# Properties

 Label 7605.2.a.bq 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: $$\Q(\zeta_{14})^+$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2535) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_{2} q^{4} + q^{5} + (\beta_1 - 1) q^{7} + ( - \beta_{2} + 2 \beta_1 - 1) q^{8}+O(q^{10})$$ q - b1 * q^2 + b2 * q^4 + q^5 + (b1 - 1) * q^7 + (-b2 + 2*b1 - 1) * q^8 $$q - \beta_1 q^{2} + \beta_{2} q^{4} + q^{5} + (\beta_1 - 1) q^{7} + ( - \beta_{2} + 2 \beta_1 - 1) q^{8} - \beta_1 q^{10} - q^{11} + ( - \beta_{2} + \beta_1 - 2) q^{14} + ( - 3 \beta_{2} + \beta_1 - 3) q^{16} + ( - 2 \beta_1 + 4) q^{17} + ( - 3 \beta_{2} + 1) q^{19} + \beta_{2} q^{20} + \beta_1 q^{22} + (\beta_{2} - \beta_1 - 1) q^{23} + q^{25} + q^{28} + (3 \beta_{2} + 2 \beta_1) q^{29} + (4 \beta_{2} - 2 \beta_1 + 1) q^{31} + (4 \beta_{2} - \beta_1 + 3) q^{32} + (2 \beta_{2} - 4 \beta_1 + 4) q^{34} + (\beta_1 - 1) q^{35} + (\beta_{2} - 4 \beta_1) q^{37} + (3 \beta_{2} - \beta_1 + 3) q^{38} + ( - \beta_{2} + 2 \beta_1 - 1) q^{40} + ( - \beta_1 - 7) q^{41} + ( - \beta_1 - 5) q^{43} - \beta_{2} q^{44} + (\beta_1 + 1) q^{46} + (4 \beta_{2} + \beta_1 + 5) q^{47} + (\beta_{2} - 2 \beta_1 - 4) q^{49} - \beta_1 q^{50} + ( - 6 \beta_{2} + 7 \beta_1 - 6) q^{53} - q^{55} + (2 \beta_{2} - 3 \beta_1 + 4) q^{56} + ( - 5 \beta_{2} - 7) q^{58} + ( - 2 \beta_{2} + 7 \beta_1 - 2) q^{59} + ( - 7 \beta_{2} + 7 \beta_1 - 8) q^{61} + ( - 2 \beta_{2} - \beta_1) q^{62} + (3 \beta_{2} - 5 \beta_1 + 4) q^{64} + (3 \beta_{2} - \beta_1 + 8) q^{67} + (2 \beta_{2} - 2) q^{68} + ( - \beta_{2} + \beta_1 - 2) q^{70} + ( - 3 \beta_{2} + 6 \beta_1 - 1) q^{71} + (7 \beta_{2} - 10 \beta_1 + 5) q^{73} + (3 \beta_{2} + 7) q^{74} + (4 \beta_{2} - 3 \beta_1 - 3) q^{76} + ( - \beta_1 + 1) q^{77} + ( - 2 \beta_{2} - 7 \beta_1 + 1) q^{79} + ( - 3 \beta_{2} + \beta_1 - 3) q^{80} + (\beta_{2} + 7 \beta_1 + 2) q^{82} + ( - 5 \beta_{2} - 2 \beta_1 + 8) q^{83} + ( - 2 \beta_1 + 4) q^{85} + (\beta_{2} + 5 \beta_1 + 2) q^{86} + (\beta_{2} - 2 \beta_1 + 1) q^{88} + (7 \beta_{2} - 4 \beta_1 + 5) q^{89} + ( - 3 \beta_{2} + \beta_1) q^{92} + ( - 5 \beta_{2} - 5 \beta_1 - 6) q^{94} + ( - 3 \beta_{2} + 1) q^{95} + ( - 12 \beta_{2} + 6 \beta_1 - 9) q^{97} + (\beta_{2} + 4 \beta_1 + 3) q^{98}+O(q^{100})$$ q - b1 * q^2 + b2 * q^4 + q^5 + (b1 - 1) * q^7 + (-b2 + 2*b1 - 1) * q^8 - b1 * q^10 - q^11 + (-b2 + b1 - 2) * q^14 + (-3*b2 + b1 - 3) * q^16 + (-2*b1 + 4) * q^17 + (-3*b2 + 1) * q^19 + b2 * q^20 + b1 * q^22 + (b2 - b1 - 1) * q^23 + q^25 + q^28 + (3*b2 + 2*b1) * q^29 + (4*b2 - 2*b1 + 1) * q^31 + (4*b2 - b1 + 3) * q^32 + (2*b2 - 4*b1 + 4) * q^34 + (b1 - 1) * q^35 + (b2 - 4*b1) * q^37 + (3*b2 - b1 + 3) * q^38 + (-b2 + 2*b1 - 1) * q^40 + (-b1 - 7) * q^41 + (-b1 - 5) * q^43 - b2 * q^44 + (b1 + 1) * q^46 + (4*b2 + b1 + 5) * q^47 + (b2 - 2*b1 - 4) * q^49 - b1 * q^50 + (-6*b2 + 7*b1 - 6) * q^53 - q^55 + (2*b2 - 3*b1 + 4) * q^56 + (-5*b2 - 7) * q^58 + (-2*b2 + 7*b1 - 2) * q^59 + (-7*b2 + 7*b1 - 8) * q^61 + (-2*b2 - b1) * q^62 + (3*b2 - 5*b1 + 4) * q^64 + (3*b2 - b1 + 8) * q^67 + (2*b2 - 2) * q^68 + (-b2 + b1 - 2) * q^70 + (-3*b2 + 6*b1 - 1) * q^71 + (7*b2 - 10*b1 + 5) * q^73 + (3*b2 + 7) * q^74 + (4*b2 - 3*b1 - 3) * q^76 + (-b1 + 1) * q^77 + (-2*b2 - 7*b1 + 1) * q^79 + (-3*b2 + b1 - 3) * q^80 + (b2 + 7*b1 + 2) * q^82 + (-5*b2 - 2*b1 + 8) * q^83 + (-2*b1 + 4) * q^85 + (b2 + 5*b1 + 2) * q^86 + (b2 - 2*b1 + 1) * q^88 + (7*b2 - 4*b1 + 5) * q^89 + (-3*b2 + b1) * q^92 + (-5*b2 - 5*b1 - 6) * q^94 + (-3*b2 + 1) * q^95 + (-12*b2 + 6*b1 - 9) * q^97 + (b2 + 4*b1 + 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} - q^{4} + 3 q^{5} - 2 q^{7}+O(q^{10})$$ 3 * q - q^2 - q^4 + 3 * q^5 - 2 * q^7 $$3 q - q^{2} - q^{4} + 3 q^{5} - 2 q^{7} - q^{10} - 3 q^{11} - 4 q^{14} - 5 q^{16} + 10 q^{17} + 6 q^{19} - q^{20} + q^{22} - 5 q^{23} + 3 q^{25} + 3 q^{28} - q^{29} - 3 q^{31} + 4 q^{32} + 6 q^{34} - 2 q^{35} - 5 q^{37} + 5 q^{38} - 22 q^{41} - 16 q^{43} + q^{44} + 4 q^{46} + 12 q^{47} - 15 q^{49} - q^{50} - 5 q^{53} - 3 q^{55} + 7 q^{56} - 16 q^{58} + 3 q^{59} - 10 q^{61} + q^{62} + 4 q^{64} + 20 q^{67} - 8 q^{68} - 4 q^{70} + 6 q^{71} - 2 q^{73} + 18 q^{74} - 16 q^{76} + 2 q^{77} - 2 q^{79} - 5 q^{80} + 12 q^{82} + 27 q^{83} + 10 q^{85} + 10 q^{86} + 4 q^{89} + 4 q^{92} - 18 q^{94} + 6 q^{95} - 9 q^{97} + 12 q^{98}+O(q^{100})$$ 3 * q - q^2 - q^4 + 3 * q^5 - 2 * q^7 - q^10 - 3 * q^11 - 4 * q^14 - 5 * q^16 + 10 * q^17 + 6 * q^19 - q^20 + q^22 - 5 * q^23 + 3 * q^25 + 3 * q^28 - q^29 - 3 * q^31 + 4 * q^32 + 6 * q^34 - 2 * q^35 - 5 * q^37 + 5 * q^38 - 22 * q^41 - 16 * q^43 + q^44 + 4 * q^46 + 12 * q^47 - 15 * q^49 - q^50 - 5 * q^53 - 3 * q^55 + 7 * q^56 - 16 * q^58 + 3 * q^59 - 10 * q^61 + q^62 + 4 * q^64 + 20 * q^67 - 8 * q^68 - 4 * q^70 + 6 * q^71 - 2 * q^73 + 18 * q^74 - 16 * q^76 + 2 * q^77 - 2 * q^79 - 5 * q^80 + 12 * q^82 + 27 * q^83 + 10 * q^85 + 10 * q^86 + 4 * q^89 + 4 * q^92 - 18 * q^94 + 6 * q^95 - 9 * q^97 + 12 * q^98

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
−1.80194 0 1.24698 1.00000 0 0.801938 1.35690 0 −1.80194
1.2 −0.445042 0 −1.80194 1.00000 0 −0.554958 1.69202 0 −0.445042
1.3 1.24698 0 −0.445042 1.00000 0 −2.24698 −3.04892 0 1.24698
 $$n$$: 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.bq 3
3.b odd 2 1 2535.2.a.bd yes 3
13.b even 2 1 7605.2.a.bz 3
39.d odd 2 1 2535.2.a.v 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2535.2.a.v 3 39.d odd 2 1
2535.2.a.bd yes 3 3.b odd 2 1
7605.2.a.bq 3 1.a even 1 1 trivial
7605.2.a.bz 3 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

 $$T_{2}^{3} + T_{2}^{2} - 2T_{2} - 1$$ T2^3 + T2^2 - 2*T2 - 1 $$T_{7}^{3} + 2T_{7}^{2} - T_{7} - 1$$ T7^3 + 2*T7^2 - T7 - 1 $$T_{11} + 1$$ T11 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 2T - 1$$
$3$ $$T^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} + 2T^{2} - T - 1$$
$11$ $$(T + 1)^{3}$$
$13$ $$T^{3}$$
$17$ $$T^{3} - 10 T^{2} + 24 T - 8$$
$19$ $$T^{3} - 6 T^{2} - 9 T + 41$$
$23$ $$T^{3} + 5 T^{2} + 6 T + 1$$
$29$ $$T^{3} + T^{2} - 44 T - 127$$
$31$ $$T^{3} + 3 T^{2} - 25 T + 29$$
$37$ $$T^{3} + 5 T^{2} - 22 T - 97$$
$41$ $$T^{3} + 22 T^{2} + 159 T + 377$$
$43$ $$T^{3} + 16 T^{2} + 83 T + 139$$
$47$ $$T^{3} - 12 T^{2} - T + 41$$
$53$ $$T^{3} + 5 T^{2} - 92 T - 83$$
$59$ $$T^{3} - 3 T^{2} - 88 T + 377$$
$61$ $$T^{3} + 10 T^{2} - 81 T - 433$$
$67$ $$T^{3} - 20 T^{2} + 117 T - 169$$
$71$ $$T^{3} - 6 T^{2} - 51 T + 307$$
$73$ $$T^{3} + 2 T^{2} - 183 T - 743$$
$79$ $$T^{3} + 2 T^{2} - 155 T + 223$$
$83$ $$T^{3} - 27 T^{2} + 152 T + 377$$
$89$ $$T^{3} - 4 T^{2} - 81 T + 421$$
$97$ $$T^{3} + 9 T^{2} - 225 T - 2241$$