# Properties

 Label 1600.2.q.a Level $1600$ Weight $2$ Character orbit 1600.q Analytic conductor $12.776$ Analytic rank $1$ 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(49,1600)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2]))

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

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

chi := DirichletCharacter("1600.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.q (of order $$4$$, degree $$2$$, 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: $$1$$ 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 16) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - i - 1) q^{3} - 2 q^{7} - i q^{9} +O(q^{10})$$ q + (-i - 1) * q^3 - 2 * q^7 - i * q^9 $$q + ( - i - 1) q^{3} - 2 q^{7} - i q^{9} + ( - i - 1) q^{11} + (i + 1) q^{13} - 2 i q^{17} + ( - 3 i + 3) q^{19} + (2 i + 2) q^{21} - 6 q^{23} + (4 i - 4) q^{27} + (3 i - 3) q^{29} + 8 q^{31} + 2 i q^{33} + (3 i - 3) q^{37} - 2 i q^{39} + (5 i - 5) q^{43} - 8 i q^{47} - 3 q^{49} + (2 i - 2) q^{51} + (5 i - 5) q^{53} - 6 q^{57} + ( - 3 i - 3) q^{59} + (9 i - 9) q^{61} + 2 i q^{63} + (5 i + 5) q^{67} + (6 i + 6) q^{69} + 10 i q^{71} - 4 q^{73} + (2 i + 2) q^{77} + 5 q^{81} + ( - i - 1) q^{83} + 6 q^{87} - 4 i q^{89} + ( - 2 i - 2) q^{91} + ( - 8 i - 8) q^{93} - 2 i q^{97} + (i - 1) q^{99} +O(q^{100})$$ q + (-i - 1) * q^3 - 2 * q^7 - i * q^9 + (-i - 1) * q^11 + (i + 1) * q^13 - 2*i * q^17 + (-3*i + 3) * q^19 + (2*i + 2) * q^21 - 6 * q^23 + (4*i - 4) * q^27 + (3*i - 3) * q^29 + 8 * q^31 + 2*i * q^33 + (3*i - 3) * q^37 - 2*i * q^39 + (5*i - 5) * q^43 - 8*i * q^47 - 3 * q^49 + (2*i - 2) * q^51 + (5*i - 5) * q^53 - 6 * q^57 + (-3*i - 3) * q^59 + (9*i - 9) * q^61 + 2*i * q^63 + (5*i + 5) * q^67 + (6*i + 6) * q^69 + 10*i * q^71 - 4 * q^73 + (2*i + 2) * q^77 + 5 * q^81 + (-i - 1) * q^83 + 6 * q^87 - 4*i * q^89 + (-2*i - 2) * q^91 + (-8*i - 8) * q^93 - 2*i * q^97 + (i - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{7}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^7 $$2 q - 2 q^{3} - 4 q^{7} - 2 q^{11} + 2 q^{13} + 6 q^{19} + 4 q^{21} - 12 q^{23} - 8 q^{27} - 6 q^{29} + 16 q^{31} - 6 q^{37} - 10 q^{43} - 6 q^{49} - 4 q^{51} - 10 q^{53} - 12 q^{57} - 6 q^{59} - 18 q^{61} + 10 q^{67} + 12 q^{69} - 8 q^{73} + 4 q^{77} + 10 q^{81} - 2 q^{83} + 12 q^{87} - 4 q^{91} - 16 q^{93} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^7 - 2 * q^11 + 2 * q^13 + 6 * q^19 + 4 * q^21 - 12 * q^23 - 8 * q^27 - 6 * q^29 + 16 * q^31 - 6 * q^37 - 10 * q^43 - 6 * q^49 - 4 * q^51 - 10 * q^53 - 12 * q^57 - 6 * q^59 - 18 * q^61 + 10 * q^67 + 12 * q^69 - 8 * q^73 + 4 * q^77 + 10 * q^81 - 2 * q^83 + 12 * q^87 - 4 * q^91 - 16 * q^93 - 2 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.00000i − 1.00000i
0 −1.00000 1.00000i 0 0 0 −2.00000 0 1.00000i 0
849.1 0 −1.00000 + 1.00000i 0 0 0 −2.00000 0 1.00000i 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
80.q even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.q.a 2
4.b odd 2 1 400.2.q.b 2
5.b even 2 1 1600.2.q.b 2
5.c odd 4 1 64.2.e.a 2
5.c odd 4 1 1600.2.l.a 2
15.e even 4 1 576.2.k.a 2
16.e even 4 1 1600.2.q.b 2
16.f odd 4 1 400.2.q.a 2
20.d odd 2 1 400.2.q.a 2
20.e even 4 1 16.2.e.a 2
20.e even 4 1 400.2.l.c 2
40.i odd 4 1 128.2.e.a 2
40.k even 4 1 128.2.e.b 2
60.l odd 4 1 144.2.k.a 2
80.i odd 4 1 64.2.e.a 2
80.j even 4 1 128.2.e.b 2
80.j even 4 1 400.2.l.c 2
80.k odd 4 1 400.2.q.b 2
80.q even 4 1 inner 1600.2.q.a 2
80.s even 4 1 16.2.e.a 2
80.t odd 4 1 128.2.e.a 2
80.t odd 4 1 1600.2.l.a 2
120.q odd 4 1 1152.2.k.b 2
120.w even 4 1 1152.2.k.a 2
140.j odd 4 1 784.2.m.b 2
140.w even 12 2 784.2.x.f 4
140.x odd 12 2 784.2.x.c 4
160.u even 8 2 1024.2.b.e 2
160.v odd 8 2 1024.2.a.e 2
160.ba even 8 2 1024.2.a.b 2
160.bb odd 8 2 1024.2.b.b 2
240.z odd 4 1 144.2.k.a 2
240.bb even 4 1 576.2.k.a 2
240.bd odd 4 1 1152.2.k.b 2
240.bf even 4 1 1152.2.k.a 2
480.br even 8 2 9216.2.a.s 2
480.ca odd 8 2 9216.2.a.d 2
560.u odd 4 1 784.2.m.b 2
560.cf even 12 2 784.2.x.f 4
560.da odd 12 2 784.2.x.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 20.e even 4 1
16.2.e.a 2 80.s even 4 1
64.2.e.a 2 5.c odd 4 1
64.2.e.a 2 80.i odd 4 1
128.2.e.a 2 40.i odd 4 1
128.2.e.a 2 80.t odd 4 1
128.2.e.b 2 40.k even 4 1
128.2.e.b 2 80.j even 4 1
144.2.k.a 2 60.l odd 4 1
144.2.k.a 2 240.z odd 4 1
400.2.l.c 2 20.e even 4 1
400.2.l.c 2 80.j even 4 1
400.2.q.a 2 16.f odd 4 1
400.2.q.a 2 20.d odd 2 1
400.2.q.b 2 4.b odd 2 1
400.2.q.b 2 80.k odd 4 1
576.2.k.a 2 15.e even 4 1
576.2.k.a 2 240.bb even 4 1
784.2.m.b 2 140.j odd 4 1
784.2.m.b 2 560.u odd 4 1
784.2.x.c 4 140.x odd 12 2
784.2.x.c 4 560.da odd 12 2
784.2.x.f 4 140.w even 12 2
784.2.x.f 4 560.cf even 12 2
1024.2.a.b 2 160.ba even 8 2
1024.2.a.e 2 160.v odd 8 2
1024.2.b.b 2 160.bb odd 8 2
1024.2.b.e 2 160.u even 8 2
1152.2.k.a 2 120.w even 4 1
1152.2.k.a 2 240.bf even 4 1
1152.2.k.b 2 120.q odd 4 1
1152.2.k.b 2 240.bd odd 4 1
1600.2.l.a 2 5.c odd 4 1
1600.2.l.a 2 80.t odd 4 1
1600.2.q.a 2 1.a even 1 1 trivial
1600.2.q.a 2 80.q even 4 1 inner
1600.2.q.b 2 5.b even 2 1
1600.2.q.b 2 16.e even 4 1
9216.2.a.d 2 480.ca odd 8 2
9216.2.a.s 2 480.br even 8 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 2T_{3} + 2$$ acting on $$S_{2}^{\mathrm{new}}(1600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 2$$
$5$ $$T^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} + 2T + 2$$
$13$ $$T^{2} - 2T + 2$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2} - 6T + 18$$
$23$ $$(T + 6)^{2}$$
$29$ $$T^{2} + 6T + 18$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 6T + 18$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 10T + 50$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 10T + 50$$
$59$ $$T^{2} + 6T + 18$$
$61$ $$T^{2} + 18T + 162$$
$67$ $$T^{2} - 10T + 50$$
$71$ $$T^{2} + 100$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 2T + 2$$
$89$ $$T^{2} + 16$$
$97$ $$T^{2} + 4$$