Embedded newform 3040.1.b.a.1329.7

• Label: 3040.1.b.a.1329.7
• Level: $3040$
• Weight: $1$
• Character: 3040.1329
• Analytic conductor: $1.517$
• Analytic rank: $0$
• Dimension: $8$
• Projective image: $D_{8}$
• CM discriminant: -95
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3040.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.51715763840\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{16})\)
Defining polynomial:	\( x^{8} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{3} \)
Twist minimal:	no (minimal twist has level 760)
Projective image:	\(D_{8}\)
Projective field:	Galois closure of 8.0.66724352000.2

Embedding invariants

Embedding label			1329.7
Root			\(0.923880 - 0.382683i\) of defining polynomial
Character	\(\chi\)	\(=\)	3040.1329
Dual form			3040.1.b.a.1329.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.84776i q^{3} -1.00000i q^{5} -2.41421 q^{9} -1.41421i q^{11} -0.765367i q^{13} +1.84776 q^{15} -1.00000i q^{19} -1.00000 q^{25} -2.61313i q^{27} +2.61313 q^{33} -1.84776i q^{37} +1.41421 q^{39} +2.41421i q^{45} +1.00000 q^{49} +1.84776i q^{53} -1.41421 q^{55} +1.84776 q^{57} -1.41421i q^{61} -0.765367 q^{65} -0.765367i q^{67} -1.84776i q^{75} +2.41421 q^{81} -1.00000 q^{95} -0.765367 q^{97} +3.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} - 8 q^{25} + 8 q^{49} + 8 q^{81} - 8 q^{95}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(1217\)	\(1921\)	\(2661\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(-1\)

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
			\(3\)	1.84776i	1.84776i	0.382683	+	0.923880i	\(0.375000\pi\)
−0.382683	+	0.923880i				\(0.625000\pi\)
\(5\)	− 1.00000i	− 1.00000i
			\(7\)	0	0	1.00000			\(0\)
−1.00000						\(\pi\)
\(11\)	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	\(-0.750000\pi\)
			0.707107	−	0.707107i	\(-0.250000\pi\)
\(13\)	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	\(-0.875000\pi\)
			0.923880	−	0.382683i	\(-0.125000\pi\)
\(17\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(19\)	− 1.00000i	− 1.00000i
			\(23\)	0	0	1.00000			\(0\)
−1.00000						\(\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(37\)	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	\(-0.625000\pi\)
			0.382683	−	0.923880i	\(-0.375000\pi\)
\(41\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(43\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(53\)	1.84776i	1.84776i	0.382683	+	0.923880i	\(0.375000\pi\)
			−0.382683	+	0.923880i	\(0.625000\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	\(-0.750000\pi\)
			0.707107	−	0.707107i	\(-0.250000\pi\)
\(67\)	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	\(-0.875000\pi\)
			0.923880	−	0.382683i	\(-0.125000\pi\)
\(71\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(73\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(79\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(83\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(89\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(97\)	−0.765367	−0.765367	−0.382683	−	0.923880i	\(-0.625000\pi\)
			−0.382683	+	0.923880i	\(0.625000\pi\)

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	3040.1.b.a.1329.7		8
4.3	odd	2		760.1.b.a.189.8	yes	8
5.4	even	2	inner	3040.1.b.a.1329.1		8
8.3	odd	2		760.1.b.a.189.7	yes	8
8.5	even	2	inner	3040.1.b.a.1329.2		8
19.18	odd	2	inner	3040.1.b.a.1329.1		8
20.3	even	4		3800.1.o.g.1101.5		8
20.7	even	4		3800.1.o.g.1101.4		8
20.19	odd	2		760.1.b.a.189.1	✓	8
40.3	even	4		3800.1.o.g.1101.3		8
40.19	odd	2		760.1.b.a.189.2	yes	8
40.27	even	4		3800.1.o.g.1101.6		8
40.29	even	2	inner	3040.1.b.a.1329.8		8
76.75	even	2		760.1.b.a.189.1	✓	8
95.94	odd	2	CM	3040.1.b.a.1329.7		8
152.37	odd	2	inner	3040.1.b.a.1329.8		8
152.75	even	2		760.1.b.a.189.2	yes	8
380.227	odd	4		3800.1.o.g.1101.5		8
380.303	odd	4		3800.1.o.g.1101.4		8
380.379	even	2		760.1.b.a.189.8	yes	8
760.189	odd	2	inner	3040.1.b.a.1329.2		8
760.227	odd	4		3800.1.o.g.1101.3		8
760.379	even	2		760.1.b.a.189.7	yes	8
760.683	odd	4		3800.1.o.g.1101.6		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.1.b.a.189.1	✓	8	20.19	odd	2
760.1.b.a.189.1	✓	8	76.75	even	2
760.1.b.a.189.2	yes	8	40.19	odd	2
760.1.b.a.189.2	yes	8	152.75	even	2
760.1.b.a.189.7	yes	8	8.3	odd	2
760.1.b.a.189.7	yes	8	760.379	even	2
760.1.b.a.189.8	yes	8	4.3	odd	2
760.1.b.a.189.8	yes	8	380.379	even	2
3040.1.b.a.1329.1		8	5.4	even	2	inner
3040.1.b.a.1329.1		8	19.18	odd	2	inner
3040.1.b.a.1329.2		8	8.5	even	2	inner
3040.1.b.a.1329.2		8	760.189	odd	2	inner
3040.1.b.a.1329.7		8	1.1	even	1	trivial
3040.1.b.a.1329.7		8	95.94	odd	2	CM
3040.1.b.a.1329.8		8	40.29	even	2	inner
3040.1.b.a.1329.8		8	152.37	odd	2	inner
3800.1.o.g.1101.3		8	40.3	even	4
3800.1.o.g.1101.3		8	760.227	odd	4
3800.1.o.g.1101.4		8	20.7	even	4
3800.1.o.g.1101.4		8	380.303	odd	4
3800.1.o.g.1101.5		8	20.3	even	4
3800.1.o.g.1101.5		8	380.227	odd	4
3800.1.o.g.1101.6		8	40.27	even	4
3800.1.o.g.1101.6		8	760.683	odd	4