# Properties

 Label 1600.2.a.h Level $1600$ Weight $2$ Character orbit 1600.a Self dual yes Analytic conductor $12.776$ 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] = [1600,2,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, 2, names="a")

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

chi := DirichletCharacter("1600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$12.7760643234$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - 2 q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 - 2 * q^7 - 2 * q^9 $$q - q^{3} - 2 q^{7} - 2 q^{9} + 5 q^{11} - 5 q^{17} + 5 q^{19} + 2 q^{21} + 6 q^{23} + 5 q^{27} - 4 q^{29} - 10 q^{31} - 5 q^{33} - 10 q^{37} + 5 q^{41} - 4 q^{43} - 8 q^{47} - 3 q^{49} + 5 q^{51} - 10 q^{53} - 5 q^{57} + 10 q^{61} + 4 q^{63} - 3 q^{67} - 6 q^{69} + 5 q^{73} - 10 q^{77} - 10 q^{79} + q^{81} + q^{83} + 4 q^{87} - 9 q^{89} + 10 q^{93} - 10 q^{97} - 10 q^{99}+O(q^{100})$$ q - q^3 - 2 * q^7 - 2 * q^9 + 5 * q^11 - 5 * q^17 + 5 * q^19 + 2 * q^21 + 6 * q^23 + 5 * q^27 - 4 * q^29 - 10 * q^31 - 5 * q^33 - 10 * q^37 + 5 * q^41 - 4 * q^43 - 8 * q^47 - 3 * q^49 + 5 * q^51 - 10 * q^53 - 5 * q^57 + 10 * q^61 + 4 * q^63 - 3 * q^67 - 6 * q^69 + 5 * q^73 - 10 * q^77 - 10 * q^79 + q^81 + q^83 + 4 * q^87 - 9 * q^89 + 10 * q^93 - 10 * q^97 - 10 * 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 −1.00000 0 0 0 −2.00000 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
$$2$$ $$1$$
$$5$$ $$1$$

## 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 1600.2.a.h 1
4.b odd 2 1 1600.2.a.r 1
5.b even 2 1 1600.2.a.s 1
5.c odd 4 2 1600.2.c.j 2
8.b even 2 1 800.2.a.g yes 1
8.d odd 2 1 800.2.a.c yes 1
20.d odd 2 1 1600.2.a.g 1
20.e even 4 2 1600.2.c.g 2
24.f even 2 1 7200.2.a.bm 1
24.h odd 2 1 7200.2.a.o 1
40.e odd 2 1 800.2.a.h yes 1
40.f even 2 1 800.2.a.b 1
40.i odd 4 2 800.2.c.c 2
40.k even 4 2 800.2.c.d 2
120.i odd 2 1 7200.2.a.bq 1
120.m even 2 1 7200.2.a.k 1
120.q odd 4 2 7200.2.f.a 2
120.w even 4 2 7200.2.f.bc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.a.b 1 40.f even 2 1
800.2.a.c yes 1 8.d odd 2 1
800.2.a.g yes 1 8.b even 2 1
800.2.a.h yes 1 40.e odd 2 1
800.2.c.c 2 40.i odd 4 2
800.2.c.d 2 40.k even 4 2
1600.2.a.g 1 20.d odd 2 1
1600.2.a.h 1 1.a even 1 1 trivial
1600.2.a.r 1 4.b odd 2 1
1600.2.a.s 1 5.b even 2 1
1600.2.c.g 2 20.e even 4 2
1600.2.c.j 2 5.c odd 4 2
7200.2.a.k 1 120.m even 2 1
7200.2.a.o 1 24.h odd 2 1
7200.2.a.bm 1 24.f even 2 1
7200.2.a.bq 1 120.i odd 2 1
7200.2.f.a 2 120.q odd 4 2
7200.2.f.bc 2 120.w even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1600))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{7} + 2$$ T7 + 2 $$T_{11} - 5$$ T11 - 5 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T - 5$$
$13$ $$T$$
$17$ $$T + 5$$
$19$ $$T - 5$$
$23$ $$T - 6$$
$29$ $$T + 4$$
$31$ $$T + 10$$
$37$ $$T + 10$$
$41$ $$T - 5$$
$43$ $$T + 4$$
$47$ $$T + 8$$
$53$ $$T + 10$$
$59$ $$T$$
$61$ $$T - 10$$
$67$ $$T + 3$$
$71$ $$T$$
$73$ $$T - 5$$
$79$ $$T + 10$$
$83$ $$T - 1$$
$89$ $$T + 9$$
$97$ $$T + 10$$