# Properties

 Label 7605.2.a.m Level $7605$ Weight $2$ Character orbit 7605.a Self dual yes Analytic conductor $60.726$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 585) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{4} + q^{5} + q^{7}+O(q^{10})$$ q - 2 * q^4 + q^5 + q^7 $$q - 2 q^{4} + q^{5} + q^{7} - 3 q^{11} + 4 q^{16} + 3 q^{17} + 4 q^{19} - 2 q^{20} + 9 q^{23} + q^{25} - 2 q^{28} + 6 q^{29} - 2 q^{31} + q^{35} + q^{37} - 3 q^{41} + 2 q^{43} + 6 q^{44} - 6 q^{47} - 6 q^{49} - 9 q^{53} - 3 q^{55} - 12 q^{59} + 5 q^{61} - 8 q^{64} + 4 q^{67} - 6 q^{68} + 9 q^{71} - 14 q^{73} - 8 q^{76} - 3 q^{77} - 7 q^{79} + 4 q^{80} + 3 q^{85} + 15 q^{89} - 18 q^{92} + 4 q^{95} - 5 q^{97}+O(q^{100})$$ q - 2 * q^4 + q^5 + q^7 - 3 * q^11 + 4 * q^16 + 3 * q^17 + 4 * q^19 - 2 * q^20 + 9 * q^23 + q^25 - 2 * q^28 + 6 * q^29 - 2 * q^31 + q^35 + q^37 - 3 * q^41 + 2 * q^43 + 6 * q^44 - 6 * q^47 - 6 * q^49 - 9 * q^53 - 3 * q^55 - 12 * q^59 + 5 * q^61 - 8 * q^64 + 4 * q^67 - 6 * q^68 + 9 * q^71 - 14 * q^73 - 8 * q^76 - 3 * q^77 - 7 * q^79 + 4 * q^80 + 3 * q^85 + 15 * q^89 - 18 * q^92 + 4 * q^95 - 5 * q^97

## 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
0 0 −2.00000 1.00000 0 1.00000 0 0 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
$$3$$ $$1$$
$$5$$ $$-1$$
$$13$$ $$1$$

## 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.m 1
3.b odd 2 1 7605.2.a.j 1
13.b even 2 1 585.2.a.e 1
39.d odd 2 1 585.2.a.f yes 1
52.b odd 2 1 9360.2.a.r 1
65.d even 2 1 2925.2.a.k 1
65.h odd 4 2 2925.2.c.m 2
156.h even 2 1 9360.2.a.bu 1
195.e odd 2 1 2925.2.a.i 1
195.s even 4 2 2925.2.c.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.a.e 1 13.b even 2 1
585.2.a.f yes 1 39.d odd 2 1
2925.2.a.i 1 195.e odd 2 1
2925.2.a.k 1 65.d even 2 1
2925.2.c.l 2 195.s even 4 2
2925.2.c.m 2 65.h odd 4 2
7605.2.a.j 1 3.b odd 2 1
7605.2.a.m 1 1.a even 1 1 trivial
9360.2.a.r 1 52.b odd 2 1
9360.2.a.bu 1 156.h even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T - 1$$
$11$ $$T + 3$$
$13$ $$T$$
$17$ $$T - 3$$
$19$ $$T - 4$$
$23$ $$T - 9$$
$29$ $$T - 6$$
$31$ $$T + 2$$
$37$ $$T - 1$$
$41$ $$T + 3$$
$43$ $$T - 2$$
$47$ $$T + 6$$
$53$ $$T + 9$$
$59$ $$T + 12$$
$61$ $$T - 5$$
$67$ $$T - 4$$
$71$ $$T - 9$$
$73$ $$T + 14$$
$79$ $$T + 7$$
$83$ $$T$$
$89$ $$T - 15$$
$97$ $$T + 5$$