# Properties

 Label 1800.4.f.n Level $1800$ Weight $4$ Character orbit 1800.f Analytic conductor $106.203$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Twist minimal: no (minimal twist has level 40) 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 9 \beta q^{7}+O(q^{10})$$ q + 9*b * q^7 $$q + 9 \beta q^{7} + 16 q^{11} - 3 \beta q^{13} - 3 \beta q^{17} + 124 q^{19} - 21 \beta q^{23} + 142 q^{29} - 188 q^{31} - 101 \beta q^{37} - 54 q^{41} + 33 \beta q^{43} + 19 \beta q^{47} + 19 q^{49} - 369 \beta q^{53} + 564 q^{59} - 262 q^{61} + 277 \beta q^{67} - 140 q^{71} + 441 \beta q^{73} + 144 \beta q^{77} + 1160 q^{79} - 321 \beta q^{83} - 854 q^{89} + 108 q^{91} + 239 \beta q^{97} +O(q^{100})$$ q + 9*b * q^7 + 16 * q^11 - 3*b * q^13 - 3*b * q^17 + 124 * q^19 - 21*b * q^23 + 142 * q^29 - 188 * q^31 - 101*b * q^37 - 54 * q^41 + 33*b * q^43 + 19*b * q^47 + 19 * q^49 - 369*b * q^53 + 564 * q^59 - 262 * q^61 + 277*b * q^67 - 140 * q^71 + 441*b * q^73 + 144*b * q^77 + 1160 * q^79 - 321*b * q^83 - 854 * q^89 + 108 * q^91 + 239*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 32 q^{11} + 248 q^{19} + 284 q^{29} - 376 q^{31} - 108 q^{41} + 38 q^{49} + 1128 q^{59} - 524 q^{61} - 280 q^{71} + 2320 q^{79} - 1708 q^{89} + 216 q^{91}+O(q^{100})$$ 2 * q + 32 * q^11 + 248 * q^19 + 284 * q^29 - 376 * q^31 - 108 * q^41 + 38 * q^49 + 1128 * q^59 - 524 * q^61 - 280 * q^71 + 2320 * q^79 - 1708 * q^89 + 216 * 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 18.0000i 0 0 0
649.2 0 0 0 0 0 18.0000i 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.n 2
3.b odd 2 1 200.4.c.a 2
5.b even 2 1 inner 1800.4.f.n 2
5.c odd 4 1 360.4.a.i 1
5.c odd 4 1 1800.4.a.bd 1
12.b even 2 1 400.4.c.a 2
15.d odd 2 1 200.4.c.a 2
15.e even 4 1 40.4.a.c 1
15.e even 4 1 200.4.a.a 1
20.e even 4 1 720.4.a.ba 1
60.h even 2 1 400.4.c.a 2
60.l odd 4 1 80.4.a.a 1
60.l odd 4 1 400.4.a.u 1
105.k odd 4 1 1960.4.a.a 1
120.q odd 4 1 320.4.a.n 1
120.q odd 4 1 1600.4.a.a 1
120.w even 4 1 320.4.a.a 1
120.w even 4 1 1600.4.a.ca 1
240.z odd 4 1 1280.4.d.b 2
240.bb even 4 1 1280.4.d.o 2
240.bd odd 4 1 1280.4.d.b 2
240.bf even 4 1 1280.4.d.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.c 1 15.e even 4 1
80.4.a.a 1 60.l odd 4 1
200.4.a.a 1 15.e even 4 1
200.4.c.a 2 3.b odd 2 1
200.4.c.a 2 15.d odd 2 1
320.4.a.a 1 120.w even 4 1
320.4.a.n 1 120.q odd 4 1
360.4.a.i 1 5.c odd 4 1
400.4.a.u 1 60.l odd 4 1
400.4.c.a 2 12.b even 2 1
400.4.c.a 2 60.h even 2 1
720.4.a.ba 1 20.e even 4 1
1280.4.d.b 2 240.z odd 4 1
1280.4.d.b 2 240.bd odd 4 1
1280.4.d.o 2 240.bb even 4 1
1280.4.d.o 2 240.bf even 4 1
1600.4.a.a 1 120.q odd 4 1
1600.4.a.ca 1 120.w even 4 1
1800.4.a.bd 1 5.c odd 4 1
1800.4.f.n 2 1.a even 1 1 trivial
1800.4.f.n 2 5.b even 2 1 inner
1960.4.a.a 1 105.k odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 324$$ T7^2 + 324 $$T_{11} - 16$$ T11 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 324$$
$11$ $$(T - 16)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 36$$
$19$ $$(T - 124)^{2}$$
$23$ $$T^{2} + 1764$$
$29$ $$(T - 142)^{2}$$
$31$ $$(T + 188)^{2}$$
$37$ $$T^{2} + 40804$$
$41$ $$(T + 54)^{2}$$
$43$ $$T^{2} + 4356$$
$47$ $$T^{2} + 1444$$
$53$ $$T^{2} + 544644$$
$59$ $$(T - 564)^{2}$$
$61$ $$(T + 262)^{2}$$
$67$ $$T^{2} + 306916$$
$71$ $$(T + 140)^{2}$$
$73$ $$T^{2} + 777924$$
$79$ $$(T - 1160)^{2}$$
$83$ $$T^{2} + 412164$$
$89$ $$(T + 854)^{2}$$
$97$ $$T^{2} + 228484$$