Properties

Label 143.1.d.b
Level 143
Weight 1
Character orbit 143.d
Self dual Yes
Analytic conductor 0.071
Analytic rank 0
Dimension 2
Projective image \(D_{5}\)
CM disc. -143
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 143 = 11 \cdot 13 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 143.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0713662968065\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.20449.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 - \beta ) q^{2} \) \( + ( -1 + \beta ) q^{3} \) \( + ( 1 - \beta ) q^{4} \) \( + ( -2 + \beta ) q^{6} \) \( + \beta q^{7} \) \(+ q^{8}\) \( + ( 1 - \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 - \beta ) q^{2} \) \( + ( -1 + \beta ) q^{3} \) \( + ( 1 - \beta ) q^{4} \) \( + ( -2 + \beta ) q^{6} \) \( + \beta q^{7} \) \(+ q^{8}\) \( + ( 1 - \beta ) q^{9} \) \(- q^{11}\) \( + ( -2 + \beta ) q^{12} \) \(- q^{13}\) \(- q^{14}\) \( + ( 2 - \beta ) q^{18} \) \( + ( 1 - \beta ) q^{19} \) \(+ q^{21}\) \( + ( -1 + \beta ) q^{22} \) \( -\beta q^{23} \) \( + ( -1 + \beta ) q^{24} \) \(+ q^{25}\) \( + ( -1 + \beta ) q^{26} \) \(- q^{27}\) \(- q^{28}\) \(- q^{32}\) \( + ( 1 - \beta ) q^{33} \) \( + ( 2 - \beta ) q^{36} \) \( + ( 2 - \beta ) q^{38} \) \( + ( 1 - \beta ) q^{39} \) \( + \beta q^{41} \) \( + ( 1 - \beta ) q^{42} \) \( + ( -1 + \beta ) q^{44} \) \(+ q^{46}\) \( + \beta q^{49} \) \( + ( 1 - \beta ) q^{50} \) \( + ( -1 + \beta ) q^{52} \) \( + ( -1 + \beta ) q^{53} \) \( + ( -1 + \beta ) q^{54} \) \( + \beta q^{56} \) \( + ( -2 + \beta ) q^{57} \) \(- q^{63}\) \( + ( -1 + \beta ) q^{64} \) \( + ( 2 - \beta ) q^{66} \) \(- q^{69}\) \( + ( 1 - \beta ) q^{72} \) \( + ( 1 - \beta ) q^{73} \) \( + ( -1 + \beta ) q^{75} \) \( + ( 2 - \beta ) q^{76} \) \( -\beta q^{77} \) \( + ( 2 - \beta ) q^{78} \) \(- q^{82}\) \( + \beta q^{83} \) \( + ( 1 - \beta ) q^{84} \) \(- q^{88}\) \( -\beta q^{91} \) \(+ q^{92}\) \( + ( 1 - \beta ) q^{96} \) \(- q^{98}\) \( + ( -1 + \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut 3q^{6} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut 3q^{6} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut -\mathstrut 2q^{11} \) \(\mathstrut -\mathstrut 3q^{12} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut 3q^{18} \) \(\mathstrut +\mathstrut q^{19} \) \(\mathstrut +\mathstrut 2q^{21} \) \(\mathstrut -\mathstrut q^{22} \) \(\mathstrut -\mathstrut q^{23} \) \(\mathstrut -\mathstrut q^{24} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut q^{26} \) \(\mathstrut -\mathstrut 2q^{27} \) \(\mathstrut -\mathstrut 2q^{28} \) \(\mathstrut -\mathstrut 2q^{32} \) \(\mathstrut +\mathstrut q^{33} \) \(\mathstrut +\mathstrut 3q^{36} \) \(\mathstrut +\mathstrut 3q^{38} \) \(\mathstrut +\mathstrut q^{39} \) \(\mathstrut +\mathstrut q^{41} \) \(\mathstrut +\mathstrut q^{42} \) \(\mathstrut -\mathstrut q^{44} \) \(\mathstrut +\mathstrut 2q^{46} \) \(\mathstrut +\mathstrut q^{49} \) \(\mathstrut +\mathstrut q^{50} \) \(\mathstrut -\mathstrut q^{52} \) \(\mathstrut -\mathstrut q^{53} \) \(\mathstrut -\mathstrut q^{54} \) \(\mathstrut +\mathstrut q^{56} \) \(\mathstrut -\mathstrut 3q^{57} \) \(\mathstrut -\mathstrut 2q^{63} \) \(\mathstrut -\mathstrut q^{64} \) \(\mathstrut +\mathstrut 3q^{66} \) \(\mathstrut -\mathstrut 2q^{69} \) \(\mathstrut +\mathstrut q^{72} \) \(\mathstrut +\mathstrut q^{73} \) \(\mathstrut -\mathstrut q^{75} \) \(\mathstrut +\mathstrut 3q^{76} \) \(\mathstrut -\mathstrut q^{77} \) \(\mathstrut +\mathstrut 3q^{78} \) \(\mathstrut -\mathstrut 2q^{82} \) \(\mathstrut +\mathstrut q^{83} \) \(\mathstrut +\mathstrut q^{84} \) \(\mathstrut -\mathstrut 2q^{88} \) \(\mathstrut -\mathstrut q^{91} \) \(\mathstrut +\mathstrut 2q^{92} \) \(\mathstrut +\mathstrut q^{96} \) \(\mathstrut -\mathstrut 2q^{98} \) \(\mathstrut -\mathstrut q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/143\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(79\)
\(\chi(n)\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
142.1
1.61803
−0.618034
−0.618034 0.618034 −0.618034 0 −0.381966 1.61803 1.00000 −0.618034 0
142.2 1.61803 −1.61803 1.61803 0 −2.61803 −0.618034 1.00000 1.61803 0
\(n\): 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
143.d Odd 1 CM by \(\Q(\sqrt{-143}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut T_{2} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{1}^{\mathrm{new}}(143, [\chi])\).