# Properties

 Label 1600.2.c.j Level $1600$ Weight $2$ Character orbit 1600.c Analytic conductor $12.776$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(449,1600)]

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, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1600.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.c (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: $$12.7760643234$$ 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: $$1$$ Twist minimal: no (minimal twist has level 800) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} - 2 i q^{7} + 2 q^{9} +O(q^{10})$$ q + i * q^3 - 2*i * q^7 + 2 * q^9 $$q + i q^{3} - 2 i q^{7} + 2 q^{9} + 5 q^{11} - 5 i q^{17} - 5 q^{19} + 2 q^{21} - 6 i q^{23} + 5 i q^{27} + 4 q^{29} - 10 q^{31} + 5 i q^{33} - 10 i q^{37} + 5 q^{41} + 4 i q^{43} - 8 i q^{47} + 3 q^{49} + 5 q^{51} + 10 i q^{53} - 5 i q^{57} + 10 q^{61} - 4 i q^{63} - 3 i q^{67} + 6 q^{69} - 5 i q^{73} - 10 i q^{77} + 10 q^{79} + q^{81} - i q^{83} + 4 i q^{87} + 9 q^{89} - 10 i q^{93} - 10 i q^{97} + 10 q^{99} +O(q^{100})$$ q + i * q^3 - 2*i * q^7 + 2 * q^9 + 5 * q^11 - 5*i * q^17 - 5 * q^19 + 2 * q^21 - 6*i * q^23 + 5*i * q^27 + 4 * q^29 - 10 * q^31 + 5*i * q^33 - 10*i * q^37 + 5 * q^41 + 4*i * q^43 - 8*i * q^47 + 3 * q^49 + 5 * q^51 + 10*i * q^53 - 5*i * q^57 + 10 * q^61 - 4*i * q^63 - 3*i * q^67 + 6 * q^69 - 5*i * q^73 - 10*i * q^77 + 10 * q^79 + q^81 - i * q^83 + 4*i * q^87 + 9 * q^89 - 10*i * q^93 - 10*i * q^97 + 10 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} + 10 q^{11} - 10 q^{19} + 4 q^{21} + 8 q^{29} - 20 q^{31} + 10 q^{41} + 6 q^{49} + 10 q^{51} + 20 q^{61} + 12 q^{69} + 20 q^{79} + 2 q^{81} + 18 q^{89} + 20 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 + 10 * q^11 - 10 * q^19 + 4 * q^21 + 8 * q^29 - 20 * q^31 + 10 * q^41 + 6 * q^49 + 10 * q^51 + 20 * q^61 + 12 * q^69 + 20 * q^79 + 2 * q^81 + 18 * q^89 + 20 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 2.00000i 0 2.00000 0
449.2 0 1.00000i 0 0 0 2.00000i 0 2.00000 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
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 1600.2.c.j 2
4.b odd 2 1 1600.2.c.g 2
5.b even 2 1 inner 1600.2.c.j 2
5.c odd 4 1 1600.2.a.h 1
5.c odd 4 1 1600.2.a.s 1
8.b even 2 1 800.2.c.c 2
8.d odd 2 1 800.2.c.d 2
20.d odd 2 1 1600.2.c.g 2
20.e even 4 1 1600.2.a.g 1
20.e even 4 1 1600.2.a.r 1
24.f even 2 1 7200.2.f.a 2
24.h odd 2 1 7200.2.f.bc 2
40.e odd 2 1 800.2.c.d 2
40.f even 2 1 800.2.c.c 2
40.i odd 4 1 800.2.a.b 1
40.i odd 4 1 800.2.a.g yes 1
40.k even 4 1 800.2.a.c yes 1
40.k even 4 1 800.2.a.h yes 1
120.i odd 2 1 7200.2.f.bc 2
120.m even 2 1 7200.2.f.a 2
120.q odd 4 1 7200.2.a.k 1
120.q odd 4 1 7200.2.a.bm 1
120.w even 4 1 7200.2.a.o 1
120.w even 4 1 7200.2.a.bq 1

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

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11} - 5$$ T11 - 5 $$T_{19} + 5$$ T19 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 5)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 25$$
$19$ $$(T + 5)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 4)^{2}$$
$31$ $$(T + 10)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T - 5)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 100$$
$59$ $$T^{2}$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2} + 9$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 25$$
$79$ $$(T - 10)^{2}$$
$83$ $$T^{2} + 1$$
$89$ $$(T - 9)^{2}$$
$97$ $$T^{2} + 100$$