# Properties

 Label 2070.2.a.w Level $2070$ Weight $2$ Character orbit 2070.a Self dual yes Analytic conductor $16.529$ 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] = [2070,2,Mod(1,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, 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("2070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2070.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: $$16.5290332184$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} + (\beta + 1) q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - q^5 + (b + 1) * q^7 + q^8 $$q + q^{2} + q^{4} - q^{5} + (\beta + 1) q^{7} + q^{8} - q^{10} + ( - \beta + 4) q^{11} + (\beta + 1) q^{13} + (\beta + 1) q^{14} + q^{16} - 3 \beta q^{17} + (3 \beta - 1) q^{19} - q^{20} + ( - \beta + 4) q^{22} + q^{23} + q^{25} + (\beta + 1) q^{26} + (\beta + 1) q^{28} + (2 \beta - 2) q^{29} + ( - 3 \beta - 1) q^{31} + q^{32} - 3 \beta q^{34} + ( - \beta - 1) q^{35} + 8 q^{37} + (3 \beta - 1) q^{38} - q^{40} + ( - 3 \beta + 6) q^{41} - 4 \beta q^{43} + ( - \beta + 4) q^{44} + q^{46} + (2 \beta - 2) q^{47} + (3 \beta - 3) q^{49} + q^{50} + (\beta + 1) q^{52} + (4 \beta + 2) q^{53} + (\beta - 4) q^{55} + (\beta + 1) q^{56} + (2 \beta - 2) q^{58} + ( - 2 \beta + 8) q^{59} + 5 \beta q^{61} + ( - 3 \beta - 1) q^{62} + q^{64} + ( - \beta - 1) q^{65} - 4 q^{67} - 3 \beta q^{68} + ( - \beta - 1) q^{70} + (\beta + 14) q^{71} + ( - 6 \beta + 8) q^{73} + 8 q^{74} + (3 \beta - 1) q^{76} + (2 \beta + 1) q^{77} + ( - 8 \beta + 4) q^{79} - q^{80} + ( - 3 \beta + 6) q^{82} + ( - 4 \beta - 2) q^{83} + 3 \beta q^{85} - 4 \beta q^{86} + ( - \beta + 4) q^{88} + (3 \beta + 4) q^{91} + q^{92} + (2 \beta - 2) q^{94} + ( - 3 \beta + 1) q^{95} + (\beta + 4) q^{97} + (3 \beta - 3) q^{98} +O(q^{100})$$ q + q^2 + q^4 - q^5 + (b + 1) * q^7 + q^8 - q^10 + (-b + 4) * q^11 + (b + 1) * q^13 + (b + 1) * q^14 + q^16 - 3*b * q^17 + (3*b - 1) * q^19 - q^20 + (-b + 4) * q^22 + q^23 + q^25 + (b + 1) * q^26 + (b + 1) * q^28 + (2*b - 2) * q^29 + (-3*b - 1) * q^31 + q^32 - 3*b * q^34 + (-b - 1) * q^35 + 8 * q^37 + (3*b - 1) * q^38 - q^40 + (-3*b + 6) * q^41 - 4*b * q^43 + (-b + 4) * q^44 + q^46 + (2*b - 2) * q^47 + (3*b - 3) * q^49 + q^50 + (b + 1) * q^52 + (4*b + 2) * q^53 + (b - 4) * q^55 + (b + 1) * q^56 + (2*b - 2) * q^58 + (-2*b + 8) * q^59 + 5*b * q^61 + (-3*b - 1) * q^62 + q^64 + (-b - 1) * q^65 - 4 * q^67 - 3*b * q^68 + (-b - 1) * q^70 + (b + 14) * q^71 + (-6*b + 8) * q^73 + 8 * q^74 + (3*b - 1) * q^76 + (2*b + 1) * q^77 + (-8*b + 4) * q^79 - q^80 + (-3*b + 6) * q^82 + (-4*b - 2) * q^83 + 3*b * q^85 - 4*b * q^86 + (-b + 4) * q^88 + (3*b + 4) * q^91 + q^92 + (2*b - 2) * q^94 + (-3*b + 1) * q^95 + (b + 4) * q^97 + (3*b - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 3 * q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8} - 2 q^{10} + 7 q^{11} + 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} + q^{19} - 2 q^{20} + 7 q^{22} + 2 q^{23} + 2 q^{25} + 3 q^{26} + 3 q^{28} - 2 q^{29} - 5 q^{31} + 2 q^{32} - 3 q^{34} - 3 q^{35} + 16 q^{37} + q^{38} - 2 q^{40} + 9 q^{41} - 4 q^{43} + 7 q^{44} + 2 q^{46} - 2 q^{47} - 3 q^{49} + 2 q^{50} + 3 q^{52} + 8 q^{53} - 7 q^{55} + 3 q^{56} - 2 q^{58} + 14 q^{59} + 5 q^{61} - 5 q^{62} + 2 q^{64} - 3 q^{65} - 8 q^{67} - 3 q^{68} - 3 q^{70} + 29 q^{71} + 10 q^{73} + 16 q^{74} + q^{76} + 4 q^{77} - 2 q^{80} + 9 q^{82} - 8 q^{83} + 3 q^{85} - 4 q^{86} + 7 q^{88} + 11 q^{91} + 2 q^{92} - 2 q^{94} - q^{95} + 9 q^{97} - 3 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 3 * q^7 + 2 * q^8 - 2 * q^10 + 7 * q^11 + 3 * q^13 + 3 * q^14 + 2 * q^16 - 3 * q^17 + q^19 - 2 * q^20 + 7 * q^22 + 2 * q^23 + 2 * q^25 + 3 * q^26 + 3 * q^28 - 2 * q^29 - 5 * q^31 + 2 * q^32 - 3 * q^34 - 3 * q^35 + 16 * q^37 + q^38 - 2 * q^40 + 9 * q^41 - 4 * q^43 + 7 * q^44 + 2 * q^46 - 2 * q^47 - 3 * q^49 + 2 * q^50 + 3 * q^52 + 8 * q^53 - 7 * q^55 + 3 * q^56 - 2 * q^58 + 14 * q^59 + 5 * q^61 - 5 * q^62 + 2 * q^64 - 3 * q^65 - 8 * q^67 - 3 * q^68 - 3 * q^70 + 29 * q^71 + 10 * q^73 + 16 * q^74 + q^76 + 4 * q^77 - 2 * q^80 + 9 * q^82 - 8 * q^83 + 3 * q^85 - 4 * q^86 + 7 * q^88 + 11 * q^91 + 2 * q^92 - 2 * q^94 - q^95 + 9 * q^97 - 3 * q^98

## 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
 −1.30278 2.30278
1.00000 0 1.00000 −1.00000 0 −0.302776 1.00000 0 −1.00000
1.2 1.00000 0 1.00000 −1.00000 0 3.30278 1.00000 0 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$+1$$
$$23$$ $$-1$$

## 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 2070.2.a.w 2
3.b odd 2 1 230.2.a.b 2
12.b even 2 1 1840.2.a.j 2
15.d odd 2 1 1150.2.a.m 2
15.e even 4 2 1150.2.b.f 4
24.f even 2 1 7360.2.a.bu 2
24.h odd 2 1 7360.2.a.bc 2
60.h even 2 1 9200.2.a.ca 2
69.c even 2 1 5290.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.b 2 3.b odd 2 1
1150.2.a.m 2 15.d odd 2 1
1150.2.b.f 4 15.e even 4 2
1840.2.a.j 2 12.b even 2 1
2070.2.a.w 2 1.a even 1 1 trivial
5290.2.a.j 2 69.c even 2 1
7360.2.a.bc 2 24.h odd 2 1
7360.2.a.bu 2 24.f even 2 1
9200.2.a.ca 2 60.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2070))$$:

 $$T_{7}^{2} - 3T_{7} - 1$$ T7^2 - 3*T7 - 1 $$T_{11}^{2} - 7T_{11} + 9$$ T11^2 - 7*T11 + 9 $$T_{13}^{2} - 3T_{13} - 1$$ T13^2 - 3*T13 - 1 $$T_{17}^{2} + 3T_{17} - 27$$ T17^2 + 3*T17 - 27 $$T_{29}^{2} + 2T_{29} - 12$$ T29^2 + 2*T29 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 3T - 1$$
$11$ $$T^{2} - 7T + 9$$
$13$ $$T^{2} - 3T - 1$$
$17$ $$T^{2} + 3T - 27$$
$19$ $$T^{2} - T - 29$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} + 2T - 12$$
$31$ $$T^{2} + 5T - 23$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} - 9T - 9$$
$43$ $$T^{2} + 4T - 48$$
$47$ $$T^{2} + 2T - 12$$
$53$ $$T^{2} - 8T - 36$$
$59$ $$T^{2} - 14T + 36$$
$61$ $$T^{2} - 5T - 75$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2} - 29T + 207$$
$73$ $$T^{2} - 10T - 92$$
$79$ $$T^{2} - 208$$
$83$ $$T^{2} + 8T - 36$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 9T + 17$$