Properties

 Label 624.2.bv.e Level $624$ Weight $2$ Character orbit 624.bv Analytic conductor $4.983$ Analytic rank $0$ Dimension $4$ 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.bv (of order $$6$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.98266508613$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{2} q^{3} + ( 1 - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + ( -1 - \zeta_{12} - \zeta_{12}^{2} ) q^{7} + ( -1 + \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{2} q^{3} + ( 1 - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + ( -1 - \zeta_{12} - \zeta_{12}^{2} ) q^{7} + ( -1 + \zeta_{12}^{2} ) q^{9} + ( 2 - 3 \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{11} + ( -3 \zeta_{12} - \zeta_{12}^{3} ) q^{13} + ( 2 + 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{15} + ( -4 - \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{17} + ( -1 - 3 \zeta_{12} - \zeta_{12}^{2} ) q^{19} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{21} + ( -3 \zeta_{12} + \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{23} + ( -2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{25} - q^{27} + ( 2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{29} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{31} + ( 1 - 3 \zeta_{12} + \zeta_{12}^{2} ) q^{33} + ( -5 - 3 \zeta_{12} + 5 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{35} + ( -4 + 7 \zeta_{12} + 2 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{37} + ( \zeta_{12} - 4 \zeta_{12}^{3} ) q^{39} + ( 12 - \zeta_{12} - 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{41} + ( 1 + 5 \zeta_{12} - \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{43} + ( 1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{45} + ( 3 - 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{47} + ( 2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{49} + ( -4 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{51} + ( -3 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{53} + ( \zeta_{12} + 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{55} + ( 1 - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{57} + 8 \zeta_{12} q^{59} + ( 4 + 3 \zeta_{12} - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{61} + ( 2 + \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{63} + ( -8 - 5 \zeta_{12} + 6 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{65} + ( 14 + \zeta_{12} - 7 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{67} + ( -1 + 3 \zeta_{12} + \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{69} + ( -1 - 3 \zeta_{12} - \zeta_{12}^{2} ) q^{71} + ( -1 + 2 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{73} + ( -4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{75} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{77} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{79} -\zeta_{12}^{2} q^{81} + ( 3 - 6 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{83} + ( 6 + 11 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{85} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{87} + ( -4 - 6 \zeta_{12} + 2 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{89} + ( -1 + 2 \zeta_{12} + 4 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{91} + ( -4 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{93} + ( -9 - 5 \zeta_{12} + 9 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{95} -6 \zeta_{12} q^{97} + ( -1 + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 6q^{7} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{3} - 6q^{7} - 2q^{9} + 6q^{11} + 6q^{15} - 8q^{17} - 6q^{19} + 2q^{23} - 8q^{25} - 4q^{27} + 2q^{29} + 6q^{33} - 10q^{35} - 12q^{37} + 36q^{41} + 2q^{43} + 6q^{45} - 6q^{49} - 16q^{51} - 12q^{53} + 6q^{55} + 8q^{61} + 6q^{63} - 20q^{65} + 42q^{67} - 2q^{69} - 6q^{71} - 4q^{75} + 24q^{79} - 2q^{81} + 36q^{85} - 2q^{87} - 12q^{89} + 4q^{91} - 12q^{93} - 18q^{95} + O(q^{100})$$

Character values

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

 $$n$$ $$79$$ $$145$$ $$209$$ $$469$$ $$\chi(n)$$ $$1$$ $$1 - \zeta_{12}^{2}$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 0.500000 + 0.866025i 0 3.73205i 0 −2.36603 1.36603i 0 −0.500000 + 0.866025i 0
49.2 0 0.500000 + 0.866025i 0 0.267949i 0 −0.633975 0.366025i 0 −0.500000 + 0.866025i 0
433.1 0 0.500000 0.866025i 0 0.267949i 0 −0.633975 + 0.366025i 0 −0.500000 0.866025i 0
433.2 0 0.500000 0.866025i 0 3.73205i 0 −2.36603 + 1.36603i 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.e even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.bv.e 4
3.b odd 2 1 1872.2.by.h 4
4.b odd 2 1 78.2.i.a 4
12.b even 2 1 234.2.l.c 4
13.e even 6 1 inner 624.2.bv.e 4
13.f odd 12 1 8112.2.a.bj 2
13.f odd 12 1 8112.2.a.bp 2
20.d odd 2 1 1950.2.bc.d 4
20.e even 4 1 1950.2.y.b 4
20.e even 4 1 1950.2.y.g 4
39.h odd 6 1 1872.2.by.h 4
52.b odd 2 1 1014.2.i.a 4
52.f even 4 1 1014.2.e.g 4
52.f even 4 1 1014.2.e.i 4
52.i odd 6 1 78.2.i.a 4
52.i odd 6 1 1014.2.b.e 4
52.j odd 6 1 1014.2.b.e 4
52.j odd 6 1 1014.2.i.a 4
52.l even 12 1 1014.2.a.i 2
52.l even 12 1 1014.2.a.k 2
52.l even 12 1 1014.2.e.g 4
52.l even 12 1 1014.2.e.i 4
156.p even 6 1 3042.2.b.i 4
156.r even 6 1 234.2.l.c 4
156.r even 6 1 3042.2.b.i 4
156.v odd 12 1 3042.2.a.p 2
156.v odd 12 1 3042.2.a.y 2
260.w odd 6 1 1950.2.bc.d 4
260.bg even 12 1 1950.2.y.b 4
260.bg even 12 1 1950.2.y.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.a 4 4.b odd 2 1
78.2.i.a 4 52.i odd 6 1
234.2.l.c 4 12.b even 2 1
234.2.l.c 4 156.r even 6 1
624.2.bv.e 4 1.a even 1 1 trivial
624.2.bv.e 4 13.e even 6 1 inner
1014.2.a.i 2 52.l even 12 1
1014.2.a.k 2 52.l even 12 1
1014.2.b.e 4 52.i odd 6 1
1014.2.b.e 4 52.j odd 6 1
1014.2.e.g 4 52.f even 4 1
1014.2.e.g 4 52.l even 12 1
1014.2.e.i 4 52.f even 4 1
1014.2.e.i 4 52.l even 12 1
1014.2.i.a 4 52.b odd 2 1
1014.2.i.a 4 52.j odd 6 1
1872.2.by.h 4 3.b odd 2 1
1872.2.by.h 4 39.h odd 6 1
1950.2.y.b 4 20.e even 4 1
1950.2.y.b 4 260.bg even 12 1
1950.2.y.g 4 20.e even 4 1
1950.2.y.g 4 260.bg even 12 1
1950.2.bc.d 4 20.d odd 2 1
1950.2.bc.d 4 260.w odd 6 1
3042.2.a.p 2 156.v odd 12 1
3042.2.a.y 2 156.v odd 12 1
3042.2.b.i 4 156.p even 6 1
3042.2.b.i 4 156.r even 6 1
8112.2.a.bj 2 13.f odd 12 1
8112.2.a.bp 2 13.f odd 12 1

Hecke kernels

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

 $$T_{5}^{4} + 14 T_{5}^{2} + 1$$ $$T_{7}^{4} + 6 T_{7}^{3} + 14 T_{7}^{2} + 12 T_{7} + 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$1 + 14 T^{2} + T^{4}$$
$7$ $$4 + 12 T + 14 T^{2} + 6 T^{3} + T^{4}$$
$11$ $$36 + 36 T + 6 T^{2} - 6 T^{3} + T^{4}$$
$13$ $$169 - T^{2} + T^{4}$$
$17$ $$169 + 104 T + 51 T^{2} + 8 T^{3} + T^{4}$$
$19$ $$36 - 36 T + 6 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$676 + 52 T + 30 T^{2} - 2 T^{3} + T^{4}$$
$29$ $$121 + 22 T + 15 T^{2} - 2 T^{3} + T^{4}$$
$31$ $$64 + 32 T^{2} + T^{4}$$
$37$ $$1369 - 444 T + 11 T^{2} + 12 T^{3} + T^{4}$$
$41$ $$11449 - 3852 T + 539 T^{2} - 36 T^{3} + T^{4}$$
$43$ $$5476 + 148 T + 78 T^{2} - 2 T^{3} + T^{4}$$
$47$ $$324 + 72 T^{2} + T^{4}$$
$53$ $$( -3 + 6 T + T^{2} )^{2}$$
$59$ $$4096 - 64 T^{2} + T^{4}$$
$61$ $$121 + 88 T + 75 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$21316 - 6132 T + 734 T^{2} - 42 T^{3} + T^{4}$$
$71$ $$36 - 36 T + 6 T^{2} + 6 T^{3} + T^{4}$$
$73$ $$3721 + 134 T^{2} + T^{4}$$
$79$ $$( 24 - 12 T + T^{2} )^{2}$$
$83$ $$4 + 104 T^{2} + T^{4}$$
$89$ $$576 - 288 T + 24 T^{2} + 12 T^{3} + T^{4}$$
$97$ $$1296 - 36 T^{2} + T^{4}$$