# Properties

 Label 624.4.a.d Level $624$ Weight $4$ Character orbit 624.a Self dual yes Analytic conductor $36.817$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 10 q^{5} + 8 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + 10 * q^5 + 8 * q^7 + 9 * q^9 $$q - 3 q^{3} + 10 q^{5} + 8 q^{7} + 9 q^{9} - 40 q^{11} + 13 q^{13} - 30 q^{15} + 130 q^{17} + 20 q^{19} - 24 q^{21} - 25 q^{25} - 27 q^{27} - 18 q^{29} + 184 q^{31} + 120 q^{33} + 80 q^{35} - 74 q^{37} - 39 q^{39} - 362 q^{41} - 76 q^{43} + 90 q^{45} + 452 q^{47} - 279 q^{49} - 390 q^{51} + 382 q^{53} - 400 q^{55} - 60 q^{57} - 464 q^{59} + 358 q^{61} + 72 q^{63} + 130 q^{65} + 700 q^{67} + 748 q^{71} + 1058 q^{73} + 75 q^{75} - 320 q^{77} + 976 q^{79} + 81 q^{81} + 1008 q^{83} + 1300 q^{85} + 54 q^{87} - 386 q^{89} + 104 q^{91} - 552 q^{93} + 200 q^{95} - 614 q^{97} - 360 q^{99}+O(q^{100})$$ q - 3 * q^3 + 10 * q^5 + 8 * q^7 + 9 * q^9 - 40 * q^11 + 13 * q^13 - 30 * q^15 + 130 * q^17 + 20 * q^19 - 24 * q^21 - 25 * q^25 - 27 * q^27 - 18 * q^29 + 184 * q^31 + 120 * q^33 + 80 * q^35 - 74 * q^37 - 39 * q^39 - 362 * q^41 - 76 * q^43 + 90 * q^45 + 452 * q^47 - 279 * q^49 - 390 * q^51 + 382 * q^53 - 400 * q^55 - 60 * q^57 - 464 * q^59 + 358 * q^61 + 72 * q^63 + 130 * q^65 + 700 * q^67 + 748 * q^71 + 1058 * q^73 + 75 * q^75 - 320 * q^77 + 976 * q^79 + 81 * q^81 + 1008 * q^83 + 1300 * q^85 + 54 * q^87 - 386 * q^89 + 104 * q^91 - 552 * q^93 + 200 * q^95 - 614 * q^97 - 360 * 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 10.0000 0 8.00000 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.d 1
3.b odd 2 1 1872.4.a.d 1
4.b odd 2 1 78.4.a.c 1
8.b even 2 1 2496.4.a.j 1
8.d odd 2 1 2496.4.a.a 1
12.b even 2 1 234.4.a.h 1
20.d odd 2 1 1950.4.a.l 1
52.b odd 2 1 1014.4.a.j 1
52.f even 4 2 1014.4.b.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.c 1 4.b odd 2 1
234.4.a.h 1 12.b even 2 1
624.4.a.d 1 1.a even 1 1 trivial
1014.4.a.j 1 52.b odd 2 1
1014.4.b.h 2 52.f even 4 2
1872.4.a.d 1 3.b odd 2 1
1950.4.a.l 1 20.d odd 2 1
2496.4.a.a 1 8.d odd 2 1
2496.4.a.j 1 8.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(624))$$:

 $$T_{5} - 10$$ T5 - 10 $$T_{7} - 8$$ T7 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T - 10$$
$7$ $$T - 8$$
$11$ $$T + 40$$
$13$ $$T - 13$$
$17$ $$T - 130$$
$19$ $$T - 20$$
$23$ $$T$$
$29$ $$T + 18$$
$31$ $$T - 184$$
$37$ $$T + 74$$
$41$ $$T + 362$$
$43$ $$T + 76$$
$47$ $$T - 452$$
$53$ $$T - 382$$
$59$ $$T + 464$$
$61$ $$T - 358$$
$67$ $$T - 700$$
$71$ $$T - 748$$
$73$ $$T - 1058$$
$79$ $$T - 976$$
$83$ $$T - 1008$$
$89$ $$T + 386$$
$97$ $$T + 614$$