# Properties

 Label 624.4.q.b Level $624$ Weight $4$ Character orbit 624.q Analytic conductor $36.817$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 624.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.8171918436$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 3 \zeta_{6} + 3) q^{3} - 9 q^{5} + 2 \zeta_{6} q^{7} - 9 \zeta_{6} q^{9} +O(q^{10})$$ q + (-3*z + 3) * q^3 - 9 * q^5 + 2*z * q^7 - 9*z * q^9 $$q + ( - 3 \zeta_{6} + 3) q^{3} - 9 q^{5} + 2 \zeta_{6} q^{7} - 9 \zeta_{6} q^{9} + ( - 30 \zeta_{6} + 30) q^{11} + (39 \zeta_{6} + 13) q^{13} + (27 \zeta_{6} - 27) q^{15} + 111 \zeta_{6} q^{17} - 46 \zeta_{6} q^{19} + 6 q^{21} + (6 \zeta_{6} - 6) q^{23} - 44 q^{25} - 27 q^{27} + ( - 105 \zeta_{6} + 105) q^{29} + 100 q^{31} - 90 \zeta_{6} q^{33} - 18 \zeta_{6} q^{35} + (17 \zeta_{6} - 17) q^{37} + ( - 39 \zeta_{6} + 156) q^{39} + ( - 231 \zeta_{6} + 231) q^{41} - 514 \zeta_{6} q^{43} + 81 \zeta_{6} q^{45} + 162 q^{47} + ( - 339 \zeta_{6} + 339) q^{49} + 333 q^{51} + 639 q^{53} + (270 \zeta_{6} - 270) q^{55} - 138 q^{57} + 600 \zeta_{6} q^{59} - 233 \zeta_{6} q^{61} + ( - 18 \zeta_{6} + 18) q^{63} + ( - 351 \zeta_{6} - 117) q^{65} + ( - 926 \zeta_{6} + 926) q^{67} + 18 \zeta_{6} q^{69} - 930 \zeta_{6} q^{71} - 253 q^{73} + (132 \zeta_{6} - 132) q^{75} + 60 q^{77} + 1324 q^{79} + (81 \zeta_{6} - 81) q^{81} - 810 q^{83} - 999 \zeta_{6} q^{85} - 315 \zeta_{6} q^{87} + (498 \zeta_{6} - 498) q^{89} + (104 \zeta_{6} - 78) q^{91} + ( - 300 \zeta_{6} + 300) q^{93} + 414 \zeta_{6} q^{95} - 1358 \zeta_{6} q^{97} - 270 q^{99} +O(q^{100})$$ q + (-3*z + 3) * q^3 - 9 * q^5 + 2*z * q^7 - 9*z * q^9 + (-30*z + 30) * q^11 + (39*z + 13) * q^13 + (27*z - 27) * q^15 + 111*z * q^17 - 46*z * q^19 + 6 * q^21 + (6*z - 6) * q^23 - 44 * q^25 - 27 * q^27 + (-105*z + 105) * q^29 + 100 * q^31 - 90*z * q^33 - 18*z * q^35 + (17*z - 17) * q^37 + (-39*z + 156) * q^39 + (-231*z + 231) * q^41 - 514*z * q^43 + 81*z * q^45 + 162 * q^47 + (-339*z + 339) * q^49 + 333 * q^51 + 639 * q^53 + (270*z - 270) * q^55 - 138 * q^57 + 600*z * q^59 - 233*z * q^61 + (-18*z + 18) * q^63 + (-351*z - 117) * q^65 + (-926*z + 926) * q^67 + 18*z * q^69 - 930*z * q^71 - 253 * q^73 + (132*z - 132) * q^75 + 60 * q^77 + 1324 * q^79 + (81*z - 81) * q^81 - 810 * q^83 - 999*z * q^85 - 315*z * q^87 + (498*z - 498) * q^89 + (104*z - 78) * q^91 + (-300*z + 300) * q^93 + 414*z * q^95 - 1358*z * q^97 - 270 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} - 18 q^{5} + 2 q^{7} - 9 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 - 18 * q^5 + 2 * q^7 - 9 * q^9 $$2 q + 3 q^{3} - 18 q^{5} + 2 q^{7} - 9 q^{9} + 30 q^{11} + 65 q^{13} - 27 q^{15} + 111 q^{17} - 46 q^{19} + 12 q^{21} - 6 q^{23} - 88 q^{25} - 54 q^{27} + 105 q^{29} + 200 q^{31} - 90 q^{33} - 18 q^{35} - 17 q^{37} + 273 q^{39} + 231 q^{41} - 514 q^{43} + 81 q^{45} + 324 q^{47} + 339 q^{49} + 666 q^{51} + 1278 q^{53} - 270 q^{55} - 276 q^{57} + 600 q^{59} - 233 q^{61} + 18 q^{63} - 585 q^{65} + 926 q^{67} + 18 q^{69} - 930 q^{71} - 506 q^{73} - 132 q^{75} + 120 q^{77} + 2648 q^{79} - 81 q^{81} - 1620 q^{83} - 999 q^{85} - 315 q^{87} - 498 q^{89} - 52 q^{91} + 300 q^{93} + 414 q^{95} - 1358 q^{97} - 540 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 - 18 * q^5 + 2 * q^7 - 9 * q^9 + 30 * q^11 + 65 * q^13 - 27 * q^15 + 111 * q^17 - 46 * q^19 + 12 * q^21 - 6 * q^23 - 88 * q^25 - 54 * q^27 + 105 * q^29 + 200 * q^31 - 90 * q^33 - 18 * q^35 - 17 * q^37 + 273 * q^39 + 231 * q^41 - 514 * q^43 + 81 * q^45 + 324 * q^47 + 339 * q^49 + 666 * q^51 + 1278 * q^53 - 270 * q^55 - 276 * q^57 + 600 * q^59 - 233 * q^61 + 18 * q^63 - 585 * q^65 + 926 * q^67 + 18 * q^69 - 930 * q^71 - 506 * q^73 - 132 * q^75 + 120 * q^77 + 2648 * q^79 - 81 * q^81 - 1620 * q^83 - 999 * q^85 - 315 * q^87 - 498 * q^89 - 52 * q^91 + 300 * q^93 + 414 * q^95 - 1358 * q^97 - 540 * 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 1.50000 + 2.59808i 0 −9.00000 0 1.00000 1.73205i 0 −4.50000 + 7.79423i 0
529.1 0 1.50000 2.59808i 0 −9.00000 0 1.00000 + 1.73205i 0 −4.50000 7.79423i 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 624.4.q.b 2
4.b odd 2 1 39.4.e.a 2
12.b even 2 1 117.4.g.b 2
13.c even 3 1 inner 624.4.q.b 2
52.i odd 6 1 507.4.a.a 1
52.j odd 6 1 39.4.e.a 2
52.j odd 6 1 507.4.a.e 1
52.l even 12 2 507.4.b.c 2
156.p even 6 1 117.4.g.b 2
156.p even 6 1 1521.4.a.c 1
156.r even 6 1 1521.4.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.a 2 4.b odd 2 1
39.4.e.a 2 52.j odd 6 1
117.4.g.b 2 12.b even 2 1
117.4.g.b 2 156.p even 6 1
507.4.a.a 1 52.i odd 6 1
507.4.a.e 1 52.j odd 6 1
507.4.b.c 2 52.l even 12 2
624.4.q.b 2 1.a even 1 1 trivial
624.4.q.b 2 13.c even 3 1 inner
1521.4.a.c 1 156.p even 6 1
1521.4.a.j 1 156.r even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(624, [\chi])$$:

 $$T_{5} + 9$$ T5 + 9 $$T_{7}^{2} - 2T_{7} + 4$$ T7^2 - 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 9$$
$5$ $$(T + 9)^{2}$$
$7$ $$T^{2} - 2T + 4$$
$11$ $$T^{2} - 30T + 900$$
$13$ $$T^{2} - 65T + 2197$$
$17$ $$T^{2} - 111T + 12321$$
$19$ $$T^{2} + 46T + 2116$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} - 105T + 11025$$
$31$ $$(T - 100)^{2}$$
$37$ $$T^{2} + 17T + 289$$
$41$ $$T^{2} - 231T + 53361$$
$43$ $$T^{2} + 514T + 264196$$
$47$ $$(T - 162)^{2}$$
$53$ $$(T - 639)^{2}$$
$59$ $$T^{2} - 600T + 360000$$
$61$ $$T^{2} + 233T + 54289$$
$67$ $$T^{2} - 926T + 857476$$
$71$ $$T^{2} + 930T + 864900$$
$73$ $$(T + 253)^{2}$$
$79$ $$(T - 1324)^{2}$$
$83$ $$(T + 810)^{2}$$
$89$ $$T^{2} + 498T + 248004$$
$97$ $$T^{2} + 1358 T + 1844164$$