# Properties

 Label 3136.2.a.by Level $3136$ Weight $2$ Character orbit 3136.a Self dual yes Analytic conductor $25.041$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 224) 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + ( - \beta + 1) q^{5} + (2 \beta + 3) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + (-b + 1) * q^5 + (2*b + 3) * q^9 $$q + (\beta + 1) q^{3} + ( - \beta + 1) q^{5} + (2 \beta + 3) q^{9} + (2 \beta - 2) q^{11} + (\beta + 3) q^{13} - 4 q^{15} + 2 \beta q^{17} + ( - \beta - 1) q^{19} - 4 q^{23} + ( - 2 \beta + 1) q^{25} + (2 \beta + 10) q^{27} - 2 \beta q^{29} + (2 \beta + 2) q^{31} + 8 q^{33} - 2 \beta q^{37} + (4 \beta + 8) q^{39} + ( - 2 \beta + 4) q^{41} + ( - 2 \beta + 2) q^{43} + ( - \beta - 7) q^{45} + ( - 2 \beta + 6) q^{47} + (2 \beta + 10) q^{51} + 10 q^{53} + (4 \beta - 12) q^{55} + ( - 2 \beta - 6) q^{57} + ( - \beta + 7) q^{59} + ( - \beta + 9) q^{61} + ( - 2 \beta - 2) q^{65} + 4 q^{67} + ( - 4 \beta - 4) q^{69} + ( - 4 \beta - 4) q^{71} + ( - 4 \beta - 6) q^{73} + ( - \beta - 9) q^{75} + (4 \beta - 4) q^{79} + (6 \beta + 11) q^{81} + ( - \beta + 7) q^{83} + (2 \beta - 10) q^{85} + ( - 2 \beta - 10) q^{87} + 6 q^{89} + (4 \beta + 12) q^{93} + 4 q^{95} + (2 \beta - 8) q^{97} + (2 \beta + 14) q^{99} +O(q^{100})$$ q + (b + 1) * q^3 + (-b + 1) * q^5 + (2*b + 3) * q^9 + (2*b - 2) * q^11 + (b + 3) * q^13 - 4 * q^15 + 2*b * q^17 + (-b - 1) * q^19 - 4 * q^23 + (-2*b + 1) * q^25 + (2*b + 10) * q^27 - 2*b * q^29 + (2*b + 2) * q^31 + 8 * q^33 - 2*b * q^37 + (4*b + 8) * q^39 + (-2*b + 4) * q^41 + (-2*b + 2) * q^43 + (-b - 7) * q^45 + (-2*b + 6) * q^47 + (2*b + 10) * q^51 + 10 * q^53 + (4*b - 12) * q^55 + (-2*b - 6) * q^57 + (-b + 7) * q^59 + (-b + 9) * q^61 + (-2*b - 2) * q^65 + 4 * q^67 + (-4*b - 4) * q^69 + (-4*b - 4) * q^71 + (-4*b - 6) * q^73 + (-b - 9) * q^75 + (4*b - 4) * q^79 + (6*b + 11) * q^81 + (-b + 7) * q^83 + (2*b - 10) * q^85 + (-2*b - 10) * q^87 + 6 * q^89 + (4*b + 12) * q^93 + 4 * q^95 + (2*b - 8) * q^97 + (2*b + 14) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^9 $$2 q + 2 q^{3} + 2 q^{5} + 6 q^{9} - 4 q^{11} + 6 q^{13} - 8 q^{15} - 2 q^{19} - 8 q^{23} + 2 q^{25} + 20 q^{27} + 4 q^{31} + 16 q^{33} + 16 q^{39} + 8 q^{41} + 4 q^{43} - 14 q^{45} + 12 q^{47} + 20 q^{51} + 20 q^{53} - 24 q^{55} - 12 q^{57} + 14 q^{59} + 18 q^{61} - 4 q^{65} + 8 q^{67} - 8 q^{69} - 8 q^{71} - 12 q^{73} - 18 q^{75} - 8 q^{79} + 22 q^{81} + 14 q^{83} - 20 q^{85} - 20 q^{87} + 12 q^{89} + 24 q^{93} + 8 q^{95} - 16 q^{97} + 28 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^9 - 4 * q^11 + 6 * q^13 - 8 * q^15 - 2 * q^19 - 8 * q^23 + 2 * q^25 + 20 * q^27 + 4 * q^31 + 16 * q^33 + 16 * q^39 + 8 * q^41 + 4 * q^43 - 14 * q^45 + 12 * q^47 + 20 * q^51 + 20 * q^53 - 24 * q^55 - 12 * q^57 + 14 * q^59 + 18 * q^61 - 4 * q^65 + 8 * q^67 - 8 * q^69 - 8 * q^71 - 12 * q^73 - 18 * q^75 - 8 * q^79 + 22 * q^81 + 14 * q^83 - 20 * q^85 - 20 * q^87 + 12 * q^89 + 24 * q^93 + 8 * q^95 - 16 * q^97 + 28 * 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
 −0.618034 1.61803
0 −1.23607 0 3.23607 0 0 0 −1.47214 0
1.2 0 3.23607 0 −1.23607 0 0 0 7.47214 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

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 3136.2.a.by 2
4.b odd 2 1 3136.2.a.bf 2
7.b odd 2 1 448.2.a.i 2
8.b even 2 1 1568.2.a.k 2
8.d odd 2 1 1568.2.a.v 2
21.c even 2 1 4032.2.a.bv 2
28.d even 2 1 448.2.a.j 2
56.e even 2 1 224.2.a.c 2
56.h odd 2 1 224.2.a.d yes 2
56.j odd 6 2 1568.2.i.m 4
56.k odd 6 2 1568.2.i.n 4
56.m even 6 2 1568.2.i.v 4
56.p even 6 2 1568.2.i.w 4
84.h odd 2 1 4032.2.a.bw 2
112.j even 4 2 1792.2.b.k 4
112.l odd 4 2 1792.2.b.m 4
168.e odd 2 1 2016.2.a.r 2
168.i even 2 1 2016.2.a.o 2
280.c odd 2 1 5600.2.a.z 2
280.n even 2 1 5600.2.a.bk 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.c 2 56.e even 2 1
224.2.a.d yes 2 56.h odd 2 1
448.2.a.i 2 7.b odd 2 1
448.2.a.j 2 28.d even 2 1
1568.2.a.k 2 8.b even 2 1
1568.2.a.v 2 8.d odd 2 1
1568.2.i.m 4 56.j odd 6 2
1568.2.i.n 4 56.k odd 6 2
1568.2.i.v 4 56.m even 6 2
1568.2.i.w 4 56.p even 6 2
1792.2.b.k 4 112.j even 4 2
1792.2.b.m 4 112.l odd 4 2
2016.2.a.o 2 168.i even 2 1
2016.2.a.r 2 168.e odd 2 1
3136.2.a.bf 2 4.b odd 2 1
3136.2.a.by 2 1.a even 1 1 trivial
4032.2.a.bv 2 21.c even 2 1
4032.2.a.bw 2 84.h odd 2 1
5600.2.a.z 2 280.c odd 2 1
5600.2.a.bk 2 280.n even 2 1

## Hecke kernels

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

 $$T_{3}^{2} - 2T_{3} - 4$$ T3^2 - 2*T3 - 4 $$T_{5}^{2} - 2T_{5} - 4$$ T5^2 - 2*T5 - 4 $$T_{11}^{2} + 4T_{11} - 16$$ T11^2 + 4*T11 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 4$$
$5$ $$T^{2} - 2T - 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4T - 16$$
$13$ $$T^{2} - 6T + 4$$
$17$ $$T^{2} - 20$$
$19$ $$T^{2} + 2T - 4$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2} - 20$$
$31$ $$T^{2} - 4T - 16$$
$37$ $$T^{2} - 20$$
$41$ $$T^{2} - 8T - 4$$
$43$ $$T^{2} - 4T - 16$$
$47$ $$T^{2} - 12T + 16$$
$53$ $$(T - 10)^{2}$$
$59$ $$T^{2} - 14T + 44$$
$61$ $$T^{2} - 18T + 76$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2} + 8T - 64$$
$73$ $$T^{2} + 12T - 44$$
$79$ $$T^{2} + 8T - 64$$
$83$ $$T^{2} - 14T + 44$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 16T + 44$$