Embedded newform 700.1.d.a.601.1

• Label: 700.1.d.a.601.1
• Level: $700$
• Weight: $1$
• Character: 700.601
• Analytic conductor: $0.349$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{3}$
• CM discriminant: -35
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$700 = 2^{2} \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	700.d (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.349345508843$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 140)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.140.1
Artin image:	$C_4\times S_3$
Artin field:	Galois closure of 12.0.1200500000000.1

Embedding invariants

Embedding label			601.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	700.601
Dual form			700.1.d.a.601.2

$q$-expansion

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{11} -1.00000i q^{13} +1.00000i q^{17} -1.00000 q^{21} -1.00000i q^{27} +1.00000 q^{29} +1.00000i q^{33} -1.00000 q^{39} +1.00000i q^{47} -1.00000 q^{49} +1.00000 q^{51} +2.00000 q^{71} +2.00000i q^{73} +1.00000i q^{77} +1.00000 q^{79} -1.00000 q^{81} +2.00000i q^{83} -1.00000i q^{87} -1.00000 q^{91} +1.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{11} - 2 q^{21} + 2 q^{29} - 2 q^{39} - 2 q^{49} + 2 q^{51} + 4 q^{71} + 2 q^{79} - 2 q^{81} - 2 q^{91}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$477$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

(See $a_n$ instead)

$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$
$2$	0	0
			$3$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
0.866025	−	0.500000i				$-0.166667\pi$
$5$	0	0
			$7$	− 1.00000i	− 1.00000i
$11$	−1.00000	−1.00000				−0.500000	−	0.866025i	$-0.666667\pi$
			−0.500000	+	0.866025i	$0.666667\pi$
$13$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
			0.866025	−	0.500000i	$-0.166667\pi$
$17$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
			−0.866025	+	0.500000i	$0.833333\pi$
$19$	0	0	1.00000			$0$
			−1.00000			$\pi$
$23$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$29$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
			0.500000	+	0.866025i	$0.333333\pi$
$31$	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	$0.5\pi$
$47$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
			−0.866025	+	0.500000i	$0.833333\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.1.d.a.601.1		2
4.3	odd	2		2800.1.f.c.2001.2		2
5.2	odd	4		140.1.h.a.69.1	✓	1
5.3	odd	4		140.1.h.b.69.1	yes	1
5.4	even	2	inner	700.1.d.a.601.2		2
7.6	odd	2	inner	700.1.d.a.601.2		2
15.2	even	4		1260.1.p.a.1189.1		1
15.8	even	4		1260.1.p.b.1189.1		1
20.3	even	4		560.1.p.a.209.1		1
20.7	even	4		560.1.p.b.209.1		1
20.19	odd	2		2800.1.f.c.2001.1		2
28.27	even	2		2800.1.f.c.2001.1		2
35.2	odd	12		980.1.n.b.129.1		2
35.3	even	12		980.1.n.b.509.1		2
35.12	even	12		980.1.n.a.129.1		2
35.13	even	4		140.1.h.a.69.1	✓	1
35.17	even	12		980.1.n.a.509.1		2
35.18	odd	12		980.1.n.a.509.1		2
35.23	odd	12		980.1.n.a.129.1		2
35.27	even	4		140.1.h.b.69.1	yes	1
35.32	odd	12		980.1.n.b.509.1		2
35.33	even	12		980.1.n.b.129.1		2
35.34	odd	2	CM	700.1.d.a.601.1		2
40.3	even	4		2240.1.p.d.769.1		1
40.13	odd	4		2240.1.p.b.769.1		1
40.27	even	4		2240.1.p.a.769.1		1
40.37	odd	4		2240.1.p.c.769.1		1
105.62	odd	4		1260.1.p.b.1189.1		1
105.83	odd	4		1260.1.p.a.1189.1		1
140.3	odd	12		3920.1.br.a.1489.1		2
140.23	even	12		3920.1.br.b.129.1		2
140.27	odd	4		560.1.p.a.209.1		1
140.47	odd	12		3920.1.br.b.129.1		2
140.67	even	12		3920.1.br.a.1489.1		2
140.83	odd	4		560.1.p.b.209.1		1
140.87	odd	12		3920.1.br.b.1489.1		2
140.103	odd	12		3920.1.br.a.129.1		2
140.107	even	12		3920.1.br.a.129.1		2
140.123	even	12		3920.1.br.b.1489.1		2
140.139	even	2		2800.1.f.c.2001.2		2
280.13	even	4		2240.1.p.c.769.1		1
280.27	odd	4		2240.1.p.d.769.1		1
280.83	odd	4		2240.1.p.a.769.1		1
280.237	even	4		2240.1.p.b.769.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.1.h.a.69.1	✓	1	5.2	odd	4
140.1.h.a.69.1	✓	1	35.13	even	4
140.1.h.b.69.1	yes	1	5.3	odd	4
140.1.h.b.69.1	yes	1	35.27	even	4
560.1.p.a.209.1		1	20.3	even	4
560.1.p.a.209.1		1	140.27	odd	4
560.1.p.b.209.1		1	20.7	even	4
560.1.p.b.209.1		1	140.83	odd	4
700.1.d.a.601.1		2	1.1	even	1	trivial
700.1.d.a.601.1		2	35.34	odd	2	CM
700.1.d.a.601.2		2	5.4	even	2	inner
700.1.d.a.601.2		2	7.6	odd	2	inner
980.1.n.a.129.1		2	35.12	even	12
980.1.n.a.129.1		2	35.23	odd	12
980.1.n.a.509.1		2	35.17	even	12
980.1.n.a.509.1		2	35.18	odd	12
980.1.n.b.129.1		2	35.2	odd	12
980.1.n.b.129.1		2	35.33	even	12
980.1.n.b.509.1		2	35.3	even	12
980.1.n.b.509.1		2	35.32	odd	12
1260.1.p.a.1189.1		1	15.2	even	4
1260.1.p.a.1189.1		1	105.83	odd	4
1260.1.p.b.1189.1		1	15.8	even	4
1260.1.p.b.1189.1		1	105.62	odd	4
2240.1.p.a.769.1		1	40.27	even	4
2240.1.p.a.769.1		1	280.83	odd	4
2240.1.p.b.769.1		1	40.13	odd	4
2240.1.p.b.769.1		1	280.237	even	4
2240.1.p.c.769.1		1	40.37	odd	4
2240.1.p.c.769.1		1	280.13	even	4
2240.1.p.d.769.1		1	40.3	even	4
2240.1.p.d.769.1		1	280.27	odd	4
2800.1.f.c.2001.1		2	20.19	odd	2
2800.1.f.c.2001.1		2	28.27	even	2
2800.1.f.c.2001.2		2	4.3	odd	2
2800.1.f.c.2001.2		2	140.139	even	2
3920.1.br.a.129.1		2	140.103	odd	12
3920.1.br.a.129.1		2	140.107	even	12
3920.1.br.a.1489.1		2	140.3	odd	12
3920.1.br.a.1489.1		2	140.67	even	12
3920.1.br.b.129.1		2	140.23	even	12
3920.1.br.b.129.1		2	140.47	odd	12
3920.1.br.b.1489.1		2	140.87	odd	12
3920.1.br.b.1489.1		2	140.123	even	12