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 $2$

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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.

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

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:

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

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