# Properties

 Label 624.4.a.g 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 39) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 12 q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 12 * q^5 - 2 * q^7 + 9 * q^9 $$q + 3 q^{3} - 12 q^{5} - 2 q^{7} + 9 q^{9} + 36 q^{11} + 13 q^{13} - 36 q^{15} - 78 q^{17} - 74 q^{19} - 6 q^{21} + 96 q^{23} + 19 q^{25} + 27 q^{27} + 18 q^{29} + 214 q^{31} + 108 q^{33} + 24 q^{35} - 286 q^{37} + 39 q^{39} - 384 q^{41} - 524 q^{43} - 108 q^{45} - 300 q^{47} - 339 q^{49} - 234 q^{51} + 558 q^{53} - 432 q^{55} - 222 q^{57} - 576 q^{59} + 74 q^{61} - 18 q^{63} - 156 q^{65} - 38 q^{67} + 288 q^{69} + 456 q^{71} - 682 q^{73} + 57 q^{75} - 72 q^{77} - 704 q^{79} + 81 q^{81} + 888 q^{83} + 936 q^{85} + 54 q^{87} - 1020 q^{89} - 26 q^{91} + 642 q^{93} + 888 q^{95} + 110 q^{97} + 324 q^{99}+O(q^{100})$$ q + 3 * q^3 - 12 * q^5 - 2 * q^7 + 9 * q^9 + 36 * q^11 + 13 * q^13 - 36 * q^15 - 78 * q^17 - 74 * q^19 - 6 * q^21 + 96 * q^23 + 19 * q^25 + 27 * q^27 + 18 * q^29 + 214 * q^31 + 108 * q^33 + 24 * q^35 - 286 * q^37 + 39 * q^39 - 384 * q^41 - 524 * q^43 - 108 * q^45 - 300 * q^47 - 339 * q^49 - 234 * q^51 + 558 * q^53 - 432 * q^55 - 222 * q^57 - 576 * q^59 + 74 * q^61 - 18 * q^63 - 156 * q^65 - 38 * q^67 + 288 * q^69 + 456 * q^71 - 682 * q^73 + 57 * q^75 - 72 * q^77 - 704 * q^79 + 81 * q^81 + 888 * q^83 + 936 * q^85 + 54 * q^87 - 1020 * q^89 - 26 * q^91 + 642 * q^93 + 888 * q^95 + 110 * 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 −12.0000 0 −2.00000 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.g 1
3.b odd 2 1 1872.4.a.m 1
4.b odd 2 1 39.4.a.a 1
8.b even 2 1 2496.4.a.f 1
8.d odd 2 1 2496.4.a.o 1
12.b even 2 1 117.4.a.a 1
20.d odd 2 1 975.4.a.e 1
28.d even 2 1 1911.4.a.f 1
52.b odd 2 1 507.4.a.c 1
52.f even 4 2 507.4.b.b 2
156.h even 2 1 1521.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.a 1 4.b odd 2 1
117.4.a.a 1 12.b even 2 1
507.4.a.c 1 52.b odd 2 1
507.4.b.b 2 52.f even 4 2
624.4.a.g 1 1.a even 1 1 trivial
975.4.a.e 1 20.d odd 2 1
1521.4.a.f 1 156.h even 2 1
1872.4.a.m 1 3.b odd 2 1
1911.4.a.f 1 28.d even 2 1
2496.4.a.f 1 8.b even 2 1
2496.4.a.o 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(624))$$:

 $$T_{5} + 12$$ T5 + 12 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 12$$
$7$ $$T + 2$$
$11$ $$T - 36$$
$13$ $$T - 13$$
$17$ $$T + 78$$
$19$ $$T + 74$$
$23$ $$T - 96$$
$29$ $$T - 18$$
$31$ $$T - 214$$
$37$ $$T + 286$$
$41$ $$T + 384$$
$43$ $$T + 524$$
$47$ $$T + 300$$
$53$ $$T - 558$$
$59$ $$T + 576$$
$61$ $$T - 74$$
$67$ $$T + 38$$
$71$ $$T - 456$$
$73$ $$T + 682$$
$79$ $$T + 704$$
$83$ $$T - 888$$
$89$ $$T + 1020$$
$97$ $$T - 110$$