# Properties

 Label 2070.2.a.u 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: $$\Q(\sqrt{5})$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{5} + \beta q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 - q^5 + b * q^7 - q^8 $$q - q^{2} + q^{4} - q^{5} + \beta q^{7} - q^{8} + q^{10} + ( - 3 \beta + 1) q^{11} + (5 \beta - 4) q^{13} - \beta q^{14} + q^{16} + (5 \beta - 3) q^{17} - 3 \beta q^{19} - q^{20} + (3 \beta - 1) q^{22} - q^{23} + q^{25} + ( - 5 \beta + 4) q^{26} + \beta q^{28} + ( - 2 \beta + 8) q^{29} + ( - 5 \beta + 6) q^{31} - q^{32} + ( - 5 \beta + 3) q^{34} - \beta q^{35} + ( - 4 \beta + 4) q^{37} + 3 \beta q^{38} + q^{40} + (7 \beta + 1) q^{41} + ( - 3 \beta + 1) q^{44} + q^{46} - 6 \beta q^{47} + (\beta - 6) q^{49} - q^{50} + (5 \beta - 4) q^{52} + (4 \beta + 2) q^{53} + (3 \beta - 1) q^{55} - \beta q^{56} + (2 \beta - 8) q^{58} + (6 \beta + 2) q^{59} + (7 \beta - 5) q^{61} + (5 \beta - 6) q^{62} + q^{64} + ( - 5 \beta + 4) q^{65} + ( - 4 \beta + 12) q^{67} + (5 \beta - 3) q^{68} + \beta q^{70} + ( - 5 \beta + 1) q^{71} + ( - 2 \beta + 2) q^{73} + (4 \beta - 4) q^{74} - 3 \beta q^{76} + ( - 2 \beta - 3) q^{77} + (4 \beta + 4) q^{79} - q^{80} + ( - 7 \beta - 1) q^{82} + ( - 8 \beta + 2) q^{83} + ( - 5 \beta + 3) q^{85} + (3 \beta - 1) q^{88} + ( - 4 \beta + 8) q^{89} + (\beta + 5) q^{91} - q^{92} + 6 \beta q^{94} + 3 \beta q^{95} + (\beta + 13) q^{97} + ( - \beta + 6) q^{98} +O(q^{100})$$ q - q^2 + q^4 - q^5 + b * q^7 - q^8 + q^10 + (-3*b + 1) * q^11 + (5*b - 4) * q^13 - b * q^14 + q^16 + (5*b - 3) * q^17 - 3*b * q^19 - q^20 + (3*b - 1) * q^22 - q^23 + q^25 + (-5*b + 4) * q^26 + b * q^28 + (-2*b + 8) * q^29 + (-5*b + 6) * q^31 - q^32 + (-5*b + 3) * q^34 - b * q^35 + (-4*b + 4) * q^37 + 3*b * q^38 + q^40 + (7*b + 1) * q^41 + (-3*b + 1) * q^44 + q^46 - 6*b * q^47 + (b - 6) * q^49 - q^50 + (5*b - 4) * q^52 + (4*b + 2) * q^53 + (3*b - 1) * q^55 - b * q^56 + (2*b - 8) * q^58 + (6*b + 2) * q^59 + (7*b - 5) * q^61 + (5*b - 6) * q^62 + q^64 + (-5*b + 4) * q^65 + (-4*b + 12) * q^67 + (5*b - 3) * q^68 + b * q^70 + (-5*b + 1) * q^71 + (-2*b + 2) * q^73 + (4*b - 4) * q^74 - 3*b * q^76 + (-2*b - 3) * q^77 + (4*b + 4) * q^79 - q^80 + (-7*b - 1) * q^82 + (-8*b + 2) * q^83 + (-5*b + 3) * q^85 + (3*b - 1) * q^88 + (-4*b + 8) * q^89 + (b + 5) * q^91 - q^92 + 6*b * q^94 + 3*b * q^95 + (b + 13) * q^97 + (-b + 6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} - 3 q^{13} - q^{14} + 2 q^{16} - q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 2 q^{23} + 2 q^{25} + 3 q^{26} + q^{28} + 14 q^{29} + 7 q^{31} - 2 q^{32} + q^{34} - q^{35} + 4 q^{37} + 3 q^{38} + 2 q^{40} + 9 q^{41} - q^{44} + 2 q^{46} - 6 q^{47} - 11 q^{49} - 2 q^{50} - 3 q^{52} + 8 q^{53} + q^{55} - q^{56} - 14 q^{58} + 10 q^{59} - 3 q^{61} - 7 q^{62} + 2 q^{64} + 3 q^{65} + 20 q^{67} - q^{68} + q^{70} - 3 q^{71} + 2 q^{73} - 4 q^{74} - 3 q^{76} - 8 q^{77} + 12 q^{79} - 2 q^{80} - 9 q^{82} - 4 q^{83} + q^{85} + q^{88} + 12 q^{89} + 11 q^{91} - 2 q^{92} + 6 q^{94} + 3 q^{95} + 27 q^{97} + 11 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8 + 2 * q^10 - q^11 - 3 * q^13 - q^14 + 2 * q^16 - q^17 - 3 * q^19 - 2 * q^20 + q^22 - 2 * q^23 + 2 * q^25 + 3 * q^26 + q^28 + 14 * q^29 + 7 * q^31 - 2 * q^32 + q^34 - q^35 + 4 * q^37 + 3 * q^38 + 2 * q^40 + 9 * q^41 - q^44 + 2 * q^46 - 6 * q^47 - 11 * q^49 - 2 * q^50 - 3 * q^52 + 8 * q^53 + q^55 - q^56 - 14 * q^58 + 10 * q^59 - 3 * q^61 - 7 * q^62 + 2 * q^64 + 3 * q^65 + 20 * q^67 - q^68 + q^70 - 3 * q^71 + 2 * q^73 - 4 * q^74 - 3 * q^76 - 8 * q^77 + 12 * q^79 - 2 * q^80 - 9 * q^82 - 4 * q^83 + q^85 + q^88 + 12 * q^89 + 11 * q^91 - 2 * q^92 + 6 * q^94 + 3 * q^95 + 27 * q^97 + 11 * 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
 −0.618034 1.61803
−1.00000 0 1.00000 −1.00000 0 −0.618034 −1.00000 0 1.00000
1.2 −1.00000 0 1.00000 −1.00000 0 1.61803 −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.u 2
3.b odd 2 1 230.2.a.c 2
12.b even 2 1 1840.2.a.l 2
15.d odd 2 1 1150.2.a.j 2
15.e even 4 2 1150.2.b.i 4
24.f even 2 1 7360.2.a.bn 2
24.h odd 2 1 7360.2.a.bh 2
60.h even 2 1 9200.2.a.bu 2
69.c even 2 1 5290.2.a.o 2

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

 $$T_{7}^{2} - T_{7} - 1$$ T7^2 - T7 - 1 $$T_{11}^{2} + T_{11} - 11$$ T11^2 + T11 - 11 $$T_{13}^{2} + 3T_{13} - 29$$ T13^2 + 3*T13 - 29 $$T_{17}^{2} + T_{17} - 31$$ T17^2 + T17 - 31 $$T_{29}^{2} - 14T_{29} + 44$$ T29^2 - 14*T29 + 44

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - T - 1$$
$11$ $$T^{2} + T - 11$$
$13$ $$T^{2} + 3T - 29$$
$17$ $$T^{2} + T - 31$$
$19$ $$T^{2} + 3T - 9$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} - 14T + 44$$
$31$ $$T^{2} - 7T - 19$$
$37$ $$T^{2} - 4T - 16$$
$41$ $$T^{2} - 9T - 41$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 6T - 36$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} - 10T - 20$$
$61$ $$T^{2} + 3T - 59$$
$67$ $$T^{2} - 20T + 80$$
$71$ $$T^{2} + 3T - 29$$
$73$ $$T^{2} - 2T - 4$$
$79$ $$T^{2} - 12T + 16$$
$83$ $$T^{2} + 4T - 76$$
$89$ $$T^{2} - 12T + 16$$
$97$ $$T^{2} - 27T + 181$$