# Properties

 Label 624.2.q.c Level $624$ Weight $2$ Character orbit 624.q Analytic conductor $4.983$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$4.98266508613$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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 39) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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 $$q + (\zeta_{6} - 1) q^{3} - q^{5} + 2 \zeta_{6} q^{7} - \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{11} + ( - \zeta_{6} - 3) q^{13} + ( - \zeta_{6} + 1) q^{15} + 7 \zeta_{6} q^{17} - 6 \zeta_{6} q^{19} - 2 q^{21} + (6 \zeta_{6} - 6) q^{23} - 4 q^{25} + q^{27} + ( - \zeta_{6} + 1) q^{29} - 4 q^{31} - 2 \zeta_{6} q^{33} - 2 \zeta_{6} q^{35} + (\zeta_{6} - 1) q^{37} + ( - 3 \zeta_{6} + 4) q^{39} + (9 \zeta_{6} - 9) q^{41} + 6 \zeta_{6} q^{43} + \zeta_{6} q^{45} - 6 q^{47} + ( - 3 \zeta_{6} + 3) q^{49} - 7 q^{51} - 9 q^{53} + ( - 2 \zeta_{6} + 2) q^{55} + 6 q^{57} - \zeta_{6} q^{61} + ( - 2 \zeta_{6} + 2) q^{63} + (\zeta_{6} + 3) q^{65} + (2 \zeta_{6} - 2) q^{67} - 6 \zeta_{6} q^{69} + 6 \zeta_{6} q^{71} + 11 q^{73} + ( - 4 \zeta_{6} + 4) q^{75} - 4 q^{77} + 4 q^{79} + (\zeta_{6} - 1) q^{81} + 14 q^{83} - 7 \zeta_{6} q^{85} + \zeta_{6} q^{87} + ( - 14 \zeta_{6} + 14) q^{89} + ( - 8 \zeta_{6} + 2) q^{91} + ( - 4 \zeta_{6} + 4) q^{93} + 6 \zeta_{6} q^{95} + 2 \zeta_{6} q^{97} + 2 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 - q^5 + 2*z * q^7 - z * q^9 + (2*z - 2) * q^11 + (-z - 3) * q^13 + (-z + 1) * q^15 + 7*z * q^17 - 6*z * q^19 - 2 * q^21 + (6*z - 6) * q^23 - 4 * q^25 + q^27 + (-z + 1) * q^29 - 4 * q^31 - 2*z * q^33 - 2*z * q^35 + (z - 1) * q^37 + (-3*z + 4) * q^39 + (9*z - 9) * q^41 + 6*z * q^43 + z * q^45 - 6 * q^47 + (-3*z + 3) * q^49 - 7 * q^51 - 9 * q^53 + (-2*z + 2) * q^55 + 6 * q^57 - z * q^61 + (-2*z + 2) * q^63 + (z + 3) * q^65 + (2*z - 2) * q^67 - 6*z * q^69 + 6*z * q^71 + 11 * q^73 + (-4*z + 4) * q^75 - 4 * q^77 + 4 * q^79 + (z - 1) * q^81 + 14 * q^83 - 7*z * q^85 + z * q^87 + (-14*z + 14) * q^89 + (-8*z + 2) * q^91 + (-4*z + 4) * q^93 + 6*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 $$2 q - q^{3} - 2 q^{5} + 2 q^{7} - q^{9} - 2 q^{11} - 7 q^{13} + q^{15} + 7 q^{17} - 6 q^{19} - 4 q^{21} - 6 q^{23} - 8 q^{25} + 2 q^{27} + q^{29} - 8 q^{31} - 2 q^{33} - 2 q^{35} - q^{37} + 5 q^{39} - 9 q^{41} + 6 q^{43} + q^{45} - 12 q^{47} + 3 q^{49} - 14 q^{51} - 18 q^{53} + 2 q^{55} + 12 q^{57} - q^{61} + 2 q^{63} + 7 q^{65} - 2 q^{67} - 6 q^{69} + 6 q^{71} + 22 q^{73} + 4 q^{75} - 8 q^{77} + 8 q^{79} - q^{81} + 28 q^{83} - 7 q^{85} + q^{87} + 14 q^{89} - 4 q^{91} + 4 q^{93} + 6 q^{95} + 2 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q - q^3 - 2 * q^5 + 2 * q^7 - q^9 - 2 * q^11 - 7 * q^13 + q^15 + 7 * q^17 - 6 * q^19 - 4 * q^21 - 6 * q^23 - 8 * q^25 + 2 * q^27 + q^29 - 8 * q^31 - 2 * q^33 - 2 * q^35 - q^37 + 5 * q^39 - 9 * q^41 + 6 * q^43 + q^45 - 12 * q^47 + 3 * q^49 - 14 * q^51 - 18 * q^53 + 2 * q^55 + 12 * q^57 - q^61 + 2 * q^63 + 7 * q^65 - 2 * q^67 - 6 * q^69 + 6 * q^71 + 22 * q^73 + 4 * q^75 - 8 * q^77 + 8 * q^79 - q^81 + 28 * q^83 - 7 * q^85 + q^87 + 14 * q^89 - 4 * q^91 + 4 * q^93 + 6 * 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.

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.c 2
3.b odd 2 1 1872.2.t.j 2
4.b odd 2 1 39.2.e.a 2
12.b even 2 1 117.2.g.b 2
13.c even 3 1 inner 624.2.q.c 2
13.c even 3 1 8112.2.a.w 1
13.e even 6 1 8112.2.a.bc 1
20.d odd 2 1 975.2.i.f 2
20.e even 4 2 975.2.bb.d 4
39.i odd 6 1 1872.2.t.j 2
52.b odd 2 1 507.2.e.c 2
52.f even 4 2 507.2.j.d 4
52.i odd 6 1 507.2.a.b 1
52.i odd 6 1 507.2.e.c 2
52.j odd 6 1 39.2.e.a 2
52.j odd 6 1 507.2.a.c 1
52.l even 12 2 507.2.b.b 2
52.l even 12 2 507.2.j.d 4
156.p even 6 1 117.2.g.b 2
156.p even 6 1 1521.2.a.a 1
156.r even 6 1 1521.2.a.d 1
156.v odd 12 2 1521.2.b.c 2
260.v odd 6 1 975.2.i.f 2
260.bj even 12 2 975.2.bb.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 4.b odd 2 1
39.2.e.a 2 52.j odd 6 1
117.2.g.b 2 12.b even 2 1
117.2.g.b 2 156.p even 6 1
507.2.a.b 1 52.i odd 6 1
507.2.a.c 1 52.j odd 6 1
507.2.b.b 2 52.l even 12 2
507.2.e.c 2 52.b odd 2 1
507.2.e.c 2 52.i odd 6 1
507.2.j.d 4 52.f even 4 2
507.2.j.d 4 52.l even 12 2
624.2.q.c 2 1.a even 1 1 trivial
624.2.q.c 2 13.c even 3 1 inner
975.2.i.f 2 20.d odd 2 1
975.2.i.f 2 260.v odd 6 1
975.2.bb.d 4 20.e even 4 2
975.2.bb.d 4 260.bj even 12 2
1521.2.a.a 1 156.p even 6 1
1521.2.a.d 1 156.r even 6 1
1521.2.b.c 2 156.v odd 12 2
1872.2.t.j 2 3.b odd 2 1
1872.2.t.j 2 39.i odd 6 1
8112.2.a.w 1 13.c even 3 1
8112.2.a.bc 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

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 2T + 4$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$T^{2} + 7T + 13$$
$17$ $$T^{2} - 7T + 49$$
$19$ $$T^{2} + 6T + 36$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} - T + 1$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + T + 1$$
$41$ $$T^{2} + 9T + 81$$
$43$ $$T^{2} - 6T + 36$$
$47$ $$(T + 6)^{2}$$
$53$ $$(T + 9)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} + 2T + 4$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$(T - 11)^{2}$$
$79$ $$(T - 4)^{2}$$
$83$ $$(T - 14)^{2}$$
$89$ $$T^{2} - 14T + 196$$
$97$ $$T^{2} - 2T + 4$$