899	0.569495	−	0.821995i	\(-0.307139\pi\)
			0.569495	+	0.821995i	\(0.307139\pi\)
\(41\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(43\)	10.3923i	1.58481i	0.609994	+	0.792406i	\(0.291172\pi\)
			−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(53\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	6.92820	0.887066	0.443533	−	0.896258i	\(-0.353725\pi\)
			0.443533	+	0.896258i	\(0.353725\pi\)
\(67\)	3.46410i	0.423207i	0.977356	+	0.211604i	\(0.0678686\pi\)
			−0.977356	+	0.211604i	\(0.932131\pi\)
\(71\)	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	\(0.698986\pi\)
\(79\)	− 4.00000i	− 0.450035i	−0.974355	−	0.225018i	\(-0.927756\pi\)
			0.974355	−	0.225018i	\(-0.0722440\pi\)
\(83\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(89\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(97\)	14.0000	1.42148	0.710742	−	0.703452i	\(-0.248359\pi\)
			0.710742	+	0.703452i	\(0.248359\pi\)

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	768.2.c.g.767.3		4
3.2	odd	2	CM	768.2.c.g.767.3		4
4.3	odd	2	inner	768.2.c.g.767.2		4
8.3	odd	2	inner	768.2.c.g.767.4		4
8.5	even	2	inner	768.2.c.g.767.1		4
12.11	even	2	inner	768.2.c.g.767.2		4
16.3	odd	4		192.2.f.b.95.3	yes	4
16.5	even	4		192.2.f.b.95.4	yes	4
16.11	odd	4		192.2.f.b.95.1	✓	4
16.13	even	4		192.2.f.b.95.2	yes	4
24.5	odd	2	inner	768.2.c.g.767.1		4
24.11	even	2	inner	768.2.c.g.767.4		4
48.5	odd	4		192.2.f.b.95.4	yes	4
48.11	even	4		192.2.f.b.95.1	✓	4
48.29	odd	4		192.2.f.b.95.2	yes	4
48.35	even	4		192.2.f.b.95.3	yes	4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.2.f.b.95.1	✓	4	16.11	odd	4
192.2.f.b.95.1	✓	4	48.11	even	4
192.2.f.b.95.2	yes	4	16.13	even	4
192.2.f.b.95.2	yes	4	48.29	odd	4
192.2.f.b.95.3	yes	4	16.3	odd	4
192.2.f.b.95.3	yes	4	48.35	even	4
192.2.f.b.95.4	yes	4	16.5	even	4
192.2.f.b.95.4	yes	4	48.5	odd	4
768.2.c.g.767.1		4	8.5	even	2	inner
768.2.c.g.767.1		4	24.5	odd	2	inner
768.2.c.g.767.2		4	4.3	odd	2	inner
768.2.c.g.767.2		4	12.11	even	2	inner
768.2.c.g.767.3		4	1.1	even	1	trivial
768.2.c.g.767.3		4	3.2	odd	2	CM
768.2.c.g.767.4		4	8.3	odd	2	inner
768.2.c.g.767.4		4	24.11	even	2	inner