# Properties

 Label 3520.2.a.bk Level $3520$ Weight $2$ Character orbit 3520.a Self dual yes Analytic conductor $28.107$ 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] = [3520,2,Mod(1,3520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3520, 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("3520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3520 = 2^{6} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3520.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: $$28.1073415115$$ 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 440) 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} - q^{5} + \beta q^{7} + (\beta + 1) q^{9} +O(q^{10})$$ q - b * q^3 - q^5 + b * q^7 + (b + 1) * q^9 $$q - \beta q^{3} - q^{5} + \beta q^{7} + (\beta + 1) q^{9} + q^{11} - 2 q^{13} + \beta q^{15} + ( - \beta + 2) q^{17} + \beta q^{19} + ( - \beta - 4) q^{21} - 2 \beta q^{23} + q^{25} + (\beta - 4) q^{27} + ( - 3 \beta - 2) q^{29} + ( - \beta - 4) q^{31} - \beta q^{33} - \beta q^{35} + (3 \beta - 2) q^{37} + 2 \beta q^{39} + 2 q^{41} + 4 \beta q^{43} + ( - \beta - 1) q^{45} + (2 \beta + 8) q^{47} + (\beta - 3) q^{49} + ( - \beta + 4) q^{51} + ( - \beta - 2) q^{53} - q^{55} + ( - \beta - 4) q^{57} + (2 \beta - 4) q^{59} + (3 \beta - 10) q^{61} + (2 \beta + 4) q^{63} + 2 q^{65} + (4 \beta - 4) q^{67} + (2 \beta + 8) q^{69} + ( - 3 \beta + 4) q^{71} - 2 q^{73} - \beta q^{75} + \beta q^{77} + 6 \beta q^{79} - 7 q^{81} + 2 \beta q^{83} + (\beta - 2) q^{85} + (5 \beta + 12) q^{87} + ( - \beta - 10) q^{89} - 2 \beta q^{91} + (5 \beta + 4) q^{93} - \beta q^{95} + (2 \beta + 2) q^{97} + (\beta + 1) q^{99} +O(q^{100})$$ q - b * q^3 - q^5 + b * q^7 + (b + 1) * q^9 + q^11 - 2 * q^13 + b * q^15 + (-b + 2) * q^17 + b * q^19 + (-b - 4) * q^21 - 2*b * q^23 + q^25 + (b - 4) * q^27 + (-3*b - 2) * q^29 + (-b - 4) * q^31 - b * q^33 - b * q^35 + (3*b - 2) * q^37 + 2*b * q^39 + 2 * q^41 + 4*b * q^43 + (-b - 1) * q^45 + (2*b + 8) * q^47 + (b - 3) * q^49 + (-b + 4) * q^51 + (-b - 2) * q^53 - q^55 + (-b - 4) * q^57 + (2*b - 4) * q^59 + (3*b - 10) * q^61 + (2*b + 4) * q^63 + 2 * q^65 + (4*b - 4) * q^67 + (2*b + 8) * q^69 + (-3*b + 4) * q^71 - 2 * q^73 - b * q^75 + b * q^77 + 6*b * q^79 - 7 * q^81 + 2*b * q^83 + (b - 2) * q^85 + (5*b + 12) * q^87 + (-b - 10) * q^89 - 2*b * q^91 + (5*b + 4) * q^93 - b * q^95 + (2*b + 2) * q^97 + (b + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 - 2 * q^5 + q^7 + 3 * q^9 $$2 q - q^{3} - 2 q^{5} + q^{7} + 3 q^{9} + 2 q^{11} - 4 q^{13} + q^{15} + 3 q^{17} + q^{19} - 9 q^{21} - 2 q^{23} + 2 q^{25} - 7 q^{27} - 7 q^{29} - 9 q^{31} - q^{33} - q^{35} - q^{37} + 2 q^{39} + 4 q^{41} + 4 q^{43} - 3 q^{45} + 18 q^{47} - 5 q^{49} + 7 q^{51} - 5 q^{53} - 2 q^{55} - 9 q^{57} - 6 q^{59} - 17 q^{61} + 10 q^{63} + 4 q^{65} - 4 q^{67} + 18 q^{69} + 5 q^{71} - 4 q^{73} - q^{75} + q^{77} + 6 q^{79} - 14 q^{81} + 2 q^{83} - 3 q^{85} + 29 q^{87} - 21 q^{89} - 2 q^{91} + 13 q^{93} - q^{95} + 6 q^{97} + 3 q^{99}+O(q^{100})$$ 2 * q - q^3 - 2 * q^5 + q^7 + 3 * q^9 + 2 * q^11 - 4 * q^13 + q^15 + 3 * q^17 + q^19 - 9 * q^21 - 2 * q^23 + 2 * q^25 - 7 * q^27 - 7 * q^29 - 9 * q^31 - q^33 - q^35 - q^37 + 2 * q^39 + 4 * q^41 + 4 * q^43 - 3 * q^45 + 18 * q^47 - 5 * q^49 + 7 * q^51 - 5 * q^53 - 2 * q^55 - 9 * q^57 - 6 * q^59 - 17 * q^61 + 10 * q^63 + 4 * q^65 - 4 * q^67 + 18 * q^69 + 5 * q^71 - 4 * q^73 - q^75 + q^77 + 6 * q^79 - 14 * q^81 + 2 * q^83 - 3 * q^85 + 29 * q^87 - 21 * q^89 - 2 * q^91 + 13 * q^93 - q^95 + 6 * q^97 + 3 * 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 2.56155 0 3.56155 0
1.2 0 1.56155 0 −1.00000 0 −1.56155 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
$$2$$ $$-1$$
$$5$$ $$+1$$
$$11$$ $$-1$$

## 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 3520.2.a.bk 2
4.b odd 2 1 3520.2.a.bp 2
8.b even 2 1 880.2.a.o 2
8.d odd 2 1 440.2.a.e 2
24.f even 2 1 3960.2.a.w 2
24.h odd 2 1 7920.2.a.bu 2
40.e odd 2 1 2200.2.a.s 2
40.f even 2 1 4400.2.a.bj 2
40.i odd 4 2 4400.2.b.t 4
40.k even 4 2 2200.2.b.i 4
88.b odd 2 1 9680.2.a.bs 2
88.g even 2 1 4840.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.e 2 8.d odd 2 1
880.2.a.o 2 8.b even 2 1
2200.2.a.s 2 40.e odd 2 1
2200.2.b.i 4 40.k even 4 2
3520.2.a.bk 2 1.a even 1 1 trivial
3520.2.a.bp 2 4.b odd 2 1
3960.2.a.w 2 24.f even 2 1
4400.2.a.bj 2 40.f even 2 1
4400.2.b.t 4 40.i odd 4 2
4840.2.a.j 2 88.g even 2 1
7920.2.a.bu 2 24.h odd 2 1
9680.2.a.bs 2 88.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3520))$$:

 $$T_{3}^{2} + T_{3} - 4$$ T3^2 + T3 - 4 $$T_{7}^{2} - T_{7} - 4$$ T7^2 - T7 - 4 $$T_{13} + 2$$ T13 + 2 $$T_{17}^{2} - 3T_{17} - 2$$ T17^2 - 3*T17 - 2 $$T_{19}^{2} - T_{19} - 4$$ T19^2 - T19 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 4$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - T - 4$$
$11$ $$(T - 1)^{2}$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} - 3T - 2$$
$19$ $$T^{2} - T - 4$$
$23$ $$T^{2} + 2T - 16$$
$29$ $$T^{2} + 7T - 26$$
$31$ $$T^{2} + 9T + 16$$
$37$ $$T^{2} + T - 38$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} - 4T - 64$$
$47$ $$T^{2} - 18T + 64$$
$53$ $$T^{2} + 5T + 2$$
$59$ $$T^{2} + 6T - 8$$
$61$ $$T^{2} + 17T + 34$$
$67$ $$T^{2} + 4T - 64$$
$71$ $$T^{2} - 5T - 32$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2} - 6T - 144$$
$83$ $$T^{2} - 2T - 16$$
$89$ $$T^{2} + 21T + 106$$
$97$ $$T^{2} - 6T - 8$$