# Properties

 Label 624.4.a.h Level $624$ Weight $4$ Character orbit 624.a Self dual yes Analytic conductor $36.817$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(1,624)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("624.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$36.8171918436$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 156) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 6 q^{5} + 4 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 6 * q^5 + 4 * q^7 + 9 * q^9 $$q + 3 q^{3} - 6 q^{5} + 4 q^{7} + 9 q^{9} - 36 q^{11} + 13 q^{13} - 18 q^{15} + 66 q^{17} - 56 q^{19} + 12 q^{21} - 96 q^{23} - 89 q^{25} + 27 q^{27} + 222 q^{29} - 260 q^{31} - 108 q^{33} - 24 q^{35} - 106 q^{37} + 39 q^{39} - 90 q^{41} - 44 q^{43} - 54 q^{45} - 168 q^{47} - 327 q^{49} + 198 q^{51} + 30 q^{53} + 216 q^{55} - 168 q^{57} - 348 q^{59} - 346 q^{61} + 36 q^{63} - 78 q^{65} + 256 q^{67} - 288 q^{69} + 168 q^{71} - 814 q^{73} - 267 q^{75} - 144 q^{77} - 200 q^{79} + 81 q^{81} - 1236 q^{83} - 396 q^{85} + 666 q^{87} + 318 q^{89} + 52 q^{91} - 780 q^{93} + 336 q^{95} - 502 q^{97} - 324 q^{99}+O(q^{100})$$ q + 3 * q^3 - 6 * q^5 + 4 * q^7 + 9 * q^9 - 36 * q^11 + 13 * q^13 - 18 * q^15 + 66 * q^17 - 56 * q^19 + 12 * q^21 - 96 * q^23 - 89 * q^25 + 27 * q^27 + 222 * q^29 - 260 * q^31 - 108 * q^33 - 24 * q^35 - 106 * q^37 + 39 * q^39 - 90 * q^41 - 44 * q^43 - 54 * q^45 - 168 * q^47 - 327 * q^49 + 198 * q^51 + 30 * q^53 + 216 * q^55 - 168 * q^57 - 348 * q^59 - 346 * q^61 + 36 * q^63 - 78 * q^65 + 256 * q^67 - 288 * q^69 + 168 * q^71 - 814 * q^73 - 267 * q^75 - 144 * q^77 - 200 * q^79 + 81 * q^81 - 1236 * q^83 - 396 * q^85 + 666 * q^87 + 318 * q^89 + 52 * q^91 - 780 * q^93 + 336 * q^95 - 502 * q^97 - 324 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 3.00000 0 −6.00000 0 4.00000 0 9.00000 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 624.4.a.h 1
3.b odd 2 1 1872.4.a.j 1
4.b odd 2 1 156.4.a.a 1
8.b even 2 1 2496.4.a.e 1
8.d odd 2 1 2496.4.a.n 1
12.b even 2 1 468.4.a.b 1
52.b odd 2 1 2028.4.a.a 1
52.f even 4 2 2028.4.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.4.a.a 1 4.b odd 2 1
468.4.a.b 1 12.b even 2 1
624.4.a.h 1 1.a even 1 1 trivial
1872.4.a.j 1 3.b odd 2 1
2028.4.a.a 1 52.b odd 2 1
2028.4.b.a 2 52.f even 4 2
2496.4.a.e 1 8.b even 2 1
2496.4.a.n 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(624))$$:

 $$T_{5} + 6$$ T5 + 6 $$T_{7} - 4$$ T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 6$$
$7$ $$T - 4$$
$11$ $$T + 36$$
$13$ $$T - 13$$
$17$ $$T - 66$$
$19$ $$T + 56$$
$23$ $$T + 96$$
$29$ $$T - 222$$
$31$ $$T + 260$$
$37$ $$T + 106$$
$41$ $$T + 90$$
$43$ $$T + 44$$
$47$ $$T + 168$$
$53$ $$T - 30$$
$59$ $$T + 348$$
$61$ $$T + 346$$
$67$ $$T - 256$$
$71$ $$T - 168$$
$73$ $$T + 814$$
$79$ $$T + 200$$
$83$ $$T + 1236$$
$89$ $$T - 318$$
$97$ $$T + 502$$