# Properties

 Label 71.1.b.a Level $71$ Weight $1$ Character orbit 71.b Self dual yes Analytic conductor $0.035$ Analytic rank $0$ Dimension $3$ Projective image $D_{7}$ CM discriminant -71 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [71,1,Mod(70,71)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(71, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("71.70");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$71$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 71.b (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: yes Analytic conductor: $$0.0354336158969$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{7}$$ Projective field: Galois closure of 7.1.357911.1 Artin image: $D_7$ Artin field: Galois closure of 7.1.357911.1

## $q$-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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} + 1) q^{4} + \beta_{2} q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{2} - 1) q^{8} + ( - \beta_1 + 1) q^{9}+O(q^{10})$$ q - b1 * q^2 + (-b2 + b1 - 1) * q^3 + (b2 + 1) * q^4 + b2 * q^5 + (b1 - 1) * q^6 + (-b2 - 1) * q^8 + (-b1 + 1) * q^9 $$q - \beta_1 q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} + 1) q^{4} + \beta_{2} q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{2} - 1) q^{8} + ( - \beta_1 + 1) q^{9} + ( - \beta_{2} - 1) q^{10} - q^{12} + (\beta_{2} - \beta_1) q^{15} + \beta_1 q^{16} + (\beta_{2} - \beta_1 + 2) q^{18} - \beta_1 q^{19} + (\beta_1 + 1) q^{20} + q^{24} + ( - \beta_{2} + \beta_1) q^{25} + (\beta_1 - 1) q^{27} + ( - \beta_{2} + \beta_1 - 1) q^{29} + q^{30} - q^{32} - \beta_1 q^{36} - \beta_1 q^{37} + (\beta_{2} + 2) q^{38} + ( - \beta_1 - 1) q^{40} + \beta_{2} q^{43} - q^{45} + ( - \beta_1 + 1) q^{48} + q^{49} - q^{50} + ( - \beta_{2} + \beta_1 - 2) q^{54} + (\beta_1 - 1) q^{57} + (\beta_1 - 1) q^{58} - \beta_{2} q^{60} + q^{71} + \beta_1 q^{72} + \beta_{2} q^{73} + (\beta_{2} + 2) q^{74} + ( - \beta_{2} + 1) q^{75} + ( - \beta_{2} - \beta_1 - 1) q^{76} + \beta_{2} q^{79} + (\beta_{2} + 1) q^{80} + (\beta_{2} - \beta_1 + 1) q^{81} - \beta_1 q^{83} + ( - \beta_{2} - 1) q^{86} + ( - \beta_1 + 2) q^{87} + ( - \beta_{2} + \beta_1 - 1) q^{89} + \beta_1 q^{90} + ( - \beta_{2} - 1) q^{95} + (\beta_{2} - \beta_1 + 1) q^{96} - \beta_1 q^{98}+O(q^{100})$$ q - b1 * q^2 + (-b2 + b1 - 1) * q^3 + (b2 + 1) * q^4 + b2 * q^5 + (b1 - 1) * q^6 + (-b2 - 1) * q^8 + (-b1 + 1) * q^9 + (-b2 - 1) * q^10 - q^12 + (b2 - b1) * q^15 + b1 * q^16 + (b2 - b1 + 2) * q^18 - b1 * q^19 + (b1 + 1) * q^20 + q^24 + (-b2 + b1) * q^25 + (b1 - 1) * q^27 + (-b2 + b1 - 1) * q^29 + q^30 - q^32 - b1 * q^36 - b1 * q^37 + (b2 + 2) * q^38 + (-b1 - 1) * q^40 + b2 * q^43 - q^45 + (-b1 + 1) * q^48 + q^49 - q^50 + (-b2 + b1 - 2) * q^54 + (b1 - 1) * q^57 + (b1 - 1) * q^58 - b2 * q^60 + q^71 + b1 * q^72 + b2 * q^73 + (b2 + 2) * q^74 + (-b2 + 1) * q^75 + (-b2 - b1 - 1) * q^76 + b2 * q^79 + (b2 + 1) * q^80 + (b2 - b1 + 1) * q^81 - b1 * q^83 + (-b2 - 1) * q^86 + (-b1 + 2) * q^87 + (-b2 + b1 - 1) * q^89 + b1 * q^90 + (-b2 - 1) * q^95 + (b2 - b1 + 1) * q^96 - b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} - q^{3} + 2 q^{4} - q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 3 * q - q^2 - q^3 + 2 * q^4 - q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9 $$3 q - q^{2} - q^{3} + 2 q^{4} - q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 3 q^{12} - 2 q^{15} + q^{16} + 4 q^{18} - q^{19} + 4 q^{20} + 3 q^{24} + 2 q^{25} - 2 q^{27} - q^{29} + 3 q^{30} - 3 q^{32} - q^{36} - q^{37} + 5 q^{38} - 4 q^{40} - q^{43} - 3 q^{45} + 2 q^{48} + 3 q^{49} - 3 q^{50} - 4 q^{54} - 2 q^{57} - 2 q^{58} + q^{60} + 3 q^{71} + q^{72} - q^{73} + 5 q^{74} + 4 q^{75} - 3 q^{76} - q^{79} + 2 q^{80} + q^{81} - q^{83} - 2 q^{86} + 5 q^{87} - q^{89} + q^{90} - 2 q^{95} + q^{96} - q^{98}+O(q^{100})$$ 3 * q - q^2 - q^3 + 2 * q^4 - q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 3 * q^12 - 2 * q^15 + q^16 + 4 * q^18 - q^19 + 4 * q^20 + 3 * q^24 + 2 * q^25 - 2 * q^27 - q^29 + 3 * q^30 - 3 * q^32 - q^36 - q^37 + 5 * q^38 - 4 * q^40 - q^43 - 3 * q^45 + 2 * q^48 + 3 * q^49 - 3 * q^50 - 4 * q^54 - 2 * q^57 - 2 * q^58 + q^60 + 3 * q^71 + q^72 - q^73 + 5 * q^74 + 4 * q^75 - 3 * q^76 - q^79 + 2 * q^80 + q^81 - q^83 - 2 * q^86 + 5 * q^87 - q^89 + q^90 - 2 * q^95 + q^96 - q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/71\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
70.1
 1.80194 0.445042 −1.24698
−1.80194 −0.445042 2.24698 1.24698 0.801938 0 −2.24698 −0.801938 −2.24698
70.2 −0.445042 1.24698 −0.801938 −1.80194 −0.554958 0 0.801938 0.554958 0.801938
70.3 1.24698 −1.80194 0.554958 −0.445042 −2.24698 0 −0.554958 2.24698 −0.554958
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.b odd 2 1 CM by $$\Q(\sqrt{-71})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.1.b.a 3
3.b odd 2 1 639.1.d.a 3
4.b odd 2 1 1136.1.h.a 3
5.b even 2 1 1775.1.d.b 3
5.c odd 4 2 1775.1.c.a 6
7.b odd 2 1 3479.1.d.e 3
7.c even 3 2 3479.1.g.e 6
7.d odd 6 2 3479.1.g.d 6
71.b odd 2 1 CM 71.1.b.a 3
213.b even 2 1 639.1.d.a 3
284.c even 2 1 1136.1.h.a 3
355.c odd 2 1 1775.1.d.b 3
355.e even 4 2 1775.1.c.a 6
497.b even 2 1 3479.1.d.e 3
497.g odd 6 2 3479.1.g.e 6
497.i even 6 2 3479.1.g.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.1.b.a 3 1.a even 1 1 trivial
71.1.b.a 3 71.b odd 2 1 CM
639.1.d.a 3 3.b odd 2 1
639.1.d.a 3 213.b even 2 1
1136.1.h.a 3 4.b odd 2 1
1136.1.h.a 3 284.c even 2 1
1775.1.c.a 6 5.c odd 4 2
1775.1.c.a 6 355.e even 4 2
1775.1.d.b 3 5.b even 2 1
1775.1.d.b 3 355.c odd 2 1
3479.1.d.e 3 7.b odd 2 1
3479.1.d.e 3 497.b even 2 1
3479.1.g.d 6 7.d odd 6 2
3479.1.g.d 6 497.i even 6 2
3479.1.g.e 6 7.c even 3 2
3479.1.g.e 6 497.g odd 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(71, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 2T - 1$$
$3$ $$T^{3} + T^{2} - 2T - 1$$
$5$ $$T^{3} + T^{2} - 2T - 1$$
$7$ $$T^{3}$$
$11$ $$T^{3}$$
$13$ $$T^{3}$$
$17$ $$T^{3}$$
$19$ $$T^{3} + T^{2} - 2T - 1$$
$23$ $$T^{3}$$
$29$ $$T^{3} + T^{2} - 2T - 1$$
$31$ $$T^{3}$$
$37$ $$T^{3} + T^{2} - 2T - 1$$
$41$ $$T^{3}$$
$43$ $$T^{3} + T^{2} - 2T - 1$$
$47$ $$T^{3}$$
$53$ $$T^{3}$$
$59$ $$T^{3}$$
$61$ $$T^{3}$$
$67$ $$T^{3}$$
$71$ $$(T - 1)^{3}$$
$73$ $$T^{3} + T^{2} - 2T - 1$$
$79$ $$T^{3} + T^{2} - 2T - 1$$
$83$ $$T^{3} + T^{2} - 2T - 1$$
$89$ $$T^{3} + T^{2} - 2T - 1$$
$97$ $$T^{3}$$