# Properties

 Label 7616.2.a.be Level $7616$ Weight $2$ Character orbit 7616.a Self dual yes Analytic conductor $60.814$ Analytic rank $0$ 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] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7616, 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("7616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7616 = 2^{6} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7616.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: $$60.8140661794$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 952) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + (\beta_1 - 1) q^{5} - q^{7} + ( - 2 \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q - b2 * q^3 + (b1 - 1) * q^5 - q^7 + (-2*b2 + b1) * q^9 $$q - \beta_{2} q^{3} + (\beta_1 - 1) q^{5} - q^{7} + ( - 2 \beta_{2} + \beta_1) q^{9} + ( - 2 \beta_1 + 2) q^{11} + (\beta_{2} - \beta_1 - 1) q^{15} - q^{17} + (2 \beta_{2} + 2) q^{19} + \beta_{2} q^{21} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{23} + (\beta_{2} - 2 \beta_1 - 1) q^{25} + ( - \beta_{2} + \beta_1 + 5) q^{27} + ( - 2 \beta_{2} + 2) q^{29} + (\beta_1 - 1) q^{31} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{33} + ( - \beta_1 + 1) q^{35} + (2 \beta_{2} + 2) q^{37} + ( - 3 \beta_{2} + 6 \beta_1 + 2) q^{41} + (\beta_{2} - 6 \beta_1) q^{43} + (3 \beta_{2} - 3 \beta_1 + 1) q^{45} + ( - 4 \beta_{2} - 4) q^{47} + q^{49} + \beta_{2} q^{51} + ( - 2 \beta_{2} + \beta_1 - 7) q^{53} + ( - 2 \beta_{2} + 4 \beta_1 - 8) q^{55} + (2 \beta_{2} - 2 \beta_1 - 6) q^{57} + ( - 2 \beta_{2} + 6 \beta_1 + 4) q^{59} + ( - 5 \beta_{2} + 4 \beta_1 - 2) q^{61} + (2 \beta_{2} - \beta_1) q^{63} + ( - 4 \beta_{2} + 7 \beta_1 + 1) q^{67} + ( - 2 \beta_{2} + 4 \beta_1 + 8) q^{69} + ( - 6 \beta_1 - 2) q^{71} + ( - \beta_{2} + 2 \beta_1 - 2) q^{73} + (3 \beta_{2} + \beta_1 - 1) q^{75} + (2 \beta_1 - 2) q^{77} - 8 q^{79} + ( - \beta_{2} - 3 \beta_1 + 2) q^{81} + (2 \beta_{2} - 8 \beta_1 + 2) q^{83} + ( - \beta_1 + 1) q^{85} + ( - 6 \beta_{2} + 2 \beta_1 + 6) q^{87} + (2 \beta_{2} + 2 \beta_1 + 4) q^{89} + (\beta_{2} - \beta_1 - 1) q^{93} + ( - 2 \beta_{2} + 4 \beta_1) q^{95} + (2 \beta_{2} + \beta_1 + 5) q^{97} + ( - 6 \beta_{2} + 6 \beta_1 - 2) q^{99}+O(q^{100})$$ q - b2 * q^3 + (b1 - 1) * q^5 - q^7 + (-2*b2 + b1) * q^9 + (-2*b1 + 2) * q^11 + (b2 - b1 - 1) * q^15 - q^17 + (2*b2 + 2) * q^19 + b2 * q^21 + (-2*b2 - 2*b1 - 2) * q^23 + (b2 - 2*b1 - 1) * q^25 + (-b2 + b1 + 5) * q^27 + (-2*b2 + 2) * q^29 + (b1 - 1) * q^31 + (-2*b2 + 2*b1 + 2) * q^33 + (-b1 + 1) * q^35 + (2*b2 + 2) * q^37 + (-3*b2 + 6*b1 + 2) * q^41 + (b2 - 6*b1) * q^43 + (3*b2 - 3*b1 + 1) * q^45 + (-4*b2 - 4) * q^47 + q^49 + b2 * q^51 + (-2*b2 + b1 - 7) * q^53 + (-2*b2 + 4*b1 - 8) * q^55 + (2*b2 - 2*b1 - 6) * q^57 + (-2*b2 + 6*b1 + 4) * q^59 + (-5*b2 + 4*b1 - 2) * q^61 + (2*b2 - b1) * q^63 + (-4*b2 + 7*b1 + 1) * q^67 + (-2*b2 + 4*b1 + 8) * q^69 + (-6*b1 - 2) * q^71 + (-b2 + 2*b1 - 2) * q^73 + (3*b2 + b1 - 1) * q^75 + (2*b1 - 2) * q^77 - 8 * q^79 + (-b2 - 3*b1 + 2) * q^81 + (2*b2 - 8*b1 + 2) * q^83 + (-b1 + 1) * q^85 + (-6*b2 + 2*b1 + 6) * q^87 + (2*b2 + 2*b1 + 4) * q^89 + (b2 - b1 - 1) * q^93 + (-2*b2 + 4*b1) * q^95 + (2*b2 + b1 + 5) * q^97 + (-6*b2 + 6*b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} - 3 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10})$$ 3 * q + q^3 - 3 * q^5 - 3 * q^7 + 2 * q^9 $$3 q + q^{3} - 3 q^{5} - 3 q^{7} + 2 q^{9} + 6 q^{11} - 4 q^{15} - 3 q^{17} + 4 q^{19} - q^{21} - 4 q^{23} - 4 q^{25} + 16 q^{27} + 8 q^{29} - 3 q^{31} + 8 q^{33} + 3 q^{35} + 4 q^{37} + 9 q^{41} - q^{43} - 8 q^{47} + 3 q^{49} - q^{51} - 19 q^{53} - 22 q^{55} - 20 q^{57} + 14 q^{59} - q^{61} - 2 q^{63} + 7 q^{67} + 26 q^{69} - 6 q^{71} - 5 q^{73} - 6 q^{75} - 6 q^{77} - 24 q^{79} + 7 q^{81} + 4 q^{83} + 3 q^{85} + 24 q^{87} + 10 q^{89} - 4 q^{93} + 2 q^{95} + 13 q^{97}+O(q^{100})$$ 3 * q + q^3 - 3 * q^5 - 3 * q^7 + 2 * q^9 + 6 * q^11 - 4 * q^15 - 3 * q^17 + 4 * q^19 - q^21 - 4 * q^23 - 4 * q^25 + 16 * q^27 + 8 * q^29 - 3 * q^31 + 8 * q^33 + 3 * q^35 + 4 * q^37 + 9 * q^41 - q^43 - 8 * q^47 + 3 * q^49 - q^51 - 19 * q^53 - 22 * q^55 - 20 * q^57 + 14 * q^59 - q^61 - 2 * q^63 + 7 * q^67 + 26 * q^69 - 6 * q^71 - 5 * q^73 - 6 * q^75 - 6 * q^77 - 24 * q^79 + 7 * q^81 + 4 * q^83 + 3 * q^85 + 24 * q^87 + 10 * q^89 - 4 * q^93 + 2 * q^95 + 13 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 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
 2.11491 −1.86081 −0.254102
0 −1.47283 0 1.11491 0 −1.00000 0 −0.830760 0
1.2 0 −0.462598 0 −2.86081 0 −1.00000 0 −2.78600 0
1.3 0 2.93543 0 −1.25410 0 −1.00000 0 5.61676 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
$$2$$ $$-1$$
$$7$$ $$1$$
$$17$$ $$1$$

## 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 7616.2.a.be 3
4.b odd 2 1 7616.2.a.bc 3
8.b even 2 1 1904.2.a.m 3
8.d odd 2 1 952.2.a.f 3
24.f even 2 1 8568.2.a.x 3
56.e even 2 1 6664.2.a.j 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
952.2.a.f 3 8.d odd 2 1
1904.2.a.m 3 8.b even 2 1
6664.2.a.j 3 56.e even 2 1
7616.2.a.bc 3 4.b odd 2 1
7616.2.a.be 3 1.a even 1 1 trivial
8568.2.a.x 3 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7616))$$:

 $$T_{3}^{3} - T_{3}^{2} - 5T_{3} - 2$$ T3^3 - T3^2 - 5*T3 - 2 $$T_{5}^{3} + 3T_{5}^{2} - T_{5} - 4$$ T5^3 + 3*T5^2 - T5 - 4 $$T_{11}^{3} - 6T_{11}^{2} - 4T_{11} + 32$$ T11^3 - 6*T11^2 - 4*T11 + 32 $$T_{19}^{3} - 4T_{19}^{2} - 16T_{19} + 56$$ T19^3 - 4*T19^2 - 16*T19 + 56

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - T^{2} - 5T - 2$$
$5$ $$T^{3} + 3T^{2} - T - 4$$
$7$ $$(T + 1)^{3}$$
$11$ $$T^{3} - 6 T^{2} + \cdots + 32$$
$13$ $$T^{3}$$
$17$ $$(T + 1)^{3}$$
$19$ $$T^{3} - 4 T^{2} + \cdots + 56$$
$23$ $$T^{3} + 4 T^{2} + \cdots + 32$$
$29$ $$T^{3} - 8T^{2} + 8$$
$31$ $$T^{3} + 3T^{2} - T - 4$$
$37$ $$T^{3} - 4 T^{2} + \cdots + 56$$
$41$ $$T^{3} - 9 T^{2} + \cdots + 1006$$
$43$ $$T^{3} + T^{2} + \cdots - 184$$
$47$ $$T^{3} + 8 T^{2} + \cdots - 448$$
$53$ $$T^{3} + 19 T^{2} + \cdots + 106$$
$59$ $$T^{3} - 14 T^{2} + \cdots + 928$$
$61$ $$T^{3} + T^{2} + \cdots - 124$$
$67$ $$T^{3} - 7 T^{2} + \cdots + 1508$$
$71$ $$T^{3} + 6 T^{2} + \cdots - 64$$
$73$ $$T^{3} + 5 T^{2} + \cdots + 2$$
$79$ $$(T + 8)^{3}$$
$83$ $$T^{3} - 4 T^{2} + \cdots - 392$$
$89$ $$T^{3} - 10 T^{2} + \cdots + 32$$
$97$ $$T^{3} - 13 T^{2} + \cdots + 46$$