# Properties

 Label 1710.2.a.w Level $1710$ Weight $2$ Character orbit 1710.a Self dual yes Analytic conductor $13.654$ 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] = [1710,2,Mod(1,1710)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1710, 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("1710.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1710.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: $$13.6544187456$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} - \beta q^{7} + q^{8} +O(q^{10})$$ q + q^2 + q^4 - q^5 - b * q^7 + q^8 $$q + q^{2} + q^{4} - q^{5} - \beta q^{7} + q^{8} - q^{10} - 4 q^{11} + (3 \beta - 2) q^{13} - \beta q^{14} + q^{16} + (\beta - 6) q^{17} - q^{19} - q^{20} - 4 q^{22} - 3 \beta q^{23} + q^{25} + (3 \beta - 2) q^{26} - \beta q^{28} + (3 \beta - 2) q^{29} - 2 \beta q^{31} + q^{32} + (\beta - 6) q^{34} + \beta q^{35} - 6 q^{37} - q^{38} - q^{40} + ( - 4 \beta - 2) q^{41} + (2 \beta - 8) q^{43} - 4 q^{44} - 3 \beta q^{46} + ( - 4 \beta + 4) q^{47} + (\beta - 3) q^{49} + q^{50} + (3 \beta - 2) q^{52} + (\beta + 2) q^{53} + 4 q^{55} - \beta q^{56} + (3 \beta - 2) q^{58} - \beta q^{59} + (2 \beta + 6) q^{61} - 2 \beta q^{62} + q^{64} + ( - 3 \beta + 2) q^{65} - \beta q^{67} + (\beta - 6) q^{68} + \beta q^{70} - 4 \beta q^{71} + ( - 3 \beta + 6) q^{73} - 6 q^{74} - q^{76} + 4 \beta q^{77} - 2 \beta q^{79} - q^{80} + ( - 4 \beta - 2) q^{82} + (2 \beta - 8) q^{83} + ( - \beta + 6) q^{85} + (2 \beta - 8) q^{86} - 4 q^{88} - 2 q^{89} + ( - \beta - 12) q^{91} - 3 \beta q^{92} + ( - 4 \beta + 4) q^{94} + q^{95} + 6 q^{97} + (\beta - 3) q^{98} +O(q^{100})$$ q + q^2 + q^4 - q^5 - b * q^7 + q^8 - q^10 - 4 * q^11 + (3*b - 2) * q^13 - b * q^14 + q^16 + (b - 6) * q^17 - q^19 - q^20 - 4 * q^22 - 3*b * q^23 + q^25 + (3*b - 2) * q^26 - b * q^28 + (3*b - 2) * q^29 - 2*b * q^31 + q^32 + (b - 6) * q^34 + b * q^35 - 6 * q^37 - q^38 - q^40 + (-4*b - 2) * q^41 + (2*b - 8) * q^43 - 4 * q^44 - 3*b * q^46 + (-4*b + 4) * q^47 + (b - 3) * q^49 + q^50 + (3*b - 2) * q^52 + (b + 2) * q^53 + 4 * q^55 - b * q^56 + (3*b - 2) * q^58 - b * q^59 + (2*b + 6) * q^61 - 2*b * q^62 + q^64 + (-3*b + 2) * q^65 - b * q^67 + (b - 6) * q^68 + b * q^70 - 4*b * q^71 + (-3*b + 6) * q^73 - 6 * q^74 - q^76 + 4*b * q^77 - 2*b * q^79 - q^80 + (-4*b - 2) * q^82 + (2*b - 8) * q^83 + (-b + 6) * q^85 + (2*b - 8) * q^86 - 4 * q^88 - 2 * q^89 + (-b - 12) * q^91 - 3*b * q^92 + (-4*b + 4) * q^94 + q^95 + 6 * q^97 + (b - 3) * 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 $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - q^{7} + 2 q^{8} - 2 q^{10} - 8 q^{11} - q^{13} - q^{14} + 2 q^{16} - 11 q^{17} - 2 q^{19} - 2 q^{20} - 8 q^{22} - 3 q^{23} + 2 q^{25} - q^{26} - q^{28} - q^{29} - 2 q^{31} + 2 q^{32} - 11 q^{34} + q^{35} - 12 q^{37} - 2 q^{38} - 2 q^{40} - 8 q^{41} - 14 q^{43} - 8 q^{44} - 3 q^{46} + 4 q^{47} - 5 q^{49} + 2 q^{50} - q^{52} + 5 q^{53} + 8 q^{55} - q^{56} - q^{58} - q^{59} + 14 q^{61} - 2 q^{62} + 2 q^{64} + q^{65} - q^{67} - 11 q^{68} + q^{70} - 4 q^{71} + 9 q^{73} - 12 q^{74} - 2 q^{76} + 4 q^{77} - 2 q^{79} - 2 q^{80} - 8 q^{82} - 14 q^{83} + 11 q^{85} - 14 q^{86} - 8 q^{88} - 4 q^{89} - 25 q^{91} - 3 q^{92} + 4 q^{94} + 2 q^{95} + 12 q^{97} - 5 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - q^7 + 2 * q^8 - 2 * q^10 - 8 * q^11 - q^13 - q^14 + 2 * q^16 - 11 * q^17 - 2 * q^19 - 2 * q^20 - 8 * q^22 - 3 * q^23 + 2 * q^25 - q^26 - q^28 - q^29 - 2 * q^31 + 2 * q^32 - 11 * q^34 + q^35 - 12 * q^37 - 2 * q^38 - 2 * q^40 - 8 * q^41 - 14 * q^43 - 8 * q^44 - 3 * q^46 + 4 * q^47 - 5 * q^49 + 2 * q^50 - q^52 + 5 * q^53 + 8 * q^55 - q^56 - q^58 - q^59 + 14 * q^61 - 2 * q^62 + 2 * q^64 + q^65 - q^67 - 11 * q^68 + q^70 - 4 * q^71 + 9 * q^73 - 12 * q^74 - 2 * q^76 + 4 * q^77 - 2 * q^79 - 2 * q^80 - 8 * q^82 - 14 * q^83 + 11 * q^85 - 14 * q^86 - 8 * q^88 - 4 * q^89 - 25 * q^91 - 3 * q^92 + 4 * q^94 + 2 * q^95 + 12 * q^97 - 5 * 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
 2.56155 −1.56155
1.00000 0 1.00000 −1.00000 0 −2.56155 1.00000 0 −1.00000
1.2 1.00000 0 1.00000 −1.00000 0 1.56155 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$+1$$
$$19$$ $$+1$$

## 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 1710.2.a.w 2
3.b odd 2 1 190.2.a.d 2
5.b even 2 1 8550.2.a.br 2
12.b even 2 1 1520.2.a.n 2
15.d odd 2 1 950.2.a.h 2
15.e even 4 2 950.2.b.f 4
21.c even 2 1 9310.2.a.bc 2
24.f even 2 1 6080.2.a.bb 2
24.h odd 2 1 6080.2.a.bh 2
57.d even 2 1 3610.2.a.t 2
60.h even 2 1 7600.2.a.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.d 2 3.b odd 2 1
950.2.a.h 2 15.d odd 2 1
950.2.b.f 4 15.e even 4 2
1520.2.a.n 2 12.b even 2 1
1710.2.a.w 2 1.a even 1 1 trivial
3610.2.a.t 2 57.d even 2 1
6080.2.a.bb 2 24.f even 2 1
6080.2.a.bh 2 24.h odd 2 1
7600.2.a.y 2 60.h even 2 1
8550.2.a.br 2 5.b even 2 1
9310.2.a.bc 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1710))$$:

 $$T_{7}^{2} + T_{7} - 4$$ T7^2 + T7 - 4 $$T_{11} + 4$$ T11 + 4 $$T_{13}^{2} + T_{13} - 38$$ T13^2 + T13 - 38 $$T_{53}^{2} - 5T_{53} + 2$$ T53^2 - 5*T53 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + T - 4$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + T - 38$$
$17$ $$T^{2} + 11T + 26$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 3T - 36$$
$29$ $$T^{2} + T - 38$$
$31$ $$T^{2} + 2T - 16$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} + 8T - 52$$
$43$ $$T^{2} + 14T + 32$$
$47$ $$T^{2} - 4T - 64$$
$53$ $$T^{2} - 5T + 2$$
$59$ $$T^{2} + T - 4$$
$61$ $$T^{2} - 14T + 32$$
$67$ $$T^{2} + T - 4$$
$71$ $$T^{2} + 4T - 64$$
$73$ $$T^{2} - 9T - 18$$
$79$ $$T^{2} + 2T - 16$$
$83$ $$T^{2} + 14T + 32$$
$89$ $$(T + 2)^{2}$$
$97$ $$(T - 6)^{2}$$