# Properties

 Label 3800.1.o.e Level $3800$ Weight $1$ Character orbit 3800.o Self dual yes Analytic conductor $1.896$ Analytic rank $0$ Dimension $3$ Projective image $D_{9}$ CM discriminant -152 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,1,Mod(1101,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3800.1101");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3800.o (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$1.89644704801$$ Analytic rank: $$0$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.8340544000000.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( - \beta_{2} + \beta_1) q^{3} + q^{4} + ( - \beta_{2} + \beta_1) q^{6} + \beta_1 q^{7} + q^{8} + ( - \beta_1 + 1) q^{9}+O(q^{10})$$ q + q^2 + (-b2 + b1) * q^3 + q^4 + (-b2 + b1) * q^6 + b1 * q^7 + q^8 + (-b1 + 1) * q^9 $$q + q^{2} + ( - \beta_{2} + \beta_1) q^{3} + q^{4} + ( - \beta_{2} + \beta_1) q^{6} + \beta_1 q^{7} + q^{8} + ( - \beta_1 + 1) q^{9} + ( - \beta_{2} + \beta_1) q^{12} + \beta_{2} q^{13} + \beta_1 q^{14} + q^{16} - \beta_{2} q^{17} + ( - \beta_1 + 1) q^{18} - q^{19} + (\beta_{2} - \beta_1 + 1) q^{21} + (\beta_{2} - \beta_1) q^{23} + ( - \beta_{2} + \beta_1) q^{24} + \beta_{2} q^{26} + ( - \beta_{2} + \beta_1 - 1) q^{27} + \beta_1 q^{28} - \beta_{2} q^{29} + q^{32} - \beta_{2} q^{34} + ( - \beta_1 + 1) q^{36} - q^{37} - q^{38} + (\beta_{2} - 1) q^{39} + (\beta_{2} - \beta_1 + 1) q^{42} + (\beta_{2} - \beta_1) q^{46} + q^{47} + ( - \beta_{2} + \beta_1) q^{48} + (\beta_{2} + 1) q^{49} + ( - \beta_{2} + 1) q^{51} + \beta_{2} q^{52} - \beta_1 q^{53} + ( - \beta_{2} + \beta_1 - 1) q^{54} + \beta_1 q^{56} + (\beta_{2} - \beta_1) q^{57} - \beta_{2} q^{58} + (\beta_{2} - \beta_1) q^{59} + ( - \beta_{2} + \beta_1 - 2) q^{63} + q^{64} - \beta_1 q^{67} - \beta_{2} q^{68} + (\beta_1 - 2) q^{69} + ( - \beta_1 + 1) q^{72} + (\beta_{2} - \beta_1) q^{73} - q^{74} - q^{76} + (\beta_{2} - 1) q^{78} + (\beta_{2} - \beta_1 + 1) q^{81} + (\beta_{2} - \beta_1 + 1) q^{84} + ( - \beta_{2} + 1) q^{87} + (\beta_1 + 1) q^{91} + (\beta_{2} - \beta_1) q^{92} + q^{94} + ( - \beta_{2} + \beta_1) q^{96} + (\beta_{2} + 1) q^{98}+O(q^{100})$$ q + q^2 + (-b2 + b1) * q^3 + q^4 + (-b2 + b1) * q^6 + b1 * q^7 + q^8 + (-b1 + 1) * q^9 + (-b2 + b1) * q^12 + b2 * q^13 + b1 * q^14 + q^16 - b2 * q^17 + (-b1 + 1) * q^18 - q^19 + (b2 - b1 + 1) * q^21 + (b2 - b1) * q^23 + (-b2 + b1) * q^24 + b2 * q^26 + (-b2 + b1 - 1) * q^27 + b1 * q^28 - b2 * q^29 + q^32 - b2 * q^34 + (-b1 + 1) * q^36 - q^37 - q^38 + (b2 - 1) * q^39 + (b2 - b1 + 1) * q^42 + (b2 - b1) * q^46 + q^47 + (-b2 + b1) * q^48 + (b2 + 1) * q^49 + (-b2 + 1) * q^51 + b2 * q^52 - b1 * q^53 + (-b2 + b1 - 1) * q^54 + b1 * q^56 + (b2 - b1) * q^57 - b2 * q^58 + (b2 - b1) * q^59 + (-b2 + b1 - 2) * q^63 + q^64 - b1 * q^67 - b2 * q^68 + (b1 - 2) * q^69 + (-b1 + 1) * q^72 + (b2 - b1) * q^73 - q^74 - q^76 + (b2 - 1) * q^78 + (b2 - b1 + 1) * q^81 + (b2 - b1 + 1) * q^84 + (-b2 + 1) * q^87 + (b1 + 1) * q^91 + (b2 - b1) * q^92 + q^94 + (-b2 + b1) * q^96 + (b2 + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 + 3 * q^9 $$3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 3 q^{9} + 3 q^{16} + 3 q^{18} - 3 q^{19} + 3 q^{21} - 3 q^{27} + 3 q^{32} + 3 q^{36} - 3 q^{37} - 3 q^{38} - 3 q^{39} + 3 q^{42} + 3 q^{47} + 3 q^{49} + 3 q^{51} - 3 q^{54} - 6 q^{63} + 3 q^{64} - 6 q^{69} + 3 q^{72} - 3 q^{74} - 3 q^{76} - 3 q^{78} + 3 q^{81} + 3 q^{84} + 3 q^{87} + 3 q^{91} + 3 q^{94} + 3 q^{98}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 + 3 * q^9 + 3 * q^16 + 3 * q^18 - 3 * q^19 + 3 * q^21 - 3 * q^27 + 3 * q^32 + 3 * q^36 - 3 * q^37 - 3 * q^38 - 3 * q^39 + 3 * q^42 + 3 * q^47 + 3 * q^49 + 3 * q^51 - 3 * q^54 - 6 * q^63 + 3 * q^64 - 6 * q^69 + 3 * q^72 - 3 * q^74 - 3 * q^76 - 3 * q^78 + 3 * q^81 + 3 * q^84 + 3 * q^87 + 3 * q^91 + 3 * q^94 + 3 * 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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1101.1
 −1.53209 1.87939 −0.347296
1.00000 −1.87939 1.00000 0 −1.87939 −1.53209 1.00000 2.53209 0
1101.2 1.00000 0.347296 1.00000 0 0.347296 1.87939 1.00000 −0.879385 0
1101.3 1.00000 1.53209 1.00000 0 1.53209 −0.347296 1.00000 1.34730 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
152.g odd 2 1 CM by $$\Q(\sqrt{-38})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.o.e yes 3
5.b even 2 1 3800.1.o.c 3
5.c odd 4 2 3800.1.b.d 6
8.b even 2 1 3800.1.o.d yes 3
19.b odd 2 1 3800.1.o.d yes 3
40.f even 2 1 3800.1.o.f yes 3
40.i odd 4 2 3800.1.b.c 6
95.d odd 2 1 3800.1.o.f yes 3
95.g even 4 2 3800.1.b.c 6
152.g odd 2 1 CM 3800.1.o.e yes 3
760.b odd 2 1 3800.1.o.c 3
760.t even 4 2 3800.1.b.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.1.b.c 6 40.i odd 4 2
3800.1.b.c 6 95.g even 4 2
3800.1.b.d 6 5.c odd 4 2
3800.1.b.d 6 760.t even 4 2
3800.1.o.c 3 5.b even 2 1
3800.1.o.c 3 760.b odd 2 1
3800.1.o.d yes 3 8.b even 2 1
3800.1.o.d yes 3 19.b odd 2 1
3800.1.o.e yes 3 1.a even 1 1 trivial
3800.1.o.e yes 3 152.g odd 2 1 CM
3800.1.o.f yes 3 40.f even 2 1
3800.1.o.f yes 3 95.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3800, [\chi])$$:

 $$T_{3}^{3} - 3T_{3} + 1$$ T3^3 - 3*T3 + 1 $$T_{7}^{3} - 3T_{7} - 1$$ T7^3 - 3*T7 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$T^{3} - 3T + 1$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 3T - 1$$
$11$ $$T^{3}$$
$13$ $$T^{3} - 3T + 1$$
$17$ $$T^{3} - 3T - 1$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} - 3T - 1$$
$29$ $$T^{3} - 3T - 1$$
$31$ $$T^{3}$$
$37$ $$(T + 1)^{3}$$
$41$ $$T^{3}$$
$43$ $$T^{3}$$
$47$ $$(T - 1)^{3}$$
$53$ $$T^{3} - 3T + 1$$
$59$ $$T^{3} - 3T - 1$$
$61$ $$T^{3}$$
$67$ $$T^{3} - 3T + 1$$
$71$ $$T^{3}$$
$73$ $$T^{3} - 3T - 1$$
$79$ $$T^{3}$$
$83$ $$T^{3}$$
$89$ $$T^{3}$$
$97$ $$T^{3}$$