# Properties

 Label 45.4.j.a Level $45$ Weight $4$ Character orbit 45.j Analytic conductor $2.655$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("45.4");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 45.j (of order $$6$$, degree $$2$$, 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: $$2.65508595026$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 54 q^{4} + 3 q^{5} - 12 q^{6} - 18 q^{9}+O(q^{10})$$ 32 * q + 54 * q^4 + 3 * q^5 - 12 * q^6 - 18 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 54 q^{4} + 3 q^{5} - 12 q^{6} - 18 q^{9} - 20 q^{10} + 90 q^{11} - 102 q^{14} - 87 q^{15} - 146 q^{16} - 8 q^{19} - 6 q^{20} + 30 q^{21} + 462 q^{24} + 71 q^{25} - 936 q^{26} - 516 q^{29} - 66 q^{30} - 38 q^{31} + 212 q^{34} - 534 q^{35} + 864 q^{36} + 330 q^{39} + 44 q^{40} + 576 q^{41} + 3288 q^{44} + 1053 q^{45} - 580 q^{46} - 4 q^{49} + 558 q^{50} + 1260 q^{51} - 3726 q^{54} + 30 q^{55} + 2430 q^{56} - 2202 q^{59} - 5052 q^{60} - 20 q^{61} + 644 q^{64} + 339 q^{65} - 5052 q^{66} + 1452 q^{69} + 636 q^{70} - 5904 q^{71} - 4080 q^{74} + 2283 q^{75} + 396 q^{76} - 218 q^{79} + 2532 q^{80} + 198 q^{81} + 4662 q^{84} - 704 q^{85} + 6108 q^{86} + 8148 q^{89} + 6408 q^{90} - 1884 q^{91} - 1078 q^{94} - 1692 q^{95} + 11874 q^{96} - 1602 q^{99}+O(q^{100})$$ 32 * q + 54 * q^4 + 3 * q^5 - 12 * q^6 - 18 * q^9 - 20 * q^10 + 90 * q^11 - 102 * q^14 - 87 * q^15 - 146 * q^16 - 8 * q^19 - 6 * q^20 + 30 * q^21 + 462 * q^24 + 71 * q^25 - 936 * q^26 - 516 * q^29 - 66 * q^30 - 38 * q^31 + 212 * q^34 - 534 * q^35 + 864 * q^36 + 330 * q^39 + 44 * q^40 + 576 * q^41 + 3288 * q^44 + 1053 * q^45 - 580 * q^46 - 4 * q^49 + 558 * q^50 + 1260 * q^51 - 3726 * q^54 + 30 * q^55 + 2430 * q^56 - 2202 * q^59 - 5052 * q^60 - 20 * q^61 + 644 * q^64 + 339 * q^65 - 5052 * q^66 + 1452 * q^69 + 636 * q^70 - 5904 * q^71 - 4080 * q^74 + 2283 * q^75 + 396 * q^76 - 218 * q^79 + 2532 * q^80 + 198 * q^81 + 4662 * q^84 - 704 * q^85 + 6108 * q^86 + 8148 * q^89 + 6408 * q^90 - 1884 * q^91 - 1078 * q^94 - 1692 * q^95 + 11874 * q^96 - 1602 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −4.37720 + 2.52718i 5.16592 + 0.559681i 8.77324 15.1957i −3.07459 10.7493i −24.0267 + 10.6054i 18.1967 10.5059i 48.2513i 26.3735 + 5.78253i 40.6234 + 39.2817i
4.2 −4.35066 + 2.51186i −1.99648 4.79730i 8.61884 14.9283i 5.42265 + 9.77726i 20.7361 + 15.8565i −4.66092 + 2.69099i 46.4074i −19.0281 + 19.1554i −48.1512 28.9166i
4.3 −3.53014 + 2.03813i −2.78227 + 4.38850i 4.30793 7.46156i −8.75492 + 6.95352i 0.877481 21.1627i 11.5553 6.67148i 2.51043i −11.5179 24.4200i 16.7339 42.3906i
4.4 −2.67377 + 1.54370i 2.05649 + 4.77188i 0.766022 1.32679i 11.1766 0.287878i −12.8649 9.58430i −27.1243 + 15.6602i 19.9692i −18.5417 + 19.6266i −29.4393 + 18.0231i
4.5 −2.31667 + 1.33753i −5.14958 0.694136i −0.422017 + 0.730954i 7.31972 8.45113i 12.8583 5.27964i 13.3501 7.70766i 23.6584i 26.0363 + 7.14902i −5.65374 + 29.3689i
4.6 −2.13376 + 1.23192i 1.50557 4.97325i −0.964722 + 1.67095i −9.64354 5.65704i 2.91415 + 12.4665i −16.6624 + 9.62007i 24.4647i −22.4665 14.9752i 27.5460 + 0.190629i
4.7 −0.680118 + 0.392666i 3.65242 3.69592i −3.69163 + 6.39408i 10.9698 + 2.15960i −1.03281 + 3.94785i 18.1117 10.4568i 12.0810i −0.319654 26.9981i −8.30875 + 2.83868i
4.8 −0.410487 + 0.236995i 4.58061 + 2.45316i −3.88767 + 6.73364i −8.32653 + 7.46116i −2.46167 + 0.0785933i 7.10710 4.10329i 7.47735i 14.9641 + 22.4739i 1.64968 5.03606i
4.9 0.410487 0.236995i −4.58061 2.45316i −3.88767 + 6.73364i −2.29829 + 10.9416i −2.46167 + 0.0785933i −7.10710 + 4.10329i 7.47735i 14.9641 + 22.4739i 1.64968 + 5.03606i
4.10 0.680118 0.392666i −3.65242 + 3.69592i −3.69163 + 6.39408i −7.35516 8.42031i −1.03281 + 3.94785i −18.1117 + 10.4568i 12.0810i −0.319654 26.9981i −8.30875 2.83868i
4.11 2.13376 1.23192i −1.50557 + 4.97325i −0.964722 + 1.67095i 9.72091 + 5.52303i 2.91415 + 12.4665i 16.6624 9.62007i 24.4647i −22.4665 14.9752i 27.5460 0.190629i
4.12 2.31667 1.33753i 5.14958 + 0.694136i −0.422017 + 0.730954i 3.65904 10.5646i 12.8583 5.27964i −13.3501 + 7.70766i 23.6584i 26.0363 + 7.14902i −5.65374 29.3689i
4.13 2.67377 1.54370i −2.05649 4.77188i 0.766022 1.32679i −5.33901 9.82319i −12.8649 9.58430i 27.1243 15.6602i 19.9692i −18.5417 + 19.6266i −29.4393 18.0231i
4.14 3.53014 2.03813i 2.78227 4.38850i 4.30793 7.46156i −1.64447 + 11.0587i 0.877481 21.1627i −11.5553 + 6.67148i 2.51043i −11.5179 24.4200i 16.7339 + 42.3906i
4.15 4.35066 2.51186i 1.99648 + 4.79730i 8.61884 14.9283i −11.1787 + 0.192474i 20.7361 + 15.8565i 4.66092 2.69099i 46.4074i −19.0281 + 19.1554i −48.1512 + 28.9166i
4.16 4.37720 2.52718i −5.16592 0.559681i 8.77324 15.1957i 10.8464 2.71197i −24.0267 + 10.6054i −18.1967 + 10.5059i 48.2513i 26.3735 + 5.78253i 40.6234 39.2817i
34.1 −4.37720 2.52718i 5.16592 0.559681i 8.77324 + 15.1957i −3.07459 + 10.7493i −24.0267 10.6054i 18.1967 + 10.5059i 48.2513i 26.3735 5.78253i 40.6234 39.2817i
34.2 −4.35066 2.51186i −1.99648 + 4.79730i 8.61884 + 14.9283i 5.42265 9.77726i 20.7361 15.8565i −4.66092 2.69099i 46.4074i −19.0281 19.1554i −48.1512 + 28.9166i
34.3 −3.53014 2.03813i −2.78227 4.38850i 4.30793 + 7.46156i −8.75492 6.95352i 0.877481 + 21.1627i 11.5553 + 6.67148i 2.51043i −11.5179 + 24.4200i 16.7339 + 42.3906i
34.4 −2.67377 1.54370i 2.05649 4.77188i 0.766022 + 1.32679i 11.1766 + 0.287878i −12.8649 + 9.58430i −27.1243 15.6602i 19.9692i −18.5417 19.6266i −29.4393 18.0231i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.4.j.a 32
3.b odd 2 1 135.4.j.a 32
5.b even 2 1 inner 45.4.j.a 32
5.c odd 4 2 225.4.e.g 32
9.c even 3 1 inner 45.4.j.a 32
9.c even 3 1 405.4.b.e 16
9.d odd 6 1 135.4.j.a 32
9.d odd 6 1 405.4.b.f 16
15.d odd 2 1 135.4.j.a 32
45.h odd 6 1 135.4.j.a 32
45.h odd 6 1 405.4.b.f 16
45.j even 6 1 inner 45.4.j.a 32
45.j even 6 1 405.4.b.e 16
45.k odd 12 2 225.4.e.g 32
45.k odd 12 2 2025.4.a.bk 16
45.l even 12 2 2025.4.a.bl 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.j.a 32 1.a even 1 1 trivial
45.4.j.a 32 5.b even 2 1 inner
45.4.j.a 32 9.c even 3 1 inner
45.4.j.a 32 45.j even 6 1 inner
135.4.j.a 32 3.b odd 2 1
135.4.j.a 32 9.d odd 6 1
135.4.j.a 32 15.d odd 2 1
135.4.j.a 32 45.h odd 6 1
225.4.e.g 32 5.c odd 4 2
225.4.e.g 32 45.k odd 12 2
405.4.b.e 16 9.c even 3 1
405.4.b.e 16 45.j even 6 1
405.4.b.f 16 9.d odd 6 1
405.4.b.f 16 45.h odd 6 1
2025.4.a.bk 16 45.k odd 12 2
2025.4.a.bl 16 45.l even 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(45, [\chi])$$.