# Properties

 Label 624.4.a.a 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 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 16 q^{5} + 8 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 16 * q^5 + 8 * q^7 + 9 * q^9 $$q - 3 q^{3} - 16 q^{5} + 8 q^{7} + 9 q^{9} + 38 q^{11} - 13 q^{13} + 48 q^{15} - 78 q^{17} + 72 q^{19} - 24 q^{21} + 52 q^{23} + 131 q^{25} - 27 q^{27} + 242 q^{29} - 76 q^{31} - 114 q^{33} - 128 q^{35} + 342 q^{37} + 39 q^{39} - 336 q^{41} - 76 q^{43} - 144 q^{45} - 94 q^{47} - 279 q^{49} + 234 q^{51} - 450 q^{53} - 608 q^{55} - 216 q^{57} - 854 q^{59} - 110 q^{61} + 72 q^{63} + 208 q^{65} + 908 q^{67} - 156 q^{69} - 838 q^{71} - 970 q^{73} - 393 q^{75} + 304 q^{77} + 352 q^{79} + 81 q^{81} - 474 q^{83} + 1248 q^{85} - 726 q^{87} - 1452 q^{89} - 104 q^{91} + 228 q^{93} - 1152 q^{95} - 562 q^{97} + 342 q^{99}+O(q^{100})$$ q - 3 * q^3 - 16 * q^5 + 8 * q^7 + 9 * q^9 + 38 * q^11 - 13 * q^13 + 48 * q^15 - 78 * q^17 + 72 * q^19 - 24 * q^21 + 52 * q^23 + 131 * q^25 - 27 * q^27 + 242 * q^29 - 76 * q^31 - 114 * q^33 - 128 * q^35 + 342 * q^37 + 39 * q^39 - 336 * q^41 - 76 * q^43 - 144 * q^45 - 94 * q^47 - 279 * q^49 + 234 * q^51 - 450 * q^53 - 608 * q^55 - 216 * q^57 - 854 * q^59 - 110 * q^61 + 72 * q^63 + 208 * q^65 + 908 * q^67 - 156 * q^69 - 838 * q^71 - 970 * q^73 - 393 * q^75 + 304 * q^77 + 352 * q^79 + 81 * q^81 - 474 * q^83 + 1248 * q^85 - 726 * q^87 - 1452 * q^89 - 104 * q^91 + 228 * q^93 - 1152 * q^95 - 562 * q^97 + 342 * 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 −16.0000 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.a 1
3.b odd 2 1 1872.4.a.p 1
4.b odd 2 1 78.4.a.b 1
8.b even 2 1 2496.4.a.p 1
8.d odd 2 1 2496.4.a.h 1
12.b even 2 1 234.4.a.j 1
20.d odd 2 1 1950.4.a.k 1
52.b odd 2 1 1014.4.a.k 1
52.f even 4 2 1014.4.b.f 2

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

 $$T_{5} + 16$$ T5 + 16 $$T_{7} - 8$$ T7 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 16$$
$7$ $$T - 8$$
$11$ $$T - 38$$
$13$ $$T + 13$$
$17$ $$T + 78$$
$19$ $$T - 72$$
$23$ $$T - 52$$
$29$ $$T - 242$$
$31$ $$T + 76$$
$37$ $$T - 342$$
$41$ $$T + 336$$
$43$ $$T + 76$$
$47$ $$T + 94$$
$53$ $$T + 450$$
$59$ $$T + 854$$
$61$ $$T + 110$$
$67$ $$T - 908$$
$71$ $$T + 838$$
$73$ $$T + 970$$
$79$ $$T - 352$$
$83$ $$T + 474$$
$89$ $$T + 1452$$
$97$ $$T + 562$$