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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.

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

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 \( T_{3}^{2} + T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(140, [\chi])\).