# Properties

 Label 3600.3.c.b Level $3600$ Weight $3$ Character orbit 3600.c Analytic conductor $98.093$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,3,Mod(449,3600)]

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, 1]))

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

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

chi := DirichletCharacter("3600.449");

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.c (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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta_1 q^{7}+O(q^{10})$$ q + 2*b1 * q^7 $$q + 2 \beta_1 q^{7} - 4 \beta_{2} q^{11} - 4 \beta_1 q^{13} + 3 \beta_{3} q^{17} - 16 q^{19} + 4 \beta_{3} q^{23} - \beta_{2} q^{29} - 44 q^{31} - 17 \beta_1 q^{37} + 11 \beta_{2} q^{41} - 20 \beta_1 q^{43} - 20 \beta_{3} q^{47} + 33 q^{49} + 9 \beta_{3} q^{53} + 8 \beta_{2} q^{59} + 50 q^{61} - 4 \beta_1 q^{67} + 12 \beta_{2} q^{71} + 8 \beta_1 q^{73} + 16 \beta_{3} q^{77} - 76 q^{79} - 28 \beta_{3} q^{83} - 3 \beta_{2} q^{89} + 32 q^{91} + 88 \beta_1 q^{97}+O(q^{100})$$ q + 2*b1 * q^7 - 4*b2 * q^11 - 4*b1 * q^13 + 3*b3 * q^17 - 16 * q^19 + 4*b3 * q^23 - b2 * q^29 - 44 * q^31 - 17*b1 * q^37 + 11*b2 * q^41 - 20*b1 * q^43 - 20*b3 * q^47 + 33 * q^49 + 9*b3 * q^53 + 8*b2 * q^59 + 50 * q^61 - 4*b1 * q^67 + 12*b2 * q^71 + 8*b1 * q^73 + 16*b3 * q^77 - 76 * q^79 - 28*b3 * q^83 - 3*b2 * q^89 + 32 * q^91 + 88*b1 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 64 q^{19} - 176 q^{31} + 132 q^{49} + 200 q^{61} - 304 q^{79} + 128 q^{91}+O(q^{100})$$ 4 * q - 64 * q^19 - 176 * q^31 + 132 * q^49 + 200 * q^61 - 304 * q^79 + 128 * q^91

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{8}^{2}$$ 2*v^2 $$\beta_{2}$$ $$=$$ $$3\zeta_{8}^{3} + 3\zeta_{8}$$ 3*v^3 + 3*v $$\beta_{3}$$ $$=$$ $$-3\zeta_{8}^{3} + 3\zeta_{8}$$ -3*v^3 + 3*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 6$$ (b3 + b2) / 6 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 6$$ (-b3 + b2) / 6

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 0 0 4.00000i 0 0 0
449.2 0 0 0 0 0 4.00000i 0 0 0
449.3 0 0 0 0 0 4.00000i 0 0 0
449.4 0 0 0 0 0 4.00000i 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d 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.c.b 4
3.b odd 2 1 inner 3600.3.c.b 4
4.b odd 2 1 450.3.b.b 4
5.b even 2 1 inner 3600.3.c.b 4
5.c odd 4 1 144.3.e.b 2
5.c odd 4 1 3600.3.l.d 2
12.b even 2 1 450.3.b.b 4
15.d odd 2 1 inner 3600.3.c.b 4
15.e even 4 1 144.3.e.b 2
15.e even 4 1 3600.3.l.d 2
20.d odd 2 1 450.3.b.b 4
20.e even 4 1 18.3.b.a 2
20.e even 4 1 450.3.d.f 2
40.i odd 4 1 576.3.e.f 2
40.k even 4 1 576.3.e.c 2
45.k odd 12 2 1296.3.q.f 4
45.l even 12 2 1296.3.q.f 4
60.h even 2 1 450.3.b.b 4
60.l odd 4 1 18.3.b.a 2
60.l odd 4 1 450.3.d.f 2
80.i odd 4 1 2304.3.h.c 4
80.j even 4 1 2304.3.h.f 4
80.s even 4 1 2304.3.h.f 4
80.t odd 4 1 2304.3.h.c 4
120.q odd 4 1 576.3.e.c 2
120.w even 4 1 576.3.e.f 2
140.j odd 4 1 882.3.b.a 2
140.w even 12 2 882.3.s.b 4
140.x odd 12 2 882.3.s.d 4
180.v odd 12 2 162.3.d.b 4
180.x even 12 2 162.3.d.b 4
220.i odd 4 1 2178.3.c.d 2
240.z odd 4 1 2304.3.h.f 4
240.bb even 4 1 2304.3.h.c 4
240.bd odd 4 1 2304.3.h.f 4
240.bf even 4 1 2304.3.h.c 4
260.l odd 4 1 3042.3.d.a 4
260.p even 4 1 3042.3.c.e 2
260.s odd 4 1 3042.3.d.a 4
420.w even 4 1 882.3.b.a 2
420.bp odd 12 2 882.3.s.b 4
420.br even 12 2 882.3.s.d 4
660.q even 4 1 2178.3.c.d 2
780.u even 4 1 3042.3.d.a 4
780.w odd 4 1 3042.3.c.e 2
780.bn even 4 1 3042.3.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 20.e even 4 1
18.3.b.a 2 60.l odd 4 1
144.3.e.b 2 5.c odd 4 1
144.3.e.b 2 15.e even 4 1
162.3.d.b 4 180.v odd 12 2
162.3.d.b 4 180.x even 12 2
450.3.b.b 4 4.b odd 2 1
450.3.b.b 4 12.b even 2 1
450.3.b.b 4 20.d odd 2 1
450.3.b.b 4 60.h even 2 1
450.3.d.f 2 20.e even 4 1
450.3.d.f 2 60.l odd 4 1
576.3.e.c 2 40.k even 4 1
576.3.e.c 2 120.q odd 4 1
576.3.e.f 2 40.i odd 4 1
576.3.e.f 2 120.w even 4 1
882.3.b.a 2 140.j odd 4 1
882.3.b.a 2 420.w even 4 1
882.3.s.b 4 140.w even 12 2
882.3.s.b 4 420.bp odd 12 2
882.3.s.d 4 140.x odd 12 2
882.3.s.d 4 420.br even 12 2
1296.3.q.f 4 45.k odd 12 2
1296.3.q.f 4 45.l even 12 2
2178.3.c.d 2 220.i odd 4 1
2178.3.c.d 2 660.q even 4 1
2304.3.h.c 4 80.i odd 4 1
2304.3.h.c 4 80.t odd 4 1
2304.3.h.c 4 240.bb even 4 1
2304.3.h.c 4 240.bf even 4 1
2304.3.h.f 4 80.j even 4 1
2304.3.h.f 4 80.s even 4 1
2304.3.h.f 4 240.z odd 4 1
2304.3.h.f 4 240.bd odd 4 1
3042.3.c.e 2 260.p even 4 1
3042.3.c.e 2 780.w odd 4 1
3042.3.d.a 4 260.l odd 4 1
3042.3.d.a 4 260.s odd 4 1
3042.3.d.a 4 780.u even 4 1
3042.3.d.a 4 780.bn even 4 1
3600.3.c.b 4 1.a even 1 1 trivial
3600.3.c.b 4 3.b odd 2 1 inner
3600.3.c.b 4 5.b even 2 1 inner
3600.3.c.b 4 15.d odd 2 1 inner
3600.3.l.d 2 5.c odd 4 1
3600.3.l.d 2 15.e even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{13}^{2} + 64$$ T13^2 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 16)^{2}$$
$11$ $$(T^{2} + 288)^{2}$$
$13$ $$(T^{2} + 64)^{2}$$
$17$ $$(T^{2} - 162)^{2}$$
$19$ $$(T + 16)^{4}$$
$23$ $$(T^{2} - 288)^{2}$$
$29$ $$(T^{2} + 18)^{2}$$
$31$ $$(T + 44)^{4}$$
$37$ $$(T^{2} + 1156)^{2}$$
$41$ $$(T^{2} + 2178)^{2}$$
$43$ $$(T^{2} + 1600)^{2}$$
$47$ $$(T^{2} - 7200)^{2}$$
$53$ $$(T^{2} - 1458)^{2}$$
$59$ $$(T^{2} + 1152)^{2}$$
$61$ $$(T - 50)^{4}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} + 2592)^{2}$$
$73$ $$(T^{2} + 256)^{2}$$
$79$ $$(T + 76)^{4}$$
$83$ $$(T^{2} - 14112)^{2}$$
$89$ $$(T^{2} + 162)^{2}$$
$97$ $$(T^{2} + 30976)^{2}$$