LMFDB - Newform orbit 45.10.b.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,10,Mod(19,45)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("45.19");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$N$	$=$	$45 = 3^{2} \cdot 5$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	45.b (of order $2$ , degree $1$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.1766126274$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 5$ x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

$f(q)$	$=$	$q + 19 \beta q^{2} - 1293 q^{4} - 625 \beta q^{5} - 14839 \beta q^{8} + 59375 q^{10} + 747689 q^{16} + 277822 \beta q^{17} + 1036316 q^{19} + 808125 \beta q^{20} - 1180124 \beta q^{23} - 1953125 q^{25} + \cdots + 766718533 \beta q^{98} +O(q^{100})$ q + 19b q^2 - 1293 * q^4 - 625b q^5 - 14839b q^8 + 59375 * q^10 + 747689 * q^16 + 277822b q^17 + 1036316 * q^19 + 808125b q^20 - 1180124b q^23 - 1953125 * q^25 + 8247368 * q^31 + 6608523b q^32 - 26393090 * q^34 + 19690004b q^38 - 46371875 * q^40 + 112111780 * q^46 + 3364516b q^47 + 40353607 * q^49 - 37109375b q^50 - 16738658b q^53 + 197894882 * q^61 + 156699992b q^62 - 244992917 * q^64 - 359223846b q^68 - 1339956588 * q^76 + 421557104 * q^79 - 467305625b q^80 + 58072888b q^83 + 868193750 * q^85 + 1525900332b q^92 - 319629020 * q^94 - 647697500b q^95 + 766718533b q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2586 q^{4} + 118750 q^{10} + 1495378 q^{16} + 2072632 q^{19} - 3906250 q^{25} + 16494736 q^{31} - 52786180 q^{34} - 92743750 q^{40} + 224223560 q^{46} + 80707214 q^{49} + 395789764 q^{61} - 489985834 q^{64}+ \cdots - 639258040 q^{94}+O(q^{100})$ 2 * q - 2586 * q^4 + 118750 * q^10 + 1495378 * q^16 + 2072632 * q^19 - 3906250 * q^25 + 16494736 * q^31 - 52786180 * q^34 - 92743750 * q^40 + 224223560 * q^46 + 80707214 * q^49 + 395789764 * q^61 - 489985834 * q^64 - 2679913176 * q^76 + 843114208 * q^79 + 1736387500 * q^85 - 639258040 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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

−

42.4853i

−1293.00

1397.54i

33181.0i

59375.0

19.2

42.4853i

−1293.00

−

1397.54i

−

33181.0i

59375.0

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	45.10.b.a	✓	2
3.b	odd	2	1	inner	45.10.b.a	✓	2
5.b	even	2	1	inner	45.10.b.a	✓	2
5.c	odd	4	2		225.10.a.i		2
15.d	odd	2	1	CM	45.10.b.a	✓	2
15.e	even	4	2		225.10.a.i		2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{2} + 1805$ T2^2 + 1805 acting on $S_{10}^{\mathrm{new}}(45, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 1805$ T^2 + 1805
$3$	$T^{2}$ T^2
$5$	$T^{2} + 1953125$ T^2 + 1953125
$7$	$T^{2}$ T^2
$11$	$T^{2}$ T^2
$13$	$T^{2}$ T^2
$17$	$T^{2} + 385925318420$ T^2 + 385925318420
$19$	$(T - 1036316)^{2}$ (T - 1036316)^2
$23$	$T^{2} + 6963463276880$ T^2 + 6963463276880
$29$	$T^{2}$ T^2
$31$	$(T - 8247368)^{2}$ (T - 8247368)^2
$37$	$T^{2}$ T^2
$41$	$T^{2}$ T^2
$43$	$T^{2}$ T^2
$47$	$T^{2} + 56599839571280$ T^2 + 56599839571280
$53$	$T^{2} + 14\!\cdots\!20$ T^2 + 1400913358204820
$59$	$T^{2}$ T^2
$61$	$(T - 197894882)^{2}$ (T - 197894882)^2
$67$	$T^{2}$ T^2
$71$	$T^{2}$ T^2
$73$	$T^{2}$ T^2
$79$	$(T - 421557104)^{2}$ (T - 421557104)^2
$83$	$T^{2} + 16\!\cdots\!20$ T^2 + 16862301603302720
$89$	$T^{2}$ T^2
$97$	$T^{2}$ T^2
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