# Properties

 Label 7605.2.a.a 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 2 q^{4} + q^{5} + 5 q^{7}+O(q^{10})$$ q - 2 * q^2 + 2 * q^4 + q^5 + 5 * q^7 $$q - 2 q^{2} + 2 q^{4} + q^{5} + 5 q^{7} - 2 q^{10} - 2 q^{11} - 10 q^{14} - 4 q^{16} - 2 q^{17} + 2 q^{20} + 4 q^{22} - 6 q^{23} + q^{25} + 10 q^{28} + 4 q^{29} - 7 q^{31} + 8 q^{32} + 4 q^{34} + 5 q^{35} - 2 q^{37} - 6 q^{41} + q^{43} - 4 q^{44} + 12 q^{46} + 8 q^{47} + 18 q^{49} - 2 q^{50} + 4 q^{53} - 2 q^{55} - 8 q^{58} - 12 q^{59} - 13 q^{61} + 14 q^{62} - 8 q^{64} - 7 q^{67} - 4 q^{68} - 10 q^{70} - 12 q^{71} + 15 q^{73} + 4 q^{74} - 10 q^{77} + 3 q^{79} - 4 q^{80} + 12 q^{82} - 8 q^{83} - 2 q^{85} - 2 q^{86} - 14 q^{89} - 12 q^{92} - 16 q^{94} - 5 q^{97} - 36 q^{98}+O(q^{100})$$ q - 2 * q^2 + 2 * q^4 + q^5 + 5 * q^7 - 2 * q^10 - 2 * q^11 - 10 * q^14 - 4 * q^16 - 2 * q^17 + 2 * q^20 + 4 * q^22 - 6 * q^23 + q^25 + 10 * q^28 + 4 * q^29 - 7 * q^31 + 8 * q^32 + 4 * q^34 + 5 * q^35 - 2 * q^37 - 6 * q^41 + q^43 - 4 * q^44 + 12 * q^46 + 8 * q^47 + 18 * q^49 - 2 * q^50 + 4 * q^53 - 2 * q^55 - 8 * q^58 - 12 * q^59 - 13 * q^61 + 14 * q^62 - 8 * q^64 - 7 * q^67 - 4 * q^68 - 10 * q^70 - 12 * q^71 + 15 * q^73 + 4 * q^74 - 10 * q^77 + 3 * q^79 - 4 * q^80 + 12 * q^82 - 8 * q^83 - 2 * q^85 - 2 * q^86 - 14 * q^89 - 12 * q^92 - 16 * q^94 - 5 * q^97 - 36 * 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 5.00000 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.a 1
3.b odd 2 1 2535.2.a.m 1
13.b even 2 1 7605.2.a.s 1
13.c even 3 2 585.2.j.b 2
39.d odd 2 1 2535.2.a.c 1
39.i odd 6 2 195.2.i.a 2
195.x odd 6 2 975.2.i.i 2
195.bl even 12 4 975.2.bb.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.a 2 39.i odd 6 2
585.2.j.b 2 13.c even 3 2
975.2.i.i 2 195.x odd 6 2
975.2.bb.f 4 195.bl even 12 4
2535.2.a.c 1 39.d odd 2 1
2535.2.a.m 1 3.b odd 2 1
7605.2.a.a 1 1.a even 1 1 trivial
7605.2.a.s 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T - 5$$
$11$ $$T + 2$$
$13$ $$T$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T + 6$$
$29$ $$T - 4$$
$31$ $$T + 7$$
$37$ $$T + 2$$
$41$ $$T + 6$$
$43$ $$T - 1$$
$47$ $$T - 8$$
$53$ $$T - 4$$
$59$ $$T + 12$$
$61$ $$T + 13$$
$67$ $$T + 7$$
$71$ $$T + 12$$
$73$ $$T - 15$$
$79$ $$T - 3$$
$83$ $$T + 8$$
$89$ $$T + 14$$
$97$ $$T + 5$$