# Properties

 Label 624.2.a.k Level $624$ Weight $2$ Character orbit 624.a Self dual yes Analytic conductor $4.983$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 624.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.98266508613$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta q^{5} + \beta q^{7} + q^{9}+O(q^{10})$$ q - q^3 + b * q^5 + b * q^7 + q^9 $$q - q^{3} + \beta q^{5} + \beta q^{7} + q^{9} + 2 q^{11} - q^{13} - \beta q^{15} + ( - 2 \beta + 2) q^{17} - \beta q^{19} - \beta q^{21} + 4 q^{23} + 3 q^{25} - q^{27} + 2 q^{29} + (\beta + 4) q^{31} - 2 q^{33} + 8 q^{35} + (2 \beta - 2) q^{37} + q^{39} + (\beta + 8) q^{41} + ( - 2 \beta - 4) q^{43} + \beta q^{45} + ( - 2 \beta + 6) q^{47} + q^{49} + (2 \beta - 2) q^{51} - 2 q^{53} + 2 \beta q^{55} + \beta q^{57} + (2 \beta - 2) q^{59} + ( - 4 \beta + 2) q^{61} + \beta q^{63} - \beta q^{65} + (\beta - 4) q^{67} - 4 q^{69} - 2 q^{71} + (2 \beta + 6) q^{73} - 3 q^{75} + 2 \beta q^{77} - 4 \beta q^{79} + q^{81} + (2 \beta + 2) q^{83} + (2 \beta - 16) q^{85} - 2 q^{87} + ( - \beta + 12) q^{89} - \beta q^{91} + ( - \beta - 4) q^{93} - 8 q^{95} + ( - 2 \beta - 2) q^{97} + 2 q^{99} +O(q^{100})$$ q - q^3 + b * q^5 + b * q^7 + q^9 + 2 * q^11 - q^13 - b * q^15 + (-2*b + 2) * q^17 - b * q^19 - b * q^21 + 4 * q^23 + 3 * q^25 - q^27 + 2 * q^29 + (b + 4) * q^31 - 2 * q^33 + 8 * q^35 + (2*b - 2) * q^37 + q^39 + (b + 8) * q^41 + (-2*b - 4) * q^43 + b * q^45 + (-2*b + 6) * q^47 + q^49 + (2*b - 2) * q^51 - 2 * q^53 + 2*b * q^55 + b * q^57 + (2*b - 2) * q^59 + (-4*b + 2) * q^61 + b * q^63 - b * q^65 + (b - 4) * q^67 - 4 * q^69 - 2 * q^71 + (2*b + 6) * q^73 - 3 * q^75 + 2*b * q^77 - 4*b * q^79 + q^81 + (2*b + 2) * q^83 + (2*b - 16) * q^85 - 2 * q^87 + (-b + 12) * q^89 - b * q^91 + (-b - 4) * q^93 - 8 * q^95 + (-2*b - 2) * q^97 + 2 * 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 $$2 q - 2 q^{3} + 2 q^{9} + 4 q^{11} - 2 q^{13} + 4 q^{17} + 8 q^{23} + 6 q^{25} - 2 q^{27} + 4 q^{29} + 8 q^{31} - 4 q^{33} + 16 q^{35} - 4 q^{37} + 2 q^{39} + 16 q^{41} - 8 q^{43} + 12 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} - 4 q^{59} + 4 q^{61} - 8 q^{67} - 8 q^{69} - 4 q^{71} + 12 q^{73} - 6 q^{75} + 2 q^{81} + 4 q^{83} - 32 q^{85} - 4 q^{87} + 24 q^{89} - 8 q^{93} - 16 q^{95} - 4 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^11 - 2 * q^13 + 4 * q^17 + 8 * q^23 + 6 * q^25 - 2 * q^27 + 4 * q^29 + 8 * q^31 - 4 * q^33 + 16 * q^35 - 4 * q^37 + 2 * q^39 + 16 * q^41 - 8 * q^43 + 12 * q^47 + 2 * q^49 - 4 * q^51 - 4 * q^53 - 4 * q^59 + 4 * q^61 - 8 * q^67 - 8 * q^69 - 4 * q^71 + 12 * q^73 - 6 * q^75 + 2 * q^81 + 4 * q^83 - 32 * q^85 - 4 * q^87 + 24 * q^89 - 8 * q^93 - 16 * q^95 - 4 * q^97 + 4 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 −1.00000 0 −2.82843 0 −2.82843 0 1.00000 0
1.2 0 −1.00000 0 2.82843 0 2.82843 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$13$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.a.k 2
3.b odd 2 1 1872.2.a.w 2
4.b odd 2 1 39.2.a.b 2
8.b even 2 1 2496.2.a.bi 2
8.d odd 2 1 2496.2.a.bf 2
12.b even 2 1 117.2.a.c 2
13.b even 2 1 8112.2.a.bm 2
20.d odd 2 1 975.2.a.l 2
20.e even 4 2 975.2.c.h 4
24.f even 2 1 7488.2.a.cl 2
24.h odd 2 1 7488.2.a.co 2
28.d even 2 1 1911.2.a.h 2
36.f odd 6 2 1053.2.e.m 4
36.h even 6 2 1053.2.e.e 4
44.c even 2 1 4719.2.a.p 2
52.b odd 2 1 507.2.a.h 2
52.f even 4 2 507.2.b.e 4
52.i odd 6 2 507.2.e.d 4
52.j odd 6 2 507.2.e.h 4
52.l even 12 4 507.2.j.f 8
60.h even 2 1 2925.2.a.v 2
60.l odd 4 2 2925.2.c.u 4
84.h odd 2 1 5733.2.a.u 2
156.h even 2 1 1521.2.a.f 2
156.l odd 4 2 1521.2.b.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 4.b odd 2 1
117.2.a.c 2 12.b even 2 1
507.2.a.h 2 52.b odd 2 1
507.2.b.e 4 52.f even 4 2
507.2.e.d 4 52.i odd 6 2
507.2.e.h 4 52.j odd 6 2
507.2.j.f 8 52.l even 12 4
624.2.a.k 2 1.a even 1 1 trivial
975.2.a.l 2 20.d odd 2 1
975.2.c.h 4 20.e even 4 2
1053.2.e.e 4 36.h even 6 2
1053.2.e.m 4 36.f odd 6 2
1521.2.a.f 2 156.h even 2 1
1521.2.b.j 4 156.l odd 4 2
1872.2.a.w 2 3.b odd 2 1
1911.2.a.h 2 28.d even 2 1
2496.2.a.bf 2 8.d odd 2 1
2496.2.a.bi 2 8.b even 2 1
2925.2.a.v 2 60.h even 2 1
2925.2.c.u 4 60.l odd 4 2
4719.2.a.p 2 44.c even 2 1
5733.2.a.u 2 84.h odd 2 1
7488.2.a.cl 2 24.f even 2 1
7488.2.a.co 2 24.h odd 2 1
8112.2.a.bm 2 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(624))$$:

 $$T_{5}^{2} - 8$$ T5^2 - 8 $$T_{7}^{2} - 8$$ T7^2 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} - 8$$
$7$ $$T^{2} - 8$$
$11$ $$(T - 2)^{2}$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 4T - 28$$
$19$ $$T^{2} - 8$$
$23$ $$(T - 4)^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} - 8T + 8$$
$37$ $$T^{2} + 4T - 28$$
$41$ $$T^{2} - 16T + 56$$
$43$ $$T^{2} + 8T - 16$$
$47$ $$T^{2} - 12T + 4$$
$53$ $$(T + 2)^{2}$$
$59$ $$T^{2} + 4T - 28$$
$61$ $$T^{2} - 4T - 124$$
$67$ $$T^{2} + 8T + 8$$
$71$ $$(T + 2)^{2}$$
$73$ $$T^{2} - 12T + 4$$
$79$ $$T^{2} - 128$$
$83$ $$T^{2} - 4T - 28$$
$89$ $$T^{2} - 24T + 136$$
$97$ $$T^{2} + 4T - 28$$