# Properties

 Label 4840.2.a.be Level $4840$ Weight $2$ Character orbit 4840.a Self dual yes Analytic conductor $38.648$ Analytic rank $0$ 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] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4840 = 2^{3} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4840.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: $$38.6475945783$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.45753625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20$$ x^6 - x^5 - 13*x^4 + 11*x^3 + 41*x^2 - 30*x - 20 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} - q^{5} + (\beta_{5} + \beta_{2} - 1) q^{7} + (\beta_{5} - \beta_{4} - 2 \beta_{2}) q^{9}+O(q^{10})$$ q - b3 * q^3 - q^5 + (b5 + b2 - 1) * q^7 + (b5 - b4 - 2*b2) * q^9 $$q - \beta_{3} q^{3} - q^{5} + (\beta_{5} + \beta_{2} - 1) q^{7} + (\beta_{5} - \beta_{4} - 2 \beta_{2}) q^{9} + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{13} + \beta_{3} q^{15} + (\beta_{5} + \beta_{3} + 2 \beta_{2} - 1) q^{17} + (\beta_{5} + \beta_{2} - \beta_1 + 2) q^{19} + (\beta_{5} + \beta_1 - 1) q^{21} + ( - 2 \beta_{4} - \beta_{2} + \beta_1 + 2) q^{23} + q^{25} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 3 \beta_1 - 1) q^{27} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1) q^{29} + (\beta_{5} - 2 \beta_{3} - 1) q^{31} + ( - \beta_{5} - \beta_{2} + 1) q^{35} + (\beta_{4} - 3 \beta_{2} + 2 \beta_1 - 1) q^{37} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} - 3) q^{39} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{41} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 1) q^{43} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{45} + (2 \beta_{4} + \beta_{2} + \beta_1 + 3) q^{47} + ( - 3 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{49} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{51} + (\beta_{4} - \beta_{2} + 2) q^{53} + (\beta_{5} - \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + \beta_1 - 3) q^{57} + (2 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{59} + ( - 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{61} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 4) q^{63} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{65} + (2 \beta_{5} + \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{67} + ( - \beta_{4} - 5 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 2) q^{69} + ( - \beta_{3} - 6 \beta_{2} - \beta_1 - 2) q^{71} + ( - 2 \beta_{5} + \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{73} - \beta_{3} q^{75} + (\beta_{4} + \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 8) q^{79} + ( - 3 \beta_{4} - 2 \beta_{3} - 10 \beta_{2} - \beta_1 - 1) q^{81} + (\beta_{5} + \beta_{4} - \beta_{3} - 4 \beta_{2} - \beta_1 + 5) q^{83} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2} + 1) q^{85} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 1) q^{87} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + 7 \beta_{2} + 2 \beta_1 + 3) q^{89} + ( - 4 \beta_{4} + 6 \beta_{2} + 1) q^{91} + (3 \beta_{5} - 2 \beta_{4} - \beta_{3} - 4 \beta_{2} + 5) q^{93} + ( - \beta_{5} - \beta_{2} + \beta_1 - 2) q^{95} + ( - 3 \beta_{5} + \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 7) q^{97}+O(q^{100})$$ q - b3 * q^3 - q^5 + (b5 + b2 - 1) * q^7 + (b5 - b4 - 2*b2) * q^9 + (b4 + b3 - b2 + 1) * q^13 + b3 * q^15 + (b5 + b3 + 2*b2 - 1) * q^17 + (b5 + b2 - b1 + 2) * q^19 + (b5 + b1 - 1) * q^21 + (-2*b4 - b2 + b1 + 2) * q^23 + q^25 + (b5 - b4 - 2*b3 - 3*b1 - 1) * q^27 + (-b4 + b3 - 2*b2 + b1) * q^29 + (b5 - 2*b3 - 1) * q^31 + (-b5 - b2 + 1) * q^35 + (b4 - 3*b2 + 2*b1 - 1) * q^37 + (-b5 + 2*b4 - b3 + 2*b2 - 3) * q^39 + (b5 + b4 - b3 + b2 - 3*b1 + 1) * q^41 + (-b5 - 2*b4 - 2*b3 - 1) * q^43 + (-b5 + b4 + 2*b2) * q^45 + (2*b4 + b2 + b1 + 3) * q^47 + (-3*b5 + b4 - b3 + b2 + b1 + 3) * q^49 + (b4 + b3 + 2*b2 + 2*b1 - 4) * q^51 + (b4 - b2 + 2) * q^53 + (b5 - b4 - 3*b3 - 4*b2 + b1 - 3) * q^57 + (2*b4 - 3*b3 + 3*b2 + b1 + 1) * q^59 + (-3*b4 + 2*b3 + 2*b2 - b1 + 2) * q^61 + (-2*b5 + b4 - b3 + b2 + 4) * q^63 + (-b4 - b3 + b2 - 1) * q^65 + (2*b5 + b3 + 2*b2 + b1 + 2) * q^67 + (-b4 - 5*b3 + 4*b2 - 3*b1 + 2) * q^69 + (-b3 - 6*b2 - b1 - 2) * q^71 + (-2*b5 + b3 - 2*b2 - b1 + 2) * q^73 - b3 * q^75 + (b4 + b3 - 4*b2 + 3*b1 - 8) * q^79 + (-3*b4 - 2*b3 - 10*b2 - b1 - 1) * q^81 + (b5 + b4 - b3 - 4*b2 - b1 + 5) * q^83 + (-b5 - b3 - 2*b2 + 1) * q^85 + (-b5 + b4 - 3*b3 + 6*b2 - 3*b1 - 1) * q^87 + (-b5 + b4 - 3*b3 + 7*b2 + 2*b1 + 3) * q^89 + (-4*b4 + 6*b2 + 1) * q^91 + (3*b5 - 2*b4 - b3 - 4*b2 + 5) * q^93 + (-b5 - b2 + b1 - 2) * q^95 + (-3*b5 + b3 + 2*b2 - 5*b1 + 7) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{3} - 6 q^{5} - 6 q^{7} + 10 q^{9}+O(q^{10})$$ 6 * q + 2 * q^3 - 6 * q^5 - 6 * q^7 + 10 * q^9 $$6 q + 2 q^{3} - 6 q^{5} - 6 q^{7} + 10 q^{9} + 6 q^{13} - 2 q^{15} - 11 q^{17} + 11 q^{19} - 2 q^{21} + 18 q^{23} + 6 q^{25} - q^{27} + 6 q^{29} + q^{31} + 6 q^{35} + 4 q^{37} - 27 q^{39} + 4 q^{41} - 3 q^{43} - 10 q^{45} + 14 q^{47} + 8 q^{49} - 31 q^{51} + 14 q^{53} + 5 q^{57} + 2 q^{59} + 4 q^{61} + 16 q^{63} - 6 q^{65} + 11 q^{67} + 8 q^{69} + 7 q^{71} + 9 q^{73} + 2 q^{75} - 36 q^{79} + 30 q^{81} + 45 q^{83} + 11 q^{85} - 25 q^{87} + q^{89} - 8 q^{91} + 55 q^{93} - 11 q^{95} + 20 q^{97}+O(q^{100})$$ 6 * q + 2 * q^3 - 6 * q^5 - 6 * q^7 + 10 * q^9 + 6 * q^13 - 2 * q^15 - 11 * q^17 + 11 * q^19 - 2 * q^21 + 18 * q^23 + 6 * q^25 - q^27 + 6 * q^29 + q^31 + 6 * q^35 + 4 * q^37 - 27 * q^39 + 4 * q^41 - 3 * q^43 - 10 * q^45 + 14 * q^47 + 8 * q^49 - 31 * q^51 + 14 * q^53 + 5 * q^57 + 2 * q^59 + 4 * q^61 + 16 * q^63 - 6 * q^65 + 11 * q^67 + 8 * q^69 + 7 * q^71 + 9 * q^73 + 2 * q^75 - 36 * q^79 + 30 * q^81 + 45 * q^83 + 11 * q^85 - 25 * q^87 + q^89 - 8 * q^91 + 55 * q^93 - 11 * q^95 + 20 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} + \nu^{4} - 13\nu^{3} - 9\nu^{2} + 35\nu ) / 10$$ (v^5 + v^4 - 13*v^3 - 9*v^2 + 35*v) / 10 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + 10\nu^{3} + 3\nu^{2} - 15\nu - 10 ) / 5$$ (-v^5 + 10*v^3 + 3*v^2 - 15*v - 10) / 5 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + \nu^{4} + 12\nu^{3} - 8\nu^{2} - 30\nu + 10 ) / 5$$ (-v^5 + v^4 + 12*v^3 - 8*v^2 - 30*v + 10) / 5 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} - \nu^{4} + 13\nu^{3} + 14\nu^{2} - 35\nu - 25 ) / 5$$ (-v^5 - v^4 + 13*v^3 + 14*v^2 - 35*v - 25) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{2} + 5$$ b5 + 2*b2 + 5 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} - 2\beta_{3} + 7\beta _1 - 1$$ b5 + b4 - 2*b3 + 7*b1 - 1 $$\nu^{4}$$ $$=$$ $$9\beta_{5} + 3\beta_{4} - \beta_{3} + 22\beta_{2} + \beta _1 + 37$$ 9*b5 + 3*b4 - b3 + 22*b2 + b1 + 37 $$\nu^{5}$$ $$=$$ $$13\beta_{5} + 10\beta_{4} - 25\beta_{3} + 6\beta_{2} + 55\beta _1 - 5$$ 13*b5 + 10*b4 - 25*b3 + 6*b2 + 55*b1 - 5

## 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
 1.87511 −2.80540 −0.444728 1.36422 3.05921 −2.04842
0 −3.03399 0 −1.00000 0 −0.865927 0 6.20511 0
1.2 0 −1.73383 0 −1.00000 0 1.25225 0 0.00617996 0
1.3 0 0.719585 0 −1.00000 0 −4.18418 0 −2.48220 0
1.4 0 0.843136 0 −1.00000 0 −4.75693 0 −2.28912 0
1.5 0 1.89070 0 −1.00000 0 2.74075 0 0.574737 0
1.6 0 3.31441 0 −1.00000 0 −0.185958 0 7.98529 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
$$2$$ $$1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## 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 4840.2.a.be 6
4.b odd 2 1 9680.2.a.cy 6
11.b odd 2 1 4840.2.a.bf 6
11.d odd 10 2 440.2.y.b 12
44.c even 2 1 9680.2.a.cx 6
44.g even 10 2 880.2.bo.j 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.y.b 12 11.d odd 10 2
880.2.bo.j 12 44.g even 10 2
4840.2.a.be 6 1.a even 1 1 trivial
4840.2.a.bf 6 11.b odd 2 1
9680.2.a.cx 6 44.c even 2 1
9680.2.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4840))$$:

 $$T_{3}^{6} - 2T_{3}^{5} - 12T_{3}^{4} + 23T_{3}^{3} + 21T_{3}^{2} - 50T_{3} + 20$$ T3^6 - 2*T3^5 - 12*T3^4 + 23*T3^3 + 21*T3^2 - 50*T3 + 20 $$T_{7}^{6} + 6T_{7}^{5} - 7T_{7}^{4} - 61T_{7}^{3} + 15T_{7}^{2} + 64T_{7} + 11$$ T7^6 + 6*T7^5 - 7*T7^4 - 61*T7^3 + 15*T7^2 + 64*T7 + 11 $$T_{13}^{6} - 6T_{13}^{5} - 25T_{13}^{4} + 141T_{13}^{3} + 177T_{13}^{2} - 738T_{13} - 151$$ T13^6 - 6*T13^5 - 25*T13^4 + 141*T13^3 + 177*T13^2 - 738*T13 - 151

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - 2 T^{5} - 12 T^{4} + 23 T^{3} + \cdots + 20$$
$5$ $$(T + 1)^{6}$$
$7$ $$T^{6} + 6 T^{5} - 7 T^{4} - 61 T^{3} + \cdots + 11$$
$11$ $$T^{6}$$
$13$ $$T^{6} - 6 T^{5} - 25 T^{4} + 141 T^{3} + \cdots - 151$$
$17$ $$T^{6} + 11 T^{5} + 6 T^{4} - 216 T^{3} + \cdots - 484$$
$19$ $$T^{6} - 11 T^{5} + 15 T^{4} + 127 T^{3} + \cdots + 55$$
$23$ $$T^{6} - 18 T^{5} + 50 T^{4} + \cdots + 9505$$
$29$ $$T^{6} - 6 T^{5} - 36 T^{4} + 19 T^{3} + \cdots - 36$$
$31$ $$T^{6} - T^{5} - 75 T^{4} - 93 T^{3} + \cdots + 2980$$
$37$ $$T^{6} - 4 T^{5} - 90 T^{4} + \cdots - 14821$$
$41$ $$T^{6} - 4 T^{5} - 176 T^{4} + \cdots - 37441$$
$43$ $$T^{6} + 3 T^{5} - 163 T^{4} + \cdots + 75284$$
$47$ $$T^{6} - 14 T^{5} - 5 T^{4} + 503 T^{3} + \cdots + 995$$
$53$ $$T^{6} - 14 T^{5} + 57 T^{4} + \cdots - 225$$
$59$ $$T^{6} - 2 T^{5} - 219 T^{4} + \cdots - 319$$
$61$ $$T^{6} - 4 T^{5} - 237 T^{4} + \cdots + 70256$$
$67$ $$T^{6} - 11 T^{5} - 66 T^{4} + \cdots - 19900$$
$71$ $$T^{6} - 7 T^{5} - 158 T^{4} + \cdots - 64476$$
$73$ $$T^{6} - 9 T^{5} - 80 T^{4} + \cdots + 3244$$
$79$ $$T^{6} + 36 T^{5} + 344 T^{4} + \cdots + 14864$$
$83$ $$T^{6} - 45 T^{5} + 679 T^{4} + \cdots - 339676$$
$89$ $$T^{6} - T^{5} - 419 T^{4} + \cdots + 179771$$
$97$ $$T^{6} - 20 T^{5} - 330 T^{4} + \cdots + 1270064$$