Properties

Label 140.2.g.a
Level $140$
Weight $2$
Character orbit 140.g
Analytic conductor $1.118$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + \zeta_{12}^{3} q^{5} + ( -2 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{6} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + ( -2 - 2 \zeta_{12}^{3} ) q^{8} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + \zeta_{12}^{3} q^{5} + ( -2 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{6} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + ( -2 - 2 \zeta_{12}^{3} ) q^{8} + ( \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{10} + ( -1 + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{11} + ( 2 + 2 \zeta_{12}^{2} ) q^{12} + ( 2 - 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{13} + ( -2 + \zeta_{12} - \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{14} + ( -1 + 2 \zeta_{12}^{2} ) q^{15} + 4 \zeta_{12}^{2} q^{16} + ( 2 - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{17} + 6 q^{19} + ( -2 + 2 \zeta_{12}^{2} ) q^{20} + ( 1 + 4 \zeta_{12}^{2} ) q^{21} + ( 2 + \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{22} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{23} + ( 2 - 4 \zeta_{12} - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{24} - q^{25} + ( -4 - \zeta_{12} + 5 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{26} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( -2 + 6 \zeta_{12}^{2} ) q^{28} + ( 1 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{29} + ( 2 - \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{30} -6 q^{31} + ( 4 - 4 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{32} + ( -2 + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{33} + ( -4 + 5 \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{34} + ( -3 + 2 \zeta_{12}^{2} ) q^{35} + ( -6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{37} + ( -6 \zeta_{12} - 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{38} + ( 3 - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{39} + ( 2 - 2 \zeta_{12}^{3} ) q^{40} + ( 2 - 4 \zeta_{12}^{2} ) q^{41} + ( 4 - 5 \zeta_{12} - 5 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{42} + 2 \zeta_{12}^{3} q^{43} + ( -4 - 2 \zeta_{12} + 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{44} + ( -4 + 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{46} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{47} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{48} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{50} + ( -3 + 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{51} + ( 6 + 4 \zeta_{12} - 6 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{52} + 2 q^{53} + ( 6 + 3 \zeta_{12} - 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{54} + ( -2 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{55} + ( 6 - 4 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{56} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{57} + ( 8 + 3 \zeta_{12} - 5 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{58} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{59} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{60} + ( -2 + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{61} + ( 6 \zeta_{12} + 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{62} + 8 \zeta_{12}^{3} q^{64} + ( 3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{65} + ( 4 + \zeta_{12} - 5 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{66} + ( -2 + 4 \zeta_{12}^{2} ) q^{67} + ( -6 + 4 \zeta_{12} + 6 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{68} + ( 2 - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{69} + ( 2 + \zeta_{12} + \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{70} + ( 2 - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{71} + ( -4 + 8 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{73} + ( -4 + 4 \zeta_{12} + 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{74} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{75} + 12 \zeta_{12} q^{76} + ( -6 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{77} + ( -6 - 3 \zeta_{12} + 9 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{78} + ( 5 - 10 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{79} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{80} -9 q^{81} + ( -4 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{82} + ( 12 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{83} + ( 2 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{84} + ( -3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{85} + ( 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{86} + ( -12 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{87} + ( 6 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{88} + ( 2 - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{89} + ( 9 + 8 \zeta_{12} - 6 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{91} + ( 4 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{92} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{93} + ( 2 + \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{94} + 6 \zeta_{12}^{3} q^{95} + ( -4 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{96} + ( 6 - 12 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{97} + ( 8 - 3 \zeta_{12} - 3 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 6q^{6} - 8q^{8} + O(q^{10}) \) \( 4q - 2q^{2} - 6q^{6} - 8q^{8} - 2q^{10} + 12q^{12} - 10q^{14} + 8q^{16} + 24q^{19} - 4q^{20} + 12q^{21} + 2q^{22} - 4q^{25} - 6q^{26} + 4q^{28} + 4q^{29} + 6q^{30} - 24q^{31} + 8q^{32} - 18q^{34} - 8q^{35} - 24q^{37} - 12q^{38} + 8q^{40} + 6q^{42} - 8q^{44} - 8q^{46} - 4q^{49} + 2q^{50} + 12q^{52} + 8q^{53} + 18q^{54} - 8q^{55} + 16q^{56} + 22q^{58} + 12q^{62} + 12q^{65} + 6q^{66} - 12q^{68} + 10q^{70} - 16q^{77} - 6q^{78} - 36q^{81} - 12q^{82} + 48q^{83} - 12q^{85} - 4q^{86} - 48q^{87} + 16q^{88} + 24q^{91} + 8q^{92} + 6q^{94} - 24q^{96} + 26q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
−1.36603 0.366025i 1.73205 1.73205 + 1.00000i 1.00000i −2.36603 0.633975i 1.73205 + 2.00000i −2.00000 2.00000i 0 0.366025 1.36603i
111.2 −1.36603 + 0.366025i 1.73205 1.73205 1.00000i 1.00000i −2.36603 + 0.633975i 1.73205 2.00000i −2.00000 + 2.00000i 0 0.366025 + 1.36603i
111.3 0.366025 1.36603i −1.73205 −1.73205 1.00000i 1.00000i −0.633975 + 2.36603i −1.73205 2.00000i −2.00000 + 2.00000i 0 −1.36603 0.366025i
111.4 0.366025 + 1.36603i −1.73205 −1.73205 + 1.00000i 1.00000i −0.633975 2.36603i −1.73205 + 2.00000i −2.00000 2.00000i 0 −1.36603 + 0.366025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.g.a 4
3.b odd 2 1 1260.2.c.a 4
4.b odd 2 1 140.2.g.b yes 4
5.b even 2 1 700.2.g.f 4
5.c odd 4 1 700.2.c.b 4
5.c odd 4 1 700.2.c.e 4
7.b odd 2 1 140.2.g.b yes 4
7.c even 3 1 980.2.o.a 4
7.c even 3 1 980.2.o.c 4
7.d odd 6 1 980.2.o.b 4
7.d odd 6 1 980.2.o.d 4
8.b even 2 1 2240.2.k.a 4
8.d odd 2 1 2240.2.k.b 4
12.b even 2 1 1260.2.c.b 4
20.d odd 2 1 700.2.g.g 4
20.e even 4 1 700.2.c.c 4
20.e even 4 1 700.2.c.f 4
21.c even 2 1 1260.2.c.b 4
28.d even 2 1 inner 140.2.g.a 4
28.f even 6 1 980.2.o.a 4
28.f even 6 1 980.2.o.c 4
28.g odd 6 1 980.2.o.b 4
28.g odd 6 1 980.2.o.d 4
35.c odd 2 1 700.2.g.g 4
35.f even 4 1 700.2.c.c 4
35.f even 4 1 700.2.c.f 4
56.e even 2 1 2240.2.k.a 4
56.h odd 2 1 2240.2.k.b 4
84.h odd 2 1 1260.2.c.a 4
140.c even 2 1 700.2.g.f 4
140.j odd 4 1 700.2.c.b 4
140.j odd 4 1 700.2.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.a 4 1.a even 1 1 trivial
140.2.g.a 4 28.d even 2 1 inner
140.2.g.b yes 4 4.b odd 2 1
140.2.g.b yes 4 7.b odd 2 1
700.2.c.b 4 5.c odd 4 1
700.2.c.b 4 140.j odd 4 1
700.2.c.c 4 20.e even 4 1
700.2.c.c 4 35.f even 4 1
700.2.c.e 4 5.c odd 4 1
700.2.c.e 4 140.j odd 4 1
700.2.c.f 4 20.e even 4 1
700.2.c.f 4 35.f even 4 1
700.2.g.f 4 5.b even 2 1
700.2.g.f 4 140.c even 2 1
700.2.g.g 4 20.d odd 2 1
700.2.g.g 4 35.c odd 2 1
980.2.o.a 4 7.c even 3 1
980.2.o.a 4 28.f even 6 1
980.2.o.b 4 7.d odd 6 1
980.2.o.b 4 28.g odd 6 1
980.2.o.c 4 7.c even 3 1
980.2.o.c 4 28.f even 6 1
980.2.o.d 4 7.d odd 6 1
980.2.o.d 4 28.g odd 6 1
1260.2.c.a 4 3.b odd 2 1
1260.2.c.a 4 84.h odd 2 1
1260.2.c.b 4 12.b even 2 1
1260.2.c.b 4 21.c even 2 1
2240.2.k.a 4 8.b even 2 1
2240.2.k.a 4 56.e even 2 1
2240.2.k.b 4 8.d odd 2 1
2240.2.k.b 4 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(140, [\chi])\):

\( T_{3}^{2} - 3 \)
\( T_{19} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( 49 + 2 T^{2} + T^{4} \)
$11$ \( 1 + 14 T^{2} + T^{4} \)
$13$ \( 9 + 42 T^{2} + T^{4} \)
$17$ \( 9 + 42 T^{2} + T^{4} \)
$19$ \( ( -6 + T )^{4} \)
$23$ \( 64 + 32 T^{2} + T^{4} \)
$29$ \( ( -47 - 2 T + T^{2} )^{2} \)
$31$ \( ( 6 + T )^{4} \)
$37$ \( ( 24 + 12 T + T^{2} )^{2} \)
$41$ \( ( 12 + T^{2} )^{2} \)
$43$ \( ( 4 + T^{2} )^{2} \)
$47$ \( ( -3 + T^{2} )^{2} \)
$53$ \( ( -2 + T )^{4} \)
$59$ \( ( -12 + T^{2} )^{2} \)
$61$ \( 576 + 96 T^{2} + T^{4} \)
$67$ \( ( 12 + T^{2} )^{2} \)
$71$ \( 16 + 56 T^{2} + T^{4} \)
$73$ \( 144 + 168 T^{2} + T^{4} \)
$79$ \( 1521 + 222 T^{2} + T^{4} \)
$83$ \( ( 132 - 24 T + T^{2} )^{2} \)
$89$ \( 576 + 96 T^{2} + T^{4} \)
$97$ \( 9801 + 234 T^{2} + T^{4} \)
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