# Properties

 Label 3645.2.a.k Level $3645$ Weight $2$ Character orbit 3645.a Self dual yes Analytic conductor $29.105$ Analytic rank $0$ Dimension $21$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3645,2,Mod(1,3645)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3645, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3645.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3645 = 3^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3645.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: $$29.1054715368$$ Analytic rank: $$0$$ Dimension: $$21$$ Twist minimal: no (minimal twist has level 135) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$21 q - 3 q^{2} + 27 q^{4} - 21 q^{5} + 12 q^{7} - 9 q^{8}+O(q^{10})$$ 21 * q - 3 * q^2 + 27 * q^4 - 21 * q^5 + 12 * q^7 - 9 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$21 q - 3 q^{2} + 27 q^{4} - 21 q^{5} + 12 q^{7} - 9 q^{8} + 3 q^{10} + 12 q^{13} + 39 q^{16} - 12 q^{17} + 24 q^{19} - 27 q^{20} + 18 q^{22} - 18 q^{23} + 21 q^{25} + 9 q^{26} + 30 q^{28} + 3 q^{29} + 24 q^{31} - 6 q^{32} + 18 q^{34} - 12 q^{35} + 24 q^{37} - 18 q^{38} + 9 q^{40} + 9 q^{41} + 42 q^{43} + 12 q^{44} + 30 q^{46} - 21 q^{47} + 39 q^{49} - 3 q^{50} + 36 q^{52} - 18 q^{53} + 30 q^{56} + 30 q^{58} + 6 q^{59} + 33 q^{61} - 18 q^{62} + 63 q^{64} - 12 q^{65} + 63 q^{67} - 36 q^{68} + 12 q^{71} + 39 q^{73} + 21 q^{74} + 48 q^{76} - 9 q^{77} + 48 q^{79} - 39 q^{80} + 24 q^{82} - 33 q^{83} + 12 q^{85} + 69 q^{86} + 54 q^{88} + 9 q^{89} + 69 q^{91} - 21 q^{92} + 30 q^{94} - 24 q^{95} + 30 q^{97} - 15 q^{98}+O(q^{100})$$ 21 * q - 3 * q^2 + 27 * q^4 - 21 * q^5 + 12 * q^7 - 9 * q^8 + 3 * q^10 + 12 * q^13 + 39 * q^16 - 12 * q^17 + 24 * q^19 - 27 * q^20 + 18 * q^22 - 18 * q^23 + 21 * q^25 + 9 * q^26 + 30 * q^28 + 3 * q^29 + 24 * q^31 - 6 * q^32 + 18 * q^34 - 12 * q^35 + 24 * q^37 - 18 * q^38 + 9 * q^40 + 9 * q^41 + 42 * q^43 + 12 * q^44 + 30 * q^46 - 21 * q^47 + 39 * q^49 - 3 * q^50 + 36 * q^52 - 18 * q^53 + 30 * q^56 + 30 * q^58 + 6 * q^59 + 33 * q^61 - 18 * q^62 + 63 * q^64 - 12 * q^65 + 63 * q^67 - 36 * q^68 + 12 * q^71 + 39 * q^73 + 21 * q^74 + 48 * q^76 - 9 * q^77 + 48 * q^79 - 39 * q^80 + 24 * q^82 - 33 * q^83 + 12 * q^85 + 69 * q^86 + 54 * q^88 + 9 * q^89 + 69 * q^91 - 21 * q^92 + 30 * q^94 - 24 * q^95 + 30 * q^97 - 15 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.72718 0 5.43749 −1.00000 0 1.54536 −9.37463 0 2.72718
1.2 −2.63794 0 4.95872 −1.00000 0 −0.0340824 −7.80492 0 2.63794
1.3 −2.57683 0 4.64004 −1.00000 0 −3.03943 −6.80293 0 2.57683
1.4 −2.34963 0 3.52074 −1.00000 0 4.64813 −3.57318 0 2.34963
1.5 −1.86636 0 1.48328 −1.00000 0 3.47631 0.964379 0 1.86636
1.6 −1.80271 0 1.24978 −1.00000 0 −3.41591 1.35244 0 1.80271
1.7 −1.68624 0 0.843417 −1.00000 0 2.05910 1.95028 0 1.68624
1.8 −1.02657 0 −0.946158 −1.00000 0 −4.10302 3.02443 0 1.02657
1.9 −0.923028 0 −1.14802 −1.00000 0 3.73011 2.90571 0 0.923028
1.10 −0.785389 0 −1.38316 −1.00000 0 1.42323 2.65710 0 0.785389
1.11 0.167968 0 −1.97179 −1.00000 0 −1.84136 −0.667132 0 −0.167968
1.12 0.258561 0 −1.93315 −1.00000 0 4.58199 −1.01696 0 −0.258561
1.13 0.299055 0 −1.91057 −1.00000 0 1.30634 −1.16947 0 −0.299055
1.14 0.821398 0 −1.32531 −1.00000 0 −2.45268 −2.73140 0 −0.821398
1.15 1.10205 0 −0.785475 −1.00000 0 −2.98606 −3.06975 0 −1.10205
1.16 1.47739 0 0.182671 −1.00000 0 3.69781 −2.68490 0 −1.47739
1.17 1.77329 0 1.14456 −1.00000 0 −2.80275 −1.51694 0 −1.77329
1.18 1.82612 0 1.33471 −1.00000 0 0.744888 −1.21490 0 −1.82612
1.19 2.36288 0 3.58321 −1.00000 0 4.48489 3.74094 0 −2.36288
1.20 2.55634 0 4.53488 −1.00000 0 −1.54869 6.48002 0 −2.55634
See all 21 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.21 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 3645.2.a.k 21
3.b odd 2 1 3645.2.a.l 21
27.e even 9 2 405.2.k.b 42
27.f odd 18 2 135.2.k.b 42
135.n odd 18 2 675.2.l.e 42
135.q even 36 4 675.2.u.d 84

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.k.b 42 27.f odd 18 2
405.2.k.b 42 27.e even 9 2
675.2.l.e 42 135.n odd 18 2
675.2.u.d 84 135.q even 36 4
3645.2.a.k 21 1.a even 1 1 trivial
3645.2.a.l 21 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{21} + 3 T_{2}^{20} - 30 T_{2}^{19} - 92 T_{2}^{18} + 372 T_{2}^{17} + 1173 T_{2}^{16} - 2477 T_{2}^{15} - 8088 T_{2}^{14} + 9657 T_{2}^{13} + 32891 T_{2}^{12} - 22665 T_{2}^{11} - 80781 T_{2}^{10} + 32123 T_{2}^{9} + \cdots - 171$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3645))$$.