# Properties

 Label 2300.2.a.k Level $2300$ Weight $2$ Character orbit 2300.a Self dual yes Analytic conductor $18.366$ 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] = [2300,2,Mod(1,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2300.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2300 = 2^{2} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2300.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: $$18.3655924649$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.321.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 1$$ x^3 - x^2 - 4*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{3} + 2 \beta_1 q^{7} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^3 + 2*b1 * q^7 + (b2 - b1 + 1) * q^9 $$q + ( - \beta_1 + 1) q^{3} + 2 \beta_1 q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + (2 \beta_1 - 2) q^{11} + (\beta_{2} - \beta_1 - 2) q^{13} + ( - 2 \beta_{2} - 2) q^{17} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{19} + ( - 2 \beta_{2} - 6) q^{21} - q^{23} + (2 \beta_{2} + \beta_1 + 2) q^{27} + ( - 3 \beta_{2} - 2 \beta_1 - 3) q^{29} + (\beta_{2} + 2 \beta_1 - 7) q^{31} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{33} - 2 \beta_1 q^{37} + (2 \beta_{2} + \beta_1 + 2) q^{39} + (4 \beta_{2} - \beta_1 + 3) q^{41} + ( - 2 \beta_{2} + 2 \beta_1) q^{43} + (3 \beta_{2} + 3 \beta_1 + 2) q^{47} + (4 \beta_{2} + 4 \beta_1 + 5) q^{49} + ( - 2 \beta_{2} + 4 \beta_1 - 4) q^{51} + (4 \beta_{2} + 2 \beta_1 - 2) q^{53} + (4 \beta_1 + 2) q^{57} + (2 \beta_{2} - 4 \beta_1 - 1) q^{59} + (2 \beta_{2} - 2 \beta_1 - 6) q^{61} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{63} + ( - 6 \beta_{2} - 4 \beta_1 - 2) q^{67} + (\beta_1 - 1) q^{69} + ( - \beta_{2} - \beta_1 - 10) q^{71} + (\beta_{2} + 2 \beta_1 + 5) q^{73} + (4 \beta_{2} + 12) q^{77} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{79} + ( - 2 \beta_{2} - \beta_1 - 2) q^{81} + (2 \beta_{2} - 2) q^{83} + ( - \beta_{2} + 6 \beta_1) q^{87} + (2 \beta_{2} - 4 \beta_1 + 4) q^{89} + ( - 2 \beta_{2} - 4 \beta_1 - 8) q^{91} + ( - \beta_{2} + 6 \beta_1 - 12) q^{93} + (2 \beta_{2} - 2 \beta_1 + 4) q^{97} + ( - 4 \beta_{2} + 4 \beta_1 - 10) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^3 + 2*b1 * q^7 + (b2 - b1 + 1) * q^9 + (2*b1 - 2) * q^11 + (b2 - b1 - 2) * q^13 + (-2*b2 - 2) * q^17 + (-2*b2 - 2*b1 - 2) * q^19 + (-2*b2 - 6) * q^21 - q^23 + (2*b2 + b1 + 2) * q^27 + (-3*b2 - 2*b1 - 3) * q^29 + (b2 + 2*b1 - 7) * q^31 + (-2*b2 + 2*b1 - 8) * q^33 - 2*b1 * q^37 + (2*b2 + b1 + 2) * q^39 + (4*b2 - b1 + 3) * q^41 + (-2*b2 + 2*b1) * q^43 + (3*b2 + 3*b1 + 2) * q^47 + (4*b2 + 4*b1 + 5) * q^49 + (-2*b2 + 4*b1 - 4) * q^51 + (4*b2 + 2*b1 - 2) * q^53 + (4*b1 + 2) * q^57 + (2*b2 - 4*b1 - 1) * q^59 + (2*b2 - 2*b1 - 6) * q^61 + (-2*b2 + 2*b1 - 8) * q^63 + (-6*b2 - 4*b1 - 2) * q^67 + (b1 - 1) * q^69 + (-b2 - b1 - 10) * q^71 + (b2 + 2*b1 + 5) * q^73 + (4*b2 + 12) * q^77 + (-2*b2 + 2*b1 - 8) * q^79 + (-2*b2 - b1 - 2) * q^81 + (2*b2 - 2) * q^83 + (-b2 + 6*b1) * q^87 + (2*b2 - 4*b1 + 4) * q^89 + (-2*b2 - 4*b1 - 8) * q^91 + (-b2 + 6*b1 - 12) * q^93 + (2*b2 - 2*b1 + 4) * q^97 + (-4*b2 + 4*b1 - 10) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} + 2 q^{7} + q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 + 2 * q^7 + q^9 $$3 q + 2 q^{3} + 2 q^{7} + q^{9} - 4 q^{11} - 8 q^{13} - 4 q^{17} - 6 q^{19} - 16 q^{21} - 3 q^{23} + 5 q^{27} - 8 q^{29} - 20 q^{31} - 20 q^{33} - 2 q^{37} + 5 q^{39} + 4 q^{41} + 4 q^{43} + 6 q^{47} + 15 q^{49} - 6 q^{51} - 8 q^{53} + 10 q^{57} - 9 q^{59} - 22 q^{61} - 20 q^{63} - 4 q^{67} - 2 q^{69} - 30 q^{71} + 16 q^{73} + 32 q^{77} - 20 q^{79} - 5 q^{81} - 8 q^{83} + 7 q^{87} + 6 q^{89} - 26 q^{91} - 29 q^{93} + 8 q^{97} - 22 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 + 2 * q^7 + q^9 - 4 * q^11 - 8 * q^13 - 4 * q^17 - 6 * q^19 - 16 * q^21 - 3 * q^23 + 5 * q^27 - 8 * q^29 - 20 * q^31 - 20 * q^33 - 2 * q^37 + 5 * q^39 + 4 * q^41 + 4 * q^43 + 6 * q^47 + 15 * q^49 - 6 * q^51 - 8 * q^53 + 10 * q^57 - 9 * q^59 - 22 * q^61 - 20 * q^63 - 4 * q^67 - 2 * q^69 - 30 * q^71 + 16 * q^73 + 32 * q^77 - 20 * q^79 - 5 * q^81 - 8 * q^83 + 7 * q^87 + 6 * q^89 - 26 * q^91 - 29 * q^93 + 8 * q^97 - 22 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3

## 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
 2.46050 0.239123 −1.69963
0 −1.46050 0 0 0 4.92101 0 −0.866926 0
1.2 0 0.760877 0 0 0 0.478247 0 −2.42107 0
1.3 0 2.69963 0 0 0 −3.39926 0 4.28799 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$+1$$

## 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 2300.2.a.k yes 3
4.b odd 2 1 9200.2.a.cc 3
5.b even 2 1 2300.2.a.j 3
5.c odd 4 2 2300.2.c.i 6
20.d odd 2 1 9200.2.a.cg 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.2.a.j 3 5.b even 2 1
2300.2.a.k yes 3 1.a even 1 1 trivial
2300.2.c.i 6 5.c odd 4 2
9200.2.a.cc 3 4.b odd 2 1
9200.2.a.cg 3 20.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2300))$$:

 $$T_{3}^{3} - 2T_{3}^{2} - 3T_{3} + 3$$ T3^3 - 2*T3^2 - 3*T3 + 3 $$T_{7}^{3} - 2T_{7}^{2} - 16T_{7} + 8$$ T7^3 - 2*T7^2 - 16*T7 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 2 T^{2} + \cdots + 3$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 2 T^{2} + \cdots + 8$$
$11$ $$T^{3} + 4 T^{2} + \cdots - 24$$
$13$ $$T^{3} + 8 T^{2} + \cdots - 27$$
$17$ $$T^{3} + 4 T^{2} + \cdots - 72$$
$19$ $$T^{3} + 6 T^{2} + \cdots - 56$$
$23$ $$(T + 1)^{3}$$
$29$ $$T^{3} + 8 T^{2} + \cdots - 257$$
$31$ $$T^{3} + 20 T^{2} + \cdots + 127$$
$37$ $$T^{3} + 2 T^{2} + \cdots - 8$$
$41$ $$T^{3} - 4 T^{2} + \cdots + 321$$
$43$ $$T^{3} - 4 T^{2} + \cdots + 168$$
$47$ $$T^{3} - 6 T^{2} + \cdots + 127$$
$53$ $$T^{3} + 8 T^{2} + \cdots + 72$$
$59$ $$T^{3} + 9 T^{2} + \cdots - 721$$
$61$ $$T^{3} + 22 T^{2} + \cdots - 72$$
$67$ $$T^{3} + 4 T^{2} + \cdots - 1176$$
$71$ $$T^{3} + 30 T^{2} + \cdots + 911$$
$73$ $$T^{3} - 16 T^{2} + \cdots - 77$$
$79$ $$T^{3} + 20 T^{2} + \cdots + 72$$
$83$ $$T^{3} + 8 T^{2} + \cdots - 8$$
$89$ $$T^{3} - 6 T^{2} + \cdots - 216$$
$97$ $$T^{3} - 8 T^{2} + \cdots + 8$$