# Properties

 Label 1600.4.a.bq Level $1600$ Weight $4$ Character orbit 1600.a Self dual yes Analytic conductor $94.403$ 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] = [1600,4,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1600.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: $$94.4030560092$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 800) 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)

 $$f(q)$$ $$=$$ $$q + 5 q^{3} + 10 q^{7} - 2 q^{9}+O(q^{10})$$ q + 5 * q^3 + 10 * q^7 - 2 * q^9 $$q + 5 q^{3} + 10 q^{7} - 2 q^{9} + 15 q^{11} - 8 q^{13} - 21 q^{17} - 105 q^{19} + 50 q^{21} + 10 q^{23} - 145 q^{27} + 20 q^{29} - 230 q^{31} + 75 q^{33} + 54 q^{37} - 40 q^{39} - 195 q^{41} - 300 q^{43} + 480 q^{47} - 243 q^{49} - 105 q^{51} - 322 q^{53} - 525 q^{57} - 560 q^{59} + 730 q^{61} - 20 q^{63} + 255 q^{67} + 50 q^{69} - 40 q^{71} + 317 q^{73} + 150 q^{77} - 830 q^{79} - 671 q^{81} + 75 q^{83} + 100 q^{87} - 705 q^{89} - 80 q^{91} - 1150 q^{93} - 1434 q^{97} - 30 q^{99}+O(q^{100})$$ q + 5 * q^3 + 10 * q^7 - 2 * q^9 + 15 * q^11 - 8 * q^13 - 21 * q^17 - 105 * q^19 + 50 * q^21 + 10 * q^23 - 145 * q^27 + 20 * q^29 - 230 * q^31 + 75 * q^33 + 54 * q^37 - 40 * q^39 - 195 * q^41 - 300 * q^43 + 480 * q^47 - 243 * q^49 - 105 * q^51 - 322 * q^53 - 525 * q^57 - 560 * q^59 + 730 * q^61 - 20 * q^63 + 255 * q^67 + 50 * q^69 - 40 * q^71 + 317 * q^73 + 150 * q^77 - 830 * q^79 - 671 * q^81 + 75 * q^83 + 100 * q^87 - 705 * q^89 - 80 * q^91 - 1150 * q^93 - 1434 * q^97 - 30 * q^99

## 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 5.00000 0 0 0 10.0000 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
$$2$$ $$-1$$
$$5$$ $$+1$$

## 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 1600.4.a.bq 1
4.b odd 2 1 1600.4.a.k 1
5.b even 2 1 1600.4.a.l 1
8.b even 2 1 800.4.a.b 1
8.d odd 2 1 800.4.a.j yes 1
20.d odd 2 1 1600.4.a.bp 1
40.e odd 2 1 800.4.a.c yes 1
40.f even 2 1 800.4.a.i yes 1
40.i odd 4 2 800.4.c.c 2
40.k even 4 2 800.4.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.4.a.b 1 8.b even 2 1
800.4.a.c yes 1 40.e odd 2 1
800.4.a.i yes 1 40.f even 2 1
800.4.a.j yes 1 8.d odd 2 1
800.4.c.c 2 40.i odd 4 2
800.4.c.d 2 40.k even 4 2
1600.4.a.k 1 4.b odd 2 1
1600.4.a.l 1 5.b even 2 1
1600.4.a.bp 1 20.d odd 2 1
1600.4.a.bq 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1600))$$:

 $$T_{3} - 5$$ T3 - 5 $$T_{7} - 10$$ T7 - 10 $$T_{11} - 15$$ T11 - 15 $$T_{13} + 8$$ T13 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 5$$
$5$ $$T$$
$7$ $$T - 10$$
$11$ $$T - 15$$
$13$ $$T + 8$$
$17$ $$T + 21$$
$19$ $$T + 105$$
$23$ $$T - 10$$
$29$ $$T - 20$$
$31$ $$T + 230$$
$37$ $$T - 54$$
$41$ $$T + 195$$
$43$ $$T + 300$$
$47$ $$T - 480$$
$53$ $$T + 322$$
$59$ $$T + 560$$
$61$ $$T - 730$$
$67$ $$T - 255$$
$71$ $$T + 40$$
$73$ $$T - 317$$
$79$ $$T + 830$$
$83$ $$T - 75$$
$89$ $$T + 705$$
$97$ $$T + 1434$$