# Properties

 Label 45.4.a.e Level $45$ Weight $4$ Character orbit 45.a Self dual yes Analytic conductor $2.655$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,4,Mod(1,45)]

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, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("45.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 45.a (trivial)

## 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: yes Analytic conductor: $$2.65508595026$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 5 q^{2} + 17 q^{4} - 5 q^{5} - 30 q^{7} + 45 q^{8}+O(q^{10})$$ q + 5 * q^2 + 17 * q^4 - 5 * q^5 - 30 * q^7 + 45 * q^8 $$q + 5 q^{2} + 17 q^{4} - 5 q^{5} - 30 q^{7} + 45 q^{8} - 25 q^{10} + 50 q^{11} - 20 q^{13} - 150 q^{14} + 89 q^{16} - 10 q^{17} - 44 q^{19} - 85 q^{20} + 250 q^{22} + 120 q^{23} + 25 q^{25} - 100 q^{26} - 510 q^{28} - 50 q^{29} + 108 q^{31} + 85 q^{32} - 50 q^{34} + 150 q^{35} - 40 q^{37} - 220 q^{38} - 225 q^{40} + 400 q^{41} + 280 q^{43} + 850 q^{44} + 600 q^{46} - 280 q^{47} + 557 q^{49} + 125 q^{50} - 340 q^{52} - 610 q^{53} - 250 q^{55} - 1350 q^{56} - 250 q^{58} + 50 q^{59} - 518 q^{61} + 540 q^{62} - 287 q^{64} + 100 q^{65} - 180 q^{67} - 170 q^{68} + 750 q^{70} + 700 q^{71} - 410 q^{73} - 200 q^{74} - 748 q^{76} - 1500 q^{77} - 516 q^{79} - 445 q^{80} + 2000 q^{82} + 660 q^{83} + 50 q^{85} + 1400 q^{86} + 2250 q^{88} - 1500 q^{89} + 600 q^{91} + 2040 q^{92} - 1400 q^{94} + 220 q^{95} - 1630 q^{97} + 2785 q^{98}+O(q^{100})$$ q + 5 * q^2 + 17 * q^4 - 5 * q^5 - 30 * q^7 + 45 * q^8 - 25 * q^10 + 50 * q^11 - 20 * q^13 - 150 * q^14 + 89 * q^16 - 10 * q^17 - 44 * q^19 - 85 * q^20 + 250 * q^22 + 120 * q^23 + 25 * q^25 - 100 * q^26 - 510 * q^28 - 50 * q^29 + 108 * q^31 + 85 * q^32 - 50 * q^34 + 150 * q^35 - 40 * q^37 - 220 * q^38 - 225 * q^40 + 400 * q^41 + 280 * q^43 + 850 * q^44 + 600 * q^46 - 280 * q^47 + 557 * q^49 + 125 * q^50 - 340 * q^52 - 610 * q^53 - 250 * q^55 - 1350 * q^56 - 250 * q^58 + 50 * q^59 - 518 * q^61 + 540 * q^62 - 287 * q^64 + 100 * q^65 - 180 * q^67 - 170 * q^68 + 750 * q^70 + 700 * q^71 - 410 * q^73 - 200 * q^74 - 748 * q^76 - 1500 * q^77 - 516 * q^79 - 445 * q^80 + 2000 * q^82 + 660 * q^83 + 50 * q^85 + 1400 * q^86 + 2250 * q^88 - 1500 * q^89 + 600 * q^91 + 2040 * q^92 - 1400 * q^94 + 220 * q^95 - 1630 * q^97 + 2785 * q^98

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
5.00000 0 17.0000 −5.00000 0 −30.0000 45.0000 0 −25.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.4.a.e yes 1
3.b odd 2 1 45.4.a.a 1
4.b odd 2 1 720.4.a.o 1
5.b even 2 1 225.4.a.a 1
5.c odd 4 2 225.4.b.b 2
7.b odd 2 1 2205.4.a.t 1
9.c even 3 2 405.4.e.b 2
9.d odd 6 2 405.4.e.n 2
12.b even 2 1 720.4.a.bc 1
15.d odd 2 1 225.4.a.h 1
15.e even 4 2 225.4.b.a 2
21.c even 2 1 2205.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.a.a 1 3.b odd 2 1
45.4.a.e yes 1 1.a even 1 1 trivial
225.4.a.a 1 5.b even 2 1
225.4.a.h 1 15.d odd 2 1
225.4.b.a 2 15.e even 4 2
225.4.b.b 2 5.c odd 4 2
405.4.e.b 2 9.c even 3 2
405.4.e.n 2 9.d odd 6 2
720.4.a.o 1 4.b odd 2 1
720.4.a.bc 1 12.b even 2 1
2205.4.a.a 1 21.c even 2 1
2205.4.a.t 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 5$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(45))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 5$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T + 30$$
$11$ $$T - 50$$
$13$ $$T + 20$$
$17$ $$T + 10$$
$19$ $$T + 44$$
$23$ $$T - 120$$
$29$ $$T + 50$$
$31$ $$T - 108$$
$37$ $$T + 40$$
$41$ $$T - 400$$
$43$ $$T - 280$$
$47$ $$T + 280$$
$53$ $$T + 610$$
$59$ $$T - 50$$
$61$ $$T + 518$$
$67$ $$T + 180$$
$71$ $$T - 700$$
$73$ $$T + 410$$
$79$ $$T + 516$$
$83$ $$T - 660$$
$89$ $$T + 1500$$
$97$ $$T + 1630$$