# Properties

 Label 624.4.a.e 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 20 q^{5} + 32 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 20 * q^5 + 32 * q^7 + 9 * q^9 $$q + 3 q^{3} - 20 q^{5} + 32 q^{7} + 9 q^{9} - 50 q^{11} - 13 q^{13} - 60 q^{15} - 30 q^{17} + 120 q^{19} + 96 q^{21} + 20 q^{23} + 275 q^{25} + 27 q^{27} + 82 q^{29} + 44 q^{31} - 150 q^{33} - 640 q^{35} - 306 q^{37} - 39 q^{39} + 108 q^{41} + 356 q^{43} - 180 q^{45} + 178 q^{47} + 681 q^{49} - 90 q^{51} + 198 q^{53} + 1000 q^{55} + 360 q^{57} - 94 q^{59} - 62 q^{61} + 288 q^{63} + 260 q^{65} + 140 q^{67} + 60 q^{69} + 778 q^{71} + 62 q^{73} + 825 q^{75} - 1600 q^{77} + 1096 q^{79} + 81 q^{81} + 462 q^{83} + 600 q^{85} + 246 q^{87} + 1224 q^{89} - 416 q^{91} + 132 q^{93} - 2400 q^{95} + 614 q^{97} - 450 q^{99}+O(q^{100})$$ q + 3 * q^3 - 20 * q^5 + 32 * q^7 + 9 * q^9 - 50 * q^11 - 13 * q^13 - 60 * q^15 - 30 * q^17 + 120 * q^19 + 96 * q^21 + 20 * q^23 + 275 * q^25 + 27 * q^27 + 82 * q^29 + 44 * q^31 - 150 * q^33 - 640 * q^35 - 306 * q^37 - 39 * q^39 + 108 * q^41 + 356 * q^43 - 180 * q^45 + 178 * q^47 + 681 * q^49 - 90 * q^51 + 198 * q^53 + 1000 * q^55 + 360 * q^57 - 94 * q^59 - 62 * q^61 + 288 * q^63 + 260 * q^65 + 140 * q^67 + 60 * q^69 + 778 * q^71 + 62 * q^73 + 825 * q^75 - 1600 * q^77 + 1096 * q^79 + 81 * q^81 + 462 * q^83 + 600 * q^85 + 246 * q^87 + 1224 * q^89 - 416 * q^91 + 132 * q^93 - 2400 * q^95 + 614 * q^97 - 450 * 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 −20.0000 0 32.0000 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.e 1
3.b odd 2 1 1872.4.a.r 1
4.b odd 2 1 78.4.a.d 1
8.b even 2 1 2496.4.a.i 1
8.d odd 2 1 2496.4.a.r 1
12.b even 2 1 234.4.a.f 1
20.d odd 2 1 1950.4.a.h 1
52.b odd 2 1 1014.4.a.d 1
52.f even 4 2 1014.4.b.e 2

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

 $$T_{5} + 20$$ T5 + 20 $$T_{7} - 32$$ T7 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 20$$
$7$ $$T - 32$$
$11$ $$T + 50$$
$13$ $$T + 13$$
$17$ $$T + 30$$
$19$ $$T - 120$$
$23$ $$T - 20$$
$29$ $$T - 82$$
$31$ $$T - 44$$
$37$ $$T + 306$$
$41$ $$T - 108$$
$43$ $$T - 356$$
$47$ $$T - 178$$
$53$ $$T - 198$$
$59$ $$T + 94$$
$61$ $$T + 62$$
$67$ $$T - 140$$
$71$ $$T - 778$$
$73$ $$T - 62$$
$79$ $$T - 1096$$
$83$ $$T - 462$$
$89$ $$T - 1224$$
$97$ $$T - 614$$