# Properties

 Label 3645.2.a.b Level $3645$ Weight $2$ Character orbit 3645.a Self dual yes Analytic conductor $29.105$ Analytic rank $1$ Dimension $3$ 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: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_{2} q^{4} + q^{5} + ( - 2 \beta_{2} - 2) q^{7} + (\beta_1 - 1) q^{8}+O(q^{10})$$ q - b1 * q^2 + b2 * q^4 + q^5 + (-2*b2 - 2) * q^7 + (b1 - 1) * q^8 $$q - \beta_1 q^{2} + \beta_{2} q^{4} + q^{5} + ( - 2 \beta_{2} - 2) q^{7} + (\beta_1 - 1) q^{8} - \beta_1 q^{10} + (3 \beta_{2} - \beta_1) q^{11} + (\beta_{2} - \beta_1) q^{13} + (4 \beta_1 + 2) q^{14} + ( - 3 \beta_{2} + \beta_1 - 2) q^{16} + (\beta_{2} - 2 \beta_1 - 4) q^{17} + ( - 2 \beta_{2} + 3 \beta_1 + 5) q^{19} + \beta_{2} q^{20} + (\beta_{2} - 3 \beta_1 - 1) q^{22} + ( - 5 \beta_{2} + 4 \beta_1) q^{23} + q^{25} + (\beta_{2} - \beta_1 + 1) q^{26} + ( - 2 \beta_1 - 4) q^{28} + (\beta_{2} - 3 \beta_1 - 1) q^{29} + ( - 2 \beta_{2} + 4 \beta_1 + 6) q^{31} + ( - \beta_{2} + 3 \beta_1 + 3) q^{32} + (2 \beta_{2} + 3 \beta_1 + 3) q^{34} + ( - 2 \beta_{2} - 2) q^{35} + (\beta_{2} - 9) q^{37} + ( - 3 \beta_{2} - 3 \beta_1 - 4) q^{38} + (\beta_1 - 1) q^{40} + (3 \beta_{2} - 4 \beta_1 - 1) q^{41} + (\beta_{2} + 2 \beta_1 - 9) q^{43} + ( - 3 \beta_{2} + 2 \beta_1 + 5) q^{44} + ( - 4 \beta_{2} + 5 \beta_1 - 3) q^{46} + ( - 2 \beta_{2} + 4 \beta_1 - 1) q^{47} + (4 \beta_{2} + 4 \beta_1 + 5) q^{49} - \beta_1 q^{50} + ( - \beta_{2} + 1) q^{52} + (5 \beta_{2} + 4) q^{53} + (3 \beta_{2} - \beta_1) q^{55} + (2 \beta_{2} - 4 \beta_1) q^{56} + (3 \beta_{2} + 5) q^{58} + ( - 3 \beta_{2} + 2 \beta_1 - 6) q^{59} + (\beta_{2} - 2 \beta_1 + 1) q^{61} + ( - 4 \beta_{2} - 4 \beta_1 - 6) q^{62} + (3 \beta_{2} - 4 \beta_1 - 1) q^{64} + (\beta_{2} - \beta_1) q^{65} + ( - 4 \beta_{2} + 5 \beta_1) q^{67} + ( - 5 \beta_{2} - \beta_1) q^{68} + (4 \beta_1 + 2) q^{70} + ( - 4 \beta_1 - 1) q^{71} + (2 \beta_{2} + 3) q^{73} + (8 \beta_1 - 1) q^{74} + (7 \beta_{2} + \beta_1 - 1) q^{76} + ( - 2 \beta_1 - 10) q^{77} + ( - \beta_{2} + 5 \beta_1 - 3) q^{79} + ( - 3 \beta_{2} + \beta_1 - 2) q^{80} + (4 \beta_{2} - 2 \beta_1 + 5) q^{82} + (4 \beta_{2} - \beta_1 - 10) q^{83} + (\beta_{2} - 2 \beta_1 - 4) q^{85} + ( - 2 \beta_{2} + 8 \beta_1 - 5) q^{86} + ( - 4 \beta_{2} + 4 \beta_1 + 1) q^{88} + (\beta_{2} + 6 \beta_1 - 8) q^{89} + (2 \beta_1 - 2) q^{91} + (5 \beta_{2} - \beta_1 - 6) q^{92} + ( - 4 \beta_{2} + 3 \beta_1 - 6) q^{94} + ( - 2 \beta_{2} + 3 \beta_1 + 5) q^{95} + ( - 2 \beta_{2} + 3 \beta_1) q^{97} + ( - 4 \beta_{2} - 9 \beta_1 - 12) q^{98}+O(q^{100})$$ q - b1 * q^2 + b2 * q^4 + q^5 + (-2*b2 - 2) * q^7 + (b1 - 1) * q^8 - b1 * q^10 + (3*b2 - b1) * q^11 + (b2 - b1) * q^13 + (4*b1 + 2) * q^14 + (-3*b2 + b1 - 2) * q^16 + (b2 - 2*b1 - 4) * q^17 + (-2*b2 + 3*b1 + 5) * q^19 + b2 * q^20 + (b2 - 3*b1 - 1) * q^22 + (-5*b2 + 4*b1) * q^23 + q^25 + (b2 - b1 + 1) * q^26 + (-2*b1 - 4) * q^28 + (b2 - 3*b1 - 1) * q^29 + (-2*b2 + 4*b1 + 6) * q^31 + (-b2 + 3*b1 + 3) * q^32 + (2*b2 + 3*b1 + 3) * q^34 + (-2*b2 - 2) * q^35 + (b2 - 9) * q^37 + (-3*b2 - 3*b1 - 4) * q^38 + (b1 - 1) * q^40 + (3*b2 - 4*b1 - 1) * q^41 + (b2 + 2*b1 - 9) * q^43 + (-3*b2 + 2*b1 + 5) * q^44 + (-4*b2 + 5*b1 - 3) * q^46 + (-2*b2 + 4*b1 - 1) * q^47 + (4*b2 + 4*b1 + 5) * q^49 - b1 * q^50 + (-b2 + 1) * q^52 + (5*b2 + 4) * q^53 + (3*b2 - b1) * q^55 + (2*b2 - 4*b1) * q^56 + (3*b2 + 5) * q^58 + (-3*b2 + 2*b1 - 6) * q^59 + (b2 - 2*b1 + 1) * q^61 + (-4*b2 - 4*b1 - 6) * q^62 + (3*b2 - 4*b1 - 1) * q^64 + (b2 - b1) * q^65 + (-4*b2 + 5*b1) * q^67 + (-5*b2 - b1) * q^68 + (4*b1 + 2) * q^70 + (-4*b1 - 1) * q^71 + (2*b2 + 3) * q^73 + (8*b1 - 1) * q^74 + (7*b2 + b1 - 1) * q^76 + (-2*b1 - 10) * q^77 + (-b2 + 5*b1 - 3) * q^79 + (-3*b2 + b1 - 2) * q^80 + (4*b2 - 2*b1 + 5) * q^82 + (4*b2 - b1 - 10) * q^83 + (b2 - 2*b1 - 4) * q^85 + (-2*b2 + 8*b1 - 5) * q^86 + (-4*b2 + 4*b1 + 1) * q^88 + (b2 + 6*b1 - 8) * q^89 + (2*b1 - 2) * q^91 + (5*b2 - b1 - 6) * q^92 + (-4*b2 + 3*b1 - 6) * q^94 + (-2*b2 + 3*b1 + 5) * q^95 + (-2*b2 + 3*b1) * q^97 + (-4*b2 - 9*b1 - 12) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10})$$ 3 * q + 3 * q^5 - 6 * q^7 - 3 * q^8 $$3 q + 3 q^{5} - 6 q^{7} - 3 q^{8} + 6 q^{14} - 6 q^{16} - 12 q^{17} + 15 q^{19} - 3 q^{22} + 3 q^{25} + 3 q^{26} - 12 q^{28} - 3 q^{29} + 18 q^{31} + 9 q^{32} + 9 q^{34} - 6 q^{35} - 27 q^{37} - 12 q^{38} - 3 q^{40} - 3 q^{41} - 27 q^{43} + 15 q^{44} - 9 q^{46} - 3 q^{47} + 15 q^{49} + 3 q^{52} + 12 q^{53} + 15 q^{58} - 18 q^{59} + 3 q^{61} - 18 q^{62} - 3 q^{64} + 6 q^{70} - 3 q^{71} + 9 q^{73} - 3 q^{74} - 3 q^{76} - 30 q^{77} - 9 q^{79} - 6 q^{80} + 15 q^{82} - 30 q^{83} - 12 q^{85} - 15 q^{86} + 3 q^{88} - 24 q^{89} - 6 q^{91} - 18 q^{92} - 18 q^{94} + 15 q^{95} - 36 q^{98}+O(q^{100})$$ 3 * q + 3 * q^5 - 6 * q^7 - 3 * q^8 + 6 * q^14 - 6 * q^16 - 12 * q^17 + 15 * q^19 - 3 * q^22 + 3 * q^25 + 3 * q^26 - 12 * q^28 - 3 * q^29 + 18 * q^31 + 9 * q^32 + 9 * q^34 - 6 * q^35 - 27 * q^37 - 12 * q^38 - 3 * q^40 - 3 * q^41 - 27 * q^43 + 15 * q^44 - 9 * q^46 - 3 * q^47 + 15 * q^49 + 3 * q^52 + 12 * q^53 + 15 * q^58 - 18 * q^59 + 3 * q^61 - 18 * q^62 - 3 * q^64 + 6 * q^70 - 3 * q^71 + 9 * q^73 - 3 * q^74 - 3 * q^76 - 30 * q^77 - 9 * q^79 - 6 * q^80 + 15 * q^82 - 30 * q^83 - 12 * q^85 - 15 * q^86 + 3 * q^88 - 24 * q^89 - 6 * q^91 - 18 * q^92 - 18 * q^94 + 15 * q^95 - 36 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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
 1.87939 −0.347296 −1.53209
−1.87939 0 1.53209 1.00000 0 −5.06418 0.879385 0 −1.87939
1.2 0.347296 0 −1.87939 1.00000 0 1.75877 −1.34730 0 0.347296
1.3 1.53209 0 0.347296 1.00000 0 −2.69459 −2.53209 0 1.53209
 $$n$$: 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 3645.2.a.b yes 3
3.b odd 2 1 3645.2.a.a 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3645.2.a.a 3 3.b odd 2 1
3645.2.a.b yes 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 3T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3645))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 3T + 1$$
$3$ $$T^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} + 6T^{2} - 24$$
$11$ $$T^{3} - 21T + 37$$
$13$ $$T^{3} - 3T - 1$$
$17$ $$T^{3} + 12 T^{2} + 39 T + 19$$
$19$ $$T^{3} - 15 T^{2} + 54 T + 17$$
$23$ $$T^{3} - 63T - 9$$
$29$ $$T^{3} + 3 T^{2} - 18 T - 37$$
$31$ $$T^{3} - 18 T^{2} + 72 T + 72$$
$37$ $$T^{3} + 27 T^{2} + 240 T + 703$$
$41$ $$T^{3} + 3 T^{2} - 36 T - 127$$
$43$ $$T^{3} + 27 T^{2} + 222 T + 503$$
$47$ $$T^{3} + 3 T^{2} - 33 T + 37$$
$53$ $$T^{3} - 12 T^{2} - 27 T + 361$$
$59$ $$T^{3} + 18 T^{2} + 87 T + 73$$
$61$ $$T^{3} - 3 T^{2} - 6 T - 1$$
$67$ $$T^{3} - 63T + 171$$
$71$ $$T^{3} + 3 T^{2} - 45 T + 17$$
$73$ $$T^{3} - 9 T^{2} + 15 T + 17$$
$79$ $$T^{3} + 9 T^{2} - 36 T - 153$$
$83$ $$T^{3} + 30 T^{2} + 261 T + 699$$
$89$ $$T^{3} + 24 T^{2} + 63 T - 969$$
$97$ $$T^{3} - 21T + 37$$