# Properties

 Label 1600.4.a.bx Level $1600$ Weight $4$ Character orbit 1600.a Self dual yes Analytic conductor $94.403$ 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] = [1600,4,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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) 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)

 $$f(q)$$ $$=$$ $$q + 8 q^{3} - 4 q^{7} + 37 q^{9}+O(q^{10})$$ q + 8 * q^3 - 4 * q^7 + 37 * q^9 $$q + 8 q^{3} - 4 q^{7} + 37 q^{9} + 12 q^{11} - 58 q^{13} - 66 q^{17} - 100 q^{19} - 32 q^{21} + 132 q^{23} + 80 q^{27} + 90 q^{29} - 152 q^{31} + 96 q^{33} - 34 q^{37} - 464 q^{39} - 438 q^{41} - 32 q^{43} - 204 q^{47} - 327 q^{49} - 528 q^{51} + 222 q^{53} - 800 q^{57} + 420 q^{59} - 902 q^{61} - 148 q^{63} + 1024 q^{67} + 1056 q^{69} - 432 q^{71} - 362 q^{73} - 48 q^{77} + 160 q^{79} - 359 q^{81} - 72 q^{83} + 720 q^{87} + 810 q^{89} + 232 q^{91} - 1216 q^{93} - 1106 q^{97} + 444 q^{99}+O(q^{100})$$ q + 8 * q^3 - 4 * q^7 + 37 * q^9 + 12 * q^11 - 58 * q^13 - 66 * q^17 - 100 * q^19 - 32 * q^21 + 132 * q^23 + 80 * q^27 + 90 * q^29 - 152 * q^31 + 96 * q^33 - 34 * q^37 - 464 * q^39 - 438 * q^41 - 32 * q^43 - 204 * q^47 - 327 * q^49 - 528 * q^51 + 222 * q^53 - 800 * q^57 + 420 * q^59 - 902 * q^61 - 148 * q^63 + 1024 * q^67 + 1056 * q^69 - 432 * q^71 - 362 * q^73 - 48 * q^77 + 160 * q^79 - 359 * q^81 - 72 * q^83 + 720 * q^87 + 810 * q^89 + 232 * q^91 - 1216 * q^93 - 1106 * q^97 + 444 * 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 −4.00000 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.bx 1
4.b odd 2 1 1600.4.a.d 1
5.b even 2 1 320.4.a.b 1
8.b even 2 1 400.4.a.b 1
8.d odd 2 1 50.4.a.c 1
20.d odd 2 1 320.4.a.m 1
24.f even 2 1 450.4.a.q 1
40.e odd 2 1 10.4.a.a 1
40.f even 2 1 80.4.a.f 1
40.i odd 4 2 400.4.c.c 2
40.k even 4 2 50.4.b.a 2
56.e even 2 1 2450.4.a.b 1
80.k odd 4 2 1280.4.d.j 2
80.q even 4 2 1280.4.d.g 2
120.i odd 2 1 720.4.a.j 1
120.m even 2 1 90.4.a.a 1
120.q odd 4 2 450.4.c.d 2
280.n even 2 1 490.4.a.o 1
280.ba even 6 2 490.4.e.a 2
280.bi odd 6 2 490.4.e.i 2
360.z odd 6 2 810.4.e.c 2
360.bd even 6 2 810.4.e.w 2
440.c even 2 1 1210.4.a.b 1
520.b odd 2 1 1690.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 40.e odd 2 1
50.4.a.c 1 8.d odd 2 1
50.4.b.a 2 40.k even 4 2
80.4.a.f 1 40.f even 2 1
90.4.a.a 1 120.m even 2 1
320.4.a.b 1 5.b even 2 1
320.4.a.m 1 20.d odd 2 1
400.4.a.b 1 8.b even 2 1
400.4.c.c 2 40.i odd 4 2
450.4.a.q 1 24.f even 2 1
450.4.c.d 2 120.q odd 4 2
490.4.a.o 1 280.n even 2 1
490.4.e.a 2 280.ba even 6 2
490.4.e.i 2 280.bi odd 6 2
720.4.a.j 1 120.i odd 2 1
810.4.e.c 2 360.z odd 6 2
810.4.e.w 2 360.bd even 6 2
1210.4.a.b 1 440.c even 2 1
1280.4.d.g 2 80.q even 4 2
1280.4.d.j 2 80.k odd 4 2
1600.4.a.d 1 4.b odd 2 1
1600.4.a.bx 1 1.a even 1 1 trivial
1690.4.a.a 1 520.b odd 2 1
2450.4.a.b 1 56.e even 2 1

## Hecke kernels

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

 $$T_{3} - 8$$ T3 - 8 $$T_{7} + 4$$ T7 + 4 $$T_{11} - 12$$ T11 - 12 $$T_{13} + 58$$ T13 + 58

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 8$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T - 12$$
$13$ $$T + 58$$
$17$ $$T + 66$$
$19$ $$T + 100$$
$23$ $$T - 132$$
$29$ $$T - 90$$
$31$ $$T + 152$$
$37$ $$T + 34$$
$41$ $$T + 438$$
$43$ $$T + 32$$
$47$ $$T + 204$$
$53$ $$T - 222$$
$59$ $$T - 420$$
$61$ $$T + 902$$
$67$ $$T - 1024$$
$71$ $$T + 432$$
$73$ $$T + 362$$
$79$ $$T - 160$$
$83$ $$T + 72$$
$89$ $$T - 810$$
$97$ $$T + 1106$$