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 disc. -71
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0354336158969\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{7}\)
Projective field Galois closure of 7.1.357911.1
Artin image size \(14\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\)

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.

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
71.b Odd 1 CM by \(\Q(\sqrt{-71}) \) yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(71, [\chi])\).