# Properties

 Label 1600.3.b.f Level $1600$ Weight $3$ Character orbit 1600.b Analytic conductor $43.597$ Analytic rank $0$ Dimension $2$ CM discriminant -20 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,3,Mod(1151,1600)]

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([1, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1600.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1600.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$43.5968422976$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 4i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - \beta q^{7} - 7 q^{9} +O(q^{10})$$ q + b * q^3 - b * q^7 - 7 * q^9 $$q + \beta q^{3} - \beta q^{7} - 7 q^{9} + 16 q^{21} - 11 \beta q^{23} + 2 \beta q^{27} - 22 q^{29} + 62 q^{41} - 19 \beta q^{43} - \beta q^{47} + 33 q^{49} + 58 q^{61} + 7 \beta q^{63} - 29 \beta q^{67} + 176 q^{69} - 95 q^{81} - 19 \beta q^{83} - 22 \beta q^{87} + 142 q^{89} +O(q^{100})$$ q + b * q^3 - b * q^7 - 7 * q^9 + 16 * q^21 - 11*b * q^23 + 2*b * q^27 - 22 * q^29 + 62 * q^41 - 19*b * q^43 - b * q^47 + 33 * q^49 + 58 * q^61 + 7*b * q^63 - 29*b * q^67 + 176 * q^69 - 95 * q^81 - 19*b * q^83 - 22*b * q^87 + 142 * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 14 q^{9}+O(q^{10})$$ 2 * q - 14 * q^9 $$2 q - 14 q^{9} + 32 q^{21} - 44 q^{29} + 124 q^{41} + 66 q^{49} + 116 q^{61} + 352 q^{69} - 190 q^{81} + 284 q^{89}+O(q^{100})$$ 2 * q - 14 * q^9 + 32 * q^21 - 44 * q^29 + 124 * q^41 + 66 * q^49 + 116 * q^61 + 352 * q^69 - 190 * q^81 + 284 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 − 1.00000i 1.00000i
0 4.00000i 0 0 0 4.00000i 0 −7.00000 0
1151.2 0 4.00000i 0 0 0 4.00000i 0 −7.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.3.b.f 2
4.b odd 2 1 inner 1600.3.b.f 2
5.b even 2 1 inner 1600.3.b.f 2
5.c odd 4 1 320.3.h.a 1
5.c odd 4 1 320.3.h.b 1
8.b even 2 1 100.3.b.c 2
8.d odd 2 1 100.3.b.c 2
20.d odd 2 1 CM 1600.3.b.f 2
20.e even 4 1 320.3.h.a 1
20.e even 4 1 320.3.h.b 1
24.f even 2 1 900.3.c.h 2
24.h odd 2 1 900.3.c.h 2
40.e odd 2 1 100.3.b.c 2
40.f even 2 1 100.3.b.c 2
40.i odd 4 1 20.3.d.a 1
40.i odd 4 1 20.3.d.b yes 1
40.k even 4 1 20.3.d.a 1
40.k even 4 1 20.3.d.b yes 1
80.i odd 4 1 1280.3.e.b 2
80.i odd 4 1 1280.3.e.c 2
80.j even 4 1 1280.3.e.b 2
80.j even 4 1 1280.3.e.c 2
80.s even 4 1 1280.3.e.b 2
80.s even 4 1 1280.3.e.c 2
80.t odd 4 1 1280.3.e.b 2
80.t odd 4 1 1280.3.e.c 2
120.i odd 2 1 900.3.c.h 2
120.m even 2 1 900.3.c.h 2
120.q odd 4 1 180.3.f.a 1
120.q odd 4 1 180.3.f.b 1
120.w even 4 1 180.3.f.a 1
120.w even 4 1 180.3.f.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.a 1 40.i odd 4 1
20.3.d.a 1 40.k even 4 1
20.3.d.b yes 1 40.i odd 4 1
20.3.d.b yes 1 40.k even 4 1
100.3.b.c 2 8.b even 2 1
100.3.b.c 2 8.d odd 2 1
100.3.b.c 2 40.e odd 2 1
100.3.b.c 2 40.f even 2 1
180.3.f.a 1 120.q odd 4 1
180.3.f.a 1 120.w even 4 1
180.3.f.b 1 120.q odd 4 1
180.3.f.b 1 120.w even 4 1
320.3.h.a 1 5.c odd 4 1
320.3.h.a 1 20.e even 4 1
320.3.h.b 1 5.c odd 4 1
320.3.h.b 1 20.e even 4 1
900.3.c.h 2 24.f even 2 1
900.3.c.h 2 24.h odd 2 1
900.3.c.h 2 120.i odd 2 1
900.3.c.h 2 120.m even 2 1
1280.3.e.b 2 80.i odd 4 1
1280.3.e.b 2 80.j even 4 1
1280.3.e.b 2 80.s even 4 1
1280.3.e.b 2 80.t odd 4 1
1280.3.e.c 2 80.i odd 4 1
1280.3.e.c 2 80.j even 4 1
1280.3.e.c 2 80.s even 4 1
1280.3.e.c 2 80.t odd 4 1
1600.3.b.f 2 1.a even 1 1 trivial
1600.3.b.f 2 4.b odd 2 1 inner
1600.3.b.f 2 5.b even 2 1 inner
1600.3.b.f 2 20.d odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1600, [\chi])$$:

 $$T_{3}^{2} + 16$$ T3^2 + 16 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 16$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 1936$$
$29$ $$(T + 22)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T - 62)^{2}$$
$43$ $$T^{2} + 5776$$
$47$ $$T^{2} + 16$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 58)^{2}$$
$67$ $$T^{2} + 13456$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 5776$$
$89$ $$(T - 142)^{2}$$
$97$ $$T^{2}$$