# Properties

 Label 624.2.c.e Level $624$ Weight $2$ Character orbit 624.c 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.98266508613$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

## Character values

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

 $$n$$ $$79$$ $$145$$ $$209$$ $$469$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.00000 0 0 0 3.46410i 0 1.00000 0
337.2 0 1.00000 0 0 0 3.46410i 0 1.00000 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.c.e 2
3.b odd 2 1 1872.2.c.e 2
4.b odd 2 1 39.2.b.a 2
8.b even 2 1 2496.2.c.d 2
8.d odd 2 1 2496.2.c.k 2
12.b even 2 1 117.2.b.a 2
13.b even 2 1 inner 624.2.c.e 2
13.d odd 4 2 8112.2.a.bv 2
20.d odd 2 1 975.2.b.d 2
20.e even 4 2 975.2.h.f 4
28.d even 2 1 1911.2.c.d 2
39.d odd 2 1 1872.2.c.e 2
52.b odd 2 1 39.2.b.a 2
52.f even 4 2 507.2.a.f 2
52.i odd 6 1 507.2.j.a 2
52.i odd 6 1 507.2.j.c 2
52.j odd 6 1 507.2.j.a 2
52.j odd 6 1 507.2.j.c 2
52.l even 12 4 507.2.e.e 4
104.e even 2 1 2496.2.c.d 2
104.h odd 2 1 2496.2.c.k 2
156.h even 2 1 117.2.b.a 2
156.l odd 4 2 1521.2.a.l 2
260.g odd 2 1 975.2.b.d 2
260.p even 4 2 975.2.h.f 4
364.h even 2 1 1911.2.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 4.b odd 2 1
39.2.b.a 2 52.b odd 2 1
117.2.b.a 2 12.b even 2 1
117.2.b.a 2 156.h even 2 1
507.2.a.f 2 52.f even 4 2
507.2.e.e 4 52.l even 12 4
507.2.j.a 2 52.i odd 6 1
507.2.j.a 2 52.j odd 6 1
507.2.j.c 2 52.i odd 6 1
507.2.j.c 2 52.j odd 6 1
624.2.c.e 2 1.a even 1 1 trivial
624.2.c.e 2 13.b even 2 1 inner
975.2.b.d 2 20.d odd 2 1
975.2.b.d 2 260.g odd 2 1
975.2.h.f 4 20.e even 4 2
975.2.h.f 4 260.p even 4 2
1521.2.a.l 2 156.l odd 4 2
1872.2.c.e 2 3.b odd 2 1
1872.2.c.e 2 39.d odd 2 1
1911.2.c.d 2 28.d even 2 1
1911.2.c.d 2 364.h even 2 1
2496.2.c.d 2 8.b even 2 1
2496.2.c.d 2 104.e even 2 1
2496.2.c.k 2 8.d odd 2 1
2496.2.c.k 2 104.h odd 2 1
8112.2.a.bv 2 13.d odd 4 2

## Hecke kernels

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

 $$T_{5}$$ $$T_{7}^{2} + 12$$ $$T_{11}^{2} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$12 + T^{2}$$
$11$ $$12 + T^{2}$$
$13$ $$13 + 2 T + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$12 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$12 + T^{2}$$
$37$ $$48 + T^{2}$$
$41$ $$48 + T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$12 + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$108 + T^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$108 + T^{2}$$
$71$ $$12 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$12 + T^{2}$$
$89$ $$48 + T^{2}$$
$97$ $$192 + T^{2}$$