# Properties

 Label 3600.3.l.d Level $3600$ Weight $3$ Character orbit 3600.l Analytic conductor $98.093$ 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] = [3600,3,Mod(1601,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3600.1601");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 3600.l (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: $$98.0928951697$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 18) 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 $$\beta = 3\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 4 q^{7}+O(q^{10})$$ q - 4 * q^7 $$q - 4 q^{7} + 4 \beta q^{11} - 8 q^{13} - 3 \beta q^{17} + 16 q^{19} + 4 \beta q^{23} - \beta q^{29} - 44 q^{31} + 34 q^{37} - 11 \beta q^{41} - 40 q^{43} + 20 \beta q^{47} - 33 q^{49} + 9 \beta q^{53} + 8 \beta q^{59} + 50 q^{61} + 8 q^{67} - 12 \beta q^{71} + 16 q^{73} - 16 \beta q^{77} + 76 q^{79} - 28 \beta q^{83} - 3 \beta q^{89} + 32 q^{91} - 176 q^{97} +O(q^{100})$$ q - 4 * q^7 + 4*b * q^11 - 8 * q^13 - 3*b * q^17 + 16 * q^19 + 4*b * q^23 - b * q^29 - 44 * q^31 + 34 * q^37 - 11*b * q^41 - 40 * q^43 + 20*b * q^47 - 33 * q^49 + 9*b * q^53 + 8*b * q^59 + 50 * q^61 + 8 * q^67 - 12*b * q^71 + 16 * q^73 - 16*b * q^77 + 76 * q^79 - 28*b * q^83 - 3*b * q^89 + 32 * q^91 - 176 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{7}+O(q^{10})$$ 2 * q - 8 * q^7 $$2 q - 8 q^{7} - 16 q^{13} + 32 q^{19} - 88 q^{31} + 68 q^{37} - 80 q^{43} - 66 q^{49} + 100 q^{61} + 16 q^{67} + 32 q^{73} + 152 q^{79} + 64 q^{91} - 352 q^{97}+O(q^{100})$$ 2 * q - 8 * q^7 - 16 * q^13 + 32 * q^19 - 88 * q^31 + 68 * q^37 - 80 * q^43 - 66 * q^49 + 100 * q^61 + 16 * q^67 + 32 * q^73 + 152 * q^79 + 64 * q^91 - 352 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1601.1
 − 1.41421i 1.41421i
0 0 0 0 0 −4.00000 0 0 0
1601.2 0 0 0 0 0 −4.00000 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.3.l.d 2
3.b odd 2 1 inner 3600.3.l.d 2
4.b odd 2 1 450.3.d.f 2
5.b even 2 1 144.3.e.b 2
5.c odd 4 2 3600.3.c.b 4
12.b even 2 1 450.3.d.f 2
15.d odd 2 1 144.3.e.b 2
15.e even 4 2 3600.3.c.b 4
20.d odd 2 1 18.3.b.a 2
20.e even 4 2 450.3.b.b 4
40.e odd 2 1 576.3.e.c 2
40.f even 2 1 576.3.e.f 2
45.h odd 6 2 1296.3.q.f 4
45.j even 6 2 1296.3.q.f 4
60.h even 2 1 18.3.b.a 2
60.l odd 4 2 450.3.b.b 4
80.k odd 4 2 2304.3.h.f 4
80.q even 4 2 2304.3.h.c 4
120.i odd 2 1 576.3.e.f 2
120.m even 2 1 576.3.e.c 2
140.c even 2 1 882.3.b.a 2
140.p odd 6 2 882.3.s.b 4
140.s even 6 2 882.3.s.d 4
180.n even 6 2 162.3.d.b 4
180.p odd 6 2 162.3.d.b 4
220.g even 2 1 2178.3.c.d 2
240.t even 4 2 2304.3.h.f 4
240.bm odd 4 2 2304.3.h.c 4
260.g odd 2 1 3042.3.c.e 2
260.u even 4 2 3042.3.d.a 4
420.o odd 2 1 882.3.b.a 2
420.ba even 6 2 882.3.s.b 4
420.be odd 6 2 882.3.s.d 4
660.g odd 2 1 2178.3.c.d 2
780.d even 2 1 3042.3.c.e 2
780.bb odd 4 2 3042.3.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 20.d odd 2 1
18.3.b.a 2 60.h even 2 1
144.3.e.b 2 5.b even 2 1
144.3.e.b 2 15.d odd 2 1
162.3.d.b 4 180.n even 6 2
162.3.d.b 4 180.p odd 6 2
450.3.b.b 4 20.e even 4 2
450.3.b.b 4 60.l odd 4 2
450.3.d.f 2 4.b odd 2 1
450.3.d.f 2 12.b even 2 1
576.3.e.c 2 40.e odd 2 1
576.3.e.c 2 120.m even 2 1
576.3.e.f 2 40.f even 2 1
576.3.e.f 2 120.i odd 2 1
882.3.b.a 2 140.c even 2 1
882.3.b.a 2 420.o odd 2 1
882.3.s.b 4 140.p odd 6 2
882.3.s.b 4 420.ba even 6 2
882.3.s.d 4 140.s even 6 2
882.3.s.d 4 420.be odd 6 2
1296.3.q.f 4 45.h odd 6 2
1296.3.q.f 4 45.j even 6 2
2178.3.c.d 2 220.g even 2 1
2178.3.c.d 2 660.g odd 2 1
2304.3.h.c 4 80.q even 4 2
2304.3.h.c 4 240.bm odd 4 2
2304.3.h.f 4 80.k odd 4 2
2304.3.h.f 4 240.t even 4 2
3042.3.c.e 2 260.g odd 2 1
3042.3.c.e 2 780.d even 2 1
3042.3.d.a 4 260.u even 4 2
3042.3.d.a 4 780.bb odd 4 2
3600.3.c.b 4 5.c odd 4 2
3600.3.c.b 4 15.e even 4 2
3600.3.l.d 2 1.a even 1 1 trivial
3600.3.l.d 2 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(3600, [\chi])$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11}^{2} + 288$$ T11^2 + 288 $$T_{13} + 8$$ T13 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 4)^{2}$$
$11$ $$T^{2} + 288$$
$13$ $$(T + 8)^{2}$$
$17$ $$T^{2} + 162$$
$19$ $$(T - 16)^{2}$$
$23$ $$T^{2} + 288$$
$29$ $$T^{2} + 18$$
$31$ $$(T + 44)^{2}$$
$37$ $$(T - 34)^{2}$$
$41$ $$T^{2} + 2178$$
$43$ $$(T + 40)^{2}$$
$47$ $$T^{2} + 7200$$
$53$ $$T^{2} + 1458$$
$59$ $$T^{2} + 1152$$
$61$ $$(T - 50)^{2}$$
$67$ $$(T - 8)^{2}$$
$71$ $$T^{2} + 2592$$
$73$ $$(T - 16)^{2}$$
$79$ $$(T - 76)^{2}$$
$83$ $$T^{2} + 14112$$
$89$ $$T^{2} + 162$$
$97$ $$(T + 176)^{2}$$