# Properties

 Label 3920.2.a.bs 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: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} - q^{5} + (\beta + 1) q^{9} +O(q^{10})$$ q - b * q^3 - q^5 + (b + 1) * q^9 $$q - \beta q^{3} - q^{5} + (\beta + 1) q^{9} - \beta q^{11} + ( - \beta - 2) q^{13} + \beta q^{15} + (\beta + 2) q^{17} + (2 \beta - 4) q^{19} + 2 \beta q^{23} + q^{25} + (\beta - 4) q^{27} + ( - 3 \beta + 2) q^{29} + (\beta + 4) q^{33} + 6 q^{37} + (3 \beta + 4) q^{39} + (2 \beta - 2) q^{41} + ( - 2 \beta - 4) q^{43} + ( - \beta - 1) q^{45} + (3 \beta - 4) q^{47} + ( - 3 \beta - 4) q^{51} + (2 \beta - 2) q^{53} + \beta q^{55} + (2 \beta - 8) q^{57} - 4 q^{59} + (6 \beta - 6) q^{61} + (\beta + 2) q^{65} + (4 \beta - 4) q^{67} + ( - 2 \beta - 8) q^{69} - 8 q^{71} + ( - 4 \beta + 6) q^{73} - \beta q^{75} + (\beta + 4) q^{79} - 7 q^{81} + 4 q^{83} + ( - \beta - 2) q^{85} + (\beta + 12) q^{87} + ( - 2 \beta - 2) q^{89} + ( - 2 \beta + 4) q^{95} + (5 \beta + 2) q^{97} + ( - 2 \beta - 4) q^{99} +O(q^{100})$$ q - b * q^3 - q^5 + (b + 1) * q^9 - b * q^11 + (-b - 2) * q^13 + b * q^15 + (b + 2) * q^17 + (2*b - 4) * q^19 + 2*b * q^23 + q^25 + (b - 4) * q^27 + (-3*b + 2) * q^29 + (b + 4) * q^33 + 6 * q^37 + (3*b + 4) * q^39 + (2*b - 2) * q^41 + (-2*b - 4) * q^43 + (-b - 1) * q^45 + (3*b - 4) * q^47 + (-3*b - 4) * q^51 + (2*b - 2) * q^53 + b * q^55 + (2*b - 8) * q^57 - 4 * q^59 + (6*b - 6) * q^61 + (b + 2) * q^65 + (4*b - 4) * q^67 + (-2*b - 8) * q^69 - 8 * q^71 + (-4*b + 6) * q^73 - b * q^75 + (b + 4) * q^79 - 7 * q^81 + 4 * q^83 + (-b - 2) * q^85 + (b + 12) * q^87 + (-2*b - 2) * q^89 + (-2*b + 4) * q^95 + (5*b + 2) * q^97 + (-2*b - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 2 q^{5} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 - 2 * q^5 + 3 * q^9 $$2 q - q^{3} - 2 q^{5} + 3 q^{9} - q^{11} - 5 q^{13} + q^{15} + 5 q^{17} - 6 q^{19} + 2 q^{23} + 2 q^{25} - 7 q^{27} + q^{29} + 9 q^{33} + 12 q^{37} + 11 q^{39} - 2 q^{41} - 10 q^{43} - 3 q^{45} - 5 q^{47} - 11 q^{51} - 2 q^{53} + q^{55} - 14 q^{57} - 8 q^{59} - 6 q^{61} + 5 q^{65} - 4 q^{67} - 18 q^{69} - 16 q^{71} + 8 q^{73} - q^{75} + 9 q^{79} - 14 q^{81} + 8 q^{83} - 5 q^{85} + 25 q^{87} - 6 q^{89} + 6 q^{95} + 9 q^{97} - 10 q^{99}+O(q^{100})$$ 2 * q - q^3 - 2 * q^5 + 3 * q^9 - q^11 - 5 * q^13 + q^15 + 5 * q^17 - 6 * q^19 + 2 * q^23 + 2 * q^25 - 7 * q^27 + q^29 + 9 * q^33 + 12 * q^37 + 11 * q^39 - 2 * q^41 - 10 * q^43 - 3 * q^45 - 5 * q^47 - 11 * q^51 - 2 * q^53 + q^55 - 14 * q^57 - 8 * q^59 - 6 * q^61 + 5 * q^65 - 4 * q^67 - 18 * q^69 - 16 * q^71 + 8 * q^73 - q^75 + 9 * q^79 - 14 * q^81 + 8 * q^83 - 5 * q^85 + 25 * q^87 - 6 * q^89 + 6 * q^95 + 9 * q^97 - 10 * 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
 2.56155 −1.56155
0 −2.56155 0 −1.00000 0 0 0 3.56155 0
1.2 0 1.56155 0 −1.00000 0 0 0 −0.561553 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.bs 2
4.b odd 2 1 245.2.a.d 2
7.b odd 2 1 560.2.a.i 2
12.b even 2 1 2205.2.a.x 2
20.d odd 2 1 1225.2.a.s 2
20.e even 4 2 1225.2.b.f 4
21.c even 2 1 5040.2.a.bt 2
28.d even 2 1 35.2.a.b 2
28.f even 6 2 245.2.e.i 4
28.g odd 6 2 245.2.e.h 4
35.c odd 2 1 2800.2.a.bi 2
35.f even 4 2 2800.2.g.t 4
56.e even 2 1 2240.2.a.bh 2
56.h odd 2 1 2240.2.a.bd 2
84.h odd 2 1 315.2.a.e 2
140.c even 2 1 175.2.a.f 2
140.j odd 4 2 175.2.b.b 4
308.g odd 2 1 4235.2.a.m 2
364.h even 2 1 5915.2.a.l 2
420.o odd 2 1 1575.2.a.p 2
420.w even 4 2 1575.2.d.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 28.d even 2 1
175.2.a.f 2 140.c even 2 1
175.2.b.b 4 140.j odd 4 2
245.2.a.d 2 4.b odd 2 1
245.2.e.h 4 28.g odd 6 2
245.2.e.i 4 28.f even 6 2
315.2.a.e 2 84.h odd 2 1
560.2.a.i 2 7.b odd 2 1
1225.2.a.s 2 20.d odd 2 1
1225.2.b.f 4 20.e even 4 2
1575.2.a.p 2 420.o odd 2 1
1575.2.d.e 4 420.w even 4 2
2205.2.a.x 2 12.b even 2 1
2240.2.a.bd 2 56.h odd 2 1
2240.2.a.bh 2 56.e even 2 1
2800.2.a.bi 2 35.c odd 2 1
2800.2.g.t 4 35.f even 4 2
3920.2.a.bs 2 1.a even 1 1 trivial
4235.2.a.m 2 308.g odd 2 1
5040.2.a.bt 2 21.c even 2 1
5915.2.a.l 2 364.h even 2 1

## Hecke kernels

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

 $$T_{3}^{2} + T_{3} - 4$$ T3^2 + T3 - 4 $$T_{11}^{2} + T_{11} - 4$$ T11^2 + T11 - 4 $$T_{13}^{2} + 5T_{13} + 2$$ T13^2 + 5*T13 + 2 $$T_{17}^{2} - 5T_{17} + 2$$ T17^2 - 5*T17 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 4$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + T - 4$$
$13$ $$T^{2} + 5T + 2$$
$17$ $$T^{2} - 5T + 2$$
$19$ $$T^{2} + 6T - 8$$
$23$ $$T^{2} - 2T - 16$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2}$$
$37$ $$(T - 6)^{2}$$
$41$ $$T^{2} + 2T - 16$$
$43$ $$T^{2} + 10T + 8$$
$47$ $$T^{2} + 5T - 32$$
$53$ $$T^{2} + 2T - 16$$
$59$ $$(T + 4)^{2}$$
$61$ $$T^{2} + 6T - 144$$
$67$ $$T^{2} + 4T - 64$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} - 8T - 52$$
$79$ $$T^{2} - 9T + 16$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} + 6T - 8$$
$97$ $$T^{2} - 9T - 86$$