# Properties

 Label 1764.2.i.c Level $1764$ Weight $2$ Character orbit 1764.i Analytic conductor $14.086$ 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] = [1764,2,Mod(373,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1764.373");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1764.i (of order $$3$$, degree $$2$$, 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: $$14.0856109166$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} + 1) q^{3} + ( - 3 \zeta_{6} + 3) q^{5} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (z + 1) * q^3 + (-3*z + 3) * q^5 + 3*z * q^9 $$q + (\zeta_{6} + 1) q^{3} + ( - 3 \zeta_{6} + 3) q^{5} + 3 \zeta_{6} q^{9} - 3 \zeta_{6} q^{11} - \zeta_{6} q^{13} + ( - 3 \zeta_{6} + 6) q^{15} + ( - 6 \zeta_{6} + 6) q^{17} - 4 \zeta_{6} q^{19} + ( - 3 \zeta_{6} + 3) q^{23} - 4 \zeta_{6} q^{25} + (6 \zeta_{6} - 3) q^{27} + (3 \zeta_{6} - 3) q^{29} - 5 q^{31} + ( - 6 \zeta_{6} + 3) q^{33} - 2 \zeta_{6} q^{37} + ( - 2 \zeta_{6} + 1) q^{39} + 3 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} + 9 q^{45} + 9 q^{47} + ( - 6 \zeta_{6} + 12) q^{51} + ( - 6 \zeta_{6} + 6) q^{53} - 9 q^{55} + ( - 8 \zeta_{6} + 4) q^{57} + 3 q^{59} + 13 q^{61} - 3 q^{65} - 7 q^{67} + ( - 3 \zeta_{6} + 6) q^{69} - 12 q^{71} + (10 \zeta_{6} - 10) q^{73} + ( - 8 \zeta_{6} + 4) q^{75} + 11 q^{79} + (9 \zeta_{6} - 9) q^{81} + (9 \zeta_{6} - 9) q^{83} - 18 \zeta_{6} q^{85} + (3 \zeta_{6} - 6) q^{87} + 6 \zeta_{6} q^{89} + ( - 5 \zeta_{6} - 5) q^{93} - 12 q^{95} + ( - 11 \zeta_{6} + 11) q^{97} + ( - 9 \zeta_{6} + 9) q^{99} +O(q^{100})$$ q + (z + 1) * q^3 + (-3*z + 3) * q^5 + 3*z * q^9 - 3*z * q^11 - z * q^13 + (-3*z + 6) * q^15 + (-6*z + 6) * q^17 - 4*z * q^19 + (-3*z + 3) * q^23 - 4*z * q^25 + (6*z - 3) * q^27 + (3*z - 3) * q^29 - 5 * q^31 + (-6*z + 3) * q^33 - 2*z * q^37 + (-2*z + 1) * q^39 + 3*z * q^41 + (-z + 1) * q^43 + 9 * q^45 + 9 * q^47 + (-6*z + 12) * q^51 + (-6*z + 6) * q^53 - 9 * q^55 + (-8*z + 4) * q^57 + 3 * q^59 + 13 * q^61 - 3 * q^65 - 7 * q^67 + (-3*z + 6) * q^69 - 12 * q^71 + (10*z - 10) * q^73 + (-8*z + 4) * q^75 + 11 * q^79 + (9*z - 9) * q^81 + (9*z - 9) * q^83 - 18*z * q^85 + (3*z - 6) * q^87 + 6*z * q^89 + (-5*z - 5) * q^93 - 12 * q^95 + (-11*z + 11) * q^97 + (-9*z + 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + 3 * q^5 + 3 * q^9 $$2 q + 3 q^{3} + 3 q^{5} + 3 q^{9} - 3 q^{11} - q^{13} + 9 q^{15} + 6 q^{17} - 4 q^{19} + 3 q^{23} - 4 q^{25} - 3 q^{29} - 10 q^{31} - 2 q^{37} + 3 q^{41} + q^{43} + 18 q^{45} + 18 q^{47} + 18 q^{51} + 6 q^{53} - 18 q^{55} + 6 q^{59} + 26 q^{61} - 6 q^{65} - 14 q^{67} + 9 q^{69} - 24 q^{71} - 10 q^{73} + 22 q^{79} - 9 q^{81} - 9 q^{83} - 18 q^{85} - 9 q^{87} + 6 q^{89} - 15 q^{93} - 24 q^{95} + 11 q^{97} + 9 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 + 3 * q^5 + 3 * q^9 - 3 * q^11 - q^13 + 9 * q^15 + 6 * q^17 - 4 * q^19 + 3 * q^23 - 4 * q^25 - 3 * q^29 - 10 * q^31 - 2 * q^37 + 3 * q^41 + q^43 + 18 * q^45 + 18 * q^47 + 18 * q^51 + 6 * q^53 - 18 * q^55 + 6 * q^59 + 26 * q^61 - 6 * q^65 - 14 * q^67 + 9 * q^69 - 24 * q^71 - 10 * q^73 + 22 * q^79 - 9 * q^81 - 9 * q^83 - 18 * q^85 - 9 * q^87 + 6 * q^89 - 15 * q^93 - 24 * q^95 + 11 * q^97 + 9 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
373.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.50000 0.866025i 0 1.50000 + 2.59808i 0 0 0 1.50000 2.59808i 0
1537.1 0 1.50000 + 0.866025i 0 1.50000 2.59808i 0 0 0 1.50000 + 2.59808i 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
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.i.c 2
3.b odd 2 1 5292.2.i.a 2
7.b odd 2 1 1764.2.i.a 2
7.c even 3 1 1764.2.j.b 2
7.c even 3 1 1764.2.l.a 2
7.d odd 6 1 36.2.e.a 2
7.d odd 6 1 1764.2.l.c 2
9.c even 3 1 1764.2.l.a 2
9.d odd 6 1 5292.2.l.c 2
21.c even 2 1 5292.2.i.c 2
21.g even 6 1 108.2.e.a 2
21.g even 6 1 5292.2.l.a 2
21.h odd 6 1 5292.2.j.a 2
21.h odd 6 1 5292.2.l.c 2
28.f even 6 1 144.2.i.a 2
35.i odd 6 1 900.2.i.b 2
35.k even 12 2 900.2.s.b 4
56.j odd 6 1 576.2.i.f 2
56.m even 6 1 576.2.i.e 2
63.g even 3 1 1764.2.j.b 2
63.h even 3 1 inner 1764.2.i.c 2
63.i even 6 1 324.2.a.a 1
63.i even 6 1 5292.2.i.c 2
63.j odd 6 1 5292.2.i.a 2
63.k odd 6 1 36.2.e.a 2
63.l odd 6 1 1764.2.l.c 2
63.n odd 6 1 5292.2.j.a 2
63.o even 6 1 5292.2.l.a 2
63.s even 6 1 108.2.e.a 2
63.t odd 6 1 324.2.a.c 1
63.t odd 6 1 1764.2.i.a 2
84.j odd 6 1 432.2.i.c 2
105.p even 6 1 2700.2.i.b 2
105.w odd 12 2 2700.2.s.b 4
168.ba even 6 1 1728.2.i.d 2
168.be odd 6 1 1728.2.i.c 2
252.n even 6 1 144.2.i.a 2
252.r odd 6 1 1296.2.a.b 1
252.bj even 6 1 1296.2.a.k 1
252.bn odd 6 1 432.2.i.c 2
315.q odd 6 1 8100.2.a.j 1
315.u even 6 1 2700.2.i.b 2
315.bn odd 6 1 900.2.i.b 2
315.bq even 6 1 8100.2.a.g 1
315.bs even 12 2 8100.2.d.h 2
315.bu odd 12 2 8100.2.d.c 2
315.bw odd 12 2 2700.2.s.b 4
315.cg even 12 2 900.2.s.b 4
504.u odd 6 1 1728.2.i.c 2
504.y even 6 1 1728.2.i.d 2
504.bf even 6 1 5184.2.a.f 1
504.bp odd 6 1 5184.2.a.e 1
504.ca even 6 1 5184.2.a.ba 1
504.cm odd 6 1 5184.2.a.bb 1
504.cw odd 6 1 576.2.i.f 2
504.cz even 6 1 576.2.i.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 7.d odd 6 1
36.2.e.a 2 63.k odd 6 1
108.2.e.a 2 21.g even 6 1
108.2.e.a 2 63.s even 6 1
144.2.i.a 2 28.f even 6 1
144.2.i.a 2 252.n even 6 1
324.2.a.a 1 63.i even 6 1
324.2.a.c 1 63.t odd 6 1
432.2.i.c 2 84.j odd 6 1
432.2.i.c 2 252.bn odd 6 1
576.2.i.e 2 56.m even 6 1
576.2.i.e 2 504.cz even 6 1
576.2.i.f 2 56.j odd 6 1
576.2.i.f 2 504.cw odd 6 1
900.2.i.b 2 35.i odd 6 1
900.2.i.b 2 315.bn odd 6 1
900.2.s.b 4 35.k even 12 2
900.2.s.b 4 315.cg even 12 2
1296.2.a.b 1 252.r odd 6 1
1296.2.a.k 1 252.bj even 6 1
1728.2.i.c 2 168.be odd 6 1
1728.2.i.c 2 504.u odd 6 1
1728.2.i.d 2 168.ba even 6 1
1728.2.i.d 2 504.y even 6 1
1764.2.i.a 2 7.b odd 2 1
1764.2.i.a 2 63.t odd 6 1
1764.2.i.c 2 1.a even 1 1 trivial
1764.2.i.c 2 63.h even 3 1 inner
1764.2.j.b 2 7.c even 3 1
1764.2.j.b 2 63.g even 3 1
1764.2.l.a 2 7.c even 3 1
1764.2.l.a 2 9.c even 3 1
1764.2.l.c 2 7.d odd 6 1
1764.2.l.c 2 63.l odd 6 1
2700.2.i.b 2 105.p even 6 1
2700.2.i.b 2 315.u even 6 1
2700.2.s.b 4 105.w odd 12 2
2700.2.s.b 4 315.bw odd 12 2
5184.2.a.e 1 504.bp odd 6 1
5184.2.a.f 1 504.bf even 6 1
5184.2.a.ba 1 504.ca even 6 1
5184.2.a.bb 1 504.cm odd 6 1
5292.2.i.a 2 3.b odd 2 1
5292.2.i.a 2 63.j odd 6 1
5292.2.i.c 2 21.c even 2 1
5292.2.i.c 2 63.i even 6 1
5292.2.j.a 2 21.h odd 6 1
5292.2.j.a 2 63.n odd 6 1
5292.2.l.a 2 21.g even 6 1
5292.2.l.a 2 63.o even 6 1
5292.2.l.c 2 9.d odd 6 1
5292.2.l.c 2 21.h odd 6 1
8100.2.a.g 1 315.bq even 6 1
8100.2.a.j 1 315.q odd 6 1
8100.2.d.c 2 315.bu odd 12 2
8100.2.d.h 2 315.bs even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 3T_{5} + 9$$ acting on $$S_{2}^{\mathrm{new}}(1764, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$T^{2} + T + 1$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 4T + 16$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$(T + 5)^{2}$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} - T + 1$$
$47$ $$(T - 9)^{2}$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$(T - 3)^{2}$$
$61$ $$(T - 13)^{2}$$
$67$ $$(T + 7)^{2}$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 10T + 100$$
$79$ $$(T - 11)^{2}$$
$83$ $$T^{2} + 9T + 81$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$T^{2} - 11T + 121$$