# Properties

 Label 2300.2.a.n Level $2300$ Weight $2$ Character orbit 2300.a Self dual yes Analytic conductor $18.366$ Analytic rank $1$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.143376304.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 12x^{4} + 22x^{2} - 6x - 1$$ x^6 - 12*x^4 + 22*x^2 - 6*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 460) 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{3} + (\beta_{3} - 2) q^{7} + (\beta_{2} + \beta_1 + 2) q^{9}+O(q^{10})$$ q + (-b2 - 1) * q^3 + (b3 - 2) * q^7 + (b2 + b1 + 2) * q^9 $$q + ( - \beta_{2} - 1) q^{3} + (\beta_{3} - 2) q^{7} + (\beta_{2} + \beta_1 + 2) q^{9} + ( - \beta_{5} + \beta_{4} - \beta_{2} + 1) q^{11} + ( - \beta_{4} - \beta_{3} + \beta_{2} + \cdots - 1) q^{13}+ \cdots + (2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 4) q^{99}+O(q^{100})$$ q + (-b2 - 1) * q^3 + (b3 - 2) * q^7 + (b2 + b1 + 2) * q^9 + (-b5 + b4 - b2 + 1) * q^11 + (-b4 - b3 + b2 - b1 - 1) * q^13 + (b5 + b2 - 1) * q^17 + (b5 - b3 + b2 + 1) * q^19 + (b5 - b4 - 2*b3 + 3*b2 - 2*b1 + 1) * q^21 - q^23 + (b4 - b3 - b2 - b1 - 3) * q^27 + (-b5 + 2*b2 + 2) * q^29 + (b5 - b4 + b3 + b1) * q^31 + (b5 - 2*b4 - b3 + b2 + b1 + 1) * q^33 + (-2*b5 + 2*b4 + b3 - 2) * q^37 + (-2*b5 + b4 + 3*b3 + b2 - 1) * q^39 + (b5 - b4 - b3 + 2*b2 - b1) * q^41 + (-b5 + 2*b4 - b3 - b2 - 1) * q^43 + (-b4 - b3 - b2 + b1 - 3) * q^47 + (b5 - 2*b3 + b2 + b1 + 4) * q^49 + (b4 + b3 - 2) * q^51 + (b5 + 2*b3 + b2 + b1 - 1) * q^53 + (-b5 + 2*b4 + 3*b3 - 3*b2 + 2*b1 - 3) * q^57 + (b5 - 3*b4 - b3 + b2 - 2*b1 - 3) * q^59 + (-3*b5 + 2*b4 + b3 - b2 + 1) * q^61 + (-3*b5 + 2*b4 + 4*b3 - b2 + b1 - 3) * q^63 + (-2*b4 - b3 + 2*b2 - 4) * q^67 + (b2 + 1) * q^69 + (-b5 + 2*b3 - 2*b2 + 2*b1 - 4) * q^71 + (-2*b5 - b2 + 2*b1 + 3) * q^73 + (2*b5 - 4*b4 - 2*b3 + 4*b2 - 4) * q^77 + (b5 + b3 + b2 + 1) * q^79 + (-b4 + 3*b3 + 2*b2 + b1 + 1) * q^81 + (3*b5 - 2*b3 + 3*b2 - 2*b1 - 3) * q^83 + (-b4 - b3 - b2 - 3*b1 - 11) * q^87 + (b5 + 2*b4 + b3 - 3*b2 - 2*b1 - 5) * q^89 + (-b5 + 3*b4 - 7*b2 + 2*b1 - 1) * q^91 + (2*b4 - 2*b3 - 3*b2 - 2*b1 + 1) * q^93 + (b5 + 2*b4 + b3 - b2 + 3*b1 - 1) * q^97 + (2*b4 + 2*b3 - 4*b2 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{3} - 9 q^{7} + 10 q^{9}+O(q^{10})$$ 6 * q - 4 * q^3 - 9 * q^7 + 10 * q^9 $$6 q - 4 q^{3} - 9 q^{7} + 10 q^{9} + 2 q^{11} - 8 q^{13} - 5 q^{17} + 4 q^{19} - 6 q^{23} - 22 q^{27} + 5 q^{29} + 9 q^{31} + 10 q^{33} - 21 q^{37} - 8 q^{39} - q^{41} - 16 q^{43} - 16 q^{47} + 19 q^{49} - 12 q^{51} + q^{53} - 12 q^{57} - 11 q^{59} - 4 q^{61} - 19 q^{63} - 25 q^{67} + 4 q^{69} - 17 q^{71} + 14 q^{73} - 20 q^{77} + 10 q^{79} + 14 q^{81} - 21 q^{83} - 64 q^{87} - 24 q^{89} - 4 q^{91} - 4 q^{97} - 16 q^{99}+O(q^{100})$$ 6 * q - 4 * q^3 - 9 * q^7 + 10 * q^9 + 2 * q^11 - 8 * q^13 - 5 * q^17 + 4 * q^19 - 6 * q^23 - 22 * q^27 + 5 * q^29 + 9 * q^31 + 10 * q^33 - 21 * q^37 - 8 * q^39 - q^41 - 16 * q^43 - 16 * q^47 + 19 * q^49 - 12 * q^51 + q^53 - 12 * q^57 - 11 * q^59 - 4 * q^61 - 19 * q^63 - 25 * q^67 + 4 * q^69 - 17 * q^71 + 14 * q^73 - 20 * q^77 + 10 * q^79 + 14 * q^81 - 21 * q^83 - 64 * q^87 - 24 * q^89 - 4 * q^91 - 4 * q^97 - 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 12x^{4} + 22x^{2} - 6x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} - 3\nu^{4} + 11\nu^{3} + 33\nu^{2} - 3\nu - 27 ) / 8$$ (-v^5 - 3*v^4 + 11*v^3 + 33*v^2 - 3*v - 27) / 8 $$\beta_{2}$$ $$=$$ $$( 3\nu^{5} + \nu^{4} - 33\nu^{3} - 11\nu^{2} + 41\nu - 7 ) / 8$$ (3*v^5 + v^4 - 33*v^3 - 11*v^2 + 41*v - 7) / 8 $$\beta_{3}$$ $$=$$ $$( -7\nu^{5} + 3\nu^{4} + 85\nu^{3} - 25\nu^{2} - 157\nu + 43 ) / 16$$ (-7*v^5 + 3*v^4 + 85*v^3 - 25*v^2 - 157*v + 43) / 16 $$\beta_{4}$$ $$=$$ $$( 9\nu^{5} + 3\nu^{4} - 107\nu^{3} - 41\nu^{2} + 195\nu + 11 ) / 16$$ (9*v^5 + 3*v^4 - 107*v^3 - 41*v^2 + 195*v + 11) / 16 $$\beta_{5}$$ $$=$$ $$( -9\nu^{5} - 3\nu^{4} + 107\nu^{3} + 25\nu^{2} - 179\nu + 53 ) / 16$$ (-9*v^5 - 3*v^4 + 107*v^3 + 25*v^2 - 179*v + 53) / 16
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{3} + \beta_1 ) / 2$$ (b4 + b3 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( -2\beta_{5} - \beta_{4} + \beta_{3} + \beta _1 + 8 ) / 2$$ (-2*b5 - b4 + b3 + b1 + 8) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{5} + 3\beta_{4} + 4\beta_{3} + 3\beta_{2} + 4\beta_1$$ b5 + 3*b4 + 4*b3 + 3*b2 + 4*b1 $$\nu^{4}$$ $$=$$ $$( -22\beta_{5} - 7\beta_{4} + 15\beta_{3} - 2\beta_{2} + 9\beta _1 + 66 ) / 2$$ (-22*b5 - 7*b4 + 15*b3 - 2*b2 + 9*b1 + 66) / 2 $$\nu^{5}$$ $$=$$ $$( 22\beta_{5} + 51\beta_{4} + 73\beta_{3} + 72\beta_{2} + 75\beta _1 + 12 ) / 2$$ (22*b5 + 51*b4 + 73*b3 + 72*b2 + 75*b1 + 12) / 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
 3.16223 −1.65047 0.420790 −0.116918 −3.08006 1.26443
0 −3.21923 0 0 0 2.43185 0 7.36343 0
1.2 0 −2.80150 0 0 0 −4.50896 0 4.84843 0
1.3 0 −1.73961 0 0 0 −3.32224 0 0.0262434 0
1.4 0 0.486391 0 0 0 1.80495 0 −2.76342 0
1.5 0 0.873449 0 0 0 −0.992530 0 −2.23709 0
1.6 0 2.40050 0 0 0 −4.41307 0 2.76241 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.n 6
4.b odd 2 1 9200.2.a.cy 6
5.b even 2 1 2300.2.a.o 6
5.c odd 4 2 460.2.c.a 12
15.e even 4 2 4140.2.f.b 12
20.d odd 2 1 9200.2.a.cx 6
20.e even 4 2 1840.2.e.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.c.a 12 5.c odd 4 2
1840.2.e.f 12 20.e even 4 2
2300.2.a.n 6 1.a even 1 1 trivial
2300.2.a.o 6 5.b even 2 1
4140.2.f.b 12 15.e even 4 2
9200.2.a.cx 6 20.d odd 2 1
9200.2.a.cy 6 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2300))$$:

 $$T_{3}^{6} + 4T_{3}^{5} - 6T_{3}^{4} - 30T_{3}^{3} + 5T_{3}^{2} + 38T_{3} - 16$$ T3^6 + 4*T3^5 - 6*T3^4 - 30*T3^3 + 5*T3^2 + 38*T3 - 16 $$T_{7}^{6} + 9T_{7}^{5} + 10T_{7}^{4} - 88T_{7}^{3} - 152T_{7}^{2} + 228T_{7} + 288$$ T7^6 + 9*T7^5 + 10*T7^4 - 88*T7^3 - 152*T7^2 + 228*T7 + 288

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 4 T^{5} + \cdots - 16$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 9 T^{5} + \cdots + 288$$
$11$ $$T^{6} - 2 T^{5} + \cdots - 256$$
$13$ $$T^{6} + 8 T^{5} + \cdots + 1184$$
$17$ $$T^{6} + 5 T^{5} + \cdots + 16$$
$19$ $$T^{6} - 4 T^{5} + \cdots + 256$$
$23$ $$(T + 1)^{6}$$
$29$ $$T^{6} - 5 T^{5} + \cdots + 11862$$
$31$ $$T^{6} - 9 T^{5} + \cdots + 916$$
$37$ $$T^{6} + 21 T^{5} + \cdots + 63216$$
$41$ $$T^{6} + T^{5} - 64 T^{4} + \cdots - 2$$
$43$ $$T^{6} + 16 T^{5} + \cdots + 91648$$
$47$ $$T^{6} + 16 T^{5} + \cdots - 464$$
$53$ $$T^{6} - T^{5} + \cdots + 12224$$
$59$ $$T^{6} + 11 T^{5} + \cdots - 360576$$
$61$ $$T^{6} + 4 T^{5} + \cdots - 171088$$
$67$ $$T^{6} + 25 T^{5} + \cdots + 7424$$
$71$ $$T^{6} + 17 T^{5} + \cdots - 16108$$
$73$ $$T^{6} - 14 T^{5} + \cdots + 5504$$
$79$ $$T^{6} - 10 T^{5} + \cdots - 128$$
$83$ $$T^{6} + 21 T^{5} + \cdots - 633344$$
$89$ $$T^{6} + 24 T^{5} + \cdots + 180432$$
$97$ $$T^{6} + 4 T^{5} + \cdots + 45728$$