# Properties

 Label 7600.2.a.m Level $7600$ Weight $2$ Character orbit 7600.a Self dual yes Analytic conductor $60.686$ 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] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7600 = 2^{4} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7600.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: $$60.6863055362$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 - q^7 - 2 * q^9 $$q + q^{3} - q^{7} - 2 q^{9} + q^{13} + 3 q^{17} - q^{19} - q^{21} + 3 q^{23} - 5 q^{27} - 3 q^{29} - 2 q^{31} + 10 q^{37} + q^{39} + 6 q^{41} + 2 q^{43} - 6 q^{49} + 3 q^{51} - 3 q^{53} - q^{57} - 3 q^{59} + 8 q^{61} + 2 q^{63} - 7 q^{67} + 3 q^{69} - 12 q^{71} + 13 q^{73} - 14 q^{79} + q^{81} + 6 q^{83} - 3 q^{87} + 6 q^{89} - q^{91} - 2 q^{93} + 10 q^{97}+O(q^{100})$$ q + q^3 - q^7 - 2 * q^9 + q^13 + 3 * q^17 - q^19 - q^21 + 3 * q^23 - 5 * q^27 - 3 * q^29 - 2 * q^31 + 10 * q^37 + q^39 + 6 * q^41 + 2 * q^43 - 6 * q^49 + 3 * q^51 - 3 * q^53 - q^57 - 3 * q^59 + 8 * q^61 + 2 * q^63 - 7 * q^67 + 3 * q^69 - 12 * q^71 + 13 * q^73 - 14 * q^79 + q^81 + 6 * q^83 - 3 * q^87 + 6 * q^89 - q^91 - 2 * q^93 + 10 * 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 1.00000 0 0 0 −1.00000 0 −2.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$1$$

## 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 7600.2.a.m 1
4.b odd 2 1 950.2.a.a 1
5.b even 2 1 1520.2.a.d 1
12.b even 2 1 8550.2.a.bd 1
20.d odd 2 1 190.2.a.c 1
20.e even 4 2 950.2.b.e 2
40.e odd 2 1 6080.2.a.h 1
40.f even 2 1 6080.2.a.p 1
60.h even 2 1 1710.2.a.d 1
140.c even 2 1 9310.2.a.o 1
380.d even 2 1 3610.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.c 1 20.d odd 2 1
950.2.a.a 1 4.b odd 2 1
950.2.b.e 2 20.e even 4 2
1520.2.a.d 1 5.b even 2 1
1710.2.a.d 1 60.h even 2 1
3610.2.a.b 1 380.d even 2 1
6080.2.a.h 1 40.e odd 2 1
6080.2.a.p 1 40.f even 2 1
7600.2.a.m 1 1.a even 1 1 trivial
8550.2.a.bd 1 12.b even 2 1
9310.2.a.o 1 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7600))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} + 1$$ T7 + 1 $$T_{11}$$ T11 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T$$
$13$ $$T - 1$$
$17$ $$T - 3$$
$19$ $$T + 1$$
$23$ $$T - 3$$
$29$ $$T + 3$$
$31$ $$T + 2$$
$37$ $$T - 10$$
$41$ $$T - 6$$
$43$ $$T - 2$$
$47$ $$T$$
$53$ $$T + 3$$
$59$ $$T + 3$$
$61$ $$T - 8$$
$67$ $$T + 7$$
$71$ $$T + 12$$
$73$ $$T - 13$$
$79$ $$T + 14$$
$83$ $$T - 6$$
$89$ $$T - 6$$
$97$ $$T - 10$$