# Properties

 Label 624.2.q.b Level $624$ Weight $2$ Character orbit 624.q Analytic conductor $4.983$ 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] = [624,2,Mod(289,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("624.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 624.q (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: $$4.98266508613$$ 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 78) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} - q^{5} - 2 \zeta_{6} q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^3 - q^5 - 2*z * q^7 - z * q^9 $$q + (\zeta_{6} - 1) q^{3} - q^{5} - 2 \zeta_{6} q^{7} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} + (3 \zeta_{6} + 1) q^{13} + ( - \zeta_{6} + 1) q^{15} - 5 \zeta_{6} q^{17} - 2 \zeta_{6} q^{19} + 2 q^{21} + ( - 6 \zeta_{6} + 6) q^{23} - 4 q^{25} + q^{27} + ( - 9 \zeta_{6} + 9) q^{29} + 4 q^{31} + 2 \zeta_{6} q^{33} + 2 \zeta_{6} q^{35} + ( - 11 \zeta_{6} + 11) q^{37} + (\zeta_{6} - 4) q^{39} + (5 \zeta_{6} - 5) q^{41} + 10 \zeta_{6} q^{43} + \zeta_{6} q^{45} - 2 q^{47} + ( - 3 \zeta_{6} + 3) q^{49} + 5 q^{51} - q^{53} + (2 \zeta_{6} - 2) q^{55} + 2 q^{57} - 8 \zeta_{6} q^{59} + 11 \zeta_{6} q^{61} + (2 \zeta_{6} - 2) q^{63} + ( - 3 \zeta_{6} - 1) q^{65} + ( - 2 \zeta_{6} + 2) q^{67} + 6 \zeta_{6} q^{69} - 14 \zeta_{6} q^{71} - 13 q^{73} + ( - 4 \zeta_{6} + 4) q^{75} - 4 q^{77} + 4 q^{79} + (\zeta_{6} - 1) q^{81} - 6 q^{83} + 5 \zeta_{6} q^{85} + 9 \zeta_{6} q^{87} + (2 \zeta_{6} - 2) q^{89} + ( - 8 \zeta_{6} + 6) q^{91} + (4 \zeta_{6} - 4) q^{93} + 2 \zeta_{6} q^{95} + 2 \zeta_{6} q^{97} - 2 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 - q^5 - 2*z * q^7 - z * q^9 + (-2*z + 2) * q^11 + (3*z + 1) * q^13 + (-z + 1) * q^15 - 5*z * q^17 - 2*z * q^19 + 2 * q^21 + (-6*z + 6) * q^23 - 4 * q^25 + q^27 + (-9*z + 9) * q^29 + 4 * q^31 + 2*z * q^33 + 2*z * q^35 + (-11*z + 11) * q^37 + (z - 4) * q^39 + (5*z - 5) * q^41 + 10*z * q^43 + z * q^45 - 2 * q^47 + (-3*z + 3) * q^49 + 5 * q^51 - q^53 + (2*z - 2) * q^55 + 2 * q^57 - 8*z * q^59 + 11*z * q^61 + (2*z - 2) * q^63 + (-3*z - 1) * q^65 + (-2*z + 2) * q^67 + 6*z * q^69 - 14*z * q^71 - 13 * q^73 + (-4*z + 4) * q^75 - 4 * q^77 + 4 * q^79 + (z - 1) * q^81 - 6 * q^83 + 5*z * q^85 + 9*z * q^87 + (2*z - 2) * q^89 + (-8*z + 6) * q^91 + (4*z - 4) * q^93 + 2*z * q^95 + 2*z * q^97 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 2 q^{5} - 2 q^{7} - q^{9}+O(q^{10})$$ 2 * q - q^3 - 2 * q^5 - 2 * q^7 - q^9 $$2 q - q^{3} - 2 q^{5} - 2 q^{7} - q^{9} + 2 q^{11} + 5 q^{13} + q^{15} - 5 q^{17} - 2 q^{19} + 4 q^{21} + 6 q^{23} - 8 q^{25} + 2 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} + 2 q^{35} + 11 q^{37} - 7 q^{39} - 5 q^{41} + 10 q^{43} + q^{45} - 4 q^{47} + 3 q^{49} + 10 q^{51} - 2 q^{53} - 2 q^{55} + 4 q^{57} - 8 q^{59} + 11 q^{61} - 2 q^{63} - 5 q^{65} + 2 q^{67} + 6 q^{69} - 14 q^{71} - 26 q^{73} + 4 q^{75} - 8 q^{77} + 8 q^{79} - q^{81} - 12 q^{83} + 5 q^{85} + 9 q^{87} - 2 q^{89} + 4 q^{91} - 4 q^{93} + 2 q^{95} + 2 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q - q^3 - 2 * q^5 - 2 * q^7 - q^9 + 2 * q^11 + 5 * q^13 + q^15 - 5 * q^17 - 2 * q^19 + 4 * q^21 + 6 * q^23 - 8 * q^25 + 2 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 + 2 * q^35 + 11 * q^37 - 7 * q^39 - 5 * q^41 + 10 * q^43 + q^45 - 4 * q^47 + 3 * q^49 + 10 * q^51 - 2 * q^53 - 2 * q^55 + 4 * q^57 - 8 * q^59 + 11 * q^61 - 2 * q^63 - 5 * q^65 + 2 * q^67 + 6 * q^69 - 14 * q^71 - 26 * q^73 + 4 * q^75 - 8 * q^77 + 8 * q^79 - q^81 - 12 * q^83 + 5 * q^85 + 9 * q^87 - 2 * q^89 + 4 * q^91 - 4 * q^93 + 2 * q^95 + 2 * q^97 - 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$$.

 $$n$$ $$79$$ $$145$$ $$209$$ $$469$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 0.866025i 0 −1.00000 0 −1.00000 + 1.73205i 0 −0.500000 + 0.866025i 0
529.1 0 −0.500000 + 0.866025i 0 −1.00000 0 −1.00000 1.73205i 0 −0.500000 0.866025i 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.q.b 2
3.b odd 2 1 1872.2.t.i 2
4.b odd 2 1 78.2.e.b 2
12.b even 2 1 234.2.h.b 2
13.c even 3 1 inner 624.2.q.b 2
13.c even 3 1 8112.2.a.x 1
13.e even 6 1 8112.2.a.bb 1
20.d odd 2 1 1950.2.i.b 2
20.e even 4 2 1950.2.z.b 4
39.i odd 6 1 1872.2.t.i 2
52.b odd 2 1 1014.2.e.d 2
52.f even 4 2 1014.2.i.e 4
52.i odd 6 1 1014.2.a.e 1
52.i odd 6 1 1014.2.e.d 2
52.j odd 6 1 78.2.e.b 2
52.j odd 6 1 1014.2.a.a 1
52.l even 12 2 1014.2.b.a 2
52.l even 12 2 1014.2.i.e 4
156.p even 6 1 234.2.h.b 2
156.p even 6 1 3042.2.a.m 1
156.r even 6 1 3042.2.a.d 1
156.v odd 12 2 3042.2.b.d 2
260.v odd 6 1 1950.2.i.b 2
260.bj even 12 2 1950.2.z.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.b 2 4.b odd 2 1
78.2.e.b 2 52.j odd 6 1
234.2.h.b 2 12.b even 2 1
234.2.h.b 2 156.p even 6 1
624.2.q.b 2 1.a even 1 1 trivial
624.2.q.b 2 13.c even 3 1 inner
1014.2.a.a 1 52.j odd 6 1
1014.2.a.e 1 52.i odd 6 1
1014.2.b.a 2 52.l even 12 2
1014.2.e.d 2 52.b odd 2 1
1014.2.e.d 2 52.i odd 6 1
1014.2.i.e 4 52.f even 4 2
1014.2.i.e 4 52.l even 12 2
1872.2.t.i 2 3.b odd 2 1
1872.2.t.i 2 39.i odd 6 1
1950.2.i.b 2 20.d odd 2 1
1950.2.i.b 2 260.v odd 6 1
1950.2.z.b 4 20.e even 4 2
1950.2.z.b 4 260.bj even 12 2
3042.2.a.d 1 156.r even 6 1
3042.2.a.m 1 156.p even 6 1
3042.2.b.d 2 156.v odd 12 2
8112.2.a.x 1 13.c even 3 1
8112.2.a.bb 1 13.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(624, [\chi])$$:

 $$T_{5} + 1$$ T5 + 1 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$T^{2} - 5T + 13$$
$17$ $$T^{2} + 5T + 25$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - 9T + 81$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} - 11T + 121$$
$41$ $$T^{2} + 5T + 25$$
$43$ $$T^{2} - 10T + 100$$
$47$ $$(T + 2)^{2}$$
$53$ $$(T + 1)^{2}$$
$59$ $$T^{2} + 8T + 64$$
$61$ $$T^{2} - 11T + 121$$
$67$ $$T^{2} - 2T + 4$$
$71$ $$T^{2} + 14T + 196$$
$73$ $$(T + 13)^{2}$$
$79$ $$(T - 4)^{2}$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 2T + 4$$
$97$ $$T^{2} - 2T + 4$$