# 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

## Newspace parameters

 Level: $$N$$ $$=$$ $$71$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 71.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.0354336158969$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ 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

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.

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

## 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.

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

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$5$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$7$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$11$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$13$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$17$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$19$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$23$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$29$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$31$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$37$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$41$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$43$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$47$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$53$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$59$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$61$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$67$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$71$ $$( 1 - T )^{3}$$
$73$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$79$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$83$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$89$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$97$ $$( 1 - T )^{3}( 1 + T )^{3}$$