# Properties

 Label 624.2.a.e Level $624$ Weight $2$ Character orbit 624.a Self dual yes Analytic conductor $4.983$ 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] = [624,2,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, 2, names="a")

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

chi := DirichletCharacter("624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$4.98266508613$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 4 q^{5} + 2 q^{7} + q^{9}+O(q^{10})$$ q + q^3 - 4 * q^5 + 2 * q^7 + q^9 $$q + q^{3} - 4 q^{5} + 2 q^{7} + q^{9} + 4 q^{11} + q^{13} - 4 q^{15} + 2 q^{17} + 2 q^{19} + 2 q^{21} + 11 q^{25} + q^{27} - 6 q^{29} + 10 q^{31} + 4 q^{33} - 8 q^{35} + 10 q^{37} + q^{39} + 8 q^{41} - 4 q^{43} - 4 q^{45} + 4 q^{47} - 3 q^{49} + 2 q^{51} - 10 q^{53} - 16 q^{55} + 2 q^{57} + 8 q^{59} - 14 q^{61} + 2 q^{63} - 4 q^{65} - 2 q^{67} - 16 q^{71} - 10 q^{73} + 11 q^{75} + 8 q^{77} + 16 q^{79} + q^{81} - 8 q^{85} - 6 q^{87} - 4 q^{89} + 2 q^{91} + 10 q^{93} - 8 q^{95} - 2 q^{97} + 4 q^{99}+O(q^{100})$$ q + q^3 - 4 * q^5 + 2 * q^7 + q^9 + 4 * q^11 + q^13 - 4 * q^15 + 2 * q^17 + 2 * q^19 + 2 * q^21 + 11 * q^25 + q^27 - 6 * q^29 + 10 * q^31 + 4 * q^33 - 8 * q^35 + 10 * q^37 + q^39 + 8 * q^41 - 4 * q^43 - 4 * q^45 + 4 * q^47 - 3 * q^49 + 2 * q^51 - 10 * q^53 - 16 * q^55 + 2 * q^57 + 8 * q^59 - 14 * q^61 + 2 * q^63 - 4 * q^65 - 2 * q^67 - 16 * q^71 - 10 * q^73 + 11 * q^75 + 8 * q^77 + 16 * q^79 + q^81 - 8 * q^85 - 6 * q^87 - 4 * q^89 + 2 * q^91 + 10 * q^93 - 8 * q^95 - 2 * q^97 + 4 * 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 1.00000 0 −4.00000 0 2.00000 0 1.00000 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 624.2.a.e 1
3.b odd 2 1 1872.2.a.s 1
4.b odd 2 1 156.2.a.a 1
8.b even 2 1 2496.2.a.o 1
8.d odd 2 1 2496.2.a.bc 1
12.b even 2 1 468.2.a.d 1
13.b even 2 1 8112.2.a.bi 1
20.d odd 2 1 3900.2.a.m 1
20.e even 4 2 3900.2.h.b 2
24.f even 2 1 7488.2.a.c 1
24.h odd 2 1 7488.2.a.d 1
28.d even 2 1 7644.2.a.k 1
36.f odd 6 2 4212.2.i.l 2
36.h even 6 2 4212.2.i.b 2
52.b odd 2 1 2028.2.a.c 1
52.f even 4 2 2028.2.b.a 2
52.i odd 6 2 2028.2.i.g 2
52.j odd 6 2 2028.2.i.e 2
52.l even 12 4 2028.2.q.h 4
156.h even 2 1 6084.2.a.b 1
156.l odd 4 2 6084.2.b.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.a 1 4.b odd 2 1
468.2.a.d 1 12.b even 2 1
624.2.a.e 1 1.a even 1 1 trivial
1872.2.a.s 1 3.b odd 2 1
2028.2.a.c 1 52.b odd 2 1
2028.2.b.a 2 52.f even 4 2
2028.2.i.e 2 52.j odd 6 2
2028.2.i.g 2 52.i odd 6 2
2028.2.q.h 4 52.l even 12 4
2496.2.a.o 1 8.b even 2 1
2496.2.a.bc 1 8.d odd 2 1
3900.2.a.m 1 20.d odd 2 1
3900.2.h.b 2 20.e even 4 2
4212.2.i.b 2 36.h even 6 2
4212.2.i.l 2 36.f odd 6 2
6084.2.a.b 1 156.h even 2 1
6084.2.b.j 2 156.l odd 4 2
7488.2.a.c 1 24.f even 2 1
7488.2.a.d 1 24.h odd 2 1
7644.2.a.k 1 28.d even 2 1
8112.2.a.bi 1 13.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(624))$$:

 $$T_{5} + 4$$ T5 + 4 $$T_{7} - 2$$ T7 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T + 4$$
$7$ $$T - 2$$
$11$ $$T - 4$$
$13$ $$T - 1$$
$17$ $$T - 2$$
$19$ $$T - 2$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T - 10$$
$37$ $$T - 10$$
$41$ $$T - 8$$
$43$ $$T + 4$$
$47$ $$T - 4$$
$53$ $$T + 10$$
$59$ $$T - 8$$
$61$ $$T + 14$$
$67$ $$T + 2$$
$71$ $$T + 16$$
$73$ $$T + 10$$
$79$ $$T - 16$$
$83$ $$T$$
$89$ $$T + 4$$
$97$ $$T + 2$$