# Properties

 Label 7600.2.a.by Level $7600$ Weight $2$ Character orbit 7600.a Self dual yes Analytic conductor $60.686$ 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] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7600 = 2^{4} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7600.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: $$60.6863055362$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.257.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 3$$ x^3 - x^2 - 4*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1900) 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.

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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ 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
 2.19869 0.713538 −1.91223
0 −1.19869 0 0 0 −1.19869 0 −1.56314 0
1.2 0 0.286462 0 0 0 0.286462 0 −2.91794 0
1.3 0 2.91223 0 0 0 2.91223 0 5.48108 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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 7600.2.a.by 3
4.b odd 2 1 1900.2.a.f 3
5.b even 2 1 7600.2.a.bj 3
20.d odd 2 1 1900.2.a.h yes 3
20.e even 4 2 1900.2.c.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.a.f 3 4.b odd 2 1
1900.2.a.h yes 3 20.d odd 2 1
1900.2.c.g 6 20.e even 4 2
7600.2.a.bj 3 5.b even 2 1
7600.2.a.by 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7600))$$:

 $$T_{3}^{3} - 2T_{3}^{2} - 3T_{3} + 1$$ T3^3 - 2*T3^2 - 3*T3 + 1 $$T_{7}^{3} - 2T_{7}^{2} - 3T_{7} + 1$$ T7^3 - 2*T7^2 - 3*T7 + 1 $$T_{11}^{3} - T_{11}^{2} - 26T_{11} - 15$$ T11^3 - T11^2 - 26*T11 - 15 $$T_{13}^{3} + T_{13}^{2} - 8T_{13} - 3$$ T13^3 + T13^2 - 8*T13 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 2 T^{2} + \cdots + 1$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 2 T^{2} + \cdots + 1$$
$11$ $$T^{3} - T^{2} + \cdots - 15$$
$13$ $$T^{3} + T^{2} - 8T - 3$$
$17$ $$T^{3} + 2 T^{2} + \cdots - 1$$
$19$ $$(T - 1)^{3}$$
$23$ $$T^{3} - 8 T^{2} + \cdots + 9$$
$29$ $$T^{3} + 7 T^{2} + \cdots - 25$$
$31$ $$T^{3} + 11 T^{2} + \cdots - 241$$
$37$ $$T^{3} - 5 T^{2} + \cdots + 405$$
$41$ $$T^{3} - 13 T^{2} + \cdots - 25$$
$43$ $$T^{3} - 11 T^{2} + \cdots + 27$$
$47$ $$T^{3} - 19 T^{2} + \cdots + 63$$
$53$ $$T^{3} + 11 T^{2} + \cdots - 27$$
$59$ $$T^{3} - 6 T^{2} + \cdots + 856$$
$61$ $$T^{3} - 7 T^{2} + \cdots - 25$$
$67$ $$T^{3} - 3 T^{2} + \cdots + 599$$
$71$ $$T^{3} - 5 T^{2} + \cdots - 123$$
$73$ $$T^{3} + 9 T^{2} + \cdots - 27$$
$79$ $$T^{3} - 18 T^{2} + \cdots + 607$$
$83$ $$T^{3} - 3 T^{2} + \cdots + 749$$
$89$ $$T^{3} - 183T - 857$$
$97$ $$T^{3} - 3 T^{2} + \cdots + 81$$