# Properties

 Label 45.4.a.b 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: no (minimal twist has level 15) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{2} + q^{4} + 5 q^{5} + 20 q^{7} + 21 q^{8}+O(q^{10})$$ q - 3 * q^2 + q^4 + 5 * q^5 + 20 * q^7 + 21 * q^8 $$q - 3 q^{2} + q^{4} + 5 q^{5} + 20 q^{7} + 21 q^{8} - 15 q^{10} + 24 q^{11} + 74 q^{13} - 60 q^{14} - 71 q^{16} - 54 q^{17} - 124 q^{19} + 5 q^{20} - 72 q^{22} + 120 q^{23} + 25 q^{25} - 222 q^{26} + 20 q^{28} + 78 q^{29} + 200 q^{31} + 45 q^{32} + 162 q^{34} + 100 q^{35} - 70 q^{37} + 372 q^{38} + 105 q^{40} - 330 q^{41} + 92 q^{43} + 24 q^{44} - 360 q^{46} + 24 q^{47} + 57 q^{49} - 75 q^{50} + 74 q^{52} - 450 q^{53} + 120 q^{55} + 420 q^{56} - 234 q^{58} - 24 q^{59} - 322 q^{61} - 600 q^{62} + 433 q^{64} + 370 q^{65} - 196 q^{67} - 54 q^{68} - 300 q^{70} + 288 q^{71} - 430 q^{73} + 210 q^{74} - 124 q^{76} + 480 q^{77} - 520 q^{79} - 355 q^{80} + 990 q^{82} - 156 q^{83} - 270 q^{85} - 276 q^{86} + 504 q^{88} - 1026 q^{89} + 1480 q^{91} + 120 q^{92} - 72 q^{94} - 620 q^{95} - 286 q^{97} - 171 q^{98}+O(q^{100})$$ q - 3 * q^2 + q^4 + 5 * q^5 + 20 * q^7 + 21 * q^8 - 15 * q^10 + 24 * q^11 + 74 * q^13 - 60 * q^14 - 71 * q^16 - 54 * q^17 - 124 * q^19 + 5 * q^20 - 72 * q^22 + 120 * q^23 + 25 * q^25 - 222 * q^26 + 20 * q^28 + 78 * q^29 + 200 * q^31 + 45 * q^32 + 162 * q^34 + 100 * q^35 - 70 * q^37 + 372 * q^38 + 105 * q^40 - 330 * q^41 + 92 * q^43 + 24 * q^44 - 360 * q^46 + 24 * q^47 + 57 * q^49 - 75 * q^50 + 74 * q^52 - 450 * q^53 + 120 * q^55 + 420 * q^56 - 234 * q^58 - 24 * q^59 - 322 * q^61 - 600 * q^62 + 433 * q^64 + 370 * q^65 - 196 * q^67 - 54 * q^68 - 300 * q^70 + 288 * q^71 - 430 * q^73 + 210 * q^74 - 124 * q^76 + 480 * q^77 - 520 * q^79 - 355 * q^80 + 990 * q^82 - 156 * q^83 - 270 * q^85 - 276 * q^86 + 504 * q^88 - 1026 * q^89 + 1480 * q^91 + 120 * q^92 - 72 * q^94 - 620 * q^95 - 286 * q^97 - 171 * 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
−3.00000 0 1.00000 5.00000 0 20.0000 21.0000 0 −15.0000
 $$n$$: 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.b 1
3.b odd 2 1 15.4.a.b 1
4.b odd 2 1 720.4.a.r 1
5.b even 2 1 225.4.a.g 1
5.c odd 4 2 225.4.b.d 2
7.b odd 2 1 2205.4.a.c 1
9.c even 3 2 405.4.e.k 2
9.d odd 6 2 405.4.e.d 2
12.b even 2 1 240.4.a.f 1
15.d odd 2 1 75.4.a.a 1
15.e even 4 2 75.4.b.a 2
21.c even 2 1 735.4.a.i 1
24.f even 2 1 960.4.a.l 1
24.h odd 2 1 960.4.a.bi 1
33.d even 2 1 1815.4.a.a 1
60.h even 2 1 1200.4.a.o 1
60.l odd 4 2 1200.4.f.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 3.b odd 2 1
45.4.a.b 1 1.a even 1 1 trivial
75.4.a.a 1 15.d odd 2 1
75.4.b.a 2 15.e even 4 2
225.4.a.g 1 5.b even 2 1
225.4.b.d 2 5.c odd 4 2
240.4.a.f 1 12.b even 2 1
405.4.e.d 2 9.d odd 6 2
405.4.e.k 2 9.c even 3 2
720.4.a.r 1 4.b odd 2 1
735.4.a.i 1 21.c even 2 1
960.4.a.l 1 24.f even 2 1
960.4.a.bi 1 24.h odd 2 1
1200.4.a.o 1 60.h even 2 1
1200.4.f.m 2 60.l odd 4 2
1815.4.a.a 1 33.d even 2 1
2205.4.a.c 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 3$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(45))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T - 20$$
$11$ $$T - 24$$
$13$ $$T - 74$$
$17$ $$T + 54$$
$19$ $$T + 124$$
$23$ $$T - 120$$
$29$ $$T - 78$$
$31$ $$T - 200$$
$37$ $$T + 70$$
$41$ $$T + 330$$
$43$ $$T - 92$$
$47$ $$T - 24$$
$53$ $$T + 450$$
$59$ $$T + 24$$
$61$ $$T + 322$$
$67$ $$T + 196$$
$71$ $$T - 288$$
$73$ $$T + 430$$
$79$ $$T + 520$$
$83$ $$T + 156$$
$89$ $$T + 1026$$
$97$ $$T + 286$$