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Properties

Label 140.1.p.b
Level 140
Weight 1
Character orbit 140.p
Analytic conductor 0.070
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM disc. -20
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 140.p (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0698691017686\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.980.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q\) \( + \zeta_{6} q^{2} \) \( + \zeta_{6}^{2} q^{3} \) \( + \zeta_{6}^{2} q^{4} \) \( -\zeta_{6} q^{5} \) \(- q^{6}\) \( -\zeta_{6}^{2} q^{7} \) \(- q^{8}\) \(+O(q^{10})\) \( q\) \( + \zeta_{6} q^{2} \) \( + \zeta_{6}^{2} q^{3} \) \( + \zeta_{6}^{2} q^{4} \) \( -\zeta_{6} q^{5} \) \(- q^{6}\) \( -\zeta_{6}^{2} q^{7} \) \(- q^{8}\) \( -\zeta_{6}^{2} q^{10} \) \( -\zeta_{6} q^{12} \) \(+ q^{14}\) \(+ q^{15}\) \( -\zeta_{6} q^{16} \) \(+ q^{20}\) \( + \zeta_{6} q^{21} \) \( -\zeta_{6} q^{23} \) \( -\zeta_{6}^{2} q^{24} \) \( + \zeta_{6}^{2} q^{25} \) \(- q^{27}\) \( + \zeta_{6} q^{28} \) \(- q^{29}\) \( + \zeta_{6} q^{30} \) \( -\zeta_{6}^{2} q^{32} \) \(- q^{35}\) \( + \zeta_{6} q^{40} \) \(- q^{41}\) \( + \zeta_{6}^{2} q^{42} \) \(+ q^{43}\) \( -\zeta_{6}^{2} q^{46} \) \( + 2 \zeta_{6} q^{47} \) \(+ q^{48}\) \( -\zeta_{6} q^{49} \) \(- q^{50}\) \( -\zeta_{6} q^{54} \) \( + \zeta_{6}^{2} q^{56} \) \( -\zeta_{6} q^{58} \) \( + \zeta_{6}^{2} q^{60} \) \( + \zeta_{6} q^{61} \) \(+ q^{64}\) \( + \zeta_{6}^{2} q^{67} \) \(+ q^{69}\) \( -\zeta_{6} q^{70} \) \( -\zeta_{6} q^{75} \) \( + \zeta_{6}^{2} q^{80} \) \( -\zeta_{6}^{2} q^{81} \) \( -\zeta_{6} q^{82} \) \(+ q^{83}\) \(- q^{84}\) \( + \zeta_{6} q^{86} \) \( -\zeta_{6}^{2} q^{87} \) \( + \zeta_{6} q^{89} \) \(+ q^{92}\) \( + 2 \zeta_{6}^{2} q^{94} \) \( + \zeta_{6} q^{96} \) \( -\zeta_{6}^{2} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut q^{10} \) \(\mathstrut -\mathstrut q^{12} \) \(\mathstrut +\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut 2q^{15} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut +\mathstrut 2q^{20} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut -\mathstrut q^{23} \) \(\mathstrut +\mathstrut q^{24} \) \(\mathstrut -\mathstrut q^{25} \) \(\mathstrut -\mathstrut 2q^{27} \) \(\mathstrut +\mathstrut q^{28} \) \(\mathstrut -\mathstrut 2q^{29} \) \(\mathstrut +\mathstrut q^{30} \) \(\mathstrut +\mathstrut q^{32} \) \(\mathstrut -\mathstrut 2q^{35} \) \(\mathstrut +\mathstrut q^{40} \) \(\mathstrut -\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut q^{42} \) \(\mathstrut +\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut q^{46} \) \(\mathstrut +\mathstrut 2q^{47} \) \(\mathstrut +\mathstrut 2q^{48} \) \(\mathstrut -\mathstrut q^{49} \) \(\mathstrut -\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut q^{54} \) \(\mathstrut -\mathstrut q^{56} \) \(\mathstrut -\mathstrut q^{58} \) \(\mathstrut -\mathstrut q^{60} \) \(\mathstrut +\mathstrut q^{61} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut q^{67} \) \(\mathstrut +\mathstrut 2q^{69} \) \(\mathstrut -\mathstrut q^{70} \) \(\mathstrut -\mathstrut q^{75} \) \(\mathstrut -\mathstrut q^{80} \) \(\mathstrut +\mathstrut q^{81} \) \(\mathstrut -\mathstrut q^{82} \) \(\mathstrut +\mathstrut 2q^{83} \) \(\mathstrut -\mathstrut 2q^{84} \) \(\mathstrut +\mathstrut q^{86} \) \(\mathstrut +\mathstrut q^{87} \) \(\mathstrut +\mathstrut q^{89} \) \(\mathstrut +\mathstrut 2q^{92} \) \(\mathstrut -\mathstrut 2q^{94} \) \(\mathstrut +\mathstrut q^{96} \) \(\mathstrut +\mathstrut q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Display 100 coefficients 10 coefficients

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 − 0.866025i −1.00000 0.500000 − 0.866025i −1.00000 0 0.500000 − 0.866025i
79.1 0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 + 0.866025i −1.00000 0.500000 + 0.866025i −1.00000 0 0.500000 + 0.866025i
\(n\): 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
20.d Odd 1 CM by \(\Q(\sqrt{-5}) \) yes
7.c Even 1 yes
140.p Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut +\mathstrut T_{3} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{1}^{\mathrm{new}}(140, [\chi])\).