Embedded newform 875.1.d.a.251.2

• Label: 875.1.d.a.251.2
• Level: $875$
• Weight: $1$
• Character: 875.251
• Analytic conductor: $0.437$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{5}$
• CM discriminant: -35
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 875 = 5^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	875.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.436681886054\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(i, \sqrt{5})\)
Defining polynomial:	\( x^{4} + 3x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.765625.1
Artin image:	$C_4\times D_5$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

Embedding invariants

Embedding label			251.2
Root			\(0.618034i\) of defining polynomial
Character	\(\chi\)	\(=\)	875.251
Dual form			875.1.d.a.251.3

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.618034i q^{3} -1.00000 q^{4} +1.00000i q^{7} +0.618034 q^{9} +0.618034 q^{11} +0.618034i q^{12} +1.61803i q^{13} +1.00000 q^{16} -1.61803i q^{17} +0.618034 q^{21} -1.00000i q^{27} -1.00000i q^{28} +1.61803 q^{29} -0.381966i q^{33} -0.618034 q^{36} +1.00000 q^{39} -0.618034 q^{44} +2.00000i q^{47} -0.618034i q^{48} -1.00000 q^{49} -1.00000 q^{51} -1.61803i q^{52} +0.618034i q^{63} -1.00000 q^{64} +1.61803i q^{68} -1.61803 q^{71} -0.618034i q^{73} +0.618034i q^{77} -0.618034 q^{79} -0.618034i q^{83} -0.618034 q^{84} -1.00000i q^{87} -1.61803 q^{91} +0.618034i q^{97} +0.381966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 2 q^{9} - 2 q^{11} + 4 q^{16} - 2 q^{21} + 2 q^{29} + 2 q^{36} + 4 q^{39} + 2 q^{44} - 4 q^{49} - 4 q^{51} - 4 q^{64} - 2 q^{71} + 2 q^{79} + 2 q^{84} - 2 q^{91} + 6 q^{99}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/875\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(626\)
\(\chi(n)\)	\(1\)	\(-1\)

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		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(3\)	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	\(-0.900000\pi\)
			0.951057	−	0.309017i	\(-0.100000\pi\)
\(5\)	0	0
			\(7\)	1.00000i	1.00000i
\(11\)	0.618034	0.618034				0.309017	−	0.951057i	\(-0.400000\pi\)
			0.309017	+	0.951057i	\(0.400000\pi\)
\(13\)	1.61803i	1.61803i	0.587785	+	0.809017i	\(0.300000\pi\)
			−0.587785	+	0.809017i	\(0.700000\pi\)
\(17\)	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	\(-0.700000\pi\)
			0.587785	−	0.809017i	\(-0.300000\pi\)
\(19\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	1.61803	1.61803	0.809017	−	0.587785i	\(-0.200000\pi\)
			0.809017	+	0.587785i	\(0.200000\pi\)
\(31\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(37\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(41\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(43\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(47\)	2.00000i	2.00000i			1.00000i	\(0.5\pi\)
					1.00000i	\(0.5\pi\)
\(53\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(59\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(61\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(67\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(71\)	−1.61803	−1.61803	−0.809017	−	0.587785i	\(-0.800000\pi\)
			−0.809017	+	0.587785i	\(0.800000\pi\)
\(73\)	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	\(-0.900000\pi\)
			0.951057	−	0.309017i	\(-0.100000\pi\)
\(79\)	−0.618034	−0.618034	−0.309017	−	0.951057i	\(-0.600000\pi\)
			−0.309017	+	0.951057i	\(0.600000\pi\)
\(83\)	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	\(-0.900000\pi\)
			0.951057	−	0.309017i	\(-0.100000\pi\)
\(89\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(97\)	0.618034i	0.618034i	0.951057	+	0.309017i	\(0.100000\pi\)
			−0.951057	+	0.309017i	\(0.900000\pi\)

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	875.1.d.a.251.2	✓	4
5.2	odd	4		875.1.c.b.874.1		2
5.3	odd	4		875.1.c.a.874.2		2
5.4	even	2	inner	875.1.d.a.251.3	yes	4
7.6	odd	2	inner	875.1.d.a.251.3	yes	4
35.13	even	4		875.1.c.b.874.1		2
35.27	even	4		875.1.c.a.874.2		2
35.34	odd	2	CM	875.1.d.a.251.2	✓	4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
875.1.c.a.874.2		2	5.3	odd	4
875.1.c.a.874.2		2	35.27	even	4
875.1.c.b.874.1		2	5.2	odd	4
875.1.c.b.874.1		2	35.13	even	4
875.1.d.a.251.2	✓	4	1.1	even	1	trivial
875.1.d.a.251.2	✓	4	35.34	odd	2	CM
875.1.d.a.251.3	yes	4	5.4	even	2	inner
875.1.d.a.251.3	yes	4	7.6	odd	2	inner