# Properties

 Label 1600.4.a.bd 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 100) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 26 q^{7} - 26 q^{9}+O(q^{10})$$ q + q^3 - 26 * q^7 - 26 * q^9 $$q + q^{3} - 26 q^{7} - 26 q^{9} + 45 q^{11} - 44 q^{13} + 117 q^{17} - 91 q^{19} - 26 q^{21} + 18 q^{23} - 53 q^{27} - 144 q^{29} - 26 q^{31} + 45 q^{33} + 214 q^{37} - 44 q^{39} - 459 q^{41} - 460 q^{43} + 468 q^{47} + 333 q^{49} + 117 q^{51} - 558 q^{53} - 91 q^{57} - 72 q^{59} + 118 q^{61} + 676 q^{63} + 251 q^{67} + 18 q^{69} - 108 q^{71} + 299 q^{73} - 1170 q^{77} + 898 q^{79} + 649 q^{81} + 927 q^{83} - 144 q^{87} + 351 q^{89} + 1144 q^{91} - 26 q^{93} + 386 q^{97} - 1170 q^{99}+O(q^{100})$$ q + q^3 - 26 * q^7 - 26 * q^9 + 45 * q^11 - 44 * q^13 + 117 * q^17 - 91 * q^19 - 26 * q^21 + 18 * q^23 - 53 * q^27 - 144 * q^29 - 26 * q^31 + 45 * q^33 + 214 * q^37 - 44 * q^39 - 459 * q^41 - 460 * q^43 + 468 * q^47 + 333 * q^49 + 117 * q^51 - 558 * q^53 - 91 * q^57 - 72 * q^59 + 118 * q^61 + 676 * q^63 + 251 * q^67 + 18 * q^69 - 108 * q^71 + 299 * q^73 - 1170 * q^77 + 898 * q^79 + 649 * q^81 + 927 * q^83 - 144 * q^87 + 351 * q^89 + 1144 * q^91 - 26 * q^93 + 386 * q^97 - 1170 * 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 −26.0000 0 −26.0000 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.bd 1
4.b odd 2 1 1600.4.a.x 1
5.b even 2 1 1600.4.a.y 1
8.b even 2 1 400.4.a.i 1
8.d odd 2 1 100.4.a.c yes 1
20.d odd 2 1 1600.4.a.bc 1
24.f even 2 1 900.4.a.p 1
40.e odd 2 1 100.4.a.b 1
40.f even 2 1 400.4.a.l 1
40.i odd 4 2 400.4.c.l 2
40.k even 4 2 100.4.c.b 2
120.m even 2 1 900.4.a.c 1
120.q odd 4 2 900.4.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.a.b 1 40.e odd 2 1
100.4.a.c yes 1 8.d odd 2 1
100.4.c.b 2 40.k even 4 2
400.4.a.i 1 8.b even 2 1
400.4.a.l 1 40.f even 2 1
400.4.c.l 2 40.i odd 4 2
900.4.a.c 1 120.m even 2 1
900.4.a.p 1 24.f even 2 1
900.4.d.a 2 120.q odd 4 2
1600.4.a.x 1 4.b odd 2 1
1600.4.a.y 1 5.b even 2 1
1600.4.a.bc 1 20.d odd 2 1
1600.4.a.bd 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1600))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} + 26$$ T7 + 26 $$T_{11} - 45$$ T11 - 45 $$T_{13} + 44$$ T13 + 44

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 26$$
$11$ $$T - 45$$
$13$ $$T + 44$$
$17$ $$T - 117$$
$19$ $$T + 91$$
$23$ $$T - 18$$
$29$ $$T + 144$$
$31$ $$T + 26$$
$37$ $$T - 214$$
$41$ $$T + 459$$
$43$ $$T + 460$$
$47$ $$T - 468$$
$53$ $$T + 558$$
$59$ $$T + 72$$
$61$ $$T - 118$$
$67$ $$T - 251$$
$71$ $$T + 108$$
$73$ $$T - 299$$
$79$ $$T - 898$$
$83$ $$T - 927$$
$89$ $$T - 351$$
$97$ $$T - 386$$