# Properties

 Label 1800.4.f.w 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: $$\Z[a_1, \ldots, a_{17}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 26 i q^{7}+O(q^{10})$$ q + 26*i * q^7 $$q + 26 i q^{7} + 59 q^{11} - 28 i q^{13} - 5 i q^{17} - 109 q^{19} - 194 i q^{23} - 32 q^{29} + 10 q^{31} - 198 i q^{37} - 117 q^{41} - 388 i q^{43} + 68 i q^{47} - 333 q^{49} - 18 i q^{53} + 392 q^{59} - 710 q^{61} - 253 i q^{67} + 612 q^{71} + 549 i q^{73} + 1534 i q^{77} - 414 q^{79} - 121 i q^{83} - 81 q^{89} + 728 q^{91} - 1502 i q^{97} +O(q^{100})$$ q + 26*i * q^7 + 59 * q^11 - 28*i * q^13 - 5*i * q^17 - 109 * q^19 - 194*i * q^23 - 32 * q^29 + 10 * q^31 - 198*i * q^37 - 117 * q^41 - 388*i * q^43 + 68*i * q^47 - 333 * q^49 - 18*i * q^53 + 392 * q^59 - 710 * q^61 - 253*i * q^67 + 612 * q^71 + 549*i * q^73 + 1534*i * q^77 - 414 * q^79 - 121*i * q^83 - 81 * q^89 + 728 * q^91 - 1502*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 118 q^{11} - 218 q^{19} - 64 q^{29} + 20 q^{31} - 234 q^{41} - 666 q^{49} + 784 q^{59} - 1420 q^{61} + 1224 q^{71} - 828 q^{79} - 162 q^{89} + 1456 q^{91}+O(q^{100})$$ 2 * q + 118 * q^11 - 218 * q^19 - 64 * q^29 + 20 * q^31 - 234 * q^41 - 666 * q^49 + 784 * q^59 - 1420 * q^61 + 1224 * q^71 - 828 * q^79 - 162 * q^89 + 1456 * 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 26.0000i 0 0 0
649.2 0 0 0 0 0 26.0000i 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.w 2
3.b odd 2 1 200.4.c.b 2
5.b even 2 1 inner 1800.4.f.w 2
5.c odd 4 1 1800.4.a.c 1
5.c odd 4 1 1800.4.a.bh 1
12.b even 2 1 400.4.c.b 2
15.d odd 2 1 200.4.c.b 2
15.e even 4 1 200.4.a.b 1
15.e even 4 1 200.4.a.j yes 1
60.h even 2 1 400.4.c.b 2
60.l odd 4 1 400.4.a.a 1
60.l odd 4 1 400.4.a.t 1
120.q odd 4 1 1600.4.a.b 1
120.q odd 4 1 1600.4.a.by 1
120.w even 4 1 1600.4.a.c 1
120.w even 4 1 1600.4.a.bz 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.4.a.b 1 15.e even 4 1
200.4.a.j yes 1 15.e even 4 1
200.4.c.b 2 3.b odd 2 1
200.4.c.b 2 15.d odd 2 1
400.4.a.a 1 60.l odd 4 1
400.4.a.t 1 60.l odd 4 1
400.4.c.b 2 12.b even 2 1
400.4.c.b 2 60.h even 2 1
1600.4.a.b 1 120.q odd 4 1
1600.4.a.c 1 120.w even 4 1
1600.4.a.by 1 120.q odd 4 1
1600.4.a.bz 1 120.w even 4 1
1800.4.a.c 1 5.c odd 4 1
1800.4.a.bh 1 5.c odd 4 1
1800.4.f.w 2 1.a even 1 1 trivial
1800.4.f.w 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 $$S_{4}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7}^{2} + 676$$ T7^2 + 676 $$T_{11} - 59$$ T11 - 59

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 676$$
$11$ $$(T - 59)^{2}$$
$13$ $$T^{2} + 784$$
$17$ $$T^{2} + 25$$
$19$ $$(T + 109)^{2}$$
$23$ $$T^{2} + 37636$$
$29$ $$(T + 32)^{2}$$
$31$ $$(T - 10)^{2}$$
$37$ $$T^{2} + 39204$$
$41$ $$(T + 117)^{2}$$
$43$ $$T^{2} + 150544$$
$47$ $$T^{2} + 4624$$
$53$ $$T^{2} + 324$$
$59$ $$(T - 392)^{2}$$
$61$ $$(T + 710)^{2}$$
$67$ $$T^{2} + 64009$$
$71$ $$(T - 612)^{2}$$
$73$ $$T^{2} + 301401$$
$79$ $$(T + 414)^{2}$$
$83$ $$T^{2} + 14641$$
$89$ $$(T + 81)^{2}$$
$97$ $$T^{2} + 2256004$$