# Properties

 Label 1600.4.a.r Level $1600$ Weight $4$ Character orbit 1600.a Self dual yes Analytic conductor $94.403$ 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] = [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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 160) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{3} - 6 q^{7} - 23 q^{9}+O(q^{10})$$ q - 2 * q^3 - 6 * q^7 - 23 * q^9 $$q - 2 q^{3} - 6 q^{7} - 23 q^{9} - 60 q^{11} + 50 q^{13} + 30 q^{17} - 40 q^{19} + 12 q^{21} - 178 q^{23} + 100 q^{27} - 166 q^{29} + 20 q^{31} + 120 q^{33} + 10 q^{37} - 100 q^{39} - 250 q^{41} + 142 q^{43} - 214 q^{47} - 307 q^{49} - 60 q^{51} + 490 q^{53} + 80 q^{57} + 800 q^{59} - 250 q^{61} + 138 q^{63} - 774 q^{67} + 356 q^{69} + 100 q^{71} + 230 q^{73} + 360 q^{77} - 1320 q^{79} + 421 q^{81} + 982 q^{83} + 332 q^{87} + 874 q^{89} - 300 q^{91} - 40 q^{93} + 310 q^{97} + 1380 q^{99}+O(q^{100})$$ q - 2 * q^3 - 6 * q^7 - 23 * q^9 - 60 * q^11 + 50 * q^13 + 30 * q^17 - 40 * q^19 + 12 * q^21 - 178 * q^23 + 100 * q^27 - 166 * q^29 + 20 * q^31 + 120 * q^33 + 10 * q^37 - 100 * q^39 - 250 * q^41 + 142 * q^43 - 214 * q^47 - 307 * q^49 - 60 * q^51 + 490 * q^53 + 80 * q^57 + 800 * q^59 - 250 * q^61 + 138 * q^63 - 774 * q^67 + 356 * q^69 + 100 * q^71 + 230 * q^73 + 360 * q^77 - 1320 * q^79 + 421 * q^81 + 982 * q^83 + 332 * q^87 + 874 * q^89 - 300 * q^91 - 40 * q^93 + 310 * q^97 + 1380 * 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 −2.00000 0 0 0 −6.00000 0 −23.0000 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.r 1
4.b odd 2 1 1600.4.a.bj 1
5.b even 2 1 320.4.a.i 1
8.b even 2 1 800.4.a.h 1
8.d odd 2 1 800.4.a.d 1
20.d odd 2 1 320.4.a.f 1
40.e odd 2 1 160.4.a.b yes 1
40.f even 2 1 160.4.a.a 1
40.i odd 4 2 800.4.c.f 2
40.k even 4 2 800.4.c.e 2
80.k odd 4 2 1280.4.d.k 2
80.q even 4 2 1280.4.d.f 2
120.i odd 2 1 1440.4.a.o 1
120.m even 2 1 1440.4.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.a 1 40.f even 2 1
160.4.a.b yes 1 40.e odd 2 1
320.4.a.f 1 20.d odd 2 1
320.4.a.i 1 5.b even 2 1
800.4.a.d 1 8.d odd 2 1
800.4.a.h 1 8.b even 2 1
800.4.c.e 2 40.k even 4 2
800.4.c.f 2 40.i odd 4 2
1280.4.d.f 2 80.q even 4 2
1280.4.d.k 2 80.k odd 4 2
1440.4.a.n 1 120.m even 2 1
1440.4.a.o 1 120.i odd 2 1
1600.4.a.r 1 1.a even 1 1 trivial
1600.4.a.bj 1 4.b odd 2 1

## Hecke kernels

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

 $$T_{3} + 2$$ T3 + 2 $$T_{7} + 6$$ T7 + 6 $$T_{11} + 60$$ T11 + 60 $$T_{13} - 50$$ T13 - 50

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T$$
$7$ $$T + 6$$
$11$ $$T + 60$$
$13$ $$T - 50$$
$17$ $$T - 30$$
$19$ $$T + 40$$
$23$ $$T + 178$$
$29$ $$T + 166$$
$31$ $$T - 20$$
$37$ $$T - 10$$
$41$ $$T + 250$$
$43$ $$T - 142$$
$47$ $$T + 214$$
$53$ $$T - 490$$
$59$ $$T - 800$$
$61$ $$T + 250$$
$67$ $$T + 774$$
$71$ $$T - 100$$
$73$ $$T - 230$$
$79$ $$T + 1320$$
$83$ $$T - 982$$
$89$ $$T - 874$$
$97$ $$T - 310$$