Properties

Label 140.1.p.a
Level 140
Weight 1
Character orbit 140.p
Analytic conductor 0.070
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM discriminant -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 \(6\), degree \(2\), minimal)

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: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.980.1
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.392000.1

$q$-expansion

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
7.c even 3 1 inner
140.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.1.p.a 2
3.b odd 2 1 1260.1.ci.b 2
4.b odd 2 1 140.1.p.b yes 2
5.b even 2 1 140.1.p.b yes 2
5.c odd 4 2 700.1.u.a 4
7.b odd 2 1 980.1.p.a 2
7.c even 3 1 inner 140.1.p.a 2
7.c even 3 1 980.1.f.c 1
7.d odd 6 1 980.1.f.d 1
7.d odd 6 1 980.1.p.a 2
8.b even 2 1 2240.1.bt.a 2
8.d odd 2 1 2240.1.bt.b 2
12.b even 2 1 1260.1.ci.a 2
15.d odd 2 1 1260.1.ci.a 2
20.d odd 2 1 CM 140.1.p.a 2
20.e even 4 2 700.1.u.a 4
21.h odd 6 1 1260.1.ci.b 2
28.d even 2 1 980.1.p.b 2
28.f even 6 1 980.1.f.a 1
28.f even 6 1 980.1.p.b 2
28.g odd 6 1 140.1.p.b yes 2
28.g odd 6 1 980.1.f.b 1
35.c odd 2 1 980.1.p.b 2
35.i odd 6 1 980.1.f.a 1
35.i odd 6 1 980.1.p.b 2
35.j even 6 1 140.1.p.b yes 2
35.j even 6 1 980.1.f.b 1
35.l odd 12 2 700.1.u.a 4
40.e odd 2 1 2240.1.bt.a 2
40.f even 2 1 2240.1.bt.b 2
56.k odd 6 1 2240.1.bt.b 2
56.p even 6 1 2240.1.bt.a 2
60.h even 2 1 1260.1.ci.b 2
84.n even 6 1 1260.1.ci.a 2
105.o odd 6 1 1260.1.ci.a 2
140.c even 2 1 980.1.p.a 2
140.p odd 6 1 inner 140.1.p.a 2
140.p odd 6 1 980.1.f.c 1
140.s even 6 1 980.1.f.d 1
140.s even 6 1 980.1.p.a 2
140.w even 12 2 700.1.u.a 4
280.bf even 6 1 2240.1.bt.b 2
280.bi odd 6 1 2240.1.bt.a 2
420.ba even 6 1 1260.1.ci.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.p.a 2 1.a even 1 1 trivial
140.1.p.a 2 7.c even 3 1 inner
140.1.p.a 2 20.d odd 2 1 CM
140.1.p.a 2 140.p odd 6 1 inner
140.1.p.b yes 2 4.b odd 2 1
140.1.p.b yes 2 5.b even 2 1
140.1.p.b yes 2 28.g odd 6 1
140.1.p.b yes 2 35.j even 6 1
700.1.u.a 4 5.c odd 4 2
700.1.u.a 4 20.e even 4 2
700.1.u.a 4 35.l odd 12 2
700.1.u.a 4 140.w even 12 2
980.1.f.a 1 28.f even 6 1
980.1.f.a 1 35.i odd 6 1
980.1.f.b 1 28.g odd 6 1
980.1.f.b 1 35.j even 6 1
980.1.f.c 1 7.c even 3 1
980.1.f.c 1 140.p odd 6 1
980.1.f.d 1 7.d odd 6 1
980.1.f.d 1 140.s even 6 1
980.1.p.a 2 7.b odd 2 1
980.1.p.a 2 7.d odd 6 1
980.1.p.a 2 140.c even 2 1
980.1.p.a 2 140.s even 6 1
980.1.p.b 2 28.d even 2 1
980.1.p.b 2 28.f even 6 1
980.1.p.b 2 35.c odd 2 1
980.1.p.b 2 35.i odd 6 1
1260.1.ci.a 2 12.b even 2 1
1260.1.ci.a 2 15.d odd 2 1
1260.1.ci.a 2 84.n even 6 1
1260.1.ci.a 2 105.o odd 6 1
1260.1.ci.b 2 3.b odd 2 1
1260.1.ci.b 2 21.h odd 6 1
1260.1.ci.b 2 60.h even 2 1
1260.1.ci.b 2 420.ba even 6 1
2240.1.bt.a 2 8.b even 2 1
2240.1.bt.a 2 40.e odd 2 1
2240.1.bt.a 2 56.p even 6 1
2240.1.bt.a 2 280.bi odd 6 1
2240.1.bt.b 2 8.d odd 2 1
2240.1.bt.b 2 40.f even 2 1
2240.1.bt.b 2 56.k odd 6 1
2240.1.bt.b 2 280.bf even 6 1

Hecke kernels

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

Hecke characteristic polynomials

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