# Properties

 Label 624.2.a.i Level $624$ Weight $2$ Character orbit 624.a Self dual yes Analytic conductor $4.983$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + 2 q^{5} + 4 q^{7} + q^{9}+O(q^{10})$$ q + q^3 + 2 * q^5 + 4 * q^7 + q^9 $$q + q^{3} + 2 q^{5} + 4 q^{7} + q^{9} - 4 q^{11} + q^{13} + 2 q^{15} + 2 q^{17} + 4 q^{21} - q^{25} + q^{27} - 10 q^{29} - 4 q^{31} - 4 q^{33} + 8 q^{35} - 2 q^{37} + q^{39} + 6 q^{41} + 12 q^{43} + 2 q^{45} + 9 q^{49} + 2 q^{51} + 6 q^{53} - 8 q^{55} - 12 q^{59} - 2 q^{61} + 4 q^{63} + 2 q^{65} + 8 q^{67} + 2 q^{73} - q^{75} - 16 q^{77} - 8 q^{79} + q^{81} - 4 q^{83} + 4 q^{85} - 10 q^{87} - 2 q^{89} + 4 q^{91} - 4 q^{93} + 10 q^{97} - 4 q^{99}+O(q^{100})$$ q + q^3 + 2 * q^5 + 4 * q^7 + q^9 - 4 * q^11 + q^13 + 2 * q^15 + 2 * q^17 + 4 * q^21 - q^25 + q^27 - 10 * q^29 - 4 * q^31 - 4 * q^33 + 8 * q^35 - 2 * q^37 + q^39 + 6 * q^41 + 12 * q^43 + 2 * q^45 + 9 * q^49 + 2 * q^51 + 6 * q^53 - 8 * q^55 - 12 * q^59 - 2 * q^61 + 4 * q^63 + 2 * q^65 + 8 * q^67 + 2 * q^73 - q^75 - 16 * q^77 - 8 * q^79 + q^81 - 4 * q^83 + 4 * q^85 - 10 * q^87 - 2 * q^89 + 4 * q^91 - 4 * q^93 + 10 * 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
 0
0 1.00000 0 2.00000 0 4.00000 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.i 1
3.b odd 2 1 1872.2.a.h 1
4.b odd 2 1 39.2.a.a 1
8.b even 2 1 2496.2.a.e 1
8.d odd 2 1 2496.2.a.q 1
12.b even 2 1 117.2.a.a 1
13.b even 2 1 8112.2.a.s 1
20.d odd 2 1 975.2.a.f 1
20.e even 4 2 975.2.c.f 2
24.f even 2 1 7488.2.a.bl 1
24.h odd 2 1 7488.2.a.by 1
28.d even 2 1 1911.2.a.f 1
36.f odd 6 2 1053.2.e.b 2
36.h even 6 2 1053.2.e.d 2
44.c even 2 1 4719.2.a.c 1
52.b odd 2 1 507.2.a.a 1
52.f even 4 2 507.2.b.a 2
52.i odd 6 2 507.2.e.b 2
52.j odd 6 2 507.2.e.a 2
52.l even 12 4 507.2.j.e 4
60.h even 2 1 2925.2.a.p 1
60.l odd 4 2 2925.2.c.e 2
84.h odd 2 1 5733.2.a.e 1
156.h even 2 1 1521.2.a.e 1
156.l odd 4 2 1521.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 4.b odd 2 1
117.2.a.a 1 12.b even 2 1
507.2.a.a 1 52.b odd 2 1
507.2.b.a 2 52.f even 4 2
507.2.e.a 2 52.j odd 6 2
507.2.e.b 2 52.i odd 6 2
507.2.j.e 4 52.l even 12 4
624.2.a.i 1 1.a even 1 1 trivial
975.2.a.f 1 20.d odd 2 1
975.2.c.f 2 20.e even 4 2
1053.2.e.b 2 36.f odd 6 2
1053.2.e.d 2 36.h even 6 2
1521.2.a.e 1 156.h even 2 1
1521.2.b.b 2 156.l odd 4 2
1872.2.a.h 1 3.b odd 2 1
1911.2.a.f 1 28.d even 2 1
2496.2.a.e 1 8.b even 2 1
2496.2.a.q 1 8.d odd 2 1
2925.2.a.p 1 60.h even 2 1
2925.2.c.e 2 60.l odd 4 2
4719.2.a.c 1 44.c even 2 1
5733.2.a.e 1 84.h odd 2 1
7488.2.a.bl 1 24.f even 2 1
7488.2.a.by 1 24.h odd 2 1
8112.2.a.s 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(624))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{7} - 4$$ T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 2$$
$7$ $$T - 4$$
$11$ $$T + 4$$
$13$ $$T - 1$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 10$$
$31$ $$T + 4$$
$37$ $$T + 2$$
$41$ $$T - 6$$
$43$ $$T - 12$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T + 12$$
$61$ $$T + 2$$
$67$ $$T - 8$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T + 8$$
$83$ $$T + 4$$
$89$ $$T + 2$$
$97$ $$T - 10$$