# Properties

 Label 1600.4.a.i 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 25) 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)

 $$f(q)$$ $$=$$ $$q - 7 q^{3} + 6 q^{7} + 22 q^{9}+O(q^{10})$$ q - 7 * q^3 + 6 * q^7 + 22 * q^9 $$q - 7 q^{3} + 6 q^{7} + 22 q^{9} + 43 q^{11} + 28 q^{13} + 91 q^{17} + 35 q^{19} - 42 q^{21} + 162 q^{23} + 35 q^{27} - 160 q^{29} + 42 q^{31} - 301 q^{33} + 314 q^{37} - 196 q^{39} - 203 q^{41} - 92 q^{43} + 196 q^{47} - 307 q^{49} - 637 q^{51} - 82 q^{53} - 245 q^{57} + 280 q^{59} + 518 q^{61} + 132 q^{63} - 141 q^{67} - 1134 q^{69} + 412 q^{71} - 763 q^{73} + 258 q^{77} + 510 q^{79} - 839 q^{81} - 777 q^{83} + 1120 q^{87} - 945 q^{89} + 168 q^{91} - 294 q^{93} + 1246 q^{97} + 946 q^{99}+O(q^{100})$$ q - 7 * q^3 + 6 * q^7 + 22 * q^9 + 43 * q^11 + 28 * q^13 + 91 * q^17 + 35 * q^19 - 42 * q^21 + 162 * q^23 + 35 * q^27 - 160 * q^29 + 42 * q^31 - 301 * q^33 + 314 * q^37 - 196 * q^39 - 203 * q^41 - 92 * q^43 + 196 * q^47 - 307 * q^49 - 637 * q^51 - 82 * q^53 - 245 * q^57 + 280 * q^59 + 518 * q^61 + 132 * q^63 - 141 * q^67 - 1134 * q^69 + 412 * q^71 - 763 * q^73 + 258 * q^77 + 510 * q^79 - 839 * q^81 - 777 * q^83 + 1120 * q^87 - 945 * q^89 + 168 * q^91 - 294 * q^93 + 1246 * q^97 + 946 * 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 −7.00000 0 0 0 6.00000 0 22.0000 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.i 1
4.b odd 2 1 1600.4.a.bs 1
5.b even 2 1 1600.4.a.bt 1
8.b even 2 1 25.4.a.b yes 1
8.d odd 2 1 400.4.a.c 1
20.d odd 2 1 1600.4.a.h 1
24.h odd 2 1 225.4.a.c 1
40.e odd 2 1 400.4.a.s 1
40.f even 2 1 25.4.a.a 1
40.i odd 4 2 25.4.b.b 2
40.k even 4 2 400.4.c.e 2
56.h odd 2 1 1225.4.a.i 1
120.i odd 2 1 225.4.a.e 1
120.w even 4 2 225.4.b.f 2
280.c odd 2 1 1225.4.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 40.f even 2 1
25.4.a.b yes 1 8.b even 2 1
25.4.b.b 2 40.i odd 4 2
225.4.a.c 1 24.h odd 2 1
225.4.a.e 1 120.i odd 2 1
225.4.b.f 2 120.w even 4 2
400.4.a.c 1 8.d odd 2 1
400.4.a.s 1 40.e odd 2 1
400.4.c.e 2 40.k even 4 2
1225.4.a.h 1 280.c odd 2 1
1225.4.a.i 1 56.h odd 2 1
1600.4.a.h 1 20.d odd 2 1
1600.4.a.i 1 1.a even 1 1 trivial
1600.4.a.bs 1 4.b odd 2 1
1600.4.a.bt 1 5.b even 2 1

## Hecke kernels

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

 $$T_{3} + 7$$ T3 + 7 $$T_{7} - 6$$ T7 - 6 $$T_{11} - 43$$ T11 - 43 $$T_{13} - 28$$ T13 - 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 7$$
$5$ $$T$$
$7$ $$T - 6$$
$11$ $$T - 43$$
$13$ $$T - 28$$
$17$ $$T - 91$$
$19$ $$T - 35$$
$23$ $$T - 162$$
$29$ $$T + 160$$
$31$ $$T - 42$$
$37$ $$T - 314$$
$41$ $$T + 203$$
$43$ $$T + 92$$
$47$ $$T - 196$$
$53$ $$T + 82$$
$59$ $$T - 280$$
$61$ $$T - 518$$
$67$ $$T + 141$$
$71$ $$T - 412$$
$73$ $$T + 763$$
$79$ $$T - 510$$
$83$ $$T + 777$$
$89$ $$T + 945$$
$97$ $$T - 1246$$