Newform orbit 45.8.b.b

• Label: 45.8.b.b
• Level: $45$
• Weight: $8$
• Character orbit: 45.b
• Analytic conductor: $14.057$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -15
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$45 = 3^{2} \cdot 5$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	45.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$14.0573261468$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-5})$
Defining polynomial:	$x^{2} + 5$
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \sqrt{-5}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + b * q^2 + 123 * q^4 + 125*b * q^5 + 251*b * q^8 - 625 * q^10 + 14489 * q^16 + 13858*b * q^17 - 11596 * q^19 + 15375*b * q^20 + 44356*b * q^23 - 78125 * q^25 + 206648 * q^31 + 46617*b * q^32 - 69290 * q^34 - 11596*b * q^38 - 156875 * q^40 - 221780 * q^46 - 614876*b * q^47 + 823543 * q^49 - 78125*b * q^50 - 792038*b * q^53 - 2774518 * q^61 + 206648*b * q^62 + 1621507 * q^64 + 1704534*b * q^68 - 1426308 * q^76 + 8763536 * q^79 + 1811125*b * q^80 + 1070488*b * q^83 - 8661250 * q^85 + 5455788*b * q^92 + 3074380 * q^94 - 1449500*b * q^95 + 823543*b * q^98
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 246 * q^4 - 1250 * q^10 + 28978 * q^16 - 23192 * q^19 - 156250 * q^25 + 413296 * q^31 - 138580 * q^34 - 313750 * q^40 - 443560 * q^46 + 1647086 * q^49 - 5549036 * q^61 + 3243014 * q^64 - 2852616 * q^76 + 17527072 * q^79 - 17322500 * q^85 + 6148760 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

19.1

	−	2.23607i
		2.23607i

−

2.23607i

0

123.000

−

279.508i

0

0

−

561.253i

0

−625.000

19.2

2.23607i

0

123.000

279.508i

0

0

561.253i

0

−625.000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15})$
3.b	odd	2	1	inner
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	45.8.b.b	✓	2
3.b	odd	2	1	inner	45.8.b.b	✓	2
5.b	even	2	1	inner	45.8.b.b	✓	2
5.c	odd	4	2		225.8.a.m		2
15.d	odd	2	1	CM	45.8.b.b	✓	2
15.e	even	4	2		225.8.a.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
45.8.b.b	✓	2	1.a	even	1	1	trivial
45.8.b.b	✓	2	3.b	odd	2	1	inner
45.8.b.b	✓	2	5.b	even	2	1	inner
45.8.b.b	✓	2	15.d	odd	2	1	CM
225.8.a.m		2	5.c	odd	4	2
225.8.a.m		2	15.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{2} + 5$ acting on $S_{8}^{\mathrm{new}}(45, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 5$
$3$	$T^{2}$
$5$	$T^{2} + 78125$
$7$	$T^{2}$
$11$	$T^{2}$