# Properties

 Label 2300.2.a.m Level $2300$ Weight $2$ Character orbit 2300.a Self dual yes Analytic conductor $18.366$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.53121.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 7x^{2} - x + 2$$ x^4 - 7*x^2 - x + 2 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + ( - \beta_{3} + \beta_1) q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10})$$ q + b1 * q^3 + (-b3 + b1) * q^7 + (b2 + 1) * q^9 $$q + \beta_1 q^{3} + ( - \beta_{3} + \beta_1) q^{7} + (\beta_{2} + 1) q^{9} + ( - \beta_{3} - \beta_1) q^{11} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{13} + (2 \beta_1 + 2) q^{17} + (\beta_{3} + \beta_1 + 2) q^{19} + 2 q^{21} - q^{23} + (\beta_{3} + 1) q^{27} + ( - \beta_{3} - \beta_{2} - 2) q^{29} + (\beta_{2} - \beta_1 + 4) q^{31} + ( - 2 \beta_{2} - 6) q^{33} + ( - 2 \beta_{2} + 2 \beta_1) q^{37} + (\beta_{3} + 2 \beta_1 + 3) q^{39} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{41} + ( - \beta_{3} - \beta_1 - 2) q^{43} + (2 \beta_{3} + \beta_{2} + 4) q^{47} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{49} + (2 \beta_{2} + 2 \beta_1 + 8) q^{51} + (2 \beta_{3} - 2) q^{53} + (2 \beta_{2} + 2 \beta_1 + 6) q^{57} + (\beta_{3} + \beta_1 + 3) q^{59} + (2 \beta_{3} + 6) q^{61} + (3 \beta_{3} - \beta_1) q^{63} + (2 \beta_{2} + 4) q^{67} - \beta_1 q^{69} + ( - \beta_{2} - 2 \beta_1 + 4) q^{71} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{73} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{77} + ( - \beta_{3} + 3 \beta_1 + 6) q^{79} + ( - 2 \beta_{2} + \beta_1 - 1) q^{81} + ( - \beta_{3} - 3 \beta_1 - 2) q^{83} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 - 3) q^{87} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{89} + (3 \beta_{3} - 2 \beta_{2} - \beta_1 + 6) q^{91} + (\beta_{3} - \beta_{2} + 6 \beta_1 - 3) q^{93} + ( - 4 \beta_{3} + 2 \beta_{2} + 2) q^{97} + (\beta_{3} - 7 \beta_1 - 2) q^{99}+O(q^{100})$$ q + b1 * q^3 + (-b3 + b1) * q^7 + (b2 + 1) * q^9 + (-b3 - b1) * q^11 + (-b3 + b2 + b1) * q^13 + (2*b1 + 2) * q^17 + (b3 + b1 + 2) * q^19 + 2 * q^21 - q^23 + (b3 + 1) * q^27 + (-b3 - b2 - 2) * q^29 + (b2 - b1 + 4) * q^31 + (-2*b2 - 6) * q^33 + (-2*b2 + 2*b1) * q^37 + (b3 + 2*b1 + 3) * q^39 + (b3 - 2*b2 - 2*b1) * q^41 + (-b3 - b1 - 2) * q^43 + (2*b3 + b2 + 4) * q^47 + (-b3 - 2*b2 + b1 - 1) * q^49 + (2*b2 + 2*b1 + 8) * q^51 + (2*b3 - 2) * q^53 + (2*b2 + 2*b1 + 6) * q^57 + (b3 + b1 + 3) * q^59 + (2*b3 + 6) * q^61 + (3*b3 - b1) * q^63 + (2*b2 + 4) * q^67 - b1 * q^69 + (-b2 - 2*b1 + 4) * q^71 + (-b3 + b2 + 2*b1 - 2) * q^73 + (-b3 - 2*b2 + b1 + 2) * q^77 + (-b3 + 3*b1 + 6) * q^79 + (-2*b2 + b1 - 1) * q^81 + (-b3 - 3*b1 - 2) * q^83 + (-b3 - b2 - 4*b1 - 3) * q^87 + (-4*b3 + 2*b2 + 2*b1) * q^89 + (3*b3 - 2*b2 - b1 + 6) * q^91 + (b3 - b2 + 6*b1 - 3) * q^93 + (-4*b3 + 2*b2 + 2) * q^97 + (b3 - 7*b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{7} + 2 q^{9}+O(q^{10})$$ 4 * q + q^7 + 2 * q^9 $$4 q + q^{7} + 2 q^{9} + q^{11} - q^{13} + 8 q^{17} + 7 q^{19} + 8 q^{21} - 4 q^{23} + 3 q^{27} - 5 q^{29} + 14 q^{31} - 20 q^{33} + 4 q^{37} + 11 q^{39} + 3 q^{41} - 7 q^{43} + 12 q^{47} + q^{49} + 28 q^{51} - 10 q^{53} + 20 q^{57} + 11 q^{59} + 22 q^{61} - 3 q^{63} + 12 q^{67} + 18 q^{71} - 9 q^{73} + 13 q^{77} + 25 q^{79} - 7 q^{83} - 9 q^{87} + 25 q^{91} - 11 q^{93} + 8 q^{97} - 9 q^{99}+O(q^{100})$$ 4 * q + q^7 + 2 * q^9 + q^11 - q^13 + 8 * q^17 + 7 * q^19 + 8 * q^21 - 4 * q^23 + 3 * q^27 - 5 * q^29 + 14 * q^31 - 20 * q^33 + 4 * q^37 + 11 * q^39 + 3 * q^41 - 7 * q^43 + 12 * q^47 + q^49 + 28 * q^51 - 10 * q^53 + 20 * q^57 + 11 * q^59 + 22 * q^61 - 3 * q^63 + 12 * q^67 + 18 * q^71 - 9 * q^73 + 13 * q^77 + 25 * q^79 - 7 * q^83 - 9 * q^87 + 25 * q^91 - 11 * q^93 + 8 * q^97 - 9 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 7x^{2} - x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 6\nu - 1$$ v^3 - 6*v - 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 6\beta _1 + 1$$ b3 + 6*b1 + 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.50653 −0.631352 0.474520 2.66337
0 −2.50653 0 0 0 −0.797915 0 3.28271 0
1.2 0 −0.631352 0 0 0 −3.16780 0 −2.60139 0
1.3 0 0.474520 0 0 0 4.21479 0 −2.77483 0
1.4 0 2.66337 0 0 0 0.750930 0 4.09352 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.m yes 4
4.b odd 2 1 9200.2.a.cm 4
5.b even 2 1 2300.2.a.l 4
5.c odd 4 2 2300.2.c.j 8
20.d odd 2 1 9200.2.a.co 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.2.a.l 4 5.b even 2 1
2300.2.a.m yes 4 1.a even 1 1 trivial
2300.2.c.j 8 5.c odd 4 2
9200.2.a.cm 4 4.b odd 2 1
9200.2.a.co 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2300))$$:

 $$T_{3}^{4} - 7T_{3}^{2} - T_{3} + 2$$ T3^4 - 7*T3^2 - T3 + 2 $$T_{7}^{4} - T_{7}^{3} - 14T_{7}^{2} + 8$$ T7^4 - T7^3 - 14*T7^2 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 7T^{2} - T + 2$$
$5$ $$T^{4}$$
$7$ $$T^{4} - T^{3} - 14 T^{2} + 8$$
$11$ $$T^{4} - T^{3} + \cdots + 120$$
$13$ $$T^{4} + T^{3} + \cdots - 17$$
$17$ $$T^{4} - 8 T^{3} + \cdots - 48$$
$19$ $$T^{4} - 7 T^{3} + \cdots + 72$$
$23$ $$(T + 1)^{4}$$
$29$ $$T^{4} + 5 T^{3} + \cdots - 93$$
$31$ $$T^{4} - 14 T^{3} + \cdots - 10$$
$37$ $$T^{4} - 4 T^{3} + \cdots + 416$$
$41$ $$T^{4} - 3 T^{3} + \cdots + 381$$
$43$ $$T^{4} + 7 T^{3} + \cdots + 72$$
$47$ $$T^{4} - 12 T^{3} + \cdots - 1242$$
$53$ $$T^{4} + 10 T^{3} + \cdots + 288$$
$59$ $$T^{4} - 11 T^{3} + \cdots + 12$$
$61$ $$T^{4} - 22 T^{3} + \cdots - 416$$
$67$ $$T^{4} - 12 T^{3} + \cdots + 992$$
$71$ $$T^{4} - 18 T^{3} + \cdots - 1800$$
$73$ $$T^{4} + 9 T^{3} + \cdots - 139$$
$79$ $$T^{4} - 25 T^{3} + \cdots + 40$$
$83$ $$T^{4} + 7 T^{3} + \cdots + 72$$
$89$ $$T^{4} - 236 T^{2} + \cdots - 3840$$
$97$ $$T^{4} - 8 T^{3} + \cdots - 1040$$