Embedded newform 2700.1.l.b.2593.1

• Label: 2700.1.l.b.2593.1
• Level: $2700$
• Weight: $1$
• Character: 2700.2593
• Analytic conductor: $1.347$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{6}$
• CM discriminant: -3
• Inner twists: $8$

Newspace parameters

Level:	$N$	$=$	$2700 = 2^{2} \cdot 3^{3} \cdot 5^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	2700.l (of order $4$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$1.34747553411$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
Defining polynomial:	$x^{4} + 9$
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.36450000.1

Embedding invariants

Embedding label			2593.1
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2700.2593
Dual form			2700.1.l.b.757.1

$q$-expansion

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{13} -2.00000i q^{19} +1.00000 q^{31} +(1.22474 - 1.22474i) q^{37} +(-1.22474 - 1.22474i) q^{43} +1.00000i q^{49} +2.00000 q^{61} +(-1.22474 + 1.22474i) q^{67} +(1.22474 + 1.22474i) q^{73} -1.00000i q^{79} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 4 q^{31} + 8 q^{61}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
							$3$		0			0
$5$		0			0
							$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
0.707107	+	0.707107i	$0.250000\pi$
$11$		0			0			−	1.00000i	$-0.5\pi$
									1.00000i	$0.5\pi$
$13$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
							−0.258819	−	0.965926i	$-0.583333\pi$
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
							0.707107	+	0.707107i	$0.250000\pi$
$19$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
								−	1.00000i	$-0.5\pi$
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
							−0.707107	+	0.707107i	$0.750000\pi$
$29$		0			0		1.00000			$0$
							−1.00000			$\pi$
$31$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
							0.500000	+	0.866025i	$0.333333\pi$
$37$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
							0.965926	−	0.258819i	$-0.0833333\pi$
$41$		0			0			−	1.00000i	$-0.5\pi$
									1.00000i	$0.5\pi$
$43$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
							−0.258819	−	0.965926i	$-0.583333\pi$
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
							0.707107	+	0.707107i	$0.250000\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.1.l.b.2593.1	yes	4
3.2	odd	2	CM	2700.1.l.b.2593.1	yes	4
5.2	odd	4	inner	2700.1.l.b.757.1	✓	4
5.3	odd	4	inner	2700.1.l.b.757.2	yes	4
5.4	even	2	inner	2700.1.l.b.2593.2	yes	4
15.2	even	4	inner	2700.1.l.b.757.1	✓	4
15.8	even	4	inner	2700.1.l.b.757.2	yes	4
15.14	odd	2	inner	2700.1.l.b.2593.2	yes	4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2700.1.l.b.757.1	✓	4	5.2	odd	4	inner
2700.1.l.b.757.1	✓	4	15.2	even	4	inner
2700.1.l.b.757.2	yes	4	5.3	odd	4	inner
2700.1.l.b.757.2	yes	4	15.8	even	4	inner
2700.1.l.b.2593.1	yes	4	1.1	even	1	trivial
2700.1.l.b.2593.1	yes	4	3.2	odd	2	CM
2700.1.l.b.2593.2	yes	4	5.4	even	2	inner
2700.1.l.b.2593.2	yes	4	15.14	odd	2	inner