# Properties

 Label 4624.2.a.n Level $4624$ Weight $2$ Character orbit 4624.a Self dual yes Analytic conductor $36.923$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4624,2,Mod(1,4624)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4624 = 2^{4} \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4624.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: $$36.9228258946$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 68) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - 2 \beta q^{5} + 3 \beta q^{7} - q^{9} +O(q^{10})$$ q + b * q^3 - 2*b * q^5 + 3*b * q^7 - q^9 $$q + \beta q^{3} - 2 \beta q^{5} + 3 \beta q^{7} - q^{9} + \beta q^{11} - 4 q^{13} - 4 q^{15} - 4 q^{19} + 6 q^{21} + \beta q^{23} + 3 q^{25} - 4 \beta q^{27} - 2 \beta q^{29} + 3 \beta q^{31} + 2 q^{33} - 12 q^{35} + 6 \beta q^{37} - 4 \beta q^{39} + 8 \beta q^{41} + 8 q^{43} + 2 \beta q^{45} + 12 q^{47} + 11 q^{49} + 6 q^{53} - 4 q^{55} - 4 \beta q^{57} - 6 \beta q^{61} - 3 \beta q^{63} + 8 \beta q^{65} + 4 q^{67} + 2 q^{69} + 5 \beta q^{71} + 3 \beta q^{75} + 6 q^{77} - 3 \beta q^{79} - 5 q^{81} - 4 q^{87} + 12 q^{89} - 12 \beta q^{91} + 6 q^{93} + 8 \beta q^{95} - \beta q^{99} +O(q^{100})$$ q + b * q^3 - 2*b * q^5 + 3*b * q^7 - q^9 + b * q^11 - 4 * q^13 - 4 * q^15 - 4 * q^19 + 6 * q^21 + b * q^23 + 3 * q^25 - 4*b * q^27 - 2*b * q^29 + 3*b * q^31 + 2 * q^33 - 12 * q^35 + 6*b * q^37 - 4*b * q^39 + 8*b * q^41 + 8 * q^43 + 2*b * q^45 + 12 * q^47 + 11 * q^49 + 6 * q^53 - 4 * q^55 - 4*b * q^57 - 6*b * q^61 - 3*b * q^63 + 8*b * q^65 + 4 * q^67 + 2 * q^69 + 5*b * q^71 + 3*b * q^75 + 6 * q^77 - 3*b * q^79 - 5 * q^81 - 4 * q^87 + 12 * q^89 - 12*b * q^91 + 6 * q^93 + 8*b * q^95 - b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 8 q^{13} - 8 q^{15} - 8 q^{19} + 12 q^{21} + 6 q^{25} + 4 q^{33} - 24 q^{35} + 16 q^{43} + 24 q^{47} + 22 q^{49} + 12 q^{53} - 8 q^{55} + 8 q^{67} + 4 q^{69} + 12 q^{77} - 10 q^{81} - 8 q^{87} + 24 q^{89} + 12 q^{93}+O(q^{100})$$ 2 * q - 2 * q^9 - 8 * q^13 - 8 * q^15 - 8 * q^19 + 12 * q^21 + 6 * q^25 + 4 * q^33 - 24 * q^35 + 16 * q^43 + 24 * q^47 + 22 * q^49 + 12 * q^53 - 8 * q^55 + 8 * q^67 + 4 * q^69 + 12 * q^77 - 10 * q^81 - 8 * q^87 + 24 * q^89 + 12 * q^93

## 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
 −1.41421 1.41421
0 −1.41421 0 2.82843 0 −4.24264 0 −1.00000 0
1.2 0 1.41421 0 −2.82843 0 4.24264 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
$$2$$ $$-1$$
$$17$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4624.2.a.n 2
4.b odd 2 1 1156.2.a.c 2
17.b even 2 1 inner 4624.2.a.n 2
17.c even 4 2 272.2.b.c 2
51.f odd 4 2 2448.2.c.d 2
68.d odd 2 1 1156.2.a.c 2
68.f odd 4 2 68.2.b.a 2
68.g odd 8 2 1156.2.e.a 2
68.g odd 8 2 1156.2.e.b 2
68.i even 16 8 1156.2.h.d 8
136.i even 4 2 1088.2.b.f 2
136.j odd 4 2 1088.2.b.e 2
204.l even 4 2 612.2.b.a 2
340.i even 4 2 1700.2.g.a 4
340.n odd 4 2 1700.2.c.a 2
340.s even 4 2 1700.2.g.a 4
476.k even 4 2 3332.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.b.a 2 68.f odd 4 2
272.2.b.c 2 17.c even 4 2
612.2.b.a 2 204.l even 4 2
1088.2.b.e 2 136.j odd 4 2
1088.2.b.f 2 136.i even 4 2
1156.2.a.c 2 4.b odd 2 1
1156.2.a.c 2 68.d odd 2 1
1156.2.e.a 2 68.g odd 8 2
1156.2.e.b 2 68.g odd 8 2
1156.2.h.d 8 68.i even 16 8
1700.2.c.a 2 340.n odd 4 2
1700.2.g.a 4 340.i even 4 2
1700.2.g.a 4 340.s even 4 2
2448.2.c.d 2 51.f odd 4 2
3332.2.b.a 2 476.k even 4 2
4624.2.a.n 2 1.a even 1 1 trivial
4624.2.a.n 2 17.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4624))$$:

 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{5}^{2} - 8$$ T5^2 - 8 $$T_{7}^{2} - 18$$ T7^2 - 18 $$T_{13} + 4$$ T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2} - 8$$
$7$ $$T^{2} - 18$$
$11$ $$T^{2} - 2$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} - 2$$
$29$ $$T^{2} - 8$$
$31$ $$T^{2} - 18$$
$37$ $$T^{2} - 72$$
$41$ $$T^{2} - 128$$
$43$ $$(T - 8)^{2}$$
$47$ $$(T - 12)^{2}$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 72$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2} - 50$$
$73$ $$T^{2}$$
$79$ $$T^{2} - 18$$
$83$ $$T^{2}$$
$89$ $$(T - 12)^{2}$$
$97$ $$T^{2}$$