# Properties

 Label 3136.2.a.bs Level $3136$ Weight $2$ Character orbit 3136.a Self dual yes Analytic conductor $25.041$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3136,2,Mod(1,3136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3136, 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("3136.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3136 = 2^{6} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3136.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: $$25.0410860739$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 196) 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{3} + \beta q^{5} + 5 q^{9}+O(q^{10})$$ q + 2*b * q^3 + b * q^5 + 5 * q^9 $$q + 2 \beta q^{3} + \beta q^{5} + 5 q^{9} + 4 q^{11} + 3 \beta q^{13} + 4 q^{15} - \beta q^{17} - 2 \beta q^{19} + 4 q^{23} - 3 q^{25} + 4 \beta q^{27} - 8 q^{29} + 8 \beta q^{33} + 8 q^{37} + 12 q^{39} + 5 \beta q^{41} - 4 q^{43} + 5 \beta q^{45} + 4 \beta q^{47} - 4 q^{51} - 10 q^{53} + 4 \beta q^{55} - 8 q^{57} - 10 \beta q^{59} - 5 \beta q^{61} + 6 q^{65} + 8 \beta q^{69} + 5 \beta q^{73} - 6 \beta q^{75} - 8 q^{79} + q^{81} + 10 \beta q^{83} - 2 q^{85} - 16 \beta q^{87} - 5 \beta q^{89} - 4 q^{95} + \beta q^{97} + 20 q^{99} +O(q^{100})$$ q + 2*b * q^3 + b * q^5 + 5 * q^9 + 4 * q^11 + 3*b * q^13 + 4 * q^15 - b * q^17 - 2*b * q^19 + 4 * q^23 - 3 * q^25 + 4*b * q^27 - 8 * q^29 + 8*b * q^33 + 8 * q^37 + 12 * q^39 + 5*b * q^41 - 4 * q^43 + 5*b * q^45 + 4*b * q^47 - 4 * q^51 - 10 * q^53 + 4*b * q^55 - 8 * q^57 - 10*b * q^59 - 5*b * q^61 + 6 * q^65 + 8*b * q^69 + 5*b * q^73 - 6*b * q^75 - 8 * q^79 + q^81 + 10*b * q^83 - 2 * q^85 - 16*b * q^87 - 5*b * q^89 - 4 * q^95 + b * q^97 + 20 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{9}+O(q^{10})$$ 2 * q + 10 * q^9 $$2 q + 10 q^{9} + 8 q^{11} + 8 q^{15} + 8 q^{23} - 6 q^{25} - 16 q^{29} + 16 q^{37} + 24 q^{39} - 8 q^{43} - 8 q^{51} - 20 q^{53} - 16 q^{57} + 12 q^{65} - 16 q^{79} + 2 q^{81} - 4 q^{85} - 8 q^{95} + 40 q^{99}+O(q^{100})$$ 2 * q + 10 * q^9 + 8 * q^11 + 8 * q^15 + 8 * q^23 - 6 * q^25 - 16 * q^29 + 16 * q^37 + 24 * q^39 - 8 * q^43 - 8 * q^51 - 20 * q^53 - 16 * q^57 + 12 * q^65 - 16 * q^79 + 2 * q^81 - 4 * q^85 - 8 * q^95 + 40 * 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
 −1.41421 1.41421
0 −2.82843 0 −1.41421 0 0 0 5.00000 0
1.2 0 2.82843 0 1.41421 0 0 0 5.00000 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
$$2$$ $$-1$$
$$7$$ $$+1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.2.a.bs 2
4.b odd 2 1 3136.2.a.br 2
7.b odd 2 1 inner 3136.2.a.bs 2
8.b even 2 1 784.2.a.m 2
8.d odd 2 1 196.2.a.c 2
24.f even 2 1 1764.2.a.l 2
24.h odd 2 1 7056.2.a.cr 2
28.d even 2 1 3136.2.a.br 2
40.e odd 2 1 4900.2.a.y 2
40.k even 4 2 4900.2.e.p 4
56.e even 2 1 196.2.a.c 2
56.h odd 2 1 784.2.a.m 2
56.j odd 6 2 784.2.i.l 4
56.k odd 6 2 196.2.e.b 4
56.m even 6 2 196.2.e.b 4
56.p even 6 2 784.2.i.l 4
168.e odd 2 1 1764.2.a.l 2
168.i even 2 1 7056.2.a.cr 2
168.v even 6 2 1764.2.k.l 4
168.be odd 6 2 1764.2.k.l 4
280.n even 2 1 4900.2.a.y 2
280.y odd 4 2 4900.2.e.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.a.c 2 8.d odd 2 1
196.2.a.c 2 56.e even 2 1
196.2.e.b 4 56.k odd 6 2
196.2.e.b 4 56.m even 6 2
784.2.a.m 2 8.b even 2 1
784.2.a.m 2 56.h odd 2 1
784.2.i.l 4 56.j odd 6 2
784.2.i.l 4 56.p even 6 2
1764.2.a.l 2 24.f even 2 1
1764.2.a.l 2 168.e odd 2 1
1764.2.k.l 4 168.v even 6 2
1764.2.k.l 4 168.be odd 6 2
3136.2.a.br 2 4.b odd 2 1
3136.2.a.br 2 28.d even 2 1
3136.2.a.bs 2 1.a even 1 1 trivial
3136.2.a.bs 2 7.b odd 2 1 inner
4900.2.a.y 2 40.e odd 2 1
4900.2.a.y 2 280.n even 2 1
4900.2.e.p 4 40.k even 4 2
4900.2.e.p 4 280.y odd 4 2
7056.2.a.cr 2 24.h odd 2 1
7056.2.a.cr 2 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3136))$$:

 $$T_{3}^{2} - 8$$ T3^2 - 8 $$T_{5}^{2} - 2$$ T5^2 - 2 $$T_{11} - 4$$ T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 8$$
$5$ $$T^{2} - 2$$
$7$ $$T^{2}$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} - 18$$
$17$ $$T^{2} - 2$$
$19$ $$T^{2} - 8$$
$23$ $$(T - 4)^{2}$$
$29$ $$(T + 8)^{2}$$
$31$ $$T^{2}$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} - 50$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} - 32$$
$53$ $$(T + 10)^{2}$$
$59$ $$T^{2} - 200$$
$61$ $$T^{2} - 50$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 50$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} - 200$$
$89$ $$T^{2} - 50$$
$97$ $$T^{2} - 2$$