# Properties

 Label 1600.4.a.e 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 32) 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)

 $$f(q)$$ $$=$$ $$q - 8 q^{3} + 16 q^{7} + 37 q^{9}+O(q^{10})$$ q - 8 * q^3 + 16 * q^7 + 37 * q^9 $$q - 8 q^{3} + 16 q^{7} + 37 q^{9} - 40 q^{11} - 50 q^{13} + 30 q^{17} + 40 q^{19} - 128 q^{21} + 48 q^{23} - 80 q^{27} + 34 q^{29} - 320 q^{31} + 320 q^{33} + 310 q^{37} + 400 q^{39} + 410 q^{41} - 152 q^{43} - 416 q^{47} - 87 q^{49} - 240 q^{51} - 410 q^{53} - 320 q^{57} - 200 q^{59} - 30 q^{61} + 592 q^{63} - 776 q^{67} - 384 q^{69} - 400 q^{71} + 630 q^{73} - 640 q^{77} + 1120 q^{79} - 359 q^{81} - 552 q^{83} - 272 q^{87} - 326 q^{89} - 800 q^{91} + 2560 q^{93} + 110 q^{97} - 1480 q^{99}+O(q^{100})$$ q - 8 * q^3 + 16 * q^7 + 37 * q^9 - 40 * q^11 - 50 * q^13 + 30 * q^17 + 40 * q^19 - 128 * q^21 + 48 * q^23 - 80 * q^27 + 34 * q^29 - 320 * q^31 + 320 * q^33 + 310 * q^37 + 400 * q^39 + 410 * q^41 - 152 * q^43 - 416 * q^47 - 87 * q^49 - 240 * q^51 - 410 * q^53 - 320 * q^57 - 200 * q^59 - 30 * q^61 + 592 * q^63 - 776 * q^67 - 384 * q^69 - 400 * q^71 + 630 * q^73 - 640 * q^77 + 1120 * q^79 - 359 * q^81 - 552 * q^83 - 272 * q^87 - 326 * q^89 - 800 * q^91 + 2560 * q^93 + 110 * q^97 - 1480 * 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 −8.00000 0 0 0 16.0000 0 37.0000 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.e 1
4.b odd 2 1 1600.4.a.bw 1
5.b even 2 1 64.4.a.e 1
8.b even 2 1 800.4.a.k 1
8.d odd 2 1 800.4.a.a 1
15.d odd 2 1 576.4.a.g 1
20.d odd 2 1 64.4.a.a 1
40.e odd 2 1 32.4.a.c yes 1
40.f even 2 1 32.4.a.a 1
40.i odd 4 2 800.4.c.b 2
40.k even 4 2 800.4.c.a 2
60.h even 2 1 576.4.a.h 1
80.k odd 4 2 256.4.b.c 2
80.q even 4 2 256.4.b.e 2
120.i odd 2 1 288.4.a.h 1
120.m even 2 1 288.4.a.i 1
280.c odd 2 1 1568.4.a.o 1
280.n even 2 1 1568.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.4.a.a 1 40.f even 2 1
32.4.a.c yes 1 40.e odd 2 1
64.4.a.a 1 20.d odd 2 1
64.4.a.e 1 5.b even 2 1
256.4.b.c 2 80.k odd 4 2
256.4.b.e 2 80.q even 4 2
288.4.a.h 1 120.i odd 2 1
288.4.a.i 1 120.m even 2 1
576.4.a.g 1 15.d odd 2 1
576.4.a.h 1 60.h even 2 1
800.4.a.a 1 8.d odd 2 1
800.4.a.k 1 8.b even 2 1
800.4.c.a 2 40.k even 4 2
800.4.c.b 2 40.i odd 4 2
1568.4.a.c 1 280.n even 2 1
1568.4.a.o 1 280.c odd 2 1
1600.4.a.e 1 1.a even 1 1 trivial
1600.4.a.bw 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1600))$$:

 $$T_{3} + 8$$ T3 + 8 $$T_{7} - 16$$ T7 - 16 $$T_{11} + 40$$ T11 + 40 $$T_{13} + 50$$ T13 + 50

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 8$$
$5$ $$T$$
$7$ $$T - 16$$
$11$ $$T + 40$$
$13$ $$T + 50$$
$17$ $$T - 30$$
$19$ $$T - 40$$
$23$ $$T - 48$$
$29$ $$T - 34$$
$31$ $$T + 320$$
$37$ $$T - 310$$
$41$ $$T - 410$$
$43$ $$T + 152$$
$47$ $$T + 416$$
$53$ $$T + 410$$
$59$ $$T + 200$$
$61$ $$T + 30$$
$67$ $$T + 776$$
$71$ $$T + 400$$
$73$ $$T - 630$$
$79$ $$T - 1120$$
$83$ $$T + 552$$
$89$ $$T + 326$$
$97$ $$T - 110$$