# Properties

 Label 900.2.i.b Level $900$ Weight $2$ Character orbit 900.i Analytic conductor $7.187$ Analytic rank $1$ 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] = [900,2,Mod(301,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("900.301");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 900.i (of order $$3$$, degree $$2$$, 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: $$7.18653618192$$ Analytic rank: $$1$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{7} - 3 q^{9}+O(q^{10})$$ q + (-2*z + 1) * q^3 + (z - 1) * q^7 - 3 * q^9 $$q + ( - 2 \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{7} - 3 q^{9} + (3 \zeta_{6} - 3) q^{11} - \zeta_{6} q^{13} - 6 q^{17} - 4 q^{19} + (\zeta_{6} + 1) q^{21} - 3 \zeta_{6} q^{23} + (6 \zeta_{6} - 3) q^{27} + (3 \zeta_{6} - 3) q^{29} - 5 \zeta_{6} q^{31} + (3 \zeta_{6} + 3) q^{33} - 2 q^{37} + (\zeta_{6} - 2) q^{39} - 3 \zeta_{6} q^{41} + (\zeta_{6} - 1) q^{43} + (9 \zeta_{6} - 9) q^{47} + 6 \zeta_{6} q^{49} + (12 \zeta_{6} - 6) q^{51} + 6 q^{53} + (8 \zeta_{6} - 4) q^{57} + 3 \zeta_{6} q^{59} + ( - 13 \zeta_{6} + 13) q^{61} + ( - 3 \zeta_{6} + 3) q^{63} - 7 \zeta_{6} q^{67} + (3 \zeta_{6} - 6) q^{69} - 12 q^{71} + 10 q^{73} - 3 \zeta_{6} q^{77} + (11 \zeta_{6} - 11) q^{79} + 9 q^{81} + (9 \zeta_{6} - 9) q^{83} + (3 \zeta_{6} + 3) q^{87} + 6 q^{89} + q^{91} + (5 \zeta_{6} - 10) q^{93} + ( - 11 \zeta_{6} + 11) q^{97} + ( - 9 \zeta_{6} + 9) q^{99} +O(q^{100})$$ q + (-2*z + 1) * q^3 + (z - 1) * q^7 - 3 * q^9 + (3*z - 3) * q^11 - z * q^13 - 6 * q^17 - 4 * q^19 + (z + 1) * q^21 - 3*z * q^23 + (6*z - 3) * q^27 + (3*z - 3) * q^29 - 5*z * q^31 + (3*z + 3) * q^33 - 2 * q^37 + (z - 2) * q^39 - 3*z * q^41 + (z - 1) * q^43 + (9*z - 9) * q^47 + 6*z * q^49 + (12*z - 6) * q^51 + 6 * q^53 + (8*z - 4) * q^57 + 3*z * q^59 + (-13*z + 13) * q^61 + (-3*z + 3) * q^63 - 7*z * q^67 + (3*z - 6) * q^69 - 12 * q^71 + 10 * q^73 - 3*z * q^77 + (11*z - 11) * q^79 + 9 * q^81 + (9*z - 9) * q^83 + (3*z + 3) * q^87 + 6 * q^89 + q^91 + (5*z - 10) * q^93 + (-11*z + 11) * q^97 + (-9*z + 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q - q^7 - 6 * q^9 $$2 q - q^{7} - 6 q^{9} - 3 q^{11} - q^{13} - 12 q^{17} - 8 q^{19} + 3 q^{21} - 3 q^{23} - 3 q^{29} - 5 q^{31} + 9 q^{33} - 4 q^{37} - 3 q^{39} - 3 q^{41} - q^{43} - 9 q^{47} + 6 q^{49} + 12 q^{53} + 3 q^{59} + 13 q^{61} + 3 q^{63} - 7 q^{67} - 9 q^{69} - 24 q^{71} + 20 q^{73} - 3 q^{77} - 11 q^{79} + 18 q^{81} - 9 q^{83} + 9 q^{87} + 12 q^{89} + 2 q^{91} - 15 q^{93} + 11 q^{97} + 9 q^{99}+O(q^{100})$$ 2 * q - q^7 - 6 * q^9 - 3 * q^11 - q^13 - 12 * q^17 - 8 * q^19 + 3 * q^21 - 3 * q^23 - 3 * q^29 - 5 * q^31 + 9 * q^33 - 4 * q^37 - 3 * q^39 - 3 * q^41 - q^43 - 9 * q^47 + 6 * q^49 + 12 * q^53 + 3 * q^59 + 13 * q^61 + 3 * q^63 - 7 * q^67 - 9 * q^69 - 24 * q^71 + 20 * q^73 - 3 * q^77 - 11 * q^79 + 18 * q^81 - 9 * q^83 + 9 * q^87 + 12 * q^89 + 2 * q^91 - 15 * q^93 + 11 * q^97 + 9 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
301.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.73205i 0 0 0 −0.500000 0.866025i 0 −3.00000 0
601.1 0 1.73205i 0 0 0 −0.500000 + 0.866025i 0 −3.00000 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.i.b 2
3.b odd 2 1 2700.2.i.b 2
5.b even 2 1 36.2.e.a 2
5.c odd 4 2 900.2.s.b 4
9.c even 3 1 inner 900.2.i.b 2
9.c even 3 1 8100.2.a.j 1
9.d odd 6 1 2700.2.i.b 2
9.d odd 6 1 8100.2.a.g 1
15.d odd 2 1 108.2.e.a 2
15.e even 4 2 2700.2.s.b 4
20.d odd 2 1 144.2.i.a 2
35.c odd 2 1 1764.2.j.b 2
35.i odd 6 1 1764.2.i.c 2
35.i odd 6 1 1764.2.l.a 2
35.j even 6 1 1764.2.i.a 2
35.j even 6 1 1764.2.l.c 2
40.e odd 2 1 576.2.i.e 2
40.f even 2 1 576.2.i.f 2
45.h odd 6 1 108.2.e.a 2
45.h odd 6 1 324.2.a.a 1
45.j even 6 1 36.2.e.a 2
45.j even 6 1 324.2.a.c 1
45.k odd 12 2 900.2.s.b 4
45.k odd 12 2 8100.2.d.h 2
45.l even 12 2 2700.2.s.b 4
45.l even 12 2 8100.2.d.c 2
60.h even 2 1 432.2.i.c 2
105.g even 2 1 5292.2.j.a 2
105.o odd 6 1 5292.2.i.c 2
105.o odd 6 1 5292.2.l.a 2
105.p even 6 1 5292.2.i.a 2
105.p even 6 1 5292.2.l.c 2
120.i odd 2 1 1728.2.i.d 2
120.m even 2 1 1728.2.i.c 2
180.n even 6 1 432.2.i.c 2
180.n even 6 1 1296.2.a.b 1
180.p odd 6 1 144.2.i.a 2
180.p odd 6 1 1296.2.a.k 1
315.q odd 6 1 1764.2.l.a 2
315.r even 6 1 1764.2.l.c 2
315.u even 6 1 5292.2.i.a 2
315.v odd 6 1 5292.2.i.c 2
315.z even 6 1 5292.2.j.a 2
315.bg odd 6 1 1764.2.j.b 2
315.bn odd 6 1 1764.2.i.c 2
315.bo even 6 1 1764.2.i.a 2
315.bq even 6 1 5292.2.l.c 2
315.br odd 6 1 5292.2.l.a 2
360.z odd 6 1 576.2.i.e 2
360.z odd 6 1 5184.2.a.f 1
360.bd even 6 1 1728.2.i.c 2
360.bd even 6 1 5184.2.a.bb 1
360.bh odd 6 1 1728.2.i.d 2
360.bh odd 6 1 5184.2.a.ba 1
360.bk even 6 1 576.2.i.f 2
360.bk even 6 1 5184.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 5.b even 2 1
36.2.e.a 2 45.j even 6 1
108.2.e.a 2 15.d odd 2 1
108.2.e.a 2 45.h odd 6 1
144.2.i.a 2 20.d odd 2 1
144.2.i.a 2 180.p odd 6 1
324.2.a.a 1 45.h odd 6 1
324.2.a.c 1 45.j even 6 1
432.2.i.c 2 60.h even 2 1
432.2.i.c 2 180.n even 6 1
576.2.i.e 2 40.e odd 2 1
576.2.i.e 2 360.z odd 6 1
576.2.i.f 2 40.f even 2 1
576.2.i.f 2 360.bk even 6 1
900.2.i.b 2 1.a even 1 1 trivial
900.2.i.b 2 9.c even 3 1 inner
900.2.s.b 4 5.c odd 4 2
900.2.s.b 4 45.k odd 12 2
1296.2.a.b 1 180.n even 6 1
1296.2.a.k 1 180.p odd 6 1
1728.2.i.c 2 120.m even 2 1
1728.2.i.c 2 360.bd even 6 1
1728.2.i.d 2 120.i odd 2 1
1728.2.i.d 2 360.bh odd 6 1
1764.2.i.a 2 35.j even 6 1
1764.2.i.a 2 315.bo even 6 1
1764.2.i.c 2 35.i odd 6 1
1764.2.i.c 2 315.bn odd 6 1
1764.2.j.b 2 35.c odd 2 1
1764.2.j.b 2 315.bg odd 6 1
1764.2.l.a 2 35.i odd 6 1
1764.2.l.a 2 315.q odd 6 1
1764.2.l.c 2 35.j even 6 1
1764.2.l.c 2 315.r even 6 1
2700.2.i.b 2 3.b odd 2 1
2700.2.i.b 2 9.d odd 6 1
2700.2.s.b 4 15.e even 4 2
2700.2.s.b 4 45.l even 12 2
5184.2.a.e 1 360.bk even 6 1
5184.2.a.f 1 360.z odd 6 1
5184.2.a.ba 1 360.bh odd 6 1
5184.2.a.bb 1 360.bd even 6 1
5292.2.i.a 2 105.p even 6 1
5292.2.i.a 2 315.u even 6 1
5292.2.i.c 2 105.o odd 6 1
5292.2.i.c 2 315.v odd 6 1
5292.2.j.a 2 105.g even 2 1
5292.2.j.a 2 315.z even 6 1
5292.2.l.a 2 105.o odd 6 1
5292.2.l.a 2 315.br odd 6 1
5292.2.l.c 2 105.p even 6 1
5292.2.l.c 2 315.bq even 6 1
8100.2.a.g 1 9.d odd 6 1
8100.2.a.j 1 9.c even 3 1
8100.2.d.c 2 45.l even 12 2
8100.2.d.h 2 45.k odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}^{2} + T_{7} + 1$$ T7^2 + T7 + 1 $$T_{11}^{2} + 3T_{11} + 9$$ T11^2 + 3*T11 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T + 1$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$T^{2} + T + 1$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$T^{2} + 5T + 25$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2} + 9T + 81$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} - 13T + 169$$
$67$ $$T^{2} + 7T + 49$$
$71$ $$(T + 12)^{2}$$
$73$ $$(T - 10)^{2}$$
$79$ $$T^{2} + 11T + 121$$
$83$ $$T^{2} + 9T + 81$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} - 11T + 121$$