Newform orbit 624.4.n.b

• Label: 624.4.n.b
• Level: $624$
• Weight: $4$
• Character orbit: 624.n
• Analytic conductor: $36.817$
• Analytic rank: $1$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$624 = 2^{4} \cdot 3 \cdot 13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	624.n (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$36.8171918436$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
Defining polynomial:	$x^{2} - x + 1$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2\cdot 3$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 3\sqrt{-3}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + b * q^3 + 20 * q^7 - 27 * q^9 + (-6*b - 35) * q^13 - 56 * q^19 + 20*b * q^21 - 125 * q^25 - 27*b * q^27 - 308 * q^31 + 84*b * q^37 + (-35*b + 162) * q^39 - 42*b * q^43 + 57 * q^49 - 56*b * q^57 - 182 * q^61 - 540 * q^63 - 880 * q^67 - 72*b * q^73 - 125*b * q^75 - 210*b * q^79 + 729 * q^81 + (-120*b - 700) * q^91 - 308*b * q^93 - 264*b * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 40 q^{7} - 54 q^{9} - 70 q^{13} - 112 q^{19} - 250 q^{25} - 616 q^{31} + 324 q^{39} + 114 q^{49} - 364 q^{61} - 1080 q^{63} - 1760 q^{67} + 1458 q^{81} - 1400 q^{91}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

Label  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

623.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

−

5.19615i

0

0

0

20.0000

0

−27.0000

0

623.2

0

5.19615i

0

0

0

20.0000

0

−27.0000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
52.b	odd	2	1	inner
156.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	624.4.n.b	yes	2
3.b	odd	2	1	CM	624.4.n.b	yes	2
4.b	odd	2	1		624.4.n.a	✓	2
12.b	even	2	1		624.4.n.a	✓	2
13.b	even	2	1		624.4.n.a	✓	2
39.d	odd	2	1		624.4.n.a	✓	2
52.b	odd	2	1	inner	624.4.n.b	yes	2
156.h	even	2	1	inner	624.4.n.b	yes	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
624.4.n.a	✓	2	4.b	odd	2	1
624.4.n.a	✓	2	12.b	even	2	1
624.4.n.a	✓	2	13.b	even	2	1
624.4.n.a	✓	2	39.d	odd	2	1
624.4.n.b	yes	2	1.a	even	1	1	trivial
624.4.n.b	yes	2	3.b	odd	2	1	CM
624.4.n.b	yes	2	52.b	odd	2	1	inner
624.4.n.b	yes	2	156.h	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(624, [\chi])$:

$T_{5}$

$T_{7} - 20$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} + 27$
$5$	$T^{2}$
$7$	$(T - 20)^{2}$
$11$	$T^{2}$