# Properties

 Label 1008.2.t.f Level $1008$ Weight $2$ Character orbit 1008.t Analytic conductor $8.049$ 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] = [1008,2,Mod(193,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 4]))

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

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

chi := DirichletCharacter("1008.193");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1008.t (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: $$8.04892052375$$ 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 126) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 2) q^{3} + 3 q^{5} + (\zeta_{6} - 3) q^{7} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10})$$ q + (-z + 2) * q^3 + 3 * q^5 + (z - 3) * q^7 + (-3*z + 3) * q^9 $$q + ( - \zeta_{6} + 2) q^{3} + 3 q^{5} + (\zeta_{6} - 3) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} + 3 q^{11} + ( - \zeta_{6} + 1) q^{13} + ( - 3 \zeta_{6} + 6) q^{15} + (3 \zeta_{6} - 3) q^{17} - 7 \zeta_{6} q^{19} + (4 \zeta_{6} - 5) q^{21} + 9 q^{23} + 4 q^{25} + ( - 6 \zeta_{6} + 3) q^{27} - 3 \zeta_{6} q^{29} + 8 \zeta_{6} q^{31} + ( - 3 \zeta_{6} + 6) q^{33} + (3 \zeta_{6} - 9) q^{35} + \zeta_{6} q^{37} + ( - 2 \zeta_{6} + 1) q^{39} + (3 \zeta_{6} - 3) q^{41} - \zeta_{6} q^{43} + ( - 9 \zeta_{6} + 9) q^{45} + ( - 5 \zeta_{6} + 8) q^{49} + (6 \zeta_{6} - 3) q^{51} + (3 \zeta_{6} - 3) q^{53} + 9 q^{55} + ( - 7 \zeta_{6} - 7) q^{57} + (2 \zeta_{6} - 2) q^{61} + (9 \zeta_{6} - 6) q^{63} + ( - 3 \zeta_{6} + 3) q^{65} - 4 \zeta_{6} q^{67} + ( - 9 \zeta_{6} + 18) q^{69} - 12 q^{71} + (11 \zeta_{6} - 11) q^{73} + ( - 4 \zeta_{6} + 8) q^{75} + (3 \zeta_{6} - 9) q^{77} + (16 \zeta_{6} - 16) q^{79} - 9 \zeta_{6} q^{81} - 9 \zeta_{6} q^{83} + (9 \zeta_{6} - 9) q^{85} + ( - 3 \zeta_{6} - 3) q^{87} - 3 \zeta_{6} q^{89} + (3 \zeta_{6} - 2) q^{91} + (8 \zeta_{6} + 8) q^{93} - 21 \zeta_{6} q^{95} + \zeta_{6} q^{97} + ( - 9 \zeta_{6} + 9) q^{99} +O(q^{100})$$ q + (-z + 2) * q^3 + 3 * q^5 + (z - 3) * q^7 + (-3*z + 3) * q^9 + 3 * q^11 + (-z + 1) * q^13 + (-3*z + 6) * q^15 + (3*z - 3) * q^17 - 7*z * q^19 + (4*z - 5) * q^21 + 9 * q^23 + 4 * q^25 + (-6*z + 3) * q^27 - 3*z * q^29 + 8*z * q^31 + (-3*z + 6) * q^33 + (3*z - 9) * q^35 + z * q^37 + (-2*z + 1) * q^39 + (3*z - 3) * q^41 - z * q^43 + (-9*z + 9) * q^45 + (-5*z + 8) * q^49 + (6*z - 3) * q^51 + (3*z - 3) * q^53 + 9 * q^55 + (-7*z - 7) * q^57 + (2*z - 2) * q^61 + (9*z - 6) * q^63 + (-3*z + 3) * q^65 - 4*z * q^67 + (-9*z + 18) * q^69 - 12 * q^71 + (11*z - 11) * q^73 + (-4*z + 8) * q^75 + (3*z - 9) * q^77 + (16*z - 16) * q^79 - 9*z * q^81 - 9*z * q^83 + (9*z - 9) * q^85 + (-3*z - 3) * q^87 - 3*z * q^89 + (3*z - 2) * q^91 + (8*z + 8) * q^93 - 21*z * q^95 + z * q^97 + (-9*z + 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + 6 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + 6 * q^5 - 5 * q^7 + 3 * q^9 $$2 q + 3 q^{3} + 6 q^{5} - 5 q^{7} + 3 q^{9} + 6 q^{11} + q^{13} + 9 q^{15} - 3 q^{17} - 7 q^{19} - 6 q^{21} + 18 q^{23} + 8 q^{25} - 3 q^{29} + 8 q^{31} + 9 q^{33} - 15 q^{35} + q^{37} - 3 q^{41} - q^{43} + 9 q^{45} + 11 q^{49} - 3 q^{53} + 18 q^{55} - 21 q^{57} - 2 q^{61} - 3 q^{63} + 3 q^{65} - 4 q^{67} + 27 q^{69} - 24 q^{71} - 11 q^{73} + 12 q^{75} - 15 q^{77} - 16 q^{79} - 9 q^{81} - 9 q^{83} - 9 q^{85} - 9 q^{87} - 3 q^{89} - q^{91} + 24 q^{93} - 21 q^{95} + q^{97} + 9 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 + 6 * q^5 - 5 * q^7 + 3 * q^9 + 6 * q^11 + q^13 + 9 * q^15 - 3 * q^17 - 7 * q^19 - 6 * q^21 + 18 * q^23 + 8 * q^25 - 3 * q^29 + 8 * q^31 + 9 * q^33 - 15 * q^35 + q^37 - 3 * q^41 - q^43 + 9 * q^45 + 11 * q^49 - 3 * q^53 + 18 * q^55 - 21 * q^57 - 2 * q^61 - 3 * q^63 + 3 * q^65 - 4 * q^67 + 27 * q^69 - 24 * q^71 - 11 * q^73 + 12 * q^75 - 15 * q^77 - 16 * q^79 - 9 * q^81 - 9 * q^83 - 9 * q^85 - 9 * q^87 - 3 * q^89 - q^91 + 24 * q^93 - 21 * q^95 + q^97 + 9 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$757$$ $$785$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 0.866025i 0 3.00000 0 −2.50000 + 0.866025i 0 1.50000 2.59808i 0
961.1 0 1.50000 + 0.866025i 0 3.00000 0 −2.50000 0.866025i 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.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.t.f 2
3.b odd 2 1 3024.2.t.a 2
4.b odd 2 1 126.2.h.b yes 2
7.c even 3 1 1008.2.q.a 2
9.c even 3 1 1008.2.q.a 2
9.d odd 6 1 3024.2.q.f 2
12.b even 2 1 378.2.h.a 2
21.h odd 6 1 3024.2.q.f 2
28.d even 2 1 882.2.h.i 2
28.f even 6 1 882.2.e.c 2
28.f even 6 1 882.2.f.g 2
28.g odd 6 1 126.2.e.a 2
28.g odd 6 1 882.2.f.i 2
36.f odd 6 1 126.2.e.a 2
36.f odd 6 1 1134.2.g.e 2
36.h even 6 1 378.2.e.b 2
36.h even 6 1 1134.2.g.c 2
63.g even 3 1 inner 1008.2.t.f 2
63.n odd 6 1 3024.2.t.a 2
84.h odd 2 1 2646.2.h.d 2
84.j odd 6 1 2646.2.e.g 2
84.j odd 6 1 2646.2.f.a 2
84.n even 6 1 378.2.e.b 2
84.n even 6 1 2646.2.f.d 2
252.n even 6 1 882.2.h.i 2
252.n even 6 1 7938.2.a.b 1
252.o even 6 1 378.2.h.a 2
252.o even 6 1 7938.2.a.t 1
252.r odd 6 1 2646.2.f.a 2
252.s odd 6 1 2646.2.e.g 2
252.u odd 6 1 882.2.f.i 2
252.u odd 6 1 1134.2.g.e 2
252.bb even 6 1 1134.2.g.c 2
252.bb even 6 1 2646.2.f.d 2
252.bi even 6 1 882.2.e.c 2
252.bj even 6 1 882.2.f.g 2
252.bl odd 6 1 126.2.h.b yes 2
252.bl odd 6 1 7938.2.a.m 1
252.bn odd 6 1 2646.2.h.d 2
252.bn odd 6 1 7938.2.a.be 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.a 2 28.g odd 6 1
126.2.e.a 2 36.f odd 6 1
126.2.h.b yes 2 4.b odd 2 1
126.2.h.b yes 2 252.bl odd 6 1
378.2.e.b 2 36.h even 6 1
378.2.e.b 2 84.n even 6 1
378.2.h.a 2 12.b even 2 1
378.2.h.a 2 252.o even 6 1
882.2.e.c 2 28.f even 6 1
882.2.e.c 2 252.bi even 6 1
882.2.f.g 2 28.f even 6 1
882.2.f.g 2 252.bj even 6 1
882.2.f.i 2 28.g odd 6 1
882.2.f.i 2 252.u odd 6 1
882.2.h.i 2 28.d even 2 1
882.2.h.i 2 252.n even 6 1
1008.2.q.a 2 7.c even 3 1
1008.2.q.a 2 9.c even 3 1
1008.2.t.f 2 1.a even 1 1 trivial
1008.2.t.f 2 63.g even 3 1 inner
1134.2.g.c 2 36.h even 6 1
1134.2.g.c 2 252.bb even 6 1
1134.2.g.e 2 36.f odd 6 1
1134.2.g.e 2 252.u odd 6 1
2646.2.e.g 2 84.j odd 6 1
2646.2.e.g 2 252.s odd 6 1
2646.2.f.a 2 84.j odd 6 1
2646.2.f.a 2 252.r odd 6 1
2646.2.f.d 2 84.n even 6 1
2646.2.f.d 2 252.bb even 6 1
2646.2.h.d 2 84.h odd 2 1
2646.2.h.d 2 252.bn odd 6 1
3024.2.q.f 2 9.d odd 6 1
3024.2.q.f 2 21.h odd 6 1
3024.2.t.a 2 3.b odd 2 1
3024.2.t.a 2 63.n odd 6 1
7938.2.a.b 1 252.n even 6 1
7938.2.a.m 1 252.bl odd 6 1
7938.2.a.t 1 252.o even 6 1
7938.2.a.be 1 252.bn odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{11} - 3$$ T11 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$(T - 3)^{2}$$
$7$ $$T^{2} + 5T + 7$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} - T + 1$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + 7T + 49$$
$23$ $$(T - 9)^{2}$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$T^{2} - 8T + 64$$
$37$ $$T^{2} - T + 1$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 3T + 9$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 2T + 4$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 11T + 121$$
$79$ $$T^{2} + 16T + 256$$
$83$ $$T^{2} + 9T + 81$$
$89$ $$T^{2} + 3T + 9$$
$97$ $$T^{2} - T + 1$$