# Properties

 Label 624.2.q.h Level $624$ Weight $2$ Character orbit 624.q Analytic conductor $4.983$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} ) q^{3} + ( -1 + \beta_{3} ) q^{5} + ( \beta_{1} - 2 \beta_{2} ) q^{7} -\beta_{2} q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{2} ) q^{3} + ( -1 + \beta_{3} ) q^{5} + ( \beta_{1} - 2 \beta_{2} ) q^{7} -\beta_{2} q^{9} + ( -2 + 2 \beta_{2} ) q^{11} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{15} + ( \beta_{1} - \beta_{2} ) q^{17} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{19} + ( -2 - \beta_{3} ) q^{21} + ( 2 - 2 \beta_{2} ) q^{23} -3 \beta_{3} q^{25} - q^{27} + ( 1 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{29} + \beta_{3} q^{31} + 2 \beta_{2} q^{33} -2 \beta_{2} q^{35} + ( -5 + \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{37} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{39} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{41} + ( \beta_{1} + 2 \beta_{2} ) q^{43} + ( \beta_{1} + \beta_{2} ) q^{45} + ( 2 + 4 \beta_{3} ) q^{47} + ( -1 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{49} + ( -1 - \beta_{3} ) q^{51} + ( 7 + 3 \beta_{3} ) q^{53} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{55} + ( 2 - 2 \beta_{3} ) q^{57} + ( 2 \beta_{1} - 8 \beta_{2} ) q^{59} + ( 2 \beta_{1} - 9 \beta_{2} ) q^{61} + ( -2 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{63} + ( 3 - 3 \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{65} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{67} -2 \beta_{2} q^{69} + 14 \beta_{2} q^{71} + ( -5 + 2 \beta_{3} ) q^{73} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{75} + ( 4 + 2 \beta_{3} ) q^{77} + ( -8 - \beta_{3} ) q^{79} + ( -1 + \beta_{2} ) q^{81} + ( 6 + 2 \beta_{3} ) q^{83} + ( -\beta_{1} - 3 \beta_{2} ) q^{85} + ( 3 \beta_{1} - \beta_{2} ) q^{87} + ( -10 - 2 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -10 - \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{91} + ( \beta_{1} + \beta_{3} ) q^{93} + ( -6 \beta_{1} - 10 \beta_{2} ) q^{95} + ( \beta_{1} + 6 \beta_{2} ) q^{97} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 6q^{5} - 3q^{7} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{3} - 6q^{5} - 3q^{7} - 2q^{9} - 4q^{11} + 2q^{13} - 3q^{15} - q^{17} + 6q^{19} - 6q^{21} + 4q^{23} + 6q^{25} - 4q^{27} - q^{29} - 2q^{31} + 4q^{33} - 4q^{35} - 11q^{37} + q^{39} - q^{41} + 5q^{43} + 3q^{45} + q^{49} - 2q^{51} + 22q^{53} + 6q^{55} + 12q^{57} - 14q^{59} - 16q^{61} - 3q^{63} - 3q^{65} + 5q^{67} - 4q^{69} + 28q^{71} - 24q^{73} + 3q^{75} + 12q^{77} - 30q^{79} - 2q^{81} + 20q^{83} - 7q^{85} + q^{87} - 18q^{89} - 27q^{91} - q^{93} - 26q^{95} + 13q^{97} + 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 5 \nu^{2} - 5 \nu + 16$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4 \beta_{2} + \beta_{1} - 4$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3} - 4$$

## Character values

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

 $$n$$ $$79$$ $$145$$ $$209$$ $$469$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$ $$1$$ $$1$$

## 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
 1.28078 − 2.21837i −0.780776 + 1.35234i 1.28078 + 2.21837i −0.780776 − 1.35234i
0 0.500000 + 0.866025i 0 −3.56155 0 0.280776 0.486319i 0 −0.500000 + 0.866025i 0
289.2 0 0.500000 + 0.866025i 0 0.561553 0 −1.78078 + 3.08440i 0 −0.500000 + 0.866025i 0
529.1 0 0.500000 0.866025i 0 −3.56155 0 0.280776 + 0.486319i 0 −0.500000 0.866025i 0
529.2 0 0.500000 0.866025i 0 0.561553 0 −1.78078 3.08440i 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.h 4
3.b odd 2 1 1872.2.t.r 4
4.b odd 2 1 39.2.e.b 4
12.b even 2 1 117.2.g.c 4
13.c even 3 1 inner 624.2.q.h 4
13.c even 3 1 8112.2.a.bk 2
13.e even 6 1 8112.2.a.bo 2
20.d odd 2 1 975.2.i.k 4
20.e even 4 2 975.2.bb.i 8
39.i odd 6 1 1872.2.t.r 4
52.b odd 2 1 507.2.e.g 4
52.f even 4 2 507.2.j.g 8
52.i odd 6 1 507.2.a.d 2
52.i odd 6 1 507.2.e.g 4
52.j odd 6 1 39.2.e.b 4
52.j odd 6 1 507.2.a.g 2
52.l even 12 2 507.2.b.d 4
52.l even 12 2 507.2.j.g 8
156.p even 6 1 117.2.g.c 4
156.p even 6 1 1521.2.a.g 2
156.r even 6 1 1521.2.a.m 2
156.v odd 12 2 1521.2.b.h 4
260.v odd 6 1 975.2.i.k 4
260.bj even 12 2 975.2.bb.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 4.b odd 2 1
39.2.e.b 4 52.j odd 6 1
117.2.g.c 4 12.b even 2 1
117.2.g.c 4 156.p even 6 1
507.2.a.d 2 52.i odd 6 1
507.2.a.g 2 52.j odd 6 1
507.2.b.d 4 52.l even 12 2
507.2.e.g 4 52.b odd 2 1
507.2.e.g 4 52.i odd 6 1
507.2.j.g 8 52.f even 4 2
507.2.j.g 8 52.l even 12 2
624.2.q.h 4 1.a even 1 1 trivial
624.2.q.h 4 13.c even 3 1 inner
975.2.i.k 4 20.d odd 2 1
975.2.i.k 4 260.v odd 6 1
975.2.bb.i 8 20.e even 4 2
975.2.bb.i 8 260.bj even 12 2
1521.2.a.g 2 156.p even 6 1
1521.2.a.m 2 156.r even 6 1
1521.2.b.h 4 156.v odd 12 2
1872.2.t.r 4 3.b odd 2 1
1872.2.t.r 4 39.i odd 6 1
8112.2.a.bk 2 13.c even 3 1
8112.2.a.bo 2 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 $$S_{2}^{\mathrm{new}}(624, [\chi])$$:

 $$T_{5}^{2} + 3 T_{5} - 2$$ $$T_{7}^{4} + 3 T_{7}^{3} + 11 T_{7}^{2} - 6 T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$( -2 + 3 T + T^{2} )^{2}$$
$7$ $$4 - 6 T + 11 T^{2} + 3 T^{3} + T^{4}$$
$11$ $$( 4 + 2 T + T^{2} )^{2}$$
$13$ $$( 13 - T + T^{2} )^{2}$$
$17$ $$16 - 4 T + 5 T^{2} + T^{3} + T^{4}$$
$19$ $$64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$( 4 - 2 T + T^{2} )^{2}$$
$29$ $$1444 - 38 T + 39 T^{2} + T^{3} + T^{4}$$
$31$ $$( -4 + T + T^{2} )^{2}$$
$37$ $$676 + 286 T + 95 T^{2} + 11 T^{3} + T^{4}$$
$41$ $$16 - 4 T + 5 T^{2} + T^{3} + T^{4}$$
$43$ $$4 - 10 T + 23 T^{2} - 5 T^{3} + T^{4}$$
$47$ $$( -68 + T^{2} )^{2}$$
$53$ $$( -8 - 11 T + T^{2} )^{2}$$
$59$ $$1024 + 448 T + 164 T^{2} + 14 T^{3} + T^{4}$$
$61$ $$2209 + 752 T + 209 T^{2} + 16 T^{3} + T^{4}$$
$67$ $$4 - 10 T + 23 T^{2} - 5 T^{3} + T^{4}$$
$71$ $$( 196 - 14 T + T^{2} )^{2}$$
$73$ $$( 19 + 12 T + T^{2} )^{2}$$
$79$ $$( 52 + 15 T + T^{2} )^{2}$$
$83$ $$( 8 - 10 T + T^{2} )^{2}$$
$89$ $$4096 + 1152 T + 260 T^{2} + 18 T^{3} + T^{4}$$
$97$ $$1444 - 494 T + 131 T^{2} - 13 T^{3} + T^{4}$$
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