# Properties

 Label 1600.2.c.f 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: $$2$$ Twist minimal: no (minimal twist has level 160) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} + i q^{7} - q^{9}+O(q^{10})$$ q + i * q^3 + i * q^7 - q^9 $$q + i q^{3} + i q^{7} - q^{9} + 4 q^{11} + 3 i q^{13} - i q^{17} + 8 q^{19} - 4 q^{21} - 3 i q^{23} + 2 i q^{27} - 2 q^{29} + 4 q^{31} + 4 i q^{33} + i q^{37} - 12 q^{39} - 10 q^{41} + i q^{43} + i q^{47} + 3 q^{49} + 4 q^{51} - i q^{53} + 8 i q^{57} - 2 q^{61} - i q^{63} - 3 i q^{67} + 12 q^{69} - 12 q^{71} + 5 i q^{73} + 4 i q^{77} + 8 q^{79} - 11 q^{81} + 5 i q^{83} - 2 i q^{87} + 6 q^{89} - 12 q^{91} + 4 i q^{93} - 5 i q^{97} - 4 q^{99} +O(q^{100})$$ q + i * q^3 + i * q^7 - q^9 + 4 * q^11 + 3*i * q^13 - i * q^17 + 8 * q^19 - 4 * q^21 - 3*i * q^23 + 2*i * q^27 - 2 * q^29 + 4 * q^31 + 4*i * q^33 + i * q^37 - 12 * q^39 - 10 * q^41 + i * q^43 + i * q^47 + 3 * q^49 + 4 * q^51 - i * q^53 + 8*i * q^57 - 2 * q^61 - i * q^63 - 3*i * q^67 + 12 * q^69 - 12 * q^71 + 5*i * q^73 + 4*i * q^77 + 8 * q^79 - 11 * q^81 + 5*i * q^83 - 2*i * q^87 + 6 * q^89 - 12 * q^91 + 4*i * q^93 - 5*i * q^97 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 8 q^{11} + 16 q^{19} - 8 q^{21} - 4 q^{29} + 8 q^{31} - 24 q^{39} - 20 q^{41} + 6 q^{49} + 8 q^{51} - 4 q^{61} + 24 q^{69} - 24 q^{71} + 16 q^{79} - 22 q^{81} + 12 q^{89} - 24 q^{91} - 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 + 8 * q^11 + 16 * q^19 - 8 * q^21 - 4 * q^29 + 8 * q^31 - 24 * q^39 - 20 * q^41 + 6 * q^49 + 8 * q^51 - 4 * q^61 + 24 * q^69 - 24 * q^71 + 16 * q^79 - 22 * q^81 + 12 * q^89 - 24 * q^91 - 8 * 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 2.00000i 0 0 0 2.00000i 0 −1.00000 0
449.2 0 2.00000i 0 0 0 2.00000i 0 −1.00000 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.f 2
4.b odd 2 1 1600.2.c.c 2
5.b even 2 1 inner 1600.2.c.f 2
5.c odd 4 1 320.2.a.e 1
5.c odd 4 1 1600.2.a.e 1
8.b even 2 1 800.2.c.a 2
8.d odd 2 1 800.2.c.b 2
15.e even 4 1 2880.2.a.d 1
20.d odd 2 1 1600.2.c.c 2
20.e even 4 1 320.2.a.b 1
20.e even 4 1 1600.2.a.t 1
24.f even 2 1 7200.2.f.g 2
24.h odd 2 1 7200.2.f.w 2
40.e odd 2 1 800.2.c.b 2
40.f even 2 1 800.2.c.a 2
40.i odd 4 1 160.2.a.a 1
40.i odd 4 1 800.2.a.i 1
40.k even 4 1 160.2.a.b yes 1
40.k even 4 1 800.2.a.a 1
60.l odd 4 1 2880.2.a.o 1
80.i odd 4 1 1280.2.d.h 2
80.j even 4 1 1280.2.d.b 2
80.s even 4 1 1280.2.d.b 2
80.t odd 4 1 1280.2.d.h 2
120.i odd 2 1 7200.2.f.w 2
120.m even 2 1 7200.2.f.g 2
120.q odd 4 1 1440.2.a.l 1
120.q odd 4 1 7200.2.a.l 1
120.w even 4 1 1440.2.a.i 1
120.w even 4 1 7200.2.a.bp 1
280.s even 4 1 7840.2.a.w 1
280.y odd 4 1 7840.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.a.a 1 40.i odd 4 1
160.2.a.b yes 1 40.k even 4 1
320.2.a.b 1 20.e even 4 1
320.2.a.e 1 5.c odd 4 1
800.2.a.a 1 40.k even 4 1
800.2.a.i 1 40.i odd 4 1
800.2.c.a 2 8.b even 2 1
800.2.c.a 2 40.f even 2 1
800.2.c.b 2 8.d odd 2 1
800.2.c.b 2 40.e odd 2 1
1280.2.d.b 2 80.j even 4 1
1280.2.d.b 2 80.s even 4 1
1280.2.d.h 2 80.i odd 4 1
1280.2.d.h 2 80.t odd 4 1
1440.2.a.i 1 120.w even 4 1
1440.2.a.l 1 120.q odd 4 1
1600.2.a.e 1 5.c odd 4 1
1600.2.a.t 1 20.e even 4 1
1600.2.c.c 2 4.b odd 2 1
1600.2.c.c 2 20.d odd 2 1
1600.2.c.f 2 1.a even 1 1 trivial
1600.2.c.f 2 5.b even 2 1 inner
2880.2.a.d 1 15.e even 4 1
2880.2.a.o 1 60.l odd 4 1
7200.2.a.l 1 120.q odd 4 1
7200.2.a.bp 1 120.w even 4 1
7200.2.f.g 2 24.f even 2 1
7200.2.f.g 2 120.m even 2 1
7200.2.f.w 2 24.h odd 2 1
7200.2.f.w 2 120.i odd 2 1
7840.2.a.e 1 280.y odd 4 1
7840.2.a.w 1 280.s even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11} - 4$$ T11 - 4 $$T_{19} - 8$$ T19 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 4$$
$19$ $$(T - 8)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T + 2)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2} + 4$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 4$$
$59$ $$T^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 36$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 100$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 100$$