# 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: $$x^{2} - x + 1$$ 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 $$\beta = 2\sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta q^{7} + q^{9}+O(q^{10})$$ q + q^3 + b * q^7 + q^9 $$q + q^{3} + \beta q^{7} + q^{9} + \beta q^{11} + (\beta - 1) q^{13} - 6 q^{17} - \beta q^{19} + \beta q^{21} + 5 q^{25} + q^{27} + 6 q^{29} + \beta q^{31} + \beta q^{33} + 2 \beta q^{37} + (\beta - 1) q^{39} - 2 \beta q^{41} + 4 q^{43} - \beta q^{47} - 5 q^{49} - 6 q^{51} + 6 q^{53} - \beta q^{57} - 3 \beta q^{59} - 2 q^{61} + \beta q^{63} + 3 \beta q^{67} - \beta q^{71} + 5 q^{75} - 12 q^{77} + 8 q^{79} + q^{81} + \beta q^{83} + 6 q^{87} - 2 \beta q^{89} + ( - \beta - 12) q^{91} + \beta q^{93} - 4 \beta q^{97} + \beta q^{99} +O(q^{100})$$ q + q^3 + b * q^7 + q^9 + b * q^11 + (b - 1) * q^13 - 6 * q^17 - b * q^19 + b * q^21 + 5 * q^25 + q^27 + 6 * q^29 + b * q^31 + b * q^33 + 2*b * q^37 + (b - 1) * q^39 - 2*b * q^41 + 4 * q^43 - b * q^47 - 5 * q^49 - 6 * q^51 + 6 * q^53 - b * q^57 - 3*b * q^59 - 2 * q^61 + b * q^63 + 3*b * q^67 - b * q^71 + 5 * q^75 - 12 * q^77 + 8 * q^79 + q^81 + b * q^83 + 6 * q^87 - 2*b * q^89 + (-b - 12) * q^91 + b * q^93 - 4*b * q^97 + b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{9} - 2 q^{13} - 12 q^{17} + 10 q^{25} + 2 q^{27} + 12 q^{29} - 2 q^{39} + 8 q^{43} - 10 q^{49} - 12 q^{51} + 12 q^{53} - 4 q^{61} + 10 q^{75} - 24 q^{77} + 16 q^{79} + 2 q^{81} + 12 q^{87} - 24 q^{91}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^9 - 2 * q^13 - 12 * q^17 + 10 * q^25 + 2 * q^27 + 12 * q^29 - 2 * q^39 + 8 * q^43 - 10 * q^49 - 12 * q^51 + 12 * q^53 - 4 * q^61 + 10 * q^75 - 24 * q^77 + 16 * q^79 + 2 * q^81 + 12 * q^87 - 24 * q^91

## 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 $$S_{2}^{\mathrm{new}}(624, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}^{2} + 12$$ T7^2 + 12 $$T_{11}^{2} + 12$$ T11^2 + 12

## Hecke characteristic polynomials

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