# Properties

 Label 1710.2.d.f Level $1710$ Weight $2$ Character orbit 1710.d Analytic conductor $13.654$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1710,2,Mod(1369,1710)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1710.1369");

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.d (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$13.6544187456$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 570) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - q^{4} - \beta_{4} q^{5} + ( - \beta_{5} - \beta_{2}) q^{7} - \beta_1 q^{8}+O(q^{10})$$ q + b1 * q^2 - q^4 - b4 * q^5 + (-b5 - b2) * q^7 - b1 * q^8 $$q + \beta_1 q^{2} - q^{4} - \beta_{4} q^{5} + ( - \beta_{5} - \beta_{2}) q^{7} - \beta_1 q^{8} + \beta_{2} q^{10} + (\beta_{5} - \beta_{2} - 2) q^{11} + ( - \beta_{5} - \beta_{4} + \cdots + 2 \beta_1) q^{13}+ \cdots + (2 \beta_{5} + 2 \beta_{4} + \cdots - 5 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 - q^4 - b4 * q^5 + (-b5 - b2) * q^7 - b1 * q^8 + b2 * q^10 + (b5 - b2 - 2) * q^11 + (-b5 - b4 + b3 - b2 + 2*b1) * q^13 + (-b4 - b3) * q^14 + q^16 + (b5 + b4 - b3 + b2) * q^17 - q^19 + b4 * q^20 + (-b4 + b3 - 2*b1) * q^22 + (b4 - b3 + 2*b1) * q^23 + (b5 + 2*b3 - b2 + 2*b1 + 1) * q^25 + (-b5 - b4 - b3 + b2 - 2) * q^26 + (b5 + b2) * q^28 + (b5 - b2 + 4) * q^29 + (-b4 - b3) * q^31 + b1 * q^32 + (b5 + b4 + b3 - b2) * q^34 + (2*b5 + b4 - b3 - 6*b1 + 2) * q^35 + (b5 + b4 - b3 + b2 - 2*b1) * q^37 - b1 * q^38 - b2 * q^40 + (2*b5 - b4 - b3 - 2*b2) * q^41 + (2*b5 + 3*b4 - 3*b3 + 2*b2 - 4*b1) * q^43 + (-b5 + b2 + 2) * q^44 + (b5 - b2 - 2) * q^46 + (b4 - b3 + 2*b1) * q^47 + (-2*b5 - 2*b4 - 2*b3 + 2*b2 - 5) * q^49 + (-2*b5 - b4 + b3 + b1 - 2) * q^50 + (b5 + b4 - b3 + b2 - 2*b1) * q^52 + 6*b1 * q^53 + (2*b5 + 3*b4 - b3 + 4*b1 + 2) * q^55 + (b4 + b3) * q^56 + (-b4 + b3 + 4*b1) * q^58 + (b5 - b4 - b3 - b2 + 4) * q^59 + (2*b5 - 2*b2 + 6) * q^61 + (b5 + b2) * q^62 - q^64 + (3*b5 + b4 + b3 + b2 - 4*b1 - 2) * q^65 + (2*b5 + 2*b4 - 2*b3 + 2*b2 - 8*b1) * q^67 + (-b5 - b4 + b3 - b2) * q^68 + (b5 + 2*b3 - b2 + 2*b1 + 6) * q^70 + (-2*b5 - 2*b4 - 2*b3 + 2*b2 - 8) * q^71 + (-2*b5 - 2*b4 + 2*b3 - 2*b2 + 4*b1) * q^73 + (b5 + b4 + b3 - b2 + 2) * q^74 + q^76 + (2*b5 + 2*b4 - 2*b3 + 2*b2 - 4*b1) * q^77 + (-b4 - b3) * q^79 - b4 * q^80 + (b5 - 2*b4 + 2*b3 + b2) * q^82 + (3*b5 - b4 + b3 + 3*b2 + 2*b1) * q^83 + (-3*b5 - b4 - b3 + b2 + 4*b1 + 2) * q^85 + (3*b5 + 2*b4 + 2*b3 - 3*b2 + 4) * q^86 + (b4 - b3 + 2*b1) * q^88 + (b4 + b3 + 8) * q^89 + (-4*b4 - 4*b3 - 8) * q^91 + (-b4 + b3 - 2*b1) * q^92 + (b5 - b2 - 2) * q^94 + b4 * q^95 + (b4 - b3 - 8*b1) * q^97 + (2*b5 + 2*b4 - 2*b3 + 2*b2 - 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4}+O(q^{10})$$ 6 * q - 6 * q^4 $$6 q - 6 q^{4} + 2 q^{10} - 16 q^{11} + 6 q^{16} - 6 q^{19} + 2 q^{25} - 8 q^{26} + 20 q^{29} - 4 q^{34} + 8 q^{35} - 2 q^{40} - 8 q^{41} + 16 q^{44} - 16 q^{46} - 22 q^{49} - 8 q^{50} + 8 q^{55} + 20 q^{59} + 28 q^{61} - 6 q^{64} - 16 q^{65} + 32 q^{70} - 40 q^{71} + 8 q^{74} + 6 q^{76} + 20 q^{85} + 12 q^{86} + 48 q^{89} - 48 q^{91} - 16 q^{94}+O(q^{100})$$ 6 * q - 6 * q^4 + 2 * q^10 - 16 * q^11 + 6 * q^16 - 6 * q^19 + 2 * q^25 - 8 * q^26 + 20 * q^29 - 4 * q^34 + 8 * q^35 - 2 * q^40 - 8 * q^41 + 16 * q^44 - 16 * q^46 - 22 * q^49 - 8 * q^50 + 8 * q^55 + 20 * q^59 + 28 * q^61 - 6 * q^64 - 16 * q^65 + 32 * q^70 - 40 * q^71 + 8 * q^74 + 6 * q^76 + 20 * q^85 + 12 * q^86 + 48 * q^89 - 48 * q^91 - 16 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( -7\nu^{5} + 10\nu^{4} - 5\nu^{3} - 30\nu^{2} - 32\nu + 13 ) / 23$$ (-7*v^5 + 10*v^4 - 5*v^3 - 30*v^2 - 32*v + 13) / 23 $$\beta_{2}$$ $$=$$ $$( -9\nu^{5} + 3\nu^{4} + 10\nu^{3} - 32\nu^{2} - 74\nu - 3 ) / 23$$ (-9*v^5 + 3*v^4 + 10*v^3 - 32*v^2 - 74*v - 3) / 23 $$\beta_{3}$$ $$=$$ $$( -10\nu^{5} + 11\nu^{4} - 17\nu^{3} - 10\nu^{2} - 72\nu - 11 ) / 23$$ (-10*v^5 + 11*v^4 - 17*v^3 - 10*v^2 - 72*v - 11) / 23 $$\beta_{4}$$ $$=$$ $$( 12\nu^{5} - 27\nu^{4} + 25\nu^{3} + 12\nu^{2} + 68\nu - 65 ) / 23$$ (12*v^5 - 27*v^4 + 25*v^3 + 12*v^2 + 68*v - 65) / 23 $$\beta_{5}$$ $$=$$ $$( -19\nu^{5} + 37\nu^{4} - 30\nu^{3} - 42\nu^{2} - 54\nu + 55 ) / 23$$ (-19*v^5 + 37*v^4 - 30*v^3 - 42*v^2 - 54*v + 55) / 23
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} - \beta _1 + 1 ) / 2$$ (b5 + b4 - b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + \beta_{2} - 4\beta_1 ) / 2$$ (b5 + b2 - 4*b1) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{5} - \beta_{4} - 3\beta_{3} + 3\beta_{2} - 4\beta _1 - 4 ) / 2$$ (b5 - b4 - 3*b3 + 3*b2 - 4*b1 - 4) / 2 $$\nu^{4}$$ $$=$$ $$( -\beta_{5} - 5\beta_{4} - 5\beta_{3} + \beta_{2} - 14 ) / 2$$ (-b5 - 5*b4 - 5*b3 + b2 - 14) / 2 $$\nu^{5}$$ $$=$$ $$( -11\beta_{5} - 11\beta_{4} - 5\beta_{3} - 5\beta_{2} + 18\beta _1 - 18 ) / 2$$ (-11*b5 - 11*b4 - 5*b3 - 5*b2 + 18*b1 - 18) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1369.1
 1.45161 + 1.45161i −0.854638 − 0.854638i 0.403032 + 0.403032i 1.45161 − 1.45161i −0.854638 + 0.854638i 0.403032 − 0.403032i
1.00000i 0 −1.00000 −2.21432 + 0.311108i 0 4.42864i 1.00000i 0 0.311108 + 2.21432i
1369.2 1.00000i 0 −1.00000 0.539189 + 2.17009i 0 1.07838i 1.00000i 0 2.17009 0.539189i
1369.3 1.00000i 0 −1.00000 1.67513 1.48119i 0 3.35026i 1.00000i 0 −1.48119 1.67513i
1369.4 1.00000i 0 −1.00000 −2.21432 0.311108i 0 4.42864i 1.00000i 0 0.311108 2.21432i
1369.5 1.00000i 0 −1.00000 0.539189 2.17009i 0 1.07838i 1.00000i 0 2.17009 + 0.539189i
1369.6 1.00000i 0 −1.00000 1.67513 + 1.48119i 0 3.35026i 1.00000i 0 −1.48119 + 1.67513i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1369.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.d.f 6
3.b odd 2 1 570.2.d.c 6
5.b even 2 1 inner 1710.2.d.f 6
5.c odd 4 1 8550.2.a.ce 3
5.c odd 4 1 8550.2.a.cq 3
15.d odd 2 1 570.2.d.c 6
15.e even 4 1 2850.2.a.bl 3
15.e even 4 1 2850.2.a.bm 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.d.c 6 3.b odd 2 1
570.2.d.c 6 15.d odd 2 1
1710.2.d.f 6 1.a even 1 1 trivial
1710.2.d.f 6 5.b even 2 1 inner
2850.2.a.bl 3 15.e even 4 1
2850.2.a.bm 3 15.e even 4 1
8550.2.a.ce 3 5.c odd 4 1
8550.2.a.cq 3 5.c odd 4 1

## Hecke kernels

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

 $$T_{7}^{6} + 32T_{7}^{4} + 256T_{7}^{2} + 256$$ T7^6 + 32*T7^4 + 256*T7^2 + 256 $$T_{11}^{3} + 8T_{11}^{2} + 8T_{11} - 16$$ T11^3 + 8*T11^2 + 8*T11 - 16 $$T_{13}^{6} + 48T_{13}^{4} + 512T_{13}^{2} + 1024$$ T13^6 + 48*T13^4 + 512*T13^2 + 1024

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{3}$$
$3$ $$T^{6}$$
$5$ $$T^{6} - T^{4} + \cdots + 125$$
$7$ $$T^{6} + 32 T^{4} + \cdots + 256$$
$11$ $$(T^{3} + 8 T^{2} + 8 T - 16)^{2}$$
$13$ $$T^{6} + 48 T^{4} + \cdots + 1024$$
$17$ $$T^{6} + 44 T^{4} + \cdots + 64$$
$19$ $$(T + 1)^{6}$$
$23$ $$T^{6} + 48 T^{4} + \cdots + 256$$
$29$ $$(T^{3} - 10 T^{2} + 20 T + 8)^{2}$$
$31$ $$(T^{3} - 16 T + 16)^{2}$$
$37$ $$T^{6} + 48 T^{4} + \cdots + 1024$$
$41$ $$(T^{3} + 4 T^{2} + \cdots - 400)^{2}$$
$43$ $$T^{6} + 284 T^{4} + \cdots + 577600$$
$47$ $$T^{6} + 48 T^{4} + \cdots + 256$$
$53$ $$(T^{2} + 36)^{3}$$
$59$ $$(T^{3} - 10 T^{2} - 4 T + 8)^{2}$$
$61$ $$(T^{3} - 14 T^{2} + \cdots + 152)^{2}$$
$67$ $$T^{6} + 304 T^{4} + \cdots + 102400$$
$71$ $$(T^{3} + 20 T^{2} + \cdots - 320)^{2}$$
$73$ $$T^{6} + 192 T^{4} + \cdots + 65536$$
$79$ $$(T^{3} - 16 T + 16)^{2}$$
$83$ $$T^{6} + 368 T^{4} + \cdots + 25600$$
$89$ $$(T^{3} - 24 T^{2} + \cdots - 400)^{2}$$
$97$ $$T^{6} + 188 T^{4} + \cdots + 87616$$