# Properties

 Label 624.1.cb.a Level $624$ Weight $1$ Character orbit 624.cb Analytic conductor $0.311$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 624.cb (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.311416567883$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 156) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.160398576.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{3} + ( 1 + \zeta_{6} ) q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{3} + ( 1 + \zeta_{6} ) q^{7} + \zeta_{6}^{2} q^{9} -\zeta_{6}^{2} q^{13} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{21} - q^{25} + q^{27} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{31} - q^{39} + \zeta_{6}^{2} q^{43} + ( 1 + \zeta_{6} + \zeta_{6}^{2} ) q^{49} + \zeta_{6}^{2} q^{61} + ( -1 + \zeta_{6}^{2} ) q^{63} + ( -1 + \zeta_{6}^{2} ) q^{67} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{73} + \zeta_{6} q^{75} - q^{79} -\zeta_{6} q^{81} + ( 1 - \zeta_{6}^{2} ) q^{91} + ( -1 + \zeta_{6}^{2} ) q^{93} + ( -1 - \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 3 q^{7} - q^{9} + O(q^{10})$$ $$2 q - q^{3} + 3 q^{7} - q^{9} + q^{13} - 2 q^{25} + 2 q^{27} - 2 q^{39} - q^{43} + 2 q^{49} - q^{61} - 3 q^{63} - 3 q^{67} + q^{75} - 2 q^{79} - q^{81} + 3 q^{91} - 3 q^{93} - 3 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$79$$ $$145$$ $$209$$ $$469$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}^{2}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 + 0.866025i 0 0 0 1.50000 0.866025i 0 −0.500000 0.866025i 0
257.1 0 −0.500000 0.866025i 0 0 0 1.50000 + 0.866025i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.e even 6 1 inner
39.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.1.cb.a 2
3.b odd 2 1 CM 624.1.cb.a 2
4.b odd 2 1 156.1.s.a 2
8.b even 2 1 2496.1.cb.b 2
8.d odd 2 1 2496.1.cb.a 2
12.b even 2 1 156.1.s.a 2
13.e even 6 1 inner 624.1.cb.a 2
20.d odd 2 1 3900.1.ca.b 2
20.e even 4 2 3900.1.br.b 4
24.f even 2 1 2496.1.cb.a 2
24.h odd 2 1 2496.1.cb.b 2
39.h odd 6 1 inner 624.1.cb.a 2
52.b odd 2 1 2028.1.s.a 2
52.f even 4 2 2028.1.o.b 4
52.i odd 6 1 156.1.s.a 2
52.i odd 6 1 2028.1.g.a 2
52.j odd 6 1 2028.1.g.a 2
52.j odd 6 1 2028.1.s.a 2
52.l even 12 2 2028.1.d.c 2
52.l even 12 2 2028.1.o.b 4
60.h even 2 1 3900.1.ca.b 2
60.l odd 4 2 3900.1.br.b 4
104.p odd 6 1 2496.1.cb.a 2
104.s even 6 1 2496.1.cb.b 2
156.h even 2 1 2028.1.s.a 2
156.l odd 4 2 2028.1.o.b 4
156.p even 6 1 2028.1.g.a 2
156.p even 6 1 2028.1.s.a 2
156.r even 6 1 156.1.s.a 2
156.r even 6 1 2028.1.g.a 2
156.v odd 12 2 2028.1.d.c 2
156.v odd 12 2 2028.1.o.b 4
260.w odd 6 1 3900.1.ca.b 2
260.bg even 12 2 3900.1.br.b 4
312.ba even 6 1 2496.1.cb.a 2
312.bg odd 6 1 2496.1.cb.b 2
780.cb even 6 1 3900.1.ca.b 2
780.cw odd 12 2 3900.1.br.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.1.s.a 2 4.b odd 2 1
156.1.s.a 2 12.b even 2 1
156.1.s.a 2 52.i odd 6 1
156.1.s.a 2 156.r even 6 1
624.1.cb.a 2 1.a even 1 1 trivial
624.1.cb.a 2 3.b odd 2 1 CM
624.1.cb.a 2 13.e even 6 1 inner
624.1.cb.a 2 39.h odd 6 1 inner
2028.1.d.c 2 52.l even 12 2
2028.1.d.c 2 156.v odd 12 2
2028.1.g.a 2 52.i odd 6 1
2028.1.g.a 2 52.j odd 6 1
2028.1.g.a 2 156.p even 6 1
2028.1.g.a 2 156.r even 6 1
2028.1.o.b 4 52.f even 4 2
2028.1.o.b 4 52.l even 12 2
2028.1.o.b 4 156.l odd 4 2
2028.1.o.b 4 156.v odd 12 2
2028.1.s.a 2 52.b odd 2 1
2028.1.s.a 2 52.j odd 6 1
2028.1.s.a 2 156.h even 2 1
2028.1.s.a 2 156.p even 6 1
2496.1.cb.a 2 8.d odd 2 1
2496.1.cb.a 2 24.f even 2 1
2496.1.cb.a 2 104.p odd 6 1
2496.1.cb.a 2 312.ba even 6 1
2496.1.cb.b 2 8.b even 2 1
2496.1.cb.b 2 24.h odd 2 1
2496.1.cb.b 2 104.s even 6 1
2496.1.cb.b 2 312.bg odd 6 1
3900.1.br.b 4 20.e even 4 2
3900.1.br.b 4 60.l odd 4 2
3900.1.br.b 4 260.bg even 12 2
3900.1.br.b 4 780.cw odd 12 2
3900.1.ca.b 2 20.d odd 2 1
3900.1.ca.b 2 60.h even 2 1
3900.1.ca.b 2 260.w odd 6 1
3900.1.ca.b 2 780.cb even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(624, [\chi])$$.

## Hecke characteristic polynomials

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