# Properties

 Label 7605.2.a.bw Level $7605$ Weight $2$ Character orbit 7605.a Self dual yes Analytic conductor $60.726$ 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] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, 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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7605 = 3^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7605.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: $$60.7262307372$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.756.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 2$$ x^3 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + ( - \beta_1 - 1) q^{7} + (2 \beta_1 + 2) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 2) * q^4 + q^5 + (-b1 - 1) * q^7 + (2*b1 + 2) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + ( - \beta_1 - 1) q^{7} + (2 \beta_1 + 2) q^{8} + \beta_1 q^{10} - 2 \beta_1 q^{11} + ( - \beta_{2} - \beta_1 - 4) q^{14} + (2 \beta_1 + 4) q^{16} + (2 \beta_{2} - \beta_1) q^{17} + ( - \beta_{2} - 4) q^{19} + (\beta_{2} + 2) q^{20} + ( - 2 \beta_{2} - 8) q^{22} - 2 \beta_{2} q^{23} + q^{25} + ( - \beta_{2} - 4 \beta_1 - 4) q^{28} + (\beta_{2} - \beta_1 - 2) q^{29} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{31} + (2 \beta_{2} + 4) q^{32} + ( - \beta_{2} + 4 \beta_1) q^{34} + ( - \beta_1 - 1) q^{35} + (2 \beta_{2} - 2) q^{37} + ( - 6 \beta_1 - 2) q^{38} + (2 \beta_1 + 2) q^{40} + ( - \beta_{2} - \beta_1) q^{41} + (\beta_{2} - \beta_1 + 3) q^{43} + ( - 8 \beta_1 - 4) q^{44} + ( - 4 \beta_1 - 4) q^{46} + (3 \beta_1 - 4) q^{47} + (\beta_{2} + 2 \beta_1 - 2) q^{49} + \beta_1 q^{50} + 2 \beta_1 q^{53} - 2 \beta_1 q^{55} + ( - 2 \beta_{2} - 4 \beta_1 - 10) q^{56} + ( - \beta_{2} - 2) q^{58} + ( - \beta_{2} - \beta_1 - 2) q^{59} + (\beta_{2} - 2 \beta_1 - 1) q^{61} + (2 \beta_{2} - 7 \beta_1 + 2) q^{62} + (4 \beta_1 - 4) q^{64} + ( - 3 \beta_{2} + 3 \beta_1 - 3) q^{67} + 14 q^{68} + ( - \beta_{2} - \beta_1 - 4) q^{70} + ( - 3 \beta_{2} + \beta_1 - 4) q^{71} + (3 \beta_1 - 7) q^{73} + (2 \beta_1 + 4) q^{74} + ( - 4 \beta_{2} - 2 \beta_1 - 16) q^{76} + (2 \beta_{2} + 2 \beta_1 + 8) q^{77} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{79} + (2 \beta_1 + 4) q^{80} + ( - \beta_{2} - 2 \beta_1 - 6) q^{82} + ( - 2 \beta_{2} - 6) q^{83} + (2 \beta_{2} - \beta_1) q^{85} + ( - \beta_{2} + 5 \beta_1 - 2) q^{86} + ( - 4 \beta_{2} - 4 \beta_1 - 16) q^{88} + ( - \beta_{2} + \beta_1 + 10) q^{89} + ( - 4 \beta_1 - 16) q^{92} + (3 \beta_{2} - 4 \beta_1 + 12) q^{94} + ( - \beta_{2} - 4) q^{95} + (\beta_{2} + 3 \beta_1 - 5) q^{97} + (2 \beta_{2} + 10) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 2) * q^4 + q^5 + (-b1 - 1) * q^7 + (2*b1 + 2) * q^8 + b1 * q^10 - 2*b1 * q^11 + (-b2 - b1 - 4) * q^14 + (2*b1 + 4) * q^16 + (2*b2 - b1) * q^17 + (-b2 - 4) * q^19 + (b2 + 2) * q^20 + (-2*b2 - 8) * q^22 - 2*b2 * q^23 + q^25 + (-b2 - 4*b1 - 4) * q^28 + (b2 - b1 - 2) * q^29 + (-3*b2 + 2*b1 - 1) * q^31 + (2*b2 + 4) * q^32 + (-b2 + 4*b1) * q^34 + (-b1 - 1) * q^35 + (2*b2 - 2) * q^37 + (-6*b1 - 2) * q^38 + (2*b1 + 2) * q^40 + (-b2 - b1) * q^41 + (b2 - b1 + 3) * q^43 + (-8*b1 - 4) * q^44 + (-4*b1 - 4) * q^46 + (3*b1 - 4) * q^47 + (b2 + 2*b1 - 2) * q^49 + b1 * q^50 + 2*b1 * q^53 - 2*b1 * q^55 + (-2*b2 - 4*b1 - 10) * q^56 + (-b2 - 2) * q^58 + (-b2 - b1 - 2) * q^59 + (b2 - 2*b1 - 1) * q^61 + (2*b2 - 7*b1 + 2) * q^62 + (4*b1 - 4) * q^64 + (-3*b2 + 3*b1 - 3) * q^67 + 14 * q^68 + (-b2 - b1 - 4) * q^70 + (-3*b2 + b1 - 4) * q^71 + (3*b1 - 7) * q^73 + (2*b1 + 4) * q^74 + (-4*b2 - 2*b1 - 16) * q^76 + (2*b2 + 2*b1 + 8) * q^77 + (-3*b2 + 2*b1 - 1) * q^79 + (2*b1 + 4) * q^80 + (-b2 - 2*b1 - 6) * q^82 + (-2*b2 - 6) * q^83 + (2*b2 - b1) * q^85 + (-b2 + 5*b1 - 2) * q^86 + (-4*b2 - 4*b1 - 16) * q^88 + (-b2 + b1 + 10) * q^89 + (-4*b1 - 16) * q^92 + (3*b2 - 4*b1 + 12) * q^94 + (-b2 - 4) * q^95 + (b2 + 3*b1 - 5) * q^97 + (2*b2 + 10) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 6 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10})$$ 3 * q + 6 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 $$3 q + 6 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 12 q^{14} + 12 q^{16} - 12 q^{19} + 6 q^{20} - 24 q^{22} + 3 q^{25} - 12 q^{28} - 6 q^{29} - 3 q^{31} + 12 q^{32} - 3 q^{35} - 6 q^{37} - 6 q^{38} + 6 q^{40} + 9 q^{43} - 12 q^{44} - 12 q^{46} - 12 q^{47} - 6 q^{49} - 30 q^{56} - 6 q^{58} - 6 q^{59} - 3 q^{61} + 6 q^{62} - 12 q^{64} - 9 q^{67} + 42 q^{68} - 12 q^{70} - 12 q^{71} - 21 q^{73} + 12 q^{74} - 48 q^{76} + 24 q^{77} - 3 q^{79} + 12 q^{80} - 18 q^{82} - 18 q^{83} - 6 q^{86} - 48 q^{88} + 30 q^{89} - 48 q^{92} + 36 q^{94} - 12 q^{95} - 15 q^{97} + 30 q^{98}+O(q^{100})$$ 3 * q + 6 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 12 * q^14 + 12 * q^16 - 12 * q^19 + 6 * q^20 - 24 * q^22 + 3 * q^25 - 12 * q^28 - 6 * q^29 - 3 * q^31 + 12 * q^32 - 3 * q^35 - 6 * q^37 - 6 * q^38 + 6 * q^40 + 9 * q^43 - 12 * q^44 - 12 * q^46 - 12 * q^47 - 6 * q^49 - 30 * q^56 - 6 * q^58 - 6 * q^59 - 3 * q^61 + 6 * q^62 - 12 * q^64 - 9 * q^67 + 42 * q^68 - 12 * q^70 - 12 * q^71 - 21 * q^73 + 12 * q^74 - 48 * q^76 + 24 * q^77 - 3 * q^79 + 12 * q^80 - 18 * q^82 - 18 * q^83 - 6 * q^86 - 48 * q^88 + 30 * q^89 - 48 * q^92 + 36 * q^94 - 12 * q^95 - 15 * q^97 + 30 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4

## 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
 −2.26180 −0.339877 2.60168
−2.26180 0 3.11575 1.00000 0 1.26180 −2.52360 0 −2.26180
1.2 −0.339877 0 −1.88448 1.00000 0 −0.660123 1.32025 0 −0.339877
1.3 2.60168 0 4.76873 1.00000 0 −3.60168 7.20336 0 2.60168
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$1$$

## 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 7605.2.a.bw 3
3.b odd 2 1 2535.2.a.ba 3
13.b even 2 1 7605.2.a.bv 3
13.e even 6 2 585.2.j.f 6
39.d odd 2 1 2535.2.a.bb 3
39.h odd 6 2 195.2.i.d 6
195.y odd 6 2 975.2.i.l 6
195.bf even 12 4 975.2.bb.k 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.d 6 39.h odd 6 2
585.2.j.f 6 13.e even 6 2
975.2.i.l 6 195.y odd 6 2
975.2.bb.k 12 195.bf even 12 4
2535.2.a.ba 3 3.b odd 2 1
2535.2.a.bb 3 39.d odd 2 1
7605.2.a.bv 3 13.b even 2 1
7605.2.a.bw 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

 $$T_{2}^{3} - 6T_{2} - 2$$ T2^3 - 6*T2 - 2 $$T_{7}^{3} + 3T_{7}^{2} - 3T_{7} - 3$$ T7^3 + 3*T7^2 - 3*T7 - 3 $$T_{11}^{3} - 24T_{11} + 16$$ T11^3 - 24*T11 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 6T - 2$$
$3$ $$T^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} + 3 T^{2} - 3 T - 3$$
$11$ $$T^{3} - 24T + 16$$
$13$ $$T^{3}$$
$17$ $$T^{3} - 42T + 98$$
$19$ $$T^{3} + 12 T^{2} + 36 T + 4$$
$23$ $$T^{3} - 48T - 96$$
$29$ $$T^{3} + 6T^{2} - 14$$
$31$ $$T^{3} + 3 T^{2} - 93 T - 363$$
$37$ $$T^{3} + 6 T^{2} - 36 T + 8$$
$41$ $$T^{3} - 24T + 26$$
$43$ $$T^{3} - 9 T^{2} + 15 T + 11$$
$47$ $$T^{3} + 12 T^{2} - 6 T - 206$$
$53$ $$T^{3} - 24T - 16$$
$59$ $$T^{3} + 6 T^{2} - 12 T - 14$$
$61$ $$T^{3} + 3 T^{2} - 21 T - 67$$
$67$ $$T^{3} + 9 T^{2} - 81 T - 351$$
$71$ $$T^{3} + 12 T^{2} - 48 T - 682$$
$73$ $$T^{3} + 21 T^{2} + 93 T - 89$$
$79$ $$T^{3} + 3 T^{2} - 93 T - 363$$
$83$ $$T^{3} + 18 T^{2} + 60 T - 168$$
$89$ $$T^{3} - 30 T^{2} + 288 T - 882$$
$97$ $$T^{3} + 15 T^{2} - 9 T - 589$$