# Properties

 Label 1800.4.f.p Level $1800$ Weight $4$ Character orbit 1800.f Analytic conductor $106.203$ 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] = [1800,4,Mod(649,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1800.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1800.f (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: $$106.203438010$$ 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: $$\Z[a_1, \ldots, a_{37}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) 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.

 $$f(q)$$ $$=$$ $$q + 6 i q^{7}+O(q^{10})$$ q + 6*i * q^7 $$q + 6 i q^{7} + 19 q^{11} + 12 i q^{13} - 75 i q^{17} + 91 q^{19} - 174 i q^{23} - 272 q^{29} - 230 q^{31} + 182 i q^{37} - 117 q^{41} + 372 i q^{43} - 52 i q^{47} + 307 q^{49} + 402 i q^{53} + 312 q^{59} + 170 q^{61} - 763 i q^{67} + 52 q^{71} - 981 i q^{73} + 114 i q^{77} - 1054 q^{79} - 351 i q^{83} + 799 q^{89} - 72 q^{91} - 962 i q^{97} +O(q^{100})$$ q + 6*i * q^7 + 19 * q^11 + 12*i * q^13 - 75*i * q^17 + 91 * q^19 - 174*i * q^23 - 272 * q^29 - 230 * q^31 + 182*i * q^37 - 117 * q^41 + 372*i * q^43 - 52*i * q^47 + 307 * q^49 + 402*i * q^53 + 312 * q^59 + 170 * q^61 - 763*i * q^67 + 52 * q^71 - 981*i * q^73 + 114*i * q^77 - 1054 * q^79 - 351*i * q^83 + 799 * q^89 - 72 * q^91 - 962*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 38 q^{11} + 182 q^{19} - 544 q^{29} - 460 q^{31} - 234 q^{41} + 614 q^{49} + 624 q^{59} + 340 q^{61} + 104 q^{71} - 2108 q^{79} + 1598 q^{89} - 144 q^{91}+O(q^{100})$$ 2 * q + 38 * q^11 + 182 * q^19 - 544 * q^29 - 460 * q^31 - 234 * q^41 + 614 * q^49 + 624 * q^59 + 340 * q^61 + 104 * q^71 - 2108 * q^79 + 1598 * q^89 - 144 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1001$$ $$1351$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
0 0 0 0 0 6.00000i 0 0 0
649.2 0 0 0 0 0 6.00000i 0 0 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 1800.4.f.p 2
3.b odd 2 1 200.4.c.g 2
5.b even 2 1 inner 1800.4.f.p 2
5.c odd 4 1 1800.4.a.l 1
5.c odd 4 1 1800.4.a.w 1
12.b even 2 1 400.4.c.m 2
15.d odd 2 1 200.4.c.g 2
15.e even 4 1 200.4.a.e 1
15.e even 4 1 200.4.a.f yes 1
60.h even 2 1 400.4.c.m 2
60.l odd 4 1 400.4.a.j 1
60.l odd 4 1 400.4.a.k 1
120.q odd 4 1 1600.4.a.v 1
120.q odd 4 1 1600.4.a.be 1
120.w even 4 1 1600.4.a.w 1
120.w even 4 1 1600.4.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.4.a.e 1 15.e even 4 1
200.4.a.f yes 1 15.e even 4 1
200.4.c.g 2 3.b odd 2 1
200.4.c.g 2 15.d odd 2 1
400.4.a.j 1 60.l odd 4 1
400.4.a.k 1 60.l odd 4 1
400.4.c.m 2 12.b even 2 1
400.4.c.m 2 60.h even 2 1
1600.4.a.v 1 120.q odd 4 1
1600.4.a.w 1 120.w even 4 1
1600.4.a.be 1 120.q odd 4 1
1600.4.a.bf 1 120.w even 4 1
1800.4.a.l 1 5.c odd 4 1
1800.4.a.w 1 5.c odd 4 1
1800.4.f.p 2 1.a even 1 1 trivial
1800.4.f.p 2 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7}^{2} + 36$$ T7^2 + 36 $$T_{11} - 19$$ T11 - 19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 36$$
$11$ $$(T - 19)^{2}$$
$13$ $$T^{2} + 144$$
$17$ $$T^{2} + 5625$$
$19$ $$(T - 91)^{2}$$
$23$ $$T^{2} + 30276$$
$29$ $$(T + 272)^{2}$$
$31$ $$(T + 230)^{2}$$
$37$ $$T^{2} + 33124$$
$41$ $$(T + 117)^{2}$$
$43$ $$T^{2} + 138384$$
$47$ $$T^{2} + 2704$$
$53$ $$T^{2} + 161604$$
$59$ $$(T - 312)^{2}$$
$61$ $$(T - 170)^{2}$$
$67$ $$T^{2} + 582169$$
$71$ $$(T - 52)^{2}$$
$73$ $$T^{2} + 962361$$
$79$ $$(T + 1054)^{2}$$
$83$ $$T^{2} + 123201$$
$89$ $$(T - 799)^{2}$$
$97$ $$T^{2} + 925444$$