# Properties

 Label 7605.2.a.v Level $7605$ Weight $2$ Character orbit 7605.a Self dual yes Analytic conductor $60.726$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

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

## 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
 0
2.00000 0 2.00000 1.00000 0 3.00000 0 0 2.00000
 $$n$$: 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.v 1
3.b odd 2 1 2535.2.a.b 1
13.b even 2 1 585.2.a.a 1
39.d odd 2 1 195.2.a.d 1
52.b odd 2 1 9360.2.a.w 1
65.d even 2 1 2925.2.a.t 1
65.h odd 4 2 2925.2.c.d 2
156.h even 2 1 3120.2.a.n 1
195.e odd 2 1 975.2.a.b 1
195.s even 4 2 975.2.c.b 2
273.g even 2 1 9555.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.d 1 39.d odd 2 1
585.2.a.a 1 13.b even 2 1
975.2.a.b 1 195.e odd 2 1
975.2.c.b 2 195.s even 4 2
2535.2.a.b 1 3.b odd 2 1
2925.2.a.t 1 65.d even 2 1
2925.2.c.d 2 65.h odd 4 2
3120.2.a.n 1 156.h even 2 1
7605.2.a.v 1 1.a even 1 1 trivial
9360.2.a.w 1 52.b odd 2 1
9555.2.a.t 1 273.g even 2 1

## Hecke kernels

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

 $$T_{2} - 2$$ T2 - 2 $$T_{7} - 3$$ T7 - 3 $$T_{11} + 5$$ T11 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T - 3$$
$11$ $$T + 5$$
$13$ $$T$$
$17$ $$T + 5$$
$19$ $$T + 2$$
$23$ $$T - 1$$
$29$ $$T + 10$$
$31$ $$T - 2$$
$37$ $$T - 3$$
$41$ $$T + 9$$
$43$ $$T + 4$$
$47$ $$T - 10$$
$53$ $$T + 9$$
$59$ $$T$$
$61$ $$T + 11$$
$67$ $$T - 4$$
$71$ $$T - 15$$
$73$ $$T + 6$$
$79$ $$T + 11$$
$83$ $$T - 8$$
$89$ $$T + 11$$
$97$ $$T - 9$$