# Properties

 Label 7600.2.a.bw Level $7600$ Weight $2$ Character orbit 7600.a Self dual yes Analytic conductor $60.686$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 475) 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.

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

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 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
 1.80194 0.445042 −1.24698
0 −0.801938 0 0 0 1.69202 0 −2.35690 0
1.2 0 0.554958 0 0 0 −3.04892 0 −2.69202 0
1.3 0 2.24698 0 0 0 1.35690 0 2.04892 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.bw 3
4.b odd 2 1 475.2.a.d 3
5.b even 2 1 7600.2.a.bn 3
12.b even 2 1 4275.2.a.bn 3
20.d odd 2 1 475.2.a.h yes 3
20.e even 4 2 475.2.b.c 6
60.h even 2 1 4275.2.a.z 3
76.d even 2 1 9025.2.a.be 3
380.d even 2 1 9025.2.a.w 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.a.d 3 4.b odd 2 1
475.2.a.h yes 3 20.d odd 2 1
475.2.b.c 6 20.e even 4 2
4275.2.a.z 3 60.h even 2 1
4275.2.a.bn 3 12.b even 2 1
7600.2.a.bn 3 5.b even 2 1
7600.2.a.bw 3 1.a even 1 1 trivial
9025.2.a.w 3 380.d even 2 1
9025.2.a.be 3 76.d even 2 1

## Hecke kernels

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

 $$T_{3}^{3} - 2T_{3}^{2} - T_{3} + 1$$ T3^3 - 2*T3^2 - T3 + 1 $$T_{7}^{3} - 7T_{7} + 7$$ T7^3 - 7*T7 + 7 $$T_{11}^{3} + T_{11}^{2} - 16T_{11} + 13$$ T11^3 + T11^2 - 16*T11 + 13 $$T_{13}^{3} + 5T_{13}^{2} + 6T_{13} + 1$$ T13^3 + 5*T13^2 + 6*T13 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 2T^{2} - T + 1$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 7T + 7$$
$11$ $$T^{3} + T^{2} - 16 T + 13$$
$13$ $$T^{3} + 5 T^{2} + 6 T + 1$$
$17$ $$T^{3} + 2 T^{2} - 15 T + 13$$
$19$ $$(T - 1)^{3}$$
$23$ $$T^{3} - 8 T^{2} + 19 T - 13$$
$29$ $$T^{3} + 7 T^{2} - 42 T - 91$$
$31$ $$T^{3} - 5 T^{2} - 36 T - 43$$
$37$ $$T^{3} + 5 T^{2} - 8 T + 1$$
$41$ $$T^{3} - T^{2} - 114 T + 421$$
$43$ $$T^{3} - 5 T^{2} - 57 T + 293$$
$47$ $$T^{3} - 3 T^{2} - 4 T + 13$$
$53$ $$T^{3} + 19 T^{2} + 55 T - 307$$
$59$ $$T^{3} + 10 T^{2} - 32 T - 328$$
$61$ $$T^{3} + 17 T^{2} + 66 T - 41$$
$67$ $$T^{3} + T^{2} - 142 T + 559$$
$71$ $$T^{3} - 19 T^{2} + 55 T + 307$$
$73$ $$T^{3} - T^{2} - 170 T + 463$$
$79$ $$T^{3} + 18 T^{2} + 87 T + 97$$
$83$ $$T^{3} - 13 T^{2} - 2 T + 139$$
$89$ $$T^{3} - 2 T^{2} - 99 T + 281$$
$97$ $$T^{3} + 5 T^{2} + 6 T + 1$$