# Properties

 Label 3920.2.a.br Level $3920$ Weight $2$ Character orbit 3920.a Self dual yes Analytic conductor $31.301$ Analytic rank $1$ 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] = [3920,2,Mod(1,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, 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("3920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3920.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: $$31.3013575923$$ Analytic rank: $$1$$ 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 245) 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.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{3} + q^{5} - 2 \beta q^{9} +O(q^{10})$$ q + (b - 1) * q^3 + q^5 - 2*b * q^9 $$q + (\beta - 1) q^{3} + q^{5} - 2 \beta q^{9} + ( - 2 \beta + 3) q^{11} + (\beta + 3) q^{13} + (\beta - 1) q^{15} + (3 \beta - 1) q^{17} - 6 q^{19} + ( - \beta - 6) q^{23} + q^{25} + ( - \beta - 1) q^{27} + ( - 4 \beta - 3) q^{29} + ( - 3 \beta - 6) q^{31} + (5 \beta - 7) q^{33} + (3 \beta - 2) q^{37} + (2 \beta - 1) q^{39} + ( - 3 \beta - 2) q^{41} - 2 q^{43} - 2 \beta q^{45} + (3 \beta + 3) q^{47} + ( - 4 \beta + 7) q^{51} + 3 \beta q^{53} + ( - 2 \beta + 3) q^{55} + ( - 6 \beta + 6) q^{57} + (3 \beta - 2) q^{59} - 2 \beta q^{61} + (\beta + 3) q^{65} + (3 \beta + 4) q^{67} + ( - 5 \beta + 4) q^{69} + ( - 2 \beta + 6) q^{71} - 6 \beta q^{73} + (\beta - 1) q^{75} + ( - 6 \beta + 7) q^{79} + (6 \beta - 1) q^{81} + (3 \beta - 1) q^{85} + (\beta - 5) q^{87} - 8 q^{89} - 3 \beta q^{93} - 6 q^{95} + (3 \beta + 9) q^{97} + ( - 6 \beta + 8) q^{99} +O(q^{100})$$ q + (b - 1) * q^3 + q^5 - 2*b * q^9 + (-2*b + 3) * q^11 + (b + 3) * q^13 + (b - 1) * q^15 + (3*b - 1) * q^17 - 6 * q^19 + (-b - 6) * q^23 + q^25 + (-b - 1) * q^27 + (-4*b - 3) * q^29 + (-3*b - 6) * q^31 + (5*b - 7) * q^33 + (3*b - 2) * q^37 + (2*b - 1) * q^39 + (-3*b - 2) * q^41 - 2 * q^43 - 2*b * q^45 + (3*b + 3) * q^47 + (-4*b + 7) * q^51 + 3*b * q^53 + (-2*b + 3) * q^55 + (-6*b + 6) * q^57 + (3*b - 2) * q^59 - 2*b * q^61 + (b + 3) * q^65 + (3*b + 4) * q^67 + (-5*b + 4) * q^69 + (-2*b + 6) * q^71 - 6*b * q^73 + (b - 1) * q^75 + (-6*b + 7) * q^79 + (6*b - 1) * q^81 + (3*b - 1) * q^85 + (b - 5) * q^87 - 8 * q^89 - 3*b * q^93 - 6 * q^95 + (3*b + 9) * q^97 + (-6*b + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 $$2 q - 2 q^{3} + 2 q^{5} + 6 q^{11} + 6 q^{13} - 2 q^{15} - 2 q^{17} - 12 q^{19} - 12 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} - 12 q^{31} - 14 q^{33} - 4 q^{37} - 2 q^{39} - 4 q^{41} - 4 q^{43} + 6 q^{47} + 14 q^{51} + 6 q^{55} + 12 q^{57} - 4 q^{59} + 6 q^{65} + 8 q^{67} + 8 q^{69} + 12 q^{71} - 2 q^{75} + 14 q^{79} - 2 q^{81} - 2 q^{85} - 10 q^{87} - 16 q^{89} - 12 q^{95} + 18 q^{97} + 16 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 + 6 * q^11 + 6 * q^13 - 2 * q^15 - 2 * q^17 - 12 * q^19 - 12 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 - 12 * q^31 - 14 * q^33 - 4 * q^37 - 2 * q^39 - 4 * q^41 - 4 * q^43 + 6 * q^47 + 14 * q^51 + 6 * q^55 + 12 * q^57 - 4 * q^59 + 6 * q^65 + 8 * q^67 + 8 * q^69 + 12 * q^71 - 2 * q^75 + 14 * q^79 - 2 * q^81 - 2 * q^85 - 10 * q^87 - 16 * q^89 - 12 * q^95 + 18 * q^97 + 16 * 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 −2.41421 0 1.00000 0 0 0 2.82843 0
1.2 0 0.414214 0 1.00000 0 0 0 −2.82843 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## 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 3920.2.a.br 2
4.b odd 2 1 245.2.a.f yes 2
7.b odd 2 1 3920.2.a.bw 2
12.b even 2 1 2205.2.a.t 2
20.d odd 2 1 1225.2.a.p 2
20.e even 4 2 1225.2.b.j 4
28.d even 2 1 245.2.a.e 2
28.f even 6 2 245.2.e.g 4
28.g odd 6 2 245.2.e.f 4
84.h odd 2 1 2205.2.a.v 2
140.c even 2 1 1225.2.a.r 2
140.j odd 4 2 1225.2.b.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.e 2 28.d even 2 1
245.2.a.f yes 2 4.b odd 2 1
245.2.e.f 4 28.g odd 6 2
245.2.e.g 4 28.f even 6 2
1225.2.a.p 2 20.d odd 2 1
1225.2.a.r 2 140.c even 2 1
1225.2.b.i 4 140.j odd 4 2
1225.2.b.j 4 20.e even 4 2
2205.2.a.t 2 12.b even 2 1
2205.2.a.v 2 84.h odd 2 1
3920.2.a.br 2 1.a even 1 1 trivial
3920.2.a.bw 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3920))$$:

 $$T_{3}^{2} + 2T_{3} - 1$$ T3^2 + 2*T3 - 1 $$T_{11}^{2} - 6T_{11} + 1$$ T11^2 - 6*T11 + 1 $$T_{13}^{2} - 6T_{13} + 7$$ T13^2 - 6*T13 + 7 $$T_{17}^{2} + 2T_{17} - 17$$ T17^2 + 2*T17 - 17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T - 1$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 6T + 1$$
$13$ $$T^{2} - 6T + 7$$
$17$ $$T^{2} + 2T - 17$$
$19$ $$(T + 6)^{2}$$
$23$ $$T^{2} + 12T + 34$$
$29$ $$T^{2} + 6T - 23$$
$31$ $$T^{2} + 12T + 18$$
$37$ $$T^{2} + 4T - 14$$
$41$ $$T^{2} + 4T - 14$$
$43$ $$(T + 2)^{2}$$
$47$ $$T^{2} - 6T - 9$$
$53$ $$T^{2} - 18$$
$59$ $$T^{2} + 4T - 14$$
$61$ $$T^{2} - 8$$
$67$ $$T^{2} - 8T - 2$$
$71$ $$T^{2} - 12T + 28$$
$73$ $$T^{2} - 72$$
$79$ $$T^{2} - 14T - 23$$
$83$ $$T^{2}$$
$89$ $$(T + 8)^{2}$$
$97$ $$T^{2} - 18T + 63$$